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quant-ph9912067	Evaluating Capacities of Bosonic Gaussian Channels	[ { "author": "A.~S. Holevo\\thanks{Permanent address: Steklov Mathematical Institute" }, { "author": "Gubkina 8" }, { "author": "117966 Moscow" }, { "author": "Russia" } ]	We show how to compute or at least to estimate various capacity-related quantities for Bosonic Gaussian channels. Among these are the coherent information, the entanglement assisted classical capacity, the one-shot classical capacity, and a new quantity involving the transpose operation, shown to be a general upper bound on the quantum capacity, even allowing for finite errors. All bounds are explicitly evaluated for the case of a one-mode channel with attenuation/amplification and classical noise.	[ { "name": "Qgaus.tex", "string": "\\documentstyle[twocolumn,aps,epsf]{revtex}\n\n\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n\\overfullrule=10pt\n\n\\begin{document}\n\\title{Evaluating Capacities of Bosonic Gaussian Channels}\n\\author{A.~S. Holevo\\thanks{Permanent address: Steklov Mathematical Institute,\nGubkina 8, 117966 Moscow, Russia} and R.~F. Werner }\n\\address{\nInstitut f\\\"ur Mathematische Physik, TU Braunschweig,\nMendelssohnstr.3, 38106 Braunschweig, Germany.\n\\\\ Electronic Mail:\na.holevo@mi.ras.ru, r.werner@tu-bs.de}\n\n\\date{December 14, 1999}\n\\maketitle\n\\begin{abstract}\nWe show how to compute or at least to estimate various\ncapacity-related quantities for Bosonic Gaussian channels. Among\nthese are the coherent information, the entanglement assisted\nclassical capacity, the one-shot classical capacity, and a new\nquantity involving the transpose operation, shown to be a general\nupper bound on the quantum capacity, even allowing for finite\nerrors. All bounds are explicitly evaluated for the case of a\none-mode channel with attenuation/amplification and classical\nnoise.\n\n\\end{abstract}\n% insert suggested PACS numbers in braces on next line\n\\pacs{03.67.-a, 02.20.Hj}\n\n\\section{Introduction}\nDuring the last years impressive progress was achieved in the\nunderstanding of the classical and quantum capacities of quantum\ncommunication channels (see, in particular, the papers \\cite{Ben}\n- \\cite{Thap}, where the reader can also find further references).\nIt appears that a quantum channel is characterized by a whole\nvariety of different capacities depending both on the kind of the\ninformation transmitted and the specific protocol used.\n\nMost of this literature studies the properties of systems and\nchannels described in finite dimensional Hilbert spaces. Recently,\nhowever, there has been a burst of interest (see e. g.\n\\cite{Braun}) in a new kind of systems, sometimes called\n``continuous variable'' quantum systems, whose basic variables\nsatisfy Heisenberg's Canonical Commutation Relations (CCR). There\nare two reasons for this new interest. On the one hand, such\nsystems play a central role in quantum optics, the canonical\nvariables being the quadratures of the field. Therefore some of\nthe current experimental realizations \\cite{Kimble} of quantum\ninformation processing are carried out in such systems. In\nparticular, the\n Bosonic Gaussian channels studied in this paper can be seen as\nbasic building blocks of quantum optical communication systems,\nallowing to build up complex operations from ``easy, linear'' ones\nand a few basic ``expensive, non-linear'' operations, such as\nsqueezers and parametric down converters.\n\nThe other reason for the interest in these systems is that in\nspite of the infinite dimension of their underlying Hilbert spaces\nthey can be handled with techniques from finite dimensional linear\nalgebra, much in analogy to the finite dimensional quantum systems\non which the pioneering work on quantum information was done.\nRoughly speaking this analogy replaces the density matrix by the\ncovariance matrix of a Gaussian state. Then operations like the\ndiagonalization of density matrices, the Schmidt decomposition of\npure states on composite systems, the purification of mixed\nstates, the computation of entropies, the partial transpose\noperation on states and channels, which are familiar from the\nusual finite dimensional setup, can be expressed once again by\noperations on finite dimensional matrices in the continuous\nvariable case. The basic framework for doing all this is not new,\nand goes under heading ``phase space quantum mechanics'' or, in\nthe quantum field theory and statistical mechanics communities,\n``quasi-free Bose systems'' \\cite{demoen}. Both authors of this\npaper have participated in the development of this subject a long\ntime ago \\cite{Hol75,Hol82,wer}. In this paper, continuing\n\\cite{Sohma} and \\cite{Hol98}, we make further contributions to\nthe study of information properties of linear Bosonic Gaussian\nchannels. We focus on the aspects essential for physical\ncomputations and leave aside a number of analytical subtleties\nrelated to infinite dimensionality and unboundedness unavoidably\narising in connection with Bosonic systems and Gaussian states.\n\nThe paper is organized as follows. In the Section~II we\nrecapitulate some notions of capacity, which are currently under\ninvestigation in the literature, and what is known about them.\nNaturally this cannot be a full review, but will be limited to\nthose quantities which we will evaluate or estimate in the\nsubsequent sections. A new addition to the spectrum of\ncapacity-like quantities is discussed in Subsection~II.B: an upper\nbound on the quantum capacity (even allowing finite errors), which\nis both simple to evaluate and remarkably close to maximized\ncoherent information, a bound conjectured to be exact. In\nSection~III we summarize the basic properties of Gaussian states.\nAlthough our main topic is channels, we need this to get an\nexplicit handle on the purification operation, which is needed to\ncompute the entropy exchange, and hence all entropy based\ncapacities. Bosonic Gaussian channels are studied in Section~IV.\nHere we introduce the techniques for determining the capacity\nquantities introduced in Section~I, deriving general formulas\nwhere possible. In the final Section~V we apply these techniques\nto the case of a single mode channel comprising\nattenuation/amplification and a classical noise. Some technical\npoints are treated in the Appendices.\n\n\\section{Notions of capacity}\n\\subsection{Basic entropy and information quantities} Consider a\ngeneral quantum system in a Hilbert space ${\\cal H=H}_{Q}$. Its\nstates are given by density operators $\\rho$ on ${\\cal H}$. A {\\it\nchannel\\/} is a transformation $\\rho \\rightarrow T[\\rho ]$ of\nquantum states of the system, which is given by a completely\npositive, trace preserving map on trace class operators. This view\nof channels corresponds to the Schr{\\\"o}dinger picture. The\nHeisenberg picture is given by the dual linear operator\n$X\\rightarrow T^[X]$ on the observables $X$, which is\ndefined by the relation\n\\[\n{\\rm Tr}T[\\rho ]X={\\rm Tr}\\rho T^[X],\n\\]\nand has to be completely positive and unit preserving (cf.\n\\cite{Hol72a}).\n\nIt can be shown (see e.g. \\cite{Kraus}) that any channel in this\nsense arises from a unitary interaction $U$ of the system with an\nenvironment described by another Hilbert space ${\\cal H}_{E}$\nwhich is initially in some state $\\rho_{E}$,\n\\[\nT[\\rho ]={\\rm Tr}_{E}U\\left( \\rho \\otimes \\rho_{E}\\right) U^,\n\\]\nwhere ${\\rm Tr}_{E}$ denotes partial trace with respect to ${\\cal H}_{E}$,\nand vice versa. The representation is not unique, and the state $\\rho_{E}$\ncan always be chosen pure, $\\rho_{E}=\|\\psi_{E}\\rangle \\langle \\psi_{E}\|$.\nThe definition of the channel has obvious generalization to the case where\ninput and output are described by different Hilbert spaces.\n\nLet us denote by\n\\begin{equation}\\label{vNentropy}\n H(\\rho )=-{\\rm Tr}\\rho \\log \\,\\rho\n\\end{equation}\n the von Neumann entropy of a density operator $\\rho $. We call $\\rho$ the\ninput state, and $T[\\rho ] $ the output state of the channel.\nThere are three important entropy quantities related to the pair\n$(\\rho ,T)$, namely, the entropy of the input state $H(\\rho)$, the\nentropy of the output state $H(T[\\rho])$, and the entropy exchange\n$H(\\rho,T)$. While the definition and the meaning of the first two\nentropies is clear, the third quantity is somewhat more\nsophisticated. To define it, one introduces the {\\it reference\nsystem}, described by the Hilbert space ${\\cal H}_{R}$, isomorphic\nto the Hilbert space ${\\cal H}_{Q}=$ ${\\cal H}$ of the initial\nsystem. Then according to \\cite{Lin}, \\cite{Sch}, there exists a\n{\\it purification} of the state $\\rho $, i.e. a unit vector $\|\\psi\n\\rangle \\in {\\cal H}_{Q}\\otimes {\\cal H}_{R}$ such that\n\\[\n\\rho ={\\rm Tr}_{R}\|\\psi \\rangle \\langle \\psi \|.\n\\]\nThe {\\it entropy exchange } is then defined as\n\\begin{equation}\\label{def-xchange}\n H(\\rho ,T)=H\\bigl((T\\otimes {\\rm id})[\|\\psi\\rangle\\langle\\psi\|]\\bigr),\n\\end{equation}\nthat is, as the entropy of the output state of the dilated channel\n$(T\\otimes {\\rm id})$ applied to the input which is purification of the state $\\rho$.\nAlternatively,\n\\[\nH(\\rho ,T)=H(\\rho_{E}'),\n\\]\nwhere $\\rho_{E}'=T_{E}[\\rho ]$ is the final state of the\nenvironment, and the channel $T_E$ from ${\\cal H}_{Q}$ to ${\\cal\nH}_{E}$ is defined as\n\\[\nT_{E}[\\rho ]={\\rm Tr}_{Q}U\\left( \\rho \\otimes \\rho_{E}\\right) U^,\n\\]\nprovided the initial state $\\rho_E$ of the environment is pure\n\\cite{Lin}, \\cite{Sch}.\n\n>From these three entropies one can construct several information\nquantities. In analogy with classical information theory, one can\ndefine {\\it quantum mutual information }between the reference\nsystem $R$ (which mirrors the input $Q$) and the output of the\nsystem $Q'$ \\cite{Lin}, \\cite {Cer} as\n\\begin{eqnarray}\\label{q-mutual}\n I(\\rho ,T)&=&H(\\rho_{R}')+H(\\rho_{Q}')\n -H(\\rho_{RQ}')\n\\nonumber\\\\\n &=&H(\\rho )+H(T[\\rho ])-H(\\rho ,T).\n\\end{eqnarray}\nThe quantity $I(\\rho ,T)$ has a number of nice and ``natural''\nproperties, in particular, positivity, concavity with respect to\nthe input state $\\rho $ and additivity for parallel channels\n\\cite{Cer}. Moreover, the maximum of $I(\\rho, T )$ with respect to\n$\\rho$ was argued recently to be equal to the {\\it\nentanglement-assisted classical capacity} of the channel\n\\cite{Thap},\\cite{Shor}, namely, the classical capacity of the\nsuperdense coding protocol using the noisy channel $T$. It was\nshown that this maximum is additive for parallel channels, the\none-shot expression thus giving the full (asymptotic) capacity.\n\nIt would be natural to compare this quantity with the (unassisted)\nclassical capacity $C(T)$ (the definition of which is outlined in\nthe next Subsection); however it is still not known whether this\ncapacity is additive for parallel channels. This makes us focus on\nthe one-shot expression, emerging from the coding theorem for\nclassical-quantum channels \\cite{Hol}\n\\begin{equation}\\label{oneshotunassist}\nC_{1}(T)=\\max \\left[ H\\Bigl( \\sum_{i}p_{i}T\\left[ \\rho_{i}\\right]\n\\Bigr)\\right. \\left.- \\sum_{i}p_{i}H\\left( T\\left[\n \\rho_{i}\\right] \\right) \\right] ,\n\\end{equation}\nwhere the maximum is taken over all probability distributions\n$\\left\\{ p_{i}\\right\\}$ and collections of density operators\n$\\left\\{ \\rho _{i}\\right\\}$ (possibly satisfying some additional\ninput constraints). $C_1(T)$ is equal to the capacity of $T$ for\nclassical information, if the coding is required to avoid\nentanglement between successive inputs to the channel. The full\ncapacity is then attained as the length $n$ of the blocks, over\nwhich encoding may be entangled goes to infinity, i.e.,\n\\begin{equation}\\label{C-from-C1}\n C(T)=\\lim_{n\\to\\infty}\\ \\frac1n C_1(T^{\\otimes n}).\n\\end{equation}\n\nAn important component of $I(\\rho ,T)$ is the {\\it coherent\ninformation}\n\\begin{equation}\\label{def-coherent}\n J(\\rho ,T)=H(T[\\rho ])-H(\\rho ,T),\n\\end{equation}\nthe maximum of which has been conjectured to be the (one-shot)\nquantum capacity of the channel $T$ \\cite{lloyd}, \\cite{Sch}. Its\nproperties are not so nice. It can be negative, its convexity\nproperties with respect to $\\rho $ are not known, and its maximum\nwas shown to be strictly superadditive for certain parallel\nchannels \\cite{smo}, hence the conjectured full quantum capacity\nmay be greater than the one-shot expression, in contrast to the\ncase of the entanglement-assisted classical capacity. In this\npaper we shall also compare this expression with a new upper bound\non the quantum capacity $Q(T)$ (as introduced e.g. in the next\nSubsection).\n\n\\subsection{A general bound on quantum channel capacity}\nIn this Subsection we will establish a general estimate on the\nquantum channel capacity, which will then be evaluated in the\nGaussian case, and will be compared with the estimates of coherent\ninformation. Let us recall first a definition of the capacity\n$Q(T)$ of a general channel $T$ for quantum information.\nIntuitively, it is the number of qubits which can be faithfully\ntransmitted per use of the channel with the best possible error\ncorrection. The standard of comparison is the ideal 1-qubit\nchannel ${\\rm id}_2$, where ${\\rm id}_n$ denotes the identity map\non the $n\\times n$-matrices. Then the {\\it quantum capacity}\n$Q(T)$ of a channel $T$ (possibly between systems of different\ntype) is defined as the supremum of all numbers $c$, which are\n``attainable rates'' in the following sense: {\\it For any pair of\nsequences $n_\\alpha,m_\\alpha$ with\n$\\lim_\\alpha(n_\\alpha/m_\\alpha)= c$ we can find encoding\noperations $E_\\alpha$ and decoding operations $D_\\alpha$ such\nthat}\n\\[\n\\Vert {\\rm id}_2^{\\otimes n_\\alpha}- D_\\alpha T^{\\otimes\nm_\\alpha}E_\\alpha\\Vert_{{\\rm cb}}\\longrightarrow0.\n\\]\nHere $\\Vert \\cdot\\Vert_{{\\rm cb}}$ is the so-called ``norm of\ncomplete boundedness''\\cite{Paulsen}, which is defined as the\nsupremum with respect to $n$ of the norms $\\Vert(T\\otimes {\\rm\nid}_n)\\Vert$. It is equal to the ``diamond metric'' introduced in\n\\cite{AhaKitNi}. We use this norm because on the one hand, it\nleads to the same capacity as analogous definitions based on other\nerror criteria (e.g., fidelities \\cite{Sch,Barn}) and, on the\nother hand, it has the best properties with respect to tensor\nproducts, which are our main concern. In particular,\n $\\Vert T\\otimes S\\Vert_{{\\rm cb}}\n =\\Vert T\\Vert_{{\\rm cb}}\\cdot\\Vert S\\Vert_{{\\rm cb}}$.\n Completely positive maps satisfy $\\Vert T\\Vert_{{\\rm cb}}=\\Vert F\\Vert$,\nwhere $F$ is the normalization operator determined by\n${\\rm Tr}(T[\\rho])={\\rm Tr}(\\rho F)$. In particular, $\\Vert T\\Vert_{{\\rm\ncb}}=1$ for any channel. We also note another kind of capacity, in\nwhich a much weaker requirement is made on the errors, namely\n\\begin{equation} \\label{errq}\n\\Vert {\\rm id}_2^{\\otimes n_\\alpha}- D_\\alpha T^{\\otimes\nm_\\alpha}E_\\alpha\\Vert_{{\\rm cb}}\\leq\\varepsilon<1\n\\end{equation}\nfor all sufficiently large $\\alpha$, and some fixed $\\varepsilon$.\nWe call the resulting capacity the {\\it $\\varepsilon$-quantum\ncapacity}, and denote it by $Q_{\\varepsilon} (T)$. Of course,\n$Q(T)\\leq Q_{\\varepsilon} (T)$, and by analogy with the classical\ncase (strong converse of Shannon's Coding Theorem) one would\nconjecture that equality always holds.\n\nThe unassisted classical capacity $C(T)$ can be defined similarly\nwith the sole difference that both the domain of encodings $E$ and\nthe range of decodings $D$ should be restricted to the state space\nof the Abelian subalgebra of operators diagonalizable in a fixed\northonormal basis. In that case there is no need to use the\ncb-norm, as it coincides with the usual norm. According to\nrecently proven strong converse to the quantum coding theorem\n\\cite{ogawa}, \\cite{winter}, $C_{\\varepsilon}(T)=C(T)$ where\n$C_{\\varepsilon}(T)$ is defined similarly to $Q_{\\varepsilon}\n(T)$.\n\nThe criterion we will formulate makes essential use of the transpose\noperation, which we will denote by the same letter $\\Theta $ in any system.\nFor matrix algebras, $\\Theta $ can be taken as the usual transpose\noperation. However, it makes no difference to our considerations, if any\nother anti-unitarily implemented symmetry (e.g. time-reversal) is chosen. In\nan abstract C-algebra setting $\\Theta $ is best taken as the ``op''\noperation, which maps every algebra to its ``opposite''. This algebra has\nthe same underlying vector space, but all products $AB$ are replaced by\ntheir opposite $BA$. Obviously, a commutative algebra is the same as\nits opposite, so on classical systems $\\Theta $ is the identity. Although\nthe transpose maps density operators to density operators, it is not an\nadmissible quantum channel, because positivity is lost, when coupling the\noperation with the identity transformation on other systems, i.e., $\\Theta $\nis not {\\it completely} positive. A similar phenomenon happens for the norm\nof $\\Theta $: we have $\\Vert \\Theta \\Vert_{{\\rm cb}}>1$ unless the system\nis classical. In fact,\n\\begin{equation}\\label{cbThetan}\n\\Vert \\Theta_{n}\\Vert_{{\\rm cb}}=n,\n\\end{equation}\nwhere $\\Theta_{n}$ denotes the transposition on the $n\\times n$-matrices\n\\cite{Paulsen}. We note that since we do not distinguish the transpose on\ndifferent systems in our notation, the observation that tensor products can\nbe transposed factor by factor is expressed by the equation $\\Theta =\\Theta\n\\otimes \\Theta $. Moreover, although for a channel $T$, the operator\n$T\\Theta $ may fail to be completely positive, $\\Theta T\\Theta $ is again a\nchannel, and, in particular, satisfies $\\Vert \\Theta T\\Theta \\Vert_{{\\rm cb}}=1$.\n\nThe main result of this Subsection is the estimate\n\\begin{equation} \\label{cbn-bound}\nQ_\\varepsilon(T)\\leq \\log\\Vert T\\Theta\\Vert_{{\\rm cb}} \\equiv Q_\\Theta(T),\n\\end{equation}\nfor any channel $T$. The proof is quite simple. Suppose\n$n_\\alpha/m_\\alpha\\rightarrow c\\leq Q_\\varepsilon(T)$, and encoding $E_\\alpha$ and decoding\n$D_\\alpha$ are as in the definition of $Q_\\varepsilon(T)$. Then by Equation~(\n\\ref{cbThetan}) we have\n\\begin{eqnarray}\n2^{n_\\alpha}\n &=& \\Vert {\\rm id}_2^{\\otimes\n n_\\alpha}\\Theta\\Vert_{{\\rm cb}} \\ \\leq\n\\nonumber \\\\\n &\\leq&\\Vert ({\\rm id}_2^{\\otimes n_\\alpha}\n - D_\\alpha T^{\\otimes m_\\alpha}E_\\alpha)\\Theta\\Vert_{{\\rm cb}}\n\\nonumber \\\\\n &&\\qquad +\\Vert D_\\alpha T^{\\otimes\n m_\\alpha}E_\\alpha\\Theta\\Vert_{{\\rm cb}}\n\\nonumber \\\\\n &\\leq& \\Vert \\Theta_{2^{n_\\alpha}}\\Vert_{{\\rm cb}}\\\n \\Vert {\\rm id}_2^{\\otimes n_\\alpha}-\n D_\\alpha T^{\\otimes m_\\alpha}E_\\alpha\\Vert_{{\\rm cb}}\n\\nonumber \\\\\n &&\\qquad +\\Vert D_\\alpha (T\\Theta)^{\\otimes m_\\alpha}\\Theta\n E_\\alpha\\Theta\\Vert_{{\\rm cb}} \\nonumber \\\\\n &\\leq& 2^{n_\\alpha}\\varepsilon\n + \\Vert T\\Theta\\Vert_{{\\rm cb}}^{m_\\alpha},\n\\nonumber\n\\end{eqnarray}\nwhere at the last inequality we have used that $D_\\alpha$ and\n $\\Theta E_\\alpha\\Theta$ are channels, and that the cb-norm is exactly tensor\nmultiplicative, so\n $\\Vert X^{\\otimes m}\\Vert_{{\\rm cb}}=\\Vert X\\Vert_{{\\rm cb}}^m$.\nHence, by taking the logarithm and dividing by $m_\\alpha$, we get\n\\[\n\\frac{n_\\alpha}{m_\\alpha}\\log2 +\\frac{\\log(1-\\varepsilon)}{m_\\alpha} \\leq\n\\log\\Vert T\\Theta\\Vert_{{\\rm cb}}.\n\\]\nIf we take base $2$ logarithms, as is customary in information\ntheory, we have $\\log2=1$. Then in the last inequality we can go\nto the limit $\\alpha\\to\\infty$, obtaining $c\\leq Q_\\Theta(T)$, and\nEquation~(\\ref{cbn-bound}) follows by taking the supremum over all\nattainable rates $c$. Note that base $2$ logarithms are built into\nthe above definition of capacity, because we are using the ideal\nqubit channel as the standard of comparison. This amounts only to\na change of units. If another base is chosen for logarithms is\nchosen, this should also be done consistently in all entropy\nexpressions, and Equation~(\\ref{cbn-bound}) holds once again\nwithout additional constants.\n\nThe upper bound $Q_{\\Theta }(T)$ computed in this way has some\nremarkable properties, which make it a capacity-like quantity in\nits own right. For example, it is exactly additive:\n\\begin{equation}\nQ_{\\Theta }(S\\otimes T)=Q_{\\Theta }(S)+Q_{\\Theta }(T), \\label{capt-add}\n\\end{equation}\nfor any pair $S,T$ of channels, and satisfies the ``bottleneck inequality''\n$Q_{\\Theta }(ST)\\leq \\min \\{Q_{\\Theta }(S),Q_{\\Theta }(T)\\}$. Moreover, it\ncoincides with the quantum capacity on ideal channels:\n$Q_{\\Theta }({\\rm id}_{n})=Q({\\rm id}_{n})=\\log\n_{2}n$, and it vanishes whenever $T\\Theta $ is completely\npositive.\nIn particular, $Q_{\\Theta}(T)=0$, whenever $T$ is {\\it separable}\nin the sense that it can be decomposed as $T=PM$ into a\nmeasurement $M$ and a subsequent preparation $P$ based on the\nmeasurement results. This follows immediately from the observation\nthat on classical systems transposition is the identity. Then\n$P\\Theta=\\Theta P\\Theta$ is a channel, and so is $MP\\Theta$.\nWe note that $Q_{\\Theta}$ is also closely related to the\nentanglement quantity\n$\\log_{2}\\Vert ({\\rm id}\\otimes \\Theta)(\\rho)\\Vert_{1}$,\n i.e., the logarithm of the trace norm of the partial\ntranspose of the density operator, which enjoys analogous\nproperties.\n\n\\section{Quantum Gaussian states}\n\\subsection{Canonical Variables and Gaussian states}\nIn this Section we recapitulate some results from \\cite{Hol75},\n\\cite{Sohma}, \\cite{Hol98} for the convenience of the reader. Our\napproach to quantum Gaussian states is based on the characteristic\nfunction of the state which closely parallels classical\nprobability \\cite{Hol82}, \\cite{wer}, and is perhaps the simplest\nand most transparent analytically. An alternative approach can be\nbased on the Wigner ``distribution function'' \\cite{agar}.\n\nLet $q_{j},p_{j}$ be the canonical observables satisfying the\nHeisenberg CCR\n\\[\n\\lbrack q_{j},p_{k}]=i\\delta_{jk}\\hbar\nI,\\;\\;[q_{j},q_{k}]=0,\\;\\;[p_{j},p_{k}]=0.\n\\]\nWe introduce the column vector of operators\n\\[\nR=[q_{1},p_{1},\\dots ,q_{s},p_{s}]^{T},\n\\]\nthe real column $2s$-vector $z=[x_{1},y_{1},\\dots ,x_{s},y_{s}]^{T}$, and\nthe unitary operators in ${\\cal H}$\n\\begin{eqnarray}\\label{Weylop}\n V(z)&=&\\exp \\,i\\sum_{j=1}^{s}(x_{j}q_{j}+y_{j}p_{j})\\\\\n &=&\\exp \\,i\\,R^{T}z.\\nonumber\n\\end{eqnarray}\nThese ``Weyl-operators'' satisfy the Weyl-Segal CCR\n\\begin{equation}\nV(z)V(z')=\\exp [\\frac{i}{2}\\Delta (z,z')]V(z+z'), \\label{weyl}\n\\end{equation}\nwhere\n\\begin{equation}\\label{sympl-form}\n \\Delta (z,z')\n =\\hbar \\sum_{j=1}^{s}(x_{j}'y_{j}-x_{j}y_{j}')\n\\end{equation}\nis the canonical symplectic form. The space $Z$ of real $2s$-vectors\nequipped with the form $\\Delta (z,z')$ is what one calls a\n{\\it symplectic vector space}.\nWe denote by\n\\begin{equation}\n\\Delta =\\left[\n\\begin{array}{ccccc}\n0 & \\hbar & & & \\\\\n-\\hbar & 0 & & & \\\\\n& & \\ddots & & \\\\\n& & & 0 & \\hbar \\\\\n& & & -\\hbar & 0\n\\end{array}\n\\right] \\label{delta}\n\\end{equation}\nthe $(2s)\\times (2s)$-skew-symmetric {\\it commutation matrix} of\ncomponents of the vector $R$, so that\n\\[\n\\Delta (z, z')= - z^{T}\\Delta z'.\n\\]\n Most of the results\nbelow are valid for the case where the commutation matrix is an\narbitrary (nondegenerate) skew-symmetric matrix, not necessarily\nof the canonical form (\\ref{delta}).\n\nA density operator $\\rho $ has {\\it finite second moments }if\n${\\rm Tr}(\\rho q_{j}^{2})<\\infty$ and ${\\rm Tr}(\\rho p_{j}^{2})<\\infty$\nfor all $j$. In this case one can define the vector {\\it mean\\/} and\nthe {\\it correlation matrix} $\\alpha$ by the formulas\n\\begin{equation}\n m={\\rm Tr}\\rho R\\;;\\;\n \\alpha -\\frac{i}{2}\\Delta ={\\rm Tr}(R-m)\\rho (R-m)^{T}.\n\\label{alpha}\n\\end{equation}\nThe mean can be an arbitrary real vector. The correlation matrix $\\alpha$\nis real and symmetric. A given $\\alpha$ is the correlation matrix\nof some state if and only if it satisfies the {\\it matrix uncertainty\nrelation}\n\\begin{equation}\n\\alpha -\\frac{i}{2}\\Delta \\geq 0. \\label{n-s condition}\n\\end{equation}\nWe denote by $\\Sigma \\left( m,\\alpha \\right)$ the set of states\nwith fixed mean $m$\\ and the correlation function $\\alpha $.\nThe density operator $\\rho $ is called {\\it Gaussian}, if its {\\it quantum\ncharacteristic function} $\\phi (z)={\\rm Tr}\\rho V(z)$ has the form\n\\begin{equation}\\label{GaussianState}\n \\phi (z)=\\exp \\left(i\\,m^{T}z-\\frac{1}{2}z^{T}\\alpha z\\right),\n\\end{equation}\nwhere $m$ is a column ($2s$)-vector and $\\alpha $\\ is a real\nsymmetric $(2s)\\times (2s)$-matrix. One then can show that $m$ is\nindeed the mean, and $\\alpha $ is the correlation matrix, and\n(\\ref{GaussianState}) defines the unique Gaussian state in\n$\\Sigma\\left(m,\\alpha\\right)$. In what follows we will be\ninterested mainly in the case $m=0$.\n\nThe correlation matrix $\\alpha$ describes a quadratic form rather\nthan an operator. Therefore its eigenvalues have no intrinsic\nsignificance, and depend on the choice of basis in $Z$. On the\nother hand, the operator $\\widehat\\alpha$ defined by $z^T\\alpha\nz=\\Delta(z,\\widehat\\alpha z)$ has a basis free meaning. In matrix\nnotation it is $\\widehat\\alpha=\\Delta^{-1}\\alpha$. This operator\nis always diagonalizable, and its eigenvalues come in pairs $\\pm\ni\\gamma_j$. Diagonalizing this operator is essentially the same as\nthe {\\it normal mode decomposition} of the phase space, when the\nform $z^T\\alpha z$ is considered as the Hamiltonian function of a\nsystem of oscillators. It leads to a decomposition of the phase\nspace into two-dimensional subspaces, such that on the $j^{\\rm\nth}$ subspace we have (in some new canonical variables ${\\tilde\nq}_j, {\\tilde p}_j$)\n\\begin{equation}\\label{one-mode-alpha}\n \\alpha=\\hbar\\left[\n \\begin{array}{ll}\\gamma_j & 0 \\\\0 & \\gamma_j\\end{array} \\right]\n\\quad;\\quad\n \\Delta=\\hbar\\left[\n \\begin{array}{ll}0&1 \\\\-1&0 \\end{array} \\right],\n\\end{equation}\nand all terms between different blocks vanish. The matrix\nuncertainty relation now requires $\\gamma_j\\geq1/2$, in which\nequality holds iff $\\rho_j$ is the pure (minimum-uncertainty)\nstate. Hence a general Gaussian state $\\rho$ is pure if and only\nif all $\\gamma_j=1/2$, or\n\\begin{equation}\n(\\Delta ^{-1}\\alpha )^{2}=-\\ \\frac{1}{4}I, \\label{pure}\n\\end{equation}\nin which case $\\Sigma \\left( m,\\alpha \\right)$ reduces to a\nsingle point.\n\n\\subsection{Gauge-invariant states}\nWe shall be interested in the particular subclass of Gaussian\nstates most familiar in quantum optics, namely, the states having\na P-representation\n\\begin{equation}\n\\rho =\\int \|\\zeta \\rangle \\langle \\zeta \|\\mu_{N}(d^{2s}\\zeta )\n\\label{p-repr}\n\\end{equation}\nwhere $\\mu_{N}(d^{2s}\\zeta )$ is the complex Gaussian probability\nmeasure with zero mean and the correlation matrix $N$. (see e.g.\n\\cite{Hel}, Sec. V, 5. II). Here $\\zeta \\in {\\bf C}^{s}$, $\|\\zeta\n\\rangle $ are the coherent vectors in ${\\cal H}$, $a\|\\zeta \\rangle\n=\\zeta \|\\zeta \\rangle $, $N$ is positive Hermitian matrix such\nthat\n\\begin{equation}\\label{def-N}\n N={\\rm Tr}\\left(a\\,\\rho \\,a^{\\dagger }\\right)\n\\end{equation}\n(we use here vector notations, where $a=[a_{1},\\dots ,a_{s}]^{T}$\nis a column vector and $a^{\\dagger }=[a_{1}^{\\dagger },\\dots\n,a_{s}^{\\dagger }]$ is a row vector) and\n$a_{j}=\\frac{1}{\\sqrt{2\\hbar }}(q_{j}+ip_{j})$.\n\nThese states respect the natural complex structure in the sense\nthat they are invariant under the gauge transformations\n$a\\rightarrow a\\exp (i\\varphi)$. As shown in \\cite{Sohma}, the\nquantum correlation matrix of such states is\n\\[\n\\alpha =\\hbar \\left[\n\\begin{array}{ll}\n{\\rm Re}N+I/2 & -{\\rm Im}N \\\\\n{\\rm Im}N & {\\rm Re}N+I/2\n\\end{array}\n\\right] ,\n\\]\nWith Pauli matrices $I_{2},\\sigma_{y}$, the real $2s\\times 2s-$ matrices of\nsuch form can be rewritten as complex $s\\times s-$ matrices, by using the\ncorrespondence\n\\[\n\\left[\n\\begin{array}{ll}\nA & -B \\\\\nB & A\n\\end{array}\n\\right] =I_{2}A+\\sigma_{y}B\\leftrightarrow A+iB,\n\\]\nwhich is an algebraic isomorphism. Obviously,\n\\[\n\\frac{1}{2}{\\rm Sp}\\left[\n\\begin{array}{ll}\nA & -B \\\\\nB & A\n\\end{array}\n\\right] ={\\rm Sp}(A+iB),\n\\]\nwhere by ``${\\rm Sp}$'' we denote the trace of matrices, as\nopposed to the trace of Hilbert space operators, which is denoted\nby ``Tr''. By using this correspondence, we have\n\\begin{equation}\n\\alpha \\leftrightarrow \\hbar (N+I/2),\\qquad \\Delta \\leftrightarrow\n-i\\hbar I, \\label{corr}\n\\end{equation}\nand\n\\begin{equation}\\label{arrow}\n\\Delta ^{-1}\\alpha \\leftrightarrow i(N+I/2).\n\\end{equation}\n\nFor the case of one degree of freedom we shall be interested in\nthe last Section, $N$ is just a nonnegative number and $\\rho $ is\nan {\\it elementary} Gaussian state with the characteristic\nfunction\n\\begin{equation}\n\\phi (z) =\\exp \\left[ -\\frac{\\hbar }{2}\\left( N+\\frac{1}{2}\\right)\n\|z\|^{2}\\right] , \\label{one-mode-cf}\n\\end{equation}\nwhere we put $\|z\|^{2}=(x^{2}+y^{2})$. This state has correlation\nmatrix of the form (\\ref{one-mode-alpha}) in the initial variables\n$q,p$, with $\\gamma = N+1/2$, and is just the temperature state of\nthe harmonic oscillator\n\\begin{equation}\\label{onemode-rho}\n \\rho_\\gamma =\\frac{1}{\\gamma +1/2}\\sum_{n =0}^{\\infty }\n \\left( \\frac{\\gamma -1/2}{\\gamma +1/2}\\right)^{n}\\ \|n\\rangle\\,\\langle n\|\n\\end{equation} in the number basis $\|n\\rangle$, with the mean photon number $N$.\n\n\\subsection{Computation of entropy}\nTo compute the von Neumann entropy of a general Gaussian state one\ncan use the normal mode decomposition. For a single mode, the\ndensity operator $\\rho_{j}$ with the correlation matrix\n(\\ref{one-mode-alpha}), setting $\\gamma_j\\equiv \\gamma$ for\nconvenience, is unitarily equivalent to the state\n(\\ref{onemode-rho}). From this one readily gets the von Neumann\nentropy $H(\\rho_\\gamma)$ by summation of the geometric series, and\nfor general Gaussian $\\rho$ by summing over normal modes.\n\nTo write the result in compact form, one introduces the function\n\\begin{eqnarray}\\label{def-g}\n g(x)&=&(x+1)\\log (x+1)-x\\log x,\\quad x>0 \\\\\n g(0)&=&0.\\nonumber\n\\end{eqnarray}\nThen\n\\begin{equation}\\label{H->g}\n H(\\rho )= \\sum_{j=1}^{s}\\ g\\left(\|\\gamma_{j}\|-\\frac{1}{2}\\right),\n\\end{equation}\nwhere $\\gamma_j$ runs over all eigenvalue pairs $\\pm i\\gamma_j$ of\n$\\Delta^{-1}\\alpha$.\n\nOne can also write this more compactly, using the following\nnotations, which we will also use in the sequel. For any\ndiagonalizable matrix $M=S{\\rm diag}(m_{j})S^{-1}$, we put ${\\rm\nabs}(M)=S{\\rm diag}(\|m_{j}\|)S^{-1}$, analogously for other\ncontinuous functions on the complex plane. Then\nequation~(\\ref{H->g}) can be written as \\cite{Sohma}\n\\begin{equation}\\label{abs}\n H(\\rho )=\\frac{1}{2}{\\rm Sp}\\ g\\left( {\\rm abs}(\\Delta ^{-1}\\alpha )\n -\\frac{I}{2}\\right) .\n\\end{equation}\nFor gauge-invariant state, by using (\\ref{arrow}), this reduces to\nthe well-known formula\n\\[\nH(\\rho )={\\rm Sp}\\ g(N).\n\\]\n\n\\subsection{Schmidt Decomposition and Purification}\nForming a composite systems out of two systems described by\nCCR-relations is very simple: one just joins the two sets of\ncanonical operators, making operators belonging to different\nsystems commute. The symplectic space of the composite system is a\ndirect sum $Z_{12}=Z_1\\oplus Z_2$, which means that elements of\nthis space are pairs $(z_1,z_2)$ with components $z_i\\in Z_i$. In\nterms of Weyl operators one can write\n$V_{12}(z_1,z_2)=V_{1}(z_1)\\otimes V_{2}(z_2)$. By definition, the\nsymplectic matrix $\\Delta_{12}$ is block diagonal with respect to\nthe decomposition $Z=Z_1\\oplus Z_2$. However, the correlation\nmatrix $\\alpha_{12}$ is block diagonal if and only if the state is\na product. The restriction of a bipartite Gaussian state $\\rho$ to\nthe first factor is determined by the expectations of the Weyl\noperators $V_1(z_1)\\otimes{\\bf1}=V_{12}(z_1,0)$, hence according to\n(\\ref{GaussianState}), by the the correlation matrix $\\alpha_1$\nwith\n $z_1^T\\alpha_1z_1=(z_1,0)^T\\alpha_{12}(z_1,0)$, which is just the\nfirst diagonal block in the block matrix decomposition\n\\begin{equation}\\label{alpha1+2}\n \\alpha_{12}=\\left[\n \\begin{array}{ll} \\alpha_1 & \\beta \\\\ \\beta^T & \\alpha_2\n \\end{array}\\right]\\quad;\\quad\n \\Delta_{12}=\\left[\n \\begin{array}{ll} \\Delta_1 & 0 \\\\ 0& \\Delta_2\n \\end{array}\\right].\n\\end{equation}\n\nAs in the case of bipartite systems with finite dimensional\nHilbert spaces there is a canonical form for {\\it pure} states of\nthe composite system, the Schmidt decomposition. Like the\ndiagonalization of a one-site density operator, it can be carried\nout for Gaussian states at the level of correlation matrices. By\nwriting out equation~(\\ref{pure}) in block matrix form, we find in\nparticular that\n\\begin{equation}\\label{intertwine}\n (\\Delta_1^{-1}\\alpha_1)(\\Delta_1^{-1}\\beta)\n =(\\Delta_1^{-1}\\beta)(\\Delta_2^{-1}\\alpha_2).\n\\end{equation}\nThus $(\\Delta_1^{-1}\\beta)$ maps eigenvectors of\n$(\\Delta_2^{-1}\\alpha_2)$ into eigenvectors of\n$(\\Delta_1^{-1}\\alpha_1)$, with the same eigenvalue. Hence the\nspectra of the restrictions are synchronized much in the same way\nas in the finite dimensional case, and all the matrices\n$\\alpha_1,\\alpha_2,\\beta$ can be diagonalized simultaneously by a\nsuitable choice of canonical coordinates. Evaluating also the\ndiagonal part of Equation~(\\ref{pure}), one gets an equation for\n$\\beta$, so that finally $\\alpha_{12}$ is decomposed into blocks\ncorresponding to (a) pure components belonging to only one\nsubsystem, and not correlated with the other, and (b) blocks of a\nstandard form, which can be written like (\\ref{alpha1+2}) with\n$\\alpha_1=\\alpha_2=\\alpha$, $\\Delta_1=\\Delta_2=\\Delta$ from\n(\\ref{one-mode-alpha}), and\n\\begin{equation}\\label{alpha12-pure}\n \\beta=\\hbar\\sqrt{\\gamma^2-\\frac{1}{4}}\\\n \\left[\\begin{array}{cc} 1&0\\\\0&-1\n \\end{array}\\right].\n\\end{equation}\nThe purification of a general Gaussian state can easily be read\noff from this, by constructing such a standard form for every\nnormal mode. In order to write $\\beta$ in operator form without\nexplicit reference to the normal mode decomposition, it is most\nconvenient to perform an appropriate reflection in the space\n$Z_2$, by which $\\beta$ becomes purely off-diagonal. Then we can\nchoose \\cite{Hol72} $\\Delta_1=\\Delta=-\\Delta_2$ and\n$\\alpha_2=\\alpha_1=\\alpha$, resulting in\n\\begin{equation}\\label{beta12}\n \\beta = - \\beta^T =\\Delta\\sqrt{-(\\Delta ^{-1}\\alpha )^{2}-I/4}.\n\\end{equation}\nThis also covers cases with $\\beta=0$ for some modes, where, strictly\nspeaking no purification would have been necessary.\nWe thus have\n\\begin{equation}\\label{da12}\n \\Delta_{12}^{-1}\\alpha_{12}=\\left[\\begin{array}{cc}\n \\Delta ^{-1}\\alpha & \\sqrt{-(\\Delta ^{-1}\\alpha )^{2}-I/4}\\\\\n \\sqrt{-(\\Delta ^{-1}\\alpha )^{2}-I/4} & - \\Delta ^{-1}\\alpha\n \\end{array}\\right].\n\\end{equation}\nIn the gauge-invariant case, we can use the correspondence\n\\begin{equation}\n\\Delta_{12}^{-1}\\alpha_{12}\\leftrightarrow \\left[\n\\begin{array}{ll}\ni(N+I/2) & \\sqrt{N^{2}+N} \\\\ \\sqrt{N^{2}+N} & -i(N+I/2)\n\\end{array}\n\\right], \\label{d-1a}\n\\end{equation} following from (\\ref{corr}).\n\n\\section{Linear Bosonic Channels}\n\\subsection{Basic Properties}\n\nThe characteristic property of the channels considered in this\npaper is their simple description in terms of phase space\nstructures. The key feature is that Weyl operators go into Weyl\noperators, up to a factor. That is, the channel map in the\nHeisenberg picture is of the form\n\\begin{equation}\n T^{}(V'(z'))=V(K^T z') f(z'), \\label{linbos}\n\\end{equation}\nwhere $K:Z\\to Z'$ is a linear map between phase spaces with\nsymplectic forms $\\Delta$ and $\\Delta'$, respectively, and $f(z')$\nis a scalar factor satisfying certain positive definiteness\ncondition to be discussed later. Because of the linearity of $K$,\nsuch channels are called {\\it linear Bosonic channels}\n\\cite{Hol72a}, and if, in addition, the factor $f$ is Gaussian,\n$T$ will be called a {\\it Gaussian channel}. In terms of\ncharacteristic functions, Equation~(\\ref{linbos}) can be written\nas\n\\begin{equation}\n\\phi '(z')= \\phi (K^{T}z') f(z') , \\label{ce}\n\\end{equation}\nwhere $\\phi$ and $\\phi'$ are the characteristic functions of\ninput state $\\rho$ and output state $T[\\rho]$, respectively.\n\nWe will make use of following key properties:\n\n(A) The dual of a linear Bosonic channel transforms any polynomial\nin the operators $R'$ into a polynomial in the $R$ of the same\norder, provided the function $f$ has derivatives of sufficiently\nhigh order. This property follows from the definition of moments\nby differentiating the relation (\\ref{linbos}) at the point $z'=0$.\n\n(B) A Gaussian channel transforms Gaussian states into Gaussian\nstates. This follows from the definition of Gaussian state\nand the relation (\\ref{ce}).\n\n(C) Linear Bosonic channels are covariant with respect to phase\nspace translations. That is if $\\rho^z=V(-\\Delta^{-1}z)\\rho\nV(-\\Delta^{-1}z)^$ is a shift of $\\rho$ by $z$, $T[\\rho]$ is\nsimilarly shifted by $Kz$.\n\nThere is a dramatic difference in the capacities of a Gaussian\nchannel for classical as opposed to quantum information. Classical\ninformation can be coded by using phase space translates of a\nfixed state as signal states, so the output signals will also be\nphase space translates of each other. Then no matter how much\nnoise the channel may add, if we take the spacing of the input\nsignals sufficiently large, the output states will also be\nsufficiently widely spaced to be distinguishable with near\ncertainty. Therefore the unconstrained {\\it classical capacity is\ninfinite}. The same would be true, of course, for a purely\nclassical channel with Gaussian noise. The classical capacity of\nsuch channels becomes an interesting quantity, however, when the\n``input power'' is taken to be constrained by a fixed value, which\nwe must take as one of the parameters defining the channel. Then\narbitrarily wide spacing of input signals is no longer an\nalternative, because an intrinsic scale for this spacing has been\nintroduced.\n\nThe remarkable fact of quantum information on Gaussian channels is\nthat such an intrinsic scale is already there: it is given by\n$\\hbar$. As we will show, the quantum information capacity is\ntypically bounded even without an energy constraint. Loosely\nspeaking, although we send arbitrarily many well distinguishable\nquantum signals through the channel, coherence in the form of\ncommutator relations is usually lost. Surprisingly, in spite of\nthe infinite classical capacity, the {\\it capacity for quantum\ninformation may be zero}, which means that even joining\narbitrarily many parallel channels with poor coherence properties\nis not good enough for sending a single qubit. This phenomenon\nwill be explained in some detail in Section V.\n\nThe choice of the scalar function $f(z')$ is crucial for the\nquantum transmission properties of the channel. Normalization of\n$T$ requires that $f(0)=1$, and it is clear that $\|f(z')\|\\leq1$\nfor all $z'$, from taking norms in (\\ref{linbos}). Beyond that, it\nis not so easy to see which choices of $f$ are compatible with the\ncomplete positivity. If $f$ decays rapidly, $T^$ maps most\noperators to operators near the identity, which means that there\nis very much noise. On the other hand, there will be a lower limit\nto the noise, depending on the linear transformation $K$. Only\nwhen $K$ is a symplectic linear map and $T$ is reversible, the\nchoice $f(z)\\equiv1$ is possible. Otherwise, there is some\nunavoidable noise.\n\nThere are two basic approaches to the determination of the\nadmissible functions $f$. The first is the familiar constructive\napproach already used in Section II, based on coupling the system\nto an environment, a unitary evolution and subsequent reduction to\na subsystem, with all of these operations in their linear\nBosonic/Gaussian form. Basically this reduces the problem to\nlinear transformations of systems of canonical operators. This\nwill be described in Subsection~B, and used for the calculation of\nentropy exchange in Subsection~C. Alternatively, one can describe\nthe admissible functions $f$ by a twisted positive definiteness\ncondition, and this will be used for evaluating the bound\n$C_\\Theta(T)$ in Subsection~D.\n\n\n\\subsection{Bosonic channels via transforming canonical operators}\n\nLet $R,R_{E}$ be vectors of canonical observables in\n${\\cal H},{\\cal H}_{E}$, with the commutation matrices $\\Delta,\\Delta_{E}$.\nConsider the linear transformation\n\\begin{equation}\nR'=KR+K_{E}R_{E,} \\label{chan}\n\\end{equation}\nwhere $K,K_{E}$ are real matrices (to simplicfy notations we write\n$R, R_{E}$ instead of $R\\otimes I_{E}, I\\otimes R_{E}$ etc.) Then\nthe commutation matrix and the correlation with respect to $R'$\nare computed via (\\ref{alpha}) with $m=0$, namely\n\\[\n\\alpha'-\\frac{i}{2}\\Delta '={\\rm Tr}R'\\rho'\nR^{\\prime T}.\n\\]\nWe apply this to the special case $\\rho'=\\rho\\otimes\\rho_E$, where\n$\\rho_{E}$ and $\\rho$ are density operators in\n${\\cal H}_{E}$ and ${\\cal H}$ with the correlation\nmatrices $\\alpha_{E}$ and $\\alpha$, respectively.\nThen using (\\ref{chan}), we obtain\n\\begin{eqnarray}\n\\quad \\Delta' &=&K\\Delta K^{T}+K_{E}\\Delta_{E}K_{E}^{T} \\nonumber\\\\\n\\alpha' &=&K\\alpha K^{T}+K_{E}\\alpha_{E}K_{E}^{T}. \\label{trans}\n\\end{eqnarray}\nOf course, the operators $R'$ need not form a complete set of\nobservables in ${\\cal H}\\otimes{\\cal H}_{E}$, but in any case $\\alpha'$\nis the correlation matrix of a system containing just the\ncanonical variables $R'$, and it is this state which we will\nconsider as the output state of the channel.\n\nFor fixed state $\\rho_E$ (state of the ``environment'') the\nchannel transformation taking the input state $\\rho$ to the output\n$\\rho'$ is described most easily in terms of characteristic\nfunctions:\n\\begin{equation}\n\\phi '(z')= \\phi (K^{T}z') \\phi _{E}\\left( K_{E}^{T}z'\\right). \\label{cE}\\\\\n\\end{equation}\nWe can write this as a linear Bosonic channel in the form\n(\\ref{ce}) with\n\\begin{equation}\\label{prefactor}\n f(z')=\\phi _{E}\\left( K_{E}^{T}z'\\right)={\\rm Tr}\\rho_{E}V_{E}( K_{E}^{T}z')\n\\end{equation}\nThus the factor $f$ is expressed in terms of the characteristic\nfunction of the initial state of the environment. Obviously, the\nchannel is Gaussian if and only if this state is Gaussian.\n\nIf we want to get the state of the environment after the channel\ninteraction, as required in the definition of exchange entropy, we have to\nsupplement the linear equation (\\ref{chan}) by a similar equation\nspecifying the environment variables $R_E'$ after the interaction:\n\\begin{eqnarray}\nR' &=&KR+K_{E}R_{E,} \\\\\nR_{E}' &=&LR+L_{E}R_{E,}\n\\end{eqnarray}\nAssuming that $Z=Z'$ and $\\Delta'=\\Delta$, one can always choose\n$L,L_E$ such that the combined transformation is canonical, i.e.,\npreserves the commutation matrix\n\\[\n\\left[\n\\begin{array}{cc}\n\\Delta & 0 \\\\\n0 & \\Delta_{E}\n\\end{array}\n\\right] .\n\\]\nThen the channel $T_{E}:\\rho \\rightarrow \\rho_{E}'$ can be defined\nby the relation\n\\[\nT_{E}^{}\\left[ V_{E}(z_{E})\\right] =V(L^{T}z_{E})\\cdot \\phi_{E}\\left(\nL_{E}^{T}z_{E}\\right) ,\n\\]\nand is thus also linear Bosonic.\n\n\\subsection{Maximization of mutual information}\n\nThe estimate for the entanglement assisted classical capacity\nsuggested by \\cite{Shor} is the maximum of the quantum mutual\ninformation (\\ref{q-mutual}) over all states satisfying an\nappropriate energy constraint. Evaluating this maximum becomes\npossible by the following result:\\footnote{ The proof of this\ntheorem was stimulated by a question posed to one of the authors\n(A.H.) by P. W. Shor.} {\\it Let $T$ be a Gaussian channel. The\nmaximum of the mutual information $I(\\rho)$ over the set of states\n$\\Sigma \\left( m,\\alpha \\right)$ with given first and second\nmoments is achieved on the Gaussian state.}\n\n{\\it Proof }(sketch). By purification (if necessary), we can\nalways assume that $\\rho_{E}$ is pure Gaussian. Then we can write\n\\[\nI(\\rho )=H(\\rho )+H(T[\\rho ])-H(T_{E}[\\rho ]).\n\\]\nLet $\\rho_{0}$ be the unique Gaussian state in $\\Sigma \\left( m,\\alpha\n\\right) $. For simplicity we assume here that $\\rho_{0}$ is nondegenerate.\nThe general case can be reduced to this by separating the pure component in\nthe tensor product decomposition of $\\rho_{0}$. The function $I(\\rho )$ is\nconcave and its directional derivative at the point $\\rho_{0}$ is (cf. \\cite\n{Shor})\n\\begin{eqnarray}\n \\nabla_{X}I(\\rho_{0})\n &=&{\\rm Tr}X(\\ln \\rho_{0}+I)+{\\rm Tr}{\\cal \\ }T[X](\\ln T[\\rho_{0}]+I)\n\\nonumber\\\\\n &&\\quad-{\\rm Tr}T_{E}[X](\\ln T_{E}[\\rho_{0}]+I).\n\\nonumber\n\\end{eqnarray}\nBy using dual maps this can be modified to\n\\begin{eqnarray}\n\\nabla_{X}I(\\rho_{0})\n &=&{\\rm Tr}X\\Bigl\\{ \\ln \\rho_{0}+T^[\\ln T[\\rho_{0}]]\n\\nonumber\\\\\n &&\\qquad\\quad -T_{E}^[\\ln T_{E}[\\rho_{0}]]+I\\Bigr\\} .\n\\label{h2}\n\\end{eqnarray}\nNow by property (B) of Gaussian channels, the operators\n$\\rho_{0},T[ \\rho_{0}],T_{E}[\\rho_{0}]$ are (nondegenerate)\nGaussian density operators, hence their logarithms are quadratic\npolynomials in the corresponding canonical variables (see Appendix\nin \\cite{Sohma}). By property (A) the expression in curly brackets\nin (\\ref{h2}) is again a quadratic polynomial in $R$, that is a\nlinear combination of the constraint operators in $\\Sigma \\left(\nm,\\alpha \\right)$. Therefore, the sufficient condition (\\ref{usl})\nin the Appendix is fulfilled and $I(\\rho)$ achieves its maximum at\nthe point $\\rho_{0}\\in \\Sigma \\left( m,\\alpha \\right) $.\n\nThis theorem implies that the maximum of $I(\\rho)$ over a set of\ndensity operators defined by arbitrary constraints on the first\nand second moments is also achieved on a Gaussian density\noperator. In particular, for an arbitrary quadratic Hamiltonian\n$H$ the maximum of $I(\\rho)$ over states with constrained mean\nenergy ${\\rm Tr}\\rho H$ is achieved on a Gaussian state. The\nenergy constraint is linear in terms of the correlation matrix:\n\\[\n{\\rm Sp}(\\epsilon \\alpha) \\leq N,\n\\]\nwhere $\\epsilon $ is the diagonal energy matrix (see \\cite{Sohma}).\n\nWhen $\\rho $ and $T$ are Gaussian, the quantities $H(\\rho )$,\n$H(T[\\rho ]),H(\\rho ,T)$ and $I(\\rho,T ), J(\\rho,T )$ can in\nprinciple be computed by using formulas (\\ref{abs}),\n(\\ref{trans}), (\\ref{da12}). Namely, $H(T[\\rho ])$ is given by\nformula (\\ref{abs}) with $\\alpha $ replaced by $\\alpha '$ computed\nvia (\\ref{trans}), and\n\\[\nH(\\rho,T)=\\frac{1}{2}{\\rm Sp}\\,g\\left({\\rm abs}(\\Delta_{12}^{-1}\\alpha_{12}')\n -\\frac{I}{2}\\right) ,\n\\]\nwhere\n\\begin{eqnarray}\n\\alpha_{12}'&=&\\left[\n\\begin{array}{cc}\n\\alpha ' & K\\beta \\\\\n\\beta^TK^T & \\alpha\n\\end{array}\n\\right]\\nonumber\\\\\n\\beta&=&\\Delta \\sqrt{-(\\Delta ^{-1}\\alpha )^{2}-I/4} \\nonumber\n\\end{eqnarray}\nis computed by inserting (\\ref{chan}) into\n\\[\n\\alpha_{12}'-\\frac{i}{2}\\Delta_{12}'={\\rm Tr}\\left(\nR',R_{2}\\right) \\rho \\left( R',R_{2}\\right) ^{T},\n\\]\nwhere $R_{2}$ are the (unchanged) canonical observables of the\nreference system.\n\n Alternatively, the entropy exchange can be\ncalculated as the output entropy $H(T_{E}[\\rho])$ if an explicit\ndescription of $T_{E}$ is available. We shall demonstrate this\nmethod in the example of one-mode channels in the Appendix.\n\n\n\\subsection{Norms of Gaussian Transformations}\nThe transposition operation on a Bosonic system can be realized as\nthe time reversal operation, i.e., the operation reversing the\nsigns of all momentum operators, while leaving the position\noperators unchanged. Obviously, the dual $T^{}$ then takes Weyl\noperators into Weyl operators. So transposition is just like a\nlinear Bosonic channel, albeit without the scalar factor $f(z') $\nin Equation~(\\ref{ce}). It is this factor which makes the\ndifference between positivity and complete positivity, and also\nenters the norm $\\Vert T\\Vert_{{\\rm cb}}$. In this Subsection we\nwill provide general criteria for deciding complete positivity and\ncomputing the norm of general linear Bosonic transformations.\n\nThese are by definition the operators $T$ acting on Weyl operators\naccording to (\\ref{linbos}) where $f(z')$ is a scalar factor. We\nwill assume for simplicity (and in view of the applications in the\nfollowing sections) that the antisymmetric form\n\\begin{equation}\n\\Delta''(z_{1},z_{2})=\\Delta' (z_{1},z_{2})-\\Delta (K^T z_{1},K^T\nz_{2}) \\label{diffsymp}\n\\end{equation}\nis non-degenerate. This makes the space $Z'$ with the form $\\Delta\n^{\\prime \\prime }$ into a phase space in its own right. With the\nintroduction of suitable canonical coordinates it becomes\nisomorphic to $(Z,\\Delta )$, so there exists an invertible linear\noperator $A:Z\\to Z$ such that $\\Delta''(z_{1},z_{2})=\\Delta\n(A^{-1}z_{1},A^{-1}z_{2})$.\n\nIf $f$ is continuous and has sufficient decay properties (which\nwill be satisfied in our applications), there is a unique trace\nclass operator $\\rho$ determined by the equation\n\\begin{equation}\n{\\rm Tr}(\\rho V(z))=f(Az). \\label{rhof}\n\\end{equation}\n{\\it Then $T$ is completely positive if and only if $\\rho $ is a\npositive trace class operator}. This is a standard result in the\ntheory of quasi-free maps on CCR-algebras \\cite{demoen}. It is\nproved by showing that both properties are equivalent to a\n``twisted positive definiteness condition'', namely the positive\ndefiniteness of all matrices of the form\n\\[\nM_{rs}=f(z_{r}-z_{s})\\,\\exp \\bigl(-\\frac{i}{2}\\Delta'\n(z_{r},z_{s})\n +\\frac{i}{2}\\Delta (K^T z_{r},K^T z_{s})\\bigr),\n\\]\nwhere $z_{1},\\ldots ,z_{n}$ are an arbitrary choice of $n$ phase\nspace points.\n\nIf $\\rho $ is a non-positive hermitian trace class operator, it\nhas a unique decomposition into positive and negative part: $\\rho\n=\\rho _{+}-\\rho_{-}$ such that $\\rho_{\\pm }\\geq 0$, and\n$\\rho_{+}\\rho_{-}=0$. Then $\|\\rho \|=\\rho_{+}+\\rho_{-}$ and the\ntrace norm is $\\Vert \\rho \\Vert _{1}={\\rm Tr}(\\rho_+)+{\\rm\nTr}(\\rho_-)$. Inserting $\\rho_{\\pm }$ into Equation~(\\ref {rhof})\ninstead of $\\rho $, we get two functions $f_{\\pm }$ on phase space\nand from Equation~(\\ref{linbos}) two linear Bosonic\ntransformations $T_{\\pm } $ with $T=T_{+}-T_{-}$. By the criterion\njust proved, $T_{+}$ and $T_{-}$ are completely positive. Hence\n\\begin{eqnarray}\n\\Vert T\\Vert_{{\\rm cb}} &\\leq &\\Vert T_{+}\\Vert_{{\\rm cb}}+\\Vert\nT_{-}\\Vert_{{\\rm cb}}=\\Vert T_{+}({\\bf 1})\\Vert +\\Vert T_{-}({\\bf\n1})\\Vert \\nonumber \\\\ &=&f_{+}(0)+f_{-}(0)={\\rm Tr}(\\rho_+)+{\\rm\nTr}(\\rho_-)\n =\\Vert \\rho \\Vert_{1}\\label{Tcb}\n\\end{eqnarray}\n\nIf the factor $f$ is a Gaussian, i.e.,\n\\begin{equation}\nf(z)=\\exp \\bigl(-\\frac{1}{2}\\,z^{T}\\beta z\\bigr) \\label{Gaussf}\n\\end{equation}\nfor some positive definite matrix $\\beta $, we can go one step\nfurther. In this case we may decompose $\\beta $ into normal modes\nwith respect to $\\Delta ^{\\prime \\prime }$, which decomposes $T$\ninto a tensor product of one-mode Gaussian transformations\n$T_{\\ell }$, for each of which $\\Vert T_{\\ell }\\Vert_{{\\rm cb}}$\nmay be computed separately by the above method. This amounts to\ncomputing the trace norm of the operator $\\rho_{\\gamma }$ given by\n(\\ref{onemode-rho}) with arbitrary positive $\\gamma$. The absolute\nvalue of $\\rho _{\\gamma }$ is obtained by taking absolute values\nof all the eigenvalues, which still makes $\\Vert \\rho_{\\gamma\n}\\Vert_{1}$ a geometric series:\n\\begin{equation}\n\\Vert \\rho_{\\gamma }\\Vert_{1}=\\frac{1}{\\gamma +1/2}\\sum_{n\n=0}^{\\infty }\n \\left\| \\frac{\\gamma -1/2}{\\gamma +1/2}\\right\|^{n}\n =\\max \\{1,\\frac{1}{2\\gamma }\\}. \\label{normrhogamma}\n\\end{equation}\nThis is all the information we need for the estimates of quantum\ncapacity in the following Section.\n\n\\section{The Case of One Mode}\n\\subsection{Attenuation/amplification channel with classical noise}\n\nThe channel we consider in this Section combines\nattenuation/amplification \\cite{Hol98} with additive classical\nnoise \\cite{Shor}. It can also be described as the most general\none-mode gauge invariant channel, or in quantum optics\nterminology, the most general one-mode channel not involving\nsqueezing. Channels of this type were also used in \\cite{clalim}\nas the basis for an analysis of the classical limit of quantum\nmechanics.\n\n Let us consider the CCR with one degree of freedom\n$a=\\frac{1}{\\sqrt{2\\hbar} }(q+ip)$, and let $a_{0}$ be another\nmode in the Hilbert space ${\\cal H}_{0}={\\cal H}_{E}$ of an\n``environment''. Let the environment be initially in the vacuum\nstate, i.e., in the state with the characteristic function\n(\\ref{one-mode-cf}) with $N=0$. Let $\\xi$ be a complex random\nvariable with zero mean and variance $N_{c}$ describing additive\nclassical noise in the channel. The linear attenuator with\ncoefficient $k<1$ and the noise $N_{c}$ is described by the\ntransformation\n\\[\n a'=ka+\\sqrt{1-k^{2}}a_{0}+\\xi\n\\]\nin the Heisenberg picture. Similarly, the linear amplifier with\ncoefficient $k>1$ is described by the transformation\n\\[\na'=ka+\\sqrt{k^{2}-1}a_{0}^{\\dagger }+\\xi .\n\\]\nIt follows that the corresponding transformations $T[\\rho ]$ of\nstates in the Schr\\\"odinger picture both have the characteristic\nfunction\n\\begin{eqnarray} \\label{atten}\n {\\rm Tr}&T[\\rho ]&V(z)= {\\rm Tr}\\rho V(kz)\n\\times\\nonumber\\\\&&\\quad\\times\n \\exp \\left[-\\frac{\\hbar}{2}\\bigl(\n \|k^{2}-1\|/2+N_{c}\\bigr)\\, \|z\|^{2}\\right].\n\\end{eqnarray}\n\nLet the input state $\\rho$ of the system be the elementary\nGaussian with characteristic function (\\ref{one-mode-cf}). Then the\nentropy of $\\rho$ is $H(\\rho)=g(N)$. From\n(\\ref{atten}) we find that the output state $T[\\rho]$ is again\nelementary Gaussian with $N$ replaced by\n\\[\nN'=k^{2} N + N'_0, \\]\n where\n\\[\n N'_0=\\max\\{0,(k^2-1)\\} +N_{c}\n\\]\nis the value of the output mean photon number corresponding to the\ninput vacuum state. Then\n\\begin{equation}\\label{out-1mode}\n H(T[\\rho])=g(N').\n\\end{equation}\n\nNow we calculate the exchange entropy $H(\\rho,T)$. The (pure)\ninput state $\\rho_{12}$ of the extended system ${\\cal\nH}_{1}\\otimes {\\cal H}_{2}$ is characterized by the $2\\times\n2-$matrix (\\ref{d-1a}). The action of the extended channel\n$(T\\otimes {\\rm id})$ transforms this matrix into\n\\[\n\\Delta_{12}^{-1}\\widetilde{\\alpha }_{12}\\leftrightarrow \\left[\n\\begin{array}{ll} i(N'+\\frac{1}{2}) & k\\sqrt{N(N+1)} \\\\\nk\\sqrt{N(N+1)} & -i(N+\\frac{1}{2}) \\end{array} \\right]. \\]\nFrom\nformula (\\ref{H->g}) we deduce\n$H(\\rho,T)=g(\|\\lambda_{1}\|-\\frac{1}{2})+g(\|\\lambda_{2}\|-\\frac{1}{2})$,\nwhere $\\lambda_{1},\\lambda_{2}$ are the eigenvalues of the complex\nmatrix in the right-hand side. Solving the characteristic equation\nwe obtain \\begin{equation} \\lambda_{1,2}=\\frac{i}{2}\\left(\n(N'-N)\\pm D\\right) , \\label{eig} \\end{equation} where\n$D=\\sqrt{\\left( N+N'+1\\right) ^{2}-4k^{2}N(N+1)}$. Hence\n\\begin{eqnarray}\\label{exch-1mode}\n H(\\rho,&&T)=\\\\\n &&=g\\left( \\frac{D+N'-N-1}{2}\\right)\n +g\\left(\\frac{D-N'+N-1}{2 }\\right).\\nonumber\n\\end{eqnarray}\n\n\nNow using the theorem of Section 5, we can calculate the quantity\n\\[\nC_{e}(T)= I(\\rho ,T)=H(\\rho ) + H(T[\\rho]) - H(\\rho ,T)\n\\]\n as a function of the\nparameters $N,k,N_{c}$, and try to compare it\nwith the one-shot unassisted classical capacity of the\nchannel $C_1 (T)$ given by expression (\\ref{oneshotunassist})\nwhere the maximum is taken over all probability distributions\n$\\left\\{ p_{i}\\right\\} $ and the collections of density operators\n$\\left\\{ \\rho _{i}\\right\\} $, satisfying the power constraint\n$\\sum_{i}p_{i}$Tr$\\rho _{i}a^{\\dagger }a\\leq N$. It is quite\nplausible, but not yet proven that this maximum is achieved on\ncoherent states with the Gaussian probability density $p(z)=\\left(\n\\pi N\\right) ^{-1}\\exp \\left( -\|z\|^{2}/N\\right) $, giving the\nvalue\n\\[\n\\underline{C}_{1}(T)=g\\left( N'\\right) -g\\left( N_{0}'\\right).\n\\]\n\nThe ratio\n\\begin{equation}\\label{gain}\n G=\\frac{C_{e}}{\\underline{C}_{1}}\n\\end{equation}\nthen gives at least an upper\nbound for the {\\it gain} of using entanglement-assisted versus\nunassisted classical capacity. In particular, when the signal mean\nphoton number $N$ tends to zero while $N'_0 >0$,\n\\begin{eqnarray}\n\\underline{C}_{1}(T)&\\sim& N k^2 \\log\\left(\\frac{N'_0\n +1}{N'_0}\\right),\\\\\n C_e (T)&\\sim& - N \\log N /(N'_0 +1),\n\\end{eqnarray}\nand $G$ tends to infinity as $ - \\log N$.\n\nThe plots of $G$ as function of $k$ for $N_{c}=0$, and as a\nfunction of $N_c$ for $k=1$ are given in Figure~1 and Figure~2,\nrespecitively. The behavior of the entropies\n$H(T[\\rho]),H(\\rho,T)$ as functions of $k$ for $N_{c}=0$ is clear\nfrom Figure~3. For all $N$ the coherent information\n$H(T[\\rho])-H(\\rho,T)$ turns out to be positive for $k>1/\\sqrt{2}$\nand negative otherwise. It tends to $-H(\\rho )$ for $k\\rightarrow\n0$, is equal to $H(\\rho)$ for $k=1$, and quickly tends to zero as\n$k\\rightarrow \\infty$ (see Figure~4).\n\n\\subsection{Estimating the quantum capacity}\nGoing back to the upper bound for quantum capacity in Section IV,\nwe see that $T$ is given by equation~(\\ref {linbos}) with $Kz=kz$\nand\n\\[\nf(z)=\\exp (-\\frac{(\|k^{2}-1\|/2+N_{c})}{2}\\ \|z\|^{2}).\n\\]\nThen $\\Delta ^{\\prime \\prime }=(1-k^{2})\\Delta $, and the operator\n$A$ mapping the symplectic form $\\Delta ^{\\prime \\prime }$ to the\nstandard form $\\Delta $ is multiplication by $\\sqrt{\|k^{2}-1\|}$,\ncombined for $k>1$ with a mirror reflection to change the sign.\nThis leaves\n\\begin{equation}\nf(Az)=\\exp \\left(-\\frac{(\|k^{2}-1\|/2+N_{c})}{2\|k^{2}-1\|}\\ \|z\|^{2}\\right),\n\\label{fLT}\n\\end{equation}\ni.e., $\\rho =\\rho_{\\gamma }$ with equations~(\\ref{onemode-rho})\nand (\\ref{rhof}), where $\\gamma =1/2+N_{c}/\|k^{2}-1\|$. This is the\nverification of the complete positivity of $T$ by the methods of\nthe above section. Of course, this is strictly speaking\nunnecessary, because $T$ was constructed explicitly as a\ncompletely positive operator in terms of its dilation in\nSubection~IV.A.\n\nBut let us now consider $T\\Theta $. It is also a Bosonic linear\ntransformation, in which $\\Theta $ only has the effect of changing\nthe sign of the symplectic form, without changing $f$. Thus\n$\\Delta ^{\\prime \\prime }=(1+k^{2})\\Delta $, and\n\\[\nf(Az)=\\exp \\left( -\\frac{(\|k^{2}-1\|/2+N_{c})}{2\|k^{2}+1\|} \|z\|^{2}\n\\right) .\n\\]\nwhich seems like a rather minor change over Equation~(\\ref{fLT}). However,\nwe now get $\\rho =\\rho_{\\gamma }$ with $\\gamma\n=(\|k^{2}-1\|/2+N_{c})/(k^{2}+1)$ which is not necessarily $\\geq 1/2$, so\n$T\\Theta $ is not necessarily completely positive. Taking the logarithm of\nEquation~(\\ref{normrhogamma}) we get\n\\begin{eqnarray}\nQ_{\\Theta }(T)&\\leq& \\max\\{0,\n\\nonumber\\\\\n &&\\ \\log_{2}(k^{2}+1)-\\log_{2}(\|k^{2}-1\|+2N_{c})\\}.\n\\label{1modBound}\n\\end{eqnarray}\nIn particular, for $\\gamma \\geq 1/2$, i.e., for $N_{c}\\geq\n(\|k^{2}+1\|-\|k^{2}-1\|)/2 =\\max \\{1,k^{2}\\}$, the capacities\n$Q_{\\Theta }(T)$, and hence $Q_{\\varepsilon }(T)$ and $Q(T)$ all\nvanish.\n\nThis upper bound on quantum capacity is interesting to compare\nwith the quantity $Q_{G}(T)=\\sup J(\\rho ,T)$, where $J(\\rho\n,T)=H(T[\\rho])-H(\\rho ,T)$, and the supremum is taken over all\n{\\it Gaussian} input states. Since the coherent information\n\\begin{eqnarray}\n J(\\rho ,T)&=& g(N')-g\\left( \\frac{D+N'-N-1}{2}\\right)-\n \\nonumber\\\\\n &&\\quad -g\\left( \\frac{D-N'+N-1}{2}\\right)\n \\label{coh-1mode}\n\\end{eqnarray}\nincreases with the input\npower $N$, we obtain\n\\begin{eqnarray}\\label{QG}\n Q_{G}(T)&=&\\lim_{N\\rightarrow \\infty }J(\\rho ,T)\\\\\n &=&\\log k^{2}-\\log \|k^{2}-1\|-\n g\\left( N_{c}/\|k^{2}-1\|\\right),\\nonumber\n\\end{eqnarray}\nwhich is in a good agreement with the upper bound\n(\\ref{1modBound})(see Figure 4).\n\n\\acknowledgments{A.H. appreciates illuminating discussion of\nfragments of the unpublished paper \\cite{Shor} with C. H. Bennett\nand P. W. Shor. He acknowledges the hospitality of R. W. in the\nInstitute for Mathematical Physics, Technical University of\nBraunschweig, which he was visiting with an A. von Humboldt\nResearch Award.}\n\n\\appendix\n\\section{Minimizing convex function of a density operator.}\n\nThere is a useful lemma in classical information theory which\ngives necessary and sufficient conditions for the global minimum\nof a convex function of probability distributions in terms of the\nfirst partial derivatives. The lemma is based on general\nKuhn-Tucker conditions and can be generalized to functions\ndepending on density operators rather than probability\ndistributions.\n\nLet $F$ be a convex function on the set of density operators\n$\\Sigma $, and $\\rho_{0}$ a density operator. In order $F$ to\nachieve minimum on $\\rho _{0}, $ it is necessary and sufficient\nthat for arbitrary density operator $\\sigma $ the convex function\n$F((1-t)\\rho_{0}+t\\sigma )$ of the real variable $t$ achieves\nminimum at $t=0$. For this, it is necessary and sufficient that\n\\begin{equation}\n\\nabla_{X}F(\\rho_{0})\\equiv \\left. \\frac{d}{dt}\\right\|_{t=0}F((1-t)\\rho\n_{0}+t\\sigma )\\geq 0, \\label{nsc}\n\\end{equation}\nwhere $X=\\sigma -\\rho_{0}$, and $\\nabla_{X}F(\\rho_{0})$ is the\ndirectional derivative of $F$ in the direction $X$, assuming that\nthe derivatives exist. If $\\sigma =\\sum_{i}p_{i}$ $\\sigma_{i}$,\nthen $\\nabla _{X}F(\\rho_{0})=\\sum_{i}p_{i}$\n$\\nabla_{X_{i}}F(\\rho_{0})$, where $X_{i}=\\sigma_{i}-$ $\\rho_{0}$.\nTherefore it is necessary and sufficient that (\\ref{nsc}) holds\nfor pure $\\sigma $.\n\nIf $(1-t)\\rho_{0}+t\\sigma \\geq 0$ for small negative $t$, then we\nsay that the direction $\\overrightarrow{\\sigma \\rho_{0}}$ is {\\it\ninner}. In that case (\\ref{nsc}) takes the form\n\\begin{equation}\n\\nabla_{X}F(\\rho_{0})=0. \\label{nsp}\n\\end{equation}\nIf $\\rho_{0}$ is nondegenerate, then the direction\n$\\overrightarrow{\\sigma \\rho_{0}}$ is inner for arbitrary pure\n$\\sigma $ in the range of $\\sqrt{\\rho_{0}}$, and the necessary and\nsufficient condition for the minimum is that (\\ref{nsp}) holds for\narbitrary such $\\sigma $.\n\nLet $A_{i},i=1,\\dots ,r$ be a collection of selfadjoint {\\it\nconstraint operators}. Assume that for some real constants\n$\\lambda_{i}$\n\\begin{equation}\n\\nabla_{X}F(\\rho_{0})={\\rm Tr}X\\sum_{i}\\lambda_{i}A_{i}. \\label{usl}\n\\end{equation}\nIt follows that the convex function $F(\\rho )-{\\rm Tr}\\rho \\sum_{i}\\lambda\n_{i}A_{i}$ achieves minimum at the point $\\rho_{0}$, hence the function\n$F(\\rho )$ achieves minimum at the point $\\rho_{0}$ under the constraints\n${\\rm Tr\\ }\\rho A_{i}={\\rm Tr\\ }\\rho_{0}A_{i},\\quad\ni=1,\\dots,r$ .\n\n\\section{Quantum signal plus classical noise.}\n\nLet us consider CCR with one degree of freedom described by one\nmode annihilation operator $a=\\frac{1}{\\sqrt{2\\hbar} }(q+ip)$, and\nconsider the transformation\n\\[\na'=a+\\xi ,\n\\]\nwhere $\\xi $ is a complex random variable with zero mean and\nvariance $N_{c}$. This is a transformation of the type\n(\\ref{chan}) with $\\Delta_{E}=0$, which describes quantum mode in\nclassical Gaussian environment. The action of the dual channel is\n\\[\nT^[f(a,a^{\\dagger })]\n =\\int f(a+z,(a+z)^{\\dagger })\\mu_{N_{c}}(d^{2}z),\n\\]\nwhere $z=\\frac{1}{\\sqrt{2\\hbar} }(x+iy)$ is now complex variable,\nand $\\mu_{N_{c}}(d^{2}z)$ is complex Gaussian probability measure\nwith zero mean and variance $N_{c}$, while the channel itself can\nbe described by the formula\n\\begin{equation}\nT[\\rho ]=\\int D(z)\\rho D(z)^\\mu_{N_{c}}(d^{2}z), \\label{spn}\n\\end{equation}\nwhere $D(z)=\\exp i\\left( za^{\\dagger }-\\bar{z}a\\right) $ is the\ndisplacement operator.\n\nThe entanglement-assisted classical capacity of the channel\n(\\ref{spn}) was first studied in \\cite{Shor} by using rather\nspecial way of purification and the computation of the entropy\nexchange. A general approach following the method of \\cite{Hol98}\nwas described in Sections IV-V; here we give an alternative\nsolution based on the computation of the environment entropy.\n\nFor this we need to extend the environment to a quantum system in\na pure state. Consider the environment Hilbert space ${\\cal\nH}_{E}=L^{2}(\\mu_{N_{c}})$ with the vector $\|\\Psi_{0}\\rangle$\ngiven by the function identically equal to 1. The tensor product\n${\\cal H}\\otimes {\\cal H}_{E}$ can be realized as the space\n$L_{{\\cal H}}^{2}(\\mu_{N_{c}})$ of $\\mu_{N_{c}}$-square integrable\nfunctions $\\psi (z)$ with values in ${\\cal H}$. Define the unitary\noperator $U$ in ${\\cal H}\\otimes {\\cal H}_{E}$ by\n\\[\n(U\\psi )(z)=D(z)\\psi (z).\n\\]\nThen\n\\[\nT[\\rho]={\\rm Tr}_{{\\cal H}_{E}}U\\left( \\rho \\otimes\n\|\\Psi_{0}\\rangle\\langle\\Psi _{0}\|\\right) U^,\n\\]\nwhile\n\\[\nT_{E}[\\rho ]={\\rm Tr}_{{\\cal H}}U\\left(\n \\rho \\otimes \|\\Psi_{0}\\rangle\\langle\\Psi_{0}\|\\right) U^.\n\\]\nThis means that $T_{E}[\\rho ]$ is an integral operator in\n$L^{2}(\\mu _{N_{c}})$ with the kernel\n\\begin{eqnarray}\n K(z,z')&=&{\\rm Tr}D(z)\\rho_{0}D(z')^\\\\\n &=&\\exp (i\\Im \\bar{z}'z-(E+1/2)\|z-z'\|^{2}).\n\\end{eqnarray}\nLet us define unitary operators $V(z_{1},z_{2})$ in $L^{2}(\\mu_{N_{c}})$\nby\n\\begin{eqnarray}\n V(z_{1},z_{2})&\\psi(z)&= \\psi (z+z_{2})\\times\\\\\n &&\\ \\times\\exp\n \\left[ i\\Re \\overline{z_{1}}(z+\\frac{z_{2}}{2})-\\frac{1}{N_{c}}\n \\Re \\overline{z_{2}}(z+\\frac{z_{2}}{2})\\right] .\n\\end{eqnarray}\nThe operators $V(z_{1},z_{2})$ satisfy Weyl-Segal CCR with two degrees of\nfreedom with respect to the symplectic form\n\\[\n\\Delta ((z_{1},z_{2}),(z_{1}',z_{2}'))\n =\\Re \\left( \\bar{z}_{1}'z_{2}-\\bar{z}_{1}z_{2}'\\right) .\n\\]\nPassing over to the real variables $x,y$ one finds the\ncorresponding commutation matrix\n\\[\n\\Delta_{E}=\\hbar\\left[\n\\begin{array}{cccc}\n0 & 0 & -1 & 0 \\\\\n0 & 0 & 0 & -1 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0\n\\end{array}\n\\right] .\n\\]\n\nThe characteristic function of the operator $T_{E}[\\rho_{0}]$ is\n\\[\n{\\rm Tr}T_{E}[\\rho_{0}]V(z_{1},z_{2})\n =\\int \\left. V(z_{1},z_{2})K(\\check{z},z)\\right\|_{\\check{z}=z}\n \\mu_{N_{c}}(d^{2}z),\n\\]\nwhere $V(z_{1},z_{2})$ acts on $K$ as a function of the argument $\\check{z}$.\nEvaluating the Gaussian integral, we obtain that it is equal to\n\\[\n\\exp \\left[ -\\frac{1}{4}\\left( N_{c}\|z_{1}\|^{2}+2N_{c}\n\\Im \\bar{z}_{1}z_{2}+\\frac{D^{2}}{N_{c}}\|z_{2}\|^{2}\\right) \\right] ,\n\\]\n(where now $D=\\sqrt{(N_c + 1)^2 + 4N_c N}$), which is Gaussian\ncharacteristic function with the correlation matrix\n\\[\n\\alpha_{E}'=\\frac{\\hbar}{2}\\left[\n\\begin{array}{cccc}\nN_{c} & 0 & 0 & N_{c} \\\\\n0 & N_{c} & -N_{c} & 0 \\\\\n0 & -N_{c} & \\frac{D^{2}}{N_{c}} & 0 \\\\\nN_{c} & 0 & 0 & \\frac{D^{2}}{N_{c}}\n\\end{array}\n\\right] .\n\\]\nThus\n\\[\n\\ \\Delta_{E}^{-1}\\alpha_{E}'=\\frac{1}{2}\\left[\n\\begin{array}{cccc}\n0 & -N_{c} & \\frac{D^{2}}{N_{c}} & 0 \\\\\nN_{c} & 0 & 0 & \\frac{D^{2}}{N_{c}} \\\\\n-N_{c} & 0 & 0 & -N_{c} \\\\\n0 & -N_{c} & N_{c} & 0\n\\end{array}\n\\right] .\n\\]\nBy using Pauli matrix $\\sigma_{y}$, we can write it as\n\\begin{eqnarray}\n\\frac{1}{2}\\left[\n\\begin{array}{cc}\n-i\\sigma_{y}N_{c} & \\frac{D^{2}}{N_{c}} \\\\\n-N_{c} & -i\\sigma_{y}N_{c}\n\\end{array}\n\\right] &&=\\\\\n=\\frac{1}{2}\\left[\n\\begin{array}{cc}\nI & 0 \\\\\n0 & \\sigma_{y}\n\\end{array}\n\\right]&& \\left[\n\\begin{array}{cc}\n-i\\sigma_{y}N_{c} & \\sigma_{y}\\frac{D^{2}}{N_{c}} \\\\\n-\\sigma_{y}N_{c} & -i\\sigma_{y}N_{c}\n\\end{array}\n\\right] \\left[\n\\begin{array}{cc}\nI & 0 \\\\\n0 & \\sigma_{y}\n\\end{array}\n\\right] ,\n\\end{eqnarray}\nhence the absolute values of the eigenvalues of\n$\\Delta_{E}^{-1}\\alpha_{E}'$ are the same as that of the matrix\n\\[\n\\left[\n\\begin{array}{cc}\niN_{c} & -\\frac{D^{2}}{N_{c}} \\\\\nN_{c} & iN_{c}\n\\end{array}\n\\right] ,\n\\]\nwhich coincide with (\\ref{eig}) in the case $k=1$.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Ben} C. 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Smolin, A. V. Thapliyal,\n\\newblock``Entanglement-assisted classical capacity of noisy quantum\nchannel,''\\newblock LANL Report no. quant-ph/9904023.\n\n\\bibitem{Braun} S. L. Braunstein, ``Squeezing as an irreducible resource'',\nLANL Report no. quant-ph/9904002.\n\n\\bibitem{Kimble} A. Furusawa, J. S{\\o}rensen, S. L. Braunstein, C. Fuchs, H. J.\nKimble, E. S. Polzik, Science, vol.{\\bf 282}, 706, 1998.\n\n\\bibitem{demoen} B. Demoen, P. Vanheuverzwijn, A. Verbeure,\n``Completely positive quasi-free maps on the CCR algebra,'' {\\it\nRep. Math. Phys.}, vol.{\\bf 15}, pp. 27-39, 1979.\n\n\\bibitem{Hol75} A.~S. Holevo, \\newblock``Some statistical problems for\nquantum Gaussian states,'' \\newblock {\\it IEEE Transactions on\nInformation Theory}, vol. {\\bf IT-21}, no.5, pp. 533-543, 1975.\n\n\\bibitem{Hol82} A.~S. Holevo, \\newblock {\\it Probabilistic and\nstatistical aspects of quantum theory}, chapter 5, North-Holland,\n1982.\n\n\\bibitem{wer} R. F. Werner, ``Quantum harmonic analysis on phase space,''\n\\newblock{\\it J. Math. Phys.}, vol. {\\bf 25}, pp. 1404-1411, 1984.\n\n\\bibitem{Sohma} A.~S. Holevo, M. Sohma and O. Hirota, \\newblock``The\ncapacity of quantum Gaussian channels ,'' \\newblock{\\it Phys. Rev. A}, vol.\n{\\bf 59}, N3, pp. 1820-1828, 1998.\n\n\\bibitem{Hol98} A. S. Holevo, ``Sending quantum information with Gaussian\nstates,'' LANL Report no. quant-ph/9809022. To appear in Proc. QCM-98, Ed.\nby M. D'Ariano, O.Hirota, P. Kumar.\n\n\\bibitem{Hol72a} A.~S. Holevo,\\newblock``Towards the mathematical theory\nof quantum communication channels,'' \\newblock {\\it Problems of Information\nTransm.}, vol. {\\bf 8}, no.1, pp. 63-71, 1972.\n\n\\bibitem{Kraus} K. Kraus, States, effects and operations, {\\it Lect.\nNotes Phys.}, vol. {\\bf 190}, 1983.\n\n\\bibitem{Lin} G. Lindblad, \\newblock``Quantum entropy and quantum\nmeasurements,'' \\newblock {\\it Lect. Notes Phys.}, vol. {\\bf 378}, Quantum\nAspects of Optical Communication, Ed. by C. Benjaballah, O. Hirota, S.\nReynaud, pp.71-80, 1991.\n\n\\bibitem{Shor} P. W. Shor et al, in preparation.\n\n\\bibitem{lloyd} S. Lloyd, ``The capacity of the noisy quantum\nchannel,'' \\newblock{\\it Phys. Rev. A}, vol. {\\bf 56}, pp. 1613,\n1997.\n\n\\bibitem{smo} D. P. DiVincenzo, P. W. Shor, J. A. Smolin, ``Quantum-channel\ncapacity of very noisy channels,''\\newblock {\\it Phys. Rev. A}, vol. {\\bf 57},\npp.830-839, 1998, LANL Report no. quant-ph/9706061.\n\n\\bibitem{Paulsen} V. I. Paulsen,\n\\newblock{\\it Completely bounded maps and dilations},\nLongman Scientific and Technical 1986\n\n\\bibitem{AhaKitNi} D. Aharonov, A. Kitaev, and N. Nisan,\n\\newblock``Quantum Circuits with Mixed States,''\n\\newblock LANL Report no. quant-ph/9806029.\n\n\\bibitem{ogawa} T. Ogawa, H. Nagaoka, ``Strong converse to the\nquantum channel coding theorem,'' LANL Report no. quant-ph/9808063\nTo appear in IEEE Trans. on Inform. Theory.\n\n\\bibitem{winter} A. Winter, ``Coding theorems and strong converse\nfor quantum channels,'' To appear in IEEE Trans. on Inform.\nTheory.\n\n\\bibitem{agar} R. Simon, M. Selvadoray, G. S. Agarwal, ``Gaussian states\nfor finite number of bosonic degrees of freedom,'' To appear in\n\\newblock{\\it Phys. Rev.}.\n\n\\bibitem{Hel} C.~W. Helstrom, \\newblock {\\it Quantum detection and\nestimation theory}, chapter 5, Academic press, 1976.\n\n\\bibitem{Hol72} A.~S. Holevo,\\newblock``Generalized free states of the\nC$^$-algebra of the CCR,'' \\newblock {\\it Theor. Math. Phys.}, vol.\n{\\bf 6}, no.1, pp. 3-20, 1971.\n\n\\bibitem{clalim} R.F. Werner,\\newblock``The classical limit of quantum\ntheory,'' LANL Report no. quant-ph/9504016.\n\\end{thebibliography}\n\n\\section*{Figures}\n\n\n\\begin{figure}\\label{Fig-Gofk}\n\n\\epsfxsize=8cm \\epsfbox{Gofk.eps} \\caption{{\\it Gain of\nentanglement assistance.}}\\small\\narrower\\noindent Gain\n(\\ref{gain}) as a function of $k$ with $N_c=0$. Parameter=input\nnoise $N$.\n\\end{figure}\n\n\\begin{figure}\\label{Fig-GofNc}\n\\epsfxsize=8cm \\epsfbox{GofNc.eps} \\caption{{\\it Gain of\nentanglement assistance.}}\\small\\narrower\\noindent Gain\n(\\ref{gain}) as a function of $N_c$ with $k=1$. Parameter=input\nnoise $N$.\n\\end{figure}\n\n\\begin{figure}\\label{Fig-OutEx}\n\\epsfxsize=8cm \\epsfbox{OutEx.eps} \\caption{{\\it\nEntropies.}}\\small\\narrower\\noindent {\\tt output} entropy from\n(\\ref{out-1mode}), {\\tt exchange} entropy from (\\ref{exch-1mode})\nwith $N_c=0$.\n\\end{figure}\n\n\\begin{figure}\\label{Fig-capty}\n\\epsfxsize=8cm \\epsfbox{capty.eps} \\caption{{\\it Bounds for\nQuantum Capacity, $N_c=0$.}}\\small\\narrower\\noindent {\\tt J}=coherent\ninformation (\\ref{coh-1mode}) with $N=.7$;\\\\ {\\tt QG}=$Q_G$= bound maximized\nover Gaussians (\\ref{QG});\\\\ {\\tt QT}=bound $Q_\\Theta$ from\ntransposition (\\ref{1modBound});\\\\ {\\tt Z}= zero at $k=1/\\sqrt2$, common\nto all curves of type {\\tt J}.\n\\end{figure}\n\n\\begin{figure}\\label{Fig-capty2} \\epsfxsize=8cm\n\\epsfbox{capty2.eps} \\caption{{\\it Gaussian maximized coherent\ninformation} $Q_G(T)$ as function of $k$ and $N_c$. The shaded\narea is the area, where $Q_\\Theta\\geq0$.}. \\end{figure}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912067.extracted_bib", "string": "{Ben C. H. Bennett, P. W. Shor, ``Quantum information theory,'' IEEE Trans. on Inform. Theory, {IT-44, N6, pp. 2724-2742, 1998." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Hol A.~S. Holevo, \\newblock``Coding theorems for Quantum Channels,'' \\newblock {Tamagawa University Research Review, No.4, 1998. LANL Report no. quant-ph/9809023." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Sch H. Barnum, M. A. Nielsen, B. Schumacher, \\newblock ``Information transmission through noisy quantum channels,''\\newblock{Phys. Rev. A, vol. {A57, pp. 4153-4175, 1998. LANL Report no. quant-ph/9702049." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Cer C. Adami and N. J. Cerf,\\newblock``Capacity of noisy quantum channels,'' \\newblock {Phys. Rev. A, vol. {A56, pp. 3470-3485, 1997; \\newblock LANL Report no. quant-ph/9609024." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Barn H. Barnum, E. Knill, M. A. Nielsen, ``On quantum fidelities and channel capacities,'' LANL Report no. quant-ph/9809. To appear in IEEE Trans. on Inform. Theory." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Thap C. H. Bennett, P. W. Shor, J. A. Smolin, A. V. Thapliyal, \\newblock``Entanglement-assisted classical capacity of noisy quantum channel,''\\newblock LANL Report no. quant-ph/9904023." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Braun S. L. Braunstein, ``Squeezing as an irreducible resource'', LANL Report no. quant-ph/9904002." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Kimble A. Furusawa, J. S{\\orensen, S. L. Braunstein, C. Fuchs, H. J. Kimble, E. S. Polzik, Science, vol.{282, 706, 1998." }, { "name": "quant-ph9912067.extracted_bib", "string": "{demoen B. Demoen, P. Vanheuverzwijn, A. Verbeure, ``Completely positive quasi-free maps on the CCR algebra,'' {Rep. Math. Phys., vol.{15, pp. 27-39, 1979." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Hol75 A.~S. Holevo, \\newblock``Some statistical problems for quantum Gaussian states,'' \\newblock {IEEE Transactions on Information Theory, vol. {IT-21, no.5, pp. 533-543, 1975." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Hol82 A.~S. Holevo, \\newblock {Probabilistic and statistical aspects of quantum theory, chapter 5, North-Holland, 1982." }, { "name": "quant-ph9912067.extracted_bib", "string": "{wer R. F. Werner, ``Quantum harmonic analysis on phase space,'' \\newblock{J. Math. Phys., vol. {25, pp. 1404-1411, 1984." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Sohma A.~S. Holevo, M. Sohma and O. Hirota, \\newblock``The capacity of quantum Gaussian channels ,'' \\newblock{Phys. Rev. A, vol. {59, N3, pp. 1820-1828, 1998." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Hol98 A. S. Holevo, ``Sending quantum information with Gaussian states,'' LANL Report no. quant-ph/9809022. To appear in Proc. QCM-98, Ed. by M. D'Ariano, O.Hirota, P. Kumar." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Hol72a A.~S. Holevo,\\newblock``Towards the mathematical theory of quantum communication channels,'' \\newblock {Problems of Information Transm., vol. {8, no.1, pp. 63-71, 1972." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Kraus K. Kraus, States, effects and operations, {Lect. Notes Phys., vol. {190, 1983." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Lin G. Lindblad, \\newblock``Quantum entropy and quantum measurements,'' \\newblock {Lect. Notes Phys., vol. {378, Quantum Aspects of Optical Communication, Ed. by C. Benjaballah, O. Hirota, S. Reynaud, pp.71-80, 1991." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Shor P. W. Shor et al, in preparation." }, { "name": "quant-ph9912067.extracted_bib", "string": "{lloyd S. Lloyd, ``The capacity of the noisy quantum channel,'' \\newblock{Phys. Rev. A, vol. {56, pp. 1613, 1997." }, { "name": "quant-ph9912067.extracted_bib", "string": "{smo D. P. DiVincenzo, P. W. Shor, J. A. Smolin, ``Quantum-channel capacity of very noisy channels,''\\newblock {Phys. Rev. A, vol. {57, pp.830-839, 1998, LANL Report no. quant-ph/9706061." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Paulsen V. I. Paulsen, \\newblock{Completely bounded maps and dilations, Longman Scientific and Technical 1986" }, { "name": "quant-ph9912067.extracted_bib", "string": "{AhaKitNi D. Aharonov, A. Kitaev, and N. Nisan, \\newblock``Quantum Circuits with Mixed States,'' \\newblock LANL Report no. quant-ph/9806029." }, { "name": "quant-ph9912067.extracted_bib", "string": "{ogawa T. Ogawa, H. Nagaoka, ``Strong converse to the quantum channel coding theorem,'' LANL Report no. quant-ph/9808063 To appear in IEEE Trans. on Inform. Theory." }, { "name": "quant-ph9912067.extracted_bib", "string": "{winter A. Winter, ``Coding theorems and strong converse for quantum channels,'' To appear in IEEE Trans. on Inform. Theory." }, { "name": "quant-ph9912067.extracted_bib", "string": "{agar R. Simon, M. Selvadoray, G. S. Agarwal, ``Gaussian states for finite number of bosonic degrees of freedom,'' To appear in \\newblock{Phys. Rev.." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Hel C.~W. Helstrom, \\newblock {Quantum detection and estimation theory, chapter 5, Academic press, 1976." }, { "name": "quant-ph9912067.extracted_bib", "string": "{Hol72 A.~S. Holevo,\\newblock``Generalized free states of the C$^*$-algebra of the CCR,'' \\newblock {Theor. Math. Phys., vol. {6, no.1, pp. 3-20, 1971." }, { "name": "quant-ph9912067.extracted_bib", "string": "{clalim R.F. Werner,\\newblock``The classical limit of quantum theory,'' LANL Report no. quant-ph/9504016." } ]
quant-ph9912068		[ { "author": "M. Schreiber" }, { "author": "%\\thanksref{ca" }, { "author": "D. Kilin" }, { "author": "%\\thanksref{ca" }, { "author": "U. Kleinekath\\\"ofer %\\thanksref{ca" } ]	Based on the reduced density matrix method, we compare two different approaches to calculate the dynamics of the electron transfer in systems with donor, bridge, and acceptor. In the first approach a vibrational substructure is taken into account for each electronic state and the corresponding states are displaced along a common reaction coordinate. In the second approach it is assumed that vibrational relaxation is much faster than the electron transfer and therefore the states are modeled by electronic levels only. In both approaches the system is coupled to a bath of harmonic oscillators but the way of relaxation is quite different. The theory is applied to the electron transfer in ${\rm H_2P-{\rm ZnP-{\rm Q$ with free-base porphyrin (${\rm H_2P$) being the donor, zinc porphyrin (${\rm ZnP$) being the bridge and quinone (${\rm Q$) the acceptor. The parameters are chosen as similar as possible for both approaches and the quality of the agreement is discussed.	[ { "name": "PAPER.tex", "string": "%\\documentclass{elsart}\n\\documentclass[11pt]{article}\n\\usepackage{latexsym,psfig,rotate,epsf,a4} \n%\\usepackage{latexsym,psfig,rotate,epsf,a4,ukleinebib3} \n%\\documentclass[twocolumn]{elsart}\n%\\renewcommand{\\baselinestretch}{1.5}\n%\\usepackage{test}\n\n\\begin{document}\n%\\begin{frontmatter}\n\n%\\title\n\\begin{center}\n{\\Large \\bf Comparison \nof two models for bridge-assisted charge transfer}\n\n\n%\\thanks[bd]{Dedicated \n%to Prof.~Dr.~Dieter~H\\\"onicke on the occasion of his\n%60th birthday.}\n%\\author{\nM. Schreiber,\n%\\thanksref{ca}}\n%\\author{\nD. Kilin,\n%\\thanksref{ca}}\n and \n%\\author{\nU. Kleinekath\\\"ofer\n%\\thanksref{ca}} \n%\\thanks[ca]\n%{Corresponding author. Fax: + 49 371 531 3143;\\\\ \n%e-mail: schreiber@physik.tu-chemnitz.de.}\n%\\address\n\n\n{\\it Institut f\\\"ur Physik, Technische Universit\\\"at, \nD-09107 Chemnitz, Germany}\n\\end{center}\n%\\journal{Journal of Luminescence}\n\\date{}\n\n\\begin{abstract}\n Based on the reduced density matrix method, we compare two different\n approaches to calculate the dynamics of the electron transfer in\n systems with donor, bridge, and acceptor. In the first approach a\n vibrational substructure is taken into account for each electronic\n state and the corresponding states are displaced along a common\n reaction coordinate. In the second approach it is assumed that\n vibrational relaxation is much faster than the electron transfer and\n therefore the states are modeled by electronic levels only. In both\n approaches the system is coupled to a bath of harmonic oscillators\n but the way of relaxation is quite different. The theory is applied\n to the electron transfer in ${\\rm H_2P}-{\\rm ZnP}-{\\rm Q}$ with\n free-base porphyrin (${\\rm H_2P}$) being the donor, zinc porphyrin\n (${\\rm ZnP}$) being the bridge and quinone (${\\rm Q}$) the acceptor.\n The parameters are chosen as similar as possible for both approaches\n and the quality of the agreement is discussed.\n\\end{abstract}\n\n%\\begin{keyword}\n% Electron transfer; Reduced density matrix; Superexchange\n%\\end{keyword}\n%\\end{frontmatter}\n\n\n\\section{Introduction}\n\nLong-range electron transfer (ET) is a very actively studied area in\nchemistry, biology, and physics; both in biological and synthetic\nsystems. Of special interest are systems with a bridging molecule\nbetween donor and acceptor. For example the primary step of charge\nseparation in the bacterial photosynthesis takes place in such a\nsystem \\cite{bixo91}. But such systems are also interesting for\nsynthesizing molecular wires \\cite{davi98}. It is known that the\nelectronic structure of the bridge component in donor-bridge-acceptor\nsystems plays a critical role \\cite{wasi92,barb96}. When the bridge\nenergy is much higher than the donor and acceptor energies, the bridge\npopulation is close to zero for all times and the bridge site just\nmediates the coupling between donor and acceptor. This mechanism is\ncalled superexchange and was originally proposed by Kramers\n\\cite{kram34} to describe the exchange interaction between two\nparamagnetic atoms spatially separated by a nonmagnetic atom. In the\nopposite limit when donor and acceptor as well as bridge energies are\ncloser than $\\sim k_{\\rm B} T$, the bridge site is actually populated and\nthe transfer is called sequential. The interplay between these two\ntypes of transfer has been investigated theoretically in various\npublications \\cite{bixo91,sumi96,felt95,okad98,schr98}.\n\nIn the present work we compare two different approaches based\non the reduced density matrix formalism. \nIn the first model\none pays attention to the fact that experiments in systems\nsimilar to the one discussed here show vibrational coherence\n\\cite{vos93,stan95}. Therefore \na vibrational substructure is introduced for each\nelectronic level within a multi-level \nRedfield theory \\cite{may92,kueh94}.\nBelow we call this the vibronic model.\nIn the second approach only\nelectronic states are taken into account because it is assumed that\nthe vibrational relaxation is much faster than the ET.\nThis model is referred to as tight-binding model below. \nIn this case\nonly the relaxation between the electronic states remains. Such a kind\nof relaxation has been phenomenologically introduced for ET by Davis\net al.\\ \\cite{davi97} and very recently derived in our group\n\\cite{schr98b,kili99} as a second order perturbation theory in the\nsystem-bath interaction similar to Redfield theory. \nThe vibronic and the tight-binding model are described in the\nnext section and compared in Section 3.\n\n\\section{Theory}\n\nFor the description of charge transfer and other dynamical processes\nin the system we introduce the Hamiltonian\n\\begin{equation}\n \\label{1}\n \\hat{H}= \\hat{H}_{\\rm S}+ \\hat{H}_{\\rm B}+ \\hat{H}_{\\rm SB},\n\\end{equation}\nwhere $\\hat{H}_{\\rm S}$ denotes the relevant system, $\\hat{H}_{\\rm B}$\nthe dissipative bath, and $\\hat{H}_{\\rm SB}$ the interaction between\nthe two. Before discussing the system part of the Hamiltonian in\nSections 2.1 and 2.2, we describe the bath and the procedure how to\nobtain the equations of motion for the reduced density matrix, because\nthis is the same for both models studied below. The bath is modeled\nby a distribution of harmonic oscillators and characterized by its\nspectral density $J(\\omega)$. Starting with a density matrix of the\nfull system, the reduced density matrix of the relevant (sub)system is\nobtained by tracing out the bath degrees of freedom \\cite{blum96}.\nWhile doing so a second-order perturbation expansion in the\nsystem-bath coupling and the Markov approximation have been applied\n\\cite{blum96}. \n\n\\subsection{Vibronic model}\nThe bridge ET system ${\\rm H_2P}-{\\rm ZnP}-{\\rm Q}$ with free-base\nporphyrin (${\\rm H_2P}$) being the donor, zinc porphyrin (${\\rm ZnP}$)\nthe bridge, and quinone (${\\rm Q}$) the acceptor is modeled by\nthree diabatic electronic potentials,\ncorresponding to the neutral excited electronic state\n$\\left\|1 \\right>=\\left\|{\\rm H_2P}^-{\\rm ZnP}-{\\rm Q}\\right>$,\nand states with charge separation\n$\\left\|2 \\right>=\\left\|{\\rm H_2P}^+-{\\rm ZnP}^--{\\rm Q}\\right>$,\n$\\left\|3 \\right>=\\left\|{\\rm H_2P}^+-{\\rm ZnP}-{\\rm Q^-}\\right>$\n(see Fig.\\ 1).\nEach of these electronic potentials has a vibrational substructure.\nThe vibrational frequency is assumed to be 1500 cm$^{-1}$ as a typical\nfrequency within carbon structures. The potentials are displaced\nalong a common reaction coordinate which represents the solvent\npolarization \\cite{marc56}. Following the reasoning of Marcus\n\\cite{marc56} the free energy differences $\\Delta G_{mn}$ \ncorresponding to the electron transfer from molecular block $n$ to $m$\n($ n=1$, $m=2,3$)\nare estimated to be \\cite{fuch96d,remp95}\n\\begin{equation}\n \\label{2}\n\\Delta G_{mn} =E_m^{\\rm ox}-E_n^{\\rm red}-E^{\\rm ex}-\n\\frac{e^2}{4\\pi\\epsilon_0\\epsilon_{\\rm s}}\\frac{1}{r_{mn}}+\\Delta G_{mn}(\\epsilon_{\\rm s})\n\\end{equation}\n with the term $\\Delta G_{mn}(\\epsilon_{\\rm s})$ correcting for the fact that the redox energies $E^{\\rm ox}_m$ and $E^{\\rm red}_n$ are measured in a reference solvent with dielectric constant $\\epsilon_{\\rm s}^{\\rm ref}$: \n \\begin{equation}\n \\label{2a}\n \\Delta G_{mn}(\\epsilon_{\\rm s})= \\frac{e^2}{4 \\pi \\epsilon_0} \n\\left(\\frac1{2r_m}+\\frac1{2r_n} \\right) \n\\left( \\frac1{\\epsilon_{\\rm s}^{\\rm }} -\\frac1{\\epsilon_{\\rm s}^{\\rm ref}} \\right).\n \\end{equation}\n The excitation energy of the donor ${\\rm H_2P} \\to {\\rm H_2P}^$ is\n denoted by $E^{\\rm ex}$. $r_n$ denotes the radius of\neither donor (1), bridge (2), or acceptor (3) \n and $r_{mn}$ the distance between two of them. They\n are estimated to be $r_1=r_2=5.5$ ${\\rm \\AA{}}$, $r_3=3.2$ ${\\rm \\AA{}}$, \n$ r_{12}=12.5$ ${\\rm \\AA{}}$, and $\n r_{13}=14.4$ ${\\rm \\AA{}}$ \\cite{fuch96d,remp95}.\n\nAlso sketched in Fig.\\ 1 are the reorganization energies \n$\\lambda_{mn}=\\lambda_{mn}^{\\rm i}+\\lambda_{mn}^{\\rm s}$.\nThese consist of an internal reorganization energy $\\lambda_{mn}^{\\rm i}$, \nwhich is\nestimated to be 0.3 eV \\cite{remp95}, and a solvent reorganization energy\n\\cite{marc56}\n\\begin{equation}\n \\label{3}\n \\lambda_{mn}^{\\rm s}=\\frac{e^2}{4 \\pi \\epsilon_0}\n\\left(\\frac1{2r_m}+\\frac1{2r_n}-\\frac1{r_{mn}}\\right) \n\\left(\\frac1{\\epsilon_{\\infty}}-\\frac1{\\epsilon_{\\rm s}}\\right)~.\n\\end{equation}\nFurther parameters are the electronic couplings between the potentials.\nFirst it should be underlined that $V_{13}=0$ because of the\nspatial separation of ${\\rm H_2P}$ and ${\\rm Q}$.\nSo there is no direct transfer between donor and acceptor.\n The other couplings are\n$V_{12}=65$ ${\\rm meV}$ and $V_{23}=2.2$ ${\\rm meV}$ \\cite{remp95}.\nThe damping is described by the\nspectral density $J(\\omega)$ of the bath. This is only needed at the \nfrequency of the vibrational transition and is determined\n$J(\\omega_{\\rm vib})/\\omega_{\\rm vib}=0.372$ \nby fitting the ET rate for the solvent methyltetrahydrofuran (MTHF).\nIn the vibronic model the spectral density\nis taken as a constant with respect to $\\epsilon_{\\rm s}$.\n\n\nNext the calculation of the dynamics is sketched.\nStarting from the Liouville equation, performing \nthe abovementioned approximations\nthe equation of motion for the\nreduced density matrix $\\rho_{\\mu{}\\nu}$ can be obtained \\cite{may92,kueh94}\n\\begin{equation}\n \\label{10}\n \\frac{\\partial}{\\partial t} \\rho_{\\mu{}\\nu}=\\frac{i}{\\hbar}\n(E_\\mu{}-E_{\\nu}) \\rho_{\\mu{}\\nu} - i \\sum_\\kappa \\{ v_{\\nu \\kappa} \n\\rho_{\\mu \\kappa} -v_{\\kappa \\mu} \\rho_{\\kappa \\nu} \\} +R_{\\mu{}\\nu}~.\n\\end{equation}\nThe index $\\mu{}$ combines the electronic quantum number $m$ and the\nvibrational quantum number $M$ of the diabatic levels $E_\\mu{}$.\n$v_{\\mu{}\\nu}=V_{mn} F_{\\rm FC}(m,M,n,N)$ comprises Franck-Condon\nfactors $F_{\\rm FC}$ and the electronic matrix elements $V_{mn}$. The third\nterm describes the interaction between the relevant system and the\nheat bath.\nEquation~(\\ref{10}) is solved numerically with the initial condition that only the\ndonor state is occupied in the beginning.\nThe population of the acceptor state\n\\begin{equation}\nP_3(t)=\\sum \\limits_M \\rho_{3M3M}(t)\n\\end{equation}\nand the ET rate\n\\begin{equation}\nk_{\\rm ET}=\\frac{P_3(\\infty)}{\\int \\limits_0^\\infty dt(1-P_3(t))}\n\\end{equation}\nare calculated by tracing out the vibrational modes.\n\n\n\\subsection{Tight-binding model}\n\nThe reasoning for the following system Hamiltonian is the assumption that\nthe vibrational excitations are relaxed on a much shorter time scale than\nthe ET time scale. Therefore only electronic states without any vibrational\nsubstructure are taken into account (see Fig.\\ 2). As a consequence \nthe relaxation during the ET process has to be described in a different\nmanner than in the previous subsection. If now relaxation takes place, it\ntakes place between the electronic states and not between vibrational\nstates within one electronic state potential surface. A similar model has\nbeen introduced phenomenologically by Davis et al.\\ \\cite{davi97} who\nsolved it in the steady state limit.\n\nThe energies of the electronic states $E_m$ are chosen to be\nthe ground states of the harmonic potentials given in the previous section.\nSo they vary with the dielectric constant. \nThe electronic coupling is fixed in two different ways. In the naive\nway they are chosen to be the same as in the vibronic model.\nBut because in the tight-binding model there is no reaction coordinate,\nin a second version we scale the electronic couplings with\nthe Franck-Condon overlap elements between the vibrational ground\nstates of each pair of electronic surfaces\n\\begin{equation}\n \\label{5}\nv_{mn}=V_{mn}F_{\\rm FC}(m,0,n,0)\n=V_{mn}\\exp{\\frac{-\|\\lambda_{mn}\|}{2 \\hbar \\omega_{\\rm vib}}}~.\n \\end{equation}\n In the vibronic model not only the free energy differences $\\Delta\n G$ but also the reorganization energies $\\lambda$ scale with the\n dielectric constant $\\epsilon_{\\rm s}$. Due to this scaling of\n $\\lambda$ the system-bath interaction is scaled with the dielectric\n constant $\\epsilon_{\\rm s}$. In the high temperature limit the\n reorganization energy is given by \\cite{weis99}\n\\begin{equation}\n \\label{4}\n \\lambda = \\hbar \\int_0^{\\infty} d\\omega\\frac{J(\\omega)}{\\omega}~.\n\\end{equation}\nThis relation is taken as motivation to\n scale the tight-binding spectral density\nwith $\\epsilon_{\\rm s}$ like the reorganization energies $\\lambda$\nin the vibronic model. \nIn the present calculations\n$\\Gamma_{21}=\\Gamma_{23}=\\Gamma$ is assumed. \nThe absolute value of the damping rate $\\Gamma$ between the \nelectronic states (see Fig.~2) is then determined by fitting the ET\nrate for the solvent MTHF to be $\\Gamma=2.8\\times{}10^{11}$ s$^{-1}$.\n \nThe advantage of the tight-binding model is the possibility to\ndetermine the transfer rate $k_{\\rm ET}$ and the final population of\nthe acceptor state either numerically or analytically. We employ the\nrotating wave approximation because we are only interested in the\nreaction rates here. For the analytic calculation three extra\nassumptions have to be made: small bridge population, the kinetic\nlimit $t\\gg{}\\Gamma^{-1}$, and the absence of initial coherence in the\nsystem. But for all situations described in this paper the\ndifferences between analytic and numerical results without the extra\nassumptions are negligible. The analytic expressions are\n\\begin{equation} \n\\label{rate}\nk_{\\rm ET}=g_{23}+\\frac{g_{23}(g_{12}-g_{32})}{g_{21}+g_{23}} \n\\end{equation}\nand \n\\begin{equation}\n\\label{population}\nP_3(\\infty)=\\frac{g_{12}g_{23}}{g_{21}+g_{23}}(k_{\\rm ET})^{-1}, \n\\end{equation} \nwhich contain both, dissipative and coherent contributions\n\\begin{equation} \n\\label{dissipation}\ng_{mn}= \nd_{mn}\n+\n\\frac{v_{mn}^2 \\sum\\limits_k (d_{mk}+d_{kn})}\n{\\hbar^2\\left\\{2\\omega^2_{mn}+\\frac12\\left[\\sum\\limits_k (d_{mk}+d_{kn}) \\right]^2\\right\\}}. \n\\end{equation}\nHerein the $d_{mn}$ are just abbreviations for $\\Gamma_{mn}\n\|n(\\omega_{mn})\|$ and $n(\\omega_{mn})$ denotes the Bose distribution\nat frequency $\\omega_{mn}=(E_m-E_n)/\\hbar$. For details and\ncomparison with the Grover-Silbey theory \\cite{silb71} as well as the\nHaken-Strobl-Reineker theory \\cite{rein82} we refer the reader to Ref.\\ \n\\cite{kili99}.\n\n\n\\section{Comparison}\nIn Fig.\\ 3 it is shown how the minima of the potential curves change\nwith varying the solvent due to the changes in Eqs.\\ (\\ref{2}) to\n(\\ref{3}). The solvents are listed\nin Table 1 together with their parameters and the results for the ET\nrates in both models.\nFor larger $\\epsilon_{\\rm s}$ the coordinates of the potential\nminima of bridge and acceptor increase while their energies decrease\nwith respect to the energy of the donor. The energy\ndifference between donor and bridge decreases with increasing\n$\\epsilon_{\\rm s}$. This makes a charge transfer more probable.\nFor small $\\epsilon_{\\rm s}$ \nthe acceptor state is higher in energy than the donor state;\nnevertheless there is a small ET rate due to coherent mixing.\n\nFor fixed $\\epsilon_{\\infty}$ the ET rate is plotted as a function of\nthe dielectric constant $\\epsilon_{\\rm s}$ in Fig.\\ 4. The ET rate in\nthe vibronic model increases strongly for small values of\n$\\epsilon_{\\rm s}$ while the increase is very small for $\\epsilon_{\\rm\n s}$ in the range between 5 and 8. The increase for small values of\n$\\epsilon_{\\rm s}$ is due to the fact that with increasing\n$\\epsilon_{\\rm s}$ the minimum of the acceptor potential moves from a\nposition higher than the minimum of the donor level to a position\nlower than the donor level. So the transfer becomes energetically\nfavorable. This can also be seen when looking at the results for the\ntight-binding model without scaling the electronic coupling with the\nFranck-Condon factor. In this case the ET rate increases almost\nlinearly with increasing $\\epsilon_{\\rm s}$. The effect missing in\nthis model is the overlap between the vibrational states. If one\ncorrects the electronic coupling in the tight-binding model by the\nFranck-Condon factor of the vibrational ground states as described in\nEq.\\ (\\ref{5}), good agreement is observed between the vibronic and\nthe tight-binding model.\n\nThe ET rate for the vibronic model shows some oscillations as a function of\n$\\epsilon_{\\rm s}$. This is due to the small density of vibrational levels\nin this model with one reaction coordinate. All three electronic potential\ncurves are harmonic and have the same frequency. So there are small maxima\nin the rate when two vibrational levels are in resonance and minima when\nthey are far off resonance. Models with more reaction coordinates do not\nhave this problem nor does the simple tight-binding model. If these\nartificial oscillations would be absent, the agreement between the results\nfor the tight-binding and the vibronic model would be even better, because\nthe rate for the vibronic model happens to have a maximum just at the\nreference point $\\epsilon_{\\rm s}=6.24$ which we have chosen to fix the spectral density, i.\\ e.\\ \nfor MTHF.\n\nThe comparison of the two models has been made assuming that the scaling of\nenergies as a function of the dielectric function is correct in the Marcus\ntheory. There have been a lot of changes to Marcus theory proposed in the\nlast years. Marcus theory assumes excess charges within cavities\nsurrounded by a polarizable medium and there one only takes the leading\norder into account. Higher order terms are included in the so called\nreaction field theory (see for example \\cite{kare97}). But to compare\ndifferent solvation models is out of the range of the present\ninvestigation. Some more details on this issue for the tight-binding model\nare given in Ref.\\ \\cite{kili99}. Here we just want to note in passing that the\neffect of scaling the system-bath interaction with $\\epsilon_{\\rm s}$, as\nassumed in the present work for the tight-binding model, has no big effect\non the ET rates.\n\n\nAs conclusion we mention that one gets good agreement for the ET rates of\nthe models with and without vibrational substructure, i.\\ e.\\ the vibronic\nand the tight-binding model, if one scales the electronic coupling with the\nFranck-Condon overlap matrix elements between the vibrational ground\nstates. The advantage of the model with electronic relaxation only is the\npossibility to derive analytic expressions for the ET rate and the final\npopulation of the acceptor state. But of course for a more realistic\ndescription of the ET transfer process in such complicated systems as\ndiscussed here, more than one reaction coordinate should be taken into\naccount. Work in this direction is in progress.\n\n\n\\section{Acknowlegements}\n We thank I. Kondov for the help with some programming as well as\n U.~\\mbox{Rempel} and E.~Zenkevich for stimulating discussions.\n Financial support of the DFG is gratefully acknowledged.\n\n\n\n\\begin{thebibliography}{10}\n\\bibitem{bixo91} M. Bixon, J. Jortner and M. E. Michel-Beyerle,\n Biochim.\\ Biophys.\\ Acta 1056 (1991) 301; Chem.\\ Phys.\\ 197 (1995)\n 389. \n\\bibitem{davi98} W. B. Davis, W. A. Svec, M. A. Ratner, and M.\n R. Wasielewski, Nature 396 (1998) 60.\\ \n\\bibitem{wasi92} M.~R.\n Wasielewski, Chem. Rev. 92 (1992) 345. \n\\bibitem{barb96} P.~F.\n Barbara, T.~J. Meyer, and M.~A.~Ratner, J. Phys. Chem. 100 (1996)\n 13148. \n\\bibitem{kram34} H. A. Kramers, Physica 1 (1934) 182.\n\\bibitem{sumi96} H.~Sumi and T.~Kakitani, Chem.\\ Phys.\\ Lett.\\ 252\n (1996) 85; H.~Sumi, J. Electroan.\\ Chem.\\ 438 (1997) 11.\n\\bibitem{felt95} A.~K.~Felts, W.~T.~Pollard, and R.~A.~Friesner, J.\n Phys.\\ Chem.\\ 99 (1995) 2929. \n\\bibitem{okad98} A. Okada, V.\n Chernyak, and S. Mukamel, J. Phys.\\ Chem.\\ A 102 (1998) 1241.\n\\bibitem{schr98} M. Schreiber, C. Fuchs, and R.~Scholz, J. Lumin.\\ \n 76\\&77 (1998) 482. \n\\bibitem{vos93} M.~H.~Vos, F.~Rappaport,\n J.-C.~Lambry, J.~Breton, and J.-L.~Martin, Nature 363 (1993) 320.\n\\bibitem{stan95} R.~J.~Stanley and S.~G.~Boxer, J. Phys.\\ Chem.\\ 99\n (1995) 859. \n\\bibitem{may92} V.~May and M.~Schreiber, Phys.\\ Rev.\\ A\n 45 (1992) 2868. \\bibitem{kueh94} O.~K\\\"uhn, V.~May, and\n M.~Schreiber, J.\\ Chem.\\ Phys.\\ 101 (1994) 10404. \n\\bibitem{davi97}\n W.~Davis, M.~Wasilewski, ~M. Ratner, V.~Mujica, and A. Nitzan, J.\n Phys. Chem. 101 (1997) 6158. \n\\bibitem{schr98b} M. Schreiber, D.\n Kilin, and U. Kleinekath\\\"ofer, in: R.~T.~Williams and W.~M.~Yen\n (Eds.), Excitonic Processes in Condensed Matter, PV 98-25, p.~99,\n The Electrochemical Society Proceedings Series, Pennington, NJ,\n 1998. \n\\bibitem{kili99} D. Kilin, U. Kleinekath\\\"ofer, and M.\n Schreiber (in preparation). \n\\bibitem{blum96} K. Blum, Density\n Matrix Theory and Applications, Plenum Press, New York, 1996, 2nd\n ed.\\ \n\\bibitem{marc56} R.~A.~Marcus, J.\\ Chem.\\ Phys.\\, 24 (1956)\n 966; R.~A.~Marcus und N.~Sutin, Biochim.\\ Biophys.\\ Acta 811 (1985)\n 265. \n\\bibitem{fuch96d} C. Fuchs, Ph.D. thesis, Technische\n Universit\\\"at Chemnitz, 1997,\n http://archiv.tu-chemnitz.de/pub/1997/0009 \n\\bibitem{remp95}\n U.~Rempel, B.~von~Maltzan, and C.~von~Borczyskowski, Chem. Phys.\n Lett. 245 (1995) 253. \n\\bibitem{weis99} U.~Weiss, Quantum\n Dissipative Systems, World Scientific, Singapore, 1999.\n\\bibitem{silb71} M.~Grover and R.~Silbey, J.\\ Chem.\\ Phys.\\ 54 (1971)\n 4843. \n\\bibitem{rein82} P. Reineker, in: G. H\\\"ohler (Ed.), Exciton Dynamics\n in Molecular Crystals and Aggregates, Springer, Berlin, 1982.\n\\bibitem{kare97} M.~Karelson, G.~H.~F.~Diercksen, in: S.~Wilson and\n G.~H.~F.~Diercksen, Problem Solving in Computational Molecular\n Science: Molecules in Different Environments, Kluwer, Dordrecht,\n 1997. \n\\bibitem{tenn99} \\mbox{Charles Tennant \\& Company (London)\n Ltd}, http://www.ctennant.co.uk/tenn04.htm \n\\bibitem{schm89}\n J.~A.~Schmidt, J.-Y.~Liu, J.~R.~Bolton, M.~D.~Archer, and\n V.~P.~Y.~Gadzepko, J. Chem. Soc. Faraday Trans. 85 (1989) 1027.\n\\end{thebibliography}\n\n\n\\newpage\n\n\\tabcolsep=0.05cm\n\\begin{table}[htp]\n \\begin{center}\n \\leavevmode\n\\begin{tabular}{l\|c\|c\|c\|c\|c\|c\|c\|c\|c}\n\\hline\nsolvent& $\\epsilon_{\\rm s}$&$\\epsilon_\\infty$&$\\Delta G_{21}$&$\\Delta G_{31}$&$\\lambda_{21}^{\\rm s}$&$\\lambda_{31}^{\\rm s}$&$\\Gamma$ &$k_{\\rm ET}^{\\rm el}$ &$k_{\\rm ET}^{\\rm vib}$ \\\\\n& && [{\\rm eV}]& [{\\rm eV}]&[{\\rm eV}]&[{\\rm eV}]&[$10^{11}$ s$^{-1}$]& [$10^8$ s$^{-1}$]&[$10^8$ s$^{-1}$] \\\\\n\\hline\n1. cyclohexane \\protect \\cite{remp95} & 2.02 & 2.00 & 0.976 & 0.393 & 0.007 & 0.012 & 0.042 & 0.181 & 0.7 \\\\\n2. toluene \\protect \\cite{tenn99} & 2.38 & 2.24 & 0.867 & 0.202 & 0.039 & 0.069 & 0.227 & 1.04 & 0.8 \\\\\n3. anisole \\protect \\cite{schm89} & 4.33 & 2.29 & 0.590 & -0.281 & 0.300 & 0.524 & 1.751 & 4.24 & 2.30 \\\\\n4. dibromoethane \\protect \\cite{schm89} & 4.78 & 2.37 & 0.558 & -0.336 & 0.312 & 0.544 & 1.817 & 4.63 & 2.45 \\\\\n5. chlorobenzene \\protect \\cite{tenn99} & 5.29 & 1.93 & 0.529 & -0.388 & 0.481 & 0.839 & 2.804 & 3.21 & 3.63 \\\\\n6. MTHF \\protect \\cite{remp95} & 6.24 & 2.00 & 0.486 & -0.462 & 0.497 & 0.868 & 2.900 & 3.59 & 3.58 \\\\\n7. methyl acetate \\protect \\cite{tenn99} & 6.68 & 1.85 & 0.471 & -0.489 & 0.571 & 0.996 & 3.328 & 2.96 & 4.15 \\\\\n8. trichloroethane \\protect \\cite{schm89} & 7.25 & 2.06 & 0.454 & -0.512 & 0.508 & 0.887 & 2.960 & 3.98 & 3.50 \\\\\n9. dichloromethane \\protect \\cite{remp95} & 9.08 & 2.03 & 0.413 & -0.590 & 0.559 & 0.977 & 3.264 & 4.00 & 3.80 \\\\\n\\hline\n\\end{tabular}\n \\caption{Parameters and obtained transfer rates \n for different solvents. The references behind\nthe names of the solvents cite the sources of $\\epsilon_{\\rm s}$\nand $\\epsilon_\\infty$.\nMTHF stands for methyltetrahydrofuran. \n$\\Gamma$ denotes the damping rate in the tight-binding model.\nThe ET rate for the solvent MTHF\nhas been used to fix the damping parameter of the models.\nThe reaction rates $k_{\\rm ET}^{\\rm el}$ were obtained using\n\\protect Eq.~(\\ref{rate})\nwithin the tight-binding model and\nthe reaction rates $k_{\\rm ET}^{\\rm vib}$ within the vibronic model.\n}\n \\label{tab1}\n \\end{center}\n\\end{table} \n\n\\newpage\n\n\n\\begin{figure}[htp]\n \\begin{center}\n\\parbox{6.0cm}{\\rule{-3cm}{.1cm}\\epsfxsize=11.0cm\\epsfbox{fig1.eps}}\n \\leavevmode\n \\caption{Electronic potentials and parameters of the vibronic model.\n The donor surface $\\left\|{\\rm H_2P}^*-{\\rm ZnP}-{\\rm Q}\\right>$\n is given by the solid line, the bridge $\\left\|{\\rm H_2P}^+-{\\rm\n ZnP}^--{\\rm Q}\\right>$ by the dashed line, and the acceptor\n $\\left\|{\\rm H_2P}^+-{\\rm ZnP}-{\\rm Q^-}\\right>$ by the dotted\n line. }\n \\label{fig1}\n \\end{center}\n\\end{figure}\n \n\\begin{figure}[htp]\n \\begin{center}\n\\parbox{6.0cm}{\\rule{-3cm}{.1cm}\\epsfxsize=11.0cm\\epsfbox{fig2.eps}}\n \\leavevmode\n \\caption{Schematic presentation of the tight-binding model.}\n \\label{fig2}\n \\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n \\begin{center}\n\\parbox{6.0cm}{\\rule{-3cm}{.1cm}\\epsfxsize=11.0cm\\epsfbox{fig3.eps}}\n \\leavevmode\n \\caption{Variation of the potential minima for different solvents.\nSquares denote the bridge minima, circles the acceptor minima.\nThe numbers correspond to the ordinal numbers in Table 1. The potentials\nare shown for solvent 6 (MTHF). }\n \\label{fig3}\n \\end{center}\n\\end{figure}\n\n\\begin{figure}[htp]\n \\begin{center}\n\\parbox{6.0cm}{\\rule{-3cm}{.1cm}\\epsfxsize=11.0cm\\epsfbox{fig4.eps}}\n \\leavevmode\n \\caption{Transfer rate as a function of the dielectric constant \n $\\epsilon_{\\rm s}$ for both models together with experimental \nresults \\protect \\cite{remp95}. The rates for the vibronic model\nare given by the circles.\nThe dashed line shows the rate for the tight-binding model with\nelectronic couplings $V_{mn}$ as in the vibronic model. The\nsolid line represents the rate for the tight-binding model\nwith $v_{mn}$ scaled as given in Eq.\\ \\protect (\\ref{5}).}\n \\label{fig4}\n \\end{center}\n\\end{figure}\n\n \n\n \n\n\n\\end{document}\n\n\n" } ]	[ { "name": "quant-ph9912068.extracted_bib", "string": "{bixo91 M. Bixon, J. Jortner and M. E. Michel-Beyerle, Biochim.\\ Biophys.\\ Acta 1056 (1991) 301; Chem.\\ Phys.\\ 197 (1995) 389." }, { "name": "quant-ph9912068.extracted_bib", "string": "{davi98 W. B. Davis, W. A. Svec, M. A. Ratner, and M. R. Wasielewski, Nature 396 (1998) 60.\\" }, { "name": "quant-ph9912068.extracted_bib", "string": "{wasi92 M.~R. Wasielewski, Chem. Rev. 92 (1992) 345." }, { "name": "quant-ph9912068.extracted_bib", "string": "{barb96 P.~F. Barbara, T.~J. Meyer, and M.~A.~Ratner, J. Phys. Chem. 100 (1996) 13148." }, { "name": "quant-ph9912068.extracted_bib", "string": "{kram34 H. A. Kramers, Physica 1 (1934) 182." }, { "name": "quant-ph9912068.extracted_bib", "string": "{sumi96 H.~Sumi and T.~Kakitani, Chem.\\ Phys.\\ Lett.\\ 252 (1996) 85; H.~Sumi, J. Electroan.\\ Chem.\\ 438 (1997) 11." }, { "name": "quant-ph9912068.extracted_bib", "string": "{felt95 A.~K.~Felts, W.~T.~Pollard, and R.~A.~Friesner, J. Phys.\\ Chem.\\ 99 (1995) 2929." }, { "name": "quant-ph9912068.extracted_bib", "string": "{okad98 A. Okada, V. Chernyak, and S. Mukamel, J. Phys.\\ Chem.\\ A 102 (1998) 1241." }, { "name": "quant-ph9912068.extracted_bib", "string": "{schr98 M. Schreiber, C. Fuchs, and R.~Scholz, J. Lumin.\\ 76\\&77 (1998) 482." }, { "name": "quant-ph9912068.extracted_bib", "string": "{vos93 M.~H.~Vos, F.~Rappaport, J.-C.~Lambry, J.~Breton, and J.-L.~Martin, Nature 363 (1993) 320." }, { "name": "quant-ph9912068.extracted_bib", "string": "{stan95 R.~J.~Stanley and S.~G.~Boxer, J. Phys.\\ Chem.\\ 99 (1995) 859." }, { "name": "quant-ph9912068.extracted_bib", "string": "{may92 V.~May and M.~Schreiber, Phys.\\ Rev.\\ A 45 (1992) 2868." }, { "name": "quant-ph9912068.extracted_bib", "string": "{kueh94 O.~K\\\"uhn, V.~May, and M.~Schreiber, J.\\ Chem.\\ Phys.\\ 101 (1994) 10404." }, { "name": "quant-ph9912068.extracted_bib", "string": "{davi97 W.~Davis, M.~Wasilewski, ~M. Ratner, V.~Mujica, and A. Nitzan, J. Phys. Chem. 101 (1997) 6158." }, { "name": "quant-ph9912068.extracted_bib", "string": "{schr98b M. Schreiber, D. Kilin, and U. Kleinekath\\\"ofer, in: R.~T.~Williams and W.~M.~Yen (Eds.), Excitonic Processes in Condensed Matter, PV 98-25, p.~99, The Electrochemical Society Proceedings Series, Pennington, NJ, 1998." }, { "name": "quant-ph9912068.extracted_bib", "string": "{kili99 D. Kilin, U. Kleinekath\\\"ofer, and M. Schreiber (in preparation)." }, { "name": "quant-ph9912068.extracted_bib", "string": "{blum96 K. Blum, Density Matrix Theory and Applications, Plenum Press, New York, 1996, 2nd ed.\\" }, { "name": "quant-ph9912068.extracted_bib", "string": "{marc56 R.~A.~Marcus, J.\\ Chem.\\ Phys.\\, 24 (1956) 966; R.~A.~Marcus und N.~Sutin, Biochim.\\ Biophys.\\ Acta 811 (1985) 265." }, { "name": "quant-ph9912068.extracted_bib", "string": "{fuch96d C. Fuchs, Ph.D. thesis, Technische Universit\\\"at Chemnitz, 1997, http://archiv.tu-chemnitz.de/pub/1997/0009" }, { "name": "quant-ph9912068.extracted_bib", "string": "{remp95 U.~Rempel, B.~von~Maltzan, and C.~von~Borczyskowski, Chem. Phys. Lett. 245 (1995) 253." }, { "name": "quant-ph9912068.extracted_bib", "string": "{weis99 U.~Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 1999." }, { "name": "quant-ph9912068.extracted_bib", "string": "{silb71 M.~Grover and R.~Silbey, J.\\ Chem.\\ Phys.\\ 54 (1971) 4843." }, { "name": "quant-ph9912068.extracted_bib", "string": "{rein82 P. Reineker, in: G. H\\\"ohler (Ed.), Exciton Dynamics in Molecular Crystals and Aggregates, Springer, Berlin, 1982." }, { "name": "quant-ph9912068.extracted_bib", "string": "{kare97 M.~Karelson, G.~H.~F.~Diercksen, in: S.~Wilson and G.~H.~F.~Diercksen, Problem Solving in Computational Molecular Science: Molecules in Different Environments, Kluwer, Dordrecht, 1997." }, { "name": "quant-ph9912068.extracted_bib", "string": "{tenn99 \\mbox{Charles Tennant \\& Company (London) Ltd, http://www.ctennant.co.uk/tenn04.htm" }, { "name": "quant-ph9912068.extracted_bib", "string": "{schm89 J.~A.~Schmidt, J.-Y.~Liu, J.~R.~Bolton, M.~D.~Archer, and V.~P.~Y.~Gadzepko, J. Chem. Soc. Faraday Trans. 85 (1989) 1027." } ]
quant-ph9912069		[]	The three-dimensional Schr\"odinger's equation is analyzed with the help of the correspondence principle between classical and quantum-mechanical quantities. Separation is performed after reduction of the original equation to the form of the classical Hamilton-Jacobi equation. Each one-dimensional equation obtained after separation is solved by the conventional WKB method. Quasiclassical solution of the angular equation results in the integral of motion $\vec M^2=(l+\frac 12)^2 \hbar^2$ and the existence of nontrivial solution for the angular quantum number $l=0$. Generalization of the WKB method for multi-turning-point problems is given. Exact eigenvalues for solvable and some ``insoluble'' spherically symmetric potentials are obtained. Quasiclassical eigenfunctions are written in terms of elementary functions in the form of a standing wave.	[ { "name": "quant-ph9912069.tex", "string": "\\documentstyle[12pt]{article}\n\\textheight 230mm\n\\textwidth 160mm\n\\topmargin -15mm\n\n\\begin{document}\n\\hspace{100mm} UICHEP-TH/99-8\n\\begin{center}\n{\\bf Quasiclassical Analysis of the Three-dimensional\n Schr\\\"odinger's Equation \\\\ and Its Solution } \\\\\n\\vspace{3mm}\n\\rm { M. N. Sergeenko } \\\\\n\\vspace{2mm}\n\\it{ The National Academy of Sciences of Belarus,\n Institute of Physics \\\\ Minsk 220072, Belarus \\ and \\\\ }\n\\it{ Department of Physics, University of Illinois at Chicago,\n Illinois 60607, USA }\n\\end{center}\n\n\\begin{abstract}\nThe three-dimensional Schr\\\"odinger's equation is analyzed with the \nhelp of the correspondence principle between classical and \nquantum-mechanical quantities. Separation is performed after \nreduction of the original equation to the form of the classical \nHamilton-Jacobi equation. Each one-dimensional equation obtained \nafter separation is solved by the conventional WKB method. \nQuasiclassical solution of the angular equation results in the \nintegral of motion $\\vec M^2=(l+\\frac 12)^2 \\hbar^2$ and the \nexistence of nontrivial solution for the angular quantum number \n$l=0$. Generalization of the WKB method for multi-turning-point \nproblems is given. Exact eigenvalues for solvable and some \n``insoluble'' spherically symmetric potentials are obtained. \nQuasiclassical eigenfunctions are written in terms of elementary \nfunctions in the form of a standing wave. \\end{abstract}\n\n\\noindent {\\bf 1. Introduction }\\\\\n\nBasic equation of quantum mechanics, the Schr\\\"odinger's wave \nequation, is usually solved in terms of special functions or \nnumerically; for several potentials, the equation is solved exactly \n\\cite{He}. The general approach to solve the Schr\\\"odinger's equation \nfor the solvable potentials\\footnote {By \"solvable\" potentials we \nmean those models for which the eigenvalue problem can be reduced to \na hypergeometric function by a suitable transformation.} is to reduce \nthis equation to the equation for hypergeometric function or some \nspecial function. To do that one needs to find first a special \ntransformation for the wave function and its arguments to reduce the \noriginal equation to the hypergeometric form. After that using \ncertain requirements (defined by boundary conditions) to the \nhypergeometric function one can write the corresponding solution for \nthe problem under consideration, i.e. the eigenfunctions and the \ncorresponding eigenvalues.\n\nThis is rather a mathematical approach to solve the eigenvalue \nproblem in quantum mechanics; the corresponding methods and solutions \nof the wave equation for some potentials have been developed long \nbefore the creation of quantum mechanics. There are several features \nof this exact mathematical method that should be clarified from the \nphysical point of view. One of them is related to the $S$-wave state. \nThe radial Schr\\\"odinger's equation has no the centrifugal term for \nthe orbital quantum number $l=0$. From the physical point of view, \nthis means that the problem does not have the left turning point. In \norder for the physical system to have a stable bound state (discrete \nspectrum) two turning points are required (see Ref. \\cite{He}, for \ninstance). However, solving the radial wave equation for $l>0$, one \nobtains energy eigenvalues for all $l$ including $l=0$. Another \nfeature is related to the angular dependence. The angular \neigenfunction for the ground state, $Y_{00}(\\theta,\\varphi)=const$, \ni.e. no nontrivial solution exists. Meanwhile, as in the case of \nradial dependence, it might be a function with no zeroes of the type \nof a standing half-wave.\n\nThere is another approach to the eigenvalue problem in quantum \nmechanics. This is the quasiclassical method which is well known and \nwidely used mainly as the Wentzel-Kramers-Brillouin (WKB) \napproximation \\cite{Fro}-\\cite{Tr} applicable in the case when the de \nBroglie wavelength, $\\lambda =h/p$ ($h=2\\pi\\hbar$), is changing \nslowly. In several cases of interest, the WKB method yields the \nexact energy levels, however, its correct application results in the \nexact energy eigenvalues for {\\em all} known solvable potentials.\n\nThe quasiclassical method is based on the correspondence principle \nbetween classical functions and operators of quantum mechanics. The \ncorrespondence principle is used to derive the wave equation in \nquantum mechanics. In Ref. \\cite{SeS} this principle has been used to \nderive the semiclassical wave equation appropriate in the \nquasiclassical region. It was shown that the standard WKB method (to \nleading order in $\\hbar$) is the appropriate method to solve this \nequation.\n\nIn this work we solve the multi-dimensional Schr\\\"odinger's equation \nby the quasiclassical method. Unlike known approaches, instead of \nmodification of one-dimensional equations obtained after separation, \nwe analyze an original multi-dimensional Schr\\\"odinger's equation and \nreduce it to the equation in canonical form (without first \nderivatives). Separation is performed after reduction of the equation \nobtained to the form of the classical Hamilton-Jacobi equation. We \nshow that the main question of the exactness of the quasiclassical \nmethod is tightly connected with the correspondence principle, i.e. \nthe form of the generalized moments obtained after separation of the \nwave equation; the moments obtained after separation have to coincide \nwith the corresponding classic generalized moments. The quantization \ncondition is written with help of the argument's principle in the \ncomplex plane that allows us to generalize the quasiclassical method \nfor multi-turning-point problems and obtain the exact energy \neigenvalues for all known solvable potentials and, also, for some \n``insoluble'' problems with more then two turning points.\n\nThe quasiclassical method reproduces not only the exact energy \nspectrum for known potentials but has new important features. One of \nthe consequences of the quasiclassical solution of the \nmulti-dimensional Schr\\\"odinger's equation is the existence of a \nnontrivial angular solution at $l=0$, $\\tilde Y_{00}^{WKB}(\\theta, \n\\varphi)$, which describes the quantum fluctuations of the angular \nmoment. This method allows us to show apparently the contribution of \nquantum fluctuations of the angular momentum into the energy of the \nground state. \\\\\n\n\\noindent {\\bf 2. Exactness of the WKB method }\\\\\n\nIt is well known that the exact eigenvalues can be defined with the \nhelp of the asymptotic solution, i.e. the exact solution and its \nasymptote correspond to the same exact eigenvalue of the problem \nunder consideration. The asymptotic solutions in quantum mechanics \ncan be obtained by the WKB method. Therefore the quasiclassical \nmethod should reproduce the exact energy spectrum.\n\nIntriguing results have been obtained with the help of the \nsupersymmetric WKB method (SWKB) \\cite{Sukh}-\\cite{Kha}, which is a \nmodification of the standard WKB quantization for obtaining the \nquasiclassical eigenvalues of nonrelativistic Hamiltonians. It was \ndemonstrated that the leading-order SWKB quantization condition in \neach and every case reproduces the exact energy eigenvalues for a \nclass of solvable potentials. For these models, solutions can be \nwritten in terms of elementary functions.\n\nSuccesses of the SWKB quantization rule have revived interest in the \noriginal WKB quantization condition. In several common applications \nthe method gives very accurate results. Proofs of varying degrees of \nrigor have been advanced that demonstrate the exactness of the \nstandard WKB quantization condition \n\\cite{Fro},\\cite{Sukh}-\\cite{SuSe}. The existing proofs of exactness \nof the WKB approximation are not entirely rigorous since the \ncorrection terms are only asymptotically valid, i.e., as $\\hbar \n\\rightarrow 0$ \\cite{Krei}. Furthermore, in the cases when a modified \nWKB integral gives the exact eigenvalues, it is not even clear which \n``correction'' must be shown to be zero.\n\nThe standard lowest-order WKB prescription reproduces the exact \nenergy levels for the harmonic oscillator in the Cartesian \ncoordinates $x$, $y$, and $z$. But just this problem is correctly \nformulated in the framework of the quasiclassical approach; in the \nCartesian coordinates, the Schr\\\"odinger's equation has the required \ncanonical form and the generalized moments for each degree of freedom \ncoincide with the corresponding classic moments. As for other \ncoordinate systems, for example spherical, the WKB method does not \nreproduce the exact energy levels unless one supplements it with \nLanger-like correction terms.\n\nFor the central potential $V(r)$, the Schr\\\"odinger's equation can be \nwritten in the spherical coordinates as\n\n\\begin{eqnarray}\n(-i\\hbar)^2\\left[\\frac 1{r^2}\\frac{\\partial }{\\partial r} \\left(r^2\n\\frac{\\partial }{\\partial r}\\right) + \\frac 1{r^2\\sin\\,\\theta}\n\\frac{\\partial }{\\partial\\theta}\\left(\\sin\\,\\theta\\frac{\\partial}\n{\\partial \\theta} \\right) + \\frac 1{r^2\\sin^2\\theta}\\frac{\\partial^2}\n{\\partial \\varphi^2}\\right]\\psi(\\vec r) = \\label{shr} \\\\\n2m[E-V(r)]\\psi(\\vec r). \\nonumber \\end{eqnarray}\nThe standard solution of this equation is the following. If one \nsubstitutes $\\psi(\\vec r) = [U(r)/r]$ $[\\Theta(\\theta)/ \n\\sqrt{\\sin\\theta}] \\Phi(\\varphi)$ into Eq. (\\ref{shr}), one obtains \n(after separation) the following three reduced one-dimensional \nequations\n\n\\begin{equation}\n\\left[\\hbar^2\\frac{d^2}{dr^2} + 2m(E-V) - \\frac{\\vec L^2}{r^2}\n\\right]U(r) = 0, \\label{rad} \\end{equation}\n\n\\begin{equation}\n\\left(\\hbar^2\\frac{d^2}{d\\theta^2} + \\vec L^2 + \\frac{\\hbar^2}4 -\n\\frac{L_z^2 - \\frac{\\hbar^2}4}{\\sin^2\\theta}\\right)\n\\Theta(\\theta) = 0, \\label{tet} \\end{equation}\n\n\\begin{equation}\n\\left(\\hbar^2\\frac{d^2}{d\\varphi^2} + L_z^2\\right)\\Phi(\\varphi) = 0.\n\\label{phi} \\end{equation}\nExact solution of Eq. (\\ref{tet}) gives, for the squared angular \nmomentum $\\vec L^2$, $\\vec L^2=l(l+1)\\hbar^2$. Application of the \nleading-order WKB quantization condition \\cite{He},\n\n\\begin{equation}\n\\int_{x_1}^{x_2}\\sqrt{p^2(x,E)}dx = \\pi\\hbar\\left(n+\\frac 12\\right),\n\\ \\ \\ \\ \\ n=0,\\,1,\\,2,\\,..., \\label{qc2}\n\\end{equation}\nto the radial Eq. (\\ref{rad}) does not reproduce the exact energy \nspectrum \\{here in (\\ref{qc2}) $x_1$ and $x_2$ are the classic \nturning points and $p^2(x,E)=2m[E-V(x)]$\\}. The problem comes from \nthe form of the centrifugal term, $l(l+1)\\hbar^2/r^2$.\n\nTo overcome this problem in particular case of the Coulomb potential, \na special techniques has been developed. In order for the \nfirst-order WKB approximation to give the exact eigenvalues, the \nquantity $l(l+1)$ in Eq. (\\ref{rad}) must be replaced by $(l+\\frac \n12)^2$ \\cite{Lang}. The reason for this modification (for the special \ncase of the Coulomb potential) was pointed out by Langer ($1937$) \n\\cite{Lang} from the Langer transformation $r = e^x$, $U(r) = \ne^{x/2}X(x)$. However, for other spherically symmetric potentials, in \norder to obtain the appropriate Langer-like correction terms, another \nspecial transformation of the wave function (w.f.) and its arguments \nis required.\n\nThere are several other related problems in the semiclassical \nconsideration of the radial Schr\\\"odinger equation (\\ref{rad}). (i) \nThe WKB solution of the radial equation is irregular at $r\\rightarrow \n0$, i.e. $R^{WKB}(r)\\propto r^\\lambda /\\sqrt r$, $\\lambda = \n\\sqrt{l(l+1)}$, whereas the exact solution in this limit is \n$R(r)\\propto r^l$. (ii) Equation (\\ref{rad}) has no the centrifugal \nterm when $l=0$, i.e. the radial problem has only one turning point \nand one can not use the WKB quantization condition (\\ref{qc2}) \nderived for two-turning-point problems. However, solving the equation \nfor $l>0$ by known exact methods one obtains energy eigenvalues for \nall $l$. (iii) The WKB solution of equation (\\ref{tet}) has \nanalogous to the radial one, incorrect behavior at $\\theta\\rightarrow \n0$: $\\Theta^{WKB}(\\theta) \\propto\\theta^\\mu$, $\\mu^2 = m^2 \n-\\hbar^2/4$, while the exact regular solution in this limit is \n$\\Theta_l^m(\\theta)$ $\\propto \\theta^{\|m\|}$. Angular eigenfunction \n$Y_{00}(\\theta,\\varphi) = const$, i.e. no nontrivial solution exists.\n\nAs practical use shows the standard leading-order WKB approximation \n{\\em always} reproduces the exact spectrum for the solvable \nspherically symmetric potentials $V(r)$ if the centrifugal term in \nthe radial Schr\\\"odinger's equation has the form $(l+\\frac \n12)^2\\hbar^2/r^2$. As will be shown below the centrifugal term of \nsuch a form can be obtained from the WKB solution of equation \n(\\ref{shr}) if separation of this three-dimensional equation has \nperformed with the help of the correspondence principle. \\\\\n\n\\noindent {\\bf 3. Separation of the Schr\\\"odinger's equation }\\\\\n\nThere are two essential features of the WKB method. First, the method \nwas developed to solve the Schr\\\"odinger's equation in canonical form \n(without first derivatives). Second, in the quasiclassical method, \nthe classic quantities such as classic momentum, classic action, \nphase, etc., are used. (For example, in the WKB quantization \ncondition, the classic generalized momentum in the phase-space \nintegral is used). For the harmonic oscillator in the Cartesian \ncoordinates, the generalized moments in the original equation and \nmoments obtained after separation coincide with the corresponding \nclassic moments. Just for this problem, the standard WKB method (in \none and multi-dimensional cases) reproduces the exact energy levels \nwithout any additional correction terms.\n\nThe generalized moments in Eqs. (\\ref{rad})-(\\ref{phi}) obtained from \nseparation of Eq. (\\ref{shr}) are different from the corresponding \nclassic moments. As a result, the WKB method does not reproduce the \nexact energy levels for the spherically symmetric potentials (unless \none supplements it with Langer-like correction terms). The reason is \nthe form of the squared angular momentum, $\\vec L^2=l(l+1)\\hbar^2$, \nwhich is obtained from solution of the equation (3). In the WKB \nmethod, in order to reproduce the exact energy spectrum, the term \n$(l+\\frac 12)^2\\hbar^2$ should be used in the centrifugal term. This \nterm, $M^2=\\vec L^2+\\hbar/4$, is in the angular equation (\\ref{tet}), \nbut is not in the radial equation (\\ref{rad}).\n\nLet us show that the term $(l+\\frac 12)^2\\hbar^2$ can be obtained \nfrom the quasiclassical solution of the reduced Schr\\\"odinger's \nequation. For this, exclude in Eq. (\\ref{shr}) the first derivatives \nthat can be easily done with the help of the following operator \nidentity:\n\n\\begin{equation}\n\\frac d{dx}g(x)\\frac d{dx} \\equiv \\left[\\sqrt{g(x)}\\frac{d^2}{dx^2} -\n\\frac{d^2}{dx^2}\\sqrt{g(x)}\\right]\\sqrt{g(x)}. \\label{opid}\n\\end{equation}\nThen, after dividing by $\\tilde\\psi(\\vec r) =\\tilde R(r)\n\\tilde\\Theta(\\theta)\\tilde\\Phi(\\varphi)$, where $\\tilde R(r) =\nrR(r)$, $\\tilde\\Theta(\\theta) = \\sqrt{\\sin(\\theta)}\\Theta(\\theta)$,\n$\\tilde\\Phi(\\varphi) = \\Phi(\\varphi)$, we obtain the equation\n\\begin{eqnarray}\n-\\hbar^2\\frac{\\tilde R_{rr}''}{\\tilde R} +\n\\frac 1{r^2}\\left(-\\hbar^2\\frac{\\tilde\\Theta_{\\theta\\theta}''}\n{\\tilde\\Theta} - \\frac{\\hbar^2}4\\right) +\n\\frac 1{r^2\\sin^2\\theta}\\left(-\\hbar^2\\frac{\\tilde\n\\Phi_{\\varphi\\varphi}''}{\\tilde\\Phi} -\n\\frac{\\hbar^2}4\\right) = \\label{rede} \\\\\n 2m[E - V(r)]. \\nonumber\n\\end{eqnarray}\nIntroducing the notations,\n\n\\begin{equation}\n\\left(\\frac{\\partial S_0}{\\partial r}\\right)^2 =\n-\\hbar^2\\frac{\\tilde R_{rr}''}{\\tilde R}, \\label{sr}\n\\end{equation}\n\n\\begin{equation}\n\\left(\\frac{\\partial S_0}{\\partial\\theta}\\right)^2 =\n-\\hbar^2\\frac{\\tilde\\Theta_{\\theta\\theta}''}{\\tilde\\Theta} -\n\\frac{\\hbar^2}4, \\label{st}\n\\end{equation}\n\n\\begin{equation}\n\\left(\\frac{\\partial S_0}{\\partial\\varphi}\\right)^2 =\n-\\hbar^2\\frac{\\tilde\\Phi_{\\varphi\\varphi}''}{\\tilde\\Phi} -\n\\frac{\\hbar^2}4, \\label{sp}\n\\end{equation}\nwe can write Eq. (\\ref{rede}) in the form of the classic \nHamilton-Jacobi equation,\n\n\\begin{equation}\n\\left(\\frac{\\partial S_0}{\\partial r}\\right)^2 + \\frac\n1{r^2}\\left(\\frac{ \\partial S_0}{\\partial\\theta }\\right)^2 +\n\\frac 1{r^2\\sin^2\\theta}\\left( \\frac{\\partial S_0}{\\partial\\varphi}\n\\right)^2 = 2m\\left[E-V(r)\\right], \\label{HJ}\n\\end{equation}\nwhere $S_0=S_0(\\vec r,E)$ is the classic action of the system.\n\nUsing the correspondence principle, we see, from Eq. (\\ref{HJ}), that \nEqs. (\\ref{sr})-(\\ref{sp}) are the squared generalized moments \nexpressed via the quantum-mechanical quantities. Now, let us separate \nequation (\\ref{HJ}). Then, taking into account Eqs. \n(\\ref{sr})-(\\ref{sp}), we obtain the following system of the \nsecond-order differential equations in canonical form\n\n\\begin{equation}\n\\left(-i\\hbar\\frac d{dr}\\right)^2\\tilde R =\n\\left[2m(E-V) - \\frac{\\vec M^2}{r^2}\\right]\\tilde R, \\label{rra}\n\\end{equation}\n\n\\begin{equation}\n\\left[\\left(-i\\hbar\\frac d{d\\theta}\\right)^2 - \\left(\\frac\\hbar\n2\\right)^2\\right]\\tilde\\Theta(\\theta) = \\left(\\vec M^2 -\n\\frac{M_z^2}{\\sin^2\\theta}\\right)\\tilde\\Theta(\\theta), \\label{rth}\n\\end{equation}\n\n\\begin{equation}\n\\left[\\left(-i\\hbar\\frac d{d\\varphi}\\right)^2 - \\left(\\frac\\hbar\n2\\right)^2\\right]\\tilde\\Phi(\\varphi) = M_z^2\\tilde\\Phi(\\varphi),\n\\label{rph} \\end{equation}\nwhere $\\vec M^2$ and $M_z^2$ are the constants of separation and, at \nthe same time, integrals of motion.\n\nEquations (\\ref{rra})-(\\ref{rph}) have the quantum-mechanical form \n$\\hat f\\psi = f\\psi$, where $f$ is the physical quantity (the squared \ngeneralized momentum) and $\\hat f$ is the corresponding operator. We \nsee that the term $\\hbar^2/4$ in the left-hand side of the equations \nis related to the squared angular momentum operator. This term \ndisappears in the leading $\\hbar$ approximation \\cite{SeS} and the \nequations (\\ref{rra})-(\\ref{rph}) can be written in the general form \nas\n\n\\begin{equation}\n\\left(-i\\hbar\\frac d{dq}\\right)^2\\psi(q) = [\\lambda^2-U(q)]\\psi(q).\n\\label{gen} \\end{equation}\nThe squared generalized moments in the right-hand sides are the same \nas the classic ones. The use of these moments in the WKB quantization \ncondition and WKB solution yields the exact energy spectra for the \ncentral-field potentials and does not result in the difficulties of \nthe WKB method mentioned above.\\\\\n\n\\noindent {\\bf 4. Solution of the Schr\\\"odinger's equation }\\\\\n\nIn this section we consider quasiclassical solution of the \nSchr\\\"odinger's equation for the spherically symmetric potentials. In \norder for the approach we consider here to be self-consistent we have \nto solve each equation obtained after separation by the same method, \ni.e. the WKB method. The quasiclassical method is general enough and \nthe WKB formulas can be written differently, i.e. on the real axis \n\\cite{He} and in the complex plane \\cite{Fro}. Most general form of \nthe WKB solution and quantization condition can be written in the \ncomplex plane.\\\\\n\n{\\bf The WKB quantization in the complex plane.} The WKB method is \nusually used to solve one-dimensional two turning point problems. \nWithin the framework of the WKB method the solvable potentials mean \nthose potentials for which the eigenvalue problem has two turning \npoints. However the WKB method can be used to solve problems with \nmore then two turning points. In this case formulation in the \ncomplex plane is the most appropriate.\n\nConsider Eqs. (\\ref{rra})-(\\ref{rph}) in the framework of the \nquasiclassical method. Solution of each of these equations [in the \ngeneral form Eq. (\\ref{gen})] we search in the form \\cite{Fro}\n\n\\begin{equation}\n\\psi_\\lambda(z) = A\\,\\exp\\left[\\frac i{\\hbar}S(z,\\lambda)\\right],\n\\label{psiS} \\end{equation}\nwhere $A$ is the arbitrary constant. The function $S(z,\\lambda)$ is \nwritten as the expansion in powers of $\\hbar$, $S(z,\\lambda)=$ \n$S_0(z,\\lambda) + \\hbar S_1(z,\\lambda) +\\frac 12\\hbar^2S_2(z,\\lambda) \n+\\dots$. In the leading $\\hbar$ approximation the WKB solution of \nEqs. (\\ref{rra})-(\\ref{rph}) can be written in the form\n\n\\begin{equation}\n\\psi^{WKB}(z) = \\frac A{\\sqrt{p(z,\\lambda)}} \\exp\\left[\\pm\\frac\ni{\\hbar}\\int_{z_0}^z\\sqrt{p^2(z,\\lambda)}dz\\right]. \\label{wfWKB}\n\\end{equation}\n\nIn quantum mechanics, quantum numbers are determined as number of \nzeroes of the w.f. in the physical region. In the complex plane, the \nnumber of zeroes $N$ of a function $y(z)$ inside the contour $C$ is \ndefined by the argument's principle \\cite{Kor,Wen}. For the w.f. \n$\\psi_\\lambda(z)$, according to this principle we have\n\n\\begin{equation}\n\\oint_C \\frac{\\psi_\\lambda^\\prime(z)}{\\psi_\\lambda(z)}dz = 2\\pi iN,\n\\label{argp} \\end{equation}\nwhere $\\psi_\\lambda^\\prime(z)$ is the derivative of the function \n$\\psi_\\lambda(z) $ over the variable $z$ [see Ref. \\cite{FF} for more \ninformation about the condition (\\ref{argp})]. Contour $C$ is chosen \nsuch that it includes cuts (therefore, zeroes of the w.f.) between \nthe turning points where $p^2(z,\\lambda)=\\lambda^2 - U(z)>0$.\n\nSubstitution of Eq. (\\ref{wfWKB}) into (\\ref{argp}) results in the\nquantization condition\n\n\\begin{equation} \\oint\\sqrt{p^2(z,\\lambda)}dz +\ni\\frac\\hbar 2\\oint\\frac{p^\\prime(z,\\lambda)}{p(z,\\lambda)}dz =\n2\\pi\\hbar N. \\label{gqc} \\end{equation}\nIn the case, when $p(z,\\lambda)$ is a smooth function of the spatial \nvariable and the equation $\\lambda^2-U(z)=0$ has two roots (turning \npoints), the quantization condition (\\ref{gqc}) takes the form\n\n\\begin{equation}\n\\oint\\sqrt{p^2(z,\\lambda)}dz = 2\\pi\\hbar\\left(N+\\frac 12\\right).\n\\label{gqc2} \\end{equation}\nIn particular, for $p^2(z,\\lambda)=\\lambda^2$, the quantization \ncondition is\n\n\\begin{equation}\n\\oint\\sqrt{p^2(z,\\lambda)}dz = 2\\pi\\hbar N. \\label{qcc}\n\\end{equation}\nIn the next section we solve the three-dimensional Schr\\\"odinger \nequation for several spherically symmetric potentials by the method \nunder consideration. \\\\\n\n{\\bf A. The angular momentum eigenvalues}.\nEquations (\\ref{rth}) and (\\ref{rph}) determine the squared angular \nmomentum eigenvalues, $\\vec M^2$, and its projection, $M_z$, \nrespectively. The quantization condition (\\ref{qcc}) appropriate to \nthe angular equation (\\ref{rph}),\n\n\\begin{equation}\n\\oint M_zd\\varphi = 2\\pi\\hbar m, \\label{qcm} \\end{equation}\ngives $M_z = \\hbar m$, $m = 0,1,2,...$ The corresponding\nquasiclassical solution is\n\n\\begin{equation}\n\\tilde\\Phi_m(\\varphi) = C_1\\,e^{im\\varphi} + C_2\\,e^{-im\\varphi},\n\\label{sph} \\end{equation}\nwhere $C_1$ and $C_2$ are the arbitrary constants.\n\nThe quantization condition (\\ref{gqc2}) appropriate to Eq.\n(\\ref{rth}) is\n\n\\begin{equation}\nI = \\oint_C\\sqrt{\\vec M^2 - \\frac{M_z^2}{\\sin^2\\theta}}d\\theta =\n2\\pi\\hbar\\left(n_{\\theta} +\\frac 12\\right), \\ \\ \\ n_{\\theta} =\n0,1,2,... \\label{Ith}\n\\end{equation}\nTo calculate the integral (\\ref{Ith}) (as other hereafter) we use the \nmethod of stereographic projection. This means that, instead of \nintegration about a contour $C$ enclosing the classical turning \npoints, we exclude the singularities outside the contour $C$, i.e., \nat $\\theta = 0$ and $\\infty $ in this particular case. Excluding \nthese infinities we have, for the integral (\\ref{Ith}), $I = I_0 + \nI_{\\infty}$. Integral $I_0 = -2\\pi M_z$, and $I_{\\infty}$ is \ncalculated with the help of the replacement $z=e^{i\\theta}$ that \ngives $I_{\\infty} = 2\\pi\\sqrt{\\vec M^2}\\equiv2\\pi M$. Therefore, $I = \n2\\pi(M - M_z)$ and we obtain, for the squared angular momentum \neigenvalues,\n\n\\begin{equation} \\vec M^2 = \\left(l+\\frac 12\\right)^2\\hbar^2,\n\\label{M2} \\end{equation}\nwhere $l=n_{\\theta}+m$. Thus the quasiclassical solution of the \nSchr\\\"odinger's equation results in the squared angular momentum \neigenvalues (\\ref{M2}). This means the centrifugal term in the radial \nEq. (\\ref{rra}) has the form $(l+\\frac 12)^2\\hbar^2/r^2$ for {\\em \nall} spherically symmetric potentials.\n\nAs known the WKB solution $\\Theta^{WKB}(\\theta)$ of the equation \n(\\ref{tet}) has incorrect asymptotes at $\\theta\\rightarrow 0$ and \n$\\pi$. At the same time, the WKB solution of Eq. (\\ref{rth}), which \ncorresponds to the eigenvalues (\\ref{M2}), has the correct asymptotic \nbehavior at these points for all $l$. So far, as the generalized \nmomentum $p(\\theta)\\simeq\\frac{\\mid m\\mid}\\theta$ at \n$\\theta\\rightarrow 0$, this gives, for the WKB solution in the \nrepresentation of the wave function $\\psi(\\vec r)$, \n$\\Theta_l^m(\\theta) = \\tilde\\Theta^{WKB} (\\theta)/ \n\\sqrt{\\sin\\,\\theta} \\propto\\theta^{\|m\|}$ which corresponds to the \nbehavior of the known exact solution $Y_{lm}(\\theta ,\\varphi)$ at \n$\\theta\\rightarrow 0$.\n\nIn the classically allowed region, where $p^2(\\theta,M)=\\vec \nM^2-M_z^2/\\sin^2\\theta >0$, the leading-order WKB solution of Eq. \n(\\ref{rth}) is\n\n\\begin{equation}\n\\tilde{\\Theta}^{WKB}(\\theta) = \\frac B{\\sqrt{p(\\theta,M)}}\n\\cos\\left[\\int_{\\theta_1}^{\\theta}p(\\theta,M)d\\theta -\\frac\\pi 4 \\right].\n\\label{solth} \\end{equation}\nThe normalized quasiclassical solution far from the turning points, \nwhere $p(\\theta,M)\\simeq (l+\\frac 12)\\hbar$, can be written in \nelementary functions as\n\n\\begin{equation}\n\\tilde{\\Theta}_l^m(\\theta) = \\sqrt{\\frac{2l+1}{\\pi(l-m +\\frac 12)}}\n\\cos\\left[\\left(l+\\frac 12\\right)\\theta +\n\\frac{\\pi}2(l-m) \\right], \\label{Thet}\n\\end{equation}\nwhere we have took into account that the phase-space integral at the \nclassic turning point $\\theta_1$ is $\\chi(\\theta_1)=$ $-\\frac\\pi \n2(n_\\theta +\\frac 12)$ and $\\chi(\\theta_2)=$ $\\frac\\pi 2(n_\\theta \n+\\frac 12)$ at $\\theta=\\theta_2$. We see that the eigenfunctions \n(\\ref{Thet}) are either symmetric or antisymmetric. The \ncorresponding WKB solution, $\\tilde Y_{lm}^{WKB}(\\theta,\\varphi)=$ \n$\\tilde{\\Theta}_l^m(\\theta)\\tilde\\Phi_m(\\varphi)$, where the \nnormalized eigenfunction $\\tilde\\Phi_m(\\varphi)=\\frac 1{\\sqrt{2\\pi}}$ \n$e^{\\pm im\\varphi}$, in the representation of the w.f. \n$\\tilde\\psi(\\vec r)$ is\n\n\\begin{equation} \\tilde Y_{lm}^{WKB}(\\theta,\\varphi) =\n\\frac 1{\\pi}\\sqrt{\\frac{l+\\frac 12}{l - m + \\frac 12}}\n\\cos\\left[\\left(l+\\frac 12\\right)\\theta + \\frac{\\pi}2(l - m)\\right]\ne^{\\pm im\\varphi}. \\label{Ythet}\n\\end{equation}\n\nRemind some results concerning the semiclassical approach in quantum \nmechanics. The general form of the semiclassical description of \nquantum-mechanical systems have been considered in Ref. \\cite{Mi}. \nIt was shown that the semiclassical description resulting from Weyl's \nassociation of operators to functions is identical with the quantum \ndescription and no information need to be lost in going from one to \nthe another. What is more \"the semiclassical description is more \ngeneral than quantum mechanical description...\" \\cite{Mi}. The \nsemiclassical approach merely becomes a different representation of \nthe same algebra as that of the quantum mechanical system, and then \nthe expectation values, dispersions, and dynamics of both become \nidentical.\n\nOne of the fundamental features of quantum mechanical systems is \nnonzero minimal energy which corresponds to quantum oscillations. The \ncorresponding w.f. has no zeroes in the physical region. Typical \nexample is the harmonic oscillator.\n\nEigenvalues of the one-dimensional harmonic oscillator are $E_n = \n\\hbar\\omega(n+\\frac 12)$, i.e. the energy of zeroth oscillations $E_0 \n= \\frac 12 \\hbar\\omega$. In three-dimensional case, in the Cartesian \ncoordinates, the eigenvalues of the oscillator are $E_n = \n\\hbar\\omega(n_x+n_y+n_z+\\frac 32)$ \\cite{Flu}, i.e. each degree of \nfreedom contributes to the energy of the ground state, $E_0 = E_{0,x} \n+ E_{0,y} + E_{0,z} = $ $\\frac 32\\hbar\\omega$. Energy of the ground \nstate should not depend on coordinate system. This means that, in the \nspherical coordinates, each degree of freedom (radial and angular) \nshould contribute to the energy of zeroth oscillations. In many \napplications and physical models a nonzero minimal angular momentum \n$M_0$ is introduced (phenomenologically) in order to obtain \nphysically meaningful result (see, for instance, Ref. \\cite{Iwa}). \nHowever, the existence of $M_0$ follows from the quasiclassical \nsolution of Eq. (\\ref{rth}) \\cite{SeS,SeF}.\n\nConsider the WKB eigenfunction (\\ref{Ythet}) for the ground state. \nSetting in (\\ref{Ythet}) $m=0$ and $l=0$, we obtain the nontrivial \nsolution in the form of a standing half-wave [remind that the \nspherical function $Y_{00}(\\theta,\\varphi)=const$],\n\n\\begin{equation}\n\\tilde Y_{00}^{WKB}(\\theta,\\varphi) = \\frac 1\\pi\\cos\\frac\\theta 2.\n\\label{Y0} \\end{equation}\nThe corresponding eigenvalue is\n\n\\begin{equation} M_0 = \\frac{\\hbar}2. \\label{M0}\n\\end{equation}\nThe eigenvalue (\\ref{M0}) contributes to the energy of zeroth \noscillations. This means that (\\ref{Y0}) can be considered as \nsolution, which describes the quantum fluctuations of the angular \nmomentum. Note, that the eigenfunction of the ground state, $\\tilde \nY_{00}^{WKB}(\\theta,\\varphi)$, is symmetric. Below, we solve the \nradial equation (\\ref{rra}) for some spherically symmetric potentials \nand show the contribution of the eigenvalue $M_0$ to the energy of \nthe ground state.\\\\\n\n{\\bf B. The Coulomb problem $V(r)=-\\frac{\\alpha}r$}.\nThe WKB quantization condition (19) appropriate to the radial \nequation (\\ref{rad}) with the Coulomb potential does not reproduce \nthe exact energy levels unless one supplements it with Langer-like \ncorrection terms. Another problem is that the radial Schr\\\"odinger's \nequation (\\ref{rad}) has no the centrifugal term at $l=0$ and one can \nnot use the WKB quantization condition (derived for two-turning-point \nproblems) to calculate the energy of the ground state directly from \nthis equation. We do not run into such a problem in case of Eqs. \n(\\ref{rra})-(\\ref{rph}). As follows from the above consideration, the \ncentrifugal term $(l+\\frac 12)^2\\hbar^2/r^2$ is the same for all \nspherically symmetric potentials and the WKB method reproduces the \nexact energy spectrum for all $l$ and $n_r$.\n\nFor the radial equation (\\ref{rra}) with the Coulomb potential, the \nWKB quantization condition (\\ref{gqc2}) is\n\n\\begin{equation} I =\\oint_C\\sqrt{2mE +\\frac{2m\\alpha}r - \\frac{\\vec\nM^2}{r^2}}dr = 2\\pi\\hbar\\left(n_r+\\frac 12\\right), \\label{Icou}\n\\end{equation}\nwhere the integral is taken about a contour $C$ inclosing the turning \npoints $r_1$ and $r_2$. Using the method of stereographic projection, \nwe should exclude the singularities outside the contour $C$, i.e. at \n$r=0$ and $\\infty$. Excluding these infinities we have, for the \nintegral (\\ref{Icou}), $I = I_0 + I_{\\infty}$, where $I_0 = 2\\pi \ni\\sqrt{-\\vec M^2}\\equiv -2\\pi M$ and $I_{\\infty} = 2\\pi i\\alpha m/ \n\\sqrt{2mE}$. The sequential simple calculations result in the exact \nenergy spectrum\n\n\\begin{equation}\n E_n = -\\frac{\\alpha^2m}{2[(n_r+\\frac 12)\\hbar + M]^2}. \\label{Ecou}\n\\end{equation}\n\nFor the energy of zeroth oscillations we have, from Eq. (\\ref{Ecou}), \n$E_0 = -\\frac 12\\alpha^2 m(\\frac{\\hbar}2 + M_0)^{-2}$, that \napparently shows the contribution of the quantum fluctuations of the \nangular momentum (see Eq. (\\ref{M0})) into the energy of the ground \nstate $E_0$ \\cite{SeF}. The radial quasiclassical eigenfunctions, \n$\\tilde R_n^{WKB}(r)$, inside the classical region $[r_1,r_2]$ far \nfrom the turning points $r_1$ and $r_2$ are written in elementary \nfunctions in the form of a standing wave \\cite{SeS},\n\n\\begin{equation}\n\\tilde R_n(r) = A\\cos\\left(\\frac 1\\hbar p_nr +\\frac\\pi 2n_r\\right),\n\\label{Rn} \\end{equation}\nwhere we have took into account that the phase-space integral \n(\\ref{Icou}) at the classic turning point $r_1$ is $\\chi(r_1)=$ \n$-\\frac\\pi 2(n_r +\\frac 12)$. Here $A$ is the normalization constant \nand $p_n$ is the eigenmomentum expressed via the energy eigenvalue \n$E_n$, $p_n = \\sqrt{2m\|E_n\|}$. The eigenfunctions (\\ref{Rn}) are \neither symmetric or antisymmetric. \\\\\n\n{\\bf C. The three-dimensional harmonic oscillator\n$V(r)=\\frac 12 m\\omega^2r^2$}.\nThe three-dimensional harmonic oscillator is another classic example \nof the exactly solvable problems in quantum mechanics. The problem \nhas $4$ turning points, $r_1$, $r_2$, $r_3$, and $r_4$, but only two \nof them, $r_3$ and $r_4$, lie in the physical region $r>0$.\n\nThe problem is usually solved with the help of the replacement \n$x=r^2$ which reduces the problem to the $2$-turning-point ($2$TP) \none. But this problem can be solved as the $4$TP problem in the \ncomplex plane. Because of importance of the oscillator potential in \nmany applications and with the purpose of further development of the \nWKB method, we shall solve the problem by two methods, on the real \naxis as $2$TP problem and then in the complex plane as $4$TP problem.\n\nConsider first the physical region $r>0$, where the problem has two\nturning points. The leading-order WKB quantization condition\n(\\ref{gqc2}) then is\n\n\\begin{equation}\nI =\\int_{r_3}^{r_4}\\sqrt{2mE - (m\\omega r)^2 -\n\\frac{\\vec M^2}{r^2}}dr = \\pi\\hbar\\left(n_r+\\frac 12\\right),\n\\label{Io} \\end{equation}\nwhere $n_r$ is the number of zeroes of the w.f. between the classic \nturning points $r_3$ and $r_4$. Integral (\\ref{Io}) is reduced to the \nabove case of the Coulomb potential with the help of the replacement \n$z=r^2$. Integration result is $I=\\pi(E/\\omega -M)/2$ and we obtain, \nfor the energy eigenvalues,\n\n\\begin{equation}\nE_n = \\omega\\left[2\\hbar\\left(n_r + \\frac 12\\right) + M\\right].\n\\label{Eosc} \\end{equation}\nSo far, as $M = (l+\\frac 12)\\hbar$, we obtain the exact energy\nspectrum for the isotropic oscillator. Energy of the ground state is \n$E_0 =\\omega(\\hbar + M_0)$, where $M_0$ is the contribution of \nquantum fluctuations of the angular momentum.\n\nEmphasize the following in this solution. The $4$TP problem has been \nsolved as the $2$TP problem; we have applied the $2$TP quantization \ncondition (\\ref{gqc2}) to the $4$TP problem that is not quite \ncorrect. We have obtained the correct result because the potential is \nsymmetric and the replacement $x=r^2$ reduces the problem to the \n$2$TP problem, i.e.``reflects'' the negative region $r<0$ (and zeroes \nof the w.f.) into the positive region. A more correct approach to \nsolve the problem would be a $4$TP quantization condition. \nFortunately, the WKB method in the complex plane allows to solve this \nproblem as the $4$TP problem.\n\nIn the complex plane, the problem has two cuts, between turning \npoints $r_1$, $r_2$ and $r_3$, $r_4$. To apply residue theory for the \nphase space integral we need to take into account all zeroes of the \nw.f. in the complex plane, i.e. the contour $C$ has to include both \ncuts. The quantization condition (\\ref{gqc}) in this case takes the \nform\n\n\\begin{eqnarray}\n\\oint_C\\left[p(r,E) +\ni\\frac\\hbar 2\\frac{p^\\prime(r,E)}{p(r,E)}\\right]dr\n\\equiv \\label{sumC} \\\\ \\oint_{C_1}\\left[p(r,E)+i\\frac\\hbar\n2\\frac{p^\\prime(r,E)}{p(r,E)}\\right]dr +\n\\oint_{C_2}\\left[p(r,E)+i\\frac\\hbar\n2\\frac{p^\\prime(r,E)}{p(r,E)}\\right]dr = 2\\pi\\hbar N, \\nonumber\n\\end{eqnarray}\nwhere $p^2(r,E)=2mE -(m\\omega r)^2 -\\vec M^2/r^2$, and $C_1$ and\n$C_2$ are the contours about the cuts at $r<0$ and $r>0$, respectively.\nThe number $N=n_{r<0}+n_{r>0}$, where $n_{r<0}$ and $n_{r>0}$ are\nthe numbers of zeroes of the w.f. at $r<0$ and $r>0$, respectively.\nFor the harmonic oscillator, because of symmetricity of the potential\nwe have $n_{r<0}=n_{r>0}=n_r$, i.e. the total number of zeroes is\n$N=2n_r$.\n\nTherefore, the quantization condition (\\ref{sumC}) for the $4$TP \nproblem takes the form,\n\n\\begin{equation}\n\\oint_Cp(r,E)dr = \\oint_{C_1}p(r,E)dr + \\oint_{C_2}p(r,E)dr =\n2\\pi\\hbar k\\left(n_r+\\frac 12\\right), \\label{oink}\n\\end{equation}\nwhere $k=2$ is the number of cuts. We can write the $4$TP \nquantization condition in this form because the effective potential \nis infinite at $r=0$. In case if the potential is finite in the whole \nregion, the quantization condition will be more complicate \n\\cite{SuSe}.\n\nThe condition (\\ref{oink}) is in agreement with the Maslov's theory. \nThis means that the right-hand side of the equation (\\ref{oink}) can \nbe written in the form\n\n\\begin{equation}\n2\\pi\\hbar k\\left(n_r+\\frac 12\\right) =\n2\\pi\\hbar\\left(N +\\frac\\mu 4\\right), \\label{qmas}\n\\end{equation}\nwhere $\\mu=2k$ is the Maslov's index, i.e. number of reflections of \nthe w.f. on the walls of the potential.\n\nIn the general case of the potential which is infinite between cuts, \nthe $2k$ turning point quantization condition is\n\n\\begin{equation}\n\\oint_C p(z,E)dz =\n2\\pi\\hbar\\sum_{i=1}^k \\left(n+\\frac 12\\right)_i \\equiv\n2\\pi\\hbar\\left(N+\\frac\\mu 4\\right), \\label{genC}\n\\end{equation}\nwhere $N=kn_i$ is the total number of zeroes of the w.f. on the $k$ \ncuts. On the real axis, for the $2k$ TP problem, the quantization \ncondition (\\ref{genC}) has the form\n\n\\begin{equation}\n\\sum_{i=1}^k\\int_{x_{1i}}^{x_{2i}}\\sqrt{p^2(z,E)}dz =\n\\pi\\hbar\\left(N+\\frac\\mu 4\\right). \\label{suxi}\n\\end{equation}\n\nBecause the harmonic oscillator potential is symmetric, integrals in \nEq. (\\ref{oink}) are identical, i.e. the quantization condition \n(\\ref{oink}) is equivalent to the $2$TP quantization condition \n(\\ref{gqc2}). The phase-space integral can be easily calculated in \nthe complex plane. For this we have to exclude the singularities at \n$r=0$ and $\\infty$ outside the contour $C$. Excluding these \ninfinities we have, for the integral (\\ref{genC}) with the isotropic \npotential, $I = I_0 + I_{\\infty}$, where $I_0 = -2\\pi M$ and integral \n$I_{\\infty}$ is calculated with the help of the replacement $r=1/z$, \n$I_\\infty=2\\pi E/\\omega$, i.e. we again obtain the exact result \n(\\ref{Eosc}) for $E_n$.\n\nConsider Eq. (\\ref{Eosc}) at $n_r=0$ and $l=0$, i.e. the energy of \nthe ground state. We have $E_0 = \\omega(\\hbar + M_0)$, where $M_0 \n=\\hbar /2$ is the contribution of the quantum fluctuations of the \nangular momentum into the energy of the ground state $E_0$. The \nradial quasiclassical eigenfunctions, $\\tilde R_n^{WKB}(r)$, in the \nregion of the classical motion far from the turning points are \nwritten analogously to the above case in the form of a standing wave \n[see Eq. (\\ref{Rn})] .\\\\\n\n{\\bf D. The Hulth\\'en potential $V(r)=-V_0e^{-r/r_0}/\n(1-e^{-r/r_0})$}.\nThe Hulth\\'en potential is of a special interest in atomic and \nmolecular physics. The potential is known as an ``insoluble'' by the \nstandard WKB method potentials, unless one supplements it with \nLanger-like corrections. The radial problem for this potential is \nusually considered at $l=0$. However, in the approach under \nconsideration, the quasiclassical method results in the nonzero \ncentrifugal term at $l=0$ and allows to obtain the analytic result \nfor all $l$.\n\nThe leading-order WKB quantization condition (\\ref{gqc2}) for the\nHulth\\'en potential is\n\n\\begin{equation}\nI=\\oint\\sqrt{2m\\left( E + V_0\\frac{e^{-r/r_0}}\n{1-e^{-r/r_0}}\\right) -\\frac{\\vec M^2}{r^2}}dr = 2\\pi\n\\hbar\\left(n_r+\\frac 12\\right). \\label{Ihul}\n\\end{equation}\nIn the region $r>0$, this problem has two turning points $r_1$ and \n$r_2$. The phase-space integral (\\ref{Ihul}) is calculated \nanalogously to the above case. Introducing the new variable $\\rho = \nr/r_0$, we calculate the contour integral in the complex plane, where \nthe contour $C$ encloses the classical turning points $\\rho_1$ and \n$\\rho_2$. Using the method of stereographic projection, we should \nexclude the infinities at $r=0$ and $\\infty$ outside the contour $C$. \nExcluding these infinities we have, for the integral (\\ref{Ihul}), \n$I=I_0+I_\\infty$, where $I_0=-2\\pi M$ and $I_\\infty$ is calculated \nwith the help of the replacement $z=e^\\rho -1$ \\cite{SeS},\n\n\\begin{eqnarray} I = \\oint \\sqrt{2mr_0^2\\left(E + V_0\n\\frac{e^{-\\rho }}{1-e^{-\\rho }}\\right) - \\frac{\\vec M^2}\n{\\rho ^2}}d\\rho = \\label{Ihul1} \\\\ -2\\pi M + 2\\pi\nr_0\\sqrt{-2m}\\left[ -\\sqrt{-E} + \\sqrt{-E+V_0}\\right]. \\nonumber\n\\end{eqnarray}\n\nSubstituting the integration result into Eq. (\\ref{Ihul}), we \nimmediately get the exact energy spectrum\n\n\\begin{equation} E_n=-\\frac 1{8mr_0^2}\\left(\\frac{2mV_0r_0^2}N -\nN\\right)^2. \\label{Ehul} \\end{equation}\nwhere $N = (n_r+\\frac 12)\\hbar + M$ is the principal quantum number. \nSetting in (\\ref{Ehul}) $M=0$, we arrive at the energy eigenvalues \nobtained from known exact solution of the Schr\\\"odinger's equation at \n$l=0$. However, in our case $M_{min}\\equiv M_0 = \\hbar/2$ at $l=0$ \nand the principal quantum number is $N = (n_r+\\frac 12)\\hbar + M_0$. \nAs in the previous examples, this apparently shows the contribution \nof the quantum fluctuations of the angular momentum into the energy \nof the ground state, $E_0$.\\\\\n\n{\\bf E. The Morse potential $V(r) = V_0[e^{-2\\alpha (r/r_0-1)}-2$\n$e^{-\\alpha (r/r_0-1)}]$}.\nThe Morse potential is usually considered as one-dimensional problem \nat $l=0$. In this case the problem has two turning points (note that \nthe left turning point, $r_1$, is negative) and can be solved \nexactly. In the general case, for $l>0$, we have an ``insoluble'' \n$4$TP problem.\n\nFor this potential, let us consider, first, the radial Schr\\\"odinger \nequation (\\ref{rad}), which does not contain the centrifugal term at \n$l=0$,\n\n\\begin{equation} \\left(-i\\hbar\\frac d{dr}\\right)^2U(r) =\n2m\\left[E-V_0 e^{-2\\alpha(r-r_0)/r_0}+2V_0e^{-\\alpha (r-r_0)/r_0}\n\\right]U(r). \\label{shMor}\n\\end{equation}\nThe first-order WKB quantization condition (\\ref{qc2}) appropriate to \nthis equation is\n\n\\begin{equation} \\int_{r_1}^{r_2}\\sqrt{2m[E -\nV_0e^{-2\\alpha (r-r_0)/r_0} + 2V_0e^{-\\alpha (r-r_0)/r_0}]}dr =\n\\pi\\hbar\\left(n_r+\\frac 12\\right). \\label{IMor}\n\\end{equation}\nIntroducing a variable $x=e^{-\\alpha (r-r_0)/r_0}$, we reduce the \nphase-space integral to the well known one. Sequential simple \ncalculations result in the exact energy eigenvalues\n\n\\begin{equation} E_n = -V_0\\left[1-\\frac{\\alpha\\hbar(n_r +\\frac 12)}\n{r_0\\sqrt{2mV_0}} \\right]^2. \\label{EMor}\n\\end{equation}\n\nNow, let us deal with Eq. (\\ref{rra}) for this potential, which \ncontains the non-vanishing centrifugal term, $\\hbar^2/4r^2$, at \n$l=0$. In this case we have an ``insoluble'' $4$TP problem. In the \ncomplex plane, the problem has two cuts ($k=2$), at $r<0$ and $r>0$, \ntherefore, we apply the $4$TP quantization condition (\\ref{oink}),\n\n\\begin{equation}\nI = \\oint_C\\sqrt{2m\\left[ E-V_0e^{-2\\alpha (r-r_0)/r_0} +\n2V_0e^{-\\alpha (r-r_0)/r_0}\\right] -\\frac{\\vec M^2}{r^2}}dr =\n4\\pi\\hbar \\left(n_r +\\frac 12\\right), \\label{IMoM}\n\\end{equation}\nwhere the contour $C$ encloses the two cuts, but does not enclose the \npoint $r=0$. To calculate this integral introduce the variable $\\rho \n= r/r_0$. Using the method of stereographic projection, we should \nexclude the singularities outside the contour $C$, i.e. at $r=0$ and \n$\\infty$. Excluding these infinities we have, for the integral \n(\\ref{IMoM}) \\cite{SeS},\n\n\\begin{equation}\nI = -2\\pi M -\\frac{2\\pi r_0}{\\alpha}\\left(\\sqrt{-2mE} -\n\\sqrt{2mV_0}\\right), \\label{IMocl} \\end{equation}\nand for the energy eigenvalues this gives\n\n\\begin{equation}\nE_n = -V_0\\left[1-\\alpha\\frac{2\\hbar(n_r+\\frac 12) +\nM}{r_0\\sqrt{2mV_0}} \\right]^2. \\label{EMorM}\n\\end{equation}\n\nSetting in (\\ref{EMorM}) $l=0$, we obtain,\n\n\\begin{equation}\nE_n = -V_0\\left[1-\\frac{\\alpha[2\\hbar(n_r+\\frac 12)+M_0}\n{r_0\\sqrt{2mV_0}} \\right]^2. \\label{EMoM0}\n\\end{equation}\nEquation (\\ref{EMoM0}) for $E_n$ is different from the expression \n(\\ref{EMor}) obtained from solution of Eq. (\\ref{rad}) for the Morse \npotential at $l=0$. This difference is caused by the nonzero \ncentrifugal term $\\hbar^2/4r^2$ in the radial equation (\\ref{rra}) at \n$l=0$. Thus we obtain two results for the Morse potential by the WKB \nmethod: the known exact eigenvalues (\\ref{EMor}) obtained from \nsolution of Eq. (\\ref{rad}) at $l=0$ and another result \n(\\ref{EMorM}) obtained from solution of Eq. (\\ref{rra}) for all \n$l$.\\\\\n\n{\\bf F. The potential $V(r) = kr +\\frac 12\\omega^2r^2$}.\nThis linear plus isotropic potential is one of the interest in \nparticle physics. The potential has four turning points and can not \nbe reduced to a hypergeometric function by a suitable transformation. \nThis multi-turning-point problem is ``insoluble'', also, by the \nstandard WKB method. However, the WKB method in the complex plane \nallows easily to solve this $4$TP problem [see, for instance, Refs. \n\\cite{SeR,KrSe}].\n\nThe problem has two cuts ($k=2$) in the complex plane between turning \npoints $r_1$, $r_2$ and $r_3$, $r_4$. Quantization condition \n(\\ref{genC}), for this problem, is\n\n\\begin{equation} I =\\oint_C\\sqrt{2mE - 2mkr - (m\\omega r)^2 -\n\\frac{\\vec M^2}{r^2}}dr = 4\\pi\\hbar\\left(n_r+\\frac 12\\right),\n\\label{Ilq} \\end{equation}\nwhere the contour $C$ includes both cuts, but includes no the point \n$r=0$. To calculate the integral (\\ref{Ilq}) we exclude the \ninfinities at $r=0$ and $\\infty$ outside the contour $C$. Excluding \nthese infinities we have, for the integral (\\ref{Ilq}),\n\n\\begin{equation} I = 2\\pi\\left[\\frac E\\omega + \\frac 1{2m\\omega}\n\\left(\\frac k{\\omega}\\right)^2 - M\\right] =\n4\\pi\\hbar\\left(n_r +\\frac 12\\right), \\label{Ilqc}\n\\end{equation}\nor, for the energy eigenvalues, this gives\n\n\\begin{equation}\nE_n = \\omega\\left[2\\hbar\\left(n_r + \\frac 12\\right) + M\\right] -\n\\frac 1{2m}\\left(\\frac k\\omega\\right)^2. \\label{Elq}\n\\end{equation}\n\nEnergy eigenvalues (\\ref{Elq}) are similar to the harmonic oscillator \nones but shifted by the constant value $-\\frac 1{2m}(k/\\omega)^2$. \nPutting in Eq. (\\ref{Elq}) $k=0$, we arrive to the eigenvalues \n(\\ref{Eosc}) for the isotropic oscillator.\n\nAnalogously one can obtain energy eigenvalues for other spherically \nsymmetric potentials. The standard leading-order WKB approximation \nappropriate to the wave equation (\\ref{rra}) yields the exact energy \neigenvalues for known solvable potentials and ``insoluble'' ones with \nmore than two turning points. This is possible because the \ncentrifugal term of the required form, $(l+\\frac 12)^2\\hbar^2/r^2$, \nhas obtained in a natural way from solution of the angular equation \n(\\ref{rth}) with the use of the same WKB method; this term is the \nsame for all central-field potentials. \\\\ \\newpage\n\n\\noindent {\\bf 5. Conclusion } \\\\\n\nConventional approach to solve the Schr\\\"odinger's equation is to \nreduce the original equation to a hypergeometric form or some special \nfunction by a suitable transformation. In each and every case, one \nneeds to find, first, a special transformation for the wave function \nand its arguments to reduce the original equation to some known \nequation. There is another way to solve the Schr\\\"odinger's equation \nwhich is simple, general for all types of problems in quantum \nmechanics, and a very efficient to solve not only two- but \nmulti-turning point problems.\n\nAlmost together with quantum mechanics an appropriate method to solve \nthe wave equation has been developed known mainly as the WKB \napproximation. This method is general for all types of problems in \nquantum mechanics, simple from the physical point of view, and its \ncorrect application results in the exact energy eigenvalues for {\\em \nall} solvable potentials.\n\nIn spite of long history no any strict rules concerning the \napplication of the WKB method to multi-dimensional problems in \nquantum mechanics have been formulated. Meanwhile, this topic is \nclosely related to the problem of exactness of the WKB method. The \nexactness of the method has proven in the literature for many \npotentials with the help of specially developed techniques, or \nimprovements, or modifications of the quasiclassical method on the \nreal axis and in the complex plane. In this work we have fulfilled \nthe quasiclassical analysis of the three-dimensional Schr\\\"odinger's \nequation. The original equation has been reduced to the form of the \nclassic Hamilton-Jacobi equation without first derivatives. \nSeparation of the equation has been performed with the help of the \ncorrespondence principle between classic and quantum-mechanical \nquantities. As a result of the separation, we have obtained the \nsystem of reduced second-order differential equations. Each of these \nequations has the correct quantum-mechanical form, $\\hat \np^2_q\\psi(q)= p^2(q)\\psi(q)$, and solved by the WKB method. We have \nstressed that the squared generalized moments, $p^2(q)$, obtained \nafter separation should coincide with the corresponding classic \nmoments. This means that the problem under consideration should \ncorrespond to a concrete classic problem.\n\nWe have shown that the Langer replacement $l(l+1)\\rightarrow (l+\\frac \n12)^2$ needed to reproduce the exact energy spectrum for the \nspherically symmetric potentials by the WKB method requires the \nmodification of the squared angular momentum in the quasiclassical \nregion. The squared angular momentum eigenvalues, $\\vec M^2=(l+\\frac \n12)^2\\hbar^2$, have obtained in our approach from solution of the \nangular wave equation in the framework of the same quasiclassical \nmethod. As a result, the centrifugal term has the form $(l+\\frac \n12)^2 \\hbar^2/ r^2$ for {\\em any} spherically symmetric potential \n$V(r)$.\n\nThe quasiclassical solution contains a more detail information in \ncomparison with known exact solution. One of consequences of the WKB \nsolution of the Schr\\\"odinger's equation is the existence of a \nnontrivial angular eigenfunction of the type of a standing half-wave \nfor the angular quantum number $l=0$. This solution has treated as \none which describes the quantum fluctuations of the angular momentum \nwith the eigenvalue $M_0=\\hbar/2$. We have shown that the quantum \nfluctuations of the angular momentum contribute to the energy of the \nground state, $E_0$.\n\nTo demonstrate efficiency of the quasiclassical method, we have \nsolved the three-dimensional Schr\\\"odinger's equation for some \ncentral-field potentials. The quasiclassical method successfully \nreproduces the exact energy spectrum not only for solvable \nspherically symmetric potentials but, also, for ``insoluble'' \npotentials with more than two turning points. The quasiclassical \neigenfunctions for the discrete spectrum have written in elementary \nfunctions in the form of a standing wave.\n\nThe remarkable features of the quasiclassical method incline us to \ntreat the leading-order WKB approximation as a special (asymptotic) \nexact method to solve the Schr\\\"odinger equation. In the \nquasiclassical approach we use the same technique for all types of \nproblems. The same simple rules formulated for two-turning problems \nwork for many turning point problems, as well. In this sense, the \nquasiclassical method is a more general in comparison with \ntraditional one with the use of techniques of the special functions.\n\n{\\it Acknowledgements}. The author thanks Prof. Uday P. Sukhatme for \nkind invitation to visit the University of Illinois at Chicago where \na part of the work has been done and, also, for useful discussions \nand valuable comments. I should also like to thank Prof. A.A. Bogush \nfor support and constant interest to this work.\n\nThis work was supported in part by the Belarusian Fund for \nFundamental Researches.\n\n\\newpage\n\\begin{thebibliography}{99}\n\n\\bibitem{He} J. Heading, {\\em An Introduction to Phase-Integral\nmethods} (John Wiley \\& Sons, Inc. New York, 1962); L. Schiff,\nQuantum mechanics (2nd edition McGraw-Hill, New York-Toronto-London,\n1955).\n\n\\bibitem{Fro} N. Fr\\\"oman and P. O. Fr\\\"oman, {\\em JWKB\nApproximation: Contributions to the Theory} (North Holland,\nAmsterdam, 1965).\n\n\\bibitem{Len} A. Lenef and S. C. Rand, Phys. Rev. A {\\bf 49}, 32\n(1994); A. Voros, Phys. Rev. A {\\bf 40}, 6814 (1989).\n\n\\bibitem{Tr} J. Treiner and H. Krivine, Ann. Phys. (N.Y.) {\\bf 170},\n406 (1986); M. V. Berry and K.E. Mount, Rep. Prog. Phys. {\\bf 35}, 315\n(1972).\n\n\\bibitem{SeS} M. N. Sergeenko, Phys. Rev. A {\\bf 53}, 3798 (1996).\n\n\\bibitem{Sukh} R. Dutt, A. Khare, and U. P. Sukhatme, Phys. Lett. B\n{\\bf 181}, 295 (1986).\n\n\\bibitem{Rep} F. Cooper, A. Khare, and U. P. Sukhatme, Phys. Rep. {\\bf\n251}, 267 (1995).\n\n\\bibitem{Kha} A. Khare, Phys. Lett. B {\\bf 161}, 131 (1985); E. Kasap,\nB. G\\\"on\\\"ul, and M. Simsek, Chem. Phys. Lett. {\\bf 172}, 499 (1990).\n\n\\bibitem{Dunh} J. L. Dunham, Phys. Rev. {\\bf 41}, 713 (1932).\n\n\\bibitem{Krei} J. B. Kreiger and C. Rosenzweig, Phys. Rev. {\\bf 164},\n713 (1967); C. Rosenzweig and J. B. Kreiger, J. Math. Phys. {\\bf 9},\n849 (1968); J. B. Kreiger, M. Lewis and C. Rosenzweig, J. Chem. Phys.\n{\\bf 44}, 2942 (1967); C. Rosenzweig and J. B. Kreiger, J. Math. Phys.\n{\\bf 9}, 849 (1968).\n\n\\bibitem{Lang} R. E. Langer, Phys. Rev. {\\bf 51}, 669 (1937).\n\n\\bibitem{Bru} A. S. Bruev, Phys. Lett. A {\\bf 161}, 407 (1992).\n\n\\bibitem{Art} G. A. Arteca, F. M. Fern\\'andez and E. A. Castro,\n Lecture notes in chemistry, Vol. 53. Large order perturbation\n theory and summation methods in quantum mechanics (Springer,\n Berlin, 1990) Ch. XVII.\n\n\\bibitem{SuSe} U. Sukhatme and M. N. Sergeenko, E-print quant-ph/9911026\n(1999).\n\n\\bibitem{Kor} G. A. Korn, T. M. Korn: {\\em Mathematical Handbook}\n(2nd, enlarged and revised edition, McGraw-Hill, New York-San\nFrancisco-Toronto-London-Sydney, 1968).\n\n\\bibitem{Wen} P. Wentzel, Z. Phys. {\\bf 38}, 518 (1926).\n\n\\bibitem{FF} N. Fr\\\"oman and P. O. Fr\\\"oman, J. Math. Phys. {\\bf\n18}, 96 (1977).\n\n\\bibitem{Mi} S. P. Misra and T. S. Shankara, J. Math. Phys. {\\bf 9},\n299 (1968).\n\n\\bibitem{Flu} S. Fl\\\"ugge, Practical Quantum Mechanics I\n(Springer-Verlag, Berlin-Heidelberg-New York, 1971).\n\n\\bibitem{Iwa} M. Iwasaki, Progr. Theor. Phys. {\\bf 91}, 139 (1994).\n\n\\bibitem{SeF} M. N. Sergeenko, Mod. Phys. Lett. A {\\bf 13}, 33\n(1998).\n\n\\bibitem{SeR} M. N. Sergeenko, Mod. Phys. Lett. A {\\bf 12}, 2859\n(1997).\n\n\\bibitem{KrSe} S. I. Kruglov and M. N. Sergeenko, Mod. Phys. Lett. A\n{\\bf 12}, 2475 (1997).\n\n\\end{thebibliography}\n\\end{document}\n\nThe Schr\\\"odinger's wave equation plays the central role in\nquantum mechanics. This equation is usually solved in terms of\nspecial functions or numerically. For several potentials, the\nequation is solved exactly\n\\cite{He}. The existing methods to solve this wave equation and\nsolutions for some potentials have been developed long before the\ncreation of quantum mechanics. These mathematical methods and\nsolutions of the Schr\\\"odinger's equation are classic now.\n\nThe WKB method was originally proposed for obtaining approximate\neigenvalues of one-dimensional two-turning-point problems in the\nlimiting case of large quantum numbers. However, from the moment of\nits appearance up to now the same old problem of the exactness of\nthe application has arisen.\n\nSetting in (\\ref{EMorM}) $M =0$, we arrive at the formula (\\ref{EMor})\nobtained from solution of Eq. (\\ref{rad}) at $l=0$. However, in\nour case $M_{min}\\equiv M_0=\\hbar/2$ at $l=0$ and the energy\neigenvalues are:\n\n\\begin{equation}\nE_n = -V_0\\left[1-\\frac{\\alpha[\\hbar(n_r+\\frac 12)+M_0}{r_0\\sqrt{2mV_0}}\n\\right]^2. \\label{EMoM0} \\end{equation}\n\nthe quantization condition (\\ref{Io}) then takes the form\n\n\\begin{equation}\nI =\\frac 12\\int_{z_3}^{z_4}\\sqrt{\\frac{2mE}z - m^2\\omega^2 -\n\\frac{\\vec M^2}{z^2}}dz = \\pi\\hbar\\left(n_r+\\frac 12\\right).\n\\label{I34} \\end{equation}\n\nThere is another approach to the eigenvalue problem in quantum\nmechanics. Almost together with quantum mechanics an appropriate\nmethod to solve the wave equation has been developed which is general\nfor all types of problems in quantum mechanics, simple from the\nphysical point of view, and its correct application results in the\nexact energy eigenvalues for {\\em all} solvable potentials.\n\nThis is\nthe quasiclassical method which is well known and widely used mainly\nas the Wentzel-Kramers-Brillouin (WKB) approximation\n\\cite{Fro}-\\cite{Tr} applicable in the case when the de Broglie\nwavelength, $\\lambda =h/p$ ($h=2\\pi\\hbar$), is changing slowly. It\nwas observed with great interest long ago that, in several cases of\ninterest, the WKB method yields the exact energy levels. But what is\nmore interesting the quasiclassical method always reproduces the\nexact energy eigenvalues." } ]	[ { "name": "quant-ph9912069.extracted_bib", "string": "{He J. Heading, {\\em An Introduction to Phase-Integral methods (John Wiley \\& Sons, Inc. New York, 1962); L. Schiff, Quantum mechanics (2nd edition McGraw-Hill, New York-Toronto-London, 1955)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Fro N. Fr\\\"oman and P. O. Fr\\\"oman, {\\em JWKB Approximation: Contributions to the Theory (North Holland, Amsterdam, 1965)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Len A. Lenef and S. C. Rand, Phys. Rev. A {49, 32 (1994); A. Voros, Phys. Rev. A {40, 6814 (1989)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Tr J. Treiner and H. Krivine, Ann. Phys. (N.Y.) {170, 406 (1986); M. V. Berry and K.E. Mount, Rep. Prog. Phys. {35, 315 (1972)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{SeS M. N. Sergeenko, Phys. Rev. A {53, 3798 (1996)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Sukh R. Dutt, A. Khare, and U. P. Sukhatme, Phys. Lett. B {181, 295 (1986)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Rep F. Cooper, A. Khare, and U. P. Sukhatme, Phys. Rep. {251, 267 (1995)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Kha A. Khare, Phys. Lett. B {161, 131 (1985); E. Kasap, B. G\\\"on\\\"ul, and M. Simsek, Chem. Phys. Lett. {172, 499 (1990)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Dunh J. L. Dunham, Phys. Rev. {41, 713 (1932)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Krei J. B. Kreiger and C. Rosenzweig, Phys. Rev. {164, 713 (1967); C. Rosenzweig and J. B. Kreiger, J. Math. Phys. {9, 849 (1968); J. B. Kreiger, M. Lewis and C. Rosenzweig, J. Chem. Phys. {44, 2942 (1967); C. Rosenzweig and J. B. Kreiger, J. Math. Phys. {9, 849 (1968)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Lang R. E. Langer, Phys. Rev. {51, 669 (1937)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Bru A. S. Bruev, Phys. Lett. A {161, 407 (1992)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Art G. A. Arteca, F. M. Fern\\'andez and E. A. Castro, Lecture notes in chemistry, Vol. 53. Large order perturbation theory and summation methods in quantum mechanics (Springer, Berlin, 1990) Ch. XVII." }, { "name": "quant-ph9912069.extracted_bib", "string": "{SuSe U. Sukhatme and M. N. Sergeenko, E-print quant-ph/9911026 (1999)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Kor G. A. Korn, T. M. Korn: {\\em Mathematical Handbook (2nd, enlarged and revised edition, McGraw-Hill, New York-San Francisco-Toronto-London-Sydney, 1968)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Wen P. Wentzel, Z. Phys. {38, 518 (1926)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{FF N. Fr\\\"oman and P. O. Fr\\\"oman, J. Math. Phys. {18, 96 (1977)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Mi S. P. Misra and T. S. Shankara, J. Math. Phys. {9, 299 (1968)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Flu S. Fl\\\"ugge, Practical Quantum Mechanics I (Springer-Verlag, Berlin-Heidelberg-New York, 1971)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{Iwa M. Iwasaki, Progr. Theor. Phys. {91, 139 (1994)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{SeF M. N. Sergeenko, Mod. Phys. Lett. A {13, 33 (1998)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{SeR M. N. Sergeenko, Mod. Phys. Lett. A {12, 2859 (1997)." }, { "name": "quant-ph9912069.extracted_bib", "string": "{KrSe S. I. Kruglov and M. N. Sergeenko, Mod. Phys. Lett. A {12, 2475 (1997)." } ]
quant-ph9912070		[]		[ { "name": "quant-ph9912070.tex", "string": "%Subj: E.Pessa, G.Vitiello, Quantum dissipation and neural net dynamics\n\n\n% published in the journal \"Bioelectrochemistry and Bioenergetics\n% 48:339-342, 1999\n\n \n \n %Latex file \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n%LEGENDA \n%1) equazioni : \\be xxxxxxxx \\lab{numero}\\ee \n%2) citazioni di formule : (\\ref{numero}) \n%3) citazioni di lavori : \\cite{QD} \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n%\\documentstyle [11pt,twocolumn]{article} \n\\documentstyle [11pt,twocolumn]{report} \n\\setlength{\\topmargin}{-1.cm} \n\\setlength{\\headsep}{1.6cm} \n\\setlength{\\evensidemargin}{.7cm} \n\\setlength{\\oddsidemargin}{.7cm} \n\\setlength{\\textheight}{21.cm} \n\\setlength{\\textwidth}{15.2cm} \n\\setcounter{section}{1} \n%\\setcounter{chapter}{1} \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\def\\ZzZ{{\\hbox{\\tenrm Z\\kern-.31em{Z}}}} \n\\def\\CcC{{\\hbox{\\tenrm C\\kern-.45em{\\vrule height.67em width0.08em depth- \n.04em \n\\hskip.45em }}}} \n\\def\\mapright#1{\\smash{\\mathop{\\longrightarrow}\\limits^{#1}}} \n\\def\\mapbelow#1{\\smash{\\mathop{\\longrightarrow}\\limits_{#1}}} \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Personal Macros %%%%%%%%%%%%%%%%%%%%%%%% \n\\newtheorem{prop}{Proposition} \n\\newtheorem{teo}{Theorem} \n\\newcommand{\\ep}{\\epsilon} \n\\newcommand{\\lab}{\\label} \n\\newcommand{\\non}{\\nonumber} \n \n\\newcommand{\\bc}{\\begin{center}} \n\\newcommand{\\ec}{\\end{center}} \n\\newcommand{\\be}{\\begin{equation}} \n\\newcommand{\\ee}{\\end{equation}} \n\\newcommand{\\bea}{\\begin{eqnarray}} \n\\newcommand{\\eea}{\\end{eqnarray}} \n\\newcommand{\\bs}{\\begin{subequations}} \n\\newcommand{\\es}{\\end{subequations}} \n\\newcommand{\\beq}{\\begin{eqalignno}} \n\\newcommand{\\eeq}{\\end{eqalignno}} \n%\\def\\bol#1{\\mbox{\\bf $#1$}} \n%\\def\\bol#1{\\mbox{\\boldmath\\tiny $#1$\\normalsize\\unboldmath}} \n%\\def\\vec#1{\\mbox{\\boldmath $#1$\\unboldmath}} \n\\def\\bol#1{{\\bf #1}} \n\\def\\vec#1{{\\bf #1}} \n \n \n\\newcommand{\\half}{\\frac{1}{2}} \n\\newcommand{\\qrt}{\\frac{1}{4}} \n \n% \n% A useful Journal macro \n\\def\\Journal#1#2#3#4{{#1} {\\bf #2}, {#3} {(#4)}} \n \n% A useful Book macro \n\\def\\Book#1#2{{\\em #1} {( #2)} } \n \n% Some useful journal names \n\\def\\AP{\\em Ann. 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Rev.} \\bf D} \n \n% Some other macros used in the sample text \n\\def\\st{\\scriptstyle} \n\\def\\sst{\\scriptscriptstyle} \n\\def\\mco{\\multicolumn} \n\\def\\epp{\\epsilon^{\\prime}} \n\\def\\vep{\\varepsilon} \n\\def\\we{\\wedge} \n\\def\\ra{\\rightarrow} \n\\def\\ko{K^0} \n\\def\\kb{\\bar{K^0}} \n\\def\\al{\\alpha} \n\\def\\ga{\\gamma} \n\\def\\om{\\omega} \n\\def\\Om{\\Omega} \n\\def\\ab{\\bar{\\alpha}} \n\\def\\lab{\\label} \n%\\setlength{\\baselineskip}{15pt} \n%\\renewcommand{\\baselinestretch}{1.5} % SPAZIATURA {1.5} \n\\begin{document} \n%\\tableofcontents % INDICE \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\thispagestyle{empty} \n \n%\\vspace{2.0cm} \n\\bc \n\\Huge{Quantum dissipation and Neural Net Dynamics} \n \n\\vspace{1.2cm} \n \n\\large{Eliano Pessa${}^{a}$ and Giuseppe Vitiello${}^{b}$} \\\\ \n\\small \n%\\bigskip \n\\bigskip \n{\\it ${}^{a}$Facolt\\`a di Psicologia, Universit\\`a di \nRoma \"Sapienza\"}\\\\ \n{\\it Via dei Marsi 78, 00185 Roma, Italia}\\\\ \n{\\it and ECONA, Interuniversity Center for Research}\\\\ \n{\\it on Cognitive Processing in Natural and Artificial Systems}\\\\ \n{\\it Roma, Italia}\\\\ \n{\\it ${}^{b}$Dipartimento di Fisica, Universit\\`a di Salerno} \\\\ \n{\\it 84100 Salerno, Italia}\\\\ \n{\\it and INFM Unit\\`a di Salerno} \\\\\n{\\it pessa@axcasp.caspur.it}\\\\\n{\\it vitiello@physics.unisa.it }\\\\ \n\\vspace{1.3cm} \n \n \n\\ec \n\\small \n%\\normalsize \n{\\bf Abstract} \nInspired by the dissipative quantum model of brain, \nwe model the states \nof neural nets in terms of collective modes by the help of the \nformalism of Quantum Field Theory. \nWe exhibit an explicit neural net \nmodel which allows to memorize a sequence of several informations \nwithout reciprocal destructive interference, namely we solve the \noverprinting problem in such a way last registered information does \nnot destroy the ones previously registered. Moreover, the net \nis able to recall not only the last registered information \nin the sequence, but also anyone of those previously registered. \n\n\n\\vspace{1.3cm} \n\\normalsize\n \n \n \nThe quantum model of brain by Umezawa and Ricciardi \\cite{UR,S1,S2,CH} \nhas attracted much \nattention in recent years. Moreover, its extension to dissipative \ndynamics \\cite{VT}, aimed to solve the long standing problem of memory \ncapacity, provides an interesting framework to study consciousness \nrelated mechanisms. On the other hand, computational neuroscience mostly \nrelies on specific activity of neural cells and of their networks, thus \nleading to a number of models and simulations of the brain activity in \nterms of neural nets, mostly based on modern methods of statistical\nmechanics and of spin glass theory \\cite{MZ,AI}. \nBesides, there is an increasing interest in the \nstudy of quantum features of network dynamics, either in connection with \ninformation processing in biological systems, or in relation with a \ncomputational strategy based on the system quantum evolution (quantum \ncomputation). \n \nInspired thus by the papers \\cite{UR,S1,S2} and \\cite{VT} \n(see also \\cite{PR,P2,YA}), \nwe explore the possibility of modeling the states \nof neural nets in terms of collective modes by the help of the \nformalism of Quantum Field Theory (QFT). \n \nWe show that the classical limit of the \ndissipative quantum brain dynamics (DQBD) \\cite{VT} \nprovides a representation of a \nneural net characterized by long range correlations among the net's \nunits. In this way we exhibit a link between DQBD and neural net \ndynamics \\cite{PV}. \n \nWe present an explicit neural net \nmodel which allows to memorize a sequence of several informations \nwithout reciprocal destructive interference, namely we solve the \noverprinting problem, i.e. last registered information does \nnot destroy the ones previously registered. The net \nis also able to recall informations registered prior to the last registered \none in the sequence. \n \nIn the following we will first introduce the general \ntheoretical background on \nwhich our neural net is modeled and then we will present some \nof its specific features and \nthe results, which, although preliminary, confirm our expectations. \n \nWe consider a three-dimensional set of $N$ interacting units (neural \nunits) sitting each one in a space-time site \n$x_{n} \\equiv ({\\bol x_{n}}, t_{n}), n = 1,2,..N$. Each unit can be in \nthe state $on~ (1)$ or $off~ (0)$. \nThe neural unit activity is characterized by the amplitude of the emitted \npulse and by the phase determined by the emission time. This suggests to us \nthat each unit can be described by a complex doublet field \n$\\psi(x_{n}) = (\\psi_{u} (x_n),~ \\psi_{d} (x_n))$, \nwith $\\psi_{u}(x_{n})$ and $\\psi_{d}(x_{n})$ complex field \ncomponents, $u$ \nand $d$ denoting the field inner degrees of freedom corresponding to $on$ \nand $off$, respectively. \n \nAt each site $x_n$ the field inner variable may assume a well specified \nvalue ($u$ or $d$). The set of these values for all the sites specifies the \n{\\it microscopic} configuration in the \n$(u-d)$-space; however, in full generality the specification of the \n{\\it macroscopic or functional state} of the net does not actually requires \nthat correspondingly one should have a unique, well definite microscopic \nconfiguration where each unit state is specified by a definite $u$ or $d$ \nvalue. In general, indeed, many distinct microscopic configurations of the \ncomponent units may correspond to the same functional state of the net. \n \nThis means that a given (equilibrium) state of the net may be well \ncompatible with fluctuations in the states of the individual component \nunits. This amounts to say that the net state is not strictly and crucially \ndependent on the specific state of each individual unit: i.e. we admit \nenough {\\it plasticity} (as contrasted with {\\it rigidity}) for \nthe net; in other words, we can say that the net macroscopic state is \nthe output, or the {\\it asymptotic} state, emerging from the microscopic \ndynamics which rules the interaction among the component units. \n \nFor large number $N$ of component units, such a picture is certainly more \n\"realistic\" than a \"rigid\" one and could also be more appropriate for a \npossible modeling of the natural brain in terms of neural nets (as it is \nwell known the brain functional activity is not strictly related with the \nactivity of each single neuron; in this paper however we will not deal \nwith modeling the natural brain). \n \nOne can view such a situation also from the \nperspective of the pulses or signals traveling on the connections among \nthe units: a traveling signal may contribute to excite or de-excite a \ncertain unit thus changing its $u$ or $d$ state. Consequently, a \nspecific state of the net, corresponding to a \ngiven {\\it dynamical} distribution of pulses on the net connections, is \nnecessarily a state for which the single unit states at each site cannot \nbe uniquely specified once for ever, due to pulse action on the units. As \na matter of fact, one should consider the unit states as non-observable, \nsince any observation on the unit may non-trivially interfere with the \ndynamics of the pulses. Only the output of such a dynamics is observable \nand this is why above we have called it the asymptotic state of the \nnet (i.e. states for time $t_n \\rightarrow \\pm \\infty$ for each $n$). \n \nSumming up, since fluctuations are allowed for \nthe states of the individual unit at each site, and thus for the basic \nfield $\\psi (x_n)$, and since, as a consequence, in the ($u-d$)-space the \nuncertainty in the identification of the \"trajectory\" representing the \nevolution of the state of the unit cannot be eliminated without strongly \ninterfering with it, we are led to treat $\\psi (x_n)$ like\na quantum field satisfying quantum dynamical equations. \n\nThe above considerations lead us to think of the neural net in terms \nsimilar to the ones usually adopted for condensed matter physics: the \nglobal behavior of the net, namely its functional state and its \nevolution, can be characterized by a (classical) macroscopic observable \nas it usually happens in solid state physics, e.g. in superconductivity, \nin ferromagnetism, etc.. Such an observable, generally called the ``order \nparameter'', is determined by the dynamics of the elementary components \nor units and by its symmetries \\cite{TFD,U2,IZ,A1}. \nIt may be considered as \na {\\it code} specifying the vacuum or ground state \\cite{UR,VT}. \n \nLike in ferromagnetism one introduces the order \nparameter \"magnetization\", in our present case we introduce the \nmacroscopic observable $\\cal M$ whose values are assumed to specify the \ninformation content of the net. We define ${\\cal M} \n\\equiv (1/2){\|(N_u - N_d)\|}$, with $N = N_u + N_d$, \nwhere $N_u$ and $N_d$ denote the number of units $on$ and $off$, \nrespectively. $\\cal M$ is the neural net order parameter which \ncharacterizes its macroscopic state. \n \nThe state ${\\cal M} =0$ is called the \"normal state\" (void of information \ncontent); the \"information states\" or ``memory states''are the ones with\n${\\cal M} \\neq0$ \n(different informations associated to different non-zero $\\cal M$ \nvalues). \n \nSince the information comes to the net {\\it from the outside}, we assume \nthat the neural net may be set into a ${\\cal M} \\neq 0$ state under the \naction of an external input (coupling of the net with the environment)\nand it remains in such a state even after the external input is not\nanymore acting on the net (the information has been recorded). In other\nwords, we assume that the interaction among the net units cannot force, by \nitself, the net into a ${\\cal M} \\neq 0$ state (i.e. prior of any\nexternal input the net remains in its normal state). This in turn means \nthat, from one side, the basic dynamics describing the interaction among \nthe units (i.e. the evolution equations for the basic $\\psi (x_n)$ field) \nmust be invariant under the $SU(2)$ group of transformations acting on \nthe doublet field $\\psi (x_n)$. On the other side, it also means that the \nground state is {\\it not} invariant under the full $SU(2)$ group of \ntransformations, i.e. the external input triggers the spontaneous\nbreakdown of the $SU(2)$ symmetry.\n \nNotice that, as observed above, there can be many configurations of the \nset of units (microscopic configurations) corresponding to a given value of \n$\\cal M$, and therefore to a given state of the net: $\\cal M$ specifies \nindeed only the difference $\|(N_u - N_d)\|$, but says nothing on \nwhich ones \nand how many are the sites $u$ and which ones and how many are those $d$; \nso that any change, or fluctuation, between the $u$ and $d$ state of the \nunits in different sites is allowed, provided the difference $\\cal M$ is \nkept constant. In this sense $\\cal M$ is a {\\it macroscopic} \nvariable. On the contrary, the $\\psi (x_n)$ fields determine the {\\it \nmicroscopic} configurations. \n \nMoreover, the quantum field dynamics \ngenerates asymptotic equilibrium states with negligible fluctuations \nof $\\cal M$. The macroscopic \n\"memory\" state of the neural net indexed by $\\cal M$ is then a {\\it \nclassical limit state} in the sense of QFT, namely the state for which the \nfluctuations in the number of certain modes is negligible with respect to \nthe number of the same kind of modes {\\it condensed} in it: in other words, \na {\\it coherent} state \\cite{KS} with respect to these modes. \nThese condensed modes are long range correlation modes. \n \nWe have in conclusion two levels of description: i) The dynamical level \nand ii) the asymptotic level. At the dynamical level the interaction \namong the neural units (represented by basic fields $\\psi (x_n)$) \nis ruled by a \ncertain set of dynamical equations, which are assumed to be invariant \nunder the $SU(2)$ group; this level is precluded to \nobservations. At the asymptotic level, the $SU(2)$ symmetry is \nspontaneously broken and the neural net state is \ncharacterized by the order parameter $\\cal M$. \n \nThe set of asymptotic fields includes the field describing \nthe non-interacting, \n\"free\" (at $t_n \\rightarrow \\pm \\infty$ for each $n$) units, \nsay $\\phi (x_n)$, which also is a complex doublet field with \n($u-d$) inner freedom, and other fields which are generated by the \ndynamics. By resorting to well known results in QFT \\cite{IZ,TFD}, \nindeed, whenever the order parameter $\\cal M$ is different from \nzero, the dynamics generates excitation fields describing {\\it long range \ncorrelations} among the units, which are therefore {\\it collective modes} \n(the Goldstone theorem). These \nlong range modes are massless and thus their condensation in the ground\nstate does not change its energy; it only produces other (ground) states\ndegenerate in the energy, different among themselves for their\ncondensation content. The stability of the memory states is thus insured.\nThe value of the order parameter $\\cal \nM$ is a measure of the collective mode condensation in the ground state. \n \nSince the net is an open system (coupled with the environment) and the\ninformation storage produces by \nitself the breakdown of the time-reversal symmetry \\cite{VT}, we consider \nthe QFT for dissipative systems and the neural net state can be then \nrecognized \\cite{PV} to be a finite temperature state of QFT. \n \nThe role of dissipation is crucial in solving the {\\it \noverprinting} problem, namely the problem of the net memory capacity: \nin a sequential information recording each information storage would \nover-impose itself to the previously recorded one, thus deleting it. \nIn the dissipative dynamics, on the \ncontrary, a large memory capacity is possible since \neach information is recorded in each of the many degenerate ground states, \nwithout destructive interference among them\n\\cite{VT}. \n \nIn other words, dissipation implies that the net overall\nstate may be represented as a superposition \nof infinitely many degenerate ground states, or memory states, each of them \nlabeled by a different code number and each of them independently \naccessible to information storage. Many information \"files\" may then \ncoexist thus allowing a huge memory capacity. \nNon-unitary equivalence among different memory states\nacts as a protection against overlap or \ninterference among different informations \\cite{VT}. \n \nIn realistic neural net, the finiteness of the \"volume\" (the number of \nneural units) and possible defect effects may spoil unitary \nnon-equivalence thus leading to information interferences and distortions. \n \nThe retrivial of information is described by \"reading off\" the mirror \nmodes of the same code number of the information to be recalled. These \nmirror modes are essentially a \"replication signal\" of the one \nresponsible for memory storage. The replication signal thus acts as \na probe by which one \"reads\" the stored information. The process of\ninformation recalling drives the net into the (macroscopic) memory state\nof code ${\\cal M}$ corresponding to that specific information to be\nretrived. \n \nWe also observe that the mirror modes may acquire an effective nonzero \nmass due to the effects of the system finite size. \nSuch an effective mass then \nintroduces a threshold in the energy to supply in order to trigger \nthe \"recall\" process. This may lead, from one side, to \"difficulties\" \nin the information retrivial; on the other side, it may act as a \n\"protection\" against unwanted perturbations and cooperate to the neural \nnet memory state stability. \n \nThe study of thermodynamic properties shows that the generator \nof time evolution of the net state is the system entropy. The stationarity \nof free energy implies the Bose distribution for the collective modes and \nthe Fermi distribution for the neural unit fields. In this way the \ntraditional activation function for neural net units is recovered. \nFor further details see \\cite{VT} and \\cite{PV}. \n \nLet us now briefly present some of the features of the \nspecific neural net model we have worked out and the related results. \nWe will give here only a qualitative description of the model. See \n\\cite{PV} for formal details. \n \nThe net dynamics is given by the Pauli-Dirac equation for a doublet \nfield $\\psi$ interacting with an external magnetic field representing \nthe external input. A spatial discretization of this equation on a two\ndimensional lattice of \n$20 \\times 20$ sites is adopted, so as to transform the original field\nequation \ninto a system of coupled ordinary differential equations. \nBesides the interaction with the external magnetic field, the \ndoublet field also interacts with the mean magnetic field \nwhich is generated over \nthe net as a response to the external input. \nWe use the Bragg-Williams approximation and the \nmean magnetic field is taken to be proportional to \nthe magnetization induced by the external \ninput with a proportionality constant given by the Weiss constant \n$\\gamma$. The dynamical equation thus presents a nonlinear term in \n$\\psi$ since the magnetization is by itself given in terms of a \nbilinear form in $\\psi$. The Weiss constant $\\gamma$ is also related \nto the mean value over the whole net of the mean values of the \nconnection strength for each site. \n \nThe practical implementation of the net was done through \nthe following steps:\n\na) after explicit representation of the $\\psi$ field components \n$\\psi_{u}$ and $\\psi_{d}$ in terms of their real and imaginary \nparts, we obtained the corresponding four equations \nfrom the Pauli-Dirac equation. \n\nb) each of the four component field variables was expressed as a \nproduct of the site activation function times the connectivity \npotential of the site itself. \n\nc) we considered the nearest neighbor approximation for the site \nconnections. \n\nd) after a spatio-temporal discretization, we associated to each\nunit (i.e. to each site) a sigmoidal activation function characterized\nby a ``temperature'' parameter. \n\ne) we assumed an independent evolution for each of the four component \nfield variables. \n\nf) we used the modulus squared value of the activation function of\neach unit to determine the time evolution of the activation dynamics of the\nunit itself.\n\ng) we used simulated annealing in each process of writing and of \nreading (recalling). \n \nThrough the implementation of the above steps we obtained a neural net\nable to record\na sequence of informations without \noverprinting (i.e. without destruction of previously registered \ninformations in the course of a subsequent registration process) \nand to able to recall anyone of the registered informations (i.e. \nnot simply the last one) under presentation of an external input \nsimilar to the one to be recalled. \n \nSuch results make us confident that a novel conceptual and \nformal scheme in neural net modeling may be introduced which is \nbased on the simulation of a quantum dynamical evolution. \n\nWe are glad to aknowledge partial support from MURST and INFM. \n \n%\\newpage \n \n\\begin{thebibliography}{99} \n \n \n\\bibitem{UR} L.M.Ricciardi and H.Umezawa, Brain physics and many-body\nproblems. {\\it Kibernetik} {\\bf 4}, 44 (1967) \n\\bibitem{S1} C.I.J.Stuart, Y.Takahashi and H.Umezawa, \nOn the stability and non-local properties of memory. {\\it J. \\ Theor. \\ \nBiol.} {\\bf 71}, 605 (1978)\n\\bibitem{S2} C.I.J. Stuart, Y. Takahashi and \nH. Umezawa, Mixed system brain dynamics: neural memory as a macroscopic\nordered state. {\\it Found. Phys.} {\\bf 9}, 301 (1979) \n\\bibitem{CH} S. Sivakami and V. Srinivasan, A model for memory,\n{\\it J. Theor. Biol.} {\\bf 102}, 287 (1983)\n\\bibitem{VT} G. Vitiello, Dissipation and memory capacity in the quantum\nbrain model. {\\it Int. J. Mod. Phys.} {\\bf 9} 973 (1995)\n\\bibitem{MZ} M.M\\'ezard, G.Parisi and M.Virasoro, {\\it Spin glass theory \nand beyond}, World Sci., Singapore 1993\n\\bibitem{AI} D.J.Amit {\\it Modeling brain functions}, \nCambridge University Press, Cambridge 1989\n\\bibitem{PR} K.H.Pribram, {\\it Languages of the brain}, Englewood Cliffs, \nNew Jersey, 1971\n\\bibitem{P2} K.H.Pribram, {\\it Brain and perception}, Lawrence Erlbaum, New \nJersey, 1991\n\\bibitem{YA} M.Jibu , K.H.Pribram \nand K.Yasue, From conscious experience to memory storage and retrivial:\nThe role of quantum brain dynamics and boson condensation of evanescent\nphotons. {\\it Int. J. Mod. Phys.} {\\bf B10}, 1735 (1996) \n\\bibitem{PV} E.Pessa and G.Vitiello, in preparation\n\\bibitem{TFD} H.Umezawa , H.Matsumoto and M.Tachiki , {\\it Thermo Field \nDynamics and Condensed States}, North-Holland, Amsterdam 1982\n\\bibitem{U2} H.Umezawa, \n{\\it Advanced field theory: micro, macro and thermal concepts}, American \nInstitute of Physics, N.Y. 1993 \n\\bibitem{IZ} C.Itzykson and J.Zuber , {\\it Quantum Field Theory}, McGraw- \nHill Inc. N.Y. 1980 \n\\bibitem{A1} P.W.Anderson , {\\it Basic Notions of Condensed Matter Physics}, \nAddison-Wesley, N.Y. 1984\n\\bibitem{KS} J.R.Klauder and E.C.Sudarshan, {\\it Fundamentals of Quantum \nOptics}, Benjamin, New York, 1968\n\n \n\\end{thebibliography} \n \n\\end{document} \n \n%\n\n" } ]	[ { "name": "quant-ph9912070.extracted_bib", "string": "{UR L.M.Ricciardi and H.Umezawa, Brain physics and many-body problems. {Kibernetik {4, 44 (1967)" }, { "name": "quant-ph9912070.extracted_bib", "string": "{S1 C.I.J.Stuart, Y.Takahashi and H.Umezawa, On the stability and non-local properties of memory. {J. \\ Theor. \\ Biol. {71, 605 (1978)" }, { "name": "quant-ph9912070.extracted_bib", "string": "{S2 C.I.J. Stuart, Y. Takahashi and H. Umezawa, Mixed system brain dynamics: neural memory as a macroscopic ordered state. {Found. Phys. {9, 301 (1979)" }, { "name": "quant-ph9912070.extracted_bib", "string": "{CH S. Sivakami and V. Srinivasan, A model for memory, {J. Theor. Biol. {102, 287 (1983)" }, { "name": "quant-ph9912070.extracted_bib", "string": "{VT G. Vitiello, Dissipation and memory capacity in the quantum brain model. {Int. J. Mod. Phys. {9 973 (1995)" }, { "name": "quant-ph9912070.extracted_bib", "string": "{MZ M.M\\'ezard, G.Parisi and M.Virasoro, {Spin glass theory and beyond, World Sci., Singapore 1993" }, { "name": "quant-ph9912070.extracted_bib", "string": "{AI D.J.Amit {Modeling brain functions, Cambridge University Press, Cambridge 1989" }, { "name": "quant-ph9912070.extracted_bib", "string": "{PR K.H.Pribram, {Languages of the brain, Englewood Cliffs, New Jersey, 1971" }, { "name": "quant-ph9912070.extracted_bib", "string": "{P2 K.H.Pribram, {Brain and perception, Lawrence Erlbaum, New Jersey, 1991" }, { "name": "quant-ph9912070.extracted_bib", "string": "{YA M.Jibu , K.H.Pribram and K.Yasue, From conscious experience to memory storage and retrivial: The role of quantum brain dynamics and boson condensation of evanescent photons. {Int. J. Mod. Phys. {B10, 1735 (1996)" }, { "name": "quant-ph9912070.extracted_bib", "string": "{PV E.Pessa and G.Vitiello, in preparation" }, { "name": "quant-ph9912070.extracted_bib", "string": "{TFD H.Umezawa , H.Matsumoto and M.Tachiki , {Thermo Field Dynamics and Condensed States, North-Holland, Amsterdam 1982" }, { "name": "quant-ph9912070.extracted_bib", "string": "{U2 H.Umezawa, {Advanced field theory: micro, macro and thermal concepts, American Institute of Physics, N.Y. 1993" }, { "name": "quant-ph9912070.extracted_bib", "string": "{IZ C.Itzykson and J.Zuber , {Quantum Field Theory, McGraw- Hill Inc. N.Y. 1980" }, { "name": "quant-ph9912070.extracted_bib", "string": "{A1 P.W.Anderson , {Basic Notions of Condensed Matter Physics, Addison-Wesley, N.Y. 1984" }, { "name": "quant-ph9912070.extracted_bib", "string": "{KS J.R.Klauder and E.C.Sudarshan, {Fundamentals of Quantum Optics, Benjamin, New York, 1968" } ]
quant-ph9912071	Half Quantization	[ { "author": "Nuno Costa Dias" }, { "author": "{Grupo de Astrof\\'{\\i" } ]	A general dynamical system composed by two coupled sectors is considered. The initial time configuration of one of these sectors is described by a set of classical data while the other is described by standard quantum data. These dynamical systems will be named half quantum. The aim of this paper is to derive the dynamical evolution of a general half quantum system from its full quantum formulation. The standard approach would be to use quantum mechanics to make predictions for the time evolution of the half quantum initial data. The main problem is how can quantum mechanics be applied to a dynamical system whose initial time configuration is not described by a set of fully quantum data. A solution to this problem is presented and used, as a guideline to obtain a general formulation of coupled classical-quantum dynamics. Finally, a quantization prescription mapping a given classical theory to the correspondent half quantum one is presented.	[ { "name": "quant-ph9912071.tex", "string": "\\documentstyle[11pt]{article}\n\\topmargin -.25in\n\\evensidemargin .0in\n\\oddsidemargin .0in\n\\textwidth 6.5in\n\\textheight 9in\n\\renewcommand{\\baselinestretch}{1}\n\n\n\\newcommand{\\R}{{\\sf R\\hspace{-0.9ex}\\rule{0.15ex}%\n{1.5ex}\\hspace{0.9ex}}}\n\n\n\\begin{document}\n\\title{Half Quantization}\n\\author{Nuno Costa Dias \\\\ {\\it Grupo de Astrof\\'{\\i}sica e Cosmologia} \\\\ {\\it Departamento de F\\'{\\i}sica - Universidade da Beira Interior}\\\\ {\\it 6200 Covilh\\~{a}, Portugal}}\n\\date{}\n\\maketitle\n\n\n\\begin{abstract}\nA general dynamical system composed by two coupled sectors is considered.\nThe initial time configuration of one of these sectors is described by a set of classical data while the other is described by standard quantum data. These dynamical systems will be named half quantum. \nThe aim of this paper is to derive the dynamical evolution of a general half quantum system from its full quantum formulation. \nThe standard approach would be to use quantum mechanics to make predictions for the time evolution of the half quantum initial data.\nThe main problem is how can quantum mechanics be applied to \na dynamical system whose initial time configuration is not \ndescribed by a set of fully quantum data.\nA solution to this problem is presented and used, as a guideline\nto obtain a general formulation of coupled classical-quantum dynamics.\nFinally, a quantization prescription mapping \na given classical theory to the correspondent half quantum one\nis presented.\n\\end{abstract}\n\n\\section{Introduction}\n\nQuasiclassical dynamics \\cite{anderson1}, hybrid dynamics \\cite{diosi1} and, in this paper, half quantum mechanics are some of the several attempts \\cite{aleksandrov,boucher,halliwell1,halliwell2} to obtain a consistent formulation of coupled classical-quantum dynamics.\nThe motivation to develop such a theory comes from a variety of different sources. The theory is expected to make important contributions to clarify the measurement procedure in quantum mechanics, where one would like to obtain an analytic description of the wave function collapse \\cite{nature,sherry,diosi2}. Closely related is the problem of developing a consistent quantization \nprocedure for closed dynamical systems \\cite{hartle1,halliwell3}. Other important applications are expected. These include semiclassical gravity, quantum field theory in curved space time and quantum cosmology \\cite{halliwell1,nature,hartle1,wald}.\n\nTwo main approaches to the problem have been followed: In \\cite{anderson1,aleksandrov,boucher} a set of axioms defining the \nquasiclassical dynamics were proposed and motivated in terms of the consistency of the thus resulting theory. On the other hand there is the deductive approach where the intention is to derive the classical-quantum dynamics from quantum mechanics \\cite{diosi1,halliwell1,halliwell2}.\n\nIn this paper we shall follow this second approach. We assume that, just like classical mechanics, half quantum mechanics is an approximate description of quantum mechanics that derives its validity from reproducing, \"in some appropriated sense\", the predictions of the underlying theory of quantum mechanics. \n\nOur approach will be as follows: A general half quantum system is composed by two coupled sectors. One of these sectors is named classical and the other quantum. The initial data for an half quantum system is given by a set of classical data $O_i(t=0)$ for the classical sector plus a standard quantum \ndata, say an initial time wave function $\|\\phi^Q>$ for the quantum sector.\nThe important issue is how can quantum mechanics be applied to a dynamical system whose initial time configuration is not described by a set of fully quantum data. \nTo solve this problem we will convert the half quantum initial data into a \nfully quantum one. More precisely, we will determine a class of wave functions $\|\\phi>$ which are consistent with the half quantum initial data. We will be able to do this by using a classicality criterion that was presented in a related paper \\cite{nuno1} and which proved to work out successfully when the intention was to study the consistency between the full classical and full quantum descriptions of a general dynamical system.\nWe can then use quantum mechanics to obtain the time evolution of this class \nof quantum initial data. The predictions of quantum mechanics, i.e. the time evolution of the class of wave functions $\|\\phi>$, will not be completely determined. This is so because we do not have a single initial data wave function, but instead we are calculating the time evolution for a class of initial data wave functions. Therefore, quantum mechanics provides a set of predictions inside an error interval.\n\nThe main result is then that these predictions might be fully recovered by an appropriated formulation of classical-quantum dynamics, which will be named \nhalf quantum mechanics. \nIn this formulation the dynamical system is not fully quantized, the classical data describing the initial time configuration of the classical sector is explicitly used\nand dynamics is obtained as the time evolution of the classical and quantum initial data. \nStill, we are able to recover the predictions of quantum mechanics for the time evolution of the class of wave functions $\|\\phi>$. This is the desired result. It means that the half quantum framework is derived as the appropriated limit \nof quantum mechanics.\n\nWe will find that the theory derived here is just the same one that was postulated by Boucher and Traschen in \\cite{boucher}. The approach however, is rather different. In that paper the theory was motivated in terms of the properties one would like to see satisfied by a theory of coupled classical-quantum dynamics.\n\nOur derivation presents some interesting features: i) it explicitly provides the degree of precision of the half quantum predictions i.e. it tells us about the degree of consistency between the half quantum and the full quantum predictions. ii) It states what type of initial data and dynamical behaviour a system should have so that it can be described by the half quantum framework. iii) It settles down a general procedure, with assumptions kept to a minimum, to develop other, eventually more consistent or better-behaved, classical-quantum dynamics frameworks. iv) It provides a half quantization procedure mapping the classical formulation of a given dynamical system to its half quantum formulation. \n\n\n\\section{From quantum mechanics to half quantum mechanics}\n\nLet us settle down the preliminaries: we are given a dynamical system with \n$N+M$ degrees of freedom. \nThe $N$ represents the number of degrees of freedom of\nthe quantum sector, while the $M$ concerns the\nclassical sector. The phase space of the classical formulation of the system \nis spanned by a set of canonical variables $\\{q_k,p_k\\}, k=1..(M+N)$ or more succinctly just designated by $O_k, k=1..2(M+N)$. The classical sector canonical variables are denoted by $(q_i,p_i), i=1..M$ or just by $O_i, i=1..2M$ and the quantum sector canonical variables by $(q_{\\alpha},p_{\\alpha}), \\alpha = (M+1)..(M+N)$ or $O_{\\alpha}, \\alpha =(2M+1)..2(M+N)$. \nThe total phase space is assumed to have a structure given by $T^{\\ast}M_1 \\otimes T^{\\ast}M_2$ where $T^{\\ast}M_1$ is the classical sector phase space \nand $T^{\\ast}M_2$ is the quantum sector phase space. \n\nBy performing the Dirac quantization \\cite{dirac1,dirac2} we obtain the quantum formulation of the dynamical system. We also supply a complete set of commuting observables (CSCO).\nWe will take the CSCO to be \n$\\{\\hat{q}_{i},\\hat{q}_{\\alpha}\\}$.\nThe set of common eigenvectors of the CSCO spans the Hilbert space ${\\cal H}\n={\\cal H}_1 \\otimes {\\cal H}_2$.\nTaking into account the structure of the Hilbert space the general \n eigenvector might be written as $\n\|k_1,...k_N>\|z_1,...z_M> $ where the $k$s are eigenvalues of the operators \n$\\hat{q}_{\\alpha}$ and the $z$s are those of the operators $\\hat{q}_i$.\n\nThe goal is now to use the full quantum formulation of the dynamical system to study the time evolution of the half quantum initial data. \nThis is far from being straightforward, the problems one encounters being\nclosely related to those that emerge when one wants to study the dynamics of a classical \nsystem using the framework of quantum mechanics \\cite{nuno1}. As in that\ncase, the first problem is how to use the \nhalf quantum initial data to produce fully quantum initial data for the quantum theory.\nThis problem shall be approached in this section.\n\n\n\\subsection{From quantum mechanics to half quantum mechanics - Kinematics}\n\nThe half quantum dynamical system is composed by two sectors. The initial \ntime configuration of one of these sectors is described by a set of classical data.\nThat is a value $O^0_i$ and an error margin $\\delta_i$ is assigned to each classical sector observable $O_i$. The aim is to convert these classical data \ninto a full quantum one, $\|\\phi^c> \\in {\\cal H}_1$. Clearly, not all wave functions $\|\\phi^c>$ will be suitable. We are looking for a class of wave functions $\|\\phi^c> \\in {\\cal H}_1$ providing a description of the initial time configuration consistent with the classical description $(O^0_i,\\delta_i)$. To obtain this class of wave functions we impose that $\|\\phi^c>$ should satisfy a set of classicality conditions that was defined and studied in \\cite{nuno1}. More precisely, we require $\|\\phi^c>$ to be $L$-order classical ($L\\in {\\cal N}$) with respect to the classical data $(O^0_i,\\delta_i)$. The higher the order of classicality $L$, the bigger will be the degree of consistency between the classical and the quantum descriptions.\nWe will not fix the value of $L$. In fact, \n$L$ is to be one of the parameters of the formalism and latter we will find \nthat its value is related to the precision of the half quantum predictions.\n\nLet us make a brief review of the definition of the classicality criterion. Let $O_k(t)$ be the classical time evolution of an arbitrary fundamental \nobservable (belonging to the classical or to the quantum sector)\nand let $S_{ia}$ be {\\it any} sequence of classical sector observables \n$S_{ia}=O_{i1},O_{i2}....O_{in}$ - associated to a sequence \n$1 \\le i_a \\le 2M, a=1..n$ ($n$ is arbitrary) -\nsuch that:\n\\begin{equation}\n\\frac{\\partial^n O_k(t)}{\\partial S_{ia}}=\n\\frac{\\partial^n O_k(t)}{\\partial O_{i1} ....\\partial O_{in}} \\not=0 ,\n\\end{equation}\nfor some $k=1..2(N+M)$. With all sequences satisfying the former relation \nwe can obtain a set of \nmixed error kets (the reader should refer to the appendix for \nthe relevant definitions):\n\\begin{equation}\n\|E_{S_{ia}}>=(\\hat{O}_{i1}-O^0_{i1})(\\hat{O}_{i2}-O^0_{i2})....\n(\\hat{O}_{in}-O^0_{in})\|\\phi^c>,\n\\end{equation}\nwhere the quantities $O^0_{ia}$ refer to the values of the\ncorresponding observables $O_{ia}$ at the initial time.\nThe classical sector initial time wave function $\|\\phi^c>$ will be {\\it 1st-order classical} \nif it satisfies:\n\\begin{equation}\n<E_{S_{ia}}\|E_{S_{ia}}> \\le (\\delta_{S_{ia}})^2 = \\delta_{{i1}}^2 \n\\delta_{{i2}}^2....\\delta_{{in}}^2 , \n\\end{equation}\nfor all the sequences $S_{ia}$ determined in (1). In the former equation \n$\\delta_{{ia}}$ are the error margins associated to the classical \ninitial data. \nNotice that given the classical initial data and its error margins the former inequalities \nconstitute a set of requirements on the functional form of the wave \nfunction $\|\\phi^c>$. To go further we consider the $L$-order sequences $S^L_{ia}=S_{ia}S_{i^{\\prime}a}...S_{i^{\\prime \\prime}a}$ constituted by $L$ arbitrary 1st-order sequences $S_{ia}$ (determined in (1)) and write the system of inequalities (3) for these sequences. If the wave function $\|\\phi^c>$ satisfies (3) for all possible $L$-order sequences then we say that $\|\\phi^c>$ is $L$-order classical. \n\nThe set of all $L$-order classical wave functions is the class of wave functions that we wanted to determine. It is worth noticing (appendix: result a)) that all $L$-order classical wave functions $\|\\phi^c>$ satisfy the following property: in the representation\nof any of the observables $\\hat{O}_i$, they have a probability of at least $p$ \nconfined to the interval $I_i=[O^0_i-\\delta_{i}/(1-p)^{1/2L}, \nO^0_i+\\delta_{i}/(1-p)^{1/2L}]$, that is $ \\sum_{a_i \\in I_i ,n}\n\|<a_i^n\|\\phi^c>\|^2 \\ge p$ for all $i=1..2M$ and where $\|a^n_i>$ is a general \neigenvector of the operator $\\hat{O}_i$ with associated eigenvalues\n$a_i$ and degeneracy index $n$. By simple inspection of the former result we notice that the higher the degree of classicality $L$, the bigger is the confinement of the probabilistic distribution function associated to $\|\\phi^c>$, around the classical intervals $[O^0_i-\\delta_i,O^0_i+\\delta_i]$. \n\nGiven the classical initial data, the degree of classicality is a statement about the quantum mechanical description of a given configuration of the dynamical system. It tell us that the wave function $\|\\phi^c>$ satisfies some properties. In the context of this paper it may be worth thinking about the classicality criterion in an equivalent but slightly different perspective: the degree of classicality can be seen as the degree of {\\it validity} of the classical description of a given configuration of the dynamical system. The classical description is {\\it valid} up to some degree $L$ if the true, physical configuration of the dynamical system is given by a wave function $\|\\phi^c>$, $L$-order classical with respect \nto that classical description.\n\nIn conclusion: the true physical configuration of the classical sector is given by a wave function $\|\\phi^c>$. However, we are only given the classical imprecise description $(O_i^0,\\delta_i)$ and thus we should not assume that we know $\|\\phi^c>$ completely. \nIf we assume that the classical initial data is $L$-order valid then any wave function belonging to the class of $L$-order classical wave functions can be, up to what we know, the true physical configuration of the system. Therefore, the classical sector initial configuration is properly described, not by a single wave function, but by the class of $L$-order classical wave functions. \n\nThe initial time configuration of the other sector of the half quantum system is described by standard quantum data. That is we supply a completely fixed initial time wave function:\n\\begin{equation}\n\|\\phi^Q>=\\sum_{k_1,...k_N} C_{k_1,...k_N} \|k_1,...k_N> .\n\\end{equation} \nThe final step is to put the two sectors together and obtain\nthe total wave function. \nTo make it simple we assume that there is no kinematical coupling between the two sectors and thus the initial data wave function is of the form:\n\\begin{equation}\n\|\\phi>=\|\\phi^Q>\|\\phi^c>=\n\\sum_{k_1,...k_N} C_{k_1,...k_N} \|k_1,...k_N>\|\\phi^c>,\n\\end{equation} \nwhere $\|\\phi^c>$ is a $L$-order classical wave function.\n\n\n\\subsection{From quantum mechanics to half quantum mechanics - Dynamics}\n\nThe goal now is to obtain the time evolution of the initial data wave \nfunction (5). To do this let us work in the Heisemberg picture and let us\ncalculate the full quantum time evolution of an arbitrary fundamental observable \n$\\hat{O}_k$:\n\\begin{equation}\n\\hat{O}_k(t)=\\sum_{n=0}^{\\infty} \\frac{1}{n!} \n\\left( \\frac{t}{i\\hbar} \\right)^n\n[...[\\hat{O}_k,\\hat{H}]...,\\hat{H}].\n\\end{equation} \nLet us designate the \ngeneral operator $\\hat{O}_k(t_0)$ just by $\\hat{A}$. The aim is then to study \nthe functional form of the initial data wave function in the representation of \n$\\hat{A}$. The first step is to write \nthe general observable $\\hat{A}$ as a sum of multiple\nproducts of the fundamental observables: \n\\begin{equation}\n\\hat{A}=\\sum_j \\hat{A}^c_j \\hat{A}^Q_j \\quad:\n\\left\\{ \\begin{array}{ccc}\n\\hat{A}^c_j=c_j \\prod_{a=1}^{n_j}\\hat{O}^{c}_{i_a(j)}\\\\\n\\hat{A}^Q_j=\\prod_{b=1}^{m_j}\\hat{O}^{Q}_{\\alpha_b(j)}\n\\end{array} \\right.\n\\end{equation}\nwhere for each $j$ the sets of \ncoefficients $i_a(j)$ and $\\alpha_b(j)$ are two \nsequences, the first one \nhaving values in $\\{1..2M\\}$ and the second one in $\\{2M+1...2(M+N)\\}$\nand $c_j$ are complex parameters that may depend on time.\nLet us proceed naively and try to obtain predictions for the outputs\nof a measurement of $\\hat{A}$. Let then $\|a_i^n>$ be the general eigenvector\nof $\\hat{A}$ with associated eigenvalue $a_i$ and degeneracy index $n$.\nUsing the standard prescription the predictions are given by the \nset of pairs $(a_i,P(a_i))$ where $P(a_i)$ is the probability of \nobtaining the value $a_i$ from a measurement of $\\hat{A}$, i.e.\n$P(a_i)=\\sum_n \|<a_i^n\|\\phi>\|^2$. \nWe easily realise that we have a problem. In fact we do not know\n$\|\\phi>$ completely and so the calculation of $P(a_i)$ is, \nto say the least, not straightforward. \n\nTo circumvent the problem we introduce a new operator $\\hat{B}$ obtained by applying a map $V_0$, named unquantization, to the operator $\\hat{A}$. This map $V_0$ is defined as a trivial extension of the full unquantization map (mapping quantum operators to full classical observables) that was defined and studied in a related paper. Let us then present the definition of $V_0$:\n\n\\bigskip\n\n\\underline{{\\bf Definition I}} {\\bf - First unquantization map} \\\\\nLet ${\\cal A}({\\cal H})$ be the algebra of linear operators \nacting on the Hilbert space ${\\cal H}={\\cal H}_1 \\otimes {\\cal H}_2$ and let ${\\cal S}$ be the algebra of $C^{\\infty}$ functionals ${\\cal S}=\\{ f:T^{\\ast}M_1 \\longrightarrow \n{\\cal A}({\\cal H}_2) \\}$. The unquantization $V_0$ is a map from ${\\cal A}({\\cal H})$ to ${\\cal S}$ that satisfies the following rules (we use the notation of (7)):\\\\\n{\\bf 1)} $V_0(\\sum_j \\hat{A}^c_j \\hat{A}_j^Q)=\\sum_j V_0(\\hat{A}^c_j)V_0(\\hat{A}^Q_j)$\\\\\n{\\bf 2)} $V_0(\\hat{A}^Q_j)=\\hat{A}^Q_j$\\\\\n{\\bf 3)} $V_0(\\hat{A}^c_j)=A^c_j$. The unquantization map that take us from $\\hat{A}_i^c$ to $A_i^c$ was defined in \\cite{nuno1} when the intention was to derive the full classical observable that corresponds to a general quantum operator. The following steps defined this procedure: \ni) $\\hat{A}^c_j$ should be expanded as a sum of a hermitian operator and an anti-hermitian one, \nii) all antisymmetric terms of $\\hat{A}^c_j$ should then be executed i.e. all the commutators present in $\\hat{A}^c_j$ should be calculated, \niii) finally, given $\\hat{A}^c_j$ displayed in an order satisfying the two previous requirements we perform the substitution of the quantum fundamental operators present in $\\hat{A}^c_j$ by the corresponding classical canonical variables, i.e. if $\\hat{A}^c_j=F(\\hat{O}_i)$ where $F$ satisfies the order requirements i) and ii) then $A^c_j=F(O_i)$.\n\nBy applying $V_0$ to $\\hat{A}$ we get:\n\\begin{equation}\n\\hat{B}=\\sum_j A^c_j \\hat{A}^Q_j .\n\\end{equation}\nNotice that $V_0(\\hat B )$ is not completely well defined ($V_0$ is not univocous). In fact there are several different orders in which we can display $\\hat{A}$ all of them satisfying the requirements i) and ii) but producing different operators $\\hat{B}$. This ambiguity will be discussed in detail in the next section. However we should point out that all future results of this section are valid for all operators $\\hat{B}$ obtained from unquantizing the same operator $\\hat{A}$.\n\nThe aim now is to use a representation induced by $\\hat{B}$ to\nobtain some knowledge about the properties of the initial\ndata wave function $\|\\phi>$ in the representation of $\\hat{A}$.\nTo do this some preliminary work is needed. \n\n\n\n\\subsubsection{Representations induced by $\\hat{B}$}\n\nLet us consider the general state $\|\\psi>=\|\\phi^c>\|\\psi^Q>$, where $\|\\phi^c>$ is the classical sector initial data wave function, that we assume to be $L$-order classical with respect to the classical initial data $(O^0_i,\\delta_i)$, and $\|\\psi^Q>$ is an arbitrary quantum sector wave function. \nThe first step will be to obtain the value of $\|<\\psi\|(\\hat A -\\hat B)^{2L}\|\\psi>\|^{1/2L}$ as a function of the half quantum initial data and of the half quantum operator. The relevance of this result will become clear latter on.\\\\\n{\\bf a)} Explicit form of $\|<\\psi\|(\\hat A -\\hat B)^{2L}\|\\psi>\|^{1/2L}$:\\\\\nLet $A_j^c=V_0(\\hat A^c_j)$. The following relation is valid up to a correction term of the order of $\\hbar^2$ \\cite{nuno1}:\n\\begin{equation}\n\\hat{A}^c_j-A^c_j=\n\\sum_{i=1}^{2M} \n\\frac{\\partial A^c_j}{\\partial O_i}(\\hat{O}_i-O_i)\n+\\frac{1}{2}\\sum_{i,k=1}^{2M}\\frac{\\partial^2 A^c_j}\n{\\partial O_i \\partial O_k}\n(\\hat{O}_i-O_i)(\\hat{O}_k-O_k) +...\n\\end{equation}\nUsing the former equation we have:\n\\begin{eqnarray}\n (\\hat{A} &-& \\hat{B})^L = \\left\\{\\sum_j(\\hat{A}^c_j-A^c_j)\\hat{A}^Q_j \\right\\}^L \\nonumber\\\\\n& = & \\left\\{ \\sum_{i=1}^{2M} \n\\frac{\\partial \\hat B}{\\partial O_i}(\\hat{O}_i-O_i)\n+\\frac{1}{2}\\sum_{i,k=1}^{2M}\\frac{\\partial^2 \\hat B}\n{\\partial O_i \\partial O_k}\n(\\hat{O}_i-O_i)(\\hat{O}_k-O_k) +...+\\sum_j c_j\\hbar^2 \\hat{\\epsilon}_j \\hat{A}^Q_j \\right\\}^L ,\n\\end{eqnarray}\nwhere we explicitly included the correction term $c_j\\hbar^2 \\hat{\\epsilon}_j$. \nUp to the lowest order we get:\n\\begin{equation}\n(\\hat{A}-\\hat{B})^L = \\sum_{i_1=1}^{2M}...\\sum_{i_L=1}^{2M} \\prod_{s =1}^L \n\\frac{\\partial \\hat B}{\\partial O_{i_s}}(\\hat{O}_{i_s}-O_{i_s})\n+...+{\\cal O}(\\hbar)^2 .\n\\end{equation}\nThe relation (9) was derived and discussed in detail in \\cite{nuno1}. There we point out that (9) is exactly valid only if $A^c_j$ is obtained from a total symmetric form of $\\hat{A}^c_j$. This is not the general case if we use the unquantization map $V_0$ to obtain \n$A^c_j$. However we also saw in \\cite{nuno1} that if we use the map $V_0$ - in which case $A_j^c$ is obtained from $\\hat{A}^c_j$ displayed in an order that does not contain antisymmetric components (see definition I) - then the difference between the two sides of eq.(9) is\ngiven by $c_j\\hbar^2 \\hat{\\epsilon}_j$, where $c_j$ is the numerical factor in $\\hat{A}_j^c$ (7) and $\\hat{\\epsilon}_j$ is the \"operator error\" proportional to a sum of products of monomials $(\\hat{O}_i -O_i)$, each of the products having at the most $n_j-2$ terms (check eq.(7) for the meaning of $n_j$). This error term was explicitly included in the expansion (10). Typically, the contribution of the term proportional to $c_j\\hbar^2$ is meaningless when compared to the terms proportional to the derivatives of $\\hat{B}$. However in some artificial examples this may not be the case. Consider for instance: $\\hat{A}=(\\hat{x}^c \\hat{y}^c \\hat{z}^c +\\hat{z}^c \\hat{y}^c \\hat{x}^c)\n\\hat{A}^Q -(\\hat{y}^c \\hat{x}^c \\hat{z}^c +\\hat{z}^c \\hat{x}^c \\hat{y}^c)\\hat{A}^Q $, \nwhere $\\hat{x}^c, \\hat{y}^c, \\hat{z}^c$ are hermitian, classical sector operators of an arbitrary system.\nWe have $V_0(\\hat{A})=xyz \\hat{A}^Q- xyz \\hat{A}^Q=0 = \\hat{B}$ and therefore, in this case, $\\hat{A}-\\hat B=\\sum_jc_j\\hbar^2\\hat{\\epsilon}_j\\hat{A}^Q_j$ which, in general, is not zero. \nThe problem lies, of course, in the order in which $\\hat{A}$ is displayed before we apply the map $V_0$. One should impose the restriction that $\\hat{A}$ can not be displayed in an order in which some unresolved commutatores are present. One easy way to check that this is the case is precisely by \ncomparing the magnitude of $\\hat{B}$ (the numerical factors in $\\hat{B}$) with the magnitude of $c_j\\hbar^2$, where the $c_j$ are the numerical factors of $\\hat{A}$. For physical relevant examples (physical Hamiltonians and observables), and namely for the time evolution of a general quantum observable, it is easy to verify if the original operator $\\hat{A}$ is in an adequate order (this is in fact the typical case), and thus upon unquatization, one has {\\it magnitude}$(\\hat{B}) \\propto c_j \\hbar^0 >> c_j\\hbar^2$. Therefore, and keeping the caution remark in mind, we shall take the result (11) to be exactly valid. \n\nTo proceed we apply the expansion (11) to the state $\|\\psi>$. \nTo the first order we have:\n\\begin{equation}\n(\\hat A -\\hat B)^{L}\|\\psi> = \\sum_{i_1=1}^{2M} ... \\sum_{i_L=1}^{2M}\n\|E(\\hat{O}_{i_1},..,\\hat{O}_{i_L},\\phi^c,O_{i_1},..,O_{i_L})> \\prod_{s=1}^L\n\\frac{\\partial \\hat{B}}{\\partial O_{i_s}} \|\\psi^Q> +...\n\\end{equation}\nUsing (12) just up to the lowest order we get:\n\\begin{eqnarray}\n\\lefteqn{ \|<\\psi\|(\\hat A -\\hat B)^{2L}\|\\psi>\| \\le }\\\\\n \\sum_{i_1=1}^{2M} .. \\sum_{i_L=1}^{2M} &&\n\\sum_{k_1=1}^{2M} .. \\sum_{k_L=1}^{2M} \n\|<E_{O_{k1}..O_{kL}} \|E_{O_{i1}..O_{iL}}>\| \n \|<\\psi^Q\|\\frac{\\partial \\hat{B}^{\\dagger}}{\\partial O_{k_1}}..\n\\frac{\\partial \\hat{B}^{\\dagger}}{\\partial O_{k_L}} \n\\frac{\\partial \\hat{B}}{\\partial O_{i_1}}.. \n\\frac{\\partial \\hat{B}}{\\partial O_{i_L}}\|\\psi^Q>\|+... \\nonumber\n\\end{eqnarray}\nand using the Shwartz inequality $L$ times, the relation (3) and disregarding the contributions of terms proportional to $\\hbar^2$ or smaller we get:\n\\begin{equation}\n\|<\\psi\|(\\hat A -\\hat B)^{2L}\|\\psi>\|^{1/2L}\n\\le \\sum_{i=1}^{2M} \n\|<\\psi^Q\|\\left(\\frac{\\partial \\hat{B}^{\\dagger}}{\\partial O_i}\\right)^L \\left(\n\\frac{\\partial \\hat{B}}{\\partial O_i}\\right)^L \|\\psi^Q>\|^{1/2L} \\delta_{i} +... \n\\end{equation}\nThe $n$-order terms (the dots in (14)) are of the general form:\n\\begin{equation}\n\\frac{1}{n!}\\sum_{i,k...,s=1}^{2M}\|<\\psi^Q\|\\left(\\frac{\\partial^n \\hat{B}^{\\dagger}}\n{\\partial O_i \\partial O_k...\\partial O_s} \\right)^L \\left( \n\\frac{\\partial^n \\hat{B}}{\\partial O_i\n\\partial O_k...\\partial O_s}\\right)^L\|\\psi^Q>\|^{1/2L} \\delta_{i} \\delta_{k}\n....\\delta_{s} .\n\\end{equation} \nThis result is valid up to any order since $\|\\phi^c>$ is $L$-order classical and so the relation $<E_{S^L_{ia}}\|E_{S^L_{ia}}>\n\\le (\\delta_{S^L_{ia}})^2$ is valid for all the sequences $S^L_{ia}$ determined in (1)\nwhich are exactly the same ones involved in the expansion of $\|<\\psi\|(\\hat A -\\hat B)^{2L}\|\\psi>\|$.\nThis is our last result concerning the value of $\|<\\psi\|(\\hat A -\\hat B)^{2L}\|\\psi>\|^{1/2L}$. \n\nTo proceed we will construct a general set of states of the form $\|\\phi^c>\|\\psi^Q>$ providing a basis to expand the initial data wave function $\|\\phi>=\|\\phi^c>\|\\phi^Q>$ (5).\nLet us start by introducing the states $\|\\psi^r_u>=\|\\phi^c>\|b^r_u>$ where $\|b^r_u>$ form a complete set of eigenstates of $\\hat B$ with degeneracy index $r$ and associated eigenvalue $b_u$. This would be the most natural set of states to be used to expand $\|\\phi>$. However, they do not provide suitable results. We will see why in the sequel. To go further we still have to construct another set of \nstates and study their properties in the representation of $\\hat{A}$. \nLet $\|\\xi_u>$ be given by:\n\\begin{equation}\n\|\\xi_u>=\|\\xi^Q_u>\|\\phi^c>=\n\\frac{1}{C_u} \\sum_{r,b_u^{\\prime} \\in I_u}<b^{\\prime r}_{u}\n\|\\phi^Q>\|b^{\\prime r}_{u}>\|\\phi^c> , \n\\end{equation}\nwhere $C_u$ is a normalisation constant, $\|b^{\\prime r}_u>$ are eigenstates of $\\hat B$ and $I_u=[b_u-I_B, b_u+I_B]$ where \n$b_u$ is named the {\\it central eigenvalue} associated to $\|\\xi_u>$ and\n$I_B$ is a constant to be supplied later and that represents\nthe spread of $\|\\xi_u>$ in the representation of $\\hat{B}$. \nWe are specially interested in a\nset of states $\|\\xi_u>$ associated to a sequence $S$ \nof eigenvalues $b_u$ of $\\hat{B}$. These eigenvalues are chosen in such a way \nthat their value grows in steps of $2I_B$. This way we guarantee \nthat firstly $<\\xi_u\|\\xi_{u^{\\prime}}>=\\delta_{u,u^{\\prime}}$ and \nsecondly, that $\|\\phi>=\\sum_{b_u \\in S}<\\xi_u\|\\phi>\|\\xi_u>$.\nThis said let us study the properties of $\|\\xi_u>$ in the representation of $\\hat{A}$,\n(the reader should refer to the appendix for the relevant definitions).\\\\\n{\\bf b)} Explicit form of the spread $\\Delta_L(\\hat{A},\\xi_u,b_u,p)$:\\\\\nThe $L$-order spread \n$\\Delta_L(\\hat{A},\\xi_u,b_u,p)=<E^L(\\hat{A},\\xi_u,b_u)\|E^L(\\hat{A},\\xi_u,b_u)>^{1/2L}\n/({1-p})^{1/2L}$ can be cast in the form:\n\\begin{equation}\n\\Delta_L(\\hat{A},\\xi_u,b_u,p)=\\frac{\|<\\xi_u\|(\\hat A - b_u)^{2L}\|\\xi_u>\|^{1/2L}}\n{(1-p)^{1/2L}} =\n\\frac{\\left\|<\\xi_u\|\\left\\{(\\hat A - \\hat B) +(\\hat B - b_u) \\right\\}^{2L}\|\\xi_u> \\right\|^{1/2L}}{(1-p)^{1/2L}}.\n\\end{equation}\nExpanding the polynomial insight the bracket and using the Schwartz inequality $L$ times to separate the terns in $(\\hat A - \\hat B)$ of those in $(\\hat B - b_u)$ we get:\n\\begin{equation}\n\\Delta_L(\\hat{A},\\xi_u,b_u,p) \\le \n\\frac{1}{(1-p)^{1/2L}}\n\\left\\{ \|<\\xi_u\|(\\hat A - \\hat B)^{2L}\|\\xi_u>\|^{1/2L} +\|<\\xi_u\|(\\hat B - b_u)^{2L}\|\\xi_u>\|^{1/2L} \\right\\},\n\\end{equation}\nwhere we disregard the contribution of terms proportional to $\\hbar^2$ or smaller.\nUsing (14) and the inequality\n$<E^L(\\hat{B},\\xi_u,b_u)\|E^L(\\hat{B},\\xi_u,b_u)> \\le I_B^{2L}$,\nwhich can easily be obtained from (16), we finally get: \n\\begin{equation}\n\\Delta_L(\\hat{A},\\xi_u,b_u,p) \\le \\frac{ \n\\sum_{i=1}^{2M} \\left\| <\\xi^Q_u\|\\left(\\frac{\\partial \\hat{B}^{\\dagger}}\n{\\partial O_i} \\right)^L \\left(\n\\frac{\\partial \\hat{B}}{\\partial O_i} \\right)^L\|\\xi_u^Q> \\right\| ^{1/2L}\n\\delta_{i} + I_B +....}{(1-p)^{1/2L}}=\\frac{\\delta_L(\\hat B)+I_B}{(1-p)^{1/2L}},\n\\end{equation}\nwhere $\\delta_L(\\hat B)$ is named the $L$-order error margin of $\\hat B$. The reason for this designation is the resemblance of the expansion (19) and the classical \nerror margin for the full classical observable $B=\\sum_j B^c_j B_j^Q$:\n$\\quad \\delta_B=\\sum_{i=1}^{2M}\|\\partial B / \\partial O_i\| \\delta_{i}+...$\nwhen the error margins associated to the \"quantum\" observables are \nidentically zero. This resemblance is also clear between the higher order\nterms of (19), which are of the form (15), and the higher order terms that are {\\it also} present in the expression of the error margin of $B$. \nFinally, notice that if we make $I_B=0$ the set of states $\|\\xi_u>$ is a set of true eigenvectores of $\\hat B$ with associated eigenvalues $b_u$ and with spread $\\Delta_L(\\hat{A},\\xi_u,b_u,p) \\le \\delta_L(\\hat B)/(1-p)^{1/2L}$.\n\nUsing the result a) from the appendix we can now state \nthat in the representation of $\\hat{A}$ \nthe state $\|\\xi_u>$ has at least a probability $p$ in the \ninterval $I=[b_u-\\Delta_L (p),b_u+\\Delta_L (p)]$, with $\\Delta_L (p)$ given by (19),\nthat is $\\sum_{n,a_i \\in I}\|<a_i^n\|\\xi_u>\|^2 \\ge p$.\n\n\n\n\n\\subsubsection{The initial data wave function in the representation \nof $\\hat{A}$}\n\nWe have completed all the preliminary work and we are now in position \nto obtain some of the properties of $\|\\phi>$ in the representation of $\\hat{A}$. \nA completely precise prediction for the probability of obtaining the \neigenvalue $a_i$ from a measurement of $\\hat{A}$ given by $P(a_i)=\n\\sum_{n}\|<\\phi\|a^n_i>\|^2$ is not possible due to \nour incomplete knowledge of $\|\\phi>$. Still, we can attempt to obtain \nfairly accurate predictions.\nWe will obtain predictions for the\nprobability of a measurement of $\\hat{A}$ yielding \nan eigenvalue $a_i \\in I_0$ where $I_0$ is an interval of \nsize at least $\\Delta_L$, with $\\Delta_L$ given by (19). \nMore precisely\nwe will be able to predict that the probability $P(a_i \\in I_0)$ is at the\n most $P_{max}(a_i \\in I_0)$ and at the least $P_{min}(a_i \\in I_0)$, \nthe error margin being a function of $L$, typically of reasonable size.\n\nLet us then consider the following three intervals: \n$I_0=[a^0-D,a^0+D]$ is \nthe interval of eigenvalues of $\\hat{A}$ for which we want \nto determine $P(a_i \\in I_0)$. $D$ is required to satisfy $D>\\Delta_L$. The two other intervals \n$I_{max}$ and $I_{min}$ will be \nneeded to majorate and minorate the former probability. \n$I_{max}$ is such that any $\|\\xi_u>$ with associated \ncentral eigenvalue $b_u \\in I_0$ has in the representation of $\\hat{A}$ \na probability of at least $p$ in $I_{max}$. For $I_{min} $ the \nstatement is that if $b_u \\in I_{min}$ then $\|\\xi_u>$ has \na probability of at least $p$ in $I_0$. Easily we see that \nthe intervals $I_{min}=[a^0-(D-\\Delta_L),a^0+(D-\\Delta_L)]$ and $I_{max}=\n[a^0-(D+\\Delta_L),a^0+(D+\\Delta_L)]$ - where $\\Delta_L=\\Delta_L(\\hat{A},\n\\xi_u,b_u,p)$ given by (19) and $b_u \\in I_0$ - will satisfy the \nformer requirements. Notice that we assumed that\n$\\Delta_L $ has a similar value for different $b_u$\nwithin $I_0$. If this is not the case all the future results \nare still valid, we just need to be more careful in constructing \nthe intervals $I_{max}$ and $I_{min}$. \nLet us then proceed. We have:\n\\begin{equation}\nP(a_i \\in I_0)=\\sum_{n,a_i \\in I_0} \|<\\phi\|a_i^n>\|^2=\n\\sum_{n, a_i \\in I_0} \\left\|\\sum_{b_u \\in S}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n\\right\|^2 , \n\\end{equation}\nwhere we have used the fact that $\|\\phi>=\\sum_{b_u \\in S} <\\xi_u\|\\phi>\|\\xi_u>$,\nwhere $S$ is the set of central eigenvalues of $\\hat{B}$, associated to the states $\|\\xi_u>$, that was presented in\nthe sequel of (16).\nWe now expand the previous expression first using the interval\n$I_{max}$:\n\\begin{eqnarray}\nP(a_i \\in I_0) & = &\n\\sum_{n, a_i \\in I_0} \\left\|\\sum_{b_u \\in I_{max} \\cap S}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n+\\sum_{b_u \\in S/I_{max}}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n\\right\|^2 \\nonumber\\\\\n& \\le & \\sum_{n, a_i} \\left\|\\sum_{b_u \\in I_{max} \\cap S}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n\\right\|^2 + \\sum_{n, a_i \\in I_0} \\left\|\\sum_{b_u \\in S/I_{max}}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\right\|^2 \\nonumber\\\\\n& + & 2 \\left\| \\sum_{n, a_i \\in I_0} \\sum_{b_u \\in I_{max} \\cap S}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\sum_{b_u \\in S/I_{max}}\n<a^n_i\|\\xi_u><\\xi_u\|\\phi> \\right\| ,\n\\end{eqnarray}\nwhere the set $S/I_{max}$ is constituted by the elements of $S$ that do \nnot belong to $I_{max}$. To derive the former expression we first wrote\nthe norm in the first summation as a product of the term inside the norm\nby its complex conjugate and then grouped the resulting terms in\na convenient way.\n\nOn the other hand, and using the interval $I_{min}$ we have: $P(a_i \\in I_0)=\n1-P(a_i \\notin I_0)$ and so:\n\\begin{eqnarray}\nP(a_i \\in I_0) & = & \n1-\\sum_{n, a_i \\notin I_0} \\left\|\\sum_{b_u \\in S/I_{min}}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n+\\sum_{b_u \\in I_{min} \\cap S}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n\\right\|^2 \\nonumber\\\\\n& \\ge & 1-\\sum_{n, a_i} \\left\|\\sum_{b_u \\in S/I_{min}}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n\\right\|^2 - \\sum_{n, a_i \\notin I_0} \\left\|\\sum_{b_u \\in I_{min} \\cap S}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\right\|^2 \\nonumber\\\\\n& - & 2 \\left\|\\sum_{n, a_i \\notin I_0} \\sum_{b_u \\in S/I_{min}}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\sum_{b_u \\in I_{min} \\cap S}\n<a^n_i\|\\xi_u><\\xi_u\|\\phi> \\right\| .\n\\end{eqnarray}\nThe right hand side of the former inequalities has three and four \nterms, respectively.\nLet us designate by $X_1,Z_1$ the second and third terms of \n(21) and by $X_2,Z_2$ the third and fourth terms of (22). \nWe will deal with each of the terms in (21) and (22) independently: \\\\\n{\\bf a)} First term of (21):\n\\begin{eqnarray}\n\\sum_{n, a_i} \\left\|\\sum_{b_u \\in I_{max} \\cap S}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n\\right\|^2 & = &\n\\sum_{n, a_i} \\sum_{b_u,b_v \\in I_{max} \\cap S}<\\phi\|\\xi_u><\\xi_v\|\\phi>\n<\\xi_u\|a^n_i><a^n_i\|\\xi_v> \\nonumber\\\\\n& = & \\sum_{b_u \\in I_{max} \\cap S} \|<\\phi\|\\xi_u>\|^2 = P(b_u \\in I_{max}),\n\\end{eqnarray}\nsince $\\sum_{n, a_i} <\\xi_u\|a^n_i><a^n_i\|\\xi_v> =\\delta_{uv}$ and\n $\\sum_{b_u \\in I_{max} \\cap S} \|<\\phi\|\\xi_u>\|^2=\n\\sum_{r,b_u \\in I_{max}} \|<\\phi\|\\psi^r_u>\|^2$ where $\|\\psi^r_u>$ are \nthe eigenstates of $\\hat{B}$.\\\\ \n{\\bf b)} First terms of (22): Just as for the previous term we have:\n\\begin{equation}\n1-\\sum_{n, a_i} \\left\|\\sum_{b_u \\in S/I_{min}}<\\phi\|\\xi_u><\\xi_u\|a^n_i> \n\\right\|^2 =1-P(b_u \\notin I_{min})=P(b_u \\in I_{min}).\n\\end{equation}\n{\\bf c)} Third term of (21) and fourth term of (22):\n\\begin{eqnarray}\n& & Z_1 = 2 \\left\| \\sum_{n, a_i \\in I_0} \\sum_{b_u \\in I_{max} \\cap S}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\sum_{b_u \\in S/I_{max}}\n<a_i^n\|\\xi_u><\\xi_u\|\\phi> \\right\| \\\\ \n& = & 2\\left\|\\sum_{n, a_i \\in I_0} \\sum_{b_u \\in I_{max} \\cap S}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> <a^n_i\| \\cdot\n\\sum_{m, a_j \\in I_0} \\sum_{b_v \\in S/I_{max}}\n<a^m_j\|\\xi_v><\\xi_v\|\\phi> \|a^m_j>\\right\| ,\\nonumber\n\\end{eqnarray}\nusing the Shwartz inequality to calculate the former inner product\nand taking result a) into account we obtain:\n\\begin{equation}\nZ_1 \\le 2P(b_u\\in I_{max})^{1/2}\n\\left(\\sum_{n, a_i \\in I_0} \\left\|\\sum_{b_u \\in S/I_{max}}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\right\|^2 \\right)^{1/2},\n\\end{equation}\nwhere the second term of the former product is exactly $X^{1/2}_1$ where\n$X_1$ is the second term in (21) to be calculated in d).\n\nFollowing exactly the same procedure we get for the last term in (22):\n\\begin{equation}\nZ_2 \\ge -2 P(b_u \\notin I_{min})^{1/2}\n\\left(\\sum_{n, a_i \\notin I_0} \\left\|\\sum_{b_u \\in I_{min} \\cap S}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\right\|^2 \\right)^{1/2},\n\\end{equation}\nwhere the second term of the right hand side of the former inequality\nis exactly $-X^{1/2}_2$ where $X_2$ is the third term of (22) \nto be calculated in d). \\\\\n{\\bf d)} Second term of (21) and third term of (22): \nThese are the last terms we need to calculate in order to obtain \nthe explicit form of the expressions (21) and (22). Let us start \nwith $X_1$:\n\\begin{eqnarray}\nX_1 &=& \\sum_{n, a_i \\in I_0} \\left\|\\sum_{b_u \\in S/I_{max}}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\right\|^2 \\nonumber\\\\ \n&\\le & \\sum_{b_u,b_v \\in S/I_{max}}\\left\|<\\phi\|\\xi_u><\\xi_v\|\\phi>\\right\|\n\\left\|\\sum_{n, a_i \\in I_0} <\\xi_u\|a^n_i><a^n_i\|\\xi_v>\\right\|.\n\\end{eqnarray}\nFor the second term of the right hand side of\nthe previous inequality, we get:\n\\begin{equation}\n\\left\|\\sum_{n, a_i \\in I_0} <\\xi_u\|a^n_i><a^n_i\|\\xi_v>\\right\|\n\\le \\left(\\sum_{n, a_i \\in I_0} \|<\\xi_u\|a^n_i>\|^2\\right)^{1/2}\n\\left(\\sum_{n, a_i \\in I_0} \|<\\xi_v\|a^n_i>\|^2\\right)^{1/2},\n\\end{equation}\nwhere we have used the Shwartz inequality to calculate\nthe inner product of the states $ \\sum_{n, a_i \\in I_0} <\\xi_u\|a^n_i><a^n_i\|$\nand $\\sum_{n, a_i \\in I_0}<a^n_i\|\\xi_v> \|a^n_i>$. \nTo proceed we notice that both $b_u, b_v \\notin I_{max}$ and \nuse the result (67) in b) from the appendix to get:\n\\begin{equation}\n\\sum_{n, a_i \\in I_0} \|<\\xi_u\|a^n_i>\|^2 \\le (1-p) \\frac{\\Delta_L (\\hat{A},\\xi_u ,b_u,p)^{2L}}{\|b_u-a\|^{2L}},\n\\end{equation}\nwhere $\\Delta_L (\\hat{A},\\xi_u ,b_u,p)$ is given by (19) and $a$ is one of the extremes of the interval $I_0$, the one that minimise the distance $\|b_u-a\|$, that is $a=a^0+D$ or \n$a=a^0-D$. \nPutting these results together we get:\n\\begin{equation}\nX_1 \\le (1-p)\n\\left\\{\\sum_{b_u \\in S/I_{max}}\\frac{\\Delta_L (\\hat{A},\\xi_u ,b_u,p)^L\n\|<\\phi\|\\xi_u>\|}{\|b_u-a\|^L}\\right\\}^2.\n\\end{equation}\nSince $\|<\\phi\|\\xi_u>\|=\|<\\phi^Q\|\\xi_u^Q>\|$ \nthe calculation of the right hand side of inequality (31) might be done explicitly \nonce the initial data of the half quantum system and the operator $\\hat B$ are given. Therefore, we can now group the former results together to obtain a prediction for $P(a_i \\in I_0)$. However, and \nsince we would like to obtain a more explicit value for $P(a_i \\in I_0)$, let us consider the simple but quite general case in which\n$\\Delta_L (\\hat{A},\\xi_u ,b_u,p)$ is approximately a constant in the range of eigenvalues where $\|<\\phi\|\\xi_u>\|$\nhave meaningful values. In this case we have: \n\\begin{equation}\nX_1 \\le (1-p)\\Delta_L(p)^{2L}\n\\left\\{\\sum_{b_u \\in S/I_{max}}\\frac{\|<\\phi\|\\xi_u>\|}{\|b_u-a\|^L}\n\\right\\}^2.\n\\end{equation}\nThe goal is to maximize the term inside the brackets to obtain the highest possible value of $X_1$. Let then $\|\\phi>=\\sum_{b_u \\in S}<\\xi_u\|\\phi>\|\\xi_u>$ and let $C_u=\|<\\xi_u\|\\phi>\|$.\nAs an intermediate step let us assume that $\|\\phi>$ spreads for an interval from $E_0=a^0+\\Delta_L +D$ (the extreme of $I_{max}$) to an arbitrary $E \\in {\\cal R}$, i.e. $C_u=0$ for $b_u \\notin [E_0,E]$. In the end we will see that the result for $X_1$ is independent of $E$ that might be taken to infinity. \nLet us proceed: since $b_u$ grows in steps of size $2I_B$,\nwe divide the former interval in sub-intervals of size $2I_B$.\nLet us say that we have $N$ such sub-intervals: $2I_B=(E-E_0)/N$.\nWe then get:\n\\begin{equation}\n\\sum_{b_u \\in S/I_{max}} \\frac{\|<\\phi\|\\xi_u>\|}{\|b_u-a\|^L}=\n\\sum_{n=0}^N \\frac{C_n}{(\\Delta_L+2nI_B)^L}=\n\\frac{1}{2I_B} \\sum_{n=0}^N 2I_B \n\\frac{C_n}{(\\Delta_L+2nI_B)^L}\n\\simeq \\frac{1}{2I_B} \\int_{E_0}^E \\frac{C(x)}{\|x-a\|^L}dx ,\n\\end{equation}\nwhere $C_n=\|<\\phi\|\\xi_n>\|$ with $\|\\xi_n>$ being the state associated \nto the central eigenvalue $b_n=a+\\Delta_L+2nI_B$. Our task now is to maximise\nthe previous integral subject to the constraint:\n\\begin{equation}\n\\sum_{n=0}^N C^2_n =1 \\Longrightarrow \\frac{1}{2I_B}\n\\int_{L_0}^L C(x)^2 d x =1.\n\\end{equation}\nTo do this we use the Lagrangian multiplier method. We put\n$L(C,\\dot{C},x)=-C/\|x-a\|^L + \\lambda C^2$\nand we write the Lagrangian equations to obtain:\n\\begin{equation}\n\\frac{\\partial L}{\\partial C}-\\frac{\\partial}{\\partial x}\n\\frac{\\partial L}{\\partial \\dot{C}}=0 \\quad \\Longleftrightarrow \\quad\nC(x)=\\frac{1}{2\\lambda\|x-a\|^L}.\n\\end{equation}\nAnd imposing the constraint (34) in $C(x)$ we get:\n\\begin{equation}\n\\lambda=\\left\\{ \\frac{1}{8(2L-1)I_B} \\left( \\frac{1}{(E_0-a)^{2L-1}}-\\frac{1}{(E-a)^{2L-1}}\n\\right) \\right\\}^{1/2},\n\\end{equation}\nsubstituting this result in (35), integrating (33) and, finally, substituting the value of this integral in (32) we get:\n\\begin{equation}\nX_1 \\le (1-p)\\frac{\\Delta_L(p)}{2(2L-1)I_B},\n\\end{equation}\nwhich is our final result for $X_1$. We see that this result could not be obtained if we had used the eigenstates $\|\\psi^r_u>$ of $\\hat B$ in which case $I_B=0$ and thus $X_1$ will not be bound.\n\nIf we proceed just along the same lines for the third term in (22) we will\nget exactly the same result, that is:\n\\begin{equation}\nX_2 = -\\sum_{n, a_i \\notin I_0} \\left\|\\sum_{b_u \\in I_{min} \\cap S}\n<\\phi\|\\xi_u><\\xi_u\|a^n_i> \\right\|^2 \n\\ge - (1-p)\\frac{\\Delta_L(p)}{2(2L-1)I_B}.\n\\end{equation}\nWe now introduce the results of a),b),c),d) into (21) and (22) and finally get:\n\\begin{equation}\nP(b_u \\in I_{min})-E_{min}\n\\le P(a_i \\in I_0) \n\\le P(b_u \\in I_{max})+E_{max},\n\\end{equation}\nwhere $P(b_u \\in I_{min})= \\sum_{r,b_u \\in I_{min}}\|<b_u^r\|\\phi^Q>\|^2$,\n$P(b_u \\in I_{max})= \\sum_{r,b_u \\in I_{max}}\|<b_u^r\|\\phi^Q>\|^2$ and\n$E_{min}$ and $E_{max}$ are given by the following expressions:\n\\begin{eqnarray}\nE_{min}=2P(b_u \\notin I_{min})^{1/2} \\left(\\frac{(1-p) \\Delta_L(p)}\n{2(2L-1)I_B}\\right)^{1/2} + \\frac{(1-p) \\Delta_L(p)}{2(2L-1)I_B}\n\\nonumber\\\\\nE_{max}=2P(b_u \\in I_{max})^{1/2} \\left(\\frac{(1-p) \\Delta_L(p)}\n{2(2L-1)I_B}\\right)^{1/2} + \\frac{(1-p) \\Delta_L(p)}{2(2L-1)I_B} .\n\\end{eqnarray}\nNotice that, given the degree of validity $L$ of the classical sector initial data, we can play with the interval $I_0$ and with the values of \n$I_B$ and $p$, which\nin turn impose a value to $\\Delta_L(p)$, to minimise the error of the \npredictions for the probabilities. \nTo get some feeling about the accuracy of the predictions let us choose some explicit\nvalues for $L, I_B$ and $p$. Let $I_B = \\delta_L(\\hat B)$ so that $\\Delta_L(p)=2I_B/ (1-p)^{1/2L}$. The errors $E_{min}$ and $E_{max}$ become: \n\\begin{eqnarray}\nE_{min}=2P(b_u \\notin I_{min})^{1/2} \\left(\\frac{(1-p)^{\\frac{2L-1}{2L}}}\n{2L-1}\\right)^{1/2} + \\frac{(1-p)^{\\frac{2L-1}{2L}}}{2L-1}\n\\nonumber\\\\\nE_{max}=2P(b_u \\in I_{max})^{1/2} \\left(\\frac{(1-p)^{\\frac{2L-1}{2L}}}\n{2L-1}\\right)^{1/2} + \\frac{(1-p)^{\\frac{2L-1}{2L}}}{2L-1}.\n\\end{eqnarray}\nLet us consider $L=1$. This means that the classical sector initial data $(O^0_i,\\delta_i)$ is first-order valid, i.e. the classical sector initial data wave function $\|\\phi^c>$ is first-order classical with respect to the classical data $(O^0_i,\\delta_i)$. Let us also choose $p=0.99$. We can then state that, in the representation of $\\hat A$, the states $\|\\xi_u>$ (16) have, at least, $99\\%$ of their probability confined to the intervals: \n$I_u=[b_u-20\\delta_1(\\hat B),b_u+20 \\delta_1(\\hat B)]$ where $\\delta_1(\\hat B)$ is given by\n(19) and is of the size of a classical error margin. Moreover:\n\\begin{equation}\nE_{min}=2\\times 0.31 \\times P(b_u \\notin I_{min})^{1/2} + 0.1 \\quad {\\rm and} \\quad\nE_{max}=2 \\times 0.31 \\times P(b_u \\in I_{max})^{1/2} +0.1 ,\n\\end{equation}\nand thus, in the worst case:\n\\begin{equation}\nP(b_u \\in I_{min})-0.72\n\\le P(a_i \\in I_0) \n\\le P(b_u \\in I_{max})+0.72 ,\n\\end{equation}\nwith the difference between the ranges of $I_{max},I_{min}$ and $I_0$ being given by $20 \\delta_1(\\hat B)$. An error of $72\\%$ is huge. The reason for such a large error lies upon the fact that the conditions imposed over $\|\\phi^c>$ are the least restrictive possible, $L=1$. In other words, the classical initial data is the least valid possible.\n\nTo see what happens when we increase the degree of validity of the classical sector initial data let us make $L=10$. We consider once again $I_B=\\delta_{10}(\\hat B)$ but, this time let us choose $p=0.99999 =1-10^{-5}$. We have $\\Delta_{10}(p=0.99999)=3.6 \\delta_{10}(\\hat B)$ and:\n\\begin{equation}\nE_{min}=0.0019 P(b_u \\notin I_{min})^{1/2} + 9.4 \\times 10^{-7} , \\quad\nE_{max}=0.0019 P(b_u \\in I_{max})^{1/2} + 9.4 \\times 10^{-7} ,\n\\end{equation}\nand thus, in the worst case:\n\\begin{equation}\nP(b_u \\in I_{min})-0.0019\n\\le P(a_i \\in I_0) \n\\le P(b_u \\in I_{max})+0.0019.\n\\end{equation}\nThat is an error of $0.19\\%$ with the difference between the ranges of $I_{max},I_{min}$ and $I_0$ decreasing to $3.6 \\delta_{10}(\\hat B)$.\n\n\n\n\\section{Half quantization}\n\nWe start by noticing that the predictions $P(a_i \\in I_0) $ and its \nerror margins could be obtained if we had knowledge of \nthe operator $\\hat{B}$ and in no way require (except to obtain \n$\\hat{B}$) the knowledge of the full quantum operator $\\hat{A}$.\nThis means that if we were able to calculate $\\hat{B}$ directly\nthen we would be able to make predictions for the evolution of the \nhalf quantum system without firstly having to obtain its full quantum\nversion. \nTherefore, the aim of this section is to obtain a framework able to provide the \noperator $\\hat{B}$ directly from the initial data of the half quantum \nsystem without requiring previous knowledge of the full quantum theory.\n\n\n\\subsection{The unquantization map}\n\nIn section 2.2 we present the first definition of the unquantization map. \nThe motivation to define $V_0$ this way was the fact that it validates the expansions (10,11)\nwhich were crucial to develop the entire approximation procedure presented in the last section.\nIt was already pointed out that the map $V_0$ is a trivial generalisation of the \nunquantization map presented in \\cite{nuno1}. Let us name this last map $V_0^c$. In fact the action of $V_0$ over \na classical sector operator is identical to the action of $V_0^c$. In that paper we saw that $V_0^c$ is just the inverse map of the Dirac quantization \\cite{dirac1, dirac2}. Taking this result into account we present a new, however equivalent, definition of the half unquantization:\n\n\\bigskip\n \n\\underline{{\\bf Definition II}} {\\bf - Second unquantization map}\\\\\nLet $\\wedge$ be the Dirac quantization map \\cite{dirac2}, $\\wedge:{\\cal A}(T^{\\ast}M_1) \\longrightarrow {\\cal A}({\\cal H}_1)$ and let $\\hat{A}=\n\\sum_j \\hat{A}_{j}^Q \\hat{A}_j^c$ where $\\hat{A}_j^Q$ and $\\hat{A}^c_j$ are arbitrary multiple products of quantum and classical sector operators, respectively (7). \nThe unquantization $\\vee$ is a map from the algebra ${\\cal A}({\\cal H})$ \nto the algebra ${\\cal S}$ (check for the definition of ${\\cal S}$ in {\\it definition I}) defined by the following rules: \\\\\n{\\bf 1)}\n$\\vee (\\hat{A})= \\sum_j \\vee (\\hat{A}_{j}^Q) \\vee (\\hat{A}_j^c)$\\\\\n{\\bf 2)}\n$\\vee(\\hat{A}_j^c)=A_i^c \\qquad \\mbox{iff} \\qquad \\wedge(A_j^c)=\\hat{A}_j^c$\\\\\n{\\bf 3)} \n$\\vee(\\hat{A}_j^Q)=\\hat{A}_j^Q$\n\nLet us study some properties of $\\vee$:\\\\\n1) The map $\\vee$ is equivalent to the map $V_0$ of definition I. This is so because the rule 2) of the definition of $\\vee$ is equivalent to the rule 3) of the definition of $V_0$. This fact was extensively discussed in \\cite{nuno1}.\\\\\n2) Just like $V_0$, the map $\\vee$ is beset by order problems. In general there are several different classical sector observables that when quantized yield the same quantum operator. Let $A_1^c \\not= A_2^c$ be two such observables, i.e. $\\wedge (A^c_1) = \\hat{A}^c$ and also $\\wedge (A^c_2) = \\hat{A}^c$. This means that $\\vee(\\hat{A}^c)=A^c_1$ but also $\\vee (\\hat{A}^c)=A^c_2$. Hence the map $\\vee$ is not univocous. \nOn the other hand, the predictions (39,40) of the last section, are made for a general quantum operator $\\hat{A}$ (for instance $\\hat{A}=\\hat{A}^c\\hat{A}^Q$) and might be obtained using any of the operators $\\hat{B}=V_0(\\hat{A})$ (or equivalently, $\\hat{B}=\\vee(\\hat{A})$).\nTherefore, the ambiguity of $\\vee$ could be problematic if the predictions obtained by using two different $\\hat{B}$ (for instance $\\hat{B}_1={A}_1^c\\hat{A}^Q$ and $\\hat{B}_2={A}_2^c\\hat{A}^Q$) were inconsistent.\n\nHowever, one can easily realise that this is not the case. The difference $\\hat{B}_2-\\hat{B}_1$ is proportional to a leading factor of $c_j\\hbar^2$ (where $c_j$ is the highest numerical coefficient of $\\hat{A}$ displayed in the orders from which $\\hat{B}_1$ and $\\hat{B}_2$ were calculated). We already saw in the sequel of (11) that the validity of the predictions of the last section rests upon the premise that the numerical factors of $\\hat{B}>>c_j\\hbar^2$ (otherwise $\\hat{B}$ can not be considered for reproducing the predictions of $\\hat{A}$). We also saw that this premise is satisfied if the original operator $\\hat{A}$, from which $\\hat{B}$ was calculated, satisfies some order conditions. Therefore, and if $\\hat{B}_2$ and $\\hat{B}_1$ are both valid operators, obtained from $\\hat{A}$ displayed in required orders, the difference $\\hat{B}_2-\\hat{B}_1$ is not meaningful when compared to the imprecision (which is proportional to the numerical factors of $\\hat B$ (19)) associated to the predictions obtained by using either $\\hat{B}_1$ or $\\hat{B}_2$. \nIn conclusion,\n$\\hat{B}_1$ and $\\hat{B}_2$ provide {\\it physical predictions} which are consistent with each other, solving the ambiguity. \\\\\n3) Unquantizing of the product of two classical sector operators: let us consider two general classical observables $B$ and $C$. To quantize $BC$ one uses the \nsymmetrization rule: $\\wedge (BC)=1/2 (\\hat{B}\\hat{C}+\\hat{C}\\hat{B})$.\nWe just use the same rule for the unquantization:\n\\begin{equation}\n\\vee(\\hat{B}\\hat{C})= \\vee \\left( \\frac{\\hat{B}\\hat{C}+\\hat{C}\\hat{B}}{2} +\n\\frac{1}{2} [\\hat{B},\\hat{C}] \\right) =BC+\\frac{1}{2}i\\hbar \\{B,C\\} .\n\\end{equation}\nNotice that the prescription is beset by order problems (comment 2).\\\\\n4) Unquantization of a self-adjoint operator:\nif $\\hat{A}=\\hat{A}^c$ then we get from rule 2):\n$\\vee(\\hat{A}^{\\dagger})=\\vee(\\hat{A})^{\\ast}$. For the case of a \ngeneral operator $\\hat{A}=\\sum_j \\hat{A}^c_j \\hat{A}^Q_j$ we have:\n\\begin{equation}\n\\vee(\\hat{A}^{\\dagger})=\\sum_j \\vee(\\hat A^c_j)^{\\ast}({\\hat{A}^Q_j})^{\\dagger}=\n\\vee(\\hat{A})^{\\dagger},\n\\end{equation}\nand if $\\hat{A}=\\hat{A}^{\\dagger}$ then $\\vee(\\hat{A})^{\\dagger}=\n\\vee(\\hat{A}^{\\dagger})=\\vee(\\hat{A})$ and so $\\vee(\\hat{A})$ is \nalso self-adjoint.\\\\\n5) Unquantizing the brackets:\nFor the simplest case of $ \\hat{A}=\\hat{A}^c$ and \n $\\hat{B}=\\hat{B}^c$, from rule 2) one immediately has:\n\\begin{equation}\n\\vee [\\hat{A},\\hat{B}]=\\vee \\left(\\wedge(i\\hbar\\{A,B\\})\\right)=i\\hbar\\{A,B\\}.\n\\end{equation}\nFor the most general case let us first put $\\hat{A}=\\hat{A}^c\\hat{A}^Q$ and \n$\\hat{B}=\\hat{B}^c\\hat{B}^Q$ which only excludes sums of operators which, using\nrule 1), are straightforward to handle.\nWe get:\n\\begin{eqnarray}\n[\\hat{A},\\hat{B}] & = & \\hat{A}^c \\hat{B}^c[\\hat{A}^Q,\\hat{B}^Q]+\n[\\hat{A}^c,\\hat{B}^c] \\hat{B}^Q \\hat{A}^Q \\nonumber \\\\ \n\\Longrightarrow \n\\vee([\\hat{A},\\hat{B}]) & = & \\vee(\\hat{A}^c \\hat{B}^c)[\\hat{A}^Q,\\hat{B}^Q]+\n\\vee([\\hat{A}^c,\\hat{B}^c]) \\hat{B}^Q \\hat{A}^Q ,\n\\end{eqnarray}\nand using (46) and (48) we get:\n\\begin{equation}\n\\vee([\\hat{A},\\hat{B}]) = A^c B^c [\\hat{A}^Q,\\hat{B}^Q]+\n\\frac{i\\hbar}{2} \\{A^c,B^c\\} ( \\hat{A}^Q \\hat{B}^Q + \\hat{B}^Q \\hat{A}^Q ).\n\\end{equation}\n\n\n\\subsection{Half quantization and half quantum mechanics}\n\n\n\nEq.(50) can be displayed in a slightly different form:\n\\begin{equation}\n\\vee([\\hat{A},\\hat{B}]) = [\\vee(\\hat{A}),\\vee(\\hat{B})]+\ni\\hbar \\{\\{\\vee(\\hat{A}),\\vee(\\hat{B})\\}\\} = (\\tilde A,\\tilde B),\n\\end{equation}\nwhere the double brackets are defined by:\n\\begin{eqnarray}\n\\{\\{\\vee(\\hat{A}),\\vee(\\hat{B})\\}\\} & = &\n\\frac{1}{2} \\{A^c,B^c\\} ( \\hat{A}^Q \\hat{B}^Q + \\hat{B}^Q \\hat{A}^Q ) \n\\nonumber \\\\\n & = & \\frac{1}{2} \\sum_i \\frac{\\partial \\tilde{A}}{\\partial q_i} \n\\frac{\\partial \\tilde{B}}{\\partial p_i}- \\frac{\\partial \\tilde{A}}\n{\\partial p_i} \n\\frac{\\partial \\tilde{B}}{\\partial q_i} + \\frac{\\partial \\tilde{B}}\n{\\partial p_i} \n\\frac{\\partial \\tilde{A}}{\\partial q_i} - \\frac{\\partial \\tilde{B}}\n{\\partial q_i} \n\\frac{\\partial \\tilde{A}}{\\partial p_i},\n\\end{eqnarray}\nand we introduced the notation \n$\\tilde{A}=\\vee(\\hat{A})$ and defined the new bracket $(\\quad ,\\quad )=[\\quad ,\\quad ] +\ni\\hbar \\{\\{\\quad ,\\quad \\}\\}$.\nThis bracket was first proposed in \\cite{aleksandrov,boucher}. The motivation to define it this way \nwas given in terms of the properties of the emerging theory, namely that it properly generalises \nboth quantum and classical mechanics. The bracket is known to be antisymmetric, multilinear\nbut it does not satisfy the Jacobi identity. This did cause much debate in the literature \\cite{anderson1,diosi1,jones,salcedo1,salcedo2}. We will come back to this problem in the conclusions. Firstly let us finish the presentation of the dynamical structure of half quantum mechanics.\n \nGiven the unquantization map of definition II we are able to define a quantization prescription mapping a general classical dynamical system to the correspondent half quantum one.\nOne just needs to specify the classical and the quantum sectors to be of the original classical theory, that\nis to provide $T^{\\ast}M_1$ and $T^{\\ast}M_2$.\nLet then ${\\cal F}$ be the algebra of observables over $T^{\\ast}M=T^{\\ast}M_1 \\otimes T^{\\ast}M_2$. The \nremaining notation is in accordance with the previous definition. \n\n\\bigskip\n\n\\underline{{\\bf Definition III}} {\\bf - Half quantization map}\\\\\nThe half quantization is defined to be the map:\n\\begin{equation}\n\\cap :{\\cal F} \\longrightarrow {\\cal S}; \\quad \\cap=\\vee \\circ \n\\wedge \\quad A \\longrightarrow \\tilde{A}=\\cap(A),\n\\end{equation}\nwhere $\\wedge : {\\cal F} \\longrightarrow {\\cal A}({\\cal H}_1 \\otimes {\\cal H}_2) $ is the Dirac quantization map and $\\vee$ is the unquantization map of definition II.\n\nThe properties of $\\bigcap$ follow directly from its definition:\nLet $A,B \\in {\\cal F}$:\\\\\n{\\bf 1)} $\n\\cap \\left( f(q_i,p_i) \\right)=f(q_i,p_i)\\hat{I}$ and $\n\\cap (q_{\\alpha}) = \\hat{q}_{\\alpha}$ , $\n\\cap (p_{\\alpha}) = \\hat{p}_{\\alpha}$, $i=1..M$, $\\alpha=(M+1)..(M+N)$\\\\\n{\\bf 2)} $\\cap$ is a linear map: $\\cap (A+bB)=\\cap(A) + b\\cap(B)$, $b \\in {\\cal C}$\\\\\n{\\bf 3)} $\\cap(f(q_i,p_i)g(q_{\\alpha},p_{\\alpha}))=f(q_i,p_i)\n\\wedge(g(q_{\\alpha},p_{\\alpha})) $\\\\\n{\\bf 4)}\n$\\cap (\\{A,B\\})=(i\\hbar)^{-1} {\\bf (}\\cap(A),\\cap(B) {\\bf )}$\\\\\n\nWe are now in position to study the theory resulting from applying the \nhalf quantization procedure to a given classical theory. \nFirst we have to choose a CSCO for\nthe quantum sector of the theory, let it be, for instance, the set \n$\\{\\hat{q}_{\\alpha}\\},\\alpha =M+1..M+N$. \nThe initial data for the classical sector is given by the set: \n$\\{q_i^0=q_i(t_0),p_i^0=p_i(t_0)\\}$ \nand the correspondent error margins $\\delta_{q_i}$\nand $\\delta_{p_i},i=1...M$. The quantum initial data is given \nby the initial data wave \nfunction $\|\\phi^Q> \\in {\\cal H}_2$.\nThe dynamical evolution of the half quantum system is determined by the \nfollowing set of equations:\n\\begin{equation}\n\\dot{\\tilde{O}}_{k} = \\cap(\\{O_k,H\\})\n = \\frac{1}{i\\hbar} {\\bf (}\\tilde{O}_{k},\\tilde{H}{\\bf )} , \n\\end{equation}\nwhere $\\tilde O_k,k=1...2(M+N)$ is any of the fundamental variables.\nThe former set of equations has the formal solutions:\n\\begin{equation} \n\\tilde{O}_{k}(t) = \\sum_{n=0}^{\\infty} \\frac{1}{n!}\n\\left(\\frac{t}{i\\hbar}\\right)^n\n{\\bf (}...{\\bf (}\\tilde{O}_{k},\\tilde{H}{\\bf )}...,\\tilde{H}{\\bf )}\n=O_k(\\hat{q}_{\\alpha},\\hat{p}_{\\alpha},q_i^0,p_i^0,t).\n\\end{equation}\nNotice that (54) is just the same set of equations as the one \nresulting from applying the unquantization map to the standard quantum evolution \nequations for the observables $\\hat{O}_{k}(t)$ and so is the solution \n(55). Hence the observables $\\tilde O_k(t)$ are just the operators $\\hat B$ we need to supply to obtain the predictions (39,40).\n\n\n\\section{Example}\n\nTo illustrate the procedure by which half quantum mechanics makes predictions for the time evolution of a given dynamical system let us consider the following system of two interacting particles described by the Hamiltonian:\n\\begin{equation}\n\\tilde H= \\frac{\\hat P^2}{2M}+\\frac{p^2}{2m} + kq\\hat P,\n\\end{equation}\nwhere $(\\hat Q,\\hat P)$ are the fundamental observables of the quantum particle of mass $M$, \n$(q,p)$ are the canonical variables of the classical particle of mass $m$ and $k$ is a coupling constant. The initial time configuration of the quantum particle is described \n by the quantum sector wave function $\|\\phi^Q>$, while the initial time configuration of the classical particle is described by the data $\\{q(0),p(0),\\delta_q(0),\\delta_p(0)\\}$.\n\nSolving the half quantum equations of motion (54) we obtain the time evolution of the fundamental observables of the half quantum system, together with the errors $\\delta_L(\\hat B)$ of the half quantum operators (19):\n\\begin{equation}\n\\left\\{ \\begin{array}{lll}\n\\tilde q(t) & = & q(0)+ \\frac{p(0)}{m} t -\\frac{k \\hat P(0)}{2m} t^2 \\\\\n\\tilde p(t) & = & p(0) - k \\hat P(0) t \\\\\n\\tilde Q(t) & = & \\hat Q(0) + \\left( \\frac{\\hat P(0)}{M} +kq(0) \\right) t \\\\\n&& + \\frac{k}{2m} p(0) t^2 -\\frac{k^2}{6m}\\hat P(0) t^3\\\\\n\\tilde P(t) & = & \\hat P(0)\n\\end{array} \\right. \\qquad , \\qquad \n\\left\\{ \\begin{array}{l}\n\\delta_L(\\tilde q(t)) = \\delta_q(0) + \|\\frac{t}{m}\| \\delta_p(0) \\\\\n\\delta_L(\\tilde p(t)) = \\delta_p(0) \\\\\n\\delta_L(\\tilde Q(t)) = \|kt\|\\delta_q(0) +\|\\frac{kt^2}{2m}\|\\delta_p(0) \\\\\n\\delta_L(\\tilde P(t)) = 0 \n\\end{array} \\right. ,\n\\end{equation}\na result that is valid for all $L \\in {\\cal N}$. The spreads are of the general form\n$\\Delta_L(\\hat B)= \\delta_L(\\hat B) /(1-p)^{1/2L}$, for $\\hat B=\\tilde q(t),\\tilde p(t), \\tilde Q(t)$ or $\\tilde P(t)$.\n\nLet $\\hat{A}$ be one of the full quantum operators $\\hat q(t),\\hat p(t), \\hat Q(t)$ or $\\hat P(t)$, the ones we would have obtained if we had performed the full quantum treatment of the system with Hamiltonian $\\hat H=\\hat P^2/2M +\\hat p^2/2m +k\\hat q \\hat P$. Let $\\hat B$ be the correspondent half quantum operator (57). Moreover, let $\|a_i^n>$ be a complete set of eigenstates of $\\hat A$ ($a_i$ is the associated eigenvalue and $n$ is the degeneracy index)\nand $\|b_u^r>$ a complete set of eigenvectores of $\\hat B$ ($b_u$ is the associated eigenvalue and $r$ is the degeneracy index). If the classical sector initial data is taken to be first-order valid ($L=1$) the half quantum predictions for the outputs of a measurement of the full quantum operator $\\hat A$ (choosing $p=0.99$ and $I_B=\\delta_1(\\hat B)$) are given by (43):\n$$\nP(b_u \\in I_{min})-0.72 \\le P(a_i \\in I_0) \\le P(b_u \\in I_{max})+0.72,\n$$ \nwhere $I_0=[a^0-D,a^0+D]$ is an arbitrary interval centred at $a^0\\in {\\cal R}$ with range $D\\ge 20 \\delta_1(\\hat B)$, the error $\\delta_1(\\hat B)$ is given by (57), $I_{max}=[a^0-(D+20\\delta_1(\\hat B)),a^0+(D+20\\delta_1(\\hat B))]$ and $I_{min}=[a^0-(D-20\\delta_1(\\hat B)),a^0+(D-20\\delta_1(\\hat B))]$. Moreover, $P(b_u \\in I_{min,max})=\\sum_{r,b_u\\in I_{min,max}}\|<\\phi^Q\|b_u^r>\|^2$.\n\nIf the classical sector initial data $\\{q(0),p(0),\\delta_q(0),\\delta_p(0)\\}$ is 10th-order valid then, as we have seen, the precision of the half quantum predictions increases considerably (let $p=0.99999$ and $I_B=\\delta_{10}(\\hat B)$):\n$$\nP(b_u \\in I_{min})-0.0019 \\le P(a_i \\in I_0) \\le P(b_u \\in I_{max})+0.0019,\n$$ \nwhere, this time, $I_{max}=[a^0-(D+3.6\\delta_{10}(\\hat B)),a^0+(D+3.6\\delta_{10}(\\hat B))]$, $I_{min}=[a^0-(D-3.6\\delta_{10}(\\hat B)),a^0+(D-3.6\\delta_{10}(\\hat B))]$ and $\\delta_{10}(\\hat B) =\\delta_1(\\hat B)$ is given by (57).\n\nClearly the former predictions are not valid in general. They are valid if the two descriptions of the classical sector initial time configuration, the classical \n$\\{q(0),p(0),\\delta_q(0),\\delta_p(0)\\}$ and the quantum $\|\\phi^c>$, satisfy some consistency conditions.\nGiven the classical initial data let us see what are the wave functions $\|\\phi^c>$ that satisfy the $L$-order classicality conditions.\n\nFollowing the procedure of section 2.1, \nthe $L=1$ fundamental sequences (1) are: \n\\begin{equation}\nS_1=q \\quad {\\rm and} \\quad S_2 = p,\n\\end{equation}\nand the $L$-order sequences:\n\\begin{equation}\nS^{(L)}=(z_1,...,z_L), \\quad z_i = q \\vee p, \\quad i=1..L,\n\\end{equation}\nand thus the $L$-order classicality condition (3) reads:\n\\begin{eqnarray}\n&& <E_{S^{(L)}}\|E_{S^{(L)}}> \\le \\delta^2_{S^{(L)}} \\quad , \\forall S^{(L)} \\quad {\\rm in \\quad (59)}\\\\\n&& \\Longleftrightarrow \\quad\n<\\phi^c\|(\\hat{z}_L-z_L(0))...(\\hat{z}_1-z_1(0))(\\hat{z}_1-z_1(0))...(\\hat{z}_L-z_L(0))\n\|\\phi^c> \\le \\delta_{z_1}(0)^2... \\delta_{z_L}(0)^2 , \\nonumber\n\\end{eqnarray}\nUsing the Shwartz inequality and disregarding the contribution of terms proportional to $\\hbar^2$, the former inequalities are reduced to:\n\\begin{equation}\n\\left\\{ \\begin{array}{l}\n <\\phi^c\|(\\hat q-q(0))^{2L}\|\\phi^c> \\le \\delta_q(0)^{2L}\\\\\n\\\\\n <\\phi^c\|(\\hat p-p(0))^{2L}\|\\phi^c> \\le \\delta_p(0)^{2L}\n\\end{array} \\right.\n\\quad \\Longleftrightarrow \\quad\n\\left\\{ \\begin{array}{l}\n\\int (q-q(0))^{2L} \|\\phi^c (q)\|^2 dq \\le \\delta_q(0)^{2L} \\\\\n\\\\\n\\int (p-p(0))^{2L} \|\\phi^c (p)\|^2 dp \\le \\delta_p(0)^{2L}\n\\end{array} \\right. \n\\end{equation}\nGiven the classical initial data $\\{q(0),p(0),\\delta_q(0),\\delta_p(0)\\}$, (61) constitute a system of inequalities to be satisfied by initial data wave function $\|\\phi^c>$. The higher the order of classicality $L$ the more restrictive is the former system.\nFor typical values of $\\delta_q(0),\\delta_p(0)$ (and choosing $L$ of reasonable size) there are many solutions of (61). Gaussian wave packets, for instance, provide well-known solutions:\n\\begin{equation}\n\\psi_c(q_0,p_0,\\Delta q, q)= \\frac{1}{(2\\Pi (\\Delta q)^2)^{1/4}} \\exp \\left\\{\n-\\frac{(q-q_0)^2}{4 (\\Delta q)^2} + i p_0 q /\\hbar \\right\\}.\n\\end{equation}\nIf we take the parameters $q_0$ and $p_0$ to be given by $q_0=q(0)$ and $p_0=p(0)$, \nand substitute $\\psi_c(q(0),p(0),\\Delta q,q)$ in (61) we get:\n\\begin{equation}\n\\frac{(2L-1)!}{2((L-1)!)}(\\Delta q)^{2L} \\le \\delta_q(0)^{2L} \\quad \\wedge \\quad\n\\frac{(2L-1)!}{2((L-1)!)}\\left(\\frac{\\hbar}{2^{1/2} \\Delta q}\\right)^{2L} \\le \\delta_p(0)^{2L}.\n\\end{equation}\nAny Gaussian wave function of the form (62), with the parameter $\\Delta q$ satisfying the inequalities (63) for a given $L$, is a $L$-order classical wave function with respect to the classical initial data $\\{q(0),p(0),\\delta_q(0),\\delta_p(0)\\}$. Hence, if the classical sector initial time configuration is in such a state then the $L$-order half quantum predictions are valid. \n\n\\section{Conclusions}\n\nThe general prescription to derive a theory of coupled classical-quantum dynamics presented in this paper might be summarised in three main steps: 1) Identification of the properties that should be satisfied by the full quantum initial data so that it might be properly described by a set of half quantum initial data (section 2, eq.(3)).\n2) Establishment of a relation between a general full quantum observable and the correspondent half quantum one so that one is able to reproduce the predictions of quantum mechanics using the half quantum operators (eq.(10)). \nThis evolves the derivation of a relation between the (central) eigenvectores of $\\hat B$ and the eigenvectores of $\\hat A$ (eq.(19)) and in the sequel of a relation between the probabilities in the representation of $\\hat B$ and of $\\hat A$ (eq.(39,40)).\n3) Finally, the derivation of a framework providing the half quantum operators without requiring previous knowledge of the full quantum theory (section 3).\n\nCertainly, there are many different ways of implementing this general plan (see for instance\n\\cite{diosi1,nujo}). In this paper we presented a particular derivation of a theory of coupled classical-quantum dynamics that was named half quantum mechanics. This theory, in the form of a set of axioms, was firstly presented in \\cite{aleksandrov,boucher}. Its properties have been extensively discussed in the literature \\cite{anderson1,diosi1,jones,salcedo1,salcedo2}. In particular, the fact that the bracket structure does not satisfy the Jacobi identity is known to be problematic, the dynamical structure displaying a set of undesirable properties (it is not unitary and time evolution does not preserve the bracket structure, just to mention two of the most intriguing). However, and despite of the fact that the internal structure of half quantum mechanics is not the most desirable, the theory was shown to provide a valid description of coupled classical-quantum dynamics in the sense that it reproduces the results of quantum mechanics in the appropriated limit. \nThe key issue in half quantum mechanics is, of course, the way in which its predictions should be interpreted. Associated to every prediction is an error margin, and within this error margin the theory is physically valid. \n\nTo finish we would like to make a few comments: \\\\\na) There is an uncertainty associated to all \npredictions made by the half quantum theory. Since we do not have a complete \nknowledge of the initial data wave function we could not expect to \nhave a complete \ndeterministic prediction, much the same to what happens in classical \nmechanics. \nAs expected, the degree of precision of the half quantum predictions is related to \nthe classicality conditions that are assumed to be satisfied by the classical sector initial data wave function or, in other words, to the degree of validity of the classical initial data.\\\\\nb) A different bracket for classical-quantum dynamics have been presented in the literature \\cite{anderson1}. The new theory was also postulated and motivated in terms of its properties. This has caused much debate over which would be the best structure for a theory of coupled classical-quantum dynamics. We would like to point out that Anderson theory might also be obtained through a procedure similar to the one presented in this paper. To do this we just have to use a slightly different unquantization map. The deductive approach will provide a way of comparing the two theories in what respects to their consistency with the full quantum description. \\\\\nc) Lastly, as a side result, we realised that the fact that the brackets do not satisfy the Jacobi identity is clearly a consequence of the fact that the unquantization map is not univocous. This might point out a path to obtain a new, better behaved theory of coupled classical-quantum dynamics \\cite{nujo}. \n\n\n\n\n\n\n\\section{Appendix - Error ket framework}\n\nThe aim of this appendix is just to present some of the\nresults of the error ket framework. For a more detailed\npresentation the reader should refer to \\cite{nuno1}.\n\nLet us start by introducing the relevant definitions. Let $\\hat{X_i},i=1..n$ \nbe a set of $n$ operators acting on the Hilbert space ${\\cal H}$ and let $\|\\psi>$ be the wave function describing the system.\n\n\\bigskip \n\n\\underline{{\\bf Definition}} {\\bf - Error Ket}\\\\\nWe define the n-order mixed error ket $\|E(\\hat{X}_1,\\hat{X}_2,...\\hat{X}_n,\n\\psi,x^0_1,x^0_2,...x^0_n)>$, \nas the quantity:\n\\begin{equation}\n\|E(\\hat{X}_1,\\hat{X}_2,...\\hat{X}_n,\\psi,x^0_1,x^0_2,...x^0_n)> \n=(\\hat{X}_1 - x_1^0) (\\hat{X}_2 - x_2^0).... (\\hat{X}_n - x_n^0) \n\|\\psi>,\n\\end{equation}\nwhere $x^0_i$ are complex numbers and the operators $\\hat{X}_i$ \ndo not need to be self-adjoint.\nThe error bra $<E(\\hat{X}_1...\\hat{X}_n,\\psi,x_1^0,...x^0_n)\|$ \nis defined according to the definition\nof the error ket. When there is no risk of confusion we will also use the notation\n$\|E_{X_1,...X_n}>$ for the mixed error ket. Moreover, when $\\hat X_1=\\hat X_2 = ... \\hat X_n =\n\\hat X$ the error (64) is named \"$n$-order error ket\" and we write it in the form: $\|E_X^n>=\|E^n(\\hat X,\\psi,x^0)>$. \\\\ \n\nLet us present some properties of the former quantity: \\\\\n{\\bf a)} The error ket provides a confinement of the wave function. \\\\\nLet $\\hat{X}$ be self-adjoint and $x^0 \\in {\\cal R}$. Given $<E^n_X\|E^n_X>$ to each \"quantity of\n probability\" $p$ we can associate an interval $I$\naround $x^0$, $I=[x^0-\\Delta_n,x^0+\\Delta_n]$, such that the probability \nof obtaining a value $x \\in I$\nfrom a measurement of $\\hat{X}$ is at least $p$. The \nsize of the interval $I$ is dependent of $\\hat{X} , \\psi$ and $x^0$ only\nthrough the value of $<E^n_X\|E^n_X>$. To the quantity $\\Delta_n=\\Delta_n(\\hat{X},\\psi,x^0,p)$ we call the $n$-order spread of the wave function. $\\Delta_n$ is given by:\n\\begin{equation}\n\\Delta_n(\\hat{X},\\psi,x^0,p)=\\left(\\frac{<E^n_X\|E^n_X>}{1-p}\\right)^{1/2n}.\n\\end{equation}\nIf $\\hat{X}$ is not self-adjoint then \nthe former result can also be obtained, but in this case $I$ is a ball of \nradius $\\Delta_n$ in the complex plane. \\\\\n{\\bf b)} Let $\\hat X$ be self-adjoit and $x^0 \\in {\\cal R}$. The former result can be restated in the following way: given $<E^n_X\|E^n_X>$\nand a distance $d$, the probability of obtaining a value $x\\notin\n[x^0-d,x^0+d]$ from a measurement $\\hat{X}$ is at the most $<E^n_X\|E^n_X>/d^{2n}$. In fact,\n(let $\|x,k>$ be a complete set of eigenvectores of $\\hat X$, where $x$ is the associated eigenvalue and $k$ is the degeneracy index):\n\\begin{eqnarray}\n& & <E^n_X\|E^n_X> = \\sum_{x,k} (x-x^0)^{2n}\|<\\psi\|x,k>\|^2 \\nonumber\\\\\n& \\ge & \\sum_{x \\notin [x^0-d,x^0+d],k} \n (x-x^0)^{2n}\|<\\psi\|x,k>\|^2 \\ge d^{2n} \\sum_{x \\notin [x^0-d,x^0+d],k} \|<\\psi\|x,k>\|^2 ,\n\\end{eqnarray}\nand this implies:\n\\begin{equation}\n\\sum_{x \\notin [x^0-d,x^0+d],k} \|<\\psi\|x,k>\|^2 \\le \\frac{<E^n_X\|E^n_X>}{d^{2n}} = \\frac{\\Delta_n(p)^{2n} (1-p)}{d^{2n}}.\n\\end{equation}\n\n\\subsection{Acknowledgements}\n\nI am very grateful to Jorge Pullin for \nencouragement and many discussions \nand suggestions concerning the present paper. \nI would also like to thank Gordon Fleming and Lee Smolin for \nseveral discussions and insights in the subject.\n\nThis work was supported by funds provided by Junta Nacional de \nInvestiga\\c{c}\\~{a}o Cient\\'{i}fica e Tecnol\\'{o}gica -- Lisbon -- Portugal, \ngrant B.D./2691/93 and by grants NSF-PHY 94-06269,\nNSF-PHY-93-96246, the Eberly Research fund at Penn State and\nthe Alfred P. 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Julia and J. Zin-Justin, Les Houches, Session LVII, (1992).\n\n\\bibitem{halliwell3} J. J. Halliwell in {\\it Fundamental Problems in Quantum Theory}, edited by D. Greenberger and A. Zeilinger, Annals of the New York Academy of Sciences, Vol. 775, 726, (1994).\n\n\\bibitem{wald} R. Wald, {\\it Quantum field theory in curve space-times\nand black hole thermodynamics}, (Chicago University Press, Chicago, 1994)\nand references therein.\n\n\\bibitem{nuno1} N.C. Dias, {\\it e-print:} quant-ph/9912034 (1999).\n\n\\bibitem{dirac1} P.A.M. Dirac, {\\it The principles of Quantum Mechanics},\n(Clarendom Press, Oxford, 1930).\n\n\\bibitem{dirac2} P.A.M. Dirac, {\\it Lectures on Quantum Mechanics},\nYeshiva University, (Academic Press, New York, 1967).\n\n\\bibitem{nujo} N. Dias, J. Prata, {\\it e-print:} quant-ph/0005019 (2000).\n\n\\bibitem{jones} K. R. W. Jones, {\\it Phys. Rev. Lett} {\\bf 76}, 4087 (1996); L. Di\\'{o}si, ibid. p4088; I. R. Senitzky ibid. p4089; A. Anderson ibib. p4089-4090.\n\n\\bibitem{salcedo1} L. L. Salcedo, {\\it Phys. Rev.} {\\bf A54}, 3657 (1996).\n\n\\bibitem{salcedo2} J. Caro, L. L. Salcedo, {\\it Phys. Rev.} {\\bf A60}, 842 (1999).\n\n\n\n\n\n\n\n\\end{thebibliography}\n\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n" } ]	[ { "name": "quant-ph9912071.extracted_bib", "string": "{anderson1 A. Anderson, {Phys. Rev. Lett. {74, 621, (1995)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{diosi1 L. Di\\'{osi, N. Gisin, W. T. Strunz, {Phys. Rev. {A61, 22108 (2000)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{aleksandrov I. V. Aleksandrov, {Z. Naturforsch. {36A, 902 (1981)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{boucher W. Boucher, J. Traschen, {Phys. Rev. {D37, 3522, (1988)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{halliwell1 J. J. Halliwell, {Phys. Rev. {D57, 2337-2348 (1998)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{halliwell2 J. J. Halliwell, {e-print: gr-qc/9808071 (1998)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{nature J. Maddox, Nature (London) {373, 469 (1995)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{sherry T. N. Sherry, E. Sudarshan, {Phys. Rev. {D18, 4580-4589 (1978)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{diosi2 L. Di\\'{osi, {e-print: quant-ph/9902087 (1999)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{hartle1 J.B. Hartle, {Spacetime quantum mechanics and the quantum mechanics of spacetime in Gravitation and Quantifications, eds B. Julia and J. Zin-Justin, Les Houches, Session LVII, (1992)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{halliwell3 J. J. Halliwell in {Fundamental Problems in Quantum Theory, edited by D. Greenberger and A. Zeilinger, Annals of the New York Academy of Sciences, Vol. 775, 726, (1994)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{wald R. Wald, {Quantum field theory in curve space-times and black hole thermodynamics, (Chicago University Press, Chicago, 1994) and references therein." }, { "name": "quant-ph9912071.extracted_bib", "string": "{nuno1 N.C. Dias, {e-print: quant-ph/9912034 (1999)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{dirac1 P.A.M. Dirac, {The principles of Quantum Mechanics, (Clarendom Press, Oxford, 1930)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{dirac2 P.A.M. Dirac, {Lectures on Quantum Mechanics, Yeshiva University, (Academic Press, New York, 1967)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{nujo N. Dias, J. Prata, {e-print: quant-ph/0005019 (2000)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{jones K. R. W. Jones, {Phys. Rev. Lett {76, 4087 (1996); L. Di\\'{osi, ibid. p4088; I. R. Senitzky ibid. p4089; A. Anderson ibib. p4089-4090." }, { "name": "quant-ph9912071.extracted_bib", "string": "{salcedo1 L. L. Salcedo, {Phys. Rev. {A54, 3657 (1996)." }, { "name": "quant-ph9912071.extracted_bib", "string": "{salcedo2 J. Caro, L. L. Salcedo, {Phys. Rev. {A60, 842 (1999)." } ]
quant-ph9912072	Nonclassical correlations of photon number and field components in the vacuum state	[ { "author": "Holger F. Hofmann and Takayoshi Kobayashi" }, { "author": "Akira Furusawa" }, { "author": "Holger F. Hofmann" }, { "author": "Department of Physics" }, { "author": "Faculty of Science" }, { "author": "%University of Tokyo" }, { "author": "7-3-1 Hongo" }, { "author": "Bunkyo-ku" }, { "author": "Tokyo113-0033" }, { "author": "Japan" } ]	It is shown that the quantum jumps in the photon number $\hat{n$ from zero to one or more photons induced by backaction evasion quantum nondemolition measurements of a quadrature component $\hat{x$ of the vacuum light field state are strongly correlated with the quadrature component measurement results. This correlation corresponds to the operator expectation value $\langle \hat{x\hat{n\hat{x\rangle$ which is equal to one fourth for the vacuum even though the photon number eigenvalue is zero. Quantum nondemolition measurements of a quadrature component can thus provide experimental evidence of the nonclassical operator ordering dependence of the correlations between photon number and field components in the vacuum state.	[ { "name": "qjump.tex", "string": " \n\\documentstyle[preprint,pra,aps]{revtex}\n%\\documentstyle[twocolumn,pra,aps]{revtex}\n\n\\begin{document}\n\n\\bibliographystyle{prsty}\n\\draft\n%\\preprint{submitted to Phys.Rev. A}\n\\tighten\n\n\\title{Nonclassical correlations of photon number and \nfield components in the vacuum state}\n\n%Nonclassical correlations in the quantum jump backaction of\n%quadrature component measurements performed on the photon vacuum}\n%\n\n\\author{Holger F. Hofmann and Takayoshi Kobayashi}\n\\address{Department of Physics, Faculty of Science, University of Tokyo,\\\\\n7-3-1 Hongo, Bunkyo-ku, Tokyo113-0033, Japan}\n\n\\author{Akira Furusawa}\n\\address{Nikon Corporation, R\\&D Headquarters,\\\\\nNishi-Ohi, Shinagawa-ku, Tokyo 140-8601, Japan}\n\n%\\author{Holger F. Hofmann\\\\Department of Physics, Faculty of Science, \n%University of Tokyo\\\\7-3-1 Hongo, Bunkyo-ku, Tokyo113-0033, Japan}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\nIt is shown that the quantum jumps in the photon number $\\hat{n}$\nfrom zero to one or more photons induced by backaction evasion quantum \nnondemolition measurements of a quadrature component $\\hat{x}$ of the \nvacuum light field state are strongly correlated with the quadrature \ncomponent measurement results.\nThis correlation corresponds to the operator expectation value\n$\\langle \\hat{x}\\hat{n}\\hat{x}\\rangle$ which is equal to one fourth for\nthe vacuum even though the photon number eigenvalue is zero.\nQuantum nondemolition measurements of a quadrature component can\nthus provide experimental evidence of the nonclassical operator \nordering dependence of the correlations between photon number and field \ncomponents in the vacuum state.\n\\end{abstract}\n\\pacs{PACS numbers:\n42.50.Lc % Quantum fluctuations, quantum noise, and quantum jumps\n42.50.Dv % Nonclassical field states; squeezed states; phase measurement\n03.65.Bz % Foundations of quantum mechanics\n}\n\n\\section{Introduction}\nOne of the main differences between quantum mechanics and classical\nphysics is the impossibility of assigning well defined values to \nall physical variables describing a system. As a consequence, all\nquantum measurements necessarily introduce noise into the system.\nA measurement which only introduces noise in those variables that \ndo not commute with the measured variable is referred to as a \nquantum nondemolition (QND) measurement \\cite{Cav80}. In most of the\ntheoretical and experimental investigations \n\\cite{Lev86,Fri92,Bru90,Hol91,Yur85,Por89,Per94}, the focus has\nbeen on the overall measurement resolution and on the reduction of \nfluctuations in the QND variable as observed in the correlation between \nthe QND measurement results and a subsequent destructive measurement \nof the QND variable. However, at finite resolution, quantum nondemolition\nmeasurements do not completely destroy the original coherence between \neigenstates of the QND variable \\cite{Imo85,Kit87}. By correlating the \nQND measurement result with subsequent destructive measurements of a \nnoncommuting variable, it is therefore possible to determine details of \nthe measurement induced decoherence \\cite{Hof20}. \n\nIn particular, QND measurements of a quadrature component of the light\nfield introduce not only noise in the conjugated quadrature component,\nbut also in the photon number of a state. By measuring a quadrature\ncomponent of the vacuum field, ``quantum jumps'' from zero photons\nto one or more photons are induced in the observed field. It is shown in\nthe following that, even at low measurement resolutions, the \n``quantum jump'' events are strongly correlated with extremely high \nmeasurement results for the quadrature component. This correlation \ncorresponds to a nonclassical relationship between the continuous\nfield components and the discrete photon number, which is directly\nrelated to fundamental properties of the operator formalism. \nThus, this experimentally observable correlation of photon number and \nfields reveals important details of the physical meaning of quantization.\n\nIn section \\ref{sec:qnd}, QND measurements of a \nquadrature component $\\hat{x}$ of the light field are discussed and\na general measurement operator $\\hat{P}_{\\delta\\! x}(x_m)$ describing \na minimum noise measurement at a resolution of $\\delta\\!x$ is derived.\nIn section \\ref{sec:vac}, the measurement operator is applied to the \nvacuum field and the measurement statistics are determined.\nIn section \\ref{sec:fundop}, the results are compared with fundamental \nproperties of the operator formalism.\nIn section \\ref{sec:ex}, an experimental realization of photon-field\ncoincidence measurements is proposed and possible difficulties are discussed.\nIn section \\ref{sec:int}, the results are interpreted in the context\nof quantum state tomography and implications for the interpretation \nof entanglement are pointed out.\nIn section \\ref{sec:concl}, the results are summarized and conclusions\nare presented.\n\n%=========================================================\n\n\\section{QND measurement of a quadrature component}\n\\label{sec:qnd}\n\nOptical QND measurements of the quadrature \ncomponent $\\hat{x}_S$ of a signal mode \n$\\hat{a}_S = \\hat{x}_S + i \\hat{y}_S$\nare realized by coupling the signal to a a meter mode \n$\\hat{a}_M = \\hat{x}_M + i \\hat{y}_M$ in such a way that the \nquadrature component $\\hat{x}_M$ of the meter mode is shifted\nby an amount proportional to the measured signal variable\n$\\hat{x}_S$. This measurement interaction \ncan be described by a unitary transformation operator, \n\\begin{equation}\n\\hat{U}_{SM} = \\exp\\left(-i\\; 2 f \\hat{x}_S\\hat{y}_M\\right),\n\\end{equation}\nwhich transforms the quadrature components of meter and signal to\n\\begin{eqnarray}\n\\label{eq:shift}\n\\hat{U}_{SM}^{-1}\\;\\hat{x}_S\\;\\hat{U}_{SM} &=& \\hat{x}_S\n\\nonumber \\\\[0.2cm]\n\\hat{U}_{SM}^{-1}\\;\\hat{y}_S\\;\\hat{U}_{SM} &=& \\hat{y}_S - f \\hat{y}_M\n\\nonumber \\\\[0.2cm]\n\\hat{U}_{SM}^{-1}\\;\\hat{x}_M\\;\\hat{U}_{SM} &=& \\hat{x}_M + f \\hat{x}_S\n\\nonumber \\\\[0.2cm]\n\\hat{U}_{SM}^{-1}\\;\\hat{y}_M\\;\\hat{U}_{SM} &=& \\hat{y}_M.\n\\end{eqnarray}\nIn general, the unitary measurement interaction operator \n$\\hat{U}_{SM}$ creates entanglement between the signal and the meter\nby correlating the values of the quadrature components.\nSuch an entanglement can be realized experimentally by squeezing\nthe two mode light field of signal and meter using optical\nparametric amplifiers (OPAs) \\cite{Yur85,Por89,Per94}. The measurement\nsetup is shown schematically in figure \\ref{setup}. Note that the \nbackaction changing $\\hat{x}_S$ is avoided by adjusting the interference\nbetween the two amplified beams. Therefore, the reflectivity of the beam\nsplitters depends on the amplification. A continuous adjustment of the \ncoupling factor $f$ would require adjustments of both the pump beam \nintensities of the OPAs and the reflectivities of the beam splitter as\ngiven in figure \\ref{setup}.\n\nIf the input state of the meter is the vacuum field state, \n$\\mid \\mbox{vac.} \\rangle$, and the signal field state is given by\n$\\mid \\Phi_S \\rangle$, then the entangled state created by the\nmeasurement interaction is given by \n\\begin{eqnarray}\n\\hat{U}_{SM}\\mid \\Phi_S; \\mbox{vac.}\\rangle &=&\n\\int d\\!x_S d\\!x_M\\; \\langle x_S \\mid \\Phi_S \\rangle \\;\n \\langle x_M - f x_S \\mid \\mbox{vac.} \\rangle\n \\; \\mid x_S; x_M \\rangle\n\\nonumber \\\\\n&=& \\int d\\!x_S d\\!x_M\\; \\left(\\frac{2}{\\pi}\\right)^{\\frac{1}{4}}\n\\exp\\left(-(x_M-f x_S)^2\\right) \\langle x_S \\mid \\Phi_S \\rangle\n \\; \\mid x_S; x_M \\rangle.\n\\end{eqnarray}\nReading out the meter variable $x_M$ removes the entanglement by\ndestroying the coherence between states with different $x_M$. \nIt is then possible to define a measurement operator $\\hat{P}_f(x_M)$\nassociated with a readout of $x_M$, which acts only on the initial\nsignal state $\\mid \\Phi_S \\rangle$. This operator is given by\n\\begin{eqnarray}\n\\langle x_S \\mid \\hat{P}_f(x_M) \\mid \\Phi_S \\rangle &=&\n\\langle x_S; x_M \\mid\\hat{U}_{SM}\\mid \\Phi_S; \\mbox{vac.}\\rangle \n\\nonumber \\\\\n&=&\n\\left(\\frac{2}{\\pi}\\right)^{\\frac{1}{4}}\n\\exp\\left(-(x_M-f x_S)^2\\right) \\langle x_S \\mid \\Phi_S \\rangle.\n\\end{eqnarray}\nThe measurement operator $\\hat{P}_f(x_M)$ multiplies the probability\namplitudes of the $\\hat{x}_S$ eigenstates with a Gaussian statistical \nweight factor given by the difference between the eigenvalue \n$x_S$ and the measurement result $x_M/f$. By defining\n\\begin{eqnarray}\nx_m &=& \\frac{1}{f} x_M\n\\nonumber \\\\\n\\delta\\!x &=& \\frac{1}{2f},\n\\end{eqnarray} \nthe measurement readout can be scaled, so that the average results\ncorrespond to the expectation value of $\\hat{x}_S$. \nThe normalized measurement operator then reads\n\\begin{equation}\n\\label{eq:project}\n\\hat{P}_{\\delta\\!x}(x_m) = \\left(2 \\pi \\delta\\!x^2\\right)^{-1/4} \n\\exp \\left(-\\frac{(x_m-\\hat{x}_S)^2}{4\\delta\\!x^2}\\right).\n\\end{equation}\nThis operator describes an ideal quantum nondemolition measurement \nof finite resolution $\\delta\\!x$. \n%\nThe probability distribution of \nthe measurement results $x_m$ is given by\n\\begin{eqnarray}\n\\label{eq:prob}\nP(x_m) &=& \\langle \\Phi_S \\mid \\hat{P}^2_{\\delta\\!x}(x_m) \\mid \\Phi_S \\rangle\n\\nonumber \\\\\n&=& \\frac{1}{\\sqrt{2\\pi \\delta\\!x^2}}\\int d\\!x_S\\; \n\\exp\\left(-\\frac{(x_S-x_m)^2}{2\\delta\\!x^2}\\right)\n\|\\langle x_S \\mid \\Phi_S \\rangle \|^2 \n. \n\\end{eqnarray}\nThus the probability distribution of measurement results is equal to\nthe convolution of $\|\\langle x_S \\mid \\Phi_S \\rangle \|^2 $ with a Gaussian\nof variance $\\delta\\! x$. The corresponding averages of $x_m$ and $x_m^2$\nare given by\n\\begin{eqnarray}\n\\label{eq:av}\n\\int d\\!x_S\\; x_m P(x_m) \n &=& \\langle \\Phi_S \\mid \\hat{x}_S \\mid \\Phi_S \\rangle\n\\nonumber \\\\\n\\int d\\!x_S\\; x_m^2 P(x_m) \n &=& \\langle \\Phi_S \\mid \\hat{x}_S^2 \\mid \\Phi_S \\rangle\n + \\delta\\!x^2. \n\\end{eqnarray}\nThe measurement readout $x_m$ therefore represents the actual value of\n$\\hat{x}_S$ within an error margin of $\\pm \\delta\\!x$.\n%\nThe signal state after the measurement is given by\n\\begin{equation}\n\\label{eq:state}\n\\mid \\phi_S(x_m)\\rangle = \\frac{1}{\\sqrt{P(x_m)}}\n\\hat{P}_{\\delta\\!x}(x_m) \\mid \\Phi_S \\rangle. \n\\end{equation}\nSince the quantum coherence between the eigenstates of $\\hat{x}_S$\nis preserved, the system state is still a pure state after the\nmeasurement. The system properties which do not commute with\n$\\hat{x}_S$ are changed by the modified statistical weight\nof each eigenstate component. Thus the physical effect of noise in\nthe measurement interaction is correlated with the measurement\ninformation obtained. \n\n%=========================================================\n\n\\section{Measurement of the vacuum field}\n\\label{sec:vac}\n\nIf the signal is in the vacuum state $\\mid \\mbox{vac.}\\rangle$,\nthen the measurement probability is a Gaussian centered around\n$x_m=0$ with a variance of $\\delta\\!x^2+1/4$,\n\\begin{equation}\n\\label{eq:vacprop}\nP(x_m)= \\frac{1}{\\sqrt{2\\pi (\\delta\\!x^2+1/4)}} \n\\exp\\left(-\\frac{x_m^2}{2 (\\delta\\!x^2+1/4)}\\right).\n\\end{equation}\nThe quantum state after the measurement is a squeezed state \ngiven by\n\\begin{equation}\n\\mid \\phi_S(x_m)\\rangle = \\int d\\!x_S\\;\n\\left(\\pi \\frac{4\\delta\\!x^2}{1+4\\delta\\!x^2}\\right)^{-\\frac{1}{4}}\n\\exp\\left(- \\frac{1+4\\delta\\!x^2}{4\\delta\\!x^2}\n \\left(x_S- \\frac{x_m}{1+4\\delta\\!x^2}\\right)^2\\right) \n \\mid x_S \\rangle.\n\\end{equation}\nThe quadrature component averages and variances of this state are \n\\begin{eqnarray}\n\\langle \\hat{x}_S \\rangle_{x_m}&=& \\frac{x_m}{1+4\\delta\\!x^2}\n\\nonumber \\\\[0.2cm]\n\\langle \\hat{y}_S \\rangle_{x_m}&=& 0\n\\nonumber \\\\[0.2cm]\n\\langle \\hat{x}_S^2 \\rangle_{x_m} - \\langle \\hat{x}_S \\rangle_{x_m}^2\n&=& \\frac{\\delta\\!x^2}{1+4\\delta\\!x^2}\n\\nonumber \\\\[0.2cm]\n\\langle \\hat{y}_S^2 \\rangle_{x_m} - \\langle \\hat{y}_S \\rangle_{x_m}^2\n&=& \\frac{1+4\\delta\\!x^2}{16\\delta\\!x^2}.\n\\end{eqnarray}\nExamples of the phase space contours before and after the measurement\nare shown in figure \\ref{xy} for a measurement resolution of $\\delta\\!x=0.5$\nand a measurement result of $x_m=-0.5$. Note that the final state is shifted\nby only half the measurement result. \n\nThe photon number expectation value after the measurement is given by\nthe expectation values of $\\hat{x}_S^2$ and $\\hat{y}_S^2$. It reads\n\\begin{eqnarray}\n\\label{eq:vacphoton}\n\\langle \\hat{n}_S \\rangle_{x_m} &=& \\langle\\hat{x}_S^2\\rangle_{x_m} \n + \\langle \\hat{y}_S^2 \\rangle_{x_m} - \\frac{1}{2}\n\\nonumber \\\\\n &=& \\frac{1}{16 \\delta\\!x^2 (1+4\\delta\\!x^2)}\n + \\frac{x_m^2}{(1+4\\delta\\!x^2)^2}.\n\\end{eqnarray}\nThe dependence of the photon number expectation value \n$\\langle \\hat{n}_S \\rangle_{x_m}$ after the measurement \non the squared measurement result $x_m^2$ describes a correlation\nbetween field component and photon number defined by\n\\begin{eqnarray}\nC(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m}) &=&\n\\int\n\\left(\\int d\\!x_m\\; x_m^2 \\langle \\hat{n}_S \\rangle_{x_m} P(x_m)\\right) \n- \n\\left(\\int d\\!x_m\\; x_m^2 P(x_m)\\right)\n\\left(\\int d\\!x_m\\; \\langle \\hat{n}_S \\rangle_{x_m} P(x_m)\\right).\n\\nonumber \\\\\n\\end{eqnarray}\nAccording to equations (\\ref{eq:vacprop}) and (\\ref{eq:vacphoton}),\nthis correlation is equal to \n\\begin{equation}\nC(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m}) = \\frac{1}{8}\n\\end{equation}\nfor measurements of the vacuum state. This result is independent of the\nmeasurement resolution. In particular, it even applies to the low resolution\nlimit of $\\delta\\!x\\to \\infty$, which should leave the original\nvacuum state nearly unchanged. It is therefore reasonable to conclude, that\nthis correlation is a fundamental property of the vacuum state, even though \nit involves nonzero photon numbers.\n%=========================================================\n\n\\section{Correlations of photon number and fields in the operator formalism}\n\\label{sec:fundop}\n\nSince the measurement readout $x_m$ represents information about\noperator variable $\\hat{x}_S$ of the system, it is possible to \nexpress the correlation \n$C(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m})$ in terms of \noperator expectation values of $\\hat{x}_S$ and $\\hat{n}_S$.\nEquation (\\ref{eq:av}) shows how the average over $x_m^2$\ncan be replaced by the operator expectation value \n$\\langle \\hat{x}_S^2 \\rangle$. Likewise, the \naverage over the product of $x_m^2$ and \n$\\langle \\hat{n}_S \\rangle_{x_m}$ can be transformed into an\noperator expression. The transformation reads\n\\begin{eqnarray}\n\\label{eq:trans}\n\\lefteqn{\n\\int d\\!x_m\\; x_m^2 \\langle \\hat{n}_S \\rangle_{x_m} P(x_m) = }\n\\nonumber \\\\[0.2cm]\n&=& \\int d\\!x_S d\\!x_S^\\prime \\left(\\frac{(x_S+x_S^\\prime)^2}{4} \n+ \\delta\\!x^2\\right)\n\\langle \\mbox{vac.}\\mid x_S \\rangle \n \\langle x_S \\mid \\hat{n}_S \\mid x_S^\\prime\\rangle \n \\langle x_S^\\prime \\mid \\mbox{vac.} \\rangle \n\\exp\\left(-\\frac{(x_S-x_S^\\prime)^2}{8\\delta\\!x^2}\\right)\n\\nonumber \\\\[0.2cm]\n&=& \\int d\\!x_m\\; \n\\left(\\frac{1}{4}\\langle \\hat{x}_S^2\\hat{n}_S + 2 \\hat{x}_S\\hat{n}_S\\hat{x}_S \n+ \\hat{n}_S\\hat{x}_S^2 \\rangle_{x_m} \n+ \\delta\\!x^2 \\langle \\hat{n}_S \\rangle_{x_m}\\right) P(x_m).\n\\end{eqnarray}\nThe average expectation value of photon number after the measurement\nis given by\n\\begin{equation}\n\\langle \\hat{n}_S \\rangle_{\\mbox{av.}} = \n\\int d\\!x_m\\; \\langle\\hat{n}_S \\rangle_{x_m} P(x_m).\n\\end{equation}\nUsing the index $\\mbox{av.}$ to denote averages over expectation values\nafter the measurement, the correlation \n$C(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m})$\nmay be expressed by the average final state expectation values as\n\\begin{equation}\nC(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m}) =\n\\left(\\frac{1}{4}\\langle \\hat{x}_S^2\\hat{n}_S \n+ 2 \\hat{x}_S \\hat{n}_S \\hat{x}_S\n+ \\hat{n}_S\\hat{x}_S^2 \\rangle_{\\mbox{av.}} \n- \\langle n_S \\rangle_{\\mbox{av.}} \n \\langle x_S^2 \\rangle_{\\mbox{av.}}\\right).\n\\end{equation}\nThe correlation observed in the measurement is therefore given by a\nparticular ordered product of operators. The most significant feature\nof this operator product is the $\\hat{x}_S\\hat{n}_S\\hat{x}_S$-term,\nin which the photon number operator $\\hat{n}_S$ is sandwiched between \nthe field operators $\\hat{x}_S$. The expectation value of \n$\\hat{x}_S\\hat{n}_S\\hat{x}_S$ of an eigenstate of $\\hat{n}_S$ does not\nfactorize into the eigenvalue of $\\hat{n}_S$ and the expectation value \nof $\\hat{x}_S^2$, because the field operators $\\hat{x}_S$ change the\noriginal state into a state with different photon number statistics. \nAccording to the commutation relations,\n\\begin{equation}\n\\hat{x}_S\\hat{n}_S\\hat{x}_S = \n\\frac{1}{2}(\\hat{x}_S^2\\hat{n}_S + \\hat{n}_S\\hat{x}_S^2) + \\frac{1}{4}.\n\\end{equation}\nTherefore, the expectation value of $\\hat{x}_S\\hat{n}_S\\hat{x}_S$\nof a photon number state is exactly $1/4$ higher than the product\nof the eigenvalue of $\\hat{n}_S$ and the expectation value of \n$\\hat{x}_S^2$. The correlation $C(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m})$\nmay then be expressed by the final state expectation values as\n\\begin{equation}\nC(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m}) =\n\\left(\\frac{1}{2}\\langle \\hat{x}_S^2\\hat{n}_S \n+ \\hat{n}_S\\hat{x}_S^2 \\rangle_{\\mbox{av.}} \n- \\langle n_S \\rangle_{\\mbox{av.}} \n \\langle x_S^2 \\rangle_{\\mbox{av.}}\\right)\n+ \\frac{1}{8}.\n\\end{equation}\nSince the additional correlation of $1/8$ does not depend on the measurement\nresolution $\\delta\\!x$, it should not be interpreted as a result of the \nmeasurement dynamics. Instead, the derivation above reveals that it \noriginates from the \noperator ordering in the quantum mechanical expression for the correlation. \nSince it is the noncommutativity of operator variables which distinguishes \nquantum physics from classical physics, the contribution of $1/8$ is a \nnonclassical contribution to the correlation of photon number and fields. \nSpecifically, it should be noted that the classical correlation of a well \ndefined variable\nwith any other physical property is necessarily zero. Only the quantum \nmechanical properties of noncommutative variables allow nonzero\ncorrelations of photon number and fields even if the field mode \nis in a photon number eigenstate. \nThe operator transformation thus reveals that the correlation \n$C(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m})$ of $1/8$ found in measurements\nof the vacuum state is a directly observable consequence of the nonclassical\noperator order dependence of correlations between noncommuting variables.\n\n%=========================================================\n\n\\section{Experimental realization: photon-field coincidence measurements}\n\\label{sec:ex}\n\nThe experimental setup required to measure the correlation between a\nQND measurement of the quadrature component $\\hat{x}_S$ and the\nphoton number after the measurement is shown in figure \\ref{setup}.\nIt is essentially identical to the setups used in previous experiments \n\\cite{Por89,Per94}. However, instead of measuring the x quadrature in the \noutput fields, it is necessary to perform a photon number \nmeasurement on the signal branch. The output of this measurement\nmust then be correlated the output from the homodyne detection of the\nmeter branch. The homodyne detection of the meter simply \nconverts a high intensity light field into a current $I_M(t)$,\nwhile the signal readout produces discreet photon detection pulse.\nThese pulses can also be described by a detection current $I_S(t)$,\nwhich should be related to the actual photon detection events by a\nresponse function $R_S(\\tau)$, such that\n\\begin{equation}\nI_S(t) = \\sum_i R_S(t-t_i),\n\\end{equation} \nwhere $t_i$ is the time of photon detection event $i$. \nAccording to the theoretical prediction discussed above, each\nphoton number detection event should be accompanied by an increase\nof noise in the homodyne detection current of the meter.\nHowever, the temporal overlap of the signal current $I_S(t)$\nand the increased noise in the meter current $I_M(t)$ is\nan important factor in the evaluation of the correlation.\nDue to the frequency filtering employed, the meter mode corresponding\nto a signal detection event is given by a filter function\nwith a width approximately equal to the inverse frequency resolution\nof the filter. For a typical filter with a Lorentzian linewidth of\n$2\\gamma$, the mode of interest\nwould read\n\\begin{equation}\n\\label{eq:mode}\n\\hat{a}_i = \n\\sqrt{\\gamma} \\int dt \\exp\\left(-\\gamma\\;\|\\;t-t_i\\;\|\\right) \\hat{a}(t).\n\\end{equation}\nThe actual meter readout should therefore be obtained by integrating\nthe current over a time of about $2/\\gamma$. For practical reasons, it\nseems most realistic to use a direct convolution of the meter\ncurrent $I_M$ and the signal current $I_S$, adjusting the response function\n$R_S(\\tau)$ to produce an electrical pulse of duration $2/\\gamma$.\nA measure of the correlation $C(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m})$\ncan then be obtained from the current correlation\n\\begin{equation}\n\\xi\\;C(x_m^2; \\langle \\hat{n}_S \\rangle_{x_m}) \n= \\overline{(I_S I_M)^2} - \\overline{I_S^2}\\;\\overline{I_M^2},\n\\end{equation}\nwhere the factor $\\xi$ denotes the efficiency of the measurement, as\ndetermined by the match between the response function $R_S(\\tau)$ and the\nfilter function given by equation (\\ref{eq:mode}). Moreover, the\nefficiency of the experimental setup may be reduced further by the\nlimited quantum efficiency of the detector. \n\nFortunately, the requirement of efficiency for the \nexperiment is not very restrictive, provided that the measurement resolution\nis so low that only few photons are created. In that case, the total\nnoise average in the meter current $I_M$ is roughly equal to the noise\naverage in the absence of a photon detection event, which is very close to \nthe shot noise limit of the homodyne detection. However, the fluctuations\nof the time averaged currents within a time\ninterval of about $1/\\gamma$ around a photon detection event in the \nsignal branch correspond to the fluctuations of the measurement \nvalues $x_m$ for a quantum jump event from zero photons to one photon.\nIn particular, the measurement result $x_m(i)$ associated with a \nphoton detection event at time $t_i$ is approximately given by\n\\begin{equation}\nx_m (i) \\approx C \\int dt R(t-t_i) I_M(t),\n\\end{equation}\nwhere $C$ is a scaling constant which maps the current fluctuations \nof a vacuum input field onto an $x_m$ variance of $\\delta\\!x^2$.\nIn the case of a photon detection event, however, the probability \ndistribution over the measurement results $x_m (i)$ is given\nby the difference between the total probability\ndistribution $P(x_m)$ and the part $P_0(x_m)$ of the probability distribution\nassociated with no photons in the signal,\n\\begin{eqnarray}\nP_{QJ}(x_m) &=& P(x_m) - P_0(x_m)\n\\nonumber \\\\[.2cm]\n&=& \\langle \\mbox{vac.}\\mid \\hat{P}_{\\delta\\!x}^2 \\mid \\mbox{vac.}\\rangle\n- \\langle \\mbox{vac.}\\mid \\hat{P}_{\\delta\\!x} \\mid \\mbox{vac.}\\rangle^2\n\\nonumber \\\\\n&=& \\frac{1}{\\sqrt{2\\pi (\\delta\\!x^2+1/4)}} \n\\exp\\left(-\\frac{x_m^2}{2 (\\delta\\!x^2+1/4)}\\right) \n\\;-\\; \\sqrt{\\frac{32\\delta\\!x^2}{\\pi(1+8\\delta\\!x^2)^2}} \n \\exp\\left(-\\frac{4}{1+8\\delta\\!x^2}x_m^2\\right)\n.\n\\end{eqnarray}\nFigure \\ref{qj} shows the results for a measurement resolution of \n$\\delta\\!x=1$, which is close to the experimentally realized resolution\nreported in \\cite{Per94}. There is only a slight difference in\n$P(x_m)$ and $P_0(x_m)$, even though the total probability of a quantum \njump to one or more photons obtained by integrating $P_{QJ}(x_m)$ \nis about 5.72\\% . The peaks of the probability \ndistribution are close to $\\pm 2$, eight times higher than the fluctuation\nof $\\hat{x}_S$ in the vacuum. \nThe measurement fluctuations corresponding to a photon detection event\nare given by\n\\begin{equation}\n\\frac{\\int d\\!x_m\\; x_m^2 P_{QJ}(x_m)}{\\int d\\!x_m P_{QJ}(x_m)}\n= \\frac{1}{4}+\\delta\\!x^2\\left(2+\\sqrt{1+\\frac{1}{8\\delta\\!x^2}}\\right)\n\\approx 3 \\delta\\!x^2. \n\\end{equation}\nFor $\\delta\\!x\\gg1$, this result is three times higher than the overall\naverage. For $\\delta\\!x=1$, the ratio between the fluctuation intensity \nof a detection event and the average fluctuation intensity of\n$1/4+\\delta\\!x^2$ is still equal to 2.65. In other words, the fluctuations\nof the measurement result $x_m$ nearly triple in the case of a quantum jump\nevent. The corresponding increase in the fluctuations of the homodyne \ndetection current $I_M$ should be detectable even at low efficiencies\n$\\xi$. Moreover, it does not matter how many photon events go undetected, \nsince the ratio has been determined relative to the overall average of\nthe meter fluctuations. It is thus possible to obtain experimental evidence\nof the fundamental correlation of field component and photon number even\nwith a rather low overall efficiency of the detector setup.\n\n%=========================================================\n\n\\section{Interpretation of the quantum jump statistics}\n\\label{sec:int}\n\nWhat physical mechanism causes the quantum jump from the zero\nphoton vacuum to one or more photons? The relationship between \nthe photon number operator and the quadrature components of the \nfield is given by\n\\begin{equation}\n\\label{eq:ndef}\n\\hat{n}_S + \\frac{1}{2} = \\hat{x}_S^2 + \\hat{y}_S^2.\n\\end{equation}\nAccording to equation (\\ref{eq:shift}) describing the measurement \ninteraction, the change in photon number $\\hat{n_S}$ should \ntherefore be caused by the change in $\\hat{y}_S$ caused by \n$\\hat{y}_M$,\n\\begin{equation}\n\\hat{U}_{SM}^{-1}\\;\\hat{n}_S\\;\\hat{U}_{SM} = \\hat{n}_S\n- 2f\\hat{y}_S\\hat{y}_M + f^2\\hat{y}_M^2.\n\\end{equation}\nThus the change in photon number does not depend explicitly on either\nthe measured quadrature $\\hat{x}_S$ or the meter variable\n$\\hat{x}_M$. Nevertheless, the meter readout shows a strong correlation \nwith the quantum jump events. In particular, the probability distribution \nof meter readout results $x_m$ for a quantum jump to one or more photons\nshown in figure \\ref{qj} has peaks at values far outside the range given\nby the variance of the vacuum fluctuations of $\\hat{x}_S$. \n\nMoreover, the correlation between readout and photon number after the \nmeasurement does not disappear in the limit of low resolution\n($\\delta\\!x\\to\\infty$). Rather, it appears to be a fundamental\nproperty of the vacuum state even before the measurement. This is confirmed \nby the operator formalism, which identifies the source of the correlation\nas the expectation value $\\langle\\hat{x}_S\\hat{n}_S\\hat{x}_S\\rangle$. \nThis expectation\nvalue is equal to $1/4$ in the vacuum, even though the photon number is \nzero. Since the operator formalism does not allow an identification of the\noperator with the eigenvalue unless it acts directly on the eigenstate,\nit is possible to find nonzero correlations even if the system is in an \neigenstate of one of the correlated variables. In particular, the action\nof the operator $\\hat{x}_S$ on the vacuum state is given by\n\\begin{equation}\n\\hat{x}_S\\mid\\mbox{vac.}\\rangle = \\frac{1}{2}\\mid n_s=1 \\rangle,\n\\end{equation}\nso the operator $\\hat{x}_S$ which should only determine the \nstatistical properties of the state with regard to the quadrature component\n$x_S$ changes the vacuum state into the one photon state. The \napplication of operators thus causes fluctuations in a variable even when the \neigenvalue of that variable is well defined. \n\nThe nature of this fluctuation might be clarified by a comparison\nof the nonclassical correlation obtained for fields and photon number \nin the vacuum with the results of quantum tomography by homodyne \ndetection\\cite{Vog89,Smi93}. In such \nmeasurements, the photon number is never obtained. Rather, the complete\nWigner distribution $W(x_S,y_S)$ can be reconstructed from the results. \nIt is therefore possible to deduce correlations between the field \ncomponents and the field intensity defined by $I = x_S^2+y_S^2$, \nwhich is the classical equivalent of\nequation (\\ref{eq:ndef}). For the vacuum, the Wigner function reads\n\\begin{equation}\n\\int d\\!x_S d\\!y_S\\; x_S^4 W_0(x_S,y_S) - (\\int d\\!x_S d\\!y_S\\; x_S^2 W_0(x_S,y_S))^2\n= 1/8.\n\\end{equation} \nThe correlation of $I$ and $x_S^2$ is given by\n\\begin{eqnarray}\n\\label{eq:wigcor}\n\\lefteqn{C(x_S^2; I) =}\n\\nonumber \\\\ &&\n\\int\n\\left(\\int d\\!x_S d\\!y_S \\; x_S^2\\;I \\; W_0(x_S,y_S)\\right) \n-\n\\left(\\int d\\!x_S d\\!y_S \\; x_S^2 \\; W_0(x_S,y_S)\\right)\n\\left(\\int d\\!x_S d\\!x_S \\; I \\; W_0(x_S,y_S)\\right) \n\\nonumber \\\\ \n&=& C(x_m^2; \\langle n_S \\rangle_{x_m})=\\frac{1}{8}.\n\\end{eqnarray}\nThus, the correlation between $I=x_S^2+y_S^2$ and $x_S^2$ described by\nthe Wigner distribution is also equal to $1/8$. In fact, the ``intensity\nfluctuations'' of the Wigner function can be traced to the same operator \nproperties that give rise to the correlations between the field measurement\nresult and the induced photon number. For arbitrary signal fields, the\ncorrelation between the squared measurement result and the photon number\nafter the measurement can therefore be derived by integrating over the \nWigner function of the signal field after the measurement interaction\naccording to equation (\\ref{eq:wigcor}).\n\nOf course the ``intensity fluctuations'' of the Wigner function \ncannot be observed directly, since any phase insensitive determination\nof photon number will reveal the well defined result of zero photons in\nthe vacuum. Nevertheless even a low resolution measurement of the quadrature \ncomponent $\\hat{x}_S$ which leaves the vacuum state nearly unchanged\nreveals a correlation of $\\hat{x}_S^2$ and $n_S$ which corresponds to the\nassumption that the measured quadrature $\\hat{x}_S$ contributes to a\nfluctuating vacuum energy. The quantum jump itself appears to draw its\nenergy not from the external influence of the measurement interaction, but\nfrom the fluctuating energy contribution $\\hat{x}_S^2$. These energy \nfluctuations could be interpreted as virtual or hidden fluctuations\nexisting only potentially until the energy uncertainty of the measurement\ninteraction removes the constraints imposed by quantization and energy\nconservation. \nIn particular, energy conservation does require that the energy for the \nquantum jump is provided by the optical parametric amplification process.\nCertainly the {\\it average} energy is supplied by the pump beam. However,\nthe energy content of the pump beam and the meter beam cannot be defined \ndue to the uncertainty principle. The pump must be coherent and the \nmeasurement of the meter field component $\\hat{x}_M$ prevents all energy\nmeasurements in that field. If it is accepted that quantum mechanical \nreality is somehow conditioned by the circumstances of the measurement, \nit can be argued that the reality of quantized photon number only exists if the\nenergy exchange of the system with the environment is controlled on the\nlevel of single quanta. Otherwise, it is entirely possible that the vacuum\nenergy might not be zero as suggested by the photon number eigenvalue,\nbut might fluctuate according to the statistics suggested by the Wigner \nfunction. \n\nEven though it may appear to be highly unorthodox at first, this \n``relaxation'' of quantization rules actually corresponds\nto the noncommutativity of the operators, and may help explain the seemingly\nnonlocal properties of entanglement associated with the \nfamous EPR paradox \\cite{EPR}. The definition of elements of reality\ngiven by EPR reads\n``{\\it If, without in any way disturbing a system, we can predict\nwith certainty (i.e., with probability equal to unity) the value of a \nphysical quantity, then there exists an element of physical reality\ncorresponding to this physical quantity.}''\nThis definition of elements of reality assumes that the eigenvalues of\nquantum states are real even if they are not confirmed in future\nmeasurements. In particular, the photon number of the vacuum would \nbe considered as a real number, not an operator, so \nthe operator correlation $\\langle\\hat{x}_S\\hat{n}_S\\hat{x}_S\\rangle$ \nshould not have any physical meaning.\nHowever, the nonzero correlation of fields and\nphoton number in the vacuum observed in the QND measurement discussed\nabove suggests that {\\it even the possibility\nof predicting the value of a physical quantity with certainty only \ndefines an element of reality if this value is directly observed\nin a measurement}. Based on this conclusion, there is no need to\nassume any ``spooky action at a distance'', or physical nonlocality,\nin order to explain Bell's inequalities \\cite{Bel64}. Instead, it is \nsufficient to point out that knowledge of the wavefunction does not provide\nknowledge of the type of measurement that will be performed.\nIn the case of spin-1/2 systems, the quantized values of spin \ncomponents are not a property inherent in the spin system, but a \nproperty of the measurement\nactually performed. To assume that spins are quantized even without \na measurement does not correspond to the implications of the operator \nformalism, since it is not correct to replace operators with their \neigenvalues. \n\nIn the same manner, the correlation discussed in this paper would be \nparadoxical if one regarded the photon number eigenvalue of zero in the\nvacuum state as an element of reality independent of the measurement\nactually performed. One would then be forced to construct mysterious\nforces changing the photon number in response to the measurement result.\nHowever, the operator formalism suggests no such hidden forces. Instead,\nthe reality of photon number quantization depends on the operator ordering\nand thus proofs to be rather fragile.\n\n%=========================================================\n\n\\section{Summary and conclusions}\n\\label{sec:concl}\n\nThe change in photon number induced by a quantum nondemolition measurement\nof a quadrature component of the vacuum is strongly correlated with\nthe measurement result. An experimental determination of this correlation\nis possible using optical parametric amplification in a setup similar \nto previously realized QND measurements of quadrature components\n\\cite{Por89,Per94}. The observed correlation corresponds to a fundamental\nproperty of the operator formalism which allows nonvanishing correlations\nbetween noncommuting variables even if the system is in an eigenstate\nof one of the variables.\n\nThe quantum jump probability reflects the properties of intensity \nfluctuations corresponding to the vacuum fluctuations of the field\ncomponents. The total correlation of fields and photon number\ntherefore reproduces the result that would be expected if there was \nno quantization. It seems that quantum jumps are a mechanism by \nwhich the correspondence between quantum mechanics and classical physics\nis ensured. The quantum jump correlation observable in the\nexperimental situation discussed above thus provides a link\nbetween the discrete nature of quantized information and the continuous\nnature of classical signals. Finite resolution QND measurements \ncould therefore provide a more detailed understanding of the nonclassical \nproperties of quantum information in the light field.\n\n\\section*{Acknowledgements}\nOne of us (HFH) would like to acknowledge support from the Japanese \nSociety for the Promotion of Science, JSPS.\n%=========================================================\n\n\\begin{thebibliography}{5}\n\\bibitem{Cav80}\nC.M. Caves, K.S. Thorne, R.W.P. Drever, V.P. Sandberg, and M. Zimmermann,\nRev. Mod. Phys. {\\bf 52}, 341 (1980).\n\n\\bibitem{Lev86}\nM.D. Levenson, R.M. Shelby, M.Reid, and D.F. Walls,\nPhys. Rev. Lett. 57, 2473 (1986).\n\n\\bibitem{Fri92}\nS.R.Friberg, S. Machida, and Y.Yamamoto,\nPhys. Rev. Lett. 69, 3165 (1992).\n\n\\bibitem{Bru90}\nM. Brune, S. Haroche, V.Lefevre, J.M. Raimond, and N.Zagury, Phys. Rev. Lett. \n{\\bf 65}, 976 (1990).\n\n\\bibitem{Hol91}\nM.J. Holland, D.F. Walls, and P. Zoller, Phys. Rev. Lett. \n{\\bf 67}, 1716 (1991).\n\n\\bibitem{Yur85}\nB. Yurke, J. Opt. Soc. Am. B {\\bf 2}, 732 (1985).\n\n\\bibitem{Por89}\nA. LaPorta, R.E. Slusher, and B. Yurke, Phys. Rev. Lett. {\\bf 62}, 28 (1989).\n\n\\bibitem{Per94}\nS.F. Pereira, Z.Y. Ou, and H.J. Kimble, Phys. Rev. Lett. {\\bf 72}, 214 (1994).\n\n\\bibitem{Imo85}\nN. Imoto, H.A. Haus, and Y. Yamamoto, Phys. Rev. A {\\bf 32}, 2287 (1985).\n\n\\bibitem{Kit87}\nM. Kitagawa, N. Imoto, and Y. Yamamoto, Phys. Rev. A {\\bf 35}, 5270 (1987).\n\n\\bibitem{Hof20}\nH.F. Hofmann, Phys.Rev. A {\\bf 61}, 033815 (2000).\n\n\\bibitem{Vog89}\nK. Vogel and H. Risken, Phys. Rev. A {\\bf 40}, 2847 (1989).\n\n\\bibitem{Smi93}\nD.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. {\\bf 70},\n1244 (1993). \n\n\\bibitem{EPR}\nA. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {\\bf 47}, 777 (1935).\n\n\\bibitem{Bel64} J.S. Bell, Physics {\\bf 1}, 195 (1964).\n\n\\end{thebibliography}\n\n%========================================================================\n\n\\begin{figure}\n\\caption{\\label{setup}\nSchematic illustration of the measurement setup for a back action evasion \nquantum nondemolition measurement of a quadrature component using optical \nparametric amplifiers (OPAs). Note that the reflectivity of the beam \nsplitters depends on the amplification achieved in the parametric \ndownconversion process. The coupling factor for the measurement is given \nby $f=(a^2-1)/a$.\n}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\\label{xy}\nVisualization of the field fluctuations before and after the measurement\nfor a measurement resolution of $\\delta\\!x=0.5$ and a measurement result\nof $x_m=-0.5$. The contours shown mark the standard deviation of the \nGaussian noise distributions. The circle represents the vacuum fluctuations.\nAfter the measurement, the x-component is shifted by $x_m/2=-0.25$ and the\nfluctuations in x are squeezed by a factor of $1/\\sqrt{2}$. The fluctuations\nin y are increased by a factor of $\\sqrt{2}$ by the noise introduced in the \nmeasurement.}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\\label{qj}\nSeparation of the probability distribution $P(x_m)$ of the measurement \nresult $x_m$ into a component $P_0(x_m)$ associated with no quantum jump\nand a component $P_{QJ}(x_m)$ associated with a quantum jump to one or more\nphotons at a measurement resolution of $\\delta\\!x=1$.\n(a) shows both $P(x_m)$ and $P_0(x_m)$, which are only slightly\ndifferent from each other. (b) shows the difference given by the quantum\njump contribution $P_{QJ}(x_m)$. The total probability of a quantum jump\nat $\\delta\\!x=1$ is 5.72\\%.}\n\\end{figure}\n\n\n%=========================================================\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912072.extracted_bib", "string": "{Cav80 C.M. Caves, K.S. Thorne, R.W.P. Drever, V.P. Sandberg, and M. Zimmermann, Rev. Mod. Phys. {52, 341 (1980)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Lev86 M.D. Levenson, R.M. Shelby, M.Reid, and D.F. Walls, Phys. Rev. Lett. 57, 2473 (1986)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Fri92 S.R.Friberg, S. Machida, and Y.Yamamoto, Phys. Rev. Lett. 69, 3165 (1992)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Bru90 M. Brune, S. Haroche, V.Lefevre, J.M. Raimond, and N.Zagury, Phys. Rev. Lett. {65, 976 (1990)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Hol91 M.J. Holland, D.F. Walls, and P. Zoller, Phys. Rev. Lett. {67, 1716 (1991)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Yur85 B. Yurke, J. Opt. Soc. Am. B {2, 732 (1985)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Por89 A. LaPorta, R.E. Slusher, and B. Yurke, Phys. Rev. Lett. {62, 28 (1989)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Per94 S.F. Pereira, Z.Y. Ou, and H.J. Kimble, Phys. Rev. Lett. {72, 214 (1994)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Imo85 N. Imoto, H.A. Haus, and Y. Yamamoto, Phys. Rev. A {32, 2287 (1985)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Kit87 M. Kitagawa, N. Imoto, and Y. Yamamoto, Phys. Rev. A {35, 5270 (1987)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Hof20 H.F. Hofmann, Phys.Rev. A {61, 033815 (2000)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Vog89 K. Vogel and H. Risken, Phys. Rev. A {40, 2847 (1989)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Smi93 D.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. {70, 1244 (1993)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{EPR A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {47, 777 (1935)." }, { "name": "quant-ph9912072.extracted_bib", "string": "{Bel64 J.S. Bell, Physics {1, 195 (1964)." } ]
quant-ph9912073	Operator Methods of \\ the Parabolic Potential Barrier	[ { "author": "Toshiki Shimbori" }, { "author": "{\\footnotesizeInstitute of Physics" }, { "author": "University of Tsukuba" } ]	The one-dimensional parabolic potential barrier dealt with in an earlier paper is re-examined from the point of view of operator methods, for the purpose of getting generalized Fock spaces.	[ { "name": "quant-ph9912073.tex", "string": "\\documentclass[12pt]{article}\n\n\\setlength{\\textheight}{21.5cm}\n\\setlength{\\textwidth}{16cm}\n\\setlength{\\topmargin}{0cm}\n\\setlength{\\oddsidemargin}{0cm}\n\\setlength{\\evensidemargin}{0cm}\n\n\\def\\bra#1{\\bigl<#1\\bigr\|}\n\\def\\ket#1{\\bigl\|#1\\bigr>}\n\\def\\bracket#1#2{\\bigl<#1\\!\\bigm\|\\!#2\\bigr>}\n\\def\\com#1#2{\\bigl[#1,\\,#2\\bigr]}\n\\def\\anticom#1#2{\\bigl\\{#1,\\,#2\\bigr\\}}\n\n\\usepackage{amsmath,amssymb}\n\n\\newcommand{\\real}{\\mathbb{R}}\n\\newcommand{\\comp}{\\mathbb{C}}\n\\newcommand{\\cS}{\\mathcal{S}} \n\\newcommand{\\Piv}{\\varPi}\n\\newcommand{\\Thetav}{\\varTheta}\n\n\\renewcommand{\\labelenumi}{(\\theenumi)}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\n\\begin{document}\n\n\\title{\\bf Operator Methods of \\\\\nthe Parabolic Potential Barrier}\n\n\\author{Toshiki Shimbori \\\\\n{\\footnotesize\\it Institute of Physics, University of Tsukuba}\\\\\n{\\footnotesize\\it Ibaraki 305-8571, Japan}}\n\n\\date{}\n\n\\maketitle\n\n\\begin{abstract}\n The one-dimensional parabolic potential barrier \n dealt with in an earlier paper is re-examined \n from the point of view of operator methods, \n for the purpose of getting generalized Fock spaces. \n\\end{abstract}\n\n\\thispagestyle{empty}\n\n\\setcounter{page}{0}\n\n\\pagebreak\n\n It is well known that the harmonic oscillator can be studied \n by a operator method~\\cite{dirac,jjs}. \n In this paper we shall show that \n the eigenvalue problem of the parabolic potential barrier~\\cite{sk} \n can be solved by a similar method. \n The generalized eigenstates of this unstable system will form \n {\\it generalized Fock spaces}. \n \n The Hamiltonian of the one-dimensional parabolic potential barrier is \n \\begin{equation}\n \\hat{H}=\\frac{1}{2m}\\hat{p}^2 -\\frac{1}{2} m\\gamma^2 \\hat{x}^2, \n \\label{2.1}\n \\end{equation}\n where $m>0$ is the mass and $\\gamma>0$ is \n proportional to the square root of the curvature at the origin $x=0$. \n The canonical coordinate and momentum $\\hat{x}$ and $\\hat{p}$ satisfy \n the commutation relations \n \\begin{equation}\n \\left.\n \\begin{gathered} \n {\\com{\\hat{x}}{\\hat{p}}\\equiv\\hat{x}\\hat{p}-\\hat{p}\\hat{x}\n =i\\hslash}, \\\\\n \\com{\\hat{x}}{\\hat{x}}=0,\\,\\,\\,\\com{\\hat{p}}{\\hat{p}}=0. \n \\end{gathered} \\right\\} \\label{2.2} \n \\end{equation}\n Written in terms of the Schr\\\"{o}dinger or \n the coordinate representatives, \n the canonical coordinate and momentum give \n \\begin{equation}\n \\left.\n \\begin{aligned}\n \\hat{x}&=x, \\\\\n \\hat{p}&=-i\\hslash\\, d/dx, \n \\end{aligned} \\right\\} \\label{2.3}\n \\end{equation} \n and the Hamiltonian \\eqref{2.1} becomes\n \\begin{equation}\n \\hat{H} =-\\frac{\\hslash^2}{2m} \\frac{d^2}{dx^2}\n -\\frac{1}{2} m\\gamma^2 x^2. \\label{2.4} \n \\end{equation}\n \n We now introduce the {\\it normal coordinates} \n \\begin{gather}\n \\hat{b}^\\pm\\equiv\\sqrt{\\frac{m\\gamma}{2\\hslash}}\n \\left(\\hat{x}\\pm\\frac{1}{m\\gamma}\\hat{p}\\right) \n =\\sqrt{\\frac{m\\gamma}{2\\hslash}} \n \\left( x\\mp\\frac{i\\hslash}{m\\gamma} \\frac{d}{dx}\\right) \n =\\frac{1}{\\sqrt{2}}\\left( \\xi\\mp i\\frac{d}{d\\xi}\\right), \\label{2.5}\\\\ \n \\intertext{where} \n \\xi\\equiv\\beta x, \\,\\,\\, \n \\beta\\equiv\\sqrt{\\frac{m\\gamma}{\\hslash}}. \\label{2.6} \n \\end{gather}\n Note that \\eqref{2.5} are defined except for \n an arbitrary phase factor. \n These operators $\\hat{b}^\\pm$ are {\\it essentially self-adjoint} on \n a Schwartz space $\\cS(\\real)$. \n (Cf. the normal coordinates $\\hat{a}$ and $\\hat{a}^\\dag$, \n being {\\it adjoint operators}, for the harmonic oscillator.) \n Further, two conditions are satisfied. \n \\begin{enumerate}\n \\item $\\cS(\\real)$ is an invariant subspace of $\\hat{b}^\\pm$. \n \\item $\\hat{b}^\\pm$ is continuous on $\\cS(\\real)$. \n \\end{enumerate} \n From the commutation relations \\eqref{2.2} we obtain \n \\begin{equation}\n \\left.\n \\begin{gathered} \n {\\com{\\hat{b}^+}{\\hat{b}^-}=-i}, \\\\ \n \\com{\\hat{b}^+}{\\hat{b}^+}=0,\\,\\,\\,\\com{\\hat{b}^-}{\\hat{b}^-}=0. \n \\end{gathered} \\right\\} \\label{2.7}\n \\end{equation}\n The first of equations \\eqref{2.7} gives the commutation relation \n connecting $\\hat{b}^+$ and $\\hat{b}^-$. \n (Cf. the commutation relation $\\com{\\hat{a}}{\\hat{a}^\\dag}=1$ \n for the harmonic oscillator.) \n One can express $\\hat{H}$ in terms of $\\hat{b}^+$ and $\\hat{b}^-$ \n and one finds \n \\begin{gather}\n \\hat{H}=-\\hslash\\gamma\\hat{N}, \\label{2.8} \n \\intertext{where}\n \\hat{N}\\equiv\\frac{1}{2}\\anticom{\\hat{b}^+}{\\hat{b}^-}\n \\equiv\\frac{1}{2}\n \\bigl(\\hat{b}^+\\hat{b}^- +\\hat{b}^-\\hat{b}^+\\bigr). \\label{2.9} \n \\end{gather}\n From \\eqref{2.7} \n \\begin{align}\n \\com{\\hat{N}}{\\hat{b}^\\pm}&=\\pm i\\hat{b}^\\pm \\label{2.10} \n \\intertext{or}\n \\com{\\hat{H}}{\\hat{b}^\\pm}&=\\mp i\\hslash\\gamma\\hat{b}^\\pm. \\label{2.11}\n \\end{align}\n Also, \\eqref{2.7} lead to\n \\begin{equation}\n \\com{\\hat{b}^\\mp}{\\bigl(\\hat{b}^\\pm\\bigr)^n} \n =\\pm in \\bigl(\\hat{b}^\\pm\\bigr)^{n-1} \\label{2.19}\n \\end{equation}\n for any positive integer $n$. \n \n We shall now work out the eigenvalue problem of $\\hat{H}$. \n Let us assume that there are {\\it standard states} $u^\\pm_0$ \n satisfying \n \\begin{equation}\n \\hat{b}^\\mp u^\\pm_0 =0. \\label{2.12}\n \\end{equation}\n If these equations are expressed in terms of representatives, \n they give us\n \\begin{equation}\n \\left(\\frac{d}{dx}\\mp i\\frac{m\\gamma}{\\hslash}x\\right)\n u^\\pm_0(x)=0 \\label{2.13}\n \\end{equation}\n with the help of \\eqref{2.5}. The solutions of \n these differential equations are \n \\begin{equation}\n u^\\pm_0(x)=B^\\pm_0 e^{\\pm im\\gamma x^2/2\\hslash}, \\label{2.14}\n \\end{equation}\n where $B^\\pm_0\\in\\comp$ are the numerical coefficients. \n These solutions $u^\\pm_0$ do not belong to a Lebesgue space $L^2(\\real)$. \n But they are {\\it generalized functions} \n in the conjugate space ${\\cS(\\real)}^\\times$ of \n the following Gel'fand triplet~\\cite{sk,bogolubov,bohm}, \n \\begin{equation}\n \\cS(\\real)\\subset L^2(\\real)\\subset{\\cS(\\real)}^\\times. \n \\label{2.15}\n \\end{equation}\n \n Let us treat the extensions $\\bigl(\\hat{b}^\\pm\\bigr)^\\times$ \n of the normal coordinates \n to the conjugate space ${\\cS(\\real)}^\\times$. \n We should be able to apply them to a generalized function \n $u\\in{\\cS(\\real)}^\\times$, \n the products $\\bigl(\\hat{b}^\\pm\\bigr)^\\times u$ \n being defined by~\\cite{bogolubov,bohm}\n $$\\bracket{v}{\\bigl(\\hat{b}^\\pm\\bigr)^\\times u}\n =\\bracket{\\hat{b}^\\pm v}{u}$$\n for all functions $v\\in\\cS(\\real)$. \n Taking the representatives, we get\n $$\\int_{-\\infty}^{\\infty}v(x)^* \n \\bigl[\\bigl(\\hat{b}^\\pm\\bigr)^\\times u\\bigr](x)dx=\n \\int_{-\\infty}^{\\infty}\\bigl[\\hat{b}^\\pm v\\bigr](x)^* u(x)dx. $$\n We can transform the right-hand sides by partial integration and get\n $$\\int_{-\\infty}^{\\infty}v(x)^* \n \\bigl[\\bigl(\\hat{b}^\\pm\\bigr)^\\times u\\bigr](x)dx=\n \\int_{-\\infty}^{\\infty}v(x)^* \\bigl[\\hat{b}^\\pm u\\bigr](x)dx, $$ \n since $v$ is a rapidly decreasing function and then \n the contributions from the limits of integration vanish. \n These give \n $$\\bracket{v}{\\bigl(\\hat{b}^\\pm\\bigr)^\\times u}\n =\\bracket{v}{\\hat{b}^\\pm u}, $$ \n showing that \n $$\\bigl(\\hat{b}^\\pm\\bigr)^\\times u=\\hat{b}^\\pm u. $$\n Thus $\\bigl(\\hat{b}^\\pm\\bigr)^\\times$ operating to a generalized function \n have the meaning of $\\hat{b}^\\pm$ operating. \n Similarly~\\cite{sk}, \n \\begin{align}\n \\hat{N}^\\times u &=\\hat{N} u \n \\intertext{or}\n \\hat{H}^\\times u &=\\hat{H} u. \n \\end{align}\n \n Let us examine physical properties of the standard states. \n The result of the operator $\\hat{N}$ applied to \n standard states $u^\\pm_0$ is \n \\begin{equation}\n \\hat{N}u^\\pm_0 =\\frac{1}{2}\\hat{b}^\\mp\\hat{b}^\\pm u^\\pm_0 \n =\\pm\\frac{i}{2}u^\\pm_0, \\label{2.16} \n \\end{equation}\n with the help of \\eqref{2.12} and \\eqref{2.7}. From \\eqref{2.8} \n \\begin{equation}\n \\hat{H}u^\\pm_0 =\\mp\\frac{i}{2}\\hslash\\gamma u^\\pm_0. \\label{2.17}\n \\end{equation}\n Thus $u^\\pm_0$ are generalized eigenstates \n of $\\hat{H}$ belonging to \n the {\\it complex energy eigenvalues} $\\mp i\\hslash\\gamma/2$. \n \n We can form \n the generalized Fock spaces of the parabolic potential barrier \n on the same lines as the harmonic oscillator~\\cite{dirac,jjs}. \n We now consider the following states: \n \\begin{equation}\n \\bigl(\\hat{b}^\\pm\\bigr)^n u^\\pm_0. \\label{2.18}\n \\end{equation}\n The result of the operator $\\hat{N}$ applied to these states is \n \\begin{equation}\n \\hat{N}\\bigl(\\hat{b}^\\pm\\bigr)^n u^\\pm_0 \n =\\pm i\\left( n+\\frac{1}{2}\\right)\n \\bigl(\\hat{b}^\\pm\\bigr)^n u^\\pm_0, \\label{2.20} \n \\end{equation}\n with the help of \\eqref{2.19} and \\eqref{2.12}. \n Here we introduce the $n$th quantum states $u^\\pm_n$, \n being a numerical multiple of \\eqref{2.18}, \n which are satisfied by \n \\begin{gather}\n \\hat{H}u^\\pm_n = E^\\pm_n u^\\pm_n, \\label{2.21} \n \\intertext{where} \n E^\\pm_n\\equiv\\mp i\\left(n+\\frac{1}{2}\\right)\\hslash\\gamma \\,\\,\\, \n \\left(n=0, 1, 2, \\cdots\\right). \\label{2.22}\n \\end{gather}\n Thus the states \\eqref{2.18} are generalized eigenstates \n of $\\hat{H}$ belonging to \n the {\\it complex energy eigenvalues} $E^\\pm_n$. \n The representatives of \n the $n$th quantum states can be obtained from \\eqref{2.14}. \n For any $f\\in {\\cS(\\real)}^\\times$ we find \n $$\\left(\\mp i\\frac{d}{d\\xi}+\\xi\\right) f(\\xi) \n =e^{\\mp i\\xi^2/2}\\left(\\mp i\\frac{d}{d\\xi}\\right)\n e^{\\pm i\\xi^2/2} f(\\xi). $$\n Thus \\eqref{2.18} give \n \\begin{align}\n \\left[\\frac{1}{\\sqrt{2}}\\left(\\mp i\\frac{d}{d\\xi}+\\xi\\right)\\right]^n \n u^\\pm_0(\\xi) \n &=\\left(\\frac{1}{\\sqrt{2}}\\right)^n e^{\\mp i\\xi^2/2} \n \\left(\\mp i\\frac{d}{d\\xi}\\right)^n e^{\\pm i\\xi^2/2} \n u^\\pm_0(\\xi) \\notag\\\\ \n &=B^\\pm_0\\left(\\frac{\\mp i}{\\sqrt{2}}\\right)^n \n e^{\\pm i\\xi^2/2} e^{\\mp i\\xi^2} \n \\frac{d^n}{d\\xi^n} e^{\\pm i\\xi^2}. \\label{2.23} \n \\end{align}\n Now define the polynomials $H^\\pm_n(\\xi)$ by~\\cite{sk}\n \\begin{equation}\n H^\\pm_n(\\xi)=\\left(\\mp i\\right)^n e^{\\mp i\\xi^2} \n \\frac{d^n}{d\\xi^n}e^{\\pm i\\xi^2}. \\label{2.24}\n \\end{equation}\n Inserting these expressions in \\eqref{2.23}, \n we get the representatives of the $n$th quantum states \n \\begin{equation}\n u^\\pm_n(x)=B^\\pm_n e^{\\pm i\\beta^2x^2/2} H^\\pm_n(\\beta x), \n \\label{2.25}\n \\end{equation}\n where $B^\\pm_n\\in\\comp$ are the numerical coefficients. \n These numerical coefficients cannot be determined by \n the normalizing condition~\\cite{sk}. \n \n Let us now see the properties of $n$th quantum states \n under a parity or a space inversion. \n The parity operator $\\hat{\\Piv}$ is a unitary operator on $\\cS(\\real)$ \n and satisfies~\\cite{jjs} \n \\begin{align}\n \\hat{\\Piv}\\hat{x}\\hat{\\Piv}^{-1}&=-\\hat{x}, \\\\\n \\hat{\\Piv}\\hat{p}\\hat{\\Piv}^{-1}&=-\\hat{p}. \n \\end{align}\n The parity operator applied to \n the normal coordinates $\\hat{b}^\\pm$ is then \n $$\\hat{\\Piv}\\hat{b}^\\pm\\hat{\\Piv}^{-1} =-\\hat{b}^\\pm $$ \n from \\eqref{2.5}. \n The result of the parity operator applied to \n the conditions \\eqref{2.12} is \n $$\\hat{\\Piv}\\hat{b}^\\mp u^\\pm_0 \n =\\hat{\\Piv}\\hat{b}^\\mp\\hat{\\Piv}^{-1}\\hat{\\Piv}u^\\pm_0 \n =-\\hat{b}^\\mp\\hat{\\Piv}u^\\pm_0 = 0, $$\n showing that $\\hat{\\Piv}u^\\pm_0$ are also standard states, i.e. \n $$\\hat{\\Piv} u^\\pm_0 = u^\\pm_0. $$ \n The phase factors are chosen unity, \n because the representatives \\eqref{2.14} are symmetrical of $x$. \n We have further \n $$\\hat{\\Piv}\\bigl(\\hat{b}^\\pm\\bigr)^n u^\\pm_0 \n =\\hat{\\Piv}\\hat{b}^\\pm\\hat{\\Piv}^{-1}\\cdots\n \\hat{\\Piv}\\hat{b}^\\pm\\hat{\\Piv}^{-1}\\hat{\\Piv} u^\\pm_0 \n =(-)^n\\bigl(\\hat{b}^\\pm\\bigr)^n u^\\pm_0, $$\n showing that \n $$\\hat{\\Piv}u^\\pm_n =(-)^n u^\\pm_n. $$\n These equations assert that \n the $n$th quantum states are eigenstates of the parity. \n \n We shall now verify a time reversal. \n The time-reversal operator $\\hat{\\Thetav}$ is \n an antiunitary operator on $\\cS(\\real)$ and satisfies~\\cite{jjs} \n \\begin{align}\n \\hat{\\Thetav}\\hat{x}\\hat{\\Thetav}^{-1}&=\\hat{x}, \\\\\n \\hat{\\Thetav}\\hat{p}\\hat{\\Thetav}^{-1}&=-\\hat{p}. \n \\end{align}\n The time-reversal operator applied to \n the normal coordinates $\\hat{b}^\\pm$ is then \n $$\\hat{\\Thetav}\\hat{b}^\\pm\\hat{\\Thetav}^{-1} =\\hat{b}^\\mp $$ \n from \\eqref{2.5}. \n The result of the time-reversal operator applied to \n the conditions \\eqref{2.12} is \n $$\\hat{\\Thetav}\\hat{b}^\\mp u^\\pm_0 \n =\\hat{\\Thetav}\\hat{b}^\\mp\\hat{\\Thetav}^{-1}\\hat{\\Thetav}u^\\pm_0 \n =\\hat{b}^\\pm\\hat{\\Thetav}u^\\pm_0 = 0, $$\n showing that $\\hat{\\Thetav}u^\\pm_0$ are also standard states, i.e. \n $$\\hat{\\Thetav} u^\\pm_0 = u^\\mp_0. $$ \n These time-reversed states contain arbitrary phase factors. \n Proceeding as before, we have \n $$\\hat{\\Thetav}\\bigl(\\hat{b}^\\pm\\bigr)^n u^\\pm_0 \n =\\hat{\\Thetav}\\hat{b}^\\pm\\hat{\\Thetav}^{-1}\\cdots\n \\hat{\\Thetav}\\hat{b}^\\pm\\hat{\\Thetav}^{-1}\\hat{\\Thetav} u^\\pm_0 \n =\\bigl(\\hat{b}^\\mp\\bigr)^n u^\\mp_0, $$\n showing that \n $$\\hat{\\Thetav}u^\\pm_n = u^\\mp_n. $$\n We see in this way that a time reversal occurs \n resulting in the interchange of the $n$th quantum states \n $u^+_n$ and $u^-_n$. \n \n Our work so far has been concerned with one instant of time. \n We shall study finally the time evolution of \n the parabolic potential barrier in the Heisenberg picture. \n The Heisenberg equations of motion are \n \\begin{equation}\n \\frac{d}{dt}\\hat{b}^\\pm(t) \n =\\frac{1}{i\\hslash}\\com{\\hat{b}^\\pm(t)}{\\hat{H}}. \\label{3.1} \n \\end{equation}\n With the help of \\eqref{2.11}, these give \n \\begin{equation}\n \\frac{d}{dt}\\hat{b}^\\pm(t)=\\pm\\gamma\\hat{b}^\\pm(t). \\label{3.2}\n \\end{equation}\n These equations can be integrated to give \n \\begin{equation}\n \\hat{b}^\\pm(t)=\\hat{b}^\\pm e^{\\pm\\gamma t}, \\label{3.3}\n \\end{equation}\n where $\\hat{b}^\\pm$ are normal coordinates \\eqref{2.5}, \n and are equal to the values of $\\hat{b}^\\pm(t)$ at time $t=0$. \n The above solutions show that \n $\\hat{N}$ or $\\hat{H}$ is also constant in the Heisenberg picture. \n The canonical coordinate and momentum \n in the Heisenberg picture are, from \\eqref{2.5} and \\eqref{3.3}, \n \\begin{equation}\n \\left.\n \\begin{aligned}\n \\hat{x}(t)\n &=\\hat{x}\\cosh\\gamma t +\\hat{p}\\sinh\\gamma t/m\\gamma, \\\\\n \\hat{p}(t)\n &=\\hat{x}m\\gamma\\sinh\\gamma t +\\hat{p}\\cosh\\gamma t \n =m\\Dot{\\Hat{x}}(t). \n \\end{aligned} \\right\\} \\label{3.7}\n \\end{equation} \n Equations \\eqref{3.7} correspond to the hyperbolic orbits \n in the classical theory. \n \n The foregoing work provides two generalized Fock spaces \n which are spanned by the tensor products of \n $\\left\\{u^+_n\\right\\}_{n=0}^{\\infty}$ and \n $\\left\\{u^-_n\\right\\}_{n=0}^{\\infty}$, respectively. \n {\\it The normal coordinates $\\hat{b}^\\pm$ are operators \n of creation or annihilation \n of a quantum of width $\\hslash\\gamma$}. \n This interpretation has been given \n by Takahashi in his book~\\cite{ashi}. \n For $\\left\\{u^\\pm_n\\right\\}_{n=0}^{\\infty}$, \n the creation operators are $\\hat{b}^\\pm$ and \n the annihilation operators are $\\hat{b}^\\mp$, \n i.e. \n the creation operator for $\\left\\{u^+_n\\right\\}_{n=0}^{\\infty}$ are \n the annihilation operator for $\\left\\{u^-_n\\right\\}_{n=0}^{\\infty}$, \n and vice versa. \n These operators vary with time \n according to the $e^{\\pm\\gamma t}$ law in the Heisenberg picture. \n Thus the parabolic potential barrier \n as a model of an unstable system will form a corner-stone \n in the quantum theory of decay. \n \n We may introduce \n the essentially self-adjoint operators $\\hat{d}^+$, $\\hat{d}^-$ \n satisfying the anticommutation relations \n \\begin{equation}\n \\left.\n \\begin{gathered}\n \\anticom{\\hat{d}^+}{\\hat{d}^-}=1,\\\\\n \\anticom{\\hat{d}^+}{\\hat{d}^+}=0,\\,\\,\\,\n \\anticom{\\hat{d}^-}{\\hat{d}^-}=0. \n \\end{gathered} \\right\\} \\tag{\\ref{2.7}$'$}\\label{4.1}\n \\end{equation}\n These relations are like \\eqref{2.7} with an anticommutator \n instead of a commutator. \n Put\n \\begin{equation}\n \\hat{N}\\equiv\\frac{i}{2}\n \\com{\\hat{d}^+}{\\hat{d}^-}, \n \\tag{\\ref{2.9}$'$}\\label{4.9} \n \\end{equation}\n the same as \\eqref{2.9}. \n We can find, by the above-mentioned method, \n that the generalized eigenstates of $\\hat{N}$ are \n only the two alternatives of a twofold degenerate state \n belonging to the complex eigenvalue $i/2$ or $-i/2$. \n \n %\\thanks\n \\section*{Acknowledgements}\n \n I would like to express my thanks to T. Kobayashi \n for many valuable suggestions in the writing of this paper.\n \n \\begin{thebibliography}{9}\n \\bibitem{dirac}\n\t P. A. M. Dirac, \n\t \\emph{The Principles of Quantum Mechanics} 4th ed. \n\t (Clarendon Press, 1958). \n\t \n \\bibitem{jjs}\n\t J. J. Sakurai, \n\t \\emph{Modern Quantum Mechanics} Rev. ed. \n\t (Addison-Wesley, 1994). \n\t \n \\bibitem{sk}\n\t T. Shimbori and T. Kobayashi, \n\t \\emph{Complex Eigenvalues of the Parabolic Potential Barrier \n\t and Gel'fand Triplet}, math-ph/9910009. \n\t \n \\bibitem{bogolubov}\n\t N. N. Bogolubov, A. A. Logunov and I. T. Todorov, \n\t \\emph{Introduction to Axiomatic Quantum Field Theory}\n\t (W. A. Benjamin, 1975). \n\t \n \\bibitem{bohm}\n\t A. Bohm and M. Gadella, \n\t \\emph{Dirac Kets, Gamow Vectors and Gel'fand Triplets} \n\t (Lecture Notes in Physics, Vol.~348, Springer, 1989). \n\t \n \\bibitem{ashi}\n\t Y. Takahashi, \n\t \\emph{Quantum Field Theory for Condensed Matter Physicists}, \n\t Vol.~1 (Baifukan, 1974). \n\n \\end{thebibliography}\n \n\\end{document}" } ]	[ { "name": "quant-ph9912073.extracted_bib", "string": "{dirac P. A. M. Dirac, The Principles of Quantum Mechanics 4th ed. (Clarendon Press, 1958)." }, { "name": "quant-ph9912073.extracted_bib", "string": "{jjs J. J. Sakurai, Modern Quantum Mechanics Rev. ed. (Addison-Wesley, 1994)." }, { "name": "quant-ph9912073.extracted_bib", "string": "{sk T. Shimbori and T. Kobayashi, Complex Eigenvalues of the Parabolic Potential Barrier and Gel'fand Triplet, math-ph/9910009." }, { "name": "quant-ph9912073.extracted_bib", "string": "{bogolubov N. N. Bogolubov, A. A. Logunov and I. T. Todorov, Introduction to Axiomatic Quantum Field Theory (W. A. Benjamin, 1975)." }, { "name": "quant-ph9912073.extracted_bib", "string": "{bohm A. Bohm and M. Gadella, Dirac Kets, Gamow Vectors and Gel'fand Triplets (Lecture Notes in Physics, Vol.~348, Springer, 1989)." }, { "name": "quant-ph9912073.extracted_bib", "string": "{ashi Y. Takahashi, Quantum Field Theory for Condensed Matter Physicists, Vol.~1 (Baifukan, 1974)." } ]
quant-ph9912074	Noised based Cipher system.	[ { "author": "Arindam Mitra" }, { "author": "V.I.P Enclave" }, { "author": "M - 403" }, { "author": "Calcutta- 700059. India." } ]	A computationally secure noised based cipher system is proposed. The advantage of this cipher system is that it operates above noise level. Therefore computationally secure communication can be done when error correction code fails. Another feature of this system is that minimum number of exhaustive key search can be made fixed.	[ { "name": "quant-ph9912074.tex", "string": "\n\\documentstyle{article}\n%\\textheight 25 true cm\n\\textwidth 15 true cm\n%\\topmargin -1.5 cm\n\\begin{document}\n\\title{Noised based Cipher system.} \\author{Arindam Mitra\n\\\\V.I.P Enclave, M - 403, Calcutta- 700059. India.\\\\}\n\\date{}\n\\maketitle\n\\begin{abstract}\\bf\nA computationally secure noised based cipher system is proposed.\nThe advantage of this cipher system is that it \noperates above noise level. Therefore computationally \nsecure communication can be done \nwhen error correction code fails. Another feature of this system is that\n minimum number of exhaustive key \nsearch can be made fixed.\n \n \\end{abstract} \n\\newpage\n\\section{}\nAll the popular classical cipher systems [1-2] are computationally secure.\nAs computational security is not \n mathematically proven, it does not always satisfy all. Nevertheless\ncomputational secure systems are used when 'cover time' is not long.\nBut these cryptosystem can not be used if the classical \ncommunication system becomes highly corrupted which classical\nerror correcting code cannot correct. \nWe shall see, the solution of the above problem lies in the noise itself.\nManipulating noise a simple cipher system is presented.\nThe cipher system works on the same coding and decoding technique developed for quantum cryptography [3].\n\n\\paragraph{}In our coding decoding procedure, \ntwo different sequences of states represent two \nbit values. The information regarding the two sequences \nare shared between the legitimate users. \nSender repeatedly and randomly uses the two sequences to\nproduce an arbitrarily long string of bits (i.e. the key)\n and receiver recovers the bit values from the shared information.\n\n\n\n\\paragraph{}\nFirstly we shall describe the error introduction strategy \nin a simple classical system.\nThe basic idea behind this protocol is that a \nsequence of one type of element \nwill be corrupted in two different ways to represent two bit values. \nThis is equivalent\nto choosing two sequences for two bit values.\nSuppose the sequence is \\\\\n$ \\it S =\\left\\{1, \\,\\, 1, \\,\\,1, \\,\\,1, \\,\\,1, \\,\\,1, \\,\\,1, \\,\\,1, \\,\\,1, \\,\\,1, \\,\\,\n1, \\,\\,1, ........\\right\\}$. This sequence can be represented by \ntwo different sub sequences (denoted by $\\alpha$ and $\\beta$) of \nelements {\\it 1}.\n$ \\it S = \\left\\{\\alpha, \\,\\,\\beta, \\,\\,\\beta, \\,\\,\\alpha, \\,\\,\\beta, \\,\\,\n\\alpha, \\,\\,\\beta, \\,\\,\\beta, \\,\\,\\alpha, \\,\\,\\alpha, \\,\\, \\beta, \\,\\,\\alpha,.......\\right\\}$,\nwhere probability of $\\alpha$ position is same with the probability of \n$\\beta$ position ($p_{\\alpha}= p_{\\beta}$) in the sequence {\\it S}. \nInformation regarding \nthe sub sequences $\\alpha$ and $\\beta$ are shared between sender and receiver.\n Apart \nfrom environmental noise these sub sequences of\n$\\alpha$ and $\\beta$ can be further corrupted by two error-introducing systems X and Y.\nConsidering natural error, two different probabilistic error $p_{x}$ and \n$p_{y}$ can be fixed by X and Y. \nFor bit {\\bf 0}, sender introduces error $p_{x}$ and $p_{y}$ in the \nsub sequences \n$\\alpha$ and $\\beta$ respectively. Now sequence {\\it S} can be \ntermed as probabilistic sequence, $S^{p}$, where each element is denoted \nby its error probability. \n$S_{0}^{p} =\\left\\{ p_{x}, \\,\\, p_{y}, \\,\\, p_{y}, \\,\\,p_{x}, \\,\n\\,p_{y}, \\,\\,p_{x}, \\,\\,p_{y}, \\,\\,p_{y}, \\,\\,p_{x}, \\,\\,p_{x}, \\,\\,p_{y}, \\,\\,p_{x}, \n....\\right\\}$ Similarly, for bit {\\bf 1}, error $p_{x}$ and $p_{y}$ \nwill be introduced in the sub sequences $\\beta$\nand $\\alpha$ respectively, then $S^{p}$ will be \n$ S_{1}^{p} =\\left\\{ p_{y}, \\,\\, p_{x}, \\,\\,p_{x}, \\,\\,p_{y}, \n\\,\\,p_{x}, \\,\\,p_{y}, \\,\\,p_{x}, \\,\\,p_{x}, \\,\\,p_{y}, \\,\\,p_{y}, \\,\\,p_{x}, \\,\\,p_{y},\n.....\\right\\}$.\n\nThe key will be:\n\\begin{eqnarray} K = \\left(\\begin{array}{ccccccccccccc}\np_{x} & p_{y} & p_{y} & p_{x} & p_{y} & p_{x} & p_{y} &p_{y} & p_{x} & p_{x} & p_{y} & p_{x} & ....\\\\\np_{y} & p_{x} & p_{x} & p_{y} & p_{x} & p_{y} & p_{x} & p_{x} & p_{y} & p_{y} & p_{x} & p_{y} & ....\\\\\np_{y} & p_{x} & p_{x} & p_{y} & p_{x} & p_{y} & p_{x} & p_{x} & p_{y} & p_{y} & p_{x} & p_{y} & ....\\\\\np_{x} & p_{y} & p_{y} & p_{x} & p_{y} & p_{x} & p_{y} & p_{y} & p_{x} & p_{x} & p_{y} & p_{x} & ....\\\\\np_{x} & p_{y} & p_{y} & p_{x} & p_{y} & p_{x} & p_{y} & p_{y} & p_{x} & p_{x} & p_{y} & p_{x} & ....\\\\\np_{y} & p_{x} & p_{x} & p_{y} & p_{x} & p_{y} & p_{x} & p_{x} & p_{y} & p_{y} & p_{x} & p_{y} & ....\\\\\n. & . & . & . & . & . & . & . & . & . & . & . & ....\\\\ \n. & . & . & . & . & . & . & . & . & . & . & . & ....\\\\ \n. & . & . & . & . & . & . & . & . & . & . & . & .... \n\\end{array}\\right) \\equiv \\left(\\begin{array}{c} \n\\bf 0 \\\\ \\bf 1 \\\\ \\bf 1 \\\\ \\bf 0 \\\\ \\bf 0 \\\\ \\bf 1 \\\\. \\\\. \\\\. \n\\end{array}\\right)\\nonumber\\end{eqnarray}\n\n\\noindent\nAll rows (columns also) are having same error $p_{z}$, where $p_{z} \n=1/2 (p_{x} + p_{y}) $. The receiver can easily distinguish the bit values\n of the corrupted sequences as he knows \nthe sub sequences $\\alpha$ and $\\beta$ provided bits are statistically distinguishable.\nBut code breaker has \nto go through the exhaustive search as he/she does not know the sub sequences \n$\\alpha$ and $\\beta$. By exhaustive search code breaker\ncan find the two \nerror levels $p_{x}$ and $p_{y}$ for two different \nsub sequences of $\\alpha$ and $\\beta$ (statistical problem of\nexact search is ignored).\nOnce code breaker knows the position of $\\alpha$ and $\\beta$, \nhe/she can easily break the code.\nOf course\n code breaker can pursue the exhaustive\nsearch with marginal number of bits. This will effectively reduce the\nnumber of search. Using the concept of pseudo-bit (fake bit), \nwe shall now discuss how to\ndefeat the minimal exhaustive search. \n\n\n\\paragraph{}\nHere {\\it S} can be represented as a composition of three sub sequence of $\\alpha$, $\\beta$ and $\\gamma$. \\\\\n So $ S= \\left\\{\\gamma, \\,\\,\\gamma, \\,\\,\\gamma, \\,\\,\\gamma, \\,\\,\\gamma, \\,\\,\\alpha, \\,\\,\\beta, \n\\,\\,\\gamma, \\,\\,\\gamma, \\,\\,\\gamma, \\,\\,\\gamma, \\,\\,\\gamma, \\,\\,\\gamma, \\,\\,\\beta, \\,\\,\\alpha, \\,\\,\\gamma, ....\\right\\}$,\nwhere $p_{\\alpha} = p_{\\beta}$, but less than $p_{\\gamma}$, the probability\nof $\\gamma$ position in the sequence of {\\it S}. \nSimilarly sender and receiver share the information of the \nsub sequences $\\alpha$, \n$\\beta$ and $\\gamma$. Apart from the devices X and\nY, we need another error introducing device Y which can produce error $p_{z}$.\nNow error $p_{z}$, $p_{x}$ and $p_{y}$ will be introduced in the sub sequences \n $\\gamma$, $\\alpha$ and $\\beta$ respectively to represent bit {\\bf 0}\nand error $p_{z}$, $p_{y}$ and $p_{x}$ will be introduced in the \nsub sequences $\\gamma$, $\\beta$ and $\\alpha$ respectively to represent bit {\\bf 1}, where\n$p_{z} =1/2 (p_{x} + p_{y}) $. \nThen the extended key is: \n\\begin{eqnarray} K^{E}= \\left(\\begin{array}{ccccccccccccccccc}\np_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{x} & p_{y} & p_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{y} & p_{x} & p_{z} & p_{z} & ....\\\\\np_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{y} & p_{x} & p_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{x} & p_{y} & p_{z} & p_{z} & ....\\\\\np_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{y} & p_{x} & p_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{x} & p_{y} & p_{z} & p_{z} & ....\\\\\np_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{x} & p_{y} & p_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{y} & p_{x} & p_{z} & p_{z} & ....\\\\\np_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{x} & p_{y} & p_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{y} & p_{x} & p_{z} & p_{z} & ....\\\\\np_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{y} & p_{x} & p_{z} & p_{z} & p_{z} & p_{z} & p_{z} & p_{x} & p_{y} & p_{z} & p_{z} & ....\\\\\n. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & ....\\\\ \n. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & ....\\\\ \n. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .... \n\\end{array}\\right) \\equiv \\left(\\begin{array}{c} \n\\bf 0 \\\\ \\bf 1 \\\\ \\bf 1 \\\\ \\bf 0 \\\\ \\bf 0 \\\\ \\bf 1 \\\\. \\\\. \\\\. \\end{array} \\right)\\nonumber\\end{eqnarray}\n\n\\noindent\nAgain the probability of error in columns (also in rows) is same \nwhich is $p_{z}$. \n Due to the presence of pseudo-elements (position of $\\gamma $), \ncode breaker's marginal statistics will have to be increased.\nIncreasing the number of pseudo-elements, we can force code breaker\nto pursue more lengthy exhaustive search.\n As $\\gamma$ sub sequence is a pseudo sequence, receiver can only measure\non $\\alpha$ and $\\beta$\nsub sequences to reveal the bit values.\n\n\\paragraph{}\nIt is trivial to mention that\nsingle sequence $S$ can be a binary sequence. Then a \nsequence of binary sources\nwill be corrupted in two different ways to produce two values.\n The error introduction strategy is also \napplicable for quantum source. But measured error \nin quantum states\nis not independent to measurements - it is a function measurement. \n \n\\paragraph{}\nLet us assume $p_{z} = 1/2 $, $p_{x} = 1/4 $ \nand $p_{y} = 3/4$ and the numbers of $\\gamma$, $\\alpha$ and $\\beta$ are\n$10^{4}$, $10^{2}$ and $10^{2}$ respectively i.e. $p_{\\gamma} : p_{\\alpha} \n: p_{\\beta} = 100:1:1$. \nSo code breaker's minimum statistics for exhaustive \nkey search should \nbe greater than 100. \nTherefore code breaker has to undertake more than $2^{100}$ operations. \n The practical advantage of this system\nis that secure message can be directly transmitted.\nAs noisy apparatus is needed, secure \ncommunications might be economical. \\\\ \n\n\n\\noindent\n{\\bf Note added:}\nUsing the concept of \"pseudo-bit\" and without using noise,\na simple unbreakable code\nhas recently been proposed [4]. \n \n\n\\small \\begin{thebibliography}{99} \n\n\n\\bibitem{be82}Beker, J. and Piper, F., 1982, Cipher systems: the protection\nof communications (London: Northwood publications).\n\n\\bibitem{hell79}Hellman, E. M. The mathematics of public-key cryptography.\n{\\em Sci. Amer.} August, 1979.\n\n\\bibitem{ari98}Mitra. A, quant-ph/9812087.\n\n\n\\bibitem{ari98}Mitra. A, physics/0008042.\n\n\n\n\\end{thebibliography}\n\n\n\\newpage\n\n \\end{document}\n" } ]	[ { "name": "quant-ph9912074.extracted_bib", "string": "{be82Beker, J. and Piper, F., 1982, Cipher systems: the protection of communications (London: Northwood publications)." }, { "name": "quant-ph9912074.extracted_bib", "string": "{hell79Hellman, E. M. The mathematics of public-key cryptography. {\\em Sci. Amer. August, 1979." }, { "name": "quant-ph9912074.extracted_bib", "string": "{ari98Mitra. A, quant-ph/9812087." }, { "name": "quant-ph9912074.extracted_bib", "string": "{ari98Mitra. A, physics/0008042." } ]
quant-ph9912075	Consistent histories and relativistic invariance in the modal interpretation of quantum mechanics	[ { "author": "Dennis Dieks" }, { "author": "Institute for the History and Foundations of Science" }, { "author": "Utrecht University" }, { "author": "P.O.Box 80.000" }, { "author": "3508 TA Utrecht" }, { "author": "The Netherlands" } ]	Modal interpretations of quantum mechanics assign definite properties to physical systems and specify single-time joint probabilities of these properties. We show that a natural extension, applying to properties at several times, can be given if a decoherence condition is satisfied. This extension defines ``histories'' of modal properties. We suggest a modification of the modal interpretation, that offers prospects of a more general applicability of the histories concept. Finally, we sketch a proposal to apply the procedure for finding histories and a many-times probability distribution to the context of algebraic quantum field theory. We show that this leads to results that are relativistically invariant.	[ { "name": "quant-ph9912075.tex", "string": "\n\\documentstyle[12pt,fleqn]{article}\n\\renewcommand{\\baselinestretch}{1}\n\\setlength{\\topmargin}{-7mm}\n\\setlength{\\textwidth}{16cm}\n\\setlength{\\textheight}{22cm}\n\\setlength{\\oddsidemargin}{-2mm}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\newcommand{\\sectiona}[1]{\\setcounter{equation}{0}\\section{#1}\n}\n\\newcommand{\\unit}{{\\sf 1 \\! I} \\mkern-6mu\n\\rule[1.55ex]{.5ex}{.08ex}}\n\\newcommand{\\nset}{{\\rm I \\! N}}\n\\newcommand{\\zset}{{\\rm Z \\!\\! Z}}\n\\newcommand{\\rset}{{\\rm I \\! R}}\n\\newcommand{\\qset}{{\\rm Q} \\mkern-9.5mu\n\\rule[-0.01ex]{.06ex}{1.5ex} \\;\n\\;}\n\\newcommand{\\cset}{{\\rm C} \\mkern-8mu\n\\rule[-0.01ex]{.06ex}{1.5ex} \\;\n\\;}\n\\newcommand{\\lh}{\\: [ \\! [ \\:}\n\\newcommand{\\rh}{\\: ] \\! ] \\:}\n\\newcommand{\\A}{\\alpha}\n\\newcommand{\\B}{\\beta}\n\\newcommand{\\C}{\\gamma}\n\\newcommand{\\D}{\\delta}\n\\newcommand{\\E}{\\epsilon}\n\\newcommand{\\F}{\\zeta}\n\\newcommand{\\G}{\\theta}\n\\newcommand{\\I}{\\iota}\n\\newcommand{\\J}{\\kappa}\n\\newcommand{\\K}{\\lambda}\n\\newcommand{\\M}{\\mu}\n\\newcommand{\\N}{\\nu}\n\\newcommand{\\X}{\\xi}\n\\newcommand{\\tr}{\\mbox{Tr}}\n\\newcommand{\\be}{\\begin{eqnarray}}\n\\newcommand{\\ee}{\\end{eqnarray}}\n\\newcommand{\\nn}{\\nonumber}\n\\newcommand{\\ket}[1]{\| #1 \\rangle}\n\\newcommand{\\bra}[1]{\\langle #1 \|}\n\\newcommand{\\proj}[1]{\| #1 \\rangle \\langle #1 \|}\n\\newcommand{\\inpr}[2]{\\langle #1 \| #2 \\rangle}\n\\begin{document}\n\n\\title{Consistent histories and relativistic invariance in the modal interpretation of\nquantum mechanics}\n\n\\author{Dennis Dieks\\\\ Institute for the History and Foundations of Science\\\\\nUtrecht University, P.O.Box 80.000 \\\\ 3508 TA Utrecht, The\nNetherlands}\n\\date{}\n\\maketitle\n\n\\begin{abstract}\nModal interpretations of quantum mechanics assign definite properties to physical\nsystems and specify single-time joint probabilities of these properties.\nWe show that a natural extension,\napplying to properties at several times, can be given if a decoherence condition is\nsatisfied. This extension defines ``histories'' of modal properties. We\nsuggest a modification of the modal interpretation, that offers prospects of\na more general applicability of the histories concept.\nFinally, we sketch a proposal to apply the procedure for finding histories and a\nmany-times\nprobability distribution to the context of algebraic quantum field\ntheory. We show that this leads to results that are relativistically invariant.\n\\end{abstract}\n\\vspace{3cm}\n\\noindent\nPACS: 03.65 \\\\\nKey words: modal interpretation, decoherence, consistent histories, relativistic\ninvariance\n\\newpage\n\n\\sectiona{Introduction}\\label{sect1}\nModal interpretations of quantum mechanics \\cite{vanfra,dieks1,healey,dieks2} \ninterpret the mathematical formalism of quantum theory in terms of\nproperties possessed by\nphysical systems---in contradistinction to interpretations that take the formalism\nas an instrument to calculate macroscopic measurement outcomes and their\nprobabilities.\nBy the statement that a system possesses a property we mean that some quantum\nmechanical observable pertaining to the system takes on a definite value. However,\nit is impossible to give all observables definite values while preserving the quantum\nmechanical relations between them (as shown by the Kochen and Specker no-go\ntheorem).\nModal\ninterpretations therefore specify a {\\em subset} of all observables; only the\nobservables in this subset are assigned definite values. It is characteristic of the\nmodal approach that this is done in a state-dependent way: the quantum mechanical\nstate of the system contains all information needed to determine the set of\ndefinite-valued observables. \nThe precise prescription for finding this set makes use of the Schmidt \nbi-orthogonal\ndecomposition of the composite state of a system plus its environment; or, more\ngenerally, of the spectral\ndecomposition\nof\nthe density operator describing a single system. \n\nMost work on the modal interpretation up to now has concentrated on properties\nat a single time---with the exception of the pioneering studies of\nBacciagaluppi and Dickson \\cite{baccia} and Vermaas \\cite{vermaas2}.\nNo proposal for a joint probability distribution of physical\nproperties at several times has been generally accepted, and it has been queried\nwhether a\nnatural joint distribution can exist at all \\cite{kent}. Moreover, the probability\ndistributions that have been proposed \\cite{baccia} suffer from the problem that\nthey are not Lorentz-invariant; and it has been argued that it is a general feature of\nmodal dynamical schemes that they single out a preferred frame of\nreference \\cite{dickson}.\n\nIn this Letter we point out that in cases in which a decoherence condition is satisfied\nthere is a natural and obvious\ncandidate for a joint many-times probability distribution (in spite of the arguments to\nthe\ncontrary in \\cite{kent}). We attempt to extend the applicability of this joint \nprobability distribution by suggesting a modification of the modal interpretation\nscheme. Finally, we consider the application of the ``histories'' idea to quantum field\ntheory (assuming decoherence again). Under certain conditions, the implementation of the idea here leads to\na Lorentz-invariant joint probability distribution of properties associated with very small\nregions (approximating\nspace-time points). \n\n\\sectiona{The modal scheme} \n\nLet $\\A$ be\nour system and let $\\B$\nrepresent its total environment (the rest of the universe).\nLet $\\A\\&\\B$ be\nrepresented by\n$\\ket{\\psi^{\\A\\B}} \\in {\\cal H}^{\\A} \\otimes {\\cal H}^{\\B}$. \nThe\nbi-orthonormal decomposition of $\\ket{\\psi^{\\A\\B}}$,\n\\be\n\\ket{\\psi^{\\A\\B}} & = & \\sum_{i} c_{i} \\: \\ket{\\psi^{\\A}_{i}}\n\\:\n\\ket{\\psi^{\\B}_{i}} \\label{eq1} \\;\\;\\; ,\n\\ee\nwith $\\inpr{\\psi^{\\A}_{i}}{\\psi^{\\A}_{j}} =\n\\inpr{\\psi^{\\B}_{i}}{\\psi^{\\B}_{j}} = \\delta_{ij}$, generates\na set of projectors operating on ${\\cal H}^{\\A}$: \n$\\{\\proj{\\psi^{\\A}_{i}} \\}_{i}$.\nIf there is no degeneracy among the numbers $\\{ \| c_{i} \|^{2}\n\\}$ this is a uniquely determined set of one-dimensional projectors.\nIf there is degeneracy, the projectors\nbelonging to one value of\n$\\{ \| c_{i} \|^{2} \\}$ can be added to form a\nmulti-dimensional projector; the thus generated new set of\nprojectors, including multi-dimensional ones, is again uniquely determined. These\nprojectors are the ones occurring in the spectral\ndecomposition of\nthe reduced density operator of $\\A$. \n\nThe modal interpretation assigns definite values to the\nsubset of all physical magnitudes that is generated by these projectors; i.e., the subset\nobtained by\nstarting with these projectors, and then including their continuous functions, real\nlinear combinations, and\nsymmetric and antisymmetric products \\cite{clifton}\n(the thus defined real, closed in norm, linear subspace of all observables constitutes\nthe set of ``well-defined'' or\n``applicable'' physical magnitudes, in Bohrian parlance). \n{\\em Which}\nvalue among the possible values of a definite magnitude\nis actually realised is not fixed by the interpretation. For\neach possible value a probability\nis specified: the\nprobability that the magnitude represented by\n$\\proj{\\psi^{\\A}_{i}}$ has\nthe value\n$1$ is given by\n$\|c_{i}\|^{2}$. In the case of degeneracy it is stipulated that\nthe magnitude\nrepresented by $\\sum_{i\\in I_{l}}\n\\proj{\\psi^{\\A}_{i}}$ has value $1$ with probability\n$\\sum_{i\\in I_{l}} \|c_{i}\|^2$ ($I_{l}$ is the index-set\ncontaining indices $j$, $k$ such that $\|c_{j}\|^{2} =\n\|c_{k}\|^{2}$).\n\nThe observation that the definite-valued projections occur in the spectral\ndecomposition of $\\A$'s density operator gives\nrise\nto a generalisation of the above scheme that is also applicable to the case in\nwhich the total system $\\A\\&\\B$\nis not represented by a pure\nstate: find $\\A$'s density operator by partial tracing from the total density operator,\ndetermine its spectral resolution\nand construct the set of definite-valued observables from the projection operators in\nthis spectral resolution\n\\cite{verm}.\n\nThe above recipe for assigning properties is meant to apply to each physical system in\na non-overlapping collection of\nsystems that together make up the total universe \\cite{baccia,dieks3}. It is easy to\nwrite down a satisfactory {\\em joint}\nprobability distribution for the properties of such a collection (or a subset of it):\n\\begin{equation}\nProb(P^{\\A}_{i}, P^{\\B}_{j},...., P^{\\G}_{k},...., P^{\\X}_{l})\n= \\langle \\Psi \| P^{\\A}_{i}.\nP^{\\B}_{j}.....P^{\\G}_{k}.....P^{\\X}_{l} \| \\Psi\n\\rangle ,\\label{prob}\n\\end{equation}\nwhere the left-hand side represents the joint probability for the\nprojectors occurring in the argument of taking the value $1$, and where $\\Psi$ is the\nstate of the total system consisting of\n$\\A$, $\\B$, $\\G$, etc. \\cite{verm}.\nIt is important for the consistency of this probability\nascription that the projection operators occurring in the\nformula all commute (which they do, since they\noperate in different non-overlapping Hilbert spaces).\n\nHowever, a full-fledged interpretation of quantum mechanics in terms of properties\nof physical systems should not only\nspecify the\nclass of definite-valued observables at each instant, and their probability distribution,\nbut should also make it clear\nwhat the probability is that a value present at one instant of time goes over\ninto a given possible value at another time; or, equivalently, what the joint probability\nis of\nvalues of definite-valued observables\nat several times. It turns out that it is not easy to find a compelling and natural \nsolution to this problem for the general case\n\\cite{baccia,vermaas2}. But in the special, though physically important, case that a\ndecoherence\ncondition is satisfied there\n{\\em is} such a natural joint probability for histories of properties, as we will now\ndiscuss.\n \n\\sectiona{Consistent histories of modal properties}\n\nThere is a natural analogue of expression (\\ref{prob}) for the case of Heisenberg\nprojection operators pertaining to\ndifferent instants of time:\n\\begin{eqnarray}\n\\lefteqn{Prob(P_{i}(t_{1}), P_{j}(t_{2}),.... P_{l}(t_{n}))\n=} \\nonumber \\\\ & & \\langle \\Psi \| P_{i}(t_{1}).\nP_{j}(t_{2}).....P_{l}(t_{n}) . P_{l}(t_{n})\n. .... P_{j}(t_{2}) . P_{i}(t_{1}) \| \\Psi \\rangle .\\label{sevtimeprob}\n\\end{eqnarray}\nThis expression is in accordance with the standard, ``Copenhagen'', prescription for\ncalculating the joint probability\nof outcomes of consecutive measurements. It agrees also with the joint distribution\nassigned to ``consistent histories'' in the consistent histories approach to the\ninterpretation of quantum mechanics \\cite{grif}. However, it should be noted that the\nprojection\noperators in (\\ref{sevtimeprob}),\npertaining to different times as they do, need not commute. As a result,\n(\\ref{sevtimeprob})\ndoes not automatically yield a consistent probability distribution. For this reason it is\nan\nessential\npart of the consistent histories approach to impose the\nfollowing decoherence condition, in\norder to guarantee that expression (\\ref{sevtimeprob}) is an ordinary\nKolmogorov probability:\n\\begin{eqnarray}\n\\lefteqn{\\langle \\Psi \| P_{i}(t_{1}).\nP_{j}(t_{2}).....P_{l}(t_{n}) .\nP_{l^{\\prime}}(t_{n}) ......\nP_{j^{\\prime}}(t_{2}) . P_{i^{\\prime}}(t_{1}) \| \\Psi \\rangle =0}\n\\nonumber \\\\ & & i \\neq\ni^{\\prime} \\vee j \\neq j^{\\prime} \\vee\n....\\vee l \\neq l^{\\prime}.\\label{consist}\n\\end{eqnarray}\n\nIn the consistent histories approach all sequences of properties which satisfy\nthe decoherence condition (\\ref{consist}) are considered. A problem of this approach\nis that the decoherence condition leaves room for many, mutually incompatible,\nfamilies of consistent properties, so that the set of definite-valued properties is not\nfully determined. By contrast, the modal interpretation provides an unequivocal rule\nto fix the definite-valued properties. It seems therefore worth-while to investigate\nwhether the two approaches can be combined by using the above probability\ndistribution for modal histories. \nHowever, in Ref.\\ \\cite{kent} it is argued that the projection operators singled out \nas definite-valued by the\nmodal interpretation will not\nsatisfy the decoherence condition (except in the very special and\nphysically\nunrealistic circumstance in which the system's properties evolve deterministically). If\nvalid, this argument would\nmake the probabilistic resources of the consistent histories approach unavailable\nto the modal interpretation. \n\nThe essential ingredient of the argument is the following (see the text\naccompanying Eq.\\ (2.6) in Ref.\\ \\cite{kent}). Consider the decoherence condition\npertaining to two instants, $t_{1}$ and $t_{2}$, and let $P_{l}(t_{2}) =\n\\proj{\\psi_{l}(t_{2})}$.\nWe then have\n$\\langle \\Psi \| P_{i}(t_{1}) \| \\psi_{l}(t_{2}) \\rangle \\langle \\psi_{l}(t_{2}) \|\nP_{i^{\\prime}}(t_{1}) \| \\Psi \\rangle = 0 $,\nif $i \\neq i^{\\prime}$, for all values of $l$. That means, the argument goes, that\neither\n$ \\langle \\Psi \| P_{i}(t_{1}) \| \\psi_{l}(t_{2}) \\rangle = 0 $ or\n$ \\langle \\psi_{l}(t_{2}) \| P_{i^{\\prime}}(t_{1}) \| \\Psi \\rangle = 0 $. From this it\nfollows that\n$\\ket{\\psi_{l}(t_{2})}$\nis orthogonal to all but one of the states $\\ket{ P_{i}(t_{1}) \\Psi}$. That would\nimply a deterministic evolution of\nproperties, and if the projectors $P_{i}(t_{1})$ are one-dimensional the properties at\ntime\n$t_{2}$ would even have to be the same as those at time $t_{1}$.\n\nThe questionable premise in this argument is \nthe presupposition that\n$ \\langle \\Psi \| P_{i}(t_{1}) \| \\psi_{l}(t_{2}) \\rangle$ and\n$ \\langle \\psi_{l}(t_{2}) \| P_{i^{\\prime}}(t_{1}) \| \\Psi \\rangle$ are numbers,\ninstead\nof a bra and a ket, respectively.\nMaking the assumption that these expressions represent numbers is equivalent to\nassuming that\n$\\ket{\\Psi}$ is an element of the Hilbert space\nof the system under consideration. This assumption does not fit in with the\nmodal approach: the modal property\nascription, as explained above, uses for $\\ket{\\Psi}$ the state of the\ncombination ``system and rest of the universe''. As we shall show in a moment,\nthe fact that the presence of an environment has always to be taken into\naccount not only invalidates the above argument, but also makes it natural and easy\nto incorporate the idea of\ndecoherence in the modal scheme so that condition (\\ref{consist})\nis satisfied. As a consequence, Eq.\\ (\\ref{sevtimeprob}) yields\na\nconsistent\njoint multi-times\nprobability distribution for modal properties in the case in which this decoherence\ncondition is fulfilled.\n\nThe notion of decoherence to be used is the following. It is a general feature of the\nmodal interpretation that if a system acquires a\ncertain property, this happens by virtue of its interaction with the environment,\nas expressed in Eq.\\ (\\ref{eq1}). As can be seen from this equation, in this process\nthe system's\nproperty\nbecomes correlated with a property of the environment. Decoherence is now defined\nto\nimply the irreversibility of this process of correlation formation: the rest of the\nuniverse retains a trace of the\nsystem's property, also at later times when the properties of the system itself may\nhave\nchanged.\nIn other words, the rest of the universe acts as a memory of the properties the system\nhas had; and decoherence guarantees that this memory remains intact. For the state\n$\\ket{\\Psi}$ this means\nthat in the Schr\\\"{o}dinger picture it can be written in the following form: \n\\be\n\\ket{\\Psi(t_{n})} & = & \\sum_{i,j,...,l} c_{i,j,...,l} \\: \\ket{\\psi_{i,j,...,l}}\n\\:\n\\ket{\\Phi_{i,j,...,l}} \\label{eqdeco} \\;\\;\\; ,\n\\ee\nwhere $\\ket{\\psi_{i,j,...,l}}$ is defined in the Hilbert space of the system,\n$\\ket{\\Phi_{i,j,...,l}}$\nin the Hilbert space pertaining to the rest of the universe, and where\n$\\langle \\Phi_{i,j,...,l} \| \\Phi_{i^{\\prime},j^{\\prime},...,l^{\\prime}} \\rangle =\n\\delta_{i i^{\\prime} j j^{\\prime}...l l^{\\prime}}$. \nIn (\\ref{eqdeco}) $l$ refers to\nthe properties $P_{l}(t_{n})$, $j$ to the properties $P_{j}(t_{2})$, $i$ to the\nproperties\n$P_{i}(t_{1})$, and so on. \n\nThe physical picture that motivates a $\\ket{\\Psi}$ of this form is that\nthe final state results from\nconsecutive measurement-like interactions, each of which is responsible for generating\nnew properties. Suppose that in the first interaction\nwith the environment the properties $ \\proj{\\A_{i}}$ become definite: then the state\nobtains the form $\\sum_{i} c_{i} \\ket{\\A_{i}} \\ket{E_{i}}$, with $\\ket{E_{i}}$\nmutually\northogonal states of the environment. In a subsequent interaction, in which the\nproperties $ \\proj{\\B_{j}}$ become definite, and in which the environment\n``remembers'' the presence of the $\\ket{\\A_{i}}$, the state is transformed into \n$\\sum_{i,j} c_{i} \\inpr{\\B_{j}}{\\A_{i}} \\ket{\\B_{j}} \\ket{E_{i,j}}$, with mutually\northogonal environment states $\\ket{E_{i,j}}$. Continuation of this series of\ninteractions eventually leads to Eq.\\ (\\ref{eqdeco}), with in this case $\n\\ket{\\psi_{i,j,...,l}} =\n\\ket{\\psi_{l}}$ (see below for a generalisation). \n\nIf this picture of consecutive measurement-like interactions applies, it follows that\nin the Heisenberg picture we have\n$P_{l}(t_{n}) ......\nP_{j}(t_{2}) . P_{i}(t_{1}) \\ket{\\Psi} = c_{i,j,...,l} \\ket{\\psi_{i,j,...,l}}\n\\ket{\\Phi_{i,j,...,l}}$. Substituting this in the expression at the left-hand side of Eq.\\\n(\\ref{consist}), and making use of the orthogonality properties of the states\n$\\ket{\\Phi_{i,j,...l}}$, we find immediately that the consistent histories\ndecoherence\ncondition (\\ref{consist}) is satisfied. As a result, expression (\\ref{sevtimeprob})\nyields a classical\nKolmogorov\nprobability distribution of the modal properties at several times.\n\n\\sectiona{A possible modification of the modal scheme}\n\nThe just-discussed decoherence scenario is based on the assumption that in\nconsecutive\ninteractions the initial state of the system is not relevant for the properties that are\ngenerated:\nwe wrote that $\\ket{\\A_{i}} \\ket{E_{i}}$ is transformed into \n$\\sum_{j} \\inpr{\\B_{j}}{\\A_{i}} \\ket{\\B_{j}} \\ket{E_{i,j}}$, with the same set\n$\\{\\ket{\\B_{j}}\\}$\nfor all values of $i$. In some physically important cases this is not \nunrealistic. The prime example is the case in which the object is subjected\nto consecutive measurements by devices designed to measure certain observables,\nregardless of the\nstate of the incoming system. A similar situation can occur in cases in which no\ninstruments\nconstructed by humans are present. For instance, interactions by means of a \npotential that depends only on the object observable $A$, with an environment with\nvery many degrees of freedom,\nwill tend to destroy the coherence between object states\nthat correspond to different $A$-values and can thus be regarded as\n(approximate) $A$-measurements, regardless of the object's initial state. Consecutive\nexposures to such environments\ncan be described by (\\ref{eqdeco}).\n\nBut in general we will have to consider the situation in which no such\nmeasurement-like interactions occur,\nand in which the type of the later interaction may depend \non the result of an earlier interaction. In this general case we\ncan represent what happens in an interaction by writing:\n\\be\n\\sum_{i} c_{i} \\ket{\\A_{i}} \\ket{E_{i}} \\longrightarrow\n\\sum_{i,j} c_{i,j} \\inpr{\\B_{i,j}}{\\A_{i}} \\ket{\\B_{i,j}} \\ket{E_{i,j}},\\label{evol}\n\\ee\nwith $\\inpr{\\B_{i,j}}{\\B_{i,j^{\\prime}}}= \\delta_{jj^{\\prime}}$. In this formula a \nbiorthogonal decomposition has been\nwritten down for each value of $i$ separately; in other words, we are looking at the\nvarious mutually orthogonal ``branches'', that result from an interaction, separately\nand look how each of them branches itself in a subsequent interaction. A series of\nconsecutive interactions described in this way again leads to\na total final state of the form (\\ref{eqdeco}). However, the biorthogonal\ndecomposition\napplied to the total right-hand side of (\\ref{evol}) would now in general\nnot yield the projectors\n$\\proj{\\B_{i,j}}$ as definite-valued observables. Therefore,\n$P_{j}(t_{2}) . P_{i}(t_{1}) \\ket{\\Psi}$, with $P(t_{1,2})$ the projectors\nthat {\\em are} singled out by the biorthogonal decomposition as\ndefinite-valued at $t_{1}$ and $t_{2}$, respectively, will not be equal to\n$ c_{i,j} \\inpr{\\B_{i,j}}{\\A_{i}} \\ket{\\B_{i,j}} \\ket{E_{i,j}}$ (the projectors\n$P_{j}(t_{2})$ will\ngenerally not commute with the projectors $\\proj{\\B_{i,j}}$). This has the\nconsequence that there is no reason to\nexpect that the decoherence condition (\\ref{consist}) will be satified. \n\nThis suggests a modification of the modal scheme, designed to guarantee that Eq.\\\n(\\ref{sevtimeprob}) remains\nvalid as the joint probability of possessed properties, even in the more general\nsituation just described.\nThe idea is to change the property assignment in such a way that the right-hand side\nof Eq.\\ (\\ref{evol})\ncomes to represent a situation in which the projectors $\\proj{\\B_{i,j}}$ {\\em are}\ndefinite-valued. Such a property ascription would be in accordance with the intuition\nthat the different branches are irrelevant to each other as long as no \nre-interference occurs. As in the many-worlds interpretation, in which we do not need\nto consider what happens in other worlds in order to describe what happens in our\nworld, it is proposed here that the ascription of properties, as a result of\ninteractions, should be done {\\em per branch}. \n \n\nTo this end, we may posit as an interpretational rule that in a composite \nstate of the form\n\\be\n\\sum_{i,j} c_{i,j} \\inpr{\\B_{i,j}}{\\A_{i}} \\ket{\\B_{i,j}} \\ket{E_{i,j}},\n\\label{newmodal}\n\\ee\nwith $\\inpr{E_{i,j}}{E_{i^{\\prime},j^{\\prime}}} = \\delta_{i i^{\\prime} j\nj^{\\prime}}$, and\n$\\inpr{\\B_{i,j}}{\\B_{i,j^{\\prime}}}= \\delta_{jj^{\\prime}}$, resulting from an\ninteraction of the form\n$\\sum_{i} c_{i} \\ket{\\A_{i}} \\ket{E_{i}} \\longrightarrow\n\\sum_{i,j} c_{i,j} \\inpr{\\B_{i,j}}{\\A_{i}} \\ket{\\B_{i,j}} \\ket{E_{i,j}}$, \nthe projectors $\\proj{E_{i,j}}$ represent properties of the environment, \nand that \nthe system has properties $\\proj{\\B_{i,j}}$, in one-to-one correlation to these\nenvironmental properties. In terms of the biorthogonal decomposition, the new\nproposal says that\nthe system's properties are determined by the separate\ndecomposition that can be written down for each value of $i$: the \nterms of a biorthogonal decomposition that do not recombine (interfere) in\nsubsequent interactions are treated as individual branches, isolated from the other\nterms. The various branches are assigned definite-valued observables through their\nown individual\nbiorthogonal decomposition.\n\nIt should be noted that the state $\\ket{\\Psi}$ of the total system, together with the\ntotal Hamiltonian, uniquely determine the properties ascribed in this scheme. This is\nbecause the Hamiltonian governs the evolution, so that the initial state---before\ninteractions started---is fixed by $\\ket{\\Psi}$ and the Hamiltonian. Further, the\nbranching that occurs in the subsequent interactions is fully determined by writing\ndown the biorthogonal decomposition (per branch) after each interaction.\n \nAccording to this way of interpreting the quantum mechanical state, the\nproperties $P_{i}(t_{n})$ that\nare assigned satisfy Eq.\\ (\\ref{consist}) and make Eq.\\ (\\ref{sevtimeprob}) a\nconsistent probability distribution (if there is decoherence, that is, as long as there is\nno interference of the several branches).\nThis brings the modal scheme closer to the consistent histories\ninterpretation, in which exactly those histories are considered for which\nEq.\\ (\\ref{sevtimeprob}) yields a consistent probability distribution. There would still\nbe a\ndistinction, though: the requirement that\nEq.\\ (\\ref{sevtimeprob}) be a consistent probability is by itself not enough\nto determine what properties are definite at the various instants---in other words,\nwhat\nthe family of consistent histories is. There can be many mutually inconsistent\nbut partially overlapping families of consistent histories \\cite{dowker}). This\nconstitutes\na problem in the\nconsistent histories approach, because the consistency condition is the only constraint\none\nhas in that approach. By contrast, in our suggested modified modal scheme we retain\nthe distinctive\nmodal feature that\nthe definite-valued observables are uniquely defined by a fixed rule. As a\nconsequence, there is only one family\nof consistent\nhistories to consider.\n\nWe do not further discuss and elaborate this suggested modification of the modal scheme\nhere, but instead turn to the question of whether the notion of histories of definite\nproperties,\nwith well-defined probabilities, can also be applied outside of the context\nof non-relativistic quantum theory, to relativistic quantum field theory.\nIt seems clear that any interpretation of quantum theory should at least have the prospect of being so applicable in\norder to be taken seriously. \n\n\n\\sectiona{Application to relativistic quantum field theory}\n\nThe traditional interpretational problems of quantum mechanics exist no less in\nquantum field theory; and as we will see shortly there are also additional problems.\nBecause quantum field theory occupies a central place in the present physical description of the world,\nany interpretation of quantum theory should at least offer prospects of leading to sensible\nresults if applied to this new context. We will therefore now briefly outline a method to implement the modal ideas\nof the foregoing to quantum field theory.\n\nAs before,\nthe central issue is that it is not obvious that the theory is about objective physical\nstates of affairs, even\nin circumstances in which no macroscopic measurements are being made. The fields\nin quantum field theory do not attach values\nof physical magnitudes to space-time points. Rather, they are fields of operators, with\na standard interpretation\nin terms of macroscopic measurement results. \nIn accordance with what we have said about the interpretation of the\nSchr\\\"{o}dinger-Heisenberg theory we\nwould like to give another meaning to the formalism, namely in terms of\nphysical systems that possess certain properties.\nWe will take as our framework the formalism of local algebraic quantum field\ntheory as explained\nin \\cite{haag}, because of its generality. In this framework, a C$^{\\ast}$-algebra of observables is associated\nto each open region\nof Minkowski space-time. What we would like to\ndo is to provide\nan interpretation in which not only operators, but also {\\em\nproperties} are assigned to\nspace-time regions. That is, we would like at least some of the observables to have\ndefinite values. This would lead\nto a picture in which it is possible to speak of objective {\\em\nevents} (if some physical\nmagnitude takes on a definite value in a certain spacetime region, this constitutes the\nevent that this magnitude has that value there and then).\n\nIf the open space-time regions could be regarded as physical systems, each one of\nthem described in a factor space of a total Hilbert space (this total Hilbert space\nwould then be the tensor product of the spaces belonging to non-overlapping subsystems),\nan immediate generalisation\nof the modal scheme would be possible. However, algebraic quantum field theory is much\nless amicable to the notion of a localised physical system than might be expected. The\nlocal algebras are of type III, and this implies that they cannot be represented as\nalgebras of bounded observables on a Hilbert space (such algebras are of type I). \nWe can therefore not take the open space-time regions and their algebras as\nfundamental, if we want an interpretation in terms of (more or less) localised systems whose properties\nwould specify an event. Such an interpretation is highly desirable, however, at least in the limiting situation in which\nclassical concepts become applicable: it should be possible, in this limiting situation, to speak of field values\nin small space-time regions.\n\nOne possible way out is to use the algebras of type I that ``lie\nbetween two local algebras''. That such type-I algebras exist is assumed in the\npostulate of the ``split property'' (\\cite{haag}, Ch.\\ V.5, \\cite{summers}). We accept this postulate and\nfocus on the algebras of type I lying between two concentric standard ``diamond'' regions\nwith radii $r$ and $r+\\epsilon$,\nrespectively, with $r$ and $\\epsilon$ small numbers. Two of such type-I algebras, defined\nwith respect to two double diamonds associated with regions \nthat have space-like separation, are independent in a strong sense: the total algebra generated by them\ncan be represented by the tensor product of the two algebras, defined on the tensor product of two representative\nHilbert spaces $\\cal{H}_{A}$ and $\\cal{H}_{B}$ of the two individual algebras separately: $\\cal{H}_{A \\& B}=\\cal{H}_{A}\n\\otimes \\cal{H}_{B}$ \\cite{summers}.\n\n\nIn this way we may\njustify the approximate validity of the notion of a small space-time region\nas a physical subsystem represented in\na factor space of the total Hilbert space. Of course, there is arbitrariness here, for example in choosing the positions\nof the double\ndiamonds in the manifold and in fixing the\nvalues of $r$ and $\\epsilon$; it is not to be expected that a sensible algebra will result if $r\\rightarrow 0$.\nThis reflects the fact that the structure of quantum field theory, despite\nfirst appearances, by itself does not provide a natural arena for a space-time picture in which physical magnitudes attach to\nsmall space-time regions. We expect therefore that a space-time interpretation,\nlike the one we are trying to construct, will not have a fundamental status but can only be employed\nin a classical limiting situation in which classical field and particle concepts become approximately\napplicable. In other words, we do not assume that actually field values {\\em are}\nassociated with space-time points, and that our description provides an approximation to that real state of affairs.\nRather, we think that in\nthe general case the ordinary space-time picture will not be possible and that the classical field and particle\nconcepts can only\nbe applied in a limiting situation; accordingly, in the general case $x$ and $t$ will be parameters that do not possess their\nusual space-time meaning. The approximate applicability of classical concepts in a limiting situation\nis perhaps connected to the presence of \ndecoherence mechanisms that\ndecouple certain $x$, $t$ regions from each other (see also below; we will need such an assumption of\ndecoherence anyway, to arrive at consistent joint probabilities). \n\nWe will now adopt the modal ideas of section 3 to assign values to the\nobservables in the just-defined type-I algebras, loosely associated with small space-time regions.\nWe will therefore assume that there is\ndecoherence in the\nsense discussed in section 3. This assumption is natural in the context of local\nquantum physics \\cite{schroer}, because\nthe physical systems in local\nquantum physics\nautomatically have an infinite environment to interact\nwith, so that decoherence and irreversibility can easily occur.\n\nThe core idea of the modal interpretation is to select a subset of definite-valued\nobservables from the algebra\nof all observables by means of an objective,\nfixed, rule. This was implemented within the framework of non-relativistic\nquantum mechanics via the selection principle\nbased on the biorthogonal decomposition of the total state or, more generally, the\nspectral resolution\nof the system's density operator.\nHere, we will analogously consider the projectors occurring in\nthe spectral resolution of the\ndensity operator (defined in a factor space) representing the partial state defined on the type-I algebra of\nobservables\nattached to the very small ``split-regions'' that approximate space-time points. At each instant, we will consider\na collection of such regions that have space-like separation with respect to each other, and can\nbe described by means of the tensor product of the individual factor spaces.\nThe selected projectors provide us with a\nbase set of definite-valued\nquantities. As before, the complete collection of definite-valued observables can be\nconstructed\nfrom this base set by closing the set under the operations of taking continuous\nfunctions, real linear combinations,\nand symmetric and antisymmetric products \\cite{clifton}. The\nprobability of projector\n$P_{l}$ having the value $1$ is $\\langle \\Psi \| P_{l} \| \\Psi \\rangle$. Subdividing (approximately) the whole\nof Minkowski space-time into a collection of non-overlapping point-like regions, that have\nspace-like separation on each simultaneity hyperplane, and\napplying the above prescription to the associated algebras, we achieve the picture\naimed\nat: to each (approximate) space-time point belong\ndefinite values of\nsome physical magnitudes, and this constitutes an event at the position and time in\nquestion.\n\nIn order to complete this picture we should specify the joint probability of events\ntaking place at different space-time\n``points''. It is natural to consider, for this purpose, a generalisation of expression\n(\\ref{sevtimeprob}). The first problem\nencountered in generalizing this expression to the relativistic context is that we no\nlonger have absolute time available\nto order the sequence $P_{i}(t_{1})$, $P_{j}(t_{2})$, ...., $P_{l}(t_{n})$. In\nMinkovski space-time we only have the\npartial ordering $y < x$ (i.e., $y$ is in the causal past of $x$) as an objective relation\nbetween space-time points. However, we can still impose a linear ordering on the\nspace-time points in any region in\nspace-time by considering equivalence classes of points which all have space-like\nseparation with respect to each other---for instance, points whose centres are on the same simultaneity hyperplane.\nOf course, there are infinitely many\nways of subdividing the region into such\nspace-like collections of points. It will have to be shown that the joint probability\ndistribution that we are going to\nconstruct is independent of the particular subdivision that is chosen.\n\nTake one particular linear time ordering of the points in a closed region of\nMinkowski space-time, for instance one generated\nby a set of parallel simultaneity hyperplanes (i.e.\\ hyperplanes that are all Minkowski-orthogonal\nto one given time-like worldline).\nLet the time parameter $t$ label very thin slices of space-time (approximating\nhyperplanes) in which ``points'' of the kind introduced above, with mutual space-like\nseparation, are located. We can now write\ndown a joint probability distribution\nfor the properties, in exactly the same form as in\nEq.\\ (\\ref{sevtimeprob}): \n\\begin{eqnarray}\n\\lefteqn{Prob(P^{\\ast}_{i}(t_{1}), P^{\\ast}_{j}(t_{2}),.... \nP^{\\ast}_{l}(t_{n}))=} \n\\nonumber \\\\ & & \\langle \\Psi \| P^{\\ast}_{i}(t_{1}).\nP^{\\ast}_{j}(t_{2}).....P^{\\ast}_{l}(t_{n}) . P^{\\ast}_{l}(t_{n})\n. .... P^{\\ast}_{j}(t_{2}) . P^{\\ast}_{i}(t_{1}) \| \\Psi \\rangle.\n\\label{sevtimeprob1}\n\\end{eqnarray} \nIn this formula the projector $P^{\\ast}_{m}(t_{l})$ represents the\nproperties of the space-time ``points'' on the ``hyperplane'' labeled by $t_{l}$.\nThat is: \n\\begin{equation}\nP^{\\ast}_{m}(t_{l}) = \\Pi_{i} P_{m_{i}}(x_{i}, t_{l}), \n\\label{product}\n\\end{equation}\nwith $\\{ x_{i} \\}$ the central positions of the point-like regions considered on the\n``hyperplane''.\nThe index $m$ is symbolic for the set of indices $\\{m_{i}\\}$.\nAs explained above, the modal proposal is that the projectors\n$P_{m_{i}}(x_{i},t_{l})$ come\nfrom the spectral decomposition of the mixed state defined on the algebra \nbelonging to the space-time ``point'' $(x_{i},t_{l})$ (we do not pursue here the\nchange\nin this if\nwe\nconsider the modification suggested in section 4). Because all the considered \npoint-like regions on the ``hyperplane'' $t_{l}$ are space-like separated from each\nother,\nthe associated projectors commute (the principle of micro-causality). This important\nfeature of local quantum physics \nguarantees that the product operator of Eq.\\ \n(\\ref{product}) is again a projection operator, so that expression \n(\\ref{sevtimeprob1}) can be treated in the same way as Eq.\\ (\\ref{sevtimeprob}).\nIn particular, we will need an additional condition to ensure that\n(\\ref{sevtimeprob1})\nyields a Kolmogorovian probability.\n\nThe decoherence condition that we are going to use is essentially the same as discussed\nin\nsection 3. Suppose that at\nspace-time ``point'' $(x,t)$ the magnitude represented by the set of projector\noperators $\\{P_{k}\\}$ is definite-valued; $P_{l}$ has value $1$, say. \nIntuitively speaking, the notion of decoherence that we invoke is \nthat in the course of the further evolution there subsists\na trace of $P_{l}(x,t)$ in the future lightcone of \n$(x,t)$. \nThat is, decoherence implies that on each space-like hyperplane \nintersecting the future lightcone of $(x,t)$ there is at least one space-time \npoint, within this lightcone, that has a property strictly correlated to\n$P_{l}$. One way of \nfulfilling this \ndecoherence \ncondition would be given by the existence of a time-like worldline (a propagating signal) \ngoing through\n$(x,t)$ (or a bundle of such worldlines), each point of which is characterized \nby the value $1$ of $P_{l}(x,t)$ (apart from evolution). \nThis makes explicit the idea\nthat the effects of the irreversible interaction responsible for decoherence \npropagate within the future lightcone of $(x,t)$. \n\nIf this decoherence condition is fulfilled, we have that on the hyperplane\n$t_{2}$ at least one of the space-time points, say $(x^{\\prime},t_{2})$,\nhas properties $\\{P^{\\prime}_{k}\\}$ correlated to $\\{P_{k}\\}$: $P^{\\prime}_{k}.P_{k^{\\prime}}\| \\Psi \\rangle = 0$\nif $k \\neq k^{\\prime}$.\n\nIn that case we not only have that \nthat $\\langle \\Psi \| P_{l}(x,t_{1}) P^{\\prime}_{k}(x^{\\prime},t_{2}) \nP^{\\prime}_{k^{\\prime}}(x^{\\prime},t_{2}) P_{l}(x,t_{1}) \| \\Psi \\rangle = 0 $ if\n$k \\neq k^{\\prime}$ (this follows simply from the orthogonality of $P_{k}$ and\n$P_{k^{\\prime}}$, expressing the incompatibility of two \ndifferent values of the same observable at one space-time point), but we also find \n$\\langle \\Psi \| P_{l}(x,t_{1}) P^{\\prime}_{k}(x^{\\prime},t_{2}) \nP^{\\prime}_{k^{\\prime}}(x^{\\prime},t_{2}) P_{l^{\\prime}}(x,t_{1}) \| \\Psi \\rangle = 0$\nif $l \\neq l^{\\prime}$, both if $k = k^{\\prime}$ and if $k \\neq k^{\\prime}$.\nThis expresses that possibilities\ncharacterized by different values of an observable remain incompatible \nin the course of time. \nAssuming decoherence for the properties of all space-time points,\nwe find the analogue\nof (\\ref{consist}):\n\\begin{eqnarray}\n\\lefteqn{\\langle \\Psi \| P^{\\ast}_{i}(t_{1}).\nP^{\\ast}_{j}(t_{2}).....P^{\\ast}_{l}(t_{n}) .\nP^{\\ast}_{l^{\\prime}}(t_{n}) ......\nP^{\\ast}_{j^{\\prime}}(t_{2}) . P^{\\ast}_{i^{\\prime}}(t_{1}) \| \\Psi \\rangle\n=0}\n\\nonumber \\\\ & & i \\neq\ni^{\\prime} \\vee j \\neq j^{\\prime} \\vee\n....\\vee l \\neq l^{\\prime}.\\label{consist1}\n\\end{eqnarray}\nThis makes (\\ref{sevtimeprob1}) a consistent Kolmogorovian \njoint probability for the joint occurrence of the events represented by\n$P^{\\ast}_{i}(t_{1})$, $P^{\\ast}_{j}(t_{2})$, ....$P^{\\ast}_{l}(t_{n})$.\n\nThe projectors $P^{\\ast}_{i}(t_{1})$, $P^{\\ast}_{j}(t_{2})$,\n....$P^{\\ast}_{l}(t_{n})$ depend for their definition on the chosen\nset of hyperplanes, labeled by $t$. Therefore\n(\\ref{sevtimeprob1})\nis not manifestly Lorentz invariant. However, the projectors $P^{\\ast}(t)$ are\nproducts of\nprojectors pertaining\nto the individual space-time points lying on the $t$-hyperplanes, so\n(\\ref{sevtimeprob1}) can\nalternatively be written in terms of these latter projectors. The specification\nof the\njoint probability of the physical quantities at all considered ``points'' in a given\nspace-time region\nrequires\n(\\ref{sevtimeprob1}) with all individual projectors appearing in it. Depending on the way in which the space-time region\nhas been\nsubdivided in space-like hyperplanes\nin the definition of $P^{\\ast}_{i}(t_{1})$, $P^{\\ast}_{j}(t_{2})$,\n....$P^{\\ast}_{l}(t_{n})$, the individual projectors occur in\ndifferent orders in this complete\nprobability specification. However, there is a lot of conventionality in this ordering.\nAll operators attached to\npoint-like regions with space-like separation commute, so that their ordering can be\narbitrarily\nchanged. The only \ncharacteristic\nof the ordering that is invariant under all these allowed permutations is that if $y <\nx$ (i.e.\\ $y$ is in the\ncausal past of $x$), $P(y)$ should appear before $P(x)$ in the expression for the joint\nprobability. But this\nis exactly the characteristic that is common to all expressions that follow from writing\nout (\\ref{sevtimeprob1}),\nstarting from all different ways of ordering events with a time parameter $t$. All\nthese expressions can therefore\nbe transformed into each other by permutations of projectors belonging to space-time\n``points'' with space-like separation.\nThe joint probability thus depends only on how the events in the space-time region\nare ordered with respect\nto the Lorentz-invariant relation $<$; it is therefore Lorentz-invariant itself. \n \n\\sectiona{Conclusion}\n\nThe main aim of this Letter has been to show that a decoherence condition, analogous to the one whose fulfilment\nis assumed in the consistent\nhistories approach,\ncan be satisfied in the modal interpretation of quantum mechanics. If the condition is\nsatisfied, a simple\nand natural joint probability can be specified for the values of definite-valued\nobservables at several times.\nThis probability expression is the same as the one proposed in the consistent histories\nscheme, or in traditional\nquantum measurement theory. It is here combined with the characteristic feature\nof the modal interpretation,\nnamely that the set of definite-valued observables at each instant is determined by an\nobjective rule that uses\nonly the form of the quantum mechanical state; this rule selects one family of\nconsistent histories. \n\nBecause quantum field theory is central in present-day theoretical physics,\nwe have proposed a way of applying the modal histories scheme to that theory.\nThis proposal was motivated by the desire to obtain, at least in the classical limiting situation,\na picture in which events occur, in small space-time regions approximating space-time points. In this way it\nbecomes possible in principle to recover the classical picture according to which field values are attached to\nspace-time points.\nWe have\nargued that the assumption of irreversible decoherence, combined with the modal\nideas, indeed offers prospects of obtaining\na Lorent-invariant account in which field magnitudes take on definite values in \npoint-like regions and in which a\nconsistent joint probability for such values can be defined. \n\n\\section*{Acknowledgement}\n\nI am indebted to Dr.\\ Klaas Landsman and Prof.\\ Miklos Redei for valuable help concerning questions\nin quantum field theory.\n\n\n\\begin{thebibliography}{99}\n\\bibitem{vanfra}\nB. van Fraassen, A modal interpretation of quantum mechanics, in: Current issues in\nquantum logic, eds. E. Beltrametti and B. van Fraassen (Plenum, New York, 1982).\n\\bibitem{dieks1}\nD. Dieks, Found.\\ Phys. 19 (1989) 1397; Phys.\\ Lett.\\ A 142 (1989) 439.\n\\bibitem{healey}\nR.\\ Healey, The Philosophy of Quantum Mechanics:\nAn Interactive Interpretation (Cambridge\nUniv.\\ Press, Cambridge, 1989).\n\\bibitem{dieks2}\nThe Modal Interpretation of Quantum Mechanics, eds.\\ D. Dieks and P.E. Vermaas\n(Kluwer Academic Publishers, Dordrecht, 1998).\n\\bibitem{kent}\nA. Kent, Phys. Lett. A 196 (1995) 313.\n\\bibitem{baccia}\nG. Bacciagaluppi and W.M. Dickson, Modal\nInterpretations with Dynamics, to appear in Found.\\ Phys.\n\\bibitem{vermaas2}\nP.E. Vermaas, Possibilities and impossibilities of modal interpretations of quantum\nmechanics (Cambridge Univ.\\ Press,\nCambridge, to appear).\n\\bibitem{dickson}\nW.M. Dickson and R. Clifton, Lorentz-invariance in modal interpretations, in: ref.\\\n\\cite{dieks2}.\n\\bibitem{verm}\nP.E. Vermaas and D. Dieks, Found.\\ Phys.\\ 25 (1995) 145.\n\\bibitem{dieks3}\nD. Dieks, Preferred factorizations and consistent property attribution, in: Quantum\nMeasurement---beyond paradox, eds. R.A.\nHealey and G. Hellman (Univ.\\ of Minneapolis Press, Minneapolis, 1998).\n\\bibitem{grif}\nR. Griffiths, J. Stat.\\ Phys.\\ 36 (1984) 219.\n\\bibitem{dowker}\nF.\\ Dowker and A.\\ Kent, J.\\ Stat.\\ Phys.\\ 82 (1996) 1575.\n\\bibitem{haag}\nR.\\ Haag, Local Quantum Physics (Springer-Verlag, Berlin, 1992).\n\\bibitem{summers}\nS.J. Summers, Rev.\\ Math.\\ Phys.\\ 2 (1990) 201.\n\\bibitem{schroer}\nB.\\ Schroer, Basic Quantum Theory and Measurement from the Viewpoint of Local\nQuantum Physics,\nquant-ph/9904072, 19 April 1999.\n\\bibitem{clifton}\nR.\\ Clifton, Beables in Algebraic Quantum Mechanics, in From Physics to\nPhilosophy: Essays in Honour of\nMichael Redhead (Cambridge Univ.\\ Press, Cambridge, 1998).\n\\end{thebibliography}\n\\end{document}\n\n\n\n\n" } ]	[ { "name": "quant-ph9912075.extracted_bib", "string": "{vanfra B. van Fraassen, A modal interpretation of quantum mechanics, in: Current issues in quantum logic, eds. E. Beltrametti and B. van Fraassen (Plenum, New York, 1982)." }, { "name": "quant-ph9912075.extracted_bib", "string": "{dieks1 D. Dieks, Found.\\ Phys. 19 (1989) 1397; Phys.\\ Lett.\\ A 142 (1989) 439." }, { "name": "quant-ph9912075.extracted_bib", "string": "{healey R.\\ Healey, The Philosophy of Quantum Mechanics: An Interactive Interpretation (Cambridge Univ.\\ Press, Cambridge, 1989)." }, { "name": "quant-ph9912075.extracted_bib", "string": "{dieks2 The Modal Interpretation of Quantum Mechanics, eds.\\ D. Dieks and P.E. Vermaas (Kluwer Academic Publishers, Dordrecht, 1998)." }, { "name": "quant-ph9912075.extracted_bib", "string": "{kent A. Kent, Phys. Lett. A 196 (1995) 313." }, { "name": "quant-ph9912075.extracted_bib", "string": "{baccia G. Bacciagaluppi and W.M. Dickson, Modal Interpretations with Dynamics, to appear in Found.\\ Phys." }, { "name": "quant-ph9912075.extracted_bib", "string": "{vermaas2 P.E. Vermaas, Possibilities and impossibilities of modal interpretations of quantum mechanics (Cambridge Univ.\\ Press, Cambridge, to appear)." }, { "name": "quant-ph9912075.extracted_bib", "string": "{dickson W.M. Dickson and R. Clifton, Lorentz-invariance in modal interpretations, in: ref.\\ \\cite{dieks2." }, { "name": "quant-ph9912075.extracted_bib", "string": "{verm P.E. Vermaas and D. Dieks, Found.\\ Phys.\\ 25 (1995) 145." }, { "name": "quant-ph9912075.extracted_bib", "string": "{dieks3 D. Dieks, Preferred factorizations and consistent property attribution, in: Quantum Measurement---beyond paradox, eds. R.A. Healey and G. Hellman (Univ.\\ of Minneapolis Press, Minneapolis, 1998)." }, { "name": "quant-ph9912075.extracted_bib", "string": "{grif R. Griffiths, J. Stat.\\ Phys.\\ 36 (1984) 219." }, { "name": "quant-ph9912075.extracted_bib", "string": "{dowker F.\\ Dowker and A.\\ Kent, J.\\ Stat.\\ Phys.\\ 82 (1996) 1575." }, { "name": "quant-ph9912075.extracted_bib", "string": "{haag R.\\ Haag, Local Quantum Physics (Springer-Verlag, Berlin, 1992)." }, { "name": "quant-ph9912075.extracted_bib", "string": "{summers S.J. Summers, Rev.\\ Math.\\ Phys.\\ 2 (1990) 201." }, { "name": "quant-ph9912075.extracted_bib", "string": "{schroer B.\\ Schroer, Basic Quantum Theory and Measurement from the Viewpoint of Local Quantum Physics, quant-ph/9904072, 19 April 1999." }, { "name": "quant-ph9912075.extracted_bib", "string": "{clifton R.\\ Clifton, Beables in Algebraic Quantum Mechanics, in From Physics to Philosophy: Essays in Honour of Michael Redhead (Cambridge Univ.\\ Press, Cambridge, 1998)." } ]
quant-ph9912076	Erratum: Asymptotic entanglement manipulations can be genuinely irreversible. [Phys. Rev. Lett. {84	[ { "author": "Micha\\l{" } ]		[ { "name": "quant-ph9912076.tex", "string": "\\documentstyle[prl,aps]{revtex}\n\n\\font\\Bbb =msbm10\n\\font\\eufm =eufm10\n\\def\\Real{{\\hbox{\\Bbb R}}} \\def\\C{{\\hbox {\\Bbb C}}}\n\n\\def\\textbf#1{{\\bf #1}}\n\\def\\tr{{\\rm Tr\\,}}\n\\def\\be{\\begin{equation}}\n\\def\\ee{\\end{equation}}\n\\def\\ben{\\begin{eqnarray}}\n\\def\\een{\\end{eqnarray}}\n\\def\\eea{\\end{array}}\n\\def\\bea{\\begin{array}}\n\\newcommand{\\bei}{\\begin{itemize}}\n\\newcommand{\\eei}{\\end{itemize}}\n\\def\\ra{\\rangle}\n\\def\\la{\\langle}\n\\def\\blacksquare{\\vrule height 4pt width 3pt depth2pt}\n\\def\\dcal{{\\cal D}}\n\\def\\pcal{{\\cal P}}\n\\def\\hcal{{\\cal H}}\n\\def\\trace{\\mbox{Tr}}\n\n\n\\def\\n{{\\otimes n}}\n\\def\\m{{\\otimes m}}\n\n\\begin{document}\n\\draft\n%\\twocolumn\n\n\\title{Erratum: Asymptotic entanglement manipulations can be genuinely \nirreversible. [Phys. Rev. Lett. {\\bf 84}, 4260 (2000)] }\n\n\n\\author{Micha\\l{} Horodecki$^{1}$,\nPawe\\l{} Horodecki$^{2}$ and Ryszard Horodecki$^{1}$}\n\n\\address{$^1$ Institute of Theoretical Physics and Astrophysics,\nUniversity of Gda\\'nsk, 80--952 Gda\\'nsk, Poland,\\\\\n$^2$Faculty of Applied Physics and Mathematics,\nTechnical University of Gda\\'nsk, 80--952 Gda\\'nsk, Poland\n}\n\n\n\n\n\\maketitle\n\n\n\\begin{abstract}\n\\end{abstract}\n\n\\pacs{}\n\nThe presented proof in Ref. \\cite{irrev} that $E_D<E_f^\\infty$ \n(irreversibility) for some Werner states, was based on an invalid \nlemma of Ref. \\cite{Rains} (cf. erratum \\cite{Rainse}) (p. 3 of the \nRef \\cite{irrev}) saying that $E_{PT}$\nis additive for Werner states. Therefore our proof of irreversibility \nis incorrect. However, the irreversibility holds, as shown recently\nin Ref. \\cite{VC}. The authors considered some family of \nbound entangled states (hence having $E_D=0$) and \nshowed that $E_f^\\infty$ for those states is nonzero. \n\nOther results of the our paper remain valid \nas they do not make use of the mentioned lemma.\nThey are:\n\\bei\n\\item Proof that $E_n=\\log\|\|\\varrho^{PT}\|\|$ is upper bound for \ndistillable entanglement\n(Lemma 2 of Ref. \\cite{irrev}). An independent proof of this fact was \nfound earlier by Werner, Benasque, 1998 [private communication], see Ref \\cite{VW}. \n\\item Proof that $E_n$ does not increase under trace-preserving \nPPT superoperators (Appendix of Ref. \\cite{irrev}).\n\\item Calculation of value of $E_{PT}$ for Werner states (eq. (16) of \nRef. \\cite{irrev}). \n\\item Calculation of the measure $E_n$ for isotropic state \nand Werner states (eq. (13) and (15) of Ref. \\cite{irrev}).\n\\eei\n\\begin{references}\n%\\bibitem[]{poczta1}\n%E-mail address: michalh@iftia.univ.gda.pl <mailto:michalh@iftia.univ.gda.pl>\n%\\bibitem[]{poczta2}\n%E-mail address:pawel@mifgate.mif.pg.gda.pl\n%\\bibitem[**]{poczta3}\n%E-mail address: fizrh@univ.gda.pl <mailto:fizrh@univ.gda.pl>\n\\bibitem{irrev}\nM. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. Lett. \n{\\bf 84}, 4260 (2000). %(quant-ph/9912076)\n\\bibitem{Rains}\nE. M. Rains, Phys. Rev. A {\\bf 60}, 179 (1999).\n\\bibitem{Rainse}\nE. M. Rains, Phys. Rev. A, {\\bf 63} 019902(E) (2000)\n\\bibitem{VC}\nG. Vidal and J. I. Cirac, quant-ph/0102036.\n\\bibitem{VW}\nG. Vidal and R. F. Werner, quant-ph/0102117.\n\\end{references}\n\\end{document}\n\n\n\n\n\n" } ]	[ { "name": "quant-ph9912076.extracted_bib", "string": "[]{poczta1 %E-mail address: michalh@iftia.univ.gda.pl <mailto:michalh@iftia.univ.gda.pl> %" }, { "name": "quant-ph9912076.extracted_bib", "string": "[]{poczta2 %E-mail address:pawel@mifgate.mif.pg.gda.pl %" }, { "name": "quant-ph9912076.extracted_bib", "string": "[**]{poczta3 %E-mail address: fizrh@univ.gda.pl <mailto:fizrh@univ.gda.pl>" }, { "name": "quant-ph9912076.extracted_bib", "string": "{irrev M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. Lett. {84, 4260 (2000). %(quant-ph/9912076)" }, { "name": "quant-ph9912076.extracted_bib", "string": "{Rains E. M. Rains, Phys. Rev. A {60, 179 (1999)." }, { "name": "quant-ph9912076.extracted_bib", "string": "{Rainse E. M. Rains, Phys. Rev. A, {63 019902(E) (2000)" }, { "name": "quant-ph9912076.extracted_bib", "string": "{VC G. Vidal and J. I. Cirac, quant-ph/0102036." }, { "name": "quant-ph9912076.extracted_bib", "string": "{VW G. Vidal and R. F. Werner, quant-ph/0102117." } ]
quant-ph9912077	The Zeno and anti-Zeno effects on decay in dissipative quantum systems\footnote{ Published in acta physica slovaca {49	[ { "author": "A. G. Kofman\\footnote{E-mail address: abraham.kofman@weizmann.ac.il" } ]	We point out that the quantum Zeno effect, i.e., inhibition of spontaneous decay by frequent measurements, is observable only in spectrally finite reservoirs, i.e., in cavities and waveguides, using a sequence of evolution-interrupting pulses or randomly-modulated CW fields. By contrast, such measurements can only accelerate decay in free space.	[ { "name": "slovac_lanl.tex", "string": "%&latex209\n% To arXiv.org\n%\\documentstyle[preprint,aps]{revtex}\n%\\documentstyle[12pt,tighten,aps,psfig,fancy]{revtex} \n\\documentstyle[12pt,tighten,aps,psfig]{revtex} \n\\def\\be{\\begin{equation}}\n\\def\\e#1{\\label{#1}\\end{equation}}\n\\def\\r#1{(\\ref{#1})}\n\n%\\pagestyle{fancyplain}\n%\\headrulewidth=0pt\n%\\lhead{}\n%\\rhead{}\n\n\\begin{document}\n\\draft\n\\title{The Zeno and anti-Zeno effects on decay in dissipative quantum \nsystems\\footnote{\nPublished in acta physica slovaca {\\bf 49}, 541-548 (1999),\nunder the title ``Decay control in dissipative quantum systems''\n}}\n\\author{A. G. Kofman\\footnote{E-mail address:\nabraham.kofman@weizmann.ac.il} and G. Kurizki\\footnote{E-mail address:\ngershon.kurizki@weizmann.ac.il}}\n\\address{Department of Chemical Physics, The Weizmann Institute of \nScience, Rehovot 76100, Israel}\n\\date{Received 10 May 1999, accepted 12 May 1999}\n\\maketitle\n\\begin{abstract}\nWe point out that the quantum Zeno effect, i.e., inhibition of \nspontaneous decay by frequent measurements, is observable only in \nspectrally finite reservoirs, i.e., in cavities and waveguides, using \na sequence of evolution-interrupting pulses or randomly-modulated CW\nfields. By contrast, such measurements can only accelerate decay in\nfree space. \n\\end{abstract}\n\\pacs{PACS: 03.65.Bz, 42.50.-p}\n\n\\section{Introduction}\n\nThe \"watchdog\" or quantum Zeno effect (QZE) is a basic\nmanifestation of the influence of measurements on the evolution of a\nquantum system. The original QZE prediction has been that {\\em \nirreversible decay} of an excited state into an open-space reservoir \ncan be inhibited \\cite{4},\nby repeated interruption of the system-reservoir coupling, which is\nassociated with measurements (e.g., the interaction of an unstable\nparticle with its environment on its flight through a bubble chamber) \n\\cite{5,6}. However, this prediction has\nnot been experimentally verified as yet! Instead, the interruption of\nRabi oscillations and analogous forms of {\\em nearly-reversible\\/}\nevolution has been at the focus of interest \n\\cite{7,8,8a,8b,8c,8d,9,10}. Tacit assumptions have been made that\nthe QZE is in principle attainable in open space, but is technically\ndifficult.\n\nWe have recently demonstrated \\cite{11} that the inhibition of {\\em\nnearly-exponential} excited-state decay by the QZE in two-level atoms,\nin the spirit of the original suggestion \\cite{4}, is amenable to\nexperimental verification in resonators. Although this task has been\nwidely believed to be very difficult, we have shown, by means of our\nunified theory of spontaneous emission into arbitrary reservoirs \n\\cite{12}, that two-level emitters in cavities or in waveguides are \nin fact adequate for radiative decay control by the QZE\\cite{11}. \nCondensed media or multi-ion traps are their analogs for vibrational \ndecay control (phonon emission) by the QZE\\cite{har98}. We have now \ndeveloped a more comprehensive view of the possibilities of \nexcited-state decay by QZE. Here we wish to demonstrate that QZE is \nindeed achievable by repeated or continuous measurements of the \nexcited state, but only in reservoirs whose spectral response {\\em \nrises up} to a frequency which does not exceed the resonance\n(transition) frequency. By contrast, in \nopen-space decay, where the reservoir response has a much higher\ncutoff, {\\em non-destructive} frequent measurements are much more\nlikely to accelerate decay, causing the anti-Zeno effect.\n\n\\section{Measurement schemes}\n\n\\subsection{Impulsive measurements (Cook's scheme)}\n\nConsider an initially excited two-level atom coupled to an {\\em\narbitrary} density-of-modes (DOM) spectrum $\\rho(\\omega)$ of the\nelectromagnetic field in the vacuum state. At time $\\tau$ its\nevolution is interrupted by a short optical pulse, which serves as an\nimpulsive quantum measurement \\cite{7,8,8a,8b,8c,8d,9,10}. Its role is\n{\\em to break the evolution coherence}, by transferring the\npopulations of the excited state $\|e\\rangle$ to an auxiliary state\n$\|u\\rangle$ which then decays back to $\|e\\rangle$ {\\em incoherently}.\n\nThe spectral response, i.e., the emission rate into this reservoir at\nfrequency $\\omega$, is \n\\begin{equation}\nG(\\omega)=\|g(\\omega)\|^2\\rho(\\omega),\n\\label{1}\n\\end{equation}\n$\\hbar g(\\omega)$ being the field-atom coupling energy.\n\nWe cast the excited-state amplitude in the form\n$\\alpha_e(\\tau)e^{-i\\omega_a\\tau}$, where $\\omega_a$ is the atomic\nresonance frequency. Restricting ourselves to sufficiently short\ninterruption intervals $\\tau$ such that $\\alpha_e(\\tau)\\simeq 1$, yet\nlong enough to allow the rotating wave approximation, we obtain\n\\begin{eqnarray}\n\\alpha_e(\\tau)&\\simeq&1-\\int_0^\\tau dt(\\tau-t)\\Phi(t)e^{i\\Delta t},\n\\label{3}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\Phi(t)=\\int_0^\\infty d\\omega G(\\omega)e^{-i(\\omega-\\omega_s)t}.\n\\label{2b}\n\\end{equation}\n$\\Delta=\\omega_a-\\omega_s$ is the detuning of the atomic resonance\nfrom the peak (or cutoff) $\\omega_s$ of $G(\\omega)$.\n\nTo first order in the atom-field interaction, the excited state\nprobability after $n$ interruptions (measurements),\n$W(t=n\\tau)=\|\\alpha_e(\\tau)\|^{2n}$, can be written as\n\\begin{equation} \nW(t=n\\tau)\\approx[2\\mbox{Re}\\alpha_e(\\tau)-1]^n\\approx e^{-\\kappa t},\n\\label{4'}\\end{equation}\nwhere\n\\begin{equation}\n\\kappa=\\frac{2}{\\tau}\\mbox{Re}[1-\\alpha_e(\\tau)]\n=\\frac{2}{\\tau}\\mbox{Re}\\int_0^\\tau dt(\\tau-t)\\Phi(t)e^{i\\Delta t}.\n\\label{4}\n\\end{equation}\nThe QZE obtains if $\\kappa$\n{\\em decreases with} $\\tau$ for sufficiently short $\\tau$. This\nessentially means that the correlation (or memory) time of the field\nreservoir is longer (or, equivalently, $\\Phi(t)$ falls off slower)\nthan the chosen interruption interval $\\tau$.\n\nEquation \\r{4} can be rewritten as \n\\be\n\\kappa=2\\pi\\int G(\\omega)\\left\\{\\frac{\\tau}{2\\pi}\n\\text{sinc}^2\\left[\\frac{(\\omega-\\omega_a)\\tau}{2}\\right]\\right\\}\nd\\omega,\n\\e{5}\nwhere the interruptions are seen to cause dephasing whose spectral\nwidth is $\\sim 1/\\tau$.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\psfig{file=gerI3.ps,width=6cm}&\n\\ \\ \\ \\ \\psfig{file=gerI4.ps,width=6cm}\\\\\n(a) & (b)\n\\end{tabular}\n\\end{center}\n\\caption{ Dependence of effective decay rate $\\kappa$\non dephasing (relaxation) spectrum\n$F(\\Delta)$ and field reservoir response with cutoff $G(\\omega)$: (a)\nLorentzian dephasing spectrum [Eq. (\\protect\\ref{14})]; \n(b) sinc-function \nspectrum [Eq. (\\protect\\ref{5}) -- impulsive measurements]. }\n\\label{fig:4}\\end{figure}\n\n\\subsection{Noisy-field dephasing: Random Stark shifts}\n\nInstead of disrupting the coherence of the evolution by a sequence of\n\"impulsive\" measurements, as above, we can achieve\nthis goal by {\\em noisy-field dephasing} of $\\alpha_e(t)$: Random\nac-Stark shifts by an off-resonant intensity-fluctuating field result\nin the replacement of Eq. (\\ref{5}) by (Fig. \\ref{fig:4})\n\\begin{equation}\n\\kappa=\\int G(\\Delta+\\omega_a){\\cal L}(\\Delta)d\\Delta,\n\\label{14}\n\\end{equation}\nHere the spectral response $G(\\Delta+\\omega_a)$ is the same as in\nEq. (\\ref{1}), whereas ${\\cal L}(\\Delta)$ is the Lorentzian-shaped\nrelaxation function of the coherence element $\\rho_{eg}(t)$, which \nfor the common dephasing model decays {\\em exponentially}. This\nLorentzian relaxation spectrum has a HWHM width \n$\\nu=\\langle\\Delta\\omega^2\\rangle\\tau_c$, the product of the \nmean-square\nStark shift and the noisy-field correlation time. The QZE condition is\nthat {\\em this width be larger than the width of\\/} $G_s(\\omega)$\n(Fig. \\ref{fig:4}). The advantage of this realization is that it does\nnot depend on $\\gamma_u$, and is realizable for any atomic\ntransition. Its importance for molecules is even greater: if we start\nwith a single vibrational level of $\|e\\rangle$, no additional levels\nwill be populated by this process.\n\n\\subsection{CW dephasing}\n\nThe random ac-Stark shifts described above cause both shifting and\nbroadening of the spectral transition. If we wish to avoid the\nshifting altogether, we may employ a CW\ndriving field that is {\\em nearly resonant} with the\n$\|e\\rangle\\leftrightarrow\|u\\rangle$ transition\\cite{7,8}. If the \ndecay rate of this transition, $\\gamma_u$, is larger than the Rabi\nfrequency $\\Omega$ of the driving field, then one can show that\n$\\kappa$ is given again by Eq. \\r{14}, where the Lorentzian\n(dephasing) width is\n\\begin{equation}\n\\nu=\\frac{2\\Omega^2}{\\gamma_u}. \n\\label{15}\\end{equation}\n\n\\subsection{Universal formula}\n\nAll of the above schemes are seen to yield the same universal\nformula for the decay rate\n\\be\n\\kappa=2\\pi\\int G(\\omega)F(\\omega-\\omega_a)d\\omega,\n\\e{8}\nwhere $F(\\omega)$ expresses the relevant measurement-induced \ndephasing (sinc- or a Lorentzian-shaped): its\nwidth relative to that of $G(\\omega)$ determines the QZE behavior.\n\n\\begin{figure}\n{\\vspace{-3cm}\n\\centerline{\\psfig{file=zeno1in.ps,width=3.375in}}\n\\vspace{-3cm}}\n\\caption{ Cavity mode with Lorentzian lineshape.\n}\\label{fig:1}\\end{figure}\n\n\\section{Applications to various reservoirs}\n\n\\subsection{Finite reservoirs: A Lorentzian line}\n\nThe simplest application of the above analysis is to the case of a\ntwo-level atom coupled to a near-resonant Lorentzian line centered at\n$\\omega_s$, characterizing a high-$Q$ cavity mode \\cite{11}. In this\ncase, \n\\be\nG_s(\\omega)=\\frac{g_s^2\\Gamma_s}{\\pi[\\Gamma_s^2+(\\omega-\\omega_s)^2]},\n\\e{9}\nwhere $g_s$ is the resonant coupling\nstrength and $\\Gamma_s$ is the linewidth (Fig. \\ref{fig:1}). \nHere $G_s(\\omega)$ stands for the sharply-varying (nearly-singular)\npart of the DOM distribution, associated with narrow cavity-mode lines\nor with the frequency cutoff in waveguides or photonic band edges. The\nbroad portion of the DOM distribution $G_b(\\omega)$ (the \"background\" \nmodes), always coincides with\nthe free-space DOM $\\rho(\\omega)\\sim\\omega^2$ at frequencies well\nabove the sharp spectral features. In an open cavity,\n$G_b(\\omega)$ represents the atom coupling to the unconfined\nfree-space modes. This\ngives rise to an {\\em exponential} decay factor in\nthe excited state probability, regardless of how short $\\tau$ is,\ni.e.,\n\\begin{equation}\n\\kappa=\\kappa_s+\\gamma_b,\n\\label{4a}\n\\end{equation}\nwhere $\\kappa_s$ is the contribution to $\\kappa$ from the\nsharply-varying modes and\n$\\gamma_b=2\\pi G_b(\\omega_a)$ is the effective rate of \nspontaneous emission into the background modes.\nIn most structures $\\gamma_b$ is comparable to the free-space decay\nrate $\\gamma_f$.\n\nIn the\nshort-time approximation, taking into account that the Fourier\ntransform of the Lorentzian $G_s(\\omega)$ is $\\Phi_s(t) =\ng_s^2e^{-\\Gamma_s t}$, Eq. (\\ref{3}) yields (without the\nbackground-modes contribution)\n\\begin{equation}\n\\alpha_e(\\tau)\\approx 1-\\frac{g_s^2}{\\Gamma_s-i\\Delta}\\left[\n\\tau+\\frac{e^{(i\\Delta-\\Gamma_s)\\tau}-1}{\\Gamma_s-i\\Delta}\\right].\n\\label{6}\n\\end{equation}\nThe QZE condition is then\n\\begin{equation}\n\\tau\\ll(\\Gamma_s+\|\\Delta\|)^{-1},g_s^{-1}.\n\\label{7a}\\end{equation}\n{\\em On resonance}, when $\\Delta=0$, Eqs. \\r{4} and (\\ref{6}) yield\n\\begin{equation}\n\\kappa_s=g_s^2\\tau.\n\\label{7}\n\\end{equation}\n\nThus the background-DOM effect cannot be modified by QZE. Only the\nsharply-varying DOM contribution $\\kappa_s$ may allow for QZE.\nOnly the $\\kappa_s$ term decreases with $\\tau$, indicating the QZE\ninhibition of the nearly-exponential decay into the Lorentzian field\nreservoir as $\\tau\\rightarrow 0$. Since $\\Gamma_s$ has dropped out of\nEq.~(\\ref{7}), the decay rate $\\kappa$ is the {\\em same} for both\nstrong-coupling ($g_s > \\Gamma_s$) and weak-coupling ($g_s \\ll\n\\Gamma_s$) regimes. Physically, this comes about since for $\\tau\\ll\ng_s^{-1}$ the energy uncertainty of the emitted photon is too large to\ndistinguish between reversible and irreversible evolutions.\n\nThe evolution inhibition, however, has rather different meaning for\nthe two regimes. In the weak-coupling regime, where, in the absence of\nthe external control, the excited-state population decays nearly\nexponentially at the rate $g_s^2/\\Gamma_s+\\gamma_b$ (at $\\Delta=0$),\none can speak about the inhibition of irreversible decay, in the\nspirit of the original QZE prediction \\cite{4}. By contrast, in the\nstrong-coupling regime in the absence of interruptions (measurements),\nthe excited-state population undergoes damped Rabi oscillations at the\nfrequency $2g_s$. In this case, the QZE slows down the evolution\nduring the first Rabi half-cycle ($0 \\leq t \\leq \\pi / 2 g_s^{-1}$),\nthe evolution on the whole becoming irreversible.\n\n\\begin{figure}\n{\\centerline{\\psfig{file=zeno2in.ps,width=2.in}}}\n\\caption{Cook's scheme for impulsive measurements.\n\\label{f5} }\n\\end{figure}\n\nA possible realization of this scheme is as follows. Within an open\ncavity the atoms repeatedly interact with a pump laser, which is\nresonant with the $\|e\\rangle\\rightarrow\|u\\rangle$ transition\nfrequency. The resulting $\|e\\rangle\\rightarrow\|g\\rangle$ fluorescence\nrate is collected and monitored as a function of the pulse repetition\nrate $1/\\tau$. Each short, intense pump pulse of duration $t_p$ and\nRabi frequency $\\Omega_p$ is followed by spontaneous decay from\n$\|u\\rangle$ back to $\|e\\rangle$, at a rate $\\gamma_u$, so as to {\\em\ndestroy the coherence} of the system evolution, on the one hand, and\n{\\em reshuffle the entire population} from $\|e\\rangle$ to $\|u\\rangle$\nand back, on the other hand (Fig. \\ref{f5}). The demand that the\ninterval between measurements significantly exceed the measurement\ntime, yields the inequality $\\tau\\gg t_p$. The above inequality can be\nreduced to the requirement $\\tau\\gg\\gamma_u^{-1}$ if the\n``measurements'' are performed with $\\pi$ pulses: $\\Omega_pt_p=\\pi,\\\nt_p\\ll\\gamma_u^{-1}$. This calls for choosing a\n$\|u\\rangle\\rightarrow\|e\\rangle$ transition with a much shorter\nradiative lifetime than that of $\|e\\rangle\\rightarrow\|g\\rangle$.\n\n\\begin{figure}\n{\\vspace{-3cm}\n\\centerline{\\psfig{file=zeno1.ps,width=3.375in}}\n\\vspace{-3cm}}\n\\caption{\nEvolution of excited-state population $W$ in\ntwo-level atom coupled to cavity mode with Lorentzian lineshape on\nresonance ($\\Delta=0$): \ncurve 1---decay to background-mode continuum\nat rate $\\gamma_b\\simeq\\gamma_f=10^6$ s$^{-1}$; curve\n3---uninterrupted decay in cavity with $F\\equiv(1-R)^{-2}=10^5$,\n$L$=15 cm, and $f$=0.02 ($\\Gamma_s=6.3\\times 10^6$ s$^{-1}$, \n$g_s=4.5\\times 10^6$ s$^{-1}$); curve 4---idem, but with \n$F=10^6$ ($\\Gamma_s=2\\times 10^6$ s$^{-1}$; damped \nRabi oscillations); curve 2---interrupted evolution along {\\em both}\ncurves 3 and 4, at intervals $\\tau=3\\times 10^{-8}$ s.\n\\label{f3} }\n\\end{figure}\n\nFigure \\ref{f3}, describing the QZE for a Lorentz line on resonance\n($\\Delta=0$), has been programmed for feasible cavity parameters:\n$\\Gamma_s=(1-R)c/L,\\ g_s=\\sqrt{cf\\gamma_f/(2L)},\\ \\gamma_b=(1-f)\n\\gamma_f$, where $R$ is the geometric-mean reflectivity of the two\nmirrors, $f$ is the fractional solid angle (normalized to $4\\pi$)\nsubtended by the confocal cavity, and $L$ is the cavity length. It\nshows, that the population of $\|e\\rangle$ decays nearly-exponentially\nwell within interruption intervals $\\tau$, but when those intervals\nbecome too short, there is significant inhibition of the decay. \nFigure \\ref{f4} shows the effect of the detuning \n$\\Delta=\\omega_a-\\omega_s$ on the\ndecay: The decay now becomes oscillatory. The interruptions now {\\em\nenhance\\/} the decay, the degree of enhancement depends on the phase\nbetween interruptions.\n\n\\begin{figure}\n{\\vspace*{-3cm}\n\\centerline{\\psfig{file=zenos.ps,width=3.375in}}\n\\vspace{-3cm}}\n\\caption{Idem, for detuning $\\Delta=10^8$s${}^{-1}$ and $F=10^6$:\ncurve 1---decay to background-mode continuum; \ncurve 2---uninterrupted free evolution;\ncurve 3---interrupted evolution at intervals \n$\\tau=5\\pi\\times 10^{-8}$ s ($\\Delta\\tau=5\\pi$); curve 4---idem, for \n$\\tau=3\\pi\\times 10^{-8}$ s ($\\Delta\\tau=3\\pi$). \n\\label{f4}}\n\\end{figure}\n\n\\subsection{Open-space reservoirs}\nThe spectral response for hydrogenic-atom radiative decay via the \n$\\vec{p}\\cdot\\vec{A}$ free-space interaction is given by\\cite{mos72}\n\\be\nG(\\omega)=\\frac{\\alpha\\omega}{[1+(\\omega/\\omega_c)^2]^4},\n\\e{10}\nwhere $\\alpha$ is the effective field-atom coupling constant and the \ncutoff frequency is\n\\be\n\\omega_{\\rm c}\\approx 10^{19}\\ \\text{s}^{-1}\\sim\\frac{c}{a_{\\rm B}}.\n\\e{11}\nUsing measurement control that produces Lorentzian broadening [Eq.\n\\r{14}] we then obtain\n\\be\n\\kappa=\\frac{\\alpha\\omega_{\\rm c}}{3}\\text{Re}\\left[\n\\frac{f(2f^4-7f^2+11)}{2(f^2-1)^3}-\\frac{6f\\ln f}{(f^2-1)^4}\n-\\frac{3i\\pi(f^2+4f+5)}{16(f+1)^4}\\right],\n\\e{12}\nwhere\n\\be\nf=\\frac{\\nu-i\\omega_a}{\\omega_{\\rm c}}.\n\\e{13}\nIn the range\n\\be\n\\nu\\ll\\omega_{\\rm c}\n\\e{16}\nwe obtain from Eq. \\r{12} the {\\em anti-Zeno effect} of \naccelerated decay. This\ncomes about due to the {\\em rising} of the spectral response\n$G(\\omega)\\approx\\alpha\\omega$ as a function of frequency (for \n$\\omega\\ll\\omega_{\\rm c}$).\nThe Zeno effect can hypothetically occur only for \n$\\nu\\agt\\omega_{\\rm c}\\sim 10^{19}$ s$^{-1}$.\nBut this range is well beyond the limit of validity of the present\nanalysis, since $\\Delta E\\sim\\hbar\\nu\\agt\\hbar\\omega_{\\rm c}$ may \nthen induce other decay channels (\"destruction\") of $\|e\\rangle$, in\naddition to spontaneous transitions to $\|g\\rangle$.\n\n\\section{Conclusions}\n\nOur unified\nanalysis of two-level system coupling to field reservoirs has revealed\nthe general optimal conditions for observing the QZE in various\nstructures (cavities, waveguides, phonon reservoirs, and photonic band\nstructures) as opposed to open space. We note that the wavefunction \ncollapse notion is not involved here, since the measurement is \nexplicitly described as an act of dephasing (coherence-breaking). \nThis analysis also clarifies that QZE cannot combat the open-space\ndecay. Rather, impulsive or continuous dephasing are much more likely\nto accelerate decay by the inverse (anti-) Zeno effect. \n\n\\begin{references}\n\n\\bibitem{4} B. Misra and E. C. G. Sudarshan, ``The Zeno paradox in\nquantum theory,'' J. Math. Phys. {\\bf 18}, 756 (1977).\n\n\\bibitem{5} J. Maddox, ``Can observations prevent decay?'', \n\\nat {\\bf 306}, 111 (1983).\n\n\\bibitem{6} A.Peres, ``Quantum limited detectors for weak classical\nsignals,'' \\prd {\\bf 39}, 2943 (1989).\n\n\\bibitem{7} \nW. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, \n``Quantum Zeno effect,'' \\pra {\\bf 41}, 2295 (1990).\n\n\\bibitem{8} P. L. Knight, ``The quantum Zeno effect,'' \\nat {\\bf\n344}, 493 (1990).\n\n\\bibitem{8a} T. Petrosky, S. Tasaki, and I. Prigogine, ``Quantum Zeno\neffect,'' \\pl A {\\bf 151}, 109 (1990).\n\n\\bibitem{8b} E. Block and P. R. Berman, ``Quantum Zeno effect and\nquantum Zeno paradox in atomic physics,'' \\pra {\\bf 44}, 1466 (1991).\n\n\\bibitem{8c} L. E. Ballentine, ``Quantum Zeno effect - comment,'' \\pra\n{\\bf 43}, 5165 (1991).\n\n\\bibitem{8d} V. Frerichs and A. Schenzle, ``Quantum Zeno effect\nwithout collapse of the wave packet,'' \\pra {\\bf 44}, 1962 (1991).\n\n\\bibitem{9} M. B. Plenio, P. L. Knight, and R. C. Thompson,\n``Inhibition of spontaneous decay by continuous measurements -\nproposal for realizable experiment,'' \\oc {\\bf 123}, 278 (1996).\n\n\\bibitem{10} A. Luis and J. Pe\\v{r}ina, ``Zeno effect in parametric\ndown-conversion,'' \\prl {\\bf 76}, 4340 (1996).\n\n\\bibitem{11} A. G. Kofman and G. Kurizki, ``Quantum Zeno effect on\natomic excitation decay in resonators,'' \\pra {\\bf 54}, R3750\n(1996).\n\n\\bibitem{12} A. G. Kofman, G. Kurizki, and B. Sherman, ``Spontaneous\nand induced atomic decay in photonic band structures,'' \\jmo\n{\\bf 41}, 353 (1994).\n\\bibitem{har98}\nG. Harel, A. G. Kofman, A. Kozhekin, and G. Kurizki,\n``Control of Non-Markovian Decay and Decoherence by Measurements and\nInterference'',\nOpt. Express {\\bf 2}, 355 (1998).\n\\bibitem{mos72}\nH. E. Moses, ``Exact electromagnetic matrix elements and exact\nselection rules for hydrogenic atoms'', Lett. Nuovo Cim. {\\bf 4}, \n51 (1972).\n\n\\end{references}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912077.extracted_bib", "string": "{4 B. Misra and E. C. G. Sudarshan, ``The Zeno paradox in quantum theory,'' J. Math. Phys. {18, 756 (1977)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{5 J. Maddox, ``Can observations prevent decay?'', \\nat {306, 111 (1983)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{6 A.Peres, ``Quantum limited detectors for weak classical signals,'' \\prd {39, 2943 (1989)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{7 W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, ``Quantum Zeno effect,'' \\pra {41, 2295 (1990)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{8 P. L. Knight, ``The quantum Zeno effect,'' \\nat {344, 493 (1990)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{8a T. Petrosky, S. Tasaki, and I. Prigogine, ``Quantum Zeno effect,'' \\pl A {151, 109 (1990)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{8b E. Block and P. R. Berman, ``Quantum Zeno effect and quantum Zeno paradox in atomic physics,'' \\pra {44, 1466 (1991)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{8c L. E. Ballentine, ``Quantum Zeno effect - comment,'' \\pra {43, 5165 (1991)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{8d V. Frerichs and A. Schenzle, ``Quantum Zeno effect without collapse of the wave packet,'' \\pra {44, 1962 (1991)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{9 M. B. Plenio, P. L. Knight, and R. C. Thompson, ``Inhibition of spontaneous decay by continuous measurements - proposal for realizable experiment,'' \\oc {123, 278 (1996)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{10 A. Luis and J. Pe\\v{rina, ``Zeno effect in parametric down-conversion,'' \\prl {76, 4340 (1996)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{11 A. G. Kofman and G. Kurizki, ``Quantum Zeno effect on atomic excitation decay in resonators,'' \\pra {54, R3750 (1996)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{12 A. G. Kofman, G. Kurizki, and B. Sherman, ``Spontaneous and induced atomic decay in photonic band structures,'' \\jmo {41, 353 (1994)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{har98 G. Harel, A. G. Kofman, A. Kozhekin, and G. Kurizki, ``Control of Non-Markovian Decay and Decoherence by Measurements and Interference'', Opt. Express {2, 355 (1998)." }, { "name": "quant-ph9912077.extracted_bib", "string": "{mos72 H. E. Moses, ``Exact electromagnetic matrix elements and exact selection rules for hydrogenic atoms'', Lett. Nuovo Cim. {4, 51 (1972)." } ]
quant-ph9912079		[]		[ { "name": "quant-ph9912079.tex", "string": "\n% PT-sym. PT osc.\n\n\\documentstyle[12pt]{article}\n%\\setlength{\\parindent}{5mm}\n\\renewcommand{\\baselinestretch}{1.5}\n\\setlength{\\headheight}{0pt}\n \\setlength{\\headsep}{0pt}\n\\setlength{\\footskip}{45pt}\n \\setlength{\\footheight}{0pt}\n \\setlength{\\textwidth}{430pt}\n \\setlength{\\textheight}{650pt}\n \\setlength{\\oddsidemargin}{10pt}\n \\def\\ha{\\mbox{$\\frac{1}{2}$}}\n\\def\\be{\\begin{equation}}\n\\def\\ee{\\end{equation}}\n\\def\\ba{\\begin{array}{c}}\n\\def\\ea{\\end{array}}\n\\def\\p{\\partial}\n\\def\\ben{$$}\n\\def\\een{$$}\n\\begin{document}\n\n\\titlepage\n\\vspace{4cm}\n\n\\begin{center}{\\Large \\bf\nNew set of exactly solvable complex potentials giving the real\nenergies }\\end{center}\n\n\\vspace{5mm}\n\n\\begin{center}\nMiloslav Znojil\n\\vspace{3mm}\n\n\\'{U}stav jadern\\'e fyziky AV \\v{C}R, 250 68 \\v{R}e\\v{z},\nCzech Republic\\\\\n\ne-mail: znojil@ujf.cas.cz\n\n\\end{center}\n\n\\vspace{5mm}\n\n\\section{Abstract}\n\nWe deform the real potential $V(x)$ of P\\\"{o}schl and Teller by a\nshift $\\varepsilon \\in (0, \\pi/2)$ of $x$ in imaginary direction.\nWe show that the new model $V(x)=F/\\sinh^2 (x-i\\,\\varepsilon) +\nG/\\cosh^2 (x-i\\,\\varepsilon)$ remains exactly solvable. Its bound\nstates are constructed in closed form. Wave functions are complex\nand proportional to Jacobi polynomials. Some of them diverge in\nthe limit $\\varepsilon \\to 0$ or $\\varepsilon \\to \\pi/2$. In\ncontrast, all their energies prove real and\n$\\varepsilon-$independent. In this sense the loss of Hermiticity\nof our family of Hamiltonians seems well counter-balanced by their\naccidental ${\\cal PT}$ symmetry.\n\n\n\\vspace{9mm}\n\n\\noindent\n PACS 03.65.Ge,\n03.65.Fd\n\n%\\vspace{9mm}\n\n%\\begin{center}\n% {\\small \\today, ptp.tex file}\n%\\end{center}\n\n\\newpage\n\n\\noindent Among all the exactly solvable models in quantum\nmechanics the one-di\\-men\\-si\\-o\\-nal Schr\\\"{o}dinger equation\n \\be\n\\left (-\\,\\frac{d^2}{dr^2} + V(r) \\right )\n \\, \\psi(r) = E \\, \\psi(r), \\ \\ \\ \\ \\ \\psi(\\pm \\infty) = 0\n\\label{SE} \\ee with one of the most elementary bell-shaped\npotentials $V^{(bs)}(r) = G/{\\cosh^2 r}$ is particularly useful.\nIts applications range from the analyses of stability and\nquantization of solitons \\cite{Bullough} to phenomenological\nstudies in atomic and molecular physics \\cite{physaa}, chemistry\n\\cite{physbb}, biophysics \\cite{physcc} and astrophysics\n\\cite{physdd}. Its appeal involves the solvability by different\nmethods \\cite{Levai} as well as a remarkable role in the\nscattering \\cite{Newton}. Its bound-state wave functions\nrepresented by Jacobi polynomials offer one of the most elementary\nillustrations of properties of the so called shape invariant\nsystems \\cite{Cooper}. The force $V^{(bs)}(r)$ is encountered in\nthe so called ${\\cal PT}$ symmetric quantum mechanics\n\\cite{Bender} where it appears as a Hermitian super-partner of a\ncomplex ``scarf\" model \\cite{RKRC}.\n\nCuriously enough, it is not too difficult to extend the exact\nsolvability of the potential $V^{(bs)}(r) $ to all its ``spiked\"\n(often called P\\\"{o}schl-Teller \\cite{Poeschl}) shape invariant\ngeneralizations\n \\be\nV^{(PT)}(r)\n= -\\frac{A(A+1)}{\\cosh^2 r} +\\frac{B(B-1)}{\\sinh^2 r}.\n\\label{sPTP}\n \\ee\nUnfortunately, as far as the one-dimensional Schr\\\"{o}dinger eq.\n(\\ref{SE}) becomes too singular at $B(B-1) \\neq 0$ the more\ngeneral force $V^{(PT)}(r) $ must be confined to the semi-axis,\ni.e., in most cases, to the $s$ wave in three dimensions with\ncoordinates $ r \\in (0, \\infty)$. This makes the ``improved\"\nP\\\"{o}schl-Teller model (\\ref{sPTP}) much less useful in practice\nsince its higher partial waves are not solvable. The impossibility\nof using eq. (\\ref{sPTP}) in three dimensions (or on the whole\naxis in one dimension at least) is felt unfortunate because the\nsingular potentials themselves are frequently needed in methodical\nconsiderations \\cite{Klauder} and in perturbation theory\n\\cite{Harrell}. They are encountered in phenomenological models\n\\cite{Sotona} and in explicit computations \\cite{Hall} but not too\nmany of them are solvable \\cite{Mathieu}. This was a strong\nmotivation of our present brief note on eq. (\\ref{SE}) +\n(\\ref{sPTP}).\n\nWe feel inspired by the pioneering letter \\cite{BB} where Bender\nand Boettcher introduced the so called ${\\cal PT}$ symmetry\n(meaning the commutativity of a complex Hamiltonian with the\nproduct of parity ${\\cal P}$ and time reversal ${\\cal T}$). They\nproposed its use as a possible source of the reality of spectra\nfor non-Hermitian Hamiltonian operators. For illustration they\nemployed the harmonic oscillator $V^{(HO)}(r) = r^2$ with the\ncomplex downward shift of its axis of coordinates,\n \\be\nr = x-i\\varepsilon, \\ \\ \\ \\ \\ \\ \\ x \\in (-\\infty,\\infty).\n\\label{shov} \\ee The ${\\cal PT}$ symmetry of their model\n$V^{(BB)}(x)=V^{(HO)}(x-ic) = x^2 - 2icx - c^2$ means its\ninvariance with respect to the simultaneous reflection $x \\to -x$\nand complex conjugation $i \\to -i$. Their example inspired their\ngeneral hypothesis that the ${\\cal PT}$ symmetry could by itself\nimply the reality of spectrum in some non-Hermitian models\n\\cite{Bender}.\n\nVarious other complex interactions have been tested and studied\nwithin this framework \\cite{review}. In particular, the\nthree-dimensional ${\\cal PT}$ symmetric harmonic oscillator of\nref. \\cite{ptho} offers us another key idea. The same shift\n(\\ref{shov}) has been employed there as a source of a {\\em\nregularization} of the strongly singular centrifugal term. As long\nas $1/(x- i\\varepsilon)^2 = (x+i\\varepsilon)^2 /\n(x^2+\\varepsilon^2)^2 $ at any $\\varepsilon \\neq 0$, this term\nremains nicely bounded in a way which is uniform with respect to\n$x$. Without any difficulties one may work with $V^{(RHO)}(x) =\nr^2(x) +\\ell(\\ell+ 1)/ r^2(x)$ on the whole real line of $x$. In\nwhat follows the same idea will be applied to the regularized\nP\\\"{o}schl-Teller-like potential\n \\ben V^{(RPT)}(x) = V^{(PT)}(x -\ni\\varepsilon), \\ \\ \\ \\ \\ \\ \\ 0 < \\varepsilon <\\pi/2.\n \\een\nThis potential is a simple function of the L\\'{e}vai's\n\\cite{Levai} variable $g(r)=\\cosh 2r$. As long as $g(x -\ni\\,\\varepsilon) = \\cosh 2x\\,\\cos 2\\varepsilon - i\\,\\sinh 2x\\,\\sin\n2\\varepsilon$, the new force is ${\\cal PT}$ symmetric on the real\nline of $x \\in (-\\infty, \\infty)$,\n \\ben V^{(RPT)}(-x)= [V^{(RPT)}(x)]^*.\n \\een\nDue to the estimates $\|\\sinh^2(x-i\\varepsilon)\|^2 = \\sinh^2 x\n\\cos^2 \\varepsilon +\\cosh^2 x \\sin^2 \\varepsilon = \\sinh^2 x\n+\\sin^2 \\varepsilon$ and $\|\\cosh^2(x-i\\varepsilon)\|^2 = \\sinh^2 x\n+\\cos^2 \\varepsilon$ the regularity of $V^{(RPT)}(x)$ is\nguaranteed for all its parameters $\\varepsilon \\in (0, \\pi/2)$.\n\nIn a way paralleling the three-dimensional oscillator the mere\nanalytic continuation of the $s-$wave bound states does not give\nthe complete solution. One must return to the original\ndifferential equation (\\ref{SE}). There we may conveniently fix $A\n+1/2=\\alpha>0$ and $B-1/2= \\beta>0$ and write\n \\be\n\\left (-\\,\\frac{d^2}{dx^2} +\n\\frac{\\beta^2-1/4}{\\sinh^2 r(x)}\n -\\frac{\\alpha^2-1/4}{\\cosh^2 r(x)} \\right )\n \\, \\psi(x) = E \\, \\psi(x), \\ \\ \\ \\ \\ \\ r(x)= x-i\\varepsilon.\n\\label{SEb}\n \\ee\nThis is the Gauss differential equation\n \\be\nz(1+z)\\,\\varphi''(z) +[c+(a+b+1)z]\\,\\varphi'(z)\n+ab\\,\\varphi(z)=0\n\\label{gauss}\n \\ee\nin the new variables\n \\ben\n\\psi(x) = z^\\mu(1+z)^\\nu\\varphi(z),\n\\ \\ \\ \\ \\ \\ \\ \\ z = \\sinh^2r(x)\n \\een\nusing the suitable re-parameterizations\n \\ben\n\\alpha^2=(2\\nu-1/2)^2,\n\\ \\ \\ \\ \\ \\ \\ \\ \\\n \\beta^2=(2\\mu-1/2)^2,\n\\ \\ \\ \\ \\ \\ \\ \\ \\\n \\een\n \\ben\n2\\mu+1/2=c, \\ \\ \\ \\ \\\n2\\mu+2\\nu=a+b, \\ \\ \\ \\ \\\nE= -(a-b)^2.\n \\een\nIn the new notation we have the wave functions\n \\be\n\\psi(x) = \\sinh^{\\tau \\beta+1/2}[r(x)]\n \\cosh^{\\sigma\\alpha+1/2}[r(x)]\\,\\varphi[z(x)]\n\\label{formula}\n \\ee\nwith the sign ambiguities $\\tau = \\pm 1$ and $\\sigma=\\pm 1$ in\n$2\\mu=\\tau \\beta+1/2$ and $2\\nu=\\sigma\\alpha+1/2$. This formula\ncontains the general solution of hypergeometric eq. (\\ref{gauss}),\n \\be\n\\varphi(z) = C_1\\ _2F_1(a,b;c;-z) + C_2z^{1-c}\\\n_2F_1(a+1-c,b+1-c;2-c;-z). \\label{gensol}\n \\ee\nThe solution should obey the complex version of the\nSturm-Liouville oscillation theorem \\cite{Hille}. In the case of\nthe discrete spectra this means that we have to demand the\ntermination of our infinite hypergeometric series. This suppresses\nan asymptotic growth of $\\psi(x)$ at $x\\to\\pm\\infty$.\n\nIn a deeper analysis let us first put $C_2=0$. We may satisfy the\ntermination condition by the non-positive integer choice of\n$b=-N$. This implies that $a=N+1 +\\sigma\\alpha +\\tau \\beta$ is\nreal and that our wave function may be made asymptotically\n(exponentially) vanishing under certain conditions. Inspection of\nthe formula (\\ref{formula}) recovers that the boundary condition\n$\\psi(\\pm \\infty) = 0$ will be satisfied if and only if\n \\ben 1 \\leq 2N+1\\leq 2N_{max}+1<-\\sigma\n\\alpha - \\tau \\beta.\n \\een The closed Jacobi polynomial\nrepresentation of the wave functions follows easily,\n \\ben\n\\varphi[z(x)] =C_1\\ \\frac{N!\\Gamma(1+\\tau \\beta)}{\\Gamma(N+1+\\tau\n\\beta)} \\ P_N^{(\\tau \\beta,\\sigma\\alpha)}[\\cosh 2r(x)].\n \\een\nThe final insertions of parameters define the spectrum of\nenergies,\n \\be\nE=-( 2N+1+\\sigma \\alpha + \\tau \\beta)^2 < 0. \\label{energy}\n \\ee\nNow we have to return to eq. (\\ref{gensol}) once more. A careful\nanalysis of the other possibility $C_1=0$ does not recover\nanything new. The same solution is obtained, with $\\tau$ replaced\nby $-\\tau$. We may keep $C_2=0$ and mark the two independent\nsolutions by the sign $\\tau$. Once we define the maximal integers\n$N_{max}^{(\\sigma,\\tau)}$ which are compatible with the inequality\n \\be\n2N_{max}^{(\\sigma,\\tau)}+1< -\\sigma\\alpha-\\tau \\beta \\label{maxes}\n \\ee\nwe get the constraint $N \\leq N_{max}^{(\\sigma,\\tau)}$. The set of\nour main quantum numbers is finite.\n\nLet us now compare our final result (\\ref{energy}) with the known\n$\\varepsilon=0$ formulae for $s$ waves \\cite{Levai}. An additional\nphysical boundary condition must be imposed in the latter singular\nlimit \\cite{conditab}. This condition fixes the unique pair\n$\\sigma = -1$ and $\\tau = +1$. Thus, the set of the $s-$wave\nenergy levels $E_N$ is not empty if and only if $\\alpha- \\beta>\n1$. In contrast, all our $\\varepsilon > 0$ potentials acquire a\nuniform bound $\|V^{(RPT}(x)\| < const < \\infty$. Due to their\nregularity, no additional constraint is needed. Our new spectrum\n$E^{(\\sigma,\\tau)}_N$ becomes richer. For the sufficiently strong\ncouplings it proves composed of the three separate parts,\n \\ben\nE^{(-,-)}_N< 0, \\ \\ \\ \\ \\ 0 \\leq N \\leq N_{max}^{(-,-)}, \\ \\ \\ \\ \\\n\\ \\alpha+ \\beta > 1,\n \\een\n \\be\nE^{(-,+)}_N<0, \\ \\ \\ \\ \\ \\ 0 \\leq N \\leq N_{max}^{(-,+)}, \\ \\ \\ \\\n\\ \\ \\alpha> \\beta + 1,\n \\ee\n \\ben\nE^{(+,-)}_N<0, \\ \\ \\ \\ \\ \\ 0 \\leq N \\leq N_{max}^{(+,-)},\n \\ \\ \\ \\ \\ \\\n\\beta > \\alpha+1.\n \\een\nThe former one is non-empty at $ A + B > 1$ (with our above\nseparate conventions $A > -1/2$ and $B > 1/2$). Concerning the\nlatter two alternative sets, they may exist either at $A> B$ or\nat $B > A+2$, respectively. We may summarize that in a parallel to\nthe ${\\cal PT}$ symmetrized harmonic oscillator of ref.\n\\cite{ptho} we have the $N_{max}^{(-,+)}+1$ quasi-odd or\n``perturbed\", analytically continued $s-$wave states (with a nodal\nzero near the origin) complemented by certain additional\nsolutions.\n\nIn the first failure of a complete analogy the number\n$N^{(-,-)}_{max}+1$ of our quasi-even states proves systematically\nhigher than $N^{(-,+)}_{max}+1$, especially at the larger\n``repulsion\" $ \\beta \\gg 1$. This is a certain paradox,\nstrengthened by the existence of another quasi-odd family which\nbehaves very non-perturbatively. Its members (with the ground\nstate $\\psi_0^{(+,-)}(x) =\\cosh^{A+1} [r(x)]\\sinh^{1-B} [r(x)]$\netc) do not seem to have any $s-$wave analogue. They are formed at\nthe prevalent repulsion $B>A+2$ which is even more\ncounter-intuitive. The exact solvability of our example enables us\nto understand this apparent paradox clearly. In a way\ncharacteristic for many ${\\cal PT}$ symmetric systems some of the\nstates are bound by an antisymmetric imaginary well. The whole\nhistory of the ${\\cal PT}$ symmetric models starts from the purely\nimaginary cubic force \\cite{Bessis} after all. A successful\ndescription of its perturbative forms $V(x) = \\omega x^2+i\\lambda\n\\,x^3$ is not so enigmatic \\cite{Graffi}, especially due to its\nanalogies with the real and symmetric $V(x) = \\omega x^2+\n\\lambda\\,x^4$ \\cite{Alvarez}. The similar mechanism creates the\nstates with $(\\sigma,\\tau)=(+,-)$ in the present example. A\nsignificant novelty of our new model $V^{(RPT)}(x)$ lies in the\ndominance of its imaginary component {\\em at the short distances},\n$x \\approx 0$. Indeed, we may expand our force to the first order\nin the small $\\varepsilon>0$. This gives the approximation\n \\be\n\\frac{1}{\\sinh^2(x-i\\varepsilon)}=\n\\frac{\\sinh^2(x+i\\varepsilon)}{(\\sinh^2x + \\sin^2\\varepsilon)^2}\n=\\frac{1}{\\sinh^2x}+2i\\varepsilon \\frac{\\cosh x}{\\sinh^3 x}\n+{\\cal O}(\\varepsilon^2).\n \\label{sini}\n \\ee\nWe see immediately the clear prevalence of the imaginary part at\nthe short distances, especially at all the negligible $A = {\\cal\nO}(\\varepsilon^2)$.\n\nAn alternative approach to the above paradox may be mediated by a\nsudden transition from the domain of a small $\\varepsilon \\approx\n0$ to the opposite extreme with $\\varepsilon \\approx \\pi/2$. This\nis a shift which changes $\\cosh x$ into $\\sinh x$ and vice versa.\nIt intertwines the role of $\\alpha$ and $ \\beta$ as a strength of\nthe smooth attraction and of the singular repulsion, respectively.\nThe perturbative/non-perturbative interpretation of both our\nquasi-odd subsets of states becomes mutually interchanged near\nboth the extremes of $\\varepsilon$.\n\nThe dominant part (\\ref{sini}) of our present model leaves its\nasymptotics comparatively irrelevant. In contrast to many other\n${\\cal PT}$ symmetric models as available in the current\nliterature our potential vanishes asymptotically,\n \\ben V^{(RPT)}(x) \\to 0, \\ \\ \\ \\ \\ \\ \\ \\ x \\to \\pm \\infty.\n \\een\nAn introduction and analysis of continuous spectra in the ${\\cal\nPT}$ symmetric quantum mechanics seems rendered possible at\npositive energies. This question will be left open here. In the\nsame spirit of a concluding remark we may also touch the problem\nof the possible breakdown of the ${\\cal PT}$ symmetry. This has\nrecently been studied on the background of the supersymmetric\nquantum mechanics \\cite{Canata}. In our present solvable example\nthe violation of the ${\\cal PT}$ symmetry is easily mimicked by\nthe complex choice of the couplings $\\alpha$ and $ \\beta$. Due to\nour closed formulae the energies will still stay real, provided\nonly that ${\\rm Im}\\ (\\sigma\\alpha+ \\tau \\beta)=0$. Unfortunately,\nthe questions of this type lie already beyond the scope of our\npresent short communication.\n\n\n\\newpage\n\\begin{thebibliography}{99}\n\n\\bibitem{Bullough}\nG. B. Whitham, Linear and Nonlinear Waves (John Wiley and Sons,\nNew York, 1974);\n\nR. Jackiw, Rev. Mod. Phys. 49, 681 (1977).\n\n\\bibitem{physaa}\nC. Eckart,\n % \"The Penetration of a Potential Barrier by Electrons\"\n Phys. Rev. 35, 1303 (1930);\n\nV. I Kukulin, V. M. Krasnopol'sky and J. Hor\\'{a}\\v{c}ek, Theory\nof resonances: Principles and Applications (Kluwer, Dordrecht,\n1989);\n\nR. Dutt, A. Gangopadhyaya, C. Rasinarin and U. Sukhatme, Phys.\nRev. A 60, 3482 (1999).\n\n\\bibitem{physbb}\n R. P. Bell,\n The Tunnel Effect in Chemistry\n (Chapman and Hall, London, 1980).\n\n\\bibitem{physcc}\n D. De Vault, Quantum Mechanical Tunneling in Biological Systems\n (Cambridge University Press, London, 1984).\n\n\\bibitem{physdd}\nH. R. Beyer, Comm. Math. Phys. 204, 397 (1999).\n\n\\bibitem{Levai}\nG. L\\'{e}vai, J. Phys. A: Math. Gen. 22, 689 (1989).\n\n\\bibitem{Newton}\nR. G. Newton, Scattering Theory of Waves and Particles\n (Springer Verlag, New York, 1982), p. 438.\n\n\\bibitem{Cooper}\nF. Cooper, A. Khare and U. Sukhatme, Phys. Rep. 251, 267 (1995).\n\n\\bibitem{Bender}\nC. M. Bender, S. Boettcher and P. N. Meisinger,\nJ. Math. Phys. 40, 2201 (1999).\n\n\\bibitem{RKRC}\nA. A. Andrianov, M. V. Ioffe, F. Cannata and J. P. Dedonder, Int.\nJ. Mod. Phys. A 14, 2675 (1999);\n\nB. Bagchi and R. Roychoudhury, preprint LANL quant-ph/9911104, to\nappear in J. Phys. A: Math. Gen.;\n\nM. Znojil, LANL preprint quant-ph/9911116.\n\n\\bibitem{Poeschl}\nG. P\\\"{o}schl and E. Teller, Z. Physik 83, 143 (1933);\n\nS. Fl\\\"{u}gge, Practical Quantum Mechanics I (Springer, Berlin,\n1971).\n\n\\bibitem{Klauder}\nL. C. Detwiler and J. R. Klauder, Phys. Rev. D 11, 1436 (1975).\n\n\\bibitem{Harrell}\nE. M. Harrell, Ann. Phys. (NY) 105, 379 (1977).\n\n\\bibitem{Sotona}\nM. Sotona and J. \\v{Z}ofka, Phys. Rev. C 10, 2646 (1974) and Czech.\nJ. Phys. B 28, 593 (1978);\n\nR. Dutt and Y. P. Varshni, J. Phys. B: At. Mol. Phys. 20, 2437 (1987);\n\nJ. Vacek, K. Konvi\\v{c}ka and P. Hobza, Chem. Phys. Lett. 220, 83 (1994).\n\n\\bibitem{Hall}\nF. M. Fern\\'{a}ndez, Phys. Lett. A 160, 511 (1991);\n\nR. Hall and N. Saad, J. Phys. A: Math. Gen. 32, 133 (1999).\n\n\\bibitem{Mathieu}\nA. Kratzer, Z. Physik 3, 289 (1920);\n\nN. A. W. Holzwarth, J. Math. Phys. 14, 191 (1973);\n\nE. Papp, Phys. Lett. A 157, 192 (1991).\n\n\\bibitem{BB}\nC. M. Bender and S. Boettcher, Phys. Rev. Lett. 24, 5243 (1988).\n\n\\bibitem{review}\nC. M. Bender and A. V. Turbiner, Phys. Lett. A 173, 442 (1993);\n\nC. M. Bender and S. Boettcher, J. Phys. A: Math. Gen. 31 (1998)\nL273;\n\nE. Delabaere and F. Pham, Phys. Lett. A 250, 25 and 29 (1998);\n\nF. Fern\\'andez, R. Guardiola, J. Ros and M. Znojil, J. Phys. A:\nMath. Gen. 31 (1998) 10105;\n\nM. Znojil, Phys. Lett. A 264, 108 (1999).\n\n\\bibitem{ptho}\nM. Znojil, Phys. Lett. A 259, 220 (1999).\n\n\\bibitem{Hille}\nE. Hille, Lectures on Ordinary Differential Equations\n(Addison-Wesley, Reading, 1969).\n\n\\bibitem{conditab}\nK. M. Case, Phys. Rev. 80, 797 (1950);\n\nL. D. Landau and E. M. Lifschitz, Quantum Mechanics (Pergamon,\nLondon, 1960), ch. V, par. 35;\n\nW. M. Frank, D. J. Land and R. M. Spector, Rev. Mod. Phys. 43, 36\n(1971);\n\nM. Znojil, LANL preprint quant-ph/9811088 and Phys. Rev. A, to\nappear.\n\n\\bibitem{Bessis}\nDaniel Bessis, private communication (1992).\n\n\\bibitem{Graffi}\nE. Caliceti, S. Graffi and M. Maioli, Commun. Math. Phys. 75, 51\n(1980).\n\n\\bibitem{Alvarez}\nG. Alvarez, J. Phys. A: Math. Gen. 27, 4589 (1995).\n\n\\bibitem{Canata}\nF. Cannata, G. Junker and J. Trost, Phys. Lett. A 246, 219 (1998).\n\n\\end{thebibliography}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912079.extracted_bib", "string": "{Bullough G. B. Whitham, Linear and Nonlinear Waves (John Wiley and Sons, New York, 1974); R. Jackiw, Rev. Mod. Phys. 49, 681 (1977)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{physaa C. Eckart, % \"The Penetration of a Potential Barrier by Electrons\" Phys. Rev. 35, 1303 (1930); V. I Kukulin, V. M. Krasnopol'sky and J. Hor\\'{a\\v{cek, Theory of resonances: Principles and Applications (Kluwer, Dordrecht, 1989); R. Dutt, A. Gangopadhyaya, C. Rasinarin and U. Sukhatme, Phys. Rev. A 60, 3482 (1999)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{physbb R. P. Bell, The Tunnel Effect in Chemistry (Chapman and Hall, London, 1980)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{physcc D. De Vault, Quantum Mechanical Tunneling in Biological Systems (Cambridge University Press, London, 1984)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{physdd H. R. Beyer, Comm. Math. Phys. 204, 397 (1999)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Levai G. L\\'{evai, J. Phys. A: Math. Gen. 22, 689 (1989)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Newton R. G. Newton, Scattering Theory of Waves and Particles (Springer Verlag, New York, 1982), p. 438." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Cooper F. Cooper, A. Khare and U. Sukhatme, Phys. Rep. 251, 267 (1995)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Bender C. M. Bender, S. Boettcher and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{RKRC A. A. Andrianov, M. V. Ioffe, F. Cannata and J. P. Dedonder, Int. J. Mod. Phys. A 14, 2675 (1999); B. Bagchi and R. Roychoudhury, preprint LANL quant-ph/9911104, to appear in J. Phys. A: Math. Gen.; M. Znojil, LANL preprint quant-ph/9911116." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Poeschl G. P\\\"{oschl and E. Teller, Z. Physik 83, 143 (1933); S. Fl\\\"{ugge, Practical Quantum Mechanics I (Springer, Berlin, 1971)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Klauder L. C. Detwiler and J. R. Klauder, Phys. Rev. D 11, 1436 (1975)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Harrell E. M. Harrell, Ann. Phys. (NY) 105, 379 (1977)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Sotona M. Sotona and J. \\v{Zofka, Phys. Rev. C 10, 2646 (1974) and Czech. J. Phys. B 28, 593 (1978); R. Dutt and Y. P. Varshni, J. Phys. B: At. Mol. Phys. 20, 2437 (1987); J. Vacek, K. Konvi\\v{cka and P. Hobza, Chem. Phys. Lett. 220, 83 (1994)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Hall F. M. Fern\\'{andez, Phys. Lett. A 160, 511 (1991); R. Hall and N. Saad, J. Phys. A: Math. Gen. 32, 133 (1999)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Mathieu A. Kratzer, Z. Physik 3, 289 (1920); N. A. W. Holzwarth, J. Math. Phys. 14, 191 (1973); E. Papp, Phys. Lett. A 157, 192 (1991)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{BB C. M. Bender and S. Boettcher, Phys. Rev. Lett. 24, 5243 (1988)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{review C. M. Bender and A. V. Turbiner, Phys. Lett. A 173, 442 (1993); C. M. Bender and S. Boettcher, J. Phys. A: Math. Gen. 31 (1998) L273; E. Delabaere and F. Pham, Phys. Lett. A 250, 25 and 29 (1998); F. Fern\\'andez, R. Guardiola, J. Ros and M. Znojil, J. Phys. A: Math. Gen. 31 (1998) 10105; M. Znojil, Phys. Lett. A 264, 108 (1999)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{ptho M. Znojil, Phys. Lett. A 259, 220 (1999)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Hille E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, 1969)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{conditab K. M. Case, Phys. Rev. 80, 797 (1950); L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Pergamon, London, 1960), ch. V, par. 35; W. M. Frank, D. J. Land and R. M. Spector, Rev. Mod. Phys. 43, 36 (1971); M. Znojil, LANL preprint quant-ph/9811088 and Phys. Rev. A, to appear." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Bessis Daniel Bessis, private communication (1992)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Graffi E. Caliceti, S. Graffi and M. Maioli, Commun. Math. Phys. 75, 51 (1980)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Alvarez G. Alvarez, J. Phys. A: Math. Gen. 27, 4589 (1995)." }, { "name": "quant-ph9912079.extracted_bib", "string": "{Canata F. Cannata, G. Junker and J. Trost, Phys. Lett. A 246, 219 (1998)." } ]
quant-ph9912080	Catalysis of entanglement manipulation for mixed states	[ { "author": "Jens Eisert and Martin Wilkens" } ]	We consider entanglement-assisted remote quantum state manipulation of bi-partite mixed states. Several aspects are addressed: we present a class of mixed states of rank two that can be transformed into another class of mixed states under entanglement-assisted local operations with classical communication, but for which such a transformation is impossible without assistance. Furthermore, we demonstrate enhancement of the efficiency of purification protocols with the help of entanglement-assisted operations. Finally, transformations from one mixed state to mixed target states which are sufficiently close to the source state are contrasted to similar transformations in the pure-state case.	[ { "name": "quant-ph9912080.tex", "string": "\n\\documentstyle[prl,tighten,aps,epsf,multicol,palatino]{revtex}\n\\begin{document}\n\n\\draft\n\n\n\\title{Catalysis of entanglement manipulation for mixed states}\n\\author{Jens Eisert and Martin Wilkens}\n\\address{Institut f{\\\"u}r Physik, Universit{\\\"a}t Potsdam, 14469 Potsdam,\nGermany}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nWe consider entanglement-assisted remote quantum \nstate manipulation of bi-partite mixed states.\nSeveral aspects are addressed:\nwe present a class of mixed states of\nrank two that can be transformed into another class of \nmixed states under entanglement-assisted local\noperations with classical communication,\nbut for which such a transformation\nis impossible without assistance.\nFurthermore, we demonstrate enhancement \nof the efficiency of purification protocols\nwith the help of entanglement-assisted operations. \nFinally, transformations from one mixed \nstate to mixed\ntarget states which are sufficiently close\nto the source state are contrasted to\nsimilar transformations in the pure-state case.\n\\end{abstract}\n\n\\pacs{PACS-numbers: 03.67.-a, 03.65.Bz}\n\n\\bigskip\\medskip\n\n\n\n\\begin{multicols}{2}\n\\narrowtext\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Introduction}\n%The feature that remotely located quantum systems may exhibit \n%quantum correlations, that is, entanglement, \n%is a particularly \n%interesting and still not fully understood aspect of quantum mechanics\n%\\cite{Contemp}. \nEntanglement between spatially separated quantum systems\nhas important implications on fundamental issues of quantum\nmechanics and forms the basis for most of \nthe practical applications of quantum information\ntheory \\cite{Contemp,Practical}. In many of\nthese applications two or more parties have direct\naccess to only\nparts of a composite quantum system, but may communicate\nby classical means and may thereby \ncoordinate their actions. \nIn the light of recent \nprogress in quantum information\ntheory entanglement is often viewed as the essential resource \nfor processing and transmitting quantum information.\n\nAs has been demonstrated in Ref.\\ \\cite{Jonathan},\nentanglement is indeed an intriguing kind of resource:\nthe mere presence of entanglement can be\nan advantage when the task it to \ntransform an initial state into a certain\nfinal state \nwith the use of local quantum operations and \nclassical communication (LOCC).\nThere are indeed target states which cannot be \nreached by LOCC starting from a particular\ninitial state, but which can be reached\nwith the assistance of a\ndistributed pair of auxiliary quantum systems in a \nparticular known state, \n%\n%\n%The class of operations that can be performed \n%under LOCC on a bipartite quantum system \n%is strictly larger \n%when additional quantum systems in an entangled state\n%are accessible than without such ``catalyst states'', \neven though these auxiliary quantum systems\nare left in {\\it exactly the same state}\\/.\n% and remain finally\n%completely uncorrelated to the quantum system of interest.\nSuch transformations are called entanglement-assisted\nLOCC operations (ELOCC).\n\nThis phenomenon is quite remarkable as the entanglement\nwhich serves as a ''catalyst'' for the otherwise\nforbidden ''reaction'' is not consumed. \nThe basis of the example given in Ref.\\ \\cite{Jonathan}\nis a criterion \npresented in Ref. \\cite{Nielsen}: \n%which offers\n%an elegant solution:\nA joint pure state corresponding to \n$\|\\psi\\rangle$ can be transformed into another \n$\|\\phi\\rangle$ with the use of LOCC if\nand only if the set of ordered Schmidt coefficients \ncharacterizing\nthe initial state is majorized \\cite{Majo} by the set\nof ordered Schmidt coefficients of the final state.\nCuriously, it is the strange class of \nELOCC operations\nthat adds a new flavor to the initial question \nraised in Ref.\\ \\cite{Nielsen},\n``What tasks may be accomplished using a given \nphysical resource?'' The class of ELOCC operations\nis in fact more powerful than LOCC even without \na concomitant consumption of the\nphysical resource entanglement \\cite{Jonathan,NewDaniel}.\n\nIn practical applications one\nwould expect to always deal with entangled mixed states \nrather than with pure states.\nUnfortunately, such a convenient tool as the majorization\ncriterion is missing in the mixed-state case, and\nthe question whether a particular entanglement transformation\nfrom one mixed state into another mixed state is possible seems \nto be much more involved \\cite{After}. \nIn mixed quantum mechanical states \nboth classical correlations and\nintrinsic quantum correlations may be present, which\nmakes the structure of mixed-state entanglement a more\ncomplex matter. A different aspect of the same problem is the\nwell known fact that a representation of a mixed state in terms\nof pure states is not uniquely defined, and\nit is essentially \nthis ambiguity that prohibits a straightforward\napplication of the majorization criterion.\n\n%However, even without a fully developed\n%theory of catalysis of\n%entanglement-assisted manipulations\n%and a theory \n%of mixed-state entanglement transformations, it can\n%be judged whether ELOCC\n%are more powerful than mere LOCC when processing mixed\n%states. \n%It is the purpose of this paper to show \n%that they actually are. \nIn this letter we demonstrate that even for mixed states the\nset of tasks that can be accomplished with \nentanglement-assisted local operations is strictly larger than the set\nof tasks which may be performed with mere LOCC.\nThis fact is not obvious a priori,\nbearing in mind that e.g.\\ \npure states and mixed states behave very differently\nas far as purification is concerned \\cite{Kent}. \nThe problem of catalysis of \nentanglement manipulation for mixed states\nwill be approached as follows:\n(i) We give a class of mixed states of rank two\nthat can be transformed into representants of\nanother class of mixed states with ELOCC but\nnot with LOCC, (ii) we show that\nthere are cases for which \nthe proportion of a certain pure state in a mixture\ncan be increased more efficiently with ELOCC\noperations than with sole LOCC, \n(iii) purification schemes are investigated for\na practically important class of mixed states, and (iv)\n``small transformations'' in the interior of the state\nspace are compared with\nsimilar entanglement manipulations in the pure-state\ncase. \n\n\n\\section{Definitions}\n%{\\it Definitions. --}\\/\nLet $\\sigma$ and $\\rho$ be states taken from\nthe state space ${\\cal S}({\\cal H})$ over ${\\cal H}$,\nwhere ${\\cal H}={\\cal H}_A\\otimes {\\cal H}_B$ is the \nHilbert space associated with a bipartite quantum system\nconsisting of parts $A$ and $B$.\nWe write in the following $\\sigma\\rightarrow\\rho$ under\nLOCC if $\\sigma$ can be transformed into $\\rho$ by applying local\ntransformations and classical communication \\cite{Nielsen}.\nA pair of states $\\rho,\\sigma$ is called\nincommensurate if both $\\sigma\\not\\rightarrow\\rho$\nand $\\rho\\not\\rightarrow\\sigma$ under LOCC.\nFor {\\it pure}\\/ states $\\sigma$ and $\\rho$ the\n(necessary and sufficient) majorization\ncriterion \nfor $\\sigma\\rightarrow\\rho$\nunder LOCC reads as \\cite{Nielsen}\n\\begin{equation}\n\t\\sum_{i=1}^k \\alpha_i\\leq \\sum_{i=1}^k \\beta_i\n\t\\,\\,\\,\\,\\,\\,\\,\\,\\,\n\t$ for all $\n\tk=1,...,N-1,\\label{maj}\n\\end{equation}\n$N=\\dim[{\\cal H}_A]=\\dim[{\\cal H}_B] $, where $\\alpha_1$, ..., $\\alpha_N$\nand $\\beta_1$, ..., $\\beta_N$ with\n$1\\geq\\alpha_1\\geq ...\\geq\\alpha_N\\geq0$ and\n$1\\geq\\beta_1\\geq...\\geq \\beta_N\\geq0$ are\nthe eigenvalues of ${\\rm tr}_A [\\sigma]$\nand ${\\rm tr}_A [\\rho]$, respectively. Such a list\nis also referred to as an ordered list.\nThe content of the conditions stated in Eq.\\ (\\ref{maj})\nis in the following abbreviated as \n${\\rm tr}_A [\\sigma]\\prec {\\rm tr}_A [\\rho]$,\nwith the majorization relation $\\prec$ \\cite{Majo}.\nAs for LOCC operations we use the notation \n$\\sigma\\rightarrow\\rho$ under ELOCC,\nif \n\\begin{equation}\n\t\\sigma\\otimes\\omega\\rightarrow\\rho\\otimes\\omega \n\\end{equation}\nfor an appropriately chosen catalyst state $\\omega$ \n\\cite{Jonathan}.\nThis state $\\omega$ is an entangled state of \nanother bi-partite quantum system. Note that in the \ncourse of the transformation this state \nremains fully unchanged.\n\n% taken\n%from the state space $\\in{\\cal S}\n%(\\tilde{\\cal H})$,\n%that is, if $\\sigma$ can be transformed into $\\rho$ by\n%some entanglement-assisted manipulation. $\\tilde{\\cal H}$\n%is the Hilbert space belonging to $\\omega$; again,\n%$\\tilde{\\cal H}$ is a tensor product \n%$\\tilde{\\cal H}=\\tilde{\\cal H}_A\\otimes\\tilde{\\cal H}_B$\n%of\n%two Hilbert spaces belonging to systems $A$ and $B$,\n%respectively.\n\n\n\n\\section{Mixed-state catalysis of entanglement manipulation}\n%{\\it Mixed-state catalysis of entanglement manipulation. --}\\/\nThe first result concerns the existence of\nincommensurate genuinely mixed states such\nthat with the use of some appropriately chosen\ncatalyst state, the initial state can be converted into \nthe final state while fully retaining the\ncatalyst state. That is, there exist \nmixed \nstates $\\sigma,\\rho\\in{\\cal S}({\\cal H})$ \nsuch that\n $\\sigma\\rightarrow \\rho$ under ELOCC\nbut not\n $\\sigma\\rightarrow \\rho$ under LOCC.\n``Genuinely'' mixed means here that \nthe projections appearing in the spectral\ndecomposition of the initial state\ncannot be locally distinguished. If this\nwere possible the \ninitial state would essentially be pure.\n\nTo see that mixed-state catalysis is possible we \nconstruct a class of states which exhibits\nthis phenomenon. For this class of \nstates the statement that \n$\\sigma\\rightarrow\\rho$ under ELOCC\nfollows immediately from \nthe theorem presented in Ref.\\ \\cite{Nielsen}.\nTo prove that such a transformation is\nimpossible under LOCC, the following \nLemma is useful.\\\\\n\n\n\\noindent{\\bf Lemma 1. --} Let $\\sigma$ and $\\rho$ \nbe mixed states of rank two of the form \n\\begin{mathletters}\n\\begin{eqnarray}\n\\sigma&=&\\lambda \|\\psi\\rangle\\langle\\psi\|\n+(1-\\lambda)\|\\eta\\rangle\\langle\\eta\|,\\\\\n\\rho&=&\\mu \n\|\\phi\\rangle\\langle\\phi\|\n+(1-\\mu)\|\\eta\\rangle\\langle\\eta\|,\\label{class}\n\\end{eqnarray}\n%\nwhere $\\mu=\\lambda \\,{\\rm tr}[\\chi]$, \n\\end{mathletters}\n\\begin{equation}\n\t\\chi=\\Pi\|\\psi\\rangle\\langle\\psi\|\\Pi,\n\\end{equation} \nand\n$\\Pi=1-\|\\eta\\rangle\\langle\\eta\|$. \n$\|\\psi\\rangle\\langle\\psi\|$ and $\|\\phi\\rangle\\langle\\phi\|$ are \nentangled pure states, while\n$\|\\eta\\rangle\\langle\\eta\|$ is a pure product state. \nFurthermore, %$\\Pi \|\\phi\\rangle\\langle\\phi\|\\Pi=\n%\|\\phi\\rangle\\langle\\phi\|$. \n$\|\\langle\\eta\|\\phi\\rangle\|^2=0$.\nThen $\\sigma\\rightarrow\\rho$ under\nLOCC implies that \n\\begin{equation}\n\t\\frac{{\\rm tr}_A [\\chi]}{{\\rm tr} [\\chi]}\n\t \\prec\n\t{\\rm tr}_A [\|\\phi\\rangle\\langle\\phi\|].\n\\end{equation}\n\n\n{\\it Proof:} \nAssume that $\\sigma\\rightarrow\\rho$ under\nLOCC.\nThe set of LOCC operations is included in the set\nof separable operations \\cite{Plenio,Rains}, that is,\ncompletely positive and trace-preserving maps\nthat can be written in the form \n$\\sigma\\longmapsto\n\\sum_i (A_i\\otimes B_i)\\sigma (A_i\\otimes B_i)^\\dagger$\nwith Kraus-operators $A_i$, $B_i$, $i=1,2,...$, acting\nin ${\\cal H}_A$ and ${\\cal H}_B$, respectively, \nwhere the trace-preserving property manifests as\n$\\sum_i A_i^\\dagger A_i=1$, $\\sum_i B_i^\\dagger B_i=1$.\nFor each $i$ the image of $\\sigma$ must be element\nin the range of $\\rho$,\n\\begin{equation} \n(A_i\\otimes B_i)\\sigma (A_i\\otimes B_i)^\\dagger\\in {\\rm range}(\\rho).\n\\end{equation} \nSince there is only a single product\nvector included in the range of $\\rho$ (which \nthen amounts to a best separable approximation in the\nsense of \\cite{Lewenstein}), \nthe state\n$\|\\psi\\rangle\\langle\\psi\|$ must be mapped on\n$\\nu \|\\phi\\rangle\\langle\\phi\|+(1-\\nu)\n\|\\eta\\rangle\\langle\\eta\|$, where $\\nu=\\mu/\\lambda$.\n%\\begin{eqnarray}\n%&&\\Pi(A_i\\otimes B_i)\|\\psi\\rangle\\langle\\psi\|(A_i\\otimes B_i)^\\dagger\n%\\Pi\\nonumber\\\\=\n%&&\\Pi(A_i\\otimes B_i)\\Pi\|\\psi\\rangle\\langle\\psi\|\\Pi(A_i\\otimes B_i)^\\dagger\n%\\Pi\n%\\end{eqnarray}\n$\\Pi (A_i\\otimes B_i) \|\\psi\\rangle=\n \\Pi (A_i\\otimes B_i) \\Pi \|\\psi\\rangle$\nfor all $i$, and hence,\n\\begin{eqnarray}\n\\nu&=&{\\rm tr}\\bigl[\n\\Pi\\sum_i (A_i\\otimes B_i) \|\\psi\\rangle\\langle\\psi\|(A_i\\otimes B_i)^\\dagger\n\\Pi\\bigr]\\nonumber\\\\\n&=&{\\rm tr}\\bigl[ \\sum_i \\Pi (A_i\\otimes B_i) \\chi (A_i\\otimes B_i)^\\dagger\n\\Pi\\bigr]\\leq{\\rm tr}[\\chi].\n\\end{eqnarray}\nAs ${\\rm tr}[\\chi]=\\nu$, it follows that\n$\\chi/{\\rm tr}[\\chi]\\longrightarrow \|\\phi\\rangle\\langle\\phi\|$\nunder LOCC, which in turn implies\nby the theorem in Ref.\\ \\cite{Nielsen}\nthat ${\\rm tr}_A [\\chi]/{\\rm tr} [\\chi]\n\t \\prec\n\t{\\rm tr}_A [\|\\phi\\rangle\\langle\\phi\|]$.\n$\\hfill{\\Box}$\n\n\nThe following one-parameter classes of states of rank two\nprovide an example of catalysis for mixed states.\n%As an example of catalysis for mixed states, take now the\n%following one-parameter classes of states of rank two:\nTake\n${\\cal H}={\\cal H}_A\\otimes {\\cal H}_B$ with\n${\\cal H}_A, {\\cal H}_B=\n\\text{span}\\{\|1\\rangle, ...,\|5\\rangle \\}$\nand let \n\\begin{mathletters}\n\\begin{eqnarray}\\label{example}\n\\sigma&=&\\lambda\|\\psi\\rangle\\langle\\psi\|+(1-\\lambda)\|55\\rangle\\langle55\|,\\\\\n\\rho&=&\\mu \|\\phi\\rangle\\langle\\phi\|\n+(1-\\mu)\|55\\rangle\\langle55\|,\n\\end{eqnarray}\nwith $\\mu=0.95\\;\\lambda$ and \n\\end{mathletters}\n\\begin{mathletters}\n\\begin{eqnarray}\n\t\|\\psi\\rangle&=&\n\t\\sqrt{0.38}\|11\\rangle+\n\t\\sqrt{0.38}\|22\\rangle+\n\t\\sqrt{0.095}\|33\\rangle\\nonumber \\\\\n\t&+&\n\t\\sqrt{0.095}\|44\\rangle+\n\t\\sqrt{0.05} \|55\\rangle,\\label{psi}\\\\\n\t\|\\phi\\rangle&=&\n\t\\sqrt{0.5}\|11\\rangle+\n\t\\sqrt{0.25}\|22\\rangle+\n\t\\sqrt{0.25}\|33\\rangle.\\label{phi}\n\\end{eqnarray}\nThese states are\nclearly included in the sets of states considered\nin Lemma 1. \n\\end{mathletters}\nMoreover, the initial state $\\sigma$ is\ngenuinely mixed. \n\nFrom Lemma 1 it follows that \n$\\sigma\\not\\rightarrow\\rho$ under LOCC\nfor all values of $\\lambda\\in(0,1]$, \nas $\\chi/{\\rm tr}[\\chi]=\|\\varphi\\rangle\\langle\\varphi\|$,\nwhere \n\\begin{equation}\\label{tild}\n\|\\varphi\\rangle=\\sqrt{0.4}\|11\\rangle+\n\\sqrt{0.4}\|22\\rangle+\n\\sqrt{0.1}\|33\\rangle+\n\\sqrt{0.1}\|44\\rangle\n\\end{equation}\nas in Ref. \\cite{Jonathan}. Hence, \n\\begin{equation}\n\t\\frac{{\\rm tr}_A [\\chi]}{{\\rm tr} [\\chi]}\n\t \\not\\prec\n\t{\\rm tr}_A [\|\\phi\\rangle\\langle\\phi\|],\n\\end{equation}\nand therefore, $\\sigma\\not\\rightarrow\\rho$\nunder LOCC.\nHowever, it can be shown that\n$\\sigma\\rightarrow\\rho$ under ELOCC. One \nmay perform a local projective von-Neumann measurement\nin system $A$\nassociated with Kraus operators \n$A_1=\\sum_{i=1}^4 \|ii\\rangle\\langle ii\|$ and\n$A_2=\|55\\rangle\\langle55\|$\nsatisfying \n%\\begin{equation}\n$A_1^\\dagger A_1+\nA_2^\\dagger A_2=1$\n%\\end{equation}\n(compare also \n\\cite{Amnesia}). \nIf one gets the outcome\ncorresponding to $A_2$, no further \noperations are applied. \nIn the other case the final state is the pure state \n$\n\|\\varphi\\rangle\\langle\\varphi\|$ given by\nEq.\\ (\\ref{tild}). \n%\\begin{equation}\n%\|\\tilde\\psi\\rangle=\\sqrt{0.4}\|11\\rangle+\n%\\sqrt{0.4}\|22\\rangle+\n%\\sqrt{0.1}\|33\\rangle+\n%\\sqrt{0.1}\|44\\rangle.\n%\\end{equation}\nAs in Ref.\\ \\cite{Jonathan} this state can be\ntransformed into $\|\\phi\\rangle\\langle\\phi\|$\nwith the help of the catalyst state $\\omega=(\\sqrt{0.4}\|66\\rangle+\n\\sqrt{0.6}\|77\\rangle)(\\sqrt{0.4}\\langle66\|+\n\\sqrt{0.6}\\langle77\|)$ \\cite{HilbertRemark},\nsince \n\\begin{equation}\n{\\rm tr}_A [\|\\varphi\\rangle\\langle\\varphi\|\n\\otimes \\omega]\\prec \n{\\rm tr}_A [\|\\phi\\rangle\\langle\\phi\|\n\\otimes \\omega].\n\\end{equation} \nFinally, \nthe classical information about the outcomes is discarded\nin order to achieve $\\rho$. \nHence, it turns out that $\\sigma\\rightarrow\\rho$ under ELOCC but \n$\\sigma\\not\\rightarrow\\rho$ under LOCC. \n\n\n\\section{Increasing the proportion of a pure state in a mixture}\n\n%{\\it Increasing the proportion of a pure state in a mixture.--}\\/\nThe possibility of catalysis of entanglement manipulations\nhas an implication \non the efficiency of attempts to \nincrease the quota of some entangled \nstate $\|\\xi\\rangle\\langle\\xi\|$ in a mixed state $\\sigma$\nby applying a trace-preserving operation. \nIndeed, such protocols can be more efficient\nwhen \nemploying ELOCC rather than exclusively using LOCC.\nMore precisely, there are \n(genuinely) mixed states $\\sigma$ and\npure states $\|\\xi\\rangle\\langle\\xi \|$ with the property that \nthe maximal average attainable value of the fidelity \nunder ELOCC\n%\n\\begin{equation}\n\tF_{\\rm ELOCC}(\\sigma,\|\\xi\\rangle\\langle\\xi\|)= \n\t\\sup_{\\rho\\in{\\cal S}_{\\rm ELOCC}^\\sigma}\n\t\\langle \\xi\|\\rho\|\\xi\\rangle\n\\end{equation}\n%\nis strictly larger than the \nmaximal attainable fidelity under LOCC,\n%\\cite{Convexity}\n\\begin{equation}\nF_{\\rm LOCC}(\\sigma,\|\\xi\\rangle\\langle\\xi\|)\n= \\sup_{\\rho\\in{\\cal S}_{\\rm LOCC}^\\sigma}\\langle \\xi\|\\rho\|\\xi\\rangle.\n\\end{equation}\nHere, ${\\cal S}_{\\rm LOCC}^\\sigma$ \nand ${\\cal S}_{\\rm ELOCC}^\\sigma$\nare the sets of states that can be reached by applying LOCC and\nELOCC, respectively, on an initial state $\\sigma$. \n\nThis statement can be proven by considering\nan initial state $\\sigma$ \nof the form specified in\nEq.\\ (\\ref{example}) with\n\\begin{eqnarray}\n\t\|\\psi\\rangle&=&\n\t\\varepsilon(\\sqrt{0.4}\|11\\rangle+\n\t\\sqrt{0.4}\|22\\rangle+\n\t\\sqrt{0.1}\|33\\rangle+\n\t\\sqrt{0.1}\|44\\rangle)\\nonumber\\\\\n\t&+&\n\t\\sqrt{1-\\varepsilon^2}\|55\\rangle,\\label{fidexample}\n\\end{eqnarray}\nand one may choose $\|\\xi\\rangle=\|\\phi\\rangle$ as in Eq.\\ (\\ref{phi}). \nClearly\n\\begin{eqnarray}\nF_{\\rm LOCC}(\\sigma,\|\\phi\\rangle\\langle\\phi\|)&\\leq&\n(1-\\lambda)\nF_{\\rm LOCC}(\|55\\rangle\\langle55\|,\|\\phi\\rangle\\langle\\phi\|)\\nonumber\\\\\n&+&\n\\lambda\nF_{\\rm LOCC}(\|\\psi\\rangle\\langle\\psi\|,\|\\phi\\rangle\\langle\\phi\|),\n\\end{eqnarray}\nas the components of the initial state $\\sigma$ are\nnot locally distinguishable, and since the achievable\nfidelity can be no better than the sum of \nboth best possible fidelities of each contribution.\nUnder LOCC\nall separable states are accessible starting from\n$\|55\\rangle\\langle55\|$.\nThe (not necessarily pure) \nseparable state closest to $\|\\phi\\rangle\\langle\\phi\|$\nwith respect to the fidelity is given by $\|11\\rangle\\langle11\|$,\nand therefore, \n$F_{\\rm LOCC}(\|55\\rangle\\langle55\|, \|\\phi\\rangle\\langle\\phi\|)=\n1/2$.\nFinally, from $F_{\\rm ELOCC}(\\sigma,\|\\phi\\rangle\\langle\\phi\|)\\geq\n\\lambda\\varepsilon^2+(1-\\lambda\\varepsilon^2)/2$ it follows that\n\\begin{equation}\nF_{\\rm ELOCC}(\\sigma,\|\\phi\\rangle\\langle\\phi\|)>\nF_{\\rm LOCC}(\\sigma,\|\\phi\\rangle\\langle\\phi\|)\n\\end{equation}\ncertainly\nholds for all $\\varepsilon\\in(\\tilde\\varepsilon,1]$, where\n\\begin{equation}\n\\tilde\\varepsilon\n=(2F_{\\rm LOCC}(\|\\psi\\rangle\\langle\\psi\|,\n\|\\phi\\rangle\\langle\\phi\|)-1 )^{1/2},\n\\end{equation}\nindependent of $\\lambda\\in(0,1)$, and for all $\\varepsilon<1$ the\ninitial state is also genuinely mixed.\n\n\n\\section{Purification procedures} \n%{\\it Purification procedures. --}\\/\nThe previous two results unambiguously indicate that\nthe class of ELOCC operations is more powerful than\nLOCC operations \nnot only on the subset of\nthe boundary of ${\\cal S}({\\cal H})$ comprising \nthe pure states, but \nalso in the interior of the set \n${\\cal S}({\\cal H})$. Albeit this facts suggests \nthat the use of supplementary catalyst states opens up\npossibilities to enhance purification procedures,\nELOCC do not necessarily imply an improved efficiency\nin practically motivated\nproblems.\nConsider the class of states studied in Ref.\\ \\cite{Kent}\n\\begin{equation}\t\n\t\\sigma=\\lambda \|\\psi\\rangle\\langle\\psi\|+(1-\\lambda)\\zeta\n\\end{equation}\nwith the property that there exists a \n$\\lambda_0\\in(0,1)$ such that $\\sigma$\nis a separable state and that every state with\na larger weight of $\|\\psi\\rangle\\langle\\psi\|$ is\nentangled. Furthermore, it is assumed that\n$\\langle\\psi\|\\zeta\|\\psi\\rangle=0$. \nThis class of states \nincludes the class of states consisting\nof a mixture of some pure state and the complete mixture\nin the corresponding state space, which is of salient\nimportance in practical applications.\nIn Ref.\\ \\cite{Kent} is has been shown that\n$\\langle\\psi\|\\rho\|\\psi\\rangle\\leq \\langle\\psi\|\\sigma\|\\psi\\rangle$\nfor all states $\\rho$ that can be reached from $\\sigma$\nwith {\\it any probability}\\/ $p>0$\n(that is, not necessarily $\\sigma\\rightarrow\\rho$ under LOCC holds),\nimplying that for this class of states the proportion of\n$\|\\psi\\rangle\\langle\\psi\|$ can not even be increased\nwith non-trace-preserving operations \\cite{MachineRemark}. \nThis is\nalso true for ELOCC operations.\n\nLet $\\sigma\\in{\\cal S}({\\cal H})$ be such a state,\nand let $\\omega\\in {\\cal S}(\\tilde{\\cal H})=\n{\\cal S}(\\tilde{\\cal H}_A\\otimes\\tilde{\\cal H}_B )$\nbe an appropriate catalyst state. The above\ntransformation then amounts to a map\n\\begin{equation}\\label{trans}\n\t\\sigma\\otimes \\omega\\longmapsto\n\t\\rho\\otimes\\omega=\n\t\\frac{\\sum_i\n\t(A_i\\otimes B_i)(\\sigma\\otimes \\omega)(A_i\\otimes B_i)^\\dagger}\n\t{{\\rm tr} [\\sum_i\n\t(A_i\\otimes B_i)(\\sigma\\otimes \\omega)(A_i\\otimes B_i)^\\dagger]},\n\t\\end{equation}\nwhere $A_i$ and $B_i$ satisfying $\\sum_i A_i^\\dagger A_i\\leq 1$ and\n$\\sum_i B_i^\\dagger B_i\\leq 1$\nact only in ${\\cal H}_A\\otimes \\tilde{\\cal H}_A$ and \n${\\cal H}_B\\otimes \\tilde{\\cal H}_B$, respectively. \nThe quantity of\ninterest is now the fidelity $F=\\langle\\psi\|\\rho\|\\psi\\rangle$\nof $\\rho$ with respect to $\|\\psi\\rangle\\langle\\psi\|$.\nIt is given by\n\\begin{eqnarray}\n\tF(\\lambda)&=&\n\t{\\rm tr}_{\\tilde{\\cal H}}\\sum_i \\bigl(\n\t\\lambda\\,\\langle\\psi\|\n\t\\left[(A_i\\otimes B_i) (\n\t\|\\psi\\rangle\\langle\\psi\|\n\t\\otimes\\omega)\n\t(A_i\\otimes B_i)^\\dagger\n\t\\right] \|\\psi\\rangle\n\t\\nonumber\\\\\n\t&+&\n\t(1-\\lambda)\\;\\langle\\psi\|\\left[\n\t(A_i\\otimes B_i) (\\zeta\\otimes\\omega)\n\t(A_i\\otimes B_i)^\\dagger\\right] \|\\psi\\rangle\\bigr)/{\\cal N},\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\t{\\cal N}=\\sum_i {\\rm tr}\\left[\n\t(A_i\\otimes B_i)\n\t((\\lambda\|\\psi\\rangle\\langle\\psi\|\n\t+(1-\\lambda)\n\t\\zeta)\t\n\t\\otimes\\omega)\n\t(A_i\\otimes B_i)^\\dagger\n\t\\right].\\nonumber\n\\end{eqnarray}\n%As for local quantum operations without a catalyst state\n%\n$dF^2(\\lambda)/d^2 \\lambda = {\\cal C}/{\\cal N}^3$ with\na number $ {\\cal C}$ independent of $\\lambda$, and\none can argue as in the case of local operations without\na catalyst state \\cite{Kent}:\nThe sign of the second derivative of the\nfunction $f(\\lambda)=\nF(\\lambda)-\\lambda$ is constant for all $\\lambda\\in(0,1)$,\nand therefore, this function is convex, concave or\nlinear. At $\\lambda=0$, $f(0)\\geq0$ as\n$f(\\lambda)\\geq-\\lambda$ for $\\lambda\\in(0,1)$, and\n$f(1)\\leq0$. $f(\\lambda_0)\\leq0$ follows from the\nfact that the map Eq.\\ (\\ref{trans}) cannot transform\nthe state pertaining to $\\lambda_0$\nto an entangled state. Hence, $f(\\lambda)\\leq0$ for\nall $\\lambda\\in[\\lambda_0,1)$, i.e., the proportion\nof $\|\\psi\\rangle\\langle\\psi\|$ can only decrease.\n\n\\section{Small transformations and catalysis for pure and mixed states}\n%\n%{\\it Small transformations and catalysis for pure and mixed states. --}\\/\nSo far, the findings in the pure state case and those\nfor mixed states have suggested a rather similar behavior of\nboth sets of states with respect to \nLOCC and ELOCC operations. However, things are\nquite different in the next issue\nconcerning the possibility to enhance the \nrange of accessible states with catalyst states\nin ``small'' transformations.\\\\\n\n\\noindent {\\bf Lemma 2. --} For all pure states $\|\\psi\\rangle\\in{\\cal H}$\nand all pure catalyst states $\|\\tilde\\psi\\rangle\\in\\tilde{\\cal H}$\nthere exists a $\\delta>0$ such that \n\\begin{eqnarray}\n\\nonumber\n\|\\psi\\rangle\\not\\rightarrow \|\\phi\\rangle{\\rm \\,\\,under\\,\\,\nLOCC}\n\\,\\,\\Rightarrow\\,\n\|\\psi\\rangle\\not\\rightarrow \|\\phi\\rangle{\\rm \\,\\,under\\,\\,\nELOCC}\n\\nonumber\n\\end{eqnarray}\nfor all $\|\\phi\\rangle\\in {\\cal H}$\nwith $\|\\langle\\psi\|\\phi\\rangle\|^2>1-\\delta$. \n\n{\\it Proof:} Let $\\alpha_1$, ..., $\\alpha_N$\nbe the ordered lists of eigenvalues of \n${\\rm tr}_A[\|\\psi\\rangle\\langle\\psi\|]$,\n$N=\\dim[{\\cal H}_A]=\\dim[{\\cal H}_B]$, \nand let $\\gamma_1$, ..., $\\gamma_M$\nbe the corresponding list of the pure catalyst state,\n$M=\\dim[\\tilde{\\cal H}_A]=\\dim[\\tilde{\\cal H}_B]$. \nLet now $\\varepsilon>0$ \nand call an $\\varepsilon$-list a list\n$\\beta_1$, ..., $\\beta_N$ with\n$1\\geq \\beta_1\\geq ...\\geq \\beta_N\\geq0$ that has the\nproperty $\|\\beta_i-\\alpha_i\|<\\varepsilon$ for all $i=1,...,N$.\nThere exists an $\\varepsilon>0$ such that\nfor all $\\varepsilon$-lists\n$\\beta_1$, ..., $\\beta_N$\nthe statement that \n$\\alpha_i\\gamma_j>\\alpha_k\\gamma_l$ for some\n$i,k\\in\\{1,...,N\\}, j,l\\in\\{1,...,M\\}$ implies that\n$\\beta_i\\gamma_j>\\beta_k\\gamma_l$. This $\\varepsilon$\nis in the following referred to as $\\tilde\\varepsilon$.\nMoreover, \nthere exists a $\\delta>0$ such that \nfor each $\|\\phi\\rangle\\in {\\cal H}$\nwith $\|\\langle\\psi\|\\phi\\rangle\|^2>1-\\delta$\nthe ordered\neigenvalues of ${\\rm tr}_A[\|\\phi\\rangle\\langle\\phi\|]$\nform a $\\tilde\\varepsilon$-list (and\nhence, for such states it is not possible that\n$\\beta_i\\gamma_j<\\beta_k\\gamma_l$ and\n$\\alpha_i\\gamma_j>\\alpha_k\\gamma_l$). \nIt follows that for all\nsuch $\|\\phi\\rangle\\in {\\cal H}$\nwith $\|\\langle\\psi\|\\phi\\rangle\|^2>1-\\delta$ \nthe majorization relation\n$\n\t{\\rm \n\ttr}_A[\|\\psi\\rangle\\langle\\psi\|\\otimes\|\\tilde\\psi\\rangle\\langle\\tilde\\psi\|]\n\t\\not\\prec\n{\\rm tr}_A[\|\\phi\\rangle\\langle\\phi\|\\otimes\n\|\\tilde\\psi\\rangle\\langle\\tilde\\psi\|]\n$ holds \nif ${\\rm tr}_A[\|\\psi\\rangle\\langle\\psi\|]\\not\\prec\n{\\rm tr}_A[\|\\phi\\rangle\\langle\\phi\|]$. Finally, this implies\nthe statement of Lemma 2.\n%\n%\n%\n%\n$\\hfill{\\Box}$\n\n\n\n\nThis is not true for mixed states, when the fidelity\nof two states $\\sigma$ and $\\rho$ is\ntaken to be\n$F(\\sigma,\\rho)= ({\\rm tr} [(\\sqrt{\\sigma}\\rho\\sqrt{\\sigma})^{1/2}])^2$\n\\cite{Uhlmann}.\nIndeed,\nthere are states $\\sigma\\in{\\cal S}({\\cal H})$ such that\nfor every $\\delta>0$ there are states $\\rho\\in{\\cal S}({\\cal H})$\nwith the property that $F(\\sigma,\\rho)>1-\\delta$ and\n$\\sigma\\not\\rightarrow\\rho$ under LOCC, but\n$\\sigma\\rightarrow\\rho$ under ELOCC. Such states can,\ne.g., be constructed using the class of states\ndefined in Eq.\\ (\\ref{example}), Eq.\\ (\\ref{psi}), and\nEq.\\ (\\ref{phi}). For any given $\\delta>0$\nthere is a sufficiently small $\\lambda>0$ such that \nthe fidelity satisfies \n$F(\\sigma,\\rho)>1-\\delta$.\n\n\nHence, quite surprisingly, in the case of \nentanglement manipulations\nfrom an initial pure state to a close pure state\nentanglement-assisted operations do not add any power to LOCC \noperations. To put it in different words, there\nis no catalysis for sufficiently close pure states. \nYet, for mixed states there can be catalysis for \nsuch close states.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Conclusion and open problems}\n%{\\it Conclusion and open problems. --}\\/\nIn this paper we have \n%shed new light on the\n%phenomenon of catalysis and have \ninvestigated the\npower of entanglement-assisted manipulation of\nentangled quantum systems in mixed states. \nInterestingly, the counterintuitive class of ELOCC \noperations has proven to be superior to mere LOCC operations\nalso in the interior of the state space, for\nwhich such strong tools as the majorization criterion\nare not available. \nYet, albeit these findings might contribute to the quest for\na better understanding of mixed-state entanglement, \nthere are numerous open problems.\nStronger criteria for the possibility of certain entanglement\ntransformation are urgently needed. \n%One might conjecture \n%that for transformations from pure states to mixed states the\n%majorization criterion stated in Eq.\\ (\\ref{maj}) provides\n%also a necessary criterion and not only a sufficient one as\n%explained in Lemma 1. \nFinally, it is the hope that this\nwork will help to explore practical applications \\cite{Barnum}\nof the strange phenomenon of catalysis.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Acknowledgements}\n%{\\it Acknowledgements. --}\\/\nWe would like to thank Martin B.\\ Plenio and Julia Kempe\nfor very helpful hints.\nWe also acknowledge fruitful discussions with \nDaniel Jonathan and the participants of the A2 Consortial \nMeeting. This work was supported\nby the European Union and the DFG.\n%\n\\begin{thebibliography}{99}\n\\bibitem{Contemp}\n\tM.B.\\ Plenio and V.\\ Vedral,\n Contemp.\\ Phys.\\ {\\bf 39}, 431 (1998);\n\tA.\\ Ekert and R.\\ Jozsa, Rev.\\ Mod.\\ Phys.\\ {\\bf 68}, 733 (1996).\n\\bibitem{Practical}\n\tD.\\ DiVincenzo, Science {\\bf 270}, 255 (1995);\n\tC.H.\\ Bennett {\\it et al.}, Phys.\\ Rev.\\ Lett.\\ {\\bf 70}, \n\t1895 (1993); \n \tA.\\ Ekert, Phys.\\ Rev.\\ Lett.\\ {\\bf 67}, 661 (1991).\n\\bibitem{Jonathan}\n\tD.\\ Jonathan and M.B.\\ Plenio, Phys.\\ Rev.\\ Lett.\\ {\\bf 83}, 3566 (1999).\n\\bibitem{Nielsen}\n\tM.A.\\ Nielsen, Phys.\\ Rev.\\ Lett.\\ {\\bf 83}, 436 (1999).\t\n\\bibitem{Majo}\n\tA.W.\\ Marshall and I.\\ Olkin, \n\t{\\it Inequalities: Theory of Majorization and its Applications}\\/ \n\t(Academic Press, New York, 1979);\n\tP.M.\\ Alberti and A.\\ Uhlmann, \n\t{\\it Stochasticity and Partial Order: Doubly Stochastic \n\tMaps and Unitary Mixing}\\/ (VEB \n\tDeutscher Verlag der Wissenschaften, Berlin, 1982).\n\\bibitem{NewDaniel}\n\tG.\\ Vidal, D.\\ Jonathan, and M.A.\\ Nielsen,\n\tquant/ph-9910099.\n\\bibitem{After}\n\tB.M.\\ Terhal and P.\\ Horodecki,\n\tquant-ph/9911117.\t\n\\bibitem{Kent}\n\tA.\\ Kent, Phys.\\ Rev.\\ Lett.\\ {\\bf 81}, 2839 (1999).\t\n%\\bibitem{Dimension}\n%\tIn this paper the dimensions of the Hilbert spaces\n%\tadjuncted with the initial and final\n%\tstates in entanglement manipulations are always\n%\ttaken to be identical.\t\n%\\bibitem{Range}\n%\tThe range of $\\rho$ is the set of $\|\\psi\\rangle\\in{\\cal H}$ \n%\tfor which there exists a $\|\\varphi\\rangle\n%\t\\in{\\cal H} $ such that $\|\\psi\\rangle=\\rho \|\\varphi\\rangle$.\n\\bibitem{Plenio}\n\tV.\\ Vedral, M.B.\\ Plenio, M.A.\\ Rippin, and P.L.\\ Knight,\n\tPhys.\\ Rev.\\ Lett.\\ {\\bf 78}, 2275 (1997).\n\\bibitem{Rains}\n E.M.\\ Rains, Phys.\\ Rev.\\ A {\\bf 60}, 173 (1999); \n\tPhys.\\ Rev.\\ A {\\bf 60}, 179 (1999).\n\\bibitem{Lewenstein}\n\tM.\\ Lewenstein and A.\\ Sanpera,\n\tPhys.\\ Rev.\\ Lett.\\ {\\bf 80}, 2261 (1998).\n\\bibitem{Amnesia}\n J.\\ Eisert, T.\\ Felbinger, \n\tP.\\ Papadopoulos, M.B.\\ Plenio,\n\tand M.\\ Wilkens, \n\tPhys.\\ Rev.\\ Lett.\\ {\\bf 84}, 1611 (2000).\n\\bibitem{HilbertRemark}\n\tThe catalyst state is a density\n\toperator on the Hilbert space $\\tilde {\\cal H}=\n\t\\tilde {\\cal H}_A\\otimes \\tilde {\\cal H}_B$, where\n\t$\\tilde {\\cal H}_A,\\tilde {\\cal H}_B =\n\t{\\text{span}}\\{\|6\\rangle,\|7\\rangle\\}$.\n%\\bibitem{Convexity}\n%\tFrom compactness considerations it follows that there\n%\tis a state $\\omega$ with $\\sigma\\rightarrow\\omega$ under\n%\tLOCC such that $\\langle\\xi\|\\omega\|\\xi\\rangle= \n%\tF_{\\rm LOCC}(\\sigma,\|\\xi\\rangle\\langle\\xi\|)$.\n\\bibitem{MachineRemark}\nIt is further assumed that the purification\nprotocol is independent of\nthe parameter $\\lambda$.\n\\bibitem{Uhlmann}\n\tA.\\ Uhlmann, Rep.\\ Math.\\ Phys.\\ {\\bf 9}, 273 (1976).\n%\\bibitem{Lo}\n%\tH.-K.\\ Lo and S.\\ Popescu, quant/ph-9707038.\n\\bibitem{Barnum}\n\t H.N.\\ Barnum, quant/ph-9910072.\n\\end{thebibliography}\n\n\n\\end{multicols}\n\n \n\\end{document}\n\n\n\n \n" } ]	[ { "name": "quant-ph9912080.extracted_bib", "string": "{Contemp M.B.\\ Plenio and V.\\ Vedral, Contemp.\\ Phys.\\ {39, 431 (1998); A.\\ Ekert and R.\\ Jozsa, Rev.\\ Mod.\\ Phys.\\ {68, 733 (1996)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Practical D.\\ DiVincenzo, Science {270, 255 (1995); C.H.\\ Bennett {et al., Phys.\\ Rev.\\ Lett.\\ {70, 1895 (1993); A.\\ Ekert, Phys.\\ Rev.\\ Lett.\\ {67, 661 (1991)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Jonathan D.\\ Jonathan and M.B.\\ Plenio, Phys.\\ Rev.\\ Lett.\\ {83, 3566 (1999)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Nielsen M.A.\\ Nielsen, Phys.\\ Rev.\\ Lett.\\ {83, 436 (1999)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Majo A.W.\\ Marshall and I.\\ Olkin, {Inequalities: Theory of Majorization and its Applications\\/ (Academic Press, New York, 1979); P.M.\\ Alberti and A.\\ Uhlmann, {Stochasticity and Partial Order: Doubly Stochastic Maps and Unitary Mixing\\/ (VEB Deutscher Verlag der Wissenschaften, Berlin, 1982)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{NewDaniel G.\\ Vidal, D.\\ Jonathan, and M.A.\\ Nielsen, quant/ph-9910099." }, { "name": "quant-ph9912080.extracted_bib", "string": "{After B.M.\\ Terhal and P.\\ Horodecki, quant-ph/9911117." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Kent A.\\ Kent, Phys.\\ Rev.\\ Lett.\\ {81, 2839 (1999). %" }, { "name": "quant-ph9912080.extracted_bib", "string": "{Dimension % In this paper the dimensions of the Hilbert spaces % adjuncted with the initial and final % states in entanglement manipulations are always % taken to be identical. %" }, { "name": "quant-ph9912080.extracted_bib", "string": "{Range % The range of $\\rho$ is the set of $\|\\psi\\rangle\\in{\\cal H$ % for which there exists a $\|\\varphi\\rangle % \\in{\\cal H $ such that $\|\\psi\\rangle=\\rho \|\\varphi\\rangle$." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Plenio V.\\ Vedral, M.B.\\ Plenio, M.A.\\ Rippin, and P.L.\\ Knight, Phys.\\ Rev.\\ Lett.\\ {78, 2275 (1997)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Rains E.M.\\ Rains, Phys.\\ Rev.\\ A {60, 173 (1999); Phys.\\ Rev.\\ A {60, 179 (1999)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Lewenstein M.\\ Lewenstein and A.\\ Sanpera, Phys.\\ Rev.\\ Lett.\\ {80, 2261 (1998)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Amnesia J.\\ Eisert, T.\\ Felbinger, P.\\ Papadopoulos, M.B.\\ Plenio, and M.\\ Wilkens, Phys.\\ Rev.\\ Lett.\\ {84, 1611 (2000)." }, { "name": "quant-ph9912080.extracted_bib", "string": "{HilbertRemark The catalyst state is a density operator on the Hilbert space $\\tilde {\\cal H= \\tilde {\\cal H_A\\otimes \\tilde {\\cal H_B$, where $\\tilde {\\cal H_A,\\tilde {\\cal H_B = {\\text{span\\{\|6\\rangle,\|7\\rangle\\$. %" }, { "name": "quant-ph9912080.extracted_bib", "string": "{Convexity % From compactness considerations it follows that there % is a state $\\omega$ with $\\sigma\\rightarrow\\omega$ under % LOCC such that $\\langle\\xi\|\\omega\|\\xi\\rangle= % F_{\\rm LOCC(\\sigma,\|\\xi\\rangle\\langle\\xi\|)$." }, { "name": "quant-ph9912080.extracted_bib", "string": "{MachineRemark It is further assumed that the purification protocol is independent of the parameter $\\lambda$." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Uhlmann A.\\ Uhlmann, Rep.\\ Math.\\ Phys.\\ {9, 273 (1976). %" }, { "name": "quant-ph9912080.extracted_bib", "string": "{Lo % H.-K.\\ Lo and S.\\ Popescu, quant/ph-9707038." }, { "name": "quant-ph9912080.extracted_bib", "string": "{Barnum H.N.\\ Barnum, quant/ph-9910072." } ]
quant-ph9912081	A Kochen-Specker Theorem for Imprecisely Specified Measurements	[ { "author": "N.\\ David Mermin" } ]	A recent claim that finite precision in the design of real experiments ``nullifies'' the impact of the Kochen-Specker theorem, is shown to be unsupportable, because of the continuity of probabilities of measurement outcomes under slight changes in the experimental configuration.	[ { "name": "quant-ph9912081.tex", "string": "\n\\documentstyle[prl,aps]{revtex}\n\n\\begin{document}\n\n\\draft\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname\n\\title{A Kochen-Specker Theorem for Imprecisely Specified Measurements}\n\\author{N.\\ David Mermin}\n\\address{Laboratory of Atomic and Solid State Physics, \nCornell University, Ithaca, NY 14853-2501}\n%\\date{\\today}\n\\maketitle\n\n\\begin{abstract}A recent claim that finite precision in the design of real\nexperiments ``nullifies'' the impact of the Kochen-Specker theorem, is\nshown to be unsupportable, because of the continuity of probabilities\nof measurement outcomes under slight changes in the experimental\nconfiguration. \\end{abstract}\n\n\\pacs{PACS numbers: 03.65.Bz, 03.67.Hk, 03.67.Lx}\n]\n\n\nThe Kochen-Specker (KS) theorem is one of the major\nno-hidden-variables theorems of quantum mechanics. It exhibits a\nfinite set of finite-valued observables with the following property:\nthere is no way to associate with each observable in the set a\nparticular one of its eigenvalues so that the eigenvalues associated\nwith every subset of mutually commuting observables obey certain\nalgebraic identities obeyed by the observables\nthemselves\\cite{ft:RMP}. Such a set of observables is traditionally\ncalled uncolorable.\n\nThe physical significance of an uncolorable set of observables stems\nfrom the fact that a simultaneous measurement of a mutually commuting\nset must yield a set of simultaneous eigenvalues, which\nare constrained to obey the algebraic identities obeyed by the\nobservables themselves. So any attempt to assign every observable\nin a KS uncolorable set a preexisting {\\it noncontextual\\/} value (a\n``hidden variable'') that is simply revealed by a measurement, will\nnecessarily assign to at least one mutually commuting subset of\nobservables a set of values specifying results that quantum mechanics\nforbids. \n\nThe term ``noncontextual'' emphasizes that the disagreement with\nquantum mechanics only arises if the value associated with each\nobservable is required to be independent of the choice of the other\nmutually commuting observables with which it is measured. {\\it\nContextual\\/} value assignments in full agreement with all quantum\nmechanical constraints can, in fact, be made. The import of the KS\ntheorem is that there exist sets of observables --- ``uncolorable\nsets'' --- for which any assignment of preexisting values must be\ncontextual if all the outcomes specified by those values are allowed\nby the laws of quantum mechanics. The theorem prohibits\nnoncontextual hidden-variable theories that agree with all the\nquantitative predictions of quantum mechanics.\n\nMeyer\\cite{ft:Meyer} and Kent\\cite{ft:Kent} have questioned the\nrelevance of the KS theorem to the outcomes of real imperfect\nlaboratory experiments, by constructing some clever noncontextual\nassignments of eigenvalues to every observable in a dense subset of\nobservables, whose closure contains the KS uncolorable observables.\nWhile no KS uncolorable set can be contained in such a dense colorable\nsubset, observables in the dense colorable subset can be found\narbitrarily close to every observable in any KS uncolorable set. This\nleads Meyer and Kent to assert that their noncontextual value\nassignments to dense sets of observables ``nullify'' the KS theorem.\nIn support of this claim they note that observables measured in an\nactual experiment cannot be specified with perfect precision so, in\nKent's words, ``no Kochen-Specker-like contradiction can rule out\nhidden variable theories indistinguishable from quantum theory by\nfinite precision measurements$\\ldots\\,$.''\\cite{ft:finite}\n\nI show below that this plausible-sounding but not entirely sharply\nformulated intuition dissolves under close scrutiny\\cite{ft:havlicek}.\nFirst I describe how the KS conclusion that quantum mechanics requires\nany assignment of pre-existing values to be contextual can be deduced\ndirectly from the data, even when one is not sure precisely which\nobservables are actually being measured. Then I identify where the\nintuition of Meyer and Kent goes astray.\n\nAt first glance it is not evident that either a KS uncolorable set or\na Meyer-Kent (MK) dense colorable set of observables is relevant when\none cannot specify to more than a certain high precision what\nobservables are actually being measured. As traditionally viewed, the\nKS theorem merely makes a point about the formal structure of quantum\nmechanics, telling us that there is no consistent way to interpret\n{\\it the theory\\/} in terms of the statistical behavior of an\nensemble, in each individual member of which every observable in the\ntheory has a unique noncontextual value waiting to be revealed by any\nappropriate measurement. Upon further reflection, however, there\nemerges a straightforward way to apply the result of the KS theorem to\nmeasurements specified with high but imperfect precision, which makes\nit evident that the theorem and its various descendants remain\nentirely relevant to imperfect experiments, while the ingenious\nconstructions of Meyer and Kent do not.\n\nLet us first rephrase the implications of the KS theorem in the ideal\ncase of perfectly specified measurements. The theorem gives a finite\nuncolorable set of observables, each with a finite number of\neigenvalues. Because the number of possible assignments of\nnoncontextual values to observables in the set is finite, no matter\nwhat probabilities are used to associate such values with the\nobservables, the assignment must give nonzero probability to at least\none mutually commuting subset of the observables having values that\ndisagree with the laws of quantum mechanics. So if noncontextual\npreexisting values existed and if one could carry out a series of\nideal experiments\\cite{ft:state} in each of which one measured a {\\it\nrandomly selected\\/} subset of perfectly defined mutually commuting\nobservables from a KS uncolorable set, then a definite nonzero\nfraction of those measurements would produce results violating the\nlaws of quantum mechanics. \n\nConversely, if an appropriately large number of such randomly selected\nmeasurements all yielded results satisfying the relevant quantum\nmechanical constraints, then in the absence of bizarre conspiratorial\ncorrelations between one's random choice of which mutually commuting\nsubset to measure and the hypothetical preexisting noncontextual\nvalues waiting to be revealed by that measurement, one would have\nestablished directly from the ideal data that there could be no\npreexisting noncontextual values.\n\nIn a real experiment, of course, the observables cannot be precisely\nspecified. The actual apparatus used to measure any mutually\ncommuting subset of an uncolorable finite set of observables will be\nslightly misaligned at all stages of the measuring process. Therefore\nthe pointer readings from which one deduces their discrete\nsimultaneous values will give slightly unreliable information about\nthe ideal observables one was trying to measure. If the misalignment\nis at the limit of ones ability to control or discern, as it will be\nin a well designed experiment, then one can and will label the\noutcomes of such a procedure with the same discrete eigenvalues used\nto label the gedanken outcomes of the ideal perfectly aligned\nmeasurement. The misalignment will only reveal itself through the\noccasional occurrence of runs with outcomes that the laws of quantum\nmechanics prohibit for a perfectly aligned\napparatus\\cite{ft:original}. But although the outcomes deduced from\nsuch imperfect measurements will occasionally differ dramatically from\nthose allowed in the ideal case, if the misalignment is very slight,\n{\\it the statistical distribution of outcomes will differ only\nslightly from the ideal case.}\n\nIt is this continuity of quantum mechanical probabilities under small\nvariations in the experimental configuration (without which quantum\nmechanics, or, for that matter, any other physical theory, would be\nquite useless) that makes the KS conclusion relevant to the imperfect\ncase. Even though the apparatus cannot be perfectly aligned, if\nquantum mechanics is correct in its quantitative predictions, then the\nfraction of runs which violate the quantum-mechanical rules applying\nto the ideal observables can be made {\\it arbitrarily small\\/} in the\nrealistic case by making the alignment sufficiently sharp. But the KS\ntheorem tells us that if the possible results of the ideal gedanken\nmeasurements were consistent with preexisting {\\it noncontextual\nvalues}, then the fraction of quantum-mechanically forbidden outcomes\nfor the real experiments would have to approach a {\\it nonzero\nlimit\\/} as the alignment became sharp.\n\nBy making the experimental misalignment sufficiently small, one can\nmake the statistics of the slightly unreliable results of the randomly\nselected realistic measurements arbitrarily close to the statistics of\nthe theoretical results of the randomly selected ideal measurements.\nTherefore a failure in the realistic case to observe {\\it sufficiently\nmany\\/} values that contradict the constraints imposed on the data by\nquantum mechanics can demonstrate, just as effectively as the failure\nto observe {\\it any\\/} such contradictions in the ideal case, that if\nthe measurements are revealing preexisting values, then those values\nmust be {\\it contextual\\/}. \n\nBecause it can be stated in terms of outcome probabilities, and\nbecause those probabilities must vary continuously with variations in\nthe experimental apparatus, the conclusion of the KS theorem are not\n``nullified'' by the finite precision with which actual measurements\ncan be specified.\n\n\\bigskip\n\nBut what about Meyer's and Kent's intuition that the dense colorable\nMK set can be used to furnish a group of nearly ideal experiments with\nnoncontextual values that agree with the constraints imposed by\nquantum mechanics on all mutually commuting ideal subsets? The\ncrucial word here is ``all''. It is impossible for a colorable set of\nobservables in one-to-one correspondence with the observables in a KS\nuncolorable set, to have mutually commuting subsets that correspond to\n{\\it every\\/} mutually commuting subset of the KS uncolorable set. At\nleast one of those subsets cannot be mutually commuting\\cite{ft:case}.\n\nThis impossibility is established by the KS theorem itself, which uses\nonly the topology of the network of links between commuting\nobservables in the full KS uncolorable set. This topology would be\npreserved by the correspondence between the nearby colorable and\nuncolorable sets, if the correspondence between their mutually\ncommuting subsets were complete. Because, as Meyer and Kent\nexplicitly show, the MK set is colorable, the correspondence cannot be\ncomplete. \n\nThus any finite MK colorable set in sufficiently close one-to-one\ncorrespondence with a finite KS uncolorable set, must necessarily lack\nthe full range of mutually commuting subsets that the KS uncolorable\none contains. There is therefore at least one mutually commuting\nsubset of the KS uncolorable set for which the MK colorable set fails\nto provide a set of values agreeing with the constraints imposed by\nquantum mechanics. It is only this deficiency that makes it possible\nto color the observables in the MK set in a noncontextual way that\nsatisfies the quantum mechanical constraints for all mutually\ncommuting subsets. But this same deficiency makes the MK set useless\nfor specifying preassigned noncontextual values agreeing with quantum\nmechanics for the outcomes of every one of the slightly imperfect\nexperiments that corresponds to measuring a mutually commuting subset\nof observables from the ideal KS uncolorable set.\n\nIf one tries to bridge this gap in the argument by associating more\nthan a single nearby MK colorable observable with some of the\nobservables in the ideal uncolorable set, one sacrifices the\nnoncontextuality of the value assignments. Nor does the MK\n``nullification'' of the KS theorem work in a toy universe in which\nthe physically allowed observables are restricted to be those in the\ndense colorable set. In such a universe measurements still could not\nbe specified with perfect precision, and one would still have to rely\non the continuity of outcome probabilities with small changes in the\nexperimental configuration to relate the theory to actual imperfect\nobservations. Under such conditions it would be highly convenient to\nintroduce fictitious observables which were the limit points of\nphysical observables in the dense colorable set, whose statistics\ncould represent to high accuracy the statistics of physically allowed\nobservables in their immediate vicinity. The KS theorem would hold\nfor these fictitious limit-point observables, and would therefore\napply by continuity to measurements of the nearby physical\nobservables, for exactly the reasons I have just described in the case\nof conventional quantum mechanics with its continuum of observables.\nIndeed, I can see no grounds other than convenience (which is, of\ncourse, so enormous as to be utterly compelling) for treating the\nwhole continuum of observables as physically real, as we actually do,\nrather than regarding it as a fictitious extension of some countable\ndense subset of observables.\n\n\nSo contrary to the claim of Meyer and Kent, the KS theorem is not\nnullified by the finite precision of real experimental setups because\nof the fundamental physical requirement that probabilities of outcomes\nof real experiments vary only slightly under slight variations in the\nconfiguration of the experimental apparatus, and because the import of\nthe theorem can be stated in terms of whether certain outcomes never\noccur, or occur a definite nonzero fraction of the time in a set of\nrandomly chosen ideal experiments. \n\nBut the elegant MK colorings of dense sets of observables make an\ninstructive contribution to our understanding of the KS theorem, by\nforcing us to recognize that a principle of continuity of physical\nprobability, obeyed by the quantum theory, plays an essential role in\nrelating the conclusions of the theorem to real experiments. Indeed,\nthe relationship between the MK dense colorable sets and the KS\nuncolorable set offers a novel perspective on why it is sensible to\nbase physics on real (as opposed to rational) numbers, in spite of the\nfinite precision of actual experimental arrangements.\n\n\n\\bigskip\n\n{\\bf Acknowledgment.} This reexamination of the physical setting of\nthe Kochen-Specker theorem was supported by the National Science\nFoundation, Grant No. PHY 9722065.\n \n\\begin{references}\n\n\n\\bibitem{ft:RMP} For an introduction to the subject from this\nperspective see N. David Mermin, Revs. Mod. Phys. {\\bf 65}, 803\n(1993).\n\n\\bibitem{ft:Meyer}\nDavid A. Meyer, Phys. Rev. Lett. {\\bf 83}, 3751 (1999).\n(quant-ph 9905080) \n\n\\bibitem{ft:Kent}\nAdrian Kent, Phys. Rev. Lett. {\\bf 83},\n3755 (1999). (quant-ph 9906006) \n\n\n\n\\bibitem{ft:finite}\n``Finite precision'' refers not to the\nreliability of the measurement outcome, but to what it is that is\nactually being measured.\n\n\\bibitem{ft:havlicek} I show how the KS conclusion {\\it\ncan\\/} be reached from the outcomes of imprecisely specified\nexperiments. Havlicek et al., quant-ph/9911040, give a complementary\ncritique of Meyer's and Kent's nullification claim, from the point of\nview of quantum logic and Bell's (nonlocality) theorem.\n\n\\bibitem{ft:state} Because the KS theorem does not depend on the state\nin which the system is prepared, the ensemble of systems on which the\nexperiments are performed need not contain identically prepared\nsystems, nor need the states of the systems be pure states.\n\n\\bibitem{ft:original} \nIn the original KS argument, for example, one\nmeasures the mutually commuting squares of three orthogonal spin\ncomponents of a spin-1 particle, and the laws of quantum mechanics\nguarantee that one must record two 1's and one 0. If, however, the\napparatus is imperfect and the three directions are not precisely\northogonal, one will occasionally get different numbers of 1's and\n0's, in apparent violation of the laws for spin-1. (If the three\ndirections are not precisely orthogonal, the relevant three\nobservables will not exactly commute, so the statistics of the\n``measurement'' will in general depend slightly on how it is carried\nout.)\n\n\n\\bibitem{ft:case} In the case of squared spin-1 components considered\nby Meyer, for example, it is impossible to satisfy the condition of\nmutual orthogonality for every trio of components in the MK colorable\nset approximating a mutually orthogonal trio of components in the KS\nuncolorable set. But a set of squared spin-components in the MK set\nthat is not associated with exactly three orthogonal directions, need\nnot satisfy the quantm-mechanical requirement that their assigned\nvalues consist of two 1's and one 0.\n\n\n\n\\end{references}\n\n\n\n\\end{document}\n\n\n" } ]	[ { "name": "quant-ph9912081.extracted_bib", "string": "{ft:RMP For an introduction to the subject from this perspective see N. David Mermin, Revs. Mod. Phys. {65, 803 (1993)." }, { "name": "quant-ph9912081.extracted_bib", "string": "{ft:Meyer David A. Meyer, Phys. Rev. Lett. {83, 3751 (1999). (quant-ph 9905080)" }, { "name": "quant-ph9912081.extracted_bib", "string": "{ft:Kent Adrian Kent, Phys. Rev. Lett. {83, 3755 (1999). (quant-ph 9906006)" }, { "name": "quant-ph9912081.extracted_bib", "string": "{ft:finite ``Finite precision'' refers not to the reliability of the measurement outcome, but to what it is that is actually being measured." }, { "name": "quant-ph9912081.extracted_bib", "string": "{ft:havlicek I show how the KS conclusion {can\\/ be reached from the outcomes of imprecisely specified experiments. Havlicek et al., quant-ph/9911040, give a complementary critique of Meyer's and Kent's nullification claim, from the point of view of quantum logic and Bell's (nonlocality) theorem." }, { "name": "quant-ph9912081.extracted_bib", "string": "{ft:state Because the KS theorem does not depend on the state in which the system is prepared, the ensemble of systems on which the experiments are performed need not contain identically prepared systems, nor need the states of the systems be pure states." }, { "name": "quant-ph9912081.extracted_bib", "string": "{ft:original In the original KS argument, for example, one measures the mutually commuting squares of three orthogonal spin components of a spin-1 particle, and the laws of quantum mechanics guarantee that one must record two 1's and one 0. If, however, the apparatus is imperfect and the three directions are not precisely orthogonal, one will occasionally get different numbers of 1's and 0's, in apparent violation of the laws for spin-1. (If the three directions are not precisely orthogonal, the relevant three observables will not exactly commute, so the statistics of the ``measurement'' will in general depend slightly on how it is carried out.)" }, { "name": "quant-ph9912081.extracted_bib", "string": "{ft:case In the case of squared spin-1 components considered by Meyer, for example, it is impossible to satisfy the condition of mutual orthogonality for every trio of components in the MK colorable set approximating a mutually orthogonal trio of components in the KS uncolorable set. But a set of squared spin-components in the MK set that is not associated with exactly three orthogonal directions, need not satisfy the quantm-mechanical requirement that their assigned values consist of two 1's and one 0." } ]
quant-ph9912082	Rotational Invariance, Phase Relationships and the Quantum Entanglement Illusion	[ { "author": "Caroline H Thompson\\cite{CHT/email+" } ]	Another Bell test ``loophole" - imperfect rotational invariance - is explored, and novel realist ideas on parametric down-conversion as used in recent ``quantum entanglement" experiments are presented. The usual quantum theory of entangled systems assumes we have rotational invariance (RI), so that coincidence rates depend on the difference only between detector settings, not on the absolute values. Bell tests, as such, do not necessarily require RI, but where it fails the presentation of results in the form of coincidence curves can be grossly misleading. Even if the well-known detection loophole were closed, the visibility of such curves would tell us nothing about the degree of entanglement! The problem may be especially relevant to recent experiments using ``degenerate type II parametric down-conversion" sources. Logical analysis of the results of many experiments suggests realist explanations involving some new physics. The systems may be more nearly deterministic than quantum theory implies. Whilst this may be to the advantage of those attempting to make use of the so-called ``Bell correlations" in computing, encryption, ``teleportation" etc., it does mean that the systems obey ordinary, not quantum, logic.	[ { "name": "ri.tex", "string": "\\documentstyle[aps,prl]{revtex}\n%\\documentstyle{article}\n \n\\parskip 2ex\n\\parindent 0in\n \n\\begin{document}\n%\\draft command makes pacs numbers print\n\\draft\n%Temporary fix to get layout of abstract right (see email, 16:4:97):\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname\n@twocolumnfalse\\endcsname\n\n\n\\title{Rotational Invariance, Phase Relationships and the Quantum Entanglement Illusion}\n\n\\author{Caroline H Thompson\\cite{CHT/email+}\\\\\n\tDepartment of Computer Science, University of Wales, \tAberystwyth, \\\\\n\tSY23 3DB, U.K.}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nAnother Bell test ``loophole\" - imperfect rotational invariance - is explored, and novel realist ideas on parametric down-conversion as used in recent ``quantum entanglement\" experiments are presented. The usual quantum theory of entangled systems assumes we have rotational invariance (RI), so that coincidence rates depend on the difference only between detector settings, not on the absolute values. Bell tests, as such, do not necessarily require RI, but where it fails the presentation of results in the form of coincidence curves can be grossly misleading. Even if the well-known detection loophole were closed, the visibility of such curves would tell us nothing about the degree of entanglement! The problem may be especially relevant to recent experiments using ``degenerate type II parametric down-conversion\" sources. Logical analysis of the results of many experiments suggests realist explanations involving some new physics. The systems may be more nearly deterministic than quantum theory implies. Whilst this may be to the advantage of those attempting to make use of the so-called ``Bell correlations\" in computing, encryption, ``teleportation\" etc., it does mean that the systems obey ordinary, not quantum, logic. \n\n\\end{abstract}\n\n\\pacs{03.65.Bz, 03.65.Sq, 03.67., 42.50.Ct}\n\\vskip2pc]\n\n\\section{Introduction}\nThough this paper is largely concerned with general considerations of the role of rotational invariance (RI) in ``EPR\" (Einstein-Podolsky-Rosen) experiments, it also introduces some entirely new hypotheses concerning frequency and phase relationships in the degenerate case of parametric down-conversion (PDC). These are covered mainly in the final part of the paper, starting at section~\\ref{Applications}. Some readers may find the detailed mathematical groundwork of section~\\ref{Rotational Invariance} redundant. \n \nThere are few references in the literature to the matter of rotational invariance, either in the theory or practice of quantum entanglement. Early tests -- the key EPR or ``Bell-test\" experiments that are commonly accepted as establishing ``quantum nonlocality\" as a fact -- involved polarised light from atomic cascade sources~\\cite{Freedman72,Aspect81+}. It seemed reasonable to assume that there was no preferred orientation of polarisation, so that coincidence rates were necessarily dependent on the {\\em difference only} between the settings of detectors, not on the particular choice of either setting individually. Even in these experiments, perhaps more attention should have been paid to testing for invariance (could the polarisation in Aspect's experiments, for example, have been biased towards the vertical -- the polarisation direction of the stimulating lasers~\\cite{Aspect83}?), but it is in later experiments, using degenerate PDC sources~\\cite{Tittel97,Weihs98,Kwiat98}, that there may have been more serious misinterpretation. \n\nAdmittedly, the actual Bell tests performed may in some cases be general ones, not dependent on rotational invariance, but published papers reinforce the psychological impact of the tests by means of graphs of coincidence rate variations. In these graphs, one of the detector settings is held fixed. The high ``visibility\" ((max - min)/(max + min)) of such graphs is understood to be corroborative evidence of entanglement. But what if all the polarisations had in fact been parallel? High -- even 100\\% -- visibility could have arisen from this cause. It is only if we know that the polarisation direction (or other hidden variable) had RI that the visibility of the curve gains any significance at all (see Appendix~\\ref{Loopholes} for yet other realist explanations of high visibility). It is not usual to conduct comprehensive tests to show that there was no bias towards particular directions, that the graphs would have been just the same had the setting of the ``fixed\" detector been chosen differently. Indeed, in some experiments~\\cite{Kwiat98,Kwiat95}, it is clear that the absolute settings have a real physical significance, and this is actively taken into account: the fixed detector is always set at $45^{\\circ}$.\n\nFailure of rotational invariance represents effectively yet another ``loophole\" in Bell test experiments, and a brief discussion of other common ones is presented in Appendix~\\ref{Loopholes}. Is it not strange, incidentally, to find Bell test violations presented as demonstrating that the logic of the quantum world is different from every-day logic -- that no realist explanation of the results is possible -- for experiments that block only {\\em one} of the possible loopholes? Such a statement would become logical only if {\\em all were blocked simultaneously}!\n\nThe ``rotational invariance loophole\" may be but a minor factor in most Bell test experiments, causing a slight increase in the visibility of the coincidence curve, but it could be an important matter of principle. I shall discuss it with special reference to the experiment by Gregor Weihs {\\em et al.} in 1998~\\cite{Weihs98}. Though it is not clear whether or not it was in fact important here -- the relevant information is not available -- the form of the experiment makes it a potentially valuable one for the direct investigation \nof the quantum theory of ``degenerate PDC''. PDC experiments of this type are currently prominent in the area of ``quantum computing\". It might be as well for those concerned to consider whether or not the purely logical, ``realist\", explanation I present might be more robust than the quantum theory (QT) entanglement story.\n\n\n\\section{Rotational Invariance}\n\\label{Rotational Invariance}\nThe two extremes for RI are the full RI case generally assumed and the case of total failure, in which one direction is preferred to the exclusion of all others. There is, however, another special case that may be important, a case that might be termed ``Binary Rotational Invariance\", in which the ``hidden variable\", instead of taking a continuous range of values, can take only two discrete ones. It can lead to false interpretations of coincidence curve visibilities (though, in all likelihood, beneficial effects so far as practical applications are concerned).\n\n\n\\subsection{Full Rotational Invariance} \nThis is the standard case, both under QT and the usual realist model. It leads to the familiar QT prediction,\n\\begin{equation}\n\\label{QT prediction}\nP_{ab}^Q = \\frac{1}{2}\\cos^2(a - b), \n\\end{equation}\nfor the coincidence rate between two `$+$' outcomes, one on each side of the experiment, or the rather less familiar basic local realist one:\n\\begin{equation}\n\\label{Furry prediction}\nP_{ab}^{BLR} = \\frac{1}{8} (1 + 2 \\cos^2 (a - b)). \n\\end{equation}\n\n%figure (from lacoincs.pic)\n\n\\begin{figure}\n\\begin{center}\n\\small % Font\n\\input{coincs.lap}\n\\end{center}\n\\caption{The principal predicted ``coincidence curves\" for the ideal case. Curve (1): quantum theory; curve (2): the basic local realist prediction.}\n\\end{figure}\n\nThis last expression comes from just a few straightforward assumptions: \n\\begin{enumerate}\n\\item {\\bf ``Locality\":} Once we have fixed the value of a ``hidden variable\" that carries all the information on correlation from the source, the actual detections at different localities are independent events. We can apply the ordinary rules of statistics to calculate the probability of detecting $A$ and $B$ together by multiplying the probabilities for $A$ and $B$ separately. \n\n\\item {\\bf Malus' Law:} The detectors are perfect and exactly ``linear\", so that Malus' Law, which in its original form tells us that the intensity of the output from a polariser is $\\cos^2$ times the input, holds for the probabilities of detection of the light pulses involved in the ``single photon\" conditions of the experiments concerned. Thus we have probabilities of detection given by the formulae:\n\\begin{eqnarray}\n\\label{probability for fixed lambda}\nf_a^+(\\lambda) & = & \\cos^2(\\lambda - a) \\\\\n& \\mbox{for the `$+$' channel and} \\nonumber \\\\\nf_a^-(\\lambda) & = & \\sin^2(\\lambda - a) \n\\end{eqnarray}\nfor the `$-$' one, for a pulse of polarisation direction $\\lambda$ on exit from a polariser (a Wollaston prism, for example -- see Appendix~\\ref{Wollaston prism}) with axis direction $a$.\n\n\\item {\\bf Rotational Invariance:} as discussed here.\n\\end{enumerate}\n\nBoth QT and this local realist model predict constant ``singles\" rates, the realist calculation being:\n\\begin{eqnarray}\n\\label{Singles formula}\np_a^+ & = & \\int^{\\pi}_0 d\\lambda \\frac{1}{\\pi} \\cos^2 (\\lambda - a) \\nonumber\n \\\\\n& = & \\frac{1}{2}. \n\\end{eqnarray}\nWe simply integrate over all $\\lambda$ values, giving each equal weight. In rather more general situations, the formula would be:\n\\begin{equation}\np_a^+ = \\int^{\\pi}_0 d\\lambda \\rho(\\lambda) f_a^+(\\lambda),\n\\end{equation}\nwhere $\\rho(\\lambda)$ is the density function giving the probability of emission at polarisation angle $\\lambda$.\n\nAssuming that polarisations of $A$ and $B$ signals are always parallel, so that values of $\\lambda$ are shared, the coincidence rates (for `$++$' coincidences) are calculated by direct integration of:\n\\begin{equation}\n\\label{Furry coincidences: basic}\nP_{ab}^{BLR} = \\int^{\\pi}_0 d\\lambda \\frac{1}{\\pi} \\cos^2 (\\lambda - a) \\cos^2 (\\lambda - b),\n\\end{equation}\nwhere $b$ is the polarisation axis direction of the second polariser. This, of course, is just a special case of:\n\\begin{equation}\n\\label{Furry coincidences: general}\nP_{ab}^{BLR} = \\int^{\\pi}_0 d\\lambda \\rho(\\lambda) f_a^+(\\lambda) f_b^+(\\lambda).\n\\end{equation}\n\nIt is worth noting in passing that a subset of the possibilities for the integrand above represents probably the most important realist model of all: that in which the detection loophole alone is responsible for Bell test violations. All that is needed is to relax the assumption of ``linear\" detectors, so that the functions $f$ are not $\\cos^2$ (indeed, the $a$ and $b$ ones need not even be the same!). If the detectors are such that the probability of detection is effectively zero for a {\\em whole range of values} near $\\lambda - a = \\frac{\\pi}{2}$, instead of just for one {\\em exact point}, the minimum of $P_{ab}^{BLR}$ will be smaller than the value $\\frac{1}{8}$ given by the standard formula and the visibility of the curve greater than the 0.5 that it predicts. It is not hard to construct examples with visibility 1. This is the ``optical\" variant of the ``missing bands\" description of the operation of the detection loophole discussed in the author's ``Chaotic Ball\" paper~\\cite{Thompson96}.\n\nTo summarise, the distinguishing features of the ``full RI\" case are (a) singles counts that do not vary with detector settings and (b) coincidence counts that are functions of just the difference, $(a - b)$, between detector settings. As will be shown below, constancy of the singles counts is not sufficient.\n\n\\subsection{Complete Rotational Invariance Failure} \nIn an atomic cascade experiment using polarised light, complete failure of RI means that all the light is emitted with just one polarisation orientation, vertical, say. It is easily seen that this means that singles rates are maximum when detectors have their axes vertical; that if one detector is held fixed and the other varied we obtain coincidence curves that all (if Malus' Law is obeyed) have minimum of zero, obtained when the variable detector is horizontal. The maximum coincidence rate depends on the setting of the ``fixed\" detector, but for all settings other than the horizontal one the {\\em visibility} is 100\\%. (The fact that visibility depends only on the minimum seems often to be forgotten. In a regime in which ``normalisation\" is customary, this is unfortunate. A ``true\" Bell test, incidentally, would involve unnormalised data only: there are no circumstances in which visibility is a satisfactory substitute.)\n\nIf experimental results were to exhibit such behaviour, they would not be taken as demonstrating entanglement, as quite clearly it would be an almost deterministic situation. Two features of the results would have ruled out entanglement: the full visibility of the singles curves and the strong sensitivity of the coincidence curves to the setting of the fixed detector. When the fixed detector was horizontal, there would have been no coincidences whatever the setting of the variable one. In all other cases the coincidence rates would vary but the maximum would depend on the fixed setting.\n\nThe formula for predicted coincidences in this ``Complete RI Failure'' case is:\n\\begin{equation}\n\\label{Complete RI Failure}\nP_{ab}^{CRIF}(\\lambda_0) = \\cos^2 (\\lambda_0 - a) \\cos^2 (\\lambda_0 - b),\n\\end{equation}\nwhere $\\lambda_0$ is the {\\em constant} (vertical, in the above discussion) value of the common hidden variable shared by all the signals. No integration is really involved in its derivation, though by use of a delta function for the weighting we can artificially frame a derivation in the standard form, following the pattern of equation~(\\ref{Furry coincidences: general}).\n\n\\subsection{``Binary Rotational Invariance\"}\nWhilst ``Complete RI Failure\" corresponds to changing the distribution of the hidden variable from the constant $\\frac{1}{\\pi}$ to a delta function centred on one particular value of $\\lambda$, the special case to be presented here (that I shall term ``Binary RI\") can be obtained by changing it to the sum of two (equal-weight) delta functions, at $\\lambda_0$ and $\\lambda_1 = \\lambda_0 + p/2$, where $p$ is the period involved. \n\nIf the hidden variable is polarisation and the source is an atomic cascade, this model may seem strange, but it possibly fits some PDC situations. All that it amounts to is that we have 50\\% vertically and 50\\% horizontally polarised signals. In many PDC experiments, however, the hidden variable is really, I assert, ``phase difference\". As explained later, there are good reasons to think that this -- in the degenerate case and if there is zero dispersion -- falls logically into two sets, differing by $\\pi$ (half the period of $2\\pi$). (Somewhat confusingly, it emerges that mathematically, after projection as in Weihs' 1998 experiment of the two components in a $45^\\circ$ direction, we find that this translates into an apparent angular difference of $\\frac{\\pi}{2}$. See Appendix~\\ref{Wollaston prism}.)\n\nIn real experiments dispersion will tend to spread the delta functions out, in extreme cases causing them to merge and eventually blend into the full RI model. An experiment that could test whether the underlying logic really is as proposed would consist of repeating that of Gregor Weihs {\\em et al.}, omitting the random number generation but using the best available filters {\\em etc.}~to reduce dispersion to a minimum. In contrast to the CRIF case, we would expect to obtain singles counts that had negligible variation with detector setting. Coincidence curves, though, would have similar characteristics to the CRIF ones, their maxima being highly sensitive to the absolute value of the setting of the ``fixed\" detector. \n\nMathematically, Binary RI means dealing with the average of two equal ensembles, one with $\\lambda = \\lambda_0 = 0$, say, and the other with $\\lambda = \\lambda_1= \\frac{\\pi}{2}$. \n\nSingles counts for given $\\lambda$ are given as usual by:\n\\begin{equation}\n\\label{Singles prob, given lambda}\nf_a (\\lambda) = \\cos^2 (\\lambda - a)\n\\end{equation}\n(omitting the `$+$' suffix in this and other similar expressions for brevity)\nbut, with $\\lambda$ equalling 0 half the time and $\\frac{\\pi}{2}$ the other, the observed singles counts averaged over the ensemble will be:\n\\begin{eqnarray}\n\\label{Singles prob, BRIF}\np_a^{BRIF} & = & \\frac{1}{2} (\\cos^2 a + \\cos^2 (\\frac{\\pi}{2} - a)) \\nonumber \\\\\n\t& = & \\frac{1}{2} (\\cos^2 a + \\sin^2 a) \\nonumber \\\\\n\t& = & \\frac{1}{2}.\n\\end{eqnarray}\n\nCoincidence counts are predicted by the realist model (integrating expression (\\ref{Furry coincidences: general})) to be:\n\\begin{equation}\n\\label{Binary RI Failure}\nP_{ab}^{BRIF} = \\frac{1}{2} (\\cos^2 a \\cos^2 b + \\sin^2 a \\sin^2 b),\n\\end{equation}\nwhich gives full visibility (as $a$ is varied) for $b = 0$ or $\\frac{\\pi}{2}$ but a constant value of $\\frac{1}{4}$ and {\\em zero} visibility for $b = \\frac{\\pi}{4}$. Clearly the curves depend on the choice of $b$, and this is the distinguishing feature to differentiate between this case and ``full RI\". \n\n\\section{Applications to Real Experiments}\n\\label{Applications}\n\n\\subsection{Introduction}\n\nReal experiments using the degenerate PDC sources under consideration are sometimes complicated, and realist models must reflect this logically. The experiments may not only have hidden variables that are partly binary and partly continuous, but may also suffer from detectors whose characteristics vary with their absolute setting! In some of Paul Kwiat's recent experiments, for example~\\cite{Kwiat98,Kwiat95}, the signals are analysed by physically rotating waveplates~\\cite{waveplates} inserted in front of polarising prisms. This means, I believe, that subensembles of quite different natures will be detected for different settings of the waveplates. \n\nFor simplicity, therefore, we consider just experiments in which there is no physical rotation. Weihs' 1998 experiment (Fig.~\\ref{Weihs' 1998 Experiment}) is of this kind, and as a further simplification we assume initially that it is dispersion-free: the pump laser operates at just one exact frequency ($2\\omega$) and induces ``downconverted\" signals of exactly half that frequency. If the mechanisms controlling the random settings are ignored, the design reduces to (almost) that of the standard EPR experiment. The fact that the settings are random is of no consequence to the current discussion (see Appendix~\\ref{Loopholes}). The major difference between this and the atomic cascade-type experiments is the nature of the hidden variable, as I shall explain.\n\n\\begin{figure}\n\t%\\begin{center}\n \\small % Font\n\t\\input{weihs.lap}\n\t%\\end{center}\n\\normalsize\n\\caption{Scheme of Weihs' 1998 experiment, omitting random number generation etc. HWP = half-wave plate and compensating crystal; FC = 500m fibre cable; WP = Wollaston prism; $D^+$, $D^-$ are photodetectors; the box marked ``Coincidences\" stands for the complete system that puts time-tags on the results, stores them, then later analyses.}\n\\label{Weihs' 1998 Experiment}\n\\end{figure}\n\nWe are dealing with ``degenerate type-II parametric down-conversion\", in which QT assumes that two simultaneous ``photons\" are emitted, one vertically and the other horizontally polarised. The vertically and horizontally polarised light is emitted in the particular case under consideration in two cones, and the light selected for the ``EPR\" experiment is taken from the two lines of intersection~\\cite{Kwiat95}. According to QT, the down-conversion process necessarily converts one pump photon into exactly two output photons, so it is necessarily assumed that we get one vertical and one horizontal and that, when there are detections in both EPR channels at the same time, one must come from a vertical and one from a horizontal signal. \n\nBut if this were so, how would the ``modulator\" produce the results it does? It functions by using a voltage change to alter the relative speeds of the vertical and horizontal components of light, and hence the alters the relative phase (see footnote 13 of ref.~\\cite{Weihs98}). If each light pulse (I use this expression in preference to ``photon\") is polarised in just one direction, then how can the relative phase have any consequence? Yet the outcome of the experiment is that the modulator setting most certainly does have effect, and I attempt to clarify how this is detected in Appendix~\\ref{Wollaston prism}, which discusses the role of the Wollaston prism. Although the two components are orthogonal, the effect of the prism is to look at projections of them onto $45^\\circ$ planes, which enables interference patterns to be formed, much as in other interferometers. The phase shift of the modulator -- the consequence of a voltage change -- plays the role of an interferometer phase shift.\n\nThus the vital difference from the QT description is, I should like to suggest, that in reality both components are present in every pulse, or, at least, in the majority of pulses that are detected as coincidences. The experiment achieves sensitivity to the modulator by enabling the projections to interfere at the prism.\n\nThe reader interested in the QT description of events is refered to the vast literature on the ``singlet state\" and EPR experiments, and to a brief comment in Appendix~\\ref{QT story}. There are some points of similarity with the realist description, but major logical differences.\n\n\\subsection{General considerations re coherence and phases in degenerate PDC}\n\nFirst, let us take a fresh look at PDC. I make the fundamental assumption that, in the degenerate case, each signal (pulse of electromagnetic oscillations) is initiated with a particular phase-relationship to the {\\em pump} phase, as well as a particular polarisation. Under QT, this is not possible, as the frequencies are assumed even in the degenerate case to be ``conjugate\"~\\cite{Conjugate outputs}, not identical. Indeed, under QT the whole picture of laser light is so different that those trained in the discipline may have difficulty following the classical ideas that seem to me to fit the facts so well. Perhaps I need to go further back, and explain that I am suggesting that the kind of laser light used in these experiments comprises a series of (long) pulses, each of one pure frequency. I see no other way to explain the observed interference effects, which imply long coherence lengths, in the original, classical, meaning of the term: the path length difference that can be introduced to a split beam such that recombination will result in interference.\n\nAs there is always some dispersion (observed spectra cover a band of frequencies), direct verification of my model is not easy, yet I have been unable to find any experimental evidence to support the QT ``coherent state\" concept~\\cite{Furusawa98}. In my model, the dispersion of the (degenerate) down-converted light is merely a matter of its comprising a series of pulses, each of pure frequency, inherited from pump laser light with similar properties\\cite{Kwiat/Chiao91}. Given a pump frequency, $2\\omega$ (which may in general vary between pulses), no conservation laws are infringed if the outputs induced by a particular pulse both have frequency exactly $\\omega$. \n\nThe phase relationships with the pump arise from the assumption that some causal mechanism is at work. In some sense, the outputs, with beats half as frequent as the pump, are able to be in phase with every alternate beat, and, since all beats are presumably equivalent, this means a natural 50-50 division into two sets, the ``even\" and the ``odd\". (The choice of even or odd might be attributed to the influence of the zero point field, as in the theory of Stochastic Electrodynamics~\\cite{Marshall88}.) The original inspiration for this idea was as an explanation for the ``induced coherence\" experiments~\\cite{Zou/Wang/Mandel91}. \n\nThe factor of importance for experiments in which both vertical and horizontal components are present in the same pulse is whether their phase sets are the same (giving phase difference zero) or opposite (phase difference $\\pi$). It is not clear whether or not the two components are always generated independently. In cases in which they are not, it is possible that all phase differences will be zero, but otherwise we would expect the phase differences to be 50\\% zero, 50\\% $\\pi$. \n\n\\subsection{A New Realist Description of Weihs' 1998 experiment}\n\nReturning now to Weihs' experiment, let us follow the progress of an individual ``double\" (V and H component) pulse on one side of the apparatus. We are, for the moment, ignoring dispersion, assuming that the two components are emitted exactly in phase, and treating the initial half-wave plate and the modulator as a single device whose effect is to modify the phase difference. \n\nWe write the equation for oscillations of the electric vector of our pulse in its initial state as:\n\\begin{equation}\n\\label{single pulse initial}\n\\psi(x) = \\frac{1}{\\sqrt{2}} (\\boldmath{i} \\cos x + \\boldmath{j} \\cos x),\n\\end{equation}\nwhere $x = \\omega t$ and $\\boldmath{i}$ and $\\boldmath{j}$ are unit vectors in the vertical and horizontal directions, as defined by the axes of the nonlinear crystal. The expression has been normalised to a wave of amplitude and intensity 1 unit.\n\nAfter passage through the waveplate and modulator, it becomes:\n\\begin{equation}\n\\label{single pulse after modulator}\n\\psi_a(x) = \\frac{1}{\\sqrt{2}} (\\boldmath{i} \\cos x + \\boldmath{j} \\cos (x + 2a)),\n\\end{equation}\nwhere $2a$ is the induced phase difference, the common phase shift having, with no loss of generality, been ignored. (More care is needed when we re-introduce frequency dispersion. $a$ is then seen to be a function of frequency, since the speed of light in the apparatus varies with frequency as well as with polarisation direction.)\n\nWe now encounter a Wollaston prism, set at $45^\\circ$. Assuming Malus' law to apply, the effect of the prism is to output the sum of the projections of the waves. The mathematics is given in Appendix~(\\ref{Wollaston prism}), and leads to the results:\n\n\\begin{eqnarray}\n\\label{single pulse probabilities}\nf_a^+(0) & = & \\cos^2 a \\nonumber \\\\\nf_a^-(0) & = & \\sin^2 a.\n\\end{eqnarray}\n\nIf all pulses were as above, with initial phase difference of zero (hidden variable $\\lambda = 0$), these patterns should appear in the `$+$' and `$-$' channels as we vary the modulator voltage. But they do not. The singles counts, we are told, have no oscillations. This is one strong indicator that what we in fact see could be the superposition of one pattern of this class and one totally out of phase, with $a$ replaced by $a + \\frac{\\pi}{2}$ (i.e. $\\lambda = \\frac{\\pi}{2}$). The two patterns could be washing each other out, producing a steady probability averaging \n\\begin{equation}\n\\frac{1}{2}(\\cos^2 a + \\sin^2 a) \\equiv \\frac{1}{2}.\n\\end{equation}\n\nNow to consider coincidences, though. Do these oscillate? The answer is ``Yes\", and, at least for the values of $b$ selected in Weihs' reported results, the visibility as $a$ is varied is high. This is the second indicator that what we are seeing is likely to contain a strong element of the ``Binary Rotational Invariance\" situation described above. If it were a pure case (which I assert might happen if frequency dispersion were negligible), the formula for coincidences would be precisely equation~(\\ref{Binary RI Failure}) above. \n\nThus the realist prediction for coincidences would be:\n\\begin{equation}\nP_{ab}^R = \\frac{1}{2} (\\cos^2 a \\cos^2 b + \\sin^2 a \\sin^2 b).\n\\end{equation}\nLet us compare this with the standard QT prediction,\n\\begin{equation}\nP_{ab}^Q = \\frac{1}{2} \\cos^2 (a - b).\n\\end{equation}\n\nExpanding the latter, we find that it can be written:\n\\begin{eqnarray}\nP_{ab}^Q & = & \\frac{1}{2} (\\cos a \\cos b + \\sin a \\sin b)^2 \\nonumber\\\\\n \t& \\equiv & P_{ab}^R + \\cos a \\cos b \\sin a \\sin b.\n\\end{eqnarray}\n\nThe two formulae are identical when the second term is zero, which may explain some of Weihs' results. \n\nThere are problems, though, in attempting a detailed reconciliation between my model and the actual experiment. Insufficient information is given in the published paper~\\cite{Weihs98 info}. Correspondence with Weihs, though very helpful, has shown that more experimentation is needed. The differences between the rival models concern the positions of the maxima and minima and the way in which the range of variation changes as we alter the ``fixed\" setting, $b$. The available information does not seem to cover either of these points reliably.\n\nAs mentioned earlier, frequency variations will blur the issue. They give rise to a second, continuous, component that is added to our hidden variable, making its distribution bimodal. This second component is the phase difference caused by variations in the pump frequency, and it can cause the Binary RI model to merge into the standard realist one, with complete rotational invariance. There are, however, sufficient clues in this and other published papers to suggest that further experimentation would be rewarding. One of the predictions of the class of models under consideration is the production in certain circumstances of skewed coincidence curves, not pure sinusoidal ones. Is there a hint of this, perhaps, in Fig.~3 of~\\cite{Tittel97-1}?\n\nNote that there are other facets of these experiments that deserve attention, and, indeed, it may be impossible to deduce the true facts without a comprehensive study of all of them. For example, realist models predict that there would (if the electronics allowed!) be some simultaneous outputs from the `$+$' and `$-$' channels of a single prism, and that the shape of the response curve of the detectors as light intensity is varied plays a critical role. The necessary investigations would entail direct challenges to the photon model of light and require critical re-evaluation of much experimental evidence. \n\n\\section{Conclusion}\n\nIt is entirely possible that the kind of entanglement we see here is, as several authors have said ~\\cite{Zeilinger99,Hardy99} no more than a change in our state of knowledge. It does not involve ``nonlocality\", but nor does it really obey the tenets of QT. It is merely a matter of using information (phase-differences) on one set of entities to select those of another identical set with certain properties. The process is possible because the correlations are real and strong, involving phase, frequency {\\em and} polarisation. Two other factors almost certainly enhance it: nonlinearity of detectors, making the detection loophole large, and failure of rotational invariance. Both factors increase the visibility of coincidence curves. Contrary to widespread belief, high visibility (over 50\\%) does {\\em not} in itself conflict with local realism.\n\nThe ideas of this paper would appear to be easily testable, and might remove all trace of mystery from Weihs' and certain other experiments. For some experiments, another interesting possibility may need to be taken into account: do some detectors integrate over time in such a way that ``interference\" can appear to occur between pulses separated in time~\\cite{Chalmers99,Kim99}?\n\nA new area of physics awaits exploration. How do the exactly-matched (but not entangled!) PDC pairs arise? A theory is needed that recognises that coincidence measurements (usually) involve individual pairs, with no time-averaging over ensembles except at a later stage. It needs to recognise that measured spectra tell us only time-averages. The rules that relate bandwidths to time-intervals are not necessarily relevant when we look at the individual pulse, which may have a much longer coherence length than is currently assumed. \n\nThere are theories to cover the general case of parametric down-conversion, but these predict that frequency variations will always be oppositely correlated, never identical. They fail to predict clearly the existence of the two phase classes of the degenerate case. The experiments that might now become transparent include those demonstrating ``induced coherence''~\\cite{Zou/Wang/Mandel91}, ``quantum erasure''~\\cite{Herzog/Kwiat/Weinfurter/Zeilinger95} and ``teleportation''~\\cite{Bouwmeester97}. \n\n\\appendix\n\\section{Other Bell Test ``Loopholes\"}\n\\label{Loopholes}\n\nThe term ``loophole\" is a euphemism for the statement that the test applied is not valid as a discriminator between QT and realism as it depends on assumptions that are not universally accepted. (The tests used in practice are, in John Bell's own words (page 60 of ref.~\\cite{Bell87}) ``more or less ad hoc extrapolations\" of his original, which would have been valid but is impractical~\\cite{Clauser/Horne74,Santos91}.) The most important loopholes are:\n\\begin{enumerate}\n\\item {\\bf The ``detection\" loophole:} An interesting paper by members of the quantum optics team in Geneva~\\cite{Gisin99-1} has drawn attention to the fact that this it is not logical to neglect this. Clauser and Horne, in their paper of 1974~\\cite{Clauser/Horne74} derived practical Bell tests that do not depend on it, but these have rarely been used. \n\nSome important papers describing the operation of the detection loophole are refs.~\\cite{Thompson96,Pearle70,Marshall/Santos/Selleri83}. In brief, it is only under quantum theory assumptions that we expect the total population of ``coincidences\", involving all four combinations of `$+$' and `$-$' outputs on the two sides, to be a fair sample of the population of emitted pairs. If this population can vary with detector setting, then clearly an estimate of probability of detection will be biased if it uses (as the standard formula does) the total number of coincidences as divisor. In realist theories, it is the exception rather than the rule for the sample to be fair in this sense. If light is a purely wave phenomenon, it is possible for photodetectors to respond nonlinearly to intensity. This, as is clear from the structure of the basic realist prediction~(\\ref{Furry coincidences: general}), can cause high coincidence curve visibilities and associated infringements of Bell tests, but the sample will not have been ``fair\". \n\nIt is not usual for experimenters to attempt a direct test for linearity~\\cite{Weihs99}. A test that is sometimes performed is to check that the total of the coincidence counts entering into the estimated ``correlation\" is constant. Failures of constancy may be masked by experimental error, however, and, in addition, the parameter values (zero and $45^{\\circ}$) most sensitive to this test are not usually explored.\n\n\\item {\\bf Subtraction of ``accidentals\":} Adjustment of the data by subtraction of ``accidentals\" biases Bell tests in favour of quantum theory. It is now recognised as not legitimate~\\cite{Tittel98-1}, but the reader should be aware that it invalidates many published results~\\cite{Thompson99}.\n\n\\item {\\bf Synchronisation problems:} There is reason to think that in a few experiments bias could be caused when the coincidence window is shorter than the some of the light pulses involved~\\cite{Thompson97}. These few include one of historical importance -- that of Freedman and Clauser, in 1972~\\cite{Freedman/Clauser72} -- which uses a test not sullied by either of the above possibilities.\n\\end{enumerate}\n\nA loophole that is notably absent from the above list is the so-called ``locality loophole\", whereby some mysterious unspecified mechanism is taken as conveying additional information between the two detectors so as to increase their correlation above the classical limit. In the view of many realists, this has never been a contender. John Bell supported Aspect's investigation of it (see page 109 of ref.~\\cite{Bell87}) and had some active involvement with the work, being on the examining board for Aspect's PhD. Gregor Weihs improved upon the work in his experiment of 1998~\\cite{Weihs98}, but nobody has ever put forward plausible ideas for the mechanism. Its properties must be quite extraordinary, as it is required to explain ``entanglement\" in a great variety of geometrical setups, including over a distance of several kilometers in the Geneva experiments of \n1997-8~\\cite{Tittel97,Tittel98-1}.\n\nThere may well be other loopholes. Vladimir Nuri~\\cite{Nuri99} is currently studying the possible consequences of the usual experimental arrangement, in which simultaneous `$+$' and `$-$' counts from both outputs of a polariser can never occur as the electronics records only one or the other. Under QT, they will not occur anyway, but under a wave theory the suppression of these counts will cause even the basic realist prediction (\\ref{Furry prediction}) to yield ``unfair sampling''. The effect is negligible if the detection efficiencies are low, however.\n\n\\section{The effect of a Wollaston prism}\n\\label{Wollaston prism}\n\nA Wollaston prism is a ``2-channel polariser\". If a plane-polarised wave is input, the output intensities are given, in the absence of losses, by Malus' Law. In other words, it is the projections onto the prism axis and the direction orthogonal to this that are output.\n\nNow in the experiment in question, the prism is set at the fixed angle of $45^\\circ$, half way between the polarisation directions of our two components. Assuming that the two components add linearly, the polariser combines the two projections, each of which will have amplitude $1/\\sqrt{2}$ times its input amplitude. As with any other case of interference, the relative phase of the two components has a striking effect on the outcome. If there are no losses, the wave function~(\\ref{single pulse after modulator}), modelling a single (two-component) wave pulse, will lead to outputs:\n\\begin{eqnarray}\n\\label{single pulse after prism}\n\\psi_a^+(x) & = & \\frac{1}{2} (\\cos x + \\cos (x + 2a))\\\\ \n& \\mbox{and} \\nonumber \\\\\n\\psi_a^-(x) & = & \\frac{1}{2} (\\cos x - \\cos (x + 2a)).\n\\end{eqnarray}\n\nUsing the trigonometric identity \n\\begin{displaymath}\n\\cos(A + B) + \\cos(A - B) \\equiv 2 \\cos A \\cos B,\n\\end{displaymath}\n\n$\\psi_a^+(x)$ can be written:\n\\begin{equation}\n\\psi_a^+(x) = \\cos (x + a) \\cos a,\n\\end{equation}\nwith a similar expression for $\\psi_a^-(x)$. \n\nThis is a plane wave whose amplitude is $\\cos a$. Effectively, it represents the interference pattern between the vertical and horizontal components of a single pulse. If our detectors are linear (detection rates proportional to intensities -- an assumption that can be challenged~\\cite{Thompson99}) and all pulses identical, the singles count rates should follow the pattern:\n\n\\begin{equation}\n\\label{single pulse detection rate}\np_a^+ = \\cos^2 a.\n\\end{equation}\n\nThe pulse in question, with zero phase difference apart from that induced by the apparatus, is to be considered as having ``hidden variable\" of zero, so that we have $f_a^+(0) = \\cos^2 a$.\n\n\n\\section{Note on QT Description of Weihs' Experiment}\n\\label{QT story}\nWeihs expresses the ``entangled state\" of the two photons (after their passage through a half-wave plate~\\cite{waveplates}) as follows: \n\\begin{equation}\n\\label{entangled}\n\|\\Psi\\rangle = 1/\\sqrt{2}(\|H\\rangle_1\|V\\rangle_2 + \ne^{i\\phi}\|V\\rangle_1\|H\\rangle_2).\n\\end{equation}\nApart from the factor $e^{i\\phi}$, this is the same ``singlet state\" formula as would apply in experiments such as Aspect's~\\cite{Aspect81+}. Whereas for Aspect's polarised light, however, the absence of an experimentally meaningful definition for vertical and horizontal directions means that the formula makes little sense, in the current experiment we can relate it quite closely to realism. Vertical and horizontal are well-defined, related to the optical axes of all parts of the apparatus, and a direct realist explanation is almost possible. There are subtle differences, though, so I shall develop realist expressions working from first principles rather then try and force a close correspondence.\n\nOne difference involves the $e^{i\\phi}$ factor. Weihs states that he is able to control the phase $\\phi$, setting it to $\\pi$. But what is this phase shift? Under the classical model I am proposing, the only phase difference that is of experimental consequence is the difference between the phases of the two components, when both are present. We can indeed change this, by means of waveplates or variable ``modulators''~\\cite{waveplates}, but the phase difference between the two possibilities represented by the two terms of the wave function~(\\ref{entangled}) is, I assert, fixed naturally. The formula implies that we get {\\em either} ($\|H\\rangle_1$ and $\|V\\rangle_2$) {\\em or} ($\|V\\rangle_1$ and $\|H\\rangle_2$), with the ability to adjust the phase between the first and second possibility. This relative phase is meaningless, as it is between events happening at different times. Hopefully, the purely realist treatment in the main text clarifies this.\n\n\n\\begin{thebibliography}{00}\n\n\\bibitem [] {CHT/email+}\nEmail address: c.h.thompson@newscientist.net\\\\\nWeb Site: http://www.aber.ac.uk/$\\tilde{\\ }$cat\n\n\\bibitem{Freedman72}\nS.~J.~Freedman, {\\em Experimental test of local hidden-variable theories} (PhD thesis (available on microfiche), University of California, Berkeley, 1972).\n\n\\bibitem{Aspect81+}\nA.~Aspect, {\\em et al.}, Physical Review Letters {\\bf 47}, 460\n (1981); {\\bf 49}, 91 (1982) and {\\bf 49}, 1804 (1982).\n\n\\bibitem{Aspect83}\nA.~Aspect, {\\em Trois tests exp\\'erimentaux des in\\'egalit\\'es de Bell par mesure de corr\\'elation de polarisation de photons}, PhD thesis No. 2674, Universit\\'e de Paris-Sud, Centre D'Orsay, (1983).\n\n\\bibitem{Tittel97}\nW.~Tittel {\\em et al.}, ``Experimental demonstration of quantum-correlations over more than 10 kilometers'', {\\em Physical Review A}, {\\bf 57}, 3229 (1997), $<$http://xxx.lanl.gov/abs/quant-ph/9707042$>$\n\n\\bibitem{Weihs98}\nG.~Weihs, {\\em et al.}, ``Violation of Bell's inequality under strict \nEinstein locality conditions'', Physical Review Letters {\\bf 81}, 5039 (1998)\nand $<$http://xxx.lanl.gov/abs/quant-ph/9910080$>$\n\n\\bibitem{Kwiat98}\nP.~G.~Kwiat {\\em et al.}, ``Ultra-bright source of polarization-entangled photons\", to appear in Phys Rev A, rapid communications. Available at \n$<$http://xxx.lanl.gov/abs/quant-ph/9810003$>$\n\n\\bibitem{Kwiat95}\nP.~G.~Kwiat {\\em et al.}, ``New High-Intensity source of \npolarization-entangled photon pairs\", Physical Review Letters {\\bf 75}, 4337 (1995).\n\n\\bibitem{Thompson96}\nC.~H.~Thompson, Foundations of Physics Letters {\\bf 9}, 357 (1996) and $<$http://xxx.lanl.gov/abs/quant-ph/9611037$>$\n\n\\bibitem{waveplates}\nWaveplates (and the ``modulator'' used in Weihs' experiment) change the relative phases of the vertical and horizontal components of light. Ordinary waveplates do this by using birefringent materials, transmitting the two components at different speeds. The precise speeds vary with frequency, so the resulting phase shifts will do so also. They do not, incidentally, produce any real ``polarisation rotation'', and this fact is recognised in footnote [13] of Weihs' paper. When light is analysed by a Wollaston prism after passage through a waveplate, it behaves very much as if its plane of polarisation had been rotated, but subtle tests should be able to reveal the distinction. Light that appears to have been rotated through $45^\\circ$ has in fact been changed from linear polarisation (phase difference $0$ or $\\pi$) to circular (phase difference $\\frac{\\pi}{2}$).\n\n\\bibitem{Conjugate outputs}\nQT assumes that in all cases of Parametric Downconversion two photons are produced and the relationship of their frequencies ($\\omega_1$ and $\\omega_2$) to the pump frequency $2\\omega$ can be expressed in the form:\n\\begin{eqnarray}\n\\omega_1 & = & \\omega + \\delta \\nonumber \\\\\n\\omega_2 & = & \\omega - \\delta.\n\\end{eqnarray}\nIn the degenerate case, $\\delta$ is assumed to be very small. See, for example, the 1997-8 Geneva experiments by Tittel {\\em et al.}.\n\n\\bibitem{Furusawa98} \nA.~Furusawa {\\em et al.}, ``Unconditional quantum teleportation\", Science {\\bf 282}, 706-709 (1998).\n\n\\bibitem{Kwiat/Chiao91}\nI am not quite alone in my hypothesis that PDC outputs could, sometimes at least, comprise a series of pure-frequency pulses. Kwiat and Chiao mention the idea, only to reject it, but I challenge the grounds for rejection. See p 591 of P.~G.~Kwiat and R.~Y.~Chiao, ``Observation of a nonclassical Berry's phase for the photon\", Physical Review Letters {\\bf 66}, 588 (1991). \n\n\\bibitem{Marshall88}\nT.~W.~Marshall, ``Stochastic Electrodynamics and the Einstein-Podolsky-Rosen Argument'' in {\\em Quantum Mechanics versus Local Realism, The \nEinstein-Podolsky-Rosen Paradox} F. Selleri ed., (Plenum Press, 1988), \npp 413-432.\n\n\\bibitem{Zou/Wang/Mandel91}\nX.~Y.~Zou {\\em et al.}, ``Induced coherence and indistinguishability in optical interference'', Physical Review Letters {\\bf 67}, 318 (1991).\n\n\\bibitem{Weihs98 info}\nThe main data, available electronically, concerns just two settings of each modulator. Coincidence curves involve further data, but though one modulator takes a continuous range of settings the other is restricted to just the two. This is insufficient to establish the extent of any failure of RI.\n\n\\bibitem{Tittel97-1}\nW.~Tittel {\\em et al.}, ``Non-local two-photon correlations using interferometers physically separated by 35 meters\", (1997)\n$<$http://xxx.lanl.gov/abs/quant-ph/9703023$>$\n\n\\bibitem{Zeilinger99}\n%{PhysicsToday99}\nA.~Zeilinger, in letters responding to Sheldon Goldstein's March 1998 article, ``Quantum theory without observers\". Physics Today, February 1999, pp11-15 and 89-92 (1999).\n\n\\bibitem{Hardy99}\nL.~Hardy, ``Disentangling nonlocality and teleportation\", (1999) $<$http://xxx.lanl.gov/abs/quant-ph/9906123$>$\n\n\\bibitem{Chalmers99}\nD.~Chalmers, private communications, (1999).\n\n\\bibitem{Kim99}\nY-H.~Kim, ``First-order interference of nonclassical light emitted spontaneously at different times\" (1999), $<$http://xxx.lanl.gov/abs/quant-ph/9911014$>$\n\n\\bibitem{Herzog/Kwiat/Weinfurter/Zeilinger95}\nT.~J.~Herzog {\\em et al.}, ``Complementarity and the quantum eraser'', Physical Review Letters {\\bf 75}, 3034 (1995).\n\n\\bibitem{Bouwmeester97}\nD.~Bouwmeester {\\em et al.}, ``Experimental Quantum Teleportation\", Nature {\\bf 390}, 575 (1997). \n\n\\bibitem{Bell87}\nJ.~A.~Bell, {\\em The Speakable and Unspeakable in Quantum Mechanics}, (Cambridge University Press, 1987).\n\n\\bibitem{Clauser/Horne74}\nJ.~F.~Clauser and M.~A.~Horne., Physical Review D {\\bf 10}, 526 (1974).\n\n\\bibitem{Santos91}\nE.~Santos, ``Does Quantum Mechanics Violate the Bell Inequalities\", Physical Review Letters {\\bf 66}, 1388 (1991).\n\n\\bibitem{Gisin99-1}\nN.~Gisin and B.~Gisin, ``A local hidden variable model of quantum correlation exploiting the detection loophole'', (1999) \n$<$http://xxx.lanl.gov/abs/quant-ph/9905018$>$\n\n\\bibitem{Pearle70} \nP.~Pearle, ``Hidden-variable example based upon data rejection'', Physical Review D {\\bf 2}, 1418-25 (1970).\n\n\\bibitem{Marshall/Santos/Selleri83}\nT.~W.~Marshall, {\\em et al.}, ``Local Realism has not been Refuted \nby Atomic-Cascade Experiments\", Physics Letters A {\\bf 98}, 5-9 (1983).\n\n\\bibitem{Weihs99}\nGregor Weihs is currently (private correspondence, 1999) investigating the detection characteristics of the photodetectors used in recent experiments.\n\n\\bibitem{Tittel98-1}\nW.~Tittel {\\em et al.}, ``Long-distance Bell-type tests using energy-time entangled photons'', (1998) $<$http://xxx.lanl.gov/abs/quant-ph/9809025$>$\n\n\\bibitem{Thompson99}\nC.~H.~Thompson, ``Subtraction of `accidentals' and the validity of Bell tests'', submitted to Physical Review A, (1999) $<$http://xxx.lanl.gov/abs/quant-ph/9903066$>$\n\n\\bibitem{Thompson97}\nC.~H.~Thompson, ``Timing, `Accidentals' and Other Artifacts in EPR \nExperiments\", (1997) $<$http://xxx.lanl.gov/abs/quant-ph/9711044$>$\n\n\n\\bibitem{Freedman/Clauser72}\nS.~J.~Freedman and J.~F.~Clauser, Physical Review Letters {\\bf 28}, 938 (1972).\n\n\\bibitem{Nuri99}\nV.~Nuri, private communications, (1999).\n\n\\end{thebibliography}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912082.extracted_bib", "string": "[*] {CHT/email+ Email address: c.h.thompson@newscientist.net\\\\ Web Site: http://www.aber.ac.uk/$\\tilde{\\ $cat" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Freedman72 S.~J.~Freedman, {\\em Experimental test of local hidden-variable theories (PhD thesis (available on microfiche), University of California, Berkeley, 1972)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Aspect81+ A.~Aspect, {\\em et al., Physical Review Letters {47, 460 (1981); {49, 91 (1982) and {49, 1804 (1982)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Aspect83 A.~Aspect, {\\em Trois tests exp\\'erimentaux des in\\'egalit\\'es de Bell par mesure de corr\\'elation de polarisation de photons, PhD thesis No. 2674, Universit\\'e de Paris-Sud, Centre D'Orsay, (1983)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Tittel97 W.~Tittel {\\em et al., ``Experimental demonstration of quantum-correlations over more than 10 kilometers'', {\\em Physical Review A, {57, 3229 (1997), $<$http://xxx.lanl.gov/abs/quant-ph/9707042$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Weihs98 G.~Weihs, {\\em et al., ``Violation of Bell's inequality under strict Einstein locality conditions'', Physical Review Letters {81, 5039 (1998) and $<$http://xxx.lanl.gov/abs/quant-ph/9910080$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Kwiat98 P.~G.~Kwiat {\\em et al., ``Ultra-bright source of polarization-entangled photons\", to appear in Phys Rev A, rapid communications. Available at $<$http://xxx.lanl.gov/abs/quant-ph/9810003$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Kwiat95 P.~G.~Kwiat {\\em et al., ``New High-Intensity source of polarization-entangled photon pairs\", Physical Review Letters {75, 4337 (1995)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Thompson96 C.~H.~Thompson, Foundations of Physics Letters {9, 357 (1996) and $<$http://xxx.lanl.gov/abs/quant-ph/9611037$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{waveplates Waveplates (and the ``modulator'' used in Weihs' experiment) change the relative phases of the vertical and horizontal components of light. Ordinary waveplates do this by using birefringent materials, transmitting the two components at different speeds. The precise speeds vary with frequency, so the resulting phase shifts will do so also. They do not, incidentally, produce any real ``polarisation rotation'', and this fact is recognised in footnote [13] of Weihs' paper. When light is analysed by a Wollaston prism after passage through a waveplate, it behaves very much as if its plane of polarisation had been rotated, but subtle tests should be able to reveal the distinction. Light that appears to have been rotated through $45^\\circ$ has in fact been changed from linear polarisation (phase difference $0$ or $\\pi$) to circular (phase difference $\\frac{\\pi{2$)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Conjugate outputs QT assumes that in all cases of Parametric Downconversion two photons are produced and the relationship of their frequencies ($\\omega_1$ and $\\omega_2$) to the pump frequency $2\\omega$ can be expressed in the form: \\begin{eqnarray \\omega_1 & = & \\omega + \\delta \\nonumber \\\\ \\omega_2 & = & \\omega - \\delta." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Furusawa98 A.~Furusawa {\\em et al., ``Unconditional quantum teleportation\", Science {282, 706-709 (1998)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Kwiat/Chiao91 I am not quite alone in my hypothesis that PDC outputs could, sometimes at least, comprise a series of pure-frequency pulses. Kwiat and Chiao mention the idea, only to reject it, but I challenge the grounds for rejection. See p 591 of P.~G.~Kwiat and R.~Y.~Chiao, ``Observation of a nonclassical Berry's phase for the photon\", Physical Review Letters {66, 588 (1991)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Marshall88 T.~W.~Marshall, ``Stochastic Electrodynamics and the Einstein-Podolsky-Rosen Argument'' in {\\em Quantum Mechanics versus Local Realism, The Einstein-Podolsky-Rosen Paradox F. Selleri ed., (Plenum Press, 1988), pp 413-432." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Zou/Wang/Mandel91 X.~Y.~Zou {\\em et al., ``Induced coherence and indistinguishability in optical interference'', Physical Review Letters {67, 318 (1991)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Weihs98 info The main data, available electronically, concerns just two settings of each modulator. Coincidence curves involve further data, but though one modulator takes a continuous range of settings the other is restricted to just the two. This is insufficient to establish the extent of any failure of RI." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Tittel97-1 W.~Tittel {\\em et al., ``Non-local two-photon correlations using interferometers physically separated by 35 meters\", (1997) $<$http://xxx.lanl.gov/abs/quant-ph/9703023$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Zeilinger99 %{PhysicsToday99 A.~Zeilinger, in letters responding to Sheldon Goldstein's March 1998 article, ``Quantum theory without observers\". Physics Today, February 1999, pp11-15 and 89-92 (1999)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Hardy99 L.~Hardy, ``Disentangling nonlocality and teleportation\", (1999) $<$http://xxx.lanl.gov/abs/quant-ph/9906123$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Chalmers99 D.~Chalmers, private communications, (1999)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Kim99 Y-H.~Kim, ``First-order interference of nonclassical light emitted spontaneously at different times\" (1999), $<$http://xxx.lanl.gov/abs/quant-ph/9911014$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Herzog/Kwiat/Weinfurter/Zeilinger95 T.~J.~Herzog {\\em et al., ``Complementarity and the quantum eraser'', Physical Review Letters {75, 3034 (1995)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Bouwmeester97 D.~Bouwmeester {\\em et al., ``Experimental Quantum Teleportation\", Nature {390, 575 (1997)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Bell87 J.~A.~Bell, {\\em The Speakable and Unspeakable in Quantum Mechanics, (Cambridge University Press, 1987)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Clauser/Horne74 J.~F.~Clauser and M.~A.~Horne., Physical Review D {10, 526 (1974)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Santos91 E.~Santos, ``Does Quantum Mechanics Violate the Bell Inequalities\", Physical Review Letters {66, 1388 (1991)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Gisin99-1 N.~Gisin and B.~Gisin, ``A local hidden variable model of quantum correlation exploiting the detection loophole'', (1999) $<$http://xxx.lanl.gov/abs/quant-ph/9905018$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Pearle70 P.~Pearle, ``Hidden-variable example based upon data rejection'', Physical Review D {2, 1418-25 (1970)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Marshall/Santos/Selleri83 T.~W.~Marshall, {\\em et al., ``Local Realism has not been Refuted by Atomic-Cascade Experiments\", Physics Letters A {98, 5-9 (1983)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Weihs99 Gregor Weihs is currently (private correspondence, 1999) investigating the detection characteristics of the photodetectors used in recent experiments." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Tittel98-1 W.~Tittel {\\em et al., ``Long-distance Bell-type tests using energy-time entangled photons'', (1998) $<$http://xxx.lanl.gov/abs/quant-ph/9809025$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Thompson99 C.~H.~Thompson, ``Subtraction of `accidentals' and the validity of Bell tests'', submitted to Physical Review A, (1999) $<$http://xxx.lanl.gov/abs/quant-ph/9903066$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Thompson97 C.~H.~Thompson, ``Timing, `Accidentals' and Other Artifacts in EPR Experiments\", (1997) $<$http://xxx.lanl.gov/abs/quant-ph/9711044$>$" }, { "name": "quant-ph9912082.extracted_bib", "string": "{Freedman/Clauser72 S.~J.~Freedman and J.~F.~Clauser, Physical Review Letters {28, 938 (1972)." }, { "name": "quant-ph9912082.extracted_bib", "string": "{Nuri99 V.~Nuri, private communications, (1999)." } ]
quant-ph9912083	Quantum Teleportation with Entangled States\\ given by Beam Splittings	[ { "author": "Karl-Heinz Fichtner" }, { "author": "%EndAName Friedrich-Schiller-Universit\\\"{a" } ]	Quantum teleportation is rigorously discussed with coherent entangled states given by beam splittings. The mathematical scheme of beam splitting has been used to study quantum communication \cite{AO2 and quantum stochastic \cite {FFL. We discuss the teleportation process by means of coherent states in this scheme for the following two cases: (1) Delete the vacuum part from coherent states, whose compensation provides us a perfect teleportation from Alice to Bob. (2) Use fully realistic (physical) coherent states, which gives a non-perfect teleportation but shows that it is exact when the average energy (density) of the coherent vectors goes to infinity.	[ { "name": "quant-ph9912083.tex", "string": "\n\\documentclass[12pt,twoside]{article}\n\\usepackage{amssymb}\n\n\\usepackage{amsmath}\n\\usepackage{theorem}\n\\parskip1ex plus0.2ex minus0.1ex\n\\def\\ts{\\textstyle}\n\\def\\dis{\\displaystyle}\n\\def\\bm{\\boldmath }\n\\def\\ubm{\\unboldmath}\n\\def\\R{\\mathbb R}\n\\def\\C{\\mathbb C}\n\\def\\1{{1\\mskip-10mu1}}\n\\def\\Lra{\\Leftrightarrow}\n\\def\\bea{\\begin{eqnarray}}\n\\def\\eea{\\end{eqnarray}}\n\\def\\bean{\\begin{eqnarray}}\n\\def\\eean{\\end{eqnarray}}\n\\def\\qed{$\\square$}\n\\newcommand{\\mfr}[1]{\\mathfrak #1}\n\\newcommand{\\mcl}[1]{\\mathcal #1}\n\\newtheorem{ftheo}{THEOREM}[section]\n\\newtheorem{fex}[ftheo]{EXAMPLE}\n\\newtheorem{fdef}[ftheo]{DEFINITION}\n\\newtheorem{fsatz}[ftheo]{SATZ}\n\\newtheorem{flemma}[ftheo]{LEMMA}\n\\newtheorem{fprop}[ftheo]{PROPOSITION}\n\\newtheorem{fcor}[ftheo]{COROLLARY}\n\\newtheorem{frem}[ftheo]{REMARK}\n\n\\begin{document}\n\n\\author{Karl-Heinz Fichtner \\\\\n%EndAName\nFriedrich-Schiller-Universit\\\"{a}t Jena \\\\\nFakult\\\"{a}t f\\\"{u}r Mathematik und Informatik \\\\\nInstitut f\\\"{u}r Angewandte Mathematik \\\\\nD-07740 Jena Deutschland \\\\\nE-mail: fichtner@minet.uni-jena.de\\\\\nand\\\\\nMasanori Ohya\\\\\nDepartment of Information Sciences\\\\\nScience University of Tokyo\\\\\nChiba 278-8510 Japan\\\\\nE-mail: ohya@is.noda.sut.ac.jp}\n\\title{Quantum Teleportation with Entangled States\\\\\ngiven by Beam Splittings}\n\\date{August 1999}\n\\maketitle\n\n\\begin{abstract}\nQuantum teleportation is rigorously discussed with coherent entangled states\ngiven by beam splittings. The mathematical scheme of beam splitting has been\nused to study quantum communication \\cite{AO2} and quantum stochastic \\cite\n{FFL}. We discuss the teleportation process by means of coherent states in\nthis scheme for the following two cases: (1) Delete the vacuum part from\ncoherent states, whose compensation provides us a perfect teleportation from\nAlice to Bob. (2) Use fully realistic (physical) coherent states, which\ngives a non-perfect teleportation but shows that it is exact when the\naverage energy (density) of the coherent vectors goes to infinity.\n\\end{abstract}\n\n\\pagestyle{myheadings}\n\\markboth{\\hfill FICHTNER AND OHYA \\hfill}\n{\\hfill TELEPORTATION AND ENTANGLED STATES \\hfill} It is the paper\n\\cite{Ben} that the quantum teleportation was first studied as a part of\nquantum cryptolgraphy \\cite{Eke}. This teleportation scheme can be\nmathematically expressed in the following steps \\cite{IOS}:\n\n\\begin{description}\n\\item[Step 0:] A girl named Alice has an unknown quantum state $\\rho $ on\n(a $N$--dimensional) Hilbert space $\\mathcal{H}_{1}$ and she was asked to\nteleport it to a boy named Bob.\n\n\\item[Step 1:] For this purpose, we need two other Hilbert spaces $\\mathcal{%\nH}_{2}$ and $\\mathcal{H}_{3}$, $\\mathcal{H}_{2}$ is attached to Alice and $%\n\\mathcal{H}_{3}$ is attached to Bob. Prearrange a so-called entangled state $%\n\\sigma $ on $\\mathcal{H}_{2}\\otimes \\mathcal{H}_{3}$ having certain\ncorrelations and prepare an ensemble of the combined system in the state $%\n\\rho \\otimes \\sigma $ on $\\mathcal{H}_{1}\\otimes \\mathcal{H}_{2}\\otimes\n\\mathcal{H}_{3}$.\n\n\\item[Step 2:] One then fixes a family of mutually orthogonal projections $%\n(F_{nm})_{n,m=1}^{N}$ on the Hilbert space $\\mathcal{H}_{1}\\otimes \\mathcal{H%\n}_{2}$ corresponding to an observable $F:=\\sum\\limits_{n,m}z_{n,m}F_{nm}$,\nand for a fixed one pair of indices $n,m$, Alice performs a first kind\nincomplete measurement, involving only the $\\mathcal{H}_{1}\\otimes \\mathcal{H%\n}_{2}$ part of the system in the state $\\rho \\otimes \\sigma $, which filters\nthe value $z_{nm}$, that is, after measurement on the given ensemble $\\rho\n\\otimes \\sigma $ of identically prepared systems, only those where $F$ shows\nthe value $z_{nm}$ are allowed to pass. According to the von Neumann rule,\nafter Alice's measurement, the state becomes\n\\[\n\\rho _{nm}^{(123)}:=\\frac{(F_{nm}\\otimes \\mathbf{1})\\rho \\otimes \\sigma\n(F_{nm}\\otimes \\mathbf{1})}{\\mathrm{tr}_{123}(F_{nm}\\otimes \\mathbf{1})\\rho\n\\otimes \\sigma (F_{nm}\\otimes \\mathbf{1})}\n\\]\nwhere $\\mathrm{tr}_{123}$ is the full trace on the Hilbert space $\\mathcal{H}%\n_{1}\\otimes \\mathcal{H}_{2}\\otimes \\mathcal{H}_{3}$.\n\n\\item[Step 3:] Bob is informed which measurement was done by Alice. This is\nequivalent to transmit the information that the eigenvalue $z_{nm}$ was\ndetected. This information is transmitted from Alice to Bob without\ndisturbance and by means of classical tools.\n\n\\item[Step 4:] Making only partial measurements on the third part on the\nsystem in the state $\\rho _{nm}^{(123)}$ means that Bob will control a state\n$\\Lambda _{nm}(\\rho )$ on $\\mathcal{H}_{3}$ given by the partial trace on $%\n\\mathcal{H}_{1}\\otimes \\mathcal{H}_{2}$ of the state $\\rho _{nm}^{(123)}$\n(after Alice's measurement)\n\\begin{eqnarray}\n\\Lambda _{nm}(\\rho ) &=&\\mathrm{tr}_{12}\\;\\rho _{nm}^{(123)} \\\\\n&=&\\mathrm{tr}_{12}\\frac{(F_{nm}\\otimes \\mathbf{1})\\rho \\otimes \\sigma\n(F_{nm}\\otimes \\mathbf{1})}{\\mathrm{tr}_{123}(F_{nm}\\otimes \\mathbf{1})\\rho\n\\otimes \\sigma (F_{nm}\\otimes \\mathbf{1)}}\n\\end{eqnarray}\nThus the whole teleportation scheme given by the family $(F_{nm})$ and the e\nntangled state $\\sigma $ can be characterized by the family $(\\Lambda _{nm})$\nof channels from the set of states on $\\mathcal{H}_{1}$ into the set of\nstates on $\\mathcal{H}_{3}$ and the family $(p_{nm})$ given by\n\\[\np_{nm}(\\rho ):=\\mathrm{tr}_{123}(F_{nm}\\otimes \\mathbf{1})\\rho \\otimes\n\\sigma (F_{nm}\\otimes \\mathbf{1})\n\\]\nof the probabilities that Alice's measurement according to the observable $F$\nwill show the value $z_{nm}$.\n\\end{description}\n\nThe teleportation scheme works perfectly with respect to a certain class $%\n\\frak{S}$ of states $\\rho $ on $\\mathcal{H}_{1}$ if the following conditions\nare fulfilled.\n\n\\begin{description}\n\\item[(E1)] For each $n,m$ there exists a unitary operator $v_{nm}:\\mathcal{%\nH}_{1}\\to \\mathcal{H}_{3}$ such that\n\\[\n\\Lambda _{nm}(\\rho )=v_{nm}\\;\\rho \\;v_{nm}^{}\\quad (\\rho \\in \\frak{S})\n\\]\n\n\\item[(E2)]\n\\[\n\\sum\\limits_{nm}p_{nm}(\\rho )=1\\quad (\\rho \\in \\frak{S})\n\\]\n\n\\item (E1) means that Bob can reconstruct the original state $\\rho $ by\nunitary keys $\\{v_{nm}\\}$ provided to him. \\newline\n\n\\item (E2) means that Bob will succeed to find a proper key with certainty.\n\\newline\n\\end{description}\n\nIn the papers \\cite{Ben,Ben2}, the authors used EPR spin pair to construct a\nteleportation model. In order to have a more handy model, we here use\ncoherent states to construct a model. One of the main points for such a\nconstruction is how to prepare the entangled state. The EPR entangled state\nused in \\cite{Ben} can be identified with the splitting of a one particle\nstate, so that the teleportation model of Bennett et al. can be described in\nterms of Fock spaces and splittings, which makes us possible to work the\nwhole teleportation process in general beam splitting scheme. Moreover to\nwork with beams having a fixed number of particles seems to be not\nrealistic, especially in the case of large distance between Alice and Bob,\nbecause we have to take into account that the beams will lose particles (or\nenergy). For that reason one should use a class of beams being insensitive\nto this loss of particles. That and other arguments lead to superpositions\nof coherent beams. \\newline\n\nIn section 2 of this paper, we construct a teleportation model being perfect\nin the sense of conditions (E1) and (E2), where we take the Boson Fock space\n$\\Gamma (L^{2}(G)):=\\mathcal{H}_{1}=\\mathcal{H}_{2}=\\mathcal{H}_{3}$ with a\ncertain class $\\rho $ of states on this Fock space.\n\nIn section 3 we consider a teleportation model where the entangled state $%\n\\sigma$ is given by the splitting of a superposition of certain coherent\nstates. Unfortunately this model doesn't work perfectly, that is, neither\n(E2) nor (E1) hold. However this model is more realistic than that in the\nsection 2, and we show that this model provides a nice approximation to be\nperfect. To estimate the difference between the perfect teleportation and\nnon-perfect teleportation, we add a further step in the teleportation scheme:\n\n\\begin{description}\n\\item[Step 5:] Bob will perform a measurement on his part of the system\naccording to the projection\n\\[\nF_{+}:=\\mathbf{1}-\|\\mathrm{exp}(0)><\\mathrm{exp}(0)\|\n\\]\nwhere $\|\\mathrm{exp}(0)><\\mathrm{exp}(0)\|$ denotes the vacuum state (the\ncoherent state with density $0$).\n\\end{description}\n\nThen our new teleportation channels (we denote it again by $\\Lambda _{nm}$)\nhave the form\n\\[\n\\Lambda _{nm}(\\rho ):=\\mathrm{tr}_{12}\\frac{(F_{nm}\\otimes F_{+})\\rho\n\\otimes \\sigma (F_{nm}\\otimes F_{+})}{\\mathrm{tr}_{123}(F_{nm}\\otimes\nF_{+})\\rho \\otimes \\sigma (F_{nm}\\otimes F_{+})}\n\\]\nand the corresponding probabilities are\n\\[\np_{nm}(\\rho ):=\\mathrm{tr}_{123}(F_{nm}\\otimes F_{+})\\,\\rho \\otimes \\sigma\n(F_{nm}\\otimes F_{+})\n\\]\nFor this teleportation scheme, (E1) is fulfilled. Furthermore we get\n\\[\n\\sum\\limits_{nm}p_{nm}(\\rho )=\\frac{(1-e^{-\\frac{d}{2}})^{2}}{1+(N-1)e^{-d}}\n\\quad \\left( \\rightarrow 1\\text{ }(d\\to +\\infty )\\right)\n\\]\nHere $N$ denotes the dimension of the Hilbert space and $d$ is the\nexpectation value of the total number of particles (or energy) of the beam,\nso that in the case of high density (or energy) $``d\\to +\\infty \"$ of the\nbeam the model works perfectly.\n\nSpecializing this model we consider in section 4 the teleportation of all\nstates on a finite dimensional Hilbert space (through the space $\\mathbf{R}%\n^{k}$). Further specialization leads to a teleportation model where Alice\nand Bob are spatially separated, that is, we have to teleport the\ninformation given by the state of our finite dimensional Hilbert space from\none region $X_{1}\\subseteq \\mathbf{R}^{k}$ into another region $%\nX_{2}\\subseteq \\mathbf{R}^{k}$ with $X_{1}\\cap X_{2}=\\emptyset $, and Alice\ncan only perform local measurements (inside of region $X_{1}$) as well as\nBob (inside of $X_{2}$). \\newline\n\n\\section{Basic Notions and Notations}\n\nFirst we collect some basic facts concerning the (symmetric) Fock space. We\nwill introduce the Fock space in a way adapted to the language of counting\nmeasures. For details we refer to \\cite{FF1,FF2,FFL,AO2,L} and other papers\ncited in \\cite{FFL}. \\newline\n\nLet $G$ be an arbitrary complete separable metric space. Further, let $\\mu $\nbe a locally finite diffuse measure on $G$, i.e. $\\mu (B)<+\\infty $ for\nbounded measurable subsets of $G$ and $\\mu (\\{x\\})=0$ for all singletons $%\nx\\in G$. In order to describe the teleportation of states on a finite\ndimensional Hilbert space through the $k$--dimensional space $\\mathbf{R}^{k}$%\n, especially we are concerned with the case\n\\begin{eqnarray}\nG &=&\\mathbf{R}^{k}\\times \\{1,\\ldots ,N\\} \\\\\n\\mu &=&l\\times \\#\n\\end{eqnarray}\nwhere $l$ is the $k$--dimensional Lebesgue measure and $\\#$ denotes the\ncounting measure on $\\{1,\\ldots ,N\\}$. \\newline\n\nNow by $M=M(G)$ we denote the set of all finite counting measures on $G$.\nSince $\\varphi \\in M$ can be written in the form $\\varphi=\\sum%\n\\limits_{j=1}^{n}\\delta _{x_{j}}$ for some $n=0,1,2,\\ldots $ and $x_{j}\\in G$\n(where $\\delta _{x}$ denotes the Dirac measures corresponding to $x\\in G$)\nthe elements of $M$ can be interpreted as finite (symmetric) point\nconfigurations in $G$. We equip $M$ with its canonical $\\sigma $--algebra $%\n\\frak{W}$ (cf. \\cite{FF1}, \\cite{FF2}) and we consider the measure $F$ by\nsetting\n\\[\nF(Y):=\\mathcal{X}_{Y}(O)+\\sum\\limits_{n\\ge 1}\\frac{1}{n!}\\int\\limits_{G^{n}}\n\\mathcal{X}_{Y}\\left( \\sum\\limits_{j=1}^{n}\\delta _{x_{j}}\\right)\n\\mu^{n}(d[x_{1},\\ldots ,x_{n}])(Y\\in \\frak{W})\n\\]\nHereby, $\\mathcal{X}_{Y}$ denotes the indicator function of a set $Y$ and $O$\nrepresents the empty configuration, i.~e., $O(G)=0$. Observe that $F$ is a $%\n\\sigma $--finite measure. \\newline\n\nSince $\\mu $ was assumed to be diffuse one easily checks that $F$ is\nconcentrated on the set of a simple configurations (i.e., without multiple\npoints)\n\\[\n\\hat{M}:=\\{\\varphi \\in M\|\\varphi (\\{x\\})\\le 1\\text{ for all }x\\in G\\}\n\\]\n\n\\begin{fdef}\n\\label{def1} $\\mathcal{M}=\\mathcal{M}(G):=L^{2}(M,\\frak{W},F)$ is called the\n(symmetric) Fock space over $G$.\n\\end{fdef}\n\nIn \\cite{FF1} it was proved that $\\mathcal{M}$ and the Boson Fock space $%\n\\Gamma (L^{2}(G))$ in the usual definition are isomorphic. \\newline\nFor each $\\Phi \\in \\mathcal{M}$ with $\\Phi \\neq 0$ we denote by $\|\\Phi >$\nthe corresponding normalized vector\n\\[\n\|\\Phi >:=\\frac{\\Phi }{\|\|\\Phi \|\|}\n\\]\nFurther, $\|\\Phi ><\\Phi \|$ denotes the corresponding one--dimensional\nprojection, describing the pure state given by the normalized vector $\|\\Phi>$%\n. Now, for each $n\\ge 1$ let $\\mathcal{M}^{\\otimes n}$ be the $n$--fold\ntensor product of the Hilbert space $\\mathcal{M}$. Obviously, $\\mathcal{M}%\n^{\\otimes n}$ can be identified with $L^{2}(M^{n},F^{n})$.\n\n\\begin{fdef}\n\\label{def2} For a given function $g:G\\to \\Bbb{C}$ the function\n\\end{fdef}\n\n\\noindent $\\mathrm{exp}\\;(g):M\\to \\Bbb{C}$ defined by\n\\[\n\\mathrm{exp}\\;(g)\\,(\\varphi ):=\\left\\{\n\\begin{array}{lll}\n1 & \\text{ if } & \\varphi =0 \\\\\n\\prod_{x\\in G,\\varphi \\left( \\left\\{ x\\right\\} \\right) >0}g(x) & & otherwise\n\\end{array}\n\\right.\n\\]\nis called exponential vector generated by $g$.\n\nObserve that $\\mathrm{exp}\\;(g)\\in \\mathcal{M}$ if and only if $g\\in L^{2}(G)\n$ and one has in this case \\newline\n$\|\|\\mathrm{exp}\\;(g)\|\|^{2}=e^{\\Vert g\\Vert ^{2}}$ and $\|\\mathrm{exp}%\n\\;(g)>=e^{-\\frac{1}{2}\\Vert g\\Vert ^{2}}\\mathrm{exp}\\;(g)$. The projection\n\\noindent \\noindent $\|\\mathrm{exp}\\;(g)><\\mathrm{exp}\\;(g)\|$ is called the\ncoherent state corresponding to $g\\in L^{2}(G)$. In the special case $%\ng\\equiv 0$ we get the vacuum state\n\\[\n\|\\mathrm{exp}(0)>=\\mathcal{X}_{\\{0\\}}\\;.\n\\]\nThe linear span of the exponential vectors of $\\mathcal{M}$ is dense in $%\n\\mathcal{M}$, so that bounded operators and certain unbounded operators can\nbe characterized by their actions on exponential vectors.\n\n\\begin{fdef}\n\\label{def3} The operator $D:\\mathrm{dom}(D)\\to \\mathcal{M}^{\\otimes 2}$\ngiven on a dense domain $\\mathrm{dom}(D)\\subset \\mathcal{M}$ containing the\nexponential vectors from $\\mathcal{M}$ by\n\\[\nD\\psi (\\varphi _{1},\\varphi _{2}):=\\psi (\\varphi _{1}+\\varphi _{2})\\quad\n(\\psi \\in \\mathrm{dom}(D),\\,\\varphi _{1},\\varphi _{2}\\in M)\n\\]\nis called compound Malliavin derivative.\n\\end{fdef}\n\nOn exponential vectors $\\mathrm{exp}\\;(g)$ with $g\\in L^{2}(G),$ one gets\nimmediately\n\\begin{equation}\nD\\;\\mathrm{exp}\\;(g)=\\mathrm{exp}\\;(g)\\otimes \\;\\mathrm{exp}\\;(g) \\label{1}\n\\end{equation}\n\n\\begin{fdef}\n\\label{def4} The operator $S:\\mathrm{dom}(S)\\to \\mathcal{M}$ given on a\ndense domain $\\mathrm{dom}\\;(S)\\subset \\mathcal{M}^{\\otimes 2}$ containing\ntensor products of exponential vectors by\n\\[\nS\\Phi (\\varphi ):=\\sum\\limits_{\\tilde{\\varphi}\\le \\varphi }\\Phi (\\tilde{%\n\\varphi},\\varphi -\\tilde{\\varphi})\\quad (\\Phi \\in \\mathrm{dom}(S),\\;\\varphi\n\\in M)\n\\]\nis called compound Skorohod integral.\n\\end{fdef}\n\nOne gets\n\\begin{equation}\n\\langle D\\psi ,\\Phi \\rangle _{\\mathcal{M}^{\\otimes 2}}=\\langle \\psi ,S\\Phi\n\\rangle _{\\mathcal{M}}\\quad (\\psi \\in \\mathrm{dom}(D),\\; \\Phi \\in \\mathrm{dom%\n}(S)) \\label{2}\n\\end{equation}\n\\begin{equation}\nS(\\mathrm{exp}\\;(g)\\otimes \\mathrm{exp}\\;(h))=\\mathrm{exp}\\;(g+h)\\quad\n(g,h\\in L^{2}(G)) \\label{3}\n\\end{equation}\nFor more details we refer to \\cite{FW}.\n\n\\begin{fdef}\n\\label{def5} Let $T$ be a linear operator on $L^{2}(G)$ with $\\Vert T\\Vert\n\\le 1$. Then the operator $\\Gamma (T)$ called second quantization of $T$ is\nthe (uniquely determined) bounded operator on $\\mathcal{M}$ fulfilling\n\\[\n\\Gamma (T)\\mathrm{exp}\\;(g)=\\mathrm{exp}\\;(Tg)\\quad (g\\in L^{2}(G))\n\\]\n\\end{fdef}\n\nClearly, it holds\n\\begin{eqnarray}\n\\Gamma (T_{1})\\Gamma (T_{2}) &=&\\Gamma (T_{1}T_{2}) \\label{4} \\\\\n\\Gamma (T^{}) &=&\\Gamma (T^{}) \\nonumber\n\\end{eqnarray}\nIt follows that $\\Gamma (T)$ is an unitary operator on $\\mathcal{M}$ if $T$\nis an unitary operator on $L^{2}(G)$.\n\n\\begin{flemma}\n\\label{def6} Let $K_{1},K_{2}$ be linear operators on $L^{2}(G)$ with\nproperty\n\\begin{equation}\nK_{1}^{}K_{1}+K_{2}^{}K_{2}=\\mathbf{1}\\;. \\label{5}\n\\end{equation}\nThen there exists exactly one isometry $\\nu _{K_{1},K_{2}}$ from $\\mathcal{M}\n$ to $\\mathcal{M}^{\\otimes 2}=\\mathcal{M}\\otimes \\mathcal{M}$ with\n\\begin{equation}\n\\nu _{K_{1},K_{2}}\\mathrm{exp}\\;(g)=\\mathrm{exp}(K_{1}g)\\otimes \\mathrm{exp}%\n(K_{2}g)\\quad (g\\in L^{2}(G)) \\label{6}\n\\end{equation}\nFurther it holds\n\\begin{equation}\n\\nu _{K_{1},K_{2}}=(\\Gamma (K_{1})\\otimes \\Gamma (K_{2}))D \\label{7}\n\\end{equation}\n(at least on $\\mathrm{dom}(D)$ but one has the unique extension). \\newline\nThe adjoint $\\nu _{K_{1},K_{2}}^{}$ of $\\nu _{K_{1},K_{2}}$ is\ncharacterized by\n\\begin{equation}\n\\nu _{K_{1},K_{2}}^{}(\\mathrm{exp}\\;(h)\\otimes \\mathrm{exp}\\;(g))=\\mathrm{%\nexp}(K_{1}^{}h+K_{2}^{}g)\\quad (g,h\\in L^{2}(G)) \\label{8}\n\\end{equation}\nand it holds\n\\begin{equation}\n\\nu _{K_{1},K_{2}}^{}=S(\\Gamma (K_{1}^{})\\otimes \\Gamma (K_{2}^{}))\n\\label{9}\n\\end{equation}\n\\end{flemma}\n\n\\begin{frem}\nFrom $K_{1},K_{2}$ we get a transition expectation $\\xi\n_{K_{1}K_{2}}:\\mathcal{M}\\otimes \\mathcal{M}\\to \\mathcal{M}$, using $\\nu\n_{K_{1},K_{2}}$ and the lifting $\\xi _{K_{1}K_{2}}^{}$ may be interpreted\nas a certain splitting (cf. \\cite{AO2}).\n\\end{frem}\n\n\\noindent \\textbf{Proof of \\ref{def6}.} We consider the operator\n\\[\nB:=S(\\Gamma (K_{1}^{})\\otimes \\Gamma (K_{2}^{}))(\\Gamma (K_{1})\\otimes\n\\Gamma (K_{2}))D\n\\]\non the dense domain $\\mathrm{dom}(B)\\subseteq \\mathcal{M}$ spanned by the\nexponential vectors. Using %%@\n(\\ref{1}), (\\ref{3}), (\\ref{4}) and (\\ref{5}) we get\n\\[\nB\\;\\mathrm{exp}\\;(g)=\\mathrm{exp}\\;(g)\\quad (g\\in L^{2}(G))\\;.\n\\]\nIt follows that the bounded linear unique extension of $B$ onto $\\mathcal{M}$\ncoincides with the unity on $\\mathcal{M}$\n\\begin{equation}\nB=\\mathbf{1}\\;. \\label{10}\n\\end{equation}\nOn the other hand, by equation (\\ref{7}) at least on $\\mathrm{dom}\\;(D),$ an\noperator $\\nu _{K_{1},K_{2}}$ is defined. Using (\\ref{2}) and (\\ref{4}) we\nobtain\n\\begin{eqnarray}\n\\Vert \\nu _{K_{1},K_{2}}\\psi \\Vert ^{2} &=&\\langle \\nu _{K_{1},K_{2}}\\psi\n,\\nu _{K_{1},K_{2}}\\psi \\rangle \\quad (\\psi \\in \\mathrm{dom}\\;(D)) \\\\\n&=&\\langle \\psi ,B\\psi \\rangle ,\n\\end{eqnarray}\nwhich implies\n\\[\n\\Vert \\nu _{K_{1},K_{2}}\\psi \\Vert ^{2}=\\Vert \\psi \\Vert ^{2}\\quad (\\psi \\in\n\\mathrm{dom}\\;(D)).\\;\n\\]\nbecause of (\\ref{10}). It follows that $\\nu _{K_{1},K_{2}}$ can be uniquely\nextended to a bounded operator on $\\mathcal{M}$ with\n\\[\n\\Vert \\nu _{K_{1},K_{2}}\\psi \\Vert =\\Vert \\psi \\Vert \\quad (\\psi \\in\n\\mathcal{M}).\n\\]\nNow from (\\ref{7}) we obtain (\\ref{6}) using (\\ref{1}) and the definition of\nthe operators of second quantization. Further, (\\ref{7}), (\\ref{3}) and (\\ref\n{4}) imply (\\ref{9}) and from (\\ref{9}) we obtain (\\ref{8}) using the\ndefinition of the operators of second quantization and equation (\\ref{3}). $%\n\\blacksquare $\n\nHere we explain fundamental scheme of beam splitting \\cite{FFL}. We define\nan isometric operator $V_{\\alpha ,\\beta }$ for coherent vectors such that\n\\[\nV_{\\alpha ,\\beta }\|\\,\\mathrm{exp}\\;(g)\\rangle =\|\\,\\mathrm{exp}\\;(\\alpha\ng)\\rangle \\otimes \|\\,\\mathrm{exp}\\,(\\beta g)\\rangle\n\\]\nwith $\\mid \\alpha \\mid ^{2}+\\mid \\beta \\mid ^{2}=1$. This beam splitting is\na useful mathematical expression for optical communication and quantum\nmeasurements \\cite{AO2}.\n\n\\begin{fex}\n\\label{def7} $\\left( \\alpha =\\beta =1/\\sqrt{2}\\text{ above}\\right) $ Let $%\nK_{1}=K_{2}$ be the following operator of multiplication on $L^{2}(G)$\n\\[\nK_{1}g=\\frac{1}{\\sqrt{2}}\\;g=K_{2}g\\quad (g\\in L^{2}(G))\n\\]\nWe put\n\\[\n\\nu :=\\nu _{K_{1},K_{2}}\n\\]\nand obtain\n\\[\n\\nu \\;\\mathrm{exp}\\;(g)=\\mathrm{exp}\\;\\left( \\frac{1}{\\sqrt{2}}g\\right)\n\\otimes \\mathrm{exp}\\;(\\frac{1}{\\sqrt{2}}\\;g)\\quad (g\\in L^{2}(G))\n\\]\n\\end{fex}\n\n\\begin{fex}\n\\label{def8} Let $L^{2}(G)=\\mathcal{H}_{1}\\oplus \\mathcal{H}_{2}$ be the\northogonal sum of the subspaces $\\mathcal{H}_{1},\\mathcal{H}_{2}$. $K_{1}$\nand $K_{2}$ denote the corresponding projections.\n\\end{fex}\n\nWe will use Example \\ref{def7} in order to describe a teleportation model\nwhere Bob performs his experiments on the same ensemble of the systems like\nAlice. \\newline\n\nFurther we will use a special case of Example \\ref{def8} in order to\ndescribe a teleportation model where Bob and Alice are spatially separated\n(cf. section 5).\n\n\\begin{frem}\n\\label{def9} The property (\\ref{5}) implies\n\\begin{equation}\n\\Vert K_{1}g\\Vert ^{2}+\\Vert K_{2}g\\Vert ^{2}=\\Vert g\\Vert ^{2}\\quad (g\\in\nL^{2}(G)) \\label{11}\n\\end{equation}\n\\end{frem}\n\n\\begin{frem}\n\\label{def10} Let $U$, $V$ be unitary operators on $L^{2}(G)$. If operators $%\nK_{1},K_{2}$ satisfy (\\ref{5}),~then the pair $\\hat{K}_{1}=UK_{1},\\ \\hat{K}%\n_{2}=VK_{2}$ fulfill (\\ref{5}).\n\\end{frem}\n\n\\section{A perfect model of teleportation}\n\n\\label{sec2} Concerning the general idea we follow the papers \\cite{IOS},\n\\cite{AO}. We fix an ONS $\\{g_{1},\\ldots ,g_{N}\\}\\subseteq L^{2}(G)$,\noperators $K_{1},K_{2}$ on $L^{2}(G)$ with (\\ref{5}), an unitary operator $T$\non $L^{2}(G)$, and $d>0$. We assume\n\\begin{equation}\nTK_{1}g_{k}=K_{2}g_{k}\\quad (k=1,\\ldots ,N), \\label{12}\n\\end{equation}\n\\begin{equation}\n\\langle K_{1}g_{k},K_{1}g_{j}\\rangle =0\\quad (k\\not{=}j;\\;k,j=1\\ldots ,N),\n\\label{13}\n\\end{equation}\nUsing (\\ref{11}) and (\\ref{12}) we get\n\\begin{equation}\n\\Vert K_{1}g_{k}\\Vert ^{2}=\\Vert K_{2}g_{k}\\Vert ^{2}=\\frac{1}{2}.\n\\label{14}\n\\end{equation}\nFrom (\\ref{12}) and (\\ref{13}) we get\n\\begin{equation}\n\\langle K_{2}g_{k},\\,K_{2}g_{j}\\rangle =0\\quad (k\\neq j\\,;\\;k,j=1,\\ldots ,N).\n\\label{15}\n\\end{equation}\nThe state of Alice asked to teleport is of the type\n\\begin{equation}\n\\rho =\\sum\\limits_{s=1}^{N}\\lambda _{s}\|\\Phi _{s}\\rangle \\langle \\Phi _{s}\|,\n\\label{16}\n\\end{equation}\nwhere\n\\begin{equation}\n\|\\Phi _{s}\\rangle =\\sum\\limits_{j=1}^{N}c_{sj}\|\\mathrm{exp}\\;(aK_{1}g_{j})-%\n\\mathrm{exp}\\;(0)\\rangle \\quad\n\\left(\\sum\\limits_{j}\|c_{sj}\|^{2}=1;s=1,\\ldots ,N\\right) \\label{17}\n\\end{equation}\nand $a=\\sqrt{d}$. One easily checks that $(\|\\mathrm{exp}\\;(aK_{1}g_{j})-%\n\\mathrm{exp}\\;(0)\\rangle )_{j=1}^{N}$ and $(\|\\mathrm{exp}\\;aK_{2}g_{j})-%\n\\mathrm{exp}\\;(0)\\rangle )_{j=1}^{N}$ are ONS in $\\mathcal{M}$. \\newline\n\nIn order to achieve that $(\|\\Phi _{s}\\rangle )_{s=1}^{N}$ is still an ONS in\n$\\mathcal{M}$ we assume\n\\begin{equation}\n\\sum\\limits_{j=1}^{N}\\bar{c}_{sj}c_{kj}=0\\quad (j\\neq\nk\\,;\\;j,k=1,\\ldots,N)\\,. \\label{18}\n\\end{equation}\nDenote $c_{s}=[c_{s1,\\ldots ,}c_{sN}]\\in \\Bbb{C}^{N}$, then $%\n(c_{s})_{s=1}^{N}$ is an CONS in $\\Bbb{C}^{N}$.\n\nNow let $(b_{n})_{n=1}^{N}$ be a sequence in $\\Bbb{C}^{N}$,\n\\[\nb_{n}=[b_{n1,\\ldots ,}b_{nN}]\n\\]\nwith properties\n\\begin{equation}\n\|b_{nk}\|=1\\quad (n,k=1,\\ldots ,N), \\label{19}\n\\end{equation}\n\\begin{equation}\n\\langle b_{n}\\,,\\;b_{j}\\rangle =0\\quad (n\\neq j\\,;\\;n,j=1,\\ldots ,N).\n\\label{20}\n\\end{equation}\nThen Alice's measurements are performed with projection\n\\begin{equation}\nF_{nm}=\|\\xi _{nm}\\rangle \\langle \\xi _{nm}\|\\quad (n,m=1,\\ldots ,N)\n\\label{21}\n\\end{equation}\ngiven by\n\\begin{equation}\n\|\\xi _{nm}\\rangle =\\frac{1}{\\sqrt{N}}\\sum\\limits_{j=1}^{N}b_{nj}\|\\mathrm{exp}%\n\\;(aK_{1}g_{j})-\\mathrm{exp}\\;(0)>\\otimes \|\\;\\mathrm{exp}\\;(aK_{1}g_{j\\oplus\nm})-\\mathrm{exp}\\;(0)\\rangle , \\label{22}\n\\end{equation}\nwhere $j\\oplus m:=j+m(\\mathrm{mod}\\;N)$. \\newline\n\nOne easily checks that $(\|\\xi _{nm}\\rangle )_{n,m=1}^{N}$ is an ONS in $%\n\\mathcal{M}^{\\otimes 2}$. Further, the state vector $\|\\xi \\rangle $ of the\nentangled state $\\sigma =\|\\xi \\rangle \\langle \\xi \|$ is given by\n\\begin{equation}\n\|\\xi \\rangle =\\frac{1}{\\sqrt{N}}\\sum\\limits_{k}\|\\mathrm{exp}\\;(aK_{1}g_{k})-\n\\mathrm{exp}\\;(0)\\rangle \\otimes \|\\mathrm{exp}\\;(aK_{2}g_{k})-\\mathrm{exp}\\;\n(0)\\rangle \\,. \\label{23}\n\\end{equation}\n\n\\begin{flemma}\n\\label{def11} For each $n,m=1,\\ldots ,N$ it holds\n\\begin{eqnarray}\n&&(F_{nm}\\otimes \\mathbf{1})(\|\\Phi _{s}\\rangle \\otimes \|\\xi \\rangle )\n\\nonumber \\\\\n&=&\\frac{1}{N}\|\\xi _{nm}\\rangle \\otimes \\sum\\limits_{j}\\bar{b}_{nj}c_{sj}\|%\n\\mathrm{exp}\\;(aK_{2}g_{j\\oplus m})-\\mathrm{exp}\\;(0)\\rangle \\text{ }%\n(s=1,\\ldots ,N) \\label{24}\n\\end{eqnarray}\n\\end{flemma}\n\n\\noindent \\textbf{Proof: }From the fact that\n\\begin{equation}\n\|\\gamma _{j}\\rangle :=\|\\mathrm{exp}\\;(aK_{1}g_{j})-\\mathrm{exp}\\;(0)\\rangle\n\\quad \\left( j=1,\\ldots ,N\\right) \\label{25}\n\\end{equation}\nis an ONS, it follows\n\\begin{equation}\n\\langle \\gamma _{\\emph{r}}\\otimes \\gamma _{\\emph{r}\\oplus m}\\,,\\;\n\\gamma_{j}\\otimes \\gamma _{k}\\rangle = \\left\\{\n\\begin{array}{ll}\n1 & \\text{if }\\emph{r}=j\\text{ and }k=\\emph{r}\\oplus m \\\\\n0 & \\text{otherwise}\n\\end{array}\n\\right. . \\label{26}\n\\end{equation}\nOn the other hand, we have\n\\begin{eqnarray}\n&&(F_{nm}\\otimes \\mathbf{1})(\|\\Phi _{s}\\rangle \\otimes \|\\xi \\rangle )\n\\nonumber \\\\\n&=&\\frac{1}{N}\\sum\\limits_{k}\\sum\\limits_{j}\\sum\\limits_{\\emph{r}}c_{sj}\\bar\n{b}_{ns}\\langle \\gamma _{\\emph{r}}\\otimes \\gamma _{\\emph{r}\\oplus m}\\,,\\;\n\\gamma _{j}\\otimes \\gamma _{k}\\rangle \\xi _{nm}\\otimes \|\\mathrm{exp}\\;\n(aK_{2}g_{k})-\\mathrm{exp}\\;(0)\\rangle . \\nonumber \\\\\n&& \\label{27}\n\\end{eqnarray}\n\\noindent Using (\\ref{26}) and (\\ref{27}), we get (\\ref{24}) \\hfill $%\n\\blacksquare$ \\newline\n\nNow we have\n\n\\begin{eqnarray}\n\\rho \\otimes \\sigma &=&\\sum\\limits_{s=1}^{N}\\lambda _{s}\|\\Phi _{s}\\rangle\n\\langle \\Phi _{s}\|\\otimes \|\\xi \\rangle \\langle \\xi \| \\label{28} \\\\\n&=&\\sum\\limits_{s=1}^{N}\\lambda _{s}\|\\Phi _{s}\\otimes \\xi \\rangle \\langle\n\\Phi _{s}\\otimes \\xi \|\\;, \\nonumber\n\\end{eqnarray}\nwhich implies\n\\begin{eqnarray}\n(F_{nm}\\otimes \\mathbf{1})(\\rho \\otimes \\sigma )(F_{nm}\\otimes \\mathbf{1})\n&=&\\sum\\limits_{s=1}^{N}\\lambda _{s}(F_{nm}\\otimes \\mathbf{1}%\n)\|\\Phi_{s}\\otimes \\xi \\rangle \\langle \\Phi _{s}\\otimes \\xi \|(F_{nm}\\otimes\n\\mathbf{\\ 1}) \\nonumber \\\\\n&=&\\sum\\limits_{s=1}^{N}\\lambda _{s}\\Vert (F_{nm}\\otimes \\mathbf{1})\n(\\Phi_{s}\\otimes \\xi )\\Vert ^{2} \\label{29} \\\\\n&&\|(F_{nm}\\otimes \\mathbf{1})(\\Phi _{s}\\otimes \\xi )\\rangle \\langle\n(F_{nm}\\otimes \\mathbf{1})(\\Phi _{s}\\otimes \\xi )\|. \\nonumber\n\\end{eqnarray}\nNote $\|\\Phi _{s}\\otimes \\xi \\rangle =\|\\Phi _{s}\\rangle \\otimes \|\\xi \\rangle $%\n. From (\\ref{12}) it follows that\n\\begin{equation}\n\\sum\\limits_{j}\\bar{b}_{nj}c_{sj}\|\\mathrm{exp}\\;(aK_{2}g_{j\\oplus m})-\n\\mathrm{exp}\\;(0)\\rangle =\\Gamma (T)\\sum\\limits_{j}\\bar{b}_{nj}c_{sj}\|%\n\\mathrm{exp} \\;(aK_{1}g_{j\\oplus m})-\\mathrm{exp}\\;(0)\\rangle . \\label{30}\n\\end{equation}\nFurther, for each $m,n\\left( =1,\\ldots ,N\\right) ,$ we have unitary\noperators $U_{m},B_{n}$ on $\\mathcal{M}$ given by\n\\begin{equation}\nB_{n}\|\\mathrm{exp}\\;(aK_{1}g_{j})-\\mathrm{exp}\\;(0)\\rangle =b_{nj}\|\\mathrm{e\nxp}\\;(aK_{1}g_{j})-\\mathrm{exp}\\;(0)\\rangle \\quad (j=1,\\ldots ,N) \\label{31}\n\\end{equation}\n\\begin{equation}\nU_{m}\|\\mathrm{exp}\\;(aK_{1}g_{j})-\\mathrm{exp}\\;(0)\\rangle =\|\\mathrm{exp}\\;\n(aK_{1}g_{j\\oplus m})-\\mathrm{exp}\\;(0)\\rangle \\quad (j=1,\\ldots ,N)\n\\label{32}\n\\end{equation}\nTherefore we get\n\\begin{equation}\n\\sum\\limits_{j}\\bar{b}_{nj}c_{sj}\|\\mathrm{exp}\\;(aK_{1}g_{j\\oplus m})-%\n\\mathrm{exp}\\;(0)\\rangle =U_{m}B_{n}^{}(\\Phi _{s}) \\label{33}\n\\end{equation}\nFrom (\\ref{30}), (\\ref{33}) and Lemma \\ref{def11} we obtain\n\\begin{equation}\n\\left( F_{nm}\\otimes \\mathbf{1}\\right) \\left( \|\\Phi _{s}\\rangle \\otimes \|\\xi\n\\rangle \\right) =\\frac{1}{N}\|\\xi _{nm}\\rangle \\otimes \\left( \\Gamma\n(T)U_{m}B_{n}^{}\|\\Phi _{s}\\rangle \\right) \\label{34}\n\\end{equation}\nIt follows\n\\begin{equation}\n\\Vert \\left( F_{nm}\\otimes \\mathbf{1}\\right) \\left( \|\\Phi _{s}\\rangle\n\\otimes \|\\xi \\rangle \\right) \\Vert ^{2}=\\frac{1}{N^{2}} \\label{35}\n\\end{equation}\nFinally from (\\ref{29}), (\\ref{34}) and (\\ref{35}) we have\n\\begin{equation}\n\\left( F_{nm}\\otimes \\mathbf{1}\\right) (\\rho \\otimes \\sigma )\n\\left(F_{nm}\\otimes \\mathbf{1}\\right) =\\frac{1}{N^{2}}F_{nm}\\otimes \\left(\n\\Gamma(T)U_{m}B_{n}^{}\\right) \\rho \\left( B_{n}U_{m}^{}\\Gamma\n(T^{})\\right) \\label{36}\n\\end{equation}\nThat leads to the following solution of the teleportation problem.\n\n\\begin{ftheo}\n\\label{def12} For each $n,m=1,\\ldots ,N$, define a channel $\\Lambda _{nm}$\nby\n\\begin{equation}\n\\Lambda _{nm}(\\rho ):=\\mathrm{tr}_{12}\\frac{\\left( F_{nm}\\otimes 1\\right)\n(\\rho \\otimes \\sigma )\\left( F_{nm}\\otimes \\mathbf{1}\\right) }{\\mathrm{tr}%\n_{123}\\left( F_{nm}\\otimes 1\\right) \\left( \\hat{\\rho}\\otimes \\sigma \\right)\n\\left( F_{nm}\\otimes \\mathbf{1}\\right) }\\quad (\\rho \\text{ normal state on }%\n\\mathcal{M}) \\label{37}\n\\end{equation}\nThen we have for all states $\\rho $ on $M$ with (\\ref{16}) and (\\ref{17})\n\\begin{equation}\n\\Lambda _{nm}(\\rho )=\\left( \\Gamma (T)U_{m}B_{n}^{}\\right) \\rho \\left(\n\\Gamma (T)U_{m}B_{n}^{}\\right) ^{} \\label{38}\n\\end{equation}\n\\end{ftheo}\n\n\\begin{frem}\n\\label{def13} In case of Example \\ref{def7} using the operators $%\nB_{n},U_{m},\\Gamma (T),$ the projections $F_{nm}$ are given by unitary\ntransformations of the entangled state $\\sigma $:\n\\begin{eqnarray}\nF_{nm} &=&\\left( B_{n}\\otimes U_{m}\\Gamma (T^{})\\right) \\sigma \\left(\nB_{n}\\otimes U_{m}\\Gamma (T^{})\\right) ^{} \\label{39} \\\\\n&&\\text{or}\\hspace{2cm} \\nonumber \\\\\n\|\\xi _{nm}\\rangle &=&\\left( B_{n}\\otimes U_{m}\\Gamma (T^{})\\right) \|\\xi\n\\rangle \\nonumber\n\\end{eqnarray}\n\\end{frem}\n\n\\begin{frem}\n\\label{def14} If Alice performs a measurement according to the following\nselfadjoint operator\n\\[\nF=\\sum\\limits_{n,m=1}^{N}z_{nm}F_{nm}\n\\]\nwith $\\{z_{nm}\|n,m=1,\\ldots ,N\\}\\subseteq \\mathbf{R}-\\{0\\},$ then she will\nobtain the value $z_{nm}$ with probability $1/N^{2}$. The sum over all this\nprobabilities is $1$, so that the teleportation model works perfectly.\n\\end{frem}\n\n\\section{A non--perfect case of Teleportation}\n\nIn this section we will construct a model where we have also channels with\nproperty (\\ref{38}). But the probability that one of these channels will\nwork in order to teleport the state from Alice to Bob is less than $1$\ndepending on the density parameter $d$ (or energy of the beams, depending on\nthe interpretation). If $d=a^{2}$ tends to infinity that probability tends\nto $1$. That is the model is asymptotically perfect in a certain sense.\n\nWe consider the normalized vector\n\\begin{eqnarray}\n\|\\eta \\rangle := &&\\frac{\\gamma }{\\sqrt{N}}\\sum\\limits_{k=1}^{N}\|\\mathrm{exp\n}\\;(ag_{k})\\rangle \\label{40} \\\\\n\\gamma := &&\\left( \\frac{1}{1+(N-1)e^{-d}}\\right) ^{\\frac{1}{2}}=\\left(\n\\frac{1}{1+(N-1)e^{-a^{2}}}\\right) ^{\\frac{1}{2}} \\nonumber\n\\end{eqnarray}\nand we replace in (\\ref{37}) the projector $\\sigma $ by the projector\n\\begin{eqnarray}\n\\tilde{\\sigma}:= &&\|\\tilde{\\xi}\\rangle \\langle \\tilde{\\xi}\| \\label{41} \\\\\n\\tilde{\\xi}:= &&\\nu _{K_{1},K_{2}}(\\eta )=\\frac{\\gamma }{\\sqrt{N}}\n\\sum\\limits_{k=1}^{N}\|\\mathrm{exp}\\;(aK_{1}g_{k})\\rangle \\otimes \|\\mathrm{ex\np}\\;(aK_{2}g_{k})\\rangle \\nonumber\n\\end{eqnarray}\nThen for each $n,m=1,\\ldots ,N,$ we get the channels on a normal state $\\rho\n$ on $\\mathcal{M}$ such as\n\\begin{eqnarray}\n\\tilde{\\Lambda}_{nm}(\\rho ):= &&\\mathrm{tr}_{12}\\frac{\\left( F_{nm}\\otimes\n\\mathbf{1}\\right) \\left( \\rho \\otimes \\tilde{\\sigma}\\right) \\left(\nF_{nm}\\otimes \\mathbf{1}\\right) }{\\mathrm{tr}_{123}\\left( F_{nm}\\otimes\n\\mathbf{1}\\right) \\left( \\rho \\otimes \\tilde{\\sigma}\\right) \\left(\nF_{nm}\\otimes \\mathbf{1}\\right) }\\quad \\label{42} \\\\[0.12in]\n\\Theta _{nm}(\\rho ):= &&\\mathrm{tr}_{12}\\frac{\\left( F_{nm}\\otimes\nF_{+}\\right) \\left( \\rho \\otimes \\tilde{\\sigma}\\right) \\left( F_{nm}\\otimes\nF_{+}\\right) }{\\mathrm{tr}_{123}\\left( F_{nm}\\otimes F_{+}\\right) \\left(\n\\rho \\otimes \\tilde{\\sigma}\\right) \\left( F_{nm}\\otimes F_{+}\\right) }\\;,\n\\label{43}\n\\end{eqnarray}\nwhere $F_{+}=\\mathbf{1}-\|\\mathrm{exp}\\;(0)\\rangle \\langle \\mathrm{exp}\\;(0)\|$\ne.~g., $F_{+}$ is the projection onto the space $\\mathcal{M}_{+}$ of\nconfigurations having no vacuum part, e.~g., orthogonal to vacuum\n\\[\n\\mathcal{M}_{+}:=\\{\\psi \\in \\mathcal{M}\|\\;\\Vert \\mathrm{exp}\\;(0)\\rangle\n\\langle \\mathrm{exp}\\;(0)\|\\psi \\Vert =0\\}\n\\]\nOne easily checks that\n\\begin{equation}\n\\Theta _{nm}(\\rho )=\\frac{F_{+}\\tilde{\\Lambda}_{nm}(\\rho )F_{+}}{\\mathrm{tr}\n\\left( F_{+}\\tilde{\\Lambda}_{nm}(\\rho )F_{+}\\right) } \\label{44}\n\\end{equation}\nthat is, after receiving the state $\\tilde{\\Lambda}_{nm}(\\rho )$ from Alice,\nBob has to omit the vacuum. \\newline\n\nFrom Theorem \\ref{def12} it follows that for all $\\rho $ with (\\ref{16})\nand (\\ref{17})\n\\[\n\\Lambda _{nm}(\\rho )=\\frac{F_{+}\\Lambda _{nm}(\\rho )F_{+}}{\\mathrm{tr}\\;\n(F_{+}\\Lambda _{nm}(\\rho )F_{+})}\n\\]\nThis is not true if we replace $\\Lambda _{nm}$ by $\\tilde{\\Lambda}_{nm}$,\nnamely, in general it does not hold\n\\[\n\\Theta _{nm}(\\rho )=\\tilde{\\Lambda}_{nm}(\\rho )\n\\]\nBut we will prove that for each $\\rho $ with (\\ref{16}), and (\\ref{17}) it\nholds\n\n\\[\n\\Theta _{nm}(\\rho )=\\Lambda_{nm}(\\rho )\n\\]\n\n\\noindent which means\n\\begin{equation}\n\\Theta _{nm}(\\rho )=(\\Gamma (T)U_{m}B_{n}^{})\\rho (\\Gamma\n(T)U_{m}B_{n}^{})^{} \\label{45}\n\\end{equation}\nbecause of Theorem \\ref{def12}. Further we will show\n\\begin{equation}\n\\mathrm{tr_{123}}\\left( F_{nm}\\otimes F_{+}\\right) \\left( \\rho \\otimes\n\\tilde{\\sigma}\\right) \\left( F_{nm}\\otimes F_{+}\\right) =\\frac{\\gamma ^{2}}{%\nN^{2}}\\left( e^{\\frac{d}{2}}-1\\right) ^{2}e^{-d} \\label{46}\n\\end{equation}\nand the sum over $n,m\\left( =1,\\ldots ,N\\right) $ gives the probability\n\\[\n\\frac{\\left( 1-e^{-\\frac{d}{2}}\\right) ^{2}}{1+(N-1)e^{-d}}\\longrightarrow 1%\n\\text{ }\\left( d\\longrightarrow \\infty \\right)\n\\]\nwhich means that the teleportation model works perfectly in the limit $%\nd\\longrightarrow \\infty $, e.~g., Bob will receive one of the states $\\Theta\n_{nm}(\\rho )$ given by (\\ref{44}). Thus we formulate the following theorem.\n\n\\begin{ftheo}\n\\label{def15} For all states $\\rho $ on $\\mathcal{M}$ with (\\ref{16}) and (%\n\\ref{17}) and each pair $n,m\\left( =1,\\ldots ,N\\right) ,$ the equations (\\ref\n{44}) and (\\ref{45}) hold. Further, we have\n\\begin{equation}\n\\sum\\limits_{n,m}\\mathrm{tr}_{123}\\left( F_{nm}\\otimes F_{+}\\right) \\left(\n\\rho \\otimes \\tilde{\\sigma}\\right) \\left( F_{nm}\\otimes F_{+}\\right) =\\frac{%\n\\left( 1-e^{-\\frac{d}{2}}\\right) ^{2}}{1+(N-1)e^{-d}}. \\label{47}\n\\end{equation}\n\\end{ftheo}\n\nIn order to prove theorem \\ref{def15}, we fix $\\rho $ with (\\ref{16}) and (%\n\\ref{17}) and start with a lemma.\n\n\\begin{flemma}\n\\label{def16} For each $n,m,s\\left( =1,\\ldots ,N\\right) ,$ it holds\n\\begin{eqnarray}\n\\left( F_{nm}\\otimes \\mathbf{1}\\right) \\left( \|\\Phi _{s}\\rangle \\otimes \|%\n\\tilde{\\xi}\\rangle \\right) &=&\\frac{\\gamma }{N}\\left( 1-e^{-\\frac{d}{2}%\n}\\right) \|\\xi _{nm}\\rangle \\otimes \\left( \\Gamma (T)U_{m}B_{n}^{}\|\\Phi\n_{s}\\rangle \\right) \\\\\n&&+\\frac{\\gamma }{N}\\left( \\frac{e^{\\frac{d}{2}}-1}{e^{d}}\\right) ^{\\frac{1}{%\n2}}\\langle b_{n},c_{s}\\rangle _{\\Bbb{C}^{N}}\\xi _{nm}\\otimes \|\\mathrm{exp}%\n\\;(0)\\rangle\n\\end{eqnarray}\n\\end{flemma}\n\n\\noindent \\textbf{Proof:} For all $k,j,\\emph{r}=1,\\ldots ,N,$ we get\n\\begin{eqnarray}\n\\alpha _{k,j,\\emph{r}}:= && \\langle \|\\mathrm{exp}\\;(aK_{1}g_{\\emph{r}})-\n\\mathrm{exp}\\;(0)\\rangle \\otimes \|\|\\mathrm{exp}\\; (aK_{1}g_{\\emph{r}\\otimes\nm})-\\mathrm{exp}\\;(0)\\rangle \\,, \\\\\n&&~\|\\mathrm{exp}\\;(aK_{1}g_{j})-\\mathrm{exp}\\;(0)\\rangle \\otimes \|\\mathrm{exp%\n}\\;(aK_{1}g_{k})\\rangle \\rangle \\\\\n&=&\\left\\{\n\\begin{array}{ll}\n\\left( \\frac{e^{\\frac{a^{2}}{2}}-1}{e^{\\frac{a^{2}}{2}}}\\right) & \\text{if }%\n\\emph{r}=j\\text{ and }k=\\emph{r}\\oplus m \\\\\n0 & \\text{otherwise}\n\\end{array}\n\\right.\n\\end{eqnarray}\nand\n\\[\n\|\\mathrm{exp}\\left( aK_{2}g_{j\\oplus m}\\right) \\rangle =e^{-\\frac{a^{2}}{2}}\n\\left( e^{\\frac{a^{2}}{2}}-1\\right) ^{\\frac{1}{2}}\|\\mathrm{exp}\\;\\left(\naK_{2}g_{j\\oplus m}\\right) -\\mathrm{exp}\\;(0)\\rangle +e^{-\\frac{a^{2}}{2}}\|\n\\mathrm{exp}\\;(0)\\rangle\n\\]\nOn the other hand, we have\n\\[\n(F_{nm}\\otimes \\mathbf{1})\\left( \|\\Phi _{s}\\rangle \\otimes \|\\tilde{\\xi}\n\\rangle \\right) =\\frac{\\gamma }{N}\\sum\\limits_{k}\\sum\\limits_{j}\\sum\\limits_\n{\\emph{r}}c_{sj}\\bar{b}_{nr}\\alpha _{k,j,\\emph{r}}\\xi _{nm}\\otimes \|\\mathrm{%\n\\ exp}\\;(aK_{2}g_{k})\\rangle\n\\]\nIt follows with $a^{2}=d$\n\\begin{eqnarray}\n\\left( F_{nm}\\otimes \\mathbf{1}\\right) \\left( \\Phi _{s}\\otimes \\tilde{\\xi}\n\\right) &=& \\frac{\\gamma }{N}\\left( e^{\\frac{d}{2}}-1\\right) e^{-\\frac{d}{2}\n}\\xi _{nm}\\otimes \\left( \\sum\\limits_{j}c_{sj}\\bar{b}_{nj}\|\\mathrm{exp}\\;\n\\left( aK_{2}g_{j\\oplus m}\\right) -\\mathrm{exp}\\;(0)\\rangle \\right) \\\\\n&&+\\frac{\\gamma }{N}\\left( e^{\\frac{d}{2}}-1\\right) ^{\\frac{1}{2}} e^{-\\frac{%\nd}{2}}\\sum\\limits_{j}c_{sj}\\bar{b}_{nj}\\xi _{nm} \\otimes \|\\mathrm{exp}%\n\\;(0)\\rangle \\\\\n&=& \\frac{\\gamma }{N}\\left( 1-e^{-\\frac{d}{2}}\\right) \\xi _{nm}\\otimes\n\\left(\\Gamma (T)U_{m}B_{n}^{}\\Phi _{s}\\right) \\\\\n&& +\\frac{\\gamma }{N} \\left( \\frac{e^{\\frac{d}{2}}-1}{e^{d}}\\right) ^{\\frac{1%\n}{2}}\\langle b_{n},c_{s} \\rangle _{\\Bbb{C}^{N}}\\xi _{nm}\\otimes \|\\mathrm{exp}%\n\\; (0)\\rangle .\\text{ }\\blacksquare\n\\end{eqnarray}\n\\hfill\\newline\n\\label{def17} If $\\rho $ is a pure state\n\\[\n\\rho =\|\\Phi _{s}\\rangle \\langle \\Phi _{s}\|\n\\]\nthen we obtain from Lemma \\ref{def16}\n\\[\n\\begin{array}{ll}\n& \\mathrm{tr}_{123}\\left( F_{nm}\\otimes \\mathbf{1}\\right) \\left( \\rho\n\\otimes \\tilde{\\sigma}\\right) \\left( F_{nm}\\otimes \\mathbf{1}\\right) \\\\\n& =\\frac{\\gamma ^{2}}{N^{2}}\\left( \\left( 1-e^{-\\frac{d}{2}}\\right) ^{2}+\n\\frac{e^{\\frac{d}{2}}-1}{e^{d}}\|\\langle b_{n},c_{s}\\rangle \|^{2}\\right) \\\\\n& =\\frac{1}{N^{2}\\left( 1+(N-1)e^{-d}\\right) } \\left( \\left( 1-e^{-\\frac{d}{\n2%\n}}\\right) ^{2}+\\frac{e^{\\frac{d}{2}}-1}{e^{d}}\|\\langle\nb_{n},c_{s}\\rangle\|^{2}\\right)\n\\end{array}\n\\]\nand\n\\[\n\\tilde{\\Lambda}_{nm}(\\rho )\\not{=}\\left( \\Gamma (T)U_{m}B_{n}^{}\\right)\n\\rho \\left( \\Gamma (T)U_{m}B_{n}^{}\\right) ^{}.\n\\]\n\\medskip Now we have \\noindent \\vspace{0cm}\n\\[\n\\Gamma (T)U_{m}B_{n}^{}\\Phi _{s}\\in \\mathcal{M}_{+}\\,,\\;\|\\mathrm{exp}\\;(0)\n\\rangle \\in \\mathcal{M}_{+}^{\\bot }\n\\]\nHence, Lemma \\ref{def16} implies\n\\[\n\\left( \\mathbf{1}\\otimes \\mathbf{1}\\otimes F_{+}\\right) \\left( F_{nm}\\otimes\n\\mathbf{1}\\right) \\left( \\Phi _{s}\\otimes \\tilde{\\xi}\\right) =\\frac{\\gamma\n} {N}\\left( 1-e^{-\\frac{d}{2}}\\right) \\xi _{nm}\\otimes \\left( \\Gamma\n(T)U_{m}B_{n}^{}\\Phi _{s}\\right)\n\\]\nthat is, we have the following Lemma\n\n\\begin{flemma}\n\\label{def18} For each $n,m,s=1,\\ldots ,N,$ it holds\n\\begin{equation}\n\\left( F_{nm}\\otimes F_{+}\\right) \\left( \\Phi _{s}\\otimes \\tilde{\\xi}\\right)\n=\\frac{\\gamma }{N}\\left( 1-e^{-\\frac{d}{2}}\\right) \\xi _{nm}\\otimes \\left(\n\\Gamma (T)U_{m}B_{n}^{}\\Phi _{s}\\right) . \\label{48}\n\\end{equation}\n\\end{flemma}\n\n\\begin{frem}\n\\label{def19} Let $K_{2}$ be a projection of the type\n\\[\nK_{2}h=h\\mathcal{X}_{X};\\text{ }h\\in L^{2}(G),\n\\]\nwhere $X\\subseteq G$ is measurable. Then (\\ref{48}) also holds if we replace\n$F_{+}$ by the projection $F_{+,X}$ onto the subspace $\\mathcal{M}_{+,X}$ of\n$\\mathcal{M}$ given by\n\\[\n\\mathcal{M}_{+,X}:=\\{\\psi \\in \\mathcal{M}\|\\psi (\\varphi )=0\\text{ if }%\n\\varphi (X)=0\\}\n\\]\nObserve that $\\mathcal{M}_{+,G}=\\mathcal{M}_{+}$.\n\\end{frem}\n\n\\noindent \\textbf{Proof of theorem \\ref{def15}}: We have assumed that $%\n\\left\n( \|\\Phi _{s}\\rangle \\right) _{s=1}^{N}$ is an ONS in $\\mathcal{M}$,\nwhich implies that $\\left( \|\\xi _{nm}\\rangle \\otimes \\left( \\Gamma\n(T)U_{m}B_{n}^{}\|\\Phi _{s}\\rangle \\right) \\right) _{s=1}^{N}$ is an ONS in $%\n\\mathcal{M}^{\\otimes 3}$. Hence we obtain the equations (\\ref{45}), (\\ref{46}%\n) and (\\ref{47}) by Lemma \\ref{def18}. This proves Theorem \\ref{def15}. $%\n\\blacksquare $\n\n\\begin{frem}\n\\label{def20} In the special case of the remark \\ref{def19}, the equations (\n\\ref{45}), (\\ref{46}) and (\\ref{47}) hold if we replace $F_{+}$ by $F_{+,X}$\nin the definition of the channel $\\Theta _{nm}$ and in (\\ref{46}), (\\ref{47}\n), that is, Bob will only perform ``local'' measurement according to the\nregion $X$, about which we will discuss more details in the next sections.\n\\end{frem}\n\n\\section{Teleportation of states inside $\\mathbf{R}^{k}$}\n\n\\label{sec4}Let $\\mathcal{H}$ be a finite-dimensional Hilbert space. We\nconsider the case $\\mathcal{H}=\\Bbb{C}^{N}=L^{2}(\\{1,\\ldots ,N\\},\\#)$\nwithout loss of generality, where $\\#$ denotes the counting measure on the\nset $\\{1,\\ldots ,N\\}$. We want to teleport states on $\\mathcal{H}$ with the\naid of the constructed channels $\\left( \\Lambda _{nm}\\right) _{n,m=1}^{N}$\nor $\\left( \\Theta _{nm}\\right) _{n,m=1}^{N}$. We fix\n\n\\begin{enumerate}\n\\item[-] a CONS $(\|j\\rangle )_{j=1}^{N}$ of $\\mathcal{H}$\n\n\\item[-] $f\\in L^{2}\\left( \\mathbf{R}^{k}\\right) $, $\\Vert f\\Vert =1$\n\n\\item[-] $d=a^{2}>0$\n\n\\item[-] $\\hat{K}_{1},\\hat{K}_{2}$ linear operators on $L^{2}\\left( \\mathbf{%\nR}^{k}\\right) $\n\n\\item[-] $\\hat{T}$ unitary operator on $L^{2}\\left( \\mathbf{R}^{k}\\right) $\n\\end{enumerate}\n\nwith two properties\n\\begin{equation}\n\\hat{K}_{1}^{}\\hat{K}_{1}f+\\hat{K}_{2}^{}\\hat{K}_{2}f=f \\label{49}\n\\end{equation}\n\\begin{equation}\n\\hat{T}\\hat{K}_{1}f=\\hat{K}_{2}f \\label{50}\n\\end{equation}\nWe put\n\\[\nG=\\mathbf{R}^{k}\\times \\{1,\\ldots ,N\\}\\,,\\;\\mu =l\\times \\#,\n\\]\nwhere $l$ is the Lebesgues measure on $\\mathbf{R}^{k}$. Then $L^{2}(G)=L^{2}\n(G,\\mu )=L^{2}(\\mathbf{R}^{k})\\otimes \\mathcal{H}.$ Further, put\n\\[\ng_{j}:=f\\otimes \|j\\rangle \\quad (j=1,\\ldots ,N)\n\\]\nThen $(g_{j})_{j=1}^{N}$ is an ONS in $L^{2}(G)$. We consider linear\noperators $K_{1},K_{2}$ on $L^{2}(G)$ with (\\ref{5}) and\n\\begin{equation}\nK_{\\emph{r}}g_{j}=\\left( \\hat{K}_{\\emph{r}}f\\right) \\otimes \|j\\rangle \\quad\n(j=1,\\ldots ,N;\\;\\emph{r}=1,2). \\label{51}\n\\end{equation}\n\n\\begin{frem}\n\\label{def21} (\\ref{51}) determines operators $K_{1},K_{2}$ on the subspace\nof $\\mathcal{M}$ spanned by the ONS $(g_{j})_{j=1}^{N}$. On the orthogonal\ncomplement, one can put for instance\n\\[\nK_{\\emph{r}}\\psi =\\frac{1}{\\sqrt{2}}\\psi\n\\]\nThen $K_{1},K_{2}$ are well defined and fulfill (\\ref{5}) because of (\\ref\n{49}). Further, one checks that (\\ref{13}) and (\\ref{15}) hold.\n\\end{frem}\n\nNow let $T$ be an unitary operator on $L^{2}(G)$ with\n\\[\nT(K_{1}g_{j})=\\left( \\hat{T}\\hat{K}_{2}f\\right) \\otimes \|j\\rangle\n\\]\nFrom (\\ref{13}) one can prove the existence of $T$ using the arguments as\nin the remark \\ref{def21}. Further, we get (\\ref{12}) from (\\ref{50}).\n\\newline\nSummarizing, we obtain that $\\{g_{1},\\ldots ,g_{N}\\}$, $K_{1},K_{2},T$\nfulfill all the assumptions required in section 2. Thus we have the\ncorresponding channels $\\Lambda _{n,m}$ ,$\\Theta _{nm}$ given by (\\ref{37})\nand (\\ref{43}) respectively. It follows that we are able to teleport a state\n$\\rho $ on $\\mathcal{M}=\\mathcal{M}(G)$ with (\\ref{16}) and (\\ref{17} ) as\nit was stated in the theorem \\ref{def12} and the theorem \\ref{def15},\nrespectively. \\newline\nIn order to teleport states on $\\mathcal{H}$ through the space $\\mathbf{R}%\n^{k}$ using the above channels, we have to consider:\n\n\\begin{description}\n\\item[first:] a ``lifting'' $\\mathcal{E}^{}$ of the states on $\\mathcal{H}$\ninto the set of states on the bigger state space on $\\mathcal{M}$ such that $%\n\\rho =\\mathcal{E}^{}(\\hat{\\rho})$ can be described by (\\ref{16}), (\\ref{17}%\n), (\\ref{18}).\n\n\\item[second:] a ``reduction'' $\\mathcal{R}$ of (normal) states on $%\n\\mathcal{M}$ to states on $\\mathcal{H}$ such that for all states $\\hat{\\rho}$\non $\\mathcal{H}$ it holds\n\\begin{equation}\n\\mathcal{R}\\left( \\left( \\Gamma (T)U_{m}B_{n}^{}\\right) \\mathcal{E}%\n^{}\\left( \\hat{\\rho}\\right) \\left( \\Gamma (T)U_{m}B_{n}^{}\\right)\n^{}\\right) =V_{nm}\\hat{\\rho}V_{nm}^{}\\quad (n,m=1,\\ldots ,N), \\label{52}\n\\end{equation}\nwhere $(V_{nm})_{n,m=1}^{N}$ are unitary operators on $\\mathcal{H}$.\n\\end{description}\n\nThat we can obtain as follows: We have already stated in section 2 that\n\\[\n\\left( \|\\mathrm{exp}\\;(aK_{1}(g_{j}))-\\mathrm{exp}\\;(0)\\rangle \\right)\n_{j=1}^{N}\\quad (\\emph{r}=1,2)\n\\]\nare ONS in $\\mathcal{M}$. We denote by $\\mathcal{M}_{\\emph{r}}$ $(\\emph{r}%\n=1,2)$ the corresponding $N$-- dimensional subspaces of $\\mathcal{M}$. Then\nfor each $\\emph{r}=1,2,$ there exists exactly one unitary operator $W_{\\emph{%\nr}}$ from $\\mathcal{H}$ onto $\\mathcal{M}_{\\emph{r}}\\subseteq \\mathcal{M}$\nwith\n\\begin{equation}\nW_{\\emph{r}}\|j\\rangle =\|\\mathrm{exp}\\;(aK_{\\emph{r}}g_{j})-\\mathrm{exp}\\;(0)\n\\rangle \\quad (j=1,\\ldots ,N) \\label{53}\n\\end{equation}\nWe put\n\\begin{equation}\n\\mathcal{E}^{}\\left( \\hat{\\rho}\\right) :=W_{1}\\hat{\\rho}W_{1}^{} \\Pi _{%\n\\mathcal{M}_{1}}\\quad \\left( \\hat{\\rho}\\text{ state on }\\mathcal{H}\\right) ,\n\\label{54}\n\\end{equation}\nwhere $\\Pi _{\\mathcal{M}_{\\emph{r}}}$ denotes the projection onto $\\mathcal{M%\n}_{\\emph{r}}$ $(\\emph{r}=1,2)$. \\newline\nDescribing the state $\\hat{\\rho}$ on $\\mathcal{H}$ by\n\\begin{equation}\n\\hat{\\rho}=\\sum\\limits_{s=1}^{N}\\lambda _{s}\|\\hat{\\Phi}_{s}\\rangle \\langle\n\\hat{\\Phi}_{s}\| \\label{55}\n\\end{equation}\nwith\n\\[\n\|\\hat{\\Phi}_{s}\\rangle =\\sum\\limits_{j=1}^{N}c_{sj}\|j\\rangle ,\n\\]\nwhere $\\left( c_{sj}\\right) _{s,j=1}^{N}$ fulfills (\\ref{18}), we obtain\nthat $\\rho =\\mathcal{E}^{}\\left( \\hat{\\rho}\\right) $ is given by (\\ref{16})\nand (\\ref{17}). \\newline\nNow, for each state $\\rho $ on $\\mathcal{M}$ we put\n\\begin{equation}\n\\mathcal{R}(\\rho ):=\\frac{W_{2}^{}\\Pi _{\\mathcal{M}_{2}}\\rho W_{2}} {%\n\\mathrm{tr}_{\\mathcal{M}}W_{2}^{}\\Pi _{\\mathcal{M}_{2}}\\rho W_{2}}\n\\label{56}\n\\end{equation}\nSince\n\\[\n\\Pi _{\\mathcal{M}_{2}}\\Gamma (T)U_{m}B_{n}^{}\\mathcal{E}^{} \\left( \\hat{%\n\\rho}\\right) \\left( \\Gamma (T)U_{m}B_{n}^{}\\right) ^{}= \\Gamma\n(t)U_{m}B_{n}^{}\\mathcal{E}^{}\\left( \\hat{\\rho}\\right) \\left(\n\\Gamma(T)U_{m}B_{n}^{}\\right) ^{},\n\\]\nwe get\n\\[\n\\mathrm{tr}_{\\mathcal{M}}W_{2}^{}\\Pi _{\\mathcal{M}_{2}}\\Gamma\n(t)U_{m}B_{n}^{}\\mathcal{E}^{}\\left( \\hat{\\rho}\\right) \\left( \\Gamma\n(T)U_{m}B_{n}^{}\\right) ^{}=1\n\\]\nand\n\\[\n\\mathcal{R}\\left( \\Gamma (T)U_{m}B_{n}^{}\\mathcal{E}^{} \\left( \\hat{\\rho}%\n\\right) \\left( \\Gamma (T)U_{m}B_{n}^{}\\right) ^{}\\right) =\nW_{2}^{}\\Gamma(T)U_{m}B_{n}^{}W_{1}\\hat{\\rho}W_{1}^{}\\Pi _{\\mathcal{M}%\n_{1}} \\left( \\Gamma(T)U_{m}B_{n}^{}\\right) ^{}W_{2}.\n\\]\nAs we have the equality\n\\[\n\\Pi _{\\mathcal{M}_{1}}\\left( \\Gamma (T)U_{m}B_{n}^{}\n\\right)^{}W_{2}=\\left( \\Gamma (T)U_{m}B_{n}^{}\\right) ^{}W_{2},\n\\]\nwhich implies\n\\[\n\\mathcal{R}\\left( \\Gamma (T)U_{m}B_{n}^{}\\mathcal{E}^{} \\left( \\hat{\\rho}%\n\\right) \\left( \\Gamma (T)U_{m}B_{n}^{}\\right) ^{}\\right) =\nW_{2}^{}\\Gamma(T)U_{m}B_{n}^{}W_{1}\\hat{\\rho}W_{1}^{} \\left(\n\\Gamma(T)U_{m}B_{n}^{}\\right) ^{}W_{2}\n\\]\nPut\n\\begin{equation}\nV_{nm}:=W_{2}^{}\\Gamma (T)V_{m}B_{n}^{}W_{1}\\quad (n,m=1,\\ldots ,N),\n\\label{57}\n\\end{equation}\nthen $V_{nm}$ $(n,m=1,\\ldots ,N)$ is an unitary operator on $\\mathcal{H}$\nand (\\ref{52}) holds. One easily checks\n\\[\nV_{nm}\|j\\rangle =\\bar{b}_{nj}\|j\\otimes m\\rangle \\quad (j,m,n=1,\\ldots ,N).\n\\]\nSummarizing these, we have the following theorem:\n\n\\begin{ftheo}\n\\label{def22} Consider the channels on the set of states on $\\mathcal{H}$\n\\begin{eqnarray}\n\\hat{\\Lambda}_{nm} &:&=\\mathcal{R}\\circ \\Lambda _{nm}\\circ \\mathcal{E}%\n^{}\\quad (n,m=1,\\ldots ,N) \\label{58} \\\\\n\\hat{\\Theta}_{nm} &:&=\\mathcal{R}\\circ \\Theta _{nm}\\circ \\mathcal{E}%\n^{}\\quad (n,m=1,\\ldots ,N) \\label{59}\n\\end{eqnarray}\nwhere $\\mathcal{R},$ $\\mathcal{E}^{},\\Lambda _{nm},\\Theta _{nm}$ are given\nby (\\ref{56}), (\\ref{54}), (\\ref{37}), (\\ref{43}), respectively. Then for\nall states $\\hat{\\rho}$ on $\\mathcal{H}$, it holds\n\\begin{equation}\n\\hat{\\Lambda}_{nm}\\left( \\hat{\\rho}\\right) =V_{nm}\\hat{\\rho}V_{nm}^{}=\\hat{%\n\\Theta}_{nm}\\left( \\hat{\\rho}\\right) \\quad (n,m=1,\\ldots ,N), \\label{60}\n\\end{equation}\nwhere $V_{nm}\\;(n,m=1,\\ldots ,N)$ are the unitary operators on $\\mathcal{H}$\ngiven by (\\ref{57}).\n\\end{ftheo}\n\n\\begin{frem}\n\\label{def23} Remember that the teleportation model according to $\\left(\n\\Lambda _{nm}\\right) _{n,m=1}^{N}$ works perfectly in the sense of the\nremark \\ref{def14}, and the model dealing with $\\left( \\Theta _{nm}\\right)\n_{n,m=1}^{N}$ was only asymptotically perfect for large $d$(i.e.,high\ndensity or high energy of the beams). They can transfer to $\\left( \\hat{%\n\\Lambda}_{n,m}\\right) $ , $\\left( \\hat{\\Theta}_{nm}\\right) $.\n\\end{frem}\n\n\\begin{fex}\n\\label{def24} We specialize\n\\[\n\\hat{K}_{1}h=\\hat{K}_{2}h=\\frac{1}{\\sqrt{2}}\\;h\\quad \\left( h\\in L^{2}(%\n\\mathbf{R}^{k})\\right) ,\\text{ }\\hat{T}=\\mathbf{1.}\n\\]\nRealizing the teleportation in this case means that Alice has to perform\nmeasurements $\\left( F_{nm}\\right) $ in the whole space $\\mathbf{R}^{k}$ and\nalso Bob (concerning $F_{+}$).\n\\end{fex}\n\n\\section{Alice and Bob are spatially separated}\n\nWe specialize the situation in section 4 as follows: We fix \\vspace{-3mm}\n\n\\begin{enumerate}\n\\item[-] $t\\in \\mathbf{R}^{k}$\n\n\\item[-] $X_{1},X_{2},X_{3}\\subseteq \\mathbf{R}^{k}$ are measurable\ndecomposition of $\\mathbf{R}^{k}$ such that $l(X_{1})\\neq 0$ and\n\\[\nX_{2}=X_{1}+t:=\\{x+t\|\\;x\\in X_{1}\\}.\n\\]\nPut\n\\[\n\\begin{array}{llll}\n\\hat{T}h(x) & := & h(x-t)\\qquad & \\left( x\\in \\mathbf{R}^{k}\\,,\\;h\\in L^{2}(%\n\\mathbf{R}^{k})\\right) \\\\\n\\hat{K}_{\\emph{r}}h & := & h\\mathcal{X}_{X_{\\emph{r}}} & \\left( \\emph{r}%\n=1,2\\,,\\;h\\in L^{2}(\\mathbf{R}^{k})\\right)\n\\end{array}\n\\]\n\\end{enumerate}\n\nand assume that the function $f\\in L^{2}\\left( \\mathbf{R}^{d}\\right) $ has\nthe properties\n\\[\nf\\mathcal{X}_{X_{2}}=\\hat{T}\\left( f\\mathcal{X}_{X_{1}}\\right) ,\\;f\\mathcal{X%\n}_{X_{3}}\\equiv 0\n\\]\nThen $\\hat{T}$ is an unitary operator on $L^{2}\\left( \\mathbf{R}^{k}\\right) $\nand (\\ref{48} ),(\\ref{49}) hold. \\newline\nUsing the assumption that $X_{1},X_{2},X_{3}$ is a measurable decomposition\nof $\\mathbf{R}^{k}$ we get immediately that\n\\[\nG_{s}:=X_{s}\\times \\{1,\\ldots ,N\\}\\quad (s=1,2,3)\n\\]\nis a measurable decomposition of $G$. It follows that $\\mathcal{M}=\\mathcal{M%\n}(G)$ is decomposed into the tensor product\n\\[\n\\mathcal{M}(G)=\\mathcal{M}(G_{1})\\otimes \\mathcal{M}(G_{2})\\otimes \\mathcal{M%\n}(G_{3}).\n\\]\n\\cite{FF1,FF2,FW}. According to this representation, the local algebras $%\n\\frak{A}(X_{s})$ corresponding to regions $X_{s}\\subseteq \\mathbf{R}^{d}$ $%\n\\left( s=1,2,3\\right) $ are given by\n\\[\n\\begin{array}{llll}\n\\frak{A}(X_{1}) & := & \\{A\\otimes \\mathbf{1}\\otimes \\mathbf{1};A\\text{\nbounded operator on }\\mathcal{M}(G_{1})\\}\\qquad & \\\\\n\\frak{A}(X_{2}) & := & \\{\\mathbf{1}\\otimes A\\otimes \\mathbf{1};A\\text{\nbounded operator on }\\mathcal{M}(G_{2})\\} & \\\\\n\\frak{A}(X_{3}) & := & \\{\\mathbf{1}\\otimes \\mathbf{1}\\otimes A;A\\text{\nbounded operator on }\\mathcal{M}(G_{3})\\} &\n\\end{array}\n\\]\nOne easily checks in our special case that\n\\[\nF_{nm}\\in \\frak{A}(X_{1})\\otimes \\frak{A}(X_{2})\\quad (n,m=1,\\ldots ,N)\n\\]\nand $\\mathcal{E}^{}\\left( \\hat{\\rho}\\right) $ gives a state on $\\frak{A}%\n(X_{1})$ (the number of particles outside of $G_{1}$ is $0$ with probabiliy $%\n1$ ). That is, Alice has to perform only local measurements inside of the\nregion $X_{1}$ in order to realize the teleportation processes described in\nsection 4 or measure the state $\\mathcal{E}^{}\\left( \\hat{\\rho}\\right) $.\nOn the other hand, $\\Lambda _{nm}\\left( \\mathcal{E}^{}\\left( \\hat{\\rho}%\n\\right) \\right) $ and $\\Theta _{nm}\\left( \\mathcal{E}^{}\\left( \\hat{\\rho}%\n\\right) \\right) $ give local states on $\\frak{A}(X_{2})$ such that by\nmeasuring these states Bob has to perform only local measurements inside of\nthe region $X_{2}$. The only problem could be that according to the\ndefinition (\\ref{43}) of the channels $\\Theta _{nm}$ Bob has to perform the\nmeasurement by $F_{+}$ which is not local. However, as we have already\nstated in the remark \\ref{def20}, this problem can be avoided if we replace $%\nF_{+}$ by $F_{+,X_{2}}\\in \\frak{A}(X_{2})$. \\newline\nTherefore we can describe the special teleportation process as follows: We\nhave a beam being in the pure state $\|\\eta \\rangle \\langle \\eta \|$ (\\ref{40}%\n) . After splitting, one part of the beam is located in the region $X_{1}$\nor will go to $X_{1}$ (cf. remark \\ref{def10}) and the other part is located\nin the region $X_{2}$ or will go to $X_{2}$. Further, there is a state $%\n\\mathcal{E}^{}\\left( \\hat{\\rho}\\right) $ localized in the region $X_{1}$.\nNow Alice will perform the local measurement inside of $X_{1}$ according to $%\nF=\\sum\\limits_{n,m}z_{nm}F_{nm}$ involving the first part of the beam and\nthe state $\\mathcal{E}^{}(\\rho )$. This leads to a preparation of the\nsecond part of the beam located in the region $X_{2}$ which can be\ncontrolled by Bob, and the second part of the beam will show the behaviour\nof the state $\\Lambda _{nm}\\left( \\mathcal{E}^{}\\left( \\hat{\\rho}\\right)\n\\right) =\\Theta _{nm}\\left( \\mathcal{E}^{}\\left( \\hat{\\rho}\\right) \\right) $\nif Alice's measurement shows the value $z_{nm}$. Thus we have teleported the\nstate $\\hat{\\rho}$ on $\\mathcal{H}$ from the region $X_{1}$ into the region $%\nX_{2}$.\n\n\\begin{thebibliography}{99}\n\\bibitem{AO} Accardi, L. and Ohya, M.: \\textit{Teleportation of\ngeneralquantum states,} Voltera Center preprint,1998.\n\n\\bibitem{AO2} Accardi L., Ohya M.: \\textit{Compound channels, transition\nexpectations and liftings}, Applied Mathematics \\& Optimization, \\textbf{39}%\n, 33--59, 1999.\n\n\\bibitem{Ben} Benneth, C. H., Brassard, G., Cr\\'{e}peau, C., Jozsa, R.,\nPeres, A. and Wotters, W.: \\textit{Teleporting an unknown quantum state\nviaDual Classical and Einstein-Podolsky- Rosen channels.} Phys. Rev. Lett.\n\\textbf{70}, 1895--1899, 1993.\n\n\\bibitem{Ben2} Bennett, C.H., G. Brassard, S. Popescu, B. Schumacher, J.A.\nSmolin, W.K. Wootters, \\textit{Purification of noisy entanglement and\nfaithful teleportation via noisy channels}, Phys. Rev. Lett. \\textbf{76}%\n,722--725 , 1996.\n\n\\bibitem{Eke} A.K. Ekert, Quantum cryptography based on Bell's theorem,\n\\emph{Phys. Rev. Lett.} \\textbf{67}, 661--663, 1991.\n\n\\bibitem{FF1} Fichtner, Karl-Heinz and Freudenberg, W.: \\textit{%\nPointprocesses and the position distrubution of infinite boson systems.} J.\nStat.Phys. \\textbf{47}, 959--978, 1987.\n\n\\bibitem{FF2} Fichtner, Karl-Heinz and Freudenberg, W.: \\textit{%\nCharacterization of states of infinite Boson systems I.-- On the\nconstruction of states.} Comm. Math. Phys. \\textbf{137}, 315--357, 1991.\n\n\\bibitem{FFL} Fichtner, K-H., Freudenberg, W. and Liebscher, V.: \\textit{%\nTime evolution and invariance of Boson systems given by beam splittings}, In\nfinite Dim. Anal., Quantum Prob. and related topics I, 511--533, 1998.\n\n\\bibitem{L} Lindsay, J. M.: \\textit{Quantum and Noncausal Stochastic\nCalculus.} Prob. Th. Rel. Fields \\textbf{97}, 65--80, 1993.\n\n\\bibitem{FW} Fichtner, K.-H. and Winkler, G.: \\textit{Generalized brownian\nmotion, point processes and stochastic calculus for random fields.} Math.\nNachr. \\textbf{161}, 291--307, 1993.\n\n\\bibitem{IOS} Inoue, K, Ohya, M. and Suyari, H.: \\textit{Characterization\nof quantum teleportation processes by nonlinear quantum mutual entropy},\nPhysica D, \\textbf{120}, 117--124, 1998.\n\\end{thebibliography}\n\n\\end{document}\n\n\n" } ]	[ { "name": "quant-ph9912083.extracted_bib", "string": "{AO Accardi, L. and Ohya, M.: Teleportation of generalquantum states, Voltera Center preprint,1998." }, { "name": "quant-ph9912083.extracted_bib", "string": "{AO2 Accardi L., Ohya M.: Compound channels, transition expectations and liftings, Applied Mathematics \\& Optimization, 39% , 33--59, 1999." }, { "name": "quant-ph9912083.extracted_bib", "string": "{Ben Benneth, C. H., Brassard, G., Cr\\'{epeau, C., Jozsa, R., Peres, A. and Wotters, W.: Teleporting an unknown quantum state viaDual Classical and Einstein-Podolsky- Rosen channels. Phys. Rev. Lett. 70, 1895--1899, 1993." }, { "name": "quant-ph9912083.extracted_bib", "string": "{Ben2 Bennett, C.H., G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W.K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett. 76% ,722--725 , 1996." }, { "name": "quant-ph9912083.extracted_bib", "string": "{Eke A.K. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett. 67, 661--663, 1991." }, { "name": "quant-ph9912083.extracted_bib", "string": "{FF1 Fichtner, Karl-Heinz and Freudenberg, W.: % Pointprocesses and the position distrubution of infinite boson systems. J. Stat.Phys. 47, 959--978, 1987." }, { "name": "quant-ph9912083.extracted_bib", "string": "{FF2 Fichtner, Karl-Heinz and Freudenberg, W.: % Characterization of states of infinite Boson systems I.-- On the construction of states. Comm. Math. Phys. 137, 315--357, 1991." }, { "name": "quant-ph9912083.extracted_bib", "string": "{FFL Fichtner, K-H., Freudenberg, W. and Liebscher, V.: % Time evolution and invariance of Boson systems given by beam splittings, In finite Dim. Anal., Quantum Prob. and related topics I, 511--533, 1998." }, { "name": "quant-ph9912083.extracted_bib", "string": "{L Lindsay, J. M.: Quantum and Noncausal Stochastic Calculus. Prob. Th. Rel. Fields 97, 65--80, 1993." }, { "name": "quant-ph9912083.extracted_bib", "string": "{FW Fichtner, K.-H. and Winkler, G.: Generalized brownian motion, point processes and stochastic calculus for random fields. Math. Nachr. 161, 291--307, 1993." }, { "name": "quant-ph9912083.extracted_bib", "string": "{IOS Inoue, K, Ohya, M. and Suyari, H.: Characterization of quantum teleportation processes by nonlinear quantum mutual entropy, Physica D, 120, 117--124, 1998." } ]
quant-ph9912084	{\flushleft {\ns E-print quant-ph/9912084	[ { "author": "D.A. Trifonov" }, { "author": "Institute for Nuclear Research and Nuclear Energetics" }, { "author": "72 Tzarigradsko chauss\\'ee" }, { "author": "Sofia" }, { "author": "Bulgaria" } ]	The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schr\"odinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard $SU(1,1)$ and $SU(2)$ coherent states are the unique states which minimize the second order characteristic inequality for the three generators. A set of states which minimize the Schr\"odinger inequality for the Hermitian components of the $su_q(1,1)$ ladder operator is also constructed. It is noted that the characteristic uncertainty relations can be written in the alternative complementary form.	[ { "name": "quant-ph9912084.tex", "string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%Paper: quant-ph/9912084 From: \"D. Trifonov\" <DTRIF@inrne.bas.bg>\n%%Date: Fri, 17 Dec 1999 15:47:21 GMT (26kb)\n%% v.5: Fri, 13 Oct 2000 (misprints in eqs. (5), (36) corrected)\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentstyle{article} %% varvi. \n%\\documentstyle[aps]{revtex} %% varvi.\n\\textwidth=16cm \\textheight=23cm %% = 17 pages\n\\voffset=-1.4cm \\hoffset=-2.2cm %% text centered\n\n%%%%%%%%%%%%%% Some definitions %%%%%%%%%%%%%%%%%%%\n\\def\\bc{\\begin{center}} \\def\\ec{\\end{center}}\n\\def\\beq{\\begin{equation}} \\def\\eeq{\\end{equation}}\n\\def\\bear{\\begin{eqnarray}} \\def\\eear{\\end{eqnarray}}\n\\def\\bt{\\begin{tabular}} \\def\\et{\\end{tabular}}\n\\def\\la{\\langle} \\def\\ra{\\rangle}\n\\def\\dg{\\dagger} \\def\\ci{\\cite}\n\\def\\lb{\\label} \\def\\ld{\\ldots}\n\\def\\hs{\\hspace} \\def\\vs{\\vspace}\n\\def\\pr{\\prime} \\def\\sm{\\small}\n\\def\\lar{\\leftarrow} \\def\\rar{\\rightarrow}\n\\def\\llar{\\longleftarrow} \\def\\lrar{\\longrightarrow}\n\\def\\td{\\tilde} \\def\\pr{\\prime}\n\\def\\pd{\\partial} \\def\\ts{\\textstyle}\n\\def\\ftn{\\footnote} \\def\\nn{\\nonumber}\n\\def\\ns{\\normalsize}\n\\def\\alf{\\alpha} \\def\\bet{\\beta} \\def\\gam{\\gamma}\n\\def\\Gam{\\Gamma} \\def\\k{\\kappa}\n\\def\\Dlt{\\Delta} \\def\\dlt{\\delta} \\def\\eps{\\epsilon}\n\\def\\lam{\\lambda} \\def\\Lam{\\Lambda} \\def\\sig{\\sigma}\n\\def\\z{\\zeta} \\def\\vphi{\\varphi} \\def\\veps{\\varepsilon}\n\\def\\ome{\\omega} \\def\\Ome{\\Omega} \\def\\tet{\\theta}\n\\def\\Tet{\\Theta} \\def\\vtet{\\vartheta}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{document}\n\n\\title{ {\\flushleft {\\ns E-print quant-ph/9912084}\\\\\n {\\ns Appeared in {\\it Geometry, Integrability and Quantization,}}\\\\[-3mm] \n {\\ns Eds. I.M. Mladenov and G.L. Naber (Coral Press, Sofia 2000), \n\t p. 257-282} \\\\[-3mm]\n {\\ns (Proc. Int. Conference, September 1-10, 1999, Varna)}\\\\[5mm]}\n \\bf The Uncertainty Way of Generalization\\\\ of Coherent States}\n\n\\author{D.A. Trifonov\\\\\n Institute for Nuclear Research and Nuclear Energetics\\\\\n 72 Tzarigradsko chauss\\'ee, Sofia, Bulgaria }\n\\maketitle\n\n\\begin{abstract}\nThe three ways of generalization of canonical coherent states are briefly\nreviewed and compared with the emphasis laid on the (minimum) uncertainty\nway. The characteristic uncertainty relations, which include the\nSchr\\\"odinger and Robertson inequalities, are extended to the case of\nseveral states. It is shown that the standard $SU(1,1)$ and $SU(2)$\ncoherent states are the unique states which minimize the second order\ncharacteristic inequality for the three generators. A set of states which\nminimize the Schr\\\"odinger inequality for the Hermitian components of the\n$su_q(1,1)$ ladder operator is also constructed. It is noted that the\ncharacteristic uncertainty relations can be written in the alternative\ncomplementary form.\n\\end{abstract}\n\n%section 1\n\\section{Introduction}\n\nCoherent states (CS) introduced in 1963 in the pioneering works by Glauber\n and Klauder \\cite{KlaSka} pervade nearly all branches of quantum physics\n(see the reviews \\ci{KlaSka}--\\ci{AliAGM}). This important\novercomplete family of states $\\{\|\\alf\\ra\\}$, $\\alf \\in {\\mathbf C}$, can be\ndefined in three equivalent ways \\cite{ZhaFenGil}: \\\\[-2mm]\n\n(D1) As the set of eigenstates of boson destruction operator (the\nladder operator) $a$:\\,\\,\n$a\|\\alf\\ra = \\alf\|\\alf\\ra,$\n\n(D2) As the orbit of the ground state $\|0\\ra$ ($a\|0\\ra = 0$) under the\naction of the unitary displacement operators $D(\\alf)=\\exp(\\alf a^\\dg -\n\\alf^a)$ (which realize ray representation of the Heisenberg--Weyl group\n$H_1$) :\\,\\, $\|\\alf\\ra = D(\\alf)\|0\\ra$.\n\n(D3) As the set of states which minimize the Heisenberg uncertainty relation\n(UR) $(\\Delta q)^2(\\Delta p)^2 \\geq 1/4$ for the Hermitian components\n$q,\\,p$ of $a$ ($a = (q+ip)/\\sqrt{2}$) with equal uncertainties:\\,\\,\n$(\\Delta q)^2(\\Delta p)^2 = 1/4,\\,\\, \\Delta q = \\Delta p$. Note that one\nrequires the minimization plus the equality of the two variances.\n\nThe overcompleteness property reads ($d^2\\alf= d{\\rm Re}\\alf\\,d{\\rm Im}\\alf$)\n\\beq\\lb{1} %eq.1\n1 = \\int\|\\alf\\ra\\la\\alf\|d\\mu(\\alf),\\quad d\\mu(\\alf) = (1/\\pi) d^2\\alf.\n\\eeq\nOne says that the family $\\{\|\\alf\\ra\\}$ resolves the unity operator with\nrespect to the measure $d\\mu(\\alf)$. The CS $\|\\alf\\ra$ should be referred as\n{\\it canonical CS} \\cite{KlaSka}. The resolution unity property (\\ref{1})\nprovides the important analytic representation (rep ), known as canonical CS\nrep or Fock--Bargman analytic rep, in\nwhich $a=d/d\\alf,\\,\\, a^\\dg = \\alf$ and the state $\|\\Psi\\ra$ is represented\nby the function $\\Psi(\\alf) = \\exp(\|\\alf\|^2/2)\\la\\alf^\|\\Psi\\ra$. In\n1963-64 Klauder \\cite{KlaSka} developed a general theory of the continuous\nreps and suggested the possibility to construct overcomplete sets\nof states using irreducible reps of Lie groups. Let us note that\nthe resolution unity property (\\ref{1}) is not a defining one for the CS\n$\|\\alf\\ra$.\n\nCorrespondingly to the definitions (D1)--(D3) there are three\ndifferent ways (methods) of generalization of the canonical CS \\ci{ZhaFenGil}:\nThe diagonalization of non-Hermitian operators (the {\\it eigenstate way}, or\nthe ladder operator method \\cite{Nie98}); The construction of Hilbert space\norbit by means of unitary operators ({\\it orbit way} or the displacement\noperator method \\cite{Nie98}); The minimization of an appropriate UR\n(the {\\it uncertainty way}). The first two methods and especially\nthe second one (the orbit method) have enjoyed a considerable attention and\nvast applications to various fields of physics \\ci{KlaSka}--\\ci{AliAGM},\nwhile the third method is receiving a significant attention only recently\n-- see \\ci{Trif94}--\\ci{Trif98a}, \\ci{FujFun}--\\ci{Bjork} and references\ntherein. It is worth noting at the point that some authors were pessimistic\nabout the possibility of effective generalization of the third\ndefining property of canonical CS.\n\nThe aim of the present paper is to consider some of the new developments in\nthe third way (the uncertainty way) and their relationship to the first two\nmethods. We show that the Robertson \\cite{SchRob} and other characteristic\ninequalities \\cite{TriDon} are those uncertainty relations which are\ncompatible with the generalizations of the ladder operator and displacement\noperator methods to the case of many observables.\n\nIn section 2 we briefly review some of the main generalizations of the first\ntwo defining properties of the canonical CS and the relationship between the\ncorresponding generalized CS. Some emphasis is laid on the family of\nsqueezed states (SS) \\ci{LouKni} and the Barut-Girardello CS (BG CS)\n\\cite{BG} and their analytic reps. The canonical SS are the unique\ngeneralization of CS for which the three definitions (D1), (D2), (D3) are\nequivalently generalized.\n\nSection 3 is devoted to the uncertainty way of generalization of CS. In\nsubsection 3.1 we consider the minimization of the Heisenberg and the\nSchr\\\"odinger UR \\cite{SchRob} for two observables and the relation of the\nminimizing states to the corresponding group-related CS \\cite{KlaSka}, on\nthe examples of $SU(2)$, $SU(1,1)$ and $SU_q(1,1)$. Here we note that the \n$SU(2)$ and $SU(1,1)$ CS with lowest (highest) weight reference vector\nminimize the Schr\\\"odinger inequality for the first two generators, while\nthe Heisenberg one is minimized in some subsets only. These group-related CS \nare particular cases of the corresponding minimizing states. A set of states\nwhich minimize the Schr\\\"odinger inequality for the Hermitian\ncomponenents of the $SU_q(1,1)$ ladder operator is also constructed.\n\nIn the subsection 3.2 the minimization of the Robertson \\cite{Rob} and the\nother {\\it characteristic UR} \\cite{TriDon} for several observables is\nconsidered. \nIn the case of the three generators (three observables) of $SU(1,1)$ (and\nthe $SU(2)$) we establish that the group-related CS with lowest (highest)\nweight reference vector are the unique states which minimize the second and\nthe third characteristic UR for the three generators simultaneously. The\ncharacteristic UR, in particular the known Robertson and the Schr\\\"odinger\nones, relate certain combinations of the second and first moment of the\nobservables in one and the same quantum state. Here we extend these\nrelations to the case of several states. States which minimize the\ncharacteristic UR are naturally called {\\it characteristic uncertainty\nstates} (characteristic US\n\\footnote[1]{Let us list the abbreviations used in the paper: CS = coherent\nstate, SS = squeezed state, UR = uncertainty relation, US = uncertainty\nstate, BG = Barut-Girardello, and rep = representation.}). The alternative\nnames could be (characteristic) intelligent states and (characteristic)\noptimal US. The extended characteristic UR are also {\\it invariant} under\nthe linear nondegenerate transformation of the observables as the\ncharacteristic ones are. It is shown that the characteristic UR can be\nwritten in the {\\it complementary form} \\cite{Bjork} in terms of two positive\nquantities less than the unity. Finally it is noted that the positive\ndefinite characteristic uncertainty functionals (for several observables)\ncan be used for the construction of distances between quantum states. In the\nAppendix the proofs of the Robertson relation (after Robertson) and of the\nuniqueness of the standard $SU(1,1)$ CS minimization of the second (and third)\norder characteristic UR are provided. \n\n%%Section 2.\n\\section{The Eigenstate and Orbit Ways}\n\nCanonical CS $\|\\alf\\ra = D(\\alf)\|0\\ra$ diagonalize the boson destruction\noperator $a$, $[a,a^\\dg]=1$. This was the first and seminal example of\ndiagonalizing of a non-Hermitian operator. We stress that the eigenstates of\n$a$ and other non-Hermitian operators in this paper are not orthogonal to\neach other -- the term \"diagonalization\" is used for brevity and in analogy\nto the case of diagonalization of Hermitian operators. The second example\nwas, to the best of our knowledge, the diagonalization of the complex\ncombination of boson lowering and raising operators $a,\\,a^\\dg$\n($\\alf\\in {\\mathbf C}$),\n\\ci{MMT}\n\\beq\\lb{\|alf;t>} %eq.2\nA(t)\|\\alf;t\\ra = \\alf\|\\alf;t\\ra,\\quad A(t) = u(t)a +\nv(t)a^\\dg = A(u,v).\n\\eeq\n%\nThe operator $A(t)$ was constructed as a non-Hermitian invariant operator\nfor the quantum varying frequency oscillator with Hamiltonian $H =\n\\left(p^2 + m^2\\ome^2(t)q^2\\right)/2m$,\\,\\, i.e. $A(t)$ had to obey the\nequation $\\pd A/\\pd t - (i/\\hbar)[A,H] = 0$ [$m$ is the mass, and $\\ome(t)$\nis the varying frequency; the case of varying mass $m(t)$ was\nreduced to that of constant mass by the time transformation \n$t \\rightarrow t^\\pr = m\\int^t d\\tau/m(\\tau)$]. \nFor that purpose the parameter $\\eps = (u-v)/\\sqrt{\\ome_0}$ was introduced \nand subjected to obey the classical oscillator equation\n\\beq\\lb{eps} %eq.3\n\\ddot{\\eps} + \\ome^2(t)\\eps = 0.\n\\eeq\nThe boson commutation relation $[A,A^\\dg]=1$ was ensured by the Wronskian\n$\\eps^\\dot{\\eps} - \\eps\\dot{\\eps}^ = 2i$. Then $\\dot{\\eps} = i(u+v)\n\\sqrt{\\ome_0}$, $\|u\|^2 - \|v\|^2 = 1$, and the invariant takes the\nform $A(t) = U(t)\\left(u(0)a + v(0)a^\\dg\\right) U^\\dg(t) \\equiv\nU(t)A(0)U^\\dg(t),$ where $U(t)$ is the evolution operator, and the\neigenstates $\|\\alf;t\\ra \\equiv \|\\alf,u(t),v(t)\\ra$ satisfy the Schr\\\"odinger\nevolution equation. One has\n\\beq\\lb{\|alf,u,v>} %eq.4\n\|\\alf,u(t),v(t)\\ra = U(t)\|\\alf,u_0,v_0\\ra,\n\\eeq\nwhere $A(0)\|\\alf,u_0,v_0\\ra = \\alf\|\\alf,u_0,v_0\\ra$ and $\|u_0\|^2 - \|v_0\|^2\n= 1$.\nThis shows that the set $\\{\|\\alf,u(t),v(t)\\ra\\}$ is an orbit through\n$\|\\alf,u_0,v_0\\ra$ of the evolution operator $U(t)$.\n\nIn the coordinate rep the wave functions $\\la q\|\\alf,u(t),v(t)\\ra$\ntake the form of an exponential of a quadratic \\ci{MMT} ($m$ is the mass\nparameter),\n\\bear\\lb{<q\|alf,u,v>} %eq.5\n\\la q\|\\alf,u,v\\ra =\\frac{(m\\ome_0/\\pi\\hbar)^{1/4}}\n{(u-v)^{1/2}}\\hspace{6.5cm}\\nn \\\\\n\\times \\exp\\left[-\\frac{m\\ome_0}{2\\hbar} \\frac{v+u}{u-v}\n\\left(q - \\left(\\frac{2\\hbar}{m\\ome_0}\\right)^{1/2}\\frac{\\alf}{u+v}\\right)^2\n- \\frac 12\\left(-\\frac{u^* + v^}{u+v}\\alf^2 + \|\\alf\|^2\\right)\\right].\n\\eear\nThese wave packets are normalized but not orthogonal to each other. \nThey are\nsolutions to the wave equation for varying frequency oscillator if\n$u=(\\eps\\sqrt{\\ome_0}-i\\dot{\\eps}/\\sqrt{\\ome_0})/2$,\n$v = -(\\eps\\sqrt{\\ome_0}+i\\dot{\\eps}/\\sqrt{\\ome_0})/2$,\nand $\\eps$ is any solution of (\\ref{eps}).\nNote that the time dependence is embedded completely in $u$ and $v$ (or,\nequivalently, in $\\eps$ and $\\dot{\\eps}$) which justifies the notation\n$\|\\alf;t\\ra = \|\\alf,u,v)\\ra$. For other systems the invariant\n$A(t)=U(t)A(0)U^\\dg(t)$ is not linear in $a$ and $a^\\dg$ and its eigenstates\nare no more of the form $\|\\alf,u,v\\ra$ \\ci{Trif93}. Therefore the term\n\"coherent states for the nonstationary oscillator\" for $\|\\alf;t\\ra =\n\|\\alf,u,v\\ra$ \\ci{MMT} is indeed adequate. Time evolution of an initial\n$\|\\alf,u_0,v_0\\ra$ for general quadratic Hamiltonian system was studied in\ngreater detail in \\ci{Yuen}, where eigenstates of $ua + va^\\dg$ were denoted\nas $\|\\alf\\ra_g$. The invariant $A(t)$ in \\ci{MMT} coincides with the boson\noperator $b(t)$ in \\ci{Yuen}.\n\nThe states (\\ref{<q\|alf,u,v>}) represent the time evolution of the canonical\nCS $\|\\alf\\ra$ if the initial conditions \\ci{MMT} $\\eps(0) =\n1/\\sqrt{\\ome_0},\\,\\, \\dot{\\eps}(0) = i\\sqrt{\\ome_0}$ are imposed (then\n$u(0)=1,\\, v(0)=0$). Under these conditions\n$\|\\alf,u(t),v(t)\\ra = U(t)\|\\alf\\ra$,\ni.e. the set of $\|\\alf,u(t),v(t)\\ra$ becomes an $SU(1,1)$\norbit through the initial CS $\|\\alf\\ra$, since the Hamiltonian of the varying\nfrequency oscillator is an element of the $su(1,1)$ algebra\nin the rep with Bargman index $k=1/4,3/4$. The $SU(1,1)$\ngenerators $K_i$ in this rep read\\,\\, ($K_\\pm = K_1\\pm iK_2$)\n\\beq\\lb{1moderep} %eq.6\nK_3 = \\frac 12 a^\\dg a + \\frac 14,\\quad K_- = \\frac 12a^2,\\quad \nK_+ = \\frac 12 a^{\\dg 2}.\n\\eeq\n%\nThe parameters $u,\\, v$ are in a direct link to the $SU(1,1)$ group\nparameters, and $\\alf$ -- to the Heisenberg--Weyl group. The whole family of\n$\|\\alf,u,v\\ra$, can be considered as an orbit through the ground state\n$\|0\\ra$ of the unitary operators of the semidirect product $SU(1,1)\\wedge\nH_1$ \\ci{Trif93}. Thus the two definitions (D1) and (D2) here are\nequivalently generalized. It has been shown \\ci{Trif93} that the third\ndefinition is also equivalently generalized on the basis of the Schr\\\"odinger\nUR (see next section).\n\nThe set $\\{\|\\alf,u,v\\ra,\\,\\,u,v - {\\rm fixed}\\}$ resolves the unity operator\nwith respect to the same measure as in the case (\\ref{1}) of canonical CS\n\\ci{MMT}: $1 = (1/\\pi)\\int d^2\\alf\\,\|\\alf,u,v\\ra\\la v,u,\\alf\|$.\n\nA second family of orthonormalized states $\|n;t\\ra = \|n,u,v\\ra$ was\nconstructed in \\cite{MMT} as eigenstates of the quadratic in $a$ and $a^\\dg$\nHermitian invariant $A^\\dg(t) A(t) = (ua + va^\\dg)^\\dg (ua + va^\\dg)$ which\nis an element of the Lie algebra $su(1,1)$. Note that any power of $A$ and\n$A^\\dg$ is also an invariant. $A^\\dg A$ coincides with the known\nErmakov--Lewis invariant. For the $N$-dimensional quadratic system there\nare $N$ linear in $a_\\mu$ and $a^\\dg_\\mu$ invariants $A_\\mu(t) =\nu_{\\mu\\nu}a_\\nu + v_{\\mu\\nu}a^\\dg_\\nu \\equiv A_\\mu(u,v)$ ($\\mu,\\nu =\n1,2,\\ldots N$), which were simultaneously diagonalized \\ci{HMMT},\n\\beq\\lb{2a} %eq.7\nA_\\mu(u,v)\|\\vec{\\alf},u,v\\ra = \\alf_\\mu\|\\vec{\\alf},u,v\\ra,\n\\eeq\nIn different notations exact solutions to the Schr\\\"odinger equation for the\nnonstationary oscillator have been previously obtained e.g. by Husimi\n\\ci{HCh} and for nonstationary general $N$-dimensional Hamiltonian by\nChernikov \\ci{HCh}, but with no reference to the eigenvalue problem of\nthe invariants $ua+va^\\dg$ and/or $(ua+va^\\dg)^\\dg(ua+va^\\dg)$. Eigenstates\nof other quadratic in $a$ and $a^\\dg$ operators were later considered in\nmany papers, the general one-mode quadratic form being diagonalized by Brif\n(see \\ci{Brif96} and references therein).\n\nBy means of the known BCH formula for the transformation $S(\\z)aS^\\dg(\\z)$\nwith\n$S(\\z) = \\exp[\\z K_+ - \\z^K_-],\\quad K_- =a^2/2,\\,\\, K_+= a^{\\dg 2}/2$,\nthe solutions $\|\\alf,u,v\\ra$ are immediately brought, up to a phase factor,\nto the form of famous Stoler states $\|\\alf,\\z\\ra = S(\\z)\|\\alf\\ra$ \\ci{Stol}:\n\\beq\\lb{Stolform} %eq.8\n\|\\alf,u,v\\ra = e^{i{\\rm arg}\\,u}\\,\\exp(\\z K_+ - \\z^K_-) \|\\alf\\ra,\n\\eeq\nwhere $\|\\z\| ={\\rm arcosh}\|u\|$ and ${\\rm arg}\\,\\z ={\\rm arg}\\,v-{\\rm arg}\\,u$.\nYuen \\ci{Yuen} called the eigenstates $\|\\alf,u,v\\ra$ of $ua+va^\\dg$ {\\it\ntwo photon CS} and suggested that the output radiation of an ideal\nmonochromatic two photon laser is in a state $\|\\alf,u,v\\ra$. In\n\\ci{Hollen} these states were named {\\it squeezed states} (SS) to reflect\nthe property of these states to exhibit fluctuations in $q$ or $p$ less\nthan those in CS $\|\\alf\\ra$. They were intensively studied in quantum\noptics and are experimentally realized (see refs in \\ci{LouKni,ZhaFenGil}).\nThe eigenstates $\|n,u,v\\ra$ of $(ua+va^\\dg)^\\dg(ua+va^\\dg)$ became known as\n squeezed Fock states ($\|n\\!=\\!0,u,v\\ra$ -- squeezed vacuum) and the\noperator $S(\\z)$ -- (canonical) {\\it squeeze operator} \\ci{LouKni,ZhaFenGil}.\nEigenstates $\|\\vec{\\alf},u,v\\ra$, eq. (\\ref{2a}), became known as multimode\n(canonical) SS.\n\nNoting that the variance $(\\Delta X)^2$ of a Hermitian operator $X$ in a\nstate $\|\\Psi\\ra$ equals zero iff $\|\\Psi\\ra$ is an eigenstate of $X$ so it\nwas suggested \\ci{Trif94} to construct SS for {\\it arbitrary two\nobservables} $X_1$ and $X_2$, in analogy to the canonical SS\n$\|\\alf,u,v\\ra$, as eigenstates of their complex combination $\\lam X_1\n+iX_2$, $\\lambda \\in {\\mathbf C}$ (or equivalently $uA + vA^\\dg$,\n$A=(X_1-iX_2)$), since if in such eigenstates $\\lambda \\rar 0$ ($\\lambda\n\\rar \\infty $) then $\\Delta X_2 \\rar 0$ ($\\Delta X_1 \\rar 0$) \\ci{Trif94}.\n\nRadcliffe \\ci{Radclif} and Arecchi et al \\ci{Gilmore} introduced and\nstudied the $SU(2)$ analog $\|\\tet,\\vphi;j\\ra$ of the states\n$\|\\alf\\!=\\!0,u,v\\ra$ in the similar form to that of Stoler states\n(\\ref{Stolform}) (the displacement operator form) ($J_\\pm = J_1\\pm iJ_2$),\n%\n\\beq\\lb{SU2CS} %eq.9\n\|\\tet,\\vphi\\ra = \\exp(\\z J_+ -\\z^J_-)\|j,-j\\ra =\n\\left(\\frac{-1}{1+\|\\tau\|^2}\\right)^j\\, e^{\\tau J_+}\|j,-j\\ra \n\\equiv \|\\tau;j\\ra,\n\\eeq\n%\nwhere $\|j,m\\ra$ ($m=-j,-j+1,\\ldots,j$, $j=1/2,1,\\ldots$) are the standard\nWigner--Dicke states, the operators $J_1$,\n$J_2$ and $J_3$ are the Hermitian generators of $SU(2)$,\n$\\tau = \\exp(-i\\vphi){\\rm tan}(\\tet/2)$, $\\z = (\\tet/2)\\exp(-i\\vphi)$ and\n$\\vphi$ and $\\tet$ are the two angles in the spherical coordinate\nsystem. The system $\\{\|\\tet,\\vphi\\ra\\}$ is overcomplete \\ci{Gilmore},\n\\beq\\lb{9} %eq.10\n1 = [(2j+1)/4\\pi]\\int d\\Ome \|\\tet,\\vphi\\ra\\la\\vphi,\\tet\|,\n\\eeq\nwhere $d\\Ome = \\sin\\tet d\\tet d\\vphi$.\nThe states $\|\\tet,\\vphi\\ra \\equiv \|\\tau;j\\ra$ are known as {\\it spin\nCS} \\ci{Radclif} or {\\it atomic CS} (Bloch states) \\ci{Gilmore}.\n\nThe results of \\ci{Radclif,Gilmore} about the $SU(2)$ CS have been extended\nto the noncompact group $SU(1,1)$ and to any Lie group $G$ as well by\nPerelomov \\ci{Perel}, who succeeded to prove the Klauder suggestion for\nconstruction of overcomplete families of states using unitary irreducible\nreps of a Lie group $G$. If $T(g)$ is an irreducible unitary\nrep of $G$, $\|\\Psi_0\\ra$ is a fixed vector in the rep\nspace, $H$ is stationary subgroup of $\|\\Psi_0\\ra$ (that is $T(h)\|\\Psi_0\\ra =\n\\exp[i\\alf(h)]\|\\Psi_0\\ra$) then the family of states $\|x\\ra = T(s(x))\n\|\\Psi_0\\ra$, where $s(x)$ is a cross section in the group fiber bundle,\n$x\\in {\\cal X} = G/H$, is overcomplete, resolving the unity with respect\nto the $G$-invariant measure on ${\\cal X}$,\n\\beq\\lb{ru} %eq.11\n1 = \\int \|x\\ra\\la x\|d\\mu(x),\\quad d\\mu(g\\cdot x) = d\\mu(x).\n\\eeq\nSuch states were called generalized CS and denoted as CS of the\ntype $\\{T(g),\\Psi_0)$\\} \\ci{Perel}. It is worth noting that an other type of\n\"generalized CS\" was previously introduced by Titulaer and Glauber\n(see the ref. in \\ci{KlaSka}) as the most general states which satisfy the\nGlauber field coherence condition. Therefore we adopt the notion\n\"group-related CS\" for the generalized CS of the type $\\{T(g),\\Psi_0\\}$\n\\ci{KlaSka}. The Perelomov $SU(1,1)$ CS $\|\\z;k\\ra$ for the discrete series\n$D^+(k)$ with the reference vector $\|\\Psi_0\\ra = \|k,k\\ra$\\,\\, ($K_-\|k,k\\ra =\n0$, $K_3\|k,k\\ra = k\|k,k\\ra$) have quite similar form to that of spin CS\n(\\ref{SU2CS}) and Stoler states (\\ref{Stolform}),\n\\beq\\lb{SU11CS} %eq.12\n\|\\z,k\\ra = \\exp(\\z K_+ - \\z^* K_-)\|k,k\\ra = (1-\|\\xi\|^2)^k\\,e^{\\xi\nK_+}\|k,k\\ra \\equiv \|\\xi;k\\ra,\n\\eeq\nwhere $\|\\xi\|= {\\rm tanh}\|\\z\|$, arg$\\xi = -{\\rm arg}\\z+\\pi$. The $SU(1,1)$\nand $SU(2)$ invariant resolution unity measures for these sets of\nstates are ($k\\geq 1/2$) \\ci{Perel}\n\\beq\\lb{measures} %eq.13\nd\\mu(\\xi) = [(2k-1)/\\pi]d^2\\xi/(1-\|\\xi\|^2)^2,\\qquad\nd\\mu(\\tau) = [(2j+1)/\\pi]d^2\\tau/(1+\|\\tau\|^2)^2.\n\\eeq\nThe $SU(1,1)$ reps with $k=1/2$ and $k=1/4$ are not square\nintegrable against the invariant measure $d\\mu(\\xi)$. The whole family of\ncanonical SS $\|\\alf,u,v\\ra$, eqs. (\\ref{\|alf;t>}), (\\ref{\|alf,u,v>}), \nremains stable (up to a phase factor) \nunder the action of unitary operators of the semidirect\nproduct $SU(1,1)\\wedge H_1$. However it does not resolve the identity\noperator with respect to the corresponding $SU(1,1)\\wedge H_1$ invariant\nmeasure \\ci{Trif93}. Noninvariant resolution unity measures for the set of\ncanonical SS were found in \\ci{Trif93,Beckers}.\nThe overcompleteness property of the CS $\|\\tau;j\\ra$ and\n$\|\\xi;k\\ra$ provide the analytic reps in the complex plain and in\nthe unit disk respectively which were successfully used by Brif \\ci{Brif97})\nfor diagonalization of the general complex combinations of the $SU(2)$ and\n$SU(1,1)$ generators. The $SU(1,1)$ analytic rep in the unit disk\nwas also considered in \\ci{Vourd90,BrifBen}.\n\nA lot of attention is paid in the physical literature, especially in\nquantum optics, to the group-related CS for $SU(2)$ and $SU(1,1)$ in their\none- and two-mode boson reps, such as the Schwinger two mode reps (see\n\\ci{KlaSka,LouKni,ZhaFenGil, LuiPer,Vourd90} and references therein), and\nthe one-mode Holstein--Primakoff reps (see e.g. \\ci{Vourd90,VouBrif} and\nreferences therein).\n\nAn extension of the group-related CS, compatible with the resolution of the\nidentity, can be obtained if the stationary subgroup $H\\subset G$ in\nGilmore--Perelomov scheme is replaced by other closed subgroup (references\n[1]-[8] in \\ci{AliAGM}). Significant progress is achieved recently\n\\ci{AliAGM} in the construction of more general type of continuous families\nof states (called also CS \\ci{AliAGM}) which satisfy the generalized\novercompleteness relation $B = \\int \|x\\ra\\la x\|d\\mu(x)$, where $B$ is a\nbounded, positive and invertible operator. When $B=1$ the Klauder definition\nof general CS (overcomplete family of states) \\ci{KlaSka} is recovered. \\\\\n\nAlong the line of generalization of the eigenvalue property (D1) of the\ncanonical CS the next step was made in 1971 by Barut and Girardello in\n\\ci{BG}, where the Weyl lowering generator $K_-$ of $SU(1,1)$ in the\ndiscrete series $D^\\pm(k)$ was diagonalized explicitly,\n\\beq\\lb{BGCS} %eq.14\nK_-\|z;k\\ra = z\|z;k\\ra,\\quad \|z;k\\ra = N_{BG}\\sum_{n = 0}^{\\infty}\n\\frac{z^{n}}{\\sqrt{n!\\Gamma(2k+n)}} \|k,k+n\\rangle.\n\\eeq\nThe family $\\{\|z;k\\ra\\}$ resolves the unity operator,\n$1 = \\int\|z;k\\ra\\la k,z\|d\\mu(z,k)$, the resolution unity measure being\n%%\n\\beq\\lb{BGmeasure} %eq.15\n d\\mu(z,k) = \\frac{2}{\\pi}\n\\left(N_{BG}\\right)^{-2} \|z\|^{2k-1} K_{2k-1}(2\|z\|)\\, d^{2} z,\n\\eeq\nwhere $K_\\nu(x)$ is the modified Bessel function of the third kind \n\\ci{Stegun}. The identity operator resolution (\\ref{BGmeasure}) provides a\nnew analytic rep in Hilbert space \\ci{BG}. The measure $d\\mu(z,k)$, eq.\n(\\ref{BGmeasure}), is {\\it not invariant} under the action of the $SU(1,1)$\non ${\\mathbf C} \\ni z$. In the Barut--Girardello (BG) rep states $\|\\Psi\\ra$\nare represented by functions $F_{BG}(z) = \\la k,z^{\\ast}\|\\Psi\\ra/ \nN_{BG}(\|z\|,k)$ which are of the growth $(1,1)$. The\northonormalized states $\|k,k+n\\ra$ are represented by monomials\n$z^n/\\sqrt{n!(2k)_n}$, $(2k)_n = \\Gamma(2k+n)/\\Gamma(2k)$. The $SU(1,1)$\ngenerators $K_\\pm$ and $K_3$ act in the space ${\\cal H}_k$ of analytic\nfunctions $F_{BG}(z)$ as linear differential operators\n\\beq\\lb{BGrep} %eq.16\nK_{+} = z, \\quad K_{-} = 2k \\frac{d}{dz} + z \\frac{d^{2}}{dz^{2}},\n \\quad K_{3} = k + z \\frac{d}{dz}.\n\\eeq\nOriginally established for the discrete series $D^+(k)$, $k=1/2,1,\\ldots$ the\nBG rep is in fact valid for any positive index $k$.\nRecently this rep has been used to diagonalize the complex\ncombination $uK_- + vK_+$ of the Weyl operators $K_\\pm$ \\ci{Trif94} and the\ngeneral element of $su(1,1)$ as well \\ci{Trif96,Trif97,Brif96,Brif97}. The\nrelations between BG rep and the Fock-Bargmann analytic rep (also\ncalled canonical CS rep) have been established in \\ci{BVM} (the\ncase of $k=1/4,\\,3/4$) and \\ci{Trif98a} (the cases of $k = 1/2,1,3/2,\n\\ldots$).\nThe BG-type analytic rep was recently extended to the\nalgebras $u(N,1)$ \\ci{FujFun} and $u(p,q)$ in their boson realizations\n\\ci{Trif98a}. The BG-type CS for these and any other (noncompact) semisimple\nLie algebra are defined \\ci{Trif98a} as common eigenstates of the mutually\ncommuting Weyl ladder operators.\n\nThe BG CS $\|z;k\\ra$ can be also defined according to the third definition\n(D3) on the basis of the Heisenberg relation for $K_1$ and $K_2$. For this\nfamily the generalization of the definition (D2) does not exist \\ci{Trif98b}.\n\nThe ladder operator method was extended to the deformed quantum oscillator\nin \\ci{Bied}, where the $q$-deformed boson annihilation operator $a_q$,\n%\n\\beq\\lb{a_q} %eq.17\na_q a^\\dg_q - q a^\\dg_q a_q = q^{-\\hat{n}},\\quad [\\hat{n},a^\\dg_q]=a^\\dg_q,\n\\quad q > 0,\n\\eeq\nhas been diagonalized, the eigenstates $\|\\alf\\ra_q$ being called\n\"$q$-CS\" or CS for the quantum Heisenberg--Weyl group $h_q(1)$,\n\\beq\\lb{\|alf>_q} %eq.18\n\|\\alf\\ra_q = {\\cal N}\\exp_q(\\alf a_q^\\dg)\|0\\ra =\n{\\cal N}\\sum_n^\\infty\\frac{\\alf^n}{\\sqrt{[n]_q!}}\|n\\ra,\\quad\n{\\cal N}=\\exp_q(-\|\\alf\|^2),\n\\eeq\nwhere $\\exp_q(x) = \\sum x^n/[n]_q!$,\\,\\,\\, $[n]_q! = [1]_q\\ldots[n]_q$, \n$a^\\dg a\|n\\ra = n\|n\\ra$ (and $a^\\dg_q a_q\|n\\ra = [n]_q\|n\\ra$). \nThe \"classical limit\" is obtained at $q=1$: $a_{q=1} = a$.\nThe $q$-SS have been constructed in the first paper of \\ci{Solom} as states\n$\|v\\ra_q$ annihilated by the linear combination $a_q + va^\\dg_q$, in\nanalogy to the case of canonical squeezed vacuum states $\|\\alf\\,=\\,0,u,v\\ra$:\n$(a_q + va^\\dg_q)\|v\\ra_q = 0$. It was noted \\ci{Solom} that both $q$-CS and\n$\|v\\ra_q$ can exhibit squeezing in the quadratures of the (ordinary) boson\noperator $a$. Group-related type CS associated with the $q$-deformed\nalgebras $su_q(2)$, $[J_-(q),J_+(q)] = - [2J_3]_q,\\quad [J_3,J_\\pm(q)] = \\pm\nJ_\\pm(q),\\lb{su2_q}$, and $su_q(1,1)$, $[K_-(q),K_+(q)] = [2K_3]_q,\\quad\n[K_3,K_\\pm(q)] = \\pm K_\\pm(q)$, in their Holstein--Primakoff realizations in\nterms of $a_q$,\n\n\\begin{eqnarray} %eq.19 %eq.20\nJ_-(q) = a_q\\sqrt{[-\\hat{n}+ 2\\k+1]_q},\\quad J_+(q) =\n\\sqrt{[-\\hat{n}+2\\k+1]_q}\\, a^\\dg_q,\\quad J_3 = \\hat{n} -\n\\k,\\lb{su2_qHP}\\\\[2mm]\nK_-(q) = a_q\\sqrt{[\\hat{n}+ 2\\k - 1]_q},\\quad K_+(q) =\n\\sqrt{[\\hat{n}+2\\k-1]_q}\\, a^\\dg_q,\\quad K_3 = \\hat{n}+\\k,\\lb{su11_qHP}\n\\end{eqnarray}\nwere constructed and discussed in \\ci{Kulish,Solom} ($\\k=1/2$ in \\ci{Kulish}\nand any $\\k$ in \\ci{Solom}). Here $[x]_q \\equiv (q^x-q^{-x})/(q-q^{-1})$.\nThese $su(2)$ and $su(1,1)$ $q$-CS are defined similarly to the ordinary\ngroup-related CS (\\ref{SU2CS}) and (\\ref{SU11CS}) with $J_i$, $K_i$, $n!$ and\n$(x)_n$ replaced by their $q$-generalizations \\ci{Kulish,Solom}. Their\novercompleteness relations (in terms of the Jackson $q$-integral) can be\nfound in \\ci{Ellin}, the corresponding resolution unity measures being the\n$q$-deformed versions of $d^2\\alf$ and (\\ref{measures}): $d\\mu_q(\\alf) =\nd^2_q\\alf/\\pi$,\n\\beq\\lb{q-measures} %eq.21\nd\\mu_q(\\tau)=\\frac{[2j+1]_q}{ _q\\la j;\\tau\|\\!\|\\tau;j\\ra_q^2}d^2_q\\tau,\\quad\nd\\mu_q(\\xi) = \\frac{[2k-1]_q}{_q\\la k;\\xi\|\\!\|\\xi;k\\ra_q^{-2}}d^2_q\\xi,\n\\eeq\nwhere $\|\\!\|\\tau;j\\ra_q = \\exp_q(\\tau J_+(q))\|j,-j\\ra$, \n $\|\\!\|\\xi;j\\ra_q = \\exp_q(\\xi K_+(q))\|k,k\\ra$. \nThe Barut-Girardello $q$-CS (eigenstates of $K_-(q)$) are constructed in the\nfirst paper of \\ci{Kulish}. The ladder operator formalism for several kinds\nof one- and two-mode boson states is considered recently in \\ci{Wang}. For\nfurther development in the field of $q$-deformed CS see e.g. \\ci{Ellin, \nSolom3Oh}. For CS related to supergroups ({\\it super-CS}) see e.g. \\ci{Fatyga}.\nThe canonical SS can be regarded as super-CS related to the orthosymplectic\nsupergroup $OSp(1/2,R)$ \\ci{Trif92}.\n\n\n%%Section 3\n\\section{The Uncertainty Way}\n\n %subsection 3.1\n \\subsection{The Heisenberg and the Schr\\\"odinger UR}\n\nCanonical CS $\|\\alf\\ra$ (and only they) minimize the Heisenberg uncertainty\nrelation with equal uncertainty of the two (dimensionless) canonical\nobservables $p$ and $q$: in $\|\\alf\\ra$ the two variances are equal and\n$\\alf$- independent, $(\\Dlt p)^2 = 1/2 = (\\Dlt q)^2$. $1/2$ is the lowest\nlevel at which the equality $(\\Dlt p)^2 = (\\Dlt q)^2$ can be maintained.\nTherefore the set of $\|\\alf\\ra$ is {\\it the set of $p$-$q$ minimum\nuncertainty states}. The CS related to any other two (or more) {\\it\nnoncanonical observables} $X_1$ and $X_2$ {\\it are not} with minimal and\nequal uncertainties -- the lowest level of the equality $(\\Dlt X_1)^2 =\n(\\Dlt X_2)^2$ can be reached on some subsets only. For example, in the\n$SU(1,1)$ CS $\|\\xi;k\\ra$ the variances of the generators $K_1$ and $K_2$ for\n$\\xi\\neq 0$ are always greater than their value in the lowest weight vector\nstate $\|k,k\\ra$: $\\Dlt K_{1,2} (\\xi) > \\Dlt K_{1,2} (0) = \\sqrt{k/2}$\n\\ci{Trif94}. The Heisenberg inequality for $K_1$ and $K_2$ is minimized in\nthe subsets of states with Re$\\xi=0$ and/or Im$\\xi=0$ only, but the\nuncertainties $\\Dlt K_{1}(\\xi)$ and $\\Dlt K_2(\\xi)$ (calculated in\n\\ci{NikTrif}) are never equal unless $\\xi = 0$. Similar is the uncertainty\nstatus of the spin CS ($SU(2)$ related CS) $\|\\tau;j\\ra$.\n\nIt turned out \\ci{Trif94} that the above $SU(1,1)$ and $SU(2)$ group related\nCS minimize, for any values of the parameters $\\xi$ and $\\tau$, the more\nprecise uncertainty inequality of Schr\\\"odinger (called also\nSchr\\\"odinger--Robertson inequality) \\ci{SchRob},\n\\beq\\lb{SUR} %eq.22\n(\\Dlt X_1)^2 (\\Dlt X_2)^2 \\geq \\frac 14\\left\|\\la[X_1,X_2]\\ra\\right\|^2 +\n(\\Dlt X_1X_2)^2,\n\\eeq\nwhere $\\la X\\ra$ is the mean value of $X$, and $\\Dlt X_1X_2 \\equiv \\la\nX_1X_2+X_2X_1\\ra/2 -\\la X_1\\ra \\la X_2\\ra$ is the covariance of $X_1$ and\n$X_2$. However the sets of states which minimize (\\ref{SUR}) for $K_{1,2}$\nand $J_{1,2}$ are much larger than the sets of the corresponding\ngroup-related CS $\|\\xi;k\\ra$ and $\|\\tau;j\\ra$ -- these larger sets have\nbeen constructed in \\ci{Trif94} as eigenstates of the general {\\it\ncomplex} combinations of the ladder operators $K_\\pm$ and $J_\\pm$\ncorrespondingly since the necessary and sufficient condition for a state\n$\|\\Psi\\ra$ to minimize (\\ref{SUR}) was realized to be the eigenvalue\nequation\n\\beq\\lb{OUS} %eq.23\n[u(X_1-iX_2) + v (X_1 + iX_2)]\\,\|\\Psi\\ra = z\|\\Psi\\ra.\n\\eeq\nThe minimizing states should be denoted by $\|z,u,v;X_1,X_2\\ra$ and called\nSchr\\\"odinger {\\it $X_1$-$X_2$ optimal uncertainty states} (optimal US). The\nother names already used in the literature are {\\it generalized (or\nSchr\\\"odinger) intelligent states} \\ci{Trif94,Trif96}, {\\it correlated CS}\n\\ci{DKM} and {\\it Schr\\\"odinger minimum uncertainty states} \\ci{Trif93}. The\nminimization of the inequality (\\ref{SUR}) for canonical $p$ and $q$ was\nconsidered in detail in \\ci{DKM}, where the minimizing states were called\n{\\it correlated CS}. The latter coincides with the canonical SS\n$\|\\alf,u,v\\ra$ \\ci{Trif93}. In the optimal US the uncertainties $\\Dlt X_1$,\n$\\Dlt X_2$ are minimal in the case of $X_1=p$, $X_2=q$ only. Therefore the\nfrequently used term \"minimum uncertainty states\"\n\\ci{Trif93,Brif96,Trif96,HilNag,Nie98,LuiPer}\nis generally not in its direct meaning. The term {\\it intelligent states}\nwas introduced in \\ci{ArChalSal} on the example of Heisenberg inequality for\n$J_{1,2}$. States $\|\\Psi\\ra$ for which the product functional\n$U[\\Psi]\\equiv (\\Dlt X_1)^2(\\Dlt X_2)^2$ is stationary under arbitrary\nvariation of $\|\\Psi\\ra$ \\ci{Jackiw} were called by Jackiw {\\it critical}.\nObviously there is no commonly accepted name for the states which minimize\nan uncertainty inequality -- the \"optimal uncertainty states\" is one more\nattempt in searching for more adequate name.\n\nIn the solutions $\|z,u,v;X_1,X_2\\ra$ to (\\ref{OUS}) the three second moments\nof $X_1$ and $X_2$ are expressed in terms of the mean of their commutator\n\\ci{Trif94} (note that in \\ci{Trif94} $\\lambda,\\,z^\\pr$ parameters were used\ninstead of $u,v,z$: $\\lambda = (v+u)/(v-u),\\,\\,z^\\pr = z/(v-u)$),\n\\beq\\lb{OUSmeans} %eq.24\n\\left.\\begin{tabular}{ll}\n$\\displaystyle\n(\\Dlt X_1)^2 = \\frac{\|u-v\|^2}{\|u\|^2-\|v\|^2}\\, C_{12},\\quad\n(\\Dlt X_2)^2 = \\frac{\|u+v\|^2}{\|u\|^2-\|v\|^2}\\, C_{12},$ & \\\\[5mm]\n$\\displaystyle\n\\Dlt X_1X_2 = \\frac{2{\\rm Im}(u^v)}{\|u\|^2-\|v\|^2}\\,C_{12},\\qquad C_{12}\n= \\frac i2\\la[X_1,X_2]\\ra.$ &\n\\end{tabular} \\right\\}\n\\eeq\nThese moments satisfy the equality in (\\ref{SUR}) identically with respect\nto $z,u,v$. From $(\\Dlt X)^2 \\geq 0$ and (\\ref{OUSmeans}) it follows that if\nthe commutator $i[X_1,X_2]$ is positive (negative) definite then normalized\neigenstates of $u(X_1-iX_2) + v (X_1 + iX_2)$ exist for $\|u\|>\|v\|$\n($\|u\|<\|v\|$) only \\ci{Trif94}. In such cases one can rescale the parameters\nand put $\|u\|^2-\|v\|^2 =1$ ($\|u\|^2-\|v\|^2 =-1$) as one normally does in the\ncanonical case of $X_1=p,\\,X_2=q$.\n\nIn order to establish the connection of $K_1$-$K_2$ and $J_1$-$J_2$\noptimal US $\|z,u,v;K_1,K_2\\ra \\equiv \|z,u,v;k\\ra$ and $\|z,u,v;J_1,J_2\\ra\n\\equiv \|z,u,v;j\\ra$ with the displacement operator method consider the\noperators\n\\bear\\lb{displaceoper} %eq.25 %eq.26\nK^\\pr_3 = \\frac{i}{2}\\sqrt{uv}\\left(uK_- + vK_+\\right),\\quad\nK^\\pr_{\\pm} = iK_3 \\mp \\left(\\sqrt{u/v}\\,K_-\n-\\sqrt{v/u}\\,K_+\\right),\\\\\nJ^\\pr_3 = \\frac{1}{2}\\sqrt{uv}\\left(uJ_- + vJ_+\\right),\\quad\nJ^\\pr_{\\pm} = J_3 \\mp \\left(\\sqrt{u/v}\\,J_- -\\sqrt{v/u}\\,J_+ \\right),\n\\eear\nwhich realize non-Hermitian reps of the algebras $su(1,1)$ and $su(2)$ with\nthe same indices $k$ and $j$. Therefore $(K^\\pr_\\pm)^n$ \\,$\\left((J^\\pr_\\pm)\n^n\\right)$ displace the eigenvalue $z$ of $uK_- + vK_+$\\, ($uJ_- + vJ_+$) by\n$\\pm n$. If one could properly define noninteger powers of $K^\\pr_\\pm$\\,\n($J^\\pr_\\pm$) (to be considered elsewhere) one might write $\|z,u,v;k\\ra =\n{\\cal N}_1(K^\\pr_\\pm)^z\|0,u,v;k\\ra$\\, ($\|z,u,v;j\\ra = \n{\\cal N}_2(J^\\pr_\\pm)^z\|0,u,v;j\\ra$), where ${\\cal N}_{1,2}$ are\nnormalization constants. \nIn slightly different notations the operators $J^\\pr_3,\\,J^\\pr_\\pm$ were\nintroduced by Rashid \\ci{Rashid}. \n\nAn important physical property of the states $\|z,u,v;X_1,X_2\\ra$\nis that they can exhibit arbitrary strong squeezing of the\nvariances of $X_1$ and $X_2$ when the parameter $v$ tend to $\\pm u$, i.e.\n$\\Dlt X_{1,2} \\lrar 0$ when $v \\lrar \\pm u$ \\ci{Trif94}. Therefore the\nfamilies of $\|z,u,v;X_1,X_2\\ra$ are the {\\it $X_1$-$X_2$ ideal SS}.\nThe canonical SS $\|\\alf,u,v\\ra$ are $p$-$q$ ideal SS, while the\ngroup-related CS $\|\\tau;j\\ra$ and $\|\\xi;k\\ra$ are not. \nExplicitly the families of $\|z,u,v;X_1,X_2\\ra$ are constructed for the\ngenerators $K_i$-$K_j$ and $J_i$-$J_j$ of $SU(1,1)$\n\\ci{Trif94,Trif96,Brif97} and $SU(2)$ \\ci{ArChalSal, Rashid,Brif97} (in\n\\ci{ArChalSal,Rashid} with no reference to the inequality (\\ref{SUR})). It\nis worth noting an important application of the $K_i$-$K_j$ and $J_i$-$J_j$\noptimal US (intelligent states) in the quantum interferometry: the $SU(1,1)$\nand $SU(2)$ optimal US which are not group-related CS can greatly improve\nthe sensitivity of the $SU(2)$ and $SU(1,1)$ interferometers as shown by\nBrif and Mann \\ci{LuiPer}. Schemes for generation of $SU(1,1)$ and $SU(2)$\noptimal US of radiation field can be found e.g. in \\ci{Trif98b,LuiPer}.\n\nSchr\\\"odinger optimal US can be constructed also for the two Hermitian\nquadratures $K_1(q),\\,K_2(q)$ ($J_1(q),\\,J_2(q)$) of the ladder\noperators of $q$-deformed $su_q(1,1)$ ($su_q(2)$). Let us consider here\nthe case of $su_q(1,1)$. The $K_1(q)$-$K_2(q)$ optimal US $\|z,u,v;k\\ra_q$\nhave to obey (\\ref{OUS}) with $X_1=K_1(q)$ and $X_2=K_2(q)$. We put\n\\beq\\lb{\|zuv;k>q} %eq.27\n\|z,u,v;k\\ra_q = {\\cal N}_q\|\\!\|z,u,v;k\\ra_q = {\\cal N}_q\\sum_n g_n(z,u,v,q,k)\n\|k,k+n\\ra,\n\\eeq\nand substitute this in (\\ref{OUS}). Using the actions\n$K_-(q)\|k,k+n\\ra = \\sqrt{[n][2k+n-1]}\|k,k+n-1\\ra$, and\n$K_+(q)\|k,k+n\\ra = \\sqrt{[n+1][2k+n]}\|k,k+n-1\\ra$\nwe get the recurrence relations for $g_n$,\n\\beq\\lb{q_recurrence} %eq.28\nu\\sqrt{[n+1][2k+n]}\\,g_{n+1} + v\\sqrt{[n+1][2k+n]}\\,g_{n-1} = zg_n.\n\\eeq\nThe solution $g_n(z,v,u,q,k)$ to these recurrence relations is a polynomial\nin $z/u$ and $v/u$,\n\\beq\\lb{g_n} %eq.29\ng_n(z,u,v,q,k) = \\sum_{m=0}^{{\\rm int}(n/2)}\np_{n,m}(k,q)\\left(\\frac{z}{u}\\right)^{n-2m}\\left(-\\frac{v}{u}\\right)\n^{m},\n\\eeq\nwhere int$(n/2)$ is the integer part of $n/2$. The particular case of $v=0$\nwas solved in \\ci{Kulish}, $g_n(z,q,k) = z^n/\\sqrt{[n]!([2k])_n}$. Here we\nwright down the solution for the subset of $z=0$,\n\\beq\\lb{z=0} %eq.30\ng_{2n+1}(u,v,q) = 0,\\quad g_{2n}(u,v,q) = \\left(-\\frac{v}{u}\\right)^n\n\\left(\\frac{[2n-1]!!\\,(\\!([2k])\\!)_{2n}}{[2n]!!\\,(\\!([2k+1])\\!)_{2n}}\\right)^\n{\\frac 12},\n\\eeq\nand for $q=1$,\n\\beq\\lb{q=1} %eq.31\ng_n(z,u,v,k) = \\left(-\\frac{l(u,v)}{2u}\\right)^n \\sqrt{\\frac{(2k)_n}{n!}}\\,\n_2F_1\\left(k+\\frac{z}{l(u,v)},-n;2k;2\\right),\n\\eeq\nwhere $l(u,v) = 2\\sqrt{-uv}$, $(\\!([x])\\!)_{2n} = [x][x+2]\\ldots[x+2n-2]$\nand $_2F_1(a,b;c;z)$ is the Gauss hypergeometric function. The normalization\ncondition is $\|v\| < \|u\|$. The BG CS are recovered at $v=0,\\,u=1$. The\nconstruction of $g_n(z,u,v,q,k)$ in the general case is postponed until the\nnext publication.\n%%\\vs{5mm}\n\n%subsection 3.2\n\\subsection{ The Robertson Inequality and the Characteristic UR}\n\nCompared to the Heisenberg uncertainty relation the Schr\\\"odinger one, eq.\n(\\ref{SUR}), has the important advantage to be {\\it invariant} under\nnondegenerate linear transformations of the two observables involved.\nIndeed the relation (\\ref{SUR}) can be rewritten in the following invariant\nform \\ci{Rob} ${\\rm det}\\,\\sigma(\\vec{X}) \\geq {\\rm det}\\, C(\\vec{X})$,\nwhere $\\vec{X}$ is the column of $X_1$ and $X_2$, $\\vec{X} = (X_1,X_2)$, and\n\\beq\\lb{sigma,C} %eq.32\nC(\\vec{X}) =\n-\\frac{i}{2}\\left(\\matrix{\\,\\,\\,0\\quad \\quad \\la[X_1,X_2]\\ra\\\\[2mm]\\cr\n\\la[X_2,X_1]\\ra\\quad \\,\\,\\,0\\quad}\\right),\n\\qquad\n\\sigma(\\vec{X}) = \\left(\n\\matrix{\\Dlt\\,\\!X_1X_1 \\quad\\Dlt\\,\\!X_1X_2\\\\[2mm]\\cr\n\\Dlt\\,\\!X_2X_1\\quad \\Dlt\\,\\!X_2X_2}\\right).\n\\eeq\n$\\sigma(\\vec{X})$ is called the uncertainty (the dispersion) matrix for\n$X_1$ and $X_2$. In order to symmetrize notations we have denoted in\n(\\ref{sigma,C}) the variance $(\\Dlt X_i)^2$ as $\\Dlt X_iX_j$. So\n$\\sigma_{ij} = \\Dlt X_i X_j$ and $C_{kj} = -(i/2)\\la[X_k,X_j]\\ra$. Under\nlinear transformations $\\vec{X} \\longrightarrow \\vec{X}^\\pr = \\Lam \\vec{X}$,\nwe have\n\\beq\\lb{sigma,C-prim} %eq.33\n\\sig^\\pr\\equiv \\sig(\\vec{X}^\\pr) = \\Lam\\sig\\Lam^T,\\qquad\nC^\\pr \\equiv C(\\vec{X}^\\pr) = \\Lam C\\Lam^T.\n\\eeq\nIt is now seen that if the transformation is non-degenerate, det$\\Lam \\neq\n0$, then the equality in the relation (\\ref{SUR}) remains invariant, i.e.\n$\\det\\sig= \\det C\\,\\,\\longrightarrow \\,\\, \\det\\sig^\\pr = \\det C^\\pr$. This\nimplies that in the canonical case of $X_1=p$, $X_2=q$ the equality in\n(\\ref{SUR}) is invariant under linear canonical transformations. The\nequality in the Heisenberg relation is not invariant under linear\ntransformations.\n\n In the Heisenberg and the Schr\\\"odinger inequalities the second moments of\n{\\it two} observables $X_{1,2}$ are involved. However two operators never\nclose an algebra [An exception is the Heisenberg--Weyl algebra $h_1$ due to\nthe fact that the third operator closing the algebra is the identity\noperator: the equality in the $p$-$q$ Schr\\\"odinger relation (but not in the\nHeisenberg one) is invariant under the linear canonical transformations].\nTherefore the equality in these uncertainty relations is not invariant under\nthe general transformations in the algebra to which $X_{1,2}$ may belong.\nFor $n$ generators of Lie algebras it is desirable to have uncertainty\nrelations invariant under algebra automorphisms, in particular under the\ncorresponding Lie group action in the algebra.\n\nSuch invariant uncertainty relations turned out to be those of Robertson\n\\ci{Rob} and of Trifonov and Donev \\ci{TriDon}. The Robertson relation for\n$n$ observables $X_1,X_2,\\ld X_n$ reads ($i,j,k = 1,2,\\ld n$)\n\\beq\\lb{RUR} %eq.34\n{\\rm det}\\,\\sigma(\\vec{X})\\,\\, \\geq \\,\\,{\\rm det}\\, C(\\vec{X}),\n\\eeq\nwhere $\\sigma_{ij} = \\Dlt X_i X_j$, and $C_{kj} = -i\\la[X_k,X_j]\\ra/2$.\nWith minor changes the Robertson proof of (\\ref{RUR}) is provided in the\nAppendix. The minimization of (\\ref{RUR}) is considered in detail in\n\\ci{Trif97}, the minimizing states being called Robertson intelligent\nstates or Robertson {\\it optimal US}. A pure state minimize (\\ref{RUR}) if\nit is an eigenstate of a real combination of the observables. For odd $n$\nthis is also a necessary condition. Robertson optimal US \nexist for a broad class of observables, the simplest example being given\nby the well known $N$-modes Glauber CS $\|\\vec{\\alf}\\ra = \|\\alf_1\\ra\\,\n\|\\alf_2\\ra,\\ld \|\\alf_N\\ra$, and by the $N$-modes canonical SS\n$\|\\vec{\\alf},u,v\\ra$ (constructed in \\ci{MMT,HMMT} with no reference to \nthe Robertson relation). A more general examle is given by the group-related\nCS $\\{T(g),\\Psi_0\\}$ when $\|\\Psi_0\\ra$ is eigenstate of a (real) Lie algebra\nelement \\ci{Trif97}. If in addition $\|\\Psi_0\\ra$ is the lowest (highest)\nweight vector (the case of semisimple Lie groups \\ci{ZhaFenGil}) then these\nCS minimize (\\ref{RUR}) for the Hermitian components of Weyl generators as\nwell \\ci{Trif97}. On the example of the $SU(2)$ and $SU(1,1)$ CS, eqs.\n(\\ref{SU2CS}) and (\\ref{SU11CS}), the above minimization properties can be\nchecked by direct calculations. In the case of one-mode and two-mode boson\nrepresentations of $su(1,1)$ the above properties mean that squeezed Fock\nstates minimize (\\ref{RUR}) for the three generators $K_i$, but \nsqueezed vacuum in addition minimizes (\\ref{SUR}) for $K_1$ and $K_2$. \n\nThe number of the Hermitian components of Weyl generators (of a semisimple\nLie group) is even. For the even number $n$ of observables the Robertson\ninequality (\\ref{RUR}) is minimized in a state $\|\\Psi\\ra$ if the latter is an\neigenstate of $n/2$ complex linear combinations of $X_j$. For these\nminimizing states the second moments of $X_i,\\,X_j$ can be expressed in\nterms of the first moments of their commutators. In that purpose and\nkeeping the analogy to the case of canonical SS (\\ref{2a}) we define\n$\\tilde{a}_\\mu = X_\\mu +iX_{\\mu+N}$ and write down the $n/2\\equiv N$ complex\ncombinations as ($\\mu,\\nu = 1,2,\\ld, N$)\n\\beq\\lb{} %eq.35\nA_\\mu(u,v) := u_{\\mu\\nu}\\tilde{a}_\\nu +\nv_{\\mu\\nu}\\tilde{a}^\\dg_\\nu = \\beta_{\\mu j}X_j ,\n\\eeq\nwhere $\\beta_{\\mu\\nu} = u_{\\mu\\nu}+v_{\\mu\\nu}$,\\, $\\beta_{\\mu,s+\\nu} =\ni(u_{\\mu\\nu} - v_{\\mu\\nu})$.\nThen after some algebra we get that in the eigenstates $\|\\vec{z},u,v\\ra$ of\n$A_\\mu(\\beta)$ the following general formula holds,\n\\bear\\lb{nOUSmeans} %eq.36\n\\sig(\\vec{X};z,u,v) = {\\cal B}^{-1}\\left( \\bt{cc} $0$ & $ \\td{C} $\\\\\n$\\td{C}^{\\rm T}$& $0$ \\et \\right)\n{\\cal B}^{-1}{}^{\\rm T},\\\\[2mm]\n\\td{C}_{\\mu\\nu} = \\frac 12\\la[A_\\mu,A_\\nu^\\dg]\\ra,\\quad \n{\\cal B} = \\left(\\bt{cc} $u+v$&$i(u-v)$\\\\\n$u^ + v^$&$i(v^ - u^)$\\et\\right). \\nn\n\\eear\nNote that $u,\\,v$ and $\\td{C}$ are $N\\times N$ matrices, $\\beta$ is an\n$N\\times n$ matrix, while ${\\cal B}$ is $n\\times n$. We suppose that\n${\\cal B}$ is not singular. For two observables, $n=2$, we have\n$\\beta_{11} = u+v$, $\\beta_{12}= i(u-v)$ and formula (\\ref{nOUSmeans})\nrecovers (\\ref{OUSmeans}).\n\nThe Robertson inequality relates the determinants of two $n\\times n$\nmatrices $\\sig$ and $C$. These are the highest order {\\it characteristic\ncoefficients} of the two matrices \\ci{Gantmaher} which are invariant under\nsimilarity transformations of the matrices. Then from (\\ref{sigma,C-prim})\nwe see that $\\det\\sig$ and $\\det C$ are invariant under the orthogonal\ntransformations of the observables. However, one can see, again from the\ntransformation law (\\ref{sigma,C-prim}), that the equality in (\\ref{RUR}) is\ninvariant under {\\it any} nondegenerate linear transformations of the\n$n$ observables. Now we recall \\ci{Gantmaher} that for an $n\\times n$ matrix\n$M$ there are $n$ invariant characteristic coefficients $C_r^{(n)}$, $r=\n1,2,\\ld, n$, defined by means of the secular equation \n\\beq\\lb{chareqn} %eq.37\n0 = \\det(M - \\lam) = \\sum_{r=0}^{n} C^{(n)}_r(M)(-\\lam)^{n-r}.\n\\eeq\n\nThe characteristic coefficients $C^{(n)}_r$ are equal to the\nsum of all principle minors ${\\cal M}(i_1,\\ldots,i_r;M)$ of order $r$. One\nhas $C^{(n)}_0 = 1$, $C^{(n)}_1 = {\\rm Tr}\\,M = \\sum m_{ii}$ and $C^{(n)}_n\n= \\det M$. For $n=3$ we have, for example, three principle minors of order\n$2$. In these notations Robertson inequality (\\ref{RUR}) reads\n$C^{(n)}_n\\left(\\sig(\\vec{X})\\right) \\geq C^{(n)}_n\\left(C(\\vec{X})\\right)$.\nIt is important to note now two points: (1) the uncertainty matrix\n$\\sig(\\vec{X})$ and the mean commutator matrix $C(\\vec{X})$ are nonnegative\ndefinite and such are all their principle minors; (2) The principle minors\nof $\\sig(\\vec{X})$ and $C(\\vec{X})$ of order $r$ can be regarded as\nuncertainty matrix and mean commutator matrix for $r$ observables $X_{i_1},\n\\ld, X_{i_r}$ correspondingly. Then all characteristic coefficients of the\ntwo matrices obey the inequalities \\ci{TriDon}\n\\beq\\lb{CUR} %eq.38\nC^{(n)}_r\\left(\\sig(\\vec{X})\\right) \\,\\geq\\,\nC^{(n)}_r\\left(C(\\vec{X})\\right),\\quad r=1,2,\\ld,n.\n\\eeq\nThese invariant relations can be called {\\it characteristic uncertainty\nrelations}. The Robertson relation (\\ref{RUR}) is one of them and can be\ncalled the $n^{\\rm th}$-order characteristic inequality.\n\nThe minimization of the {\\it first order} inequality in (\\ref{CUR}),\nTr$\\,\\sig(\\vec{X}) = {\\rm Tr}\\,C(\\vec{X})$, can occur in the case of\ncommuting operators only since Tr\\,$C(\\vec{X})\\equiv 0$. Important examples \nof minimization of the {\\it second order} inequality were pointed out in\n\\ci{TriDon} -- the spin and quasi spin CS $\|\\tau;j\\ra$ and $\|\\xi;k\\ra$\nminimize the second order characteristic inequality for the three\ngenerators $J_{1,2,3}$ and $K_{1,2,3}$ correspondingly. We have already\nnoted that these group-related CS minimize the {\\it third order}\ninequalities too, so their characteristic minimization \"ability\" is\nmaximal. \nThe analysis of the solutions of the eigenvalue equation $[uK_- + vK_+ +\nwK_3]\\,\|\\Psi\\ra = z\|\\Psi\\ra$ shows (see Appendix) that the CS $\|\\xi;k\\ra$\nare {\\it the unique states} which minimize simultaneously the second and the\nthird order characteristic inequalities for $K_{1,2,3}$ and there are no\nstates which minimize the second order inequality only. Thus the\nminimization of the characteristic inequalities (\\ref{CUR}) of order $r < n$\ncan be used for {\\it finer classification} of group-related CS with\nsymmetry. It turned out (see the Appendix) that the uniqueness of these \nstates follows also \nfrom the requirement to minimize simultaneously (\\ref{RUR}) for the three\ngenerators and (\\ref{SUR}) for the Hermitian components of $K_-$. \\\\[0mm]\n\nAll the above characteristic inequalities \\footnote{Let us note that other\ntypes of uncertainty relations, e.g. the entropic and the\nparameter-based ones, are also considered in the literature \\cite{drugi}.}\nrelate combinations \n$C_r^{(n)}(\\sig(\\vec{X};\\rho))$ of second moments of $X_1,\\ld,X_n$ in a\n(generally mixed) state $\\rho$ to the combinations\n$C_r^{(n)}(C(\\vec{X};\\rho))$ of first moments of their commutators in the\nsame state. It turned out that these relations can be extended to the case\nof {\\it several state} in the following way. From the derivation of the\ncharacteristic inequalities (\\ref{CUR}) (see Appendix) one can deduce that\nthey are valid for any nonnegative definite matrix ${\\cal S}+i{\\cal C}$ with\n${\\cal S}$ nonnegative definite and symmetric and ${\\cal C}$ --\nantisymmetric. Well, the finite sum $\\sum_md_m\\sig_m$, $d_m\\geq 0$, of\nnonnegative and symmetric matrices is nonnegative and symmetric, and the\nfinite sum of antisymmetric matrices is again antisymmetric. And if \n$\\sig_m+iC_m\\geq 0$ their finite sum is also nonnegative. Thus we obtain the\n{\\it extended} characteristic uncertainty inequalities \n%%\n\\beq\\lb{extendCUR} %eq.39\nC_r^{(n)}\\left({\\textstyle\\sum_m}d_m\\sig_m\\right) \\,\\geq\\,\nC_r^{(n)}\\left({\\textstyle\\sum_m}d_m C_m\\right),\n\\eeq\nwhere $d_m$ are arbitrary real nonnegative parameters.\n Here $\\sig_m$ and $C_m$, $m=1,2,\\ld$, may be the uncertainty and the mean\ncommutator matrices for $\\vec{X}$ in states $\\rho_m$ or the uncertainty\nand the mean commutator matrices of different sets of $n$ observables\n$\\vec{X}^{(m)}$ in the same state $\\rho$. For $r=n$ in (\\ref{extendCUR}) we\nhave the extension of the Robertson relation to the case of several states\nand/or several sets of $n$ observables. In the first case the\nextension reads\n%%\n\\beq\\lb{extendRUR} %eq.40\n{\\rm det}\\left({\\textstyle\\sum_m}d_m\\sig(\\vec{X},\\rho_m)\\right)\\, \\geq\\,\n{\\rm det}\\left({\\textstyle\\sum_m}d_mC(\\vec{X},\\rho_m)\\right).\n\\eeq\nSince $\\det\\sum \\sig_m\\neq \\sum\\det\\sig_m$ these are indeed {\\it new}\nuncertainty inequalities, which extend the Robertson one to several states.\nWe note that the extended relations (\\ref{extendCUR}), (\\ref{extendRUR}) are\n{\\it invariant} under the nondegenerate linear transformations of the\noperators $X_1,\\ld,X_n$. If the latter span a Lie algebra then we obtain the\ninvariance of (\\ref{extendCUR}) under the Lie group action in the algebra.\nIf for several states $\|\\psi_m\\ra$, $m=1,2,\\ld$, the inequality\n(\\ref{extendRUR}) is minimized, then it is minimized also for the \ngroup-related CS $U(g)\|\\psi_m\\ra$ as well, $U(g)$ being the unitary rep of\nthe group $G$. In the simplest case of two observables $X,\\,Y$ and two\nstates $\|\\psi_{1,2}\\ra$ which minimize Schr\\\"odinger inequality (\\ref{SUR})\neq. (\\ref{extendRUR}) produces\n\\bear\\lb{extendSUR} %eq.41\n\\frac 12 \\left[\\sig_{XX}(\\psi_1)\\sig_{YY}(\\psi_2) +\n\\sig_{XX}(\\psi_2)\\sig_{YY}(\\psi_1)\\right] - \\sig_{XY}(\\psi_1)\\sig_{XY}(\\psi_2)\n\\nn \\\\\n\\geq -\\frac 14\\la\\psi_1\|[X,Y]\|\\psi_1\\ra \\la\\psi_2\|[X,Y]\|\\psi_2\\ra,\n\\eear\nwhere, for convenience, $\\sig_{XX}(\\psi)$ denotes the variance of $X$ in\n$\|\\psi\\ra$ and $\\sig_{XY}(\\psi)$ denotes the covariance. The more\ndetailed analysis (to be presented elsewhere) shows that this\nuncertainty relation holds for every two states. For $\\psi_1=\\psi_2$\nthe {\\it new inequality} (\\ref{extendSUR}) recovers that of Shcr\\\"odinger.\nOne can easily verify (\\ref{extendSUR}) for $p$ and $q$ and any two Fock\nstates $\|n\\ra$ and/or Glauber CS $\|\\alf\\ra$ for example. The relation is\nminimized in two squeezed states $\|\\alf_1,u,v\\ra$ and\n$\|\\alf_2,u,v\\ra$, Im$(uv^)=0$. Looking at (\\ref{extendSUR}) and (\\ref{SUR})\none feels that, to complete the symmetry between states and observables, the\nthird inequality is needed (for one observable and two states), namely\n\\beq\\lb{extendSUR2} %eq.42\n\\sig_{XX}(\\psi_1)\\sig_{XX}(\\psi_2) \\geq\n\\left\|\\la\\psi_2\|X^2\|\\psi_1\\ra\\right\|^2-\\sig_{XX}(\\psi_1)\n\\la\\psi_2\|X\|\\psi_2\\ra^2 - \\sig_{XX}(\\psi_2)\\la\\psi_1\|X\|\\psi_1\\ra^2. \n\\eeq\nRelations (\\ref{SUR}) and (\\ref{extendSUR2}) both follow from the Schwarz\ninequality, while (\\ref{extendSUR}) is different.\\\\\n\nIt is worth noting that every extended characteristic inequality\ncan be written down in terms of two new positive quantities the sum of\nwhich is not greater than unity. Indeed, let us put\n\\beq\\lb{P_r} %eq.43\nC_r^{(n)}(\\sig(\\vec{X},\\rho)) = \\alf_r (1-P_r^2),\n\\eeq\nwhere $0\\leq P_r^2 \\leq 1$ (i.e. $1-P_r^2 \\leq 1$) and $\\alf_r \\neq\n0$. For $r=n$ eq. (\\ref{P_r}) reads (omitting index $r=n$)\ndet$\\sig(\\vec{X},\\rho) = \\alf\\,(1-P^2)$. $\\alf_r$ may be viewed as scaling\nparameters. Then we can put $C_r^{(n)}(C(\\vec{X},\\rho)) = \\alf_r V_r^2$ and\nobtain from (\\ref{CUR}) the inequality for $P_r$ and $V_r$\n\n\\beq\\lb{CUR2} %eq.44\nP_r^2(\\vec{X},\\rho) + V_r^2(\\vec{X},\\rho) \\leq 1,\\quad r=1,\\ld,n.\n\\eeq\nThe equality in (\\ref{CUR2}) corresponds to the equality in (\\ref{CUR}) (or\n(\\ref{extendCUR})). For every set of observables $X_1,\\ld, X_n$ the\nnonnegative quantities $P_r,\\,V_r$ are functionals of the state $\\rho$ (or\nof $\\rho_1,\\rho_2,\\ld$ in the case of extended inequalities\n(\\ref{extendCUR})). These can be called {\\it complementary quantities} and\nthe form (\\ref{CUR2}) of the extended characteristic relations -- {\\it\ncomplementary form}. Let us note that $P_r$ and $V_r$ are not uniquely\ndetermined by the characteristic coefficients of $\\sig$ and $C$. They depend\non the choice of the scaling parameter $\\alf_r$. In the case of bounded\noperators $X_i$ (say spin components) the characteristic coefficients of\n$\\sig$ and $C$ are also bounded. In that case $\\alf_r$ can be taken as the\ninverse maximal value of $C_r^{(n)} (\\sig)$. In the very simple case of one\nstate and two operators with only two eigenvalues each the complementary\ncharacteristic inequality (\\ref{CUR2}) was recently considered in the\nimportant paper by Bjork et al \\ci{Bjork}. In this particular case the\nmeaning of the complementary quantities $P$ and $V$ was elucidated to be\nthat of the {\\it predictability} ($P$) and the {\\it visibility} ($V$) in the\n{\\it welcher weg} experiment \\ci{Bjork}.\n\nFinally we note that as functionals of the states $\\rho$ the characteristic\ncoefficients of positive definite uncertainty matrix $\\sig(\\vec{X})$ (then\nthe coefficients $C_r(\\sig(\\vec{X},\\rho))$ are all positive), can be used\nfor the construction of {\\it distances} between quantum states. One\npossible series of such (Euclidean type) distances\n$D^2_r[\\rho_1,\\rho_2;\\vec{X}]$ is \\ci{TriDon2}\n\\bear\\lb{dist} %eq.45 \nD^2_r[\\rho_1,\\rho_2] = C_r(\\sig(\\vec{X},\\rho_1)) +\nC_r(\\sig(\\vec{X},\\rho_2)) - 2\\left(C_r(\\sig(\\vec{X},\\rho_1))\nC_r(\\sig(\\vec{X},\\rho_2))\\right)^{\\frac 12}\\nn\\\\\n\\times \\, g(\\rho_1,\\rho_2),\\quad\n\\eear\nwhere $g(\\rho_1,\\rho_2)$ is any nonnegative functional of $\\rho_1,\\rho_2$,\nsuch that\\, $0\\leq g(\\rho_1,\\rho_2)\\leq 1$\\,\\, and \\,$\\,\\rho_1 = \\rho_2\n\\Leftrightarrow g=1$.\\, A known simple such functional ($g$-type\nfunctional) is \n$g(\\rho_1,\\rho_2) = \n{\\rm Tr}(\\rho_1\\rho_2)/\\sqrt{{\\rm Tr}(\\rho_1^2) {\\rm Tr}(\\rho_2^2)}$. \nBy means of (\\ref{extendSUR2}) with any observable $X$ such that $X\|\\psi\\ra\n\\neq 0$ (continuous or strictly positive $X$, for example) we can construct a\nnew $g$-type functional \n\\beq\\lb{g[]} %eq.46\ng(\\psi_1,\\psi_2;X) = \\frac{\\left\|\\la\\psi_2\|X^2\|\\psi_1\\ra\\right\|}\n{\\sqrt{\\la\\psi_1\|X^2\|\\psi_1\\ra\\la\\psi_2\|X^2\|\\psi_2\\ra}},\n\\eeq\nwhich can be used for distance constructions, the simplest distance \nbeing $D^2 = 2\\left(1 - g(\\psi_1,\\psi_2;X)\\right)$.\nSeveral other $g$-type functionals are also possible \\ci{TriDon2}.\nThe uncertainty matrix $\\sig(\\vec{X})$ is positive for examples in the case\nof $X_i$ being the quadratures component of $N$ $q$-deformed boson annihilation\noperators $a_{q,\\mu}$ with positive $q$ \\ci{Trif97}.\n\\vs{5mm}\n\n\\section{Conclusion}\n We have briefly reviewed and compared the three ways of generalization of\ncanonical coherent states (CS) with the emphasis laid on the uncertainty (the\nthird) way. The Robertson inequality and the other characteristic relations\nfor several operators \\ci{TriDon} are those uncertainty inequalities which\nbring together the three ways of generalization on the level of many\nobservables. The equalities in these relations for the group generators are\ninvariant under the group action in the Lie algebra. From the Robertson\ninequality minimization conditions \\ci{Trif97} it follows that all\ngroup-related CS whose reference vector is eigenstate of an element of the\ncorresponding Lie algebra do minimize the Robertson relation (\\ref{RUR}).\nThe minimization of the other characteristic inequalities (\\ref{CUR}) can be\nused for {\\it finer classification} of group-related CS with symmetry. Along\nthese lines we have shown that $SU(1,1)$ CS with lowest weight reference\nvector $\|k,k\\ra$ are the unique states which minimize the second order\ncharacteristic inequality for the three $SU(1,1)$ generators. Also, these\nare the unique states to minimize simultaneously the Robertson inequality\nfor the three generators and the Schr\\\"odinger one for the Hermitian\ncomponents of the ladder operator $K_-$. These statements are valid for \nthe $SU(2)$ CS with the lowest (highest) reference vector $\|j,\\mp j\\ra$ as\nwell. They can be extended to the case of semisimple Lie groups. \n\nIn all so far considered characteristic uncertainty inequalities (the\nSchr\\\"odinger and Robertson relations are characteristic ones) two or more\nobservables and {\\it one state} are involved. It turned out that these\nrelations, for any $n$ observables, are extendable to the case of two or\nmore states. We also have shown that the (extended) characteristic\ninequalities can be written down in the {\\it complementary form} in terms of\ntwo positive quantities less than unity. In the case of two observables with\ntwo eigenvalues each these {\\it complementary quantities} were recently\nproved \\ci{Bjork} to have the meaning of the predictability and visibility\nin the welcher weg experiment. The notion of \"characteristic complementary\nquantities\" might be useful in treating complicated quantum systems. It\nwas also noted that the characteristic coefficients of positive definite\nuncertainty matrices can be used for the construction of distances between\nquantum states.\n\n\n\\section{Appendix}\n\n\\subsection{ Robertson Proof of the Relation\n $\\det \\sig \\geq \\det C$}\n\nSince the derivation of the characteristic (\\ref{CUR}) and the extended\ncharacteristic uncertainty inequalities (\\ref{extendCUR}) is based on the\nRobertson relation (\\ref{RUR}) here we provide the proof of (\\ref{RUR})\nfollowing Robertson' paper \\ci{Rob} with some modern notations. Let\n$X_1,X_2,\\ld,X_n$ be Hermitian operators, and $\|\\psi\\ra$ be a pure state.\nConsider the squared norm of the composite state $\|\\psi^\\pr\\ra = \\sum_j \\alf_j\n(X_j - \\la X_j\\ra)\|\\psi\\ra$, where $\\alf_j$ are arbitrary complex\nparameters. One has\n\\beq\\lb{\|psi\|} %eq.47\n\\la\\psi^\\pr\|\\psi^\\pr\\ra = \\sum_{jk} \\alf^_k\\alf_j\n\\la\\psi\|(X_k-\\la X_k\\ra)(X_j-\\la X_j\\ra)\|\\psi\\ra\n= \\sum_{k,j}\\alf_k^S_{kj}\\alf_j \\equiv {\\cal S}(\\vec{\\alf}^,\\vec{\\alf}),\n\\eeq\nwhere the matrix elements $S_{kj}$ are\n$S_{kj} = \\la\\psi\|(X_k-\\la X_k\\ra)(X_j-\\la X_j\\ra)\|\\psi\\ra =\n\\sig_{jk} + i C_{jk}$.\nWe see that $S =\\sig + i C$, where $\\sig$ and $C$ are the uncertainty and\nthe mean commutator matrices of the operators $X_1,\\ld,X_n$ in the state\n$\|\\psi\\ra$ (see eq. (\\ref{sigma,C})). In Hilbert space we have\n$\\la\\psi^\\pr\|\\psi^\\pr\\ra = 0$ iff $\|\\psi^\\pr\\ra = \\sum_j \\alf_j(X_j - \\la\nX_j\\ra) \|\\psi\\ra =0$, which means that $\|\\psi\\ra$ is an eigenstate of the\ncomplex combination of $X_j$. Thus the form ${\\cal S}$ is nonnegative\ndefinite, which means that the $n\\times n$ matrix $S=\\sig +iC$ is\nnonnegative: all its principle minors are nonnegative \\ci{Gantmaher}, in\nparticular $\\det S > 0$. For the case of two operators, $n=2$, one can\neasily verify that\n\\beq\\lb{detS} %eq.48\n0\\leq\\det S =\\det(\\sig +iC) = \\det\\sig -\\det C,\\quad (n=2\\,\\,\\,{\\rm only}).\n\\eeq\nThis proves the Robertson relation for two observables which was also\nderived by Schr\\\"odinger \\ci{SchRob} using the Schwarz inequality.\nThe property (\\ref{detS}) is due to the symmetricity of $\\sig$ and\nantisymmetricity of $C$ and is valid for $n=2$ only.\n\nFor odd $n$, $n\\geq 1$, the Robertson inequality $\\det\\sig \\geq \\det C$\nis trivial, since the determinant of an antisymmetric matrix of odd\ndimension vanishes identically. For even $n=2N$ and $n>2$ we follow the\nproof of Robertson \\ci{Rob}, using however some notions from the present\nmatrix theory \\ci{Gantmaher}. One considers the regular sheaf (bundle)\nof the matrices $\\sig$ and $\\eta = iC$, $\\eta - \\lam\\sig$, supposing\n$\\sig > 0$. There exist congruent transformation (by means of the so called\nsheaf principle matrix $Z$, $\\det Z\\neq 0$), which brings both\nmatrices to the diagonal form -- $\\sig$ to the unit matrix, $\\sig^\\pr =\nZ^T\\sig T = 1$ and $\\eta^\\pr = {\\rm diag}\\{\\lam_1,\\lam_2,\\ld,\\lam_{2N}\\}$,\nwhere $\\lam_i$ are the $2N$ roots of the secular equation\n$\\det(\\eta-\\lam\\sig) = 0$. The product of all roots equals\n$\\det\\eta/\\det\\sig$. From $\\det (\\eta-\\lam\\sig) = \\det(\\eta-\\lam\\sig)^T =\n\\det (\\eta+\\lam\\sig)$ (since $\\eta^T=-\\eta$ and $n=2N$) it follows that the\npolynomial $\\det(\\eta-\\lam\\sig)$ contains only even powers of $\\lam$,\n$\\det(\\eta-\\lam\\sig) = \\det\\eta + \\ld + (-\\lam)^{2N}\\det\\sig = 0.$ This\nmeans that the $2N$ real roots $\\lam_j$ are equal and opposite in pairs.\nDenoting positive routs as $\\lam_\\mu$, $\\mu = 1,\\ld,N$ and negative roots as\n$\\lam_{\\mu+N} = -\\lam_\\mu$ one writes\\\\[-3mm]\n\\beq\\lb{detC} %eq.49\n\\det\\eta = (-1)^N\\det C = (-1)^N\\prod_\\mu\\lam_\\mu^2\\,\\det\\sig.\n\\eeq\\\\[-2mm]\nOn the other hand the Hermitian matrix $\\sig + \\eta = \\sig +iC$ is positive\ndefinite and after the diagonalization takes the form \n\\beq\\lb{s^pr+eta^pr} %eq.50\n\\sig^\\pr +\\eta^\\pr = {\\rm diag}\\{1+\\lam_1,\\ld,\n1+\\lam_{2N}\\} = {\\rm diag}\\{1+\\lam_1,1-\\lam_1,\\ldots,1+\\lam_N,1-\\lam_N\\}.\n\\eeq\n%\\beq\\lb{detS3}\n%\\det(\\sig+\\eta) = (\\det Z)^{-2}\\prod_\\mu^N (1-\\lam_\\mu^2) > 0,\n%\\eeq\nThe diagonal matrix $\\sig^\\pr +\\eta^\\pr$ is again nonnegative definite,\ni.e. all the elements on the diagonal are nonnegative, which implies that\n$\\lam^2_\\mu \\leq 1$, $\\mu = 1,\\ld, N$. Then eq. (\\ref{detC}) yields the\nRobertson inequality $\\det \\sig \\geq \\det C$. End of the proof. \\\\[0mm]\n\n{\\bf Remarks:} \n(a) Robertson considered the case of pure states only. However one can\nsee from the proof that his relation holds for mixed states as well; \n(b) It is seen from the above proof that the inequality $\\det\n\\sig \\geq \\det C$ holds for any two real matrices $C$ and $\\sig$, one of\nwhich is antisymmetric ($C$), the other -- symmetric and nonnegative\ndefinite and such that Hermitian matrix $\\sig + iC$ is again nonnegative;\n(c) If the matrices $\\sig_j$ and $C_j$, $j=1,2,\\ld,m$, obey the requirements\nof (b) then $\\det(\\sig_1+\\sig_2+\\ld) \\geq \\det(C_1+C_2+\\ld)$ since (as one\ncan easily prove) the sum of nonnegative $\\sig_j + iC_j$ is again a\nnonnegative matrix. These observations have been used in establishing the\nextended characteristic relations (\\ref{extendCUR}) for several\nstates and in formulating the remark (a) as well.\n\\vs{3mm}\n\n\\subsection{The $SU(1,1)$ CS $\|\\xi;k\\ra$ are the Unique States\n Which Minimize the Characteristic Inequalities for the Three Generators}\n\nFor the three generators $K_i$ of $SU(1,1)$ there are two nontrivial\ncharacteristic uncertainty inequalities corresponding to $r=n=3$ and\n$r=n-1=2$ in (\\ref{CUR}). The third order characteristic UR is minimized in\na pure state $\|\\psi\\ra$ iff $\|\\psi\\ra$ is an eigenstate of a real\ncombination of $K_i$, i.e. iff $\|\\psi\\ra = \|z,u,v,w;k\\ra$ obey the equation\n\\beq\\lb{CUS3} %eq.51\n[uK_- + vK_+ + wK_3]\\,\|z,u,v,w;k\\ra = z\|z,u,v,w;k\\ra\n\\eeq\nwith real $w$ and $v=u^$. The second order characteristic UR is\nminimized iff $\|\\psi\\ra$ is an eigenstate of complex combinations\nof all three pair $K_i$-$K_j$ simultaneously, i.e. iff\n\\beq\\lb{CUS2} %eq.52\n\\left.\\begin{tabular}{l}\n$[u_1K_- + v_1K_+ +w_1K_3]\\,\|\\psi\\ra = z_1\|\\psi\\ra,\\quad w_1=0,$\\\\\n$\\,[u_2K_- + v_2K_+ +w_2K_3]\\,\|\\psi\\ra = z_2\|\\psi\\ra,\\quad v_2=u_2,\\,\nw_2\\neq0,$\\\\ $\\,[u_3K_- + v_3K_+ +w_3K_3]\\,\|\\psi\\ra = z_3\|\\psi\\ra,\\quad\nv_3=-u_3,\\, w_3\\neq0,$\n\\end{tabular}\\right\\}\n\\eeq\nwhere the complex parameters\n$u_1,\\,v_1,\\,u_2,\\,w_2,\\,u_3,$ and $w_3$ shouldn't vanish and\n$z_1,\\,z_2,\\,z_3$ may be arbitrary. \nTo solve this system it is convenient to use BG analytic rep (\\ref{BGrep}).\nLet us start with the first equation in (\\ref{CUS2}). Its normalizable\nsolutions $\|z_1,u_1,v_1;k\\ra$ for $k=1/2,1,\\ld$ were found in \\ci{Trif94}.\nThey are normalizable for $\|u_1\|>\|v_1\|$ only and in BG rep have the form (up\nto the normalization constant)\n\\beq\\lb{K1-K2CUS} %eq.53\n \\Phi_{z_1}(\\eta;u_1,v_1) = e^{-\\eta\\sqrt{-v_1/u_1}}\\,_1F_1\\left(k+\\frac{z_1/2}\n{\\sqrt{-u_1v_1}};2k;2\\eta\\sqrt{-v_1/u_1}\\right),\n \\eeq\nwhere (for any $\|u_1\|>\|v_1\|$) the eigenvalue $z_1$ is arbitrary complex\nnumber. Here the complex \nvariable in the BG rep (\\ref{BGrep}) is denoted by $\\eta$. For $k<1/2$ a\nsecond normalizable solution exist of the form\n\\beq\\lb{secondsol} %eq.54\n \\Phi^\\pr_{z_1}(\\eta;u_1,v_1) = \\eta^{1-2k}\\,e^{-\\eta\\sqrt{-v_1/u_1}}\\,\n _1F_1\\left(\\frac{z_1/2} {\\sqrt{-u_1v_1}}-k+1;2(1-k);2\\eta\\sqrt{-v_1/u_1}\n \\right),\n\\eeq\nIn order to obtain second order $SU(1,1)$ characteristic US we have to subject \nthe solution (\\ref{K1-K2CUS}) to obey the rest two equations in (\\ref{CUS2}).\nLet us try to obey the second one.\nSince $u_1\\neq 0$ we can write $K_-\|z_1,u_1,v_1;k\\ra = (z_1-v_1K_+)\n\|z_1,u_1,v_1;k\\ra/u_1$ and substitute into the second equation to obtain\n\\beq\\lb{K_3\|zuv>} %eq.55\nK_3\|z_1,u_1,v_1;k\\ra = \\frac{1}{w_2}[z_2-\\frac{u_2}{u_1}z_1\n+(v_1\\frac{u_2}{u_1}-u_2)K_+]\|z_1,u_1,v_1;k\\ra.\n\\eeq\nIn BG rep (\\ref{BGrep}) this is a first order equation which the function\n(\\ref{K1-K2CUS}) has to obey. By equating the coefficients of the terms\nproportional to $\\eta^n$, $n=0,1,\\ld$, we obtain after some manipulations the\nnecessary conditions (a) $k + z_1/2\\sqrt{-u_1v_1} = 0$; (b) $k = z_2/w^2 -\nu_2z_1/u_1w_2$ and (c) $u_2(1-v_1/u_1) = w_2\\sqrt{-v_1/u_1}/2$. \nThe first condition requires the relation between the parameters\n$z_1,\\,u_1,\\,v_1$ and reduces the \"wave function\" (\\ref{K1-K2CUS}) to\n\\beq\\lb{CSunique} %eq.56\n\\Phi_{z_1}(\\eta;u_1,v_1) = \\exp\\left[-\\eta\\sqrt{-v_1/u_1}\\right],\n\\eeq \nwhich is just the CS $\|\|\\xi;k\\ra$ in BG rep with $\\xi = -\\sqrt{-v_1/u_1}$.\nThe second condition is always satisfied by $z_2 = kw_2 + u_2z_1/u_1$,\n$u_2,\\,w_2$ remaining arbitrary. Thus {\\it it is the CS $\|\\xi;k\\ra$\nonly}, $k=1/2,1,\\ld$, which minimize simultaneously the Schr\\\"odinger\ninequality for $K_1,\\,K_2$ and $K_1,\\,K_3$.\n\nNext it is a simple (but not short) exercise to check that\n$\\exp\\left[-\\eta\\sqrt{-v_1/u_1}\\right]$ satisfy the third equation in\n(\\ref{CUS2}) with\n$w_3 = w_2(z_3u_1-iu_3z_1)/(u_1z_2-u_2z_1)$,\n$z_3 = i(u_3/u_2u_1)\\,(u_1+v_1z_1)/(u_1z_2-u_2z_1) + iu_3z_1/u_1$,\n ($u_2,z_2,\\,u_3$ being free) and the eigenvalue eq. (\\ref{CUS3}) with\n$v=u^$ and real $w$, $w = (-uv_1 + u^u_1)/\\sqrt{-u_1v_1} = w(u_1,v_,u)$.\nOne can see that for every given $\\xi=-\\sqrt{-v_1/u_1}$ the equation\nIm$[w(u_1,v_1,u)] = 0$ can be solved with respect to $u$, the solution being\nnot unique: $u = \|u\|\\exp\\left[\\pi/4 - {\\rm arg}\\xi/2\\right]$, $\|u\|$ being\narbitrary. So the family of CS $\|\\xi;k\\ra$ is the unique family of states\nwhich minimize the third and the second order characteristic US\nsimultaneously. If we subject the function (\\ref{K1-K2CUS}) directly to \n(\\ref{CUS3}) we will get again (\\ref{CSunique}).\n\nIn the case of $SU(1,1)$ characteristic US for $K_i$ in rep (\\ref{1moderep})\n($k = 1/4,3/4$) we have to consider the two solutions\n(\\ref{K1-K2CUS}) and (\\ref{secondsol}). The consideration gives no new\nresult - again the eqs. (\\ref{CUS3}) and (\\ref{CUS2}) are satisfied by\n$\\exp\\left[-\\eta\\sqrt{-v_1/u_1}\\right]$ only.\n\nSimilar results can be obtained for the minimization of (\\ref{RUR}) \nin CS $\|\\tau;j\\ra$ using for example their own analytic representation\nand the results of paper \\ci{Brif97}.\n\n\\subsection{Acknowledgment}\nThis work is partially supported by the National Science Fund of the \nBulgarian Ministry of Education and Science, Grant No F-644/1996.\n\n\\begin{thebibliography}{99}\n\\def\\bi{\\bibitem}\n\n\\bi{KlaSka}\n Klauder J.R. and Skagerstam B.-S., {\\it Coherent States -- Applications\n\tin Physics and Mathematical Physics}, World Scientific,\n Singapore, 1985. {\\footnotesize A comprehensive list of references\n on coherent states is available in this book with reprints of\n selected papers, in particular of 1963 papers of Glauber, Klauder\n and Sudarshan}.\n\n\\bi{LouKni}\n\t Loudon R. and Knight P., {\\it Squeezed light}, J. Mod. Opt. {\\bf 34}\n\t (1987) 709-759.\n\n\\bi{ZhaFenGil}\n Zhang W.-M., Feng D.H. and Gilmore R., {\\it Coherent states: theory \n\t and some applications}, Rev. Mod. Phys. {\\bf 62} (1990) 867-924.\n\n\\bi{AliAGM}\n Tareque Ali S., Antoine J.-P., Gazeau J.-P. and Mueler U.A.,\n {\\it Coherent states and their generalizations: a mathematical overview},\n\t Rev. Math. Phys. {\\bf7}(7) (1995) 1013-1104.\n\n\\bi{Nie98} \n Nieto M.M., {\\it The discovery of squeezed states --- in 1927},\n\t In: Proc. 5th Int. Conf. on Squeezed States and Uncertainty\n\t Relations, Eds. Han D., Janszky J., Kim I.S. and\n Man'ko V.I., NASA/CP-1998-206855, Maryland, 1998 [E-print\n quant-ph/9708012].\n\n\\bi{Trif93} Trifonov D.A.,\n {\\it Completeness and geometry of Schr\\\"odinger minimum uncertainty states},\n J. Math. Phys. {\\bf34} (1993) 100-110.\n\n\\bi{Trif94} Trifonov D.A.,\n {\\it Generalized intelligent states and squeezing},\n J. Math. Phys. {\\bf35} (1994) 2297-2308.\n\n\\bi{Brif96} Brif C.,\n {\\it Two-photon algebra eigenstates. A unified approach to squeezing},\n \t Ann. Phys. {\\bf 251} (1996) 180-207.\n\n\\bi{Brif97} Brif C.,\n {\\it SU(2) and SU(1,1) algebra eigenstates: a unified analytic approach\n to coherent and intelligent states},\n Int. J. Theor. Phys. {\\bf36} (1997) 1651-1682.\n\n\\bi{Trif97} Trifonov D.A., {\\it Robertson intelligent states},\n J. Phys. A {\\bf30} (1997) 5941-5957.\n\n\\bi{Trif98a} Trifonov D.A., {\\it Barut-Girardello coherent states for\n\t $u(p,q)$ and $sp(N,R)$ and their macroscopic superpositions},\n J. Phys. A {\\bf31} (1998) 5673-5696.\n\t%% [E-prints quant-th/9706028, quant-th/9711066]\n\n\\bi{Trif98b}\n\t Trifonov D.A., {\\it On the squeezed states for $n$ observables},\n Phys. Scripta {\\bf58} (1998) 246-255.\n\t %% [E-prints quant-th/9705001].\n\n\\bi{FujFun} Fujii K. and Funahashi K.,\n {\\it Extension of the Barut-Girardello coherent state and path integral},\n J. Math. Phys. {\\bf38} (1997) 4422-4434.\n\t [E-prints quant-th/9704011, quant-th/9708041].\n\n\\bi{TriDon} Trifonov D.A. and Donev S.G.,\n {\\it Characteristic uncertainty relations}, J. Phys. A {\\bf31} \n (1998) 8041-8047.\n\n\\bi{Bjork}\n Bj\\\"ork G., S\\\"oderholm J., Trifonov A., Tsegaye T. and Karlson A.,\n\t {\\it Complementarity and uncertainty relations},\n\t Phys. Rev. A {\\bf60} (1999) 1874-1882.\n\n\\bi{SchRob} Schr\\\"odinger E., {\\it Zum Heisenbergschen\n Unsch\\\"arfeprinzip}, In: Sitzungsberichte Preus. Acad. Wiss.,\n\t\t Phys.-Math. Klasse, p. 296-303 (Berlin 1930);\n Robertson H.P., {\\it A general formulation of the uncertainty principle\n\tand its classical interpretation},\tPhys. Rev. {\\bf 35}(5) (1930) 667-667.\n\n\\bi{Rob} Robertson H.P., {\\it An indeterminacy relation for several\n observables and its classical interpretation},\n Phys. Rev. {\\bf 46}(9) (1934) 794-801.\n\n\\bi{MMT} Malkin I.A., Man'ko V.I. and Trifonov D.A.,\n {\\it Invariants and evolution of coherent states of charged particle in\n\t a time-dependet magnetic field}, Phys. Lett. A {\\bf30} (1969) 414-415;\n Malkin I.A., Man'ko V.I. and Trifonov D.A., {\\it Coherent states and\n transition probabilities in a time-dependent electromagnetic field},\n Phys. Rev. D {\\bf2} (1970) 1371-1385.\n\n\\bi{Yuen} Yuen H., {\\it Two-photon coherent states of the radiation\n field}, Phys. Rev. A{\\bf 13} (1976) 2226-2243.\n\n\\bi{HCh} Husimi K., {\\it Miscellanea in elementary quantum mechanics},\n Progr. Theor. Phys. {\\bf9} (1953) 381-402;\n\t Chernikov N.A., Zh. Exp. Theor. Fiz. {\\bf53} (1967) 1006-1017.\n\n\\bi{HMMT} Holz A., Lett. N. Cimento A{\\bf 4} (1970) 1319-1321;\n Malkin I.A., Man'ko V.I. and Trifonov D.A.,\n {\\it Dynamical symmetry of nonstationary systems}, N. Cimento A {\\bf4}\n (1971) 773-793; Malkin I.A. and Man'ko V.I.,\n {\\it Coherent states and excitation of n-dimensional nonstationary forced\n oscillator}, Phys. Lett. A {\\bf32} (1970) 243-244.\n\n\\bi{Stol} Stoler D.A.,\n {\\it Equivalent classes of minimum uncertainty packets}, Phys. Rev.\n\tD {\\bf1} (1970) 3217-3219.\n\n\\bi{Perel} Perelomov A.M., {\\it Coherent states for arbitrary Lie\n group}, Commun. Math. Phys. {\\bf26} (1972) 222-236.\n\n\\bi{Hollen} Hollenhorst J.N.,\n {\\it Quantum limits on resonant-mass gravitational-radiation detection},\n\t Phys. Rev. D {\\bf19} (1979) 1669-1679.\n\n\\bi{Radclif} Radcliffe J.M., {\\it Some properties of coherent spin\n states}, J. Phys. A {\\bf4} (1971) 313-323.\n\n\\bi{Gilmore} Arecchi F.T., Courtens E., Gilmore R., and Thomas H.,\n {\\it Atomic coherent states in quantum optics},\n Phys. Rev. A {\\bf6}, (1972) 2211-2237. %%% (Dec 1972).\n\n\\bi{Beckers} %% H x) SU(1,1) CS and res. unity measure:\n Beckers J. and Debergh N., {\\it On generalized coherent\n states with maximal symmetry for the harmonic oscillator},\n\t J. Math. Phys. {\\bf30} (1989) 1739-1743.\n\n\\bi{BVM}\n Brif C., Vourdas A. and Mann A., {\\it Analytic representations based on\n $SU(1,1)$ coherent states and their applications},\n J. Phys. A {\\bf29} (1996) 5873-5886.\n\n\\bi{BG} Barut A.O. and Girardello L., {\\it New \"coherent\" states\n associated with noncompact groups}, Commun. Math. Phys. {\\bf21}\n\t (1971) 41-55.\n\n\\bi{Trif96}\n Trifonov D.A., {\\it Algebraic coherent states and\n squeezing}, E-print quant-ph/9609001.\n\n\\bi{Stegun} {\\it Handbook of mathematical functions}, edited by\n\t\tAbramowitz M. and Stegun I.A., National bureau of standards, 1964.\n\t\t(Russian translation, Nauka, Moscow, 1979).\n\n\\bi{HilNag} %% sob. sast. na njakoi ua^2+va^{\\dg 2}+wa^\\dg a:\n\tBergou J.A., Hillery M. and Yu D., {\\it Minimum uncertainty states for\n amplitude-squared squeezing: Hermite polynomial states},\n\tPhys. Rev. A{\\bf43} (1991) 515-520;\n Nagel B., {\\it Higher power SS, Jacobi matrices,\n\t\tand the hamburger moment problem}, E-print quant-ph/9711028.\n\n\\bi{LuiPer}\n\tLuis A. and Perina J., {\\it SU(2) coherent states in parametric\n down-conversion}, Phys. Rev. A {\\bf53} (1996) 1886-1893;\n Brif C. and Mann A., {\\it Nonclassical interferometry with intelligent\n light}, Phys. Rev. A {\\bf54} (1996) 4505-4518.\n\n\\bi{Vourd90} Vourdas A., {\\it $SU(2)$ and $SU(1,1)$ phase states},\n Phys. Rev. A {\\bf41} (1990) 1653-1661;\n %% (two-mode su2, su11 representations used for the phase states)\n Vourdas A., {\\it Analytic representations in the unit disk and applications\n to phase states and squeezing}, Phys. Rev. A {\\bf45} (1992) 1943-1950.\n %% (Perelomov \|z;k> are called negative binomial states. For k=1/2\n %% one gets thermal distribution, two-mode su11 representations noted).\n\n\\bi{VouBrif} Vourdas A., {\\it Coherent states on the $m$-sheeted covering\n group of $SU(1,1)$}, J. Math. Phys. {\\bf34} (1993) 1223-1235;\n Brif C., {\\it Photon states associated with Holstein-Primakoff realization\n of $SU(1,1)$ Lie algebra}, Quant. Semiclass. Opt. {\\bf7} (1995) 803-34;\n Wang X.G. and Fu H.C. {\\it Negative binomial states of the radiation \n\tfield and their excitations are nonlinear coherent states},\n Mod. Phys. Lett. B {\\bf13} (1999) 617-623. \n\n\n\\bi{BrifBen} Brif C. and Ben-Aryeh Y.,\n {\\it $SU(1,1)$ intelligent states: analytic representation in the unit\n disk}, J. Phys. A {\\bf27} (1995) 8185-8195.\n\n\\bi{Bied}\n Biedenharn L.C., {\\it The quantum group $SU_q(2)$ and a $q$-analoge of the\n\tboson operators}, J. Phys. A {\\bf22} (1989) L873-L878;\n Macfarlane A.J., {\\it On $q$-analogue of the quantum harmonic oscillator\n\tand the quantum group $SU(2)_q$}, J. Phys. A {\\bf22} (1989) 4581-4588,\n\n\\bi{Kulish}\n Chainchian M., Ellinas D. and Kulish P.P., {\\it Quantum algebra as the\n dynamical symmetry of the deformed Jaynes-Cummings model},\n Phys. Rev. Lett. {\\bf65} (1990) 980-983;\n Kulish P.P. and Damaskinsky E.V., {\\it On the $q$-oscillator in quantum\n algebra $su_q(1,1)$}, J. Phys. A {\\bf23} (1990) L415-L419.\n\n\\bi{Solom} Solomon A.I. and Katriel J., {\\it On $q$-squeezed states},\n J. Phys. A {\\bf23} (1990) L1209-L1212;\n Solomon A.I. and Katriel J., {\\it Generalized $q$-bosons and their squeezed\n states}, J. Phys. A {\\bf24} (1991) 2093-2105,\n\n\\bi{Ellin} D. Ellinas, {\\it On coherent states and q-deformed\n algebras}, E-print hep-th/9309072 (Presented at the\n 'International Symposium on Coherent States', June 1993, USA).\n\n\\bi{Solom3Oh}\n Mcdermott R.J. and Solomon A.I.,\n {\\it Squeezed states parametrized by elements of noncommutative algebras},\n Czechoslovak J. Phys. {\\bf46} (1996) 235-241;\n Oh P. and Rim C., {\\it The $q$-deformed oscillator representations and their\n coherent states of the $su(1,1)$ algebra},\n Rep. Math. Phys. {\\bf40} (1997) 285-293.\n\n\\bi{Wang} Wang X.G., {\\it Ladder operator formalisms and generally deformed\n oscillator algebraic structures of quantum states in Fock space},\n\t\t E-print quant-ph/9911114. \n\t\t \n\\bi{Fatyga} Fatyga B.W., Kostelecky V.A., Nieto M.M. and Truax D.R.,\n {\\it Supercoherent states}, Phys. Rev. D {\\bf43} (1991) 1403-1412;\n El Gradechi A.M. and Nieto L.M., {\\it Supercoherent states, super K\\\"ahler\n geometry and geometric quantization}, Commun. Math. Phys. {\\bf175}\n\t (1996) 521-564.\n\n\\bi{Trif92}\n Trifonov D.A., {\\it Rimannian and supersymmetric properties of squeezed\n and correlated states}.\n\t In: Quantization and Coherent States Methods, Eds Ali S.T.,\n Mladenov I.M. and Odzijewicz A., W. Scientific, Singapore, 1993.\n %%(Proc. XI Workshope on Geom. Methods in Physics, Bialowieza, 1992)\n\n\n\\bi{NikTrif}\n\t\tNikolov B.A. and Trifonov D.A., {\\it On the dynamics of generalized\n\t\tcoherent states. II. Classical equations of motion},\n\t\tCommun. JINR E2-81-798 (Dubna, 1981).\n\n\\bi{Jackiw} Jackiw R., {\\it Minimum uncertainty product,\n\t\tnumber--phase uncertainty product and coherent states},\n J. Math. Phys. {\\bf9} (1968) 339-346;\n\t\tWeigert S., {\\it Landscape of uncertainty in Hilbert space for\n\t\tone-particle states}, Phys. Rev. A {\\bf53}(4) (1996) 2084-2088.\n\n\\bi{ArChalSal} Aragone C., Chalbaud E. and Salamo S., {\\it\n On intelligent spin states}, J. Math. Phys. {\\bf17} (1976) 1963-1971;\n\tRushin S. and Ben-Aryeh Y., {\\it Minimum uncertainty states for angular\n momentum operators}, Phys. Lett. A{\\bf58} (1976) 207-208.\n\n\\bi{Rashid} Rashid M.A., {\\it The intelligent states I. Group-theoretic\n study and the computation of matrix elements},\n J. Math. Phys. {\\bf19} (1978) 1391-1396.\n\n\\bi{DKM} Dodonov V.V., Kurmyshev E.V. and Man'ko V.I.,\n {\\it Generalized uncertainty relation and correlated coherent states},\n\t Phys. Lett. A{\\bf79} (1980) 150-152;\n Dodonov V.V. and Man'ko V.I., {\\it Invariants and correlated states of\n nonstationary systems}, Trudy FIAN {\\bf183} (1987) 71-181\n (Nauka, Moscow, 1987 and Nuova Science, Commack, N.Y., 1988).\n\n\\bi{Gantmaher} Gantmaher F.R., {\\it Teoria matrits}, Nauka, Moscow, 1975.\n\n\\bi{drugi} Braunstein S.L., Caves C.M. and Milburn G.J. \n {\\it Generalized uncertainty relations: Theory, examples, and Lorentz\n invariance}, E-print quant-ph/9507004;\n Dodonov V.V. and Man'ko V.I., {\\it Generalized uncertainty relations in\n quantum mechanics}, Trudy FIAN {\\bf183} (1987) 5-70\n (Nauka, Moscow, 1987 and Nuova Science, Commack, N.Y., 1988).\n\n\\bi{TriDon2} Trifonov D.A. and Donev S.G., {\\it Polarized Euclidean type\n distances between quantum states and observables}, Preprint-TH-99/3.\n\n\\end{thebibliography}\n\\vspace{5mm}\n\n{\\footnotesize \nMisprints in v. 4 (here, in v. 5, corrected):\n \nIn eq.(5): $\\frac{2\\hbar}{m} \\lrar \\left(\\frac{2\\hbar}{m\\ome_0}\\right)^{1/2}$,\n\nIn eq.(36): $[A_\\mu,A_\\nu] \\lrar [A_\\mu,A_\\nu^\\dg]$.}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912084.extracted_bib", "string": "\\bi{KlaSka Klauder J.R. and Skagerstam B.-S., {Coherent States -- Applications in Physics and Mathematical Physics, World Scientific, Singapore, 1985. {\\footnotesize A comprehensive list of references on coherent states is available in this book with reprints of selected papers, in particular of 1963 papers of Glauber, Klauder and Sudarshan. \\bi{LouKni Loudon R. and Knight P., {Squeezed light, J. Mod. Opt. {34 (1987) 709-759. \\bi{ZhaFenGil Zhang W.-M., Feng D.H. and Gilmore R., {Coherent states: theory and some applications, Rev. Mod. Phys. {62 (1990) 867-924. \\bi{AliAGM Tareque Ali S., Antoine J.-P., Gazeau J.-P. and Mueler U.A., {Coherent states and their generalizations: a mathematical overview, Rev. Math. Phys. {7(7) (1995) 1013-1104. \\bi{Nie98 Nieto M.M., {The discovery of squeezed states --- in 1927, In: Proc. 5th Int. Conf. on Squeezed States and Uncertainty Relations, Eds. Han D., Janszky J., Kim I.S. and Man'ko V.I., NASA/CP-1998-206855, Maryland, 1998 [E-print quant-ph/9708012]. \\bi{Trif93 Trifonov D.A., {Completeness and geometry of Schr\\\"odinger minimum uncertainty states, J. Math. Phys. {34 (1993) 100-110. \\bi{Trif94 Trifonov D.A., {Generalized intelligent states and squeezing, J. Math. Phys. {35 (1994) 2297-2308. \\bi{Brif96 Brif C., {Two-photon algebra eigenstates. A unified approach to squeezing, Ann. Phys. {251 (1996) 180-207. \\bi{Brif97 Brif C., {SU(2) and SU(1,1) algebra eigenstates: a unified analytic approach to coherent and intelligent states, Int. J. Theor. Phys. {36 (1997) 1651-1682. \\bi{Trif97 Trifonov D.A., {Robertson intelligent states, J. Phys. A {30 (1997) 5941-5957. \\bi{Trif98a Trifonov D.A., {Barut-Girardello coherent states for $u(p,q)$ and $sp(N,R)$ and their macroscopic superpositions, J. Phys. A {31 (1998) 5673-5696. %% [E-prints quant-th/9706028, quant-th/9711066] \\bi{Trif98b Trifonov D.A., {On the squeezed states for $n$ observables, Phys. Scripta {58 (1998) 246-255. %% [E-prints quant-th/9705001]. \\bi{FujFun Fujii K. and Funahashi K., {Extension of the Barut-Girardello coherent state and path integral, J. Math. Phys. {38 (1997) 4422-4434. [E-prints quant-th/9704011, quant-th/9708041]. \\bi{TriDon Trifonov D.A. and Donev S.G., {Characteristic uncertainty relations, J. Phys. A {31 (1998) 8041-8047. \\bi{Bjork Bj\\\"ork G., S\\\"oderholm J., Trifonov A., Tsegaye T. and Karlson A., {Complementarity and uncertainty relations, Phys. Rev. A {60 (1999) 1874-1882. \\bi{SchRob Schr\\\"odinger E., {Zum Heisenbergschen Unsch\\\"arfeprinzip, In: Sitzungsberichte Preus. Acad. Wiss., Phys.-Math. Klasse, p. 296-303 (Berlin 1930); Robertson H.P., {A general formulation of the uncertainty principle and its classical interpretation, Phys. Rev. {35(5) (1930) 667-667. \\bi{Rob Robertson H.P., {An indeterminacy relation for several observables and its classical interpretation, Phys. Rev. {46(9) (1934) 794-801. \\bi{MMT Malkin I.A., Man'ko V.I. and Trifonov D.A., {Invariants and evolution of coherent states of charged particle in a time-dependet magnetic field, Phys. Lett. A {30 (1969) 414-415; Malkin I.A., Man'ko V.I. and Trifonov D.A., {Coherent states and transition probabilities in a time-dependent electromagnetic field, Phys. Rev. D {2 (1970) 1371-1385. \\bi{Yuen Yuen H., {Two-photon coherent states of the radiation field, Phys. Rev. A{13 (1976) 2226-2243. \\bi{HCh Husimi K., {Miscellanea in elementary quantum mechanics, Progr. Theor. Phys. {9 (1953) 381-402; Chernikov N.A., Zh. Exp. Theor. Fiz. {53 (1967) 1006-1017. \\bi{HMMT Holz A., Lett. N. Cimento A{4 (1970) 1319-1321; Malkin I.A., Man'ko V.I. and Trifonov D.A., {Dynamical symmetry of nonstationary systems, N. Cimento A {4 (1971) 773-793; Malkin I.A. and Man'ko V.I., {Coherent states and excitation of n-dimensional nonstationary forced oscillator, Phys. Lett. A {32 (1970) 243-244. \\bi{Stol Stoler D.A., {Equivalent classes of minimum uncertainty packets, Phys. Rev. D {1 (1970) 3217-3219. \\bi{Perel Perelomov A.M., {Coherent states for arbitrary Lie group, Commun. Math. Phys. {26 (1972) 222-236. \\bi{Hollen Hollenhorst J.N., {Quantum limits on resonant-mass gravitational-radiation detection, Phys. Rev. D {19 (1979) 1669-1679. \\bi{Radclif Radcliffe J.M., {Some properties of coherent spin states, J. Phys. A {4 (1971) 313-323. \\bi{Gilmore Arecchi F.T., Courtens E., Gilmore R., and Thomas H., {Atomic coherent states in quantum optics, Phys. Rev. A {6, (1972) 2211-2237. %%% (Dec 1972). \\bi{Beckers %% H x) SU(1,1) CS and res. unity measure: Beckers J. and Debergh N., {On generalized coherent states with maximal symmetry for the harmonic oscillator, J. Math. Phys. {30 (1989) 1739-1743. \\bi{BVM Brif C., Vourdas A. and Mann A., {Analytic representations based on $SU(1,1)$ coherent states and their applications, J. Phys. A {29 (1996) 5873-5886. \\bi{BG Barut A.O. and Girardello L., {New \"coherent\" states associated with noncompact groups, Commun. Math. Phys. {21 (1971) 41-55. \\bi{Trif96 Trifonov D.A., {Algebraic coherent states and squeezing, E-print quant-ph/9609001. \\bi{Stegun {Handbook of mathematical functions, edited by Abramowitz M. and Stegun I.A., National bureau of standards, 1964. (Russian translation, Nauka, Moscow, 1979). \\bi{HilNag %% sob. sast. na njakoi ua^2+va^{\\dg 2+wa^\\dg a: Bergou J.A., Hillery M. and Yu D., {Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states, Phys. Rev. A{43 (1991) 515-520; Nagel B., {Higher power SS, Jacobi matrices, and the hamburger moment problem, E-print quant-ph/9711028. \\bi{LuiPer Luis A. and Perina J., {SU(2) coherent states in parametric down-conversion, Phys. Rev. A {53 (1996) 1886-1893; Brif C. and Mann A., {Nonclassical interferometry with intelligent light, Phys. Rev. A {54 (1996) 4505-4518. \\bi{Vourd90 Vourdas A., {$SU(2)$ and $SU(1,1)$ phase states, Phys. Rev. A {41 (1990) 1653-1661; %% (two-mode su2, su11 representations used for the phase states) Vourdas A., {Analytic representations in the unit disk and applications to phase states and squeezing, Phys. Rev. A {45 (1992) 1943-1950. %% (Perelomov \|z;k> are called negative binomial states. For k=1/2 %% one gets thermal distribution, two-mode su11 representations noted). \\bi{VouBrif Vourdas A., {Coherent states on the $m$-sheeted covering group of $SU(1,1)$, J. Math. Phys. {34 (1993) 1223-1235; Brif C., {Photon states associated with Holstein-Primakoff realization of $SU(1,1)$ Lie algebra, Quant. Semiclass. Opt. {7 (1995) 803-34; Wang X.G. and Fu H.C. {Negative binomial states of the radiation field and their excitations are nonlinear coherent states, Mod. Phys. Lett. B {13 (1999) 617-623. \\bi{BrifBen Brif C. and Ben-Aryeh Y., {$SU(1,1)$ intelligent states: analytic representation in the unit disk, J. Phys. A {27 (1995) 8185-8195. \\bi{Bied Biedenharn L.C., {The quantum group $SU_q(2)$ and a $q$-analoge of the boson operators, J. Phys. A {22 (1989) L873-L878; Macfarlane A.J., {On $q$-analogue of the quantum harmonic oscillator and the quantum group $SU(2)_q$, J. Phys. A {22 (1989) 4581-4588, \\bi{Kulish Chainchian M., Ellinas D. and Kulish P.P., {Quantum algebra as the dynamical symmetry of the deformed Jaynes-Cummings model, Phys. Rev. Lett. {65 (1990) 980-983; Kulish P.P. and Damaskinsky E.V., {On the $q$-oscillator in quantum algebra $su_q(1,1)$, J. Phys. A {23 (1990) L415-L419. \\bi{Solom Solomon A.I. and Katriel J., {On $q$-squeezed states, J. Phys. A {23 (1990) L1209-L1212; Solomon A.I. and Katriel J., {Generalized $q$-bosons and their squeezed states, J. Phys. A {24 (1991) 2093-2105, \\bi{Ellin D. Ellinas, {On coherent states and q-deformed algebras, E-print hep-th/9309072 (Presented at the 'International Symposium on Coherent States', June 1993, USA). \\bi{Solom3Oh Mcdermott R.J. and Solomon A.I., {Squeezed states parametrized by elements of noncommutative algebras, Czechoslovak J. Phys. {46 (1996) 235-241; Oh P. and Rim C., {The $q$-deformed oscillator representations and their coherent states of the $su(1,1)$ algebra, Rep. Math. Phys. {40 (1997) 285-293. \\bi{Wang Wang X.G., {Ladder operator formalisms and generally deformed oscillator algebraic structures of quantum states in Fock space, E-print quant-ph/9911114. \\bi{Fatyga Fatyga B.W., Kostelecky V.A., Nieto M.M. and Truax D.R., {Supercoherent states, Phys. Rev. D {43 (1991) 1403-1412; El Gradechi A.M. and Nieto L.M., {Supercoherent states, super K\\\"ahler geometry and geometric quantization, Commun. Math. Phys. {175 (1996) 521-564. \\bi{Trif92 Trifonov D.A., {Rimannian and supersymmetric properties of squeezed and correlated states. In: Quantization and Coherent States Methods, Eds Ali S.T., Mladenov I.M. and Odzijewicz A., W. Scientific, Singapore, 1993. %%(Proc. XI Workshope on Geom. Methods in Physics, Bialowieza, 1992) \\bi{NikTrif Nikolov B.A. and Trifonov D.A., {On the dynamics of generalized coherent states. II. Classical equations of motion, Commun. JINR E2-81-798 (Dubna, 1981). \\bi{Jackiw Jackiw R., {Minimum uncertainty product, number--phase uncertainty product and coherent states, J. Math. Phys. {9 (1968) 339-346; Weigert S., {Landscape of uncertainty in Hilbert space for one-particle states, Phys. Rev. A {53(4) (1996) 2084-2088. \\bi{ArChalSal Aragone C., Chalbaud E. and Salamo S., {On intelligent spin states, J. Math. Phys. {17 (1976) 1963-1971; Rushin S. and Ben-Aryeh Y., {Minimum uncertainty states for angular momentum operators, Phys. Lett. A{58 (1976) 207-208. \\bi{Rashid Rashid M.A., {The intelligent states I. Group-theoretic study and the computation of matrix elements, J. Math. Phys. {19 (1978) 1391-1396. \\bi{DKM Dodonov V.V., Kurmyshev E.V. and Man'ko V.I., {Generalized uncertainty relation and correlated coherent states, Phys. Lett. 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quant-ph9912085		[]		[ { "name": "comment.tex", "string": "\n\\documentstyle[aps,twocolumn,pra,psfig]{revtex}\n\n\\begin{document}\n\n{\\bf Comment on ``Demonstration of the Casimir Force in the 0.6 to 6 $\\mu$m \nRange''} \n\nIn a recent Letter \\cite{Lamoreaux97}, Lamoreaux reports a measurement \nof the Casimir force for distances in the 0.6 to 6 $\\mu$m range. The force has \nbeen measured between a flat and a spherical plate. Both plates are coated with \na layer of Cu, covered with an additional 0.5 $\\mu$m thick layer \nof Au. The author compares his experimental data to theoretical predictions \n\\cite{Lamoreaux98} and reports an agreement at 5\\% between theory and experiment\nif a pure Cu surface is assumed. Since his theoretical evaluation \n\\cite{Lamoreaux99} gives very different \nvalues for the Casimir force between Au surfaces and Cu surfaces, there\nresults a net discrepancy between expected and experimentally observed values.\n\nWe have recently recalculated the Casimir force between metallic mirrors and\nobtained results differing significantly from \\cite{Lamoreaux98,Lamoreaux99}, in \nparticular \nfor Au mirrors. \nDetails about the evaluation procedure, the interpolation and \nextrapolation of optical data, and the numerical integration techniques\nare given in \\cite{Lambrecht00}. Here, we restrict our attention on the \nAu/Cu problem underlined by Lamoreaux.\n\nThe upper graph of figure \\ref{fig} shows the imaginary part of the dielectric \nconstant $\\varepsilon ^{\\prime \\prime} (\\omega )$ as a function of frequency \n$\\omega$ for Au and Cu. All optical data are taken from \n\\cite{Palik}. At low frequencies they are extrapolated by a Drude model which is \nconsistent with present theoretical knowledge of optical properties of metals \nand, at the same time, fits quite nicely higher frequency optical data. \nSince the optical response functions are very similar for Au and Cu,\nthe Casimir forces evaluated from these functions are expected to be \nnearly equal.\n\nIn the experiment, the Casimir force is measured in the plane-sphere geometry. \nTheoretically it is evaluated by using the proximity force \ntheorem. We do not discuss here the validity of this approximation but\nfocus our attention on the effect of finite conductivity. \nWe calculate the reduction factor $\\eta$ (notation of \\cite{Lamoreaux98}; \nnotation \n$\\eta_E$ in \\cite{Lambrecht00}) of the force in the plane-sphere \ngeometry as the reduction factor of the energy evaluated in the \nplane-plane configuration.\nThe frequency dependent reflection coefficients are derived from the dielectric \nconstant, using causality relations, and $\\eta$ is then deduced\nthrough numerical integrations. \n\nThe lower graph of figure \\ref{fig} shows \n$\\eta$ for Au and Cu with, as expected, equal values at better than 1\\% in the \nrange of \ndistances studied in the experiment. This contradicts theoretical values \nobtained by Lamoreaux. For Au at $0.6~\\mu$m our value \n$\\eta = 0.87$ exceeds by 12\\% the value $\\eta = 0.78$ given in \n\\cite{Lamoreaux98}, while \nat the same distance the values for Cu are compatible within 2\\%.\n\nThis result clears up the Au/Cu discrepancy pointed out in \n\\cite{Lamoreaux98}. \nBesides this specific difficulty, more work is needed, on both the experimental \nand theoretical side, to reach an accurate agreement between theoretical\nexpectation and experimental measurements of Casimir force (see \n\\cite{Lambrecht00}\nand references therein).\n\n\\begin{figure}[tbh]\n\\centerline{\\psfig{figure=eps2.eps,width=7cm}}\n\\centerline{\\psfig{figure=PhiE_tot.eps,width=7cm}}\n\\caption{The imaginary part of the dielectric constant as function of \nfrequency (upper graph) and the reduction of the Casimir energy between \nplane metallic reflectors with respect to plane perfect\nmirrors as a function of distance (lower graph) for Au (solid line) and Cu \n(dashed line).}\n\\label{fig}\n\\end{figure}\n\n\\vspace{0.5cm}\n\\noindent\nAstrid Lambrecht and Serge Reynaud\\\\\n\\hspace{4.8mm}Laboratoire Kastler Brossel \\cite{LKB}\\\\\n\\hspace{4.8mm}Campus Jussieu, case 74 \\\\\n\\hspace{4.8mm}75252 Paris Cedex 05, France\n\n\\vspace{0.5cm}\n\\noindent\nPACS numbers: 12.20 Fv, 07.07 Mp, 03.70 +k\n\n\n\\begin{references}\n\n\\bibitem{Lamoreaux97} S.K. Lamoreaux {\\it Phys. Rev. Lett.} {\\bf 78} 5\n(1997)\n\n\\bibitem{Lamoreaux98} S.K. Lamoreaux, Erratum {\\it Phys. Rev. Lett.} \n{\\bf 81} 5475 (1998)\n\n\\bibitem{Lamoreaux99} S.K. Lamoreaux {\\it Phys. Rev.} {\\bf A59} R3149\n(1999)\n\n\\bibitem{Lambrecht00} A. Lambrecht and S. Reynaud {\\it Eur. Phys. J.} {\\bf D}, \nto appear (quant-ph/9907105)\n\n\\bibitem{Palik} {\\it Handbook of Optical Constants of Solids} E.D. Palik\ned. (Academic Press, New York 1995); {\\it Handbook of Optics II} (McGraw-Hill, \nNew York,\n1995)\n\n\\bibitem[]{LKB} Ecole Normale Sup\\'{e}rieure, Universit\\'{e} Pierre\net Marie Curie, et Centre National de la Recherche Scientifique\n\n\\end{references}\n\n\\end{document}\n\n\n" } ]	[ { "name": "quant-ph9912085.extracted_bib", "string": "{Lamoreaux97 S.K. Lamoreaux {Phys. Rev. Lett. {78 5 (1997)" }, { "name": "quant-ph9912085.extracted_bib", "string": "{Lamoreaux98 S.K. Lamoreaux, Erratum {Phys. Rev. Lett. {81 5475 (1998)" }, { "name": "quant-ph9912085.extracted_bib", "string": "{Lamoreaux99 S.K. Lamoreaux {Phys. Rev. {A59 R3149 (1999)" }, { "name": "quant-ph9912085.extracted_bib", "string": "{Lambrecht00 A. Lambrecht and S. Reynaud {Eur. Phys. J. {D, to appear (quant-ph/9907105)" }, { "name": "quant-ph9912085.extracted_bib", "string": "{Palik {Handbook of Optical Constants of Solids E.D. Palik ed. (Academic Press, New York 1995); {Handbook of Optics II (McGraw-Hill, New York, 1995)" }, { "name": "quant-ph9912085.extracted_bib", "string": "[*]{LKB Ecole Normale Sup\\'{erieure, Universit\\'{e Pierre et Marie Curie, et Centre National de la Recherche Scientifique" } ]
quant-ph9912087	TELEPORTATION OF\ GENERAL\ QUANTUM\ STATES	[ { "author": "Luigi Accardi$\\dagger $ and Masanori Ohya$\\ddagger $" }, { "author": "$\\dagger$ Graduate School of Polymathematics" }, { "author": "Nagoya University" }, { "author": "Chi\\-ku\\-sa--ku" }, { "author": "Na\\-go\\-ya" }, { "author": "464--01" }, { "author": "Japan" }, { "author": "and Centro V. Volterra" }, { "author": "Universit\\`{a" } ]		[ { "name": "quant-ph9912087.tex", "string": "\\documentclass[12pt]{article}\n\\usepackage{amssymb}\n\\usepackage{amsmath}\n\\usepackage{theorem}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{satz}[theorem]{Satz}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{remark}[theorem]{Remark}\n\\begin{document}\n\\author{Luigi Accardi$\\dagger $ and Masanori Ohya$\\ddagger $ \\\\\n$\\dagger$ Graduate School of Polymathematics,\\\\\nNagoya University, Chi\\-ku\\-sa--ku,\\\\\nNa\\-go\\-ya, 464--01, Japan,\\\\\nand Centro V. Volterra, Universit\\`{a} degli Studi di\\\\\nRoma ``Tor Vergata'' -- 00133 Rome, Italy\\\\\nE-mail: accardi@volterra.mat.uniroma2.it,\\\\\nand accardi@math.nagoya-u.ac.jp\\\\\n$\\ddagger $Department of Information Sciences\\\\\nScience University of Tokyo\\\\\nNoda City, Chiba 278-8510, Japan\\\\\nE-mail: ohya@is.noda.sut.ac.jp}\n\\title{TELEPORTATION OF\\ GENERAL\\ QUANTUM\\ STATES }\n\\date{}\n\\maketitle\n\n\\footnotetext{\n\\noindent {Invited talk to the: International Conference on quantum\ninformation and computer, Meijo University 1998}}\n\n\\section{Introduction}\n\nQuantum teleportation has been introduced by Benett et al. \\cite{BBCJPW}\nand discussed by a number of authors in the framework of the singlet state \\cite{BBPSSW}. Recently, a rigorous formulation of the teleportation problem of arbitrary quantum states by means of quantum channel was given in \\cite{IOS} based on the general channel theoretical formulation of the quantum gates introduced in \\cite{OW}. In this note we discuss a generalization of the scheme proposed in \\cite{IOS} and we give a general method to solve the teleportation problem in spaces of arbitrary finite dimensions.\n\n\\section{Formulation of the problem}\n\nThe set of all quantum states on a Hilbert space ${\\cal H}$, identified\nto the set of the density operators, is denoted by ${\\cal S}({\\cal H}), $ namely, \n\\[\n{\\cal S}({\\cal H})\\equiv \\{\\rho \\in B({\\cal H})\\ ;\\ \\rho ^{}=\\rho\n,\\ \\rho \\geq 0,\\ \\mathrm{tr}\\rho =1\\} \n\\]\nwhere $B({\\cal H})$ is the set of all bounded operators on ${\\cal H}$.\n\nThe following is a generalization of the channel theoretical approach to the\nteleportation problem proposed by \\cite{IOS}:\n\n\\begin{itemize}\n\\item[{STEP 0}] : Alice has a unknown quantum state $\\rho ^{(1)}$ on a\nHilbert space ${\\cal H}_{1}$, and she wants to teleport it to Bob.\n\n\\item[{STEP 1}] :Two auxiliary Hilbert spaces ${\\cal H}_{2}$ and ${\\cal H}_{3}$, attached to Alice and to Bob respectively, are introduced.\n\nOne fixes a set of (entangled) states\n\n\\begin{equation}\n\\sigma _{k}^{(23)}\\in {\\cal S}\\left( {\\cal H}_{2}\\otimes {\\cal H}_{3}\\right) \\label{2.1}\n\\end{equation}\nin the (Alice, Bob)--space, having certain prescribed correlations and one\nprepares an ensemble of the combined system $\\left( 1,2,3\\right) $ in the\nstate\n\\end{itemize}\n\n\\begin{equation}\n\\rho _{k}^{\\left( 123\\right) }\\equiv \\rho ^{\\left( 1\\right) }\\otimes \\sigma\n_{k}^{23} \\label{2.2}\n\\end{equation}\non the space ${\\cal H}_{1}\\otimes {\\cal H}_{2}\\otimes {\\cal H}_{3}$\n\n\\begin{itemize}\n\\item[{STEP 2}] : One then fixes a family of mutually orthogonal\nprojections \n\\begin{equation}\n\\left\\{ F_{k}^{(12)}\\right\\} \\label{2.3}\n\\end{equation}\non the Hilbert space ${\\cal H}_{1}\\otimes {\\cal H}_{2}$, corresponding\nto an observable $A:=\\sum_{k}\\lambda _{k}F_{k}^{(12)}$ and having fixed one index $k$, Alice performs a first kind incomplete measurement, involving only the $(1,2)$ system, which filters the (arbitrarily chosen) value $\\lambda_{k}$, i.e. after the measurement on the given ensemble (\\ref{2.2}), of identically prepared systems, only those with \n$A=\\lambda_{k}$ are allowed to pass. According to quantum mechanics, after Alice's measurement, the state of the $(1,2,3)$ system becomes \n\\begin{equation}\n\\rho _{k}^{(123)}:={\\frac{\\left( F_{k}^{(12)}\\otimes 1_{3}\\right) \\rho\n_{k}^{(123)}\\left( F_{k}^{(12)}\\otimes 1_{3}\\right) }{\\mathrm{tr}_{123}\\left( F_{k}^{(12)}\\otimes 1_{3}\\right) \\rho _{k}^{(123)}\\left(\nF_{k}^{(12)}\\otimes 1_{3}\\right) }} \\label{2.4}\n\\end{equation}\nwhere $\\mathrm{tr}_{123}$ is the full trace on the Hilbert space ${\\cal H}_{1}\\otimes {\\cal H}_{2}\\otimes {\\cal H}_{3}.$\n\n\\item[{STEP 3}] : Bob is informed which measurement was done by Alice. This\nis equivalent to transmit the information that the $k$--th eigenvalue was\nchosen. This information is transmitted from Alice to Bob without\ndisturbance and by means of completely classical tools (e.g. telephone).\n\n\\item[{STEP 4}] : By making partial measurements on the system $3$, that\nis, on the system corresponding to the auxiliary space related to him and\nnot to the original system, Bob can obtain the state $\\rho ^{(3)}$ induced\nby the state (\\ref{2.4}) and reconstruct the state $\\rho ^{(1)}$ on the\nsystem $1$by unitary keys provided to him. Notice that this state $\\rho\n^{(3)}$is unknown by Alice unless the ensemble (\\ref{2.2}) has been prepared\nby her.\n\\end{itemize}\n\n\\medskip\n\nHere we must distinguish two cases:\n\n\\begin{itemize}\n\\item[{(I)}] Bob can perform his experiments on the same ensemble of\nsystems found by Alice as a result of her measurement\n\n\\item[{(II)}] Bob and Alice are spatially separated so that the situation\nof case (I) is not realizable. In this case Bob has to prepare an ensemble\nof systems in the state $\\rho ^{(123)}$ and therefore also this state has to\nbe transmitted by Alice by means of classical communication.\n\\end{itemize}\n\nThe crucial point of the construction is that, knowing the information\ntransmitted by Alice about which measurement was done by her, Bob is able to\nreconstruct in a unique way the original state $\\rho ^{(1)}$, of system $1$,\nfrom the state $\\rho ^{(3)}$, of system $3$. \\medskip When the state $\\sigma\n_{k}^{(23)}$ is independent of $k$, the above problem reduces to the channel\ntheoretical formulation of \\cite{IOS}. \\medskip The above procedure can be\nrealized by a channel (dual of a completely positive map) \n\\[\n\\Lambda _{k}^{}:{\\cal S}\\left( {\\cal H}_{1}\\right) \\longrightarrow \n{\\cal S}\\left({\\cal H}_{3}\\right) \n\\]\ncomposed of the following four channels:\n\n\\begin{description}\n\\item[(i)] A trivial (i.e. product) lifting in the sense of \\cite{AO} \n\\begin{equation}\n\\gamma _{k}^{}:{\\cal S}\\left( {\\cal H}_{1}\\right) \\longrightarrow \n{\\cal S}\\left( {\\cal H}_{1}\\otimes {\\cal H}_{2}\\otimes {\\cal H}_{3}\\right) \\label{2.5}\n\\end{equation}\n\n\\begin{equation}\n\\gamma _{k}^{}\\left( \\rho ^{(1)}\\right) =\\rho ^{(1)}\\otimes \\sigma\n_{k}^{(23)},\\qquad \\forall \\rho ^{(1)}\\in {\\cal S}\\left( {\\cal H}_{1}\\right) \\label{2.6}\n\\end{equation}\nexpresses the independent coupling of the initial state $\\rho ^{(1)}$ with\nthe state $\\sigma _{k}^{(23)}$.\n\n\\item[(ii)] The second step is described by a measurement type channel \n\\begin{equation}\n\\pi _{k}^{}:{\\cal S}\\left( {\\cal H}_{1}\\otimes {\\cal H}_{2}\\otimes \n{\\cal H}_{3}\\right) \\longrightarrow {\\cal S}\\left( {\\cal H}_{1}\\otimes {\\cal H}_{2}\\otimes {\\cal H}_{3}\\right) \\label{2.7}\n\\end{equation}\nof the form \n\\[\n\\pi _{k}^{}\\left( \\rho _{k}^{(123)}\\right) :={\\frac{\\left(\nF_{k}^{(12)}\\otimes 1_{3}\\right) \\rho _{k}^{(123)}\\left( F_{k}^{(12)}\\otimes\n1_{3}\\right) }{\\mathrm{tr}_{123}\\left( F_{k}^{(12)}\\otimes 1_{3}\\right) \\rho\n_{k}^{(123)}\\left( F_{k}^{(12)}\\otimes 1_{3}\\right) },}\n\\]\nwhere $\\rho _{k}^{(123)}\\in {\\cal S}\\left( {\\cal H}_{1}\\otimes \n{\\cal H}_{2}\\otimes {\\cal H}_{3}\\right) $, corresponding to an\nincomplete first kind measurement describing the state change determined by\nAlice's filtering of the eigenvalue $\\lambda _{k}$ of the observable $A$\n\n\\item[(iii)] The third step is defined by the channel $a^{}:{\\cal S}\\left( {\\cal H}_{1}\\otimes {\\cal H}_{2}\\otimes {\\cal H}_{3}\\right)\n\\longrightarrow {\\cal S}\\left( {\\cal H}_{3}\\right) $ defined by \n\\[\n\\rho _{k}^{(3)}=a^{}\\left( \\rho _{k}^{(123)}\\right) =\\mathrm{tr}_{12}\\rho\n_{k}^{(123)},\\qquad \\forall \\rho _{k}^{(123)}\\in {\\cal S}\\left( {\\cal H}_{1}\\otimes {\\cal H}_{2}\\otimes {\\cal H}_{3}\\right) .\n\\]\nHere $\\mathrm{tr}_{12}$ is the partial trace on the Hilbert space ${\\cal H}_{1}\\otimes {\\cal H}_{2}$ \n\\[\n<\\Phi _{1},\\mathrm{tr}_{12}\\,Q\\Phi _{2}>\\,\\,\\equiv \\sum\\limits_{n}<\\Psi\n_{n}\\otimes \\Phi _{1},\\,Q\\Psi _{n}\\otimes \\Phi _{2}>,\\;\\; Q\\in B({\\cal H}_{1}\\otimes {\\cal H}_{2}\\otimes {\\cal H}_{3})\n\\]\nfor any CONS $\\{\\Psi _{n}\\}$ of ${\\cal H}_{1}\\otimes {\\cal H}_{2}$ and\nany $\\Phi _{1},\\Phi _{2}\\in {\\cal H}_{3}$. This channel $a^{}$\ncorresponds to Bob's partial measurement over the system $3$ and describes\nthe reduction, from the state $\\rho ^{(123)}$ obtained after Alice's\nmeasurement, to the state $\\rho ^{(3)}$, obtained by Bob. Thus the whole\nteleportation process above is written by the channel \n\\[\n\\Lambda _{k}^{}:{\\cal S}\\left( {\\cal H}_{1}\\right) \\longrightarrow \n{\\cal S}\\left( {\\cal H}_{3}\\right) \n\\]\n\\[\n\\Lambda _{k}^{}\\equiv \\Lambda _{k,A\\to B}^{}\\equiv a^{}\\circ \\pi_{k}^{}\\circ \\gamma ^{}\n\\]\n\nThe above subscript ``$k$'' means that the channels $\\Lambda _{k}^{}$ and $\\Lambda _{k,A\\to B}^{}$ depend on the choice of Alice's measurement $F_{k}^{(12)}$. More precisely, $\\forall \\rho ^{(1)}\\in {\\cal S}\\left( \n{\\cal H}_{1}\\right) $ \n\\begin{equation}\n\\Lambda _{k}^{}\\rho ^{(1)}\\equiv \\mathrm{tr}_{12}\\left[ {\\frac{\\left(\nF_{k}^{(12)}\\otimes 1_{3}\\right) \\left( \\rho ^{(1)}\\otimes \\sigma\n_{k}^{(23)}\\right) \\left( F_{k}^{(12)}\\otimes 1_{3}\\right) }{\\mathrm{tr}_{123}\\left( F_{k}^{(12)}\\otimes 1_{3}\\right) \\left( \\rho ^{(1)}\\otimes\n\\sigma _{k}^{(23)}\\right) \\left( F_{k}^{(12)}\\otimes 1_{3}\\right) }}\\right] \n\\label{2.8}\n\\end{equation}\nNote that the channel $\\Lambda _{k}^{}$ is generally non linear.\n\\end{description}\n\nWith these notations we can formulate the general mathematical problem of\nteleportation as follows. \\bigskip\n\n\\bigskip Given the initial Hilbert space ${\\cal H}_{1}$, find:\n\n\\begin{itemize}\n\\item[{(1)}] two auxiliary Hilbert spaces ${\\cal H}_{2}$, ${\\cal H}_{3}$\n\n\\item[{(2)}] a family of entangled states $\\sigma _{k}^{(23)}$ on ${\\cal H}_{2}\\otimes {\\cal H}_{3}$\n\n\\item[{(3)}] a family of mutually orthogonal projections $\\left\\{\nF_{k}^{(12)}\\right\\} $ acting on ${\\cal H}_{1}\\otimes {\\cal H}_{2}$\n\n\\item[{(4)}] for each $k$ a unitary operator $U_{k}$ such that the\nassociated unitary channel \n\\[\nu_{k}^{}:{\\cal S}\\left( {\\cal H}_{3}\\right) \\longrightarrow {\\cal S}\\left( {\\cal H}_{1}\\right) \n\\]\n\\[\nu_{k}^{}\\left( \\rho ^{(3)}\\right) =U_{k}\\rho ^{(3)}U_{k}^{}\\quad \\forall\n\\rho ^{(3)}\\in {\\cal S}\\left( {\\cal H}_{3}\\right) \n\\]\nso that it satisfies the identity \n\\begin{equation}\n\\Lambda _{k}^{}\\rho ^{(1)}=U_{k}^{}\\rho ^{(1)}U_{k} \\label{2.9}\n\\end{equation}\nfor any $k$ and for any state $\\rho ^{(1)}\\in {\\cal S}({\\cal H}_{1})$\nor at least for $\\rho ^{(1)}$ in a preassigned subset of ${\\cal S}({\\cal H}_{1})$. \\bigskip \n\\end{itemize}\n\nIf the conditions (2), (3), (4)above are replaced by the weaker ones:\n\n\\begin{itemize}\n\\item[{(2')}] a single entangled state $\\sigma ^{(23)}$acting on ${\\cal H}_{1}\\otimes {\\cal H}_{2}$\n\n\\item[{(3')}] a single projection $F^{(12)}$ acting on ${\\cal H}_{1}\\otimes {\\cal H}_{2}$\n\n\\item[{(4')}] a single unitary operator $U$ such that the identity \n\\begin{equation}\n\\Lambda ^{}\\rho ^{(1)}=U^{}\\rho ^{(1)}U \\label{2.10}\n\\end{equation}\nholds for any state $\\rho ^{(1)}\\in {\\cal S}({\\cal H}_{1})$ and the\nchannel determined by $\\sigma ^{(23)}$ and $F^{(12)},$ \\bigskip then we\nspeak of the \\mbox{weak teleportation problem}.\n\\end{itemize}\n\nThe connection between the weak and the general teleportation problem is the\nfollowing. Given a family $\\{\\sigma _{k}^{(23)},F_{k}^{(12)},U_{k}\\}$ of\nsolutions of the weak teleportation problem for each k such that the\nprojections $F_{k}^{(12)}$ are mutually orthogonal, then this family\nprovides a solution of the general teleportation problem. In the following\nsection, we shall solve the weak teleportation problem, and then we shall\nuse this result to solve the general teleportation problem.\n\n\\section{Solution of the weak teleportation problem}\n\nIn the notations of the previous section, we shall assume that \n\\[\nN=\\dim {\\cal H}_{1}<+\\infty \n\\]\nUnder this assumption we shall look for a solution of the weak teleportation\nproblem in which \n\\[\nN=\\dim {\\cal H}_{1}=\\dim {\\cal H}_{2}=\\dim {\\cal H}_{3}\n\\]\n\\begin{equation}\n\\sigma ^{(23)}=\|\\psi \\rangle \\langle \\psi \| \\label{3.1}\n\\end{equation}\n\\begin{equation}\nF:=\|\\xi \\rangle \\langle \\xi \| \\label{3.2}\n\\end{equation}\nwhere $\\psi \\in {\\cal H}_{2}\\otimes {\\cal H}_{3}$ and $\\xi \\in \n{\\cal H}_{1}\\otimes {\\cal H}_{2}$ are unit vectors. In the following\nwe identify $F$ with \n\\begin{equation}\nF=\|\\xi \\rangle \\langle \\xi \|\\otimes 1_{3}\\in \\Pr oj({\\cal H}_{1}\\otimes \n{\\cal H}_{2}\\otimes {\\cal H}_{3}) \\label{3.3}\n\\end{equation}\nand we look for a unitary transformation $U:{\\cal H}_{3}\\to {\\cal H}_{1}$ such that for any density matrix $\\rho \\in {\\cal S}({\\cal H}_{1})\n$ one has \n\\begin{equation}\nU\\cdot {\\frac{tr_{12}(F(\\rho \\otimes \|\\psi \\rangle \\langle \\psi \|)F)}{tr(F(\\rho \\otimes \|\\psi \\rangle \\langle \\psi \|)F)}}U^{}\\,=\\rho \\label{3.4}\n\\end{equation}\n\nUnder these assumptions, let us fix three arbitrary orthonormal bases: \n\\begin{equation}\n(\\varepsilon _{\\alpha })\\qquad \\mbox{of}\\qquad {\\cal H}_{3} \\label{3.5}\n\\end{equation}\n\n\\begin{equation}\n(\\varepsilon _{h}^{\\prime })\\qquad \\mbox{of}\\qquad {\\cal H}_{2}\n\\label{3.6}\n\\end{equation}\n\n\\begin{equation}\n(\\varepsilon _{n}^{\\prime \\prime })\\qquad \\mbox{of}\\qquad {\\cal H}_{1}\n\\label{3.7}\n\\end{equation}\n\n\\bigskip\n\n\\begin{proposition}\nIn the notations and assumptions of this section, fix\nan arbitrary $N\\times N$ complex unitary matrix $(\\lambda _{\\gamma \\alpha })$\nand define \n\\begin{equation}\n\\psi :=\\sum \\lambda _{h\\alpha }\|\\varepsilon _{h}^{\\prime }\\rangle \\otimes\n\|\\varepsilon _{\\alpha }\\rangle \\in {\\cal H}_{2}\\otimes {\\cal H}_{3}\n\\label{3.8}\n\\end{equation}\n\\begin{equation}\n\\xi ={\\frac{1}{N^{1/2}}}\\,\\sum_{\\mu }\\varepsilon _{\\mu }^{\\prime \\prime\n}\\otimes \\varepsilon _{\\mu }^{\\prime } \\label{3.9}\n\\end{equation}\n\\end{proposition}\n\nThen if $U:{\\cal H}_{3}\\to {\\cal H}_{1}$ is the unique unitary\noperator such that \n\\begin{equation}\n\\sum_{h}\\overline{\\lambda }_{h\\alpha }\\varepsilon _{h}^{\\prime \\prime\n}=U\\varepsilon _{\\alpha } \\label{3.10}\n\\end{equation}\n(existing because of our assumption on the $\\lambda _{k\\beta }$'s), the\ntriple $(\\psi ,\\xi ,U)$ satisfies \n\\begin{equation}\ntr_{12}(F(\\rho \\otimes \|\\psi \\rangle \\langle \\psi \|)F)={\\frac{1}{N}}U^{}\\rho U \\label{3.11}\n\\end{equation}\nfor any choice of the density operator $\\rho \\in {\\cal S}({\\cal H}_{1})\n$ and with $F$ given by (\\ref{3.2}), (\\ref{3.3}). \n\n\\bigskip\n\n\\noindent \n{\\bf Proof.} Notice that under the conditions (\\ref{3.8}) and (\\ref{3.9}), we use in a crucial way the finite dimensionality of ${\\cal H}_{2}$ and ${\\cal H}_{3}$. In particular \n\\begin{equation}\n\\Vert \\psi \\Vert ^{2}=\\sum_{h,\\alpha }\|\\lambda _{h\\alpha\n}\|^{2}=\\sum_{h}1=\\dim {\\cal H}_{2}=\\dim {\\cal H}_{3} \\label{3.12}\n\\end{equation}\n\nWe do not normalize $\\psi $ because in all the formulae the corresponding\nrank one projection will enter both in the numerator and in the denominator.\nFor $F$ as in (\\ref{3.3}), one has, using from now on the convenction of\nsummation over repeated indices \n\\[\nF={\\frac{1}{N}}\|\\varepsilon _{\\mu }^{\\prime \\prime }\\rangle \\langle\n\\varepsilon _{\\mu ^{\\prime }}^{\\prime \\prime }\|\\otimes \|\\varepsilon _{\\mu\n}^{\\prime }\\rangle \\langle \\varepsilon _{\\mu ^{\\prime }}^{\\prime }\|\\otimes\n1_{3},\n\\]\ntherefore \n\\[\nF(\\rho \\otimes \|\\psi \\rangle \\langle \\psi \|)F=\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }F(\\rho \\otimes \|\\varepsilon _{h}^{\\prime }\\rangle \\langle\n\\varepsilon _{k}^{\\prime }\|)F\\otimes \|\\varepsilon _{\\alpha }\\rangle \\langle\n\\varepsilon _{\\beta }\|=\n\\]\n\\[\n={\\frac{1}{N^{2}}}\\,\|\\varepsilon _{\\mu }^{\\prime \\prime }\\rangle \\langle\n\\varepsilon _{\\mu ^{\\prime }}^{\\prime \\prime },\\rho \\varepsilon _{\\nu\n}^{\\prime \\prime }\\rangle \\langle \\varepsilon _{\\nu ^{\\prime }}^{\\prime\n\\prime }\|\\otimes \|\\varepsilon _{\\mu }^{\\prime }\\rangle \\langle \\varepsilon\n_{\\mu ^{\\prime }}^{\\prime },\\varepsilon _{h}^{\\prime }\\rangle \\langle\n\\varepsilon _{k}^{\\prime },\\varepsilon _{\\nu }^{\\prime }\\rangle \\langle\n\\varepsilon _{\\nu ^{\\prime }}^{\\prime }\|\\otimes \|\\varepsilon _{\\alpha\n}\\rangle \\langle \\varepsilon _{\\beta }\|=\n\\]\n\\[\n={\\frac{1}{N^{2}}}\\,\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }\\delta\n_{\\mu ^{\\prime },h}\\delta _{k,\\nu }\\langle \\varepsilon _{\\mu ^{\\prime\n}}^{\\prime \\prime },\\rho \\varepsilon _{\\nu }^{\\prime \\prime }\\rangle\n\|\\varepsilon _{\\mu }^{\\prime \\prime }\\rangle \\langle \\varepsilon _{\\nu\n^{\\prime }}^{\\prime \\prime }\|\\otimes \|\\varepsilon _{\\mu }^{\\prime }\\rangle\n\\langle \\varepsilon _{\\nu ^{\\prime }}^{\\prime }\|\\otimes \|\\varepsilon\n_{\\alpha }\\rangle \\langle \\varepsilon _{\\beta }\|=\n\\]\n\\[\n={\\frac{1}{N^{2}}}\\,\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }\\langle\n\\varepsilon _{h}^{\\prime \\prime },\\rho \\varepsilon _{k}^{\\prime \\prime\n}\\rangle \|\\varepsilon _{\\mu }^{\\prime \\prime }\\rangle \\langle \\varepsilon\n_{\\nu ^{\\prime }}^{\\prime \\prime }\|\\otimes \|\\varepsilon _{\\mu }^{\\prime\n}\\rangle \\langle \\varepsilon _{\\nu ^{\\prime }}^{\\prime }\|\\otimes\n\|\\varepsilon _{\\alpha }\\rangle \\langle \\varepsilon _{\\beta }\|\n\\]\n\nTaking the trace over ${\\cal H}_{1}\\otimes {\\cal H}_{2},$ this gives \n\\[\n={\\frac{1}{N}}\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }\\langle\n\\varepsilon _{h}^{\\prime \\prime },\\rho \\varepsilon _{k}^{\\prime \\prime\n}\\rangle \|\\varepsilon _{\\alpha }\\rangle \\langle \\varepsilon _{\\beta }\|={\\frac{1}{N}}\\langle \\overline{\\lambda }_{h\\alpha }\\varepsilon _{h}^{\\prime\n\\prime },\\rho \\overline{\\lambda }_{k\\beta }\\varepsilon _{k}^{\\prime \\prime\n}\\rangle \|\\varepsilon _{\\alpha }\\rangle \\langle \\varepsilon _{\\beta }\|\n\\]\nThus, with the choice of $U$ given by (10), (2) becomes \n\\[\ntr_{12}(F(\\rho \\otimes \|\\psi \\rangle \\langle \\psi \|)F)={\\frac{1}{N}}\\langle\n\\varepsilon _{\\alpha },U^{}\\rho U\\varepsilon _{\\beta }\\rangle \|\\varepsilon_{\\alpha }\\rangle \\langle \\varepsilon _{\\beta }\|={\\frac{1}{N}}U^{}\\rho U\n\\]\nwhich is (\\ref{3.11}). \n\n\\section{Uniqueness of the key}\n\nIn this section we discuss the uniqueness of the key uintary operator.\n\n\\begin{proposition}\nLet $\\rho=\\sum p_{\\gamma}P_{\\gamma}$ be the spectral decomposition of $\\rho \\in {\\cal H}_{1}$. Then if $U$ and $V$ satisfy equation\n (\\ref{2.9}) with the above $\\rho$, then there exists a unitary operator $W$ from ${\\cal H}_{1}$ to ${\\cal H}_{1}$ such that $VU^{}=\\Sigma W_{\\gamma }$ with $W_{\\gamma }\\equiv P_{\\gamma }WP_{\\gamma }$. Moreover, the equality $W_{\\gamma }W_{\\gamma }^{}=\\delta _{\\gamma \\gamma ^{\\prime }}P_{\\gamma}$ is satisfied.\n\\end{proposition}\n\n{\\bf Proof.} Suppose $U$ and $V$ are two solutions of equation $1_{3}$,\nthen \n\\[\nU^{}\\rho U=V^{}\\rho V\n\\]\nor, equivalently \n\\begin{equation}\nVU^{}\\rho =\\rho VU^{} \\label{4.1}\n\\end{equation}\nThis means that $VU^{}\\equiv W:{\\cal H}_{1}\\to {\\cal H}_{1}$ is in\nthe commutant of $\\rho $. Since $W$ is a unitary operator commuting with $\\rho$, \n\\[\nWP_{\\gamma }=P_{\\gamma }W\n\\]\nis satisfied. Therefore \n\\[\nW=\\sum P_{\\gamma }WP_{\\gamma }=\\sum W_{\\gamma }:W_{\\gamma }P_{\\gamma\n}=P_{\\gamma }W_{\\gamma }=W_{\\gamma }.\n\\]\nThe equalities \n\\[\n1=W^{}W=\\sum_{\\gamma \\gamma ^{\\prime }}W_{\\gamma }^{}W_{\\gamma ^{\\prime\n}}=\\sum_{\\gamma \\gamma ^{\\prime }}W_{\\gamma }^{}P_{\\gamma }P_{\\gamma\n^{\\prime }}W_{\\gamma }=\\sum_{\\gamma }W_{\\gamma }^{}W_{\\gamma }.\n\\]\nimply \n\\[\nW_{\\gamma }W_{\\gamma ^{\\prime }}^{}=\\delta _{\\gamma \\gamma ^{\\prime\n}}P_{\\gamma }.\\;\\;\\hfill \n\\]\n\\bigskip \n\n\\begin{corollary}\n\\noindent Let ${\\cal H}$ be an Hilbert space of arbitrary dimensions. If $U$ and $V$ are two solutions of the weak teleportation problem corresponding\nto the same $F$ and $\\psi ,$ then they coincide up to multiplication by a\nnumber of modulus one. \n\\end{corollary}\n\n\\bigskip\n\n\\noindent \n{\\bf Proof.} Equation (\\ref{4.1}) above implies that in this\ncase the operator $VU^{}$ commutes with all density operators, since they\nare unitary, it follows that $V=e^{i\\theta }U$ for some real number $\\theta $.\n\n\\section{A necessary condition}\n\nIn the notations and assumptions of the previous section, let us suppose\nthat the normalized state vector $\\xi $ has the form with some constants $\\left\\{ t_{\\mu }\\right\\} $ \n\\begin{equation}\n\\xi =\\sum_{\\mu }t_{\\mu }\\varepsilon _{\\mu }^{\\prime \\prime }\\otimes\n\\varepsilon _{\\mu }^{\\prime } \\label{5.1}\n\\end{equation}\nand let us look for the conditions under which the map (\\ref{3.4}) becomes\nlinear. \\bigskip \n\n\\bigskip\n\n\\begin{proposition}\nGiven $\\xi $ as in (\\ref{5.1}) and $F$ as in (\\ref{3.2}) the trace \n\\[\ntr_{123}(F(\\rho \\otimes \|\\psi \\rangle \\langle \\psi \|F)\n\\]\nis independent of $\\rho $ if and only if the coefficients $t_{k}$ of $\\xi $\nhave the following form \n\\begin{equation}\nt_{k}=e^{i\\theta _{k}}/\\sqrt{N} \\label{5.2}\n\\end{equation}\nIn particular, if the condition (\\ref{5.2}) above is satisfied, then the map\n(\\ref{3.4}) linearizes. \n\\end{proposition}\n\n\\bigskip\n\n\\noindent \n{\\bf Proof.} One has: \n\\[\nF(\\rho \\otimes \|\\psi \\rangle \\langle \\psi \|)F=\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }F(\\rho \\otimes \|\\varepsilon _{h}^{\\prime }\\rangle \\langle\n\\varepsilon _{k}^{\\prime }\|)F\\otimes \|\\varepsilon _{\\alpha }\\rangle \\langle\n\\varepsilon _{\\beta }\| \n\\]\n\\[\n=\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }t_{\\mu }\\overline{t}_{\\mu\n^{\\prime }}t_{\\nu }\\overline{t}_{\\nu ^{\\prime }}\|\\varepsilon _{\\mu }^{\\prime\n\\prime }\\rangle \|\\varepsilon _{\\mu }^{\\prime }\\rangle \\langle \\varepsilon\n_{\\mu ^{\\prime }}^{\\prime }\|(\\rho \\otimes \|\\varepsilon _{h}^{\\prime }\\rangle\n\\langle \\varepsilon _{k}^{\\prime }\|)\|\\varepsilon _{\\nu }^{\\prime \\prime\n}\\rangle \|\\varepsilon _{\\nu }^{\\prime }\\rangle \\langle \\varepsilon _{\\nu\n^{\\prime }}^{\\prime \\prime }\|\\langle \\varepsilon _{\\nu ^{\\prime }}^{\\prime\n}\|\\otimes \|\\varepsilon _{\\alpha }\\rangle \\langle \\varepsilon _{\\beta }\| \n\\]\n\\[\n=\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }t_{\\mu }\\overline{t}_{\\mu\n^{\\prime }}t_{\\nu }\\overline{t}_{\\nu ^{\\prime }}\|\\varepsilon _{\\mu }^{\\prime\n\\prime }\\rangle \\langle \\varepsilon _{\\mu ^{\\prime }}^{\\prime \\prime },\\rho\n,\\varepsilon _{\\nu }^{\\prime \\prime }\\rangle \\langle \\varepsilon _{\\nu\n^{\\prime }}^{\\prime \\prime }\|\\otimes \|\\varepsilon _{\\mu }^{\\prime }\\langle\n\\varepsilon _{\\mu ^{\\prime }}^{\\prime },\\varepsilon _{h}^{\\prime }\\rangle\n\\langle \\varepsilon _{k}^{\\prime },\\varepsilon _{\\nu }^{\\prime }\\rangle\n\\langle \\varepsilon _{\\nu ^{\\prime }}^{\\prime }\|\\otimes \|\\varepsilon\n_{\\alpha }\\rangle \\langle \\varepsilon _{\\beta }\| \n\\]\n\\[\n=\\lambda _{h\\alpha }\\overline{\\alpha }_{k\\beta }t_{\\mu }\\overline{t}_{h}t_{k} \\overline{t}_{\\nu },\\langle \\varepsilon _{h}^{\\prime },\\rho\n\\varepsilon _{k}^{\\prime \\prime }\\rangle \|\\varepsilon _{\\mu }^{\\prime \\prime\n}\\rangle \\langle \\varepsilon _{\\nu ^{\\prime }}^{\\prime \\prime }\|\\otimes\n\|\\varepsilon _{\\mu }^{\\prime }\\rangle \\langle \\varepsilon _{\\nu ^{\\prime\n}}^{\\prime }\|\\otimes \|\\varepsilon _{\\alpha }\\rangle \\langle \\varepsilon\n_{\\beta }\| \n\\]\n\nTracing over ${\\cal H}_{1}\\otimes {\\cal H}_{2},$ \n\\[\n\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }t_{\\mu }\\overline{t}_{h}t_{k} \\overline{t}_{\\mu }\\langle \\varepsilon _{h}^{\\prime \\prime },\\rho\n\\varepsilon _{k}^{\\prime \\prime }\\rangle \|\\varepsilon _{\\alpha }\\rangle\n\\langle \\varepsilon _{\\beta }\| \n\\]\nSince $\\sum_{\\mu }\|t_{\\mu }\|^{2}=1$, this is equal to \n\\[\n\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\beta }\\overline{t}_{h}t_{k}\\langle\n\\varepsilon _{h}^{\\prime \\prime },\\rho \\varepsilon _{k}^{\\prime \\prime\n}\\rangle \|\\varepsilon _{\\alpha }\\rangle \\langle \\varepsilon _{\\beta }\| \n\\]\n\nTaking the ${\\cal H}_{3}$--trace, we find \n\\[\n\\lambda _{h\\alpha }\\overline{\\lambda }_{k\\alpha }\\overline{t}_{h}t_{k}\\langle \\varepsilon _{h}^{\\prime \\prime },\\rho \\varepsilon\n_{k}^{\\prime \\prime }\\rangle =\|t_{h}\|^{2}\\langle \\varepsilon _{h}^{\\prime\n\\prime },\\rho \\varepsilon _{h}^{\\prime \\prime }\\rangle \n\\]\nbecause of the unitarity of $\\lambda _{\\alpha ,\\beta }$. \\bigskip If follows\nthat the problem linearizes if and only if \n\\[\n\\sum_{k}\|t_{k}\|^{2}\|\\varepsilon _{k}^{\\prime \\prime }\\rangle \\langle\n\\varepsilon _{k}^{\\prime \\prime }\|=c=\\mbox{constant}\n\\]\nand this is equivalent to: \n\\[\n\|t_{k}\|^{2}=c\\ ;\\quad \\forall \\,k=1,\\dots ,N\n\\]\nConsequently $c=N$ and (\\ref{5.2}) follows. \\hfill \n\n\\section{{Solution of the general teleportation problem: the case $N=2^{2m}$}\n}\n\nThe remark at the end of section 2 and the result of section 4 show that, in\norder to apply the solution of the weak teleportation problem to the\nsolution of the general one, one has to solve the the following problem:\nfind an o.n. basis $f_{n}$ of a finite dimensional Hilbert space ${\\cal H}\n$, identified to $\\mathbf{C}^{N}$, such that one can determine the phases $\n\\sigma _{\\alpha ,j}$ ($j=1,\\ldots ,N$) so that the vectors \n\\[\ng_{\\alpha }:={\\frac{1}{N^{1/2}}}\\sum_{j=1}^{N}e^{i\\sigma _{\\alpha ,j}}f_{j} \n\\]\nare still an o.n. basis of $\\mathbf{C}^{N}$. If we further restrict the\ncondition requiring that $e^{i\\sigma _{\\alpha ,j}}=\\pm 1$, then by\nconsidering the two vectors $\\psi _{\\alpha }=\\sum_{j=1}^{N}\\varepsilon\n_{j}^{\\alpha }f_{j}$, $\\psi _{\\beta }=\\sum_{j=1}^{N}\\varepsilon _{j}^{\\beta\n}f_{j}$ with $\\varepsilon _{j}^{\\alpha },$ $\\psi _{\\beta }=\\pm 1$, the\nconditions \n\\[\n0=\\langle \\psi _{\\alpha },\\psi _{\\beta }\\rangle =\\sum_{j}\\varepsilon\n_{j}^{\\alpha }\\varepsilon _{j}^{\\beta } \n\\]\nshow that a necessary condition for an affirmative answer to the above\nproblem is that $N$ is an even number. \\bigskip\n\n\\bigskip\n\n\\begin{proposition}\n\\noindent If $N=2^{m}$ for some $m\\in \\mathbf{N,}$ then the answer to the\nabove problem is affirmative.\n\\end{proposition}\n\n\\bigskip\n\n\\noindent \n{\\bf Proof.} The statement is true for $m=1$. Assume by\ninduction that it is true for $m$ and consider $\\mathbf{C}^{2^{2m+1}}\\equiv \n\\mathbf{C}^{2^{2m}}\\otimes \\mathbf{\\ C}^{2}\\equiv \\mathbf{C}^{2^{m}}\\oplus \n\\mathbf{C}^{2^{m}}$.\n\nLet $(\\psi _{\\alpha })$ $(\\alpha =1,\\dots ,2^{m})$ be an o.n. basis which\nsolves the problem for $\\mathbf{C}^{2^{m}}$. Then clearly the set of vectors \n\\[\n{\\frac{1}{\\sqrt{2}}}\\,\\left( \n\\begin{array}{l}\n{\\psi _{\\alpha }} \\\\ \n{+\\psi _{\\alpha }}\n\\end{array}\n\\right) ,\\quad {\\frac{1}{\\sqrt{2}}}\\,\\,\\left( \n\\begin{array}{l}\n{\\psi _{\\alpha ^{\\prime }}} \\\\ \n{-\\psi _{\\alpha ^{\\prime }}}\n\\end{array}\n\\right) \n\\]\nis an o.n. basis of $\\mathbf{C}^{2^{m+1}}$ in $Q_{2^{m+1}}$.\n\nA more explicit solution of the problem is obtained as follows. We fix \n\\[\nN=2^{m} \n\\]\nand we choose \n\\[\nf_{\\nu }:=e_{\\nu _{1}}\\otimes \\dots \\otimes e_{\\nu _{m}}\\ ;\\quad \\nu =(\\nu\n_{1},\\dots ,\\nu _{m})\\in \\{0,1\\}^{m} \n\\]\n\\[\n\\nu _{j}\\in \\{0,1\\}\\ ;\\quad j=1,\\dots ,m\\ ;\\quad e_{1}=\\left( \n\\begin{array}{l}\n{1} \\\\ \n{0}\n\\end{array}\n\\right) ,\\ e_{0}=\\left( \n\\begin{array}{l}\n{0} \\\\ \n{1}\n\\end{array}\n\\right) \n\\]\n\nWe know that \n\\begin{equation}\ng_{0}:={\\frac{1}{\\sqrt{2}}}\\,(e_{1}+e_{0})={\\frac{1}{\\sqrt{2}}}\\,\\left( \n\\begin{array}{l}\n{1} \\\\ \n{1}\n\\end{array}\n\\right) ;\\quad g_{1}:={\\frac{1}{\\sqrt{2}}}\\,(e_{1}-e_{0})={\\frac{1}{\\sqrt{2}}}\\,\\left( \n\\begin{array}{l}\n{1} \\\\ \n-{1}\n\\end{array}\n\\right) \n\\end{equation}\nis an o.n. basis of $\\mathbf{C^{2}}$ so that \n\\begin{equation}\ng_{\\alpha }:=g_{\\alpha _{1}}\\otimes \\dots \\otimes g_{\\alpha _{m}}\\quad\n;\\qquad \\alpha :=(\\alpha _{1},\\dots ,\\alpha _{m})\\in \\{0,1\\}^{m}\n\\end{equation}\nis an o.n. basis of $\\bigotimes\\limits_{1}^{m}\\mathbf{C}^{2}=\\mathbf{C}^{2^{m}}$. However we have \n\\begin{eqnarray}\ng_{\\alpha _{1}}\\otimes \\dots \\otimes g_{\\alpha _{m}} &=&{\\frac{1}{2^{m/2}}}\\,\\bigotimes\\limits_{j=1}^{m}(e_{1}+(-1)^{\\alpha _{j}}e_{0}) \\nonumber \\\\\n{} &=&{\\frac{1}{2^{m/2}}}\\,\\sum_{\\nu _{1},\\dots ,\\nu\n_{m}}(-1)^{\\sum\\limits_{j=1}^{m}(1-\\nu _{j})\\alpha _{j}}e_{\\nu _{1}}\\otimes\n\\dots \\otimes e_{\\nu _{m}} \\nonumber \\\\\n{} &=&{\\frac{1}{2^{m/2}}}\\,\\sum_{\\nu \\in \\{0,1\\}^{m}}(-1)^{\\sigma _{\\alpha\n}(\\nu )}e_{\\nu }=g_{\\alpha }\\quad \\quad \\quad \\quad \\quad \\quad \\quad \n\\label{6.3}\n\\end{eqnarray}\nwhere \n\\[\n\\sigma _{\\alpha }(\\nu ):=\\sum_{j=1}^{m}(1-\\nu _{j})\\alpha _{j}.\\hfill \n\\]\n\nIn the notations of section 3, let $N=2^{m}$ for some $m\\in \\mathbf{N}$ and\nlet the orthonormal bases $(\\varepsilon _{\\alpha }^{\\prime })$ of ${\\cal H}_{2}$ and $(\\varepsilon _{\\alpha }^{\\prime \\prime })$ of ${\\cal H}_{1}$\nin (\\ref{3.6}), (\\ref{3.7}) be two copies of the basis $e_{\\nu _{1}}\\otimes\n\\dots \\otimes e_{\\nu _{m}}$, described in Proposition 6.1. Let $(\\varepsilon\n_{\\alpha })$ be an arbitrary orthonormal basis of ${\\cal H}_{3}$ and let $\\psi $ and $U$ be as in Proposition 3.1. Then, it is easy to see the\nfollowimg corollary.\n\n\\bigskip\n\n\\begin{corollary}\n\\noindent If the vector $\\xi _{\\alpha }$ for each $\\alpha \\in \\{0,1\\}^{m}$\nis defined by \n\\begin{equation}\n\\xi _{\\alpha }:={\\frac{1}{2^{m/2}}}\\,\\sum_{\\mu \\in \\{0,1\\}^{m}}(-1)^{\\sigma\n_{\\alpha }(\\nu )}\\varepsilon _{\\mu }^{\\prime }\\varepsilon _{\\mu }^{\\prime\n\\prime }\n\\end{equation}\nthen, for each $\\alpha \\in \\{0,1\\}^{m}$, the triple $\\{\\psi ,U,\\xi _{\\alpha\n}\\}$ solves the weak teleportation problem and the projections \n\\[\nF_{\\alpha }:=\|\\xi _{\\alpha }\\rangle \\langle \\xi _{\\alpha }\|\\otimes 1_{3}\n\\]\nare mutually orthogonal. \n\\end{corollary}\n\n\\medskip\n\n\\begin{remark}\n\\noindent In the above construction, nothing prevents the possibility of\nchoosing a different unitary matrix $(\\lambda _{j,k})$ for each $\\alpha \\in\n\\{0,1\\}^{m}$ so that the triple $\\{\\psi _{\\alpha },U_{\\alpha },\\xi _{\\alpha\n}\\}$ solves the teleportation problem in the general formulation of section\n2. \n\\end{remark}\n\nWe here notice that the BBCJPW scheme provides nice examples to our results.\nIn their scheme, $\\sigma ^{(23)}$ is given by an EPR spin pair in a singlet\nstate such as \n\\[\n\\sigma ^{(23)}=\|\\psi \\rangle \\langle \\psi \|, \n\\]\nwhere\n\n\\[\n\|\\psi \\rangle =c\|\\uparrow ^{(2)}\\rangle \\otimes \|\\downarrow ^{(3)}\\rangle\n+d\|\\downarrow ^{(2)}\\rangle \\otimes \|\\uparrow ^{(3)}\\rangle ,\\quad \\left\|\nc\\right\| ^{2}+\\left\| d\\right\| ^{2}=1. \n\\]\nwith the spin up vector $\|\\uparrow \\rangle :=$ $e_{1}=\\left( \n\\begin{array}{l}\n{1} \\\\ \n{0}\n\\end{array}\n\\right) $ and the spin down vector $\|\\downarrow \\rangle :=e_{0}=\\left( \n\\begin{array}{l}\n{0} \\\\ \n{1}\n\\end{array}\n\\right) .$ There, Alice's measurement $F_{k}^{(12)}$ is chosen in the\npartition of the identity $\\left\\{ F_{k}^{(12)};k=1,2,3,4\\right\\} $ ;\n\n\\begin{eqnarray}\nF_{1}^{(12)} &=&\|\\xi ^{(-)}\\rangle \\langle \\xi ^{(-)}\|,\\ F_{2}^{(12)}=\|\\xi\n^{(+)}\\rangle \\langle \\xi ^{(+)}\|,\\ \\\\\nF_{3}^{(12)} &=&\|\\zeta ^{(-)}\\rangle \\langle \\zeta ^{(-)}\|,\\\nF_{4}^{(12)}=\|\\zeta ^{(+)}\\rangle \\langle \\zeta ^{(+)}\|\n\\end{eqnarray*}\nwith \n\\begin{eqnarray}\n\|\\xi ^{(-)}\\rangle &=&\\sqrt{\\frac{1}{2}}\\left( \|\\uparrow ^{(1)}\\rangle\n\\otimes \|\\downarrow ^{(2)}\\rangle -\|\\downarrow ^{(1)}\\rangle \\otimes\n\|\\uparrow ^{(2)}\\rangle \\right) \\nonumber \\\\\n\|\\xi ^{(+)}\\rangle &=&\\sqrt{\\frac{1}{2}}\\left( \|\\uparrow ^{(1)}\\rangle\n\\otimes \|\\downarrow ^{(2)}\\rangle +\|\\downarrow ^{(1)}\\rangle \\otimes\n\|\\uparrow ^{(2)}\\rangle \\right) \\nonumber \\\\\n\|\\zeta ^{(-)}\\rangle &=&\\sqrt{\\frac{1}{2}}\\left( \|\\uparrow ^{(1)}\\rangle\n\\otimes \|\\uparrow ^{(2)}\\rangle -\|\\downarrow ^{(1)}\\rangle \\otimes\n\|\\downarrow ^{(2)}\\rangle \\right) \\nonumber \\\\\n\|\\zeta ^{(+)}\\rangle &=&\\sqrt{\\frac{1}{2}}\\left( \|\\uparrow ^{(1)}\\rangle\n\\otimes \|\\uparrow ^{(2)}\\rangle +\|\\downarrow ^{(1)}\\rangle \\otimes\n\|\\downarrow ^{(2)}\\rangle \\right) . \\nonumber\n\\end{eqnarray}\n\nThe unitary (key) operators $U_{k}$ ($k=1,2,3,4)$ are given as follows \n\\[\n\\begin{array}{l}\nU_{1}\\equiv \|\\uparrow ^{(1)}\\rangle \\langle \\uparrow ^{(3)}\| +\|\\downarrow\n^{(1)}\\rangle \\langle \\downarrow ^{(3)}\| \\\\ \nU_{2}\\equiv \|\\uparrow ^{(1)}\\rangle \\langle \\uparrow ^{(3)}\| -\|\\downarrow\n^{(1)}\\rangle \\langle \\downarrow ^{(3)}\| \\\\ \nU_{3}\\equiv \|\\uparrow ^{(1)}\\rangle \\langle \\downarrow ^{(3)}\| +\|\\downarrow\n^{(1)}\\rangle \\langle \\uparrow ^{(3)}\| \\\\ \nU_{4}\\equiv \|\\uparrow ^{(1)}\\rangle \\langle \\downarrow ^{(3)}\| -\|\\downarrow\n^{(1)}\\rangle \\langle \\uparrow ^{(3)}\|.\n\\end{array}\n\\]\n\nIn \\cite{BBCJPW}, they discussed a more general example in which $\\sigma_{23}=\|\\psi\\rangle \\langle \\psi\|$ $\\in {\\cal S}({\\cal H}_{2}\\otimes {\\cal H}_{3})$ with ${\\cal H}_{2}={\\cal H}_{3})= \n\\mathbf{C}^N$ and \n\\[\n\|\\psi \\rangle =\\sum_{j=0}^{N-1}\\frac{1}{\\sqrt{N}}\|j\\rangle \\otimes \|j\\rangle\n.\\quad \n\\]\nIn this case, Alice's measurement is performed with projections of the form\n\n\\[\nF_{nm}^{(12)}=\|\\xi _{nm}\\rangle \\langle \\xi _{nm}\| \n\\]\nwhere \n\\[\n\|\\xi _{nm}\\rangle =\\frac{1}{\\sqrt{N}}\\sum_{j=0}^{N-1}e^{\\frac{2\\pi ijn}{N}}\|j\\rangle \\otimes \|\\left( j+m\\right) \\ \\mathrm{mod}\\ N\\rangle . \n\\]\n\nThese states are examples of the entangled states and Alice's projections in\nour general teleportation scheme.\n\nFinally we note that the channels constructed by the above states are\nlinear, and they could show the teleportation of the initial state attached\nto ${\\cal H}_{1}$.\n\n\\begin{thebibliography}{9}\n\\bibitem{AO} Accardi L., Ohya M.: Compound channels, transition\nexpectations and liftings, \\emph{Applied Mathematics} \\& \\emph{Optimization}. 39 (1999) 33--59 Volterra preprint N. 75 (1991).\n\n\\bibitem{AC97} Accardi L. An open system approach to quantum computers in:\nQuantum Communication and Measurement, O. Hirota, A.S. Holevo, C.M. Caves\n(eds.), Plenum Press (1997) 387-393 Preprint Volterra N. 267 (1997)\n\n\\bibitem{BBCJPW} Bennett C.H., Brassard G., Cr\\'{e}peau C., Jozsa R., Peres\nA., Wootters W. K.:Teleporting an unknown quantum state via Dual Classical\nand Einstein-Podolsky-Rosen channels, \\emph{Phys. Rev. Lett.} \n{\\bf 70}, pp.1895--1899 (1993) .\n\n\\bibitem{BBPSSW} Bennett C.H., Brassard G., Popescu S.: Schumacher B.,\nSmolin J.A, Wootters W.K, Purification of noisy entanglement and faithful\nteleportation via noisy channels, \\emph{Phys. Rev. Lett.} {\\bf 76}, \npp.722--725(1996).\n\n\\bibitem{OW} Ohya M., Watanabe N.: On Mathematical treatment of\nFredkin-Toffoli-Milburn gate, \\emph{Physica D, } {\\bf 120}, pp.206-213\n(1998).\n\n\\bibitem{IOS} Inoue K., Ohya M., Suyari H.: Characterization of quantum\nteportation processes by nonlinear quantum mutual entropy, \\emph{Physica D, }\n{\\bf 120}, pp.117-124 (1998).\n\n\\bibitem{MYOY} Motoyoshi A., Yamaguchi K., Ogura T., Yoneda T.: A set of\nconditions for teleportation without resort to von Neumann's projection\npostulate, \\emph{Progress of Theoretical Physics}, No.5, pp.819-824 (1997).\n\n\\bibitem{Tur} Turton R.: ''The Quantum Dot'', Oxford U.P. (1996).\n\\end{thebibliography}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912087.extracted_bib", "string": "{AO Accardi L., Ohya M.: Compound channels, transition expectations and liftings, Applied Mathematics \\& Optimization. 39 (1999) 33--59 Volterra preprint N. 75 (1991)." }, { "name": "quant-ph9912087.extracted_bib", "string": "{AC97 Accardi L. An open system approach to quantum computers in: Quantum Communication and Measurement, O. Hirota, A.S. Holevo, C.M. Caves (eds.), Plenum Press (1997) 387-393 Preprint Volterra N. 267 (1997)" }, { "name": "quant-ph9912087.extracted_bib", "string": "{BBCJPW Bennett C.H., Brassard G., Cr\\'{epeau C., Jozsa R., Peres A., Wootters W. K.:Teleporting an unknown quantum state via Dual Classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. {70, pp.1895--1899 (1993) ." }, { "name": "quant-ph9912087.extracted_bib", "string": "{BBPSSW Bennett C.H., Brassard G., Popescu S.: Schumacher B., Smolin J.A, Wootters W.K, Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett. {76, pp.722--725(1996)." }, { "name": "quant-ph9912087.extracted_bib", "string": "{OW Ohya M., Watanabe N.: On Mathematical treatment of Fredkin-Toffoli-Milburn gate, Physica D, {120, pp.206-213 (1998)." }, { "name": "quant-ph9912087.extracted_bib", "string": "{IOS Inoue K., Ohya M., Suyari H.: Characterization of quantum teportation processes by nonlinear quantum mutual entropy, Physica D, {120, pp.117-124 (1998)." }, { "name": "quant-ph9912087.extracted_bib", "string": "{MYOY Motoyoshi A., Yamaguchi K., Ogura T., Yoneda T.: A set of conditions for teleportation without resort to von Neumann's projection postulate, Progress of Theoretical Physics, No.5, pp.819-824 (1997)." }, { "name": "quant-ph9912087.extracted_bib", "string": "{Tur Turton R.: ''The Quantum Dot'', Oxford U.P. (1996)." } ]
quant-ph9912089	Separability of entangled q-bit pairs	[ { "author": "Berthold-Georg Englert$^{\\ast\\dagger" } ]	% The state of an entangled q-bit pair is specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a $3\times3$-dyadic. There are easy-to-use criteria to check whether a given pair of 3-vectors plus a dyadic specify a 2--q-bit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state. Some progress has been made in the search for analytical expressions for the degree of separability. We report, in particular, the answer in the case of vanishing 3-vectors.\\[0.5\baselineskip] PACS numbers: 89.70.+c, 03.65.Bz	[ { "name": "quant-ph9912089.tex", "string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% %\n% LaTeX source file in REVTeX style for submission %\n% to the LANL preprint server. %\n% %\n% This paper has been submitted to % \n% Journal of Modern Optics %\n% as a contribution to the Proceedings of the %\n% Workshop on Entanglement and Decoherence, held %\n% at Gargnano/Italy, 20-25 September 1999. %\n% %\n% This file replaces quant-ph/9912089 as submitted %\n% originally. Alterations concern Section IV.C. %\n% %\n% Original version sent/received: 18 December 1999 %\n% This corrected version sent/rec'd: 27 January 2000 %\n% %\n% All correspondence should be directed to: %\n% Berthold-Georg Englert %\n% MPI fur Quantenoptik %\n% Hans-Kopfermann-Str. 1 %\n% 85748 Garching %\n% Germany %\n% %\n% phone: +49-89-32905-731 %\n% fax: +49-89-32905-200 %\n% email: bge@mpq.mpg.de %\n% %\n% An ASCII table is at the end of the document. % \n% %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\documentstyle[pra,aps,preprint,floats]{revtex} \n\n\n%%%%%%%%%% makros defined here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\makeatletter\n\\newcommand{\\DoS}{{\\mathcal{S}}}\n%\n\\newcommand{\\row}[1]%\n{\\mathord{\\buildrel{\\lower3pt%\n\\hbox{$\\scriptscriptstyle\\rightarrow$}}\\over 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modulus\n%\n\\newcommand{\\magn}[1]{%\n{\\mathchoice{\\magn@D{#1}}{\\magn@T{#1}}{\\magn@S{#1}}{\\magn@SS{#1}}}}\n%\n\\newlength{\\@xx}\\newlength{\\@yy}\\newlength{\\@zz}\n\\setlength{\\@zz}{0.055em}\n%\n\\newcommand{\\magn@D}[1]{%\n\\settoheight{\\@xx}{$\\displaystyle\\left\|#1\\right\|$}%\n\\settodepth{\\@yy}{$\\displaystyle\\left\|#1\\right\|$}%\n\\addtolength{\\@xx}{\\@yy}%\n{\\,\\rule[-\\@yy]{\\@zz}{\\@xx}\\,#1\\,\\rule[-\\@yy]{\\@zz}{\\@xx}\\,}}\n%\n\\newcommand{\\magn@T}[1]{%\n\\settoheight{\\@xx}{$\\textstyle\\left\|#1\\right\|$}%\n\\settodepth{\\@yy}{$\\textstyle\\left\|#1\\right\|$}%\n\\addtolength{\\@xx}{\\@yy}%\n{\\,\\rule[-\\@yy]{\\@zz}{\\@xx}\\,#1\\,\\rule[-\\@yy]{\\@zz}{\\@xx}\\,}}\n%\n\\newcommand{\\magn@S}[1]{%\n\\settoheight{\\@xx}{$\\scriptstyle\\left\|#1\\right\|$}%\n\\settodepth{\\@yy}{$\\scriptstyle\\left\|#1\\right\|$}%\n\\addtolength{\\@xx}{\\@yy}%\n{\\,\\rule[-\\@yy]{\\@zz}{\\@xx}\\,#1\\,\\rule[-\\@yy]{\\@zz}{\\@xx}\\,}}\n%\n\\newcommand{\\magn@SS}[1]{%\n\\settoheight{\\@xx}{$\\scriptscriptstyle\\left\|#1\\right\|$}%\n\\settodepth{\\@yy}{$\\scriptscriptstyle\\left\|#1\\right\|$}%\n\\addtolength{\\@xx}{\\@yy}%\n{\\,\\rule[-\\@yy]{\\@zz}{\\@xx}\\,#1\\,\\rule[-\\@yy]{\\@zz}{\\@xx}\\,}}\n%\n\\newcommand{\\olmagn}[1]{\\magn{\\raisebox{0pt}[0pt][0pt]{$#1$}}}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\makeatother\n%%%%%%% end of makro definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\tighten\n\n\\begin{document}\n\n\\title{Separability of entangled q-bit pairs}\n\\author{Berthold-Georg Englert$^{\\ast\\dagger}$ \nand Nasser Metwally$^{\\ddagger}$}\n\\address{%\n$^\\ast$Max-Planck-Institut f\\\"ur Quantenoptik, \nHans-Kopfermann-Strasse 1, 85748 Garching, Germany\\protect\\\\\n$^\\dagger$Abteilung f\\\"ur Quantenphysik, Universit\\\"at Ulm,\nAlbert-Einstein-Allee~1, 89081 Ulm, Germany\\protect\\\\\n$^\\ddagger$Sektion Physik, Universit\\\"at M\\\"unchen, \nTheresienstra\\ss{}e 37, 80333 M\\\"unchen, Germany}\n\\date{18 December 1999; 25 January 2000}\n\n\\maketitle\n\\begin{abstract}%\nThe state of an entangled q-bit pair is specified \nby 15 numerical parameters that are naturally regarded\nas the components of two 3-vectors and a $3\\times3$-dyadic. \nThere are easy-to-use criteria to check whether a given pair of 3-vectors\nplus a dyadic specify a 2--q-bit state; and if they do, whether the\nstate is entangled; and if it is, whether it is a separable state.\nSome progress has been made in the search for analytical\nexpressions for the degree of separability. \nWe report, in particular, the answer in the case of \nvanishing 3-vectors.\\\\[0.5\\baselineskip]\nPACS numbers: 89.70.+c, 03.65.Bz\n\\end{abstract}\n\n\\renewcommand{\\thefootnote}{(\\alph{footnote})}\n\n\n\\section{Introduction}\\label{sec:intro}\n%\nA q-bit is, in general terms, a binary quantum alternative, for which there\nare many different physical realizations.\nFamiliar examples include the binary alternatives \nof a Stern-Gerlach experiment (``spin up'' or ``spin down''); \nof a photon's helicity (``left handed'' or ``right handed''); \nof two-level atoms (``in the upper state'' or ``in the lower one'');\nof Young's double-slit set-up (``through this slit'' or \n``through that slit''); \nof Mach-Zehnder interferometers (``reflected at the entry beam splitter'' or\n``transmitted at it''); \nand of Ramsey interferometers (``transition in the first zone'' or \n``in the second zone'').\n\nThe actual physical nature of the q-bits in question is irrelevant, however,\nfor the issues dealt with in this paper.\nWe are remarking on entangled states of two q-bits, and as far as the somewhat\nabstract mathematical properties are concerned, all q-bits are equal.\nIn particular, the two q-bits in question could be of quite different kinds,\none the spin-$\\frac12$ degree of freedom of a silver atom, say, the other a\nphoton's helicity.\nIt is even possible, and of experimental relevance \n\\cite{DNRa,DNRb,KSE,SKE,BBER},\nthat both q-bits are carried by the same physical object: the which-way\nalternative of an atom (photon, neutron, \\dots) passing through an \ninterferometer could represent one q-bit, for instance, while its polarization\n(or another internal degree of freedom) is the other.\n\nEntangled q-bit pairs are the basic vehicle of proposed quantum communication\nschemes, envisioned quantum computers, and the like.\nAccordingly, a thorough understanding of the 2--q-bit states they can be in is\nhighly desirable.\n\nWhereas the possible states of a single q-bit are easily classified with the\naid of a 3-vector (the Bloch vector in one physical context, the Poincar\\'e\nvector in another, and analogs of both in general --- we shall speak of Pauli\nvectors), the classification of the states of entangled q-bit pairs has not\nbeen fully achieved as yet. \nThe obvious reason is the richness of the state space, which is parameterized by\ntwo 3-vectors, one for each q-bit, and a $3\\times3$-dyadic that represents \nexpectation values of joint observables, so that 15 real numbers are necessary\nto specify an arbitrary 2--q-bit state.\nA first important division is the one into entangled states and disentangled\nones; a second distinguishes entangled states that are separable from the\nnon-separable ones (technical definitions are given in Sec.\\ \\ref{sec:notation}\nbelow). \nThe latter ones differ from each other by various properties. \nAmong them is the \\emph{degree of separability}, which we would like\nto express in terms of the said 15 parameters (or rather of the 9 relevant\nones among them, see Sec.\\ \\ref{sec:invariants}).\n\nIn the present paper, which is a progress report in spirit, we'll be content\nwith an exposition of the formalism we employ and a concise presentation of\nsome results of particular interest.\nA more technical account will be given elsewhere \\cite{E+Mprep}.\n \n\n\\section{Notation, terminology, and other preparatory remarks}\n\\label{sec:notation}\n%\nAnalogs of Pauli's spin operators are, as usual, used for the description of\nthe individual q-bits: the set $\\sigma_x$, $\\sigma_y$, $\\sigma_z$ for the\nfirst q-bit, and $\\tau_x$, $\\tau_y$, $\\tau_z$ for the second.\nUpon introducing corresponding sets of unit vectors --- $\\row{e_x}$,\n$\\row{e_y}$, $\\row{e_z}$ and $\\row{n_x}$, $\\row{n_y}$, $\\row{n_z}$,\nrespectively, each set orthonormal and right-handed --- we form the vector\noperators\n\\begin{equation}\\label{eq:Pauli-ops}\n\\row{\\sigma}=\\sum_{\\alpha=x,y,z}\\sigma_{\\alpha}\\row{e_{\\alpha}}\\,,\\qquad\n\\row{\\tau}=\\sum_{\\beta=x,y,z}\\tau_{\\beta}\\row{n_{\\beta}}\\,.\n\\end{equation}\nWe emphasize that the two three-dimensional vector spaces thus introduced are\nunrelated and they may have nothing to do with the physical space.\nEven if the q-bits should consist of the spin-$\\frac12$ degrees of freedom of\ntwo electrons, say, so that an identification with the physical space would be\nnatural, we could still define the $x$, $y$, and $z$ directions independently\nfor both q-bits.\n\nBook keeping is made considerably easier if one distinguishes row vectors from\ncolumn vectors, related to each other by transposition.\nWe write\n\\begin{equation}\\label{eq:row-col}\n\\col{\\sigma}=\\trans{\\row{\\sigma}}\\,,\\qquad\n\\row{\\tau}=\\trans{\\col{\\tau}}\\,,\\qquad\\mbox{et cetera} \n\\end{equation}\nwith a self-explaining notation. \nScalar products, such as $\\olexpect{\\row{\\sigma}}\\cdot\\col{\\sigma}$ and\n$\\row{\\tau}\\cdot\\olexpect{\\col{\\tau}}$ involve a row and a column of the same\ntype; products of the ``column times row'' kind are dyadics, for which\n\\begin{equation}\\label{eq:dyad-ex}\n\\col{\\sigma}\\row{\\tau}=\n\\sum_{\\alpha,\\beta=x,y,z}\n\\col{e_{\\alpha}}\\sigma_{\\alpha}\\tau_{\\beta}\\row{n_{\\beta}}\n\\end{equation}\nis an important example; it is a column of $e$-type combined with a row of\n$n$-type. \n\nThe statistical operators, the \\emph{states} for short, for the two q-bits\nthemselves are given by\n\\begin{equation}\\label{eq:rho1+2}\n\\rho_1=\\frac{1}{2}\\left(1+\\row{s}\\cdot\\col{\\sigma}\\right)\\,,\\qquad\n\\rho_2=\\frac{1}{2}\\left(1+\\row{\\tau}\\cdot\\col{t}\\right)\n\\end{equation}\nwith\n\\begin{equation}\\label{eq:Pauli}\n\\row{s}=\\expect{\\row{\\sigma}}\\,,\\qquad \\col{t}=\\expect{\\col{\\tau}}\\,,\n\\end{equation}\nrespectively.\nAn arbitrary joint 2--q-bit state,\n\\begin{equation}\\label{eq:Rho}\n\\Rho=\\frac{1}{4}\\left(1+\\row{s}\\cdot\\col{\\sigma}+\\row{\\tau}\\cdot\\col{t}\n+\\row{\\sigma}\\cdot\\dyadic{C}\\cdot\\col{\\tau}\\right) \n\\end{equation}\ninvolves the \\emph{cross dyadic}\n\\begin{equation}\\label{eq:cross}\n \\dyadic{C}=\\expect{\\col{\\sigma}\\row{\\tau}}\n\\end{equation}\nin addition to the \\emph{Pauli vectors} $\\row{s}$ and $\\col{t}$.\nThe 15 expectation values that constitute $\\row{s}$, $\\col{t}$, and\n$\\dyadic{C}$ can be obtained by measuring 5 well chosen 2--q-bit observables,\nsuch as the ones specified in Table \\ref{tbl:5obs}.\nThese 5 observables are pairwise complementary and thus represent\nan optimal set in the sense of Wootters and Fields \\cite{WF}.\nOr, as Brukner and Zeilinger would put it, the left column of Table\n\\ref{tbl:5obs} lists ``a complete set of five pairs of complementary\npropositions''~\\cite{BZ}.\n\n \n\\begin{table}[!t]\n\\caption[Aa]{\\label{tbl:5obs}%\nA minimal set of five 2--q-bit observables whose measurement supplies all 15\nparameters that characterize the state $\\Rho$ of Eq.\\ (\\ref{eq:Rho}).\n\\rule[-12pt]{0pt}{5pt}}\n\\begin{tabular}{cc}\n\\multicolumn{2}{c}{The observable}\\\\\nwhich identifies the joint eigenstates of\n&\ndetermines the three expectation values\\\\\n\\hline\n$\\sigma_x$ and $\\tau_x$ &\n$\\olexpect{\\sigma_x}$, $\\olexpect{\\tau_x}$, $\\olexpect{\\sigma_x\\tau_x}$ \\\\\n$\\sigma_y$ and $\\tau_y$ &\n$\\olexpect{\\sigma_y}$, $\\olexpect{\\tau_y}$, $\\olexpect{\\sigma_y\\tau_y}$ \\\\\n$\\sigma_z$ and $\\tau_z$ &\n$\\olexpect{\\sigma_z}$, $\\olexpect{\\tau_z}$, $\\olexpect{\\sigma_z\\tau_z}$ \\\\\n$\\sigma_x\\tau_y$ and $\\sigma_y\\tau_z$ &\n$\\olexpect{\\sigma_x\\tau_y}$, $\\olexpect{\\sigma_y\\tau_z}$, \n$\\olexpect{\\sigma_z\\tau_x}$ \\\\\n$\\sigma_y\\tau_x$ and $\\sigma_z\\tau_y$ &\n$\\olexpect{\\sigma_y\\tau_x}$, $\\olexpect{\\sigma_z\\tau_y}$, \n$\\olexpect{\\sigma_x\\tau_z}$ \n\\end{tabular}\n\\end{table}\n\n\nPartial traces,\n\\begin{equation}\\label{eq:part-tr}\n\\rho_1=\\tr{2}{\\Rho}\\,,\\qquad\\rho_2=\\tr{1}{\\Rho}\n\\end{equation}\nextract $\\rho_1$ and $\\rho_2$ from $\\Rho$, of course.\nThe difference between the product state $\\rho_1\\rho_2$ and the actual one,\n\\begin{equation}\\label{eq:Edyad}\n\\Rho-\\rho_1\\rho_2=\\frac{1}{4}\\row{\\sigma}\\cdot\\dyadic{E}\\cdot\\col{\\tau}\\,, \n\\end{equation}\ninvolves the \\emph{entanglement dyadic} $\\dyadic{E}$, given by\n\\begin{equation}\\label{eq:Edyad-expl}\n\\dyadic{E}=\\dyadic{C}-\\col{s}\\row{t}=\\expect{\\col{\\sigma}\\row{\\tau}}\n-\\expect{\\col{\\sigma}}\\expect{\\row{\\tau}}\\,.\n\\end{equation}\nThe state $\\Rho$ is \\emph{entangled} if ${\\dyadic{E}\\neq0}$. \n\nAn entangled state $\\Rho$ can be a mixture of disentangled ones,\n\\begin{equation}\\label{eq:mix}\n\\Rho=\\sum_kw_k\\frac{1}{2}\\left(1+\\row{s_k}\\cdot\\col{\\sigma}\\right)\n\\frac{1}{2}\\left(1+\\row{\\tau}\\cdot\\col{t_k}\\right)\n\\qquad\\mbox{with $w_k>0$ and $\\displaystyle\\sum_kw_k=1$,}\n\\end{equation}\nin which case it is \\emph{separable}.\nThe correlations associated with an entangled, but separable state are not of\na quantum nature and can be understood classically.\n\nAccording to the findings of Lewenstein and Sanpera \\cite{LS}, any\n2--q-bit state $\\Rho$ can be written as a mixture of a separable state $\\sep$\nand a non-separable pure state $\\pure$ ${[\\,=\\pure^2\\,]}$,\n\\begin{equation}\\label{eq:LS}\n \\Rho=\\lambda\\sep+(1-\\lambda)\\pure\\qquad\\mbox{with ${0\\leq\\lambda\\leq1}$.} \n\\end{equation}\nAs a rule, there are many different such \\emph{LS decompositions} with varying\nvalues of $\\lambda$.\nAmong them is the unique optimal decomposition, the one with the largest\n$\\lambda$ value,\n\\begin{equation}\\label{eq:LSopt}\n\\Rho=\\DoS\\sepopt+(1-\\DoS)\\pureopt\\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:DoS}\n\\DoS=\\max\\{\\lambda\\} \n\\end{equation}\nis the \\emph{degree of separability} possessed by $\\Rho$; the value $\\DoS=0$\nobtains only if $\\Rho$ itself is a non-separable pure state. \nThe number $\\DoS$ measures to which extent the correlations associated with \n$\\Rho$ are classical; in rough terms, a state $\\Rho$ is the more useful for\nquantum communication purposes, the smaller its degree of separability.\n\nTherefore, we would like to express $\\DoS$ and $\\pureopt$ in terms of the\nPauli vectors $\\row{s}$, $\\col{t}$ and the cross dyadic $\\dyadic{C}$ that\nspecify the state $\\Rho$.\nWe are still searching for the general answer, but for a number of important\nspecial cases the problem is solved already.\nWe report some of this partial progress below.\n\nWhereas it is relatively easy to find LS decompositions for a given state\n$\\Rho$, it is usually rather difficult to check whether a certain\ndecomposition is the optimal one.\nHere is what's involved (for ${\\lambda>0}$):\n\\begin{equation}\\label{eq:LSpairing}\n\\parbox{0.7\\columnwidth}{%\nIf $\\Rho=\\lambda\\sep+(1-\\lambda)\\pure$ is the optimal decomposition, then\\\\ \n\\begin{tabular}{@{\\qquad}rp{0.5\\columnwidth}@{}}\n(a)& {\\raggedright%\nthe state $(1+\\varepsilon)^{-1}\\left(\\sep+\\varepsilon\\pure\\right)$\\\\ \nis non-separable for $\\varepsilon>0$;}\\\\\nand (b)& {\\raggedright%\nthe state $\\sep+(1/\\lambda-1)\\left(\\pure-\\pure'\\right)$\\\\ \nis either non-positive or non-separable\\\\ for each $\\pure'\\neq\\pure$.}\n\\end{tabular}}\n\\end{equation}\nOnly $\\sep$ and $\\pure$ of the optimal decomposition (when ${\\lambda=\\DoS}$)\nmeet both criteria.\nUnfortunately, their verification is rather complicated even in seemingly\nsimple cases.\n\nSince the infinitesimal neighborhood of $\\pure$ is critical in\n(\\ref{eq:LSpairing}), the actual value of $1/\\lambda-1$ is irrelevant and, as\na consequence, we note an important pairing property:\n\\begin{equation}\\label{eq:pair2}\n\\parbox{0.7\\columnwidth}{%\nIf $\\Rho_{\\lambda}=\\lambda\\sep+(1-\\lambda)\\pure$ is the optimal LS\ndecomposition for one value of $\\lambda$ in the range $0<\\lambda<1$,\nthen it is optimal also for all other $\\lambda$ values.}\n\\end{equation}\nObviously, a systematic method for identifying all $\\sep$s that pair with a\ngiven $\\pure$, or vice versa, would be quite helpful, but we are not aware of\none presently.\n\n\n\\section{Invariants and inequalities}\\label{sec:invariants}\n%\nThe freedom to choose $\\col{e_x}$, $\\col{e_y}$, $\\col{e_z}$ and $\\row{n_x}$,\n$\\row{n_y}$, $\\row{n_z}$ to our liking means that unitary transformations that\naffect only $\\col{\\sigma}$, or only $\\row{\\tau}$, or both separately, turn a\ngiven $\\Rho$ into a physically equivalent one.\nIn terms of the Pauli vectors and the cross dyadic, such \\emph{local}\ntransformations are of the form\n\\begin{equation}\\label{eq:local}\n \\col{s}\\to\\dyadic{O}_{\\rm ee}\\cdot\\col{s}\\,,\\qquad\n\\row{t}\\to\\row{t}\\cdot\\dyadic{O}_{\\rm nn}\\,,\\qquad\n\\dyadic{C}\\to\\dyadic{O}_{\\rm ee}\\cdot\\dyadic{C}\\cdot\\dyadic{O}_{\\rm nn}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:Oee+Onn}\n\\dyadic{O}_{\\rm ee}=\\col{e_1}\\row{e_x}+\\col{e_2}\\row{e_y}+\\col{e_3}\\row{e_z}\\,,\n\\qquad\n\\dyadic{O}_{\\rm nn}=\\col{n_x}\\row{n_1}+\\col{n_y}\\row{n_2}+\\col{n_z}\\row{n_3} \n\\end{equation}\nare orthogonal unimodular dyadics that relate the $x,y,z$ description to the\n$1,2,3$ one.\nSince each of them needs 3 parameters for its specification, there must be\n${9=15-(3+3)}$ independent combinations of $\\row{s}$, $\\col{t}$, and\n$\\dyadic{C}$ that are invariant under (\\ref{eq:local}).\nThese are\\footnote{\\label{fn:Spur}%\nWe write $\\Spur{\\ }$ for the trace of a dyadic in order to avoid confusion\nwith quantum mechanical traces such as \n$C_{xy}=\\olexpect{\\sigma_x\\tau_y}=\\tr{1\\&2}{\\sigma_x\\tau_y\\Rho}$.} \n\\begin{equation}\\label{eq:loc-invs}\n\\begin{array}{rcl@{\\quad}rcl}\na^{(2)}_1&=&\\Spur{\\trans{\\dyadic{C}}\\cdot\\dyadic{C}}\\,, &\na^{(2)}_2&=&\\row{s}\\cdot\\col{s}\\,,\\qquad\na^{(2)}_3=\\row{t}\\cdot\\col{t}\\,, \\\\[2ex]\na^{(3)}_1&=&\\determ{\\dyadic{C}}\\,, &\na^{(3)}_2&=&\\row{s}\\cdot\\dyadic{C}\\cdot\\col{t}\\,, \\\\[2ex]\na^{(4)}_1&=&\\Spur{\\left(\\trans{\\dyadic{C}}\\cdot\\dyadic{C}\\right)^2}\\,, &\na^{(4)}_2&=&\\row{s}\\cdot\\sub{C}\\cdot\\col{t}\\,, \\\\\na^{(4)}_3&=&\\row{s}\\cdot\\dyadic{C}\\cdot\\trans{\\dyadic{C}}\\cdot\\col{s}\\,, &\na^{(4)}_4&=&\\row{t}\\cdot\\trans{\\dyadic{C}}\\cdot\\dyadic{C}\\cdot\\col{t}\\,, \n\\end{array}\n\\end{equation}\nwhere the dyadic $\\sub{C}$ consists of the subdeterminants of $\\dyadic{C}$.\nAll other local invariants can be expressed in terms of the nine $a^{(n)}_m$s.\nImportant examples are the determinant of the entanglement dyadic,\n\\begin{equation}\\label{eq:detE}\n\\determ{\\dyadic{E}}=\\determ{\\dyadic{C}}-\\row{s}\\cdot\\sub{C}\\cdot\\col{t}\n=a^{(3)}_1-a^{(4)}_2\\,, \n\\end{equation}\nand the trace of the modulus of the cross dyadic,\n\\begin{equation}\\label{eq:Sp-modC}\n\\Spur{\\magn{\\dyadic{C}}}\n=\\Spur{\\left(\\trans{\\dyadic{C}}\\cdot\\dyadic{C}\\right)^{1/2}}\n=\\sqrt{\\zeta_1}+\\sqrt{\\zeta_2}+\\sqrt{\\zeta_3}\\,, \n\\end{equation}\nwhere $\\zeta_1$, $\\zeta_2$, and $\\zeta_3$ are the three roots of the cubic\nequation \n\\begin{equation}\\label{eq:cubic}\n\\zeta^3-a^{(2)}_1\\zeta^2+\\frac{1}{2}\\left[\\left(a^{(2)}_1\\right)^2\n-a^{(4)}_1\\right]\\zeta-\\left(a^{(3)}_1\\right)^2=0\\,.\n\\end{equation}\n\nAdmixing the totally chaotic state $\\chaos=\\frac{1}{4}$ to the given $\\Rho$,\n\\begin{equation}\\label{eq:Rhox}\n\\Rho_x=(1-x)\\chaos+x\\Rho\\qquad\\mbox{with ${0\\leq x\\leq1}$}\\,, \n\\end{equation}\namounts to\n\\begin{equation}\\label{eq:Rhox'}\n\\row{s}\\to x\\row{s}\\,,\\qquad\\col{t}\\to x\\col{t}\\,,\n\\qquad\\dyadic{C}\\to x\\dyadic{C}\\,. \n\\end{equation}\nThe resulting scaling of the local invariants is\n\\begin{equation}\\label{eq:invs-scal}\n a^{(n)}_m\\to x^n a^{(n)}_m\\,,\n\\end{equation}\nwhich is the reason for the grouping in (\\ref{eq:loc-invs}).\n\nIn addition to the local transformations (\\ref{eq:local}), there are also the\n\\emph{global} ones that represent arbitrary unitary transformations of the\nstate $\\Rho$.\nExcept for the eigenvalues of $\\Rho$, nothing is left unchanged.\nIn view of the restriction $\\tr{1\\&2}{\\Rho}=1$, there must be 3 global\ninvariants.\nA convenient choice is\n\\begin{equation}\\label{eq:As}\n\\begin{array}{rcl}\nA_2&=&2\\left(a^{(2)}_1+a^{(2)}_2+a^{(2)}_3\\right)\\,,\\\\[2ex]\nA_1&=&8\\left(a^{(3)}_2-a^{(3)}_1\\right)\\,,\\\\[2ex]\nA_0&=&\\left(a^{(2)}_1\\right)^2-2a^{(2)}_1\\left(a^{(2)}_2+a^{(2)}_3\\right)\n-\\left(a^{(2)}_2-a^{(2)}_3\\right)^2 \\\\ &&\n-2a^{(4)}_1-8a^{(4)}_2+4a^{(4)}_3+4a^{(4)}_4\\,,\n\\end{array}\n\\end{equation}\nwhich scale in accordance with $A_k\\to x^{4-k}A_k$ under (\\ref{eq:Rhox'}).\nThe $A_k$s are significant because they are the coefficients in the quartic\nequation\n\\begin{equation}\\label{eq:quartic}\n\\kappa^4-A_2\\kappa^2+A_1\\kappa-A_0=0 \n\\end{equation}\nthat determines the eigenvalues of $\\Rho$: If $\\kappa$ is a solution of\n(\\ref{eq:quartic}), then ${(1-\\kappa)/4}$ is an eigenvalue of $\\Rho$.\nThe absence of the cubic term reflects the unit trace of $\\Rho$.\n\nSince $\\Rho$ is hermitian, all roots of (\\ref{eq:quartic}) are real by\nconstruction, and ${\\Rho\\geq0}$ implies the inequalities\n\\begin{equation}\\label{eq:Rho>0}\nA_2-A_1+A_0\\leq1\\,,\\qquad 2A_2-A_1\\leq4\\,,\\qquad A_2\\leq6\\,. \n\\end{equation}\nThey enable one to check whether a given set of $\\row{s}$, $\\col{t}$,\n$\\dyadic{C}$ actually defines a state $\\Rho$.\n\nThe global reflection\n\\begin{equation}\\label{eq:glob-refl}\n\\row{s}\\to-\\row{s}\\,,\\qquad\\col{t}\\to-\\col{t}\\,,\\qquad\\dyadic{C}\\to\\dyadic{C} \n\\end{equation}\nhas no effect on the local invariants (\\ref{eq:loc-invs}), and therefore%\n\\footnote{\\label{fn:barP}%\nIn the studies by Hill and Wootters \\cite{HW,Woo} of what they call\n``entanglement of formation'' the state $\\bRho$ plays a central role; in\nparticular the eigenvalues of $\\olmagn{\\sqrt{\\bRho}\\sqrt{\\Rho}}$ are of interest.}\n\\begin{equation}\\label{eq:barRho}\n\\bRho=\\frac{1}{4}\\left(1-\\row{s}\\cdot\\col{\\sigma}-\\row{\\tau}\\cdot\\col{t}\n+\\row{\\sigma}\\cdot\\dyadic{C}\\cdot\\col{\\tau}\\right) \n\\end{equation}\nhas the same eigenvalues as $\\Rho$ and also the same degree of separability\n$\\DoS$.\nMixtures of both,\n\\begin{equation}\\label{eq:Rhoy}\n\\Rho_y=\\frac{1+y}{2}\\Rho+\\frac{1-y}{2}\\bRho\n=\\frac{1}{4}\\left(1+y\\row{s}\\cdot\\col{\\sigma}+y\\row{\\tau}\\cdot\\col{t}\n+\\row{\\sigma}\\cdot\\dyadic{C}\\cdot\\col{\\tau}\\right)\\,,\n\\end{equation}\n(with ${-1\\leq y\\leq1}$) have degrees of separability $\\DoS_y$ that cannot be\nless than that of $\\Rho$ and $\\bRho$,\n\\begin{equation}\\label{eq:Sy}\n\\DoS_y\\geq\\DoS\\,,\n\\end{equation}\nwhich is a useful piece of information because everything is known for the\n${y=0}$ case, see Sec.\\ \\ref{ssec:genWern1} below.\n\nThe partial reflection\n\\begin{equation}\\label{eq:part-refl}\n\\row{s}\\to-\\row{s}\\,,\\qquad\\col{t}\\to\\col{t}\\,,\\qquad\\dyadic{C}\\to-\\dyadic{C} \n\\end{equation}\nis a non-unitary transformation of $\\Rho$, which is turned into\n\\begin{equation}\\label{eq:tRho}\n\\tRho=\\frac{1}{4}\\left(1-\\row{s}\\cdot\\col{\\sigma}+\\row{\\tau}\\cdot\\col{t}\n-\\row{\\sigma}\\cdot\\dyadic{C}\\cdot\\col{\\tau}\\right)\\,. \n\\end{equation}\nPeres \\cite{Per} observed that ${\\tRho\\geq0}$ if $\\Rho$ is separable, and his\nconjecture that $\\Rho$ is separable if ${\\tRho\\geq0}$ was proven by M., P.,\nand R. Horodecki \\cite{MPRHor}:\n\\begin{equation}\\label{eq:P3H}\n\\parbox{0.6\\columnwidth}{%\nA 2--q-bit state $\\Rho$ is separable if its $\\tRho$ is non-negative, \\\\ \nand only then.\n}\n\\end{equation}\nNow, since (\\ref{eq:part-refl}) affects only two of the nine local invariants\n(\\ref{eq:loc-invs}), namely $a^{(3)}_1$ and $a^{(4)}_2$ whose sign changes,\nthe positivity conditions (\\ref{eq:Rho>0}) are immediately translated into\ncorresponding conditions for $\\tRho$, and we arrive at this statement:\n\\begin{equation}\\label{eq:tRho>0}\n\\parbox{0.5\\columnwidth}{\\raggedright%\nIf\\qquad\\ $A_2-A_1+A_0\\leq1+16\\determ{\\dyadic{E}}$\\\\\nand\\qquad\\ $2A_2-A_1\\leq4+16\\determ{\\dyadic{C}}$\\\\\nthen $\\Rho$ is separable; if one of the inequalities\\\\ is violated, then $\\Rho$\nis not separable.}\n\\end{equation}\nIt is therefore a straightforward matter to check whether a certain $\\Rho$ is\nseparable (${\\DoS=1}$) or not (${\\DoS<1}$).\n\nWith the aid of a local transformation (\\ref{eq:local}), one can bring a given\n$\\Rho$ into a generic form.\nA standard one refers to the bases for which the cross dyadic is diagonal,\n\\begin{equation}\\label{eq:diagC}\n\\dyadic{C}=\\sum_{\\alpha,\\beta=x,y,z}\n\\col{e_{\\alpha}}C_{\\alpha\\beta}\\row{n_{\\beta}}\n=\\pm\\sum_{k=1}^3\\col{e_k}c_k\\row{n_k}\\qquad\n\\mbox{for}\\ \\left\\{\n\\begin{array}{l}\n\\determ{\\dyadic{C}}\\geq0\\,,\\\\[1ex]\\determ{\\dyadic{C}}<0\\,,\n\\end{array}\\right.\n\\end{equation}\nwhere the $c_k$s are the square roots of the $\\zeta_k$s in (\\ref{eq:Sp-modC}),\nordered in accordance with \n\\begin{equation}\\label{eq:charvals}\n c_1\\geq c_2\\geq c_3\\geq0\n\\end{equation}\nby convention.\nThen, the moduli $\\olmagn{\\dyadic{C}}$ and $\\olmagn{\\trans{\\dyadic{C}}}$ of\n$\\dyadic{C}$ and $\\trans{\\dyadic{C}}$ as well as $\\sub{C}$ have simple\nappearances, too,\n\\begin{equation}\\label{eq:mod+sub}\n\\begin{array}{c}\\displaystyle\n\\magn{\\dyadic{C}}=\\sum_{k=1}^3\\col{n_k}c_k\\row{n_k}\\,,\\qquad\n\\magn{\\trans{\\dyadic{C}}}=\\sum_{k=1}^3\\col{e_k}c_k\\row{e_k}\\,,\\\\[2ex]\n\\sub{C}=\\col{e_1}c_2c_3\\row{n_1}+\\col{e_2}c_3c_1\\row{n_2}\n+\\col{e_3}c_1c_2\\row{n_3}\\,,\n\\end{array}\n\\end{equation}\nso that\n\\begin{equation}\\label{eq:C=Omod}\n\\dyadic{C}=\\pm\\dyadic{O}_{\\rm en}\\cdot\\magn{\\dyadic{C}}\n=\\pm\\magn{\\trans{\\dyadic{C}}}\\cdot\\dyadic{O}_{\\rm en}\\,, \n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:Oen}\n \\dyadic{O}_{\\rm en}=\\sum_{k=1}^3\\col{e_k}\\row{n_k} \n\\end{equation}\nis an orthogonal unimodular dyadic.\n\nFor example, a pure state $\\pure$ has $A_2-A_1+A_0=1$, $2A_2-A_1=4$, $A_2=6$\nand its generic form is \n\\begin{equation}\\label{eq:generic-pure}\n\\pure=\\frac{1}{4}\\left(1+p\\sigma_1-p\\tau_1\n-\\sigma_1\\tau_1-q\\sigma_2\\tau_2 -q\\sigma_3\\tau_3\\right)\n\\end{equation}\nwith\n\\begin{equation}\\label{eq:sigma_k,tau_k}\n\\sigma_k=\\row{\\sigma}\\cdot\\col{e_k}\\,,\\qquad\n\\tau_k=\\row{n_k}\\cdot\\col{\\tau}\\qquad\n\\mbox{for ${k=1,2,3}$} \n\\end{equation}\nand ${0\\leq p\\leq1}$, ${q\\equiv\\sqrt{1-p^2}}$.\nThus, up to local transformations, pure states are characterized by a single\nparameter, namely the common length of the Pauli vectors,\n${p=(\\row{s}\\cdot\\col{s})^{1/2}}{=(\\row{t}\\cdot\\col{t})^{1/2}}$.\nA pure state is separable if ${p=1}$, not separable if ${p<1}$.\nFor ${p=0}$, one has the so-called Bell states\n\\begin{equation}\\label{eq:Bell}\n\\Bell=\\frac{1}{4}\n\\left(1-\\row{\\sigma}\\cdot\\dyadic{O}_{\\rm en}\\cdot\\col{\\tau}\\right) \n\\end{equation}\nwith $\\dyadic{O}_{\\rm en}$ as in (\\ref{eq:Oen}).\n\n\n\\section{Special cases}\\label{sec:special}\n%\n\\subsection{Werner states}\\label{ssec:Wern}\n%\nThe so-called Werner states \\cite{Wern} are (pseudo-)mixtures of Bell states\nand the chaotic state,\n\\begin{equation}\\label{eq:Wern0}\n\\Wern=(1-x)\\chaos+x\\Bell=\\frac{1}{4}\n\\left(1-x\\row{\\sigma}\\cdot\\dyadic{O}_{\\rm en}\\cdot\\col{\\tau}\\right)\\,,\n\\end{equation}\nwhere $\\Wern\\geq0$ requires $-\\frac{1}{3}\\leq x\\leq1$ since the eigenvalues\nof $\\Wern$ are $\\frac{1}{4}(1+3x)$ and $\\frac{1}{4}(1-x)$, the latter being\nthree-fold. \nHere one has\n\\begin{equation}\\label{eq:Wern1}\n\\row{s}=0\\,,\\qquad\\col{t}=0\\,,\\qquad\\dyadic{C}=-x\\dyadic{O}_{\\rm en} \n\\end{equation}\nand finds%\n\\footnote{\\label{fn:LSnumerics}%\nThe numerical findings of Lewenstein and Sanpera \\cite{LS} agree well with\nthis analytical result.}\n\\begin{equation}\\label{eq:Wern2}\n\\DoS=\\left\\{\\begin{array}{c@{\\quad\\mbox{if}\\quad}l}\n1& -\\frac{1}{3}\\leq x \\leq\\frac{1}{3}\\,,\\\\[1ex]\n\\frac{3}{2}(1-x) & \\frac{1}{3}< x \\leq1\\,,\n\\end{array}\\right.\n\\end{equation}\nfor the degree of separability.\nThe pure state of the optimal LS decomposition is the Bell state that appears\nin (\\ref{eq:Wern0}).\n\n\\subsection{Generalized Werner states of the first kind}\\label{ssec:genWern1}\n%\nStates $\\Rho$ for which\n\\begin{equation}\\label{eq:1stWern0}\n\\row{s}=0\\,,\\qquad\\col{t}=0\\,,\\qquad\n\\dyadic{C}=\\pm\\dyadic{O}_{\\rm en}\\cdot\\magn{\\dyadic{C}}\\quad\\mbox{arbitrary} \n\\end{equation}\nrepresent a first generalization of the Werner states (\\ref{eq:Wern0}).\nThe $y=0$ states of (\\ref{eq:Rhoy}) are among them.\n\nThe eigenvalues of\n$\\row{\\sigma}\\cdot\\dyadic{O}_{\\rm en}\\cdot\\magn{\\dyadic{C}}\\cdot\\col{\\tau}$ are\n${c_1+c_2-c_3}$, ${c_1-c_2+c_3}$, ${-c_1+c_2+c_3}$, and ${-c_1-c_2-c_3}$ with\nthe $c_k$s as in (\\ref{eq:diagC}), and the positivity of\n\\begin{equation}\\label{eq:1stWern1}\n \\WernA=\\frac{1}{4}\n\\left(1\\pm\\row{\\sigma}\\cdot\\dyadic{O}_{\\rm en}\\cdot\\magn{\\dyadic{C}}\n\\cdot\\col{\\tau}\\right)\n\\end{equation}\nthen requires that the triplet $(c_1,c_2,c_3)$ --- which is not a 3-vector ---\nis inside the tetrahedron that R. and M. Horodecki speak of in Ref.\\\n\\cite{RMHor}. \n\nThe degree of separability of a state $\\WernA$ is given by\n\\begin{equation}\\label{eq:1stWern2}\n\\DoS=\\left\\{\\begin{array}{c@{\\quad\\mbox{if}\\quad}lcl}\n1& \\determ{\\dyadic{C}}\\geq0 &\\mbox{or}& \\Spur{\\magn{\\dyadic{C}}}\\leq1\\,,\\\\[1ex]\n\\frac{3}{2}-\\frac{1}{2}\\Spur{\\magn{\\dyadic{C}}} &\n \\determ{\\dyadic{C}}<0 &\\mbox{and}& \\Spur{\\magn{\\dyadic{C}}}>1\\,,\n\\end{array}\\right.\n\\end{equation}\nand the pure state of the optimal LS decomposition is the Bell state\n(\\ref{eq:Bell}) with $\\dyadic{O}_{\\rm en}$ from (\\ref{eq:1stWern0}).\n\n\\subsection{Generalized Werner states of the second kind}\\label{ssec:genWern2}\n%\nA second generalization of the Werner states is obtained by replacing the Bell\nstate in (\\ref{eq:Wern0}) by an arbitrary pure state with $0<p,q<1$ in\n(\\ref{eq:generic-pure}). \nThen one has\n\\begin{equation}\\label{eq:2ndWern0}\n\\WernB=\\frac{1+3x}{4}\\pure+\\frac{1-x}{4}\\left(1-\\pure\\right)\\,. \n\\end{equation}\nUpon denoting by $q_0$ the $q$ parameter of the pure state in the optimal LS\ndecomposition, one gets\n\\begin{equation}\\label{eq:2ndWern1}\n\\DoS=\\left\\{\\begin{array}{c@{\\quad\\mbox{if}\\quad}l}\n1& -\\frac{1}{3}\\leq x \\leq (1+2q)^{-1}\\,,\\\\[1ex]\n\\displaystyle\n1-\\frac{(1+2q)x-1}{2q_0} & (1+2q)^{-1}< x \\leq1\\,,\n\\end{array}\\right.\n\\end{equation}\nand $q_0=\\sqrt{1-p_0^2}$ is the largest value that obeys\n\\begin{equation}\\label{eq:2ndWern2}\n\\frac{1+x-2xpp_0}{q_0}\\leq\\left(qx-\\frac{1-x}{2}\\right)\n+\\left(qx-\\frac{1-x}{2}\\right)^{-1}(x-x^2p^2)\\,.\n\\end{equation}\nThis gives ${q_0>q}$ for ${x<1}$ and ${q_0\\to q}$ in the limit $x\\to1$; \nthe extreme value $q_0=1$ is reached if $x$ is in the range\n\\begin{equation}\\label{eq:2ndWern3}\n \\frac{1}{1+2q}<x\\leq\\frac{3/4}{q-1/4+\\sqrt{(1-q)(1+q/2)}}\\,,\n\\end{equation}\nand then a Bell state shows up in the optimal LS decomposition.\n\n\\subsection{States of rank 2}\\label{ssec:rank2}\n%\nA state $\\Rho$, for which $A_2-A_1+A_0=1$ and $2A_2-A_1=4$, has eigenvalues\n$0$ (two-fold), $(1+x)/2$, and $(1-x)/2$ with $x^2=(A_2-2)/4\\leq1$. \nFor $x^2<1$, such a $\\Rho$ is of rank 2.\nIts generic form is\n\\begin{equation}\\label{eq:rk2-0}\n\\rktwo=\\frac{1}{2}\\left(\\Sigma_0+x_1\\Sigma_1+x_2\\Sigma_2+x_3\\Sigma_3\\right)\n\\qquad\\mbox{with $x_1^2+x_2^2+x_3^2=x^2$}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:rk2-1}\n\\Sigma_0=\\frac{1}{2}\\left(1\n+\\sigma_3\\cos\\gamma_1\\cos\\gamma_2\n+\\tau_3\\sin\\gamma_1\\sin\\gamma_2\n+\\sigma_1\\tau_1\\sin\\gamma_1\\cos\\gamma_2\n+\\sigma_2\\tau_2\\cos\\gamma_1\\sin\\gamma_2\\right)\n\\end{equation}\nprojects onto the two-dimensional subspace in question.\nBy convention, the parameters $\\gamma_1$ and $\\gamma_2$ are such that\n${\\pi/2\\geq\\gamma_1\\geq\\gamma_2\\geq0}$. \nThey also appear in the expressions for $\\Sigma_{1,2,3}$,\n\\begin{equation}\\label{eq:rk2-2}\n\\begin{array}{rcl}\n\\Sigma_1&=&\\displaystyle\\frac{1}{2}\\left(\n\\sigma_1\\sin\\gamma_1\n+\\tau_1\\cos\\gamma_2\n+\\sigma_1\\tau_3\\sin\\gamma_2\n+\\sigma_3\\tau_1\\cos\\gamma_1\\right)\\,,\\\\[1ex]\n\\Sigma_2&=&\\displaystyle\\frac{1}{2}\\left(\n\\sigma_2\\sin\\gamma_2\n+\\tau_2\\cos\\gamma_1\n+\\sigma_2\\tau_3\\sin\\gamma_1\n+\\sigma_3\\tau_2\\cos\\gamma_2\\right)\\,,\\\\[1ex]\n\\Sigma_3&=&\\displaystyle\\frac{1}{2}\\bigl(\n\\sigma_3\\sin\\gamma_1\\sin\\gamma_2\n+\\tau_3\\cos\\gamma_1\\cos\\gamma_2\\\\ \n&&\\displaystyle\\phantom{\\frac{1}{2}\\bigl(}\n-\\sigma_1\\tau_1\\cos\\gamma_1\\sin\\gamma_2\n-\\sigma_2\\tau_2\\sin\\gamma_1\\cos\\gamma_2\n+\\sigma_3\\tau_3\\bigr)\\,,\n\\end{array}\n\\end{equation}\nwhich are analogs of Pauli's spin operators for the subspace defined by\n$\\Sigma_0$.\nTheir basic algebraic properties are\n\\begin{equation}\\label{eq:rk2-3}\n\\begin{array}{c}\\displaystyle\n\\Sigma_0\\Sigma_k=\\Sigma_k\\qquad\\mbox{for ${k=0,1,2,3}$}\\,,\\\\[1ex]\n\\displaystyle\n\\Sigma_j\\Sigma_k=\\delta_{jk}\\Sigma_0+i\\sum_{l=1}^3\\epsilon_{jkl}\\Sigma_l\n\\qquad\\mbox{for ${j,k=1,2,3}$}\\,.\n\\end{array}\n\\end{equation}\n\nThe pure rank-2 states (\\ref{eq:rk2-0}) have $x_1^2+x_2^2+x_3^2=1$. \nIf $\\sin\\gamma_1\\cos\\gamma_2=0$, which is to say that \n${\\pi/2=\\gamma_1=\\gamma_2}$ or ${\\gamma_1=\\gamma_2=0}$, \nthen all the states (\\ref{eq:rk2-0}) are separable; \notherwise the separable ones have $x_2=0$,\n$x_3=\\tan\\gamma_2/\\tan\\gamma_1\\equiv\\cos(2\\vartheta)$ and\n$\\magn{x_1}\\leq\\sin(2\\vartheta)$ with $0\\leq\\vartheta\\leq\\pi/4$.\nFor $\\gamma_1>\\gamma_2$ there are two separable pure states, for\n$\\pi/2>\\gamma_1=\\gamma_2>0$ (and thus $\\vartheta=0$) there is only one.\nEquivalent observations about rank-2 states have been made by Sanpera, Tarrach,\nand Vidal \\cite{STV}.\n\nFor $\\sin\\gamma_1\\cos\\gamma_2>0$, the pairing of (\\ref{eq:LSpairing}) and\n(\\ref{eq:pair2}) leads to pairs of three different kinds, viz.\\\n\\begin{equation}\\label{eq:rk2-4}\n\\begin{tabular}{rl}\n(a)& $\\pure$ with $x_1=0$ \\& $\\sep$ with $\\magn{x_1}<\\sin(2\\vartheta)$\\,,\\\\\n(b)& $\\pure$ with $x_1\\geq0$ \\& $\\sep$ with $x_1=\\sin(2\\vartheta)$\\,,\\\\\n(c)& $\\pure$ with $x_1\\leq0$ \\& $\\sep$ with $x_1=-\\sin(2\\vartheta)$\\,.\n\\end{tabular}\n\\end{equation}\nFor a given rank-2 state (\\ref{eq:rk2-0}) this means the following.\nIf the inequality\n\\begin{equation}\\label{eq:rk2-5}\n\\left[(1+x_3)\\sin\\vartheta-\\magn{x_1}\\cos\\vartheta\\right] \n\\left[(1-x_3)\\cos\\vartheta-\\magn{x_1}\\sin\\vartheta\\right]\n\\leq x_2^2\\sin\\vartheta\\cos\\vartheta \n\\end{equation}\nholds, then \n\\begin{equation}\\label{eq:rk2-6}\n\\DoS=\\frac{(1-x^2)/2}{1-x_3\\cos(2\\vartheta)-\\magn{x_1}\\sin(2\\vartheta)}\n\\end{equation}\nand the pairs (\\ref{eq:rk2-4})(b) or (\\ref{eq:rk2-4})(c) apply for ${x_1>0}$ and\n${x_1<0}$, respectively.\nIf (\\ref{eq:rk2-5}) is violated, then the optimal LS decomposition involves\npair (\\ref{eq:rk2-4})(a) and\n\\begin{equation}\\label{eq:rk2-7}\n\\DoS=\\frac{1}{\\sin^2(2\\vartheta)}\\left(1-x_3\\cos(2\\vartheta)\n-\\sqrt{[x_3-\\cos(2\\vartheta)]^2+[x_2\\sin(2\\vartheta)]^2}\\,\\right) \n\\end{equation}\nis the degree of separability.\n\n\n\\section{Outlook}\\label{sec:outlook}\n%\nSince any arbitrary 2--q-bit state $\\Rho$ is a mixture of two rank-2 states, the\ncomplete solution of the rank-2 case can be used in an iterative manner to\narrive at LS decompositions of a given $\\Rho$. \nIt is hoped that the optimal decomposition can be found this way, and we shall\nreport results in due course.\n\n\n\\section{Acknowledgments}\n%\nWe are grateful for very helpful discussions with H.-J. Briegel and I. Cirac.\nBGE would like to thank H. Rauch and his collaborators at the Atominstitut in\nVienna, where part of this work was done, for the hospitality they\nprovided, and the Technical University of Vienna for financial support.\nBGE would also like to thank M. O. Scully and the physics faculty at Texas\nA\\&M University, where another part of this work was done, for their\nhospitality and financial support. \nNM would like to thank the Egyptian government for granting a fellowship.\n\n\\begin{references}\n\\bibitem{DNRa}\nS. D\\\"urr, T. Nonn, and G. Rempe,\n\\nat \\textbf{395}, 33 (1998).\n\\bibitem{DNRb}\nS. D\\\"urr, T. Nonn, and G. Rempe,\n\\prl \\textbf{81}, 5705 (1998).\n\\bibitem{KSE}\nP. G. Kwiat, P. D. D. Schwindt, and B.-G. Englert,\nin \\textit{Mysteries, Puzzles, and Paradoxes in Quantum Mechanics},\nedited by R. Bonifacio (AIP CP461, 1999).\n\\bibitem{SKE}\nP. D. D. Schwindt, P. G. Kwiat, and B.-G. Englert,\n\\pra \\textbf{60}, 4285 (1999).\n\\bibitem{BBER}\nG. Badurek, R. J. Buchelt, B.-G. Englert, and H. Rauch,\nNucl.\\ Instrum.\\ Meth.\\ A, in print.\n\\bibitem{E+Mprep}\nB.-G. Englert and N. Metwally, in preparation.\n\\bibitem{WF}\nW. K. Wootters and B. D. Fields, \nAnn.\\ Phys.\\ (NY) \\textbf{191}, 363 (1989).\n\\bibitem{BZ}\n\\v{C}.\\ Brukner and A. Zeilinger,\n\\prl \\textbf{83}, 3345 (1999).\n\\bibitem{LS}\nM. Lewenstein and A. Sanpera,\n\\prl \\textbf{80}, 2261 (1998).\n\\bibitem{HW}\nS. Hill and W. K. Wootters,\n\\prl \\textbf{78}, 5022 (1997).\n\\bibitem{Woo}\nW. K. Wootters,\n\\prl \\textbf{80}, 2245 (1998).\n\\bibitem{Per}\nA. Peres,\n\\prl \\textbf{77}, 1413 (1996).\n\\bibitem{MPRHor}\nM. Horodecki, P. Horodecki, R. Horodecki,\n\\pl \\textbf{A223}, 1 (1996). \n\\bibitem{Wern}\nR. F. Werner, \n\\pra \\textbf{40}, 4277 (1989).\n\\bibitem{RMHor}\nR. Horodecki and M. Horodecki,\n\\pra \\textbf{54}, 1838 (1996).\n\\bibitem{STV}\nA. Sanpera, R. Tarrach, and G. Vidal,\n\\pra \\textbf{58}, 826 (1998).\n\\end{references}\n\n\\vfill\n\n\\setlength{\\fboxsep}{10pt}\n\\begin{center}\n\\framebox{\\parbox{0.75\\columnwidth}{% \n\\begin{center}\nThis paper has been submitted to\\\\\nJournal of Modern Optics\\\\ \nas a contribution to the\\\\ \nProceedings of the Workshop on Entanglement and Decoherence,\\\\ \nheld at Gargnano/Italy, 20-25 September 1999.\n\\end{center}}} \n\\end{center}\n\n\\vfill\n\n\\end{document}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%% ASCII table %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Upper-case A B C D E F G H I J K L M N O P Q R S T U V W X Y Z\n% Lower-case a b c d e f g h i j k l m n o p q r s t u v w x y z\n% Digits 0 1 2 3 4 5 6 7 8 9\n% Exclamation ! Double quote \" Hash (number) #\n% Dollar $ Percent % Ampersand &\n% Acute accent ' Left paren ( Right paren )\n% Asterisk Plus + Comma ,\n% Minus - Point . Solidus /\n% Colon : Semicolon ; Less than <\n% Equals = Greater than > Question mark ?\n% At @ Left bracket [ Backslash \\\n% Right bracket ] Circumflex ^ Underscore _\n% Grave accent ` Left brace { Vertical bar \|\n% Right brace } Tilde ~\n%%%%%%%%%%%%%%%%%%%%%%%%%% 72-characters line %%%%%%%%%%%%%%%%%%%%%%%%%%\n" } ]	[ { "name": "quant-ph9912089.extracted_bib", "string": "{DNRa S. D\\\"urr, T. Nonn, and G. Rempe, \\nat 395, 33 (1998)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{DNRb S. D\\\"urr, T. Nonn, and G. Rempe, \\prl 81, 5705 (1998)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{KSE P. G. Kwiat, P. D. D. Schwindt, and B.-G. Englert, in Mysteries, Puzzles, and Paradoxes in Quantum Mechanics, edited by R. Bonifacio (AIP CP461, 1999)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{SKE P. D. D. Schwindt, P. G. Kwiat, and B.-G. Englert, \\pra 60, 4285 (1999)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{BBER G. Badurek, R. J. Buchelt, B.-G. Englert, and H. Rauch, Nucl.\\ Instrum.\\ Meth.\\ A, in print." }, { "name": "quant-ph9912089.extracted_bib", "string": "{E+Mprep B.-G. Englert and N. Metwally, in preparation." }, { "name": "quant-ph9912089.extracted_bib", "string": "{WF W. K. Wootters and B. D. Fields, Ann.\\ Phys.\\ (NY) 191, 363 (1989)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{BZ \\v{C.\\ Brukner and A. Zeilinger, \\prl 83, 3345 (1999)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{LS M. Lewenstein and A. Sanpera, \\prl 80, 2261 (1998)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{HW S. Hill and W. K. Wootters, \\prl 78, 5022 (1997)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{Woo W. K. Wootters, \\prl 80, 2245 (1998)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{Per A. Peres, \\prl 77, 1413 (1996)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{MPRHor M. Horodecki, P. Horodecki, R. Horodecki, \\pl A223, 1 (1996)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{Wern R. F. Werner, \\pra 40, 4277 (1989)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{RMHor R. Horodecki and M. Horodecki, \\pra 54, 1838 (1996)." }, { "name": "quant-ph9912089.extracted_bib", "string": "{STV A. Sanpera, R. Tarrach, and G. Vidal, \\pra 58, 826 (1998)." } ]
quant-ph9912090		[]	The finite conductivity corrections to the Casimir force in two configurations are calculated in the third and fourth orders in relative penetration depth of electromagnetic zero oscillations into the metal. The obtained analytical perturbation results are compared with recent computations. Applications to the modern experiments are discussed.	[ { "name": "condcor.tex", "string": "\\documentstyle[12pt,epsfig]{article}\n\\textwidth 165mm\n\\textheight 240mm\n\\topmargin =-0.5cm\n\\oddsidemargin =1.2cm\n\\language=1\n\\pagestyle{myheadings}\n\\renewcommand{\\thempfootnote}{\\fnsymbol{mpfootnote}}\n\\renewcommand{\\thefootnote}{\\arabic{footnote}}\n\\renewcommand{\\thesection}{\\arabic{section}.}\n\n\\begin{document}\n\\def \\beq {\\begin{equation}}\n\\def \\eeq {\\end{equation}}\n\\def \\eeqn {\\end{equation}\\noindent}\n\\def \\bes {\\begin{eqnarray}}\n\\def \\ees {\\end{eqnarray}}\n\\def \\eesn {\\end{eqnarray}\\noindent}\n\\def \\nn {\\nonumber}\n\\def\\mum{\\,$\\mu$m}\n\n\\large\n\n\\begin{center}\n{\\Large {\\bf Higher order conductivity corrections\n\\\\ \nto the Casimir force }}\n\\vskip 6mm\n\n\n\n{\\large\nV.B.~Bezerra, \nG.L.~Klimchitskaya\\footnote{On leave from: North-West Polytechnical\nInstitute, St.~Petersburg, Russia},\nV.M.~Mostepanenko\\footnote{On leave from: A.~Friedmann\nLaboratory for Theoretical Physics, St.Petersburg, \nRussia}}\\\\[5mm]\n\n{\\normalsize\n{\\it Departamento de F\\'{\\i}sica, Universidade Federal da Para\\'{\\i}ba, \\\\\nCaixa Postal 5008, CEP 58059-970\nJo\\~{a}o Pessoa, Pb-Brazil}}\n\n\\end{center}\n\\vskip 6mm\n\\begin{abstract}\n The finite conductivity corrections to the Casimir force \nin two configurations are calculated in \nthe third and fourth orders in relative penetration depth\nof electromagnetic zero oscillations into the metal.\nThe obtained analytical perturbation results are\ncompared with recent computations. Applications to\nthe modern experiments are discussed.\n\\end{abstract}\n\\vskip 8mm\nPACS codes: 03.70.+k, 12.20.Ds, 78.20.-e \\hfill \\\\\nKeywords: Casimir effect, finite conductivity\ncorrections, plasma model \n\\\\[10mm]\nCorresponding author: V.M. Mostepanenko, Departamento de F\\'{\\i}sica, \\\\UFPB, Caixa Postal 5008, CEP~58.059-970 \nJo\\~{a}o Pessoa, Pb-Brazil\\\\\nTel: 55(83)216-7529 \\hfill \\\\\nFax: 55(83)216-7422 \\hfill \\\\\nE-mail: mostep@fisica.ufpb.br \\hfill\n\\newpage\n\\section{Introduction}\n\n\\hspace{\\parindent}\nIn 1998 just 50 years have passed after the publication\nof the famous paper by Casimir [1] (see the Proceedings\n[2] especially devoted to this event). Accidentally the\nexperimental interest in the Casimir effect was\nrekindled at the same time. In Ref.[3] the torsion\npendulum was used to measure the Casimir force between\n$Cu$ plus $Au$ coated quartz plate, and a spherical lens\nin a distance range from 0.6\\mum\\ to 6\\mum. The accuracy\nof order 5\\% was claimed in [3] for the agreement of the\nmeasurement with theory. In Refs.[4--6] the Casimir\nforce between $Al$ plus $Au/Pd$ coated disk and a sphere\nwas measured for surface separations between 0.1\\mum\\ \nto 0.9\\mum\\ using the Atomic Force Microscope. The\ndeviation between theory and experiment was shown to be\nof around 1\\% at the smallest surface separation [4,6,7].\nThe obtained experimental results and the extent of their\nagreement with theory were used to establish stronger\nconstraints for the parameters of hypothetical long-range\ninteractions predicted by the unified gauge theories,\nsupersymmetry and supergravity [8--11].\n\nTo be confident that data fit theory at a level of about\nseveral percent, the different corrections to the ideal\nexpression for the Casimir force should be taken into\naccount. The main contribution is given by the corrections\ndue to finite conductivity of the boundary metal, its\nroughness and due to non-zero temperature (see [12] for\nreview). Experimental data of [3] do not support the\npresence of any of these corrections although they should\ncontribute at a level of 5\\%. By contrast, in [4,6] the\nsurface roughness and finite conductivity corrections are\nof great concern (the temperature corrections are\nnegligible in the measurement range of [4,6]).\n\nThe subject of the present paper is the calculation of\nhigher order finite conductivity corrections to the\nCasimir force in relative penetration depth of\nelectromagnetic zero oscillations into the metal. We\nconsider configurations of two plane parallel plates and\na sphere above a plate. The first order finite conductivity\ncorrection was found in [13] for configuration of two plane\nparallel plates with an error in numerical coefficient\ncorrected in [14]. Later the correct result was reobtained\nin [15]. Second order correction was firstly found in [16]\n(see also [12]). It was modified for the configuration of\na sphere above a disk in [17] by the use of Proximity\nForce Theorem (PFT) [18]. The results of [16,17] for \nthe Casimir\nforce up to the second power in relative penetration\ndepth are in common use when discussing the recent\nexperiments (see, e.g., [4,6,7,19--21]). In [7,10] the\nthird and the fourth order corrections were obtained\napproximately from the interpolation formula. They\nallowed to achieve the excellent agreement between theory\nand experiment.\n\nIn [19] numerical calculation of the Casimir force with\naccount of finite conductivity has been attempted based\non the tabulated data for the comp\\-lex dielectric\npermittivity as a function of frequency. The same\ncomputation was repeated in [21] with the diverged\nresults. The reason of these dif\\-fe\\-ren\\-ces \nwas interpreted\nin [21] as the invalid manipulation of optical data in\n[19]. Our analytical calculation of higher order \nconductivity corrections agrees with the results of [21] \nin the application range of perturbation approach. As\nshown below the perturbation results obtained in the\ncontext of plasma model are valid with rather high\naccuracy when the distance between the test bodies is\nlarger than the plasma wavelength (not much larger as\nadvocated in [20]). \nThis gives the\npossibility, in some instance, to use the plasma model\nfor the distances of order or even less than the\ncharacteristic absorption wavelength of test body material.\n\nThe paper is organized as follows. In Sec.~2 the general\nfinite results for the Casimir energy density and force\nare briefly presented. Sec.~3 contains derivation of the\nthird and fourth order conductivity corrections. \nIn Sec.~4 the\nobtained perturbation results are compared with numerical\ncalculations. Sec.~5 contains conclusions and discussion.\n\n\\section{Casimir energy density and force between\nrealistic materials}\n\n\\hspace{\\parindent}\nLet us consider two semi-infinite solids with dielectric\npermittivity $\\varepsilon(\\omega)$ separated by a \nplane-parallel gap of width $a$. The surfaces of the bodies\nare planes $z=0, a$. The Casimir energy density\nand force acting between these bodies can be found most\nsimply following [22,23] (see also [24] for the\nmultilayered walls). Some additional clarification is\ngiven here in the case in which the finite energy\ndensity rather than force is the subject of interest.\n\nThe mode frequencies of electromagnetic field are found\nfrom Maxwell equations supplemented by the standard\nboundary conditions at $z=0, a$. Two types of such\nfrequencies \n$\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1,2)}$\n(surface modes [25]) corresponding to two polarizations \nof the electric field are the solutions of the equations\n\\bes\n&&\n\\Delta{\\!}^{(1)}(\n\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1)})\n\\equiv\\varepsilon^2 K_{\\varepsilon}^2\ne^{-K_{\\varepsilon}a}\\left[\n\\left(K_{\\varepsilon}+\\varepsilon K\\right)^2\ne^{Ka}-\n\\left(K_{\\varepsilon}-\\varepsilon K\\right)^2\ne^{-Ka}\\right]=0,\\nn\\\\\n&&\n\\Delta{\\!}^{(2)}(\n\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(2)})\n\\equiv K_{\\varepsilon}^2\ne^{-K_{\\varepsilon}a}\\left[\n\\left(K_{\\varepsilon}+ K\\right)^2 e^{Ka}-\n\\left(K_{\\varepsilon}- K\\right)^2 e^{-Ka}\n\\right]=0.\n\\label{1}\n\\eesn\nHere the following notations are introduced\n\\beq\nK^2=k^2-\\frac{\\omega^2}{c^2}, \\qquad\nK_{\\varepsilon}^2=k^2-\\varepsilon\\frac{\\omega^2}{c^2},\n\\label{2}\n\\eeqn\nand {\\boldmath$k$} is the two-dimensional \npropagation vector\nin the $xy$-plane.\n\nThe infinite zero-point energy of the electromagnetic field\nbetween the plates is given by [23,24]\n\\beq\nE(a)=\\frac{1}{2}\\hbar\n\\sum\\limits_{\\mbox{\\footnotesize{\\boldmath$k$}},n}\n\\left(\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1)}\n+\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(2)}\n\\right),\n\\label{3}\n\\eeqn\nwhere the sum in continuous index {\\boldmath$k$} is\nactually an integral. Introducing the length $L$ for the\n$x, y$ sides of the plates we obtain the vacuum energy\ndensity\n\\beq\n{\\cal{E}}(a)=\\frac{E(a)}{L^2}=\n\\frac{\\hbar}{4\\pi}\n\\int\\limits_{0}^{\\infty}\\! k\\,dk\n\\sum\\limits_{n}\n\\left(\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1)}\n+\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(2)}\n\\right),\n\\label{4}\n\\eeqn\nwhich is also infinite.\n\nSummation in (\\ref{4}) over the solutions of (\\ref{1})\ncan be performed with the help of the argument principle\n\\beq\n\\sum\\limits_{n}\n\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1,2)}\n=\\frac{1}{2\\pi i}\\left[\n\\int\\limits_{i\\infty}^{-i\\infty}\\!\\!\n\\omega d\\ln\\Delta{\\!}^{(1,2)}(\\omega)+\n\\int\\limits_{C{\\!}_{+}}\\!\n\\omega d\\ln\\Delta{\\!}^{(1,2)}(\\omega)\\right],\n\\label{5}\n\\eeqn\nwhere contour $C{\\!}_{+}$ is a right semicircle of\ninfinite radius in complex $\\omega$-plane with a center \nat the origin (note that the functions \n$\\Delta{\\!}^{(1,2)}(\\omega)$ have no poles).\n\nThe integral over $C{\\!}_{+}$ can be simply calculated when\nit is considered that \n\\beq\n\\lim\\limits_{\\omega\\to\\infty}\\varepsilon(\\omega)=1,\n\\qquad\n\\lim\\limits_{\\omega\\to\\infty}\n\\frac{d\\varepsilon(\\omega)}{d\\omega}=0\n\\label{6}\n\\eeqn\nalong the arbitrary radial direction in complex plane.\nThe result (infinite) does not depend on $a$. It is\ngiven by\n\\beq\n\\int\\limits_{C{\\!}_{+}}\\!\n\\omega d\\ln\\Delta{\\!}^{(1,2)}(\\omega)=\n3\\int\\limits_{C{\\!}_{+}}\\!\nd\\omega.\n\\label{7}\n\\eeq\n\nIntroducing a new variable $\\xi=-i\\omega$ in (\\ref{5}),\n(\\ref{7}) and performing a partial integration one\nobtains\n\\beq\n\\sum\\limits_{n}\n\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1,2)}\n=\\frac{1}{2\\pi}\n\\int\\limits_{-\\infty}^{\\infty}\\!\\!\n\\xi d\\ln\\Delta{\\!}^{(1,2)}(i\\xi)+\n\\frac{3}{2\\pi}\n\\int\\limits_{C{\\!}_{+}}\\!\nd\\xi.\n\\label{8}\n\\eeq\n\nNow let us turn to the removing of divergencies (this\nimportant point was not discussed in [22--24]). It is\napparent that for the infinitely remote plates the\nregularized physical vacuum energy density should\nvanish [12]. In the limit $a\\to\\infty$ we have from\n(\\ref{8})\n\\beq\n\\left(\\sum\\limits_{n}\n\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1,2)}\n\\right)_{\\infty}\n=\\frac{1}{2\\pi}\n\\int\\limits_{-\\infty}^{\\infty}\\!\\!\n\\xi d\\ln\\Delta{\\!}_{\\ \\infty}^{(1,2)}(i\\xi)+\n\\frac{3}{2\\pi}\n\\int\\limits_{C{\\!}_{+}}\\!\nd\\xi.\n\\label{9}\n\\eeqn\nwhere by the use of (\\ref{1}), it follows\n\\beq\n\\Delta{\\!}_{\\ \\infty}^{(1)}=\n\\varepsilon^2 K_{\\varepsilon}^2\ne^{-K_{\\varepsilon}a}\n\\left(K_{\\varepsilon}+\\varepsilon K\\right)^2\ne^{Ka},\n\\quad\n\\Delta{\\!}_{\\ \\infty}^{(2)}=\n K_{\\varepsilon}^2 e^{-K_{\\varepsilon}a}\n\\left(K_{\\varepsilon}+ K\\right)^2 e^{Ka}.\n\\label{10}\n\\eeq\n\nFor a regularized quantity the result is\n\\beq\n\\left(\\sum\\limits_{n}\n\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1,2)}\n\\right)_{reg}\\equiv\n\\sum\\limits_{n}\n\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1,2)}\n-\\left(\\sum\\limits_{n}\n\\omega_{\\mbox{\\footnotesize{\\boldmath$k$}},n}^{(1,2)}\n\\right)_{\\infty}=\n\\frac{1}{2\\pi}\n\\int\\limits_{-\\infty}^{\\infty}\\!\\!\nd\\xi \\ln\\frac{\\Delta{\\!}^{(1,2)}\n(i\\xi)}{\\Delta{\\!}_{\\ \\infty}^{(1,2)}(i\\xi)},\n\\label{11}\n\\eeqn\nwhere we are guided by the argument of infinitely remote\nplates. \n\nSubstituting the regularized quantities (\\ref{11}) into\n(\\ref{4}) instead of (\\ref{8}) we obtain the final\nexpression for the Casimir energy density between plates\n\\beq\n{\\cal{E}}_{reg}(a)=\\frac{\\hbar}{4\\pi^2}\n\\int\\limits_{0}^{\\infty}\\! kdk\n\\int\\limits_{0}^{\\infty}\\! d\\xi\\left[\n\\ln\\tilde\\Delta{\\!}^{(1)}(i\\xi)+\n\\ln\\tilde\\Delta{\\!}^{(2)}(i\\xi)\\right],\n\\label{12}\n\\eeqn\nwhere\n\\beq\n\\tilde\\Delta{\\!}^{(1)}\\equiv\n\\frac{\\Delta{\\!}^{(1)}}{\\Delta{\\!}_{\\ \\infty}^{(1)}}=1-\n\\frac{(K_{\\varepsilon}-\\varepsilon K)^2}{(K_{\\varepsilon}\n+\\varepsilon K)^2}e^{-2Ka},\n\\quad\n\\tilde\\Delta{\\!}^{(2)}\\equiv\n\\frac{\\Delta{\\!}^{(2)}}{\\Delta{\\!}_{\\ \\infty}^{(2)}}=1-\n\\frac{(K_{\\varepsilon}- K)^2}{(K_{\\varepsilon}\n+ K)^2}e^{-2Ka},\n\\label{13}\n\\eeqn\nand also use was made of the fact\nthat $\\tilde\\Delta{\\!}^{(1,2)}$\nare even functions of $\\xi$.\n\nNotice that in [23] no finite expression for the energy\ndensity was obtained. In [24] the omission of infinities\nwas performed implicitly without a physical\n justification. To illustrate this, in [24] instead of\nEqs.(\\ref{1}) the result of their division by the terms\ncontaining $\\exp(Ka)$ was used in spite of the fact that\non $C_{\\! +}$ such operation is the division by infinity.\nFortunately, this operation did not influence the final\nresult for the energy density obtained in [24] which is\nperfectly correct.\n\nOne can obtain the Casimir force between plates from\n(\\ref{12})\n\\bes\n&&\nF_p(a)=-\\frac{\\partial {\\cal{E}}_{reg}(a)}{\\partial a}=\n-\\frac{\\hbar}{2\\pi^2}\n\\int\\limits_{0}^{\\infty}\\! kdk\n\\int\\limits_{0}^{\\infty}\\! d\\xi K\\left\\{\\left[\n\\frac{(K_{\\varepsilon}+\\varepsilon K)^2}{(K_{\\varepsilon}\n-\\varepsilon K)^2}e^{2Ka}-1\\right]^{-1}\\right.\n\\nn\\\\\n&&\\phantom{aaaaaaaaaaaaaaaaaaaaaa}\n+\\left.\\left[\n\\frac{(K_{\\varepsilon}+ K)^2}{(K_{\\varepsilon}\n- K)^2}e^{2Ka}-1\\right]^{-1}\\right\\},\n\\label{14}\n\\eesn\nwhich is exactly equivalent to Lifshitz result [23,26].\nUsing PFT [18] we obtain from \n(\\ref{12}) the Casimir force acting between a plane plate\nand a spherical lens or a sphere which is given by\n\\beq\nF_l(a)=2\\pi R{\\cal{E}}_{reg}(a)=\\frac{\\hbar R}{2\\pi}\n\\int\\limits_{0}^{\\infty}\\! kdk\n\\int\\limits_{0}^{\\infty}\\! d\\xi\\left[\n\\ln\\tilde\\Delta{\\!}^{(1)}(i\\xi)+\n\\ln\\tilde\\Delta{\\!}^{(2)}(i\\xi)\\right].\n\\label{15}\n\\eeq\n\nBoth Eqs.(\\ref{14}) and (\\ref{15}) are used below to\ncalculate higher order conductivity corrections to the\nCasimir force between realistic metals in two\nconfigurations under consideration.\n\n\\section{Higher order conductivity corrections}\n\n\\hspace{\\parindent}\nIt is common knowledge that the dominant contribution to\nthe Casimir force comes from frequencies \n$\\xi\\sim c/a$. We consider the micrometre domain with\n$a$ from a few tenths of a micrometre to around a\nhundred micrometers. Here the dominant frequencies are\nof visible light and infrared optics. In this domain, the\nplasma model works well and the dielectric permittivity\nof a metal can be presented as\n\\beq\n\\varepsilon(\\omega)=1-\\frac{\\omega_p^2}{\\omega^2},\n\\qquad\n\\varepsilon(i\\xi)=1+\\frac{\\omega_p^2}{\\xi^2},\n\\label{16}\n\\eeqn\nwhere the plasma frequency $\\omega_p$ is different for\ndifferent metals.\n\nThe case of plane parallel plates will be our initial\nconcern. Introducing new variables $p$ and $x$\naccording to\n\\beq\nk^2=\\frac{\\xi^2}{c^2}(p^2-1),\\qquad\n\\xi=\\frac{cx}{2pa}\n\\label{17}\n\\eeqn\nwe transform Eq.(\\ref{14}) into the form\n\\beq\nF_p(a)=\n-\\frac{\\hbar c}{32\\pi^2 a^4}\n\\int\\limits_{0}^{\\infty}\\! x^3dx\n\\int\\limits_{1}^{\\infty}\\! \\frac{dp}{p^2}\n\\left\\{\\left[\n\\frac{(s+p\\varepsilon)^2}{(s-p\\varepsilon )^2}\ne^{x}-1\\right]^{-1}+\\left[\n\\frac{(s+ p)^2}{(s-p)^2}e^{x}-1\\right]^{-1}\\right\\},\n\\label{18}\n\\eeqn\nwhere \n\\beq\ns\\equiv\\sqrt{\\varepsilon -1+p^2}.\n\\label{19}\n\\eeq\n\nLet us expand the expression under the integral with \nrespect\nto $p$ in powers of a small parameter\n\\beq\n\\alpha\\equiv\\frac{\\xi}{\\omega_p}=\n\\frac{c}{2\\omega_p a}\\cdot \\frac{x}{p}=\n\\frac{\\delta_0}{a}\\cdot\\frac{x}{2p},\n\\label{20}\n\\eeqn\nwhere $\\delta_0=\\lambda_p/(2\\pi)$ is the effective\npenetration depth of the electromagnetic oscillations into\nthe metal. Note that in terms of this parameter\n$\\varepsilon(\\omega)= 1+ (1/\\alpha^2)$.\n\nAfter the straightforward calculations one obtains\n\\bes\n&&\n\\left[\n\\frac{(s+p\\varepsilon)^2}{(s-p\\varepsilon )^2}\ne^{x}-1\\right]^{-1}=\n\\frac{1}{e^x-1}\\left[\\vphantom{\\frac{A}{p^4}}\n1-\\frac{4A}{p}\\alpha +\n\\frac{8A}{p^2}(2A-1)\\alpha^2\\right.\n\\nn\\\\\n&&\\phantom{aaaaaaaa}\n+\\frac{2A}{p^3}(-6+32A-32A^2+2p^2-p^4)\\alpha^3\n\\label{21}\\\\\n&&\\phantom{aaaaaaaa}\\left.\n+\\frac{8A}{p^4}(2A-1)(2-16A+16A^2-2p^2+p^4)\\alpha^4\n+O(\\alpha^5)\\right],\n\\nn\n\\eesn \nwhere $A\\equiv e^x/(e^x-1)$.\n\nIn perfect analogy, the other contribution from (\\ref{18})\nis \n\\bes\n&&\n\\left[\n\\frac{(s+p)^2}{(s-p)^2}\ne^{x}-1\\right]^{-1}=\n\\frac{1}{e^x-1}\\left[\n1-4Ap\\alpha +\n8A(2A-1)p^2\\alpha^2\\right.\n\\nn\\\\\n&&\\phantom{aaaaaaaa}\n+2A(-5+32A-32A^2)p^3\\alpha^3\n\\label{22}\\\\\n&&\\phantom{aaaaaaaa}\\left.\n+8A(1+18A-48A^2+32A^3)p^4\\alpha^4\n+O(\\alpha^5)\\right]\n\\nn\n\\eesn \n(note that this expression actually does not depend on\n$p$ due to (\\ref{20})).\n\nAfter substitution of (\\ref{21}), (\\ref{22}) into\n(\\ref{18}) all integrals with respest to $p$ have the\nform $\\int_{0}^{\\infty}dpp^{-k}$ with $k\\geq 2$ and are\ncalculated immediately. The integrals with respect to\n$x$ have the form\n\\beq\n\\int\\limits_{0}^{\\infty}dx\n\\frac{x^ne^{mx}}{(e^x-1)^{m+1}}\n\\label{23}\n\\eeqn\nand can be easily calculated with the help of [27].\nSubstituting their values into (\\ref{18}) we obtain\nafter some transformations the Casimir force between\nmetallic plates with finite conductivity corrections\nup to the fourth power in relative penetration depth\n\\bes\n&&\nF_p(a)=F_p^{(0)}(a)\\left[\n1-\\frac{16}{3}\\frac{\\delta_0}{a}+\n24\\frac{\\delta_0^2}{a^2}-\n\\frac{640}{7}\\left(1-\\frac{\\pi^2}{210}\\right)\n\\frac{\\delta_0^3}{a^3}\\right.\n\\nn\\\\\n&&\n\\phantom{aaaaaaaaaaaaa}\\left.+\n\\frac{2800}{9}\\left(1-\\frac{163\\pi^2}{7350}\\right)\n\\frac{\\delta_0^4}{a^4}\\right],\n\\label{24}\n\\eesn\nwhere $F_p^{(0)}(a)\\equiv -(\\pi^2\\hbar c)/(240a^4)$.\n\nAs was mentioned in the Introduction, the first order\ncorrection in (\\ref{24}) was obtained in [13--15].\n The second order correction was\nobtained in [16] (see also [12]). The third and fourth\norder corrections which are obtained here are important\nfor the recent Casimir force measurements (see Sec.4).\n\nNow let us turn to the configuration of a lens or a sphere\nabove a plate. Introducing the new variable (\\ref{17})\ninto (\\ref{15}) we get the Casimir force \n\\beq\nF_l(a)=\n\\frac{\\hbar cR}{16\\pi a^3}\n\\int\\limits_{0}^{\\infty}\\! x^2dx\n\\int\\limits_{1}^{\\infty}\\! \\frac{dp}{p^2}\n\\left\\{\\ln\\left[1-\n\\frac{(s-p\\varepsilon)^2}{(s+p\\varepsilon )^2}\ne^{-x}\\right]+\\ln\\left[1-\n\\frac{(s-p)^2}{(s+p)^2}e^{-x}\\right]\\right\\}.\n\\label{25}\n\\eeq\n\nBearing in mind the further expansions it is convenient\nto perform in (\\ref{25}) integration by parts with respect\nto $x$. The result is\n\\bes\n&&\nF_l(a)=\n-\\frac{\\hbar cR}{48\\pi a^3}\n\\int\\limits_{0}^{\\infty}\\! x^3dx\n\\int\\limits_{1}^{\\infty}\\! \\frac{dp}{p^2}\n\\left[\n\\frac{(s-p\\varepsilon)^2-(s+p\\varepsilon )^2\n\\frac{\\partial}{\\partial x}\\frac{(s-\np\\varepsilon)^2}{(s+p\\varepsilon )^2}}{(s+\np\\varepsilon)^2 e^x-(s-p\\varepsilon )^2}\n\\right.\n\\nn\\\\\n&&\\phantom{aaaaaaaaaaaaaaaa}\\left.+\n\\frac{(s-p)^2-(s+p)^2\\frac{\\partial}{\\partial x}\n\\frac{(s-p)^2}{(s+p)^2}}{(s+p)^2 e^x-(s-p)^2}\n\\right].\n\\label{26}\n\\ees\n\nThe expansion of the first term under the integral in\npowers of the parameter $\\alpha$ \nintroduced in (\\ref{20}) is\n\\bes\n&&\n\\frac{(s-p\\varepsilon)^2-(s+p\\varepsilon )^2\n\\frac{\\partial}{\\partial x}\\frac{(s-\np\\varepsilon)^2}{(s+p\\varepsilon )^2}}{(s+\np\\varepsilon)^2 e^x-(s-p\\varepsilon )^2}\n=\n\\frac{1}{e^x-1}\\left\\{\\vphantom{\\frac{A}{p^4}}\n1+\\frac{4}{px}(1-Ax)\\alpha \\right.\n\\nn\\\\\n&&\\phantom{aaaaa}\n+\n\\frac{8A}{p^2x}(-2-x+2Ax)\\alpha^2\n+\\frac{2}{p^3x}\\left[\\vphantom{A^2}\n2-6p^2+3p^4\\right.\n\\label{27}\\\\\n&&\\phantom{aaaaa}\\left.\n+Ax(-6+32A-32A^2+2p^2-p^4)\n+16A(2A-1)\\right]\\alpha^3\n\\nn\\\\\n&&\\phantom{aaaaa}\n+\\frac{8A}{p^4x}\\left[-8+32A-32A^2+8p^2-4p^4\\right.\n\\nn\\\\\n&&\\phantom{aaaaaaaa}\\left.\\left.+\nx(2A-1)(2-16A+16A^2-2p^2+p^4)\\right]\\alpha^4\n+O(\\alpha^5)\n\\vphantom{\\frac{A}{p^4}}\\right\\}.\n\\nn\n\\ees \n\nIn the same way for the second term under the integral\nof (\\ref{26}) one obtains\n\\bes\n&&\n\\frac{(s-p)^2-(s+p)^2\\frac{\\partial}{\\partial x}\n\\frac{(s-p)^2}{(s+p)^2}}{(s+p)^2 e^x-(s-p)^2}\n=\n\\frac{1}{e^x-1}\\left[\n1+\\frac{4}{x}(1-Ax)p\\alpha \\right.\n\\label{28}\\\\\n&&\\phantom{aaaaa}\n+\n\\frac{8A}{x}(-2-x+2Ax)p^2\\alpha^2\n+\\frac{2}{x}\\left(\\vphantom{A^2}\n-1-16A+32A^2-5Ax\\right.\n\\nn\\\\\n&&\\phantom{aaaaa}\n\\left.+32A^2x-32A^3x\\right)\np^3\\alpha^3\n+\\frac{8A}{x}\\left(-4+32A-32A^2-x\\right.\n\\nn\\\\\n&&\\phantom{aaaaaaaa}\\left.\\left.\n+18Ax-48A^2x+32A^3x\\right)p^4\\alpha^4\n+O(\\alpha^5)\n\\right].\n\\nn\n\\ees \n\nSubstituting (\\ref{27}), (\\ref{28}) into (\\ref{26})\nwe firstly calculate integrals with respect to $p$.\nAll integrals with respect to $x$ are of the form\n(\\ref{23}). Calculating them we come to the\nfollowing result after long but straightforward \ncalculations\n\\bes\n&&\nF_l(a)=F_l^{(0)}(a)\\left[\n1-4\\frac{\\delta_0}{a}+\n\\frac{72}{5}\\frac{\\delta_0^2}{a^2}-\n\\frac{320}{7}\\left(1-\\frac{\\pi^2}{210}\\right)\n\\frac{\\delta_0^3}{a^3}\\right.\n\\nn\\\\\n&&\\phantom{aaaaaaaaaaaaaaaa}\n\\left.+\n\\frac{400}{3}\\left(1-\\frac{163\\pi^2}{7350}\\right)\n\\frac{\\delta_0^4}{a^4}\\right],\n\\label{29}\n\\eesn\nwhere $F_l^{(0)}(a)\\equiv -(\\pi^3\\hbar cR)/(360a^3)$.\nNote that the first order correction from (\\ref{29})\nwas firstly published in [3] and the second order one\n in [17].\n\nAlthough the results (\\ref{24}) and (\\ref{29}) for two\nconfigurations were obtained\nindependently they can be tied by the use of PFT.\nBy way of example, the energy density\nassociated with the fourth order contribution \nin (\\ref{24}) is\n\\beq\nE_p^{(4)}(a)=\n\\int\\limits_{a}^{\\infty}F_p^{(4)}(a)da=\n-\\frac{5\\pi^2\\hbar c }{27}\n\\left(1-\\frac{163\\pi^2}{7350}\\right)\n\\frac{\\delta_0^4}{a^7}.\n\\label{30}\n\\eeqn\nThen the fourth order contribution to the force between\na plate and a lens \n\\beq\nF_l^{(4)}(a)=2\\pi RE_p^{(4)}(a)=\n-\\frac{10\\pi^3\\hbar c R}{27a^3}\n\\left(1-\\frac{163\\pi^2}{7350}\\right)\n\\frac{\\delta_0^4}{a^4}\n\\label{31}\n\\eeqn\n agrees with (\\ref{29}). The other coefficients\nof (\\ref{29}) can be verified in the same way.\n\n\\section{Comparision with numerical calculations}\n\n\\hspace{\\parindent}\nIn this section we consider the application range of the\nexpressions (\\ref{24}) and (\\ref{29}) for the Casimir force\nwhich take into account higher order conductivity\ncorrections. It is apparent that the greater the distance\n$a$ between the test bodies, the more exact are the\nperturbation formulas obtained up to the fourth power in\nsmall parameter $\\delta_0/a$. Let us compare the\ncorrection to the Casimir force between two plane parallel\nplates given by Eq.~(\\ref{24}) with the numerical\nresults. These results were obtained in \\cite{21} for\nthree metals ($Au,\\>Cu$ and $Al$) by the numerical\nintegration of the formulas which are equivalent to\n(\\ref{18}). In doing so the tabulated data [28]\nfor the complex dielectric permittivity was used. The\nquantity $\\varepsilon(i\\xi)$ was obtained through the\nimaginary part of dielectric permittivity \n by the use of dispersion\nrelation \\cite{26}.\n\nIn Fig.~1a, the solid line represents computational\nresults of Ref.~\\cite{21} for \n$F_p/F_p^{(0)}$ in\ncase of $Al$ depending on distance between the plates\n$a$. The short-dashed line is obtained from \nEq.~(\\ref{24}) with the value of plasma wavelength\n$\\lambda_p^{Al}=98\\,$nm; the long-dashed line takes\naccount the terms of (\\ref{24}) up to the second power\nonly. It is seen that (\\ref{24}) is in\nexcellent agreement with computational results of \\cite{21}\nfor all $a\\geq\\lambda_p^{Al}$. For example, for\n$a=0.1\\,\\mu$m, $0.5\\,\\mu$m and $3\\,\\mu$m it follows from\n(\\ref{24}) that $F_p/F_p^{(0)}=0.56$, 0.85 and 0.97 \nwhich can be compared with computations of\n\\cite{21}: 0.55, 0.85, and 0.96, respectively.\n\nIn Fig.~1b, the analogical results for $Cu$ and $Au$ are\nshown. The dashed lines were obtained with\n$\\lambda_p^{Cu,Au}=132\\,$nm. For the typical distances\nindicated above it follows from (\\ref{24}) that\n$F_p/F_p^{(0)}=0.60$, 0.81 and 0.96 which can be\ncompared with the values: 0.48, 0.81 and 0.96 \\cite{21}.\nThe difference in the first values is due to\n$\\lambda_p^{Cu,Au}>100\\,$nm, i.e. (\\ref{24}) is not\napplicable for $a=100\\,$nm in case of $Cu$ and $Au$.\nFor $a\\geq\\lambda_p^{Cu,Au}$\nthe results agree perfectly well.\n Note that the values\nof plasma wavelength \n$\\lambda_p=c\\sqrt{\\pi m}/(e\\sqrt{N})$, where $m$ is the\neffective mass of conduction electrons, $N$ is their\ndensity are known not very precisely. For $Al$, usually\n$\\lambda_P^{Al}=100\\,$nm is used \\cite{28}. For $Au$ and\n$Cu$ the value $\\lambda_p^{Cu,Au}=136\\,$nm was estimated\nrecently \\cite{21}. We used a bit different values which\nprovide the smallest rms deviation between\nthe computational results and the ones obtained from\n(\\ref{24}) (in \\cite{28a} $\\lambda_p^{Cu}=132\\,$nm). \nThe values of $F_p/F_p^{(0)}$ at typical\ndistances do not depend on the change\nof $\\lambda_p$ for 2--3 percent which is the uncertainty\nof the current information regarding $\\lambda_p$.\n\nNow let us turn to the Casimir force between a plate and\na lens. The numerical results were obtained in \\cite{21}\nby the integration of equation equivalent to (\\ref{25}).\nIn~Fig.~2a, the results for $Al$ bodies are shown, and\nin Fig.~2b --- for $Cu$ or $Au$ ones. Solid lines\nrepresent computations of \\cite{21}, short- and\nlong-dashed ones are obtained from Eq.~(\\ref{29}) used in\nfull or up to the second power terms.\nIn both figures the fourth-order perturbation results are\nin excellent agreement with computations for all\n$a\\geq\\lambda_p$. At the distances\n$a=0.1\\,\\mu$m, $0.5\\,\\mu$m and $3\\,\\mu$m in \nthe case of $Al$\nwe have $F_l/F_l^{(0)}=0.62$, 0.89, 0.98 from\nEq.~(\\ref{29}) and 0.63, 0.88, 0.97 from \\cite{21}.\nFor $Cu$ and $Au$ Eq.~(\\ref{29}) gives \n $F_l/F_l^{(0)}=0.59$, 0.85, 0.97 in agreement\nwith the values: 0.55, 0.85, 0.97 \\cite{21}.\n\nAs was mentioned in the Introduction the computation of\nfinite conductivity corrections to the Casimir force by\nthe use of tabulated data was firstly performed in\n\\cite{19}. It should be emphasized that our analytical\nresults are in contradiction with \\cite{19}.\nBy way of example, at $a=0.5\\,\\mu$m for $Au$ and $Cu$\none can find in \\cite{19}\n $F_p/F_p^{(0)}=0.657$ and 0.837 correspondingly \nwhereas according to our results $F_p/F_p^{(0)}=0.81$\nfor both metals. At the same distance and metals for\n a lens above a plate \n$F_l/F_l^{(0)}=0.719$ and 0.874 \\cite{19} whereas from\nEq.~(\\ref{29}) one gets \n $F_l/F_l^{(0)}=0.85$. Our results, however, are in good\nagreement with the alternative computations of \\cite{21}\nsupporting the conclusion of \\cite{21} that the\nmanipulation of optical data in \\cite{19} is invalid.\n\nIt might be well to compare also the exact third and\nfourth order conductivity corrections obtained above\nwith the approximate ones obtained by the use of\ninterpolation formula \\cite{7,10}. To take one example, \nfor the force between a lens and a plate the coefficients\nnear the third and fourth order corrections in\ninterpolation formula are $-50.67$ and $+177.33$ (compare\nwith $-43.57$ and $+104.13$ from (\\ref{29})). For the\nsmallest separations $a=120\\,$nm in experiment \\cite{4}\nand $\\delta_0/a\\approx 0.13$ for $Al$ this leads to \nthe 0.5\\% difference only in the results obtained by the\ninterpolation formula \\cite{7} and by (\\ref{29}).\n\n\\section{Conclusions and discussion}\n\n\\hspace{\\parindent}\nIn the above the third and the fourth order corrections to\nthe Casimir force due to finite conductivity of the \nmetal were calculated analytically in configurations of\ntwo plane parallel plates and a spherical lens (or a\nsphere) above a plate. The Casimir forces (\\ref{24}),\n(\\ref{29}) are in\nexcellent agreement with computations of \\cite{21} based \non the tabulated data for the complex dielectric\npermittivity for all distances larger than the effective\nplasma wavelength of the test body metal. \nWhat this means is\nthat the results (\\ref{24}), (\\ref{29}) can be reliably\nused even for the distances $a$ less than the\ncharacteristic absorption wavelength $\\lambda_0$ \n if $\\lambda_p<\\lambda_0$ (this is a case, e.g.,\nfor $Au$ and $Cu$, which are characterized by\n$\\lambda_0\\approx 500\\,$nm or for Cr with\n$\\lambda_p\\approx 314\\,$nm, $\\lambda_0\\approx 600\\,$nm\n\\cite{29}).\n\nTo obtain the higher order conductivity corrections we\nhave used the plasma model representing dielectric\npermittivity by the Eq.~(\\ref{16}). This model does not\ntake into account relaxation processes. However, the\nrelaxation parameter is much smaller than the plasma\nfrequency. As was shown in \\cite{21}, relaxation could\nplay some role only for large distances between plates\n$a\\gg\\lambda_p$ and even there the variation of the\ncorrections to the Casimir force due to it is smaller\nthan 2\\%. If to take into account that for so large\ndistances the corrections themselves decrease very quickly\nit becomes evident that the influence of relaxation can\nbe neglected.\n\nIn conclusion we would like to stress that both the results\n(\\ref{24}) and (\\ref{29}) \nare of the same accuracy in spite of\nthe fact that the PFT was used in\n(\\ref{15}) to obtain (\\ref{29}). The thing is that this\ntheorem is equivalent to the addition method of\ncalculation of the Casimir force which leads to the\nerror no larger than $10^{-2}$\\% for small deviations\nfrom plane parallel geometry \\cite{12,30}. What this means\nis that for a sphere or spherical lens of large curvature\nradius $R\\gg a$ the additional error introduced by the\nuse of PFT is negligible. Therefore it is possible to\nconclude that formulas like (\\ref{24}),\n(\\ref{29}) and the analogical expressions for the other\ncorrections to the Casimir force can be reliably used \nfor confronting theory and experiment at a level of\n1\\% accuracy.\n\n\\section{Acknowledgements}\n\nV.B.B.~wishes to thank Conselho Nacional de Desenvolvimento\nCient\\'{\\i}fico e Tecnol\\'{o}gico (CNPq) for partial\nfinancial support. G.L.K.~and V.M.M.~are grateful to the\nDepartment of Physics of the Federal University of\nPara\\'{\\i}ba (Brazil) for hospitality.\n\n\\newpage\n\\begin{thebibliography}{99}\n\\bibitem{1}\nH.~B.~G.~Casimir, \n { Proc. Kon. Nederl. Akad. \nWet.}\n{51} (1948) 793.\n\\bibitem{2}\nThe Casimir Effect 50 Years Later, ed. M.~Bordag, World\nScientific, Singapore, 1999.\n\\bibitem{3}\nS.~K.~Lamoreaux, Phys. Rev. Lett. 78 (1997) 5;\n{81} (1998) 5475.\n\\bibitem{4}\nU.~Mohideen, A.~Roy, \n{ Phys. Rev. Lett.} \n{81} (1998) 4549.\n\\bibitem{5}\n A.~Roy, U.~Mohideen,\nPhys. Rev. Lett. {82} (1999) 4380.\n\\bibitem{6}\nA.~Roy, C.-Y.~Lin, U.~Mohideen, \nPhys. Rev. D 60 (1999) 111101.\n\\bibitem{7}\nG.~L.~Klimchitskaya, A.~Roy, U.~Mohideen,\nV.~M.~Mostepanenko, \nPhys. Rev. A 60 (1999) 3487.\n\\bibitem{8}\nG.~L.~Klimchitskaya, E.~R.~Bezerra~de~Mello,\n V.M.~Mostepanenko,\n Phys. Lett. A {236} (1997) 280.\n\\bibitem{9}\n M.~Bordag, B.~Geyer, G.~L.~Klimchitskaya,\nV.~M.~Mostepanenko, \nPhys. Rev. {D 58} (1998) 075003.\n\\bibitem{10}\n M.~Bordag, B.~Geyer, G.~L.~Klimchitskaya,\nV.~M.~Mostepanenko, Phys. Rev. {D 60} (1999)\n055004.\n\\bibitem{11}\nJ.~C.~Long, H.~W.~Chan, J.~C.~Price, Nucl. Phys.\n{ B 539} (1999) 23.\n\\bibitem{12}\n V.~M.~Mostepanenko, N.~N.~Trunov, \n{The Casimir Effect and its Applications}, \nClarendon Press, Oxford, 1997.\n\\bibitem{13}\nI.~E.~Dzyaloshinskii, E.~M.~Lifshitz, L.~P.~Pitaevskii,\nSov. Phys. Uspekhi 4 (1961) 153.\n\\bibitem {14}\nC.~M.~Hargreaves, { Proc. Kon. Nederl. Acad. \nWet.}\n{ B68} (1965) 231.\n\\bibitem {15}\nJ.~Schwinger, L.~L.~DeRaad,~Jr., K.~A.~Milton,\n{ Ann. Phys.} {115} (1978) 1.\n\\bibitem {16}\nV.~M.~Mostepanenko, N.~N.~Trunov, {Sov. J. Nucl.\nPhys.} { 42} (1985) 818.\n\\bibitem{17}\nV.~B.~Bezerra, G.~L.~Klimchitskaya, C.~Romero,\nMod. Phys. Lett. A { 12} (1997) 2623.\n\\bibitem {18}\nJ.~Blocki, J.~Randrup, W.~J.~Swiatecki, C.~F.~Tsang,\n{Ann. Phys.} {105} (1977) 427.\n\\bibitem{19}\nS.~K.~Lamoreaux, Phys. Rev. A 59 (1999) R3149.\n\\bibitem{20}\nS.~K.~Lamoreaux, e-print quant-ph/9907076.\n\\bibitem{21}\nA.~Lambrecht, S.~Reynaud, e-print quant-ph/9907105.\n\\bibitem{22}\nN.~G.~van Kampen, B.~R.~A.~Nijboer, K.~Schram,\nPhys. Lett. A 26 (1968) 307. \n\\bibitem{23}\nP.~W.~Milonni, {The Quantum Vacuum}, Academic Press,\nSan Diego, 1994.\n\\bibitem{24}\nF.~Zhou, L.~Spruch, Phys. Rev. A 52 (1995) 297.\n\\bibitem{25}\nG.~Barton, Rep. Prog. Phys. 42 (1979) 963.\n\\bibitem{26}\nE.~M.~Lifshitz, L.~P.~Pitaevskii, \n{Statistical Physics, Part 2}, Pergamon Press,\nOxford, 1980.\n\\bibitem{27}\nI.~S.~Gradshteyn, I.~M.~Ryzhik, \n{Table of Integrals, Series and Products},\nAcademic Press, New York, 1980.\n\\bibitem{28}\n{Handbook of Optical Constants of Solids}, ed.\nE.\\ D.\\ Palik, Academic Press, New York, 1995.\n\\bibitem{28a}\nH.~Ehrenreich, H.~R.~Philipp, Phys. Rev. 128 (1962) 1622.\n\\bibitem{29}\nP.~H.~G.~M.~van Blockland, J.~T.~G.~Overbeek, J. Chem.\nSoc. Faraday Trans. 74 (1978) 2637.\n\\bibitem{30}\n M.~Bordag, G.~L.~Klimchitskaya,\nV.~M.~Mostepanenko, \nInt. J. Mod. Phys. {A 10} (1995) 2661. \n\\end{thebibliography}\n\n\\newpage\n\\large\n\\begin{center} {\\Large \\bf Figure captions} \\end{center}\n\\begin{tabular}{l p{142mm}}\n& \\\\\n{\\bf Fig.1.} &\n Correction factors to the Casimir force \n$F_p/F_p^{(0)}$ in configuration\nof two plane parallel plates for $Al$ (a) and $Cu$ or $Au$\n(b) bodies in dependence of distance\nmeasured in $\\mu$m. Solid lines\nrepresent the results of computations [21], short-\nand long-dashed lines are obtained by the Eq.~(\\ref{24})\nup to the fourth and the second power respectively.\\\\\n{\\bf Fig.2.} &\n Correction factors to the Casimir force \n$F_l/F_l^{(0)}$ in configuration\nof a lens (sphere) above a plate \nfor $Al$ (a) and $Cu$ or $Au$\n(b) bodies in dependence of distance\nmeasured in $\\mu$m. Solid lines\nrepresent the results of computations [21], short-\nand long-dashed lines are obtained by the Eq.~(\\ref{29})\nup to the fourth and the second power respectively.\n\\end{tabular}\n\\newpage\n\\begin{figure}[ht]\n\\renewcommand{\\thefigure}{1,\\alph{figure}}\n\\centerline{\\epsffile{fig1a.eps}}\n\\caption{}\n\\end{figure}\n\\newpage\n\\begin{figure}[ht]\n\\renewcommand{\\thefigure}{1,\\alph{figure}}\n\\centerline{\\epsffile{fig1b.eps}}\n\\caption{}\n\\end{figure}\n\\setcounter{figure}{0}\n\\newpage\n\\begin{figure}[ht]\n\\renewcommand{\\thefigure}{2,\\alph{figure}}\n\\centerline{\\epsffile{fig2a.eps}}\n\\caption{}\n\\end{figure}\n\\newpage\n\\begin{figure}[ht]\n\\renewcommand{\\thefigure}{2,\\alph{figure}}\n\\centerline{\\epsffile{fig2b.eps}}\n\\caption{}\n\\end{figure}\n\\end{document}\n\n" } ]	[ { "name": "quant-ph9912090.extracted_bib", "string": "{1 H.~B.~G.~Casimir, { Proc. Kon. Nederl. Akad. Wet. {51 (1948) 793." }, { "name": "quant-ph9912090.extracted_bib", "string": "{2 The Casimir Effect 50 Years Later, ed. M.~Bordag, World Scientific, Singapore, 1999." }, { "name": "quant-ph9912090.extracted_bib", "string": "{3 S.~K.~Lamoreaux, Phys. Rev. Lett. 78 (1997) 5; {81 (1998) 5475." }, { "name": "quant-ph9912090.extracted_bib", "string": "{4 U.~Mohideen, A.~Roy, { Phys. Rev. Lett. {81 (1998) 4549." }, { "name": "quant-ph9912090.extracted_bib", "string": "{5 A.~Roy, U.~Mohideen, Phys. Rev. Lett. {82 (1999) 4380." }, { "name": "quant-ph9912090.extracted_bib", "string": "{6 A.~Roy, C.-Y.~Lin, U.~Mohideen, Phys. Rev. D 60 (1999) 111101." }, { "name": "quant-ph9912090.extracted_bib", "string": "{7 G.~L.~Klimchitskaya, A.~Roy, U.~Mohideen, V.~M.~Mostepanenko, Phys. Rev. A 60 (1999) 3487." }, { "name": "quant-ph9912090.extracted_bib", "string": "{8 G.~L.~Klimchitskaya, E.~R.~Bezerra~de~Mello, V.M.~Mostepanenko, Phys. Lett. A {236 (1997) 280." }, { "name": "quant-ph9912090.extracted_bib", "string": "{9 M.~Bordag, B.~Geyer, G.~L.~Klimchitskaya, V.~M.~Mostepanenko, Phys. Rev. {D 58 (1998) 075003." }, { "name": "quant-ph9912090.extracted_bib", "string": "{10 M.~Bordag, B.~Geyer, G.~L.~Klimchitskaya, V.~M.~Mostepanenko, Phys. Rev. {D 60 (1999) 055004." }, { "name": "quant-ph9912090.extracted_bib", "string": "{11 J.~C.~Long, H.~W.~Chan, J.~C.~Price, Nucl. Phys. { B 539 (1999) 23." }, { "name": "quant-ph9912090.extracted_bib", "string": "{12 V.~M.~Mostepanenko, N.~N.~Trunov, {The Casimir Effect and its Applications, Clarendon Press, Oxford, 1997." }, { "name": "quant-ph9912090.extracted_bib", "string": "{13 I.~E.~Dzyaloshinskii, E.~M.~Lifshitz, L.~P.~Pitaevskii, Sov. Phys. Uspekhi 4 (1961) 153." }, { "name": "quant-ph9912090.extracted_bib", "string": "{14 C.~M.~Hargreaves, { Proc. Kon. Nederl. Acad. Wet. { B68 (1965) 231." }, { "name": "quant-ph9912090.extracted_bib", "string": "{15 J.~Schwinger, L.~L.~DeRaad,~Jr., K.~A.~Milton, { Ann. Phys. {115 (1978) 1." }, { "name": "quant-ph9912090.extracted_bib", "string": "{16 V.~M.~Mostepanenko, N.~N.~Trunov, {Sov. J. Nucl. Phys. { 42 (1985) 818." }, { "name": "quant-ph9912090.extracted_bib", "string": "{17 V.~B.~Bezerra, G.~L.~Klimchitskaya, C.~Romero, Mod. Phys. Lett. A { 12 (1997) 2623." }, { "name": "quant-ph9912090.extracted_bib", "string": "{18 J.~Blocki, J.~Randrup, W.~J.~Swiatecki, C.~F.~Tsang, {Ann. Phys. {105 (1977) 427." }, { "name": "quant-ph9912090.extracted_bib", "string": "{19 S.~K.~Lamoreaux, Phys. Rev. A 59 (1999) R3149." }, { "name": "quant-ph9912090.extracted_bib", "string": "{20 S.~K.~Lamoreaux, e-print quant-ph/9907076." }, { "name": "quant-ph9912090.extracted_bib", "string": "{21 A.~Lambrecht, S.~Reynaud, e-print quant-ph/9907105." }, { "name": "quant-ph9912090.extracted_bib", "string": "{22 N.~G.~van Kampen, B.~R.~A.~Nijboer, K.~Schram, Phys. Lett. A 26 (1968) 307." }, { "name": "quant-ph9912090.extracted_bib", "string": "{23 P.~W.~Milonni, {The Quantum Vacuum, Academic Press, San Diego, 1994." }, { "name": "quant-ph9912090.extracted_bib", "string": "{24 F.~Zhou, L.~Spruch, Phys. Rev. A 52 (1995) 297." }, { "name": "quant-ph9912090.extracted_bib", "string": "{25 G.~Barton, Rep. Prog. Phys. 42 (1979) 963." }, { "name": "quant-ph9912090.extracted_bib", "string": "{26 E.~M.~Lifshitz, L.~P.~Pitaevskii, {Statistical Physics, Part 2, Pergamon Press, Oxford, 1980." }, { "name": "quant-ph9912090.extracted_bib", "string": "{27 I.~S.~Gradshteyn, I.~M.~Ryzhik, {Table of Integrals, Series and Products, Academic Press, New York, 1980." }, { "name": "quant-ph9912090.extracted_bib", "string": "{28 {Handbook of Optical Constants of Solids, ed. E.\\ D.\\ Palik, Academic Press, New York, 1995." }, { "name": "quant-ph9912090.extracted_bib", "string": "{28a H.~Ehrenreich, H.~R.~Philipp, Phys. Rev. 128 (1962) 1622." }, { "name": "quant-ph9912090.extracted_bib", "string": "{29 P.~H.~G.~M.~van Blockland, J.~T.~G.~Overbeek, J. Chem. Soc. Faraday Trans. 74 (1978) 2637." }, { "name": "quant-ph9912090.extracted_bib", "string": "{30 M.~Bordag, G.~L.~Klimchitskaya, V.~M.~Mostepanenko, Int. J. Mod. Phys. {A 10 (1995) 2661." } ]
quant-ph9912091	Topological Chern indices in molecular spectra	[ { "author": "F. Faure\\protect\\( ^{1" } ]	Topological Chern indices are related to the number of rotational states in each molecular vibrational band. Modification of the indices is associated to the appearance of ``band degeneracies'', and exchange of rotational states between two consecutive bands. The topological dynamical origin of these indices is proven through a semi-classical approach, and their values are computed in two examples. The relation with the integer quantum Hall effect is briefly discussed.	[ { "name": "chern_mol.tex", "string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%% F. F. 15 Dec 1999 %%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%% B. Z. 14 Dec 1999 %%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%% F. F. 12 Dec 1999 %%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%% F. F. 6 Dec 1999 %%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%% B. Z. 2 Dec 1999 %%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%% F. F. 1 Dec 1999 %%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%% from B. Z. 29 Nov 1999 %%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%% based on F.F file ``modeleSJ'' 25 Nov 1999 %%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentstyle[prl,twocolumn,aps,epsfig]{revtex}\n\\begin{document}\n\n\n\\title{Topological Chern indices in molecular spectra}\n\n\n\\author{F. Faure\\protect\$ ^{1}\\protect \$ and B. Zhilinskii\\protect\$ ^{2}\\protect \$}\n\n\n\\address{\\protect\$ ^{1}\\protect \$ L.P.M.M.C. (Maison des Magist�res Jean Perrin,\nCNRS) BP 166, 38042 Grenoble Cedex 9, France. \\\\\n faure@labs.polycnrs-gre.fr \\\\\n \\protect\$ ^{2}\\protect \$ Universit\\'{e} du Littoral, MREID, 145 av. M.\nSchumann, 59140 Dunkerque , France \\\\\n zhilin@univ-littoral.fr }\n\n\n\\date{\\today{}}\n\n\\maketitle\n\\draft\n\n\\begin{abstract}\nTopological Chern indices are related to the number of rotational states in\neach molecular vibrational band. Modification of the indices is associated to\nthe appearance of ``band degeneracies'', and exchange of rotational states\nbetween two consecutive bands. The topological dynamical origin of these indices\nis proven through a semi-classical approach, and their values are computed in\ntwo examples. The relation with the integer quantum Hall effect is briefly discussed. \n\\end{abstract}\n\\pacs{PACS numbers: 03.65.Sq; 02.40.-k; 31.15.Gy;}\n\n\\newcommand{\\R}{I\\! \\! R}\n\n\\newcommand{\\C}{I\\! \\! \\! \\! C}\n\n\\newcommand{\\Z}{Z\\! \\! \\! Z}\n\n\\newcommand{\\N}{I\\! \\! N}\n\n\\bigskip\n\nTopological numbers play an important role in many area of physics \\cite{thouless2},\nbut their appearance in molecular physics and especially in rovibrational problems\nhas not been systematically appreciated so far. Simple molecular systems typically\nallow adiabatic separation of vibrational and rotational motion. For non-degenerate\nisolated electronic state (this is the case of ground state for most molecules)\nthe rovibrational energy level system consists of vibrational bands, each associated\nwith one or several degenerate vibrational states. If the rovibrational coupling\nis not too strong, further splitting of the rovibrational structure into sub-bands\ncan be clearly seen. The well-known example is the splitting of the triply degenerate\nvibrational structure for a spherically symmetrical molecule into three sub-bands\ndue to the first-order Coriolis interaction \\cite{Landau,BiL}. Within each\nsub-band formed by \$ 2j+1-C \$ levels, with respectively \$ C=+2,0,-2 \$,\nall energy levels are usually characterized by the quantum number \$ j \$ of\nthe total angular momentum, and by another quantum number \$ R=j+C/2 \$ which\ncharacterizes the coupling of \$ j \$ with the vibrational angular momentum.\n\nIn this letter we show that the integer \$ C \$ can be defined in much more\ngeneral situation as an additional quantum number having a precise topological\nmeaning, namely a Chern index, whose construction will be explained below. This\nindex is defined in the classical limit of the rotational motion. It can be\nassociated with any vibrational band presented in the energy level pattern of\nmolecular systems. Theorem (\\ref{e:formule2}) relates this topological index\nto the number of rotational states within the band. A modification of the index\nis associated with the formation of a contact (a degeneracy) between two consecutive\nvibrational bands, and is shown to generically imply an exchange of one rotational\nstate between the two bands.\n\nSuch a relation was first conjectured in 1988 \\cite{EurophL} after the study\nof the simple model (\\ref{HpolJSsimpl}), and a number of effective Hamiltonians\nreconstructed from experimental data (see Ref.\\cite{MolEx,ZhBr,BrZh} for the\nmolecular examples: SiH\$ _{4} \$, CD\$ _{4} \$, SnH\$ _{4} \$, CF\$ _{4} \$,\nMo(CO)\$ _{6} \$). The universal character of the redistribution phenomena\nand its relevance to integer Hall effect was discussed on several occasions\n\\cite{Zhil,Bel,Zeld}.\n\nTo explain the physical phenomenon and to prepare the formulation of a rigorous\nstatement let us consider a toy problem which involves two quantum angular momenta\n\$ \\mathbf{J} \$ and \$ \\mathbf{S} \$, with fixed modulus \$ {\\mathbf{J}}^{2}=j(j+1) \$\nand \$ {\\mathbf{S}}^{2}=s(s+1) \$ with \$ j,s \$ integer or half-integer:\n\$ \\mathbf{J} \$ acts in the space \$ {\\mathcal{H}}_{j} \$ of dimension \$ (2j+1) \$\nwhich is the irreducible representation space of the \$ SU(2) \$ group. Similarly,\n\$ \\mathbf{S} \$ acts in the space \$ {\\mathcal{H}}_{s} \$ of dimension \$ (2s+1) \$.\nThe total space is \$ {\\mathcal{H}}_{\\textrm{tot}}={\\mathcal{H}}_{j}\\otimes {\\mathcal{H}}_{s} \$\nwith dimension \$ (2j+1)(2s+1) \$.\n\nThe most general quantum Hamiltonian \$ \\widehat{H}({\\mathbf{S}},{\\mathbf{J}}/j) \$\nwe will consider is a hermitian operator acting in \$ \\mathcal{H}_{\\textrm{tot}} \$\nand its action in space \$ {\\mathcal{H}}_{j} \$ is supposed to be expressed\nin terms of the operators \$ {\\mathbf{J}}/j \$. The factor \$ 1/j \$ is introduced\nhere to ensure the existence of the classical limit for \$ j\\rightarrow \\infty \$.\nAn extremely simple form of \$ \\hat{H} \$ is \n\\begin{eqnarray}\n\\hat{H}=(1-t)S_{z}\\, \\, +\\frac{t}{j}(\\mathbf{J}\\cdot \\mathbf{S}), & \\label{HpolJSsimpl} \n\\end{eqnarray}\n with \$ j>s \$, \$ t\\in \\R \$, which was used initially in Ref.\\cite{EurophL}\nto study the redistribution phenomenon and further in Ref.\\cite{PhysL} to establish\nits relation with the classical monodromy. We use this Hamiltonian (\\ref{HpolJSsimpl})\nto illustrate the strict formulation of our result (\\ref{e:formule2}), but\nits validity extends to a general \$ \\widehat{H}({\\mathbf{S}},{\\mathbf{J}}/j) \$.\n\nIn the two extremes limits \$ t=0 \$ (no ``spin-orbit'' coupling), and \$ t=1 \$\n(``spin-orbit'' coupling), the energy level spectrum of (\\ref{HpolJSsimpl})\nshows different patterns of energy levels into bands indexed by \$ g \$. For\n\$ t=0 \$ all energies \$ E_{g}=g \$, \$ g\\in \\{-s,..,+s\\} \$ appear with\nthe same multiplicities \$ N_{g}=(2j+1) \$. For \$ t=1 \$ the spectrum is\nsplit into degenerate multiplets characterized by different eigenvalues of the\ncoupled angular momentum \$ {\\mathbf{N}}^{2}=({\\mathbf{J}}+{\\mathbf{S}})^{2} \$.\nAs in the case of standard spin-orbit coupling with \$ j>s \$ there are \$ 2s+1 \$\ndifferent levels \$ E_{g}=[n(n+1)-j(j+1)-s(s+1)]/(2j) \$, \$ g=n-j\\in \\{-s,\\ldots ,+s\\} \$,\nwith different multiplicities \$ N_{g}=(2j+1)+2g. \$ The two different limiting\ncases for the structures of the \$ (2s+1) \$ bands suggest to introduce a new\nquantum number \$ C_{g} \$ associated with the value of \$ N_{g} \$ within\neach band. \n\nTo define \$ C_{g} \$ for a general Hamiltonian \$ \\widehat{H}({\\mathbf{S}},{\\mathbf{J}}/j) \$\nwe assume that \$ j\\gg s \$, so that it is physically reasonable to consider\n\$ {\\mathbf{J}}_{cl}=\\mathbf{J} \$ as classical, whereas \$ \\mathbf{S} \$\nremains quantum. The classical dynamics for \$ {\\mathbf{J}}_{cl} \$ can be\ndefined through the \$ SU(2) \$ coherent states \$ \|{\\mathbf{J}}_{cl}\\rangle \$\n\\cite{ec1}. The classical phase space for \$ {\\mathbf{J}}_{cl} \$ is the sphere\n\$ S_{j}^{2} \$. From \$ d{\\mathbf{J}}_{cl}/dt=\\partial _{{\\mathbf{J}}_{cl}}H_{cl}\\times {\\mathbf{J}}_{cl} \$\nand because of the factor \$ 1/j \$ in Eq.(\\ref{HpolJSsimpl}), \$ {\\mathbf{J}}_{cl} \$\ncorresponds to a slow dynamical variable compared to \$ \\mathbf{S} \$, and\nthe Born-Oppenheimer approximation suggests to consider for each \$ {\\mathbf{J}}_{cl} \$,\nthe Hermitian operator \$ \\hat{H}_{s}({\\mathbf{J}}_{cl})=\\langle {\\mathbf{J}}_{cl}\|\\hat{H}\|{\\mathbf{J}}_{cl}\\rangle \$\nacting on \$ {\\mathcal{H}}_{s} \$, with spectrum \$ \\hat{H}_{s}({\\mathbf{J}}_{cl})\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle =E_{g,{\\mathbf{J}}_{cl}}\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle \$,\n\$ g\\in \\{-s,..,+s\\} \$. We suppose that for each \$ {\\mathbf{J}}_{cl} \$,\nthe \$ (2s+1) \$ eigenvalues are isolated: \$ E_{-s,{\\mathbf{J}}_{cl}}<E_{-s+1,{\\mathbf{J}}_{cl}}<\\ldots <E_{+s,{\\mathbf{J}}_{cl}} \$.\nThis is the generic situation, because degeneracies are of codimension 3, and\n\$ {\\mathbf{J}}_{cl}\\in S_{j}^{2} \$ is only two-dimensional.\n\nFor each level \$ g \$, let us note \$ [\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle ] \$\nthe eigenvector defined up to a multiplication by a phase \$ e^{i\\alpha } \$.\nThe application \$ {\\mathbf{J}}_{cl}\\rightarrow [\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle ] \$\ndefines then a \$ U(1) \$ fiber bundle over the sphere \$ S_{j}^{2} \$, which\nis the set of the all possible phases \$ \\alpha \$ for every values of \$ {\\mathbf{J}}_{cl} \$.\nThe topology of this bundle is characterized by a Chern number \$ C_{g}\\in \\Z \$\n\\cite{nakahara}. \$ C_{g} \$ reveals the possible global twist of the fiber\nof phases \$ \\alpha \$ over the sphere \$ S_{j}^{2} \$, in the same way the\nwell known M\\\"{o}bius strip is the real line fiber bundle over the circle \$ S^{1} \$\nwith a global twist \$ +1 \$.\n\\begin{figure}\n\\par\\centering \\resizebox{0.8\\columnwidth}{!}{\\includegraphics{bundle.eps} } \\par{}\n\n\n\\caption{The Chern index \\protect\\protect\$ C_{g}\\protect \\protect \$ expresses the\ntwist made by the circles for the phases \\protect\\protect\$ \\alpha \\protect \\protect \$\nof the eigen-vectors \\protect\\protect\$ e^{i\\alpha }\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle \\protect \\protect \$\nover the sphere of spin \\protect\\protect\$ {\\mathbf{J}}_{cl}\\protect \\protect \$,\nin the same way the M\\\"{o}bius strip has a twist made by the lines over a circle.}\n\\end{figure}\n\n\nNote that \n\\begin{equation}\n\\label{eq/sum}\n\\sum _{g=-s}^{+s}C_{g}=0,\n\\end{equation}\n just because of the additivity of Chern indices, and because the eigenvectors\n\$ (\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle ) \$ span the space \$ {\\mathcal{H}}_{s} \$\nwhich does not depend on \$ {\\mathbf{J}}_{cl} \$.\n\nThe Born Oppenheimer approach tells that the total spectrum \$ \\hat{H}\|\\phi _{i}\\rangle =E_{i}\|\\phi _{i}\\rangle ,i\\in \\{1,..,(2s+1)(2j+1)\\} \$\nin \$ \\mathcal{H}_{\\textrm{tot}} \$, can be represented as formed in \$ 2s+1 \$\ngroups of levels (bands) with eigen-functions of each group spanning vector\nspaces: \$ L_{g}\\subset {\\mathcal{H}}_{\\textrm{tot}},\\quad g\\in \\{-s,\\ldots ,+s\\}. \$\nWe precise now the number of levels in each group:\n\n\\textbf{Theorem.}\n\n\\emph{For a general Hamiltonian \$ \\hat{H}({\\mathbf{S}},{\\mathbf{J}}/j) \$,\nand for \$ j \$ large enough then \$ \\dim L_{g}=\\dim {\\mathcal{H}}_{j}-C_{g}, \$\nso}\n\n\n\\begin{equation}\n\\label{e:formule2}\nN_{g}=(2j+1)-C_{g}.\n\\end{equation}\n The number \$ N_{g} \$ of states \$ \|\\phi _{i}\\rangle \$ in each band \$ L_{g} \$\nof \$ \\mathcal{H}_{\\textrm{tot}} \$ is given by the quantum number \$ j \$\nand an additional quantum number \$ C_{g} \$, namely the topological Chern\nindex.\n\nA few remarks are in order here. Since \$ (2j+1) \$ is the number of quanta\nin the classical phase space \$ S_{j}^{2} \$ for spin \$ {\\mathbf{J}}_{cl} \$,\nEq.(\\ref{e:formule2}) looks like a Weyl formula with a correction. The index\n\$ C_{g} \$ has been defined and can be computed in ``a semi-quantal'' approach\nwhere \$ {\\mathbf{J}}_{cl} \$ is considered as a classical variable and \$ \\mathbf{S} \$\nquantum. Nevertheless Eq.(\\ref{e:formule2}) provides an information on the\nfull quantum problem: the spectrum of \$ \\hat{H} \$. Finally the topological\nnature of \$ C_{g} \$ reveals a qualitative and robust property of the spectrum\nof \$ \\hat{H} \$, stable under perturbations, provided no degeneracy appears\nbetween consecutive bands. One can say that \$ C_{g} \$ expresses a topological\ncoupling between the dynamical variables \$ \\mathbf{J} \$ and \$ \\mathbf{S} \$.\n\nTwo approaches to the computation of the Chern indices of different bands will\nbe suggested. The first one we use below is algebraic. The second one uses the\nBerry's connection, and is based on the curvature formula \\cite{nakahara}.\nThis last formula could be useful for numerical computations.\n\nThe algebraic calculation is based on the geometric interpretation of the Chern\nindex \$ C_{g} \$ as the total intersection number between the one-dimensional\ncurve \$ [\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle ]_{{\\mathbf{J}}_{cl}} \$ in \$ P({\\mathcal{H}}_{s}) \$\nwith the hyperplane \$ (\|\\psi _{0}\\rangle )^{\\bot }=\\left\\{ \|\\varphi \\rangle \\, \\textrm{such}\\, \\, \\textrm{that}\\, \\langle \\varphi \|\\psi _{0}\\rangle =0\\right\\} \$.\nHere \$ \|\\psi _{0}\\rangle \\in {\\mathcal{H}}_{s} \$ is arbitrary. Each intersection\nhas number \$ \\sigma =+1 \$ (\$ -1 \$) if the curve orientation is compatible\n(incompatible) with the orientation of the hyperplane. \$ C_{g} \$ is the sum\nof these intersection numbers \\cite{harris1}.\n\nThe application of this algebraic method of calculation to the two limiting\ncases of Hamiltonian (\\ref{HpolJSsimpl}) is immediate. For \$ t=0 \$ the Hamiltonian\n\$ \\hat{H}_{s}({\\mathbf{J}}_{cl})=S_{z} \$ does not depend on \$ {\\mathbf{J}}_{cl} \$,\nthe application \$ {\\mathbf{J}}_{cl}\\rightarrow [\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle ] \$\nis constant, the topology of the bundle is trivial with zero Chern index: \$ C_{g}=0 \$,\nin accordance with Eq.(\\ref{e:formule2}). For \$ t=1 \$ we have \n\\[\n\\hat{H}_{s}({\\mathbf{J}}_{cl})=\\langle {\\mathbf{J}}_{cl}\|\\hat{H}\|{\\mathbf{J}}_{cl}\\rangle =\\frac{1}{j}({\\mathbf{J}}_{cl}\\cdot {\\mathbf{S}}),\\]\n with \$ {\\mathbf{J}}_{cl} \$ a vector on the sphere \$ S_{j}^{2} \$, and\n\$ {\\mathbf{S}} \$ a vectorial operator in \$ {\\mathcal{H}}_{s} \$. The eigenvector\n\$ \|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle \$ is easily obtained from the eigenvector\n\$ \|m_{s}=g\\rangle \\propto S_{+}^{(s+g)}\|0\\rangle \$ of \$ S_{z} \$ by a\nrotation which transforms the \$ z \$ axis to the \$ {\\mathbf{J}}_{cl} \$\naxis on the sphere. It is convenient to choose a coherent state \$ \|{\\mathbf{S}}_{0,cl}\\rangle \$\nas the reference state \$ \|\\psi _{0}\\rangle \$, where \$ {\\mathbf{S}}_{0,cl} \$\nis an arbitrary classical spin \$ \\mathbf{S} \$. An intersection of the curve\n\$ [\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle ]_{{\\mathbf{J}}_{cl}} \$with the hyperplane\n\$ (\|\\psi _{0}\\rangle )^{\\bot } \$ is then given by the equation \$ \\langle {\\mathbf{S}}_{0,cl}\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle =0 \$\nwhich has a very simple interpretation: the point \$ {\\mathbf{S}}_{0,cl} \$\nis a zero of the Husimi representation \$ \\textrm{Hus}({\\mathbf{S}}_{cl})=\\left\| \\langle {\\mathbf{S}}_{cl}\|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle \\right\| ^{2} \$\nof the state \$ \|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle \$ on the sphere \$ S^{2}_{s} \$\n(the classical phase space of \$ {\\mathbf{S}} \$), shown in figure \\ref{Fig1}.\nThe Husimi representation has \$ (s-g) \$ zeros inside the (oriented) trajectory,\nand \$ (s+g) \$ zeros outside \\cite{zero2}. When the axis \$ {\\mathbf{J}}_{cl} \$\nmoves with direct orientation on the whole sphere, the \$ (s-g) \$ zeros pass\nwith positive orientation over the fixed point \$ {\\mathbf{S}}_{0,cl} \$, giving\n\$ \\sigma _{1}=s-g \$, whereas the \$ s+g \$ zeros pass with negative orientation\nover \$ {\\mathbf{S}}_{0,cl} \$, giving \$ \\sigma _{2}=-(s+g) \$. So \$ C_{g}=\\sigma _{1}+\\sigma _{2}=-2g \$\nin accordance with Eq.(\\ref{e:formule2}).\n\\begin{figure}\n\\par\\centering \\resizebox{0.8\\columnwidth}{!}{\\includegraphics{2husimi_mJ.eps} } \\par{}\n\n\n\\caption{\\label{Fig1} The Husimi distribution for the state \\protect\\protect\\protect\$ \|\\psi _{g,{\\mathbf{J}}_{cl}}\\rangle \\protect \\protect \\protect \$\non the sphere \\protect\\protect\\protect\$ S_{s}^{2}\\protect \\protect \\protect \$\nfor spin \\protect\\protect\\protect\$ {\\mathbf{S}}\\protect \\protect \\protect \$.\n\\protect\\protect\$ {\\mathbf{S}}_{0,cl}\\protect \\protect \$ is a reference point. }\n\\end{figure}\n\n\nIn this last example, the variation of Chern indices \$ \\Delta C_{g}=-2g \$\noccurs at \$ t=1/2 \$, with a degeneracy between the bands at \$ {\\mathbf{J}}_{cl}=(0,0,-j) \$,\ngiving \$ \\hat{H}_{s}({\\mathbf{J}}_{cl})=0 \$. In the case of two bands (\$ s=1/2 \$,\n\$ g=\\pm 1/2 \$) then \$ \\Delta C_{g=\\pm }=\\mp 1 \$, and the two bands have\na conical contact at the degeneracy. In the vicinity of the degeneracy, and\nfor \$ j\\rightarrow \\infty \$, we observe that \$ [J_{-},J_{+}]=-2J_{z}\\simeq 2j \$,\nso \$ a=J_{-}/\\sqrt{2j} \$, \$ a^{+}=J_{+}/\\sqrt{2j} \$ fulfilled the harmonic\noscillator commutation relations. In the basis \$ \|\\pm \\rangle =\|m_{s}=\\pm 1/2\\rangle \$\nof \$ {\\mathcal{H}}_{s} \$, the expression of \$ \\hat{H} \$ can be simplified\nand gives \n\\begin{equation}\n\\label{e:modele}\n\\hat{H}=\\frac{1}{\\sqrt{2j}}\\left( \\begin{array}{cc}\n-\\tilde{t} & a\\\\\na^{+} & \\tilde{t}\n\\end{array}\\right) ,\n\\end{equation}\n with \$ \\tilde{t}=(2t-1)\\sqrt{2j} \$. We also scales the energy with \$ \\tilde{E}=E\\sqrt{2j} \$,\nand note \$ \|n\\rangle \\propto a^{+n}\|0\\rangle \$. The stationary equation\n\$ \\hat{H}\|\\phi \\rangle =E\|\\phi \\rangle \$ can easily be solved, giving for\n\$ n=1,2,\\ldots \$, \$ \|\\phi _{n}^{\\pm }\\rangle =\|n\\rangle \|-\\rangle +\\sqrt{n}/(\\tilde{E}_{n}^{\\pm }+\\tilde{t})\|n-1\\rangle \|+\\rangle \$\nwith \$ \\tilde{E}_{n}^{\\pm }=\\pm \\sqrt{n+\\tilde{t}^{2}} \$ , and one single\nstate \$ \|\\phi _{0}\\rangle =\|0\\rangle \|-\\rangle \$ with \$ \\tilde{E}_{0}=\\tilde{t} \$.\nFigure \\ref{fig:spectre} shows this spectrum with the simplified expressions\nof \$ \|\\phi _{n}^{\\pm }\\rangle \$ obtained for large \$ \\left\| \\tilde{t}\\right\| \$.\nThese simplified expressions involving the states \$ \|n\\rangle \$ of the harmonic\noscillator express the quantized modes for small oscillations near the extrema\nof the two bands \\cite{clusterInd}. We clearly observe the exchange of one\nstate in the spectrum, giving \$ \\Delta N_{g=\\pm }=\\pm 1 \$. As a consequence,\n\$ \\Delta N_{g}+\\Delta C_{g}=0 \$. This gives for each band a conserved quantity,\nnamely \$ (N_{g}+C_{g}) \$. The variation \$ \\Delta C_{+}=-1 \$ can also be\nconsidered as ``the topological charge'' associated to the degeneracy of the\n\$ 2\\times 2 \$ matrix (\\ref{e:modele}) where \$ a_{cl}=x_{cl}-ip_{cl} \$\nis a classical variable. {[}\$ \\Delta C_{+}=-1 \$ is also obtained from Eq.\n(\\ref{e:formule C+-}), for a little sphere in space \$ (x_{cl},p_{cl},\\tilde{t}) \$\ncentered at the degeneracy \$ (0,0,0) \${]}.\n\\begin{figure}\n\\par\\centering \\resizebox{0.8\\columnwidth}{!}{\\includegraphics{spectre.eps} } \\par{}\n\n\n\\caption{\\label{fig:spectre}Spectrum of rotational states for the Hamiltonian (\\ref{e:modele}).\n\\protect\$ \\tilde{t}=0\\protect \$ corresponds a conical contact (degeneracy)\nbetween the two bands, with a topological charge \\protect\\protect\$ \\Delta C_{+}=-1\\protect \\protect \$,\nand an exchange of one rotational state, \\protect\\protect\$ \\Delta N_{+}=+1\\protect \\protect \$.}\n\\end{figure}\n\n\nFor a general Hamiltonian \\emph{\$ \\hat{H}({\\mathbf{S}},{\\mathbf{J}}/j) \$}\nthe very simple model Eq.(\\ref{e:modele}) provides the general mechanism for\nthe exchange of one state in the vicinity of every degeneracy between two consecutive\nbands. In the trivial case \$ \\hat{H}_{0}=S_{z} \$, we have computed the value\nof the conserved quantity: \$ N_{g}+C_{g}=2j+1 \$. We deduce that it is still\ncorrect when \$ \\hat{H}_{0} \$ is deformed to \\emph{\$ \\hat{H}({\\mathbf{S}},{\\mathbf{J}}/j) \$},\nproving formula (\\ref{e:formule2}).\n\nThe second example, addresses the Chern indices of the two-state model corresponding\nto \$ s=1/2 \$, i.e. to \$ \\dim {\\mathcal{H}}_{s}=2 \$. In a given fixed basis\n, say \$ \|m_{s}=\\pm 1/2\\rangle \$, the matrix of \$ \\hat{H}_{s}({\\mathbf{J}}_{cl}) \$\nhas the form: \n\\begin{equation}\n\\label{e:2x2matrix}\n\\hat{H}_{s}\\equiv \\left( \\begin{array}{cc}\nh_{11}({\\mathbf{J}}_{cl}) & h_{12}({\\mathbf{J}}_{cl})\\\\\n\\overline{h_{12}}({\\mathbf{J}}_{cl}) & h_{22}({\\mathbf{J}}_{cl})\n\\end{array}\\right) .\n\\end{equation}\n\n\nThis matrix will give two vibrational sub-bands with Chern indices \$ C_{-} \$\nand \$ C_{+} \$. In Ref.\\cite{fred1}, it is shown that the Chern indices have\nthe following property: Let \$ {\\mathbf{J}}^{} \$ be the zeros of \$ h_{12}({\\mathbf{J}}_{cl}) \$,\nand \$ \\sigma ({\\mathbf{J}}^{}) \$, their degree defined as follows: take\na small direct circle around \$ {\\mathbf{J}}^{} \$; its image by \$ h_{12}({\\mathbf{J}}_{cl}) \$\nis a closed curve around \$ 0 \$ with \$ \\sigma ({\\mathbf{J}}^{}) \$ turns.\nDefine the set \$ {\\mathcal{S}}^{+}=\\left\\{ {\\mathbf{J}}_{cl}\\in S^{2}_{j}\\textrm{ such }\\, \\, \\textrm{ that }\\left( h_{22}({\\mathbf{J}}_{cl})-h_{11}({\\mathbf{J}}_{cl})\\right) >0\\right\\} \$.\nThen \n\\begin{eqnarray}\nC_{+}=\\sum _{J^{}\\in {\\mathcal{S}}^{+}}\\sigma ({\\mathbf{J}}^{})=-C_{-}.\\label{e:formule C+-} \n\\end{eqnarray}\n\n\nA direct consequence of this property is that a change of Chern index can only\noccur when simultaneously \$ h_{12}({\\mathbf{J}}^{})=h_{22}({\\mathbf{J}}^{})-h_{11}({\\mathbf{J}}^{})=0. \$\nThis corresponds to a degeneracy in the spectrum of \$ \\hat{H}_{s} \$, and\na conical contact of the two bands.\n\nFormula (\\ref{e:formule C+-}) can be applied to a Hamiltonian describing the\nrotational structure of the doubly degenerated vibrational state of a tetrahedral\n(or octahedral) spherical top molecule \\cite{boris1}. The most general Hamiltonian,\ntaken up to the third degree in \$ \\mathbf{J} \$, has the form (\\ref{e:2x2matrix})\nwith\n\n\\begin{eqnarray}\nh_{12}({\\mathbf{J}})=\\left( J_{x}^{2}-J_{y}^{2}\\right) /j^{2}+iXJ_{x}J_{y}J_{z}/j^{3}, & & \\\\\nh_{22}({\\mathbf{J}})-h_{11}({\\mathbf{J}})=\\left( 3J_{z}^{2}-j(j+1)\\right) /j^{2}, & & \n\\end{eqnarray}\n with parameter \$ X\\in \\R \$. In this case the set \$ {\\mathcal{S}}^{+} \$\nincludes all points around north and south pole for which \$ J_{z}>\\frac{1}{\\sqrt{3}}\\sqrt{j(j+1)} \$\nor \$ J_{z}<\\frac{-1}{\\sqrt{3}}\\sqrt{j(j+1)} \$. As \$ h_{12}({\\mathbf{J}})=0 \$\nfor simultaneously \$ J_{x}=\\pm J_{y} \$, and \$ J_{x}J_{y}J_{z}=0 \$ there\nare two points \$ J^{}\\in {\\mathcal{S}}^{+} \$: the north pole \$ J_{z}=j \$\nand the south pole \$ J_{z}=-j \$. We consider \$ {\\mathbf{J}} \$ going through\na closed path surrounding each \$ J^{} \$ to calculate \$ \\sigma ({\\mathbf{J}}^{*}) \$.This\ngives for north and for the south poles \$ \\sigma _{north}=\\sigma _{south}=2\\, \\textrm{sign}(X) \$,\nso that \$ C_{+}=4\\, \\textrm{sign}(X) \$ and \$ C_{-}=-C_{+} \$.\n\nThis calculation explains why the rotational structure of doubly degenerate\nvibrational state is generally split into two sub-bands with respectively \$ 2j+5 \$\nand \$ 2j-3 \$ levels \\cite{mich,ZhBr}. In our current approach the appearance\nof two bands with Chern indices \$ \\pm 4 \$ for the Hamiltonian (\\ref{e:2x2matrix})\nis due to the formation of eight degeneracies {[}equivalent by symmetry{]} between\nthe two vibrational bands at \$ X=0 \$ parameter value. More generally the\ncharacterization of the rovibrational structure of molecules and its possible\nmodification under the variation of some physical parameters like total angular\nmomentum can be done systematically by using Chern indices as topological quantum\nnumbers. In Ref.\\cite{BrZh} similar effect was discussed without explicit introduction\nof Chern indices.\n\nThe molecular application studied in this Letter was mainly inspired by the\nrole played by topological Chern indices in the integer quantum Hall effect\n\\cite{chern3}. In this context, Chern indices describe the topology of Floquet\nbands \$ (\\mathbf{k})\\rightarrow [\|\\psi _{g}(\\mathbf{k})\\rangle ] \$ where\n\$ \\mathbf{k} \$ is the Bloch wave vector, and give a quantum Hall conductance\n\$ \\sigma _{g}=(e^{2}/h)C_{g} \$ under the hypothesis of adiabatic motion of\n\$ \\mathbf{k} \$ when a weak electrical field is applied. Contrary to Eq.(\\ref{eq/sum}),\ntheir sum for a given Landau level is \$ \\sum _{g}C_{g}=+1 \$, because of the\nnon trivial topology of the quantum space \\cite{chern3}. Many properties of\nthe Chern indices are similar: a change \$ \\Delta C_{g}=\\pm 1 \$ is related\nto a conical degeneracy between consecutive bands. The application of semi-classical\ncalculations of \$ C_{g} \$ done for the Hall conductance in Ref.\\cite{fred3},\nwill be the subject of future work.\n\nIn summary we have discussed the role of Chern quantum numbers to molecular\nspectroscopy. The interpretation of good integer quantum numbers associated\nwith rotational structure of different vibrational bands in terms of topological\nChern numbers has naturally a wide applicability. These indices can be introduced\nany time when the adiabatic separation of variables enables one to split the\nglobal structure into bands associated with the ``fast motion'' and their\ninternal structure described by a ``slow motion'' on a compact phase space.\nWe have considered here only the problem when the dimension of the classical\nphase space formed by the slow variable \$ {\\mathbf{J}}_{cl} \$ is 2. Only\nthe first Chern class appears in this case. Extension to higher dimension requires\nmore delicate physical interpretation and more sophisticated mathematical tools\nin relation with the index theorem of Atiyah-Singer. In molecular spectroscopy\nmany problems with intra-molecular dynamics are known in great detail. They\ncan be used to test the applicability of this new concept.\n\nWe gratefully acknowledge Y. Colin de Verdiere for stimulating discussions and\nBart Van-Tiggelen for helpful comments.\n\n\\begin{references} \n\n\\bibitem{thouless2} D. J. 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quant-ph9912092	Nonlinear Matter Wave Dynamics with a Chaotic Potential	[ { "author": "S.~A.~Gardiner$^{1" } ]	We consider the case of a cubic nonlinear Schr\"{odinger equation with an additional chaotic potential, in the sense that such a potential produces chaotic dynamics in classical mechanics. We derive and describe an appropriate semiclassical limit to such a nonlinear Schr\"{odinger equation, using a semiclassical interpretation of the Wigner function, and relate this to the hydrodynamic limit of the Gross-Pitaevskii equation used in the context of Bose-Einstein condensation. We investigate a specific example of a Gross-Pitaevskii equation with such a chaotic potential: the one-dimensional delta-kicked harmonic oscillator, and its semiclassical limit. We explore the feasibility of experimental realization of such a system in a Bose-Einstein condensate experiment, giving a concrete proposal of how to implement such a configuration, and considering the problem of condensate depletion.	[ { "name": "nonlinmat.tex", "string": "\\documentstyle[aps,twocolumn,epsfig,amssymb]{revtex}\n%\\documentstyle[aps,twocolumn,epsfig,amssymb,bbm]{revtex}\n%\\documentstyle[preprint,aps,epsfig,amssymb,bbm]{revtex}\n\\tolerance = 10000\n\n\\begin{document}\n\n\\draft\n\\onecolumn\n\\title{Nonlinear Matter Wave Dynamics with a Chaotic Potential}\n\n\\author{S.~A.~Gardiner$^{1}$,\nD.~Jaksch$^{1}$, R.~Dum$^{1,2}$,\nJ.~I.~Cirac$^{1}$, \nand P.~Zoller$^{1}$}\n\\address{$^{1}$Institut f{\\\"u}r Theoretische Physik,\nUniversit{\\\"a}t Innsbruck, A-6020\nInnsbruck, Austria\\\\\n$^{2}$Ecole Normale Sup\\'{e}rieure, Laboratoire Kastler Brossel, 24, Rue\nLhomond, F-75231 Paris Cedex 05, France}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nWe consider the case of a cubic nonlinear Schr\\\"{o}dinger equation with an \nadditional chaotic potential, in the sense that such a potential produces\nchaotic dynamics in classical mechanics. \nWe derive and describe \nan appropriate semiclassical limit to such a nonlinear Schr\\\"{o}dinger equation,\nusing a semiclassical interpretation of the Wigner function, and relate this to the\nhydrodynamic limit of the Gross-Pitaevskii equation used in the context of \nBose-Einstein condensation. We investigate\na specific example of a Gross-Pitaevskii equation with such a chaotic potential: \nthe one-dimensional delta-kicked\nharmonic oscillator, and its semiclassical limit. We explore the feasibility of \nexperimental realization of such a\nsystem in a Bose-Einstein condensate experiment, giving a concrete proposal of how to\nimplement such a configuration, and considering the problem of\ncondensate depletion.\n\\end{abstract}\n\n\\pacs{PACS numbers: \n03.75.-b, %matter waves\n05.45.-a, %Nonlinear dynamics\n%05.45.Mt, %Semiclassical chaos/ quantum chaos\n03.65.Bz, %foundations\n42.50.Vk %mechanical effects of light on atoms\n}\n\n\\section{Introduction}\nChaos in classical Hamiltonian systems, most simply thought of as the extreme\nsensitivity of trajectories in phase space to initial conditions, making long \nterm predictions extremely\ndifficult, is by now broadly understood \\cite{reichl,gutzwiller}. \nMore recently, the field of quantum chaos, for our purpose meaning the study \nof quantum mechanical equivalents of classical chaotic systems, has been the \nsubject of much investigation \\cite{reichl,gutzwiller,haake}. From this it \ndoes seem that the dynamics of quantum mechanical systems can be divided into \nregular and irregular subsets, with distinct differences between the two, just \nas is the case in classical mechanics. For example, due to the unitarity of \nthe evolution of the state vector, there can be no equivalent of sensitivity \nto initial conditions in the Hilbert space, but there appears to be an \nequivalent sensitivity to perturbation which distinguishes\nquantum chaotic motion \\cite{peres}.\nA certain amount of understanding has thus been achieved, although there are still\nunresolved problems, in particular how to extract classical chaos from quantum\nmechanics \\cite{zurek}.\n\nQuantum dynamics are determined by the Schr\\\"{o}dinger equation. A\nseemingly natural extension is to ask what happens when we take a quantum chaotic \nSchr\\\"{o}dinger equation, and add some kind of nonlinearity. This is \nsomething which has been much less studied \\cite{nonlinchaos}, and is certainly \nof more than academic\ninterest; such equations do appear in nature, for example the Gross-Pitaevskii\nequation in the field of Bose-Einstein condensation \\cite{bosecond,review}, \nand also in the field of\nnonlinear optics \\cite{nonlinearopt}. There are thus experimentally accessible \nsystems in which such\nchaotic effects may manifest themselves.\n\nIt is also interesting to note that just as in the case of quantum mechanics, where\n if\none takes the limit $\\hbar\\rightarrow 0$ one expects to regain classical dynamics,\none can also carry out this limit for nonlinear Schr\\\"{o}dinger equations. This\nproduces equations reminiscent of classical hydrodynamics, an \ninterpretation\nalso extensively used in the theoretical study of Bose-Einstein condensates\n\\cite{hydro}. This interconnection of different kinds of dynamics is displayed\nschematically in Fig.~(\\ref{connections}).\n\n\nWe will firstly be concerned with effective ``single particle'' systems. That is to\nsay, where one takes a linear single particle Schr\\\"{o}dinger equation with a \npotential which is known to produce chaotic dynamics in Hamilton's equations of \nmotion, and adds a nonlinearity to it. The Gross-Pitaevskii equation (for\nexample) describes the\ncollective dynamics of huge numbers of particles, but may nevertheless be \nthought of as an effective single particle wave equation. We later consider\ncorrections to this interpretation, taking into account more fully the many body\ndynamics.\n\nIt is thus of general interest to determine how effects of classical chaos and\nquantum chaos manifest themselves in the dynamics of nonlinear Schr\\\"{o}dinger \nequations, and to what extent the dynamics of the nonlinear Schr\\\"{o}dinger\nequation can be explained by motion in the hydrodynamic limit, as determined by\nthe hydrodynamic equations. \n\n\\section{Generalities}\n\\subsection{Gross-Pitaevskii Equation}\n\\label{gpeint}\nIn this paper we will consider explicitly only one dimensional systems, although\nthe analytic results presented can easily be generalized to two or three spatial\ndimensions. To simplify things further, we consider only the cubic nonlinearity \nexplicitly, the simplest\nnonlinearity possible, resulting in the one dimensional\nGross-Pitaevskii equation, well known in the context of Bose-Einstein\ncondensation:\n\\begin{equation}\ni\\hbar\\frac{\\partial}{\\partial t}\\varphi=\n-\\frac{\\hbar^{2}}{2m}\\frac{\\partial^{2}}{\\partial x^{2}}\\varphi\n+V(x,t)\\varphi +u\|\\varphi\|^{2}\\varphi,\n\\label{gpe}\n\\end{equation}\nwhere $\\varphi(x,t)$ is the wavefunction and $u$ the strength of the\nnonlinearity. Again, the analytic results here can easily be generalized to more\ncomplicated nonlinearities. Such a simplified system demonstrates all the main\nfeatures of a nonlinear Schr\\\"{o}dinger equation, and is perfectly adequate for\nillustrative purposes. This kind of \nsimplified system is in fact experimentally accessible,\nfor example in a Bose-Einstein condensate experiment, as will be shown in\nSection \\ref{physicalmodel}.\n\n\\subsection{Hydrodynamic Equations}\nIt is tempting to think of the hydrodynamic equations as the semiclassical limit\nof the Gross-Pitaevskii equation. This turns out to be not quite so, as will be\nshown in Sec.~\\ref{wigner}. We nevertheless sketch out the standard derivation\nof the hydrodynamic equations, in order to set notation, and so that later we\ncan point out the differences between the hydrodynamic limit and the genuine\nsemiclassical limit, which we will derive using Wigner functions.\n\nWe rewrite the Gross-Pitaevskii equation Eq.~(\\ref{gpe}) \nusing the density $\\rho$ and a momentum field $P$ \\cite{velocity}, \ndefined in terms of the \nwavefunction\n$\\varphi=\\sqrt{\\rho}e^{iS/\\hbar}$ as:\n\\begin{eqnarray}\n\\rho&=&\|\\varphi\|^{2},\\\\\n\\rho P&=&\\frac{\\hbar}{2i}\\left[\n\\varphi^{}\\frac{\\partial}{\\partial x}\\varphi\n-\\left(\\frac{\\partial}{\\partial x}\\varphi^{}\\right)\\varphi\n\\right] = \\rho\\frac{\\partial}{\\partial x}S.\n\\end{eqnarray}\nThe resulting equation of motion for the density is\n\\begin{equation}\n\\frac{\\partial}{\\partial t}\\rho=\n-\\frac{\\partial}{\\partial x}\\left(\nP\\rho\n\\right).\\label{continuity}\n\\end{equation}\nBefore moving to the equation of motion for $P$, \nwe first consider the equation\nfor $S$, which is\n\\begin{equation}\n\\frac{\\partial}{\\partial t}S=\n-\\frac{1}{2m}\\left(\\frac{\\partial}{\\partial x} S\\right)^{2}\n-V(x,t)- u\\rho+\n\\frac{\\hbar^{2}}{2m\\sqrt{\\rho}}\n\\frac{\\partial^{2}}{\\partial x^{2}}\\sqrt{\\rho}.\n\\label{prehj}\n\\end{equation}\nThe equation \nof motion for the momentum field $P$ is exactly the spatial derivative of\n Eq.~(\\ref{prehj}): \n\\begin{equation}\n\\frac{\\partial}{\\partial t}P=\n-\\frac{\\partial}{\\partial x}\\left[\n\\frac{P^{2}}{2m}+\nV(x,t)+u\\rho-\n\\frac{\\hbar^{2}}{2m\\sqrt{\\rho}}\n\\frac{\\partial^{2}}{\\partial x^{2}}\\sqrt{\\rho}\n\\right].\\label{premomentumfield}\n\\end{equation}\n\nTaking the hydrodynamic limit \\cite{review,hydro} consists of abandoning the term in\nEq.~(\\ref{premomentumfield}) proportional to $\\hbar^{2}$, generally justified by\nclaiming that the density $\\rho$ is sufficiently smooth for its derivatives\nto be insignificant, resulting in\n\\begin{equation}\n\\frac{\\partial}{\\partial t}P=\n-\\frac{\\partial}{\\partial x}\\left[\n\\frac{P^{2}}{2m}+\nV(x,t)+u\\rho\n\\right].\\label{momentumfield}\n\\end{equation}\nClearly, to get Eq.~(\\ref{momentumfield}), we have discarded all quantum\ncharacter of the Gross-Pitaevskii equation. Also note that if the corresponding\nterm is abandoned in Eq.~(\\ref{prehj}) in the case where $u=0$ and $V$ is time\nindependent, we get\nthe Hamilton-Jacobi equation for a single particle in the potential $V$, with\nthe interpretation that $\\partial S/\\partial x$ is the canonically conjugate\nmomentum to the coordinate $x$ \\cite{goldstein}. \n\nThis seems to indicate that the hydrodynamic\nequations Eqs.~(\\ref{continuity},\\ref{momentumfield}) might be an \nequivalent ``classical'' limit to the Gross-Pitaevskii equation with finite $u$\n[Eq.~(\\ref{gpe})]. As previously stated however, this turns out to be not \nquite so, as we shall soon see.\n\n\\section{Wigner Function Dynamics}\n\\label{wigner}\n\\subsection{Expansion in $\\hbar$}\nWe wish to carry out a consistent expansion of Eq.~(\\ref{gpe}) around $\\hbar$,\nin order to clearly separate classical from quantum dynamics, and to provide\norder by order corrections. We will do this by considering the dynamics of \nthe Wigner function\n$W$, which is exactly equivalent to the wavefunction $\\varphi$, in the sense\nthat all information about the wavefunction is contained within its Wigner\nrepresentation.\n\nWe define the Wigner function (for a pure state) as\n\\begin{equation}\nW(x,p) = \\frac{1}{2\\pi\\hbar}\\int_{-\\infty}^{\\infty}d\\tau\ne^{-ip\\tau/\\hbar} \\varphi^{}(x-\\tau/2)\\varphi(x+\\tau/2).\n\\label{wignerdef}\n\\end{equation}\nIt is well known that the dynamics of the Wigner function of a single particle \nto lowest order give\nsimply the classical Liouville equation of a distribution of noninteracting\nparticles \\cite{zurek}. The exact expression to all orders in $\\hbar$ for the \ntime evolution\nof the Wigner function $W$ is given by:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}W &=& \n\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{(2s+1)!} \n\\left(\\frac{\\hbar}{2}\\right)^{2s}\n\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}H\n\\frac{\\partial^{2s+1}}{\\partial p^{2s+1}}W\n-\\frac{\\partial }{\\partial p}H\n\\frac{\\partial}{\\partial x} W\n\\label{singleparticle}\n\\end{eqnarray}\nwhere $H$ is the single particle classical Hamiltonian function. How to \nobtain this expression is sketched in Appendix \\ref{wignerapp}. Setting\n$\\hbar=0$ we see we do indeed get the classical Liouville equation\n\\begin{equation}\n\\frac{\\partial}{\\partial t}W= \n\\frac{\\partial}{\\partial x}H\n\\frac{\\partial}{\\partial p}W\n-\\frac{\\partial }{\\partial p}H\n\\frac{\\partial}{\\partial x} W,\n\\label{singleliouville}\n\\end{equation}\nso long as the initial Wigner function can in fact be interpreted as a classical\nprobability density (i.e.\\ is non-negative). If we have as a classical\nLiouville density a delta\ndistribution, $W(x,p)=\\delta(x-x_{0})\\delta(p-p_{0})$,\nwe regain classical point dynamics. \nOne can think of\na point particle being regained from quantum mechanics if we have \na coherent state centred at $x=x_{0}$ and $p=p_{0}$ and let $\\hbar\\rightarrow\n0$, causing the Wigner function to tend to just such a delta distribution.\n\nIt is worth mentioning that although we talk blithely about letting $\\hbar$ tend\nto zero, this is in fact physically meaningless. As $\\hbar$ is a constant, we\nmust in fact expand around some scaling parameter to do with the characteristic\naction scales of the problem at hand, such that at some point the quantum\ncorrections should be completely dominated, at least for some characteristic\ntime \\cite{zurek}. Generally some appropriate parameter presents itself, as will\nbe shown in the model we present in Section \\ref{model}, and expansions where it\nis stated that the limit $\\hbar\\rightarrow 0$ is explored should be interpreted\nin this manner.\n\nWhat we now wish to do is to take an\nequivalent limit to that presented in\nEqs.~(\\ref{singleparticle},\\ref{singleliouville}) \nfor the Gross-Pitaevskii equation, with the object of\ngetting some kind of Liouville equation with the nonlinearity taken into \naccount. The full expansion of the Wigner function dynamics\ngoverned by Eq.~(\\ref{gpe}) in terms of $\\hbar$ turns out to be:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}W&=&\n-\\frac{\\partial }{\\partial p}H\n\\frac{\\partial }{\\partial x}W+\n\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{(2s+1)!}\n\\left(\\frac{\\hbar}{2}\\right)^{2s}\n\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}\n\\left[H+u\\rho\\right]\n\\frac{\\partial^{2s+1}}{\\partial p^{2s+1}}W,\n\\label{wignerexp}\n\\end{eqnarray}\nwhere we have the density\n\\begin{equation}\n\\rho(x) = \\int_{-\\infty}^{\\infty}dp'W(x,p')=\|\\varphi(x)\|^{2},\n\\label{wigdens}\n\\end{equation}\nexactly as in the hydrodynamic equations,\nEqs.~(\\ref{continuity},\\ref{momentumfield}). \nThe result of Eq.~(\\ref{wignerexp}) \nis\noutlined in Appendix \\ref{wignerapp}.\n\nIf we take only the zeroth term in the infinite sum, we do indeed\nobtain a kind of Liouville\nequation\n\\begin{equation}\n\\frac{\\partial }{\\partial t}W=\n\\frac{\\partial }{\\partial x}H_{\\rho}\n\\frac{\\partial }{\\partial p }W\n-\\frac{\\partial }{\\partial p}H_{\\rho}\n\\frac{\\partial }{\\partial x}W,\n\\label{liouville}\n\\end{equation}\nwhere\n\\begin{equation}\nH_{\\rho}=\\frac{p^{2}}{2m}+V(x,t)+u\\rho,\n\\end{equation}\ni.e.\\ there is an additional ``potential'' proportional to the\ndensity of the distribution in position space. This can be interpreted as a\nlarge number of classical particles initially placed in phase space according to some\nkind of distribution function and\ninteracting repulsively with one another, i.e.\\ as a kind of {\\em non-ideal\\/}\ngas. If $u$ is large we would\ngenerally expect large numbers of such particles concentrated heavily in some\ncell in position space to tend to drive one another apart, meaning that large\nvalues of $\\rho$ should in the long term be heavily disfavoured.\n\n\\subsection{Hydrodynamics Related to Wigner Function Dynamics}\n\\label{hydrowig}\nHydrodynamic equations can also be derived from the equation of motion for \nthe Wigner function Eq.~(\\ref{wignerexp}), and if one expects the hydrodynamic equations to describe a \nsemiclassical limit of the Gross-Pitaevskii equation, this should be consistent with the \nsemiclassical limit described by the Liouville-like equation of Eq.~(\\ref{liouville}). In this section\nwe conclusively show this not to be the case, and explain why this is so.\n\nIn terms of the Wigner function, $P$ is\ndefined by:\n\\begin{equation}\n\\rho(x) P(x) = \\int_{-\\infty}^{\\infty}dp p W(x,p),\n\\end{equation}\nwhere $\\rho(x)$ has already been defined by Eq.~(\\ref{wigdens}). $P$ is thus\nseen to be simply the first order momentum moment of the Wigner function. It\nturns out to be useful to define higher order moments as well:\n\\begin{equation}\n\\rho(x) P_{n}(x) = \\int_{-\\infty}^{\\infty}dp p^{n} W(x,p).\n\\end{equation}\n\nThe derivation of the equation of motion for $\\rho$ is carried out in Appendix\n\\ref{hydroapp}, and is exactly the continuity equation of\nEq.~(\\ref{continuity}), correct to all orders in $\\hbar$. The equation of motion\nfor $P$, again to all orders in $\\hbar$, turns out to be\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}P&=&-\n\\frac{\\partial }{\\partial x}[V(x,t)+u\\rho]\n-\\frac{1}{\\rho m}\nP_{2}(x)+\n\\frac{P}{\\rho m}\\frac{\\partial}{\\partial x}(\\rho P) \\nonumber\\\\\n&=&-\\frac{\\partial}{\\partial x}\\left[V(x,t)+u\\rho\n+\\frac{P_{2}}{2m}\n\\right]\n-\\frac{1}{\\rho m}\n\\frac{\\partial }{\\partial x}(\\sigma_{p}^{2}\\rho),\n\\label{protohydro}\n\\end{eqnarray}\nwhere\n$\n\\sigma_{p}^{2}(x) = P_{2}(x)-P(x)^{2}\n$\nis the variance of the Wigner function in $p$ at a given point in $x$.\n\nExcept for the term involving $\\sigma_{p}^{2}$, Eq.~(\\ref{protohydro}) is\nidentical to the hydrodynamic equation Eq.~(\\ref{momentumfield}). However, it \ncan be seen that Eqs.~(\\ref{continuity},\\ref{protohydro}) do not form a closed\nsystem, as the equation of motion for $P(x)$ refers to the higher order moment\n$P_{2}(x)$. There is in fact, as shown in Appendix \\ref{hydroapp}, an infinite\nchain of differential equations for the moments $P_{n}(x)$ \\cite{lill}:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}P_{n}(x)&=&\\frac{P_{n}(x)}{\\rho}\n\\frac{\\partial}{\\partial x}[\\rho P(x)]\n-\\frac{1}{\\rho m}\\frac{\\partial}{\\partial x}[\\rho P_{n+1}(x)]\n-nP_{n-1}(x)\\frac{\\partial }{\\partial x}[V(x,t)+u\\rho]\n\\nonumber \\\\ &&\n-\\sum_{s=1}^{n-1}\\left\\{\\frac{(\\hbar/2)^{2s}n!}{(2s+1)![n-(s+1)]!}\nP_{n-(s+1)}(x)\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}[V(x,t)+u\\rho]\\right\\}.\n\\label{motionmom}\n\\end{eqnarray}\nIn each equation the quantum corrections are described by the\nsum, but there is also an\ninfinite chain of classical corrections; the second term of\nEq.~(\\ref{motionmom}) refers to the higher order moment $P_{n+1}(x)$. To get\nthe second hydrodynamic equations, Eq.~(\\ref{momentumfield}), in\nclosed form from Eq.~(\\ref{protohydro}), we must additionally \n make the zeroth order moment \napproximation,\n\\begin{equation}\nP_{n}(x)=P(x)^{n}.\n\\end{equation}\nIn Appendix~\\ref{hydroapp} this is treated in a little more detail.\n\nIn order to reach the ``hydrodynamic limit'', it is necessary\nto kill off all the quantum corrections, but there\nis in fact a much more drastic approximation than only taking the limit\n$\\hbar\\rightarrow 0$, as a whole chain of classical corrections must be abandoned at\nthe same time. The reason for the failure of the hydrodynamic equations as a\nsemiclassical limit can be seen by examining our initial reasoning more closely.\nThis was based partly on a\ncorrespondence between the hydrodynamic limit of the linear Schr\\\"{o}dinger\nequation and the equivalent Hamilton-Jacobi equation, however this also\nimplicitly assumes that the {\\em interpretation\\/} of the quantum wavefunction \ntends to a classical {\\em point}. The Liouville dynamics given by\nEqs.~(\\ref{singleliouville},\\ref{liouville}) describe the motion of\nclassical {\\em distributions}. As has already been mentioned, in the case of \nno nonlinearity ($u=0$) one can connect the two classical cases by considering a\ndistribution of the form $W=\\delta(x-x_{0})\\delta(p-p_{0})$, but when one is\nconsidering a case where the dynamics are influenced by the \ndensity in position space \n$\\rho$, this is clearly meaningless.\n\n\nThe correct semiclassical limit described in terms of moment equations \nis thus described by the following system:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}\\rho &=&\n -\\frac{1}{m}\\frac{\\partial}{\\partial x}[\\rho P(x)],\\\\\n\\frac{\\partial}{\\partial t}P_{n}(x)&=&\\frac{P_{n}(x)}{\\rho}\n\\frac{\\partial}{\\partial x}[\\rho P(x)]\n-\\frac{1}{\\rho m}\\frac{\\partial}{\\partial x}[\\rho P_{n+1}(x)]\n-nP_{n-1}(x)\\frac{\\partial }{\\partial x}[V(x,t)+u\\rho],\n\\end{eqnarray}\nwhere we must include every value of $n$. \nAll of this is accounted for in Eq.~(\\ref{liouville}).\nIt seems clear that\nEq.~(\\ref{liouville}) is a simpler way of describing the correct classical\nlimit, and is almost certainly easier to integrate numerically.\n\nThe purpose of comparing a nonlinear Schr\\\"{o}dinger equation with its\nsemiclassical limit is that it explicitly\nremoves the wave-like or quantum behaviour, allowing us to see what there is\nthat is specifically ``quantum'' about the dynamics of the\n nonlinear Schr\\\"{o}dinger equation\nunder consideration.\n\n\\section{Model}\n\\label{model}\n\\subsection{The Delta-Kicked Harmonic Oscillator}\nTo gain insight into the general problem, it is useful\nto take\na simple test system, which is (a) accessible experimentally, and (b) amenable \nto numerical attack. \nThe system chosen is the one dimensional \n{\\em delta-kicked harmonic oscillator},\nwhich has been studied both classically \n\\cite{classharm,stochastic,web,symmetry} and quantum\nmechanically \\cite{quantharm,borgonovi,frasca,hogg}. The total potential for \nthe classical Hamiltonian consists of a standard harmonic potential perturbed by\na time dependent kicking potential:\n\\begin{equation}\nV(x,t)=\\frac{m\\omega^{2}x^{2}}{2}+K\\cos(kx)\\sum_{n=-\\infty}^{\\infty}\n\\delta(t-n\\tau),\n\\end{equation}\nwhere $x$ is the position, $m$ is the particle\nmass, $\\omega$ the harmonic frequency, $K$ the kick strength, $k$ the\nwavenumber, and $\\tau$ the time interval between kicks. \n\n\\subsection{Scaling}\n\\label{textscaling}\nIn our model, there are two basic\nparameters: the kick strength $K$, and the strength of the\nnonlinearity $u$. Additionally there is $\\hbar$, which we have expanded around\nin Section \\ref{wigner}. The parameters $K$ and $u$ need to be rescaled \nso that they remain equivalent in different regimes, as determined by a scaling\nparameter which takes the place of $\\hbar$. In\nthe case of the delta-kicked harmonic oscillator there is a natural dimensionless\nscaling parameter, which is $\\eta$, the Lamb-Dicke parameter. \n\\begin{equation}\n\\eta = k\\sqrt{\\frac{\\hbar}{2m\\omega}}\n\\label{eta}\n\\end{equation}\nIt should also\nbe pointed out that $\\eta$ is a real physical magnitude, which really can be\nadjusted in the laboratory, unlike $\\hbar$. We\ncall the \ndimensionless kicking\nstrength $\\kappa$, \nand the dimensionless nonlinearity strength $\\upsilon$:\n\\begin{eqnarray}\n\\kappa&=&\\frac{Kk^{2}}{\\sqrt{2} m\\omega^{2}},\\label{kappa}\\\\\n\\upsilon&=& \\frac{uk^{3}}{2\\sqrt{2}m\\omega^{2}}.\\label{upsilon}\n\\end{eqnarray}\nIt is shown in Appendix \\ref{scalingapp} that $\\kappa$ and $\\upsilon$ \nhave an equivalent effect on the overall dynamics for any\nvalue of $\\eta$.\n\nIf, as is often the case when the trapping potential is harmonic, \nthe Gross-Pitaevskii equation has been rescaled in\nterms of harmonic coordinates $(\\hat{x}_{h}=\\sqrt{m\\omega/\\hbar}\\hat{x}$,\n$\\hat{p}_{h}=\\hat{p}/\\sqrt{m\\hbar\\omega})$, then it can be written in terms of these\ndimensionless parameters as:\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t_{h}}\\varphi &=&\n-\\frac{1}{2}\\frac{\\partial^{2}}{\\partial x_{h}^{2}}\\varphi\n+V(x_{h},t_{h})\\varphi +\\frac{\\upsilon}{\\eta^{3}}\|\\varphi\|^{2}\\varphi,\n\\label{NLSEscaled}\\\\\nV(x_{h},t_{h})&=&\\frac{x_{h}^{2}}{2}+\\frac{\\kappa}{\\sqrt{2}\\eta^{2}}\n\\cos(\\sqrt{2}\\eta x)\\sum_{n=-\\infty}^{\\infty}\n\\delta(t_{h}-n\\tau_{h}).\n\\label{potentialscaled}\n\\end{eqnarray}\nThe wavefunctions have been rescaled so that they are properly\nnormalized with respect to the harmonic position coordinate, and the \ntime evolution is with respect to the dimensionless \ntime $t_{h}=\\omega t$. \nIt is this form of the Gross-Pitaevskii equation that we use in our numerical \nsimulations.\n\n\\section{Model Phase Space Dynamics}\n\\subsection{Classical Point Dynamics}\n\\label{poinsec}\nThe dynamics of a classical point particle in a delta-kicked harmonic potential\nhave been described fairly extensively elsewhere \n\\cite{classharm,stochastic,web,symmetry}. \nBriefly,\nwe choose a value for $\\tau_{h}$. For a given $\\tau$ there is only one free\nparameter which affects the phase space dynamics: $\\kappa$.\nThere is a resonance condition $\\tau_{h} = 2\\pi r/q$ ($r/q$ is a positive\nrational, where $q>2$), whereby \nthere are interconnecting channels of chaotic dynamics in the\nphase space \\cite{web,symmetry}, the thickness of which depends on the kick\nstrength $\\kappa$ \\cite{web}. For $\\kappa$ not too large, these form\nan Arnol'd stochastic web which spreads through all of phase \nspace, and has a characteristic $q$ symmetry. \nFor large $\\kappa$, one observes global chaos. \nNote that\nArnol'd diffusion \\cite{arnold} can occur in systems of less than two\ndimensions when the conditions for the KAM (Kolmogorov, Arnol'd, Moser) \ntheorem \\cite{reichl,kam} are not fulfilled, as is the case here \n\\cite{stochastic,web,symmetry}.\n\nHere [and also in the following numerical work on the Gross-Pitaevskii equation\nEq.~(\\ref{gpe})\nand Liouville equation Eq.~(\\ref{liouville})] \nwe consider the case where $\\tau_{h}=2\\pi/6$ and $\\kappa=1$.\n%In Fig.~\\ref{poincare}(a), the unstable initial condition is where\n%$\\tilde{x}=\\sqrt{2}\\pi$ and $\\tilde{p}=0$.\n%, the stable initial condition where $\\tilde{x}=2\\sqrt{2}\\pi$ and $\\tilde{p}=0$. \nThe scaled position and momentum are defined as\n\\begin{eqnarray}\n\\tilde{x}&=&\\frac{kx}{\\sqrt{2}}=\\eta x_{h},\\label{dimpos}\\\\\n\\tilde{p}&=&\\frac{kp}{\\sqrt{2}m\\omega}=\\eta p_{h}.\\label{dimmom}\n\\end{eqnarray}\nThese scaled variables are chosen so that the phase space dynamics of a\nclassical point particle described in terms of them are affected only by\n$\\kappa$ and \n$\\tau_{h}$. As can be seen, they correspond \nexactly to the scaled harmonic position and momentum when\n$\\eta=1$.\n\nIt can be seen in Fig.~\\ref{poincare} that the phase space, in this case \nhaving a 6 symmetry, consists of a\nstochastic web of chaotic dynamics, where an initial condition can spread \nthroughout phase space, enclosing\ncells of stable dynamics. An trajectory initially inside one of these stable \ncells will generally be held in a ring of six cells, equidistant from the\ncentre, for all time (with the exception\nof the particle initially in the central cell, where it stays)\n\\cite{classharm}.\n\n\\subsection{Gross-Pitaevskii Equation}\nIn this section and in Sec.~\\ref{liouvillesec}, we always work with the \nharmonically scaled position $x_{h}$ and momentum $p_{h}$ \nand with the dimensionless time\n$t_{h}$. For the sake of brevity we omit the $h$ subscript, and thus write these\nvariables simply as $x$, $p$, and $t$ (or $\\tau$).\n\nWe integrate numerically, using a split operator method, the\nGross-Pitaevskii equation as given in Eq.~(\\ref{NLSEscaled}) considering only\nthe harmonic potential for periods of time of length $\\tau$, \npunctuated by the exact mapping\n\\begin{equation}\n\\varphi(x,t^{+})=e^{-i\\kappa\\cos(\\sqrt{2}\\eta x)/\\sqrt{2}\\eta^{2}}\n\\varphi(x,t^{-})\n\\end{equation}\nwhich accounts for the effect of the instantaneous delta kicks. \nThis was carried out for various values\nof $\\upsilon$ and $\\eta$, where $\\kappa=1$ and $\\tau = 2\\pi/6$\nin every case. \n\nWe have calculated\nthe {\\em time averaged\\/} Wigner function, by which we mean the average of all\nthe Wigner functions determined just before each delta kick, for 100 kick\nperiods. The initial wavefunctions are displaced ground states. That is, the\nground state of the Gross-Pitaevskii equation is determined\nnumerically, for each value of $\\upsilon$. We then locate the centre of the\nwavefunction at a point which is in a regular or chaotic region of the the {\\em\nclassical single particle\\/} phase space. ``Unstable'' initial wave-packets are\ncentred at $x=\\sqrt{2}\\pi/\\eta$ (harmonic units), and ``stable'' initial\nwave-packets at $x=2\\sqrt{2}\\pi/\\eta$. The initial wavefunctions are thus\ncentred exactly either in the middle of a cell in phase space, or in an area \ndominated by web dynamics.\nThese displaced states are the natural equivalent of coherent states for a cubic\nnonlinear Schr\\\"{o}dinger equation. Just like coherent states, the density profile\nkeeps its shape in a simple harmonic potential as it oscillates back and forth. \nThis oscillating excitation is the so-called Kohn mode \\cite{kohn}. \n\nFirstly we show, in Fig.~(\\ref{uzero}), the case of no nonlinearity, for the\nsake of reference. In this case the initial conditions are simply coherent\nstates. Note that because it is possible for the Wigner function to \nhave negative values, the colour representing zero is in general different in\neach pseudocolour plot. Thus, in each plot there is a ``background'' colour,\nwhich represents zero, with a superimposed \npattern made up of darker and lighter shades.\nNotice that for $\\eta=1$, the unstable initial condition\n[Fig.~\\ref{uzero}(a)] appears to move through\nphase space following the stochastic web, whereas the stable initial condition\n[Fig.~\\ref{uzero}(b)] \nsimply circles around phase space, as would an initial coherent state in a\nsimple harmonic potential. The wavefunction is clearly somewhat deformed (in the\ncase of a harmonic potential we would see perfect circles) but is otherwise well\nlocalized and well behaved. In the case of $\\eta=2$, one might be forgiven for\nthinking that whether the initial condition is ostensibly stable or not is of\nnegligible importance. The fact that $\\eta$ is larger has the effect that the\nphase space is smaller compared to the size of the initial wavefunction (as\nplotted here, using harmonic units), \nand also quantum corrections play a bigger role (see Appendix \\ref{scalingapp}), \nleading to the\n``tunneling'' seen in Fig.~\\ref{uzero}(d), through classically forbidden areas\nof phase space. This tunneling can take place because the eigenstates of the\nFloquet operator $\\hat{F}$ \ndescribing the period from just before one kick to just before\nthe next,\n\\begin{equation}\n\\hat{F}=e^{-i(\\hat{x}^{2}+\\hat{p}^{2})\\tau/2}e^{-i\\kappa\\cos(\\sqrt{2}\\eta\n\\hat{x})/\\sqrt{2}\\eta^{2}},\n\\end{equation}\nare highly delocalized \\cite{quantharm,borgonovi,frasca}, as is described \nin Appendix~\\ref{antilocalapp}.\n\nIn Fig.~\\ref{utenth} equivalent plots are shown when a nonlinearity of\n$\\upsilon=0.1$ is added to the Gross-Pitaevskii equation.\nIt can be seen that this does not make very much difference to the phase space\ndynamics compared to no nonlinearity (Fig.~\\ref{uzero}), which is not really\nunexpected.\n\nWhen, as shown in Fig.~\\ref{uone}, a nonlinearity of $\\upsilon=1$ is added to\nthe Gross-Pitaevskii equation, it can be seen that this does make a difference.\nIntriguingly, given that the interaction potential is more strongly repulsive, \nthe phase space dynamics appear to be more strongly localized. In the case of an\nunstable initial condition [Fig.~\\ref{uone}(a] and \\ref{uone}(c)] the web\nstructure is noticeably reduced, and whereas in Fig.~\\ref{utenth}(d) there was\nsignificant tunneling leading to a very delocalized phase space distribution,\nin Fig.~\\ref{uone}(d) this has effectively disappeared. \n\nIn Fig.~\\ref{uten}, where $\\upsilon=10$, this is even more marked. Where\n$\\eta=1$, in the case of an unstable initial condition [Fig.~\\ref{uten}(a)], \ndensity seems to be\nconcentrated around a ``ring'' in phase space, based around how far out in phase\nspace the initial condition was. Where $\\eta=2$ \n[Fig.~\\ref{uten}(c,d)], whether the initial condition is ostensibly stable or\nunstable, we see only six symmetrically placed round blobs of density,\nanalogous to a coherent state in a harmonic potential.\n\n\\subsection{Liouville Equation}\n\\label{liouvillesec}\nHere we wish to investigate the semiclassical limit of the dynamics of the\nGross-Pitaevskii equation with a delta-kicked harmonic oscillator potential\n[Eq.~(\\ref{NLSEscaled})].\nThe appropriate dynamics are described in general by Eq.~(\\ref{liouville}). As\nwith the Gross-Pitaevskii equation, in our case this can be carried out by\nconsidering only the harmonic potential for periods of time $\\tau$, punctuated\nby an exact map describing the momentum kick.\n\nEquation (\\ref{liouville}) can be\nqualitatively determined numerically by taking an ensemble of starting points\nfrom some desired distribution, using Hamilton's equations of motion to\ndetermine the trajectories, and using the numerically determined coarse-grained\ndensity for the overall potential governing the motion of the individual points.\nObviously the coarse-grained density must be determined sufficiently frequently\nso that between times when it is determined, it does not change enough to\nhave a very significant effect on the dynamics. This is in some sense analogous\nto the split-step method we have used to integrate the nonlinear\nSchr\\\"{o}dinger equation, where as the time steps shrink to length zero, the\napproximate solution converges (in principle) to the exact solution. \n\nIn each case the initial distribution is chosen by determining the ground state\nof the harmonic potential Gross-Pitaevskii equation (for \nappropriate $\\upsilon$ and $\\eta$), shifting it so that the centre of the \nwavefunction is\nat an unstable or stable fixed point (in the classical, single particle sense),\ncalculating the Wigner function, and interpreting this as a classical\nprobability distribution in $x$ and $p$. The ground state Wigner function in the\ncase of a harmonic potential is\nalways strictly nonnegative, so one can always do this.\n\nNote that although $\\eta$ does not enter into the dynamics of\nEq.~(\\ref{liouville}) directly, by the above recipe it does enter by way of the\nchoice of the initial condition, which affects the effective potential due to\nthe distribution's density in position space, and so\non. The time averaged density distribution plots in \nFigs.~\\ref{lutenth}--\\ref{luten} are chosen to have\ninitial conditions and scaling exactly equivalent to the time-averaged Wigner\nfunction plots shown in Figs.~\\ref{utenth}--\\ref{uten}.\n\nIn Fig.~\\ref{lutenth} we see the density distribution averaged over 100 kicks\nfor the case where $\\upsilon=0.1$. The dynamics are essentially similar to those\nshow in Fig.~\\ref{poincare} for various single trajectories, and we observe a\nmuch lesser degree of distribution through phase space when compared to the full\nGross-Pitaevskii equation (see Fig.~\\ref{utenth}). In particular we see no\ntunneling in Fig.~\\ref{lutenth}(d), compared to Fig.~\\ref{utenth}(d).\nThe dynamics in the cases of ``unstable'' initial conditions perhaps do not \nappear to be very strongly chaotic. Remember that only 100 kicks have been \napplied, and that in the case of the single particle classical delta-kicked harmonic\noscillator, there are {\\em slow\\/} chaotic dynamics along the stochastic \nweb \\cite{classharm,stochastic,web}, with an overall tendency to diffuse ``outwards'' \nin phase space. We have examined the case of 100 kicks only\nin order to directly compare with the the numerically determined Gross-Pitaevskii \ndynamics.\n\n\nIf we examine Fig.~\\ref{luone}, which shows analogous dynamics to \nFig.~\\ref{lutenth} for the case that $\\upsilon=1$, we observe some\nincreased spreading out through phase space, still contained within the\ncharacteristic cells formed by the stochastic web in the case of the stable\ninitial condition for $\\eta=1$. In the case of $\\eta=2$ The initial distribution\nseems too large for the cells, and even in the stable case there is some\ndiffusion outwards through phase space.\n\n\nFinally we consider the case where $\\upsilon = 10$, shown in Fig.~\\ref{luten}.\nThere is significant additional diffusion through phase space for the unstable\ninitial condition, compared to the cases of $\\upsilon=0.1$ (Fig.~\\ref{lutenth})\nand $\\upsilon=1$ (Fig.~\\ref{luone}). Even for the supposedly stable initial\ncondition there is some density which has found its way onto the stochastic web,\nand appears to be diffusing outward. Nevertheless, the basic structure of the\nsingle particle stochastic web appears to be retained.\n\nThere thus appears to be a clear trend, where the larger the interaction\nparameter $\\upsilon$, the greater the degree of diffusion outward through phase\nspace, but nevertheless along routes typical for single particle dynamics. \nThis has a simple explanation: when $\\upsilon$ is large and the distribution is\nhighly localized, the distribution tends to push itself apart. After this\ninitial explosion through phase space (actively encouraged in the unstable parts\nof phase space) the contribution by the density to the effective potential is\nsmall, and so the by now thinly spread distribution undergoes local dynamics\nequivalent to single noninteracting classical particles, chaotic or stable,\ndepending on the location in phase space.\n\n\\subsection{Interpretation}\n\\subsubsection{Overview}\nThe most interesting thing shown by these numerical experiments, is the\nconclusive demonstration that the localization observed in \nFigs.~\\ref{uone}, \\ref{uten} is due to interference effects, caused by terms of \nhigher order in $\\hbar$ in Eq.~(\\ref{wignerexp}) (or more correctly,\nhigher order in $\\eta^{2}$, as shown in Appendix.~\\ref{scalingapp}). The\nintuitive picture of a stronger repulsive interaction driving the Wigner\nfunction/Liouville distribution apart, is fulfilled in the semiclassical limit,\nbut breaks down when all ``quantum'' corrections are accounted for.\n\nThe increasing degree of localization shown with increasing $\\upsilon$ in the\nGross-Pitaevskii dynamics can also be\nqualitatively explained. As is shown in Appendix \\ref{antilocalapp}, in the case of\nlinear Schr\\\"{o}dinger equation dynamics, the Floquet eigenstates are highly\ndelocalized, due to extra symmetries connected to the fact that the wavefunction is\nkicked exactly six times per oscillation period. \nThe presence of delocalized eigenstates means that the wavefunction tends to spread\nthroughout phase space with ease; along the stochastic web if the initial condition\nis in a classically unstable part of phase space, and possibly by tunneling from cell\nto cell (promoted by large $\\eta$) if the wavefunction is initially in a stable part\nof phase space.\nWith increasing $\\upsilon$ this\nsymmetry is more and more perturbed, to a point where this ability to spread freely\nthrough phase space is lost. Interference effects due to higher order terms of the\ndensity in\nEq.~(\\ref{wignerexp}), \n%probably due to the quasi random appearance of the\n%Gross-Pitaevskii wavefunction position density profile, \nact to hold the wavefunction\ntogether, in contrast to the Liouville type dynamics described by\nEq.~(\\ref{liouville}).\n\n\\subsubsection{Density in Position Space}\nOn this note it is instructive to look at the kinds of densities actually \nproduced. We consider the final wavefunction, produced after 100 kicks, at a time just before\na hypothetical 101st kick. In Fig.~\\ref{xtenth} we see plots of $\|\\varphi(x)\|^2$ for the case \n$\\upsilon=0.1$. Unsurprisingly for the unstable cases, and also for the stable case where \n$\\eta=2$, the states are highly delocalized in position space, with a great deal of fine \nstructure. In Fig.~\\ref{xone}, this has substantially changed; the densities which were \nvery complex are now much simplified, and even the stable initial condition for $\\eta=1$\nappears to have less structure when $\\upsilon=1$ compared to $\\upsilon=0.1$. When $\\upsilon$\nis increased to 10, as shown in Fig.~\\ref{xten}, there is still some structure to the \ndensities where $\\eta=1$, wheras in the case where $\\eta=2$ there appears now to be none.\n\nObviously much more radical change is induced for the case of $\\eta=2$ when increasing\n$\\upsilon$. Bearing in mind that $\\eta^{2}$ is our effective $\\hbar$, it is clear from \nEq.~(\\ref{wignerexp}) that higher order derivatives in the effective potential $V(x,t)+u\\rho(x)$\nwill be more strongly emphasized (see also Appendix \\ref{scalingapp}). Between kicks, the non-Liouville\ncorrections are due only to $\\rho(x)$, as the derivatives of $x^{2}$ vanish.\n\nConsidering the cases of Figs.~\\ref{xten}(a) and \\ref{xten}(b) in particular, one might ask what \nthere is about these densities which seemingly so totally dominates the dynamics. We consider the initial\nstate, which is simply a shifted ground state. The ground state of the Gross-Pitaevskii equation lie \nsomewhere between the cases of a Gaussian (no nonlinearity) and the Thomas-Fermi limit \\cite{review}, which \nis essentially an inverted parabola (large nonlinearity). With regard to the parameters we have chosen\nto use, the degree of ``Thomas-Fermi-ness'' is proportional to $\\upsilon/\\eta^{3}$. In the \nThomas-Fermi limit, there are no higher order derivatives of $\\rho$. A Gaussian however, has an infinite\nnumber of derivatives. For Figs.~\\ref{xten}(a,b), $\\upsilon/\\eta^{3}=1.25$ only.\nThe initial state density is thus more Gaussian than Paraboloid, and the large value of the effective \n$\\hbar$ ensures that corrections due to the inevitable higher order derivatives are substantial.\n\nBriefly:\nthe application of a kick scrambles the phase of the position representation of a wavefunction; instantaneously\nthe density in position space is unaffected however. When looking at Eq.~(\\ref{wignerexp}) we see that corrections\ndue to higher order derivatives of $\\rho$ will be emphasized for larger effective $\\hbar$, in our case\n$\\eta^{2}$. The effect of these corrections appears to be a strong tendency for the {\\em shape\\/} of the\nwavefunction to be preserved.\n\nIn this work, we have not really explored the regime of very large nonlinearities. In view of the fact that in the\nThomas-Fermi limit for the ground state there are no corrections to the Liouville-like equation of \nEq.~(\\ref{liouville}), it is possible that the kind of very pronounced localization observed for the case of\n$\\eta=2$ might again be suppressed for much larger $\\upsilon$.\n\n\\subsubsection{Density in Momentum Space}\nFor the sake of comparison, in Figs.~(\\ref{ptenth},\\ref{pone},\\ref{pten}) we show the corresponding momentum\ndensities to the position densities of Figs.~(\\ref{xtenth},\\ref{xone},\\ref{xten}). The densities in position \nand momentum space essentially correspond, in that complex structure in one indicates complex structure in the other.\nThis is not surprising, if we consider the kinds of Wigner functions displayed in \nFigs.~(\\ref{utenth},\\ref{uone},\\ref{uten}).\n\n\\section{Physical Model: Driven Bose-Einstein Condensate}\n\\label{physicalmodel}\n\\subsection{Introduction}\nA series of pioneering experiments investigating quantum chaos with atom-optical \nsystems has been carried out by Raizen and co-workers \\cite{raizen}, mainly for a quantum \nrealization of the delta-kicked rotor. We take a similar approach;\na possible physical realization of the delta-kicked harmonic oscillator,\nconsisting of a single trapped ion periodically driven by a laser, has been\ndescribed in \\cite{ionchaos}. This can in principle be readily extended to a\nperiodically driven Bose-Einstein condensate.\n\n\\subsection{Single Particle}\nWe begin by regarding a single two level atom. \nIn the $x$ direction, it is trapped in a harmonic potential of frequency\n$\\omega$, and driven time dependently by a laser field of Rabi frequency\n$\\Omega(t)$, wavenumber $k$, and frequency $\\omega_{L}$. We disregard motional\ndegrees of freedom in the\n$y$ and $z$ directions as being presently uninteresting, and arrive at the \nfollowing Hamiltonian operator:\n\\begin{eqnarray}\n\\hat{H}&=&\n\\frac{\\hat{p}^{2}}{2m}+\\frac{m\\omega^{2}\\hat{x}^{2}}{2}\n+\\frac{\\hbar}{2}\\{\\omega_{0}(\|e\\rangle\\langle e\|-\|g\\rangle\n\\langle g\|)\n+\\cos(k\\hat{x})[\\Omega(t)e^{-i\\omega_{L}t}\|e\\rangle\\langle g\|+\\mbox{H.c.}]\\}.\n\\end{eqnarray}\nIn a rotating frame defined by \n\\begin{equation}\n\\hat{U}=\\exp[-i\\omega_{L}t(\|e\\rangle\\langle e\|-\|g\\rangle \\langle g\|)/2],\n\\end{equation}\nand in the limit of large detuning\n$\n\|\\Delta\|=\|\\omega_{L}-\\omega_{0}\|\\gg\|\\Omega(t)\|\n$,\n$\|e\\rangle $ can be adiabatically eliminated to give, after transformation to an\nappropriate rotating frame,\n\\begin{equation}\n\\hat{H}=\n\\frac{\\hat{p}^{2}}{2m}+\\frac{m\\omega^{2}\\hat{x}^{2}}{2}\n+\\frac{\\hbar}{2}\\frac{\\Omega(t)^{2}}{4\\Delta}\n[\\cos(2k\\hat{x})+1]\|g\\rangle\\langle g\|.\n\\end{equation}\nThe laser is periodically switched on and off, giving a series of short \npulses, approximated by Gaussians:\n\\begin{equation}\n\\Omega(t)^{2}=\\Omega^{2}\\sum_{n=-\\infty}^{\\infty}e^{-(t-n\\tau)^{2}/\\sigma^{2}},\n\\end{equation}\nwhich approximate a series of delta kicks in the limit $\\sigma\\rightarrow 0$.\nNote also that we require $\\sigma\\gg 1/\\Delta$, otherwise the laser is too\nspectrally broad. Thus, we have finally\n\\begin{eqnarray}\n\\hat{H}&=&\n\\frac{\\hat{p}^{2}}{2m}+\\frac{m\\omega^{2}\\hat{x}^{2}}{2}\n+\\frac{\\hbar\\sigma\\sqrt{\\pi}\\Omega^{2}}{8\\Delta}\n[\\cos(2k\\hat{x})+1]\|g\\rangle\\langle g\|\\sum_{n=-\\infty}^{\\infty}\\delta(t-n\\tau).\n\\end{eqnarray}\nBecause we assume that the atom is always in electronic state $\|g\\rangle$, the\n$\|g\\rangle \\langle g\|$ operator can be effectively abandoned. The extra\n$+1$ \nsimply adds a global phase, which can easily be accounted for,\nand so this can be further simplified to:\n\\begin{eqnarray}\n\\hat{H}&=&\n\\frac{\\hat{p}^{2}}{2m}+\\frac{m\\omega^{2}\\hat{x}^{2}}{2}\n+\\frac{\\hbar\\sigma\\sqrt{\\pi}\\Omega^{2}}{8\\Delta}\n\\cos(2k\\hat{x})\\sum_{n=-\\infty}^{\\infty}\\delta(t-n\\tau).\n\\end{eqnarray}\nThis is exactly the Hamiltonian for the quantum delta-kicked Harmonic oscillator,\nexcept that we have $\\cos(2k\\hat{x})$ instead of $\\cos(k\\hat{x})$. As far as\nscaling is concerned, this means we must in turn consider $\\eta'=2\\eta$ \ninstead of $\\eta$ as the appropriate dimensionless parameter.\n\n\\subsection{Many Particles}\nIt is clear that if we consider a many particle system, then the above derivation\nis independent of any particle-particle interactions which do not change the\ninternal states of the atoms. \nWe thus consider the model \nHamiltonian of a weakly interacting Bose gas, in second\nquantized form:\n\\begin{equation}\n\\hat{H}=\\int_{-\\infty}^{\\infty}d\\vec{x}\n\\hat{\\Psi}^{\\dagger}(\\vec{x})\n\\left[\n-\\frac{\\hbar^{2}}{2m}\\nabla^{2}+V(\\vec{x},t) + \n\\frac{g}{2}\\hat{\\Psi}^{\\dagger}(\\vec{x})\n\\hat{\\Psi}(\\vec{x})\n\\right]\n\\hat{\\Psi}(\\vec{x}),\n\\end{equation}\nwhere $\\hat{\\Psi}\n$ is the particle field operator,\n$g=4\\pi\\hbar^{2}a_{s}/m$, and $a_{s}$ is the $s$-wave scattering length. \nWe take\n$V(\\vec{x},t)$ to be\n\\begin{equation}\nV(\\vec{x},t)=V(x,t)+\\frac{m\\omega_{r}^{2}}{2}(y^{2}+z^{2}),\n\\end{equation}\nwhere the potential in the $x$ direction is exactly that derived above, i.e.\n\\begin{equation}\nV(x,t)=\\frac{m\\omega^{2}x^{2}}{2} +\n\\frac{\\hbar\\sigma\\sqrt{\\pi}\\Omega^{2}}{8\\Delta}\n\\cos(2kx)\\sum_{n=-\\infty}^{\\infty}\\delta(t-n\\tau).\n\\end{equation}\n\nWe assume the radial frequency $\\omega_{r}$ to be very large compared to the\naxial frequency $\\omega$ (cigar shaped\ntrapping configuration), and thus assume that every particle is in the harmonic\noscillator ground state in $y$ and $z$. With this assumption we can integrate\nover $y$ and $z$, reducing to a\nsingle dimension:\n\\begin{equation}\n\\hat{H}=\\int_{-\\infty}^{\\infty}dx\n\\hat{\\Psi}^{\\dagger}(x)\n\\left[\n-\\frac{\\hbar^{2}}{2m}\\frac{\\partial^{2}}{\\partial x^{2}}+V(x,t) + \n\\frac{g_{1d}}{2}\\hat{\\Psi}^{\\dagger}(x)\n\\hat{\\Psi}(x)\n\\right]\n\\hat{\\Psi}(x)\n,\n\\end{equation}\nwhere $g_{1d}=m\\omega g/2\\pi\\hbar=2\\hbar\\omega_{r}a_{s}$.\n\n\n\\subsection{Asymptotic Expansion}\nUsing the particle number conserving formalism of Castin and Dum \\cite{castin}, we split the \nfield operator $\\hat{\\Psi}\n$ of the many particle system into a condensate part \nand a non-condensate part:\n\\begin{equation}\n\\hat{\\Psi}(x,t)=\\varphi_{\\mbox{\\scriptsize ex}}(x,t)\n\\hat{a}_{\\varphi_{\\mbox{\\tiny ex}}}(t)+\\delta\\hat{\\Psi}(x,t),\n\\end{equation}\nwhere $\\varphi_{\\mbox{\\scriptsize ex}}$ is the {\\em exact\\/} condensate wave\nfunction, and $\\delta\\hat{\\Psi}$ describes the non-condensate particles.\nIntroducing the operator\n\\begin{equation}\n\\hat{\\Lambda}_{\\mbox{\\scriptsize ex}}(x,t)=\\frac{1}{\\sqrt{\\hat{N}}}\n\\hat{a}_{\\varphi_{\\mbox{\\tiny ex}}}^{\\dagger}(t)\\delta\\hat{\\Psi}(x,t),\n\\end{equation}\nit is possible to make asymptotic expansions of\n$\\hat{\\Lambda}_{\\mbox{\\scriptsize ex}}(x,t)$, \n$\\varphi_{\\mbox{\\scriptsize ex}}(x,t)$, such that\n\\begin{eqnarray}\n\\hat{\\Lambda}_{\\mbox{\\scriptsize ex}}&=&\\hat{\\Lambda}+\n\\frac{1}{\\sqrt{\\hat{N}}}\\hat{\\Lambda}^{(1)}+\n\\frac{1}{\\hat{N}}\\hat{\\Lambda}^{(2)}+\\cdots,\\\\\n\\varphi_{\\mbox{\\scriptsize ex}}&=&\\varphi+\n\\frac{1}{\\sqrt{\\hat{N}}}\\varphi^{(1)}+\n\\frac{1}{\\hat{N}}\\varphi^{(2)}+\\cdots,\n\\end{eqnarray}\nwhere $\\hat{N}$ is the total particle number operator.\n\nThus, to lowest order, the condensate particles are described by $\\varphi(x)$.\nThe time evolution of this can be shown to be given by the Gross-Pitaevskii \nequation \\cite{castin}, which in \nour case is\n\\begin{equation}\ni\\hbar\\frac{\\partial}{\\partial t}\\varphi=\n-\\frac{\\hbar^{2}}{2m}\\frac{\\partial^{2}}{\\partial x^{2}}\\varphi+V(x,t)\\varphi+\nNg_{1d}\|\\varphi\|^{2}\\varphi,\n\\label{physicalgp}\n\\end{equation}\nwhere $N$ is the total number of particles.\nIn turn, the non-condensate particles are described to lowest order\nby $\\hat{\\Lambda}(x,t)$.\n\nThe Gross-Pitaevskii equation which we have arrived at in Eq.~(\\ref{physicalgp})\n can be rewritten in terms\nof the dimensionless parameters $\\eta'$, $\\kappa$, and $\\upsilon$, as described\nin Sec.~\\ref{textscaling}, where\n\\begin{eqnarray}\n\\eta'&=&k\\sqrt{\\frac{2\\hbar}{m\\omega}},\n\\label{etaprime}\n\\\\\n\\kappa&=&\\frac{\\hbar k^{2}\\sigma\\sqrt{\\pi/2}\\Omega^{2}}{2m\\omega \\Delta},\n\\label{kappafull}\n\\\\\n\\upsilon&=&\\frac{8\\hbar Nk^{3}\\omega_{r}a_{s}}{\\sqrt{2}m\\omega^{2}}.\n\\label{upsilonfull}\n\\end{eqnarray}\n\n\\subsection{Non-Condensate Particles}\nThe mean number of the non-condensate\nparticles is given by $\\langle \\delta\\hat{N} \\rangle =\n\\langle \\delta\\hat{\\Psi}^{\\dagger} \\delta\\hat{\\Psi}\\rangle$, which to lowest\norder may be described by \n$\\langle \\hat{\\Lambda}^{\\dagger}\\hat{\\Lambda}\\rangle$. In turn, \n$\\hat{\\Lambda}^{\\dagger}$ and $\\hat{\\Lambda}$ can be expanded as\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\\hat{\\Lambda}(x,t)\\\\\n\\hat{\\Lambda}^{\\dagger}(x,t)\n\\end{array}\n\\right)\n=\n\\sum_{k=1}^{\\infty}\n\\hat{b}_{k}\n\\left(\n\\begin{array}{c}\nu_{k}(x,t)\\\\\nv_{k}(x,t)\n\\end{array}\n\\right)+\n\\sum_{k=1}^{\\infty}\n\\hat{b}_{k}^{\\dagger}\n\\left(\n\\begin{array}{c}\nv_{k}^{}(x,t)\\\\\nu_{k}^{}(x,t)\n\\end{array}\n\\right).\n\\end{equation}\nwhich gives rise to the following equation describing the mean number of\nnon-condensate particles to lowest order in the perturbation expansion:\n\\begin{equation}\n\\langle \\delta\\hat{N}(t)\\rangle = \\sum_{k=1}^{\\infty}\n\\langle \\hat{b}_{k}^{\\dagger}\\hat{b}_{k}\\rangle\n\\langle u_{k}(t)\|u_{k}(t)\\rangle +\n\\langle \\hat{b}_{k}^{\\dagger}\\hat{b}_{k}+1\\rangle\n\\langle v_{k}(t)\|v_{k}(t)\\rangle.\n\\label{noncond}\n\\end{equation}\n\n\nThe $\\hat{b}_{k}$ are time-independent \\cite{castin}. \nWe see that the time-dependence of \nEq.~(\\ref{noncond}) is thus contained\ncompletely within $\\langle u_{k} \|u_{k} \\rangle$, \n$\\langle v_{k} \|v_{k} \\rangle$. \nA system initially prepared at temperature $T$\nhas \n$\\langle b_{k}^{\\dagger} b_{k} \\rangle= [\\exp(E_{k}/k_{B}T)]^{-1}$, and so, if\nwe take the limit $T\\rightarrow 0$, we get\n\\begin{equation}\n\\langle \\delta\\hat{N}(t)\\rangle = \\sum_{k=1}^{\\infty}\n\\langle v_{k}(t)\|v_{k}(t)\\rangle.\n\\end{equation}\nWe thus wish to study the dynamics of $\|v_{k}(t)\\rangle$ to get some idea of the\nchange in the number of non-condensate particles, in an analogous fashion to the \nwork of Castin and Dum when investigating the behaviour of a condensate held in\na time dependent isotropic harmonic potential \\cite{castindumdeplete}. Note that\nbecause the Gross-Pitaevskii equation is {\\em nonlinear}, it is possible to have\nchaos in the sense of exponential sensitivity to initial conditions within the\nHilbert space. If this is the case, the above estimate of \n$\\langle \\delta\\hat{N}(t)\\rangle$ will grow automatically, due to the fact that\nthis estimate is essentially from a linearization around the Gross-Pitaevskii\nsolution \\cite{castindumdeplete}. Thus the {\\em rate\\/} of growth of \nthis estimate of $\\langle \\delta\\hat{N}(t)\\rangle$ is similar to the Lyapunov\nexponent for the divergence of trajectories in phase space for discrete \nclassical systems.\n\nThe dynamics of the\n$\|u_{k}(t)\\rangle$ and \n$\|v_{k}(t)\\rangle$ are given by\n\\begin{equation}\ni\\hbar\\frac{d}{dt}\n\\left(\n\\begin{array}{c}\n\|u_{k}(t)\\rangle\\\\\n\|v_{k}(t)\\rangle\n\\end{array}\n\\right)=\n{\\cal L}(t)\n\\left(\n\\begin{array}{c}\n\|u_{k}(t)\\rangle\\\\\n\|v_{k}(t)\\rangle\n\\end{array}\n\\right),\n\\label{Luveq}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\cal L}(t)=\n\\left(\n\\begin{array}{cc}\n\\hat{H}_{\\mbox{\\scriptsize GP}}(t)\n+ Ng_{\\mbox{\\scriptsize 1d}}\\hat{Q}(t)\|\\varphi(\\hat{x},t)\|^{2}\\hat{Q}(t)\n& Ng_{\\mbox{\\scriptsize 1d}}\\hat{Q}(t)\\varphi(\\hat{x},t)^{2} \\hat{Q}^{}(t)\\\\\n-Ng_{\\mbox{\\scriptsize 1d}}\\hat{Q}^{}(t)\\varphi(\\hat{x},t)^{2}\\hat{Q}(t)\n& -\\hat{H}_{\\mbox{\\scriptsize GP}}\n-Ng_{\\mbox{\\scriptsize 1d}}\\hat{Q}^{}(t)\|\\varphi(\\hat{x},t)\|^{2}\\hat{Q}^{}(t)\n\\end{array}\n\\right)\n\\label{CasDumL}\n\\end{equation}\nand where we have defined the Gross-Pitaevskii ``Hamiltonian'',\n\\begin{equation}\n\\hat{H}_{\\mbox{\\scriptsize GP}}(t)=\\frac{\\hat{p}^{2}}{2m}+V(\\hat{x},t)+ \nu \|\\varphi(\\hat{x},t)\|^{2}-\\xi(t).\n\\end{equation}\nThe phase factor $\\xi(t)$ is equal to the ground state chemical potential $\\mu$\nwhen $\\varphi(x,t)$ is the Gross-Pitaevskii equation ground state, for a\nharmonic potential.\nThe projection operators $\\hat{Q}$, $\\hat{Q}^{}$ are given by\n\\begin{eqnarray}\n\\hat{Q}&=&%{\\mathbbm{1}}\n1-\|\\varphi\\rangle \\langle \\varphi\|,\n\\\\\n\\hat{Q}^{}&=&%{\\mathbbm{1}}\n1-\|\\varphi^{}\\rangle \\langle \\varphi^{}\|,\n\\end{eqnarray}\nwhere $\|\\varphi^{}\\rangle$ is defined by \n$\\langle x\|\\varphi^{}\\rangle=\\varphi^{}(x)=\\langle \\varphi\|x\\rangle$.\n\n\\subsection{Dynamics of $\\langle \\delta\\hat{N}(t)\\rangle$}\nTo determine how $\\langle \\delta\\hat{N}(t)\\rangle$ changes over time we need to\ndetermine the dynamics of $\|v_{k}(t)\\rangle$, which are coupled to the dynamics\nof $\|u_{k}(t)\\rangle$ through Eq.~(\\ref{Luveq}). We thus need to integrate\nEq.~(\\ref{Luveq}), and\nto integrate Eq.~(\\ref{Luveq}), we need as initial conditions\n$\|u_{k}(0)\\rangle$, $\|v_{k}(0)\\rangle$.\n\n\nThe initial conditions $\|u_{k}(0)\\rangle$, $\|v_{k}(0)\\rangle$, for \n$\\varphi(x)$ in the \nground state for a harmonic potential\nare determined by diagonalizing ${\\cal L}$ where $\\varphi(\\hat{x},t)$ is chosen\nto correspond to the Gross-Pitaevskii equation ground state, for a harmonic\npotential, and $\\xi(t)=\\mu$. For this we need to determine the\nground state condensate wavefunction $\\varphi(x)$ and the ground state chemical\npotential $\\mu$. This is achieved by propagating the Gross-Pitaevskii equation\nin imaginary time, where we use a split-operator method. \n\nWe then determine ${\\cal L}$ in the position representation\nwhere $\\varphi(x,t)$ is the previously determined\nground state and $\\xi(t)=\\mu$. We use a Fourier grid \\cite{fouriergrid} \nto describe $\\hat{p}^{2}$ in the position representation. We then diagonalize\n${\\cal L}$ numerically, and gain as the resultant set of eigenvectors\n\\begin{equation}\n\\left\\{\n\\left(\n\\begin{array}{c}\nu_{k}(x)\\\\\nv_{k}(x)\n\\end{array}\n\\right),\n\\left(\n\\begin{array}{c}\nv_{k}^{}(x)\\\\\nu_{k}^{}(x)\n\\end{array}\n\\right),\n\\left(\n\\begin{array}{c}\n\\varphi(x)\\\\\n0\n\\end{array}\n\\right),\n\\left(\n\\begin{array}{c}\n0\\\\\n\\varphi^{}(x)\n\\end{array}\n\\right)\n\\right\\},\n\\end{equation}\nwith eigenvalues\n$\\{ E_{k}, -E_{k}, 0 , 0\\}$, respectively \\cite{castin}. \nThese eigenvectors must be properly\nnormalized \\cite{castin}, so that\n\\begin{equation}\n\\int_{-\\infty}^{\\infty}dx u_{k}^{}(x)u_{k'}(x) -\n\\int_{-\\infty}^{\\infty}dx v_{k}^{}(x)v_{k'}(x)=\\delta_{kk'}\n\\end{equation}\n\nOur initial condition for the Gross-Pitaevskii equation is in general a shifted\nground state, that is, we take the ground state wavefunction, and\ninstantaneously translate it in position space, otherwise altering nothing.\nPhysically, this could be achieved by almost instantaneously translating the\ncentre of the harmonic potential, so that $x^{2}\\rightarrow (x-a)^{2}$.\nInstantaneously, this would leave the Gross-Pitaevskii wavefunction and the\n$u_{k}(x)$, $v_{k}(x)$ modes unchanged.\nIf we then re-express everything in terms of $x'=x-a$, we end up with the same\n{\\em equations\\/} in terms of $x'$ as we had initially in terms of $x$, but the\n{\\em wavefunctions} are transformed: \n$\n\\{ \\varphi(x), u_{k}(x),v_{k}(x)\\}\\rightarrow\n \\{ \\varphi(x'+a), u_{k}(x'+a),v_{k}(x'+a)\\}\n$.\n\nThus, if the initial Gross-Pitaevskii wavefunction is simply a shifted ground\nstate, then the appropriate initial $u_{k}$, $v_{k}$ are correspondingly shifted\nfrom those determined from ${\\cal L}$ for the ground state condensate\nwavefunction. This set of initial conditions is in fact somewhat special;\nas previously mentioned, the density profile of $\\varphi(x)$ remains unchanged as it\noscillates back and forth (without kicks), the same is also true of\n$u_{k}(x)$ and $v_{k}(x)$.\n\nOnce we\nhave the initial conditions we can start integrating Eq.~(\\ref{Luveq}).\n\n\\subsection{Numerical Results}\n\\label{vkevol}\nWe integrated numerically Eq.~(\\ref{Luveq}) for the first fifteen $u_{k}(x)$,\n$v_{k}(x)$ pairs over the time span of 100 kicks, using a split operator method\ndescribed in some detail in Appendix~\\ref{intLapp}, parallel to\nnumerical integration of the Gross-Pitaevskii equation, also using a split\noperator method. Just before each kick each of the inner products \n$\\langle v_{k}\|v_{k}\\rangle$ were determined, which are plotted against \ntime in\nFigs.~\\ref{vkkoneone}--\\ref{vkktwoten}, for\nvarious parameter regimes we have already investigated the Gross-Pitaevskii\ndynamics of. The ``stable'' and ``unstable'' initial conditions referred to \nare those of\nthe initial Gross-Pitaevskii wavefunction [which in turn determines the initial\nconditions of each of the $u_{k}(x)$, $v_{k}(x)$ modes], and are exactly those\ntaken in the integrations of the Gross-Pitaevskii equation described in\nSec.~\\ref{gpeint}. To reiterate, the data presented in the plots in this\nsection correspond exactly to the phase space plots presented in\nSec.~\\ref{gpeint} for the appropriate values of $\\upsilon$ and $\\eta'$, with\nregards to the initial condition. Figs.~\\ref{vkkoneone},\\ref{vkktwoone}\ncorrespond to Figs.~\\ref{uone},\\ref{luone}, and\nFigs.~\\ref{vkkoneten},\\ref{vkktwoten}\ncorrespond to Figs.~\\ref{uten},\\ref{luten}.\n\nIn Fig.~(\\ref{vkkoneone}), where $\\eta'=1$ and $\\upsilon=1$, we see a marked difference between the ``stable'' and\n``unstable'' cases. In unstable case we see much greater growth of the \n$\\langle v_{k}\|v_{k}\\rangle$. Interestingly, the $k=1$ mode in the stable\ncase does not on average seem to grow at all, instead undergoing quasiregular \noscillations in time. The leading terms are also different; $k=1$ for the\nunstable case, and $k=2$ in the unstable case.\n\n\nCompared to Fig.~\\ref{vkkoneone}, the ``stable'' and ``unstable'' cases shown in\nFig.~\\ref{vkktwoone} (where the only difference is that $\\eta'=2$), appear\ncomparatively similar. In particular there does not seem to be a great deal more\ngrowth of the $\\langle v_{k}\|v_{k}\\rangle$ in the unstable case when compared\nto the stable case. \n\n\n\nWe see the same pattern repeated in Figs.~\\ref{vkkoneten} and \\ref{vkktwoten},\nwhere $\\upsilon$ is now 10. In Fig.~\\ref{vkkoneten} the \n$\\langle v_{k}\|v_{k}\\rangle$ very rapidly grow in the unstable case when\ncompared to the stable case, whereas in Fig.~\\ref{vkktwoten}, where $\\eta'=2$, the\ndifference is not nearly so marked (and in any case the growth of the \n$\\langle v_{k}\|v_{k}\\rangle$ is generally less). \nThis reflects in some sense the observed\nWigner function dynamics in Sec.~\\ref{gpeint}, where there does not seem to be\nsuch a strong qualitative difference between the ``unstable'' and ``stable''\ncases where $\\eta'=2$ for any value of $\\upsilon$, in contrast to the cases \nwhere $\\eta'=1$. \nOne should bear in mind that although the dimensionless\nnonlinearity strength $\\upsilon$ is the same in both Figs.~\\ref{vkkoneone} and\n\\ref{vkktwoone}, the actual repulsive interaction $Nu_{1d}$ is proportional \nto $\\upsilon/\\eta'^{3}$. \nOne might argue then\nthat one would expect that there is \ngenerally less depletion from the wavefunction described by the \nGross-Pitaevskii equation. The evolution of $\\varphi(x,t)$ is also important\nhowever:\n$\\upsilon/\\eta'^{3}=1$ where $\\upsilon=1$ and $\\eta'=1$ is not that different from\n$\\upsilon/\\eta'^{3}=1.25$ where $\\upsilon=10$ and $\\eta'=2$, but the evolutions of the \n$\\langle v_{k}\|v_{k}\\rangle$ are.\nThere appears to be some correspondence between the Gross-Pitaevskii phase space\ndynamics shown in Figs.~\\ref{uone},\\ref{uten} and the evolutions of the \n$\\langle v_{k}\|v_{k}\\rangle$, in that when there is a significant difference\nbetween the ``stable'' and ``unstable'' cases, this shows up in the dynamics of\nthe $\\langle v_{k}\|v_{k}\\rangle$ corresponding to these different cases. Also a\nmore ``smooth'' phase space plot (as for $\\eta'=2$ compared to $\\eta'=1$ in\nFigs.~\\ref{uone},\\ref{uten}) appears to correspond to a more ``smooth''\nevolution of the $\\langle v_{k}\|v_{k}\\rangle$\n(Figs.~\\ref{vkktwoone},\\ref{vkktwoten} compared with \nFigs.~\\ref{vkkoneone},\\ref{vkkoneten}). As the equation describing the time\nevolution of the $\|u_{k}\\rangle,\|v_{k}\\rangle$ pairs is essentially the same as\nthat describing the evolution of linearized orthogonal perturbations of the\nGross-Pitaevskii wavefunction \\cite{castin}, this is not unexpected.\n\n\n\\subsection{Comparison with Experimental Parameters}\nWe first examine our best estimate for $\\langle \\delta\\hat{N}(t)\\rangle$, which is\n$\\sum_{k=1}^{15}\\langle v_{k}(t)\|v_{k}(t)\\rangle$, where $t$ is expressed as the\nnumber of kicks. In Fig.~\\ref{growthone} this\nis plotted for each case where $\\upsilon=1$ against the number of kicks, and in\nFig.~\\ref{growthten} for $\\upsilon=10$.\nInterestingly, for $\\upsilon=1$ and $\\eta'=2$, total growth appears to be almost \nexactly linear in time, after a short buildup period; as noted before, growth does\nnot appear to be that different when comparing the ``stable'' and ``unstable'' \ncases. For $\\eta'=1$ however, there is a clear and substantial difference\nbetween the two cases.\n\n\nWhen $\\upsilon$ is increased to 10, as shown in Fig.~\\ref{growthten}, growth becomes\nmore erratic. We see that for the ``unstable'' case where $\\eta'=1$, \n$\\sum_{k=1}^{15}\\langle v_{k}\|v_{k}\\rangle$ ends up being very large, making it\nunlikely that an experiment for this parameter regime would follow Gross-Pitaevskii\ndynamics. The general pattern observed in Fig.~\\ref{growthone} is repeated here,\nbut with larger numbers. Note however, that the beginnings of a clear\ndifferentiation\nbetween the degree of growth for the ``stable'' and ``unstable'' cases when \n$\\eta'=2$ appear to be occurring; in both cases growth is certainly not linear\nwith time.\n\nOverall, our results can be interpreted as similar to those obtained in \n\\cite{castindumdeplete} for the case of a time dependent harmonic potential. When one would \nexpect classical chaotic behaviour, one observes rapid growth of the \n$\\langle v_{k}\|v_{k}\\rangle$.\n\nTo examine the behaviour of a possible experimental realization of this scheme,\nwe consider Rubidium 87, which has an $s$ wave scattering length of\n$a_{s}=5.1\\times10^{-9}m$ \\cite{rubidium}, \nand Sodium 23 ($a_{s}=2.75\\times10^{-9}m$) \\cite{sodium}.\nSubstituting Eq.~(\\ref{etaprime}) into Eq.~(\\ref{upsilonfull}), we can rewrite\n$\\upsilon$, so that\n\\begin{equation}\n\\upsilon=\\sqrt{\\frac{m}{\\hbar\\omega}}2N\\omega_{r}a_{s}\\eta'^{3}\n\\label{usefulupsilon}\n\\end{equation}\nis expressed in terms of $\\eta'$, which is more convenient for our purposes.\nUsing\nEq.~(\\ref{usefulupsilon}), we get as a general relation for the number of\nparticles\n$N=\\lambda \\sqrt{\\omega}/\\omega_{r}$, where\n\\begin{equation}\n\\lambda = \n\\sqrt{\\frac{\\hbar}{m}}\\frac{\\upsilon}{2a_{s}\\eta'^{3}}\n\\end{equation}\nThe values of $\\lambda$ in units of $s^{-1/2}$ \nfor the parameter regimes we have investigated are\nsummarized in Table~\\ref{atomdata}. \n\nWe let $\\omega_{r}= 10\\omega$, \nremembering that we should have $\\omega_{r}$ significantly bigger than $\\omega$,\nwe take this to be a reasonable minimum, bearing in mind that\n the values of the harmonic potential ground state\nchemical potential $\\mu$ lie \nbetween 0.55 and 3.11 in units of $\\hbar\\omega$, \nas shown in Table~\\ref{atomdata}. We then get\n$N=\\nu/\\sqrt{\\omega_{r}}$, where $\\nu=\\lambda\\sqrt{1/10}$. Numerical\nvalues for $\\nu$ in units of $s^{-1/2}$, where $\\omega_{r}= 10\\omega$ \nare also displayed in Table.~\\ref{atomdata}. \nIn principle this leaves us one free parameter to tweak; the smaller the radial\nfrequency, the larger $N$ can be, and the less significant the effect of the growth\nof the number of particles not described by the Gross-Pitaevskii equation. This would\nmean that we could reasonably expect to describe the dynamics of the particles\nlargely with the Gross-Pitaevskii equation, with small corrections accounted for by\nEq.~(\\ref{Luveq}).\n\nIn practice trapping frequencies for alkali atoms such as Rubidium and Sodium \nlie between about 1 and 100 Hertz. The growth of $\\sum_{k=1}^{15}\\langle\nv_{k}\|v_{k}\\rangle $ in the ``unstable'' case where $\\upsilon=10$, $\\eta'=1$ is thus \nfar too high for this simplest interpretation of the real dynamics. The cases \nwhere $\\eta'=2$ look more promising, and here in fact the interesting effect of\nnonlinearity induced localization within phase space of the Gross-Pitaevskii \nwavefunction is even more pronounced. Also note that even for a small\nnonlinearity of $\\upsilon=1$, there is still a pronounced difference in the\nGross-Pitaevskii equation phase space dynamics (see Fig.~\\ref{uone}) compared to\nthe case where there is no nonlinearity (Fig.~\\ref{uzero}), for both $\\eta'=1$\nand $\\eta'=2$, and here the numbers also seem more promising for the\nnonlinearity induced localizing effect to be observed, corresponding to our\nnumerical integrations of the Gross-Pitaevskii equation.\n\n\\section{Conclusions}\nWe have derived explicitly an appropriate semiclassical limit for a general\ncubic nonlinear Schr\\\"{o}dinger equation, or Gross-Pitaevskii equation, and find\nit to be a Liouville type equation, with a term involving the density in\nposition space. We have shown how and why this differs \nfrom the hydrodynamic limit of the Gross-Pitaevskii equation. \nIn particular, this derivation shows how an eccentric wavefunction \n$\\varphi(x)$ can produce large deviations \nfrom this semiclassical limit, through higher order corrections involving \nderivatives of the density $\\rho(x)=\|\\varphi(x)\|^{2}$, in addition to \neffects due to an unusual potential. We have investigated\nnumerically a simple test system, the one-dimensional delta-kicked harmonic\noscillator, studying the dynamics of the Gross-Pitaevskii equation and the\nappropriate Liouville type equation. We have found for moderate \nnonlinearity strengths that there is a localization effect explicitly \ndue to interferences caused by the nonlinearity. We have outlined a possible\nexperimental implementation of such a system in a Bose-Einstein condensate\nexperiment, and have investigated numerically to what degree the\nGross-Pitaevskii equation describes correctly the dynamics of the bulk of the\nparticles for certain test cases. From this we have determined a lowest order \nestimate for the growth\nin the number of non-condensate particles. We have found that for this system \nthis depends strongly on the\nparameter regime of $\\eta'$ and $\\upsilon$ under study, and that this seems to\ncorrespond to the kinds of phase space dynamics observed in the Gross-Pitaevskii\nequation. We have compared the numbers obtained with realistic experimental\nparameters for condensates formed from sodium or rubidium atoms.\n\n\\section{Acknowledgements}\nWe thank\nJ.~R.~Anglin, for helping clear up a number of points on the work\nin Sec.~\\ref{hydrowig}, M.~G.~Raizen, Th.~Busch, and K.~M.~Gheri, \nfor discussions, and D.~A.~Steck for bringing reference \n\\cite{borgonovi} to our attention.\nWe also thank the Austrian Science Foundation, and the European Union TMR network\nERBFMRX-CT96-0002.\n\n\\begin{appendix}\n\\section{Derivation of Wigner function dynamics}\n\\label{wignerapp}\n\\subsection{Definitions}\nDefining the Wigner function for a pure state as\n\\begin{equation}\nW(x,p) = \\frac{1}{2\\pi\\hbar}\\int_{-\\infty}^{\\infty}d\\tau\ne^{-ip\\tau/\\hbar} \\varphi^{}(x-\\tau/2)\\varphi(x+\\tau/2),\n\\label{wignertwo}\n\\end{equation}\nwe take the time derivative\n\\begin{equation}\n\\frac{\\partial}{\\partial t}W(x,p) = \n\\frac{\\partial}{\\partial t}W(x,p)_{\\mbox{\\scriptsize SP}}+\\frac{\\partial}{\\partial t}W(x,p)_{\\mbox{\\scriptsize NL}},\n\\end{equation}\nwhere we have split up the differential equation into a part which is governed\nby the single particle linear dynamics (SP), and a part which is governed by\nthe nonlinearity (NL). \n\n\\subsection{Single-particle dynamics}\nThe single particle dynamics are described by:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}W(x,p)_{\\mbox{\\scriptsize SP}}&=&\n\\frac{i}{2\\pi\\hbar^{2}}\n\\int_{-\\infty}^{\\infty}d\\tau e^{-i\\tau p/\\hbar}\n\\left[\n\\langle \\varphi \|\\hat{H}\|x-\\tau/2\\rangle\\langle x +\\tau/2\|\\varphi\\rangle \n-\n\\langle\\varphi\|x-\\tau/2\\rangle\\langle x+\\tau/2\|\\hat{H}\|\\varphi\\rangle\n\\right].\n\\end{eqnarray}\nThe expansion we desire is exactly that used by\nZurek and Paz in investigating the quantum-classical boundary \\cite{zurek}, and\nis based on work originally carried out by Moyal\\cite{moyal} \nand Wigner\\cite{wigner}:\n\\begin{eqnarray}\n\\frac{\\partial }{ \\partial t}W(x,p)_{\\mbox{\\scriptsize SP}} &=&\n\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{(2s+1)!} \n\\left(\\frac{\\hbar}{2}\\right)^{2s}\n\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}H\n\\frac{\\partial^{2s+1}}{\\partial p^{2s+1}}W\n-\\frac{\\partial}{\\partial p}H\n\\frac{\\partial }{\\partial x}W.\n\\label{appsingleparticle}\n\\end{eqnarray}\n\n\\subsection{Nonlinear Dynamics}\nFor a simple cubic nonlinearity $u\|\\varphi\|^{2}\\varphi$, we can express\n$\\partial W(x,p)_{\\mbox{\\scriptsize NL}}/\\partial t$ as\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}W(x,p)_{\\mbox{\\scriptsize NL}}&=&\n\\frac{iu}{2\\pi\\hbar^{2}}\n\\int_{-\\infty}^{\\infty}d\\tau \\left\\{e^{-i\\tau p/\\hbar}\n\\int_{-\\infty}^{\\infty}dp'\\left[\nW(x-\\tau/2,p')-W(x+\\tau/2,p')\n\\right]\n\\int_{-\\infty}^{\\infty}dp''e^{i\\tau p''/\\hbar}W(x,p'')\\right\\}.\n\\label{nonlin}\n\\end{eqnarray}\nWe expand $W(x-\\tau/2,p')-W(x+\\tau/2,p')$\nas a McLaurin series:\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial t}W(x,p)_{\\mbox{\\scriptsize NL}}&=&-\\frac{iu}{\\pi\\hbar^{2}}\n\\sum_{s=0}^{\\infty}\\frac{(1/2)^{2s+1}}{(2s+1)!}\n\\int_{-\\infty}^{\\infty}dp'\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}W(x,p')\n\\int_{-\\infty}^{\\infty}d\\tau e^{-i\\tau p/\\hbar}\n\\int_{-\\infty}^{\\infty}dp''\\tau^{2s+1}e^{i\\tau p''/\\hbar}W(x,p'').\n\\end{eqnarray}\nUsing the chain rule and Fourier's integral theorem, we arrive at\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial t}W(x,p)_{\\mbox{\\scriptsize NL}}&=&\n-i\\frac{u}{\\pi\\hbar^{2}}\n\\sum_{s=0}^{\\infty}\\frac{(-\\hbar/2i)^{2s+1}}{(2s+1)!}\n\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}\n\\left[\\int_{-\\infty}^{\\infty}dp'W(x,p')\\right]\n\\frac{\\partial^{2s+1}}{\\partial p^{2s+1}}W(x,p).\n\\label{expandnonlin}\n\\end{eqnarray}\n\\subsection{Combined Result}\nCombining Eqs.~(\\ref{appsingleparticle}) and (\\ref{expandnonlin}), we get the\nWigner function dynamics to all orders in $\\hbar$ of the cubic nonlinear\nSchr\\\"{o}dinger equation with arbitrary potential, in one dimension \n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}W&=&\n\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{(2s+1)!}\n\\left(\\frac{\\hbar}{2}\\right)^{2s}\n\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}\n\\left[H+u\\rho\\right]\n\\frac{\\partial^{2s+1}}{\\partial p^{2s+1}}W\n-\\frac{\\partial }{\\partial p}H\\frac{\\partial }{\\partial x}W,\n\\label{appwignerexp}\n\\end{eqnarray}\nwhich has as its semiclassical limit ($\\hbar\\rightarrow 0$) a Liouville-like\nequation:\n\\begin{equation}\n\\frac{\\partial}{\\partial t}W=\n\\frac{\\partial}{\\partial x}\n\\left[H+u\\rho\\right]\n\\frac{\\partial}{\\partial p}W\n-\\frac{\\partial }{\\partial p}H\n\\frac{\\partial }{\\partial x}W,\n\\label{pseudohydro}\n\\end{equation}\nwhere $\\rho$ is the Wigner function integrated over $p$, as defined in\nEq.~(\\ref{wigdens}).\nThis derivation can be easily generalized for other nonlinearities and to two and\nthree dimensions.\n\n\\section{Re-derivation of the hydrodynamic equations}\n\\label{hydroapp}\n\\subsection{Definitions}\nThe density $\\rho$ has already been defined in terms of the \nWigner function by Eq.~(\\ref{wigdens}). The quantity $P$\nis defined in terms of the Wigner function as\n\\begin{equation}\n\\rho P = \\int_{-\\infty}^{\\infty}dp p W.\n\\label{momentum}\n\\end{equation}\n\n\\subsection{Regaining the First Hydrodynamic Equation}\nThe equation of motion for $\\rho$ is given by\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial t}\\rho &=&\n\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{(2s+1)!}\n\\left(\\frac{\\hbar}{2}\\right)^{2s}\n\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}\\left[H+u\\rho\\right]\n\\int_{-\\infty}^{\\infty}dp\\frac{\\partial^{2s+1}}{\\partial p^{2s+1}}W\n-\\int_{-\\infty}^{\\infty}dp\\frac{\\partial}{\\partial x} W\n\\frac{\\partial}{\\partial p}H.\n\\end{eqnarray}\nDue to the fact that $W(x,p)$ and all of its derivatives are equal to zero at\n$x=\\pm\\infty$, something we make frequent use of, \nthis simplifies to the continuity equation\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial t}\\rho \n&=&-\\frac{1}{m}\\frac{\\partial }{\\partial x}(\\rho P),\n\\label{appcontinuity}\n\\end{eqnarray}\nusing the definition of Eq.~(\\ref{momentum}).\n\n\\subsection{Equations for Higher Order Moments}\nWe now turn to the equation of motion for $P$. We have, \nfrom Eq.~(\\ref{momentum})\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial t}P&=&\\frac{1}{\\rho}\n\\int_{-\\infty}^{\\infty}dp p \n\\left\\{\n\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{(2s+1)!}\n\\left(\\frac{\\hbar}{2}\\right)^{2s}\n\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}[V(x,t)+u\\rho]\n\\frac{\\partial^{2s+1} }{\\partial p^{2s+1}}W-\n\\frac{p}{m}\n\\frac{\\partial}{\\partial x}W\n\\right\\}\n+\\frac{P}{\\rho m}\n\\frac{\\partial }{\\partial x}(\\rho P).\n\\label{prehydro}\n\\end{eqnarray}\nThe integral of the Wigner function over $p$,\n$\\int_{-\\infty}^{\\infty} dp p \\partial^{2s+1} W/\\partial p^{2s+1}$, is \nequal to $\\rho$ when $s=0$, and is otherwise\nequal to zero. We therefore have\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}P&=&-\n\\frac{\\partial }{\\partial x}[V(x,t)+u\\rho]\n-\\frac{1}{\\rho m}\n\\frac{\\partial }{\\partial x}\\left(\\int_{-\\infty}^{\\infty}dp p^{2}W\\right)\n+\\frac{P}{\\rho m}\\frac{\\partial}{\\partial x}(\\rho P).\n\\label{lasthydro}\n\\end{eqnarray}\nClearly Eq.~(\\ref{appcontinuity}) and Eq.~(\\ref{prehydro})\ndo not form a closed system of equations, due to the presence of the second\norder moment $P_{2}(x)$, where\n\\begin{equation}\nP_{n}(x) = \\frac{1}{\\rho(x)}\\int_{-\\infty}^{\\infty}dp p^{n} W(x,p).\n\\label{mommoment}\n\\end{equation}\nIt is relatively simple to derive a chain of equations\nof motion for all $P_{n}(x)$:\n\\begin{equation}\n\\frac{\\partial }{\\partial t}P_{n}(x)=\n\\frac{1}{\\rho}\\int_{-\\infty}^{\\infty} dp p^{n}\\frac{\\partial}{\\partial t}W\n-\\frac{P_{n}(x)}{\\rho}\\frac{\\partial}{\\partial t}\\rho.\n\\end{equation}\nSubstituting in Eqs.~(\\ref{appwignerexp},\\ref{appcontinuity}),\nwe get as the general form:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}P_{n}(x)&=&\\frac{P_{n}(x)}{\\rho m}\n\\frac{\\partial}{\\partial x}[\\rho P(x)]\n-\\frac{1}{\\rho m}\\frac{\\partial}{\\partial x}[\\rho P_{n+1}(x)]\n-nP_{n-1}(x)\\frac{\\partial }{\\partial x}[V(x,t)+u\\rho]\n\\nonumber \\\\ &&\n-\\sum_{s=1}^{n-1}\\left\\{\\frac{(\\hbar/2)^{2s}n!}{(2s+1)![n-(s+1)]!}\nP_{n-(s+1)}(x)\\frac{\\partial^{2s+1}}{\\partial x^{2s+1}}[V(x,t)+u\\rho]\\right\\}.\n\\label{motionmoment}\n\\end{eqnarray}\nThe system of equations Eqs.~(\\ref{appcontinuity},\\ref{motionmoment}), where $n$\nranges from $1$ to $\\infty$, thus describes the full dynamics of the\nGross-Pitaevskii equation, Eq.~(\\ref{gpe}) \\cite{lill}. \n\n\\subsection{Regaining the Second Hydrodynamic Equation}\nWe consider a set of solutions of the moments where \n$P_{n}(x)=P(x)^{n}$. \nTaking Eq.~(\\ref{motionmoment}) and setting $\\hbar=0$, i.e.\nignoring all quantum corrections, we substitute this solution in, which after\ndifferentiation results in:\n\\begin{eqnarray}\nnP(x)^{n-1}\\frac{\\partial}{\\partial t}P(x)&=&\n-\\frac{nP(x)^{n}}{m}\\frac{\\partial}{\\partial x}P(x)\n-nP(x)^{n-1}\\frac{\\partial }{\\partial x}[V(x,t)+u\\rho],\n\\end{eqnarray}\nwhere we can immediately carry out cancellations, to finally arrive at\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}P(x)&=&-\\frac{\\partial }{\\partial x}\\left[\n\\frac{P(x)^{2}}{2m}\nV(x,t)+u\\rho\\right],\n\\end{eqnarray}\nwhich is the second hydrodynamic equation, Eq.~(\\ref{momentumfield}). Thus\nhydrodynamic equations describing dynamics in the hydrodynamic limit\n\\cite{review,hydro} are valid \nwhenever $\\hbar\\rightarrow 0$ and $P_{n}(x)=P(x)^{n}$.\nThis condition can be expressed in terms of Liouville distributions as\n\\begin{equation}\n\\frac{1}{\\rho}\\int_{\\infty}^{\\infty} dp p^{n}(x)W(x,p)\n=\n\\left[\n\\frac{1}{\\rho}\n\\int_{\\infty}^{\\infty}\ndp p W(x,p)\n\\right]^{n},\n\\end{equation}\nwhich is in general fulfilled for $W(x,p)=\\rho(x)\\delta[p-p_{0}(x)]$, where\n$p_{0}(x)$ is some single valued function of $x$.\n\n\\section{More Scaling}\n\\label{scalingapp}\nAs dimensionless parameters we have\n$\\eta$, $\\kappa$, and \n$\\upsilon$, defined in Eqs.~(\\ref{eta},\\ref{kappa},\\ref{upsilon}), respectively.\nWe have as dimensionless coordinate and canonically conjugate momentum the\nvariables of Eqs.~(\\ref{dimpos},\\ref{dimmom}), and use the dimensionless time\n$t_{h}=\\omega t$.\nUsing this, we can write the dimensionless single\nparticle Hamiltonian\nfunction as\n\\begin{eqnarray}\n\\tilde{H}&=&\\frac{\\tilde{p}^{2}}{2}+\\tilde{V}(\\tilde{x},t_{h})\n \\label{dlessham}\\\\\n\\tilde{V}(\\tilde{x},t_{h})&=&\n\\frac{\\tilde{x}^{2}}{2}\n+\\frac{\\kappa}{\\sqrt{2}}\\cos(\\sqrt{2}\\tilde{x})\\sum_{n=-\\infty}^{\\infty}\n\\delta(t_{h}-n\\tau_{h}),\n\\end{eqnarray}\nthe Gross-Pitaevskii equation, Eq.~(\\ref{gpe}), as\n\\begin{equation}\ni\\frac{\\partial}{\\partial t_{h}}\\tilde{\\varphi}=\n-\\frac{\\eta^{2}}{2}\\frac{\\partial^{2}}{\\partial \\tilde{x}^{2}}\\tilde{\\varphi}\n+\\frac{1}{\\eta^{2}}\\tilde{V}(\\tilde{x},\\tilde{t})\\tilde{\\varphi} \n+\\frac{\\upsilon}{\\eta^{2}}\|\\tilde{\\varphi}\|^{2}\\tilde{\\varphi},\n\\end{equation}\nand the equation of motion for the Wigner function as\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t_{h}}\\tilde{W}&=&\n\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{(2s+1)!}\n\\left(\\frac{\\eta^{2}}{2}\\right)^{2s}\n\\frac{\\partial^{2s+1}}{\\partial \\tilde{x}^{2s+1}}\n\\left[\\tilde{H}+\\upsilon\\tilde{\\rho}\\right]\n\\frac{\\partial^{2s+1}}{\\partial \\tilde{p}^{2s+1}}\\tilde{W}\n-\\frac{\\partial }{\\partial \\tilde{p}}\\tilde{H}\n\\frac{\\partial }{\\partial \\tilde{x}}\\tilde{W}.\n\\label{etawigexp}\n\\end{eqnarray}\nThe wavefunction, Wigner function, and density, have been rescaled so that they are \nproperly normalized: \n\\begin{eqnarray}\n\\tilde{\\varphi}&=&\\sqrt{\\sqrt{2}/k}\\varphi,\\\\\n\\tilde{W}&=& \\frac{2m\\omega^{2}}{k^{2}}W,\\\\\n\\tilde{\\rho}&=&\\int_{-\\infty}^{\\infty}d\\tilde{p}\\tilde{W}.\n\\end{eqnarray} \nIn the expansion shown in Eq.~(\\ref{etawigexp})\nit can clearly be seen that if $\\eta$ is varied, then this is\ncompletely independent of all other rescaled quantities. We see thus that\n$\\eta^{2}$ is an appropriate expansion parameter, and that the other\ndimensionless parameters $\\kappa$ and $\\upsilon$ are correctly scaled to be\nindependent of the expansion parameter. \nIf one takes only the zero order term in the sum, $\\eta$ drops out\ncompletely.\n\n\\section{Crystal Symmetry and Non-Localization}\n\\label{antilocalapp}\n\\subsection{Classical Background}\nConsider the classical delta kicked harmonic oscillator described in\nEq.~\\ref{dlessham}.\nThe symmetry properties of this system have been extensively investigated by\nZaslavsky and coworkers \\cite{classharm,stochastic,web,symmetry}; we\nrecapitulate some of this to provide context.\n\nOne can determine a kick to kick mapping terms of \n$\\alpha=(\\tilde{x}+i\\tilde{p})/\\sqrt{2}$:\n\\begin{equation}\n\\alpha_{n+1}=\\left[\n\\alpha_{n}+i\\frac{\\kappa}{\\sqrt{2}}\\sin(\\alpha_{n}+\\alpha_{n}^{})\n\\right]e^{-i\\omega\\tau}.\n\\end{equation}\n If \n$\\omega\\tau=2\\pi r/q$, then we can write the mapping after $q$ kicks as\n\\begin{equation}\n\\alpha_{n+q}=\\alpha_{n}+i\\frac{\\kappa}{\\sqrt{2}}\n\\sum_{k=0}^{q-1}\\sin(\\alpha_{n+k}+\\alpha_{n+k}^{})e^{i2\\pi kr/q}.\n\\label{qmap}\n\\end{equation}\nKeeping terms in $\\kappa$ up to first order only, we observe\nan {\\em approximate\\/} rotational $q$ symmetry in phase space \n\\cite{stochastic,symmetry};\nif we substitute $\\alpha_{n}$ with $\\beta_{n}=\\alpha_{n}e^{i2\\pi l/q},\nl\\in{\\Bbb{Z}}$,\nwe end up with $\\beta_{n+q}=\\alpha_{n+q}e^{i2\\pi l/q}$.\nThere can also be\na translational symmetry in phase space, i.e. \n$\n\\beta_{n}=\\alpha_{n} + \\gamma \\Rightarrow \\beta_{n+q}=\\alpha_{n+q} +\n\\gamma, \\gamma \\in {\\Bbb{C}}.\n$\nNote that it is only possible \nto combine a rotational $q$ symmetry with translational symmetry when \n$q\\in q_{c}=\\{1,2,3,4,6\\}$ \\cite{tessellate}.\n\nTranslational symmetry demands\n\\begin{eqnarray}\n\\sum_{k=0}^{q-1}\\sin(\\alpha_{n+j}+\\alpha_{n+j}^{})e^{i2\\pi kr/q}\n&=&\\sum_{k=0}^{q-1}\\sin(\\beta_{n+j}+\\beta_{n+j}^{})e^{i2\\pi kr/q},\n\\end{eqnarray}\nwhich in turn implies\n$\\beta_{n+j}+\\beta_{n+k}^{}=\\alpha_{n+k}+\\alpha_{n+k}^{}+\n2\\pi\\l_{k};\\forall\\; k,l_{k}\\in{\\Bbb{Z}}$. Thus, Eq.~(\\ref{qmap}) for\n$\\beta_{n+q}$ can be\nsimplified to\n\\begin{eqnarray}\n\\beta_{n+q}&=&\\alpha_{n}+\\gamma_{0}+\ni\\frac{\\kappa}{\\sqrt{2}}\n\\sum_{k=0}^{q-1}\\sin(\\alpha_{n+k}+\\alpha_{n+k}^{}+\\gamma_{k} +\\gamma_{k}^{})\ne^{i2\\pi kr/q},\n\\end{eqnarray}\nwhere $\\gamma_{k}=\\gamma e^{-i2\\pi kr/q}$. The condition for translational \nsymmetry\nis thus reduced to $\\gamma_{k}+ \\gamma_{k}^{}= 2\\pi l_{k}$, which implies\n\\begin{equation}\nl_{k}=l_{0}\\cos(2\\pi kr/q)-i\\frac{\\gamma-\\gamma^{}}{2\\pi}\\sin(2\\pi k r/q).\n\\label{classinteg}\n\\end{equation}\nIf we now let $k_{\\pm}=q/2\\pm m$ or $(q\\pm m)/2$, depending on whether or not\n$q$ is even, we get\n\\begin{eqnarray}\n\\cos(2\\pi k_{+}r/q)&=&\\frac{l_{k_{+}}+l_{k_{-}}}{2l_{0}}\\in{\\Bbb{Q}},\n\\label{gammaone}\\\\\ni\\frac{\\gamma-\\gamma^{}}{\\pi}\\sin(2\\pi k_{+} r/q)&=&l_{k_{-}}-l_{k_{+}}\n\\in{\\Bbb{Z}}.\n\\label{gammatwo}\n\\end{eqnarray}\nThis implies that $\\cos(2\\pi/q)\\in {\\Bbb{Q}}$, and it is known that this can\nonly be true if $q\\in q_{c}=\\{1,2,3,4,6\\}$ \\cite{algebra}. This directly \nimplies that $\\cos(2\\pi k r/q)\\in {\\Bbb{Q}}\\;\\forall \\;k,r \\in {\\Bbb{Z}}$. \nThere is is thus an {\\em exact\\/}\ntranslational or {\\em crystal\\/} symmetry in phase space, \nfor $q\\in q_{c}$ only. There are an infinite number of\nvalues of $\\gamma$ for which this applies, \ndeterminable from Eqs.~(\\ref{gammaone},\\ref{gammatwo}).\n\n\\subsection{Quantum Expression}\nBroadly following the treatment of Borgonovi and Rebuzzini \\cite{borgonovi}, we\nconsider the unitary displacement operator \n$\nD(\\alpha)=e^{\\alpha\\hat{a}^{\\dagger}-\\alpha^{}\\hat{a}}\n=e^{i(\\varpi\\hat{x}-\\xi\\hat{p})}\n$ \\cite{dan}.\nThe operators $\\hat{a}^{\\dagger}$ and $\\hat{a}$ are the quantum harmonic\noscillator creation and annihilation operators, and\nthe operators $\\hat{x}$, $\\hat{p}$, are scaled in harmonic\nunits.\nThe displacement operator acting on a wavefunction is a quantum \nanalogue to translating a classical point particle in phase space.\nWe now consider the Floquet operator\n$\n\\hat{F}=e^{-i(\\hat{a}^{\\dagger}\\hat{a}+1/2)\\omega\\tau}\ne^{-i\\kappa\\cos[\\eta(\\hat{a}+\\hat{a}^{\\dagger})]/\\sqrt{2}\\eta^{2}}\n$\nand determine the commutation properties of it with the displacement operator.\n\nUsing elementary properties of coherent states \\cite{dan}, it can be seen that\n\\begin{eqnarray}\nD(\\alpha)\\hat{F}^{q}\n&=&\\prod_{k=0}^{q-1}\\left\\{\ne^{-i(\\hat{a}^{\\dagger}\\hat{a}+1/2)2\\pi r/q}\ne^{-i\\kappa\n\\cos[\\eta(\\hat{a}+\\hat{a}-\\alpha_{k}-\\alpha_{k}^{})]/\\sqrt{2}\\eta^{2}}\n\\right\\}D(\\alpha),\n\\end{eqnarray}\nwhere $\\alpha_{k}=\\alpha e^{i 2\\pi k r/q}$. The product of Floquet operators\n$\\hat{F}^{q}$ corresponds to the mapping of Eq.~(\\ref{qmap}) which we used to\ninvestigate classical symmetry properties.\n\nThus, $D(\\alpha)$ commutes with $\\hat{F}^{q}$ if\n$\\eta(\\alpha_{k}+\\alpha_{k}^{})=\\sqrt{2}\\eta\\xi_{k}=2\\pi l_{k},\\quad l_{k}\\in \n{\\Bbb{Z}} \\;\\forall\\; \nk$. Using this we arrive at, similarly to the \nderivation of Eq.~(\\ref{classinteg}), \n\\begin{equation}\nl_{k}=l_{0}\\cos(2\\pi kr/q)-i\\frac{(\\alpha-\\alpha^{})\\eta}{\\sqrt{2}\\pi}\\sin(2\\pi kr/q).\n\\end{equation}\nAnalogously to the classical case, \n$D(\\alpha)$ commutes with $\\hat{F}^{q}$ if and only if $q\\in q_{c}$. \nThis implies that for $q\\in q_{c}$, the eigenstates of $\\hat{F}^{q}$ are\ninvariant under certain displacements, of which there are an \ninfinite number, and are thus extended. Localization is not expected to take \nplace, similarly to the case of quantum\nresonances in a delta-kicked rotor \\cite{frasca,resonances}.\n\n\\section{Integration of the ${\\cal L}$ Equation.}\n\\label{intLapp}\nFrom \\cite{castin}, we know that\n\\begin{equation}\ni\\hbar \\frac{d}{dt}\n\\left(\n\\begin{array}{c}\n\|u_{k}(t)\\rangle \\\\\n\|v_{k}(t)\\rangle\n\\end{array}\n\\right)\n=\n{\\cal L}\n\\left(\n\\begin{array}{c}\n\|u_{k}(t)\\rangle \\\\\n\|v_{k}(t)\\rangle\n\\end{array}\n\\right),\n\\end{equation}\nand that the corresponding time evolution operator\n\\begin{equation}\n{\\cal U}(t)=\n\\left(\n\\begin{array}{cc}\n\\hat{Q}(t) & 0 \\\\\n0 & \\hat{Q}^{}(t)\n\\end{array}\n\\right)\n{\\cal U}_{\\mbox{\\scriptsize GP}}(t)\n\\left(\n\\begin{array}{cc}\n\\hat{Q}(0) & 0 \\\\\n0 & \\hat{Q}^{}(0)\n\\end{array}\n\\right).\n\\end{equation}\nThe operator ${\\cal U}_{\\mbox{\\scriptsize GP}}(t)$\nis the time evolution operator corresponding to \n${\\cal L}_{\\mbox{\\scriptsize GP}}(t)$, given by\n\\begin{equation}\n{\\cal L}_{\\mbox{\\scriptsize GP}}(t)=\n\\left(\n\\begin{array}{cc}\nV(\\hat{x},t)+ 2u \|\\varphi(\\hat{x},t)\|^{2} + \\hat{p}^{2}/2m \n& u \\varphi(\\hat{x},t)^{2} \\\\\n-u\\varphi(\\hat{x},t)^{2}\n& -V(\\hat{x},t)- 2u \|\\varphi(\\hat{x},t)\|^{2}-\\hat{p}^{2}/2m\n\\end{array}\n\\right).\n\\end{equation}\n\nIn our case, the potential is that of the delta-kicked harmonic oscillator.\nIntegrating between kicks, we consider $V(\\hat{x})$\n time independent. Note however, that ${\\cal L}_{\\mbox{\\scriptsize GP}}(t)$ is still in principle\n time dependent through $\\varphi(x,t)$. Thus,\ntaking very small time steps $\\Delta t$, the evolution is given \napproximately by\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\|U_{k}(t+\\Delta t)\\rangle \\\\\n\|V_{k}(t+\\Delta t)\\rangle\n\\end{array}\n\\right)\n\\approx\ne^{-i{\\cal L}_{\\mbox{\\scriptsize GP}}(t)\\Delta t/\\hbar}\n\\left(\n\\begin{array}{c}\n\|U_{k}(t)\\rangle \\\\\n\|V_{k}(t)\\rangle\n\\end{array}\n\\right).\n\\end{equation}\nThe time evolution operator \n$e^{-i{\\cal L}_{\\mbox{\\scriptsize GP}}(t)\\Delta t/\\hbar}$ can be split \ninto position dependent and momentum\ndependent parts, and the time evolution was then \ndetermined using a split operator method, of which there are\nmany variations \\cite{bandrauk}. We set $\|U_{k}(0)\\rangle=\|u_{k}(0)\\rangle$\nand $\|V_{k}(0)\\rangle=\|v_{k}(0)\\rangle$, and determined\n$\|u_{k}(t)\\rangle$ and\n$\|v_{k}(t)\\rangle$ \nfrom\n$\|U_{k}(t)\\rangle$ and\n$\|V_{k}(t)\\rangle$\nby projection, just before each kick .\n\nThe effect of a kick is given by:\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nu_{k}(x,t^{+}) \\\\\nv_{k}(x,t^{+})\n\\end{array}\n\\right)\n=\n\\left(\n\\begin{array}{c}\ne^{-i\\kappa\\cos(\\sqrt{2}\\eta x)/\\sqrt{2}\\eta^{2}}u_{k}(x,t^{-}) \\\\\ne^{i\\kappa\\cos(\\sqrt{2}\\eta x)/\\sqrt{2}\\eta^{2}}v_{k}(x,t^{-})\n\\end{array}\n\\right).\n\\end{equation}\nIn\nSec.~\\ref{vkevol}, the procedure outlined\nabove was used, in conjunction with numerical integration of the Gross-Pitaevskii\nequation, also by a split operator method with matching time steps. \n\n\\end{appendix}\n\n\\begin{references}\n\n\\bibitem{reichl}\nL.~E.~Reichl \n{\\em The Transition to Chaos In Conservative Classical Systems: Quantum\nManifestations\\/} (Springer-Verlag, New York 1992).\n\n\\bibitem{gutzwiller}\nM.~C.~Gutzwiller \n{\\em Chaos in Classical and Quantum Mechanics} (Springer-Verlag, Berlin 1990).\n\n\\bibitem{haake}\nF.~Haake, \n{\\em Quantum Signatures of Chaos\\/} (Springer-Verlag, Berlin 1991).\n\n\\bibitem{peres}\nA.~Peres, in {\\em Quantum Chaos: Proceedings of the Adriatico Research\nConference on Quantum Chaos}, edited by \nH.~A.~Cerdeira, R.~Ramaswamy, M.~C.~Gutzwiller, and G.~Casati \n(World Scientific, Singapore, 1991); \nA.~Peres, \n{\\em Quantum Theory: Concepts and Methods\\/} (Kluwer Academic Publishers, \nDordrecht 1993); \nfor alternative treatments see also\nR.~Schack and C.~M.~Caves\nPhys.\\ Rev.\\ E {\\bf 53}, 3257 (1996);\nR.~Schack and C.~M.~Caves\nPhys.\\ Rev.\\ E {\\bf 53}, 3387 (1996); \nand\nG.~Garcia de Polavieja,\nPhys.\\ Rev.\\ A {\\bf 57}, 3184 (1998).\n\n\\bibitem{zurek}\nW.~H.~Zurek and J.~P.~Paz,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 72}, 2508 (1994);\nW.~H.~Zurek and J.~P.~Paz,\nNuov.\\ Cim.\\ B {\\bf 110}, 611 (1995).\n\n\n\\bibitem{nonlinchaos}\nN.~Finlayson, K.~J.~Blow, L.~J.~Bernstein, and K.~W.~Delong,\nPhys.\\ Rev.\\ A {\\bf 48}, 3863 (1993);\nF.~Benvenuto, G.~Casati, A.~Pikovsky, and D.~L.~Shepelyansky,\nPhys.\\ Rev.\\ A {\\bf 44}, R3423 (1994);\nB.~M.~Herbst and M.~J.~Ablowitz,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 18}, 2065 (1989).\n\n\\bibitem{bosecond}\nG.~Baym and C.~Pethick, \nPhys.\\ Rev.\\ Lett.\\ {\\bf 76}, 6 (1996);\n\n\\bibitem{review}\nF.~Dalfovo, S.~Giorgini, L.~P.~Pitaevskii, and S.~Stringari,\nRev.\\ Mod.\\ Phys.\\ {\\bf 71}, 463 (1999).\n\n\\bibitem{nonlinearopt}\nY.~R.~Shen \n{\\em Principles of Nonlinear Optics\\/} (Wiley \\& Sons, New York 1984);\nR.~W.~Boyd \n{\\em Nonlinear Optics\\/} (Academic Press, San Diego 1992).\n\n\\bibitem{hydro}\nS.~Stringari,\nPhys.\\ Rev.\\ A {\\bf 58}, 2385 (1998);\nM.~Fliesser, A.~Csord\\'{a}s, P.~Sz\\'{e}pfalusy, and R.~Graham,\nPhys.\\ Rev.\\ A {\\bf 56}, R2533 (1997);\nS.~Stringari,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 77}, 2360 (1996).\n\n\\bibitem{velocity}\nIt is more conventional to describe the hydrodynamic equations in terms of a\nvelocity field $V=P/m$, as in \\cite{review,hydro}. 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In the context of quantum chaos\nsignificant experimental work has been carried out \non microwave driven hydrogen \\cite{misc}\nand mesoscopic solid state systems \\cite{fromhold}.\n\n\\bibitem{ionchaos}\nS.~A.~Gardiner, J.~I.~Cirac, and P.~Zoller, \nPhys.\\ Rev.\\ Lett.\\ {\\bf 79}, 4790 (1997).\n\n\\bibitem{castin}\nWe use the formalism of Y.~Castin and R.~Dum, \nPhys.\\ Rev.\\ A {\\bf 57}, 3008 (1998),\nan analogous formalism is presented in\nC.~W.~Gardiner,\nPhys.\\ Rev.\\ A {\\bf 56}, 1414 (1997);\n see also \\cite{castindumdeplete}.\n\n\\bibitem{castindumdeplete}\nY.~Castin and R.~Dum, \nPhys.\\ Rev.\\ Lett.\\ {\\bf 79}, 3553 (1997).\n\n\\bibitem{fouriergrid}\nC.~C.~Marston and G.~G.~Balint-Kurti,\nJ. 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The parameter $u$ represents the\nstrength of the nonlinearity.}\n\\label{connections}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=ppFig2.ps,width=85mm} \n\\end{center}\n\\caption{Poincar\\'{e} sections of the phase space dynamics of the classical\ndelta-kicked harmonic oscillator. \n(a) Single unstable initial condition forming a stochastic web spreading \nthrough phase space. \n(b) Close up of the phase space, showing the closed curves characteristic \nof regular dynamics. In both cases\n$\\tau_{h}=2\\pi/6$, $\\kappa=1$.}\n\\label{poincare}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=ppFig3.ps,width=85mm} \n\\end{center}\n\\caption{Pseudocolour plot of time averaged Wigner functions when \n$\\upsilon=0$, i.e.\\ {\\em linear\\/} Schr\\\"{o}dinger equation dynamics, \nin the two cases of: \n$\\eta=1$, for (a) unstable initial condition,\n(b) stable initial condition; \n$\\eta=2$, for (c) unstable initial condition,\n(d) stable initial condition.\nPosition and\nmomentum are scaled in harmonic units, and black means large\nand positive.}\n\\label{uzero}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=ppFig4.ps,width=85mm} \n\\end{center}\n\\caption{\nAs for Fig.~\\ref{uzero}, where $\\upsilon=0.1$.}\n\\label{utenth}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=ppFig5.ps,width=85mm} \n\\end{center}\n\\caption{\nAs for Fig.\\ref{uzero}, where $\\upsilon=1$.}\n\\label{uone}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=ppFig6.ps,width=85mm} \n\\end{center}\n\\caption{\nAs for Fig.~\\ref{uzero}, where $\\upsilon=10$.}\n\\label{uten}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=ppFig7.ps,width=85mm} \n\\end{center}\n\\caption{Pseudocolour plot of time averaged distributions undergoing Liouville\ndynamics when \n$\\upsilon=0.1$, in the two cases of: $\\eta=1$, \nfor (a) unstable initial condition, (b)\nstable initial condition;\n$\\eta=2$, \nfor (a) unstable initial condition, (b)\nstable initial condition.\nBlack means large\nand positive.}\n\\label{lutenth}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=ppFig8.ps,width=85mm} \n\\end{center}\n\\caption{As for Fig.~\\ref{lutenth} when \n$\\upsilon=1$.}\n\\label{luone}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=ppFig9.ps,width=85mm} \n\\end{center}\n\\caption{As for Fig.~\\ref{lutenth} when \n$\\upsilon=10$.}\n\\label{luten}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure10.eps,width=85mm}\n\\end{center}\n\\caption{Plots of $\|\\varphi(x)\|^{2}$ after the application of 100 kicks and where\n$\\upsilon=0.1$, in the cases of:\n$\\eta=1$, for (a) unstable initial condition, (b) stable initial condition; and $\\eta=2$,\n for (c) unstable initial condition, (d) stable initial condition.}\n\\label{xtenth}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure11.eps,width=85mm}\n\\end{center}\n\\caption{As for Fig.~\\ref{xtenth}, but for $\\upsilon=1$.}\n\\label{xone}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure12.eps,width=85mm}\n\\end{center}\n\\caption{As for Fig.~\\ref{xtenth}, but for $\\upsilon=10$.}\n\\label{xten}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure13.eps,width=85mm}\n\\end{center}\n\\caption{Plots of $\|\\varphi(p)\|^{2}$ after the application of 100 kicks and where $\\upsilon=0.1$, \nin the cases of: $\\eta=1$, for (a) unstable initial condition, (b) stable initial condition; and $\\eta=2$,\n for (c) unstable initial condition, (d) stable initial condition.}\n\\label{ptenth}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure14.eps,width=85mm}\n\\end{center}\n\\caption{As for Fig.~\\ref{ptenth}, but for $\\upsilon=1$.}\n\\label{pone}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure15.eps,width=85mm}\n\\end{center}\n\\caption{As for Fig.~\\ref{ptenth}, but for $\\upsilon=10$.}\n\\label{pten}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure16.eps,width=170mm} \n\\end{center}\n\\caption{Semilog plot of change in $\\langle v_{k}\|v_{k}\\rangle$ with respect \nto \nthe number of kicks $n$,\nfor $k=1,\\ldots,15$:\n$k=1$ solid line,\n$k=2$ dotted line,\n$k=3$ dashed-dotted line,\n$k=4$ dashed line,\n$k=5$ circles,\n$k=6$ crosses,\n$k=7$ pluses,\n$k=8$ squares,\n$k=9$ diamonds,\n$k=10$ downward pointing triangles,\n$k=11$ upward pointing triangles,\n$k=12$ left pointing triangles,\n$k=13$ right pointing triangles,\n$k=14$ pentagrams,\n$k=15$ hexagrams, where\n$\\eta'=1$, and $\\upsilon=1$. \n(a) shows data for the ``unstable'' initial condition, where the leading term\nafter 100 kicks is for $k=1$, \n(b) shows data for the ``stable'' initial \ncondition, where the leading term corresponds to $k=2$.}\n\\label{vkkoneone}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure17.eps,width=170mm} \n\\end{center}\n\\caption{As for Fig.\\ref{vkkoneone}, except that\n$\\eta'=2$, $\\upsilon=1$. \nIn (a) the leading term is for $k=3$, in (b) for $k=6$.}\n\\label{vkktwoone}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure18.eps,width=170mm} \n\\end{center}\n\\caption{As for Fig.\\ref{vkkoneone}, except that\n$\\eta'=1$, $\\upsilon=10$. \nIn (a) the leading term is for $k=1$, in (b) for $k=1,5$.}\n\\label{vkkoneten}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure19.eps,width=170mm} \n\\end{center}\n\\caption{As for Fig.\\ref{vkkoneone}, except that\n$\\eta'=2$, $\\upsilon=10$. In (a) the leading term is for $k=4$, in (b) for\n$k=6$.}\n\\label{vkktwoten}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure20.eps,width=85mm} \n\\end{center}\n\\caption{Plots of $\\sum_{k=1}^{15}\\langle v_{k}\|v_{k}\\rangle$ against the number\nof kicks $n$, where\n$\\upsilon=1$, in the two cases of:\n$\\eta'=1$, for (a) unstable initial condition,\n(b) stable initial condition;\n$\\eta'=2$, (c) unstable initial condition.\n(d) stable initial condition.\n}\n\\label{growthone}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=figure21.eps,width=85mm} \n\\end{center}\n\\caption{Corresponds exactly to Fig.~\\ref{growthone}, except that\n$\\upsilon=10$.}\n\\label{growthten}\n\\end{figure}\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{c\|c\|c\|\|c\|c\|\|c\|c}\n\\multicolumn{3}{c\|\|}{\\mbox{}} \n& \n\\multicolumn{2}{c\|\|}{Na$^{23}$} &\n\\multicolumn{2}{c}{Rb$^{87}$}\n\\\\ \n\\hline\n$\\upsilon$ & $\\eta'$ & $\\mu$ & $\\lambda$ & $\\nu$ & $\\lambda$ & $\\nu$ \n\\\\ \n\\hline\\hline\n 1 & 1 & $0.87\\hbar\\omega$ & $9.55\\times 10^{3}s^{-1/2}$ & $3.02\\times 10^{3}s^{-1/2}$ & \n $2.65\\times 10^{3}s^{-1/2}$ & $8.38\\times 10^{2}s^{-1/2}$\\\\\n\\cline{2-7}\n & 2 & $0.55\\hbar\\omega$ &$1.19\\times 10^{3}s^{-1/2}$ & $3.77\\times 10^{2}s^{-1/2}$ & \n $3.31\\times 10^{2}s^{-1/2}$ & $1.05\\times 10^{2}s^{-1/2}$\\\\\n\\hline\n 10 & 1 & $3.11\\hbar\\omega$ & $9.55\\times 10^{4}s^{-1/2}$ & $3.02\\times 10^{4}s^{-1/2}$ & \n $2.65\\times 10^{4}s^{-1/2}$ & $8.38\\times 10^{3}s^{-1/2}$\\\\\n\\cline{2-7}\n & 2 & $0.95\\hbar\\omega$ & $1.19\\times 10^{4}s^{-1/2}$ & $3.77\\times 10^{3}s^{-1/2}$ & \n $3.31\\times 10^{3}s^{-1/2}$ & $1.05\\times 10^{3}s^{-1/2}$\\\\\n\\end{tabular}\n\\end{center}\n\\caption{Values of $\\lambda$ and $\\nu$ for Sodium 23 and Rubidium 87, when in the\nparameter regimes of $\\upsilon$ and $\\eta'$ under study. 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quant-ph9912093	Optical Holonomic Quantum Computer	[ { "author": "Jiannis Pachos\\footnote{Electronic address: pachos@isiosf.isi.it" } ]	In this paper the idea of holonomic quantum computation is realized within quantum optics. In a non-linear Kerr medium the degenerate states of laser beams are interpreted as qubits. Displacing devices, squeezing devices and interferometers provide the classical control parameter space where the adiabatic loops are performed. This results into logical gates acting on the states of the combined degenerate subspaces of the lasers, producing any one qubit rotations and interactions between any two qubits. Issues such as universality, complexity and scalability are addressed and several steps are taken towards the physical implementation of this model.	[ { "name": "optic.tex", "string": "\\documentstyle[prl,aps,preprint,epsfig,epsf,floats]{revtex}\n\n\\clubpenalty=10000\n\\widowpenalty=10000\n\\brokenpenalty=10000\n\\interdisplaylinepenalty=5000\n\\predisplaypenalty=10000\n\\postdisplaypenalty=100\n\\tolerance=100\n\n\\begin{document} % End of preamble and beginning of text.\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\bq}{\\begin{eqnarray}}\n\\newcommand{\\eq}{\\end{eqnarray}}\n\\newcommand{\\Sc}{Schr\\\"odinger\\,\\,}\n\\newcommand{\\Sp}{\\,\\,\\,\\,}\n\\newcommand{\\no}{\\nonumber\\\\}\n\\newcommand{\\tr}{\\text{tr}}\n\\newcommand{\\p}{\\partial}\n\\newcommand{\\la}{\\lambda}\n\\newcommand{\\La}{\\Lambda}\n\\newcommand{\\G}{{\\cal G}}\n\\newcommand{\\D}{{\\cal D}}\n\\newcommand{\\E}{{\\cal E}}\n\\newcommand{\\W}{{\\bf W}}\n\\newcommand{\\de}{\\delta}\n\\newcommand{\\al}{\\alpha}\n\\newcommand{\\bi}{\\beta}\n\\newcommand{\\ep}{\\varepsilon}\n\\newcommand{\\ga}{\\gamma}\n\\newcommand{\\epp}{\\epsilon}\n\\newcommand{\\vep}{\\varepsilon}\n\\newcommand{\\th}{\\theta}\n\\newcommand{\\om}{\\omega}\n\\newcommand{\\si}{\\sigma}\n\\newcommand{\\J}{{\\cal J}}\n\\newcommand{\\pr}{\\prime}\n\\newcommand{\\ka}{\\kappa}\n\\newcommand{\\TH}{\\mbox{\\boldmath${\\theta}$}}\n\\newcommand{\\DE}{\\mbox{\\boldmath${\\delta}$}}\n\\newcommand{\\lan}{\\langle}\n\\newcommand{\\ran}{\\rangle}\n\\newcommand{\\Hol}{\\text{Hol}}\n\\newcommand{\\cp}{{\\bf CP}}\n\\newcommand{\\spp}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\n\n\\bibliographystyle{unsrt}\n \n\\setcounter{page}{0}\n\\def\\footnoterule{\\kern-3pt \\hrule width\\hsize \\kern3pt}\n\\tighten\n\\title{\nOptical Holonomic Quantum Computer \n}\n\\author{Jiannis Pachos\\footnote{Electronic address: pachos@isiosf.isi.it} and Spiros Chountasis\\footnote{Electronic address: spiros@isiosf.isi.it}\\\\\n{~}\n}\n\n\\address{\nInstitute for Scientific Interchange Foundation, \\\\\nVilla Gualino, Viale Settimio Severo 65, I-10133 Torino, Italy\\\\\n{~}\n}\n\n\\date{October 1999}\n\n\\maketitle\n\n\\thispagestyle{empty}\n\n\\begin{abstract}\nIn this paper the idea of holonomic quantum computation is realized within quantum optics. In a non-linear Kerr medium the degenerate states of laser beams are interpreted as qubits. Displacing devices, squeezing devices and interferometers provide the classical control parameter space where the adiabatic loops are performed. This results into logical gates acting on the states of the combined degenerate subspaces of the lasers, producing any one qubit rotations and interactions between any two qubits. Issues such as universality, complexity and scalability are addressed and several steps are taken towards the physical implementation of this model.\n\\end{abstract}\n\n%\\vspace*{\\fill}\n\n%\\newpage\n\n\\section{Introduction}\n\nHolonomic transformations have been recently proposed \\cite{ZARA} and more extensively studied \\cite{PAZARA} as logical gates for quantum computation \\cite{QC}. The idea formal as it may apear at a first glance is not confined to a purely theoretical shere, but has the challenging possibility of experimental implementation. Towards this purpose we employ existing devices of quantum optics, such as displacing and squeezing devices and interferometers acting on laser beams in a non-linear medium. A different setting of optical quantum computer has been reported in \\cite{Chuang}. The attempt to apply the abstract idea of holonomic quantum computation (HQC) to a physical system has lead us to deal with and clarify some theoretical problems of HQC such as universality, complexity (tensor product structure of qubits) and scalability. On the other hand, the experimental setup of the scheme proposed may prove to be a possible even though challenging task for the experimenters.\n\nThe basic idea of HQC is related with the geometrical phases \\cite{SHWI} generated by the \nisospectral transformations of an $n$-fold\ndegenerate Hamiltonian, $H_0$, so its presentation can be given in a geometrical form \\cite{ZARA}.\nInitially, quantum information is encoded in the $n$ dimensional degenerate eigenspace $\\cal C$ of $H_0$, \nwith eigenvalue $ E_0$. The operator\n$H_0$ is considered to belong to the family ${\\cal F}=\\{H_\\sigma={\\cal U}(\\sigma)\\,H_0\\,\n{\\cal U}^\\dagger(\\sigma);\\sigma\\in{\\cal M}\\}$ of Hamiltonians unitarily (${\\cal U}^\\dagger (\\sigma)={\\cal U}^{-1}(\\sigma)$) equivalent and therefore isospectral with $H_0$, where $H_0=H_{\\sigma_0}$ for some $\\sigma_0 \\in {\\cal M}$.\nas $\\sigma$ ranges over the control manifold $\\cal M$ no energy level crossing occurs.\nThe $\\sigma$'s represent the classical ``control'' parameters that one uses in order\nto manipulate the encoded states $\|\\psi\\rangle\\in{\\cal C}$.\nLet $C$ be a {\\em loop} in the control manifold $\\cal M$.\nWhen the loop $C$ is slowly gonne through, then no transition among different energy levels occurs\nand the evolution is adiabatic, i.e. ${\\cal F}$ is faithfully realized by the experimental setup. \nIf $\|\\psi\\rangle_{in}\\in{\\cal C}$ is an initial state in the degenerate \neigenspace, at the end of the loop it becomes\n$ \|\\psi\\rangle _{out}=e^{i\\,E_0\\,T}\\, \\Gamma_{A}(C) \\,\|\\psi\\rangle_{in}$. \nThe first factor is just an overall dynamical phase which in the following \nwill be omitted by a redefinition of the energy levels, taking $E_0=0$. \nThe second contribution is the holonomy $\\Gamma_{A}(C)\\in U(n)$, \nand is a result of the non-trivial topology of the {\\em bundle} of eigenspaces \nover $\\cal M$. By introducing the Wilczek-Zee connection \\cite{WIZE}\n\\begin{equation}\nA_{\\sigma_i}^{\\bar \\rho \\rho}:= \\langle \\bar \\rho \|{\\cal U}^\\dagger(\\sigma)\n\\,{\\partial \\over \\partial\\sigma_i}\\,{\\cal U}(\\sigma)\|\\rho\\rangle \\,\\, ,\n\\label{conn}\n\\end{equation}\nwhere $A_{\\sigma_i}^{\\bar \\rho \\rho}$ is the $(\\bar \\rho, \\rho)$ matrix element of the $\\sigma_i$ \ncomponent of the connection,\none finds $\\Gamma_{A}(C) ={\\bf{P}}\\exp \\int_C A$, \\cite{SHWI}, where ${\\bf{P}}$ denotes path ordering.\nThe set Hol$(A):=\\{\\Gamma_{A}(C);\\forall C\\in{\\cal M}\\} \\subset U(n)$ is known as the holonomy group\n\\cite{NAK}. In the case where it coincides with the whole unitary\ngroup $U(n)$ the connection $A$ is called {\\em irreducible} \\cite{ZARA}.\nThe transformations $\\Gamma_A(C)$ for suitable $C$'s can be used as logical gates for the HQC.\n\nWe shall focus on quantum optics, a well established area of quantum physics, in which the developed technology is quite mature as a possible venue for practical implementation of HQC. The model we study here includes laser beams moving through non-linear Kerr media, and acted on by displacing and squeezing devices and interferometers. This implementation has the merit that it gives direct answers to several problems which were raised in the theoretical study of HQC {\\cite{PAZARA}.\n\nIn Chapter II we present the schematic theoretical description of the quantum optical components employed for HQC. This includes the non-linear Kerr medium, the one and two mode displacing and squeezing devices as well as their effect on the states of laser beams. In Chapter III we construct the non-Abelian Berry connection, the field strength and the holonomies related with this optical setup. A model with $SU(2)$ interferometers is also given as an alternative tool for classical control, and its holonomies are calculated resorting to the non-Abelian Stokes theorem. In Chapter IV the connection between the experimental components of quantum optics and the theoretical requirements for HQC is described. A numerical simulation is finally reported indicating the reliability of the logical gates with respect to the scale resources of the HQC. In the Conclusions the quantum computation characteristics of this model are discussed and issues like the universality, complexity and scalability are addressed.\n\n\\section{The Quantum Optical Model}\n\nIn the following we shall exploit the advanced tools of quantum optics in order to implement a specific HQC model. All the components used here are thoroughly analyzed in the optics literature \\cite{Kral} and experimentally realized by employing such devices as beam splitters, frequency converters, four wave mixers, and others.\n\n\\subsection{Kerr medium Hamiltonian and Degenerate States}\n\nIn order to perform holonomic computation we shall employ the nonlinear interaction Hamiltonian produced by a Kerr medium\n\\bq\n&&\nH_I=\\hbar X n(n-1) \\,\\, ,\n\\nonumber \n\\eq\nwith $n=a^\\dagger a$ the number operator, $a$ and $a^{\\dagger}$ being the usual bosonic annihilation and creation operators respectively, and $X$ a constant proportional to the third order nonlinear susceptibility, $\\chi^{(3)}$, of the medium. Degenerate eigenstates of $H_I$ are $\|0\\ran$ and $\|1\\ran$ ($\\{\|\\nu\\ran ; \\nu=0,1,... \\}$ denoting the Fock basis of number eigenstates $n\|\\nu\\ran=\\nu \|\\nu \\ran$). In the case of two laser beams, with annihilation operators $a_1$ and $a_2$ respectively, the total Hamiltonian is given by the sum \n\\bq\n&&\nH_I^{12}=\\hbar X n_1(n_1-1) + \\hbar X n_2 (n_2-1) \\,\\, .\n\\nonumber\n\\eq\nIts degenerate eigenstates are the tensor product of the eigenstates of each subsystem: $\|i_1j_2\\ran$ $:=\|i_1\\ran \\otimes \|j_2 \\ran$ for $i_1,j_2=0,1$ with $\|i_1\\ran$ and $\|j_2\\ran$ the degenerate states of each beam. Accordingly, the unitary transformations acting on the system are given by the tensor product of the transformations on each individual subsystem. For example, the transformation of a system (Hamiltonian and states) of two lasers when one beam is transformed by $U_1$ is given by the tensor product $U_{12}=U_1\\otimes {\\bf 1}$. These rules can be applied to build up a system with $m$ lasers. In this case the subspace of Fock states on which we restrict in order to apply the adiabaticity theorem has as basis vectors the degenerate states $\|0_l\\ran$ and $\|1_l\\ran$ for each laser labelled by $l$. The general state of the system of $m$ lasers is given by $\|\\rho_1...\\rho_m\\ran=\|\\rho_1\\ran \\otimes...\\otimes \|\\rho_m\\ran$ where $\\rho_l$ could be zero or one, for $l=1,...,m$. On this space of states the code can be written. \nWe have good reasons to\nbelieve that the problem of the generation of stable Fock states will be\novercome, as suggested by some recent developments \\cite{Hong}. \n\n\\subsection{One and Two Laser-Qubit Transformations}\n\nOn state $\|\\psi\\ran$ of a laser beam with annihilation operator $a$, the following operators can act\n\\bq\n&&\n\\text{\\it Displacer:} \\spp \\spp D(\\la)=\\exp(\\la a^\\dagger-\\bar \\la a) \\,\\, ,\n\\nonumber\n\\eq\nwhere $\\la$ is an arbitrary complex parameter. The displacing device that implements $D(\\la)$ is a simple device that performs a linear amplification to the light field components.\n\\bq\n&&\n\\text{\\it Squeezer:} \\spp \\spp S(\\mu)=\\exp(\\mu {a^\\dagger}^2-\\bar \\mu a^2) \\,\\, ,\n\\nonumber\n\\eq\nwhere $\\mu$ is an arbitrary complex parameter. The squeezing operator $S(\\mu)$ can be implemented in the laboratory by a degenerate parametric amplifier.\n\nThe transformation operators $D(\\la)$ and $S(\\mu)$ acting on a single laser beam will result, after a closed loop is performed in their parameter space, into rotations in the state space spanned by $\|0\\ran$ and $\|1\\ran$, according to the adiabatic theorem.\n\nThe displacer $D(\\la)$, transforms the operators $a$, $a^{\\dagger}$ and \nany analytic function thereof $f(a,a^{\\dagger})$, for any choice of parameters $\\lambda$,\nas follows \\cite{Bishop}\n\\begin{eqnarray}\n&\nD(\\lambda)aD^{\\dagger}(\\lambda) = a-\\lambda \\,\\,\\, ,\\,\\,\\,\\,\\,\\,\\,\\,\\, D(\\lambda)a^{\\dagger}D^{\\dagger}(\\lambda) = a^{\\dagger}-\\bar \\lambda \\,\\, ,\n&\n\\no \\no\n&\nD(\\lambda)f(a,a^{\\dagger})D^{\\dagger}(\\lambda) =f(a-\\lambda,a^{\\dagger}-\\bar \\lambda) \\,\\, .\n&\n\\nonumber\n\\end{eqnarray} \nSimilarly for the squeezing operator\n\\begin{eqnarray}\n&\nS(\\mu)aS^{\\dagger}(\\mu) = \\cosh(2r)a+e^{-i \\theta} \\sinh(2r)a^{\\dagger} \\,\\, ,\n&\n\\no \\no\n&\nS(\\mu)a^{\\dagger}S^{\\dagger}(\\mu) = e^{i \\theta} \\sinh(2r)a+\\cosh(2r)a^{\\dagger} \\,\\, ,\n&\n\\no \\no\n&\nS(\\mu)f(a,a^{\\dagger})S^{\\dagger}(\\mu) = f\\left(S(\\mu)aS^{\\dagger}(\\mu),S(\\mu)a^{\\dagger}\nS^{\\dagger}(\\mu)\\right) \\,\\, ,\n&\n\\nonumber\n\\end{eqnarray} \nwhere $\\mu= r e^{i \\theta}$, with $r>0$ and $-\\pi < \\theta \\leq \\pi$.\n\nOn the general state of two lasers $\|\\psi_{12}\\ran=\|\\psi_1\\ran\\otimes\|\\psi_2\\ran$ with corresponding annihilation operators $a_1$ and $a_2$, the following operators can act\n\\bq\n&&\n\\text{\\it Two mode squeezer:} \\spp \\spp M(\\zeta)=\\exp(\\zeta a_1^\\dagger a_2^\\dagger -\\bar \\zeta a_1 a_2) \\,\\, .\n\\nonumber\n\\eq\nThe operator $M(\\zeta)$, can be implemented in the laboratory by a non-degenerate parametric amplifier.\n\\bq\n&&\n\\text{\\it Two mode displacer:} \\spp \\spp N(\\xi)=\\exp(\\xi a_1^\\dagger a_2 -\\bar \\xi a_1 a_2^\\dagger) \\,\\, .\n\\nonumber\n\\eq\n$M(\\zeta)$ and $N(\\xi)$ are the transformations between two laser beams that produce, after performing adiabatically a loop in their parametric space, coherent transformations in the two qubit state space spanned by $\|00\\ran$, $\|01\\ran$, $\|10\\ran$ and $\|11\\ran$. \n\nThese transformations on the states of the laser beams can be produced by $SU(2)$ or $SU(1,1)$ interferometers \\cite{Yurke}, according to the algebra which their generators belong to. For instance, each one of $\\la a^\\dagger-\\bar \\la a$ and $\\xi a_1^\\dagger a_2 -\\bar \\xi a_1 a_2^\\dagger$ belongs to an $su(2)$ algebra, while $\\mu {a^\\dagger}^2-\\bar \\mu a^2$ and $\\zeta a_1^\\dagger a_2^\\dagger -\\bar \\zeta a_1 a_2$ belong into (different) $su(1,1)$ algebras \\cite{Perelomov}.\n\n\\section{Application to The Holonomic Theory}\n\nThe non-Abelian Berry connection, $A$, is generated by the topological structure of the bundle of the degenerate sub-spaces. It determines the way to perform a parallel transport of the degenerate eigenstates along an adiabatically spanned loop. In this section we shall show that a complete set of holonomies of $A$ can be {\\it explicitly} calculated for our model.\n\n\\subsection{The Connection $A$ \\label{AAA}}\n\nWe initially perform the following polar decomposition of the control variables\n\\bq\n&&\n\\la=r_0e^{i\\th_0}\\,\\, , \\,\\,\\, \\mu=r_1e^{i\\th_1} \\,\\, , \\,\\,\\, \\zeta=r_2e^{i\\th_2} \\,\\, , \\,\\,\\, \\xi =r_3e^{i\\th_3} \\,\\, .\n\\nonumber\n\\eq\nWe obtain the connection, $A$, from (\\ref{conn}), parametrizing the control manifold by the set of real variables introduced above ${\\cal M}:=\\{r_i,\\th_i; \\, i=0,...,3\\}$ with elements $\\sigma_i\\in{\\cal M}$, where we take ${\\cal U}(\\sigma)=D(\\la)S(\\mu)$ for the one laser transformations and ${\\cal U}(\\sigma)=N(\\xi)M(\\zeta)$ for transformations between two lasers. We have\n\n\\[ \\begin{array}{cc}\nA_{r_0}=& \n \\left[ \\begin{array}{ccc} 0 & -(e^{-i\\th_0}\\cosh 2r_1-e^{i(\\th_0+\\th_1)} \\sinh 2r_1) \\\\\n \t\t \t\te^{i\\th_0}\\cosh 2r_1-e^{-i(\\th_0+\\th_1)} \\sinh 2r_1 & 0 \\\\\n\\end{array} \\right] \\,\\, , \n\\end{array}\\]\n\n\\[ \\begin{array}{cc}\nA_{\\th_0}=& \n \\left[ \\begin{array}{ccc} ir_0^2 & ir_0(e^{-i\\th_0}\\cosh 2r_1+e^{i(\\th_0+\\th_1)} \\sinh 2r_1) \\\\\n \t\t \t\tir_0(e^{i\\th_0}\\cosh 2r_1+e^{-i(\\th_0+\\th_1)} \\sinh 2r_1)& ir_0^2 \\\\\n\\end{array} \\right] \\,\\, .\n\\end{array}\\]\nFor the connection components, $A_{r_0}$ and $A_{\\th_0}$, it is more convenient to use for the variables $\\la$ the decomposition $\\la=x+iy$, with $x$ and $y$ real, resulting into the following components of the connection\n\\[ \\begin{array}{cc}\nA_x=\\cos\\th_0 A_{r_0}-{\\sin \\th_0 \\over r_0}A_{\\th_0}=& \n \\left[ \\begin{array}{ccc} -iy & -(\\cosh 2r_1 - e^{i\\th_1} \\sinh 2r_1) \\\\\n \t\t \t\t\\cosh 2r_1-e^{-i\\th_1} \\sinh 2r_1& -iy \\\\\n\\end{array} \\right] \\,\\, ,\n\\end{array}\\]\n\n\\[ \\begin{array}{cc}\nA_y=\\sin \\th_0 A_{r_0} + {\\cos \\th_0 \\over r_0} A_{\\th_0}=& \n \\left[ \\begin{array}{ccc} ix & i(\\cosh 2r_1 + e^{i\\th_1} \\sinh 2r_1) \\\\\n \t\t \t\ti(\\cosh 2r_1 + e^{-i\\th_1} \\sinh 2r_1) & ix \\\\\n\\end{array} \\right] \\,\\, ,\n\\end{array}\\]\n\n\\[ \\begin{array}{ccc}\nA_{r_1}=& \n \\left[ \\begin{array}{ccc} 0 & 0 \\\\\n \t\t \t 0 & 0 \\\\\n\\end{array} \\right] \\,\\,\\,\\,\\,\\, ,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n%\\end{array}\\]\n%\n%\\[ \\begin{array}{cc}\nA_{\\th_1}=& \n \\left[ \\begin{array}{ccc} 1 & 0 \\\\\n \t\t \t 0 & 3 \\\\\n\\end{array} \\right] {i \\over 4} (\\cosh 4 r_1 -1) \\,\\, ,\n\\end{array}\\]\n\n\\[ \\begin{array}{ccc}\nA_{r_2}=& \n \\left[ \\begin{array}{cccc} 0 & 0 & 0 & -e^{-i\\th_2}\\\\\n \t\t \t 0 & 0 & 0 & 0 \\\\\n\t\t\t\t0 & 0 & 0 & 0 \\\\\n\t\t\t\te^{i\\th_2} & 0 & 0 & 0 \\\\ \n\\end{array} \\right] \\,\\,\\,\\,\\,\\, ,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\nA_{r_3}=& \n \\left[ \\begin{array}{cccc} 0 & 0 & 0 & 0 \\\\\n \t\t \t 0 & 0 & -e^{-i\\th_3} & 0 \\\\\n\t\t\t\t0 & e^{i\\th_3} & 0 & 0 \\\\\n\t\t\t\t0 & 0 & 0 & 0 \\\\ \n\\end{array} \\right] (2 \\cosh ^2 r_2 -1) \\,\\, .\n\\end{array}\\]\nThe components $A_{\\th_2}$ and $A_{\\th_3}$ have more complicated forms that we shall not give explicitly here as they are not necessary for performing universal quantum computation \\cite{UG}.\n\n\\subsection{The Commutators and the Field Strengths $F$\\label{comm}}\n\nIn order to be able to calculate the holonomies \\cite{PAZARA} it is convenient to consider loops $C$ on the planes $(\\sigma_i,\\sigma_j)$\nin ${\\cal M}$,\\footnote{Note that the components ($r_0$,$\\th_0$) have been \nreplaced by ($x$,$y$).} on which the two components of the connection commute with each other \nyet giving a non-trivial holonomy (i.e. they have non-zero field strength component, $F_{\\sigma_i \\sigma_j}=\\p_{\\sigma_i} \nA_{\\sigma_j}-\\p_{\\sigma_j}A_{\\sigma_i}+[A_{\\sigma_i},A_{\\sigma_j}]$).\nIndeed, for $\\hat \\sigma_i$, $i=1,2,3$, denoting the Pauli matrices, we have\n\\bq\n&\n[A_x,A_{r_1}]=0\\,\\,\\,\\,\\, {\\text{with}} \\,\\,\\,\\, \n\\left. F_{xr_1} \\right\|_{\\th_1=0}=-2 i {\\hat \\sigma}_2 \\exp(-2 r_1) \\,\\, ,\n&\n\\no \\no\n&\n[A_y,A_{r_1}]=0\\,\\,\\,\\,\\, {\\text{with}} \\,\\,\\,\\, \n\\left. F_{yr_1} \\right\|_{\\th_1=0}=-2 i {\\hat \\sigma}_1 \\exp(2 r_1) \\,\\, ,\n&\n\\no \\no\n&\n[A_{r_1},A_{\\th_1}]=0\\,\\,\\,\\,\\, {\\text{with}} \\,\\,\\,\\, \nF_{r_1 \\th_1}= - i \\hat{s}_3 \\sinh 4 r_1 \\,\\, ,\n&\n\\no \\no\n&\n[A_{r_2},A_{r_3}]=0\\,\\,\\,\\,\\, {\\text{with}} \\,\\,\\,\\, \n\\left. F_{r_2 r_3} \\right\|_{\\th_2=\\th_3=0}=-2i \\hat \\sigma_2^{12}\\sinh 2 r_2 \\,\\, ,\n&\n\\no \\no\n&\n[A_{r_2},A_{r_3}]=0\\,\\,\\,\\,\\, {\\text{with}} \\,\\,\\,\\, \n\\left. F_{r_2 r_3} \\right\|_{\\th_2=0, \\th_3=3\\pi/2}=-2i \\hat{\\sigma}_1^{12} \\sinh 2 r_2 \\,\\, ,\n&\n\\nonumber\n\\eq\nwhere\n\\bq\n&&\n\\hat{s}_3:= - \\left[ \\begin{array}{cccc} 1 & 0 \\\\\n \t \t\t \t 0 & 3 \\\\\n\\end{array} \\right]\n\\Sp , \\Sp\n\\hat \\sigma _2^{12}:=\n\\left[ \\begin{array}{cccc} 0 & 0 & 0 & 0 \\\\\n \t\t \t 0 & 0 & -i & 0 \\\\\n\t\t\t\t0 & i & 0 & 0 \\\\\n\t\t\t\t0 & 0 & 0 & 0 \\\\\n\\end{array} \\right]\n\\Sp \\text{and} \\Sp\n\\hat \\sigma _1^{12}:=\n\\left[ \\begin{array}{cccc} 0 & 0 & 0 & 0 \\\\\n \t\t \t 0 & 0 & 1 & 0 \\\\\n\t\t\t\t0 & 1 & 0 & 0 \\\\\n\t\t\t\t0 & 0 & 0 & 0 \\\\\n\\end{array} \\right] \\,\\, .\n\\nonumber\n\\eq\nThe above conditions that are satisfied on the planes $\\left.(x,r_1)\\right._{\\th_1=0}$, $\\left.(y,r_1)\\right._{\\th_1=0}$, $(r_1,\\th_1)$, $\\left.(r_2,r_3)\\right._{\\th_2=\\th_3=0}$ and $\\left.(r_2,r_3)\\right._{\\th_2=0, \\th_3=3\\pi/2}$ allow for the explicit calculation of the holonomies for paths restricted on such planes. \n\n\\subsection{The Holonomies $\\Gamma_A(C)$}\n\nIn order to perform universal quantum computation it is necessary to produce at least two independent unitary gates \\cite{Lloyd}. In the following we shall present holonomic gates, which involve (any) one qubit rotations and a special class of (any) two qubit transformations. In detail we have\n\\bq\n&\nC_I \\in \\left. (x,r_1)\\right._{\\th_1=0} \\,\\,\\,\\text{gives}\\,\\,\\, \\Gamma_A(C_I)=\\exp -i\\hat \\sigma_1 \\Sigma_I \\,\\,\\, \\text{with} \\,\\,\\, \\Sigma_I:=\\int_{\\Sigma(C_I)} \\!dxdr_1 2 e^{-2r_1} \\,\\, ,\n&\n\\no \\no\n&\nC_{II} \\in \\left. (y,r_1)\\right._{\\th_1=0} \\,\\,\\, \\text{gives} \\,\\,\\, \\Gamma_A(C_{II})=\\exp -i\\hat \\sigma_2 \\Sigma_{II} \\,\\,\\, \\text{with} \\,\\,\\, \\Sigma_{II}:=\\int_{\\Sigma(C_{II})} \\! dydr_1 2 e^{2r_1} \\,\\, ,\n&\n\\no \\no\n&\nC_{III} \\in (r_1,\\th_1) \\,\\,\\, \\text{gives} \\,\\,\\, \\Gamma_A(C_{III})=\\exp -i\\hat{\\tilde{\\sigma}}_3 \\Sigma_{III} \\,\\,\\, \\text{with} \\,\\,\\, \\Sigma_{III}:=\\int_{\\Sigma(C_{III})} \\! dr_1d\\th_1 \\sinh 4r_1 \\,\\, ,\n&\n\\no \\no\n&\nC_{IV} \\in \\left. (r_2,r_3)\\right._{\\th_2=\\th_3=0} \\,\\,\\,\\text{gives} \\,\\,\\,\\Gamma_A(C_{IV})=\\exp -i\\hat \\sigma_2^{12} \\Sigma_{IV} \\,\\,\\, \\text{with} \\,\\,\\, \\Sigma_{IV}:=\\int_{\\Sigma(C_{IV})} \\! dr_2dr_3 2 \\sinh 2 r_2 \\,\\, ,\n&\n\\no \\no\n&\nC_{V} \\in \\left. (r_2,r_3)\\right._{\\th_2=0, \\th_3=3\\pi/2} \\,\\,\\,\\text{gives} \\,\\,\\, \\Gamma_A(C_{V})=\\exp -i\\hat \\sigma_1^{12} \\Sigma_{V} \\,\\,\\, \\text{with} \\,\\,\\, \\Sigma_{V}:=\\int_{\\Sigma(C_{V})} \\! dr_2dr_3 2 \\sinh 2 r_2 \\,\\, ,\n&\n\\no \\nonumber\n\\eq\nwhere $\\Sigma(C_\\rho)$ with $\\rho=I,...,V$ is the surface on the relevant submanifold $(\\sigma_i,\\sigma_j)$ of ${\\cal M}$ whose boundary is the path $C_\\rho$. The hyperbolic functions in these integrals stem out of the geometry of the $su(1,1)$ manifold associated with the relative control submanifold. The $\\Gamma_A(C)$'s thus generated belong either in the $U(2)$ or $U(4)$ group. Considering the tensor product structure of our system these rotations represent in the $2^m$ space of $m$ qubits respectively single qubit rotations and two qubit interactions, thus resulting into a universal set of logical gates. Their explicit constructions are similar to those presented in \\cite{PAZARA} for the ${\\bf CP}^n$ model.\n\n\\subsection{The $SU(2)$ Control Manifold \\label{su2}}\n\nIn what follows we discuss the employment of $SU(2)$ interferometer as control devices \\cite{Yurke} for producing holonomies. For $a_1$ and $a_2$ the annihilation operator of two different laser beams, consider the Hermitian operators\n\\be\nJ_x={1 \\over 2} (a_1^{\\dagger} a_2 +a_2^\\dagger a_1)\n\\,\\, , \\,\\,\\,\nJ_y=-{i \\over 2} (a_1^{\\dagger} a_2 -a_2^\\dagger a_1)\n\\,\\, , \\,\\,\\,\nJ_z={1 \\over 2} (a_1^{\\dagger} a_1 -a_2^\\dagger a_2)\n\\label{JJJ}\n\\ee\nand\n\\bq\n&&\nN=a_1^{\\dagger} a_1 +a_2^\\dagger a_2=n_1+n_2 \\,\\, .\n\\nonumber\n\\eq\nThe operators (\\ref{JJJ}) satisfy the commutation relations for the Lie algebra of $SU(2)$; $[J_x,J_y]=iJ_z$, $[J_y,J_z]=iJ_x$, $[J_z,J_x]=iJ_y$. The operator $N$, which is proportional to the free Hamiltonian of two laser beams, commutes with all of the $J$'s. On the other hand, however, the Kerr Hamiltonian does not commute with the $J$'s, allowing for the possibility that $SU(2)$ interferometers be used as transformation controllers in view of the holonomic computation. \n\nFrom these operators we obtain the unitaries, $U_x(\\al)=\\exp(i\\al J_x)$, $U_y(\\beta)= \\exp (i \\beta J_y)$ and $U_z(\\gamma)=\\exp(i\\gamma J_z)$. For the degenerate state space of two laser beams spanned by $\|i_1 j_2\\ran$, we have from (\\ref{conn}) and for ${\\cal U}=U_x(\\al)U_y(\\beta)U_z(\\gamma)$ the following connection components\n\\be\nA_\\al ={i \\over 2}\n \\left[ \\begin{array}{cccc} 0 & 0 & 0 & 0 \\\\\n\t\t\t\t0 & \\sin \\beta & \\cos \\beta e^{i\\gamma} & 0 \\\\\n\t\t\t\t0 & \\cos \\beta e^{-i \\gamma} & -\\sin \\beta & 0 \\\\\n\t\t\t\t0 & 0 & 0 & 0 \\\\\n\\end{array} \\right] \n,\\,\\,\\,\nA_\\beta=-{1 \\over 2}\n \\left[ \\begin{array}{cccc} 0 & 0 & 0 & 0 \\\\\n\t\t\t\t0 & 0 & e^{i\\gamma} & 0 \\\\\n\t\t\t\t0 & -e^{-i \\gamma} & 0 & 0 \\\\\n\t\t\t\t0 & 0 & 0 & 0 \\\\\n\\end{array} \\right] \n,\\,\\,\\,\nA_\\gamma=-{i \\over 2}\n \\left[ \\begin{array}{cccc} 0 & 0 & 0 & 0 \\\\\n\t\t\t\t0 & 1 & 0 & 0 \\\\\n\t\t\t\t0 & 0 & -1 & 0 \\\\\n\t\t\t\t0 & 0 & 0 & 0 \\\\\n\\end{array} \\right] .\n\\label{conn1}\n\\ee\nThese components do not commute with each other, when projected on planes with non-trivial field strength. Hence, it is not possible to employ again the method used in the previous section to calculate the holonomies of paths in the three dimensional control parameter space, $(\\al, \\beta,\\gamma)$. Instead, for this purpose we may employ the non-Abelian Stokes theorem \\cite{Karp}. The extra limitation, now, for the choice of the path comes from the constraint that, apart from being confined on a special two dimensional subspace, it has to have the shape of an orthogonal parallelogram with two sides lying along the coordinate axis. This will facilitate the extraction of an analytic result from the Stokes theorem. Though, experimentally, this restriction poses additional (possibly minor) difficulties, theoretically it leads to the very interesting possibility of a direct calculation of non-Abelian holonomies without resorting to their Abelian substructures. Note that the application of the Stokes theorem for the evaluation of the holonomies in the previous section gives the same results, as expected.\n\nTo state the non-Abelian Stokes theorem let us first present some preliminaries, where a few simplifications are introduced, as its general form will not be necessary in the present work. \n\n\\begin{figure}[h]\n \\epsffile{loop.eps}\n \\caption[contour]{\\label{loop}\nThe loop $C$ for the non-Abelian Stokes theorem.\n }\n\\end{figure}\n\nConsider the Wilson loop (holonomy), $W={\\bf P}\\exp \\oint_C A$, of the loop $C$ given in Fig. \\ref{loop}, with connection $A$, made out of the Wilson lines $W_i$ for $i=1,...,4$, as $W=W_4 W_3 W_2 W_1$. $(\\sigma,\\tau)$ is a reparametrization of the plane where the loop $C$ lies. Define $T^{-1}(\\sigma,\\tau)=W_4 W_3$. Then, for $F_{\\sigma\\tau}$ the field strength of the connection $A$ on the plane $(\\sigma,\\tau)$, $W$ is given in terms of a surface integral\n\\bq\n&&\nW={\\bf P}_\\tau e^{\\int_\\Sigma T^{-1}(\\sigma,\\tau)F_{\\sigma \\tau}(\\sigma,\\tau)T(\\sigma,\\tau) d\\sigma d\\tau} \\,\\, ,\n\\nonumber\n\\eq\nwhere ${\\bf P}_\\tau$ is the path ordered symbol with respect only to the $\\tau$ variable, contrary to the usual path ordering symbol {\\bf P}, which is with respect to both variables, $\\sigma$ and $\\tau$. Here $F_{\\sigma \\tau}(\\sigma,\\tau)=-\\p_\\sigma A_\\tau +\\p_\\tau A_\\sigma+[A_\\sigma , A_\\tau]$.\n\nFrom the connection given in (\\ref{conn1}) the following holonomies are derived. For a closed rectangular loop $C_1\\in (\\al,\\beta)$-plane with coordinates $(0,0),(\\al \\! = \\! \\pi,0),(\\al \\! = \\! \\pi,\\beta),(0,\\beta)$ we obtain the following unitary transformation\n\\be\n\\Gamma_A(C_1)=\\exp (-i2\\beta \\hat \\sigma_2^{12}) \\,\\, .\n\\label{C1}\n\\ee\nIn addition for a rectangular loop $C_2\\in (\\al,\\gamma)$-plane with coordinates $(0,0),(\\al \\! = \\! \\pi,0),(\\al \\! = \\! \\pi,\\gamma),(0,\\gamma)$ we obtain the holonomy\n\\be\n\\Gamma_A(C_2)=\\exp (-i2\\gamma \\hat \\sigma_3^{12}) \\,\\, ,\n\\ee\nwhere the matrix $\\hat \\sigma_3^{12}$ is defined similarly to $\\hat \\sigma_1^{12}$ and $\\hat \\sigma_2^{12}$ in Subsection \\ref{comm}. These operations can be implemented by using interferometers between {\\it any} two laser beams. Note that the coefficients in front of the matrices in the unitaries are areas on spheres spanned by the angles $\\al$ and $\\beta$ or $\\al$ and $\\gamma$. This is consistent with the geometry of $SU(2)$. \n\nThese two matrices can produce any unitary transformation of {\\it one} qubit encoded in a sub-space of states of the two laser beams spanned by $\\{\|01\\ran,\|10\\ran\\}$. In other words, we need two laser beams to encode one qubit, contrary to previous construction. As these transformations can be performed between any two beams, we can generate interaction transformations between two qubits, resulting finally (together with the one qubit rotations) into a universal set of transformations. For example the SWAP two qubit gate given by\n\\bq\n&&\nU_{SWAP}= \\left[ \\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\\n\t\t\t \t\t0 & 0 & 1 & 0 \\\\\n\t\t\t\t\t0 & 1 & 0 & 0 \\\\\n\t\t\t\t\t0 & 0 & 0 & 1 \\\\\n\\end{array} \\right] \\,\\, ,\n\\nonumber\n\\eq\nis achieved as follows. On four arbitrary laser beams $1,2,3$ and $4$ with $\\{\|01\\ran,\|10\\ran\\}_{1,2}$ encoding the one qubit and $\\{\|01\\ran,\|10\\ran\\}_{3,4}$ encoding the other, we may act with $\\left.\\Gamma_A(C_1)\\right\|_{\\beta={\\pi \\over 4}}$ between beams $1$ and $3$ and with $\\left.\\Gamma_A(C_1)\\right\|_{\\beta={3 \\pi \\over 4}}$ between $2$ and $4$ producing eventually the $U_{SWAP}$ gate. The loop $C_1$ is defined as in (\\ref{C1}).\n\nThis model facilitates the physical implementation as it will be seen in the following section.\n\n\\section{Towards experimental implementation}\n\nWe address here the task of combining the theoretical requirements of HQC together with the features of the ``experimental'' components described in the previous two sections. While the Abelian holonomies have been produced in the laboratory by various means, the non-Abelian ones are more complicated. However, the holonomies calculated above, require successive restrictions on two dimensional planes of the control parameter space, quite in the same way as one needs to do to generate Abelian Berry phases. This constructive method may prove experimentally advantageous for performing and measuring non-Abelian holonomies. A survey over some Berry phase experiments in optics is given below.\n\n\\subsection{Various Abelian Berry Phase Setups}\n\nPhotons can be seen as massless spin-1 bosons. This characteristic has been the\ndriving force \nfor the optical manifestation of the Berry phase with respect to the polarization \nquantum numbers \\cite{Simon}.\nNecessary condition for the generation of this phase factor is the\nadiabatic\nchange of the direction of the photon propagation. Various optical experiments\nhave been performed. Results at the classical level have been reported in \\cite{Chiao}, for the\ncase of a single mode in a wounded optical fiber, whereas quantum mechanically, in \\cite{Kwiat},\nthe case of a single photon has been treated.\nOf special importance, for our case, is the latter experiment where the\nBerry phase has been observed\nat quantum optical level. In this case the incident\nlight is prepared in an entanglent state \n\\bq\n&&\n\|\\psi\\rangle_{in} = \\int A(E')\|n \\rangle_{E'} \|n\\rangle_{E-E'} dE' \\,\\, ,\n\\nonumber\n\\eq\nwhere $A(E')=A(E-E')$ is the complex probability amplitude for finding one\nphoton with an \nenergy $E'$ ($\|n =1\\rangle_{E'}$) or with an energy $E-E'$ ($\|n=1\\rangle_{E-E'}$).\nThis type of states can be \nproduced in the lab by driving a single-mode ultraviolet laser\ninto a $\\chi ^{(2)}$ nonlinear optical crystal. A Michelson interferometer\nhas been used for the observation of the phase in the output state. It was\nfound that the output state (photons in essentially $n$=1 Fock states) had\nan extra phase factor due to the optical-path-length difference $\\Delta L$\nof the interferometer plus the contribution of the Berry phase. \nThe form of such state is given by\n\\bq\n&&\n\|\\psi\\rangle_{out} = \\frac{1}{\\sqrt{2}} \\int A(E')\| n\\rangle_{E'}\n\|n\\rangle_{E-E'} \\{1+\\exp[i\\phi (E-E')]\\} dE' \\,\\, ,\n\\nonumber\n\\eq\nwhere $\\phi (E-E') = 2\\pi\\Delta L / \\lambda_{E-E'} + \\phi_{Berry}$, with $\\phi_{Berry}$ the geometrical phase predicted theoretically.\n\nRecently, an alternative approach to the geometric phase \nhas been considered \\cite{Jackiw}, through squeezed\nstates of photons. Squeezed states have been \nfound considerably interesting in the field of quantum optics for \nvarious reasons, as for example, the noise reduction which is\nnecessary for practical applications with noise sensitivity. \n \nDisplacement and squeezing give different contributions to the Berry\nphase of the Fock states $\|\\nu\\ran$.\nFor the case of squeezing one finds that this contribution is given \nby\n\\bq\n&&\n\\phi_{Berry}^n = \\frac{2n+1}{4} \\oint \\left( \\cosh 4r_1 -1 \\right)dr_1 \\,\\, .\n\\nonumber\n\\eq\nSuch Berry phase agrees with the form of the diagonal connection $A_{\\theta_1}$ in Subsection \\ref{AAA}, as it was to be expected. On the other hand if we perform a loop in the control parameters of the displacing device we expect the following Berry phase to arise\n\\bq\n&&\n\\phi_{Berry}^n = \\oint (ydx-xdy) \\,\\, .\n\\nonumber\n\\eq\nThe equivalent connection of displacing in the Kerr medium ($A_x$ and $A_y$ with $r_1=0$, in Subsection \\ref{AAA}) are non-diagonal matrices, whose holonomy cannot be calculated easily. In fact, as it is observed by the numerical simulations in the following, the phase factors produced in front of $\|0\\ran$ and $\|1\\ran$ are not equal, due to the off-diagonal elements of $A_x$ and $A_y$. This effect is related with the degeneracy structure of each model.\n\n\\subsection{Free Hamiltonian and Kerr Medium}\n\nIn the previous sections we used the Kerr non-linear Hamiltonian in order to produce the degenerate eigenspace spanned by $\|0\\ran$ and $\|1\\ran$. The full Hamiltonian of the system is the combined one of the free photons and the non-linear medium, i.e. $H_{Tot}=H_{Free}+H_{Kerr}=\\hbar \\omega n +\\hbar X n(n-1)$. Of course the first part lifts the degeneracy of $\|0\\ran$ and $\|1\\ran$ destroying the basic requirements for the holonomic computation. In order to overcome this problem we resort to the following constructions.\n\nConsidering $H_{Free}$ as unperturbed Hamiltonian and the non-linear part as the interaction term we may move to the interaction picture of the full system, with $H_I= H_{Kerr}$. The rotation to the interaction picture may be incorporated in the devices used for the external control resulting in a redefinition of their control parameters.\n\nAlternatively, we may define a one dimensional lattice with points on the trajectory of the laser. As the free Hamiltonian is acting only on the state $\|1\\ran$ changing its phase by $e^{-iH_{Free} \\Delta t}=e^{-i \\hbar \\omega \\Delta x /c}$, with $c$ the speed of light, we may single out the points $x_k =2\\pi c k /(\\hbar \\omega)$ for $k$ integers. On these points the phase is trivial and it does not contribute to the state. Hence, $\|0\\ran$ and $\|1\\ran$ are degenerate on this lattice \\cite{Kitano}. \n\nIn Subsection \\ref{su2} we have introduced $SU(2)$ interferometers as control devices. The $su(2)$ operators commute with $H_{Free}$ allowing the effect of the free Hamiltonian to factorize out of the whole control procedure. At the end of the algorithm the detectors may be placed on a point of the degenerate lattice in order to avoid the dynamical phase produced by $H_{Free}$ on the states $\|0\\ran$ and $\|1\\ran$. Even though in the $SU(2)$ model each qubit is encoded with the help of two laser beams increasing in this way the necessary resources, it overcomes the problem of the degeneracy in the most efficient way.\n\n\\subsection{Holonomies and Devices}\n\nFor the implementation of the continuous adiabatic loops we should adopt the kick method described in \\cite{PAZARA}, \\cite{VIO} and \\cite{Vitali}. A general state $\|\\psi\\ran$ in the degenerate eigenspace of $H_0=H_{Kerr}$ is given as a linear combination of $\|0\\ran$ and $\|1\\ran$. Under an isospectral cyclic evolution of the Hamiltonian in the family ${\\cal F}$, the evolution operator acting on $\|\\psi\\ran$ is given by the $2 \\times 2$ submatrix in the upper left corner of\n\\bq\n&&\nU(0,T)={\\bf T} \\exp -i \\int ^T_0 {\\cal U}(\\sigma (t))H_0 {\\cal U}^\\dagger(\\sigma(t))dt \\,\\, .\n\\nonumber\n\\eq\nThis evolution takes place from time $0$ to time $T$ and, for performing a closed loop, we demand $\\sigma(0)=\\sigma(T)$. By dividing the time interval, $[0,T]$, into $m$ equal segments $\\Delta t$ we may approximate the above operator by\n\\bq\n&&\nU(0,T)\\approx{\\bf T} \\prod_{i=1}^{m} {\\cal U}_ie^{-iH_0 \\Delta t} {\\cal U}_i^\\dagger \\,\\,\\,\\,\\,\\,\\,\\, \\text{with} \\,\\,\\,\\, {\\cal U}_i={\\cal U}(\\sigma_i)={\\cal U}(\\sigma(t_i)) \\,\\, .\n\\nonumber\n\\eq\nAssuming the evolutions ${\\cal U}_{i+1}^{\\dagger}{\\cal U}_i$ to be a very small rotation and restricting to evolutions which remain in the zeroth degenerate eigenspace we might once more derive the holonomy operator $\\Gamma_A(C)$ for $A$ defined in (\\ref{conn}). We prefer instead to see what the effect of finitely many devices would be, when acting on the space of states of the qubits (the lasers). \n\nFor the sake of concreteness we work out examples in terms of displacing devices $D(\\la)$, performing a closed loop in their control parameters $\\la$. This is shown in Fig. \\ref{trig}, where for simplicity the least possible number of displacing devices (three) for performing a closed loop has been considered. Two displacing unitaries are combined as $D(\\lambda)D(\\lambda')=\\exp{(i \\Im (\\lambda \\bar \\lambda'))} D(\\la+\\la')$. The physical process behind this is as follows. On the state $\|\\psi\\ran$ first acts a displacing device with unitary $D^\\dagger(\\la_1)$, taking it to the point $\\la_1$. Then, the evolution operator of the Kerr Hamiltonian acts for a time interval $\\Delta t= T/3$ $U(\\Delta t)=\\exp( -i H_0 \\Delta t)$. This effect is achieved by propagating the beam inside a Kerr medium. Then, the evolution $D^{\\dagger}(\\la_2)D(\\la_1)$ is performed. This is achieved, with a single displacing device, given (up to an overall phase factor that will cancel at the end) by $D(\\la_1-\\la_2)$. After exiting the displacing device (we are at point $\\la_2$) the beam enters a Kerr medium for time $\\Delta t$ and then the procedure is repeated until we come back to the point $\\la_1$ and the beam enters once more the Kerr medium. Finally, the state is thus displaced by $D(\\la_1)$. This loop may be transported to any other place of the control parameter complex plane by acting at the beginning and at the end of this procedure with the appropriate displacing unitary (device).\n\\begin{figure}[h]\n \\epsffile{trig.eps}\n \\caption[contour]{\\label{trig}\nThe triangular (and polygonal) loop $C$ on the complex plane of the displacing control parameters, $\\la$, approximating the circle.\n }\n\\end{figure}\nIn this case the evolution operator is approximated by \n\\bq\n&&\nU(0,T)\\approx D(\\la_1) \\left( U(\\Delta t;0) U(\\Delta t;\\tilde \\la_1 +\\tilde \\la_2 ) U(\\Delta t; \\tilde \\la_1) U(\\Delta t;0) \\right) D^\\dagger(\\la_1) \\,\\, ,\n\\nonumber\n\\eq\nwhere $U(\\Delta t;\\tilde \\la) =D(\\tilde \\la) U(\\Delta t) D^\\dagger(\\tilde \\la)$, $\\tilde \\la_i=\\la_{i+1}-\\la_i$ and $\\la_4=\\la_1$.\n\nAccording to the above analysis, we proceed to the numerical simulation of a system with various numbers of displacing devices represented by different polygons on the control complex plane (see Fig. \\ref{trig}). We start with a pentagon which demands five displacers. Fig. \\ref{sub} represents the absolute values of the (0,0), (0,1), (1,0), (1,1) elements of the evolution operator $U(0,T)$ as functions of the number of displacers used to approximate a cyclic evolution. These are the relevant elements for the evolution of the states in the degenerate eigenspace describing a qubit. The parameters involved are taken to be $T=0.1$ and $\\hbar X=1$, with the radius of the circle equal to 1. The initial point is taken to be the origin of the complex plane rather than $\\la_1$, or in other words we do not perform the initial and final displacings by $D^\\dagger(\\la_1)$ and $D(\\la_1)$.\n\\begin{figure}[h]\n \\epsffile{sub4.ps}\n \\caption[contour]{\\label{sub}\nThe absolute values $\|U_{00}(0,T)\|$, $\|U_{01}(0,T)\|$, $\|U_{10}(0,T)\|$ and $\|U_{11}(0,T)\|$ as functions of the number of sides of the polygons.\n }\n\\end{figure}\nIn the table below are depicted the percent deviations of those values obtained with 5, 10, 20 and 26 displacers with respect to the ones obtained with 100 displacers.\n\\begin{center}\n\\begin{tabular}{\|c\|\|c\|c\|c\|c\|} \\hline\n & 5 & 10 & 20 & 26 \\\\ \\hline \\hline\n\\Sp 00 \\Sp & \\Sp 0.2419 \\% \\Sp & \\Sp 0.0595 \\% \\Sp & \\Sp 0.0149 \\% \\Sp & \\Sp 0.0099 \\% \\Sp \\\\ \\hline\n\\Sp 01 \\Sp & 0.9119 \\% & 0.2260 \\% & 0.0558 \\% & 0.0186 \\% \\\\ \\hline\n\\Sp 10 \\Sp & 0.9119 \\% & 0.2260 \\% & 0.0558 \\% & 0.0186 \\% \\\\ \\hline\n\\Sp 11 \\Sp & 1.6763 \\% & 0.4061 \\% & 0.0760 \\% & 0.0269 \\% \\\\ \\hline\n\\end{tabular}\n\\end{center}\nWe see that with 26 displacers the error is of the order of 1 in $10^4$ acceptable for quantum computation with error correction. This provides an indication for the necessary number of devices needed in order to reproduce faithfully the holonomic adiabatic loop. \n\n\\section{Conclusions}\n\nThe implementation of HQC in the frame of quantum optics has provided novel insight into many technical aspects of the theory. Moreover, the components demanded for it are widely used in the laboratories. The possibility of overcoming the difficulties in combining them in the appropriate way for obtaining holonomies is an open problem to be faced by experimenters. \n\nIn summary the main quantum computational features we observed in our model are the following. First, the {\\it universality} condition is proven explicitly, stemming out of the ability to construct holonomies representing any possible logical gate. This is achieved by combining one qubit rotations (realized by displacing and squeezing devices) and two qubit transformations (by interferometers) between any two qubits. Second, the setup exhibits quantum entanglement, having built in tensor product structure as it consists of a multi partite system. This resolves the problem of {\\it complexity} posed in \\cite{PAZARA} which is one of the main features which make QC's more efficient than classical ones. Third, the degenerate space of the Hamiltonian eigenstates, which is used to write the code is constructed out of laser beams each with a two dimensional degenerate space. So the demand of using a big degenerate space to write useful codes is performed not by resorting to one system with very large degeneracy, which is almost impossible to realize in nature, but by adding up the 2-dimensional subspaces of the lasers. This is the characteristic of {\\it scalability} of the proposed model. Fourth, the chosen loops associated with the given holonomies are restricted on specific planes $(\\sigma_i,\\sigma_j)$ of two control parameters $\\sigma_i$ and $\\sigma_j$, exactly in the same way as used for the production of {\\it Abelian Berry phases}. The latter has been verified in several theoretical and experimental applications in optics \\cite{Simon,Chiao,Kwiat} and elsewhere \\cite{ELSE}. From these phase transformations $U(1)$ of different components of the system we are able to obtain with proper combinations any desired $U(2^m)$ transformation. Since there exist experimental measurements of the Berry phase, it is plausible to expect the implementation of the $U(2^m)$ holonomic transformations.\n\nA further final advantage of the holonomic setup is that it is confined in the degenerate eigenspace produced by $\\{\|0\\ran,\|1\\ran\\}$, describing one qubit. Entanglement of these states with the non-degenerate ones in the course of application of the logical gates does not occur due to the adiabaticity requirement. The initial control operators we use here, $D$, $S$, $M$ and $N$ in general mix all the states of the Fock space, but at the end of the loop, only rotations between the degenerate eigenstates will be accounted for.\n\nThe possibility to observe the proposed holonomies in the laboratory or even perform specific logical gates is a demanding task and an open question for the future.\n\n\\section{Acknowledgements}\n\nWe would like to thank Mario Rasetti, Paolo Zanardi and Matteo Paris for inspiring conversations. This work was supported in parts by TMR Network under the condract no. ERBFMRXCT96 - 0087.\n\n\\begin{references}\n\n\\bibitem{ZARA} P. Zanardi and M. Rasetti, to appear in Phys. Lett. A, quant-ph/9904011. For related works see \nA. Kitaev, quant-ph/9707021; J. A. Jones, V. Vedral, A. Ekert and G. Castagnoli, quant-ph/9910052;\nK. Fujii, quant-ph/9910069.\n\n\\bibitem{PAZARA} J. Pachos, P. Zanardi and M. Rasetti, \nto appear in Phys. Rev. A (Rapid Comm.), quant-ph/9907103.\n\n\\bibitem{QC} For reviews, see D.P. DiVincenzo, {\\sl Science} {\\bf\n270}, 255 (1995); A. Steane, Rep. Prog. Phys. {\\bf 61}, 117 (1998).\n\n\\bibitem{Chuang} I. L. Chuang and Y. Yamamoto, quant-ph/9505011.\n\n\\bibitem{SHWI} For a review see, {\\em Geometric Phases in Physics}, A. Shapere and F. Wilczek, Eds.\nWorld Scientific (1989).\n\n\\bibitem{WIZE} F. Wilczek and A. Zee, Phys. Rev. Lett. {\\bf {52}}, 2111 (1984).\n\n\\bibitem{NAK} M. Nakahara, {\\em Geometry, Topology and Physics}, IOP Publishing Ltd. (1990).\n\n\\bibitem{Kral} V. Buzek and P.L Knight, in Progress in Optics XXXIV, E. Wolf (North\nHolland, Amsterdam) (1995); P. Kral, Phys. Rev. A, {\\bf 42}, 4177 (1990), J. Mod. Opt. {\\bf 37}, 889 (1990);\nC. F. Lo, Phys. Rev. A {\\bf 43}, 404 (1991); M. G. A. Paris, Phys. Lett. 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A (Rapid Comm.), quant-ph/9907103." }, { "name": "quant-ph9912093.extracted_bib", "string": "{QC For reviews, see D.P. DiVincenzo, {\\sl Science {270, 255 (1995); A. Steane, Rep. Prog. Phys. {61, 117 (1998)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Chuang I. L. Chuang and Y. Yamamoto, quant-ph/9505011." }, { "name": "quant-ph9912093.extracted_bib", "string": "{SHWI For a review see, {\\em Geometric Phases in Physics, A. Shapere and F. Wilczek, Eds. World Scientific (1989)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{WIZE F. Wilczek and A. Zee, Phys. Rev. Lett. {52, 2111 (1984)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{NAK M. Nakahara, {\\em Geometry, Topology and Physics, IOP Publishing Ltd. (1990)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Kral V. Buzek and P.L Knight, in Progress in Optics XXXIV, E. Wolf (North Holland, Amsterdam) (1995); P. Kral, Phys. Rev. A, {42, 4177 (1990), J. Mod. Opt. {37, 889 (1990); C. F. Lo, Phys. Rev. A {43, 404 (1991); M. G. A. Paris, Phys. Lett. A, {217, 78 (1996)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Hong J. I. Cirac, R. Blatt, A. S. Parkins and P. Zoller, Phys. Rev. Lett., {70, 762 (1993); T. Pellizzari and H. Ritsch, Phys. Rev. Lett., {72, 3973 (1994); T. Pellizzari and H. Ritsch, Phys. Rev. Lett., {72, 3973 (1994); M. G. A. Paris, M. B. Plenio, S. Bose, D. Jonathan and G. M. D'Ariano, quant-ph/9911036." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Bishop R. F. Bishop and A. Vourdas, J. Phys. A, {20, 3743 (1987), Phys. Rev. A, {50, 4488 (1994)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Yurke B. Yurke, S. L McCall and J. R. Klauder, Phys. Rev. A, {33, 4033 (1986); C. Brif and A. Mann, Phys. Rev. A, {54, 4505 (1996); C. Brif and Y. Ben-Aryeh, Quant. Semiclass. Opt., {8, 1 (1996)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Perelomov A. Perelomov, {\\em Generalized Coherent States and their Applications, Springer-Verlag (1986)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{UG D. Deutsch, A. Barenco and A. Ekert, Proc. R. Soc. London { A, {449, 669 (1995); D.P. Di Vincenzo, Phys. Rev. A, {50, 1015 (1995)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Lloyd S. Lloyd, Phys. Rev. Lett., {75, 346 (1995)." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Karp R. Karp, F. Mansouri and J. Rno, to appear in Jour. Math. Phys., hep-th/9910173." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Simon A. Simon, Phys. Rev. Lett., {51, 2167 (1983); J. N. Ross, Opt. Quantum Electron. {16, 455 (1984); P. Facchi and S. Pascazio, submitted to Acta Physica Slovaca, quant-ph/9904082." }, { "name": "quant-ph9912093.extracted_bib", "string": "{Chiao R. Y. Chiao and Y-S. Wu, Phys. Rev. Lett., {57, 933 (1986); A. Tomita and R. Y. Chiao, Phys. Rev. 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Lett., {56, 1003 (1986); G. Delacr\\'etaz et al, Phys. Rev. Lett., {56, 2598 (1986)." } ]
quant-ph9912094	Coherent states for a particle on a sphere	[ { "author": "K Kowalski and J Rembieli\\'nski" } ]	The coherent states for a particle on a sphere are introduced. These states are labelled by points of the classical phase space, that is the position on the sphere and the angular momentum of a particle. As with the coherent states for a particle on a circle discussed in Kowalski K {\em et al\/ 1996 {\em J. Phys. A {29 4149, we deal with a deformation of the classical phase space related with quantum fluctuations. The expectation values of the position and the angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space. The correctness of the introduced coherent states is illustrated by an example of the rotator.	[ { "name": "coherent.tex", "string": "%Format: LaTeX\n\\documentstyle[fleqn,iopfts,12pt]{ioplppt}\n\\jl{1}\n\\textheight 8.8in\n%\\voffset=-2.5cm \\hoffset=.5cm\n\\voffset=.5cm \\hoffset=.5cm %PS\n\\eqnobysec\n\\begin{document}\n\\title{Coherent states for a particle on a sphere}\n\\author{K Kowalski and J Rembieli\\'nski}\n\\address{Department of Theoretical Physics, University\nof \\L\\'od\\'z, ul.\\ Pomorska 149/153,\\\\ 90-236 \\L\\'od\\'z,\nPoland}\n\\begin{abstract}\nThe coherent states for a particle on a sphere are introduced.\nThese states are labelled by points of the classical phase space, that is\nthe position on the sphere and the angular momentum of a particle.\nAs with the coherent states for a particle on a circle discussed in\nKowalski K {\\em et al\\/} 1996 {\\em J. Phys. A} {\\bf 29} 4149, we deal\nwith a deformation of the classical phase space related with quantum\nfluctuations. The expectation values of the position and the\nangular momentum in the coherent states are regarded as the best\npossible approximation of the classical phase space. The correctness\nof the introduced coherent states is illustrated by an example of\nthe rotator.\n\\end{abstract}\n\\pacs{02.20.Sv, 02.30.Gp, 02.40.-k, 03.65.-w, 03.65.Sq}\n\\section{Introduction}\nIt has become a clich\\'e to say that coherent states abound in\nquantum physics \\cite{1}. Moreover, it turns out that they can also\nbe applied in the theory of quantum deformations \\cite{2} and even\nin the theory of classical dynamical systems \\cite{3}.\n\nIn spite of the fact that the problem of the quantization of a\nparticle motion on a sphere is at least seventy years old, there still\nremains an open question concerning \nthe coherent states for a particle on a sphere. Indeed, the\ncelebrated spin coherent states introduced by Radcliffe \\cite{4} and\nPerelomov \\cite{5} are labelled by points of a sphere, i.e., the\nelements of the configuration space. On the other hand, it seems\nthat as with the standard coherent states, the coherent states for a\nparticle on a sphere should be marked with points of the phase\nspace rather than the configuration space.\n\nThe aim of this work is to introduce the coherent states for a quantum\nparticle on the sphere $S^2$, labelled by points of\nthe phase space, that is the cotangent bundle $T^S^2$. The construction\nfollows the general scheme introduced in \\cite{6} for the case of the motion\nin a circle, based on the polar decomposition of the operator\ndefining via the eigenvalue equation the coherent states. From the technical\npoint of view our treatment utilizes both the\nBarut-Girardello \\cite{7} and Perelomov approach \\cite{5}. Namely, as with\nthe Barut-Girardello approach the coherent states are defined as the\neigenvectors of some non-Hermitian operators. On the other hand, in\nanalogy to the Perelomov formalism those states are generated from\nsome ``vacuum vector'', nevertheless in opposition to the\nPerelomov group-theoretic construction, the coherent states are\nobtained by means of the non-unitary action.\n\nIn section 2 we recall the construction of the coherent states for\na particle on a circle. Sections 3--6 are devoted to the\ndefinition of the coherent states for a particle on a sphere\nand discussion of their most important properties. For an easy\nillustration of the introduced approach we study in section 7 the\ncase with the free motion on a sphere.\n\\section{Coherent states for a particle on a circle}\nIn this section we recall the basic properties of the coherent\nstates for a particle on a circle introduced in \\cite{6}.\nConsider the case of the free motion in a circle. For the sake of\nsimplicity we assume that the particle has unit mass and it moves in\na unit circle. The classical Lagrangian is\n\\begin{equation}\n%<2.1>\nL = \\hbox{$\\scriptstyle1\\over2$}\\dot \\varphi^2,\n\\end{equation}\nso the angular momentum canonically conjugate to the angle $\\varphi$\nis given by\n\\begin{equation}\n%<2.2>\nJ = \\frac{\\partial L}{\\partial \\dot \\varphi}=\\dot \\varphi,\n\\end{equation}\nand the Hamiltonian can be written as\n\\begin{equation}\n%<2.3>\nH = \\hbox{$\\scriptstyle1\\over2$}J^2.\n\\end{equation}\nEvidently, we have the Poissson bracket of the form\n\\begin{equation}\n%<2.4>\n\\{\\varphi,J\\} = 1,\n\\end{equation}\nimplying accordingly to the rules of the canonical\nquantization the commutator\n\\begin{equation}\n%<2.5>\n[\\hat\\varphi,\\hat J] = i,\n\\end{equation}\nwhere we set $\\hbar=1$. The operator $\\hat\\varphi$ does not take\ninto account the topology of the circle and (2.5) needs very subtle\nanalysis. The better candidate to represent the position of the\nquantum particle on the unit circle is the unitary operator $U$\n\\begin{equation}\n%<2.6>\nU = e^{i\\hat\\varphi}.\n\\end{equation}\nIndeed, the substitution $\\hat\\varphi\\to\\hat\\varphi+2n\\pi$ does not\nchange $U$, i.e.\\ $U$ preserves the topology of the circle. The operator\n$U$ leads to the algebra\n\\begin{equation}\n%<2.7>\n[\\hat J,U] = U,\n\\end{equation}\nwhere $U$ is unitary. Consider the eigenvalue equation\n\\begin{equation}\n%<2.8>\n\\hat J\|j\\rangle = j\|j\\rangle.\n\\end{equation}\nUsing (2.7) and (2.8) we find that the operators $U$ and\n$U^\\dagger $ are the ladder operators, namely\n\\numparts\n%<2.9>\n\\begin{eqnarray}\nU\|j\\rangle &=& \|j+1\\rangle,\\\\\nU^\\dagger \|j\\rangle &=& \|j-1\\rangle.\n\\end{eqnarray}\n\\endnumparts\nDemanding the time-reversal invariance of representations of the\nalgebra (2.7) we conclude \\cite{6} that the eigenvalues $j$ of the operator\n$\\hat J$ can be only integer (boson case) or half-integer (fermion case).\n\n\nWe define the coherent states $\|\\xi\\rangle$ for a particle on a circle\nby means of the eigenvalue equation\n\\begin{equation}\n%<2.10>\nZ\|\\xi\\rangle = \\xi\|\\xi\\rangle,\n\\end{equation}\nwhere $\\xi$ is complex. In analogy to the eigenvalue equation\nsatisfied by the standard coherent states $\|z\\rangle$ \\cite{8,9} with complex\n$z$, of the form\n\\begin{equation}\n%<2.11>\ne^{i\\hat a}\|z\\rangle = e^{iz}\|z\\rangle,\n\\end{equation}\nwhere $\\hat a\\sim \\hat q+i\\hat p$ is the standard Bose\nannihilation operator and $\\hat q$ and $\\hat p$ are the\nposition and momentum operators, respectively, we set\n\\begin{equation}\n%<2.12>\nZ := e^{i(\\hat \\varphi + i\\hat J)}.\n\\end{equation}\nHence, making use of the Baker-Hausdorff formula we get\n\\begin{equation}\n%<2.13>\nZ = e^{-\\hat J + \\hbox{$\\frac{1}{2}$}}U.\n\\end{equation}\nWe remark that the complex number $\\xi$ should parametrize the\ncylinder which is the classical phase space for the particle\nmoving in a circle. The convenient parametrization of $\\xi$\nconsistent with the form of the operator $Z$ such that\n\\begin{equation}\n%<2.14>\n\\xi = e^{-l + i\\varphi}.\n\\end{equation}\narises from the deformation of the circular cylinder by means of the\ntransformation\n\\begin{equation}\n%<2.15>\nx=e^{-l}\\cos\\varphi,\\qquad y=e^{-l}\\sin\\varphi,\\qquad z=l.\n\\end{equation}\nThe coherent states $\|\\xi\\rangle$ can be represented as\n\\begin{equation}\n%<2.16>\n\|\\xi\\rangle = e^{-(\\ln\\xi) \\hat J}\|1\\rangle,\n\\end{equation}\nwhere\n\\begin{equation}\n%<2.17>\n\|1\\rangle =\n\\sum_{j=-\\infty}^{\\infty}e^{-\\frac{j^2}{2}}\|j\\rangle.\n\\end{equation}\nThe coherent states satisfy\n\\begin{equation}\n%<2.18>\n\\frac{\\langle \\xi\|\\hat J\|\\xi\\rangle}{\\langle\n\\xi\|\\xi\\rangle}\\approx l,\n\\end{equation}\nwhere the maximal error arising in the case $l\\to0$ is of order\n$0.1$ per cent and we have the exact equality in the case with $l$\ninteger or half-integer. Therefore, $l$ can be identified with the\nclassical angular momentum. Furthermore, we have\n\\begin{equation}\n%<2.19>\n\\frac{\\langle \\xi\|U\|\\xi\\rangle}{\\langle \\xi\|\\xi\\rangle}\n \\approx\ne^{-\\frac{1}{4}}e^{i\\varphi}.\n\\end{equation}\nIt thus appears that the average value of $U$ in the normalized\ncoherent state does not belong to the unit circle. On introducing\nthe relative average of $U$ of the form\n\\begin{equation}\n%<2.20>\n\\frac{\\langle U\\rangle_{\\xi}}{\\langle U\\rangle_{\\eta}} :=\n\\frac{\\langle \\xi\|U\|\\xi\\rangle}{\\langle \\eta\|U\|\\eta\\rangle},\n\\end{equation}\nwhere $\|\\xi\\rangle$ and $\|\\eta\\rangle$ are the normalized\ncoherent states, we find\n\\begin{equation}\n%<2.21>\n\\frac{\\langle U\\rangle_{\\xi}}{\\langle U\\rangle_1}\n\\approx e^{i\\varphi}.\n\\end{equation}\nFrom (2.21) it follows that that the relative expectation\nvalue $\\langle U\\rangle_{\\xi}/\\langle U\\rangle_1$ is the\nmost natural candidate to describe the average position of a\nparticle on a circle and $\\varphi$ can be regarded as the\nclassical angle.\n\n\nWe remark that the coherent states on the circle have been\nrecently discussed by Gonz\\'ales {\\em et al\\/} \\cite{10}. In spite\nof the fact that they formally generalize the coherent states\ndescribed above, the ambiguity of the definition of those states\nmanifesting in their dependence on some extra parameter, can be\navoided only by demanding the time-reversal invariance mentioned\nearlier, which leads precisely to the coherent states introduced in\n\\cite{6}. Since the time-reversal symmetry seems to be fundamental\none for the motion of the classical particle in a circle and makes\nthe quantization unique, therefore the generalization of the coherent\nstates discussed in \\cite{10} which does not preserve that symmetry is\nof interest rather from the mathematical point of view.\n\nHaving in mind the properties of the standard coherent states one\nmay ask about the minimalization of the Heisenberg uncertainty\nrelations by the introduced coherent states for a particle\non a circle. In our opinion, in the case with the compact manifolds\nthe minimalization of the Heisenberg uncertainty relations is not an\nadequate tool for the definition of the coherent states. A\ncounterexample can be easily deduced from (2.7), (2.8) and (2.9).\nIndeed, taking into account (2.8) and (2.9) we find that for the\neigenvectors $\|j\\rangle$'s of the angular momentum $\\hat J$ the\nequality sign is attended in the Heisenberg uncertainty relations\nimplied by (2.7) such that\n\\begin{equation}\n%<2.22>\n(\\Delta \\hat J)^2\\ge\\frac{1}{4}\\frac{\|\\langle U \\rangle\|^2}{1-\|\\langle U\n\\rangle\|^2}.\n\\end{equation}\nMore precisely, for these states (2.22) takes the form $0=0$.\nOn the other hand, the vectors $\|j\\rangle$'s are clearly rather poor\ncandidate for the coherent states. In our opinion the fact that the\ncoherent states are ``the most classical'' ones is better described\nby the following easily proven formulae:\n\\begin{eqnarray}\n(\\Delta \\hat J)^2&\\approx& {\\rm const},\\\\\n\\frac{\\langle U^2\\rangle}{\\langle U\\rangle^2}&\\approx&{\\rm const},\n\\end{eqnarray}\nwhere the approximations are very good ones. In fact, these\nrelations mean that the quantum variables $\\hat J$ and $U$ are at\npractically constant ``distance'' from their classical counterparts\n$\\langle \\hat J\\rangle$ and $\\langle U\\rangle$, respectively, and\ntherefore the quantum observables and the corresponding expectation\nvalues connected to the classical phase space are mutually related.\nWe point out that in the case with the standard coherent states for\na particle on a real line we have the exact formulae\n\\begin{eqnarray}\n%<2.24>\n(\\Delta \\hat p)^2&=&{\\rm const},\\\\\n(\\Delta \\hat q)^2&=&{\\rm const}.\n\\end{eqnarray}\nIt seems to us that the approximative nature of the relations (2.23)\nand (2.24) is related to the compactness of the circle.\n\\section{Unitary representations of the $e(3)$ algebra and quantum\nmechanics on a sphere}\nOur experience with the case of the circle discussed in the previous\nsection indicates that in order to introduce the coherent states we\nshould first identify the algebra adequate for the study of the\nmotion on a sphere. The fact that the algebra (2.7) referring to the\ncase with the circle $S^1$ is equivalent to the $e(2)$ algebra, where\n$E(2)$ is the group of the plane consisting of translations and rotations,\n\\begin{equation}\n%<3.1>\n[\\hat J,X_\\alpha]={\\rm i}\\varepsilon_{\\alpha\\beta}X_\\beta,\n\\qquad [X_\\alpha,X_\\beta]=0,\\qquad \\alpha,\\,\\beta=1,\\,2,\n\\end{equation}\nrealized in a unitary irreducible representation by Hermitian operators\n\\begin{equation}\n%<3.2>\nX_1=r(U+U^\\dagger)/2,\\qquad X_2=r(U-U^\\dagger)/2{\\rm i},\n\\end{equation}\nwhere the Casimir is\n\\begin{equation}\n%<3.3>\nX_1^2+X_2^2=r^2,\n\\end{equation}\nand $\\varepsilon_{\\alpha\\beta}$ is the anti-symmetric tensor,\nindicates that the most natural algebra for the case with the\nsphere $S^2$ is the $e(3)$ algebra such that\n\\begin{equation}\n%<3.4>\n[J_i,J_j]={\\rm i}\\varepsilon_{ijk}J_k,\\qquad [J_i,X_j]={\\rm i}\n\\varepsilon_{ijk}X_k,\\qquad [X_i,X_j]=0,\\qquad i,\\,j,\\,k=1,\\,2,\\,3.\n\\end{equation}\nIndeed, the algebra (3.4) has two Casimir operators given in a unitary\nirreducible representation by\n\\begin{equation}\n%<3.5>\n{\\bi X}^2=r^2,\\qquad {\\bi J}\\bdot{\\bi X}=\\lambda,\n\\end{equation}\nwhere dot designates the scalar product. Therefore, as with the generators\n$X_\\alpha $, $\\alpha=1,\\,2$, describing the position of a particle\non the circle, the generators $X_i$, $i=1,\\,2,\\,3$, can be regarded as\nquantum counterparts of the Cartesian coordinates of the points of the sphere\n$S^2$ with radius $r$. We point out that unitary irreducible\nrepresentations of (3.4) can be labelled by $r$ and the new scale\ninvariant parameter $\\zeta =\\frac{\\lambda }{r}$. It is clear that $\\zeta $\nis simply the projection of the angular momentum ${\\bi J}$ on the\ndirection of the radius vector of a particle. Since we did not find\nany denomination for such an entity in the literature, therefore we have\ndecided to call $\\zeta $ the {\\em twist\\/} of a particle.\n\nLet us now recall the basic properties of the unitary\nrepresentations of the $e(3)$ algebra. The $e(3)$ algebra expressed with\nthe help of operators $J_3$, $J_\\pm=J_1\\pm {\\rm i}J_2$, $X_3$ and $X_\\pm=X_1\\pm\n{\\rm i}X_2$, takes the form\n\\numparts\n\\begin{eqnarray}\n%<3.6>\n[J_+,J_-] &=& 2J_3,\\qquad [J_3,J_\\pm]=\\pm J_\\pm,\\\\\n{}[J_\\pm ,X_\\mp] &=& \\pm 2X_3,\\qquad [J_\\pm,X_\\pm]=0,\\qquad [J_\\pm ,X_3]=\\mp X_\\pm,\\\\\n{}[J_3,X_\\pm] &=& \\pm X_\\pm,\\qquad [J_3,X_3]=0,\\\\\n{}[X_+,X_-] &=& [X_\\pm,X_3]=0.\n\\end{eqnarray}\n\\endnumparts\nConsider the irreducible representation of the above algebra in the\nangular momentum basis spanned by the common eigenvectors\n$\|j,m;r,\\zeta\\rangle$ of the operators ${\\bi J}^2=J_+J_-+J_3^2-J_3$,\n$J_3$, ${\\bi X}^2$ and ${\\bi J}\\bdot{\\bi X}/r$\n\\numparts\n\\begin{eqnarray}\n%<3.7>\n&&{\\bi J}^2 \|j,m;r,\\zeta\\rangle = j(j+1) \|j,m;r,\\zeta\\rangle,\\qquad J_3\n\|j,m;r,\\zeta\\rangle=m\|j,m;r,\\zeta\\rangle,\\\\\n&&{\\bi X}^2 \|j,m;r,\\zeta\\rangle=r^2 \|j,m;r,\\zeta\\rangle,\\qquad\n({\\bi J}\\bdot{\\bi X}/r) \|j,m;r,\\zeta\\rangle=\\zeta\|j,m;r,\\zeta\\rangle,\n\\end{eqnarray}\n\\endnumparts\nwhere $-j\\le m\\le j$. Recall that\n\\begin{equation}\n%<3.8>\nJ_\\pm \|j,m;r,\\zeta\\rangle=\\sqrt{(j\\mp m)(j\\pm m+1)}\\,\|j,m\\pm 1;r,\\zeta\\rangle.\n\\end{equation}\nThe operators $X_\\pm$ and $X_3$ act on the vectors $\|j,m;r,\\zeta\\rangle$ in the\nfollowing way:\n\\numparts\n\\begin{eqnarray}\n%<3.9>\nX_+ \|j,m;r,\\zeta\\rangle\n&=&-\\frac{r\\sqrt{(j+1)^2-\\zeta^2}\\sqrt{(j+m+1)(j+m+2)}}\n{(j+1)\\sqrt{(2j+1)(2j+3)}}\|j+1,m+1;r,\\zeta\\rangle\\nonumber\\\\\n&&{}+\\frac{\\zeta r\\sqrt{(j-m)(j+m+1)}}{j(j+1)}\|j,m+1;r,\\zeta\\rangle\\nonumber\\\\\n&&{}+\\frac{r\\sqrt{j^2-\\zeta^2}\\sqrt{(j-m-1)(j-m)}}{j\\sqrt{(2j-1)(2j+1)}}\n\|j-1,m+1;r,\\zeta\\rangle,\\\\\nX_- \|j,m;r,\\zeta\\rangle\n&=&\\frac{r\\sqrt{(j+1)^2-\\zeta^2}\\sqrt{(j-m+1)(j-m+2)}}\n{(j+1)\\sqrt{(2j+1)(2j+3)}}\|j+1,m-1;r,\\zeta\\rangle\\nonumber\\\\\n&&{}+\\frac{\\zeta r\\sqrt{(j-m+1)(j+m)}}{j(j+1)}\|j,m-1;r,\\zeta\\rangle\\nonumber\\\\\n&&{}-\\frac{r\\sqrt{j^2-\\zeta^2}\\sqrt{(j+m-1)(j+m)}}{j\\sqrt{(2j-1)(2j+1)}}\n\|j-1,m-1;r,\\zeta\\rangle,\\\\\nX_3 \|j,m;r,\\zeta\\rangle\n&=&\\frac{r\\sqrt{(j+1)^2-\\zeta^2}\\sqrt{(j-m+1)(j+m+1)}}\n{(j+1)\\sqrt{(2j+1)(2j+3)}}\|j+1,m;r,\\zeta\\rangle\\nonumber\\\\\n&&\\fl\\fl{}+\\frac{\\zeta rm}{j(j+1)}\|j,m;r,\\zeta\\rangle+\n\\frac{r\\sqrt{j^2-\\zeta^2}\\sqrt{(j-m)(j+m)}}{j\\sqrt{(2j-1)(2j+1)}}\|j-1,m;r,\\zeta\n\\rangle.\n\\end{eqnarray}\n\\endnumparts\nAn immediate consequence of (3.9) is the existence of the minimal\n$j=j_{\\rm min}$ satisfying\n\\begin{equation}\n%<3.10>\nj_{\\rm min}=\|\\zeta\| .\n\\end{equation}\nThus, it turns out that in the representation defined by (3.9) the twist\n$\\zeta $ can be only integer or half integer. We finally write down\nthe orthogonality and completeness conditions satisfied by the\nvectors $\|j,m;r,\\zeta\\rangle$ such that\n\\begin{eqnarray}\n%<3.11>\n&&\\langle j,m;r,\\zeta\|j',m';r,\\zeta\\rangle=\\delta_{jj'}\\delta_{mm'},\\\\\n&&\\sum_{j=\|\\zeta\|}^{\\infty}\\sum_{m=-j}^{j}\n\|j,m;r,\\zeta\\rangle\\langle j,m;r,\\zeta\|=I,\n\\end{eqnarray}\nwhere $I$ is the identity operator.\n\\section{Definition of coherent states for a particle on a sphere}\nNow, an experience with the circle indicates that one should identify by means\nof the $e(3)$ algebra an analogue of the unitary operator $U$ (2.6),\nrepresenting the position of a particle on a sphere. To do\nthis, let us recall that a counterpart of the ``position'' $e^{{\\rm\ni}\\varphi}$ on the circle $S^1$ is a unit length imaginary quaternion\nwhich can be represented with the help of the Pauli matrices\n$\\sigma_i$, $i=1,\\,2,\\,3$, as\n\\begin{equation}\n%<4.1>\n\\eta = {\\rm i}{\\bi n}\\bdot{\\bsigma},\n\\end{equation}\nwhere ${\\bi n}^2=1$. Notice that $\\eta$ is simply an element of the\n$SU(2)$ group and it is related to the $S^2\\approx SU(2)/U(1)$\nquotient space. Therefore the most natural choice for the\n``position operator'' of a particle on a sphere is to set\n\\begin{equation}\n%<4.2>\nV=\\hbox{$\\scriptstyle 1\\over r$}\\bsigma\\bdot{\\bi X},\n\\end{equation}\nwhere $X_i$, $i=1,\\,2,\\,3$\nobey (3.4) and (3.9) and we have omitted for convenience the imaginary\nfactor i. Furthermore, let us introduce a version of the Dirac matrix\noperator \\cite{11}\n\\begin{equation}\n%<4.3>\nK := -(\\bsigma\\bdot{\\bi J}+1).\n\\end{equation}\nObserve that\n\\begin{equation}\n%<4.4>\nV^\\dagger=V,\\qquad K^\\dagger=K.\n\\end{equation}\nMaking use of the operators $V$ and $K$ we can write the relations\ndefining the $e(3)$ algebra in the space of the unitary irreducible\nrepresentation introduced above as\n\\numparts\n\\begin{eqnarray}\n%<4.5>\n({\\rm Tr}\\bsigma K)^2 &=& 4K(K+1),\\\\\n{}[K,V]_+ &=& {\\rm Tr}KV,\\\\\nV^2&=&I,\n\\end{eqnarray}\n\\endnumparts\nwhere ${\\rm Tr}A=A_{11}+A_{22}$, and the subscript ``+'' designates the\nanti-commutator. In particular,\n\\begin{equation}\n%<4.6>\n{\\rm Tr}KV=-2{\\bi J}\\bdot{\\bi X}/r=-2\\zeta .\n\\end{equation}\nIt should also be noted that in view of (4.4) and (4.5{\\em c}) $V$\nsatisfies the unitarity condition $V^\\dagger V=I$.\n\nWe now introduce the vector operator ${\\bi Z}$ generating, via the eigenvalue\nequation analogous to (2.10), the coherent states for a particle on\na sphere $S^2$. The experience with the circle (see eq.\\ (2.13)) suggests the\nfollowing form of the ``polar decomposition'' for the matrix operator\ncounterpart $Z$ of the operator ${\\bi Z}$:\n\\begin{equation}\n%<4.7>\nZ=e^{-K}V.\n\\end{equation}\nIndeed, it is easy to see that in the case of the circular motion in\nthe equator defined semiclassically by $J_1=J_2=0$ and $X_3=0$, $Z$\nreduces to the diagonal matrix operator with $Z$ given by (2.13) and its\nHermitian conjugate on the diagonal. Furthermore, using (4.5{\\em b})\nwe find\n\\begin{equation}\n%<4.8>\nZ-Z^{-1} = 2\\zeta K^{-1}\\sinh K.\n\\end{equation}\nMotivated by the complexity of the problem we now restrict to the\nsimplest case of the twist $\\zeta=0$ when (4.8) takes the form\n\\begin{equation}\n%<4.9>\nZ^2=I.\n\\end{equation}\nIn the following we confine ourselves to the case $\\zeta=0$.\nThe general case with arbitrary $\\zeta\\ne0$ will be discussed in a\nseparate work. Besides (4.9) we have also remarkably simple\nrelation (4.5{\\em b}) referring to $\\zeta=0$ such that\n\\begin{equation}\n%<4.10>\n[K,V]_+ = 0.\n\\end{equation}\nNotice that the\ncase $\\zeta=0$ is the ``most classical'' one. Indeed, the projection\nof the angular momentum onto the direction of the radius vector should\nvanish for the classical particle on a sphere. It should also be noted\nthat in view of (3.10) $j$'s and $m$'s labelling the basis vectors\n$\|j,m;r,\\zeta\\rangle$ are integer in the case of the twist $\\zeta =0$.\nWe finally point out that the condition $\\zeta=0$ ensures the\ninvariance of the irreducible representation of the $e(3)$ algebra\nunder time inversions and parity transformations which\nchange the sign of the product ${\\bi J}\\bdot{\\bi X}$. Clearly\ndemanding the time-reversal or the parity invariance when $\\zeta\\ne0$ one\nshould work with representations involving both $\\zeta$ and $-\\zeta$.\n\n\nWe now return to (4.7). Making use of (4.10) and the fact that the\nmatrix operator $V$ in view of (4.2) is traceless one we obtain for $\\zeta=0$\n\\begin{equation}\n%<4.11>\n{\\rm Tr}Z=0.\n\\end{equation}\nHence,\n\\begin{equation}\n%<4.12>\nZ = \\bsigma\\bdot{\\bi Z}.\n\\end{equation}\nTaking into account (4.9) we get from (4.12)\n\\begin{equation}\n%<4.13>\n{\\bi Z}^2=1,\n\\end{equation}\nand\n\\begin{equation}\n%<4.14>\n[Z_i,Z_j]=0,\\qquad i,j=1,\\,2,\\,3.\n\\end{equation}\nAs with (4.2) describing in the matrix language the position of a quantum\nparticle on a sphere, the matrix operator (4.12) can be only interpreted as a\nconvenient arrangement of the operators $Z_i$ generating the coherent\nstates, simplifying the algebraic analysis of the problem. Accordingly, we\ndefine the coherent states for a quantum mechanics on a sphere in terms of\noperators $Z_i$, as the solutions of the eigenvalue equation such that\n\\begin{equation}\n%<4.15>\n{\\bi Z} \|{\\bi z}\\rangle = {\\bi z} \|{\\bi z}\\rangle,\n\\end{equation}\nwhere in view of (4.13) ${\\bi z}^2=1$. What is ${\\bi Z}$ ? Using\n(4.7), (4.2), (4.3) and setting $\\zeta=0$, we find after some calculation\n\\begin{eqnarray}\n%<4.16>\n{\\bi Z} &=&\\left(\\frac{e^{\\frac{1}{2}}}{\\sqrt{1+4{\\bi J}^2}}{\\rm\nsinh}\\hbox{$\\scriptstyle 1\\over2 $}\\sqrt{1+4{\\bi\nJ}^2}+e^{\\frac{1}{2}}{\\rm cosh}\\hbox{$\\scriptstyle 1\\over2 $}\n\\sqrt{1+4{\\bi J}^2}\\right){{\\bi X}\\over r}\\nonumber\\\\\n&&{}+{\\rm i}\\left(\\frac{2e^{\\frac{1}{2}}}{\\sqrt{1+4{\\bi J}^2}}{\\rm sinh}\n\\hbox{$\\scriptstyle 1\\over2 $}\\sqrt{1+4{\\bi J}^2}\\right){\\bi\nJ}\\times{{\\bi X}\\over r}.\n\\end{eqnarray}\nWe remark that $Z_i$ have the structure resembling\nthe standard annihilation operators. In fact, one can easily check\nthat it can be written as a combination\n\\begin{equation}\n%<4.17>\n{\\bi Z}=a{\\bi X}+{\\rm i}b{\\bi P},\n\\end{equation}\nof the ``position operator'' ${\\bi X}$ and the ``momentum'' ${\\bi P}$, where\nthe coefficients $a$ and $b$ are functions of ${\\bi J}^2$. We finally point\nout that derivation of the operator ${\\bi Z}$ (4.16) without the knowledge of\nthe matrix operator $Z$ seems to be very difficult task.\n\\section{Construction of the coherent states}\nIn this section we construct the coherent states specified by the eigenvalue\nequation (4.15). On projecting (4.15) on the basis vectors\n$\|j,m;r\\rangle\\equiv\|j,m;r,0\\rangle$ and using (3.7{\\em a}), (3.8) and (3.9) with\n$\\zeta=0$ we arrive at the system of linear difference equations satisfied by\nthe Fourier coefficients of the expansion of the coherent state\n$\|{\\bi z}\\rangle$ in the basis $\|j,m;r\\rangle$. The direct solution of such\nsystem in the general case seems to be difficult task. Therefore, we adopt the\nfollowing technique. We first solve the eigenvalue equation for ${\\bi z}=\n{\\bi n}_3=(0,0,1)$, and then generate the coherent states from the vector\n${\\bi n}_3$ using the fact (see (4.16)) that ${\\bi Z}$ is a vector operator. As\ndemonstrated in the next section the case with ${\\bi z}={\\bi n}_3$ refers to\n${\\bi x}=(0,0,1)$ and ${\\bi l}={\\bf 0}$, where ${\\bi x}$ is the\nposition and ${\\bi l}$ the angular momentum, respectively, i.e., the particle\nresting on the ``North Pole'' of the sphere. Let us write down the\neigenvalue equation (4.15) for ${\\bi z}={\\bi n}_3$\n\\begin{equation}\n%<5.1>\n{\\bi Z} \|{\\bi n}_3\\rangle={\\bi n}_3 \|{\\bi n}_3\\rangle.\n\\end{equation}\nUsing the following relations which can be easily derived with the help\nof (4.16), (3.7{\\em a}), (3.8) and (3.9) with $\\zeta =0$:\n\\numparts\n\\begin{eqnarray}\n%<5.2>\nZ_1 \|j,m;r\\rangle\n&=&-\\frac{1}{2}e^{-j-1}\\sqrt{\\frac{(j+m+1)(j+m+2)}{(2j+1)(2j+3)}}\n\|j+1,m+1;r\\rangle\\nonumber\\\\\n&&{}+\\frac{1}{2}e^j\\sqrt{\\frac{(j-m-1)(j-m)}{(2j-1)(2j+1)}}\n\|j-1,m+1;r\\rangle\\nonumber\\\\\n&&+\\frac{1}{2}e^{-j-1}\\sqrt{\\frac{(j-m+1)(j-m+2)}{(2j+1)(2j+3)}}\n\|j+1,m-1;r\\rangle\\nonumber\\\\\n&&{}-\\frac{1}{2}e^j\\sqrt{\\frac{(j+m-1)(j+m)}{(2j-1)(2j+1)}}\n\|j-1,m-1;r\\rangle,\\\\\nZ_2 \|j,m;r\\rangle\n&=&\\frac{{\\rm i}}{2}e^{-j-1}\\sqrt{\\frac{(j+m+1)(j+m+2)}{(2j+1)(2j+3)}}\n\|j+1,m+1;r\\rangle\\nonumber\\\\\n&&{}-\\frac{{\\rm i}}{2}e^j\\sqrt{\\frac{(j-m-1)(j-m)}{(2j-1)(2j+1)}}\n\|j-1,m+1;r\\rangle\\nonumber\\\\\n&&+\\frac{{\\rm i}}{2}e^{-j-1}\\sqrt{\\frac{(j-m+1)(j-m+2)}{(2j+1)(2j+3)}}\n\|j+1,m-1;r\\rangle\\nonumber\\\\\n&&{}-\\frac{{\\rm i}}{2}e^j\\sqrt{\\frac{(j+m-1)(j+m)}{(2j-1)(2j+1)}}\n\|j-1,m-1;r\\rangle,\\\\\nZ_3 \|j,m;r\\rangle\n&=&e^{-j-1}\\sqrt{\\frac{(j-m+1)(j+m+1)}{(2j+1)(2j+3)}}\n\|j+1,m;r\\rangle\\nonumber\\\\\n&&{}+e^j\\sqrt{\\frac{(j-m)(j+m)}{(2j-1)(2j+1)}}\|j-1,m;r\\rangle,\n\\end{eqnarray}\n\\endnumparts\nit can be easily checked that the solution to (5.1) is given by\n\\begin{equation}\n%<5.3>\n\|{\\bi\nn}_3\\rangle=\\sum_{j=0}^{\\infty}e^{-\\frac{1}{2}j(j+1)}\\sqrt{2j+1}\|j,0;r\\rangle.\n\\end{equation}\nNow, using the commutator\n\\begin{equation}\n%<5.4>\n[{\\bi w}\\bdot{\\bi J},{\\bi Z}]=-{\\rm i}{\\bi w}\\times{\\bi Z},\n\\end{equation}\nwhere ${\\bi w}\\in{\\Bbb C}^3$, we generate the complex rotation of\n${\\bi Z}$\n\\begin{equation}\n%<5.5>\ne^{{\\bi w}\\bdot{\\bi J}}{\\bi Z}e^{-{\\bi w}\\bdot{\\bi J}}=\n\\cosh\\sqrt{{\\bi w}^2}\\,{\\bi Z}-{\\rm i}\\frac{\\sinh\\sqrt{{\\bi w}^2}}\n{\\sqrt{{\\bi w}^2}}\n{\\bi w}\\times{\\bi Z}+\\frac{1-\\cosh\\sqrt{{\\bi w}^2}}{{\\bi w}^2}{\\bi w}\n({\\bi w}\\bdot{\\bi Z}).\n\\end{equation}\nTaking into account (5.5) and (4.15) we find that the coherent states\ncan be expressed by\n\\begin{equation}\n%<5.6>\n\|{\\bi z}\\rangle = e^{{\\bi w}\\bdot{\\bi J}}\|{\\bi n}_3\\rangle,\n\\end{equation}\nwhere ${\\bi w}$ is given by\n\\begin{equation}\n%<5.7>\n{\\bi w}=\\frac{{\\rm arccosh}z_3}{\\sqrt{1-z_3^2}}{\\bi z}\\times{\\bi n}_3.\n\\end{equation}\nIt thus appears that the coherent states can be written as\n\\begin{equation}\n%<5.8>\n\|{\\bi z}\\rangle = \\exp\\left[\\frac{{\\rm arccosh}z_3}{\\sqrt{1-z_3^2}}\n({\\bi z}\\times{\\bi n}_3)\\bdot{\\bi J}\\right]\n\|{\\bi n}_3\\rangle.\n\\end{equation}\nWe remark that the discussed coherent states are generated\nanalogously as in the case of the circle described by the equation\n(2.16). The formula (5.8) can be furthermore written in the form\n\\begin{equation}\n%<5.9>\n\|{\\bi z}\\rangle = e^{\\mu J_-}e^{\\gamma J_3}e^{\\nu J_+} \|{\\bi\nn}_3\\rangle,\n\\end{equation}\nwhere\n\\begin{equation}\n%<5.10>\n\\mu =\\frac{z_1+{\\rm i}z_2}{1+z_3},\\qquad \\nu=\\frac{-z_1+{\\rm\ni}z_2}{1+z_3},\\qquad \\gamma =\\ln\\frac{1+z_3}{2}.\n\\end{equation}\nFinally, eqs.\\ (5.9), (5.3), (3.7{\\em a}) and (3.8) taken together yield\nthe following formula on the coherent states:\n\\begin{equation}\n%<5.11>\n\\fl \|{\\bi z}\\rangle =\\sum_{j=0}^{\\infty}e^{-\\frac{1}{2}j(j+1)}\n\\sqrt{2j+1}\\sum_{m=0}^{j}\\frac{\\nu^m}{m!}\\frac{(j+m)!}{(j-m)!}\ne^{\\gamma m}\\sum_{k=0}^{j+m}\\frac{\\mu^k}{k!}\n\\sqrt{\\frac{(j-m+k)!}{(j+m-k)!}} \|j,m-k;r\\rangle,\n\\end{equation}\nwhere $\\mu ,\\,\\nu$ and $\\gamma$ are expressed by (5.10) and ${\\bi\nz}^2=1$. Taking into account the identities\n\\begin{equation}\n%<5.12>\n\\sum_{s=0}\\sp{n}\\frac{(s+k)!}{(s+m)!s!(n-s)!}z^s=\\frac{k!}{m!n!}\\,\\,\n{}_2F_1(-n,k+1,m+1;-z),\n\\end{equation}\n\\begin{equation}\n%<5.13>\nC_n^\\alpha(x) =\\frac{\\Gamma(n+2\\alpha)}{\\Gamma(n+1)\\Gamma(2\\alpha)}\n\\,{}_2F_1(-n,n+2\\alpha,\\alpha+\\hbox{$\\scriptstyle 1\\over2 $};\n\\hbox{$\\scriptstyle 1\\over2 $}(1-x)),\n\\end{equation}\nwhere ${}_2F_1(a,b,c;z)$ is the hypergeometric function,\n$C_n^\\alpha(x)$ are the Gegenbauer polynomials and $\\Gamma(x)$ is\nthe gamma function, we obtain\n\\begin{equation}\n%<5.14>\n\\fl \\langle j,m;r\|{\\bi z}\\rangle = e^{-\\frac{1}{2}j(j+1)}\\sqrt{2j+1}\\,\n\\frac{(2\|m\|)!}{\|m\|!}\\sqrt{\\frac{(j-\|m\|)!}{(j+\|m\|)!}}\\left(\n\\frac{-\\varepsilon(m)z_1+{\\rm i}z_2}{2}\\right)^{\|m\|} C_{j-\|m\|}^{\|m\|+\\frac{1}{2}}\n(z_3),\n\\end{equation}\nwhere $\\varepsilon(m)$ is the sign of $m$. Let us recall in the context of\nthe relations (5.14) that the polynomial dependence of the projection of\ncoherent states onto the discrete basis vectors, on the complex numbers\nparametrizing those states is one of their most characteristic properties.\nClearly, the polynomials (5.14) should span via the ``resolution of\nthe identity operator'' the Fock-Bargmann representation. We recall\nthat existence of such representation is one of the most important\nproperties of coherent states. The problem of finding the\nFock-Bargmann representation in the discussed case of the coherent\nstates for a particle on a sphere is technically complicated and it will be\ndiscussed in a separate work. Finally, notice that the coherent states\n$\|{\\bi z}\\rangle$ are evidently stable under rotations.\n\\section{Coherent states and the classical phase space}\nWe now show that the introduced coherent states for a quantum\nparticle on a sphere are labelled by points of the classical phase\nspace, that is $T^S^2$. Referring back to eq.\\ (4.16) and\ntaking into account the fact that the classical limit corresponds to\nlarge $j$'s, we arrive at the following parametrization of ${\\bi z}$\nby points of the phase space:\n\\begin{equation}\n%<6.1>\n{\\bi z}=\\cosh\|{\\bi l}\|\\,\\frac{{\\bi x}}{r}+{\\rm i}\\frac{\\sinh\|{\\bi l}\|}\n{\|{\\bi l}\|}\\,{\\bi l}\\times \\frac{{\\bi x}}{r},\n\\end{equation}\nwhere the vectors ${\\bi l},\\,{\\bi x}\\in{\\Bbb R}^3$, fulfil\n${\\bi x}^2=r^2$ and ${\\bi l}\\bdot{\\bi x}=0$, i.e.,\nwe assume that ${\\bi l}$ is the classical angular momentum and ${\\bi x}$ is the\nradius vector of a particle on a sphere. In accordance with the\nformulae (4.15) and (4.13) the vector ${\\bi z}$ satisfies ${\\bi z}^2=1$.\nThus, the vector ${\\bi z}$ is really parametrized by the points $({\\bi x},{\\bi l})$\nof the classical phase space $T^S^2$.\n\n\nConsider now the expectation value of the angular momentum operator ${\\bi J}$\nin a coherent state. The explicit formulae which can be derived\nwith the help of (3.7{\\em a}), (3.8), (3.12) and (5.14) are too complicated\nto reproduce them herein. From computer simulations it follows that\n\\begin{equation}\n%<6.2>\n\\langle{\\bi J}\\rangle_{\\bi z} =\\frac{\\langle {\\bi z}\|{\\bi J}\|{\\bi z}\\rangle}{\\langle {\\bi\nz}\|{\\bi z}\\rangle}\\approx{\\bi l}.\n\\end{equation}\nNevertheless, in opposition to the case with the circular motion, the\napproximate relation (6.2) does not hold for practically arbitrary small\n$\|{\\bi l}\|$. Namely, we have found that whenever $\|{\\bi l}\|\\sim1$, then\n(6.2) is not valid. Note that returning to dimension entities in\nthe formulae like (3.6) we measure $\|{\\bi l}\|$ in the units of\n$\\hbar$, so in the physical units we deal rather with ${\\bi L}=\\hbar\n{\\bi l}$. For $\|{\\bi l}\|\\ge10$ the relative error $\|(\\langle\nJ_i\\rangle_{\\bi z}-l_i)/\\langle J_i\\rangle_{\\bi z}\|$, $i=1,\\,2,\\,3$, is small.\nMore precisely, if $\|{\\bi l}\|\\sim10$, then $\|(\\langle\nJ_i\\rangle_{\\bi z}-l_i)/\\langle J_i\\rangle_{\\bi z}\|\\sim$1 per cent. In other\nwords, in the case of the motion on a sphere, the quantum fluctuations are not\nnegligible for $\|{\\bi L}\|\\sim$1 $\\hbar$ and the description based on the\nconcept of the classical phase space is not adequate one. However, it\nmust be borne in mind that the condition $\|{\\bi L}\|\\ge$ 10 $\\hbar$, when (6.2)\nholds is not the same as the classical limit $\|{\\bi l}\|\\to\\infty$. We only\npoint out that $10\\,\\hbar\\approx 10^{-33}\\,{\\rm J}\\cdot{\\rm s}$. It thus appears\nthat the parameter ${\\bi l}$ in (6.2) can be identified with the classical angular\nmomentum divided by $\\hbar$.\n\nWe now study the role of the parameter ${\\bi x}$ in (6.1). As with\nthe momentum operator ${\\bi J}$ the explicit relations obtained by\nmeans of (3.9) with $\\zeta=0$, (3.12) and (5.14) are too\ncomplicated to write them down herein. The computer simulations\nindicate that\n\\begin{equation}\n%<6.3>\n\\langle{\\bi X}\\rangle_{\\bi z}=\\frac{\\langle{\\bi z}\|{\\bi X}\|{\\bi z}\\rangle}\n{\\langle {\\bi z}\|{\\bi z}\\rangle}\\approx e^{-\\frac{1}{4}}{\\bi x}.\n\\end{equation}\nIt seems that the formal resemblance of the formula (6.3) and (2.19)\nreferring to the case with the circular motion is not accidental one.\nThe range of application of (6.3) is the same as for (6.2), i.e., $\|{\\bi\nl}\|\\ge10$. Because of the term $e^{-\\frac{1}{4}}$, it appears that\nthe average value of ${\\bi X}$ does not belong to the sphere with\nradius $r$. Proceeding analogously as in the case of the circle we introduce\nthe relative average value of ${\\bi X}$ of the form\n\\begin{equation}\n%<6.4>\n\\langle\\!\\langle X_i\\rangle\\!\\rangle_{\\bi z}=\\frac{\\langle X_i\\rangle_{\\bi z}}\n{\\langle X_i\\rangle_{{\\bi w}_i}},\\qquad i=1,\\,2,\\,3,\n\\end{equation}\nwhere $\|{\\bi w}_i\\rangle$ is a coherent state with\n\\begin{equation}\n%<6.5>\n{\\bi w}_k=\\cosh\|{\\bi l}\|{\\bi n}_k+{\\rm i}\\frac{\\sinh\|{\\bi l}\|}{\|{\\bi l}\|}\n{\\bi l}\\times{\\bi n}_k,\\qquad k=1,\\,2,\\,3,\n\\end{equation}\nwhere ${\\bi n}_k$ is the unit vector along the $k$ coordinate axis\nand ${\\bi l}$ is the same as in (6.1). In view of (6.3) and (6.4) we have\n\\begin{equation}\n%<6.6>\n\\langle\\!\\langle {\\bi X}\\rangle\\!\\rangle_{\\bi z}\\approx{\\bi x}.\n\\end{equation}\nTherefore, the relative expectation value $\\langle\\!\\langle {\\bi X}\\rangle\\!\n\\rangle_{\\bi z}$ seems to be the most natural one to describe the average\nposition of a particle on a sphere.\n\nWe have thus shown that the parameter ${\\bi x}$ can be immediately\nrelated to the classical radius vector of a particle on a sphere. As with the\ncase of the circular motion (see formulae (2.18) and (2.21)), we\ninterpret the relations (6.2) and (6.6) as the best possible\napproximation of the classical phase space. In this sense the\ncoherent states labelled by points of such deformed phase space are\nclosest to the classical ones. The quantum fluctuations which are\nthe reason of the approximate nature of (6.2) and (6.6) are in our\nopinion a characteristic feature of quantum mechanics on a sphere.\n\nWe finally remark that the discussion of the Heisenberg uncertainty\nrelations analogous to that referring to the circle (see section 2)\ncan be performed also in the case with the coherent states for a\nparticle on a sphere. For example a counterpart of the\nformula (2.22) is\n\\begin{equation}\n%<6.7>\n(\\Delta {\\bi J})^2\\ge\\frac{1}{2}\\frac{\\frac{1}{2}{\\rm Tr}\\langle\nV\\rangle^2}{1-\\frac{1}{2}{\\rm Tr}\\langle V\\rangle^2},\n\\end{equation}\nwhere according to eq.\\ (4.2) we have $\\langle V\\rangle=\\frac{1}{r}\n\\bsigma\\bdot\\langle {\\bi X}\\rangle$. Such discussion as\nwell as the detailed analysis of the Heisenberg uncertainty relations\nfor the quantum mechanics on a compact manifold will be the subject of\na separate paper which is in preparation.\n\\section{Simple application: the rotator}\nWe now illustrate the actual treatment by the example of a free\ntwist 0 particle on a sphere, i.e.\\ the rotator. The corresponding\nHamiltonian is given by\n\\begin{equation}\n%<7.1>\n\\hat H=\\hbox{$\\scriptstyle 1\\over2 $}{\\bi J}^2.\n\\end{equation}\nBy (3.7{\\em a}) the normalized solution of the Schr\\\"odinger equation\n\\begin{equation}\n%<7.2>\n\\hat H \|E\\rangle = E \|E\\rangle\n\\end{equation}\ncan be expressed by\n\\begin{equation}\n%<7.3>\n\|E\\rangle= \|j,m;r\\rangle,\\qquad E=\\hbox{$\\scriptstyle 1\\over2 $}j(j+1).\n\\end{equation}\nWe now discuss the distribution of the energies in the coherent\nstate. The computer simulations indicate that the function\n\\begin{equation}\n%<7.4>\np_{j,m}({\\bi x},{\\bi l})=\\frac{\|\\langle j,m;r\|{\\bi z}\\rangle\|^2}{\\langle\n{\\bi z}\|{\\bi z}\\rangle},\\qquad -j\\le m\\le j,\n\\end{equation}\ndetermined by (5.14) and (6.1), which gives the probability\nof finding the system in the state $\|j,m;r\\rangle$, when the system\nis in the normalized coherent state $\|{\\bi z}\\rangle/\\sqrt{\\langle{\\bi\nz}\|{\\bi z}\\rangle}$, has the following properties. For fixed\ninteger $m=l_3$ the function $p_{j,m}$ has a maximum at $j_{\\rm max}$\ncoinciding with the integer nearest to the positive root of the equation\n\\begin{equation}\n%<7.5>\nj(j+1)={\\bi l}^2,\n\\end{equation}\n(see Fig.\\ 1). Thus, it turns out that the parameter $\\frac{1}{2}{\\bi l}^2$\ncan be regarded as the energy of the particle. Further, for fixed integer $j$ in $p_{j,m}({\\bi\nx},{\\bi l})$ (see Fig.\\ 2), such that (7.5) holds,\nthe function $p_{j,m}$ has a maximum at $m_{\\rm max}$ coinciding\nwith the integer nearest to $l_3$. It thus appears that the parameter $l_3$\ncan be identified with the projection of the momentum on the $x_3$ axis.\n\\section{Conclusion}\nIn this work we have introduced the coherent states for a quantum\nparticle on a sphere. An advantage of the formalism used\nis that the coherent states are labelled by points of the classical\nphase space. The authors have not found alternative constructions\nof coherent states for a quantum mechanics on a sphere preserving\nthis fundamental property of coherent states. As pointed out in\nSec.\\ 6, the quantum fluctuations arising in the case of the motion\non a sphere are bigger than those taking place for the circular\nmotion. This observation is consistent with the appearance of the\nadditional degree of freedom for the motion on a sphere. We\nremark that as with the particle on a circle, we deal within\nthe actual treatment with the deformation of the classical phase\nspace expressed by the approximate relations (6.2) and (6.6). We\nalso point out that besides (6.2) and (6.6) the quasi-classical\ncharacter of the introduced coherent states is confirmed by the\nbehaviour of the distribution of the energies investigated in\nsection 7. It seems that the approach introduced in this paper is not\nrestricted to the study of the quasi-classical aspects of the quantum\nmotion on a sphere. For example, the results of this work would be of\nimportance in the theory of quantum chaos. In fact, in this theory the kicked\nrotator is one of the most popular model systems. Because of the\nwell known difficulties in the analysis of the Heisenberg uncertainty relations\noccuring in the case with observables having compact spectrum like\nthe position operator ${\\bi X}$ satisfying the $e(3)$ algebra (3.4) we\nhave not studied them herein. The analysis of the Heisenberg\nuncertainty relations as well as the discussion of the case with a\nnonvanishing twist will be performed in future work.\n\\section{References}\n\\begin{thebibliography}{VV}\n\\bibitem{1}Klauder J R and Skagerstam B S 1985 {\\em Coherent\nStates--Applications in Physics and Mathematical Physics} (World\nScientific: Singapore)\n\\bibitem{2}Kowalski K and Rembieli\\'nski J 1993 {\\em J. Math. Phys.} {\\bf\n34} 2153\n\\bibitem{3}Kowalski K 1994 {\\em Methods of Hilbert Spaces in the Theory\nof Nonlinear Dynamical Systems} (World Scientific: Singapore)\n\\bibitem{4}Radcliffe J M 1971 {\\em J. Phys. A} {\\bf 4} 313\n\\bibitem{5}Perelomov A M 1972 {\\em Commun. Math. Phys.} {\\bf 26} 222;\nPerelomov A M 1986 {\\em Generalized Coherent States and Their\nApplications} (Springer: Berlin)\n\\bibitem{6}Kowalski K, Rembieli\\'nski J and Papaloucas L C 1996 {\\em J.\nPhys. A} {\\bf 29} 4149\n\\bibitem{7}Barut A O and Girardello L 1971 {\\em Commun. Math. Phys.} {\\bf\n21} 41\n\\bibitem{8}Glauber R J 1963 {\\em Phys. Rev} {\\bf 130} 2529; {\\bf\n131} 2766\n\\bibitem{9}Klauder J R 1963 {\\em J. Math. Phys.} {\\bf 4} 1055\n\\bibitem{10}Gonz\\'ales J A and del Olmo M A 1998 {\\em J. Phys. A}\n{\\bf 31} 8841\n\\bibitem{11}Biedenharn L C and Louck J D 1981 {\\em Angular Momentum in\nQuantum Physics. Theory and Application.} (Addison-Wesley:\nMassachusetts)\n\\end{thebibliography}\n\\Figures\n\\begin{figure}\n\\caption{The plot of $\\ln p_{j,m}$ (see (7.5)), with fixed\n$m=0$ and ${\\bi z}$ given by (6.1), where ${\\bi\nx}=(0.412,0.412,0.812)$ and ${\\bi l}=(8.124,-8.124,0)$. Since\n${\\bi l}^2=132$, therefore $j_{\\rm max}=11$ coincides with the\npositive root of Eq.\\ (7.5).}\n\\label{fig1}\n%\\end{figure}\n%\\begin{figure}\n\\caption{The plot of $\\ln p_{j,m}$ with ${\\bi x}=(0.411,0.911,0.036)$\nand ${\\bi l}=(-17.490,7.490,10)$. The fixed $j=21$ corresponds to the\npositive root of (7.5), where ${\\bi l}^2=462$. The fuction has the\nmaximum at $m_{\\rm max}=l_3=10$.}\n\\label{fig2}\n%\\end{figure}\n%\\begin{figure}\n\\end{figure}\n\\end{document}\n" } ]	[ { "name": "quant-ph9912094.extracted_bib", "string": "{1Klauder J R and Skagerstam B S 1985 {\\em Coherent States--Applications in Physics and Mathematical Physics (World Scientific: Singapore)" }, { "name": "quant-ph9912094.extracted_bib", "string": "{2Kowalski K and Rembieli\\'nski J 1993 {\\em J. Math. Phys. {34 2153" }, { "name": "quant-ph9912094.extracted_bib", "string": "{3Kowalski K 1994 {\\em Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems (World Scientific: Singapore)" }, { "name": "quant-ph9912094.extracted_bib", "string": "{4Radcliffe J M 1971 {\\em J. Phys. A {4 313" }, { "name": "quant-ph9912094.extracted_bib", "string": "{5Perelomov A M 1972 {\\em Commun. Math. Phys. {26 222; Perelomov A M 1986 {\\em Generalized Coherent States and Their Applications (Springer: Berlin)" }, { "name": "quant-ph9912094.extracted_bib", "string": "{6Kowalski K, Rembieli\\'nski J and Papaloucas L C 1996 {\\em J. Phys. A {29 4149" }, { "name": "quant-ph9912094.extracted_bib", "string": "{7Barut A O and Girardello L 1971 {\\em Commun. Math. Phys. {21 41" }, { "name": "quant-ph9912094.extracted_bib", "string": "{8Glauber R J 1963 {\\em Phys. Rev {130 2529; {131 2766" }, { "name": "quant-ph9912094.extracted_bib", "string": "{9Klauder J R 1963 {\\em J. Math. Phys. {4 1055" }, { "name": "quant-ph9912094.extracted_bib", "string": "{10Gonz\\'ales J A and del Olmo M A 1998 {\\em J. Phys. A {31 8841" }, { "name": "quant-ph9912094.extracted_bib", "string": "{11Biedenharn L C and Louck J D 1981 {\\em Angular Momentum in Quantum Physics. Theory and Application. (Addison-Wesley: Massachusetts)" } ]
quant-ph9912095	Quantum noise in optical fibers I: \\ stochastic equations	[ { "author": "P. D. Drummond$^{1" } ]	\noindent We analyze the quantum dynamics of radiation propagating in a single mode optical fiber with dispersion, nonlinearity, and Raman coupling to thermal phonons. We start from a fundamental Hamiltonian that includes the principal known nonlinear effects and quantum noise sources, including linear gain and loss. Both Markovian and frequency-dependent, non-Markovian reservoirs are treated. This allows quantum Langevin equations to be calculated, which have a classical form except for additional quantum noise terms. In practical calculations, it is more useful to transform to Wigner or +$P$ quasi-probability operator representations. These result in stochastic equations that can be analyzed using perturbation theory or exact numerical techniques. The results have applications to fiber optics communications, networking, and sensor technology.	[ { "name": "ramanI11.tex", "string": "%% This LaTeX-file was created by <drummond> Sun Nov 21 16:12:33 1999\n%% LyX 1.0 (C) 1995-1999 by Matthias Ettrich and the LyX Team\n\n%% Do not edit this file unless you know what you are doing.\n\\documentstyle[osa,eqsecnum,manuscript]{revtex}\n\n\\begin{document}\n\n\\tightenlines\n\\title{Quantum noise in optical fibers I: \\\\\n stochastic equations}\n\n\n\\author{P. D. Drummond$^{1}$ and J. F. Corney$^{1,2}$ }\n\n\n\\address{$^{1}$Department of Physics, The University of Queensland, St. Lucia, QLD 4072, Australia\\\\\n$^{2}$Department of Mathematical Modelling, Technical University of Denmark, DK-2800 Lyngby, Denmark}\n\n\n\\date{\\today{}}\n\n\\maketitle\n\\begin{abstract}\n\\noindent We analyze the quantum dynamics of radiation propagating in a single mode optical\nfiber with dispersion, nonlinearity, and \nRaman coupling to thermal phonons. We start from a fundamental\nHamiltonian that includes the principal known nonlinear effects and quantum\nnoise sources, including linear gain and loss. Both Markovian and frequency-dependent,\nnon-Markovian reservoirs are treated. This allows quantum Langevin equations\nto be calculated, which have a classical form except for additional quantum\nnoise terms. In practical calculations, it is more useful to transform to Wigner\nor +$P$ quasi-probability operator representations. These result in stochastic\nequations that can be analyzed using perturbation theory or exact numerical\ntechniques. The results have applications to fiber optics communications, networking, and sensor technology. \n\\end{abstract}\n%\\pacs{03.75.Fi,05.30.Jp,32.80.Pj,42.50.Lc,42.50.Dv}\n\n%OSA journals require OCIS numbers instead of PACS, so I have picked out a few:\n\\pacs{060.4510, 270.5530, 270.3430, 190.4370, 190.5650, 060.2400}\n\n\n\\section{Introduction}\n\nThe propagation of electromagnetic radiation through optical fibers is the central\nparadigm of optical communications and sensor technology. It is also a novel\nphysical system, due to the materials processing of fused silica, that leads\nto single-mode behaviour with extremely low losses. Over short distances (depending\non the pulse intensity) the well-known nonlinear Schr\\\"{o}dinger equation can\ndescribe most optical fibers with great accuracy, and leads to soliton behaviour,\nas well as to many other effects. Over longer distances, a number of reservoir\neffects intervene, including attenuation, Raman scattering, and the use of amplifiers\nand filters to compensate for losses. At the quantum level, both the original\nnonlinearity and the additional couplings to reservoirs can lead to quantum\nnoise - which modifies the predictions of the classical nonlinear Schr\\\"{o}dinger\nequation.\n\nIn this paper, we analyze the effects of quantum noise in fiber\noptics. This extends and explains in more detail earlier theoretical work in\nthis area, which led to the first prediction\\cite{s14,2} and measurement\\cite{Solexp}\nof intrinsic quantum noise effects in optical solitons. The theory given here\nincludes a detailed derivation of the relevant quantum Hamiltonian. We include\nquantum noise effects due to nonlinearities, Raman reservoirs and Brillouin\nscattering. The Raman/Brillouin noise is modeled using a multiple Lorentzian\nfit to measured fluorescence data, in order to estimate the Raman gain coefficients.\nBoth gain and loss effects are included. This treatment is unified with theory\nof gain/loss reservoirs, which was also predicted\\cite{s35} and observed\\cite{s53}\nto have large effects on soliton propagation. All these reservoirs are treated\nwithout using the Markovian approximation, in order to accurately treat the\nfrequency-dependent reservoirs found in practical applications.\n\nThe purpose of this work is to lay the foundations of practical methods for calculating and numerically simulating quantum effects in nonlinear optical fibers. These are significant\nwhenever quantum-limited behaviour is important in communications, sensing,\nor measurement with optical fibers.\n\nWe introduce the basic quantum Hamiltonian for an optical fiber in section \\ref{NSM}.\nThis gives a Heisenberg equation of motion which reduces to the nonlinear Schr\\\"{o}dinger\nequation in the classical limit. The equation of motion is extended in section\n\\ref{RH} to include Raman and Brillouin effects, with gain and absorption processes\nconsidered in section \\ref{GA}. The complete Heisenberg equation in section\n\\ref{CHE} is the central result of this paper. Stochastic partial differential\nequations can be derived from the quantum equation, using the phase-space methods\noutlined in section \\ref{PSM}. Applications of these methods to practical examples\nare reserved for a following paper (QNII)\\cite{DruCor99b}.\n\n\n\\section{Nonlinear Schr\\\"{o}dinger Model}\n\n\\label{NSM} The interaction between photons in a fiber is mediated through\nthe dielectric material constituting the fiber. The coupling to the dielectric\nintroduces frequency dependent and time delayed behaviour. The complete Hamiltonian\nand its derivation have been given in the literature\\cite{s14,s208,s21,s90,Drumhill};\nwe will only go over the salient points here. The starting point is a Lagrangian\nthat generates the classical Maxwell's equations for a one-dimensional dielectric\nwaveguide, and that gives a Hamiltonian corresponding to the dielectric energy\\cite{s208}:\n\\begin{eqnarray}\nH_{D}=\\int dV\\left[ \\frac{1}{2\\mu }\|{\\mathbf{B}}\|^{2}+\\int ^{t}_{t_{0}}{\\mathbf{E}}(t')\\cdot \\dot{\\mathbf{D}}(t')dt'\\right] \\, \\, , & \\label{Hamtn1} \n\\end{eqnarray}\n where the electric field \$ \\mathbf{E}=(\\mathbf{D}-\\mathbf{P})/\\epsilon _{0} \$\nincludes the polarization response of the dielectric to an incident electric\ndisplacement \$ \\mathbf{D} \$. The field variables are then quantized by introducing\nequal-time commutators between the canonical coordinates \$ \\mathbf{D} \$ and\n\$ \\mathbf{B} \$. We note that, of course, it is also possible to make other\nchoices of canonical momenta. This choice corresponds to a dipole-coupled\\cite{PZL},\nrather than minimal-coupled fundamental Lagrangian. While different Lagrangians\nare canonically equivalent, the present choice - originally introduced\\cite{Hillery}\nby Hillery and Mlodinow in applications to dielectric theory - has the advantage\nof comparative simplicity. The Lagrangian must produce both the correct energy\\cite{Bloembergen}\nand Maxwell's equations, otherwise the conjugate momenta will contain an arbitrary\nscaling, leading to incorrect commutation relations\\cite{Hillery,s208}.\n\n\n\\subsection{Fiber-optic Hamiltonian}\n\nThe optical fiber treated will be a single transverse mode fiber with dispersion\nand nonlinearity. Since boundary effects are usually negligible in experiments,\nit is useful to first take the infinite volume limit, which effectively replaces\na summation over wave-vectors with the corresponding integral. We will start\nwith a single polarization direction (i.e., a polarization preserving fiber).\nThe more general case is summarized elsewhere\\cite{Drummond}, and will be treated in detail\nsubsequently. The basic normally ordered, nonlinear Hamiltonian for the fiber in this case\nis\\cite{s208}: \n\\begin{equation}\n\\label{eq. A1}\n\\widehat{H}_{F}=\\int dk\\hbar \\omega (k)\\widehat{a}^{\\dagger }(k)\\widehat{a}(k)-\\int d^{3}{\\mathbf{x}}\\left\\{ \\left[ \\frac{\\Delta \\chi ^{(1)}({\\mathbf{x}})}{2\\epsilon (\\omega _{0})}\\right] :\|\\widehat{\\mathbf{D}}\|^{2}({\\mathbf{x}}):+\\, \\left[ \\frac{\\chi ^{(3)}({\\mathbf{x}})}{4\\epsilon ^{3}(\\omega _{0})}\\right] :\|\\widehat{\\mathbf{D}}\|^{4}({\\mathbf{x}}):\\right\\} \\, .\n\\label{Ham_f}\n\\end{equation}\n Here \$ \\omega (k) \$ is the angular frequency of modes with wave-vector \$ k \$,\ndescribing the \\textit{linear} polariton excitations in the fiber, including\ndispersion. We will assume that \$ \\omega (k) \$ describes the average linear\nresponse of the fiber, in the limit of a spatially uniform environment. If the\nfiber is spatially nonuniform, then it is necessary to add additional inhomogeneous\nterms to the Hamiltonian, of generic form \$ \\Delta \\chi ^{(1)}(\\mathbf{x}) \$.\nAs usual, \$ \\epsilon (\\omega _{0}) \$ is the mode-average dielectric permittivity\nat a carrier frequency \$ \\omega _{0}=\\omega (k_{0}), \$ while \$ \\widehat{a}(k) \$\nis an annihilation operator defined so that \n\\begin{equation}\n\\label{eq. A2}\n\\left[ \\widehat{a}(k'),\\widehat{a}^{\\dagger }(k)\\right] =\\delta (k-k')\\, \\, .\n\\end{equation}\n The coefficient \$ \\chi ^{(3)}(\\mathbf{x}) \$ is the nonlinear coefficient\narising when the electronic polarization field is expanded as a function of\nthe electric displacement, in the commonly used Bloembergen\\cite{Bloembergen}\nnotation (the units are S.I. units, following current standard usage). This\nmay vary along the longitudinal position on the fiber, if the fiber has a variable\ncomposition. In terms of modes of the waveguide, and neglecting modal dispersion,\nthe electric displacement field operator \$ \\widehat{\\mathbf{D}}(\\mathbf{x}) \$\nis: \n\\begin{eqnarray}\n{\\widehat{\\mathbf{D}}({\\mathbf{x}})}=i\\int dk\\left[ \\frac{\\hbar k\\epsilon (\\omega (k))v(k)}{4\\pi }\\right] ^{\\frac{1}{2}}\\widehat{a}(k){\\mathbf{u}}({\\mathbf{r}})\\exp(ikx)+h.c.\\, \\, ,\n\\end{eqnarray}\n where: \n\\begin{equation}\n\\label{eq. A3}\n\\int d^{2}{\\mathbf{r}}\|{\\mathbf{u}}({\\mathbf{r}})\|^{2}=1\\, \\, .\n\\end{equation}\n Here \$ v(k)=\\partial \\omega (k)/\\partial k \$ is the group velocity. The function \$ {\\mathbf{u}}({\\mathbf{r}}) \$ gives the transverse mode structure. Although a general mode structure can be included, for the purposes of this paper we could equally well assume a square wave-guide of area \$ A \$, which gives\n\$ {\\mathbf{u}}({\\mathbf{r}})\\simeq {\\mathbf{e}}_{y}/\\sqrt{A} \$. We note here\nthat the above mode expansion for a dispersive medium is a rather general one,\nand has been worked out both from macroscopic quantization\\cite{s208}, and\nfrom a microscopic model\\cite{Drumhill} with an arbitrary number of electronic\nor phonon resonances.\n\nIn the infinite volume limit, the polariton field is defined by noting that\nthe annihilation and creation operators can be related to a quantum field using:\n\\begin{equation}\n\\label{eq. A4}\n\\widehat{\\Psi }(t,x)=\\frac{1}{\\sqrt{2\\pi }}\\int dk\\, \\widehat{a}(t,k)\\exp[i(k-k_{0})x+i\\omega _{0}t]\\, \\, .\n\\end{equation}\n This photon-density operator \$ \\widehat{\\Psi }(t,x) \$ is the slowly varying\nfield annihilation operator for the linear quasi-particle excitations of the\ncoupled electromagnetic and polarization fields traveling through the fiber\\cite{s21}.\nThe nonzero equal-time commutations relations for these Bose operators are\n\\begin{eqnarray}\n\\left[ \\widehat{\\Psi }(t,x),\\widehat{\\Psi }^{\\dagger }(t,x')\\right] =\\delta (x-x')\\, \\, .\n\\end{eqnarray}\n\nAs shown in earlier treatments\\cite{2}, the Hamiltonian [Eq.~(\\ref{Ham_f})] can now be rewritten approximately as: \n\\begin{equation}\n\\widehat{H}_{F}=\\hbar \\int dx\\int dx'\\omega (x,x')\\widehat{\\Psi }^{\\dagger }(t,x)\\widehat{\\Psi }(t,x')-\\frac{\\hbar }{2}\\int dx\\chi ^{E}(x)\\widehat{\\Psi }^{\\dagger 2}(t,x)\\widehat{\\Psi }^{2}(t,x)\\, \\, .\n\\label{ham_2f}\\end{equation}\nHere we have introduced the kernel \$ \\omega (x,x') \$, which is the linear\ndielectric component of the Hamiltonian, and a nonlinear coupling term $\\chi_E(x)$. This kernel is then Taylor expanded around \$ k=k_{0} \$, and approximated to quadratic order in \$ (k-k_{0}) \$, by:\n\\begin{eqnarray}\n\\omega (x,x') & = & \\int \\frac{dk}{2\\pi }\\omega (k)\\exp[i(k-k_{0})(x-x')]-\\frac{1}{2}k_{0}v(k_{0})\\int d^{2}{\\mathbf{r}}\\Delta \\chi ^{(1)}({\\mathbf{x}})\|{\\mathbf{u}}({\\mathbf{r}})\|^{2}\\delta (x-x')\\nonumber \\\\\n & \\simeq & [\\omega _{0}+\\Delta \\omega (x)]\\delta (x-x')+\\int \\frac{dk}{4\\pi }\\big [i\\omega' _{0}(\\partial _{x'}-\\partial _{x})+\\omega'' _{0}(\\partial _{x}\\partial _{x'})+\\cdots \\big ]\\exp[ik(x-x')]\\, \\, .\n\\end{eqnarray}\n\n\nIn writing down Eq.~(\\ref{ham_2f}), we have assumed that the frequency dependence in the nonlinear coupling can be neglected, which is a good approximation for relatively narrow band-widths. The nonlinear term\nis often called the \$ \\chi ^{(3)} \$ effect, so named because it arises from\nthe third order term in the expansion of the polarization field in terms of\nthe electric field\\cite{s05}. It causes an electronic contribution \$ n_{2e} \$\nto the intensity dependent refractive index, where: \$ n=n_{0}+In_{2}=n_{0}+I(n_{2e}+n_{2p}) \$.\nThus we define \$ \\chi ^{E} \$, in units of \$ [m/s] \$, as: \n\\begin{equation}\n\\label{eq. A5}\n\\chi ^{E}(x)\\equiv \\left[ \\frac{3\\hbar w^{2}_{0}v(k_{0})^{2}}{4\\epsilon (\\omega _{0})c^{2}}\\right] \\int d^{2}{\\mathbf{r}}\\chi ^{(3)}({\\mathbf{x}})\|{\\mathbf{u}}({\\mathbf{r}})\|^{4}\\equiv \\left[ \\frac{\\hbar n_{2e}(x)\\omega _{0}^{2}v^{2}}{{A}c}\\right] \\, \\, .\n\\end{equation}\n Here \$ {A}=[\\int d^{2}{\\mathbf{r}}\|{\\mathbf{u}}({\\mathbf{r}})\|^{4}]^{-1} \$\nis the effective modal cross-section, and \$ n_{2e} \$ is the refractive index\nchange per unit field intensity due to electronic transitions. This is less\nthan the total value observed for \$ n_{2} \$, since phonon contributions have\nyet to be included.\n\nThe free evolution part of the total Hamiltonian, which will be removed in subsequent calculations, just describes the carrier frequency \$ \\omega _{0} \$.\nThis is not needed in Heisenberg picture calculations for \$ \\widehat{\\Psi }(t,x) \$,\nsince it is cancelled by the slowly varying field definition. Next, on partial\nintegration of the derivative terms and Fourier transforming, the resulting\ninteraction Hamiltonian \$ \\widehat{{H}}_{F}' \$ describing the evolution of\n\$ \\widehat{\\Psi } \$ in the slowly varying envelope and rotating-wave approximations\nis:\n\\begin{eqnarray}\n\\widehat{{H}}_{F}' & = & \\widehat{{H}}_{F}-\\int dk\\hbar \\omega _{0}\\widehat{a}^{\\dagger }(k)\\widehat{a}(k)\\nonumber \\\\\n & = & \\frac{\\hbar }{2}\\int _{-\\infty }^{\\infty }dx\\left[ \\Delta \\omega (x)\\widehat{\\Psi }^{\\dagger }\\widehat{\\Psi }+\\frac{iv}{2}\\left( \\nabla \\widehat{\\Psi }^{\\dagger }\\widehat{\\Psi }-\\widehat{\\Psi }^{\\dagger }\\nabla \\widehat{\\Psi }\\right) +\\frac{\\omega ''}{2}\\nabla \\widehat{\\Psi }^{\\dagger }\\nabla \\widehat{\\Psi }-\\frac{\\chi ^{E}(x)}{2}\\widehat{\\Psi }^{\\dagger 2}\\widehat{\\Psi }^{2}\\right] \\, \\, .\n\\label{ham_fd}\n\\end{eqnarray}\n\n\nFor simplicity, only quadratic dispersion is included here. However, the extension\nto higher-order dispersion is relatively straightforward. This can be achieved\nby including higher-order terms in the expansion of the dielectric kernel, or\nelse by treating the dispersion as part of the reservoir response function -\nas in following sections. The response function approach has the advantage that\na completely arbitrary polarization response can be included, and transformations\nto a different frame of reference are simplified. If part of the dielectric\nresponse is treated using response functions, then this part of the measured\nrefractive index must be excluded from the free Hamiltonian, to avoid double-counting.\n\n\n\\subsection{Heisenberg equation}\n\nFrom the interaction Hamiltonian [Eq.~\\ref{ham_fd}], we find the following Heisenberg\nequation of motion for the field operator propagating in the \$ +x \$ direction:\n\\begin{equation}\n\\label{eq. A6}\n\\left( v\\frac{\\partial }{\\partial x}+\\frac{\\partial }{\\partial t}\\right) \\widehat{\\Psi }(t,x)=\\left[ -i\\Delta \\omega (x)+\\frac{i\\omega ''}{2}\\frac{\\partial ^{2}}{\\partial x^{2}}+i\\chi ^{E}(x)\\widehat{\\Psi }^{\\dagger }(t,x)\\widehat{\\Psi }(t,x)\\right] \\widehat{\\Psi }(t,x)\\quad .\n\\end{equation}\n This is the quantum nonlinear Schr\\\"{o}dinger equation in the laboratory frame\nof reference, \nwhich is completely equivalent to the theory of a\n Bose gas of massive quasi-particles with an effective mass of \$\\hbar/\\omega '' \$ and an average velocity\nof \$ v \$, for photons near to the carrier frequency of interest. It includes\nthe possibility that the dielectric constant (i.e., the linear response) has\na spatial variation, through the term \$ \\Delta \\omega (x) \$ .\n\nWe note here that it is occasionally assumed that operators obey equal-space,\nrather than equal-time commutation relations. This cannot be exactly true, since\ncommutators in an interacting quantum field theory are only well-defined at\nequal times. At different times, it is possible for a causal effect to propagate\nto a different spatial location, which can therefore change the unequal-time\ncommutator. In other words, imposing free-field equal-space but unequal-time\ncommutators is not strictly compatible with causality. The assumption of equal-space\ncommutators may be used as an approximation under some circumstances, provided\ninteractions are weak. In this paper, we will use standard equal-time commutators.\n\n\n\\section{Raman Hamiltonian}\n\n\\label{RH} To the Hamiltonian given in Eq.~(\\ref{ham_fd}) must be added couplings to linear\ngain, absorption and phonon reservoirs\\cite{CD,s55,s76}. The gain and absorption\nreservoirs are discussed at length in section \\ref{GA}. The phonon field consists\nof thermal and spontaneous excitations in the displacement of atoms from their\nmean locations in the dielectric lattice. Although previous quantum treatments\nof Raman scattering have been given \\cite{s34}, it is necessary to modify these\nsomewhat in the present situation. The Raman interaction energy\\cite{Levenson,CD}\nof a fiber, in terms of atomic displacements from their mean lattice positions,\nis known to have the form: \n\\begin{eqnarray}\n{H}_{R}=\\frac{1}{2}\\sum _{j}\\eta _{j}^{R}\\vdots {\\mathbf{D}}(\\bar{\\mathbf{x}}^{j}){\\mathbf{D}}(\\bar{\\mathbf{x}}^{j})\\delta {\\mathbf{x}}^{j}+\\frac{1}{2}\\sum _{ij}\\kappa _{ij}:\\delta {\\mathbf{x}}^{i}\\delta {\\mathbf{x}}^{j}\\, \\, .\\label{6} \n\\end{eqnarray}\n Here \$ {\\mathbf{D}}(\\bar{\\mathbf{x}}^{j}) \$ is the electric displacement\nat the j-th mean atomic location \$ \\bar{\\mathbf{x}}^{j},\\; \\delta {\\mathbf{x}}^{j} \$\nis the atomic displacement operator, \$ \\eta _{j}^{R} \$ is a Raman coupling\ntensor, and \$ \\kappa _{ij} \$ represents the short-range atom-atom interactions.\n\nIn order to quantize this interaction with atomic positions using our macroscopic\nquantization method, we must now take into account the existence of a corresponding\nset of phonon operators, \$ \\widehat{b}(\\omega ,x) \$ and \$ \\widehat{b}^{\\dagger }(\\omega ,x) \$.\nThese operators diagonalize the atomic displacement Hamiltonian in each fiber segment,\nand have well-defined eigen-frequencies. There are calculations\\cite{BellDean70}\nof the frequency spectrum and normal modes of vibration for vitreous silica,\nusing physical models based on the random network theory of disordered systems.\nThe computed vibrational frequency spectrum is remarkably similar to the observed\nRaman gain profile\\cite{e78}. The phonon-photon coupling induces Raman transitions\nand scattering from acoustic waves (the Brillouin effect) resulting in extra\nnoise sources and an additional contribution to the nonlinearity. The initial\nstate of phonons is thermal, with \$ n_{th}(\\omega )=\\left[ \\exp {(\\hbar \\omega /kT)}-1\\right] ^{-1} \$.\n\n\n\\subsection{Hamiltonian and Heisenberg equations}\n\nIn terms of these phonon operators, the fiber Hamiltonian in the interaction\npicture and within the rotating wave approximation for a single polarization\nis\\cite{CD} \$ \\widehat{{H}}'=\\widehat{{H}}_{R}+\\widehat{{H}}_{F}' \$, where\nwe have introduced a Raman interaction Hamiltonian: \n\\begin{equation}\n\\widehat{{H}}_{R}=\\hbar \\int _{-\\infty }^{\\infty }dx\\int _{0}^{\\infty }d\\omega \\left\\{ \\widehat{\\Psi }^{\\dagger }(x)\\widehat{\\Psi }(x)R(\\omega ,x)\\left[ \\widehat{b}(\\omega ,x)+\\widehat{b}^{\\dagger }(\\omega ,x)\\right] +\\omega \\widehat{b}^{\\dagger }(\\omega ,x)\\widehat{b}(\\omega ,x)\\right\\} \\, \\, .\n\\end{equation}\n Here, the atomic vibrations within the silica structure of the fiber are modeled\nas a continuum of localized oscillators, and are coupled to the radiation modes\nby a Raman transition with a real frequency dependent coupling \$ R(\\omega ,x) \$.\nThis coupling could be nonuniform in space, and is determined empirically through\nmeasurements of the Raman gain spectrum\\cite{CD}. The atomic displacement is\nproportional to \$ \\widehat{b}+\\widehat{b}^{\\dagger } \$, where the phonon\nannihilation and creation operators, \$ \\widehat{b} \$ and \$ \\widehat{b}^{\\dagger } \$,\nhave the equal-time commutations relations \n\\begin{eqnarray}\n\\left[ \\widehat{b}(t,\\omega ,x),\\widehat{b}^{\\dagger }(t,\\omega ',x')\\right] =\\delta (x-x')\\delta (\\omega -\\omega' )\\, \\, .\n\\end{eqnarray}\n Thus the Raman excitations are treated as an inhomogeneously broadened continuum\nof modes, localized at each longitudinal location \$ x \$. GAWBS (Guided Wave\nAcoustic Brillouin Scattering)\\cite{s795,s80,s109,s20} is a special case of this, in the\nlow-frequency limit. Since neither Raman nor Brillouin excitations are completely\nlocalized, this treatment requires a frequency and wave-number cut-off, so that\nthe field operator \$ \\widehat{\\Psi } \$ is slowly varying on the phonon scattering\ndistance scale. The corresponding coupled nonlinear operator equations are: \n\\begin{eqnarray}\n\\left( v\\frac{\\partial }{\\partial x}+\\frac{\\partial }{\\partial t}\\right) \\widehat{\\Psi }(t,x) & =i & \\left[ -\\Delta \\omega (x)+\\frac{\\omega'' }{2}\\frac{\\partial ^{2}}{\\partial x^{2}}+\\chi ^{E}(x)\\widehat{\\Psi }^{\\dagger }(t,x)\\widehat{\\Psi }(t,x)\\right] \\widehat{\\Psi }(t,x)\\nonumber \\\\\n & - & i\\left\\{ \\int ^{\\infty }_{0}R(\\omega ,x)\\left[ \\widehat{b}(t,\\omega ,x)+\\widehat{b}^{\\dagger }(t,\\omega ,x)\\right] d\\omega \\right\\} \\widehat{\\Psi }(t,x)\\, \\, ,\\nonumber \\\\\n\\frac{\\partial }{\\partial t}\\widehat{b}(t,\\omega ,x) & = & -i\\omega \\widehat{b}(t,\\omega ,x)-iR(\\omega ,x)\\widehat{\\Psi }^{\\dagger }(t,x)\\widehat{\\Psi }(t,x)\\, \\, .\n\\end{eqnarray}\n\nIn summary, the original theory of nonlinear quantum field propagation is now\nextended to include both the the electronic\nand the Raman nonlinearities. The result is a modified Heisenberg equation with a delayed\nnonlinear response to the field due to the Raman coupling. On integrating the\nRaman reservoirs, one obtains: \n\\begin{eqnarray}\n\\left( v\\frac{\\partial }{\\partial x}+\\frac{\\partial }{\\partial t}\\right) \\widehat{\\Psi }(t,x) & = & i\\left[-\\Delta \\omega (x)+\\frac{\\omega ^{\\prime \\prime }}{2}\\, \\frac{\\partial ^{2}}{\\partial {x}^{2}}+\\int _{0-}^{\\infty }dt'\\, \\chi (t',x)[\\widehat{\\Psi }^{\\dagger }\\widehat{\\Psi }](t-t',x)+\\widehat{\\Gamma }^{R}(t,x)\\right]\\widehat{\\Psi }(t,x)\\, ,\\nonumber \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\chi (t,x) & = & \\chi ^{E}(x)\\delta (t)+2\\Theta (t)\\int _{0}^{\\infty }R^{2}(\\omega ,x)\\sin (\\omega t)d\\omega \\nonumber \\\\\n\\widehat{\\Gamma }^{R}(t,x) & = & -\\int ^{\\infty }_{0}R(\\omega ,x)\\left[ \\widehat{b}(t,\\omega ,x)+\\widehat{b}^{\\dagger }(t,\\omega ,x)\\right] d\\omega, \n\\end{eqnarray}\nin which we have defined \$ \\Theta (t) \$ as the step function.\n\n\nThe operators $\\widehat{\\Gamma }^{R}$ and $\\widehat{\\Gamma }^{R\\dagger}$ are stochastic, with Fourier transforms\ndefined using the normal Fourier transform conventions for field operators:\n\\begin{eqnarray}\n\\widehat{\\Gamma }^{R}(\\omega ,x) & = & \\frac{1}{\\sqrt{2\\pi }}\\int dt\\exp (i\\omega t)\\widehat{\\Gamma }^{R}(t,x)\\\\\n\\widehat{\\Gamma }^{R\\dagger }(\\omega ,x) & = & \\frac{1}{\\sqrt{2\\pi }}\\int dt\\exp (-i\\omega t)\\widehat{\\Gamma }^{R}(t,x).\n\\end{eqnarray}\nThe frequency-space correlations are given by: \n\\begin{eqnarray}\n\\langle \\widehat{\\Gamma }^{R\\dagger }(\\omega ',x')\\, \\widehat{\\Gamma }^{R}(\\omega ,x)\\rangle & = & 2\\chi ^{\\prime \\prime }(x,\|\\omega \|)\\left[n_{th}({\|\\omega \|})+\\Theta (-\\omega )\\right]{\\delta (x-x')}{\\delta (\\omega -\\omega' )}.\n\\end{eqnarray}\nIn this expression, we introduce a Raman amplitude gain of \$ \\chi ^{\\prime \\prime } \$\nper unit photon flux, equal to the imaginary part of the Fourier transform of\n\$ \\chi (t,x) \$, so that: \$ \\chi ^{\\prime \\prime }(x,\|\\omega \|)=\\pi R^{2}(x,\|\\omega \|) \$.\nHere we use the Bloembergen normalization for response function Fourier transforms,\n\\begin{equation}\n\\widetilde{\\chi }(\\omega ,x)=\\int dt\\exp (i\\omega t)\\chi (t,x)\\, \\, ,\n\\end{equation}\n which does not have the \$ \\sqrt{2\\pi } \$ factor included.\n \n Of some significance is the physical interpretation of the correlation functions,\nwhich can be regarded as directly contributing to the normally ordered spectrum\nof the transmitted field. Given a cw carrier, the correlations when \$ \\omega \$ is positive\ncorrespond to an anti-Stokes (blue-shifted) spectral term, which is clearly\nzero unless the thermal phonon population is appreciable. However, when \$ \\omega \$ is negative, the theta function term indicates that the Stokes (red-shifted) spectral term is nonzero, due to spontaneous Stokes photons emitted even at zero temperature.\n\n\n\\subsection{Raman gain measurements}\n\nThe measured intensity gain due to Raman effects at a given relative frequency\n\$ \\omega \$ per unit length, per unit carrier photon flux \$ I_{0}=v\\langle \\widehat{\\Psi }^{\\dagger }\\widehat{\\Psi }\\rangle \$,\nis:\n\\begin{equation}\n\\frac{1}{I_{0}}\\frac{\\partial \\ln I}{\\partial x}=-2\\chi ^{\\prime \\prime }(\\omega ,x)/v^{2}\\, .\n\\end{equation}\nHere the gain is positive for Stokes-shifted frequencies (\$ \\omega <0 \$\n), and negative for anti-Stokes (\$ \\omega >0 \$), as one would expect. This\nrelationship allows the coupling to be estimated from measured Raman gain and\nfluorescence properties. The simplest way to achieve this goal is to expand\nthe Raman response function in terms of a multiple-Lorentzian model, which can\nthen be fitted to observed Raman fluorescence data using a nonlinear least-squares\nfit. We therefore expand: \n\\begin{eqnarray}\n\\chi (t,x)=\\chi ^{E}(x)\\delta (t)+\\chi (x)\\Theta (t)\\sum _{j=0}^{n}F_{j}\\delta _{j}\\exp(-\\delta _{j}t)\\sin (\\omega _{j}t)\\, . & \n\\end{eqnarray}\nFor normalization purposes, we have introduced \$ \\chi (x) \$, which is defined\nas the total effective nonlinear phase-shift coefficient per unit time and photon\ndensity (in units of \$ rad.m/s \$), obtained from the low-frequency nonlinear\nrefractive index. This is given in terms of the electronic or fast-responding\nnonlinear coefficient, \$ \\chi ^{E}(x) \$ , together with the Raman contribution,\nby integrating over time: \n\\begin{equation}\n\\label{chie}\n\\chi (x)=\\chi ^{E}(x)+2\\int ^{\\infty }_{0}\\int ^{\\infty }_{0}R^{2}(\\omega ,x)\\sin (\\omega t)d\\omega dt\\, .\n\\end{equation}\n \n\nIn the above expansion, \$ F_{j} \$ are a set of dimensionless Lorentzian\nstrengths, and \$ \\omega _{j} \$ and \$ \\delta _{j} \$ are the resonant frequencies\nand widths respectively, of the effective Raman resonances at each frequency.\nTo improve convergence, the Lorentzian strength parameters are not constrained\nto be positive. The \$ j=0 \$ Lorentzian models the Brillouin contribution\nto the response function. In general, all of these parameters could be \$ x \$-dependent,\nbut we will often assume that they are constant in space for notational simplicity.\nThe values for an \$ n=10 \$ fit \nin the case of a typical fused silica fiber \nare given in Table \\ref{lorfit},\nincluding an estimate of the effective Brillouin contribution averaged\r\nover the individual Brillouin scattering peaks. \nThe coefficient\nof the electronic nonlinearity is now obtained explicitly in terms of the total\nnonlinear refractive index: \n\\begin{eqnarray}\n\\chi ^{E}(x)=\\frac{\\hbar (1-f)n_{2}\\omega _{0}^{2}v^{2}}{{A}c}, & \n\\end{eqnarray}\n where \$ \\omega _{0} \$ is the carrier frequency, \$ {A} \$ is the effective\ncross-sectional area of the traveling mode, and \$ f \$ is the fraction of\nthe nonlinearity due to the Raman gain, which has been estimated using the procedure\noutlined above: \n\\begin{eqnarray}\nf & = & \\frac{\\chi ^{R}}{\\chi }=\\frac{2}{\\chi }\\int _{0}^{\\infty }dt\\int _{0}^{\\infty }d\\omega R^{2}(\\omega ,x)\\sin (\\omega t)\\nonumber \\\\\n & \\simeq & 0.2.\n\\end{eqnarray}\n\n\nA result of this model is that the phonon operators do not have white noise\nbehaviour. In fact, this colored noise property is significant enough to invalidate\nthe usual Markovian and rotating-wave approximations, which are therefore not\nused in the phonon equations. Of course, the photon modes may also be in a thermal\nstate of some type. However, thermal effects are typically much more important\nat the low frequencies that characterize Raman and Brillouin scattering, than\nthey are at optical frequencies. In addition, if the input is a photon field\ngenerated by a laser, any departures from coherent statistics will be rather\nspecific to the laser type, instead of having the generic properties of thermal\nfields.\n\nFinally, there is another effect which has been so far neglected. This is the ultra-low\nfrequency tunneling due to lattice defects\\cite{Perlmutter}. As this is not strictly\nlinear, it can not be included accurately in our macroscopic Hamiltonian. Despite\nthis, the effects of this \$ 1/f \$ type noise may be included approximately\nfor any predetermined temperature. This can be achieved by simply modifying\nthe refractive index perturbation term so that it becomes \$ \\Delta \\omega (t,x) \$\nand generates the known low-frequency refractive index fluctuations.\n\n\n\\section{Gain and absorption}\n\n\\label{GA} In silica optical fibers, there is a relatively flat absorption\nprofile, with a minimum absorption coefficient of approximately \$ 0.2dB/km \$\nin the vicinity of the commonly used communications wavelengths of around \$ \\lambda =1.5\\mu m \$.\nThis effect can be compensated for by the use of fiber laser amplifiers, resulting\nin nearly zero net absorption over a total link that includes both normal and\namplified fiber segments. In practical terms, this situation leads to an approximately\nuniform fiber environment, provided the net gain and loss are spatially modulated\nmore rapidly than than the pulse dispersion length. These additional effects\nneed to be included within the present Hamiltonian model, in order to have a\nfully consistent quantum theory. For wide-band communications systems, either\nwith time-domain multiplexing or frequency-domain multiplexing, it can become\nnecessary to include the frequency-dependence and spatial variation of the gain\nand loss terms. This is especially true if spectral filters are included in\nthe fiber.\n\n\n\\subsection{Absorbing reservoirs}\n\nThe absorption reservoir is modeled most simply by a coupling to a continuum\nof harmonic oscillators at resonant frequency \$ \\omega . \$ In the interaction\npicture used here, the Hamiltonian term causing rapidly varying operator evolution\nof the reservoir at the carrier frequency \$ \\omega _{0} \$ is subtracted,\nleaving: \n\\begin{equation}\n\\widehat{H}_{A}'=\\hbar \\int _{-\\infty }^{\\infty }dx\\int _{0}^{\\infty }d\\omega \\left\\{ [\\widehat{\\Psi }(x)\\widehat{a}^{\\dagger }(\\omega ,x)A(\\omega ,x)+h.c.\\, ]+(\\omega -\\omega _{0})\\widehat{a}^{\\dagger }\\widehat{a}(\\omega ,x)\\right\\} \\, \\, ,\n\\end{equation}\n where $A(\\omega, x)$ provides the frequency dependent coupling between the radiation modes and the absorption reservoirs. The reservoir annihilation and creation operators, \$ \\widehat{a} \$\nand \$ \\widehat{a}^{\\dagger } \$, have the commutation relations \n\\begin{eqnarray}\n\\left[ \\widehat{a}(\\omega ,x),\\widehat{a}^{\\dagger }(\\omega ',x')\\right] =\\delta (x-x')\\delta (\\omega -\\omega' )\\, \\, .\n\\end{eqnarray}\n\n\nThe equations for the absorbing photon reservoirs can be integrated immediately.\nThe photon reservoir variable, for instance, obeys: \n\\begin{equation}\n\\frac{\\partial }{\\partial t}\\widehat{a}(t,\\omega ,x)=-i(\\omega -\\omega _{0})\\widehat{a}(t,\\omega ,x)-iA(\\omega ,x)\\widehat{\\Psi }(t,x)\\, \\, .\n\\end{equation}\n Hence, the solutions are: \n\\begin{equation}\n\\widehat{a}(t,\\omega ,x)=\\widehat{a}(t_{0},\\omega ,x)\\, \\exp[-i(\\omega -\\omega _{0})(t-t_{0})]-iA(\\omega ,x)\\int ^{t}_{t_{0}}\\exp[-i(\\omega -\\omega _{0})(t-t')]\\widehat{\\Psi }(t',x)dt'\\, \\, ,\n\\end{equation}\n with initial correlations for the reservoir variables in the far past \$ (t_{0}\\, \\rightarrow \\, -\\infty ) \$\ngiven by: \n\\begin{eqnarray}\n\\langle \\widehat{a}^{\\dagger }(t_{0},\\omega ,x)\\widehat{a}(t_{0},\\omega ',x')\\rangle & = & n_{th}(\\omega )\\delta (x-x')\\delta (\\omega -\\omega' )\\, \\, ,\\nonumber \\\\\n\\langle \\widehat{a}(t_{0},\\omega ,x)\\widehat{a}^{\\dagger }(t_{0},\\omega ',x')\\rangle & = & [n_{th}(\\omega )+1]\\delta (x-x')\\delta (\\omega -\\omega' )\\, \\, .\n\\end{eqnarray}\n\n\nThe solution for \$ \\widehat{a}(t,\\omega ,x) \$ is substituted into the Heisenberg\nequation for the field evolution, giving rise to an extra time-dependent term,\nof the form: \n\\begin{eqnarray}\n-i\\int _{0}^{\\infty }A^{}(\\omega ,x)\\widehat{a}(t,\\omega ,x)d\\omega & = & -\\int _{0}^{\\infty }d\\omega \|A(\\omega ,x)\|^{2}\\int ^{t}_{t_{0}}dt'\\exp[-i(\\omega -\\omega _{0})(t-t')]\\widehat{\\Psi }(t',x)\\nonumber \\\\\n & - & i\\int _{0}^{\\infty }d\\omega A^{}(\\omega ,x)\\exp[-i(\\omega -\\omega _{0})(t-t_{0})]\\widehat{a}(t_{0},\\omega ,x)\\nonumber \\\\\n & = & -\\int _{0}^{\\infty }dt''\\, \\gamma ^{A}(t'',x)\\widehat{\\Psi }(t-t'',x)+\\widehat{\\Gamma }^{A}(t,x)\\, ,\n\\end{eqnarray}\n where \$ t''=t-t' \$ , and the response function and reservoir terms are obtained\nmost simply by extending the lower limit on the frequency integral to \$ -\\infty , \$\nintroducing only an infinitesimal error in the process, so that: \n\\begin{eqnarray}\n\\gamma ^{A}(t,x) & \\approx & \\Theta (t)\\int _{-\\infty }^{+\\infty }d\\omega \\, \|A(\\omega ,x)\|^{2}\\, \\exp[-i(\\omega -\\omega _{0})t]\\nonumber \\\\\n\\widehat{\\Gamma }^{A}(t,x) & = & -i\\int _{0}^{\\infty }d\\omega A^{\\dagger }(\\omega ,x)\\exp[-i(\\omega -\\omega _{0})(t-t_{0})]\\widehat{a}(t_{0},\\omega ,x).\\label{ga_res}\n\\end{eqnarray}\n\n\nThe response function integral represents a deterministic or `drift' term to\nthe motion, with a Fourier transform of: \n\\begin{equation}\n\\widetilde{\\gamma }^{A}(\\omega ,x)=\\int \\gamma ^{A}(t,x)\\exp(i\\omega t)dt=\\gamma ^{A}(\\omega ,x)+i\\gamma ^{A\\prime \\prime }(\\omega ,x)\\, \\, ,\n\\end{equation}\n so that the amplitude loss rate is:\n\\begin{equation}\n\\gamma ^{A}(\\omega ,x)={\\pi }\\, \|A(\\omega _{0}+\\omega ,x)\|^{2}\\, \\, .\n\\end{equation}\nIn the case of a spatially uniform reservoir with a flat spectral density, the\nWigner-Weiskopff approximation (neglecting frequency shifts) gives a uniform\nMarkovian loss term with: \n\\begin{eqnarray}\n\\gamma ^{A}(t) & \\approx & \\widetilde{\\gamma }^{A}\\delta (t)\\, \\, ,\n\\end{eqnarray}\n where the average amplitude loss coefficient is: \n\\begin{eqnarray}\n\\widetilde{\\gamma }^{A}=\\widetilde{\\gamma }^{A}(0) & = & \\int _{0}^{\\infty }\\int _{-\\infty }^{+\\infty }dtd\\omega \\, \|A(\\omega )\|^{2}\\, \\exp[-i(\\omega -\\omega _{0})t]\\nonumber \\\\\n & = & \\gamma ^{A}+i\\gamma ^{A\\prime \\prime }\\, .\n\\end{eqnarray}\nThis approximation, known as the Markov approximation, is generally rather accurate\nfor the absorbing reservoirs, whose response does not typically vary fast with\nfrequency. An exception to this rule would be any case involving resonant impurities\nin the fiber, or very short pulses whose bandwidth is comparable to the frequency-scale\nof absorption changes.\n\nThe second quantity in Eq.~(\\ref{ga_res}), \$ \\widehat{\\Gamma }^{A}(t,x) \$, behaves like a stochastic\nterm due to the random initial conditions. Neglecting the frequency dependence\nof the thermal photon number, the corresponding correlation functions are \n\\begin{eqnarray}\n{\\langle }\\widehat{\\Gamma }^{A}(t,x)\\widehat{\\Gamma }^{A\\dagger }(t',x'){\\rangle }\\; & = & \\int _{0}^{\\infty }d\\omega \|A(\\omega ,x)\|^{2}\\exp[-i(\\omega -\\omega _{0})(t-t')][n_{th}(\\omega )+1]\\delta (x-x')\\nonumber \\\\\n & \\approx & [\\gamma ^{A}(t-t',x)+\\gamma ^{A}(t'-t,x)][n_{th}(\\omega _{0})+1]\\, \\delta (x-x')\\, ,\n\\end{eqnarray}\n and: \n\\begin{eqnarray}\n{\\langle }\\widehat{\\Gamma }^{A\\dagger }(t',x')\\widehat{\\Gamma }^{A}(t,x){\\rangle }\\; & = & \\int _{0}^{\\infty }d\\omega \|A(\\omega ,x)\|^{2}\\exp[-i(\\omega -\\omega _{0})(t-t')]n_{th}(\\omega )\\delta (x-x')\\nonumber \\\\\n & \\approx & [\\gamma ^{A}(t-t',x)+\\gamma ^{A}(t'-t,x)]n_{th}(\\omega _{0})\\, \\delta (x-x')\\, .\n\\end{eqnarray}\nAt optical or infra-red frequencies, it is a good approximation to set \$ n_{th}(\\omega _{0})=0 \$.\nOn Fourier transforming the noise sources, one then obtains: \n\\begin{equation}\n{\\langle }\\widehat{\\Gamma }^{A}(\\omega ,x)\\widehat{\\Gamma }^{A\\dagger }(\\omega ',x'){\\rangle }=2\\gamma ^{A}(\\omega ,x)\\delta ({x}-{x}^{\\prime })\\delta ({\\omega }-{\\omega }^{\\prime }).\n\\end{equation}\nAgain taking the simplifying case of a spatially uniform reservoir in the Wigner-Weiskopff\nlimit, this reduces to: \n\\begin{eqnarray}\n{\\langle }\\widehat{\\Gamma }^{A}(t,x)\\widehat{\\Gamma }^{A\\dagger }(t',x'){\\rangle }\\; & = & 2\\gamma ^{A}\\delta (t-t')\\delta (x-x')\\nonumber \\\\\n{\\langle }\\widehat{\\Gamma }^{A\\dagger }(t,x)\\widehat{\\Gamma }^{A}(t',x'){\\rangle }\\; & = & 0.\n\\end{eqnarray}\n\n\nNote that the dimensions for the amplitude relaxation rates are \$ [\\gamma ^{A}]\\, =\\, s^{-1}. \$\nIt is easy to show that \$ 2\\gamma ^{A}/v \$ corresponds to the usual linear\nabsorption coefficient for fibers during propagation. A typical measured absorption\nfigure in current fused silica communications fibers is \$ 0.2dB/km \$ in the\nminimum region of absorption (near \$ \\lambda =1.5\\mu m \$). The corresponding\nabsorption coefficient is \$ 2\\gamma ^{A}/v\\, \\simeq \\, 2.3\\times 10^{-5}m^{-1}. \$\nTo the extent that this effect is wavelength (and hence frequency) dependent,\nthe resulting dispersion can be included as well, giving rise to a complete\nresponse function \$ \\gamma ^{A}(t) \$ for absorption. Non-Markovian effects\nlike this can either be neglected completely -- which is a good approximation for slowly varying\nabsorption in undoped fiber -- or else included in the correlation functions of the\nreservoirs as given above.\n\nThe physical meaning of the reservoir operator spectral correlations is best\nunderstood by considering the effect of these terms on photodetection,\nwhich according to photodetection theory, means a normally ordered field\ncorrelation. This involves normally ordered reservoir correlations to lowest\norder. Since these are zero, we conclude that the absorbing reservoirs essentially\nadd no quantum noise that is observable via normal photodetection methods.\n\n\n\\subsection{Waveguide laser amplifiers}\n\n\\label{sec:Amp} The equations for gain or laser reservoirs are generally more\ncomplex, involving the nonlinear response of atomic impurities added to provide\nsome gain in the fiber medium. This also involves a pump process (usually from\na semiconductor laser) to maintain the lasing atoms in an inverted state. In\nthe case of silica fibers, a commonly used lasing transition is provided by \nerbium impurities\\cite{Mears}. The effect of these gain reservoirs\nis typically to introduce new types of dispersion, owing to the frequency dependence\nof the gain\\cite{Desurvire}. In addition, there are new nonlinear effects,\ndue to the effects of saturation - which in turn depend on the pumping intensity.\n\nIt is possible to develop a detailed theory of erbium laser amplifiers. However,\nthis paper will treat the simplest possible quantum theory of a traveling-wave\nquantum-limited laser amplifier. More details of the quantum theory, including\nnonlinear effects, are treated elsewhere\\cite{DrumRayn}. However, the simple\ntheory presented here provides a microscopic justification for observed quantum\nnoise effects in fiber amplifier chains. In particular, it reproduces the results\nof the phenomenological theory of Gordon and Haus\\cite{s35}, which is known\nto give predictions in accord with soliton transmission experiments. The resulting\n``Gordon-Haus jitter'' can be reduced through the use of filtering techniques.\nAssuming that the laser amplifier is polarization-insensitive, we again omit\nthe polarization index. The reservoir variable \$ \\widehat{\\sigma }_{\\mu }=\|1\\rangle _{\\mu }\\langle 2\|_{\\mu } \$\nis an atomic transition operator, which induces a near-resonant atomic transition\nfrom an upper to a lower state, with two-level transitions having an assumed\ndensity of \$ \\rho (\\omega ,x) \$ in position and resonant angular frequency\n\$ \\omega \$ .\n\nThese quantum effects are modeled here by including gain reservoirs in the Hamiltonian, coupled by a frequency dependent term $G(\\omega,x)$ to the radiating field. Here the\ngain terms \$ \\widehat{\\sigma }^{\\pm }(\\omega ,x,t) \$ represent the raising\nand lowering Pauli field operators, for two-level lasing transitions at frequency\n\$ \\omega \$. In more detail, we have gain given by an interaction Hamiltonian:\n\\begin{equation}\n\\widehat{H}_{G}'=\\hbar \\int _{-\\infty }^{\\infty }dx\\int _{0}^{\\infty }d\\omega \\left\\{ [\\widehat{\\Psi }\\widehat{\\sigma }^{+}(\\omega ,x)G(\\omega ,x)+h.c.]+\\frac{\\omega -\\omega _{0}}{2}\\sigma ^{z}(\\omega ,x)\\right\\} \\, \\, ,\n\\end{equation}\n where the atomic raising and lowering field operators, \$ \\widehat{\\sigma }^{\\pm } \$,\nare defined in terms of discrete Pauli operators, by: \n\\begin{eqnarray}\n\\widehat{\\sigma }^{+}(\\omega ,x,t) & = & \\frac{1}{\\sqrt{\\rho (\\omega ,x)}}\\sum _{\\mu }\|2\\rangle \\langle 1\|_{\\mu }\\exp(-i\\omega _{0}t)\\delta (x-x_{\\mu })\\delta (\\omega -\\omega _{\\mu })\\, ,\\nonumber \\\\\n\\widehat{\\sigma }^{-}(\\omega ,x,t) & = & \\frac{1}{\\sqrt{\\rho (\\omega ,x)}}\\sum _{\\mu }\|1\\rangle \\langle 2\|_{\\mu }\\exp(i\\omega _{0}t)\\delta (x-x_{\\mu })\\delta (\\omega -\\omega _{\\mu })\\, ,\\nonumber \\\\\n\\widehat{\\sigma }^{z}(\\omega ,x,t) & = & \\frac{1}{\\rho (\\omega ,x)}\\sum _{\\mu }[\|2\\rangle \\langle 2\|-\|1\\rangle \\langle 1\|]_{\\mu }\\delta (x-x_{\\mu })\\delta (\\omega -\\omega _{\\mu })\\, .\n\\end{eqnarray}\nThese operators are in general time-dependent in the Heisenberg picture, but have the equal-time commutation relations:\n\\begin{eqnarray}\n\\left[ \\widehat{\\sigma }^{+}(t,\\omega ,x),\\widehat{\\sigma }^{-}(t,\\omega ',x')\\right] =\\widehat{\\sigma }^{z}(t,\\omega ,x) & \\delta (x-x')\\delta (\\omega -\\omega' ). & \n\\end{eqnarray}\n In the limit of complete inversion, with linear response and pure inhomogeneous\nbroadening, \n\\begin{equation}\n\\frac{\\partial }{\\partial t}\\widehat{\\sigma }^{-}(t,\\omega ,x)=-i(\\omega -\\omega _{0})\\widehat{\\sigma }^{-}(t,\\omega ,x)+i\\widehat{\\sigma }^{z}(t,\\omega ,x)G(\\omega ,x)\\widehat{\\Psi }(t,x)\\, \\, .\n\\end{equation}\n Hence, the solutions in the amplifier case are: \n\\begin{eqnarray}\n\\widehat{\\sigma }^{-}(t,\\omega ,x)&=&\\widehat{\\sigma }^{-}(t_{0},\\omega ,x)\\, \\exp[-i(\\omega -\\omega _{0})(t-t_{0})]\\nonumber \\\\ &+&iG(\\omega ,x)\\int ^{t}_{t_{0}}\\exp[-i(\\omega -\\omega _{0})(t-t')]\\widehat{\\sigma }^{z}(t'\\omega ,x)\\widehat{\\Psi }(t',x)dt'\\, \\, .\n\\end{eqnarray}\n With complete inversion, \$ \\langle \\widehat{\\sigma }^{z}(t_{0},\\omega ,x)\\rangle =1 \$\n, so the initial correlations for the reservoir variables in the far past \$ (t_{0}\\, \\rightarrow \\, -\\infty ) \$\nare given by: \n\\begin{eqnarray}\n\\langle \\widehat{\\sigma }^{+}(t_{0},\\omega ,x)\\widehat{\\sigma }^{-}(t_{0},\\omega ',x')\\rangle & = & \\delta (x-x')\\delta (\\omega -\\omega' )\\, \\, ,\\nonumber \\\\\n\\langle \\widehat{\\sigma }^{-}(t_{0},\\omega ,x)\\widehat{\\sigma }^{+}(t_{0},\\omega ',x')\\rangle & = & 0\\, \\, .\n\\end{eqnarray}\n \n We substitute the solution for \$ \\widehat{\\sigma }^{-}(t,\\omega ,x) \$ into\nthe Heisenberg equation for the field evolution, assuming no depletion of the\ninversion, and trace over the atomic gain reservoirs. This gives rise to an\nextra time-dependent term, of the form: \n\\begin{eqnarray}\n-i\\int _{0}^{\\infty }G^{}(\\omega ,x)\\widehat{\\sigma }^{-}(t,\\omega ,x)d\\omega & = & \\int _{0}^{\\infty }d\\omega \|G(\\omega ,x)\|^{2}\\int ^{t}_{t_{0}}dt'\\exp[-i(\\omega -\\omega _{0})(t-t')]\\widehat{\\Psi }(t',x)\\nonumber \\\\\n & - & i\\int _{0}^{\\infty }d\\omega G^{}(\\omega ,x)\\exp[-i(\\omega -\\omega _{0})(t-t_{0})]\\sigma ^{-}(t_{0},\\omega ,x)\\nonumber \\\\\n & = & \\int _{0}^{\\infty }dt''\\, \\gamma ^{G}(t'',x)\\widehat{\\Psi }(t-t'',x)+\\widehat{\\Gamma }^{G}(t,x),\n\\end{eqnarray}\n where \$ t''=t-t' \$, as before. This gives: \n\\begin{eqnarray}\n\\gamma ^{G}(t,x) & \\approx & \\Theta (t)\\int _{-\\infty }^{+\\infty }d\\omega \\, \|G(\\omega ,x)\|^{2}\\, \\exp[-i(\\omega -\\omega _{0})t] \\nonumber \\\\\n\\widehat{\\Gamma }^{G}(t,x) & \\approx & -i\\int _{-\\infty }^{\\infty }d\\omega G^{}(\\omega ,x)\\exp[-i(\\omega -\\omega _{0})(t-t_{0})]\\sigma ^{-}(t_{0},\\omega ,x)\\,\\, .\n\\end{eqnarray}\n Fourier transforming the response function gives: \n\\begin{eqnarray}\n\\widetilde{\\gamma }^{G}(\\omega ,x)=\\int \\gamma ^{G}(t,x)\\exp(i\\omega t)dt=\\gamma ^{G}(\\omega ,x)+i\\gamma ^{G\\prime \\prime }(\\omega ,x)\\, \\, , & \n\\end{eqnarray}\n and the (real) resonant amplitude gain coefficient is: \n\\begin{eqnarray}\n\\gamma ^{G}(\\omega ,x)={\\pi }\\, \|G(\\omega +\\omega _{0},x)\|^{2}\\, \\, .\n\\end{eqnarray}\n \n \nAs with the loss case, \$ \\widehat{\\Gamma }^{G}(t,x) \$ behaves like a stochastic\nterm due to the random initial conditions. The corresponding correlation functions\nare \n\\begin{eqnarray}\n{\\langle }\\widehat{\\Gamma }^{G\\dagger }(t',x')\\widehat{\\Gamma }^{G}(t,x){\\rangle }\\; & = & \\int _{0}^{\\infty }d\\omega \|G(\\omega ,x)\|^{2}\\exp[i(\\omega -\\omega _{0})(t-t')]\\delta (x-x')\\nonumber \\\\\n & = & [\\gamma ^{G}(t-t',x)+\\gamma ^{G^{}}(t'-t,x)]\\, \\delta (x-x')\\, \\, .\n\\end{eqnarray}\nFourier transforming these noise sources gives: \n\\begin{equation}\n{\\langle }\\widehat{\\Gamma }^{G\\dagger }(\\omega ',x')\\widehat{\\Gamma }^{G}(\\omega ,x){\\rangle }=2\\gamma ^{G}(\\omega ,x)\\delta ({x}-{x}^{\\prime })\\delta ({\\omega }-{\\omega }^{\\prime })\n\\end{equation}\nTaking the uniform fiber in the Wigner-Weiskopff limit as before, so \$ \\gamma ^{G}=\\gamma ^{G}(0,x) \$,\nthis reduces to: \n\\begin{eqnarray}\n{\\langle }\\widehat{\\Gamma }^{G}(t,x)\\widehat{\\Gamma }^{G\\dagger }(t',x'){\\rangle }\\; & = & 0\\nonumber \\\\\n{\\langle }\\widehat{\\Gamma }^{G\\dagger }(t,x)\\widehat{\\Gamma }^{G}(t',x'){\\rangle }\\; & = & 2\\gamma ^{G}\\, \\delta (t-t')\\delta (x-x').\n\\end{eqnarray}\n\n\nThe dimensions for the amplitude gain are \$ [\\gamma ^{G}]\\, =\\, s^{-1}. \$\nOn Fourier transforming, the response function can be related to the measured\nintensity gain \$ 2{\\mbox {\\textrm{Re}}}[\\widetilde{\\gamma }^{G}(\\omega ,x)/v] \$\nat any frequency offset \$ \\omega \$, relative to the carrier frequency \$ \\omega _{0} \$.\nThis allows one to obtain the linear gain coefficient for fibers during propagation.\nSince measured laser gain figures can be much greater than the absorption, it\nis possible to compensate for fiber absorption with relatively short regions\nof gain. The results presented here are only valid in the linear gain regime.\nMore generally, a functional Taylor expansion up to at least third order in\nthe field would be needed to represent the full nonlinear response of the laser\namplifier, together with additional quantum noise terms.\n\nFinally, it is necessary to consider the result of incomplete inversion of an\namplifier. Here, the noninverted atoms give rise to absorption, not gain, and\nwill generate additional quantum-noise and absorption response terms. These\nmust be treated as in the previous section, including non-Markovian effects\nif the absorption line is narrow-band. An important consequence is that the\nmeasured gain only gives the difference between the gain and the loss. This\ndoesn't cause any problems with the deterministic response function -- but it\ndoes cause difficulties in determining the amplifier quantum noise levels, which\ncan only be uniquely determined through spontaneous fluorescence measurements.\nObviously, the lowest quantum noise levels occur when all the lasing transitions\nare completely inverted.\n\n\nThe physical meaning of the reservoir operator spectral correlations \nfor the amplifier case is clearly quite different to the case of the absorber.\nIf we consider the effect of these terms on photodetection as before,\nwhich means a normally ordered field correlation, we should look again at\nthe normally ordered correlations of the reservoirs. Since these are \nno longer zero, we conclude that the amplifying reservoirs emit\nfluorescent photons due to spontaneous emission over the amplifier bandwidth.\n\n\n\n\\section{Combined Heisenberg Equations}\n\n\\label{CHE} Coupling linear gain and absorption reservoirs in this way to the \nRaman-modified Heisenberg equation leads\nto a generalized quantum nonlinear Schr\\\"{o}dinger equation. Such equations are \nsometimes called quantum Langevin\nequations. In the present case of a single polarization, the resulting field\nequations are:\n\\begin{eqnarray}\n\\left( v\\frac{\\partial }{\\partial x}+\\frac{\\partial }{\\partial t}\\right) \\widehat{\\Psi }(t,x) & = & -\\int _{0}^{\\infty }dt'\\, \\gamma (t',x)\\widehat{\\Psi }(t-t',x)+\\widehat{\\Gamma }(t,x)\\nonumber \\\\\n & + & i\\left[\\frac{\\omega ^{\\prime \\prime }}{2}\\, \\frac{\\partial ^{2}}{\\partial {x}^{2}}+\\int _{0-}^{\\infty }dt'\\, \\chi (t')[\\widehat{\\Psi }^{\\dagger }\\widehat{\\Psi }](t-t',x)+\\widehat{\\Gamma }^{R}(t,x)\\right]\\widehat{\\Psi }(t,x).\n\\label{RMQ}\\end{eqnarray}\n In this equation, \n\\begin{eqnarray}\n\\gamma (t,x)=\\gamma ^{A}(t,x)-\\gamma ^{G}(t,x)+i\\Delta \\omega (x)\\delta (t)\n\\end{eqnarray}\n is a net linear response function due to a coupling to linear gain/absorption\nreservoirs, including the effects of a spatially varying refractive index. This\ncan be Fourier transformed, giving: \$ \\widetilde{\\gamma }(\\omega ,x)={\\gamma }(\\omega ,x)+i\\gamma ^{\\prime \\prime }(\\omega ,x) \$,\nwhere \$ {\\gamma }(\\omega ,x)<0 \$ for gain, and \$ {\\gamma }(\\omega ,x)>0 \$\nfor absorption. Similarly, \$ \\widehat{\\Gamma }(t,x) \$ is the linear quantum\nnoise due to gain and absorption. The actual measured intensity gain at frequency\n\$ \\omega +\\omega _{0} \$ is given in units of \$ [m^{-1}] \$ , by: \n\\begin{equation}\n\\frac{\\partial \\ln I}{\\partial x}=2(\\gamma ^{G}(\\omega ,x)-\\gamma ^{A}(\\omega ,x))/v\\, .\n\\end{equation}\n\nThe stochastic terms have the correlations\n\\begin{eqnarray}\n\\langle \\widehat{\\Gamma }^{R\\dagger }(\\omega ',x')\\, \\widehat{\\Gamma }^{R}(\\omega ,x)\\rangle & = & 2\\chi ^{\\prime \\prime }(x,\|\\omega \|)[n_{th}({\|\\omega \|})+\\Theta (-\\omega )]{\\delta (x-x')}{\\delta (\\omega -\\omega' )}\\nonumber \\\\\n\\langle \\widehat{\\Gamma }^{\\dagger }(\\omega ',x')\\, \\widehat{\\Gamma }(\\omega ,x)\\rangle & = & 2\\gamma ^{G}(\\omega ,x)\\delta ({x}-{x}^{\\prime })\\delta ({\\omega }-{\\omega }^{\\prime })\\nonumber \\\\\n\\langle \\widehat{\\Gamma }(\\omega ,x)\\, \\widehat{\\Gamma }^{\\dagger }(\\omega' ,x')\\rangle & = & 2\\gamma ^{A}(\\omega ,x)\\delta ({x}-{x}^{\\prime })\\delta ({\\omega }-{\\omega }^{\\prime })\\, ,\n\\end{eqnarray}\nwhere we have introduced minimal linear quantum noise terms $\\Gamma$ and $\\Gamma^{\\dagger}$ for the gain/absorption\nreservoirs, and where thermal photons have been neglected (since usually \$ \\hbar \\omega _{0}>>kT \$, as explained in section \\ref{GA}). Equation (\\ref{RMQ}) can be easily generalized to include nonlinear absorption\nor laser saturation effects, relevant to amplifiers with intense fields, but\nthese terms are omitted here for simplicity.\n\n\nThis complete Heisenberg equation gives a consistent quantum theoretical description\nof dispersion, nonlinear refractive index, Raman/GAWBS scattering, linear gain,\nand absorption. It is important to notice that the reservoir correlations have\na simple physical interpretation, especially in the zero-temperature limit.\nNormally ordered noise correlations occur when there is gain, anti-normally\nordered correlations when there is absorption. This is the reason why the normally ordered\nRaman noise correlations vanish at zero temperature for positive frequencies.\nAt low temperatures, Raman processes only cause absorption to occur at positive\ndetunings from a pump frequency. Thermal correlations have a more classical\nbehaviour, and occur for both types of operator ordering.\n\nIt is often useful to do calculations in a simpler model, in which we include\nthe effects of uniform gain and loss in a moving frame. This can either be carried\nout using a standard moving frame (\$ x_{v}=x-vt \$), or with a propagative\ntime (\$ t_{v}=t-x/v \$ ) as in the original Gordon-Haus calculations. For\npropagative calculations, it is most convenient to use photon flux operators\n\\begin{equation}\n\\widehat{\\Phi }(t_{v},x)=\\sqrt{v}\\widehat{\\Psi }(t,x)\\, .\n\\end{equation}\n For long pulses, assuming a uniform gain/loss response in the frequency domain,\nthe propagative transformation gives the following approximate equations: \n\\begin{eqnarray}\n\\frac{\\partial }{\\partial x}\\widehat{\\Phi }(t_{v},x) & = & -\\int _{0}^{\\infty }dt_{v}'\\, \\frac{\\gamma (t_{v}',x)}{v}\\widehat{\\Phi }(t_{v}-t_{v}',x)+\\widehat{\\Gamma }(t)/\\sqrt{v}\\nonumber \\\\\n & + & i\\biggl [-\\frac{k^{\\prime \\prime }}{2}\\, \\frac{\\partial ^{2}}{\\partial {t_{v}}^{2}}+\\int _{0-}^{\\infty }dt'\\, \\frac{\\chi (t_{v}')}{v^{2}}[\\widehat{\\Phi }^{\\dagger }\\widehat{\\Phi }](t_{v}-t_{v}',x)+\\frac{1}{v}\\widehat{\\Gamma }^{R}\\biggr ]\\widehat{\\Phi }(t_{v},x)\\, .\n\\end{eqnarray}\nIn addition, if the pulses are narrow-band compared to the gain and loss bandwidths,\nand the reservoirs are uniform, then the gain and absorption reservoirs are\nnearly delta-correlated, with \n\\begin{eqnarray}\n\\langle \\widehat{\\Gamma }^{\\dagger }(t,x_{v})\\, \\widehat{\\Gamma }(t',x_{v}')\\rangle & = & 2\\gamma ^{G}\\delta ({x_{v}}-{x_{v}}^{\\prime })\\delta (t-t')\\nonumber \\\\\n\\langle \\widehat{\\Gamma }(t,x_{v})\\, \\widehat{\\Gamma }^{\\dagger }(t',x_{v}')\\rangle & = & 2\\gamma ^{A}\\delta ({x_{v}}-{x_{v}}^{\\prime })\\delta (t-t').\n\\end{eqnarray}\n\n\nIt is essentially this set of approximate equations that corresponds to those\nused to predict the soliton\\cite{e36} self-frequency shift\\cite{s348} and related\neffects\\cite{s35} in soliton propagation, except for the omission of the Raman reservoir\nterms.\n\n\n\\section{Phase-space methods}\n\n\\label{PSM} The Heisenberg equations are not readily soluble in their present form.\nTo generate numerical equations for analytic calculations or for simulation,\noperator representation theory can be used. There is more than one possible\nmethod, depending on which phase-space representation is used. The positive-\$ P \$\nrepresentation, for example, produces exact results\\cite{s14,s21,CD} provided phase-space boundary terms are negligible, while a\ntruncated Wigner representation\\cite{s23,s12} gives approximate results \nthat are valid in the limit of large photon number.\nIt is important to note that the Wigner method represents symmetrically ordered\nrather than normally ordered operator products, and so has finite quantum noise\nterms even for a vacuum field. These can be thought of as corresponding to the\nshot noise detected in a homodyne or local-oscillator measurement, while the\npositive-\$ P \$ representation represents normally ordered operators, and\ntherefore corresponds to direct-detection noise.\n\nEither technique can be used for this problem, each with its characteristic\nadvantages and disadvantages. The positive-\$ P \$ representation, although\nexact, uses an enlarged phase-space which therefore takes longer to simulate\nnumerically. It only includes normally ordered noise and initial conditions,\nand this is an advantage in some cases, since the resulting noise is zero in\nthe vacuum state. The Wigner technique is simpler, and for large mode occupations,\nits results are accurate enough for many purposes. However, it has the drawback\nthat it includes symmetrically ordered vacuum fluctuations.\n\nFirst, we expand the field operators in terms of operators for the free-field\nmodes. Applying the appropriate operator correspondences to the master equation\nfor the reduced density operator \$ \\widehat{\\rho }_{\\Psi} \$ in which the reservoir\nmodes have been traced over, namely \n\\begin{eqnarray}\n\\widehat{\\dot{\\rho }}_{\\Psi }={\\textrm{Tr}}_{R}\\widehat{\\dot{\\rho }}={\\textrm{Tr}}_{R}\\frac{1}{i\\hbar }\\left[ \\widehat{H},\\widehat{\\rho }\\right] ,\n\\end{eqnarray}\n gives a functional equation for the corresponding operator representation.\n\nIn the positive-$P$ case, the equation is defined on a functional phase-space\nof double the classical dimensions, so that a complete expansion in terms of\na coherent-state basis \$ \|\\Psi \\rangle \$ is obtained:\n\\begin{equation}\n\\hat{\\rho }_{\\Psi}(t)=\\int \\int P(t,\\Psi ,\\overline{\\Psi })\\, \\frac{\|\\Psi \\rangle \\langle \\overline{\\Psi }\|}{\\langle \\overline{\\Psi }\|\\Psi \\rangle }d[\\Psi ]\\, d[\\overline{\\Psi }]\\, .\n\\end{equation}\n The resulting Fokker-Planck equation for the positive distribution \$ P(t,\\Psi ,\\overline{\\Psi }) \$\nhas only second order derivative terms. Sometimes the notation \$ \\Psi ^{+}=\\overline{\\Psi }^{} \$is\nused to indicate the stochastic field that corresponds to the hermitian conjugate\nof \$ \\Psi \$.\n\nThe equation for the Wigner function \$ W(t,\\Psi ) \$ also contains third and\nfourth order derivative terms, which may be neglected at large photon number.\nThe resultant Fokker-Planck equation in either case, can be converted into equivalent \nIto stochastic equations for the phase space variables $\\Psi$ (and $\\overline{\\Psi}$). \nPhysical quantities can be calculated by forming the average over many stochastic realizations, or\npaths, in phase-space. For example, in the positive-$P$ representation, $\\langle\\overline{\\Psi}^\\Psi\\rangle_{\\rm stochastic} = \\langle\\widehat\\Psi^{\\dagger} \\widehat\\Psi\\rangle_{\\rm quantum}$, while in the Wigner representation, $\\langle\\Psi^\\Psi\\rangle_{\\rm stochastic} = \\frac{1}{2}\\langle\\widehat\\Psi^{\\dagger} \\widehat\\Psi + \\widehat\\Psi \\widehat\\Psi^{\\dagger}\\rangle_{\\rm quantum}$.\n\nIt should be clear from this that the positive-$P$ representation directly generates an intensity corresponding to the usual normally ordered intensity that is detected in direct photodetection. The Wigner representation, however, generates an intensity result that includes some vacuum fluctuations. In a computer simulation with a finite number $M$ of modes, we must correct the Wigner result by subtracting $M/2$ from any simulated photon number, or $vM/2$ from any calculated photon flux, in order to obtain the direct photodetection result. For the calculation of a homodyne measurement, the Wigner method will give the most directly suitable result with symmetric ordering. In this case it is the positive-$P$ representation that will need correction terms added to it. Once these corrections are made, either method will give similar results, although the sampling error may not be identical.\n\n\\subsection{Modified nonlinear Schr\\\"{o}dinger equation}\n\nStandard custom in fiber optics applications\\cite{e36} involves using the propagative\nreference frame with the normalized variables: \$ \\tau =(t-x/v)/t_{0} \$ and\n\$ \\zeta =x/x_{0} \$, where \$ t_{0} \$ is a typical pulse duration used for\nscaling purposes, and \$ x_{0}=t_{0}^{2}/\|k''\|\\sim 1km \$ for dispersion shifted\nfiber. This change of variables is useful only when slowly varying second order\nderivatives involving \$ \\zeta \$ can be neglected, which occurs for \$ vt_{0}/x_{0}\\ll 1 \$.\nFor typical values of the parameters used in experiments, this inequality is\noften well-satisfied (\$ vt_{0}\\sim 10^{-4}m \$). To make it simpler to compare\nwith this usage, we will make the same transformation for the stochastic equations\nthat are equivalent to our complete operator equations, and scale the variables\nused in a dimensionless form.\n\nFor definiteness, we will now focus on the spatially uniform case. The resultant\nequation, which includes gain and loss, is a Raman-modified nonlinear Schr\\\"{o}dinger\n(NLS) equation with stochastic noise terms: \n\\begin{eqnarray}\n\\frac{\\partial }{\\partial \\zeta }\\phi (\\tau ,\\zeta ) & = & -\\int ^{\\infty }_{-\\infty }d\\tau' g(\\tau -\\tau' )\\phi (\\tau' ,\\zeta )+\\Gamma (\\tau ,\\zeta )\\nonumber \\\\\n & + & \\left[\\pm \\frac{i}{2}\\frac{\\partial ^{2}\\phi }{\\partial \\tau ^{2}}+i\\int ^{\\infty }_{-\\infty }d\\tau' h(\\tau -\\tau' )\\phi ^{}(\\tau' ,\\zeta )\\phi (\\tau' ,\\zeta )+\\Gamma ^{R}(\\tau ,\\zeta )\\right]\\phi (\\tau ,\\zeta ),\n\\end{eqnarray}\n where \$ \\phi =\\Psi \\sqrt{vt_{0}/\\overline{n}} \$ is a dimensionless photon\nfield amplitude. The photon flux is \$ \|\\phi \|^{2}\\overline{n}/t_{0} \$, and\n\$ \\overline{n}=\|k''\|{A}c/(n_{2}\\hbar \\omega _{c}^{2}t_{0})=v^{2}t_{0}/\\chi x_{0} \$\nis the typical number of photons in a soliton pulse of width \$ t_{0} \$,\nfor scaling purposes. The positive sign in front of the second derivative term\napplies for anomalous dispersion (\$ k''<0 \$), which occurs for longer wavelengths,\nand the negative sign applies for normal dispersion (\$ k''>0 \$). A similar\nequation is obtained in the positive-$P$ case, except that \$ \\phi ^{} \$and\n\$ \\Gamma ^{R}(\\tau ,\\zeta ) \$ are replaced by non-complex-conjugate fields\ndenoted \$ \\phi ^{+} \$ and \$ \\Gamma ^{R+}(\\tau ,\\zeta ) \$ respectively:\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial \\zeta }\\phi ^{+} (\\tau ,\\zeta ) & = & -\\int ^{\\infty }_{-\\infty }d\\tau' g^(\\tau -\\tau' )\\phi ^{+} (\\tau' ,\\zeta )+\\Gamma ^{+} (\\tau ,\\zeta )\\nonumber \\\\\n & + & \\left[\\mp \\frac{i}{2}\\frac{\\partial ^{2}\\phi ^{+} }{\\partial \\tau ^{2}}-i\\int ^{\\infty }_{-\\infty }d\\tau' h ^{}(\\tau -\\tau' )\\phi (\\tau' ,\\zeta )\\phi ^{+} (\\tau' ,\\zeta )+\n \\Gamma ^{+ R}(\\tau ,\\zeta )\\right]\\phi ^{+} (\\tau ,\\zeta ),\n\\end{eqnarray}\nThe equations in \$ \\phi \$ and \$ \\phi^{+} \$ both have the same\nadditive noises and identical mean values, only differing by the independent\nparts of the multiplicative noise sources - which therefore generate nonclassical\nquantum statistics.\n\nThe causal linear response function \$ g(\\tau ) \$ is defined as: \n\\begin{eqnarray}\ng(\\tau )=\\frac{\\gamma (\\tau t_{0})x_{0}}{v}\\, . & \n\\end{eqnarray}\nIf the Fourier transform of this function is \$ \\widetilde{g}(\\Omega )=g(\\Omega )+ig'(\\Omega ) \$,\nthen we can relate this to dimensionless intensity gain $\\alpha ^{G}(\\Omega )$ and loss $\\alpha ^{A}(\\Omega )$, at a relative\n(dimensionless) detuning of \$ \\Omega \$ , by: \n\\begin{equation}\n2g(\\Omega )=\\alpha ^{A}(\\Omega )-\\alpha ^{G}(\\Omega )\\, \\, .\n\\end{equation}\n\n\nThe causal nonlinear response function \$ h(\\tau ) \$ is normalized so that\n\$ \\int h(\\tau )d\\tau =1 \$, and it includes both electronic and Raman nonlinearities:\n\\begin{eqnarray}\nh(\\tau )=h^{E}(\\tau )+h^{R}(\\tau )=\\frac{\\overline{n}x_{0}\\chi (\\tau t_{0})}{v^{2}}\\, . & \n\\end{eqnarray}\n The Raman response function \$ h^{R}(\\tau ) \$ causes effects like the soliton\nself-frequency shift\\cite{s348}. The response function Fourier transform is given by:\n\\begin{equation}\n\\widetilde{h}(\\Omega )=\\int dt\\exp (i\\Omega \\tau )h(\\tau )=h^{\\prime }(\\Omega )+ih^{\\prime \\prime }(\\Omega ).\n\\end{equation}\nThis definition has the property that the value of $\\widetilde{h}(\\Omega )=\\widetilde{h}(\\omega t_0 )$ is a dimensionless number, which depends on the frequency $\\omega$ only, independent of the time-scale used for normalization. The Raman gain, whose spectrum has been extensively\nmeasured\\cite{e78}, can be modeled as a sum of \$ n \$ Lorentzians, as\nexplained in section \\ref{RH} and as illustrated in Fig.~1. \n\nThis expansion as \$ n \$ Lorentzians gives a response function of the form \n\\begin{eqnarray}\nh^{R}(t/t_{0})=\\Theta (t)\\sum _{j=0}^{n}F_{j}\\delta _{j}t_{0}\\exp(-\\delta _{j}t)\\sin (\\omega _{j}t), & \n\\end{eqnarray}\n It is most convenient to express these in terms of dimensionless parameters\n\$ \\Omega _{j}=\\omega _{j}t_{0} \$ and \$ \\Delta _{j}=\\delta _{j}t_{0} \$,\ngiving: \n\\begin{eqnarray}\nh^{R}(\\tau )=\\Theta (\\tau )\\sum _{j=0}^{n}F_{j}\\Delta _{j}\\exp(-\\Delta _{j}\\tau )\\sin (\\Omega _{j}\\tau ).\n\\end{eqnarray}\n Here \$ \\Delta _{j} \$ are the equivalent dimensionless widths (corresponding\nto damping), and the \$ \\Omega _{j} \$ are the dimensionless center frequencies,\nall in normalized units. It is useful to compare these results with the dimensionless\nRaman gain \$ \\alpha ^{R}(\\Omega ) \$ , normalized following Gordon\\cite{s348},\nwhich uses a characteristic time-scale of \$ t_{0} \$ . The relationship of\nmacroscopic coupling \$ R(\\omega ) \$ to measured Raman gain \$ \\alpha ^{R}(\\Omega ) \$\nis \$ R^{2}(\\omega )={\\chi \\alpha ^{R}(\\omega t_{0})}/{2\\pi } \$. It follows\nthat the dimensionless gain function is \n\\begin{equation}\n\\alpha ^{R}(\\Omega )=2\|h^{\\prime \\prime }(\\Omega )\|\\, \\, .\n\\end{equation}\n\nThese stochastic partial differential equations can be discretized and, without\nany further approximation, can be numerically simulated\\cite{s23,Werner1997}\nusing a split-step Fourier integration routine. The equations include all the\ncurrently known noise physics significant in soliton propagation, including\neffects like the soliton self-frequency shift. Guided acoustic wave Brillouin\nscattering \\cite{s795,s109,s80} noise sources are included in the Raman gain\nfunction. These have little effect on the position of an isolated soliton, but\nare important for long-range soliton collision effects\\cite{s20} that occur\nin pulse-trains.\n\n\n\\subsection{Initial conditions }\n\nThe initial conditions for the calculations could involve any required quantum\nstate, if the +$P$ representation is used. In the case of the Wigner equations,\nonly a subset of possible states can be represented with a positive probability\ndistribution. The usual initial condition is the multi-mode coherent state,\nsince this is the simplest model for the output of mode-locked lasers. In general,\nthere could be extra technical noise. We note that the choice of a coherent\nstate is the simplest known model of a laser sources. To represent this in the\npositive-$P$ distribution is simple; one just takes \n\\begin{equation}\n\\phi _{P}(\\tau ,0)=[\\phi _{P}^{+}(\\tau ,0)]^{}=\\langle \\widehat{\\phi }(\\tau ,0)\\rangle.\n\\end{equation}\n In the Wigner case, which corresponds to symmetric operator ordering, one must\nalso include complex quantum vacuum fluctuations, in order to correctly represent\noperator fields. For coherent inputs, the Wigner vacuum fluctuations are Gaussian,\nand are correlated as \n\\begin{eqnarray}\n\\langle \\phi _{W}(\\tau ,0)\\rangle & = & \\langle \\widehat{\\phi }(\\tau ,0)\\rangle \\nonumber \\\\\n\\langle \\Delta \\phi _{W}(\\tau ,0)\\Delta \\phi _{W}^{}(\\tau' ,0)\\rangle & = & \\frac{1}{2\\overline{n}}\\delta (\\tau -\\tau' ).\n\\end{eqnarray}\n We note that these equations imply that an appropriate correction is made for\nlosses at the input interface, so that the mean-field evolution is known at\nthe fiber input face.\n\n\n\\subsection{Wigner noise}\n\nBoth fiber loss and the presence of a gain medium contribute quantum noise to\nthe equations in this symmetrically ordered representation. The complex gain/absorption\nnoise enters the Wigner equation through an additive stochastic term \$ \\Gamma \$,\nwhose correlations are obtained by averaging the normally and anti-normally\nordered reservoir correlation functions given previously, together with appropriate\nvariable changes. This symmetrically ordered noise source is present for both\ngain and loss reservoirs. Thus, \n\\begin{equation}\n\\label{gain_{c}or}\n\\langle \\Gamma (\\Omega ,\\zeta )\\Gamma ^{}(\\Omega' ,\\zeta' )\\rangle =\\frac{(\\alpha ^{G}(\\Omega )+\\alpha ^{A}(\\Omega ))}{2\\overline{n}}\\delta (\\zeta -\\zeta' )\\delta (\\Omega -\\Omega' ),\n\\end{equation}\n where \$ \\Gamma (\\Omega ,\\zeta ) \$ is the Fourier transform of the noise\nsource: \n\\begin{eqnarray}\n\\Gamma (\\Omega ,\\zeta ) & = & \\frac{1}{\\sqrt{2\\pi }}\\int ^{\\infty }_{-\\infty }d\\tau \\Gamma (\\tau ,\\zeta )\\exp(i\\Omega \\tau )\\nonumber \\\\\n\\Gamma ^{}(\\Omega ,\\zeta ) & = & \\frac{1}{\\sqrt{2\\pi }}\\int ^{\\infty }_{-\\infty }d\\tau \\Gamma ^{}(\\tau ,\\zeta )\\exp(-i\\Omega \\tau ).\n\\end{eqnarray}\n Similarly, the real Raman noise, which appears as a multiplicative stochastic\nvariable \$ \\Gamma ^{R} \$, has correlations \n\\begin{equation}\n\\label{Raman_{c}or}\n\\langle \\Gamma ^{R}(\\Omega ,\\zeta )\\Gamma ^{R}(\\Omega' ,\\zeta' )\\rangle =\\frac{\\alpha ^{R}(\|{\\Omega }\|)}{\\overline{n}}\\left[ n_{th}(\|\\Omega \|/t_{0})+\\frac{1}{2}\\right] \\delta (\\zeta -\\zeta' )\\delta (\\Omega -\\Omega' )\\, .\n\\end{equation}\n\n\nThus the Raman noise is strongly temperature-dependent, but it also contains\na spontaneous component which provides vacuum fluctuations even at \$ T=0 \$.\nSince the spontaneous component can occur through coupling to either a gain\nor a loss reservoir, in a symmetrically ordered representation, it is present\nfor both positive and negative frequency detunings.\n\nIt must be remembered here that the noise terms in the Wigner representation\ndo not correspond to normally ordered correlations, and so have no direct\ninterpretation in terms of photodetection experiments. Any predictions made\nwith this method of calculation need to be corrected by subtracting the appropriate\ncommutators, to convert the results into a normally ordered form. This is the\nreason why there is no obvious distinction between the amplifier and absorber cases.\n\n\n\\subsection{+$P$ noise}\n\nThe positive P-representation is a useful alternative strategy, because it does\nnot require truncation of higher order derivatives in a Fokker-Planck equation,\nand corresponds directly to observable normally ordered, time-ordered operator\ncorrelations. It has no vacuum fluctuation terms. Provided the phase-space boundary\nterms are negligible, one can then obtain a set of c-number stochastic differential\nequations in a phase-space of double the usual classical dimensions. These are\nvery similar to the classical equations. Here the additive stochastic term is\nas before, except it \\textit{only} depends on the gain term \$ \\alpha ^{G} \$;\nthe conjugate term \$ \\Gamma ^{} \$ is used in the \$ \\phi ^{+} \$ equation:\n\\begin{eqnarray}\n\\langle \\Gamma (\\Omega ,\\zeta )\\, \\Gamma ^{}(\\Omega ',\\zeta ^{\\prime })\\rangle =\\frac{\\alpha ^{G}(\\Omega )}{\\overline{n}}\\delta (\\zeta -\\zeta' )\\delta (\\Omega -\\Omega' ) .\n\\end{eqnarray}\n Since this representation is normally ordered, the only noise sources present\nare due to the gain reservoirs. There is no vacuum noise term for the absorbing\nreservoirs, because absorption simply maps a coherent state into another coherent\nstate.\n\nThe complex terms \$ \\Gamma ^{R} \$, \$ \\Gamma ^{R+} \$ include both Raman\nand electronic terms (through \$ {h}'(\\Omega ) \$). As elsewhere in this paper,\nwe regard \$ \\Gamma ^{R+}(\\Omega ,\\zeta ) \$ as a hermitian conjugate Fourier\ntransform (with the opposite sign frequency exponent):\n\\begin{eqnarray}\n\\Gamma ^{+}(\\Omega ,\\zeta ) & = & \\frac{1}{\\sqrt{2\\pi }}\\int ^{\\infty }_{-\\infty }d\\tau \\Gamma ^{+}(\\tau ,\\zeta )\\exp(-i\\Omega \\tau ).\n\\end{eqnarray}\nThis quantity is not the same as \$ \\Gamma ^{R}(\\Omega ,\\zeta ) \$, since it involves a noise source that is in general independent. In some cases, where classical noise is dominant (and nonclassical squeezing is negligible), we can ignore this fact, and approximately set \$ \\Gamma ^{R+}(\\Omega ,\\zeta ) =\\Gamma ^{R}(\\Omega ,\\zeta ) \$. More generally, we obtain the following results:\r\n\\begin{eqnarray}\n\\langle \\Gamma ^{R}(\\Omega ,\\zeta )\\, \\Gamma ^{R}(\\Omega ',\\zeta ')\\rangle & = & \\delta (\\zeta -\\zeta ')\\, \\delta (\\Omega +\\Omega ')\\left\\{ \\left[n_{th}(\|\\Omega \|/t_{0})+1/2\\right]\\alpha ^{R}(\|\\Omega \|)-i\\, {h}'(\\Omega )\\right\\} /{\\overline{n}}\\nonumber \\\\\n\\langle \\Gamma ^{R+}(\\Omega ',\\zeta ')\\, \\Gamma ^{R}(\\Omega ,\\zeta )\\rangle & = & \\delta (\\zeta -\\zeta ')\\, \\delta (\\Omega -\\Omega ')\\left[n_{th}(\|\\Omega \|/t_{0})+\\Theta (-\\Omega )\\right]\\alpha ^{R}(\|\\Omega \|)/{\\overline{n}}.\n\\end{eqnarray}\nThis equation is an expected result, since it states that when \$ \\Omega <0 \$\nthe spectral intensity of noise due to the Stokes process, in which a photon\nis down-shifted in frequency by an amount \$ \\Omega \$ with the production\nof a phonon of the same frequency, is proportional to \$ n_{th}^{}+1. \$ However\nthe anti-Stokes process in which a phonon is absorbed (\$ \\Omega >0 \$) is\nonly proportional to \$ n_{th}^{}. \$ Thus, at low temperatures the only direct\nnoise effect is that due to the Stokes process, which can be interpreted physically\nas originating in spontaneous photon emission, detectable through photodetection.\n\nAs one might expect, the two forms of equation are identical at high phonon\noccupation numbers, when classical noise is so large that it obscures the differences\ndue to the operator orderings of the two representations. Another, less obvious,\nresult is that the two equations have identical additive noise sources, provided\nthe gain and loss are balanced. To understand this, we can see that in the absence\nof any net gain or loss, the differences in the operator correlations due to\nordering is a constant, contained in the initial conditions.\n\nHowever, when gain and loss are not equal, the additive noise sources are quite different.\nIn particular, the Wigner representation has noise contributions from both\ntypes of reservoir. On the other hand, the normally ordered +$P$ method only\nleads to additive noise when there is a real fluorescent field present, which\nis detectable through photodetection. This corresponds physically to some kind\nof gain, either due to the presence of an amplifier, or through Raman effects.\n\nIn general, the Wigner and +$P$ reservoir correlations are obtainable simply by\nexamining the expectation values of the Heisenberg reservoir terms, with symmetric\nand normal ordering respectively. The additional term proportional to \$ {h}'(\\Omega ) \$\nin the +$P$ noise terms is due to dispersive nonlinear effects, and gives rise\nto a nonclassical noise source which is responsible for the observed quantum\nsqueezing effects. Extensions required to treat polarization dependent Raman scattering are given elsewhere\\cite{Drummond}.\n\n\n\\section{Conclusions}\n\n\\label{Cs} Our major conclusion is that quantum noise effects due to the intrinsic\nfinite-temperature phonon reservoirs and finite bandwidth amplification or absorption can be readily modeled using stochastic equations. The equations themselves\nhave the usual classical form, together with correction terms that we can describe as quantum noise terms. The precise form of the correction terms depends in detail\non the representation employed (although this difference is purely due to operator\nordering), as well as the physical origin of the reservoir couplings. These\ncorrection terms can be non-Markovian or nonuniform in space. The generation\nof the corresponding stochastic noises is a straightforward numerical procedure,\nand generally much simpler than the use of noncommuting operators. By contrast,\nthe original operator equations have no practical numerical solution in most\ncases, due to the exponential growth of the dimension of the underlying Hilbert\nspace with the number of modes and photons involved.\n\nDetailed applications to short-pulse soliton communications will be given in\na subsequent paper\\cite{DruCor99b}. In general, the increasing bandwidth, reducing pulse energies\nand greater demands placed on fiber communications and sensors mean that these\nquantum limits are becoming increasingly important. Already, limits set by quantum\namplifiers are known to have great significance in long-distance laser-amplified\ncommunications systems. We note that the quantum theory given here also establishes\nthe levels of quantum noise in silica fibers in more general situations. Examples\nof this are for dispersion-managed fiber communications \\cite{Smith1997,Lakoba1999},\nand for fiber ring lasers with relatively low gain\\cite{Collings1998}. Similarly,\nthese equations set the limits for experiments using spectral filtering and\nrelated techniques to generate sub-shot-noise pulses\\cite{Fri1996,Werner1999PRA}\nin optical fibers.\n\n\\acknowledgments\n\nWe would like to acknowledge helpful comments on this paper by Wai S. Man.\n\n\\begin{thebibliography}{10}\n\\bibitem{s14} S.~J. Carter, P.~D. 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Corney, ``Quantum noise in optical fibers {II}: {R}aman jitter in soliton communications,'' {\\em submitted to Journal of the Optical Society of America B}. \n\\bibitem{s208}P.~D. Drummond, ``Electromagnetic quantization in dispersive inhomogeneous\nnonlinear dielectrics,'' Physical Review A \\textbf{42}, 6845--6857 (1990). \n\\bibitem{s21} P.~D. Drummond and S.~J. Carter, ``Quantum-Field Theory of Squeezing in Solitons,''\nJournal of the Optical Society of America B \\textbf{4}, 1565--1573 (1987); P.~D.\nDrummond, S.~J. Carter, and R.~M. Shelby, ``Time Dependence of Quantum Fluctuations\nin Solitons,'' Optics Letters \\textbf{14}, 373--375 (1989).\n\\bibitem{s90} B. Yurke and M.~J. Potasek, ``Solution to the Initial Value Problem for the\nQuantum Nonlinear Schroedinger Equation,'' Journal of the Optical Society of\nAmerica B \\textbf{6}, 1227--1238 (1989). \n\\bibitem{Drumhill} P. D. Drummond and M. Hillery, ``Quantum theory of dispersive electromagnetic\nmodes'', Phys. Rev. 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A \\textbf{60}, R781--R784 (1999). \n\\begin{figure}\n\n\\caption{The parallel polarization Raman gain \\protect\\protect\\protect\$ \|\\Im \\{\\tilde{h}(\\omega t_0 )\\}\| = \|h''(\\omega t_0 )\|\\protect \\protect \\protect \$\nfor the 11-Lorentzian model (continuous lines) and the single-Lorentzian model\n(dashed lines), for a temperature of \\protect\\protect\\protect\$ T=300K\\protect \\protect \\protect \$. }\n\\end{figure}\n\\begin{table}\n\n\\caption{Fitting parameters for the 11-Lorentzian model of the Raman gain function \\protect\\protect\\protect\$ h^{R}(t/t_{0})\\protect \\protect \\protect \$.\nAll frequencies are in T.rad/s.}\n\\begin{tabular}{\|c\|c\|c\|c\|}\n\\hline \n\$ j \$&\n \$ F_{j} \$&\n \$ \\omega _{j} \$&\n \$ \\delta _{j} \$\\\\\n\\hline \n0 &\n 0.16 &\n 0.005 &\n 0.005 \\\\\n\\hline \n1 &\n -0.3545 &\n 0.3341 &\n 8.0078 \\\\\n\\hline \n2 &\n 1.2874 &\n 26.1129 &\n 46.6540 \\\\\n\\hline \n3 &\n -1.4763 &\n 32.7138 &\n 33.0592 \\\\\n\\hline \n4 &\n 1.0422 &\n 40.4917 &\n 30.2293 \\\\\n\\hline \n5 &\n -0.4520 &\n 45.4704 &\n 23.6997 \\\\\n\\hline \n6 &\n 0.1623 &\n 93.0111 &\n 2.1382 \\\\\n\\hline \n7 &\n 1.3446 &\n 99.1746 &\n 26.7883 \\\\\n\\hline \n8 &\n -0.8401 &\n 100.274 &\n 13.8984 \\\\\n\\hline \n9 &\n -0.5613 &\n 114.6250 &\n 33.9373 \\\\\n\\hline \n10 &\n 0.0906 &\n 151.4672 &\n 8.3649 \\\\\n\\hline \n\\end{tabular}\n\\label{lorfit}\n\\end{table}\n\n\\end{thebibliography}\n\\end{document}\n\n" } ]	[ { "name": "quant-ph9912095.extracted_bib", "string": "{s14 S.~J. Carter, P.~D. Drummond, M.~D. Reid, and R.~M. Shelby, ``Squeezing of Quantum Solitons,'' Physical Review Letters 58, 1841--1844 (1987)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{2 P.~D. Drummond and S.~J. Carter, ``Quantum-Field Theory of Squeezing in Solitons,'' Journal of the Optical Society of America B 4, 1565--1573 (1987)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{Solexp M. Rosenbluh and R. M. Shelby, ``Squeezed Optical Solitons'', Phys. Rev. Lett. 66, 153--156 (1991); P.D. Drummond, R. M. Shelby, S. R. Friberg and Y. Yamamoto, ``Quantum solitons in optical fibres,'' Nature 365, 307--313 (1993)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{s35 J.~P. Gordon and H.~A. Haus, ``Random Walk of Coherently Amplified Solitons in Optical Fiber Transmission,'' Optics Letters 11, 665--667 (1986)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{s53 H.~A. Haus and W.~S. Wong, ``Solitons in Optical Communications,'' Reviews of Modern Physics 68, 423--444 (1996)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{DruCor99b P.~D. Drummond and J.~F. Corney, ``Quantum noise in optical fibers {II: {Raman jitter in soliton communications,'' {\\em submitted to Journal of the Optical Society of America B." }, { "name": "quant-ph9912095.extracted_bib", "string": "{s208P.~D. Drummond, ``Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,'' Physical Review A 42, 6845--6857 (1990)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{s21 P.~D. Drummond and S.~J. Carter, ``Quantum-Field Theory of Squeezing in Solitons,'' Journal of the Optical Society of America B 4, 1565--1573 (1987); P.~D. Drummond, S.~J. Carter, and R.~M. Shelby, ``Time Dependence of Quantum Fluctuations in Solitons,'' Optics Letters 14, 373--375 (1989)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{s90 B. Yurke and M.~J. Potasek, ``Solution to the Initial Value Problem for the Quantum Nonlinear Schroedinger Equation,'' Journal of the Optical Society of America B 6, 1227--1238 (1989)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{Drumhill P. D. Drummond and M. Hillery, ``Quantum theory of dispersive electromagnetic modes'', Phys. Rev. A 59, 691--707 (1999)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{PZL E. Power, S. Zienau, ``Coulomb gauge in nonrelativistic quantum electrodynamics and the shape of spectral lines'', Philos. Trans. Roy. Soc. Lond. A251, 427 -454 (1959); R. Loudon, The Quantum Theory of Light (Clarendon Press, Oxford, 1983)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{Hillery M. Hillery and L. D. Mlodinow, ``Quantization of electrodynamics in nonlinear dielectric media'', Phys. Rev. A30, 1860 (1984)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{Bloembergen N. 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Yu, ``General Quantum Theory of Nonlinear Optical-Pulse Propagation,'' Physical Review A 51, 817--829 (1995); S.-S. Yu and Y. Lai, ``Impacts of Self-Raman Effect and Third-Order Dispersion on Pulse Squeezed State Generation Using Optical Fibers,'' Journal of the Optical Society of America B 12, 2340--2346 (1995)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{s34T. von Foerster and R.~J. Glauber, ``Quantum Theory of Light Propagation in Amplifying Media,'' Physical Review A 3, 1484--1511 (1971); I. A. Walmsley and M. G. Raymer, ``Observation of Macroscopic Quantum Fluctuations in Stimulated {Raman Scattering,'' Phys. Rev. Lett. 50, 962--965 (1983)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{LevensonM. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic Press, New York, 1982)." }, { "name": "quant-ph9912095.extracted_bib", "string": "{BellDean70P. Dean, ``The Vibrational Properties of Disordered Systems: Numerical Studies'', Rev. Mod. 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quant-ph9912096	Quantum noise in optical fibers II: \\ Raman jitter in soliton communications	[ { "author": "J. F. Corney$^{1" }, { "author": "2" } ]	\noindent The dynamics of a soliton propagating in a single-mode optical fiber with gain, loss, and Raman coupling to thermal phonons is analyzed. Using both soliton perturbation theory and exact numerical techniques, we predict that intrinsic thermal quantum noise from the phonon reservoirs is a larger source of jitter and other perturbations than the gain-related Gordon-Haus noise, for short pulses ($ \lesssim 1ps $), assuming typical fiber parameters. The size of the Raman timing jitter is evaluated for both bright and dark (topological) solitons, and is larger for bright solitons. Because Raman thermal quantum noise is a nonlinear, multiplicative noise source, these effects are stronger for the more intense pulses needed to propagate as solitons in the short-pulse regime. Thus Raman noise may place additional limitations on fiber-optical communications and networking using ultrafast (subpicosecond) pulses.	[ { "name": "ramanII11.tex", "string": "\\documentstyle[osa,eqsecnum,manuscript]{revtex}\n\\tightenlines\n\\begin{document}\n\n\n\\title{Quantum noise in optical fibers II: \\\\\nRaman jitter in soliton communications}\n\n\n\\author{J. F. Corney$^{1,2}$ and P. D. Drummond$^{1}$}\n\n\n\n\\address{$^{1}$Department of Physics, The University of Queensland, St. Lucia, QLD 4072, Australia \\\\\n$^{2}$Department of Mathematical Modelling, Technical University of Denmark, DK-2800 Lyngby, Denmark}\n\n\n\\date{\\today{}}\n\n\\maketitle\n\\begin{abstract}\n\n\n\\noindent The dynamics of a soliton propagating in a single-mode optical fiber\nwith gain, loss, and Raman coupling to thermal phonons is analyzed. Using both soliton perturbation\ntheory and exact numerical techniques, we predict that intrinsic thermal quantum\nnoise from the phonon reservoirs is a larger source of jitter and\nother perturbations than the gain-related Gordon-Haus noise, for short pulses (\$ \\lesssim 1ps \$), assuming typical fiber parameters. The size of the\nRaman timing jitter is evaluated for both bright and dark (topological) solitons,\nand is larger for bright solitons. Because Raman thermal quantum noise is a\nnonlinear, multiplicative noise source, these effects are stronger for the more\nintense pulses needed to propagate as solitons in the short-pulse regime. Thus Raman\nnoise may place additional limitations on fiber-optical communications and networking\nusing ultrafast (subpicosecond) pulses.\n\\end{abstract}\n%\\pacs{03.75.Fi,05.30.Jp,32.80.Pj,42.50.Lc,42.50.Dv}\n\n%OSA journals require OCIS numbers instead of PACS, so I have picked out a few:\n\\pacs{060.4510, 270.5530, 270.3430, 190.4370, 190.5650, 060.2400}\n\n\n\\section{Introduction}\n\nIn this paper, we analyze in some detail the effects of Raman noise on solitons.\nIn particular we derive approximate analytic expressions and provide further\ndetail for the precise numerical results published earlier\\cite{s147}. The\nmotivation for this study is essentially that coupling to phonons is one property\nof a solid medium that definitely does not obey the nonlinear Schr\\\"{o}dinger\nequation. The presence of Raman interactions plays a major role in perturbing the fundamental\nsoliton behaviour of the nonlinear Schr\\\"{o}dinger equation in optical fibers. This perturbation is in addition to the more straightforward gain/loss effects that produce the well-known Gordon-Haus effect\\cite{s35}.\n\nThe complete derivation of the quantum theory for optical fibers is given in\nan earlier paper\\cite{DruCor99a}, denoted (QNI). That paper presented a detailed derivation of the quantum Hamiltonian, and included quantum noise effects due to nonlinearities,\ngain, loss, Raman reservoirs and Brillouin scattering. Phase-space techniques allowed the quantum Heisenberg equations of motion to be mapped onto stochastic partial differential equations. The result was a generalized nonlinear Schr\\\"odinger equation, which can be solved numerically or with perturbative analytical techniques.\n\nThe starting point for this paper is the phase-space equation for the case of a single polarization mode, obtained using a truncated Wigner representation, which is accurate in the limit of large\nphoton number. We use both soliton perturbation theory and numerical integration of\nthe phase-space equation to calculate effects on soliton propagation of all known quantum noise sources, with good agreement between the two methods.\n\nOur main result is that the Raman noise due to thermal phonon reservoirs is\nstrongly dependent on both temperature and pulse intensity. At room temperature,\nthis means that Raman jitter and phase noise become steadily more important\nas the pulse intensity is increased, which occurs when a shorter soliton pulse\nis required for a given fiber dispersion. Using typical fiber parameters, we\nestimate that Raman-induced jitter is more important than the well-known Gordon-Haus jitter for pulses shorter than about one picosecond. Although we do not analyze\nthis in detail here, we note that similar perturbations may occur during the\ncollision of short pulses in a frequency-multiplexed environment.\n\n\n\\section{Raman-Schr\\\"{o}dinger Model}\n\\label{PT}\nWe begin with the Raman-modified stochastic nonlinear Schr\\\"odinger equation [Eq.\\ (6.3) of (QNI)], obtained using the Wigner representation, for simplicity:\n\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial \\zeta }\\phi (\\tau ,\\zeta ) & = & -\\int ^{\\infty }_{0 }d\\tau' g(\\tau -\\tau' )\\phi (\\tau' ,\\zeta )+\\Gamma (\\tau ,\\zeta )+\\nonumber \\\\\n & + & i\\left[\\pm \\frac{1}{2}\\frac{\\partial ^{2}\\phi }{\\partial \\tau ^{2}}+\\int ^{\\infty }_{0 }d\\tau' h(\\tau -\\tau' )\\phi ^{}(\\tau' ,\\zeta )\\phi (\\tau' ,\\zeta )+i\\Gamma ^{R}(\\tau ,\\zeta )\\right]\\phi (\\tau ,\\zeta ) \\, \\, .\n\\label{RMS}\n\\end{eqnarray}\nHere \$ \\phi =\\Psi \\sqrt{vt_{0}/\\overline{n}} \$ is a dimensionless photon\nfield amplitude, while $\\tau =(t-x/v)/t_{0}$ and\n$\\zeta =x/x_{0}$, where \$ t_{0} \$ is a typical pulse duration used for\nscaling purposes and $ x_{0}=t_{0}^{2}/\|k''\|$ is a characteristic dispersion length. The group velocity $v$ and the dispersion relation $k''$ are calculated at the carrier frequency $\\omega_0$. \n\nApart from a cut-off dependent vacuum noise,\nthe photon flux is \${ \\cal J}= \|\\phi \|^{2}\\overline{n}/t_{0} \$, where\n\$ \\overline{n}=\|k''\|{\\cal A}c/(n_{2}\\hbar \\omega _{0}^{2}t_{0})=v^{2}t_{0}/\\chi x_{0} \$\nis the typical number of photons in a soliton pulse of width \$ t_{0} \$, again\nfor scaling purposes. In this definition, the fibre is assumed to have a modal cross-sectional area ${\\cal A}$ and a change in refractive index per unit intensity of $n_2$. The\npositive sign in front of the second derivative term applies for anomalous dispersion (\$ k''<0 \$), and the negative sign applies for normal dispersion (\$ k''>0 \$). The functions $g$ and $h$ are gain/loss and Raman scattering response functions respectively, while $\\Gamma$ and $\\Gamma^R$ are stochastic terms, discussed below.\n\n Similar, but more accurate, equations occur with the positive-$P$ representation, although in this case, the phase-space dimension is doubled. In order to simplify the calculations further, we assume that gain and loss in\nthe fiber are broadband relative to the soliton bandwidth, and balance exactly. This requires that the amplifier\nsections in the fiber are sufficiently close together (of the order of the soliton scaling length or\nless) so that the soliton can propagate without distortion\\cite{e375}. \n\nFor the analytic calculations, we also assume that the Raman nonlinear response function is instantaneous\non the timescale of the soliton width. This is equivalent to assuming that\nthe phonon modes are heavily damped, and means that\nthe Raman coupling leads to only incoherent scattering of the propagating radiation.\nWhile this approximation neglects the well-known self-frequency shift\\cite{MooWonHau94,AtiMysChrGal,s11}, we find that the self-frequency shift by itself is not a major cause of jitter for the distance scales we consider here. This assumption can be improved at the expense of\nmore complicated analytic calculations. However, the full equations are used in the\nnumerical simulations, which agree quite well with our analytic predictions.\n\nThe Raman-modified equation then reduces to \n\\begin{eqnarray}\n\\frac{\\partial }{\\partial \\zeta }\\phi (\\tau ,\\zeta )=\\left[ \\pm \\frac{i}{2}\\frac{\\partial ^{2}}{\\partial \\tau ^{2}}+i\\phi ^{}(\\tau ,\\zeta )\\phi (\\tau ,\\zeta )\\right] \\phi (\\tau ,\\zeta )+\\Gamma^{C} (\\tau ,\\zeta )\\, \\, , & \\label{NLSE} \n\\end{eqnarray}\n where the bracketed term represents the usual nonlinear Schr\\\"odinger (NLS) equation in normalized, propagative form. The combined noise sources have been grouped together\nas \n\\begin{eqnarray}\n\\Gamma^{C} (\\tau ,\\zeta )=\\Gamma (\\tau ,\\zeta ) +i\\Gamma ^{R}(\\tau ,\\zeta )\\phi (\\tau ,\\zeta )\\, \\, . & \n\\end{eqnarray}\n\n\n\\subsection{Initial conditions and quantum evolution}\n\nEquation (\\ref{NLSE}) is a complex-number equation that can accurately represent quantum operator evolution through the inclusion of various noise sources. In the absence of any noise sources, this equation reduces to the classical nonlinear Schr\\\"odinger equation. This deterministic limit corresponds to taking $\\overline{n} \\rightarrow \\infty$. As well as the noise sources explicitly appearing in Eq. (\\ref{NLSE}), there must be noise in the initial conditions to properly represent a quantum state in the Wigner representation. Regardless of the initial quantum state chosen, there must be at least minimal level of initial fluctuations in $\\phi$ to satisfy Heisenberg's uncertainty principle. We choose to begin with a multimode coherent state, which contains this minimal level of initial quantum noise and which is an accurate model of mode-locked laser output. This is also the simplest model for the output of mode-locked lasers, and we note that, in general, there could be extra technical noise. For coherent inputs, the Wigner vacuum fluctuations are Gaussian, and are correlated as \n\\begin{eqnarray}\n\\langle \\Delta \\phi(\\tau ,0)\\Delta \\phi^{}(\\tau' ,0)\\rangle & = & \\frac{1}{2\\overline{n}}\\delta (\\tau -\\tau' ).\n\\end{eqnarray}\n\nPhysical quantities can be calculated from this phase-space simulation by averaging products of $\\phi$ and $\\phi^$ over many stochastic trajectories. In this Wigner representation, these stochastic averages correspond to the ensemble averages of symmetrically-ordered products of quantum operators, such as those representing homodyne measurements and other measurements of phase. \n\n\\subsection{Wigner noise}\n\nBoth fiber loss and the presence of a gain medium each contribute quantum noise\nto the equations in the symmetrically-ordered Wigner representation. The complex gain/absorption\nnoise enters the nonlinear Sch\\\"odinger (NLS) equation through an additive stochastic term \$ \\Gamma \$, whose correlations are:\n\\begin{eqnarray}\n\\langle \\Gamma(\\Omega ,\\zeta )\\Gamma ^{}(\\Omega' ,\\zeta' )\\rangle =\\frac{(\\alpha ^{G}+\\alpha ^{A})}{2\\overline{n}}\\delta (\\zeta -\\zeta' )\\delta (\\Omega +\\Omega' ), & \\label{gain_{c}or} \n\\end{eqnarray}\n where \$ \\Gamma(\\Omega ,\\zeta ) \$ is the Fourier transform of the noise\nsource: \n\\begin{eqnarray}\n\\Gamma(\\Omega ,\\zeta )=\\frac{1}{\\sqrt{2\\pi }}\\int ^{\\infty }_{-\\infty }d\\tau \\Gamma(\\tau ,\\zeta )\\exp(i\\Omega \\tau).\n\\end{eqnarray} \nThe dimensionless intensity gain and loss are given by $\\alpha^{G}$ and $\\alpha ^{A}$, respectively.\n\nSimilarly, the real Raman noise, which appears as a multiplicative stochastic\nvariable \$ \\Gamma ^{R} \$, has correlations \n\\begin{eqnarray}\n\\langle \\Gamma ^{R}(\\Omega ,\\zeta )\\Gamma ^{R}(\\Omega' ,\\zeta' )\\rangle =\\frac{1}{\\overline{n}}\\delta (\\zeta -\\zeta' )\\delta (\\Omega +\\Omega' )\\left[ n_{th}(\\Omega )+\\frac{1}{2}\\right] \\alpha ^{R}({\\Omega }), & \\label{Raman_{c}or} \n\\end{eqnarray}\n where the thermal Bose distribution is given by \$ n_{th}(\\Omega )=\\left[ \\exp {(\\hbar \|\\Omega \|/k_BTt _{0})}-1\\right] ^{-1} \$ and where $\\alpha ^{R}({\\Omega })$ is the Raman gain, whose profile is given in Fig.~1 of (QNI).\nThus the Raman noise is strongly temperature dependent, but it also contains\na spontaneous component which provides vacuum fluctuations even at \$ T=0 \$. \n\nAs the $\\overline{n}$ dependence of all the noise correlations show, the classical limit of these quantum calculations is the deterministic nonlinear Schr\\\"odinger equation. The problem of jitter in soliton communications is an example of how intrinsic quantum features can have a direct macroscopic consequence, even in a way that impinges on current developments of applied technology. There are, of course, classical contributions to jitter, such as noise arising from technical sources. However, it is the jitter contributions from essentially quantum processes, namely spontaneous emission in fibre amplifiers, that is the current limiting factor in soliton based communications systems. Other jitter calculations rely on a classical formulation with an empirical addition of amplifier noise, and important predictions of the Gordon-Haus effect have been obtained. Nevertheless, this quantum treatment presented here of all known noise sources is necessary to determine the limiting effects of other intrinsic noise sources, which become important for shorter pulses and longer dispersion lengths. \n\nIn the absence of the noise sources, the phase-space equations have stationary solutions\nin the form of bright (\$ + \$) or dark (\$ - \$) solitons.\nSolitons are solitary waves in which the effects of dispersion are balanced\nby nonlinear effects, to produce a stationary pulse that is robust in the presence\nof perturbations. We note here that, in reality, the Raman response function is noninstantaneous, which causes a redshift in the soliton frequency. This soliton self-frequency shift is a deterministic effect, and\nso can be neglected in the treatment of noise effects, to a first approximation.\nThe accuracy of this approximation will be evident in the subsequent comparison\nof analytic with numerical results. The numerical results all include the complete\nnonlinear response function, rather than the approximate instantaneous form\ngiven above. \n\nExcessive self-frequency shift may cause problems when finite bandwidth elements are used. However, it has been shown\\cite{SFS_comp} that bandwidth-limited gain can in fact cancel the effect of the Raman redshift, by pulling the soliton back towards the centre of the spectral band. In the simulations we show in this paper, the total redshift is estimated to be $\\Delta f \\simeq 0.02 {\\rm THz}$, which is small compared to the total width of the gain spectrum in typical fiber laser amplifiers ($\\Delta \\nu \\simeq 3 {\\rm THz}$)\\cite{s05}. \n\n\n\\section{Perturbation Theory}\n\nWe now proceed to derive the approximate analytic expressions for the effects\nof noise on soliton jitter, using soliton perturbation theory\\cite{s24,s53,s55,s58,s65,s83}, for both bright and dark solitons. \n\n\n\\subsection{Bright solitons}\n\nThe stationary soliton of Eq.\\ (\\ref{NLSE}) for anomalous dispersion is: \n\\begin{eqnarray}\n\\phi _{\\textrm{bright}}(\\tau ,\\zeta )=A{\\textrm{sech}}[A\\tau -q(\\zeta )]\\exp[iV\\tau +i\\theta (\\zeta )],\\label{bright_{s}ol} \n\\end{eqnarray}\n where \$ \\partial q/\\partial \\zeta =VA \$ and \$ \\partial \\theta /\\partial \\zeta =(A^{2}-V^{2})/2 \$,\nwith amplitude A and velocity V. Following the method presented by Haus et al\\cite{s53,s55},\nwe treat the effects of the noise terms as perturbations around a soliton solution\nwhose parameters vary slowly with \$ \\zeta \$: \n\\begin{eqnarray}\n\\phi (\\tau ,\\zeta )=\\overline{\\phi }(\\tau ,\\zeta )+\\Delta \\phi (\\tau ,\\zeta ),\\label{pert} \n\\end{eqnarray}\n where the unperturbed soliton solution is given by: \n\\begin{eqnarray}\n\\overline{\\phi }(\\tau ,\\zeta )=A(\\zeta ){\\textrm{sech}}[A(\\zeta )\\tau -q(\\zeta )]\\exp[iV(\\zeta )\\tau +i\\theta (\\zeta )] & \n\\end{eqnarray}\n for a bright soliton. Substituting Eq.\\ (\\ref{pert}) into Eq.\\ (\\ref{NLSE})\ngives the following linearized equation (first order in \$ \\Delta \\phi (\\tau ,\\zeta ) \$)\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial \\zeta }\\Delta \\phi (\\tau ,\\zeta )=\\left[ \\pm \\frac{i}{2}\\frac{\\partial ^{2}}{\\partial \\tau ^{2}}+i2\\overline{\\phi}^{}(\\tau ,\\zeta )\\overline{\\phi }(\\tau ,\\zeta )\\right] \\Delta \\phi (\\tau ,\\zeta )+i\\overline{\\phi }(\\tau ,\\zeta )^{2}\\Delta \\phi ^{}(\\tau ,\\zeta )+\\overline{\\Gamma }(\\tau ,\\zeta )\\, \\, , & \\label{linear} \n\\end{eqnarray}\n where the linearized noise source \$ \\overline{\\Gamma }(\\tau ,\\zeta ) \$ is\ndefined as: \n\\begin{eqnarray}\n\\overline{\\Gamma }(\\tau ,\\zeta )=\\Gamma(\\tau ,\\zeta ) +i\\Gamma ^{R}(\\tau ,\\zeta )\\overline{\\phi }(\\tau ,\\zeta )\\, \\, . & \n\\end{eqnarray}\n\n\nNow we wish to determine the evolution of the soliton parameters as a function\nof propagation distance \$ \\zeta \$. To do this, we expand the perturbation\nin terms of the soliton parameters plus a continuum term: \n\\begin{eqnarray}\n\\Delta \\phi (\\tau ,\\zeta ) & = & \\sum _{i}\\frac{\\partial \\overline{\\phi }(\\tau ,\\zeta )}{\\partial P_{i}}\\Delta P_{i}+\\Delta \\phi _{c}(\\tau ,\\zeta )\\nonumber \\\\\n & = & \\sum _{i}f_{P_{i}}\\Delta P_{i}+\\Delta \\phi _{c}(\\tau ,\\zeta ).\\label{expan} \n\\end{eqnarray}\n where \$ P_{i}\\in \\{V,q,A,\\theta \\} \$. The projection functions for each\nparameter are \n\\begin{eqnarray}\nf_{A} & = & \\left[ \\frac{1}{A}-\\tau \\tanh (A\\tau -q)\\right] \\overline{\\phi },\\nonumber \\\\\nf_{q} & = & \\tanh (A\\tau -q)\\overline{\\phi },\\nonumber \\\\\nf_{V} & = & i\\tau \\overline{\\phi },\\nonumber \\\\\nf_{\\theta } & = & i\\overline{\\phi }.\n\\end{eqnarray}\nSince the linearised equation Eq.\\ (\\ref{linear}) is not self-adjoint, these eigenfunctions are not orthogonal. In order to select out the evolution of particular parameters, we therefore choose an alternative set of functions:\n\\begin{eqnarray}\n\\underline{f_{A}} & = &\\overline{\\phi },\\nonumber \\\\\n\\underline{f_{q}} & = & \\tau \\overline{\\phi },\\nonumber \\\\\n\\underline{f_{V}} & = & i\\tanh (A\\tau -q)\\overline{\\phi },\\nonumber \\\\\n\\underline{f_{\\theta }} & = & i\\tau \\tanh (A\\tau -q)\\overline{\\phi }.\n\\end{eqnarray}\nThese are the eigenfunctions of the adjoint equation Eq.\\ (\\ref{linear}), and obey the orthogonality\ncondition \n\\begin{eqnarray}\n\\Re \\left\\{ \\int _{-\\infty }^{\\infty }d\\tau f_{P_{i}}\\underline{f_{P_{j}}}^{}\\right\\} =\\delta _{i,j}.\n\\end{eqnarray}\nSubstituting the Taylor expansion [Eq.\\ (\\ref{expan})] into the linearized\nequation [Eq.\\ (\\ref{linear})] and using the functions \$ \\underline{f_{P_{i}}} \$ to project out\nparticular parameters shows that the growth of fluctuations in position $\\Delta q$ is governed by\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial \\zeta }\\Delta q(\\zeta ) &=& A\\Delta V(\\zeta ) + \\Gamma_q(\\zeta ) \\nonumber \\\\\n\\frac{\\partial }{\\partial \\zeta }\\Delta V(\\zeta ) &=& \\Gamma_V(\\zeta ),\n\\end{eqnarray}\nwhere we have taken the unperturbed velocity to be zero: $V = 0$. The stochastic terms are defined as\n\\begin{eqnarray}\\Gamma _{P_{i}}(\\zeta )=\\Re \\left\\{ \\int _{-\\infty }^{\\infty }d\\tau \\underline{f_{p}}^{}(\\zeta )\\overline{\\Gamma }(\\tau ,\\zeta )\\right\\} . & \\label{sp} \n\\end{eqnarray}\nHere we have assumed that the perturbations in the continuum $\\phi _{c}$ are orthogonal to the \$ \\underline{f_{P_{i}}} \$. This depends on such perturbations dispersing sufficiently rapidly away from the region around the soliton. In fact, any nonsoliton perturbation will disperse and would also move away from the soliton, since the group velocity for any linear perturbations will be different to the propagation velocity of the soliton. \n\nWe wish to find the growth of fluctuations in position \$ q(\\zeta ) \$. Because\nthe position depends on the soliton frequency \$ V \$, the contributions arising\nfrom both \$ \\Gamma _{q} \$ and \$ \\Gamma _{V} \$ must be considered. Firstly,\n\\begin{eqnarray}\n\\Gamma _{q}(\\zeta ) & = & \\Re \\left\\{ \\int _{-\\infty }^{\\infty }d\\tau A\\tau {\\textrm{sech}}(A\\tau -q)\\exp(-iV\\tau -i\\theta )\\overline{\\Gamma} (\\tau ,\\zeta )\\right\\} \\nonumber \\\\\n & = & \\int _{-\\infty }^{\\infty }d\\tau A\\tau {\\textrm{sech}}(A\\tau -q)\\Re \\{\\exp(-iV\\tau -i\\theta )\\Gamma \\},\n\\end{eqnarray}\n and \n\\begin{eqnarray}\n\\Gamma _{V}(\\zeta ) & = & \\Re \\left\\{ \\int _{-\\infty }^{\\infty }d\\tau A (-i){\\textrm{sech}}(A\\tau -q)\\tanh (A\\tau -q)\\exp(-iV\\tau -i\\theta )\\overline{\\Gamma} (\\tau ,\\zeta )\\right\\} \\nonumber \\\\\n & = & \\int _{-\\infty }^{\\infty }d\\tau A{\\textrm{sech}}(A\\tau -q)\\tanh (A\\tau -q)\\left[ A{\\textrm{sech}}(A\\tau -q)\\Gamma ^{R}+\\Im \\{\\exp(-iV\\tau -i\\theta )\\Gamma \\}\\right] .\n\\end{eqnarray}\nFrom this we can calculate the growth of the fluctuations in velocity: \n\\begin{eqnarray}\n\\Delta V(\\zeta ) & = & \\Delta V(0)+\\int _{0}^{\\zeta }d\\zeta' \\Gamma _{V}(\\zeta ')\\nonumber \\\\\n & = & \\Re \\left\\{ \\int _{-\\infty }^{\\infty }\\Delta \\phi (\\tau ,\\zeta )\\underline{f_{V}}^{}d\\tau \\right\\} +\\int _{0}^{\\zeta }d\\zeta' \\Gamma _{V}(\\zeta ').\\label{delV} \n\\end{eqnarray}\n Using the noise correlations calculated above, the correlations in the velocity\nfluctuations can now be calculated: \n\\begin{eqnarray}\n\\langle \\Delta V(\\zeta )\\Delta V^{}(\\zeta' )\\rangle & = & \\bigl <{\\Delta V(0)\\Delta V^{}(0)}\\bigr >+\\int _{0}^{\\zeta }\\int _{0}^{\\zeta' }d\\zeta''d \\zeta''' \\langle \\Gamma _{V}(\\zeta'' )\\Gamma _{V}^{}(\\zeta''' )\\rangle \\nonumber \\\\\n & = & \\frac{A}{6\\overline{n}}+\\left[ \\frac{\\alpha ^{G}A}{3\\overline{n}}+\\frac{2A^2{\\mathcal{I}}(t_{0})}{\\overline{n}}\\right] \\zeta \\quad \\quad \\zeta <\\zeta' ,\n\\end{eqnarray}\n where the overlap integral \$ {\\mathcal{I}}(t_{0}) \$ is defined as \n\\begin{eqnarray}\n{\\mathcal{I}}(t_{0})=\\int ^{\\infty }_{-\\infty }\\int ^{\\infty }_{-\\infty }d\\tau d\\tau' \\tanh (\\tau ){\\textrm{sech}}^{2}(\\tau )\\tanh (\\tau' ){\\textrm{sech}}^{2}(\\tau' ){\\tilde{\\mathcal{F}}}(\\tau/A -\\tau'/A ).\\label{calI} \n\\end{eqnarray}\n Here \$ {\\tilde{\\mathcal{F}}}(\\tau ) \$ is the inverse Fourier transform of\nthe fluorescence \$ {\\mathcal{F}}(\\Omega )=\\frac{1}{2}[n_{th}({\\Omega })+\\frac{1}{2}]\\alpha ^{R}({\\Omega }). \$\n\nThe correlations in position fluctuations correspond to the jitter in arrival\ntimes, because we have chosen a propagative reference frame. The jitter therefore\nfeeds off position fluctuations as well as noise entering through the velocity:\n\\begin{eqnarray}\n & & \\langle \\Delta q(\\zeta )\\Delta q^{}(\\zeta' )\\rangle =\\langle \\Delta q(0)\\Delta q^{}(0)\\rangle \\nonumber \\\\\n & + & \\int _{0}^{\\zeta}\\int _{0}^{\\zeta'}d\\zeta'' d\\zeta''' \\left[ A^{2}\\langle \\Delta V(\\zeta'' )\\Delta V^{}(\\zeta''' )\\rangle +\\langle \\Gamma _{q}(\\zeta'' )\\Gamma _{q}^{}(\\zeta''' )\\rangle \\right] \\quad \\quad \\zeta <\\zeta' .\n\\end{eqnarray}\n Thus the timing jitter is \n\\begin{eqnarray}\n\\langle [\\Delta \\tau (\\zeta )]^{2}\\rangle & = & \\langle \\Delta q(\\zeta )\\Delta q^{}(\\zeta )\\rangle \\nonumber \\\\\n & = & \\frac{\\pi ^{2}}{24\\overline{n}}+\\frac{\\pi ^{2}\\alpha ^{G}}{12\\overline{n}}\\zeta +\\frac{A^3}{6\\overline{n}}\\zeta ^{2}+\\left[ \\frac{\\alpha ^{G}A^3}{9\\overline{n}}+\\frac{2A^4{\\mathcal{I}}(t_{0})}{3\\overline{n}}\\right] \\zeta ^{3},\\label{jtr} \n\\end{eqnarray}\n which contains cubic terms due to the gain and Raman couplings, and also slower\ngrowing terms due to the initial vacuum fluctuations and amplifier noise.\n\nWe note that an alternative method that exploits conserved quantities in the NLS equation is often used\\cite{s35,s65,s83} for deriving the\ntiming jitter. The linearised approach that we have presented has the advantage that derivatives of products of stochastic variables do not appear. With such derivatives, the normal rules of calculus do not apply. Rather, the rules of Ito stochastic calculus must be observed, leading to extra drift terms. \n\\subsection{Dark solitons}\n\nFibers in the normal dispersion regime can support dark soliton solutions, so\ncalled since they correspond to a dip in the background intensity\\cite{s50}:\n\\begin{eqnarray}\n\\phi _{\\textrm{dark}}(\\tau ,\\zeta ) & = & \\phi _{0}\\sqrt{1-A^{2}{\\textrm{sech}}^{2}[\\phi _{0}A\\tau -q(\\zeta )]}\\exp[i\\theta (\\zeta )]\\exp[i\\sigma (\\zeta ,\\tau )],\\nonumber \\\\\n\\sigma (\\zeta ,\\tau ) & = & \\arcsin \\left\\{ \\frac{A\\tanh [\\phi _{0}A\\tau -q(\\zeta )]}{\\sqrt{1-A^{2}{\\textrm{sech}}^{2}[\\phi _{0}A\\tau -q(\\zeta )]}}\\right\\} ,\n\\end{eqnarray}\n where \$ d\\theta /d\\zeta =\\phi _{0}^{2} \$, \$ dq/d\\zeta =A\\sqrt{1-A^{2}}\\phi _{0}^{2} \$\nand \$ \\phi _{0} \$ is the amplitude of the background field. The size of the intensity\ndip at the center of the soliton is given by \$ A \$, with the intensity going\nto zero in a black \$ \\tanh (\\tau ) \$ soliton, for which \$ A=1 \$. Dark\nsolitons are classed as topological solitons, because they connect two background\npulses of different phase. The total phase difference between the boundaries\nis \$ \\psi =2\\arcsin (A) \$.\n\nThe nonvanishing boundary conditions of the dark pulse complicate the perturbation calculation of jitter variance. To ensure that all relevant integrals take on finite values, we impose periodic boundary conditions at $\\tau = \\pm \\tau_l$, which are taken to infinity at the end of the calculation. These boundary conditions require a soliton solution of the form\n\\begin{eqnarray}\n\\phi _{\\textrm{dark}}(\\tau ,\\zeta ) & = & \\phi _{0}\\exp[i\\theta (\\zeta ) - i\\kappa\\tau]\\left\\{ \\cos{\\frac{\\psi}{2}}+i\\sin{\\frac{\\psi}{2}}\\tanh [\\phi _{0}\\tau\\sin{\\frac{\\psi}{2}} -q(\\zeta )]\\right\\} ,\n\\end{eqnarray}\nwith a wavenumber offset $\\kappa = \\frac{1}{\\tau_l}\\arctan\\left[ \\tan{\\frac{\\psi}{2}}\\tanh(\\phi_0\\tau_l\\sin{\\frac{\\psi}{2}})\\right]$. The perturbation theory now proceeds in a similar fashion to the bright soliton case, except that we can greatly simplify the calculation if the unperturbed solution is taken to be a black soliton, i.e. $\\psi = \\pi$. \n\nThe projection functions for the soliton parameters $P_{i}\\in \\{\\psi,q,\\phi_0,\\theta \\}$ are \n\\begin{eqnarray}\nf_{\\theta} & = & -\\phi_0\\tanh(\\phi_0 \\tau-q)\\exp(i\\theta-i\\kappa\\tau)\\nonumber \\\\\nf_{\\phi_0} & = & i[\\tanh(\\phi_0 \\tau-q) +\\phi_0\\tau{\\textrm{sech}}^2(\\phi_0 \\tau-q)]\\exp(i\\theta-i\\kappa\\tau)\\nonumber \\\\\nf_{q} & = & -i\\phi_0{\\textrm{sech}}^2(\\phi_0 \\tau-q)\\exp(i\\theta-i\\kappa\\tau)\\nonumber \\\\\nf_{\\psi} & = & \\phi_0[\\beta_1\\phi_0 \\tau\\tanh(\\phi_0 \\tau-q) - \\frac{1}{2}]\\exp(i\\theta-i\\kappa\\tau),\n\\end{eqnarray}\nwhere $\\beta_1 = 1/[2\\phi_0\\tau_l\\tanh(\\phi_0\\tau_l)]$. For the required adjoint functions, we choose:\n\\begin{eqnarray}\n\\underline{f_{q}} & = & \\frac{-i3\\gamma_q}{4}{\\textrm{sech}}^2(\\phi_0 \\tau-q)\\exp(i\\theta-i\\kappa\\tau)\\nonumber \\\\\n\\underline{f_{\\psi}} & = & \\frac{\\gamma_{\\psi}}{\\beta_1-1}{\\textrm{sech}}^2(\\phi_0 \\tau-q)\\exp(i\\theta-i\\kappa\\tau),\n\\end{eqnarray}\nwhere $\\gamma_q = 4/(3\\int^{\\phi_0\\tau_l}_{-\\phi_0\\tau_l}dt {\\textrm{sech}}^4t)$ and $\\gamma_{\\psi} = (\\beta_1-1)/\\int^{\\phi_0\\tau_l}_{-\\phi_0\\tau_l}dt (\\beta_1 t\\tanh{t}{\\textrm{sech}}^2{t} - {\\textrm{sech}}^2{t}/2)$. The orthogonality condition is now\n\\begin{eqnarray}\n\\Re \\left\\{ \\int _{-\\tau_l +q/\\phi_0}^{\\tau_l +q/\\phi_0}d\\tau f_{P_{i}}\\underline{f_{P_{j}}}^{}\\right\\} =\\delta _{i,j}.\n\\end{eqnarray}\n\nOnce again, the adjoint functions can be used in the linearized equation [Eq.\\ (\\ref{linear})] to determine how the fluctuations in position evolve:\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial \\zeta }\\Delta q(\\zeta ) &=& \\frac{\\phi_0\\beta_2}{2}\\Delta \\psi(\\zeta ) + \\Gamma_q(\\zeta ) \\nonumber \\\\\n\\frac{\\partial }{\\partial \\zeta }\\Delta \\psi(\\zeta ) &=& \\Gamma_{\\psi}(\\zeta ),\n\\end{eqnarray}\nwhere $\\beta_2 = \\phi_0\\tanh(\\phi_0 \\tau_l) - 1/\\tau_l$. Here we see how the fluctuations in phase produce fluctuations in position. The stochastic term in the equation for $\\psi$ evaluates to \n\\begin{eqnarray}\n\\Gamma _{\\psi}(\\zeta ) & = & \\Re \\left\\{ \\int _{-\\tau_l +q/\\phi_0 }^{\\tau_l +q/\\phi_0 }d\\tau \\underline{f_{\\psi}}^{}(\\zeta )\\overline{\\Gamma }(\\tau ,\\zeta )\\right\\} \\nonumber \\\\\n & = & \\int _{-\\tau_l +q/\\phi_0 }^{\\tau_l +q/\\phi_0}d\\tau \\frac{\\gamma_{\\psi}}{\\beta_1-1}{\\textrm{sech}}^2(\\phi_0 \\tau -q)\\left[ \\Re \\{\\exp(-i\\theta+i\\kappa\\tau)\\Gamma \\} - \\phi_0 \\tanh(\\phi_0\\tau -q)\\Gamma ^{R}\\right],\n\\end{eqnarray}\nfrom which the correlations of the phase fluctuations can be calculated: \n\\begin{eqnarray}\n\\langle \\Delta \\psi(\\zeta )\\Delta \\psi^{}(\\zeta' )\\rangle & = & \\bigl <{\\Delta \\psi(0)\\Delta \\psi^{}(0)}\\bigr >+\\int _{0}^{\\zeta }\\int _{0}^{\\zeta' }d\\zeta''d \\zeta''' \\langle \\Gamma _{\\psi}(\\zeta'' )\\Gamma _{\\psi}^{}(\\zeta''' )\\rangle \\nonumber \\\\\n & = & \\frac{{\\gamma_\\psi}^2}{3\\overline{n}\\gamma_q\\phi_0}+\\left[ \\frac{2\\alpha ^{G}{\\gamma_\\psi}^2}{3\\overline{n}\\gamma_q(\\beta_1-1)^2\\phi_0}+\\frac{2{\\gamma_\\psi}^2{\\mathcal{I}}_{\\tau_l}(t_{0})}{\\overline{n}(\\beta_1-1)^2}\\right] \\zeta \\quad \\quad \\zeta <\\zeta' ,\n\\end{eqnarray}\n where the overlap integral \$ {\\mathcal{I}}_{\\tau_l}(t_{0}) \$ is now defined as \n\\begin{eqnarray}\n{\\mathcal{I}}_{\\tau_l}(t_{0})=\\int ^{\\phi_0\\tau_l}_{-\\phi_0\\tau_l }\\int ^{\\phi_0\\tau_l}_{-\\phi_0\\tau_l }d\\tau d\\tau' \\tanh (\\tau ){\\textrm{sech}}^{2}(\\tau )\\tanh (\\tau' ){\\textrm{sech}}^{2}(\\tau' ){\\tilde{\\mathcal{F}}}(\\tau/\\phi_0 -\\tau'/\\phi_0 ). \n\\end{eqnarray}\n\nThe leading order terms for the fluctuations in position are thus\n\\begin{eqnarray}\n\\langle \\Delta q(\\zeta )\\Delta q^{}(\\zeta)\\rangle &=& \\frac{\\phi_0^2\\beta_2^2}{4}\\int _{0}^{\\zeta}\\int _{0}^{\\zeta'}d\\zeta'' d\\zeta''' \\langle \\Delta \\psi(\\zeta'' )\\Delta \\psi^{*}(\\zeta''' )\\rangle \\nonumber \\\\\n&=& \\frac{\\phi_0\\beta_2^2{\\gamma_{\\psi}}^2}{12 \\overline{n}\\gamma_q}z^2 + \\left[ \\frac{\\alpha ^{G}\\phi_0\\beta_2^2{\\gamma_{\\psi}}^2}{18\\overline{n}\\gamma_q(\\beta_1-1)^2} +\\frac{{\\mathcal{I}}_{\\tau_l}(t_{0})\\phi_0^2\\beta_2^2{\\gamma_{\\psi}}^2}{6\\overline{n}}\\right] \\zeta ^{3}.\n\\end{eqnarray}\nBy taking the limit $\\tau_l \\rightarrow \\infty$, we find the leading order terms in the jitter growth for a black soliton to be\n\\begin{eqnarray}\n\\langle [\\Delta \\tau (\\zeta )]^{2}\\rangle =\\frac{\\phi_0^3}{12\\overline{n}}\\zeta ^{2}+\\left[ \\frac{\\alpha ^{G}\\phi_0^3}{18\\overline{n}}+\\frac{{\\mathcal{I}}(t_{0})\\phi_0^4}{6\\overline{n}}\\right] \\zeta ^{3},\n\\end{eqnarray}\nwhere the overlap integral \$ {\\mathcal{I}}(t_{0}) \$ is as defined in Eq.\\ (\\ref{calI}).\nAs in the anomalous dispersion regime, the vacuum fluctuations contribute to\nquadratic growth in the jitter variance, and gain and Raman fluctuations contribute\nto cubic growth. However, the size of the jitter is smaller than that in the\nbright soliton case, for the same propagation distance \$ \\zeta \$. The contribution\nfrom the vacuum and gain terms is one half and the contribution from the Raman\nterm is one quarter of that in Eq.\\ (\\ref{jtr}), giving dark solitons some\nadvantage over their bright cousins. \\label{dksol}\n\n\n\\section{Scaling properties}\n\n\\label{SP} In summary, there are three different sources of noise in the soliton,\nall of which must be taken into account for small pulse widths. These noise\nsources contribute to fluctuations in the velocity parameter, which lead to\nquadratic or cubic growth in the timing-jitter variance for single-pulse propagation.\nThe noise sources also produce other effects, such as those effected through\nsoliton interactions, but we will not consider these here.\n\nEach of the noise sources has different characteristic scaling properties, which\nare summarized as follows:\n\n\n\\subsection{Vacuum Fluctuations}\n\nThe vacuum fluctuations cause diffusion in position which is important for small\npropagation distances. There are position fluctuations even at the initial position,\nsince the shot noise in the arrival time of individual coherent-state photons\ngives an initial fluctuation effect. After propagation has started, this initial\nposition fluctuation is increased by the additional variance in the soliton\nvelocity, due essentially to randomness in the frequency domain.\n\nFor bright solitons the resulting soliton timing variance is given by\\cite{s24}\n\\begin{eqnarray}\n\\langle [\\Delta \\tau (\\zeta )]^{2}\\rangle _{I}=\\frac{\\pi ^{2}}{24\\overline{n}}+\\frac{1}{6\\overline{n}}\\zeta ^{2}\\quad \\quad {(\\textrm{bright})}. & \n\\end{eqnarray}\n For purposes of comparison, note that \$ \\overline{N}=2\\overline{n} \$ is\nthe mean photon number for a \$ {\\textrm{sech}}(\\tau ) \$ soliton. Numerical\ncalculations confirm that for \$ \\tanh (\\tau ) \$ dark solitons, the variance\nwas about one half the bright-soliton value, as predicted by the analysis in\noutlined in Sec.\\ \\ref{dksol}: \n\\begin{eqnarray}\n\\langle [\\Delta \\tau (\\zeta )]^{2}\\rangle _{I}=\\frac{\\pi ^{2}}{48\\overline{n}}+\\frac{1}{12\\overline{n}}\\zeta ^{2}\\quad \\quad {(\\textrm{dark})}. & \n\\end{eqnarray}\n This shot-noise effect, which occurs without amplification, is simply due to\nthe initial quantum-mechanical uncertainty in the position and momentum of the\nsoliton. Because of the Heisenberg uncertainty principle, the soliton momentum\nand position cannot be specified exactly. This effect dominates the Gordon-Haus\neffect over propagation distances less than a gain length. However, for short\npulses, this distance can still correspond to many dispersion lengths - thus\ngenerating large position jitter. We note that there are also initial fluctuations\nin the background continuum, which may feed into the soliton parameters as the soliton \npropagates. This lessor effect is included in the numerical \ncalculation; a comparison of the numerical results with the analytic results \nconfirms that the initial fluctuations in the soliton parameters account for almost all\nof the shot-noise contribution to the jitter. \n\n\n\\subsection{Gordon-Haus noise}\n\nAs is well known, the noise due to gain and loss in the fiber produces the Gordon-Haus\neffect, which is currently considered the major limiting factor in any long-distance\nsoliton-based communications system using relatively long (\$ >10ps \$) pulses.\nAmplification with mean intensity gain \$ \\alpha ^{G} \$, chosen to compensate\nfiber loss, produces a diffusion (or jitter) in position. Unless other measures\nare taken, for sufficiently small amplifier spacing\\cite{e375} and at large\ndistances this is given by\\cite{s35,s65,s45}\n\\begin{eqnarray}\n\\langle [\\Delta \\tau (\\zeta )]^{2}\\rangle _{GH} & \\simeq & \\frac{\\alpha ^{G}}{9\\overline{n}}\\zeta ^{3}\\quad \\quad {(\\textrm{bright})},\\nonumber \\\\\n\\langle [\\Delta \\tau (\\zeta )]^{2}\\rangle _{GH} & \\simeq & \\frac{\\alpha ^{G}}{18\\overline{n}}\\zeta ^{3}\\quad \\quad {(\\textrm{dark})},\n\\end{eqnarray}\n in which the linearly growing terms have been neglected.\n\nAnother effect of the amplifier noise is to introduce an extra noise term via\nthe fluctuations in the Raman-induced soliton self-frequency shift. This term\nscales as the fifth power of distance and hence will become important for long\npropagation distances. This combined effect of spontaneous emission noise and\nthe Raman intrapulse scattering has been dealt with by others\\cite{MooWonHau94}.\nThe full phase-space equation [Eq. (\\ref{RMS})] models this accurately, since it includes the delayed\nRaman nonlinearity, and the effect would be seen in numerical simulations\ncarried out over long propagation distances.\n\n\n\\subsection{Raman noise}\n\nA lesser known effect are the fluctuations in velocity that arise from the Raman\nphase-noise term \$ \\Gamma ^{R} \$ in Eq.\\ (\\ref{NLSE}). Like the Gordon-Haus\neffect, this Raman noise generates a cubic growth in jitter variance: \n\\begin{eqnarray}\n\\langle [\\Delta \\tau (\\zeta )]^{2}\\rangle _{R} & = & \\frac{2{\\mathcal{I}}(t_{0})}{3\\overline{n}}\\zeta ^{3}\\quad \\quad {(\\textrm{bright})},\\nonumber \\\\\n\\langle [\\Delta \\tau (\\zeta )]^{2}\\rangle _{R} & = & \\frac{{\\mathcal{I}}(t_{0})}{6\\overline{n}}\\zeta ^{3}\\quad \\quad {(\\textrm{dark})}.\n\\end{eqnarray}\n where \$ {\\mathcal{I}}(t_{0}) \$ is the integral defined in Eq.\\ (\\ref{calI})\nthat indicates the spectral overlap between the pulse spectrum and the Raman\nfluorescence. The mean-square Raman induced timing jitter has a cubic growth\nin both cases, but the dark soliton variance is one quarter of that of the bright\nsoliton.\n\nThe magnitude of this Raman jitter can be found by evaluating \$ {\\mathcal{I}}(t_{0}) \$\nnumerically, or else using an analytic approximation. An accurate model of the\nRaman gain, on which \$ {\\mathcal{I}}(t_{0}) \$ depends, requires a multi-Lorentzian\nfit to the experimentally measured spectrum\\cite{RamanGain}. A fit with 11 Lorentzians was used\nin the numerical simulations, including 10 Lorentzians to accurately model the measured\ngain and fluorescence. One extra Lorentzian was used at low frequencies to model GAWBS\n(Guided Wave Acoustic Brillouin Scattering); this has a relatively small effect on\nan isolated soliton, except to cause phase noise.\n\nFor analytic work, however, a single-Lorentzian model\\cite{s76} can\nsuffice for approximate calculations. A plot of the Raman gain profile $\\alpha ^R(\\Omega )$ for both models is given in (QNI), along with a table of the fitting parameters for the multi-Lorentzian model. The spectral features of the Raman noise correlations are determined directly from the Raman fluorescence function ${\\mathcal{F}}(\\Omega )$, which we plot in Fig.\\ \\ref{solspect}. For the single-Lorentzian model, the fluorescence spectrum is approximately flat at low frequencies: \n\\begin{eqnarray}\n{\\mathcal{F}}(\\Omega ) & = & \\frac{1}{2}\\left[ n_{th}(\\Omega )+\\frac{1}{2}\\right] \\alpha ^R(\\Omega )\\nonumber \\\\\n & \\simeq & \\frac{2F_{1}\\Omega _{1}\\delta _{1}^{2}k_{B}T}{(\\Omega _{1}^{2}+\\delta _{1}^{2})^{2}\\hbar }={\\mathcal{F}}(0),\n\\end{eqnarray}\n which greatly simplifies the Raman correlations. As Fig.\\ \\ref{solspect}\nindicates, the spectral overlap of \$ {\\mathcal{F}}(\\Omega ) \$ with a \$ t_{0}=1ps \$\nsoliton occurs in this low frequency region. Thus the white noise approximation\nfor the Raman correlations is good for solitons of this pulse width and larger.\nFor smaller pulse widths, not only is the Raman contribution to the noise larger\ndue to the greater overlap, but the colored nature of the correlations must\nbe taken into account.\n\nIn the single-Lorentzian model, \$ {\\mathcal{I}}(t_{0}\\rightarrow \\infty )\\simeq \\frac{4}{15}{\\mathcal{F}}(0) \$,\nwhich gives \\label{sol_approx}\n\\begin{eqnarray}\n{\\langle [\\Delta t(x)]^{2}\\rangle }_{R} & \\simeq & \\frac{8\|k''\|^{2}n_{2}\\hbar \\omega_{0}^{2}{\\mathcal{F}}(0)}{45{\\cal A}ct_{0}^{3}}x^{3} = \\frac{8t_{0}^{2}{\\mathcal{F}}(0)}{45{\\overline{n}}}\\left( \\frac{x}{x_{0}}\\right) ^{3}\\quad \\quad {(\\textrm{bright})},\\nonumber \\\\\n{\\langle [\\Delta t(x)]^{2}\\rangle }_{R} & \\simeq & \\frac{2\|k''\|^{2}n_{2}\\hbar \\omega_{0}^{2}{\\mathcal{F}}(0)}{45{\\cal A}ct_{0}^{3}}x^{3} = \\frac{2t_{0}^{2}{\\mathcal{F}}(0)}{45{\\overline{n}}}\\left( \\frac{x}{x_{0}}\\right) ^{3}\\quad \\quad {(\\textrm{dark})}.\n\\end{eqnarray}\n At a temperature of \$ 300K \$, \$ {\\mathcal{F}}(0)=4.6\\times10 ^{-2} \$\nwhen a single Lorentzian centered at 12 THz with fitting parameters \$ F_{1}=0.7263 \$,\n\$ \\delta _{1}=20\\times10 ^{12}t_{0} \$ and \$ \\Omega _{1}=75.4\\times10 ^{12}t_{0} \$\nis used.\n\n\n\\section{Numerical results}\n\n\\label{NR} \\label{numres} More precise results can be obtained by numerically integrating the original Wigner phase-space equation, Eq. (\\ref{RMS}), which includes the full time-delayed nonlinear Raman response function. The results for \$ t_{0}=500fs \$\nbright and dark solitons are shown in Figs.\\ \\ref{ol}(a) and \\ref{ol}(b),\nrespectively. The gain and photon number were chosen to be \$ G=\\alpha ^{G}/x_{0}=4.6\\times {10}^{-5}m^{-1}\\, (0.2dB/km) \$\nand \$ \\overline{n}=4\\times 10^{6} \$, with \$ x_{0}=440m \$. These values\nare based on \$ {\\cal A}=40[\\mu m]^{2} \$, \$ k''=0.57[ps]^{2}/km \$, and \$ n_{2}=2.6\\times10 ^{-20}[m]^{2}/W \$\nfor a dispersion-shifted fiber\\cite{e34}. These numerical calculations use\nthe multiple-Lorentzian model of the Raman response function shown in Fig.\\ \\ref{solspect},\nwhich accurately represents the detailed experimental response function. \n\nThe numerical method is based on the split-step idea\\cite{Drummond1983}\nas adapted to Raman propagation\\cite{s23}. Noise is treated using a central\ndifference technique appropriate to stochastic equations\\cite{Drummond1991},\nwith the necessary adaptations required to treat a partial stochastic differential\nequation \\cite{Werner1997}. All calculations were duplicated using two different\nspace steps, but with the same underlying noise sources, in order to calculate\ndiscretization error. Sampling error was also estimated using standard central\nlimit theorem procedure over a large ensemble of noise sources. Tests on time steps\nand window sizes were also carried out to ensure there were no errors from these\nsources.\n\nThe initial conditions consist of a coherent laser pulse injected into the fiber.\nIn the Wigner representation, this minimum uncertainty state leads to the initial\nvacuum fluctuations. The numerical calculation thereby includes the full effect of\nthese zero-point fluctuations, including the noise that appears in the \nbackground continuum and in the soliton parameters. We take the initial \npulse shape in the anomalous dispersion\nregime to be a fundamental bright soliton, with \$ A=1 \$ and \$ V=\\theta =q=0 \$.\nSuch a soliton can be experimentally realized with a sufficiently intense pulse,\nwhich will reshape into a soliton or soliton train. The nonsoliton part of\nthe wave will disperse and any extra solitons will move away at different velocities\nfrom the fundamental soliton. The numerical simulation in the normal dispersion\nregime used two black solitons of opposite phase chirp (\$ A=\\pm1 \$), so\nthat the field amplitude at either boundary was the same. This phase matching\nensured the stability of the numerical algorithm, which assumes periodic boundary\nconditions.\n\nThe position jitter at a given propagation time was calculated by combining\nthe waveform with a phase-matched local-oscillator pulse that had a linear chirp\nin amplitude, and integrating the result to give the soliton position. This\nhomodyne measurement involves symmetrically ordered products, so the Wigner\nrepresentation will give the correct statistics. The variance in soliton position\nwas then calculated from a sample of 1000 trajectories. For this small distance\nof propagation (\$ \\simeq 10km \$), the jitter variance due to the initial\nnoise is twice the Gordon-Haus jitter, but for larger distances, the cubic effects\nare expected to dominate.\n\nFor ultrafast pulses, the Raman jitter dominates the Gordon-Haus jitter (by\na factor of two in the \$ 500fs \$ bright soliton case) and will continue to\ndo so even for long propagation distances. For short propagation distances,\nthe Gordon-Haus effect is not exactly cubic, because of neglected terms in the\nperturbation expansion, which give a linear (as opposed to cubic) growth in\nthe jitter variance. However, there are no such terms in the Raman case. The\nanalytic Raman results are also shown in the figures, which show that our approximate\nformula gives a reasonable fit to the numerical data even for subpicosecond\npulses. Using this approximate formula, the relative size of the two effects\nscales as \n\\begin{eqnarray}\n\\frac{{\\langle [\\Delta t(x)]^{2}\\rangle }_{R}}{{\\langle [\\Delta t(x)]^{2}\\rangle }_{GH}} & = & \\frac{6{\\mathcal{I}}(t_{0})\|k''\|}{G{t_{0}}^{2}}=\\frac{6{\\mathcal{I}}(t_{0})}{Gx_{0}}\\simeq \\frac{8{\\mathcal{F}}(0)}{5Gx_{0}}\\quad \\quad {(\\textrm{bright})},\\nonumber \\\\\n\\frac{{\\langle [\\Delta t(x)]^{2}\\rangle }_{R}}{{\\langle [\\Delta t(x)]^{2}\\rangle }_{GH}} & = & \\frac{3{\\mathcal{I}}(t_{0})\|k''\|}{G{t_{0}}^{2}}=\\frac{3{\\mathcal{I}}(t_{0})}{Gx_{0}}\\simeq \\frac{4{\\mathcal{F}}(0)}{5Gx_{0}}\\quad \\quad {(\\textrm{dark})},\n\\end{eqnarray}\n These equations show why experiments to date\\cite{e34}, which have used longer\npulses (\$ t_{0}>1ps \$) and dispersion-shifted fiber, have not detected the\nRaman-noise contribution to the jitter. The Raman jitter exceeds the Gordon-Haus\njitter for bright solitons with periods \$ x_{0}<1.5km \$. Dark solitons, on\nthe other hand, have an enhanced resistance to the Raman noise, which means\nthat a shorter period is needed before the Raman jitter will become important.\n\nThe total jitter, which corresponds to the realistic case where all three noise\nsources are active, is also shown in Fig.\\ \\ref{ol}, and, in the bright soliton\ncase, is about a factor of three larger than the ordinary Gordon-Haus effect,\nover the propagation distance shown. The physical origin of these quantum noise\nsources cannot easily be suppressed. The initial vacuum-induced timing jitter\nis caused by the shot-noise variance in the soliton guiding frequency. The physical\norigin of the Raman jitter is in frequency shifts due to soliton phase modulation\nby the ever-present quantum and thermal phonon fields in the fiber medium.\n\nThe numerical method should give accurate results far beyond the distance shown in \nFig. \\ref{ol}, provided that the transverse and propagative resolutions are made large \nenough. The equations generated in the Wigner method should remain valid up to $\\zeta \n\\sim \\sqrt{\\overline{n}}$, which corresponds to about $1000 km$. When the Wigner equations\ncan no longer be trusted, the positive-$P$ equations will still give accurate results. This \npaper has not analysed any multisoliton effects, although the numerical method does simulate \ninteractions between solitons. This just requires the initial conditions and simulation window width to \nbe set up according to whether interactions are to be considered or not. We \nhave not included third-order dispersion in our model, which would become important for very \nsmall pulses ($t_0 \\simeq 100 fs$), but it could easily be included into the equations for numerical simulation.\n\nThe approximate analytic results are most limited probably by their exclusion of Raman intrapulse effects, such as\nthe deterministic self-frequency shift and the amplifier jitter that feeds through this. Approximate calculations\\cite{MooWonHau94} of the self-frequency shift jitter variance show that it grows as the fifth power of distance. With our parameters and the measured value of the Raman time constant\\cite{AtiMysChrGal}, it would become larger than the usual Gordon-Haus effect at $x \\simeq 100 km$, or about 10 times the propagation distance shown in Fig. \\ref{ol}. Using standard techniques\\cite{MooWonHau94}, the perturbation theory presented in this paper could be extended to include the self-frequency shift contributions (from both the amplifier noise and Raman phase noise) to the total jitter.\n\n\n\\section{Conclusions}\n\n\\label{Cs} Our major conclusion is that quantum noise effects due to the intrinsic\nfinite-temperature phonon reservoirs are a dominant source of fluctuations in\nphase and arrival time, for subpicosecond solitons. For longer solitons, Raman\neffects are reduced when compared to the Gordon-Haus jitter from the laser gain\nmedium that is needed to compensate for losses. The reason for this is the smaller\nintensity of the pulse, and therefore the reduced Raman couplings that occur\nfor longer solitons -- which are less intense than shorter solitons with the\nsame dispersion. The ratio can be calculated simply from the product \$ Gx_{0} \$,\nwhich gives the gain per soliton length. A smaller \$ x_{0} \$ corresponds\nto a shorter, more intense soliton and hence a larger Raman noise; while a larger\n\$ G \$ corresponds to increased laser gain, with larger spontaneous noise.\n\nAt a given pulse duration and fiber length, a strategy for testing this prediction\nwould be to use short pulses with dispersion-shifted fiber having an increased\ndispersion, since this increases the relative size of the Raman jitter. The\nphysical reason for this is very simple. Solitons have an intensity which increases\nwith dispersion if everything else is unchanged. At the same time, the multiplicative\nphase noise found in Raman propagation is proportional to intensity, and hence\nbecomes relatively large compared to the additive Gordon-Haus noise due to amplification.\nFor large enough dispersion, the temperature-dependent Raman jitter should become\nreadily observable at short enough distances so that amplification is unnecessary.\nThis would give a completely unambiguous signature of the effect we have calculated.\nA mode-locked fiber soliton laser would be a suitable pulse source, owing to\nthe very short (\$ 64fs \$), quiet pulses\\cite{Yu1997,Namiki1996} that are\nobtainable.\n\n\n\n\\acknowledgments\n\nWe would like to acknowledge helpful comments on this paper by Wai S. Man.\n\n\\begin{references}\n\n\\bibitem{s147} J.~F. Corney, P.~D. Drummond, and A. Liebman, ``Quantum noise limits to terabaud\ncommunications,'' Optics Communications \\textbf{140}, 211--215 (1997). \n\n\\bibitem{s35} J.~P. Gordon and H.~A. 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Mollenauer, Optics Letters \\textbf{14}, 1284 (1989). \n\n\n\\end{references}\n\\begin{figure}\n\\caption{Spectrum of the fluorescence function \\protect\\protect\$ {\\mathcal{F}}(\\omega )\\protect \n\\protect \$ for the 11-Lorentzian model (continuous lines) and the single-Lorentzian model\n(dashed lines), for a temperature of \\protect\\protect\$ T=300K\\protect \\protect \$.\nAlso shown is the spectrum of a \\protect\\protect\$ t_{0}=1ps\\protect \\protect \$\nsoliton.}\n\n\\label{solspect}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Timing jitter in \\protect\\protect\$ t_{0}=500fs\\protect \\protect \$ bright\n(a) and dark (b) solitons due to initial quantum fluctuations (circles), Gordon-Haus\neffect (crosses) and Raman noise (plus signs). The asterisks give the total\njitter and the continuous line gives the approximate analytic results for the\nRaman jitter. }\n\n\\label{ol}\n\\end{figure}\n\n\n\\end{document}\n\n" } ]	[ { "name": "quant-ph9912096.extracted_bib", "string": "{s147 J.~F. Corney, P.~D. Drummond, and A. Liebman, ``Quantum noise limits to terabaud communications,'' Optics Communications 140, 211--215 (1997)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s35 J.~P. Gordon and H.~A. Haus, ``Random walk of coherently amplified solitons in optical fiber transmission,'' Optics Letters 11, 665--667 (1986)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{DruCor99a P.~D. Drummond and J.~F. Corney, ``Quantum noise in optical fibers {I: Stochastic equations, {\\em submitted to Journal of the Optical Society of America B." }, { "name": "quant-ph9912096.extracted_bib", "string": "{e375 L.~F. Mollenauer, J.~P. Gordon, and M.~N. Islam, ``Soliton propagation in long fibers with periodically compensated loss,'' IEEE Journal of Quantum Electronics 22, 157--173 (1986)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{MooWonHau94 J.~D. Moores, W.~S. Wong, and H.~A. Haus, ``Stability and timing maintenance in soliton transmission and storage rings,'' Optics Communications 113, 153--175 (1994)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{AtiMysChrGal A.~K. Atieh, P. Myslinski, J. Chrostowski and P. Galko, ``Measuring the Raman time constant ($T_R$) for soliton pulses in standard single-mode fiber,'', Journal of Lightwave Technology 17, 216--221 (1999)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s11 D.-M. Baboiu, D. Mihalache, and N.-C. Panoiu, ``Combined influence of amplifier noise and intra-pulse Raman scattering on the bit-rate limit of optical fiber communication systems,'' Optics Letters 20, 1865--1867 (1995); D. Mihalache, L.-C. Crasovan, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, ``Timing jitter of femtosecond solitons in monomode optical fibers,'' Optical Engineering 35, 1611--1615 (1996); D. Wood, ``Constraints on the bit rates in direct detection optical communication systems using linear or soliton pulses,'' Journal of Lightwave Technology 8, 1097--1106 (1990)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{SFS_comp D. Shenoy and A. Puri, ``Compensation for the soliton self-frequency shift and the third-order dispersion using bandwidth-limited optical gain,'' \\oc 113, 410--406 (1995); S.~V. Chernikov and S.~M.~J. Kelly, ``Stability of femtosecond solitons in optical fibres influenced by optical attenuation and bandwidth limited gain,'' Electronics Letters 28, 238--240 (1992)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s05 G.~P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic Press, 1995), p 475." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s24 P.~D. Drummond and W. Man, ``Quantum noise in reversible soliton logic,'' Optics Communications 105, 99--103 (1994)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s53 H.~A. Haus and W.~S. Wong, ``Solitons in optical communications,'' Reviews of Modern Physics 68, 423--444 (1996)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s55 F.~X. Kartner, D.~J. Dougherty, H.~A. Haus, and E.~P. Ippen, ``Raman noise and soliton squeezing,'' Journal of the Optical Society of America B 11, 1267--1276 (1994)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s58 D.~J. Kaup, ``Perturbation theory for solitons in optical fibers,'' Physical Review A 42, 5689--5694 (1990)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s65 Y.~S. Kivshar, M. Haelterman, P. Emplit, and J.~P. Hamaide, ``Gordon-Haus effect on dark solitons,'' Optics Letters 19, 19--21 (1994); Y.~S. Kivshar, ``Dark solitons in nonlinear optics,'' IEEE Journal of Quantum Electronics {29, 250--264 (1993)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s83 I.~M. Uzunov and V.~S. Gerdjikov, ``Self-frequency shift of dark solitons in optical fibers,'' Physical Review A 47, 1582--1585 (1993)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s50 A. Hasegawa and F. Tappert, ``Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,'' Applied Physics 23, 171--172 (1973)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{s45 J. Hamaide, P. Emplit, and M. Haelterman, ``Dark-soliton jitter in amplified optical transmission systems,'' Optics Letters 16, 1578--1580 (1991)." }, { "name": "quant-ph9912096.extracted_bib", "string": "{RamanGain R.~H. Stolen, C. Lee, and R.~K. 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quant-ph9912097	Bose Condensates with 1/r Interatomic Attraction: Electromagnetically Induced ``Gravity''	[ { "author": "D. O'Dell$^{(1)" } ]	We show that particular configurations of intense off-resonant laser beams can give rise to an attractive $1/r$ interatomic potential between atoms located well within the laser wavelength. Such a ``gravitational-like'' interaction is shown to give stable Bose condensates that are self-bound (without an additional trap) with unique scaling properties and measurably distinct signatures.	[ { "name": "papero.tex", "string": "\\documentstyle[twocolumn,aps,prl,psfig,epsf,epsfig]{revtex}\n%\\documentstyle[aps,preprint,psfig,epsf,epsfig]{revtex}\n\\draft\n\\begin{document}\n\\title{Bose Condensates with 1/r Interatomic Attraction:\nElectromagnetically Induced ``Gravity''}\n\\author{\nD. O'Dell$^{(1)}$ \\and S. Giovanazzi$^{(1)}$ \n\\and G. Kurizki$^{(1)}$ \\and V. M. Akulin$^{(2)}$ }\n\\address{\n(1) Department of Chemical Physics, Weizmann Institute of Science,\n76100 Rehovot, Israel\\\\\n(2) Laboratoire Aim\\'{e} Cotton, CNRS II, Orsay Cedex 91405, France}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nWe show that particular configurations of intense off-resonant\nlaser beams can give rise to\nan attractive $1/r$ interatomic potential between atoms located\nwell within the laser wavelength.\nSuch a ``gravitational-like'' interaction is shown to give stable\nBose condensates that are \\emph{self-bound} (without an additional\ntrap) with unique scaling properties and measurably distinct \nsignatures.\n\\end{abstract}\n\\pacs{PACS: 03.75.Fi,34.20.Cf,34.80.Qb,04.40.-b}\n\n\n\n \n\n\nIn the atomic Bose-Einstein condensates (BECs) \ncreated thus far \\cite{anderson95} the atoms interact only at \nvery short distance in good correspondence with the hard sphere model.\nThe majority of the properties of these dilute gases \ncan be understood by taking into account only two-body collisions \nwhich are characterized by the s-wave scattering length \\cite{dalfovo99}.\nA number of groups \\cite{tiesinga92} have investigated the fascinating\npossibility of\nchanging the magnitude and sign of the s-wave scattering length\nusing external fields. The resulting condensates retain\nthe essential hard-sphere, s-wave, nature\nof the interatomic interaction.\n\n\nHere we wish to introduce a qualitatively new regime of cold atoms\nin which the atom-atom interactions are attractive and have a very long range,\nvarying as $r^{-1}$ \\cite{goral99}.\n\n\nWe shall demonstrate that a stable BEC with attractive $r^{-1}$\ninteractions\nis achievable by irradiating the atoms with\nintense, \\emph{off-resonant}, electromagnetic fields.\nThe atoms are then coupled\nvia the dipoles that are induced by these external fields\n(in contrast to those induced by the random vacuum field\nresponsible for the van der Waals-London interaction,\nwhich varies as $r^{-6}$, and leads to the usual\nhard sphere description) \\cite{thirunamachandran80}.\n\n\nSuch an $r^{-1}$ attractive potential can simulate gravity\nbetween quantum particles. Remarkably, this potential gives an\ninteratomic attraction\n(depending on the laser intensity and wavelength) which\ncan be some 17 orders of magnitude greater than \ntheir gravitational interaction at the same distance.\n\nThis suggests it might be possible\nto study gravitational effects, normally only important on the\nstellar scale, in the laboratory. \nParticularly interesting is the possibility of experimentally\n emulating Boson stars \n\\cite{ruffini69}: gravitationally bound condensed Boson configurations of \nfinite volume, in which the zero-point kinetic energy balances the \ngravitational attraction and thus \\emph{stabilizes} the system against \ncollapse.\n\n\n\nIn this letter we shall discuss the interplay of the usual hard-core \ninteratomic potential with an electromagnetically induced \n``gravitational'' one on a BEC using a variational \nmean-field approximation (MFA).\nTwo new physical regimes with unique scaling properties emerge where the BEC\nis self-bound (no trap required).\n\n\n\nHow can one realize the $r^{-1}$ potential between neutral atoms? \nConsider the dipole-dipole interaction energy induced by the presence\nof external electromagnetic radiation of intensity $I$, wavevector\n${\\bf q}$, and polarization $\\hat{{\\bf e}}$. This energy can be written\n(in S.I. units) in terms of cartesian components $i,j$\n\\cite{thirunamachandran80}\n\\begin{equation}\nU({\\bf r}) =\n\\left(\\frac{ I}{4 \\pi c \\varepsilon_{0}^{2}}\\right)\n\\alpha^{2}(q) \\hat{{\\bf e}}_{i}^{\\ast} \\hat{{\\bf e}}_{j} V_{ij}(q, {\\bf r})\n\\cos ({\\bf q} \\cdot {\\bf r}). \\label{eq:tpot}\n\\end{equation}\nHere ${\\bf r}$ is the interatomic axis, $\\alpha(q)$ the isotropic,\ndynamic, polarizability of the atoms at frequency $cq$, and\n$V_{ij}$ is the retarded dipole-dipole interaction tensor\n\\begin{eqnarray}\nV_{ij}=\\frac{1}{r^{3}} & \\Big[ \\big(\\delta_{ij}-3\\hat{{\\bf r}}_{i}\n\\hat{{\\bf r}}_{j}\\big) \\big( \\cos q r +qr \\sin q r \\big) & \\nonumber \\\\\n& -\\big(\\delta_{ij}-\\hat{{\\bf r}}_{i}\\hat{{\\bf r}}_{j}\\big)q^{2}r^{2}\n\\cos qr \\Big] & \\label{eq:retarded-dip-int}\n\\end{eqnarray}\nwhere $\\hat{{\\bf r}}_{i}=r_{i}/r$.\nFor a \\emph{fixed orientation} of the interatomic axis with respect to the\nexternal field, (\\ref{eq:tpot}) and (\\ref{eq:retarded-dip-int}) give the \nwell known $r^{-3}$ variation of the interaction energy at near-zone \nseparations ($qr \\ll 1$). \nThe near zone limit of $U({\\bf r})$ is \\emph{strongly anisotropic}. \nIt was noted by Thirunamachandran \\cite{thirunamachandran80}\nthat when an \\emph{average over all orientations} of the interatomic\naxis with respect to the incident radiation direction is taken,\nthe static dipolar part of the coupling (i.e., the instantaneous,\nnon-retarded part $r^{-3}\\big(\\delta_{ij}-3\\hat{{\\bf r}}_{i}\n\\hat{{\\bf r}}_{j}\\big)$) vanishes. \nThe remaining `transverse' part is, in the near-zone, an attractive $r^{-1}$\npotential. It is weaker by a factor of $(qr)^{2}$ than the $r^{-3}$\nterm.\n\n\n \nHowever, thus far no scheme has been suggested wherein \nan average over all orientations is guaranteed for cold gases. \nWe shall consider a spatial configuration of external fields which \n\\emph{enforces the `averaging out'} of the $r^{-3}$ interactions.\nA simple combination which ensures the suppression of the $r^{-3}$\ninteraction while retaining the weaker $r^{-1}$ attraction\nin the near-zone, uses three orthogonal circularly polarized\nlaser beams pointing along $\\hat{{\\bf x}},\\hat{{\\bf y}},\\hat{{\\bf z}}$ \n(`a triad'---see Fig.\\ 1).\nLet us momentarily ignore interference between the three beams, and only\nconsider the sum of their intensities. \nIn the near zone one can Taylor-expand Eqs.\\ (\\ref{eq:tpot}) \nand (\\ref{eq:retarded-dip-int}) \nin powers of the small quantity $qr$.\nUsing the identity \n$\\hat{{\\bf e}}^{\\ast (\\pm)}_{i}({\\bf q}) \\hat{{\\bf e}}^{(\\pm)}_{j}\n({\\bf q})=\\frac{1}{2}\n\\left( \\left(\\delta_{ij}-\\hat{{\\bf q}}_{i} \\hat{{\\bf q}}_{j} \\right)\n\\pm {\\mathrm{i}} \\epsilon_{ijk}\\hat{{\\bf q}}_{k} \\right)$, with $+(-)$\ncorresponding to left(right) circular polarizations,\ntogether with Eqs.\\ (\\ref{eq:tpot}) and (\\ref{eq:retarded-dip-int}),\nthe triad can be shown to give rise to the (near-zone) $r^{-1}$ \npair potential\n\\begin{eqnarray}\n\\lefteqn{U({\\bf r}) = \n-\\frac{3 I q^{2} \\alpha^{2}}{(16 \\pi c \\varepsilon_{0}^{2})} \n\\times} & \\quad & \\quad \\nonumber \\\\\n \\quad & \\quad & \\frac{1}{r} \\Big[\\frac{7}{3} +\n\\left(\\sin \\theta \\cos \\phi \\right)^{4} +\n\\left(\\sin \\theta \\sin \\phi \\right)^{4} +\n\\left( \\cos \\theta \\right)^{4} \n\\Big]. \n\\label{eq:triad}\n\\end{eqnarray}\nNote that this interaction is attractive for any orientation\n $(\\theta,\\phi)$ of ${\\bf r}$\nrelative to the beams as long as the polarizability $\\alpha(q)$ is\nreal.\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[h]\n\\vspace{-1.6cm}\n\\begin{center}\n\\centerline{\\psfig{figure=fig1.ps,height=16cm}}\n\\end{center}\n\\vspace{-12.4cm}\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\n\\begin{caption}\n{\\textbf{(a):} Schematic depiction of a triad of lasers incident\nupon an ensemble of atoms. \nThis triad generates the attractive $r^{-1}$ potential given by Eq.\\ (3), \n whose magnitude has the angular dependence shown\nin \\textbf{(b)}.}\n\\end{caption}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\n\nIf one wishes, the angular anisotropy in (\\ref{eq:triad}) can be \ncancelled to give a purely radial $r^{-1}$ potential\nby combining a number of such triads with different orientations.\nIt is convenient to define the orientation of each triad by\nthe Euler angles $(\\alpha,\\beta,\\gamma)$ \n\\cite{mathews-walker}, namely, a rotation of $\\alpha$ about\nthe $\\hat{{\\bf z}}$ axis, followed by a rotation of $\\beta$ about\nthe new $\\hat{{\\bf y}}$ axis and finally a rotation of $\\gamma$ about the\nfinal $\\hat{{\\bf z}}$ axis.\nOne configuration that cancels the anisotropy completely\nuses 6 triads (18 laser beams)\nrotated through the following Euler angles:\n$(0,\\pi/4,\\pi/8)$, $(0, \\pi/4, -\\pi/8)$, $(0, \\pi/4, 3\\pi/8)$,\n$(0, \\pi/4 ,-3 \\pi/8)$, $(0,0,\\pi/8)$, $(0,0, -\\pi/8)$.\nThe last two triads should be of half the intensity $I$ of the others.\nThen the interatomic potential becomes\n\\begin{equation}\nU({\\bf r}) = -\n\\frac{11}{4 \\pi}\n\\frac{I q^{2} \\alpha^{2}}{c \\varepsilon_{0}^{2}}\n\\frac{1}{r} = -\\frac{u}{r}\\;. \n\\label{eq:supertriads}\n\\end{equation}\n\n\n\n\nThe main difficulty in realizing the near-zone $r^{-1}$ potential is that the\n$r^{-3}$ interaction survives due to the\n\\textit{interference} between different pairs of beams, whose contribution is\nproportional to the product of their respective field amplitudes.\nThis difficulty can be overcome if one introduces frequency shifts\nbetween the laser beams.\nSpreading the frequencies $\\omega_{n}$ of the $E_n$\nlaser fields ($n=1,2,3$ for one triad or $n=1,2 \\dots 18$ for six triads) in\nintervals about the central frequency makes the $r^{-3}$\ninterference terms\nin the interaction energy $\\propto E_n E^{\\ast}_{n'}$, ($n\\neq n'$),\noscillate at the difference frequencies $\| \\omega_{n}-\\omega_{n'} \|$.\nIf these difference frequencies are much higher than the other\nrelevant frequencies\n(e.g., collective oscillation frequencies)\nthen the contribution of the interference terms\nto the mean field potential averages to zero.\nTypically these conditions hold for\n $\| \\omega_{n}-\\omega_{n'} \| \\geq 10^4$ Hz.\nAngular misalignment errors, $\\delta$, between the orthogonal beams\nshould satisfy $\\delta\\ll q\\, L$ \n(where $L$ is the mean radius of the condensate)\nand intensity fluctuations should satisfy\n$\\Delta I/I \\ll q \\,L$, in order to ensure the $r^{-3}$ cancellation\nfor the non-interfering terms $\\propto \\sum_{n} \| E_{n} \|^{2}$.\nAlthough these oscillating $r^{-3}$ terms do not contribute\nto the mean field potential, they can eject atoms from the condensate,\nbut this process can be strongly reduced, \nas will be discussed at the end of this letter.\n\n\nA \\emph{lower bound} on the magnitude of the\n$r^{-1}$ attraction is obtained by using\nthe static rather than dynamic polarizability, which for sodium atoms,\n say, has the value $24.08 \\times 10^{-24} {\\mathrm{cm}}^{3}$. \nThus, for a strongly off-resonant light intensity of\n $I = 10^{4}$ ${\\mathrm{Watts/cm}}^{2}$ one finds\n$-u/r$ $\\approx$ $-2 \\times 10^{-19} {\\mathrm{eV}}$, at $r = 100$nm, \nthe mean separation in a typical BEC.\nThis is only around $10^{-4}$ of the\nmagnitude of the van der Waals-London dispersion energy at this distance.\nHowever in a system of many atoms the $r^{-1}$ potential acts \nover the entire sample whereas the van der Waals-London interaction is \nonly effective for nearest-neighbors and so the $r^{-1}$ contribution\nto the total energy can become important.\n\n\nOur treatment of the many particle problem \nis based on a two-body potential \n$V({\\bf r})=4 \\pi a \\hbar^{2} \\delta({\\bf r})/m - u/r$,\nwhere the first term is the the pseudo-potential arising from the \ns-wave scattering ($a$ is the s-wave scattering length\nand $m$ the atomic mass).\nIn order to write $V$ in this form we require that the $-u/r$ potential\nbe sufficiently weak (compared with the mean kinetic energy per particle)\nso as not to affect the short-range hard-sphere scattering.\nThis requirement certainly holds if \n$a_* \\gg \\lambda_{DB} \\gg a$, where \n$\\lambda_{DB}$ is the de Broglie wavelength and\n$a_=h^2/m u$ is the Bohr radius associated with the \ngravitational-like coupling $u$.\nWith the values given above $a_$ $\\sim 10^3$ m, \nwhilst for a typical BEC $\\lambda_{DB} \\sim\n10^{-5}-10^{-3}$ m and \n$a\\sim 3$ nm.\n\n\nConsider now the application of this two-body potential\nto a trapped dilute BEC gas well below the critical temperature. \nWe assume that the condensate initially occupies\na fraction of the wavelength of the laser so that the near-zone\ncondition is valid (lasers in the far infrared, or microwave sources\nwould satisfy this condition).\nThe ``zero-temperature'' many-body problem leads, within\nthe MFA, to the following equation for\nthe order parameter $\\Psi ({\\bf R},t)$\n\\begin{equation}\ni \\hbar \\frac{\\partial \\Psi ({\\bf R},t)}{\\partial t} =\n\\left[\n- \\frac{\\hbar^{2}}{2m} \\nabla^{2}\n+ V_{\\mathrm{ext}}({\\bf R})\n+ V_{\\mathrm{H}}({\\bf R})\n\\right] \\Psi({\\bf R},t) \\label{21}\n\\end{equation}\nwhere $V_{\\mathrm{ext}}(R)= m\\omega_{0}^2 R^2/2$ is for simplicity\nan isotropic trap potential\n(which can be set to zero in certain cases---see below), and\n $V_{\\mathrm{H}}({\\bf R})$ is the self-consistent Hartree potential\n\\begin{equation}\nV_{\\mathrm{H}}({\\bf R})=\n\\frac{4 \\pi a \\hbar^{2}}{m} \\mid \\Psi({\\bf R},t) \\mid^{2}\n- u\\int d^3 R^{\\prime}\n\\frac{ \\mid \\Psi({\\bf R^{\\prime}},t) \\mid^{2}}\n{\\mid {\\bf R}^{\\prime}- {\\bf R}\\mid} \\;. \\label{22}\n\\end{equation}\nThe order parameter $\\Psi( {\\bf R},t)$\nis normalized so that $\\int d^{3} R \\,\\,\|\\Psi ( {\\bf R},t) \|^2$\n$=N$,\nwith $N$ the total number of atoms. \nThe usual Gross-Pitaevskii (GP) equation \\cite{dalfovo99} is \nrecovered in the limit when $u=0$.\nThe MFA is valid when the system is dilute, i.e.\\,\n$\\rho a^{3} \\ll 1$, with $\\rho$ the density. \nAn additional condition on the MFA validity is that the\n$r^{-1}$ potential is weak, and this can be expressed as $\\rho a_^3 \\gg 1$.\nThis constraint, as in the related problem of the charged Boson gas\n\\cite{foldy61}, means that many atoms must be present within the \ninteraction volume $a_^3$.\nHowever, it is in fact the diluteness condition that turns out to be\nthe stricter of the two, as one can check from the MFA\n results for the density (see Table 1 below).\n\n\nAn analytical estimate for the mean radius \nof the N-atom condensate\ncan be given using the following variational wave function\n\\begin{equation}\n\\Psi_{\\lambda}(R)=N^{\\frac 12}(\\pi \\lambda^2 l^2_{0})^{-{3\\over4}}\n\\exp (-R^2/2\\lambda^2 l^2_{0} )\n\\label{eq:ansatz}\n\\end{equation}\nwhere $l_0=\\sqrt{\\hbar/m\\omega_{0}}$.\nThe variational parameter $\\lambda$ is proportional to the root mean\nsquare radius through $\\sqrt{\\langle R^2 \\rangle }\n = \\sqrt{3/2}\\ \\lambda l_{0} $.\nThis parameter is obtained by minimizing\nthe variational mean field energy\n\\begin{equation}\n\\frac{H(\\lambda)}{N\\;\\hbar\\omega_0}=\n\\frac{3}{4}\n\\left(\n\\lambda^{-2}\n+\\lambda^{2}\n- 2 \\tilde{u} \\,\\lambda^{-1}\n+ \\frac 23 \\tilde{s} \\,\\lambda^{-3}\n\\right)\n\\label{varene}\n\\end{equation}\nwhere we have chosen the dimensionless $\\tilde{u}$ (proportional to the \n``gravity'' strength $u$)\nand $\\tilde{s}$ (proportional to s-wave scattering length $a$) to be\n\\begin{eqnarray}\n\\tilde{u} &=& \n% 10.49 \n\\pi\\sqrt{32\\pi/9}\n\\; \\; (N l_{0} / a_{}) \\; \n\\label{eq:eps}\\nonumber \\\\\n\\tilde{s} &=& \n%0.7979 \n\\sqrt{2/\\pi}\n\\; \\; (N a/l_{0}) \\;.\n\\label{eq:eta}\n\\end{eqnarray}\nThe numerical factors \nare chosen to make the equation for $\\lambda$ simple\n\\begin{equation}\n- \\lambda^{-4} + 1 + \\tilde{u} \\,\\lambda^{-3}\n- \\tilde{s} \\,\\lambda^{-5} = 0 \\;. \\label{vareq}\n\\end{equation}\nThis equation is equivalent to requiring\nthat the variational solution satisfies the\nfollowing virial relation:\n$- T + V_{\\mathrm{ext}} - \\frac{1}{2} U_{u}\n- \\frac{3}{2} U_{s}$ $= 0$,\nwhere $T$, $V_{ext}$, $U_{u}$ and $U_{s}$ are the kinetic energy,\nthe harmonic trap potential energy, and the internal energies due to the\n$-u/r$ and hard-sphere interatomic potentials, respectively. \nThis relation can be obtained from scaling considerations\n(see Ref. \\cite{dalfovo96} for the case $u=0$).\n\n\n\nThe general asymptotic properties of the ground state solutions\nof Eqs.\\ (\\ref{21}) and (\\ref{22}), \nas a function of $(\\tilde{u},\\tilde{s})$,\nare summarized in the ``phase diagram'' of Fig.\\ 2a\nfor positive scattering lengths. \nIn this diagram\nthere are four asymptotic regions:\nThe non-interacting ideal region\n($I$) and the ordinary Thomas-Fermi region ($TF-O$)\nare dominated by the balance of\nthe external trap potential with, respectively, the kinetic energy\nand the repulsive s-wave scattering, and so\nare not sensitive to the $-u/r$ potential.\nThe regions $G$ and $TF-G$, which represent two \\emph{new physical regimes}\nfor atomic BECs, are controlled by the balance\nof the gravity-like potential with either the kinetic energy ($G$) or\nthe s-wave scattering ($TF-G$).\nNeither region is sensitive to the external trap, so that\nwe can adiabatically turn it off ($V_{\\mathrm{ext}}=0$)\nand access either the $G$ \nor the $TF-G$ region.\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[h]\n\\vspace{-2cm}\n\\begin{center}\n\\centerline{\\psfig{figure=fig2.ps,height=14cm}}\n\\end{center}\n\\vspace{-9cm}\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\n\\begin{caption}\n{\\textbf{(a)}:\nContour plot of $\\log (\\lambda)$,\nwhere $\\lambda$ is the condensate radius,\nin the parameter space $\\log(\\tilde{u})$ \nversus $\\log(\\tilde{s})$: \ndarker shade corresponds to smaller $\\lambda$.\nThe border separating the $TF-O$ and $TF-G$ regions is given by\n$\\tilde{s} = \\tilde{u}^{5/3}$ \nand that separating the $TF-G$ and $G$ regions by\n$\\tilde{s} = \\tilde{u}^{-1}$. \n\\textbf{(b)}:\nMean energies per particle for large $\\tilde{u}$\n(no external trap) as a function of the condensate radius.\nCurves are plotted for positive as well as negative values\nof the scattering strength $\\tilde{s}/\\tilde{u}$.\nFor $\\tilde{s}/\\tilde{u} \\le -1/4$ there is no minimum for \na finite radius.\nThe energy and radii units are $\\tilde{u}^2\\hbar\\omega_0$ and \n$l_0/\\tilde{u}$, respectively.}\n\\end{caption}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%% F I G U R E %%%%%%%%%%%%%%%%%%%%%%\nExperimentally, direct signatures of the $r^{-1}$ interaction\ncome from the radius $\\lambda$ and the release energy $E_{rel} = T + U_{s}$.\nThe release energy is the kinetic energy that can be \nmeasured after the expansion occurring due to \nswitching off the external trap and the laser fields \\cite{dalfovo99}.\nTable 1 summarizes these quantities \nas well as the peak density $\\rho_{\\mathrm{max}}$ in the four regions.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%% T A B L E %%%%%%%%%%%%%%%%%%%%%%\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{lllll} \n & $G$ & $TF-G$ & $TF-O$ & $I$ \\\\ \\cline{2-5} \n\\raisebox{.0ex}[3.5ex]{defn.:} & \n $\\tilde{u} \\gg 1$ & $\\tilde{s} \\ll \\tilde{u}^{5/3}$\n & $\\tilde{s} \\gg 1$ &\n $\\tilde{u} \\ll 1$ \\\\ \n & $\\tilde{s} \\ll 1/\\tilde{u}$ & $\\tilde{s} \\gg 1/\\tilde{u}$ \n & $\\tilde{s} \\gg \\tilde{u}^{5/3}$ & $\\tilde{s} \\ll 1$ \\\\ \\cline{2-5}\n\\raisebox{.0ex}[2.5ex]{$\\lambda$:} & $1/\\tilde{u}$ &\n $(\\tilde{s}/\\tilde{u})^{1/2}$\n & $\\tilde{s}^{-1/5} $ & 1 \\\\ \\cline{2-5}\n\\raisebox{.0ex}[3.5ex]{$E_{\\mathrm{rel}}/ \\hbar \\omega_{0}$:} &\n $\\frac{3}{4} \\tilde{u}^{2}N$ \n & $\\frac{1}{2} \\tilde{u}^{3/2} \\tilde{s}^{-1/2}N$\n & $\\frac{1}{2} \\tilde{s}^{2/5}N$\n & $\\frac{3}{4}N$ \\\\ \n & $\\propto \\; \\; N^{3}$ & $ \\propto \\; \\; N^{2} $\n & $ \\propto \\; \\; N^{7/5}$ \n & $\\propto \\; \\; N $ \\\\ \\cline{2-5}\n\\raisebox{.0ex}[3.5ex]{$\\rho_{\\mathrm{max}}:$} \n & $\\frac{\\sqrt{32}^{3} \\pi^{3} N^{4}}{27 a_{}^{3}} $ \n & $\\frac{2 \\pi^{2} N}{\\sqrt{a a_{}}^{3}}$ \n & $\\frac{15^{2/5} N^{2/5}}{8\\pi a^{3/5} l_0^{12/5}}$\n & $ \\frac{N}{\\sqrt{\\pi}^{3}l_{0}^{3}}$\n\\end{tabular}\n\\end{center}\n\\begin{caption}\n{A comparison of the four asymptotic regions.\nThe radius, $\\lambda$, and the release energy, $E_{\\mathrm{rel}}$,\nare discussed in the text. $\\rho_{\\mathrm{max}}$ is the peak density\n(at the center) of the condensate.}\n\\end{caption}\n\\end{table}\n%\\vspace{0.2cm}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWe now focus on the properties of the two new regimes:\na) In the $TF-G$ region an analytic solution \nfor the ground state of Eqs.\\\n(\\ref{21}) and (\\ref{22}) is given by\n\\begin{equation}\n \\Psi_{TF-G} ({\\bf R}) = \\frac{\\sqrt{N}}{2 R_{0}}\n\\sqrt{\\frac{\\sin(\\pi R/R_{0})}{R}}\n\\Theta(R_{0} - R) \\; \\label{eq:tf}\n\\end{equation}\nwhere $R_{0}= \\sqrt{a \\ a_{}}/2 $.\nContrary to the ordinary Thomas-Fermi limit of the GP equation,\n the size of the condensate is\nfixed by the ratio of the coupling constants, $4 \\pi a \\hbar^{2}/m$\nand $u$, and is\n\\textit{independent of $N$}. \nb) \nThe $G$ region, where only the $r^{-1}$ attraction and \nkinetic energy play a r\\^{o}le, is of particular interest\nsince our system is then equivalent to a Boson-star \n(a system of gravitating Bosons) \\cite{ruffini69}\nin the non-relativistic regime.\nThe mean field equations in this region are also identical to those\ndescribing a single particle moving in the \ngravitational field generated by its own wavefunction \\cite{penrose98}.\nIn both cases smooth bound solutions have been shown to exist \n\\cite{ruffini69,penrose98}.\nThis establishes the possibility of a stable \\textit{self-bound}\n(no external trap) $r^{-1}$ condensate.\n\n\nThe gravitational-like attraction does not induce ``collapse'' of the \ncondensate, since, at short radii, it is always weaker than the kinetic energy.\nThis can be seen from the scaling of the kinetic energy ($\\lambda^{-2}$) \nversus that of the $r^{-1}$ potential ($\\lambda^{-1}$) \nin Eq. (\\ref{varene}) and Fig.\\ 2b.\nBy contrast, this kind of instability can occur for \\emph{negative} \nscattering lengths \n\\cite{tiesinga92,goral99}\nwhen $N$ exceeds a critical number ($N_{cr} \\approx 0.6 \\times \|a\|/l_0$)\nbecause the mean energy due to scattering ($\\lambda^{-3}$) is dominant\n at small radii.\nThe $u/r$ attraction does reduce, when combined with the attractive scattering,\nthe critical number to $N_{cr}\\approx 0.17 \\times \\sqrt{a_*/\|a\|}$\n (see Fig.\\ 2b for the critical case with $\\tilde{s}\\tilde{u}=-1/4$).\n\n\n\nFinally, we estimate the losses of $G$ or $TF-G$ condensates due to\nthe $r^{-3}$ oscillating \\emph{interfering} terms discussed above.\nConsider one of the possible oscillating interfering terms \n$A({\\bf r}) \\cos(\\Omega t)$, where\n$A(x,y,z) = - 3 \\,u\\, \\frac{x\\, y\\,}{q^2\\, r^5}$ \nand $\\Omega$ is the difference in frequency between the two \ninterfering lasers.\nUsing Fermi's golden rule one can derive an expression \nfor the rate of depletion of the condensate density $\|\\Psi\|^2$ \ndue to creation of a pair of quasiparticles\nof opposite momenta \n(with $k\\approx \\pm \\sqrt{m\\Omega / \\hbar}$)\nin the ideal homogeneous Bose gas \n\\begin{equation}\n\\frac{d\\,\|\\Psi\|^2}{dt} = -\\frac{\\overline{\|A(k)\|^2}}{6\\pi}\n \|\\Psi\|^4 \\left(\\frac{m}{\\hbar^2}\\right)^{3/2}\n\\sqrt{\\frac{\\Omega}{\\hbar}}\n\\label{eq.13}\n\\end{equation}\nwhere $\\overline{\|A(k)\|^2}$ is the \nangular average of the square of the Fourier transform \nof $A({\\bf r})$ ($ 0.1418\\, u^2/q^4 $).\nFor our purposes it is sufficient to apply Eq.\\ (\\ref{eq.13})\nat each point ${\\bf R}$ ($\|\\Psi\|^2=\|\\Psi\|^2({\\bf R})$).\nWe then find the following approximations:\n$d\\, N_0/dt \\approx \\tilde{u}^{5}\n\\sqrt{\\Omega\\omega_0} / (q l_0)^4$ in the $G$ region\nand $d\\, N_0/dt \\approx \\tilde{u}^{7/2} \\tilde{s}^{-3/2}\n\\sqrt{\\Omega\\omega_0} / (q l_0)^4 $ in the $TF-G$ region.\nThese expressions can be used to find conditions\nsuch that these loss rates are smaller than, say, the trap oscillation\nfrequency $\\omega_0$. Taking, e.g.,\n${\\Omega} \\approx 2\\pi\\times 10^4 $ s$^{-1}$, \n${\\omega_0}\\approx 2\\pi\\times 10^2$ s$^{-1}$, $ q l_0 \\approx 1$,\n$\\tilde{u} \\approx 5$, we obtain:\nfor the $G$ region \n$d\\;N_0/dt \\approx 6 \\times 10^{4}{\\omega_0}$, i.e. we need more than\n$10^5$ atoms;\nfor the $TF$ region (e.g., $ \\tilde{s} \\approx 1$), \n$d\\;N_0/dt \\approx 6 \\times 10^{3}{\\omega_0}$, i.e. we need more than\n$10^4$ atoms.\n\n\nTo conclude, the laser-induced attractive $r^{-1}$ interaction\ncan give rise to stable condensates with unique static properties.\nTheir stability, long lifetime\n(low loss rates incurred by the $r^{-3}$ oscillating terms) and\nlack of sensitivity to alignment errors or amplitude noise of\nthe laser beams makes the experimental realization of such\ncondensates rather likely.\nTheir fascinating analogy with gravitating quantum systems whose gravitational\ninteraction can be enormously enhanced by the field merits further\nresearch.\n\n\nSupport is acknowledged from ISF, Minerva and BSF (G.K.,S.G.),\nthe Royal Society (D.O.) and Arc-en-Ciel (V.A.,G.K.).\nValuable discussions with G.Hose are acknowledged by G.K..\n\n\n\n\n\\begin{thebibliography}{10}\n\n\n\n\\bibitem{anderson95}\n{M.H. Anderson \\textit{et al.\\ }, Science, {\\bf 269}, 198 (1995);\nC.C. Bradley \\textit{et al.\\ }, {P}hys. {R}ev. {L}ett.,\n{\\bf 75},\n1687 (1995);\nK.B. Davis \\textit{et al.\\ }, {P}hys. {R}ev. {L}ett., {\\bf 75},\n3969 (1995).}\n\n\\bibitem{dalfovo99}\n{L.{P}. {P}itaevskii, {S}ov. {P}hys. {JETP}, {\\bf 13}, 451 (1961); \n {E}.{P}. {G}ross, Nuovo Cimento {\\bf 20}, 454 (1961); \n J. Math. Phys. {\\bf 4}, 195 (1963);\nF. Dalfovo, S. Giorgini, L.~{P}.~{P}itaevskii and S. Stringari, \nRev. Mod. Phys. {\\bf 71}, 463 (1999); Parkins and Walls, \n{P}hys. {R}ep. {\\bf 303}, 1 (1998).}\n\n\\bibitem{tiesinga92}\n{E. {T}iesinga, {B}.{J}. {V}erhaar, and {H}.{T}.{C}. Stoof,\n{P}hys. {R}ev. {A}, {\\bf 46}, R1167 (1992);\nP.O. Fedichev \\textit{et al.\\ }, {P}hys. {R}ev. {L}ett.,\n{\\bf 77}, 2913 (1996); Yu. Kagan, E.L. Surkov and G.V. Shlyapnikov,\n{P}hys. {R}ev. {L}ett., {\\bf 79}, 2604 (1997);\nJ.L. Bohn and P.S. Julienne, {P}hys. {R}ev. {A}, {\\bf 56}, 1486 (1997);\nS. Inouye \\textit{et al.\\ }, {N}ature, {\\bf 392}, 151 (1998);\nPh. Courteille \\textit{et al.\\ }, {P}hys. {R}ev. {L}ett., {\\bf 81}, 69 (1998);\nM. Marinescu and L. You, {P}hys. {R}ev. {L}ett., {\\bf 81}, 4596 (1998).}\n\n\\bibitem{goral99}\n{For $r^{-3}$ magnetic-dipole interactions in condensates, see\nK. G{\\'{o}}ral, K. Rz{\\c{a}}{\\.{z}}ewski and T. Pfau,\n eprint: cond-mat/9907308 (1999).}\n\n\\bibitem{thirunamachandran80}\n{T. Thirunamachandran, {M}ol. Phys., {\\bf 40}, 393 (1980);\nD.P.~Craig and T.~Thirunamachandran,\n\\textit{Molecular Quantum Electrodynamics} (Academic Press,\nLondon, 1984).}\n\n\\bibitem{ruffini69}\n{R. Ruffini and S. Bonazzola, {P}hys. {R}ev., {\\bf 187}, 1767 (1969);\nG.~Ingrosso and D.~Grasso and R.~Ruffini, {A}stron. {A}strophys.,\n{\\bf 248}, 481 (1991); \nP. Jetzer, {P}hys. {R}ep., {\\bf 220}, 163 (1992).}\n\n\\bibitem{mathews-walker}\n{J. Mathews, R.L. Walker, \\textit{Mathematical Methods of Physics},\n(Addison-Wesley, New York, 2nd Edition, 1970).}\n\n\\bibitem{foldy61}\n{L.L. Foldy, {P}hys. {R}ev., {\\bf 124}, 649 (1961);\nA.S.~Alexandrov and W.H.~Beere, {P}hys. {Rev}. {B},\n{\\bf 51}, 5887 (1995).}\n\n\\bibitem{dalfovo96}\n{F. Dalfovo and S. Stringari, {P}hys. {R}ev. {A},\n{\\bf 53}, 2447 (1995).}\n\n\\bibitem{penrose98}\n{R. Penrose, {P}hil. {T}rans. {R}oy. {S}oc. A, {\\bf 356}, 1 (1998);\nI.M. Moroz, R. Penrose and P. Tod, {C}las. {Q}uan. {G}rav.\n{\\bf 15}, 2733 (1998).}\n\n\\end{thebibliography}\n\n\\end{document}\n\n" } ]	[ { "name": "quant-ph9912097.extracted_bib", "string": "{anderson95 {M.H. Anderson et al.\\ , Science, {269, 198 (1995); C.C. Bradley et al.\\ , {Phys. {Rev. {Lett., {75, 1687 (1995); K.B. Davis et al.\\ , {Phys. {Rev. {Lett., {75, 3969 (1995)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{dalfovo99 {L.{P. {Pitaevskii, {Sov. {Phys. {JETP, {13, 451 (1961); {E.{P. {Gross, Nuovo Cimento {20, 454 (1961); J. Math. Phys. {4, 195 (1963); F. Dalfovo, S. Giorgini, L.~{P.~{Pitaevskii and S. Stringari, Rev. Mod. Phys. {71, 463 (1999); Parkins and Walls, {Phys. {Rep. {303, 1 (1998)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{tiesinga92 {E. {Tiesinga, {B.{J. {Verhaar, and {H.{T.{C. Stoof, {Phys. {Rev. {A, {46, R1167 (1992); P.O. Fedichev et al.\\ , {Phys. {Rev. {Lett., {77, 2913 (1996); Yu. Kagan, E.L. Surkov and G.V. Shlyapnikov, {Phys. {Rev. {Lett., {79, 2604 (1997); J.L. Bohn and P.S. Julienne, {Phys. {Rev. {A, {56, 1486 (1997); S. Inouye et al.\\ , {Nature, {392, 151 (1998); Ph. Courteille et al.\\ , {Phys. {Rev. {Lett., {81, 69 (1998); M. Marinescu and L. You, {Phys. {Rev. {Lett., {81, 4596 (1998)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{goral99 {For $r^{-3$ magnetic-dipole interactions in condensates, see K. G{\\'{oral, K. Rz{\\c{a{\\.{zewski and T. Pfau, eprint: cond-mat/9907308 (1999)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{thirunamachandran80 {T. Thirunamachandran, {Mol. Phys., {40, 393 (1980); D.P.~Craig and T.~Thirunamachandran, Molecular Quantum Electrodynamics (Academic Press, London, 1984)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{ruffini69 {R. Ruffini and S. Bonazzola, {Phys. {Rev., {187, 1767 (1969); G.~Ingrosso and D.~Grasso and R.~Ruffini, {Astron. {Astrophys., {248, 481 (1991); P. Jetzer, {Phys. {Rep., {220, 163 (1992)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{mathews-walker {J. Mathews, R.L. Walker, Mathematical Methods of Physics, (Addison-Wesley, New York, 2nd Edition, 1970)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{foldy61 {L.L. Foldy, {Phys. {Rev., {124, 649 (1961); A.S.~Alexandrov and W.H.~Beere, {Phys. {Rev. {B, {51, 5887 (1995)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{dalfovo96 {F. Dalfovo and S. Stringari, {Phys. {Rev. {A, {53, 2447 (1995)." }, { "name": "quant-ph9912097.extracted_bib", "string": "{penrose98 {R. Penrose, {Phil. {Trans. {Roy. {Soc. A, {356, 1 (1998); I.M. Moroz, R. Penrose and P. Tod, {Clas. {Quan. {Grav. {15, 2733 (1998)." } ]
quant-ph9912098	Local environment can enhance fidelity of quantum teleportation	[ { "author": "Piotr Badzi{\\c{a" } ]	We show how an interaction with the environment can enhance fidelity of quantum teleportation. To this end, we present examples of states which cannot be made useful for teleportation by any local unitary transformations; nevertheless, after being subjected to a dissipative interaction with the local environment, the states allow for teleportation with genuinely quantum fidelity. The surprising fact here is that the necessary interaction does not require any intelligent action from the parties sharing the states. In passing, we produce some general results regarding optimization of teleportation fidelity by local action. We show that bistochastic processes cannot improve fidelity of two-qubit states. We also show that in order to have their fidelity improvable by a local process, the bipartite states must violate the so-called reduction criterion of separability.	[ { "name": "quant-ph9912098.tex", "string": "\\documentstyle[pra,aps]{revtex}\n\\begin{document}\n\\title{Local environment can enhance fidelity of quantum teleportation}\n\\author{Piotr Badzi{\\c{a}}g$^{1,}$\\cite{poczta}, Micha\\l {} Horodecki$^{2,}$\n\\cite{poczta1}, Pawe\\l{} Horodecki$^{3,}$\\cite{poczta2} and Ryszard Horodecki $%\n^{2,}$\\cite{poczta3}}\n\\address{$^1$ Department of Mathematics and Physics,\nMalardalens Hogskola, S-721 23 Vasteras, Sweden, \\\\\n$^2$ Institute of Theoretical Physics and Astrophysics,\nUniversity of Gda\\'nsk, 80--952 Gda\\'nsk, Poland,\\\\\n$^3$Faculty of Applied Physics and Mathematics,\nTechnical University of Gda\\'nsk, 80--952 Gda\\'nsk, Poland\\\\\n}\n\\maketitle\n\n\\begin{abstract}\nWe show how an interaction with the environment can enhance fidelity of\nquantum teleportation. To this end, we present examples of states which\ncannot be made useful for teleportation by any local unitary\ntransformations; nevertheless, after being subjected to a dissipative\ninteraction with the local environment, the states allow for teleportation\nwith genuinely quantum fidelity. The surprising fact here is that the\nnecessary interaction does not require any intelligent action from the\nparties sharing the states. In passing, we produce some general results\nregarding optimization of teleportation fidelity by local action. We show\nthat bistochastic processes cannot improve fidelity of two-qubit states. We\nalso show that in order to have their fidelity improvable by a local\nprocess, the bipartite states must violate the so-called reduction criterion\nof separability.\n\\end{abstract}\n\n\\section{Introduction}\n\nQuantum teleportation \\cite{Bennett_tel} is fundamentally important as an\noperational test of the presence and the strength of entanglement. Moreover,\na recent series of beautiful experiments \\cite{exp}, which realized\nteleportation in practice, opened a window for a wide range of its possible\ntechnological applications.\n\nIn this paper, teleportation is understood as any strategy which uses local\nquantum operations and classical communication (LOCC) \\cite{Bennett_pur} to\ntransmit an unknown state via a shared pair. In an ideal teleportation\nscheme, the EPR-channel is constituted by a pure, maximally entangled\nbipartite state:\n\\begin{equation}\n\\psi _{-}={\\frac{1}{\\sqrt{2}}}(\|01\\rangle -\|10\\rangle ).\n\\end{equation}\nThe state is shared by a sender (Alice) and a receiver (Bob). By sharing $%\n\\psi _{-}$ with Alice, Bob can produce an {\\em exact} replica of another\n(input) state originally held by Alice. In reality, however, interactions\nwith the environment and imperfections of preparation result in Alice and\nBob sharing a state which is always mixed. Consequently, at Bob's end, the\nteleported state can only be a distorted copy of the input initially held by\nAlice. Moreover, if the bipartite state is mixed too much, it will not\nprovide for any better transmission fidelity than that of an ordinary\nclassical communication channel \\cite{Popescu94}. To do better than a\nclassical channel, the shared quantum state must be entangled. A natural\nquestion then is \\cite{Popescu94}: can any entangled state provide better\nthan classical fidelity of teleportation?\n\nEarly attempts to answer this question, concentrated on the characterization\nof the states which can offer non-classical fidelity within the original\nteleportation scheme supplemented by local unitary rotations. Henceforth we\nwill call such a scheme the {\\em standard teleportation scheme} (STS).\nFidelity of teleportation achievable in STS is uniquely determined by the\nbipartite state's {\\it fully entangled fraction}. It was defined in \\cite\n{huge} as\n\\begin{equation}\nf(\\varrho )=\\max_{\\psi }\\langle \\psi \|\\varrho \|\\psi \\rangle . \\label{fully}\n\\end{equation}\nIn the definition, the maximum is taken over all maximally entangled states $%\n\\psi $ i.e. over $\\psi =U_{1}\\otimes U_{2}\\psi _{+}$, where\n\\begin{equation}\n\\psi _{+}={\\frac{1}{\\sqrt{d}}}\\sum_{i=1}^{d}\|i\\rangle \|i\\rangle \\label{plus}\n\\end{equation}\n$U_{1}$ and $U_{2}$ are unitary transformations. Later, it was shown that in\norder to be useful for STS, the states acting on a Hilbert space $%\nC^{d}\\otimes C^{d}$ must have $f>1/d$ \\cite{tel,single}. Moreover, it was\nshown that no {\\it bound entangled} state (see \\cite{bound}) can offer\nbetter fidelity than classical communication \\cite{Linden,single}. Somewhat\nearlier, in Refs. \\cite{Massar,Kent}, the authors identified a class of\nstates which do not permit any increase of $f$, neither by any trace\npreserving (TP) LOCC nor even by some less restricted non-TP LOCC actions.\nMixtures of a maximally mixed state and $\\psi _{+}$ \\cite{Popescu94,Werner}\nbelong, among others, to this class.\n\nOne could then be tempted to speculate that $f$ could not be increased by\nany TP LOCC operations. If so, then STS would be a unique\nteleportation scheme in the sense that no other scheme would provide better\nfidelity than STS. On the other hand, one could still suspect that by some\nintelligent, sophisticated LOCC operation, Alice and Bob would be able to\nincrease $f$ for some states anyway. An important question was then to be\nanswered:\n\n{\\em Is it possible to design a teleportation scheme, for which at least\nsome states with }$f\\leq 1/d${\\em \\ would give non-classical fidelity? }\n\nIn this paper, we answer this question by presenting a class of two-qubit\nstates with $f\\leq 1/2$, which can, nevertheless, be used for teleportation\nwith non-classical fidelity. For that, however, one has to allow for some\n{\\em dissipative} interaction between the states and their local environment\nfirst. This means that dissipation, which is usually associated with\ndecoherence and destruction of teleportation, increases $f$ of some\ninitially non-teleporting states to above $1/2$. In other words, some states\ncan produce non-classical fidelity within the original teleportation scheme\nbut only after being 'corrupted' by the environment !\n\nTo our knowledge, this is a previously unknown effect. In particular, it is\ndifferent than that used in the so called {\\it filtering} method of\nimproving some of the states' parameters \\cite{Gisin,conc}. Filtering\nincludes a selection process based on a {\\it readout} of measurement\noutcomes. In our examples, on the other hand, Alice and Bob do not need to\nknow the outcomes at all. Hence, in particular, unlike filtering, the actions in our\nexamples are entirely trace preserving.\n\nWe begin our presentation by recalling some of the general results on\noptimal teleportation fidelity in Sect.\\ref{sec2}(c.f. Ref. \\cite{single}).\nThis allows us to conclude that an optimal teleportation scheme should\ninclude maximization of $f$ by means of TP LOCC operations. Then, in Sect.\n\\ref{sec3} we put the problem in the context of increasing $f$ by the maps\nof the form $I\\otimes \\Lambda $. We can limit the possible successful maps\nby showing that, e.g., for two qubits, the bistochastic processes cannot do\nthe job. We also show that the states with $f$ improvable by $I\\otimes\n\\Lambda $ action must violate the so called {\\it reduction criterion}.\nSubsequently, in Sect. \\ref{sec4}\\ we present the examples of states, for\nwhich $f$ can be non-trivially increased by TP LOCC\noperations. The paper ends with the summary of the results and the\nconclusions in Sect. \\ref{sec5}.\n\n\\section{Optimal fidelity in a general teleportation scheme}\n\n\\label{sec2} Let Alice and Bob share a pair of particles in a given state $%\n\\varrho $ acting on a Hilbert space ${\\cal H}_{A}\\otimes {\\cal H}%\n_{B}=C^{d}\\otimes C^{d}$. Additionally, let Alice have a third particle in\nan unknown pure state $\\psi \\in {\\cal H}_{C}=C^{d}$ to be teleported. In the\nmost general teleportation scheme, Bob and Alice apply some trace preserving\n(TP) (hence without selection of the ensemble) LOCC operation ${\\cal T}$ to\nthe particles which they share and to the third (Alice's) particle. After\nthe operation is completed, the final state of Bob's particle (from the\npair) is\n\\begin{equation}\n\\varrho _{Bob}^{\\psi }={\\rm Tr}_{A,C}\\left[ {\\cal T}(\|\\psi \\rangle \\langle\n\\psi \|\\otimes \\varrho )\\right] . \\label{Bob}\n\\end{equation}\nThe resulting mapping of the input state (the state of the third particle)\nonto $\\varrho _{Bob}({\\psi })$ establishes a {\\it teleportation channel} $%\n\\Lambda $ (it depends on both, ${\\cal T}$ and $\\varrho $):\n\\begin{equation}\n\\Lambda (\|\\psi \\rangle \\langle \\psi \|)=\\varrho _{Bob}(\\psi ).\n\\end{equation}\nThe aim of teleportation is to bring $\\varrho _{Bob}(\\psi )$\\ as close to $%\n\|\\psi \\rangle \\langle \\psi \|$\\ as possible. A useful measure of the quality\nof teleportation is then provided by teleportation's {\\em fidelity} \\cite\n{Popescu94}\n\\begin{equation}\n{\\cal F}=\\overline{\\langle \\psi \|\\varrho _{Bob}(\\psi )\|\\psi \\rangle }.\n\\label{fidelity}\n\\end{equation}\nFidelity is a function of map $\\Lambda $ and, like $\\Lambda $,\\ it depends\non both, teleporting state $\\varrho $ and the strategy of teleportation $%\n{\\cal T}$ . One can show \\cite{single} that in the standard teleportation\nscheme, the maximal fidelity achievable from a given bipartite state $%\n\\varrho $ is\n\\begin{equation}\n{\\cal F}={\\frac{fd+1}{d+1}}\n\\end{equation}\nwhere $f$ is the fully entangled fraction of $\\rho $\\ given by formula (\\ref\n{fully}). To achieve this fidelity, Alice and Bob have to rotate their\nrespective parts of the teleporting state $\\rho $ so that the maximum of\nformula (2) is attained on singlet $\\psi _{-}$. The original teleportation\nscheme applied with the rotated bipartite state $\\rho $\\ will now produce\nthe maximal fidelity (8).\n\nIf, on the other hand, Alice and Bob do not share any quantum state, then\ntheir best strategy is \\cite{Popescu94}:\n\n\\begin{enumerate}\n\\item[(i)] Alice performs an optimal measurement of the system to be\nteleported and sends the outcome to Bob (classically).\n\n\\item[(ii)] On the basis of her results, Bob tries to reconstruct the state.\n\\end{enumerate}\n\nThe optimal teleportation fidelity for this strategy is equal to the optimal\nfidelity of the state estimation for a single system. It is given by \\cite\n{Massar95,single}\n\\begin{equation}\n{\\cal F}_{cl}={\\frac{2}{1+d}}.\n\\end{equation}\nOne can easily see now that, in order to perform better than classical\ncommunication, STS needs bipartite states with $f>1/d$. With $f\\leq 1/d$,\nAlice and Bob can just as well discard their bipartite state and communicate\nclassically.\n\nThere is no reason why STS should represent the most efficient teleportation\nscheme using states with $f>1/d$. One can show, however, that the optimal\nteleportation scheme (OTS) is a generalization of STS \\cite{single}. OTS\nconsists of two steps:\n\n\\begin{enumerate}\n\\item[(i)] Alice and Bob try to maximize $f$\\ by applying TP\nLOCC (not necessarily unitary) operations to the original state $\\varrho $.\n\n\\item[(ii)] They apply STS using the transformed state.\n\\end{enumerate}\n\nLet then $f_{max}(\\varrho )$ denote the maximal $f$ attainable from $\\varrho\n$ by means of TP LOCC operations. The maximal teleportation fidelity from\nstate $\\varrho $ is then given by \\cite{single}\n\\begin{equation}\n{\\cal F}_{max}={\\frac{f_{max}d+1}{d+1}}.\n\\end{equation}\nThus, to find the optimal teleportation fidelity for a given bipartite state\n$\\rho $, one must find $f_{max}$. In other words, the fidelity of STS can be\nimproved if:\n\n\\begin{enumerate}\n\\item $f$ can be increased by LOCC,\n\n\\item The final $f$ is in quantum region i.e. it is greater than $1/d$.\n\\end{enumerate}\n\nHenceforth, when referring to a process of increasing $f$, we will\nunderstand it as increasing so that the final value is above $1/d$ (Within\nthe range $f\\leq 1/d$, the fully entangled fraction can be increased\nrelatively easily. This, however, does not produce any better fidelity than $%\n{\\cal F}_{cl}$).\n\n\\section{\\label{3}Some general results on improving ${\\cal F}$ by local\nintractions}\n\n\\label{sec3}\n\n\\subsection{A simplified formula for maximal $f$ attainable by local\ninteraction}\n\nWhen local TP transformations are used to increase $f$ of a general\nbipartite state $\\varrho \\in C^{d}\\otimes C^{d}$, then the best attainable\nresult is\n\\begin{equation}\nf_{A}=\\max_{\\Lambda }\\mbox{Tr}\\left( (\\Lambda \\otimes I)\\varrho P_{+}\\right)\n. \\label{fa}\n\\end{equation}\nThe maximum is here taken over all TP completely positive (CP) maps $\\Lambda\n$ and $P_{+}=\|\\psi _{+}\\rangle \\langle \\psi _{+}\|$, with $\\psi _{+}$ given\nby (\\ref{plus}). Stinespring decomposition of $\\Lambda $\\ gives \\cite{Kraus}\n\\begin{equation}\n\\Lambda (\\cdot )=\\sum_{i}V_{i}(\\cdot )V_{i}^{\\dagger }\n\\end{equation}\nwith $\\sum_{i}V_{i}^{\\dagger }V_{i}=I$. Moreover, we can utilize the fact\nthat $A\\otimes I\\psi _{+}=I\\otimes A^{T}\\psi _{+}$ \\cite{Jozsa} (superscript\n$T$ denotes transposition in basis $\\left\\{ \|i\\rangle \\right\\} $) and\nrewrite formula (\\ref{fa}) as\n\\begin{equation}\nf_{A}=\\max_{\\Gamma }\\mbox{Tr}\\left( \\varrho (I\\otimes \\Gamma )P_{+}\\right) ,\n\\end{equation}\nwith\n\\begin{equation}\n\\Gamma (\\cdot )=\\sum_{i}W_{i}(\\cdot )W_{i}^{\\dagger }\n\\end{equation}\nand $W_{i}=V_{i}^{\\ast }$ (the star denotes complex conjugation). Naturally,\nlike $\\Lambda $, $\\Gamma $ is trace preserving, too.\n\nWe can now recall that there is an isomorphism between the TP CP maps and\nthe bipartite states with one subsystem maximally mixed. The isomorphism is\ngiven by\n\\begin{equation}\n\\varrho ^{\\prime }=(I\\otimes \\Lambda )P_{+}.\n\\end{equation}\nThus, for any TP CP map, the corresponding state has a maximally mixed\nsubsystem $A$ and for any state with a maximally mixed subsystem $A$, there\nexists a map that realizes it via the above formula. Consequently, we can\nobtain the following form for $f_{A}$\n\\begin{equation}\nf_{A}(\\varrho )=\\max_{\\varrho ^{\\prime }}\\mbox{Tr}(\\varrho \\varrho ^{\\prime\n}), \\label{general}\n\\end{equation}\nwhere the maximum is taken over all states $\\varrho ^{\\prime }$ with\nmaximally mixed subsystem $A$. An analogous formula holds for $f_{B}$. In\ngeneral, the values $f_{A}$ and $f_{B}$ are likely to be different from one\nanother.\n\nFormula (\\ref{general}) allows for identification of those maps which\ndefinitely cannot improve $f$. Take, for instance, the maps describing the\naction of random external fields \\cite{Alicki}. They are of the form\n\\begin{equation}\n\\Lambda (\\cdot )=\\sum_{i}p_{i}U_{i}(\\cdot )U_{i}^{\\dagger }, \\label{random}\n\\end{equation}\nwith $U_{i}$ denoting unitary transformations. The corresponding $\\varrho\n^{\\prime }=(I\\otimes \\Lambda )P_{+}$ is a mixture of maximally entangled\nvectors. Consequently, $\\mbox{Tr}(\\varrho \\varrho ^{\\prime })$ cannot exceed\n$f(\\varrho )$ which is equal to the maximal overlap of $\\varrho $ with one\nmaximally entangled vector.\n\nIn addition to preserving trace, maps (\\ref{random}) preserve the identity,\ni.e. $\\Lambda (I)=I$. Maps preserving both the trace and the identity are\ncalled bistochastic. In general, the class of bistochastic maps can be wider\nthan the class specified by (\\ref{random}). For two qubits, however, the two\nclasses coincide. To see this, one can note that, in general, the set of\nstates corresponding to the set of bistochastic maps via the isomorphism\nconsists of the states with {\\it both} subsystems maximally mixed. For\ntwo-qubit systems such states are mixtures of maximally entangled vectors\n\\cite{inf}. Each such vector can be written as $I\\otimes U\\psi _{+}$ for\nsome unitary $U$. Hence, the maps corresponding to mixtures of such vectors\nare mixtures of unitary maps. Thus, for two qubits the bistochastic maps\ncannot increase $f$. One may conjecture that this should be the case in\nhigher dimensions, too.\n\n\\subsection{Increasing $f$ by local actions and the reduction criterion for\nseparability}\n\nLet us now derive some constraints for the states with $f$ improvable by\nlocal interaction. A state suitable for a teleportation channel must be\nentangled, i.e., it must be impossible to represent it by a mixture of\nproduct states \\cite{Werner}.\n\\begin{equation}\n\\varrho \\neq \\sum_{i}p_{i}\\varrho _{i}\\otimes \\tilde{\\varrho}_{i}.\n\\end{equation}\nSuch states violate different separability criteria. Here, we consider the\nso called {\\it reduction criterion} for separability. It is given by the\nfollowing conditions satisfied by all separable states \\cite{cerf,xor}:\n\\begin{equation}\n\\varrho _{A}\\otimes I-\\varrho \\geq 0,\\quad I\\otimes \\varrho _{B}-\\varrho\n\\geq 0. \\label{kryt}\n\\end{equation}\nThe inequalities mean that the operators on the left hand sides must be {\\it %\npositive}, i.e., they must have nonnegative eigenvalues only. In a two-qubit\ncase, the reduction criterion is equivalent to separability (hence it is\nalso a sufficient condition for separability), while it becomes a weaker\n``detector'' of entanglement in higher dimensions. In other words, there\nexist non-separable (entangled) states in higher dimensions which do not\nviolate the reduction criterion.\n\nSuppose now that for some state $\\varrho $ one has $f_{A}(\\varrho\n)>f(\\varrho )$, i.e., $f$ can be improved by a local TP operation on\nsubsystem $A$. Naturally, we require that the improvement is non-trivial,\ni.e., $f_{A}>1/d$. We will show now that this condition implies violation of\nthe reduction criterion. Indeed, since $f_{A}>1/d$, then there exists a\nstate $\\varrho ^{\\prime }$ whose one subsystem (say, $\\varrho _{A}^{\\prime }$%\n) has maximal entropy and:\n\\begin{equation}\n\\mbox{Tr}(\\varrho \\varrho ^{\\prime })>1/d. \\label{ineq}\n\\end{equation}\nMaximum entropy means that $\\varrho _{A}^{\\prime }=I/d$. This implies $%\n\\mbox{Tr}((\\varrho _{A}\\otimes I)\\varrho ^{\\prime })=\\mbox{Tr}\\left( \\varrho\n_{A}\\varrho _{A}^{\\prime }\\right) =1/d$. By putting this into inequality (%\n\\ref{ineq}), we obtain\n\\begin{equation}\n\\mbox{Tr}\\left( (\\varrho _{A}\\otimes I-\\varrho )\\varrho ^{\\prime }\\right) <0\n\\end{equation}\nThe trace of a composition of two positive operators is nonnegative.\nOperator $\\varrho ^{\\prime }$ is positive. Consequently, in order to satisfy\nthe last inequality, the operator $\\varrho _{A}\\otimes I-\\varrho $ cannot be\npositive.\n\nSince all the entangled two-qubit states violate the reduction criterion,\nthe condition for improvability of $f$ derived above, does not put any new\nrestrictions on the class of states with improvable $f$ here \\cite\n{Massar,Kent}. Nevertheless, the condition should be useful while\ninvestigating bipartite states in more dimensions. This is because not all\nthe entangled states there violate the reduction criterion.\n\n%Finally note that the reduction criterion consists of two conditions\n%involving different reductions. There can be an asymetry: a state may\n%violate only one of these conditions.\n\n\\section{Beating the standard teleportation scheme}\n\n\\label{sec4}Before showing how to do better than STS, we will still need to\nintroduce some methods of dealing with the fully entangled fraction of\ntwo-qubit states.\n\n\\subsection{Fully entangled fraction in the Hilbert-Schmidt representation}\n\nAn arbitrary state of a two-qubit system can be represented as\n\\begin{equation}\n\\varrho ={\\frac{1}{4}}(I\\otimes I+\\bbox {r\\cdot \\sigma}\\otimes I+I\\otimes %\n\\bbox {s\\cdot \\sigma}+\\sum_{m,n=1}^{3}t_{nm}\\sigma _{n}\\otimes \\sigma _{m}).\n\\end{equation}\nHere, $I$ stands for the identity operator, ${\\bbox r}$ and ${\\bbox s}$\nbelong to $R^{3}$, $\\{\\sigma _{n}\\}_{n=1}^{3}$ are standard Pauli matrices, $%\n\\bbox{\nr\\cdot\\sigma}=\\sum_{i=1}^{3}r_{i}\\sigma _{i}$. Coefficients $t_{mn}={\\rm Tr}%\n(\\rho \\sigma _{n}\\otimes \\sigma _{m})$ form a real $3\\times 3$ matrix later\ndenoted by $T$. Note that $\\bbox r$ and $\\bbox s$ are local parameters as\nthey determine the reductions of~$\\varrho $:\n\\begin{eqnarray}\n\\varrho _{1} &\\equiv &{\\rm Tr}_{{\\cal H}_{2}}\\varrho ={\\frac{1}{2}}(I+\\bbox{\nr\\cdot\\sigma}), \\nonumber \\\\\n\\varrho _{2} &\\equiv &{\\rm Tr}_{{\\cal H}_{1}}\\varrho ={\\frac{1}{2}}(I+%\n\\bbox{s\\cdot\\sigma}). \\label{redukcje}\n\\end{eqnarray}\nMatrix $T$ , on the other hand, is responsible for the correlations\n\\begin{equation}\nE(\\bbox{a},\\bbox{b})\\equiv \\text{Tr}(\\varrho \\bbox{a\\cdot\\sigma}\\otimes %\n\\bbox{b\\cdot\\sigma})=(\\bbox a,T\\bbox b).\n\\end{equation}\nOne can notice now, that for any two-qubit state $\\varrho $, one can find a\nproduct unitary transformation $U_{1}\\otimes U_{2}$ which will transform $%\n\\varrho $ to a form with {\\it diagonal} $T$. This statement follows from the\nfact that for any $2\\times 2$\\ unitary transformation $U$, there is a unique\n$3\\times 3$\\ rotation $O$ such that \\cite{thir2}\n\\begin{equation}\nU\\bbox{ \\hat n\\cdot\\sigma}U^{\\dagger }=(O\\bbox{\\hat n})\\bbox{\\cdot\\sigma}.\n\\label{pawel}\n\\end{equation}\nNow, if a state is subjected to a $U_{1}\\otimes U_{2}$ transformation, the\nparameters $\\bbox r,\\bbox s$ and $T$ are transformed into\n\\begin{eqnarray}\n&&\\bbox r^{\\prime }=O_{1}\\bbox r, \\nonumber \\\\\n&&\\bbox s^{\\prime }=O_{2}\\bbox s, \\nonumber \\\\\n&&T^{\\prime }=O_{1}TO_{2}^{\\dagger }.\n\\end{eqnarray}\nwith $O_{i}$'s corresponding to $U_{i}$'s via formula (\\ref{pawel}). Thus,\nfor every two-qubit state $\\rho $, we can always find such $U_{1}$ and $%\nU_{2} $ so that the corresponding rotations will diagonalize $T$ \\cite\n{correl}. Moreover, by selecting suitable rotations, one can make $t_{11}$\nand $t_{22}$ non-positive. In what follows, the states with diagonal $T$ and\n$t_{11},t_{22}\\leq 0$ will be called {\\em canonical.}\n\nFor the states with diagonal matrix $T$ (hence also for the canonical\nstates), the fully entangled fraction is given by (c.f.\\cite{hab})\n\\begin{equation}\nf=\\left\\{\n\\begin{array}{l}\n{\\frac{1}{4}}(1+{\\sum_{i}\|t_{ii}\|})\\quad {\\rm \\ if\\ }\\quad {\\rm det}T\\leq 0\n\\\\[1mm]\n{\\frac{1}{4}}\\left( 1+{\\max_{i\\not=k\\not=j}(\|t_{ii}\|+\|t_{jj}\|-\|t_{kk}\|)}%\n\\right) {\\rm \\ if\\ }{\\rm \\ det}T>0\n\\end{array}\n\\right. . \\label{max}\n\\end{equation}\nOne can show now \\cite{inf,hab} that if ${\\rm det}T\\geq 0$, then $f\\leq 1/2$%\n, i.e., $f$ belongs to the classical region. Thus, while analyzing $f$ in\nthe quantum region, it will be convenient to investigate a relatively simple\nfunction $N(\\varrho )$, instead of a more involved matrix $T$. Function $%\nN(\\varrho )$ is given by\n\\begin{equation}\nN(\\varrho )=\\sum_{i}\|t_{ii}\|.\n\\end{equation}\nIt has the following important properties:\n\n\\begin{enumerate}\n\\item $f(\\varrho )={\\frac{1}{4}}(1+N(\\varrho ))$ for $f\\geq {\\frac{1}{2}}$\n\n\\item $N(\\varrho )\\leq 1$ if and only if $f\\leq {\\frac{1}{2}}$\n\\end{enumerate}\n\nIt then contains all the information necessary to analyze $f$.\n\n\\subsection{The canonical form in terms of the matrix elements}\n\nBy applying the formula for $t_{ij}$, one can easily show that diagonality\nof $T$ is equivalent to the following conditions for the matrix elements of $%\n\\varrho $ written in the standard basis ($\|1\\rangle =\|00\\rangle $, $%\n\|2\\rangle =\|01\\rangle $ etc.):\n\\begin{eqnarray}\n&&\\varrho _{12}=\\varrho _{34} \\\\\n&&\\varrho _{14}=\\varrho _{32} \\\\\n&&\\varrho _{23}\\text{\\ and\\ \\ }\\varrho _{14}\\text{ are real}.\n\\end{eqnarray}\n%The matrix elements of $\\varrho $ are written here\nMoreover, since $t_{11}=2(\\varrho _{14}+\\varrho _{23})$ and $%\nt_{22}=2(\\varrho _{23}-\\varrho _{14})$, the condition $t_{11},t_{22}\\leq 0$\nis equivalent to\n\\begin{eqnarray}\n&&\\varrho _{23}\\leq 0 \\\\\n&&\|\\varrho _{23}\|\\geq \\left\| \\varrho _{14}\\right\|\n\\end{eqnarray}\nThus, any state $\\varrho $ can be locally rotated to a form with matrix\nelements satisfying the above constraints. This gives the following\nexpression for $N(\\varrho )$:\n\\begin{equation}\nN(\\varrho )=\|1-2(\\varrho _{22}+\\varrho _{33})\|-2\\varrho _{23}.\n\\end{equation}\nNow, for\n\\begin{equation}\n\\varrho _{22}+\\varrho _{33}\\geq {\\frac{1}{2}} \\label{singlet-cond}\n\\end{equation}\nwe have $t_{33}\\leq 0$ hence ${\\rm det}T\\leq 0$. Consequently, by eq. (\\ref\n{max}) the fully entangled fraction is given by\n\\begin{equation}\nf(\\varrho )={\\frac{1}{4}}(1+N(\\varrho ))={\\frac{1}{2}}(\\varrho _{22}+\\varrho\n_{33}-2\\varrho _{23}).\n\\end{equation}\nThen, with $-2\\varrho _{23}$ large enough, one has $f\\geq 1/2$ and $f$ \\ is\nattained on singlet $\\psi _{-}$: $f=\\langle \\psi _{-}\|\\varrho \|\\psi\n_{-}\\rangle $.\n\n\\subsection{A local action which improves $f$.}\n\nWith the canonical form of $\\varrho $\\ at hand, it is not all that difficult\nto eventually find examples of states with improvable $f$. After some\ntrials, we focused our attention on a simple family of states which in their\ncanonical form have $\\varrho _{24}=\\varrho _{13}=0$:\n\\begin{equation}\n\\varrho =\\left[\n\\begin{array}{cccc}\n\\varrho _{11} & 0 & 0 & \\varrho_{14} \\\\\n0 & \\varrho _{22} & -p_{23} & 0 \\\\\n0 & -p_{23} & \\varrho _{33} & 0 \\\\\n\\varrho_{14} & 0 & 0 & \\varrho _{44}\n\\end{array}\n\\right]\n\\end{equation}\nHere $p_{23}\\geq 0$ and $\\varrho_{14}$ is real. We assumed also that $%\n\\varrho $ satisfies the condition (\\ref{singlet-cond}) and that $p_{23}\\geq\n(1-\\varrho _{22}-\\varrho _{33})/2$, so that the state has $f=\\langle \\psi\n_{-}\|\\varrho \|\\psi _{-}\\rangle \\geq 1/2$. Explicitly, $f$ is given by\n\\begin{equation}\nf(\\varrho )={\\frac{1}{2}}(\\varrho _{22}+\\varrho _{33}+2p_{23}).\n\\end{equation}\n\nWe know (see Sec.\\ref{sec3}) that bistochastic maps cannot improve $f$. So,\nto improve it, we must try a non-bistochastic map. A possible simple\ncandidate is, e.g., a map which acts on Bob's qubit and transforms it as\nfollows:\n\\begin{equation}\n\\varrho _{B}\\rightarrow \\tilde{\\varrho}_{B}=\\Lambda (\\varrho )=W_{0}\\varrho\n_{B}W_{0}^{\\dagger }+W_{1}\\varrho _{B}W_{1}^{\\dagger } \\label{action}\n\\end{equation}\nwhere the operators $W_{i}$ are given by\n\\begin{equation}\nW_{1}=\\left[\n\\begin{array}{cc}\n1 & 0 \\\\\n0 & \\sqrt{p}\n\\end{array}\n\\right] ,\\quad W_{2}=\\left[\n\\begin{array}{cc}\n0 & \\sqrt{1-p} \\\\\n0 & 0\n\\end{array}\n\\right]\n\\end{equation}\nIt is easy to check that $W_{i}$'s satisfy $W_{1}^{\\dagger\n}W_{1}+W_{2}^{\\dagger }W_{2}=I$, hence the operation is trace preserving.\nMoreover, one can notice that $\\Lambda $ can be regarded as resulting from\nthe interaction of a two-level atom (Bob's qubit) with electromagnetic field\n(an environment). Such an interaction produces the following transitions:\n\\begin{equation}\n\|0\\rangle _{a}\|0\\rangle _{e}\\rightarrow \|0\\rangle _{a}\|0\\rangle _{e}\n\\end{equation}\n\\begin{equation}\n\|1\\rangle _{a}\|0\\rangle _{e}\\rightarrow \\sqrt{p}\|0\\rangle _{a}\|1\\rangle _{e}+%\n\\sqrt{1-p}\|1\\rangle _{a}\|0\\rangle _{e}.\n\\end{equation}\nwhere the subscripts $a$ and $e$ denote atomic and field states\nrespectively. The parameter $p$ is then interpreted as the probability of\nphoton emission from the atom in its upper state $\|1\\rangle _{a}$. This kind\nof interaction is called the {\\it amplitude damping channel} and one can\ncheck \\cite{Preskill} that, if repeatedly applied to a qubit, it produces an\nexponential decay characteristic to spontaneous emission. The completely\npositive map $\\Lambda $ is then obtained from the amplitude damping channel\nby tracing out the environment variables \\cite{Kraus}.\n\nLet us then put $\\sqrt{p}=\\sin \\theta $ and apply transformation (\\ref\n{action}) to Bob's part of the total (2-qubit) system. The 2-qubit operator\ncorresponding to $W_{i}$\\ is $A_{i}\\equiv I\\otimes W_{i}$ and, consequently,\nwe obtain\n\\begin{equation}\n\\varrho \\rightarrow \\varrho ^{\\prime }=A_{1}\\varrho A_{1}^{\\dagger\n}+A_{2}\\varrho A_{2}^{\\dagger } \\label{oper}\n\\end{equation}\nwith\n\\begin{equation}\nA_{1}=\\left[\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 & \\cos \\theta & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & \\cos \\theta\n\\end{array}\n\\right]\n\\end{equation}\nand\n\\begin{equation}\nA_{2}=\\left[\n\\begin{array}{cccc}\n0 & \\sin \\theta & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\sin \\theta \\\\\n0 & 0 & 0 & 0\n\\end{array}\n\\right] .\n\\end{equation}\nNote that like the original state $\\varrho $, the new state $\\tilde{\\varrho}$\nis in its canonical form, too.\n\\begin{equation}\n\\ \\kern-6mm\\tilde{\\varrho}=\\left[\n\\begin{array}{cccc}\n\\varrho _{11}+\\varrho _{22}\\sin ^{2}\\theta & 0 & 0 & \\varrho _{14}\\cos \\theta\n\\\\\n0 & \\varrho _{22}\\cos ^{2}\\theta & -p_{23}\\cos ^{2}\\theta & 0 \\\\\n0 & -p_{23}\\cos ^{2}\\theta & \\varrho _{33}+\\varrho _{44}\\sin ^{2}\\theta & 0\n\\\\\n\\varrho _{14}\\cos \\theta & 0 & 0 & \\varrho _{44}\\cos ^{2}\\theta\n\\end{array}\n\\right]\n\\end{equation}\nThe change of $f$ associated with the transformation is now given by $\\Delta\n_{B}=\\langle \\psi _{-}\|\\tilde{\\varrho}\|\\psi _{-}\\rangle -f(\\varrho )$. A\nsimple calculation shows that\n\\begin{equation}\n\\Delta _{B}=\\left( 1-\\cos \\theta \\right) \\left[ \\frac{1+\\cos \\theta }{2}%\n\\left( \\varrho _{44}-\\varrho _{22}\\right) -p_{23}\\right] . \\label{3a}\n\\end{equation}\nHere, the index $B$ indicates that Bob's qubit has been transformed. One can\ncheck that if one transforms Alice's qubit instead of Bob's then the\nresulting $\\Delta _{A}$ is given by\n\\begin{equation}\n\\Delta _{A}=\\left( 1-\\cos \\theta \\right) \\left[ \\frac{1+\\cos \\theta }{2}%\n\\left( \\varrho _{44}-\\varrho _{33}\\right) -p_{23}\\right] . \\label{3b}\n\\end{equation}\nFinally, one can swap places of $1$ and $\\cos \\theta $ on the diagonal of\nthe first transformation matrix $A_{1}$ and adjust $A_{2}$ accordingly.\nThis, translated into changes of $f$, result in expressions like (\\ref{3a})\nand (\\ref{3b}) but with $\\varrho _{44}$ substituted by $\\varrho _{11}$. In\nother words, single qubit, trace preserving transformations like that\ndefined by (\\ref{oper}) can improve fidelity of states in form (29) provided\nthat\n\\begin{equation}\n\\left[ \\max \\left( \\varrho _{11},\\varrho _{44}\\right) -\\min \\left( \\varrho\n_{22},\\varrho _{33}\\right) \\right] -p_{23}\\geq 0. \\label{eq4}\n\\end{equation}\nThe maximal increase $\\Delta =\\max \\{\\Delta _{A},\\Delta _{B}\\}$ achievable\nin this way is\n\\begin{equation}\n\\Delta =\\frac{\\left[ \\max \\left( \\varrho _{11},\\varrho _{44}\\right) -\\min\n\\left( \\varrho _{22},\\varrho _{33}\\right) -p_{23}\\right] ^{2}}{2\\left[ \\max\n\\left( \\varrho _{11},\\varrho _{44}\\right) -\\min \\left( \\varrho _{22},\\varrho\n_{33}\\right) \\right] }\n\\end{equation}\nTo obtain a more clear picture of the situation, let us write the diagonal\nelements of $\\varrho $ as:\n\\begin{equation}\n\\varrho _{11}=\\frac{1-\\varepsilon -\\gamma }{4}\\quad \\varrho _{44}=\\frac{%\n1-\\varepsilon +\\gamma }{4}\n\\end{equation}\n\\begin{equation}\n\\quad \\varrho _{22}=\\frac{1+\\varepsilon -\\delta }{4}\\quad \\varrho _{33}=%\n\\frac{1+\\varepsilon +\\delta }{4}\n\\end{equation}\nTo satisfy $\\left( \\varrho _{22}+\\varrho _{33}+2p_{23}\\right) \\geq 1$ (so\nthat $f(\\varrho )=\\langle \\psi _{-}\|\\varrho \|\\psi _{-}\\rangle \\geq 1/2$),\none needs a non-negative $\\varepsilon $ and :\n\\begin{equation}\n\\frac{1-\\varepsilon }{4}\\leq p_{23}\\leq \\frac{1}{4}\\sqrt{\\left(\n1+\\varepsilon \\right) ^{2}-\\delta ^{2}}.\n\\end{equation}\n(the upper limit for $p_{23}$\\ guaranties positivity of $\\varrho $). Thus,\nthe method improves $f$ on states with $0<\\varepsilon <1$ and $\\left\| \\gamma\n\\right\| +\\left\| \\delta \\right\| -2\\varepsilon >4\\,p_{23}$. One can easily\ncheck that in this class, the ''most improvable'' border state ($%\n4\\,p_{23}=1-\\varepsilon $, i.e., $f=1/2$) is\n\\begin{equation}\n\\varrho ={\\frac{1}{2}}\\left[\n\\begin{array}{cccc}\n0 & 0 & 0 & 0 \\\\\n0 & 3-2\\sqrt{2} & 1-\\sqrt{2} & 0 \\\\\n0 & 1-\\sqrt{2} & 1 & 0 \\\\\n0 & 0 & 0 & 2\\sqrt{2}-2\n\\end{array}\n\\right] \\label{stan}\n\\end{equation}\nSince $f(\\varrho )=1/2$ then standard teleportation scheme using $\\varrho $\\\ndoes not offer any better fidelity than classical. On the other hand, if we\ntransform $\\varrho $ by transformation (\\ref{oper}) with $\\cos \\theta =(%\n\\sqrt{2}-1)/(4\\sqrt{2}-5)$ (this choice maximizes $\\Delta $), then the new\nstate still satisfies the condition (\\ref{singlet-cond}), and we obtain $f(%\n\\tilde{\\varrho})\\approx 0.53>1/2$. The new state can than be used for\nteleportation with non-classical fidelity\n\\begin{equation}\n{\\cal F}\\approx {\\frac{2.06}{3}}>{\\frac{2}{3}}\n\\end{equation}\nIn other words, the state $\\varrho $ gets ``better'' when corrupted by\nenvironment. The improvement is small, nevertheless it is significant. It\nchanges the character of the state: from non-teleporting to teleporting.\n\nWhile analyzing this result, one may notice that the states with the fully\nentangled fraction improvable by the map (\\ref{oper}) form a rather\nrestricted class. In particular, this map cannot increase the entangled\nfraction of states like\n\\[\n{\\varrho}=\\frac{1}{2}\|\\psi _{-}\\rangle \\langle \\psi _{-}\|+\\frac{1}{2}%\n\|00\\rangle \\langle 00\|.\n\\]\nIt would then be very interesting to provide a complete characterization the\nclass of states which allow to improve fidelity by some local process, as\nwell as the class of local processes capable to improve fidelity for some\nstates. This task is, however, beyond the scope of this paper.\n\n\\section{Conclusions}\n\n\\label{sec5}We have examined the problem of optimal teleportation fidelity\nwith given bipartite quantum states. To this end, we investigated a\npossibility of increasing the fully entangled fraction by means of trace\npreserving LOCC operations and discovered a class of LOCC operations\\ \\\nwhich non-trivially increase $f$ on some of the two-qubit states. To a\nsurprise, the successful operations do not represent any sophisticated\naction of Alice or Bob. Instead, they result from a common (dissipative)\ninteraction between the teleporting state and the local environment. The\nunexpected conclusion then is that a dissipative interaction, normally\nassociated with the destruction of quantum teleportation, can sometimes\nfacilitate it.\n\nP.B. acknowledges stimulating discussions with Richard Bonner and Benjamin\nBaumslag. M.H., P.H. and R.H. are supported by Polish Committee for\nScientific Research, contract No. 2 P03B 103 16. P.B. is partially\nsupported by Svenska Institutet, project ML2000.\n\\begin{references}\n\\bibitem[\\#]{poczta} E-mail address: piotr.badziag@mdh.se\n\n\\bibitem[]{poczta1} E-mail address: michalh@iftia.univ.gda.pl\n\n\\bibitem[{{{{}}}}]{poczta2} E-mail address:pawel@mifgate.mif.pg.gda.pl\n\n\\bibitem[{{{{}}}}]{poczta3} E-mail address: fizrh@univ.gda.pl\n\n\\bibitem{Bennett_tel} C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A.\nPeres and W. K. Wootters, Phys. Rev. Lett. {\\bf 70}, 1895 (1993).\n\n\\bibitem{exp} D. Bouwmeester, J.-W. Pan, K. Mattle, M. Elbl, H. Weinfurter\nand A. Zeilinger, Nature (London) {\\bf 390}, 575 (1997); D. Boschi, S.\nBrance, F. De Martini, L. Hardy and S. Popescu, Phys. Rev. Lett. {\\bf 80},\n1121 (1998); A. Furusawa {\\it et al.}, Science {\\bf 282}, 706 (1998).\n\n\\bibitem{Bennett_pur} C. H. Bennett, G. Brassard, S. Popescu, B.\nSchumacher, J. Smolin and W. K. Wootters: Phys. Rev. Lett. {\\bf 76}, 722\n(1996).\n\n\\bibitem{Popescu94} S. Popescu, Phys. Rev. Lett. {\\bf 72}, 797 (1994).\n\n\\bibitem{huge} C. H. Bennett, D. P. Di Vincenzo, J. Smolin and W. K.\nWootters, Phys. Rev. A {\\bf 54}, 3814 (1997).\n\n\\bibitem{tel} R. Horodecki, M. Horodecki and P. Horodecki, Phys. Lett. A\n{\\bf 222}, 21 (1996).\n\n\\bibitem{single} M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A\n{\\bf 60} 1888 (1999).\n\n\\bibitem{bound} M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev.\nLett. {\\bf 80}, 5239 (1998); P. Horodecki, Phys. Lett. A, {\\bf 232}, 333\n(1997).\n\n\\bibitem{Linden} N. Linden and S. Popescu, Phys. Rev. A {\\bf 59}, 137\n(1999).\n\n\\bibitem{Massar} N. Linden, S. Massar and S. Popescu, Phys. Rev. Lett. {\\bf %\n81}, 3279 (1998).\n\n\\bibitem{Kent} A. Kent, {\\it ibid} {\\bf 81}, 2839 (1998).\n\n\\bibitem{Werner} R. F. Werner, Phys. Rev. A {\\bf 40}, 4277 (1989).\n\n\\bibitem{Gisin} N. Gisin, Phys. Lett. A {\\bf 210}, 151 (1996).\n\n\\bibitem{conc} C. H. Bennett, H. J. Bernstein, S. Popescu and B.\nSchumacher, Phys. Rev. A {\\bf 53}, 2046 (1996).\n\n\\bibitem{Massar95} S. Massar and S. Popescu, Phys. Rev. Lett. {\\bf 74},\n1259 (1995).\n\n\\bibitem{Kraus} K. Kraus, {\\it States, Effects and Operations: Fundamental\nNotions of Quantum Theory} (Wiley, New York, 1991).\n\n\\bibitem{Jozsa} R. Jozsa, J. Mod. Opt. {\\bf 41}, 2315 (1994).\n\n\\bibitem{Alicki} R. Alicki and K. Lendi, {\\it Quantum Dynamical Semigroups\nand Applications}, Lecture Notes in Physics, vol. 286 (Springer, Berlin,\n1987).\n\n\\bibitem{inf} R. Horodecki and M. Horodecki, Phys. Rev. A {\\bf 54}, 1838\n(1996).\n\n\\bibitem{cerf} N. Cerf, C. Adami and R. M. Gingrich, Phys. Rev. {\\bf 60},\n898 (1999).\n\n\\bibitem{xor} M. Horodecki and P. Horodecki, Phys. Rev. A {\\bf 59}, 4206\n(1999).\n\n\\bibitem{thir2} W. Thirring, {\\it Lehrbuch der Mathematischen Physik},\n(Springer-Verlag, Wien, New York, 1980).\n\n\\bibitem{correl} R. Horodecki and P. Horodecki, Phys. Lett. A {\\bf 210},\n227 (1996).\n\n\\bibitem{hab} R. Horodecki, {\\it Correlation and information-theoretic\naspects of quantum nonseparability of mixed states}, Uniwersytet\nGda\\'{n}ski, Gda\\'{n}sk, 1996.\n\n\\bibitem{Preskill} J. Preskill, //http:\nwww.theory.caltech.edu/people/preskill/ph229.\n\\end{references}\n\n\\end{document}\n--\n**************************************************************\n Michal Horodecki \t \t\n Institute of Theoretical \t\t\t\t \t\n Physics and Astrophysics\t\t\t\t \n University of Gdansk\t\t\t\t \t\n ul. Wita Stwosza 57 \t \t\n 80-952 Gdansk, Poland e-mail: michalh@iftia.univ.gda.pl \n***************************************************************\n\n" } ]	[ { "name": "quant-ph9912098.extracted_bib", "string": "[\\#]{poczta E-mail address: piotr.badziag@mdh.se" }, { "name": "quant-ph9912098.extracted_bib", "string": "[]{poczta1 E-mail address: michalh@iftia.univ.gda.pl" }, { "name": "quant-ph9912098.extracted_bib", "string": "[{{{{]{poczta2 E-mail address:pawel@mifgate.mif.pg.gda.pl" }, { "name": "quant-ph9912098.extracted_bib", "string": "[{{{{**]{poczta3 E-mail address: fizrh@univ.gda.pl" }, { "name": "quant-ph9912098.extracted_bib", "string": "{Bennett_tel C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. Wootters, Phys. Rev. Lett. {70, 1895 (1993)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{exp D. Bouwmeester, J.-W. Pan, K. Mattle, M. Elbl, H. Weinfurter and A. Zeilinger, Nature (London) {390, 575 (1997); D. Boschi, S. Brance, F. De Martini, L. Hardy and S. Popescu, Phys. Rev. Lett. {80, 1121 (1998); A. Furusawa {et al., Science {282, 706 (1998)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Bennett_pur C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin and W. K. Wootters: Phys. Rev. Lett. {76, 722 (1996)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Popescu94 S. Popescu, Phys. Rev. Lett. {72, 797 (1994)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{huge C. H. Bennett, D. P. Di Vincenzo, J. Smolin and W. K. Wootters, Phys. Rev. A {54, 3814 (1997)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{tel R. Horodecki, M. Horodecki and P. Horodecki, Phys. Lett. A {222, 21 (1996)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{single M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A {60 1888 (1999)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{bound M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. Lett. {80, 5239 (1998); P. Horodecki, Phys. Lett. A, {232, 333 (1997)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Linden N. Linden and S. Popescu, Phys. Rev. A {59, 137 (1999)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Massar N. Linden, S. Massar and S. Popescu, Phys. Rev. Lett. {% 81, 3279 (1998)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Kent A. Kent, {ibid {81, 2839 (1998)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Werner R. F. Werner, Phys. Rev. A {40, 4277 (1989)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Gisin N. Gisin, Phys. Lett. A {210, 151 (1996)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{conc C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher, Phys. Rev. A {53, 2046 (1996)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Massar95 S. Massar and S. Popescu, Phys. Rev. Lett. {74, 1259 (1995)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Kraus K. Kraus, {States, Effects and Operations: Fundamental Notions of Quantum Theory (Wiley, New York, 1991)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Jozsa R. Jozsa, J. Mod. Opt. {41, 2315 (1994)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Alicki R. Alicki and K. Lendi, {Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics, vol. 286 (Springer, Berlin, 1987)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{inf R. Horodecki and M. Horodecki, Phys. Rev. A {54, 1838 (1996)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{cerf N. Cerf, C. Adami and R. M. Gingrich, Phys. Rev. {60, 898 (1999)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{xor M. Horodecki and P. Horodecki, Phys. Rev. A {59, 4206 (1999)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{thir2 W. Thirring, {Lehrbuch der Mathematischen Physik, (Springer-Verlag, Wien, New York, 1980)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{correl R. Horodecki and P. Horodecki, Phys. Lett. A {210, 227 (1996)." }, { "name": "quant-ph9912098.extracted_bib", "string": "{hab R. Horodecki, {Correlation and information-theoretic aspects of quantum nonseparability of mixed states, Uniwersytet Gda\\'{nski, Gda\\'{nsk, 1996." }, { "name": "quant-ph9912098.extracted_bib", "string": "{Preskill J. Preskill, //http: www.theory.caltech.edu/people/preskill/ph229." } ]
quant-ph9912099	The Space-time Origin of Quantum Mechanics: Covering Law	[ { "author": "George Svetlichny\\thanks{ Departamento de Matem\\protect\\'atica" }, { "author": "Pontif\\protect\\'{\\protect\\i" } ]	Lorentz covariance imposed upon a quantum logic of local propositions for which all observers can consistently maintain state collapse descriptions, implies a condition on space-like separated propositions that if imposed on generally commuting ones would lead to the covering law, and hence to a hilbert-space model for the logic. Such a generalization can be argued if state preparation can be conditioned to space-like separated events using EPR-type correlations. This suggests that the covering law is related to space-time structure, though a final understanding of it, through a self-consistency requirement, will probably require quantum space-time.	[ { "name": "quant-ph9912099.tex", "string": "\\newtheorem{postulate}{Postulate}\n\\newtheorem{theorem}{Theorem}\n\\documentclass[12pt]{article}\n\n\\font\\gothic=eufm10 at 12pt\n\\font\\blackboard=msbm10 at 12pt\n\\def\\bbf#1{\\hbox{\\blackboard #1}}\n\\def\\goth#1{\\hbox{\\gothic #1}}\n\n\n\n\\def\\cA{{\\cal A}}\n\\def\\cB{{\\cal B}}\n\\def\\cD{{\\cal D}}\n\\def\\cG{{\\cal G}}\n\\def\\cH{{\\cal H}}\n\\def\\cL{{\\cal L}}\n\\def\\cO{{\\cal O}}\n\\def\\cP{{\\cal P}}\n\\def\\cS{{\\cal S}}\n\\def\\gA{{\\goth A}}\n\\def\\x{\\bowtie}\n\\def\\ssa{{\\em sub specie aeternitatis \\/}}\n\\def\\Tr{{\\rm Tr}}\n\\begin{document}\n\\title{The Space-time Origin of Quantum Mechanics: \n Covering Law}\n\\date{\\today}\n\\author{George Svetlichny\\thanks{\nDepartamento de Matem\\protect\\'atica,\nPontif\\protect\\'{\\protect\\i}cia Universidade Cat\\protect\\'olica,\nRio de Janeiro, Brazil \\newline e-mail: svetlich@mat.puc-rio.br}}\n\\maketitle\n\\begin{abstract}\nLorentz covariance imposed upon a quantum\nlogic of local propositions for which all observers can consistently\n maintain state collapse\ndescriptions, implies a condition\non space-like separated propositions that if imposed on generally\ncommuting ones would lead to the covering law, and hence to a hilbert-space\nmodel for the logic. Such a\ngeneralization can be argued if state preparation can be \nconditioned to space-like separated events using EPR-type correlations. \nThis suggests that the covering law is related to space-time structure,\nthough a final understanding of it, through a self-consistency requirement,\nwill probably require quantum space-time. \n\\end{abstract}\n\n\n\n\\section{Introduction}\n\nThe origin of hilbert-space quantum theory has been a\nnagging question ever since its creation.\nAxiomatic approaches, by which one\nattempts to derive the hilbert-space formalism from postulates\nwhose content is supposed to be clear and whose truth is supposed to\nbe compelling, have only had limited\nsuccess. Even if progressive clarity has been achieved, \n the truth of the axioms never seems compelling. \nSomething is missing, and the formalism\ncontinues to mystify. We shall here attempt to dispel part of\nthis mystery by arguing that space-time considerations provide\nmotivation for adopting some of the axioms that are\nhard to justify otherwise.\n\nThough there are many axiomatizations of hilbert-space quantum\nmechanics, we shall here focus on one,\nthe well-known and much investigated Piron's \\cite{piron}``quantum logic\". \nThe main object of\nconsideration is a complete atomic orthomodular lattice of ``physical\npropositions\". To have a generalized hilbert-space model one has to\nassume, among others, an axiom called the ``covering law.\" \nIt is this \nlaw that has received considerable attention, being\nthe most controversial of the ingredients. \n\n\nThere are many attempts to reduce the covering law to clearer\nand more compelling physical statements, generally by introducing further\nstructures into the quantum logic, such as measurements, transition\nprobabilities, propensities, etc.\nWe show here that some such structures provide us\nwith means of deriving necessary conditions on the quantum\nlogic if it is to describe a Lorentz-covariant\ntheory. Generalizations of such conditions are then seen to be\nsufficient to derive the covering law and thereby a Piron-type\nhilbert-space model.\n\nThere are three main ingredients in our argument. The first is the\nexistence of Heisenberg-like physical states that suffer \n``collapse\"-type transformations upon measurements. The second is\nLorentz covariance, which, beyond the usual group-action type formulation,\n includes also\nwhat we call ``covariance\nof objectivity'' (Postulates \\ref{pos:oi} and \\ref{pos:oii}) \nof \nsection \\ref{sec:stm}. \nThese state roughly that if a state is\nprepared by a measuring apparatus with space-like separated\nparts then it has the usual covariance properties with respect to local\nobservables in regions that are future time-like to all the parts of the\nmeasuring apparatus. An immediate consequence of this assumption is a\ncondition on space-like separated propositions which, if applied to any\ncommuting ones, would imply the covering law in existing axiomatic\nschemes. This suggests that the whole covering law may have a space-time\norigin. To reach such a conclusion however, one has to somehow \nrelate time-like\nand space-like situations, which leads to the third major ingredient, \nthat there are \nsufficiently many states with EPR-type correlations to be able to\nprepare arbitrary states conditioned to space-like separated events, as\nis the case for ordinary relativistic quantum mechanics. \n\nOur approach is also of an axiomatic character, and so too suffers\nfrom the shortcomings we attribute to all such attempts. \nTo its merit, it does clarify\nthe nature of the final\nphysical basis behind quantum mechanics. In particular, the third \n assumption suggests that a final justification could only come\nthrough some form of generalized quantum gravity\nwhere the light cone is not a fundamental but an emergent and object.\n\nWe have already argued for a space-time origin of the quantum\nformalism \n(Svetlichny \\cite{svet3}). There we use the\nhypothesis that it is impossible to communicate superluminally\n(ISC).\nNow the use of ISC to deduce constraints on physical theories must\nbe considered at best heuristic, for ISC must be traceable to more\nbasic considerations. In fact, in theories such as quantum gravity,\nwhere the light cone is an emergent object, ISC\nitself must be emergent.\nIt is thus imperative that we try to\nre-establish the putative connection between hilbert space and\nlorentzian space-time in a way that makes no appeal to signals. \nTo this end we must set up some of the machinery of what could\nbe called {\\em relativistic quantum logic}, quantum logic\nsubject to the requirements of special relativity. \n\nRelativistic quantum logic is relatively new, about\ntwo\ndecades old. The paper of Mittelstaedt \\cite{mitt1} could be said to be one\nof the first pioneering works published on the subject. \nThe formalism\nwas applied to the analysis of the Einstein-Podolsky-Rosen experiment by\nMittelstaedt \\cite{mitt2} and by Mittelstaedt and Stachow\n\\cite{mitt3}. The approach\nis based on the dialogical (dialog logic) view of physical propositions\nupon which relativistic restrictions are applied in the form of\nspatio-temporal validity regions. The work of Neumann and Werner\n\\cite{neumann}\nis an elaboration of Ludwig's \\cite{ludwig} measurement axiomatics. A \ncausality\npostulate is introduced for systems prepared in a space-time region and\nrecorded in a space-like separated region. Examples are presented but no\nconsequences are derived. The author's own first ideas were also\ndeveloping around the same time, but in contrast to the above mentioned\nworks were inspired mainly by local algebraic quantum field theory\nintroduced originally by Haag and Kastler (see Haag \\cite{haag}). \nIt is this\nviewpoint that we follow in this paper. One should also mention the\nwork of Mugur-Sch\\\"achter \\cite{mugur1,mugur2} which though not\nexplicitly ``relativistic'' is undeniably spatio-temporal and thus\nrelated.\n\nThough the idea of relativistic quantum logic is\nnot new, the application that we have in mind, to seek a\nspace-time basis for the covering law, {\\em is} new. \nIt seems that to do so, we must incorporate\nstructures that go beyond the usual ortho-algebraic ones. \nAs a guide we try to\nadhere as much as possible to notions current within the usual\n``Copenhagen\" interpretation and accepted by the greater part of the\nphysics community. In doing so, we do not\nadvocate this interpretation nor claim that\nit is in some sense correct, only that it provides a set of principles\nthat are sufficiently characteristic of quantum mechanics to be an\ninteresting and familiar starting point. \n\n We thus assume the usual notion of {\\em ensemble\\/}.\nEnsembles of physical systems give rise to representational elements in\nsome abstract set of ``physical states\", which for conventional quantum\nmechanics is the set of {\\em density matrices\\/}, positive hilbert space\ntrace-class operators of trace one.\nEnsembles may consist of {\\em subensembles \\/} in which case these form\nwell defined fractions given by a real number in \$[0,1]\$. \nEnsembles and subensembles are to be considered as\npotential ontological entities capable of partial realization. Generally\nensembles are partially realized by repetition of preparation procedures\nand subensembles identified by the occurrence of some physical results\nduring preparation. The subensemble fraction is assumed to be\napproximated by the frequency of occurrence of the corresponding result.\nBesides ensembles of physical systems we can consider ensembles of\nmeasurements or experiments. These are partially realized by carrying\nout the corresponding acts a large number of times in such a way that\nthey do not interfere with each other nor are interfered with by other\nacts and events in the universe. For this to make sense one must assume\nthat one can individuate the necessary physical systems and the\nexperimental apparatus in a way that warrants neglect of external\ninfluences, and posit some type of relativity theory by which act\nperformed in different regions of space-time may be considered as\nperforming the same experiment. Also for each one of these\nexperiments it is usual to use a mathematical model in which the\ncorresponding experiment is the only thing existing in all of\nspace-time. This is a deliberate idealization of the isolation of the\nexperiment from external influences. We shall tacitly subscribe to all\nsuch usual idealizations and conventions which underlie an ``ensemble\"\ninterpretation of a physical theory.\n\n \nIn the recently introduced ``consistent histories\" approach to quantum\nmechanics \\cite{hartle,omnes}, many of the usual assumptions about\n``physical states\" become considerably weakened, especially concerning\nthe ``collapse\" of the state due to measurements. A relativistic quantum\nlogic based on a consistent histories viewpoint would proceed in\na radically different direction, and at first sight would not lead to the\nsame conclusions, and so provide no justification for the covering law.\nWe have suggested elsewhere \\cite{bialo} \nthat quantum gravity would renormalize any such\ntheory to one in which the covering law holds, as this would be a\nfixed point in a self-consistency requirement, however the argument used \nis still\nrather sketchy, and so will not be considered here, except for a few\nremarks toward the end. Based on this\nhowever, we feel that the present ``collapse-biased\" considerations are\npertinent to a final explanation and must be taken into account.\n\nWe must call attention to the distinction between ultimate physical\nfacts and physical descriptions. Physical facts include at least such\nuncontroversial happenings as counter clicks, collisions,\nsupernovas, etc., about which all observers agree. Descriptions are\nformal tools needed to deal with facts. The distinction is not at all\nclear-cut for one generally tries to include among the facts inferred\nobjects such as the earth's interior, and these may be argued by others\nto be just descriptive constructs that coordinate the true\nuncontroversial facts. One may maintain that elementary particles are\njust formal objects we have invented to provide a more visualizable\ndescription of the surprisingly complex and subtle antics of\nmacroscopic bodies. In Feynman-Wheeler electrodynamics there are no\nelectromagnetic fields, only charged bodies interacting along light-like\nintervals. If such a theory is taken as true, then the usual\nelectromagnetic fields become just remote and formal descriptive\nparaphrases of the facts. A physical theory must make some declarations\nas to what is factual and what is descriptive, it must make some\n``ontological\" commitment, though part of one category may slide over to\nthe other as one changes the postulated relation of the descriptive\nelements to what are considered ultimate facts. Our concern in this\npaper is with classes of theories in which certain {\\em descriptions}\ncan be consistently maintained regardless of their relation to true\nultimate facts. Descriptions belong to observers and are often\nframe-dependent. Ultimate facts are self-subsisting and have nothing to\ndo with frames. All observers must agree upon them. Relativistic\ntheories relate ultimate facts placing them in equivalence classes under\nthe action of an appropriate relativity group. Frame-dependent\ndescriptions and group action must coexist. This places constraints on\nthe possible theories. It is some of these constraints that we try to\nexplicit.\n\nOur exposition, though roughly of an axiomatic nature, will gloss over\nmathematical details of purely technical type so as not to overburden\nthe principal conceptual structure, whose presentation is the aim of\nthis paper. We do not claim that we've found compelling reasons for\nquantum mechanics to be the way it is. We do claim that we've found\na set of physical assumptions that \ncan guide a more physically motivated \naxiomatics. That such a set of\nassumptions exists is worthy of note even if some of them can be\nseriously questioned in isolation.\n Our results must be considered in any attempt at\nunification of space-time with quantum mechanics.\n\n\n\\section{Projection Rule and Objective Mixtures}\n\nLet a quantum state be represented by a density matrix \$\\rho\$\nand perform an ideal measurement represented by a self-adjoint operator\n\$A\$, which for simplicity's sake we assume has a discrete spectrum. \nThus \$A = \\sum \\lambda P_\\lambda\$, where the\nsum is over distinct eigenvalues \$\\lambda\$, and the\n\$P_\\lambda\$ are spectral projectors.\nThe projection rule states that outcome \$\\lambda\$ occurs with\nfrequency \$\\Tr(\\rho P_\\lambda)\$ and if this not zero, then the state\nafter the measurement is given by\n\$\\rho_\\lambda = P_\\lambda\n\\rho P_\\lambda / \\Tr(\\rho P_\\lambda)\$. We have here a ``beam-splitter\"\ninterpretation of the measurement: the resulting states \$\\rho_\\lambda\$\nfor different values of \$\\lambda\$ are maintained separate either\nformally (by conditioning further measurements or even data analysis to\nparticular outcomes of the current one), or even physically by guiding\nthe resultant states into different spatially separated regions. One\ncan however disregard which outcome occurs and consider each instance of\nany of the post-measurement states\n as being an instance of a single state which\nwould now be represented by \$\\rho_A = \\sum_\\lambda {\\rm\nTr}(\\rho P_\\lambda) \\rho_\\lambda = \\sum_\\lambda P_\\lambda \\rho\nP_\\lambda\$. This is usually referred to as an ``incoherent\nmixture\" of the resulting states \$\\rho_\\lambda\$. One cannot reverse\nthis, just from the density matrix \$\\rho_A\$, there is no way of\ndetermining the constituent components \$\\rho_\\lambda\$ and the\ncorresponding frequencies \$\\Tr(\\rho P_\\lambda)\$. Even if we seek pure\ncomponents, a non-extreme density matrix \$\\rho\$ can be decomposed in an\ninfinite number of ways into a convex combination of extreme matrices,\nthat is, there are\ninfinitely many Borel probability measures \$\\mu\$ with support in the\nsubset \$\\cP\$ of extreme points such that \$\\rho = \\int p\\, d\\mu (p)\$.\nThis of course\nraises a much ventilated controversy: given a non-extreme density matrix\n\$\\rho\$, is any among its infinite integral representations as a convex\ncombination of extreme points somehow better, or even objectively or\nontologically ``correct\"? If some principle is assumed by which a\nunique representation is singled out, we shall call this representation\nan {\\em objective mixture}, and to distinguish it from a purely\nmathematical integral representation, we shall use an indexed equality\nsign \$=_o\$ for the former. Thus \$\\rho =_o \\int p \\, d\\mu(p)\$ \nmeans that\nit is {\\em this} representation that is singled out by the postulated\nprinciple. We use the term ``objective\" to give a deliberate bias to the\nnotion, as we envisage that such unique representations have their roots\nin some objective reality. In particular, \nprior measurements may provide a basis for such\nrepresentations.\n\nOne often sees another type of pure-to-mixed state transformation, \nthe partial trace. For any density matrix \$\\rho\$ defined in a tensor\nproduct hilbert space \$H_1 \\otimes H_2\$ one can \ndefine the {\\em partial\ntrace\\/} \$\\rho^{(1)}=\\Tr_{H_2}\\rho\$, a density matrix in \$H_1\$,\ndefined by requiring that for any bounded operator \$A\$ in \$H_1\$ one\nhas \$\\Tr(\\rho^{(1)}A)= \\Tr(\\rho (A \\otimes I_{H_2}))\$.\nIn general even if \$\\rho\$ is a pure state,\nthe partial trace \$\\rho^{(1)}_{\\psi}\$ is not.\nWhat is\nusually said about this last situation is that \$\\rho^{(1)}_{\\psi}\$\nrepresents an ensemble of first members, in an {\\em undetermined\\/} state,\nof a pure ensemble of a two-member composite system. Thus if we write\n\$\\psi = \\sum_i \\alpha_i \\otimes \\beta_i\$ where \$\\alpha_i \\in H_1\$ and\n\$\\beta_i \\in H_2\$ then one can conceive \$\\psi\$ as representing a\ncomposite system with two components, the states of one of which are\nrepresented in \$H_1\$ and of the other in \$H_2\$. If one now makes a\nmeasurement represented by the self adjoint operator \$A\$ only on the\nfirst component, the expected value is \$\\Tr(\\rho^{(1)}_{\\psi}A)\$ and so\nas far as the measurements on the {\\em first} component are concerned,\nthe system acts as though the first component is in a mixed stated given by\n\$\\rho^{(1)}_{\\psi}\$. The conventional wisdom concerning this situation\nis however that this is always a mathematical description and no\nobjective mixture \$\\rho^{(1)}_{\\psi}\$ of any kind is present \n(except for the very particular\ncase of the partial trace being extreme). The\n``partial trace\" state is considered to be ontologically different from\nthe other types of ensembles. The prototypical example of this situation\nis the singlet state of a two photon \nsystem. If no measurement is made on one of\nthe photons, then the other one, in so far as it could be construed as a\nseparate entity, is considered to be ``unpolarized\", that is, in no\ndefinite state of polarization. In fact, \nif any polarizer is placed in front of it, the probability\nis always one half that it will pass through. One must point out that to\nbe ``unpolarized\" is {\\em not\\/} a possible state that a photon may be\nin, as any one-photon state is always in some state of polarization. Thus\nto talk about an ``unpolarized\" photon is to employ a (useful) \nmetaphor concerning\nthe presence of an entangled state involving several photons. \n\nA rather strong principle that leads to objective mixtures \ncould be called ``primacy of pure states\": given a mixed state, then any\ngiven physical instance of such a state is in fact a physical instance\nof a unique pure state which possibly varies from instance to\ninstance. Primacy is given to pure states and mixed states arise\nthrough mere ensemble mixtures and do not represent new irreducible\nontological entities. This is usually the attitude upheld in elementary\ntextbooks on quantum mechanics especially for systems comprised of a\nsmall number of particles, as one can easily prepare mixtures that {\\it\nprima facie\\/} seem to obey it. One could seriously question it for\nmesoscopic and larger systems. We have argued (Svetlichny\n\\cite{svet1,svet2})\nthat its negation can and does lead to interesting possibilities as\nthere are combinatorial hidden-variable models in which mixed states\nviolate this principle, opening up a new approach to the distinction\nbetween the classical and the quantum. A explicit revocation of the\nprimacy of pure states can be found in \nCzachor's \\cite{czachor} proposals for non-linear quantum mechanics, in\nwhich \ndensity matrix evolution is not reducible to evolution\nof its component mixtures, the non-uniqueness of which is behind the\ncausality problems of non-linear deformations of quantum theory.\nWe shall see below that the\nprinciple cannot be universally upheld along with special relativity and\nusual notions of causality. Nevertheless, its simplicity makes it a\nuseful heuristic device and it does bear examination on two grounds: 1)\nit becomes relevant once one contemplates alternative physical theories,\nand 2) it's a useful starting point for seeking weaker criteria for\nobjective mixtures.\n\nUnder the primacy of pure states, if \$\\rho =_o \\int p \\, d\\mu(p)\$\nand \$\\cA\$ an observable whose expected value in a pure state \$p\$ is\n\$<\\!\\!\\cA\\!\\!>_p\$ then the expected value \$<\\!\\!\\cA\\!\\!>_\\rho\$ in state \n\$\\rho\$ has to be\n\$\\int <\\!\\!\\cA\\!\\!>_p \\, d\\mu(p)\$. For a conventional quantum mechanical\nobservable represented by a self adjoint operator \$A\$ one \nhas \\linebreak\n\$<\\!\\!\\cA\\!\\!>_p = \\Tr(pA)\$ and then \$<\\!\\!\\cA\\!\\!>_\\rho = \n\\int \\Tr(pA)\\, \nd\\mu(p) = \\Tr\\left(\\left( \\int p \\,\nd\\mu(p)\\right)A\\right) = {\\rm Tr}(\\rho A)\$ by the \nlinearity of the trace and the\noperator A. The representation as an objective mixture drops out\nand any other representation would lead to the same observable\nconsequences. Operationally, there is no\nobservable difference between two different representations, and some\nmaintain that objective reality is related to the density matrix itself\nand representations as convex combinations of pure states is a purely\nmathematical affair. If however one wants to depart from ordinary\nlinear quantum mechanics, the question of representation of mixtures\nbecomes crucial as the integral formula for \$<\\!\\!\\cA\\!\\!>_\\rho\$ could \nvery well\ndepend on the representation used in which case some form of objective\nmixtures has to be maintained. Furthermore, maintaining some such\nversion does lead to very interesting and important consequences as one\nis then able to influence ``objective reality\" at a location space-like\nto one's own through EPR-type long-range quantum correlations. \nConsider the singlet state of a\ntwo-photon system when the two photons are space-like separated and,\n say, traveling along opposite arms of an EPR-type apparatus. If we now\nperform an observation represented by a non-degenerate quantum\nobservable \$A\$ with normalized eigenvectors \$\\psi_1\$ and \$\\psi_2\$ on\none arm of the apparatus, then by the projection postulate and strict\ncorrelations in the singlet state, the state on the other arm\nimmediately after the measurement is an equal mixture of \$\\psi_1\$ and\n\$\\psi_2\$ represented by the density matrix \${1 \\over 2}I = {1 \\over\n2}(\\psi_1, \\cdot)\\psi_1 + {1 \\over 2}(\\psi_2, \\cdot)\\psi_2\$. Now, if we\nbelieve in objective mixtures and rewrite this with \$=_o\$, we see that\nby changing the observable \$A\$ to one with a different eigenbasis, we\nimmediately change the objective mixture on the other arm. This ``action\nat a distance\" upon supposed objective mixtures has been \nextensively discussed\never since the original EPR paper (Einstein, Podolsky, and Rosen,\n\\cite{EPR})\nand has recently been used to derive a series of strong constraints on\npossible alternatives conventional quantum mechanics \n(Gisin \\cite{gisin1,gisin2,gisin3,gisin4}; Pearle\n\\cite{pearle1,pearle2}; Svetlichny \\cite{svet3}). Such\nconstraints stem from the fact that if one is not careful, such\nalternative theories will allow for humanly controlled superluminal\nsignals and hence supposed \ndifficulties with special relativity. We shall\nhenceforth refer to these results as the {\\em ISC constraints}. \nTheories in which objective mixtures as the result of measurement are \nallowed descriptive elements are strongly constrained and so \nit behooves us to try to give this notion some solid\nfoundation.\n\n\nLet us therefore assume that states collapse by measurements to\nobjective mixtures and see what this may mean. Consider again a source\nof singlet two-photon states which then travel in opposing arms of an\nEPR-type apparatus. Assume we are in a reference frame, \nthe rest frame, in which the apparatus and the source\nis at rest so that the two correlated photons are always at equal\ndistances along the two arms. Put a vertically oriented linear\npolarizer at some distance\nalong one arm, call it arm {\\it 1\\/} and mark the other arm, call it arm\n{\\it 2\\/}, at the same distance without first placing anything in the\nway of the photon. By the usual arguments we must now conclude that just\nbeyond the mark one has an objective mixture of vertically and\nhorizontally polarized photons in equal proportions. We shall also\nconsider a frame, the moving frame, in which\na photon on arm {\\it 2\\/} reaches the mark before its mate reaches the\nlinear polarizer. In this frame, just beyond the mark, the photons are\nstill unpolarized and an objective mixture of linearly polarized photons\ncomes into being further down the arm. Thus the objective mixture\ndescription is frame dependent. In itself, frame dependence is not a \ndefect and we face it all the time.\nA static magnetic field, viewed from a moving frame, becomes a\nmagnetic and an electric field, so the presence or absence of an\nelectric field is frame dependent. However, this type of behavior is\neasily explained by the notion of covariance under a group\nrepresentation while the frame dependence of objective mixtures is of a\ncompletely different nature.\nTry now\nto give meaning to the statement that in the rest frame, just after the\nmark, there is an objective mixture of vertically and horizontally\npolarized photons. At first glance one might say that each photon would\npass with certainty either through a vertically or horizontally oriented\nlinear polarizer and that this is not true for any other ``filters\" that\ncan be placed in its path (uniqueness of objective mixtures). Let us\ncall this the {\\em passage criterion}. Note that in this situation the\ncriterion is counterfactual for we have no way of knowing which\npolarization any individual photon has, but if we {\\em did} know, it\n{\\em would} pass through an appropriately\noriented polarizer. \nNow call upon Maxwell's demon's cousin, the quantum demon. This being\nhas knowledge of quantum mechanical systems that cannot be achieved by\nany humanly constructed apparatus, in our case the knowledge missing in\nthe counterfactual criterion. Place now a linear polarizer just after\nthe mark and have the quantum demon rotate it through a sequence of\nhorizontal and vertical orientations in such a manner as to pass all the\nphotons from the objective mixture that impinges upon it. How does this\nsituation look from the moving frame? In this case the demon is twirling\nhis polarizer in front of {\\em unpolarized} photons and even so he is\ncapable of letting all of them pass through. What's responsible for this\nstrange fact? We can of course try to blame the linear polarizer that\nsits at arm {\\it 1\\/} but the event of a photon impinging there is to\nthe future of its mate impinging upon the demon's polarizer. This looks\nlike inverted causal order, but it's inverted order at space-like\nseparation and so could be deemed innocuous (though there is the danger\nthat concatenating two such could lead to time-like retrograde\ncausality).\nAlso it's not surprising\nthat such inverted causal order appears, as we have already posited\nsomething like ``action at a distance\" for manipulating distant\nobjective mixtures and a Lorentz transformation can turn this into\naction into the past. What is often desired of theories that show such\napparent causal anomalies (such as tachyon theories) is that the \ncausal\norder of events can be {\\em reinterpreted} as again to follow a\nstrict temporal order. This would make the notion of cause and effect\nframe dependent, but in the end the usual notions of causality can be\nmaintained in any frame. Let us call this desideratum upon physical\ntheories ``strict temporal causality\". If we assume this then we cannot\nblame the polarizer on arm {\\it 1\\/} and we must assume that even a beam\nof unpolarized photons is an objective mixture of vertically and\nhorizontally polarized photons. However we can restart the whole\nargument now with a circular polarizer on arm {\\it 1\\/} and conclude\nthat a beam of unpolarized photons is an objective mixture of left and\nright circularly polarized photons. Similarly for any other type of\npolarizer. This contradicts the whole idea of objective mixtures as\nuniqueness is important, and we must state that the conjunction of\nspecial relativity, strict temporal causality, and the counterfactual\npassage criterion for objective mixtures is contradictory. Note\nhowever that primacy of pure states implies the passage criterion\n(counterfactual or not) and so the notion of primacy must be abandoned\nas a universal principle if we want to keep the other two.\nOne way to weaken the passage criterion (or the primacy of pure states)\nis to abandon the uniqueness requirement. This is not a desirable step\nfor us as this would undermine the ISC constraints. However such a step\ndoes bring its insights. We would then admit that an unpolarized photon\nalready has a well defined value for any of its possible polarizations.\nWe are now faced with a hidden-variable theory where each relevant quantum\nobservable has a well defined value. We know that any such theory to be\nsuccessful must be contextual and non-local (Redhead \\cite{redhead}). It is \nthe\nappearance of non-locality in this context that is indicative. Primacy\nof pure states and the passage criterion seem at first to be local in\nnature, however let us examine them from an operational point of view.\nConfronted with a given physical instance of a mixed state we call upon\nour quantum demon to tell us which particular instance of a pure state,\nrepresented by a normalized vector \$\\psi\$, we are dealing with. How can\nwe be sure that the demon tells the truth? We test the state with the\nquestion represented by the hermitian projector \$P_\\psi\$ upon the\none-dimensional space spanned by \$\\psi\$. If the test fails, the demon\nlied, if it passes we're still not sure but this is the best we can do.\nIf after a very long run all tests pass, we have strong statistical\nevidence to believe the demon is truthful. Now comes the crux of the\nmatter: in any relativistic field theory, \$P_\\psi\$ is {\\em not} a local\nobservable. We shall treat this some paragraphs below, but given this, we\nsee that {\\em the notion of objective mixtures cannot be operationally a\nlocal notion\\/}, and it is the contradictory attempts to treat it as such\nthat leads to major difficulties.\n\n\nOn the other hand, in the rest frame, the photons just beyond the mark\nseem to behave exactly like an objective mixture of linearly polarized\nphotons because if we place a linear polarizer in a horizontal\norientation just beyond the mark, we get exact coincidence with what\nhappens at the other polarizer.\nIn the moving frame the roles of the arms are reversed\nto maintain strict temporal causality but the same description applies.\nAnother argument for objective mixtures is that we can reproduce the\nquantum demon's exploit if we delay the photon on arm {\\it 2\\/}\nsufficiently. At some point before the mark on arm {\\it 2\\/}\nplace a mirror that reflects the\nphoton to a second distant mirror which then reflects it back to the arm\nat a position further down the arm from the first mirror but still before \nthe mark,\nand at this position place a third mirror that redirects the photon\nagain outward along the arm. Let the distant mirror be so removed\nthat information about what happened at the polarizer at arm\n{\\it 1\\/} has time to reach an observer stationed at the mark before the\ndetoured photon reaches him. The observer then uses this information to\nrotate a linear polarizer just beyond the mark so as to allow all the\nphotons to pass through. Of course now the event of the photon passing\nthrough the polarizer on arm {\\it 2\\/} is inside the future light\ncone of the event of its mate passing through the polarizer on arm {\\it\n1\\/} and this then remains true in all frames and we cannot invoke the\nargument presented above\nwhich knocked down the counterfactual passage criterion.\nSo it seems we have all\nreason to believe in objective mixtures in this situation. In fact, the\npassage criterion is now factual and shows indeed that one deals with an\nobjective mixture of linearly polarized photons. What happens then as\nthe photon takes its detour. Does its ontological status changes from\nunpolarized to polarized somewhere along its path? If so, when does this\nhappen? If we give a negative answer to the first question then the\nphotons have been polarized all along including at points of space-like\nseparation and by our previous argument we fall back into a particular\nhidden variable theory which we are trying to avoid, so the\nanswer must be positive. A natural answer then to the second question would\nbe that the change happens when the photon reaches a point along its\npath that is light-like to the event of its mate encountering the\npolarizer at arm {\\it 1\\/}. But now this means that a fundamental\nquantum mechanical feature changes merely due to a spatio-temporal\narrangement. At space-like separation, given our bias for strict\ntemporal causality, we can never maintain the (counterfactual)\npassage criterion of objective mixtures, but as soon as it becomes\nlight-like, the passage criterion is applicable (and factual) and shows\nthat one has an objective mixture of linearly polarized photons. This\nin itself demonstrates that quantum mechanics is linked to\nspace-time structure for the photon becomes polarized just by\npenetrating a certain light cone. In other words, {\\em besides the \nmeasuring\napparatus, space-time itself participates in state collapse\\/}.\n\nThis is in stark contrast with Galileian covariant theories. In such\ntheories the primacy of pure states can be maintained universally\nand hence the passage criterion for objective mixtures can be used\nin all circumstances.\n\nNow we must look again at the original space-like situation. We saw\nwe cannot use the counterfactual passage criterion yet we would like\nto maintain some version of objective mixtures. The answer that\noffers itself is that whereas for time-like situations objective\nmixtures are ultimate physical facts, for space-like separations\nthey are physical descriptions. By this we do not mean they are\narbitrary or ``unreal\" but that the criteria for their choice can\ndepend on specifics of experimental arrangements and inertial\nframes. Thus we should not be surprised that if we place\n a circular polarizer\nplaced just beyond the mark, in the rest frame there are (according\nto the {\\em description}) linearly polarized photons impinging on\nit, whereas in the moving frame (again according to the {\\em\ndescription})\nthere are no linearly polarized photons anywhere near it.\nWe can base our criterion for objective mixtures on a\nprevious measurement event, according to the time order in the given frame. \nThus in the rest frame it is the linear\npolarizer on arm {\\it 1\\/} that determines the objective mixtures in the \ntime\ninterval between one photon having reached its polarizer and its mate\nthe other polarizer, and in the moving frame it is the circular\npolarizer on arm {\\it 2\\/}. It is also part of the conventional wisdom that\nthis is a consistent way of proceeding and that observers in either\nframe, each one using his own description, will agree as to their\npredictions about ultimate physical facts.\n\nRelativity theory forces us to abandon a naive picture of\nprimacy of pure states and to adopt a sort of hybrid view in which\nobjective mixtures are ultimate physical facts in some situations\nand physical descriptions in others. In a fixed frame one\nsituation blends into the other without apparent discontinuity as\nsoon as certain spatio-temporal relations are achieved.\n\nThe paradoxical nature of objectifying too much the state description\nafter measurement in relativistic theories was also pointed out by Mielnik\n\\cite{mielnik} who concludes that state-reduction and relativity are\nmutually inconsistent. This is a surprising conclusion as relativistic\nquantum field theory is highly successful. He is led\nto this by however by tacitly admitting results of counterfactual\nexperiments. Applied to the singlet two-photon state\nconsidered above, his reasoning would lead to the same hidden-variable \ntheory that the\ncounterfactual passage criterion does. Disallowing such counterfactual\ndefiniteness blocks the contradiction. A different resolution of\nMielnik's paradox is given by Finkelstein \\cite{fink}.\n\n\nReturn again to the projection postulate and let \$A\$ be a\nself adjoint operator with discrete spectrum and spectral\ndecomposition \$A = \\sum \\lambda P_\\lambda\$. Let the normalized vector\n\$\\psi\$ represent \nthe state\nupon which the observation is performed, and let \$\\psi_\\lambda =\nP_\\lambda \\psi / \|\|P_\\lambda \\psi\|\|\$ whenever \$\|\|P_\\lambda\n\\psi\|\| \\neq 0\$. We now write \$\\rho_A =_o \\sum \|\|P_\\lambda \\psi\|\|^2\n(\\psi_\\lambda,\\cdot)\\psi_\\lambda\$ to indicate that \$\\rho_A\$ occurred \nthrough a\nprevious\npreparation (measurement) in which, had the ``beam splitter\"\nviewpoint been adopted, the \$\\psi_\\lambda\$ would describe the pure\nstate in each ``beam\". Objective mixtures are then tokens of\npreparations. Whether such a mixture is actual and complies with the\npassage criterion or descriptive, depends now on some\nspatio-temporal situation. In all cases however one has a\n{\\em correlation } criterion. If immediately after observing \$A\$ we\nperform a test of whether the resulting state is \$\\psi_\\lambda\$ \n(by using the orthogonal projector\nonto this vector as the observable), then\nthe test is satisfied if and only if \$\\lambda\$\noccurs (exact correlation). This of course is due to the orthogonality of the\n\$\\psi_\\lambda\$ for different \$\\lambda\$. This is not true for any other \nset of\none-dimensional projectors associated to those outcomes \$\\lambda\$ for\nwhich \$P_\\lambda \\psi \\neq 0\$. Thus the correlation\ncriterion picks out a unique convex combination of pure states and\nso is a legitimate basis for objective mixtures. Being an\noperational criterion (at least for ensembles)\nit is about as ``objective\"\nas one can wish. It works both in space-like and time-like\nsituations. Its major disadvantage is that it doesn't apply to the\nstate in itself but involves the preparation that produced the\nstate. Such knowledge of the preparation procedure can be used to\nseparate the beams again if the original measurement was not of the\nbeam splitter type. Just measure \$A\$ again (or any other observable\nfor which the \$\\psi_\\lambda\$ are eigenstates) and adopt the beam\nsplitter attitude.\n\n\nFor future reference, let us examine now what happens when we perform a \nsecond measurement\nrepresented by a self-adjoint operator $B$ with discrete\nspectrum\nand spectral decomposition $B = \\sum \\mu Q_\\mu$. We now have $(\\rho_A)_B\n= \\sum_\\mu Q_\\mu \\rho_A Q_\\mu = \\sum_{\\mu \\lambda} (Q_\\mu\nP_\\lambda\\psi, \\cdot)Q_\\mu P_\\lambda\\psi$. Now can this be interpreted as\nan {\\sl objective} mixture according to the correlation criterion?\nIf by this we mean that each distinct (unnormalized) final state $Q_\\mu\nP_\\lambda\\psi$ is uniquely correlated to a set of outcomes, then yes. If\nhowever we want that there be a test for each final state which passes\nif and only if that state is produced, or if we want to ``separate the\nbeams'' as was done for a single measurement above,\nthen the distinct final states\nmust be orthogonal. We\nshall consider the objective mixture description legitimate in this\ncase. This latter situation is always\nrealized for any initial state $\\psi$ whenever $A$ and $B$ commute. \n\n\nHow should one interpret then the ISC constraints? They must now be read\nin the following manner: in any theory for which the objective mixture\ncriterion is the presence of an adequate previous measurement and for\nwhich such {\\em descriptions} form a consistent logical system leading\nto frame independent predictions of ultimate physical facts, the results\nof the references hold. So reinterpreted, the validity of the works is\nmaintained and they still provide strong criteria for selecting\ntheories, but these theories are to be chosen among special types.\nThis is an important insight, as one can now see under what conditions\nISC may not constrain a theory or constrain it less. Thus if ``state\ncollapse\" as a descriptive element, along with ISC, argues strongly for\na linear theory (as is expounded in \\cite{gisin2,svet3}), theories\nwithout this descriptive element, such as the consistent-histories\napproach to quantum mechanics, may possibly be made non-linear without\nviolating ISC. We argue in this direction in \\cite{svetlichny:quantum, \nsvet5} and\ntake up this point later in this paper.\n\n\n\\section{Locality and Purity}\n\nSince the idea of objective mixtures involves decomposition into\npure states, we must examine the nature of pure states in\nrelativistic quantum mechanics. There are roughly two\napproaches to this, via fields (Streater\nand Wightman \\cite{streater}) and via\nalgebras of observables (Haag \\cite{haag}).\n\nThe algebraic approach is closer in spirit\nto what we are contemplating here as it introduces {\\em algebras of\nlocal observables}, that is, it associates to each bounded region \$\\cO\$ of\nspace-time an algebra \$\\gA(\\cO)\$ of observables that correspond to\nexperiments that can be executed in \$\\cO\$. In relativistic quantum logic\nwe would be interested in propositions that can be tested\nin \$\\cO\$ and so should in some natural way be related to the algebra\n\$\\gA(\\cO)\$. It is\nhowever rather difficult to come across examples of local algebras\nexcept through quantum fields and so we present here one possible\nconstruction. Assume for simplicity that we have a real (uncharged)\nscalar relativistic Wightman field \$\\Phi\$.\nSuch a field is an\noperator-valued distribution in the sense that there is a fixed dense\ndomain \$\\cD\$ such for any \$f \\in \\cS({\\bbf R}^4)\$ there is an \nessentially\nself-adjoint\noperator \$\\Phi(f)\$ on the invariant domain \$\\cD\$ and such that\nfor all \$\\phi, \\psi \\in \\cD\$ the map\n\$f \\mapsto (\\phi,\\Phi(f)\\psi)\$ defines a tempered distribution.\nConsider now the operators \$\\Phi(f)\$\nfor \${\\rm supp}\\,f \\subset \\cO\$ for some bounded region of space-time \n\$\\cO\$. Let\n\$\\gA^c(\\cO)\$ be the set of bounded operators \$A\$ such that \$A\\cD \n\\subset\n\\cD\$ and which commute on \${\\cal D}\$ with all the operators \$\\Phi(f)\$\nintroduce above. Define\nthe von-Neumann algebra \$\\gA(\\cO)\$ as \$(\\gA^c(\\cO))'\$, the\ncommutant of \$\\gA^c(\\cO)\$. The algebra \$\\gA(\\cO)\$ is then taken to be\nthe algebra of observables in \$\\cO\$. An alternative definition would be to \ntake for\n\$\\gA(\\cO)\$ the von-Neumann algebra generated by the bounded functions of\nthe \$\\Phi(f)\$, that is by those operators of the form \$F(\\Phi(f))\$ where\n\$F\$ is a real Borel-measurable bounded function on \${\\bbf R}\$.\nThe hermitian projectors in \$\\gA(\\cO)\$ should then correspond to the\ntestable proposition in \$\\cO\$. Now it is a know fact (Araki\n\\cite{araki}, Haag\n\\cite{haag}) that the von Neumann algebra \$\\gA(\\cO)\$ is of type III. This \nmeans\nthat it contains no finite-dimensional projector and in particular no\none-dimensional projector. One-dimensional projectors are indicator\npropositions for pure states, that is if \$P = (\\psi,\\cdot)\\psi\$ \nfor a normalized vector \$\\psi\$, and\n\$\\rho\$ is a density matrix, then one has \$\\Tr(\\rho P) = 1\$ if and only\nif \$\\rho = (\\psi,\\cdot)\\psi\$. It is these projectors that must be used\nto test for pure states, and therefore purity of states is not a local\nnotion. This is the basic insight that relativistic quantum field theory\nprovides for quantum logic. A consequence of this is that the notion of\nobjective mixtures becomes a non-local notion, and in particular the\ncorrelation criterion, as it involves testing for pure states, is a\nnon-local criterion. Now if one cannot test for purity by projectors in\n\$\\gA(\\cO)\$ for a bounded region \$\\cO\$, it is also highly plausible that\none can test for pure states by projectors in the algebra associated to\nany {\\em time slice}: \$\\cO = \\{(x,y,z,t)\| t_1 < t < t_2\\}\$. Such an\nalgebra is defined by an appropriate limiting procedure in term of the\nalgebras associated to bounded regions contained in the time slice.\nFor free\nfields any time-slice algebra is just \$B(H)\$, the full operator algebra\nof the physical hilbert space. In general one expects the field to obey,\nin some appropriate sense, a hyperbolic differential equation and so the\nfield values at any point can be determined from their values in a time\nslice. This would mean that the time-slice algebra coincides with the\nalgebra associated to all of space-time, which again should be \$B(H)\$\nfor reasonable theories.\n\nSuppose now that we perform a measurement in a bounded space-time\nregion \$\\cO\$ upon a pure Heisenberg state represented by a normalized \nvector\n\$\\psi\$.\nConsider at a space-like point to \$\\cO\$ two observers, one\nfor whom, according to the time variable in his frame, the\nmeasurement is still to happen and another one for whom the\nmeasurement has already taken place. As they fly by each other they\nexchange notes, each one indicating what the quantum state is. One\nsays it's pure, the other one mixed. This apparent contradiction\ndisappears when one realizes that an operational definition of\npurity is not local. Each one's assessment depends on a time-slice\nwhich for one is prior to the experiment and\nposterior for the other. Thus in relativistic quantum mechanics, in\nthe presence of measurements, whether a state is pure or not is\nframe dependent. If both observers are however in the future light\ncone of all points of \$\\cO\$, then their assessments of the state\nagree, both will say it is mixed and their respective descriptions\nshould differ merely by the action of a representation of the\nLorentz group, that is by usual Lorentz covariance. There is a point\nof consistency here: in the region where the observers do not\nagree, no local observable should be able to distinguish between the\ntwo conflicting descriptions. This in fact is the case, for let the\nexperiment be\nrepresented by a self-adjoint operator \$A\$ with discrete spectral\ndecomposition \$A=\\sum \\lambda P_\\lambda\$ and let \$B\$ be a\nself-adjoint operator pertaining to a space-like\nseparated local algebra. Let us say the first observer's description\nis that \$B\$ is being observed on \$\\psi\$. The expected value then\nwould be \$(\\psi, B\\psi)\$. Assuming normal Lorentz covariance, the\nsecond observer describes the situation as observing \$U(g)^BU(g)\$\nupon \$U(g)^\\rho_AU(g)\$ where \$U(g)\$ is a unitary operator\nrepresenting the element \$g\$ of the Lorentz group that connects the\ntwo observers. This last description gives a mean value of \$\n\\Tr(U(g)^BU(g)U(g)^\\rho_AU(g)) = \\Tr(B\\rho_A) = \\Tr(B \\sum\nP_\\lambda P_\\psi P_\\lambda)\$\nwhere \$P_\\psi = (\\psi, \\cdot)\\psi\$. By the\nlinearity and permutation symmetry of the trace this is \$\\sum\n\\Tr(P_\\lambda B P_\\lambda P_\\psi)\$. Now in all relativistic\nfield theories, local observables in space-like separated regions\ncommute, so one has \$P_\\lambda B P_\\lambda = P_\\lambda^2 B =\nP_\\lambda B\$ and since \$\\sum P_\\lambda = I\$, the expected value is\n\$\\Tr(BP_\\psi) = (\\psi, B \\psi)\$ exactly as for the first\nobserver. Hence the discrepancy in description of purity of states,\ndue to the non-local nature of the correlation criterion, has no\neffect on locally observable quantities and this is precisely what\naccounts for the consistency of this criterion.\n\nWe are now in position to abstract from the above situation in\nconventional quantum mechanics and introduce a sketch of an axiom system\nfor a measurement theory in relativistic quantum logic.\n\n\n\\section{Propositions, Properties, States, and Ensembles}\n\n\nThe basic ingredient of a ``quantum logic\" approach to a physical theory\nis a pair \$(\\cL, \\cS )\$ where \$\\cL \$ is an orthomodular poset\n(usually a lattice) of physical {\\em propositions\\/}, and \$\\cS \$ an\nabstract convex set whose elements correspond to physical {\\em\nstates\\/}. The set \$\\cP \$ of extreme points of \$\\cS\$ correspond to\n{\\em pure states\\/}. The relation between \$\\cL\$ and \$\\cS\$ is usually\nthat elements of \$\\cS\$ are \$\\sigma\$-additive probability measures on\n\$\\cL\$, and given \$s \\in \\cS\$ and \$a \\in \\cL\$ the number \$s(a)\$\ncorresponds to the {\\em probability\\/} that the proposition \$a\$ be\ntrue in the state \$s\$. We shall not necessarily adhere to such a view\nthough much of the literature adopts it. We shall not here go into\ndetails about how one operationally prepares pure states nor determines\ntheir purity except in so far as is needed to consider the space-time\nrelations involved.\n\nOne generally sees two concepts of physical state that. The\nmost common one is that of an {\\em instantaneous state \\/}, and a\nphysical system is supposed to at each time instant {\\em be\\/} in some\ninstantaneous state. The other notion is that of a state \\ssa in which\ncase the whole temporal history from \$t= -\\infty\$ to \$t=+\\infty\$ is\nsubsumed in the notion. Of course this is an idealization as normally a\nstate is prepared at some instant and destroyed at some future instant\nboth by processes foreign to its ``normal\" isolated temporal evolution.\nThus to maintain a view \\ssa one has to rely on some deterministic\nevolution extendible to both temporal infinities. This also means that\nthe states is considered as a self-subsisting entity. The ability to\nextend its evolution to a time prior to its creation, means that it could\nhave been created at a different time and so it has no knowledge of its\ncreation, and the ability to extend its evolution to a time beyond its\ndestruction means that it has no presage of its demise. Such a physical\nstate is an ontological entity all to itself. The two views coexist in\nordinary quantum mechanics whereby the instantaneous state view is\nmaintained in the Schr{\\\"o}dinger picture and the other in the Heisenberg\npicture. The \\ssa view is more convenient for space-time description as\nthe notion of instantaneous state, even for normal time evolution,\nbrings in frame-related considerations due to the frame-dependent nature\nof the notion of ``instant\". Thus we maintain the \\ssa viewpoint in this\npaper. Of course even this view cannot entirely avoid frame-related\nnotions as this type of state is allowed to undergo change through the\nmeasurement process or other external interventions. Such changes are\ngenerally held to be instantaneous in some frame and so if the state is\nconceived as having a space-time extent, a frame dependent change of\ndescription is involved. We shall in fact be interested in states having\nsufficiently large space-time extents to be able to perform independent\nmeasurements at spatially separated distances.\n\nJust as in quantum mechanics, we shall assume the notion of ensembles\nat least in so far as they can be partially realized by repetitions of\npreparation procedures. The notions of subensemble and subensemble\nfraction is also maintained. Ensembles are represented by elements of\n\$\\cS\$ and we admit, just as in quantum mechanics, that the same element\nof \$\\cS\$ could very well represent many ontologically distinct ensembles\nso one should not conflate the two.\n\n\\section{Measurements and Objective Mixtures}\n\nWe shall assume at least that \$\\cL\$ and \$\\cS\$ are related through\nmeasurement. By an {\\em instrument\\/} \$I\$ we shall mean an\nexhaustive n-tuple \$(a_1,\\dots,a_n) \\in \\cL^n\$ of mutually exclusive\npropositions; that is, \$a_i \\perp a_j\$ for \$i \\neq j\$ and\n\$\\bigvee_{i=1}^n a_i = 1\$. We shall assume that such n-tuples\ncorrespond to physical measurements with \$n\$ mutually exclusive and\nexhaustive outcomes; that is, when such a measurement procedure is\nexecuted, one and only one of the outcomes occurs. We shall now make\na series of assumptions concerning the act of measurement and\ndiscuss them later.\n\n\n\\begin{postulate}[M1 -- Frequency] \\label{pos:m1}\nGiven an instrument \$I=(a_1,\\dots,a_n)\$ and a state \$s\$, then associated\nto an ensemble of measurements of \$I\$ in \$s\$, is a {\\em frequency\nfunction} \$\\omega_i^I(s)\$ where \$\\omega_i^I(s) \\geq 0\$ and \n\$\\sum_{i=1}^n\n\\omega_i^I(s) = 1\$.\n\\end{postulate}\nWe've used the term ``frequency function\" as a neutral alternative to\n``probability\" or ``propensity\" as there is no need to enter into\ninterpretational questions at this moment. Of course, the subensembles\nof measurements corresponding to the occurrences of distinct \$a_i\$ do not\noverlap.\n\\begin{postulate}[M2 -- State transformation] \\label{pos:m2}\nA state subject to a measurement undergoes a transformation to\nanother state representable by an element of \$\\cS\$. Let \$\\pi^I : \\cS\n\\rightarrow \\cS\$ be the map representing this transformation.\n\\end{postulate}\nWe are adopting here what in the quantum mechanical case we called the\n``incoherent mixture\" view of measurement.\n\\begin{postulate}[M3 -- Subensemble] \\label{pos:m3}\nGiven a state \$s\$ the transformed state \$\\pi^I s\$ consists of\nsubensembles corresponding to each particular outcome of the\nmeasurement. Each such subensemble is a fraction of the total ensemble\ngiven by the frequency function. Thus there are partial maps \$\\pi^I_i\$\nsuch that \$\\pi^I s = \\sum_{i=1}^n \\omega_i^I(s) \\pi^I_i s\$. The\nexpression \$\\pi^I_is\$ is considered to be defined only when\n\$\\omega_i^I(s) \\neq 0\$.\n\\end{postulate}\nThis assumption allows us now to also adopt the ``beam splitter\" view of\nmeasurements.\n\\begin{postulate}[M4 -- Ideality] \\label{pos:m4}\nMeasurements are ideal, in the sense that for any pure state \$p\$ and\nany instrument \$I\$ one has that \$\\pi^I_ip\$ is also pure, when\ndefined.\n\\end{postulate}\n\\begin{postulate}[M5 -- Objectivity] \\label{pos:m5}\nThe ensemble produced by a measurement is an objective mixtures of the\nsubensembles corresponding to the individual outcomes. That is,\n\$\\pi^I s =_o \\sum_{i=1}^n \\omega_i^I(s) \\pi^I_i s\$.\n\\end{postulate}\nIn particular, if one performs a measurement on a pure state, then by\nM4 and M5 one obtains an objective mixture of pure states.\n\nWe leave open as to what exactly is the criterion for objective\nmixtures. The precise nature of this criterion is not as\nrelevant as some of its desirable properties to which we shall draw\nattention in due time. We mention, for the sake of concreteness, a\npossible correlation-type criterion just as in quantum\nmechanics:\n\\begin{quote}(Correlation Criterion of Objectivity) The criterion\nfor an observer in the coordinate future to the measurement by an instrument \n\$I\$ on a state \$s\$ to maintain \$\\pi^I s =_o \\sum_{i=1}^n \\alpha_i s_i\$\nwith \$\\alpha_i \\neq 0\$ is the availability \nof dicotomic instruments \n\$J^{(j)}=(b^{(j)}_1, b^{(j)}_2)\$ such that \n\$\\omega_1^{J^{(j)}}(\\pi^I_i(s)) = \\delta_{ij}\$, and\n a strict correlation of the first outcome of\n \$J^{(j)}\$ with the \$j\$-th outcome of \$I\$, in which case\n \$\\alpha_i=\\omega^I_i(s)\$ and \$s_i=\\pi^I_i(s)\$. \n\\end{quote}\nThis criterion would suffice for what follows but others could probably\ndo just as well.\n\n\n\n\nOne consequence of assuming the existence of objective mixtures is that\nif \$s =_o \\int p \\, d\\mu(p)\$ then \$\\omega_i^I(s) = \\int \\omega_i^I(p) \\,\nd\\mu(p)\$ and so it's enough to know the frequency function only in pure\nstates, and we can assume the functions \$\\omega_i^I\$ are affine.\nThe same goes for the map \$\\pi^I\$.\n\n\nHow plausible are these assumptions? Taken together they express our\nability to individuate physical system by appropriate ``filtering\"\nthrough measurements and then to perform statistical experiments on\nsituations so created. This constitutes the basis of normal physical\nexperimental practice, so to negate this is to radically change our view\nof the statistical nature of physical phenomena and our experimental\naccess to them. Of course these assumptions are already idealized, and\none could argue with the details of each one individually, but some such\nset must be postulated to even begin a formulation of a statistical\nscience. Part of the above assumptions incorporate the notion of \nself-subsisting physical states, that is the \\ssa view. This is not a\nlogically necessary ingredient of a physical theory, ingrained as it may\nbe. All a physical theory must be able to do is\npredict the joint probabilities of events, and the mediation of these by\nphysical states under evolution is not a necessity. As was mentioned\nbefore, the consistent histories approach does not make use of such a\nnotion, and the preceding assumptions would have to be modified if one\nwere to axiomatize such a viewpoint.\n\n\\section{Space-time Structure of Measurements}\\label{sec:stm}\n\n\nThe relation between a physical theory \$(\\cL,\\cS)\$ and space-time\nstructure generally comes in through external considerations. Ordinary\nhilbert-space quantum mechanics admits both Galileian and Lorentz\ncovariance; such considerations only enter through unitary\nrepresentations of an appropriate group and have no expression in the\nfundamental formalism as such. In our context we cannot do much better\nwhile some basic mathematical questions have yet to be settled. We\nassume lorentzian space-time, and in relation to the physical theory we\nassume a series of postulates generalizing some of the usual external\nconnections already seen in hilbert-space theory.\n\n\\begin{postulate}[S1 -- Localization] \\label{pos:s1}\nGiven a bounded space-time region \$\\cO\$ then there is a\nsub-orthomodular-poset \$\\cL(\\cO) \\subset \\cL\$ corresponding to\npropositions testable in \$\\cO\$.\n\\end{postulate}\nThis assumption reflects the notion that experiments are essentially\nbounded in space-time. Usually one also feels that propositions\nreferring to unbounded regions should only be admitted as\nidealizations, that is, limits of local ones. Thus one could postulate\nthat the union of the subsets \$\\cL(\\cO)\$ is join-dense in \$\\cL\$ which\nis then viewed as a set of ``quasi-local\" propositions. Quantum field\ntheory teaches us that one should not assume about \$\\cL(\\cO)\$ properties\nthat one usually postulates about \$\\cL\$; thus while the latter is often\ntaken to be atomic and atomistic, this should not be the case for the\nlocal posets.\n\n\nWe say an instrument \$I = (a_1,\\dots,a_n)\$ {\\em belongs} to a region\n\$\\cO\$ if \$a_i \\in \\cL(\\cO)\$ for each \$i\$. For two regions \$\\cO_1, \n\\cO_2\$\nwe write \$\\cO_1 \\x \\cO_2\$ in case they are space-like separated, that\nis,\nevery point of one is space-like to every point of the other. For\n\$a_1,a_2 \\in \\cL\$ we write \$a_1 \\x a_2\$ in case \$a_1 \\in \\cL(\\cO_1)\$ and\n\$a_2\n\\in \\cL(\\cO_2)\$ for some regions \$\\cO_1 \\x \\cO_2\$. For two instruments\n\$I\$ and \$J\$ we write \$I \\x J\$ in case they belong to space-like\nseparated regions.\n\n\\begin{postulate}[S2 -- Locality] \\label{pos:s2}\nIf \$\\cO_1 \\x \\cO_2\$ then every element of \$\\cL(\\cO_1)\$ commutes with\nevery element of \$\\cL(\\cO_2)\$.\n\\end{postulate}\nIt is customary in the lattice-theoretic approach to equate\ncommutativity of propositions with their commensurability. It is\nlikewise customary to assume that space-like separated regions are\ncausally disjoint (locality or causality assumption). This then is\na major assumption relating space-time structure and the poset of\nproposition. We shall write \$a \\leftrightarrow b\$ whenever \$a\$ and \$b\$\ncommute. It is interesting to mention that in Mittelstaedt's\n\\cite{mitt1}\nscheme, commutativity of space-like separated propositions is necessary\nfor logical consistency.\n\nIf \$I = (a_1,\\dots,a_n)\$ and \$J = (b_1,\\dots,b_m)\$, are two instruments\nsuch that \$a_i \\leftrightarrow b_j\$ for all \$i\$ and \$j\$, in particular\nif \$I \\x J\$, then we can form a new instrument \$I \\wedge J = (a_i \\wedge\nb_j)_{i=1,\\dots,n;j=1,\\dots,m}\$\n\nNow the execution of an experiment consists of physical acts leading to\nphysical results. This in itself has nothing to do with our description\nof the experiment nor consequently with the adoption of any particular\nreference frame for space-time. The propositions being tested however do\nhave something to do with reference frames. If an observer finds at time\n\$t_0\$ that a proposition \$a\$ is true about a state \$s\$, then the truth\nof \$a\$ is aspatial, hence to be considered as such at all points of\nspace. If, as is customary, one considers that a proposition can {\\em\nbecome\\/} true at a time instant \$t_0\$ then it must become true\ninstantly and simultaneously at all space points. An observer in a\ndifferent frame has a different plane of simultaneity so his\npropositions become true in a different manner. This frame dependence\nof truth values and becoming-true is not necessarily in conflict with a\nframe-independent physics. The mutual consistency of the two however,\ngiven other requirements, can and does lead to constraints on physical\ntheories.\n\n\n\n\\begin{postulate}[S3 -- Measurement instant] \\label{pos:s3}\nIf an experiment corresponding to an instrument belonging to a region\n\$\\cO\$ has been executed, then every observer assigns a unique time\ninstant at which the experiment is considered to be realized and at\nwhich instant one of the propositions related to the instrument becomes\ntrue and the others false. The plane of simultaneity of this instant\nintersects \$\\cO\$.\n\\end{postulate}\n\nThis assumption seems to be generally, even if grudgingly, accepted. The\nexperimenter often doesn't have control over the instant in question\nwhich is somehow decided by the physical processes that the experiment\nunleashes, yet such an instant is generally identified. Even more often\nthe case is that the realization of the experiment is associated to some\nunique event (counter click, for instance) and the instant is just the\ntime coordinate of this event in the given frame. In this case different\nobservers can agree on the same event, which lies in\n\$\\cO\$. There are of course important apparent exceptions to the single\nevent viewpoint such as coincidence experiments, but even in this case\nsome maintain that the experiment is only over when all the information\nis gathered in some single recording device, such as a brain, in which\ncase one falls back on the single event hypothesis. We shall not however\nadhere to this viewpoint.\n\n\nConsider now two experiments executed in space-like separated regions.\nLet \$I = (a_1,\\dots,a_n)\$ and \$J = (b_1,\\dots,b_m)\$ be the corresponding\ninstruments. Suppose that an observer assigns the same time instant \$t_0\$\nto the realization of both experiments. Then, by the commensurability of\nthe two instruments, at \$t_0\$ the observer maintains not only that one of\nthe \$a_i\$ becomes true and that one of the \$b_j\$ becomes true but also\nthat one of the \$a_i \\wedge b_j\$ becomes true. As far as the observer is\nconcerned, the two separate realizations of \$I\$ and \$J\$ is\nindistinguishable from a single realization of \$I \\wedge J\$. This\nindistinguishability is our next assumption.\n\n\n\\begin{postulate}[S4 -- Confluence of simultaneous measurements] \n\\label{pos:s4}\nIf an observer assigns the same time instant to the realization of two\nspace-like separated experiments, these can be treated equivalently as\nthe realization of a single experiment whose outcomes are the\nconjunctions of the outcomes of the separate experiments.\n\\end{postulate}\nAnother observer in a different frame would generally see a time\ninterval between the two realizations, either \$I\$ first, followed by\n\$J\$, or vice-versa. In this case he must consider the two experiments\nas consecutive and for him the becoming-true of propositions related\nto the two experiments do not occur simultaneously. If physics is\nframe-independent then the two observers must agree about ultimate\nphysical facts. For this to be the case, the pair \$(\\cL, \\cS)\$ must\nsatisfy certain constraints.\n\nWhat we still lack are assumptions that express the supposed frame\nindependence of ultimate physical facts. Just as in hilbert-space\ntheory, we introduce this via group action.\n\n\\begin{postulate}[C -- Lorentz covariance] \\label{pos:c}\nLet \$\\cG\$ be the Poincar\\'e Group. There are actions \$(g,a) \\mapsto\n\\lambda_g a\$ of \$\\cG\$ on \$\\cL\$ and \$(g, s) \\mapsto \\sigma_g s\$ on\n\$\\cS\$ such that:\n\\begin{enumerate}\n\\item For all \$g \\in \\cG\$, \$\\lambda_g\$ is an orthomodular-poset \nisomorphism.\n\\item \$\\lambda_g \\cL(\\cO) \\subset \\cL(g(\\cO))\$\n\\item For all \$g \\in \\cG\$, the action \$s \\mapsto \\sigma_g s\$ is\naffine and \$\\sigma_g (\\cP ) \\subset \\cP \$.\n\\item Given an instrument \$I = (a_1,\\dots,a_n)\$, let \$\\lambda_g I =\n (\\lambda_g a_1,\\dots,\\lambda_g a_n)\$. One then has:\n\\begin{eqnarray}\n\\omega_i^I(s) &=& \\omega_i^{\\lambda_g I}(\\sigma_g s) \\\\\n\\pi^{\\lambda_g I}_i \\sigma_g p &=& \\sigma_g \\pi^I_i p\n\\end{eqnarray}\n\\end{enumerate}\n\\end{postulate}\nFor simplicity's sake we shall write \$g\\cdot a\$ and \$g\\cdot s\$ \ninstead of\n\$\\lambda_g a\$ and \$\\sigma_g s\$.\n\nMost of this assumption is a fairly straightforward rendition of rather\nstandard covariance conditions on physical theories. There are two ways\nof understanding the group action. The first, or ``passive\" view, is\nthat if an observer describes an experimental procedure as that of\nexecuting \$I\$ upon state \$s\$, then another observer whose frame is\nobtained from that of the first one by action of \$g\$ will describe the\nsame procedure as that of executing \$g^{-1}\\cdot I\$ \nupon \$g^{-1}\\cdot s\$. The\n``active\" view states that there is another procedure (whose\nexecution has a clear operational relation to the execution of the first\none) described by \$g\\cdot I\$ and \$g\\cdot s\$ \nby the first observer and whose\ndescription by the second observer is by \$I\$ and \$s\$. The equivalence of\nthe two views is the essence of relativistic theories.\n\nOne possible strengthening of C3 would be to assert that objective\nmixtures map to objective mixtures, that is, if \$s =_o \\int p \\,\nd\\mu(p)\$, then \$g\\cdot s =_o \\int g\\cdot p \\, d\\mu(p)\$. This is\n{\\it prima facie \\/} a natural assumption, and we shall formulate \na version of it. However \nwe shall also need a more subtle manifestation of covariance in relation\nto objective mixtures. Since our only assumption concerning objective\nmixtures is that they come about through measurements and since the\ncorresponding ``collapse\" is frame dependent, covariance\nof\nobjective mixtures is a rather more involved concept since whatever\ncriterion for objective mixtures one may adopt, one should not think of\nit as a local criterion as the quantum field theory case teaches us. One\nmust imagine that such a criterion would utilize something like a\ntime-slice region, idealized say to a space-like hyperplane. In some\nregions two observers would disagree even as to which (if any) of the\nrelevant experiments have been carried out and in which order, leading\nto objective mixtures not related by the action of the Poincar\\'e group.\nObjective mixtures as\nobserver related descriptions would have varying relation to physical\nfacts. Thus the simple relation for objective mixtures given in the\nbeginning of this paragraph can be maintained as referring to the\nsate in question only by a pair of observers that are both in the future\nlight cone of all the relevant measurement processes that enter into the\nobjective mixture criterion. \n\n\\begin{postulate}[O~I -- Covariance of objectivity I]\\label{pos:oi}\nIf \$s =_o \\int p \\,\nd\\mu(p)\$, then \$g\\cdot s =_o \\int g\\cdot p \\, d\\mu(p)\$.\nThe right hand side of these equations refers to the state in question\nas seen by two observers related by the element \$g\$ of the\nPoincar\\'e group provided both are in the time-like future of all events\ninvolved in the objective mixture criterion. \n\\end{postulate}\n\nLet now \$I =(a_1,\\dots,a_n)\$ and \$J=(b_1,\\dots,b_m)\$ be two instruments\nand assume \$I \\x J\$. Consider now an observer, call him Observer 1,\nwho assigns the same time instant to the realization of both experiments\nupon a pure state \$p\$ (note well: we do not mean that each experiment\nacts on its own ``copy\" of \$p\$ but each on the {\\em same \\/} spatially\nextended state \$p\$). According to S4 we can view this as a realization\nof \$I \\wedge J\$ and so by the measurement assumptions the state is\ntransformed into \$s =_o \\sum_{i,j}\\omega_{i,j}^{I \\wedge J}(p)\\pi^{I\n\\wedge J}_{i,j} p\$. Another observer, call her Observer 2, will\ndescribe the situation as the realization of \$g\\cdot I\$ upon \$g\\cdot\np\$ followed by a realization of \$g\\cdot J\$ upon the resulting state\nof the first measurement. The first experiment produces a state \$s_1\n=_o \\sum_i \\omega_i^{g\\cdot I}(g\\cdot p)\\pi^{g\\cdot I}_i g\\cdot p\$ and\nthis is transformed by the second experiment into (note the lack of the\nobjective equality sign) \$s_2 = \\sum_{j,i}\\omega_j^{g\\cdot\nJ}(\\pi^{g\\cdot I}_i g\\cdot p) \\omega_i^{g\\cdot I}(g\\cdot p) \\pi^{g\\cdot\nJ}_j\\pi^{g\\cdot I}_i g\\cdot p\$. Our next assumption is that this is in\nfact an objective mixture:\n\n\n\\begin{postulate}[O~II -- Covariance of objectivity II]\\label{pos:oii}\\ \\ \n\\begin{description}\n\\item {\\rm a)} \$s_2 =_o \\sum_{j,i}\\omega_j^{g\\cdot J}(\\pi^{g\\cdot I}_i\ng\\cdot p) \\omega_i^{g\\cdot I}( g\\cdot p) \\pi^{g\\cdot J}_j\\pi^{g\\cdot\nI}_i g\\cdot p\$.\n\\item{\\rm b)} \$s_2 = g\\cdot s\$.\n\\end{description}\n\\end{postulate}\n\nThis is a subtle point. One has no grounds on the basis of previous\nassumptions or analyses to claim that the expression for \$s_2\$ is an\nobjective mixture and in fact one needs an additional argument to adopt\nthis hypothesis (it is true that M5 implies that \$s_2\$ is an objective\nmixture of certain {\\em non-extreme} states, but this is not what we are\nsaying here, and M5 does not imply O~II). One saw in the quantum mechanical\ndiscussion that two successive measurements with commuting instruments\nlead to an objective mixture of the type expressed by \$s_2\$ according\nto the correlation criterion. One could be tempted to use this as\nguideline and adopt a similar hypothesis in the quantum logic case,\nseeing that space-like separated instruments commute. This however has\nno more compelling reason to be than the covering law itself and we do\nnot view O~II as a statement about commutativity but about covariance. As\nhas been pointed out, observers in different frames may even sustain\nobjective mixture descriptions not related by the action of an element\nof the Poincar\\'e group. Such disagreements as to the {\\em\ndescription\\/} of the state, objective as they may be, should of course\nnot be detectable by local observations just as in the quantum field\ntheory case, and this leads to definite constraints on the theory, but\nthese as we shall see will be automatically satisfied. Let us however\nconcentrate on the set of points that are future time-like to all events\ninvolved in the measurement process. As we found out in the quantum\nmechanical case, in this region, the passage criterion for objective\nmixtures can be maintained and actually carried out as we can have\nknowledge of the measurement results. This means that we can actually\nuse the ``beam-splitter\" view of measurements and consider conditioning\nfurther measurements to the outcomes of the past measurements \$I\$ and\n\$J\$. Now if we condition to outcome \$a_i\$ and \$b_j\$, then these\noutcomes are ultimate facts whether they are simultaneous or not. In the\nfuture region we expect, by the very notion of Lorentz covariance, that\nthe state defined by this ``beam\" should behave in a purely covariant\nfashion, that is, obeying C, when tested by local observables. As we\nmove further and further into the future, local observables in this\nregion can have increasingly greater space-like extent and\nasymptotically can approach time-slice observables. As these are\npresumed to be involved in the objective mixture criterion, these are\nthen also subject to C. In particular, if Observer 1 finds the state to\nbe \$s_{ij}\$ and if Observer 2 is related to Observer 1 by and element\n\$g\$ of the Poincar\\'e group, she should find the state to be\n\$g^{-1}\\cdot s_{ij}\$. Thus we can argue that the given representation for\n\$s_2\$ should be an objective mixture purely on the grounds of Lorentz\ncovariance. We should also have \$s_2 = g\\cdot s\$. This type of\ncovariance is if course somewhat different from the one expressed in C\nas that one makes no reference to measurements. Observer 1 has a\ndifferent geometrical relation to the measurements than Observer 2 as in\nhis situation there is only one plane of simultaneity related to the\nmeasurements, whereas for her there are two. The geometrical structures\nare not group transforms of each other. The frame-dependent description\nof the measurement process is in contrast with the conventional\ncovariance behavior expressed by C. It is thus not surprising that a\nfull expression of covariance should include statements concerning the\nmeasurement process, and O~II is such a statement. Another way to motivate\nthis assumptions is to argue that any measurement process, occurring as\nit does in an interaction of a system with a macroscopic apparatus, is\nalready likely to involve space-like separated events. Thus any\nphysically reasonable system obeying C already must presuppose something\nlike O~II.\n\n\n\nFrom \$s_2 = g\\cdot s\$ and Postulates \\ref{pos:c} and \\ref{pos:oi}, \none has the following equality of\nobjective mixtures: \n\\[\n\\sum_{i,j}\\omega_{i,j}^{I \\wedge J}(p)\\pi^{I\n\\wedge J}_{i,j} p = \\sum_j \\sum_i \\omega_j^J(\\pi^{I}_i p) \\omega_i^I(p)\n\\pi^{J}_j\\pi^{I}_i p,\n\\] \nand thus, the final pure components and the\ncorresponding fractions must coincide. Assuming that the set of pure\nstates is sufficiently large to distinguish the functions \$\\omega\$ and\n\$\\pi\$ one comes to the following consequence:\n\n\n\\begin{theorem}\nGiven that \$I \\x J\$ where \$I\n= (a_1,\\dots,a_n)\$ and \$J=(b_1,\\dots,b_m)\$ then:\n\\begin{eqnarray}\\label{eq:one}\n\\forall p \\in \\cP,\\quad \\omega_{i,j}^{I \\wedge J}(p) &=&\n\\omega_j^J(\\pi^I_i p)\\omega_i^I(p) \\\\ \\label{eq:two}\n\\pi^{I \\wedge J}_{i,j} &=& \\pi^J_j \\pi^I_i.\n\\end{eqnarray}\n\\end{theorem}\n\nThese then are the constraints that any measurement theory of the type\ndescribed here must satisfy if it is to be able to describe\nLorentz-covariant physics.\n\nOne should note that in the two equations of the theorem, the left-hand\nside is the same if \$I\$ and \$J\$ are interchanged. This leads\nimmediately therefore to the following relations:\n\\begin{eqnarray} \\label{eq:three}\n\\forall p \\in \\cP,\\quad \\omega_j^J(\\pi^I_i p)\\omega_i^I(p)\n&= &\n\\omega_i^I(\\pi^J_j p)\\omega_j^J(p),\\\\ \\label{eq:four}\n \\pi^J_j \\pi^I_i &=& \\pi^I_i \\pi^J_j.\n\\end{eqnarray}\nThese relations of course could have been independently derived from an\nargument similar to ours if we considered two frames in which the\ntemporal order of experiments \$I\$ and \$J\$ are opposite, without\nconsidering a frame in which they are simultaneous.\n\nNow we come to the covering law. What is remarkable is that in various\naxiomatic schemes that have been proposed to replace the covering\nlaw with more ``physical\" assumptions, these coincide with or are\nreadily deduced from one or both of equations (\\ref{eq:one}--\\ref{eq:two}) \nor even the weaker\nresults (\\ref{eq:three}--\\ref{eq:four}), \nwith the difference that they are postulated for \$I\n\\leftrightarrow J\$ and not only for \$I \\x J\$. We have been somewhat more\ngeneral in this exposition than what is normally assumed, as usually\nthe \$\\omega_i^I(p)\$ and \$\\pi^I_i p\$ are postulated not to depend\non \$I\$ but only on the proposition \$a_i\$ in question. This is the\nabsence of certain type of contextuality.\n\n\nBearing this in mind we thus see that the ``Compatibility Postulate\" in\nGuz \\cite{guz1} is a direct consequence of (\\ref{eq:two}) stated for\ncommuting pairs; axiom F4 in Guz \\cite{guz2} follows from (\\ref{eq:one}), \nagain stated for commuting pairs, as can be found in\nSvetlichny \\cite{svet3}. Pool's \\cite{pool1,pool2} derivation of the \nsemimodularity of\nquantum logic, necessary for a hilbert-space interpretation, follows a\ndifferent chain of reasoning but is clearly related to our results. The\naxioms of the first paper lead to the equivalence of\ncommutativity of propositions with a form of equation (\\ref{eq:four}) \nwhile in the\nsecond paper semimodularity is derived from a further\nassumption which in our scheme is M4. Nicolas Gisin (private\ncommunications) has likewise derived the covering law from a form of\nequation (\\ref{eq:one}) stated for propensities and has \nindependently postulated a\npossible relation between the covering law and space-time structure.\n\nLet us now consider the constraints that express the fact that local\nobservables must not distinguish between two objective mixtures not\nrelated by an element of the group action but that are legitimately\nmaintained by two observers who disagree as to which experiments have\nalready been carried out. Suppose that in some region \$\\cO\$ an\nexperiment is performed which the first observer, space-like to \$\\cO\$,\nsays has already happened and consisted of an observation by an\ninstrument \$I\$ upon a pure state \$p\$. He would say the new state is\n\$\\sum_i\\omega^I_i(p)\\pi^I_ip\$. Another observer flying past him would\nclaim that since the experiment has not been performed in his frame, the\nstate is \$g\\cdot p\$ where \$g\$ is the element of the Lorentz group\nthat relates the two observers. Suppose now that a local experiment is\nperformed in a region \$\\cO'\$ space-like to \$\\cO\$. The first observer\ndescribes this by an instrument \$J\$ and the frequency of the \$j\$-th\nresult is thus given by \$\\sum_i\\omega^I_i(p)\\omega^J_j(\\pi^I_ip)\$.\nThis by (3) is \$\\sum_i\\omega^I_i(\\pi^J_jp)\\omega^J_j(p) =\n\\omega^J_j(p)\$. The second observer will assign frequency\n\$\\omega^{g\\cdot J}_j(g\\cdot p)\$ to the same result but this by C is\nalso \$\\omega^J_j(p)\$ and so the two observers will agree as to the\nfrequencies of outcomes of local experiments and the two descriptions\nare consistent just as in the quantum mechanical case. Finally we\nmention that (\\ref{eq:one}) also implies that no superluminal signal can\nbe sent through long-range correlation by mere change of a local\nmeasuring device. The frequency assigned to the \$i\$-th result of\ninstrument \$I\$, under lack of knowledge of the outcome of instrument\n\$J\$, is by the right-hand side of (\\ref{eq:one}) just\n\$\\sum_j\\omega_j^J(\\pi^I_i p)\\omega_i^I(p) = \\omega^I_i(p)\$ which is\nindependent of instrument \$J\$ just as in the quantum-mechanical case\nand so no signal of this type is possible.\n\n\n\n\n\n\\section{Bridging the Light Cone}\n\nOne sees that in some of the existing schemes the crucial axiom that\nleads to the covering law can be substituted by Lorentz covariance and\nan appeal to a principle generalizing necessary conditions on space-like separated\npropositions to merely commuting ones. \nLet us first examine the purely {\\em logical\\/}\n nature of such a principle which\ncan be stated as follows:\n\\begin{quote}(E -- Equivalence of commutativities) \n{\\em Let \$Q(a,b)\$ be two-place predicate concerning \$\\cL\$ then, \\[(a \\x b\n \\Rightarrow Q(a,b)) \\Rightarrow (a \\leftrightarrow b \\Rightarrow\n Q(a,b))\\]}\n\\end{quote}\n\n\nAssuming the metatheoretic principle E, the covering law can be\ndeduced from Lorentz covariance and axioms of generally less\ncontroversial nature. The covering law would thus be compelling if one\ncan turn E compelling.\nOne has to be a bit careful though about this principle, one should \nprobably not\nmaintain it for all possible \$Q\$. If for \$Q(a,b)\$ \nwe take \$a \\x b\$\nthen we deduce \$(a \\leftrightarrow b) \\Leftrightarrow (a \\x b)\$ which is\nlikely too strong. It may be that all commutativity in physics can be\nreduced to space-like commutativity, but one should see this in detail\nand not through a metatheoretic principle. \n Is there reason to believe E? One way that E could be\ntrue is if there is a symmetry group on \$\\cL\$ by which if \$a\$ and \$b\$\nare local and \$a\n\\leftrightarrow b\$ then there is an element \$\\phi\$ of this group such\nthat \$\\phi(a) \\x \\phi(b)\$. This would establish E for group invariant\npredicates and a join-dense set of propositions, which would probably be \nsufficient. In ordinary quantum\nmechanics the full unitary group of hilbert space is apparently such a\nsymmetry group. There seems to be no compelling reason however to\nbelieve in such a group. Given a complete lack of any\nmathematical development of lattice-theoretic approaches ``localized\"\ninto space-time regions and with Poincar\\'e group action, ``local\nrelativistic quantum logic\" in other words, it's even hard to say how\nrestrictive E is. From the physical side one lacks any real\nunderstanding of commutativity of propositions except for space-like\nseparated ones where causality arguments seem compelling. Thus the spin\nand orbital angular momenta of a particle commute, but we see this in a\npurely formal way: the corresponding operators act on different tensor\nfactors. We cannot say we have a true physical understanding of this.\nWithout this understanding, the covering law is still not compelled.\n\nTo better understand what can be physically \ninvolved in assumptions such as E, it is necessary to examine more\nclosely the meaning of commutativity, which formalizes the notion of\ncompatibility. \nFor the rest of this paragraph we assume a more primitive notion of an\ninstrument than that used in the previous section. \nAn instrument \$I\$ corresponds to some procedure or physical construct\nwhich leads to a finite set of outcomes \$x_1,\\dots,x_n\$ but we do\nnot assume that these are necessarily represented by some algebraic\nelements. We can still talk about states \$p\$, frequencies\n\$\\omega_i^I(p)\$ and projections \$\\pi^I_i p\$ as before.\nOperationally, we say that an instruments \$I\$ \nwith outcome set \$x_1,\\dots,x_n\$ and \nanother one \$J\$ with outcome set \$y_1,\\dots,y_m\$\nare compatible if there is a compound instrument \$K\$ with outcome\nset \$(x_i,y_j),\\, i=1,\\dots,n,\\, j=1,\\dots,m\$ whose outcome\nfrequencies in any state has as marginals the outcome frequencies of\n\$I\$ and \$J\$ in the same state. The actual physical construction of\n\$K\$ may have in general no clear relation to the physical\nconstructions\nof \$I\$ and \$J\$, but in certain cases it is customary to consider\n\$K\$ as just \$I\$ and \$J\$ physically coexisting and consider the\noutcomes of \$K\$ as coincidences of outcomes of \$I\$ and \$J\$. This\nis the case when \$I\\x J\$ and also when \$I\$ and \$J\$ belong\nto limited space-time regions in which all points of one are future\ntime-like to all points of the other. In ordinary quantum mechanics,\nthe compound observation of compatible observations that succeed\ntemporally are taken to be just these successive individual\nmeasurements. Two instruments that satisfy (\\ref{eq:one}) for both\norders of \$I\$ and \$J\$ on the right hand side are compatible for then\nthe compound instrument \$K\$ is \$I\\wedge J\$. \nIndeed one has for the two marginals:\n\$\\sum_j \\omega_{i,j}^{I \\wedge J}(p) = \n\\sum_j\\omega_j^J(\\pi^I_i p)\\omega_i^I(p) = \\omega_i^I(p)\$ and \n\$\\sum_i \\omega_{i,j}^{I \\wedge J}(p) = \n\\sum_i\\omega_i^I(\\pi^J_j p)\\omega_j^J(p) = \\omega_j^J(p)\$.\nFor space-like separated instruments one can argue for both orders on\nthe right-hand side of (\\ref{eq:one}) \nfrom covariance arguments as there are frames in\nwhich the measurements occur in either temporal order. In the time-like\ncase one cannot use this argument. Let us consider the case in which\n\$J\$ belongs to a space-time region all points of which are\nfuture time-like in relation to each point of the region to which \$I\$\nbelongs. One can then consider the instrument \$J\\circ I\$ \nconsisting of the\nsuccessive measurements by \$I\$ and then by \$J\$. By our postulates\none has immediately \$\\omega_{i,j}^{J\\circ I}(p) = \n\\omega_j^J(\\pi^I_i p)\\omega_i^I(p)\$ and so (\\ref{eq:one}) holds for one\norder by definition even if \$J\$ and \$I\$ are not compatible. Now if\n\$I\$ and \$J\$ {\\em are\\/} compatible, then conventional wisdom deems\n\$J\\circ I\$ to be the compound instrument \$K\$, and so \nhalf of (\\ref{eq:one}) is trivially true. The difficulty then is in\narguing for the other half, that is, \$\\omega_{i,j}^{J\\circ I}(p) = \n\\omega_i^I(\\pi^J_j p)\\omega_j^J(p)\$. The operational verification of\nthis cannot now be simply done by successive measurements as the right\nhand side now has an \$I\$ measurement on a state conditioned by a\nfuture time-like \$J\$ measurement. \nThe frequencies \$\\omega_j^J(p)\$ can be\nestablished just from \$J\$ measurements, but then to verify the \nfrequencies \$\\omega_i^I(\\pi^J_j p)\$ one needs to be able to create\nstates \$\\pi^J_j p\$ prior to the execution of \$I\$, that is one needs\nan instrument \$L\$ with \$m\$ outcomes, \nbelongs to a region in the temporal past of\n\$I\$, whether time-like or space-like, and a state \$p'\$ such that \n\$\\pi^L_jp'=\\pi^J_j p\$. One can then verify (\\ref{eq:one}). Thus \nthe operational verification that this\nequation holds for generally commutative instruments involves an\nassumption concerning the producibility of given states by actions in\ndistinct and possibly widely separated regions\nand so transcends purely covariant considerations. It seems that to\nbe able to perform the generalization mentioned in the beginning of this\nsection one needs a {\\em physical\\/} assumption going beyond covariance.\n\nStandard relativistic quantum theory provides a clue by the presence of\nlong-range correlated states, that is, states of the EPR type. As a\nsimplified version of this situation suppose you want to study\nright-hand circularly polarized photons. One way is to simply put an\nappropriate filter in front of a light source and those photons that get\nthrough are of the right kind and so can be observed at will. Another\nequivalent way is to set up an EPR-type arrangement that creates singlet\ntwo-photon states with the individual photons flying off in opposite\ndirections. Put now the same filter on the {\\em distant\\/} arm of the\nEPR apparatus and {\\em nothing\\/} on the near arm. Observe at will. Half\nof the photons observed are right-hand circularly polarized and half are\nin the orthogonal left-hand circularly polarized state, and as the\nmeasurements are done, there is no way of knowing which is which. If all\none wants however is analysis of experimental outcomes, this is no\nproblem, just wait enough time that the results (passage through the\nfilter or not) at the distant arm of each photon pair are available\n(typical correlation experiment situation) and simply throw out all the\nexperimental data for the instances where the distant photon did not\npass through the filter. This provides you with data now of just the\nright-hand circularly polarized photons at the near arm. The fact that\nthese two experimental procedures are equivalent is a feature of\nordinary quantum mechanics and depends on the existence of a particular\nentangled state, the two-photon singlet. We now show that this\nsimplified situation is quite general. Assume we are working with a\nlocal relativistic quantum field theory of the Haag type in a hilbert\nspace \$\\cH\$. Let \$I=(P_1,\\dots,P_n)\$ and \$J=(Q_1,\\dots,Q_m)\$ be\ntwo compatible instruments belonging to two limited space-time regions,\n\$\\cO_I\$ and \$\\cO_J\$ respectively, which need not be space-like\nseparated. Let \$\\Psi\$ be a pure state. Consider now an element \$g\$\nof the Poincar\\'e group such that \$\\cO_J'=g(\\cO_J)\$ is space-like\nseparated from both the original regions. Let \$U(g)\$ be the unitary\nsymmetry operator associated to \$g\$, and let\n\$J'=(Q_1',\\dots,Q_m')\$, where \$Q_j'=U(g)Q_jU(g)^*\$, be the\ntransformed instrument. One can choose \$\\cO_J'\$ in such a \nmanner \\cite{haag, schroer} that \$\\cH\$ decomposes into a tensor product\n\$\\cH=\\cH_1\\otimes\\cH_2\$ with \$P_i = \\tilde P_i\\otimes I\$, \$Q_j =\n\\tilde Q_j\\otimes I\$, and \$Q_j'=I\\otimes \\tilde Q_j'\$. The projectors\nof all three instruments commute among themselves. Let \$\\Lambda\$ be\nthe set of triples \$ijk\$ such that \$P_iQ_jQ_k'\\cH\\neq \\{0\\}\$, and\nfor \$ijk\\in \\Lambda\$ let \$e_{ijk\\alpha},\\,\\alpha\\in A(ijk)\$ be an\northonormal basis for \$P_iQ_jQ_k'\\cH\$. One has \$\\Psi =\n\\sum_{ijk\\in \\Lambda}\\sum_{\\alpha\\in A(ijk)}\n\\psi_{ijk\\alpha}e_{ijk\\alpha}\$. Now for fixed \$ij\$ one can find\nan index \$p\$ such that \$ipj\\in \\Lambda\$, for otherwise one would have\n\$\\sum_pP_iQ_pQ_j'=P_iQ_j'=0\$ which is impossible given the tensor\nproduct decomposition. Choose \$\\beta\\in A(ipj)\$, set \n\\[\\psi_{ipj\\beta}'=\\sqrt{\\sum_{\\{k\\,\|\\,ijk\\in \\Lambda\\}}\n\\sum_{\\alpha\\in A(ijk)}\|\\psi_{ijk\\alpha}\|^2}\\]\nand set all other components \$\\psi_{ikj\\alpha}'=0\$ for \$(k,\\alpha)\n\\neq (p,\\beta)\$. Obviously \n\$\\Psi' =\n\\sum_{ijk\\in \\Lambda}\\sum_{\\alpha\\in A(ijk)}\n\\psi_{ijk\\alpha}'e_{ijk\\alpha}\$ is another normalized state vector. One\neasily verifies \$\|\|P_iQ_k'\\Psi'\|\|^2=\|\|P_iQ_k\\Psi\|\|^2\$.\n In\nstandard quantum mechanics one has \$\\omega_i^I(\\Psi)=\|\|P_i\\Psi\|\|^2\$\nand \$\\pi^I_i\\Psi=P_i\\Psi/\|\|P_i\\Psi\|\|\$, whenever\n\$\\omega_i^I(\\Psi)\\neq0\$. \nWe have from our construction that \$\\omega_{i,j}^{I\n\\wedge J'}(\\Psi')=\\omega_{i,j}^{I \\wedge J}(\\Psi)\$,\n\$\\omega_j^{J'}(\\pi^I_i \\Psi')=\\omega_j^J(\\pi^I_i \\Psi)\$ and\n\$\\omega_i^I(\\Psi') =\\omega_i^I(\\Psi)\$. So the validity of\n(\\ref{eq:one}) for the pair of instruments \$(I,J)\$ can be deduced\nfrom the validity for the pair \$(I,J')\$ of {\\em space-like\nseparated\\/} instruments {\\em due to the existence of \nthe state \$\\Psi'\$\\/}.\nNote that no distinction is made between the two instruments so that the\nright-hand side of (\\ref{eq:one}) holds for any order. \nThe validity of (\\ref{eq:one}) for space-like separated instruments\nfollows basically from Lorentz covariance and other assumptions of this\npaper and so is not peculiar to standard quantum theory. The existence\nof \$\\Psi'\$ has to be viewed however as an additional physical\nassumption, that happens to be true in standard quantum theory and is in\nsome sense also characteristic of it. \nThis state is in general one that has long-range correlations of\nthe EPR-type. We can now envisage a more physically plausible version of \nprinciple E:\n\\begin{postulate}[EP - Equivalence of conditioning] For any pair of\ninstruments \$(I,J)\$ with \$I\\leftrightarrow J\$ and pure state \$p\$ \nthere is an instrument \$J'\$ such that \$J' \\x I\$, and a pure state\n\$p'\$ such that the joint experiment \$(I,J)\$ on the state \$p\$ is\nequivalent to the joint experiment \$(I,J')\$ on \$p'\$. That is, \n\$\\omega_{i,j}^{I\\wedge J'}(p')=\\omega_{i,j}^{I \\wedge J}(p)\$, \n\$\\omega_j^{J'}(\\pi^I_i p')=\\omega_j^J(\\pi^I_i p)\$ and \n\$\\omega_i^I(p') =\\omega_i^I(p)\$\n\\end{postulate}\n\nWe call this equivalence of conditioning since it allows one to prepare\nstates conditioned to outcomes at space-like separations as was the case\nfor the simplified photon polarization experiment. With this additional postulate, as\nwas pointed out before, one can now deduce the covering law\nin some of the existing\naxiomatization schemes. \n\n\n\\section{Conclusions}\n\nOne of the most intriguing features of quantum mechanics is its\nuniversality. All phenomena, to the extent that their quantum behavior\ncan be exhibited experimentally, are subject to the same general\nformalism. The postulated connection of quantum mechanics to space-time\nstructure makes this understandable. The measurements to which the above\ndiscussion refer could be {\\em any} measurements. To the extend that any\nmeasurement takes place in space-time, it must exhibit universal quantum\nbehavior. The universality of quantum mechanics is a reflection of the\nuniversality of space-time as the arena for our experiments.\n\nTo be able to deduce such universality one however has to be able to\nmake some assumption such as EP compelling. Now why should EP be\ncompelling? It is unlikely that one can find an argument on purely\nformal grounds or by appeal to ``reasonableness\" of any kind just as\nsuch appeals are ultimately unconvincing in all the axiomatic approaches\nto quantum mechanics. Space-like and time-like situations are in\nlogically distinct domains and any relation between them must come from\nsome realm in which this distinction is weakened. The only existing\nconsiderations of this sort come from what is loosely known as ``quantum\ngravity\". In fact, at this point one perceives a fundamental difficulty\nwith the whole argument of this paper. One has started with a definitely\nclassical view of space-time and traced out a route which leads to a\nuniversal mechanics. But now space-time phenomena themselves must, by\nuniversality, be subject to the same mechanics, which distorts the\noriginal starting point. Space-time itself must be quantum mechanical.\nThere is a self-consistency question. The constraints that space-time\nstructure places on mechanics must govern the structure of space-time\nitself. Unfortunately ``quantum-space-time quantum logic\" is just a\nglimmer of an idea at this moment, more remote than the ``relativistic\nquantum logic\" that we've embarked upon. In any case, in all present-day\napproaches to quantum gravity, the rigid structure of the light cone\ndisappears and the usual notions of space-like and time-like are\nemergent and not fundamental. In such a context it is perfectly\nunderstandable that relations between space-like and time-like\nsituations arise out of a more basic theory in which such a\ndistinction is not fundamental. \n \nOn a more technical side, we note that the above argument utilizes more\nthe lorentzian causal structure of space-time and the relativity of\nsimultaneity than the exact details of Lorentz covariance. Thus one can\nexpect that a similar result can be obtained for curved space-time as\nwell. Strict Lorentz covariance should therefore be viewed as a\nsimplifying assumption for these preliminary studies. One also has the\nawkwardness of deriving a global feature (covering law) through local\nconsiderations. One knows from algebraic quantum field theory that local\nvon Neumann algebras of observables have a unique form (Haag \\cite{haag}), \nthey\nare all type \$\\hbox{III}_1\$ hyperfinite factors for a causal diamond\n\$\\cO\$ (intersection of a forward and a backward light cone). It would be\nmore in keeping with the local approach to try to deduce that \$\\cL(\\cO)\$\nis isomorphic to the projection lattice of a type \$\\hbox{III}_1\$\nhyperfinite factor and then only secondarily argue for the covering law\nthrough global considerations. This of course requires a\nlattice-theoretic characterization of such factors, which to our\nknowledge is not available.\n\n\n\n\n\n\n\n\n\\section{Acknowledgment}\n\nThe author thanks professor Nicolas Gisin for helpful correspondence.\nSpecial thanks go to the Mathematics Department of Rutgers University\nfor its hospitality during the author's stay there where part of this\nwork was done. This research was financially supported by the Secretaria\nde Ci\\^encia e Tecnologia (SCT) and the Conselho Nacional de\nDesenvolvimento Cient\\'\\i fico e Tecnol\\'ogico (CNPq), both agencies of\nthe Brazilian government.\n\n%\\vfill\n%\\eject\n\n\n\n\n\n\\begin{thebibliography}{xx}\n\n\\bibitem{piron}Piron,~C. (1976) ``Foundations of Quantum Physics\",~W.~A. \nBenjamin,\nInc., London.\n\n\\bibitem{svet3}Svetlichny,~G. (1998) {\\sl Foundations of Physics}, {\\bf 28}, \n131; quant-ph/9511002.\n\n\\bibitem{mitt1}Mittelstaedt,~P. (1983a) {\\em International Journal of \nTheoretical\nPhysics\\/} {\\bf 22}, 293.\n\n\\bibitem{mitt2}Mittelstaedt,~P. (1983b) {\\em Proceedings of the International \nSymposium on the Foundations of \nQuantum\nMechanics\\/} Tokyo, pp. 251-255\n\n\\bibitem{mitt3}Mittelstaedt,~P. and Stachow~E.~W. (1983) {\\em International \nJournal of\nTheoretical Physics\\/} {\\bf 22}, 517.\n\n\\bibitem{neumann}Neumann,~N. and Werner,~R. (1983) {\\em International Journal \nof\nTheoretical Physics\\/} {\\bf 22}, 781.\n\n\\bibitem{ludwig}Ludwig,~G. (1983) ``Foundations of Quantum Mechanics\", \nSpringer, New York.\n\n\\bibitem{haag}Haag,~R. (1992) ``Local Quantum Physics\", Springer Verlag, \nBerlin.\n\n\\bibitem{mugur1}Mugur-Sch\\\"acter,~M. (1991) {\\em Foundations of Physics\\/} \n{\\bf 21}, 1387.\n\n\\bibitem{mugur2}Mugur-Sch\\\"acter,~M. (1992) {\\em Foundations of Physics\\/} \n{\\bf 22}, 235.\n\n\\bibitem{hartle} Hartle,~J.~B., ``Spacetime Quantum Mechanics and the\nQuantum Mechanics of\nSpacetime\", in\n{\\em 1992 Les Houches Ecole d'\\'et\\'e, Gravitation et\nQuantifications\\/}\n\n\\bibitem{omnes} Omn\\'es,~R., {\\em The Interpretation of Quantum Mechanics},\nPrinceton University Press, (1994)\n\n\\bibitem{bialo}Svetlichny,~G. (1999) ``Space-time Structure and Quantum\nMechanics\" to appear in the proceedings of the XXX\${}^{\\it th}\$\nWorkshop on Geometric Methods in Physics, {\\em Coherent States,\nQuantization and Gravity\\/}, Bia\\l{o}wie\\.za, Poland, 1998.\n\n\n\\bibitem{svet1}Svetlichny,~G. (1990) {\\em Foundations of Physics \\/}\n{\\bf 20}, 635 - 650.\n\n\\bibitem{svet2}Svetlichny,~G. (1992) {\\em International Journal of\nTheoretical Physics} {\\bf 31}, 1797.\n\n\\bibitem{czachor}Czachor,~M., quant-ph/9501008\n\n\n\\bibitem{EPR}Einstein,~A., Podolsky,~B. and Rosen,~N. (1935) {\\em Physical \nReview\\/}\n{\\bf 47}, 777.\n\n\\bibitem{gisin1}Gisin,~N. (1984a) {\\em Physical Review Letters\\/} {\\bf 52}, \n1657.\n\n\\bibitem{gisin2}Gisin,~N. (1984b) {\\em Physical Review Letters\\/} {\\bf 53} \n1776.\n\n\\bibitem{gisin3}Gisin,~N. (1989) {\\em Helvetica Physica Acta \\/} {\\bf 62} \n363.\n\n\\bibitem{gisin4}Gisin,~N. (1990) {\\em Physics Letters A\\/} {\\bf 143}, 1.\n\n\\bibitem{pearle1}Pearle,~P. (1984) {\\em Physical Review Letters\\/} {\\bf 53}, \n1775.\n\n\\bibitem{pearle2}Pearle,~P. (1986) {\\em Physical Review D\\/} {\\bf 33}, 2240.\n\n\\bibitem{redhead}Redhead,~M. (1987) ``Incompleteness, Nonlocality, and \nRealism: A\nProlegomenon to the Philosophy of Quantum Mechanics\", Claredon Press,\nOxford.\n\n\\bibitem{mielnik}Mielnik,~B. (1990) {\\em Foundation of Physics\\/} {\\bf 20}, \n745.\n\n\\bibitem{fink}Finkelstein,~J. (1992) {\\em Foundation of Physics Letters\\/} \n{\\bf 5},\n383.\n\n\\bibitem{svetlichny:quantum} Svetlichny,~G., ``Quantum Evolution and\nSpace-Time Structure\" in H.-D.~Doebner, V.~K.~Dobrev, and\nP.~Nattermann, eds., {\\em Nonlinear, Deformed and Irreversible Quantum\nSystems. Proceedings of the International Symposium on Mathematical\nPhysics, Arnold Sommerfeld Institute, 15-19 August 1994, Clausthal,\nGermany\\/}, p. 246, World Scientific, Singapore, 1995, or in a slightly\nexpanded form in\nquant-ph/9512004.\n\n\\bibitem{svet5}Svetlichny,~G., ``On Relativistic Non-linear Quantum \nMechanics\" in M.~Shkil,\nA.~Nikitin, V.~Boyko, editors,\n{\\sl Proceedings of the Second International Conference\n``Symmetry in Nonlinear Mathematical Physics.\nMemorial Prof. W. Fushchych Conference\"},\nInstitute of Mathematic of the National Academy of Sciences of Ukraine,\nKiev, Ukraine, 1997, Vol. 2, pp. 262--269.\n\n\n\n\\bibitem{streater}Streater,~R.~R. and Wightman,~A.~S. (1964) ``PCT, Spin and \nStatistics\nand all that\", Benjamin, New York.\n\\bibitem{araki}Araki,~H. (1964) {\\em Progress in Theoretical Physics\\/} {\\bf 32}, 956.\n\n\\bibitem{guz1}Guz,~W. (1979) {\\em Reports on Mathematical Physics \\/} {\\bf 16}, 125.\n\n\\bibitem{guz2}Guz,~W. (1980) {\\em Reports on Mathematical Physics \\/} {\\bf 17}, 385.\n\n\\bibitem{pool1}Pool,~J.~C.~T., (1968a) {\\em Communications in Mathematical\nPhysics\\/}\n{\\bf 9}, 118.\n\n\\bibitem{pool2}Pool,~J.~C.~T., (1968b) {\\em Communications in Mathematical\nPhysics\\/}\n{\\bf 9}, 212.\n\n\\bibitem{schroer} Schroer,~B., ``New Concepts in Particle Physics from \nSolution of an Old Problem\" hep-th/9908021\n\n\\end{thebibliography}\n\\end{document}\n\n\n\n\n" } ]	[ { "name": "quant-ph9912099.extracted_bib", "string": "{pironPiron,~C. (1976) ``Foundations of Quantum Physics\",~W.~A. Benjamin, Inc., London." }, { "name": "quant-ph9912099.extracted_bib", "string": "{svet3Svetlichny,~G. (1998) {\\sl Foundations of Physics, {28, 131; quant-ph/9511002." }, { "name": "quant-ph9912099.extracted_bib", "string": "{mitt1Mittelstaedt,~P. (1983a) {\\em International Journal of Theoretical Physics\\/ {22, 293." }, { "name": "quant-ph9912099.extracted_bib", "string": "{mitt2Mittelstaedt,~P. (1983b) {\\em Proceedings of the International Symposium on the Foundations of Quantum Mechanics\\/ Tokyo, pp. 251-255" }, { "name": "quant-ph9912099.extracted_bib", "string": "{mitt3Mittelstaedt,~P. and Stachow~E.~W. (1983) {\\em International Journal of Theoretical Physics\\/ {22, 517." }, { "name": "quant-ph9912099.extracted_bib", "string": "{neumannNeumann,~N. and Werner,~R. (1983) {\\em International Journal of Theoretical Physics\\/ {22, 781." }, { "name": "quant-ph9912099.extracted_bib", "string": "{ludwigLudwig,~G. (1983) ``Foundations of Quantum Mechanics\", Springer, New York." }, { "name": "quant-ph9912099.extracted_bib", "string": "{haagHaag,~R. (1992) ``Local Quantum Physics\", Springer Verlag, Berlin." }, { "name": "quant-ph9912099.extracted_bib", "string": "{mugur1Mugur-Sch\\\"acter,~M. (1991) {\\em Foundations of Physics\\/ {21, 1387." }, { "name": "quant-ph9912099.extracted_bib", "string": "{mugur2Mugur-Sch\\\"acter,~M. (1992) {\\em Foundations of Physics\\/ {22, 235." }, { "name": "quant-ph9912099.extracted_bib", "string": "{hartle Hartle,~J.~B., ``Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime\", in {\\em 1992 Les Houches Ecole d'\\'et\\'e, Gravitation et Quantifications\\/" }, { "name": "quant-ph9912099.extracted_bib", "string": "{omnes Omn\\'es,~R., {\\em The Interpretation of Quantum Mechanics, Princeton University Press, (1994)" }, { "name": "quant-ph9912099.extracted_bib", "string": "{bialoSvetlichny,~G. (1999) ``Space-time Structure and Quantum Mechanics\" to appear in the proceedings of the XXX\${^{th\$ Workshop on Geometric Methods in Physics, {\\em Coherent States, Quantization and Gravity\\/, Bia\\l{owie\\.za, Poland, 1998." }, { "name": "quant-ph9912099.extracted_bib", "string": "{svet1Svetlichny,~G. (1990) {\\em Foundations of Physics \\/ {20, 635 - 650." }, { "name": "quant-ph9912099.extracted_bib", "string": "{svet2Svetlichny,~G. (1992) {\\em International Journal of Theoretical Physics {31, 1797." }, { "name": "quant-ph9912099.extracted_bib", "string": "{czachorCzachor,~M., quant-ph/9501008" }, { "name": "quant-ph9912099.extracted_bib", "string": "{EPREinstein,~A., Podolsky,~B. and Rosen,~N. (1935) {\\em Physical Review\\/ {47, 777." }, { "name": "quant-ph9912099.extracted_bib", "string": "{gisin1Gisin,~N. (1984a) {\\em Physical Review Letters\\/ {52, 1657." }, { "name": "quant-ph9912099.extracted_bib", "string": "{gisin2Gisin,~N. (1984b) {\\em Physical Review Letters\\/ {53 1776." }, { "name": "quant-ph9912099.extracted_bib", "string": "{gisin3Gisin,~N. (1989) {\\em Helvetica Physica Acta \\/ {62 363." }, { "name": "quant-ph9912099.extracted_bib", "string": "{gisin4Gisin,~N. (1990) {\\em Physics Letters A\\/ {143, 1." }, { "name": "quant-ph9912099.extracted_bib", "string": "{pearle1Pearle,~P. (1984) {\\em Physical Review Letters\\/ {53, 1775." }, { "name": "quant-ph9912099.extracted_bib", "string": "{pearle2Pearle,~P. (1986) {\\em Physical Review D\\/ {33, 2240." }, { "name": "quant-ph9912099.extracted_bib", "string": "{redheadRedhead,~M. (1987) ``Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics\", Claredon Press, Oxford." }, { "name": "quant-ph9912099.extracted_bib", "string": "{mielnikMielnik,~B. (1990) {\\em Foundation of Physics\\/ {20, 745." }, { "name": "quant-ph9912099.extracted_bib", "string": "{finkFinkelstein,~J. (1992) {\\em Foundation of Physics Letters\\/ {5, 383." }, { "name": "quant-ph9912099.extracted_bib", "string": "{svetlichny:quantum Svetlichny,~G., ``Quantum Evolution and Space-Time Structure\" in H.-D.~Doebner, V.~K.~Dobrev, and P.~Nattermann, eds., {\\em Nonlinear, Deformed and Irreversible Quantum Systems. Proceedings of the International Symposium on Mathematical Physics, Arnold Sommerfeld Institute, 15-19 August 1994, Clausthal, Germany\\/, p. 246, World Scientific, Singapore, 1995, or in a slightly expanded form in quant-ph/9512004." }, { "name": "quant-ph9912099.extracted_bib", "string": "{svet5Svetlichny,~G., ``On Relativistic Non-linear Quantum Mechanics\" in M.~Shkil, A.~Nikitin, V.~Boyko, editors, {\\sl Proceedings of the Second International Conference ``Symmetry in Nonlinear Mathematical Physics. Memorial Prof. W. Fushchych Conference\", Institute of Mathematic of the National Academy of Sciences of Ukraine, Kiev, Ukraine, 1997, Vol. 2, pp. 262--269." }, { "name": "quant-ph9912099.extracted_bib", "string": "{streaterStreater,~R.~R. and Wightman,~A.~S. (1964) ``PCT, Spin and Statistics and all that\", Benjamin, New York." }, { "name": "quant-ph9912099.extracted_bib", "string": "{arakiAraki,~H. (1964) {\\em Progress in Theoretical Physics\\/ {32, 956." }, { "name": "quant-ph9912099.extracted_bib", "string": "{guz1Guz,~W. (1979) {\\em Reports on Mathematical Physics \\/ {16, 125." }, { "name": "quant-ph9912099.extracted_bib", "string": "{guz2Guz,~W. (1980) {\\em Reports on Mathematical Physics \\/ {17, 385." }, { "name": "quant-ph9912099.extracted_bib", "string": "{pool1Pool,~J.~C.~T., (1968a) {\\em Communications in Mathematical Physics\\/ {9, 118." }, { "name": "quant-ph9912099.extracted_bib", "string": "{pool2Pool,~J.~C.~T., (1968b) {\\em Communications in Mathematical Physics\\/ {9, 212." }, { "name": "quant-ph9912099.extracted_bib", "string": "{schroer Schroer,~B., ``New Concepts in Particle Physics from Solution of an Old Problem\" hep-th/9908021" } ]
quant-ph9912100	{\Large Quantum Computing, NP-complete Problems \\ and \\ Chaotic Dynamics	[ { "author": "Masanori Ohya and Igor V. Volovich \\thanks{% Permanent address: Steklov Mathematical Institute" }, { "author": "Gubkin St.8" }, { "author": "GSP-1" }, { "author": "117966" }, { "author": "Moscow" }, { "author": "Russia" }, { "author": "volovich@mi.ras.ru" } ]	An approach to the solution of NP-complete problems based on quantum computing and chaotic dynamics is proposed. We consider the satisfiability problem and argue that the problem, in principle, can be solved in polynomial time if we combine the quantum computer with the chaotic dynamics amplifier based on the logistic map. We discuss a possible implementation of such a chaotic quantum computation by using the atomic quantum computer with quantum gates described by the Hartree-Fock equations. In this case, in principle, one can build not only standard linear quantum gates but also nonlinear gates and moreover they obey to Fermi statistics. This new type of entaglement related with Fermi statistics can be interesting also for quantum communication theory.	[ { "name": "ovnpq.tex", "string": "\\documentclass[12pt,thmsa]{article}\n\\usepackage{amsfonts}\n\\usepackage{graphicx}\n\\begin{document}\n\n\\title{{\\Large \\textbf{Quantum Computing, NP-complete Problems \\\\\nand \\\\\nChaotic Dynamics}}}\n\\author{Masanori Ohya and Igor V. Volovich \\thanks{%\nPermanent address: Steklov Mathematical Institute, Gubkin St.8, GSP-1,\n117966, Moscow, Russia, volovich@mi.ras.ru} \\\\\n%EndAName\n\\\\\nDepartment of Information Sciences\\\\\nScience University of Tokyo\\\\\nNoda City, Chiba 278-8510, Japan\\\\\ne-mail: ohya@is.noda.sut.ac.jp}\n\\date{}\n\\maketitle\n\n\\begin{abstract}\nAn approach to the solution of NP-complete problems based on quantum\ncomputing and chaotic dynamics is proposed. We consider the satisfiability\nproblem and argue that the problem, in principle, can be solved in\npolynomial time if we combine the quantum computer with the chaotic dynamics\namplifier based on the logistic map. We discuss a possible implementation of\nsuch a chaotic quantum computation by using the atomic quantum computer with\nquantum gates described by the Hartree-Fock equations. In this case, in\nprinciple, one can build not only standard linear quantum gates but also\nnonlinear gates and moreover they obey to Fermi statistics. This new type of\nentaglement related with Fermi statistics can be interesting also for\nquantum communication theory.\n\\end{abstract}\n\n\\newpage\n\n\\section{Introduction}\n\nThere are important problems such as the napsack problem, the traveling\nsalesman problem, the integer programming problem, the subgraph isomorphism\nproblem, the satisfiability problem that have been studied for decades and\nfor which all known algorithms have a running time that is exponential in\nthe length of the input. These five problems and many other problems belong\nto the set of \\textbf{NP}-complete problems. Any problem that can be solved\nin polynomial time on a nondeterministic Turing machine is polynomially\ntransformed to an \\textbf{NP}-complete problem \\cite{GJ}.\n\nMany \\textbf{NP}-complete problems have been identified, and it seems that\nsuch problems are very difficult and probably exponential. If so, solutions\nare still needed, and in this paper we consider an approach to these\nproblems based on quantum computers and chaotic dynamics.\n\nIt is widely believed that quantum computers are more efficient than\nclassical computers. In particular Shor \\cite{Sho} gave a remarkable quantum\npolynomial-time algorithm for the factoring problem. However, it is unknown\nwhether this problem is \\textbf{NP}-complete.\n\nThe computational power of quantum computers has been explored in a number\nof papers. Bernstein and Vasirani \\cite{BV} proved that \\textbf{BPP}$%\n\\subseteq $\\textbf{BQP}$\\subseteq $ \\textbf{PSPACE}. Here \\textbf{BPP }\nstands for the class of problems efficiently solvable in the classical\nsense, i.e., the class of problems that can be solved in polynomial time by\nprobabilistic Turing machines with error probability bounded by 1/3 for all\ninputs. The quantum analogue of the class \\textbf{BPP }is the class \\textbf{%\nBQP} which is the class of languages that can be solved in polynomial time\nby quantum Turing machines with error probability bounded by 1/3 for all\ninputs.\n\nThe question whether \\textbf{NP}$\\subseteq $\\textbf{BQP, }i.e., can quantum\ncomputers solve \\textbf{NP}-complete problems in polynomial time, was\nconsidered in \\cite{BBBV}. It was proved in \\cite{BBBV} that relative to an\noracle chosen uniformly at random, with probability 1, the class \\textbf{NP}\ncan not be solved on a quantum Turing machine in time o$\\left(\n2^{n/2}\\right) .$ An oracle is a special subroutine call whose invocation\nonly costs unit time. This result does not rule out the possibility that \n\\textbf{NP}$\\subseteq $\\textbf{BQP }but it does establish that there is no\nblack-box approach to solving \\textbf{NP}-complete problems in polynomial\ntime on quantum Turing machines. We would like to mention that these results\nare not immediately applicable to the chaotic quantum computer which we\nconsider in this paper.\n\nFor a recent discussion of computational complexity in quantum computing see \n\\cite{FR,Cle,HHZ}. Mathematical features of quantum computing and quantum\ninformation theory are summarized in \\cite{O1}. A possibility to exploit\nnonlinear quantum mechanics so that the class of problems \\textbf{NP} may be\nsolved in polynomial time has been considered by Abrams and Lloyd in \\cite\n{AL}. It is mentioned in \\cite{AL} that such nonlinearity is purely\nhypotetical; all known experiments confirm the linearity of quantum\nmechanics.\n\nThe satisfiability problem (SAT), which is \\textbf{NP}-complete problem, has\nbeen considered in quantum computing in \\cite{OM}. It was shown in \\cite{OM}\nthat the SAT problem can be solved in polynomial time by using a quantum\ncomputer under the assumption that a special superposition of two orthogonal\nvectors can be physically detected . The problem one has to overcome here is\nthat the output of computations could be a very small number and one needs\nto amplify it to a reasonable large quantity.\n\nIn this paper we propose that chaotic dynamics plays a constructive role in\ncomputations. Chaos and quantum decoherence are considered normally as the\ndegrading effects which lead to an unwelcome increase of the error rate with\nthe input size. However, in this paper we argue that under some\ncircumstances chaos can play a constructive role in computer science. In\nparticular we propose to combine quantum computer with the chaotic dynamics\namplifier. We will argue, by using the consideration from \\cite{OM}, that\nsuch a chaotic quantum computer can solve the SAT problem in polynomial time.\n\nAs a possible specific implementation of chaotic quantum computations we\ndiscuss the recently proposed atomic quantum computer \\cite{Vol1}. It is\nproposed in \\cite{Vol1} to use a \\textit{single }atom as a quantum computer.\nOne can implement a single qubit in atom as a one-particle electron state in\nthe self-consistent field approximation and multi-qubit states as the\ncorresponding multi-electron states represented by the Slater determinant.\n\nA possible realization of the standard quantum gates in the atomic quantum\ncomputer by using the electron spin resonance has been discussed in \\cite\n{Vol1}. In this paper we argue that in the atomic quantum computer one can\nbuild also \\textit{nonlinear} quantum gates because the dynamics of the\nmulti-electron atom in the very good approximation is described by nonlinear\nHartree-Fock equations.\n\nThe tensor product structure of states is very important for computations\nand the multielectron atom admits such a structure. More exactly, instead of\nthe standard tensor product used in quantum computing we have to use the\nSlater determinant to take into account the Fermi statistics.The standard\ncomputational basis in quantum computing does not have Bose or Fermi\nsymmetry. In the atomic case we have to make an appropriate modification of\nquantum gates to take into account Fermi statistics and this leads to a new\ntype of entanglement related with Fermi statistics.\n\nSuch Fermi or Bose entanglement could be interesting also for quantum\ncommunication theory, in particular for quantum teleportation\n\\cite{AO,FO}.\n\n\\section{SAT Problem}\n\nLet $\\left\\{ x_{1},\\cdots ,x_{n}\\right\\} $ be a set of Boolean variables, $%\nx_{i}=0$ or $1.$ Then the set of the Boolean variables $\\left\\{ x_{1}, \n\\overline{x}_{1},\\cdots ,x_{n},\\overline{x}_{n}\\right\\} $ with or without\ncomplementation is called the set of \\textit{literals.} A formula, which is\nthe product (AND) of disjunctions (OR) of literals is said to be in the \n\\textit{product of sums }(POS) form. For example, the formula \n\\[\n\\left( x_{1}\\vee \\overline{x}_{2}\\right) \\left( \\overline{x}_{1}\\right)\n\\left( x_{2}\\vee \\overline{x}_{3}\\right) \n\\]\nis in POS form. The disjunctions $\\left( x_{1}\\vee \\overline{x}_{2}\\right)\n,\\left( \\overline{x}_{1}\\right) ,\\left( x_{2}\\vee \\overline{x}_{3}\\right) $\nhere are called \\textit{clauses.} A formula in POS form is said to be \n\\textit{satisfiable }if there is an assignment of values to variables so\nthat the formula has value 1. The preceding formula is satisfiable when $%\nx_{1}=0,$ $x_{2}=0,$ $x_{3}=0.$\n\n\\textbf{Definition}(SAT Problem). The satisfiability problem (SAT) is to\ndetermine whether or not a formula in POS form is satisfiable.\n\nThe following analytical formulation of SAT problem is useful. We define a\nfamily of Boolean polynomials $f_{\\alpha }$, indexed by the following data.\nOne $\\alpha $ is a set \n\\[\n\\alpha =\\left\\{ S_{1},...,S_{N},T_{1},...,T_{N}\\right\\} , \n\\]\nwhere $S_{i},T_{i}\\subseteq \\left\\{ 1,...,n\\right\\} ,$ and $f_{\\alpha }$ is\ndefined as \n\\[\nf_{\\alpha }(x_{1},\\cdots ,x_{n})=\\prod_{i=1}^{N}\\left( 1+\\prod_{a\\in\nS_{i}}(1+x_{a})\\prod_{b\\in T_{i}}x_{b}\\right) . \n\\]\n\nWe assume here the addition modulo 2. The SAT problem now is to determine\nwhether or not there exists a value of $\\mathbf{x}=(x_{1},\\cdots ,x_{n})$\nsuch that $f_{\\alpha }(\\mathbf{x})=1.$\n\n\\section{Quantum Algorithm}\n\nWe will work in the $\\left( n+1\\right) $-tuple tensor product Hilbert space $%\n\\mathcal{H\\equiv }$ $\\otimes _{1}^{n+1}$\\textbf{C}$^{2}$ with the\ncomputational basis \n\\[\n\\left\| x_{1},...,x_{n},y\\right\\rangle =\\otimes _{i=1}^{n}\\left\|\nx_{i}\\right\\rangle \\otimes \\left\| y\\right\\rangle \n\\]\nwhere $x_{1},...,x_{n},$ $y=0$ or $1.$ We denote $\\left\|\nx_{1},...,x_{n},y\\right\\rangle =\\left\| \\mathbf{x},y\\right\\rangle .$ The\nquantum version of the function $f(\\mathbf{x})=f_{\\alpha }(\\mathbf{x})$ is\ngiven by the unitary operator $U_{f}\\left\| \\mathbf{x},y\\right\\rangle =\\left\| \n\\mathbf{x},y+f(\\mathbf{x})\\right\\rangle .$ We assume that the unitary matrix \n$U_{f}$ can be build in the polynomial time, see \\cite{OM}. Now let us use\nthe usual quantum algorithm:\n\n(i) By using the Fourier transform produce from $\\left\| \\mathbf{0,}\n0\\right\\rangle $ the superposition \n\\[\n\\left\| v\\right\\rangle =\\frac{1}{\\sqrt{2^{n}}} \\sum_{\\mathbf{x}}\\left\| \n\\mathbf{x},0\\right\\rangle . \n\\]\n\n(ii) Use the unitary matrix $U_{f}$ to calculate $f(\\mathbf{x}):$%\n\\[\n\\left\| v_{f}\\right\\rangle =U_{f}\\left\| v\\right\\rangle =\\frac{1}{\\sqrt{2^{n}}}\n\\sum_{\\mathbf{x}}\\left\| \\mathbf{x},f(\\mathbf{x})\\right\\rangle \n\\]\nNow if we measure the last qubit, i.e., apply the projector $P=I\\otimes\n\\left\| 1\\right\\rangle \\left\\langle 1\\right\| $ to the state $\\left\|\nv_{f}\\right\\rangle ,$ then we obtain that the probability to find the result \n$f(\\mathbf{x})=1$ is $\\left\\\| P\\left\| v_{f}\\right\\rangle \\right\\\|\n^{2}=r/2^{n}$ where $r$ is the number of roots of the equation $f(\\mathbf{x}\n)=1.$ For small $r $ the probability is very small and this means we in fact\ndon't get an information about the existence of the solution of the equation \n$f(\\mathbf{x} )=1.$ Let us simplify our notations. After the step (ii) the\nquantum computer will be in the state \n\\[\n\\left\| v_{f}\\right\\rangle =\\sqrt{1-q^{2}}\\left\| \\varphi _{0}\\right\\rangle\n\\otimes \\left\| 0\\right\\rangle +q\\left\| \\varphi _{1}\\right\\rangle \\otimes\n\\left\| 1\\right\\rangle \n\\]\nwhere $\\left\| \\varphi _{1}\\right\\rangle $ and $\\left\| \\varphi\n_{0}\\right\\rangle $ are normalized $n$ qubit states and \n$q=\\sqrt{r/2^{n}}.$\nEffectively our problem is reduced to the following $1$ qubit problem. We\nhave the state \n\\[\n\\left\| \\psi \\right\\rangle =\\sqrt{1-q^{2}}\\left\| 0\\right\\rangle +q\\left\|\n1\\right\\rangle \n\\]\nand we want to distinguish between the cases $q=0$ \n(i.e. very small $q$) and $q>0$. To this end we\npropose to employ chaotic dynamics.\n\n\\section{Chaotic Dynamics}\n\nVarious aspects of classical and quantum chaos have been the subject of\nnumerious studies, see \\cite{O2} and ref's therein.The investigation of\nquantum chaos by using quantum computers has been proposed in \\cite{\nGCZ,Sch,KM}. Here we will argue that chaos can play a constructive role in\ncomputations.\n\nChaotic behaviour in a classical system usually is considered as an\nexponential sensitivity to initial conditions. It is this sensitivity we\nwould like to use to distinquish between the cases $q=0$ and $q>0$ from the\nprevious section.\n\nConsider the so called logistic map which is given by the equation \n\\[\nx_{n+1}=ax_{n}(1-x_{n}),~~~x_{n}\\in \\left[ 0,1\\right] .\n\\]\n\n\\noindent \\noindent \\noindent The properties of the map depend on the\nparameter $a.$ If we take, for example, $a=3.71,$ then the Lyapunov exponent\nis positive, the trajectory is very sensitive to the initial value and one\nhas the chaotic behaviour \\cite{O2}. It is important to notice that if the\ninitial value $x_{0}=0,$ then $x_{n}=0$ for all $n.$\n\n\\begin{center}\n\\includegraphics{logistic.eps}\n\\end{center}\n\nIt is known \\cite{Deu} that any classical algorithm can be implemented on\nquantum computer. Our stochastic quantum computer will be consisting from\ntwo blocks. One block is the ordinary quantum computer performing\ncomputations with the output $\\left\| \\psi \\right\\rangle =\\sqrt{1-q^{2}}%\n\\left\| 0\\right\\rangle +q\\left\| 1\\right\\rangle $. The second block is a \n\\textit{quantum} computer performing computations of the \\textit{classical}\nlogistic map. This two blocks should be connected in such a way that the\nstate $\\left\| \\psi \\right\\rangle $ first be transformed into the density\nmatrix of the form \n\\[\n\\rho =q^{2}P_{1}+\\left( 1-q^{2}\\right) P_{0} \n\\]\nwhere $P_{1}$ and $P_{0}$ are projectors to the states $\\left\|\n1\\right\\rangle $ and $\\left\| 0\\right\\rangle .$ This connection is in fact\nnontrivial and actually it should be considered as the third block. One has\nto notice that $P_{1}$ and $P_{0}$ generate an Abelian algebra which can be\nconsidered as a classical system. In the second block the density matrix $%\n\\rho $ above is interpreted as the initial data $\\rho _{0}$ for the logistic\nmap \n\\[\n\\rho _{n+1}=a\\rho _{n}(1-\\rho _{n}) \n\\]\nAfter one step, the state $\\rho _{1}$ becomes \n\\[\n\\rho _{1}=aq^{2}(1-q^{2})I, \n\\]\nwhere $I$ is the identity matrix on $\\Bbb{C}^{2}.$ In paricular, if one has $%\nq=0$ then $\\rho _{0}=P_{0}$ and we obtain $\\rho _{n}=P_{0}$ for all $n.$\nOtherwise the stochastic dynamics leads to the amplification of the small\nmagnitude $q$ in such a way that it can be detected. As is seen in Fig.1, we\ncan easily amplify the small $q$ in several steps, i.e., within about ten\ntimes measurements as in Shor's algorithm. \nThe transition from $\\rho _{n}$ to $%\n\\rho _{n+1}$ is nonlinear and can be considered as a discrete Heisenberg\nevolution of the variable $x_{n}$.\n\nOne can think about various possible implementations of the idea of using\nchaotic dynamics for computations. Below we discuss how one can realize\nnonlinear quantum gates on atomic quantum computer.\n\n\\section{\\noindent Atomic Quantum Computer}\n\nMany current proposals for the realization of quantum computer such as NMR,\nquantum dots and trapped ions are based on the using of an atom or an ion as\none qubit, see \\cite{CZ,GC,BLD,EJ}. In these proposals a quantum computer\nconsists from several atoms, and the coupling between them provides the\ncoupling between qubits necessary for a quantum gate. It was proposed in \n\\cite{Vol1} that a \\textit{single} atom can be used as a quantum computer.\nOne can implement a single qubit in atom as a one-particle electron state in\nthe self-consistent field approximation and multi-qubit states as the\ncorresponding multi-electron states represented by the Slater determinant.\nSo, to represent 10 qubits one can use an atom with 10 electrons and to\nrepresent 50 qubits one has to control only around 50 levels in an atom with\n50 electrons.\n\nA possible realization of the standard quantum gates in the atomic quantum\ncomputer by using the electron spin resonance has been discussed in \\cite\n{Vol1}. In this paper we propose that in the atomic quantum computer one can\nbuild also \\textit{nonlinear} quantum gates because the dynamics of the\nmulti-electron atom in the very good approximation is described by nonlinear\nHartree-Fock equations. Therefore it follows from \\cite{OM} and the\nconsiderations in this paper that the atomic quantum computer can solve the\nSAT problem in polynomial time.\n\nIt is well known that in atomic physics the concept of the individual state\nof an electron in an atom is accepted and one proceeds from the Hartree-Fock\nself-consistent field approximation, see for example \\cite{Sob}. The state\nof an atom is determined by the set of the states of the electrons. Each\nstate of the electron is characterized by a definite value of its orbital\nangular momentum $l$, by the principal quantum number $n$ and by the values\nof the projections of the orbital angular momentum $m_{l}$ and of the spin $%\nm_{s}$ on the $z$-axis. In the Hartree-Fock central field approximation the\nenergy of an atom is completely determined by the assignment of the electron\nconfiguration, i.e., by the assignment of the values of $n$ and $l$ for all\nthe electrons.\n\nThe tensor product structure of states is very important for computations.\nFortunately a multielectron atom admits such a structure. More exactly,\ninstead of the standard tensor product used in quantum computing we have to\nuse the Slater determinant to take into account the Fermi statistics.The\nstandard computational basis in quantum computing does not have Bose or\nFermi symmetry. In the atomic case we have to make an appropriate\nmodification of quantum gates to take into account Fermi statistics and this\nleads to a new type of entanglement related with Fermi statistics.\n\nAn application of the electron spin resonance (ESR) to process the\ninformation encoded in the hyperfine splitting of atomic energy levels and\nto build standard linear quantum gates has been considered in \\cite{Vol1}.\nIn this paper we suggest that in atomic quantum computer one can build also \n\\textit{nonlinear} quantum gates described by the Hartree-Fock equations.\n\nThe Hamiltonian for the $N-$ particle system has the form \n\\[\nH=\\sum_{i=1}^{N}\\left( -\\frac{\\nabla _{i}^{2}}{2m_{i}}+v(r_{i})\\right)\n+\\sum_{i<j}V(r_{i},r_{j}) \n\\]\nIn the Hartree-Fock method one takes the $N-$ particle wave function in the\nform of the Slater determinant \n\\[\n\\Psi (t,r_{1},...,r_{N})=Antisym(\\Phi _{1}(t,r_{1})...\\Phi _{N}(t,r_{N})) \n\\]\nHere the one-particle wave functions $\\Phi _{i}(t,r_{i})$ satisfy the\nnonlinear Hartree-Fock equations which have the form of nonlinear\nSchrodinger equation \n\\[\ni\\frac{\\partial \\Phi _{i}(t,r)}{\\partial t}=\\mathit{H}(\\Phi )\\Phi _{i}(t,r) \n\\]\nwhere \n\\[\n\\mathit{H}(\\Phi )\\Phi _{i}(t,r)=\\left( -\\frac{\\nabla ^{2}}{2m_{i}}+v(r)+%\n\\mathcal{U}_i(t,r)\\right) \\Phi _{i}(t,r)-\\int dr^{\\prime }\\mathcal{W}%\n(t,r^{\\prime },r)\\Phi _{i}(t,r^{\\prime }) \n\\]\nand \n\\[\n\\mathcal{U}_i\\left( t,r\\right) =\\sum_{j\\neq i}\\int dr^{\\prime }\\Phi\n_{j}^{}\\left( t,r^{\\prime }\\right) V\\left( r^{\\prime },r\\right) \\Phi\n_{j}\\left( t,r^{\\prime }\\right) , \n\\]\n\n\\[\n\\mathcal{W}\\left( t,r^{\\prime },r\\right) = \\sum_{j}\\Phi _{j}^{}\\left(\nt,r^{\\prime }\\right) V\\left( r^{\\prime },r\\right) \\Phi _{j}\\left(\nt,r\\right) \n\\]\n\nIf we consider only the spin dependent part of the wave function of the\none-electron state \n\\[\n\\varphi =\\left( \n\\begin{array}{c}\n\\varphi _{0}\\left( t\\right) \\\\ \n\\varphi _{1}\\left( t\\right)\n\\end{array}\n\\right) , \n\\]\nthen one can get the nonlinear equation of the form \n\\[\ni\\frac{\\partial \\varphi }{\\partial t}=A\\varphi +B\\left( \\varphi \\right)\n\\varphi . \n\\]\nHere $A$ is a $2\\times 2$ matrix and the matrix $B$ depends on $\\varphi .$\nBy using this equation one can describe nonlinear quantum gate. The\nnonlinearity can be tuned by means of magnetic field.\n\n\\section{Conclusion}\n\nThe complexity of the quantum algprithm for the SAT problem has been\nconsidered in \\cite{OM} where it was shown that one can build the unitary\nmatrix $U_{f}$ in the polynomial time. We have also to consider the number\nof steps in the classical algorithm for the logistic map performed on\nquantum computer. It is the probabilistic part of the construction and one\nhas to repeat computations several times to be able to distingish the cases $%\nq=0$ and $q>0.$ Thus it seems that the chaotic quantum computer can solve\nthe SAT problem in polynominal time. \n\nIn conclusion, in this paper the\nchaotic quantum computer is proposed. It combines the ordinary quantum\ncomputer with quantum chaotic dynamics amplifier which can be implemented by\nusing the atomic quantum computer. We argued that such a device can be\npowerful enough to solve the \\textbf{NP}-complete problem \nin polynomial time.\n\n\\begin{thebibliography}{99}\n\\bibitem{GJ} M. Garey and D. Johnson, \\textit{Computers and Intractability\n- a guide to the theory of NP-completeness}, Freeman, 1979.\n\n\\bibitem{Sho} P.W. Shor, \\textit{Algorithm for quantum computation :\nDiscrete logarithm and factoring algorithm}, Proceedings of the 35th Annual\nIEEE Symposium on Foundation of Computer Science, pp.124-134, 1994.\n\n\\bibitem{BV} E. Bernstein and U. Vazirani, \\textit{Quantum Complexity Theory%\n}, in: Proc. of the 25th Annual ACM Symposium on Theory of Comuting, (ACM\nPress, New York,1993), pp.11-20.\n\n\\bibitem{BBBV} C. H. Bennett, E. Bernstein, G. Brassard, U. Vazirani,%\n\\newline\n\\textit{Strengths and Weaknesses of Quantum Computing}, quant-ph/9701001\n\n\\bibitem{FR} L. Fortnow and J. Rogers, \\textit{Complexity Limitations on\nQuantum Computation}, cs.CC/9811023.\n\n\\bibitem{Cle} R. Cleve, \\textit{An Introduction to Quantum \nComplexity Theory%\n}, quant-ph/9906111.\n\n\\bibitem{HHZ} E. Hemaspaandra, L.A. Hemaspaandra and M. Zimand,\\newline\n\\textit{Almost-Everywhere Superiority for Quantum Polynomial Time},\nquant-ph/9910033.\n\n\\bibitem{O1} M. Ohya, \\textit{Mathematical Foundation of Quantum Computer,}\nMaruzen Publ. Company, 1998\n\n\\bibitem{AL} D. S. Abrams and S. Lloyd, \\textit{Nonlinear quantum mechanics\nimplies polynomial-time solution for NP-complete and \\#P problems,}\nquant-ph/9801041.\n\n\\bibitem{OM} M. Ohya and N. Masuda, \\textit{NP problem in Quantum Algorithm,%\n} quant-ph/9809075.\n\n\\bibitem{Vol1} I.V. Volovich, \\textit{Atomic Quantum Computer,}\nquant-ph/9911062.\n\n\\bibitem{O2} M. Ohya, \\textit{Complexities and Their Applications to\nCharacterization of Chaos,} Int. Journ. of Theoret. Physics, 37 (1998) 495.\n\n\\bibitem{GCZ} S.A. Gardiner, J.I. Cirac and P. Zoller, Phys. Rev. Lett.\n79(1997) 4790.\n\n\\bibitem{Sch} R. Schack, Phys. Rev. A57 (1998) 1634; T. Brun and R. Schack,\nquant-ph/9807050.\n\n\\bibitem{KM} I. Kim and G. Mahler, \\textit{Quantum Chaos in Quantum Turing\nMachine,} quant-ph/9910068.\n\n\\bibitem{Deu} D. Deutsch, \\textit{Quantum theory, the Church-Turing\nprinciple and the universal quantum computer,} Proc. of Royal Society of\nLondon series A, \\textbf{400}, pp.97-117, 1985.\n\n\\bibitem{CZ} J.I. Cirac and P. Zoller, Phys. Rev. Lett., 74 (1995) 74.\n\n\\bibitem{GC} N.A. Gershenfeld and I.L. Chuang, Science, 275 (1997) 350.\n\n\\bibitem{BLD} G. Burkard, D. Loss and D.P. DiVincenzo, cond-mat/9808026.\n\n\\bibitem{EJ} A. Ekert and R. Jozsa, \\textit{Quantum computation and Shor's\nfactoring algorithm,} Reviews of Modern Physics, \\textbf{68}\nNo.3,pp.733-753, 1996.\n\n\\bibitem{Sob} I.I. Sobelman, \\textit{Atomic Spectra and Radiative\nTransitions,} Springer-Verlag, 1991.\n\n\\bibitem{AO} Accardi, L. and Ohya, M.: \\textit{Teleportation of\ngeneral quantum states,} quant-ph/9912087.\n\n\\bibitem{FO} Fichtner, K.-H. and Ohya, M.:\\textit{Quantum Teleportation\nwith Entangled States given by Beam Splittings, } quant-ph/9912083.\n\\end{thebibliography}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912100.extracted_bib", "string": "{GJ M. Garey and D. Johnson, Computers and Intractability - a guide to the theory of NP-completeness, Freeman, 1979." }, { "name": "quant-ph9912100.extracted_bib", "string": "{Sho P.W. Shor, Algorithm for quantum computation : Discrete logarithm and factoring algorithm, Proceedings of the 35th Annual IEEE Symposium on Foundation of Computer Science, pp.124-134, 1994." }, { "name": "quant-ph9912100.extracted_bib", "string": "{BV E. Bernstein and U. Vazirani, Quantum Complexity Theory% , in: Proc. of the 25th Annual ACM Symposium on Theory of Comuting, (ACM Press, New York,1993), pp.11-20." }, { "name": "quant-ph9912100.extracted_bib", "string": "{BBBV C. H. Bennett, E. Bernstein, G. Brassard, U. Vazirani,% \\newline Strengths and Weaknesses of Quantum Computing, quant-ph/9701001" }, { "name": "quant-ph9912100.extracted_bib", "string": "{FR L. Fortnow and J. Rogers, Complexity Limitations on Quantum Computation, cs.CC/9811023." }, { "name": "quant-ph9912100.extracted_bib", "string": "{Cle R. Cleve, An Introduction to Quantum Complexity Theory% , quant-ph/9906111." }, { "name": "quant-ph9912100.extracted_bib", "string": "{HHZ E. Hemaspaandra, L.A. Hemaspaandra and M. Zimand,\\newline Almost-Everywhere Superiority for Quantum Polynomial Time, quant-ph/9910033." }, { "name": "quant-ph9912100.extracted_bib", "string": "{O1 M. Ohya, Mathematical Foundation of Quantum Computer, Maruzen Publ. Company, 1998" }, { "name": "quant-ph9912100.extracted_bib", "string": "{AL D. S. Abrams and S. Lloyd, Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and \\#P problems, quant-ph/9801041." }, { "name": "quant-ph9912100.extracted_bib", "string": "{OM M. Ohya and N. Masuda, NP problem in Quantum Algorithm,% quant-ph/9809075." }, { "name": "quant-ph9912100.extracted_bib", "string": "{Vol1 I.V. Volovich, Atomic Quantum Computer, quant-ph/9911062." }, { "name": "quant-ph9912100.extracted_bib", "string": "{O2 M. Ohya, Complexities and Their Applications to Characterization of Chaos, Int. Journ. of Theoret. Physics, 37 (1998) 495." }, { "name": "quant-ph9912100.extracted_bib", "string": "{GCZ S.A. Gardiner, J.I. Cirac and P. Zoller, Phys. Rev. Lett. 79(1997) 4790." }, { "name": "quant-ph9912100.extracted_bib", "string": "{Sch R. Schack, Phys. Rev. A57 (1998) 1634; T. Brun and R. Schack, quant-ph/9807050." }, { "name": "quant-ph9912100.extracted_bib", "string": "{KM I. Kim and G. Mahler, Quantum Chaos in Quantum Turing Machine, quant-ph/9910068." }, { "name": "quant-ph9912100.extracted_bib", "string": "{Deu D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. of Royal Society of London series A, 400, pp.97-117, 1985." }, { "name": "quant-ph9912100.extracted_bib", "string": "{CZ J.I. Cirac and P. Zoller, Phys. Rev. Lett., 74 (1995) 74." }, { "name": "quant-ph9912100.extracted_bib", "string": "{GC N.A. Gershenfeld and I.L. Chuang, Science, 275 (1997) 350." }, { "name": "quant-ph9912100.extracted_bib", "string": "{BLD G. Burkard, D. Loss and D.P. DiVincenzo, cond-mat/9808026." }, { "name": "quant-ph9912100.extracted_bib", "string": "{EJ A. Ekert and R. Jozsa, Quantum computation and Shor's factoring algorithm, Reviews of Modern Physics, 68 No.3,pp.733-753, 1996." }, { "name": "quant-ph9912100.extracted_bib", "string": "{Sob I.I. Sobelman, Atomic Spectra and Radiative Transitions, Springer-Verlag, 1991." }, { "name": "quant-ph9912100.extracted_bib", "string": "{AO Accardi, L. and Ohya, M.: Teleportation of general quantum states, quant-ph/9912087." }, { "name": "quant-ph9912100.extracted_bib", "string": "{FO Fichtner, K.-H. and Ohya, M.:Quantum Teleportation with Entangled States given by Beam Splittings, quant-ph/9912083." } ]
quant-ph9912101	Observation of radiation pressure exerted by evanescent waves	[ { "author": "D. Voigt" }, { "author": "B.T. Wolschrijn" }, { "author": "R. Jansen" }, { "author": "N. Bhattacharya" }, { "author": "R.J.C. Spreeuw" }, { "author": "and H.B. van Linden van den Heuvell" } ]	We report a direct observation of radiation pressure, exerted on cold rubidium atoms while bouncing on an evanescent-wave atom mirror. We analyze the radiation pressure by imaging the motion of the atoms after the bounce. The number of absorbed photons is measured for laser detunings ranging from {190~MHz to {1.4~GHz and for angles from {0.9~mrad to {24~mrad above the critical angle of total internal reflection. Depending on these settings, we find velocity changes parallel with the mirror surface, ranging from 1 to {18~cm/s. This corresponds to 2 to 31 photon recoils per atom. These results are independent of the evanescent-wave optical power.	[ { "name": "radpre2c.tex", "string": "%\\documentstyle[aps,preprint,epsf]{revtex}\n\\documentstyle[aps,twocolumn,epsf]{revtex}\n\n\n\\begin{document}\n\\draft\n\n\\title{Observation of radiation pressure exerted by evanescent waves}\n\n\\author{D. Voigt, B.T. Wolschrijn, R. Jansen, N. Bhattacharya,\n\tR.J.C. Spreeuw, and H.B. van Linden van den Heuvell}\n\n\\address{Van der Waals-Zeeman Institute, University of Amsterdam, \\\\\n Valckenierstraat 65, 1018 XE Amsterdam, the Netherlands\\\\\n e-mail: spreeuw@wins.uva.nl}\n\n\\date{\\today}\n\\maketitle\n\n\n\\begin{abstract}\n\nWe report a direct observation of radiation pressure, exerted on cold rubidium\natoms while bouncing on an evanescent-wave atom mirror. We analyze the\nradiation pressure by imaging the motion of the atoms after the bounce.\nThe number of absorbed photons is measured for laser detunings ranging from\n{190~MHz} to {1.4~GHz} and for angles from {0.9~mrad} to {24~mrad} above the\ncritical angle of total internal reflection. Depending on these settings,\nwe find velocity changes parallel with the mirror surface, ranging from\n1 to {18~cm/s}. This corresponds to 2 to 31 photon recoils per atom.\nThese results are independent of the evanescent-wave optical power.\n\n\\end{abstract}\n\n\n\\pacs{32.80.Lg, 42.50.Vk, 03.75.-b}\n% 32.80.Lg; Mechanical effects of light on atoms, molecules, and ions\n% 42.50.Vk: Mechanical effects of light on atoms, molecules, electrons, and ions\n% 03.75.-b: Matter waves\n\n\n\\section{Introduction}\n\nAn evanescent wave (EW) appears whenever an electromagnetic wave undergoes\ntotal internal reflection at a dielectric interface.\nThe EW is characterized by an electric field amplitude that decays\nexponentially with the distance to the interface.\nThe decay length is on the order of the (reduced) optical wavelength.\nCook and Hill \\cite{CooHil82} proposed to use the EW as a mirror for slow\nneutral atoms, based on the ``dipole force''.\nEW mirrors have since become an important tool in atom optics\n\\cite{AdaSigMly94}.\nThey have been demonstrated for atomic beams at grazing incidence\n\\cite{BalLetOvc87} and for ultracold atoms at normal incidence\n\\cite{KasWeiChu90}.\n\nMost experimental work so far, has been concentrated on the reflective\nproperties, {i.e.} the change of the atomic motion {\\em perpendicular}\nto the surface \\cite{SeiAdaBal94}.\nThis is dominated by the dipole force due to the strong gradient of the\nelectric field amplitude.\nIn the present paper we report on our measurement of the force {\\em parallel}\nto the surface.\nIt has been mentioned already in the original proposal \\cite{CooHil82}\nthat there should be such a force.\nThe propagating component of the wavevector leads to a spontaneous scattering\nforce, ``radiation pressure'' \\cite{GorAsh80,Coo80}. \nWe present here the first direct observation of radiation pressure exerted by\nevanescent waves on cold atoms.\nPreviously, a force parallel to the surface has been observed for\nmicrometer-sized dielectric spheres \\cite{KawSug92}.\n\nIn our experiment, we observe the trajectory of a cloud of cold rubidium atoms\nfalling and bouncing on a horizontal EW mirror. The radiation pressure appears\nas a change in horizontal velocity during the bounce. We study the average\nnumber of scattered photons per atom as a function of the detuning and angle of\nincidence of the EW. The latter varies the ``steepness'' of the optical\npotential.\n\nDue to its short extension, on the order of the optical wavelength, an EW\nmirror is a promising tool for efficient loading of low-dimensional optical\natom traps in the vicinity of the dielectric surface\n\\cite{SurfaceTraps,SprVoiWol}. In these schemes, spontaneous optical\ntransitions provide dissipation and cause the mirror to be ``inelastic'',\nsuch that the final atomic phase-space density increases. This might open a\nroute towards quantum degenerate gases, which does not use evaporative cooling\n\\cite{KetDurSta99}, and could potentially yield ``atom lasers''\n\\cite{BECAtomLasers} which are open, driven systems out of thermal equilibrium,\nsimilar to optical lasers. It is this application of EW mirrors which drives\nour interest in experimental control of the photon scattering of bouncing\natoms.\n\nThis article is structured as follows. In {Sec.\\,II} we summarize the\nproperties of the EW potential, including the photon scattering of a bouncing\natom. In {Sec.\\,III} we describe our experimental setup and our observation of\nradiation pressure. We investigate its dependence on the angle of incidence and\nlaser detuning. Finally, we discuss several systematic errors.\n\n\n\\section{The {evanescent-wave} atomic mirror}\n\n\\subsection{Optical dipole potential}\n\nOur evanescent wave is created by total internal reflection at a glass surface\nin vacuum \\cite{Hec87}. We choose the $z$-direction as the surface normal and\nthe $xz$ plane as the plane of incidence (see {Fig.~\\ref{fig:SetupFigure}}).\nThe EW can be written as\n$E\\left({\\rm\\bf r}\\right)\\propto\\exp(i\\,{\\rm\\bf k}\\cdot{\\rm\\bf r})$,\nwhere ${\\rm\\bf k}=(k_x, 0, i \\kappa)$.\nThe wavevector has a propagating component along the surface,\n$k_x=k_0 n \\sin \\theta$, where $k_0=2\\pi/\\lambda_0$ is the vacuum wave number,\n$n$ is the refractive index and $\\theta$ the angle of incidence.\nNote, that $k_x>k_0$, since $\\theta$ is larger than the critical angle\n$\\theta_c=\\arcsin n^{-1}$.\nThe wavevector component perpendicular to the surface is imaginary,\nwith $\\kappa=k_0\\sqrt{n^2\\sin ^{2}\\theta -1}$.\n\n\\begin{figure}[t]\n \\centerline{\\epsfxsize=8.0cm\\epsffile{radpre2cfig1.eps}}\n \\vspace{0.5cm}\n \\caption{Evanescent-wave atom mirror.\n (a)\tConfiguration in the rubidium vapor cell:\n\tmagneto-optical trap (MOT), right-angle prism with refractive index $n$\n\t({6.6~mm} below the MOT, {gravity$\\parallel$$z$}), evanescent-wave beam\n\t(EW), camera facing from the side (CCD, in $y$ direction),\n\tresonant fluorescence probe beam from above (P).\n (b)\tConfocal relay telescope for adjusting the angle of incidence $\\theta$.\n\tThe lenses $L_{1,2}$ have equal focal length, {$f=75~$mm}.\n\tA translation of $L_1$ by a distance $\\Delta a$ changes the angle of\n\tincidence by $\\Delta\\theta=\\Delta a/{f n}$.\n\tThe position of the EW spot remains fixed.\n M is a mirror.}\n \\label{fig:SetupFigure}\n\\end{figure}\n\nThe optical dipole potential for a {two-level} atom at a distance $z$ above the prism\ncan then be written as \t${\\cal U}_{dip}(z)={\\cal U}_0 \\exp(-2\\kappa z)$.\nIn the limit of large laser detuning, $\\left\|\\delta\\right\|\\gg\\Gamma$,\nand low saturation, $s_0\\ll 1$, the maximum potential at the prism surface is\n${\\cal U}_0=\\hbar\\delta s_0$/2, with a saturation parameter \n$s_0\\simeq(\\Gamma/{2\\delta})^2\\,T I/{I_0}$ \\cite{CohDupGry92}.\nHere, {$I_0=1.65~$mW/cm$^2$} is the saturation intensity for rubidium and\n{$\\Gamma=2\\pi\\times 6.0~$MHz} is the natural linewidth. The intensity of the\nincoming beam in the glass substrate is given as $I$. It is {\\em enhanced} by\na factor $T$, that ranges between 5.4 and 6.0 for our TM polarized EW\n\\cite{FresnelCoefficients}. The detuning of the laser frequency $\\omega_L$ with\nrespect to the atomic transition frequency\n$\\omega_0$ is defined as $\\delta=\\omega_L -\\omega_0$. Thus a ``blue'' detuning,\n$\\delta>0$, yields an exponential potential barrier for incoming atoms.\nGiven an incident atom with momentum $p_i$, a classical turning point of the\nmotion exists if the barrier height exceeds the kinetic energy $p_i^2/2M$ of\nthe atom.\n\nFor a purely optical potential, the barrier height is given by ${\\cal U}_0$.\nIn reality, the potential is also influenced by gravity and the attractive\n{van der Waals} potential,\n \\begin{equation}\n {\\cal U}={\\cal U}_{dip}+{\\cal U}_{grav}+{\\cal U}_{vdW}.\n \\label{eq:TotalPotential}\n \\end{equation}\nThe gravitational potential ${\\cal U}_{grav}(z)\\propto z$ can be neglected on\nthe length scale of the EW decay length $\\xi\\equiv 1/\\kappa$. The van der Waals\npotential ${\\cal U}_{vdW}(z)\\propto\\left(k_0 z\\right)^{-3}$ significantly\nlowers the maximum potential close to the prism and thus decreases the\neffective mirror surface on which atoms still can bounce. This effect has been\ndemonstrated by Landragin {\\it et al.} \\cite{LanCouLab96}.\n\n\n\\subsection{Photon scattering by bouncing atoms}\n\nThe photon scattering rate of a two-level atom in steady state at low\nsaturation can be written as\n$\\Gamma'=s\\Gamma/2=(\\Gamma/\\hbar\\delta)\\,{\\cal U}_{dip}$ \\cite{CohDupGry92}.\nAn atom bouncing on an EW mirror sees a time dependent saturation\nparameter $s(t)$.\nAssuming that the excited state population follows adiabatically,\nwe can integrate the scattering rate along an atom's trajectory to get the\nnumber of scattered photons,\n \\begin{equation}\n N_{scat}=\\int{\\Gamma'(t)}dt\n =\\frac{\\Gamma}{\\hbar\\delta}\n \\int_{-p_i}^{+p_i}{(\\frac{{\\cal U}_{dip}}{-\\partial_z {\\cal U}})}dp.\n \\label{eq:PathIntegral}\n \\end{equation}\nFor a purely optical potential, ${\\cal U}\\propto\\exp(-2\\kappa z)$, this leads\nto an analytical solution:\n \\begin{equation}\n N_{scat}=\\frac{\\Gamma}{\\delta}\\frac{p_i}{\\hbar\\kappa}.\n \\label{eq:AnalyticalSolution}\n \\end{equation}\nThe ``steepness'' of the\noptical potential is determined by $\\kappa$.\nThe steeper the potential,\nthe shorter the time an atom spends in the light field,\nand the lower $N_{scat}$.\nNote that $N_{scat}$ is independent of ${\\cal U}_0$,\nas a consequence of the $\\exp(-2\\kappa z)$ shape of the potential.\nThis is in fact the reason why a {two-level} description of the atoms is appropriate.\nStrictly speaking, for a realistic atom\nthe potential strength depends on the magnetic sublevel $m_F$\nthrough the {Clebsch-Gordan} coefficients.\nFor example, our {EW} is {TM}-polarized which,\nclose to the critical angle, is approximately linear.\nEffectively, for every $m_F$ sublevel,\n${\\cal U}_0$ is then multiplied by the square of a Clebsch-Gordan coefficient,\nranging from 1/3 to 3/5 (for a $F=2\\rightarrow F'=3$ transition).\nRemarkably though,\nthe value of $N_{scat}$ remains unaffected and the same for all atoms.\n\nOne expects that an absorbed photon gives a recoil momentum $p_{rec}=\\hbar k_x$\nto the atom, directed along the propagating component of the EW. Experimentally\nwe observe this effect by the altered horizontal velocity of atom clouds after\nthe bounce. The spontaneous emission of photons leads to heating of the cloud\nand thus to thermal expansion \\cite{LanLabHen96,HenMolKaiA97}.\n\nIn principle, $N_{scat}$ is changed if other than optical forces are present.\nFor example, the van der Waals attraction tends to ``soften'' the potential\nand thus to increase $N_{scat}$. We investigated this numerically and found it\nto be negligible in our present experiment.\n\n\n\\section{The observation of radiation pressure}\n\n\\subsection{Experimental setup}\n\nOur experiment is performed in a rubidium vapor cell. We trap about $10^{7}$\natoms of {$^{87}$Rb} in a magneto-optical trap (MOT) and subsequently cool them\nin optical molasses to {10~$\\mu$K}. The MOT is centered {6.6~mm} above the\nhorizontal surface of a right-angle BK7 prism\n($n=1.51$, $\\theta_c=41.4^{\\circ}$ \\cite{MellesGriotPrism}),\nas shown in {Fig.\\,\\ref{fig:SetupFigure}a.\n\nThe EW beam emerges from a single-mode optical fiber, is collimated and\ndirected to the prism through a relay telescope\n(see {Fig.\\,\\ref{fig:SetupFigure}b}).\nThe angle of incidence $\\theta$ is controlled by the vertical displacement\n$\\Delta a$ of the first telescope lens, $L_1$.\nThis lens directs the beam, whereas the second lens, $L_2$,\nimages it to a {\\em fixed} spot at the prism surface.\nA displacement $\\Delta a$ leads to a variation in $\\theta$,\ngiven by $\\Delta\\theta=\\Delta a/{n f}$.\nThe focal length of both lenses is {$f=75~$mm}.\nThe beam has a minimum waist of {$335~\\mu$m} at the surface\n($1/e^2$ intensity radius).\nWe checked the beam collimation and found it almost diffraction limited with\na divergence half-angle of less than {1~mrad}.\nWe use TM polarization for the EW because it yields a stronger dipole\npotential than a TE polarized beam of the same power.\nClose to the critical angle, the ratio of potential heights is\napproximately $n^2$ \\cite{FresnelCoefficients}.\n\nFor the EW, an injection-locked single mode laser diode\n({\\it Hitachi} HL7851G98) provides us with up to {28~mW} of optical power\nbehind the fiber.\nIt is seeded by an external grating stabilized diode laser,\nlocked to the {$^{87}$Rb} hyperfine transition\n$5S_{1/2}(F=2)\\rightarrow 5P_{3/2}(F'=3)$ of the $D_2$ line ({780 nm}),\nwith a natural linewidth {$\\Gamma=2\\pi\\times 6.0~$MHz}.\nThe detuning $\\delta$ with respect to this transition defines the optical\npotential ${\\cal U}_{dip}$ for atoms that are released from the MOT in the\n$F=2$ ground state.\nWe adjust detunings up to {$2\\pi\\times 500~$MHz} by frequency shifting the\nseed beam from resonance, using an acousto-optic modulator.\nFor larger detunings we unlock the seed laser and set its frequency manually,\naccording to the reading of an optical spectrum analyzer with {1~GHz}\nfree spectral range.\n\nAtoms that have bounced on the EW mirror are detected by induced fluorescence\nfrom a pulsed probe beam, resonant with the $F=2\\rightarrow F'=3$ transition.\nThe probe beam travels in the vertical downward direction and has a diameter\nof {10~mm}.\nThe fluorescence is recorded from the side, in the $y$ direction,\nby a digital frame-transfer CCD camera ({\\it Princeton Instruments}).\nThe integration time is chosen between {0.1~ms} and {1~ms} and is matched to\nthe duration of the probe pulse.\nEach camera image consists of $400\\times 400$ pixels, that were\nhardware-binned on the CCD chip in groups of four pixels.\nThe field of view is {$10.2\\times10.2~$mm$^2$} with a spatial resolution of\n{$51~\\mu$m} per pixel.\n\nA typical timing sequence of the experiment is as follows.\nThe MOT is loaded from the background vapor during {1~s}.\nAfter {4~ms} of polarization gradient cooling in optical molasses the atoms\nare released in the $F=2$ ground state by closing a shutter in the cooling\nlaser beams.\nThe image capture is triggered with a variable time delay after releasing\nthe atoms.\nDuring the entire sequence, the EW laser is permanently on.\nIn addition, a permanent repumping beam, tuned to the\n$5S_{1/2}(F=1)\\rightarrow 5P_{1/2}(F'=2)$ transition of the $D_1$ line\n({795~nm}), counteracts optical pumping to the $F=1$ ground state by the\nprobe. We observed no significant influence on the performance of the EW mirror\nby the repumping light.\n\nWe measure the trajectories of bouncing atoms by taking a series of images with\nincremental time delays. A typical series with increments of {10~ms} between\nthe images is shown in {Fig.\\,\\ref{fig:TimeSequence}}. Our detection destroys\nthe atom cloud, so a new sample was prepared for each image. The exposure time\nwas {0.5~ms}. Each image has been averaged over 10 shots. The fourth image\nshows the cloud just before the average bouncing time, {$\\bar{t_b}=36.7~$ms},\ncorresponding to the fall height of {6.6~mm}. In later frames we see the atom\ncloud bouncing up from the surface. Close to the prism, the fast vertical\nmotion causes blurring of the image. Another cause of vertical blur is motion\ndue to radiation pressure by the probe pulse. The horizontal motion of the\nclouds was not affected by the probe. We checked this by comparing with images\ntaken with considerably shorter probe pulses of {0.1~ms} duration.\n\n\\begin{figure}[t]\n \\centerline{\\epsfxsize=8.0cm\\epsffile{radpre2cfig2.eps}}\n \\vspace{0.5cm}\n \\caption{Fluorescence images of a bouncing atom cloud.\n\tThe first image was taken {5~ms} after releasing the atoms from the MOT.\n\tThe configuration of prism and evanescent wave is illustrated by the\n\tschematic (Field of view: {$10.2\\times 10.2~$mm$^2$}).} \n \\label{fig:TimeSequence}\n\\end{figure}\n\n\n\\subsection{Results}\n\nRadiation pressure in the evanescent wave was observed by analyzing the\nhorizontal motion of the clouds. From the camera images we determine the center\nof mass position of the clouds to about $\\pm1$ pixel accuracy.\nIn {Fig.\\,\\ref{fig:Trajectories}}, we plot the horizontal position vs. time\nelapsed since release from the MOT. We find that the horizontal motion is\nuniform before and after the bounce. The horizontal velocity changes suddenly\nduring the bounce as a consequence of scattering EW photons. The change in\nvelocity is obtained from a linear fit.\n\n\\begin{figure}[thb]\n \\centerline{\\epsfxsize=9.0cm\\epsffile{radpre2cfig3.eps}}\n \\vspace{0.5cm}\n\\caption{Horizontal motion of bouncing atom clouds.\n\tThe center of mass position is plotted vs. time elapsed since release.\n\tThe bounces occur at {36.7~ms} (vertical dotted line). \n (a)\tThe EW decay length is varied as $\\xi\\left(\\theta\\right)=$ \n\t$1.87,~1.03,~0.79,~0.67,~0.59,~0.53~\\mu$m\n\t(from large to small change in velocity).\n\tThe detuning is $44~\\Gamma$ and the optical power is {19~mW}.\n (b)\tComparison of two values of EW optical power, {19~mW} (solid points)\n\tand {10.5~mW} (open points). The detuning is $31~\\Gamma$\n\tand the EW decay lengths are {$\\xi=1.87~\\mu$m} (large velocity change)\n\tand {$0.67~\\mu$m} (small change). Solid lines indicate linear fits.}\n\\label{fig:Trajectories}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centerline{\\epsfxsize=9.0cm\\epsffile{radpre2cfig4.eps}}\n \\vspace{0.5cm}\n \\caption{Radiation pressure on bouncing atoms expressed as number of absorbed\n\tphotons, ${\\cal N}_{scat}$.\n (a)\tDetuning $\\delta$ varied for {$\\xi=2.8~\\mu$m} (open points) and\n\t{$0.67~\\mu$m} (solid points).\n (b)\tEW decay length $\\xi$ varied for $\\delta =44~\\Gamma$.\n\tThe laser power was {19~mW}. The thin solid line is a linear fit\n\tthrough the first four data points.\n\tTheoretical predictions: two-level atom\n\t(see Eq.(\\ref{eq:AnalyticalSolution}), thick solid lines).\n\tRubidium excited-state hyperfine structure and saturation taken into\n\taccount\t(dashed lines).}\n\\label{fig:Recoils}\n\\end{figure}\n\nIn {Fig.\\,\\ref{fig:Recoils}}, we show how the radiation pressure depends on the\nlaser detuning $\\delta$ and on the angle of incidence $\\theta$. The fitted\nhorizontal velocity change has been expressed in units of the EW photon recoil,\n$p_{rec}=\\hbar k_0 n\\sin\\theta$, with {$\\hbar k_0/M=5.88~$mm/s}.\n\nIn {Fig.\\,\\ref{fig:Recoils}a}, the detuning is varied from {188-1400~MHz}\n($31-233~\\Gamma$).\nTwo sets of data are shown, taken for two different angles,\n{$\\theta=\\theta_c+0.9~$mrad} and {$\\theta_c+15.2~$mrad}.\nThis corresponds with an EW decay length {$\\xi(\\theta)=2.8~\\mu$m} and\n{$0.67~\\mu$m}, respectively.\nWe find that the number of scattered photons is inversely proportional to\n$\\delta$, as expected.\nThe predictions based on {Eq.\\,(\\ref{eq:AnalyticalSolution})} are indicated in\nthe figure (solid lines).\n\nIn {Fig.\\,\\ref{fig:Recoils}b}, the detuning was kept fixed at $44~\\Gamma$ and\nthe angle of incidence was varied between {0.9~mrad} and {24.0~mrad} above the\ncritical angle $\\theta_c$. This leads to a variation of\n$\\xi\\left(\\theta\\right)$ from {$2.8~\\mu$m} to {$0.53~\\mu$m}. Here also, we find\na linear dependence on $\\xi\\left(\\theta\\right)$. We see clearly, that a steep\noptical potential, {i.e.} a small decay length, causes less radiation pressure\nthan a shallow potential.\n\nThe observed radiation pressure ranges from 2 to 31 photon recoils per atom.\nNote, that we separate this subtle effect from the faster vertical motion,\nin which atoms enter the optical potential with a momentum of\n$p_i\\simeq 61~p_{rec}$.\n\nIn {Fig.\\,\\ref{fig:Trajectories}b}, we compare trajectories for {$19\\pm1~$mW}\nand {$10.5\\pm0.5~$mW} optical power in the EW.\nAs expected from {Eq.\\,(\\ref{eq:AnalyticalSolution})}, there is no significant\ndifference in horizontal motion. For a decay length of {$\\xi=2.78~\\mu$m}\n({$0.67~\\mu$m}) {\\em both} power settings lead to essentially the same\nradiation pressure, that is $25\\pm3$ ($13\\pm2$) scattered photons for {19~mW},\nand $23\\pm2$ ($11\\pm1$) photons for {10.5~mW}. The optical power only\ndetermines the effective mirror surface and thus the fraction of bouncing\natoms. This is also visible in the horizontal width of bouncing clouds.\nThe bouncing fraction scales with the intensity and the detuning as\n$\\propto\\ln (I/\\delta)$. We observe typical fractions of {13\\,\\%} for\n$\\delta=44~\\Gamma$. For a given optical power, there is an upper limit for the\ndetuning, above which no bounce can occur.\nFor the data in {Fig.\\,\\ref{fig:Recoils}a}, the threshold is calculated as\n{$\\delta_{th}=6.5~$GHz} ({8.1 GHz}) for {$\\xi=2.8~\\mu$m} ({$0.67~\\mu$m}).\nAnother threshold condition is implied by the van der Waals interaction,\nand yields a lower limit for the EW decay length $\\xi$.\nFor {Fig.\\,\\ref{fig:Recoils}b} this lower limit is calculated as\n{$\\xi_{th}=116~$nm}, {i.e.} {$\\theta_{th}=(\\theta_c+0.59~$rad$)$}.\n\n\n\\subsection{Systematic errors and discussion}\n\nAccording to {Eq.~(\\ref{eq:AnalyticalSolution})}, the radiation pressure\nshould be inversely proportional to both $\\delta$ and $\\kappa(\\theta)$.\nAs shown in {Fig.\\,\\ref{fig:Recoils}}, we find deviations from this\nexpectation in our experiment, particularly in the $\\kappa$ dependence.\nA linear fit to the data for {$\\xi<1~\\mu$m}, extrapolates to an offset of\napproximately 3 photon recoils in the limit $\\xi\\rightarrow 0$\n(thin solid line in {Fig.\\,\\ref{fig:Recoils}b}). The vertical error bars on the\ndata include statistical errors in the velocity determination from the cloud\ntrajectories as well as systematic errors. We discuss several possible\nsystematic errors, namely\n(i) the geometric alignment,\n(ii) the EW beam angle calibration and collimation,\n(iii) diffusely scattered light,\n(iv) the van der Waals atom-surface interaction,\n(v) excited state contributions to the optical potential,\n(vi) and saturation effects.\n\n(i) Geometrical misalignments give rise to systematic errors in the radiation\npressure measurements. For example, a tilt of the prism causes a horizontal\nvelocity change even for specularly reflected atoms. We checked the prism\nalignment and found it tilted {$12\\pm5~$mrad} from horizontal. This corresponds\nto an offset of $1.5\\pm0.6$ recoils on $N_{scat}$. In addition, the atoms are\n``launched'' from the MOT with a small initial horizontal velocity which we\nfound to correspond to less than $\\pm0.4$ recoils for all our data.\nFrom {Fig.\\,\\ref{fig:Trajectories}}, we see that the extrapolated trajectories\nat the bouncing time $\\bar{t_b}$ do not start from the horizontal position\nbefore the bounce. We attribute this to a horizontal misalignment of the MOT\nwith respect to the EW spot. Obviously, there is a small displacement of the EW\nspot at the prism surface, when adjusting $\\theta$ by means of the lens $L_1$\n(see {Fig.\\,\\ref{fig:SetupFigure}b}).\nSince the finite sized EW mirror reflects only part of the thermally expanding\natom cloud, such a displacement selects a nonzero horizontal velocity for\nbouncing atoms. We corrected for those alignment effects in the radiation\npressure data of {Fig.\\,\\ref{fig:Recoils}}.\nFor small radiation pressure values, the systematic error due to\nalignment is the dominant contribution in the vertical error bar.\n\n(ii) The uncertainty in the EW angle with respect to the critical angle is\nexpressed by the horizontal error bars. We determined $\\theta-\\theta_c$ within\n{$\\pm0.2~$mrad} by monitoring the optical power transmitted through the prism\nsurface, while tuning the angle $\\theta$ from below to above $\\theta_c$. Close\nto the critical angle, the decay length $\\xi(\\theta)$ diverges, and thus the\nerror bar on $\\xi$ becomes very large.\nAlso the diffraction-limited divergence of the EW beam may become significant.\nIt causes part of the optical power to propagate into the vacuum. In addition,\nthe optical potential is governed by a whole distribution of decay lengths.\nThus the model of a simple exponential optical potential\n$\\propto\\exp{\\left(-2\\kappa z\\right)}$ might not be valid and cause the\ndisagreement of our data with the prediction by\n{Eq.\\,(\\ref{eq:AnalyticalSolution})}. \nFor larger angles, {i.e.} {$\\xi(\\theta)<1~\\mu$m}, the effect of the beam\ndivergence is negligible. This we could verify by numerical analysis.\n\n(iii) Light from the EW can diffusely scatter and propagate into the vacuum\ndue to roughness of the prism surface. We presume this is the reason for the\nextrapolated offset of 3 photon recoils in the radiation pressure\n({Fig.\\,\\ref{fig:Recoils}b}). A preferential light scattering in the direction\nof the EW propagating component can be explained, if the power spectrum of the\nsurface roughness is narrow compared to $1/\\lambda$ \\cite{HenMolKaiA97}.\nThe effect of surface roughness on bouncing atoms has previously been observed\n\\cite{LanLabHen96} as a broadening of atom clouds by the roughness of the\n{\\em dipole potential}.\nIn our case, we observe a change in center of mass motion of the clouds\ndue to an increase in the {\\em spontaneous scattering force}.\nSuch a contribution to the radiation pressure due to surface roughness\nvanishes in the limit of large\ndetuning $\\delta$. Thus, we find no significant offset in\n{Fig.\\,\\ref{fig:Recoils}a}. Scattered light might also be the reason for the\nsmall difference in radiation pressure for the two distinct EW power settings,\nshown in {Fig.\\,\\ref{fig:Trajectories}b}. Lower optical power implies slightly\nless radiation pressure.\n\n(iv) As stated above, the van der Waals interaction ``softens'' the mirror\npotential.\nThis makes bouncing atoms move longer in the light field,\nthus enhancing photon scattering.\nWe investigated this numerically by integrating the scattering rate along the\natoms' path, including the van der Waals contribution to the mirror potential.\nEven for our shortest decay parameter of $0.53~\\mu$m, the number of scattered\nphotons would increase only about {0.8\\,\\%} compared with\n{Eq.\\,(\\ref{eq:AnalyticalSolution})}, which we cannot resolve experimentally\n\\cite{vdWaalsSample}.\n\n(v) In a two-level atom model, the scattering rate can be expressed\nin the dipole potential as $\\Gamma'=(\\Gamma/{\\hbar\\delta})\\,{\\cal U}_{dip}$.\nThis is no longer true if we take into account the excited state manifold\n$F'=\\{0,1,2,3\\}$ of {$^{87}$Rb}.\nBeside $F'=3$, also $F'=2$ contributes significantly to the mirror potential,\nwhereas it does not much affect the scattering rate.\nWith an EW detuning of $44\\,\\Gamma$, this results in a number of scattered\nphotons, $N_{scat}$, typically $9\\,\\%$ lower than expected for a {two-level}\natom, {Eq.\\,(\\ref{eq:AnalyticalSolution})}.\nHere we averaged over contributions from distinct magnetic sublevels.\n\n(vi) In order to investigate the influence of saturation on the number of\nscattered photons, $N_{scat}$, we solved the optical Bloch equations\nnumerically. A bouncing atom encounters the EW as a fast varying light pulse\n$I(t)\\propto~$sech$^2(\\kappa p_i t/M)$, with a typical duration between 3 and\n{$10~\\mu$s}. By integrating the time-dependent scattering rate for an atom\nbouncing with an EW detuning of $44\\,\\Gamma$, we find\napproximately $7\\,\\%$ less scattered photons compared with the unsaturated\nexpression of {Eq.\\,(\\ref{eq:AnalyticalSolution})}.\nNote, that the bounces occur sufficiently slow to preserve adiabaticity.\nIn {Fig.\\,\\ref{fig:Recoils}}, we show predicted curves,\ncorrected for hyperfine structure and saturation (dashed solid lines).\n\n\n\\section{Conclusions}\n\nWe have directly observed radiation pressure that is exerted on rubidium atoms\nwhile bouncing on an evanescent-wave atom mirror. We did so by analyzing the\nbouncing trajectories. The radiation pressure is directed parallel to the\npropagating component of the EW, {i.e.} parallel to the interface.\nWe observe 2 to 31 photon recoils per atom\nper bounce. We find the radiation pressure to be independent of the optical\npower in the EW, as expected from the exponential character of the EW.\n\nThe inverse proportionality to both the EW detuning and the angle of incidence\nis in reasonable agreement with a simple two-level atom calculation, using\nsteady state expressions for the EW optical potential and the photon scattering\nrate. The agreement improved when also the excited state hyperfine structure\nand saturation effects were taken into account. The measured number of photon\nrecoils as a function of decay length $\\xi$ indicates an offset of approximately\n3 recoils in the limit of a very steep EW potential. We assume this is due to\nlight that is diffusely scattered due to roughness of the prism surface, but\nretains a preferential forward direction parallel with the EW propagating\ncomponent.\n\nWith sufficient resolution, it should be possible to resolve the discrete\nnature of the number of photon recoils and also their magnitude,\n$\\hbar k_x>\\hbar k_0$ \\cite{MatTakHir98}. \nOur technique could also be used to observe quantum electrodynamical effects\nfor atoms in the vicinity of a surface, such as radiation pressure out of the\ndirection of the propagating EW component \\cite{HenCou98}.\n\n\n\\section{Acknowledgments}\n\nWe wish to thank E.C. Schilder for help with the experiments. \nThis work is part of the research program of the ``Stichting voor Fundamenteel\nOnderzoek van de Materie'' (Foundation for the Fundamental Research on Matter)\nand was made possible by financial support from the ``Nederlandse Organisatie\nvoor Wetenschappelijk Onderzoek'' (Netherlands Organization for the Advancement\nof Research). R.S. has been financially supported by the Royal Netherlands\nAcademy of Arts and Sciences.\n\n\n\\bibliographystyle{prsty}\n\n\\begin{thebibliography}{10}\n\n\\bibitem{CooHil82}\n R.J. Cook and R.K. Hill, Opt. Comm. {\\bf 43}, 258 (1982).\n\n\\bibitem{AdaSigMly94}\n C.S. Adams, M. Sigel, and J. Mlynek, Phys. Rep. {\\bf 240}, 143 (1994).\n\n\\bibitem{BalLetOvc87}\n V.I. Balykin, V.S. Letokhov, Yu.B. Ovchinnikov, and A.I. Sidorov,\n Pis'ma Zh. Eksp. Teor. Fiz. {\\bf 45}, 282 (1987)\n [JETP Lett. {\\bf 45}, 353 (1987)].\n\n\\bibitem{KasWeiChu90}\n M.A. Kasevich, D.S. Weiss, and S. Chu, Opt. Lett. {\\bf 15}, 607 (1990).\n\n\\bibitem{SeiAdaBal94}\n W. Seifert, C.S. Adams, V.I. Balykin, C. Heine, Yu. Ovchinnikov, and J. Mlynek,\n Phys. Rev. A {\\bf 49}, 3814 (1994).\n\n\\bibitem{GorAsh80}\n J.P. Gordon and A. Ashkin, Phys. Rev. A {\\bf 21},1606 (1980).\n\n\\bibitem{Coo80}\n R.J. Cook, Phys. Rev. A {\\bf 22}, 1078 (1980).\n\n\\bibitem{KawSug92}\n S. Kawata and T. Sugiura, Opt. Lett. {\\bf 17}, 772 (1992).\n\n\\bibitem{MatTakHir98}\n T. Matsudo, Y. Takahara, H. Hori, and T. Sakurai, Opt. Comm. {\\bf 145}, 64 (1998).\n \n\\bibitem{SurfaceTraps}\n Yu. B. Ovchinnikov, S.V. Shul'ga, and V.I. Balykin,\n J. Phys. B: At. Mol. Opt. Phys. {\\bf 24}, 3173 (1991);\n P. Desbiolles and J. Dalibard, Optics Comm. {\\bf 132}, 540 (1996);\n Yu.B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. {\\bf 79}, 2225 (1997);\n W.L. Power, T. Pfau, and M. Wilkens, Optics Comm. {\\bf 143}, 125 (1997);\n H. Gauck, M. Hartl, D. Schneble,H. Schnitzler, T. Pfau, and J. Mlynek,\n Phys. Rev. Lett. {\\bf 81}, 5298 (1998).\n\n\\bibitem{SprVoiWol}\n R.J.C. Spreeuw, D. Voigt, B.T. Wolschrijn, and H.B. van Linden van den Heuvell,\n [quant-ph/9911017 (1999)].\n\n\\bibitem{KetDurSta99}\n For a recent review on {Bose-Einstein} condensation see\n W. Ketterle, D.S. Durfee, and D.M. Stamper-Kurn,\n in {\\em Making, probing and understanding Bose-Einstein condensates},\n Proc. of the Int. School of Physics ``Enrico Fermi,''\n ed. by M. Inguscio, S. Stringari, and C. Wieman [cond-mat/9904034 (1999)].\n\n\\bibitem{BECAtomLasers}\n {M.-O.} Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, and W. Ketterle,\n Phys. Rev. Lett. {\\bf 78}, 582 (1997);\n B.P. Anderson and M.A. Kasevich, Science {\\bf 282}, 1686 (1998);\n E. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S.L. Rolston, and W.D. Phillips,\n Science {\\bf 283}, 1706 (1999);\n I. Bloch, T.W. {H{\\\"a}nsch}, and T. Esslinger,\n Phys. Rev. Lett. {\\bf 82}, 3008 (1999).\n\n\\bibitem{Hec87}\n E. Hecht, {\\em Optics} (Addison-Wesley, 1987).\n\n\\bibitem{CohDupGry92}\n C.N. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,\n {\\em Atom-Photon Interactions} (Wiley, New York, 1992).\n\n\\bibitem{FresnelCoefficients}\n The enhancement factor of the EW ``intensity'' is calculated for TM and TE\n polarization as\n $T_{TM}={4 n \\cos^2\\theta\\,(2 n^2 \\sin^2\\theta-1)}\n /{(\\cos^2\\theta+n^2(n^2 \\sin^2\\theta-1))}$ and\n $T_{TE}={4 n \\cos^2\\theta}/{(n^2-1)}$.\n\n\\bibitem{LanCouLab96}\n A. Landragin, J.-Y. Courtois, G. Labeyrie, N. Vansteenkiste, C.I. Westbrook, and A. Aspect,\n Phys. Rev. Lett. {\\bf 77}, 1464 (1996).\n\n\\bibitem{LanLabHen96}\n A. Landragin, G. Labeyrie, C. Henkel, R. Kaiser, N. Vansteenkiste, C.I. Westbrook, and\n A. Aspect, Opt. Lett. {\\bf 21}, 1591 (1996).\n\n\\bibitem{HenMolKaiA97}\n C. Henkel, K. M{\\o}lmer, R. Kaiser, N. Vansteenkiste, C.I. Westbrook, and A. Aspect,\n Phys. Rev. A {\\bf 55}, 1160 (1997).\n\n\\bibitem{MellesGriotPrism}\n {\\it Melles Griot}, high precision prism, order no. 01\\,PRB\\,009. \n We have cut it to a size of {$10\\times 10\\times 4$ mm$^3$}.\n\n\\bibitem{vdWaalsSample}\n As an example where van der Waals interaction is significant: For\n {$\\delta=1~$GHz}, {$\\xi=370~$nm}, and {2.5~mW} optical power, the calculated\n number of scattered photons per bounce increases from 1.09 to 1.13 due to the\n van der Waals interaction.\n \n\\bibitem{HenCou98}\n C. Henkel and J.-Y. Courtois, Eur. Phys. J. D {\\bf 3}, 129 (1998).\n\n\\end{thebibliography}\n\n\\end{document}\n\n" } ]	[ { "name": "quant-ph9912101.extracted_bib", "string": "{CooHil82 R.J. Cook and R.K. Hill, Opt. Comm. {43, 258 (1982)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{AdaSigMly94 C.S. Adams, M. Sigel, and J. Mlynek, Phys. Rep. {240, 143 (1994)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{BalLetOvc87 V.I. Balykin, V.S. Letokhov, Yu.B. Ovchinnikov, and A.I. Sidorov, Pis'ma Zh. Eksp. Teor. Fiz. {45, 282 (1987) [JETP Lett. {45, 353 (1987)]." }, { "name": "quant-ph9912101.extracted_bib", "string": "{KasWeiChu90 M.A. Kasevich, D.S. Weiss, and S. Chu, Opt. Lett. {15, 607 (1990)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{SeiAdaBal94 W. Seifert, C.S. Adams, V.I. Balykin, C. Heine, Yu. Ovchinnikov, and J. Mlynek, Phys. Rev. A {49, 3814 (1994)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{GorAsh80 J.P. Gordon and A. Ashkin, Phys. Rev. A {21,1606 (1980)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{Coo80 R.J. Cook, Phys. Rev. A {22, 1078 (1980)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{KawSug92 S. Kawata and T. Sugiura, Opt. Lett. {17, 772 (1992)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{MatTakHir98 T. Matsudo, Y. Takahara, H. Hori, and T. Sakurai, Opt. Comm. {145, 64 (1998)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{SurfaceTraps Yu. B. Ovchinnikov, S.V. Shul'ga, and V.I. Balykin, J. Phys. B: At. Mol. Opt. Phys. {24, 3173 (1991); P. Desbiolles and J. Dalibard, Optics Comm. {132, 540 (1996); Yu.B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. {79, 2225 (1997); W.L. Power, T. Pfau, and M. Wilkens, Optics Comm. {143, 125 (1997); H. Gauck, M. Hartl, D. Schneble,H. Schnitzler, T. Pfau, and J. Mlynek, Phys. Rev. Lett. {81, 5298 (1998)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{SprVoiWol R.J.C. Spreeuw, D. Voigt, B.T. Wolschrijn, and H.B. van Linden van den Heuvell, [quant-ph/9911017 (1999)]." }, { "name": "quant-ph9912101.extracted_bib", "string": "{KetDurSta99 For a recent review on {Bose-Einstein condensation see W. Ketterle, D.S. Durfee, and D.M. Stamper-Kurn, in {\\em Making, probing and understanding Bose-Einstein condensates, Proc. of the Int. School of Physics ``Enrico Fermi,'' ed. by M. Inguscio, S. Stringari, and C. Wieman [cond-mat/9904034 (1999)]." }, { "name": "quant-ph9912101.extracted_bib", "string": "{BECAtomLasers {M.-O. Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, and W. Ketterle, Phys. Rev. Lett. {78, 582 (1997); B.P. Anderson and M.A. Kasevich, Science {282, 1686 (1998); E. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S.L. Rolston, and W.D. Phillips, Science {283, 1706 (1999); I. Bloch, T.W. {H{\\\"ansch, and T. Esslinger, Phys. Rev. Lett. {82, 3008 (1999)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{Hec87 E. Hecht, {\\em Optics (Addison-Wesley, 1987)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{CohDupGry92 C.N. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, {\\em Atom-Photon Interactions (Wiley, New York, 1992)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{FresnelCoefficients The enhancement factor of the EW ``intensity'' is calculated for TM and TE polarization as $T_{TM={4 n \\cos^2\\theta\\,(2 n^2 \\sin^2\\theta-1) /{(\\cos^2\\theta+n^2(n^2 \\sin^2\\theta-1))$ and $T_{TE={4 n \\cos^2\\theta/{(n^2-1)$." }, { "name": "quant-ph9912101.extracted_bib", "string": "{LanCouLab96 A. Landragin, J.-Y. Courtois, G. Labeyrie, N. Vansteenkiste, C.I. Westbrook, and A. Aspect, Phys. Rev. Lett. {77, 1464 (1996)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{LanLabHen96 A. Landragin, G. Labeyrie, C. Henkel, R. Kaiser, N. Vansteenkiste, C.I. Westbrook, and A. Aspect, Opt. Lett. {21, 1591 (1996)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{HenMolKaiA97 C. Henkel, K. M{\\olmer, R. Kaiser, N. Vansteenkiste, C.I. Westbrook, and A. Aspect, Phys. Rev. A {55, 1160 (1997)." }, { "name": "quant-ph9912101.extracted_bib", "string": "{MellesGriotPrism {Melles Griot, high precision prism, order no. 01\\,PRB\\,009. We have cut it to a size of {$10\\times 10\\times 4$ mm$^3$." }, { "name": "quant-ph9912101.extracted_bib", "string": "{vdWaalsSample As an example where van der Waals interaction is significant: For {$\\delta=1~$GHz, {$\\xi=370~$nm, and {2.5~mW optical power, the calculated number of scattered photons per bounce increases from 1.09 to 1.13 due to the van der Waals interaction." }, { "name": "quant-ph9912101.extracted_bib", "string": "{HenCou98 C. Henkel and J.-Y. Courtois, Eur. Phys. J. D {3, 129 (1998)." } ]
quant-ph9912102	Field quantization by means of a single harmonic oscillator	[ { "author": "Marek~Czachor" } ]	A new scheme of field quantization is proposed. Instead of associating with different frequencies different oscillators we begin with a single oscillator that can exist in a superposition of different frequencies. The idea is applied to the electromagnetic radiation field. Using the standard Dirac-type mode-quantization of the electromagnetic field we obtain several standard properties such as coherent states or spontaneous and stimulated emission. As opposed to the standard approach the vacuum energy is finite and does not have to be removed by any ad hoc procedure.	[ { "name": "quant-ph9912102.tex", "string": "\\tolerance = 10000\n%\\documentstyle[aps,pra,twocolumn]{revtex}\n\\documentstyle[aps,pra]{revtex}\n%\\documentstyle[aps,preprint]{revtex}\n%\\textwidth18cm\n\\begin{document}\n\\draft\n\n\n\\newcommand{\\pp}[1]{\\phantom{#1}}\n\\newcommand{\\be}{\\begin{eqnarray}}\n\\newcommand{\\ee}{\\end{eqnarray}}\n\\newcommand{\\ve}{\\varepsilon}\n\\newcommand{\\vs}{\\varsigma}\n\\newcommand{\\Tr}{{\\,\\rm Tr\\,}}\n\\newcommand{\\pol}{\\frac{1}{2}}\n\n\\title{\nField quantization by means of a single harmonic oscillator\n}\n\\author{Marek~Czachor}\n\\address{\nKatedra Fizyki Teoretycznej i Metod Matematycznych\\\\\nPolitechnika Gda\\'{n}ska,\nul. Narutowicza 11/12, 80-952 Gda\\'{n}sk, Poland\\\\\nand\\\\\nArnold Sommerferld Instit\\\"ut f\\\"ur Mathematische Physik\\\\\nTechnische Universit\\\"at Clausthal, 38678 Clausthal-Zellerfeld, Germany}\n\\maketitle\n\n\n\\begin{abstract}\nA new scheme of field quantization is proposed. Instead of\nassociating with different frequencies different oscillators we begin\nwith a single oscillator that can exist in a superposition of\ndifferent frequencies. \nThe idea is applied to the electromagnetic radiation field.\nUsing the standard Dirac-type mode-quantization of the electromagnetic\nfield we obtain several standard properties such as coherent states\nor spontaneous and stimulated emission. As opposed to the\nstandard approach the vacuum energy is finite and does not have to be\nremoved by any ad hoc procedure. \n\\end{abstract}\n\n%\\narrowtext\n\n\n\n\n\\section{Harmonic oscillator in superposition of frequencies}\n\nThe standard quantization of a harmonic oscillator is based on \nquantization of $p$ and $q$ but $\\omega$ is a parameter. To have,\nsay, two different frequencies one has to consider two independent \noscillators. On the other hand, it is evident that there exist\noscillators which are in a {\\it superposition\\/} of different\nfrequencies. The example is an oscillator wave packet associated with\ndistribution of center-of-mass momenta. \n\nThis simple observation raises the question of the role of\nsuperpositions of frequencies for a description of a single harmonic\noscillator. We know that frequency is typically associated with an\neigenvalue of some Hamiltonian or, which is basically the same, with\nboundary conditions. A natural way of incorporating different\nfrequencies into a single harmonic oscillator is by means of the\n{\\it frequency operator\\/} \n\\be\n\\Omega=\\sum_{\\omega_k,j_k}\\omega_k\|\\omega_k,j_k\\rangle\n\\langle \\omega_k,j_k\|\n\\ee\nwhere all $\\omega_k\\geq 0$. \nFor simplicity we have limited the discussion to the discrete\nspectrum but it is useful to include from the outset the possibility\nof degeneracies. The corresponding Hamiltonian is defined by\n\\be\nH\n&=&\n\\hbar\\Omega\\otimes \\frac{1}{2}\\big(a^{\\dag}a+aa^{\\dag}\\big)\n\\ee\nwhere $a=\\sum_{n=0}^\\infty\\sqrt{n+1}\|n\\rangle\\langle n+1\|$. \nThe eigenstates of $H$ are $\|\\omega_k,j_k,n\\rangle$ and satisfy \n\\be\nH\|\\omega_k,j_k,n\\rangle= \\hbar\\omega_k\\Big(n+\\frac{1}{2}\\Big)\n\|\\omega_k,j_k,n\\rangle.\n\\ee\nThe standard case of the oscillator whose frequency is just $\\omega$ \ncoresponds either to $\\Omega=\\omega\\bbox 1$ or to the subspace\nspanned by $\|\\omega_k,j_k,n\\rangle$ with fixed $\\omega_k=\\omega$. \nIntroducing the operators \n\\be\na_{\\omega_k,j_k}=\|\\omega_k,j_k\\rangle\\langle \\omega_k,j_k\|\\otimes a\n\\ee\nwe find that \n\\be \nH&=&\n\\frac{1}{2}\\sum_{\\omega_k,j_k}\\hbar\\omega_k\n\\Big(a_{\\omega_k,j_k}^{\\dag}a_{\\omega_k,j_k} \n+\na_{\\omega_k,j_k}a_{\\omega_k,j_k}^{\\dag}\\Big).\n\\ee\nThe algebra of the oscillator is\n\\be\n{[a_{\\omega_k,j_k},a_{\\omega_l,j_l}^{\\dag}]}&=&\n\\delta_{\\omega_k\\omega_l}\\delta_{j_kj_l}\n\|\\omega_k,j_k\\rangle\\langle\\omega_k,j_k\|\\otimes \\bbox 1\\\\\na_{\\omega_k,j_k}a_{\\omega_l,j_l}&=&\n\\delta_{\\omega_k\\omega_l}\\delta_{j_kj_l}(a_{\\omega_k,j_k})^2\\\\\na_{\\omega_k,j_k}^{\\dag}a_{\\omega_l,j_l}^{\\dag}&=&\n\\delta_{\\omega_k\\omega_l}\\delta_{j_kj_l}(a_{\\omega_k,j_k}^{\\dag})^2.\n\\ee\nThe dynamics in the Schr\\\"odinger picture is given by \n\\be\ni\\hbar\\partial_t\|\\Psi\\rangle\n&=&\nH \|\\Psi\\rangle=\n\\hbar\\Omega\\otimes \\big(a^{\\dag}a+\\frac{1}{2}\\bbox 1\\big)\n\|\\Psi\\rangle.\n\\ee\nIn the Heisenberg picture we obtain the important formula\n\\be\na_{\\omega_k,j_k}(t)\n&=&\ne^{iHt/\\hbar}a_{\\omega_k,j_k}e^{-iHt/\\hbar}\\\\\n&=&\n\|\\omega_k,j_k\\rangle\\langle \\omega_k,j_k\|\n\\otimes\ne^{-i\\omega_k t} a\n=\ne^{-i\\omega_k t} a_{\\omega_k,j_k}(0).\n\\label{exp}\n\\ee\nTaking a general state\n\\be\n\|\\psi\\rangle=\\sum_{\\omega_k,j_k,n}\\psi(\\omega_k,j_k,n)\|\\omega_k,j_k\\rangle\n\|n\\rangle\n\\ee\nwe find that the average energy of the oscillator \nis \n\\be\n\\langle H\\rangle=\n\\langle\\psi\|H\|\\psi\\rangle\n=\n\\sum_{\\omega_k,j_k,n}\|\\psi(\\omega_k,j_k,n)\|^2\n\\hbar\\omega_k\\Big(n+\\frac{1}{2}\\Big).\n\\ee\nThe average clearly looks as an average energy of an {\\it ensemble of\ndifferent and independent oscillators\\/}. The ground state of the\nensemble, i.e. the one with $\\psi(\\omega_k,j_k,n>0)=0$ \nhas energy \n\\be\n\\langle H\\rangle=\\frac{1}{2}\\sum_{\\omega_k,j_k}\|\\psi(\\omega_k,j_k,0)\|^2\n\\hbar\\omega_k<\\infty.\n\\ee\nThe result is not surprising but still quite remarkable if one thinks\nof the problem of field quantization. \n\nThe very idea of quantizing the electromagnetic field, as put\nforward by Born, Heisenberg, Jordan \\cite{BHJ} and Dirac \\cite{D},\nis based on the observation that the mode decomposition of the\nelectromagnetic energy is analogous to the energy of an ensemble of\nindependent harmonic oscillators. In 1925, after the work of\nHeisenberg, it was clear what to do: One had to replace each\nclassical oscillator by a quantum one. But since each oscillator had\na definite frequency, to have an infinite number of different\nfrequencies one needed an infinite number of oscillators. \nThe price one payed for this assumption was the \ninfinite energy of the electromagnetic vacuum. \n\nThe infinity is regarded as an ``easy\" one since one can get rid of\nit by redefining the Hamiltonian and removing the infinite term. \nThe result looks correct and many properties typical of a {\\it\nquantum\\/} harmonic oscillator are indeed observed in electromagnetic\nfield. However, once we remove the infinite term by the procedure of\n``normal reordering\" the resulting Hamiltonian is no longer {\\it\nphysically\\/} equivalent to the one of the harmonic oscillators. For\na single oscillator we can indeed add any finite number and the new\nHamiltonian will describe the same physics. But having two or more\nsuch oscillators we cannot remove the ground state energies by a single\nshift of energy: Each oscillator has to be shifted by a different\nnumber and, accordingly, we change the energy differences between the\nlevels of the global Hamiltonian describing the multi-oscillator\nsystem. And this is not just ``shifting the origin of the energy\nscale\". Alternatively, one can add up all the ground state corrections\nand remove the overall energy shift by a different choice of the origin of\nthe energy scale. This would have been acceptable if the shift\nwere {\\it finite\\/}. Subtraction of infinite terms is in mathematics \nas forbidden as division by zero. (Example:\n$1+\\infty=2+\\infty\\Rightarrow 1=2$ is as justified as \n$1\\cdot 0=2\\cdot 0\\Rightarrow 1=2$.) \n\nThe oscillator which can exist in superpositions of different\nfrequencies is a natural candidate as a starting point for Dirac-type\nfield quantization. \nWe do not need to remove the ground state energy since in the Hilbert\nspace of physical states the correction is finite. The question we\nhave to understand is whether one can obtain the well known quantum\nproperties of the radiation field by this type of quantization.\n\n\n\n\\section{Field operators: Free Maxwell fields}\n\nThe energy and momentum operators of the field are defined in analogy to $H$\nfrom the previous section\n\\be\nH &=& \n\\sum_{s,\\kappa_\\lambda}\\hbar \\omega_\\lambda\n\|s,\\vec \\kappa_\\lambda\\rangle \\langle s,\\vec \\kappa_\\lambda\|\n\\otimes \\frac{1}{2}\\Big(a^{\\dag}a+a a^{\\dag}\\Big)\\\\\n&=& \n\\frac{1}{2}\\sum_{s,\\kappa_\\lambda}\\hbar \\omega_\\lambda\n\\Big(a_{s,\\kappa_\\lambda}^{\\dag}a_{s,\\kappa_\\lambda} \n+a_{s,\\kappa_\\lambda} a_{s,\\kappa_\\lambda}^{\\dag}\\Big)\\\\\n\\vec P &=& \n\\sum_{s,\\kappa_\\lambda}\\hbar \\vec \\kappa_\\lambda\n\|s,\\vec \\kappa_\\lambda\\rangle \\langle s,\\vec \\kappa_\\lambda\|\n\\otimes \\frac{1}{2}\\Big(a^{\\dag}a+a a^{\\dag}\\Big)\\\\\n&=& \n\\frac{1}{2}\\sum_{s,\\kappa_\\lambda}\\hbar \\vec \\kappa_\\lambda\n\\Big(a_{s,\\kappa_\\lambda}^{\\dag}a_{s,\\kappa_\\lambda} \n+a_{s,\\kappa_\\lambda} a_{s,\\kappa_\\lambda}^{\\dag}\\Big)\n\\ee\nwhere $s=\\pm 1$ corresponds to circular polarizations. Denote \n$P=(H/c,\\vec P)$ and $P\\cdot x=Ht-\\vec P\\cdot \\vec x$.\nWe employ the standard Dirac-type definitions for mode quantization in\nvolume $V$\n\\be\n\\hat{\\vec A}(t,\\vec x)\n&=&\n\\sum_{s,\\kappa_\\lambda}\n\\sqrt{\\frac{\\hbar}{2\\omega_\\lambda V}}\n\\Big(a_{s,\\kappa_\\lambda}\ne^{-i\\omega_\\lambda t} \\vec e_{s,\\kappa_\\lambda}\ne^{i\\vec \\kappa_\\lambda\\cdot \\vec x}\n+\na^{\\dag}_{s,\\kappa_\\lambda}e^{i\\omega_\\lambda t} \n\\vec e^{\\,}_{s,\\kappa_\\lambda}\ne^{-i\\vec \\kappa_\\lambda\\cdot \\vec x}\n\\Big)\\\\\n&=&\ne^{iP\\cdot x/\\hbar} \\hat{\\vec A} e^{-iP\\cdot x/\\hbar}\\\\\n\\hat{\\vec E}(t,\\vec x)\n&=&\ni\\sum_{s,\\kappa_\\lambda }\n\\sqrt{\\frac{\\hbar\\omega_\\lambda}{2V}}\n\\Big(\na_{s,\\kappa_\\lambda}e^{-i\\omega_\\lambda t} \ne^{i\\vec \\kappa_\\lambda\\cdot \\vec x}\n\\vec e_{s,\\kappa_\\lambda}\n-\na^{\\dag}_{s,\\kappa_\\lambda}e^{i\\omega_\\lambda t} \ne^{-i\\vec \\kappa_\\lambda\\cdot \\vec x}\n\\vec e^{\\,}_{s,\\kappa_\\lambda}\n\\Big)\\\\\n&=&\ne^{iP\\cdot x/\\hbar} \\hat{\\vec E} e^{-iP\\cdot x/\\hbar}\\\\\n\\\\\n\\hat{\\vec B}(t,\\vec x)\n&=&\ni\\sum_{s,\\kappa_\\lambda }\n\\sqrt{\\frac{\\hbar\\omega_\\lambda}{2V}}\n\\vec n_{\\kappa_\\lambda}\n\\times \n\\Big(a_{s,\\kappa_\\lambda}e^{-i\\omega_\\lambda t} \ne^{i\\vec \\kappa_\\lambda\\cdot \\vec x}\n\\vec e_{s,\\kappa_\\lambda}\n-\na^{\\dag}_{s,\\kappa_\\lambda}e^{i\\omega_\\lambda t} \ne^{-i\\vec \\kappa_\\lambda\\cdot \\vec x}\n\\vec e^{\\,}_{s,\\kappa_\\lambda}\n\\Big)\\\\\n&=&\ne^{iP\\cdot x/\\hbar} \\hat{\\vec B} e^{-iP\\cdot x/\\hbar}.\n\\ee\nNow take a state (say, in the Heisenberg picture)\n\\be\n\|\\Psi\\rangle\n&=&\n\\sum_{s,\\vec \\kappa_\\lambda,n}\\Psi_{s,\\vec \\kappa_\\lambda,n}\n\|s,\\vec \\kappa_\\lambda,n\\rangle\\\\\n&=&\n\\sum_{s,\\vec \\kappa_\\lambda}\\Phi_{s,\\vec \\kappa_\\lambda}\n\|s,\\vec \\kappa_\\lambda\\rangle\|\\alpha_{s,\\vec \\kappa_\\lambda}\\rangle\n\\ee\nwhere $\|\\alpha_{s,\\vec \\kappa_\\lambda}\\rangle$ form a family of \ncoherent states:\n\\be\na\|\\alpha_{s,\\vec \\kappa_\\lambda}\\rangle\n=\n\\alpha_{s,\\vec \\kappa_\\lambda}\n\|\\alpha_{s,\\vec \\kappa_\\lambda}\\rangle\n\\ee\nThe averages of the field operators are \n\\be\n\\langle\\Psi\|\\hat{\\vec A}(t,\\vec x)\|\\Psi\\rangle\n&=&\n\\sum_{s,\\kappa_\\lambda }\|\\Phi_{s,\\vec \\kappa_\\lambda}\|^2\n\\sqrt{\\frac{\\hbar}{2\\omega_\\lambda V}}\n\\Big(\n\\alpha_{s,\\kappa_\\lambda}\ne^{-i\\kappa_\\lambda\\cdot x} \n\\vec e_{s,\\kappa_\\lambda}\n+\n\\alpha^_{s,\\kappa_\\lambda}\ne^{i\\kappa_\\lambda\\cdot x}\n\\vec e^{\\,}_{s,\\kappa_{\\lambda}}\n\\Big)\\\\\n\\langle\\Psi\|\\hat{\\vec E}(t,\\vec x)\|\\Psi\\rangle\n&=&\n\\sum_{s,\\kappa_\\lambda }\|\\Phi_{s,\\vec \\kappa_\\lambda}\|^2\n\\sqrt{\\frac{\\hbar\\omega_\\lambda}{2V}}\n\\Big(\n\\alpha_{s,\\kappa_\\lambda}(0)e^{-i\\kappa_\\lambda\\cdot x}\n\\vec e_{s,\\kappa_\\lambda}\n-\n\\alpha^_{s,\\kappa_\\lambda}(0)e^{i\\kappa_\\lambda\\cdot x}\n\\vec e^{\\,}_{s,\\kappa_{\\lambda}}\n\\Big)\\\\\n\\langle\\Psi\|\\hat{\\vec B}(t,\\vec x)\|\\Psi\\rangle\n&=&\ni\\sum_{s,\\kappa_\\lambda }\|\\Phi_{s,\\vec \\kappa_\\lambda}\|^2\n\\sqrt{\\frac{\\hbar\\omega_\\lambda}{2V}}\n\\Big(\n\\alpha_{s,\\kappa_\\lambda}e^{-i\\kappa_\\lambda\\cdot x}\n\\vec n_{\\kappa_\\lambda}\n\\times \n\\vec e_{s,\\kappa_\\lambda}\n-\n\\alpha^_{s,\\kappa_\\lambda}e^{i\\kappa_\\lambda\\cdot x}\n\\vec n_{\\kappa_\\lambda}\n\\times \n\\vec e^{\\,}_{s,\\kappa_{\\lambda}}\n\\Big)\n\\ee\nThese are just the classical fields. More precisely, the fields look\nlike averages \nof monochromatic coherent states with probabilities \n$\|\\Phi_{s,\\vec \\kappa_\\lambda}\|^2$. The energy-momentum operators\nsatisfy also the standard relations\n\\be\nH\n&=&\n\\frac{1}{2}\n\\int_V d^3x\n\\Big(\n\\hat{\\vec E}(t,\\vec x)\\cdot \\hat{\\vec E}(t,\\vec x)\n+\n\\hat{\\vec B}(t,\\vec x)\\cdot \\hat{\\vec B}(t,\\vec x)\n\\Big),\\\\\n\\vec P\n&=&\n\\int_V d^3x \\hat{\\vec E}(t,\\vec x)\\times \\hat{\\vec B}(t,\\vec x).\n\\ee\nIt should be stressed, however, that these relations have a\ncompletely different mathematical origin than in the usual formalism\nwhere the integrals are necessary in order to make plane waves\ninto an orthonormal basis. Here orthogonality follows from the\npresence of the projectors in the definition of\n$a_{s,\\kappa_\\lambda}$ and the integration in itself is {\\it\ntrivial\\/} since\n\\be\n\\hat{\\vec E}(t,\\vec x)\\cdot \\hat{\\vec E}(t,\\vec x)\n+\n\\hat{\\vec B}(t,\\vec x)\\cdot \\hat{\\vec B}(t,\\vec x)\n&=&\n\\hat{\\vec E}\\cdot \\hat{\\vec E}\n+\n\\hat{\\vec B}\\cdot \\hat{\\vec B}\\\\\n\\hat{\\vec E}(t,\\vec x)\\times \\hat{\\vec B}(t,\\vec x)\n&=&\n\\hat{\\vec E}\\times \\hat{\\vec B}.\n\\ee\nTherefore the role of the integral is simply to produce the factor\n$V$ which cancels with $1/V$ arising from the term $1/\\sqrt{V}$\noccuring in the mode decomposition of the fields. \nTo end this section let us note that \n\\be\n\\langle \\Psi\|H\|\\Psi\\rangle\n&=&\n\\sum_{s,\\kappa_\\lambda }\n\\hbar\\omega_\\lambda \n\|\\Phi_{s,\\kappa_\\lambda}\|^2\n\\Big(\n\|\\alpha_{s,\\kappa_\\lambda}\|^2\n+\\frac{1}{2}\n\\Big)\\\\\n\\langle \\Psi\|\\vec P\|\\Psi\\rangle\n&=&\n\\sum_{s,\\kappa_\\lambda }\n\\hbar\\vec \\kappa_{\\lambda}\n\|\\Phi_{s,\\kappa_\\lambda}\|^2\n\\Big(\n\|\\alpha_{s,\\kappa_\\lambda}\|^2\n+\\frac{1}{2}\n\\Big).\n\\ee\nThe contribution from the vacuum fluctuations is nonzero but {\\it finite\\/}. \n\n\\section{Spontaneous and stimulated emission}\n\nThe next test we have to perform is to check the examples that were\nresponsible for the success of Dirac's quantization in atomic\nphysics. It is clear that no differences are expected to occur for single-mode\nproblems such as the Jaynes-Cummings model. \nIn what follows we will therefore concentrate on spontaneous and\nstimulated emission from two-level atoms.\n \nBeginning with the dipole and rotating wave approximations \nwe arrive at the Hamiltonian \n\\be\nH\n&=&\n\\frac{1}{2}\\hbar\\omega_0\\sigma_3\n+\n\\frac{1}{2}\\sum_{s,\\vec \\kappa_\\lambda }\\hbar\\omega_\\lambda \n\\Big(\na_{s,\\vec \\kappa_\\lambda}^{\\dag}a_{s,\\vec \\kappa_\\lambda} \n+a_{s,\\vec \\kappa_\\lambda}a_{s,\\vec \\kappa_\\lambda}^{\\dag}\\Big)\n+\n\\hbar\\omega_0d\\sum_{s,\\vec \\kappa_\\lambda }\n\\Big(\ng_{s,\\vec \\kappa_\\lambda}\na_{s,\\vec \\kappa_\\lambda} \\sigma_+\n+\ng_{s,\\vec \\kappa_\\lambda}^\na_{s,\\vec \\kappa_\\lambda}^{\\dag}\\sigma_-\n\\Big)\n\\ee\nwhere $d\\vec u=\\langle + \|\\hat{\\vec d}\|-\\rangle$ is the matrix\nelement of the dipole\nmoment evaluated between the excited and ground states, \nand $g_{s,\\vec \\kappa_\\lambda}=i\\sqrt{\\frac{1}{2\\hbar\\omega_\\lambda V}} \n\\vec e_{s,\\vec \\kappa_\\lambda}\\cdot \\vec u$. The Hamiltonian represents a\ntwo-level atom located at $\\vec x_0=0$. \n\nThe Hamiltonian in the interaction picture has the well known form\n\\be\nH_I\n&=&\n\\hbar\\omega_0d\\sum_{s,\\vec \\kappa_\\lambda }\n\\Big(\ng_{s,\\vec \\kappa_\\lambda}e^{i(\\omega_0-\\omega_\\lambda)t}\na_{s,\\vec \\kappa_\\lambda} \\sigma_+\n+\ng_{s,\\vec \\kappa_\\lambda}^e^{-i(\\omega_0-\\omega_\\lambda)t}\na_{s,\\vec \\kappa_\\lambda}^{\\dag}\\sigma_-\n\\Big).\n\\ee\nConsider the initial state \n\\be\n\|\\Psi(0)\\rangle\n&=&\n\\sum_{s',\\vec \\kappa_{\\lambda'},m}\\Psi_{s',\\vec \\kappa_{\\lambda'},m}\n\|s',\\vec \\kappa_{\\lambda'},m,+\\rangle\\nonumber\\\\\n&=&\n\\sum_{s',\\vec \\kappa'_{0}}\n\\Psi_{s',\\vec \\kappa'_{0},0}\n\|s',\\vec \\kappa'_{0},0,+\\rangle\n+\n\\sum_{s',\\vec \\kappa'_{n}}\\Psi_{s',\\vec \\kappa'_{n},n}\n\|s',\\vec \\kappa'_{n},n,+\\rangle.\n\\ee\nThe states corresponding to $n=0$ play a role of a\nvacuum. As a \nconsequence the vacuum is not represented here by a unique vector,\nbut rather by a subspace of the Hilbert space of states. It is also\nclear that the energy of this vacuum may be nonzero since no normal\nordering of observables is necessary. \n\nUsing the first-order time-dependent perturbative expansion we arrive at\n\\be\n\|\\Psi(t)\\rangle\n&=&\n\|\\Psi(0)\\rangle\n\\nonumber\\\\\n&\\pp =&\n+\n\\omega_0d\n\\sum_{s,\\vec \\kappa_0 }\n\\frac{e^{-i(\\omega_0-\\omega_{\\lambda_0})t}-1}{\\omega_0-\\omega_\\lambda}\n\\Psi_{s,\\vec \\kappa_{\\lambda_0},0}\ng_{s,\\vec \\kappa_{\\lambda_0}}^\n\|s,\\vec \\kappa_{\\lambda_0},1,-\\rangle\\nonumber\\\\\n&\\pp =&\n+\n\\omega_0d\n\\sum_{s,\\vec \\kappa_n }\n\\frac{e^{-i(\\omega_0-\\omega_{\\lambda_n})t}-1}{\\omega_0-\\omega_{\\lambda_n}}\n\\Psi_{s,\\vec \\kappa_{n},n}\n\\sqrt{n+1}\ng_{s,\\vec \\kappa_n}^*\n\|s,\\vec \\kappa_{n},n+1,-\\rangle.\n\\ee\nOne recognizes here the well known contributions from spontaneous\nand stimulated emissions. It should be stressed that although the\nfinal result looks familiar, the mathematical details behind the calculation\nare different from what we are accustomed to. For example, instead of\n\\be\na_{s_1,\\vec \\kappa_1}^{\\dag}\|s,\\vec \\kappa,m\\rangle\n\\sim \|s_1,\\vec \\kappa_1,1;s,\\vec \\kappa,m\\rangle,\n\\ee\nwhich would hold in the standard formalism for $\\vec\\kappa_1\\neq\n\\vec\\kappa$, we get simply\n\\be\na_{s_1,\\vec \\kappa_1}^{\\dag}\|s,\\vec \\kappa,m\\rangle=0,\n\\ee\na consequence of \n$a_{s_1,\\vec \\kappa_1}^{\\dag}a_{s,\\vec \\kappa}^{\\dag}=0$. \n\n\\acknowledgements\n\nThis work was done mainly during my stay in Arnold Sommerfeld\nInstitute in Clausthal. I gratefully acknowledge a support from the\nAlexander von Humboldt Foundation. \n\n\\begin{references}\n\\bibitem{BHJ}M. Born, W. Heisenberg, and P. Jordan, Z. Phys. {\\bf\n35}, 557 (1925)\n\\bibitem{D}P. A. M. Dirac, Proc. Roy. Soc. A {\\bf 112}, 661 (1926); \nibid. {\\bf 114}, 243 (1927).\n\\end{references}\n\\end{document}\n\n\n" } ]	[ { "name": "quant-ph9912102.extracted_bib", "string": "{BHJM. Born, W. Heisenberg, and P. Jordan, Z. Phys. {35, 557 (1925)" }, { "name": "quant-ph9912102.extracted_bib", "string": "{DP. A. M. Dirac, Proc. Roy. Soc. A {112, 661 (1926); ibid. {114, 243 (1927)." } ]
quant-ph9912103		[]	In this paper the quantum version of the source coding theorem is obtained for a completely ergodic source. This result extends Schumacher's quantum noiseless coding theorem for memoryless sources. The control of the memory effects requires some earlier results of Hiai and Petz on high probability subspaces. Our result is equivalently considered as a compression theorem for noiseless stationary channels.	[ { "name": "quant-ph9912103.tex", "string": "% 2000.01.15 javitottxc\n% 2000.03.30. javitott\n% 2001.02.20. javitott\n% 2001.03.09. \n\\documentclass[12pt]{article}\n\\usepackage{amssymb}\n%\\usepackage{amsmath}\n\\hoffset -1.3cm\n\\voffset -1.7cm\n\\textwidth=15.8cm\\textheight=23.5cm\\parindent=15pt\\parskip=3pt\n\\addtolength{\\evensidemargin}{-0.03\\textwidth}\n\\addtolength{\\oddsidemargin}{-0.03\\textwidth}\n\\addtolength{\\textwidth}{0.06\\textwidth}\n\\addtolength{\\topmargin}{-0.04\\textheight}\n\\addtolength{\\textheight}{0.07\\textheight}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}{Proposition}[section]\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{pelda}[theorem]{Example}\n\\newtheorem{conj}[theorem]{Conjecture}\n\\newtheorem{remark}{Remark}\n\\renewcommand{\\thesection}{\\Roman{section}.}\n\\renewcommand{\\thetheorem}{\\arabic{theorem}.}\n\\renewcommand{\\theproposition}{.}\n\\def\\tr{{\\rm Tr}\\,}\n\\def\\ot{\\otimes}\n\\def\\L{\\Lambda}\n\\def\\vfi{\\varphi}\n\\def\\eps{\\varepsilon}\n\\def\\id{\\mbox{id}}\n\\def\\iK{{\\cal K}}\n\\def\\iH{{\\cal H}}\n\\def\\iA{{\\cal A}}\n\\def\\iB{{\\cal B}}\n\\def\\bbbn{\\mathbb{N}}\n\\def\\bbbc{\\mathbb{C}}\n\\def\\bbbz{\\mathbb{Z}}\n\\def\\<{\\langle}\n\\def\\>{\\rangle}\n\\def\\eps{\\varepsilon}\n\\def\\proof{\\noindent{\\bf Proof: }}\n\\def\\half{\\frac{1}{2}}\n\\def\\iff{\\Longleftrightarrow}\n\\def\\ran{\\rm Range}\n\\def\\bQ{$\\spadesuit$}\n\\def\\qed{~~{\\bf QED}}\n\\def\\raw{\\rightarrow}\n\\def\\an {\\otimes_{i=1entco}^{n} \\mathcal{A} }\n\\def\\ot{ \\otimes_{i=1}^{n} }\n\\def\\av{ \\otimes_{i=- \\infty}^{+ \\infty} \\mathcal{A} }\n\\def\\iAn{{\\iA}^{\\otimes n}}\n\\def\\iAv{{\\iA}^{\\otimes \\infty}}\n\\def\\vphi{\\varphi}\n\\def\\vfin{\\vfi^{\\otimes n}}\n\\def\\pr{ \\mathbb{P}(\\iAv)}\n\\def\\prn{ \\mathbb{P}(\\iAn)}\n\\def\\supp{\\mbox{supp}}\n\n\\begin{document}\n\\ \\vskip 1cm\n\\centerline{\\LARGE Stationary quantum source coding}\n\\bigskip\n\\bigskip\n\\centerline{\\Large D\\'enes Petz\\footnote{Supported by the Hungarian National \nFoundation for Scientific Research grant no. OTKA T 032662, e-mail: \npetz@math.bme.hu.} and\nMil\\'an Mosonyi\\footnote{E-mail: mosonyi@chardonnay.math.bme.hu.}}\n\\bigskip\n\\centerline{Mathematical Institute}\n\\centerline{Budapest University of Technology and Economics}\n\\centerline{H-1521 Budapest XI. Sztoczek u.\\ 2, Hungary}\n\\bigskip\n\\begin{abstract}\nIn this paper the quantum version of the source coding theorem is obtained \nfor a completely ergodic source. This result extends Schumacher's\nquantum noiseless coding theorem for memoryless \nsources. The control of the memory effects requires some earlier results of \nHiai and Petz on high probability subspaces. Our result is equivalently\nconsidered as a compression theorem for noiseless stationary channels.\n\\end{abstract}\n\n\\section{Introduction}\n\nAlthough it is difficult to define a discipline, to give some idea we\ncan say that the objective of quantum information theory is the\ntransmission and manipulation of information stored in systems obeying\nquantum mechanics. A quantum channel has a source that emits systems in \nquantum states to the channel. For example, the source could be a laser\nthat emits individual monochromatic photons and the channel could be an \noptical fiber. The noisy signal output of the channel arrives at the \nreceiver. In principle, there are two very different problems about \nquantum channels. The sender has a quantum system in an unknown state \nand wants to have the receiver to end up with a similar system in the\nsame state. In this case we speak of a pure quantum channel which has a \nquantum mechanical input and output. On the other hand, one might want \nto use quantum states to carry classical information, roughly \nspeaking a sequence of zeros and ones. Now both the input and the output are\nclassical, however there is a quantum mechanical section inbetween. The classical\ninformation is encoded into a quantum state and this is sent down the channel. \nThe higher the channel noise is, the more redundant the encoding must be in order \nto restore the original signal at the reciever, where the quantum signal is \nconverted into classical information. In this paper we do not deal with the problem \nhow such a scheme can be realistically implemented; practical quantum encoding and\ndecoding requires sophisticated ability to manipulating quantum states. However,\nwe are interested in the amount of classical information getting through the channel\nwhich is assumed to be noiseless. It was emphasized already by Shannon that\na computer memory is a communication channel. (Quantum or classical depends on\nthe type of the computer.) In an optimal situation the computer memory is free of \nany noise and this is the case we are concentrating on in the present paper. We \nwant to consider rather general noiseless quantum channels (with possibly memory\neffects but strong ergodic properties) and our aim is to discuss the quantum source\ncoding theorem. As a general reference on quantum information theory we\nsuggest the recent book \\cite{N-Ch} but the really necessary definitions\nare given below.\n \nTo each classical input message $x_i$ there corresponds a signal state $\\vfi_i$ of \nthe quantum communication system. The quantum states $\\vfi_i$ are functioning as\ncodewords of the messages. The signal states $\\vfi_i$ could be pure and orthogonal\nin the sense of quantum mechanics but for example in quantum cryptography \nnonorthogonal states are used intentionally in order to avoid eavesdropping. At\nthe moment we do not impose any condition on the signal states, they could be\narbitrary pure or mixed states. In the stochastic model of communication, one\nassumes that each input message $x_i$ appears with certain probability. Let $p_{ji}$\nbe the probability that the message $x_i$ is sent and $y_j$ is recieved. The joint\ndistribution $p_{ji}$ yields marginal probability distributions $p_i$ and $q_j$\non the set of input and output messages. According to Shannon the mutual information\n$$\nI=\\sum_{i,j} p_{ji}\\log {p_{ji} \\over p_i q_j }\n$$\nmeasures the amount of information going through the channel from Alice to Bob. Of\ncourse, the relation of $I$ to the quantum encoding and decoding should be made\nclear. This comes next.\n\nThe message $x_i$ has {\\it a priori} probability $p_i$ and the mixed quantum state\nof the channel is\n$$\n\\vfi=\\sum_i p_i \\vfi_i .\n$$\nThis might be considered as the statistical operator of the {\\it mesagge ensemble},\nfor example when $\\vfi_i$ is a pure state $\|i\\>\\<i\|$, then $\\vfi=\\sum_i p_i \n\|i\\>\\<i\|$ acts on the input Hilbert space $\\iH$. The distribution of the output is \ndetermined by a measurement, which is nothing else but a physical word for \ndecoding. To each output message there corresponds an obsevable $A_j$ on the\noutput Hilbert space $\\iK$. It is customary to assume that $0 \\le A_j$,\n$\\sum_j A_j=\\id$ ($\\id$ stands for the identity operator) and $p_{ji}=p_i \\vfi_i(A_j)$.\nThe so-called {\\it Kholevo bound} (\\cite{Ho1}) provides an upper bound \non the amount of information accessible to Bob in terms of von Neumann entropies:\n$$\nI \\le S(\\vfi)-\\sum_i p_i S(\\vfi_i) \n$$\n(When $\\lambda_1,\\lambda_2, \\dots$ are the eigenvalues of the statistical operator\nof a quantum state $\\psi$, then $S(\\psi)=-\\sum_k \\lambda_k \\log(\\lambda_k)$.) In \nparticular, if all signal states $\\vfi_i$ are pure, then $S(\\vfi_i)=0$ and\nwe have $I\\le S(\\vfi)$. In this way the von Neumann entropy gets an information\ntheoretical interpretation. Kholevo's bound is actually not very strong, it is \nattained only in trivial situations (\\cite{OPW}).\n\nThe basic problem of communication theory is to maximize the amount of information\nreceived by Bob from Alice. However, up to now this problem is not well-posed\nin our discussion yet. Let us deal with messages of length $n$, they are \n$n$-term-sequences of $0$ and $1$. (So the size of this message set is $2^n$.)\nFor each message length $n$ we carry out the above procedure of coding and decoding\nand the amount of information going through the channel is $I_n$. Since $I_n$\nis presumably proportional to $n$, the good information quantity is $I_n/n$,\nthat is, the transmitted information per letter. Since Shannon's theory is not only\nstochastic but asymptotic as well, we are going to let $n$ to $\\infty$. In this\nway we need to repeat the above information transmission scheme for each $n$. The\nmesagge set, the input Hilbert space $\\iH^{(n)}$, our coding, the channel state\n$\\vfi^{(n)}$, the output Hilbert space $\\iK^{(n)}$ and the observables applied in \nthe measurement are all depending on the parameter $n$. \n\nThe subject of the present paper is faithful signal transmission, which bears the\nname noiseless channel. In place of faithful transmission, one can think of \ninformation storage. In this case the aim is to use the least possible number of \nHilbert space dimension per signal for coding. The new feature of the noiseless\nchannel we are studying is the memory effect. Mathematicaly this means that the\nchannel state (of the $n$-fold channel) is not of product type but we assume \nstationarity and good ergodic properties. In Section 2 we use the standard \nformalism of statistical mechanics to describe such a channel. It turns out that\nthe mean von Neumann entropy, familiar also from statistical mechanics, gives\nthe optimal coding rate. The proof of our main result, Theorems \\ref{thmpoz} and\n\\ref{thmneg}, is similar to the proof\npresented in \\cite{JSch} for Schumacher's coding theorem, however instead of\ntypical sequences we use the high probability subspace of strongly ergodic\nstationary states, a subject studied by Hiai and Petz in \\cite{HP1}. We note\nfor the interested reader that most of the concepts used in the present paper\nare treated in details in the monograph \\cite{OP}.\n \n\\section{An infinite system setting of the source}\n\nIf $\\iH$ is a finite dimensional Hilbert space then\n$(A,B) \\mapsto \\tr(A^* B)$ defines an inner product on $\\iB(\\iH)$, so for\nevery linear functional $\\vfi$ on $\\iB(\\iH)$ there exists a unique $D_{\\vfi}\n\\in \\iB(\\iH)$ with the property $\\vfi(A)=\\tr(D_{\\vfi} A)$. When\n$\\vfi$ is a state then $D_{\\vfi}$ is the corresponding density matrix.\nLet $X^n$ denote the set of all messages of length $n$. If $x^n \\in X^n$ is a message\nthen a quantum state $\\vfi(x^n)$ of the $n$-fold quantum system is corresponded with\nit. The Hilbert space of the $n$-fold system is the $n$-fold tensor product \n$\\iH^{\\otimes n}$ and $\\vfi(x^n)$ has a statistical operator $D(x^n)$. If messages\nof length $n$ are to be transmitted then our quantum source should be put in\nthe state $\\vfi_n=\\sum_{x^n} p(x^n) \\vfi(x^n)$ with statistical operator\n$D_n=\\sum_{x^n} p(x^n) D(x^n)$, where $p(x^n)$ is the probability of the message\n$x^n$. Since we want to let $n \\to \\infty$, it is reasonable to view all the \n$n$-fold systems as subsystems of an infinite one. In this way we can conveniently\nuse a formalism standard in statistical physics, see Chap. 15 of \\cite{OP}. \n\nLet an infinitely extended system be considered over the lattice $\\bbbz$ of integers.\nThe observables confined to a lattice site $k\\in\\bbbz$ form the selfadjoint part of \na finite dimensional matrix algebra $\\iA_k$, that is the set of all operators\nacting on the finite dimensional space $\\iH$. It is assumed that the\nlocal observables in any finite subset $\\L\\subset\\bbbz$ are those of the finite \nquantum system\n$$\n\\iA_\\L=\\mathop{\\otimes}_{k\\in\\L}\\iA_k.\n$$\nThe quasilocal algebra $\\iA$ is the norm completion of the normed algebra\n$\\iA_\\infty=\\cup_{\\L}\\iA_\\L$, the union of all local algebras $\\iA_\\L$ associated \nwith finite intervals $\\L\\subset\\bbbz$.\n\nA state $\\vfi$ of the infinite system is a positive normalized functional $\\iA \\to \n\\bbbc$. It does not make sense to associate a statistical operator to a state\nof the infinite system in general. However, $\\vfi$ restricted to a finite dimensional\nlocal algebra $\\iA_\\L$ admits a density matrix $D_\\L$. We regard the algebra \n$\\iA_{[1,n]}$ as the set of all operators acting on the $n$-fold tensor product space\n$\\iH^{\\otimes n}$. Moreover, we assume that the density $D_n$ from the first part \nof this section is identical with $D_{[1,n]}$. Under this assumptions we call the\nstate $\\vfi$ the state of the (infinite) channel. Roughly speaking, all the\nstates used in the transmission of messages of length $n$ are marginals of this \n$\\vfi$. Coding, transmission and decoding could be well formulated using the states\n$\\vfi_n\\equiv \\vfi_{[1,n]}$. However, it is more convenient to formulate our setting\nin the form of an infinite system, particularly because we do not want to assume\nthat the channel state $\\vfi$ is a product type. This corresponds to the possibility\nthat our quantum source has a memory effect. \n\nThe right shift on the set $\\bbbz$ induces a transformation $\\gamma$ on $\\iA$. A \nstate $\\vfi$ is called {\\it stationary} if $\\vfi \\circ \\gamma=\\vfi$. The state $\\vfi$\nis called {\\it ergodic} if it is an extremal point in the set of stationary states.\nMoreover, $\\vfi$ is {\\it completely ergodic} when it is an extreme point for every\n$m \\in \\bbbn$ in the convex set of all states $\\psi$ such that $\\psi\\circ \\gamma^m \n=\\psi$. By a {\\it completely ergodic stationary quantum source} we simply mean a \ncompletely ergodic stationary state $\\vfi$ of the infinite system $\\iA$. Of course,\na stationary product state, corresponding to a memoryless channel, is completely\nergodic. The emphasis is put to other states here.\n\nBelow we show an example of a completely ergodic stationary \nquantum source from the context of algebraic states. For the\ndetails see the original paper \\cite{HP2}.\n \n\\begin{pelda}\nLet $\\iA := M_3(\\mathbb{C})$, $\\iB :=M_2(\\mathbb{C})$, moreover let\n$\\{E_{ij} \\}_{i,j=1}^3$ be the usual matrix units of $M_3(\\mathbb{C})$.\nSet\n$$\nV_1:= \\left[\\matrix{ {1 \\over \\sqrt{2}} & 0 \\cr 0 & 0 }\\right],\\quad\nV_2:= \\left[\\matrix{ 0 & 0 \\cr {1 \\over \\sqrt{2}} & 0 }\\right],\\quad\nV_3:= \\left[\\matrix{ 0 & 1 \\cr 0 & 0 }\\right].\n$$\nThen $\\sum_{i=1}^3 V_i^V_i=I_{\\iB}$. \n\nLet $\\rho$ be a state on $\\cal B$ with density matrix \n$$\n\\left[\\matrix{ {2 \\over 3} & 0 \\cr 0 & {1 \\over 3} }\\right].\n$$\nDefine $\\Sigma:\\iA \\otimes \\iB \\to \\iB$ by $\\Sigma(E_{ij}\\otimes x):= \nV_i^ x V_j$. It is easy to check that $\\Sigma$ is a completely positive \nunital map and $\\rho (\\Sigma(I_{\\iA}\\otimes x))=\n\\rho(x),x\\in \\iB$. \n\nThen the algebraic state $\\vfi$ generated by $(\\iB,\\Sigma, \\rho)$ is given by \n$$\n\\vfi(E_{i_1 j_1}\\otimes \\dots \\otimes E_{i_n j_n})=\\rho(V_{i_1}^\\dots V_{i_n}^V_{j_n}\n\\dots V_{j_1}).\n$$\nIt is shown in \\cite{HP2} that $\\vfi$ is completely ergodic. Of course, it is \nnot a product state.\n\\end{pelda}\n\nIt is well-known in quantum statistical mechanics that due to the subadditivity\nof the von Neumann entropy (proven first in \\cite{LR} by Lieb and Ruskai) the limit\n$$\n\\lim_{n\\to +\\infty} \\frac{1}{n} S(\\vfi_n)= \\inf \\frac{1}{n} S(\\vfi_n)=:h\n$$\nexists for any stationary state and this quantity is called the {\\it mean entropy} \nof $\\vfi$. (See \\cite{OP} for a textbook treatment of the subject or \\cite{Pe} \nfor some related properties of the mean entropy.)\n\n\n\\section{Source coding}\n\nFor a while we fix a message length $n$ and we denote by $d$ the dimension of the\nHilbert space $\\iH$. Assume that our $n$-fold composite quantum system is operating\nas a quantum source and emits the quantum states $D^{(1)}, D^{(2)}, \\dots,\nD^{(m)}$ with a-priory probabilities $p_1,p_2,\\dots, p_m$. (Therefore the state\nof the system is $D_n=\\sum_i p_i D^{(i)}$.) By source coding we mean an association\n$$\nD^{(i)}\\mapsto \\tilde{D}^{(i)},\n$$\nwhere $\\tilde{D}^{(i)}$ is some other statistical operator on the Hilbert space \n$\\iH^{\\otimes n}$. (This definition allows $D^{(i)}=D^{(j)}$ but\n$\\tilde{D}^{(i)}\\ne \\tilde{D}^{(j)}$, however in the coding constructed\nin the proof of Theorem 1 this cannot happen.)\n\nWe denote by $\\iK_n$ the subspace spanned by the eigenvectors \ncorresponding to all nonzero eigenvalues of all statistical operators \n$\\tilde{D}^{(i)}$, $1 \\le i \\le m$. The goal of source coding is to keep the \ndimension of $\\iK_n$ to be small and to fulfil some fidelity criterium. \n(A mathematically demanding survey about quantum coding is the paper \\cite{Ho}.)\nThe {\\it source coding rate}\n$$\n\\limsup_{n\\to \\infty} \\frac{\\log \\dim (\\iK_n)}{n} \n$$\nexpresses the resolution of the encoder in qubits per input symbol. (It is actually\nmore precise to speak about ``qunats'' per input symbol, but the difference is\nonly a constant factor.) \n\nThe distortion measure is a number which allows us to compare the goodness or \nbadness of communication sytems. The {\\it fidelity} of the coding scheme was\nintroduced by Schumacher (\\cite{Sch}):\n$$\nF:=\\sum_{i} p_i \\tr D^{(i)} \\tilde{D}^{(i)},\n$$\nwhere $p_i$ is a probaility distribution on the input and $\\tilde{D}^{(i)}$ is the \ndensity used to encode the density $D^{(i)}$. Note that $0\\le F\\le 1$ and $F=1$ if \nand only if $D^{(i)}=\\tilde{D}^{(i)}$ are pure states.\n\nFirst we present our positive source coding theorem for a completely ergodic\nsource. The result says that the source coding rate may approach the mean\nentropy while we can keep the fidelity arbitrarily good.\n\n\\begin{theorem}\\label{thmpoz}\nLet $\\iH$ be a finite dimensional Hilbert space, and\n$\\vfi$ be a completely ergodic state on $B(\\iH)^{\\otimes \\infty}$.\nThen for every $\\eps, \\delta>0$ there exists $n_{\\varepsilon,\\delta}\n\\in \\bbbn$ such that for $n \\ge n_{\\varepsilon,\\delta}$ there is\na subspace $\\iK_n(\\varepsilon,\\delta)$ of $\\iH^{\\otimes n}$ such that\n\\begin{itemize}\n\\item[(i)] $\\log \\dim \\iK_n(\\varepsilon,\\delta) < n(h+\\delta)$ and\n\\item[(ii)] for every extremal decomposition $D_n=\\sum_{i=1}^{m} p_i {D}^{(i)}$ one can \nfind an encoding ${D}^{(i)}\\mapsto \\tilde{D}^{(i)}$ with density matrices \n$\\tilde{D}^{(i)}$ supported in $\\iK_n(\\varepsilon,\\delta)$ \nsuch that the fidelity $F:=\\sum_{i=1}^{m} p_i \\tr D^{(i)} \\tilde{D}^{(i)}$ \nexceeds $1-\\varepsilon$.\n\\end{itemize}\n\\end{theorem}\n\nThe negative part of the coding theorem tells that the source coding rate\ncannot exceed the mean entropy when the fidelity is good.\n\n\\begin{theorem}\\label{thmneg}\nLet $\\iH$ be a finite dimensional Hilbert space, and\n$\\vfi$ be a completely ergodic state on $B(\\iH)^{\\otimes \\infty}$.\nThen for every $\\delta>0$ there exist $0 < \\eta< 1$ and $n_{\\delta}\n\\in \\bbbn$ such that for $n \\ge n_{\\delta}$ \n\\begin{itemize}\n\\item[(i)] for all subspaces $\\iK_n$ of $\\iH^{\\otimes n}$\nwith the property\n$\\log$ dim $\\iK_n \\le n(h-\\delta)$ and\n\\item[(ii)] for every decomposition $D_n=\\sum_{i=1}^{m} \np_i {D}^{(i)}$ and for\nevery encoding ${D}^{(i)}\\mapsto \\tilde{D}^{(i)}$ with density matrices \n$\\tilde{D}^{(i)}$ supported in $\\iK_n$, the fidelity\n$F:=\\sum_{i=1}^{m} p_i \\tr D^{(i)} \\tilde{D}^{(i)}$ is smaller than \n$\\eta$.\n\\end{itemize}\n\\end{theorem}\n\nThe detailed proofs are given in the next section of the paper. Now we make\nsome comments on the fidelity $F$. It is possible that $F<1$ although\n$D^{(i)}=\\tilde{D}^{(i)}$. This fact might suggest to use another concept\nof fidelity. Since $D^{1/2}\\ge {D}$ holds for a density matrix, we have\n\\begin{eqnarray}\n\\tr {D_1}^{1/2}{D_2}^{1/2}&= & \\tr {D_1}^{1/4}{D_2}^{1/2} {D_1}^{1/4}\n\\ge \\tr {D_1}^{1/4}{D_2} {D_1}^{1/4}=\\tr {D_2}^{1/2}D_1^{1/2} {D_2}^{1/2}\n\\\\ & \\ge &\\tr {D_2}^{1/2}{D_1} {D_2}^{1/2}= \\tr {D_1}{D_2}.\n\\end{eqnarray}\nThis implies that\n$$\nF':=\\sum_{i} p_i \\tr \\big[ D^{(i)}\\big]^{1/2} \\big[\\tilde{D}^{(i)}\\big]^{1/2}\n\\ge F.\n$$\nBoth our positive and negative source\ncoding theorems hold if $F$ is replaced by $F'$. (In case of Theorem \\ref{thmpoz}\nthis follows from the inequality $F' \\ge F$ and in the proof of Theorem \\ref{thmneg}\nwe will show $F' \\le \\eta $.)\n\n\\section{High probability subspace}\n\nThe proof of Shannon's original source coding theorem is based on the \ntypical sequences (\\cite{CsK}, Chap. 1). The quantum extension of this result \nobtained\nby Schumacher still benefits from the classical result. When the channel state\nis a product, the densities $D_n$ commute and simultanous diagonalization is \npossible. If the memory effects are present, then these densities do not\ncommute and in some sense we are in a really quantum mechanical non-commutative\nsituation. Nevertheless, the high probability subspace can be used but new \ntechniques are required.\n\nLet $\\iK$ be a Hilbert space and $D$ be a density matrix on $\\iK$. $D$ has a Schatten\ndecomposition $D=\\sum_i \\lambda_i \|f_i\\>\\<f_i\|$, where $\|f_i\\>$'s are eigenvectors\nand the eigenvalues $\\lambda_i$ are numbered decreasingly: $\\lambda_1\\ge \\lambda_2 \n\\ge \\dots $. Choose and fix $0 < \\eps <1$. Let $n(\\eps)$ be the smallest integer\nsuch that\n$$\n\\sum_{i=1}^{n(\\eps)} \\lambda_i \\ge 1 -\\eps\\, .\n$$ \nThe subspace $HP(D,\\eps)$ spanned by the eigenvectors $\|f_1\\>,\\dots, \|f_{n(\\eps)}\\>$\nis called the {\\it high probability subspace} corresponding to the level $\\eps$. \nNote that $HP(D,\\eps)$ is not completely well-defined, if there are \nmultiplicities in the spectrum of $D$, then the Schatten decomposition is \nnot unique. However, the dimension $n(\\eps)$ of $HP(D,\\eps)$ is determined. \nThe term ``high probability subspace'' is borrowed from the monograph \\cite{Gr} \nand its role in macroscopic uniformity was discussed in \\cite {HP3}.\n\nIn the following, $\\vfi$ will be a completely ergodic state on $\\iAv$.\nFor $\\eps \\in$ (0,1) let \n$$\n\\beta_{\\eps,n} : = \\inf\\{\\log \\tr_n (q)) \\colon q \\in \\prn , \\vfi_n(q) \n\\geq 1-\\eps\\},\n$$\nwhere $\\prn$ denotes the set of projections of $\\iAn$. ($\\exp \\beta_{\\eps,n}$ is\nthe dimension of the high probability subspace.) It was shown in \n\\cite{HP1} (and formulated in terms of relative entropy) that\n\\begin{eqnarray}\n\\limsup_{n \\to +\\infty} \\frac{1}{n}\\beta_{\\varepsilon,n} &\\leq & h, \\label{(1)}\\\\\n\\liminf_{n \\to +\\infty} \\frac{1}{n}\\beta_{\\varepsilon,n} & \\geq &\n\\frac{1}{1-\\varepsilon} h -\\frac{\\eps}{1-\\eps} \\log d. \\label{(2)}\n\\end{eqnarray}\nFrom this one can deduce the following\n\n\\begin{proposition}\\label{thmhp} For every positive $\\delta$ \n\\begin{itemize}\n\\item[(i)] and for every positive $\\varepsilon$\nthere exists $N_{\\varepsilon, \\delta} \\in \\bbbn$ \nsuch that for every $n>N_{\\varepsilon,\\delta}$ there exists a projection \n$q_n(\\eps,\\delta)$ in $\\iAn$ such that\n$$\n\\log(\\tr_n (q_n(\\eps,\\delta)))<n(h+\\delta)\\quad \\hbox{and}\\quad \\vfi_n(q) \\geq 1-\\varepsilon,\n$$\n\\item[(ii)] there exists $1 > \\eta >0$ and $N_{\\delta} \\in \\bbbn$ such\nthat for every $n>N_{\\delta}$ and for every projection $q$ in $\\iAn$\n$$\n\\log(\\tr_n (q))\\le n(h-\\delta),\n$$\nimplies $\\vfi_n(q)\\le \\eta$.\n\\end{itemize}\n\\end{proposition}\n\nPart (i) of the Proposition is a plain reformulation of (\\ref{(1)}). In order to see\n(ii) we first note that \n$$\n\\frac{1}{\\eta}h-\\frac{1-\\eta}{\\eta}\\log d \\to h \\quad \\mbox{as}\\quad\n\\eta\\to 1\\, .\n$$\nHence given $\\delta>0$ we choose $0<\\eta<1$ such that\n$$\n\\frac{1}{\\eta}h-\\frac{1-\\eta}{\\eta}\\log d > h-\\delta\\,.\n$$\nNext we replace $1-\\eta$ by $1-\\delta$ in (\\ref{(2)}):\n\\begin{equation}\\label{eps}\n\\liminf_{n \\to +\\infty} {1 \\over n}\\inf\\{\\log \\tr_n (q)) \\colon \nq \\in \\prn , \\vfi_n(q) \n\\geq 1-\\eta\\} \\ge {1 \\over \\eta}h-{1-\\eta \\over \\eta}\\log d >h-\\delta.\n\\end{equation}\nIn this way we arrived at (ii).\n\nNext we prove the source coding theorem.\n\n{\\it Proof of Theorem \\ref{thmpoz}}:\nUse part (i) of the Proposition and set $q_n:=q_n(\\eps/2,\\delta)$ ,\n$\\iK_n(\\varepsilon,\\delta):=\\hbox{Ran}\\, q_n$, where\n$n>n(\\eps,\\delta):=N_{\\eps/2,\\delta}$. Given an extremal decomposition $D_n=\n\\sum_{i=1}^{k} p_i D^{(i)}$, that is $D^{(i)}=\|x_i\\>\\<x_i\|$ for some\nvectors $x_i$, we construct the coding densities $\\tilde{D}^{(i)}$. Let \n$$\n\\tilde{x_i}:=\\frac{q_n x_i}{\\\|q_n x_i \\\|}, \\quad \\alpha_i:=\\\|q_n x_i \\\|,\n\\quad \\beta_i:=\\\|(I-q_n) x_i \\\|\n$$\nand let $x$ be any unit vector such that $q_n x=x$. Then we set \n$$\n\\tilde{D}^{(i)}:=\\alpha_i^2 \|\\tilde{x_i}\\>\\<\\tilde{x_i}\|+\\beta_i^2\|x\\>\\<x\|.\n$$\nSince $\\tilde{x_i},x \\in \\iK_n(\\varepsilon, \\delta)$, we have $\\supp \n\\tilde{D}^{(i)} \\subset \\iK_n(\\varepsilon, \\delta)$. Furthermore,\n\\begin{eqnarray}\n\\tr {D}^{(i)}\\tilde{D}^{(i)}&=&\\<x_i,\\tilde{D}^{(i)} x_i\\>=\n\\alpha_i^2 \|\\<x_i\|\\tilde{x_i}\\>\|^2+\\beta_i^2 \|\\<x_i,x\\>\|^2 \\\\ &\\geq&\n\\alpha_i^2 \|\\<x_i\|\\tilde{x_i}\\>\|^2 = \\alpha_i^4 \\geq 2\\alpha_i^2-1\\\\ &=&\n2\\tr q_n {D}^{(i)}-1.\n\\end{eqnarray}\nWe need to sum over $i$:\n$$\n\\sum_i p_i \\tr {D}^{(i)}\\tilde{D}^{(i)}\\ge \\sum_i p_i \\big(2\\tr q_n \n{D}^{(i)}-1\\big)=2\\tr D_n q_n-1=2 \\vfi_n (q_n)-1 \\ge 1 - \\eps.\n$$\n\n{\\it Proof of Theorem \\ref{thmneg}}:\nFor the given $\\delta$ we choose $\\eta$ and $n(\\delta)$\naccording to the Proposition. Let $q$ be the projection onto the subspace \n$\\iK_n$. We want to use the Schwarz inequality in the form\n$$\n\\Big\|\\sum_i p_i \\tr x_i y_i\\Big\| \\le\n\\Big[\\sum_i p_i \\tr x_i^* x_i\\Big]^{1/2} \n\\Big[\\sum_i p_i \\tr y_i^* y_i\\Big]^{1/2}\n$$\nfor $x_i=[D^{(i)}]^{1/2} q$ and $y_i=[\\tilde{D}^{(i)}]^{1/2}$. \nSince $[\\tilde{D}^{(i)}]^{1/2}=q[\\tilde{D}^{(i)}]^{1/2}$ follows from \nthe hypothesis, we have \n\\begin{eqnarray}\nF'&=&\\sum_{i=1}^{m} p_i \\tr [D^{(i)}]^{1/2}[ \\tilde{D}^{(i)}]^{1/2}\n= \\sum_{i=1}^{m} p_i \\tr [D^{(i)}]^{1/2}q[ \\tilde{D}^{(i)}]^{1/2}\\\\\n&\\le & \\Big[\\sum_{i=1}^{m} p_i \\tr D^{(i)} q \\Big]^{1/2}\n\\Big[\\sum_{i=1}^{m} p_i \\tr \\tilde{D}^{(i)} \\Big]^{1/2}\\\\\n&=& \\vphi_n (q)^{1/2} \\le \\sqrt{\\eta}.\n\\end{eqnarray}\nThis estimate completes the proof.\n\nIt is known that for strongly mixing algebraic states (\\cite{HP2})\nand for ergodic Gibbs states (\\cite{HP3})\n$$\n\\lim_{n \\to +\\infty} \\frac{1}{n} \\beta_{\\eps,n}=h\n$$ \nand in this case the negative part of the coding theorem holds in a stronger\nform:\nFor every $\\eps, \\delta>0$ there exists $n_{\\varepsilon,\\delta}\n\\in \\bbbn$ such that for $n \\ge n_{\\varepsilon,\\delta}$ \nfor all subspaces $\\iK_n$ of $\\iH^{\\otimes n}$ \nwith the property $\\log$ dim $\\iK_n < n(h-\\delta)$ and\nfor every decomposition $D_n=\\sum_{i=1}^{m} \np_i {D}^{(i)}$ and for\nevery encoding ${D}^{(i)}\\mapsto \\tilde{D}^{(i)}$ with density matrices \n$\\tilde{D}^{(i)}$ supported in $\\iK_n(\\varepsilon,\\delta)$, the fidelity \n$F:=\\sum_{i=1}^{m} p_i \\tr D^{(i)} \\tilde{D}^{(i)}$ is smaller than \n$\\varepsilon$.\n\nThere is a seemingly slight difference between the two theorems. The \nstatistical operator $D_n$ has an extremal decomposition in the first\none and arbitrary decomposition in the second. The difference between\nthe pure and mixed message ensemble is discussed in the recent paper \n\\cite{jozsa}.\n \n\n\\section{Discussion}\nIn this paper a theory of quantum source coding subject to a fidelity criterion\nor quantum data compression is presented. The minimum of the source coding\nrate is studied under the conditions that Schumacher's fidelity must exceed\n$1-\\varepsilon$ and the quantum mechanical state of the channel has a strong\nergodic property. This latter condition allows many states with memory effect.\nFor the mathematical model and in the proof of the main result techniques\nof quantum statistical mechanics are used. We prove that the minimal source\ncoding rate is the mean entropy of the channel state, and, to some extent,\nit is independent of the message ensemble. \n\n\\section{Acknowledgement}\nThe authors thank to Prof. O.E. Barndorff-Nielsen for an invitation to a\nworkshop held at MaPhySto, to Prof. A. Holevo and Dr. S. Furuichi for \ncomments on the first draft of the paper.\n\n\\begin{thebibliography}{9}\n\n\\bibitem{N-Ch}\nM.A. Nielsen, I.L. Chuang, {\\it Quantum Computation and Quantum\nInformation}, Cambridge University Press, 2000.\n\n\\bibitem{Ho1}\nA.S. Holevo, Some estimates for the amount of information transmittable by a \nquantum communication channel, Problems Inf. Transmission, {\\bf 9}(1973), 177--183\n \n\\bibitem{OPW}\nM. Ohya, D. Petz, N. Watanabe, On capacities of quantum channels,\nProb. Math. Stat. {\\bf 17}(1997), 179--196\n\n\\bibitem{JSch} \nR. Jozsa, B. Schumacher, A new proof of the quantum noiseless coding theorem,\nJ. Modern Optics, 1994.\n\n\\bibitem{HP1}\nF. Hiai, D. Petz, The proper formula for relative entropy and its \nasymptotics in quantum probability, Commun. Math. Phys. {\\bf 143}(1991), 99-114.\n\n\\bibitem{OP}\nM. Ohya, D. Petz, {\\it Quantum Entropy and Its Use}, Springer-Verlag, \nHeidelberg, 1993.\n\n\\bibitem{HP2} \nF. Hiai, D. Petz, Entropy density for algebraic states, J. Functional Anal. \n{\\bf 125}(1994), 287--308.\n\n\\bibitem{LR} \nE.H. Lieb, M.B. Ruskai, Proof of the strong subadditivity of quantum \nmechanical entropy, J. Math, Phys. {\\bf 14}(1973), 1938--1941\n\n\\bibitem{Pe}\nD. Petz, Entropy density in quantum statistical mechanics and information\ntheory, in {\\it Contributions in Probability}, ed. C. Cecchini, 221--226, \nForum, Udine, 1996.\n\n\\bibitem{Ho}\nA.S. Holevo, Quantum coding theorems, Russian Math. Surveys, {\\bf 53}(1998), \n1295--1331\n\n\\bibitem{Sch}\nB. Schumacher, Quantum coding, Phys. Rev. A {\\bf 51}(1995), 2738--2747\n\n\\bibitem{CsK}\nI. Csisz\\'ar, J. K\\\"orner, Information theory. Coding theorems for discrete\nmemoryless systems, Akad\\'emiai Kiad\\'o, Budapest, 1981\n\n\\bibitem{Gr}\nW.T. Grandy, Jr., {\\it Foundations of Statistical Mechanics.\nVolume I: Equilibrium Theory}, D. Reidel, Dordrecht, 1987\n\n\\bibitem{HP3} \nF. Hiai, D. Petz,\nEntropy densities for Gibbs states of quantum spin systems, Rev. Math. Phys. \n{\\bf 5}(1994), 693--712\n\n\\bibitem{jozsa}\nH. Barnum, C.M. Caves, C.A. Fuchs, R. Jozsa, B.W. Schumacher,\nOn quantum coding for ensembles of mixed states, quant-ph/0008024\n\n\n\n\\end{thebibliography}\n\n\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\nm\n\n\n\n\n" } ]	[ { "name": "quant-ph9912103.extracted_bib", "string": "{N-Ch M.A. Nielsen, I.L. Chuang, {Quantum Computation and Quantum Information, Cambridge University Press, 2000." }, { "name": "quant-ph9912103.extracted_bib", "string": "{Ho1 A.S. Holevo, Some estimates for the amount of information transmittable by a quantum communication channel, Problems Inf. Transmission, {9(1973), 177--183" }, { "name": "quant-ph9912103.extracted_bib", "string": "{OPW M. Ohya, D. Petz, N. Watanabe, On capacities of quantum channels, Prob. Math. Stat. {17(1997), 179--196" }, { "name": "quant-ph9912103.extracted_bib", "string": "{JSch R. Jozsa, B. Schumacher, A new proof of the quantum noiseless coding theorem, J. Modern Optics, 1994." }, { "name": "quant-ph9912103.extracted_bib", "string": "{HP1 F. Hiai, D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability, Commun. Math. Phys. {143(1991), 99-114." }, { "name": "quant-ph9912103.extracted_bib", "string": "{OP M. Ohya, D. Petz, {Quantum Entropy and Its Use, Springer-Verlag, Heidelberg, 1993." }, { "name": "quant-ph9912103.extracted_bib", "string": "{HP2 F. Hiai, D. Petz, Entropy density for algebraic states, J. Functional Anal. {125(1994), 287--308." }, { "name": "quant-ph9912103.extracted_bib", "string": "{LR E.H. Lieb, M.B. Ruskai, Proof of the strong subadditivity of quantum mechanical entropy, J. Math, Phys. {14(1973), 1938--1941" }, { "name": "quant-ph9912103.extracted_bib", "string": "{Pe D. Petz, Entropy density in quantum statistical mechanics and information theory, in {Contributions in Probability, ed. C. Cecchini, 221--226, Forum, Udine, 1996." }, { "name": "quant-ph9912103.extracted_bib", "string": "{Ho A.S. Holevo, Quantum coding theorems, Russian Math. Surveys, {53(1998), 1295--1331" }, { "name": "quant-ph9912103.extracted_bib", "string": "{Sch B. Schumacher, Quantum coding, Phys. Rev. A {51(1995), 2738--2747" }, { "name": "quant-ph9912103.extracted_bib", "string": "{CsK I. Csisz\\'ar, J. K\\\"orner, Information theory. Coding theorems for discrete memoryless systems, Akad\\'emiai Kiad\\'o, Budapest, 1981" }, { "name": "quant-ph9912103.extracted_bib", "string": "{Gr W.T. Grandy, Jr., {Foundations of Statistical Mechanics. Volume I: Equilibrium Theory, D. Reidel, Dordrecht, 1987" }, { "name": "quant-ph9912103.extracted_bib", "string": "{HP3 F. Hiai, D. Petz, Entropy densities for Gibbs states of quantum spin systems, Rev. Math. Phys. {5(1994), 693--712" }, { "name": "quant-ph9912103.extracted_bib", "string": "{jozsa H. Barnum, C.M. Caves, C.A. Fuchs, R. Jozsa, B.W. Schumacher, On quantum coding for ensembles of mixed states, quant-ph/0008024" } ]
quant-ph9912104	Automatic Quantum Error Correction.	[ { "author": "Jeff P Barnes and Warren S Warren" }, { "author": "Department of Chemistry" }, { "author": "Princeton University" }, { "author": "Princeton" }, { "author": "NJ 08544-1009" } ]	Criteria are given by which dissipative evolution can transfer populations and coherences between quantum subspaces, without a loss of coherence. This results in a form of quantum error correction that is implemented by the joint evolution of a system and a cold bath. It requires no external intervention and, in principal, no ancilla. An example of a system that protects a qubit against spin-flip errors is proposed. It consists of three spin 1/2 magnetic particles, and three modes of a resonator. The qubit is the triple quantum coherence of the spins, and the photons act as ancilla. This article is a greatly expanded version of a letter submitted to {Physical Review Letters.	[ { "name": "Qec.tex", "string": "%\n% style\n%\n\\documentclass[10pt,onecolumn]{article}\n\\usepackage[dvips]{graphicx}\n\\usepackage{multicol}\n\\pagestyle{plain}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\oddsidemargin}{-0.0in}\n\\setlength{\\evensidemargin}{-0.0in}\n\\pagenumbering{arabic}\n\\renewcommand{\\baselinestretch}{1.0}\n% this is so I don't have to put up with\n% annoying float placement but can still ref figures\n\\newcounter{no_float_fig}\n%\n%\n\\begin{document}\n%\n\\title{Automatic Quantum\nError Correction.}\n\\author{Jeff P Barnes and Warren S Warren \\\\\nDepartment of Chemistry, Princeton University, \\\\\nPrinceton, NJ 08544-1009 }\n\\date{ (\\today) }\n\\maketitle\n%\n\\begin{abstract}\nCriteria are given by which dissipative evolution\ncan transfer populations and coherences between\nquantum subspaces, without a loss of coherence.\nThis results in a form of quantum error correction\nthat is implemented by the joint evolution of a system\nand a cold bath. It requires no external intervention\nand, in principal, no ancilla. \nAn example of a system that protects a qubit\nagainst spin-flip errors is proposed.\nIt consists of three spin 1/2 magnetic particles,\nand three modes of a resonator. The qubit is the\ntriple quantum coherence of the spins, and the\nphotons act as ancilla. This article is a greatly\nexpanded version of a letter submitted to \n{\\it Physical Review Letters}.\n\\end{abstract}\n%\nPACS number(s): 03.67.Lx\n%\n% sections\n%\n\\setcounter{page}{1}\n\\section{Introduction}\n%\nQuantum computation is of interest because\nalgorithms have been discovered with a significant\nspeed-up over any classical algorithm \\cite{shor,grover},\nalthough these may be unique cases \\cite{beals}.\nIt is very likely that any physical implementation of\na quantum computation will require some form of active\nquantum error correction. Quantum error correcting codes\n(QECC) have been devised \\cite{shor,steane,qecc5,qecc7,preskill} \nand experimentally demonstrated \\cite{qecnmr} \nthat can protect a set of states, the codewords, \nagainst a set of errors. QECC is similar in spirit\nto quantum erasure experiments \\cite{scully}, \nbut with the twist that one is not allowed\nto manipulate the environment. The surprising fact is\nthat one can still disentangle the codewords from the\nenvironment, by transferring the entanglement to another\nset of states, the ancilla.\n\\par\nHowever, implementing QECC is a formidable task.\nThere is a high premium placed on using as few qubits\n(two-level systems) as possible, \nbecause as quantum systems grow in size, \nthe number of transitions to be manipulated, unwanted\nthermal effects \\cite{warren}, and decoherence rates\n\\cite{giulini} all increase exponentially.\nBut, to take a specific example, the fault-tolerant\nerror detection and repair of even a single qubit\ncan require 15 physical qubits, 5 to store the\ntwo codewords, and 10 of which must be in \nknown states of zero entropy \\cite{qecc5}.\nIn addition, \ndepending upon how one counts a ``logic gate'',\nas many as 28 coherent manipulations of\npairs of qubits are required for each \nrepair, because \n``measuring the stabilizer'' means finding \nthe eigenvalues of operators such as \n$I_{x1}I_{x2}I_{z3}I_{z5}$\n(see Fig. (2) of Ref. \\cite{qecc5}).\nSuch control over a 32,768-level system \nis a daunting task, even for a highly \ncoherent spectroscopy such as NMR.\nAlthough the efficiency of QECC improves for\nlarger computations, a physical scale-up factor \nof 22 is still required to factorize a \nthousand-digit number \\cite{steane2}.\nPart of the difficulty stems\nfrom the need to know which error \nhas struck, in order to repair it.\nThis is because different errors \nrotate the codeword states about \nseparate axes in Hilbert space.\nBy containing information\nabout which error occurred, \nthe ancilla also provide a \nconditional axis about which rotation\ncan coherently repair an error. \nAlthough this seems like an air-tight\nargument, there is another way to approach\nQECC, which we will explore here.\n\\par\nTo begin with, note how curious it is \nthat QECC can assign a \nunique status to the codewords. \nWhile a classical probability \nspace inherently contains a privileged basis, \nHilbert space does not, and this difference \nhas some striking consequences \\cite{bell}.\nIn order to function, QECC requires access to ancilla\nin a state of zero entropy \\cite{nielsen}, which\nsuggests that one could view QECC as a controlled\ncooling of the system. It is dissipative evolution \nthat adds classical aspects back into Hilbert space.\nA large body of work exists that model a diverse range \nof relaxation phenomena in magnetic \\cite{slichter} \nand optical resonance \\cite{walls,zubairy}. \nThey use Lindblad equations of motion \\cite{giulini}.\nAn earlier approach that did use a Lindblad equation\n\\cite{paz} implemented QECC as a limit \nof very fast external manipulation.\nIn contrast, we seek an approach that is distinct\nfrom the concepts of error detection and repair.\n\\par\nWe show here that dissipation can be used to implement\nan ``automatic quantum error correction'' (AQEC),\nso called because error correction results exclusively\nfrom the joint evolution of a system coupled to a cold,\nMarkovian bath. No intervention is required by the\nprogrammer, and in theory, no ancilla are required,\nalthough this would be unlikely in practice.\nClearly, dissipation can be used to stabilize two \ndistinct states of a quantum system that could\nstore a classical bit of information. \nWhat is not obvious, is \nwhether such a system could also\nhold a qubit, since dissipation \nusually destroys coherence.\nThe key ideas are to use codewords such that\nerrors must add energy to the system, and\nto set up the evolution of the system such\nthat excitation and environmental entanglements\nare expelled from distinct codewords in a\nsymmetric way. This prevents the bath\nfrom gaining information on the codewords,\nand thus coherence can be maintained.\nIn the last section, we outline a system\nthat utilizes three magnetic spin 1/2 particles,\nand three photons, to implement an AQEC that\nprotects against spin-flip errors.\nIt requires only well understood interactions\nfrom magnetic resonance spectroscopy, and\nis intended to show that AQEC has potentially\nreal-world applications.\n%\n\\section{A simple QECC example.}\n%\nTo begin, let us review the idea of\nquantum error correction by way of\na simple method that protects a single \nquantum state against \nenvironmental entanglements \\cite{knill}.\nThe idea is similar to that of a \nquantum eraser experiment \\cite{scully},\nbut with the twist that one is not allowed to \ninteract the environment. \nThree two-level systems, labeled as $S$,\n$A$ and $E$, are initially in the state\n$( a\|1_S \\rangle + b \|0_S \\rangle )\n\|0_E \\rangle \|0_A \\rangle$.\nThe goal is to keep $S$ in its current state.\nAn interaction between $S$ and $E$ creates\nthe new state $( a \|1_S\\rangle \|p_E\\rangle\n+ b \|0_S \\rangle \| q_E\\rangle )\|0_A\\rangle$.\nThe environment is scattered \ninto two states, $\|p_E \\rangle$ and $\|q_E \\rangle$.\nWhen $\|\\langle p_E \| q_E \\rangle \| < 1$,\nthe final state of $E$ depends upon the initial\nstate of $S$, so they are entangled.\nIf $\| p_E\\rangle = -\|q_E\\rangle$, the\nphase of $S$ has been flipped.\nTo repair $S$, first note that the\nentangled state can be written as:\n\\[\n\\frac{1}{2} \\: \\Bigg\\{\n\\bigg( a\|1_S \\rangle + b\|0_S \\rangle \\bigg)\n\\bigg( \|p_E \\rangle + \|q_E \\rangle \\bigg)\n+ \\bigg( a\|1_S \\rangle - b\|0_S \\rangle \\bigg)\n\\bigg( \|p_E \\rangle - \|q_E \\rangle \\bigg)\n\\Bigg\\} \|0_A \\rangle\n\\]\nSuppose we can externally manipulate the qubits.\nConditionally flip $A$, if the sign of the\nstate $S$ is flipped from what we expect it\nto be. In the language of QECC, this is\n``measuring the stabilizer'', or ``detecting\nthe error''. $A$ serves as the memory.\nNext, flip the sign of $S$, \nconditional on if $A$ detected an error. \nThis is ``repairing the state''.\nBoth of these actions are unitary \ntransforms on $S$ and $A$ only; the environment\nis never directly manipulated.\nAfter measuring the error, the new state is:\n\\[\n\\frac{1}{2} \\Bigg\\{\n\\bigg( a \|1_S \\rangle + b\|0_S \\rangle \\bigg)\n\\bigg( \|p_E \\rangle + \|q_E \\rangle \\bigg) \|0_A \\rangle\n+ \\bigg( a \|1_S \\rangle - b\|0_S \\rangle \\bigg)\n\\bigg(\|p_E \\rangle - \|q_E \\rangle \\bigg) \|1_A \\rangle\n\\Bigg\\},\n\\]\nand then, after the repair,\n\\[\n\\bigg( a\|1_S \\rangle + b\|0_S \\rangle \\bigg)\n\\frac{1}{2} \\Bigg\\{\n\\bigg( \|p_E \\rangle + \|q_E \\rangle \\bigg) \|0_A \\rangle\n+ \\bigg( \|p_E \\rangle - \|q_E \\rangle \\bigg) \|1_A \\rangle\n\\Bigg\\}\n\\]\nThe original state of $S$ has re-emerged!\nThe entanglement between $S$ and $E$ was \ntransferred to be between $A$ and $E$,\nwithout ever touching $E$. In order for this\nscheme to work, it is crucial that $A$ is initially\nin a single pure state, or in a state of zero entropy.\nWe expect that we can achieve this by cooling $A$\ndown to 0 $^\\circ$K by the third law of thermodynamics.\nHowever, there are systems such as protons in ice\nor frustrated spin lattices \\cite{ramirez} that\nare postulated to violate the third law.\nSince cooling these systems still leaves them\nin a state of non-zero entropy, one should avoid\nusing them as ancilla.\n%\n\\subsection{A dynamical re-formulation of the above example.}\n%\nThe next step is to transform the\nabove error correcting method into the\nlanguage of a quantum system, relaxing\ntowards equilibrium.\nWe will make use of the \noperator formalism of NMR \\cite{ernst}.\nThe qubit states are $\|0\\rangle$ and $\|1\\rangle$,\nand the projection operators are\n$I_\\alpha = \|1\\rangle \\langle 1\|$ and\n$I_\\beta = \|0\\rangle \\langle 0\|$.\nThe raising and lowering operators are\n$I_+ = \|1\\rangle \\langle 0\|$ and\n$I_- = \|0\\rangle \\langle 1\|$, respectively.\nThe Hermitian Pauli operators are\n$I_x = (I_+ + I_-)/2$,\n$I_y = (I_+ - I_-)/2i$, and\n$I_z = (I_\\alpha - I_\\beta)/2$,\nand $\\vec{I}$ is the vector\nformed by them. The subscript\nalso indicates which spin is acted upon,\nso $I_{n,x}$ acts only on spin $n$.\nThis section is similar to that of Ref. \\cite{paz}, \nbut with the difference that\nthe measurement of the syndrome \nand the repair process are treated more\nexplicitly.\n\\par\nThe Hamiltonian of Eq. (\\ref{eqn:H1}).\nis designed to continuously implement the\nexample of the last section. To keep $S$ in \nthe state $\|1_S\\rangle$, first flip $A$ at\nthe rate $d$, if $S$ departs from $\|1_S\\rangle$.\nIf $A$ has flipped, then $S$ is flipped at a rate $r$. \nTo complete the process, $A$ is cooled at a rate\n$c$ by interaction with a bath\nof harmonic oscillators with a broad spectral\nresponse. If the bath temperature is low\nin comparison to the separation of the levels\nof $A$, then the density matrix $\\rho$ \nevolves as \\cite{giulini,walls},\n%\n\\begin{equation} \\begin{array}{c}\nH = r ( I_{A,\\beta} + I_{A,\\alpha} I_{S,x} )\n+ d ( I_{S,\\alpha} + I_{S,\\beta} I_{A,x} ) \\\\\n\\frac{\\displaystyle \\partial \\rho }\n{\\displaystyle \\partial t} =\n-i [ H, \\rho ] - c ( I_{A,\\alpha} \\rho + \n\\rho I_{A,\\alpha} - 2 I_{A,-} \\, \\rho \\, I_{A,+} )\n\\end{array} \\label{eqn:H1} \\end{equation} \n%\nWe suppose the errors occur rapidly in comparison\nto the system dyanmics, so they are modeled as\ninstantaneous transforms. However, slower\ninteractions can also be corrected \\cite{paz}.\n\\par\nDissipative evolution is usually handled in Liouville\nspace, where $\\rho$ is a vector, and transformations\nlike $-i[H,\\rho]$ are matrix-vector multiplications \n\\cite{superoperator}. These matrices are called\nsuperoperators, since they operate on operators.\nEq. (\\ref{eqn:H1}) becomes a set of linear\ndifferential equations, $\\dot{\\rho}$ = \n$\\Gamma \\rho$. The elements of $\\Gamma$ are\nthen indexed by how they transform the populations \nand coherences of an orthonormal set that spans\nthe Hilbert space: each element of $\\Gamma$\ntransforms a $\|j\\rangle \\langle k\|$ to \na $\|n\\rangle \\langle m\|$.\nWhen $c=0$, the evolution is unitary, and $\\Gamma$ \nhas two kinds of eigenvalues:\n$\\lambda$ = 0, corresponding to populations of \neigenstates of $H$, $\|n\\rangle \\langle n\|$, and \n$\\lambda$ = $\\pm i \\beta$, corresponding to \ncoherences $\|n\\rangle \\langle m\|$ and \n$\|m\\rangle \\langle n\|$. Unfortunately, Liouville\nspace also increases the problem size:\n$N$ qubits now require an evolution superoperator\nwith $4^N$ eigenstates. \n\\par\nWhen evolution is dissipative,\n$\\Gamma$ is not a symmetric matrix. It can still\nbe written as the outer product of its right and left\neigenvectors, $\\Gamma$ = \n$\\sum \\lambda_n \\vec{r}_n \\otimes \\vec{l}_n$,\nbut in general the $\\vec{r}_n$ are not orthogonal.\nHowever, $\\vec{l}_n \\cdot \\vec{r}_m$ = $\\delta_{nm}$, \nwhich allows one to formally solve the\nequation of motion for an operator as $\\vec{x}(t)$ = \n$\\sum \\vec{r}_n ( \\vec{l}_n \\cdot \\vec{x}(0) \\, )\n\\exp(\\lambda_n t)$. The structure of Eq. (\\ref{eqn:H1}) \nimplies that $\\Gamma$ conserves $\\mbox{tr}(\\rho(t)\\,)$, \nbut a pure state will not necessarily remain pure.\n\\par\nDoes the system of Eq. (\\ref{eqn:H1}) work?\nFig. (\\ref{fig:graph}) plots the real parts of the\neigenvalues of $\\Gamma$ of Eq. (\\ref{eqn:H1})\nfor various values of $d$, $r$ and $c$.\nThe only stable states of $\\Gamma$ will\nbe those with $\\Re e(\\lambda_n)=0$, and there is \nonly one such state: $\|1_S 0_A \\rangle \\langle 1_S 0_A \|$.\nThis is how dissipative evolution can confer \na privileged status on a state.\n\\par\nA numerical integration of the system dynamics\nalso shows this. Fig. (\\ref{fig:graph2}) plots \nthe linear entropy, defined as\n$0 \\le \\mbox{tr}(\\rho(t)-\\rho^2 (t) \\, ) \\le 1$,\nof $\\rho(t)$, starting from the corrupted state \n$\|0_S 0_A \\rangle \\langle 0_S 0_A \|$.\nThe rate at which the error is repaired is dominated\nby the eigenvalue of $\\Gamma$ with the least negative,\nbut nonzero, real part. Curiously, a larger $c$ \nis counterproductive, as it traps the state \ninto a cycle:\n\\[ \|0_S 0_A \\rangle \n\\begin{array}{c} \\mbox{detect}\\rightarrow \\\\\n\\leftarrow \\mbox{cool} \\end{array}\n\|0_S 1_A \\rangle \\;\\; \\mbox{repair} \\rightarrow \n\\;\\; \|1_S 1_A \\rangle \\;\\; \n\\mbox{cool} \\rightarrow \\;\\;\n\|1_S 0_A \\rangle\n\\]\n%\n\\par\n%\n\\includegraphics[angle=0,width=3in,height=2.5in]\n{suplineig.eps}\n\\hfill\n\\includegraphics[angle=0,width=3in,height=2.5in]\n{linent.eps}\n\\par\n\\parbox{3.0in}{\n\\refstepcounter{no_float_fig} \\label{fig:graph}\n\\textbf{Fig. \\ref{fig:graph}.}\nThe real parts of the 16 eigenvalues \nof the superoperator $\\Gamma$\nfor some values of $d$, $r$ and $c$.\nNote that there is only a single stable state.}\n\\hfill\n\\parbox{3in}{\n\\refstepcounter{no_float_fig} \\label{fig:graph2}\n\\textbf{Fig. \\ref{fig:graph2}.} \nThe linear entropy\nduring the continuous error correction,\nfor $d$, $r$, $c$ = 1, and doubling each\nparameter separately. The starting state\nis $\\rho = \|0_S 0_A \\rangle \\langle 0_S 0_A \|$.}\n\\par\n%\n%\n\\section{Conditions for AQEC.}\n%\n\\subsection{Repairing the Populations.}\n%\nWe now show what conditions\nare necessary in order that\ndissipative evolution can automatically\nprotect a subspace of codewords against a\ngiven set of errors.\nWe first suppose that the system obeys\na Lindblad equation of motion.\nIn general, this is not a trivial assumption \\cite{giulini}.\nThe most speculative condition in deriving\na Lindblad equation is that the system \nand the bath initially factorize.\nCuriously, it can be justified here on the grounds\nthat a properly working error correction should drive\nthe system to this state.\nA more troublesome condition is that if degenerate\ntransitions are coupled to the Markov bath,\nthey must couple to orthogonal bath modes \\cite{cooling,vankampen}.\n\\par\nTwo conditions can be stated immediately.\nWe are assuming that evolution for a sufficient\ntime, $T$, can repair any error.\nThus, $\\exp(\\Gamma \\, T)$ must be a\nrepair superoperator. Necessary and sufficient\nconditions for its existence are known from QECC \\cite{knill}.\nUnder these conditions, the original codeword\npopulations and coherences might be transported\nelsewhere in Hilbert space, but they are not\ndestroyed. \nThe second condition is that the codewords must\nbe immune from the influence of the bath, or that\nthey form a decoherence free subspace with respect\nto the system / bath coupling \\cite{lidar}.\n\\par\nAn example is instructive. Suppose that\nwe are interested in protecting a two-codeword\nsystem against spin-flip errors.\nThe system is split into two groups of qubits, \n$S$ and $A$, where the $A$ are continuously cooled.\nThis is not necessary for the general argument, \nwhich can be formulated entirely in terms of the \neigenvalues of $\\Gamma$.\nHowever, it simplifies the physical interpretation.\nThe system evolution is given by \\cite{walls,zubairy}\n%\n\\begin{equation}\n\\frac{\\partial \\rho}{\\partial t} =\n-i [ \\, H, \\rho \\, ] - \n\\sum_n^{\\mbox{ancilla}}\nc_n \\bigg( I_{n,\\alpha} \\rho +\n\\rho I_{n,\\alpha} - 2 I_{n,-}\n\\rho I_{n,+} \\, \\bigg)\n\\label{eqn:lindblad} \\end{equation}\n%\nwhere $H$ acts on both the $S$ and $A$.\nThe second term irreversibly draws population\nfrom the $\|1\\rangle$ states of the ancilla\nspins, and places it in the $\|0\\rangle$ states.\nChoosing codewords of the form \n$\|\\psi_n \\rangle \|0_A \\, \\rangle$, for which\n(1) the $A$ are in their ground states, and \n(2) the codewords are eigenstates of $H$,\nwill satisfy the criteria for the \ndecoherence-free subspace.\n\\par\nThe need for the QECC conditions \ncan be seen as follows. Suppose\nwe choose the two-$S$ states\n$\|00\\rangle$ and $\|11\\rangle$ \nas the codewords. But then the errors \n$I_{1,x}\|00\\rangle$ and $I_{2,x}\|11\\rangle$ \nboth result in the same state, $\|10\\rangle$.\nUnder the Markov approximation, \nthe system can not know \nwhich codeword was the original codeword, \nand so $\\Gamma$ can not repair these errors.\nBut the three-$S$ states \n$\|000\\rangle$ and $\|111\\rangle$ will work,\nsince the spaces spanned by all\nthe errors acting on each codeword are now disjoint.\nThus, errors should transfer separate codewords\ninto disjoint subspaces.\n\\par\nNow consider how to repair the codeword\npopulations. The set of errors acting on a\ncodeword, and all the further states that the\ncorrupted codeword evolves into under $\\Gamma$,\nform a subspace, as indicated in Fig.~(\\ref{fig:example}).\nCall this subspace the ``funnel'' associated\nwith the codeword, but excluding the codeword state\nitself. The name is suggestive of its role\nin AQEC. The QECC conditions already require the\ninitially excited states to be disjoint between\nseparate codewords. Thus, if we add the\nthird condition that $\\Gamma$ draws all\npopulation from each funnel state into its associated\ncodeword, and transfers no amplitude between funnels,\nthen the codeword populations are repaired.\n%\n\\par\n\\parbox[t]{3in}{\n\\includegraphics[angle=0,width=3in,height=2.5in]\n{example.eps}}\n\\hfill\n\\parbox[b][2.5in][c]{3in}{\n\\refstepcounter{no_float_fig} \\label{fig:example}\n\\textbf{Fig. \\ref{fig:example}.} \nThe level diagram of a hypothetical\nsystem. The errors (solid arrows) transfer\namplitude into the disjoint funnels associated\nwith each codeword. The three levels between\nthe brackets form the funnel for the labeled\ncodeword to the right. Dissipative cooling\nof selected transitions (dashed lines) then\nreturns the populations to their original\ncodewords.}\n%\n%\n\\par\nAn important difference between the usual\nmethod by which QECC is implemented, and AQEC,\nhas emerged. By placing the burden of the\nrepair on the system / bath coupling, AQEC,\nin theory, requires no ancilla. \nConsider how to repair the error $I_{1,x}$,\nacting on the three-$S$ and two-$A$ codewords\n$\|000,00\\rangle$ and $\|111,00\\rangle$.\nSuppose that the unnormalized states\n$\|100,00\\rangle \\pm \|000,10\\rangle$ and\n$\|011,00\\rangle \\pm \|111,10\\rangle$\nare eigenstates of $H$.\nThe corrupted state $\|100,00\\rangle$\nnow periodically becomes the state\n$\|000,10\\rangle$, where cooling of the\nfirst ancilla returns it to the codeword $\|000,00\\rangle$.\nHowever, we can not choose\n$\|010,00\\rangle \\pm \|000,10\\rangle$ and\n$\|101,00\\rangle \\pm \|111,10\\rangle$ as\neigenstates of $H$ in order to repair\nthe error $I_{2,x}$, because they are not\northogonal to the first set.\nIt seems that we must choose\n$\|010,00\\rangle\\pm\|000,01\\rangle$ and\n$\|101,00\\rangle\\pm\|111,01\\rangle$ to\nrepair $I_{2,x}$, and\n$\|001,00\\rangle\\pm\|000,11\\rangle$ and\n$\|110,00\\rangle\\pm\|111,11\\rangle$ to\nrepair $I_{3,x}$.\nIn this strategy, we must be able\nto distinguish between the different\nerrors in order to be able to repair\nthem, and counting $\\epsilon$ different\nerrors requires log$_2\\epsilon$ binary\ndigits (or ancilla qubits).\n\\par\nBut the third condition for AQEC only\nrequires that $H$ mix the states\n$\|100,00\\rangle$, $\|010,00\\rangle$, and\n$\|001,00\\rangle$ with $\|000,10\\rangle$,\neven if only partially! Because cooling\nirreversibly draws probably away from the\nexcited ancilla, any degree of mixing will\ndo. In other words, when the 4$\\times$4\nblock of $H$ corresponding to the above\nstates is diagonalized, each eigenstate\nshould have a non-zero projection onto\nthe state $\|000,10\\rangle$, and similarly\nfor the other codeword.\nWe still require at least one ancilla here,\nbecause the system was split into $S$ and\n$A$, and only the $A$ are cooled.\nIf a bath / system coupling is found that\ndirectly cools the funnel to codeword\ntransitions as in Fig.~(\\ref{fig:example}),\nthen no ancilla are necessary.\nHowever, not all the conditions for AQEC have\nbeen stated yet. The rest of these come\nfrom the seemingly bizarre notion that we can use \ndissipation to restore a coherence.\n%\n\\subsection{Repairing the Coherences.}\n%\nThe QECC conditions ensure that the codeword\ncoherences are transferred, but not ``measured'',\nby the environment. AQEC must transfer them back.\nWhat happens to coherences during dissipative\nevolution is a subtle point, which is best explored\nby way of a comprehensive example.\nCodewords of the form $\| \\psi_n \\rangle \|00\\rangle$,\nwith two ancilla, will serve this purpose.\nThe environment, $\|e\\rangle$, is initially\nunentangled with the computer. An interaction, $U$, \ncan entangle the system so that \\cite{barnes}\n%\n\\begin{equation} \\begin{array}{c}\nU \\bigg( a_0 \| \\psi_0 \\rangle \|00\\rangle + \na_1 \| \\psi_1 \\rangle \|00\\rangle \n+ a_2 \| \\psi_2 \\rangle \|00\\rangle \n+ a_3 \| \\psi_3 \\rangle \|00\\rangle + \\cdots \\bigg)\n\| e \\rangle = \\\\\n\\;\\;\\: a_0 \\bigg(\nu_0^{(0)} \| \\phi_0^{(0)} \\rangle \|00\\rangle \| e_0^{(0)} \\rangle +\nu_0^{(1)} \| \\phi_0^{(1)} \\rangle \|00\\rangle \| e_0^{(1)} \\rangle + \nu_0^{(2)} \| \\phi_0^{(2)} \\rangle \|00\\rangle \| e_0^{(2)} \\rangle + \n\\cdots \\bigg) \\\\\n+ a_1 \\bigg( \nu_1^{(0)} \| \\phi_1^{(0)} \\rangle \|00\\rangle \| e_1^{(0)} \\rangle +\nu_1^{(1)} \| \\phi_1^{(1)} \\rangle \|00\\rangle \| e_1^{(1)} \\rangle +\nu_1^{(2)} \| \\phi_1^{(2)} \\rangle \|00\\rangle \| e_1^{(2)} \\rangle + \n\\cdots \\bigg) \\\\\n+ a_2 \\bigg(\nu_2^{(0)} \| \\phi_2^{(0)} \\rangle \|00\\rangle \| e_2^{(0)} \\rangle +\nu_2^{(1)} \| \\phi_2^{(1)} \\rangle \|00\\rangle \| e_2^{(1)} \\rangle +\nu_2^{(2)} \| \\phi_2^{(2)} \\rangle \|00\\rangle \| e_2^{(2)} \\rangle + \n\\cdots \\bigg) \\\\\n+ a_3 \\bigg(\nu_3^{(0)} \| \\phi_3^{(0)} \\rangle \|00\\rangle \| e_3^{(0)} \\rangle +\nu_3^{(1)} \| \\phi_3^{(1)} \\rangle \|00\\rangle \| e_3^{(1)} \\rangle +\nu_3^{(2)} \| \\phi_3^{(2)} \\rangle \|00\\rangle \| e_3^{(2)} \\rangle +\n\\cdots \\bigg) \\\\\n\\end{array} \\label{eqn:error} \\end{equation}\n%\nAfter the error, the amplitude originally\nin each codeword is spread throughout \nits funnel. While the funnel states\n$\\{ \|\\phi_n^{(k)}\\rangle \\}$ can be chosen\nas an orthogonal set for each $n$, this is\nnot true in general for the $\\{ \|e_n^{(k)}\\rangle \\}$.\n\\par\nAs yet, there is no constraint on either how separate\ncodewords can excite the ancilla qubits, or how the\ndynamics of the repair should proceed. Suppose\n$H$ uses the first ancilla to repair the codewords\n$n$ = 0 and 1, the second ancilla to repair \n$n$=2, and both ancilla to repair $n$=3.\nThat is, $H$ mixes each $\|\\phi_0^{(k)}\\rangle\|00\\rangle$\nwith $\|\\psi_0\\rangle\|10\\rangle$, and so on.\nLet us follow an argument\nanalogous to the ``quantum jump'' \napproach \\cite{zubairy}.\nThe relaxation process is divided up into small\ntime steps, $\\Delta t$, during which the system \nand bath evolve separately. \nAt the end of each interval, a fraction \nof the amplitude in each excited ancilla state\njumps into a de-excited state.\nTagging on two more qubits to represent two modes\nof the cold bath, at the end of each time\ninterval, a fraction of the amplitudes make\nthe following jumps:\n%\n\\[ \\begin{array}{ccc}\n\|\\psi_0\\rangle \|10\\rangle \|e_0^{(k)} \\rangle \|00\\rangle &\n\\rightarrow &\n\|\\psi_0\\rangle \|00\\rangle \|e_0^{(k)} \\rangle \|10\\rangle \\\\\n\|\\psi_1\\rangle \|10\\rangle \|e_1^{(k)} \\rangle \|00\\rangle &\n\\rightarrow &\n\|\\psi_1\\rangle \|00\\rangle \|e_1^{(k)} \\rangle \|10\\rangle \\\\\n\|\\psi_2\\rangle \|01\\rangle \|e_2^{(k)} \\rangle \|00\\rangle &\n\\rightarrow &\n\|\\psi_2\\rangle \|00\\rangle \|e_2^{(k)} \\rangle \|01\\rangle \\\\\n\|\\psi_3\\rangle \|11\\rangle \|e_3^{(k)} \\rangle \|00\\rangle &\n\\rightarrow & \\!\\!\\! \\left\\{\n\\begin{array}{c}\n\|\\psi_3\\rangle \|10\\rangle \|e_3^{(k)} \\rangle \|01\\rangle \\\\\n\|\\psi_3\\rangle \|01\\rangle \|e_3^{(k)} \\rangle \|10\\rangle \\\\\n\\end{array} \\right. \\\\\n\|\\psi_3\\rangle \|10\\rangle \|e_3^{(k)} \\rangle \|00\\rangle &\n\\rightarrow &\n\|\\psi_3\\rangle \|00\\rangle \|e_3^{(k)} \\rangle \|10\\rangle \\\\\n\|\\psi_3\\rangle \|01\\rangle \|e_3^{(k)} \\rangle \|00\\rangle &\n\\rightarrow &\n\|\\psi_3\\rangle \|00\\rangle \|e_3^{(k)} \\rangle \|01\\rangle \\\\\n\\end{array} \\]\nThe entire process is repeated until a time $T$, \nwhen the relaxation process is complete.\n\\par\nThe heart of the argument relies on the idea\nthat, in the limit of a large number of cold bath\nmodes interacting with the ancilla, it\nis very likely that different ancilla that de-excite\nat different times, will transfer their excitation\nto orthogonal modes of the bath. Once excited,\nthese modes do not further influence the evolution\nof the computer, {\\it i.e.} there is no back-reaction\nfrom the bath. In this case, after equilibrium\nis reached, the final wavefunction is given by:\n%\n\\begin{equation}\n\\begin{array}{c}\n\\;\\;\\: a_0 \|\\psi_0\\rangle \|00\\rangle \nu_0^{(0)} \|e_0^{(0)} \\rangle \\bigg(\nc_0^{(0)} (\\Delta t)\|100000 \\rangle + \nc_0^{(0)} (2\\Delta t)\|010000 \\rangle +\nc_0^{(0)} (3\\Delta t)\|001000 \\rangle + \\cdots \\bigg) \\\\\n+ a_0 \| \\psi_0 \\rangle \|00\\rangle \nu_0^{(1)} \|e_0^{(1)} \\rangle \\bigg(\nc_0^{(1)} (\\Delta t) \|100000 \\rangle +\nc_0^{(1)} (2\\Delta t) \|010000 \\rangle +\nc_0^{(1)} (3\\Delta t) \|001000 \\rangle + \\cdots \\bigg) \\\\\n+ a_0 \| \\psi_0 \\rangle \|00\\rangle \nu_0^{(2)} \|e_0^{(2)} \\rangle \\bigg(\nc_0^{(2)} (\\Delta t) \|100000 \\rangle +\nc_0^{(2)} (2\\Delta t) \|010000 \\rangle +\nc_0^{(2)} (3\\Delta t) \|001000 \\rangle + \\cdots \\bigg) \\\\\n\\cdots \\\\\n+ a_1 \|\\psi_1\\rangle \|00\\rangle \nu_1^{(0)} \|e_1^{(0)} \\rangle \\bigg(\nc_1^{(0)} (\\Delta t)\|100000 \\rangle + \nc_1^{(0)} (2\\Delta t)\|010000 \\rangle +\nc_1^{(0)} (3\\Delta t)\|001000 \\rangle + \\cdots \\bigg) \\\\\n+ a_1 \|\\psi_1\\rangle \|00\\rangle \nu_1^{(1)} \|e_1^{(1)} \\rangle \\bigg(\nc_1^{(1)} (\\Delta t)\|100000 \\rangle + \nc_1^{(1)} (2\\Delta t)\|010000 \\rangle +\nc_1^{(1)} (3\\Delta t)\|001000 \\rangle + \\cdots \\bigg) \\\\\n\\cdots \\\\\n+ a_2 \|\\psi_2\\rangle \|00\\rangle \nu_2^{(0)} \|e_2^{(0)} \\rangle \\bigg( \nc_2^{(0)} (\\Delta t)\|000100 \\rangle +\nc_2^{(0)} (2\\Delta t)\|000010 \\rangle +\nc_2^{(0)} (3\\Delta t)\|000001 \\rangle + \\cdots \\bigg) \\\\\n+ a_2 \|\\psi_2\\rangle \|00\\rangle\nu_2^{(1)} \|e_2^{(1)} \\rangle \\bigg( \nc_2^{(1)} (\\Delta t)\|000100 \\rangle +\nc_2^{(1)} (2\\Delta t)\|000010 \\rangle +\nc_2^{(1)}(3\\Delta t)\|000001 \\rangle + \\cdots \\bigg) \\\\\n\\cdots \\\\\n+ a_3 \|\\psi_3\\rangle \|00\\rangle \nu_3^{(0)} \|e_3^{(0)} \\rangle \\bigg( \nc_3^{(0)} (\\Delta t,2\\Delta t)\|100010\\rangle +\nc_3^{(0)} (2\\Delta t,\\Delta t)\|010100\\rangle +\nc_3^{(0)} (\\Delta t,3\\Delta t)\|010001\\rangle \\\\\n\\hspace{2cm} +\nc_3^{(0)} (3\\Delta t,\\Delta t)\|001100\\rangle +\nc_3^{(0)} (2\\Delta t,3\\Delta t)\|010001\\rangle +\nc_3^{(0)} (3\\Delta t,2\\Delta t)\|001010\\rangle + \\cdots \\bigg) \\\\\n\\end{array} \\label{eqn:jump} \\end{equation}\n%\nFor a funnel that uses a single ancilla,\nthe $c_n^{(k)}(m\\Delta t)$ are the amplitude\nto start in the state \n$\|\\phi_n^{(k)}\\rangle\|00\\rangle\|e_0^{(k)}\\rangle$,\nand transfer an excitation to the bath at $m\\Delta t$.\nFormally, it can be constructed from the system\npropagator, $\\exp(-i H m\\Delta t / \\hbar)$, and\nmatrix elements of the system / bath interaction.\nUsing more than one excited ancilla results\nin a two-time dependence for the $c$.\nAll these functions approach zero for $t \\rightarrow T$,\ndue to the irreversible loss of amplitude from the\nfunnel states at earlier times.\n\\par\nThe important point is that the $c_n^{(k)}(m\\Delta t)$,\nfor different $n$ and $m$, uniquely label orthogonal\nmodes of the bath. To see the consequences of this,\nform $\\rho$ from Eq. (\\ref{eqn:jump}) by tracing out\nthe bath and environment. The populations look\nlike this:\n%\n\\begin{equation}\n\|\\psi_0 \\rangle \|00\\rangle \\langle \\psi_0 \| \\langle 00\| \\,\n\|a_0\|^2 \\times \\sum_m\n\| u_0^{(0)} c_0^{(0)}(m\\Delta t)\|e_0^{(0)}\\rangle +\nu_0^{(1)} c_0^{(1)}(m\\Delta t)\|e_0^{(1)}\\rangle + \\cdots \|^2\n\\label{eqn:popprod} \\end{equation}\n%\nwith similar expressions for the other codewords.\nBecause of the earlier conditions on $\\Gamma$,\nthe populations must be repaired (there is no where\nelse for the populations to go).\nThus, the sum in Eq. (\\ref{eqn:popprod}) is one.\nHowever, from Eq. (\\ref{eqn:jump}), it is easy\nto see that the coherence $\|\\psi_2\\rangle \\langle \\psi_0\|$\nis zero! Using orthogonal ancilla states between\n$n$=0 and 2 resulting in these codewords\nbecoming entangled with orthogonal bath modes.\nWhat has happened, is that the \npattern of excitation in the bath \ncan be used to determine the probability \nto be in each codeword. Using orthogonal\nancilla leaves a separate pattern of excitation\nbehind, which means the bath has gained \ninformation about the system, and\ncoherence is irreversibly lost \\cite{giulini}.\nThus, another condition for AQEC is that \nexcitation should be symmetrically removed\nfrom separate funnels.\n\\par\nThe final condition comes from examining the coherence\n%\n\\[\n\|\\psi_0 \\rangle \|00\\rangle \\langle \\psi_1 \| \\langle 00\| \\,\na_0^\\star a_1 \\times \n\\] \\begin{equation} \n\\sum_m \\left(\nu_0^{(0)} c_0^{(0)}(m\\Delta t) \|e_0^{(0)} \\rangle +\nu_0^{(1)} c_0^{(1)}(m\\Delta t) \|e_0^{(1)} \\rangle + \\cdots\n\\right)^\\dag \\left(\nu_1^{(0)} c_1^{(0)}(m\\Delta t) \|e_1^{(0)} \\rangle +\nu_1^{(1)} c_1^{(1)}(m\\Delta t) \|e_1^{(1)} \\rangle + \\cdots\n\\right)\n\\label{eqn:xprod} \\end{equation}\n%\nThe sum is the inner product of two vectors,\nindexed by $m$, whose elements are \nenvironmental wavefunctions. Each vector\nindividually has a unit norm, so by the Swartz\ninequality, the sum is one if the inner product\nof each element is maximum. Thus, the final\ncriteria for AQEC is to have\n$\\sum_k u_n^{(k)} c_n^{(k)}(m\\Delta t)\|e_n^{(k)}\\rangle$\n=$\\sum_k u_q^{(k)} c_q^{(k)}(m\\Delta t)\|e_q^{(k)}\\rangle$,\nfor each pair of codewords $n$ and $q$, and\nat each time $m\\Delta t$.\nPhysically, we are again preventing the bath\nfrom gaining information about the codewords.\nIn this case, however, the information would be\ntransferred by the pattern of environmental\nentanglements with the bath, instead of the\nexcitation.\n\\par\nThis last requirement is similar\nto the phase-matching requirement for\nfrequency mixing in non-linear optical\nmaterials \\cite{boyd}.\nConsider the following contrived example. \nA qubit suffers an error, \n$\\alpha \|0\\rangle +\n\\beta \|1 \\rangle$ $\\rightarrow$\n$\\alpha \|2\\rangle + \n\\beta \|3\\rangle$.\nThese four states have frequencies\n$\\omega_0$, $\\omega_1$, $\\omega_2$\nand $\\omega_3$, respectively.\nThe original coherence between $\|0\\rangle$\nand $\|1\\rangle$ has been transferred\nto a new set of states. \nThere are no ancilla to this repair; instead,\nrelaxation symmetrically drives \n$\|2\\rangle \\rightarrow \|0\\rangle$\nand $\|3\\rangle \\rightarrow \|1\\rangle$ \nat a steady rate. Therefore, we can write \n$c_0(t) = \\sqrt{\\gamma} \\exp(-i \\{\n\\omega_2 t + \\omega_0 (T-t) \\} - \\gamma t/2)$\nand $c_1(t) = \\sqrt{\\gamma} \\exp(-i \n\\{ \\omega_3 t + \\omega_1 (T-t) \\} -\\gamma t/2)$.\nThe original coherence gains a factor of \n$\\int c_0^\\star(t) c_1(t) dt$ =\n$\\exp(i(\\omega_0 - \\omega_1)T) \\times \\gamma\n/ (\\gamma - i( \\omega_0 - \\omega_1 -\n\\omega_2 + \\omega_3) )$.\nWhen $\\omega_0-\\omega_2$ =\n$\\omega_1-\\omega_3$,\nthe dynamics between the separate\nfunnels is indistinguishable as far as the\nbath can discern, and a full repair results.\n\\par\nTo summarize, the following are sufficient\nconditions for AQEC, although they may not all\nbe necessary. In particular, it is likely that\nthe Markov assumption could be relaxed.\n(\\textbf{1}) The system obeys a Lindblad equation\nof motion, with an evolution superoperator $\\Gamma$\n\\cite{giulini}. If cooling occurs on\ndegenerate transitions, they must be coupled to\northogonal modes of the bath \\cite{cooling,vankampen}.\n(\\textbf{2}) The eigenstates of $H$ consist of\ncodewords, $\|\\psi_n\\rangle$, the funnel subspaces\nassociated with each codeword, $\\{ \|\\phi_n^{(p)}\\rangle \\}$,\nand the rest. The codewords obey the conditions\nof QECC \\cite{knill,nielsen}, and errors transform\ncodewords only into their associated funnels.\nIf the errors are available as joint system / environmental\ntransforms, $U$, then check whether \n$\\langle \\phi_n^{(k)}\|U\|\\psi_n\\rangle$ =\n$\\langle \\phi_q^{(k)}\|U\|\\psi_q\\rangle$ for all\n$n\\ne q$, and for some labeling, $k$, of the\nfunnel states.\n(\\textbf{3}) The codewords form a decoherence free\nsubspace with respect to the bath \\cite{lidar}.\n(\\textbf{4}) $\\Gamma$ does not transfer\namplitude between funnels. All funnel populations\ndecay under $\\Gamma$ into their associated\ncodeword populations.\n(\\textbf{5}) The dynamics under $\\Gamma$ within\neach codeword-funnel subspace are identical.\nIf ancilla are used, they must be excited symmetrically\nbetween separate codewords.\nThe last conditions, which are the novel aspect of\nthis approach, are necessary in order to repair\nthe coherences of the codewords using dissipation.\nAlternatively, one could replace the last three\nconditions with criteria on the eigenvalues and\nleft and right eigenstates of $\\Gamma$.\n%\n%\n\\section{Some Numerical Simulations of AQEC.}\n%\n\\subsection{The Single Codeword Model.}\n%\nThis section provides\na better idea of how AQEC works\nby examining the behavior of a few \nnumerical simulations.\nRecall the example of keeping $S$ in\nthe state $\|1_S\\rangle$. We now see that\nwe could have used \n(the states are ordered as\n$\|0_S 0_A \\, \\rangle$, \n$\|0_S 1_A \\, \\rangle$,\n$\|1_S 0_A \\, \\rangle$ \nand $\|1_S 1_A \\, \\rangle$):\n\\begin{equation}\nH / \\hbar = \\left( \\begin{array}{cccc}\n\\omega_{00} & 0 & 0 & \\mu \\\\\n0 & \\omega_{01} & 0 & 0 \\\\\n0 & 0 & \\omega_{10} & 0 \\\\\n\\mu^\\star & 0 & 0 & \\omega_{11} \\end{array} \\right)\n\\;\\;\\;\\; \\mbox{instead of} \\;\\;\\;\\;\nH / \\hbar =\n\\left( \\begin{array}{cccc}\nr & d & 0 & 0 \\\\\nd & 0 & 0 & r \\\\\n0 & 0 & d & 0 \\\\\n0 & r & 0 & d \\end{array} \\right).\n\\label{eqn:hamone} \\end{equation}\n%\nAs previously, $\\Gamma$ has one\nzero eigenvalue, $\|1_S 0_A \\rangle \\langle 1_S 0_A\|$.\nHowever, the path by which\nerror correction occurs is different:\n$\|0_S 0_A \\, \\rangle \n\\leftrightarrow \n\|1_S 1_A \\, \\rangle \n\\rightarrow \n\|1_S 0_A \\, \\rangle$.\nThis results in a more efficient repair,\nas seen by comparison of Fig.~(\\ref{fig:graph2}) \nto Fig.~(\\ref{fig:linent2}).\nThe phase of $\\mu$, and the parameters\n$\\omega_{01}$ and $\\omega_{10}$, are irrelevant, \nbut as $\\Delta \\omega$ = $\\omega_{11}-\\omega_{00}$ \nincreases, the first step becomes less efficient. \nHowever, it isn't crucial \nthat $\\omega_{00}$ = $\\omega_{11}$ exactly.\nThe less optimized parameters slow down,\nbut do not halt, the correction process.\nNote that, in contrast to QECC,\na spin-flip error at $A$ is removed \nwithout ever influencing $S$.\n\\par\n%\n\\parbox{3.0in}{\n\\includegraphics[angle=0,width=3in,height=2.5in]\n{linent2.eps}}\n\\parbox{3.2in}{\n\\refstepcounter{no_float_fig} \\label{fig:linent2}\n\\textbf{Fig. \\ref{fig:linent2}.}\nThe linear entropy as a function of time\nduring dissipative AQEC for a single stable state.\nThe starting state is $\\rho$ =\n$\|0_S 0_A \\rangle \\langle 0_S 0_A \|$.\nThe more rapidly that $H$ can mix the\ncorrupted state with an excited ancilla,\nand then cool the ancilla, the more rapid\nthe repair. Non-zero values of $\\Delta \\omega$,\nor small values of $c$, lead to a slower repair.}\n%\n\\par\n%\n\\subsection{The Two-Codeword, Spin-flip Correcting Model.}\n%\nLet us re-examine the system that \nprotects against spin-flip errors,\nusing three $S$ and two $A$ qubits.\nThe codewords are $\|000,00\\rangle$ and\n$\|111,00\\rangle$. Parameterize the\nsystem $H$ as shown in Fig.~(\\ref{fig:parms}):\n%\n\\par\n%\n\\parbox[b][3in][c]{3.2in}{\n\\[ \\left( \\begin{array}{c\|ccc\|ccc}\n\\omega_0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline\n0 & \\omega_{e1} & \\gamma_{12} & \\gamma_{13} & \n\\mu_{11} & \\mu_{12} & \\mu_{13} \\\\\n0 & \\gamma_{12}^\\star & \\omega_{e2} & \n\\gamma_{23} & \\mu_{21} & \\mu_{22} & \\mu_{23} \\\\\n0 & \\gamma_{13}^\\star & \\gamma_{23}^\\star & \\omega_{e3} & \n\\mu_{31} & \\mu_{32} & \\mu_{33} \\\\ \\hline\n0 & \\mu_{11}^\\star & \\mu_{21}^\\star & \\mu_{31}^\\star & \n\\omega_{c1} & \\kappa_{12} & \\kappa_{13} \\\\\n0 & \\mu_{12}^\\star & \\mu_{22}^\\star & \\mu_{32}^\\star & \n\\kappa_{12}^\\star & \\omega_{c2} & \\kappa_{23} \\\\\n0 & \\mu_{13}^\\star & \\mu_{23}^\\star & \\mu_{33}^\\star & \n\\kappa_{13}^\\star & \\kappa_{23}^\\star & \\omega_{c3}\n\\end{array} \\right) \\;\\;\\; \\begin{array}{c} \n\|000,00\\rangle \\\\ \|001,00\\rangle \\\\ \|010,00\\rangle \\\\ \n\|100,00\\rangle \\\\ \|000,01\\rangle \\\\ \|000,10\\rangle \\\\ \n\|000,11\\rangle \\end{array} \\]\n%\n\\refstepcounter{no_float_fig} \\label{fig:parms}\n%\n\\textbf{Fig. \\ref{fig:parms}.}\nA parameterized $H$ for a\ntwo-codeword AQEC with three $S$ and two $A$.\n$H$ is block diagonal, with\nthe two blocks parameterized as shown\nabove (the lines provide a guide for the eye,\nwith a listing of the order of the states for\nthe first funnel / codeword combination).\nThe blocks must be identical, to within a constant\noffset along the diagonal, so the dynamics between\nthe funnels appears indistinguishable.}\n\\parbox[b][3in][t]{3.0in}{\n\\includegraphics[angle=0,width=3in,height=3in]\n{pic_of_H.eps} }\n%\n\\par\nThe $\\gamma$ mix the different error states,\nthe $\\kappa$ mix the excited\nancilla states, and the $\\mu$ mix\nthe errors with the excited ancilla states.\nPrevious implementations of QECC kept\nthe $\\mu$ matrix diagonal and the \n$\\gamma$, $\\kappa$ = 0, so that separate errors\nexcited orthogonal ancilla states.\nFor AQEC, we must check that each of the\nsix eigenstates of the funnels have some non-zero\nprojection along a state with excited ancilla\nso that population is not trapped in a funnel.\n\\par\nEq.~(\\ref{eqn:examples}) shows some examples for $H$.\nAll three sets properly repair spin-flip errors, but\nset (C), which is nearest in spirit to QECC, \nimplements the most rapid repair.\nFor the simulations,\nonly the 14 total codeword and funnel states\nare used in the numerical simulations, so\n$\\Gamma$ is $196 \\times 196$ in size.\nThe matrix exponential routine of MATLAB \n\\cite{matlab} was\nused to produce $\\exp(\\Gamma \\, t)$. \nThe initial $\\rho$ is found by tracing\nthe environment out from the initial\nerror state, $\|\\Psi\\rangle \|e_0\\rangle$ +\n$I_{1,x} \|\\Psi\\rangle \|e_1\\rangle$ +\n$I_{2,x} \|\\Psi\\rangle \|e_2\\rangle$ +\n$I_{3,x} \|\\Psi\\rangle \|e_3\\rangle$,\nwhere $\|\\Psi\\rangle$ = $(1/\\sqrt{2})\|000,00\\rangle$\n+ $(\\exp(i\\pi/3)/\\sqrt{2})\|111,00\\rangle$.\nThis state allows us to check whether \nthe coherence phase is properly recovered.\nIn general, the environmental overlaps\n$\\langle e_n \| e_m \\rangle$\ncould be any complex numbers subject to\n$\\sum_n \\langle e_n \| e_n \\rangle = 1$\nand $\|\\langle e_n \| e_m \\rangle\|^2$ $\\le$ \n$\\langle e_n \| e_n \\rangle \\langle e_m \| e_m \\rangle$.\n%\n\\begin{equation} A \\;\n\\left( \\begin{array}{c\|ccc\|ccc}\n10 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 2 & 1 & 0 & 1 & 0 & 0 \\\\\n0 & 1 & 2 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 2 & 0 & 0 & 0 \\\\\n\\hline\n0 & 1 & 0 & 0 & 2 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 2 & 0 \\\\ \n0 & 0 & 0 & 0 & 0 & 0 & 2\n\\end{array} \\right)\n\\;\\;\\; B \\;\n\\left( \\begin{array}{c\|ccc\|ccc}\n10 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 2 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 2 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 2 & 1 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 1 & 2 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 2 & 1 \\\\ \n0 & 0 & 0 & 0 & 0 & 1 & 2\n\\end{array} \\right)\n\\;\\;\\; C \\;\n\\left( \\begin{array}{c\|ccc\|ccc}\n10 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 2 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 2 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 2 & 0 & 0 & 1 \\\\\n\\hline\n0 & 1 & 0 & 0 & 2 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 2 & 0 \\\\ \n0 & 0 & 0 & 1 & 0 & 0 & 2\n\\end{array} \\right)\n\\label{eqn:examples} \\end{equation}\n\\par\nSet Eq.~(\\ref{eqn:examples},A) \nuses a single ancilla to correct \nthe three independent spin-flip errors. \nFig.~(\\ref{fig:twoa5})\nshows the recovery of the codeword populations \nand coherences for a spin-flip error at each $S$,\nand for a spin-flip error with a set of randomly\nchosen environmental overlaps, $\\langle e_n \| e_m \\rangle$,\nas given in Eq.~(\\ref{eqn:overlaps}).\n%\n\\begin{equation}\n\\left( \\begin{array}{cccc}\n.10 & -.7+.2i & 0 & -.3-.3i \\\\\n& .41 & .3+.7i & .4-.2i \\\\\n& & .27 & .8+.3i \\\\\n& & & .22 \\end{array} \\right)\n\\label{eqn:overlaps} \\end{equation}\n%\nThe $H$ of Eq. (\\ref{eqn:examples},A)\nexcites an ancilla only for the\nfirst spin-flip error. It repairs the other spin-flip\nerror by mixing all the errors together. The numerical\nsimulations shown in Fig.~(\\ref{fig:twoa5}) show this\nprocess in detail. Note that if the all three spin-flips\nentangle the system with the environment, then \nthe linear entropy of the system is initially non-zero.\n%\n\\par\n\\parbox{3.0in}{\n\\includegraphics[angle=0,width=3.0in,height=2.5in]{twoa5p.eps}\n} \\hfill\n\\parbox{3.0in}{\n\\includegraphics[angle=0,width=3.0in,height=2.5in]{twoa5c.eps}\n} \\par\n\\parbox{3.0in}{\n\\includegraphics[angle=0,width=3.0in,height=2.5in]{twoa5s.eps}\n} \\hfill\n\\parbox{3.0in}{ \n\\refstepcounter{no_float_fig} \\label{fig:twoa5}\n\\textbf{Fig. \\ref{fig:twoa5}.}\nRepair of the populations (upper left),\ncoherences (upper right), and the linear\nentropy (lower), for errors in which each\n$S$ alone flips, or there is a correlated\nflip of all the spins whose environmental\noverlaps are given by Eq. (\\ref{eqn:overlaps}).\nAll three graphs have the same legend.\nThe repair process is a linear dynamics, so \nit also repairs correlated single spin-flips.\nPopulations and coherences of both codewords\nfollow identical paths.\nThe parameter set is Eq. (\\ref{eqn:examples},A),\nbut with the second codeword / funnel \noffset by $\\Delta \\omega$ = 1\nin order to show the oscillation of the\ncoherences. The cooling rates were\n$c_1, c_2$ = 1.} \\par\n%\nThus, AQEC can expell the information about\nwhich error occurred at the same time as the\nerror is repaired. Eq.~(\\ref{eqn:examples},B) \nis another example of this.\nIt mixes together all the errors, and all the\nexcited ancilla states. Because it does so symmetrically\nbetween the separate codewords, the errors are repaired.\nFig.~(\\ref{fig:twoa1}) shows the populations \nfor all the funnel states after the first spin is flipped.\nIt can be observed that all the states are transiently\nexcited: the state vector ``swirls around'' in each\nfunnel as the repair occurs.\n%\n\\par\n\\parbox{3.0in}{\n\\includegraphics[angle=0,width=3in,height=2.5in]\n{twoa1.eps}} \\hfill\n\\parbox{3.0in}{\n\\refstepcounter{no_float_fig} \\label{fig:twoa1}\n\\textbf{Fig. \\ref{fig:twoa1}.}\nThe populations of the codewords (at back),\nof the three different spin-flip states \n(middle three), and of the three orthogonal\nexcited ancilla states (forward three),\nduring a repair after the first spin is flipped.\nThe parameter set is Eq. (\\ref{eqn:examples},B),\nwith cooling rates $c_1, c_2$ = 1.\nIt is permissible under AQEC to mix together\ndifferent errors during the repair, so long\nas the dynamics between the separate codeword / funnel\nsubspaces is indistinguishable to the bath.}\n\\par\n%\nWhat happens when condition (\\textbf{5}) is violated?\nThere are two possibilities: excite the ancilla\nasymmetrically between the codewords, or have different\ndynamics between the two codeword / funnel subspaces.\nFig.~(\\ref{fig:twoa2}) shows the first case, for which\nEq.~(\\ref{eqn:examples},A) was used, but modified for\nthe funnel surrounding $\|111,00\\rangle$ by setting\n$\\mu_{11}=\\mu_{12}=1/\\sqrt{2}$. The cooling rates\nwere $c_1, c_2$ = 1, and the error was $I_{1,x}$.\nThe partially orthogonal\nancilla states do not allow $\\Gamma$ to return\n$\\rho$ to a pure state. In fact, if $\\mu_{11}$ were\nset to zero for the second codeword, then there would be\nno element in $\\Gamma$ to transfer \n$\|000,10\\rangle \\langle 111,01\|$ to\n$\|000,00\\rangle \\langle 111,00\|$. \nThe bath gains information about the system through\nexcitation. The coherence asymptotically approaches\n$0.3530 \\exp(i\\pi(0.3333)\\,)$, with correct phase\nbut low magnitude.\n\\par\n%\n%\n\\parbox{3.0in}{\n\\includegraphics[angle=0,width=3in,height=2.5in]\n{twoa2.eps}} \\hfill\n\\parbox{3.0in}{\n\\includegraphics[angle=0,width=3in,height=2.5in]\n{twoa3.eps} }\n\\par\n\\parbox{3.0in}{ \n\\refstepcounter{no_float_fig} \\label{fig:twoa2}\n\\textbf{Fig. \\ref{fig:twoa2}.}\nA partial repair\ndue to the use of partially orthogonal\nexcited ancilla states, \n$\|10\\rangle$ and $(\|10\\rangle+\|01\\rangle)/\\sqrt{2}$,\nbetween the two codewords.\nThe populations (solid line) are repaired,\nbut the coherence magnitudes are not (dashed line).}\n\\hfill\n\\parbox{3.0in}{\n\\refstepcounter{no_float_fig} \\label{fig:twoa3}\n\\textbf{Fig. \\ref{fig:twoa3}.}\nA partial repair resulting \nfrom a dissimilar dynamics between the two\ncodeword / funnel subspaces. The populations\nare repaired (solid line), but the coherence\nmagnitudes are not (dashed lines).}\n\\par\n\\vspace{0.1in}\n%\nThe second possibility is shown in Fig.~(\\ref{fig:twoa3}).\nHere, the ancilla are excited symmetrically,\nbut the dynamics between the funnels\nis not equivalent, and coherence is again lost.\nThe parameter set is Eq.~(\\ref{eqn:examples},C),\nbut $\\mu_{11}$=2 for the second funnel,\nso the mixing was more rapid.\nThe cooling rates were $c_1, c_2$ = 1, and the\nerror was $I_{1,x}$. Again, the population\nis repaired, but some coherence is lost.\n%\n\\section{A Proposed Test System.}\n%\nWe now give a physically realizable example\nof an AQEC system that implements Shor's three-qubit,\nmajority code against spin-flip errors \\cite{shor}.\nIt can not repair phase-flip errors, so it is not\nsuitable for a quantum computer. It serves to\nillustrate a method by which to find systems\nsuitable for AQEC. It is also encouraging that a\nsystem can be found without making recourse to exotic\ninteractions.\n\\par\nOur strategy is to find a system that obeys multiple\nconservation laws that the errors violate.\nConsider a system with an observable $A$\nsuch that $[H,A]=0$, and an error $E$ where $[A,E]=E$.\nThe simultaneous eigenstates of $H$ and $A$,\n$\|\\epsilon,a\\rangle$, have the property that\n$A E \|a\\rangle$ = $(a+1)E\|a\\rangle$.\nThus, choosing codewords with $a$ = 0 and 2\ngives rise to funnel states with $a$ = 1 and 3.\nIf, in addition, there is a unitary $B$ such that\n$B\|\\epsilon,a\\rangle = \|\\epsilon,a+2\\rangle$,\nthen the funnel states can be mapped onto one\nanother, and their dynamics are equivalent.\n\\par\nLet three spin 1/2 particles be lined up along\nthe $z$ axis, in a zero static magnetic field.\nThey interact by point dipolar\n$D_{nm}(I_{n,x}I_{m,x}+I_{n,y}I_{m,y}-2I_{n,z}I_{m,z})$\nand exchange\n$J_{nm}(I_{n,x}I_{m,x}+I_{n,y}I_{m,y}+I_{n,z}I_{m,z})$\nterms \\cite{slichter}.\nDipolar interactions decrease with distance\nas $r^{-3}$, so for equally spaced spins,\n$D_{12}$=$D_{23}$=$8D_{13}$=$\\zeta$,\nwhere $\\zeta$ can be as large as 0.1 cm$^{-1}$\n\\cite{esr}.\n\\par\nAssuming the dipolar interactions dominate,\nthe level diagram of the spins is given in\nFig.~(\\ref{fig:spins},A). The spins attempt to\nmutually align, resulting in ground states\nof $\|000\\rangle$ and $\|111\\rangle$. These\nare the codewords. A spin-flip error, $I_{n,x}$,\nis equivalent to rotating a spin by $\\pi$\nabout the $x$ axis. Since this requires work\nagainst the dipolar field, dissipation can\nrepair these errors. The funnels come from\nthe conservation of the spin angular momentum\nabout the $z$ axis, $\\sum_n I_{n,z}$, which\nhas eigenvalues denoted as $m_z$. An error\nchanges $m_z \\rightarrow m_z \\pm 1$.\nThe codewords have $m_z = \\pm 3/2$.\nThe funnel surrounding $\|000\\rangle$\n(levels A-C of Fig.~(\\ref{fig:spins},A) )\nhas $m_z = -1/2$, and the funnel\nsurrounding $\|111\\rangle$ has $m_z = +1/2$\n(levels D-F).\n\\par\n\\vspace{0.2in}\n\\parbox{4.0in}{\n\\includegraphics[angle=0,width=4in,height=2.0in]\n{qec_dipole.eps} } \\hfill\n\\parbox{2.0in}{ \n\\refstepcounter{no_float_fig} \\label{fig:spins}\n\\textbf{Fig. \\ref{fig:spins}.}\n(\\textbf{A}) The level diagram for the three spin system.\nThe ground states are the codewords. States\nA-C with $m_z=-1/2$ form the funnel for $\|000\\rangle$.\nThe dashed lines show the dipole-allowed transitions \nfor the first funnel, with symmetric transitions\nfor the second funnel.\n(\\textbf{B}) The spectrum of dipole-allowed transitions.\nThe starred lines represent the funnel to codeword\ntransitions that should be cooled.}\n\\par\n\\vspace{0.2in}\nSpontaneous emission of a photon with an $x$ polarized\n$B$ field will symmetrically de-excite the degenerate\nfunnels. Thus, we could use photons as the ancilla\nfor this system. There are several advantages to this\napproach. First, it is difficult to selectively\ncool a single spin, but it is easy to cool an\nelectromagnetic resonator. Second, the validity\nof the Markov approximation for a damped resonator mode\nis better understood \\cite{scully,walls}.\nA $y$-polarized $B$ field will anti-symmetrically\nde-excite the funnels, so an error $I_{n,y}$ will be\n``repaired'' to the phase-flip error $I_{n,z}$.\nThis phase-flip error can not be repaired, because\nit requires no work to rotate a spin about the $z$\naxis for this system.\n\\par\nThere are two last difficulties to overcome.\nThe dashed lines of Fig.~(\\ref{fig:spins},A) show\nthe dipole-allowed transitions for the $m_z = -1/2$\nstates. Their strengths are proportional to matrix\nelements of the operator $\\sum_n I_{n,x}$.\nPoint dipolar interactions alone are not sufficient\nfor AQEC, because $\\langle B\|\\sum_n I_{n,x}\|000\\rangle$\nis always zero, no matter how the spins are positioned\nalong the $z$ axis. This unwanted symmetry is broken\nby setting $J_{12}$ = $J_{13}$ = 0 and $J_{23}=0.2\\zeta$.\nActually, any $0 < \|J_{23}\| \\le 0.5\\zeta$ will work.\nUsing the above parameters gives rise to the spectrum\nof Fig.~(\\ref{fig:spins},B). The second difficulty\nis that, besides spontaneous emission of $y$-polarized\n$B$-field photons, there are also transitions between\nthe funnels. We wish to cool only the starred transitions\n(at $0.64\\zeta$, $1.03\\zeta$, and $2.39\\zeta$), but\nnot the un-starred ones \n(at $0.39\\zeta$, $1.36\\zeta$ and $1.75\\zeta$).\n\\par\nThis can be achieved by placing the spins at the\ncenter of a resonator whose modes are only resonant\nwith the starred transitions.\nConsider a rectangular, conducting cavity\nof linear dimensions $a$, $b$, and $d$.\nResonances are indexed as transverse electric\n(TE$_{mnp}$, where $n+m>0$, $p > 0$) and transverse magnetic \n(TM$_{mnp}$, where $n,m>0$, $p \\ge 0 $) modes \\cite{gandhi},\nwith frequencies $\\omega_{mnp}$ =\n$\\sqrt{(m/a)^2+(n/b)^2+(p/d)^2}$ in units \nof cm$^{-1}$ if the cavity lengths are in cm.\nEach mode produces either a linearly\npolarized electric or magnetic field, \nor no field, at the center.\nOne can invert the above to find that a\nresonator with dimensions $a$=2.32/$\\zeta$, \n$b$=0.87/$\\zeta$, and $d$=4.28/$\\zeta$,\nhas TE$_{102}$, TE$_{104}$, and TE$_{122}$\nmodes resonant with the starred transitions.\nEach mode has an $x$ polarized $B$ field at the\nspins. Another 29 modes exist with $\\omega < 2.5\\zeta$.\nOf those that produce $B$ fields at the spins,\nthe nearest to a funnel-funnel transition is TE$_{302}$, \nwhich is offset by 0.018$\\zeta$ from the C-E\ntransition. A resonator Q $\\gg 76$ is required\nto suppress emission of this transition.\nFor $\\zeta \\approx 0.1$ cm$^{-1}$, microwave\nresonators can achieve this goal. This larger \n$\\zeta$ is also desirable because the cold bath \nmust satisfy $T \\ll (hc/k)\\zeta \\approx$\n0.1 K, a not outrageous requirement.\n%\n\\section{Discussion}\nAQEC borrows the same structure for storing information\nas in QECC, but implements the error correction in a\ndifferent way. In NMR terminology, the qubit of the\nabove system is hidden in the triple quantum coherence\nof the spins. The novel aspect is that dissipation can\nbe used to directly repair not only the codeword populations,\nbut also the coherences. The criteria for this is\nsimply summed up by demanding that excitation, and environmental\nentanglements, be expelled from the codewords in a symmetric\nmanner.\n\\par\nEspecially interesting is the possibility that an AQEC qubit\nexists that can protect against both spin- and phase-flip errors.\nExchange interactions may prove more useful in this regard.\nBeing isotropic interactions, they resist the rotation of a\nspin about any axis. One difficulty with using only exchange\ninteractions, is that dipole-allowed transitions vanish, so a\nsymmetric de-excitation of all the funnels by photons becomes\nproblematic. Another open question is how AQEC behaves when\nit is scaled up to large numbers of codewords. It is, however,\nhelpful to contemplate an error correction scheme that requires\nno additional burden to the programmer.\n%\n\\section{Conclusions}\nConditions are given by which the dissipative evolution of a\nsystem, coupled to a cold Markovian bath, can be used to\nimplement automatic quantum error correction. The new condition,\nnecessary to repair codeword coherences, requires a symmetric\nde-excitation of separate codewords, and an equivalent\ndynamics between the different codeword / funnel subspaces.\nThey resemble the conditions of phase-matching in nonlinear\noptics. A test case, that of Shor's majority-code against\nspin-flip errors \\cite{shor}, is proposed. It utilizes\nwell known dipolar and exchange interactions between spins,\nand dipole-allowed transitions with the modes of a resonator.\n%\n\\subsection{Acknowledgements}\nWe gratefully acknowledge support from\nthe Air Force Office of Scientific Research.\n%\n%\n\\begin{thebibliography}{99}\n% 1\n\\bibitem{shor} P. Shor, in {\\it Proceedings, 35th Annual\nSymposium on Foundations of Computer Science.} \nS. Goldwasser, Ed. (IEEE Press, New York, 1994), pp.56-65.\n% 2\n\\bibitem{grover} L. Grover, {\\it Phys. Rev. Lett.} {\\bf 79}, 325 (1997).\n% 3\n\\bibitem{beals} R. Beals, H. Buhrman, R. Cleve, M. Mosca,\nR. de Wolf, e-print quant-ph/9802049.\n% 4\n\\bibitem{steane} A. M. Steane, in {\\it Introduction \nto Quantum Computing and Information} H.-K. Lo, \nS. Popescu, T. Spiller, Eds. \n(World Scientific, Singapore, 1998), pp.184-212.\n% 5\n\\bibitem{qecc5} A. M. Steane, \n{\\it Phys. Rev. Lett.} {\\bf 78}, 2252 (1997).\n% 6\n\\bibitem{qecc7} A. R. Calderbank, P. W. Shor, \n{\\it Phys. Rev. A} {\\bf 54}, 1098 (1996).\n% 7\n\\bibitem{preskill} J. Preskill, in {\\it Introduction\nto Quantum Computing and Information} H.-K. Lo,\nS. Popescu, T. Spiller, Eds. (World Scientific,\nSingapore, 1998). pp.213-269.\n% 8\n\\bibitem{qecnmr} D. Cory, M. Price, W. Maas,\nE. Knill, R. Laflamme, W. Zurek, T. Havel,\nS. Somaroo, {\\it Phys. Rev. Lett.} {\\bf 81}, 2152 (1998).\n% 9\n\\bibitem{scully} M. O. Scully, B.-G. Englert,\nH. Walther, {\\it Nature} {\\bf 351}, 111 (1991). \n% 10\n\\bibitem{warren} W. S. Warren, \n{\\it Science} {\\bf 277}, 1688 (1997).\n% 11\n\\bibitem{giulini} D. Giulini, E. Joos, C. Kiefer,\nJ. Kupsch, I.-O. Stamatescu, and H. D. Zeh,\n{\\it Decoherence and\nthe Appearance of a Classical World in Quantum\nTheory} (Springer, Berlin, 1996), chap. 7.\n% 12\n\\bibitem{steane2} A. M. Steane, \n{\\it Nature} {\\bf 399}, 124 (1999).\n% 13\n\\bibitem{bell} J. S. Bell, {\\it Physics} {\\bf 1}, 195 (1964);\nJ. F. Clauser, M. A. Horne, {\\it Phys. Rev. D} {\\bf 10},\n526 (1974); A. Garg, N. D. Mermin, {\\it Found. Phys.} {\\bf 14},\n1 (1984).\n% 14\n\\bibitem{nielsen} M. A. Nielsen, C. M. Caves, B. Schumacher,\nH. Barnum. {\\it Proc. R. Soc. Lond. A} {\\bf 454}, 277 (1998).\n% 15\n\\bibitem{slichter} C. P. Slichter, {\\it Principles of Magnetic\nResonance, Third Edition}. (Springer-Verlag, Berlin,\n1990), chap. 5.\n% 16\n\\bibitem{walls} D. F. Walls, G. J. Milburn, \n{\\it Quantum Optics} (Springer, Berlin, 1994),\nchap. 6, 10.\n% 17\n\\bibitem{zubairy} M. O. Scully, M. S. Zubairy,\n{\\it Quantum Optics} (Cambridge University,\nCambridge, 1997), chap. 8.5. \n% 18\n\\bibitem{paz} J. P. Paz, W. H. Zurek,\n{\\it Proc. R. Soc. Lond. A} {\\bf 454}, 355 (1998).\n% 19\n\\bibitem{knill} E. Knill, R. Laflamme,\n{\\it Phys. Rev. A} {\\bf 55}, 900 (1997).\n% 20\n\\bibitem{ramirez} A. P. Ramirez, A. Hayashi, B. S. Shastry,\n{\\it Nature} {\\bf 399}, 333 (1999).\n% 21\n\\bibitem{ernst} R. R. Ernst, G. Bodenhausen, and A. Wokun,\n{\\it Principals of Nuclear Magnetic Resonance in One and\nTwo Dimensions.} (Clarendon Press, Oxford, 1987).\n% 22\n\\bibitem{superoperator} C. N. Banwell,\nH. Primas, {\\it Mol. Phys.} {\\bf 6}, 225 (1963).\n% 23\n\\bibitem{cooling} The typical derivation of the master\nequation for cooling multiple qubits \\cite{walls} gives\na choice of whether to couple each qubit symmetrically\nto the modes of the bath, or to couple separate qubits to\nseparate modes. The later choice leads to the form we\nuse. The former choice leads to qubit interaction terms\nsuch as $I_{1,+}I_{2,-}$, implying that a transfer of\nexcitation back from the bath can occur \\cite{vankampen}.\nIt seems likely, although unproven to the author's knowledge,\nthat typical cooling methods ({\\it e.g.}, contact with liquid\nHe) involve a range of coupling symmetries, so that qubit-qubit\ninteractions through the bath can be neglected.\n% 24\n\\bibitem{vankampen} N. G. Van Kampen, Physica A\n{\\bf 147}, 165 (1987).\n% 25\n\\bibitem{lidar} D. A. Lidar, I. L. Chuang, K. B. Whaley,\nPhys. Rev. Lett. {\\bf 81}, 2594 (1998).\n% 26\n\\bibitem{barnes} J. P. Barnes, W. S. Warren,\nPhys. Rev. A, in press; e-print quant-ph/9902084.\n% 27\n\\bibitem{boyd} R. W. Boyd, {\\it Nonlinear Optics}\n(Academic Press, San Deigo, 1992). Chapter 2.7.\n% 28\n\\bibitem{matlab} MATLAB 5.3.0. The MathWorks, Inc.,\n3 Apple Hill Drive, Natick, MA 01760-2098.\n% 29\n\\bibitem{esr} A. Bencini, D. Gatteschi,\n{\\it EPR of Exchange Coupled Systems}.\n(Springer-Verlag, Berlin, 1990), chap. 2.\n%\n\\bibitem{gandhi} O. Gandhi, {\\it Microwave Engineering and\nApplications} (Pergamon, New York, 1981), chap. 8.2.\n%\n\\end{thebibliography}\n%\n\\end{document}\n" } ]	[ { "name": "quant-ph9912104.extracted_bib", "string": "{shor P. Shor, in {Proceedings, 35th Annual Symposium on Foundations of Computer Science. S. Goldwasser, Ed. (IEEE Press, New York, 1994), pp.56-65. % 2" }, { "name": "quant-ph9912104.extracted_bib", "string": "{grover L. Grover, {Phys. Rev. Lett. {79, 325 (1997). % 3" }, { "name": "quant-ph9912104.extracted_bib", "string": "{beals R. Beals, H. Buhrman, R. Cleve, M. Mosca, R. de Wolf, e-print quant-ph/9802049. % 4" }, { "name": "quant-ph9912104.extracted_bib", "string": "{steane A. M. Steane, in {Introduction to Quantum Computing and Information H.-K. Lo, S. Popescu, T. Spiller, Eds. (World Scientific, Singapore, 1998), pp.184-212. % 5" }, { "name": "quant-ph9912104.extracted_bib", "string": "{qecc5 A. M. Steane, {Phys. Rev. Lett. {78, 2252 (1997). % 6" }, { "name": "quant-ph9912104.extracted_bib", "string": "{qecc7 A. R. Calderbank, P. W. Shor, {Phys. Rev. A {54, 1098 (1996). % 7" }, { "name": "quant-ph9912104.extracted_bib", "string": "{preskill J. Preskill, in {Introduction to Quantum Computing and Information H.-K. Lo, S. Popescu, T. Spiller, Eds. (World Scientific, Singapore, 1998). pp.213-269. % 8" }, { "name": "quant-ph9912104.extracted_bib", "string": "{qecnmr D. Cory, M. Price, W. Maas, E. Knill, R. Laflamme, W. Zurek, T. Havel, S. Somaroo, {Phys. Rev. Lett. {81, 2152 (1998). % 9" }, { "name": "quant-ph9912104.extracted_bib", "string": "{scully M. O. Scully, B.-G. Englert, H. Walther, {Nature {351, 111 (1991). % 10" }, { "name": "quant-ph9912104.extracted_bib", "string": "{warren W. S. Warren, {Science {277, 1688 (1997). % 11" }, { "name": "quant-ph9912104.extracted_bib", "string": "{giulini D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H. D. Zeh, {Decoherence and the Appearance of a Classical World in Quantum Theory (Springer, Berlin, 1996), chap. 7. % 12" }, { "name": "quant-ph9912104.extracted_bib", "string": "{steane2 A. M. Steane, {Nature {399, 124 (1999). % 13" }, { "name": "quant-ph9912104.extracted_bib", "string": "{bell J. S. Bell, {Physics {1, 195 (1964); J. F. Clauser, M. A. Horne, {Phys. Rev. D {10, 526 (1974); A. Garg, N. D. Mermin, {Found. Phys. {14, 1 (1984). % 14" }, { "name": "quant-ph9912104.extracted_bib", "string": "{nielsen M. A. Nielsen, C. M. Caves, B. Schumacher, H. Barnum. {Proc. R. Soc. Lond. A {454, 277 (1998). % 15" }, { "name": "quant-ph9912104.extracted_bib", "string": "{slichter C. P. Slichter, {Principles of Magnetic Resonance, Third Edition. (Springer-Verlag, Berlin, 1990), chap. 5. % 16" }, { "name": "quant-ph9912104.extracted_bib", "string": "{walls D. F. Walls, G. J. Milburn, {Quantum Optics (Springer, Berlin, 1994), chap. 6, 10. % 17" }, { "name": "quant-ph9912104.extracted_bib", "string": "{zubairy M. O. Scully, M. S. Zubairy, {Quantum Optics (Cambridge University, Cambridge, 1997), chap. 8.5. % 18" }, { "name": "quant-ph9912104.extracted_bib", "string": "{paz J. P. Paz, W. H. Zurek, {Proc. R. Soc. Lond. A {454, 355 (1998). % 19" }, { "name": "quant-ph9912104.extracted_bib", "string": "{knill E. Knill, R. Laflamme, {Phys. Rev. A {55, 900 (1997). % 20" }, { "name": "quant-ph9912104.extracted_bib", "string": "{ramirez A. P. Ramirez, A. Hayashi, B. S. Shastry, {Nature {399, 333 (1999). % 21" }, { "name": "quant-ph9912104.extracted_bib", "string": "{ernst R. R. Ernst, G. Bodenhausen, and A. Wokun, {Principals of Nuclear Magnetic Resonance in One and Two Dimensions. (Clarendon Press, Oxford, 1987). % 22" }, { "name": "quant-ph9912104.extracted_bib", "string": "{superoperator C. N. Banwell, H. Primas, {Mol. Phys. {6, 225 (1963). % 23" }, { "name": "quant-ph9912104.extracted_bib", "string": "{cooling The typical derivation of the master equation for cooling multiple qubits \\cite{walls gives a choice of whether to couple each qubit symmetrically to the modes of the bath, or to couple separate qubits to separate modes. The later choice leads to the form we use. The former choice leads to qubit interaction terms such as $I_{1,+I_{2,-$, implying that a transfer of excitation back from the bath can occur \\cite{vankampen. It seems likely, although unproven to the author's knowledge, that typical cooling methods ({e.g., contact with liquid He) involve a range of coupling symmetries, so that qubit-qubit interactions through the bath can be neglected. % 24" }, { "name": "quant-ph9912104.extracted_bib", "string": "{vankampen N. G. Van Kampen, Physica A {147, 165 (1987). % 25" }, { "name": "quant-ph9912104.extracted_bib", "string": "{lidar D. A. Lidar, I. L. Chuang, K. B. Whaley, Phys. Rev. Lett. {81, 2594 (1998). % 26" }, { "name": "quant-ph9912104.extracted_bib", "string": "{barnes J. P. Barnes, W. S. Warren, Phys. Rev. A, in press; e-print quant-ph/9902084. % 27" }, { "name": "quant-ph9912104.extracted_bib", "string": "{boyd R. W. Boyd, {Nonlinear Optics (Academic Press, San Deigo, 1992). Chapter 2.7. % 28" }, { "name": "quant-ph9912104.extracted_bib", "string": "{matlab MATLAB 5.3.0. The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098. % 29" }, { "name": "quant-ph9912104.extracted_bib", "string": "{esr A. Bencini, D. Gatteschi, {EPR of Exchange Coupled Systems. (Springer-Verlag, Berlin, 1990), chap. 2. %" }, { "name": "quant-ph9912104.extracted_bib", "string": "{gandhi O. Gandhi, {Microwave Engineering and Applications (Pergamon, New York, 1981), chap. 8.2. %" } ]
quant-ph9912105	Entangled state quantum cryptography: Eavesdropping on the Ekert protocol	[ { "author": "D. S. Naik$^{1" } ]	Using polarization-entangled photons from spontaneous parametric downconversion, we have implemented Ekert's quantum cryptography protocol. The near-perfect correlations of the photons %(when measured in the appropriate basis) allow the sharing of a secret key between two parties. The presence of an eavesdropper is continually checked by measuring Bell's inequalities. We investigated several possible eavesdropper strategies, including pseudo-quantum non-demolition measurements. In all cases, the eavesdropper's presence was readily apparent. We discuss a procedure to increase her detectability.	[ { "name": "EKERTXXX.TEX", "string": "\n\\documentstyle[twocolumn,aps,prl,graphics,epsfig,floats]{revtex}\n\\topmargin -0.5in\n\\textheight 9.3in\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{document}\n\\draft\n\\twocolumn\n\\wideabs{\n\\title{Entangled state quantum cryptography: \nEavesdropping on the Ekert protocol}\n\\vskip -0.8 cm\n\\author{D. S. Naik$^{1}$, C. G. Peterson$^{1}$, A. G. White$^{1,2}$,\nA. J. Berglund$^{1}$, and P. G. Kwiat$^{1}$\n%\\thanks{Please address correspondence to: Kwiat@lanl.gov.}\n}\n\\address{$^{1}$ Physics Division, P-23, Los Alamos National Laboratory,\nLos Alamos, New Mexico 87545, USA}\n\\address{$^{2}$ Department of Physics, University of Queensland,\nBrisbane, Queensland 4072, AUSTRALIA}\n\\date{Submitted to Phys. Rev. Lett, Oct. 17, 1999; resubmitted \n12/22/99}\n\n\\maketitle\n%\\vskip -0.1cm\n\\begin{abstract}\nUsing polarization-entangled photons from spontaneous\nparametric downconversion, we have implemented Ekert's quantum\ncryptography protocol. The near-perfect correlations of the photons \n%(when measured in the appropriate basis) \nallow the sharing of a secret\nkey between two parties. The presence of an eavesdropper is\ncontinually checked by measuring Bell's inequalities. We investigated\nseveral possible eavesdropper strategies, including pseudo-quantum\nnon-demolition measurements. In all cases, the eavesdropper's presence \nwas readily apparent. We discuss a procedure to increase her detectability.\n\n\\end{abstract}\n\\draft\n%\\vspace {1 cm}\n\\pacs{PACS numbers:\n03.67.Dd, 03.65.-a, 42.79.Sz, 03.65.Bz}}\\narrowtext\n%q crypt,q. information, optical communication, foundations\n%\\vspace{-1 cm} \nThe emerging field of quantum information science\naims to use the nonclassical features of quantum systems to achieve\nperformance in communications and computation that is superior to that\nachievable with systems based solely on classical physics. For \nexample, current methods \nof public-key cryptography base their security on the supposed\n(but unproven) computational difficulty in solving certain problems,\ne.g., finding the prime factors of large numbers -- these problems have\nnot only been unproven to be difficult, but actually been shown to be\ncomputational ``easy'' in the context of quantum computation\n\\cite{Shor}. In contrast, it is now generally accepted that \ntechniques of quantum\ncryptography can allow completely secure communications between\ndistant parties \\cite{BB84,Ekert,Bennett,B92}. \n%, with the inherent security guaranteed by physical laws of nature. \nSpecifically, by using single\nphotons to distribute a secret random cryptographic key, one can\nensure that no eavesdropping goes unnoticed; more precisely, one can\nset rigid upper bounds on the possible information known to a\npotential eavesdropper, based on measured error rates, and %then\nuse appropriate methods of ``privacy amplification'' to reduce this\ninformation to an acceptable level \\cite{privacy}.\n\nSince its discovery, quantum cryptography has been demonstrated by a\nnumber of groups using weak coherent states, both in fiber-based\nsystems \\cite{fiber_exps} and in free space arrangements\n\\cite{Bennett_exp,freespace}. These experiments are \nprovably secure against all eavesdropping attacks based \non presently available technology; however, there are \ncertain conceivable attacks to which they are %might be \nvulnerable, as sometimes the pulses used necessarily \ncontain more than one photon -- an eavesdropper could in \nprinciple use these events to gain information \nabout the key without introducing any extra errors \\cite{loophole}. \nUse of true single-photon sources can close this potential \nsecurity loophole; and while the loophole still exists when \nusing {\\it pairs} of photons as from parametric down-conversion \n(because occasionally there will be \n{\\it double} pairs), it has been shown that they may allow secure \ntransmissions over longer distances \\cite{Lutkenhaus}.\n\nWhile a number of groups use correlated photon pairs to study nonlocal\ncorrelations (via tests of Bell's inequalities \n\\cite{Bell,Bellexps,Kwiat}), and their\npossible application for quantum cryptography \\cite{Gisin,Sasha}, to\nour knowledge no results explicitly using entangled photons in a\nquantum cryptographic protocol have been reported in the\nliterature\\cite{Jennewein}. It is now well established that one\ncannot employ these nonlocal correlations \nfor superluminal signaling\\cite{eberhard}. \nNevertheless, Ekert showed that one can use the\ncorrelations to establish a secret random key between two parties, as\npart of a completely secure cryptography protocol \\cite{Ekert}.\n\nIn our version of the Ekert protocol, ``Alice'' and ``Bob'' each\nreceive one photon of a polarization-entangled pair in the state\n$\|\\phi^{+}\\rangle = (\|H_{1}H_{2}\\rangle +\n\|V_{1}V_{2}\\rangle)/\\sqrt{2}$, where $H$ ($V$) represents horizontal\n(vertical) polarization. They each respectively measure the\npolarization of their photons in the bases $(\|H_{1}\\rangle + e^{i\n\\alpha}\|V_{1}\\rangle)$ and $(\|H_{2}\\rangle + e^{i\n\\beta}\|V_{2}\\rangle)$, where $\\alpha$ and $\\beta$ randomly take on the\nvalues $\\alpha_1\\!=\\!45^{\\circ}, \\alpha_2\\!=\\!90^{\\circ},\n\\alpha_3\\!=\\!135^{\\circ}, \\alpha_4\\!=\\!180^{\\circ};\n\\beta_1\\!=\\!0^{\\circ}, \\beta_2\\!=\\!45^{\\circ}, \\beta_3\\!=\\!90^{\\circ},\n\\beta_4\\!=\\!135^{\\circ}$. \nThey then disclose by public discussion which bases were used, but\nnot the measurement results.\nFor the state $\|\\phi^{+}\\rangle$, the\nprobabilities for a coincidence between Alice's detector 1 (or\ndetector 1', which detects the orthogonally-polarized photons) and\nBob's detector 2 (2') are given by\n\\begin{eqnarray}\n\\rm{P}_{12}(\\alpha,\\beta)\n &=&\n\\rm{P}_{1'2'}(\\alpha,\\beta)\n = (1 + cos(\\alpha + \\beta))/4\\\\\n\\rm{P}_{12'}(\\alpha,\\beta)\n &=&\n\\rm{P}_{1'2}(\\alpha,\\beta)\n = (1 - cos(\\alpha + \\beta))/4 \\,\\,.\\nonumber\n\\end{eqnarray}\nNote that \nwhen $\\alpha + \\beta= 180^{\\circ}$, \nthey will have completely correlated results, which\nthen constitute the quantum cryptographic key. As indicated in \nTable~\\ref{tab:settings}, the results from other \ncombinations are revealed and \nused in two independent tests of Bell's inequalities, to\ncheck the presence of an intermediate eavesdropper (``Eve''). Here we \n\\begin{table}[!b]\n\\caption{Distribution of data dependent on Alice's and Bob's\nrespective phase settings $\\alpha_{i}$ and $\\beta_{i}$ (see text\nfor details).} \n\\begin{center}\n\\begin{tabular}{cc\| c c c c }\n & \t\t& & Alice & & \\\\\n & \t\t& $\\alpha_{1}$ & $\\alpha_{2}$ & $\\alpha_{3}$\n& $\\alpha_{4}$ \\\\* \\hline\n &$\\beta_{1}$\t&S\t\t& --\t\t& S\n\t\t& QKey \\\\\n Bob &$\\beta_{2}$\t&--\t\t& $S'$\t\t& QKey\n\t\t& $S'$ \\\\\n &$\\beta_{3}$\t&S\t\t&QKey\t\t& S\n\t\t& -- \\\\\n &$\\beta_{4}$\t&QKey\t\t& $S'$\t\t& --\n\t\t& $S'$ \\\\\n\\end{tabular}\n\\end{center}\n\\label{tab:settings}\n%\\vspace {-0.3cm}\n\\end{table}\n\\noindent\n\\begin{figure}[t!]\n \\vspace {-0.3 cm}\n\\begin{center}\n\\epsfxsize=\\columnwidth\n\\epsfbox{XXXFIG1.EPS}\n\\end{center}\n% \\footnotesize\n\\caption{Schematic of quantum cryptography system. 351.1nm light from\nan Argon ion laser is used to pump two perpendicularly-oriented\nnonlinear optical crystals (BBO). The resultant entangled photons are\nsent to Alice and Bob, who each analyze them in one of four randomly\nchosen bases. The eavesdropper, if present, was incorporated using\neither a polarizer or a decohering birefringent plate (both\norientable, and in some cases with additional wave plates to allow\nanalysis in arbitrary elliptical polarization bases [Fig. 2a,c]).}\n\\label{fig:setup}\n\\end{figure}\n\\noindent \npresent an experimental realization of this protocol, and look at the\neffect of various eavesdropping strategies.\n\nWe prepare the polarization-entangled state using the process of\nspontaneous parametric down-conversion \nin a nonlinear crystal\\cite{Kwiat}. In brief, \ntwo identically-cut adjacent crystals\n(beta-barium-borate, BBO) are oriented with their optic axes in planes\nperpendicular to each other (Fig. 1). A $45^{\\circ}$-polarized pump\nphoton is then equally likely to convert in either crystal. Given the\ncoherence and high spatial overlap (for our 0.6mm-thick crystals)\nbetween these two processes, the photon pairs are then created in the\nmaximally-entangled state $\|\\phi^{+}\\rangle$. Alice's and Bob's\nanalysis systems each consist of a randomly driven liquid crystal [LC]\n(to set the applied phase shift), a half wave plate [HWP] (with optic\naxis at $22.5^{\\circ}$), and a calcite Glan-Thompson prism [PBS].\nPhotons from the horizontal and vertical polarization outputs of each\nprism are detected (after narrow-band interference filters) using\nsilicon avalanche photodiodes (EG\\&G SPCM-AQ's, efficiency $\\sim\n60\\%$, dark count $< 400s^{-1}$). The correlated detector signals are\nsynchronized and temporally discriminated through AND gates. Because\nof the narrow gate window (5 ns), the rate of accidental coincidences\n(resulting from multiple pairs or background counts) is only\n$10^{-5}s^{-1}$. From separate computers, Alice and Bob control their\nrespective LC's with synchronously clocked arbitrary waveform\ngenerator cards \\cite{random}. A coincident event triggers a\ndigitizer, which records the LC states, and the outputs from each of\nthe four detector pairs \\cite{key}.\n\nBecause the total rate of coincidences between Alice's and Bob's\ndetectors was typically 5000 s$^{-1}$, the probability of having at\nleast one pair of photons during the collection time window of 1 ms was\n99\\%. Of course, there was then also a high probability of more than\none pair being detected within the window (96\\%). Because the phase\nsetting remains unchanged during a collection window, muliple pairs\ncould conceivably give extra information to a potential eavesdropper.\nWe avoided this problem by keeping \nonly the first event in any given window.\nAssuming that Alice and Bob each have completely isolated measurement\nsystems (i.e., there is no way for an eavesdropper to learn about the\nmeasurement parameters $\\alpha$ and $\\beta$ by sending in extra\nphotons of her own), this system is secure even though no rapid\nswitching is employed, since only $\\sim$1 photon pair event is used for\nany particular $\\alpha$-$\\beta$ setting \\cite{doubles}. \nGiven the 22ms cycle period\ndetermined by the liquid crystals \\cite{liquidcrystals}, the maximum\nrate of data collection in our system is 45.4Hz. The usable rate is\nslightly less, because the LC voltages were occasionally in transition\nwhen the digitizer read them, yielding an ambiguous determination of\nthe actual phase setting. Typically we collected 40 useful pairs per\nsecond.\n\nAs seen in Table I, only 1/4 of the data actually contributes to the\nraw cryptographic key; half the data are used to test Bell's inequalities;\nand 1/4 are not used at all \\cite{threephases}. In four independent\nruns of $\\sim$10 minutes each, we obtained a total of 24252 secret key\nbits (see Table II), corresponding to a raw bit rate of 10.1s$^{-1}$; \nthe corresponding bit error rate (BER) was $3.06\\pm 0.11\\%$\\cite{BER}. If \nwe attribute this BER (conservatively estimated as 3.4\\%) \nentirely to an eavesdropper, we\nshould assume she has knowledge of up to 0.7\\% + (4/$\\sqrt(2)* \n3.4\\% = 10.3\\%$ ($\\sim$2500 bits) of the key, \nwhere the 0.7\\% comes from possible double-pair \nevents \\cite{doubles}, and the second term assumes an intercept-resend\nstrategy (see \\cite{Bennett_exp}). We must then perform sufficient privacy\namplification to reduce this to an acceptable level.\nAfter running an error detection procedure on our raw key material,\n17452 error-free bits remained. Using appropriate privacy\namplification techniques \\cite{privacy}, this was further compressed\nto 12215 useful secret bits (a net bit rate of 5.1s$^{-1}$); \nthe residual information available to any\npotential eavesdropper is then $2^{-(17452-12215-2500)}/\\ln 2$, i.e.,\n$\\ll 1$ bit \\cite{Bennett_exp}.\n\nIn contrast to nearly all tests of Bell's inequalities previously\nreported, instead of using linear polarization\nanalyses (i.e., in the equatorial plane of the Poincar\\'{e} sphere),\nwe used elliptical polarization analysis (i.e., on the plane\ncontaining the circularly polarized poles of the sphere and the $\\pm\n45^{\\circ}$ linearly polarized states). In particular, we measured\nthe Bell parameters\\cite{CHSH}:\n\\begin{eqnarray}\n\\rm{S} = -E(\\alpha_{1},\\beta_{1})\n+E(\\alpha_{1},\\beta_{3})+E(\\alpha_{3},\n\\beta_{1})+E(\\alpha_{3},\\beta_{3})\\\\\nS'\\rm{} = E(\\alpha_{2},\\beta_{2})\n +E(\\alpha_{2},\\beta_{4})+E(\\alpha_{4},\\beta_{2})\n-E(\\alpha_{4},\\beta_{4})\\;,\\nonumber\n\\end{eqnarray}\nwhere\nE($\\alpha,\\!\\beta$) = $\\frac{\\rm{R}_{12}\n(\\alpha,\\!\\beta)+\\rm{R}_{1'2'}(\\alpha,\\!\\beta)\n -\\rm{R}_{12'}(\\alpha,\\!\\beta)-\\rm{R}_{1'2}(\\alpha,\\!\\beta)}\n {\\rm{R}_{12}(\\alpha,\\!\\beta)+\\rm{R}_{1'2'}(\\alpha,\\!\\beta)\n +\\rm{R}_{12'}(\\alpha,\\!\\beta)+\\rm{R}_{1'2}(\\alpha,\\!\\beta)}$,\nand the R's are the various coincidence counts between Alice's and\nBob's detectors. For any local realistic theory $\|$S$\|$,$\|S'\|\\le2$,\nwhile for the combinations of $\\alpha$ and $\\beta$ \nindicated in Table \n1, the quantum mechanically expected values of $\|$S$\|$,$\|S'\|$ are\n$2\\sqrt{2}$. In a typical 10 minute run of our system, we observed\nS=$-2.67\\pm 0.04$ and $S'$=$-2.65\\pm 0.04$; for the 40 minutes of\ncollected data, our combined values were S=$-2.665\\pm 0.019$,\n$S'$=$-2.644\\pm 0.019$, each a 34-$\\sigma$ violation of Bell's\ninequality. It is expected (and demonstrated experimentally; see\nbelow) that the presence of an eavesdropper will reduce the observed\nvalues of $\|$S$\|$,$\|S'\|$. In fact, if the eavesdropper measures one\nphoton from every pair, then $\|S_{eve}\| \\le \\sqrt{2}$\\cite{Ekert}. \nBecause we observed high values of $\|$S$\|$,$\|S'\|$,\nin our system the presence of an eavesdropper could thus be detected\nin $\\sim$1s of data collection (the time interval for which our\n$\|$S$\|$,$\|S'\|$ exceed $\\sqrt{2}$ by 2$\\sigma$). Of course one could\nsimilarly use the BER as a check for a potential eavesdropper, who\nintroduces a minimum BER of 25\\% if she measures every photon;\nhowever, this requires sacrificing some of the cryptographic key to\naccurately determine the BER.\n\\begin{table}[!b]\n \\vspace {-0.1cm}\n\\caption{100 bits of typical shared quantum key data for Alice (A) and\nBob (B), generated using the Ekert protocol. Italic entries indicate\nerrors; our average BER was 3.06\\%.}\n\\begin{center}\n\\begin{tabular}{c }\nA: 1111100101010110100110000 1010011100110{\\it 1}11010100000\\\\\nB: 1111100101010110100110000 1010011100100{\\it 0}11010101000\\\\\n\\hline\nA: 100010010100001{\\it 0}100111101 110100100101{\\it 0}101010010111\\\\\nB: 110010010100001{\\it 1}100111101 110100100101{\\it 1}101010010111\\\\\n\\end{tabular}\n\\end{center}\n\\label{tab:data}\n\\vspace {-0.3cm}\n\\end{table}\n\\noindent \n\nIn investigating the effects of the presence of an eavesdropper there\nare two main difficulties. First, there are various possible strategies; and\nsecond, we always assume that Eve has essentially perfect equipment\nand procedures, \n%and hence is superior to both Alice's and Bob's capabilities, \nwhich of course is experimentally impossible to\nimplement. Hence, we can at best simulate the effects she would\nhave; we did this for two particular intercept/resend\neavesdropping strategies. In the first, we make a\nstrong filtering measurement of the polarization, in some basis $\\chi$, and\nsend on the surviving photons to Bob. The simulated \neavesdropper thus \nmakes the projective measurement $\|\\chi\\rangle\\langle\\chi\|$. The\neffect on the measured value of S and $S'$ and the BER depend strongly\non what eavesdropping basis $\|\\chi\\rangle$ is used \n\\cite{Bennett_exp}.\nTheoretical predictions and results for bases in three\northogonal planes in the Poincar{\\'e} sphere are shown in Figure 2.\n\nThe second eavesdropping strategy examined was a quantum\nnon-demolition (QND) measurement \\cite{Werner}. \nQND measurements of {\\it optical}\nphoton number and polarization are presently impossible. \nIn fact, precisely for this reason current quantum\ncryptography implementations {\\it are} secure, even though they employ\nweak optical pulses (with {\\it average} photon number/pulse less than\n1) \\cite{Durt_reply}. Nevertheless, the ideal of quantum cryptography\nis that it can be made secure against {\\it any} physically possible\neavesdropping strategies; hence, it is desirable to test any system\nagainst as many strategies as possible.\n\nAlthough appropriate QND measurements cannot be performed at present,\nit is well known that their {\\it effect} is to produce a random phase\nbetween the eigenstates of the measurement, in turn due to the\nentanglement of these states with the readout quantum system. We can\nexactly simulate this effect by inserting, in Bob's path, a\nbirefringent element that separates the extraordinary and ordinary\ncomponents of the photon wavepacket by more than the coherence length\n($\\sim 140\\mu$m, determined by the interference filters before the\ndetectors); the result is a completely random phase between these\npolarization \n\\begin{figure}[t!]\n \\vspace {-0.2 cm}\n\\begin{center}\n\\epsfxsize=\\columnwidth\n%\\vspace{10.5 cm}\n\\epsfbox{XXXFIG2.EPS}\n\\end{center}\n% \\footnotesize\n\\caption{Data and theory showing the effect of an eavesdropper on S\nand BER for various attack bases (as $S'$ closely agrees with S, it is\nomitted for clarity). Diamonds represent strong measurements, made\nwith a polarizer; circles represent QND attacks, simulated with a\n3mm-thick BBO crystal; error bars are within the points. The attacks\nbases are: a. $\|H\\rangle + e^{i\\phi}\|V\\rangle$; b.\n$\\cos\\theta\|H\\rangle + \\sin\\theta\|V\\rangle$; and c.\n$\|45^{\\circ}\\rangle + e^{i\\psi}\|$-$45^{\\circ}\\rangle$; the actual\nmeasurement points in these bases are illustrated on the inset\nPoincar{\\'e} spheres. The measured average values with no\neavesdropper are indicated by unbroken grey lines, the broken lines\nrepresent the maximum classical value of $\|$S$\|$.}\n \\vspace {-0.1 cm}\n\n\\end{figure}\n\\noindent\ncomponents, just as if a QND measurement had been made on\nthem. Mathematically, the effect of the eavesdropper is to make a\nprojective measurement $\|\\chi\\rangle\\langle\\chi\|+ e^{i\n\\langle\\xi\\rangle} \|\\chi^{\\perp}\\rangle\\langle\\chi^{\\perp}\|$, where\n$\\langle\\xi\\rangle$ represents a random phase. Note that the\ntheoretical predictions are identical with that for the strong\npolarization measurement. The experimental data are also shown in\nFig. 2.\n\nWe see immediately that the optimal bases for eavesdropping lie in the\nsame plane (on the Poincar{\\'e} sphere) as the bases employed by Alice\nand Bob -- for this case, the probability that the eavesdropper causes\nan error is ``only'' 25\\% per intercepted bit, and the $\|$S$\|$ value\nis $\\sqrt{2}$ (Fig. 2a). On the other hand, if the eavesdropper does\nnot know the plane of the measurement bases, and uses, e.g., random\nmeasurements in an orthogonal plane, her {\\it average} probability of\nproducing an error climbs to 32.5\\%/bit, and the average value of\n$\|$S$\|$ drops to $1/\\sqrt{2}$. This suggests a strategy for improved\nsecurity: \nAlice and Bob should choose bases corresponding to at least two (and\nideally three) orthogonal planes, thereby ``magnifying'' the presence of an\neavesdropper (at least one implementing the sort of strong projective\nor QND-like measurements strategies investigated here) \nabove the usual 25\\%/bit error probability. Quantitative \ntheoretical investigations of such a strategy, known as the ``six-state''\nprotocol, support these claims \\cite{Bruss}.\n\nAn eavesdropper could also examine only a {\\it fraction} \nof the photons, thus reducing her induced BER and increasing\nthe S value measured by Alice and Bob, at the expense of her own\nknowledge of the cryptographic key. For example, if she measures (in\nthe optimal basis) less than 58.6\\% of the photons, S $> 2$ and the\ncorresponding BER $< 15$\\%, but Eve's knowledge of the key will be\nless than Bob's (and privacy amplification techniques will\nstill permit generation of a secret key) \n\\cite{Gisin2,11limit}.\n\nIn summary, we have implemented the Ekert quantum cryptography\nprotocol using entangled photon pairs. For this proof-of-principle\nexperiment, Alice and Bob were situated side by side on the same\noptical table, obviously not the optimal configuration for useful\ncryptography. Nevertheless, our system demonstrates the essential\nfeatures of the Ekert protocol, and moreover, we believe is the first\nto {\\it experimentally} investigate the effect of a physical\nintermediate eavesdropper \\cite{exp_eve}. We see no bar to extending\nthe transmission distance to hundreds of meters \\cite{freespace} or\neven to earth-to-satellite distances \\cite{Richard}. \n%(perhaps with adaptive optics \n%to compensate for turbulence) \n%% \n % One could\n % imagine a scenario where the source is on a satellite, which both\n % Alice and Bob can see, though they might not have direct line of sight\n % to each other. The satellite first generates a random key with each\n % of them, then publicly communicates to Bob which bits of his key need\n % to be flipped so that it agrees with Alice's. In this way one can\n % produce an entanglement-based quantum cryptographic key between two\n % parties which are not in direct communication with each other.\n %%\n\nWe gratefully acknowledge the laboratory assistance of S. Lopez, the\nerror correction/privacy amplification programs written by E.\nTwyeffort, and very helpful discussions with R. Hughes and N. \nLutkenhaus. \n\n$^{}$Please address correspondence to: Kwiat@lanl.gov.\n\n\\vspace {-0.5 cm}\n\\begin{references}\n\\vspace {-1.6 cm}\n\n\\bibitem{Shor} P. W. Shor, SIAM Review, {\\bf 41}, 303 (1999).\n\n\\bibitem{BB84} C. H. Bennett and G. Brassard, in {\\it Proc. of the\nIEEE Int. Conf. on Computers, Systems, and Signal Processing,\nBangalore, India} (IEEE, New York, 1984), p. 175.\n\n\\bibitem{Ekert} A. K. Ekert, Phys. Rev. Lett, {\\bf 67}, (1991).\n\n\\bibitem{Bennett} C. H. Bennett, G. Brassard, and N. D. Mermin, Phys.\nRev. Lett. {\\bf 68}, 557 (1992).\n\n\\bibitem{B92} C. H. Bennett,Phys. Rev. Lett. {\\bf 68}, 3121 (1992).\n\n\\bibitem{privacy} \nU. M. Maurer, IEEE Trans. Inform. Theory {\\bf 39}, 773 (1993); C. H.\nBennett, G. Brassard, C. Crepeau, and U. M. Maurer, {\\it ibid}. {\\bf\n41}, 1915 (1995); and references therein.\n\n\\bibitem{fiber_exps} P. D. Townsend, J. G. Rarity, and P. R. Tapster,\nElectron. Lett, {\\bf 29}, 1291 (1993); P. D. 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Lett. {\\bf 81}, 3563 (1998); G. Weihs, {\\it et al}.,\nPhys. Rev. Lett. {\\bf 81}, 5039 (1998).\n\n\\bibitem{Kwiat} P. G. Kwiat, {\\it et al.}, Phys. Rev. A\n{\\bf 60}, R773 (1999).\n\n\\bibitem{Gisin} J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys.\nRev. Lett. {\\bf 82}, 2594 (1999).\n \n\\bibitem{Sasha} A. V. Sergienko, {\\it et al}., Phys. Rev. A {\\bf 60},\nR2622 (1999).\n\n\\bibitem{Jennewein} During completion of our work, related results\nwere reported by T. Jennewein {\\it et al.}, \nto appear in Phys. Rev. Lett.; and more recently by \nW. Tittel {\\it et al.}, in quant-ph/9911109.\n\n\\bibitem{eberhard} See, for instance, P. H. Eberhard and R. R. Ross,\nFound. Phys. Lett. {\\bf 2}, 127 (1989).\n\n\\bibitem{random} The random pulse sequences to drive the LC's were\ngenerated by LabView on each of Alice's and Bob's separate computers.\n%Ideally, a {\\it physical} random number source would be used, but \nFor our comparatively short data strings, the pseudo-random numbers were\nadequate.\n\n\\bibitem{key} In order to average out the effect of different detector\nefficiencies, different detectors represented ``0'' and ``1''\ndepending on the phase settings, i.e., detector 1 (2') represented a\n``0'' for $\\alpha_{1},\\alpha_{3}$ ($\\beta_{4},\\beta_{2}$), but a ``1''\nfor $\\alpha_{2},\\alpha_{4}$ ($\\beta_{3},\\beta_{1})$. The resulting\nkey contained 49\\% ``1''s.\n\n\\bibitem{doubles}\nThere is a small contribution ($\\sim$ 0.7\\% per key bit) of events \nin which a double pair was emitted within the detection time \n(conservatively estimated as the 5ns gate window + 35ns dead time). \nIn principle an eavesdropper could learn the value of these key bits, \nusing methods similar to those for weak pulse schemes \\cite{loophole}. \n%This extra information gain is removed by suitable privacy amplification. \n\n\\bibitem{liquidcrystals} Heating the LC's to $35^{\\circ}$, and\napplying ``overshoot'' voltages, improved the switching times to under\n19 ms for all transitions; photons were collected in the following 1\nms. The cycle time could be improved to nanoseconds by using, e.g.,\nelectro-optic modulators instead of LC's.\n\n\\bibitem{threephases}\nIf Alice and Bob use only {\\it three} settings and the standard\nBell's inequality (as proposed by Ekert \\cite{Ekert}), \n4/9 of the pairs go to testing Bell's inequality, and 2/9 to the key.\n%If desired, one could skew the probabilities to have a greater\n%fraction of key material and correspondingly fewer photon pairs\n%devoted to testing Bell's inequalities.\n\n\\bibitem{BER} The BER is defined as the number of errors divided by\nthe total size of the cryptographic key; BER = 0.5 implies Alice and\nBob have completely uncorrelated strings.\n\n\n\\bibitem{CHSH} J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,\nPhys. Rev. Lett. {\\bf 23}, 880 (1969); A. Garuccio and V. A.\nRapisarda, Nuovo Cim. A {\\bf 65}, 269 (1981).\n\n\\bibitem{Werner} M. Werner and G. Milburn, Phys. Rev. A \n{\\bf 47}, 639 (1993).\n\n\\bibitem{Durt_reply} \nW. T. Buttler {\\it et al}., Phys. Rev. Lett. {\\bf 83}, 2477 (1999).\n\n\\bibitem{Bruss}\nD. Bru\\ss, Phys. Rev. Lett. {\\bf 81}, 3018 (1999); \nH. Bechmann-Pasquinucci and N. Gisin, Phys. Rev. A {\\bf 59}, 4238 \n(1999).\n\n\\bibitem{Gisin2} N. Gisin and B. Huttner, Phys. Lett. A {\\bf 228}, 13\n(1997); C. A. Fuchs {\\it et al}., Phys. Rev. A {\\bf 56}, 1163 (1997).\n\n\\bibitem{11limit}\nInformation lost during error correction actually reduces the \nBER ``safety'' threshold to $< 11$\\% \\cite{Lutkenhaus}.\n\n\\bibitem{exp_eve}\nThe very first quantum cryptography experiment \\cite{Bennett_exp}\nsimulated the effect of an intermediate eavesdropper by letting her\n``borrow'' Alice's and Bob's apparatus, and by introducing her in the\npost-detection computation.\n\n\\bibitem{Richard} R. Hughes and J. Nordholt, Phys. World {\\bf 12}, 31\n(1999).\n\n\\end{references}\n\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912105.extracted_bib", "string": "{Shor P. W. Shor, SIAM Review, {41, 303 (1999)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{BB84 C. H. Bennett and G. Brassard, in {Proc. of the IEEE Int. Conf. on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, New York, 1984), p. 175." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Ekert A. K. Ekert, Phys. Rev. Lett, {67, (1991)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Bennett C. H. Bennett, G. Brassard, and N. D. Mermin, Phys. Rev. Lett. {68, 557 (1992)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{B92 C. H. Bennett,Phys. Rev. Lett. {68, 3121 (1992)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{privacy U. M. Maurer, IEEE Trans. Inform. Theory {39, 773 (1993); C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer, {ibid. {41, 1915 (1995); and references therein." }, { "name": "quant-ph9912105.extracted_bib", "string": "{fiber_exps P. D. Townsend, J. G. Rarity, and P. R. Tapster, Electron. Lett, {29, 1291 (1993); P. D. Townsend, {ibid. {30, 809 (1994); A. Muller {et al., Appl. Phys. Lett. {70, 793 (1997); R. J. Hughes, G. L. Morgan, and C. G. Peterson, J. Mod. Opt., to appear (1999)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Bennett_exp C. H. Bennett, {et al., J. Cryptol. {5, 3 (1992)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{freespace B.\\, Jacobs and J.\\,D.\\,Franson, Opt. Lett. {21, 1854 (1996); W. T. Buttler, {et al., Phys. Rev. Lett. {81, 3283 (1998)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{loophole T. Durt, Phys. Rev. Lett. {83, 2476 (1999); N. Lutkenhaus, Acta. Phys. Slov. {49, 549 (1999); G. Brassard, T. Mor, and B. C. Sanders, quant-ph/9906074." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Lutkenhaus N. Lutkenhaus, Phys. Rev. A {59, 3301 (1999); see also, N. Lutkenhaus, quant-ph/9910093." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Bell J. S. Bell, Physics 1, 195 (1965)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Bellexps P. R. Tapster, J. G. Rarity, and P. C. M Owens, Phys. Rev. Lett. {73, 1923 (1994); W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Phys. Rev. Lett. {81, 3563 (1998); G. Weihs, {et al., Phys. Rev. Lett. {81, 5039 (1998)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Kwiat P. G. Kwiat, {et al., Phys. Rev. A {60, R773 (1999)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Gisin J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys. Rev. Lett. {82, 2594 (1999)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Sasha A. V. Sergienko, {et al., Phys. Rev. A {60, R2622 (1999)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Jennewein During completion of our work, related results were reported by T. Jennewein {et al., to appear in Phys. Rev. Lett.; and more recently by W. Tittel {et al., in quant-ph/9911109." }, { "name": "quant-ph9912105.extracted_bib", "string": "{eberhard See, for instance, P. H. Eberhard and R. R. Ross, Found. Phys. Lett. {2, 127 (1989)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{random The random pulse sequences to drive the LC's were generated by LabView on each of Alice's and Bob's separate computers. %Ideally, a {physical random number source would be used, but For our comparatively short data strings, the pseudo-random numbers were adequate." }, { "name": "quant-ph9912105.extracted_bib", "string": "{key In order to average out the effect of different detector efficiencies, different detectors represented ``0'' and ``1'' depending on the phase settings, i.e., detector 1 (2') represented a ``0'' for $\\alpha_{1,\\alpha_{3$ ($\\beta_{4,\\beta_{2$), but a ``1'' for $\\alpha_{2,\\alpha_{4$ ($\\beta_{3,\\beta_{1)$. The resulting key contained 49\\% ``1''s." }, { "name": "quant-ph9912105.extracted_bib", "string": "{doubles There is a small contribution ($\\sim$ 0.7\\% per key bit) of events in which a double pair was emitted within the detection time (conservatively estimated as the 5ns gate window + 35ns dead time). In principle an eavesdropper could learn the value of these key bits, using methods similar to those for weak pulse schemes \\cite{loophole. %This extra information gain is removed by suitable privacy amplification." }, { "name": "quant-ph9912105.extracted_bib", "string": "{liquidcrystals Heating the LC's to $35^{\\circ$, and applying ``overshoot'' voltages, improved the switching times to under 19 ms for all transitions; photons were collected in the following 1 ms. The cycle time could be improved to nanoseconds by using, e.g., electro-optic modulators instead of LC's." }, { "name": "quant-ph9912105.extracted_bib", "string": "{threephases If Alice and Bob use only {three settings and the standard Bell's inequality (as proposed by Ekert \\cite{Ekert), 4/9 of the pairs go to testing Bell's inequality, and 2/9 to the key. %If desired, one could skew the probabilities to have a greater %fraction of key material and correspondingly fewer photon pairs %devoted to testing Bell's inequalities." }, { "name": "quant-ph9912105.extracted_bib", "string": "{BER The BER is defined as the number of errors divided by the total size of the cryptographic key; BER = 0.5 implies Alice and Bob have completely uncorrelated strings." }, { "name": "quant-ph9912105.extracted_bib", "string": "{CHSH J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. {23, 880 (1969); A. Garuccio and V. A. Rapisarda, Nuovo Cim. A {65, 269 (1981)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Werner M. Werner and G. Milburn, Phys. Rev. A {47, 639 (1993)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Durt_reply W. T. Buttler {et al., Phys. Rev. Lett. {83, 2477 (1999)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Bruss D. Bru\\ss, Phys. Rev. Lett. {81, 3018 (1999); H. Bechmann-Pasquinucci and N. Gisin, Phys. Rev. A {59, 4238 (1999)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Gisin2 N. Gisin and B. Huttner, Phys. Lett. A {228, 13 (1997); C. A. Fuchs {et al., Phys. Rev. A {56, 1163 (1997)." }, { "name": "quant-ph9912105.extracted_bib", "string": "{11limit Information lost during error correction actually reduces the BER ``safety'' threshold to $< 11$\\% \\cite{Lutkenhaus." }, { "name": "quant-ph9912105.extracted_bib", "string": "{exp_eve The very first quantum cryptography experiment \\cite{Bennett_exp simulated the effect of an intermediate eavesdropper by letting her ``borrow'' Alice's and Bob's apparatus, and by introducing her in the post-detection computation." }, { "name": "quant-ph9912105.extracted_bib", "string": "{Richard R. Hughes and J. Nordholt, Phys. World {12, 31 (1999)." } ]
quant-ph9912106	Atom Chips	[ { "author": "Ron Folman$^1$" }, { "author": "Peter Kr\\\"uger$^1$" }, { "author": "Donatella Cassettari$^1$" }, { "author": "Bj\\\"orn Hessmo$^{1" }, { "author": "2" } ]	Atoms can be trapped and guided using nano-fabricated wires on surfaces, achieving the scales required by quantum information proposals. These Atom Chips form the basis for robust and widespread applications of cold atoms ranging from atom optics to fundamental questions in mesoscopic physics, and possibly quantum information systems.	[ { "name": "AtomChip_preprint.tex", "string": "%\\documentstyle[prl,aps,epsf,floats]{revtex}\n\\documentstyle[preprint,aps,epsf,floats]{revtex}\n\n\n\\newcommand{\\infig}[2]{\\begin{center}\\hspace{0mm}\\mbox{\\input epsf\n\\epsfxsize#2\\epsfbox{#1}}\\end{center}}\n\n\\def\\lb{\\label}\n\n\\begin{document}\n\n% \\draft command makes pacs numbers print\n\\draft\n\n%\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname\n\n\\title{Atom Chips}\n\\author{Ron Folman$^1$, Peter Kr\\\"uger$^1$, Donatella Cassettari$^1$, Bj\\\"orn\nHessmo$^{1,2}$, Thomas Maier$^3$, J\\\"org Schmiedmayer$^1$}\n\n\\address{$^1$Institute f\\\"ur Experimentalphysik,\nUniversit\\\"at Innsbruck, A-6020 Innsbruck, Austria\\\\\n$^2$Department of Quantum Chemistry, Uppsala University, S-75120\nUppsala, Sweden\\\\\n$^3$Institute f\\\"ur Festk\\\"orperelektronik, Floragasse 7, A-1040 Wien, Austria}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\nAtoms can be trapped and guided using nano-fabricated wires on\nsurfaces, achieving the scales required by quantum information\nproposals. These Atom Chips form the basis for robust and\nwidespread applications of cold atoms ranging from atom optics to\nfundamental questions in mesoscopic physics, and possibly quantum\ninformation systems.\n\\end{abstract}\n\n% insert suggested PACS numbers in braces on next line\n\\pacs{PACS number(s): 03.75.Be, 03.65.Nk}\n\n\\vskip1pc\n\n%]\n\n%\\narrowtext\nIn mesoscopic quantum electronics, electrons move {\\em inside}\nsemiconductor structures and are manipulated using potentials where\nat least one dimension is comparable to the de-Broglie wavelength\nof the electrons \\cite{imry,moti}. Similar potentials can be\ncreated for neutral atoms moving microns {\\em above} surfaces,\nusing nano-fabricated charged and current carrying structures\n\\cite{schm,joerg2,hind}. Surfaces carrying such structures form\nAtom Chips which, for coherent matter wave optics, may form the\nbasis for a variety of novel applications and research tools,\nsimilar to what integrated circuits are for electronics.\n\nIn this work we make use of the magnetic interaction $V_{mag} =\n-\\vec{\\mu} \\cdot \\vec{B}$ based on the coupling of the atomic\nmagnetic moment $\\vec{\\mu}$ to the magnetic field $\\vec{B}$ to\ntrap and manipulate atoms close to the surface of an Atom Chip.\nThe trapping potentials are created by superposing a homogeneous\nmagnetic bias field with the field generated by a thin current\ncarrying wires. The trap depth is given by the homogeneous field,\nthe gradients and curvatures by the magnetic fields from the wire\n\\cite{joerg2,joerg1}.\n\nWe have previously reported on the manipulation of neutral atoms\nusing thin (down to below $1 \\mu m$) charged wires \\cite{Den98}\nand current carrying wires (down to $25 \\mu m$) to form guides\n\\cite{joerg1,Schm92}, beam splitters \\cite{joerg2}, and Z or U\nshaped 3-dimensional traps \\cite{joerg3}. These structures were\nfree standing.\n\nThe next step was to turn to surface mounted wires\n\\cite{SurfaceMounting} which was recently achieved for large\nstructures \\cite{fortagh,surface}. However,the full potential in\nsurface mounted atom optics lies in the robust miniaturization\ndown to the mesoscopic scale. Such a move is primarily motivated\nby the theoretically required scale needed to achieving\nentanglement with neutral atoms through controlled collisions\n\\cite{zoller} or cavity QED \\cite{QED}, entanglement being the\nbasic building block for quantum information devices.\n\nHere we present such a nanofabricated device with which the\nrequired ground state size of less than $100nm$ was achieved. This\nis a first step towards our vision, the realization of a fully\nintegrated {\\em Atom Chip}. We start by describing the chip and\nthe experimental setup, followed by a presentation of the results.\nFinally, we discuss potential applications and future\nperspectives.\n\nThe chip we have used in this work is made of a $2.5 \\, \\mu m$\ngold layer placed on a $600\\mu m$ thick GaAs substrate\n\\cite{maier}. The gold layer is patterned using nano-fabrication\ntechnology. The scale limit of the process used is well below\n$100 \\, nm$.\n\nIn figure 1a we present the main elements of the chip design used\nin the work described here. Each of the large U-shaped wires,\ntogether with a bias field, creates a quadrupole field, which may\nbe used to form a Magneto-Optical-Trap (MOT) on the chip as well\nas a magnetic trap. Both U-shaped wires together may be used to\nform a strong magnetic trap in order to 'load' atoms into the\nsmaller structures, or as an on-board (i.e. without need for\nexternal coils) bias field, for guides and traps created by the\nthin wire running between them. The thin wires are $10\\mu m$ wide\nand depending on the contact used, may form a U-shaped or a\nZ-shaped magnetic trap or a magnetic guide. The chip wires are all\ndefined by boundaries of $10 \\, \\mu m$ wide etchings in which the\nconductive gold has been removed. This leaves the chip as a gold\nmirror (with $10 \\, \\mu m$ etchings) and it can be used to reflect\nthe laser beams for the MOT during the cooling and collecting of\natoms. Figure 1b presents the mounted chip before it is introduced\ninto the vacuum chamber. In addition, a U-shaped $1 \\, mm$ thick\nwire, capable of carrying up to $20 \\, A$ of current, has been put\nunderneath the chip in order to assist with the loading of the\nchip. Its location and shape are identical to those of one of the\n$200\\mu m$ U-shaped wires and it differs only in the amount of\ncurrent it can carry.\n\nThe chip assembly (Fig. 1b) is then mounted inside a vacuum chamber\nused for atom trapping experiments, with optical access for the\nlaser beams and the observation cameras and with the possibility of\napplying the desired bias fields (Fig. 2). For a more detailed\ndescription of the apparatus and the atom trapping procedure, see\n\\cite{joerg2,joerg1,denschlag}.\n\n\nThe experimental procedure for loading cold atoms into the small traps on the\nchip is the following:\n\nIn the first step typically $10^8$ $^{7}Li$ atoms are loaded from\nan effusive atomic beam into a MOT \\cite{LiMOT}. Because the atoms\nhave to be collected a few millimeters away from a surface we use\na 'reflection' MOT \\cite{reflect}. Thereby, the 6 laser beams\nneeded for the MOT are formed from 4 beams by reflecting two of\nthem off the chip surface (Fig.~\\ref{fig:pict2}). Hence atoms\nabove the chip actually encounter six light beams. To assure a\ncorrect magnetic field configuration needed for the formation of a\nMOT, one of the reflected light beams has to be in the axis of the\nMOT coils. Figure 3a shows a top view of the chip and the\nreflection MOT sitting above the U-shaped wires.\n\nThe large external quadrupole coils are then switched off while\nthe current in the U-shaped wire underneath the chip is switched\non (up to $16A$), together with an external bias field ($8G$).\nThis forms a nearly identical, but spatially smaller, quadrupole\nfield as compared to the fields of the large coils. The atoms are\nthus transferred to a secondary MOT which by construction is\nalways well aligned with the chip (Fig. 3b). By changing the bias\nfield, the MOT can be shifted close to the chip surface\n(typically, $2 \\, mm$). The laser power and detuning are changed\nto further cool the atoms, giving us a sample with a temperature\nbelow $200 \\, \\mu K$.\n\nIn the next step, the laser beams are switched off and the\nquadrupole field serves as a magnetic trap in which the low field\nseeking atoms are attracted to the minimum of the field. Without\nthe difficulties of near surface shadows hindering the MOT, the\nmagnetic trap can now be lowered further towards the surface of\nthe chip (Fig. 3c). This is simply done by increasing the bias\nfield (up to $19 \\, G$). Atoms are now close enough so that they\ncan be trapped by the chip fields. The loading of the chip has\nbegun.\n\nNext, 2A are sent through each of the two $200 \\, \\mu m$ U-shaped\nwires on the chip and the current in the U-shaped wire located\nunderneath the chip is ramped down to zero. This procedure brings\nthe atoms even closer to the chip, compresses the trap\nconsiderably, and transfers the atoms to a magnetic trap formed by\nthe currents in the chip. The distances of the atoms from the\nsurface are now typically a few hundred microns (Fig. 3d).\n\nFinally, the $10 \\, \\mu m$ wire trap is loaded in much the same\nway. It first receives a current of $300 \\, mA$. Then the current\nin both the U-shaped wires is ramped down to zero\n(Fig.~\\ref{fig:compression}). Atoms are now typically a few tens\nof microns above the surface (Fig. 3e).\n\n%Overall we could transfer of up to 50\\% of the atoms trapped in\n%the large magnetic quadrupole into the smallest chip trap.\n\nThese guides and traps can be further compressed by ramping up the\nbias magnetic field. In this process we typically achieve\ngradients of $>25 \\, kG/cm$. By applying a bias field of $40 \\,\nG$ and a current of $200 \\, mA$ in the $10 \\, \\mu m$ wire we\nachieve trap parameters with a transverse ground state size below\n100 nm and frequencies of above $100 \\, kHz$ (as required by the\nquantum computation proposals \\cite{zoller}).\n\n\nBy running the current through a longer $10\\mu m$ section of the\nthin wire, we turn the magnetic trap into a guide, and atoms could\nbe observed expanding along it (Fig. 3f).\n\nIn an additional experiment we used the thick wires on the chip to\ncreate an {\\em on chip} bias fields for the trapping. In the\nexperiment this is done by sending current through the two U-traps\nin the opposite direction with respect to the current in the $10\n\\, \\mu m$ wire, which creates a magnetic field parallel to the\nchip surface. Hence, we demonstrate trapping of atoms on a self\ncontained chip.\n\nIn these small traps, the atom gas can be compressed to the point\nwhere direct visual observation is difficult. In such a case, we\nobserve those atoms after guiding or trapping, by 'pulling' them up\nfrom the surface into a less compressed wire trap (by increasing the\nwire current or decreasing the bias field).\n\nDuring the transfer from the large magnetic trap to the small 10\n$\\mu m$ trap the density of the atomic cloud is increased by up to\na factor 350. As the trap is compressed, the temperature of the\natoms rises, and if in this course the trapping potential is not\ndeep enough atoms are lost. In our case, the trap depth is\nuniquely determined by the bias field used, which leads to depths\n$E=-m_Fg_F\\mu_B\|B\|$ ranging between $ \\sim 6 \\, MHz$ ($ \\sim 0.25\n\\, mK$) for the $8G$ bias field and $\|m_F\|=1$ to $\\sim 70 \\, MHz$\n($ \\sim 3 \\, mK$) for the $50G$ bias field and $\|m_F\|=2$. This\nadiabatic heating and the finite trap depth limited the transfer\nefficiency for atoms from the large magnetic quadrupole into the\nsmallest chip trap to $<$50 \\%.\n\nSince we use an trapped atomic sample consisting of 3 different\nspin states ($\|F=2, m_F=2\\rangle$, $\|F=2, m_F=1\\rangle$, and\n$\|F=1, m_F=-1\\rangle$) the large compression also increases the\nrate for inelastic two body spin flip collisions dramatically.\nThis rate is for our Li sample similar to the elastic collision\nrate \\cite{cote} and is therefore a good estimate of the\nachievable collision rates in a polarized sample. From measured\ndecay curves we estimate the collision rate to be in the order of\n20 $s^{-1}$ for atoms in a typical small chip trap. This estimate\nof the scattering rate in the small chip traps is supported by the\nobservation that the atoms expand very fast into the wire guide,\nindicating that energy gained from the transverse compression of\nthe trap is transformed efficiently into longitudinal velocity at\na very high rate.\n\nThe above shows that the concept of an Atom Chip clearly works. We\nhave demonstrated that a wide variety of magnetic potentials may\nbe realized with simple wires on surfaces. Wires together with a\nbias field can produce quadrupole fields for a MOT, 3D minima for\ntrapping, and 2D minima for guiding. Furthermore it is very easy\nto manipulate the center of the trap and its width. We have shown\nthat loading such an atom trap $\\mu m$ above the surface does not\npresent a major problem and trap parameters with a transverse\nground state size below $100 \\, nm$ and frequencies of above $100\n\\, kHz$ have been achieved. In addition we could trap atoms\nexclusively with the chip fields, creating the required bias\nfields 'on board'. Last but not least, it has been shown that\nstandard nano-fabrication techniques and materials may be utilized\nto build these Atom Chips. The wires on the surface can stand\nsufficiently high current densities ($>10^6 \\, A/cm^2$) in vacuum\nand at room temperature. Together with the scaling laws of these\ntraps \\cite{joerg2,joerg1,joerg3}, this will allow us to use much\nthinner wires and reach traps with ground state sizes of $10 \\,\nnm$ and trap frequencies in the MHz range.\n\n\nWe conclude with a long term outlook. In this work we have\nsuccessfully realized a step which is but one of many still needed.\nA final integrated Atom Chip, should have a reliable source of cold\natoms, for example a BEC \\cite{BEC}, with an efficient loading\nmechanism, single mode guides for coherent transportation of atoms,\nnano-scale traps, movable potentials allowing controlled collisions\nfor the creation of entanglement between atoms, extremely high\nresolution light fields for the manipulation of individual atoms,\nand internal state sensitive detection to read out the result of the\nprocesses that have occurred (e.g. the quantum computation). All of\nthese, including the bias fields and probably even the light\nsources, could be on-board a self-contained chip. This would\ninvolve sophisticated 3D nano-fabrication and the integration of a\ndiversity of electronic and optical elements, as well as extensive\nresearch into fundamental issues such as decoherence near a surface.\nSuch a robust and easy to use device, would make possible advances\nin many different fields of quantum optics: from applications in\natom optics \\cite{AtomOptics} such as clocks and sensors to\nimplementations of quantum information processing and communication\n\\cite{QIPC}.\n\nWe would like to thank A. Chenet, A. Kasper and A. Mitterer for\nhelp in the experiments. Atom chips used in the preparation of\nthis work and in the actual experiments were fabricated at the\nInstitut f\\\"{u}r Festk\\\"{o}rperelektronik, Technische\nUniversit\\\"{a}t Wien, Austria, and the Sub-micron center, Weizmann\nInst. of Science, Israel. We thank E.Gornik, C. Unterrainer and I.\nBar-Joseph of these institutions for their assistance. Last but\nnot least, we gratefully acknowledge P. Zoller and T. Calarco who\nare responsible for the theoretical vision. This work was\nsupported by the Austrian Science Foundation (FWF), project\nS065-05 and SFB F15-07, the Jubil\\\"aums Fonds der\n\\\"Osterreichischen Nationalbank, project 6400, and by the European\nUnion, contract Nr. TMRX-CT96-0002. B.H. acknowledge financial\nsupport form Svenska Institutet.\n\n%\\vspace{-5mm}\n\\begin{references}\n%\\vspace{-15mm}\n\n\\bibitem{imry}\n Y. Imry, {\\em Introduction to Mesoscopic Physics},\n Oxford University Press, Oxford 1987.\n\n\\bibitem{moti}\n E. Buks, R. Schuster, M. Heilblum, D. Mahalu, V. Umansky,\n Nature 391, 871-874 (1998).\n\n\\bibitem{schm}\n J. Schmiedmayer, Eur. Phs. J. D 4, 57 (1998).\n\n\\bibitem{joerg2}\n J. Denschlag, D. Cassettari, A. Chenet, S.Schneider, J.\n Schmiedmayer, Appl. Phys. B {\\bf 69}, 291 (1999).\n\n\\bibitem{hind}\n E.A. Hinds, I.G. Hughes, J. Phys. D {\\bf 32}, R119 (1999).\n\n\\bibitem{joerg1}\n J. Denschlag, D. Cassettari, J. Schmiedmayer, Phys. Rev.\n Lett. \\textbf{82}, 2014 (1999).\n\n\\bibitem{Den98}\n J. Denschlag, G. Umshaus, J. Schmiedmayer, Phys. Rev. Lett.\n {\\bf 81}, 737 (1998).\n\n\\bibitem{Schm92}\n J. Schmiedmayer in {\\it XVIII International Conference on\n Quantum Electronics:} Technical Digest, edited by G.~Magerl (Technische\n Universit\\\"{a}t Wien, Vienna 1992), Series 1992, Vol. 9, 284 (1992); Appl.\n Phys. B {\\bf 60}, 169 (1995); Phys. Rev. A {\\bf 52}, R13 (1995).\n\n\\bibitem{joerg3}\n A. Haase, D. Cassettari, B. Hessmo, J. Schmiedmayer,\n Submitted to Phys. Rev. A.\n\n\\bibitem{SurfaceMounting}\n J.D. Weinstein, K. Libbrecht, Phys. Rev. A {\\bf 52}, 4004 (1995);\n M. Drndic {\\em et al.}, Appl. Phys. Lett. {\\bf 72}, 2906 (1998);\n\n\\bibitem{fortagh}\n Fortagh {\\em et al.} Phys. Rev. Lett. {\\bf 81}, 5310 (1998);\n\n\\bibitem{surface}\n J. Reichel, W. H\\\"ansel, T.W. H\\\"ansch,\n Phys. Rev. Lett. {\\bf 83}, 3398 (1999);\n D. M\\\"uller {\\em et al.}, Pre-print Physics/9908031;\n N. H. Dekker, {\\em et al.}, Los Alamos e-print archive: physics/9908029;\n\n\\bibitem{zoller}\n D. Jaksch {\\em et al.}, Phys. Rev. Lett {\\bf 81}, 3108 (1998);\n D. Jaksch {\\em et al.}, Phys. Rev. Lett {\\bf 82}, 1975 (1999);\n H.J.Briegel {\\em et al.}, J. Mod. Optics, (in print 1999);\n T.Calarco {\\em et al.}, Phys. Rev. A (in print 1999),\n (quant-ph/9905013).\n\n\\bibitem{QED}\n S. J. Van Enk, J. I. Cirac, P. Zoller, Science {\\bf 279},\n 205 (1998)\n\n\\bibitem{denschlag}\n J. Denschlag, PhD Thesis, Universit\\\"at Innsbruck (1998).\n\n\\bibitem{maier}\n The chip is produced using standard nano-fabrication methods. A deta\\'{\\i}led account\n will be given in: T. Maier {\\em et al.}, to be published.\n\n\\bibitem{LiMOT}\n The atoms are loaded into the MOT for $20s$ out of an effusive\n beam at a red laser detuning of $25MHz$ and a total laser power of\n about $150mW$. An electro-optic modulator produces sidebands of $812\n MHz (30\\%)$ one of which is used as a repumper. To increase the\n loading rate we use an additional slower beam ($20mW$, $100MHz$ red\n detuned) directed through the MOT into the oven. The MOT is\n typically $1mm$ in diameter (FWHM) and has a temperature of\n T $\\sim$ 200 $\\mu$K which corresponds to a velocity of about $0.5m/s$.\n\n\\bibitem{reflect}\n K.I. Lee, J.A. Kim, H.R. Noh, W. Jhe, Opt. Lett. {\\bf 21}, 1177 (1996).\n\n\\bibitem{cote} Robin C\\^ot\\'e, private communication.\n\n\\bibitem{BEC}\n M. H. Anderson {\\em et al.}, Science 269, 198 (1995);\n K. B. Davis {\\em et al.}, Phys. Rev. Lett. {\\bf 75}, 3969 (1995);\n C. C. Bradley {\\em et al.}, Phys. Rev. Lett. {\\bf 75}, 1687\n (1995); For an extensive list of references see:\n http://amo.phy.gasou.edu/bec.html/bibliography.html\n\n\\bibitem{AtomOptics}\n For an overview see: C.S. Adams, M. Sigel, J. Mlynek,\n Phys. Rep. \\textbf{240}, 143 (1994);\n \\emph{Atom Interferometry},\n edited by P. Berman (Academic Press, 1997).\n %, and references therein.\n\n\\bibitem{QIPC}\n {\\em The Physics of Quantum Information} edited by, D.\n Bouwmeester, A. Ekert, and A. Zeilinger, Springer-Verlag 2000.\n\n\n\\end{references}\n\n\\begin{figure}\n \\infig{Fig1.eps}{\\columnwidth}\n \\vspace{1cm}\n \\caption{(a) A schematic of the chip surface design. For simplicity,\n only wires used in the experiment are shown. The wide wires are\n $200 \\, \\mu m$ wide while the thin wires are $10 \\, \\mu m$ wide.\n The insert shows an electron microscope image of the surface and\n its $10 \\, \\mu m$ wide etchings defining the wires. (b) The\n mounted chip before it is introduced into the vacuum chamber.}\n \\label{fig:pict1}\n\\end{figure}\n\n\\newpage\n\n\\begin{figure}\n \\infig{Fig2.eps}{0.8 \\columnwidth}\n \\vspace{1cm}\n \\caption{Experimental setup: Four circularly polarized light\n beams enter the chamber; two are counter propagating parallel to\n the surface of the chip, while the two others, impinging on the\n surface of the chip at $45$ degrees, are reflected by the gold\n layer. The chip, the underlying U-wire\n trap, and the bias field, are oriented in such a manner as to\n provide a quadrupole field with the same orientations as the MOT\n coils. The oven, the effusive beam, and the two cameras observing\n the atomic cloud are also shown.}\n \\label{fig:pict2}\n\\end{figure}\n\n\\newpage\n\n\\begin{figure}\n \\infig{Fig3.eps}{0.5 \\columnwidth}\n \\caption{(color) Experiments with an Atom Chip: column (i) shows the view\n from the top (camera 1), column (ii) the front view (camera 2) and\n (iii) a schematic of the wire configuration. Current carrying\n wires are highlighted in red. The front view shows two images:\n the upper is the actual atom cloud and the lower is the reflection\n on the gold surface of the chip. The distance between both images\n is an indication of the distance of the atoms from the chip\n surface. Rows (a)-(f) show the various steps of the experiments.\n (a)-(d) show the step wise process of loading atoms onto the chip\n while (e) and (f) show atoms in a microscopic trap and propagating\n in a guide. The pictures of the magnetically trapped atomic cloud\n are obtained by fluorescence imaging using a short laser pulse\n (typically $0.5 \\, ms$)}\n \\label{fig:pict3}\n\\end{figure}\n\n\\newpage\n\n\n\\begin{figure}\n \\infig{Fig4.eps}{0.9 \\columnwidth}\n \\vspace{2cm}\n \\caption{Transfer from a large trap formed by two U-shaped wires\n to one thin wire: The position of the surface mounted wires and\n equipotential lines for the trapping potential are shown.\n i) The first picture: the large 200 $\\mu m$ U-traps carry a current\n of 2A and the small 10 $\\mu m$ wire 300 mA.\n ii) The second picture shows an intermediate stage in the transfer\n to the 10 $\\mu m$ trap. The current in the large U-traps is\n decreased to 0.5 A.\n iii) The large U-traps are now switched off and the transfer to\n the small 10 $\\mu m$ trap is complete.\n iv) By increasing the bias field the trap can be compressed\n further.}\n \\label{fig:compression}\n\\end{figure}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912106.extracted_bib", "string": "{imry Y. Imry, {\\em Introduction to Mesoscopic Physics, Oxford University Press, Oxford 1987." }, { "name": "quant-ph9912106.extracted_bib", "string": "{moti E. Buks, R. Schuster, M. Heilblum, D. Mahalu, V. Umansky, Nature 391, 871-874 (1998)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{schm J. Schmiedmayer, Eur. Phs. J. D 4, 57 (1998)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{joerg2 J. Denschlag, D. Cassettari, A. Chenet, S.Schneider, J. Schmiedmayer, Appl. Phys. B {69, 291 (1999)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{hind E.A. Hinds, I.G. Hughes, J. Phys. D {32, R119 (1999)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{joerg1 J. Denschlag, D. Cassettari, J. Schmiedmayer, Phys. Rev. Lett. 82, 2014 (1999)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{Den98 J. Denschlag, G. Umshaus, J. Schmiedmayer, Phys. Rev. Lett. {81, 737 (1998)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{Schm92 J. Schmiedmayer in {XVIII International Conference on Quantum Electronics: Technical Digest, edited by G.~Magerl (Technische Universit\\\"{at Wien, Vienna 1992), Series 1992, Vol. 9, 284 (1992); Appl. Phys. B {60, 169 (1995); Phys. Rev. A {52, R13 (1995)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{joerg3 A. Haase, D. Cassettari, B. Hessmo, J. Schmiedmayer, Submitted to Phys. Rev. A." }, { "name": "quant-ph9912106.extracted_bib", "string": "{SurfaceMounting J.D. Weinstein, K. Libbrecht, Phys. Rev. A {52, 4004 (1995); M. Drndic {\\em et al., Appl. Phys. Lett. {72, 2906 (1998);" }, { "name": "quant-ph9912106.extracted_bib", "string": "{fortagh Fortagh {\\em et al. Phys. Rev. Lett. {81, 5310 (1998);" }, { "name": "quant-ph9912106.extracted_bib", "string": "{surface J. Reichel, W. H\\\"ansel, T.W. H\\\"ansch, Phys. Rev. Lett. {83, 3398 (1999); D. M\\\"uller {\\em et al., Pre-print Physics/9908031; N. H. Dekker, {\\em et al., Los Alamos e-print archive: physics/9908029;" }, { "name": "quant-ph9912106.extracted_bib", "string": "{zoller D. Jaksch {\\em et al., Phys. Rev. Lett {81, 3108 (1998); D. Jaksch {\\em et al., Phys. Rev. Lett {82, 1975 (1999); H.J.Briegel {\\em et al., J. Mod. Optics, (in print 1999); T.Calarco {\\em et al., Phys. Rev. A (in print 1999), (quant-ph/9905013)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{QED S. J. Van Enk, J. I. Cirac, P. Zoller, Science {279, 205 (1998)" }, { "name": "quant-ph9912106.extracted_bib", "string": "{denschlag J. Denschlag, PhD Thesis, Universit\\\"at Innsbruck (1998)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{maier The chip is produced using standard nano-fabrication methods. A deta\\'{\\iled account will be given in: T. Maier {\\em et al., to be published." }, { "name": "quant-ph9912106.extracted_bib", "string": "{LiMOT The atoms are loaded into the MOT for $20s$ out of an effusive beam at a red laser detuning of $25MHz$ and a total laser power of about $150mW$. An electro-optic modulator produces sidebands of $812 MHz (30\\%)$ one of which is used as a repumper. To increase the loading rate we use an additional slower beam ($20mW$, $100MHz$ red detuned) directed through the MOT into the oven. The MOT is typically $1mm$ in diameter (FWHM) and has a temperature of T $\\sim$ 200 $\\mu$K which corresponds to a velocity of about $0.5m/s$." }, { "name": "quant-ph9912106.extracted_bib", "string": "{reflect K.I. Lee, J.A. Kim, H.R. Noh, W. Jhe, Opt. Lett. {21, 1177 (1996)." }, { "name": "quant-ph9912106.extracted_bib", "string": "{cote Robin C\\^ot\\'e, private communication." }, { "name": "quant-ph9912106.extracted_bib", "string": "{BEC M. H. Anderson {\\em et al., Science 269, 198 (1995); K. B. Davis {\\em et al., Phys. Rev. Lett. {75, 3969 (1995); C. C. Bradley {\\em et al., Phys. Rev. Lett. {75, 1687 (1995); For an extensive list of references see: http://amo.phy.gasou.edu/bec.html/bibliography.html" }, { "name": "quant-ph9912106.extracted_bib", "string": "{AtomOptics For an overview see: C.S. Adams, M. Sigel, J. Mlynek, Phys. Rep. 240, 143 (1994); Atom Interferometry, edited by P. Berman (Academic Press, 1997). %, and references therein." }, { "name": "quant-ph9912106.extracted_bib", "string": "{QIPC {\\em The Physics of Quantum Information edited by, D. Bouwmeester, A. Ekert, and A. Zeilinger, Springer-Verlag 2000." } ]
quant-ph9912107	Quantum feedback control and classical control theory	[ { "author": "Andrew C. Doherty\\footnote{Present address: Norman Bridge Laboratory of Physics 12-33" }, { "author": "California Institute of Technology" }, { "author": "Pasadena CA 91125. Email: dohertya@its.caltech.edu" }, { "author": "Salman Habib" }, { "author": "Kurt Jacobs" }, { "author": "Hideo Mabuchi" }, { "author": "Sze M. Tan" } ]	We introduce and discuss the problem of quantum feedback control in the context of established formulations of classical control theory, examining conceptual analogies and essential differences. We describe the application of state-observer based control laws, familiar in classical control theory, to quantum systems and apply our methods to the particular case of switching the state of a particle in a double-well potential.	[ { "name": "fc2fn2.tex", "string": "\\documentstyle[aps,epsfig,multicol]{revtex}\n\n\\newcommand{\\bq}{\\begin{equation}}\n\\newcommand{\\eequ}{\\end{equation}}\n\\newcommand{\\bqa}{\\begin{eqnarray}}\n\\newcommand{\\eqa}{\\end{eqnarray}}\n\\newcommand{\\nn}{\\nonumber}\n\\newcommand{\\ms}[1]{\\mbox{\\scriptsize #1}}\n\\newcommand{\\dg}{^\\dagger}\n\\newcommand{\\smallfrac}[2]{\\mbox{$\\frac{#1}{#2}$}}\n\\newcommand{\\la}{\\langle}\n\\newcommand{\\ra}{\\rangle}\n\\newcommand{\\ket}[1]{\| {#1} \\ra}\n\\newcommand{\\bra}[1]{\\la {#1} \|}\n\\newcommand{\\inprod}[2]{\\la {#1} \| {#2} \\ra}\n\\newcommand{\\outprod}[2]{\| {#1} \\ra \\lar {#2} \|}\n%\\newcommand{\\dbd}[1]{\\frac{\\partial}{\\partial #1}}\n\\newcommand{\\ioh}{-\\frac{i}{\\hbar}}\n\\newcommand{\\oh}{-\\frac{1}{\\hbar^2}}\n\\newcommand{\\sch}{Schr\\\"odinger }\n\\newcommand{\\hei}{Heisenberg }\n\\newcommand{\\htn}{Hamiltonian }\n\\newcommand{\\htnn}{Hamiltonian}\n\\newcommand{\\half}{\\smallfrac{1}{2}}\n\\newcommand{\\bl}{{\\bigl(}}\n\\newcommand{\\br}{{\\bigr)}}\n\n\\begin{document}\n\\draft\n\n\\title{Quantum feedback control and classical control theory}\n\n\\author{Andrew C. Doherty\\footnote{Present address: \nNorman Bridge Laboratory of Physics 12-33,\nCalifornia Institute of Technology, Pasadena CA 91125. Email: \ndohertya@its.caltech.edu}}\n\\address{Department of Physics, University of Auckland,\nAuckland, Private Bag 92019, New Zealand} \n\\author{Salman Habib, Kurt Jacobs}\n\\address{T-8, Theoretical Division, Los Alamos National\nLaboratory, Los Alamos, New Mexico 87545} \n\\author{Hideo Mabuchi}\n\\address{Norman Bridge Laboratory of Physics 12-33,\nCalifornia Institute of Technology, Pasadena CA 91125}\n\\author{Sze M. Tan}\n\\address{Department of Physics, University of Auckland,\nAuckland, Private Bag 92019, New Zealand} \n\\maketitle\n\n\\begin{abstract}\nWe introduce and discuss the problem of quantum feedback control in\nthe context of established formulations of classical control theory,\nexamining conceptual analogies and essential differences. We describe\nthe application of state-observer based control laws, familiar in\nclassical control theory, to quantum systems and apply our methods to\nthe particular case of switching the state of a particle in a\ndouble-well potential.\n\\end{abstract}\n\n\\pacs{03.65.Bz,45.80.+r,02.50.-r,03.67.-a}\n\n\\begin{multicols}{2}\n\n\\section{Introduction}\n\nExperimental technology, particularly in the fields of cavity\nQED~\\cite{CQED}, ion trapping~\\cite{ion} and Bose-Einstein\ncondensation~\\cite{andrews1996a}, has now developed to the point where\nindividual quantum systems can be monitored continuously with very low\nnoise and may be manipulated rapidly on the time-scales of the system\nevolution. It is therefore natural to consider the possibility of\ncontrolling individual quantum systems in real time using\nfeedback~\\cite{qfb1}. In this paper we consider the problem of\nfeedback control at the quantum limit. In a fully quantum mechanical\nfeedback control theory the quantum dynamics of the system and the\nback-action of measurements must both be taken into account.\n\nThe major theoretical challenge of extending feedback control to the\nquantum mechanical regime is to describe properly the back-action of\nmeasurement on the evolution of individual quantum systems.\nFortunately, the formalism of quantum measurement, and particularly\nthat of the continuous observation of quantum systems, is now\nsufficiently well developed to provide a general framework in which to\nask salient questions about this new subject of {\\em quantum feedback\ncontrol}. In fact, the formulation that results from this theory is\nsufficiently similar to that of classical control theory that the\nexperience gained there provides valuable insights into the\nproblem. However, there are also important differences which render\nthe quantum problem potentially more complex. In this paper we\ndescribe a fairly general formulation of the classical feedback\ncontrol problem, and compare it with a similarly general quantum\nfeedback control problem. This allows us to examine ways in which the\nclassical problem may be mapped to the quantum problem, to provide\ninsight, and to show when results from the classical theory may be\napplied directly to the control of quantum systems. This will also\nallow us to highlight the essential features of the quantum problem\nwhich distinguish it from classical feedback control.\n\nThe field of quantum-limited feedback was introduced \n%in the mathematical physics literature by Belavkin~\\cite{}, \n%and in the physics literature \nby Wiseman and Milburn~\\cite{qfb1}, who considered the instantaneous\nfeedback of some measured photocurrent onto the dynamics of a quantum\nsystem. The master equation for the resulting evolution was then\nMarkovian. In this work we are interested in more general schemes in\nwhich some arbitrary functional of the entire history of the\nmeasurement results can be used to alter the system evolution. The\nresulting dynamics of the system is then non-Markovian, however the\ndynamics of the system and controller remain Markovian. As we shall\nsee this is completely analogous to the situation in classical control\ntheory.\n\nThe Wiseman-Milburn theory has been applied to the generation of\nsub-shot noise photocurrents through feedback and the affect of the\nin-loop light on the fluorescence of an\natom~\\cite{taubman1995a}. Other proposed applications include the\nprotection and generation of non-classical states of the light\nfield~\\cite{slosser1995a}\nand the manipulation of the motional state of atoms or the mirrors of\noptical cavities~\\cite{dunningham1997}. In\nrelated work Hofman {\\it et al}~\\cite{hofman1998a} consider the\npreparation and preservation of states of a two-level atom through\nhomodyne detection and feedback in a slightly different\nformalism. Finally the so-called `dynamical decoupling' of a quantum\nsystem from its environment has been\ndiscussed~\\cite{viola1999a} which protects states of the\nsystem of interest from the effects of coupling to the environment in\nsituations in which it is possible to manipulate the system on times\nshort compared to the correlation time of the environment. This is the\nopposite limit to the Wiseman-Milburn theory which considers feedback\nslow on the time scale of the bath correlations but fast on the\ntime scales of the dissipative or non-linear dynamics. This work\nadopts, as we do, ideas from classical control theory, in this case\nthe so-called bang-bang control, to open quantum systems. There is an\nextensive literature on the application of classical control\ntechniques such as optimal control to {\\em closed} quantum systems, a\nuseful entry point into this literature being Ref.~\\cite{warren1993a}.\n \nAlthough the task of determining useful functionals of the measurement\ncurrent may seem daunting we argue that much progress can be made by\nadopting the lessons of the classical theory of state estimation and\ncontrol. In particular it is helpful to break the feedback control\nprocess into two steps --- the propagation of some estimate of the\nstate of the system given the history of the measurement results, and\nthe use of this state estimate at a given time in order to calculate\nappropriate control inputs to affect the dynamics of the system at\nthat time. This approach has already yielded results for the optimal\ncontrol of observed linear quantum systems~\\cite{BelavkinLQG,DJ}.\n\nA simple example of an experiment in quantum optics in which similar\ncontrol strategies have already been employed is the work of Cohadon\n{\\em et al.}~\\cite{CHP}. In this experiment the aim is to damp the\nthermal motion of the end mirror of a high finesse optical cavity. A\nvery high precision interferometric measurement is made of the\nmirror's position and the resulting signal is filtered in an\nappropriate way to generate an estimate of the current mirror\nmomentum. This momentum-estimate signal is then used to modulate the\nlaser power of a laser driving the back of the mirror in order to\nexert a radiation pressure force in the opposite direction to the\nmirror momentum, thus reducing the effective temperature of the\nmirror. In fact the considerable thermal noise in the experiment means\nthat the back-action noise is not significant and so an essentially\nclassical treatment of the feedback is sufficient. In this paper we\nwish to consider a relatively general description of this kind of\nfeedback technique in a way that explicitly takes into account the\nquantum mechanical back-action noise and will thus be relevant to\nexperiments such as~\\cite{CQED} where truly quantum control is a near\nfuture possibility.\n\nIn the next section we describe the classical feedback control problem\nwell-known in classical control theory, while in Section III we\nintroduce a formulation of the quantum problem and examine conceptual\nanalogies between the two. We consider optimization of the control\nstrategy and discuss the quantum equivalent of the Bellman equation,\nbeing a general statement of the quantum optimal control problem in a\ndynamic programming form. In Section IV we consider the possibility of\nmaking precise mappings between the classical and quantum problems,\nand examine when the quantum problem may be addressed using the\nclassical theory directly. In Section V we consider the classical\nconcept of observability and discuss ways in which this may be defined\nfor quantum systems. In Section VI we consider the application of\nsub-optimal control strategies developed for non-linear classical\nsystems to quantum systems. As an example we consider controlling the\nstate of a particle in a double-well potential in the presence of\nnoise. Section VII concludes.\n\n\\section{Classical Feedback Control}\n\\label{CFC}\nIn this section we consider the classical feedback control\nproblem~\\cite{cfb1,Maybeck,cfb2,Ben,residual}. It is not\nour intention here to be completely general, since the control problem\nis a very broad one. We will consider explicitly only continuous time\nsystems, and these driven by Gaussian noise. Since most of what we say\nwill apply also to discrete systems, and those driven by other kinds\nof noise sources, little is lost by this restriction.\n\nThe problem which classical feedback control theory addresses consists\nof the following: A given dynamical system, driven by noise, and\nmonitored imperfectly, is driven also by some input(s) with the\nintention of controlling it, and these inputs are allowed to be a\nfunction of the results of the observations performed on the\nsystem. The dynamics of the system may be written as\n\\begin{equation}\nd{\\bf x} = {\\bf F}({\\bf x},{\\bf u})dt + {\\cal G}({\\bf x},{\\bf u})\\cdot {\\bf\ndW} , \n\\end{equation}\nwhere ${\\bf x}$ is the state of the system (a vector consisting of the\nessential dynamical variables), ${\\bf u}$ is a set of externally\ncontrollable inputs to the system, ${\\bf dW}$ is a set of Wiener\nincrements, and $t$ is time. Note that since ${\\bf x}$ and ${\\bf dW}$\nare vectors, ${\\cal G}$ is a matrix. In this paper we follow the terminology\nof the quantum optics community and refer to the system of interest\nthat is to be controlled as simply the {\\em system}. In the control\ntheory literature this is often termed the {\\em process}. Hence the\nnoise driving the system is often referred to as the {\\em process\nnoise}. To avoid confusion it may be useful to bear in mind that in\nthe control theory literature it is common to use the term system to\nrefer to all the parts of the control problem --- the process, the\ncontrol loop and all the noise and other inputs. The observation\nprocess is usually written as\n\\begin{equation}\n {\\bf dy} = {\\bf H}({\\bf x},t) dt + {\\cal R}(t)\\cdot{\\bf dV} ,\n\\end{equation}\nwhere ${\\bf dV}$, referred to as the {\\em observation} noise, is\nanother set of Wiener increments which may or may not be correlated\nwith the noise driving the system, ${\\bf dW}$.\n\nThe process of feedback control involves choosing the inputs ${\\bf\nu}$, at each time $t$, as some function of the entire history of the\nobservation process ${\\bf dy}$ and of the initial conditions. To\ncomplete the specification of a given control problem, one must define\na {\\em cost function}, which specifies the desired behavior, and the\n`cost' associated with deviations from this behavior. An important \ngoal of control theory is then to specify ${\\bf u}$ such that the cost\nfunction is minimized. Such a result is referred to as {\\em optimal\ncontrol}.\n\nAs a general principle we can say that as our knowledge regarding the\nstate of the system at any given time becomes better, so too does the\nefficacy of the feedback algorithm, since we can better determine the\nappropriate feedback. Hence the question of state-estimation (that is,\nthe determination of our best estimate of the state from the results\nof the measurement process) arises naturally in this context. In the\nfullest description, one can decide upon a probability density,\n$P({\\bf x})$, that describes one's complete initial state of knowledge\nof the dynamical variables ${\\bf x}$, and then determine how this\ndensity evolves due to the system dynamics and the continual\nobservation. The equation governing this {\\em a posteriori}\nprobability density is called the Kushner-Stratonovitch (KS) equation,\nbeing\n\\bqa dP & = & -\\sum_{i=1}^{n} \\frac{\\partial}{\\partial x_i} (F_i\nP) dt \\nn \\\\\n & & + \\frac{1}{2} \\sum_{i=1}^{n} \\sum_{j=1}^{n}\n\\frac{\\partial^2}{\\partial x_i \\partial x_j} ([{\\cal GG}^T]_{ij} P) dt \\label{KSE} \\\\ \n& & + [{\\bf H(x,t)} - \\langle {\\bf H}(x,t)\\rangle]^T \n({\\cal RR}^T)[{\\bf dy} - \\langle {\\bf H}(x,t)\\rangle dt] P. \\nn\n\\eqa \nHere we have written the elements of ${\\bf x}$ and ${\\bf F}$ as\n$x_i$ and $F_i$ respectively, $[{\\cal GG}^T]_{ij}$ denotes the $ij^{th}$ \nelement of the matrix ${\\cal GG}^T$, and $\\langle \\ldots \\rangle$ is the\nexpectation value with respect to $P$ at the current time. With the\nexclusion of the final term, this is merely the Fokker-Planck equation\nfor (unconditional) evolution of the noise-driven system. It is the\nfinal term which takes into account the effect of the measurement on\nour state of knowledge. Note that as a result of the terms involving\n$\\langle {\\bf H}(x,t)\\rangle$ this is a non-linear equation for the\nprobability distribution. Here we have made the usual assumption that \nthe process and measurement noises are decorrelated. The stochastic \nprocess which drives the KS\nequation is the difference between the actual measured values, ${\\bf\ndy}$, and the value one {\\em expected} to measure, $\\langle {\\bf\nH}(x,t)\\rangle$. This is referred to as the {\\em residual}, or {\\em\ninnovation}. Since the conditioned probability distribution is the\noptimal estimate of the state that may be obtained from the\nmeasurement record, the residual has zero mean and is uncorrelated\nwith the conditioned probability distribution. Note that the residual\nis distinct from both the process noise and the measurement noise.\n\nIt is worth mentioning that it is also possible to write a linear\nequation for the conditional probability density $P$, if we relax the\nrequirement that $P$ be normalized. The resulting equation, which may\nbe found in e.g. Ref.~\\cite{Ben}, is called the Zakai equation.\n\nFor linear systems driven by Gaussian noise, the KS equation becomes\nparticularly simple, with initially Gaussian densities remaining\nGaussian. As a result closed equations of motion for the means (being\nalso the `best', or maximum {\\em a posteriori} estimates of the system\nstate) and variances can be obtained. Evolving these moment equations\nis then much simpler than trying to keep track of an entire\ndistribution.\n\nIn addition, for linear systems the classical optimal control problem\nis essentially solved. Under the assumption of a cost function\nquadratic in the dynamical variables, the optimal control law involves\nmaking ${\\bf u}$ a linear function of the best estimate of the\ndynamical variables, and the equation for determining this function\nmay be given explicitly in terms of the (in this case linear)\nfunctions ${\\bf F}$ and ${\\cal G}$. Moreover, the solution of the linear\nproblem possesses certain important properties which make it\nparticularly simple: It satisfies the {\\em separation theorem}, which\nstates that the optimal control law depends on only one estimate of\nthe state~\\cite{cfb1,cfb2,residual} --- in this case the mean of the\n{\\em a posteriori} probability distribution. There is no advantage in\nmodifying the control law based on the uncertainty of the current\nstate estimate. The linear problem also satisfies {\\em certainty\nequivalence}. This means that the optimal control strategy is the same\nas it would be even if there was no noise driving the system and the\nstate of the system were known exactly; in the stochastic problem the\noptimal state estimate simply takes the place of the system state in\nthe deterministic problem. Furthermore the linear problem is {\\em\nneutral}, which means that the choice of controls does not affect the\naccuracy of the state estimate. If the action of the controller\naffects the uncertainty about the state of the system as the well as\nthe evolution of the system itself this is termed {\\em dual effect}.\n\nFor non-linear systems the situation is very different. Non-linear\nsystems may satisfy only a few of the above conditions, or none at\nall. Few exact results exist for optimal control strategies. True\noptimal estimation almost invariably requires the integration of the\nfull KS equation, something which is impractical for real-time\napplications. Therefore it is generally necessary to develop good\napproximate, but nevertheless sub-optimal, estimation and control\nstrategies, and many approaches to this problem have been\ndeveloped. In Section~\\ref{subopt} we will consider similar approaches\nto the quantum problem where integration of an optimal estimate of the\nsystem state may also be impractical in real time.\n\nAnother reason for employing nominally sub-optimal feedback control is\nto account for uncertainty in the model. If parameters of the model of\nthe system are not in fact well known then the control that is optimal\nfor the nominal model may in fact be a very poor control loop for\nmodels with similar but not identical values of the parameters. This\nproblem can be particularly pronounced in systems with large numbers\nof degrees of freedom and the solution of this problem is the domain\nof {\\em robust control}~\\cite{RAOC}. Another control technique commonly \nused in practice is pole-placement for which quantum mechanical \nanalogues could also be developed.\n\n\\section{Quantum Feedback Control}\n\n\\subsection{Continuous Quantum Measurement}\n\\label{sec:contqmeas}\nThe model of the control problem introduced above makes sense in\nclassical physics --- however it is implicitly assumed that it is\npossible to extract information about the state of the system without\ndisturbing it. This is not a valid assumption in quantum mechanics,\nand hence in describing any experiment on a quantum system it is\nnecessary to consider carefully, as well as the quantum dynamics of\nthe system, the coupling of the system of interest with the measuring\napparatus. To provide a similarly useful formulation of quantum\nfeedback control we require a model of quantum continuous measurement\nwith a similarly wide applicability to the classical model of the\nprevious section. In recent years, in the field of quantum optics,\nwhere continuous quantum measurements are realized experimentally, a\nformalism was developed to accurately describe such \nmeasurements~\\cite{MeasP,GPZ,Carm,WMhom}, and\nit was realized later that this description was identical to that\ndeveloped in the mathematical physics literature using more abstract\nreasoning~\\cite{MeasM,BelavkinLQG}. This formalism appears \nto fill the role for quantum systems\nthat the classical formulation introduced above plays for classical\nsystems. In order to describe noise in quantum systems we will employ\nthe master equation formalism and because the measurement of the\nsystem requires some coupling to the external world the continuous\nmeasurement of a quantum system also requires the consideration of\nmaster equations of a particular type.\n\nIf we denote the state of the quantum system that we are concerned\nwith controlling as $\\rho$ and the system Hamiltonian as $H$, then the\neffect of measurement and environmental noise may be included by\nadding two Lindblad terms to the master equation for $\\rho$:\n\\begin{equation}\n \\dot{\\rho} = -i[H,\\rho] + {\\cal D}[Q]\\rho + {\\cal D}[c]\\rho\n\\label{me1}\n\\end{equation}\nwhere $D[A]\\rho \\equiv (2A\\rho A^\\dagger - A^\\dagger A\\rho - \\rho\nA^\\dagger A)/2$ for an arbitrary operator $A$. When $A$ is Hermitian\nthis reduces to $D[A]\\rho=-[A,[A,\\rho]]/2$. The term $D[Q]\\rho$\ndescribes the {\\em unconditional} evolution resulting from a\ncontinuous measurement where the interaction of the measuring device\nand the system is via the system operator $Q$. If $Q$ is Hermitian,\nthen it describes a continuous measurement of the observable\ncorresponding to $Q$. By unconditional evolution we mean that the\nmaster equation describes our state of knowledge if we make the\nmeasurement but throw away the information (the measurement record).\nIt is therefore the result of averaging over all the possible final\nstates resulting from the measurement history. Similarly, averaging\nover the measurement results in the classical Kushner-Stratonovitch\nequation results in a Fokker-Planck equation for the probability\ndistribution of the state. The second term of the master equation,\n$D[c]\\rho$, describes the effect of noise due to the environment.\nSince it has the same form as that of the unconditional measurement\nevolution, it is always possible to view it as the result of a\nmeasurement to which we have no access. Similarly, it is always\npossible to view the measurement process as an interaction with an\nenvironment (bath) where we are performing measurements on the bath to\nobtain the information, producing a continuous measurement on the\nsystem.\n\nAssociated with any given history of measurement results will be a\nconditioned state, $\\rho_{\\text{c}}$, being the observer's actual\nstate of knowledge resulting from recording the (continuous) series of\nmeasurement outcomes. The evolution of the conditioned state is\nreferred to as a quantum {\\em trajectory}. If one conditions on the \nmeasurement of the observable $Q$, the master equation (Eq.(\\ref{me1})) \nbecomes~\\cite{WMhom}\n\\begin{equation}\n \\label{SME1}\n d\\rho_{\\text{c}}\n = dt{\\cal L}_{0}\\rho_{\\text{c}} + dt{\\cal D}[Q]\\rho_{\\text{c}} +\n {\\cal H}[Q]\\rho_{ \\text{c}} dW + {\\cal D}[c]\\rho,\n\\end{equation}\nwhich is described as a Stochastic Master Equation (SME).\nHere $\\cal{H}$ is defined by\n\\begin{equation}\n {\\cal H}[\\Lambda]\\rho = \\Lambda\\rho + \\rho \\Lambda^\\dagger - \\mbox{Tr}[(\\Lambda + \\Lambda^\\dagger)\\rho]\\rho.\n\\end{equation}\nThe measurement process is given in terms of the process $dW$ by\n\\begin{equation}\n dy = \\mbox{Tr}[ \\left(Q+Q^{\\dagger}\\right) \\rho] dt + dW .\n\\label{dy1}\n\\end{equation}\nHere $dW$ is a Wiener increment, and we see that there is a close\nsimilarity between the quantum measurement process and the classical\nmeasurement process. It should be remembered that for a fixed master\nequation, it is, in fact, possible to alter ones measurements to\nobtain different SME's. This is referred to as choosing a different\n{\\em unraveling} of the master equation. In general the SME (and\ntherefore the measurement process) may be driven by Poisson noise as\nwell as Wiener noise. We will return to this point later when we\nconsider feedback.\n\nIn the classical description of state estimation, it is the\nconditional probability density, whose evolution is governed by the\nKushner-Stratonovitch equation, that describes the observer's complete\nstate of knowledge. The conditional probability density contains the\nprobabilities for the outcomes of all measurements which may be\nperformed on the system. In quantum mechanics it is the density matrix\nthat may be used to calculate probability distributions for arbitrary\nmeasurements on the system. It is therefore the conditional density\nmatrix which replaces the conditional probability density in quantum\nstate estimation theory, and it is the SME which is the analogue of\nthe Kushner-Stratonovitch equation, being the propagator for the\noptimal estimate of the quantum mechanical state given the history of\nthe measurement current $I_{[t_{0},t)}=\\{dy(t')/dt:t_{0}\\leq t'<t\\}$.\nJust as in the classical problem a residual process ($dW$)\nuncorrelated with the state estimate arises. This zero mean noise\nprocess is again the difference between the actual measurement result\nand the result expected on the basis of previous measurements.\n\nWe also note that if one allows the conditional density matrix to be\nunnormalized, it is possible to write the SME as a {\\em linear}\nstochastic master equation. This then, is the equivalent of the Zakai\nequation of classical state-estimation, which is a linear equation\npropagating an unnormalized {\\em a posteriori} probability\ndistribution.\n\n%%\n%% keep?\n%%\nThe SME~(\\ref{SME1}), like any other master equation, may be unraveled\ninto trajectories of pure states obeying a stochastic evolution. This\ninvolves imagining that it is in fact possible to make some kind of\ncomplete measurement on the bath and that the results of these\nmeasurements are known to the observer. In that case we would have\ncomplete information about the system, so that an initial pure state\nwould remain pure, and we could write the stochastic master equation\ninstead as a Stochastic Shr\\\"{o}dinger Equation (SSE) for the state\nvector. The result is\n\\bqa \nd\|\\psi\\rangle & = & \\left( -iH dt + \\left[ Q-\\smallfrac{1}{2}\\langle\nQ+Q^{\\dagger}\\rangle \\right] dW_0 \\right) \|\\psi\\rangle \\nn \\\\\n& & + \\sum_j \\left(c_j-\\smallfrac{1}{2}\\langle\n c_j+c_{j}^{\\dagger}\\rangle \\right) dW_j \|\\psi\\rangle \\\\\n& & - \\smallfrac{1}{2}\\left(Q^{\\dagger}Q -\\langle\n Q+Q^{\\dagger}\\rangle Q + \\smallfrac{1}{4}\\langle\n Q+Q^{\\dagger}\\rangle^2 \\right)dt \|\\psi\\rangle \\nn \\\\ \n& & - \\smallfrac{1}{2}\\sum_{j} \\left( c_{j}^{\\dagger}c_{j} -\n\\langle c_j+c_{j}^{\\dagger}\\rangle c_j + \\smallfrac{1}{4} \\langle\n c_j+c_{j}^{\\dagger}\\rangle^2 \\right) dt \|\\psi\\rangle, \\nn\n\\eqa\nwhere the notation $\\langle a \\rangle \\equiv \\langle \\psi \| a \| \\psi\n\\rangle$ was used. Here $Q$ is once again the measured observable, and \nthis time we have included an arbitrary number of noise sources, \n$c_j$, rather than merely a single noise source (determined previously \nby the operator $c$). Of the Wiener processes, $dW_0$ results from the measurement process of the real observer (measuring the observable $Q$), \nand the $dW_j$ from the fictitious measurements on the bath. Many of \nthese unravelings are possible depending on what measurements are \nimagined to be performed on the bath (for example a Poisson process \nmight be used, for any of the noise sources, rather than a Wiener \nprocess), the property that all unravelings will have in common is \nthat the average of the SSE over many realizations will produce the \ncorrect SME. It turns out that the measurement process is now given by\n\\begin{equation}\n dy = \\langle\\psi\|Q+Q^{\\dagger}\|\\psi\\rangle dt + dW_0\n\\label{dy2}\n\\end{equation}\nBy comparing Eqs. (\\ref{dy1}) and (\\ref{dy2}), we see that for a given\nrealization of the measurement process $dy$, since in general\n$\\mbox{Tr}[\\left(Q+Q^{\\dagger}\\right)\\rho] \\not=\n\\langle\\psi\|Q+Q^{\\dagger}\|\\psi\\rangle$, the processes $dW$ and $dW_0$\nare {\\em not the same}.\n\nSince the SSE is an equation for the state vector chosen such that the\naverage over all trajectories correctly reproduces the SME, the\nequivalent classical object would be a stochastic equation for the\nstate vector ${\\bf x}$ such that the average reproduced the KS\nequation. Such an equation can certainly be constructed, with the\nintroduction of fictitious noise sources corresponding to $dW_j$ in\nthe SSE introduced above. The use of stochastic differential equations\nto propagate Fokker-Planck equations is well known in classical\ntheories; the Kushner-Stratonovitch equation is simply a non-linear,\nstochastic Fokker-Planck equation for the {\\em a posteriori}\nprobability distribution. It should be noted that these fictitious\nnoises do not correspond to the process noise.\n%%\n%% if we take it out it means only removing one item from the table\n%%\n\nWhile we have presented quantum analogies here for many of the objects\nin classical state-estimation, we have not presented analogies for the\nobjects that describe the underlying classical system, being the\nclassical state vector, process noise, and measurement noise. Such\nanalogies may be made at the cost of replacing the state vector,\nprocess noise and measurement noise by operators in appropriate\nHilbert spaces. This requires the formulation of the problem in terms\nof quantum stochastic differential equations (QSDE's). Space prevents\nus from examining this in detail here, and the reader is referred to\nthe work of Gardiner {\\em et al}\\ for a discussion of QSDE's in the context of continuous measurement \\cite{GPZ}. In Table 1 we include\nthe analogous quantities which result from such an analysis along with\nthe tentative analogies we have discussed in detail in this section.\n\n\\end{multicols}\n\\begin{table}\n\\caption{Quantum/Classical Analogies in State-Estimation}\n\\begin{tabular}{\|l@{\\hspace{0.2cm}}\|l@{\\hspace{0.7cm}}\|}\n {\\bf Classical State Estimation} & {\\bf Quantum State Estimation} \\\\ \\hline\n A Posteriori Probability Distribution & Conditioned Density Matrix \\\\\n Kushner-Stratonovitch Equation & Non-linear Stochastic Master Equation \\\\\n Zakai Equation & Linear Stochastic Master Equation \\\\\n Innovation/Residual process & Quantum residuals ($dy - \\langle\n Q+Q^{\\dagger}\\rangle dt$)\\\\\n Fokker-Planck Equation for {\\em a priori} distribution & Master Equation \\\\\n Fictitious noise to simulate KS Eq. using SDE & Fictitious noise to\n simulate SME using SSE \\\\\n State Vector & Operators for System Observables \\\\\n Process noise & Bath Noise Operators \\\\\n Measurement noise & Meter Field Noise Operators\\\\\n\\end{tabular}\n\\label{tbl1}\n\\end{table}\n\\begin{multicols}{2}\n\n% & \\omit\\span transverse\\kern24pt \\\\\n% & with FS \\\\\n\n\\subsection{Controlled Quantum Systems}\n\nThe goal of feedback control of quantum systems will be to use the\ncontinuous stream of measurement results to prepare some desired state\nor enforce some desired evolution of the system. In the classical\nformulation this involves effectively altering the system Hamiltonian\nby adding the control inputs ${\\bf u}$, which are functions of the\nmeasurement record. Quantum mechanically the equivalent action is to\nmake the Hamiltonian $H$ a function of the measurement record. In an\nactual experiment the variation of the Hamiltonian involves the\nmodulation of classical parameters such as external DC fields,\nlaser phases and driving strengths.\n\nHowever, while feedback control of the system Hamiltonian is\nsufficient to cover the full classical control problem, it is not\nsufficient in the quantum case. This is because, in general, the\nquantum measurement process changes the dynamics of the system.\nConsequently the formulation of the full quantum feedback control\nproblem must also allow for the possibility that the measurement\nprocess is also changed as a result of the observations. There are two\ndistinct possibilities for the modification of the measurement. The\nfirst is to control the coupling between the system and the bath (i.e.\nchange the operator Q) and we might refer to this as altering the\nmeasured observable, or altering the measurement interaction. The\nsecond is that even for a fixed system-environment coupling one can\ncontrol the nature of the measurements made on the bath. Since in this\ncase the master equation describing the unconditional evolution\nremains the same, but the trajectories change, we may may refer to\nthis as altering the measurement unraveling. Such adaptive measurements~\\cite{adapt} may have distinct advantages in the \nsetting of quantum control.\n\n%One example of varying the coupling operator such that $Q=Q(D)$ in the\n%case of measuring the light emerging from an optical cavity is\n%altering the cavity decay rate on the time-scale of the system\n%dynamics. This, while possible in principle, is not achievable in\n%current experiments. However it is worth considering this possibility\n%since there are models for which the full dynamics of the system may\n%be approximated in some limit by the time varying decay rate of an\n%oscillator. In particular Parkins and Kimble have developed a model of\n%an ion trapped inside a high finesse optical cavity in the strong\n%coupling regime that reduces to the decay of the oscillator motion at\n%a rate which is readily controllable by the modulation of a laser that\n%is driving the ion through the side of the cavity~\\cite{PK}. Moreover\n%in the important example of continuous interferometric measurement of\n%position the coupling of the system to the field and therefore the\n%sensitivity of the measurement is typically a function of some laser\n%power~\\cite{MJW} and this may also be varied during the course of the\n%experiment. It may be possible to develop experimentally relevant\n%models in which more complicated modifications of the coupling\n%operator $Q$ are possible.\n\nIn a general feedback scheme, the three tools of control (the\nHamiltonian, the measured observable and the measurement unraveling)\nare chosen to be some integral of the measurement record. In\nparticular, for state-observer based control, at each point in time\nthey are chosen to be a function of the best estimate of the state of\nthe system at that time (which is also, naturally, an integral of the\nmeasurement record). Note that in the situation considered by Wiseman\nand Milburn it is only the measurement result at the latest (most\nrecent) time which is used in the feedback. This\nleads to various complications since the feedback must always act\nafter the measurement and so it is necessary to be very careful of\nthis ordering when deriving stochastic master equations. It is\nimportant to note that so long as the kernel of the integral of the\nmeasurement record is not singular and concentrated at the latest\ntime, these complications do not arise (for the same reason that they\ndo not arise in classical control theory). Certainly, the integral\nrequired to obtain the optimal state-estimate is not singular (since\nit results from integrating the SME), and this remains true in all\ncases we consider here (such as the sub-optimal strategy in\nSection~\\ref{subopt}).\n\nWith the addition of feedback the various terms in the SME are in general\nfunctionals of the measurement record up to the latest time $t$. \n%The general feedback SME therefore has the form\n%\\begin{eqnarray}\n% d\\rho_{\\text{c}} & = & [H(\\{dy\\}),\\rho_{\\text{c}}] dt\n% + \\sum_i {\\cal D}[c_i(\\{dy\\})] \\rho_{\\text{c}} + \n% {\\cal H}[c_i(\\{dy\\})]\\rho_{\\text{c}} dW_i \\nn \\\\\n% & & .\n%\\end{eqnarray}\nIn general, this new SME is not Markovian. However, in the special\ncase in which the tools of control are chosen to be a function of the\noptimal state-estimate (i.e. $\\rho_{\\text{c}}(t)$), it follows\nimmediately that this SME is Markovian. Since it follows from the\nQuantum Bellman equation (derived below) that the optimal control\nstrategy may always be achieved when using a function of the best\nestimate, it follows that the optimal control strategy can always be\nachieved with an SME that is Markovian. The master equation that\nresults from averaging over the SME trajectories however, will in\ngeneral not be Markovian. In the Wiseman-Milburn scheme even the\nMarkovian nature of the master equation is preserved, but that is not\nthe case here.\n\n%In this way the operators $H$ will become adapted random processes.\n%They will depend only on the noise increments at earlier times and\n%thus the relevant quantum stochastic integrals will remain well\n%defined. The exact dependence will be termed the strategy, following\n%Belavkin, but for the moment we do not wish to specify it. We will\n%just assume that given the initial state and the output field the\n%operators $D_{k}$ are completely determined. A linear example is\n%\\begin{equation}\n% \\label{eq:control}\n% D_{k}(t)=\\int^{t}_{t_{0}} a(t,s)dX(s).\n%\\end{equation}\n%for some real function $a(t,s)$. The situation considered by the\n%Wiseman Milburn feedback theory is described by a singular $a$: $a =\n%\\delta(t-s)$. In this case the measured current is fed back only at\n%the instant it was observed, without the contribution of the current\n%at earlier times. This leads to various complications since the\n%feedback must always act after the measurement and so it is necessary\n%to be very careful of this ordering when deriving QSDE's or stochastic\n%master equations. However, in this case, so long as $a$ is not\n%singular and concentrated at $s=t$, it is in fact only the output\n%operators at earlier times which are affecting the dynamics of the\n%system and so it is not necessary to be concerned about such questions\n%here. It is not necessarily the case that this linear dependence on\n%the output field is the only form that is relevant. For example, if\n%the output currents are to some extent filtered into a finite\n%bandwidth there is not a problem with considering strategies which\n%depend non-linearly on the measured current. We will see below that\n%the optimal strategy may always be expressed in a useful fashion as\n%some function of the optimal state estimate.\n\n%When it comes to considering the SME for the state of the system\n%conditioned on the history of the measured photocurrent, the\n%projection of the output field onto particular values will also lead\n%to the projection of the control operators onto specific real number\n%values. Thus the operators $D_{k}(t)$ in the QSDE will be replaced by\n%real number noise processes which are correlated with the measured\n%photocurrent$I_{[t_{0},t)}$. This will be the case because, as noted\n%above, the operators $D_{k}$ will all commute with the output field\n%quadratures and so for each measurement current $I_{[t_{0},t)}$ the\n%control operators will themselves be projected onto particular\n%eigenstates which are completely determined for any given strategy by\n%the history of the measurement current and the initial state of the\n%system. We will label the resulting classical noise processes $d_{k}$.\n%Since the unraveling may be changed as a result of the measurement, in\n%general the SME will be driven by Wiener and Poisson noise, and SME\n%will be of the form\n%\\begin{eqnarray}\n% d\\rho_{\\text{c}} & = & {\\cal L}_{0}(\\{d_k\\})\\rho_{\\text{c}} dt\n% + \\sum_i {\\cal L}_{1i}(\\{d_k\\})\\rho_{\\text{c}}dW_i %\\nn \\\\\n% & & + \\sum_i %{\\cal L}_{2i}(\\{d_k\\})\\rho_{\\text{c}}dN_i.\n%\\end{eqnarray}\n%where the ${\\cal L}_{1i}$ and ${\\cal L}_{2i}$ are non-linear\n%superoperators.\n\n\\subsection{Quantum Optimal Control: the Quantum Bellman Equation}\n\nClassically, the optimal control problem can be written in a form\nwhich is, at least in principle, amenable to solution via the method\nof dynamic programming (to be explained below). This form is \ncalled the Bellman Equation, and\none can also write an equivalent quantum Bellman equation. This was\nfirst done by Belavkin~\\cite{BelBell,BelavkinLQG,Belavkin99}, but since the\ntreatment in~\\cite{BelBell} is very abstract, and since neither\noptimization over unravelings, nor the possibility of ensemble\ndependent cost functions were mentioned there, we feel it worthwhile\nderiving this equation here using a simpler, although less rigorous\nmethod.\n\nTo define an optimal control problem we must specify a cost function\n$f(\\rho (t),u(t),t)$, which defines how far the system is from the\ndesired state, how much this `costs', and how much a given control\n`costs' to implement. The problem then involves finding the control\nwhich minimizes the value of the cost function integrated over the\ntime during which the control is acting. The important point to note\nis that the cost function can almost always be written as a function\nof the conditional density matrix followed by an average over\ntrajectories. This is because the density matrix determines completely\nthe probabilities of all future measurements that can be made on the\nsystem, and consequently captures completely the future behavior of\nthe system as far as future observers are concerned (given that the\ndynamics are known, of course), which is what one almost always wants\nto control.\n\nThe possible exceptions to this rule come when one is interested in\n preserving or manipulating unknown information which has been encoded \nin the system by a previous observer who prepared it in one of a known \nensemble of states. Thus as far as the second observer is concerned the \nstate of the system is found by averaging over these states with the \nweighting appropriate to the ensemble. However, in this case it may \nwell be sensible to use a cost function that depends on the ensemble \nas well as this density matrix~\\cite{fuchs}. It remains a topic for \nfuture work to determine whether problems such as this will constitute \nan important application of quantum feedback control. We will restrict \nourselves here to what might be referred to as `orthodox' control \nobjectives in which it is only the future behavior of the system which \nis important, and this is captured by cost functions which depend only \non the density matrix (ensemble independent cost functions).\n\nThe general statement of our optimal control problem may therefore be\nwritten as\n\\begin{eqnarray}\n {\\cal C} = \\left< \\int_0^T f(\\rho_{\\text{c}}(t),u(t),t) dt +\n f_{\\text{f}}(\\rho_{\\text{c}}(T),T) \\right> . \n\\end{eqnarray}\nHere ${\\cal C}$ denotes the total average cost for a given control\nstrategy $u(t)$, $f$ is the cost function up until the final time $T$, $f_{\\text{f}}$ is the cost function associated with the final state, \nand $\\langle\\ldots\\rangle$\ndenotes the average over all trajectories. The solution is given by\nminimizing ${\\cal C}$ over $u(t)$, to obtain the minimal cost ${\\cal\nC}^$, and resulting optimal strategy $u^(t)$. Note that the values\nof $u$ will be different for different trajectories. In this\nformulation a cost is specified at each point in time, with the total\ncost merely the integral over time, and an allowance is explicitly\nmade for extra weighting to be given to the cost of the state at the\nfinal time. It is crucial that the cost function takes this `local in\ntime' form in order that it be rewritten as a Bellman equation.\n\nTo derive the quantum Bellman equation we will consider the problem to\nbe discrete in time, since this provides the clearest treatment. In\nany case the continuous limit may be taken at the end of the\nderivation, if the result is desired. In this case, dividing the\ninterval $[0,T]$ into N steps, the cost function consists of a sum of\nthe costs at times $t_i=t_1,\\ldots,t_{N+1}$, with $t_{N+1}=T$ denoting\nthe final time. The idea of dynamic programming (which results from \nthe Bellman equation) is that if the period\nof control is broken into two steps, then the optimal control during\nthe second step must be the control that would be chosen by optimizing\nover the later time period alone given the initial state reached after\nthe first step. This allows the optimal control to be calculated from\na recursion relation that runs backwards from the final time, or in the\ncontinuous-time case from a backwards time differential equation. To \nderive the Bellman equation one proceeds as follows.\n\nTrivially, at the final time, given the state $\\rho(T)$, the minimal\ncost is merely the final cost, so ${\\cal C}^(t_{N+1}) =\nf_{\\text{f}}(\\rho(T),T)$. Next, stepping back to the time $t_{N}$, \nthe total cost-to-go, given the state $\\rho(t_N)$ is\n\\begin{eqnarray}\n {\\cal C}(t_N) & = & f(\\rho_{\\text{c}}(t_N),u(t_N),t_N) \\Delta t \\\\\n& + & \\int f_{\\text{f}}(\\rho(T),T)\nP_{\\ms{c}}(\\rho(T)\|\\rho_{\\text{c}}(t_N),u(t_N)) d\\rho(T) \\nn \n\\end{eqnarray}\nwhere $P_{\\ms{c}}$ is the conditional probability density for the\nstate at time $T$ given the state $\\rho_{\\text{c}}(t_N)$, which is\nconditioned on any earlier measurement results and controls, and the\ncontrol $u(t_N)$ at time $t_N$, so that the integral is simply the\nconditional expectation value of the cost at the final time. Note that\nthe choice of the control $u(t_N)$ may depend on the measurement\nresult at $t_N$ and that the conditional probability density is\nconditioned not only on the chosen value of $u(t_N)$ but also on the\nmeasurement result at $t_N$. Since, $f_{\\text{f}}(\\rho(T),T)$ is ${\\cal\nC}(t_{N+1})$, we have\n\\begin{eqnarray}\n {\\cal C}(t_N) & = & \\min_{u(t_N)} \\biggl[\n f(\\rho(t_N)_{\\text{c}},u(t_N),t_N) \\Delta t\n \\biggr.\\label{bell1} \\\\\n & + & \\biggl. \\int {\\cal C}(t_{N+1}) P_{\\ms{c}} (\\rho(t_{N+1})\|\\rho(t_N)_{\\text{c}},u(t_N)) d\\rho(t_{N+1}) \\biggr] \\nn\n\\end{eqnarray}\nThe important step comes when we consider the total cost-to-go at the\nthird-to-last time $t_{N-1}$. This time there are three terms in the\nsum. Nevertheless, using the Chapman-Kolmogorov equation for the\nconditional probability densities, it is straightforward to write the\nequation for ${\\cal C}(t_{N-1})$ in {\\em precisely} the same form as\nthat for ${\\cal C}(t_N)$: it is simply Eq.~(\\ref{bell1}) with $N$\nreplaced with $N-1$. In fact, this equation holds for {\\em every}\n${\\cal C}(t_i), i=1,\\ldots,N$.\n\nFrom this point, the crucial fact that results in the Bellman equation\nis this: since the conditional probability densities are positive\ndefinite, it follows that the minimum of ${\\cal C}(t_i)$ is only\nobtained by choosing ${\\cal C}(t_{i+1})$ to be minimum. We can\ntherefore write a backwards-in-time recursion relation for the minimum\ncost, being \n\\begin{eqnarray} {\\cal C}^(t_i) & = & \\min_{u(t_i)}\n\\biggl[ f(\\rho_{\\text{c}}(t_i),u(t_i),t_i) \\Delta t\n\\biggr. \\label{bellq} \\\\ & + & \\biggl. \\int {\\cal C}^*(t_{i+1})\nP_{\\ms{c}} (\\rho(t_{i+1})\|\\rho_{\\text{c}}(t_i),u(t_i)) d\\rho(t_{i+1})\n\\biggr] \\nn ,\n\\end{eqnarray}\nwhich is the discrete time version of the Bellman equation. In words,\nthis states that an optimal strategy has the property that, whatever\nany initial states and decisions, all remaining decisions must\nconstitute an optimal strategy with regard to the state that results\nfrom the first decision, which is referred to as the `optimality\nprinciple'. \n\nThe quantum Bellman equation confirms the intuitive result that any\noptimal quantum control strategy concerned only with the future\nbehavior of the system is a function only of the conditional density\nmatrix, and further, that the strategy at time $t$ is only a function\nof the conditioned density matrix at that time.\n\nThe procedure of stepping back through successive time steps from the\nfinal time to obtain the optimal strategy is referred to as dynamic\nprogramming. This could be used, at least in principle, to solve the\nproblem numerically. In practice it will be useful to employ some\napproximate strategy. Much progress in this direction has been made\nfor closed quantum systems, see for example Ref.~\\cite{botina1997a}.\n\n%Exact solutions of the Bellman equation have been found for certain\n%special cases. \n\n% Could say something about solutions here.\n% Also about the quantum problem being more complex because change\n% unraveling. \n\n\\section{Classical Analogies for the Quantum Control Problem}\n\nIn the preceding sections we have examined the conceptual mappings\nbetween the elements of the classical and quantum control problems. In\nthis section we want to examine the possibility of making such a\nmapping precise. That is, to address the question of if and when it is\npossible to model a given quantum control problem exactly as a\nclassical control problem. When this is possible it allows the quantum\nproblem in question to be solved using the relevant classical methods.\n\nOne can always formulate a given quantum control problem using the\nquantum Bellman equation, but the different cost functions will be\nmotivated by different control objectives, and to formulate an\nequivalent classical control problem we should examine these objects\nof control. For example, as the object of control one might focus on\nthe expectation values of a set of observables, the state-vector of\nthe quantum system, or the entire set of density matrix elements\ndescribing one's state of knowledge. Once we have a vector of\nquantities to control, we can ask whether, if we identify this set of\nquantities with the classical object of control (being the system\nstate vector ${\\bf x}$), there exists an identical classical control\nproblem. In what follows we examine when this can be achieved for the\nthree objects of control we have mentioned.\n\n\\subsection{Correspondence using physical observables}\n\nIn this case we wish to control a vector consisting of the expectation\nvalues of a set of observables (or, more precisely, the {\\em\nconditional} expectation values of a set of observables). To formulate\nan equivalent classical problem we identify these with the conditional\nexpectation values of the classical vector ${\\bf x}$, and ask whether\nthere exists a classical problem corresponding to a given quantum\nproblem. It is immediately clear that in general there will not be,\nbecause the conditional joint probability density (e.g.\\ the Wigner\nfunction) for the quantum observables will in general not be positive\ndefinite, while the classical equivalent is forced to be. However, it\nturns out that whenever both the quantum dynamics and the measurement\nis linear in the observables, and the measurement process (unraveling)\nis Gaussian, there exists an identical linear classical problem driven\nby Gaussian noise, and therefore the quantum problem reduces to a\nclassical one. This is possible because in this case the quantum\ndynamics preserves the positivity of the joint conditional probability\ndensity.\n\nThe simplest example of this is the quantum single particle in a\nquadratic potential. The equivalent classical control problem is that\nfor a single classical particle subject to the same potential, driven\nby Gaussian noise, and with an imperfect measurement on whatever\nobservable is being measured in the quantum problem. Because it is the\nexpectation values of quantum observables which correspond physically\nwith the classical dynamical variables ${\\bf x}$, we can denote this\nformulation as using a physical correspondence between the quantum and\nclassical systems. Because the equivalent classical problem is linear,\nit provides immediately an analytic solution to the optimal quantum\ncontrol problem for those cost functions for which solutions have been\nfound for the classical problem. Solutions exist for cost functions\nthat are quadratic in the classical variables (the so-called\nLinear-Quadratic-Gaussian (LQG) theory) and also those exponential in\nthe variables (Linear-Exponential-Gaussian (LEG) theory). A detailed\ntreatment of this analogy, and the resulting quantum LQG theory is\ngiven in Ref.~\\cite{DJ}, and a rigorous mathematical treatment using a\ndifferent approach may be found in Ref.~\\cite{BelavkinLQG}.\n\nAn interesting feature of this quantum-classical control analogy is\nthat for {\\em non-linear} quantum systems it transforms smoothly from\na quantum control problem (not amenable to a classical formulation) to\na classical control problem across the quantum-to-classical\ntransition: from a number of numerical studies, it is now clear that\ncontinuously observed quantum systems behave as classical systems in\nthe classical regime (even in the absence of any source of decoherence\nother than the measurement process)~\\cite{Percival}. By the\nclassical regime we mean the regime in which macroscopic objects\nexist, with $\\hbar$ small compared to the classical action, and this\ntherefore provides an explanation for the emergence of classical\nmechanics from quantum mechanics. This has an immediate connection to\nthe problem of feedback control in quantum systems since feedback\ncontrolled systems are observed systems (and the ones we are\ninterested in here are continuously observed). Since it is the\nexpectation values of the physical observables which behave as the\nclassical observables in the classical regime, in this regime the\nabove procedure will provide an effective equivalent classical control\nproblem. Effective non-linear classical control strategies will\ntherefore work in the classical regime, and a natural question to ask\nis then how they perform as the system makes the quantum-to-classical\ntransition, and especially, whether such classical control strategies\nwill still work deep in the quantum regime. We explore this question\nin Section~\\ref{subopt}.\n\n\\subsection{Correspondence using the quantum state vector}\n\nIn this case it is the quantum state-vector $\|\\psi\\rangle$ which is\nthe object of control, and so we wish to see whether we can form an\nequivalent classical problem with ${\\bf x}$ identified as the state\n$\|\\psi\\rangle$. In the classical case our state of knowledge is\ndescribed by the probability density, $P({\\bf x})$, so that in order\nto pursue a classical formulation we must consider a probability\ndensity over the states, $P_{\\ms{q}}(\|\\psi\\rangle)$. However, there\nare important differences between the roles of $P$ and $P_q$. While in\nthe classical case a complete knowledge of $P$ is required to predict\nthe results of measurements performed on the system, in the quantum\ncase it is only the density matrix which is required, being the set of\nsecond moments of $P_q$:\n\\begin{equation}\n \\rho = \\int d\|\\psi\\rangle P_{\\ms{q}}(\|\\psi\\rangle)\n \|\\psi\\rangle\\langle\\psi\| ,\n\\end{equation}\nTwo important consequences of this are the following. First, that\nbecause it is only the set of second moments that characterize our\nstate of knowledge, many different densities $P_q$ may be chosen to\ncorrespond to this state of knowledge, and in particular, these can\nhave different modes or means. Since the classical best estimate is\nusually defined as a mode (maximum a posteriori estimator) or a mean,\nwe must immediately conclude that there is no quantum `best estimate'\nfor the state vector in the classical sense. Referring back to Section\n\\ref{CFC} then, it follows that there are no separable quantum control\nproblems when it is the state-vector that is the object of control.\nNevertheless, this does not rule out the possibility that it might be\nuseful to construct definitions of quantum `best estimates' for the\nstate vector in the development of sub-optimal control laws.\n\nSecond, because the equation which propagates our state of knowledge\nis an equation for the density matrix, the quantum problem\nautomatically has moment closure. In general, the term moment closure\nmeans that the equation for the evolution of some finite set of\nmoments of the conditional probability density can be written only in\nterms of themselves, without coupling to the infinite set of higher\nmoments. In a sense, this fact introduces a simplification into the\nquantum problem.\n\nTo obtain a classical model one requires that there exists a noise\ndriven classical system, with state vector ${\\bf x}$, such that the\nequation of motion for ${\\bf x}$, along with the continuous\nobservation, whatever it may be, gives a conditional probability\ndensity, the second moments of which obey the quantum SME. We now\npresent strong evidence to suggest that this is, in fact, not\npossible. That is, there exists {\\em no} observed classical system\nthat reproduces the SME, and consequently it is not possible to think\nof the quantum measurement process as a classical estimation process\non the state vector. Note that this is not directly connected to the\nHeisenberg uncertainty principle: the quantum state vector can be\ndetermined completely during the observation process, just as can the\nclassical state. Nevertheless, the processes are fundamentally\ndifferent.\n\nTo see this first consider the equation for the second moments that\nresults from the the KS equation (Eq.~(\\ref{KSE})), for time \ninvariant linear observations on a time invariant linear system. \nIn this case ${\\bf F} = {\\cal F}{\\bf x}$, \n${\\bf H} = {\\cal H}{\\bf x}$ and ${\\cal F}$, \n${\\cal H}$, ${\\cal G}$ and ${\\cal R}$ are constant matrices. \nThe equation for the second moments may be written\n\\bqa\ndC & = & [C {\\cal F}^\\dagger + {\\cal F} C] - {\\cal GG}^\\dagger dt + \\\\\n& + & \\langle {\\bf x} \\rangle {\\bf dW}^\\dagger \\sqrt{{\\cal RR}^T}{\\cal H}\n(C - \\langle {\\bf x}\\rangle \\langle {\\bf x}^\\dagger\\rangle) \\nn \\\\\n& + & (C - \\langle {\\bf x}\\rangle \\langle {\\bf x}^\\dagger\\rangle)\n{\\cal H}^\\dagger \\sqrt{{\\cal RR}^T} {\\bf dW} \\langle {\\bf x}^\\dagger\\rangle \\nn \n\\eqa\nwhere $C = \\langle {\\bf x} {\\bf x}^\\dagger\\rangle$ is the matrix of\nsecond moments. While the terms involving $F$ reproduce the commutator\nfor the Hamiltonian evolution of the density matrix (with the choice\n$F = -iH$), as expected, the deterministic and stochastic terms\nresulting from the observation are quite different. In particular, the\ndeterministic part is constant (i.e. not a function of $C$), and the\nstochastic part depends upon the first moments. The first moments\nthemselves obey a stochastic equation, where the deterministic part is\ngiven by $F$. We therefore cannot choose a linear classical estimation\nproblem directly equivalent to the quantum problem. If we consider\nclassical systems with non-linear deterministic dynamics, then the\ndeterministic motion fails to match the quantum evolution, which is\nstrictly linear. If one chooses the noise or the measurement process\nto be non-linear, then, in general, the moment closure is lost.\n\nWe can gain some insight into the difference between quantum and\nclassical estimation by considering the change in the quantum\nprobability density, $P(\|\\psi\\rangle)$, upon the result of a\nmeasurement. Given a measurement described by the POVM $\\sum\\Omega_y\n\\Omega_y^\\dagger$, and an initial density matrix $\\rho$, the\npost-measurement density matrix is given by $\\rho' = \\Omega_y \\rho\n\\Omega_y^\\dagger/{\\rm Tr}(\\rho \\Sigma_y^{\\dagger}\\Sigma_y)$. Writing\nthis in terms of $P(\|\\psi\\rangle)$, we have the post measurement\ndensity for result $y$ as\n\\begin{equation}\n P'(\|\\psi_y\\rangle)\n = \\frac{1}{N}P(y\|\|\\psi\\rangle)P(\|\\psi\\rangle) \\left\|\n \\frac{d\|\\psi_y\\rangle}{d\|\\psi\\rangle}\\right\| \\end{equation} where \\begin{equation}\n \|\\psi_y\\rangle = \\frac{\\Omega_y\|\\psi\\rangle}{\\sqrt{\\langle \\psi\n \|\\Omega_y^\\dagger\\Omega_y\|\\psi \\rangle}}\n\\end{equation}\nand $P(y\|\|\\psi\\rangle)$ is the conditional probability for the result\n$y$ given the state $\|\\psi\\rangle$, with $N$ a normalization. In\ncontrast to this, the classical result is simply Bayes' rule, being\n\\begin{equation}\n P'({\\bf x}) = \\frac{1}{N} P({\\bf x})P(y\|{\\bf x})\n\\end{equation}\n\nWe see that the quantum result is Bayes rule, with the addition of a\nnon-linear transformation of the states, since if we set\n$\|\\psi_y\\rangle = \|\\psi\\rangle$ for all $\|\\psi\\rangle$ in the quantum\nrule, we recover the classical Bayes rule. This is the sense in which\nwe can view the quantum measurement process as an active process,\nsince it is equivalent to a classical (passive) measurement process,\nwith the addition of an (active) transformation of the states.\n\n\\subsection{Correspondence using the density matrix}\n\nIn this case one considers the elements of the (conditional) density\nmatrix as the vector to control. Since the density matrix\ncharacterizes our state of knowledge, by definition we always know\nwhat it is. Consequently the SME becomes the fundamental dynamical\nequation, and there is no longer any estimation in the control\nproblem. This is exactly analogous to considering the conditional\nprobability density of the classical control problem as the object of\ncontrol. Since there is no estimation the control problem is\nautomatically a classical one, and all the techniques of classical\ncontrol theory can be applied. However, the problem is necessarily\nnon-linear since the SME is non-linear.\n\n\\section{Observability and controllability}\n\nObservability and controllability are two key concepts in classical\ncontrol theory, and here we want to examine ways in which they may be\nextended to the quantum domain. They are useful because they indicate\nthe existence of absolute limits to observation and control in some\nsystems. If it is not possible to completely determine the state of a\nsystem given a chosen measurement or to prepare an arbitrary state of\nthe system given the chosen control Hamiltonian then this will place\nsevere limitations on the feedback control of that system. It is\nimportant to note that the definitions of observability and\ncontrollability apply classically to noiseless systems (that is,\nsystems with neither process nor measurement noise), although they are\nrelevant for stochastic systems, and it is these systems in which we\nare naturally interested here.\n\nConsider the concept of observability. A system is defined to be\nobservable if the initial state of the system can be determined from\nthe time history of the output (i.e. the measurements made on the\nsystem from the initial time onwards)~\\cite{RAOC}. It follows that in\nan observable system, {\\em every} element in the (classical)\nstate-vector affects at least one element in the output vector, so\nthat the relation can be inverted to obtain the initial state from the\noutputs. If one considers adding process and measurement noise, then\nobservability is still a useful concept, because it tells us that the\noutputs, while corrupted by noise, nevertheless provide information\nabout {\\em every} element in the state-vector. Consequently, given\nimprecise initial knowledge of the state, we can expect our knowledge\nof all the elements to improve with time. For an unobservable system,\nthere will be at least one state element about which the measurement\nprovides no information. The simplest example of this is a free\nparticle in which the momentum is observed. Since the position never\naffects the momentum, any initial uncertainty in the position will not\nbe reduced by the measurement. Note that observability is a joint\nproperty of a system and the kind of measurement that is being made\nupon it.\n\nIt is interesting that there are at least two inequivalent ways in\nwhich this concept of observability may be applied to a measured\nquantum system, and these result from the choice of making an analogy\neither in terms of the quantum state-vector, or a set of quantum\nobservables. First consider observability defined in terms of a set\nof observables. The concept of observability applies in this case to\nwhether or not the output contains information about all the physical\nobservables in question. A simple example once again consists of the\nsingle particle, in which we can use the position and momentum as the\nrelevant set of observables. If we consider the observation of the\nposition, then the system is observable: the output contains\ninformation about both the position and momentum since the momentum\ncontinually affects the position. As a result a large initial\nuncertainty in both variables is reduced during the\nobservation. Naturally this is eventually limited by the uncertainty\nprinciple. The conditioned state may eventually become pure but there\nwill be a finite limiting variance in the measured quantity since this\nstate must obey the uncertainty relations. In linear systems the\nmeasurement back action noise has a role rather similar to process\nnoise in a classical system since process noise also leads to non-zero\nlimiting variances of the measured property of the state. This kind of\nbehavior is discussed in Ref.~\\cite{purity}.\n\nIf we consider alternatively the measurement of momentum on a quantum\nfree particle, the system is unobservable, in exactly the same fashion\nas the classical system is unobservable, since the momentum provides\nno information about the position. It is not entirely coincidental\nthat in quantum mechanics momentum is a Quantum Non-Demolition (QND)\nobservable of the free particle while classically momentum measurement\nof a free particle does not constitute an observable system. This is\nclearly a general result: when it is a QND observable that is\nobserved, the system is always unobservable. This follows from the\nfact that a QND observable is defined as one that commutes with the\nHamiltonian. Since it commutes with the Hamiltonian, no other system\nobservable can appear in its equation of motion, with the result that\nits observation can provide no information about any other\nobservable. There will however be measurements on systems which while they are not\nclassically observable are also not QND measurements.\n\nAn alternative way to define quantum observability is in terms of the\nstate-vector. In this case the question of observability concerns\nwhether or not the output contains information about all the elements\nof the quantum state vector. Consider a quantum system in which the\nobservation is the only source of noise. Then, if the system is\nobservable with respect to a particular measurement, as time proceeds\none obtains increasingly more information about all the elements of\nthe state vector, and the conditioned state tends to a pure state as\n$t\\rightarrow\\infty$. For an unobservable system, any initial\nuncertainty in at least one state vector element remains, even in the\nlong time limit. A simple example of a system that is observable in\nthis sense is the measurement of momentum on a free particle (recall\nthat this is {\\em unobservable} in the previous sense). In this case\nit is a simple matter to calculate the time evolution of the purity of the conditioned state\n(using, for example, the method in Ref.~\\cite{EvOp}), to verify that\nthe system is observable. An example of an unobservable system is a\nset of two non-interacting spins, in which it is an observable of only\none of the spins that is measured. In this case, while the state of\nthe measured spin may become pure, clearly the state of the joint\nsystem can remain mixed for a suitable choice of initial state.\n\nA key factor which differs between these examples is that in the\nobservable case the measured quantity (being the momentum) has a\nnon-degenerate eigenspectrum, whereas in the unobservable case the\nmeasured quantity (being any observable of the first spin) has\ndegenerate eigenvalues when written as an operator on the full (two\nspin) system. It is clear that in the case that the measured\nobservable commutes with the system Hamiltonian the non-degeneracy of\nthe eigenvalues of the observable is a necessary and sufficient\ncondition for observability in this sense. Writing the evolution of\nthe system as multiplication by a series of measurement operators\nalternating with unitary operators (due to the Hamiltonian evolution),\nthe measurement operators may be combined together since they commute\nwith the unitary operators, and it is readily shown that as\n$t\\rightarrow\\infty$, one is left with a projection onto the basis of\nthe measured observable. If the eigenvalues of the observable are all\ndifferent, then the measurement results distinguish the resulting\neigenvector, and the result is a pure state. However, if any two of\nthe eigenvectors are degenerate, the measurement results will not\ndistinguish those two states. Consequently, if the system exists\ninitially in a mixture of these two states it will remain so for all\ntime. Whether this continues to be true in the general case remains an\nopen question.\n\n%For the general case, in which the measured observable does not\n%commute with the Hamiltonian, there are compelling reasons to believe\n%that a measurement by an observable with a non-degenerate\n%eigenspectrum always generates a pure state in the long time limit,\n%providing observability. To see why this is so consider again the\n%evolution of a measured system as an alternating series of operations\n%by measurement operators and unitary operators~\\cite{EvOp}. First note\n%that the measurement operators, when acting on any state, cannot on \n%average decrease the purity of the state~\\cite{entangle}.\n%Next one notes that the unitary operators due to the Hamiltonian\n%cannot alter the purity, and hence cannot undo the change in purity\n%effected by the measurement operators. Finally, since in the absence\n%of the Hamiltonian the system is reduced to a pure state for {\\em all}\n%initial states, all states of the system are `vulnerable' to the\n%action of the measurement; therefore, no matter what the Hamiltonian\n%evolution, it cannot rotate the system in such a way as to `protect'\n%it from the action of the measurement.\n\nWe need not consider controllability in any detail here, since this\nhas been considered elsewhere. The controllability of quantum\nmechanical systems --- that is, whether the interaction Hamiltonians\navailable are able to prepare an arbitrary state of a quantum system\n--- has been considered by applying directly the ideas of classical\ncontrol theory \\cite{huang1983a}. Interestingly, this has a new\ninterpretation in quantum computation. The gates of the computer must\nbe able to perform an arbitrary unitary operation on the register of\nqubits; a set of gates with this property is termed universal. Since\nit may perform arbitrary unitary operations a universal quantum\ncomputer may prepare any desired state of the system from any given\ninitial state. The conditions for controllability of a quantum system\nwere therefore rediscovered as the conditions for universality of a\nquantum computer \\cite {lloyd1995a}.\n\n%CONTROLLABILITY -- REALLY NEED ONLY CONSIDER CLOSED SYSTEMS HERE, DONE\n%BY CONTROL THEORISTS A WHILE AGO, DONE BY QUANTUM COMPUTER PEOPLE\n%RECENTLY (A UNIVERSAL QUANTUM COMPUTER IS JUST A CONTROLLABLE QUANTUM\n%SYSTEM, FINITE AND INFINITE HILBERT SPACES HAVE BEEN CONSIDERED)\n\n\\section{Sub-optimal estimation and control for a non-linear quantum\nsystem} \n\\label{subopt}\n\nHere we examine the application of sub-optimal estimation and control\nlaws, developed for non-linear classical systems, to the corresponding\nquantum systems, where the objects of control are the expectation\nvalues of physical observables. This gives a simple initial example of\nthe use of state observer based control systems outside of the regime\nof linear systems considered in Ref.~\\cite{DJ}. Since, for this\nparticular control objective, it is possible to completely solve the\nproblem of the feedback control of linear quantum systems using\nclassical methods for linear systems, and since continuously observed\nnon-linear quantum systems in the classical regime are clearly\namenable to classical control strategies, it remains to examine the\neffectiveness of classical non-linear control strategies for quantum\nsystems deep in the quantum regime. For non-linear systems, optimal\nestimation involves integration of the KS equation for classical\nsystems, and the SME for quantum systems. For real time control this\nis almost always computationally impractical, so that it is important\nto develop simpler (sub-optimal) algorithms which are sufficiently\naccurate.\n\nIt is important to note that the use of a sub-optimal estimation\nalgorithm also makes the task of simulating the controlled quantum\nsystem computationally less expensive. This is because it allows the\nsystem, including control, to be simulated using an SSE rather than\nthe full SME. The reason for this is that regardless of whether the\nobserver is dynamically changing the inputs to the system the SSE\ncorrectly simulates the SME --- the full SME need only be integrated\nif the actual conditioned state is required to calculate the sequence\nof controls. As a result, to simulate a controlled quantum system, one\nneed only integrate the sub-optimal estimator, if one is available,\nand the SSE for the system.\n\nHere we use as an example system a particle in a double well potential\nwith the control objective of keeping the particle in a given well,\nand switching it from one well to the other when desired, in the\npresence of a coupling to an (infinitely) high temperature bath. As\ndiscussed in previous sections, the first important choice in such a\nproblem is that of the measurement, as this should be chosen so as not\nto cause any unwanted dynamics (i.e. it should not force the particle\naway from the desired states) and since it is the position of the\nparticle that is to be controlled, a position measurement is a\nsensible choice.\n\nVarious approximate estimators have been developed for classical\nsystems, and these usually involve a moment truncation of the KS\nequation. For example, one can assume that the conditional probability\ndensity will remain Gaussian, and truncate the moments accordingly.\nMore generally, for a given control problem certain characteristics of\nthe conditional probability density might be known, and motivate\nanother approximation. In both the classical and the quantum\nmechanical systems it is a reasonable expectation that the conditioned\nstates will remain Gaussian for sufficiently strong position\nmeasurement which is the regime we will investigate here.\n\nFor the purposes of feedback control we will assume that the observer\nhas the ability to apply a linear force to the double well, so the\nfeedback Hamiltonian is proportional to $x$. When the quantum state is\nclose to Gaussian, quantum dynamics follows closely the equivalent\nclassical dynamics, and we can expect non-linear classical control\nstrategies to work. The strategy we will apply is that of linearized\nLQG optimal control. In this method, for each time-step, the system\ndynamics are linearized about the current state-estimate, and the\ncorresponding optimal LQG strategy is chosen for the next\ntime-step. In this way the control is always `locally optimal'.\nClearly the key requirement for the strategy we have outlined is that\nthe conditioned state remains closely Gaussian during the\nevolution. The control will fail if the measurement fails to maintain\nthe Gaussian distribution, or if the measurement only maintains a\nGaussian at the expense of introducing an intolerable amount of noise.\n\nThe Hamiltonian for the system is\n\\begin{equation}\n H = \\half p^2 - A x^2 + B x^4,\n\\end{equation}\nwhere we have set the particle mass to unity. We will also use\n$\\hbar=1$. The resulting SME is\n\\begin{eqnarray}\n d\\rho_{\\text{c}} & = & -i[H + H_{\\ms{fb}},\\rho_{\\text{c}}]dt\n + 2\\beta {\\cal D}[x]\\rho_{\\text{c}}dt \\nonumber \\\\\n & & + 2k{\\cal D}[x]\\rho_{\\text{c}}dt +\n \\sqrt{2k}{\\cal H}[x]\\rho_{ \\text{c}}dW. \\label{twellsme}\n\\end{eqnarray}\nwhere $k$ gives the strength of the position measurement, and $\\beta$\nthe strength of the thermal noise. On any given trajectory the\ncorresponding measured current is $I(t)=dQ(t)/dt$ where $dQ(t) =\n\\text{Tr}(x\\rho_{\\text{c}}(t))+dW(t)$. The feedback Hamiltonian is\n$H_{\\ms{fb}} = -ux$ where $u$ is a function of the history of the\nphotocurrent described below.\n\n\\begin{figure}\n\\centerline{\\psfig{file=twinfig.eps,width=3.25in,height=2.8in}}\n\\caption{\\narrowtext Behavior of a particle under the\nestimation/feedback control scheme outlined in the text. (a) The\ntarget position (blue line), the `true' mean position obtained from\nthe SSE simulation (red line), and the estimated position (magenta\nline). (b) The control strength (size of applied force) as a function\nof time. The various units are $X_{\\mbox{\\scriptsize s}} = \n\\sqrt{\\hbar/(m\\nu)}$, $u_{\\mbox{\\scriptsize s}} = \\nu\\sqrt{\\hbar m\\nu}$ \nand $\\tau = 1/\\nu$, where $m$ is the mass of the \nparticle and $\\nu$ is an arbitrary frequency. In the text we have set \n$\\hbar = m = \\nu = 1$, so that all quantities are dimensionless.}\n\\label{fig1}\n\\end{figure}\n\nThe estimator chosen is a variational solution of the SME: it is the\nGaussian state closest to the actual conditioned state which may be\nobtained by integrating the SME. This approach to the approximate solution of the SME appears in~\\cite{Hal}. This is a more realistic estimator\nfor use in control than the SME since it only requires the integration of five\nstochastic differential equations. The approximate solution is a\nGaussian mixed state which may be characterized by its mean position\n$\\langle x\\rangle$ and momentum $\\langle p\\rangle$ and symmetric\nsecond order moments $V_x,V_p,C$ the position and momentum variance\nand the symmetric covariance $C = (1/2)\\langle xp + px\\rangle -\n\\langle x \\rangle \\langle p\\rangle $ respectively.\n\\begin{eqnarray}\n \\label{eq:csme}\n d\\langle x\\rangle & = & \\langle p\\rangle dt + 2\\sqrt{2k} V_x dV\n \\\\ \n d\\langle p\\rangle & = & -4B\\langle x\\rangle^3dt + 2A\\langle\n x\\rangle dt - 12B\\langle x\\rangle V_x dt \\nn \\\\ \n & & + 2\\sqrt{2k}C dV + u dt, \\\\\n \\dot{V}_x & = & 2 C - 8k V_x^2 \\\\\n \\dot{V}_p & = & -24 B \\langle x\\rangle^2 C + 4 A C - 24 B C V_x \\nn\n \\\\ \n & & + 2 (k+\\beta) \\hbar^2 - 8 k C^2 \\\\\n \\dot{C} & = & V_p - 12 B \\langle x\\rangle^2 V_x + 2 A V_x \\nn \\\\\n & & - 12 B V_x^2 - 8 k C V_x\n\\end{eqnarray}\nwhere $dV=dQ -\\langle x\\rangle dt$. Thus from an initial state the\nobserver may propagate this Gaussian estimate of the true conditioned\nstate given a particular measurement record. Note that since the full\nSME is not in fact integrated the noise processes $dW$and $dV$ are not\nthe same. In our pure state trajectory simulations we perform the\nstochastic integration of Eq.~(\\ref{twellsme}) for different\nrealizations of the Wiener increments $dW$ that in turn determine, for\neach trajectory, values of $dQ$ that are used to integrate the five\nestimator equations. In order to obtain equations for pure states it\nis also necessary to introduce a second Wiener increment to account\nfor the thermal noise as described in Section~\\ref{sec:contqmeas}.\n\nThe state estimate is then used to determine the values of $u$. Under\nlinearized LQG control $u=u_1+u_2+u_3$ where\n\\begin{eqnarray}\n u_0 & = & 2 A \\langle x\\rangle - 4 B \\langle x\\rangle^3 \\\\\n u_1 & = & - \\tilde{u} (\\langle x\\rangle - x_0) \\\\\n u_2 & = & - (\\sqrt{2 \\tilde{u} + \\Gamma}) (\\langle p\\rangle - p_0)\n \\\\ \n \\tilde{u} & = & \\partial_{\\langle x \\rangle} u_0 +\n \\sqrt{[\\partial_{\\langle x \\rangle} u_0]^2 + \\Gamma}. \n\\end{eqnarray}\nThe current target points in phase space are $x_0$ and $p_0$. Here\n$\\Gamma$ is a `free' parameter which one chooses to set the overall\nstrength of the feedback.\n\nAs a particular example we choose $A=2$ and $B=A/18$, which puts the\ntwo minima at $\\pm 3$, with a well depth of $13.5$. Since we set\n$\\hbar=1$, this puts the problem deep in the quantum regime, since the\npotential varies considerably over the phase space area $\\hbar$.\nBecause of this, the density (Wigner function) for the particle is\nforced to be broad on the scale of the occupiable phase space, which\nis a key limiting factor in the problem. We choose $\\beta=0.1$, which\ngives a thermal heating rate $d\\langle E\\rangle/dt=0.1$. Due to the\nthermal heating, feedback control is essential to maintain a desired\nbehavior. In implementing the sub-optimal estimation and control\nstrategy described above, we have the choice of measurement strength\n$k$ and feedback strength $\\Gamma$. We find that it is possible to\nobtain a fairly effective control with a choice of $k=0.3$ and\n$\\Gamma=100$. A resulting trajectory for the system, given a target\nposition that switches between the well minima is shown in\nfigure~\\ref{fig1}, along with the strength of the linear force applied\nas a result of the control strategy. To evaluate the efficacy of the\ncontrol, we also plot the RMS deviation of the average position from\nthe target position, and plot this in Figure 2. We see from this that\nthe system achieves the target position within an average error of\n$\\pm 0.6$. When the target is switched, the system relaxes to the\ndesired value with a time constant of $\\sim 3$.\n\n\\begin{figure}\n\\centerline{\\psfig{file=avxdev.eps,width=3.25in,height=2.8in}}\n\\caption{\\narrowtext RMS deviation of the position from the target\nvalue as a function of time. This was obtained by averaging over 1000\ntrajectories. The units are $X_{\\mbox{\\scriptsize s}} = \n\\sqrt{\\hbar/(m\\nu)}$ and $\\tau = 1/\\nu$, where $m$ is the mass of the \nparticle and $\\nu$ is an arbitrary frequency. In the text we have \nset $\\hbar = m = \\nu = 1$, so that all quantities are dimensionless.}\n\\label{fig2}\n\\end{figure}\n\nWhile this strategy is fairly effective, it is limited by specifically\nquantum effects. In order to maintain a Gaussian state in the presence\nof the non-linear potential the combined effect of the thermal noise\nand measurement must be sufficiently strong, and this results in\nunwanted heating which must be countered by the feedback. While this\nis a limitation of the Gaussian estimator, there is still a more\nfundamental limitation. In the presence of noise, the measurement must\nbe sufficiently strong in order to obtain sufficient information about\nthe system to control it. In this case we found we needed a\nmeasurement strength three times that of the noise, resulting in the\ncorresponding heating. Naturally, these quantum limiting features are\nultimately due to the size of $\\hbar$; as $\\hbar$ decreases, the\nmeasurement induced heating rate, as well as the rate at which the\nWigner function deforms from Gaussian, is reduced. It is to be\nexpected that with the use of more sophisticated estimation\ntechniques, and more subtle quantum control strategies, the simple\nmethod we have outlined here can be beaten, possibly significantly,\nand the development of such techniques constitutes a central problem\nfor future work in quantum feedback control.\n\n%Questions: over what region of parameter space does this work (this\n%question has a quantum element and a classical element in that the\n%classical problem also becomes non-Gaussian and requires linearizing\n%while the quantum system will display uniquely quantum behavior such\n%as tunneling in some regime,\n%secondly what is the best measurement strength for a given set of\n%parameters since changing the measurement strength will change the\n%achievable cost, this second consideration is entirely a feature of\n%the quantum model. Since classically there is no back action noise it\n%is always better to have a more accurate measurement.\n\n\\section{Conclusion}\nIn this paper we have argued that it is useful to consider quantum\nfeedback control in the light of methods developed in classical\ncontrol theory. In order to do this it is important to understand the\nrelationship between the two theories. We began by comparing the\nformulations of these theories, in order to identify conceptual\nanalogies. We then considered three ways in which the quantum control\nproblem could be formally mapped to the classical problem, and\ndiscussed if and when these formulations may be addressed directly\nwith the classical theory.\n\nAs an example, we applied the ideas presented here to the control of\nthe position of a single quantum particle in a non-linear potential\ndeep in the quantum regime. In this case we fixed both the measurement\nobservable (system/environment coupling) and the unraveling, and\nconsidered the use of sub-optimal estimation and control\nstrategies. While this approach was fairly effective, it is clearly\nlimited by quantum effects.\n\nAs experimental techniques improve, and quantum technology becomes\nincreasingly relevant in practical applications, we can anticipate\nthat questions of quantum feedback control will become increasingly\nimportant. It is clear that most questions regarding the optimal\nobservables, unravelings, and control strategies required for quantum\nfeedback control problems, and the effectiveness of sub-optimal\nestimation algorithms, are as yet unanswered, and that this field\npresents a considerable theoretical challenge for future work.\n\n\\section{acknowledgements}\nSH, KJ and HM would like to thank Tanmoy Bhattacharya, Chris Fuchs and\nHoward Barnum for helpful discussions. This research was performed in\npart using the resources located at the Advanced Computing Laboratory\nof Los Alamos National Laboratory.\n\n\\end{multicols}\n\\widetext\n\\begin{multicols}{2}\n\n\\begin{thebibliography}{10}\n\\vspace{-1.6cm}\n\\bibitem{CQED} H.~Mabuchi, J.~Ye, and H.J. Kimble, Appl. Phys. B, 1095\n(1999), Eprint: quant-ph/9805076.\n\n\\bibitem{ion} B.E. King, C.S. Wood, C.J. Myatt, Q.A. Turchette,\nD.~Leibfried, W.M. Itano, C.~Monroe, and D.J. Wineland,\nPhys. Rev. Lett. {\\bf 81}, 1525 (1998), Eprint: quant-ph/9803023.\n\n\\bibitem{andrews1996a} M.R.~Andrews {\\it et al}, Science {\\bf 273}, 84\n(1996).\n\n\\bibitem{qfb1} H.M. Wiseman and G.J. Milburn, Phys. Rev. Lett. {\\bf\n70}, 548 (1993); Phys. Rev. A {\\bf 49}, 2133 (1994); {\\em ibid} {\\bf\n49}, 5159(E) (1994); {\\em ibid} {\\bf 50}, 4428(E) (1994).\n\n\\bibitem{taubman1995a} M.S. Taubman, H.M. Wiseman, D.E. McClelland,\nand H.-A. Bachor, J. Opt. Soc. Am. B {\\bf 12}, 1792 (1995);\nH.M. Wiseman, Phys. Rev. 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A {49, 2133 (1994); {\\em ibid {49, 5159(E) (1994); {\\em ibid {50, 4428(E) (1994)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{taubman1995a M.S. Taubman, H.M. Wiseman, D.E. McClelland, and H.-A. Bachor, J. Opt. Soc. Am. B {12, 1792 (1995); H.M. Wiseman, Phys. Rev. Lett. {81, 3840 (1998), Eprint: quant-ph/9805077." }, { "name": "quant-ph9912107.extracted_bib", "string": "{slosser1995a J.J. Slosser and G.J. Milburn, Phys. Rev. Lett. {75, 418 (1995); P.~Tombesi and D.~Vitali, Appl. Phys. B {60, S69 (1995); Phys. Rev. A {51, 4913 (1995); P. Goetsch, P. Tombesi, and D. Vitali, Phys. Rev. A {54, 4519 (1996); D.B. Horoshko and S.Ya. Kilin, Phys. Rev. Lett. {78, 840 (1997)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{dunningham1997 J.A. Dunningham, H.M. Wiseman, and D.F. Walls, Phys. Rev. A {55, 1398 (1997); S. Mancini and P. Tombesi, Phys. Rev. A {56, 2466 (1997); S. Mancini, D. Vitali, and P. Tombesi, Phys. Rev. Lett. {80, 688 (1998), Eprint: quant-ph/9802034." }, { "name": "quant-ph9912107.extracted_bib", "string": "{hofman1998a H.F. Hofman, G. Mahler, and O. Hess, Phys. Rev. A {57, 4877 (1998)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{viola1999a E. Knill, L. Viola and S. Lloyd, Phys. Rev. Lett. {82, 2417 (1999); L. Viola and S. Lloyd, Phys. Rev. A {58, 2733 (1998)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{warren1993a H. Rabitz W.S. Warren and M. Dahleh, Science {259, 1581 (1993)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{BelavkinLQG V.P. Belavkin, {\\em Non-demolition measurement and control in quantum dynamical systems, In: Information complexity and control in quantum physics, ed. A. Blaquiere, S. Diner and G. Lochak, (Springer-Verlag, New York, 1987)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{DJ A.C.~Doherty and K.~Jacobs, Phys. Rev. A {60, 2700 (1999), Eprint: quant-ph/9812004." }, { "name": "quant-ph9912107.extracted_bib", "string": "{CHP P.~Cohadon, A.~Heidmann, and M.~Pinard, Phys.Rev.Lett. {83 3174 (1999), Eprint: quant-ph/9903094." }, { "name": "quant-ph9912107.extracted_bib", "string": "{cfb1 O.L.R. Jacobs, {\\em Introduction to Control Theory, (Oxford University Press, Oxford, 1993)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{Maybeck P.S. Maybeck, {\\em Stochastic Models, Estimation and Control, volumes II and III, (Academic Press, New York, 1982)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{cfb2 P.~Whittle, {\\em Optimal Control, (John Wiley \\& Sons, Chichester, 1996)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{Ben A.~Bensoussan, {\\em Stochastic Control of Partialy Observable Systems, (Cambridge University Press, Cambridge, 1992)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{residual B.D.O. Anderson and J.B. Moore, {\\em Optimal Control: Linear Quadratic Methods, (Prentice-Hall, Englewood Cliffs, New Jersey, 1990)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{RAOC K.~Zhou, J.C. Doyle, and K.~Glover, {\\em Robust and Optimal Control, (Prentice Hall, New Jersey, 1996)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{MeasP H.J. Carmichael, S. Singh, R. Vyas and P.R. Rice, Phys. Rev. A {39, 1200 (1989); G.C. Hegerfeldt and T.S. Wilser, in {\\em Proceedings of the II International Wigner Symposium, Goslar, Germany, July 1991, ed. H.D. D\\\"obner, W. Scherer and F. Schr\\\"ock (World Scientific, Singapore, 1992); J. Dalibard, Y. Castin and K. M\\/olmer, Phys. Rev. Lett. {68, 580 (1992)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{GPZ C.W. Gardiner, A.S. Parkins and P. Zoller, Phys. Rev. A {46, 4363 (1992)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{Carm H.J. Carmichael, {\\em An Open Systems Approach to Quantum Optics, Lecture Notes in Physics m18, (Springer-Verlag, Berlin, 1993)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{WMhom H.M. Wiseman and G.J. Milburn, Phys. Rev. A {47, 642 (1993)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{MeasM M.D. Srinivas and E.B. Davies, Optica Acta {28, 981 (1981); N. Gisin, Phys. Rev. Lett. {52, 1657 (1984); L. Diosi, Phys. Lett. A {114, 451 (1986)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{adapt H.M. Wiseman, Phys. Rev. A {75, 4587 (1995)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{BelBell V.P. Belavkin, {\\em Automat. Remote Control, {44 178 (1983)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{Belavkin99 V.P. Belavkin, Rep. Math. Phys. {43, 405 (1999)." }, { "name": "quant-ph9912107.extracted_bib", "string": "{fuchs C.A. Fuchs and C.M. Caves, Phys. Rev. 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quant-ph9912108	Complementarity between Position and Momentum \\ as a Consequence of Kochen-Specker Arguments	[ { "author": "Rob Clifton" } ]	We give two simple Kochen-Specker arguments for complementary between the position and momentum components of spinless particles, arguments that are identical in structure to those given by Peres and Mermin for spin-1/2 particles.	[ { "name": "quant-ph9912108.tex", "string": "%\\documentstyle[preprint,aps,multicol,amssymb,amsfonts]{revtex}\n%\\documentstyle[preprint,aps,amssymb,amsfonts]{revtex}\n\\documentstyle[aps,pra,multicol,amssymb,amsfonts]{revtex}\n\n\\newcommand{\\alg}[1]{\\mbox{$\\mathcal{#1}$}}\n\n\\begin{document}\n\\title{Complementarity between Position and Momentum \\\\ as a \nConsequence of \nKochen-Specker Arguments}\n\\author{Rob Clifton}\n\\address{Departments of Philosophy and History and Philosophy of\nScience, \\\\ 10th floor, Cathedral of\nLearning, University of\nPittsburgh, \\\\ Pittsburgh, PA\\ 15260, U.S.A. \\\\ email: rclifton+@pitt.edu}\n\n\n\\maketitle\n\\begin{abstract} We give two simple Kochen-Specker arguments for \ncomplementary between the position and momentum components of spinless \nparticles, arguments that are identical in structure \nto those given by Peres and Mermin for spin-1/2 particles. \n\\end{abstract} \\draft\n\\pacs{PACS numbers: 03.65.Bz}\n\n\\begin{multicols}{2}\n\\section{Introduction}\n\nComplementarity is the idea that mutually exclusive pictures are \nneeded for a complete description of quantum-mechanical reality. The \nparadigm example is the complementarity \nbetween particle and wave (or `spacetime' and `causal') pictures, \nwhich Bohr took to be \nreflected in the uncertainty \nrelation $\\Delta x\\Delta p\\geq\\hbar$. Bohr saw this relation as defining the \nlatitude of applicability of the concepts of position and momentum to \na single system, not just as putting a limit on our ability to predict the values of both \nposition and momentum within an ensemble of identically prepared systems.\nFurthermore, right at the start of his celebrated reply to the \nEinstein-Podolsky-Rosen (EPR) argument \nagainst the completeness of quantum theory \\cite{epr}, Bohr \nconfidently asserted:\n``it is never possible, in the description \nof the state of a mechanical system, to attach definite values to both \nof two canonically conjugate variables'' \\cite{bohr}.\nCritics have often pointed out that complementarity does not \nlogically follow from the uncertainty relation without making the \npositivistic assumption that position and momentum can only be \nsimultaneously defined if their values can be simultaneously measured \nor predicted \n\\cite{popper}. \nHowever, we shall show here how direct Kochen-Specker arguments for complementarity between \nposition and momentum can be given that are entirely independent of \nthe uncertainty relation and its interpretation.\n\nThe aim of a Kochen-Specker argument is to establish that a certain \nset of observables of a quantum system cannot have\nsimultaneously definite values that respect the functional \nrelations between compatible observables within the \nset \\cite{ks}. \nLet $\\alg{O}$ be a collection of bounded self-adjoint \noperators (acting on some Hilbert space) containing the identity $I$ and both $AB$ and $\\lambda A+\\mu B$ \n($\\lambda,\\mu\\in\\mathbb{R}$), \nwhenever $A,B\\in\\alg{O}$ and $[A,B]=0$. Kochen \nand Specker called such a \nstructure\na \\emph{partial} algebra because there is no \nrequirement that $\\alg{O}$ contain \n\\emph{arbitrary} self-adjoint functions of its members (such \nas $i[A,B]$ or $A+B$, when $[A,B]\\not=0$). They then assumed that \nan assignment of values $[\\cdot]:\\alg{O}\\rightarrow\\mathbb{R}$ to the observables \nin $\\alg{O}$ should at least be a \\emph{partial} homomorphism,\nrespecting linear combinations and products of \\emph{compatible} observables \nin $\\alg{O}$. That is, whenever $A,B\\in\\alg{O}$ and $[A,B]=0$, \n\\begin{equation} \\label{eq:constraints}\n \\left[AB\\right] = \\left[A\\right]\\left[B\\right],\\ \n \\left[\\lambda A+\\mu B\\right] = \\lambda \\left[A\\right]+\\mu \n \\left[B\\right],\\ [I]=1.\n \\end{equation}\n Clearly these constraints \n are motivated by analogy with classical physics, in which all \n physical magnitudes (functions on phase space) trivially commute, \n and possess values (determined by points of phase space) that respect \n their functional relations. (The requirement that $[I]=1$ is only needed to avoid triviality; for\n $[I^{2}]=[I]=[I]^{2}$ \n implies $[I]=0$ or $1$, and if we took $[I]=0$, it would follow \n that \n $[A]=[AI]=[A][I]=0$ for all $\\alg{A}\\in\\alg{O}$.) \n \n Constraints (\\ref{eq:constraints}) are not entirely out of place \n in quantum theory. For example, any common eigenstate $\\Psi$ of a collection of observables $\\alg{O}$ \n automatically \n defines a \n partial homomorphism, given by assigning to each \n $A\\in\\alg{O}$ the eigenvalue $A$ \n has in state $\\Psi$. \n Difficulties --- called Kochen-Specker contradictions or \n obstructions \\cite{hamilton} --- arise when \n not all observables in \n $\\alg{O}$ share a common eigenstate. In that case, there is no \n guarantee that \n value assignments on all the commutative subalgebras of $\\alg{O}$ \n can be extended to a partial homomorphism on $\\alg{O}$ as a whole. Should such an \n extension exist, one could be justified in thinking of the \n noncommuting \n observables in $\\alg{O}$ as having simultaneously definite values, \n notwithstanding that\n a quantum state may not permit all their values to be predicted \n with certainty. But should some \n particular collection of observables $\\alg{O}$ not possess \n \\emph{any} partial homomorphisms, the natural response would be to\n concede to Bohr that the observables \n in $\\alg{O}$ ``transcend the scope of classical physical \n explanation'' and cannot be discussed using ``unambiguous language with suitable application of the \n terminology of classical physics'' \\cite{bohr2}. \n That is, one would have strong reasons for \n taking the noncommuting observables in $\\alg{O}$ to be mutually \n complementary.\n \n Bell \n \\cite{ks} has emphasized one other way to escape \n an obstruction with respect to some set of \n observables $\\alg{O}$. One could still take all $\\alg{O}$'s observables to \n have definite values by allowing the value of a particular \n $A\\in\\alg{O}$ to be a function of the context in which $A$ is \n measured. Thus, suppose $\\alg{O}_{1},\\alg{O}_{2}\\subseteq\\alg{O}$ are \n two different commuting subalgebras both containing $A$, where \n $[\\alg{O}_{1},\\alg{O}_{2}]\\not=0$. Then if\n $[\\cdot]_{1},[\\cdot]_{2}$ are homomorphisms on these subalgebras \n such that $[A]_{1}\\not=[A]_{2}$, one could \n interpret \n this difference in values (the obstruction) \n as signifying that the measured result for $A$ \n has to depend on whether it is measured along with the observables in $\\alg{O}_{1}$ \n or those in $\\alg{O}_{2}$. \n Such \n value assignments to the observables in $\\alg{O}$ are called \n contextual, because the context in which an observable is measured is allowed to \n influence what outcome is obtained \\cite{ks}. For \n example, Bohm's theory is \n contextual in exactly this sense \\cite{bohm}. \n On the other hand (as Bell himself\n was quick to observe), complementarity also demands a kind of \n contextualism: in some \n contexts it is appropriate to assign a system a \n definite position, and in \n other contexts, a definite momentum. The difference from\n Bohm is that Bohr takes the definiteness of the values of observables \n \\emph{itself} to be a function of context. \n And this makes \\emph{all} the difference \n in cases where value contextualism can only be \n enforced by making the measured value of an observable nonlocally depend on \n whether an observable of another \n spacelike-separated \n system is measured. We shall see below that complementarity between position \n and momentum can only be avoided by embracing such nonlocality. \n \n Numerous Kochen-Specker obstructions have been identified in the \n literature, and their practical and theoretical \n implications continue to be analyzed \n \\cite{cabello}. \n While obstructions cannot\n occur for observables sharing a common eigenstate, failure to \n possess a \n common eigenstate does not suffice for an obstruction. \n As Kochen and \n Specker themselves showed, \n the partial algebra generated by all components of a \n spin-1/2 particle possesses plenty of partial homomorphisms. \n But for particles with higher spin, or \n collections of more than one spin-1/2 \n particle, obstructions can occur, perhaps the simplest being \n those identified by Peres \\cite{peres} in the case of two spin-1/2 particles, \n and Mermin\\cite{mermin} in the case of three. Obstructions for sets that contain\n functions of position and momentum \n observables \\emph{have} been identified \\cite{fleming}, but \n additional observables need to be invoked that weaken the case for\n complementarity between position and momentum alone. \n In the arguments below, we shall only need simple \n \\emph{continuous} functions of \n the individual position and momentum components of a system. \n Though all our observables have purely continuous spectra, obstructions \n arise in \n exactly the \n same way that they do in the arguments given by Peres and Mermin \n for the spin-1/2 case. And because our \n obstructions depend only on the structure of the Weyl \n algebra, they immediately extend to relativistic\n quantum field theories, which are constructed out of representations \n of the Weyl \n algebra \\cite{wald}. \n \n \\section{The Weyl Algebra}\n \n Let $\\vec{x}=(x_{1},x_{2},x_{3})$ and \n $\\vec{p}=(p_{1},p_{2},p_{3})$ be the unbounded position and \n momentum operators for \n three degrees of freedom. We cannot extract a Kochen-Specker \n contradiction directly out of these operators, since \n domain questions prevent them from defining a simple \n algebraic structure. However, we may just as well consider the collection of all \n bounded, continuous, self-adjoint \nfunctions of $x_{1}$, and, similarly, the same set \nof functions in each of the variables $x_{2},x_{3},p_{1},p_{2},p_{3}$. Taking $\\alg{O}$ \nto be the partial algebra of observables generated by all these \nfunctions (obtained by taking compatible products and linear \ncombinations thereof), we shall show that $\\alg{O}$ does not possess any partial \nhomomorphisms. \n\nOur arguments are greatly simplified by employing the following \nmethod, \nanalogous to simplifying a problem in real analysis by passing to \nthe complex plane. Assuming that $\\alg{O}$ \\emph{does} possess a partial \nhomomorphism $[\\cdot]:\\alg{O}\\rightarrow\\mathbb{R}$ (an assumption we shall eventually have to \ndischarge), we can extend this mapping to the set \n$\\alg{O}_{\\mathbb{C}}\\equiv\\alg{O}+i\\alg{O}$ in a well-defined manner, by taking\n$[X]\\equiv[\\Re(X)]+i[\\Im(X)]\\in \\mathbb{C}$, \nwhere $\\Re(X)$ and $\\Im(X)$ are the unique real and \nimaginary parts of $X$. Now, if we consider any pair of \ncommuting unitary operators \n$U,U'\\in\\alg{O}_{\\mathbb{C}}$, then since $U,U^{},U',U'^{}$ pairwise commute, \nthe four self-adjoint operators\n\\begin{eqnarray}\n\\Re(U)=(U+U^{})/2,\\ \\ & \\Im(U)=i(U^{}-U)/2, \\\\\n \\Re(U')=(U'+U'^{})/2,\\ \\ & \n\\Im(U')=i(U'^{}-U')/2, \n\\end{eqnarray}\nwhich must lie in $\\alg{O}_{\\mathbb{C}}$, also pairwise commute. Thus\n\\begin{eqnarray} \\nonumber\nUU' & = & \\Re(U)\\Re(U')-\\Im(U)\\Im(U') \\\\\n& & +i(\\Re(U)\\Im(U')+\\Im(U)\\Re(U'))\\in\\alg{O}_{\\mathbb{C}},\n\\end{eqnarray}\n using the fact that $\\alg{O}$ is a partial algebra. In \n addition, \n\\begin{eqnarray}\n[UU'] & = & \n[\\Re(U)\\Re(U')-\\Im(U)\\Im(U')] \\nonumber \\\\\n& & +i[\\Re(U)\\Im(U')+\\Im(U)\\Re(U')], \\\\\n& = & \n[\\Re(U)][\\Re(U')]-[\\Im(U)][\\Im(U')] \\nonumber \\\\ \\label{eq:next}\n& & +i([\\Re(U)][\\Im(U')]+[\\Im(U)][\\Re(U')]), \\\\\n& = & ([\\Re(U)]+i[\\Im(U)])([\\Re(U')]+i[\\Im(U')]) \\\\\n& = & [U][U'],\n\\end{eqnarray}\nusing the fact that $[\\cdot]$ is a partial \nhomomorphism in step (\\ref{eq:next}). \nSo we have established that the following product rule must hold \nin $\\alg{O}_{\\mathbb{C}}$:\n\\begin{eqnarray}\n& U,U'\\in\\alg{O}_{\\mathbb{C}}\\ \\ \\&\\ \\ & [U,U']=0 \\nonumber \\\\\n \\Rightarrow \\ & UU'\\in \\alg{O}_{\\mathbb{C}}\\ \\ \\&\\ \\ & \n[UU']=[U][U'].\n\\end{eqnarray}\nHenceforth, we shall only this need this simple product rule, together \nwith $[\\pm I]=\\pm 1$. Our \nobstructions will manifest themselves as contradictions obtained by \napplying the product rule to compatible unitary operators in \n$\\alg{O}_{\\mathbb{C}}$. \n\nTo see what operators those are, we first \nrecall the definition of the Weyl algebra for three degrees of \nfreedom. \nConsider the two families of unitary operators \ngiven by\n\\begin{equation} \\label{eq:Weyl}\nU_{\\vec{a}}=e^{-i\\vec{a}\\cdot\\vec{x}/\\hbar},\\ \nV_{\\vec{b}}=e^{-i\\vec{b}\\cdot\\vec{p}/\\hbar},\\ \n\\vec{a},\\vec{b}\\in \\mbox{$\\mathbb{R}$}^{3}.\n\\end{equation}\nThese operators act on any wavefunction $\\Psi\\in \nL_{2}(\\mbox{$\\mathbb{R}$}^{3})$ as \n \\begin{equation} \\label{eq:one}\n (U_{\\vec{a}}\\Psi)(\\vec{x}) = \n e^{-i\\vec{a}\\cdot\\vec{x}/\\hbar}\\Psi(\\vec{x}),\\ \n (V_{\\vec{b}}\\Psi)(\\vec{x}) = \\Psi(\\vec{x}-\\vec{b}), \n \\end{equation}\nand satisfy the Weyl form of the canonical \ncommutations relations $[x_{j},p_{k}]=\\delta_{jk}i\\hbar I$, \n\\begin{equation} \\label{eq:nocommute}\nU_{\\vec{a}}V_{\\vec{b}} = e^{-i\\vec{a}\\cdot\\vec{b}/\\hbar}V_{\\vec{b}}U_{\\vec{a}}.\n\\end{equation}\nThe Weyl algebra (which is independent of the representation in \n(\\ref{eq:one})) is just the $C^{*}$-algebra \ngenerated by the two families of unitary operators in \n(\\ref{eq:Weyl}) subject to the commutation relation \n(\\ref{eq:nocommute}). \n\n$\\alg{O}_{\\mathbb{C}}$ is properly contained in the \nWeyl algebra. Indeed, writing $U_{a_{j}}$ ($\\equiv e^{-ia_{j}x_{j}/\\hbar}$)\n for the $j$th component of the operator \n$U_{\\vec{a}}$, and similarly $V_{b_{k}}$ ($\\equiv e^{-ib_{k}p_{k}/\\hbar}$), all nine \nof these \ncomponent generators of the Weyl algebra lie in\n$\\alg{O}_{\\mathbb{C}}$, because their real and imaginary parts, cosine \nand sine functions of the $x_{j}$'s and $p_{k}$'s, lie in $\\alg{O}$. \nBy the \nproduct rule, $\\alg{O}_{\\mathbb{C}}$ also contains the products of \ncompatible unitary operators for different degrees of \nfreedom, as well as compatible products of those products. But, \nunlike the full Weyl algebra, $\\alg{O}_{\\mathbb{C}}$ \ndoes not contain incompatible products, like $U_{a_{j}}V_{b_{j}}$ when \n$a_{j}b_{j}\\not=2n\\pi\\hbar$ ($n\\in \\mathbb{Z}$). Nevertheless, $\\alg{O}_{\\mathbb{C}}$ is all we need to exhibit \nobstructions. The key is that we can choose values for the components \nof $\\vec{a},\\vec{b}$ so \nthat, for $j=1$ to $3$, $a_{j}b_{j}=(2n+1)\\pi\\hbar$. In that case, we immediately obtain \nfrom (\\ref{eq:nocommute}) \nthe \\emph{anti}-commutation rule\n\\begin{equation} \\label{eq:anti}\n[U_{\\pm a_{j}},V_{\\pm b_{j}}]_{+}=0=[U_{\\mp a_{j}},V_{\\pm b_{j}}]_{+},\n\\end{equation}\nwhich, together with the product rule, will generate the required \nobstructions. \n\n\\section{Obstructions for Two and Three Degrees of Freedom}\n\nWe first limit ourselves to continuous functions of the four observables \n$x_{1},x_{2},p_{1},p_{2}$, extracting a contradiction in exactly the \nway Peres \\cite{peres} does for a pair of spin-1/2 particles. \n A first application of the product rule in $\\alg{O}_{\\mathbb{C}}$ \n yields\n \\begin{eqnarray} \\label{eq:first'}\n \\left[U_{-a_{1}}U_{a_{2}}\\right] & = & \\left[U_{-a_{1}}\\right]\\left[U_{a_{2}}\\right], \\\\\n \\left[U_{a_{1}}V_{b_{2}}\\right] & = & \\left[U_{a_{1}}\\right]\\left[V_{b_{2}}\\right], \\\\\n\\left[V_{b_{1}}U_{-a_{2}}\\right] & = & \\left[V_{b_{1}}\\right]\\left[U_{-a_{2}}\\right], \\\\\n\\left[V_{-b_{1}}V_{-b_{2}}\\right] & = & \\left[V_{-b_{1}}\\right]\\left[V_{-b_{2}}\\right]. \\label{eq:last'}\n\\end{eqnarray}\nMultiplying equations (\\ref{eq:first'})--(\\ref{eq:last'}) together, and using one \nfurther (trivial) application of the product rule\n\\begin{equation} \\label{eq:trivial}\n [U_{a_{j}}][U_{-a_{j}}]=[I]=1=[V_{b_{k}}][V_{-b_{k}}],\n \\end{equation}\n one obtains\n\\begin{equation} \\label{eq:square'}\n\\left[U_{-a_{1}}U_{a_{2}}\\right]\\left[V_{-b_{1}}V_{-b_{2}}\\right]\n\\left[U_{a_{1}}V_{b_{2}}\\right]\\left[V_{b_{1}}U_{-a_{2}}\\right] = 1.\n \\end{equation}\n However, because of the anti-commutation rule (\\ref{eq:anti}), the first pair of product \n operators occurring in (\\ref{eq:square'}) actually \\emph{commute}, as do the second \n pair of product operators. \n Hence we may make a further application of the \n product rule to (\\ref{eq:square'}) to get \n\\begin{equation} \\label{eq:square''}\n \\left[U_{-a_{1}}U_{a_{2}}V_{-b_{1}}V_{-b_{2}}\\right]\n \\left[U_{a_{1}}V_{b_{2}}V_{b_{1}}U_{-a_{2}}\\right] = 1.\n \\end{equation}\n Again, due to the anti-commutation rule, the two remaining (four-fold) product operators occurring \n in (\\ref{eq:square''}) commute, and their product is \n $-I$. Thus, a final application of the \n product rule to (\\ref{eq:square''}) yields the contradiction \n $[-I]=-1=1$.\n \n Notice that this obstruction remains for \n any given nonzero values for $a_{1}$ and $a_{2}$, provided only that we choose \n $b_{1,2}=(2n+1)\\pi\\hbar/a_{1,2}$. The obstruction would vanish if, \n instead, we chose any of the numbers $a_{1},a_{2},b_{1},b_{2}$ to be zero. \n When $a_{1}=a_{2}=0$ or $b_{1}=b_{2}=0$, this is to be expected, since \n one would then no longer be attempting to assign values to nontrivial functions \n of \\emph{both} the positions and momenta. However, the breakdown of the \n argument when either $a_{2}$ or $b_{2}$ is zero does not necessarily \n mean that a more complicated argument could not be given \n for position-momentum complementarity by invoking \n only a \\emph{single} degree of freedom. \n \n As Mermin \\cite{mermin} has emphasized (for the spin-1/2 analogue of \n the above argument), one can get by without \n independently assuming the existence of values for the two commuting unitary operators \n occurring in (\\ref{eq:square''}), and \n thereby strengthen the argument. For we can suppose that the quantum state \n of the system is an eigenstate of these operators, with \n eigenvalues that \\emph{necessarily} multiply to -1. \n Using (\\ref{eq:one}), a wavefunction $\\Psi$ will \n be an eigenstate of both products in (\\ref{eq:square''}) just in case \n \\begin{eqnarray} \\label{eq:hi}\n e^{i(a_{1}x_{1}-a_{2}x_{2})/\\hbar}\\Psi(x_{1}+b_{1},x_{2}+b_{2})=c\\Psi(x_{1},x_{2}), \\\\\n -e^{-i(a_{1}x_{1}-a_{2}x_{2})/\\hbar}\\Psi(x_{1}-b_{1},x_{2}-b_{2})=c'\\Psi(x_{1},x_{2}), \n \\label{eq:hi'}\n \\end{eqnarray}\n for some $c,c'\\in\\mathbb{C}$. We should not \n expect there to be a \\emph{normalizable} wavefunction satisfying (\\ref{eq:hi}) and (\\ref{eq:hi'}),\n because the commuting product operators in (\\ref{eq:square''}) have purely continuous spectra. \n But if we allow ourselves the idealization of using Dirac \n states (which can be approximated arbitrarily closely by \n elements of $L_{2}(\\mbox{$\\mathbb{R}$}^{2})$), and just choose \n $a_{1}=a_{2}$ for simplicity, then the two-dimensional delta function \n $\\delta(x_{1}-x_{2}-x_{0})$ --- an improper eigenstate of the \n relative position operator $x_{1}-x_{2}$ with `eigenvalue' \n $x_{0}\\in\\mathbb{R}$ --- provides a simple solution to the above \n equations. However, this\n state cannot also be used to independently justify the assignment of values \n to the operators $U_{a_{1}}V_{b_{2}}$ and $V_{b_{1}}U_{-a_{2}}$\n occurring in (\\ref{eq:square'}), which do not have \n $\\delta(x_{1}-x_{2}-x_{0})$ as an eigenstate. \n \n It is ironic that $\\delta(x_{1}-x_{2}-x_{0})=\\delta(p_{1}+p_{2})$ is exactly \n the state of two spacelike-seperated particles that EPR invoked to argue \\emph{against} \n position-momentum complementarity. So in a sense the EPR argument \n carries the seeds of its own destruction. For suppose we follow their \n reasoning by invoking locality and the strict correlations entailed \n by the EPR state between $x_{1}$ and $x_{2}$, and between $p_{1}$ and \n $p_{2}$, \n to argue for the existence of noncontextual \n values for all four positions and momenta. Then all eight component unitary \n operators we employed above must have definite noncontextual values, since \n their real and imaginary parts are simple functions of those \n $x$'s and $p$'s. It \n is then a small step to conclude that the four product operators in (\\ref{eq:square'}) \n should also have\n definite noncontextual values satisfying the product rule, and from there \n contradiction follows. This final step cannot itself be \n justified by appeal to locality, for the four product observables \n in (\\ref{eq:square'}) do not \n pertain to either particle on its own and, hence, a measurement \n context for any one of these operators (i.e., their self-adjoint real and \n imaginary parts) necessarily requires a joint measurement undertaken on both \n particles \\cite{mermin}. Still, the above argument sheds an entirely new \n light on the nonclassical features of the original EPR \n state, which have hitherto only been discussed from a statistical \n point of view \\cite{bell}.\n \n Our second argument employs all three degrees of freedom, extracting a contradiction in exactly the \nway Mermin \\cite{mermin} does for three spin-1/2 particles.\n Again, a first application of the product rule in $\\alg{O}_{\\mathbb{C}}$ \n yields\n \\begin{eqnarray} \n\\left[U_{a_{1}}V_{-b_{2}}V_{-b_{3}}\\right] \n& \n= & \\left[U_{a_{1}}\\right]\\left[V_{-b_{2}}\\right]\\left[V_{-b_{3}}\\right], \\label{eq:first} \\\\ \t\n\\left[V_{-b_{1}}U_{a_{2}}V_{b_{3}}\\right] & = & \n\\left[V_{-b_{1}}\\right]\\left[U_{a_{2}}\\right]\\left[V_{b_{3}}\\right], \\\\\n\\left[V_{b_{1}}V_{b_{2}}U_{a_{3}}\\right] \n& = & \n\\left[V_{b_{1}}\\right]\\left[V_{b_{2}}\\right]\\left[U_{a_{3}}\\right], \\\\\n \\left[U_{-a_{1}}U_{-a_{2}}U_{-a_{3}}\\right] \n & = & \\left[U_{-a_{1}}\\right]\\left[U_{-a_{2}}\\right]\n \\left[U_{-a_{3}}\\right]. \\label{eq:last}\n \\end{eqnarray} \n Multiplying (\\ref{eq:first})--(\\ref{eq:last}) together, again \n using (\\ref{eq:trivial}), \n yields\n \\begin{eqnarray} \\nonumber\n \\left[U_{a_{1}}V_{-b_{2}}V_{-b_{3}}\\right]\\left[V_{-b_{1}}U_{a_{2}}V_{b_{3}}\\right] & & \\\\ \\label{eq:square}\n \\left[V_{b_{1}}V_{b_{2}}U_{a_{3}}\\right]\\left[U_{-a_{1}}U_{-a_{2}}U_{-a_{3}}\\right] & = & 1.\n \\end{eqnarray}\n But now, exploiting the anti-commutation rule once again, the four \n product operators occurring\n in square brackets in (\\ref{eq:square}) pairwise commute, and \n their product is easily seen to be \n $-I$. \n So one final application of the \n product rule to (\\ref{eq:square}) once more yields the \n contradiction $[-I]=-1=1$.\n \nAs before, we may interpret the $x$'s and $p$'s as the positions \n and momenta of three spacelike-separated particles. And we can\n avoid independently assuming values for the four \n products in (\\ref{eq:square}) by taking the state of the \n particles to be a simultaneous (improper) eigenstate of these \n operators --- exploiting that state's strict correlations and \n EPR-type reasoning from locality to motivate values for all \n the component operators. (The reader is invited to use (\\ref{eq:one}) \n to determine the set of all such common \n eigenstates, which are new position-momentum analogues of the \n Greenberger-Horne-Zeilinger state \\cite{mermin}.) \n This time, the \\emph{only} way to prevent contradiction is to \n introduce contextualism to distinguish, for example, the \n value of $U_{a_{1}}$ as it occurs in (\\ref{eq:first}) from the \n value this operator (or rather its inverse) receives in \n (\\ref{eq:last}) in the context of different \n operators for particles $2$ and $3$ --- forcing the values \n of $\\sin a_{1}x_{1}$ and $\\cos a_{1}x_{1}$ to \n \\emph{nonlocally} depend on whether position or momentum observables for \n $1$ and $2$ are measured.\nBohr of course denied\n that there could be any such nonlocal ``mechanical'' \n influence, but only ``an influence on the very conditions which \n define'' which of the two mutually complementary pictures available \n for each system can be unambiguously employed \\cite{bohr}. \n \nThe author would like to thank All Souls College, Oxford for support, and Paul Busch, Jeremy Butterfield, and \nHans Halvorson for helpful \ndiscussions.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{epr} A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {\\bf \n47}, 777 (1935).\n\n\\bibitem{bohr} N. Bohr, Phys. Rev. {\\bf \n48}, 696 (1935).\n\n\\bibitem{popper} K. R. Popper, \\emph{Quantum Theory and the Schism in \nPhysics} (London, Hutchinson, 1982).\n\n\\bibitem{ks} S. Kochen and E. P. Specker, J. Math. Mech. {\\bf 17}, 59 \n(1967); J. S. Bell, Rev. Mod. Phys. {\\bf 38}, 447 (1966); N. D. Mermin, Rev. Mod. Phys. {\\bf 65}, 803 (1993).\n\n\\bibitem{hamilton} C. J. Isham and J. Butterfield, Int. J. Theor. \nPhys. {\\bf 38}, 827 (1999); J. Hamilton, {\\tt \nquant-ph/9912018}.\n\n\\bibitem{bohr2} N. Bohr, in P. A. Schilpp (ed.), \\emph{Albert \nEinstein, Philosopher-Scientist} (Library of Living Philosophers, \nEvanston) pp. 201--241.\n\n\\bibitem{bohm} D. Bohm, Phys. Rev. {\\bf 85}, 66, 180 (1952); R. Clifton and C. Pagonis, Found. Phys. {\\bf 25}, \n281 (1995).\n\n\\bibitem{cabello} \nJ. Bub, \\emph{Interpreting the Quantum World} \n(Cambridge, Cambridge University Press, 1997);\nI. Pitowsky, J. Math. Phys. {\\bf 39}, 218 (1998); A. \nCabello, G. Garcia-Alcaine, Phys. Rev. Lett. {\\bf 80}, 1797 (1998); D. \nMeyer, Phys. Rev. Lett. {\\bf 83}, 3751 (1999); A. Kent, Phys. Rev. \nLett. {\\bf 83}, 3755 (1999); R. Clifton and A. Kent, {\\tt \nquant-ph/9908031}. \n\n\\bibitem{peres} A. Peres, Phys. Lett. A {\\bf 151}, 107 (1990); Found. \nPhys. {\\bf 22}, 357 (1992); \\emph{Quantum Theory: Concepts and \nMethods} (Dordrecht, Kluwer, 1993).\n\n\\bibitem{mermin} N. D. Mermin, Phys. Rev. Lett. {\\bf 65}, 3373 \n(1990); D. \nGreenberger, M. Horne, A. Zeilinger, and A. Shimony, Am. J. Phys. \n{\\bf 58}, 1131 (1990). \n\n\\bibitem{fleming} G. Fleming, Ann. (N. Y.) Acad. Sci. {\\bf 755}, \n646; J. Zimba, Found. Phys. Lett. {\\bf 11}, 503 (1998) \n(1995).\n\n\\bibitem{wald} R. Wald, \\emph{Quantum Field Theory in Curved \nSpacetime and Black Hole Thermodynamics} (Chicago, Univ. of Chicago \nPress, 1994).\n\n\\bibitem{bell} J. S. Bell, Ann. (N. Y.) Acad. Sci. {\\bf 480}, 263 \n(1986); O. Cohen, Phys. Rev. A {\\bf 56}, 3484 (1997); \nK. Banaszek and K. W\\'{o}dkiewicz, Phys. Rev. A {\\bf 58}, \n4345 (1998); {\\tt \nquant-ph/9904071}. \n\n\\end{thebibliography}\n\\end{multicols}\n\\end{document}" } ]	[ { "name": "quant-ph9912108.extracted_bib", "string": "{epr A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {47, 777 (1935)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{bohr N. Bohr, Phys. Rev. {48, 696 (1935)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{popper K. R. Popper, Quantum Theory and the Schism in Physics (London, Hutchinson, 1982)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{ks S. Kochen and E. P. Specker, J. Math. Mech. {17, 59 (1967); J. S. Bell, Rev. Mod. Phys. {38, 447 (1966); N. D. Mermin, Rev. Mod. Phys. {65, 803 (1993)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{hamilton C. J. Isham and J. Butterfield, Int. J. Theor. Phys. {38, 827 (1999); J. Hamilton, {\\tt quant-ph/9912018." }, { "name": "quant-ph9912108.extracted_bib", "string": "{bohr2 N. Bohr, in P. A. Schilpp (ed.), Albert Einstein, Philosopher-Scientist (Library of Living Philosophers, Evanston) pp. 201--241." }, { "name": "quant-ph9912108.extracted_bib", "string": "{bohm D. Bohm, Phys. Rev. {85, 66, 180 (1952); R. Clifton and C. Pagonis, Found. Phys. {25, 281 (1995)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{cabello J. Bub, Interpreting the Quantum World (Cambridge, Cambridge University Press, 1997); I. Pitowsky, J. Math. Phys. {39, 218 (1998); A. Cabello, G. Garcia-Alcaine, Phys. Rev. Lett. {80, 1797 (1998); D. Meyer, Phys. Rev. Lett. {83, 3751 (1999); A. Kent, Phys. Rev. Lett. {83, 3755 (1999); R. Clifton and A. Kent, {\\tt quant-ph/9908031." }, { "name": "quant-ph9912108.extracted_bib", "string": "{peres A. Peres, Phys. Lett. A {151, 107 (1990); Found. Phys. {22, 357 (1992); Quantum Theory: Concepts and Methods (Dordrecht, Kluwer, 1993)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{mermin N. D. Mermin, Phys. Rev. Lett. {65, 3373 (1990); D. Greenberger, M. Horne, A. Zeilinger, and A. Shimony, Am. J. Phys. {58, 1131 (1990)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{fleming G. Fleming, Ann. (N. Y.) Acad. Sci. {755, 646; J. Zimba, Found. Phys. Lett. {11, 503 (1998) (1995)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{wald R. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago, Univ. of Chicago Press, 1994)." }, { "name": "quant-ph9912108.extracted_bib", "string": "{bell J. S. Bell, Ann. (N. Y.) Acad. Sci. {480, 263 (1986); O. Cohen, Phys. Rev. A {56, 3484 (1997); K. Banaszek and K. W\\'{odkiewicz, Phys. Rev. A {58, 4345 (1998); {\\tt quant-ph/9904071." } ]
quant-ph9912109	Time of arrival \\ through interacting environments: \\ Tunneling processes	[ { "author": "Ken-Ichi Aoki\\footnote{Electronic address : aoki@hep.s.kanazawa-u.ac.jp" } ]	We discuss the propagation of wave packets through interacting environments. Such environments generally modify the dispersion relation or shape of the wave function. To study such effects in detail, we define the distribution function $P_{X(T)$, which describes the arrival time $T$ of a packet at a detector located at point $X$. We calculate $P_{X(T)$ for wave packets traveling through a tunneling barrier and find that our results actually explain recent experiments. We compare our results with Nelson's stochastic interpretation of quantum mechanics and resolve a paradox previously apparent in Nelson's viewpoint about the tunneling time.	[ { "name": "arrivalvf.tex", "string": "\\documentstyle[a4,12pt,epsf]{article}\n\\begin{document}\n\\title{\\bf Time of arrival \\\\\nthrough interacting environments: \\\\\nTunneling processes}\n\\author{Ken-Ichi Aoki\\footnote{Electronic address : \naoki@hep.s.kanazawa-u.ac.jp}, \n~Atsushi Horikoshi\\footnote{Electronic address : \nhorikosi@hep.s.kanazawa-u.ac.jp}, \n~and ~Etsuko Nakamura\\footnote{Electronic address : \netsuko@hep.s.kanazawa-u.ac.jp}\\\\\n\\\\\nInstitute for Theoretical Physics, Kanazawa University, \\\\\nKakuma-machi Kanazawa 920-1192, Japan}\n\\date{July 2000}\n \\maketitle\n\\vspace{-100mm}\n\\begin{flushright}\nquant-ph/9912109\\\\\nKANAZAWA/99-14\n\\end{flushright}\n\\vspace{80mm}\n%%%%%%%%%%%%%%%%%%%%%%%% Abstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\begin{abstract}\n We discuss the propagation of wave packets \n through interacting environments. \n Such environments generally modify the dispersion \n relation or shape of the wave function. \n To study such effects in detail, \n we define the distribution function $P_{X}(T)$, \n which describes the arrival time $T$ of a packet \n at a detector located at point $X$. We calculate $P_{X}(T)$ for\n wave packets traveling through a tunneling barrier and find \n that our results actually explain recent experiments. \n We compare our results with Nelson's stochastic interpretation of\n quantum mechanics and resolve a paradox previously apparent in \n Nelson's viewpoint about the tunneling time. \n\\end{abstract}\n%%%%%%%%%%%%%%%%%%%%%%%% Section 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\section{Introduction}\n\\quad We are interested in the behavior of quantum particles, \nthat is, wave packets propagating through interacting environment. \nIn general, there are two types of environment. \nOne is the ordinary medium (plasma, dielectric, etc.) \nwhich consists of ``matter'' [1-4]. \nThe other is the nontrivial structure of the vacuum due to field\ntheoretical fluctuations \\cite{lpt}\nor effects of quantum gravity \\cite{aemns,gar}. In both cases, \nthe presence of such environments will modify the dispersion \nrelation of particles, $E=f(p)$, or modify the shape of the wave packet. \nObservation of the arrival time of particles through such environments is\na way to see the effects of these modifications.\n%\nRecently, these effects have been tested in two fields, astrophysics and\nquantum optics.\n%\nThe first is the observation of arrival times of photons from distant\nastrophysical sources such as $\\gamma$-ray bursters. \nSeveral models of quantum gravity suggest \nthat the velocity of light has an effective energy dependence \ndue to the modified dispersion relation \ninduced by the nontrivial structure of space-time at distances \ncomparable to the Planck length. \nTo confirm this effect, it is necessary to observe a certain difference\nof the arrival time of photons with different energies, and \n$\\gamma$-ray bursters work for this purpose \\cite{aemns}. \nAs a result, a lower bound on the energy scale of quantum gravity \nis obtained \\cite{sch}. \nThe second recent test is observation of tunneling of photons. \nChiao and co-workers constructed an elaborate stadium for the race between \nphotons propagating in the vacuum and through an optical barrier, \nand measured their arrival times \\cite{exp1,exp2}.\nThey found that the photon tunneling through the barrier arrived \nat the goal earlier than the other photon traveling in the vacuum. \nAlthough this result implies superluminal velocity \nof the tunneling photon, it does not mean causality violation, \nbecause in this case the group velocity itself \ndoes not transport any information at all. \nThe apparent superluminality results from reshaping of wave packets \nwhile tunneling. \nSimilar phenomena can be found in absorbing media \\cite{tfi}.\nAnyway, in both experiments, measurement of the arrival time \nof wave packets plays an essential role. \\par \n%\nHowever, \nthere is no clear definition of arrival time in quantum mechanics.\nThis has its root in the well-known fact that time is not an operator \nbut a parameter in quantum mechanics. \nThough many authors have attempted to define an operator \nof arrival time and construct its eigenstates, \na satisfactory formulation has not yet been obtained [10-25].\n%\\cite{ab,dm,leav,ml,grt,gian,aopru,mlp,muga,ljpu,bspm,lf,kirpol,kw} \nIn this article we define a distribution function $P_{X}(T)$, \nwhich describes the arrival time of packets at a detector located \nat point $X$. \nIn terms of $P_{X}(T)$, we can compute a mean arrival time\n$\\left\\langle T\\right\\rangle _{X}$. \nOf course we assume an ideal detector and our definition of $P_{X}(T)$\nmight not exactly correspond to the physical measurement process. \nHowever, concrete calculation of $P_{X}(T)$ shows us clearly \nthe dynamical properties of propagation of packets \nthrough interacting environments. \\par \n%\n\nWe investigate the arrival time distribution $P_{X}(T)$ numerically\nfor nonrelativistic massive particles traveling through a potential \nbarrier in one space dimension, that is, tunneling processes. \nThis might be a simple model for the experiment \nof Chiao and co-workers. \nIn this case the existence of a potential barrier $V(x)$ \ncauses reflection and transmission of packets; \ntherefore the behavior of $P_{X}(T)$ will be highly nontrivial, \ndepending on various parameters. \nHow to deal with time in tunneling processes is also known \nas the tunneling time problem. \nThe problem arises from the paradox that a particle \nunder a potential greater than the particle's energy seems to move \nwith a purely imaginary velocity. \nIn recent developments of nanotechnology, \nthe study of the tunneling time has great significance \nbecause it might enable us to estimate the response \ntime of nanodevices \\cite{npt}. \nVarious approaches to the tunneling time have been proposed \nby many authors [27-36];\n%\\cite{w,b,h-s,l-m,aagi,yamada,bkr} \nhowever, it seems difficult to define it uniquely\n\\footnote{``The systematic projector approach'' has been proposed as a\nunifying theory of the various times proposed so far \\cite{muga3}.}.\nTherefore we need to define effective tunneling times \nfor each system and each purpose. \nWe have no intention of wrestling with the general theory \nof tunneling time now; therefore, we restrict ourselves to analyzing \nthe time of appearance of the packet in the exit \nof the potential barrier and how it moves after that. \nThese two notions determining the arrival time difference \nhave usually been confused.\nIn this article we will distinguish them clearly. \\par \n%\nFinally we consider the real-time stochastic interpretation \nof quantum mechanics introduced by Nelson \\cite{nel}. \nSince it utilizes the real-time trajectories of quantum particles \nas sample paths, we can construct an appropriate time distribution \nfrom ensemble of sample paths. \nThis is why Nelson's approach is expected to be effective \nfor time problems in quantum mechanics. \nIn particular, it is interesting to attack the tunneling time problem \nfrom this approach because we can trace the particle's real-time motion\neven under the tunneling potential.\nActually it has been found that the tunneling particle ``hesitates'' \nin front of the barrier \\cite{ohba}. \nThis property seems paradoxical because it implies \nthat the particle tunneling through the barrier should always \nbe delayed compared with the free one due to this hesitation and it \nseems contradictory to the advancement of the peak of the wave packet\nas seen in the experiment of Chiao and co-workers. \nIs it a real paradox?\\par\nIt is clear that Nelson's approach can reproduce \nany physical quantities of the usual quantum mechanics \nby averaging them about the sample path ensemble. \nHowever, there is no reason that any ``observables'' classically\ndefined in Nelson's stochastic procedures should have corresponding\nquantities in the standard quantum mechanics. \nWe will compute the arrival time distribution in Nelson's approach \nand compare it with our $P_{X}(T)$. \nThen we clarify the real physical meaning of the ``hesitation'' \nand show that there is no paradox at all. Furthermore, we mention that\nNelson's interpretation can explain the characteristic behavior of\n$\\left\\langle T\\right\\rangle _{X}$ for tunneling particles very well.\n%%%%%%%%%%%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\section{Definition of the arrival time distribution}\n\\quad First we will briefly review previous attempts \nto define a time of arrival operator and their difficulties. \nIn the 1960s, Aharonov and Bohm quantized the representation \nof the classical arrival time for the free particle \nat a point $X=0$ \\cite{ab},\n\\begin{eqnarray}\n T=-m\\frac{x}{p} ~~\\rightarrow ~~\\hat{T}=-\\frac{m}{2}\n\\left(\\hat{x}\\frac{1}{\\hat{p}}+\\frac{1}{\\hat{p}}\\hat{x}\\right).\\label{(1)}\n\\end{eqnarray}\nHere $x$ and $p$ are the initial position and momentum, respectively, \nwhere we work in the Heisenberg picture. Because $\\hat{T}$ satisfies \n$\\left[\\hat{T},\\hat{H}\\right]=i\\hbar$, it seems a good definition.\nWe construct its eigenstates \n$\\hat{T}\\left\|T\\right\\rangle =T\\left\|T\\right\\rangle$.\n\\footnote{In order to obtain a complete set, one needs two\neigenstates $\\left\|T,\\pm\\right\\rangle$ for every value of $T$\n\\cite{muga}.}\nHowever, these eigenstates turn out to be not orthogonal,\n\\begin{eqnarray}\n \\left\\langle p\|T\\right\\rangle &\\propto&[\\theta(p)+i\\theta(-p)]\n \\sqrt{p}e^{ip^2T/2m\\hbar},\\label{(2)} \\\\\n \\left\\langle T\|T'\\right\\rangle &\\propto&\\delta(T-T')\n -\\frac{i}{\\pi}{\\rm P}\\frac{1}{T-T'},\\label{(3)} \n\\end{eqnarray}\nwhere P represents Cauchy's principal value. \nThat is, $\\hat{T}$ is not Hermitian. \nThe origin of difficulty is the singular behavior of $\\hat{T}$ at $p=0$. \nRecently the regularization of $\\hat{T}$ with an infrared momentum\ncut off \\cite{grt} and an interpretation by means of\nthe positive-operator-valued measure were proposed \\cite{gian}. \nHowever, the validity of this procedure is not clear \\cite{aopru,mlp}. \nIn the first place, there is no one-to-one correspondence \nbetween the operator representation in quantum theory\nand the classical representation, and it becomes more complicated \nfor interacting cases [19-22].\n%\\cite{muga,ljpu,bspm} \n\\par\nNow we will not insist on defining an arrival time operator; \nrather, we try to construct an arrival time distribution directly.\nWe suppose that there is a detector on the path along the motion \nof wave packets and it counts the particle according to the value \nof the wave function $\\psi (X,t)$ at every time $t=T$. \nSupposing the detector is ideal, we directly define \n{\\it the arrival time distribution} $P_{X}(T)$ from $\\psi (X,T)$,\n\\begin{eqnarray}\n &&P_{X}(T)dT=\\frac{\\rho_{X}(T)dT}{\\displaystyle\n \\int_0^{\\infty}dT \\rho_{X}(T)},\\quad\n\\rho_{X}(T)=\\left\|\\psi(X,T)\\right\|^2. \\label{(4)} \n\\end{eqnarray}\n\\par\nAlthough Eq. (\\ref{(4)}) looks like a trivial definition in our picture,\nwe will derive it, clarifying our system setup and assumptions.\nWe consider a system consisting of a particle and a detector \nlocated at $x=X$. \nIf there is no interaction between them, the system Hamiltonian $H_0$ \nand the system state $\\left \|\\Psi\\right\\rangle$ are given by \n\\begin{eqnarray}\nH_0&=&H_{\\rm p}\\otimes {\\bf 1}+{\\bf 1}\\otimes H_{\\rm D}, \\label{(a5)}\\\\\n\\left \|\\Psi\\right\\rangle &=&\n\\left \|\\psi\\right\\rangle\\otimes\\left \|D\\right\\rangle ,\\label{(a6)}\n\\end{eqnarray}\nwhere $H_{\\rm p}$ is the particle Hamiltonian, \n$\\left \|\\psi\\right\\rangle$ is the particle state, \nand similarly $H_{\\rm D}$ and $\\left \|D\\right\\rangle$ \nare those of the detector. \nWe define the total Hamiltonian $H$ by adding the interaction\nHamiltonian $H_{\\rm I}$ between the particle and the detector,\n\\begin{equation}\nH=H_0+H_{\\rm I},\\quad H_{\\rm I}=gV_{\\rm p}(x)\\otimes V_{\\rm D}.\n\\label{(a7)}\n\\end{equation}\nFor simplicity, we consider a detector \nwhose state consists essentially of two components, \n\\begin{eqnarray}\n \\left \|\\downarrow\\right\\rangle\n =\\left(\\begin{array}{c}\n 0\\\\\n 1\n\t\\end{array}\\right),{\\rm unreacted},\n \\qquad\n \\left \|\\uparrow\\right\\rangle\n =\\left(\\begin{array}{c}\n\t 1\\\\\n 0\n\t \\end{array}\\right),{\\rm reacted}. \\label{(a8)}\n\\end{eqnarray}\nCorresponding to this representation, we set the interaction potentials\n\\begin{eqnarray}\n V_{\\rm p}(x)=\\delta (x-X), \\label{(a9)}\\\\\n V_{\\rm D}=\\left(\\begin{array}{cc}\n\t\t 0 & 1\\\\\n 1 & 0 \n\t \\end{array}\\right), \\label{(a10)}\n\\end{eqnarray}\nwhich induces a transition $\\left \|\\downarrow\\right\\rangle \\Rightarrow \nV_{\\rm D}\\left \|\\downarrow\\right\\rangle=\\left \|\\uparrow\n\\right\\rangle$. \nThis choice of $V_D$ should be meaningful only \nin the first order of $g$.\n\\par\nNow we consider the time evolution of the system from $t=0$ to $T$. \nWe prepare the initial state\n$\\left \|D(0)\\right\\rangle=\\left \|\\downarrow\\right\\rangle$ \nand evaluate a quantity $R_X(T)$, which is the probability \nthat the state $\\left \|D(t)\\right\\rangle$ is found \nto be $\\left \|\\uparrow\\right\\rangle$ when $t=T$. \nFrom $R_X(T)$, we get $P_X(T)\\Delta T$, \nwhich is the probability that the transition \n$\\left \|\\downarrow\\right\\rangle \\Rightarrow \n \\left \|\\uparrow\\right\\rangle$ \noccurs in a time interval $[T,T+\\Delta T]$, that is,\n\\begin{equation}\nP_X(T)\\Delta T=R_X(T+\\Delta T)-R_X(T)\n\\stackrel{\\rm \\Delta T\\to 0}{\\sim}P_X(T)dT=dR_X(T).\\label{(a11)} \n\\end{equation}\n\\par Next we evaluate $R_X(T)$ in terms \nof the particle wave function $\\psi(x,t)$.\nWe now assume that the detector reacts only once incoherently, \nand therefore we calculate only in the first order of $g$. \nAdopting the interaction picture, the time evolution of the\nstate can be represented as follows in the first order of $g$: \n\\begin{eqnarray}\n \\left \|\\Psi (T)\\right\\rangle _{\\rm I}\n &=&{\\rm T}e^{-i/\\hbar \\int_0^{T}\\!dt\\,gV_{\\rm pI}(x,t)\n \\otimes V_{\\rm DI}(t)}\n \\left \|\\psi (0)\\right\\rangle _{\\rm I}\n \\otimes\\left \|D(0)\\right\\rangle _{\\rm I} \\nonumber\\\\\n &\\simeq&\\left \|\\psi (0)\\right\\rangle _{\\rm I}\n \\otimes\\left \|D(0)\\right\\rangle _{\\rm I} \\nonumber \\\\\n &&-\\frac{i}{\\hbar}\\int_0^{T}\\!dt\\,g\n V_{\\rm pI}(x,t)\\left \|\\psi (0)\\right\\rangle _{\\rm I}\n \\otimes V_{\\rm DI}(t)\\left \|D (0)\\right\\rangle _{\\rm I}\n \\nonumber \\\\\n &\\equiv& \\left \|\\Psi (0)\\right\\rangle _{\\rm I}\n +\\overline{\\left \|\\Psi (T)\\right\\rangle _{\\rm I}}\n ,\\label{(a14)}\n\\end{eqnarray}\nwhere T represents the time ordered product. \n$\\left \|\\Psi (0)\\right\\rangle _{\\rm I}$ is the undetected state \nand $\\overline{\\left \|\\Psi (T)\\right\\rangle _{\\rm I}}$ \nis the detected state, which is written in the Schr\\\"odinger picture as\n\\begin{eqnarray}\n \\overline{\\left \|\\Psi (T)\\right\\rangle}\n &=&-\\frac{i}{\\hbar}\n \\int_0^{T}\\!dt\\,\\left[e^{-iH_{\\rm p}(T-t)/\\hbar}gV_{\\!\\rm p}(x)\n e^{-iH_{\\rm p}t/\\hbar}\\right]\n \\left \|\\psi (0)\\right\\rangle \\nonumber \\\\\n &&~\\qquad\\quad\\otimes\\left[e^{-iH_{\\rm D}(T-t)/\\hbar}V_{\\rm D}\n e^{-iH_{\\rm D}t/\\hbar}\\right]\n \\left \|D(0)\\right\\rangle \\nonumber\\\\\n &\\equiv&-\\frac{i}{\\hbar}\n \\int_0^{T}\\!dt\\,\\left \|\\psi (T;t)\\right\\rangle\n \\otimes\\left \|D(T;t)\\right\\rangle,\\label{(a21)}\n\\end{eqnarray}\nwhere we introduced \n\\begin{eqnarray}\n \\left \|\\psi (T;t)\\right\\rangle\n &\\equiv&\\left[e^{-iH_{\\rm p}(T-t)/\\hbar}gV_{{\\rm p}}(x)\n e^{-iH_{\\rm p}t/\\hbar}\\right]\n \\left \|\\psi (0)\\right\\rangle, \\label{(a22)}\\\\\n \\left \|D(T;t)\\right\\rangle\n &\\equiv&\\left[e^{-iH_{\\rm D}(T-t)/\\hbar}V_{{\\rm D}}\n e^{-iH_{\\rm D}t/\\hbar}\\right]\n \\left \|D(0)\\right\\rangle. \\label{(a23)} \n\\end{eqnarray} \nWe obtain $R_X(T)$ in terms of the norm of the detected state,\n\\begin{equation}\nR_X(T)=\\overline{\\left\\langle\\Psi (T)\\right.}\n \|\\overline{\\left .\\Psi (T)\\right\\rangle},\\label{(a24)} \n\\end{equation}\nunder our approximation of weak coupling.\nNow we apply a macroscopic decoherence condition,\n\\begin{equation}\n \\left\\langle D(T;t_1)\\right.\|\\left .D(T;t_2)\\right\\rangle\n =\\delta (t_1-t_2).\\label{(a25)} \n\\end{equation}\nThis means that the states reacted at different times are orthogonal \nto each other, that is, once the detection process occurs, \nthe total state effectively loses its coherence \nand looks like a mixed state. \nOf course it is not possible to satisfy this condition by working\nin the two-dimensional Hilbert space in Eq. (\\ref{(a8)}). \nWe should describe the detector by means of an\ninfinite-dimensional Hilbert space to realize decoherence \neffectively \\cite{hall}. \nHowever, we can avoid this assumption by\n``switching on'' the interaction Hamiltonian during different small time\nintervals in repeated experiments, instead of using the finite time\ninterval $[0,T]$ and differentiating with respect to $T$.\\par\n\nUnder this condition, \nthe evaluation of $R_X(T)$ and $P_X(T)$ is straightforward as follows: \n\\begin{eqnarray}\n R_X(T)=\\overline{\\left\\langle\\Psi (T)\\right.}\n \|\\overline{\\left .\\Psi (T)\\right\\rangle}\n &=&\\frac{1}{\\hbar ^2}\\int_0^{T}\\!dt_1\\int_0^{T}\\!dt_2\\,\n \\left\\langle\\psi (T;t_1)\|\\psi(T;t_2)\\right\\rangle\n \\delta (t_1-t_2) \\nonumber\\\\\n &=&\\frac{1}{\\hbar ^2}\\int_0^{T}\\!dt\\,\n \\left\\langle\\psi (T;t)\|\\psi(T;t)\\right\\rangle, \\label{(a26)}\n\\end{eqnarray}\n\\begin{eqnarray}\n P_X(T)=\\frac{\\partial}{\\partial T}R_X(T)\n &=&\\frac{\\partial}{\\partial T}\n \\overline{\\left\\langle\\Psi (T)\\right.}\n \|\\overline{\\left .\\Psi (T)\\right\\rangle} \n =\\frac{1}{\\hbar ^2}\n \\left\\langle\\psi (T;T)\|\\psi(T;T)\\right\\rangle \\nonumber \\\\\n &=&\\frac{g^2}{\\hbar ^2}\n \\left\\langle\\psi (0)\\left\|e^{iH_{\\rm p}T/\\hbar}\n V^{\\dagger}_{\\rm p}(x)V_{\\rm p}(x)\n e^{-iH_{\\rm p}T/\\hbar}\\right\|\\psi(0)\n \\right\\rangle \\nonumber \\\\\n &=&\\frac{g^2}{\\hbar ^2}\\delta (0)\\left\|\\psi (X,T)\\right\|^2\n ,\\label{(a27)}\n\\end{eqnarray}\nwhere at the last step we inserted the complete set \n($\\int dx \\left\|x\\right\\rangle \\left\\langle x\\right \|$) three times. \nAlthough the divergent $\\delta (0)$ seems to break the validity \nof our formulation, \nwe can remove this singularity by replacing the $\\delta$ function \nin Eq. ({\\ref{(a9)}}) with a smeared function.\nWe normalize the right hand side of Eq. ({\\ref{(a27)}}) \nto get our expression for the arrival time distribution $P_{X}(T)$.\nUsing $P_{X}(T)$ we define {\\it the mean arrival time} \n$\\left\\langle T\\right\\rangle _{X}$,\n\\begin{equation}\n \\left\\langle T\\right\\rangle _{X}=\\int _0^{\\infty}T P_{X}(T)dT\n .\\label{(5)} \n\\end{equation}\nBecause $P_{X}(T)$ and $\\left\\langle T\\right\\rangle _{X}$ \nhave simple and general expressions, \nwe can calculate them easily even for interacting cases.\nOur $P_X(T)$ is often called the ``presence time distribution''\nbecause of its behavior in the classical limit \\cite{muga1}.\nIn order to avoid confusion, we should make clear that\nthe distribution $P_X(T)$ may not be interpreted as\nthe probability distribution of a quantum mechanical time observable. \nIt is an effective distribution describing a ``relative probability.''\n\\par\nOf course, our definition of arrival time distribution Eq. ({\\ref{(4)}}) is \nnot a unique one. Considering a different system setup, some people\nhave proposed a definition using the current $J_{X}(T)$ \ninstead of $\\rho_{X}(T)=\\left\|\\psi\\right\|^2$ \\cite{dm},\n\\begin{eqnarray}\n &&P_{X}^{c}(T)dT=\\frac{J_{X}(T)dT}{\\displaystyle\n \\int_0^{\\infty}dT J_{X}(T)},\\quad\n J_{X}(T)=\\frac{\\hbar}{m}{\\rm Im}\\left.\\left(\\psi^{}\\frac{\\partial\n \\psi}{\\partial x}\\right)\\right\|_{x=X}.\n \\label{(4J)} \n\\end{eqnarray}\nThis definition has the serious problem \nthat $J_{X}(T)$ can be negative in some cases, \nfor example, detection before the potential barrier. \nTherefore we cannot identify $P_{X}^{c}(T)$ as a probability distribution.\nAs for detection beyond the potential barrier as we discuss below, \n$J_{X}(T)$ might effectively maintain positivity and\nactually the behavior of $P_{X}^{c}(T)$ is found to be similar to ours. \n%%%%%%%%%%%%%%%%%%%%%%%% Section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\section{Calculation of $P_X(T)$ for tunneling particles}\n\\quad Now let us calculate $P_{X}(T)$ and $\\left\\langle\nT\\right\\rangle _{X}$ for nonrelativistic massive particles traveling\nthrough a potential barrier $V(x)$ in one dimension. \nThis is a simple model of tunneling processes\nsuch as the experiment of Chiao and co-workers . \nSolving the time dependent Schr\\\"odinger equation with some initial\nconditions, we can get $\\psi(x,t)$. \nExcept for the free case it is difficult to solve the partial\ndifferential equation analytically, and therefore we solve it numerically.\nWe now employ a discretization scheme known as the Crank-Nicholson method, \nwhich conserves the norm of $\\psi(x,t)$ even with a finite\ndiscrete time step \\cite{cal3}.\nWe work with the units $m=\\hbar=1$ and for the initial\ncondition we prepare a Gaussian wave packet,\n\\begin{equation}\n \\psi(x,0)=\\left(\\frac{1}{\\pi\\sigma^2}\\right)^{1/4}\n e^{-(x-x_0)^2/2\\sigma^2}e^{ik_0(x-x_0)}\\label{(6)},\n\\end{equation}\nwhose mean energy is $\\langle E\\,\\rangle=k_0^{\\,2}/2+1/4\\sigma ^2$, \nand we set a time independent square potential barrier $V(x)$ \non a section $[0,d]$. \nFor simplicity, in this article we work with a unique initial packet.\nWe fix the central wave number $k_0=2$ and in this unit we set\nthe width of the initial packet in the configuration space \n$\\sigma =10(2/k_0)$ and the center of initial packet $x_0=-50(2/k_0)$. \nAll quantities that have a dimension of time are measured \nby the $(4/k_0^2)$ unit.\nHereafter we will omit the units of the numerical values.\nWe change two parameters of the barrier potential $V(x)$: \nthe width $d$ and the height $h$, and also the detector location $X$. \n\\par\n\\begin{figure}[htb] \n\\hspace{0mm}\n \\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{snap.eps}\n\\vspace{-5mm}\n \\caption{Snapshots of the wave function squared at various times}\n \\label{fig:tunnel} \n }\n\\hspace{2mm} \n\\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{dist.eps}\n\\vspace{-5mm}\n \\caption{The arrival time distribution for detection at $X=50 [2/k_0]$}\n \\label{fig:detect1} \n }\n\\end{figure}\nLet us begin by watching the motion of wave packets\nwith an $h=1.1\\langle E\\,\\rangle$, $d=1.5$ potential.\nThe snapshots of the motion are shown in Fig. \\ref{fig:tunnel}, \nin which the free packet motion is also shown for comparison. \nBoth packets spread due to the dispersive properties \nthat come from their own masses. \nThe packet moving through the potential barrier experiences reflection \nand transmission and the peak of the transmitted part will often advance\ncompared to the free packet. \nIt is usually explained that this is because the higher momentum\ncomponents of the packet preferably go through the barrier and they\npropagate faster than the lower momentum parts due to their dispersive\nproperties. \nThat is, the advancement results from reshaping of the transmitted packet.\nHowever, tracing the peak of the packet is often difficult \nbecause near the barrier the peak cannot be clearly identified. \nTherefore we must use more well-defined quantities, \n$P_{X}(T)$ and $\\left\\langle T\\right\\rangle _{X}$. \n\\par\n\n\\begin{quote}\n {\\bf Analysis 1: Detection at $X=50$} \n\\end{quote} \n\\quad \nNow let us calculate $P_{X}(T)$ and $\\left\\langle T\\right\\rangle _{X}$\nat $X=50$ with an $h=2\\langle E\\,\\rangle$, $d=4$ potential.\nIn Fig. \\ref{fig:detect1}, the arrival time distributions $P_{X}(T)$ \nare plotted for the free and tunneling particles \nand the mean arrival times $\\left\\langle T\\right\\rangle _{X}$ \nare shown by dashed lines. \nThe remarkable feature of $P_{X}(T)$ is the stretched tail \nand the shift of the peak caused by spread of the packet. \nFor the free case, $\\left\\langle T\\right\\rangle _{X}=50.13$ is later than \n$T=50$, which is expected from the group velocity of the free packet, \nand ``the peak of $P_{X}(T)$'' $=49.94$ is earlier than $T=50$. \nIt is also seen that, because of the packet's reshaping, $P_{X}(T)$ \nfor the tunneling particle has a narrower shape than the free one \nand ``$\\left\\langle T\\right\\rangle _{X}$ for the tunneling particle'' \n$=47.65$ is earlier than the free one. \nHowever, it should be noted that only one detection far from the barrier \ncannot describe the dynamics of packets since we should discriminate\neffects in and out of the potential barrier. \nTherefore we investigate detection at various points.\n\\begin{quote}\n {\\bf Analysis 2: Detection at various points} \n\\end{quote}\n\\quad \nWhen we try to give a definite answer to the so-called \ntunneling time problem, we might have to calculate the difference \n$\\Delta =\\left\\langle T\\right\\rangle _{d}-\\left\\langle\nT\\right\\rangle _{0}$, since this problem demands that we answer the\nquestion ``How long does it take for the particle to tunnel across the\nbarrier?'' However the difference $\\Delta$ does not make much sense \nbecause, as we can see in Fig. \\ref{fig:tunnel}, the shape \nof the packet is oscillating frequently at the entrance of the barrier \nand it is difficult to distinguish between the tunneling packet \nand the reflected one, that is, \n$\\left\\langle T\\right\\rangle _{0}$ is not a good physical quantity. \nOn the other hand, \nthe packet has a relatively clear shape at the exit of the barrier.\nTherefore we can analyze what time the packet will appear at \nthe exit of the barrier and how it moves after that. \\par\n\\vspace{5mm}\n\\begin{figure}[htb] \n\\hspace{0mm}\n \\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{avet.eps}\n\\vspace{-5mm}\n \\caption{The mean arrival time for detection at various points }\n \\label{fig:avet} \n }\n\\hspace{2mm} \n\\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{km.eps}\n\\vspace{-5mm}\n \\caption{The mean momentum of the transmitted packet}\n \\label{fig:km} \n }\n\\end{figure}\nWe calculate $\\left\\langle T\\right\\rangle _{X}$ \nfor the exit of the barrier and several points after that: \n$\\left\\langle T\\right\\rangle _{X}$ at $X=d,10,20,30,40,50$, with $d=4$, \n$h=0.5\\langle E\\,\\rangle,1.1\\langle E\\,\\rangle,2\\langle E\\,\\rangle$\nbarrier potentials (Fig. \\ref{fig:avet}). \nWe can see two remarkable features in this figure. \nThe first is that for high barriers $\\left\\langle T\\right\\rangle _{d}$ \nis earlier than in the free case, but it is later for low barriers. \nThat is, it seems that the transmitted packet arrives at the barrier \nexit earlier than the free one for tunneling dominated cases. \nThese are regarded as effects in the barrier.\nThe second feature is that, after passing the barrier, \nthe tunneling packet moves with a constant mean velocity larger than \nthat of the corresponding free packet. \nThis is an effect outside the barrier. \nThese two types of effect are combined to cause nontrivial behaviors \nof the arrival time. For example, in the case of $d=4$, \n$h=0.5\\langle E\\,\\rangle$, the tunneling packet arrives \nat the barrier exit $X=4$ later than the free packet; \nhowever, after exiting the barrier, the tunneling packet \ncatches up with the free one and overtakes it at $X\\simeq 15$. \nAfter all, it depends on $X$ which arrives at $X$ earlier, \nthe tunneling or the free packet.\\par\nWe can see the second effect clearly in the Fourier transformed\nform of the transmitted packet \\cite{aagi},\n\\begin{eqnarray}\n\\psi(x,t) = (4\\pi\\sigma ^2)^{1/4}\\int \\frac{dk}{2\\pi}\ne^{-\\sigma ^2(k-k_0)^2/2}\|T_k\| e^{i\\theta} e^{i[k(x-x_0)-\\omega\nt]},\\label{(27)}\n\\end{eqnarray}\nwhere $T_k$ is the transmission amplitude and $\\theta$ is the phase. \nUsing an analytically obtained $\|T_k\|$, the mean momentum $k_m$ can be\ncalculated for the transmitted packet. Results for several potential\nconditions are shown in Fig. \\ref{fig:km}. For ``high'' barriers, \na wider barrier gives a larger mean momentum in the region $d:[0,4]$. \nThis is because as $d$ grows the $\|T_k\|$ support shifts to \nthe higher momentum side. \nTherefore the statement ``Higher momentum components of the packet \npreferably go through the barrier'' applies indeed.\nThis kind of ``acceleration'' effect is found in other areas of physics \n\\cite{obe}.\\par\n\\begin{quote}\n {\\bf Analysis 3: Detection at the barrier exit $X=d$}\n\\end{quote}\n\\quad\nTo see the in-barrier effects more definitely, \nwe calculate the difference between mean arrival times \nfor the tunneling packet and the free one at the barrier exit $X=d$,\n\\begin{eqnarray}\n\\Delta T&\\equiv&\\left\\langle T\\right\\rangle _{d}^{\\rm tunnel}\n-\\left\\langle T\\right\\rangle _{d}^{\\rm free}.\n\\end{eqnarray}\nResults for the same potential\nconditions as in Fig. \\ref{fig:km} are shown in Fig. \\ref{fig:deltat}.\nAt first we see that in the small $d$ region $\\Delta T$ is positive, \nthat is, the tunneling packet gets behind the free one, \nfor any potential height. However, as $d$ increases, \n$\\Delta T$ shows different behaviors according to the potential height.\n\\par\n\\begin{figure}[htb]\n\\begin{center}\n \\epsfxsize=122mm\n \\epsfysize=122mm\n \\leavevmode\n\\epsfbox{vm.eps}\n\\vspace{-5mm}\n \\caption{The difference between mean arrival times at the barrier exit\n $X=d$ for the tunneling packet and the free one:\n$\\Delta T\\equiv\\left\\langle T\\right\\rangle _{d}^{\\rm tunnel}\n-\\left\\langle T\\right\\rangle _{d}^{\\rm free}$. $\\Delta T _{\\varphi}$\n(shown by dots) are the same quantity calculated by the stationary\nphase method \nfor each potential.}\n \\label{fig:deltat}\n\\end{center}\n\\end{figure}\nRoughly speaking, for a ``low'' barrier $\\Delta T$ almost stays positive \nbut for a ``high'' barrier $\\Delta T$ becomes negative.\nThe ``high'' barrier means that the tunneling modes dominate in the\ntransmitted packet. In the large $d$ region, $\\Delta T$ is negative, \nthat is, the tunneling packet goes ahead of the free one \nfor the tunneling dominated case. \nWe also see a strange behavior where $\\Delta T$ changes sign twice \nand finally becomes positive. \nThe typical case in Fig. \\ref{fig:deltat} is \nthe $h=1.1\\langle E\\,\\rangle$ barrier. \nWe understand this effect as follows. For a very wide barrier, \nover-the-barrier modes dominate in thetransmitted packet \n($\\omega _m=k_m^2/2>h$). That is, as in the ``low'' barrier case, \n$\\Delta T$ becomes positive again.\n\\par\nIn Fig. \\ref{fig:deltat} we also plotted an analogous quantity \n$\\Delta T_{\\varphi}$ calculated by the stationary phase method. \nWe define $\\Delta T_{\\varphi}$ as follows:\n\\begin{eqnarray}\n\\tau _{\\varphi}&\\equiv\n&\\left.\\frac{d\\theta}{d\\omega}\\right\|_{\\omega=\\omega_m},\\\\\n&&\\nonumber\\\\\n\\Delta T_{\\varphi}&\\equiv&\\left\n( \\frac{1}{v_g(k_m)}-\\frac{1}{v_g(k_0)}\\right)(d-x_0)+\\tau _{\\varphi},\n\\end{eqnarray}\nwhere $\\theta$ is the phase shift of the transmitted wave defined in\nEq. (\\ref {(27)}) and $v_g(\\cdot)$ are the group velocities\n$v_g(k_m)=\\left.(d\\omega/dk)\\right\|_{k=k_m}=k_m$, $v_g(k_0)=k_0$. \nIn the ordinary tunneling time problem context $\\tau _{\\varphi}$ \nis called the phase time. As seen in Fig. \\ref{fig:deltat},\nalthough $\\Delta T_{\\varphi}$ has good agreement with our $\\Delta T$ \nin the small $d$ region, as $d$ increases, the difference becomes clear \nfor $h\\simeq\\langle E\\,\\rangle$ barriers.\nThis is because for such barriers the momentum distribution\n$e^{-\\sigma ^2 (k-k_0)^2/2}\|T_k\|$ is no longer symmetric \nwith respect to $k_m$, and the packet's peak given by \nthe stationary phase method loses physical significance.\n\\par\n%\\vspace{7mm}\nWe would like to close this section by referring to the relationship \nbetween our results and the experiment by Chiao and co-workers, \nthat is, tunneling of the massless photon.\nOf course our model does not describe the propagation of photons, \nand we now mention only the qualitative behavior. \nBecause the energy of the photon in the vacuum is \nexactly proportional to its momentum, \nthe group velocity of the photon after tunneling is constant $c$. \nTherefore we get an $X$ independent constant value of the difference \n$\\Delta T=\\left\\langle T\\right\\rangle _{X}^{\\rm tunnel}\n-\\left\\langle T\\right\\rangle _{X}^{\\rm free}$ at any $X\\geq d$. \nSince their experimental setup is the tunneling dominated one, \nit may correspond to our model with high and medium wide\nbarriers. Then our results are consistent with their experimental\nobservation that the tunneling photon arrives earlier than the free photon. \n%%%%%%%%%%%%%%%%%%%%%%%% Section 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\section{Nelson's stochastic interpretation}\n\\quad\nNow we consider the stochastic interpretation of quantum mechanics\nintroduced by Nelson. \nThis approach interprets the motion of particles in quantum mechanics \nas ``real-time'' stochastic processes \\cite{nel}.\nNelson substituted the coordinate variable $x(t)$ \nfor a stochastic variable performing the Brownian motion \nin a certain drift force field.\nThe time evolution of $x(t)$ is described by the Ito-type stochastic\ndifferential equation,\n\\begin{equation}\n dx(t)=b\\left(x(t),t\\right)dt+dw(t),\\label{(7)}\n\\end{equation}\nwhere $b(x,t)$ is the so-called drift term,\ngiven by the ordinary Schr\\\"odinger wave function $\\psi (x,t)$ as\n\\begin{equation}\n b(x,t)=\\frac{\\hbar}{m}\\frac{\\partial}{\\partial x}\n \\left({\\rm Im+Re}\\right){\\rm ln}\\psi (x,t). \\label{(8)}\n\\end{equation} \nThe Gaussian noise $dw$ characterizes the stochastic behavior\nand should have the following statistical properties:\n\\begin{eqnarray}\n \\langle dw(t)\\rangle=0, \\quad \n \\langle dw(t)dw(t)\\rangle=\\frac{\\hbar}{m}dt.\\label{(10)}\n\\end{eqnarray}\nStarting with an initial distribution of $x(0)$ we solve Eq. (\\ref {(7)})\nand obtain sample paths. \nAveraging a physical variable with these sample paths, \nwe can calculate the expectation value \nfor the ordinary probability distribution $\\left\|\\psi\n(x,t)\\right\|^2$. In this approach,\nwe are able to observe ``trajectories'' of real-time motion of a particle, \nthat is, to describe the quantum mechanical time evolution \nby a classical stochastic process. \n\\par Thus in Nelson's approach \nit may be possible to understand an imaginary-time process \nsuch as tunneling in real-time language. \nIt was pointed out that the tunneling particle ``hesitates'' \nin front of the barrier as seen in Fig. \\ref{fig:paths} \\cite{ohba}. \nThis fact was understood to imply that the particle tunneling \nthrough the barrier should always be delayed compared with the free one \nbecause of this hesitation. \nIs it contradictory to our results? \nNelson's approach can reproduce physical quantities \nin standard quantum mechanics, and there cannot be any conflict.\n\\par\n\\begin{figure}[htb] \n\\hspace{0mm}\n\\vspace{0mm} \n \\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{path.eps}\n\\vspace{-5mm}\n \\caption{Typical sample paths with hesitation}\n \\label{fig:paths} \n }\n\\hspace{2mm} \n\\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{neldis.eps}\n\\vspace{-5mm}\n \\caption{Comparison of two methods~~~~~~}\n \\label{fig:neldis} \n }\n\\end{figure}\n\nNow we analyze the mean arrival time in Nelson's stochastic interpretation. \nOne intuitive idea of defining the arrival time for a sample path is to\nmeasure the time for a path to reach a detecting point for the first time:\n``the first time counting scheme'' \\cite{hasi}. \nHowever, this notion has no counterpart in the physical quantities \nof standard quantum mechanics. \nWe have to work with the probability of existence of paths at a point\n(or a section) at some definite time. \nThe difference between these two notions is that the latter counts \nthe possibility of a path going beyond the point \nand coming back to it at the measuring time.\n\\par \nWe define a probability function $\\rho_{X}^{\\rm N}(T)$, \n\\begin{eqnarray}\n\\rho_X^{\\rm N}(T)~dx=\\frac{n(X,T)}{N},\\label{(12)}\n\\end{eqnarray}\nwhere $N$ is the total number of sample paths \nand $n(X,T)$ is the number of sample paths that exist in $[X,X+dx]$ \nat time $T$. As stressed before, \nwe will count the number of paths passing a target point \nover and over again, i.e., we now employ ``the multiple counting scheme.'' \nWith this scheme, \nwe define the arrival time distribution $P_{X}^{\\rm N}(T)$\nand the mean arrival time $\\left\\langle T\\right\\rangle _{X}^{\\rm N}$ \nof the particle in Nelson's stochastic interpretation,\n\\begin{eqnarray}\n &&P_{X}^{\\rm N}(T)~dT=\n \\frac{\\rho_X^{\\rm\n N}(T)~dT}{{\\displaystyle\\int_0^{\\infty}dT\\rho_X^{\\rm N}(T)}}\n ,\\label{(13)}\\\\ \n &&\\left\\langle T\\right\\rangle _{X}^{\\rm N}\n =\\int _0^{\\infty}T P^{\\rm N}_{X}(T)dT.\\label{(14)}\n\\end{eqnarray}\n\\par\nWe calculate $P_X^{\\rm N}(T)$ and $\\left\\langle T\\right\\rangle_X^{\\rm N}$ \nwith an $h=2\\left\\langle E\\right\\rangle$, $d=1$ barrier \nby solving Eq. ({\\ref{(7)}}) to get $N=10^6$ sample paths.\nThe result is shown in Fig. \\ref{fig:neldis}.\nThe distribution $P_X^{\\rm N}(T)$ agrees with $P_X(T)$ very well; \ntherefore $\\left\\langle T\\right\\rangle _X^{\\rm N}$ agrees with \n$\\left\\langle T\\right\\rangle _X$. \nThe distribution given by Nelson's approach exactly reproduces \nour previous results, just as expected. \nOf course, if we employ the first time counting scheme, \n$P_X^{\\rm N}(T)$ shifts to an earlier time region \nand therefore $\\left\\langle T\\right\\rangle _X^{\\rm N}$ is smaller \nthan $\\left\\langle T\\right\\rangle _X$.\n\\par\n\\begin{figure}[htb] \n\\hspace{0mm}\n \\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{path05.eps}\n\\vspace{-5mm}\n \\caption{Sample paths for $\\Delta T>0$}\n \\label{fig:path05} \n }\n\\hspace{2mm} \n\\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{path10.eps}\n\\vspace{-5mm}\n \\caption{Sample paths for $\\Delta T<0$}\n \\label{fig:path10} \n }\n\\end{figure}\n\\par\n%\\vspace{-3mm}\n\\vspace{0mm}\n\\begin{figure}[htb] \n\\hspace{0mm}\n \\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{drift05.eps}\n\\vspace{-5mm}\n \\caption{Drift velocity for $\\Delta T>0$}\n \\label{fig:drift05} \n }\n\\hspace{2mm} \n\\parbox{70mm}{\n \\epsfxsize=70mm \n \\epsfysize=70mm\n \\leavevmode\n\\epsfbox{drift10.eps}\n\\vspace{-5mm}\n \\caption{Drift velocity for $\\Delta T<0$}\n \\label{fig:drift10} \n }\n\\end{figure}\n\nThen, we should answer the paradoxical question, \n``Why does a hesitating particle arrive earlier than the free one?'' \nTo answer this question, let us compare the two cases of\n$\\Delta T>0$ and $\\Delta T<0$. \nFirst we show typical sample paths for two cases, \n$\\Delta T>0~$($\\Delta T=0.084$), $h=2\\left\\langle E\\right\\rangle$, \n$d=0.5$ in Fig. \\ref{fig:path05} and $\\Delta T<0~$($\\Delta T=-0.138$), \n$h=2\\left\\langle E\\right\\rangle$, $d=1$ in Fig. \\ref{fig:path10}. \nIn both figures we also plot the average position \nof the free sample paths $\\left\\langle x\\right\\rangle_{\\rm free}$.\nWe should pay attention to the point $(x,t)=(0,25)$ \nbecause $\\left\\langle x\\right\\rangle_{\\rm free}$ arrives in $x=0$ at $t=25$. \nThe two figures, Fig. \\ref{fig:path05} and Fig. \\ref{fig:path10}, \nmake a remarkable contrast. \nThat is, although in Fig. \\ref{fig:path05} even the paths \nthat arrive at $x=0$ later than $t=25$ can pass through the barrier, \nin Fig. \\ref{fig:path10} essentially only the paths \nthat arrived at $x=0$ earlier than $t=25$ can go through it. \nThe ``hesitation'' property is seen in both cases. \nIn Fig. \\ref{fig:path10}, however, even with ``hesitation,'' \nthe averaged tunneling path can appear at the barrier exit $x=d$ \nearlier than the averaged free path \nbecause the tunneling path arrived at $x=0$ much earlier \nthan $\\left\\langle x\\right\\rangle_{\\rm free}$. \nThis is the key to the mystery between hesitation and advancement.\n\\par Well, \nwhy do the tunneling paths conduct themselves in such a strange way? \nIn the first place, why does hesitation occur?\nThe reason is hidden in the time dependence of the drift velocity\n$b(x,t)$. We show $b(x,t)$ for the same conditions discussed above,\nespecially near the potential barrier \n(Fig. \\ref{fig:drift05} and Fig. \\ref{fig:drift10}).\nIn the foreground of the barrier, according to the interference \nof the incident packet and the reflected packet, \n$b(x,t)$ oscillates frequently and becomes null many times. \nEspecially near the barrier entrance $x=0$, \n$b(x,t)$ changes from positive to negative, where the particle \nis ``trapped.'' \nThese effects cause the path's hesitation. \n\\par At earlier times, \n$b(x,t)$ is almost always positive value but at later times, \nit becomes almost always negative. In Fig. \\ref{fig:drift10}, \nthis tendency is extreme and realization of the tunneling path is \nmuch rarer than in Fig. \\ref{fig:drift05}. \nThis is the reason why the early arrived paths tend to pass the barrier \nmore easily. \nAfter all, there is no inconsistency between our results \nand the hesitation behavior in Nelson's interpretation.\n\\par \nFurthermore, Nelson's interpretation provides us \nan intuitive explanation of our results. \nLet us consider the high potential barrier case. \nIt is important that every transmitted path hesitates to some extent. \nIn the small $d$ region, because of the high transmission rate, \neven a path arriving at $x\\simeq 0$ relatively late can pass the barrier, \nand as a result we find $\\Delta T>0$. As $d$ increases, \nthe transmission rate becomes lower and only the paths arriving \nat $x\\simeq 0$ earlier can penetrate the barrier, \nand as a result we find $\\Delta T<0$. Finally, as $d$ becomes very large, \nthe paths arriving at $x\\simeq 0$ very early hesitate \nthere for a very long time;\n therefore $\\Delta T$ becomes positive again.\n\\par\nOf course, we must recall that the ``path'' in Nelson's view \nnever corresponds to a real particle in ordinary quantum mechanics, \nand the explanation we gave above is just an interpretation. \nThe same is true for an interpretation by the Bohm trajectory \n\\cite{leav,ml}.\nIt may be interesting to regard the path as a physical one \nand to calculate various quantities that cannot be calculated \nin ordinary quantum mechanics (the tunneling time \n$\\Delta^{\\rm N}=\\left\\langle T\\right\\rangle _{d}^{\\rm N}\n-\\left\\langle T\\right\\rangle _{0}^{\\rm N}$, \nquantities calculated in the first time counting scheme, etc.). \nAlthough these attempts may give us deeper insights into quantum dynamics,\nthe validity and significance of them have not been argued much so far \n\\cite{ohba2,hino}.\n%%%%%%%%%%%%%%%%%%%%%%%% Section 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\section{Summary}\n\\quad Supposing an ideal detector, \nwe defined simple expressions for the arrival time distribution $P_X$ \nand the mean arrival time $\\left\\langle T\\right\\rangle_X$, \nand applied them to analysis of wave packet tunneling. \nWe defined $\\Delta T\\equiv\\left\\langle T\\right\\rangle _{d}^{\\rm tunnel}\n-\\left\\langle T\\right\\rangle _{d}^{\\rm free}$ and calculated it for\nvarious barrier conditions and showed the barrier effects clearly. \nIn the small $d$ region, $\\Delta T$ is always positive, \nbut as $d$ increases, $\\Delta T >0$ for the over-the-barrier case \nand $\\Delta T <0$ for the tunneling case. \nAfter tunneling, the packet usually moves faster than the free one \nbecause it preferentially consists of the higher momentum modes \nof the incident packet.\nThe barrier works as an acceleration filter in a sense.\nWe also clarified that the stationary phase method gives a good\napproximation to our results, particularly in the small $d$ region.\n\\par\nWe also confirmed that the stochastic interpretation \nintroduced by Nelson reproduces our results. \nFurthermore, we clarified how the ``hesitation'' of the tunneling paths \nin Nelson's picture is consistent with the advancement \nof the tunneling packet.\nThe key observation is that the paths arriving at the barrier earlier \nthan the free mean paths tend to penetrate the barrier more easily.\nWe pointed out that this property can be explained by the time dependence \nof the drift velocity $b(x,t)$ and found that the behavior of $\\Delta T$ \nis intuitively understandable with Nelson's language.\n%\\thanks\n\\section{Acknowledgments}\n\\pagestyle{myheadings} \n\\quad \nWe would like to thank T. Hashimoto, E. M. Ilgenfritz, K. Imafuku, \nK. Morikawa, I. Ohba, T. Tanizawa, H. Terao, and M. Ueda \nfor fruitful and encouraging discussions and suggestions. \nWe are also grateful to J. G. Muga for comments and for \ncalling our attention to some relevant references.\n%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\begin{thebibliography}{99}\n \\bibitem{exp1}\n Quantum Optical Studies of Tunneling and other Superluminal Phenomena\\\\\n R. Y. Chiao and A. M. Steinberg, \n Phys. Scr. {\\bf T76}, 61 (1998).\n \\bibitem{exp2}\n Measurement of the Single-Photon Tunneling Time\\\\\n A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, \n Phys. Rev. Lett. {\\bf 71}, 708 (1993).\n \\bibitem{exp3}\n An atom optics experiment to investigate faster-than-light tunneling\\\\\n A. M. Steinberg, S. Myrskog, H. S. Moon, H. A. Kim, J. Fox, \n and J. B. Kim, \n Ann. Phys. (Leipzig) {\\bf 7}, 593 (1998).\n \\bibitem{exp4}\n Light speed reduction to 17 metres per second\n in an ultracold atomic gas\\\\ \n L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, \n Nature (London) {\\bf 397}, 594 (1999).\n \\bibitem{lpt}\n Speed of light in non-trivial vacua\\\\\n J. I. Latorre, P. Pascual, and R. Tarrach,\n Nucl. Phys. B {\\bf 437}, 60 (1995).\n \\bibitem{aemns}\n Tests of quantum gravity from observations of $\\gamma$-ray bursts\\\\\n G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, \n and S. Sarkar, \n Nature (London) {\\bf 393}, 763 (1998).\n \\bibitem{gar}\n Quantum Evolution in Space-Time Foam\\\\\n L. J. Garay,\n Int. J. Mod. Phys. A {\\bf 14}, 4079 (1999).\n \\bibitem{sch}\n Severe Limits on Variations of the Speed of Light with Frequency\\\\\n B. E. Schaefer,\n Phys. Rev. Lett. {\\bf 82}, 4964 (1999).\n \\bibitem{tfi}\n Propagation of a Gaussian wave packet in an absorbing medium\\\\\n M. Tanaka, M. Fujiwara, and H. Ikegami,\n Phys. Rev. A {\\bf 34}, 4851 (1986).\n \\bibitem{ab}\n Time in the Quantum Theory and the Uncertainty Relation\n for Time and Energy\\\\\n Y. Aharonov and D. Bohm, \n Phys. Rev. {\\bf 122}, 1649 (1961).\n \\bibitem{dm}\n Tunneling-time probability distribution\\\\\n R. S. Dumont and T. L. Marchioro,\n Phys. Rev. A {\\bf 47}, 85 (1993).\n \\bibitem{leav}\n Arrival time distributions\\\\\n C. R. Leavens, \n Phys. Lett. A {\\bf 178}, 27 (1993).\n \\bibitem{ml}\n Distributions of delay times and transmission times \n in Bohm's causal interpretation of quantum mechanics \\\\\n W. R. McKinnon and C. R. Leavens, \n Phys. Rev. A {\\bf 51}, 2748 (1995).\n \\bibitem{muga2}\n Time of Arrival in Quantum Mechanics\\\\\n J. G. Muga, S. Brouard, and D. Macias, \n Ann. phys. (N.Y.) {\\bf 240}, 351 (1995). \n \\bibitem{grt}\n Time-of-arrival in quantum mechanics\\\\\n N. Grot, C. Rovelli, and R. S. Tate, \n Phys. Rev. A {\\bf 54}, 4676 (1996).\n \\bibitem{gian}\n Positive-Operator-Valued Time Observable in Quantum Mechanics\\\\\n R. Giannitrapani,\n Int. J. Theor. Phys. {\\bf 36}, 1575 (1997).\n \\bibitem{aopru}\n Measurement of time of arrival in quantum mechanics\\\\\n Y. Aharonov, J. Oppenheim, S. Popescu, B. Reznik, and W. G. Unruh, \n Phys. Rev. A {\\bf 57}, 4130 (1998).\n \\bibitem{mlp}\n Space-time properties of free-motion time-of-arrival eigenfunctions\\\\\n J. G. Muga, C. R. Leavens, and J. P. Palao, \n Phys. Rev. A {\\bf 58}, 4336 (1998).\n \\bibitem{muga}\n Arrival time in quantum mechanics\\\\\n V. Delgado and J. G. Muga, \n Phys. Rev. A {\\bf 56}, 3425 (1997).\n \\bibitem{muga1}\n The time of arrival concept in quantum mechanics\\\\\n J. G. Muga, R. Sala, and J. P. Palao, \n Superlattices Microstruct. {\\bf 23}, 833 (1998). \n \\bibitem{ljpu}\n Time of arrival through a quantum barrier\\\\\n J. Leon, J. Julve, P. Pitanga, and F. J. de Urries, \n e-print quant-ph/9903060.\n \\bibitem{bspm}\n Time-of-arrival distribution for arbitrary potentials \n and Wiger's time-energy uncertainty relation\\\\\n A. D. Baute, R. S. Mayato, J. P. Palao, and J. G. Muga, \n e-print quant-ph/9904055.\n \\bibitem{lf}\n Transmisson time of wave packets through tunneling barriers\\\\\n Y. E. Lozovik and A. V. Filinov, \n Zh. Eksp. Teor. Fiz. {\\bf 115}, 1872 (1999)[JETP {\\bf 88}, 1026 (1999)]. \n \\bibitem{kirpol}\n Travel Time of a Quantum Particle through a Given Domain\\\\\n A. I. Kirillov and E. V. Polyachenko, \n Theor. Math. Phys. {\\bf 118}, 41 (1999).\n \\bibitem{kw}\n Operational time of arrival in quantum phase space\\\\\n P. Kochanski and K. Wodkiewicz, \n Phys. Rev. A {\\bf 60}, 2689 (1999).\n \\bibitem{hall}\n Arrival Times in Quantum Theory from an Irrevesible Detector Model\\\\\n J. J. Halliwell, \n Prog. Theor. Phys. {\\bf 102}, 707 (1999). \n \\bibitem{npt}\n Coherent control of macroscopic quantum states\n in a single-Cooper-pair box\\\\\n Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, \n Nature (London) {\\bf 398}, 786 (1999).\n \\bibitem{obe}\n Observation of Quantum Accelerator Modes\\\\\n M. K. Oberthaler, R. M. Godun, M. B. d'Arcy, G. S. Summy, \n and K. Burnett, \n Phys. Rev. Lett. {\\bf 83}, 4447 (1999).\n \\bibitem{w}\n Lower Limit for the Energy Derivative of the Scattering Phese Shift\\\\\n E. P. Wigner, \n Phys. Rev. {\\bf 98}, 145 (1955).\n \\bibitem{b}\n Larmor precession and the traversal time for tunneling\\\\\n M. B\\\"uttiker, \n Phys. Rev. B {\\bf 27}, 6178 (1983).\n \\bibitem{h-s}\n Tunneling times: a critical review \\\\\n E. H. Hauge and J. A. St{\\o}vneng, \n Rev. Mod. Phys. {\\bf 61}, 917 (1989).\n \\bibitem{l-m}\n Barrier interaction time in tunneling \\\\\n R. Landauer and Th. Martin, \n Rev. Mod. Phys. {\\bf 66}, 217 (1994).\n \\bibitem{muga3}\n Systematic approach to define and classify quantum transmission and\n reflection times\\\\\n S. Brouard, R. Sala, and J. G. Muga, \n Phys. Rev. A {\\bf 49}, 4312 (1994).\n \\bibitem{aagi}\n Tunneling time through a rectangular barrier\\\\\n V. M. de Aquino, V. C. Aguilera-Navarro, M. Goto, and H. Iwamoto,\n Phys. Rev. A {\\bf 58}, 4359 (1993).\n \\bibitem{yamada}\n Speakable and Unspeakable in the Tunneling Time Problem\\\\\n N. Yamada,\n Phys. Rev. Lett. {\\bf 83}, 3350 (1999).\n \\bibitem{bkr}\n Variational approach to the tunneling-time problem\\\\\n C. Bracher, M. Kleber, and M. Riza,\n Phys. Rev. A {\\bf 60}, 1864 (1999).\n \\bibitem{nel}\n Derivation of the Schr\\\"odinger Equation from Newtonian Mechanics\\\\\n E. Nelson, \n Phys. Rev. {\\bf 150}, 1079 (1966).\n \\bibitem{ohba}\n Tunneling time based on the quantum diffusion process approach \\\\\n K. Imafuku, I. Ohba, and Y. Yamanaka, \n Phys. Lett. A {\\bf 204}, 329 (1995).\n \\bibitem{hasi}\n Comments on Nelson's Quantum Stochastic Process\\\\\n \|Tunneling time and domain structure\|\\\\\n T. Hashimoto,\n in {\\it Quantum Information}, edited by T. Hida and K. Sait\\^o\n (World Scientific, Singapore, 1999).\n \\bibitem{ohba2}\n Effects of inelastic scattering on tunneling time based on the\n generalized diffusion process approach \\\\\n K. Imafuku, I. Ohba, and Y. Yamanaka, \n Phys. Rev. A {\\bf 56}, 1142 (1997).\n \\bibitem{hino}\n Measurement of Larmor precession angles of tunneling neutrons \\\\\n M. Hino, N. Achiwa, S. Tasaki, T. Ebisawa, T. Kawai, T. Akiyoshi,\n and D. Yamazaki, \n Phys. Rev. A {\\bf 59}, 2261 (1999).\n \\bibitem{cal3}\n Quantum Mechanics Simulations\\\\\n J. R. Hiller, I. D. Johnston, and D. F. Styer, \n (John Wiley \\& Sons, New York, 1995). \n\\end{thebibliography}\n\\end{document}\n%%%%%%%%%%%%%%%%%%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n" } ]	[ { "name": "quant-ph9912109.extracted_bib", "string": "{exp1 Quantum Optical Studies of Tunneling and other Superluminal Phenomena\\\\ R. Y. Chiao and A. M. Steinberg, Phys. Scr. {T76, 61 (1998)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{exp2 Measurement of the Single-Photon Tunneling Time\\\\ A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. {71, 708 (1993)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{exp3 An atom optics experiment to investigate faster-than-light tunneling\\\\ A. M. Steinberg, S. Myrskog, H. S. Moon, H. A. Kim, J. Fox, and J. B. Kim, Ann. Phys. (Leipzig) {7, 593 (1998)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{exp4 Light speed reduction to 17 metres per second in an ultracold atomic gas\\\\ L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature (London) {397, 594 (1999)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{lpt Speed of light in non-trivial vacua\\\\ J. I. Latorre, P. Pascual, and R. Tarrach, Nucl. Phys. B {437, 60 (1995)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{aemns Tests of quantum gravity from observations of $\\gamma$-ray bursts\\\\ G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, and S. Sarkar, Nature (London) {393, 763 (1998)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{gar Quantum Evolution in Space-Time Foam\\\\ L. J. Garay, Int. J. Mod. Phys. A {14, 4079 (1999)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{sch Severe Limits on Variations of the Speed of Light with Frequency\\\\ B. E. Schaefer, Phys. Rev. Lett. {82, 4964 (1999)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{tfi Propagation of a Gaussian wave packet in an absorbing medium\\\\ M. Tanaka, M. Fujiwara, and H. Ikegami, Phys. Rev. A {34, 4851 (1986)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{ab Time in the Quantum Theory and the Uncertainty Relation for Time and Energy\\\\ Y. Aharonov and D. Bohm, Phys. Rev. {122, 1649 (1961)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{dm Tunneling-time probability distribution\\\\ R. S. Dumont and T. L. Marchioro, Phys. Rev. A {47, 85 (1993)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{leav Arrival time distributions\\\\ C. R. Leavens, Phys. Lett. A {178, 27 (1993)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{ml Distributions of delay times and transmission times in Bohm's causal interpretation of quantum mechanics \\\\ W. R. McKinnon and C. R. Leavens, Phys. Rev. A {51, 2748 (1995)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{muga2 Time of Arrival in Quantum Mechanics\\\\ J. G. Muga, S. Brouard, and D. Macias, Ann. phys. (N.Y.) {240, 351 (1995)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{grt Time-of-arrival in quantum mechanics\\\\ N. Grot, C. Rovelli, and R. S. Tate, Phys. Rev. A {54, 4676 (1996)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{gian Positive-Operator-Valued Time Observable in Quantum Mechanics\\\\ R. Giannitrapani, Int. J. Theor. Phys. {36, 1575 (1997)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{aopru Measurement of time of arrival in quantum mechanics\\\\ Y. Aharonov, J. Oppenheim, S. Popescu, B. Reznik, and W. G. Unruh, Phys. Rev. A {57, 4130 (1998)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{mlp Space-time properties of free-motion time-of-arrival eigenfunctions\\\\ J. G. Muga, C. R. Leavens, and J. P. Palao, Phys. Rev. A {58, 4336 (1998)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{muga Arrival time in quantum mechanics\\\\ V. Delgado and J. G. Muga, Phys. Rev. A {56, 3425 (1997)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{muga1 The time of arrival concept in quantum mechanics\\\\ J. G. Muga, R. Sala, and J. P. Palao, Superlattices Microstruct. {23, 833 (1998)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{ljpu Time of arrival through a quantum barrier\\\\ J. Leon, J. Julve, P. Pitanga, and F. J. de Urries, e-print quant-ph/9903060." }, { "name": "quant-ph9912109.extracted_bib", "string": "{bspm Time-of-arrival distribution for arbitrary potentials and Wiger's time-energy uncertainty relation\\\\ A. D. Baute, R. S. Mayato, J. P. Palao, and J. G. Muga, e-print quant-ph/9904055." }, { "name": "quant-ph9912109.extracted_bib", "string": "{lf Transmisson time of wave packets through tunneling barriers\\\\ Y. E. Lozovik and A. V. Filinov, Zh. Eksp. Teor. Fiz. {115, 1872 (1999)[JETP {88, 1026 (1999)]." }, { "name": "quant-ph9912109.extracted_bib", "string": "{kirpol Travel Time of a Quantum Particle through a Given Domain\\\\ A. I. Kirillov and E. V. Polyachenko, Theor. Math. Phys. {118, 41 (1999)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{kw Operational time of arrival in quantum phase space\\\\ P. Kochanski and K. Wodkiewicz, Phys. Rev. A {60, 2689 (1999)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{hall Arrival Times in Quantum Theory from an Irrevesible Detector Model\\\\ J. J. Halliwell, Prog. Theor. Phys. {102, 707 (1999)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{npt Coherent control of macroscopic quantum states in a single-Cooper-pair box\\\\ Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Nature (London) {398, 786 (1999)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{obe Observation of Quantum Accelerator Modes\\\\ M. K. Oberthaler, R. M. Godun, M. B. d'Arcy, G. S. Summy, and K. Burnett, Phys. Rev. Lett. {83, 4447 (1999)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{w Lower Limit for the Energy Derivative of the Scattering Phese Shift\\\\ E. P. Wigner, Phys. Rev. {98, 145 (1955)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{b Larmor precession and the traversal time for tunneling\\\\ M. B\\\"uttiker, Phys. Rev. B {27, 6178 (1983)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{h-s Tunneling times: a critical review \\\\ E. H. Hauge and J. A. St{\\ovneng, Rev. Mod. Phys. {61, 917 (1989)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{l-m Barrier interaction time in tunneling \\\\ R. Landauer and Th. Martin, Rev. Mod. Phys. {66, 217 (1994)." }, { "name": "quant-ph9912109.extracted_bib", "string": "{muga3 Systematic approach to define and classify quantum transmission and reflection times\\\\ S. Brouard, R. Sala, and J. G. Muga, Phys. Rev. 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quant-ph9912110	Experiments on Sonoluminescence: Possible Nuclear and QED Aspects and Optical Applications	[ { "author": "V.B.\\" }, { "author": "Belyaev$^{(1)" } ]	Experiments aimed at testing some hypothesis about the nature of Single Bubble Sonoluminescence are discussed. A possibility to search for micro-traces of thermonuclear neutrons is analyzed, with the aid of original low-background neutron counter operating under conditions of the deep shielding from Cosmic and other sources of background. Besides, some signatures of QED-contribution to the light emission in SBSL are under the consideration, as well as new approaches to probe a temperature inside the bubble. An applied-physics portion of the program is presented also, in which an attention is being paid to single- and a few-pulse light sources on the basis of SBSL.	[ { "name": "quant-ph9912110.tex", "string": "\\documentstyle[12pt]{aipproc}\n\\textwidth 148mm\n\\textheight 219mm\n\\topmargin -1.7cm\n%\\oddsidemargin -5mm\n\\linewidth 6mm\n\\parskip=0mm\n\\begin{document}\n\\title{Experiments on Sonoluminescence: Possible Nuclear and QED Aspects\nand Optical Applications}\n\n\n\\author{V.B.\\,Belyaev$^{(1)}$, Yu.Z.\\,Ionikh$^{(2)}$, M.B.\\,Miller$^{(3)}$,\nA.K.\\,Motovilov$^{(1)}$, A.V.\\,Sermyagin$^{(3)}$, A.A.\\,Smolnikov$^{(1,4)}$,\nYu.A.\\,Tolmachev$^{(2)}$}%\n\n\\address{\\small $^{(1)}$Joint Institute for Nuclear Research, Bogolubov Laboratory\nof Theoretical Physics, \\mbox{141980 Dubna, Moscow reg., Russia}\\\\\n$^{(2)}$Institute of Physics of the St.\\,Petersburg State University,\n \\mbox{198904 Peterhof, St.\\,Petersburg reg., Russia}\\\\\n$^{(3)}$Institute of Physical and Engineering\nProblems\\thanks{E-mail: iftp@dubna.ru}\n \\mbox{P.\\,O.\\,Box\\,39, 141980 Dubna, Moscow reg., Russia}\\\\\n$^{(4)}$Institute for Nuclear Research, Baksan Neutrino Observatory,\n \\mbox{117312 Moscow, Russia}}\n\n%\\lefthead{LEFT head}\n%\\righthead{RIGHT head}\n\\maketitle\n\n\\begin{abstract}\nExperiments aimed at testing some hypothesis about the nature\nof Single Bubble Sonoluminescence are discussed.\nA possibility to search for micro-traces of thermonuclear neutrons\nis analyzed, with the aid of original low-background neutron\ncounter operating under conditions of the deep shielding from\nCosmic and other sources of background. Besides, some signatures\nof QED-contribution to the light emission in SBSL are under\nthe consideration, as well as new approaches to probe a\ntemperature inside the bubble. An applied-physics portion of\nthe program is presented also, in which an attention is being\npaid to single- and a few-pulse light sources on the basis of SBSL.\n\\end{abstract}\n\n\\section{Introduction}\n\n\nA lack of a complete explanation of some\nunusual characteristics of sonoluminescence came to be a source of\na few exotic suggestions about its nature. For our\nstudies we selected the most intriguing ones that are somehow\nin line with a general\nscientific stream in\nDubna research center.\nSearch for predicted nuclear and quantum electrodynamic effects in\nsonoluminescence is an aim of described experiments that are now in\na stage of preparation.\n\n\n\n\\section{Experimental approaches}\n\n\n\n\\paragraph{Hot-plasma hypotesis.} It was predicted that very high\ntemperatures are possible at the\nmoment of a collapse of the sonoluminescencing bubble. Under\nsome special mode of upscaling sonoluminescence (a strong short\npressure pulse should be added to the ultrasound standing wave)\nthe temperature presumably\nreach a level of observable traces of\nthermonuclear fusion in the D+D system.\n The additional\nshort pressure\npulse with a magnitude of several bars are to be synchronous\nwith the sonoluminescence\nflash. If the system contains deuterium dissolved in heavy\nwater (D$_2$O) a neutron yield is expected. A value about 0.1\nnph is predicted for some optimal conditions~\\cite{Moss}. Measurements of\nthis low neutron rate are planned to be\ndone by means of a triple-coincidence method using an\noriginal\nneutron counter. The neutron spectrometer was designed taking into account\nrequirements for\nminimizing the $\\gamma$-ray and random coincidence backgrounds~\\cite{12}. It is a\ncalorimeter based on a liquid organic scintillator-thermalizer with\n$^3$He\nproportional counters of thermalized neutrons distributed uniformly over the\nvolume. The energy of thermalized neutrons is transformed into light signals\nin a scintillation detector. The signals from proportional counters provide\na `neutron label' of an event.\nThe triple\ncoincidences are to be sorted by a following algorithm:\n\n{\\it(signal from the sonoluminescence-light flash) \\&\n(the scintillator flash in moderator) \\&\n(the signal from the $^3$He counter)}.\n\nThe measurements are supposed to be performed\nin the underground laboratory of the Baksan Neutrino Observatory of\nthe Institute for Nuclear Research of Russian Academy of\nSciences, Caucasus. A shield from cosmic rays in this Lab is\nabout 5,000\\,m of w.\\,e. %\nUnder these\nconditions a sensitivity of\nabout 0.01 nph can be reached for about a three-month measurement cycle. The main\ncomponents of necessary devices and equipment are already in our\ndisposal, including the sonoluminescence devices and the neutron\ncounter.\n\n\n\n\nThe system of intensive pressure pulsing is under construction.\nCertainly, many efforts\nare necessary to modify the experimental setup and accommodate\nit for the measurements at Baksan. To reach the most possible\nsensitivity in these experiments it would be reasonable to use a\nso-called few-bubble-sonoluminescence (FBSL) regime when several single\nbubbles are trapped within a higher harmonic modes of acoustic\nresonator as it was reported in~\\cite{Geisler}.\nWe have observed\nconcurrently lighting SL-bubbles in the second harmonic\nunder some special\nboundary conditions which have yet to be specified~\\cite{Belyaev}.\n\n\\paragraph{QED-hypotesis.} Another idea connects the sonoluminescence\nto the energy of zero vibrations of the vacuum (Casimir\nenergy)~\\cite{13}. To test this idea the following two types of\nexperiments are designed.\n\n{\\it(1) Transforming a short wave part of the sonoluminescence\nspectrum to the region of $\\lambda$ higher than the water\nabsorption edge.} To this end, certain specific\nluminofores should be selected, and among them perhaps tiny dispersive\npowder of crystal r\\\"ontgenoluminofores would be promising as an\ninteraction with lattice is essential in this case. Certainly,\npossible influence of those suspensions on cavitation properties\nof water\nhave to be clarified beforehand.\nIf the hyposesis on the\nvacuum-fluctuation nature of the sonoluminescence phenomenon is\nvalid then no short wave emission with $\\lambda<200$\\,nm is\nexpected.\n\n{\\it(2) Angular-correlation measurements of the coincident photons in the\nvicinity of 180$^\\circ$ .} The QED model for the\nsonoluminescence predicts the emission of time-correlated pairs\nof photons flying away in opposite directions.\n\\vskip 0.5cm\nSome other experiments are considered also. In particular,\n\n{\\it (1) Direct testing the so-called dissociation\nhypothesis (DH) of the\nsonoluminescence.} According to DH, when the\nstable single bubble sonoluminescence conditions have been created,\nthe inert gas alone remains inside the bubble. Due to high temperature\ninside the bubble all other components (nitrogen and\noxygen) undergo the intensive chemical interaction with each\nother and with water. This results in nitrogen oxides\n(NO, NO$_2$) and NH$_3$~\\cite{14}.\nAt present only indirect evidence for the DH has been\nfound~\\cite{15,16}. For direct measurement of the\n products of dissociation a small SL cell,\n completely closed to the atmosphere, and containing a relatively\n small volume of water, will be needed. Long-time runs can be\n accomplished via computer-controlled monitoring of the system ~\\cite{MATULA PC}.\n\n {\\it (2) Measurements of spatial distribution of the light in water.}\nMeasurements of the time-averaged spectral distribution of the\nradiation emitted by the bubble will be done\nin the presence of the\nluminofore\nadditives that will be used in the procedure of\nspectrum-trasformation experiments. The aim of the experiments is to infer the\nsource brightness\nin the short UV range by means of taking\ninto account the diffusion of UV emission in the water solutions or suspensions of the\nabove luminofores, and, by\ncomparing them with\npredictions of quasi-stationary thermal source model,\nestimates of plasma temperature are expected to be obtained~\\cite{17}.\n\n {\\it (3) Study of near-IR spectrum of the emission of Xe-doped bubble.}\n We will try to search for spectral lines of Xe similar to distinctive line\n emissions\n observed in high pressure xenon lamps.\n\n\\section{Development of new type of super fast pulse light sources}\n\nOne of the most remarkable features of SBSL flashes is\ntheir brief duration.\nThe most recent measurements show that\nmany parameters,\nsuch as the nature and concentration of gases,\ntemperature, pressure, resonance performances of the SL instrument,\netc., influence the temporal and other properties. For example,\nlarger flasks operating at lower frequencies, cause the bubble to\nemit more light~\\cite{18}. It is important that\nthe light pulse duration remains the same within the limit of a few\npicosecond for\nwhole wavelength range .\n \\vskip 0.5cm\n\nGoals of this part of experiments are:\n\n\n(1) Studying parameters controlling the duration of light\npulses and other temporal parameters of SBSL radiation.\n(2) Investigation of correlation in intensity of flashes.\nIn this experiment the statistical properties of the intensity\nof the light flashes will be studied to determine the short-term\n and long-term aspects of SL. The synchronicity between\nflashes has\nbeen shown to be remarkably high~\\cite{19}. It is interesting whether the\nintensity distribution is as narrow, as well.\n (3) Development of SL-light sources of single light pulses.\n To this end the Kerr cells will be applied. Regular and relatively\n rare repetition of the flashes makes possible using the\n photo-shutters with a limited temporal resolution to obtain\n the single pulses with temporal performances determined by\n primary SBSL-properties.\n (4) Development of methods to generate various series\n of the single light\n pulses.\n (5) On this basis, developing simple, inexpensive light\nsources for physical research, fist of all for the\ntime-resolution calibration of fast photodetectors (PMT and the\nlike).\n\n\\section{Acknowledgments}\nThe authors are grateful to Prof.\\,L.\\,A.\\,Crum and Dr.\\,T.\\,J.\\,Matula\nfor very fruitful discussions, and to Prof.\\,W.\\,Lauterborn\nfor his interest to this program.\\\\\n\\par\nThis work was supported in part by the Russian Foundation for Basic\nResearch, Grant No.\\ 98--02--16884.\n\n\\begin{references}\n\\bibitem{Moss} W.\\,C.\\,Moss, D.\\,B.\\,Clarke, J.\\,W.\\,White, D.\\,A.\\,Yang,\n{\\it Phys.\\ Lett}\\ {\\bf A211}, 69 (1996).\n\\bibitem{12} J.\\,N.\\,Abdurashitov, V.\\,N.\\,Gavrin, G.\\,D.\\,Efimov, A.\\,V.\\,Kalikhov,\nA.\\,A.\\,Shikhin, and V.\\,E.\\,Yants, {\\it Instruments and\nExperimental Techniques},\\ {\\bf 40} , No 6, 741 (1997). Translated\nfrom {\\it Pribory i Tekhnika Eksperimenta},\\ No 6, 5 (1997).\n\\bibitem{Geisler} R.\\,Geisler, T.\\,Kurz, W.\\,Lauterborn, {\\it 15-th International Symposium\non Nonlinear Acoustics, G\\\"ottingen, Germany, September 1-4, 1999: Symposium\nProgram and Book of Abstracts}, 74 (1999).\n\\bibitem{Belyaev} V.\\,Belyaev, M.\\,Miller, A.\\,Sermyagin, submitted to {\\it 7-th Symposium of Japan\nSociety of Sonochemistry, 29-30 Oct. 1998, Yonezava, Japan}, unpublished.\n\\bibitem{13} J.\\,Schwinger, {\\it Proc. Natl.\\ Acad.\\ Sci.\\ U.S.A.}\\ {\\bf\n 89}, 4091 (1992); {\\bf 89}, 11118 (1992); {\\bf 91}, 6473 (1994).\n\\bibitem{14} D.\\,Lohse, M.\\,Brenner, T.\\,Dupont, S.\\,Hilgenfeldt, and B.\\,\n Johnston, {\\it Phys.\\ Rev.\\ Lett.}\\ {\\bf 78}, 1359 (1997).\n\\bibitem{15} T.\\,J.\\,Matula and L.\\,A.\\,Crum, {\\it Phys.\\ Rev.\\ Lett.}\\ {\\bf 80}, 865 (1998).\n\\bibitem{16} J.\\,A.\\,Ketterling and R.\\,E.\\,Apfel, {\\it Phys.\\ Rev.\\ Lett.}\\ {\\bf 81}, 4991 (1998).\n\\bibitem{MATULA PC}T.\\,J.\\,Matula, {\\it private communication}.\n\\bibitem{17} N.\\,N.\\,Bezuglov, Yu.\\,A.\\,Tolmachev, unpublished.\n\\bibitem{18} T.\\,J.\\,Matula, {\\it private communication}.\n\\bibitem{19} B.\\,P.\\,Barber and S.\\,J.\\,Putterman, {\\it Nature}\\ {\\bf 352}, 318 (1991).\n\\end{references}\n\\end{document}\n" } ]	[ { "name": "quant-ph9912110.extracted_bib", "string": "{Moss W.\\,C.\\,Moss, D.\\,B.\\,Clarke, J.\\,W.\\,White, D.\\,A.\\,Yang, {Phys.\\ Lett\\ {A211, 69 (1996)." }, { "name": "quant-ph9912110.extracted_bib", "string": "{12 J.\\,N.\\,Abdurashitov, V.\\,N.\\,Gavrin, G.\\,D.\\,Efimov, A.\\,V.\\,Kalikhov, A.\\,A.\\,Shikhin, and V.\\,E.\\,Yants, {Instruments and Experimental Techniques,\\ {40 , No 6, 741 (1997). Translated from {Pribory i Tekhnika Eksperimenta,\\ No 6, 5 (1997)." }, { "name": "quant-ph9912110.extracted_bib", "string": "{Geisler R.\\,Geisler, T.\\,Kurz, W.\\,Lauterborn, {15-th International Symposium on Nonlinear Acoustics, G\\\"ottingen, Germany, September 1-4, 1999: Symposium Program and Book of Abstracts, 74 (1999)." }, { "name": "quant-ph9912110.extracted_bib", "string": "{Belyaev V.\\,Belyaev, M.\\,Miller, A.\\,Sermyagin, submitted to {7-th Symposium of Japan Society of Sonochemistry, 29-30 Oct. 1998, Yonezava, Japan, unpublished." }, { "name": "quant-ph9912110.extracted_bib", "string": "{13 J.\\,Schwinger, {Proc. Natl.\\ Acad.\\ Sci.\\ U.S.A.\\ {89, 4091 (1992); {89, 11118 (1992); {91, 6473 (1994)." }, { "name": "quant-ph9912110.extracted_bib", "string": "{14 D.\\,Lohse, M.\\,Brenner, T.\\,Dupont, S.\\,Hilgenfeldt, and B.\\, Johnston, {Phys.\\ Rev.\\ Lett.\\ {78, 1359 (1997)." }, { "name": "quant-ph9912110.extracted_bib", "string": "{15 T.\\,J.\\,Matula and L.\\,A.\\,Crum, {Phys.\\ Rev.\\ Lett.\\ {80, 865 (1998)." }, { "name": "quant-ph9912110.extracted_bib", "string": "{16 J.\\,A.\\,Ketterling and R.\\,E.\\,Apfel, {Phys.\\ Rev.\\ Lett.\\ {81, 4991 (1998)." }, { "name": "quant-ph9912110.extracted_bib", "string": "{MATULA PCT.\\,J.\\,Matula, {private communication." }, { "name": "quant-ph9912110.extracted_bib", "string": "{17 N.\\,N.\\,Bezuglov, Yu.\\,A.\\,Tolmachev, unpublished." }, { "name": "quant-ph9912110.extracted_bib", "string": "{18 T.\\,J.\\,Matula, {private communication." }, { "name": "quant-ph9912110.extracted_bib", "string": "{19 B.\\,P.\\,Barber and S.\\,J.\\,Putterman, {Nature\\ {352, 318 (1991)." } ]
quant-ph9912111	Thermostatistics of $q$-deformed boson gas	[ { "author": "A. Lavagno $^a$ and P. Narayana Swamy $^b$" } ]	We show that a natural realization of the thermostatistics of $q$-bosons can be built on the formalism of $q$-calculus and that the entire structure of thermodynamics is preserved if we use an appropriate Jackson derivative in place of the ordinary thermodynamics derivative. This framework allows us to obtain a generalized $q$-boson entropy which depends on the $q$-basic number. We study the ideal $q$-boson gas in the thermodynamic limit which is shown to exhibit Bose-Einstein condensation with higher critical temperature and discontinuous specific heat.	[ { "name": "qs.tex", "string": "\\documentstyle[preprint,aps,tighten,epsfig]{revtex}\n%\\documentstyle[preprint,aps]{revtex}\n\n\\def\\q1{{q^{-1}}}\n\\def\\qq1{{q-q^{-1}}}\n\\def\\djk{{\\partial_x^{(q)}}}\n\\def\\dqq{{{\\cal D}^{(q)}}}\n\\def\\nq{{n_{i}}}\n\n\\begin{document}\n\\draft\n\\title{Thermostatistics of $q$-deformed boson gas}\n\\author{A. Lavagno $^a$ and P. Narayana Swamy $^b$}\n\\address{\n$^a$ Dipartimento di Fisica and INFN, Politecnico di Torino, Torino, \nI-10129, Italy \\\\\n$^b$ Physics Department, Southern Illinois University,\nEdwardsville, IL 62026, USA}\n\\maketitle\n\n\\begin{abstract}\nWe show that a natural realization of the thermostatistics of $q$-bosons can be built on the formalism of $q$-calculus and that the entire structure of thermodynamics is preserved if we use an appropriate Jackson derivative in place of the ordinary thermodynamics derivative. This framework allows us to obtain a generalized $q$-boson entropy which depends on the $q$-basic number. We study the ideal $q$-boson gas in the thermodynamic limit which is shown to exhibit Bose-Einstein condensation with higher critical temperature and discontinuous specific heat. \n\\end{abstract}\n\n%\\vspace{0.5cm}\n\n%\\noindent\n\\pacs{PACS numbers: 05.30.-d; 05.20.-y; 05.70.-a}\n\n\n\\section{Introduction}\n\nThe spin-statistics theorem represents one of the fundamental principles of physics and establishes a strict connection between quantum mechanics of many-body systems and quantum statistical mechanics. The complete symmetrization or antisymmetrization of the many body wave function (or the commutation-anticommutation relations in the language of second quantization) reflects the contrasting nature of bosons and fermions. Such quantum many body statistical behavior affects the number of possible states of the system corresponding to the set of occupation numbers and consequently the collective statistical mechanics description. \n\nThe power of the statistical mechanics lies not only in the derivation of the general laws of thermodynamics but also in determining the meaning of all the thermodynamic functions in terms of the microscopic interparticle interaction \nand in providing a collective description of the equilibrium many body system by means of the macroscopic variables such as pressure and internal energy. \n\nIn the recent past there has been increasing emphasis in quantum statistics different from the standard bosons and fermions. Since the pioneering work of Gentile and Green \\cite{genti,green}, there have been many extensions beyond the standard statistics (such as parastatistics, fractional statistics, quon statistics, anyon statistics and quantum groups) which have become topics of great interest because of the wide range of applications envisaged, from cosmic strings and black holes to the fractional quantum Hall effect and anyonic physics in condensed matter \\cite{wil}. \n\nIn the literature there are two principal methods of introducing an intermediate statistical behavior. The first is to deform the quantum algebra of the commutation-anticommutation relations thus deforming the exchange factor between permuted particles. The second method is based on modifying the number of ways of assigning particles to a collection of states and thus the statistical weight of the many-body system. \nThe two methods are related but a full connection \nbetween the quantum mechanics approach and the statistical mechanics approach \nis possible only with the simultaneous knowledge of both. \n\nOne interesting realization of the first approach is the study of exactly solvable statistical systems which has led to a new algebra, the $q$-deformed algebra of creation and annihilation operators, usually called $q$-bosons ($q$-fermions) or $q$-oscillators and related to the general theory of quantum groups \\cite{bie,mac}. Many recent investigations in the theory of $q$-bosons have provided much insight into both the mathematical development and the $q$-deformed thermodynamics \\cite{lee,su,tus,song,nar,kan,vok,ubri,rodi}. \nHowever, we believe that a fully consistent formulation connecting the statistical mechanics and the thermodynamics (i.e., thermostatistics) of $q$-bosons has been lacking. \nIn particular it is desirable to derive an explicit expression for the entropy of the $q$-bosons, which plays a central role in the thermostatistics of the system and in the information theory. \nIt is important to show that the full structure of thermodynamics of $q$-bosons is preserved and the closed loop of thermodynamic relations is satisfied. This is a nontrivial task because there is no {\\it a priori} reason that the thermodynamic relations be automatically preserved for the $q$-deformed structures. \n\nA remarkable example is the Tsallis nonextensive statistics \\cite{tsallis}, based on a generalization of the Boltzmann-Gibbs entropy, where the thermodynamic functions such as entropy and internal energy, are deformed but the whole structure of thermodynamics is preserved. Although Tsallis nonextensive thermodynamics is inspired by the (multi)fractal property of a system and does not embody quantum group theory, many papers are devoted to the formal analogies between $q$-oscillators and nonextensive statistics \\cite{tsa,abe,joh,ubri2}. The reason for this connection has to do with the common language of the two deformed theories which is the $q$-calculus. \n\nThe $q$-calculus was introduced at the beginning of this century by Jackson \\cite{jack} in the study of the basic hypergeometric function and it plays a central role in the representation of the quantum groups \\cite{exton}. In fact it has been shown that it is possible to obtain a ``coordinate\" realization of the Fock space of the $q$-oscillators by using the deformed Jackson derivative (JD) \\cite{flo,fink}. Moreover we observe that it has recently been shown that the JD can be identified with the generators of fractal and multifractal sets with discrete symmetries \\cite{erz}.\n\nSince the thermodynamic functions of nonextensive statistics are deformed by using the framework of $q$-calculus, we expect $q$-calculus to play an important role also in $q$-boson thermostatistics. \n\nIt is the purpose of this paper to show that a fully consistent thermostatistics of $q$-boson gas can be obtained by using an appropriate Jackson derivative rule in the standard thermodynamics relations. In this framework, the whole structure of thermodynamics is preserved and this enables us to derive all the thermodynamic quantities such as the entropy, internal energy and the distribution function in the $q$-deformed theory. Special attention is paid to the study of the ideal $q$-boson gas and the phenomenon of $q$-boson condensation. \n\nThis paper is organized as follows. In Sec. II we review the $q$-boson algebra and outline the modification of the standard boson theory brought about by the $q$-calculus. In Sec. III we determine the distribution function of the $q$-boson gas by utilizing the standard definition of the thermal average of an operator. In Sec. IV we introduce a consistent prescription for the use of the Jackson derivative in the thermodynamic relations. This allows us to obtain in Sec. V the generalized entropy for $q$-bosons and to derive this from the deformed statistical weight. Sec. VI describes the behavior of the ideal $q$-boson gas and the phenomenon of $q$-boson condensation. We report our conclusions in Sec. VII.\n\n\n\\section{$q$-boson algebra and its realizations}\n\nWe shall briefly review the theory of $q$-deformed bosons defined by the $q$-Heisenberg algebra of creation and annihilation operators of bosons introduced by Biedenharn and McFarlane \\cite{bie,mac}, derivable through a map from $SU(2)_q$. The \n$q$-boson algebra is determined by the following commutation relations for \n$a$, $a^{\\dag}$ and the number operator $N$, thus (for simplicity we omit the particle index)\n\n\\begin{equation}\n[a,a]=[a^\\dag,a^\\dag]=0 \\; , \\ \\ \\ aa^\\dag-q a^\\dag a =1 \\; , \n\\end{equation}\n\\begin{equation}\n[N,a^\\dag]= a^\\dag \\; , \\ \\ \\ [N,a]=-a\\; .\n\\end{equation}\n\nThe $q$-Fock space spanned by the orthornormalized eigenstates $\\vert n\\rangle$\nis constructed according to\n\\begin{equation}\n\\vert n\\rangle=\\frac{(a^\\dag)^n}{\\sqrt{[n]!}} \\vert 0\\rangle \\; , \n\\ \\ \\ a\\vert 0\\rangle=0 \\; ,\n\\label{fock}\n\\end{equation}\nwhere the $q$-basic factorial is defined as \n\\begin{equation}\n[n]!=[n] [n-1] \\cdots [1]\n\\label{brnf}\n\\end{equation}\nand the $q$-basic number $[x]$ is defined in terms of the $q$-deformation parameter \n\\begin{equation}\n[x]=\\frac{q^x-1}{q-1}\\; .\n\\label{bn}\n\\end{equation}\n\nFor the following discussion it is worth observing that the $q$-basic number satisfies the non-additivity property\n\\begin{equation}\n[x+y]=[x]+[y]+(q-1)\\, [x]\\,[y] \\; .\n\\label{nadd}\n\\end{equation}\nIn the limit $q \\rightarrow 1$, the $q$-basic number $[x]$ reduces to the ordinary number $x$ and all the above relations reduce to the standard boson relations.\n\nThe actions of $a$, $a^\\dag$ on the Fock state $\\vert n \\rangle$ are given by\n\\begin{eqnarray}\na^\\dag \\vert n\\rangle &=& [n+1]^{1/2} \\vert n+1\\rangle\\; , \\\\\na \\vert n\\rangle&=&[n]^{1/2} \\vert n-1\\rangle \\; ,\\\\\nN \\vert n\\rangle&=&n\\vert n\\rangle \\; .\n\\end{eqnarray}\nFrom the above relations, it follows that\n$a^\\dag a=[N]$, $aa^\\dag=[N+1]$. \n\nWe observe that the Fock space of the $q$-bosons has the same structure as the standard bosons but with the replacement $n!\\rightarrow [n]!$ . Moreover the number operator is not $a^\\dag a$ but can be expressed as the nonlinear functional relation $N=f(a^\\dag a)$ which can be explicitly written formally in the closed form\n\\begin{equation}\nN=\\frac{1}{\\log q} \\log\\Big (1+(q-1) a^\\dag a \\Big )\\; .\n\\label{nop}\n\\end{equation}\n\nThe transformation from Fock observables to the configuration space (Bargmann holomorphic representation) may be accomplished by choosing \\cite{flo,fink}\n\n\\begin{equation}\na^\\dag=x \\; , \\ \\ \\ a=\\djk \\; , \n\\label{jd}\n\\end{equation}\nwhere $\\djk$ is the Jackson derivative (JD) \\cite{jack}\n\\begin{equation}\n\\djk f(x)=\\frac{f(qx)-f(x)}{x\\,(q-1)}\\; ,\n\\end{equation}\nwhich reduces to the ordinary derivative when $q$ goes to unity and therefore, the JD occurs naturally in $q$-deformed structures \\cite{exton}. \n\n\n\\section{Thermal averages and statistical distribution for $q$-boson gas}\n\nSeveral investigators have studied the equilibrium statistical mechanics of the gas of non-interacting $q$-bosons \\cite{lee,su,tus,song,nar,kan,vok,ubri,rodi}. We shall now briefly discuss some of the important results from these studies before introducing our formulation of the thermostatistics of $q$-deformed bosons. \n\nIn the grand canonical ensemble, the Hamiltonian of the non-interacting boson gas is expected to have the following form \\cite{lee,su,tus,song}\n\\begin{equation}\nH=\\sum_i (\\epsilon_i-\\mu) \\, N_i\\; ,\n\\label{ha}\n\\end{equation}\nwhere the index $i$ is the state label, $\\mu$ is the chemical potential and $\\epsilon_i$ is the kinetic energy in the state $i$ with the number operator $N_i$. It should be mentioned that the form of the Hamiltonian is not unique in the literature, where some authors introduce the Hamiltonian which involves the basic number $[N_i]$. The advantage of the form in Eq.(\\ref{ha}) is that it describes clearly the number of particles in the spectrum by an integer number and will allow us to generalize the laws of thermodynamics in a simple manner. \n\nThe thermal average of an operator is written in the standard form\n\\begin{equation}\n\\langle {\\cal O}\\rangle=\\frac{ Tr \\left ({\\cal O} \\, e^{-\\beta H} \\right )}{\\cal Z}\\; ,\n\\end{equation}\nwhere $\\cal Z$ is the grand canonical partition function defined as \n\\begin{equation}\n{\\cal Z}=Tr \\left ( e^{-\\beta H} \\right )\\; ,\n\\label{pf}\n\\end{equation}\nand $\\beta = 1/T$. Henceforward we shall set Boltzmann constant to unity.\nLet us observe that the structure of the density matrix $\\rho = e^{-\\beta H}$ and the thermal average are undeformed. As a consequence, the structure of the partition function is also unchanged. We emphasize that this is not a trivial assumption because its validity implicitly amounts to an unmodified structure of the Boltzmann-Gibbs entropy,\n\\begin{equation}\nS=\\log W \\; ,\n\\end{equation}\nwhere $W$ stands for the number of states of the system corresponding to the set of occupation numbers $\\{ n_i \\}$.\nObviously the number $W$ is modified in the $q$-deformed case.\nIt may be pointed out that in the case of nonextensive $q$-deformed Tsallis statistics, the structure of the entropy is deformed via the logarithm function \\cite{tsallis}.\n\nBy using the definition in Eq.(\\ref{bn}) of the $q$-basic number, the mean value of the occupation number $\\nq$ can be calculated starting from the relation \n\\begin{equation}\n[n_{i}]=\\frac{1}{\\cal Z}\\, Tr\\left ( e^{-\\beta H} a^\\dag_i a_i\\right ),\n\\end{equation}\nand after applying the cyclic property of the trace and using the $q$-boson algebra, it is easy to show that \\cite{lee,su,tus}\n\\begin{equation}\n\\frac{[n_{i}]}{[n_{i}+1]}=e^{-\\beta (\\epsilon_i-\\mu)} \\; .\n\\label{nqbr}\n\\end{equation}\nThe explicit expression for the mean occupation number can be obtained by using the following property of the basic number, \n\\begin{equation}\n[n_{i}+1]=q\\, [n_{i}]+1 \\; ,\n\\label{brnp}\n\\end{equation}\nand hence for $q$ real, \n\\begin{equation}\nn_{i}=\\frac{1}{\\log q} \n\\log\\left (\\frac{z^{-1}e^{\\beta\\epsilon_i}-1}{ z^{-1}e^{\\beta\\epsilon_i}-q}\n\\right) \\; ,\n\\label{nqi}\n\\end{equation}\nwhere $z = e^{\\beta \\mu}$ is the fugacity. It is easy to see that the above equation reduces to the standard Bose-Einstein distribution when $q\\rightarrow 1$. The total number of particles is given by $N=\\sum_i\\,n_i$.\n\n\\section{Jackson derivatives in $q$-thermodynamics relations}\n\nFrom the definition of the partition function, Eq.(\\ref{pf}), and the Hamiltonian, Eq.(\\ref{ha}), it follows that the logarithm of the partition function has the same structure as that of the standard boson\n\\begin{equation}\n\\log {\\cal Z}=-\\sum_i \\log (1-z e^{-\\beta\\epsilon_i}) \\; .\n\\end{equation}\nThis is due to the fact that we have chosen the Hamiltonian to be a linear function of the number operator but it is not linear in $a^\\dag a$ as seen from Eq.(\\ref{nop}). For this reason,\nthe standard thermodynamic relations in the usual form are ruled out. It is verified, for instance, that \n\\begin{equation}\nN\\ne z \\, \\frac{\\partial}{\\partial z} \\log {\\cal Z}\\; .\n\\end{equation}\n\nAs the coordinate space representation of the $q$-boson algebra is realized by the introduction of the JD (see Eq.(\\ref{jd})), we stress that the key point of the $q$-deformed thermostatistics is in the observation that the ordinary thermodynamics derivative with respect to $z$, must be replaced by the JD \n\\begin{equation}\n\\frac{\\partial}{\\partial z} \\Longrightarrow {\\cal D}^{(q)}_z \\; ,\n\\end{equation}\nwhere we have defined ${\\cal D}^{(q)}_z$ as the Jackson derivative up to a constant (which goes to unity when $q \\rightarrow 1$)\n\\begin{equation}\n{\\cal D}^{(q)}_z = \\frac{q-1}{\\log q}\\, \\partial^{(q)}_z \\; .\n\\end{equation}\nConsequently, the number of particles in the $q$-deformed theory can be derived from the relation \n\\begin{equation}\nN=z \\; {\\cal D}^{(q)}_z \\log {\\cal Z}\\equiv \\sum_i n_i \\; ,\n\\label{num}\n\\end{equation}\nwhere $n_i$ is the mean occupation number expressed in Eq.(\\ref{nqi}).\n\nThe usual Leibniz chain rule is ruled out for the JD and therefore derivatives encountered in thermodynamics must be modified according to the following prescription. First we observe that the JD applies only with respect to the variable in the exponential form such as $z=e^{\\beta \\mu}$ or $y_i=e^{-\\beta \\epsilon_i}$. Therefore for the $q$-deformed case, any thermodynamic derivative of functions which depend on $z$ or $y_i$ must be converted to derivatives in one of these variables by using the ordinary chain rule and then applying the JD with respect to the exponential variable. For example, the internal energy in the $q$-deformed case can be written as\n\\begin{equation}\nU=-\\left. \\frac{\\partial}{\\partial\\beta} \\log {\\cal Z} \\right \|_z=\\sum_i \\frac{\\partial y_i}{\\partial\\beta} \\, {\\cal D}^{(q)}_{y_i}\\log(1-z\\,y_i) \\; .\n\\label{int}\n\\end{equation}\nIn this case we obtain the correct form of the internal energy\n\\begin{equation}\nU=\\sum_i \\epsilon_i \\, \\nq\\; ,\n\\label{un}\n\\end{equation}\nwhere $n_i$ is the mean occupation number expressed in Eq.(\\ref{nqi}).\n\nThis prescription is a crucial point of our approach because this allows us to maintain the whole structure of thermodynamics and the validity of the Legendre transformations in a fully consistent manner.\n\n\\section{Entropy of the $q$-boson gas and the deformed statistical weight}\n\nIn light of the above discussion, we have the recipe to derive the entropy of the $q$-bosons which leads to \n\\begin{eqnarray}\nS=-\\left. \\frac{\\partial\\Omega}{\\partial T}\\right \|_\\mu &&\\equiv \n\\log {\\cal Z} +\\beta\\sum_i\\left.\\frac{\\partial\\kappa_i}{\\partial\\beta}\\right\|_\\mu \n{\\cal D}^{(q)}_{\\kappa_i}\\log(1-\\kappa_i)\\nonumber\\\\\n&&=\\log {\\cal Z} +\\beta U-\\beta\\mu N \\; ,\n\\end{eqnarray}\nwhere $\\kappa_i=z\\, e^{-\\beta\\epsilon_i}$, \n$U$ and $N$ are the modified functions expressed in Eqs.(\\ref{int}) and (\\ref{num}) and $\\Omega = - T \\log {\\cal Z}$ is the thermodynamic potential. \n\nUsing Eqs.(\\ref{nqbr})-(\\ref{nqi}), after some manipulations, we obtain the entropy as follows \n\\begin{equation}\nS=\\sum_i \\Big \\{ -\\nq \\, \\log\\, [\\nq]+(\\nq+1)\\, \\log\\, [\\nq+1]-\\nq \\, \\log q\n\\Big \\} \\; .\n\\label{entro}\n\\end{equation}\n\nThe above entropy goes over to the standard boson entropy in the limit $q \\rightarrow 1$. It has the compact form which resembles the entropy of the standard boson but with the appearance of the $q$-basic numbers, $[n_i]$ and $[n_i+1]$, in the argument of the logarithmic function and in the presence of the last term, $-\\nq\\log q$, which follows from non-additivity property of the $q$-basic number. In fact, using Eqs.(\\ref{bn}) and (\\ref{brnp}) \ncan be re-expressed as \n\\begin{equation}\n\\nq\\, \\log q=\\log\\, ([\\nq+1]-[\\nq])\\; .\n\\end{equation}\n\nThe expression for the entropy is very relevant to the statistical information about the number of possible states occupied by the $q$-bosons and gives us the desired connection between the deformed quantum algebra and the quantum statistical behavior. It is interesting to observe that in the classical limit, the entropy does not reduce to the standard Boltzmann-Gibbs entropy ($S = - \\sum_i n_i \\log n_i) $, but remains deformed, except in the limit $q \\rightarrow 1$. This result is similar to the case of Greenberg's infinite statistics and the quantum Boltzmann distribution obtained as a particular case of quon statistics \\cite{greenberg}. The meaning of this is that the deformation exhibited in the entropy transcends the quantum nature but is built into the theory, somewhat similar to the case of nonextensive Tsallis statistics \\cite{tsallis}. The origin of the connection between the two different deformations ($q$-deformed quantum groups and nonextensive statistics) is beyond the scope of this paper and will be reported elsewhere. \n\nIn order to assure consistency, we must now show that the extremization of the entropy with fixed internal energy and number of particles leads to the correct $q$-boson distribution function. The extremum condition can be written as\n\\begin{equation}\n\\delta \\, \\Big ( S-\\beta U+\\beta\\mu N \\Big )=0 \\; ,\n\\label{extr}\n\\end{equation}\nwhere $\\beta$ and $\\beta\\mu$ plays the role of Lagrange multipliers. \n\nTo perform such extremization in the $q$-boson case, we assume that \nthe mean occupational number depends on the energy only as a function of $y_i=e^{-\\beta\\epsilon_i}$, $S=S[n(y_i)]$. Following our prescription described in Sec. IV on the use of JD, \nthe above extremization condition can be written as\n\\begin{equation}\n{\\cal D}^{(q)}_{y_i} \\, \\Big ( S-\\beta U+\\beta\\mu N \\Big ) \\, \n\\delta y_i=0\\; .\n\\end{equation}\nEmploying Eqs. (\\ref{num}), (\\ref{un}) and (\\ref{entro}), and carrying out the JD, the extremization condition reduces to \n\\begin{eqnarray}\n&&n(q y_i) \\left ( \\log\\frac{[ n(q y_i)+1]}{q\\,[n(q y_i)]}-\\tilde{\\epsilon}_i \\right )- \nn(y_i) \\left ( \\log\\frac{[ n(y_i)+1]}{[n(y_i)]}-\\tilde{\\epsilon}_i \\right )+ \\nonumber\\\\\n&&n(y_i) \\log q- \\log\\frac{[n(y_i)+1]}{[n(q y_i)+1]}=0 \\; ,\n\\label{nqy}\n\\end{eqnarray}\nwhere $\\tilde{\\epsilon}_i=\\beta (\\epsilon_i-\\mu)$. \n\nThe algebraic simplification of the above equation is intractable because of the complexity of the property of $q$-basic numbers. However, it is possible to determine the solution of the equation by observing that for any function $f(x)$, there exists a functional relationship\n\\begin{equation}\nq^{f(x)}=\\frac{[f(x)+1]}{[f(qx)+1]} \\ \\ \\Longleftrightarrow \\ \\ \n\\frac{[f(qx)+1]}{[f(qx)]}=\\, q \\, \\frac{[f(x)+1]}{[f(x)]}\\; .\n\\end{equation}\nThe notation $\\Longleftrightarrow $ used here denotes that one relation implies the other and vice versa. The validity of the first relation in the above equation eliminates the last two terms in Eq.(\\ref{nqy}) and the validity of the second relation implies that the quantities in parenthesis in Eq.(\\ref{nqy}) are equal, and since $ n(qy_i) \\ne n(y_i)$, for $q \\ne 1$, it follows that Eq.(\\ref{nqy}) is satisfied if \n\\begin{equation}\n\\frac{[n(y_i)+1]}{[n(y_i)]}=e^{\\tilde{\\epsilon}_i}\\; .\n\\label{ny34}\n\\end{equation}\n\nThe above relation is equivalent to Eq.(\\ref{nqbr}) which implies the mean occupational number $n_i$ of Eq.(\\ref{nqi}).\n\nAs discussed earlier, the entropy provides the information about the statistical weight $W$ which will be deformed in the case of $q$-boson particles. To investigate this deformation we begin with the basic relation for the entropy\n\\begin{equation}\nS=\\log W_q \\; ,\n\\label{sw}\n\\end{equation}\nwhere $W_q$ is the deformed statistical weight. Just as the ordinary factorial $n!$ is replaced by the $q$-basic factorial $[n]!$ in the construction $q$-Fock space (see Eq.(\\ref{fock})), we assume that this substitution also prevails in the expression for the statistical weight and hence we require\n\\begin{equation}\nW_q=\\prod_i \\frac{[\\nq+g_i-1]!}{[\\nq]! \\, [g_i-1]!}\\; ,\n\\end{equation}\nwhere $g_i$ denotes the number of subcell levels. \nThe reason for this modification lies in the definition of the binomial coefficient in the $q$-combinatorial calculus \\cite{exton}. \n\nObserving that $[n]!$ for large $n$, is given by the \n$q$-Stirling approximation for $q>1$ (see appendix for the explicit derivation)\n\\begin{equation}\n\\log\\,[n]!\\approx n\\, \\log\\,[n]-\\frac{n^2}{2} \\, \\log q\\; ,\n\\end{equation}\nthe entropy (\\ref{sw}) can be written as\n\\begin{equation}\nS=\\sum_i \\left \\{ \\nq \\, \\log\\, \\frac{[\\nq+g_i]}{[\\nq]}+g_i \\, \\log\\, \\frac{[\\nq+g_i]}{[g_i]}-\\nq\\,g_i\\, \\log q \\right\\} \\; .\n\\label{entro2}\n\\end{equation}\n\nThis is similar to the structure of the entropy given by Eq.(\\ref{entro}) and therefore the extremization procedure can be carried out as was done before and derive the same condition as in Eq.(\\ref{ny34}) except for the factor $g_i$. We observe, however, that the partition operation into subcells is not rigorously true in this context because of the nonextensive property (nonadditivity of the $q$-basic number) of the expression for the entropy in Eq.(\\ref{entro}). For this reason, the mean occupation number derived from Eq.(\\ref{entro2}) is not rigorously proportional to the factor $g_i$. The nonextensivity implies that the result for the mean occupation number is not entirely independent of the manner in which the energy levels of the particles are grouped into cells.\n\n\\section{Ideal $q$-Bose gas and $q$-boson condensation}\n\nWe shall now proceed to study the thermodynamic behavior of an ideal $q$-Bose gas and the phenomenon of $q$-boson condensation. For a large volume (and a large number of particles), the sum over all single particle energy states can be transformed to an integral over the energy, as follows\n\\begin{equation}\n\\sum_i f(x_i) \\ \\ \\Longrightarrow \\ \\ \n\\frac{2}{\\sqrt{\\pi}} \\, \\frac{V}{\\lambda^3} \\int_0^\\infty \\!\\!\\! dx \\; x^{1/2} \\, f(x)\\; ,\n\\end{equation}\nwhere $x=\\beta\\epsilon$, $\\epsilon = p^2/2m$ is the kinetic energy and \n$\\lambda = h/(2\\pi m T)^{1/2}$ is the thermal wavelength.\n\nWe anticipate that the ground state will be associated with macroscopically large occupation number rather than a zero weight due to $q$-boson condensation. For this reason we need to isolate the ground state and include the contribution from all the other states in the integral. The number density of particles can thus be written as\n\\begin{equation}\n\\frac{N}{V}= \\frac{2}{\\sqrt{\\pi}} \\, \\frac{1}{\\lambda^3} \\int_0^\\infty \\!\\!\\! dx \\; x^{1/2} \\, \\frac{1}{\\log q} \\, \n\\log \\left (\\,\\frac{z^{-1}e^x-1}{ z^{-1}e^x-q} \\right)+\\frac{n_0}{V}\\; ,\n\\end{equation}\nwhere $n_0$ is the mean occupational number of the zero momentum state \n\\begin{equation}\nn_0= \\frac{1}{\\log q} \\, \n\\log \\left (\\,\\frac{1-z}{1-q\\, z} \\right)\\; .\n\\end{equation}\n\nAs in the standard boson case, we need to set the range of fugacity $z$ which will correspond to non-negative occupation number. In the case of $q$-bosons we see that the condition is $ z < 1/q$ for $q>1$ and $z<1$ for $q<1$. It should be pointed out that we also have to require the existence of the JD of the mean occupation number which is encountered in the calculation of thermodynamic quantities such as the specific heat and this changes the upper bound of the fugacity $z$. We thus find the correct condition to be $z< z_q$, where we have defined \n\\begin{equation}\nz_q=\\cases{ q^{-2} &if $q>1$ ; \\cr 1 &if $q<1$ .\\cr} \n\\label{zq}\n\\end{equation}\n\nWe will have $q$-boson condensation when the critical combination of density and temperature occurs such that the fugacity will reach its maximum value $z=z_q$. \n\nFollowing the prescription of the JD in the $q$-deformed thermodynamics derivatives, we obtain the expression for pressure above the critical point \n\\begin{equation}\n\\left.\\frac{P}{T}\\right\|_>=\\frac{1}{\\lambda^3} \\; g_{_{5/2}} (z,q) \\; ,\n\\label{presa}\n\\end{equation}\nand below the critical point we have \n\\begin{equation}\n\\left.\\frac{P}{T}\\right\|_<=\\frac{1}{\\lambda^3} \\; g_{_{5/2}} (z_q,q) \\; .\n\\label{presb}\n\\end{equation}\nSimilar expression can be found for the number of particles above the critical point\n\\begin{equation}\n\\left.\\frac{N}{V}\\right\|_>=\\frac{1}{\\lambda^3} \\; g_{_{3/2}} (z,q) \\; ,\n\\label{totnpa}\n\\end{equation}\nand below the critical point we have\n\\begin{equation}\n\\left.\\frac{N}{V}\\right\|_<=\\frac{n_0}{V}+\n\\frac{1}{\\lambda^3} \\; g_{_{3/2}} (z_q,q) \\; .\n\\label{totnpb}\n\\end{equation}\n\nIn the above equations we have defined the $q$-deformed $g_{_n}(z,q)$ functions as\n\\begin{eqnarray}\ng_{_n}(z,q)&=&\\frac{1}{\\Gamma (n)} \\int_0^\\infty \\!\\!\\! dx \\; x^{n-1} \n\\frac{1}{\\log q} \\log\\left (\\,\\frac{z^{-1}e^x-1}{ z^{-1}e^x-q} \\right) \\nonumber\\\\\n&\\equiv& \\frac{1}{\\log q} \\left ( \n\\sum_{k=1}^{\\infty} \\frac{(zq)^k}{k^{n+1}} - \n\\sum_{k=1}^{\\infty} \\frac{z^k}{k^{n+1}} \\right ) \\; .\n\\label{gn}\n\\end{eqnarray}\n\nIn the limit $q\\rightarrow 1$, the deformed $g_n(z,q)$ functions reduce to the standard $g_n(z)$. In Fig. 1 and 2 we present the behavior of $g_{_{3/2}}(z,q)$ and $g_{_{5/2}}(z,q)$ as a function of $z$ for different values of the parameter $q$. \n\nThe internal energy can be calculated considering the thermodynamic limit of Eq.(\\ref{int}) by means of the JD recipe. Using the expression for the pressure, Eq.(\\ref{presa}), it is easy to verify that as in the undeformed case, \nthe following well-known relation is satisfied for the $q$-bosons,\n\\begin{equation}\nU=\\frac{3}{2} \\, PV \\; .\n\\end{equation}\n\nWe can calculate the critical temperature by using the same method as in the standard boson case. Comparing the ratio of the critical temperature $T_c^q$ of the $q$-deformed gas with that of the standard boson $T_c$ at the same density, we find\n\\begin{equation}\n\\frac{T_c^q}{T_c}=\\left ( \\frac{g_{_{3/2}}(1)}{g_{_{3/2}}(z_q,q)} \\right )^{2/3} \\; ,\n\\end{equation}\nwhere $g_{_{3/2}}(1)=2.61$ is the value of the undeformed function when $z=1$.\nIn Fig. 3 we show the plot of the above ratio as a function $q$. We observe that the critical temperature of the $q$-boson is always higher than the standard boson and for $q>1$ there is a rapid increase of the critical temperature $T^q_c$ for small values of $q$. For example, for $q=1.01$, $T^q_c$ increases by $18\\%$ and for $q=1.1$, $T^q_c$ increases by $75\\%$ with respect to the standard value. \n\nApplying the thermodynamic limit to the entropy of the $q$-boson in Eq.(\\ref{entro}), we obtain the entropy per unit volume above the critical point with a structure similar to that of the standard boson,\n\\begin{equation}\n\\left.\\frac{S}{V}\\right\|_>=\\frac{1}{\\lambda^3} \\left(\\,\\frac{5}{2} \\; g_{_{5/2}} (z,q)- g_{_{3/2}} (z,q) \\log z \\right) \\; ,\n\\label{entroa}\n\\end{equation}\nand below the critical point\n\\begin{equation}\n\\left.\\frac{S}{V}\\right\|_<= \\frac{5}{2}\\, \\frac{1}{\\lambda^3}\\, g_{_{5/2}} (z_q,q) \\; .\n\\label{entrob}\n\\end{equation}\n\nLet us observe that the generalized $q$-boson entropy obeys the third law of thermodynamics. In fact, in Eq.(\\ref{entrob}), $g_{_{5/2}} (z_q,q)$ has a finite value that depends on $q$ and the entropy approaches zero in the limit of zero temperature.\n\nAs in the ordinary Bose condensation it is possible to show that in the $q$-boson condensation also a Clausius-Clapeyron equation holds and first order phase transition occurs. In fact it is easy to see that below the critical point the following equation is satisfied \n\\begin{equation}\n\\left.\\frac{dP}{dT}\\right\|_<=\\frac{L_q}{T \\, v_c} \\; ,\n\\end{equation}\nwhere $v_c$ is the critical specific volume, defined as\n\\begin{equation}\nv_c=\\frac{\\lambda^3}{ g_{_{3/2}}(z_q,q)}\\; ,\n\\end{equation}\n$L_q$ is the q-deformed latent heat given by \n\\begin{equation}\nL_q=T \\, \\Delta s= \\frac{5}{2} \\; T \\; \\frac{g_{_{5/2}} (z_q,q)}{g_{_{3/2}}(z_q,q)} \\; ,\n\\end{equation}\nand where $\\Delta s$ is the difference in specific entropy across the transition region.\n\nWe now proceed to calculate the heat capacity of the $q$-boson gas, starting from the thermodynamic definition \n\\begin{equation}\nC_v=\\left. \\frac{\\partial U}{\\partial T}\\right\|_{V,N}\\; .\n\\label{cvt}\n\\end{equation}\n\nFor this purpose we first need the derivative of the fugacity with respect to $T$ (or $\\beta$), keeping $V$ and $N$ constant. To apply the JD prescription described earlier in Sec. IV, we start from the expression for the total number of particles, Eq.(\\ref{num}) and the identity (since the number of particles is kept constant) \n\\begin{equation}\n\\frac{\\partial}{\\partial \\beta}\\sum_i \\log \\left ( \\frac{1- \\kappa_i}{1-q \\kappa_i} \\right ) = 0 \\; ,\n\\end{equation}\nwhere $\\kappa_i= z \\, e^{- \\beta \\epsilon_i}$. This identity can be rewritten according to our JD recipe as\n\\begin{equation}\n\\sum_i \\, \\frac{\\partial \\kappa_i}{\\partial \\beta}\\, {\\cal D}_{\\kappa_i}^{(q)}\\log \n\\left ( \\frac{1- \\kappa_i}{1-q \\kappa_i} \\right ) = 0\\; ,\n\\end{equation}\nand now evaluating in the limit $V\\rightarrow\\infty$, we obtain\n\\begin{equation}\n\\left. \\frac{1}{z}\\, \\frac{\\partial z}{\\partial\\beta}\\right\|_{V,N}=\\frac{3}{2}\\; \\frac{1}{\\beta} \\; \\frac{{\\cal D}^{(q)}_z g_{_{5/2}} (z,q)}{ {\\cal D}^{(q)}_z g_{_{3/2}} (z,q)} \\; .\n\\label{dzb}\n\\end{equation}\n\nWe shall now proceed to calculate the heat capacity. Using the discrete expression of internal energy (\\ref{int}), Eq.(\\ref{cvt}) can be expressed as\n\\begin{equation}\nC_v=-\\beta^2 \\sum_i \\, \\epsilon_i\\,\\frac{\\partial \\kappa_i}{\\partial \\beta}\\; \\frac{1}{\\log q} {\\cal D}_{\\kappa_i}^{(q)}\\log \n\\left ( \\frac{1- \\kappa_i}{1-q \\kappa_i} \\right ) \\; .\n\\end{equation}\nCarrying out the limit $V\\rightarrow\\infty$ and utilizing Eqs.(\\ref{totnpa}) and (\\ref{dzb}), we obtain the following expression for the specific heat per particle above the critical point\n\\begin{equation}\n\\left. \\frac{C_v}{N}\\right\|_>= \\frac{15}{4}\\; \\frac{z\\, {\\cal D}^{(q)}_z g_{_{7/2}} (z,q)}{ g_{_{3/2}} (z,q)}-\\frac{9}{4}\\; \\frac{z\\, {\\cal D}^{(q)}_z g_{_{5/2}} (z,q)}{ g_{_{3/2}} (z,q)}\\; \\frac{{\\cal D}^{(q)}_z g_{_{5/2}} (z,q)}{ {\\cal D}^{(q)}_z g_{_{3/2}} (z,q)} \\; ,\n\\label{cva}\n\\end{equation}\nand similarly the specific heat below the critical point\n\\begin{equation}\n\\left. \\frac{C_v}{N}\\right\|_<=\\frac{15}{4}\\; \\frac{z\\, {\\cal D}^{(q)}_z g_{_{7/2}} (z,q)\|_{z=z_q} }{\\lambda^3/v}\\; ,\n\\label{cvb}\n\\end{equation}\nwhere $v$ is the specific volume below the critical temperature that can be expressed, by means of Eq.(\\ref{totnpb}), in terms of the critical temperature $T_c$ as follows\n\\begin{equation}\n\\frac{v}{\\lambda^3}=\\frac{1}{ g_{_{3/2}} (z,q) }\\, \\left ( \\frac{T}{T^q_c}\\right )^{3/2} \\; .\n\\end{equation}\n\nThe above expressions have the same structure as that of the undeformed boson but the difference arises from the property \n$z \\, {\\cal D}^{(q)}_z g_{_n}(z,q)\\ne g_{_{n-1}}(z,q)$, \nwhere the equality is true for ordinary derivative only. From this observation it is easy to see that in the limit $q\\rightarrow 1$ the specific heat reduces to the well-known undeformed result. \n\nAs usual, the classical limit can be achieved considering the limit $z\\rightarrow 0$. In this limit the deformed $g_{_n}(z,q)$ functions reduce to \n\\begin{equation}\ng_n(z,q)\\rightarrow \\frac{q-1}{\\log q} \\; z \\; ,\n\\end{equation}\nand from Eq.(\\ref{cva}), the ``classical\" limit of the specific heat per particle number reduces to \n\\begin{equation}\n\\left.\\frac{C_v}{N}\\right\|_{cl}=\\frac{3}{2} \\; \\frac{q-1}{\\log q} \\; .\n\\end{equation}\nAs discussed before in the context of the entropy, the $q$-deformation persists also in the classical limit. \n\nWe expect small deviations from undeformed behavior in the experimental observables, therefore only values of $q$ close to standard value $q=1$ are physically significant. Small deformation leads to a negligible departure from the high temperature limit of the specific heat but implies sharp deformation of the behavior of the specific heat in the range of the critical temperature. \nTo exemplify this feature we plot in Fig. 4 the specific heat as a function of $T/T_c^q$ for $q=1.05$. We have chosen a value of $q$ in the range $q>1$ because this region appears particularly interesting with a higher critical temperature for small $q$ (see Fig. 3). For this value of $q$ the critical temperature $T^q_c$ is increased by $48\\%$ relative to the standard boson case. \nWe observe that for $q\\ne 1$, the specific heat shows a discontinuous $\\lambda$ point behavior. This is a characteristic of $q$-deformation as has been observed in other investigations \\cite{ubri,rodi}. \n\nUsing Eqs.(\\ref{cva}) and (\\ref{cvb}) we can calculate the jump in the specific heat $\\Delta (C_v/N)$ at the critical temperature as a function of $q$. In Fig. 5 we plot this behavior. We observe that the jump is an increasing function of $q$. \n\nAlthough the model that we have investigated is based on the Hamiltonian of noninteracting particles, we note that the jump in the specific heat is of the order of the experimental value in the case of Bose condensation in $^{87}$Rb atoms \\cite{ensher}.\n\n\n\\section{Conclusion}\nThe outstanding problem in the theory of $q$-bosons has been the lack of a demonstration that the thermodynamic relations follow from the $q$-calculus framework. \n\nIn this paper, we have shown that the whole structure of thermodynamics is preserved if the ordinary derivatives are replaced by the Jackson derivatives following the prescription described in Sec. IV. We establish a fully consistent set of relations between the thermodynamic functions (partition function, internal energy, mean occupation number) and this enables us to derive the entropy of $q$-bosons. The $q$-deformed entropy so obtained has been shown to follow from the deformed statistical weight as a consequence of the $q$-combinatorial calculus known in the literature \\cite{exton}. This result represents a close connection between the quantum deformed algebra and the quantum statistical approach.\n\nThe expression for the entropy is nonextensive because of the non-additive property (\\ref{nadd}) of the $q$-basic numbers. We find that for $q \\ne 1$, the entropy remains deformed in the classical limit as is also true of the other thermodynamic functions. This can be understood by observing that the deformation arises from quantum groups but the nature of the deformation is inherently contained in the theory. A similar feature is found in the nonextensive Tsallis statistics and infinite statistics where the deformation persists in the classical limit \\cite{tsallis,greenberg}. \n\nIn this framework, we have studied the basic properties of the ideal $q$-Bose gas in the thermodynamic limit and the phenomenon of $q$-boson condensation. We find that the critical temperature of the $q$-boson is always higher than that of the standard boson. The behavior of the specific heat exhibits a discontinuity at the transition point, which is in qualitative agreement, for values of $q$ close to unity, with experimental data in the case of a dilute gas of Rubidium atoms \\cite{ensher}. \n\nWe observe that for an ideal Bose gase the specific heat is continuous. On the basis of the Ginsburg-Landau theory of $\\lambda$-points, a discontinuous behavior of the specific heat implies a broken symmetry in the transition characterized by an order parameter. The deformation of the algebra in $q$-boson theory implies a broken permutation symmetry of the standard boson wave function. Therefore, the recent experimental data \\cite{ensher} can be interpreted as an indication of the effects due to $q$-deformation in Bose-Einstein condensation, where the order parameter of the phase transition depends on $q$. \n\nAlthough we employed the non-symmetric $q$-deformation in this investigation, all the results can be easily extended by using the symmetric $q$-calculus $(q \\leftrightarrow q^{-1})$. We have confined our study to the $q$-deformation of bosons. It may be worthwhile to investigate the theory of $q$-fermions in this framework.\n\nOur theoretical framework and the results appear to provide a deeper insight into the behavior of the $q$-boson gas. We believe that the results derived here may be relevant to future investigations, and may be of interest from theoretical as well as experimental point of view.\n\n\\vspace{.2in}\n\n\\noindent\n{\\bf Acknowledgments}\n\\vspace{.1 in}\n\nWe are grateful to P. Quarati for encouragement and useful discussions. One of us (A.L.) would like to thank the Physics Department of Southern Illinois University for warm hospitality where this work was done. \n\n\n\\appendix\n\\section{}\n\nHere we present a derivation for the approximation of $q$-basic factorial, $[n]!$ for large $n$, which is the analog of the Stirling approximation. This is employed in the derivation of the entropy in Sec. V. We limit our discussion to the case of $q>1$. \n\nStarting from the definition (\\ref{brnf}) of the $[x]!$ and using the property (\\ref{brnp}) of the $q$-basic number, it is possible write any product factor contained in the $[x]!$ as follows:\n\n\\begin{eqnarray}\n&&[n] \\ \\ \\ \\ \\ \\, = \\ \\ \\ \\ \\, [n] \\nonumber\\\\\n&&[n-1]=q^{-1} \\, [n] -q^{-1}\\nonumber\\\\\n&&[n-2]=q^{-2} \\, [n] -q^{-2}-q^{-1}\\nonumber\\\\\n&&[n-3]=q^{-3} \\, [n] -q^{-3}-q^{-2}-q^{-1}\\nonumber\\\\\n&&\\ \\ \\ \\ \\vdots \\nonumber\\\\\n&&[n-k]=q^{-k} \\, [n] -q^{-k}-q^{-k+1}- \\cdots -q^{-1}\\nonumber\\\\\n&&\\ \\ \\ \\ \\vdots \\nonumber\\\\\n&&[1] \\ \\ \\ \\ \\ \\, =q^{-n+1} \\, [n] -q^{-n+1}-q^{-n+2}- \\cdots -q^{-1}\\; .\n\\label{appefat}\n\\end{eqnarray}\nFor $n\\gg 1$, the leading term is seen to be \n\\begin{equation}\n[n]!\\approx \\, [n]^n \\; q^{-\\sum_{k=0}^{n-1} {\\displaystyle k}} \\; \n\\left ( 1-\\frac{n}{q^n}\\right )\\; ,\n\\end{equation}\nwhere the first term arises from the product of the first term in each of the equations (\\ref{appefat}) and the second term is the result of the sum of the dominant corrections. \n\nThe above equation can be rewritten as\n\\begin{equation}\n[n]!\\approx \\, [n]^n \\; q^{-n(n-1)/2} \\left ( 1-\\frac{n}{q^n}\\right )\\; .\n\\end{equation}\nTaking the logarithm on both sides, we have \n\\begin{equation}\n\\log[n]!\\approx n\\, \\log[n]-\\frac{n^2}{2} \\, \\log q -\\frac{n}{q^n}\\; ,\n\\label{stir}\n\\end{equation}\nwhere we observe that the last term, which follows from the approximation: $\\log(1-n/q^n)\\approx -n/q^n$, is very small and significant only for $q$ very close to unity ($\\vert q-1\\vert <10^{-3}$). We neglect this term in the derivation of the entropy in Sec. V.\n\nWe have verified numerically that the derived $q$-Stirling approximation \nis very good for large $n$. For example, for $q=1.5$ it is correct to an error of $1.8\\%$ for $n=100$, $0.2\\%$ for $n=1000$ and $0.04\\%$ for $n=5000$.\n\n\n\\begin{thebibliography}{30}\n\n\\bibitem{genti}\nG. Gentile, Nuovo Cimento {\\bf 17}, 493 (1940).\n\\bibitem{green}\nH.S. Green, Phys. Rev. {\\bf 90}, 270 (1953).\n\\bibitem{wil}\n{\\it Fractional Statistics and Anyon Superconductivity}, ed. F. Wilczek, World Scientific, Singapore, 1990; \n{\\it Common trends in Condensed matter and high energy physics}, eds. L. Alvarez-Gaum\\'e, A. Devoto, S. Fubini and C. Trugenberger, North-Holland, Amsterdam, 1993.\n\\bibitem{bie}\nL.Biedenharn, J. Phys. A {\\bf 22}, L873 (1989).\n\\bibitem{mac}\n\nA. Macfarlane, J. Phys. A {\\bf 22}, 4581 (1989).\n\n\\bibitem{lee}\n\nC.R. Lee and J.P. Yu, Phys. Lett. A {\\bf 150}, 63 (1990).\n\\bibitem{su}\nG. Su and M. Ge, Phys. Lett. A {\\bf 173}, 17 (1993).\n\\bibitem{tus}\nJ.A. Tuszynski {\\it et al.}, Phys. Lett. A {\\bf 175}, 173 (1993). \n\\bibitem{song}\nH.S. Song, S.X. Ding and I. An, J. Phys. A {\\bf 26}, 5197 (1993).\n\\bibitem{nar}\nP. Narayana Swamy, Int. J. Mod. Phys. B {\\bf 10}, 683 (1996).\n\\bibitem{kan}\nG. Kaniadakis, A. Lavagno and P. Quarati, Phys. Lett A {\\bf 227}, 227 (1997).\n\\bibitem{vok}\nS. Vokos and C. Zachos, Mod. Phys. Lett. A {\\bf 9}, 1 (1994)\nand other references cited in this paper.\n\n\\bibitem{ubri}\nM.R. Ubriaco, Phys. Rev. E {\\bf 57}, 179 (1998).\n\\bibitem{rodi}\n\nM. Rego-Monteiro, I. Roditi, L. Rodrigues, Phys. Lett. A {\\bf 188}, 11 (1994). \n\\bibitem{tsallis}\nC. Tsallis, J. Stat. Phys. {\\bf 52}, 479 (1988). \nSee also Braz. J. Phys. {\\bf 29} (1999), special issue dedicated to nonextensive\nstatistical mechanics and thermodynamics.\n\\bibitem{tsa}\nC. Tsallis, Phys. Lett. A {\\bf 195}, 539 (1994).\n\\bibitem{abe}\nS. Abe, Phys. Lett. A {\\bf 224}, 326 (1997).\n\\bibitem{joh}\nR.S. Johal, Phys. Rev. E {\\bf 58}, 4147 (1998); \nPhys. Lett. A {\\bf 258}, 15 (1999).\n\\bibitem{ubri2}\nM.R. Ubriaco, Phys. Rev. E {\\bf 60}, 165 (1999).\n\\bibitem{jack}\nF. Jackson, Mess. Math. {\\bf 38}, 57 (1909). \n\\bibitem{exton}\nH. Exton, {\\it $q$-Hypergeometric Functions and Applications}, John Wiley and Sons, New York 1983.\n\\bibitem{flo}\nE.G. Floratos, J. Phys. Math. {\\bf 24}, 4739 (1991).\n\\bibitem{fink}\nR. J. Finkelstein, Int. J. Mod. Phys. A {\\bf 13}, 1795 (1998).\n\\bibitem{erz}\nA. Erzan and J.P. Eckmann, Phys. Rev. Lett. {\\bf 78}, 3245 (1997).\n\\bibitem{greenberg}\nO.W. Greenberg, Phys. Rev. Lett. {\\bf 64}, 705 (1990); \nPhys. Rev. D {\\bf 43}, 4111 (1991).\n\\bibitem{ensher}\nJ. Ensher et al., Phys. Rev. Lett. {\\bf 77}, 4984 (1996).\n\n\\end{thebibliography}\n\n\\begin{figure}[htb]\n\\mbox{\\epsfig{file=fig1.eps,width=0.95\\textwidth}}\n\\caption[]{The behavior of $g_{_{3/2}} (z,q)$ as a function of $z$ for different values of $q$. The value $q=1$ corresponds to the standard $g_{_{3/2}}(z)$ boson function. For $q>1$ the upper bound of $z$ is $1/q^2$ and for $q<1$ it is unity \n(see Eq.(\\ref{zq})) \ndue to the existence of the JD of the $g_{_{n}} (z,q)$.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\mbox{\\epsfig{file=fig2.eps,width=0.95\\textwidth}}\n\\caption[]{Same as Fig. 1 for the function $g_{_{5/2}} (z,q)$.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\mbox{\\epsfig{file=fig3.eps,width=0.95\\textwidth}}\n\\caption[]{The ratio $T^q_c/T_c$ of the deformed critical temperature $T^q_c$ and the undeformed ($q=1$) $T_c$ as a function of $q$.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\mbox{\\epsfig{file=fig4.eps,width=0.95\\textwidth}}\n\\caption[]{The specific heat $C_v/N$ as a function of $T/T^q_c$ for $q=1.05$.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\mbox{\\epsfig{file=fig5.eps,width=0.95\\textwidth}}\n\\caption[]{The jump in the specific heat $\\Delta (C_v/N)$ at the critical temperature $T^q_c$ as a function of $q$.}\n\\end{figure}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912111.extracted_bib", "string": "{genti G. Gentile, Nuovo Cimento {17, 493 (1940)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{green H.S. Green, Phys. Rev. {90, 270 (1953)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{wil {Fractional Statistics and Anyon Superconductivity, ed. F. Wilczek, World Scientific, Singapore, 1990; {Common trends in Condensed matter and high energy physics, eds. L. Alvarez-Gaum\\'e, A. Devoto, S. Fubini and C. Trugenberger, North-Holland, Amsterdam, 1993." }, { "name": "quant-ph9912111.extracted_bib", "string": "{bie L.Biedenharn, J. Phys. A {22, L873 (1989)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{mac A. Macfarlane, J. Phys. A {22, 4581 (1989)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{lee C.R. Lee and J.P. Yu, Phys. Lett. A {150, 63 (1990)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{su G. Su and M. Ge, Phys. Lett. A {173, 17 (1993)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{tus J.A. Tuszynski {et al., Phys. Lett. A {175, 173 (1993)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{song H.S. Song, S.X. Ding and I. An, J. Phys. A {26, 5197 (1993)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{nar P. Narayana Swamy, Int. J. Mod. Phys. B {10, 683 (1996)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{kan G. Kaniadakis, A. Lavagno and P. Quarati, Phys. Lett A {227, 227 (1997)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{vok S. Vokos and C. Zachos, Mod. Phys. Lett. A {9, 1 (1994) and other references cited in this paper." }, { "name": "quant-ph9912111.extracted_bib", "string": "{ubri M.R. Ubriaco, Phys. Rev. E {57, 179 (1998)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{rodi M. Rego-Monteiro, I. Roditi, L. Rodrigues, Phys. Lett. A {188, 11 (1994)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{tsallis C. Tsallis, J. Stat. Phys. {52, 479 (1988). See also Braz. J. Phys. {29 (1999), special issue dedicated to nonextensive statistical mechanics and thermodynamics." }, { "name": "quant-ph9912111.extracted_bib", "string": "{tsa C. Tsallis, Phys. Lett. A {195, 539 (1994)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{abe S. Abe, Phys. Lett. A {224, 326 (1997)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{joh R.S. Johal, Phys. Rev. E {58, 4147 (1998); Phys. Lett. A {258, 15 (1999)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{ubri2 M.R. Ubriaco, Phys. Rev. E {60, 165 (1999)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{jack F. Jackson, Mess. Math. {38, 57 (1909)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{exton H. Exton, {$q$-Hypergeometric Functions and Applications, John Wiley and Sons, New York 1983." }, { "name": "quant-ph9912111.extracted_bib", "string": "{flo E.G. Floratos, J. Phys. Math. {24, 4739 (1991)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{fink R. J. Finkelstein, Int. J. Mod. Phys. A {13, 1795 (1998)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{erz A. Erzan and J.P. Eckmann, Phys. Rev. Lett. {78, 3245 (1997)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{greenberg O.W. Greenberg, Phys. Rev. Lett. {64, 705 (1990); Phys. Rev. D {43, 4111 (1991)." }, { "name": "quant-ph9912111.extracted_bib", "string": "{ensher J. Ensher et al., Phys. Rev. Lett. {77, 4984 (1996)." } ]
quant-ph9912112		[]	\noindent Quantum measurement theory is a perplexing discipline fraught with paradoxes and dichotomies. Here we discuss a gedanken experiment that uses a popular testbed - namely, a coupled double quantum dot system - to revisit intriguing questions about the collapse of wavefunctions, irreversibility, objective reality and the actualization of a measurement outcome.	[ { "name": "quantum_measurement.tex", "string": "\\documentstyle[11pt,epsf]{article}\n\\textheight=9.2in\n\\oddsidemargin=0in\n\\textwidth=6.5in\n\\headheight=0pt\n\\headsep=0pt\n\\topmargin=0in\n\n\n\\newcommand{\\spone}{1.1}\n\\newcommand{\\sptwo}{1.6}\n\\newcommand{\\singlespace}{\\edef\\baselinestretch{\\spone}\\Large\\normalsize}\n\\newcommand{\\doublespace}{\\edef\\baselinestretch{\\sptwo}\\Large\\normalsize}\n\n\n\n\\begin{document}\n\n\n\n\\doublespace\n\n\\begin{center}\n{\\large \\bf The Double Quantum Dot Feline Cousin of Schr\\\"odinger's Cat:\nAn Experimental Testbed for a Discourse on Quantum Measurement Dichotomies}\n\\end{center}\n\n\\medskip\n\n\\begin{center}\n{\\bf S. Bandyopadhyay$\\footnote{Corresponding author. E-mail: bandy@quantum1.unl.edu}$}\\\\\n{\\it Department of Electrical Engineering,\nUniversity of Nebraska,\nLincoln, Nebraska 68588-0511, USA}\n\\end{center}\n\n\\medskip\n\n\n\\begin{abstract}\n\\noindent Quantum measurement theory is a perplexing discipline \nfraught with paradoxes and dichotomies. Here we discuss a\ngedanken experiment that uses a popular testbed - namely, a coupled double quantum dot system - to revisit intriguing\nquestions about the collapse of wavefunctions, irreversibility,\nobjective \nreality \nand the actualization of a measurement outcome.\n\\end{abstract}\n\n\\twocolumn\n\n\nQuantum measurement theory is a sub discipline replete with many \nsubtleties of \nquantum mechanics. Its basic underpinning can be summarized by\na fundamental and yet profound question: when and \nhow does a pure state, descriptive of a quantum\nsystem entangled with a measuring apparatus (also a quantum system), \nevolve into a \nmixed state that results in distinguishable outcomes of the \nmeasurement. Since in standard quantum mechanics, \n no unitary time evolution can cause a pure state to \nevolve into a mixed state there is essentially no cookbook\n``quantum recipe''' to forge distinguishable \noutcomes \\cite{home_book, weinberg}in quantum measurement.\n\nA number of formalisms that augment the standard mathematical \nframework of quantum mechanics \\cite{ghirardi, penrose} provide\na dynamical description of the measurement process in \nterms of an actual transition of a pure state into a \nmixed state. This has been termed ``collapse of a \nwave function''. However, even if we accept the \naugmented mathematical framework,\nsome mysteries still remain.\nHow does the collapse occur? Is it a discrete event in\ntime or is it a continuous process? Is the collapse observer-dependent\n(i.e. it happens only when an observer decides to look at the \noutcome of a quantum measurement) or does the outcome materialize at some \ntime independent of the observer? In this short communication,\nwe re-visit these issues in the context of a \npopular quantum system that illustrates many of the subtleties \nin quantum measurement theory.\n\nConsider a double quantum dot system coupled by a translucent \ntunnel barrier.\nThe conduction band diagram is shown in Fig. 1(a).\nThe two quantum dot materials are identical in all respects \nexcept in their elastic constants. That is, electrons cannot\ndistinguish between them, but {\\it phonons can}.\n\\begin{figure}\n\\epsfxsize=5.8in\n\\epsfysize=2.4in\n\\centerline{\\epsffile{schrodinger_cat.ps}}\n\\caption{(a) The conduction band profile of two semiconductor \nquantum dots with an intervening tunnel barrier. All subbands \nare aligned in energy to allow resonant tunneling of electrons\nbetween the two dots. The only difference in the material \nof the two dots is in their elastic constants. (b) The experimental \nset-up.}\n\\end{figure}\nAn electron is introduced into the ground state of the system\nand exists in a coherent superposition of two states $\|1>$ and $\|2>$\n\\begin{equation}\n\\psi = {{1}\\over{\\sqrt{2}}}( \|1> + \|2> )\n\\end{equation}\nwhere $\|1>$ is a semi-localized wave function in the left dot\nand $\|2>$ is a semi-localized wave function in the right dot.\nA weakly coupled point detector in the vicinity\nof one of the dots can tell whether that \ndot is occupied by the electron or the other one is.\nThis experimentally realizable system has been studied in the \ncontext of the quantum \nmeasurement problem by a number of authors \\cite{gurvitz,\nkorotkov, stodolsky} recently.\n\nWe now summarize three different viewpoints regarding the \nquantum measurement problem. The orthodox viewpoint \nassociated with the Copenhagen interpretation is epitomized\nby Von-Neumann:\nthe wave function collapses when an observer chooses to look at the \ndetector and gain knowledge about where the electron is \\cite{von-neumann}.\nThis is an observer-dependent reality and has been \nmuch discussed in the context of the Schr\\\"odinger cat paradox. A different viewpoint \nespoused by a number of researchers \\cite{bohm, gisin, omnes,\nghirardi} is predicated on objective reality. It can be briefly stated as follows: once a measurement\noutcome is actualized, it remains ``out there'' forever to \nbe inspected by an observer at {\\it any} subsequent time\nwithout changing the outcome. The outcome does not depend \non when, or if at all, the observer inspects it, and does \nnot change once actualized. Home and Chattopadhyay \\cite{home} have\nsuggested an experiment involving UV-exposed DNA molecules\nto empirically determine at what {\\it instant} an outcome is actualized\nand the result recorded in a stable and discernible form for perpetuity. \nA third viewpoint \\cite{gurvitz} claims that there may be no\nsuch precise {\\it instant}. The pure state may {\\it\ngradually} evolve towards a mixed state and concomitantly\ndecoherence begins \nto set in, but the system may\nnever quite completely decohere in a finite time (we define complete decoherence\nas the state in which the off-diagonal terms of the 2$\\times$2 density matrix\nassociated with Equation (1) vanish). \nThe off-diagonal terms may decay with time owing\nto the interaction with the detector (and this may slow\ndown the {\\it wiederkehr} quantum oscillation between the states $\|1>$ \nand $\|2>$ \n- the so-called quantum Zeno effect) but the \noff-diagonal terms need not {\\it ever} vanish completely. This has been\ntermed a ``continuous collapse''. Korotkov\n\\cite{korotkov} claims that continuous measurement need not\ncause {\\it any} decoherence or collapse (i.e, the off-diagonal\nterms need not decay at all because of the interaction\nwith the detector) if continuous knowledge \nof the measurement result at all stages of detection is used \nto faithfully reconstruct the pure state. These three viewpoints\nare quite disparate and cannot be reconciled easily.\n\nWe suggest a simple gedanken experiment to resolve some of\n these conflicting viewpoints. Consider\nthe situation when we have two independent\ndetectors capable of detecting which dot is occupied by the \nelectron in Fig. 1. The detectors are independent in the \nsense that they are located vast distances apart and initially there\nis no coupling between them. One detector is the weakly\ncoupled point detector (see Fig. 1b) in the vicinity of a dot capable of \nfairly non-invasive measurement which causes at most\ngradual collapse a l\\'a Gurvitz. The other detector is \na phonon detector located far away. Suppose that when the\nelectron is in the right dot it emits a {\\it zero energy}\nacoustic phonon which has a finite wave vector and hence a finite\nmomentum. It also has a finite group velocity.\nSuch phonons do not typically exist in bulk \nmaterials, but exist in quantum confined structures \nlike wires \\cite{svizhenko} and dots. The emitted phonon\nhas different wave vectors depending on whether the \nemission took place in the left dot or the right dot\nbecause elastic constants (and hence the phonon dispersion\nrelations) in the two dots are different. When the phonon\narrives at the detector, it is absorbed by an electron\nand by measuring the momentum imparted to the electron\n(or equivalently the associated current), one can tell whether the\nphonon came from the left dot or the right dot. Thus, monitoring\nthe current in the phonon detector will constitute a ``measurement''.\nLet us say that the phonon was emitted at time $t$ = 0$\\footnote{\nIt may bother the reader that Heisenberg's Uncertainty Principle\nis being violated in this thought experiment. If the phonon\nhas precisely zero energy, how can we say that it is emitted\nat exactly time t=0? The answer is that at time t=0, we are \nnot {\\it measuring} the energy. If we ever wanted to measure the\nphonon's energy, we could take forever. If indeed Heisenberg's\nUncertainty Principle were relevant here, then {\\it all}\nelastic collisions (e.g. electron-impurity collision) will\ntake forever. Yet we can calculate an effective scattering\ntime for an electron impurity collision from Fermi's\nGolden Rule.}$ and it arrives at the phonon detector at \ntime $t$ = $t_1$. The detector\nfinds that the phonon came from the right dot.\n\n\nIf the viewpoint of objective reality \\cite{home, bohm, gisin, omnes,\nghirardi} is correct, then the actualization of the outcome took \nplace at time $t$ = 0. Thereafter, the electron will be \nalways found in the right dot.\nWe can empirically\npinpoint this instant at a later time\n$t$ $>$ 0 (actually at $t$ $\\geq$ $t_1$)\nsince we can determine $t_1$, the time of flight of the \nphonon between the dot and the phonon detector. We simply have to know\nthe distance between the dot and the detector and the phonon group \nvelocity to know $t_1$. Thus when the phonon detector registers the phonon,\nwe will know that the actualization took place $t_1$ units of \ntime prior to the registration event.\nAdditionally, if we know the time $t$ = -$t_2$ when the \nelectron was injected into the double dot system, then \nwe can find out how long thereafter the actualization of \nthe outcome took place (this time is simply $t_2$). This is\nsimilar to what Home and Chattopadhyay had proposed to achieve\nin their UV-exposed DNA system \\cite{home}.\n\nWe now come to the central issue. Between the time $t$ = 0 and \n$t$ = $t_1$ (i.e. while the phonon is in flight), the observer\n(phonon detector) is still ignorant of the outcome,\nbut the actualization of the measurement \\cite{home}\nhas supposedly already taken place. During this critical \ntime period, the\nweakly coupled point detector tries to \n{\\it continuously} determine which dot is occupied. \nIf the {\\it observer-independent} viewpoint is correct,\nthen the electron will be always found in the right dot. \n But, if the {\\it observer-dependent} \nviewpoint is correct \\cite{von-neumann}, then the \nSchr\\\"odinger cat is in suspended animation between \n$t$ = 0 and $t$ = $t_1$ since the observer (phonon \ndetector) has not registered any phonon yet. Consequently,\n the almost non-invasive point detector (which takes a very long\ntime to destroy the superposition acting alone) should have a\nnon-zero probability of finding the electron in the left dot.\nTo ensure that these are the only two possible scenarios, we will\nallow the maximum latitude.\nFor instance, we will assume: (i) the quantum oscillation\nperiod between the two dots ({\\it wiederkehr}) is much smaller than the time of flight $t_1$\nand the Zeno effect \\cite{misra} is negligible because of the weak \ncoupling with the non-invasive point detector,\n(ii) the emission of zero energy phonon does not alter the electron's\nenergy and hence does not subsequently disallow resonant tunneling between \nthe quantum dots, and (iii) the remote phonon detector is unaware of the set-up\nbefore time $t$ = $t_1$ and hence cannot influence events before \ntime $t$ = $t_1$ (causality). Thus, if the point detector ever finds the \nelectron in the left dot between $t$ = 0 and $t$ = $t_1$, the \nobjective reality (observer-independent) viewpoint will be suspect.\nIn this pathological example, the difference between the observer-dependent \nand observer-independent viewpoint can be simply stated thus:\nin the first viewpoint, the collapse took place at $t$ = $t_1$ and in\nthe second viewpoint, it took place at $t$ = 0. As long as any non-invasive\ndetector in the timeframe $t$ = 0 till $t$ = $t_1$ finds the electron\nin the left dot and the phonon detector at time $t_1$ \nfinds the electron to have \nemitted the phonon in the right dot, we will know that the ``collapse'' \ndid not take place at $t$ = 0 which would then contradict the \nobserver independent viewpoint. We will then be forced to admit \nthat perhaps collapse ultimately takes place in the sensory\nperception of the observer \\cite{aicardi}. This is currently\na contentious topic.\n\nAn interesting question is whether the phonon \nemission is a collapse event. There is no energy \ndissipation involved in emitting a zero-energy phonon,\nbut energy dissipation is not necesssary for collapse since \n{\\it elastic} interaction of an electron with a magnetic \nimpurity that causes a change in the internal degree of \nfreedom of the scatterer (say, spin flip) constitutes \neffective collapse. ``Creation'' of a phonon is certainly changing\nits internal degrees of freedom in a major way and therefore\nshould be viewed as a collapse event within the framework\nof standard models.\n\nBut what if the point detector will find the \nelectron in the left dot {\\it after} time $t$ = $t_1$ when \nthe phonon detector has already determined that the electron\ncollapsed in the right dot. This will\nmake standard collapse models suspect \\cite{leggett} since \nwe must then admit that the phonon emission did not cause a collapse.\n Complete collapse is an irreversible \nevent (equivalent to saying that the Zeno time is infinite).\nHowever the third viewpoint of Gurvitz \\cite{gurvitz}\nguarantees that the electron will be ultimately delocalized\n(and hence found in the left dot with a non-zero\nprobability) if we make a continuous measurement with the point detector.\nIn contrast, if frequent repeated measurements are made, then the Zeno\neffect guarantees that the opposite will happen; the electron\nwill become more localized in one dot as the frequency of \nobservation is increased. Thus, there is an essential \ndichotomy when one considers the fact that a continuous measurement\nis really the ultimate limit of frequent repeated measurements\nand yet they make opposite predictions. It is \nnot clear how this dichotomy will be ultimately resolved. \n\nIn this communication, we have proposed a gedanken experiment \nto resolve some of the \\\\\ndichotomies between the myriad viewpoints\npermeating quantum measurement theory. Experiments such as the one \nproposed here\nwill soon be within the reach of modern technology. Hopefully,\nthey will shed new light on this fascinating topic. \n\n\n\\bigskip\n\n%\\bibliographystyle{/d/gady/Style/yzaip}\n%\\bibliography{/d/rr/latexlib/one}\n \n\n\\begin{thebibliography}{10}\n\n\\bibitem{home_book}\nD. Home, {\\it Conceptual Foundations of Quantum Physics - An Overview\nFrom Modern Perspectives} (Plenum, New York, 1997).\n\n\\bibitem{weinberg}\nS. Weinberg, {\\it Dreams of a Final Theory} (Vintage, London, 1993).\n\n\\bibitem{ghirardi}\nG. C. Ghiradi, A. Rimini and T. Weber, {\\it Phys. Rev. D}, {\\bf\n34}, 470 (1986); G. C. Ghiradi, R. Grassi and A, Rimini, {\\it Phys. Rev. A},\n{\\bf 42}, 1057 (1990)\n\n\\bibitem{penrose}\nR. Penrose, {\\it Gen. Rel. and Gravit.}, {\\bf 28}, 581 (1996).\n\n\n\n\n\\bibitem{gurvitz}\nS. A. Gurvitz, {\\it Phys. Rev. B}, {\\bf 56}, 15215 (1997); e-print\nquant-ph 9806050\n\n\\bibitem{korotkov}\nA. N. Korotkov, e-print quant-ph 9808026\n\n\n\\bibitem{stodolsky}\nL. Stodolsky, e-print quant-ph 9805081\n\n\\bibitem{von-neumann}\nJ. Von Neumann, {\\it Mathematische Grundlagen der Quantenthorie}\n(Springer, Berlin, 1931).\n\n\\bibitem{bohm}\nD. Bohm and B. J. Hiley, {\\it The Undivided Universe} (Routledge,\nLondon, 1993).\n\n\\bibitem{gisin}\nN. Gisin in {\\it Fundamental Problems in Quantum Theory}, Ed. D. M. Greenberger\nand A. Zeilinger (Annals of the New York Academy of Sciences, New\nYork, 1995), 524.\n\n\n\n\\bibitem{omnes}\nR. Omnes, {\\it The Interpretation of Quantum Mechanics}, (Princeton\nUniversity Press, Princeton, 1994).\n\n\n\n\n\n\\bibitem{home}\nD. Home and R. Chattopadhyay, {\\it Phys. Rev. Lett.}, {\\bf 76},\n2836 (1996).\n\n\\bibitem{svizhenko}\nA. Svizhenko, A. Balandin, S. Bandyopadhyay and M. A. Stroscio,\n{\\it Phys. Rev. B}, {\\bf 57}, 4687 (1998).\n\n\\bibitem{misra}\nB. Misra and E. C. G. Sudarshan, {\\it J. Math. Phys.}, {\\bf 18}, 756 (1977);\nR. A. Harris and L. Stodolsky, {\\it Phys. Lett.}, {\\bf B 116}, 464 (1982);\nC. Priscilla, R. Onofrio and U. Tambini, {\\it Ann. Phys.},\n{\\bf 248}, 95 (1996).\n\n\\bibitem{aicardi}\nF. Aicardi, F. Borsellino, G. C. Ghirardi and R. Grassi, {\\it Found. Phys.\nLett.}, {\\bf 4}, 109 (1991).\n\n\\bibitem{leggett}\nA. J. Leggett, in {\\it Nanostructure Physics and Fabrication}, Eds. M. A. Reed and W. P. Kirk (Academic Press, Boston, 1989).\n\n\\end{thebibliography}\n\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912112.extracted_bib", "string": "{home_book D. Home, {Conceptual Foundations of Quantum Physics - An Overview From Modern Perspectives (Plenum, New York, 1997)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{weinberg S. Weinberg, {Dreams of a Final Theory (Vintage, London, 1993)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{ghirardi G. C. Ghiradi, A. Rimini and T. Weber, {Phys. Rev. D, {34, 470 (1986); G. C. Ghiradi, R. Grassi and A, Rimini, {Phys. Rev. A, {42, 1057 (1990)" }, { "name": "quant-ph9912112.extracted_bib", "string": "{penrose R. Penrose, {Gen. Rel. and Gravit., {28, 581 (1996)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{gurvitz S. A. Gurvitz, {Phys. Rev. B, {56, 15215 (1997); e-print quant-ph 9806050" }, { "name": "quant-ph9912112.extracted_bib", "string": "{korotkov A. N. Korotkov, e-print quant-ph 9808026" }, { "name": "quant-ph9912112.extracted_bib", "string": "{stodolsky L. Stodolsky, e-print quant-ph 9805081" }, { "name": "quant-ph9912112.extracted_bib", "string": "{von-neumann J. Von Neumann, {Mathematische Grundlagen der Quantenthorie (Springer, Berlin, 1931)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{bohm D. Bohm and B. J. Hiley, {The Undivided Universe (Routledge, London, 1993)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{gisin N. Gisin in {Fundamental Problems in Quantum Theory, Ed. D. M. Greenberger and A. Zeilinger (Annals of the New York Academy of Sciences, New York, 1995), 524." }, { "name": "quant-ph9912112.extracted_bib", "string": "{omnes R. Omnes, {The Interpretation of Quantum Mechanics, (Princeton University Press, Princeton, 1994)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{home D. Home and R. Chattopadhyay, {Phys. Rev. Lett., {76, 2836 (1996)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{svizhenko A. Svizhenko, A. Balandin, S. Bandyopadhyay and M. A. Stroscio, {Phys. Rev. B, {57, 4687 (1998)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{misra B. Misra and E. C. G. Sudarshan, {J. Math. Phys., {18, 756 (1977); R. A. Harris and L. Stodolsky, {Phys. Lett., {B 116, 464 (1982); C. Priscilla, R. Onofrio and U. Tambini, {Ann. Phys., {248, 95 (1996)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{aicardi F. Aicardi, F. Borsellino, G. C. Ghirardi and R. Grassi, {Found. Phys. Lett., {4, 109 (1991)." }, { "name": "quant-ph9912112.extracted_bib", "string": "{leggett A. J. Leggett, in {Nanostructure Physics and Fabrication, Eds. M. A. Reed and W. P. Kirk (Academic Press, Boston, 1989)." } ]
quant-ph9912113	Coherent information analysis of quantum channels in simple quantum systems	[ { "author": "B.\\ A.\\ Grishanin and V.\\ N.\\ Zadkov\\thanks{zadkov@comsim1.ilc.msu.su" } ]	The coherent information concept is used to analyze a variety of simple quantum systems. Coherent information was calculated for the information decay in a two-level atom in the presence of an external resonant field, for the information exchange between two coupled two-level atoms, and for the information transfer from a two-level atom to another atom and to a photon field. The coherent information is shown to be equal to zero for all full-measurement procedures, but it completely retains its original value for quantum duplication. Transmission of information from one open subsystem to another one in the entire closed system is analyzed to learn quantum information about the forbidden atomic transition via a dipole active transition of the same atom. It is argued that coherent information can be used effectively to quantify the information channels in physical systems where quantum coherence plays an important role.	[ { "name": "paper.tex", "string": "%%\n%% Coherent information analysis of quantum channels\n%% in simple quantum systems\n%% by Boris Grishanin and Victor Zadkov\n%%\n%% International Laser Center and Department of Physics\n%% M.V.Lomonosov Moscow State University\n%% Moscow 119899, Russia\n%%\n%% Phone: 007(095)939-51-73\n%% Fax: (007)095)939-31-13\n%% E-mail: zadkov@comsim1.ilc.msu.su\n%%\n%%\n%% Corrected version, May 30, 2000\n%%\n\n\\documentstyle[pra,aps,epsf,preprint]{revtex}\n\n\n\\newcommand{\\ket}[1]{\\mathop{\\left\| #1 \\right\\rangle}\\nolimits}\n\\newcommand{\\bra}[1]{\\mathop{\\left\\langle #1 \\right\|}\\nolimits}\n\\newcommand{\\braket}[2]{\\mathop{\\left\\langle #1 \\left\| #2 \\right.\n\\right\\rangle}\\nolimits}\n\\newcommand{\\avr}[1]{\\mathop{\\left\\langle #1\\right\\rangle}\\nolimits}\n\n\n\\begin{document}\n\\draft\n\n\\tighten % comment when submitting\n\n%\\wideabs{\n\\title{Coherent information analysis of quantum channels in\nsimple quantum systems}\n\\author{B.\\ A.\\ Grishanin and\nV.\\ N.\\ Zadkov\\thanks{zadkov@comsim1.ilc.msu.su}}\n\\address{International Laser Center and Department of Physics\\\\\nM.\\ V.\\ Lomonosov Moscow State University, 119899 Moscow, Russia}\n\\date{March 15, 2000}\n\\maketitle\n\n\\begin{abstract}\nThe coherent information concept is used to analyze a variety of\nsimple quantum systems. Coherent information was calculated for\nthe information decay in a two-level atom in the presence of an\nexternal resonant field, for the information exchange between two\ncoupled two-level atoms, and for the information transfer from a\ntwo-level atom to another atom and to a photon field. The coherent\ninformation is shown to be equal to zero for all full-measurement\nprocedures, but it completely retains its original value for\nquantum duplication. Transmission of information from one open\nsubsystem to another one in the entire closed system is analyzed\nto learn quantum information about the forbidden atomic transition\nvia a dipole active transition of the same atom. It is argued that\ncoherent information can be used effectively to quantify the\ninformation channels in physical systems where quantum coherence\nplays an important role.\n\\end{abstract}\n\\pacs{PACS numbers: 03.65.Bz, 03.65.-w, 89.70.+c}\n%}\n\n\\narrowtext\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe concept of noisy quantum channel may be used in many\ninformation-carrying applications, such as quantum communication,\nquantum cryptography, and quantum computers \\cite{qc-book}.\nShannon's theory of information\n\\cite{shannon48,shannon49,gallagher68,grover91} is a purely\nclassical one and cannot be applied to quantum mechanical systems.\nTherefore, much recent work has been done on quantum analogues of\nthe Shannon theory\n\\cite{schumacher96,schum96,bennett96,lloyd96,barnum98,preskill}.\nThe {\\em coherent information} introduced in\n\\cite{schum96,lloyd96} is suggested to be analogous to the concept\nof {\\em mutual information} in classical information theory. It is\ndefined by\n\\begin{equation}\n\\label{Ic}\nI_c=S_{\\rm out}-S_e,\n\\end{equation}\n\n\\noindent where $S_{\\rm out}$ is the entropy of the information\nchannel output and $S_e$ is the {\\em entropy exchange}\n\\cite{schumacher96,lloyd96} taken from the channel reservoir. If\n$S_{\\rm out}-S_e>0$, then, expressed in {\\em qubits}, $I_c$ describes a\nbinary logarithm of the Hilbert space dimension, all states of\nwhich are transmitted with the probability $p=1$ in the limit of\ninfinitely large ergodic ensembles. Otherwise, we set $I_c=0$.\n\nThe validity of the coherent information concept was proved in\n\\cite{lloyd96,barnum98}, and it was used successfully for\nquantifying the resources needed to perform physical tasks.\nCoherent information is expected to be as universal as its\nclassical analogue, Shannon information, and it characterizes a\nquantum information channel regardless of the nature of both\nquantum information and quantum noise. In contrast to Shannon\ninformation in classical physics, however, coherent information is\nexpected to play a more essential role in quantum physics. The\ncapacity of information channels in classical physics can be\nestimated, in most cases, even without relying on any information\ntheory, at least within an order of magnitude. This, however, is\nnot feasible in quantum physics and the coherent information\nconcept, or a similar concept, must be used to quantify the\ninformation capacity of the channel. An analysis of the quantum\ninformation potentially available in physical systems is\nespecially important for planning experiments in new fields of\nphysics, such as quantum computations, quantum communications, and\nquantum cryptography\\cite{qc-book,preskill}, where the coherent\ninformation of the quantum channel determines its potential\nefficiency.\n\nIn this paper, we apply the coherent information concept to an\nanalysis of the quantum information exchange between two systems,\nwhich in general case may have essentially different Hilbert\nspaces. For this purpose, we must specify the noisy quantum\ninformation channel and its corresponding superoperator $\\cal S$,\nwhich transforms the initial state of the first system into the\nfinal state of another system. A classification scheme for\npossible quantum channels connecting two quantum systems is shown\nin Fig.\\ \\ref{fig1} \\cite{note}. In addition to the two-time\nchannels shown in the figure, we consider also their one-time\nanalogues. Two-time quantum channels are widely used in quantum\ncommunications and measurements, whereas one-time quantum channels\nare appropriate for quantum computing and quantum teleportation.\n\nThe paper is organized as follows. In section\n\\ref{sec:definitions}, we explain key definitions and review\nsuperoperator representation technique, which is used throughout\nthe paper. In the following sections we consider a variety of\nquantum channels that correspond to the classification scheme\nshown in Fig.\\ \\ref{fig1}. Section \\ref{sec:onequbit} discusses\nthe coherent information transfer between quantum states of a\ntwo-level atom (TLA) in a resonant laser field at two time\ninstants (Fig.\\ \\ref{fig1}a). The same type of quantum channel\n($1\\to1$) can be considered for a system that contains two (or\nmore) subsystems. This case is analyzed in section\n\\ref{sec:qubit-intra}, using a spinless model of the hydrogen atom\nas an example. Coherent information transfer between two different\nquantum systems is considered in section \\ref{sec:twoqs}. The\nanalysis includes coherent information transfer between (i) two\nunitary coupled TLAs (Fig.\\ \\ref{fig1}b), (ii) two TLAs coupled\nvia the measuring procedure (Fig.\\ \\ref{fig1}b), (iii) an\narbitrary system and its duplication (Fig.\\ \\ref{fig1}c), (iv) a\nTLA in the free space photon field (Fig.\\ \\ref{fig1}b), and (v)\ntwo TLAs via the free space photon field (Fig.\\ \\ref{fig1}b).\nFinally, section \\ref{sec:conclusions} concludes with a summary of\nour results.\n\n\n\\section{Key definitions and calculation technique}\n\\label{sec:definitions}\n\n\\subsection{Notations and superoperator representation technique}\n\\label{subsec:notations}\n\nThis subsection explains key notations and briefly reviews the symbolic\nsuperoperator representation technique \\cite{gKE79}, which is especially\nconvenient for mathematical treatment of coherent information transmission\nthrough a noisy quantum channel.\n\nThe most general symbolic representation of a superoperator is defined by\nthe expression\n\\begin{equation}\n\\label{genS}\n{\\cal S}=\\sum\\hat s_{kl}\\bra{k}\\odot\\ket{l},\n\\end{equation}\n\n\\noindent where the substitution symbol $\\odot$ must be replaced\nby the transforming operator variable and $\\bra{k}$ is an arbitrarily\nchosen vector basis in Hilbert space $H$, to which the transformed\noperators are applied. In order to correctly apply this\ntransformation to a density matrix, operators $\\hat s_{kl}$ must\nobey the positivity condition for the block operator $\\hat S=(\\hat\ns_{kl})$ \\cite{positivity} and orthonormalization condition\n\\begin{equation}\n\\label{snorm}\n{\\rm Tr}\\,\\hat s_{kl}=\\delta_{kl},\n\\end{equation}\n\n\\noindent which provides normalization for all normalized operators\n$\\hat\\rho$ with ${\\rm Tr}\\,\\hat\\rho=1$.\n\nUsing symbolic representation (\\ref{genS}), one can easily\nrepresent the production of superoperators ${\\cal S}_1$, ${\\cal\nS}_2$, which constitutes a symbolic representation of the\nsuperoperator algebra. For $\\hat s_{kl}=\\ket{k}\\bra{l}$ it results\nto the identity superoperator, ${\\cal I}$, and for $\\hat s_{kl}\n=\\ket{k}\\bra{k} \\delta_{kl}$---to the quantum reduction\nsuperoperator ${\\cal R} =\\sum \\ket{k} \\bra{k} \\odot\n\\ket{k}\\bra{k}$. The case of $\\hat s_{kl}= \\delta_{kl}$ represents\nthe trace superoperator ${\\rm Tr}\\odot,$ which is a linear\nfunctional in the density matrix space. The correspondence between\nthe matrix representation $S=(S_{mn})$ of the superoperator ${\\cal\nS}$ in orthonormalized operator basis $\\hat e_k$ and its symbolic\nrepresentation (\\ref{genS}) is given by\n\\begin{equation}\n\\label{skl}\n\\hat s_{kl}= {\\cal S}(\\ket{k}\\bra{l})=\n\\sum\\limits_{mn}S_{mn} \\bra{l}\\hat e_n\\ket{k}\\hat e_m\n\\end{equation}\n\n\\noindent and can be easily checked by substituting it in Eq.\\\n(\\ref{genS}) and comparing with the standard definition of matrix\nelements ${\\cal S}\\hat e_n=\\sum_m S_{mn}\\hat e_m$.\n\n\n\\subsection{The calculation of coherent information}\n\\label{subsec:general}\n\nThe entropy exchange in Eq.\\ (\\ref{Ic}) for the coherent\ninformation is defined as\n\\begin{equation}\n\\label{Se}\nS_e=S(\\hat\\rho_\\alpha),\\quad S(\\hat\\rho)=-{\\rm\nTr}\\,\\hat\\rho\\log_2\\hat\\rho,\n\\end{equation}\n\n\\noindent where the joint input-output density matrix $\\hat\\rho_\\alpha$\nis given, in accordance with \\cite{lloyd96,conjugation}, by\n\\begin{equation}\n\\label{inout}\n\\hat\\rho_\\alpha=\\sum\\limits_{ij}\\,{\\cal S} (\\ket{\\rho_i} \\bra{\\rho_j})\n\\otimes \\ket{\\bar{\\rho}_i}\\bra{\\bar{\\rho}_j}.\n\\end{equation}\n\n\\noindent Here $\\ket{\\rho_i}=\\hat\\rho_{\\rm in}^{1/4}\\ket{i}$ are\nthe transformed eigenvectors of the input density matrix\n$\\hat\\rho_{\\rm in}=\\sum p_i\\ket{i}\\bra{i}$, bar symbol stands for\ncomplex conjugation, and ${\\cal S}$ is the channel input-output\nsuperoperator, so that the output density matrix $\\hat\\rho_{\\rm\nout}={\\cal S} \\hat \\rho_{\\rm in}$. Using superoperator\nrepresentation (\\ref{genS}) within the above defined eigen basis\n$\\ket{i}$, the density matrix (\\ref{inout}) takes the form:\n\\begin{equation}\n\\label{inouts}\n\\hat\\rho_\\alpha=\\sum\\limits_{ij}(p_i^{}p_j^{})^{1/4}\\,\\hat s_{ij}^{}\n\\otimes \\ket{\\bar{\\rho}_i}\\bra{\\bar{\\rho}_j},\n\\end{equation}\n\n\\noindent where operators $\\hat s_{ij}$ represent the states of\nthe output. Both the input and output marginal density matrices\nare given by the trace over the corresponding complementary\nsystem: $\\hat \\rho_{\\rm out} ={\\rm Tr}_{\\rm in} \\hat\\rho_\\alpha$,\n$\\hat{\\bar{\\rho}}_{\\rm in}={\\rm Tr}_{\\rm out} \\hat\\rho_\\alpha$. Finally,\nthe coherent information (\\ref{Ic}) can be calculated, keeping in\nmind that $S_{\\rm out}=S(\\hat\\rho_{\\rm out})$.\n\n\n\\subsubsection{Two-time coherent information for two quantum systems}\n\\label{subsubsec:twosystem}\n\nFor the coherent information transfer between two quantum systems\nthrough the quantum channels shown in Figs \\ref{fig1}b,c ($1\\to2$ or\n$1\\to(1+2)$), the initial joint density matrix must be taken in the\nproduct form $\\hat\\rho_{1+2}= \\hat\\rho_{\\rm in}\\otimes\\hat\\rho_2$, where\n$\\hat\\rho_{\\rm in}=\\hat\\rho_1$ and $\\hat\\rho_2$ are the initial\nmarginal density matrices, the first one being an input. For the\n$1\\to2$ quantum channel, the output is the state of the second system,\nsince a transformation on these two systems is made and a certain\namount of information is transmitted into the second system from the\ninitial state of the first one.\n\nThe dynamical evolution of the joint (1+2) system is given by a\nsuperoperator ${\\cal S}_{1+2}$ and the corresponding channel\ntransformation superoperator, which converts $\\hat\\rho_{\\rm\nout}={\\cal S}\\hat\\rho_{\\rm in}$, can be written as\n$$\n{\\cal S}={\\rm Tr}_1\\,{\\cal S}_{1+2}(\\odot\\otimes\\hat\\rho_2),\n$$\n\n\\noindent where the trace is taken over the final state of the first\nsystem. The transformation is described in terms of Eq.\\ (\\ref{genS})\nfor the joint system as\n\\begin{equation}\n\\label{skl12}\n{\\cal S}=\\sum\\limits_{k\\kappa\\;l\\lambda}\\sum\\limits_n \\bra{n}\n\\hat s_{k\\kappa,l\\lambda} \\ket{n}\\bra{\\kappa}{\\hat\\rho_2}\\ket{\\lambda}\n\\bra{k} \\odot\\ket{l},\n\\end{equation}\n\n\\noindent where the product basis $\\ket{k}\\ket{\\kappa}$ is used and\nindexes $k$, $\\kappa$ stand for the first and second quantum\nsystems, respectively. The operator coefficients $\\hat s_{kl}$ in Eq.\\\n(\\ref{genS}) now take the form:\n\\begin{equation}\n\\label{sklnew}\n\\hat s_{kl}=\\sum\\limits_{\\kappa\\lambda}\\sum\\limits_n \\bra{n}\n\\hat s_{k\\kappa,l\\lambda}\n\\ket{n}\\bra{\\kappa}{\\hat\\rho_2}\\ket{\\lambda}.\n\\end{equation}\n\n\\noindent Superoperator ${\\cal S}$ depends on both the dynamical transformation\n${\\cal S}_{1+2}$ and the initial state $\\hat\\rho_2$, and couples the\ninitial state of the first system with the final state of the second\nsystem.\n\n\n\\subsubsection{One-time coherent information}\n\\label{subsubsec:one-time}\n\nOne-time information quantities can be easily calculated if the\ncorresponding joint density matrix is known. In the case of a\nsingle system, the corresponding channel is described by the\nidentity superoperator ${\\cal I}$. For the joint input-output\ndensity matrix (\\ref{inout}), we get a pure state $\\hat \\rho_\\alpha\n= \\sum_i \\ket{\\rho_i}\\ket{\\rho_i} \\sum_j\\bra{\\rho_j} \\bra{\\rho_j}$\nand then calculate the entropy exchange $S_e=0$ and the coherent\ninformation $I_c= S_{\\rm out}=S_{\\rm in}$. In the case of two\nsystems, the input-output density matrix is the joint density\nmatrix $\\hat \\rho_{1+2}$, and the corresponding coherent\ninformation in system 2 on system 1 at time $t$ is $I_c(t)\n= S[\\hat\\rho_2(t)]- S[\\hat \\rho_{1+2}(t)]$. In the case of unitary\ndynamics and a pure initial state of the second system, all initial\neigenstates $\\ket{i}$ of the first system\ntransform into the corresponding orthogonal set $\\Psi_i(t)$ of the\n(1+2) system, so that the joint entropy is time-independent and\nthe coherent information yields $I_c(t)= S[\\hat\\rho_2(t)] - S[\\hat\n\\rho_1(0)]$. If the initial state of the first system is also a\npure state, we get simply $I_c(t)=S[\\hat \\rho_2(t)]$. For the\nTLA case, this simply yields $I_c=1$ qubit, if a maximally entangled\nstate of two-atom qubits is achieved.\n\n\n\\section{TLA in a resonant laser field}\n\\label{sec:onequbit}\n\nIn this section, we discuss the coherent information transfer\nbetween the quantum states of a TLA in a resonant laser field at\ntwo time instants (Fig.\\ \\ref{fig1}a). Such quantum channel with\npure dephasing in the absence of an external field was considered\nin \\cite{lloyd96}. In a more general case, coherent information,\nbased on the joint input-output density matrix (\\ref{inout}), can\nbe readily calculated by using the matrix representation technique\nfor the relaxation dynamics superoperator. An interesting question\nis how the coherent information depends on the applied resonance\nfield.\n\nThe field changes the relaxation rates of the TLA. These rates are\npresented with the real parts of the eigenvalues $\\lambda_k$ of the\ndynamical Liouvillian ${\\cal L}={\\cal L}_r + {\\cal L}_E$ of the TLA,\nwhere ${\\cal L}_r$ and ${\\cal L}_E$ stand for the relaxation and field\ninteraction Liouvillians. For simplicity, we will consider here\nrelaxation caused only by pure dephasing, combined with the laser field\ninteraction. The corresponding Liouvillian matrix in the basis of $\\hat\ne_k=\\{\\hat I, \\hat \\sigma_3, \\hat\\sigma_1, \\hat\\sigma_2\\}$ reads\n\\begin{equation}\n\\label{Lm}\nL=\\left(\n\\begin{array}{cccc}\n 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\Omega \\\\\n 0 & 0 & -\\Gamma & 0 \\\\\n 0 & -\\Omega & 0 & -\\Gamma\n\\end{array}\n\\right),\n\\end{equation}\n\n\\noindent where $\\Gamma$ is the pure dephasing rate, $\\Omega$ is the\nRabi frequency, and $\\hat \\sigma_1$, $\\hat \\sigma_2$, $\\hat \\sigma_3$\nare the Pauli matrices. The eigenvalues of the matrix (\\ref{Lm}) can be\nreadily calculated and are given by\n$$\n\\lambda_k=\\{0, -\\Gamma,-(\\Gamma+ \\sqrt{\\Gamma^2 -4\\Omega^2})/2,\n-(\\Gamma-\\sqrt{\\Gamma^2- 4\\Omega^2}) /2\\}.\n$$\n\n\\noindent These values are affected by the resonant laser field with\nrespect to the unperturbed values $0,$ $\\Gamma$, which also affects the\nresonant fluorescence spectrum of the TLA. At $\\Omega>\\Gamma/2$ it results\nin so-called Mollow-triplet structure, centered at the\ntransition frequency, which has been predicted theoretically \\cite{mollow}\nand subsequently confirmed experimentally \\cite{ezekil}.\n\nFrom the information point of view, the resonant laser field might\nreduce the coherent information decay rate and, therefore, lead to the\nincrease of information, although this information gain could intuitively be\nexpected only from the laser-induced reduction of the\nrelaxation rates of the relaxation superoperator ${\\cal L}_r$ itself\n\\cite{pestov73,lisitsa75,burnett82,gJETP83}.\n\nCalculating the matrix of the evolution superoperator ${\\cal\nS}=\\exp({\\cal L}t)$ and using its corresponding representation\n(\\ref{genS}), the joint density matrix may be calculated\nanalytically (\\ref{inout}). Then (with the help of Eqs (\\ref{Se}),\n(\\ref{Ic})), the coherent information left in the TLA's state at\ntime $t$ may be calculated about its initial state. This state is\nchosen in the form of the maximum entropy density matrix\n$\\hat\\rho_0=\\hat I/2$. The results of our calculations are\npresented in Fig.\\ \\ref{fig2}. They show the typical\nthreshold-type dependence of the coherent information versus time,\nwhich is determined by the loss of coherence in the system. Also,\nthe coherent information does not increase with an increase of the\nlaser field intensity, as might be expected. The coherent\ninformation even decreases as the Rabi frequency increases.\n\nIn addition, the results show a singularity in the first\nderivative of the coherent information dependence at time $t=0$,\nwhich is a characteristic feature of the starting point of the\ndecay of coherent quantum information. Initially, the input-output\ndensity matrix (\\ref{inout}) of the TLA is a pure state\n$\\hat\\rho_\\alpha=\\Psi\\Psi^+$ with the input-output wave function\n$\\Psi=\\sum\\sqrt{p_i} \\ket{i} \\ket{i}$. Its eigenvalues $\\lambda_k$\nand the probabilities of the corresponding eigenstates are all\nequal to zero, except for the eigenstate corresponding to $\\Psi$.\nDue to the singularity of the entropy function\n$-\\sum\\lambda_k\\log\\lambda_k$ at $\\lambda_k=0$ the derivative of\nthe corresponding exchange entropy also shows a logarithmic\nsingularity.\n\nAnother interesting feature of coherent information is its\ndependence on the initial (input) state $\\hat\\rho_{in}$. If it were\npossible, $\\hat\\rho_{in}$ might be chosen in the form of the\neigenoperator\n$$\n\\hat\\rho_{\\rm in}=\\sum\\limits_{l=1}^4\\ket{k_{\\min}}_l\\hat e_l\n$$\n\n\\noindent of the Liouvillian, where $\\ket{k_{\\min}}$ is the\neigenvector corresponding to the minimum value\n$\|\\Re{e}\\lambda_k\|>0$ of the matrix $L$. Yet the vector\n$\\ket{k_{\\min}}$ is equal to $\\{0, (\\Gamma+\\sqrt{\\Gamma^2 -\n4\\Omega^2})/2\\Omega, 0, 1\\}$, which corresponds to the linear\nspace of operators with zero trace due to the zero value of the\nfirst component. Therefore, the coherent information decay rate\ncannot be reduced by reducing the corresponding decay of atomic\ncoherence.\n\n\n\\section{Coherent information transfer between two subsystems of\nthe same quantum system}\n\\label{sec:qubit-intra}\n\nIn this section we investigate the quantum channel ($1\\to1$, Fig.\\\n\\ref{fig1}a) between two open subsystems $A$ and $B$ of a closed\nsystem $A+B$ having a common Hilbert space ${\\rm sp}\\,(H_A,H_B)$,\nwhere $H_A$ and $H_B$ are the Hilbert subspaces of the subsystems\n$A$ and $B$, respectively.\n\nIn classical information theory, this situation corresponds to the\ntransmission of part $A\\subset X$ of the values of an input random\nvariable $x\\in X$. The situation where a receiver receives no message\nis also informative and means that $x$ belongs to the supplement\nof $A$, $x\\in\\bar A$. It can be described by the {\\em choice}\ntransformation ${\\cal C}=P_A+P_0(1-P_A)$, where $P_A$ is the\nprojection operator from $X$ onto the subset $A$, $P_A x=x$ for\n$x\\in A$ and $P_A x=\\O$ (empty set) for $x\\in\\bar A$, $P_0$ is the\nprojection from $X$ onto an independent single-point set $X_0$,\nand $P_0 x=X_0$. This transformation corresponds to the classical\n{\\em reduction} channel, resulting in information loss only if\n$\\bar A$ is not a single point. If $\\bar A$ is a single point, we are\nable to get a maximum of one bit of information, for $\\bar A$ can provide\nanother point of the bit, so that for an input bit we have no loss\nof information.\n\nIn quantum mechanics, the corresponding reduction channel is\nrepresented as the choice superoperator\n\\begin{equation}\n\\label{choice}\n{\\cal C}=\\hat P_A \\odot\\hat P_A+\\ket{0}\\bra{0}{\\rm Tr}(1-\\hat P_A)\\odot\n(1-\\hat P_A),\n\\end{equation}\n\n\\noindent where state $\\ket0$ is a quantum analogue of the\nclassical single-point set, which is separate from all other states.\nEq.\\ (\\ref{choice}) defines a positive and trace-preserving transformation,\nwhich can appropriately describe coherent information transfer\nbetween subsets of the entire system. The last term in Eq.\\\n(\\ref{choice}) represents the total norm preservation, if all the\nstates outside the output $B$-set are included. In our case, these\nstates are included in the incoherent $\\ket0\\bra0$ form, which in\ncontrast to the classical one-bit analogue of a TLA yields no coherent\ninformation due to the complete destruction of the coherence.\n\nConsidering the coherent information transmitted from part $A$ to\npart $B$ of the system, which evolves in time, we deal with the\nchannel superoperator\n\\begin{equation}\n\\label{ABflow}\n{\\cal S}_{AB}={\\cal C}_B{\\cal S}_0(t){\\cal C}_A,\\quad {\\cal S}_0(t)=\nU(t)\\odot U^{-1}(t)\n\\end{equation}\n\n\\noindent with $U(t)$ being the time evolution unitary operator. Here\nthe input choice superoperator ${\\cal C}_A$ is shown just to define the\ntotal channel superoperator, regardless of the input density matrix.\nOtherwise, ${\\cal C}_A$ is already accounted in the input density matrix\n$\\hat\\rho_{\\rm in}$, defined as the operator in the corresponding subspace\n$H_A$ of the total Hilbert space $H$.\n\nLet us assume that the dynamical evolution of the system is determined and\nthe Bohr frequencies $\\omega_k$ and the corresponding eigenstates\n$\\ket{k}$ are found. Then, representing the projectors in terms of the\ncorresponding input $\\ket{\\psi_l}$ and output $\\ket{\\varphi_m}$ wave\nfunctions, Eq.\\ (\\ref{ABflow}) gives the specified time evolution form\n\\begin{eqnarray}\n\\label{SABt}\n&{\\cal S}_{AB}(t)=\\displaystyle \\sum_{ll'\\in A}\\left[\\hat\ns_{ll'}(t)+\\ket0\\bra0\\sum_{m\\notin B}\\braket{\\varphi_m}{\\psi_l(t)}\n\\braket{\\psi_{l'}(t)}{\\varphi_m} \\right]\\bra{\\psi_l} \\odot\\ket{\\psi_{l'}},&\\nonumber\\\\\n&\\hat s_{ll'}(t)= \\displaystyle\\sum_{mm'\\in B} \\braket{\\varphi_m}{\\psi_l(t)}\n\\braket{\\psi_{l'}(t)}{\\varphi_{m'}}\\,\\ket{\\varphi_m}\\bra{\\varphi_{m'}},&\\\\\n&\\ket{\\psi_l(t)}=\\sum_{k}e^{-i\\omega_kt} \\braket{k}{\\psi_l}\\ket{k}.&\\nonumber\n\\end{eqnarray}\n\n\\noindent Let us consider the case of the orthogonal subsets of\ninput/output wave functions, which is of special interest. Then, if\nthere is only one common state $\\ket{\\phi}$ in the sets $\\ket{\\psi_l}$,\n$\\ket{\\varphi_m}$ and $U(t_0)=1$ holds for some $t_0$, we get\n$$\n{\\cal\nS}_{AB}(t_0)=\\ket{\\phi} \\bra{\\phi}\\odot\\ket{\\phi} \\bra{\\phi}+\\ket{0}\n\\bra{0}\\sum \\limits_{\\varphi_m\\ne\\phi}\\bra{\\varphi_m} \\odot\n\\ket{\\varphi_m},\n$$\n\n\\noindent which means that the quantum system is reduced into a\nclassical bit of the states $\\ket{\\phi}$ and $\\ket0$ and no\ncoherent information is stored in the subsystem $B$. Nevertheless,\nif the eigenstates $\\ket{k}$ of $U(t)$ do not coincide with the\ninput/output states $\\ket{\\psi_l}$, $\\ket{\\varphi_m}$ the coherent\ninformation will increase with the time evolution. Hence, the information\ncapacity of the channel is determined by quantum coupling of the input and\noutput.\n\nTo illustrate the coherent information transfer through the\nquantum channel considered in this section, let us analyze a\ntypical intra-atomic channel between two two-level systems formed\nof two pairs of orthogonal states $A=\\{\\ket{\\psi_0},\n\\ket{\\psi_1}\\}$ and $B= \\{\\ket{\\psi_0}, \\ket{\\psi_2}\\}$ of the\nsame atom. A spinless model of the hydrogen atom could serve as\nsuch a system (Fig.\\ \\ref{fig3}): $\\psi_0$ is the ground $s$-state\nwith $n=1$, $\\psi_{1,2}$ are the $s$-state with $l=0,$ $m=0$ and\n$p$-state with $l=1,$ $m=0$ of the first excited state with $n=2$,\nrespectively.\n\nIn the absence of an external field, this quantum channel transmits no\ncoherent information, as the $l=0,$ $m=0$ and $l=1,$ $m=0$ states\nare uncoupled. In the presence of an external electric field applied\nalong the $Z$-axis, the considered two out of four initially\ndegenerated states with $n=2$ are split, due to the Stark shift\ninto the new eigenstates $\\ket1=(\\ket{\\psi_1}+ \\ket{\\psi_2})/\n\\sqrt2$, $\\ket2=(\\ket{ \\psi_1}- \\ket{\\psi_2})/\\sqrt2$. The\ninput $l=0$ state oscillates with the Stark shift frequency:\n$\\ket{\\psi_1(t)}= \\cos(\\omega_st) \\ket{\\psi_1}+\n\\sin(\\omega_st)\\ket{\\psi_2}$. Therefore, due to the applied\nelectric field, the input state becomes coupled to the output state,\nwhich carries the coherent information.\n\nFor our model, Eq.\\ (\\ref{SABt}) presents the $\\hat s_{kl}$-operators\nin the form of a $3\\times3$-matrix, where the third column and row\nintroduce the phantom ``vacuum'' state $\\ket0$:\n\\begin{eqnarray}\n&\\hat s_{11}=\\left(\\begin{array}{ccc}\n 1 & 0 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{array}\\right),\\quad\n\\hat s_{12}=\\left(\\begin{array}{ccc}\n 0 & \\sin\\omega_s t & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{array}\\right), & \\\\\n&\\hat s_{21}=\\left(\\begin{array}{ccc}\n 0 & 0 & 0 \\\\\n \\sin\\omega_s t & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{array}\\right),\\quad\n\\hat s_{22}=\\left(\\begin{array}{ccc}\n 0 & 0 & 0 \\\\\n 0 &\\sin^2\\omega_s t & 0 \\\\\n 0 & 0 & \\cos^2\\omega_s t\n\\end{array}\\right).&\n\\end{eqnarray}\n\n\\noindent Zero values of $\\hat s_{12}$, $\\hat s_{21}$ correspond to the\nabsence of coherent information at $t=0$ or to the absence of\ncoupling. Choosing the input matrix in the maximum entropy form $\\hat\n\\rho_{\\rm in}=\\hat I/2$, we get the corresponding joint input-output\nmatrix in the form\n$$\n\\hat\\rho_\\alpha=\\left(\\begin{array}{cccccc}\n\\displaystyle\n \\frac{1}{2} & 0 & 0 &\\displaystyle \\frac{x}{2} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\displaystyle \\frac{x}{2} & 0 & 0 & \\displaystyle\\frac{x^2}{2} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 &\\displaystyle \\frac{1-x^2}{2}\n\\end{array}\\right),\n$$\n\n\\noindent where $x=\\sin\\omega_s t$ and the output density matrix\n$\\hat\\rho_{\\rm out}$ is diagonal with the diagonal elements $1/2$,\n$x^2/2$, and $(1-x^2)/2$.\n\nCalculating non-zero eigenvalues $(1\\pm x^2)/2$ of $\\hat\\rho_\\alpha$\nand the entropies $S_{\\rm out}$, $S_\\alpha$, we get the coherent\ninformation\n$$\\\nI_c=[(1 + x^2)\\log_2(1+x^2)-x^2 \\log_2(x^2)] /2.\n$$\n\n\\noindent This function is positive except for $x=0$, where the\ncoherent information is equal to zero, and its maximum is equal to\n1 qubit at $x=\\pm1$, e.g., for the precession angle $\\omega_s\nt=\\pm\\pi/2$. Thus coherent information on the state of the\nforbidden transition is available, in principle, from a dipole\ntransition via Stark coupling. Its time-averaged value is\n$\\avr{I_c}=0.46$ qubit.\n\nThis forbidden transition was discussed in\n\\cite{alekseev,moskalev} as a potential source of information on\nspatial symmetry breaking caused by the weak neutral current\n\\cite{weinberg,salam}. For example, if $I_c= 0$, only the\nincoherent impact of the forbidden transition (by means of the\nground state population $n_0$) remains and provides a\nclassical-type of information on the interactions that cannot be\nobserved directly. In this case, only one\nparameter---population---can be potentially measured, while exact\nknowledge of the phase of the transition demands $I_c=1$.\n\n\n\\section{Coherent information transfer between two quantum systems}\n\\label{sec:twoqs}\n\nIn recent years, a few results have been published related to\ncoherent information transfer in a system of two TLAs, including\ndiscussion of the problem from the entanglement measure viewpoint\n\\cite{hill97} and the ``eavesdropping problem'' \\cite{niu99}. A\nnumber of different experiments have been proposed to study\ncontrolled entanglement between two atoms \\cite{brennen,trieste}.\nFrom the informational point of view, the coherent information\ntransmitted in the system of two TLAs connected by a quantum\nchannel depends both on the specific quantum channel\ntransformation and the initial states of the TLAs. For the latter,\nit seems reasonable to assume that they can be represented by the\nproduct of the independent states of each TLA: $\\hat\\rho_{1+2}\n=\\hat\\rho_{\\rm in} \\otimes\\hat \\rho_{2}$.\n\nIn this section, we present a systematic treatment of the coherent\ninformation transfer between two different quantum systems. The\nanalysis includes coherent information transfer between (i) two unitary\ncoupled TLAs (subsection \\ref{subsec:twoqb}), (ii) two TLAs coupled via\nthe measuring procedure (subsection \\ref{subsec:twoqbm}), (iii) an\narbitrary system and its duplicate (subsection\n\\ref{subsec:duplication}), (iv) a TLA and the free space photon field\n(subsection \\ref{subsec:at-f}), and (v) two TLAs coupled via the free space\nphoton field (subsection \\ref{subsec:at-f-at}).\n\n\n\\subsection{Two unitary coupled TLAs}\n\\label{subsec:twoqb}\n\nLet us first examine a deterministic noiseless quantum channel\nconnecting two TLAs (Fig.\\ \\ref{fig1}b). Such a channel can be\ndescribed by the unitary two-TLA transformation, which is defined\nby the matrix elements $U_{ki,k'i'}$ with $k,i,k',i'=1,2$. Then, the\nchannel transformation superoperator ${\\cal S}$ describing the\ntransformation $\\hat\\rho_{\\rm in}\\to \\hat\\rho_{\\rm\nout}=\\hat\\rho_2'$ can be written in terms of the substitution\nsymbol (see Eq.\\ (\\ref{genS})), with operators $\\hat{s}_{kl}=\n\\sum_{\\mu\\nu}S_{kl,\\mu\\nu} \\ket{\\mu}\\bra{\\nu}$, represented with\nthe matrix elements of ${\\cal S}$ (in accordance with Eqs\n(\\ref{skl}), (\\ref{sklnew})), in the following form:\n\\begin{equation}\n\\label{Sklij}\nS_{kl,\\mu\\nu}=\\sum_{m\\alpha\\beta}\\rho_{2\\alpha\\beta}^{}U_{m\\mu,k\\alpha}^{}\nU_{m\\nu,l\\beta}^.\n\\end{equation}\n\n\\noindent The relation ${\\rm Tr}\\,\\hat{s}_{kl}=\\sum_\\mu\nS_{kl,\\mu\\mu} = \\delta_{kl}$ is valid here and ensures the correct\nnormalization condition, whereas the positivity of the block\nmatrix\n$$ (\\hat{s}_{kl})=\\left(\\begin{array}{cc}\n \\hat{s}_{11} & \\hat{s}_{12}\\\\\n \\hat{s}_{21} & \\hat{s}_{22}\n\\end{array}\\right)\n$$\n\n\\noindent ensures the positivity of ${\\cal S}$.\n\nFor the no-entanglement transformation $U=U_1\\otimes U_2$, Eqs\n(\\ref{genS}), (\\ref{Sklij}) yield ${\\cal S}=\\hat\\rho_{2}'{\\rm\nTr}\\;\\odot$, which means that the initial state $\\hat\\rho_1$ of\nthe first TLA transfers into the final state, which is not\nentangled with the state $\\hat\\rho_2'=U_2 \\hat\\rho_2U_2^+$ of the\nsecond TLA.\n\nWe can simplify Eq.\\ (\\ref{Sklij}) by considering a pure state\n$\\hat\\rho_2$, so that together with an arbitrary choice of\nno-entanglement transformation $U$, it seems reasonable to consider\na special case of the pure state: $\\rho_{2\\alpha\\beta}=\n\\delta_{\\alpha\\beta} \\delta_{\\alpha\\alpha_0}$. Keeping also in\nmind that $S_{kl,\\mu\\nu}$ is linear on the density matrix $\\hat\\rho_2$\nand the coherent information $I_c$ is a convex function of $\\cal\nS$ \\cite{barnum98}, Eq.\\ (\\ref{Sklij}) simplifies to\n\\begin{equation}\n\\label{Sklij1}\nS_{kl,\\mu\\nu}=\\sum_{m}U_{m\\mu,k\\alpha_0}^{} U_{m\\nu,l\\alpha_0}^,\n\\end{equation}\n\n\\noindent which means that the quantum channel is described only by\nthe unitary transformation $U$. Here the summation is taken over only\nthe states $\\ket{m}$ of the first TLA after the coupling\ntransformation.\n\nThe coherent information transmitted in systems of two unitary\ncoupled TLAs with $\\hat\\rho_{\\rm in}=\\hat I/2$ and\n$\\left(\\hat\\rho_2\\right)_{12}=\\sqrt{\\left(\\hat\\rho_2\\right)_{11}\\left[1-\n\\left(\\hat\\rho_2\\right)_{11}\\right]}$ is shown in Fig.\\\n\\ref{fig4}. A convex function of $\\hat\\rho_2$ is shown, which has\nthe maximum on the border,\n$\\rho_{11}=\\left(\\hat\\rho_2\\right)_{11}=0,1$. As in the case of a\nsingle TLA, the behavior of the coherent information preserves the\ntypical threshold-type dependency on the coupling angle, which\ndetermines the degree of the coherent coupling of two TLAs with\nrespect to the independent fluctuations of the second TLA.\n\n\n\\subsection{Two TLAs coupled via the measuring procedure}\n\\label{subsec:twoqbm}\n\nHere we will discuss a specific type of quantum channel\nconnecting two TLAs \\cite{TTLA}, where the\nsuperoperator $\\cal S$ is defined by the {\\em measuring procedure}, which\nimplements a different approach to the quantum information\n\\cite{chen99} called {\\em measured information}.\n\nWe start with a channel formed of two identical two-level\nsystems. In terms of wave function, the corresponding {\\em full\nmeasurement} transformation of the first TLA state is defined as\n\\begin{equation}\n\\label{meas}\n\\psi\\otimes\\varphi\\to\\sum a_i\\ket{\\phi_i}\\ket{\\phi_i},\\quad\na_i=\\braket{\\phi_i}{\\psi}.\n\\end{equation}\n\n\\noindent This transformation provides full entanglement of some basis states\n$\\ket{\\phi_i}$, which do not depend on the initial state $\\varphi$\nof the second TLA. The latter serves as a measuring device, yet\nfully preserves information on the basis states of the first\nsystem state $\\psi=\\sum a_i\\ket{\\phi_i}$. Eq.\\ (\\ref{meas}), being\na deterministic transformation of the wave function, is\nneither linear nor unitary transformation with respect to $\\varphi$ and,\ntherefore, cannot represent a true deterministic transformation.\nThe corresponding representation in terms of the two-TLA density\nmatrices has the form:\n\\begin{equation}\n\\label{rho12}\n\\hat\\rho_{12}\\to\\sum\\limits_i\\sum\\limits_j\\bra{\\phi_i}\\bra{\\phi_j}\\hat\n\\rho_{12}\\ket{\\phi_j}\\ket{\\phi_i}\\, \\ket{\\phi_i}\\ket{\\phi_i}\\bra{\\phi_i}\n\\bra{\\phi_i}.\n\\end{equation}\n\n\\noindent This representation is linear on $\\hat\\rho_{12}$ and satisfies the\nstandard conditions of physical feasibility\n\\cite{barnum98,kraus83}, i.e completely positive and trace\npreserving. This matrix is in the form of $\\sum p_i\\ket{\\phi_i}\n\\ket{\\phi_i}\\bra{\\phi_i}\\bra{\\phi_i}$, so that $S(\\hat\n\\rho_{12})=S(\\hat\\rho_2)$. Due to the classical nature of the\ninformation represented here only with the classical indexes $i$\nand in accordance with the equations of section \\ref{sec:definitions}, the\nsingle-instant coherent information is zero.\n\nIn the case of a two-time channel, the superoperator for the quantum channel\nconnecting two TLAs can be readily derived from Eq.\\ (\\ref{genS})\nwith $\\hat s_{kl}=\\ket{\\phi_k}\\bra{\\phi_k} \\delta_{kl}$,\n$\\bra{k}\\to\\bra{\\phi_k}$, and $\\ket{k}\\to\\ket{\\phi_k}$. After\ncalculating the trace over the first TLA and replacing\n$\\hat\\rho_{12}$ with the substitution symbol $\\odot$, the\nequation takes the form:\n\\begin{equation}\n\\label{dirmeas}\n{\\cal M}=\\sum\\limits_k\\hat P_k {\\rm Tr}_1^{}\\hat E_k\\,\\odot.\n\\end{equation}\n\n\\noindent Here $\\hat P_k=\\ket{\\phi_k}\\bra{\\phi_k}$ are the orthogonal\nprojectors representing the eigenstates of the ``pointer'' variable of\nthe second TLA and $\\hat E_k=\\ket{\\phi_k}\\bra{\\phi_k}$ is the\northogonal expansion of the unit (orthogonal map) formed of the same projectors.\nThis orthogonal map determines here the quantum-to-classical reduction\ntransformation ${\\rm Tr}_1\\hat E_k \\, \\odot= \\bra{\\phi_k}\n\\odot \\ket{\\phi_k}$, which represents the procedure of getting classical\ninformation $k$ from the first system.\nApplying the transformation (\\ref{dirmeas}) to $\\hat\\rho_{\\rm\nin}$ and using Eq.\\ (\\ref{inout}) for the respective output and\ninput-output density matrices, we get\n\\begin{equation}\n\\label{rhodirmeas}\n\\hat\\rho_{\\rm out}= \\sum\\limits_k \\tilde p_k \\ket{\\phi_k}\n\\bra{\\phi_k},\\quad \\hat\\rho_\\alpha= \\sum\\limits_k \\tilde p_k\n\\ket{\\phi_k} \\ket{\\pi_k} \\bra{\\pi_k}\\bra{\\phi_k},\n\\end{equation}\n\n\\noindent where $\\tilde p_k= \\bra{\\phi_k} \\hat\\rho_{\\rm in}\\ket{\\phi_k}\n= \\sum_i p_i \|\\braket{\\phi_k} {i}\|^2$ are the eigenvalues\nof the reduced density matrix and $\\ket{\\pi_k}= \\sum_i\\sqrt{p_i/\\tilde\np_k}\\braket{\\phi_k} {i}\\ket{\\bar{i}}$ are the normalized modified input\nstates coupled with the output states $\\ket{\\phi_k}$ after the\nmeasurement procedure. It is important to note (as it follows from\nEq.\\ (\\ref{rhodirmeas})) that there is no coherent information in the system\nbecause vectors $\\ket{\\phi_k}$ are orthogonal and therefore the\nentropies of the density matrices (\\ref{rhodirmeas}) are obviously the\nsame. Conversely, the measured information introduced in \\cite{chen99}\nis not equal to zero in this case.\n\nWe can easily generalize our result for a more general case of the\nquantum channel, when the second system has a different structure\nfrom the first and, therefore, they occupy different Hilbert\nspaces. This difference leads to the replacement of the basis states\n$\\ket{\\phi_i}$ of the second system in our previous results with\nanother orthogonal set $\\ket{\\varphi_i}= V\\ket{\\phi_i}$, where $V$\nis an isometric transformation from the Hilbert state $H_1$ of the\nfirst system to the different Hilbert space $H_2$ of the second\nsystem. After simple straightforward calculations, the final\nresult is the same---there is no coherent information\ntransmitted through the quantum channel. This result is a natural\nfeature of coherent information, in contrast to other information\napproaches (see, for example Ref. \\cite{chen99}).\n\nIt is interesting to discuss more general measuring-type\ntransformations, for instance, the indirect (generalized)\nmeasurement procedure. This procedure was first applied to the\nproblems of optimal quantum detection and measurement in\n\\cite{helstr70} and then, in a form of non-orthogonal\nexpansion of unit $\\hat{\\cal E} (d\\lambda)$, in \\cite{gTK73}\n($\\hat{\\cal E} (d\\lambda)$ is equivalent to the positive\noperator-valued measure, POVM, used in the semiclassical version of\nquantum information and measurement theory\n\\cite{preskill,helstrom,peres93}). This indirect measuring\ntransformation results from averaging a direct measuring\ntransformation applied, not to the system of interest, but to its\ncombination with an auxiliary independent system. The\nindirect-measurement superoperator in the general form can be written\nas\n\\begin{equation}\n\\label{indmf}\n{\\cal M}=\\sum\\limits_q\\hat P_q{\\rm Tr}\\,\\hat{\\cal E}_q\\odot,\n\\end{equation}\n\n\\noindent where $\\hat P_q$ are the arbitrary orthogonal projectors and\n$\\hat{\\cal E}_q$ is the general-type non-orthogonal expansion of the\nunit in $H$ space (POVM). Note that $\\hat{\\cal E}_q=\\ket{\\varphi_q}\n\\bra{\\varphi_q}$ is a specific ``pure'' type of POVM, first used in\nquantum detection and estimation theory \\cite{helstr70}. The latter\ndescribes the full measurement in $H\\otimes H_a$ for the\nsingular choice of the initial auxiliary system density matrix\n$\\rho^a_{bc}= \\delta_{b0} \\delta_{bc}$.\n\nThe information transfer from the initial density matrix to the final\noutput state is represented in Eq.\\ (\\ref{indmf}) via the coupling\nprovided by indexes $q$. Because the number $N_q$ of $q$ values can be\ngreater than ${\\rm Dim}\\,H$, it seems reasonable to suggest that some\noutput coherent information is left about the input state. The\ncorresponding output and input-output density matrices are given by\n\\begin{equation}\n\\label{rhoalpha}\n\\hat\\rho_{\\rm out}=\\sum\\limits_q\n\\tilde p_q \\hat P_q,\\quad \\hat\\rho_\\alpha=\\sum\\limits_{qij} \\sqrt{p_i\np_j}\\bra{j}\\hat{\\cal E}_q\\ket{i}\\hat P_q\\otimes\\ket{\\bar{i}}\\bra{\\bar{j}},\n\\end{equation}\n\n\\noindent where $\\tilde p_q={\\rm Tr}\\,\\hat{\\cal E}_q\\hat\\rho_{\\rm in}$\nare the state probabilities given by the indirect measurement.\n\nIn the case of full indirect measurement, it can be easily\ninferred theoretically or confirmed by numerical calculations for\nparticular examples that no coherent information is available. The\nproof is based on the quantum analogue \\cite{schum96} of the\nclassical data processing theorem and the above discussed result\non a full direct measurement. Therefore, in order to get non-zero\ncoherent information, a class of incomplete (soft) measurements\nmust be implemented, which are subject to more detailed quantum\ninformation analyzis.\n\n\n\\subsection{Quantum duplication procedure}\n\\label{subsec:duplication}\n\nIn the previous subsection, we demonstrated that the classical-type\nmeasuring procedure defined by the transformation (\\ref{rho12})\ncompletely destroys the coherent information transmitted through the\nquantum channel. Here we will consider a modified transformation for\nthe quantum channel shown in Fig.\\ \\ref{fig1}c, which preserves the coherent\ninformation:\n$$\n\\hat\\rho_{12} \\to\\hat\\rho_{12}'= \\sum \\limits_{ij} \\bra{\\phi_i} {\\rm\nTr}_2\\hat\\rho_{12}\\ket{\\phi_j} \\ket{\\phi_i}\\ket{\\phi_i}\n\\bra{\\phi_j}\\bra{\\phi_j}.\n$$\n\n\\noindent In this equation off-diagonal matrix elements of the input density\nmatrix $\\hat\\rho_1=\\hat\\rho_{\\rm in}$ are taken into account, which\npreserves the phase connections between different $\\phi_i$.\n\nFor the initial density matrix of a product type $\\hat\\rho_{\\rm\nin}\\otimes \\hat\\rho_2$, in terms of $\\hat \\rho_{\\rm in} \\to\\hat\n\\rho_{12}'$ transformation from $H$ to $H\\otimes H$, the corresponding\nsuperoperator has the form:\n\\begin{equation}\n\\label{rho12coh}\n{\\cal Q}=\\sum\\limits_{ij} \\ket{\\phi_i}\\ket{\\phi_i}\\bra{\\phi_j} \\bra{\\phi_j}\n\\bra{\\phi_i}\\odot\\ket{\\phi_j}.\n\\end{equation}\n\n\\noindent This superoperator defines the coherent measuring\ntransformation, in contrast to the incoherent transformation discussed in\n\\cite{chen99}. The coherent measuring transformation converts\n$\\hat\\rho_{\\rm in}$ into $\\hat\\rho_2$-independent state\n\\begin{equation}\\label{rhooutQ}\n\\hat\\rho_{\\rm out}=\\rho_{12}'=\\sum_{ij} \\bra{\\phi_i}\\hat\\rho_{\\rm in}\n\\ket{\\phi_j} \\ket{\\phi_i}\\ket{\\phi_i} \\bra{\\phi_j} \\bra{\\phi_j},\n\\end{equation}\n\n\\noindent which results in duplication of the input eigenstates\n$\\phi_i$ into the same states of the pointer variable $\\hat k=\\sum_k k\n\\ket{\\phi_k}\\bra{\\phi_k}$. Pure states of the input are transformed\ninto the pure states of the joint (1+2)-system by doubling the pointer\nstates:\n$$\n\\psi\\to\\sum\\limits_i \\braket{\\phi_i}{\\psi}\\ket{\\phi_i}\\ket{\\phi_i}.\n$$\n\n\\noindent This mapping is similar to the mapping given by Eq.\\\n(\\ref{meas}). Of course, only the input states $\\psi$ equal to the\nchosen pointer basis states $\\phi_k$ are duplicated without\ndistortion because it is impossible to transmit non-orthogonal\nstates using only orthogonal ones. The entropy of the output state\nwith a density matrix (\\ref{rhooutQ}) having the same matrix\nelements as $\\hat\\rho_{\\rm in}$, is evidently the same as the\ninput state, $S_{\\rm out}=S_{\\rm in}=S[\\hat\\rho_{\\rm in}]$, due to\nthe preservation of the coherence of all pure input states.\n\nFor the joint input-output states, the transformation (\\ref{rho12coh})\nyields the corresponding density matrix (\\ref{inout}) in $H\\otimes\nH\\otimes H$ space:\n\\begin{equation}\n\\label{rhoalphaQ}\n\\hat\\rho_\\alpha=\\sum\\limits_{kl}\\ket{\\phi_k}\\ket{\\phi_k}\\bra{\\phi_l}\n\\bra{\\phi_l} \\otimes\\sqrt{\\tilde p_k\\tilde p_l}\\ket{\\chi_k}\\bra{\\chi_l},\n\\end{equation}\n\n\\noindent where $\\tilde p_k$, $\\ket{\\chi_k}$ are the same as\nabove, providing an expansion of the input density matrix in the\nform $\\hat\\rho_{\\rm in}=\\sum_k \\tilde p_k \\ket{\\chi_k}\n\\bra{\\chi_k}$. Taking into account that the first tensor product\nterm in Eq.\\ (\\ref{rhoalphaQ}) is a set of transition projectors\n$\\hat P_{kl},\\, \\hat P_{kl}\\hat P_{mn}=\\delta_{lm}\\hat P_{kn}$, we\ncan apply easily proven algebraic rules valid for a scalar\nfunction $f$: $$ f(\\sum_{kl}\\hat P_{kl} \\otimes \\hat\nR_{kl})=\\sum_{kl}\\hat P_{kl}\\otimes f(\\hat R)_{kl}, $$\n\n\\noindent where $\\hat R=(\\hat R_{kl})$ is the block matrix and\n${\\rm Tr}\\, f(\\sum_{kl}\\hat P_{kl} \\otimes \\hat R_{kl})={\\rm\nTr}\\,f(\\hat R)$. Here $\\hat R= \\left(\\sqrt{\\tilde p_k\\tilde\np_l}\\ket{\\chi_k} \\bra{\\chi_l}\\right)$, and it is simply\n$\\ket{\\ket{\\chi}}\\bra{\\bra{\\chi}}^+$ with $\\ket{\\ket{\\chi}}_{ki} =\n\\sqrt{\\tilde p_k}\\chi_{ki}$, a vector in the $H\\otimes H$ space.\nAll eigenvalues $\\lambda_k$ of this matrix are equal to zero,\nexcept one value corresponding to the eigenvector\n$\\ket{\\ket{\\chi}}$.\n\nCalculation of the exchange entropy gives $S_e=0$, and, therefore,\n$I_c=S_{\\rm in}$. Consequently, the coherent duplication does not\nreduce the input information transmitted through the 1$\\to$(1+2)\nchannel, nor does it matter whether the register $\\hat k$ is\ncompatible with the input density matrix, $[\\hat k,\\hat\\rho_{\\rm\nin}]=0$, or not.\n\nIf the channel is reduced to the one shown in Fig.\\ \\ref{fig1}b\nand discussed in the previous subsection, by taking in Eq.\\\n(\\ref{rhooutQ}) trace either over the first or the second system,\nwe evidently come to the measurement procedure discussed in\nsubsection \\ref{subsec:twoqbm}. As a result, we can conclude that\nthe coherent information is strictly associated with the joint\nsystem but not with its subsystems. This natural property could\nbe used in quantum error correction algorithms\n\\cite{errorcorr} or for producing stable entangled states\n\\cite{lasphys}.\n\n\n\\subsection{TLA-to-vacuum field channel}\n\\label{subsec:at-f}\n\nIn this subsection, we analyze the quantum channel between a TLA\nand a vacuum electromagnetic field (Fig.\\ \\ref{fig1}b), which is an\nextension of the TLA in an external laser field, as considered in\nsection \\ref{sec:onequbit}.\n\nFor this analysis, we will use a reduced model of the field, which\nis based on the reduction of the Hilbert space of the field in the\nFock representation (Fig.\\ \\ref{fig5}). The problem, therefore, is\nreduced to that of the interaction of a two-level system with\ncontinuous multi-mode oscillator systems \\cite{Cohen92}, a\nspecific case of which is the interaction of an atom with the free\nphoton field. However, to analyze the information in the system\n(atom+field), we do not need to consider the specific dependence\nof the wave function $\\psi_0({\\bf k},\\lambda)$ of the field photon\non the wave vector (including polarization), because only its\ntotal probability and phase are significant.\n\nIn the basis of the free atomic and field states for the vacuum's initial\nstate $\\alpha_0=0$, we get from Eq.\\ (\\ref{Sklij1})\n$$\nS_{kl,\\mu\\nu}=\\sum_m U_{m\\mu,k0}^{} U_{m\\nu,l0}^.\n$$\n\n\\noindent Greek letters are used to distinguish the photon field\nindexes, which in the general case include both the number of\nphotons and their space or momentum coordinates. Matrix elements\nof this superoperator calculated via the atom-to-field unitary\nevolution matrix $U_{m\\mu,k0}$ coefficients (Table \\ref{table1})\nare shown in Table \\ref{table2}.\n\nThe choice of $\\psi_0({\\bf k},\\lambda)$ as a basis for the photon\nfield \\cite{addnote} reduces the matrix of operator $S_{kl,\\mu\\nu}$ to the\nnon-operator matrix transformation, which in terms of $\\hat\ns_{kl}$ matrices has the form:\n\\begin{equation}\n\\label{Sklmnr}\n\\begin{array}{cc}\n\\hat s_{11}=\n\\left(\\begin{array}{cc}1&0\\\\0&0\\end{array}\\right),&\n\\hat s_{12}=\n\\left(\\begin{array}{cc}0&\\displaystyle\\left(1-e^{-\\gamma t}\\right)^{1/2}\\\\\n0&0\\end{array}\\right),\\\\\n\\hat s_{21}=\n\\left(\\begin{array}{cc}0&0\\\\\\displaystyle\\left(1-e^{-\\gamma t}\n\\right)^{1/2}&0\\end{array}\\right),&\n\\hat s_{22}=\n\\left(\\begin{array}{cc}e^{-\\gamma t}&0\\\\0&1-e^{-\\gamma t}\\end{array}\\right),\n\\end{array}\n\\end{equation}\n\n\\noindent where $\|c_1\|^2=\\exp(-\\gamma t)$ describes the population\ndecay of the totally populated initial excited state of the atom and\n$\\int\\sum \|\\psi_0({\\bf k},\\lambda)\|^2 {\\rm d}{\\bf k}=1-\\exp(-\\gamma t)$ is the\nprobability a photon will be detected. From Eq.\\ (\\ref{Sklmnr}), it follows\nthat the structure of the photon field plays no role, and the transmitted\ninformation defined by the input-output density matrix depends only on\nthe photon emission probability by time $t$. The reduction of the\nphoton field (only the photon numbers $\\mu,\\nu=0,1$ were taken into\naccount) leads to the conclusion that the photon states also are equivalent to\nthose of a two-level system.\n\nApplying the transformation (\\ref{Sklmnr}) to the input atom density matrix\n$$\n\\hat\\rho_{\\rm in}=\n\\left(\\begin{array}{cc}\n \\rho_{11} & \\rho_{12} \\\\\n \\rho_{12} & 1-\\rho_{11}\n\\end{array}\\right),\n$$\n\n\\noindent restricted to the real off-diagonal matrix elements, we get\nthe output density matrix\n$$ \\hat\\rho_{\\rm out}=\\left(\\begin{array}{cc}\n \\rho_{11}+\\rho_{22}e^{-\\gamma t} & \\rho_{12}\n \\displaystyle\\left(1-e^{-\\gamma t}\\right)^{1/2} \\\\\n \\rho_{12}\\displaystyle\\left(1-e^{-\\gamma t}\\right)^{1/2} &\n \\rho_{22}(1-e^{-\\gamma t})\n\\end{array}\\right)\n$$\n\n\\noindent and for $\\rho_{12}=0$ the respective input-output density matrix\n$$\n\\hat\\rho_\\alpha= \\left(\n\\begin{array}{cc\|cc}\n\\rho_{11} & 0 & 0 & \\displaystyle\\left[\\rho_{11}\\rho_{22}\n(1-e^{-\\gamma t})\\right]^{1/2}\\\\\n0 & \\rho_{22}e^{-\\gamma t} & 0 & 0 \\\\ \\hline\n0 & 0 & 0 & 0 \\\\\n\\displaystyle\\left[\\rho_{11}\\rho_{22}(1-e^{-\\gamma t})\n\\right]^{1/2} & 0 & 0 & \\rho_{22}(1-e^{-\\gamma t})\n\\end{array}\\right).\n$$\n\nFor $t\\to\\infty$ this expression yields a pure atom-photon state,\nwhich converts incoherent fluctuations of the atomic states,\nforming the incoherent ensemble, to equivalent coherent\nfluctuations of the photon states. The corresponding eigenvalues\nare $\\lambda_\\alpha= \\{0,0,1-\\rho_{22}\\exp(-\\gamma t),\\rho_{22}\n\\exp(-\\gamma t)\\}$. Non-zero values are equal to the probabilities\nof the atomic states at time $t$. For the output (photon) density\nmatrix $\\hat\\rho_{\\rm out}$ the eigenvalues are $\\lambda_{\\rm\nout}=\\{\\rho_{22}[1-\\exp(-\\gamma t)],1-\\rho_{22}[1- \\exp(-\\gamma\nt)]\\}$, which are the probability that a photon will be emitted or\nnot. These sets of eigenvalues determine the eigen probabilities\nof the joint input-output and marginal output matrices. The\ncoherent information, defined by the difference of the\ncorresponding entropies, then takes the form:\n\\begin{equation}\n\\label{Ican}\n\\begin{array}{ll}\nI_c = & x\\rho_{\\rm _{22}} \\log_2(x \\rho_{22})-(1-\\rho_{22}+x \\rho_{22})\n\\log_2[1-(1-x) \\rho_{22}]+\\\\ &(1-x \\rho_{22}) \\log_2(1-x \\rho_{22})-\n(1-x)\\rho_{22} \\log_2(\\rho_{22}-x \\rho_{22}),\n\\end{array}\n\\end{equation}\n\n\\noindent where $x=\\exp(-\\gamma t)$. This formula is valid for $I_c>0$,\notherwise, $I_c=0$. The corresponding critical point is $\\exp(-\\gamma\nt)= 1/2$, the time when the probability $1-\\rho_{22} [1- \\exp(-\\gamma\nt)]$ of finding no photon is equal to the population of the lower\natomic state $1-\\rho_{22}\\exp(-\\gamma t)$.\n\nThe results for calculating the coherent information are shown in\nFig.\\ \\ref{fig6} for two specific cases: $\\rho_{12}=0$ (Fig.\\\n\\ref{fig6}a) and $\\rho_{11}=1/2$, $0\\le\\rho_{12}\\le1/2$ (Fig.\\\n\\ref{fig6}b). One can see from Fig.\\ \\ref{fig6}a that the coherent\ninformation is symmetrical with respect to the population\n$\\rho_{11}$ around the symmetry point $\\rho_{11}=1/2$. Increasing\nthe excited state population $\\rho_{22}=1-\\rho_{11}$ and the\ncorresponding photon emission yield does not increase the coherent\ninformation, because of the reduction of the source\nentropy, which determines the potential maximum value of the\ncoherent information. For the same reason, the coherent\ninformation decreases when there is a non-zero coherent\ncontribution to the initial maximum entropy atom state and\ncompletely vanishes for the pure coherent initial state (Fig.\\\n\\ref{fig6}b).\n\nIn accordance with section \\ref{sec:definitions} and because of\nthe purity of the initial field state, one-time information\nis equal to the difference of the entropies of the photon field\nonly, represented by $\\hat\\rho_{\\rm out}$, and the initial atomic\nstate, represented by $\\hat\\rho_{\\rm in}$. For a pure initial\nstate, expressed in the form of the excited atom state $\\ket2$,\nand for $0<t<\\infty$, we always get non-zero information\n$I_c=-x\\log_2 x-(1-x)\\log_2 (1-x)$ that yields 1 qubit for\n$x=1/2$, when the excited state population is equal to the\nprobability a photon will be emitted.\n\n\n\\subsection{The transmission of coherent information between two atoms\nvia a free space field}\n\\label{subsec:at-f-at}\n\nIn this subsection, we will consider the quantum channel when\ninformation is transmitted from one atom to another via the free\nspace field (Fig.\\ \\ref{fig1}b). Suppose that the second atom is\ninitially in the ground state. In addition, we will restrict\nourselves here to the long time scale approximation, in which the\neffects of the discrete nature of the retarding electromagnetic\ninteraction are neglected\n\\cite{fermi32,hamilton49,heitler49,milonni74}. Under such\nrestrictions and approximations we have the Dicke problem\n\\cite{dicke54}, for which the well-known solution for the atomic\nstate in the form of two decaying symmetric and antisymmetric\nDicke states $\\ket{s}=(\\ket1\\ket2+\\ket2\\ket1)/\\sqrt2,$\n$\\ket{a}=(\\ket1\\ket2- \\ket2\\ket1)/\\sqrt2$ and the stable vacuum\nstate $\\ket0=\\ket1\\ket1$ can be written as:\n\\begin{eqnarray}\n\\label{dickedyn} &c_s(t)=c_s(0)\\exp[-(\\gamma_s/2+i\\Lambda)t],&\\nonumber\\\\\n&c_a(t)=c_a(0)\\exp[-(\\gamma_a/2-i\\Lambda)t],&\\\\\n&c_0(t)=c_0(0)+\\left[c_s(0)^2+c_a(0)^2-c_s(t)^2-c_a(t)^2\\right]^{1/2}\ne^{i\\xi(t)}.&\\nonumber\n\\end{eqnarray}\n\n\\noindent Here $c_0(t)$ is the amplitude of the stable vacuum\ncomponent $\\ket1\\ket1$, which has an incoherent contribution due to the\nspontaneous radiation transitions from the excited two-atomic\nstates, $\\xi(t)$ is the homogeneously distributed random phase,\n$\\gamma_{s,a}$ and $\\Lambda$ are their decay rate and coupling\nshift, respectively, and $c_{s,a}$ are the amplitudes of the Dicke\nstates.\n\nIn terms of the products of the individual atomic states\n$\\ket{i}\\ket{j}$ for the corresponding initial amplitudes\n$c_{12}(0)=0,$ $c_{22}(0)=0$ the system's dynamics is described,\naccording to the Dicke dynamics (\\ref{dickedyn}), by the following\nequations:\n\\begin{eqnarray}\n&c_{11}(t)=c_{11}(0)+f(t)e^{i\\xi(t)}c_{21}(0), \\quad c_{21}(t)=\nf_s(t)c_{21}(0),&\\\\ & c_{12}(t)=f_a(t)c_{12}(0),\\quad c_{22}(t)=0,&\\\\\n&f(t)=\\left\\{1-[\\exp(- \\gamma_st)+ \\exp(-\\gamma_at)]/2\\right\\}^{1/2},&\\\\\n&f_s(t)= \\{\\exp[-(\\gamma_s/2+ i\\Lambda)t]+\\exp[-(\\gamma_a/2-i\\Lambda)t]\\}/2,&\\\\\n&f_a(t)=\\{\\exp[-(\\gamma_s/2 +i\\Lambda)t]-\\exp[-(\\gamma_a/2-i\\Lambda)t]\\}/2.&\n\\end{eqnarray}\n\n\\noindent Applying these formulas to the input operators\n$c_{k1}(0)c_{l1}^(0) \\ket{k}\\bra{l}$ of the first atom and then\naveraging the output over the final states of the first atom and the\nfield fluctuations (the latter is represented here only with $\\xi(t)$),\nwe get the symbolic channel superoperator transformation $\\hat\n\\rho^{(1)} (0)\\to\\hat\\rho^{(2)}(t)={\\cal S}(t)\\hat\\rho^{(1)}(0)$ and\ncorresponding $\\hat s_{kl}$ operators in the form:\n\\begin{eqnarray}\n\\label{at-f-at-dyn}\n&\\begin{array}{ccl}\n{\\cal S}(t)&=&\\ket1\\bra1\\odot\\ket1\\bra1+\\left[f(t)^2+\|f_s(t)\|^2\\right]\\ket1\n\\bra2\\odot\\ket2\\bra1+\|f_a(t)\|^2\\ket2\\bra2\\odot\\ket2\\bra2\\\\\n & &+f_a(t)\\ket2\\bra2\\odot\\ket1\\bra1+f_a^(t)\\ket1\\bra1\\odot\\ket2\\bra2,\n\\end{array}& \\nonumber \\\\\n&\\hat s_{11}=\\left(\\begin{array}{cc}1&0\\\\0&0\\end{array}\\right),\\quad\n\\hat s_{12}=\\left(\\begin{array}{cc}0&f_a^(t)\\\\0&0\\end{array}\\right), &\\\\\n&\\hat s_{21}=\\left(\\begin{array}{cc}0&0\\\\f_a(t)&0\\end{array}\\right),\\quad\n\\hat s_{22}=\\left(\\begin{array}{cc}f(t)^2+\|f_s(t)\|^2&0\\\\0&\|f_a(t)\|^2 \\nonumber\n\\end{array}\\right).\n\\end{eqnarray}\n\nTo further elucidate this problem, let us now discuss the case of two\nidentical atoms having parallel dipole moments aligned perpendicular\nto the vector connecting the atoms. Here only two dimensionless\nparameters are essential: dimensionless time, $\\gamma t$, where\n$\\gamma$ is the free atom's decay rate, and dimensionless distance,\n$\\varphi=k_0R$, where $R$ is the interatomic distance and $k_0$ is the\nwave vector at the atomic frequency. Then, the dimensionless two-atomic\ndecay rates and the short distance dipole-dipole shift are given by\n\\cite{trieste,lasphys,milonni74}:\n$$\n\\gamma_{s,a}/\\gamma=1\\pm g\\quad \\text{and}\\quad\n\\Lambda/\\gamma= (3/4)/\\varphi^3,\n$$\n\n\\noindent respectively, with $g=(3/2)(\\varphi^{-1} \\sin\\varphi +\n\\varphi^{-2} \\cos\\varphi- \\varphi^{-3}\\sin\\varphi)$.\n\nThe coherent information may be calculated as previously described in subsection\n\\ref{subsec:at-f} by replacing $\\exp(-\\gamma t)$ with\n$f(t)^2+\|f_s(t)\|^2$ in Eq.\\ (\\ref{Sklmnr}). Then, the operators $\\hat\ns_{kl}$ in Eq.\\ (\\ref{Sklmnr}) become similar to the corresponding\noperators in Eq.\\ (\\ref{at-f-at-dyn}). The coherent information is\ngiven by the same Eq.\\ (\\ref{Ican}) with $x=f(t)^2+ \|f_s(t)\|^2$, which,\nhowever, now has (in contrast with a single-atom case considered in\n\\ref{subsec:at-f}) new qualitative features arising from the specific\noscillatory dependence of $\|f_{s,a}(t)\|^2$ on the interatomic distance\n$\\varphi$.\n\nIf there were no oscillations from the quasi-electrostatic\ndipole-dipole coupling, i.e. as in the case of $\\Lambda=0$, the\ncoherent information would always be equal to zero, because the\nthreshold $x<0.5$ would not be achieved. Parameter $(1-x)$\ncorresponds to the population of the excited state of the second\natom for the initial state $\\ket2$ of the first atom, and for the\noptimal value $\\rho_{22}=1/2$ of its initial population (from the\ninformation point of view), we have $1-x\\le1/4$ and $x\\ge3/4$.\nOscillations in $\|f_a(t)\|^2$ lead to the interference between the\ntwo decaying Dicke components, so that the maximum of the\npopulation $n_2=1-x$ goes to the larger values, maximally up to\n$n_2=1$, and the coherent information becomes a non-zero value.\n\nFunctions $n_2(\\varphi, \\gamma t)$ and $I_c(\\varphi,\\gamma t)$,\ncalculated with Eq.\\ (\\ref{Ican}) are shown in Fig.\\ \\ref{fig7}.\nFor the considered geometry, they serve as the universal measures\nfor a system of two atoms independent of their frequency or dipole\nmoments.\n\nAs can be seen from Fig.\\ \\ref{fig7}a, the population decreases\nrapidly versus time because of the decay of the short-lived Dicke\ncomponent. Both the population and the coherent information (Fig.\\\n\\ref{fig7}b) show strong oscillations at smaller interatomic\ndistances $\\varphi$. At $\\varphi\\to0$ the long-lived Dicke state\nyields an essential population even at infinitely long times, but\nit does not yield any coherent information after the total decay\nof the other short-lived Dicke state.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper, we have shown that the coherent information concept\ncan be used effectively to quantify the interaction between two\nreal quantum systems, which in general case may have essentially\ndifferent Hilbert spaces, and to elucidate the role of quantum\ncoherence specific for the joint system.\n\nFor a TLA in a resonant laser field, coherent information in the\nsystem does not increase as the intensity of the external field\nincreases, unless the external field modifies the relaxation\nparameters.\n\nAs an example of information transmission between the subsystems\nof a whole system, the hydrogen atom was considered. The coherent\ninformation in the atom was shown to transfer from the forbidden\natomic transition to the dipole active transition in an external\nelectric field, due to coupling through Stark splitting.\n\nFor two unitary coupled TLAs, the maximum value $I_c=1$ qubit of\nthe coherent information was shown to be achieved for a complete\nunitary entanglement of two TLAs and $I_c=0$, for any kind of\nmeasuring procedure discussed in subsection \\ref{subsec:twoqbm}.\n\nFor the information exchange between a TLA and a free-space vacuum\nphoton field via spontaneous emission, the coherent information\nwas shown to reach a non-zero value at the threshold point of the\ndecay exponent $\\exp(-\\gamma t)$ equal to 1/2, when the\nprobability of finding no photon is equal to the population of the\nlower atomic state. At its maximum, the coherent information can\nreach the value of $I_c=1$ qubit.\n\nFor the information transfer between two atoms via vacuum field,\nwhen the atoms are located at a distance of the order of their\ntransition wavelength, the coherent information was shown to be a\nnon-zero value, only because of the coherent oscillations of the\nDicke states, which originate from the dipole-to-dipole short\ndistance electrostatic-like $\\sim1/R^3$ interaction. In contrast,\nthe semiclassical information received from the quantum detection\nprocedure results from the population correlations \\cite{lasphys}.\n\n\n\\acknowledgements\n\nThis work was partially supported by the programs ``Fundamental\nMetrology\", ``Physics of Quantum and Wave Phenomena\", and\n``Nanotechnology\" of the Russian Ministry of Science and\nTechnology. The help of C.~M.~Elliott in preparing the manuscript\nis much appreciated.\n\n\n%%\n%% List of References\n%%\n\\begin{references}\n\n\\bibitem{qc-book}\nC. P. Williams and S. H. Clearwater, {\\em Explorations in Quantum\nComputing} (Telos/Springer-Verlag, New York, 1998).\n\n\\bibitem{shannon48}\nC. E. Shannon, Bell System Tech. J. {\\bf 27}, 379 (1948).\n\n\\bibitem{shannon49}\nC. E. Shannon and W. Weaver, {\\em The Mathematical Theory of\nCommunication} (University of Illinois Press, Urbana, 1949).\n\n\\bibitem{gallagher68}\nR. G. Gallagher, {\\em Information Theory and Reliable Communication}\n(John Wiley and Sons, New York, 1968).\n\n\\bibitem{grover91}\nT. M. Grover and J. A. Thomas, {\\em Elements of Information Theory}\n(John Wiley and Sons, New York, 1991).\n\n\\bibitem{schumacher96}\nB. W. Schumacher, Phys. Rev. A {\\bf 54}, 2614 (1996).\n\n\\bibitem{schum96}\nB. W. Schumacher and M. A. Nielsen, Phys. Rev. A {\\bf 54}, 2629 (1996).\n\n\\bibitem{bennett96}\nC. H. Bennett, D. DiVincenzo, J. Smolin, and W. K. Wootters, Phys. Rev.\nA {\\bf 54}, 3824 (1996).\n\n\\bibitem{lloyd96}\nS. Lloyd, Phys. Rev. A {\\bf 55}, 1613 (1997).\n\n\\bibitem{barnum98}\nH. Barnum, M. A. Nielsen, and B. Schumacher, Phys. Rev. A {\\bf 57},\n4153 (1998).\n\n\\bibitem{preskill}\nJ. Preskill, {\\em Lecture notes on Physics 229: Quantum information and\ncomputation}, located at\nhttp://www.theory.caltech.edu/people/preskill/ph229/.\n\n\\bibitem{note}\nIt is worthwhile to note a qualitative difference between these channels in\nconnection with the physical casuality principle \\cite{einstein}:\nspecific restrictions due to the casuality principle are essential only\nfor the channels $1\\to2$ and $1\\to(1+2)$ because of the spatial\nlocality of the subsystems 1, 2. With this, analysis of a system of two\natoms interacting with the vacuum photon field, given below, could be\ncomplemented by the relativistic retardation.\n\n\\bibitem{gKE79}\nB. A. Grishanin, Sov. J. Quantum Electron. {\\bf 9}, 827 (1979).\n\n\\bibitem{positivity}\nTo check the complete positivity \\cite{kraus83}, one has to\nintroduce the operators $\\hat s_{kl}\\otimes\\hat1$.\n\n\\bibitem{conjugation}\nComplex conjugation is added here to make the expression invariant\nin respect to the rotations in the eigen subspaces corresponding\nto the degenerate eigenvalues $p_i$. This correction has no effect\nfor the matrices $\\hat\\rho_{\\rm in}$ with real matrix elements and\nnondegenerate spectrum.\n\n\\bibitem{mollow}\nB. R. Mollow, Phys. Rev. {\\bf 188}, 1969 (1969).\n\n\\bibitem{ezekil}\nF. Wu, R. Grove, and S. Ezekiel, Phys. Rev. Lett. {\\bf 35}, 1426\n(1975).\n\n\\bibitem{pestov73}\nE. G. Pestov and S. G. Rautian, Sov. Phys. JETP {\\bf 37}, 1025 (1973).\n\n\\bibitem{lisitsa75}\nV. S. Lisitsa and S. I. Yakovlenko, Sov. Phys. JETP {\\bf 41}, 233\n(1975).\n\n\\bibitem{burnett82}\nK. Burnett, J. Cooper, P. D. Kleiber, and A. Ben-Reuven, Phys. Rev. A\n{\\bf 25}, 1345 (1982).\n\n\\bibitem{gJETP83}\nB. A. Grishanin, Sov. Phys. JETP {\\bf 58}, 262 (1983).\n\n\\bibitem{alekseev}\nV. A. Alekseev, B. Ya. Zel'dovich, I. I. Sobelman, Sov. Phys.---Usp. {\\bf\n19}, 207 (1976).\n\n\\bibitem{moskalev}\nA. N. Moskalev, R. M. Ryndin, I. B. Khriplovich, Sov. Phys.---Usp. {\\bf\n19}, 220 (1976).\n\n\\bibitem{weinberg}\nS. Weinberg, Phys. Rev. Lett. {\\bf 19}, 1264 (1967).\n\n\\bibitem{salam}\nA. Salam, in {\\em Proc. of the 8th Nobel Symp.} (Stockholm, 1968), p.\n367.\n\n\\bibitem{hill97}\nS. Hill and W. K. Wooters, LANL e-print quant-ph/9703041 (1997).\n\n\\bibitem{niu99}\nC.-S. Niu and R. B. Griffits, LANL e-print quant-ph/9810008 (1999).\n\n\\bibitem{brennen}\nG. K. Brennen, I. H. Deutsch, and P. S. Jessen, LANL e-print\nquant-ph/9910031 (1999).\n\n\\bibitem{trieste}\nI. V. Bargatin, B. A. Grishanin, and V. N. Zadkov, Fortschr. Phys.\n(Progress of Physics) {\\bf 48}, 631 (2000); LANL e-print\nquant-ph/9903056 (1999).\n\n\\bibitem{TTLA}\nIn fact, the results of this section are valid not only for two\nTLAs, but also for any two quantum systems having Hilbert spaces of\nfinite dimensions.\n\n\\bibitem{chen99}\nY.-X. Chen, LANL e-print quant-ph/9906037 (1999).\n\n\\bibitem{kraus83}\nK. Kraus, {\\em States, Effects, and Operations} (Springer Verlag,\nBerlin, 1983).\n\n\\bibitem{helstr70}\nC. W. Helstrom, J. W. S. Liu, and J. P. Gordon, Opt. Commun. {\\bf 58},\n1578 (1970).\n\n\\bibitem{gTK73}\nB. A. Grishanin, Izv. Akad. Nauk SSSR, Ser. Tekh. Kiber. {\\bf 11},\nno. 5, 127 (1973).\n\n\\bibitem{helstrom}\nC. W. Helstrom, {\\em Quantum detection and estimation theory} (Academic\nPress, New York, 1976).\n\n\\bibitem{peres93}\nA. Peres, {\\em Quantum Theory: Concepts and Methods} (Kluwer Academic,\nDordrecht, 1993).\n\n%\\bibitem{nocloning}\n%W. K. Wooters and W. H. Zurek, Nature {\\bf 299}, 802 (1982).\n\n\\bibitem{errorcorr}\nP. W. Shor, Phys. Rev. A {\\bf 52}, 2493 (1995).\n\n\\bibitem{lasphys}\nB. A. Grishanin and V. N. Zadkov, Laser Physics {\\bf 8}, 1074 (1998);\nLANL e-print quant-ph/9906069 (1999).\n\n\\bibitem{Cohen92}\nC. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, {\\em Atom-Photon\nInteractions} (Wiley, New York, 1992).\n\n\\bibitem{addnote}\nIt is important to emphasize here that there are no restrictions\non the manipulations with the photon field, so that the emitted\nphoton appears to be a coherent signal.\n\n\\bibitem{fermi32}\nE. Fermi, Rev. Mod. Phys. {\\bf 4}, 87 (1932).\n\n\\bibitem{hamilton49}\nJ. Hamilton, Proc. R. Soc. London A {\\bf 62}, 12 (1949).\n\n\\bibitem{heitler49}\nW. Heitler and S. T. Ma, Proc. Irish. Acad. A {\\bf 52}, 109 (1949).\n\n\\bibitem{milonni74}\nP. W. Milonni and P. L. Knight, Phys. Rev. A {\\bf 10}, 1096 (1974).\n\n\\bibitem{dicke54}\nR. H. Dicke, Phys. Rev. {\\bf 93}, 9 (1954).\n\n\\bibitem{einstein}\nA. Einstein, in {\\em Albert Einstein: Philosopher, Scientist},\nedited by P. A. Schilpp (Evanston, 1970), p. 85.\n\n\n\n%\\bibitem{notequbit}\n%As a physical implementation of a qubit, one can consider any two-level\n%physical system, for example, two-level atom (TLA). For more details,\n%see Ref.\\ \\cite{qc-book}.\n\n\\end{references}\n\n%%\n%% Tables\n%%\n\\begin{table}\n\\caption{Unitary (atom+field) to (atom+field) transformation\n$U_{m\\mu,k\\alpha}$ for the vacuum initial photon field state, where\nindexes $m,k$ stand for atomic quanta and $\\mu,\\alpha$---for the\nnumber of photons. Long dash symbol stays for the elements not involved\ninto the calculated terms $S_{kl,\\mu\\nu}$ (Table \\ref{table2}).}\n\\begin{tabular}{ccccc}\n~~~~~~$m\\mu$ & \\multicolumn{1}{c}{00} & \\multicolumn{1}{c}{01}\n& \\multicolumn{1}{c}{10} & \\multicolumn{1}{c}{11} \\\\\n\\unitlength1pt\n\\begin{picture}(-3,0)\n\\put(0,21){\\line(1,-1){24}}\n\\end{picture}\n$k\\alpha$ \\\\ \\hline\n00& 1 & 0 & 0 & 0 \\\\\n01& --- & --- & --- & --- \\\\\n10& 0 & $\\psi_0({\\bf k},\\lambda)$ & $c_1$ & 0 \\\\\n11& --- & --- & --- & ---\n\\end{tabular}\n\\label{table1}\n\\end{table}\n\n\\begin{table}\n\\caption{Atom-to-field transformation $S_{kl,\\mu\\nu}$, which defines\n$\\ket{k}\\bra{l}\\to\\ket{\\mu}\\bra{\\nu}$ superoperator transformation.\nIndexes $k,l$ stand for atomic quanta and $\\mu,\\nu$---for the\nnumber of photons.}\n\\begin{tabular}{ccccc}\n~~~~~~~$\\mu\\nu$ & \\multicolumn{1}{c}{00} & \\multicolumn{1}{c}{01}\n& \\multicolumn{1}{c}{10} & \\multicolumn{1}{c}{11} \\\\\n\\unitlength1pt\n\\begin{picture}(-2,0)\n\\put(0,21){\\line(1,-1){24}}\n\\end{picture}\n$kl$ \\\\ \\hline\n00& 1 & 0 & 0 & 0 \\\\\n01& 0 & 0 & $\\psi_0({\\bf k},\\lambda)$ & 0 \\\\\n10& 0 & $\\psi_0^+({\\bf k},\\lambda)$ & 0 & 0 \\\\\n11&$\|c_1\|^2$&0&0&$\\psi_0({\\bf k},\\lambda)\\psi_0^+({\\bf k}',\\lambda')$\n\\end{tabular}\n\\label{table2}\n\\end{table}\n\n%%\n%% Figures with Figure Captions\n%%\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=5.cm\\epsfclipon\\leavevmode\\epsffile{fig1.eps}\n\\end{center}\n\\caption{Classification of possible quantum channels connecting two\nquantum systems. $1\\to1$, information is transmitted from the initial\nstate of the system to its final state (a); $1\\to2$, information is\ntransmitted from subsystem 1 of the system (1+2) to subsystem 2\nof the system (b); $1\\to(1+2)$, information is transmitted from\nsubsystem 1 of the system (1+2) to the whole system (1+2) (c).}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfysize=5.cm \\epsfclipon \\leavevmode \\epsffile{fig2.eps}\n\\end{center}\n\\caption{The coherent information transmitted between the states of the\nTLA at two time instants, $t=0$ and $t>0$, versus time $\\Gamma t$ and\nthe Rabi frequency $\\Omega/\\Gamma$ (both are dimensionless).}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=5.cm \\epsfclipon \\leavevmode \\epsffile{fig3.eps}\n\\end{center}\n\\caption{A spinless model of the hydrogen atom. The information\nchannel is made of the input forbidden $nlm\\to n'l'm'$ transition\n100--200 and the output dipole active 100--210 transition.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=6.cm\\epsfclipon\\leavevmode\\epsffile{fig4.eps}\n\\end{center}\n\\caption{The coherent information transmitted between two unitary\ncoupled TLAs versus population $\\rho_{11}$ of the diagonal initial\ndensity matrix of the second TLA and the coupling precession angle\n$\\Omega t$.} \\label{fig4}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=5.5cm \\epsfclipon\\leavevmode\\epsffile{fig5.eps}\n\\end{center}\n\\caption{Structure of the joint Hilbert space of the (atom+field)\nsystem. For the vacuum initial field state, both atomic states and only\ntwo Fock states of the field ($\\ket0$ and $\\ket1$) are involved in the\ndynamics of the joint system (atom+field). The dynamics is entirely\ndefined by just two states, $\\ket0_a\\ket1_{{\\bf k}\\lambda}$ and\n$\\ket1_a\\ket0$, which are described by $\\psi_0({\\bf k},\\lambda)$ and\n$c_1$, respectively.}\n\\label{fig5}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=6.cm\\epsfclipon\\leavevmode\\epsffile{fig6a.eps}\n\\epsfxsize=6.cm\\epsfclipon\\leavevmode\\epsffile{fig6b.eps}\n\\end{center}\n\\caption{The coherent information transmitted in the atom-to-field\nquantum channel versus the dimensionless time $\\gamma t$ and input\natomic density matrix, which is taken either diagonal with the ground\nstate matrix element $\\rho_{11}$ (a) or as the sum of $\\hat I/2$ and the real\n(``cosine-type'') coherent contribution of the off-diagonal elements\n$\\rho_{12}\\hat\\sigma_1$ (b).}\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=6.cm \\epsfclipon \\leavevmode \\epsffile{fig7a.eps}\n\\epsfxsize=6.cm \\epsfclipon \\leavevmode \\epsffile{fig7b.eps}\n\\end{center}\n\\caption{Excited state population of the second atom (a) and the\ncoherent information (b) in a system of two atoms interacting via the\nfree space field versus time $\\gamma t$ and the interatomic distance\n$\\varphi=\\omega_0R/c$ (both are dimensionless). The input density\nmatrix is diagonal with the ground state matrix element\n$\\rho_{22}=1/2$.} \\label{fig7}\n\\end{figure}\n\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912113.extracted_bib", "string": "{qc-book C. P. Williams and S. H. Clearwater, {\\em Explorations in Quantum Computing (Telos/Springer-Verlag, New York, 1998)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{shannon48 C. E. Shannon, Bell System Tech. J. {27, 379 (1948)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{shannon49 C. E. Shannon and W. Weaver, {\\em The Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{gallagher68 R. G. Gallagher, {\\em Information Theory and Reliable Communication (John Wiley and Sons, New York, 1968)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{grover91 T. M. Grover and J. A. Thomas, {\\em Elements of Information Theory (John Wiley and Sons, New York, 1991)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{schumacher96 B. W. Schumacher, Phys. Rev. A {54, 2614 (1996)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{schum96 B. W. Schumacher and M. A. Nielsen, Phys. Rev. A {54, 2629 (1996)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{bennett96 C. H. Bennett, D. DiVincenzo, J. Smolin, and W. K. Wootters, Phys. Rev. A {54, 3824 (1996)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{lloyd96 S. Lloyd, Phys. Rev. A {55, 1613 (1997)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{barnum98 H. Barnum, M. A. Nielsen, and B. Schumacher, Phys. Rev. A {57, 4153 (1998)." }, { "name": "quant-ph9912113.extracted_bib", "string": "{preskill J. Preskill, {\\em Lecture notes on Physics 229: Quantum information and computation, located at http://www.theory.caltech.edu/people/preskill/ph229/." }, { "name": "quant-ph9912113.extracted_bib", "string": "{note It is worthwhile to note a qualitative difference between these channels in connection with the physical casuality principle \\cite{einstein: specific restrictions due to the casuality principle are essential only for the channels $1\\to2$ and $1\\to(1+2)$ because of the spatial locality of the subsystems 1, 2. With this, analysis of a system of two atoms interacting with the vacuum photon field, given below, could be complemented by the relativistic retardation." }, { "name": "quant-ph9912113.extracted_bib", "string": "{gKE79 B. A. Grishanin, Sov. J. 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quant-ph9912114	``PARTIAL'' FIDELITIES	[ { "author": "Armin Uhlmann" } ]	For pairs $\omega$, $\rho$, of density operators on a finite dimensional Hilbert space of dimension $d$ I call $k$-fidelity the $d - k$ smallest eigenvalues of $\| \sqrt{\omega \sqrt{\rho \|$. $k$-fidelities are jointly concave in $\omega, \rho$. This follows from representing them as infima over linear functions. For $k=0$ known properties of fidelity and transition probability are reproduced. Partial fidelities characterize equivalence classes of pairs of density operators which are partially ordered in a natural way.	[ { "name": "quant-ph9912114.tex", "string": "%\\NeedsTeXFormat{LaTeX2e}\n\\documentclass[12pt]{article}\n%\\usepackage{showkeys}\n\\usepackage{latexsym}\n\\hfuzz=10pt\n\n\\newcommand{\\cA}{{\\mathcal A}}\n\\newcommand{\\cB}{{\\mathcal B}}\n\\newcommand{\\cH}{{\\mathcal H}}\n\\newcommand{\\1}{{\\mathbf 1}}\n\\newcommand{\\rP}{{\\rm PAIRS}}\n\\newcommand{\\ra}{{\\rm rank}}\n\\newcommand{\\T}{{\\rm Tr}}\n\n\n\\begin{document}\n\n\\title{ ``PARTIAL'' FIDELITIES}\n\n\\author{Armin Uhlmann }\n\n\\date{Institut f.~Theoretische Physik, Universit\\\"at Leipzig\\\\\nAugustusplatz 10/11, D-04109 Leipzig}\n\n\\maketitle\n\n\\begin{abstract}\nFor pairs $\\omega$, $\\rho$,\nof density operators on a finite dimensional Hilbert\nspace of dimension $d$ I call $k$-fidelity the $d - k$\nsmallest eigenvalues of $\| \\sqrt{\\omega} \\sqrt{\\rho} \|$.\n$k$-fidelities are jointly concave in $\\omega, \\rho$. This\nfollows from representing them as infima over linear functions.\nFor $k=0$ known properties of fidelity and transition\nprobability are reproduced. Partial fidelities\ncharacterize equivalence classes of pairs of density operators\nwhich are partially ordered in a natural way.\n\\end{abstract}\n\n\\section{Introduction}\nFor two unit vectors, $\\psi$ and $\\varphi$, of an Hilbert space\nthe quantity $\|\\langle \\psi, \\varphi\\rangle\|^2$ is their transition\nprobability. It is the squared modulus of their transition amplitude,\n$\\langle \\psi, \\varphi\\rangle$. Assume the state of the quantum\nsystem is $\|\\psi\\rangle\\langle\\psi\|$.\nA von Neumann measurement, designed\nto decide whether the quantum system is in the state\n$\|\\varphi\\rangle\\langle\\varphi\|$, prepares this state\nwith probability $\|\\langle \\psi, \\varphi\\rangle\|^2$.\\\\\nNotice further that two pairs of unit vectors are unitarily\nequivalent iff they enjoy equal transition probabilities.\n\nAll that becomes more complex, \\cite{fidel}, if two density\noperators, $\\rho_1$ and $\\rho_2$, are considered on a Hilbert space\n$\\cH$, and the quantum system is in state, say, $\\rho_1$. The algebra\nof operators on $\\cH$ will be called $\\cB$.\nOne can choose vectors $\\psi_j$ in the direct product $\\cH \\otimes \\cH$\nsuch that\n\\begin{equation} \\label{pf1}\n\\T \\, A \\rho_j = \\langle \\psi_j, (A \\otimes \\1) \\psi_j \\rangle,\n\\quad A \\in \\cB, \\, j = 1, 2\n\\end{equation}\nThe transition probability between $\\psi_1$ and $\\psi_2$ is not\ndetermined by the pair $\\rho_1$, $\\rho_2$. But running through all\npossible arrangements (\\ref{pf1}), the numbers $\|\\langle \\psi_2,\n\\psi_1\\rangle\|^2$ fill completely an interval $[0, p]$ of real\nnumbers. The largest one, the upper bound of this interval, is\ncalled {\\em transition probability between $\\rho_1$ and $\\rho_2$}\nand is denoted by $P(\\rho_1, \\rho_2)$. Thus a von Neumann\nmeasurement in $\\cH \\otimes \\cH$ can cause a transition $\\rho_1\n\\mapsto \\rho_2$ with a probability bounded by $P(\\rho_1, \\rho_2)$.\nThe bound can be reached by suitable measurements in the\nlarger system.\\\\\nNow I call attention to possibilities to\ncharacterize $P$ intrinsically, i.~e. without leaving the quantum\nsystem in question. The first one comes rather directly from\n(\\ref{pf1}). Let us call {\\em transition functional from $\\rho_1$\nto $\\rho_2$} every linear functional on $\\cB$ of the form\n\\begin{equation} \\label{pf2}\nA \\longrightarrow \\T \\, \\nu A :=\n\\langle \\psi_2, (A \\otimes \\1) \\psi_1 \\rangle\n\\end{equation}\nwhich arises from a setting (\\ref{pf1}). The operators $\\nu$ may be\ncalled {\\em transition operators from $\\rho_1$ to $\\rho_2$}.\nGenerally, $\\nu$ is not Hermitian: Exchanging the roles of\n$\\psi_1$ and $\\psi_2$ the operator $\\nu$ becomes its Hermitian\nadjoint, $\\nu^$.\\\\\nNow (\\ref{pf2}) is a transition functional if and only if\n\\begin{equation} \\label{pf3}\n\| \\, \\T \\, A_1 \\nu A_2^ \\, \|^2 \\leq ( \\T \\, A_1 \\rho_1 A_1^* ) \\,\n( \\T \\, A_2 \\rho_2 A_2^* ), \\quad A_i \\in \\cB\n\\end{equation}\nand it follows from the definition of $P$ that\n\\begin{equation} \\label{pf4}\nP(\\rho_2, \\rho_1) = \\max \| \\, \\T \\, \\nu \\, \|^2\n\\end{equation}\nwhere one takes the maximum over all transition operators from\n$\\rho_1$ to $\\rho_2$.\nCalculating the maximum in (\\ref{pf4}) is a standard exercise\nwith a well known outcome. Before writing it down\nI would like to explain the following:\\\\\nThe transition probability is separately concave in every one of\nits arguments.\nHowever, taking the root of $P$, the concavity properties become\ndramatically enhanced: $\\sqrt{P}$ is jointly concave\n\\cite{AlUh:84}.\nIn the following the square root of the transition probability\nwill be called {\\em fidelity} and will be denoted by $F$,\nessentially following a proposal\nof Richard Jozsa\\footnote{Jozsa introduced the word {\\em fidelity} for\nthe transition probability. Its present usage is not unique.\nI think\nthe peculiar properties of $\\sqrt{P}$ need an extra notation\nanyway.}. Thus\n\\begin{equation} \\label{pf5}\nF(\\omega, \\rho) := \\sqrt{P(\\omega, \\rho)} = \\T \\bigl( \\rho^{1/2}\n\\omega \\rho^{1/2} \\bigr)^{1/2}\n\\end{equation}\nThe assertion that $F$ is jointly concave is seen from\n\\begin{equation} \\label{pf6}\nF(\\omega, \\rho) = {1 \\over 2} \\inf\n\\bigl( \\T \\, A \\omega + \\T \\, A^{-1} \\rho \\bigr),\n\\quad A > 0, \\, \\, A^{-1} \\in \\cB\n\\end{equation}\nwhich is the finite dimensional version of a representation of\n$\\sqrt{P} = F$ as an infimum of linear functionals, valid for\npairs of states on von Neumann and on C$^$-algebras, see\n\\cite{AU99}. The representation is related to another one of\nequal generality estimating $P(\\omega, \\rho)$ from above by\nthe product of\nTr$\\omega A$ and Tr$\\rho A^{-1}$, with $A$ an invertible positive\noperator, see \\cite{inf1} for a partial result and \\cite{inf2}\nfor the C$^$-case in full generality. For finite dimensions\nthese well know results are reproduced by setting $k = 0$ in\nthe expressions (\\ref{infk2}) and (\\ref{infk3}) below.\n\nAs a matter of fact, the equality of $F$ (or of $P$) for two\npairs of density operators do not imply their unitary equivalence.\nThis pleasant feature, valid for pure states, is missing for\nmixed ones.\nLooking at (\\ref{pf6}) one may wonder whether it is\nnot possible to get a whole series of concave invariants by taking\nother suitable sets of operators than the invertible positive\noperators in the expression (\\ref{pf6}). To give an affirmative\nanswer belongs to the issues of the present paper. By\nthe partial fidelities one gets a reasonable classification of\npairs of density operators, coarser than unitary\nequivalence would give.\n\n\nAll what follows remain in finite dimensions.\nBy modifying certain settings and by adding new arguments,\nPeter M.~Alberti,\n\\cite{Alberti}, was able to extend essential parts of what follows\nto von Neumann Algebras.\nHis results are particularly satisfying for type II$_1$.\n\n\\section{$k$--Fidelities}\nLet $\\cH$ be a finite-dimensional Hilbert space and $d = \\dim \\cH$.\\\\\nThe {\\em spectrum}, spec$(A)$, of an operator $A$ is the\nfamily of roots of the polynomial $\\det(A - \\lambda \\1)$ counted\nwith their correct multiplicities. If the spectrum is real\nwe assume the set spec$(A)$ {\\em decreasingly ordered}.\nThis convention applies to every diagonalizable operator\nwith real eigenvalues\nand in particular to every Hermitian one. Consider now\n\\begin{equation} \\label{spectrum}\n\\hbox{spec}\\bigl((\\sqrt{\\omega} \\rho \\sqrt{\\omega})^{1/2}\\bigr)\n=\n\\hbox{spec}\\bigl((\\sqrt{\\rho} \\omega \\sqrt{\\rho})^{1/2}\\bigr)\n=\n\\{ \\, \\lambda_1 \\geq \\lambda_2 \\geq \\dots, \\geq \\lambda_d \\, \\}\n\\end{equation}\nso that, according to (\\ref{pf5}),\nthe sum of the lambdas is the fidelity. The spectrum (\\ref{spectrum})\nis equal to the ordered singular numbers of\n$\\sqrt{\\omega} \\sqrt{\\rho}$ and of $\\sqrt{\\rho} \\sqrt{\\omega}$.\\\\\nI define partial fidelities simply by summing\nup parts of the spectrum (\\ref{spectrum}).\\\\\n{\\bf Definition:} \\, For $0 \\leq k \\leq d-1$\n\\begin{equation} \\label{kfidel}\nF_k(\\omega, \\rho) := \\sum_{j > k} \\lambda_j, \\quad\nk=0, 1, \\dots, d-1 .\n\\end{equation}\nIf $k \\geq d$ then $F_k = 0$. For the time being\n$F_k$ will be called {\\em $k$-th partial fidelity}, or simply\n{\\em $k$-fidelity} of the pair $\\omega$ and $\\rho$.\\\\\nAn important point is to add: I do not necessarily require\nthat $\\rho$ and $\\omega$ have trace one. Indeed, on a finite\ndimensional Hilbert space (\\ref{kfidel}) is naturally defined for\npairs from the cone of positive operators and\n\\begin{equation} \\label{scale}\n\\sqrt{c} \\, F_k(\\omega, \\rho)\n= F_k(c \\omega, \\rho) = F_k(\\omega, c \\rho), \\quad c > 0\n\\end{equation}\nfor positive real numbers $c$. Notice the properties:\\\\\na) $F_k$ is symmetric in its arguments.\\\\\nb) $F_0$ is just the fidelity $F$.\\\\\nc) For pairs of pure density operators it is $F_k = 0$ for $k > 0$.\\\\\nd) If $F_k \\neq 0$ then rank$(\\omega) > k$ and\nrank$(\\rho) > k$ necessarily.\\\\\ne) $F_k$ is unitarily invariant, i.e.~invariant by the\nsimultaneous transformation\n$\\rho \\to U \\rho U^$ , $\\omega \\to U \\omega U^$.\\\\\nHowever, a deeper justification for the definition above is in\\\\\n{\\bf Theorem 1}\\\\\n{\\em The partial fidelities are concave functions of the pairs }\n$\\{ \\omega, \\rho \\}$ :\n\\begin{equation} \\label{concave}\n\\sum_j p_j F_k(\\omega_j, \\rho_j) \\leq F_k(\\sum_j p_j \\omega_j,\n\\sum_i p_i \\rho_i)\n\\end{equation}\n{\\em for any probability vector $p_1, p_2, \\dots$ and arbitrary\npairs $\\{ \\omega_i, \\rho_i \\}$.} $\\Box$\n\n\\noindent\nThe theorem is a consequence of a new relation representing $F_k$\nas an infimum of linear functionals quite similar to (\\ref{pf6}).\nIt estimates partial fidelities linearly as close as possible\nfrom above. To get the announced representation\nI am going to define the set PAIRS which consists\nof all pairs $\\{A, B\\}$ of positive Hermitian\noperators, $A$, $B$, such that\n\\begin{equation} \\label{pairs1}\n ABA = A, \\quad BAB = B\n\\end{equation}\nLet $\\{A, B\\}$ be such a pair. It follows $(AB)^2 = AB$ immediately.\nBecause $Q = AB$ is a product of two positive operators it is\ndiagonalizable. On the other hand we see $Q^2 = Q$ so that its\nspectrum consists of zeros and ones. Therefore, the trace of $Q$\nis equal to the rank of $Q$. Now (\\ref{pairs1}) says $QA=A$ and\n$BQ=B$ implying that the ranks of $A$ and $B$ cannot be larger\nthan the rank of $Q$. Now $Q=AB$ shows that neither the rank of\n$A$ nor the rank of $B$ can be less than that of $Q$. Altogether\nwe have:\\\\\n{\\bf Lemma 1:} \\, For all $\\{A, B\\} \\in \\rP$\n\\begin{equation} \\label{pairs2}\n\\ra(A) = \\ra(B) = \\ra(AB) = {\\rm Tr} \\, AB\n\\end{equation}\nis an integer called {\\em rank of the pair} $\\{A, B\\}$. $\\Box$\\\\\n{\\bf Definition:} PAIRS$_m$ consists of all pairs from PAIRS of\nrank $m$.\\\\ The promised representation of the $k$-th fidelities\nis in the following theorem.\\\\\n\n\\noindent\n {\\bf Theorem 2}\\\\\n {\\em Let $m + k = \\dim \\cH$. Then}\n\\begin{equation} \\label{infk2}\nF_k(\\omega, \\rho) = {1 \\over 2} \\, \\inf \\Bigl( \\,\n\\T \\, A \\omega + \\T \\, B \\rho \\Bigr), \\quad\n\\{A, B \\} \\in \\rP_m.\n\\end{equation}\n\n\\noindent\nOne can deduce from (\\ref{infk2}) the following inequality:\n\\begin{equation} \\label{infk3}\nF_k(\\omega, \\rho)^2 = \\inf\n\\Bigl(\\T \\, A \\omega \\Bigr) \\Bigl(\\T \\, B \\rho \\Bigr),\n\\quad \\{A, B \\} \\in \\rP_m.\n\\end{equation}\nThe point is that with $\\{A, B\\}$ also\n$\\{\\lambda A, \\lambda^{-1} B\\}$ is contained in PAIRS$_m$\nfor $\\lambda >0$. After this trivial substitution, the right hand\nside of (\\ref{infk2}) is of the form $\\lambda a + \\lambda^{-1}b$.\nTaking the infimum of over $\\lambda$ results in $2\\sqrt{ab}$,\nand (\\ref{infk3}) is derived from (\\ref{infk2}). (\\ref{infk3}),\nsuitably reformulated, is known on C$^$-algebras if $k=0$,\nsee \\cite{inf2}.\n\nTheorem 1 is a consequence of theorem 2.\nWe shall prove theorem 2 in the next section, at first assuming\n$\\rho$ invertible, (which would be sufficient for theorem 1).\nThen, by continuity arguments, we can allow for all $\\omega$ and $\\rho$.\nHowever, before going into the proof,\nwe have to look at a \"hidden\" symmetry of the $k$-fidelities.\n\n\n\n\n\n\\section{The symmetry group of the $k$-fidelities}\nLet us denote by $\\Gamma$\nthe multiplicative group of all invertible operators acting on $\\cH$.\nWith $X \\in \\Gamma$ we define the {\\em $X$-transform} of a pair\n$\\{ \\omega, \\rho \\}$ by\n\\begin{equation} \\label{transf1}\n\\{ \\omega, \\rho \\}^X := \\{ X \\omega X^, (X^{-1})^* \\rho X^{-1} \\}\n\\end{equation}\nThe transformations create orbits of $\\Gamma$ in the set\nof pairs. Two pairs, $\\{ \\omega, \\rho \\}$\nand $\\{ \\omega', \\rho' \\}$, are called {\\em $\\Gamma$-equivalent}\niff there is $X \\in \\Gamma$ such that\n$\\{ \\omega', \\rho' \\}$ is the $X$-transform\nof $\\{ \\omega, \\rho \\}$.\\\\\n{\\bf Lemma 2:} \\,\nThe $k$-fidelities of $\\Gamma$-equivalent pairs are equal\nfor every $k$\n\\begin{equation} \\label{inv1}\nF_k(\\omega, \\rho) = F_k(\\omega', \\rho') \\, \\hbox{ if } \\,\n\\{ \\omega', \\rho' \\} = \\{ \\omega, \\rho \\}^X\n\\end{equation}\nFor the proof we start with the identity\n$$\n(X \\omega X^) (X^{-1})^ \\rho X^{-1} = X \\omega \\rho X^{-1}\n$$\nsaying that the spectrum of $\\omega \\rho$ is an invariant for\n$\\Gamma$-equivalent pairs. It suffices to show that\n\\begin{equation} \\label{spectrum2}\n\\hbox{spec}(\\omega \\rho) = \\hbox{spec}(\\rho \\omega)\n= \\{ \\, \\lambda_1^2, \\lambda_2^2, \\dots, \\lambda_d^2 \\, \\}\n\\end{equation}\nfollows from (\\ref{spectrum}). By\n$\\sqrt{\\rho} (\\omega \\rho) (\\sqrt{\\rho})^{-1} =\n\\sqrt{\\rho} \\omega \\sqrt{\\rho}$ this is true for invertible $\\rho$.\nThe assumption of invertibility can be removed\nby continuity.$\\Box$\\\\\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nSubstituting in (\\ref{transf1}) $\\rho = X^* \\tau X$ we may rewrite\n(\\ref{inv1}) as\n\\begin{equation} \\label{inv1a}\nF_k(\\omega, X^* \\tau X) = F_k( X \\omega X^, \\tau)\n\\end{equation}\nIn (\\ref{inv1a}), by continuity, also $X$ need not be invertible.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nOnly for the purpose of the the following proof\nwe abbreviate\nthe right-hand-side of (\\ref{infk2}) by $G_k(\\omega, \\rho)$,\n$$\nG_k(\\omega, \\rho) := {1 \\over 2} \\, \\inf \\Bigl( \\,\n\\T \\, A \\omega + \\T \\, B \\rho \\Bigr), \\quad\n\\{A, B \\} \\in \\rP_m.\n$$\nWe observe that also $G_k$ allows for $\\Gamma$-invariance.\nIndeed, with a pair $\\{A, B \\}$ its transform\n\\begin{equation} \\label{transf2}\n\\{A, B \\}_X := \\{ X^ A X, X^{-1} B (X^{-1})^* \\}\n\\end{equation}\nis also in PAIRS$_m$, and we get the trace identities\n$$\n\\T \\omega A = \\T \\omega' A', \\quad \\T \\rho B = \\T \\rho' B'\n$$\nwhenever\n$$\n\\{A', B' \\} = \\{A, B \\}_X, \\quad \\{ \\omega', \\rho' \\} =\n\\{ \\omega, \\rho \\}^X\n$$\nTherefore,\n$$\nG_k(\\omega, \\rho) = G_k(\\omega', \\rho') \\, \\hbox{ if } \\,\n\\{ \\omega', \\rho' \\} = \\{ \\omega, \\rho \\}^X\n\\eqno(a)\n$$\nNow we start the {\\em proof of theorem 2}. As we just have seen,\n{\\em both} sides of (\\ref{infk2}) do not change along\na $\\Gamma$-orbit of $X$-transforms (\\ref{transf1}).\n\n\nStep 1 in the proof is to show $F_k \\leq G_k$. To this end we first\nassume a pair $\\{ \\omega, \\rho \\}$ with an\ninvertible $\\rho$. We transform the given pair according to\n(\\ref{transf1}) by $X = \\sqrt{\\rho}$.\nThe new pair is $\\{ \\omega', \\1 \\}$ with\n$\\omega' = \\sqrt{\\rho} \\omega \\sqrt{\\rho}$. By (a) and lemma 2\nit suffices to prove $F_k \\leq G_k$ for the new\npair, i.~e. we estimate $G_k(\\omega', \\1)$ from below.\\\\\nWe choose a pair $\\{ A, B \\}$ from\nPAIRS$_m$ arbitrarily. Let $\\phi_1, \\phi_2, \\dots$ be an eigenbasis of\n$A$ and $a_1, a_2, \\dots$ the corresponding eigenvalues. By\nsandwiching $ABA=A$ between these eigenvectors of $A$ one gets\n$$\n\\langle \\phi_i, B \\phi_k \\rangle =\na_i^{-1} \\delta_{ik} \\, \\, \\hbox{for} \\, \\, i, k \\geq m\n$$\nNow we can write\n$$\n\\T \\, A \\omega' + \\T\n\\, B = \\sum_1^m a_i \\langle \\phi_i, \\omega' \\phi_i \\rangle + \\sum_1^m\na_i^{-1} + \\sum_{j > m} \\langle \\phi_j, B \\phi_j \\rangle\n$$\nWith\npositive reals $a$ and $x$ it holds $ax + a^{-1} \\geq 2 \\sqrt{x}$.\nUsing that inequality to estimate the first two sums from below and\nneglecting the last term, we arrive at\n$$\n\\T \\, A \\omega' + \\T \\, B \\geq\n2 \\sum_1^m \\sqrt{ \\langle \\phi_i, \\omega' \\phi_i \\rangle }\n$$\nThe square root is concave.\nHence, see \\cite{A+U}, equ. 1-46,\n$$\n\\sum_1^m \\sqrt{\\langle \\phi_i, \\omega' \\phi_i \\rangle }\n\\geq \\sum_1^m \\langle \\phi_i,\n\\sqrt{\\omega'} \\phi_i \\rangle \\geq F_k(\\omega, \\rho)\n$$\nThe last\ninequality sign is an estimation of the $m$ smallest eigenvalues (due to\nFan and Horn) and respecting $F_k(\\omega', \\1) = F_k(\\omega, \\rho)$.\nIt results\n\\begin{equation} \\label{below}\n{1 \\over 2} \\, \\inf \\Bigl( \\, \\T \\, A \\omega + \\T \\, B \\rho \\Bigr) \\geq\nF_k(\\omega, \\rho)\n\\end{equation}\nat first for pairs $\\{\\omega', \\1 \\}$\nand then, by $\\Gamma$-invariance, for all pairs $\\{ \\omega, \\rho \\}$\nwith invertible $\\rho$. However, both sides of (\\ref{below}) are\ncontinuous in $\\omega$ and $\\rho$. Thus step one terminates in\nthe validity of\n(\\ref{below}) for all $\\omega$ and all $\\rho$. The inequality is\nequivalent to $G_k \\geq F_k$.\\\\\nIn step 2\nwe show $G_k \\leq F_k$ at first for invertible $\\rho$. As above we\nreduce the problem by $\\Gamma$-invariance to that of a pair\nconsisting of $\\omega'$ and $\\1$. We now choose\n$\\phi_1, \\dots, \\phi_m$ to be eigenvectors of $\\omega'$ belonging\nto the $m$ smallest eigenvectors\nof $\\omega'$. The latter are $\\lambda_{k+1}^2, \\dots, \\lambda_{k+m}^2$.\nby lemma 2. Define\n$$\nA' = \\sum_1^m a_i \|\\phi_i\\rangle\\langle\\phi_i\|, \\quad\nB' = \\sum_1^m a_i^{-1} \|\\phi_i\\rangle\\langle\\phi_i\|\n$$\nand consider\n$$\n\\T A' \\omega' + \\T B' =\n\\sum_{j=1}^m a_j \\lambda_{j+k}^2 + \\sum_1^m a_j^{-1}\n$$\nIf $\\lambda_{j+k} > 0$ we choose $a_j = \\lambda_{j+k}^{-1}$.\nOtherwise we set $a_j = c^{-1} > 0$ arbitrarily. If $n$ of the $m$\neigenvalues $\\lambda_{j+k}$ are zero, then our convention implies\n$$\n\\T A' \\omega' + \\T B' = 2 \\sum_{j=1}^m \\lambda_{j+k} + nc =\n2 F_k(\\omega', \\1) + nc\n$$\nand, hence, $G_k \\leq F_k + nc$.\nSince $c$ can be made arbitrarily small we arrive at the wanted\ninequality $G_k \\leq F_k$. Now, relying on $\\Gamma$-invariance\n(\\ref{inv1}) and (a), the inequality is shown true\nfor all pairs of invertible density operators.\\\\\nCombining step one and two we see:\n$F_k(\\omega, \\rho) = G_k(\\omega, \\rho)$ if both arguments\nare invertible. Hence $F_k$ is concave for these pairs.\nBut $F_k$ is a continuous function of $\\omega$ and $\\rho$\nby (\\ref{kfidel}). Therefore, $F_k$ is jointly concave\nand theorem 1 is valid.\\\\\nBut one knows that a concave\nfunction is semi-continuous from below, see \\cite{Roc70}, theorem 10.2,\nwhere semi-continuity from above is stated for convex functions.\nBecause $F_k$ is continuous and concave it dominates every function\nwhich is concave and coincides for convexly inner points with $F_k$.\nThis means $F_k \\geq G_k$ always.\nNow step one of the proof provides $F_k = G_k$. $\\Box$\n\n\n\n\\section{Equivalence and partial order}\nIt is tempting to collect pairs of positive (density) operators\ninto equivalence classes according to their partial fidelities.\nFor the purpose of the present paper we call two pairs\n{\\em equivalent}, and we write\n\\begin{equation} \\label{sim}\n\\{ \\omega, \\rho \\} \\sim \\{ \\omega', \\rho' \\},\n\\end{equation}\niff their $k$-fidelities are equal,\n$F_k(\\omega, \\rho) = F_k(\\omega', \\rho')$\nfor $k = 0, 1, \\dots, d-1$. The relation $\\sim$ is an\nequivalence relation. Notice that\n$\\{ \\omega, \\rho \\} \\sim \\{ \\rho, \\omega, \\}$.\nGenerally, an equivalence class\ncontains a lot of $\\Gamma$-orbits. But there is an important\nexception:\\\\\n{\\bf Lemma 3:} \\,\nIf both operators, $\\omega$ and $\\rho$, are invertible,\nthe equivalence class of $\\{\\omega, \\rho\\}$ consists exactly\nof all pairs $\\{\\omega, \\rho\\}^X$, $X \\in \\Gamma$ .$\\Box$\n\nProof: The assumption is valid if and only if 0 does not belong\nto the eigenvalues (\\ref{spectrum}). This takes place if the\nsmallest one is different from zero, hence iff $F_{d-1} \\neq 0$.\nHence, if the assumption of the lemma is valid for one member\nof an equivalence class, then it is true for all members.\nLet $\\{\\omega_1, \\rho_1\\}$ be in the equivalence class\nof $\\{\\omega, \\rho\\}$.\nTransforming the latter by\n$X = \\sqrt{\\rho}$ and the former by $X_1 = \\sqrt{\\rho_1}$\n by the receipt (\\ref{transf1}) results in accordingly\n$\\Gamma$-equivalent pairs\n$\\{\\omega', \\1 \\}$ and $\\{\\omega'_1, \\1 \\}$. Being in the same\nequivalence class, $\\omega'$ and $\\omega'_1$\nhave to have equal eigenvalues and they are even unitarily\nequivalent. Thus all the pairs considered belong to the same\n$\\Gamma$-orbit. $\\Box$\n\nLet us write\n$\\{\\omega', \\rho' \\} \\leq \\{\\omega, \\rho \\}$ if both,\n$\\omega - \\omega'$ and $\\rho - \\rho'$, are positive\noperators. A simple example\nis as follows: Write $\\omega = \\omega' + \\omega_0$,\n$\\rho = \\rho' + \\rho_0$, and assume orthogonality between\n$\\omega'$ and $\\rho_0$ and between $\\rho'$ and $\\omega_0$, i.~e.\n$\\omega_0 \\rho' = 0$, $\\rho_0 \\omega' = 0$. Then\n$\\{ \\omega, \\rho \\}$ and $\\{ \\omega', \\rho' \\}$\nbelong to the same equivalence class.\nTo see what we can learn from\n$\\{\\omega', \\rho' \\} \\leq \\{\\omega, \\rho \\}$\ngenerally, we proceed in two steps,\n$\\{\\omega', \\rho' \\} \\leq \\{\\omega, \\rho' \\}$ and\n$\\{\\omega, \\rho' \\} \\leq \\{\\omega, \\rho \\}$. Consider the\n second one. It implies\n$\\sqrt{\\omega} \\rho'\\sqrt{\\omega} \\leq \\sqrt{\\omega} \\rho \\sqrt{\\omega}$\nand, because taking the square root does not destroy the\ninequality,\n$$\n\\bigl(\\sqrt{\\omega} \\rho'\\sqrt{\\omega} \\bigr)^{1/2}\n\\leq\n\\bigl(\\sqrt{\\omega} \\rho \\sqrt{\\omega} \\bigr)^{1/2}\n$$\nThe sums of its $m$ smallest eigenvalues, which are the partial\nfidelities, obey the same inequality. Further, if the traces\nof both positive operators happen to be equal, the operators\nthemselves have to be equal one to another. In repeating\nthe arguments for the first step and combining both, we arrive\nat\\\\\n{\\bf Lemma 4:} \\, If\n\\begin{equation} \\label{L41}\n\\{ \\omega, \\rho \\} \\geq \\{ \\omega', \\rho' \\} \\,\n\\end{equation}\nthen\n\\begin{equation} \\label{pgeq}\nF_k(\\omega, \\rho) \\geq F_k(\\omega', \\rho'),\n\\quad k = 0, 1, \\dots, d-1\n\\end{equation}\nIf in addition to (\\ref{L41}) $F(\\omega, \\rho) = F(\\omega', \\rho')$\nis true, then all partial fidelities must be equal in pairs, and the\ntwo pairs belong to the same equivalence class:\n$\\{ \\omega, \\rho \\} \\sim \\{ \\omega', \\rho' \\}$. $\\Box$\n\n\nGiven $\\omega$, $\\rho$,\nAlberti \\cite{mpairs} has shown, even in the C$^$-category,\nthat there is one and only one pair $\\{ \\omega_0, \\rho_0 \\}$\nwhich enjoys the same transition probability, (and, therefore,\nthe same fidelity), and which is minimal with\nrespect to $\\geq$. This {\\em minimal pair} satisfies\n$$\n\\{ \\omega_0, \\rho_0 \\} \\leq \\{ \\omega', \\rho' \\}\n$$\nwhenever\n$$\n\\{ \\omega', \\rho' \\} \\leq \\{ \\omega, \\rho \\} \\,\\, \\hbox{ and } \\,\\,\nF(\\omega', \\rho' ) = F(\\omega, \\rho )\n$$\nis valid.\\\\\nWe see that every equivalence class contains a minimal\npair and, therefore, a $\\Gamma$-orbit of minimal pairs. It is\ntempting to believe that there is only one minimal $\\Gamma$-orbit\nin every equivalence class of pairs. But I do not know whether\nthat conjecture is true.\\\\\n\nNow one may go a step further, anticipating the ideas of\nmajorization, \\cite{M+O}, or those of partially ordering orbits\nbelonging to certain classes of transformations, \\cite{A+U}.\nTo do so, let us call\n $\\{ \\omega_1, \\rho_1 \\}$ {\\em F--dominated} by\n $\\{ \\omega_2, \\rho_2 \\}$ iff\n\\begin{equation} \\label{F-dom}\nF_k(\\omega_1, \\rho_1) \\leq F_k(\\omega_2, \\rho_2), \\quad\nk = 0, 1, 2, \\dots\n\\end{equation}\nFrom theorem 1 we get the following\\\\\n{\\bf Corollary:} \\,\nIf $\\{ \\omega_2, \\rho_2 \\}$ is contained in the convex hull of\nthe $\\sim$equivalence class of $\\{ \\omega_1, \\rho_1 \\}$ then\n(\\ref{F-dom}) takes place. $\\Box$\n\nWe thus get a new partial ordering (or majorization tool) for pairs\nof positive (density) operators which seems worthwhile to\ninvestigate.\nThere is a link, indeed a morphism, to singular number majorization.\nDenote by sing$(B)$ the decreasingly ordered singular numbers of\nthe operator $B$, that is\n$$\n \\hbox{sing}(B) = \\hbox{spec}(\\sqrt{B^B}) =\n\\hbox{spec}(\\sqrt{BB^}) = \\hbox{sing}(B^),\n$$\nand by sing$[B]$ the set of all operators $C$ with\nsing$(C)$ = sing$(B)$, the {\\em singular number class} of $B$.\nIn particular\n$$\n\\hbox{spec}((\\sqrt{\\omega} \\rho \\sqrt{\\omega})^{1/2}) =\n\\hbox{sing}(\\sqrt{\\rho} \\sqrt{\\omega})\n$$\nThere are many useful rules governing the partial order of the\nsingular number classes, see \\cite{M+O}, 9.E and \\cite{A+U},\n2.4 (theorem 2-8), 2.5. With them one easily proves\\\\\n{\\bf Lemma 5:} \\, The following items are mutually equivalent:\\\\\na) $\\{ \\omega_1, \\rho_1 \\}$ is F--dominated by\n $\\{ \\omega_2, \\rho_2 \\}$.\\\\\nb) $\\sqrt{\\omega_2} \\sqrt{\\rho_2}$ is contained in the convex hull\nof sing$[\\sqrt{\\omega_1} \\sqrt{\\rho_1}]$.\\\\\nc) There are finitely many operators $A_i$, $B_i$, all with operator\nnorms not exceeding 1, such that\n$$\n\\sqrt{\\omega_2} \\sqrt{\\rho_2} = \\sum A_i \\sqrt{\\omega_1}\n\\sqrt{\\rho_1} B_i\n$$\n\n\\section{More about PAIRS}\nIt is our aim to get some insight into the structure of PAIRS.\nLet $\\{ A, B\\} \\in$ PAIRS$_m$ with $0 < m \\leq d = \\dim \\cH$.\n$k$ is defined by $k+m=d$.\nLet us write $A$, $B$ as block matrices with respect to an\neigenvector basis of $A$ as in the proof of theorem 2.\nThen, with is a positive $m \\times m$ matrix $A_{11}$,\n\\begin{equation} \\label{block1}\nA = \\pmatrix{A_{11} & 0 \\cr 0 & 0 \\cr}, \\quad\nB = \\pmatrix{B_{11} & B_{12} \\cr B_{21} & B_{22} \\cr}\n\\end{equation}\nHere $B_{11}$ is $m \\times m$, $B_{12}$ is $m \\times k$,\nand $B_{22}$ is $k \\times k$.\n The equation $ABA=A$ results in\n$B_{11} = A_{11}^{-1}$. Having this in mind, one gets from $BAB=B$\n\\begin{equation} \\label{block2}\nB_{11} = A_{11}^{-1}, \\quad B_{22} = B_{21} A_{11} B_{12}, \\quad\nB_{12}^* = B_{21}\n\\end{equation}\nNotice that $B_{12}$ can be chosen arbitrarily: Given the first\nmember, $A$, of the pair, $B$ depends freely on $km$ complex\nparameters.\n\nThere is a further representation of the pairs in PAIRS$_m$.\nCall $2m$ vectors, $\\psi_1, \\psi_2, \\dots, \\psi_m$,\n$\\varphi_1, \\varphi_2, \\dots, \\varphi_m$, a\n{\\em bi-orthogonal system of length $m$} if\n$$\n \\langle \\psi_i, \\varphi_j \\rangle = \\delta_{ij}, \\quad\ni, j = 1, 2, \\dots, m\n\\eqno(A)\n$$\nTogether with $m$ positive numbers, $a_1, a_2, \\dots, a_m$,\nwe obtain from (A) a pair of operators\n$$\nA = \\sum a_i \|\\psi_i \\rangle\\langle \\psi_i\|, \\quad\nB = \\sum a_i^{-1} \|\\varphi_i \\rangle\\langle \\varphi_i\|\n\\eqno(B)\n$$\nfor which $ABA=A$ and $BAB=B$ can be\nchecked. Let us prove that {\\em every pair from} PAIRS$_m$\n{\\em can be gained by this procedure.}\n\nLet $\\{ A, B \\} \\in$ PAIRS$_m$. Because $AB$ is diagonalizable\nwith eigenvalues 0 and 1, there is\n$X \\in \\Gamma$ such that $X AB X^{-1}$ is a Hermitian projection\noperator. Hence the operators\n$$\nA_1 = X A X^, \\quad B_1 = (X^{-1})^ B X^{-1}\n$$\ncommute. Therefore there is a\nrepresentation\n$$\nA_1 = \\sum a_i \|\\phi_i \\rangle\\langle \\phi_i\|, \\quad\nB_1 = \\sum a_i^{-1} \|\\phi_i \\rangle\\langle \\phi_i\|\n$$\nwith $m$ orthonormal vectors $\\phi_1, \\dots, \\phi_m$. But\n$$\n\\psi_i = X^{-1} \\phi_i, \\quad \\varphi_i = X^* \\phi_i\n$$\nis bi-orthogonal with length $m$. Transforming $A_1$\nand $B_1$ back to $A$ and $B$ gives the desired representation\nof the pair.\\\\\nThe bi-orthonormal system (A) of (B) can be chosen {\\em balanced} :\n$$\n\\langle \\psi_i, \\psi_i\\rangle = \\langle \\varphi_i, \\varphi_i\\rangle,\n\\quad i = 1, \\dots, m\n\\eqno(C)\n$$\nIndeed, the necessary changes in the norms can be compensated\nby adjusting the $a_i$. Now we insert (B) into the right hand side of\n(\\ref{infk2}) and observe\n$$\n\\T \\, A \\omega + \\T \\, B \\rho =\n\\sum_1^m \\bigl( a_j < \\psi_j \| \\omega \| \\psi_j > +\n a_j^{-1} < \\varphi_j \| \\varrho \| \\varphi_j > \\bigr)\n$$\nBy varying the free parameters $a_j$ we arrive at\\\\\n{\\bf Theorem 3}\\\\\n {\\em Let $m + k = \\dim \\cH$. Then}\n\\begin{equation} \\label{infk4}\nF_k(\\omega, \\rho) = \\inf \\sum_{i=1}^m\n\\sqrt{\\langle \\psi_i, \\omega \\psi_i\\rangle\n\\langle \\varphi_i, \\rho \\varphi_i\\rangle}\n\\end{equation}\n{\\em where the infimum runs through all balanced bi-orthogonal\nsystems of length} $m$. $\\Box$\n\nFinally, assume the infimum in (\\ref{infk2}) is attained by\n$\\{A, B\\} \\in$ PAIRS$_m$,\n\\begin{equation} \\label{minimum1}\nF_k(\\omega, \\rho) = {1 \\over 2}\n\\Bigl( \\, \\T \\, A \\omega + \\T \\, B \\rho \\Bigr),\n\\quad m + k = \\dim \\cH\n\\end{equation}\nIf we vary the minimizing pair, the first variation must vanish,\n$$\n\\Bigl({d \\over ds} \\Bigr)_{s=0} c(s), \\quad c(s) =\n(\\T \\, A_s \\omega + \\T \\, B_s \\rho)\n$$\nwhere, with $X_s = \\exp s Y$ and any operator $Y$,\n$$\nA_s = X_s^* A X_s, \\quad B_s = X_s^{-1} B (X_s^)^{-1}\n$$\nWe perform the first derivative and obtain\n$$\n\\Bigl({d \\over ds} \\Bigr)_{s=0} A_s = Y^ A + A Y,\n\\quad\n\\Bigl({d \\over ds} \\Bigr)_{s=0} B_s = - Y B - B Y^\n$$\nAfter inserting in $\\dot c(0) = 0$ and an rearrangement it results\n$$\n{\\rm Tr} \\, Y (A \\rho - \\omega B)^ +\n{\\rm Tr} \\, Y^* (A \\rho - \\omega B) = 0\n$$\nAs $Y$ could be chosen arbitrarily, we arrive at\n\\begin{equation} \\label{minimum2}\n A \\rho = \\omega B\n\\end{equation}\nas a necessary condition for the validity of (\\ref{minimum1}).\n\nIs there $\\{A, B\\} \\in$ PAIRS$_m$ fulfilling (\\ref{minimum1})\nand minimizing (\\ref{infk2})? If we can $\\Gamma$-transform\n$\\omega, \\rho$, to the form $\\tau, \\tau$, we are\ndone. Indeed, we then can choose a projection operator $P_m$\nonto the $m$ smallest eigenvalues of $\\tau$ and we get\n$$\nF_k(\\tau, \\tau) = {\\rm Tr} \\, P_m \\tau, \\quad\n\\{P_m, P_m \\} \\in \\rP_m\n$$\ni.~e. the problem is solved in that case. Now, if $\\omega$\nand $\\rho$ are both invertible, there is a unique positive $X$\nsuch that\n\\begin{equation} \\label{minimum3}\nX \\omega X = X^{-1} \\rho X^{-1} := \\tau, \\quad X > 0\n\\end{equation}\nThe choice (\\ref{minimum3}) ensures (\\ref{minimum1}) with\n$A = XP_mX$, $B = X^{-1} P_m X^{-1}$. To get $X$ one has to\nsolve $X^2 \\omega X^2 = \\rho$. There is\na unique positive solution $X$ which is the square root\nof the geometric mean \\cite{PW75} between $\\rho$ and $\\omega^{-1}$.\n$$\nX^2 = \\omega^{-1/2} \\bigl( \\omega^{1/2} \\rho \\omega^{1/2} \\bigr)^{1/2}\n\\omega^{-1/2}\n$$\nas one can convince oneself by inserting into $X^2 \\omega X^2 = \\rho$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Acknowledgement}\nI like to thank P.~M.~Alberti for several good advices and\n B.~Crell, C.~Fuchs, H.~Narnhofer, and W.~Thirring for\nstimulating discussions.\nI am thankful for support to the Erwin Schr\\\"odinger Institute for\nMathematical Physics, Vienna, and, during the Workshop on Complexity,\nComputation and the physics of information,\nto the Isaac Newton Institute for\nMathematical Sciences, Cambridge, and to the European Science\nFoundation.\n\n\n\n\\begin{thebibliography}{}\n\n\\bibitem{fidel}\nA.~Uhlmann, {\\em Rep. Math. Phys.} {\\bf 9} 273 (1976);\nR.~Jozsa, {\\em J. Mod. Opt.} {\\bf 41} 2315 (1994);\nCh.~A.~Fuchs and C.~M.~Caves,\n{\\em Open Sys.} \\& {\\em Inf. Dyn.} {\\bf 3} 345 (1995).\n\n\\bibitem{AlUh:84}\nP.~M. Alberti and A.~Uhlmann.\n{Transition probabilities on $C^$- and $W^$-algebras}.\nIn: H.~Baumg{\\\"a}rtel, G.~La{\\ss}ner, A.~Pietsch, and A.~Uhlmann,\n editors, {\\em {Proceedings of the Second International Conference on Operator\n Algebras, Ideals, and Their Applications in Theoretical Physics, Leipzig\n 1983}}, pages 5--11, Leipzig, 1984. BSB B.G.Teubner Verlagsgesellschaft.\nTeubner-Texte zur Mathematik, Bd.67.\n\n\n\\bibitem{AU99}\nP.~M.~Alberti and A.~Uhlmann,\nOn Bures-Distance and $^$--Algebraic Transition Probability\nbetween Inner Derived Positive Linear Forms over W$^*$--Algebras.\nLU-ITP 1999/011, to appear in: Acta Applicandae Mathematicae\n\n\\bibitem{inf1}\nH.~Araki and G.~A.~Raggio, {\\em Lett. Math. Phys.} {\\bf 6} 237 (1982),\n\n\\bibitem{inf2}\nP.~M.~Alberti, {\\em Lett. Math. Phys.} {\\bf 7} 25 (1983).\n\n\n\\bibitem{Alberti}\nP.~M.~Alberti, private communication.\n\n\\bibitem{Roc70}\nR.~T.~Rockafellar,\n{\\it Convex Analysis},\nPrinceton University Press, 1970.\n\n\\bibitem{mpairs}\nP.~M.~Alberti, {\\em Wiss. Z. KMU Leipzig} {\\bf MNR 39} 579 (1990);\nP.~M.~Alberti, {\\em Z. Anal. Math.} {\\bf 11} 293 (1992);\nV.~Heinemann, Thesis, Leipzig 1991\n\n\\bibitem{M+O}\nA.~W.~Marshall and I.~Olkin: {\\em Inequalities: Theory of Majorization\nand Its Applications.} Mathematics in Science and Engineering/143,\nAcademic Press, New York 1979\n\n\\bibitem{A+U}\nP.~M.~Alberti and A.~Uhlmann: {\\em Stochasticity and Partial Order.}\nVEB Deutscher Verlag der Wissenschaften, Berlin 1981, {\\em and}\\\\\n(M.~Hazewinkel, ed.),\nMathematics and its Applications/9, D.~Reidel Publ.~Company,\nDordrecht 1982\n\n\\bibitem{PW75}\nW.~Pusz and L.~Woronowicz, {\\em Rep. Math. Phys.} {\\bf 8} 159 (1975)\n\n\n\n\\end{thebibliography}\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912114.extracted_bib", "string": "{fidel A.~Uhlmann, {\\em Rep. Math. Phys. {9 273 (1976); R.~Jozsa, {\\em J. Mod. Opt. {41 2315 (1994); Ch.~A.~Fuchs and C.~M.~Caves, {\\em Open Sys. \\& {\\em Inf. Dyn. {3 345 (1995)." }, { "name": "quant-ph9912114.extracted_bib", "string": "{AlUh:84 P.~M. Alberti and A.~Uhlmann. {Transition probabilities on $C^$- and $W^$-algebras. In: H.~Baumg{\\\"artel, G.~La{\\ssner, A.~Pietsch, and A.~Uhlmann, editors, {\\em {Proceedings of the Second International Conference on Operator Algebras, Ideals, and Their Applications in Theoretical Physics, Leipzig 1983, pages 5--11, Leipzig, 1984. BSB B.G.Teubner Verlagsgesellschaft. Teubner-Texte zur Mathematik, Bd.67." }, { "name": "quant-ph9912114.extracted_bib", "string": "{AU99 P.~M.~Alberti and A.~Uhlmann, On Bures-Distance and $^$--Algebraic Transition Probability between Inner Derived Positive Linear Forms over W$^$--Algebras. LU-ITP 1999/011, to appear in: Acta Applicandae Mathematicae" }, { "name": "quant-ph9912114.extracted_bib", "string": "{inf1 H.~Araki and G.~A.~Raggio, {\\em Lett. Math. Phys. {6 237 (1982)," }, { "name": "quant-ph9912114.extracted_bib", "string": "{inf2 P.~M.~Alberti, {\\em Lett. Math. Phys. {7 25 (1983)." }, { "name": "quant-ph9912114.extracted_bib", "string": "{Alberti P.~M.~Alberti, private communication." }, { "name": "quant-ph9912114.extracted_bib", "string": "{Roc70 R.~T.~Rockafellar, {Convex Analysis, Princeton University Press, 1970." }, { "name": "quant-ph9912114.extracted_bib", "string": "{mpairs P.~M.~Alberti, {\\em Wiss. Z. KMU Leipzig {MNR 39 579 (1990); P.~M.~Alberti, {\\em Z. Anal. Math. {11 293 (1992); V.~Heinemann, Thesis, Leipzig 1991" }, { "name": "quant-ph9912114.extracted_bib", "string": "{M+O A.~W.~Marshall and I.~Olkin: {\\em Inequalities: Theory of Majorization and Its Applications. Mathematics in Science and Engineering/143, Academic Press, New York 1979" }, { "name": "quant-ph9912114.extracted_bib", "string": "{A+U P.~M.~Alberti and A.~Uhlmann: {\\em Stochasticity and Partial Order. VEB Deutscher Verlag der Wissenschaften, Berlin 1981, {\\em and\\\\ (M.~Hazewinkel, ed.), Mathematics and its Applications/9, D.~Reidel Publ.~Company, Dordrecht 1982" }, { "name": "quant-ph9912114.extracted_bib", "string": "{PW75 W.~Pusz and L.~Woronowicz, {\\em Rep. Math. Phys. {8 159 (1975)" } ]
quant-ph9912115	The $\bdelta$-deformation of the Fock space	[ { "author": "Krzysztof Kowalski and Jakub Rembieli\\'nski" } ]	A deformation of the Fock space based on the finite difference replacement for the derivative is introduced. The deformation parameter is related to the dimension of the finite analogue of the Fock space.	[ { "name": "quant-ph9912115.tex", "string": "%Format: LaTeX\n\\documentstyle[fleqn,iopfts,12pt]{ioplppt}\n\\jl{1}\n\\textheight 8.8in\n%\\voffset=-2.5cm \\hoffset=.5cm\n\\voffset=.5cm \\hoffset=.5cm %PS\n\\eqnobysec\n\\begin{document}\n\\title{The $\\bdelta$-deformation of the Fock space}\n\\author{Krzysztof Kowalski and Jakub Rembieli\\'nski}\n\\address{\\it Department of Theoretical Physics, University\nof \\L\\'od\\'z, ul. Pomorska 149/153,\\\\ 90-236 \\L\\'od\\'z, Poland}\n\\begin{abstract}\nA deformation of the Fock space based on the finite difference\nreplacement for the derivative is introduced. The deformation\nparameter is related to the dimension of the finite analogue\nof the Fock space.\n\\end{abstract}\n\\pacs{02.20.Sv, 03.65.Fd}\n\\section{Introduction}\nIn recent years there has been a growing interest to discretizations\nof quantum mechanics based on the finite difference replacement for\nthe derivative. This is motivated by the well-known speculations\nthat below the Planck scale the conventional notions of space and\ntime break down and the new discrete structures are likely to\nemerge. This has echoes in the arguments put forward in string\ntheory and quantum gravity. We also mention the technical reasons\nfor the application of discrete models. Let us only recall the\nlattice gauge theories. As a matter of fact the connection has been shown\nin ref.\\ 1 between ordinary quantum mechanics on a equidistant\nlattice, where the the role of the derivative is played by the\nforward or backward discrete derivative, and {\\em q}-deformations\nutilizing the Jackson derivative, nevetheless no explicit form of\nthe corresponding deformation of the Fock space has been provided in\nref.\\ 1. On the other hand, there are indications \\cite{2} that\napproaches based on the central difference operator are more\nadequate for discretization of quantum mechanics than those using\nasymmetric forward or backward discrete derivatives.\n\nIn this paper we introduce a deformation of the Fock space, such\nthat the creation and annihilation operators are elements of the\nquotient field of the deformed Heisenberg algebra generated by the\nusual position operator and the central difference operator. The\ndeformation parameter $\\delta$ describing the fixed coordinate\nspacing is naturally related to the dimension of the\nfinite-dimensional space which can be regarded as an analogue\nof the Fock space. In the formal limit $\\delta\\to 0$ we arrive at\nthe infinite-dimensional space coinciding with the usual Fock space.\n\\newpage\n\\section{The $\\bdelta$-deformation of the Heisenberg algebra}\nAs mentioned in the introduction there are indications that\ndiscretizations of quantum mechanics should involve the central\ndifference operator such that\n\\begin{equation}\n%<2.1>\n\\Delta_\\delta f(x)=\\frac{f(x+\\delta)-f(x-\\delta)}{2\\delta}.\n\\end{equation}\nFurthermore, it seems to us that the most natural candidate for the\nposition operator in any discretized version of quantum mechanics is\nthe standard one of the form\n\\begin{equation}\n%<2.2>\n\\hat xf(x)=xf(x).\n\\end{equation}\nIn order to close the algebra satisfied by the operators\n$\\Delta_\\delta $ and $\\hat x$ we introduce the operator $I_\\delta $\ndefined by\n\\begin{equation}\n%<2.3>\nI_\\delta f(x)=\\frac{f(x+\\delta )+f(x-\\delta )}{2}.\n\\end{equation}\nIt follows that\n\\begin{equation}\n%<2.4>\n[\\Delta_\\delta ,\\hat x]=I_\\delta,\\qquad [I_\\delta ,\\hat\nx]=\\delta^2\\Delta_\\delta ,\\qquad [I_\\delta ,\\Delta_\\delta ]=0.\n\\end{equation}\nEvidently,\n\\begin{equation}\n%<2.5>\n\\Delta_\\delta=\\hbox{$\\scriptstyle{\\rm i}\\over\\delta $}\\sin\\delta\\hat p,\\qquad I_\\delta\n=\\cos\\delta\\hat p,\n\\end{equation}\nwhere $\\hat p=-{\\rm i}\\frac{d}{dx}$ is the usual momentum operator,\nso the contraction of the algebra (2.4) referring to $\\delta\\to 0$, is\nthe usual Heisenberg algebra\n\\begin{equation}\n%<2.6>\n[\\hat x,\\hat p]={\\rm i}I.\n\\end{equation}\nUsing (2.1), (2.2) and (2.3) we find easily the following Casimir\noperator for the algebra (2.4):\n\\begin{equation}\n%<2.7>\nI_\\delta^2-\\delta^2\\Delta_\\delta^2=1.\n\\end{equation}\n\nWe now discuss the representations of the algebra (2.4). We first observe\nthat (2.4) can be related to the following deformation of the $e(2)$\nalgebra (A.3) (see appendix):\n\\begin{equation}\n%<2.8>\n[J,U_\\delta ]=\\delta U_\\delta ,\n\\end{equation}\nwhere $U_\\delta $ is unitary, by means of the relations such that\n\\numparts\n\\begin{eqnarray}\n%<2.9>\n\\hat x &=& J,\\\\\n\\Delta_\\delta &=& -\\hbox{$\\scriptstyle1\\over2\\delta$}(U_\\delta\n-U_\\delta^\\dagger),\\\\\nI_\\delta &=& \\hbox{$\\scriptstyle1\\over2$}(U_\\delta + U_\\delta^\\dagger).\n\\end{eqnarray}\n\\endnumparts\nConsider the representation of (2.8) spanned by eigenvectors of the Hermitian\noperator $J$. Taking into account (2.8) and (A.5) we find\n\\begin{equation}\n%<2.10>\nJ\|j\\delta \\rangle=j\\delta \|j\\delta \\rangle.\n\\end{equation}\nHence, with the help of (2.8) we get\n\\begin{equation}\n%<2.11>\nU_\\delta \|j\\delta \\rangle= \|(j+1)\\delta \\rangle,\\qquad\nU_\\delta^\\dagger \|j\\delta \\rangle= \|(j-1)\\delta \\rangle.\n\\end{equation}\nEquations (2.9)--(2.11) taken together yield\n\\numparts\n\\begin{eqnarray}\n%<2.12>\n\\hat x \|j\\delta \\rangle &=& j\\delta \|j\\delta \\rangle,\\\\\n\\Delta_\\delta \|j\\delta \\rangle &=& -\\hbox{$\\scriptstyle1\\over2\\delta\n$}( \|(j+1)\\delta \\rangle- \|(j-1)\\delta \\rangle),\\\\\nI_\\delta \|j\\delta \\rangle &=& \\hbox{$\\scriptstyle1\\over2$}(\n\|(j+1)\\delta \\rangle+ \|(j-1)\\delta \\rangle).\n\\end{eqnarray}\n\\endnumparts\nLet us now specialize to the case with integer $j$ (see appendix).\nIn view of the form of eq.\\ (2.12a) it turns out that the operator\n$\\hat x$ really describes the position of a particle on equidistant\nlattice with the fixed coordinate spacing $\\delta $. The\ncompleteness condition satisfied by the vectors $ \|j\\delta\\rangle$\ncan be written as\n\\begin{equation}\n%<2.13>\n\\sum_{j=-\\infty}^{\\infty}\\delta \|j\\delta \\rangle\\langle j\\delta \|=I.\n\\end{equation}\nThe relation (2.13) leads to the realization of the\nabstract Hilbert space of states specified by the inner product\n\\begin{equation}\n%<2.14>\n\\langle f\|g\\rangle=\\sum_{j=-\\infty}^{\\infty}\\langle f\|j\\delta \\rangle\n\\langle j\\delta\n\|g\\rangle\\delta=\\sum_{j=-\\infty}^{\\infty}f^(j\\delta)g(j\\delta)\\delta ,\n\\end{equation}\nwhere $f(j\\delta)=\\langle j\\delta\|f\\rangle$. The action of operators\nin the representation (2.14) is of the following form:\n\\numparts\n\\begin{eqnarray}\n%<2.15>\n\\hat xf(j\\delta) &=& j\\delta f(j\\delta),\\\\\n\\Delta_\\delta f(j\\delta) &=& \\hbox{$\\scriptstyle1\\over2\\delta\n$}[f((j+1)\\delta)-f((j-1)\\delta)],\\\\\nI_\\delta f(j\\delta) &=& \\hbox{$\\scriptstyle1\\over2$}[f((j+1)\\delta)+\nf((j-1)\\delta)].\n\\end{eqnarray}\n\\endnumparts\n\nWe now study the representation generated by eigenvectors\n$\|\\varphi\\rangle_\\delta $, $\\varphi\\in{\\Bbb R}$, of the unitary operator \n$U_\\delta$ such that\n\\begin{equation}\n%<2.16>\nU_\\delta \|\\varphi\\rangle_\\delta =e^{-{\\rm i}\\delta\\varphi}\n\|\\varphi\\rangle_\\delta .\n\\end{equation}\nIt follows immediately from (2.9) and (2.16) that\n\\numparts\n\\begin{eqnarray}\n%<2.17>\n\\Delta_\\delta \|\\varphi\\rangle_\\delta &=& \\hbox{$\\scriptstyle{\\rm\ni}\\over \\delta $}\\sin\\delta \\varphi \|\\varphi\\rangle_\\delta ,\\\\\nI_\\delta \|\\varphi\\rangle_\\delta &=& \\cos\\delta \\varphi\n\|\\varphi\\rangle_\\delta .\n\\end{eqnarray}\n\\endnumparts\nThe completeness of the vectors $ \|\\varphi\\rangle_\\delta $ can be\nexpressed by\n\\begin{equation}\n%<2.18>\n\\frac{1}{2\\pi}\\int\\limits_{-\\frac{\\pi}{\\delta }}^{\\frac{\\pi}{\\delta }}\n \|\\varphi\\rangle_\\delta {}_\\delta\\langle \\varphi \|=I.\n\\end{equation}\nThe resolution of the identity (2.18) gives rise to the functional\nrepresentation of vectors\n\\begin{equation}\n%<2.19>\n\\langle f\|g\\rangle =\\frac{1}{2\\pi}\\int\\limits_{-\\frac{\\pi}{\\delta }}^{\\frac{\\pi}{\\delta }}\nf^(\\varphi)g(\\varphi)d\\varphi,\n\\end{equation}\nwhere $f(\\varphi)=\\langle\\varphi\|f\\rangle$, and we have omitted for\nbrevity the dependence of $f(\\varphi)$ on $\\delta $. The operators\nact in the representation (2.19) as follows:\n\\numparts\n\\begin{eqnarray}\n%<2.20>\n\\hat xf(\\varphi) &=& {\\rm i}\\frac{d}{d\\varphi}f(\\varphi),\\\\\n\\Delta_\\delta f(\\varphi) &=& \\hbox{$\\scriptstyle {\\rm i}\\over\\delta $}\n\\sin\\delta\\varphi f(\\varphi),\\\\\nI_\\delta f(\\varphi) &=& \\cos\\delta\\varphi f(\\varphi).\n\\end{eqnarray}\n\\endnumparts\nOur purpose now is to analyze the contraction $\\delta\\to0$ of the\nrepresentations (2.14) and (2.19) introduced above. Taking into\naccount (2.16), (2.13) and (2.11) we find that the passage from the\nrepresentation spanned by the vectors $ \|j\\delta\\rangle$ and that\ngenerated by the vectors $ \|\\varphi\\rangle_\\delta$ can be described\nby the kernel\n\\begin{equation}\n%<2.21>\n\\langle j\\delta \|\\varphi \\rangle_\\delta = e^{{\\rm i}j\\delta \\varphi}.\n\\end{equation}\nEquations (2.18) and (2.21) taken together yield\n\\begin{equation}\n%<2.22>\n\\langle j\\delta\|j'\\delta\\rangle =\n\\frac{1}{2\\pi}\\int\\limits_{-\\frac{\\pi}{\\delta }}^{\\frac{\\pi}{\\delta }}\ne^{{\\rm i}(j-j')\\delta\\varphi}d\\varphi =\n\\frac{\\sin\\pi(j-j')}{\\pi(j-j')\\delta}.\n\\end{equation}\nTherefore\n\\begin{equation}\n%<2.23>\n\\langle j\\delta\|j'\\delta\\rangle = \\hbox{$\\scriptstyle\n1\\over\\delta$}\\delta_{jj'},\n\\end{equation}\nwhenever $\\delta\\ne 0$. On the other hand, defining the continuum\nlimit as\n\\begin{equation}\n%<2.24>\nj\\to\\infty,\\qquad \\delta\\to0,\\qquad j\\delta = {\\rm const} = x,\n\\end{equation}\nand using the well known formula on the Dirac delta function\n\\begin{equation}\n%<2.25>\n\\delta(x) = \\lim_{\\alpha\\to\\infty}\\frac{1}{\\pi}\\frac{\\sin\\alpha x}{x},\n\\end{equation}\nwe find that (2.22) takes the form\n\\begin{equation}\n%<2.26>\n\\lim\\limits_{\\scriptstyle j,\\,j'\\to\\infty,\\,\\,\\delta\\to0\n\\atop\\scriptstyle j\\delta=x,\\,j'\\delta=x'}\\langle j\\delta\|j'\\delta\\rangle\n=\\delta(x-x').\n\\end{equation}\nHence, we get\n\\begin{equation}\n%<2.27>\n\\lim\\limits_{\\scriptstyle j\\to\\infty,\\,\\,\\delta\\to0\n\\atop\\scriptstyle j\\delta=x} \|j\\delta\\rangle = \|x\\rangle,\n\\end{equation}\nwhere $ \|x\\rangle$, $x\\in{\\Bbb R}$, are the usual normalized\neigenvectors of the position operator for a quantum mechanics on a\nreal line. This observation is consistent with the fact that for\n$\\delta\\to0$ the sum from (2.14) is simply the integral sum for the\nscalar product in $L^2({\\Bbb R},dx)$. By (2.15) and (2.24) it is\nalso evident that in the limit $\\delta\\to0$ we arrive at the\nHeisenberg algebra (2.6). We have thus shown that the\ncontraction referring to $\\delta\\to0$ of the representation of the\nalgebra (2.4) given by (2.14) and (2.15) coincides with the standard\ncoordinate $L^2$ representation of the Heisenberg algebra (2.6).\nAnalogously, we have\n\\begin{equation}\n%<2.28>\n{}_\\delta\\langle\\varphi \|\\varphi'\\rangle_\\delta =\n\\sum_{j=-\\infty}^{\\infty}e^{-{\\rm i}j\\delta(\\varphi -\\varphi')}\\delta.\n\\end{equation}\nTherefore,\n\\begin{equation}\n%<2.29>\n\\lim_{\\delta\\to0}{}_\\delta\\langle\\varphi\n\|\\varphi'\\rangle_\\delta=2\\pi\\delta(\\varphi -\\varphi'),\n\\end{equation}\nand we can identify\n\\begin{equation}\n%<2.30>\n\\lim_{\\delta\\to0} \|\\varphi\\rangle_\\delta = \\sqrt{2\\pi} \|p\\rangle,\n\\end{equation}\nwhere $p=\\varphi$, and $\|p\\rangle$, $p\\in{\\Bbb R}$, are the normalized\neigenvectors of the momentum operator. Further, in view of (2.20)\nthe case $\\delta\\to0$ really corresponds to the Heisenberg algebra\n(2.6). So the representation specified\nby (2.19) coincides in the limit $\\delta\\to0$ with the standard momentum\nrepresentation. We conclude that the introduced deformation works\nboth on the level of the algebra and the representation.\n\\section{The $\\bdelta$-deformation of the Heisenberg-Weyl algebra}\nIn this section we study the $\\delta$-deformation of the\nHeisenberg-Weyl algebra satisfied by the Bose creation and annihilation\noperators. Let us introduce the following family of operators:\n\\begin{equation}\n%<3.1>\nA(s)=\\hbox{$\\scriptstyle1\\over\\sqrt{2}$}[\\hat\nx+(1-\\delta^2s)\\Delta_\\delta I_\\delta^{-1}],\\qquad A^\\dagger (s)=\n\\hbox{$\\scriptstyle1\\over\\sqrt{2}$}[\\hat\nx-(1-\\delta^2s)\\Delta_\\delta I_\\delta^{-1}],\n\\end{equation}\nwhere $s=0,1,\\ldots.$ Clearly, these operators reduce to the\nstandard Bose creation and annihilation operators in the limit\n$\\delta\\to0$. We point out that then $A(s)$ and $A^\\dagger(s)$ do\nnot depend on $s$. Notice that in view of (2.9) $A^\\dagger(s)$ is really \nthe Hermitian conjugate of $A(s)$. It should also be noted that in\nthe representation (2.20) the action of the operator $I_{\\delta}^{-1}$ is\nsimply the multiplication by ${\\rm sec}\\,\\delta\\varphi$. We now\nseek the vectors $ \|s\\rangle$ and functions $\\alpha(s)$ and\n$\\beta(s)$, satisfying\n\\begin{equation}\n%<3.2>\nA(s) \|s\\rangle=\\alpha(s) \|s-1\\rangle,\\qquad A^\\dagger(s)\n\|s\\rangle=\\beta(s) \|s+1\\rangle,\\qquad s=0,1,\\ldots .\n\\end{equation}\nIn other words, we are looking for the $\\delta$-deformation of\nvectors spanning the occupation number representation. Using the\nfollowing form of the Casimir (2.7) which can be obtained with the\nhelp of (3.1):\n\\begin{equation}\n%<3.3>\nA(s+1)A^\\dagger(s)-A^\\dagger(s-1)A(s)=(1-\\delta^2s)I,\n\\end{equation}\nwhere $I$ is the unit operator, we get\n\\begin{equation}\n%<3.4>\n\\alpha(s+1)\\beta(s)-\\alpha(s)\\beta(s-1)=1-\\delta^2s.\n\\end{equation}\nHence, setting $\\alpha(0)=0$ and solving the elementary recurrence\n(3.4) we obtain\n\\begin{equation}\n%<3.5>\n\\alpha(s)\\beta(s-1)=s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1).\n\\end{equation}\nThe following solution of (3.5) consistent with the limit values\n$\\alpha(s)=\\sqrt{s}$ and $\\beta(s)=\\sqrt{s+1}$, corresponding to\n$\\delta=0$, when $ \|s\\rangle$ span the usual occupation number\nrepresentation can be guessed easily:\n\\begin{equation}\n%<3.6>\n\\alpha(s)=\\sqrt{s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)},\\qquad\n\\beta(s)=\\sqrt{s+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s+1)},\n\\end{equation}\nso we have\n\\begin{equation}\n%<3.7>\nA(s) \|s\\rangle=\\sqrt{s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)}\n\|s-1\\rangle,\\qquad A^\\dagger(s) \|s\\rangle=\\sqrt{s+1-\n\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s+1)} \|s+1\\rangle.\n\\end{equation}\nNow, by virtue of\n\\begin{equation}\n%<3.8>\n\\langle\ns\|A^\\dagger(s)A(s)\|s\\rangle=[s-\\hbox{$\\scriptstyle\\delta^2\\over2$}\ns(s-1)]\\langle s-1\|s-1\\rangle\\ge0,\n\\end{equation}\nwe see that the sequence of $s$ and thus $ \|s\\rangle$ should\ntruncate. The only possibility left is to set\n\\begin{equation}\n%<3.9>\n\\delta^2=\\frac{1}{s_{\\rm max}}.\n\\end{equation}\nIndeed, by (3.1) we then have\n\\begin{equation}\n%<3.10>\nA(s_{\\rm max})=A^\\dagger(s_{\\rm\nmax})=\\hbox{$\\scriptstyle1\\over\\sqrt{2}$}\\hat x.\n\\end{equation}\nUsing this and (3.7), we find\n\\begin{equation}\n%<3.11>\n \|s_{{\\rm max}}+1\\rangle= \|s_{{\\rm max}}-1\\rangle,\n\\end{equation}\nwhere $ \|s_{\\rm max}+1\\rangle=A^\\dagger(s_{\\rm max}) \|s_{\\rm max}\\rangle$.\nWe have thus shown that instead of $\\delta$ we can use the parameter\n$s_{{\\rm max}}$ exceeding by one the dimension of the system of\nvectors $\\{\|s\\rangle\\}_{0\\le s\\le s_{{\\rm max}}}$. Such systems for\n$s_{\\rm max}=1$, $s_{\\rm max}=2$ and so on, can be interpreted as a\nfinite-dimensional analogues of the usual infinite-dimensional Fock space. \nThe latter evidently refers to the case with $s_{\\rm max}=\\infty$, when\n$\\delta=0$.\n\nWe now discuss the algebra satisfied by the operators (3.1), that is\nthe $\\delta$-deformation of the Heisenberg-Weyl algebra. Taking\ninto account (3.7) we get\n\\numparts\n\\begin{eqnarray}\n%<3.12>\nA(s')&=&\\left[1-\\frac{\\delta^2(s-s')}{2(\\delta^2s-1)}\\right]A(s)+\n\\frac{\\delta^2(s-s')}{2(\\delta^2s-1)}A^\\dagger(s),\\\\\nA^\\dagger(s')&=&\\frac{\\delta^2(s-s')}{2(\\delta^2s-1)}A(s)+\n\\left[1-\\frac{\\delta^2(s-s')}{2(\\delta^2s-1)}\\right]A^\\dagger(s),\n\\qquad s<s_{\\rm max}.\n\\end{eqnarray}\n\\endnumparts\nMaking use of (2.4), (3.1), (3.12) and the following form of the Casimir\n(2.7), which can be easily derived with the help of (3.1):\n\\begin{equation}\n%<3.13>\n\\delta^2[A(s)-A^\\dagger(s)]^2=2(1-\\delta^2s)(1-I_\\delta^{-2}),\n\\end{equation}\nwe arrive at the commutation relations such that\n\\numparts\n\\begin{eqnarray}\n%<3.14>\n&&\\fl [A(s),A^\\dagger(s')]=[1-\\hbox{$\\scriptstyle\\delta\n^2\\over2$}(s+s')]I_\\delta^{-2},\\\\\n&&\\fl [A(s),A(s')]=[A^\\dagger(s'),A^\\dagger(s)]=\\hbox{$\\scriptstyle\\delta\n^2\\over2$}(s'-s)I_\\delta^{-2},\\qquad s,s'\\le s_{\\rm max},\\\\\n&&\\fl [A(s),I_\\delta^{-2k}]=[A^\\dagger(s),I_\\delta^{-2k}]=[A(s_{\\rm max}),\nI_\\delta^{-2k}]=k\\frac{\\delta^2}{1-\\delta^2s}B_k(s),\\quad s<s_{\\rm max},\\\\\n&&\\fl [A(s),B_k(s')]=[A^\\dagger(s),B_k(s')]\\nonumber\\\\\n&&\\fl\\quad{}=2k(1-\\delta^2s')I_\\delta^{-2k}-\n(2k+1)(1-\\delta^2s')I_\\delta^{-2(k+1)},\\\\\n&&\\fl [B_k(s),I_\\delta^{-2l}]=[I_\\delta^{-2k},I_\\delta^{-2l}]=\n[B_k(s),B_l(s')]=0,\\,\\, s,s'\\le s_{\\rm max},\\,\\, k,l=1,2,\\ldots,\n\\end{eqnarray}\n\\endnumparts\nwhere $B_k(s)=[A(s)-A^\\dagger(s)]I_\\delta^{-2k}$.\nWe remark that due to the commutator (3.14d) the algebra (3.14) is infinite\ndimensional. It should also be noted that in view of the following\nrelation:\n\\begin{equation}\n%<3.15>\nA(s)=(1-\\hbox{$\\scriptstyle\\delta^2s\\over2$})A(0)+\n\\hbox{$\\scriptstyle\\delta^2s\\over2$}A^\\dagger(0),\\qquad 0\\le s\\le\ns_{\\rm max},\n\\end{equation}\nwhich is an immediate consequence of (3.1), $A(s)$, $A^\\dagger(s)$ and\n$B_k(s)$ can be regarded as a discrete curve in the algebra\ngenerated by $A(0)$, $A^\\dagger(0)$, $I_\\delta^{-2k}$ and $B_k(0)$\nof the form\n\\numparts\n\\begin{eqnarray}\n%<3.16>\n&&[A(0),A^\\dagger(0)]=I_\\delta^{-2},\\\\\n&&[A(0),I_\\delta^{-2k}]=[A^\\dagger(0),I_\\delta^{-2k}]=k\\delta^2B_k(0),\\\\\n&&[A(0),B_k(0)]=[A^\\dagger(0),B_k(0)]=2kI_\\delta^{-2k}-\n(2k+1)I_\\delta^{-2(k+1)},\\\\\n&&[B_k(0),I_\\delta^{-2l}]=[I_\\delta^{-2k},I_\\delta^{-2l}]=[B_k(0),B_l(0)]=0,\n\\qquad k,l=1,2,\\ldots .\n\\end{eqnarray}\n\\endnumparts\nOf course, both (3.14) and (3.16) reduce to the Heisenberg-Weyl algebra\nin the limit $\\delta\\to0$, that is $s_{\\rm max}\\to\\infty$.\n\\section{The $\\bdelta$-deformation of the Fock space}\nWe now discuss the $\\delta$-deformation of the Fock space expressed\nby (3.7) in a more detail. We first observe that the generation of\nthe states $\|s\\rangle$, with $s\\ge1$, from the ``vacuum vector''\n$\|0\\rangle$ can be described with the help of the second equation of\n(3.7) by\n\\begin{equation}\n%<4.1>\n\|s\\rangle=\\left(\\prod\\limits_{s'=0}^{s-1}\n\\frac{1}{\\sqrt{s'+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}s'(s'+1)}}\n\\right)A^\\dagger(s-1)\\cdots A^\\dagger(1)A^\\dagger(0) \|0\\rangle,\\quad\n0<s\\le s_{\\rm max}.\n\\end{equation}\nThe vectors $\|s\\rangle$ are not orthonormal. In fact, using (3.12a)\nwith $s'=s+1$, and (3.7) we find\n\\begin{eqnarray}\n%<4.2>\n&&\\fl\\fl\\,\\,\\delta^2\\sqrt{s+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s+1)}\n\\sqrt{s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)}\\langle s-1\|s+1\\rangle\n=-2(\\delta^2s-1)[s+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s+1)]\\langle\ns\|s\\rangle\\nonumber\\\\\n&&\\fl\\fl\\quad{}+[\\delta^2+2(\\delta^2s-1)][s+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}\ns(s+1)]\\langle s+1\|s+1\\rangle.\n\\end{eqnarray}\nFurther, calculating the expectation value of the Casimir (3.13) in\nthe state $\|s\\rangle$ with the use of (3.14a) for $s=s'$, and taking\ninto account (4.2), we obtain\n\\begin{eqnarray}\n%<4.3>\n&&\\fl\\delta^2\\sqrt{s+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s+1)}\n\\sqrt{s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)}\\langle s+1\|s-1\\rangle\n=2(\\delta^2s-1)\n[s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)]\\langle\ns\|s\\rangle\\nonumber\\\\\n&&\\fl\\quad{} + [\\delta^2-2(\\delta^2s-1)][s-\\hbox{$\\scriptstyle\\delta^2\\over2$}\ns(s-1)]\\langle s-1\|s-1\\rangle.\n\\end{eqnarray}\nEquating right-hand sides of (4.2) and (4.3) we finally arrive at the\nfollowing recursive formula on the squared norm of $\|s\\rangle$:\n\\begin{eqnarray}\n%<4.4>\n&&2(\\delta^2s-1)(2s+1-\\delta^2s^2)\\langle s\|s\\rangle+[\\delta^2-\n2(\\delta^2s-1)][s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)]\n\\langle s-1\|s-1\\rangle\\nonumber\\\\\n&&\\quad{}-[\\delta^2+2(\\delta^2s-1)][s+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}\ns(s+1)]\\langle s+1\|s+1\\rangle=0,\\qquad s\\le s_{\\rm max}.\n\\end{eqnarray}\nA straightforward calculation shows that the recurrence (4.4) can be\nwritten in a more convenient form such that\n\\numparts\n\\begin{eqnarray}\n%<4.5>\n&&\\fl\\langle 1\|1\\rangle=\\frac{2}{2-\\delta^2}\\langle 0\|0\\rangle,\\\\\n&&\\fl\\langle s\|s\\rangle=\\frac{\\delta^2}{[2(\\delta^2s-1)-\\delta ^2][s-\n\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)]}\\sum_{s'=0}^{s-2}\n(\\delta ^2s'-1)\\langle s'\|s'\\rangle\\nonumber\\\\\n&&\\fl\\quad{}+\n\\left(1+\\frac{\\delta^2[\\delta^2(s-1)-1]}{[2(\\delta^2s-1)-\n\\delta^2][s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)]}\\right)\n\\langle s-1\|s-1\\rangle,\\qquad 2\\le s\\le s_{\\rm max}.\n\\end{eqnarray}\n\\endnumparts\nFinally, eqs.\\ (3.12a) and (3.7) taken together yield\n\\begin{eqnarray}\n%<4.6>\n&&\\fl\\fl\\,\\sqrt{s-\\hbox{$\\scriptstyle\\delta^2\\over2$}s(s-1)}\\langle s\|s'\\rangle\n=\\left[1-\\frac{\\delta^2(s'-s+1)}{2(\\delta^2s'-1)}\\right]\n\\sqrt{s'-\\hbox{$\\scriptstyle\\delta^2\\over2$}s'(s'-1)}\\langle s-1\|s'-1\\rangle\n\\nonumber\\\\\n&&\\fl\\fl\\!{}+\\frac{\\delta^2(s'-s+1)}{2(\\delta^2s'-1)}\n\\sqrt{s'+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}s'(s'+1)}\\langle s-1\n\|s'+1\\rangle,\\!\\!\\!\\quad 0<s\\le s_{\\rm max},\\!\\!\\!\\!\\!\\quad 0\\le s'\n<s_{\\rm max}.\n\\end{eqnarray}\nThe equations (4.5) and (4.6) form the closed system which enables to\ncalculate the inner product $\\langle s\|s'\\rangle$ for arbitrary\n$s,s'\\le s_{\\rm max}$. In particular, utilizing the relation\n\\begin{equation}\n%<4.7>\n\\langle s\|s+1\\rangle=0,\\qquad s\\le s_{\\rm max},\n\\end{equation}\nimplied by (4.6) and using recursively (4.6) we find that\n\\begin{equation}\n%<4.8>\n\\langle s\|s'\\rangle=0, \\qquad s,s'\\le s_{\\rm max},\n\\end{equation}\nwhere $s$ is even and $s'$ is odd.\n\nWe finally discuss the concrete realization of the introduced\n$\\delta$-deformation of the abstract Fock space in the\nrepresentation (2.19). On using (2.20) and (3.7) we arrive at the\nfollowing system:\n\\numparts\n\\begin{eqnarray}\n%<4.9>\n\\left[\\frac{d}{d\\varphi}+(1-\\delta^2s)\\frac{1}{\\delta}{\\rm tg}\\delta\n\\varphi\\right]f_s(\\varphi)&=&-{\\rm i}\\sqrt{2}\\sqrt{s-\\hbox{$\\scriptstyle\\delta^2\\over2$}\ns(s-1)}\\,f_{s-1}(\\varphi),\\\\\n\\left[\\frac{d}{d\\varphi}-(1-\\delta^2s)\\frac{1}{\\delta}{\\rm tg}\\delta\n\\varphi\\right]f_s(\\varphi)&=&-{\\rm i}\\sqrt{2}\\sqrt{s+1-\\hbox{$\\scriptstyle\\delta^2\\over2$}\ns(s+1)}\\,f_{s+1}(p),\n\\end{eqnarray}\n\\endnumparts\nwhere $f_s(\\varphi)=\\langle \\varphi\|s\\rangle$. We remark that the\nsystem (4.9) is the special case of the more general one\n\\numparts\n\\begin{eqnarray}\n%<4.10>\n\\left[\\frac{d}{d\\varphi}+k(s,\\varphi)\\right]f_s(\\varphi)&=&-{\\rm i}\\mu(s)\nf_{s-1}(\\varphi),\\\\\n\\left[\\frac{d}{d\\varphi}-k(s,\\varphi)\\right]f_s(\\varphi)&=&-{\\rm i}\\nu(s)\nf_{s+1}(\\varphi).\n\\end{eqnarray}\n\\endnumparts\nIt can be easily checked that (4.10) is equivalent to\n\\numparts\n\\begin{eqnarray}\n%<4.11>\n\\left[\\frac{d}{dx}+k(s,x)\\right]y_s(x)&=&\\mu(s)\ny_{s-1}(x),\\\\\n\\left[-\\frac{d}{dx}+k(s,x)\\right]y_s(x)&=&\\nu(s)\ny_{s+1}(x).\n\\end{eqnarray}\n\\endnumparts\nThe system (4.11) was studied by Jannussis {\\em et al} \\cite{3} in the\ncontext of the generalization of the Infeld-Hull method of factorization\nin the case of the harmonic oscillator. Analyzing the compatibility of the\ntwo second order differential equations implied by (4.11) they showed\nthat besides the periodic solution there exists the following one:\n\\begin{equation}\n%<4.12>\nk(s,x)=a{\\rm ctg}(ax+\\theta)\\,s-\\frac{b}{a}{\\rm ctg}(ax+\\theta)+\n\\frac{c}{\\sin(ax+\\theta)},\n\\end{equation}\nprovided\n\\begin{equation}\n%<4.13>\n\\mu(s)\\nu(s-1)=-a^2s(s-1)+2bs+\\lambda,\n\\end{equation}\nwhere $a$, $b$, $c$, $\\theta$ and $\\lambda$ are are arbitrary\nconstants. A look at (4.12), (4.13), (4.9) and (3.5) is enough to\nconclude that the actual treatment refers to the case with\n$a=\\delta$, $b=1$, $c=0$, $\\theta=\\pi/2$ and $\\lambda=0$. We point\nout that within the formalism introduced herein the second order\nequations implied by (4.9) are simply the realization of the\nabstract equations\n\\numparts\n\\begin{eqnarray}\n%<4.14>\nA(s+1)A^\\dagger(s)\|s\\rangle &=& \\beta(s)\\alpha(s+1)\|s\\rangle,\\\\\nA^\\dagger(s-1)A(s)\|s\\rangle &=& \\alpha(s)\\beta(s-1)\|s\\rangle.\n\\end{eqnarray}\n\\endnumparts\nin the representation (2.19). The compatibility of the eqs.\\ (4.14)\nis ensured by the Casimir (3.3). In this sense the actual approach\ncan be interpreted as an abstract form of the Infeld-Hull factorization\nmethod.\n\nWe now return to (4.9). Using (4.9a) and the limit\n\\begin{equation}\n%<4.15>\n\\lim\\limits_{\\delta\\to0}\\,(\\cos\\delta \\varphi)^{\\frac{1}{\\delta^2}}=\ne^{-\\frac{\\varphi^2}{2}},\n\\end{equation}\nwe find\n\\begin{equation}\n%<4.16>\nf_0(\\varphi)=\\pi^{-\\frac{1}{4}}(\\cos\\delta \\varphi)^{\\frac{1}{\\delta^2}}.\n\\end{equation}\nFurthermore, utilizing (4.9b) and\n\\begin{equation}\n%<4.17>\n\\frac{d}{d\\varphi}(\\cos\\delta \\varphi)^{\\frac{1}{\\delta^2}}=-\\frac{{\\rm\ntg}\\delta \\varphi}{\\delta }(\\cos\\delta\n\\varphi)^{\\frac{1}{\\delta^2}},\\qquad \\frac{d}{d\\varphi}\\left(\\frac{{\\rm\ntg}\\delta \\varphi}{\\delta }\\right)=1+\\delta^2\\left(\\frac{{\\rm\ntg}\\delta \\varphi}{\\delta }\\right)^2,\n\\end{equation}\nwe get\n\\begin{equation}\n%<4.18>\nf_s(\\varphi)=\\frac{\\pi^{-\\frac{1}{4}}(-{\\rm i})^s}{(\\sqrt{2})^s}\n\\left(\\prod\\limits_{s'=0}^{s-1}\\frac{1}{\\sqrt{s'+1-\n\\hbox{$\\scriptstyle\\delta^2\\over2$}s'(s'+1)}}\\right)\nH^{(\\delta)}_s\\left(\\frac{{\\rm tg}\\delta \\varphi}{\\delta}\\right)\n(\\cos\\delta \\varphi)^{\\frac{1}{\\delta^2}},\n\\end{equation}\nwhere $\\quad 1\\le s\\le s_{\\rm max}$, and $H^{(\\delta)}_s(x)$ are\nthe polynomials satisfying the recurrence\n\\begin{eqnarray}\n%<4.19>\nH^{(\\delta)}_{s+1}(x)&=&(2-\\delta^2s)xH^{(\\delta)}_s(x)-(1+\\delta^2x^2)\nH^{(\\delta ){}'}_s(x),\\nonumber\\\\\nH^{(\\delta )}_0(x)&=&1,\n\\end{eqnarray}\nwhere the prime designates the differentiation with respect to $x$. Of\ncourse, $H^{(\\delta)}_s(x)$ are simply the $\\delta$-deformation\nof the usual Hermite polynomials refering to the limit $\\delta\\to0$,\ni.e.\\ $s_{\\rm max}\\to\\infty$. The first few $\\delta$-deformed\nHermite polynomials are of the form\n\\begin{eqnarray}\n%<4.20>\n\\fl H^{(\\delta)}_0(x)&=&1,\\nonumber\\\\\n\\fl H^{(\\delta)}_1(x)&=&2x,\\nonumber\\\\\n\\fl H^{(\\delta)}_2(x)&=&4(1-\\delta^2)x^2-2,\\nonumber\\\\\n\\fl H^{(\\delta)}_3(x)&=&8(1-\\delta^2)(1-2\\delta^2)x^3-12(1-\\delta^2)x,\\nonumber\\\\\n\\fl H^{(\\delta)}_4(x)&=&16(1-\\delta^2)(1-2\\delta^2)(1-3\\delta^2)x^4\n-48(1-\\delta ^2)(1-2\\delta ^2)x^2+12(1-\\delta^2).\n\\end{eqnarray}\nAs with the standard Hermite polynomials the general formula on the\n$\\delta$-deformed ones can be derived such that\n\\begin{eqnarray}\n%<4.21>\n\\fl H^{(\\delta)}_0(x)&=&1,\\nonumber\\\\\n\\fl H^{(\\delta)}_s(x)&=&\\sum_{j=0}^{\\left[\\hbox{$\\scriptstyle\ns\\over2$}\\right]}\n(-1)^j\\frac{s!}{j!(s-2j)!}2^{s-2j}\\left[\\prod\\limits_{s'=0}^{s-j-1}\n(1-\\delta^2s')\\right]x^{s-2j},\\qquad 1\\le s\\le s_{\\rm max},\n\\end{eqnarray}\nwhere $[y]$ is the biggest integer in $y$.\n\nWe finally write down\nthe following formula on the matrix elements $\\langle s\|s'\\rangle$\nimplied by (2.18) and (4.18):\n\\begin{equation}\n%<4.22>\n\\langle s\|s'\\rangle =\n\\frac{1}{2\\pi}\\int\\limits_{-\\frac{\\pi}{\\delta}}^{\\frac{\\pi}{\\delta}}\nf_s^(\\varphi)f_{s'}(\\varphi)d\\varphi,\n\\end{equation}\nwhere $f_s(\\varphi)$ is given by (4.16) and (4.18). The calculation\nof the integral from (4.22) for arbitrary $s,\\,s'$ seems to be more\ncomplicated than the solution of the recurrences (4.5) and (4.6).\nIt should be noted however that (4.22) enables to calculate the\nsquared norm of the ``vacuum vector'' $ \|0\\rangle$ parametrizing\nsolutions of (4.5) and (4.6). Namely, we find\n\\begin{equation}\n%<4.23>\n\\langle 0\|0\\rangle =\n\\frac{1}{2\\pi^\\frac{3}{2}}\\int\\limits_{-\\frac{\\pi}{\\delta}}^{\\frac{\\pi}\n{\\delta}}\n(\\cos\\delta\\varphi)^\\frac{2}{\\delta^2}d\\varphi = \\sqrt{\\frac{s_{\\rm\nmax}}{\\pi}}\\,\\frac{(2s_{\\rm max}-1)!!}{(2s_{\\rm max})!!}=\n\\frac{\\sqrt{s_{\\rm\nmax}}}{\\pi}\\,\\frac{\\Gamma(s_{\\rm max}+\\frac{1}{2})}{\\Gamma(s_{\\rm\nmax}+1)},\n\\end{equation}\nwhere $\\delta^2s_{\\rm max}=1$ and $\\Gamma(x)$ is the gamma\nfunction.\n\\section{Conclusion}\nWe have introduced in this work the deformation of the Fock space\nbased on the utilization of the central difference operator instead\nof the usual derivative. It should be mentioned that there exist\nalternative approaches for discretization of quantum mechanics\nrelying on finite difference representations of the usual Heisenberg\n\\cite{4} or Heisenberg-Weyl algebra \\cite{5}. Nevertheless, the general problem\nwith them is the interpretation of the nonequivalence of the\nobtained representations of the canonical commutation relations and\nthe standard Schr\\\"odinger one. Some problems with the spectrum of\noperators within such approaches have been also reported \\cite{4}.\nWe also recall the discretization of the harmonic oscillator\nintroduced in \\cite{6} relying on the replacement of the Hermite\npolynomials with the Kravchuk polynomials in a discrete variable\nas well as the finite-dimensional counterpart of the Fock space\nspanned by the eigenvectors of the phase operator discussed in\n\\cite{7}. In analogy with the actual treatment in both approaches\ntaken up in \\cite{6} and \\cite{7} the standard infinite-dimensional\nFock space refers to the formal limit $N\\to\\infty$, where $N$ is \ndimension of the finite-dimensional discrete version of the Fock space.\nMoreover, in the case with the discretization described in \\cite{6} one can\nrecognize a counterpart of the parameter $\\delta$ specified by (3.9)\nsuch that $\\delta\\simeq N^{-\\frac{1}{2}}$. Nevertheless, besides of\nthose similarities we have also serious differences. For example,\nin opposition to the operators (3.1) the generalizations of the Bose\noperators introduced in \\cite{6} do not depend on the index\nlabelling the basis of the finite-dimensional analogue of the\nFock space. On the other hand, the alternatives to the number\nstates discussed in \\cite{7} form the orthonormal set. This is not\nthe case for the states $ \|s\\rangle$ described herein.\nLast but not least we point out that besides of quantum mechanics the\nresults of this paper would be of importance in the theory of\ndifferential equations. We only recall the abstract form of the\nInfeld-Hull method of factorization described by the equations (4.14)\nand (3.3).\n\\appendix\n\\section{}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\nHere we briefly discuss the basic properties of the $e(2)$ algebra.\nConsider the $e(2)$ algebra\n\\begin{equation}\n%<a.1>\n[J,X]={\\rm i}Y,\\qquad [J,Y]=-{\\rm i}X,\\qquad [X,Y]=0.\n\\end{equation}\nThe Casimir operator for (A.1) is of the following form:\n\\begin{equation}\n%<a.2>\nX^2+Y^2=r^2.\n\\end{equation}\nMaking use of (A.1) and (A.2) we arrive at the following form of the\nalgebra (A.1):\n\\begin{equation}\n%<a.3>\n[J,U]=U,\n\\end{equation}\nwhere\n\\begin{equation}\n%<a.4>\nU=\\hbox{$\\scriptstyle 1\\over r$}(X+{\\rm i}Y)\n\\end{equation}\nis unitary. Consider the eigenvalue equation\n\\begin{equation}\n%<a.5>\nJ \|j\\rangle = j \|j\\rangle.\n\\end{equation}\nFrom equations (A.3) and (A.5) it follows that the operators $U$ and\n$U^\\dagger$ act on the vectors $ \|j\\rangle$ as the rising and\nlowering operator, respectively, that is\n\\begin{equation}\n%<a.6>\nU \|j\\rangle = \|j+1\\rangle,\\qquad U^\\dagger \|j\\rangle = \|j-1\\rangle.\n\\end{equation}\nTaking into account (A.6) we find that the whole basis {$\n\|j\\rangle$} of the Hilbert space of states can be generated from the\nunique ``vacuum vector'' $ \|j_0\\rangle$, where $j_0\\in[0,1]$. The\nnon-equivalent irreducible representations of the commutation\nrelations (A.3) are labelled by different $j_0$. We remark that the\nalgebra (A.3) is the most natural for the study of a quantum\nparticle on a circle \\cite{8}. In such a case $J$ represents the angular\nmomentum and the unitary operator $U$ describes the position of a\nparticle on a unit circle. We now demand the time-reversal\ninvariance of the algebra (A.3). Having in mind the interpretation\nof $J$ as the angular momentum this leads to\n\\begin{eqnarray}\n%<a.7>\nTJT^{-1} &=& -J,\\\\\nTUT^{-1} &=& U^{-1},\n\\end{eqnarray}\nwhere $T$ is the anti-unitary operator of time inversion. Using\n(A.5)--(A.8) we obtain\n\\begin{equation}\n%<a.9>\nT \|j\\rangle = \|-j\\rangle.\n\\end{equation}\nAs an immediate consequence of (A.9), we find that $T$ is well\ndefined on the Hilbert space of states generated by the vectors $\n\|j\\rangle$ if and only if the spectrum of $J$ is symmetric with\nrespect to zero. Hence, in view of (A.6) the only possibility left\nis $j_0=0$ or $j_0=\\frac{1}{2}$. Obviously, $j_0=0$\n($j_0=\\frac{1}{2}$) implies integer (half-integer) eigenvalues $j$.\n\nHowever, in this work we interpret $J$ as the\nposition operator for a quantum particle on a lattice. Accordingly,\nthe operator $T$ from (A.7) should be replaced with a unitary parity\noperator $P$ and the invariance of (A.3) under parity transformation\ndemanded. In that case the relations (A.7) and (A.8) (with $T$\nreplaced by $P$) and their consequences ($j$ integer or half-integer)\nremain unchanged.\n\\newpage\n\\section*{References}\n\\begin{thebibliography}{VV}\n\\bibitem{1}\nDimakis A and M\\\"uller-Hoissen F 1992 {\\em Phys. Lett. B} {\\bf 295} 242 \n\\bibitem{2}\nG\\'orski A Z and Szmigielski J 1998 {\\em J. Math. Phys.} {\\bf 39} 545 \n\\bibitem{3}\nJannussis A, Karagannis G, Kanagopoulos P and Brodimos G 1983 {\\em Lett. Nuovo\nCim.} {\\bf 38} 155 \n\\bibitem{4}\nDimakis A, M\\\"uller-Hoissen F and Striker T 1995 {\\em Umbral calculus,\ndiscretization, and quantum mechanics on a lattice}, report GOET-TP\n96/95 \n\\bibitem{5}\nSmirnov Y M and Turbiner A V 1995 {\\em Modern Physics Letters A} {\\bf 10}\n1795\n\\bibitem{6} \nAtakishiyev N M and Suslov S K 1991 {\\em Theor. Math. Phys.} {\\bf\n85} 1055\n\\bibitem{7}\nPegg D T and Barnett S M 1988 {\\em Europhys. Lett.} {\\bf 6} 483\n\\bibitem{8}\nKowalski K, Rembieli\\'nski J and Papaloucas L C 1996 {\\em J. Phys. A:Math.\nGen.} {\\bf 29} 4149 \n\\end{thebibliography}\n\\end{document}\n\n\n" } ]	[ { "name": "quant-ph9912115.extracted_bib", "string": "{1 Dimakis A and M\\\"uller-Hoissen F 1992 {\\em Phys. Lett. B {295 242" }, { "name": "quant-ph9912115.extracted_bib", "string": "{2 G\\'orski A Z and Szmigielski J 1998 {\\em J. Math. Phys. {39 545" }, { "name": "quant-ph9912115.extracted_bib", "string": "{3 Jannussis A, Karagannis G, Kanagopoulos P and Brodimos G 1983 {\\em Lett. 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quant-ph9912116	Complete separability and Fourier representations of n-qubit states	[ { "author": "Arthur O. Pittenger \\dag \\footnote[3]{Present address:The Centre of Quantum Computation" }, { "author": "Clarendon Laboratory" }, { "author": "Oxford University" } ]	Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and sufficient conditions for full separability for a particular set of $n$-qubit states whose densities all satisfy the Peres condition.	[ { "name": "quant-ph9912116.tex", "string": "\n\\documentstyle[pra,aps]{revtex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%TCIDATA{OutputFilter=LATEX.DLL}\n%TCIDATA{LastRevised=Sun Dec 19 22:10:17 1999}\n%TCIDATA{<META NAME=\"GraphicsSave\" CONTENT=\"32\">}\n%TCIDATA{CSTFile=revtex.cst}\n\n\\draft\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{corollary}{Corollary}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\n\\begin{document}\n\\title{Complete separability and Fourier representations of n-qubit \nstates}\n\\author{Arthur O. Pittenger \\dag \\footnote[3]{Present address:The Centre of\nQuantum\nComputation, Clarendon Laboratory,\n Oxford University} and Morton H. Rubin \\ddag}\n\\address{\\dag Department of Mathematics and Statistics,\nUniversity of Maryland, Baltimore County,\nBaltimore, MD 21228-5398}\n\\address{\\ddag Department of Physics,\nUniversity of Maryland, Baltimore County,\nBaltimore, MD 21228-5398}\n\\date{December 22, 1999}\n\\maketitle\n\n\\begin{abstract}\nNecessary conditions for separability are most easily expressed in the\ncomputational basis, while sufficient conditions are most conveniently\nexpressed in the spin basis. We use the Hadamard matrix to define the\nrelationship between these two bases and to emphasize its \ninterpretation as a Fourier transform. We then prove a general sufficient\ncondition for complete\nseparability in terms of the spin coefficients and give necessary and\nsufficient conditions for the complete separability of a class of\ngeneralized Werner densities. As a further application of the theory, we\ngive necessary and sufficient conditions for full separability for a\nparticular set of $n$-qubit states whose densities all satisfy the Peres\ncondition. \n\\end{abstract}\n\n\\pacs{03.65.Bz, 03.65.Ca, 03.65.Hk}\n\nThe study of non-classical correlations has led to a number of suprising\nresults arising from the existence of entangled states of separated\nsubsystems \\cite{EPR,GHZ,Bell,teleportation}. This has led to renewed\ninterest in the study of entanglement itself \\cite{niel,vidal} as well as in\napplications such as quantum information theory and quantum communication \n\\cite{Bennett1}. Before one can effectively use entanglement, it is \nnecessary to determine if a given state $\\rho $ actually has entangled subsystems.\nIt is this ``separability'' problem with which we concern ourselves in\nthis paper.\n\nThere exists a useful, general necessary condition for separability\\cite\n{Peres} and a theoretical necessary and sufficient condition \\cite{H1}, but\nno operational necessary and sufficient conditions, and as a result\nattention has tended to focus on classes of densities \\cite{Lew,Dur,Schack}.\nIn this paper, we record a useful variant of the Peres (necessary) condition\nand prove a new sufficient condition for full separability of mixed states\nof a system composed of n-qubits. To do that, we highlight the roles of the\ncomputational basis, composed of projections and raising and lowering\noperators, and the spin basis, composed of the identity and the real Pauli\nmatrices. We derive a change of basis formula which facilitates changing\nfrom one basis to another and apply these insights to obtain the general\nsufficient condition for separability and to obtain both necessary and\nsufficient conditions for a particular class of states satisfying the Peres\ncondition. In a separate paper we will show how these ideas generalize to\nhigher dimensional states.\n\nA state defined on the Hilbert space ${\\cal {H}}_{A_{1}}\\otimes {\\cal {H}}%\n_{A_{2}}$ is said to be separable if it can be written as \n\\begin{equation}\n\\rho =\\sum_{a}p(a)\\rho \\left( a\\right) =\\sum_{a}p(a)\\rho ^{A_{1}}(a)\\otimes\n\\rho ^{A_{2}}(a), \\label{sep rho}\n\\end{equation}\nwhere $\\rho \\left( a\\right) =\\rho ^{A_{1}}(a)\\otimes \\rho ^{A_{2}}(a)$ with $%\n\\rho ^{A_{k}}\\left( a\\right) $ a state on ${\\cal {H}}_{A_{k}}$, and the $%\np(a) $ are positive numbers that sum to one. Peres showed that a necesary\ncondition for a density matrix of such a bipartite system to be separable is\nfor its partial transpose to be a density matrix \\cite{Peres}, where the\npartial transpose $\\rho ^{T_{1}}$ of $\\rho $ is defined by $\\langle\na\\;b\|\\rho ^{T_{1}}\|a^{\\prime }\\;b^{\\prime }\\rangle =\\langle a^{\\prime\n}\\;b\|\\rho \|a\\;b^{\\prime }\\rangle $. ($\\rho ^{T_{2}}$ is defined\nanalogously.) For $2\\otimes 2$ and $2\\otimes 3$ systems this condition is\nalso sufficient \\cite{H1}. The Peres condition is basis\nindependent, but to facilitate applications we derive a weaker\nversion which is most usefully expressed in the computational basis. This \nresult is based on the positivity of the subsystem states, the assumed degree\nof separability and the Cauchy-Schwarz inequality. First, we introduce some\nnotation. Let $j$ denote an n-long binary index vector and let $\\tilde{0}$\nand $\\tilde{1}$ stand for the n-bit numbers consisting of all $0^{\\prime }s$\nand all $1^{\\prime }s$, respectively. The binary complement of $j$ will be\ndenoted by $\\bar{j}=\\tilde{1}\\oplus j$ where the addition is $mod\\,2$. We\nshall write $j=j^{1}j^{2}$ to mean that $j$ is the concatenation of $j^{1}$\nand $j^{2}$. Now assume $\\rho $ has the form given in (\\ref{sep rho}). Then \n\\begin{eqnarray}\n\\sqrt{\\langle j\|\\rho \|j\\rangle }\\sqrt{\\langle k\|\\rho \|k\\rangle } &=&\\left[\n\\sum_{a}p\\left( a\\right) \\rho _{j,j}\\left( a\\right) \\right] ^{1/2}\\left[\n\\sum_{a}p\\left( a\\right) \\rho _{k,k}\\left( a\\right) \\right] ^{1/2} \\nonumber\n\\\\\n&\\geq &\\sum_{a}p(a)\\sqrt{\\rho _{j,j}(a)\\rho _{k,k}(a)} \\nonumber \\\\\n&=&\\sum_{a}p(a)\\sqrt{\\rho _{j^{1},j^{1}}^{A_{1}}(a)\\rho\n_{k^{1},k^{1}}^{A_{1}}(a)}\\sqrt{\\rho _{j^{2},j^{2}}^{A_{2}}(a)\\rho\n_{k^{2},k^{2}}^{A_{2}}(a)} \\\\\n&\\geq &\\sum_{a}p(a)\|\\rho _{j^{1},k^{1}}^{A_{1}}(a)\|\|\\rho\n_{k^{2},j^{2}}^{A_{2}}(a)\| \\nonumber \\\\\n&\\geq &\|\\langle j^{1}k^{2}\|\\rho \|k^{1}j^{2}\\rangle \|, \\label{prop1}\n\\end{eqnarray}\nwhere the first inequality is the Cauchy-Schwarz inequality, the middle\nequality reflects the assumed separability, and the second inequality\nfollows from the positivity of the subsystem density matrices. Note that\nthere are four expressions possible for the last term.\n\nFor n-qubit systems, we shall be particularly interested in fully separable\nsystems which have no quantum correlations between any pair of qubits \\cite\n{Dur}. Specifically, an n-qubit density matrix $\\rho ^{\\left[ n\\right] }$ is\nfully separable on $H^{\\left[ n\\right] }$, the tensor product of $n$ two\ndimensional spaces, if \n\\begin{equation}\n\\rho ^{\\left[ n\\right] }=\\sum_{a}p(a)\\rho ^{\\left( 1\\right) }(a)\\otimes\n\\cdots \\otimes \\rho ^{\\left( n\\right) }(a), \\label{fully sep}\n\\end{equation}\nwhere $\\rho ^{\\left( k\\right) }(a)$ are qubit density matrices. If $k=\\bar{j}\n$, the arguments of (\\ref{prop1}) give \n\\begin{equation}\n\\min_{j}\\sqrt{\\langle j\|\\rho \|j\\rangle \\langle \\bar{j}\|\\rho \|\\bar{j}\\rangle }%\n\\geq \\max_{u}\|\\langle u\|\\rho \|\\bar{u}\\rangle \|. \\label{prop1a}\n\\end{equation}\n\nWe apply this result to the n-qubit Werner state \\cite{Werner}. Define the\ngeneralized GHZ states indexed by $j=0j_{2\\cdots }j_{n}$ as \n\\begin{equation}\n\|\\Psi ^{\\pm }(j)\\rangle =(\|j\\rangle \\pm \|\\bar{j}\\rangle )/\\sqrt{2},\n\\label{Psi}\n\\end{equation}\nand the generalized Werner states as \n\\begin{equation}\nW^{\\pm \\lbrack n]}(s,j)=\\frac{1-s}{2^{n}}I_{n}+s\\rho ^{\\pm }(j)\n\\label{Werner state}\n\\end{equation}\nwhere $\\rho ^{\\pm }(j)=\|\\Psi ^{\\pm }(j)\\rangle \\langle \\Psi ^{\\pm }(j)\|$ and \n$I_{n}$ is the identity matrix on ${\\cal H}^{[n]}=\\bigotimes_{{}}^{n}{\\cal H}%\n_{2}$. Then choosing $j$ judiciously in (\\ref{prop1a}), it is easy to see\nthat a necessary condition for the Werner state to be fully separable is\nthat $s\\leq 1/(2^{n-1}+1)$. We shall show below that this condition is also\nsufficient by using the spin basis representation to give a fully separable\nexpression of these states when $s\\leq 1/(2^{n-1}+1)$ \\cite{Dur,Schack}.\n\nWe can also apply (\\ref{prop1a}) to a convex set of n-qubit states which\nincludes the generalized Werner states. Let ${\\cal {D}}^{\\left[ n\\right] }$\ndenote the set of density matrices on ${\\cal H}^{[n]}$of the form \n\\begin{equation}\n\\rho (t)=\\sum_{j=\\tilde{0}}^{\\jmath _{m}}(t_{j}^{+}\\rho\n^{+}(j)+t_{j}^{-}\\rho ^{-}(j))\\qquad \\sum_{j=\\tilde{0}}^{\\jmath\n_{m}}(t_{j}^{+}+t_{j}^{-})=1 \\label{rho t}\n\\end{equation}\nwhere $t_{j}^{\\pm }\\geq 0$ and $j_{m}=01\\ldots 1$. The only non-zero \nelements of these density matrices are on the main\npositive and negative diagonals when the matrix is expressed in the\ncomputational basis $\\bigotimes_{{}}^{n}\\{\|0\\rangle ,\|1\\rangle \\}$, \nthus the Peres condition is equivalent\nto (\\ref{prop1a}). In\naddition to the generalized Werner densities, ${\\cal {D}}^{\\left[ n\\right] }$\nalso contains the states invariant with respect to depolarization: $%\nt^{+}(j)=t^{-}(j)$ for all $j\\neq \\tilde{0}$. Recall that depolarization is\ncarried out by averaging over the application of an arbitrary rotation $%\nexp(i\\phi _{r}\\sigma _{z})$ to each qubit, subject to $\\sum_{r}\\phi\n_{r}=2\\pi $, followed by spin-flip of all the qubits \\cite{Dur,Bennett2}.\nThis subset is the set of density matrices with non-zero entries appearing\nonly on the main diagonal and in the upper and lower corners. Note \nthat the set $\\{\\rho ^{\\pm}(j)\\}$ is the set of extreme points of ${\\cal \n{D}}^{[n]}$, and for\nstates in ${\\cal {D}}^{\\left[ n\\right] }.$ If ${\\cal {D}}_{s}^{[n] }$ denotes\nthe convex subset of fully separable states in ${\\cal {D}}^{\\left[ n\\right] }\n$, a necessary condition for $\\rho (t)\\in {\\cal {D}}_{s}^{\\left[ n\\right] }$\nis \n\\begin{equation}\n\\min_{j}(t_{j}^{+}+t_{j}^{-})\\geq \\max_{u}\|t_{u}^{+}-t_{u}^{-}\|.\n\\label{rho t prop1}\n\\end{equation}\n\nWe proved the necessary condition (\\ref{rho t prop1}) in the computational\nbasis; however, to find sufficient conditions we shall work in the spin\nbasis. To see why this is appropriate, we first prove a useful sufficiency\ncondition which is expressed entirely in terms of the Pauli matrices.\n\n\\begin{theorem}\nLet $M_{n}$ be a set of $n$ unit vectors, $M_{n}=\\{{\\bf m}_{1},\\cdots ,{\\bf m%\n}_{n}\\}$ and define the usual scalar product of the Pauli matrices with $%\n{\\bf m}$, $\\sigma _{{\\bf m}}=\\sigma \\cdot {\\bf m}$. Then the density matrix \n\\begin{equation}\n\\rho ^{\\pm }(M_{n})=\\frac{1}{2^{n}}(\\sigma _{0}\\otimes \\cdots \\otimes \\sigma\n_{0}\\pm \\sigma _{{\\bf m}_{1}}\\otimes \\cdots \\otimes \\sigma _{{\\bf m}_{n}})\n\\label{appendix1}\n\\end{equation}\non ${\\cal H}^{[n]}$ is fully separable.\n\\end{theorem}\n\nThis is easily proved using induction. The densities $P^{\\pm }({\\bf m}%\n)=(\\sigma _{0}\\pm \\sigma _{{\\bf {m}}})/2$ on ${\\cal H}_{2}$ are projections\nand are trivially separable. Suppose that $\\rho ^{\\pm }(M_{n-1})$ is fully\nseparable on ${\\cal H}^{[n-1]}$. Then \n\\begin{eqnarray}\n\\rho ^{\\pm }(M_{n}) &=&\\frac{1}{2^{n}}(\\sigma _{0}\\otimes \\cdots \\otimes\n\\sigma _{0}\\otimes \\lbrack P^{+}({\\bf m}_{n})+P^{-}({\\bf m}_{n})]\\pm \\sigma\n_{{\\bf m}_{1}}\\otimes \\cdots \\otimes \\sigma _{{\\bf m}_{n-1}}\\otimes \\lbrack\nP^{+}({\\bf m}_{n})-P^{-}({\\bf m}_{n})]) \\nonumber \\\\\n&=&\\frac{1}{2}[\\rho ^{\\pm }(M_{n-1})\\otimes P^{+}({\\bf m}_{n})+\\rho ^{\\mp\n}(M_{n-1})\\otimes P^{-}({\\bf m}_{n})], \\label{appendix2}\n\\end{eqnarray}\nwhere all the upper signs and all the lower signs go together, is completely\nseparable on ${\\cal H}^{[n]}$, completing the proof.\nThis particular set of separable states has the property that the only\nnon-zero (classical) correlation is among $n$ Pauli spin matrices. For $n=2$\nthis corresponds to the case studied in reference \\cite{H2}, $\\rho =(\\sigma\n_{0}\\otimes \\sigma _{0}+\\sum\\limits_{j,k}T_{jk}\\sigma _{j}\\otimes \\sigma\n_{k})/4,$ and in our case the $T$ matrix is of rank one and $T={\\bf m}%\n_{1}\\otimes {\\bf m}_{2}.$\n\nTo express a density in the spin basis, given its definition\nin the computational basis, we need a change of basis formula to relate the\ncoefficients in the two bases. This is a standard exercise, but we do it in\na non-conventional way to emphasize its structure as a Fourier transform. On\nthe Hilbert space ${\\cal {H}}_{2}$ introduce the operators $%\nE_{a,b}=\|a\\rangle \\langle b\|$, where $a,b=0,1$, and index the Pauli matrices\nusing binary notation, so $I_{2}=\\sigma _{00},\\sigma _{x}=\\sigma _{01},$\netc. Both of these sets form a basis of operators on the space of qubit\nstates. Write the spin basis $S$ and the adjusted basis $A$ in $2\\times 2$\narrays as \n\\begin{equation}\n\\left( S\\right) =\\left[ \n\\begin{array}{cc}\n\\sigma _{00} & \\sigma _{01} \\\\ \n\\sigma _{11} & i\\sigma _{10}\n\\end{array}\n\\right] \\qquad \\left( A\\right) =\\left[ \n\\begin{array}{cc}\nE_{0,0} & E_{0,1} \\\\ \nE_{1,1} & E_{1,0}\n\\end{array}\n\\right] \\qquad H=\\left[ \n\\begin{array}{cc}\n1 & 1 \\\\ \n1 & -1\n\\end{array}\n\\right] \\label{SAH}\n\\end{equation}\nwhere $H$ is the Hadamard matrix and the elements of $\\left( S\\right) $ are\nthe (real) Pauli matrices. Note that $A_{j,k}=E_{j,j\\oplus k}$ and $%\nS_{j,k}=\\sigma _{j,j\\oplus k}.$ In addition to the unconventional labelling\nin (\\ref{SAH}), it is necessary to work with real Pauli matrices, which is\nwhy the factor of $i$ appears. It is then easy to check that $%\nS_{j,k}=\\sum_{r}H\\left( j,r\\right) A_{r,k}$, which we record as \n\\begin{equation}\n(S)=H\\cdot (A). \\label{transform}\n\\end{equation}\nThe sets $\\{S_{j,k}/\\sqrt{2}\\}$ and $\\left\\{ {A_{j,k}}\\right\\} $ each\nform an orthonormal basis of operators on ${H}_{2}$ where we use the\ntrace inner product $\\left\\langle B,C\\right\\rangle =tr(B^{\\dag }C)$.\nTherefore, for any $\\rho $ \n\\begin{equation}\n\\rho =\\sum_{j,k}a_{j,k}A_{j,k}=\\frac{1}{2}\\sum_{j,k}s_{j,k}S_{j,k},\n\\label{coeff}\n\\end{equation}\nwhere $s_{j,k}=tr(S_{j,k}^{\\dag }\\rho ), a_{j,k}=tr( A_{j,k}^{\\dag\n}\\rho) =\\rho _{j,k\\oplus j}$ and it follows that \n\\begin{equation}\n\\left( s\\right) =H\\cdot \\left( a\\right) . \\label{spincoef}\n\\end{equation}\n\nThe use of the Hadamard matrix allows an easy generalization to tensor\nproduct spaces. Define $A_{j,k}^{[n]}$ and $S_{j,k}^{[n]}$ in the usual way: \n$S_{j,k}^{[n]}=S_{j_{1},k_{1}}\\otimes \\cdots \\otimes S_{j_{n},k_{n}}$\nso, for example, $S_{\\tilde{0},\\tilde{0}}^{[n]}=I_{n}.$ It then follows\neasily from $(S)=H\\cdot (A)$ that \n\\begin{equation}\n(S^{[n]})=H^{[n]}(A^{[n]}). \\label{Sn}\n\\end{equation}\nThe two sets of $2^{2n}$ operators $\\{S_{j,k}^{[n]}/\\sqrt{2^{n}}\\}$ and $%\n\\{A_{j,k}^{[n]}\\}$ each form an orthonormal basis on the Hilbert space $%\n{\\cal L}\\left( H^{\\left[ n\\right] }\\right) $ of linear operators acting on $%\n{\\cal H}^{[n]}$ where the inner product on ${\\cal L}\\left( H^{\\left[ n\\right]\n}\\right) $ is $\\left\\langle B,C\\right\\rangle =tr(B^{\\dag }C)$. Therefore, we\ncan express an arbitrary n-qubit density matrix in the form \n\\begin{equation}\n\\rho =\\frac{1}{2^{n}}\\sum_{j,k}s_{j,k}S_{j,k}^{[n]}=%\n\\sum_{j,k}a_{j,k}A_{j,k}^{[n]} \\label{nqubit density}\n\\end{equation}\nwith $[s]=H^{[n]}[a],$ the analogue of (\\ref{spincoef}). As an example\nof the notation, both the matrices $[s]$ and $[a]$ for a density matrix \nin the set defined in (\\ref{rho t}) have zeros everwhere\nexcept in the first and last columns. Equation (\\ref{transform}) can also be interpreted\nas a two dimensional Fourier transform of the computational basis which\ndefines the spin basis, and there is a natural generalization to $d$--dimensions \nusing finite Fourier transforms \\cite{Pittenger}.\n\nAs a first application we use the spin representation to show the density\nmatrices (\\ref{Werner state}) are fully separable for $s=1/(2^{n-1}+1)$. \nConsider $W^{+[n]}(s,\\tilde{0})$, but the result is independent of\nwhich $j$--state we choose. In terms of the adjusted basis, \n\\[\nW^{+[n]}(s,\\tilde{0})=\\frac{1-s}{2^{n}}I_{n}+\\frac{s}{2}\\left( A_{\\tilde{0},\\tilde{0}}^{[n]}+A_{%\n\\tilde{1}\\tilde{0}}^{[n]}+A_{\\tilde{0},\\tilde{1}}^{[n]}+A_{\\tilde{1},\\tilde{1}%\n}^{[n]}\\right) . \n\\]\nThe first two terms in the brackets are diagonal projections and are\ntherefore fully separable. We write the last two terms in the spin\ncoordinates. The only non-zero spin coefficients are in the last column, and\n\\[\ns_{j,\\tilde{1}}=\\frac{s}{2}\\left( 1+\\left( -1\\right) ^{j\\odot 1}\\right) , \n\\]\nwhere we have used $H_{j,k}^{[n]}=(-1)^{j\\odot k}$ with ${j\\odot k}$\ndenoting the binary scalar product. Define the set of $2^{n-1}$ elements $%\nInd=\\left\\{ j:j\\odot \\tilde{1}=\\sum_{r} j_{r}=0\\,mod\\,2\\right\\} .$ It follows\nfrom some easy algebra, that adding and subtracting a\nterm proportional to the identity $I_{n}=S_{\\tilde{0},\\tilde{0}}^{[n]}$\ngives \n\\[\nW^{+[n]}(s,\\tilde{0})=(\\frac{1-s}{2^{n}}-\\frac{s}{2})S_{\\tilde{0},\\tilde{0}}^{[n]}+s%\n\\frac{1}{2}\\left( A_{\\tilde{0},\\tilde{0}}^{[n]}+A_{\\tilde{1},\\tilde{0}%\n}^{[n]}\\right) +s\\sum_{j\\in Ind}\\frac{1}{2^{n}}\\left( S_{\\tilde{0},\\tilde{0}%\n}^{[n]}+S_{j,\\tilde{1}}^{[n]}\\right) . \n\\]\nNotice that $j\\in Ind$ means that there are an even number of factors of $%\nS_{1,1}=i\\sigma _{y}$ in $S_{j,\\tilde{1}}^{[n]}$, so that $S_{j,\\tilde{1}%\n}^{[n]}$ is Hermitian. The reason for adding and subtracting the identity is\nthat (\\ref{appendix1}) shows each term in the summation on the right is\nfully separable. To guarantee that $W^{+[n]}(s,\\tilde{0})$ is a density matrix, the\ncoefficient of the first term must be non-negative, forcing $s\\leq\n1/(2^{n-1}+1)$ and concluding the proof that $W^{\\pm \\lbrack n]}(s,j)$ is\nfully separable if and only if $s\\leq 1/(2^{n-1}+1).$ This result may be\ncompared with those obtained earlier in \\cite{Dur} and \\cite{Schack}.\n\nWe next use the spin representation to establish a new and general\nsufficient condition for full separability. We introduce a norm on densities which\nis expressed in terms of the spin coefficients.\n\n\\begin{theorem}\nIf the spin coefficients $s_{j,k}$ of a density $\\rho $ on ${\\cal H}^{[n]}$\nsatisfy $\\left\\\| \\rho \\right\\\| _{1}\\equiv \\sum\\limits_{\\left( j,k\\right)\n\\neq \\left( 0,0\\right) }\\nolimits\|s_{j,k}\|\\leq 1$, then $\\rho $ is fully\nseparable.\n\\end{theorem}\n\nSince $\\left( -i\\right) ^{j\\odot k}S_{j,k}^{\\left[ n\\right] }$ is Hermitian, \n$i^{j\\odot k}s_{j,k}$ must be real. Now use (\\ref{nqubit\ndensity}) to write \n\\[\n\\rho =\\left( 1-\\left\\\| \\rho \\right\\\| _{1}\\right) \\frac{1}{2^{n}}S_{\\tilde{0},%\n\\tilde{0}}^{[n]}+\\sum_{\\left( j,k\\right) \\neq \\left( 0,0\\right) }\\nolimits%\n\\left\| s_{j,k}\\right\| \\frac{1}{2^{n}}\\left( \nS_{\\tilde{0},\\tilde{0}}^{[n]}+v_{j,k}\n\\left( -i\\right) ^{j\\odot k}S_{j,k}^{\\left[ n\\right] }\\right) ,\n\\]\nwhere $v_{j,k}$ is the sign of $i^{j\\odot k}s_{j,k}.$ Again (\\ref{appendix1}) applies and gives full separability for $\\rho $.\nThis guarantees that there is a neighborhood of the completely random state $%\nS_{\\tilde{0},\\tilde{0}}^{[n]}/2^{n}$ in which all the densities are\nseparable, and in particular that every density with $\|s_{j,k}\|\\leq 1/\\left(\n2^{2n}-1\\right) $ is fully separable, giving the analogous result in \\cite\n{Braunstein} as a corollary. \n\nIf $\\rho =W^{+[n]}(s(n), j),$ \nwith $s(n) =1/(2^{n-1}+1)$, then\n$\\left\\\| \\rho \\right\\\|_{1}=(2^{n}-1)/(2^{n-1}+1).$ \nThus condition $\\left\\\| \\rho \\right\\\| _{1}\\leq 1$ is sharp for $n=2$\nbut may be too restrictive for larger $n.$ One can take advantage of the\nspecial structure of a class of densities to obtain more refined conditions.\nFor example (\\ref{rho t prop1}) is also sufficient for the states \nin${\\cal D}^{[n]}$ invariant with respect to depolarization. Consider \nalso the following subset of ${\\cal D}^{\\left[ n\\right] }.$\nLet $t^{\\pm (j)}=(1-s)/2^{n-1}+su_{j}^{\\pm }$ so that $\\sum\\limits_{j=\\tilde{%\n0}}^{j_{m}}(u_{j}^{+}+u_{j}^{-})=1$ (recall $j_{m}=01\\ldots 1$), and let $\\mu (s)=(1-s)S_{\\tilde{0},%\n\\tilde{0}}^{[n]}/2^{n}+s\\rho (u).$ Using the same approach that was used\nwith the Werner densities, we find that $\\mu (s)$ is fully separable provided $s\\leq \\left(\n1+2^{n-1}\\sum\\limits_{j=\\tilde{0}}^{j_{m}}\|u_{j}^{+}-u_{j}^{-}\|)\\right) \n^{-1}.$\n\n\nAs our final result, we show that for any $n\\geq 2$ and $\\epsilon >0$ there\nexists a density $\\rho $ on ${\\cal H}^{[n]}$ which is not fully separable\nbut which has $\\left\\\| \\rho \\right\\\| _{1}<1+\\epsilon $. Thus, the bound of\nthe theorem is not only the best possible in general but also the best\npossible for each value of $n$. As part of the proof we give necessary and\nsufficient conditions for full separability for a class of densities \n${\\tilde {\\cal{D}}}_{c}^{\\left[ n\\right] }$, each of which satisfies the Peres\ncondition. Define first the subset ${\\cal {D}}_{c}^{[n] }$ of ${\\cal {D}}^{[n] }$\nwith all diagonal elements\nequal: $\\rho ^{\\lbrack n]}(t)_{j,j}=(t^{+}(j)+t^{-}(j))/2=1/2^{n}$. In the\ncomputational coordinates $\\rho ^{\\lbrack n]}(t)$ is constant down the main\ndiagonal and the only non-zero entries are on the main negative diagonal.\nEach such density matrix satisfies (\\ref{rho t prop1}) and In spin coordinates has the form \n\\begin{equation}\n\\rho ^{\\lbrack n]}=\\frac{1}{2^{n}}(S_{\\tilde{0},\\tilde{0}}^{[n]}+\\sum_{j=%\n\\tilde{0}}^{\\tilde{1}}s_{j,\\tilde{1}}S_{j,\\tilde{1}}^{[n]}).\n\\label{rho special}\n\\end{equation}\nIf $\\rho ^{\\lbrack n]}$ is fully separable, it can be expressed in the form (%\n\\ref{fully sep}) with $\\rho (a,j_{k})=(\\sigma _{0}+\\sigma \\cdot {\\bf m}%\n(a,j_{k}))/2$ where ${\\bf m}(a,j_{k})$ is a unit vector in the $x$--$y$\nplane. Let $\\theta (a,j_{k})$ be the angle ${\\bf m}(a,j_{k})$ makes with the \n$x$--axis. Then from (\\ref{fully sep}) the non-zero off-diagonal elements\nare \n\\begin{equation}\n\\rho _{j,\\tilde{j}}^{[n]}=\\sum_{a}p(a)\\frac{1}{2^{n}}\\exp\n(i\\sum_{r=0}^{n-1}(-1)^{j_{r}}\\theta (a,j_{r})). \\label{rho j barj}\n\\end{equation}\nAll the matrix elements of the matrices in ${\\cal {D}}^{\\left[ n\\right] }$\nare real, so the densities in ${\\cal {D}}_{c}^{\\left[ n\\right] }$ satisfy \n\\begin{equation}\n\\sum_{a}p(a)\\frac{1}{2^{n}}\\cos (\\sum_{r=0}^{n-1}(-1)^{j_{r}}\\theta\n(a,j_{r})) =\\rho _{j,\\bar{j}}^{[n]}. \\label{cos}\n\\end{equation}\nIt is immediate from (\\ref{cos}) that if $\\rho _{j,\\bar{j}}^{[n]}=1/2^{n}$ \nfor some $j,$ then for all $a$ \n\\begin{equation}\n\\sum_{r=0}^{n-1}(-1)^{j_{r}}\\theta (a,j_{r})=0\\quad mod\\;2\\pi. \\label{angle}\n\\end{equation}\n\nWe use (\\ref{angle}) to define necessary and sufficient conditions for\nfull separability for a subset ${\\tilde {\\cal{D}}}_{c}^{\\left[ n%\n\\right] }$ of ${\\cal {D}}_{c}^{\\left[ n\\right] }$. Included in \n${\\tilde{\\cal{D}}}_{c}^{\\left[ n\\right] }$ are non-separable density \nmatrices with \n$\\left\\\| \\rho \\right\\\| _{1}$ arbitrarily close to $1$, confirming the\nassertion that $\\left\\\| \\rho \\right\\\| _{1}\\leq 1$ cannot be improved for $%\nn\\times n$ densities. To illustrate the ideas with minimal notational\nclutter, we work with the case $n=3$. Let $\\rho _{000,111}=\\rho\n_{001,110}=1/8$. Then (\\ref{angle}) implies that for all $a$, \n\\[\n\\theta \\left( a,1\\right) +\\theta \\left( a,2\\right) +\\theta \\left( a,3\\right)\n=0\\;mod\\,2\\pi \\hspace{0.1in}\\text{and}\\hspace{0.1in}\\theta \\left( a,1\\right)\n+\\theta \\left( a,2\\right) -\\theta \\left( a,3\\right) =0\\;mod \\,2\\pi ,\n\\]\nso that $\\theta \\left( a,3\\right) =0$ and $\\theta \\left( a,2\\right) =-\\theta\n\\left( a,1\\right) $. But then it follows that a {\\it necessary} condition\nfor full separability is \n\\[\n\\rho _{010,101}=\\rho _{011,100}=\\frac{1}{8}\\sum_{a}p\\left( a\\right) \\cos\n(2\\theta \\left( a,1\\right) ).\n\\]\nDefine ${\\tilde {\\cal{D}}}_{c}^{\\left[ 3\\right] }$ as the set of states with\nthe additional restrictions: if $c$ and $d$ satisfy $-1/8\\leq\nc,d\\leq 1/8$, then $t^{\\pm }\\left( 010\\right) =\\frac{1}{8}\\pm c$ and $t^{\\pm\n}\\left( 011\\right) =\\frac{1}{8}\\pm d$. Thus the states in ${\\tilde \n{\\cal{D}}}_{c}^{\\left[ 3\\right] }$ have the form\n\\begin{equation}\n\\rho (t(c,d))=\\left[ \n\\begin{array}{cccccccc}\n1/8 & 0 & 0 & 0 & 0 & 0 & 0 & 1/8 \\\\ \n0 & 1/8 & 0 & 0 & 0 & 0 & 1/8 & 0 \\\\ \n0 & 0 & 1/8 & 0 & 0 & c & 0 & 0 \\\\ \n0 & 0 & 0 & 1/8 & d & 0 & 0 & 0 \\\\ \n0 & 0 & 0 & d & 1/8 & 0 & 0 & 0 \\\\ \n0 & 0 & c & 0 & d & 1/8 & 0 & 0 \\\\ \n0 & 1/8 & 0 & c & 0 & 0 & 1/8 & 0 \\\\ \n1/8 & 0 & 0 & 0 & 0 & 0 & 0 & 1/8\n\\end{array}\n\\right] \n\\end{equation}\nIt follows that $\\rho \\left( t(c,d)\\right) $ is not fully separable if $%\nc\\neq d$. Using spin coordinates, it is easy to establish that \n$\\left\\\| \\rho \\left( t(c,d)\\right) \\right\\\| =1+4\\left\| c-d\\right\|, $ and thus \n$\\rho \\left( t(c,d)\\right) $ is fully separable if $c=d$. Since $\\left\\\|\n\\rho \\left( t(c,d)\\right) \\right\\\| $ can be made arbitrarily close to $1$,\nwe have shown $\\left\\\| \\rho \\right\\\| _{1}\\leq 1$ is sharp for $n=3$. The\nargument for larger $n$ is similar, and we omit the details. \n\n\\begin{proposition}\nLet $\\tilde{\\cal{D}}_{c}^{\\left[ n\\right] }$ denote the subset of densities in $%\n{\\cal {D}}_{c}^{\\left[ n\\right] }$ which, in addition to being constant on\nthe main diagonal, have $t^{+}=1/(2^{n-1})$ and $t^{-}=0$ for the\nfirst (and last) $2^{n-2}$ positions on the main negative diagonal, $t^{\\pm\n}=\\frac{1}{2^{n}}\\pm c$ on the next $2^{n-3}$ positions and $t^{\\pm }=\\frac{%\n1}{2^{n}}\\pm d$ on the remaining positions, where $-1/2^{n}\\leq\nc,d\\leq 1/2^{n}$. Then every density in $\\tilde{\\cal{D}}_{c}^{\\left[ n%\n\\right] }$ satisfies the Peres condition and is fully separable if and only\nif $c=d$. Given $\\epsilon >0$, there exist densities in $\\tilde{\\cal{D}}_{c}^{%\n\\left[ n\\right] }$ which are not fully separable and $\\left\\\| \\rho \\right\\\|\n_{1}<1+\\epsilon $.\n\\end{proposition}\n\n\\acknowledgements\nA. O. Pittenger gratefully acknowledges the hospitality of the Centre for\nQuantum Computation at Oxford University and support from UMBC and the\nNational Security Agency. M. H. Rubin wishes to thank the Office of Naval\nResearch, the National Security Agency and the U.S. Army Research Office for\nsupport of this work.\n\n\\begin{references}\n\\bibitem{EPR} A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {\\bf 47},\n777 (1935)\n\\bibitem{GHZ} D. M. Greenberger, M. Horne, and A. Zeilinger, in Bell's\nTheorem, {\\it Quantum Theory, and Conceptions of the Universe}, M. Kaftos\ned., (Kluwer, Dordrect, 1989); D. M. Greenberger, M. Horne, A. Shimony, and\nA. Zeilinger, Am. J. Phys. {\\bf 50}, 1131 (1990).\n\\bibitem{Bell} J. S. Bell, Physics {\\bf 1}, 195 (1964).\n\\bibitem{teleportation} C. H. Bennett, et. al., Phys. Rev. Lett. {\\bf 70},\n1895 (1993).\n\\bibitem{niel} M. A. Nielsen, Phys. Rev. Lett. {\\bf 83}, 436 (1999).\n\\bibitem{vidal} G. Vidal, ``Entanglement monotones'', LANL quant-ph/9807077\nv2 (Mar 1999)\n\\bibitem{Bennett1} C. H. Bennett and P.W. Shor, IEEE Trans. on Information\nTheory {\\bf 44}, 2724 (1998).\n\\bibitem{Peres} A. Peres, Phys. Rev. Lett. {\\bf 77}, 1413 (1996).\n\\bibitem{H1} M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A \n{\\bf 223}, 1 (1996).\n\\bibitem{Lew} Lewenstein, J. I. Cirac, and S. Karnas, quant-ph/9903012.\n\\bibitem{Dur} W. D\\\"{u}r, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. \n{\\bf 83}, 3562 (1999).\n\\bibitem{Schack} R. Schack and C. M. Caves, quant-ph/9904109 v2.\n\\bibitem{Werner} R. F. Werner, Phys. Rev. A {\\bf 40}, 4277 (1989).\n\\bibitem{Bennett2} C. H. Bennett {\\it et al.}, Phys. Rev. Lett. {\\bf 76},\n722 (1996).\n\\bibitem{H2} R. Horodecki and P. Horodecki, Phys. Lett. A {\\bf 210},\n1(1996).\n\\bibitem{Pittenger} A.O. Pittenger and M. H. Rubin, to appear\n\\bibitem{Braunstein} S. L. Braunstein, {\\it et. al.}, Phys. Rev. Lett. {\\bf %\n83},1054 (1999).\n\\end{references}\n\n\\end{document}\n\n" } ]	[ { "name": "quant-ph9912116.extracted_bib", "string": "{EPR A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {47, 777 (1935)" }, { "name": "quant-ph9912116.extracted_bib", "string": "{GHZ D. M. Greenberger, M. Horne, and A. Zeilinger, in Bell's Theorem, {Quantum Theory, and Conceptions of the Universe, M. Kaftos ed., (Kluwer, Dordrect, 1989); D. M. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. {50, 1131 (1990)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{Bell J. S. Bell, Physics {1, 195 (1964)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{teleportation C. H. Bennett, et. al., Phys. Rev. Lett. {70, 1895 (1993)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{niel M. A. Nielsen, Phys. Rev. Lett. {83, 436 (1999)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{vidal G. Vidal, ``Entanglement monotones'', LANL quant-ph/9807077 v2 (Mar 1999)" }, { "name": "quant-ph9912116.extracted_bib", "string": "{Bennett1 C. H. Bennett and P.W. Shor, IEEE Trans. on Information Theory {44, 2724 (1998)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{Peres A. Peres, Phys. Rev. Lett. {77, 1413 (1996)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{H1 M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A {223, 1 (1996)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{Lew Lewenstein, J. I. Cirac, and S. Karnas, quant-ph/9903012." }, { "name": "quant-ph9912116.extracted_bib", "string": "{Dur W. D\\\"{ur, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. {83, 3562 (1999)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{Schack R. Schack and C. M. Caves, quant-ph/9904109 v2." }, { "name": "quant-ph9912116.extracted_bib", "string": "{Werner R. F. Werner, Phys. Rev. A {40, 4277 (1989)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{Bennett2 C. H. Bennett {et al., Phys. Rev. Lett. {76, 722 (1996)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{H2 R. Horodecki and P. Horodecki, Phys. Lett. A {210, 1(1996)." }, { "name": "quant-ph9912116.extracted_bib", "string": "{Pittenger A.O. Pittenger and M. H. Rubin, to appear" }, { "name": "quant-ph9912116.extracted_bib", "string": "{Braunstein S. L. Braunstein, {et. al., Phys. Rev. Lett. {% 83,1054 (1999)." } ]
quant-ph9912117	Quantum Cryptography with Entangled Photons	[ { "author": "Thomas Jennewein" }, { "author": "Christoph Simon" }, { "author": "Gregor Weihs" }, { "author": "Harald Weinfurter\\dag" }, { "author": "and Anton Zeilinger" } ]	By realizing a quantum cryptography system based on polarization entangled photon pairs we establish highly secure keys, because a single photon source is approximated and the inherent randomness of quantum measurements is exploited. We implement a novel key distribution scheme using Wigner's inequality to test the security of the quantum channel, and, alternatively, realize a variant of the BB84 protocol. Our system has two completely independent users separated by $360$~m, and generates raw keys at rates of $400$ -- $800$~bits/second with bit error rates arround $3$\%.	[ { "name": "Cryptog.tex", "string": "%\\documentstyle[preprint,aps]{revtex}\n%\\documentstyle[preprint,aps,graphicx]{revtex}\n\\documentstyle[prb,aps,twocolumn,graphicx]{revtex}\n\n\\newcommand{\\ket}[1]{\|#1\\rangle}\n\\newcommand{\\bra}[1]{\\langle#1\|}\n\n\\newcommand{\\sfrac}[2]{\n \\textstyle\n \\frac{#1}{#2}\n \\displaystyle}\n\n\n\n\n\\begin{document}\n\n\n\\title{Quantum Cryptography with Entangled Photons}\n\n \\author{Thomas Jennewein, Christoph Simon, Gregor Weihs, \\\\\n Harald Weinfurter\\dag, and Anton Zeilinger}\n\n\n \\address{Institut f\\\"{u}r Experimentalphysik, Universit\\\"{a}t Wien, \\\\\n Boltzmanngasse 5, A--1090 Wien, Austria \\\\\n \\dag Sektion Physik, Universit\\\"{a}t M\\\"{u}nchen, \\\\\n Schellingstr. 4/III, D-80799 M\\\"{u}nchen, Germany\\ddag }\n\n\n\n\n\\date{\\today}\n\n\\maketitle\n\n\n\\thispagestyle{empty}\n\n\n\n\n\n\\begin{abstract}\n\n\n\n\nBy realizing a quantum cryptography system based on polarization\nentangled photon pairs we establish highly secure keys, because a\nsingle photon source is approximated and the inherent randomness\nof quantum measurements is exploited. We implement a novel key\ndistribution scheme using Wigner's inequality to test the security\nof the quantum channel, and, alternatively, realize a variant of\nthe BB84 protocol. Our system has two completely independent users\nseparated by $360$~m, and generates raw keys at rates of $400$ --\n$800$~bits/second with bit error rates arround $3$\\%.\n\n\\end{abstract}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n% body of paper here\n\n%\\section{Introduction}\n\\vspace*{1cm}\n\nThe primary task of cryptography is to enable two parties\n(commonly called Alice and Bob) to mask confidential messages\nsuch, that the transmitted data are illegible to any unauthorized\nthird party (called Eve). Usually this is done using shared secret\nkeys. However, in principle it is always possible to intercept\nclassical key distribution unnoticedly. The recent development of\nquantum key distribution \\cite{BB84} can cover this major loophole\nof classical cryptography. It allows Alice and Bob to establish\ntwo completely secure keys by transmitting single quanta (qubits)\nalong a quantum channel. The underlying principle of quantum key\ndistribution is that nature prohibits to gain information on the\nstate of a quantum system without disturbing it. Therefore, in\nappropriately designed schemes, no tapping of the qubits is\npossible without showing up to Alice and Bob. These secure keys\ncan be used in a One-Time-Pad protocol \\cite{Vernam}, which makes\nthe entire communication absolutely secure.\n\nTwo well known concepts for quantum key distribution are the BB84\nscheme and the Ekert scheme. The BB84 scheme \\cite{BB84} uses\nsingle photons transmitted from Alice to Bob, which are prepared\nat random in four partly orthogonal polarization states:\n$0^\\circ$, $45^\\circ$, $90^\\circ$, $135^\\circ$. If Eve tries to\nextract information about the polarization of the photons she will\ninevitably introduce errors, which Alice and Bob can detect by\ncomparing a random subset of the generated keys.\n\nThe Ekert scheme \\cite{Ekert} is based on entangled pairs and uses\nBell's inequality \\cite{Bell} to establish security. Both Alice\nand Bob receive one particle out of an entangled pair. They\nperform measurements along at least three different directions on\neach side, where measurements along parallel axes are used for key\ngeneration and oblique angles are used for testing the inequality.\nIn \\cite{Ekert}, Ekert pointed out that eavesdropping inevitably\naffects the entanglement between the two constituents of a pair\nand therefore reduces the degree of violation of Bell's\ninequality. While we are not aware of a general proof that the\nviolation of a Bell inequality implies the security of the system,\nthis has been shown \\cite{otherprot} for the BB84 protocol adapted\nto entangled pairs and the CHSH inequality \\cite{chsh}.\n\nIn any real cryptography system, the raw key generated by Alice\nand Bob contains errors, which have to be corrected by classical\nerror correction \\cite{errorcorr} over a public channel.\nFurthermore it has been shown that whenever Alice and Bob share a\nsufficiently secure key, they can enhance its security by privacy\namplification techniques \\cite{privacy}, which allow them to\ndistill a key of a desired security level.\n\n\n\n\n\n\n\nA range of experiments have demonstrated the feasibility of\nquantum key distribution, including realizations using the\npolarization of photons \\cite{polexp} or the phase of photons in\nlong interferometers \\cite{phaseexp}. These experiments have a\ncommon problem: the sources of the photons are attenuated laser\npulses which have a non-vanishing probability to contain two or\nmore photons, leaving such systems prone to the so called beam\nsplitter attack \\cite{cohpulse}.\n\nUsing photon pairs as produced by parametric down-conversion\nallows us to approximate a conditional single photon source\n\\cite{singlephot} with a very low probability for generating two\npairs simultaneously and a high bit rate \\cite{twopairs}.\nMoreover, when utilizing entangled photon pairs one immediately\nprofits from the inherent randomness of quantum mechanical\nobservations leading to purely random keys.\n\n\n\nVarious experiments with entangled photon pairs have already\ndemonstrated that entanglement can be preserved over distances as\nlarge as 10~km \\cite{entexp}, yet none of these experiments was a\nfull quantum cryptography system. We present in this paper a\ncomplete implementation of quantum cryptography with two users,\nseparated and independent of each other in terms of Einstein\nlocality and exploiting the features of entangled photon pairs for\ngenerating highly secure keys.\n\n\nIn the following we will describe the variants of the Ekert scheme\nand of the BB84 scheme which we both implemented in our\nexperiment, based on polarization entangled photon pairs in the\nsinglet state\n\\begin{equation}\n\\ket{\\Psi^-}=\\frac{1}{\\sqrt{2}}[\\ket{H}_A\\ket{V}_B-\\ket{V}_A\\ket{H}_B]\n\\:,\n\\end{equation}\nwhere photon $A$ is sent to Alice and photon $B$ is sent to Bob,\nand $H$ and $V$ denote the horizontal and vertical linear\npolarization respectively. This state shows perfect\nanticorrelation for polarization measurements along parallel but\narbitrary axes. However, the actual outcome of an individual\nmeasurement on each photon is inherently random. These perfect\nanticorrelations can be used for generating the keys, yet the\nsecurity of the quantum channel remains to be ascertained by\nimplementing a suitable procedure.\n\nOur first scheme utilizes Wigner's inequality \\cite{Wigner} for\nestablishing the security of the quantum channel, in analogy to\nthe Ekert scheme which uses the CHSH inequality. Here Alice\nchooses between two polarization measurements along the axes\n$\\chi$ and $\\psi$, with the possible results $+1$ and $-1$, on\nphoton $A$ and Bob between measurements along $\\psi$ and $\\omega$\non photon $B$. Polarization parallel to the analyzer axis\ncorresponds to a $+1$ result, and polarization orthogonal to the\nanalyzer axis corresponds to $-1$.\n\nAssuming that the photons carry preassigned values determining the\noutcomes of the measurements $\\chi, \\psi, \\omega$ and also\nassuming perfect anticorrelations for measurements along parallel\naxes, it follows, that the probabilities for obtaining $+1$ on\nboth sides, $p_{++}$, must obey Wigner's inequality:\n\\begin{equation}\np_{++}(\\chi,\\psi) + p_{++}(\\psi,\\omega) - p_{++}(\\chi,\\omega) \\geq\n0 \\:. \\label{w_ineq}\n\\end{equation}\n\n\nThe quantum mechanical prediction $p^{qm}_{++}$ for these\nprobabilities at arbitrary analyzer settings $\\alpha$ (Alice) and\n$\\beta$ (Bob) measuring the $\\Psi^-$ state is\n\\begin{equation}\np^{qm}_{++}(\\alpha,\\beta)=\\sfrac{1}{2} \\sin^2 \\left( \\alpha-\\beta\n\\right) \\:.\n\\end{equation}\n\nThe analyzer settings $\\chi=-30^\\circ$, $\\psi=0^\\circ$, and\n$\\omega=30^\\circ$ lead to a maximum violation of Wigner's\ninequality~(\\ref{w_ineq}):\n\\begin{eqnarray}\n& & p^{qm}_{++}(-30^\\circ,0^\\circ) + p^{qm}_{++}(0^\\circ,30^\\circ)\n- p^{qm}_{++}(-30^\\circ,30^\\circ) = \\nonumber \\\\ & & =\n\\sfrac{1}{8}+\\sfrac{1}{8}-\\sfrac{3}{8} = -\\sfrac{1}{8} \\geq 0 \\:.\n\\end{eqnarray}\n\nAs Wigner's inequality is derived assuming perfect\nanticorrelations, which are only approximately realized in any\npractical situation, one should be cautious in applying it to test\nthe security of a cryptography scheme. When the deviation from\nperfect anticorrelations is substantial, Wigner's inequality has\nto be replaced by an adapted version \\cite{ryff}.\n\nIn order to implement quantum key distribution, Alice and Bob each\nvary their analyzers randomly between two settings, Alice:\n$-30^\\circ,0^\\circ$ and Bob: $0^\\circ,30^\\circ$\n(Figure~\\ref{settings}a). Because Alice and Bob operate\nindependently, four possible combinations of analyzer settings\nwill occur, of which the three oblique settings allow a test of\nWigner's inequality and the remaining combination of parallel\nsettings (Alice$=0^\\circ$ and Bob$=0^\\circ$) allows the generation\nof keys via the perfect anticorrelations, where either Alice or\nBob has to invert all bits of the key to obtain identical keys.\n\nIf the measured probabilities violate Wigner's inequality, then\nthe security of the quantum channel is ascertained, and the\ngenerated keys can readily be used. This scheme is an improvement\non the Ekert scheme which uses the CHSH inequality and requires\nthree settings of Alice's and Bob's analyzers for testing the\ninequality and generating the keys. From the resulting nine\ncombinations of settings, four are taken for testing the\ninequality, two are used for building the keys and three are\nomitted at all. However in our scheme each user only needs two\nanalyzer settings and the detected photons are used more\nefficiently, thus allowing a significantly simplified experimental\nimplementation of the quantum key distribution.\n\nAs a second quantum cryptography scheme we implemented a variant\nof the BB84 protocol with entangled photons, as proposed in\nReference \\cite{BBM}. In this case, Alice and Bob randomly vary\ntheir analysis directions between $0^\\circ$ and $45^\\circ$\n(Figure~\\ref{settings}b). Alice and Bob observe perfect\nanticorrelations of their measurements whenever they happen to\nhave parallel oriented polarizers, leading to bitwise\ncomplementary keys. Alice and Bob obtain identical keys if one of\nthem inverts all bits of the key. Polarization entangled photon\npairs offer a means to approximate a single photon situation.\nWhenever Alice makes a measurement on photon $A$, photon $B$ is\nprojected into the orthogonal state which is then analyzed by Bob,\nor vice versa. After collecting the keys, Alice and Bob\nauthenticate their keys by openly comparing a small subset of\ntheir keys and evaluating the bit error rate.\n\n\nThe experimental realization of our quantum key distribution\nsystem is sketched in Figure~\\ref{setup}. Type-II parametric\ndown-conversion in $\\beta$-barium borate \\cite{kwiat} (BBO),\npumped with an argon-ion laser working at a wavelength of $351$~nm\nand a power of $350$~mW, leads to the production of polarization\nentangled photon pairs at a wavelength of $702$~nm. The photons\nare each coupled into $500$~m long optical fibers and transmitted\nto Alice and Bob respectively, who are separated by $360$~m.\n\nAlice and Bob both have Wollaston polarizing beam splitters as\npolarization analyzers. We will associate a detection of parallel\npolarization ($+1$) with the key bit 1 and orthogonal detection\n($-1$) with the key bit 0. Electro-optic modulators in front of\nthe analyzers rapidly switch (rise time $<15$~ns, minimum\nswitching interval $100$ ns) the axis of the analyzer between two\ndesired orientations, controlled by quantum random signal\ngenerators \\cite{rngpaper}. These quantum random signal generators\nare based on the quantum mechanical process of splitting a beam of\nphotons and have a correlation time of less than $100$~ns.\n\nThe photons are detected in silicon avalanche photo diodes\n\\cite{cova}. Time interval analyzers on local personal computers\nregister all detection events as time stamps together with the\nsetting of the analyzers and the detection result. A measurement\nrun is initiated by a pulse from a separate laser diode sent from\nthe source to Alice and Bob via a second optical fiber. Only after\na measurement run is completed, Alice and Bob compare their lists\nof detections to extract the coincidences. In order to record the\ndetection events very accurately, the time bases in Alice's and\nBob's time interval analyzers are controlled by two rubidium\noscillators. The stability of each time base is better than 1~ns\nfor one minute. The maximal duration of a measurement is limited\nby the amount of memory in the personal computers (typically one\nminute).\n\nOverall our system has a measured total coincidence rate of $\\sim\n1700 \\mathrm{s}^{-1}$ , and a singles rate of $\\sim 35 000\n\\mathrm{s}^{-1}$ . From this, one can estimate the overall\ndetection efficiency of each photon path to be 5~\\% and the pair\nproduction rate to be $7\\cdot 10^5 \\mathrm{s}^{-1}$. Our system is\nvery immune against a beam splitter attack because the ratio of\ntwo-pair events is only $\\sim 3 \\cdot 10^{-3}$, where a two-pair\nevent is the emission of two pairs within the coincidence window\nof $4$~ns. The coincidence window in our experiment is limited by\nthe time resolution of our detectors and electronics, but in\nprinciple it could be reduced to the coherence time of the\nphotons, which is usually of the order of picoseconds.\n\n\n\nIn realizing the quantum key distribution based on Wigner's\ninequality, Alice's analyzer switch randomly with equal frequency\nbetween $-30^{\\circ}$ and $0^{\\circ}$, and Bob's analyzer between\n$0^{\\circ}$ and $30^{\\circ}$. After a measurement, Alice and Bob\nextract the coincidences for the combinations of settings of\n$(-30^{\\circ},30^{\\circ})$, $(-30^{\\circ},0^{\\circ})$ and\n$(0^{\\circ},30^{\\circ})$, and calculate each probability. E.g. the\nprobability $p_{++}(0^\\circ,30^\\circ)$ is calculated from the\nnumbers of coincident events $C_{++}$, $C_{+-}$, $C_{-+}$,\n$C_{--}$ measured for this combination of settings by\n\\begin{equation}\np_{++}(0^\\circ,30^\\circ)= \\frac{C_{++}} {C_{++}+C_{+-}+C_{-+}+\nC_{--}}.\n\\end{equation}\nWe observed in our experiment that the left hand side of\ninequality (\\ref{w_ineq}) evaluated to $-0.112 \\pm 0.014$. This\nviolation of (\\ref{w_ineq}) is in good agreement with the\nprediction of quantum mechanics and ensures the security of the\nkey distribution. Hence the coincident detections obtained at the\nparallel settings $(0^{\\circ},0^{\\circ})$, which occur in a\nquarter of all events, can be used as keys. In the experiment\nAlice and Bob established $2162$~bits raw keys at a rate of\n$420$~bits/second \\cite{bias}, and observed a quantum bit error\nrate of $3.4$~\\%.\n\nIn our realization of the BB84 scheme, Alice's and Bob's analyzers\nboth switch randomly between $0^{\\circ}$ and $45^{\\circ}$. After a\nmeasurement run, Alice and Bob extract the coincidences measured\nwith parallel analyzers, $(0^{\\circ},0^{\\circ})$ and\n$(45^{\\circ},45^{\\circ})$, which occur in half of the cases, and\ngenerate the raw keys. Alice and Bob collected $\\sim 80000$ bits\nof key at a rate of $850$ bits/second, and observed a quantum bit\nerror rate of $2.5$~\\%, which ensures the security of the quantum\nchannel.\n\nFor correcting the remaining errors while maintaining the secrecy\nof the key, various classical error correction and privacy\namplification schemes have been developed \\cite{errorcorr}. We\nimplemented a simple error reduction scheme requiring only little\ncommunication between Alice and Bob. Alice and Bob arrange their\nkeys in blocks of $n$ bits and evaluate the bit parity of the\nblocks (a single bit indicating an odd or even number of ones in\nthe block). The parities are compared in public, and the blocks\nwith agreeing parities are kept after discarding one bit per block\n\\cite{discard}. Since parity checks only reveal odd occurrences of\nbit errors, a fraction of errors remains. The optimal block length\n$n$ is determined by a compromise between key losses and remaining\nbit errors. For a bit error rate $p$ the probability for $k$ wrong\nbits in a block of $n$ bits is given by the binomial distribution\n$P_n(k)= {n \\choose k} p^k (1-p)^{n-k}$.\n\nNeglecting terms for three or more errors and accounting for the\nloss of one bit per agreeing parity, this algorithm has an\nefficiency\n%\\begin{equation}\n$\\eta(n)=(1-P_n(1))(n-1)/n$,\n%\\end{equation}\ndefined as the ratio between the key sizes after parity check and\nbefore. Finally, under the same approximation as above, the\nremaining bit error rate $p'$ is\n%\\begin{equation}\n$p'= (1-P_n(0)-P_n(1))(2/n)$.\n%\\end{equation}\nOur key has a bit error rate $p=2.5$~\\%, for which $\\eta(n)$ is\nmaximized at $n=8$ with $\\eta(8)=0.7284$, resulting in\n$p'=0.40$~\\%. Hence, from $\\sim 80000$~bits of raw key with a\nquantum bit error rate of $2.5$~\\%, Alice and Bob use $10$~\\% of\nthe key for checking the security and the remaining $90$~\\% of the\nkey to distill $49984$ bits of error corrected key with a bit\nerror rate of $0.4$\\%. Finally, Alice transmits a $43200$ bit\nlarge image to Bob via the One-Time-Pad protocol, utilizing a\nbitwise XOR combination of message and key data\n(Figure~\\ref{venus}).\n\n\nIn this letter we presented the first full implementation of\nentangled state quantum cryptography. All the equipment of the\nsource and of Alice and Bob has proven to operate outside shielded\nlab-environments with a very high reliability. While further\npractical and theoretical investigations are still necessary, we\nbelieve that this work demonstrates that entanglement based\ncryptography can be tomorrow's technology.\n\n\nThis work was supported by the Austrian Science Foundation FWF\n(Projects No.~S6502, S6504 and F1506), the Austrian Academy of\nSciences, and the TMR program of the European Commission (Network\ncontract No.~ERBFMRXCT96-0087).\n\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{BB84}\nC. H. Bennett and G. Brassard, Proc. Internat. Conf. Computer\nSystems and Signal Processing, Bangalore, pp. 175 (1984). C. H.\nBennett, G. Brassard, and A. Ekert, Scientific American, pp. 26,\nOctober 1992.\n\n\\bibitem{Vernam} In this\nclassical cryptographic protocol the message is combined with a\nrandom key string of the same size as the message to form an\nencoded message which cannot be deciphered by any statistical\nmethods. G.S. Vernam, J. Am. Inst. Elec. Eng. {\\bf 55}, 109\n(1926).\n\n\n\n\\bibitem{Ekert}A.K. Ekert, Phys. Rev. Lett. {\\bf 67}, 661 (1991).\n\n\\bibitem{Bell}J. S. Bell, Physics (Long Island City, N.Y.) {\\bf 1},\n195 (1965).\n\n\\bibitem{otherprot} C. Fuchs, N. Gisin, R. B.\nGriffiths, C. S. Niu, and A. Peres, Phys. Rev. A {\\bf 56}, 1163\n(1997).\n\n\\bibitem{chsh} J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,\n Phys. Rev. Lett. {\\bf 23}, 880 (1969).\n\n\\bibitem{errorcorr} C.H. Bennett and G. Brassard, J. Cryptology 5, 3 (1992).\n\n\\bibitem{privacy} C.H. Bennett, G. Brassard, C. Cr\\'{e}peau, and\nU.M. Maurer, IEEE Trans. Inf. Theo. {\\bf 41}, 1915 (1995).\n\n\n\\bibitem{polexp}C. H. Bennett, F. Bessette, G. Brassard, L. Savail, and J. Smolin,\nJ. Cryptology {\\bf 5}, 3 (1992); A. Muller, J. Breguet, and N.\nGisin, Europhys. Lett. {\\bf 23}, 383 (1993); J.D. Franson and B.C.\nJacobs, Electron. Lett. {\\bf 31}, 232 (1995); W. T. Buttler, R. J.\nHughes, P. G. Kwiat, S. K. Lamoreaux, G. G. Luther, G. L. Morgan,\nJ. E. Nordholt, C. G. Peterson, and C. M. Simmons, Phys. Rev.\nLett. {\\bf 81}, 3283 (1998).\n\n\n\\bibitem{phaseexp}C. Marand and P.D. Townsend, Opt. Lett. {\\bf\n20}, 1695 (1995); R.J. Hughes, G.G. Luther, G.L. Morgan, C.G.\nPeterson, and C. Simmons, Lect. Notes in Comp. Sci. {\\bf 1109},\n329 (1996); A. Muller, T. Herzog, B. Huttner, W. Tittel, H.\nZbinden, and N. Gisin, Appl. Phys. Lett. {\\bf 70}, 793 (1997).\n\n\\bibitem{cohpulse} N. L\\\"{u}tkenhaus, G. Brassard, T. Mor, and B.C.\nSanders, to be published.\n\n\n\\bibitem{singlephot}\nOne photon of the pair can be used as a trigger for finding the\nother photon of the pair,\n provided that the probability of producing two pairs at a single time can be neglected.\nP. Grangier, G. Roger, and A. Aspect, Europhys. Lett., {\\bf 1}, 4,\n173 (1986); J.G. Rarity, P.R. Tapster, and E. Jakeman, Opt.Comm.\n{\\bf 62}, 201 (1987).\n\n\\bibitem{twopairs} Note also that in our case the beam splitter attack is less\neffective than for coherent pulses, because even when two pairs\nare produced simultaneously, Eve does not gain any information in\nthose cases where Alice and Bob detect photons belonging to the\nsame pair, because then the photon detected by Eve originates from\na different pair and is completely uncorrelated to Alice's and\nBob's photons.\n\n\\bibitem{entexp}P.R. Tapster, J.G. Rarity, and P.C.M. Owens,\nPhys. Rev. Lett. {\\bf 73}, 1923 (1994); W. Tittel, J. Brendel, H.\nZbinden, and N. Gisin, Phys. Rev. Lett. {\\bf 81}, 3563 (1998); G.\nWeihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger,\nPhys. Rev. Lett. {\\bf 81}, 5039 (1998).\n\n\n\n\\bibitem{Wigner}E.P. Wigner, Am. J. Phys., {\\bf 38}, 1005\n(1970).\n\n\\bibitem{ryff} M. Zukowski, private communication; L. C. Ryff, Am. J. Phys. {\\bf 65}(12), 1197 (1997).\n\n\\bibitem{BBM}C.H. Bennett, G. Brassard, and N.D. Mermin, Phys. Rev. Lett.\n{\\bf 68}, 557 (1992).\n\n\n\n\\bibitem{kwiat}\nP. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V.\nSergienko, and Y. H. Shih, Phys. Rev. Lett. {\\bf 75}, 4337\n(1995).\n\n\\bibitem{rngpaper} T. Jennewein, U. Achleitner, G. Weihs, H.\nWeinfurter, and A. Zeilinger, to appear in Rev. Sci. Instr.\n\n\\bibitem{cova}\nS. Cova, M. Ghioni, A. Lacaita, C. Samori, and F. Zappa, Appl.\nOpt., {\\bf 35}, 1956 (1996).\n\n\n\n\\bibitem{bias} Note that it would be simple to bias the frequencies of analyzer\ncombinations to increase the production rate of the keys.\n\n\\bibitem{discard}\nRemoval of one bit erases the information about the blocks\ncontained in the (public) parities.\n\n\n\\bibitem{Bitmap}\nWindows-BMP format containing $60 \\times 90$ pixel, 8 bit color\ninformation per pixel: $43200$ bit of picture information. The\nfile includes some header information and a color table, making\nthe entire picture file $51840$ bit. We encrypted only the\npicture information, leaving the file header and the color table\nunchanged.\n\n\\bibitem{Venusdate}\nThe ``Venus'' von Willendorf was found in $1908$ at Willendorf in\nAustria and presently resides in the Naturhistorisches Museum,\nVienna. Carved from limestone and dated $24.000$--$22.000$ BC, she\nrepresents an icon of prehistoric art.\n\n\n\n\\end{thebibliography}\n\n\n\n\n\\begin{figure}\n\n\\includegraphics[width=\\columnwidth]{figure1.eps}\n\n\\caption{Settings for Alice's and Bob's analyzers for realizing\nquantum key distribution based either on (a) Wigner's inequality\nor (b) the BB84 protocol. The angular coordinates are referenced\nto the propagation direction of the particle.}\n\n\\label{settings}\n\n\\end{figure}\n\n\n\n\\begin{figure}\n\n\n\\includegraphics[width=\\columnwidth]{figure2.eps}\n\n\\caption{The polarization entangled photons are transmitted via\noptical fibers to Alice and Bob, who are separated by $360$~m, and\nboth photons are analyzed, detected and registered independently.\nAfter a measurement run the keys are established by Alice and Bob\nthrough classical communication over a standard computer network.}\n\n\\label{setup}\n\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure3.eps}\n\n \\caption{The $49984$ bit large keys generated by the BB84 scheme are used to\n securely transmit an image \\protect\\cite{Bitmap} (a) of the\n``Venus von Willendorf''\\protect\\cite{Venusdate} effigy. Alice\nencrypts the image via bitwise XOR operation with her key and\ntransmits the encrypted image (b) to Bob via the computer\nnetwork. Bob decrypts the image with his key, resulting in (c)\nwhich shows only few errors due to the remaining bit errors in\nthe keys.}\n\n\\label{venus}\n\n\\end{figure}\n\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912117.extracted_bib", "string": "{BB84 C. H. Bennett and G. Brassard, Proc. Internat. Conf. Computer Systems and Signal Processing, Bangalore, pp. 175 (1984). C. H. Bennett, G. Brassard, and A. Ekert, Scientific American, pp. 26, October 1992." }, { "name": "quant-ph9912117.extracted_bib", "string": "{Vernam In this classical cryptographic protocol the message is combined with a random key string of the same size as the message to form an encoded message which cannot be deciphered by any statistical methods. G.S. Vernam, J. Am. Inst. Elec. Eng. {55, 109 (1926)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{EkertA.K. Ekert, Phys. Rev. Lett. {67, 661 (1991)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{BellJ. S. Bell, Physics (Long Island City, N.Y.) {1, 195 (1965)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{otherprot C. Fuchs, N. Gisin, R. B. Griffiths, C. S. Niu, and A. Peres, Phys. Rev. A {56, 1163 (1997)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{chsh J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. {23, 880 (1969)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{errorcorr C.H. Bennett and G. Brassard, J. Cryptology 5, 3 (1992)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{privacy C.H. Bennett, G. Brassard, C. Cr\\'{epeau, and U.M. Maurer, IEEE Trans. Inf. Theo. {41, 1915 (1995)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{polexpC. H. Bennett, F. Bessette, G. Brassard, L. Savail, and J. Smolin, J. Cryptology {5, 3 (1992); A. Muller, J. Breguet, and N. Gisin, Europhys. Lett. {23, 383 (1993); J.D. Franson and B.C. Jacobs, Electron. Lett. {31, 232 (1995); W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, G. G. Luther, G. L. Morgan, J. E. Nordholt, C. G. Peterson, and C. M. Simmons, Phys. Rev. Lett. {81, 3283 (1998)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{phaseexpC. Marand and P.D. Townsend, Opt. Lett. {20, 1695 (1995); R.J. Hughes, G.G. Luther, G.L. Morgan, C.G. Peterson, and C. Simmons, Lect. Notes in Comp. Sci. {1109, 329 (1996); A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, Appl. Phys. Lett. {70, 793 (1997)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{cohpulse N. L\\\"{utkenhaus, G. Brassard, T. Mor, and B.C. Sanders, to be published." }, { "name": "quant-ph9912117.extracted_bib", "string": "{singlephot One photon of the pair can be used as a trigger for finding the other photon of the pair, provided that the probability of producing two pairs at a single time can be neglected. P. Grangier, G. Roger, and A. Aspect, Europhys. Lett., {1, 4, 173 (1986); J.G. Rarity, P.R. Tapster, and E. Jakeman, Opt.Comm. {62, 201 (1987)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{twopairs Note also that in our case the beam splitter attack is less effective than for coherent pulses, because even when two pairs are produced simultaneously, Eve does not gain any information in those cases where Alice and Bob detect photons belonging to the same pair, because then the photon detected by Eve originates from a different pair and is completely uncorrelated to Alice's and Bob's photons." }, { "name": "quant-ph9912117.extracted_bib", "string": "{entexpP.R. Tapster, J.G. Rarity, and P.C.M. Owens, Phys. Rev. Lett. {73, 1923 (1994); W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Phys. Rev. Lett. {81, 3563 (1998); G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. {81, 5039 (1998)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{WignerE.P. Wigner, Am. J. Phys., {38, 1005 (1970)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{ryff M. Zukowski, private communication; L. C. Ryff, Am. J. Phys. {65(12), 1197 (1997)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{BBMC.H. Bennett, G. Brassard, and N.D. Mermin, Phys. Rev. Lett. {68, 557 (1992)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{kwiat P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. {75, 4337 (1995)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{rngpaper T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, to appear in Rev. Sci. Instr." }, { "name": "quant-ph9912117.extracted_bib", "string": "{cova S. Cova, M. Ghioni, A. Lacaita, C. Samori, and F. Zappa, Appl. Opt., {35, 1956 (1996)." }, { "name": "quant-ph9912117.extracted_bib", "string": "{bias Note that it would be simple to bias the frequencies of analyzer combinations to increase the production rate of the keys." }, { "name": "quant-ph9912117.extracted_bib", "string": "{discard Removal of one bit erases the information about the blocks contained in the (public) parities." }, { "name": "quant-ph9912117.extracted_bib", "string": "{Bitmap Windows-BMP format containing $60 \\times 90$ pixel, 8 bit color information per pixel: $43200$ bit of picture information. The file includes some header information and a color table, making the entire picture file $51840$ bit. We encrypted only the picture information, leaving the file header and the color table unchanged." }, { "name": "quant-ph9912117.extracted_bib", "string": "{Venusdate The ``Venus'' von Willendorf was found in $1908$ at Willendorf in Austria and presently resides in the Naturhistorisches Museum, Vienna. Carved from limestone and dated $24.000$--$22.000$ BC, she represents an icon of prehistoric art." } ]
quant-ph9912118	A Fast and Compact Quantum Random Number Generator	[ { "author": "Thomas Jennewein" }, { "author": "Ulrich Achleitner\\dag" }, { "author": "Gregor Weihs" }, { "author": "Harald Weinfurter\\ddag" }, { "author": "and Anton Zeilinger" } ]	We present the realization of a physical quantum random number generator based on the process of splitting a beam of photons on a beam splitter, a quantum mechanical source of true randomness. By utilizing either a beam splitter or a polarizing beam splitter, single photon detectors and high speed electronics the presented devices are capable of generating a binary random signal with an autocorrelation time of $11.8$~ns and a continuous stream of random numbers at a rate of 1~Mbit/s. The randomness of the generated signals and numbers is shown by running a series of tests upon data samples. The devices described in this paper are built into compact housings and are simple to operate.	[ { "name": "Rng.tex", "string": "\\documentstyle[preprint,prb,aps,graphicx]{revtex}\n\n\n%\\usepackage{graphicx}\n\n%\\usepackage{float}\n\n\n\n\n%\\newcommand{\\unit}[1]{\\; \\mathrm{#1}}\n%\\newcommand{\\ket}[1]{\|#1\\rangle}\n%\\newcommand{\\bra}[1]{\\langle#1\|}\n\n\n\n\n\n\n\n\\begin{document}\n\n\n\\title{A Fast and Compact Quantum Random Number\nGenerator}\n\n \\author{Thomas Jennewein, Ulrich Achleitner\\dag, Gregor Weihs, \\\\\n Harald Weinfurter\\ddag, and Anton Zeilinger}\n\n\n \\address{Institut f\\\"{u}r Experimentalphysik, Universit\\\"{a}t Wien, \\\\\n Boltzmanngasse 5, A--1090 Wien, Austria \\\\\n \\dag Institut f\\\"{u}r Experimentelle Anaesthesie,\n Universit\\\"{a}tsklinik f\\\"{u}r Anaesthesie und Intensivmedizin,\n Anichstra{\\ss}e 35, A--6020 Innsbruck, Austria \\\\\n \\ddag Sektion Physik, Ludwig-Maximilians-Universit\\\"{a}t Muenchen\\\\\n Schellingstr. 4/III D-80799 M\\\"{u}nchen, Germany}\n\n\\date{\\today}\n\n\\maketitle\n\n\n\\thispagestyle{empty}\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%\\abstract{ We present the realization of a physical quantum random\n%generator which has a maximum toggle rate of $170$~MHz and\n%an autocorrelation time of less than $14$~ns. The source of\n%randomness is either coherent beam splitting of photons or the\n%polarization measurement of photons polarized at $45^\\circ$ in\n%respect to the axis of a two channel polarization analyzer. The\n%photons in the two output beams are detected with fast photo\n%multipliers and processed with high speed digital electronics. By\n%periodically sampling this random signal, our device can produce\n%random numbers at rates well above $1$~MBit/s. }\n\n\\begin{abstract}\nWe present the realization of a physical quantum random number\ngenerator based on the process of splitting a beam of photons on\na beam splitter, a quantum mechanical source of true randomness.\nBy utilizing either a beam splitter or a polarizing beam splitter,\nsingle photon detectors and high speed electronics the presented\ndevices are capable of generating a binary random signal with an\nautocorrelation time of $11.8$~ns and a continuous stream of\nrandom numbers at a rate of 1~Mbit/s. The randomness of the\ngenerated signals and numbers is shown by running a series of\ntests upon data samples. The devices described in this paper are\nbuilt into compact housings and are simple to operate.\n\\end{abstract}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n% body of paper here\n\n\\newpage\n\n\\section{Introduction}\nRandom numbers are a vital ingredient in many applications ranging,\nto name some examples, from computational methods such as Monte Carlo\nsimulations and programming \\cite{AGA89}, over to the large field of\ncryptography for generating of crypto code or masking messages, as\nfar as to commercial applications like lottery games and slot\nmachines \\cite{MA93,MACD,Maurer90,Ellison}. Recently the range of\napplications requiring random numbers was extended with the\ndevelopment of quantum cryptography and quantum information\nprocessing \\cite{BB92}. Yet a novelty is the application for which\nthe random number generator presented in this paper was developed\nfor: an experiment regarding the entanglement of two particles, a\nfundamental concept within quantum theory \\cite{gregor}. Firstly this\nexperiment demanded the generation of random signals with an\nautocorrelation time of $<100$~ns. Secondly, for the clarity of the\nexperimental results, it was necessary that true (objective)\nrandomness was implemented.\n\nThe range of applications using random numbers has lead both to\nthe development of various random number generators as well as to\nthe means for testing the randomness of their output. Generally\nthere are two approaches of random number generation, the pseudo\nrandom generators which rely on algorithms that are implemented on\na computing device, and the physical random generators that\nmeasure some physical observable expected to behave randomly.\n\nPseudo random generators are based on algorithms or even a\ncombination of algorithms and have been highly refined in terms of\nrepetition periods\\cite{KU94} ($2^{800}$) and robustness against\ntests for randomness\\cite{MA93}. But the inherent algorithmic\nevolution of pseudo random generators is an essential problem in\napplications requiring unpredictable numbers as the unguessability\nof the numbers relies on the randomness of the seeding of the\ninternal state. Dependent on the intended application this can be\na drawback. The requirements of our specific implementation were\neven such, that the use of a pseudo random number generator was in\nitself already ruled out by its deterministic nature.\n\nPhysical random generators use the randomness or noise of a physical\nobservable, such as noise of electronic devices, signals of\nmicrophones, etc. \\cite{Ellison}. Many such physical sources utilize\nthe behavior of a very large and complex physical systems which have\na chaotic, yet at least in principle deterministic, evolution in\ntime. Due to the many unknown parameters of large systems their\nbehavior is taken for true randomness. Still, purely classical\nsystems have a deterministic nature over relevant time scales, and\nexternal influences into the random generator may remain hidden.\n\n\nCurrent theory implies that the only way to realize a clear and\nunderstandable physical source of randomness is the use of elementary\nquantum mechanical decisions, since in the general understanding the\noccurrence of each individual result of such a quantum mechanical\ndecision is objectively random (unguessable, unknowable). There\nexists a range of such elementary decisions which are suitable\ncandidates for a source of randomness. The most obvious process is\nthe decay of radioactive nucleus ($^{85}\\mathrm{Kr}$,\n$^{60}\\mathrm{Co}$) which has already been used \\cite{IS56,WA96}.\nHowever, the handling of radioactive substances demands extra\nprecautions, especially at the radioactivity required by the\nswitching rates of our envisaged random signals. Optical processes\nsuitable as a source of randomness are the splitting of single photon\nbeams \\cite{AC97}, the polarization measurement of single photons,\nthe spatial distribution of laser speckles \\cite{MA91,MA86} or the\nlight-dark periods of a single trapped ion's resonance fluorescence\nsignal \\cite{IT87,SA86}. But only the first two of the mentioned\noptical processes are fast enough and, in addition, do not require an\noverwhelming technical effort in their realization. Thus we developed\na physical quantum mechanical random generator based on the splitting\nof a beam of photons with an optical 50:50 beam splitter or by\nmeasuring the polarization of single photons with a polarizing beam\nsplitter.\\cite{AC97}\n\n\\section{Theory of Operation}\n\nThe principle of operation of the random generator is shown in\nFigure~\\ref{prinzip}. For the case of the $50:50$ beam splitter (BS)\n(Figure~\\ref{prinzip}(a)), each individual photon coming from the\nlight source and traveling through the beam splitter has, for itself,\nequal probability to be found in either output of the beam splitter.\nIf a polarizing beam splitter (PBS) is used\n(Figure~\\ref{prinzip}(b)), then each individual photon polarized at\n$45^{\\circ}$ has equal probability to be found in the H (horizontal)\npolarization or V (vertical) polarization output of the polarizer.\nAnyhow, quantum theory predicts for both cases that the individual\n``decisions'' are truly random and independent of each other. In our\ndevices this feature is implemented by detecting the photons in the\ntwo output beams with single photon detectors and combining the\ndetection pulse in a toggle switch (S), which has two states, 0 and\n1. If detector D1 fires, then the switch is flipped to state 0 and\nleft in this state until a detection event in detector D2 occurs,\nleaving the switch in state 1, until a event in detector D1 happens,\nand S is set to state 0. (Figure~\\ref{prinzip}(c)). In the case that\nseveral detections occur in a row in the same detector, then only\nthe first detection will toggle the switch S into the corresponding\nstate, and the following detections leave the switch unaltered.\nConsequently, the toggling of the switch between its two states\nconstitutes a binary random signal with the randomness lying in the\ntimes of the transitions between the two states. In order to avoid\nany effects of photon statistic of the source or optical interference\nonto the behaviour of the random generator the light source should be\nset to produce $\\ll 1$ photon per coherence time.\n\n\n\n\\section{Realization of the Device}\nFigure~\\ref{schaltbild} shows the circuit diagram of the physical\nquantum random generator. The light source is a red light emitting\ndiode (LED) driven by an adjustable current source (AD586 and TL081)\nwith maximally $110\\ \\mu\\mathrm{A}$. Due to the very short coherence\nlength of this kind of source ($<1$~ps) it can be ascertained, that\nmost of the time there are no photons present within the coherence\ntime of the source, thus eliminating effects of source photon\nstatistics or optical interference. The light emerging from the LED\nis guided through a piece of pipe to the beam splitter, which can be\neither a 50:50 beam splitter or a polarizing beam splitter. In the\nlatter case the photons are polarized beforehand with polarization\nfoil (POL) at $45^\\circ$ with respect to the axis of the dual channel\npolarization analyzer (PBS). The photons in the two output beams are\ndetected with fast photo multipliers\\cite{pmt} (PM1, PM2). The PMs\nare enclosed modules which contain all necessary electronics as well\nas a generator for the tube voltage, and thus only require a $+12$~V\nsupply. The tube voltages can be adjusted with potentiometers (TV1,\nTV2) for\n optimal detection pulse rates and pulse amplitudes . The\noutput signals are amplified in two Becker\\&Hickl amplifier\nmodules (A) and transmitted to the signal electronics which is\nrealized in emitter-coupled-logic (ECL). The detector pulses are\nconverted into ECL signals by two comparators (MC1652) in\nreference to adjustable threshold voltages set by potentiometers,\n(RV1, RV2). The actual synthesis of the random signal is done\nwithin a RS--flip-flop (MC10EL31) as PM1 triggers the {\\bf\nS}--input and PM2 triggers the {\\bf R}--input of the flip-flop.\nThe output of this flip-flop toggles between the high and low\nstate dependent upon whether the last detection occurred in PM1 or\nPM2. Finally the random signal is converted from ECL to TTL logic\nlevels (MC10EL22) for further usage.\n\nIn order to generate random numbers on a personal computer the\nsignal from the random generator is sampled periodically and\naccumulated in a 32~bit wide shift register\n(Figure~\\ref{sampling}). Every 32 clock cycles the contents of the\nshift register are transferred in parallel to a personal computer\nvia a fast digital I/O board. In this way a continuous stream of\nrandom numbers is transferred to a personal computer.\n\n\n\\section{Testing the Randomness of the Device}\nUp to now, no general definition of randomness exists and discussions\nstill go on. Two reasonable and widely accepted conditions for the\nrandomness of any binary sequence is its being ``chaotic'' and\n``typical''. The first of these concepts was introduced by Kolmogorov\nand deals with the algorithmic complexity of the sequence, while the\nsecond originates from Martin-L\\\"{o}v and says that no particular random\nsequence must have any features that make it distinguishable from all\nrandom sequences \\cite{CO91b,US90}. With pseudorandom generators it\nis always possible to predict all of their properties by more or less\nmathematical effort, due to the fact of knowing their algorithm. Thus\none may easily reject their randomness from a rigorous point of view.\nIn contrast, the mostly desired feature of a true random generator,\nits ``truth'', bears the principal impossibility of ever describing\nsuch a generator completely and proving its randomness beyond any\ndoubt. This could only be done by recording its random sequence for\nan infinite time. One is obviously limited experimentally to finite\nsamples taken out of the infinite random sequence. There are lots of\nempirical tests, mostly developed in connection with certain Monte\nCarlo simulation problems, for testing the randomness of such finite\nsamples \\cite{MA93,LE92}. The more tests one sample passes, the\nhigher we estimate its randomness. We estimate a test for randomness\nthe better, the smaller or more hidden the regularities may be that\nit can detect \\cite{AC97,VA95}.\n\n\nAs the range of tests for the randomness of a sequence is almost\nunlimited we must find tests which can serve as an appropriate\nmeasure of randomness according to the specific requirements of\nour application. Since the experiment that our random generators\nare designed for demands random signals at a high rate, we focus\non the time the random generators take to establish a random state\nof its signal starting from a point in time where the output state\nand the internal state of the generator may be known.\n\nWe will briefly describe the relatively intuitive tests that will\nbe applied to data samples taken from the random generator, which\nwe consider to be sufficient in qualifying the device for its use\nin the experiment.\n\n\\begin{enumerate}\n\n\n\\item {\\bf Autocorrelation Time of the Signal:} For a binary sequence as\nproduced by our random generator the autocorrelation function\nexhibits an exponential decay of the form:\\cite{mandel}\n\\begin{equation}\nA(\\tau)=A_0\\mathrm{e}^{-2R\|\\tau\|},\n\\end{equation}\nwhere $R$ is the average toggle rate of the signal, $A_0$ is the\nnormalization constant and $\\tau$ is the delay time. Per\ndefinition the autocorrelation time is given by\n\\begin{equation}\n\\tau_{ac}=\\frac{1}{2R}. \\label{autocorrelationtime}\n\\end{equation}\nThe autocorrelation function is a measure for the average\ncorrelation between the signal at a time $t$ and later time\n$t+\\tau$.\n\n\n\\item {\\bf Internal Delay within the Device:} The internal delay time\nwithin the device between the emission of a photon and its effect on\nthe output signal. This internal delay time is the minimal time the\ngenerator needs to establish a truly random state of its output.\n\n\n\\item {\\bf Equidistribution of the Signal:} This is the most obvious and\nsimple test of randomness of our device, as for random generator\nthe occurrence of each event must be equally probable. Yet, by\nitself the equidistribution is not a criterion for the randomness\nof a sequence.\n\n\\item {\\bf Distribution of Time Intervals between Events:} The transitions\nof the signal generated by our system are independent of any\npreceding events and signals within the device. For such a\nPoissonian process the time intervals between successive events\nare distributed exponentially in the following way:\n\\begin{equation}\np(T)=p_0\\mathrm{e}^{-T/T_0},\n\\end{equation}\nwhere $p(T)$ is the probability of a time interval $T$ between two\nevents, $T_0=1/R$ is the mean time interval and is the reciprocal\nvalue of the average toggle rate $R$ defined earlier and $p_0$ is\nthe normalization constant. The evaluation of $p(T)$ for a data\nsample taken from our generator shows directly for which time\nintervals the independence between events is ascertained and for\nwhich time intervals the signal is dominated by bandwidth limits\nor other deficiencies within the system.\n\n\n\\item {\\bf Further Illustrative Tests of Randomness:}\nThese statistical tests will be applied to samples of random\nnumbers produced by the random generator in order to illustrate\nthe functionality of the device. For the application our random\ngenerators are designed for these statistical measures are not as\nimportant as the tests described above, and the tests proposed\nhere represent just a tiny selection of possible test. Yet, these\ntests allow a cautious comparison of random numbers produced with\nour device with random numbers taken from other sources. The code\nfor the evaluation evaluation of these tests was developed in\n\\cite{AC97}.\n\n\\begin{enumerate}\n\n\\item {\\bf Equidistribution and Entropy of $n$--Bit Blocks:} Provided\nthat the sample data set is sufficiently long, all possible\n$n$--bit blocks (where $n$ is the length of the block) should\nappear with equal probability within the data set. A direct, but\ninsufficient, way of determining the equidistribution of a data\nset is to evaluate the mean value of all $n$--bit blocks, which\nshould be $(2^n-1)/2$. This will give the same result for any\nsymmetric distribution. The distribution of $n$--bit blocks of a\ndata set corresponds to the entropy, a value which is often used\nin the context of random number analysis. The entropy is defined\nas:\n\\begin{equation}\nH_n=-\\sum_np_i\\log_2p_i\n\\end{equation}\nand is expressed in units of bits. $p_i$ is the empirically\ndetermined probability for finding the $i$--th block. For a set of\nrandom numbers a block of the length $n$ should produce $n$ bits\nof entropy. In the case of bytes, which are blocks of 8 bits, the\nentropy of these blocks should be 8 bits.\n\n\n\n\\item {\\bf Blocks of $n$ Zeros or Ones:} Another test for the\nrandomness of a set of bits is the counting of blocks of\nconsecutive zeros or ones. Each bit is equally likely a zero as a\none, therefore the probability of finding blocks of $n$\nconcatenated zeros or ones should be proportional to a $2^{-n}$\nfunction.\n\n\\item {\\bf Monte-Carlo estimation of $\\pi$:} A pretty way of\ndemonstrating the quality of a set of random numbers produced by a\nrandom generator is a simple Monte Carlo estimation of $\\pi$. The\nidea is to map the numbers onto points on a square with a quarter\ncircle fitted into the square and count the points which lay\nwithin the quarter circle. The ratio of the number of points lying\nin the circle and the total number of points is an estimation of\n$\\pi$.\n\n\\end{enumerate}\n\n\\end{enumerate}\n\n\\section{Operation of the Device}\nTwo random generators were each built into single width NIM-module\n(dimensions: $25193\\mathrm{cm}^3$) in order to match our existing\nequipment. The optical beam splitter, the two photo multipliers and\nthe pulse amplifiers are mounted on a base-plate within the modules\nand the electronics is realized on printed circuit boards. The random\ngenerator modules require only a standard voltage supply of $\\pm6$~V\nand $+12$~V. The random signal generators were configured either with\na 50:50 beam splitter or with a polarizing beam splitter as the\nsource of randomness. In both cases they performed equally well. Yet,\nthe polarization measurement of the photons offers the advantage of\nadjusting the division ratio of the photons in the two beams by\nslightly rotating the polarization foil sitting just in front of the\nbeam splitter. The results presented here were all obtained from a\nrandom signal generator configured with the polarizing beam splitter.\n\nAfter warm up the devices require a little adjustment for maximum\naverage toggle rate and equidistribution of the output signal. The\naverage toggle rate of the random signals is checked with a\ncounter and the equidistribution is checked by sampling the signal\na couple of thousand times and counting the occurrences of zeros\nand ones. These measures are both optimized by trimming the\nreference voltage of the discriminators (RV1, RV2) and by\nadjusting the tube voltages of the photo multipliers (TV1, TV2).\nThe maximum average toggle rate of the random signals at the\noutput of the random number generators is $34.8$~MHz. Once the\ndevices are set up in this way they run stably for many hours.\n\nTypically the PMs produce output pulses with an amplitude of\nmaximally $50$~mV at a width of $2$~ns. The rise and fall time of\nthe signals produced by the random number generators is $3.3$~ns.\nAs it turns out, this limit is set by the output driver stage of\nthe electronics. The transition times of the internal ECL signals\nwas measured to be less then $1$~ns, which is in accordance with\nthe specifications of this ECL logic.\n\n\\section{Performance of the Device}\nThe time delay between the emission of a photon from the light\nsource and its effect on the output signal after running through the\ndetectors and the electronics was measured by using a pulsed light\nsource instead of the continuous LED and observing the electronic\nsignals within the generator on an oscilloscope. The total time delay\nbetween a light pulse and its effect on the output was $75$~ns, and\nconsists of $20$~ns time delay in the light source, light path and\nthe detection, $20$~ns time delay in the amplifiers and cables and\n$35$~ns time delay in the main electronics.\n\nIn order to evaluate the autocorrelation time of the signal\nproduced by the device, signal traces consisting of 15000 points\nwere recorded on a digital storage oscilloscope for three\ndifferent average toggle rates. The sampling rate of the\noscilloscope was $500$~MS/s for the toggle rates of $34.8$~MHz\nand $26$~MHz and $250$~MS/s for the toggle rate of $16$~MHz. The\nautocorrelation function for each trace was evaluated on a\npersonal computer and the autocorrelation time $\\tau_{AC}$ was\nextracted by fitting an exponential decay model to these\nfunctions. (Figure~\\ref{autocorrelation}) The resulting\nautocorrelation times are $11.8 \\pm 0.2$~ns ($14.4$~ns) for the\n$34.8$~MHz signal, $16.0 \\pm 0.6$~ns ($19.2$~ns) for the $26$~MHz\nsignal and $30.7 \\pm 0.5$~ns ($31.3$~ns) for the $16$~MHz signal\nwhich are comparable with the autocorrelation times calculated\nwith expression~(\\ref{autocorrelationtime}) from the average\ntoggle rate $R$, given in parenthesis.\n\nThe time difference between successive toggle events of the random\nsignal is measured with a time interval counter. The start input\nof the counter is triggered by the positive transition, and the\nstop input is triggered by the negative transition of the signal.\nFigure~\\ref{deltat} shows the distribution of $10^6$ time\nintervals for a random signal with an average signal toggle rate\nof $26$~MHz. For times $<3$~ns the transition time of the\nelectronics between the two logical states becomes evident as a\ncutoff in the distribution. For intervals of up to $35$~ns some\nwiggles of the distribution are apparent. This is most likely due\nto ringing of the signals on the transmission line. For times\n$>35$~ns the distribution approaches an exponential decay\nfunction. The spike at $96$~ns was identified as an artifact of\nthe counter due to an internal time marker.\n\n\nAs described earlier, our device can produce random numbers by\nperiodically sampling the signal and cyclically transferring the\ndata to a personal computer. Our personal computer (Pentium\nprocessor, $120$~MHz, $144$~MB RAM, running LabView on Win95)\nmanages to register sets of random numbers up to a size of\n$15$~MByte in a single run at a maximum sample rate of $30$~MHz.\nIn order to obtain independent and evenly distributed random\nnumbers, the sampling period must be well above the\nautocorrelation time of the random signal. We observed that for a\nsignal autocorrelation time of roughly $20$~ns a sampling rate of\n$1$~MHz suffices for obtaining ``good'' random numbers.\n\n\nAll data samples used for the following evaluations consisted of\n$80\\cdot10^6$~bits produced in continuous runs with a $1$~MHz bit\nsampling frequency. Figure~\\ref{blocks}(a) depicts the\ndistribution of blocks with $8$~bit length within a data sample.\nThis distribution approaches an even distribution, but still shows\nsome non-statistical deviations, such as a peak at in the center\nand some symmetric deviations. Possibly this is due to a yet to\nhigh sampling rate and a slight misadjustment of the generator.\nThe distribution of blocks of $n$ concatenated zeros and ones\nwithin a sample should be proportional to a $2^{-n}$--function.\n(Figure~\\ref{blocks}(b)) The slopes of the logarithmically scaled\ndistributions are measured with a linear fit and are $-0,29725 \\pm\n0,00121$ for the $n$--zero blocks and $-0.30299 \\pm 0.00138$ in\nthe case of the $n$--one blocks. Ideally, the slopes should both\nbe equal to $-\\log(2)= -0,30103$. The deviation can be understood\nas a consequence of minor differences in the probabilities of\nfinding a zero or a one at the output of the generator, again due\nto misadjustment of the generator.\n\n\nThe mean value of $8$-bit blocks, the entropy for $8$-bit blocks\nand the Monte Carlo estimation of $\\pi$ are evaluated for a data\nsample produced by our random generator and compared to data\nsamples taken from the Marsaglia CD--ROM \\cite{MACD} and a sample\ndata set built with the Turbo C++ random function \\cite{AC97}.\n(Table~\\ref{table})\n\nThe results in Table~\\ref{table} are in favor of our device but\nthe numbers must be treated with caution, as they represent only a\ncomparison of single samples which may not be representative.\n\n\n\n\\section{Discussion and Outlook}\nThe experimental results presented in the chapter above gives\nstrong support to the expectation, that our physical quantum\nrandom generator is capable of producing a random binary sequence\nwith an autocorrelation time of $12$~ns and internal delay time of\n$75$~ns. This underlines the suitability of these devices for\ntheir use in our specific experiment demanding random signal\ngenerators with a time for establishing a random output state to\nbe less than $<100$~ns, which is easily achieved by the physical\nquantum random generators presented in this paper.\n\nThe high speed of our random generators is made possible by the\nimplementation of state of the art technology using fast single\nphoton detectors as well as high speed electronics. Moreover, the\ncollection of tests applied to the signals and random numbers\nproduced with our quantum random generator demonstrate the quality\nof randomness that is obtained by using a fundamental quantum\nmechanical decision as a source of randomness.\n\nSome methods for enhancing the performance, be it in terms of signal\nequidistribution and/or autocorrelation time, can be foreseen. For\ninstance, a different method for generating the random signal would\nbe that each of the PM's toggles a $\\frac{1}{2}$-divider which\nresults in evenly distributed signals. These signals could be\ncombined in an XOR-gate in order to utilize the quantum randomness of\nthe polarization analyzer, but fully keeping the equidistribution of\nthe signal. A reduction of the signal autocorrelation time is\npossible by optimizing the signal electronics for speed (e.g. using\nECL signals throughout the design). Further, it is simple to\nparallelize several such random generators within one single device,\nas there is no crosstalk between the subunits, since the elementary\nquantum mechanical processes are completely independent and\nundetermined. Hence designing a physical quantum random number\ngenerator capable of producing true random numbers at rates\n$>100$~MBit/s or even above 1~GBit/s is a feasible task \\cite{LA93}.\n\nWe believe that random generators designed around elementary\nquantum mechanical processes will eventually find many\napplications for the production of random signals and numbers,\nsince the source of randomness is clear and the devices operate in\na straightforward fashion.\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Acknowledgement}\nThis work was supported by the Austrian Science Foundation (FWF),\nproject S6502, by the U.S. NSF grant no. PHY 97-22614, and by the\nAPART program of the Austrian Academy of Sciences.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{thebibliography}{99}\n\n\\bibitem{AGA89} J. Alspector, B. Gupta, and R. B. Allen,\n Adv. Neural Inform. Proc. Syst. {\\bf 1,} 748 (1989).\n\n\\bibitem{MA93}G. Marsaglia,\n %{\\em Monkey Tests for Random Number Generators},\n Computers \\& Mathematics with Applications, {\\bf 9,} 1 (1993).\n\n\\bibitem{MACD}G. Marsaglia,\n {\\em The Marsaglia Random Number CDROM},\n Department of Statistics and Supercomputer\n Computations Research Institute, Florida Sate University (1995).\n\n\\bibitem{Maurer90}U. Maurer,\n %{\\em A Universal Statistical Test for Random Bit Generators},\n Proceedings of CRYPTO '90, Santa Barbara CA (1990).\n\n\\bibitem{Ellison}C. M. Ellison,\n %{\\em Cryptographic Random Numbers}\n published on the internet:\n {\\tt www.clark.net/pub/cme/P1363/ranno.html} (1995).\n\n\\bibitem{BB92} For a review see: C. H. Bennett et al.,\n %{\\em Quantum Cryptography},\n Scientific American {\\bf 257}(10), 50 (1992).\n\n\\bibitem{gregor} G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A.\n Zeilinger,\n %{\\em A Bell Experiment under Strict Einstein Locality Conditions},\n Phys. Rev. Lett. {\\bf 81,} 5039 (1998).\n\n\n\\bibitem{KU94}\n Matsumoto, Kurita, ACM Transactions on Modelling and Computer\n Simulation {\\bf 4}(3), 254 (1997).\n\n\n%\\bibitem{HA64} R.L.T. Hampton,\n% {\\em A hybrid analog-digital pseudo-random noise generator},\n% Proceedings - Spring Joint Computer Conference, 287, (1964).\n%\n%\\bibitem{ER89} T. Erber, P. Hammerling, G. Hockney, M. Porrati,\n% S. Putterman,\n% {\\em Resonance Fluorescence and Quantum Jumps in Single\n% Atoms: Testing the Randomness of Quantum Mechanics},\n% Annals of Physics 190, 254-309 (1989).\n\n%\n%\\bibitem{CO91a} T. M. Cover, J. A. Thomas,\n% {\\em Elements of Information Theory},\n% John Wiley \\& Sons, (1991).\n%\n%\n%\\bibitem{MA84} G. Marsaglia,\n% {\\em A Current View of Random Number Generators},\n% in Proceedings of the Conference: \\'{Y}Computer Science\n% and Statistics: 16th Symposium of the Interface\\'{Y},\n% Atlanta, 1984;\n% Published by Elsevier Press.\n\n\n\\bibitem{IS56}\n M. Isida, H. Ikeda,\n %{\\em Random Number Generator},\n Ann. Inst. Stat. Math.\n Tokyo {\\bf 8,} 119 (1956).\n\n\\bibitem{WA96} J. Walker,\n %email:{\\tt kelvin\\@fourmilab.ch},\n published on the internet:\n {\\tt www.fourmilab.ch/hotbits/how.htm} (1996).\n\n\\bibitem{AC97} U. Achleitner,\n %{\\em Zufall und Photonen am Strahlteiler},\n Diploma Thesis, Innsbruck University (1997).\n\n\\bibitem{MA91} A. J. Martino, G. M. Morris,\n %{\\em Optical random number generator based on photoevent locations},\n Applied Optics {\\bf 30,} 981 (1991).\n\n\\bibitem{MA86} G. M. Morris, Opt. Engin. {\\bf 24,} 86 (1985); J. Marron, A. J. Martino, G. M. Morris,\n %{\\em Generation of random arrays using clipped laser speckle},\n Applied Optics {\\bf 25,} 26 (1986).\n\n\\bibitem{IT87} W. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J.\n Wineland,\n %{\\em Relative Decay Rates in Hg+ from Observation of Quantum Jumps in a Single Ion},\n Phys. Rev. Lett. {\\bf 59,} 2732 (1987).\n\n\\bibitem{SA86} Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek,\n %{\\em Observation of Quantum Jumps},\n Phys. Rev. Lett. {\\bf 57,} 1696 (1986).\n\n\\bibitem{pmt} Photomultiplier module, type HP5783-P, selected for single\n photon detection, Hamamatsu Photonics Deutschland GmbH,\n D-82211 Herrsching am Ammersee, Germany.\n\n%\\bibitem{QC95} Manual of ComScire\n% QNG Model J20KP produced by The Quantum World Corporation, P.O.\n% Box 1930, Boulder, Co. 80306-1930, Fax: (303)443-2450.\n\n\\bibitem{CO91b} A. Compagner,\n %{\\em Definitions of Randomness},\n Am. J. Phys. {\\bf 59,} 700 (1991).\n\n\\bibitem{US90} V. A. Uspekhi, A. L. Semenov, A. K. Shen,\n% {\\em Can an individual sequence of zeros and ones be random?},\n Russian Math. Surveys {\\bf 45}(1), 121 (1990).\n\n\\bibitem{LE92} P. L'Ecuyer,\n %{\\em Testing Random Number Generators},\n Proceedings of the 1992 Winter Simulation Conference,\n {\\bf 305,} IEEE Press (1992).\n\n\n\\bibitem{VA95} L. Vattulainen, T. Ala-Nissila,\n %{\\em Mission Impossible: Find a random pseudorandom number generator},\n Computers in Physics {\\bf 9,} 500 (1995).\n\n\n\\bibitem{mandel} See e.g.: L. Mandel and E. Wolf,\n %{\\em Optical Coherence and Quantum Optics},\n Cambridge University Press (1995).\n\n\\bibitem{LA93} P. Lalanne et al.,\n Opt. Engin. {\\bf 32}, 1904 (1993).\n\n\n\\end{thebibliography}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n\n\\addtocounter{page}{2}\n\n\n\\begin{figure}\n\n\\includegraphics{figure1.eps}\n\n\\vspace{0.7cm}\n\n\\caption{The source of randomness within our device is the\nsplitting of a weak light beam. This is realized with a 50:50\noptical beam splitter BS (a) or a polarizing beam splitter PBS\nwhere the incoming light is polarized with POL at $45^{\\circ}$\nwith respect to the PBS (b). The photon detections of the\ndetectors D1 and D2 in the two output paths toggle the switch S\nbetween its two states (c). This produces a randomly alternating\nbinary signal OUT.}\n\n\\label{prinzip}\n\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\n\\begin{figure}\n\n\n\\includegraphics[width=14cm]{figure2.eps}\n\n\\vspace{0.7cm}\n\n\\caption{Circuit diagram of the physical quantum random generator.\nTo the left is the light source (LED) and the configuration of the\nbeam splitter (BS/PBS) and the two photo multipliers (PM1, PM2).\nThe detection pulses of the PMs are turned into standard logic\npulses with discriminators (MC1652) and combined in the\nRS--flip-flop (MC10EL31) to generate the random signal.}\n\n\\label{schaltbild}\n\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{figure}\n\n\\includegraphics{figure3.eps}\n\n\\vspace{0.7cm}\n\n\\caption{Schematic diagram of the circuit for transferring random\nnumbers to a personal computer at a constant bit rate.}\n\n\\label{sampling}\n\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{figure}\n\n\\includegraphics{figure4.eps}\n\n\\vspace{0.7cm}\n\n\\caption{The autocorrelation functions computed from traces of the\nrandom signal with different average signal toggle rates. The\ngiven autocorrelation times are obtained with an exponential decay\nfit.}\n\n\\label{autocorrelation}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\n\\begin{figure}\n\n\\includegraphics{figure5.eps}\n\n\\vspace{0.7cm}\n\n\\caption{Distribution of $10^6$ time intervals between successive\ntransitions of the random signal with an average toggle rate of\n$26$~MHz. This distribution follows an exponential decay function\nfor times $>35$~ns. (The spike at $96$~ns is from the counter\nitself.) }\n\n\\label{deltat}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\\begin{figure}\n\n\\includegraphics{figure6.eps}\n\n\\vspace{0.7cm}\n\n\\caption{Diagram (a): distribution of bytes within $80\\cdot10^6$\nrandom bits produced with the random number generator. Diagram\n(b): the occurrence of blocks of concatenated zeros and ones\nwithin the same set of random numbers.}\n\n\\label{blocks}\n\n\\end{figure}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\\begin{table}\n\n\n\\begin{tabular}{c\|c\|c\|c\|c}\n & QRNG & Bits52 & Canada & C++ \\\\\n \\hline \\hline\n Mean & $127.50$ & $127.50$ & $126.58$ & $127.22$\\\\\n Entropy & $7.999965$ & $7.999982$ & $7.997665$ & $7.81118$\\\\\n $\\pi$ & $3.14017$ & $3.14367$ & $3.15789$ & $3.15761$\n\n\\end{tabular}\n\n\\vspace{0.7cm}\n\n\\caption{Evaluation of tests of randomness for data samples taken\nfrom a selection of sources. QRNG: data set generated with our\nphysical quantum random generator, Bits52: taken from the\nMarsaglia CD-ROM \\protect\\cite{MACD}, data set generated with a\ncombination of pseudo random generators, Canada: taken from the\nMarsaglia CD-ROM, data set generated with a commercial physical\nrandom generator, C++: data set generated with the pseudo random\ngenerator within Turbo C++ (Borland Inc.) \\protect\\cite{AC97}.}\n\n\\label{table}\n\n\\end{table}\n\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912118.extracted_bib", "string": "{AGA89 J. Alspector, B. Gupta, and R. B. Allen, Adv. Neural Inform. Proc. Syst. {1, 748 (1989)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{MA93G. Marsaglia, %{\\em Monkey Tests for Random Number Generators, Computers \\& Mathematics with Applications, {9, 1 (1993)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{MACDG. Marsaglia, {\\em The Marsaglia Random Number CDROM, Department of Statistics and Supercomputer Computations Research Institute, Florida Sate University (1995)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{Maurer90U. Maurer, %{\\em A Universal Statistical Test for Random Bit Generators, Proceedings of CRYPTO '90, Santa Barbara CA (1990)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{EllisonC. M. Ellison, %{\\em Cryptographic Random Numbers published on the internet: {\\tt www.clark.net/pub/cme/P1363/ranno.html (1995)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{BB92 For a review see: C. H. Bennett et al., %{\\em Quantum Cryptography, Scientific American {257(10), 50 (1992)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{gregor G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, %{\\em A Bell Experiment under Strict Einstein Locality Conditions, Phys. Rev. Lett. {81, 5039 (1998)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{KU94 Matsumoto, Kurita, ACM Transactions on Modelling and Computer Simulation {4(3), 254 (1997). %" }, { "name": "quant-ph9912118.extracted_bib", "string": "{HA64 R.L.T. Hampton, % {\\em A hybrid analog-digital pseudo-random noise generator, % Proceedings - Spring Joint Computer Conference, 287, (1964). % %" }, { "name": "quant-ph9912118.extracted_bib", "string": "{ER89 T. Erber, P. Hammerling, G. Hockney, M. Porrati, % S. Putterman, % {\\em Resonance Fluorescence and Quantum Jumps in Single % Atoms: Testing the Randomness of Quantum Mechanics, % Annals of Physics 190, 254-309 (1989). % %" }, { "name": "quant-ph9912118.extracted_bib", "string": "{CO91a T. M. Cover, J. A. Thomas, % {\\em Elements of Information Theory, % John Wiley \\& Sons, (1991). % % %" }, { "name": "quant-ph9912118.extracted_bib", "string": "{MA84 G. Marsaglia, % {\\em A Current View of Random Number Generators, % in Proceedings of the Conference: \\'{YComputer Science % and Statistics: 16th Symposium of the Interface\\'{Y, % Atlanta, 1984; % Published by Elsevier Press." }, { "name": "quant-ph9912118.extracted_bib", "string": "{IS56 M. Isida, H. Ikeda, %{\\em Random Number Generator, Ann. Inst. Stat. Math. Tokyo {8, 119 (1956)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{WA96 J. Walker, %email:{\\tt kelvin\\@fourmilab.ch, published on the internet: {\\tt www.fourmilab.ch/hotbits/how.htm (1996)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{AC97 U. Achleitner, %{\\em Zufall und Photonen am Strahlteiler, Diploma Thesis, Innsbruck University (1997)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{MA91 A. J. Martino, G. M. Morris, %{\\em Optical random number generator based on photoevent locations, Applied Optics {30, 981 (1991)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{MA86 G. M. Morris, Opt. Engin. {24, 86 (1985); J. Marron, A. J. Martino, G. M. Morris, %{\\em Generation of random arrays using clipped laser speckle, Applied Optics {25, 26 (1986)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{IT87 W. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J. Wineland, %{\\em Relative Decay Rates in Hg+ from Observation of Quantum Jumps in a Single Ion, Phys. Rev. Lett. {59, 2732 (1987)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{SA86 Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, %{\\em Observation of Quantum Jumps, Phys. Rev. Lett. {57, 1696 (1986)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{pmt Photomultiplier module, type HP5783-P, selected for single photon detection, Hamamatsu Photonics Deutschland GmbH, D-82211 Herrsching am Ammersee, Germany. %" }, { "name": "quant-ph9912118.extracted_bib", "string": "{QC95 Manual of ComScire % QNG Model J20KP produced by The Quantum World Corporation, P.O. % Box 1930, Boulder, Co. 80306-1930, Fax: (303)443-2450." }, { "name": "quant-ph9912118.extracted_bib", "string": "{CO91b A. Compagner, %{\\em Definitions of Randomness, Am. J. Phys. {59, 700 (1991)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{US90 V. A. Uspekhi, A. L. Semenov, A. K. Shen, % {\\em Can an individual sequence of zeros and ones be random?, Russian Math. Surveys {45(1), 121 (1990)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{LE92 P. L'Ecuyer, %{\\em Testing Random Number Generators, Proceedings of the 1992 Winter Simulation Conference, {305, IEEE Press (1992)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{VA95 L. Vattulainen, T. Ala-Nissila, %{\\em Mission Impossible: Find a random pseudorandom number generator, Computers in Physics {9, 500 (1995)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{mandel See e.g.: L. Mandel and E. Wolf, %{\\em Optical Coherence and Quantum Optics, Cambridge University Press (1995)." }, { "name": "quant-ph9912118.extracted_bib", "string": "{LA93 P. Lalanne et al., Opt. Engin. {32, 1904 (1993)." } ]
quant-ph9912119	Nuclear Teleportation	[ { "author": "$^{\\dagger" } ]	Until recently, only science-fiction authors ventured to use a term teleportation. However, in the last few years, on the eve of upcoming new millennium, the situation changed very much. The present report gives a synopsis of main concepts in this area. The readers will be able to make sure that paradoxical phenomena in the microcosm give a possibility to demonstrate the exchange of properties between microobjects, removed at a very large distance from each other, when no forces act between them. A new experimental scheme with hydrogen and helium nuclei is proposed. It is expected that the results of these experiments will be considered as teleportation of nuclear properties of atoms of the simplest chemical elements. A problem of teleportation of the more palpable cargo is left to the physics of the more distant future. \smallskip	[ { "name": "nuctel_jrc.tex", "string": "\\documentstyle[12pt,aps,epsfig,epsf,floats]{revtex}%%{,prlrevtex}\n\\topmargin 0pt\n\\baselineskip=20pt\n\\vsize=490pt\n\\tighten\n\\begin{document}\n\\title{Nuclear Teleportation }\n\\author\n{$^{\\dagger}$\\,B.\\,F. Kostenko\\,\\thanks{e-mail: kostenko@cv.jinr.ru}\n\\and $^{\\dagger}$\\,V.\\,D. Kuznetsov\n\\and $^{\\ddagger}$\\,M.\\,B. Miller\n\\and \\mbox{$^{\\ddagger}$\\,A.\\,V. Sermyagin}\\and \\mbox{and $^{\\dagger}$\\,D.\\,V. Kamanin}}\n\\address{$^{\\dagger}$ Joint Institute for Nuclear Research\\\\%\nDubna Moscow Region 141980, Russia\\\\\n$^{\\ddagger}$Institute of Physical and Technology Problems\\\\ P.O.\\,Box 39 Dubna Moscow\nRegion 141980, Russia\\\\}\n\\maketitle\n\\begin{abstract}\nUntil recently, only science-fiction authors ventured to use a term teleportation. However,\nin the last few years, on the eve of upcoming new millennium, the situation changed very much. The present report\ngives a synopsis of main concepts in this area. The readers will be able to make sure that paradoxical\nphenomena in the microcosm give a possibility to demonstrate the exchange of properties between microobjects,\nremoved at a very large distance from each other, when no forces act between them. A new experimental scheme\nwith hydrogen and helium nuclei is proposed. It is expected that the results of these experiments will be\nconsidered as\nteleportation of\nnuclear properties of atoms of the simplest chemical elements. A problem of teleportation of\nthe more palpable\ncargo is left to the physics of the more distant future.\n\\smallskip\n\\end{abstract}\n\n\\subsection{Introduction}\n\nIt was in the middle of twenties that an analysis of transportation of soya beans on\nthe Chinese Eastern Railway was carried out. It appeared that counter transportation\nconstituted a greater share of the total cargo traffic. Then an original procedure of processing\n the cargoes was invented: in the number of cases it was possible to deliver bean lots\n to recipients from the nearest stations, where at that time there was a sufficient amount\n of beans of a corresponding category, intended, though, to be sent to some other and more\n remote points. Economy of a rolling stock and other advantages for the railway were obvious.\n The history fails to mention how this innovation ended. Probably the complicated events\n on the CER in the beginning of the thirties put an end to the promising experiment.\n Nevertheless, this was perhaps a first attempt to realize the supertransportation of dry\n substances, or particulate solids.\n\n\\bigskip\nThe process of teleportation (commonly accepted term for supertransportation) according to\nusual understanding is reduced to moving through space in such a way that\nthe object to be transported disappears at one spot of space and reappears exactly at the\nsame time in some other point. It is well understood that it is not necessary to move\nthrough the space the matter the object is composed of. It is enough to extract\nan exact information about inner properties of the object, then transmit this information\nto a predetermined place, and use it afterward to reconstruct the initial\nobject from a stuff that comes to hand at the point of destination. Thus the teleportation\nresults in disappearing of the object with its initial properties in the initial place and\n the identical object to reappear in another place. Without disappearing it would not be the\n teleportation, but merely a reproduction, i.\\,e. a creation of a new identical specimen, or\n a copy of the object. Let us look how physicists cope with this problem.\n\\subsection{Action-at-a-distance (teleporting information?)}\nIn 1935 Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen (EPR) developed a gedanken\nexperiment to show as they thought a defect in quantum mechanics (QM)\\,\\cite{EPR,Bohr}. This\nexperiment has obtained the\nname of the EPR-paradox, and essence of the paradox is as follows.\nThere are two particles that interacted with each other for some time and have constituted a\nsingle\nsystem. Within the framework of QM that system is described by a certain wave function.\nWhen the\n interaction of the particles is finished and they flew far away from\n each other,\n these two particles are still described by the same wave function as\n before. However, individual states\n of each separate\n particle are completely unknown, moreover, definite individual properties do not exist in principle\n as quantum mechanics postulates dictate.\n It is only after one of the particles is registered by a particle-detection system that the states\n arise to existence\n for both particles. Furthermore, these states are generated instantly and simulteneously regardless of\n the distance\nbetween the particles at the moment. This scheme is used to be considered sometimes as\nteleportation of information possible at a speed higher than that of light.\nThe real (not only \"gedanken\") experiments on teleportation of\ninformation, in the sense of EPR-effect, or \"a spooky-action-at-a-distance\", as A. Einstein\ncalled it,\nwere carried out only 30-35 years later, in the\nseventies-eighties\\,\\cite{Aspect,Clauser}. Experimenters, however, managed to achieve full\nand\ndefinite success only with photons (quanta of visible light),\nthough, experiments with atoms\\,\\cite{atoms} and protons (nuclei of hydrogen)\nwere also performed\\,\\cite{proton}.\nFor the case of photons, the experiments were carried out for various distances between\nthe members of the EPR-pairs in the moment of registration.\nThe EPR-correlation between the complementary photons was shown to survive\nup to as large distances as more than ten kilometers from one to another\nphoton\\,\\cite{10 kilometers}.\nIn the case of protons, the experiment was carried out only for much\n smaller distances (of about a\nfew centimeters) and a condition of so-called causal separation, $\\Delta x>c\\Delta t$,\nwas not met. Thus, it was not fully convincing, as have been recognized by the\nauthors of the work themselves.\n\n\n\\subsection{Teleporting photon-quantum state (or the light\nquantum itself?)}\n\n\n\nA next step in this way that suggested itself was not merely\n \"action-at-a-distance\",\n but the teleportation at least of a quantum state\nfrom one quantum\n object to another. In spite of the successful experiments with the net\n EPR-effect, it was\n thought until\n recently that even this kind of teleportation is at best\n a long way in the\n future, if at all.\nAt first sight it seems that the Heisenberg uncertainty principle forbids the\nfirst necessary step of the teleportation\nprocedure: the extraction of complete information about the inner properties of the quantum\nobject. This is because of the impossibility to obtain simultaneously the exact\nvalues of so-called complementary\nvariables of a quantum microscopic object (e.\\,g., spatial coordinates and momenta).\nNevertheless, in 1993, a group of physicists (C.\\,Bennet and his colleagues) managed\nto get round this\ndifficulty\\,\\cite{Bennett}.\n They showed that full quantum information is not necessary for the process of transferring\n quantum states from one\n object to another which are at an arbitrary large distance from each other. Besides,\n they proposed that\n a\n so-called EPR-channel of communication has to be created on the basis\n of the EPR-pair of two\n quantum object\n (let it be photons B and C, shown in FIG.\\,1).\n\\begin{figure}[h]\n\\includegraphics[width=\\textwidth]{Fotc.eps}\n\\caption{Illustration of a general idea of how the teleportation can be realized. Here A is\na light photon we want to pass to a destination place, B and C, representing an EPR-pair\n of photons, constitute\na so-called quantum transmission channel. As a result,\ndefinite properties of A are destroyed completely at the zone of scanning, and at another place\n we\nhave photon with the properties A had just before it met intermediary object B (\"vehicle\").\nNote\nthat the vehicle first contacts with (so to say \"visits\") the C-photon to which the \"cargo\" has to be\ntransported, and only later it calls A to take the cargo from it!}\n\\label{cntrate}\n\\end{figure}\n After they have interacted in a way to form a\n single system\n decaying afterward,\n the photon B is directed to a \"point of departure\", where it meets A in a device\n (a registration\n system) arranged in a mode to \"catch\" only those events, in which B\n will appear in the state, leaving no choice to its \"EPR-mate\" but\n to take the state A had\n initially -- before the\n interaction with B in the detector at the \"point of departure\".\n This experimental technique is very fine but well known to\n those skilled in the EPR-art. The conservation laws of general physics are the basis of the\n procedure realizing the system with a given selective sensitivity.\n A result of all these manipulations is that particle C gets\n something from A. It is only the quantum state. Unfortunately, not a soya bean,\n but all the same it is something.\nWhat is important from the point of view of QM, is the\ndisappearing of A in the place, notified in FIG.\\,1 as a \"Zone\nof scanning\" (ZS). That is, the procedure of interaction of B and A\nphotons destroyed A photon, in a sense that of two photons outgoing from\nthe ZS no one has definite properties of A. They constitute a new\nEPR-pair of photons, which only as a whole has the definite quantum\nstate, the individual components of the pair are deprived of these\nproperties. Thus, the photon A disappears at ZS. Exactly at the\nsame moment the photon C obtains the properties A had in the\nbeginning. Once it has happened, in view of the principle of identity\nof elementary particles, we can say\nthat A, disappearing at ZS, reappears at another place, i.\\,e., the\nteleportation is accomplished.\n This process has several paradoxical features. In spite of the absence of\n contacts between objects (particles, photons) A and C, A manages to pass its properties to C.\n It may be arranged in such a way,\n that\n the distance from A to C is large enough to prevent any exchange of signals between A and C.\n And last,\n but not\n least of interest, in contrast to the transportation of ordinary material cargo,\n when a delivery vehicle\n first visits\n the\n sender to collect the cargo from it, in the case of cargo as subtle as quantum properties,\n it is delivered in a backward fashion. Here the photon\n B plays a role of the delivery vehicle, and we can see\n that\n B first visits (interacts with) the recipient (photon C) and only after that it travels\n to the sender (A)\n for the cargo.\n\n Finally, to\n reconstruct\ninitial object completely, it is necessary to fix a time moment when\nthe interaction of A and B\noccurred\n(the moment of the arrival of the \"vehicle\" to the departure \"station\"\nafter it visits the\nrecipient), and\naccomplish the required\nexperimental\ndata\n processing in due manner. The task of recording the moment of (A-B)-interaction\n and using it in the data analysis together with the information transmitted by a quantum\n EPR-channel\n requires\n one more channel of communication, an ordinary or classical\n transmission line. Receiving information that A and B to form a\n new EPR--pair (using a classical telecommunication line), an observer\n in the point of destination may be sure that the properties of C\n are identical to those of A before the teleportation.\n\n\\bigskip\nThe new idea was immediately recognized as extremely important and a\n few groups of\nexperimenters\nset\n forth concurrently to implement it. Nevertheless, it took more than four years to overcome\n all technology obstacles\n in the way to realize the project\\,\\cite{Zeilinger,DBOSCHI}.\n This is because every experiment in this\n field, being a record by itself, is always one step farther beyond the limits of experimental state of the art\n achieved before.\n\n\n\\subsection{Start with protons}\n\n\nAn analysis of the problem carried out by authors of the present\nexperimental project\n which is now in a stage of preparation\ntakes them to a conclusion\nthat the experimental setups and instruments developed for usual, though the most\nmodern,\nnuclear-physics studies\n(high-current accelerators\nof protons and heavier nuclei, liquid\\cite{Liquid} and polarized\\cite{Polarized} hydrogen\ntargets,\n multi-parameter near $2\\pi$-geometry -- i.\\,e. semi-spherical\n \\mbox{aperture --} facility for\nparticle detection named\n\"Fobos\" at Flerov Laboratory of Nuclear Reaction of the Joint Institute for\nNuclear Research\\cite{FOBOS}),\nallow one to design\n a new way to perform the teleportation of the \"heavy\" matter\n (i.\\,e., with\n non-zero mass at rest), with\n prospects to realize the project\nin a short time.\nThus, the teleportation of the protons\n(nucleus of hydrogen atoms) could be achieved in about a year, and it would\ntake about two years to prepare the teleportation of more heavy nuclei,\ne.g., $^3$He. The concept of\n measurements consists in recording signals entering two independent\nbut strictly synchronized memory devices with the aim to select\nafterwards only those events that for sure appeared to be\ncausally separate, for even the most rapid signal (light) could not\nconnect them.\n\\begin{figure}[h]\n\\includegraphics[width=\\textwidth]{protc.eps}\n\\caption{Layout of the experiments on proton teleportation. p$_0$ is an\ninitial proton from\n the accelerator,\nLH$_2$ - liquid hydrogen target, p${_2}$p${_3}$ - entangled EPR-pair, PH$_2$ - polarized\n hydrogen target,\nC - carbon target\noperating as an analyzer of the polarization of protons by a sign of scattering angle (left-right asymmetry),\n F-1 and F-2 - large-aperture position-sensitive particle detectors (so-called Fobos-facilities). Proton\n spin-state is being teleported from the PH$_2$-target placed at x$_0$ to the point x$_1$.\n It can be arranged\n that no\n signal from x$_0$ has enough time to reach point x$_1$ before p$_2$ obtains properties\n of p$_1$ at a\n moment t$_1$. That fact is justified by the detection system F-1/F-2 connected with\n a data-processing center by\n usual communication lines. K is a point, where the spin of p${_2}$ gets a definite\n orientation: just the same,\n that one of the protons p$_1$ in the PH$_2$-target had\n before the scattering of p${_3}$ from it; the proton p$_1$ loses its definite quantum\n state, as it forms a new\n EPR-pair together with the scattered proton p${_3}$.}\n\\label{prod}\n\\end{figure}\nFIG.\\,2 shows the layout of the experiment on teleportation of spin states of\nprotons from a\npolarized target PH$_2$ into the point of destination (target C). A proton beam p$_0$ of\na suitable\nenergy within the 20-50 MeV range\n bombards the liquid-hydrogen target LH$_2$. According to\n the known experimental data, the scattering in\nthe target LH$_2$ in the direction of a second target (i.\\,e., at the c.m. angle\n $\\theta\\simeq 90^\\circ$)\nwithin a few percent occurs through a\nso-called singlet intermediary state, characterized by a zero total spin of the\ntwo-proton system\\,\\cite{proton}.\nThus, the outgoing p$_2$ and p$_3$ protons present a two-proton\nentangled\nsystem and\nare fully analogous to the EPR-correlated photons\nused for transmitting\ninformation via the quantum communication channel in the experiments\non the teleportation of\n\"massless\" matter\n(light photons), as it was discussed in the preceding section. One of\nthe scattered protons,\np$_2$, then travels to the point of destination\n(target-analyzer C), while the other,\np$_3$, comes to a point where the teleportation is expected to be\nstarted, i.\\,e., to the\nPH$_2$-target. The latter is used as a source of particles we are going to teleport.\nIn this sense, protons within this\ntarget play the same role as photons A in the above section. There\nare two features differentiating the case of protons from that of photons.\nFirst, protons p$_1$ are within the motionless target (and, thus, they\nare motionless themselves) where their density is greater; besides,\nthe protons within the PH$_2$-target have quite a\ndefinite quantum state, determined by a direction of polarization.\nThe last circumstance allows one to perform the experiment under\ncontrollable conditions, i.\\,e., this gives the possibility to check the\nexpected result of the teleportation action.\n In the case\nwhen the scattering in the\npolarized target PH$_2$ occurs under the same kinematics conditions\nas in the target\nLH$_2$ (i.\\,e., at the c.m. angle\n$\\theta\\simeq 90^\\circ$),\nthe total spin of the particles p$_1$\nand p$_3$ must also be equal to zero after collision.\nTo detect these events,\na removable circular module F-1 of the facility \"Fobos\" is supposed to\nbe used, thus, the\ndetection\nefficiency\nis hoped to be much enhanced. According to QM, if all the above conditions\nare provided,\nthe\nprotons\nreaching a point K suddenly receive the same spin projections as the protons in\nthe\npolarized target PH$_2$ have. Therefore, the teleportation of the spin\nstates from\n the PH$_2$-target\nto the recipient p$_2$ really takes place at the point K.\nThus, if the coincidence mode of the detection is provided via any\nclassical channel, then a\n strong\ncorrelation has to take place between polarization direction in the\ntarget\nPH$_2$ and the\ndirection of the\ndeflection of p$_2$-protons scattered in the carbon target C. C plays a\n role of the\nanalyzer of polarization: the protons are deflected to the left or to\nthe right depending on\nsign of their polarization, i.\\,e., the orientation of the proton\nspin that\ncan have only two alternatives (along or opposite to a given direction\n\\cite{analyzer}).\nThe second module of \"Fobos\", designated F-2 in the FIG.\\,2, crowns the procedure of\nteleportation, as it indicates the proton scattering direction in the\ncarbon target C, and hence,\n its\npolarization.\n\nIf we succeeded to make a distance between\nthe detectors F-1 and F-2 to be sufficiently large, then it would be\npossible to meet the important criteria of the space-like interval\n(causal independence) between the events of the \"departure\" of the\nquantum state from the\nPH$_2$-target and \"arrival\" of this \"cargo\" to the recipient\n(p$_2$-proton) at the point K. To prevent any exchange of signals\nbetween the points PH$_2$ and K, it is essential to choose appropriate\nproportions of some time and space segments, indicated in FIG.2.\nNamely, we have to obtain $S>ct_{12}$, where $t_{12} =\|t_{F1}-t_{F2}\|$.\nHere $t_{F1}$ and $t_{F2}$ are moments of registration of signals\nfrom the corresponding detectors F-1 and F-2 (their arrival at the\ndata collection-processing center). For simplicity, we neglected\na time of flight of the protons from K to\nC, and from the PH$_2$- and C-targets to the detectors F-1 and F-2,\nrespectively.\n\n\\subsection*{Conclusion}\nFinally, referring to the principle of identity of elementary particles\nof the same kind with\nthe same\nquantum characteristics, i.\\,e. the protons in our case, we can say that protons from a\npolarized target PH$_2$ are\ntransmitted to the destination point C (through the point K). Thus,\nin the nearest future, teleportation of\nprotons can come from the domain of dreams and fiction to the\nreality in physicists' laboratories.\n\n\n\\smallskip\n\nRemembering that\nthe above soybeans contains not only protons but as well proteins,\nsomebody\nperhaps feels\ndisillusioned.\nHowever, we should not be stingy, something should be left for physics\nof the third\nmillennium.\n\n\\bigskip\n\nThe work was supported in part by the\nRussian Foundation for Basic Research, projects nr. 99-01-01101.\n\n\\begin{thebibliography}{99}\n\\bibitem{EPR} A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description\n of Physical Reality be Considered Complete? Physical Review {\\bf 41}, 777 (1935).\n\\bibitem{Bohr} N.\\,Bohr, Can Quantum-Mechanical Description\n of Physical Reality be Considered Complete? Physical Review {\\bf 48}, 696 (1935).\n\\bibitem{Aspect} A. Aspect, J. Dalibard, G. Roger, Experimental Test of Bell's Inequalities Using\n Time-Varying Analyzers, Phys. Rev. Lett., Vol.{\\bf 49}, No.25 1804 (1982).\n\\bibitem{Clauser} J.F. Clauser, A. Shimony, Bell's Theorem: Experimental\n Tests and Implications, Rep. Prog. Phys., {\\bf 41}, 1881 (1978).\n\\bibitem{atoms} E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J.M. Raimond,\n and S. Haroche, Generation of Einstein-Podolsky-Rosen Pairs of Atoms,\n Phys. Rev. Lett. {\\bf 79}, 1 (1997).\n\\bibitem{proton} M. Lamehi-Rachti and W. Mittig, Quantum Mechanics and Hidden\n Variables: A Test of Bell's Inequality by the Measurement\n of the Spin Correlation in Low-Energy Proton-Proton\n Scattering, Phys. Rev. D, Vol. {\\bf 14}, No. 10 2543 (1976).\n\\bibitem{10 kilometers} W. Tittel, J. Brendel, H. Zbinden, N. Gisin,\n Violation of Bell Inequalities by Photons More than 10 km\n Apart, Phys. Rev. Lett., Vol. {\\bf 81}, No. 17 3563 (1998).\n\\bibitem{Bennett} C.\\,H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W.\\,K.\n Wooters, Teleporting an Unknown Quantum State via Dual Classical and\n Einstein-Podolsky-Rosen\n Channels, Phys. Rev. Lett. {\\bf 70}, 1895 (1993).\n\\bibitem{Zeilinger} D. Bauwmeester, J.\\,W. Pan, K. Mattle,\nM. Eibl, H. Weinfurter, and A. Zeilinger, Experimental quantum teleportation, Nature, Vol. {\\bf 390} 575\n(1997).\n\\bibitem{DBOSCHI} D. Boschi, S. Branca, F. De Martini, L. Hardy,\n and S. Popescu, Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual\n Classical\nand Einstein-Podolsky-Rosen Channels, Physical Review Letters,\nVol. {\\bf 80} No.\\,6 1121 (1998).\n\\bibitem{FOBOS} H.\\,-G. Ortlepp et al., FOBOS Collaboration,\nScientific Report 1995/1996 Heavy Ion Physics, B.\\,I. Pustylnik (Ed.),\nJINR E7-97-206 JINR Dubna, Russia, p.\\,236.\n\\bibitem{Liquid} M.\\,A. van Uden, R.\\,L.\\,J. van der Meer, Th.\\,S. Bauer, M. Bron,\nR. Buis, P.\\,J.\\,M. de Groen, Y. Lefevere, G.\\,J.\\,L. Nooren, H. Postma,\nG. van der Steenhoven, H.\\,W. Willering, The HARP Liquid Hydrogen System,\nNuclear Instruments And Methods In Physics Research Sect. A Vol. {\\bf 424}\nNo.2-3 580 (1999).\n\\bibitem{Polarized} D.\\,J. Crabb and W. Meyer, Solid Polarized\nTargets for Nuclear and Particle Physics Experiments, Annu. Rev. Nucl. Part.,\nSci., {\\bf 47}, 67 (1997).\n\\bibitem{analyzer} C. Tschal\\\"ar, C.\\,J. Batty and\nA.\\,I. Kilvington, A Polarization Analyzer for 40- to 50-MeV\nProtons, Nuclear Instruments and Methods, Vol. {\\bf 78} 141\n(1970).\n\n\\end{thebibliography}\n\\newpage\n\n\\newpage\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912119.extracted_bib", "string": "{EPR A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality be Considered Complete? Physical Review {41, 777 (1935)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{Bohr N.\\,Bohr, Can Quantum-Mechanical Description of Physical Reality be Considered Complete? Physical Review {48, 696 (1935)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{Aspect A. Aspect, J. Dalibard, G. Roger, Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett., Vol.{49, No.25 1804 (1982)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{Clauser J.F. Clauser, A. Shimony, Bell's Theorem: Experimental Tests and Implications, Rep. Prog. Phys., {41, 1881 (1978)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{atoms E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J.M. Raimond, and S. Haroche, Generation of Einstein-Podolsky-Rosen Pairs of Atoms, Phys. Rev. Lett. {79, 1 (1997)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{proton M. Lamehi-Rachti and W. Mittig, Quantum Mechanics and Hidden Variables: A Test of Bell's Inequality by the Measurement of the Spin Correlation in Low-Energy Proton-Proton Scattering, Phys. Rev. D, Vol. {14, No. 10 2543 (1976)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{10 kilometers W. Tittel, J. Brendel, H. Zbinden, N. Gisin, Violation of Bell Inequalities by Photons More than 10 km Apart, Phys. Rev. Lett., Vol. {81, No. 17 3563 (1998)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{Bennett C.\\,H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W.\\,K. Wooters, Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels, Phys. Rev. Lett. {70, 1895 (1993)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{Zeilinger D. Bauwmeester, J.\\,W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Experimental quantum teleportation, Nature, Vol. {390 575 (1997)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{DBOSCHI D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels, Physical Review Letters, Vol. {80 No.\\,6 1121 (1998)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{FOBOS H.\\,-G. Ortlepp et al., FOBOS Collaboration, Scientific Report 1995/1996 Heavy Ion Physics, B.\\,I. Pustylnik (Ed.), JINR E7-97-206 JINR Dubna, Russia, p.\\,236." }, { "name": "quant-ph9912119.extracted_bib", "string": "{Liquid M.\\,A. van Uden, R.\\,L.\\,J. van der Meer, Th.\\,S. Bauer, M. Bron, R. Buis, P.\\,J.\\,M. de Groen, Y. Lefevere, G.\\,J.\\,L. Nooren, H. Postma, G. van der Steenhoven, H.\\,W. Willering, The HARP Liquid Hydrogen System, Nuclear Instruments And Methods In Physics Research Sect. A Vol. {424 No.2-3 580 (1999)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{Polarized D.\\,J. Crabb and W. Meyer, Solid Polarized Targets for Nuclear and Particle Physics Experiments, Annu. Rev. Nucl. Part., Sci., {47, 67 (1997)." }, { "name": "quant-ph9912119.extracted_bib", "string": "{analyzer C. Tschal\\\"ar, C.\\,J. Batty and A.\\,I. Kilvington, A Polarization Analyzer for 40- to 50-MeV Protons, Nuclear Instruments and Methods, Vol. {78 141 (1970)." } ]
quant-ph9912120		[]		[ { "name": "quant-ph9912120.tex", "string": "%\\\\\n%Title: Life-time and hierarchy of memory in the dissipative quantum\n%model of brain\n%Authors: Eleonora Alfinito and Giuseppe Vitiello \n%Comments: 4pages, no figures, paper accepted for publication in the JCIS\n%2000 Proceedings\n\n%\\\\\n% Some recent developments of the dissipative quantum model of \n% brain are reported. 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Rev.} \\bf D} \n \n% Some other macros used in the sample text \n\\def\\st{\\scriptstyle} \n\\def\\sst{\\scriptscriptstyle} \n\\def\\mco{\\multicolumn} \n\\def\\epp{\\epsilon^{\\prime}} \n\\def\\vep{\\varepsilon} \n\\def\\we{\\wedge} \n\\def\\ra{\\rightarrow} \n\\def\\ko{K^0} \n\\def\\kb{\\bar{K^0}} \n\\def\\al{\\alpha} \n\\def\\ga{\\gamma} \n\\def\\om{\\omega} \n\\def\\Om{\\Omega} \n\\def\\ab{\\bar{\\alpha}} \n\\def\\lab{\\label} \n%\\setlength{\\baselineskip}{15pt} \n%\\renewcommand{\\baselinestretch}{1.5} % SPAZIATURA {1.5} \n\\begin{document} \n\n\n\\thispagestyle{empty} \n \n%\\vspace{2.0cm} \n\\bc \n\n{\\Large{ \\bf Life-time and hierarchy of memory in \nthe dissipative quantum model of brain}}\n\n\\vspace{8mm}\n\n\\large{Eleonora Alfinito and Giuseppe Vitiello} \\\\ \n\\small \n%\\bigskip \n%\\bigskip \n\n{\\it Dipartimento di Fisica, Universit\\`a di Salerno} \\\\ \n{\\it 84100 Salerno, Italia and INFM Unit\\`a di Salerno} \\\\\n\\indent {\\it alfinito@physics.unisa.it}\\\\\n\\indent {\\it vitiello@physics.unisa.it }\\\\ \n\\vspace{1.3cm} \n\n \n\\ec \n\\small \n\n\\normalsize\n \n \n\n%%%%%%%%%%%%%%%%%%%%%% revtex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%\\documentstyle[prd,aps,twocolumn,epsf,floats,amsfonts,amssymb,amsmath]{revtex}\n\n\n\n%\n\\newcommand{\\bib}{\\bibitem} \n\n\n\n%\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname]\n\n\n\n%%\\maketitle\n\n\n\n%\\noindent {\\Large{ \\bf Life-time and hierarchy of memory in \n%the dissipative quantum model of brain}}\n\n\n%\\vspace{8mm} \n\n%\\vspace{0.2cm} \n\n%\\large{Eleonora Alfinito and Giuseppe Vitiello} \\\\ \n%\\small \n%\\bigskip \n%\\bigskip \n\n%{\\it Dipartimento di Fisica, Universit\\`a di Salerno} \\\\ \n%{\\it 84100 Salerno, Italia and INFM Unit\\`a di Salerno} \\\\\n%\\indent {\\it alfinito@physics.unisa.it}\\\\\n%\\indent {\\it vitiello@physics.unisa.it }\\\\ \n%\\vspace{1.3cm} \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n \nIn this report we will consider some recent developments of the quantum\nmodel of brain\\cite{UR,S1,S2,CH} which include dissipative\ndynamics\\cite{VT} and time dependent frequency of the electric dipole\nwave quanta (dwq)\\cite{AV}. The dissipative quantum model of brain has\nbeen recently investigated\\cite{PV} also in relation with the\npossibility of modeling neural networks exhibiting collective dynamics\nand long range correlations among the net units. The study of such a\nquantum dynamical features for neural nets is of course of great\ninterest either in connection with computational neuroscience, either in\nconnection with quantum computational strategies based on quantum\nevolution (quantum computation). On the other hand, further developments\nof the quantum model of brain, on which here we do not report, show\nattractive features also related with the r\\^ole of microtubules in\nbrain activity\\cite{PR,P2,YA}.\n\nIn previous works\\cite{UR,S1,S2,VT} it has been considered the case of\ntime independent frequencies associated to each dwq. A more general\ncase is the one where time dependent frequencies are also\nconsidered. Such a case is of course more appropriate to realistic\nsituations where the dwq may undergo a number of fluctuating\ninteractions with other quanta and then their characteristic frequency\nmay accordingly change in time. The study of the dissipative model with\ntime dependent frequency leads to a number of interesting new features\nsome of which we will briefly discuss in the following.\n\nLet us first summarize few aspects of the quantum brain model. We\nremind that the observable specifying the ordered state is called the\norder parameter and acts as a macroscopic variable for the system. The\norder parameter is specific of the kind of symmetry into play and may\nthus be considered as a code specifying the vacuum or ground state of\nthe system. The value of the order parameter is related with the\ndensity of condensed Goldstone bosons in the vacuum and specifies the\nphase of the system with relation to the considered symmetry. Since\nphysical properties are different for different phases, also the value\nof the order parameter may be considered as a code number specifying\nthe system state.\n\nIn the quantum model of brain the information storage function is\nrepresented by the coding of the ground state through the\ncondensation of the dipole wave quanta.\n\nSuppose a vacuum of specific code number has been selected by the\nprinting of a specific information. The brain then sets in that state\nand no other vacuum state is successively accessible for recording\nanother information, unless the external stimulus carrying the new\ninformation produces a phase transition to the vacuum specified by\nthe new code number. This will destroy the previously stored\ninformation, we have {\\it overprinting}: vacua labeled by different\ncode numbers are accessible only through a sequence of phase\ntransitions from one to another one of them.\n \nIt can be shown\\cite{VT} that by taking into account the fact that the\nbrain is an open system one may reach a solution to the problem of\nmemory capacity: infinitely many vacua are accessible to memory\nprinting in such a way that in a sequential information recording the\nsuccessive printing of information does not destroy the previously\nrecorded ones; a huge memory capacity is thus achieved.\n \nIn the quantum brain model spontaneous breakdown of dipole rotational\nsymmetry is triggered by the coupling of the brain with external\nstimuli. Let us remark that {\\it once} the dipole rotational symmetry\nhas been broken (and information has thus been recorded), {\\it then,\nas a consequence}, time-reversal symmetry is also broken: {\\it Before}\nthe information recording process, the brain can in principle be in\nanyone of the infinitely many (unitarily inequivalent) vacua. {\\it\nAfter} information has been recorded, the brain state is completely\ndetermined and the brain cannot be brought to the state configuration\nin which it was {\\it before} the information printing occurred. Thus,\nthe same fact of getting information introduces {\\it the arrow of\ntime} into brain dynamics. Due to memory printing process time\nevolution of the brain states is intrinsically irreversible. Getting\ninformation introduces a partition in the time evolution, it\nintroduces the {\\it distinction} between the past and the future, a\ndistinction which did not exist {\\it before} the information recording.\n\nLet us now illustrate in more details how the quantum dissipation\nformalism\\cite{VT} allows to solve the overprinting problem in the\nquantum model of brain.\n \nThe mathematical treatment of quantum dissipation requires the\n\"doubling\"of the degrees of freedom of the dissipative system. Let\n$A_{\\kappa}$ and ${\\tilde A}_{\\kappa}$ denote the dwq mode and its\n\"doubled mode\", respectively. The $\\tilde A$ mode is the \"time-reversed\nmirror image\" of the $A$ mode and represents the environment mode. Let\n${\\cal N}_{A_{\\kappa}}$ and ${\\cal N}_{{\\tilde A}_{\\kappa}}$ denote the\nnumber of ${A_{\\kappa}}$ modes and ${\\tilde A}_{\\kappa}$ modes,\nrespectively.\n \nTaking into account dissipativity requires that the memory state,\nidentified with the vacuum ${\|0>}_{\\cal N}$ , is a condensate of {\\it\nequal number} of $A_{\\kappa}$ and ${\\tilde A}_{\\kappa}$ modes, for any\n$\\kappa$ : such a requirement ensures in fact that the flow of the\nenergy exchanged between the system and the environment is\nbalanced. Thus, the difference between the number of tilde and non-tilde\nmodes must be zero: ${\\cal N}_{A_{\\kappa}} - {\\cal N}_{{\\tilde\nA}_{\\kappa}} = 0$, for any $\\kappa$. Note that the label ${\\cal N}$ in\nthe vacuum symbol ${\|0>}_{\\cal N}$ specifies the set of integers\n$\\{{\\cal N}_{A_{\\kappa}}, ~for~any~ \\kappa \\}$ which indeed defines the\n\"initial value\" of the condensate, namely the {\\it code} number\nassociated to the information recorded at time $t_{0} = 0$. Note now\nthat the requirement ${\\cal N}_{A_{\\kappa}} - {\\cal N}_{{\\tilde\nA}_{\\kappa}} = 0, $ for any $ \\kappa$, does not uniquely fix the set\n$\\{{\\cal N}_{A_{\\kappa}}, ~for~any~ \\kappa \\}$. Also ${\|0>}_{\\cal N'}$\nwith ${\\cal N'} \\equiv \\{ {\\cal N'}_{A_{\\kappa}}; {\\cal\nN'}_{A_{\\kappa}} - {\\cal N'}_{{\\tilde A}_{\\kappa}} = 0, ~for~any~\n\\kappa \\}$ ensures the energy flow balance and therefore also\n${\|0>}_{\\cal N'}$ is an available memory state: it will correspond\nhowever to a different code number $(i.e. {\\cal N'})$ and therefore to\na different information than the one of code ${\\cal N}$. Thus,\ninfinitely many memory (vacuum) states, each one of them corresponding\nto a different code $\\cal N$, may exist: A huge number of sequentially\nrecorded informations may {\\it coexist} without destructive\ninterference since infinitely many vacua ${\|0>}_{\\cal N}$, for all\n$\\cal N$, are {\\it independently} accessible in the sequential\nrecording process. The \"brain (ground) state\" may be represented as\nthe collection (or the superposition) of the full set of memory states\n${\|0>}_{\\cal N}$, for all $\\cal N$.\n \nIn summary, one may think of the brain as a complex system with a huge\nnumber of macroscopic states (the memory states). The degeneracy\namong the vacua ${\|0>}_{\\cal N}$ plays a crucial r\\^ole in solving\nthe problem of memory capacity. The dissipative dynamics introduces\n$\\cal N$-coded \"replicas\" of the system and information printing can\nbe performed in each replica without destructive interference with\npreviously recorded informations in other replicas. In the\nnondissipative case the \"$\\cal N$-freedom\" is missing and consecutive\ninformation printing produces overprinting.\n \nWe remind that it does not exist in the infinite volume limit any\nunitary transformation which may transform one vacuum of code ${\\cal\nN}$ into another one of code ${\\cal N'}$: this fact, which is a typical\nfeature of QFT, guarantees that the corresponding printed informations\nare indeed {\\it different} or {\\it distinguishable} informations (\n$\\cal N$ is a {\\it good} code) and that each information printing is\nalso {\\it protected} against interference from other information\nprinting (absence of {\\it confusion} among informations). The effect\nof finite (realistic) size of the system may however spoil unitary\ninequivalence. In the case of open systems, in fact, transitions among\n(would be) unitary inequivalent vacua may occur (phase transitions) for\nlarge but finite volume, due to coupling with external\nenvironment. The inclusion of dissipation leads thus to a picture\nof the system \"living over many ground states\" (continuously\nundergoing phase transitions). It is to be noticed that even very weak\n(although above a certain threshold) perturbations may drive the system\nthrough its macroscopic configurations. In this way, occasional\n(random) weak perturbations are recognized to play an important r\\^ole\nin the complex behavior of the brain activity. The possibility of\ntransitions among different vacua is a feature of the model which is not\ncompletely negative: smoothing out the exact unitary inequivalence among\nmemory states has the advantage of allowing the familiar phenomenon of\nthe \"association\" of memories: once transitions among different memory\nstates are \"slightly\" allowed the possibility of associations\n(\"following a path of memories\") becomes possible. Of course, these\n\"transitions\" should only be allowed up to a certain degree in order to\navoid memory \"confusion\" and difficulties in the process of storing\n\"distinct\" informations.\n \nAccording to the original quantum brain model\\cite{UR}, the recall\nprocess is described as the excitation of dwq modes under an external\nstimulus which is \"essentially a replication signal\" of the one\nresponsible for memory printing. When dwq are excited the brain\n\"consciously feels\" the presence of the condensate pattern in the\ncorresponding coded vacuum. The replication signal thus acts as a probe\nby which the brain \"reads\" the printed information. Since the\nreplication signal is represented in terms of ${\\tilde\nA}$-modes\\cite{VT} these modes act in such a reading as the \"address\"\nof the information to be recalled. In this connection, we also observe\nthat the dwq may acquire an effective non-zero mass due to the effects\nof the system finite size. Such an effective mass will then act as a\nthreshold in the excitation energy of dwq so that, in order to trigger\nthe recall process an energy supply equal or greater than such a\nthreshold is required. When the energy supply is lower than the required\nthreshold a \"difficulty in recalling\" may be experienced. At the same\ntime, however, the threshold may positively act as a \"protection\"\nagainst unwanted perturbations (including thermalization) and\ncooperate to the stability of the memory state. In the case of zero\nthreshold any replication signal could excite the recalling and the\nbrain would fall in a state of \"continuous flow of memories\".\n \nWe observe that we are considering memory states associated to the\nground states and therefore of long life-time. These states have a\nfinite (although long) life-time because of dissipativity. Then, at\nsome time $t = \\tau$, conveniently larger than the memory life-time,\nthe memory state $\|0>_{\\cal N}$ is reduced to the \"empty\" vacuum\n$\|0>_{0}$ where ${\\cal N}_{\\kappa} = 0$ for all $\\kappa$: the\ninformation has been {\\it forgotten}. At the time $t = \\tau$ the state\n$\|0>_{0}$ is available for recording a new information. It is\ninteresting to observe that in order to not completely forget certain\ninformation, one needs to \"restore\" the ${\\cal N}$ code, namely to\n\"refresh\" the memory by {\\it brushing up} the subject (external\nstimuli maintained memory). We thus see how another familiar feature of\nmemory can find a possible explanation in the dissipative quantum model\nof brain.\n\nIt can be shown that the evolution of the memory state is controlled\nby the entropy variations: this feature indeed reflects the\nirreversibility of time evolution (breakdown of time-reversal symmetry)\ncharacteristic of dissipative systems, namely the choice of a\nprivileged direction in time evolution (arrow of time). Moreover, the\nstationary condition for the free energy functional leads to recognize\nthe memory state $\|0(t)>_{\\cal N}$ to be a finite temperature\nstate\\cite{U2}. The dissipative quantum brain model thus also brings\nus to the possibility of thermodynamic considerations in the brain\nactivity.\n\nUntil now we have considered the case of time independent dwq\nfrequencies. In the case of time dependent frequencies one\nfinds\\cite{AV} that dwq of different momentum $k$ acquire different\nlife-time values. Modes with longer life-time are the ones with higher\nmomentum. Since the momentum goes as the reciprocal of the distance over\nwhich the mode can propagate, this means that modes with shorter range\nof propagation will survive longer. On the contrary, modes with longer\nrange of propagation will decay sooner. The scenario becomes then\nparticularly interesting since this mechanism may produce the formation\nof ordered domains of finite different sizes with different degree of\nstability: smaller domains would be the more stable ones. Remember now\nthat the regions over which the dwq propagate are the domains where\nordering (i.e. symmetry breakdown) is produced. Thus we arrive at the\ndynamic formation of a hierarchy (according to their life-time or\nequivalently to their sizes) of ordered domains. Since in our case\n\"ordering\" corresponds to the recording process, we have that the\nrecording of specific information related to dwq of specific momentum\n$k$ may be \"localized\" in a domain of size proportional to $1/k$, and\nthus we also have a dynamically generated hierarchy of memories. This\nmight fit some neurophysiological observations by which some specific\nmemories seem to belong to certain regions of the brain and some other\nmemories seem to have more diffused localization. We thus see how the\ndissipative quantum dynamics leads to a dynamic organization of the\nmemories in space (i.e. in their domain of localization) and in time\n(i.e. in their persistence or life-time).\n \nOne more remark has to do with the \"competition\" between the frequency\nterm and the dissipative term in the dwq equation. Such a competition\n(i.e. which term dominates over the other one) may result in a smoothing\nout of the dissipative term, which may become then even negligible,\nor, on the contrary, in the enhancement of its dynamical r\\^ole. In the\nformer case, according to the discussion in the present chapter, we\nshould expect a lowering of the memory capacity which could manifest in\na sensible \"confusion\" of memories, a difficulty in memorizing, a\ndifficulty in recalling, namely those \"pathologies\" arising from the\nlost of the many degenerate and inequivalent vacua due to the lost of\ndissipativity. In the latter case, the memory capacity would be\nmaintained high, but the hierarchy of memory life-time would result in a\nstrong inhibition of small momenta, i.e. the number of smaller domains\nwould be greater than the number of larger ones. Since smaller domains\nhave longer life-time, we see that more persistent memories would be\nfavored with respect to short-range memories. Which in some\ncircumstances also corresponds to\na commonly experienced phenomenon. To such a situation\nwould correspond an higher degree of localization of memories, but also\nmore sensible finite volume effects corresponding to effective non-zero\nmass for the dwq and therefore again difficulties in recalling.\n \nThe competition or the balance between the frequency term and the\ndissipative term in the dwq equation is controlled by parameters whose\nvalues cannot be fixed by the model dynamics. They have to be given by\nsome external input of biochemical nature. It is quite interesting that\nthe dissipative quantum model contains such freedom in setting the\nvalues for such parameters. We recall that Takahashi, Stuart and Umezawa\ndid suggested that the formation of ordered domains could play a\nsignificant r\\^ole in establishing the bridge between the basic dynamics\nand the biomolecular phenomenology. Work along this direction is in\nprogress.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\n\\begin{thebibliography}{9999}\n\n\n\\bibitem{UR} L.M.Ricciardi and H.Umezawa, Brain physics \nand many-body\nproblems. {\\it Kibernetik} {\\bf 4}, 44 (1967)\n\\bibitem{S1} \nC.I.J.Stuart, Y.Takahashi and H.Umezawa, \nOn the stability and \nnon-local \nproperties of memory. {\\it J. Theor. Biol.} {\\bf 71}, 605 (1978)\n\\bibitem{S2} C.I.J.Stuart, Y.Takahashi and \nH. Umezawa, Mixed system brain dynamics: neural memory as a macroscopic\nordered state, {\\it Found. Phys.} {\\bf 9}, 301 (1979) \n\\bibitem{CH} S. Sivakami and V. Srinivasan, \nA model for memory,\n{\\it J. Theor. Biol.} {\\bf 102}, 287 (1983)\n\\bibitem{VT} G. Vitiello, Dissipation and memory capacity in the quantum\nbrain model. {\\it Int. J. Mod. Phys.} {\\bf 9} 973 (1995)\n\\bibitem{AV} E. Alfinito and G. Vitiello, in preparation\n\\bibitem{PV} E.Pessa and G.Vitiello, Quantum dissipation and neural \nnet dynamics, {\\it Biolectrochemistry and bioenergetics} {\\bf 48}, 339 (1999)\n\\bibitem{PR} K.H.Pribram, {\\it Languages of the brain}, \nEnglewood Cliffs, \nNew Jersey, 1971\n\\bibitem{P2} K.H.Pribram, {\\it Brain and perception}, Lawrence\nErlbaum, New Jersey, 1991\n\\bibitem{YA} M.Jibu , K.H.Pribram \nand K.Yasue, From conscious experience to memory storage and retrivial:\nThe role of quantum brain dynamics and boson condensation of evanescent\nphotons. {\\it Int. J. Mod. Phys.} {\\bf B10}, 1735 (1996) \n\\bibitem{U2} H.Umezawa, \n{\\it Advanced field theory: micro, macro and thermal concepts}, \nAmerican Institute of Physics, N.Y. 1993\n \n\\end{thebibliography} \n \n\\end{document} \n\n" } ]	[ { "name": "quant-ph9912120.extracted_bib", "string": "%\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname] %%\\maketitle %\\noindent {\\Large{ Life-time and hierarchy of memory in %the dissipative quantum model of brain %\\vspace{8mm %\\vspace{0.2cm %\\large{Eleonora Alfinito and Giuseppe Vitiello \\\\ %\\small %\\bigskip %\\bigskip %{Dipartimento di Fisica, Universit\\`a di Salerno \\\\ %{84100 Salerno, Italia and INFM Unit\\`a di Salerno \\\\ %\\indent {alfinito@physics.unisa.it\\\\ %\\indent {vitiello@physics.unisa.it \\\\ %\\vspace{1.3cm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this report we will consider some recent developments of the quantum model of brain\\cite{UR,S1,S2,CH which include dissipative dynamics\\cite{VT and time dependent frequency of the electric dipole wave quanta (dwq)\\cite{AV. The dissipative quantum model of brain has been recently investigated\\cite{PV also in relation with the possibility of modeling neural networks exhibiting collective dynamics and long range correlations among the net units. The study of such a quantum dynamical features for neural nets is of course of great interest either in connection with computational neuroscience, either in connection with quantum computational strategies based on quantum evolution (quantum computation). On the other hand, further developments of the quantum model of brain, on which here we do not report, show attractive features also related with the r\\^ole of microtubules in brain activity\\cite{PR,P2,YA. In previous works\\cite{UR,S1,S2,VT it has been considered the case of time independent frequencies associated to each dwq. A more general case is the one where time dependent frequencies are also considered. Such a case is of course more appropriate to realistic situations where the dwq may undergo a number of fluctuating interactions with other quanta and then their characteristic frequency may accordingly change in time. The study of the dissipative model with time dependent frequency leads to a number of interesting new features some of which we will briefly discuss in the following. Let us first summarize few aspects of the quantum brain model. We remind that the observable specifying the ordered state is called the order parameter and acts as a macroscopic variable for the system. The order parameter is specific of the kind of symmetry into play and may thus be considered as a code specifying the vacuum or ground state of the system. The value of the order parameter is related with the density of condensed Goldstone bosons in the vacuum and specifies the phase of the system with relation to the considered symmetry. Since physical properties are different for different phases, also the value of the order parameter may be considered as a code number specifying the system state. In the quantum model of brain the information storage function is represented by the coding of the ground state through the condensation of the dipole wave quanta. Suppose a vacuum of specific code number has been selected by the printing of a specific information. The brain then sets in that state and no other vacuum state is successively accessible for recording another information, unless the external stimulus carrying the new information produces a phase transition to the vacuum specified by the new code number. This will destroy the previously stored information, we have {overprinting: vacua labeled by different code numbers are accessible only through a sequence of phase transitions from one to another one of them. It can be shown\\cite{VT that by taking into account the fact that the brain is an open system one may reach a solution to the problem of memory capacity: infinitely many vacua are accessible to memory printing in such a way that in a sequential information recording the successive printing of information does not destroy the previously recorded ones; a huge memory capacity is thus achieved. In the quantum brain model spontaneous breakdown of dipole rotational symmetry is triggered by the coupling of the brain with external stimuli. Let us remark that {once the dipole rotational symmetry has been broken (and information has thus been recorded), {then, as a consequence, time-reversal symmetry is also broken: {Before the information recording process, the brain can in principle be in anyone of the infinitely many (unitarily inequivalent) vacua. {After information has been recorded, the brain state is completely determined and the brain cannot be brought to the state configuration in which it was {before the information printing occurred. Thus, the same fact of getting information introduces {the arrow of time into brain dynamics. Due to memory printing process time evolution of the brain states is intrinsically irreversible. Getting information introduces a partition in the time evolution, it introduces the {distinction between the past and the future, a distinction which did not exist {before the information recording. Let us now illustrate in more details how the quantum dissipation formalism\\cite{VT allows to solve the overprinting problem in the quantum model of brain. The mathematical treatment of quantum dissipation requires the \"doubling\"of the degrees of freedom of the dissipative system. Let $A_{\\kappa$ and ${\\tilde A_{\\kappa$ denote the dwq mode and its \"doubled mode\", respectively. The $\\tilde A$ mode is the \"time-reversed mirror image\" of the $A$ mode and represents the environment mode. Let ${\\cal N_{A_{\\kappa$ and ${\\cal N_{{\\tilde A_{\\kappa$ denote the number of ${A_{\\kappa$ modes and ${\\tilde A_{\\kappa$ modes, respectively. Taking into account dissipativity requires that the memory state, identified with the vacuum ${\|0>_{\\cal N$ , is a condensate of {equal number of $A_{\\kappa$ and ${\\tilde A_{\\kappa$ modes, for any $\\kappa$ : such a requirement ensures in fact that the flow of the energy exchanged between the system and the environment is balanced. Thus, the difference between the number of tilde and non-tilde modes must be zero: ${\\cal N_{A_{\\kappa - {\\cal N_{{\\tilde A_{\\kappa = 0$, for any $\\kappa$. Note that the label ${\\cal N$ in the vacuum symbol ${\|0>_{\\cal N$ specifies the set of integers $\\{{\\cal N_{A_{\\kappa, ~for~any~ \\kappa \\$ which indeed defines the \"initial value\" of the condensate, namely the {code number associated to the information recorded at time $t_{0 = 0$. Note now that the requirement ${\\cal N_{A_{\\kappa - {\\cal N_{{\\tilde A_{\\kappa = 0, $ for any $ \\kappa$, does not uniquely fix the set $\\{{\\cal N_{A_{\\kappa, ~for~any~ \\kappa \\$. Also ${\|0>_{\\cal N'$ with ${\\cal N' \\equiv \\{ {\\cal N'_{A_{\\kappa; {\\cal N'_{A_{\\kappa - {\\cal N'_{{\\tilde A_{\\kappa = 0, ~for~any~ \\kappa \\$ ensures the energy flow balance and therefore also ${\|0>_{\\cal N'$ is an available memory state: it will correspond however to a different code number $(i.e. {\\cal N')$ and therefore to a different information than the one of code ${\\cal N$. Thus, infinitely many memory (vacuum) states, each one of them corresponding to a different code $\\cal N$, may exist: A huge number of sequentially recorded informations may {coexist without destructive interference since infinitely many vacua ${\|0>_{\\cal N$, for all $\\cal N$, are {independently accessible in the sequential recording process. The \"brain (ground) state\" may be represented as the collection (or the superposition) of the full set of memory states ${\|0>_{\\cal N$, for all $\\cal N$. In summary, one may think of the brain as a complex system with a huge number of macroscopic states (the memory states). The degeneracy among the vacua ${\|0>_{\\cal N$ plays a crucial r\\^ole in solving the problem of memory capacity. The dissipative dynamics introduces $\\cal N$-coded \"replicas\" of the system and information printing can be performed in each replica without destructive interference with previously recorded informations in other replicas. In the nondissipative case the \"$\\cal N$-freedom\" is missing and consecutive information printing produces overprinting. We remind that it does not exist in the infinite volume limit any unitary transformation which may transform one vacuum of code ${\\cal N$ into another one of code ${\\cal N'$: this fact, which is a typical feature of QFT, guarantees that the corresponding printed informations are indeed {different or {distinguishable informations ( $\\cal N$ is a {good code) and that each information printing is also {protected against interference from other information printing (absence of {confusion among informations). The effect of finite (realistic) size of the system may however spoil unitary inequivalence. In the case of open systems, in fact, transitions among (would be) unitary inequivalent vacua may occur (phase transitions) for large but finite volume, due to coupling with external environment. The inclusion of dissipation leads thus to a picture of the system \"living over many ground states\" (continuously undergoing phase transitions). It is to be noticed that even very weak (although above a certain threshold) perturbations may drive the system through its macroscopic configurations. In this way, occasional (random) weak perturbations are recognized to play an important r\\^ole in the complex behavior of the brain activity. The possibility of transitions among different vacua is a feature of the model which is not completely negative: smoothing out the exact unitary inequivalence among memory states has the advantage of allowing the familiar phenomenon of the \"association\" of memories: once transitions among different memory states are \"slightly\" allowed the possibility of associations (\"following a path of memories\") becomes possible. Of course, these \"transitions\" should only be allowed up to a certain degree in order to avoid memory \"confusion\" and difficulties in the process of storing \"distinct\" informations. According to the original quantum brain model\\cite{UR, the recall process is described as the excitation of dwq modes under an external stimulus which is \"essentially a replication signal\" of the one responsible for memory printing. When dwq are excited the brain \"consciously feels\" the presence of the condensate pattern in the corresponding coded vacuum. The replication signal thus acts as a probe by which the brain \"reads\" the printed information. Since the replication signal is represented in terms of ${\\tilde A$-modes\\cite{VT these modes act in such a reading as the \"address\" of the information to be recalled. In this connection, we also observe that the dwq may acquire an effective non-zero mass due to the effects of the system finite size. Such an effective mass will then act as a threshold in the excitation energy of dwq so that, in order to trigger the recall process an energy supply equal or greater than such a threshold is required. When the energy supply is lower than the required threshold a \"difficulty in recalling\" may be experienced. At the same time, however, the threshold may positively act as a \"protection\" against unwanted perturbations (including thermalization) and cooperate to the stability of the memory state. In the case of zero threshold any replication signal could excite the recalling and the brain would fall in a state of \"continuous flow of memories\". We observe that we are considering memory states associated to the ground states and therefore of long life-time. These states have a finite (although long) life-time because of dissipativity. Then, at some time $t = \\tau$, conveniently larger than the memory life-time, the memory state $\|0>_{\\cal N$ is reduced to the \"empty\" vacuum $\|0>_{0$ where ${\\cal N_{\\kappa = 0$ for all $\\kappa$: the information has been {forgotten. At the time $t = \\tau$ the state $\|0>_{0$ is available for recording a new information. It is interesting to observe that in order to not completely forget certain information, one needs to \"restore\" the ${\\cal N$ code, namely to \"refresh\" the memory by {brushing up the subject (external stimuli maintained memory). We thus see how another familiar feature of memory can find a possible explanation in the dissipative quantum model of brain. It can be shown that the evolution of the memory state is controlled by the entropy variations: this feature indeed reflects the irreversibility of time evolution (breakdown of time-reversal symmetry) characteristic of dissipative systems, namely the choice of a privileged direction in time evolution (arrow of time). Moreover, the stationary condition for the free energy functional leads to recognize the memory state $\|0(t)>_{\\cal N$ to be a finite temperature state\\cite{U2. The dissipative quantum brain model thus also brings us to the possibility of thermodynamic considerations in the brain activity. Until now we have considered the case of time independent dwq frequencies. In the case of time dependent frequencies one finds\\cite{AV that dwq of different momentum $k$ acquire different life-time values. Modes with longer life-time are the ones with higher momentum. Since the momentum goes as the reciprocal of the distance over which the mode can propagate, this means that modes with shorter range of propagation will survive longer. On the contrary, modes with longer range of propagation will decay sooner. The scenario becomes then particularly interesting since this mechanism may produce the formation of ordered domains of finite different sizes with different degree of stability: smaller domains would be the more stable ones. Remember now that the regions over which the dwq propagate are the domains where ordering (i.e. symmetry breakdown) is produced. Thus we arrive at the dynamic formation of a hierarchy (according to their life-time or equivalently to their sizes) of ordered domains. Since in our case \"ordering\" corresponds to the recording process, we have that the recording of specific information related to dwq of specific momentum $k$ may be \"localized\" in a domain of size proportional to $1/k$, and thus we also have a dynamically generated hierarchy of memories. This might fit some neurophysiological observations by which some specific memories seem to belong to certain regions of the brain and some other memories seem to have more diffused localization. We thus see how the dissipative quantum dynamics leads to a dynamic organization of the memories in space (i.e. in their domain of localization) and in time (i.e. in their persistence or life-time). One more remark has to do with the \"competition\" between the frequency term and the dissipative term in the dwq equation. Such a competition (i.e. which term dominates over the other one) may result in a smoothing out of the dissipative term, which may become then even negligible, or, on the contrary, in the enhancement of its dynamical r\\^ole. In the former case, according to the discussion in the present chapter, we should expect a lowering of the memory capacity which could manifest in a sensible \"confusion\" of memories, a difficulty in memorizing, a difficulty in recalling, namely those \"pathologies\" arising from the lost of the many degenerate and inequivalent vacua due to the lost of dissipativity. In the latter case, the memory capacity would be maintained high, but the hierarchy of memory life-time would result in a strong inhibition of small momenta, i.e. the number of smaller domains would be greater than the number of larger ones. Since smaller domains have longer life-time, we see that more persistent memories would be favored with respect to short-range memories. Which in some circumstances also corresponds to a commonly experienced phenomenon. To such a situation would correspond an higher degree of localization of memories, but also more sensible finite volume effects corresponding to effective non-zero mass for the dwq and therefore again difficulties in recalling. The competition or the balance between the frequency term and the dissipative term in the dwq equation is controlled by parameters whose values cannot be fixed by the model dynamics. They have to be given by some external input of biochemical nature. It is quite interesting that the dissipative quantum model contains such freedom in setting the values for such parameters. We recall that Takahashi, Stuart and Umezawa did suggested that the formation of ordered domains could play a significant r\\^ole in establishing the bridge between the basic dynamics and the biomolecular phenomenology. Work along this direction is in progress. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\begin{thebibliography{9999" }, { "name": "quant-ph9912120.extracted_bib", "string": "{UR L.M.Ricciardi and H.Umezawa, Brain physics and many-body problems. {Kibernetik {4, 44 (1967)" }, { "name": "quant-ph9912120.extracted_bib", "string": "{S1 C.I.J.Stuart, Y.Takahashi and H.Umezawa, On the stability and non-local properties of memory. {J. Theor. Biol. {71, 605 (1978)" }, { "name": "quant-ph9912120.extracted_bib", "string": "{S2 C.I.J.Stuart, Y.Takahashi and H. Umezawa, Mixed system brain dynamics: neural memory as a macroscopic ordered state, {Found. Phys. {9, 301 (1979)" }, { "name": "quant-ph9912120.extracted_bib", "string": "{CH S. Sivakami and V. Srinivasan, A model for memory, {J. Theor. Biol. {102, 287 (1983)" }, { "name": "quant-ph9912120.extracted_bib", "string": "{VT G. Vitiello, Dissipation and memory capacity in the quantum brain model. {Int. J. Mod. Phys. {9 973 (1995)" }, { "name": "quant-ph9912120.extracted_bib", "string": "{AV E. Alfinito and G. Vitiello, in preparation" }, { "name": "quant-ph9912120.extracted_bib", "string": "{PV E.Pessa and G.Vitiello, Quantum dissipation and neural net dynamics, {Biolectrochemistry and bioenergetics {48, 339 (1999)" }, { "name": "quant-ph9912120.extracted_bib", "string": "{PR K.H.Pribram, {Languages of the brain, Englewood Cliffs, New Jersey, 1971" }, { "name": "quant-ph9912120.extracted_bib", "string": "{P2 K.H.Pribram, {Brain and perception, Lawrence Erlbaum, New Jersey, 1991" }, { "name": "quant-ph9912120.extracted_bib", "string": "{YA M.Jibu , K.H.Pribram and K.Yasue, From conscious experience to memory storage and retrivial: The role of quantum brain dynamics and boson condensation of evanescent photons. {Int. J. Mod. Phys. {B10, 1735 (1996)" }, { "name": "quant-ph9912120.extracted_bib", "string": "{U2 H.Umezawa, {Advanced field theory: micro, macro and thermal concepts, American Institute of Physics, N.Y. 1993" } ]
quant-ph9912122	Optimal signal ensembles	[ { "author": "Benjamin Schumacher$^{(1)" } ]		[ { "name": "optimal.tex", "string": "\\documentclass[12pt]{article}\n\n\\usepackage{graphicx}\n\n\\newcommand{\\tr}{\\mbox{Tr} \\, }\n\\newcommand{\\ket}[1]{\\left \| #1 \\right \\rangle}\n\\newcommand{\\bra}[1]{\\left \\langle #1 \\right \|}\n\\newcommand{\\amp}[2]{\\left \\langle #1 \\left \| #2 \\right. \\right \\rangle}\n\n\\newcommand{\\proj}[1]{\\ket{#1} \\! \\bra{#1}}\n\\newcommand{\\outerprod}[2]{\\ket{#1} \\! \\bra{#2}}\n\\newcommand{\\ave}[1]{\\left \\langle #1 \\right \\rangle}\n\\newcommand{\\superop}{{\\cal E}}\n\\newcommand{\\unity}{\\mbox{\\bf I}}\n\\newcommand{\\hilbert}{{\\cal H}}\n\\newcommand{\\relent}[2]{{\\cal D}\\left ( #1 \|\| #2 \\right )}\n\\newcommand{\\supp}{\\mbox{supp} \\, }\n\\newcommand{\\avail}{\\mbox{$\\cal A$}}\n\\newcommand{\\bigsum}[1]{{\\displaystyle \\sum_{#1}}}\n\n\\begin{document}\n\n\\title{Optimal signal ensembles}\n\\author{Benjamin Schumacher$^{(1)}$\n and Michael D. Westmoreland$^{(2)}$}\n\\maketitle\n\\begin{center}\n{\\sl\n$^{(1)}$Department of Physics, Kenyon College, Gambier, OH 43022 USA \\\\\n$^{(2)}$Department of Mathematical Sciences, Denison University,\n Granville, OH 43023 USA }\n\\end{center}\n\n\\section{Abstract}\n\nClassical messages can be sent via a noisy quantum channel in\nvarious ways, corresponding to various choices of ensembles of\nsignal states of the channel. Previous work by Holevo and by \nSchumacher and Westmoreland relates the capacity of the channel\nto the properties of the signal ensemble. Here we describe some\nproperties characterizing the ensemble that maximizes the capcity,\nusing the relative entropy ``distance'' between density operators\nto give the results a geometric flavor.\n\n\\section{Communication via quantum channels}\n\nSuppose Alice wishes to send a (classical) message to Bob, using a\nquantum system as the communication channel. Alice prepares the\nsystem in the ``signal state'' $\\rho_k$ with probability $p_k$, so\nthat the ensemble of states is described by an average density\noperator $\\rho = \\bigsum{k} p_k \\, \\rho_k$.\nBob makes a measurement of a ``decoding observable''\non the system and uses the result to infer which signal state\nwas prepared. The choice of system preparation (represented\nby the index $k$) and Bob's measurement outcome are the input\nand output of a classical communication channel.\n\nHolevo \\cite{holevo1} proved (as Gordon \\cite{gordon} and\nLevitin \\cite{levitin} had previously conjectured) that the\nmutual information between the input and output of this channel,\nregardless of Bob's choice of decoding observable, can never\nbe greater than $\\chi$, where\n\\begin{equation}\n \\chi = S(\\rho) - \\sum_{k} p_k S(\\rho_k) \\label{chidef}\n\\end{equation}\nwhere $S(\\rho) = - \\tr \\rho \\log \\rho$ is the von Neumann\nentropy of the density operator $\\rho$.\n\nMore recently, it has been shown by Holevo \\cite{holevo2} and by\nSchumacher and Westmoreland \\cite{noisy} that the Holevo bound is\nasymptotically achievable. That is, if Alice uses many copies of\nthe same channel, preparing long code words of signal states, and\nif Bob chooses an entangled decoding observable, Alice can convey\nto Bob up to $\\chi$ bits of information per use of the channel,\nwith arbitrarily low probability of error. (This fact was\nfirst shown for pure state signals in \\cite{hjsww}.)\n\nSuppose the channel is a noisy one described by a superoperator\n$\\superop$. Then if Alice prepares the input signal state\n$\\rho_k$, Bob will receive the output signal state\n$\\superop (\\rho_k)$. It is the ensemble of output\nsignal states that determines the capacity of the channel.\nEffectively, the superoperator $\\superop$ restricts the set\nof signals that Alice can present to Bob for decoding.\nIf $\\cal B$ is the set of all density operators, then\nAlice's efforts can only produce output states in the\nset $\\avail = \\superop ( {\\cal B} )$.\n\nIn this paper we will consider the problem of maximizing $\\chi$\nfor ensembles of states drawn from a given set\n$\\avail$ of available states.\nThis includes the problem of maximizing $\\chi$ for the\noutputs of a noisy channel, if $\\avail$ is chosen to be the set\nof possible channel outputs. In this case, $\\avail$ will be a\nconvex set; but we will not need the convexity of $\\avail$ for\nmany of our results.\n\n\n\\section{Relative entropy}\n\nIf $\\rho$ and $\\sigma$ are density operators, then the\n{\\em relative entropy} of $\\rho$ with respect to $\\sigma$\nis defined to be\n\\begin{equation}\n\\relent{\\rho}{\\sigma} = \\tr \\rho \\log \\rho - \\tr \\rho \\log \\sigma .\n \\label{relentdef}\n\\end{equation}\nHere are three important points about the relative entropy:\n\\begin{itemize}\n\\item $\\relent{\\rho}{\\sigma} \\geq 0$, with equality if and\n\tonly if $\\rho = \\sigma$.\n\\item Strictly speaking, $\\relent{\\rho}{\\sigma}$ is defined\n\tonly if $\\supp \\rho \\subseteq \\supp \\sigma$\n\t(where ``$\\supp \\rho$'' is the support of the operator\n\t$\\rho$). If this is not the case, then we take\n\t$\\relent{\\rho}{\\sigma} = \\infty$. For\n example, if $\\rho$ and $\\sigma$ are distinct pure states,\n the relative entropy is always infinite.\n\\item The relative entropy is jointly convex in its arguments:\n \\begin{equation}\n \\relent{p_{1} \\rho_{1} + p_{2} \\rho_{2}}{p_{1} \\sigma_{1}\n + p_{2} \\sigma_{2}} \\leq\n p_{1} \\relent{\\rho_{1}}{\\sigma_{1}} +\n p_{2} \\relent{\\rho_{2}}{\\sigma_{2}}\n \\end{equation}\n for $p_{1}, p_{2} \\geq 0$ with $p_{1} + p_{2} = 1$.\n\tFrom this fact it also follows that the relative entropy\n\tis convex in each of its arguments.\n\\end{itemize}\nThe relative entropy plays a role in the asymptotic distinguishability\nof quantum states by measurement \\cite{distinguish}, and has been used\nto develop measures of quantum entanglement \\cite{entangle}\n\nIt is often convenient to think of the relative entropy\n$\\relent{\\rho}{\\sigma}$ as a ``directed distance'' from $\\sigma$\nto $\\rho$, even though it lacks some of the properties of a true metric.\nThis view of the relative entropy will let us give a geometric\ninterpretation to our results.\n\nSuppose as before we have an ensemble of signal states in\nthe available set $\\avail$, in which $\\rho_k$ appears\nwith probability $p_k$. It is easy to verify that the\nHolevo bound $\\chi$ can be given in terms of the relative entropy:\n\\begin{equation}\n\\chi = \\sum_k p_k \\relent{\\rho_k}{\\rho} . \\label{chirelent}\n\\end{equation}\nThat is, $\\chi$ is just the average of the relative\nentropy of the members of the signal ensemble with respect\nto the average signal state.\n\n\n\\section{The optimal signal ensemble}\n\nTo maximize the information capacity of the channel,\nAlice will want to choose a signal ensemble that maximizes $\\chi$.\nWe will denote the maximum of $\\chi$ for a given set\n$\\avail$ of available states by $\\chi^{}$. Any ensemble\nof signal states that achieves this value\nof the Holevo bound will be called an {\\em optimal} signal ensemble.\n\nIf the set of available states $\\avail$ is a closed convex set,\nthen we can always take an optimal ensemble to be composed of\nextreme points of $\\avail$---that is, states which cannot be written\nas convex sums of other states in $\\avail$.\nTo see this, suppose we have an ensemble of $\\avail$-states\nwith average state $\\rho$, and further suppose that $\\rho_{k}$\nis a member of the ensemble that is not an extreme point. This\nmeans that there are states $\\rho_{k0}$ and $\\rho_{k1}$ in $\\avail$\nsuch that\n\\begin{equation}\n \\rho_{k} = q_{0} \\rho_{k0} + q_{1} \\rho_{k1}\n\\end{equation}\nfor probabilities $q_{0}$ and $q_{1}$ that sum to unity. By the\nconvexity\nof the relative entropy,\n\\begin{equation}\n \\relent{\\rho_{k}}{\\rho} \\leq q_{0} \\relent{\\rho_{k0}}{\\rho}\n + q_{1} \\relent{\\rho_{k1}}{\\rho} .\n\\end{equation}\nSince $\\chi$ is the average of the relative entropies, we will never\nmake\n$\\chi$ smaller by replacing $\\rho_{k}$ (with probability $p_{k}$) by\n$\\rho_{k0}$ and $\\rho_{k1}$ (with probabilities $p_{k} q_{0}$ and\n$p_{k} q_{1}$, respectively) in the ensemble. Thus, at least\none optimal ensemble will be composed of extreme points of $\\avail$.\n\nFor noisy channels, this means that pure state inputs to the channel\nare optimal -- that is, it never increases $\\chi$ to use\nmixed states as inputs. This fact was shown in \\cite{noisy}.\n\nA second and very surprising fact was discovered by Fuchs\n\\cite{fuchs}. The quantity $\\chi$ is a measure of\nthe distinguishability of an ensemble of signal states.\nIf we wish to maximize the distinguishability of the\noutput signals of a noisy channel, we might imagine that we should\nalways maximize the distinguishability\nof the input signals---i.e., choose an orthogonal set\nof input states. But this intuition turns out to be false.\n\nSome insight can be gained by examining a specific counter-example.\nOur quantum system is a spin, and $\\ket{\\uparrow}$ and\n$\\ket{\\downarrow}$\nrepresent eigenstates of $S_{z}$. The spin is subject\nto ``amplitude damping'',\nso that an initial density operator $\\rho$ evolves into a density\noperator\n\\begin{equation}\n \\rho' = \\superop(\\rho) = A_{1} \\rho A_{1}^{\\dagger}\n + A_{2} \\rho A_{2}^{\\dagger}\n\\end{equation}\nwhere $A_{1} = \\sqrt{1-\\lambda} \\proj{\\uparrow} + \\proj{\\downarrow}$\nand\n$A_{2} = \\sqrt{\\lambda} \\outerprod{\\downarrow}{\\uparrow}$,\nand $0 \\leq \\lambda \\leq 1$.\nThe result of this operation is, for instance, to leave the state\n$\\ket{\\downarrow}$ unchanged but to cause\n$\\ket{\\uparrow}$ to decay to $\\ket{\\downarrow}$\nwith probability $\\lambda$. We choose $\\lambda = 1/2$.\nA diagram of\nthis process in the Bloch sphere is found in Figure~\\ref{fuchs}.\n\nIf we consider only orthogonal input signal ensembles, the maximum\n$\\chi$ is obtained for an equally weighted ensemble of\n$\\ket{\\rightarrow}$ and $\\ket{\\leftarrow}$,\nfor which $\\chi = 0.4567$ bits.\nBut a non-orthogonal ensemble of the states\n$\\ket{\\phi_0}$ and $\\ket{\\phi_1}$ can achieve\n$0.4717$ bits, where the angle in\nHilbert space between the two inputs is about 80$^{\\circ}$.\n\nWhy is this? Recall that $\\chi$ is the average relative entropy\n``distance''\nfrom the average signal state to the individual signal states.\nThis distance function grows\nlarger near the boundary of the Bloch\nsphere--so that, for example, the relative entropy distance between\ndistinct pure states is infinite. Thus, despite the appearance in\nFigure~\\ref{fuchs}, the relative entropy distances for the ensemble\nof $\\rho_{0}$ and $\\rho_{1}$ are {\\em greater} than those for the\nensemble of $\\rho_{\\rightarrow}$ and $\\rho_{\\leftarrow}$.\n\n\n\\section{Changing the ensemble}\n\nIn this section we will prove some useful results that\nwill enable us to further characterize the optimal ensembles\nfor a given set $\\avail$ of available states.\n\nSuppose as before that the signal state $\\rho_{k} \\in \\avail$\nappears in our ensemble with probability $p_{k}$,\nyielding an average state $\\rho$.\nLet $\\sigma$ be some other density operator,\nwhich we will call the ``alternate'' state.\nThen we can calculate the average relative entropy distance\nof the signal states from $\\sigma$:\n\\begin{eqnarray}\n\\sum_{k} p_k \\relent{\\rho_k}{\\sigma}\n & = & \\sum_{k} p_k \\left ( \\tr \\rho_k \\log \\rho_k\n - \\tr \\rho_k \\log \\sigma \\right ) \\nonumber \\\\\n & = & \\sum_{k} p_k \\left ( \\tr \\rho_k \\log \\rho_k\n - \\tr \\rho_k \\log \\rho \\right ) \\nonumber \\\\\n & & + \\left ( \\tr \\rho \\log \\rho\n - \\tr \\rho \\log \\sigma \\right ) \\nonumber \\\\\n & & \\nonumber \\\\\n & = & \\sum_k p_k \\relent{\\rho_k}{\\rho} + \\relent{\\rho}{\\sigma}\n \\nonumber \\\\\n\\sum_{k} p_k \\relent{\\rho_k}{\\sigma}\n & = & \\chi + \\relent{\\rho}{\\sigma} \\label{coverish} .\n\\end{eqnarray}\nThis useful identity, first given by Donald\\cite{donald},\nhas a number of implications. For example,\n\\begin{itemize}\n\\item For any ensemble and any $\\sigma$,\n \\begin{equation}\n \\sum_k p_k \\relent{\\rho_k}{\\sigma} \\geq \\chi\n\t\\label{donaldcorollary}\n \\end{equation}\n with equality if and only if $\\sigma = \\rho$.\n\\item From the previous point it follows that\n \\begin{equation}\n \\chi = \\min_{\\sigma} \\left ( \\sum_k p_k\n\t\t\\relent{\\rho_k}{\\sigma} \\right )\n \\end{equation}\n where the minimum is taken over all density operators\n\t$\\sigma$.\n\\end{itemize}\n\nNow we will use our identity to consider how the value of $\\chi$\nwould change if we were to modify our ensemble. In particular, we\ncan introduce a new state $\\rho_{0}$ with probability $\\eta$,\nshrinking the other probabilities to maintain normalization. We\nmay conveniently refer to our ensembles as the ``original'' and\n``modified'' ensembles, as summarized in the following table:\n\\begin{center}\n\\begin{tabular}{lcc}\nensemble & original & modified \\\\\nsignal states & $\\rho_{k}$ & $\\rho_{k}, \\rho_{0}$\n\\\\\nprobabilities & $p_{k}$ & $(1-\\eta) p_{k},\n\\eta$ \\\\\naverage state & $\\rho$ & $\\rho'$ \\\\\nHolevo bound & $\\chi$ & $\\chi'$\n\\end{tabular}\n\\end{center}\nwhere\n\\begin{eqnarray}\n\\rho' & = & (1 - \\eta) \\rho + \\eta \\rho_{0} \\\\\n\\chi & = & \\sum_{k} p_{k} \\relent{\\rho_{k}}{\\rho} \\\\\n\\chi' & = & (1-\\eta)\\sum_{k} p_{k} \\relent{\\rho_{k}}{\\rho'} \\,\\, +\n\\,\\,\n \\eta \\relent{\\rho_{0}}{\\rho'} .\n\\end{eqnarray}\nWe wish to find how the Holevo bound changes -- that is, we wish to\nmake an estimate of $\\Delta \\chi = \\chi' - \\chi$.\n\nBegin with the expression for $\\chi'$ and apply\nEquation~\\ref{coverish}, choosing the original ensemble\nand letting the modified average state\n$\\rho'$ play the role of the alternate state. This yields\n\\begin{eqnarray}\n\\chi' & = & (1 - \\eta) \\left ( \\chi + \\relent{\\rho}{\\rho'}\n\t\t\\right )\n + \\eta \\relent{\\rho_0}{\\rho'} \\\\\n & = & \\chi + \\eta \\left ( \\relent{\\rho_0}{\\rho'}\n\t\t- \\chi \\right )\n + (1-\\eta) \\relent{\\rho}{\\rho'} \\\\\n\\Delta \\chi\n & = & \\eta \\left ( \\relent{\\rho_0}{\\rho'} - \\chi \\right )\n + (1-\\eta) \\relent{\\rho}{\\rho'} .\n\\end{eqnarray}\nTherefore,\n\\begin{equation}\n\\Delta \\chi \\geq \\eta \\left ( \\relent{\\rho_0}{\\rho'} - \\chi \\right ) .\n \\label{lower}\n\\end{equation}\nThis gives us a lower bound for $\\Delta \\chi$.\n\nTo obtain an upper bound, we apply Equation~\\ref{coverish} to the\nmodified ensemble, with the original average state $\\rho$ playing\nthe role of the alternate state.\n\\begin{eqnarray}\n\\chi' + \\relent{\\rho'}{\\rho}\n & = & (1-\\eta) \\left( \\sum_{k} p_k \\relent{\\rho_k}{\\rho}\n\t\t\\right ) + \\eta \\relent{\\rho_0}{\\rho} \\\\\n & = & (1-\\eta) \\chi + \\eta \\relent{\\rho_0}{\\rho} \\\\\n\\chi' - \\chi & = & \\eta \\left ( \\relent{\\rho_0}{\\rho}\n\t\t- \\chi \\right ) - \\relent{\\rho'}{\\rho}\n\\end{eqnarray}\nAnd so we obtain\n\\begin{equation}\n\\Delta \\chi \\leq \\eta \\left ( \\relent{\\rho_0}{\\rho}\n\t\t- \\chi \\right ) .\n \\label{upper}\n\\end{equation}\nIn deriving this inequality, we obviously assume that\n$\\supp \\rho_0 \\subseteq \\supp \\rho$. But if this\nis not the case, then the inequality still holds\nin the sense that the right-hand side is infinite.\n\nIt is easy to generalize these results to a situation in which\nwe modify the ensemble by adding many states.\nSuppose the states $\\rho_{0a}$ are added with probabilities\n$\\eta q_a$ (where the $q_a$'s form a probability distribution).\nThen the above results would become\n\\begin{equation}\n\\eta \\left( \\sum_{k} q_a \\relent{\\rho_{0a}}{\\rho'} - \\chi \\right )\n \\leq \\Delta \\chi \\leq\n \\eta \\left ( \\sum_{k} q_a \\relent{\\rho_{0a}}{\\rho}\n\t\t- \\chi \\right ).\n \\label{manystate}\n\\end{equation}\nAll of our subsequent results still\nhold in this more general situation,\nbut to simplify the discussion we will phrase our arguments\nin terms of ``single state'' modifications of a given ensemble.\n\nFinally, consider states $\\rho_0$ and $\\rho$, and let\n$\\rho' = (1-\\eta) \\rho + \\eta \\rho_0$.\nThen $\\relent{\\rho_0}{\\rho'}$ exists and is finite for\n$0 < \\eta \\leq 1$,\nand\n\\begin{itemize}\n\\item If $\\supp \\rho_0 \\subseteq \\supp \\rho$, then\n $\\relent{\\rho_0}{\\rho'} \\rightarrow \\relent{\\rho_0}{\\rho}$\n as $\\eta \\rightarrow 0$.\n\\item Otherwise, $\\relent{\\rho_0}{\\rho'} \\rightarrow \\infty$ as\n $\\eta \\rightarrow 0$.\n\\end{itemize}\nWe see that Equations \\ref{lower} and \\ref{upper} are fairly\n``tight'' lower and upper bounds for $\\Delta \\chi$,\nbecause (informally speaking) the two expressions\napproach one another as $\\eta$ approaches zero.\n\n\n\\section{Properties of optimal ensembles}\n\nFor a given set ${\\cal A}$ of available states (e.g., the\noutputs of a noisy channel),\nlet $\\rho_k$ and $p_k$ be the members and probabilities\nof the ensemble of ${\\cal A}$-states for which $\\chi$ takes\non its maximum value. Call this the ``$\\chi$-optimal ensemble'',\nand let $\\rho^{\\ast}$ be the average state of this ensemble.\nDenote $\\max \\chi$ by $\\chi^{\\ast}$.\nThe $\\chi$-optimal ensemble has a number\nof important properties.\n\n\\noindent\n\\begin{description}\n\\item[Existence.] If the letter states are outputs of a\nnoisy channel in a finite-dimensional Hilbert space,\nthen a $\\chi$-maximizing ensemble exists.\n\n {\\bf Proof:} The key result can found in \\cite{uhlmann}:\nLet $\\avail$ be a convex, compact subset\n$\\avail$ of density operators\non a Hilbert space of finite dimension $d$, and let\n$\\rho$ be in $\\avail$.\nIf the set of extremal elements of $A$\nis compact then for any $\\rho \\in A$ there\nexists an ensemble of states $\\{ \\rho_k \\} \\subset A$\nwith $\\rho = \\sum p_k \\rho_k$\nthat maximizes $\\chi$ over the set\nof all ensembles whose average state is $\\rho$.\nIn other words, there exist optimal signal ensembles\nfor a given average state $\\rho$.\nBy Caratheodory's Theorem, since the Hilbert space\nhas $d$ dimensions, then there are optimal ensembles\n(in this sense) with no more than $d^2$ states.\n\nWe see that the conditions for the result from \\cite{uhlmann}\nare met. The set of states $\\avail$ that are possible outputs\nof the channel is a convex, compact set with a compact set of\nextremal points. For any average state $\\rho$ in $\\avail$,\nwe can find a $\\rho$-fixed optimal ensemble with $d^2$ or\nfewer elements.\nThus, in order to maximize $\\chi$ over all possible ensembles,\nwe only need to consider the set of ensembles with no more\nthan $d^2$ elements drawn from $\\avail$. As this is a\nfinite cartesian product of a compact set, it is compact.\nAs $\\chi$ is a continuous function, it must achieve its\nmaximum in this set of ensembles. Thus, the existence of\nan optimal ensemble of states in $\\avail$ is assured.\n\n\\item[Maximal distance property.] For any state $\\rho_0$\n\tin ${\\cal A}$,\n \\begin{equation}\n \\relent{\\rho_0}{\\rho^{\\ast}} \\leq \\chi^{\\ast} .\n \\end{equation}\n\n{\\bf Proof.} We assume the existence of a state $\\rho_0$\nwith $\\relent{\\rho_0}{\\rho^{\\ast}} > \\chi^{\\ast}$.\n(We allow for the possibility that\n$\\relent{\\rho_0}{\\rho^{\\ast}}$\nis infinite.) Since $\\relent{\\rho_0}{\\rho'} \\rightarrow\n\\relent{\\rho_0}{\\rho^{\\ast}}$ as $\\eta \\rightarrow 0$,\nwe can find a value of $\\eta$ so that $\\relent{\\rho_0}{\\rho'} >\n\\chi^{\\ast}$. Then by Equation~\\ref{lower},\n\\begin{displaymath}\n \\Delta \\chi \\geq\n \\eta \\left ( \\relent{\\rho_0}{\\rho'} - \\chi^{\\ast} \\right )\n > 0 .\n\\end{displaymath}\nThat is, we can increase $\\chi$ by including $\\rho_0$ in the\nsignal ensemble, which is a contradiction.\n\\item[Maximal support property.] For a $\\chi$-optimal ensemble,\n $\\supp \\rho^{\\ast} = \\supp {\\cal A}$.\n\t(By ``$\\supp {\\cal A}$'' we mean the smallest\n\tsubspace that contains $\\supp \\rho_k$ for any\n\t$\\rho_k \\in {\\cal A}$.) In other words, any\n $\\chi$-optimal ensemble ``covers'' the support of the set\n of available states.\n\n{\\bf Proof.} This is a corollary to the maximum distance\nproperty. If there were a state $\\rho_{0} \\in {\\cal A}$\nso that $\\supp \\rho_0$ were not contained in $\\supp\n\\rho^{\\ast}$, then $\\relent{\\rho_0}{\\rho^{\\ast}}$\nwould be infinite.\n\n\\item[Sufficiency of maximal distance property.] Suppose\n\twe have an ensemble with average state $\\rho$ and\n\ta particular value of $\\chi$, and suppose that\n \\begin{displaymath}\n \\relent{\\rho_0}{\\rho} \\leq \\chi\n \\end{displaymath}\n for all $\\rho_0 \\in {\\cal A}$. Then this must be a\n\t$\\chi$-optimal ensemble.\n\tThat is, the only ensembles that have the maximal\n distance property are $\\chi$-optimal ensembles.\n\n{\\bf Proof:} If we add a state $\\rho_0$ with probability $\\eta$\nto the ensemble, then from Equation~\\ref{upper}\n \\begin{displaymath}\n \\Delta \\chi \\leq \\eta \\left ( \\relent{\\rho_0}{\\rho}\n\t\t- \\chi \\right ) \\leq 0\n \\end{displaymath}\nso that we cannot increase $\\chi$. (By Equation~\\ref{manystate},\nthe same would hold if we were to add several\ndifferent states instead of only one.) Thus,\n$\\chi = \\chi^{\\ast}$.\n\n\\item[Equal distance property.] Suppose $\\rho_k$ is a member\n\tof a $\\chi$-optimal ensemble with probability\n\t$p_k \\neq 0$. Then\n \\begin{equation}\n \\relent{\\rho_k}{\\rho^{\\ast}} = \\chi^{\\ast} .\n \\end{equation}\n In other words, all of the non-zero members of a $\\chi$-optimal\n ensemble have the same relative entropy ``distance'' with\n respect to the average state $\\rho^{\\ast}$.\n\n{\\bf Proof:} This is another corollary to the maximal distance\nproperty. If $\\relent{\\rho_k}{\\rho^{\\ast}} < \\chi^{\\ast}$\nfor any $\\rho_k$ with $p_k \\neq 0$, then the average relative\nentropy cannot equal $\\chi^{\\ast}$.\n\n\\item[Min-max formula for $\\chi^{\\ast}$.] From the above\n\tproperties, we can show the following formula:\n \\begin{equation}\n \\chi^{\\ast} = \\min_{\\rho} \\left (\n \\max_{\\rho_0} \\relent{\\rho_{0}}{\\rho}\n\t\t\t\\right ) ,\n\t\t\\label{minmax}\n \\end{equation}\n where the maximum is taken over all\n\t$\\rho_{0} \\in {\\cal A}$ and the minimum is taken over\n\tall average states $\\rho$ of ensembles\n of ${\\cal A}$-states.\n\n{\\bf Proof:} We first show that, for any state $\\sigma$, the\nquantity $\\displaystyle \\max_{\\rho_0} \\relent{\\rho_{0}}{\\sigma}$\nis an upper bound for the value of $\\chi$ for any possible\nensemble. By Equation~\\ref{donaldcorollary}, we find that\n\\begin{displaymath}\n\t\\chi \\leq \\sum_{k} p_{k} \\relent{\\rho_{k}}{\\sigma}\n\t\t\\leq \\max_{\\rho_0} \\relent{\\rho_{0}}{\\sigma} .\n\\end{displaymath}\nThis will also hold for an optimal signal ensemble,\nfor which $\\chi = \\chi^{\\ast}$. Thus,\n\\begin{displaymath}\n\t\\chi^{\\ast} \\leq \\min_{\\rho} \\left (\n \\max_{\\rho_0} \\relent{\\rho_{0}}{\\rho}\n\t\t\t\\right ) .\n\\end{displaymath}\nNext we note that the maximal distance property implies that\n\\begin{displaymath}\n\t\\chi^{\\ast} = \\max_{\\rho_0} \\relent{\\rho_{0}}{\\rho^{\\ast}} ,\n\\end{displaymath}\nfrom which we can see that\n\\begin{displaymath}\n\t\\chi^{\\ast} \\geq \\min_{\\rho} \\left (\n \\max_{\\rho_0} \\relent{\\rho_{0}}{\\rho}\n\t\t\t\\right ) .\n\\end{displaymath}\nThese two inequalities establish the formula in Equation~\\ref{minmax}.\n\n\\end{description}\n\nThese properties provide strong characterizations of an\noptimal signal ensemble for a quantum channel. Equation~\\ref{minmax},\nfor example, shows that $\\chi^{\\ast}$ can be calculated as a purely\n``geometric'' property of the set $\\avail$, without direct reference\nto any ensemble. We believe that our results are likely to prove\nuseful in further investigations of the efficient use\nof quantum resources to transmit classical messages.\n\n\\section{Acknowledgements}\n\nWe would like to thank A. Uhlmann for helpful and enlightening\ncomments, particularly about the existence of an\noptimal ensemble. We also had useful conversations with\nT. Cover, C. A. Fuchs, A. S. Holevo, V. Vedral\nand W. K. Wootters. Most of these discussions took place in\nconnection with the programme on ``Complexity, Computation\nand the Physics of Information'' at the Isaac Newton Institute\nin Cambridge (England) during the summer of 1999.\nThis programme was sponsored in part by the European\nScience Foundation. One of us (BS) gratefully acknowledges\nthe support of a Rosenbaum Fellowship at the Isaac Newton\nInstitute to participate in this programme.\n\n\n\\begin{thebibliography}{99}\n%\n\\bibitem{holevo1} A.~S.~Kholevo, Probl. Peredachi Inf. {\\bf 9}, 3\n (1973) [Probl. Inf. Transm. (USSR) {\\bf 9}, 110 (1973)].\n\\bibitem{gordon} J.~P.~Gordon, in {\\em Quantum Electronics\n and Coherent Light, Proceedings of the International School\n of Physics ``Enrico Fermi,'' Course XXXI},\n edited by P.~A.~Miles (Academic, New York, 1964), pp. 156-181.\n\\bibitem{levitin} L.~B.~Levitin, ``On the quantum measure of the\n amount of information,'' in {\\em Proceedings of the IV National\n Conference on Information Theory}, Tashkent, 1969, pp.~111--115\n (in Russian); ``Information Theory for Quantum Systems,''\n in {\\em Information, Complexity, and Control in Quantum Physics},\n edited by A.~Blaqui\\`ere, S.~Diner, and G.~Lochak (Springer, Vienna,\n 1987).\n\\bibitem{holevo2} A.~S.~Holevo, IEEE Trans. Inform. Theory {\\bf 44},\n 269 (1998).\n\\bibitem{noisy} B.~Schumacher and M.~Westmoreland, Phys. Rev. A\n {\\bf 51}, 2738 (1997).\n\\bibitem{hjsww} P.~ Hausladen, R.~Josza, B.~Schumacher,\n M.~Westmoreland and W.~K.~Wootters, Phys. Rev. A {\\bf 54},\n 1869 (1996).\n\\bibitem{distinguish} F.~Hiai and D.~Petz, Comm. Math. Phys.\n {\\bf 143}, 99 (1991). V.~Vedral, M.~B.~Plenio, K.~Jacobs and\n P.~L.~Knight, Phys. Rev. A {\\bf 56}, 4452 (1997).\n\\bibitem{entangle} V.~Vedral, M.~B.~Plenio, M.~A.~Rippin and\n P.~L.~Knight, Phys. Rev. Lett. {\\bf 78}, 2275 (1997).\n\\bibitem{fuchs} C.~A.~Fuchs, Phys. Rev. Lett. {\\bf 79}, 1162 (1997).\n\\bibitem{donald} M.~J.~Donald, Math. Proc. Cam. Phil. Soc.\n {\\bf 101}, 363 (1987).\n\\bibitem{uhlmann} A.~Uhlmann, Open Sys. and Inf. Dynamics {\\bf 5},\n 209 (1998).\n%\n\\end{thebibliography}\n\n\\newpage\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=5in]{fuchs.eps}\n\\end{center}\n\\caption{Bloch sphere diagram for amplitude damping.\n The highest value of $\\chi$ for a set of orthogonal\n input signals is attained by an equally weighted mixture\n of $\\ket{\\rightarrow}$ and $\\ket{\\leftarrow}$, but\n the non-orthogonal input signals $\\ket{\\phi_0}$\n and $\\ket{\\phi_1}$ yield\n a larger value of $\\chi$.}\n\\label{fuchs}\n\\end{figure}\n\n\n\n\\end{document}\n" } ]	[ { "name": "quant-ph9912122.extracted_bib", "string": "{holevo1 A.~S.~Kholevo, Probl. Peredachi Inf. {9, 3 (1973) [Probl. Inf. Transm. (USSR) {9, 110 (1973)]." }, { "name": "quant-ph9912122.extracted_bib", "string": "{gordon J.~P.~Gordon, in {\\em Quantum Electronics and Coherent Light, Proceedings of the International School of Physics ``Enrico Fermi,'' Course XXXI, edited by P.~A.~Miles (Academic, New York, 1964), pp. 156-181." }, { "name": "quant-ph9912122.extracted_bib", "string": "{levitin L.~B.~Levitin, ``On the quantum measure of the amount of information,'' in {\\em Proceedings of the IV National Conference on Information Theory, Tashkent, 1969, pp.~111--115 (in Russian); ``Information Theory for Quantum Systems,'' in {\\em Information, Complexity, and Control in Quantum Physics, edited by A.~Blaqui\\`ere, S.~Diner, and G.~Lochak (Springer, Vienna, 1987)." }, { "name": "quant-ph9912122.extracted_bib", "string": "{holevo2 A.~S.~Holevo, IEEE Trans. Inform. Theory {44, 269 (1998)." }, { "name": "quant-ph9912122.extracted_bib", "string": "{noisy B.~Schumacher and M.~Westmoreland, Phys. Rev. A {51, 2738 (1997)." }, { "name": "quant-ph9912122.extracted_bib", "string": "{hjsww P.~ Hausladen, R.~Josza, B.~Schumacher, M.~Westmoreland and W.~K.~Wootters, Phys. Rev. A {54, 1869 (1996)." }, { "name": "quant-ph9912122.extracted_bib", "string": "{distinguish F.~Hiai and D.~Petz, Comm. Math. Phys. {143, 99 (1991). V.~Vedral, M.~B.~Plenio, K.~Jacobs and P.~L.~Knight, Phys. Rev. A {56, 4452 (1997)." }, { "name": "quant-ph9912122.extracted_bib", "string": "{entangle V.~Vedral, M.~B.~Plenio, M.~A.~Rippin and P.~L.~Knight, Phys. Rev. Lett. {78, 2275 (1997)." }, { "name": "quant-ph9912122.extracted_bib", "string": "{fuchs C.~A.~Fuchs, Phys. Rev. Lett. {79, 1162 (1997)." }, { "name": "quant-ph9912122.extracted_bib", "string": "{donald M.~J.~Donald, Math. Proc. Cam. Phil. Soc. {101, 363 (1987)." }, { "name": "quant-ph9912122.extracted_bib", "string": "{uhlmann A.~Uhlmann, Open Sys. and Inf. Dynamics {5, 209 (1998). %" } ]
solv-int9912001	Construction of variable mass sine-Gordon and other novel inhomogeneous quantum integrable models	[ { "author": "Anjan Kundu \\footnote {email: anjan@tnp.saha.ernet.in" } ]	%--------------------------------------------------- The inhomogeneity of the media or the external forces usually destroy the integrability of a system. We propose a systematic construction of a class of quantum models, which retains their exact integrability inspite of their explicit inhomogeneity. Such models include variable mass sine-Gordon model, cylindrical NLS, spin chains with impurity, inhomogeneous Toda chain, the Ablowitz-Ladik model etc. \medskip %PACS numbers 03.65.Fd, 02.20.Sv, 02.30.Jr, 11.10.Lm	[ { "name": "solv-int9912001.tex", "string": "%Proc of 'NEEDS99, Greece 20-30,1999 (J Nonlinear Math Phys..) 31.10\n%:Anjan Kundu {email: anjan@tnp.saha.ernet.in} \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%=========================== titlepage =================================\n%========================================================================\n% my paperhead \n% use lateX, compile twice.\n%========================================================================\n % \\documentstyle[romp31]{article}\n \\documentstyle[12pt]{article}\n\\oddsidemargin -.3cm \n\\textheight 22.5cm \n\\textwidth 16.8cm\n%\\def\\theequation{\\thesection.\\arabic{equation}}\n%\\renewcommand{\\baselinestretch}{1.1}\n\\renewcommand{\\baselinestretch}{1.24}\n\\def\\ll{\\label}\n\\def\\re{\\ref}\n\\def\\c{\\cite}\n\\def\\b{\\begin}\n\\def\\La{\\Lambda}\n\\def\\r1{(\\ref{$1})}\n\\def\\ot{\\otimes}\n\\def\\nn{\\nonumber}\n\\def\\sn{\\rm sn}\n\\def\\pa{\\partial}\n\\def\\kap{\\kappa}\n\\def\\ms{\\medskip}\n\\def\\cR{\\cal R}\n\\def\\cF{\\cal F}\n\\def\\cP{\\cal P}\n\\def\\ti{\\tilde}\n\\def\\cn{\\rm cn}\n\\def\\dn{\\rm dn}\n\\def\\ga{\\gamma}\n\\def\\ep{\\epsilon}\n\\def\\th{\\theta}\n\\def\\ba{\\begin{array}{c}}\n\\def\\e{\\end}\n\\def\\sk{\\smallskip}\n\\def\\ea{\\end{array}}\n\\def\\pr{\\prod}\n\\def\\ni{\\noindent}\n\\def\\si{\\sigma}\n\\def\\da{\\dagger}\n\\def\\De{\\Delta}\n\\def\\de{\\delta}\n\\def\\bet{\\beta}\n\\def\\ov{\\over}\n\\def\\ha{{1\\over 2}}\n\\def\\qr{{1\\over 4}}\n\\def\\l{\\left}\n\\def\\l({\\left(}\n\\def\\r){\\right)}\n\\def\\r{\\right}\n\\def\\rw{\\rightarrow}\n\\def\\om{\\omega}\n\\def\\la{\\lambda}\n\\def\\al{\\alpha}\n\\def\\sec{\\section}\n\\def\\be{\\begin{equation}}\n\\def\\bc{\\begin{center}}\n\\def\\ec{\\end{center}}\n\\def\\bit{\\begin{itemize}}\n\\def\\eit{\\end{itemize}}\n\\def\\ee{\\end{equation}}\n\\def\\ed{\\end{document}}\n\\def\\bea{\\begin{eqnarray}}\n\\def\\eea{\\end{eqnarray}}\n\\def\\efr{\\end{flushright}}\n\\def\\nn{\\nonumber\\\\}\n%======================== journal macros ===============================\n\\begin{document}\n\\title{Construction of variable mass sine-Gordon and other novel \n inhomogeneous quantum\n integrable models\n}\n%\\vskip 1cm\n\n\n\\author{\nAnjan Kundu \\footnote {email: anjan@tnp.saha.ernet.in} \\\\ \n Saha Institute of Nuclear Physics, \n Theory Group \\\\\n 1/AF Bidhan Nagar, Calcutta 700 064, India.\n }\n\\maketitle\n%\n\\vskip 1 cm\n\n\n%\\vskip 4 cm\n%\\e{itemize}\n\\begin{abstract} \n%---------------------------------------------------\nThe inhomogeneity of the media or the external forces usually destroy\nthe integrability of a system. We propose a \nsystematic construction of a class of quantum models, \nwhich retains their exact integrability inspite of their \nexplicit inhomogeneity.\nSuch models include\n variable mass sine-Gordon model, \ncylindrical NLS, spin chains with impurity, \ninhomogeneous Toda chain, the Ablowitz-Ladik model etc.\n\\medskip\n %PACS numbers 03.65.Fd, 02.20.Sv, 02.30.Jr, 11.10.Lm\n\\end{abstract}\n\n\\smallskip\n\n%\\tableofcontents\n%\\newpage\n \n\\section{Introduction}\n\\setcounter{equation}{0}\n\nThe physical systems often encounter the inhomogeneity in the form of\nimpurities, defects or the density fluctuations in the media or it\n can enter as variable magnetic fields or other external\nforces.\n Such inhomogeneities \ncan depend on space as well as time variables and can appear as \nexplicit space-time dependent coefficients in the Hamiltonian. \nThey\n usually destroy the integrability of a system\n making its analytic study almost impossible \\c{msg}.\nHowever there are several examples of \nclassical models like deformed MKDV \\c{burtsev} or NLS \\c{CL},\nspherical or cylindrical symmetric \nNLS \\c{rl}, inhomogeneous Ablowitz-Ladik (AL)\nmodel \\c{konotop} etc.\n where the exact integrability could be retained along \nwith their Lax operators, inspite of the presence of space-time\ndependent coefficients in their evolution equations.\nNevertheless, in case of quantum models there seems to be no\n systematic \nattempts to explore such possibilities,\n except certain construction for the impurity chains \\c{impchain} .\n%\n%--------------\n\n\nWe propose here a scheme for introducing inhomogeneities in the known\nquantum integrable models, which retains their integrability and allows\nexplicit construction of their $R$-matrix as well as\n the Lax operators. This systematic scheme is based on a novel quadratic\n algebra derived from the quantum Yang--Baxter\nequation,\n where its Casimir operators play the role of the\ninhomogeneity parameters and the proper realisation of other\ngenerators construct the Lax operator of the model. The $R$-matrix\nsimply corresponds to that of the standard homogeneous model. Applying this\nprocedure one first constructs quantum models with inhomogeneity on discrete\nlattices, which preserves the exact integrability\n and then taking the continuum limit builds the corresponding field\nmodels. Thus one obtains a series of new quantum integrable \ninhomogeneous models like\na variable mass quantum sine-Gordon model, cylindrical quantum\n NLS, \ninhomogeneous Toda and Ablowitz Ladik chains. It also provides a\ndifferent way of introducing integrable impurities in \n the spin chain models. \n\n\\section{The generating scheme}\n\\setcounter{equation}{0}\n\n We start with the quadratic algebra \\c{kunprl99} \n\\be\n [S^3,S^{\\pm}] = \\pm S^{\\pm} , \\ \\ \\ [ S^ {+}, S^{-} ] =\n \\left ( M^+\\sin (2 \\al S^3) + {M^- } \\cos\n( 2 \\al S^3 ) \\right){1 \\over \\sin \\al}, \\quad [M^\\pm, \\cdot]=0,\n\\ll{nlslq2a}\\ee\nand the quantum L-operator\n\\be\nL_t{(\\xi)} = \\left( \\begin{array}{c}\n \\xi{c_1^+} e^{i \\al S^3}+ \\xi^{-1}{c_1^-} e^{-i \\al S^3}\\qquad \\ \\ \n2 \\sin \\al S^- \\\\\n \\quad \n2 \\sin \\al S^+ \\qquad \\ \\ \\xi{c_2^+}e^{-i \\al S^3}+ \n\\xi^{-1}{c_2^-}e^{i \\al S^3}\n \\end{array} \\right), \\quad\n \\xi=e^{i \\alpha \\la}. \\ll{nlslq2} \\ee\nwhere \n$ M^\\pm=\\pm \\sqrt {\\pm 1} ( c^+_1c^-_2 \\pm\nc^-_1c^+_2 ) $ are the Casimir operators \n of algebra (\\re {nlslq2a}). \nTaking \n the well known trigonometric $R(\\la)$-matrix solution \\c{kulskly} \nalong with (\\re{nlslq2}) the quadratic algebra (\\re{nlslq2a}) \n can be shown to be equivalent to the \n quantum Yang Baxter relation \\c{kulskly} $R(\\la-\\mu) L(\\la)\\otimes \nL(\\mu)= (I \\otimes L(\\mu) ) \\otimes ( L(\\la)\\otimes I) R(\\la-\\mu)$.\nTherefore the associated L-operator \n(\\re {nlslq2}) may serve as the generating Lax operator for \nthe quantum integrable models belonging to the relativistic or the\nanisotropic class of models, the parameter $q=e^{i \\al}$ \nplaying the role of the deformation parameter \\c{tarunpr99}. The \nintegrable inhomogeneities are\n introduced in fact through different\n representations of the Casimir operators \n$c^\\pm_a$ by choosing their eigenvalues \n as position and time dependent functions.\n\nAt the undeformed limit $q=e^\\al \\to 1$ or equivalently at $\\al \\to 0$\nall the entries in the above scheme, i.e.\n algebra (\\re {nlslq2a}), L-operator \n(\\re {nlslq2}) and the trigonometric $R$-matrix \n are reduced to their corresponding rational forms.\nThe reduced algebra is simplified but still represents a quadratic algebra: \n\\be [ s^+ , s^- ]\n= 2m^+ s^3 +m^-,\\ \\ \\ \\ \n ~ [s^3, s^\\pm] = \\pm s^\\pm \n \\ll{k-alg} \\ee\nwith $m^+=c_1^0c_2^0,\\ \\ m^-= c_1^1c_2^0+c_1^0c_2^1$ as\nthe new central elements.\nNote that both (\\re {nlslq2a}) and (\\re {k-alg})\n are Hopf algebras with explicit coproduct structure, counit, antipode etc.\n\\c{tarunpr99}. \n\n\n\nDue to $\\al \\to 0$ and $\\xi \\to 1+ \\al \n\\la $ the L-operator takes the form\n \\be\nL_r{(\\la)} = \\left( \\begin{array}{c}\n {c_1^0} (\\la + {s^3})+ {c_1^1} \\ \\ \\quad \n s^- \\\\\n \\quad \ns^+ \\quad \\ \\ \nc_2^0 (\\la - {s^3})- {c_2^1}\n \\end{array} \\right), \\ll{LK} \\ee\n with spectral parameter $\\la$ and \n the quantum\n $R$-matrix is reduced to its well known rational form \\c{kulskly}. \n Remarkably, our scheme with these reduced entries \nbelonging to the \nrational class becomes suitable \nfor generating quantum integrable nonrelativistic \n models with inhomogeneity. \n\n\\section{Inhomogeneous quantum integrable models}\n\n\\subsection{Variable mass sine-Gordon model}\n\nSince this is a relativistic model we have to use the objects belonging to\nthe trigonometric class.\nThrough canonical operators $u,p$ a representation of \n(\\ref{nlslq2a})\n may be given by \n\\be\n S^3=u, \\ \\ \\ S^+= e^{-i p}g(u),\\ \\ \\ \n S^-= g(u)e^{i p},\n\\ll{ilsg}\\ee\nwhere\nthe operator function\n\\be g (u)= \\left ( 1-\\sin \\al u (M^+ \\sin \\al (u+1) )\n \\right )^{\\ha} { 1 \\ov \\sin \\al } \\ll{g}\\ee\nBy choosing the eigenvalues of the Casimirs as\n$ M_j^+=- (\\De m_j)^2, M_j^-=0$\nand inserting (\\re {ilsg}) in (\\ref{nlslq2}) \none gets a quantum integrable lattice model involving bosonic operators \nand the inhomogeneity parameter $m_j. $ Comparing with the well known\nresult \\c{korepinsg} we may conclude that the model thus constructed is \na generalisation of the exact lattice version of the quantum sine-Gordon\nmodel. \nFor going to the \n continuum limit we may scale $p$ by lattice constant $\\Delta$\nand take the limit $\\De \\to 0.$ As a result one derives from \n(\\ref{nlslq2}) the Lax operator of the sine-Gordon field model\n\\be{\\cal L}= im({u_t } \\si ^3 +( k_1 \\cos u \\si ^1 +k_0 \\sin u \\si ^2) \n, \\ll{Lsg}\\ee \nwhere\nthe mass parameter $m=m(x,t)$ now \nis not a constant as in the standard case, but\nan arbitrary function of $x,t$. The variable mass\nalso enters the Hamiltonian of this novel \n { sine-Gordon \n model} as \n\\be{\\cal H}= \\int dx \\left [ m(x,t) (u_t)^2 + (1/m(x,t)) (u_x)^2 +\n 8(m_0-m(x,t)\n\\cos (2 \\al u )) \\right], \\ll{msg1}\\ee \nwhich is integrable both at classical and the quantum level for the \n arbitrary mass\nfunction $m(x,t)$.\n Note that if the mass is independent of time and depends only on the space\ncoordinate: $m=m(x)$, one can\nformally convert the evolution equation into the standard sine-Gordon\nthrough a coordinate change:$ x \\to X=\\int^x m(y) dy $ and can find its\nexact\nsoliton solution\nas\n\\be u=2\\tan^{-1}[ \\exp (\\gamma \\int^x m(y) dy +v t)], \\ll{sol} \\ee \nwhich exhibit intriguing structure depending on the choice of the\nmass-function $m(x)$. \n Such \nvariable mass sine-Gordon equations may arise in physical situations\n\\c{msg} and therefore the related exact results become important.\n\\subsection\n{Inhomogeneous NLS model}\nNonlinear Schr\\\"odinger equation belongs to the nonrelativistic class.\nTherefore we should use the rational $R$-matrix and the rational L-operator\n(\\re{LK}) with suitable realisations of algebra (\\re{k-alg}).\nA simple such realisation may be given by \nconsidering site-dependent values for central elements in (\\ref{LK}) and in\nthe\n generalized HPT \n \\be\n s^3=s-N, \\ \\ \\ \\ s^+= g_0(N) \\psi, \\ \\ \\ s^-= \\psi^\\dag g_0(N)\n, \\ \\ \\ \\ g_0^2(N)=m^-+m^+ (2s -N), \\ \\ N=\\psi^\\dag \\psi.\n\\ll{ilnls} \\ee\nThis exactly integrable \nquantum discrete model is an inhomogeneous generalisation of the known\nlattice NLS \\c{korepinsg}. In the continuum limit one may introduce \n the inhomogeneity by choosing the eigenvalues of the central elements as\n$$ c^1_1={1 \\ov \\De} +f, c^1_2= -({1 \\ov \\De} -f), c^0_1=-c^0_2=g$$\nwith $f$ and $g$ being space-time dependent arbitrary functions.\nThe Lax operator of the field model would be given formally \nby that of the NLS model, where \nthe constant spectral parameter should be replaced by $\\ti \\la= g \\la + f$\nand the field variables by $\\psi /\\sqrt g.$\n Particular choice of these \nfunctions as $f={4 x \\ov t}, g={1 \\ov t} $ would yield integrable \n { cylindrical NLS}\n \\c{rl} like equation at the quantum level.\n%\\be i\\sqrt{g} \\dot \\psi= \\psi_{xx}+(f_x-f^2) \\psi+ 2 \|\\psi\|^2\\psi\\ll{cnls}\\ee\n\n\\subsection\n{Inhomogeneous toda chain}\nAnother interesting \n realisation of algebra \n(\\re{k-alg}) may be given by\n\\be\n s^3 =-ip, \\ s^\\pm= \\al e^{\\mp u } \\ \\ \n \\ll{dtl1} \\ee\nwith $m^\\pm=0, $ which leads to the construction\nof quantum \nToda chain model. A consistent choice of the Casimir eigenvalues like \n $c^0_2=c^1_2=0$ together with $ c^0_1$ and $c^1_1 $ taken as space-time dependent\ncoefficients $c^0_j(t)$ and $c^1_j(t) ,$ would now result a novel quantum\nintegrable \n inhomogeneous Toda chain given by the \n Hamiltonian\n \\be \\ H= \\sum_j (p_j +{c^1_j \\ov\n c^0_j})^2+{1 \\ov c^0_jc^0_{j+1}} e^{u_j-u_{j+1}}. \n\\ll{todah}\\ee\nFor $ c^1_j=0$ the evolution equation can be written down in an interesting\n compact form\n\\be \n{\\partial^2 \\ov \\partial t^2}\n u_j=\ne^{u^+_j-u^-_{j+1}}-e^{u^+_{j-1}-u^-_{j}},\\ll{ihtoda}\\ee\nwhere $u^\\pm_j(t)=u_j(t)\\pm \\phi_j (t),$ with $\\phi_j(t)$ being an\n arbitrary function inducing inhomogeneity in the system. \nNote that, when the $c'$s are time independent\ncoefficients such inhomogeneities can not be removed through gauge\ntransformation or variable change.\n\n\n Using similar procedure one may construct impurity spin chains\n in a different way \nby seeking various spin\noperator realisations of the Lax operators (\\ref{nlslq2}) or (\\ref{LK})\nat the impurity sides.\n\n\n\n%\\end{document}\n\n\n\\section {Concluding remarks}\n\nThus we have prescribed a systematic scheme for \n constructing a novel series of inhomogeneous quantum\nintegrable models belonging to the\n lattice as well as the field models\nof both relativistic ( $q\\neq 1$) and nonrelativistic \n ($q=1$) class\n along with their corresponding classical counterparts. \nThe scheme is based on an algebraic approach, \nwhere the generators through different realisations construct \nnonlinear functions of field operators and the Casimir operators \n with space-time dependent eigenvalues introduce inhomogeneity \ninto the system. \nIn our scheme\none also obtains automatically \nthe Lax operators and the $R$-matrices of the \nmodels constructed.\n\n\n\\ni {\\bf Acknowledgment}: \n\nI thank the\nHumboldt Foundation, Germany and the organisers of NEEDS99 for financial\nsupport. \n%\\newpage\n\\begin{thebibliography}{99}\n%1-----------------\n\\bibitem{msg} Malomed et al Phys. Rev. B41 (1990) 11271; Phys. Lett. A\n144(1990) 351\n\nN. Sanchez et al eprint patt-sol/ 9401005\n\nP. Woafo, J. Phys. Soc. Japan, 67 (1998) 3734\n\n D. Sen and S. Lal, e-print cond-mat/9811330\n\\bibitem{burtsev} S. P. Burtsev, V. E. Zakharov and A. V. Mikhailov,\nTeor . Mat. Fiz. 70 (1987) 323\n\\bibitem{CL} H. H. Chen and C. S. Liu, Phys. Rev. Lett. 37 (1976) 693\n\nP. A. Horvatthy and J C Yera, {\\it The variable coefficient NLS equation and\nthe conformal properties of non-relativistic space time}, preprint 18/3/99\n(submitted to J. Nonlinear Math. Phys.)\n\n\\bibitem{rl} R. Radha and M. Lakshmanan, Chaos Solitons\nand Fractals, 4 (1994) 181\n\\bibitem{konotop} \nM. Bruschi, D. Levi and O. Ragnisco, Nuove Cimento, 53A (1979) 21\n;\nV. V. Konotop, O. A. Chubaikalo and L. Vazquez, Phys. Rev. E 48 (1993) 563\n\\bibitem{impchain} \nP. Schmitteckert, P. Schwab and U. Eckern, Erophys. Lett. 30 (1995) 543\n\n H. P. Eckle, A. Punnoose and R. R\\\"omer,Erophys. Lett. 39 (1997) 293\n\nG. Bed\\\"urftig, F. Essler and H. Frahm, Nucl. Phys. B489 (1997) 697\n\n\\bibitem{kunprl99}\n Anjan Kundu, Phys. Rev. Lett., 82 (1999) 3936\n\\bibitem{kulskly} P. Kulish and E. K. Sklyanin,\nLect. Notes in Phys. {\\bf 151} (ed. J. Hietarinta et al, Springer, 1982), 61.\n\\bibitem{tarunpr99} Anjan Kundu, {\\it\nAlgebraic construction of quantum integrable models\nincluding inhomogeneous models},\nProc of the Annual Conf. of Math. Rev., Tarun, Poland ( May ,1999)\n\\bibitem{korepinsg} A. G. Izergin and V. E. Korepin, \n{ Nucl. Phys.} {\\bf B 205} [FS 5] 401 (1982)\n%\\bibitem{rl}\n%R. Radha and M. Lakshmanan, J. Phys. {\\bf A 28} 6977 (1995) \n\\end{thebibliography}\n \\end{document}\n\\ed\n" } ]	[ { "name": "solv-int9912001.extracted_bib", "string": "{msg Malomed et al Phys. Rev. B41 (1990) 11271; Phys. Lett. A 144(1990) 351 N. Sanchez et al eprint patt-sol/ 9401005 P. Woafo, J. Phys. Soc. Japan, 67 (1998) 3734 D. Sen and S. Lal, e-print cond-mat/9811330" }, { "name": "solv-int9912001.extracted_bib", "string": "{burtsev S. P. Burtsev, V. E. Zakharov and A. V. Mikhailov, Teor . Mat. Fiz. 70 (1987) 323" }, { "name": "solv-int9912001.extracted_bib", "string": "{CL H. H. Chen and C. S. Liu, Phys. Rev. Lett. 37 (1976) 693 P. A. Horvatthy and J C Yera, {The variable coefficient NLS equation and the conformal properties of non-relativistic space time, preprint 18/3/99 (submitted to J. Nonlinear Math. Phys.)" }, { "name": "solv-int9912001.extracted_bib", "string": "{rl R. Radha and M. Lakshmanan, Chaos Solitons and Fractals, 4 (1994) 181" }, { "name": "solv-int9912001.extracted_bib", "string": "{konotop M. Bruschi, D. Levi and O. Ragnisco, Nuove Cimento, 53A (1979) 21 ; V. V. Konotop, O. A. Chubaikalo and L. Vazquez, Phys. Rev. E 48 (1993) 563" }, { "name": "solv-int9912001.extracted_bib", "string": "{impchain P. Schmitteckert, P. Schwab and U. Eckern, Erophys. Lett. 30 (1995) 543 H. P. Eckle, A. Punnoose and R. R\\\"omer,Erophys. Lett. 39 (1997) 293 G. Bed\\\"urftig, F. Essler and H. Frahm, Nucl. Phys. B489 (1997) 697" }, { "name": "solv-int9912001.extracted_bib", "string": "{kunprl99 Anjan Kundu, Phys. Rev. Lett., 82 (1999) 3936" }, { "name": "solv-int9912001.extracted_bib", "string": "{kulskly P. Kulish and E. K. Sklyanin, Lect. Notes in Phys. {151 (ed. J. Hietarinta et al, Springer, 1982), 61." }, { "name": "solv-int9912001.extracted_bib", "string": "{tarunpr99 Anjan Kundu, {Algebraic construction of quantum integrable models including inhomogeneous models, Proc of the Annual Conf. of Math. Rev., Tarun, Poland ( May ,1999)" }, { "name": "solv-int9912001.extracted_bib", "string": "{korepinsg A. G. Izergin and V. E. Korepin, { Nucl. Phys. {B 205 [FS 5] 401 (1982) %" }, { "name": "solv-int9912001.extracted_bib", "string": "{rl %R. Radha and M. Lakshmanan, J. Phys. {A 28 6977 (1995)" } ]
solv-int9912002		[]	We develop a quantum Lax scheme for IRF models and difference versions of Calogero-Moser-Sutherland models introduced by Ruijsenaars. The construction is in the spirit of the Adler-Kostant-Symes method generalized to the case of face Hopf algebras and elliptic quantum groups with dynamical R-matrices.	[ { "name": "solv-int9912002.tex", "string": " %% below is the ASI LATEX format for the Proceedings of\n %% `Quantum Theory and Symmetries' (Goslar, 18-22 July 1999)\n %% (World Scientific, 2000),\n %% edited by H.-D. Doebner, V.K. Dobrev, J.-D. Hennig and W. Luecke\n\n %% You can make a copy of this file and just fill in your\n %% contribution replacing appropriately.\n\n\n \\documentstyle[12pt,amssymb]{article}\n\n%\\usepackage{amssymb}\n\\newtheorem{thm}{Theorem}[section]\n\\def\\ha{\\mbox{$\\cal H$}}\n\\def\\la{\\langle}\n\\def\\ra{\\rangle}\n\\def\\i{^{-1}}\n\\def\\p{^\\phi}\n\\def\\ie{\\emph{i.e.}}\n\\def\\eg{\\emph{e.g.}}\n\\def\\eq{\\begin{equation}}\n\\def\\en{\\end{equation}}\n\\def\\ot{\\otimes}\n\\def\\id{\\mbox{\\small id}}\n\\def\\no{\\mbox{\\footnotesize \\#}}\n\n\\def\\Z{\\mathbb{Z}}\n\\def\\C{\\mathbb{C}}\n\\def\\Re{\\mathbb{R}}\n\\def\\R{{\\cal R}}\n\n\n \\hfuzz=10pt\n \\pagestyle{empty}\n \\textheight 8.5in \\textwidth 6in\n %\\textheight 22.5cm \\textwidth 16cm\n \\normalbaselineskip=12pt\n \\normalbaselines\n \\oddsidemargin 0.5cm\n \\evensidemargin 0.5cm\n \\topmargin -1cm\n\n \\begin{document}\n \\begin{center}\n \\vspace{1.0cm}\n\n {\\LARGE{\\bf Quantum Lax scheme for \nRuijsenaars models}}\n\n \\vskip 1.5cm\n\n {\\large {\\bf Branislav Jur\\v co${}^{}$\\footnote{e-mail: jurco@mpim-bonn.mpg.de} \nand Peter Schupp${}^{*}$ }}\n\n \\vskip 0.5 cm\n\n${}^$Max-Planck-Institut f\\\"ur Mathematik\\\\Vivatgasse 7\\\\\nD-53111 Bonn, Germany\\\\[1ex]\n%\n${}^{*}$Sektion Physik\\\\\nUniversit\\\"at M\\\"unchen\\\\\nTheresienstr.\\ 37\\\\\nD-80333 M\\\"unchen, Germany\n\n \\end{center}\n\n \\vspace{1 cm}\n\n \\begin{abstract}\n We develop a quantum Lax scheme for IRF models \nand difference versions of Calogero-Moser-Sutherland\nmodels introduced by Ruijsenaars. The construction is in the spirit of the\nAdler-Kostant-Symes method generalized\nto the case of face\nHopf algebras and elliptic quantum groups with dynamical\nR-matrices.\n \\end{abstract}\n\n \\vspace{1 cm}\n\n \\section{Introduction}\n\n The Hamiltonian of the (relativistic) Ruijsenaars model \\cite{Rui}\nfor two particles on a line with coordinates $x_1$ and $x_2$ acts on a wave function \nin a manifestly non-local way as\n\\eq\n\\ha \\,\\psi(\\lambda) = \n\\frac{\\theta(\\frac{c \\eta}{2} \n- \\lambda)}{\\theta(-\\lambda)}\\psi(\\lambda - \\eta) \n+ \\frac{\\theta(\\frac{c \\eta}{2} \n+ \\lambda)}{\\theta(\\lambda)}\\psi(\\lambda + \\eta), \\label{Rui}\n\\en\nwhere $\\lambda = x_1 - x_2$ is the relative coordinate, $c \\in \\C$ is a coupling \nconstant, $\\eta$ is the\nrelativistic deformation parameter, and $\\theta$ is the Jacobi theta function.\nThe Hamiltonian apparently contains shift operators\n$t_i$ that generate a one-dimensional\ngraph:\n\\begin{center}\n\\unitlength 0.50mm\n\\linethickness{0.4pt}\n\\thicklines\n\\begin{picture}(164.00,20.00)\n\\put(40.00,10.00){\\circle{2.00}}\n\\put(100.00,10.00){\\circle{2.00}}\n\\put(73.00,10.00){\\vector(1,0){24.00}}\n\\put(67.00,10.00){\\vector(-1,0){24.00}}\n\\put(70.00,7.00){\\makebox(0,0)[ct]{$\\lambda$}}\n\\put(84.00,11.00){\\makebox(0,0)[cb]{$t_1$}}\n\\put(55.00,11.00){\\makebox(0,0)[cb]{$t_2$}}\n\\put(10.00,10.00){\\circle{2.00}}\n\\put(130.00,10.00){\\circle{2.00}}\n\\put(70.00,10.00){\\circle{2.00}}\n\\put(70.00,10.00){\\makebox(0,0)[cc]{$\\times$}}\n\\thinlines\n\\put(0.00,10.00){\\line(1,0){140.00}}\n\\put(142.00,10.00){\\line(1,0){2.00}}\n\\put(146.00,10.00){\\line(1,0){2.00}}\n\\put(150.00,10.00){\\line(1,0){2.00}}\n\\put(-12.00,10.00){\\line(1,0){2.00}}\n\\put(-8.00,10.00){\\line(1,0){2.00}}\n\\put(-4.00,10.00){\\line(1,0){2.00}}\n\\end{picture}\n%\\input{shift.pic}\n\\end{center}\nRelative to a fixed vertex $\\lambda$ the\nvertices of this (ordered) graph \nare at points $\\eta\\cdot\\Z \\in \\Re$.\nThis picture generalizes arbitrary ordered graphs, which we \nshall consider in the following.\nWe will continue to\nfix a vertex \n$\\lambda$ to avoid a continuous\nfamily of disconnected graphs.\n\n\\subsection{Face algebras}\n \nThe Hilbert space of the model \nare vector spaces on paths of fixed length on the ordered graph.\nIts operators are elements of\na Face Algebra $F$ \\cite{Hay1}. This is a novel mathematical structure that\nnaturally incorporates the complicated shifts of the Ruijsenaars model.\nThere are two commuting projection operators $e^i, e_i \\in F$ \nonto bra's and ket's corresponding to each vertex $i$:\n$\ne_i e_j = \\delta_{ij} e_i$, $e^i e^j = \\delta_{ij} e^i$,\n$\\sum e_i = \\sum e^i = 1$.\n$F$ has a coalgebra structure such that $e^i_j \\equiv e^i e_j = e_j e^i$ is a\ncorepresentation:\n\\eq\n\\Delta(e^i_j) = {\\textstyle\\sum}_k e^i_k \\ot e^k_j,\n\\quad \\epsilon(e^i_j) = \\delta_{ij}; \\qquad \\Rightarrow \\quad\n\\Delta(1) = \\sum_k e_k \\ot e^k \\neq 1 \\ot 1 .\n\\en\nThe latter is a key feature of face algebras and weak \n$C^$-Hopf\nalgebras.\nThe matrix indices of $e^i_j$ are vertices,\n\\ie\\ paths of length zero. In the given setting it is natural to also\nallow paths of fixed length on a finite\noriented graph $\\cal G$\nas matrix indices. \nWe shall use capital letters \nto label paths. A path $P$ has an origin (source) $\\cdot P$,\nan end (range) $P \\cdot$ and a length \n$\\no P$. \nTwo paths $Q$, $P$ can be concatenated to form a new path\n$Q \\cdot P$, if the end of the first path coincides with the start of\nthe second path, \\ie\\ if \n$Q \\cdot = \\cdot P$.\nThe symbols\n$T^A_B$, where $\\no A = \\no B = \\no A' \\geq 0$,\nwith relations\n\\eq\n\\Delta\\left( T^A_B \\right) = \\sum_{A'} T^A_{A'} \\ot T^{A'}_B, \\qquad\n\\epsilon( T^A_B ) = \\delta_{A B} \\label{eps}, \\qquad\nT^A_B T^C_D = \\delta_{A\\cdot,\\cdot C} \\delta_{B\\cdot,\\cdot D} T^{A \\cdot\nC}_{B \\cdot D} \\label{tt}\n\\en\nspan an object that obeys the axioms of a face algebra.\nRelations (\\ref{eps}) make $T^A_B$ a\ncorepresentation; and the last expression is the rule for combining\nrepresentations.\nThe axioms of a face algebra can be found in \\cite{Hay1}.\n%\n\\paragraph{\\it Pictorial representation:}\n$$T^A_B \\,\\sim\\;\n\\unitlength1mm\n\\begin{picture}(10,10)(0,4)\\small\n\\put(0,0){\\line(1,0){10}}\n\\put(0,10){\\line(1,0){10}}\n\\multiput(0,0)(0,1){10}{\\line(0,1){0.5}}\n\\multiput(10,0)(0,1){10}{\\line(0,1){0.5}}\n\\put(0,5){\\vector(0,1){1}}\n\\put(10,5){\\vector(0,1){1}}\n\\put(5,0){\\vector(1,0){1}}\n\\put(5,10){\\vector(1,0){1}}\n\\put(5,-1){\\makebox(0,0)[ct]{$B$}}\n\\put(5,11){\\makebox(0,0)[cb]{$A$}}\n\\end{picture}\n\\quad\nT^A_B T^C_D \\,\\sim\\;\n\\begin{picture}(20,10)(0,4)\\small\n\\put(0,0){\\line(1,0){10}}\n\\put(0,10){\\line(1,0){10}}\n\\multiput(0,0)(0,1){10}{\\line(0,1){0.5}}\n\\multiput(10,0)(0,1){10}{\\line(0,1){0.5}}\n\\put(0,5){\\vector(0,1){1}}\n\\put(10,5){\\vector(0,1){1}}\n\\put(5,0){\\vector(1,0){1}}\n\\put(5,10){\\vector(1,0){1}}\n\\put(5,-1){\\makebox(0,0)[ct]{$B$}}\n\\put(5,11){\\makebox(0,0)[cb]{$A$}}\n\\put(10,0){\\line(1,0){10}}\n\\put(10,10){\\line(1,0){10}}\n\\multiput(20,0)(0,1){10}{\\line(0,1){0.5}}\n\\put(20,5){\\vector(0,1){1}}\n\\put(15,0){\\vector(1,0){1}}\n\\put(15,10){\\vector(1,0){1}}\n\\put(15,-1){\\makebox(0,0)[ct]{$D$}}\n\\put(15,11){\\makebox(0,0)[cb]{$C$}}\n\\end{picture}\\quad\n\\Delta T^A_B = \\sum_{A'} T^A_{A'} \\otimes T^{A'}_B \\,\\sim\\;\n\\begin{picture}(15,10)(0,9)\\small\n\\put(0,20){\\line(1,0){10}}\n\\multiput(0,10)(0,1){10}{\\line(0,1){0.5}}\n\\multiput(10,10)(0,1){10}{\\line(0,1){0.5}}\n\\put(0,15){\\vector(0,1){1}}\n\\put(10,15){\\vector(0,1){1}}\n\\put(5,20){\\vector(1,0){1}}\n\\put(5,21){\\makebox(0,0)[cb]{$A$}}\n\\put(0,0){\\line(1,0){10}}\n\\put(0,10){\\line(1,0){10}}\n\\multiput(0,0)(0,1){10}{\\line(0,1){0.5}}\n\\multiput(10,0)(0,1){10}{\\line(0,1){0.5}}\n\\put(0,5){\\vector(0,1){1}}\n\\put(10,5){\\vector(0,1){1}}\n\\put(5,0){\\vector(1,0){1}}\n\\put(5,10){\\vector(1,0){1}}\n%\\put(5,11){\\makebox(0,0)[cb]{\\scriptsize$\\sum A'$}}\n\\put(5,-1){\\makebox(0,0)[ct]{$B$}}\n\\end{picture}\\vspace{4mm}\n$$\nThe dashed paths indicate the $F$-space(s). Inner paths are summed over.\n\nIt is convenient to work with\nan abstract, universal $T$, which is the\ncanonical element $T_{12}$ of $U \\ot F$, \nwhere $U$ is the dual of $F$ via the\npairing $\\la \\: , \\: \\ra$.\\footnote{Here \nand in the following we \nshall frequently suppress the\nthe second index of $T$; it corresponds to the $F$-space.\nThe displayed expressions are\nshort for $\\la T_{12},f\\ot \\id\\ra = f \\in F$,\n$\\la T_{12} T_{13}, f \\ot \\id \\ot \\id\\ra \n= \\Delta\\, f \\in F \\ot F$\nand $\\la T_{13} T_{23}, f\\ot g\\ot \\id\\ra = f g \\in F$.}\n\\[ \n\\la T , f \\ra = f, \\quad \\la T_1 \\ot T_1, f \\ra = \\Delta\\,f, \\quad\n\\la T_1 T_2 , f \\ot g \\ra = f g ; \\quad f , g \\in F\n\\]\n\n\nA face \\emph{Hopf} algebra has an antipode $S$ which is denoted by a tilde\nin the universal tensor\nformalism: $\\la \\tilde T, f\\ra = S(f)$. \nAn ordinary Hopf algebra is a special case of a Face Hopf algebra with a single\nvertex.\nOrdinary matrix indices correspond\nto closed loops in that case.\n\n\n\\subsection{Boltzmann weights}\n\nThe axioms \\cite{Hay1} for a quasitriangular face algebra are similar to\nthose of a quasitriangular Hopf algebra; there is a universal\n$R \\in U \\ot U$ that controls the non-cocommutativity of the coproduct in $U$\nand the non-commutativity of the\nproduct in $F$,\n\\eq\nR T_1 T_2 = T_2 T_1 R , \\quad \\tilde R T_2 T_1 = T_1 T_2 \\tilde R,\n\\quad \\tilde R \\equiv (S \\ot \\id)(R),\n\\label{rtt}\n\\en\nhowever the antipode of $R$ is\nno longer inverse of $R$ but rather \n$\\tilde R R = \\Delta(1)$.\nThe numerical \n``$R$-matrix'' obtained by contracting $R$ with two face corepresentations\nis given by the face Boltzmann weight $W$:\n\\[ \n\\la R , T^A_B \\ot T^C_D \\ra \\; = \\; R^{A C}_{B D} \n\\;\\equiv\\; W\\Big( {C {B \\atop A} D} \\Big)\n\\quad\\sim\\qquad\n\\unitlength1mm\n\\begin{picture}(10,7)(-2,4)\\small\n\\put(0,0){\\line(1,0){10}}\n\\put(0,0){\\line(0,1){10}}\n\\put(10,10){\\line(0,-1){10}}\n\\put(10,10){\\line(-1,0){10}}\n\\put(0,5){\\vector(0,-1){1}}\n\\put(10,5){\\vector(0,-1){1}}\n\\put(5,0){\\vector(1,0){1}}\n\\put(5,10){\\vector(1,0){1}}\n\\put(5,-1){\\makebox(0,0)[ct]{$A$}}\n\\put(5,11){\\makebox(0,0)[cb]{$B$}}\n\\put(-1,5){\\makebox(0,0)[rc]{$C$}}\n\\put(11,5){\\makebox(0,0)[lc]{$D$}}\n\\end{picture}\\]\\vspace{2mm}\n\n\\noindent The Boltzmann weight is zero unless $C\\cdot A$ and $B\\cdot D$\nare valid paths with common source and range.\nFor $f \\in F$ there are two algebra homomorphisms\n$F \\rightarrow U$:\n\\eq\nR^+(f) = \\la R, f \\ot \\id\\ra ,\\qquad R^-(f) = \\la \\tilde R, \\id \\ot f\\ra . \\label{rplus}\n\\en\n$R$ satisfies the Yang\\-Baxter Equation\n$R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$.\nContracted \nwith $T^A_B \\ot T^C_D \\ot T^E_F$ this expression yields\na numerical Yang-Baxter equation with the following pictorial representation\n\\cite{Bax}:\n\\begin{center}\n\\unitlength 0.50mm\n\\linethickness{0.4pt}\n\\begin{picture}(145.00,46.00)\n\\put(0.00,20.00){\\line(3,-4){15.00}}\n\\put(15.00,0.00){\\line(1,0){25.00}}\n\\put(40.00,0.00){\\line(3,4){15.00}}\n\\put(55.00,20.00){\\line(-3,4){15.00}}\n\\put(40.00,40.00){\\line(-1,0){25.00}}\n\\put(15.00,40.00){\\line(-3,-4){15.00}}\n\\put(90.00,20.00){\\line(3,-4){15.00}}\n\\put(105.00,0.00){\\line(1,0){25.00}}\n\\put(130.00,0.00){\\line(3,4){15.00}}\n\\put(145.00,20.00){\\line(-3,4){15.00}}\n\\put(130.00,40.00){\\line(-1,0){25.00}}\n\\put(105.00,40.00){\\line(-3,-4){15.00}}\n\\put(0.00,20.00){\\line(1,0){25.00}}\n\\put(25.00,20.00){\\line(3,4){15.00}}\n\\put(25.00,20.00){\\line(3,-4){15.00}}\n\\put(105.00,40.00){\\line(3,-4){15.00}}\n\\put(120.00,20.00){\\line(-3,-4){15.00}}\n\\put(120.00,20.00){\\line(1,0){25.00}}\n\\put(15.00,40.00){\\vector(-3,-4){9.00}}\n\\put(0.00,20.00){\\vector(3,-4){9.00}}\n\\put(15.00,40.00){\\vector(1,0){15.00}}\n\\put(0.00,20.00){\\vector(1,0){15.00}}\n\\put(15.00,0.00){\\vector(1,0){15.00}}\n\\put(25.00,20.00){\\vector(3,-4){9.00}}\n\\put(40.00,40.00){\\vector(-3,-4){9.00}}\n\\put(40.00,40.00){\\vector(3,-4){9.00}}\n\\put(55.00,20.00){\\vector(-3,-4){9.00}}\n\\put(90.00,20.00){\\vector(3,-4){9.00}}\n\\put(105.00,40.00){\\vector(-3,-4){9.00}}\n\\put(105.00,40.00){\\vector(3,-4){9.00}}\n\\put(120.00,20.00){\\vector(-3,-4){9.00}}\n\\put(130.00,40.00){\\vector(3,-4){9.00}}\n\\put(145.00,20.00){\\vector(-3,-4){9.00}}\n\\put(105.00,40.00){\\vector(1,0){15.00}}\n\\put(120.00,20.00){\\vector(1,0){15.00}}\n\\put(105.00,0.00){\\vector(1,0){15.00}}\n\\put(20.00,30.00){\\makebox(0,0)[cc]{$R_{13}$}}\n\\put(40.00,20.00){\\makebox(0,0)[cc]{$R_{23}$}}\n\\put(20.00,10.00){\\makebox(0,0)[cc]{$R_{12}$}}\n\\put(105.00,20.00){\\makebox(0,0)[cc]{$R_{23}$}}\n\\put(125.00,10.00){\\makebox(0,0)[cc]{$R_{13}$}}\n\\put(125.00,30.00){\\makebox(0,0)[cc]{$R_{12}$}}\n\\put(72.00,20.00){\\makebox(0,0)[cc]{$=$}}\n\\put(27.00,42.00){\\makebox(0,0)[cb]{$B$}}\n\\put(27.00,-2.00){\\makebox(0,0)[ct]{$A$}}\n\\put(5.00,30.00){\\makebox(0,0)[rb]{$E$}}\n\\put(5.00,10.00){\\makebox(0,0)[rt]{$C$}}\n\\put(50.00,10.00){\\makebox(0,0)[lt]{$F$}}\n\\put(50.00,30.00){\\makebox(0,0)[lb]{$D$}}\n\\put(117.00,42.00){\\makebox(0,0)[cb]{$B$}}\n\\put(117.00,-2.00){\\makebox(0,0)[ct]{$A$}}\n\\put(95.00,30.00){\\makebox(0,0)[rb]{$E$}}\n\\put(95.00,10.00){\\makebox(0,0)[rt]{$C$}}\n\\put(140.00,10.00){\\makebox(0,0)[lt]{$F$}}\n\\put(140.00,30.00){\\makebox(0,0)[lb]{$D$}}\n\\end{picture}\n\\end{center}\n\\vspace{1ex}\nThe inner edges are paths that are summed over. \n\n\\section{Quantum factorization}\n\nCocommutative functions, like the trace of the T-matrix, \nprovide mutually commutative operators including the Hamiltonian. \nThe following theorem gives for\nthe case of face Hopf algebras what has become known as the\n``Main theorem'' for the solution by factorization of the equations\nof motion \\cite{AKS,RS,SJ,PB}:\n\n\\begin{thm}[Main theorem for face algebras]\n\\mbox{ }\n\n\\begin{enumerate}\n\\item[(i)] The set of cocommutative functions, $I$, is a commutative\nsubalgebra of $F$.\n\\item[(ii)] The Heisenberg \nequations of motion defined by a Hamiltonian $\\ha \\in I$\nare of Lax form\n$i \\frac{d T}{dt} = \\left[ M^\\pm , T \\right]$,\nwith $M^\\pm = 1 \\ot \\ha - m_\\pm \\in U_\\pm \\ot F$, \n$m_\\pm = R^\\pm(\\ha_{(2)}) \\ot \\ha_{(1)}$; see (\\ref{rplus}).\n\\item[(iii)] Let $g_\\pm(t) \\in U_\\pm \\ot F$\nbe the solutions to the factorization problem\n$g_-^{-1}(t) g_+(t) = \\exp( i t (m_+ - m_-)) \\: \\in \\: U \\ot F ,$\nthen \n$T(t) = g_\\pm(t) T(0) g_\\pm(t)^{-1} $\nsolves the Lax equation; $\\:g_\\pm(t)$ are given by \n$g_\\pm(t) = \\exp(-it(1\\ot h))\\,\\exp(it(1\\ot h - M^\\pm(0))$\nand are the solutions to the differential equation\n$i \\frac{d}{dt}g_\\pm(t) =\nM^\\pm(t) g_\\pm(t), \\; g_\\pm(0) = 1 . $\n\\end{enumerate}\n\\end{thm}\n\n\\section{Dynamical operators}\n\nLike we\nmentioned in the introduction we are interested in the action of\nthe Hamiltonian with respect to a fixed vertex.\nWe fix a vertex with the help of $e^\\lambda, e_\\lambda \\in F$.\nR-matrices with a fixed vertex are {\\it Dynamical $R$-matrices}:\n$\nR_{12}(\\lambda) \\equiv \\la R , T_1(\\lambda) \\ot T_2 \\ra,\n$\nwhere $T(\\lambda)^A_B$ is zero unless the range (end) of path $A$ is equal to the\nfixed vertex $\\lambda$:\n\\eq\nT(\\lambda)^A_B \\; = \\; T^A_B \\, e^\\lambda\n\\quad\\sim\\quad\n\\unitlength1mm\n\\begin{picture}(10,8)(0,4)\\small\n\\put(0,0){\\line(1,0){10}}\n\\put(0,10){\\line(1,0){10}}\n\\multiput(0,0)(0,1){10}{\\line(0,1){0.5}}\n\\multiput(10,0)(0,1){10}{\\line(0,1){0.5}}\n\\put(0,5){\\vector(0,1){1}}\n\\put(10,5){\\vector(0,1){1}}\n\\put(5,0){\\vector(1,0){1}}\n\\put(5,10){\\vector(1,0){1}}\n\\put(10,10){\\makebox(0,0)[cc]{$\\times$}}\n\\put(11,10){\\makebox(0,0)[lb]{$\\lambda$}}\n\\put(5,-1){\\makebox(0,0)[ct]{$B$}}\n\\put(5,11){\\makebox(0,0)[cb]{$A$}}\n\\end{picture}\n\\en\\vspace{2mm}\n\n\\noindent The formalism naturally includes shifts, as can e.g.\\\nbe seen in the {\\it Dynamical $RTT$-equation} \\cite{Fel}\n\\eq\nR_{12}(\\lambda) T_{1}(\\lambda - h_2) T_{2}(\\lambda)\n= T_{2}(\\lambda - h_1) T_{1}(\\lambda) R_{12}(\\lambda - h_3) .\n\\en\nand the {\\it Dynamical Yang-Baxter equation} \\cite{Ger}\n\\eq\nR_{12}(\\lambda) R_{13}(\\lambda - h_2) R_{23}(\\lambda)\n= R_{23}(\\lambda - h_1) R_{13}(\\lambda) R_{12}(\\lambda - h_3) .\n\\en\nThe shift operators $h_1$, $h_2$, $h_3$\ncan be expressed in terms of $e^\\mu$, $e_\\eta$ and\ntheir duals $E^\\mu$, $E_\\eta$,\nif we assume a (local) embedding of the vertices of the\ngraph in $\\C^n$.\n\nThe Hamiltonian of the\nRuijsenaars model is the trace of a $T$-matrix, in an appropriate\nrepresentation.\nIt can be written as a sum of operators that act in\nsubspaces corresponding to paths ending in a vertex\n$\\lambda$:\n$\n\\ha = \\sum_\\lambda \\ha(\\lambda)\n$\nwith\n$\\ha(\\lambda) = \\ha e^\\lambda = \\sum T(\\lambda)^Q_Q$.\nThe pictorial representation of the Hamiltonian is two closed dashed\npaths ($F$-space) connected by paths $Q$ of fixed length\nthat are summed over. In $\\ha(\\lambda)$ the end of path\n$Q$ is fixed. When we look at a representation on Hilbert space the\npaths $Q$ with endpoint $\\lambda$ that appear in the component\n$\\ha(\\lambda)$ of the Hamiltonian $\\ha$ will shift the argument\nof a state $\\psi(\\lambda)$\ncorresponding to the vertex $\\lambda$ to a new vertex corresponding to the\nstarting point of the path $Q$, exactly as in (\\ref{Rui}).\n\nGiven the Boltzmann weights \\cite{Has,Jim}, the Main Theorem provides Lax operators\nand Lax equations for the Ruijsenaars model \\cite{JS}.\n\n\n \\section{Acknowledgments}\n \nWe would like to thank Pavel Winternitz for\ninterest and support, and Koji Hasegawa and Jan Felipe\nvan Diejen for interesting discussions. B. J. would like to thank the Alexander von Humboldt Foundation for support.\n\n \\begin{thebibliography}{**}\n\n\\bibitem{Fel} Felder, G.: Elliptic quantum groups. In Proc. XI$^{th}$ \nInternational Congress of Mathematical Physics, D. Iagolnitzer (ed.), \nBoston: International Press, 1995\n\n\\bibitem{Bax} Baxter, R.J.: Exactly solved models in statistical physics.\nNew York: Academic Press, 1982\n\n\\bibitem{Ger} Gervais, J.L., Neveu, A.,\nNucl. Phys. {\\bf B 238}, 125 (1984)\n\n\\bibitem{Rui} Ruijsenaars, S.N.M.,\nComm. Math. Phys. {\\bf 110}, 191 (1987) \n\n\\bibitem{Has} Hasegawa, K.,\nComm. Math. Phys. {\\bf 187}, 289 (1997)\n\n\n\\bibitem{Hay1} Hayashi, T.,\nPubl. RIMS, Kyoto University {\\bf 32}, 351 (1996)\n\n\\bibitem{AKS} Adler, M.,\nInv. Math. {\\bf 50},\n219 (1979);\nKostant, B.,\nInv. Math. {\\bf 159}, 13 (1980)\n\n\\bibitem{RS} Reyman, A. G., Semenov-Tian-Shansky, M. A.: Group theoretical\nmethods in the theory of finite-dimensional integrable systems. In Encyclopedia\nof mathematical sciences, Dynamical systems VII:\nSpringer 1993\n\n\\bibitem{SJ} Jur\\v co, B., Schlieker, M.,\nComm. Math. Phys {\\bf 185}, 397 (1997)\n\n\\bibitem{PB} Jur\\v co, B., Schupp, P., Int. J. Mod. Phys. {\\bf A 12}, \n5735 (1997) \n\n\n\\bibitem{Jim} Jimbo, M., Miwa, T., Okado, M.,\nNucl. Phys. {\\bf B 300}, 74 (1988)\n\n\\bibitem{JS} Jur\\v co, B., Schupp, P.,\nJ. Math. Phys. {\\bf 39}, 3577 (1998)\n \n \\end{thebibliography}\n\n \\end{document}\n" } ]	[ { "name": "solv-int9912002.extracted_bib", "string": "{Fel Felder, G.: Elliptic quantum groups. In Proc. XI$^{th$ International Congress of Mathematical Physics, D. Iagolnitzer (ed.), Boston: International Press, 1995" }, { "name": "solv-int9912002.extracted_bib", "string": "{Bax Baxter, R.J.: Exactly solved models in statistical physics. New York: Academic Press, 1982" }, { "name": "solv-int9912002.extracted_bib", "string": "{Ger Gervais, J.L., Neveu, A., Nucl. Phys. {B 238, 125 (1984)" }, { "name": "solv-int9912002.extracted_bib", "string": "{Rui Ruijsenaars, S.N.M., Comm. Math. Phys. {110, 191 (1987)" }, { "name": "solv-int9912002.extracted_bib", "string": "{Has Hasegawa, K., Comm. Math. Phys. {187, 289 (1997)" }, { "name": "solv-int9912002.extracted_bib", "string": "{Hay1 Hayashi, T., Publ. RIMS, Kyoto University {32, 351 (1996)" }, { "name": "solv-int9912002.extracted_bib", "string": "{AKS Adler, M., Inv. Math. {50, 219 (1979); Kostant, B., Inv. Math. {159, 13 (1980)" }, { "name": "solv-int9912002.extracted_bib", "string": "{RS Reyman, A. G., Semenov-Tian-Shansky, M. A.: Group theoretical methods in the theory of finite-dimensional integrable systems. In Encyclopedia of mathematical sciences, Dynamical systems VII: Springer 1993" }, { "name": "solv-int9912002.extracted_bib", "string": "{SJ Jur\\v co, B., Schlieker, M., Comm. Math. Phys {185, 397 (1997)" }, { "name": "solv-int9912002.extracted_bib", "string": "{PB Jur\\v co, B., Schupp, P., Int. J. Mod. Phys. {A 12, 5735 (1997)" }, { "name": "solv-int9912002.extracted_bib", "string": "{Jim Jimbo, M., Miwa, T., Okado, M., Nucl. Phys. {B 300, 74 (1988)" }, { "name": "solv-int9912002.extracted_bib", "string": "{JS Jur\\v co, B., Schupp, P., J. Math. Phys. {39, 3577 (1998)" } ]
solv-int9912003		[]		[ { "name": "solv-int9912003.tex", "string": "\\documentstyle[12pt]{article}\n\\topmargin=0cm\\textheight=23.cm\\textwidth=17.cm\n\\oddsidemargin=-0.25cm\n\\evensidemargin=-0.25cm\n\\begin{document}\n\\thispagestyle{empty}\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\sect}[1]{\\setcounter{equation}{0}\\section{#1}}\n\\newcommand{\\vs}[1]{\\rule[- #1 mm]{0mm}{#1 mm}}\n\\newcommand{\\hs}[1]{\\hspace{#1mm}}\n\\newcommand{\\mb}[1]{\\hs{5}\\mbox{#1}\\hs{5}}\n\\newcommand{\\bea}{\\begin{eqnarray}}\n\\newcommand{\\eea}{\\end{eqnarray}}\n\\newcommand{\\wt}[1]{\\widetilde{#1}}\n\\newcommand{\\ux}[1]{\\underline{#1}}\n\\newcommand{\\ov}[1]{\\overline{#1}}\n\\newcommand{\\sm}[2]{\\frac{\\mbox{\\footnotesize #1}\\vs{-2}}\n {\\vs{-2}\\mbox{\\footnotesize #2}}}\n\\newcommand{\\prt}{\\partial}\n\\newcommand{\\eps}{\\epsilon}\n\\newcommand{\\p}[1]{(\\ref{#1})}\n\\newcommand{\\R}{\\mbox{\\rule{0.2mm}{2.8mm}\\hspace{-1.5mm} R}}\n\\newcommand{\\Z}{Z\\hspace{-2mm}Z}\n\\newcommand{\\cd}{{\\cal D}}\n\\newcommand{\\cg}{{\\cal G}}\n\\newcommand{\\ck}{{\\cal K}}\n\\newcommand{\\cw}{{\\cal W}}\n\\newcommand{\\vj}{\\vec{J}}\n\\newcommand{\\vl}{\\vec{\\lambda}}\n\\newcommand{\\vz}{\\vec{\\sigma}}\n\\newcommand{\\vt}{\\vec{\\tau}}\n\\newcommand{\\vw}{\\vec{W}}\n\\newcommand{\\poiss}{\\stackrel{\\otimes}{,}}\n\\newcommand{\\tx}{\\theta_{12}}\n\\newcommand{\\tb}{\\overline{\\theta}_{12}}\n\\newcommand{\\zw}[1]{{#1 \\over Z_{12}}}\n%\\newcommand{\\zwtwo}{{z_{12}}}\n%\\newcommand{\\ux}{\\underline}\n\\newcommand{\\sqp}{{\\alpha}}\n\\newcommand{\\sqm}{{\\overline\\alpha}}\n\\newcommand{\\nn}{\\nonumber \\\\}\n% REVUES POUR BIBLIO\n\n\\newcommand{\\NP}[1]{Nucl.\\ Phys.\\ {\\bf #1}}\n\\newcommand{\\PLB}[1]{Phys.\\ Lett.\\ {B \\bf #1}}\n\\newcommand{\\PLA}[1]{Phys.\\ Lett.\\ {A \\bf #1}}\n\\newcommand{\\NC}[1]{Nuovo Cimento {\\bf #1}}\n\\newcommand{\\CMP}[1]{Commun.\\ Math.\\ Phys.\\ {\\bf #1}}\n\\newcommand{\\PR}[1]{Phys.\\ Rev.\\ {\\bf #1}}\n\\newcommand{\\PRL}[1]{Phys.\\ Rev.\\ Lett.\\ {\\bf #1}}\n\\newcommand{\\MPL}[1]{Mod.\\ Phys.\\ Lett.\\ {\\bf #1}}\n\\newcommand{\\BLMS}[1]{Bull.\\ London Math.\\ Soc.\\ {\\bf #1}}\n\\newcommand{\\IJMP}[1]{Int.\\ J.\\ Mod.\\ Phys.\\ {\\bf #1}}\n\\newcommand{\\JMP}[1]{Jour.\\ Math.\\ Phys.\\ {\\bf #1}}\n\\newcommand{\\LMP}[1]{Lett.\\ Math.\\ Phys.\\ {\\bf #1}}\n\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\\newpage\n\\setcounter{page}{0}\n\\pagestyle{empty}\n\\vs{12}\n\\begin{center}\n{\\LARGE {\\bf $N=4$ Sugawara construction on $\\widehat{sl(2\|1)}$,}}\n\\\\ {\\LARGE{\\bf $\\widehat{sl(3)}$ and mKdV-type superhierarchies\n}}\\\\[0.8cm]\n\n\\vs{10} {\\large E. Ivanov$^{a,1}$, S. Krivonos$^{a,2}$ and F.\nToppan$^{b,3}$} ~\\\\ \\quad \\\\ {\\em {~$~^{(a)}$ JINR-Bogoliubov\nLaboratory of Theoretical Physics,}}\\\\ {\\em 141980 Dubna, Moscow\nRegion, Russia}~\\quad\\\\ {\\em ~$~^{(b)}$ DCP-CBPF,}\\\\ {\\em Rua\nXavier Sigaud 150, 22290-180, Urca, Rio de Janeiro, Brazil}\n\\end{center}\n\\vs{6}\n\n\\centerline{ {\\bf Abstract}}\n\n\\vspace{0.3cm} \\noindent The local Sugawara constructions of the\n``small'' $N=4$ SCA in terms of supercurrents of $N=2$ extensions\nof the affine $\\widehat{sl(2\|1)}$ and $\\widehat{sl(3)}$ algebras are\ninvestigated. The associated super mKdV type hierarchies\ninduced by $N=4$ SKdV ones are defined. In the $\\widehat{sl(3)}$ case the\nexistence of\ntwo non-equivalent Sugawara constructions is found. The ``long''\none involves\nall the affine $\\widehat{sl(3)}$ currents, while the ``short'' one deals\nonly with those from the subalgebra $\\widehat{sl(2)\\oplus u(1)}$. As a\nconsequence, the $\\widehat{sl(3)}$-valued affine superfields carry two\nnon-equivalent mKdV type super hierarchies induced by the\ncorrespondence between ``small'' $N=4$ SCA and $N=4$ SKdV\nhierarchy. However, only the first hierarchy possesses genuine global\n$N=4$ supersymmetry. We discuss peculiarities of the realization\nof this $N=4$ supersymmetry on the affine supercurrents.\n\n\\vs{6} \\vfill \\rightline{CBPF-NF-046-99} \\rightline{JINR E2-99-302}\n\\rightline{ solv-int/9912003} {\\em E-Mail:\\\\ 1)\neivanov@thsun1.jinr.ru\\\\ 2) krivonos@thsun1.jinr.ru\\\\\n3) toppan@cbpf.br}\n\\newpage\n\\pagestyle{plain}\n\\renewcommand{\\thefootnote}{\\arabic{footnote}}\n\\setcounter{footnote}{0}\n\n\\section{Introduction}\n\nIn the last several years integrable hierarchies of non-linear\ndifferential equations have been intensely explored, mainly in\nconnection with the discretized two-dimensional gravity theories\n(matrix models) \\cite{DGZ} and, more recently, with the\n$4$-dimensional super Yang-Mills theories in the\nSeiberg-Witten approach \\cite{SW}.\n\nA vast literature is by now available on the construction and\nclassification of the hierarchies. In the bosonic case the understanding\nof integrable hierarchies in $1+1$ dimensions is to a large extent\ncomplete. Indeed, a generalized Drinfeld-Sokolov scheme \\cite{DS}\nis presumably capable to accommodate all known bosonic\nhierarchies.\n\nOn the other hand, due to the presence of even and odd fields, the\nsituation for supersymmetric extensions remains in many respects\nunclear. Since a fully general supersymmetric Drinfeld-Sokolov\napproach to the superhierarchies is still lacking, up to now they\nwere constructed using all sorts of the available tools. These\ninclude, e.g., direct methods, Lax operators of both scalar and\nmatrix type, bosonic as well as fermionic, coset construction,\netc. \\cite{MRK}-\\cite{Top}.\n\nIn \\cite{IKT} a general Lie-algebraic framework for the $N=4$\nsuper KdV hierarchy \\cite{{DI},{DG2},{IK},{DG1}} and, hopefully, for its\nhypothetical higher conformal spin counterparts (like $N=4$\nBoussinesq) has been proposed. It is based upon a generalized\nSugawara construction on the $N=2$ superextended affine\n(super)algebras which possess a hidden (nonlinearly realized)\n$N=4$ supersymmetry. This subclass seemingly consists of $N=2$\naffine superextensions of both the bosonic algebras with the\nquaternionic structure listed in \\cite{SSTP} and proper\nsuperalgebras having such a structure. In its simplest version\n\\cite{IKT}, the $N=4$ Sugawara construction relates affine\nsupercurrents taking values in the $sl(2)\\oplus u(1)$ algebra to\nthe ``minimal'' (or ``small'') $N=4$ superconformal algebra\n($N=4$ SCA) which provides the second Poisson structure for the\n$N=4$ super KdV hierarchy. The Sugawara-type transformations are\nPoisson maps, i.e. they preserve the Poisson-brackets structure of\nthe affine (super)fields. Therefore for any Sugawara\ntransformation which maps affine superfields, say, onto the\nminimal $N=4$ SCA, the affine supercurrents themselves inherit an\nintegrable hierarchy which is constructed using the tower of the\n$N=4$ SKdV hamiltonians in involution.\nSuch $N=4$ hierarchies realized on the affine supercurrents\ncan be interpreted as generalized mKdV-type superhierarchies.\nThe simplest example, the combined $N=4$ mKdV-NLS hierarchy\nassociated with the affine $N=2 \\;\\;\\widehat{sl(2)\\oplus u(1)}$\nsuperalgebra, was explicitly constructed in \\cite{IKT}.\n\nIn the case of higher-dimensional $N=4$ affine superalgebras this\nsort of Sugawara construction is expected to yield additional\n$N=4$ multiplets of currents which would form, together with those\nof $N=4$ SCA (both ``minimal'' and ``large''), more general\nnonlinear $N=4$ superalgebras of the $W$ algebra type.\nRespectively, new SKdV (or super Boussinesq) type hierarchies with\nthese conformal superalgebras as the second Poisson structures can\nexist, as well as their mKdV type counterparts associated with the\ninitial $N=4$ affine superalgebras. Besides, the linear $N=4$ SCAs\ncan be embedded into a given affine superalgebra in different\nways, giving rise to a few non-equivalent mKdV-type superhierarchies\nassociated with the same KdV-type superhierarchy.\n\nIn this paper we describe non-equivalent $N=4$ Sugawara\nconstructions for the eight-dimensional affine\n(super)algebras $N=2 \\;\\;\\widehat{sl(2\|1)}$ and $N=2 \\;\\;\\widehat{sl(3)}$.\nThese algebras are natural candidates for the higher-rank affine\nsuperalgebras with hidden $N=4$ supersymmetry, next in complexity to\nthe simplest $\\widehat{sl(2)\\oplus u(1)}$ case treated in ref. \\cite{IKT}.\n\nThe results can be summarized as follows.\n\nIn the $\\widehat{sl(2\|1)}$ case there are no\nother {\\em local} Sugawara constructions leading to the ``small''\n$N=4$ SCA besides the one which proceeds from the\nbosonic $\\widehat{sl(2)\\oplus u(1)}$ subalgebra supercurrents.\nThe $\\widehat{sl(2\|1)}$ affine\nsupercurrents carry a unique mKdV type hierarchy,\nthe evolution equations for the extra four superfields\nbeing induced from their Poisson brackets with\nthe $N=4$ SKdV hamiltonians constructed from the\n$sl(2)\\oplus u(1)$-valued supercurrents.\nThe full hierarchy possesses by construction the\nmanifest $N=2$ supersymmetry and also reveals some extra exotic ``$N=2$\nsupersymmetry''. These two yield the standard\n$N=4$ supersymmetry only on the $\\widehat{sl(2)\\oplus u(1)}$\nsubset of currents (``standard'' means closing on $z$\ntranslations). Actually, such an extra $N=2$ supersymmetry\nis present in {\\it any} $N=2$ affine (super)algebra with a\n$\\widehat{sl(2)\\oplus u(1)}$ subalgebra. As the result,\nneither the $N=2$ $\\widehat{sl(2\|1)}$ superalgebra itself, nor\nthe above-mentioned mKdV hierarchy reveal the genuine $N=4$\nsupersymmetry.\n\nThe $\\widehat{sl(3)}$ case is more interesting since it admits\nsuch an extended supersymmetry. In this case, besides the\n``trivial'' $N=4$ SCA based on the $\\widehat{sl(2)\\oplus u(1)}$\nsubalgebra, one can define an extra $N=4$ SCA containing the full\n$N=2$ stress-tensor and so involving all affine $\\widehat{sl(3)}$\nsupercurrents \\footnote{In what follows we name the corresponding\nSugawara construction ``long'' $N=4$ Sugawara, as opposed to the\n``short'' one based on the $\\widehat{sl(2)\\oplus u(1)}$\nsubalgebra.}. We have explicitly checked that no other\nnon-equivalent local $N=4$ Sugawaras exist in this case. The\nsupercurrents of the second $N=4$ SCA generate global $N=4$\nsupersymmetry closing in the standard way on $z$-translations.\nThe defining relations of the $N=2$ $\\widehat{sl(3)}$ algebra are\ncovariant under this supersymmetry, so it is actually $N=4$\nextension of $\\widehat{sl(3)}$, similarly to the\n$\\widehat{sl(2)\\oplus u(1)}$ example. In the original basis,\nwhere the affine currents satisfy nonlinear constraints, the\nhidden $N=2$ supersymmetry transformations are essentially\nnonlinear and mix all the currents. After passing, by means of a\nnon-local field redefinition, to the basis where the constraints\nbecome the linear chirality conditions, the supercurrents split\ninto some invariant subspace and a complement which transforms\nthrough itself and the invariant subspace. In other words, they\nform a not fully reducible representation of the $N=4$\nsupersymmetry. This phenomenon was not previously encountered in\n$N=4$ supersymmetric integrable systems. We expect it to hold\nalso in higher rank $N=2$ affine superalgebras with the hidden\n$N=4$ structure.\n\nThe ``long'' Sugawara gives rise to a new mKdV type hierarchy\nassociated with the $N=4$ SKdV one. Thus the $\\widehat{sl(3)}$\naffine supercurrents provide an example of a Poisson structure\nleading to two non-equivalent mKdV-type hierarchies, both associated\nwith $N=4$ SKdV, but recovered from the ``short'' and,\nrespectively, ``long'' $N=4$ Sugawara constructions. Only the\nsecond hierarchy possesses global $N=4$ supersymmetry.\n\nAs a by-product, we notice the existence of another sort of super mKdV\nhierarchies associated with both affine superalgebras considered.\nThey are related to the so-called ``quasi'' $N=4$ SKdV hierarchy \n\\cite{{DGI},{DG2}}\nwhich still possesses the ``small'' $N=4$ SCA as the second\nPoisson structure but lacks global $N=4$ supersymmetry.\nIn the $\\widehat{sl(3)}$ case there also exist two non-equivalent\n``quasi'' super mKdV hierarchies generated through the ``short'' and\n`long'' Sugawara constructions.\n\nLike in \\cite{IKT}, in the present paper we use the $N=2$ superfield\napproach with the manifest linearly realized $N=2$ supersymmetry.\nThe results are presented in the language of\nclassical OPEs between $N=2$ supercurrents, which is equivalent\nto the Poisson brackets formalism used in \\cite{IKT}.\nWhen evaluating these $N=2$ OPEs, we systematically exploit the\nMathematica package of ref. \\cite{KT}.\n\n\n\\section{$N=2$ conventions and the minimal $N=4$ SCA}\n\nHere we fix our notation and present the $N=2$ superfield\nPoisson brackets structure of the ``minimal'' (``small'')\n$N=4$ superconformal algebra (in the OPE language).\n\nThe $N=2$ superspace is parametrized by the coordinates\n$Z\\equiv \\left\\{ z, \\theta , {\\overline \\theta}\\right\\}$,\nwith $\\left\\{ \\theta , {\\overline \\theta} \\right\\}$\nbeing Grassmann variables. The (anti)-chiral $N=2$\nderivatives $D, {\\overline D}$ are defined as\n\\begin{eqnarray}\nD = \\frac{\\partial}{\\partial \\theta}\n-\\frac{1}{2}{\\overline \\theta} \\partial_z \\;,\\;\\;\n{\\overline D} = \\frac{\\partial}{\\partial\n{\\overline \\theta}} - \\frac{1}{2}\\theta\\partial_z\\; ,\\;\\;\nD^2 = {\\overline D}{}^2 = 0 \\; , \\;\\;\n\\{ D, {\\overline D}\\} = -\\partial_z \\; \\;. \\label{Dcomm}\n\\end{eqnarray}\n\nIn the $N=2$ superfield notation the minimal $N=4$ SCA\nis represented by the spin $1$ general superfield $J(Z)$ and two\n(anti)-chiral spin $1$ superfields $W$, ${\\ov W}$\n($DW = {\\ov D}\\, {\\ov W} =0$), with the following\nclassical OPE's\n\\begin{eqnarray}\n{\\ux {J(1)J(2)}} &=& {2\\over {Z_{12}}^2} - {{\\tx \\tb}\\over {Z_{12}}^2}\nJ - \\zw{\\tb} {\\ov D}J +\\zw{ \\tx} DJ -\\zw{ \\tx \\tb} J' \\;, \\nonumber\\\\\n{\\ux {J(1)W(2)}} &=& -{{\\tx\\tb}\\over {Z_{12}}^2} W - \\zw{2} W -\\zw{\\tb} {\\ov\nD}W -\\zw{\\tx\\tb} W' \\;, \\nonumber\\\\\n{\\ux {J(1){\\ov W}(2)}} &=& -{{\\tx\\tb}\\over {Z_{12}}^2} {\\ov W} + \\zw{2} {\\ov W}\n+\\zw{\\tx} D{\\ov W} -\\zw{ \\tx\\tb} {\\ov W}' \\;, \\nonumber\\\\\n{\\ux {W(1){\\ov W}(2)}} &=& {{\\tx\\tb}\\over {Z_{12}}^3 } - {1\\over {Z_{12}}^2}\n- {{\\frac{1}{2} \\tx\\tb} \\over {Z_{12}}^2} J +\\zw {\\tb } {\\ov D} J +\\zw{1}\n J\\; . \\label{n4sca}\n\\end{eqnarray}\nHere\n$Z_{12} =\nz_1 -z_2+\\frac{1}{2}\\left( \\theta_1{\\overline\\theta}_2\n-\\theta_2{\\overline\\theta}_1\\right)$, $\\tx=\\theta_1-\\theta_2$, $\\tb\n={\\overline\\theta}_1-{\\overline\\theta}_2$, and the superfields\nin the r.h.s. are evaluated at the point $(2)\\equiv (\\,z_2, \\theta_2,\n{\\overline\\theta}_2\\,) $.\n\n\\section{The superaffinization of the $sl(2\|1)$ superalgebra}\n\nIn this and next Sections we follow the general $N=2$ superfield\nsetting for $N=2$ extensions of affine (super)algebras \\cite{HS,AIS}.\n\nThe $N=2$ $\\widehat{sl(2\|1)}$ superalgebra is generated by four fermionic\nand four bosonic superfields, respectively\n($H, {\\overline H}, F, {\\overline F}$) and ($S, {\\overline S}, R,\n{\\overline R}$).\n\\par\nThe superfields $H, {\\overline H}$ are associated with the Cartan\ngenerators\nof $sl(2\|1)$ and satisfy the (anti)chiral constraints\n\\begin{eqnarray}\n{\\overline D}\\, {\\overline H} = D H = 0 \\; \\label{chir23}\n\\end{eqnarray}\nwhile the remaining superfields are associated with the root\ngenerators of $sl(2\|1)$. In particular $F, {\\overline F}$ are\nrelated to the bosonic ($\\pm$)-simple roots and, together with\n$H, {\\ov H}$, close on the superaffine ${\\widehat {sl(2)\\oplus u(1)}}$\nsubalgebra. The extra superfields satisfy the non-linear chiral\nconstraints\n\\begin{eqnarray}\n&& {\\overline D}\\,{\\overline R} =0 \\;, \\quad\n{\\overline D}\\, {\\overline F} = {\\overline H}\\,{\\overline F} \\; , \\quad\n{\\overline D}\\, {\\overline S} = -{\\overline F}\\,{\\overline R}\n+{\\overline H}\\,{\\overline S}\\; , \\nonumber \\\\\n&& DR = HR \\;, \\quad DF = - H F\\; ,\n \\quad DS = F R \\;.\\label{cond23}\n\\end{eqnarray}\nThe full set of OPEs defining the classical\n$N=2$ superaffine ${\\widehat{sl(2\|1)}}$ algebra is given by\n\\begin{eqnarray}\n&&{\\underline{H(1){\\overline H}(2)}} = {{\\frac{1}{2}\\tx\\tb}\\over {Z_{12}}^2}\n- {1\\over Z_{12}}\\;, \\;\n{\\underline{H(1)F(2)}} = \\zw{\\tb} {F}\\;, \\;\n{\\underline{H(1){\\overline F}(2)}}= - \\zw{\\tb} {\\overline F}\\; , \\;\n\\nonumber\\\\\n&&{\\ux {H(1)S(2)}} = \\zw{\\tb} S\\;,\\;\n{\\ux {H(1) {\\ov S}(2)}} = -\\zw{\\tb} {\\ov S}\\;, \\;\n{\\ux {{\\ov H} (1)F(2)}} = \\zw{\\tx} F\\nonumber\\; , \\;\n{\\ux {{\\ov H}(1){\\ov F}(2)}} = -\\zw{\\tx} {\\ov F}\\; , \\nonumber \\\\\n&& {\\ux {{\\ov H}(1) R(2)}} = -\\zw{\\tx} R\\; , \\;\n{\\ux {{\\ov H}(1){\\ov R}(2)}} = \\zw {\\tx }{\\ov R}\\;, \\nonumber\\\\\n&& {\\ux {F(1){\\ov F}(2)}} = {{\\frac{1}{2}\\tx\\tb}\\over {Z_{12}}^2}\n-\\zw{1 -\\tb {\\ov H} - \\tx H - \\tx\\tb\n( F{\\ov F} + H{\\ov H} + {\\ov D} H)} \\;,\n\\nonumber\\\\\n&&{\\ux{F(1)S(2)}} = -\\zw{\\tx\\tb} FS\\;,\\;\n{\\ux {F(1){\\ov S}(2)}} \n= \\zw{ \\tb {\\ov R} +\\tx\\tb (F {\\ov S} +H{\\ov R})}\\;, \\nonumber\\\\\n&&{\\ux {F(1){R}(2)}} = -\\zw{\\tb S + \\tx\\tb HS}\\;,\\;\n{\\ux {{\\ov F} (1)S(2)}} = -\\zw{\\tx R + \\tx\\tb {\\ov H} R}\\;,\\nonumber\\\\\n&&{\\ux {{\\ov F}(1) R(2)}} = \\zw{\\tx\\tb} R{\\ov F}\\;, \\;\n{\\ux {{\\ov F}(1) {\\ov R}(2)}} = \\zw{ \\tx {\\ov S} -\\tx\\tb\n({\\ov F}\\,{\\ov R} -{\\ov H}\\,{\\ov S})}\\;,\\nonumber\\\\\n&&{\\ux {S(1){\\ov S}(2)}} = -{{\\frac{1}{2}\\tx\\tb}\\over {Z_{12}}^2}\n+\\zw{1 -\\tb {\\ov H} -\\tx\\tb (F{\\ov F}- R{\\ov R})} \\;,\\;\n{\\ux{S(1)R(2)}} = -\\zw{\\tx\\tb} SR\\;, \\nonumber\\\\\n&&{\\ux {S(1){\\ov R}(2)}} = \\zw{\\tx F +\\tx\\tb {\\ov D}F}\\;, \\;\n{\\ux {{\\ov S}(1)R(2)}} = \\zw{\\tb {\\ov F}\n+\\tx\\tb ( R{\\ov S} + H {\\ov F} - D{\\ov F})}\\;,\\nonumber\\\\\n&&{\\ux {R(1){\\ov R}(2)}} = -{{\\frac{1}{2}\\tx\\tb}\\over {Z_{12}}^2}\n+ \\zw{1 + \\tx H +\\tx\\tb {\\ov D} H} \\;.\\label{sope23}\n\\end{eqnarray}\nAll other OPEs are vanishing. The superfields\nin the r.h.s. are evaluated at the point (2).\n\nThere is only one local Sugawara realization of $N=4$ SCA associated\nwith this affine $sl(2\|1)$ superalgebra. It is explicitly given\nby the relations\n\\begin{equation} \\label{sl21N4}\nJ = {\\ov D} H + D {\\ov H} + H{\\ov H} + F {\\ov F}\\; ,\\;\nW = D{\\ov F}\\; , \\;\n{\\ov W} = {\\ov D} F \\; .\n\\end{equation}\nIt involves only the superfields ($H, {\\ov H}, F, {\\ov F}$)\nwhich generate just the ${\\widehat{sl(2)\\oplus u(1)}}$-superaffine\nsubalgebra. It can be checked that no Sugawara construction\ninvolving all the $sl(2\|1)$ superfields exists in this case. The\n$N=4$ SKdV hamiltonians constructed from the superfields \\p{sl21N4}\nproduce an mKdV type hierarchy of the evolution equations for the\n$\\widehat{sl(2\|1)}$ supercurrents through the OPE relations \\p{sope23}.\n\nNote that the supercurrents \\p{sl21N4} generate\nglobal non-linear automorphisms of $N=2$ $\\widehat{sl(2\|1)}$\n(preserving both\nthe OPEs \\p{sope23} and the constraints \\p{cond23}), such that their\nalgebra formally coincide with the $N=4$ supersymmetry algebra.\nHowever, these\nfermionic transformations close in a standard way on $z$-translations\nonly on the ${\\widehat{sl(2)\\oplus u(1)}}$ subset. On the rest of\naffine supercurrents they yield complicated\ncomposite objects in the closure. It is of course a consequence of\nthe fact that the true $z$-translations of all supercurrents are generated\nby the full $N=2$ stress-tensor on the affine superalgebra, while\n$N=4$ SCA \\p{sl21N4} contains the stress-tensor on a subalgebra.\nSo this fermionic automorphisms symmetry cannot be viewed\nas promoting the manifest $N=2$ supersymmetry\nto $N=4$ one \\footnote{This\nkind of odd automorphisms is inherent to any $N=2$ affine algebra or\nsuperalgebra containing ${\\widehat{sl(2)\\oplus u(1)}}$ subalgebra.}.\nThus the $N=2$ superaffine $\\widehat{sl(2\|1)}$ algebra as a whole\npossesses no hidden $N=4$ structure, as distinct from its\n${\\widehat{sl(2)\\oplus u(1)}}$ subalgebra. This obviously implies\nthat the super mKdV hierarchy induced on the full set of\nthe $\\widehat{sl(2\|1)}$ supercurrents through the Sugawara construction\n\\p{sl21N4} is not $N=4$ supersymmetric as well.\n\n\\section{The superaffine ${\\widehat{sl(3)}}$ algebra}\n\nThe superaffinization of the $sl(3)$ algebra is spanned by eight\nfermionic $N=2$ superfields subjected to non-linear (anti)chiral\nconstraints. We denote these superfields $H, F, R, S$ (their antichiral\ncounterparts are ${\\ov H}, {\\ov F}, {\\ov R}, {\\ov S}$). The\n${\\widehat{sl(2)\\oplus u(1)}}$ subalgebra is represented by $H, {\\ov\nH}, S, {\\ov S}$. As before the Cartan subalgebra\nis represented by the standard (anti)chiral $N=2$\nsuperfields $H, {\\overline H}$\n\\begin{eqnarray}\n&& D H = {\\ov D}\\, {\\ov H} = 0~. \\label{chirH}\n\\end{eqnarray}\nThe remaining supercurrents are subject to\nthe non-linear constraints:\n\\begin{eqnarray}\n&&DS = - HS\\;, \\quad D F = -\\sqm H F + SR \\;, \\quad\n DR = \\sqp H R \\;, \\nonumber \\\\\n&&{\\ov D}\\, {\\ov S} = {\\ov H}\\,{\\ov S}\\; ,\\quad\n{\\ov D}\\, {\\ov F} = \\sqp {\\ov H}\\,{\\ov F} -{\\ov S}\\,{\\ov R}\\; , \\quad\n{\\ov D}\\,{\\ov R} = - \\sqm {\\ov H}\\,{\\ov R}\\;, \\label{cond3}\n\\end{eqnarray}\nwhere\n\\begin{equation} \\label{param}\n\\sqp= \\frac{1+i\\sqrt{3}}{2} \\;, \\quad \\sqm= \\frac{1-i\\sqrt{3}}{2} \\;.\n\\end{equation}\n\nThe non-vanishing OPEs of the classical $N=2$ superaffine\n${\\widehat{sl(3)}}$ algebra read:\n\\begin{eqnarray}\n&&{\\ux{H(1) {\\ov H}(2)}} = {{\\frac{1}{2}\\tx\\tb}\\over {Z_{12}}^2}- \\zw{1} \\;,\\;\n{\\ux {H(1) F(2)}} = \\zw{\\sqp\\tb} F \\;,\\;\n{\\ux {H(1) {\\ov F}(2)}} = -\\zw{\\sqp \\tb} {\\ov F}\\;, \\nonumber\\\\\n&&{\\ux {H(1) S(2) }} = \\zw{\\tb} S\\;, \\;\n{\\ux {H(1) {\\ov S}(2)}} = -\\zw{\\tb} {\\ov S}\\; ,\\;\n{\\ux {H(1) R(2)}}= -\\zw{\\sqm \\tb} R\\; ,\\;\n{\\ux {H(1) {\\ov R}(2)}} = \\zw{\\sqm \\tb} {\\ov R}\\; ,\\nonumber\\\\\n&&{\\ux {{\\ov H}(1) F(2)}} = \\zw{\\sqm \\tx} F\\; , \\;\n{\\ux {{\\ov H}(1) {\\ov F}(2)}} = -\\zw{\\sqm \\tx} {\\ov F}\\; , \\;\n{\\ux {{\\ov H}(1) S(2)}} = \\zw{ \\tx} S\\; , \\;\n{\\ux {{\\ov H}(1) {\\ov S}(2)}} = - \\zw{ \\tx} {\\ov S}\\; , \\nonumber\\\\\n&&{\\ux {{\\ov H}(1) R(2)}} = -\\zw{ \\sqp \\tx} R \\; , \\;\n{\\ux {{\\ov H}(1) {\\ov R} (2)}} = \\zw{\\sqp \\tx} {\\ov R}\\; , \\nonumber\\\\\n&&{\\ux { F(1){\\ov F}(2)}} = {{\\frac{1}{2}\\tx\\tb}\\over {z_{12}}^2}\n-\\zw{1 - \\sqp \\tb {\\ov H} -\\sqm \\tx H -\n \\tx \\tb\n( F {\\ov F} + H {\\ov H} + R {\\ov R}+ S{\\ov S} +\\sqm {\\ov D} H)}\\;,\n\\nonumber\\\\\n&&{\\ux {F(1) S(2) }} = \\zw{\\sqp \\tx \\tb} FS \\; , \\;\n{\\ux {F(1){\\ov S}(2)}}= \\zw{ \\tx R + \\tx \\tb ({\\ov D} R +\n\\sqm F{\\ov S} - {\\ov H} R )}\\; ,\\nonumber\\\\\n&&{\\ux { F(1)R(2)}} = \\zw{ \\sqm \\tx\\tb} FR\\; , \\;\n{\\ux {F(1){\\ov R}(2)}} = -\\zw {\\tx S +\\tx\\tb ( {\\ov D} S -\\sqp F{\\ov R}\n+\\sqm {\\ov H} S)}\\; ,\\nonumber\\\\\n&&{\\ux { {\\ov F}(1) S(2)}} = -\\zw{ \\tb {\\ov R } - \\tx \\tb\n( H{\\ov R} -\\sqp {\\ov F} S + D {\\ov R} )} \\; , \\;\n{\\ux { {\\ov F}(1) {\\ov S}(2)}} = -\\zw{\\sqm \\tx\\tb}\n{\\ov F}\\,{\\ov S}\\; , \\nonumber\\\\\n&&{\\ux { {\\ov F} (1) R(2)}} = \\zw{ \\tb {\\ov S} - \\tx \\tb( D {\\ov S}-\n \\sqp H {\\ov S} +\\sqm {\\ov F} R)} \\; , \\;\n{\\ux { {\\ov F} (1) {\\ov R} (2)}} \n= -\\zw{ \\sqp\\tx \\tb} {\\ov F} \\,{\\ov R}\\;, \\nonumber\\\\\n&& {\\ux {S(1) {\\ov S} (2)}} = {{\\frac{1}{2}\\tx\\tb}\\over{Z_{12}}^2} -\n\\zw{1- \\tb {\\ov H} - \\tx H - \\tx \\tb\n(S{\\ov S} + H {\\ov H} + {\\ov D} H )}\\;,\\; \\nonumber\\\\\n&&{\\ux { S(1) R(2)}} = -\\zw{ \\tb F + \\tx \\tb (H F - \\sqm S R )} \\;, \\;\n{\\ux {S(1){\\ov R}(2)}} = -\\zw{\\sqm \\tx\\tb} S{\\ov R} \\; ,\\nonumber\\\\\n&&{\\ux{{\\ov S}(1) R(2)}} = \\zw{\\sqp \\tx\\tb} {\\ov S} R\\; , \\;\n{\\ux{{\\ov S}(1){\\ov R}(2)}} = \\zw{ \\tx {\\ov F} + \\tx \\tb ({\\ov H}\\,\n{\\ov F} -\\sqp {\\ov S}\\, {\\ov R})}\\; ,\\nonumber\\\\\n&&{\\ux { R(1){\\ov R}(2)}} = \n{{\\frac{1}{2}\\tx\\tb}\\over {Z_{12}}^2} -\\zw{1 + \\sqm \\tb\n {\\ov H} +\\sqp \\tx H - \\tx\\tb ( H{\\ov H} + R {\\ov R}\n-\\sqp {\\ov D} H)} \\;. \\label{sl3}\n\\end{eqnarray}\n\nThere exist two non-equivalent ways to embed the affine supercurrents\ninto the minimal $N=4$ SCA via a local Sugawara construction. One\nrealization, like in the $\\widehat{sl(2\|1)}$ case, corresponds to the\n``short'' Sugawara construction based solely upon the\n$\\widehat{sl(2)\\oplus u(1)}$\nsubalgebra. The second one,\nwhich in what follows is referred to as the ``long'' Sugawara\nconstruction, involves {\\it all} the $sl(3)$-valued affine supercurrents.\nThis realization corresponds to a new globally $N=4$\nsupersymmetric hierarchy realized on the full set of superaffine\n$\\widehat{sl(3)}$ supercurrents. Thus the set of superfields generating the\nsuperaffine ${\\widehat{sl(3)}}$ algebra supplies the first known example of a\nPoisson-brackets structure carrying two non-equivalent hierarchies of the\nsuper mKdV type associated with $N=4$ SKdV hierarchy.\n\nThe two Sugawara realizations are respectively given by:\n\n{\\em i)} in the ``short'' case,\n\\begin{equation} \\label{short}\nJ = D{\\ov H} + {\\ov D} H + H {\\ov H} + S {\\ov S}\\; ,\\quad\nW = D {\\ov S}\\; , \\quad\n{\\ov W} = {\\ov D}S \\;,\n\\end{equation}\n\n{\\em ii)} in the ``long'' case\n\\begin{equation} \\label{long}\nJ= H{\\ov H} + F {\\ov F} + R {\\ov R} + S {\\ov S}\n+ \\sqm{\\ov D} H +\\sqp D{\\ov H}\\; , \\quad\nW = D {\\ov F}\\; , \\quad\n{\\ov W} = {\\ov D}F \\;.\n\\end{equation}\n\nTheir Poisson brackets (OPEs) are given by the relations (\\ref{n4sca}).\n\n\n\\section{$N=4$ supersymmetry}\n\nLike in the $\\widehat{sl(2\\vert 1)}$ case, the ``short'' Sugawara\n$N=4$ supercurrents \\p{short} do not produce the true global\n$N=4$ supersymmetry for the entire set of the affine\nsupercurrents, yielding it only for the $\\widehat{sl(2)\\oplus\nu(1)}$ subset. At the same time, the ``long'' Sugawara \\p{long}\ngenerates such a supersymmetry. In the $z, \\theta, \\bar{\\theta} $\nexpansion of the supercurrents $J, W, {\\ov W}$ the global\nsupersymmetry generators are present as the coefficients of the\nmonomials $\\sim \\theta / z$. {}From $J$ there come out the\ngenerators of the manifest linearly realized $N=2$ supersymmetry,\nwhile those of the hidden $N=2$ supersymmetry appear from $W, {\\ov\nW}$. The precise form of the hidden supersymmetry transformations\ncan then be easily read off from the OPEs \\p{sl3}:\n\\begin{eqnarray}\n\\delta H & = & {\\ov\\epsilon} \\left( HF -\\sqp \\, SR \\right)\n + \\epsilon \\sqp \\, D{\\ov F} \\; , \\qquad\n\\delta {\\ov H} = {\\ov\\epsilon}\\, \\sqm \\, {\\ov D} F\n -\\epsilon \\left( {\\ov H}\\,{\\ov F} -\\sqm \\,{\\ov S}\\,{\\ov R}\n \\right) \\;, \\nn\n\\delta F &=& -\\epsilon\\left( \\sqp D{\\ov H}+F{\\ov F}+H{\\ov H}+\n R{\\ov R}+S{\\ov S}\\right)\\,,\n\\delta{\\ov F} = -{\\ov \\epsilon}\\left(\n \\sqm\\,{\\ov D}H+F{\\ov F}+H{\\ov H}+\n R{\\ov R}+S{\\ov S}\\right)\\;, \\nn\n\\delta S & = & -{\\ov \\epsilon} \\sqp\\, FS-\\epsilon\\left(\n D{\\ov R}-\\sqp\\,{\\ov F}S+H{\\ov R}\\right)\\,, \\;\n\\delta {\\ov S} = -{\\ov \\epsilon} \\left(\n {\\ov D} R +\\sqm \\, F{\\ov S}-{\\ov H}R \\right)\n +\\epsilon \\sqm\\, {\\ov F}\\,{\\ov S}\\;, \\nn\n\\delta R & = & -{\\ov \\epsilon}\\,\\sqm\\,FR + \\epsilon\\left( D{\\ov S}\n + \\sqm\\,{\\ov F}R -\\alpha \\, H{\\ov S}\\right)\\,,\\;\n\\delta {\\ov R} = {\\ov \\epsilon}\\left( {\\ov D} S-\n \\sqp \\, F{\\ov R} +\\sqm\\,{\\ov H} S \\right)\n + \\epsilon\\alpha \\,{\\ov F}\\,{\\ov R} \\;. \\label{hidN2}\n\\end{eqnarray}\nHere $\\epsilon, {\\ov \\epsilon}$ are the corresponding odd\ntransformation parameters. One can check that these\ntransformations have just the same standard closure in terms of\n$\\partial_z $ as the manifest $N=2$ supersymmetry\ntransformations, despite the presence of nonlinear terms. Also it\nis straightforward to verify that the constraints \\p{cond3} and\nthe OPEs \\p{sl3} are covariant under these transformations.\n\nLet us examine the issue of reducibility of the set of the $N=2$\n$\\widehat{sl(3)}$ currents with respect to the full $N=4$ supersymmetry.\nIn the $\\widehat{sl(2)\\oplus u(1)}$ case the involved currents form an\nirreducible $N=4$ multiplet which is a nonlinear version of the\nmultiplet consisting of two chiral (and anti-chiral) $N=2$\nsuperfields \\cite{IK}. In the given case one can expect that eight $N=2$\n$\\widehat{sl(3)}$ currents form a reducible multiplet which can be divided\ninto a sum of two irreducible ones, each involving four superfields\n(a pair of chiral and anti-chiral superfields together with\nits conjugate). However, looking at the r.h.s. of \\p{hidN2}, it is\ndifficult to imagine how this could be done in a purely algebraic\nand local way. Nevertheless, there is a non-local redefinition of the\nsupercurrents which partly makes this job. As the first step one\nintroduces a prepotential for the chiral superfields $H, {\\ov H}$\n\\bea\nH = DV, \\quad {\\ov H} = -{\\ov D}\\,{\\ov V}\n\\eea\nand chooses a gauge for $V$ in which it is expressed\nthrough $H, {\\ov H}$ \\cite{Egau}\n\\bea\nV &=& - \\partial^{-1}({\\ov D} H + \\sqm \\, D{\\ov H})\\;,\\quad\n{\\ov V} = \\partial^{-1}(D {\\ov H} + \\sqp \\, {\\ov D}H) \\;, \\quad\nV = - \\sqm {\\ov V}\\;, \\label{relV} \\\\\n\\delta V &=& \\sqp (\\bar \\epsilon F -\\epsilon {\\ov F}) \\;, \\qquad\n\\delta {\\ov V} = \\sqm (\\bar \\epsilon F -\\epsilon {\\ov F})\n\\;. \\label{tranV}\n\\eea\nUsing this newly introduced quantity, one can pass to the\nsupercurrents which satisfy the standard chirality conditions\nfollowing from the original constraints \\p{chirH}, \\p{cond3}\nand equivalent to them\n\\bea S &=& \\exp\\{-V\\}\\tilde{S}\\;, \\quad {\\ov S} =\n\\exp\\{\\sqp V\\}{\\ov {\\tilde{S}}}\n\\;, \\quad\nR = \\exp\\{\\sqp V\\}\\tilde{R}\\;, \\quad {\\ov R} = \\exp\\{-V\\}{\\ov {\\tilde{R}}}\\;,\n\\nonumber \\\\\nF &=& \\exp\\{-\\sqm V\\}[\\tilde{F} -\n\\partial^{-1}{\\ov D}(\\tilde{S}\\tilde{R}) + \\partial^{-1}D\n({\\ov {\\tilde{S}}}\\,{\\ov {\\tilde{R}}})]\\;,\\nn\n{\\ov F} &=& \\exp\\{-\\sqm V\\}[{\\ov {\\tilde{F}}} -\n\\partial^{-1}{\\ov D}(\\tilde{S}\\tilde{R}) + \\partial^{-1}D\n({\\ov {\\tilde{S}}}\\,{\\ov {\\tilde{R}}})]\\;,\n\\label{redef2}\n\\eea\n\\bea\nD\\tilde{S} = D \\tilde{R} = D \\tilde{F} = 0\\;, \\qquad\n{\\ov D}{\\ov {\\tilde{S}}} = {\\ov D}{\\ov {\\tilde{R}}} =\n{\\ov D}{\\ov {\\tilde{F}}} =\n0\\;. \\label{chirSRF}\n\\eea\nThe $N=4$ transformation rules \\p{hidN2} are radically simplified in\nthe new basis\n\\bea\n&& \\delta \\tilde{S} = -\\epsilon D {\\ov {\\tilde{R}}} \\;,\\quad\n\\delta {\\ov {\\tilde{S}}} = -\\bar\\epsilon {\\ov D} \\tilde{R} \\;,\\quad\n\\delta \\tilde{R} = \\epsilon D {\\ov {\\tilde{S}}} \\;,\\quad\n\\delta {\\ov {\\tilde{R}}} = \\bar\\epsilon {\\ov D} \\tilde{S} \\;, \\nn\n&& \\delta \\tilde{F} = \\epsilon D {\\ov D} (\\exp\\{\\sqm V\\})\\;, \\qquad\n\\delta {\\ov {\\tilde{F}}} = -\\bar\\epsilon {\\ov D}D\n(\\exp\\{\\sqm V\\})\\;, \\nn\n&& \\delta (\\exp\\{\\sqm V\\}) = \\bar\\epsilon \\tilde{F} -\\epsilon {\\ov\n{\\tilde{F}}} -(\\bar\\epsilon - \\epsilon)\n\\partial^{-1}[{\\ov D}(\\tilde{S}\\tilde{R}) - D\n({\\ov {\\tilde{S}}}\\,{\\ov {\\tilde{R}}})] \\;.\\label{tranVn} \\eea We\nsee that the supercurrents $\\tilde{S}\\;, {\\ov {\\tilde{S}}}\\;,\n\\tilde{R}\\;, {\\ov {\\tilde{R}}}$ form an irreducible $N=4$\nsupermultiplet, just of the kind found in \\cite{IK}. At the same\ntime, the superfields $V, \\tilde{F}\\;, {\\ov {\\tilde{F}}}$ do not\nform a closed set: they transform through the former multiplet. We\ndid not succeed in finding the basis where these two sets of\ntransformations entirely decouple from each other. So in the\npresent case we are facing a new phenomenon consisting in that the\n$N=2 \\;\\;\\widehat{sl(3)}$ supercurrents form a not fully reducible\nrepresentation of $N=4$ supersymmetry. The same can be anticipated\nfor higher rank affine supergroups with a hidden $N=4$ structure.\nOne observes that putting the supercurrents $\\tilde{S}\\;, {\\ov\n{\\tilde{S}}}\\;, \\tilde{R}\\;, {\\ov {\\tilde{R}}}$ (or their\ncounterparts in the original basis) equal to zero is the\ntruncation consistent with $N=4$ supersymmetry. After this\ntruncation the remaining supercurrents $H,F, {\\ov H}, {\\ov F}$ form\njust the same irreducible multiplet as in the\n$\\widehat{sl(2)\\oplus u(1)}$ case \\cite{IKT}.\n\nNote that the above peculiarity does not show up at the level of\nthe composite supermultiplets like \\p{long}. Indeed, it is straightforward to\nsee that the supercurrents in \\p{long} form the same irreducible\nrepresentation as in the $\\widehat{sl(2)\\oplus u(1)}$ case \\cite{IKT}\n\\bea\n\\delta J = -\\epsilon {\\ov D} W - \\bar \\epsilon D{\\ov W}\\;, \\qquad\n\\delta W = \\bar \\epsilon D J\\;, \\qquad\n\\delta {\\ov W} = \\epsilon {\\ov D} J\\;. \\label{JWtran}\n\\eea\nAnother irreducible multiplet is comprised by the following\ncomposite supercurrents\n\\bea\n\\hat{J} &=& H{\\ov H} + F{\\ov F} + S{\\ov S} +R{\\ov R}\\;, \\nn\n\\quad \\hat{W} &=& DF = -\\sqm HF +SR \\;, \\; \\hat{{\\ov W}} =\n{\\ov D}\\,{\\ov F} = \\sqp {\\ov H}\\,{\\ov F} - {\\ov S}\\,{\\ov R}~.\n\\label{top}\n\\eea\nUnder \\p{hidN2} they transform as\n\\bea\n\\delta \\hat{J} = -\\epsilon D \\hat{{\\ov W}} -\n\\bar \\epsilon {\\ov D} \\hat{W}\\;, \\qquad\n\\delta \\hat{W} = \\epsilon D \\hat{J}\\;, \\qquad\n\\delta \\hat{{\\ov W}} = \\bar\\epsilon {\\ov D} \\hat{J}\\;.\n\\label{JWtran2}\n\\eea\nThe OPEs of these supercurrents can be checked to generate another\n``small'' $N=4$ SCA with zero central charge, i.e. a topological\n``small'' $N=4$ SCA. The same SCA was found in the\n$\\widehat{sl(2)\\oplus u(1)}$ case \\cite{IKT}. This SCA\nand the first one together close on the ``large'' $N=4$ SCA in some particular\nrealization \\cite{RASS,IKT}. Thus the\n$N=2 \\;\\;\\widehat{sl(3)}$ affine\nsuperalgebra provides a Sugawara type construction for this\nextended SCA as well. It would be of interest to inquire whether this\nsuperalgebra conceals in its enveloping algebra any other SCA containing\n$N=4$ SCA as a subalgebra, e.g., possible $N=4$ extensions of nonlinear\n$W_n$ algebras.\n\n\\section{$N=4$ mKdV-type hierarchies}\n\nBoth two non-equivalent $N=4$ Sugawara constructions, eqs.\n({\\ref{short}) and (\\ref{long}), define Poisson maps. As a\nconsequence, the superaffine $sl(3)$-valued supercurrents inherit\nall the integrable hierarchies associated with $N=4$ SCA.\n\nThe first known example of hierarchy with $N=4$ SCA\nas the Poisson structure is $N=4$ SKdV\nhierarchy (see \\cite{DI}). The densities of the lowest hamiltonians from\nan infinite sequence of the corresponding superfield hamiltonians\nin involution, up to an overall normalization factor, read\n\\begin{eqnarray}\n{\\cal H}_1 &=& J\\nonumber\\\\ {\\cal H}_2 &=& -{1\\over 2}( J^2 - 2 W{\\ov\nW})\\nonumber \\\\\n\\relax {\\cal H}_3 &=&\n{1\\over 2} (J [ D, {\\overline D}] J + 2 W {\\overline W}' +{2\\over 3} J^3\n- 4 J W{\\ov W})~.\n\\end{eqnarray}\nHere the $N=2$ superfields $J$, $W$, ${\\ov W}$ satisfy the\nPoisson brackets (\\ref{n4sca}).\n\nLet us concisely denote by $\\Phi_a$, $a=1,2,...,8$,\nthe $\\widehat{sl(3)}$-valued superfields $H,F,R,S$ together with the barred\nones. Their evolution equations which, by construction, are\ncompatible with the $N=4$ SKdV flows, for the $k$-th flow\n($k=1,2,...$) are written as\n\\begin{eqnarray}\n\\relax\n\\frac{\\partial}{\\partial t_k}\\Phi_a (X,t_k) &=& \\{ \\int dY\n{\\cal H}_k (Y, t_k) , \\Phi_a (X,t_k)\\}~.\n\\end{eqnarray}\nThe Poisson bracket here is given by the superaffine $\\widehat{sl(3)}$\nstructure (\\ref{sl3}), with $X, Y$ being two different ``points'' of\n$N=2$ superspace.\n\nThe identification of the superfields $J$, $W$, $\n{\\ov W}$ in terms of the affine supercurrents\ncan be made either via eqs. (\\ref{short}), i.e. the\n``short'' Sugawara, or via eqs. (\\ref{long}),\nthat is the ``long'' Sugawara. Thus the same $N=4$ SKdV\nhierarchy proves to produce two non-equivalent\nmKdV type hierarchies for the affine supercurrents, depending on\nthe choice of the underlying Sugawara construction.\nThe first hierarchy is $N=2$ supersymmetric, while the other one\ngives a new example of globally $N=4$ supersymmetric hierarchy.\n\nLet us briefly outline the characteristic features of\nthese two hierarchies.\n\nIt is easy to see that\nfor the superfields $H, {\\ov H}, S, {\\ov S}$ corresponding to the\nsuperaffine algebra $\\widehat{sl(2)\\oplus u(1)}$ as\na subalgebra in $\\widehat{sl(3)}$, the ``short'' hierarchy coincides\nwith $N=4$ NLS-mKdV hierarchy of ref. \\cite{IKT}.\nFor the remaining $\\widehat{sl(3)}$ supercurrents one gets\nthe evolution equations in the ``background'' of the\nbasic superfields just mentioned.\n\nNew features are revealed while examining the ``long'', i.e. $N=4$\nsupersymmetric $\\widehat{sl(3)}$ mKdV hierarchy. It can be easily\nchecked that for all non-trivial flows $(k \\geq 2)$ the evolution\nequations for any given superfield $\\Phi_a$ necessarily contain in the\nr.h.s. the whole set of eight $\\widehat{sl(3)}$ supercurrents.\nIn this case the previous $N=4$ NLS-mKdV hierarchy can also be\nrecovered. However, it is obtained in a less trivial\nway. Namely, it is produced only after coseting out\nthe superfields $R, S$ and ${\\ov R}, {\\ov S}$, i.e. those associated\nwith the simple roots of $sl(3)$ (as usual, the passing to the\nDirac brackets is required in this case). As was mentioned in the\npreceding Section, this truncation preserves the global $N=4$\nsupersymmetry.\n\nLet us also remark that, besides the two mKdV\nhierarchies carried by the superaffine $\\widehat{sl(3)}$ algebra\nand discussed so far, this Poisson bracket structure also carries\nat least one extra pair of non-equivalent hierarchies of the mKdV type\npossessing only global $N=2$ supersymmetry.\nIt was shown in \\cite{DGI} (see also \\cite{DG2}) that the enveloping \nalgebra of $N=4$ SCA\ncontains, apart from an infinite abelian subalgebra corresponding\nto the genuine $N=4$ SKdV hierarchy, also an infinite abelian\nsubalgebra formed by the hamiltonians in involution associated\nwith a different hierarchy referred to as the ``quasi'' $N=4$ SKdV one.\nThis hierarchy admits only a global $N=2$ supersymmetry\nand can be thought of as an integrable extension of the $a=-2$, $N=2$ SKdV\nhierarchy. In \\cite{DGI} there was explicitly found a non-polynomial\nMiura-type transformation which in a surprising way relates $N=4$ SCA\nto the non-linear $N=2$ super-$W_3$ algebra. This transformation\nmaps the ``quasi'' $N=4$ SKdV hierarchy onto the $\\alpha=-2$, $N=2$\nBoussinesq hierarchy. Since these results can be\nrephrased in terms of the Poisson brackets structure alone, and the\nsame is true both for our ``short'' (\\ref{short}) and ``long''\n(\\ref{long}) Sugawara constructions, it immediately follows that the\nsuper-affine $\\widehat{sl(3)}$ superfields also carry two non-equivalent\n``quasi'' $N=4$ SKdV structures and can be mapped in two non-equivalent\nways onto the $\\alpha=-2$, $N=2$ Boussinesq hierarchy.\n\n\\section{Conclusions}\nIn this work we have investigated the local Sugawara\nconstructions leading to the $N=4$ SCA expressed in terms of the\nsuperfields corresponding to the $N=2$ superaffinization of the\n$sl(2\|1)$ and the $sl(3)$ algebras. We have shown that the\n$\\widehat{sl(3)}$ case admits a non-trivial $N=4$ Sugawara\nconstruction involving all eight affine supercurrents and\ngenerating the hidden $N=4$ supersymmetry of $N=2\n\\;\\;\\widehat{sl(3)}$ algebra. This property has been used to\nconstruct a new $N=4$ supersymmetric mKdV hierarchy associated\nwith $N=4$ SKdV. Another mKdV hierarchy is obtained using the\n$N=4$ Sugawara construction on the subalgebra\n$\\widehat{sl(2)\\oplus u(1)}$. Thus the $N=2$ $\\widehat{sl(3)}$\nalgebra was shown to provide the first example of a Poisson\nbrackets structure carrying two non-equivalent integrable mKdV type\nhierarchies associated with the $N=4$ SKdV one. Also, the\nexistence of two non-trivial $N=2$ supersymmetric mKdV-type\nhierarchies associated with the same superaffine Poisson structure\nand ``squaring'' to the quasi $N=4$ SKdV hierarchy of ref.\n\\cite{DGI} was noticed.\n\nAn interesting problem is to generalize the two Sugawara constructions\nto the full quantum case and to find out (if existing) an $N=4$ analog\nof the well-known GKO coset construction \\cite{GKO} widely used in\nthe case of bosonic affine algebras. It is also of importance to perform\na more detailed analysis of the enveloping algebra of $N=2$\n$\\widehat{sl(3)}$ with the aim to list all irreducible composite $N=4$\nsupermultiplets and to study possible $N=4$ extended $W$ type algebras\ngenerated by these composite supercurrents. At last, it still remains\nto classify all possible $N=2$ affine superalgebras admitting the\nhidden $N=4$ structure, i.e. $N=4$ affine superalgebras. As is clear from\nthe two available examples ( $\\widehat{sl(2)\\oplus u(1)}$ and\n$\\widehat{sl(3)}$ ) a sufficient condition of the existence of such\na structure on the given affine superalgebra is the possibility to\ndefine $N=4$ SCA on it via the corresponding ``long'' Sugawara\nconstruction, with the full $N=2$ stress-tensor included.\n\n\\vskip1cm\n\\noindent{\\Large{\\bf Acknowledgments}} \\\\\n\n\\noindent F.T. wishes to express his gratitude to the\nJINR-Bogoliubov Laboratory of Theoretical Physics, where this work\nhas been completed, for the kind hospitality. E.I. and S.K.\nacknowledge a support from the grants RFBR-99-02-18417,\nINTAS-96-0308 and INTAS-96-538.\n\\vspace{0.3cm}\n\\section*{Appendix: the second flow of the ``long'' $\\widehat{sl(3)}$\n$N=4$ mKdV}\n\n\\vspace{0.1cm}\nFor completeness we present here the evolution equations for the\nsecond flow of the ``long''\n$\\widehat{sl(3)}$ mKdV hierarchy (it is the first non-trivial flow).\nWe have \\footnote{ In order to save space and to avoid an unnecessary\nduplication we present the equations only for the\nnon-linear chiral sector.}\n\\begin{eqnarray}\n{\\dot H} &=& - 2\\partial^2 H- 2\\alpha ( 2HD\\partial\n\\overline{H}+\\partial HD\\overline{H}-SD\\partial\\overline{S}\n-\\partial S D\\overline{S}- R D\\partial \\overline{R}-\\partial R\nD\\overline{R})-\\nonumber\\\\ && -4{\\overline \\alpha} \\partial H\n\\overline{D}H + 2\\alpha\\overline{(F}S\\partial R +\n\\overline{F}\\partial S R - H S\\partial\\overline{S} - D\\overline{F}\nS\\overline{D}R+D\\overline{F}\\overline{D}SR) -\n\\nonumber\\\\&&-2{\\overline\\alpha} HR\n\\partial \\overline{R}- 2(1+\\alpha)(H\\partial S\n\\overline{S}+\\partial H S\\overline{S}) -2(1-{\\overline{\\alpha}}) (H\\partial R\n\\overline{R}+\\partial H R\\overline{R})+\\nonumber\\\\&&\n+2(2H\\overline{D}FD\\overline{F}+H\\overline{D}RD\\overline{R}+\nH\\overline{D}SD\\overline{S}\n-2\\overline{D}HFD\\overline{F}-\n\\overline{D}HRD\\overline{R}-\\overline{D}HSD\\overline{S}\n-2H\\partial F \\overline{F} -\\nonumber\\\\&&- 2\\partial\nHF\\overline{F})\n+2\\alpha (2H\\overline{H}FD\\overline{F}+2HD\\overline{H}F\\overline{F}+\nS\\overline{S} RD\\overline{R}+SD\\overline{S}R\\overline{R})\n-\\nonumber\\\\&&-2{\\overline{\\alpha}}(H\\overline{H}RD\n\\overline{R}+HD\\overline{H}R\\overline{R}+\n\\overline{H}D\\overline{F}SR\n-D\\overline{H}\\overline{F}SR)+\\nonumber\\\\ &&\n+2(2HF\\overline{S}D\\overline{R}-2HF\nD\\overline{S}\\overline{R}-H\\overline{F}S\\overline{D}R\n+H\\overline{F}\\overline{D}SR\n+H\\overline{H}SD\\overline{S}+HD\\overline{H}S\\overline{S}+\n\\overline{D}H\\overline{F}SR)-\\nonumber\\\\ &&\n-2\\alpha H\\overline{H}\\overline{F}SR -\n2HS\\overline{S}R\\overline{R}\\nonumber\\\\\n{\\dot S} &=&\n2{\\overline\\alpha}(D\\overline{H}\\partial S -\\overline{D}H\\partial S +\nD\\partial \\overline{H}S -\\partial H\\overline{D}S) -\n2\\overline{D}\\partial H S - 2\\partial FD\\overline{R}-\\nonumber\\\\\n&& -2\\alpha(2H\\partial\n\\overline{H}S+SR\\partial\\overline{R}+S\\partial\nR\\overline{R}+\\partial H \\overline{H}S)-\\nonumber\\\\ &&\n-2{\\overline\\alpha}\n(H\\overline{D}FD\\overline{R}-\\overline{D}HFD\\overline{R}+\\partial\nS S\\overline{S}) -\\nonumber\\\\ && -2(F\\overline{F}\\partial S\n+FD\\overline{F}\\overline{D}S + H\\overline{H}\\partial S\n+HD\\overline{H}\\overline{D}S -H\\partial\nF\\overline{R}-S\\overline{D}RD\\overline{R}+S\\overline{D}SD\\overline{S}+\n2\\overline{D}SRD\\overline{R}+\\nonumber\\\\ && +\\partial S\nR\\overline{R}) +2\\alpha (FSD\\overline{S}R\n-FS\\overline{S}D\\overline{R}) -2{\\overline\\alpha}\n(HF\\overline{F}\\overline{D}S\n+H\\overline{D}HF\\overline{R}+D\\overline{H}F\\overline{F}S+\\nonumber\\\\\n&& +\\overline{H}FD\\overline{F}S)\n+2(1+\\alpha)(\\overline{F}S\\overline{D}SR +\nH\\overline{D}SR\\overline{R})-2 (\n2HS\\overline{D}R\\overline{R}+2HS\\overline{D}S\\overline{S}-2H\n\\overline{D}F\\overline{F}S-\n\\nonumber\\\\ &&\n-H\\overline{D}H\\overline{H}S+\\overline{D}HF\\overline{F}S)\n+2\\alpha H\\overline{H}F\\overline{F}S\n-2{\\overline\\alpha} H\\overline{H}SR\\overline{R}+2HFS\\overline{S}R\\nonumber\\\\\n{\\dot R} &=& 2\\alpha (\\overline{D}\\partial H R -\nD\\overline{H}\\partial R) -2{\\overline\\alpha}(\\overline{D}H\\partial R\n+\\partial H \\overline{D}R) -2(D\\partial \\overline{H}R -\\partial\nFD\\overline{S}) +\\nonumber\\\\ && +2\\alpha (H\\partial\nF\\overline{S}-\\partial R R\\overline{R}) +2{\\overline\\alpha}\n(H\\overline{D}FD\\overline{S}-2H\\partial\\overline{H}R-S\\partial\\overline{S}R-\n\\overline{D}HFD\\overline{S}-\\nonumber\\\\ && -\\partial\nH\\overline{H}R -\\partial S\\overline{S}R)-2(F\\overline{F}\\partial R\n+FD\\overline{F}\\overline{D}R + H\\overline{H}\\partial R +H\nD\\overline{H}\\overline{D}R +R\\overline{D}RD\\overline{R} +S\n\\overline{S}\\partial R +\\nonumber\\\\ &&+ 2\nSD\\overline{S}\\overline{D}R-\\overline{D}SD\\overline{S}R ) +\n2\\alpha(2HR\\overline{D}R\n\\overline{R}-2H\\overline{D}F\\overline{F}R-H\\overline{D}H\n\\overline{H}R-2H\\overline{D}S \\overline{S}R +\\nonumber\\\\\n&&+\\overline{D}HF\\overline{F}R) + 2{\\overline\\alpha}\n(F\\overline{S}RD\\overline{R}+FD\\overline{S}R\\overline{R}-HF\n\\overline{F}\\overline{D}R)\n+2(1+{\\overline\\alpha})(\\overline{F}SR\\overline{D}R) -\\nonumber\\\\\n&&-2(1+\\alpha )HS\\overline{S}\\overline{D}R\n-2(H\\overline{D}HF\\overline{S}-\\overline{H}FD\\overline{F}R\n-D\\overline{H}F\\overline{F}R) +\\nonumber\\\\ &&+2\\alpha\n(HF\\overline{S}R\\overline{R}-H\\overline{H}S\\overline{S}R)\n+2{\\overline\\alpha} H\\overline{H}F\\overline{F}R \\nonumber\\\\ {\\dot F}\n&=& 2\\partial^2 F -4{\\alpha}D\\overline{H}\\partial F\n-4{\\overline\\alpha}( \\overline{D}H\\partial F +\\overline{D}\\partial H\nF) +2(\\overline{D}S\\partial R -S\\overline{D}\\partial R\n+\\overline{D}\\partial S R-\\nonumber\\\\ &&\n -\\partial S\\overline{D}R)-2{\\overline\\alpha} (4HF\\overline{D}F\\overline{F}\n -2H\\overline{D}H\\overline{H}F+\\overline{H}FRD\\overline{R}-\nD\\overline{H}FR\\overline{R})\n -\\nonumber\\\\\n &&\n -2\\alpha\\overline{D}HFS\\overline{S}-2(1+{\\overline\\alpha})\n(H\\overline{D}FS\\overline{S}+\n HF\\overline{D}S\\overline{S})\n +2i\\sqrt{3}(HF\\overline{D}R\\overline{R}+H\\overline{D}FR\\overline{R})+\n\\nonumber\\\\\n && +2 (2F\\overline{F}S\\overline{D}R -2F\\overline{F}\\overline{D}SR\n +H\\overline{H}S\\overline{D}R -H\\overline{H}\\overline{D}SR +\\overline{H}\nFSD\\overline{S}+2SR\\overline{D}R\\overline{R}-2S\\overline{D}S\\overline{S}R\n+\\nonumber\\\\ && +2\\overline{D}F\\overline{F}SR +\\overline{D}H FR\n\\overline{R}+\\overline{D}H\\overline{H}SR\n-D\\overline{H}FS\\overline{S})+\\nonumber\\\\ &&+2\\alpha\n(D\\overline{H}S\\overline{D}R-D\\overline{H}\\overline{D}SR) -\n2{\\overline\\alpha}\\partial \\overline{H}SR\n-2(FR\\partial\\overline{R}+FS\\partial\\overline{S}+2F\\overline{D}FD\n\\overline{F}\n+\\nonumber\\\\ && +F\\overline{D}RD\\overline{R}+\nF\\overline{D}SD\\overline{S}+2H\\overline{H}\\partial F + 2 HD\n\\overline{H}\\overline{D}F +2H\\partial\\overline{H}F +\\overline{D}FR\nD\\overline{R}+\\overline{D}FSD\\overline{S}+\\nonumber\\\\&&+ 2\\partial\nF F\\overline{F}+2\\partial F R\\overline{R}+2\\partial F\nS\\overline{S}) +2(1+\\alpha) H\\overline{H}FR\\overline{R}\n+2(1+{\\overline\\alpha}) H \\overline{H}FS\\overline{S}~.\\nonumber\n\\end{eqnarray}\nThe parameters $\\alpha, \\bar\\alpha$ have been defined in eq. \\p{param}.\n\n\n\\vskip1cm\n\\begin{thebibliography}{99}\n\\bibitem{DGZ} P. Di Francesco, P. Ginsparg, J. Zinn-Justin, Phys.\nReports {\\bf 254} (1995) 1.\n\\bibitem{SW} N. Seiberg, E. Witten, Nucl. Phys.\n{\\bf B 426} (1994) 19; \\\\\nA. Gorskii, I. Krichever, A. Marshakov, A. Mironov, A. Morozov,\nPhys. Lett. {\\bf B 355} (1995) 466.\n\\bibitem{DS} V. Drinfeld, V. Sokolov, J. Sov. Math. {\\bf 30} (1984) 1975.\n\\bibitem{MRK} Yu. I. Manin, A.O. Radul, Commun. Math.\nPhys. {\\bf 98} (1985) 65; \\\\\nP.P. Kulish, Lett. Math. Phys {\\bf 10} (1985) 87.\n\\bibitem{Mat} P. Mathieu, Phys. Lett. {\\bf B 203} (1988) 287; \\\\\nC.A. Laberge, P. Mathieu, Phys. Lett. {\\bf B 215} (1988) 718; \\\\\nP. Labelle, P. Mathieu, J. Math. Phys. {\\bf 89} (1991) 923.\n\\bibitem{InKn} T. Inami, H. Kanno, Commun. Math. Phys. {\\bf 136}\n(1991) 519; Int. J. Mod. Phys. {\\bf A 7} Suppl. 1A (1992) 419.\n\\bibitem{DG1} F. Delduc, L. Gallot, J. Math. Phys. {\\bf 39} (1998) 4729.\n\\bibitem{DG2} F. Delduc, L. Gallot, Commun. Math. Phys. {\\bf 190} (1997) 395.\n\\bibitem{dubna} L. 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Lett. {\\bf A 231} (1997) 75.\n\\bibitem{SSTP} Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen,\nPhys. Lett. {\\bf B 206} (1988) 71.\n\\bibitem{DGI} F. Delduc, L. Gallot, E. Ivanov, Phys. Lett. {\\bf B 396}\n(1997) 122.\n\\bibitem{KT} S. Krivonos, K. Thielemans, Class. Quant. Grav. {\\bf 13}\n(1996) 2899.\n\\bibitem{HS} C.M. Hull, B. Spence, Phys. Lett. {\\bf B 241} (1990) 357.\n\\bibitem{AIS} C. Ahn, E. Ivanov, A. Sorin, Commun. Math. Phys. {\\bf 183}\n(1997) 205.\n\\bibitem{Egau} E. Ivanov, Mod. Phys. Lett. {\\bf A 13} (1998) 2855.\n\\bibitem{RASS} M. Ro\\v{c}ek, C. Ahn, K. Schoutens, A. Sevrin,\nSuperspace WZW Models and Black Holes, in: Workshop on Superstrings\nand Related Topics, Trieste. Aug. 191,IASSNS-HEP-91/69, ITP-SB-91-49,\nLBL-31325, UCB-PTH-91/50.\n\\bibitem{GKO} P. Goddard, A. Kent, D. Olive, Phys. Lett.\n{\\bf B 152} (1985) 88.\n\n\\end{thebibliography}\n\\end{document}\n\n\n\n\n" } ]	[ { "name": "solv-int9912003.extracted_bib", "string": "{DGZ P. Di Francesco, P. Ginsparg, J. Zinn-Justin, Phys. Reports {254 (1995) 1." }, { "name": "solv-int9912003.extracted_bib", "string": "{SW N. Seiberg, E. Witten, Nucl. Phys. {B 426 (1994) 19; \\\\ A. Gorskii, I. Krichever, A. Marshakov, A. Mironov, A. Morozov, Phys. Lett. {B 355 (1995) 466." }, { "name": "solv-int9912003.extracted_bib", "string": "{DS V. Drinfeld, V. Sokolov, J. Sov. Math. {30 (1984) 1975." }, { "name": "solv-int9912003.extracted_bib", "string": "{MRK Yu. I. Manin, A.O. Radul, Commun. Math. Phys. {98 (1985) 65; \\\\ P.P. Kulish, Lett. Math. Phys {10 (1985) 87." }, { "name": "solv-int9912003.extracted_bib", "string": "{Mat P. Mathieu, Phys. Lett. {B 203 (1988) 287; \\\\ C.A. Laberge, P. Mathieu, Phys. Lett. {B 215 (1988) 718; \\\\ P. Labelle, P. Mathieu, J. Math. Phys. {89 (1991) 923." }, { "name": "solv-int9912003.extracted_bib", "string": "{InKn T. Inami, H. Kanno, Commun. Math. Phys. {136 (1991) 519; Int. J. Mod. Phys. {A 7 Suppl. 1A (1992) 419." }, { "name": "solv-int9912003.extracted_bib", "string": "{DG1 F. Delduc, L. Gallot, J. Math. Phys. {39 (1998) 4729." }, { "name": "solv-int9912003.extracted_bib", "string": "{DG2 F. Delduc, L. Gallot, Commun. Math. Phys. {190 (1997) 395." }, { "name": "solv-int9912003.extracted_bib", "string": "{dubna L. Bonora, S. Krivonos, A. Sorin, Nucl. Phys. {B 477 (1996) 835; Lett. Math. Phys. {45 (1998) 63; Phys. Lett. {A 240 (1998) 201;\\\\ L. Bonora, S. Krivonos, Mod. Phys. Lett. {A 12 (1997) 3037." }, { "name": "solv-int9912003.extracted_bib", "string": "{KS S. Krivonos, A. Sorin, Phys. Lett. {A 251 (1999) 109." }, { "name": "solv-int9912003.extracted_bib", "string": "{DG3 F. Delduc, L. Gallot, J. Nonlinear Math. Phys. {V6, N3 (1999) 332." }, { "name": "solv-int9912003.extracted_bib", "string": "{Pop Z. Popowicz, Phys. Lett. {A 194 (1994) 375; J. Phys. {A 29 (1996) 1281; {ibid {30 (1997) 7935; Phys. Lett. {B 459 (1999) 150." }, { "name": "solv-int9912003.extracted_bib", "string": "{BD J.C. Brunelli, A. Das, J. Math. Phys. {36 (1995) 268." }, { "name": "solv-int9912003.extracted_bib", "string": "{Top F. Toppan, Int. J. Mod. Phys. {A 10 (1995) 895." }, { "name": "solv-int9912003.extracted_bib", "string": "{IKT E. Ivanov, S. Krivonos, F. Toppan, Phys. Lett. {B 405 (1997) 85." }, { "name": "solv-int9912003.extracted_bib", "string": "{DI F. Delduc, E. Ivanov, Phys. Lett. {B 309 (1993) 312; \\\\ F. Delduc, E. Ivanov, S. Krivonos, J. Math. Phys. {37 (1996) 1356." }, { "name": "solv-int9912003.extracted_bib", "string": "{IK E. Ivanov, S. Krivonos, Phys. Lett. {A 231 (1997) 75." }, { "name": "solv-int9912003.extracted_bib", "string": "{SSTP Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen, Phys. Lett. {B 206 (1988) 71." }, { "name": "solv-int9912003.extracted_bib", "string": "{DGI F. Delduc, L. Gallot, E. Ivanov, Phys. Lett. {B 396 (1997) 122." }, { "name": "solv-int9912003.extracted_bib", "string": "{KT S. Krivonos, K. Thielemans, Class. Quant. Grav. {13 (1996) 2899." }, { "name": "solv-int9912003.extracted_bib", "string": "{HS C.M. Hull, B. Spence, Phys. Lett. {B 241 (1990) 357." }, { "name": "solv-int9912003.extracted_bib", "string": "{AIS C. Ahn, E. Ivanov, A. Sorin, Commun. Math. Phys. {183 (1997) 205." }, { "name": "solv-int9912003.extracted_bib", "string": "{Egau E. Ivanov, Mod. Phys. Lett. {A 13 (1998) 2855." }, { "name": "solv-int9912003.extracted_bib", "string": "{RASS M. Ro\\v{cek, C. Ahn, K. Schoutens, A. Sevrin, Superspace WZW Models and Black Holes, in: Workshop on Superstrings and Related Topics, Trieste. Aug. 191,IASSNS-HEP-91/69, ITP-SB-91-49, LBL-31325, UCB-PTH-91/50." }, { "name": "solv-int9912003.extracted_bib", "string": "{GKO P. Goddard, A. Kent, D. Olive, Phys. Lett. {B 152 (1985) 88." } ]
solv-int9912004	Integrable Couplings of Soliton Equations by Perturbations\\ I. A General Theory and Application to the KdV Hierarchy	[ { "author": "Wen-Xiu Ma" } ]	A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale perturbations can be taken and thus higher dimensional integrable couplings can be presented. The theory is applied to the KdV soliton hierarchy. Infinitely many integrable couplings are constructed for each soliton equation in the KdV hierarchy, which contain integrable couplings possessing quadruple Hamiltonian formulations and two classes of hereditary recursion operators, and integrable couplings possessing local $2+1$ dimensional bi-Hamiltonian formulations and consequent $2+1$ dimensional hereditary recursion operators.	[ { "name": "solv-int9912004.tex", "string": "\\documentclass[a4paper,11pt]{article}\n%\\documentstyle[11pt]{article}\n\\usepackage{latexsym}\n\\usepackage{rotating}\n\\usepackage{amsmath}\n\\author {Wen-Xiu Ma }\n\\title\n{Integrable Couplings of Soliton Equations by Perturbations\\\\\nI. A General Theory and Application to the KdV Hierarchy}\n\\setlength{\\parindent}{20pt}\n\\setlength{\\parskip}{6pt plus 2pt minus 1 pt}\n\\frenchspacing\t\n\\date{\\nonumber}\n%\\maketitle\n\\setlength{\\textwidth}{15cm}\n\\setlength{\\textheight}{222mm}\n%\n\\setlength{\\oddsidemargin}{3mm}\n\\setlength{\\evensidemargin}{3mm}\n\\setlength{\\topmargin}{-15mm} \n\n\\begin{document}\n\n\\setlength{\\baselineskip}{17pt}\n\\maketitle\n\n\\\n\\vskip 1cm \n\n\\begin{abstract}\nA theory for constructing integrable couplings of soliton equations\nis developed by using various perturbations around solutions of perturbed \nsoliton equations being analytic with respect to a small perturbation \nparameter. Multi-scale perturbations can be taken and thus higher \ndimensional integrable couplings can be presented. The theory is applied \nto the KdV soliton hierarchy. Infinitely many integrable couplings are \nconstructed for each soliton equation in the KdV hierarchy, which contain \nintegrable couplings possessing quadruple Hamiltonian formulations \nand two classes of hereditary recursion operators, and integrable \ncouplings possessing local $2+1$ dimensional bi-Hamiltonian formulations \nand consequent $2+1$ dimensional hereditary recursion operators.\n \n\\end{abstract}\n%\\tableofcontents\n\n%\\begin{center}\n\\vskip 1cm\n\\noindent {\\bf Running title:} Integrable couplings by perturbations I.\n%Key words:\n\n\n\\vskip 2cm \n\n\\noindent 1991 {\\it Mathematics Subject Classification}: 58F07, 35Q53. \n\n\\noindent {\\it Key words and phrases}: Integrable coupling,\nMulti-scale perturbation, Hereditary recursion operator,\nLax representation, Bi-Hamiltonian formulation, KdV hierarchy.\n\n\\newcommand{\\R}{\\mbox{\\rm I \\hspace{-0.9em} R}}\n\n\\def \\J {\\hat{J}_N}\n\\def \\la {\\lambda}\n\\def \\La {\\Lambda}\n\\def \\be {\\beta}\n\\def \\al{\\alpha}\n\\def \\del{\\delta}\n\\def \\Del{\\Delta}\n\\def \\al{\\alpha}\n\\def \\vare{\\varepsilon}\n\n\\def \\part {\\partial}\n\\def \\h1 \\hat{\\eta }\n\\def \\be {\\begin{equation}}\n\\def \\ee {\\end{equation}}\n\\def \\ba {\\begin{array}}\n\\def \\ea {\\end{array}}\n\\def\\bea{\\begin{eqnarray}}\n\\def\\eea{\\end{eqnarray}}\n\n\\newcommand{\\eqnsection}{\n \\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n \\makeatletter\n \\csname $addtoreset\\endcsname\n \\makeatother}\n\\eqnsection\n\n\\newtheorem{lemma}{Lemma}[section]\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{definition}{Definition}[section]\n\n\\newpage\n\\section{Introduction}\n\\setcounter{equation}{0}\n\nIntegrable couplings are a quite new interesting aspect in the field of \nsoliton theory \\cite{Fuchssteiner-book1993}. It originates from an \ninvestigation on centerless Virasoro symmetry algebras of integrable \nsystems or soliton equations. The Abelian parts of those Virasoro \nsymmetry algebras correspond to isospectral flows from higher order \nLax pairs and the non-Abelian parts, to non-isospectral flows from \nnon-isospectral Lax pairs \\cite{Ma-JPM1992,Ma-JPAPLA199293}.\nIf we make a given system of soliton equations and\neach time part of Lax pairs of its hierarchy to be\nthe first component and the second component of a new system respectively,\nthen such a new system will keep the same structure of \nVirasoro symmetry algebras as the old one. Therefore this can\nlead to a hierarchy of integrable couplings for the original system.\n\nMathematically, the problem of integrable couplings may be expressed as:\n{\\it for a given integrable system of evolution equations $u_t=K(u)\n%=K(u,u_x,\\cdots)\n,$ how can we construct a non-trivial system of evolution equations\nwhich is still integrable and includes $u_t=K(u)$ as a sub-system?}\n\nTherefore, up to a permutation (note that we can put some components\nof $u_t=K(u)$ seperately),\nwe actually want to construct a new bigger integrable system as follows\n\\begin{equation} \\left\\{ \\begin{array} {l} u_t=K(u),\\\\ v_t=S(u,v),\n\\end{array} \\right. \\label{couplingsystem}\\end{equation}\nwhich should satisfy the non-triviality condition\n$ \\part S /\\part [u] \\ne 0.$\nHere $[u]$ denotes a vector consisting of all derivatives of $u$ \nwith respect to a space variable. For example, we have \n$[u]=(u,u_x,u_{xx},\\cdots)$ in the case of $1+1$ dimensions.\nThe non-triviality condition guarantees that trivial diagonal\nsystems with $S(u,v)=c K(v)$ are excluded, where $c $ is an arbitrary\nconstant.\n\nThere are two facts which have a direct relation to\nthe study of integrable couplings.\n%New system (\\ref{couplingsystem}) is meaningful and important, although it\n%is triangular. This can be observed from the following two facts.\nFirst, all possible methods for constructing integrable couplings\nwill tell us how to extend integrable systems,\nfrom small to large and from simple to complicated,\nand/or how to hunt for new integrable systems, which are probably difficult\nto find in other ways. The corresponding theories may also provide useful\ninformation for completely classifying integrable systems in whatever\ndimensions. Secondly, the symmetry problem of integrable systems can be\nviewed as a special case of integrable couplings. \nStrictly speaking, if a system of evolution equations $u_t=K(u)$\nis integrable, then a new system (called a perturbation system)\nconsisting of the original system and its linearized system\n\\begin{equation}\n\\left \\{ \\begin{array} {l} u_t=K(u),\\\\ v_t=K'(u)[v],\n\\end{array} \\right. \\nonumber\n\\end{equation} \nmust be still integrable \\cite{Fuchssteiner-book1993}.\nThe second part of the above new system is exactly the system that\nall symmetries need to satisfy, but new system itself is \na special integrable coupling of the original system $u_t=K(u)$.\nGenerally, the search for the approximate solutions\n$\\hat {u}_N=\\sum_{i=0}^N \\varepsilon ^i \\eta _i,\\ N \\ge 1$,\nof physical interest to a given system $u_t=K(u)$ \ncan be cast into a study of the general standard perturbation systems\n$\\eta _{it}=\\frac 1{i!} \\frac {\\part ^iK(\\hat{u}_N)}{\\part \\varepsilon ^i},\n\\ 0\\le i\\le N$. These perturbation systems were proved to form\nintegrable couplings of the original system $u_t=K(u)$\n\\cite{LakshmananT-JMP1985,MaF-CSF1996}, the simplest case of which\nis the above system associated with the symmetry problem.\nThis fact is also a main motivation for us to\nconsider the problem of integrable couplings.\n\nHowever the standard perturbation systems above\nare just special examples of integrable couplings.\nThey keep the spatial dimensions of given integrable systems invariant\nand only the perturbations around solutions of\nunperturbed integrable systems have been considered.\nWe already know \\cite{Ma-Needs98} that it is possible to extend\nthe standard perturbation systems and to change the spatial dimensions,\nin order to make more examples of integrable couplings.\nThe question we want to ask here is how to do generally,\nor what related theory we can develop.\nIn this paper, we would like to provide our partial answer to this extensive\nquestion, by establishing a theory on the multi-scale perturbation systems\nof perturbed integrable systems.\nAn approach for extending the standard perturbation systems\nand for enlarging the spatial dimensions by perturbations will be proposed.\n\nLet us now introduce our basic \nnotation and conception, some notation of which comes from\nRefs. \n\\cite{FuchssteinerF-PD1981,Oevel-JMP1988,Fuchssteiner-book1993,MaF-CSF1996}. \nLet $M=M(u)$ be a suitable manifold possessing a manifold variable $u$,\nwhich is assumed to be a column vector of $q$ functions of $t\\in \\R$ and\n$x\\in \\R^p$ with $t$ playing the role of time and $x$ representing\nposition in space. We are concerned with coupling systems\nby perturbations and thus need\nto introduce another bigger suitable manifold \n$\\hat {M}_N=\\hat {M}_N(\\hat {\\eta }_N)$ possessing a manifold variable\n$\\hat {\\eta }_N =(\\eta _0^T,\\eta _1^T,\\cdots,\\eta _N^T)^T,$ $ N\\ge0$, \nwhere $\\eta _i$, $0\\le i\\le N,$ are also assumed to be\ncolumn function vectors of the same dimension as $u$\nand $T$ means the transpose of matrices. Assume that \n$T(M), T(\\hat{M}_N)$ denote the tangent bundles on $M$ and $\\hat{M}_N$,\n $T^(M), T^(\\hat{M}_N)$ denote the cotangent bundles\n on $M$ and $\\hat{M}_N$, and $C^\\infty (M)$, $C^\\infty (\\hat{M}_N)$\ndenote the spaces of smooth functionals on $M$ and $ \\hat{M}_N$, respectively.\nMoreover let $T^r_s(M)$ be the s-times co- and r-times contravariant \ntensor bundle and $T^r_s\|_u(M)$, \nthe space of s-times co- and r-times contravariant \ntensors at $u\\in M$. We use $X(u)$ (not $X\|_u$) to denote a tensor\n of $X\\in T_s^r(M)$ at $u\\in M$ but sometimes we omit the point $u$ \nfor brevity if there is no confusion. Note that four linear operators\n$\\Phi :T(M)\\to T(M)$, $\\Psi :T^(M)\\to T^(M)$, $\nJ :T^(M)\\to T(M)$, $\\Theta :T(M)\\to T^(M)$\nmay be identified with the second-degree tensor fields\n$T_{\\Phi}\\in T^1_1(M),\\ T_{\\Psi}\\in T^1_1(M),\\\nT_{J}\\in T^2_0(M),\\ T_{\\Theta }\\in T^0_2(M)$\nby the following relations\n\\[\\begin{array} {l} T_\\Phi (u)(\\al (u),K(u))=<\\al (u), \\Phi (u)K(u)>,\\ \\al \n\\in T^(M),\\ K\\in T(M),\n\\vspace {1mm}\\\\\n T_\\Psi (u)(\\al (u),K(u))=<\\Psi (u)\\al (u), K(u)>,\\ \\al \n\\in T^(M),\\ K\\in T(M),\n\\vspace {1mm}\\\\\nT_J (u)(\\al (u),\\beta (u))=<\\al (u), J (u)\\beta (u)>,\\ \\al ,\\beta \\in T^(M),\n\\vspace {1mm}\\\\\nT_\\Theta (u)(K(u),S(u))=<\\Theta (u)K(u), S(u)>,\\ K,S\\in T(M),\n \\end{array} \\]\nwhere $<\\cdot,\\cdot>$ denotes the duality between cotangent vectors and\ntangent vectors. \n\nOf fundamental importance is\nthe conception of the Gateaux derivative, which provides a tool\nto handle various tensor fields. For a tensor field $X\\in T_s^r(M)$,\nits Gateaux derivative at a direction $Y\\in T(M)$ is defined by\n\\begin{equation} X'(u)[Y(u)]=\\left.\\frac {\\part X(u+\\varepsilon \nY(u))}{\\part \\varepsilon }\\right\|_{\\varepsilon =0}.\\end{equation} \nFor those operators between the tangent bundle and \nthe cotangent bundle,\ntheir Gateaux derivatives may be given similarly or \nby means of their tensor fields.\nThe commutator of two vector fields $K,S\\in T(M)$ and the adjoint map\n$\\textrm{ad}_K:T(M)\\to T(M)$ are commonly defined by\n\\begin{equation} [K,S](u)=K'(u)[S(u)]-S'(u)[K(u)],\\ \n\\textrm{ad}_KS=[K,S].\\label{commutatorofvectorfield}\\end{equation}\nNote that there are some authors who use the other commutator\n\\[ [K,S](u)=S'(u)[K(u)]-K'(u)[S(u)].\\] \nIt doesn't matter of course but each type has many proponents and\nhence one must be careful of plus and minus signs in reading various sources.\n\nThe conjugate operators of operators between the tangent bundle and \nthe cotangent bundle are determined in terms of the duality\nbetween cotangent vectors and tangent vectors. For instance, \nthe conjugate operator $\\Phi ^\\dagger :T^(M)\\to T^(M)$\nof an operator $\\Phi :T(M)\\to T(M)$\nis established by\n\\[ <\\Phi ^\\dagger (u)\\al (u) ,K(u) >=<\\al (u),\\Phi(u) K(u)>,\n\\ \\al \\in T^(M),\\ K\\in T(M).\\]\nIf an operator $J:T^(M)\\to T(M)$ (or $\\Theta :T(M)\\to T^(M)$)\nplus its conjugate operator is equal to zero, then it is called \nto be skew-symmetric. \n\n\\begin{definition} \\label{def:gradientfield}\nFor a functional $\\tilde H \\in C^\\infty (M)$, \nits variational derivative $\\frac{\\delta \\tilde{H}}{\\delta u}\n\\in T^(M)$\nis defined by \n \\[<\\frac{\\delta \\tilde {H}}{\\delta u}(u),K(u)>\n%= <\\frac{\\delta \\tilde{H}(u)}{\\delta u},K(u)>\n=\\tilde{H}'(u)[K(u)],\\ K\\in T(M).\n\\]\nIf for $\\gamma \\in T^(M)$ there exists a functional $\\tilde{H}\\in\nC^\\infty (M)$ so that\n\\[ \\frac {\\delta \\tilde{H}}{\\delta u}=\\gamma,\\\n\\textrm{i.e.,}\\ \\tilde{H}'(u)[K(u)]=<\\gamma (u)\n,K(u)>,\\ K\\in T(M),\\label{gradient}\\]\nthen $\\gamma \\in T^(M)$ is called a gradient field with \na potential $\\tilde{H}$.\n\\end{definition}\n\nA cotangent vector field \n$\\gamma \\in T^(M)$ is a gradient field if and only if \n\\begin{eqnarray} && (d\\gamma)(u)(K(u),S(u))\\nonumber \\\\\n&:=& <\\gamma '(u)[K(u)],S(u)>-\n<\\gamma '(u)[S(u)],K(u)>=0,\\ K,S\\in T(M).\n\\label{defofdifferentialofcotangentvector}\\end{eqnarray} \nIf $\\gamma \\in T^(M)$ is gradient, then its potential $\\tilde{H}$\nis given by \n\\[ \\tilde{H}(u)=\\int _0^1<\\gamma (\\la u),u>d\\lambda.\\]\n\n\\begin{definition} For a linear operator \n$\\Phi :T(M)\\to T(M)$ and a vector field $K\\in T(M)$,\nthe Lie derivative $L_K\\Phi:T(M)\\to T(M) $\nof $\\Phi $ with respect to $K$ is defined by\n\\begin{equation} \n(L_K\\Phi )(u)S(u)=\\Phi '(u)[K(u)]S(u)-K'(u)[\\Phi (u)S(u)]\n+\\Phi (u)K'(u)[S(u)],\\ S\\in T(M).\\end{equation}\n\\end{definition}\n\nAn equivalent form of the Lie derivative is\n\\begin{equation} (L_K\\Phi)(u)S(u)=\\Phi (u)[K(u),S(u)]-[K(u),\\Phi (u) S(u)],\n\\label{anotherformforLiederivativeofPhi}\\end{equation}\nwhere $\\Phi :T(M)\\to T(M)$, $K,S\\in T(M)$, and the commutator\n$[\\cdot,\\cdot]$ is defined by (\\ref{commutatorofvectorfield}).\n\n\\begin{definition}\nA linear operator $\\Phi :T(M)\\to T(M)$ is called a recursion operator \nof $u_t=K(u),\\ K\\in T(M),$ if for all $S\\in T(M)$ and $u\\in M$,\nwe have\n\\begin{equation} \\frac {\\part \\Phi(u)}{\\part t}S(u)+\n\\Phi '(u)[K(u)]S(u)-K'(u)[\\Phi (u)S(u)]+\\Phi (u)K'(u)[S(u)]=0.\\end{equation} \n\\label{def:recursionoperator}\n\\end{definition}\n\nObviously a recursion operator $\\Phi :T(M)\\to T(M)$ of \na system $u_t=K(u),\\ K\\in T(M),$ \ntransforms a symmetry into another symmetry of the same system $u_t=K(u)$.\nTherefore it is very useful\nin constructing the corresponding\nsymmetry algebra of a given system\nand its existence is regarded as an\nimportant characterizing property for integrability of the system\nunder study.\n \n\\begin{definition}\n\\label{def:hereditaryoperator}\nA linear operator $\\Phi :T(M)\\to T(M)$ is called a hereditary\noperator or to be hereditary\n\\cite{Fuchssteiner-NATMA1979}, if the following equality holds\n\\begin{equation}\\begin{array} {l} \\Phi '(u)[\\Phi(u) K(u)]S(u)-\\Phi(u) \n\\Phi '(u)[K(u)]S(u)\n\\vspace{1mm}\\\\ \\ \n-\\Phi '(u)[\\Phi (u)S(u)\n]K(u)\n+\\Phi(u) \\Phi '(u)[S(u)]K(u)=0\\end{array}\\label{hereditarydefinition}\n \\end{equation} \nfor all vector fields $K,S\\in T(M)$.\n\\end{definition}\n\nFor a linear operator $\\Phi :T(M)\\to T(M)$, \nThe above equality (\\ref{hereditarydefinition})\ncan be replaced with either of the following equalities:\n\\[\\begin{array}{l} (L_{\\Phi K})(u)\\Phi (u) =\\Phi (u)(L_K\\Phi)(u) ,\n\\ K\\in T(M),\\vspace{2mm} \\\\\n\\Phi ^2(u)[K(u),S(u)]+[\\Phi (u) K(u),\\Phi (u)S(u)]\n\\vspace{2mm} \\\\\n\\quad -\\Phi(u)\\{[K(u),\\Phi (u)S(u)]+[\\Phi (u)K(u),S(u)]\\}=0,\\ K,S\\in T(M).\n\\end{array}\\]\nIt follows directly from (\\ref{anotherformforLiederivativeofPhi}) that\nthese two equalities are equivalent to each other.\nWe point out that hereditary operators have two remarkable properties.\nFirst, if $\\Phi:T(M)\\to T(M)$ is\nhereditary and $L_K\\Phi =0, \\ K\\in T(M)$,\nthen we have $[\\Phi ^mK,\\Phi ^nK]=0,\\ m,n\\ge 0$ (see, for example, \n\\cite{Fuchssteiner-NATMA1979,Fokas-SAM1987,Gu-book1995}). Therefore, \nwhen a system $u_t=K(u),\\ K\\in T(M),$ possesses\na time-independent hereditary recursion operator $\\Phi:T(M)\\to T(M) $,\na hierarchy of vector fields\n $\\Phi ^nK, \\ n\\ge 0$, are all symmetries and commute with each other.\nSecondly, if the conjugate operator $\\Psi =\\Phi ^\\dagger $ of \na hereditary operator $\\Phi :T(M)\\to T(M)$ maps a gradient field \n$\\gamma \\in T^(M)$ into another gradient field, then $\\Psi ^n\\gamma ,\\\n n\\ge0$, are all gradient fields (see, for example, \\cite{Olver-book1986}).\n\n\\begin{definition} \nA linear skew-symmetric operator\n$J:T^(M)\\to T(M)$ is called a Hamiltonian operator or to be Hamiltonian,\nif for all $\\alpha , \\beta , \\gamma \\in T^(M)$, we have\n\\begin{equation} <\\alpha ,J'(u)[J(u)\\beta ]\\gamma >\n+\\textrm{cycle}(\\alpha , \\beta , \\gamma )=0.\\end{equation}\nIts Poisson bracket is defined by \n\\begin{equation} \\{\\tilde{H}_1,\\tilde{H}_2\\}_J(u)\n=<\\frac {\\delta \\tilde{H}_1}{\\delta u}(u),J(u)\n\\frac {\\delta \\tilde{H}_2}{\\delta u}(u)>,\\end{equation}\nwhere $\\tilde{H}_1,\\, \\tilde{H}_2 \\in C^\\infty (M)$.\nA pair of operators $J,M:T^(M)\\to T(M)$ is called a Hamiltonian\npair, if $cJ+d M$ is always Hamiltonian for any constants $c,d $.\n\\end{definition}\n\nWhen $J:T^(M)\\to T(M)$ is Hamiltonian, we have \\cite{GelfandD-FAA1979}\n\\[ J(u)\\frac {\\delta \\{\\tilde {H}_1,\\tilde{H}_2\\}_J }{\\delta u}(u)\n=[J(u)\\frac {\\delta \\tilde{H}_1}{\\delta u}(u),\nJ(u)\\frac {\\delta \\tilde{H}_2}{\\delta u}(u)], \\]\nwhere $\\tilde{H}_1,\\tilde{H}_2 \\in C^\\infty (M)$.\nThis implies that the operator $J\\frac{\\delta }{\\delta u}$\nis a Lie homomorphism from the Poisson algebra to the vector field algebra.\nMoreover if $J,M:T^(M)\\to T(M)$ is a Hamiltonian pair\nand $J$ is invertible, then $\\Phi =MJ^{-1}:T(M)\\to T(M)$ defines a \nhereditary operator \n\\cite{GelfandD-FAA1979,FuchssteinerF-PD1981}.\n\n\\begin{definition}A linear skew-symmetric\noperator $\\Theta :T(M)\\to T^(M)$ is called a\nsymplectic operator or to be symplectic, if for all $K,S,T\\in T(M)$, we have\n\\begin{equation} <K(u),\\Theta '(u)[S(u)]T(u)>+\\textrm{cycle}(K,S,T)=0.\n\\end{equation}\n\\end{definition}\n\nIf $\\Theta :T(M)\\to T^(M)$ is a symplectic operator, then its\nsecond-degree tensor field $T_\\Theta \\in T_2^0(M)$ can be expressed as\n\\[ T_\\Theta =d\\gamma \\ \\textrm{with}\\ \n <\\gamma (u),K(u)>=\\int _0^1<\\Theta (\\la u)\\la u, K(u)>\\, d\\lambda ,\n\\ K\\in T(M),\\]\nwhere $d\\gamma $ is determined by (\\ref{defofdifferentialofcotangentvector}).\nIt is not difficult to verify that the inverse of a symplectic operator\nis Hamiltonian if it exists and vice versa. We also mention that \nHamiltonian and symplectic operators can be defined only \nin terms of Dirac structures \\cite{Dorfman-book1993}.\n\n\\begin{definition}\nA system of evolution equations $u_t=K(u),\\ K\\in T(M),$ is called\na Hamiltonian system or to be Hamiltonian, if there exists a functional\n$\\tilde{H}\\in C^\\infty (M)$ so that\n\\begin{equation} u_t=K(u)=J(u)\\frac{\\delta \\tilde{H}}\n{\\delta u}(u).\\end{equation}\nIt is called a bi-Hamiltonian system, if there exist two functionals\n$\\tilde{H}_1,\\tilde{H}_2\\in C^\\infty (M)$ and a Hamiltonian pair\n$J,M:T^(M)\\to T(M)$ so that\n\\begin{equation}u_t=K(u)=J(u)\\frac{\\delta \\tilde{H}_1}{\\delta u}(u)\n=M(u)\\frac{\\delta \\tilde{H}_2}{\\delta u}(u).\\end{equation} \n\\end{definition}\n\nThere is the other kind of Hamiltonian systems, which \ncan be defined by symplectic operators. However, the above definition \nhas more advantages in handling symmetries and conserved functionals.\nFor a Hamiltonian system $u_t=J(u)\\frac{\\delta \\tilde{H}}\n{\\delta u}(u)$, the linear operator\n$J\\frac{\\delta }{\\delta u}$ maps a conserved\nfunctional into a symmetry. For a bi-Hamiltonian system, \nwe will be able to recursively construct infinitely many commuting\nsymmetries and conserved functionals for the system, if either of two\nHamiltonian operators is invertible \\cite{Magri-JMP1978}. \n\nIn what follows, we would like to develop a theory for constructing\nintegrable couplings of soliton equations, by analyzing integrable \nproperties of the perturbation systems resulted from perturbed soliton \nequations by multi-scale perturbations. The paper is organized as follows.\nIn Section 2, we first establish general explicit structures of\nhereditary operators, Hamiltonian operators and symplectic operators \nunder the multi-scale perturbations. We will go on to show that \nthe perturbations preserves complete integrability, by establishing \nvarious integrable properties of the resulting perturbation systems, such as\nhereditary recursion operator structures, Virasoro symmetry algebras,\nLax representations, zero curvature representations, Hamiltonian formulations \nand so on. In Section 3, the whole theory will be applied to the KdV equations\nas illustrative examples. This leads to infinitely many integrable couplings \nof the KdV equations, which include Hamiltonian integrable couplings\npossessing two different hereditary recursion operators and local\nbi-Hamiltonian integrable couplings in $2+1$ dimensions. Finally,\nsome concluding remarks are given in Section 5.\n\n\\section{Integrable couplings by perturbations}\n\\setcounter{equation}{0}\n\n\\subsection{Triangular systems by perturbations}\n\nLet us take a perturbation series for any $N\\ge 0$ and $r\\ge 0$:\n\\begin{equation}\n\\hat{ u}_N=\\sum_{i=0}^N \\varepsilon ^i\\eta _i , \\\n\\eta _i=\\eta _i(y_0, y_1, y_2,\\cdots,y_r,t),\\\ny_i=\\varepsilon ^i x, \\ t\\in \\R,\\ x\\in \\R ^p,\\ 0\\le i\\le N,\n\\label{multiplescaleperturbationseries}\n\\end{equation}\nwhere $\\varepsilon $ is a perturbation parameter\nand $\\eta _i,\\ 0\\le i\\le N,$ are assumed to be column vectors of \n$q$ dimensions as before. When $r\\ge 1$, \n(\\ref{multiplescaleperturbationseries}) is really a multi-scale \nperturbation series. We fix a perturbed vector field \n$K=K(\\varepsilon)\\in T(M)$ which is required to be \nanalytic with respect to $\\varepsilon$. Let us introduce\n\\begin{equation} K^{(i)}=K^{(i)}(\\hat{\\eta }_N)=\n\\bigr( K(u,\\varepsilon )\\bigl)^{(i)}(\\hat {\\eta }_N)= \\frac 1 {i!}\n\\left.\\frac {\\part ^i K(\\hat {u}_N,\\varepsilon)}{\\part \\varepsilon ^i}\n\\right \| _{\\varepsilon =0}, \\\n 0\\le i\\le N,\n\\end{equation}\nwhere $\\hat {\\eta }_N =(\\eta _0^T,\\eta _1^T,\\cdots,\\eta _N^T)^T$\nas before, and then define the $N$-th order perturbation vector field on \n$\\hat {M}_N$:\n\\begin{equation} (\\textrm{per}_NK)(\\hat {\\eta }_N)=\n\\hat {K}_N(\\hat {\\eta }_N)=(K^{(0)T}(\\hat {\\eta }_N),K^{(1)T}(\\hat {\\eta }_N)\n,\\cdots,K^{(N)T}(\\hat {\\eta }_N))^T. \\label{perturbationvectorfield}\n\\end{equation} \nHere the vector fields on $M$ are viewed as column vectors of\n$q$ dimensions, and the vector fields on $\\hat{M}_N$, column vectors\nof $q(N+1)$ dimensions, as they are normally handled. \nSince we have\n\\[ \\left.\\frac {\\part ^i K(\\hat{u}_i,\\varepsilon)}{\\part \\varepsilon ^i}\n\\right\|_{\\varepsilon =0}=\n\\left.\\frac {\\part ^i K(\\hat{u}_j,\\varepsilon)}{\\part \\varepsilon ^i}\n\\right\|_{\\varepsilon =0}, \\\n\\hat{u}_i=\\sum_{k=0}^i\\varepsilon ^k\n\\eta _k,\\ \\hat{u}_j=\\sum_{k=0}^j\\varepsilon ^k\n\\eta _k,\\ 0\\le i\\le j\\le N, \\]\nit is easy to find that\n\\begin{equation}\\hat {K}_N(\\hat{\\eta}_N) =(K^{(0)T}(\\hat{\\eta}_0),K^{(1)T}\n(\\hat{\\eta}_1),\\cdots,K^{(N)T}(\\hat{\\eta}_N))^T,\n\\ \\hat{\\eta }_i=(\\eta _0^T,\\eta _1^T,\\cdots, \\eta _i^T)^T,\\ 0\\le i\\le N.\n\\label{triangularpropertyofpvf}\n\\end{equation}\nThus the perturbation vector field $\\textrm{per}_NK=\\hat{K}_N\n\\in T(\\hat{M}_N)$ has a specific property that\nthe $i$-th component depends only on\n$\\eta _0,\\eta _1,\\cdots , \\eta _i$,\nnot on any $\\eta _j,\\ j>i$.\n\nLet us now consider a system of perturbed evolution equations \n\\begin{equation}\nu_{t}=K(u,\\varepsilon ),\\ K=K(\\varepsilon)\\in T(M),\n\\label{initialperturbedsystem}\\end{equation}\nwhere $K(\\varepsilon)$ is assumed to be analytic with respect to \n$\\varepsilon$, as an initial system that we start from. \nIt is obvious that the following perturbed system\n\\begin{equation}\n\\hat {u}_{Nt}=K({\\hat{u}_N},\\varepsilon )\n+\\textrm{o}(\\varepsilon ^N)\\ \\ \\textrm{or}\\ \\ \n\\hat {u}_{Nt} \\equiv K({\\hat{u}_N},\\varepsilon )\n\\ \\pmod{\\varepsilon ^N},\\end{equation}\nleads equivalently to a bigger system of evolution equations\n\\begin{equation}\n \\hat {\\eta }_{Nt}=\\hat {K}_{N}({\\hat{\\eta}_N} ),\\ \\ \\textrm{i.e.,}\\ \\ \n\\eta _{it}=\\left.\n\\frac1 {i!} \\frac {\\part ^iK({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^i}\\right\|_{\\varepsilon =0},\\ 0\\le i\\le N,\n\\label{multiplescaleperturbationsystem}\n\\end{equation}\nwhere $\\hat {u}_N$ is defined by (\\ref{multiplescaleperturbationseries}).\nConversely, a solution $\\hat{\\eta }_N$ of the bigger system \n(\\ref{multiplescaleperturbationsystem}) gives rise to an approximate solution\n$\\hat{u}_N$ of the initial system (\\ref{initialperturbedsystem}) \nto a precision $\\textrm{o}(\\varepsilon ^N)$.\nThe resulting bigger system (\\ref{multiplescaleperturbationsystem})\nis called an $N$-th order perturbation system of \nthe initial perturbed system (\\ref{initialperturbedsystem}),\nand it is a triangular system, owing to (\\ref{triangularpropertyofpvf}).\nWe will analyze its integrable properties by exposing structures \nof other perturbation objects.\n \n\\subsection{Symmetry problem}\n\nLet us shed more right on an remarkable relation between the symmetry problem \nand integrable couplings. Assume that a system $u_t=K(u),\\ K\\in T(M),$ is\ngiven. Then its linearized system reads as $v_t=K'(u)[v]$. What the symmetry\nproblem requires to do is to find vector fields $S\\in T(M)$ which satisfy\nthis linearized system, i.e., $(S(u))_t=K'(u)[S(u)]$ when $u_t=K(u)$.\nTherefore $(u^T,(S(u))^T)$ solves the following coupling system\n\\begin{equation}\\left\\{ \\begin{array} {l} u_t=K(u), \\vspace{2mm} \\\\ \nv_t=K'(u)[v], \\end{array} \\right. \\label{symmetrysystem} \\end{equation} \nif $S\\in T(M)$ is found to be a symmetry of $u_t=K(u)$. \nThis system (\\ref{symmetrysystem}) has been carefully considered upon\nintroducing the perturbation bundle \\cite{Fuchssteiner-book1993}. \nIt is the first-order standard perturbation system of $u_t=K(u)$,\nintroduced in Ref. \\cite{MaF-CSF1996}. Since it keeps complete integrability,\nit provides us with an integrable coupling of the original system $u_t=K(u)$.\nTherefore the symmetry problem is viewed as a sub-case of general\nintegrable couplings.\n\nThe commutator of the vector fields of the form $(K(u),A(u)v)^T$ with\n$A(u)$ being linear has a nice structure:\n\\[ \\Bigl[ \\Bigl( \\begin{array}{c} K(u)\\vspace{2mm}\\\\ A(u)v \\end{array}\\Bigr),\n\\Bigl( \\begin{array}{c} S(u)\\vspace{2mm}\\\\ B(u)v \\end{array}\\Bigr)\\Bigr]=\n\\Bigl( \\begin{array}{c} [K(u),S(u)]\\vspace{2mm}\\\\\n\\lbrack\\!\\lbrack A(u), B(u) \\rbrack\\!\\rbrack v \\end{array}\\Bigr), \\]\nwhere the commutator $[K(u),S(u)]$ is given by (\\ref{commutatorofvectorfield})\nand the commutator $\\lbrack\\!\\lbrack A(u), B(u) \\rbrack\\!\\rbrack$ \nof two linear operators $A(u),B(u)$ is defined by\n\\[ \\lbrack\\!\\lbrack A(u), B(u) \\rbrack\\!\\rbrack\n=A'(u)[S(u)]-B'(u)[K(u)]+A(u)B(u)-B(u)A(u), \\]\nwhich was used to analyze algebraic structures of Lax operators in\n\\cite{Ma-JPA1992}. Moreover for linearized operators, we can have \n\\begin{equation} \n\\lbrack\\!\\lbrack K'(u),S'(u) \\rbrack\\!\\rbrack =T'(u),\\ T=[K,S],\\ K,S\\in T(M),\n\\label{lieproductoflinearizeoperators}\n\\end{equation}\nwhich will be shown later on.\n\n\\subsection{Candidates for integrable couplings}\n\nLet us illustrate the idea of how to construct candidates \nfor integrable couplings by perturbations. \nAssume that an unperturbed system is given by \n\\begin{equation}u_t=K(u),\\ K\\in T(M),\\label{unperturbedsystem}\\end{equation}\nand we want to construct its integrable couplings. To this end,\nlet us choose a simple perturbed system \n\\begin{equation}u_t=K(u)+\\varepsilon K(u),\n\\label{firstorderperturbedinitialsystem}\\end{equation}\nwhich is analytic with respect to $\\varepsilon $ of course,\nas an initial system. Obviously this system doesn't change integrable \nproperties of the original system (\\ref{unperturbedsystem}). \nIn practice, we can have lots of choices of such perturbed systems.\nFor example, if the system (\\ref{unperturbedsystem})\nhas a symmetry $S\\in T(M)$, then we can choose either\n$u_t=K(u)+\\varepsilon S(u)$ or $u_t=K(u)+\\varepsilon ^2S(u)$ \nas another initial perturbed system. According to the definition\nof the perturbation systems in (\\ref{multiplescaleperturbationsystem}),\nthe first-order perturbation system of the above perturbed system\n(\\ref{firstorderperturbedinitialsystem}) reads as\n\\begin{equation}\\left \\{ \\begin{array}{l} \\eta _{0t}=K(\\eta _0),\\vspace{2mm}\\\\\n\\eta _{1t}=K'(\\eta _0)[\\eta _1]+K(\\eta _0). \\end{array}\\right. \n\\label{nonstandardfirstorderperturbationsystem}\\end{equation}\nThis coupling system is a candidate that we want to \nconstruct for getting integrable couplings of the original system \n$u_t=K(u)$. In fact, we will verify that the perturbation defined by \n(\\ref{multiplescaleperturbationseries}) preserves complete integrability. \nTherefore the above coupling system \n(\\ref{nonstandardfirstorderperturbationsystem}) is an integrable coupling \nof the original system $u_t=K(u)$, provided that \n$u_t=K(u)$ itself is integrable. The realization of more integrable couplings,\nsuch as local $2+1$ dimensional bi-Hamiltonian systems, can be found from \nan application to the KdV hierarchy in the next section.\n\n\\subsection{Structures of perturbation operators}\n\nRather than working with concrete examples, we would like to \nestablish general structures for three kinds of perturbation operators.\nThe following three theorems will show us how to construct them explicitly. \nFor the proof of the theorems, we first need to prove a basic result \nabout the Gateaux derivative of the perturbation tensor fields.\n\n\\begin{lemma} \\label{lemma:gateaxderivativeofptf}\nLet $X=X(\\varepsilon )\\in T^r_s(M)$ be analytic with respect to $\\varepsilon$\nand assume that the vector \nfield $\\bar {S}_N=(S_0^T,S_1^T,\\cdots,S_N^T)^T\\in T(\\hat{M}_N)$,\nwhere all sub-vectors $S_i,\\ 0\\le i\\le N$, are of the same dimension. Then\nthe following equalities hold: \n\\begin{equation}\n\\Bigl (\\left.\\frac {\\part ^iX \n(\\hat{u}_N,\\varepsilon )}{\\part \\varepsilon ^i}\\right\|_{\\varepsilon =0}\n\\Bigr )'(\\hat {\\eta }_N)[\\bar S_N]=\\left.\n\\frac{\\part ^i}{\\part \\varepsilon ^i}\\right\|_{\\varepsilon =0\n}X'(\\hat {u}_N,\\varepsilon )\\bigl[\\sum_{\nj=0}^N\\varepsilon ^jS_j\\bigr],\\ 0\\le i\\le N.\n\\label{gateaxderivativeofptf}\n\\end{equation}\n\\end{lemma}\n{\\bf Remark:} Note that in (\\ref{gateaxderivativeofptf}), we have adopted\nthe notation\n\\begin{equation}X'(\\hat{u}_N,\\varepsilon)[K(u)]\n=(X(\\hat{u}_N,\\varepsilon))'(\\hat{u}_N) [K(u)],\\ X=X(\\varepsilon)\n\\in T^r_s(M),\\ K\\in T(M), \\end{equation}\nin order to save space. The same notation will be used in\nthe remainder of the paper.\n\n\\noindent {\\bf Proof:}\nLet us first observe Taylor series\n\\[ X(\\hat {u}_N,\\varepsilon)=\\sum_{i=0}^N\\frac {\\varepsilon ^i}{i!}\n\\left.\\frac{\\part ^i X(\\hat {u}_N,\\varepsilon)}{\\part \\varepsilon ^i}\n\\right \|_{\\varepsilon =0}+\\textrm{o}(\\varepsilon ^N).\\]\nIt follows that\n\\[(X(\\hat{u}_N,\\varepsilon))'({\\hat{\\eta}_N} )[\\bar{S}_N]\n=\\sum_{i=0}^N\\frac {\\varepsilon ^i}{i!}\n\\Bigl(\\left.\\frac {\\part ^iX(\\hat{u}_N,\\varepsilon)}{\\part \\varepsilon \n^i}\\right\|_{\\varepsilon =0}\\Bigr)'({\\hat{\\eta}_N} )[\n\\bar{S}_N]+\\textrm{o}(\\varepsilon ^N).\\]\nSecondly, we can compute that \n\\[(X(\\hat{u}_N,\\varepsilon))'({\\hat{\\eta}_N} )[\\bar{S}_N]=\n\\left.\\frac {\\part }{\\part \\delta}\\right\|_{\\delta =0}X(\\hat {u}_N+\\delta\n\\sum_{j=0}^N\\varepsilon ^jS_j,\\varepsilon)=\nX'(\\hat {u}_N ,\\varepsilon)\\Bigl [ \\sum_{j=0}^N\\varepsilon ^j S_j\\Bigr].\\]\nA combination of the above two equalities\nleads to the required equalities in\n(\\ref{gateaxderivativeofptf}), again according to Taylor series.\nThe proof is completed. \n$\\vrule width 1mm height 3mm depth 0mm$\n\n\\begin{theorem} \\label{thm:perturbationhereditaryoperator}\nIf the operator $\\Phi =\\Phi (\\varepsilon):T(M)\\to T(M)$ being analytic with\nrespect to $\\varepsilon$ is hereditary, then the following operator\n${\\hat{\\Phi}_N}: T(\\hat{M}_N)\\to T(\\hat{M}_N)$ defined by\n\\begin{eqnarray}\n&&(\\textrm{per}_N\\Phi)(\\hat{\\eta }_N)= {\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\n\\nonumber\n\\\\&=&\\left[ \\bigl({\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\\bigr)_{ij}\n\\right]_{i,j=0,1,\\cdots,N}\n=\\left[ \\frac 1 {(i-j)!}\\left.\\frac {\\part ^{i-j}\n\\Phi ({\\hat{u}_N},\\varepsilon )}{\\part \\varepsilon ^{i-j}}\n\\right\|_{\\varepsilon =0}\n \\right]_{q(N+1)\\times q(N+1)}\\nonumber\n\\\\ &= & \\left[ \\begin{array}{cccc}\n\\Phi (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}& & &0\\vspace{1mm}\\\\ \n\\frac {1}{1!}\\left.\\frac {\\part \\Phi ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon }\\right\| _{\\varepsilon =0}&\n\\Phi (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}\n & &\\vspace{1mm} \\\\ \\vdots & \\ddots & \\ddots &\\vspace{1mm} \\\\\n\\frac {1}{N!}\\left.\\frac {\\part ^N \\Phi ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^N } \\right\|_{\\varepsilon =0}&\\cdots &\n\\frac {1}{1!}\\left.\\frac {\\part \\Phi ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon }\\right\|_{\\varepsilon =0}&\n\\Phi (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}\n\\end{array}\n\\right]\\label{newhere}\n\\end{eqnarray}\nis also hereditary, where $\\hat{u}_N$ is a perturbation series\ndefined by (\\ref{multiplescaleperturbationseries}).\n\\end{theorem}\n{\\bf Proof:} Let $\\bar{ K}_N=(K_0^T,K_1^T,\\cdots,K_N^T)^T,$ $\n\\bar{ S}_N=(S_0^T,S_1^T,\\cdots,S_N^T)^T\\in T(\\hat{M}_N)$,\nwhere the sub-vectors $K_i,S_i,\\ 0\\le i\\le N$, are of the same dimension.\nSince $\\hat {\\Phi}_N(\\hat {\\eta }_N)$ is obviously linear,\nwe only need to prove that\n\\begin{eqnarray}\n&&{\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )[{\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\\bar{ K}_N]\n\\bar{ S}_N-{\\hat{\\Phi}_N}({\\hat{\\eta}_N} ){\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )\n[\\bar{ K}_N]\\bar{ S}_N\\nonumber\\\\\n &&-{\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )[{\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\n \\bar{ S}_N]\\bar{ K}_N+{\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\n {\\hat{\\Phi}_N}'({\\hat{\\eta}_N} ) [\\bar{ S}_N]\\bar{ K}_N=0, \\label{heequiv}\n\\end{eqnarray}\naccording to Definition \\ref{def:hereditaryoperator}.\nIn what follows, we are going to prove this equality.\n\nFirst, we immediately obtain the $i$-th element of the vector field\n${\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\\bar{K}_N $ and the element in the $(i,j)$\nposition of the matrix ${\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )[\\bar{K}_N]$:\n \\begin{eqnarray} ({\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\\bar{K}_N)_i=\\sum_{j=0}^i\n\\frac1 {(i-j)!}\\left.\\frac {\\part ^{i-j}\\Phi ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^{i-j}}\n \\right\|_{\\varepsilon =0} K_j,\\ 0\\le i\\le N,&&\\nonumber\\\\\n\\bigl({\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )[\\bar{K}_N]\\bigr)_{ij}=\n\\frac1 {(i-j)!}\\left(\\left.\n\\frac {\\part ^{i-j}\\Phi (\\hat {u}_N,\\varepsilon)}{\\part \\varepsilon ^{i-j}}\n \\right\|_{\\varepsilon =0}\\right)'({\\hat{\\eta}_N} )[\\bar{K}_N]&&\\nonumber\\\\\n\\qquad =\\frac1 {(i-j)!}\\left.\\frac {\\part ^{i-j}}{\\part \\varepsilon ^{i-j}}\n \\right\|_{\\varepsilon =0}\\Bigl(\\Phi '({\\hat{u}_N},\\varepsilon )\\Bigl[\n\\sum_{k=0}^N\\varepsilon ^kK_k\\Bigr]\\Bigr),\\ 0\\le i,j\\le N , \n&&\\nonumber\\end{eqnarray}\n the last equality of which follows from Lemma\n \\ref{lemma:gateaxderivativeofptf}.\n\nNow we can compute the $i$-th element of ${\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )\n[{\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\\bar{K}_N]\\bar{S}_N$ as follows:\n\\begin{eqnarray}\n& &({\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )\n[{\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\\bar{K}_N]\\bar{S}_N)_i\\nonumber\\\\&=&\n\\sum_{j=0}^i\n\\frac1 {(i-j)!}\\left.\\frac {\\part ^{i-j}}{\\part \\varepsilon ^{i-j}}\n \\right\|_{\\varepsilon =0}\n\\Phi '({\\hat{u}_N},\\varepsilon )\\Bigl[\n\\sum_{k=0}^N\\varepsilon ^k \\sum_{l=0}^k \n\\frac1 {(k-l)!}\\left.\\frac {\\part ^{k-l}\\Phi ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^{k-l}} \\right\|_{\\varepsilon =0}K_l\n\\Bigr]S_j\\nonumber\\\\\n& = &\\sum_{j=0}^i\n\\frac1 {(i-j)!}\\left.\\frac {\\part ^{i-j}}{\\part \\varepsilon ^{i-j}}\n \\right\|_{\\varepsilon =0} \\Phi '({\\hat{u}_N},\\varepsilon )\\Bigl[\n\\sum_{l=0}^N\\varepsilon ^l \\sum_{k=l}^N \n\\frac{\\varepsilon ^{k-l}} {(k-l)!}\\left.\\frac \n{\\part ^{k-l}\\Phi ({\\hat{u}_N},\\varepsilon )}{\\part \\varepsilon ^{k-l}}\n \\right\|_{\\varepsilon =0}K_l\n\\Bigr]S_j\\nonumber\\\\\n& =&\\sum_{j=0}^i\n\\frac1 {(i-j)!}\\left.\\frac {\\part ^{i-j}}{\\part \\varepsilon ^{i-j}}\n \\right\|_{\\varepsilon =0}\\Phi '({\\hat{u}_N},\\varepsilon )\\Bigl[\n\\sum_{l=0}^N\\varepsilon ^l \\bigl(\\Phi ({\\hat{u}_N},\\varepsilon )\n+\\textrm{o}(\\varepsilon ^{N-l})\\bigr)K_l\n\\Bigr]S_j\\nonumber\\\\\n& =&\\sum_{j=0}^i\n\\frac1 {(i-j)!}\\left.\\frac {\\part ^{i-j}}{\\part \\varepsilon ^{i-j}}\n \\right\|_{\\varepsilon =0}\\Phi '({\\hat{u}_N},\\varepsilon )\\Bigl[\n\\sum_{l=0}^N\\varepsilon ^l \\Phi ({\\hat{u}_N},\\varepsilon )K_l\n\\Bigr]S_j\\nonumber\\\\\n&=&\\sum_{0\\le j+l\\le i}\n\\frac1 {(i-j-l)!}\\left.\\frac {\\part ^{i-j-l}}{\\part \\varepsilon ^{i-j-l}}\n \\right\|_{\\varepsilon =0}\\Phi '({\\hat{u}_N},\\varepsilon )\n\\Bigl[\\Phi ({\\hat{u}_N},\\varepsilon )K_l\\Bigr]S_j,\\\n0\\le i\\le N.\\nonumber\n\\end{eqnarray}\nOn the other hand, we can compute the $i$-th element of\n${\\hat{\\Phi}_N}({\\hat{\\eta}_N} ){\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )\n[\\bar {K}_N]\\bar {S}_N$ as follows:\n\\begin{eqnarray}\n&&\\bigl({\\hat{\\Phi}_N}({\\hat{\\eta}_N} ){\\hat{\\Phi}_N}'\n({\\hat{\\eta}_N} )[\\bar {K}_N]\\bar {S}_N\\bigr)_i\\nonumber\n\\\\&=& \\sum_{j=0}^i\\frac1 {(i-j)!}\\left.\n\\frac {\\part ^{i-j}\\Phi ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^{i-j}} \\right\|_{\\varepsilon =0}\\sum_{k=0}^j\n\\frac1 {(j-k)!}\\left.\\frac {\\part ^{j-k}}{\\part \\varepsilon ^{j-k}}\n \\right\|_{\\varepsilon =0}\\Phi '({\\hat{u}_N},\\varepsilon )\n\\Bigl[\\sum_{l=0}^N\\varepsilon ^lK_l\\Bigr]S_k\\nonumber\\\\\n& =&\\sum_{j=0}^i\\frac1 {(i-j)!}\\left.\n\\frac {\\part ^{i-j}\\Phi ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^{i-j}} \\right\|_{\\varepsilon =0}\n\\sum_{ k=0}^j\\sum_{l=0}^{j-k}\n\\frac1 {(j-k-l)!}\\left.\\frac {\\part ^{j-k-l}}{\\part \\varepsilon ^{j-k-l}}\n \\right\|_{\\varepsilon =0}\\Phi '({\\hat{u}_N} ,\\varepsilon)[K_l]S_k\\nonumber\\\\\n& =&\\sum_{k=0}^i\\sum_{j=k}^i\\sum_{l=0}^{j-k}\n\\frac1 {(i-j)!(j-k-l)!}\\left.\\frac {\\part ^{i-j}\\Phi ({\\hat{u}_N},\\varepsilon )\n}{\\part \\varepsilon ^{i-j}} \\right\|_{\\varepsilon =0}\n\\left.\\frac {\\part ^{j-k-l}}{\\part \\varepsilon ^{j-k-l}}\n \\right\|_{\\varepsilon =0}\\Phi '({\\hat{u}_N} ,\\varepsilon)[K_l]S_k\\nonumber\\\\\n& =&\\sum_{k=0}^i\\sum_{l=0}^{i-k}\\sum_{j=k+l}^i\n\\frac1 {(i-j)!(j-k-l)!}\\left.\\frac {\\part ^{i-j}\\Phi ({\\hat{u}_N},\\varepsilon )\n}{\\part \\varepsilon ^{i-j}} \\right\|_{\\varepsilon =0}\\left.\n\\frac {\\part ^{j-k-l}}{\\part \\varepsilon ^{j-k-l}}\n \\right\|_{\\varepsilon =0}\\Phi '({\\hat{u}_N} ,\\varepsilon)[K_l]S_k\\nonumber\\\\\n& =&\\sum_{k=0}^i\\sum_{l=0}^{i-k} \n\\frac1 {(i-k-l)!}\\left.\\frac {\\part ^{i-k-l}}{\\part \\varepsilon ^{i-k-l}}\n \\right\|_{\\varepsilon =0}\\Bigl(\\Phi ({\\hat{u}_N},\\varepsilon )\n\\Phi '({\\hat{u}_N},\\varepsilon ) [K_l]S_k\\Bigr)\\nonumber\\\\\n& =&\\sum_{0\\le k+l\\le i}\n\\frac1 {(i-k-l)!}\\left.\\frac {\\part ^{i-k-l}}{\\part \\varepsilon ^{i-k-l}}\n \\right\|_{\\varepsilon =0}\\Bigl(\\Phi ({\\hat{u}_N},\\varepsilon )\n\\Phi '({\\hat{u}_N},\\varepsilon ) [K_l]S_k\\Bigr),\\ 0\\le i\\le N.\\nonumber\n\\end{eqnarray} \nTherefore, it follows from the hereditary property of $\\Phi(u,\\varepsilon)$\nthat each element in the left-hand side of\n(\\ref{heequiv}) is equal to zero, which means that (\\ref{heequiv}) is true. \nThe proof is completed.\n$\\vrule width 1mm height 3mm depth 0mm$\n\n\\begin{theorem} \\label{thm:perturbationHamiltonianoperator}\nIf the operator $J=J(\\varepsilon) :T^(M)\\to T(M)$ being analytic\nwith respect to $\\varepsilon$ is Hamiltonian,\n then the following operator\n ${\\hat{J}_N}:T^(\\hat{M}_N)\\to T(\\hat{M}_N)$ defined by\n\\begin{eqnarray}&&(\\textrm{per}_NJ)(\\hat{\\eta }_N)=\n{\\hat{J}_N}({\\hat{\\eta}_N} )\\nonumber\n\\\\\n&=&\\left[ \\bigl({\\hat{J}_N}({\\hat{\\eta}_N} )\\bigr)_{ij}\\right]_{i,j=0,1,\n\\cdots,N} =\\left[\n\\frac 1 {(i+j-N)!}\\left.\\frac {\\part ^{i+j-N}J ({\\hat{u}_N} ,\\varepsilon)}\n{\\part \\varepsilon ^{i+j-N}}\n\\right\|_{\\varepsilon =0}\n \\right]_{q(N+1)\\times q(N+1)}\\nonumber\n\\\\\n &= &\n\\left[ \\begin{array}{cccc}\n0& & & J (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}\n\\\\ & & J (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}&\n\\frac 1{1!}\\left.\n\\frac {\\part J ({\\hat{u}_N} ,\\varepsilon )}{\\part \\varepsilon }\n\\right\|_{\\varepsilon =0} \\\\\n &\\begin{turn}{45}\\vdots\\end{turn} &\\begin{turn}{45}\\vdots\\end{turn}& \n\\vdots\\\\ J (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}&\\frac 1{1!}\\left.\n\\frac {\\part J ({\\hat{u}_N} ,\\varepsilon )}{\\part \\varepsilon }\n\\right\|_{\\varepsilon =0}&\\cdots&\n\\frac {1}{N!}\\left.\\frac {\\part ^N J ({\\hat{u}_N} ,\\varepsilon )}\n{\\part \\varepsilon ^N }\n\\right\|_{\\varepsilon =0} \n\\end{array}\n\\right] \\label{msperHamiltonianoperator}\n\\end{eqnarray}is also Hamiltonian,\nwhere $\\hat{u}_N$ is a perturbation series\ndefined by (\\ref{multiplescaleperturbationseries}).\n\\end{theorem}\n{\\bf Proof:} \nLet $\\bar{\\al }_N=(\\al _0^T,\\al _1^T,\\cdots,\\al _N^T)^T$, $\\bar{\\beta\n }_N=(\\beta _0^T,\\beta _1^T,\n\\cdots,\\beta _N^T)^T$,\n$\\bar{\\gamma }_N=(\\gamma _0^T,\\gamma _1^T,\n\\cdots,\\gamma _N^T)^T\\in T^(\\hat{M}_N)$,\nwhere the sub-vectors\n$\\al _i,\\beta _i,\\gamma _i,\\ 0\\le i\\le N$, are of the same dimension.\nIt suffices to prove that\n\\begin{equation}\n<\\bar{\\al }_N,\\hat {J}_N'({\\hat{\\eta}_N} )[\\hat {J}_N\n\\bar{\\beta }_N]\\bar{ \\gamma }_N>+\\textrm{cycle}\n(\\bar{\\al }_N,\\bar{\\beta }_N,\\bar{\\gamma }_N)=0,\n\\end{equation}\nsince there is no problem on the linearity and the skew-symmetric property \nfor the perturbation operator $\\hat {J}_N(\\hat{\\eta }_N)$.\n\nFirst, based on Lemma \\ref{lemma:gateaxderivativeofptf}, we can calculate\nthe element in the $(i,j)$ position of the matrix $\\hat{J}_N'({\\hat{\\eta}_N})\n[\\hat {J}_N ({\\hat{\\eta}_N} )\\bar{\\beta }_N]$ as follows:\n\\begin{eqnarray} \n&&\\bigl(\\hat {J}_N '({\\hat{\\eta}_N} )[\\hat {J}_N ({\\hat{\\eta}_N} )\n\\bar{\\beta }_N] \\bigr)_{ij}\\nonumber\\\\\n& =& \\frac 1 {(i+j-N)!}\\left.\\frac {\\part ^{i+j-N}}{\\part \n\\varepsilon ^{i+j-N}} \\right\|_{\\varepsilon =0} {J} '({\\hat{u}_N},\n\\varepsilon )\\Bigl[ \\sum _{l=0}^N\n\\varepsilon ^l (\\hat {J}_N \\bar{\\beta }_N)_l\\Bigr]\\nonumber\\\\\n&= &\\frac 1 {(i+j-N)!}\\left.\\frac {\\part ^{i+j-N}}\n{\\part \\varepsilon ^{i+j-N}}\\right\|_{\\varepsilon =0} {J} '({\\hat{u}_N},\n\\varepsilon )\\Bigl[ \\sum _{l=0}^N\\varepsilon ^l \n \\sum_{k=N-l}^N \\frac 1 {(k+l-N)!}\\left.\\frac {\\part ^{k+l-N}\n {J}({\\hat{u}_N},\\varepsilon )}{\\part \\varepsilon ^{k+l-N}}\n\\right\|_{\\varepsilon =0}\\beta _k\n\\Bigr]\\nonumber\\\\ &=&\\frac 1 {(i+j-N)!}\\left.\\frac\n{\\part ^{i+j-N}}{\\part \\varepsilon ^{i+j-N}}\n\\right\|_{\\varepsilon =0}J '({\\hat{u}_N},\\varepsilon )\\Bigl[\\sum_{k=0}^N\n\\varepsilon ^{N-k}\\bigl(\\sum_{l=N-k}\n^{N}\\frac {\\varepsilon ^{k+l-N}}{(k+l-N)!}\\left.\\frac{\\part ^{k+l-N}J\n({\\hat{u}_N} ,\\varepsilon )} {\\part \\varepsilon ^{k+l-N}}\n\\right\|_{\\varepsilon =0}\\bigr)\\beta _k\\Bigr]\\nonumber\\\\\n&=&\\frac 1 {(i+j-N)!}\\left.\\frac {\\part ^{i+j-N}}{\\part \n\\varepsilon ^{i+j-N}}\n\\right\|_{\\varepsilon =0}{J}'({\\hat{u}_N},\\varepsilon )\\Bigl[\\sum_{k=0}^N\n\\varepsilon ^{N-k}\\bigl(J(\\hat{u}_N ,\\varepsilon )\n+\\textrm{o}(\\varepsilon ^k)\\bigr)\\beta _k\\Bigr]\\nonumber\\\\\n&=&\\frac 1 {(i+j-N)!}\\left.\\frac {\\part ^{i+j-N}}\n{\\part \\varepsilon ^{i+j-N}}\n\\right\|_{\\varepsilon =0} {J} '({\\hat{u}_N},\\varepsilon )\n\\Bigl[\\sum_{k=0}^N\\varepsilon ^{N-k}\nJ({\\hat{u}_N},\\varepsilon )\\beta _k\\Bigr]\\nonumber\\\\\n&=&\\sum_{k=0}^N\\frac 1 {(i+j-N)!}\\left.\\frac {\\part ^{i+j-N}}\n{\\part \\varepsilon ^{i+j-N}}\n\\right\|_{\\varepsilon =0}\\bigl(\\varepsilon ^{N-k}\n{J} '({\\hat{u}_N},\\varepsilon )\\bigl[\nJ({\\hat{u}_N},\\varepsilon )\\beta _k\\bigr]\\bigr)\\nonumber\\\\\n&=&\\sum_{k=2N-(i+j)}^N\\frac 1 {(i+j+k-2N)!}\\left.\n\\frac {\\part ^{i+j+k-2N}}{\\part \\varepsilon ^{i+j+k-2N}}\n\\right\|_{\\varepsilon =0} {J} '({\\hat{u}_N},\\varepsilon )\\bigl[\nJ({\\hat{u}_N},\\varepsilon )\\beta _k\\bigr],\\ 0\\le i,j\\le N.\\nonumber\n\\end{eqnarray}\n\nIn what follows, let us give the remaining proof for the case of\n\\begin{equation} \\eta _i=\\eta _i(y_0,y_1,t)=\\eta _i(x,\\varepsilon x,t),\n\\ 0\\le i\\le N. \\label{twoscaleperturbationcase}\\end{equation}\nSuppose that the duality between cotangent vectors and tangent vectors\nis given by \n\\begin{equation}<\\al , K> = \\int_{\\R^p} \\al ^TK\\, dx,\\ x\\in \\R^p,\\ \n\\alpha \\in T^(M),\\ K\\in T(M). \\end{equation} \nLet us consider the case of $x\\in \\R$ without loss of generality.\nFor brevity, we set \n\\begin{equation}\nF_{ijk}(\\bar{\\al }_N,\\bar{\\beta }_N,\\bar{\\gamma }_N,\n\\hat{\\part } _x)=\\Bigl(\\al _i^T {J} '({\\hat{u}_N},\\varepsilon )\\bigl[\nJ({\\hat{u}_N},\\varepsilon )\\beta _k\\bigr]\\gamma _j\n +\\textrm{cycle}(\\al _i,\\beta _k,\\gamma _j)\\Bigr),\\ 0\\le i,j,k\\le N,\n\\end{equation}\nwhere $\\hat{\\part }_x=\\part _{y_0}+\\varepsilon \\part _{y_1}$,\nowing to (\\ref{twoscaleperturbationcase}), and we assume\nthat the original Hamiltonian operator $J(u,\\varepsilon)$ \ninvolves the differential operator $\\part _x$. Then we can have\n\\begin{eqnarray} &&\n<\\bar{\\al }_N,\\hat {J}_N'({\\hat{\\eta}_N} )[\\hat {J}_N \\bar{\\beta }_N]\n\\bar{ \\gamma }_N>+\\textrm{cycle}(\\bar{\\al }_N,\\bar{\\beta }_N,\n\\bar{\\gamma }_N)\n\\nonumber\\\\\n&=& \\int _{-\\infty }^\\infty \\int _{-\\infty }^\\infty\n\\sum_{2N\\le i+j+k\\le 3N}\\frac 1 {(i+j+k-2N)!}\\left.\n\\frac {\\part ^{i+j+k-2N}}{\\part \\varepsilon ^{i+j+k-2N}}\n\\right\|_{\\varepsilon =0}\nF_{ijk}(\\bar{\\al }_N,\\bar{\\beta }_N,\\bar{\\gamma }_N,\n\\hat{\\part } _x)\\, dy_0dy_1. \\nonumber \\end{eqnarray} \nIn order to apply the Jacobi identity of $J(u,\\varepsilon)$,\nwe make a dependent variable transformation\n\\begin{equation}\ny_0=p,\\ y_1=q+\\varepsilon p,\n\\end{equation} \nfrom which it follows that\n\\begin{equation}\\part _p=\\part _{y_0}+\\varepsilon \\part _{y_1},\\ \n\\part _q=\\part _{y_1}. \\end{equation} \nNow we can continue to compute that \n\\begin{eqnarray} &&\n<\\bar{\\al }_N,\\hat {J}_N'({\\hat{\\eta}_N} )[\\hat {J}_N \\bar{\\beta }_N]\n\\bar{ \\gamma }_N>+\\textrm{cycle}(\\bar{\\al }_N,\\bar{\\beta }_N,\n\\bar{\\gamma }_N) \\nonumber \\\\ &=&\n \\int _{-\\infty }^\\infty \\int _{-\\infty }^\\infty\n\\sum_{2N\\le i+j+k\\le 3N}\\frac 1 {(i+j+k-2N)!}\\left.\n\\frac {\\part ^{i+j+k-2N}}{\\part \\varepsilon ^{i+j+k-2N}}\n\\right\|_{\\varepsilon =0}\\times \\nonumber \\\\ && \\qquad \nF_{ijk}(\\bar{\\al }_N,\\bar{\\beta }_N,\\bar{\\gamma }_N,{\\part }_p)\\left\|\n\\rm{det}\\left[\\begin{array} {cc} \\frac {\\part y_0}{\\part p}&\n\\frac {\\part y_0}{\\part q}\\vspace{2mm}\\\\\n\\frac {\\part y_1}{\\part p}&\\frac {\\part y_1}{\\part q}\n \\end{array} \\right]\n\\right\|\\, dpdq\n\\nonumber \\\\\n&=& \\int _{-\\infty }^\\infty \\Bigl (\n\\sum_{2N\\le i+j+k\\le 3N}\\frac 1 {(i+j+k-2N)!}\\left.\n\\frac {\\part ^{i+j+k-2N}}{\\part \\varepsilon ^{i+j+k-2N}}\n\\right\|_{\\varepsilon =0} \\int _{-\\infty }^\\infty\nF_{ijk}(\\bar{\\al }_N,\\bar{\\beta }_N,\n\\bar{\\gamma }_N,\\part _p) \\, dp \\Bigr) \\,dq \\quad \\nonumber\n\\\\ &=&\n\\int _{-\\infty }^\\infty 0 \\,dq=0.\\nonumber\n\\end{eqnarray}\nIn the last but one step,\nwe have utilized the Jacobi identity of $J(u,\\varepsilon)$.\n\nThe method used here for showing the Jacobi identity can be extended \nto the other cases of the perturbations. Therefore the required result\nis proved. $\\vrule width 1mm height 3mm depth 0mm$\n\nSimilarly, we can show the following structure for the perturbation\nsymplectic operators.\n\n\\begin{theorem}\nIf the operator $\\Theta=\\Theta (\\varepsilon ) :T(M)\\to T^(M)$\nbeing analytic with respect to $\\varepsilon $ is symplectic,\nthen the following operator\n$\\hat{\\Theta }_N :T(\\hat{M}_N)\\to T^(\\hat{M}_N)$ defined by\n\\begin{eqnarray}&&(\\textrm{per}_N\\Theta )(\\hat{\\eta }_N)=\n\\hat{\\Theta }_N ({\\hat{\\eta}_N} )\\nonumber\n\\\\\n&=&\\left[ \\bigl(\\hat{\\Theta }_N ({\\hat{\\eta}_N} )\\bigr)_{ij}\n\\right]_{i,j=0,1,\\cdots,N}\n=\\left[\n\\frac 1 {(N-i-j)!}\\left.\\frac {\\part ^{N-i-j}\\Theta ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^{N-i-j}}\n\\right\|_{\\varepsilon =0}\n \\right]_{q(N+1)\\times q(N+1)}\\nonumber\n\\\\\n &= &\n\\left[ \\begin{array}{cccc}\n\\frac {1}{N!}\\left.\\frac {\\part ^N \\Theta ({\\hat{u}_N},\\varepsilon )}{\\part \n\\varepsilon ^N }\n\\right\|_{\\varepsilon =0}\n&\\cdots &\\frac 1{1!}\\left.\n\\frac {\\part \\Theta ({\\hat{u}_N} ,\\varepsilon)}{\\part \\varepsilon }\n\\right\|_{\\varepsilon =0} \n & \\Theta (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}\\vspace{1mm}\\\\ \\vdots &\n\\begin{turn}{45}\\vdots\\end{turn}&\n\\begin{turn}{45}\\vdots\\end{turn}& \\vspace{1mm} \\\\\n\\frac {1}{1!}\\left.\\frac {\\part ^N \\Theta ({\\hat{u}_N},\\varepsilon )}{\\part \n\\varepsilon ^N }\n\\right\|_{\\varepsilon =0} \n &\\Theta (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}\n & & \\vspace{2mm}\\\\ \\Theta (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}& & & 0\n\\end{array}\n\\right]\\label{newsym}\n\\end{eqnarray}is also symplectic,\nwhere $\\hat{u}_N$ is a perturbation series\ndefined by (\\ref{multiplescaleperturbationseries}).\n\\end{theorem}\n\n\\subsection{Integrable Properties}\n% of the perturbation equations}\n\\label{integrableproperties}\n\nIn this sub-section, we study integrable properties of \nthe perturbation systems defined by\n(\\ref{multiplescaleperturbationsystem}), which include recursion \nhereditary operators, $K$-symmetries (i.e., time independent symmetries), \nmaster-symmetries, Lax representations\nand zero curvature representations, Hamiltonian formulations and etc.\nSimultaneously we establish explicit structures for constructing \nother perturbation objects such as spectral problems,\nHamiltonian functionals, and cotangent vector fields. \n\n\\begin{theorem}\\label{thm:perturbationrecursionoperator}\nLet $K=K(\\varepsilon )\\in T(M)$ be analytic\nwith respect to $\\varepsilon$ and assume that $\\Phi =\\Phi (\\varepsilon):\nT(M)\\to T(M)$ is a recursion operator of $u_t=K(u,\\varepsilon )$.\nThen the operator ${\\hat{\\Phi}_N}: T(\\hat {M}_N)\\to T(\\hat {M}_N)$ \ndetermined by (\\ref{newhere}) is a recursion operator of the perturbation \nsystem $\\hat {\\eta }_{Nt}={\\hat{K}_N} ({\\hat{\\eta}_N} )$ defined by \n(\\ref{multiplescaleperturbationsystem}).\nTherefore if $u_t=K(u,\\varepsilon)$ has a hereditary recursion operator \n$\\Phi (u,\\varepsilon)$, then the perturbation system $\\hat {\\eta }_{Nt}=\n{\\hat{K}_N} ({\\hat{\\eta}_N} )$ has a hereditary recursion operator \n${\\hat{\\Phi}_N}({\\hat{\\eta}_N} )$.\n\\end{theorem}\n{\\bf Proof:} Let $\\bar{S}_N=(S_0^T,S_1^T,\\cdots,S_N^T)^T\\in T(\\hat {M}_N)$,\nwhere the sub-vectors $S_i,\\ 0\\le i\\le N$, are of the same dimension.\nBy Lemma \\ref{lemma:gateaxderivativeofptf}, we can compute that\n\\begin{eqnarray}\n&& \n\\left(\\left. \\frac {\\part ^k\\Phi ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^k}\n\\right\|_{ \\varepsilon=0}\\right)'({\\hat{\\eta}_N} )[{\\hat{K}_N} ]=\\left.\\frac \n{\\part ^k}{\\part \\varepsilon ^k}\\right\|_{\\varepsilon =0}\n\\Phi '(\\hat {u}_N,\\varepsilon)\\Bigl[\\sum_{j=0}^N\\varepsilon ^j K^{(j)}\\Bigr]\n\\nonumber\\\\\n &=& \\left.\\frac {\\part ^k}{\\part \\varepsilon ^k}\\right\|_{\\varepsilon =0}\n\\Phi '(\\hat {u}_N,\\varepsilon)\\bigl[K(\\hat {u}_N,\\varepsilon )\n+\\textrm{o}(\\varepsilon ^N)\\bigr]=\\left.\n\\frac {\\part ^k \\Phi '(\\hat {u}_N,\\varepsilon)[K(\\hat {u}_N,\\varepsilon )]}\n{\\part \\varepsilon ^k}\\right\|_{\\varepsilon =0}\n,\\ 0\\le k\\le N,\\nonumber \\end{eqnarray}\nand \n\\begin{eqnarray}\n&&(K^{(i)})'({\\hat{\\eta}_N} )[\\bar{S}_N]\n=\\frac 1 {i!}\\left.\\frac {\\part ^i}{\\part \\varepsilon ^i}\n\\right\|_{\\varepsilon =0}K'({\\hat{u}_N},\\varepsilon )\\Bigl[\\sum _{k=0}^N\n\\varepsilon ^kS_k\\Bigr]\\nonumber\\\\ &=& \\sum_{j=0}^i\\frac1 {(i-j)!}\n\\left.\\frac {\\part ^{i-j} K'({\\hat{u}_N},\\varepsilon )\n[S_j]}{\\part \\varepsilon ^{i-j}}\n\\right\|_{\\varepsilon =0},\\ 0\\le i\\le N.\\nonumber\n\\end{eqnarray}\nTherefore, immediately from the first equality above, we obtain\nthe $i$-th element of ${\\hat{\\Phi}_N}' ({\\hat{\\eta}_N} )\n[{\\hat{K}_N} ]\\bar{S}_N$ as follows: \n\\begin{equation}\n\\bigl( {\\hat{\\Phi}_N}' ({\\hat{\\eta}_N} )[{\\hat{K}_N} ]\\bar{S}_N \\bigr)_i=\n\\sum_{j=0}^i\\frac1 {(i-j)!}\\left.\n\\frac{\\part ^{i-j} (\\Phi '({\\hat{u}_N},\\varepsilon )\n[K({\\hat{u}_N} ,\\varepsilon )]S_j) }\n{\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0},\\ \n0\\le i\\le N.\\label{recursion1}\n\\end{equation}\nBased on the second equality above, we can make the following computation:\n\\begin{eqnarray}\n&&\\bigl({\\hat{K}_N} '({\\hat{\\eta}_N} )[{\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\n\\bar {S}_N]\\bigr)_i \\nonumber\\\\\n&=&\\sum_{k=0}^i\\frac1 {(i-k)!}\\left.\\frac{\\part ^{i-k} }\n{\\part \\varepsilon ^{i-k}}\\right\|_{\\varepsilon =0}\\bigl(\nK'({\\hat{u}_N},\\varepsilon )[({\\hat{\\Phi}_N}({\\hat{\\eta}_N} )\\bar{S}_N)_k]\n \\bigr)\\nonumber\\\\\n &=&\\sum_{k=0}^i\\frac1 {(i-k)!}\n\\left.\\frac{\\part ^{i-k} }{\\part \\varepsilon ^{i-k}}\n\\right\|_{\\varepsilon =0}K'({\\hat{u}_N} ,\\varepsilon )\n\\Bigl[\\sum_{j=0}^k\\frac1 {(k-j)!}\n\\left.\\frac{\\part ^{k-j} \\Phi ({\\hat{u}_N},\\varepsilon )\n }{\\part \\varepsilon ^{k-j}}\\right\|_{\\varepsilon =0}S_j\\Bigr]\\nonumber\\\\\n&=& \\sum_{j=0}^i\\sum_{k=j}^i\\frac 1{(i-k)!(k-j)!}\\left.\\frac {\\part ^{i-k}}{\n\\part \\varepsilon ^{i-k}}\\right\|_{\\varepsilon =0}\nK'({\\hat{u}_N},\\varepsilon )\\Bigl[\\left.\n\\frac{\\part ^{k-j} \\Phi ({\\hat{u}_N} ,\\varepsilon )\n }{\\part \\varepsilon ^{k-j}}\\right\|_{\\varepsilon =0}S_j\\Bigr]\\nonumber\\\\\n&=& \n\\sum_{j=0}^i\\sum_{k=j}^i\\frac 1{(i-j)!}\\left.\\frac {\\part ^{i-j}}{\n\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0}\\Bigl(\\frac {\\varepsilon \n^{k-j}}{(k-j)!}K'({\\hat{u}_N} ,\\varepsilon )\\Bigl[\n\\left.\n\\frac{\\part ^{k-j} \\Phi ({\\hat{u}_N} ,\\varepsilon )\n }{\\part \\varepsilon ^{k-j}}\\right\|_{\\varepsilon =0}S_j\\Bigr]\\Bigr)\n\\nonumber\\\\ &=& \n\\sum_{j=0}^i\\frac 1{(i-j)!}\\left.\\frac {\\part ^{i-j}}{\n\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0}\nK'({\\hat{u}_N},\\varepsilon )\n\\Bigl[\\sum_{k=j}^i\\frac {\\varepsilon ^{k-j}}{(k-j)!}\n\\left.\\frac{\\part ^{k-j} \\Phi ({\\hat{u}_N} ,\\varepsilon )\n }{\\part \\varepsilon ^{k-j}}\\right\|_{\\varepsilon =0}S_j\\Bigr]\\nonumber\\\\\n &=&\\sum_{j=0}^i\\frac1 {(i-j)!}\n\\left.\\frac{\\part ^{i-j} \n }{\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0}\nK'({\\hat{u}_N},\\varepsilon )[\n\\Phi ({\\hat{u}_N} )S_j+\\textrm{o}(\\varepsilon ^{i-j})]\\nonumber\\\\\n&=&\\sum_{j=0}^i\\frac1 {(i-j)!}\\left.\n\\frac{\\part ^{i-j} \\bigl(K'({\\hat{u}_N},\\varepsilon )\n[\\Phi ({\\hat{u}_N},\\varepsilon )S_j]\\bigr)\n }{\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0}, \\ 0\\le i\\le N;\n\\label{recursion2}\n\\end{eqnarray}\n\\begin{eqnarray}\n&& \\bigl({\\hat{\\Phi}_N}({\\hat{\\eta}_N} ){\\hat{K}_N} '({\\hat{\\eta}_N} )\n[\\bar{S}_N]\\bigr)_i\\nonumber\\\\\n&=&\\sum_{k=0}^i\\frac1 {(i-k)!}\n\\left.\\frac{\\part ^{i-k} \\Phi ({\\hat{u}_N},\\varepsilon )\n }{\\part \\varepsilon ^{i-k}}\\right\|_{\\varepsilon =0}\n\\sum_{j=0}^k\\frac1 {(k-j)!}\n\\left.\\frac{\\part ^{k-j} K'({\\hat{u}_N},\\varepsilon )[S_j]\n }{\\part \\varepsilon ^{k-j}}\\right\|_{\\varepsilon =0}\n\\nonumber\\\\\n&=&\\sum_{j=0}^i\\sum_{k=j}^i\\frac1 {(i-k)!(k-j)!}\n\\left.\n\\frac{\\part ^{i-k} \\Phi ({\\hat{u}_N},\\varepsilon )\n }{\\part \\varepsilon ^{i-k}}\\right\|_{\\varepsilon =0}\n\\left.\n\\frac{\\part ^{k-j} K'({\\hat{u}_N} ,\\varepsilon )[S_j]\n }{\\part \\varepsilon ^{k-j}}\\right\|_{\\varepsilon =0}\\nonumber\\\\\n&=&\\sum_{j=0}^i\\frac1 {(i-j)!}\n\\left.\\frac{\\part ^{i-j} \\bigl(\\Phi ({\\hat{u}_N},\\varepsilon )\nK'({\\hat{u}_N} ,\\varepsilon )[S_j]\\bigr)\n }{\\part \\varepsilon ^{i-k}}\\right\|_{\\varepsilon =0},\\ 0\\le i\\le N.\n\\label{recursion3} \\end{eqnarray}\nIt follows directly from the above \nthree equalities above (\\ref{recursion1}), (\\ref{recursion2}) and\n(\\ref{recursion3}) that\n\\[\\frac {\\part \\hat {\\Phi}_N}{\\part t}(\\hat{\\eta }_N) \\bar {S}_N+\n {\\hat{\\Phi}_N}'({\\hat{\\eta}_N} )[{\\hat{K}_N}(\\hat{\\eta }_N) ]\n\\bar{S}_N-{\\hat{K}_N} '({\\hat{\\eta}_N} )\n[{\\hat{\\Phi}_N}(\\hat{\\eta }_N)\\bar{S}_N]\n+{\\hat{\\Phi}_N}(\\hat{\\eta }_N){\\hat{K}_N} '({\\hat{\\eta}_N} ) \n[\\bar{S}_N]=0.\\]\nAccording to Definition \\ref{def:recursionoperator},\nthis implies that the perturbation operator \n$\\hat{\\Phi}_N (\\hat {\\eta }_N)$ defined by (\\ref{newhere}) \nis a recursion operator\nof $\\hat {\\eta }_{Nt}=\\hat {K}_N (\\hat{\\eta }_N)$. A combination \nwith Theorem \\ref{thm:perturbationhereditaryoperator} gives \nrise to the proof of the second required conclusion. The proof \nis finished. $\\vrule width 1mm height 3mm depth 0mm$\n\n\\begin{theorem} Let $K=K(\\varepsilon),S=S(\\varepsilon)\\in T(M)$\nbe analytic with respect to $\\varepsilon$. For two perturbation vector fields\n${\\hat{K}_N},\\, {\\hat{S}_N}\\in T(\\hat{M}_N)$ defined by \n(\\ref{perturbationvectorfield}), there exists the following relation:\n\\begin{equation}\n[{\\hat{K}_N} ({\\hat{\\eta}_N} ), {\\hat{S}_N} ({\\hat{\\eta}_N} )]\n=({\\hat{K}_N} )'({\\hat{\\eta}_N} )[{\\hat{S}_N} ({\\hat{\\eta}_N} )]\n-({\\hat{S}_N} )'({\\hat{\\eta}_N} )[{\\hat{K}_N} ({\\hat{\\eta}_N} )]=\n\\hat {T}_N({\\hat{\\eta}_N} ),\\label {pvfrelation}\n\\end{equation}\nwhere $\\hat {T}_N\\in (\\hat{M}_N)$ is the perturbation vector field of the \nvector field $ T(\\varepsilon)=[K(\\varepsilon),S(\\varepsilon)]$, \ndefined by (\\ref{perturbationvectorfield}). Furthermore we can have\nthe following:\n\n\\noindent (1) \nif $\\sigma =\\sigma(\\varepsilon)\\in T(M)$\nis an $n$-th order master-symmetry of the perturbed system\n$u_t=K(u,\\varepsilon)$,\nthen $\\hat{\\sigma}_N\n%=(\\sigma ^{(0)T},\\sigma ^{(1)T},\\cdots,\\sigma ^{(N)T})^T\n\\in T(\\hat {M}_N)$ defined by (\\ref{perturbationvectorfield})\nis an $n$-th order master-symmetry of the perturbation\nsystem $\\hat {\\eta}_{Nt}={\\hat{K}_N} ({\\hat{\\eta}_N} )$ defined by\n(\\ref{multiplescaleperturbationsystem});\n\n\\noindent (2) the perturbation\nsystem $\\hat{\\eta }_{Nt}={\\hat{K}_N} ({\\hat{\\eta}_N} )$ \ndefined by (\\ref{multiplescaleperturbationsystem}) possesses\nthe same structure of symmetry algebras as \nthe original perturbed system $u_t=K(u,\\varepsilon)$.\n\\end{theorem}\n{\\bf Proof:} \nAs usual, assume that \n\\[ \\hat {S }_i=(S^{(0)T},S^{(1)T},\\cdots,S^{(i)T})^T, \n\\ \\hat {\\eta }_i=({\\eta }_0^T,{\\eta }_1^T,\\cdots, {\\eta }_i^T)^T,\\\n\\hat {u}_i=\\sum_{k=0}^i\\varepsilon ^k\\eta _k,\\ 0\\le i\\le N.\\] \nBy the definition of the Gateaux derivative, we first have \n\\begin{eqnarray} \n&&(K(\\hat {u}_i,\\varepsilon ))'(\\hat {\\eta}_i)[\\hat{S}_i]=\\left.\n\\frac {\\part }{\\part \\delta}\\right\|_{\\delta =0}K(\\hat {u}_i\n+\\delta \\sum_{k=0}^i\\varepsilon ^kS^{(k)},\\varepsilon )\n\\nonumber\\\\\n&=& \\left.\\frac {\\part }{\\part \\delta}\\right\|_{\\delta =0}K(\\hat {u}_i+\\delta \nS(\\hat {u}_i,\\varepsilon )+\\delta \\textrm{o}(\\varepsilon ^i),\\varepsilon )\n\\nonumber\\\\\n&=& K'(\\hat {u}_i,\\varepsilon )[S(\\hat {u}_i,\\varepsilon )] \n+\\textrm{o}(\\varepsilon ^i),\\ 0\\le i\\le N.\\nonumber\n\\end{eqnarray}\nLet us apply the equality above to the following Taylor series \n\\[\nK(\\hat {u}_i,\\varepsilon )=\\sum_{k=0}^i\\frac{\\varepsilon ^k}{k!}\n\\left.\\frac{\\part ^k K(\\hat {u}_i,\\varepsilon) }{\\part \\varepsilon ^k}\n\\right \|_{\\varepsilon =0}+\\textrm{o}(\\varepsilon ^i),\\ 0\\le i\\le N,\\]\nand then we arrive at \n\\[K'(\\hat{u}_i,\\varepsilon)[S(\\hat{u}_i,\\varepsilon)]=\n\\sum_{k=0}^i\\frac {\\varepsilon ^k}{k!} \\Bigl(\\left .\\frac \n{\\part ^kK(\\hat{u}_i,\\varepsilon)}{\\part \\varepsilon ^k}\n\\right\|_{\\varepsilon=0}\\Bigr)' (\\hat {\\eta }_i)[\\hat{S}_i]\n+\\textrm{o}(\\varepsilon ^i),\\ 0\\le i\\le N. \\]\nTaking the $i$-th derivative with respect to $\\varepsilon $ leads to \n\\begin{equation} \n\\bigr(K'(u,\\varepsilon)[S(u,\\varepsilon )]\\bigl)^{(i)}(\\hat {\\eta }_i)\n=\\Bigr(\\bigr(K(u,\\varepsilon )\n\\bigl)^{(i)}\n\\Bigl)'(\\hat {\\eta }_i)[\\hat {S }_i],\\ 0\\le i\\le N.\n\\label{pvfcomponentrelation}\n\\end{equation}\n\nNow it follows from (\\ref{pvfcomponentrelation}) that \nfor the $i$-th element of $\\hat{T}_N$ we have \n\\begin{eqnarray} &&(T(u,\\varepsilon ))^{(i)}=\n\\bigl(K'(u,\\varepsilon )[S(u,\\varepsilon )]\\bigr)^{(i)}(\\hat {\\eta }_i)-\n\\bigl(S'(u,\\varepsilon )[K(u,\\varepsilon )]\\bigr)^{(i)}\n(\\hat {\\eta }_i)\\nonumber\\\\ &=&\n\\bigl(\\bigl(K(u,\\varepsilon )\\bigr)^{(i)}\\bigr)'(\\hat {\\eta }_i )\n[\\hat {S }_i]-\\bigl(\\bigl(S(u,\\varepsilon )\\bigr)\n^{(i)}\\bigr)'(\\hat {\\eta }_i )[\\hat {K}_i]\\nonumber\\\\\n &=&\n\\bigl({\\hat{K}_N} '({\\hat{\\eta}_N} )[\\hat {S}_N]\\bigr)_i-\\bigl({\\hat{S}_N}\n '({\\hat{\\eta}_N} )[\\hat {K}_N]\\bigr)_i\n,\\ 0\\le i\\le N.\\nonumber\n\\end{eqnarray} \nThis shows that (\\ref{pvfrelation}) holds. All other results\nis a direct consequence of (\\ref{pvfrelation}). \nThe proof is completed. $\\vrule width 1mm height 3mm depth 0mm$\n\nThe relation (\\ref{pvfrelation}) implies that the perturbation series \n(\\ref{multiplescaleperturbationseries}) \nkeeps the Lie product of vector fields invariant.\nIn particular, the second component of (\\ref{pvfrelation}) yields \nthe Lie product property (\\ref{lieproductoflinearizeoperators}) \nof linearized operators. In what follows, we will go on to\nconsider Lax representations and zero curvature representations\nfor the perturbation system defined by\n(\\ref{multiplescaleperturbationsystem}).\nIn our formulation below, we will adopt the following notation for\nthe perturbation of a spectral parameter $\\lambda $:\n\\begin{equation} \\hat {\\lambda }_N=\\sum_{i=0}^N \\varepsilon\n^i \\mu _i ,\\ \\hat {\\mu }_N =(\\mu _0,\\mu _1,\\cdots ,\\mu _N)^T ,\n\\label{perturbationofspectralparameter}\n\\end{equation}\nwhich is quite similar to the notation for the perturbation of the potential\n$u$. Here $\\mu _i,\\ 0\\le i\\le N,$ will be taken as the spectral parameters\nappearing in the perturbation spectral problems. A customary symbol \n$\\bigtriangledown _x\\lambda ,\\ x\\in \\R ^p,$ will still be used to \ndenote the gradient of the spectral parameter $\\lambda $ with respect to $x$.\n\n\\begin{theorem} \\label{pelaxrepresentation}\nLet $K=K(\\varepsilon)\\in T(M)$ be analytic with respect to\n$\\varepsilon$. Assume that \nthe system $u_t=K(u,\\varepsilon)$ has an isospectral Lax representation\n\\begin{equation}\\left\\{ \\begin{array} {l}L(u,\\varepsilon)\\phi =\\lambda \\phi, \n\\vspace{2mm}\\\\ \\phi _t=A(u,\\varepsilon)\\phi, \n\\end{array} \\right.\\ (\\bigtriangledown _x\\lambda =0,\\ x\\in \\R ^p),\\ \n\\textrm{i.e.,}\\ \n(L(u,\\varepsilon))_t=[A(u,\\varepsilon),L(u,\\varepsilon)], \n\\label{laxrepresentationofu_t=K(uvarepsilon)}\\end{equation} \nwhere $L$ and $A$ are two $s\\times s$ matrix differential operators\nbeing analytic with respect to $u$ and $\\varepsilon$.\nDefine the perturbation spectral operator $\\hat {L}_{N}$ and the perturbation \nLax operator $\\hat {A}_{N}$ by \n\\begin{eqnarray}\n&&(\\textrm{per}_NB)(\\hat {\\eta }_N)=\n\\hat {B}_{N}({\\hat{\\eta}_N} )\\nonumber\n\\\\ &=&\\left[\\bigl(\\hat {B}_{N}({\\hat{\\eta}_N} )\\bigr)_{ij}\n\\right]_{i,j=0,1,\\cdots,N}=\\left[\\left.\n\\frac 1 {(i-j)!}\\frac {\\part ^{i-j}B ({\\hat{u}_N} ,\\varepsilon)}\n{\\part \\varepsilon ^{i-j}} \\right\|_{\\varepsilon =0}\n \\right]_{s(N+1)\\times s(N+1)}\\nonumber\n\\\\ &= &\n\\left[ \\begin{array}{cccc}B (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}\n& & &0\\\\ \\left.\n\\frac {1}{1!}\\frac {\\part B ({\\hat{u}_N} ,\\varepsilon)}{\\part \\varepsilon }\n\\right\|_{\\varepsilon =0}&B (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0} & \n& \\\\ \\vdots & \\ddots & \\ddots & \\\\\n \\left. \\frac {1}{N!}\\frac {\\part ^N B ({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^N} \\right\|_{\\varepsilon =0}&\\cdots &\\left.\n\\frac {1}{1!}\\frac {\\part B ({\\hat{u}_N},\\varepsilon )}{\\part \\varepsilon }\n\\right\|_{\\varepsilon =0}&B (\\hat{u}_N,\\varepsilon)\|_{\\varepsilon=0}\n\\end{array}\\right], \\ B=L,A,\\qquad\\ \\ \n\\end{eqnarray}\nwhere $\\hat{u}_N$ is given by (\\ref{multiplescaleperturbationseries}).\nThen under the condition for the spectral operator $L$ that\n\\begin{equation}\n \\textrm{if}\\ L'(\\hat{u}_N)[S(\\hat{u}_N)]=\\textrm{o}(\\varepsilon ^N),\\\n S\\in T(M),\\ \n\\textrm{then}\\ S(\\hat{u}_N)=\\textrm{o}(\\varepsilon ^N),\n\\label{injectiveconditionuptovarepsilon^NforL}\\end{equation} \nthe $N$-th order perturbation system $\\hat {\\eta }_{Nt}=\n{\\hat{K}_N} ({\\hat{\\eta}_N} )$ defined by \n(\\ref{multiplescaleperturbationsystem}) has the following isospectral \nLax representation\n\\begin{equation} (\\hat {L}_N({\\hat{\\eta}_N} ))_{t}\n=[\\hat {A}_{N}({\\hat{\\eta}_N} ),\\hat {L}_{N}({\\hat{\\eta}_N} )],\n\\label{perturbationlaxequation} \\end{equation} \nwhich is the compatibility condition of \nthe following perturbation spectral problem\n\\begin{equation}\n\\left\\{ \\begin{array} {l}\\hat {L}_N({\\hat{\\eta}_N} )\\hat{\\phi}_N =\\lambda\n\\hat{\\phi}_N, \\vspace{2mm}\\\\\n\\hat{\\phi}_{Nt}=\\hat {A}_{N}({\\hat{\\eta}_N} )\\hat{\\phi}_N, \\end{array} \\right.\n(\\bigtriangledown _{y_0} \\lambda =\\\n\\bigtriangledown _{y_1} \\lambda =\\cdots =\n\\bigtriangledown _{y_r} \\lambda =0),\n\\label{perturbationlaxspectralproblem1}\n\\end{equation}\nor the following perturbation spectral problem\n%in the case of $\\eta _i=\\eta_i(x,y)=\n%\\eta _i(x,\\varepsilon x),\\ x\\in \\R ,\\ 0\\le i\\le N$,\n\\begin{equation}\n\\left\\{ \\begin{array} {l}\\hat {L}_N({\\hat{\\eta}_N} )\\hat{\\phi}_N =\\Lambda\n\\hat{\\phi}_N, \\vspace{2mm}\\\\ \\hat{\\phi}_{Nt}=\\hat {A}_{N}\n({\\hat{\\eta}_N} )\\hat{\\phi}_N, \\end{array} \\right. \n\\label{perturbationlaxspectralproblem2}\n\\end{equation}\nwhere the matrix $\\Lambda $ reads as\n\\begin{equation}\\Lambda =\\left [ \\begin{array} {cccc} \\mu _0I_s & & &\n\\vspace{2mm}\\\\\n\\mu _1 I_s&\\mu _0I_s & & \\vspace{2mm}\\\\\n\\vdots & \\ddots & \\ddots & \\vspace{2mm}\\\\\n\\mu _N I_s&\\cdots &\\mu _1I_s & \\mu _0I_s \n\\end{array} \\right],\\ I_s=\\textrm{diag}({\\underbrace\n{1,1,\\cdots,1} _{s} }), \\end{equation} \nwith the spectral parameter $\\mu _i,\\ 0\\le i\\le N$, satisfying \n\\begin{equation}\n\\sum_{k+l=i}\\bigtriangledown _{y_k}\\mu _l=0,\\ 0\\le i\\le N.\n \\label{generalconditionsofspectralparameters}\n\\end{equation} \n\\end{theorem}\n{\\bf Proof:} \nWe first observe that the perturbed system\n\\begin{equation}\\hat {u}_{Nt}=K(\\hat{u}_N,\\varepsilon)+\\textrm{o}\n(\\varepsilon ^N),\\label{pertuebationsystemuptovarepsilon^N}\\end{equation}\nwhich engenders precisely the perturbation system\n$\\hat{\\eta }_{Nt}= \\hat {K}_N(\\hat {\\eta }_N)$ defined by\n(\\ref{multiplescaleperturbationsystem}).\nNoting that $L(u,\\varepsilon),\\,A(u,\\varepsilon)$ are analytic \nwith respect to $u$ and $\\varepsilon$, it follows from \n(\\ref{laxrepresentationofu_t=K(uvarepsilon)}) that \n(\\ref{pertuebationsystemuptovarepsilon^N}) is equivalent to the following\n\\begin{equation}\n\\left.\\frac{\\part ^{k}}{\\part \\varepsilon ^{k}}\\right\|_{\\varepsilon =0}\n\\Bigl((L(\\hat{u}_N,\\varepsilon))_t-[A(\\hat{u}_N,\\varepsilon),\nL(\\hat{u}_N,\\varepsilon)]\\Bigr)=0,\\ 0\\le k\\le N,\n\\label{laxrep0toN}\n\\end{equation}\nby use of (\\ref{injectiveconditionuptovarepsilon^NforL}).\n\nWhat we want to prove next is that (\\ref{laxrep0toN}) is equivalent to \n(\\ref{perturbationlaxequation}). \nLet us compute the elements of the differential operator \nmatrix $[\\hat {A}_N(\\hat{\\eta }_N), \\hat{L}_N(\\hat{\\eta }_N)]$.\nIt is obvious that $[\\hat {A}_N(\\hat{\\eta }_N), \\hat{L}_N(\\hat{\\eta }_N)]$ \nis lower triangular, that is to say,\n\\[ ([\\hat {A}_N(\\hat{\\eta }_N), \\hat{L}_N(\\hat{\\eta }_N)])_{ij}=0,\n\\ 0\\le i<j\\le N.\\]\nFor the other part of \n$[\\hat {A}_N(\\hat{\\eta }_N), \\hat{L}_N(\\hat{\\eta }_N)]$, we can compute that\n\\begin{eqnarray}\n(\\hat {A}_N(\\hat{\\eta }_N) \\hat{L}_N(\\hat{\\eta }_N))_{ij}\n&=&\\sum_{k=j}^i\\frac{1}{(i-k)!}\\left.\\frac{\n\\part ^{i-k}A(\\hat{u}_N,\\varepsilon)}{\\part \\varepsilon \n^{i-k}}\\right\|_{\\varepsilon =0}\n\\frac{1}{(k-j)!}\\left.\\frac{\n\\part ^{k-j}L(\\hat{u}_N,\\varepsilon)}\n{\\part \\varepsilon ^{k-j}}\\right\|_{\\varepsilon =0}\\nonumber\\\\\n&=& \\frac 1{(i-j)!}\\sum_{k=j}^i{ i-j \\choose i-k}\n\\left.\\frac{\\part ^{i-k}A(\\hat{u}_N,\\varepsilon)}{\\part\n \\varepsilon ^{i-k}}\\right\|_{\\varepsilon =0}\n\\left.\\frac{\\part ^{k-j}L(\\hat{u}_N,\\varepsilon)}\n{\\part \\varepsilon ^{k-j}}\\right\|_{\\varepsilon =0}\\nonumber\n\\\\\n&=& \\frac 1{(i-j)!}\\left.\\frac{\n\\part ^{i-j}A(\\hat{u}_N,\\varepsilon)L(\\hat{u}_N,\\varepsilon)}\n{\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0},\\ \n0\\le j\\le i\\le N, \\nonumber\n\\end{eqnarray}\nwhere the $i\\choose j$ are the binomial coefficients.\nIn the same way, we can obtain\n\\[(\\hat {L}_N(\\hat{\\eta }_N) \\hat{A}_N(\\hat{\\eta }_N))_{ij}=\n\\frac 1{(i-j)!}\\left.\\frac{\n\\part ^{i-j}L(\\hat{u}_N,\\varepsilon)A(\\hat{u}_N,\\varepsilon)}\n{\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0},\n\\ 0\\le j\\le i\\le N.\\]\nTherefore we arrive at\n\\[([\\hat {A}_N(\\hat{\\eta }_N), \\hat{L}_N(\\hat{\\eta }_N)])_{ij}=\n\\frac 1{(i-j)!}\\left.\\frac{\n\\part ^{i-j}[A(\\hat{u}_N,\\varepsilon),L(\\hat{u}_N,\\varepsilon)]}\n{\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0},\n\\ 0\\le j\\le i\\le N.\\]\nNow it is easy to find that (\\ref{laxrep0toN}) is equivalent to\n(\\ref{perturbationlaxequation}). Therefore the perturbation system\ndefined by (\\ref{multiplescaleperturbationsystem}) has the Lax\nrepresentation (\\ref{perturbationlaxequation}). \n\nLet us now turn to the perturbation spectral problems\n(\\ref{perturbationlaxspectralproblem1}) and\n(\\ref{perturbationlaxspectralproblem2}).\nObviously, the compatibility condition of the perturbation spectral problem\n(\\ref{perturbationlaxspectralproblem1}) is the Lax equation\n(\\ref{perturbationlaxequation}),\nsince the spectral parameter $\\lambda $ doesn't vary whatever the spatial\nvariables change. Therefore let us consider \nthe compatibility condition of the perturbation spectral problem\n(\\ref{perturbationlaxspectralproblem2}).\n%in the case of $\\eta _i=\\eta_i(x,y)=\\eta _i(x,\\varepsilon x),\n%\\ x\\in \\R ,\\ 0\\le i\\le N$.\nFirst, we want to prove that\n\\begin{equation}\\Lambda \\hat {A}_N(\\hat{\\eta }_N)=\n \\hat {A}_N(\\hat{\\eta }_N)\\Lambda ,\n\\label{commutativepropertyofLambdahatA{N}}\\end{equation} \nif the spectral parameters $\\mu _i,\\ 0\\le i\\le N,$ satisfy\n(\\ref{generalconditionsofspectralparameters}).\nNotice that the condition (\\ref{generalconditionsofspectralparameters})\non the spectral parameters $\\mu _i,\\ 0\\le i\\le N$, is required by\n\\[\\hat {\\bigtriangledown }_x\\hat {\\lambda }_N=\\textrm{o}(\\varepsilon ^N),\n\\ \\hat {\\bigtriangledown }_x=\\sum_{i=0}^r\\varepsilon ^i \\bigtriangledown\n_{y_i},\\ \\hat {\\lambda }_N= \\sum_{i=0}^N\\varepsilon ^i\\mu _i, \\]\nwhich is a perturbation version of $\\bigtriangledown _x\\lambda =0$.\nTherefore we have \n\\[A(\\hat {u}_N,\\varepsilon, \\hat {\\bigtriangledown }_x)\n\\hat {\\lambda }_N=\\hat {\\lambda }_N \nA(\\hat {u}_N,\\varepsilon, \\hat {\\bigtriangledown }_x)\n+\\textrm{o}(\\varepsilon ^N).\\]\nThis guarantees that\n\\[ \\frac 1{(i-j)!} \n\\left.\\frac{\\part ^{i-j}}{\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0}\n\\Bigl(A(\\hat {u}_N,\\varepsilon, \\hat {\\bigtriangledown}_x)\n\\hat {\\lambda }_N\\Bigr)\n=\\frac 1{(i-j)!} \n\\left.\\frac{\\part ^{i-j}}{\\part \\varepsilon ^{i-j}}\\right\|_{\\varepsilon =0}\n\\Bigl(\\hat {\\lambda }_N A(\\hat {u}_N,\\varepsilon,\n\\hat {\\bigtriangledown }_x)\\Bigr),\\ 0\\le j\\le i\\le N,\n\\]\nwhich exactly means that \nthe equality (\\ref{commutativepropertyofLambdahatA{N}}) holds.\nNow we can compute from (\\ref{perturbationlaxspectralproblem2}) that\n\\[ (\\hat {L}_N(\\hat{\\eta}_N))_{t}\\hat{\\phi}_N+\\hat {L}_N(\\hat{\\eta}_N)\n\\hat {A}_N(\\hat{\\eta}_N)\\hat {\\phi}_N=\n\\Lambda \\hat {A}_N(\\hat{\\eta}_N)\\hat {\\phi }_N=\\hat {A}_N(\\hat{\\eta}_N)\\Lambda\n\\hat {\\phi }_N=\\hat {A}_N(\\hat{\\eta}_N)\\hat {L}_N(\\hat{\\eta}_N)\\hat {\\phi}_N.\n\\] \nIt follows that the compatibility condition of the perturbation spectral\nproblem (\\ref{perturbationlaxspectralproblem2}) is also\nthe Lax equation (\\ref{perturbationlaxequation}). The proof is completed.\n$\\vrule width 1mm height 3mm depth 0mm$\n\nThe perturbation spectral operator $\\hat {L}_N$ is very similar\nto the perturbation recursion operator $\\hat {\\Phi }_N$, in spite of\ndifferent orders of matrices. Actually,\nwe may take any recursion operator $\\Phi $ as a spectral operator and \nthe system $u_t=K(u)$ can have a Lax representation $\\Phi _t=[\\Phi ,K']$.\nThis Lax representation is usually non-local, because\nmost recursion operators are intego-differential.\n%characteristic of the recursion operator $\\Phi$.\nWe also remark that two perturbation spectral problems above are\nrepresented for the same perturbation system defined by\n(\\ref{multiplescaleperturbationsystem}),\nwhich involve different conditions on the spectral parameters.\nFor the case of\n\\begin{equation} \\hat {u}_N=\\sum_{i=0}^N\\varepsilon ^i\\eta _i(x,y,t)=\n\\sum_{i=0}^N\\varepsilon ^i\\eta _i(x,\\varepsilon x,t), \\ x\\in \\R ,\n\\label{specificcaseofperturbation} \\end{equation}\nthe condition (\\ref{generalconditionsofspectralparameters}) can be\nreduced to \n\\begin{equation} \\mu _{0x}=0,\\ \\mu _{ix}+\\mu _{i-1,y}=0, \\ 1\\le i\\le N.\n \\label{conditionsofspectralparameters}\n\\end{equation}\nIn the following theorem,\na similar result is shown for zero curvature \nrepresentations of the perturbation systems defined by \n(\\ref{multiplescaleperturbationsystem}).\n\n\\begin{theorem} \n\\label{thm:zcrepofpe}\nLet $K=K(\\varepsilon)\\in T(M)$ be analytic with respect to $\\varepsilon $.\nAssume that the initial system $u_t=K(u,\\varepsilon)$ has \nan isospectral zero curvature representation\n\\begin{equation} \\left \\{\\begin{array} {l} \\phi _x=\nU(u,\\lambda ,\\varepsilon)\\phi, \\vspace{2mm}\n\\\\ \\phi _t=V(u,\\lambda ,\\varepsilon)\\phi ,\\end{array} \\right.\n\\ (\\lambda _x=0,\\ x\\in \\R),\n\\end{equation}\n\\begin{equation} \\textrm{i.e.,}\\ \n(U(u,\\lambda ,\\varepsilon))_t-(V(u,\\lambda ,\\varepsilon))_x+[U(u,\n\\lambda ,\\varepsilon),V(u,\\lambda ,\\varepsilon)]=0, \\end{equation}\nwhere $U$ and $V$ are two $s\\times s$ matrix differential \n(sometimes multiplication) operators being analytic with respect to \n$u$, $\\lambda $ and $\\varepsilon$. Define two perturbation matrix \ndifferential operators $\\hat {U}_{N}$ and $\\hat {V}_{N}$ by\n\\begin{eqnarray}\n&&(\\textrm{per}_N W )(\\hat {\\eta }_N)=\\hat { W }_{N}({\\hat{\\eta}_N},\n\\hat {\\mu }_N )= \\hat { W }_{N}({\\hat{\\eta}_N} )\\nonumber\n\\\\\n&=&\\left[\\bigl(\\hat { W }_{N}({\\hat{\\eta}_N} )\\bigr)_{ij}\n\\right]_{i,j=0,1,\\cdots,N}=\\left[\\left.\n\\frac 1 {(i-j)!}\\frac {\\part ^{i-j} W ({\\hat{u}_N},\n\\hat{\\lambda }_N ),\\varepsilon}{\\part \\varepsilon ^{i-j}}\n\\right\|_{\\varepsilon =0}\n \\right]_{s(N+1)\\times s(N+1)}\\nonumber\n\\\\ &= &\n\\left[ \\begin{array}{cccc}\n W (\\hat{\\eta}_N, \\hat {\\lambda }_N,\\varepsilon )\n\\Bigl.\\Bigr\|_{\\varepsilon=0} & & &0\\vspace{1mm}\\\\ \\left.\n\\frac {1}{1!}\\frac {\\part W ({\\hat{u}_N}, \\hat {\\lambda }_N,\\varepsilon )}\n{\\part \\varepsilon } \\right\|_{\\varepsilon =0}&\n W (\\hat{\\eta}_N, \\hat {\\lambda }_N ,\\varepsilon)\n \\Bigl.\\Bigr\|_{\\varepsilon=0} & &\\vspace{1mm} \\\\\n\\vdots & \\ddots & \\ddots & \\vspace{1mm}\\\\\n \\left. \\frac {1}{N!}\\frac {\\part ^N W\n ({\\hat{u}_N}, \\hat {\\lambda }_N,\\varepsilon )} {\\part \\varepsilon ^N}\n\\right\|_{\\varepsilon =0}&\\cdots &\\left.\n\\frac {1}{1!}\\frac {\\part W ({\\hat{u}_N},\\hat {\\lambda }_N ,\\varepsilon )}\n{\\part \\varepsilon } \\right\|_{\\varepsilon =0}&\n W (\\hat{\\eta}_N,\\hat {\\lambda }_N,\\varepsilon )\n\\Bigl.\\Bigr\|_{\\varepsilon=0} \\end{array}\n\\right], \\qquad\\quad\n\\end{eqnarray}\nwhere $W=U,V$, and $\\hat{u}_N$ and $\\hat{\\lambda }_N$ are given by \n(\\ref{specificcaseofperturbation}) and \n(\\ref{perturbationofspectralparameter}). \nThen under the condition for the spectral operator $U$ that \n\\begin{equation}\n \\textrm{if}\\ U'(\\hat{u}_N)[S(\\hat{u}_N)]=\\textrm{o}(\\varepsilon ^N),\\\nS\\in T(M),\\ \\textrm{then}\\ S(\\hat{u}_N)=\\textrm{o}(\\varepsilon ^N),\n\\label{injectiveconditionuptovarepsilonNforU}\n\\end{equation}\nthe $N$-th order perturbation system $\\hat {\\eta }_{Nt}={\\hat{K}_N} \n({\\hat{\\eta}_N} )$ defined by (\\ref{multiplescaleperturbationsystem})\nhas the following isospectral zero curvature representation\n\\begin{equation}\n\\left\\{\\begin{array} {l} \\displaystyle\n{\\sum_{i=0}^r} \\Pi ^i \\hat {\\phi }_{Ny_i}\n=\\hat {U}_N(\\hat {\\eta }_N,\\hat{\\mu}_N ) \\hat {\\phi }_N,\n\\vspace{2mm}\\\\ \\hat {\\phi }_{Nt}=\\hat {V}_N(\\hat {\\eta }_N,\\hat{\\mu}_N )\n\\hat {\\phi }_N,\\end{array} \\right.\n\\label{perturbationzcspectralproblem}\\end{equation}\n\\begin{equation}\\textrm{i.e.,} \\ (\\hat {U}_N({\\hat{\\eta}_N} ))_{t}\n-\\sum_{i=0}^r \\Pi ^i (\\hat {V}_{N}(\\hat{\\eta }_N))_{y_i}\n+[\\hat {U}_{N}({\\hat{\\eta}_N} ),\n\\hat {V}_{N}({\\hat{\\eta}_N} )]=0,\n\\label{perturbationzcrep} \\end{equation} \nwhere the matrix $\\Pi $ is defined by\n\\begin{equation}\n \\Pi =\\left [\\begin{array} {cc} 0&0\\vspace{2mm}\\\\\nI_{sN} &0 \\end{array} \\right ]_{s(N+1)\\times s(N+1)},\\ \nI_{sN}=\\textrm{diag}(\\underbrace{I_s,\\cdots ,I_s}_{N})\n=\\textrm{diag}(\\underbrace{1,\\cdots ,1}_{sN}),\\end{equation}\nand the spectral parameters $\\mu_i,\\ 0\\le i\\le N,$\nsatisfy\n \\begin{equation} \\sum_{k+l=i}\\part _{y_k}\\mu _l=0,\\ 0\\le i\\le N.\n\\label{generalconditionsofspectralparameters2} \\end{equation}\n\\end{theorem}\n{\\bf Proof:}\nNote that by use of (\\ref{injectiveconditionuptovarepsilonNforU}),\nthe zero curvature equation \n\\[ ((U({u},\\lambda ,\\varepsilon))_t-(V({u},\\lambda ,\\varepsilon))_x\n+[U({u},{\\lambda },\\varepsilon),V( {u},\\lambda ,\\varepsilon)]=0\\]\nfor the system $u_t=K(u,\\varepsilon)$ yields an equivalent representation\n\\begin{eqnarray} && (U(\\hat {u}_N,\\hat {\\lambda }_N,\\varepsilon))_t\n-\\sum_{i=0}^r \\varepsilon ^i (V(\\hat {u}_N,\\hat {\\lambda }_N,\n\\varepsilon))_{y_i} +\\left[U(\\hat {u}_N,\\hat {\\lambda }_N,\\varepsilon),\nV(\\hat {u}_N,\\hat {\\lambda }_N,\\varepsilon)\\right] \\nonumber \\\\\n&\\equiv &\nU'(\\hat {u}_N)[K(\\hat{u}_N,\\varepsilon)]\n- \\sum_{i=0}^r \\varepsilon ^i (V(\\hat {u}_N,\\hat {\\lambda }_N,\n\\varepsilon))_{y_i}\n+[U(\\hat {u}_N,\\hat {\\lambda }_N,\\varepsilon),\nV(\\hat {u}_N,\\hat {\\lambda }_N,\\varepsilon)] \\nonumber \\\\\n&\\equiv & \\textrm{o}(\\varepsilon ^N) \\ \\pmod{\\varepsilon ^N}\n\\label{basicpezcrep} \\end{eqnarray}\nfor the perturbation system $\\hat{\\eta }_{Nt}=\\hat {K}_N(\\hat {\\eta }_N)$.\nIn order to recover $\\hat {u}_{Nt}=K(\\hat {u}_N,\\varepsilon)+\\textrm{o}\n(\\varepsilon^N) $ from (\\ref{basicpezcrep}), we need to keep \nthe spectral property $\\lambda _x=\\part _x\\lambda =0$ under the perturbation\nup to a precision $\\textrm{o}(\\varepsilon ^N)$.\nThis requires \n\\[ \\hat {\\part }_x \\hat {\\lambda }_N=\\textrm{o}(\\varepsilon ^N),\\\n\\hat{\\part }_x= \\sum _{i=0}^r \\varepsilon ^i\\part _{y_i}, \\hat {\\lambda }_N=\n\\sum_{i=0}^N \\varepsilon ^i \\mu _i, \\]\nwhich generates (\\ref{generalconditionsofspectralparameters2}).\nSimilar to the proof of Theorem \\ref{pelaxrepresentation},\ndifferentiating the above equation (\\ref{basicpezcrep}) with respect to\n$\\varepsilon$ up to $N$ times leads to \nthe zero curvature equation (\\ref{perturbationzcrep}), and \nconversely, we have (\\ref{basicpezcrep}) if (\\ref{perturbationzcrep}) holds.\nTherefore the perturbation system\n$\\hat{\\eta }_{Nt}=\\hat {K}_N(\\hat {\\eta }_N)$ has an isospectral zero \ncurvature representation (\\ref{perturbationzcrep}).\n \nThe other thing that we need to prove is that the zero curvature equation\n(\\ref{perturbationzcrep}) is exactly the compatibility condition\nof the perturbation spectral problem (\\ref{perturbationzcspectralproblem}).\nFrom the first system of (\\ref{perturbationzcspectralproblem}), we have \n\\[ \\sum_{i=0}^r\\Pi ^i\\hat{\\phi}_{Ny_it}=\\hat {U}_{Nt}\\hat{\\phi}_N\n+\\hat {U}_N\\hat{\\phi}_{Nt}.\n \\]\nFrom the second system of (\\ref{perturbationzcspectralproblem}), we obtain \n\\[\\hat{\\phi}_{Nty_i}=\\hat {V}_{Ny_i}\\hat{\\phi}_N+\\hat {V}_N\n\\hat{\\phi}_{Ny_i}, \\ 0\\le i\\le r. \\]\nA combination of the above equalities yields \n\\begin{equation}\n\\sum_{i=0}^r\\Pi ^i(\\hat{V}_{Ny_i}\\hat{\\phi}_N+\\hat {V}_N\\hat{\\phi}_{Ny_i})\n=\\hat {U}_{Nt}\\hat{\\phi}_N+\\hat {U}_N\\hat {V}_N\n\\hat{\\phi}_N. \\label{hatVNhatUN}\\end{equation}\nOn the other hand, we have\n\\begin{equation}\n\\sum_{i=0}^N \\Pi ^i \\hat {V}_N \\hat{\\phi}_{Ny_i}\n=\\sum_{i=0}^N \\hat {V}_N \\Pi ^i\\hat{\\phi} _{Ny_i}\n=\\hat {V}_N\\hat {U}_N\\hat{\\phi}_N,\\label{HatVNtimesspatialpart}\n\\end{equation}\nby using $\\Pi \\hat {V}_N=\\hat {V}_N\\Pi $ and the first system of \n(\\ref{perturbationzcspectralproblem}).\nIt follows from (\\ref{hatVNhatUN}) and (\\ref{HatVNtimesspatialpart}) that \nthe zero curvature equation (\\ref{perturbationzcrep}) is \nthe compatibility condition of the perturbation spectral problem\n(\\ref{perturbationzcspectralproblem}). The proof is completed.\n$\\vrule width 1mm height 3mm depth 0mm$\n\nIf we consider the specific case of the perturbation defined by\n(\\ref{specificcaseofperturbation}), then the perturbation\nspectral problem and the perturbation zero curvature\nequation, defined by (\\ref{perturbationzcspectralproblem})\nand (\\ref{perturbationzcrep}), will be simplified to\n\\begin{equation}\n\\left\\{\\begin{array} {l} \\hat {\\phi }_{Nx}+\\Pi \\hat {\\phi }_{Ny}=\n\\hat {U}_N(\\hat {\\eta }_N,\\hat{\\mu}_N ) \\hat {\\phi }_N,\n\\vspace{2mm}\\\\ \\hat {\\phi }_{Nt}=\\hat {V}_N(\\hat {\\eta }_N,\\hat{\\mu}_N )\n\\hat {\\phi }_N,\\end{array} \\right.\n\\label{specificperturbationzcspectralproblem}\\end{equation}\nand \n\\begin{equation} (\\hat {U}_N({\\hat{\\eta}_N} ))_{t}\n-(\\hat {V}_N({\\hat{\\eta}_N} ))_x\n-\\Pi (\\hat {V}_{N}(\\hat{\\eta }_N))_y +[\\hat {U}_{N}({\\hat{\\eta}_N} ),\n\\hat {V}_{N}({\\hat{\\eta}_N} )]=0,\n\\label{specificperturbationzcrep} \\end{equation} \nrespectively. The involved spectral parameters $\\mu _i,\\ 0\\le i\\le N,$\nneed to satisfy a reduction (\\ref{conditionsofspectralparameters})\nof the general condition (\\ref{generalconditionsofspectralparameters2}).\n\n\\begin{theorem} \\label{thm:perturbationbi-Hamiltonianformulation}\nLet $K=K(\\varepsilon)\\in T(M)$ be analytic with respect to\n$\\varepsilon$.\nAssume that the initial system $u_t=K(u,\\varepsilon)$ \npossesses a Hamiltonian formulation\n\\[ u_t=K(u,\\varepsilon)=J(u,\\varepsilon)\\frac {\\delta \\tilde{H}}\n{\\delta u}(u,\\varepsilon), \n%\\tilde{H}=\\int H\\,dx, \n\\]\nwhere $J:T^(M)\\to T(M)$ is a Hamiltonian operator and \n$\\tilde{H}\\in C^\\infty (M)$ is a Hamiltonian functional.\nThen the perturbation system $\\hat {\\eta} _{Nt}={\\hat{K}_N} \n({\\hat{\\eta}_N} )$ defined by (\\ref{multiplescaleperturbationsystem}) \nalso possesses a Hamiltonian formulation \n\\begin{equation}\n\\hat {\\eta} _{Nt}={\\hat{K}_N} ({\\hat{\\eta}_N} )=\n{\\hat{J}_N} ({\\hat{\\eta}_N} )\\frac \n{\\delta ( \\textrm{per}_N \\tilde{H}) }{\\delta \\hat{\\eta}_N }\n({\\hat{\\eta}_N} ),\n%\\ \\tilde{\\Hat {H}}_N({\\hat{\\eta}_N} )=\\int \\hat {H}_N\\,dx,\n\\end{equation} \nwhere the Hamiltonian operator ${\\hat{J}_N} ({\\hat{\\eta}_N} )$ \nis determined by (\\ref{msperHamiltonianoperator}) \nand the Hamiltonian functional $\\textrm{per}_N \\tilde{H}\n=\\Hat {\\tilde{{H}}}_N\\in C^\\infty (\\hat{M}_N)$ is defined by\n\\begin{equation}\n(\\textrm{per}_N\\tilde{H})(\\hat{\\eta }_N)=\n\\Hat {\\tilde{{H}}}_N({\\hat{\\eta}_N} )= \n\\frac 1{N!}\\left. \\frac {\\part ^N \\tilde{H}({\\hat{u}_N} ,\\varepsilon)}\n{\\part \\varepsilon ^N}\\right \|_{\\varepsilon =0}.\\label{constantsformular}\n\\end{equation}\nThe corresponding Poisson bracket has the property\n\\begin{equation}\\{ \\textrm{per}_N\\tilde{H}_{1},\\textrm{per}_N \\tilde{H}_{2}\\}\n_{\\hat {J}_{N}}=\\textrm{per}_N \n\\{\\tilde{H}_1,\\tilde{H}_2\\}\n_J,\\ \\tilde{H}_1,\\tilde{H}_2\\in C^\\infty (M).\n\\label{Poissonpro}\\end{equation} \nMoreover the perturbation systems \n$\\hat {\\eta}_{Nt}={\\hat{K}_N} ({\\hat{\\eta}_N} )$\ndefined by (\\ref{multiplescaleperturbationsystem})\npossesses a multi-Hamiltonian formulation\n\\[\n\\hat {\\eta} _{Nt}={\\hat{K}_N} ({\\hat{\\eta}_N} )=\n\\hat{J}_{1N}({\\hat{\\eta}_N} )\\frac{\\delta (\\textrm{per}_N\\tilde{H}_1)}\n{\\delta {\\hat{\\eta}_N} }({\\hat{\\eta}_N} )=\n\\cdots=\n\\hat{J}_{mN}({\\hat{\\eta}_N} )\\frac{\\delta (\\textrm{per}_N\\tilde{H}_{m})}\n{\\delta {\\hat{\\eta}_N} }({\\hat{\\eta}_N} ),\n\\]\nif $u_t=K(u,\\varepsilon)$ possesses an analogous\n multi-Hamiltonian formulation\n\\[u_t=K(u,\\varepsilon )=J_1(u,\\varepsilon )\\frac{\\delta \\tilde{H}_1\n}{\\delta u}(u ,\\varepsilon )=\n%J_2(u)\\frac{\\delta {H_2}(u )}{\\delta u}=\n\\cdots=\nJ_m(u,\\varepsilon )\\frac{\\delta \\tilde{{H}}_m}{\\delta u}(u ,\\varepsilon ).\\]\n\\end{theorem}\n{\\bf Proof:} Assume that $\\gamma (\\varepsilon) \n=\\frac {\\delta \\tilde{H}}{\\delta u}(\\varepsilon )\\in T^(M)$. \nLet us observe that\n\\begin{eqnarray}\n\\eta _{it}&=& \\frac1 {i!}\\left.\\frac {\\part ^i(J({\\hat{u}_N},\\varepsilon )\n\\gamma ({\\hat{u}_N},\\varepsilon ))}{\\part \\varepsilon ^i}\n\\right\|_{\\varepsilon =0}\\nonumber \\\\\n&=& \\sum_{j=0}^i\\frac 1{j!(i-j)!}\n\\left.\\frac {\\part ^{i-j} J({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^{i-j}}\n\\right\|_{\\varepsilon =0}\\left.\\frac {\\part ^j \\gamma \n({\\hat{u}_N},\\varepsilon )}{\\part \\varepsilon ^j}\n\\right\|_{\\varepsilon =0},\\ 0\\le i\\le N. \\nonumber \n\\end{eqnarray}\nThus, noting the structure of $\\hat {J}_N$, we can represent \nthe perturbation system as follows \n\\begin{equation}\n\\hat {\\eta}_{Nt}={\\hat{K}_N} ({\\hat{\\eta}_N} )={\\hat{J}_N}\n (\\hat {\\eta}_N)\\hat {\\gamma }_N(\\hat {\\eta}_N),\n\\label{newhamiltonianstru}\n\\end{equation}\nwhere the cotangent vector field \n$\\hat {\\gamma }_N\\in T^(\\hat {M}_N)$ reads as\n\\begin{eqnarray} && \\hat {\\gamma }_N(\\hat {\\eta}_N)=\n\\Bigl( \\frac 1 {N!}\\left.\\frac \n{\\part ^N\\gamma ^T({\\hat{u}_N} ,\\varepsilon )}\n{\\part \\varepsilon ^N}\\right\|_{\\varepsilon =0}\n,\\frac 1 {(N-1)!}\\left.\\frac \n{\\part ^{N-1}\\gamma ^T({\\hat{u}_N},\\varepsilon )}{\\part \\varepsilon ^{N-1}}\n\\right\|_{\\varepsilon =0}\n, \\nonumber \\\\ &&\\quad \\qquad \\qquad \n\\cdots, \\frac 1 {1!}\\left.\\frac \n{\\part \\gamma ^T({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon }\\right\|_{\\varepsilon =0}\n, \\gamma ^T(\\hat{u}_N,\\varepsilon )\|_{\\varepsilon =0}\\Bigr)^T.\\label{pcvf}\n\\end{eqnarray} \nLet us check whether this cotangent vector field\n $\\hat {\\gamma }_N$ is a gradient field. If it is gradient,\nthe corresponding potential functional has to be the following\n\\begin{eqnarray} &&\\tilde{H}_N({\\hat{\\eta}_N} )\n=\\int _0^1<\\hat {\\gamma }_N(\\lambda {\\hat{\\eta}_N} ), {\\hat{\\eta}_N} >\\,\nd\\lambda \\nonumber\\\\&=&\n\\int_0^1\\sum_{i=0}^N\\frac 1{i!}<\\left.\\frac {\\part ^i\\gamma (\\lambda\n{\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^i}\\right\|_{\\varepsilon =0},\\eta _{N-i}>\\, \nd\\lambda\\nonumber\n\\\\&=&\\frac 1 {N!}\\left.\\frac {\\part ^N}{\\part \\varepsilon ^N}\n\\right\|_{\\varepsilon =0}\n\\int _0^1<\\gamma (\\lambda {\\hat{u}_N},\\varepsilon ), {\\hat{u}_N} >\\,\nd\\lambda =\\frac 1 {N!}\\left.\\frac\n {\\part ^N \\tilde{H}({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^N}\\right\|_{\\varepsilon =0}.\n\\nonumber \n\\end{eqnarray}\nThe cotangent vector field $\\hat {\\gamma }_N$ is indeed a gradient field,\nbecause we can show that \n\\begin{equation}\n\\hat {\\gamma }_N({\\hat{\\eta}_N} )=\\frac {\\delta (\\textrm{per}_N\\tilde{H})\n}{\\delta {\\hat{\\eta}_N} } ({\\hat{\\eta}_N} ).\n\\label{pvector}\\end{equation}\nAccording to Definition \\ref{def:gradientfield} and using \nLemma \\ref{lemma:gateaxderivativeofptf},\nfor any $S_i\\in T(M(\\eta _i))$ we can compute that\n\\begin{eqnarray} &&<\\frac {\\delta }{\\delta \\eta _i }\\,\\Bigl( \\frac 1 {N!}\n\\left.\\frac {\\part ^N \\tilde{H}({\\hat{u}_N},\\varepsilon )}\n{\\part \\varepsilon ^N}\\right\|_{\\varepsilon =0}\n\\Bigr), S_i(\\eta_i) >=\\Bigl( \\frac 1 {N!}\n\\left.\\frac {\\part ^N \\tilde{H}({\\hat{u}_N} ,\\varepsilon )}\n{\\part \\varepsilon ^N}\\right\|_{\\varepsilon =0}\n\\Bigr)'(\\eta _i)[S_i(\\eta_i) ]\\nonumber\\\\&=&\n\\frac 1 {N!}\n\\left.\\frac {\\part ^N }{\\part \\varepsilon ^N}\\right\|_{\\varepsilon =0}\n\\tilde{H}'({\\hat{u}_N},\\varepsilon )\n[\\varepsilon ^iS_i(\\eta_i) ]=\\frac 1 {(N-i)!}\n\\left.\\frac {\\part ^{N-i} }{\\part \\varepsilon ^{N-i}}\n\\right\|_{\\varepsilon =0}\\tilde{H}'({\\hat{u}_N} ,\\varepsilon )[S_i(\\eta_i) ]\n\\nonumber\\\\\n&=&\\frac 1 {(N-i)!}\\left.\\frac {\\part ^{N-i} }\n{\\part \\varepsilon ^{N-i}}\\right\|_{\\varepsilon =0}<\\frac \n{\\delta \\tilde{H}}\n{\\delta {\\hat{u}_N}}({\\hat{u}_N},\\varepsilon ), S_i(\\eta_i) >=\n\\frac 1 {(N-i)!}\\left.\\frac {\\part ^{N-i} }\n{\\part \\varepsilon ^{N-i}}\\right\|_{\\varepsilon =0}\n<\\gamma ({\\hat{u}_N},\\varepsilon ), S_i(\\eta_i) >\n\\nonumber\\\\\n&=&<\\frac 1 {(N-i)!}\n\\left.\\frac {\\part ^{N-i}\\gamma ({\\hat{u}_N},\\varepsilon ) }\n{\\part \\varepsilon ^{N-i}}\\right\|_{\\varepsilon =0}, \nS_i(\\eta_i) >,\\ 0\\le i\\le N.\n\\nonumber \n\\end{eqnarray}\nThis equality implies that (\\ref{pvector}) holds. It follows that \nthe perturbation system (\\ref{newhamiltonianstru}) is a Hamiltonian system.\n\nLet us now turn to \nthe property (\\ref{Poissonpro}) for the Poisson bracket. \nSet $\\gamma _1 (\\varepsilon)\n =\\frac {\\delta \\tilde{H}_{1}}{\\delta u}(\\varepsilon),\\gamma _2 (\\varepsilon)\n=\\frac {\\delta \\tilde{H}_{2}}{\\delta u}(\\varepsilon)\n\\in T^(M)$. In virtue of (\\ref{pvector}), we can make the computation\n\\begin{eqnarray}&& \\{ \\textrm{per}_N\\tilde{H}_{1},\\textrm{per}_N\n\\tilde{H}_{2}\\}\n_{\\hat{J}_{N}}(\\hat{\\eta}_{N})=<\\frac {\\delta (\\textrm{per}_N\n\\tilde{H}_{1}) }{\\delta \\hat{\\eta}_{N}}(\\hat{\\eta}_{N}),\n\\hat{J}_{N}(\\hat{\\eta}_{N})\\frac {\\delta (\\textrm{per}_N\n\\tilde{H}_{2})}{\\delta \\hat{\\eta}_{N}}(\\hat{\\eta}_{N})>\n\\nonumber\\\\&=&\n\\sum_{i=0}^N<\\frac 1 {(N-i)!}\\left.\n\\frac {\\part ^{N-i}\\gamma _1 (\\hat {u}_N,\\varepsilon )}{\\part \\varepsilon ^\n{N-i}}\\right\|_{\\varepsilon =0},\\sum_{j=N-i}^N\\frac 1 {(i+j-N)!}\\times\n\\nonumber\\\\&&\\qquad\\quad\n \\left.\n\\frac {\\part ^{i+j-N}J(\\hat {u}_N,\\varepsilon)}{\\part \\varepsilon ^\n{i+j-N}}\\right\|_{\\varepsilon =0}\\frac 1 {(N-j)!}\n\\left.\\frac {\\part ^{N-j}\\gamma _2 (\\hat {u}_N,\\varepsilon )}\n{\\part \\varepsilon ^{N-j}}\\right\|_{\\varepsilon =0}>\n\\nonumber\\\\&=&\n\\sum_{i=0}^N<\\frac 1 {(N-i)!}\\left.\n\\frac {\\part ^{N-i}\\gamma _1 (\\hat {u}_N,\\varepsilon )}{\\part \\varepsilon ^\n{N-i}}\\right\|_{\\varepsilon =0}, \\frac 1 {i!}\\left.\n\\frac {\\part ^{i}(J (\\hat {u}_N,\\varepsilon )\n\\gamma _2 (\\hat {u}_N,\\varepsilon ))}\n{\\part \\varepsilon ^{i}}\\right\|_{\\varepsilon =0}>\n\\nonumber\\\\&=&\n\\frac 1{N!}\\left.\\frac \n{\\part ^N}{\\part \\varepsilon ^N}\\right\|_{\\varepsilon =0}\n<\\gamma _1 (\\hat {u}_N,\\varepsilon ),J(\\hat {u}_N,\\varepsilon )\n\\gamma _2 (\\hat {u}_N,\\varepsilon )>=\n(\\textrm{per}_N \\{\\tilde{H}_1,\\tilde{H}_2\\}_J)(\\hat{\\eta}_{N}). \\nonumber\n\\end{eqnarray}\nThis shows that the property (\\ref{Poissonpro}) holds for the Poisson bracket.\n\nFurther, noting the structure of the perturbation Hamiltonian operators,\na multi-Hamiltonian formulation may readily be established for\nthe perturbation system. Therefore the proof is completed.\n$\\vrule width 1mm height 3mm depth 0mm$\n\n\\label{theory}\n\nWe should realize that two formulas (\\ref{constantsformular}) and\n(\\ref{pcvf}) provide the explicit structures for \nthe perturbation Hamiltonian functionals \nand the perturbation cotangent vector fields.\nThe whole theory above can be applied to all soliton hierarchies\nand thus various interesting perturbation systems including\nhigher dimensional integrable couplings may be presented.\nIn the next section, we will however be only concerned with \nan application of the theory to the KdV soliton hierarchy. \n\n\\section{Application to the KdV hierarchy}\n\\setcounter{equation}{0}\n\nLet us consider the case of the KdV hierarchy\n\\begin{equation} u_t=K_n=K_n(u)=\n(\\Phi (u))^n u_x,\\ \\Phi=\\Phi (u)=\\part _x^2 + 2u_x\\part _x^{-1}+4u,\n\\ n\\ge 0. \\label{KdVhierarchy}\\end{equation}\nExcept the first linear equation $u_t=u_x$, each equation \n$u_t=K(u)$ $(n\\ge 1)$ can be written as the following bi-Hamiltonian equation\n\\cite{Magri-JMP1978}\n\\begin{equation} u_t=K_n=J \\frac {\\delta {\\tilde H}_{n} }{\\delta u}=\nM\\frac {\\delta {\\tilde H}_{n-1} }{\\delta u}. \\end{equation} \nThe corresponding Hamiltonian pair and Hamiltonian functionals read as\n\\begin{eqnarray} && J=\\part _x, \\ M=M(u)=\\part _x^3+2(\\part _xu+u\\part _x),\n\\label{HamiltonianpairforKdV}\n\\\\ && {\\tilde H}_n=\\int H_n\\,dx ,\\\nH_n= H_n(u)=\\int_0^1uf_n(\\lambda u)\\, d\\lambda,\\ f_n=\\Psi ^nu, \\ n\\ge 0,\n\\end{eqnarray} \nwhere $\\Psi=\\Phi ^\\dagger=\\part _x^2 +4u-2\\part _x^{-1}u_x$. \nTherefore each equation in the KdV hierarchy (\\ref{KdVhierarchy})\nhas infinitely many commuting symmetries $\\{K_m\\}_{m=0}^\\infty$\nand conserved densities $\\{H_m\\}_{m=0}^\\infty$.\n\nThe second equation in the hierarchy (\\ref{KdVhierarchy}) gives the \nfollowing KdV equation\n\\begin{equation} u_t=u_{xxx}+6uu_x, \\label{KdVeq} \\end{equation}\nwhich serves as a well-known model of soliton phenomena. Its many\nremarkable properties were reviewed by Miura \\cite{Miura-SIAMR1976}.\nIn our discussion, we are concerned only with bi-Hamiltonian formulations\nand consequent symmetries and conserved densities.\nThe bi-Hamiltonian formulation of the KdV equation (\\ref{KdVeq}) \ncan be written down \n\\begin{equation}u_t=J\\frac {\\delta \\tilde { H}_1}{\\delta u}=M \\frac \n{\\delta \\tilde {H}_0}{\\delta u}\n\\label{biHamiltonianformulationofKdV}\\end{equation} \nwith\n%the Hamiltonian pair $\\{J,M\\}$ is determined by (\\ref{HamiltonianpairforKdV})\ntwo Hamiltonian functionals\n%$ \\tilde { H}_0$, $\\tilde { H}_1$ are given by \n\\begin{equation}\n\\tilde { H}_0=\\int H_0\\,dx =\\int \\frac 12 u^2\\,dx,\\ \\tilde { H}_1=\n\\int H_1\\,dx= \\int (\\frac 12 uu_{xx}+u^3)\\, dx.\n\\end{equation}\nIt has also an isospectral zero curvature representation \n$U_t-V_x+[U,V]=0$ with\n\\begin{equation} U=\\left[\\ba {cc} 0& -u-\\lambda \n\\vspace{2mm} \\\\ 1&0 \\ea \\right],\\\nV=\\left[\\ba {cc} u_x&-u_{xx}-2u^2+2\\lambda u+4\\lambda ^2 \\vspace{2mm} \\\\\n2u-4\\lambda & -u_x\\ea \\right],\\label{zcrepofKdV}\n\\end{equation} \nwhere $\\lambda $ is a spectral parameter (see \\cite{AblowitzS-book1981}\nfor more information). \nThese two properties will be used to construct bi-Hamiltonian formulations \nand zero curvature representations for the related perturbation systems. \n\nIn order to apply the general idea of constructing integrable couplings\nto the KdV equations, let us start from the following perturbed equation\n\\begin{equation}u_t=K^{\\rm{per}}(u)=\\sum_{i=0}^\\infty \\alpha _i\n\\varepsilon ^i S_i(u), \\label{gKdVpie}\\end{equation} \nwhere $\\alpha _i $ are arbitrary constants and\nthe $S_i$ are taken from zero function and $K_n,\\,n\\ge 0$,\nso that the series (\\ref{gKdVpie}) terminates.\nTo obtain integrable couplings of the $n$-th order KdV equation $u_t=K_n$,\nwe need to fix $S_0=K_n$.\nVarious integrable couplings can be generated by making the perturbation\ndefined by (\\ref{multiplescaleperturbationseries}). In what follows, \nwe would only like to present some illustrative examples.\n\n\\subsection{Standard perturbation systems}\n\nFirst of all, let us choose the $n$-th order KdV equation itself as\nan initial equation:\n\\[ u_t=K^{\\rm{per}}(u)=K_n(u) \\]\nfor each $n\\ge 1$.\nIn this case, the single scale perturbation $\\hat{u}_N=\\sum_{i=0}^N\n\\varepsilon ^i\\eta _i(x,t)$ leads to a type of integrable couplings:\n\\begin{equation}\\hat {\\eta }_{Nt}=\\hat {K}_{nN}(\\hat {\\eta }_N),\\ N\\ge 0,\n\\label{perturbationsystemofnthKdV}\\end{equation} \nwhich are called the standard perturbation systems of $u_t=K_n$\nand have been discussed in \\cite{TamizhmaniL-JPA1983,MaF-PLA1996}.\nThese systems have the following bi-Hamiltonian formulations \\cite{MaF-PLA1996}\n\\begin{equation}\\hat {\\eta }_{Nt}= \\hat {K}_{nN}(\\hat {\\eta }_{N})=\n\\hat {\\Phi }_{N}^n\\hat {\\eta }_{Nx}=\n\\hat {J }_{N}\\frac {\\delta ( \\textrm{per}_N \\tilde {{H}}\n_{n})}{\\delta \\hat {\\eta }_N}=\n\\hat {M }_{N}\\frac {\\delta (\\textrm{per}_N \\tilde {H}_{n-1})}\n{\\delta \\hat {\\eta }_N} ,\\end{equation}\nwhere the Hamiltonian functionals $\\textrm{per}_N\\tilde{H }_{n}$,\nthe hereditary recursion operator $\\hat {\\Phi }_{N}$ and\nthe Hamiltonian pair $\\{\\hat {J }_{N},\\hat {M }_{N}\\}$ are given by\n\\begin{eqnarray} && \\textrm{per}_N \\tilde{H}_{n} = \\frac 1{N!}\\frac\n{\\part ^N \\tilde {H}_n (\\hat {u}_N)}{\\part \\varepsilon ^N },\n\\label{hat{H}{nN}} \\\\ &&\n\\hat {\\Phi }_{N} = \\left [\n\\begin{array} {cccc} \\Phi_0(\\eta _0) & & & 0\\vspace{2mm}\\\\\n\\Phi_1(\\eta _1)&\\Phi _0(\\eta _0) & & \\vspace{2mm}\\\\\n\\vdots & \\ddots & & \\vspace{2mm}\\\\\n\\Phi _N(\\eta _N)& \\cdots &\\Phi _1(\\eta _1)&\\Phi _0(\\eta _0)\n\\end{array} \\right ],\n \\\\\n&& \\hat {J }_{N}=\n\\left [ \\begin{array} {cccc} 0& & & \\part _x\n\\vspace{2mm}\\\\ & &\\part _x & \\vspace{2mm}\\\\\n& \\begin{turn} {45}\\vdots \\end{turn} & & \\vspace{2mm}\\\\\n\\part _x & & & 0\n\\end{array} \\right ],\\ \n\\hat {M }_{N} = \\left [\\begin{array} {cccc} 0& & & M_0(\\eta _0)\n\\vspace{2mm}\\\\ & &M_0(\\eta _0)& M_1(\\eta _1)\\vspace{2mm}\\\\\n& \\begin{turn} {45}\\vdots \\end{turn}& \\begin{turn} {45}\\vdots \\end{turn}\n& \\vdots \\vspace{2mm}\\\\\nM_0(\\eta _0) &M_1(\\eta _1) & \\cdots &M_N(\\eta _N) \n\\end{array} \\right ], \\qquad \\ \\, \n\\end{eqnarray} \nwith\n\\begin{equation}M_i=M_i(\\eta _i)=\\delta_{i0}\\part _x^3\n+2(\\part _x\\eta _i+\\eta _i\\part _x),\n\\ \\Phi _i=\n\\Phi_i(\\eta _i)=\\delta_{i0}\\part _x^2+2(\\part _x\\eta _i\\part _x^{-1}\n+\\eta _i) ,\\ 0\\le i\\le N . \\label{Mi(etai)andPhii(etai)}\\end{equation} \nMoreover they have infinitely many commuting symmetries \n$\\{\\hat {K}_{mN}\\}_{m=0}^\\infty$ and \nconserved densities $\\{\\hat {H}_{mN}\\}_{m=0}^\\infty$.\n\nWe list the first two standard perturbation systems of \nthe KdV equation (\\ref{KdVeq}):\n\\begin{eqnarray} &&\\left \\{\\begin{array} {l} \n\\eta _{0t}=\\eta _{0xxx}+6\\eta _{0}\\eta _{0x}, \\vspace{2mm}\\\\\n\\eta _{1t}=\\eta _{1xxx}+6(\\eta _{0}\\eta _{1})_x;\\end{array} \\right.\n\\label{firstorderpsofKdV}\\\\ && \n\\left \\{\\begin{array} {l} \n\\eta _{0t}=\\eta _{0xxx}+6\\eta _{0}\\eta _{0x}, \\vspace{2mm}\\\\\n\\eta _{1t}=\\eta _{1xxx}+6(\\eta _{0}\\eta _{1})_x,\\vspace{2mm} \\\\\n\\eta _{2t}=\\eta _{2xxx}+6\\eta _{1}\\eta _{1x}+6(\\eta _{0}\\eta _{2})_x.\n\\end{array} \\right. \\label{secondorderpsofKdV}\n\\end{eqnarray} \nThe first-order perturbation system (\\ref{firstorderpsofKdV}) has \nthe following bi-Hamiltonian formulation \n\\begin{equation}\n\\hat{\\eta}_{1t} = \\left[ \\begin{array} {cc} 0&\\part _x \\vspace{2mm}\\\\ \n\\part _x&0 \\end{array} \\right]\n\\frac {\\delta (\\textrm{per}_1 \\tilde {H}_{1})} {\\delta \\hat {\\eta} _1}=\n\\left[ \\begin{array} {cc} 0 &\\part _x^3 +2\\eta _{0x}+4\\eta _0\\part _x\n \\vspace{2mm}\\\\ \\part _x^3 +2\\eta _{0x}+4\\eta _0\\part _x\n& 2\\eta _{1x}+4\\eta _1\\part _x \\end{array} \\right]\n\\frac {\\delta (\\textrm{per}_1\\tilde{H}_{0})}{\\delta \\hat {\\eta} _1}\n\\end{equation}\nwith $\\hat {\\eta} _1= (\\eta _0,\\eta _1)^T$\nand the Hamiltonian functionals\n\\begin{eqnarray} &&\n\\textrm{per}_1\\tilde{H}_{0}=\\int \\hat {H}_{01}\\,dx,\\ \n\\hat {H}_{01}=\\eta _0\\eta _1,\n\\nonumber \\\\ && \\textrm{per}_1\\tilde{H}_{1}\n=\\int \\hat {H}_{11}\\,dx,\\\n\\hat {H}_{11}=\n\\frac 12 (\\eta _0 \\eta _{1xx}+\\eta _{0xx}\\eta _1)+3\\eta _0^2\\eta _1.\n\\nonumber \\end{eqnarray} \nThe second-order perturbation system (\\ref{secondorderpsofKdV}) has \nthe following bi-Hamiltonian formulation\n\\begin{equation}\n \\hat {\\eta }_{2t}=\\hat {J}_2\n\\frac {\\delta (\\textrm{per}_2 \\tilde{H}_{1})}{\\delta \\hat {\\eta }_2}=\n\\hat{M}_2\n\\frac {\\delta (\\textrm{per}_2\\tilde{H}_{0})}{\\delta \\hat {\\eta }_2},\\ \n\\hat{\\eta} _2 =(\\eta _0,\\eta _1,\\eta _2)^T\n\\end{equation} \nwith the Hamiltonian functionals\n\\begin{eqnarray} &&\\textrm{per}_2\\tilde{H}_{0}=\\int \\hat {H}_{02}\\,dx,\\ \n\\hat H_{02}= \\eta _0\\eta _2+\\frac 12 \\eta _1^2 ,\n\\nonumber \\\\&& \\textrm{per}_2\\tilde{H}_{1}=\\int \\hat {H}_{12}\\,dx,\\ \n \\hat H_{12}=\n \\frac 12 (\\eta _0\\eta _{2xx}+\\eta _1\\eta _{1xx}+\\eta _{0xx}\\eta _2)+3\n\\eta _0\\eta _1^2+3\\eta _0^2\\eta _2.\\nonumber\n\\end{eqnarray} \n\nAnother example that we want to show is the first-order standard \nperturbation system of the fifth order KdV equation $u_t=K_2(u)$:\n\\begin{equation} \n\\left\\{ \\begin{array} {l} \n\\eta _{0t}=\\eta _{0,5x}+10\\eta _0\\eta _{0xxx}+20\\eta _{0x}\\eta _{0xx}\n+30\\eta _0^2\\eta _{0x},\n\\vspace{2mm}\\\\\n\\eta _{1t}=\\eta _{1,5x}+10 \\eta _{0xxx}\\eta _1+10 \\eta _0\\eta _{1xxx}+\n20\\eta _{0xx}\\eta _{1x}+20\\eta _{0x}\\eta _{1xx}+60\\eta _{0}\\eta _{0x}\\eta _1\n+30 \\eta _0^2\\eta _{1x},\n\\end{array} \\right.\n\\end{equation}\nwhere $\\eta _{0,5x}$ and $\\eta _{1,5x}$, as usual, stand for \nthe fifth order derivatives of $\\eta _0$ and $\\eta _1$ with respect to $x$.\nIt has the following bi-Hamiltonian formulation \n\\begin{equation} \\hat{\\eta }_{2t}=\\hat {J}_1\\frac {\\delta \n(\\textrm{per}_1\\tilde{H}_2)}{\\delta \\hat{\\eta }_1}=\n\\hat {M}_1\\frac {\\delta \n(\\textrm{per}_1\\tilde{H}_1)}{\\delta \\hat{\\eta }_1},\n \\end{equation}\nwhere the Hamiltonian function $\\textrm{per}_1\\tilde{H}_1$\nis given as before and the Hamiltonian function \n $\\textrm{per}_1\\tilde{H}_2$, \ngiven as follows \n\\begin{equation} \n\\textrm{per}_1\\tilde{H}_2 = \\frac12 \\eta _{0xxxx}\\eta _1+\\frac1 2\\eta _0\n\\eta _{1xxxx}+\\frac {20}3 \\eta _0\\eta _{0xx}\\eta _1 +\\frac {10}3 \n\\eta _0^2 \\eta _{1xx}+\\frac 53 \\eta _{0x}^2 \\eta _1+\n\\frac {10}3 \\eta _0\\eta _{0x}\\eta _{1x} +\\frac {40}3 \\eta _{0}^3 \\eta _1.\n\\nonumber \n \\end{equation}\n\n\\subsection{Nonstandard perturbation systems}\n\nSecondly, let us choose\na perturbed equation \n\\[ u_t=K^{\\rm{per}}(u,\\varepsilon)=K_n+\\alpha \\varepsilon K_n, \\ \n\\alpha =\\textrm{const.},\\ \\alpha \\ne 0,\\]\nas an initial equation for each $n\\ge 1$.\nThis equation can be viewed as\n\\begin{eqnarray} u_t=K^{\\rm{per}}(u,\\varepsilon)& =&\nJ \\frac {\\delta ({\\tilde {H}}_n\n+ \\alpha \\varepsilon {\\tilde {H}}_{n}) }{\\delta u} =\nM \\frac {\\delta ({\\tilde {H}}_{n-1}\n+ \\alpha \\varepsilon \\tilde {H}_{n-1}) }{\\delta u} \\nonumber \\\\ &=&\n(J+ \\alpha \\varepsilon J) \\frac { \\delta {\\tilde {H}} _{n} }{\\delta u}\n= (M+ \\alpha \\varepsilon M) \\frac {\\delta \\tilde {H}_{n-1} }{\\delta u}\n.\\end{eqnarray} \nTherefore the corresponding perturbation systems also\nhave quadruple Hamiltonian formulations. We focus on the \nfirst-order perturbation system under the single scale perturbation.\nIt has the quadruple Hamiltonian formulation\n\\begin{equation}\\hat {\\eta }_{1t} =\n\\hat {J}_{1}^{(1)}\\frac {\\delta (\\textrm{per}_1\\tilde {H}_{n}^{(1)})}\n{\\delta \\hat{\\eta }_{1}}\n=\\hat {M}_{1}^{(1)}\\frac {\\delta (\\textrm{per}_1\\tilde{H}_{n-1}^{(1)})}\n{\\delta \\hat{\\eta }_{1}}\n=\\hat {J}_{1}^{(2)}\\frac {\\delta (\\textrm{per}_1\\tilde{H}_{n}^{(2)})}\n{\\delta \\hat{\\eta }_{1}}\n=\\hat {M}_{1}^{(2)}\\frac {\\delta (\\textrm{per}_1\\tilde {H}_{n-1}^{(2)})}\n{\\delta \\hat{\\eta }_{1}}, \\label{nonstandardperturbationsystemofnthKdV}\n\\end{equation} \nnamely,\n\\begin{eqnarray} \n\\hat {\\eta }_{1t} & = &\n\\left[ \\begin{array} {cc } 0& \\part \\vspace{2mm}\\\\ \\part & 0 \\end{array} \n\\right ]\n\\frac{\\delta (\\textrm{per}_1\\tilde{ {H}}_{n}+ \\alpha \n\\tilde{ {H}}_{n}(\\eta _0))}\n{\\delta \\hat {\\eta }_1 }\n=\\left[ \\begin{array} {cc } 0& M_0 \\vspace{2mm}\\\\ M_0 & M_1 \\end{array} \n\\right ]\n\\frac{\\delta (\\textrm{per}_1\\tilde{{H}}_{n-1}+ \\alpha \\tilde{{H}}_{n-1}\n(\\eta _0))} {\\delta \\hat {\\eta }_1 }\n\\nonumber \\\\ &= &\n\\left[ \\begin{array} {cc } 0& \\part \\vspace{2mm}\\\\ \\part & \\alpha \\part \n\\end{array} \\right ]\n\\frac{\\delta (\\textrm{per}_1\\tilde{{H}}_{n})}{\\delta \\hat {\\eta }_1} \n=\\left[ \\begin{array} {cc } 0& M_0 \\vspace{2mm}\\\\ M_0 & M_1+\\alpha M_0 \n\\end{array} \\right ]\n\\frac{\\delta (\\textrm{per}_1\\tilde{{H}}_{n-1})}{\\delta \\hat {\\eta }_1 },\n\\label{11perturbationsystem}\\end{eqnarray} \nwhere the functionals $ \\textrm{per}_1\\tilde{{H}}_{n},\\,\n\\textrm{per}_1\\tilde{{H}}_{n-1}$ and the operators\n$M_i$ are defined by (\\ref{hat{H}{nN}}) and\n(\\ref{Mi(etai)andPhii(etai)}), respectively.\nSince two Hamiltonian operators $\\hat {J}_{1}^{(1)}$\nand $\\hat {J}_{1}^{(2)}$ are invertible, we can obtain five hereditary\nrecursion operators for the equation\n(\\ref{nonstandardperturbationsystemofnthKdV}):\n\\begin{eqnarray} \n&& \\hat {J}_{1}^{(2)} (\\hat {J}_{1}^{(1)} )^{-1}=\\left[\\begin{array}\n {cc}\n1&0 \\vspace{2mm}\\\\ \\alpha & 1 \\end{array} \\right],\\\n\\hat {J}_{1}^{(1)} (\\hat {J}_{1}^{(2)} )^{-1}=\\left[\\begin{array} {cc}\n1&0 \\vspace{2mm}\\\\ -\\alpha & 1 \\end{array} \\right], \\nonumber \\\\ &&\n\\hat {M}_{1}^{(1)} (\\hat {J}_{1}^{(2)} )^{-1}=\\left[\\begin{array} {cc}\n\\Phi _0 &0 \\vspace{2mm}\\\\ \\Phi _1-\\alpha \\Phi _0& \\Phi _0 \\end{array} \n\\right],\\\n\\hat {M}_{1}^{(2)} (\\hat {J}_{1}^{(1)} )^{-1}=\\left[\\begin{array} {cc}\n\\Phi _0 &0 \\vspace{2mm}\\\\ \\Phi _1+\\alpha \\Phi _0& \\Phi _0 \\end{array} \n\\right],\n\\nonumber \\\\ && \n\\hat {M}_{1}^{(1)} (\\hat {J}_{1}^{(1)} )^{-1}\n=\\hat {M}_{1}^{(2)} (\\hat {J}_{1}^{(2)} )^{-1}\n=\\left[\\begin{array} {cc}\n\\Phi _0 &0 \\vspace{2mm}\\\\ \\Phi _1 & \\Phi _0 \\end{array} \\right],\n\\nonumber \\end{eqnarray}\nwhere the operators $\\Phi_i$ are defined by (\\ref{Mi(etai)andPhii(etai)}).\nThese operator structures suggest two classes of hereditary\nrecursion operators for the equation\n(\\ref{nonstandardperturbationsystemofnthKdV})\n\\begin{equation}\n\\hat {\\Phi }_{1}^{(1)}(\\beta )=\\left[\\begin{array} {cc}\n\\beta _0 &0 \\vspace{2mm}\\\\ \\beta _1& \\beta _0 \\end{array} \\right],\\\n\\hat {\\Phi }_{1}^{(2)}(\\beta )=\\left[\\begin{array} {cc}\n\\beta _0\\Phi _0&0 \\vspace{2mm}\\\\ \\beta _0\\Phi _1 +\\beta _1\\Phi _0& \\beta _0\n\\Phi _0 \\end{array} \\right],\n\\label{tworos} \\end{equation} \nwhere $\\beta =(\\beta _0,\\beta _1)^T$ with the $\\beta _i$ being\narbitrary constants.\nThey are really hereditary operators and recursion operators\nfor the equation (\\ref{nonstandardperturbationsystemofnthKdV}),\nwhich can be verified by direct computation or by viewing them as\nthe first-order perturbation operators of the initial operators $\\beta _0\n+\\beta _1\\varepsilon$ and $\\beta _0\\Phi +\\beta _1 \\varepsilon \\Phi$. \nTherefore the integrable coupling\n(\\ref{nonstandardperturbationsystemofnthKdV})\nof the $n$-th order KdV equation $u_t=K_n(u)$ possesses \ntwo classes of hereditary recursion operators defined by (\\ref{tworos}). \nThese two classes of operators have the property \n\\begin{equation}{\\Phi }_{1}^{(1)}(\\beta ){\\Phi }_{1}^{(2)}(\\gamma ) =\n{\\Phi }_{1}^{(2)}(\\beta ){\\Phi }_{1}^{(1)}(\\gamma )=\n\\left[\\begin{array} {cc}\n\\beta _0\\gamma _0 \\Phi _0&0 \\vspace{2mm}\\\\ \\beta _0\\gamma _0\\Phi _1 \n+(\\beta _0\\gamma _1+\\beta _1\\gamma _0)\\Phi _0&\n\\beta _0\\gamma _0\\Phi _0 \\end{array} \\right],\n\\end{equation} \nfor any two constant vectors $\\beta =(\\beta _0,\\beta _1)^T$ and $\\gamma =\n(\\gamma _0,\\gamma _1)^T$, which also shows that their product \ncan not constitute completely new recursion operators.\n\nWe can also start from the perturbed KdV type equation\n\\begin{equation}u_t=K^{\\textrm{per}}\n(u,\\varepsilon)= K_n+\\al \\varepsilon ^jK_{i_j},\\ \n\\al =\\textrm{const.}, \\ \\alpha \\ne 0,\n \\end{equation} \nwhere $i_j$ is a natural number.\nLet us illustrate the idea of construction by the following specific example\n\\begin{equation}u_t=K^{\\textrm{per}}(u,\\varepsilon)\n= K_n + \\al \\varepsilon ^2 K_{n+1}\\ (n\\ge 1),\n \\label{K_n+alphavarepsilonK_{n+1}} \n\\end{equation}\nwhich can be viewed as a tri-Hamiltonian system:\n\\begin{equation}\n u_t=K^{\\textrm{per}} =J\\frac {\\delta (\\tilde{H}_n+\\alpha \\varepsilon ^2\n\\tilde {H}_{n+1}) }{\\delta u}\n= (J+\\alpha \\varepsilon ^2M)\\frac {\\delta \\tilde{H}_n }{\\delta u}\n=M\\frac {\\delta (\\tilde{H}_{n-1}+\\alpha \\varepsilon ^2\n\\tilde {H}_{n}) }{\\delta u}.\n\\end{equation}\nTherefore, according to Theorem \n\\ref{thm:perturbationbi-Hamiltonianformulation}, \nthe second-order perturbation system of the the perturbed system \n(\\ref{K_n+alphavarepsilonK_{n+1}})\n\\begin{equation}\\left \\{ \n\\begin{array} {l} \\eta _{0t } = {K}_{n}(\\eta _0), \\vspace{2mm}\\\\\n\\eta _{1t} =K_n'(\\eta _0)[\\eta _1], \\vspace{2mm}\\\\\n \\eta _{2t}= \\frac 12\\left.\\frac\n {\\part ^2 K_n(\\hat{u}_2) }{\\part \\varepsilon ^2}\n \\right\|_{\\varepsilon =0}+ \\alpha K_{n+1}(\\eta _0)\\ea\n\\right. \\label{biggerK_npe} \\end{equation} \npossesses the following tri-Hamiltonian formulation\n\\begin{equation}\n\\hat {\\eta }_{2t}=\n\\hat {J}_2^{(1)}\\frac {\\delta \\tilde{{H}}_n^{(1)}}{\\delta \\hat {\\eta }_2}=\n\\hat{J}_2^{(2)}\\frac {\\delta \\tilde{{H}}_n^{(2)}}{\\delta \\hat {\\eta }_2}=\n\\hat{J}_2^{(3)}\n\\frac {\\delta \\tilde{{H}}_n^{(3)}}{\\delta \\hat {\\eta }_2},\\\n\\hat{\\eta} _2=(\\eta _0,\\eta _1,\\eta _2)^T\n\\end{equation} \nwith a triple of Hamiltonian operators \n\\begin{equation}\n\\hat {J}_2^{(1)}=\n\\left[\\ba {ccc}0&0&\\part _x \\vspace{2mm}\\\\ 0&\\part _x& 0\\vspace{2mm}\\\\\n\\part _x& 0& 0\\ea \\right],\\ \n%\\nonumber \\\\ &&\n\\hat {J}_2^{(2)}=\n\\left[\\ba {ccc}0&0&\\part _x \\vspace{2mm}\\\\ 0&\\part _x& 0\\vspace{2mm}\\\\\n\\part _x& 0& \\alpha M_0\\ea \\right],\\ \n\\hat {J}_2^{(3)}=\n\\left[\\ba {ccc}0&0&M_0 \\vspace{2mm}\\\\ 0&M_0& M_1\\vspace{2mm}\\\\\nM_0 & M_1& M_2\\ea \\right] \n%\\nonumber \n\\end{equation}\nand the corresponding three Hamiltonian functionals\n\\begin{equation} \\left\\{ \\begin{array}{l}\n\\tilde{H}_n^{(1)} (\\hat{\\eta } _2)= (\\textrm{per}_2\\tilde{{H}}_{n})\n(\\hat {\\eta }_2)+ \\alpha \\tilde {H}_{n+1}(\\eta _0),\\vspace{2mm} \\\\\n \\tilde{H}_n^{(2)} (\\hat{\\eta } _2)= (\\textrm{per}_2\\tilde{{H}}_{n})\n(\\hat {\\eta }_2),\\vspace{2mm}\\\\ \n \\tilde{H}_n^{(3)} (\\hat{\\eta } _2)= (\\textrm{per}_2\\tilde{{H}}_{n-1})\n(\\hat {\\eta }_2)+ \\alpha \\tilde {H}_{n}(\\eta _0).\n \\end{array}\\right. \\end{equation}\nSimilarly, the perturbation system (\\ref{biggerK_npe}) \nhas also two classes of hereditary recursion operators: \n\\begin{equation}\n\\hat {\\Phi }_2^{(1)}(\\beta )= \\left[ \\begin{array}{ccc} \\beta _0& 0& 0 \\vspace{2mm}\\\\ \n\\beta _1& \\beta_0& \\vspace{2mm}\\\\ \n\\beta _2& \\beta _1& \\beta_0 \\end{array} \\right],\\ \n\\hat {\\Phi }_2^{(2)}(\\beta )\n= \\left[ \\begin{array}{ccc} \\beta _0\\Phi _0& 0& 0 \\vspace{2mm}\\\\ \n\\beta _0\\Phi _1+\\beta _1\\Phi _0 & \\beta_0\\Phi _0& \\vspace{2mm}\\\\ \n\\beta _0\\Phi _2+\\beta _1\\Phi _1+\\beta _2\\Phi _0\n& \\beta _0\\Phi _1+\\beta _1\\Phi _0 & \\beta_0\\Phi _0 \\end{array} \\right],\n\\end{equation}\nwhere the operators $\\Phi_i$ are defined by (\\ref{Mi(etai)andPhii(etai)})\nand $\\beta =(\\beta _0,\\beta _1,\\beta _2)^T$ is a constant vector.\n\nLet us fix $n=1$ and then the system (\\ref{biggerK_npe}) gives \nan integrable coupling of the KdV equation (\\ref{KdVeq}),\nwhich possesses the following tri-Hamiltonian formulation\n\\begin{equation}\n\\hat {\\eta }_{2t}=\n\\hat {J}_2^{(1)}\\frac {\\delta \\tilde{{H}}_1^{(1)}}{\\delta \\hat {\\eta }_2}=\n\\hat{J}_2^{(2)}\\frac {\\delta \\tilde{{H}}_1^{(2)}}{\\delta \\hat {\\eta }_2}=\n\\hat{J}_2^{(3)}\n\\frac {\\delta \\tilde{{H}}_1^{(3)}}{\\delta \\hat {\\eta }_2}\n\\end{equation}\nwith three Hamiltonian functionals\n\\begin{equation} \\left\\{ \\begin{array} {l}\n\\tilde{H}_1^{(1)} (\\hat{\\eta } _2)=\n \\frac 12 (\\eta _0\\eta _{2xx}+\\eta _1\\eta _{1xx}+\\eta _{0xx}\\eta _2)+3\n\\eta _0\\eta _1^2+3\\eta _0^2\\eta _2\n \\vspace{2mm} \\\\ \\qquad \\qquad\\quad \n+\\alpha (\\frac12 \\eta _0\\eta _{0xxxx}+\\frac {10}3\\eta _0^2\\eta _{0xx}\n+\\frac 53 \\eta _0\\eta _{0x}^2+\\frac {10}3 \\eta _0^4),\n \\vspace{2mm} \\\\\n \\tilde{H}_1^{(2)} (\\hat{\\eta } _2)=\n \\eta _0\\eta _2+\\frac12 \\eta _1^2\n +\\alpha (\\frac12 \\eta _0\\eta _{0xx}+\\eta _0^3),\n \\vspace{2mm} \\\\\n \\tilde{H}_1^{(3)} (\\hat{\\eta } _2)=\n \\frac 12 (\\eta _0\\eta _{2xx}+\\eta _1\\eta _{1xx}+\\eta _{0xx}\\eta _2)+3\n\\eta _0\\eta _1^2+3\\eta _0^2\\eta _2.\n\\end{array} \\right. \\end{equation}\n\nIn order to distinguish the standard perturbation systems\ndefined by (\\ref{perturbationsystemofnthKdV}),\nthe integrable couplings of the $n$-th order KdV equation $u_t=K_n$, defined by\n(\\ref{nonstandardperturbationsystemofnthKdV}) and (\\ref{biggerK_npe}),\nare called the non-standard perturbation systems.\nInterestingly, each of these systems has both a local multi-Hamiltonian \nformulation and two classes of hereditary recursion operators.\n\n\\subsection{2+1 dimensional integrable couplings}\n\nThirdly, let us consider a case of bi-scale perturbations\n(\\ref{specificcaseofperturbation}), i.e., \n\\begin{equation}\\hat {u}_N\n=\\sum_{i=0}^N\\varepsilon ^i \\eta _{i}, \\ \\eta _i=\\eta _i(x,y,t), \\ \ny=\\varepsilon x. \\nonumber\n%\\label{twoscaleperturbationseries} \n\\end{equation} \nIn order to present explicit results for integrable couplings, \nwe take the KdV equation\n(\\ref{KdVeq}) as an illustrative example, due to its simplicity.\nWe recall that the KdV equation (\\ref{KdVeq})\nhas the bi-Hamiltonian formulation (\\ref{biHamiltonianformulationofKdV})\nand the Lax pair (\\ref{zcrepofKdV}). \n\nLet us introduce the bi-scale perturbation series above \n%(\\ref{specificcaseofperturbation})\ninto the KdV equation (\\ref{KdVeq}) and equate powers of $\\varepsilon $. \nAs a $N$-th order approximation, \nwe obtain a $2+1$ dimensional perturbation systems of evolution equations\n\\begin{equation} \n\\left \\{\n\\begin{array}{l}\n\\eta _{0t_1}=\\eta _{0xxx}+6\\eta _0\\eta _{0x},\\\\\n\\eta _{1t_1}=\\eta _{1xxx}+3\\eta _{0xxy}+6(\\eta _0\\eta _{1})_x\n+6\\eta _0\\eta _{0y},\\\\\n\\eta _{2t_1}=\\eta _{2xxx}+3\\eta _{1xxy}+3\\eta _{0xyy}+6(\\eta _0\n\\eta _{2})_x\n+6\\eta _1\\eta _{1x}+6(\\eta _0\\eta _{1})_y,\\\\\n\\eta _{jt_1}=\\eta _{jxxx}+3\\eta _{j-1,xxy}+3\\eta _{j-2,xyy}\n+\\eta _{j-3,yyy}\n\\\\ \\ \\qquad +6\\Bigl(\\sum_{i=0}^j\\eta _i\\eta _{j-i,x}+\\sum_{i=0}^{j-1}\n\\eta _i\\eta _{j-i-1,y}\n\\Bigr),\\ 3\\le j\\le N.\\end{array}\\right.\n\\label{2+1KdVperturbationsystem}\\end{equation} \nThis system has been already presented in \\cite{MaF-PLA1996}.\nIt follows from our general theory that it gives an integrable coupling\nof the KdV equation (\\ref{KdVeq}).\n\nIn what follows,\nwe would like to propose a bi-Hamiltonian formulation and the consequent\nhereditary recursion operator for the system\n(\\ref{2+1KdVperturbationsystem}). To the end,\nwe first need to compute a perturbation Hamiltonian pair by Theorem\n\\ref{thm:perturbationHamiltonianoperator}:\n\\begin{eqnarray} \n&& \\hat{J}_N = \\left [ \\begin{array} {ccccc} 0 & & & & \\part _x \\\\\n& & & \\part _x &\\part _y \\\\\n & & \\part _x &\\part _y &0 \\\\\n& \\begin{turn}{45}\\vdots\\end{turn} \n& \\begin{turn}{45}\\vdots\\end{turn} &\n \\begin{turn}{45}\\vdots\\end{turn}& \\vdots \\\\\n\\part _x & \\part _y & 0 & \\cdots & 0\n \\end{array} \\right], \\label{2+1firstHamiltonianoperator} \\\\\n&& \\hat{M}_N = \\left[ \\begin{array} {ccccc} \n0 & & & & P(\\varepsilon )\|_{\\varepsilon =0}\\vspace{2mm} \\\\\n & & & P(\\varepsilon )\|_{\\varepsilon =0}&\\frac {1}{1!}\\left.\n\\frac {\\part P(\\varepsilon )}{\\part \\varepsilon }\\right.\\Bigl.\\Bigr\n\|_{\\varepsilon =0}\\vspace{2mm} \\\\\n& & P(\\varepsilon )\|_{\\varepsilon =0}&\\frac {1}{1!}\\left.\n\\frac {\\part P(\\varepsilon )}{\\part \\varepsilon }\\right.\\Bigl.\\Bigr\n\|_{\\varepsilon =0}&\n\\frac {1}{2!}\\left.\n\\frac {\\part ^2P(\\varepsilon )}{\\part \\varepsilon ^2}\n\\right.\\Bigl.\\Bigr \|_{\\varepsilon =0}\\vspace{2mm} \\\\\n&\\begin{turn}{45}\\vdots\\end{turn}\n& \\begin{turn}{45}\\vdots\\end{turn} &\\begin{turn}{45}\\vdots\\end{turn}\n &\\vdots \\vspace{2mm} \\\\\n P(\\varepsilon )\|_{\\varepsilon =0}&\n \\frac {1}{1!}\\left.\n\\frac {\\part P(\\varepsilon )}{\\part \\varepsilon }\\right.\\Bigl.\\Bigr\n\|_{\\varepsilon =0}\n& \\frac {1}{2!}\\left.\n\\frac {\\part ^2P(\\varepsilon )}{\\part \\varepsilon ^2}\\right.\\Bigl.\\Bigr\n\|_{\\varepsilon =0}\n & \\cdots & \\frac {1}{N!}\\left.\\frac {\\part ^N P(\\varepsilon )}\n{\\part \\varepsilon ^N}\\right.\\Bigl.\\Bigr \|_{\\varepsilon =0} \n \\end{array} \\right) ,\\qquad\\quad\n \\label{2+1secondHamiltonianoperator}\n \\end{eqnarray} \nwhere the differential operator $P(\\varepsilon )$ represents \n\\[P(\\varepsilon )=(\\part _x+\\varepsilon \\part _y)^3 +2[(\n\\part _x+\\varepsilon \\part _y)\\hat {u}_N+\\hat {u}_N(\\part _x\n+\\varepsilon \\part _y)].\\]\nThe explicit expressions for various derivatives\nof $P(\\varepsilon )$ with respect to $\\varepsilon $ \ncan be obtained as follows:\n\\begin {equation}\\left \\{\\begin{array}{l}\nP(\\varepsilon )\|_{\\varepsilon =0} =\\part _x ^3+2(\\part _x\\eta _0+\\eta _0\n\\part _x), \\vspace{2mm}\\\\ \n \\frac {1}{1!}\\left. \n\\frac {\\part P(\\varepsilon )} {\\part \\varepsilon }\\right.\\Bigl.\\Bigr\n\|_{\\varepsilon =0}\n=3\\part _x ^2\\part _y +2(\\part _x\\eta _1+\\eta _1\\part _x)+\n2(\\part _y\\eta _0+\\eta _0\\part _y), \\vspace{2mm}\\\\ \n \\frac {1}{2!}\\left.\\frac {\\part ^2P(\\varepsilon )}\n{\\part \\varepsilon ^2}\\right.\\Bigl.\\Bigr \|_{\\varepsilon =0}\n=3\\part _x \\part _y ^2+2(\\part _x\\eta _2+\\eta _2\n\\part _x) +2(\\part _y\\eta _1+\\eta _1\\part _y), \\vspace{2mm}\\\\ \n \\frac {1}{3!}\\left . \\frac {\\part ^3 \nP(\\varepsilon )}{\\part \\varepsilon ^3}\\right.\\Bigl.\\Bigr \|_{\\varepsilon =0}\n=\\part _y ^3 +2(\\part _x\\eta _3+\\eta _3\n\\part _x) +2(\\part _y\\eta _2+\\eta _2\\part _y), \\vspace{2mm}\\\\ \n \\frac {1}{i!}\n\\left. \\frac {\\part ^iP (\\varepsilon )}\n{\\part \\varepsilon ^i}\\right.\\Bigl.\\Bigr \|_{\\varepsilon =0}\n=2(\\part _x\\eta _i+\\eta _i \\part _x) +2(\\part _y\\eta _{i-1}\n+\\eta _{i-1}\\part _y), \\ 4\\le i\\le N,\n\\end{array}\\right. \\end{equation} \nwhich gives rise to an explicit expression for the Hamiltonian operator\n$\\hat{M}_N$. Secondly, we need to compute the Hamiltonian functionals for \nthe system (\\ref{2+1KdVperturbationsystem}). Note that\n$\\part _x \\to \\part _x+\\varepsilon \\part _y$, and thus,\nunder the perturbation (\\ref{specificcaseofperturbation}), we have \n\\[u_{xx}\\to \\sum_{i=0}^N \\varepsilon ^i(\\eta _{ixx}+2\\varepsilon \\eta _{ixy}\n+\\varepsilon ^2\\eta _{iyy}).\\]\nFurther, by Theorem \\ref{thm:perturbationbi-Hamiltonianformulation},\nwe obtain two perturbation Hamiltonian functionals:\n\\begin{eqnarray} \\textrm{per}_N\\tilde{H}_{0}& =&\n\\iint \\frac {1}{N!} \\frac {\\part ^N {H}_0(\\hat {u}_N)}\n{\\part \\varepsilon ^N}\\Bigl.\\Bigr \|_{\\varepsilon =0} \\,dxdy\n=\\iint \\frac12 \\sum_{i=0}^N\\eta _i \\eta _{N-i}\\, dxdy,\n\\label{2+1firstHamiltonianfunctional} \\\\ \n \\textrm{per}_N \\tilde {{H}}_{1}&=&\n \\iint\\frac {1}{N!}\\frac {\\part ^N {H}_1(\\hat {u}_N)}\n {\\part \\varepsilon ^N} \\Bigl.\\Bigr \|_{\\varepsilon =0} \\,dxdy\n =\\iint \\bigl[\\frac12 \\sum_{i+j=N}\\eta _i \\eta _{jxx} \\bigr .\\nonumber \\\\\n && \\bigl. +\\sum_{i+j=N-1}\\eta _i\\eta _{jxy}+\n\\frac12 \\sum_{i+j=N-2}\\eta _i\\eta _{jyy}\n+\\sum_{i+j+k=N}\\eta _i\\eta _j\\eta _k \\bigr ]\\, dxdy.\n\\label{2+1secondHamiltonianfunctional}\n\\end{eqnarray} \nNow the following bi-Hamiltonian formulation for the system\n(\\ref{2+1KdVperturbationsystem}) becomes clear:\n\\begin{equation} \\hat {\\eta }_{Nt}\n=\\hat {J}_N \\frac {\\delta (\\textrm{per}_N\\tilde{H}_1)}{\\delta \\hat{\\eta }_N}\n=\\hat {M}_N \\frac {\\delta (\\textrm{per}_N\\tilde{H}_0)}{\\delta \\hat{\\eta }_N},\n \\label{2+1bi-Hamiltoniansystem}\\end{equation} \nwhere $\\hat {J}_N,\\hat {M}_N,\\textrm{per}_N\\tilde{H}_0$ and $\n\\textrm{per}_N\\tilde{H}_1$ are defined by\n(\\ref{2+1firstHamiltonianoperator}), (\\ref{2+1secondHamiltonianoperator}),\n(\\ref{2+1firstHamiltonianfunctional}) and\n(\\ref{2+1secondHamiltonianfunctional}), respectively.\nIt should be realized that the $2+1$ dimensional bi-Hamiltonian system\n(\\ref{2+1bi-Hamiltoniansystem}) is local, because the Hamiltonian \npair $\\{\\hat{J}_N,\\hat{M}_N\\}$ involves \nonly the differential operators $\\part _x$ and $\\part _y$.\n\nTheorem \\ref{thm:perturbationrecursionoperator} guarantees the existence\nof a hereditary recursion operator for the system\n(\\ref{2+1KdVperturbationsystem}).\nIt is of interest to get its explicit expression.\nNote that the first Hamiltonian operator $\\hat {J}_N$ has an invertible\noperator\n\\begin{equation} (\\hat {J}_N)^{-1}=\\left[ \\begin{array} {cccc} P_N &P_{N-1}& \\cdots\n& P_0 \\vspace{2mm}\\\\\nP_{N-1}& &\\begin{turn}{45}\\vdots\\end{turn} & \\vspace{2mm}\\\\\n\\vdots &\\begin{turn}{45}\\vdots\\end{turn} & & \\vspace{2mm}\\\\\nP_0& & &0 \\end{array} \\right], \\end{equation} \nwhere the operators $P_i$ are defined by\n\\begin{equation} \nP_0=\\part _x^{-1},\\ P_{1}=-\\part _x^{-2}\\part _y,\\ \\cdots,\\ P_i=\n(-1)^{i}\\part _x^{-i-1}\\part _y^{i} ,\\ \\cdots, \\\nP_N=(-1)^N\\part _x^{-N-1}\\part _y ^N.\n\\end{equation}\nTherefore, the corresponding hereditary recursion operator is determined \nby $\\hat {\\Phi }_N=\\hat {M}_N\\hat {J}_N^{-1}$, but it can also be computed \ndirectly by Theorem \\ref{thm:perturbationhereditaryoperator}:\n\\begin{equation}\n\\hat{\\Phi }_N = \\left[ \\begin{array} {cccc} \n\\Phi(\\hat{u}_N)\|_{\\varepsilon =0}\n & & & 0 \\\\\n\\frac 1 {1!}\\frac {\\part \\Phi(\\hat{u}_N)}{\\part \\varepsilon}\n\\Bigl.\\Bigr\|_{\\varepsilon =0} &\\Phi(\\hat{u}_N)\|_{\\varepsilon =0} & & \\\\\n\\vdots & \\ddots & \\ddots & \\\\\n\\frac 1 {N!}\\frac {\\part ^N\\Phi(\\hat{u}_N)}{\\part \\varepsilon ^N}\n\\Bigl.\\Bigr\|_{\\varepsilon =0} \n &\\cdots & \\frac 1 {1!}\\frac {\\part \\Phi(\\hat{u}_N)}{\\part \\varepsilon}\n\\Bigl.\\Bigr\|_{\\varepsilon =0} & \\Phi(\\hat{u}_N)\|_{\\varepsilon =0} \\ea\n\\right ]. \\label{2+1dimensionalrecursionoperator}\n\\end{equation}\nHere the operator $\\Phi(\\hat{u}_N)$ is defined by\n\\[\\Phi(\\hat{u}_N)= (\\part _x+\\varepsilon \\part _y)^2+2(\\hat {u}_{Nx}+\n\\varepsilon \\hat {u}_{Ny})(\\part _x+\\varepsilon \\part _y)^{-1} +\n4\\hat {u}_N,\\]\nand thus its $N+1$ derivatives with respect to $\\varepsilon $ are found to be \n\\begin{equation} \\left\\{ \\begin{array} {l} \n\\Phi (\\hat {u}_N) \\bigl.\\bigr\|_{\\varepsilon =0}=\n\\part _x ^2+2\\eta _{0x}\\part _x^{-1}+4\\eta _0,\n\\vspace{2mm} \\\\ \n\\frac 1 {1!}\\frac {\\part \\Phi (\\hat {u}_N) }{\\part\n\\varepsilon } \\Bigl.\\Bigr\|_{\\varepsilon =0}=\n2\\part _x\\part _y+2(\\eta _{1x}+\\eta _{0y})\\part _x^{-1}\n-2\\eta _{0x}\\part _x^{-2}\\part _y+4\\eta _1,\n\\vspace{2mm} \\\\ \n\\frac 1 {2!}\\frac {\\part ^2 \\Phi (\\hat {u}_N) }{\\part\n\\varepsilon ^2} \\Bigl.\\Bigr\|_{\\varepsilon =0}=\n\\part _y^2+2(\\eta _{2x}+\\eta _{1y})\\part _x^{-1}\n-2(\\eta _{1x}+\\eta _{0y})\\part _x^{-2}\\part _y\n+2\\eta _{0x}\\part _x^{-3}\\part _y^2+4\\eta _2,\n\\vspace{2mm} \\\\ \n\\frac 1 {k!}\\frac {\\part ^k\\Phi (\\hat {u}_N) }{\\part\n\\varepsilon ^k} \\Bigl.\\Bigr\|_{\\varepsilon =0}=\n\\sum_{i+j=k}(-1)^j(2\\eta _{ix}+\\eta _{i-1,y})\n\\part _x^{-j-1}\\part _y^j+4\\eta_k,\n\\ 3\\le k\\le N,\\end{array}\\right. \n\\label{concreteexpressionforderivativeofhatPhiN}\n\\end{equation} \nwhere we accept $\\eta _{-1}=0$.\n\nLet us now show the corresponding zero curvature representation\nfor the $2+1$ dimensional perturbation \nsystem (\\ref{2+1KdVperturbationsystem}). By Theorem \\ref{thm:zcrepofpe}\nor (\\ref{specificperturbationzcrep}),\nthe zero curvature representation for the system \n(\\ref{2+1KdVperturbationsystem}) can be given by \n\\be\n\\hat {U}_{Nt}-\\hat {V}_{Nx}-\\Pi \\hat{V}_{Ny}+[\\hat {U}_N,\\hat {V}_N]=0,\n\\label{zcrepof2+1perturbationsystem} \\end{equation} \nwhere three matrices $\\Pi $, \n$\\hat {U}_N$ and $\\hat {V}_N$ read as \n\\begin{eqnarray} &&\n\\Pi =\\left [\\begin{array} {cc} 0&0\\vspace{2mm}\\\\\nI_{2N} &0 \\end{array} \\right ]_{2(N+1)\\times 2(N+1)},\\ \nI_{2N}=\\textrm{diag}(\\underbrace{I_2,\\cdots ,I_2}_{N})\n=\\textrm{diag}(\\underbrace{1,\\cdots ,1}_{2N}),\n\\\\ &&\n\\hat{U}_N=\\left[\\begin{array} {cccc} U_0 & & & 0 \\vspace{2mm}\\\\\nU_1 & U_0 & & \\vspace{2mm}\\\\ \n\\vdots &\\ddots &\\ddots & \\vspace{2mm}\\\\\nU_N &\\cdots &U_1&U_0 \\end{array} \\right],\n\\ \\hat{V}_N=\\left[\\begin{array} {cccc} V_0 & & & 0 \\vspace{2mm}\\\\\nV_1 & V_0 & & \\vspace{2mm}\\\\ \n\\vdots &\\ddots &\\ddots & \\vspace{2mm}\\\\\nV_N &\\cdots &V_1&V_0 \\end{array} \\right],\n\\end{eqnarray} \nwith the $U_i,V_i$ being determined by\n\\begin{eqnarray} && U_i\n =\\left.\\frac 1{i!}\\frac {\\part ^i U(\\hat {u}_N,\\hat {\\lambda}_N)}\n{\\part \\varepsilon ^i} \\right\|_{\\varepsilon=0}\n =\\left[\\begin{array} {cc} 0&- \\eta _i-\\mu _i\\vspace{2mm} \\\\\n\\delta _{i0} &0 \\end{array} \\right],\n \\ 0\\le i\\le N,\n\\\\ && V_i\n =\\left.\\frac 1{i!}\\frac {\\part ^i V(\\hat {u}_N,\\hat {\\lambda}_N)}\n{\\part \\varepsilon ^i} \\right\|_{\\varepsilon=0}\n =\\left[\\begin{array} {cc} \\eta _{ix}+\\eta _{i-1,y}&\nQ_i\n\\vspace{2mm} \\\\ \n 2\\eta _i-4\\mu _i & -\\eta _{ix}-\\eta _{i-1,y} \\end{array} \\right],\n \\ 0\\le i\\le N,\\\\\n&& Q_i=\n- \\eta _{ixx}-2\\eta _{i-1,xy}-\\eta _{i-2,yy}-2\\sum_{k+l=i}(\\eta _k\\eta _l\n-\\mu _k\\eta _l-2\\mu _k\\mu _l), \\ 0\\le i\\le N,\\quad \n\\end{eqnarray} \nwhere we accept that $\\eta _{-1}=\\eta _{-2}=0$, and $U,V$ are defined by\n(\\ref{zcrepofKdV}).\nOf course, we require the condition (\\ref{conditionsofspectralparameters}) \non the involved spectral parameters \n$\\mu _i$, $0\\le i\\le N$,\n%\\[ \\mu _{0x}=0,\\ \\mu _{ix}+\\mu _{i-1,y}=0,\\ 1\\le i\\le N, \\]\nin order to guarantee the equivalence between the system\n(\\ref{2+1KdVperturbationsystem}) and the zero curvature equation\n(\\ref{zcrepof2+1perturbationsystem}). \n\nIn particular, the first-order bi-scale perturbation system\n\\begin{equation}\\left \\{\\begin{array}{l}\n\\eta _{0t}=\\eta _{0xxx}+6\\eta _0\\eta _{0x},\\vspace{2mm}\\\\\n\\eta _{1t}=\\eta _{1xxx}+3\\eta _{0xxy}+6(\\eta _0\\eta _{1})_x\n+6\\eta _0\\eta _{0y},\n\\end{array}\\right. \\label{2+1firstorderperturbationsystemofKdV}\n\\end{equation}\nhas a local $2+1$ dimensional bi-Hamiltonian formulation \n\\begin{eqnarray} && \\hat{\\eta }_{1t}\n =\\hat {J}_1\\frac {\\delta (\\textrm{per}_1\\tilde{H}_{1})}\n{\\delta \\hat {\\eta }_1}=\n\\hat {M}_1\\frac {\\delta (\\textrm{per}_1\\tilde{H}_{0})}\n{\\delta \\Hat{ {\\eta} }_1},\\\n\\hat{\\eta }_1= \\left[\\begin{array} {c} \\eta _{0}\\vspace{2mm} \\\\ \\eta _{1}\n\\end{array} \\right],\n \\\\ &&\n\\hat {J}_1= \\left[\\begin{array} {cc} 0 & \\part _x \\vspace{2mm} \\\\ \n\\part _x &\\part _y \\end{array} \\right],\\ \n\\hat {M}_1= \\left[\\begin{array}\n{cc} 0&\\part _x^3 +2\\eta _{0x}+4\\eta _0\\part _x\n\\vspace{2mm} \\\\ \\part _x^3 +2\\eta _{0x}+4\\eta _0\\part _x &\n3\\part _x^2\\part _y +2\\eta _{1x}+2\\eta _{0y}+4\\eta _1\\part _x\n+4\\eta _0\\part _y\n\\end{array} \\right],\\qquad \\quad\\\\ &&\n\\textrm{per}_1\\tilde{H}_{0}=\\iint \\eta _0\\eta _1\\,dxdy,\\\n\\textrm{per}_1\\tilde{H}_{1}=\\iint (\\frac 12\n\\eta _0\\eta _{1xx}+\\eta _0\\eta _{0xy}+\\frac 12 \\eta _1\\eta _{0xx}\n+3\\eta _0^2\\eta _1)\\,dxdy.\n \\end{eqnarray} \nHere the extended variables $\\eta _0(x,y,t) $ and $\n\\eta _1(x,y,t)$ are taken as a potential vector $\\hat {\\eta }_1$.\nMoreover the above Hamiltonian pair yields\na hereditary recursion operator in $2+1$ dimensions\n\\begin{equation}\\hat {\\Phi }_{1}(\\hat {\\eta }_1) = \n\\left[\\begin{array} {cc} \\part _x^2 +2\\eta _{0x}\\part _x^{-1} +4\\eta _0& 0\n\\vspace{2mm} \\\\\n2\\part _x\\part _y-2\\eta _{0x}\\part _x^{-2}\\part _y\n+2(\\eta _{1x}+\\eta _{0y})\\part _x^{-1}+4\\eta _1\n& \\part _x^2 +2\\eta _{0x}\\part _x^{-1} +4\\eta _0\n \\end{array} \\right].\n\\end{equation} \nThe system (\\ref{2+1firstorderperturbationsystemofKdV}) was\nfurnished in \\cite{MaF-PLA1996},\nits Painlev\\'e property and zero curvature representation\nwere discussed by Sakovich \\cite{Sakovich-JNMP1998}, and its localized\nsoliton-like solutions were found in \\cite{ZhouM-preprint}.\nAll these properties show that the system \n(\\ref{2+1firstorderperturbationsystemofKdV}) is a good example of typical \nsoliton equations in $2+1$ dimensions.\n\n\\section{Concluding remarks}\n\\setcounter{equation}{0}\n\n\nWe have developed a theory for constructing integrable couplings \nof soliton equations by perturbations. The symmetry problem is viewed as a \nspecial case of integrable couplings.\nThe general structures of hereditary recursion operators, Hamiltonian\noperators, symplectic operators, Hamiltonian formulations etc.\nhave been established under the multi-scale perturbations.\nThe perturbation systems have richer structures\nof Lax representations and zero curvature\nrepresentations than the original systems.\nFor example, in the higher dimensional\ncases, the involved spectral parameters $\\mu _i,\\ 0\\le i\\le N,$\nmay vary with respect to the spatial variables,\nbut they need to satisfy some conditions, for example,\n \\[ \\mu_{0x}=0,\\ \\mu _{ix}+\\mu _{i-1,y}=0,\\ 1\\le i\\le N,\\] \nin the $2+1$ dimensional case of the perturbation\n\\[ \\hat {u}_n=\\sum_{i=0}^N \\varepsilon ^i\\eta_i (x,y,t)=\n \\sum_{i=0}^N \\varepsilon ^i\\eta _i(x,\\varepsilon x,t),\\ x\\in\\R .\n \\]\nThe resulting theory has been applied to the KdV soliton hierarchy\nand thus various integrable couplings are presented for\neach soliton equation in the KdV hierarchy.\nThe obtained integrable couplings of the original KdV equations \nhave infinitely many commuting symmetries and conserved densities. \nLinear combinations of the KdV hierarchy containing a small\nperturbation parameter may yield much more interesting integrable couplings.\nFor example, the KdV type systems of soliton equations \npossessing both multi-Hamiltonian formulations and two classes of\nhereditary recursion operators have been presented and what's more,\nlocal $2+1$ dimensional bi-Hamiltonian systems of the KdV type with\nhereditary structures have also been constructed. \n\nOur success in extending the standard perturbation cases\nto the non-standard cases and the higher dimensional cases are based on \nthe following two simple ideas. First, we \nchose the perturbed systems as initial systems\nto generate integrable couplings for given integrable systems.\nThe method of construction is similar to that in \n\\cite{MaF-CSF1996}.\nOnly a slight difference is that new initial \nsystems themselves involve a small perturbation parameter, but\nimportantly, such initial perturbed systems take effect \nin getting new integrable couplings.\nIn particular, our result showed that \n the following non-standard perturbation system\n\\begin{equation} \\left \\{ \\begin{array} {l} u_t=K(u),\\\\ v_t=K'(u)[v]+K(u),\n\\end{array} \\right. \\end{equation}\nkeeps complete integrability. Therefore, this\nalso provides us with an extension\nof integrable systems. Secondly, we took the multi-scale perturbations,\nby which higher dimensional integrable couplings can be presented.\nIndeed, the multi-scale perturbations enlarge the spatial dimensions\nand keeps complete integrability of the system under study.\nA concrete example of integrable couplings resulted from the multi-scale \nperturbations is the following system\n\\[ \\left \\{\\begin{array}{l}\n\\eta _{0t}=\\eta _{0xxx}+6\\eta _0\\eta _{0x},\\vspace{2mm}\\\\\n\\eta _{1t}=\\eta _{1xxx}+3\\eta _{0xxy}+6(\\eta _0\\eta _{1})_x\n+6\\eta _0\\eta _{0y},\n\\end{array}\\right. \n\\]\nwhich has been proved to be a local bi-Hamiltonian system.\n\nA kind of reduction of the standard multi-scale\nperturbations defined by (\\ref{multiplescaleperturbationseries})\nmay be taken, which can be generally represented as\n\\[\n\\hat{ u}_N=\\sum_{j=0}^N \\varepsilon ^{i_j}\\eta _j , \\\n\\eta _i=\\eta _i(y_0, y_1, y_2,\\cdots,y_r,t),\\\ny_j=\\varepsilon ^{i_j'} x, \\ t\\in \\R,\\ x\\in \\R ^p,\\ 0\\le i\\le N,\n\\]\nwhere the $i_j,i_j'$ can be any two finite sets of natural numbers.\nThis kind of perturbations can be generated from\nthe standard perturbations (\\ref{multiplescaleperturbationseries}),\nif some dependent variables $\\eta_i$ are chosen to be zero and\nthe other dependent variables are assumed to be independent of some dependent\nvariables $y_i$.\nThey yield more specific integrable couplings.\nThere is another interesting problem related integrable couplings.\nCould one reduce the spatial dimensions of a given integrable system\nwhile formulating integrable couplings? \nIf the answer is yes, it is of interest to find some ways\nto construct such kind of integrable couplings, i.e.,\nto hunt for the second part $S(u,v)$ with $v$ being less dimensional than $u$\nto constitute integrable systems with the original system\n$u_t=K(u)$.\n\nThere exist some important works to deal with\nasymptotic analysis and asymptotic integrability \n\\cite{Wong-book1989,FokasL-PRL1996,Kodama-book1997,DegasperisP-preprint1998},\nto which the study of the perturbation systems may be helpful.\nIt is also worthy mentioning that our $2+1$ dimensional\nhereditary recursion operators,\nfor example, the operators defined by (\\ref{2+1dimensionalrecursionoperator})\nand (\\ref{concreteexpressionforderivativeofhatPhiN}), \nare of the form described only by independent variables involved.\nThus they are a supplement to \na class of recursion operators in $2+1$ dimensions\ndiscussed by Zakharov and Konopelchenko \\cite{ZakharovK-CMP1984},\nand a class of the extended recursion operators in $2+1$ dimensions\nincluding additional independent variables, introduced\nby Santini and Fokas \\cite{SantiniF-CMP1988} and Fokas and Santini\n\\cite{FokasS-CMP1988}. The other properties such \nas B\\\"acklund transformations, bilinear forms and soliton solutions\nmight be found for the resulting perturbation systems.\nA remarkable Miura transformation \\cite{Miura-JMP1968} \nmight also be introduced for the \n$2+1$ dimensional perturbation systems (\\ref{2+1KdVperturbationsystem}),\nwhich will lead to new $2+1$ dimensional integrable systems of the MKdV type.\nAll these problems will be analyzed in a further publication.\n\n\\vskip 1mm\n\\noindent{\\bf Acknowledgments:} \nThe author would like to thank the City University of Hong Kong and the \nResearch Grants Council of Hong Kong\nfor financial support. \n\n\\setlength{\\baselineskip}{15pt}\n\n\\small\n\\begin{thebibliography}{99}\n\n\\bibitem{Fuchssteiner-book1993} B. Fuchssteiner,\n{\\it Coupling of completely integrable systems: the perturbation bundle},\n in: {Applications of Analytic and\n Geometric Methods to Nonlinear Differential Equations},\n (P. A. Clarkson, ed.) Kluwer, Dordrecht, 1993, pp. 125--138.\n\\bibitem{Ma-JPM1992} W. X. Ma, {\\it Lax representations and Lax operator \n algebras of isospectral and nonisospectral hierarchies of evolution \n equations},\n J. Math. Phys. {\\bf 33} (1992), 2464--2476.\n\\bibitem{Ma-JPAPLA199293} W. X. Ma, {\\it An approach for \n constructing nonisospectral hierarchies of evolution equations}, \n J. Phys. A: Math. Gen. {\\bf 25} (1992), L719--L726; \n {\\it A simple scheme for generating nonisospectral flows from zero \n curvature representation}, Phys. Lett. A {\\bf 179} (1993), 179--185.\n\\bibitem{LakshmananT-JMP1985} M. Lakshmanan and K. M. Tamizhmani,\n{\\it Lie-B\\\"acklund symmetries of certain nonlinear evolution equations \n under perturbation around solutions}, \n J. Math. Phys. {\\bf 26} (1985), 1189--1200.\n\\bibitem{MaF-CSF1996} W. X. Ma and B. Fuchssteiner, \n {\\it Integrable theory of the perturbation equations},\n Chaos, Solitons $\\&$ Fractals {\\bf 7} (1996), 1227--1250.\n\\bibitem{Ma-Needs98}W. X. Ma, {\\it Bi-Hamiltonian structures of triangular\n systems by perturbations}, preprint (1998).\n\\bibitem{FuchssteinerF-PD1981} B. Fuchssteiner and A. S. Fokas,\n {\\it Symplectic structures, their B\\\"acklund transformations and \n hereditary symmetries}, Physica D {\\bf 4} (1981), 47--66.\n\\bibitem{Oevel-JMP1988}W. Oevel,\n {\\it Dirac constraints in field theory: lifts of\n Hamiltonian systems to the\n cotangent bundle}, J. Math. Phys. {\\bf 29} (1988), 210--219.\n\\bibitem{Fuchssteiner-NATMA1979}B. Fuchssteiner, {\\it \n Application of hereditary symmetries to nonlinear evolution equations}, \n Nonlinear Analysis TMA {\\bf 3} (1979), 849--862.\n\\bibitem{Fokas-SAM1987} A. S. Fokas, \n {\\it Symmetries and integrability}, \n Studies in Appl. Math. {\\bf 77} (1987), 253--299.\n\\bibitem{Gu-book1995}C. H. Gu, ed.\n {\\it Soliton Theory and Its Applications},\n Springer-Verlag, Berlin, 1995.\n\\bibitem{Olver-book1986} P. J. Olver, {\\it Applications of Lie Groups\n to Differential Equations}, Springer-Verlag, New York, 1986.\n\\bibitem{GelfandD-FAA1979}I. M. Gelfand and I. Ya. Dorfman,\n {\\it Hamiltonian operators and algebraic structures related to them}, \n Func. Anal. Appl. {\\bf 13} (1979), 248--262. \n\\bibitem{Dorfman-book1993} I. Ya. Dorfman, {\\it Dirac Structures and \n Integrability of Nonlinear Evolution Equations}, \n John Wiley \\& Sons, Chichester, 1993.\n\\bibitem{Magri-JMP1978}F. Magri,\n {\\it A simple model of the integrable Hamiltonian\n equation}, J. Math. Phys. {\\bf 19} (1978), 1156--1162.\n\\bibitem{Ma-JPA1992} W. X. Ma, {\\it The algebraic structures of \n isospectral Lax operators and applications to integrable equations},\n J. Phys. A: Math. Gen. {\\bf 25} (1992), 5329--5343.\n\\bibitem{Miura-SIAMR1976}\n R. M. Miura, {\\it The Korteweg-de Vries equation: a survey of results},\n SIAM Rev. {\\bf 18} (1976), %no. 3, \n 412--459. \n\\bibitem{AblowitzS-book1981} M. J. Ablowitz and\n H. Segur, {\\it Solitons and the Inverse Scattering Transform},\n SIAM, Philadelphia, 1981.\n\\bibitem{TamizhmaniL-JPA1983} K. M. Tamizhmani and M. Lakshmanan,\n {\\it Complete integrability of the Korteweg-de Vries\n equation under perturbation around its solution: \n Lie-B\\\"acklund symmetry approach}, \n J. Phys. A: Math. Gen. {\\bf 16} (1983), 3773--3782.\n\\bibitem{MaF-PLA1996} W. X. Ma and B. Fuchssteiner, \n {\\it The bi-Hamiltonian structure of \n the perturbation equations of the KdV hierarchy},\n Phys. Lett. A {\\bf 213} (1996), 49--55.\n\\bibitem{Sakovich-JNMP1998}S. Yu. Sakovich,\n {\\it On integrability of a $(2+1)$-dimensional perturbed KdV equation},\n J. Nonlinear Math. Phys. {\\bf 5} (1998), 230--233.\n\\bibitem{ZhouM-preprint} Z. X. Zhou and W. X. Ma,\n {\\it Darboux transformations for a 2+1 dimensional matrix \n Gelfand-Dickey system}, preprint, 1998.\n\\bibitem{Wong-book1989}R. Wong, {\\it Asymptotic Approximations of Integrals}, \n Computer Science and Scientific Computing, Academic Press, Boston, 1989.\n\\bibitem{FokasL-PRL1996}A. S. Fokas and Q. M. Liu, \n {\\it Asymptotic integrability\n of water waves}, Phys. Rev. Lett. {\\bf 77} (1996), \n %no. 12,\n 2347--2351.\n\\bibitem{Kodama-book1997}Y. Kadama and A. V. Mikhailov,\n {\\it Obstacles to Asymptotic Integrability}, in: Algebraic Aspects of\n Integrable Systems: in memory of Irene Dorfman, \n Progr. Nonlinear Differential Equations Appl. 26 \n (A. S. Fokas\n and I. M. Gelfand. eds.), Birkh\\\"auser, Boston, 1997, pp. 173--204.\n\\bibitem{DegasperisP-preprint1998}A. Degasperis and M. Procesi,\n {\\it A test of asymptotic integrability of $1+1$ wave equations},\n preprint, 1998.\n\\bibitem{ZakharovK-CMP1984}\n V. E. Zakharov and B. G. Konopelchenko, {\\it \n On the theory of recursion operator}, Comm. Math.\n Phys. {\\bf 94} (1984), \n %no. 4, \n 483--509.\n\\bibitem{SantiniF-CMP1988} P. M.\n Santini and A. S. Fokas, {\\it Recursion operators and \n bi-Hamiltonian structures in multidimensions. I.},\n Comm. Math. Phys. {\\bf 115} (1988), %no. 3, \n 375--419. \n\\bibitem{FokasS-CMP1988}\n A. S. Fokas and P. M. Santini, \n {\\it Recursion operators and bi-Hamiltonian structures \n in multidimensions. II.},\n Comm. Math. Phys. {\\bf 116} (1988), %no. 3, \n 449--474. \n\\bibitem{Miura-JMP1968}R. M. Miura, \n {\\it Korteweg-de Vries equation and generalizations. \n I. A remarkable explicit nonlinear transformation}, \n J. Math. Phys. {\\bf 9} (1968), 1202--1204.\n\n\\end{thebibliography}\n\n\\vskip 5mm\n\n\\noindent \nDEPARTMENT OF MATHEMATICS, CITY UNIVERSITY OF HONG KONG, KOWLOON, HONG KONG\n\n\\noindent Email: mawx@math.cityu.edu.hk\n\n\n\n\\end{document}\n\n\n\n\n" } ]	[ { "name": "solv-int9912004.extracted_bib", "string": "{Fuchssteiner-book1993 B. Fuchssteiner, {Coupling of completely integrable systems: the perturbation bundle, in: {Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, (P. A. Clarkson, ed.) Kluwer, Dordrecht, 1993, pp. 125--138." }, { "name": "solv-int9912004.extracted_bib", "string": "{Ma-JPM1992 W. X. Ma, {Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations, J. Math. Phys. {33 (1992), 2464--2476." }, { "name": "solv-int9912004.extracted_bib", "string": "{Ma-JPAPLA199293 W. X. Ma, {An approach for constructing nonisospectral hierarchies of evolution equations, J. Phys. A: Math. Gen. {25 (1992), L719--L726; {A simple scheme for generating nonisospectral flows from zero curvature representation, Phys. Lett. A {179 (1993), 179--185." }, { "name": "solv-int9912004.extracted_bib", "string": "{LakshmananT-JMP1985 M. Lakshmanan and K. M. Tamizhmani, {Lie-B\\\"acklund symmetries of certain nonlinear evolution equations under perturbation around solutions, J. Math. Phys. {26 (1985), 1189--1200." }, { "name": "solv-int9912004.extracted_bib", "string": "{MaF-CSF1996 W. X. Ma and B. Fuchssteiner, {Integrable theory of the perturbation equations, Chaos, Solitons $\\&$ Fractals {7 (1996), 1227--1250." }, { "name": "solv-int9912004.extracted_bib", "string": "{Ma-Needs98W. X. Ma, {Bi-Hamiltonian structures of triangular systems by perturbations, preprint (1998)." }, { "name": "solv-int9912004.extracted_bib", "string": "{FuchssteinerF-PD1981 B. Fuchssteiner and A. S. Fokas, {Symplectic structures, their B\\\"acklund transformations and hereditary symmetries, Physica D {4 (1981), 47--66." }, { "name": "solv-int9912004.extracted_bib", "string": "{Oevel-JMP1988W. Oevel, {Dirac constraints in field theory: lifts of Hamiltonian systems to the cotangent bundle, J. Math. Phys. {29 (1988), 210--219." }, { "name": "solv-int9912004.extracted_bib", "string": "{Fuchssteiner-NATMA1979B. Fuchssteiner, {Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Analysis TMA {3 (1979), 849--862." }, { "name": "solv-int9912004.extracted_bib", "string": "{Fokas-SAM1987 A. S. Fokas, {Symmetries and integrability, Studies in Appl. Math. {77 (1987), 253--299." }, { "name": "solv-int9912004.extracted_bib", "string": "{Gu-book1995C. H. Gu, ed. {Soliton Theory and Its Applications, Springer-Verlag, Berlin, 1995." }, { "name": "solv-int9912004.extracted_bib", "string": "{Olver-book1986 P. J. Olver, {Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986." }, { "name": "solv-int9912004.extracted_bib", "string": "{GelfandD-FAA1979I. M. Gelfand and I. Ya. Dorfman, {Hamiltonian operators and algebraic structures related to them, Func. Anal. Appl. {13 (1979), 248--262." }, { "name": "solv-int9912004.extracted_bib", "string": "{Dorfman-book1993 I. Ya. Dorfman, {Dirac Structures and Integrability of Nonlinear Evolution Equations, John Wiley \\& Sons, Chichester, 1993." }, { "name": "solv-int9912004.extracted_bib", "string": "{Magri-JMP1978F. Magri, {A simple model of the integrable Hamiltonian equation, J. Math. Phys. {19 (1978), 1156--1162." }, { "name": "solv-int9912004.extracted_bib", "string": "{Ma-JPA1992 W. X. Ma, {The algebraic structures of isospectral Lax operators and applications to integrable equations, J. Phys. A: Math. Gen. {25 (1992), 5329--5343." }, { "name": "solv-int9912004.extracted_bib", "string": "{Miura-SIAMR1976 R. M. Miura, {The Korteweg-de Vries equation: a survey of results, SIAM Rev. {18 (1976), %no. 3, 412--459." }, { "name": "solv-int9912004.extracted_bib", "string": "{AblowitzS-book1981 M. J. Ablowitz and H. Segur, {Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981." }, { "name": "solv-int9912004.extracted_bib", "string": "{TamizhmaniL-JPA1983 K. M. Tamizhmani and M. Lakshmanan, {Complete integrability of the Korteweg-de Vries equation under perturbation around its solution: Lie-B\\\"acklund symmetry approach, J. Phys. A: Math. Gen. {16 (1983), 3773--3782." }, { "name": "solv-int9912004.extracted_bib", "string": "{MaF-PLA1996 W. X. Ma and B. Fuchssteiner, {The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy, Phys. Lett. A {213 (1996), 49--55." }, { "name": "solv-int9912004.extracted_bib", "string": "{Sakovich-JNMP1998S. Yu. Sakovich, {On integrability of a $(2+1)$-dimensional perturbed KdV equation, J. Nonlinear Math. Phys. {5 (1998), 230--233." }, { "name": "solv-int9912004.extracted_bib", "string": "{ZhouM-preprint Z. X. Zhou and W. X. Ma, {Darboux transformations for a 2+1 dimensional matrix Gelfand-Dickey system, preprint, 1998." }, { "name": "solv-int9912004.extracted_bib", "string": "{Wong-book1989R. Wong, {Asymptotic Approximations of Integrals, Computer Science and Scientific Computing, Academic Press, Boston, 1989." }, { "name": "solv-int9912004.extracted_bib", "string": "{FokasL-PRL1996A. S. Fokas and Q. M. Liu, {Asymptotic integrability of water waves, Phys. Rev. Lett. {77 (1996), %no. 12, 2347--2351." }, { "name": "solv-int9912004.extracted_bib", "string": "{Kodama-book1997Y. Kadama and A. V. Mikhailov, {Obstacles to Asymptotic Integrability, in: Algebraic Aspects of Integrable Systems: in memory of Irene Dorfman, Progr. Nonlinear Differential Equations Appl. 26 (A. S. Fokas and I. M. Gelfand. eds.), Birkh\\\"auser, Boston, 1997, pp. 173--204." }, { "name": "solv-int9912004.extracted_bib", "string": "{DegasperisP-preprint1998A. Degasperis and M. Procesi, {A test of asymptotic integrability of $1+1$ wave equations, preprint, 1998." }, { "name": "solv-int9912004.extracted_bib", "string": "{ZakharovK-CMP1984 V. E. Zakharov and B. G. Konopelchenko, {On the theory of recursion operator, Comm. Math. Phys. {94 (1984), %no. 4, 483--509." }, { "name": "solv-int9912004.extracted_bib", "string": "{SantiniF-CMP1988 P. M. Santini and A. S. Fokas, {Recursion operators and bi-Hamiltonian structures in multidimensions. I., Comm. Math. Phys. {115 (1988), %no. 3, 375--419." }, { "name": "solv-int9912004.extracted_bib", "string": "{FokasS-CMP1988 A. S. Fokas and P. M. Santini, {Recursion operators and bi-Hamiltonian structures in multidimensions. II., Comm. Math. Phys. {116 (1988), %no. 3, 449--474." }, { "name": "solv-int9912004.extracted_bib", "string": "{Miura-JMP1968R. M. Miura, {Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. {9 (1968), 1202--1204." } ]
solv-int9912005	Generalized KP hierarchy:\\ M\"obius Symmetry, Symmetry Constraints \\and Calogero-Moser System	[ { "author": "L.V. Bogdanov\\thanks{ L.D. Landau ITP" }, { "author": "Kosygin str. 2" }, { "author": "Moscow 117940" }, { "author": "Russia; e-mail Leonid@landau.ac.ru" } ]		[ { "name": "solv-int9912005.tex", "string": "\\documentstyle[12pt]{article}\n\\newcommand{\\nn}{\\nonumber}\n\\def\\bbox#1{{\\bf #1}}\n%\\def\\bbox#1{{\\mathbf{#1}}}\n\\def\\text#1{{\\rm #1}}\n\\newcommand{\\wt}{\\widetilde}\n\\newcommand{\\D}{{\\underline{\\Delta}}}\n\n% A useful Journal macro\n\\def\\Journal#1#2#3#4{{#1} {#2} (#4) #3}\n\\def\\Book#1#2#3#4{{#1}, #2 (#3, #4)}\n\\def\\Proc#1#2#3#4#5#6{{#1}, in: #3 (#5, #6) p.~#4}\n\n% Some useful journal names\n\\def\\NCA{Nuovo Cimento}\n\\def\\NIM{Nucl. Instrum. Methods}\n\\def\\NIMA{{Nucl. Instrum. Methods} A}\n\\def\\NPB{{Nucl. Phys.} B}\n\\def\\PLB{{Phys. Lett.} B}\n\\def\\PRL{Phys. Rev. Lett.}\n\\def\\PRD{{Phys. Rev.} D}\n\\def\\ZPC{{Z. Phys.} C}\n\\def\\PLA{{Phys. Lett.} A}\n\\def\\JMP{J. Math. Phys.}\n\\def\\LOMI{Zap. Nauchn. Semin. LOMI}\n\\def\\FAP{Funkt. Anal. Ego Pril.}\n\\def\\PHD{Physica D}\n\\def\\LMP{Lett. Math. Phys.}\n% Some other macros used in the sample text\n\\def\\st{\\scriptstyle}\n\\def\\sst{\\scriptscriptstyle}\n\\def\\mco{\\multicolumn}\n\\def\\epp{\\epsilon^{\\prime}}\n\\def\\vep{\\varepsilon}\n\\def\\ra{\\rightarrow}\n\\def\\ppg{\\pi^+\\pi^-\\gamma}\n\\def\\vp{{\\bf p}}\n\\def\\ko{K^0}\n\\def\\kb{\\bar{K^0}}\n\\def\\al{\\alpha}\n\\def\\ab{\\bar{\\alpha}}\n\\def\\be{\\begin{equation}}\n\\def\\ee{\\end{equation}}\n\\def\\bea{\\begin{eqnarray}}\n\\def\\eea{\\end{eqnarray}}\n\\def\\CPbar{\\hbox{{\\rm CP}\\hskip-1.80em{/}}}%temp replacemt due to no font\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%BEGINNING OF TEXT\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{document}\n\n\\title{Generalized KP hierarchy:\\\\\nM\\\"obius Symmetry, Symmetry Constraints\n\\\\and Calogero-Moser System}\n\n\\author{L.V. Bogdanov\\thanks{\nL.D. Landau ITP, Kosygin str. 2,\nMoscow 117940, Russia; e-mail Leonid@landau.ac.ru}\n\\hspace{0.1em} and B.G. Konopelchenko\n\\thanks{\nDipartimento di Fisica dell' Universit\\`a\nand Sezione INFN, 73100 Lecce, Italy}}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% You may repeat \\author \\address as often as necessary %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\maketitle\n\\abstract{Analytic-bilinear approach is used to study\ncontinuous and discrete\nnon-isospectral symmetries of the generalized KP\nhierarchy.\nIt is shown that M\\\"obius symmetry transformation\nfor the singular manifold equation leads to continuous or discrete\nnon-isospectral symmetry of the basic (scalar or multicomponent KP)\nhierarchy connected with binary B\\\"acklund transformation. A more\ngeneral class of multicomponent M\\\"obius-type symmetries is studied.\nIt is demonstrated that symmetry constraints of KP hierarchy\ndefined using multicomponent M\\\"obius-type symmetries give rise\nto Calogero-Moser system.}\n\n\\section{Introduction}\nIt is a pleasure for us to dedicate this paper to Prof.\nV.E. Zakharov 60th birthday. The technique used in this work\n(analytic-bilinear approach, see \\cite{AB1}, \\cite{AB2})\ntakes its origin in the ideas of the $\\bar{\\partial}$-dressing\nmethod developed by V.E. Zakharov and S.V. Manakov \\cite{dbar}.\nIt is interesting to note that the first paper formulating\nbasic ideas of analytic-bilinear approach \\cite{NLS} was\npublished in the volume dedicated to 55th birthday of\nProf. Zakharov.\n\nAnalytic-bilinear approach formalizes important features of the\n$\\bar{\\partial}$-dressing method connected with construction\nof integrable equations, leaving aside some details of the scheme\nof generating special classes of solutions. In this form the method\nbecomes close to Grassmannian approach \\cite{Sato}, \\cite{Wilson},\nthus filling the gap between $\\bar{\\partial}$-dressing method\nand more abstract Grassmannian approach,\npreserving at the same time some useful structures characteristic\nof the original $\\bar{\\partial}$-dressing method.\n\nIn this work we use analytic-bilinear approach to study\ncontinuous and discrete\nnon-isospectral symmetries of the generalized KP\nhierarchy. We demonstrate that M\\\"obius symmetry on the\nlevel of KP singular manifold equations (KPSM) hierarchy,\nbinary B\\\"acklund transformations of KP hierarchy\nand solitonic transformations of the $\\tau$-functions\nthrough the Date-Jimbo-Kashiwara-Miwa vertex operator\nare different manifestations of the same discrete symmetry.\nConsidering continuous non-isospectral symmetries, we show that\nCalogero-Moser system can be obtained through the symmetry constraint\nof KP hierarchy.\n\n\\section{Generalized KP Hierarchy}\nFirst we give a sketch of the picture of generalized KP\nhierarchy in frame of analytic-bilinear approach;\nfor details we refer to \\cite{AB1}, \\cite{AB2}.\n\nThe formal starting point is Hirota bilinear identity\nfor Cauchy-Baker-Akhiezer function,\n\\bea\n\\oint \\chi(\\nu,\\mu;g_1)g_1(\\nu)g_2^{-1}(\\nu)\n\\chi(\\lambda,\\nu;g_2)d\\nu=0\\,\\quad \\lambda\\,,\\mu\\in D.\n\\label{HIROTA0}\n\\eea\nHere $\\chi(\\lambda,\\mu;g)$ (the Cauchy kernel)\nis a function of two complex variables\n$\\lambda ,\\mu\\in \\bar D$, where $D$ is a unit disc,\nand a functional of the loop group\nelement $g\\in\\Gamma^+$, i.e., of a\ncomplex-valued function analytic and having\nno zeros in $\\bbox{C}\\setminus D$, equal to 1 at infinity;\nthe integration goes over the\nunit circle . By definition, the function $\\chi(\\lambda,\\mu)$\npossesses the following analytical properties: as $\\lambda\n\\rightarrow\\mu$, $\\chi\\rightarrow (\\lambda-\\mu)^{-1}$ and\n$\\chi(\\lambda,\\mu)$ is an analytic function of\ntwo variables $\\lambda,\\mu \\in \\bar D$ for $\\lambda\\neq\\mu$.\nThe function $\\chi(\\lambda,\\mu;g)$ is a solution to\n(\\ref{HIROTA0}) if it possesses specified analytic properties\nand satisfies (\\ref{HIROTA0}) for all $\\lambda,\\mu \\in D$ and\nsome class of loops $g\\in\\Gamma^+$.\n\nIn another form, more similar to standard Hirota bilinear identity,\nthe identity (\\ref{HIROTA0}) can be written as\n\\be\n\\oint \\psi(\\lambda,\\nu;g_2)\n\\psi(\\nu,\\mu;g_1)d\\lambda=0 ,\n\\ee\nwhere\n$$\n\\psi(\\lambda,\\mu,g)=g(\\lambda)\\chi(\\lambda,\\mu,g)g^{-1}(\\mu).\n$$\nWe call the function $\\psi(\\lambda,\\mu;g)$ a Cauchy-Baker-Akhiezer\nfunction.\n\nHirota bilinear identity (\\ref{HIROTA0})\nincorporates the standard Hirota bilinear identity\nfor the Baker-Akhiezer (BA) and dual (adjoint) Baker-Akhiezer function\nof the KP hierarchy.\nIndeed, let us introduce\nthese functions by the formulae\n\\bea\n\\psi(\\lambda;g)=g(\\lambda)\\chi(\\lambda;0),\n\\nn\\\\\n\\wt\\psi(\\mu;g)=g^{-1}(\\mu)\\chi(0;\\mu).\n\\nn\n\\eea\nThen for the Baker-Akhiezer function\n$\\psi(\\lambda;g)$ and the dual Baker-Akhiezer\nfunction\n$\\wt\\psi(\\mu;g)$,\ntaking the identity (\\ref{HIROTA0})\nat $\\lambda=\\mu=0$, we get the usual form of the Hirota bilinear identity\n\\be\n\\oint \\wt\\psi(\\nu;g_2)\n\\psi(\\nu,;g_1)d\\nu=0.\n%\\label{HIROTA}\n\\ee\nThe only minor difference from the standard setting here is that\nwe define the BA and dual BA function in the neighborhood of zero,\nnot in the neighborhood of infinity.\n\nThere are three different types of integrable\ndiscrete equations implied by identity (\\ref{HIROTA0}),\nthat, in the continuous limit, lead\nto the KP hierarchy in the usual form (in terms of potentials),\nto the modified KP hierarchy and to the hierarchy of the singular manifold\nequations. They arise for different types of functions\nconnected with the Cauchy-Baker-Akhiezer function\nsatisfying Hirota bilinear\nidentity (see the derivation in \\cite{AB1}, \\cite{AB2}).\n\nOn the first level,\nwe have the equations for the diagonal of the regularized\nCauchy kernel taken at zero (the potential)\n$$\nu(g)=\\left(\\chi(\\lambda,\\mu;g)-\n(\\lambda-\\mu)^{-1}\\right)_{\\lambda=0\\,,\\mu=0}\\,,\n$$\non the second level, the equations for the\nBaker-Akhiezer and dual Baker-Akhiezer type wave\nfunctions (the modified equations)\n$$\n\\Psi(g)=\\int\\psi(\\lambda,g) \\rho(\\lambda)d\\lambda\\,,\n$$\n$$\n\\widetilde \\Psi(g)=\n\\int\\widetilde\\rho(\\mu)\\widetilde\\psi(\\mu,g) d\\mu\\,,\n$$\nand on the third level -- the equations for the\nCauchy-Baker-Akhiezer type wave function\n\\bea\n\\Phi(g)=\\oint\\!\\!\\oint (\\psi(\\lambda,\\mu;g))\n\\rho(\\lambda)\n\\wt\\rho(\\mu) d\\lambda\\,d\\mu,\n\\nn\n\\eea\nwhere $\\rho(\\lambda)$,\n$\\wt\\rho(\\mu)$ are some arbitrary weight functions.\nThe equations of all three levels\npossess an infinite number of commuting symmetries\nand form in some sense a hierarchy of integrable\ndiscrete equations represented in the form of the general equation\nlabeled by three continuous parameters (the lattice parameters).\n\nTo present discrete equations forming three levels\nof generalized KP hierarchy, we introduce difference\nand shift operators\n$$\nT_a f(g)=f(g\\times g_a^{-1}),\n$$\n$$\n\\Delta_a =T_a-1,\n$$\n$$\n\\widetilde T_a f(g)=T_a^{-1} f(g)=f(g\\times g_a),\n$$\n$$\n\\widetilde\\Delta_a =1-\\widetilde T_a,\n$$\nwhere elementary rational loop $g_a(\\nu)$\nis defined as\n$$\ng_a(\\nu):=\n{\\frac{\\nu-a\n}{\\nu}}\n$$\nWe will use shift and difference operators with different values of\nlattice parameter $a=a_i$ denoting\n$$\nT_i=T_{a_i}.\n$$\n\nThe first level of the generalized KP hierarchy is\nformed by equations for the potential\n$u$\n\\bea\n\\sum_{(ijk)} \\epsilon_{ijk}T_k \\left({\\Delta_i\\over a_i} u\n- uT_iu\\right)=0,\n\\label{KPa0}\n\\eea\nwhere $i\\neq j\\neq k\\neq i$; $i,j,k\\in\\{1,2,3\\}$,\nsummation goes over different permutations of indices.\nHirota bilinear identity (\\ref{HIROTA0}) also implies\nlinear equations\n\\begin{eqnarray}\n&&\n\\left({\\Delta_i\\over a_i}-{\\Delta_j\\over a_j} \\right) \\widetilde\\Psi(g)\n=\n%\\nonumber\\\\&&\n\\left((T_i-T_j)u(g)\\right)\\widetilde\\Psi(g)\n\\label{compa}\n\\,\n\\end{eqnarray}\nand\n\\bea\n&&\n\\left({\\widetilde\\Delta_i\\over a_i} -\n{\\widetilde\\Delta_j\\over a_j} \\right)\\Psi(g)\n=\n%\\nonumber\\\\&&\n\\left((\\widetilde T_i-\\widetilde T_j)u(g)\\right)\n\\Psi(g)\n\\label{compb}\n\\,,\n\\end{eqnarray}\nthat generate equation (\\ref{KPa0})\nas compatibility condition\n\nThe second\nlevel of the generalized hierarchy is formed\nby equations for the wave functions\nof linear operators producing equations of the first\nlevel as compatibility conditions (the modified equations).\nIt splits into two\nparts: equations for the dual wave function $\\wt\\Psi$\n\\bea\n\\sum_{(ijk)}\\epsilon_{ijk}a_ja_kT_k\\left(\\widetilde \\Psi^{-1}\n({T_i}\\widetilde \\Psi)\n\\right)=0,\n\\label{KPwavea1}\n\\eea\nand equations for the wave functions $\\Psi$\n\\bea\n\\sum_{(ijk)}\\epsilon_{ijk}a_ja_k\\wt T_k\\left(\\Psi^{-1}\n({\\wt T_i}\\Psi)\n\\right)=0.\n\\label{KPwaveb1}\n\\eea\n\nThe third level represents equations for the\nwave functions $\\Phi$ of linear operators of the second level\n(the singular manifold type equations)\n\\bea\n(T_j\\Delta_i \\Phi)(T_k\\Delta_j \\Phi)(T_i\\Delta_k \\Phi)=\n(T_j\\Delta_k \\Phi)(T_k\\Delta_i \\Phi)(T_i\\Delta_j \\Phi).\n\\label{singmana}\n\\eea\nOne could expect\nthis chain to continue, but the wave functions\nof linear operators of the third level coincide\nwith the wave functions of linear operators of the first level,\nand so the chain closes.\n\nEquations of the second and third levels of the hierarchy and linear\nproblems for them can be derived from the set of simple equations\nthat directly follows from identity (\\ref{HIROTA0}), namely\n\\bea\n{\\Delta_i\\over a_i}\\Phi=\\wt\\Psi T_i\\Psi\n\\label{Phi1}\n\\eea\nand, equivalently,\n\\bea\n{\\wt\\Delta_i\\over a_i}\\Phi=\\Psi \\wt T_i\\wt \\Psi.\n\\label{Phi2}\n\\eea\n\nTo reproduce second and third levels of the generalized KP hierarchy\nfrom the first,\ntaking as a basic object equation (\\ref{KPa0})\nand without reference to bilinear identity,\nit is enough to\nnotice that\nlinear equations\n(\\ref{compa}) and (\\ref{compb}) imply that there exists a\nfunction $\\Phi(\\bbox{x})$ satisfying equations (\\ref{Phi1}),\n(\\ref{Phi2}).\nIndeed, using equations (\\ref{compa}), (\\ref{compb})\nit is easy to check that\ncross-differences for the set of equations (\\ref{Phi1}), (\\ref{Phi2})\nare equal, and the function $\\Phi$ is well-defined on the\nlattice (and through some limit also as a function of continuous\nvariables).\n\nConnections between three different levels of the hierarchy\nof discrete equations may be described in terms of\nMiura maps and Combescure symmetry transformations,\nwhich are in some sense complementary.\n\n\\section{From Discrete Equations to the Continuous Hierarchy}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe loop $g\\in\\Gamma^+$ can be parametrized by the infinite\nset of complex variables $x_i$, $1\\leq i\\leq\\infty$,\n$$\ng(\\lambda)=\\exp(\\sum_{i=1}^{\\infty} x_i\\lambda^{-i}),\n$$\nand then the functionals of $g$ may be considered as\nfunctions of the infinite set of variables\n$$\n\\bbox{x}=\\{x_1,x_2,\\dots,x_n,\\dots\\}.\n$$\nThe transformation operators $T_i$\nnow look like\n%\\begin{equation}\n$T_i:{\\bf x}\\rightarrow{\\bf x}+[a_i]$,\n%\\label{shifts}\n%\\end{equation}\nwhere\n$[a]_n={1\\over n}a^n$. To compactify the notations, we will also\nuse scaled difference operators\n$$\\D_i=a_i^{-1}\\Delta_i,\\quad\n\\wt\\D_i=a_i^{-1}\\wt\\Delta_i.$$\n%the shift vector $[a]$ is characterized by the property\n%$\n%\\sum_{k=1}^{\\infty}[a]_k\\lambda^{-k}=-\\ln\\left(1-{a\\over\\lambda}\\right)$.\nThe transformation operators $T_a$ can be represented in terms\nof differential operators in the form\n$\nT_a=\\sum_{n=1}^{\\infty}a^n p_n(\\widetilde\\partial)\n$,\n$\n\\widetilde T_a=\\sum_{n=1}^{\\infty}a^n p_n(-\\widetilde {\\partial})\n$,\nwhere\n$\n{\\widetilde\\partial} =\\left({\\frac{\\partial}{\n\\partial x_1}},{\\frac{1}{2}}{\\frac{\\partial}{\\partial x_2}}, \\dots,\n{\\frac{1\n}{n}}{\\frac{\\partial}{\\partial x_n}},\\dots\\right)\n$,\nand $p_i$ are the Schur polynomials generated by the\nrelation\n$\n\\exp\\left(\\sum_{n=1}^{\\infty} \\lambda^n x_n\\right)=\n\\sum_{n=0}^{\\infty}p_i(\\bbox{x})\\lambda^n\n$.\nFor the first three continuous variables we will use the notations\n$x=x_1$, $y=x_2$, $t=x_3$.\n\nTo demonstrate that the discrete form of KP hierarchy (\\ref{KPa0})\nwritten in terms of continuous variables as\n\\begin{equation}\n\\sum_{(ijk)} \\epsilon_{ijk}T_k \\left({\\Delta_i\\over a_i} u(\\bbox{x})\n- u(\\bbox{x})T_iu(\\bbox{x})\\right)=0\n\\label{KPa1}\n\\end{equation}\ngenerates equations of KP hierarchy in the standard form, we consider\nexpansion of this equation into powers of parameters $a_i\\,,a_j\\,,a_k$.\nThe zeroth order of expansion\nof equation (\\ref{KPa1}) into powers of the\nparameter $a_i$ gives the equation\ncontaining two discrete transformations and one partial derivative\n\\bea\n&&\nT_k \\left(\\partial_x u\n- uu\\right)-\n\\left({\\D_k} u\n- uT_ku\\right)\n\\nn\\\\&&\n\\qquad+\\left({\\D_j} u\n- uT_ju\\right)-\nT_j \\left(\\partial_x u\n- u u\\right)\n\\nn\\\\&&\n\\qquad\\qquad+\nT_j \\left({\\D_k} u\n- uT_ku\\right)-\nT_k \\left({\\D_j} u\n- uT_ju\\right)=0.\n\\label{KPdiscrete1}\n\\eea\nThe first order of expansion of equation (\\ref{KPdiscrete1})\ninto the powers of the parameter $a_j$\nrepresents an equation containing\npartial derivatives over two continuous variables and one discrete\ntransformation,\n\\bea\n&&\n\\left(\\mbox{${1\\over2}$}\\left(\\partial_y+\\partial_x^2\\right) u\n- u\\partial_x u\\right)-\n\\partial_x \\left(\\partial_x u\n- u u\\right)\n\\nn\\\\&&\n\\qquad+\n\\partial_x \\left({\\D_k} u\n- uT_ku\\right)-\nT_k \\left(\\mbox{${1\\over2}$}\\left(\\partial_y+\\partial_x^2\\right) u\n- u\\partial_x u\\right)=0.\n\\label{KPdiscrete2}\n\\eea\nThe final step is to take the first nontrivial\norder of the expansion into the powers of $a_k$\n(the second order) to get\nthe potential form of the KP equation\n\\begin{equation}\n\\partial_x\\left(u_t-\\mbox{${1\\over 4}$}u_{xxx}+\n\\mbox{${3\\over2}$}(u_x)^2\\right)=\\mbox{${3\\over4}$}\nu_{yy}\\,,\n\\label{KP00}\n\\end{equation}\nwhich reduces to standard KP equation for\nthe function $v=-2\\partial_x u$.\n\nThe higher orders of expansion of equation\n(\\ref{KPdiscrete2}) will give us the higher equations of the\nKP hierarchy. This sequence of equations should be used recursively\nto get equations containing only partial derivatives\nover the highest order time, $\\partial_x$ and $\\partial_y$.\n\nThe interpretation of the chain of equations we have derived\ndepends on the\nchoice of the basic equation (i.e., in some\nsense on the point of reference).\n\nA standard way is to take continuous equation\n(\\ref{KP00}) (or, rather, the KP hierarchy\nin the form of PDEs)\nas a basic system. Then the interpretation\nof the other objects is:\n1) equation (\\ref{KPdiscrete2}) defines a B\\\"acklund\ntransformation for equation (\\ref{KPa1}),\n2) equation (\\ref{KPdiscrete1}) is a superposition\nprinciple for two B\\\"acklund transformations,\n3) discrete equation (\\ref{KPa1})\nprovides an {\\em algebraic} superposition principle\nfor three B\\\"acklund transformations.\n\nOn the other hand, the discrete equation\n(\\ref{KPa1}) (in other words, the discrete form of the\nKP hierarchy)\nmay be treated as a basic system as well.\nThen formula (\\ref{KPdiscrete1}) is a lowest order continuous\nsymmetry for this system, equation (\\ref{KPdiscrete2})\nis a superposition principle for two continuous symmetries,\nand equation (\\ref{KP00}) is a superposition\nprinciple for three continuous symmetries of different orders.\n\nLinear problems (\\ref{compa}), (\\ref{compb})\ngenerating the discrete form of the KP\nhierarchy (\\ref{KPa1}) as compatibility conditions\nin terms of continuous variables look like\n\\begin{eqnarray}\n&&\n({\\D_i}-{\\D_j}) \\widetilde\\Psi(\\bbox{x})\n=\n%\\nonumber\\\\&&\n((T_i-T_j)u(\\bbox{x}))\\widetilde\\Psi(\\bbox{x})\n\\label{KPbasea13}\n\\,,\n\\\\\n&&\n({\\widetilde\\D_i} -\n{\\widetilde\\D_j})\\Psi(\\bbox{x})\n=\n%\\nonumber\\\\&&\n((\\widetilde T_i-\\widetilde T_j)u(\\bbox{x}))\n\\Psi(\\bbox{x})\n\\label{KPbaseb13}\n\\,.\n\\end{eqnarray}\nBoth the set of equations (\\ref{KPbasea13}) and the dual set\n(\\ref{KPbaseb13}) imply the same equation (\\ref{KPa1}). Expansion\nof these linear equations into powers of parameters gives standard\nlinear problems for KP hierarchy.\n\nWe will not use equations of the\nsecond level of generalized KP hierarchy in the present work.\nEquation for $\\Phi$ (\\ref{singmana})\n(discrete form of KP singular manifold equation\nhierarchy) in terms of continuous variables looks like\n\\bea\n&&\n(T_j\\Delta_i \\Phi(\\bbox{x}))(T_k\\Delta_j \\Phi(\\bbox{x}))\n(T_i\\Delta_k \\Phi(\\bbox{x}))\n\\nn\\\\\n&&\n\\qquad=\n(T_j\\Delta_k \\Phi(\\bbox{x}))(T_k\\Delta_i \\Phi(\\bbox{x}))\n(T_i\\Delta_j \\Phi(\\bbox{x})),\n\\label{KPSM}\n\\eea\nand, performing expansion into powers of parameters, we get\nthe chain of equations connecting discrete and continuous case,\n\\begin{equation}\n(T_j \\Phi_x)(T_k\\Delta_j\\Phi)\\Delta_k\\Phi=\n(T_k\\Phi_x)(T_j\\Delta_k\\Phi)\\Delta_j\\Phi,\n\\label{SM2-1}\n\\end{equation}\n%\\hfill{\\tiny SM2-1}\\\\\n\\begin{equation}\n{\\partial\\over \\partial x}\n\\ln \\left({1\\over \\Phi_x}\n{\\Delta \\Phi\\over a}\\right)=\n{1\\over 2}\\Delta\\left({\\Phi_y+\\Phi_{xx}\\over\n\\Phi_x}\\right)\n\\label{SM1-2},\n\\end{equation}\n%\\hfill{\\tiny SM1-2}\\\\\n\\begin{eqnarray}\n&&\\Phi_t=\\mbox{${1\\over4}$}\\Phi_{xxx}+\\mbox{${3\\over8}$} {\\frac{\n\\Phi_y^2-\\Phi_{xx}^2}{\\Phi_x}}+ \\mbox{${3\\over4}$}\\Phi_x W_y ,\n\\quad W_x={\n\\frac{\\Phi_y}{\\Phi_x}}\\,.\n\\label{singman1}\n\\end{eqnarray}\n%\\hfill{\\tiny singman}\\\\\nThe last equation first arose in Painleve analysis of the KP\nequation as a\nsingular manifold equation \\cite{Weiss}.\n\nThe interpretation of this chain of equations is similar to\nthe interpretation given\nfor equation (\\ref{KPa1}).\n\nThus the integrable discrete equations\nwritten in terms of elementary\nrational loops encode the continuous hierarchy,\nthe B\\\"acklund transformations and different\ntypes of superposition principles for them,\nand the discrete linear equations generate a\nhierarchy of linear problems,\nDarboux transformations and superposition principles for them.\n\n\\section{Discrete and Continuous Non-Iso\\-spec\\-tral Symmetries}\nThe dynamics defined by Hirota bilinear\nidentity (\\ref{HIROTA0}) is connected with operator of\nmultiplication by loop group element $g\\in \\Gamma^+$;\nthis dynamics can be interpreted in terms of commuting flows\ncorresponding to infinite number of `times'\n$x_n$. A general idea of introduction additional\n(in general, non-commutative) symmetries is to consider\nmore general operators $\\hat R$ on the unit circle.\nLet us introduce symmetric bilinear form\n$$\n(f\|g)=\\oint f(\\nu)g(\\nu)d(\\nu).\n$$\nIn terms of this form identity (\\ref{HIROTA0})\nlooks like\n\\bea\n(\\chi(\\dots,\\mu;g_1)g_1(\\dots)\|\ng_2^{-1}(\\dots)\\chi(\\lambda,\\dots;g_2))=0\\,\\quad \\lambda\\,,\\mu\\in D,\n\\label{HIROTA01}\n\\eea\nor, for Cauchy-Baker-Akhiezer function $\\psi(\\lambda,\\mu;g)$,\n\\bea\n(\\psi(\\dots,\\mu;g_1)\|\n\\psi(\\lambda,\\dots;g_2))=0\\,\\quad \\lambda\\,,\\mu\\in D,\n\\label{HIROTA02}\n\\eea\nwhere by dots we denote the argument which is involved into\nintegration.\nLet some CBA function $\\psi(\\lambda,\\mu;g)$ satisfying Hirota\nbilinear identity be given.\nWe define symmetry transformation connected with\narbitrary invertible\nlinear operator $\\hat R$ in the space of functions on the\nunit circle by the equations\n\\bea\n(\\wt\\psi(\\dots,\\mu;g_1)\|\\hat R\|\n\\psi(\\lambda,\\dots;g_2))=0,\n\\nn\\\\\n(\\psi(\\dots,\\mu;g_1)\|\\hat R^{-1}\|\n\\wt\\psi(\\lambda,\\dots;g_2))=0.\n\\nn\n\\eea\nIt is possible to show that if both these equations\nfor the transformed CBA function $\\wt\\psi(\\lambda,\\mu;g)$\nare solvable, then the solution for them is the same\n(and unique), and it satisfies identity (\\ref{HIROTA02}).\nIn this case the symmetry transformation connected with\noperator $\\hat R$ is correctly defined.\nIt is also possible to define one-parametric groups of\ntransformations by the equation\n\\bea\n(\\psi(\\dots,\\mu;g_1,\\Theta_1)\|\\exp((\\Theta_1-\\Theta_2)\\hat r)\|\n\\psi(\\lambda,\\dots;g_2,\\Theta_2))=0.\n\\label{H1}\n\\eea\nTaking the generators $\\hat r_{mn}=\\lambda^n\\partial_\\lambda^m$,\nwe get noncommutative\nsymmetries in the form proposed by Orlov and Shulman \\cite{Orlov}.\nIn our work we will consider non-isospectral symmetries connected with\noperators with degenerate kernel, and, in particular,\ngenerators with the kernel of the form\n\\be\nr_{\\alpha\\beta}(\\nu,\\nu')={1\\over 2\\pi\\text{i}} \\delta(\\alpha-\\nu)\n\\delta(\\beta-\\nu'),\n\\label{delta}\n\\ee\nwhere $\\alpha,\\beta$ belong to the unit circle, or, more generally,\n\\be\nr_{\\rho\\wt\\rho}(\\nu,\\nu')={1\\over 2\\pi\\text{i}}\\wt\\rho(\\nu')\\rho(\\nu),\n\\label{rho}\n\\ee\nwhere for simplicity we put\n$$\n(\\wt\\rho\|\\rho)=0.\n$$\n{\\bf Remark.} To make a transformation from the\ngenerators $\\hat r_{\\alpha\\beta}$ to the generators\n$\\hat r_{mn}$ used by Orlov and Shulman, it is enough to note\nthat operator $\\hat r_{\\alpha\\beta}$ can be represented\nas a composition of the shift operator\n$T_{\\beta-\\alpha}:\\nu\\rightarrow \\nu+\\beta-\\alpha$\nand operator of multiplication by the function\n$\\delta(\\nu-\\alpha)$. Then, expanding shift\noperator and $\\delta$-function into powers of parameters,\nit is possible to make a transformation from one set\nof generators to the other.\n\nUsing simple identity\n$$\n\\exp(\\Theta_{\\alpha\\beta} \\hat r_{\\alpha\\beta})=\nI+\\Theta_{\\alpha\\beta} \\hat r_{\\alpha\\beta},\n$$\nwhich is satisfied due to nilpotence of the generators,\nand performing integration in the equation (\\ref{H1}) taken for\n$g_1=g_2$,\nwhich in this case reads\n\\bea\n\\oint\\oint d\\nu d\\nu'\n\\psi(\\nu,\\mu;g,\\Theta_1)(\\delta(\\nu-\\nu')+{\\Theta_1-\\Theta_2\\over\n2\\pi\\text{i}}\\delta(\\beta-\\nu')\\delta(\\alpha-\\nu))&&\n\\nn\\\\\n%\\qquad\\qquad\n\\times\n\\psi(\\lambda,\\nu';g,\\Theta_2))=0,&&\n\\label{H2}\n\\eea\nwe get equation for the CBA function\n\\bea\n&&\n\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta}+\\Delta\\Theta_{\\alpha\\beta})\n=\n\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta})\n\\nn\\\\\n&&\n\\qquad\n+\\Delta\\Theta_{\\alpha\\beta}\n{\\psi(\\lambda,\\beta;\\bbox{x},\\Theta_{\\alpha\\beta})\n\\psi(\\alpha,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta}+\\Delta\n\\Theta_{\\alpha\\beta})}.\n\\label{Delta}\n\\eea\nIt is possible to resolve this equation and express\n$\\psi(\\lambda,\\mu;\\bbox{x},\n\\Theta_{\\alpha\\beta}+\\Delta\\Theta_{\\alpha\\beta})$ through\n$\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta})$.\nFirst we take equation (\\ref{Delta}) at $\\lambda=\\alpha$\nand get the expression for\n$\\psi(\\alpha,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta}+\\Delta\\Theta_{\\alpha\\beta})$,\n\\bea\n\\psi(\\alpha,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta}+\\Delta\\Theta_{\\alpha\\beta})=\n{\\psi(\\alpha,\\mu;\n\\bbox{x},\\Theta_{\\alpha\\beta})\\over 1-\\Delta\\Theta_{\\alpha\\beta}\n\\psi(\\alpha,\\beta;\\bbox{x},\\Theta_{\\alpha\\beta})}.\n\\label{Delta1}\n\\eea\nSubstituting (\\ref{Delta1}) into (\\ref{Delta}), we\nfinally get\n\\bea\n&&\n\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta}+\\Delta\\Theta_{\\alpha\\beta})=\n\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta})\n\\nn\\\\&&\\qquad\\qquad\n+\\Delta\\Theta_{\\alpha\\beta}\n{\\psi(\\lambda,\\beta;\\bbox{x},\n\\Theta_{\\alpha\\beta})\\psi(\\alpha,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta}\n)\\over1-\\Delta\\Theta_{\\alpha\\beta}\n\\psi(\\alpha,\\beta;\\bbox{x},\\Theta_{\\alpha\\beta})}.\n\\label{Delta2}\n\\eea\nIn particular, this formula expresses\nthe function $\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta})$ through\nthe initial data $\\psi_0(\\lambda,\\mu;\\bbox{x})=\n\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta}=0)$, thus giving explicit\nformula for the action of non-isospectral symmetry connected\nwith the generator (\\ref{delta}) on the CBA function.\n\nFormula (\\ref{Delta2}) can be rewritten as\n\\bea\n&&\n\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta})\n\\nn\\\\&&\\qquad\n=\n\\psi_0(\\lambda,\\mu;\\bbox{x})\n{1-\\Theta_{\\alpha\\beta}{\\text{det}_{\\alpha\\beta}\\psi_0(\\lambda,\\mu;\\bbox{x})\n\\over \\psi_0(\\lambda,\\mu;\\bbox{x})}\n\\over 1-\\Theta_{\\alpha\\beta}\\psi(\\alpha,\\beta;\\bbox{x})},\n\\eea\nwhere\n\\bea\n\\text{det}_{\\alpha\\beta}\\psi_0(\\lambda,\\mu;\\bbox{x})=\n\\text{det}\\left(\n\\begin{array}{cc}\n\\psi_0(\\lambda,\\mu;\\bbox{x})&\\psi_0(\\lambda,\\beta;\\bbox{x})\\\\\n\\psi_0(\\alpha,\\mu;\\bbox{x})&\\psi_0(\\alpha,\\beta;\\bbox{x})\n\\end{array}\n\\right).\n\\eea\nRecalling determinant formula for the transformation\nof CBA function under the action of a rational loop\n(see \\cite{NLS}, \\cite{AB1}),\n\\bea\n\\psi_0(\\alpha,\\beta;\\bbox{x}+[\\mu]-[\\lambda])\n={\\text{det}_{\\lambda\\mu}\\psi_0(\\alpha,\\beta;\\bbox{x})\n\\over \\psi_0(\\lambda,\\mu;\\bbox{x})},\n\\eea\nwe get another representation of the transformation\n(\\ref{Delta2}),\n\\bea\n\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta})=\n\\psi_0(\\lambda,\\mu;\\bbox{x})\n{1-\\Theta_{\\alpha\\beta}\\psi_0(\\alpha,\\beta;\\bbox{x}+[\\mu]-[\\lambda])\n\\over 1-\\Theta_{\\alpha\\beta}\\psi_0(\\alpha,\\beta;\\bbox{x})}.\n\\eea\nComparing this formula with the formula connecting the CBA\nfunction and the $\\tau$-function (which in fact defines\nthe $\\tau$-function through the CBA function)\n\\bea\n\\psi(\\lambda,\\mu,\\bbox{x})=\n{g(\\lambda)g(\\mu)^{-1}}{1\\over\n\\lambda-\\mu}{\\tau(\\bbox{x}+[\\mu]-[\\lambda]) \\over\\tau(\\bbox{x})},\n\\eea\nwe come to the conclusion that the $\\tau$-function corresponding\nto the transformed CBA function\n$\\psi(\\lambda,\\mu;\\bbox{x},\\Theta_{\\alpha\\beta})$\nis given by the expression\n\\bea\n\\tau(\\bbox{x},\\Theta_{\\alpha\\beta})=\n\\tau_0(\\bbox{x})(1-\\Theta_{\\alpha\\beta}\\psi_0(\\alpha,\\beta;\\bbox{x})).\n\\label{tautrans}\n\\eea\nThus we have explicitly defined action of non-isospectral symmetry with\nthe generator (\\ref{delta}) on KP $\\tau$-function. Transformation\n(\\ref{tautrans}) coincides with the solitonic transformation\nof the $\\tau$-function defined through Date-Jimbo-Kashiwara-Miwa\nvertex operator \\cite{Date}.\nBelow we will demonstrate that in terms of potential\nthis is just a binary B\\\"acklund transformation, and for some\nchoice of KPSM solution $\\Phi(\\bbox{x})$ this is a M\\\"obius\ntransformation.\n\nFor the function $\\Phi_{\\alpha\\beta}=\\psi(\\alpha,\\beta;\\bbox{x})$\nsatisfying\nsingular manifold equation (\\ref{KPSM}) from the formula (\\ref{Delta2})\nwe get especially\nsimple transformation,\n\\bea\n\\Phi_{\\alpha\\beta}(\\bbox{x},\\Theta_{\\alpha\\beta})=\n{\\Phi^0_{\\alpha\\beta}(\\bbox{x})\\over1-\n\\Theta_{\\alpha\\beta}\n\\Phi^0_{\\alpha\\beta}(\\bbox{x})},\n\\label{Delta3}\n\\eea\nand this is nothing more then one-parametric subgroup\nof the M\\\"obius group. Taking this formula\nat $\\Theta_{\\alpha\\beta}\\rightarrow\\infty$, we get (up to a constant)\ntransformation of inversion $\\Phi_{\\alpha\\beta}\\rightarrow\n\\Phi^{-1}_{\\alpha\\beta}$.\n\nIt is easy to check that the same\nderivation holds for the generators (\\ref{rho})\n$\\hat r_{\\rho\\tilde\\rho}$, and in this case we get M\\\"obius\ntransformation\n\\bea\n\\Phi(\\bbox{x},\\Theta)=\n{\\Phi^0(\\bbox{x})\\over1-\n\\Theta\n\\Phi^0(\\bbox{x})},\n\\label{Delta4}\n\\eea\nfor the solution of KPSM equation (\\ref{KPSM})\ncorresponding to the weight functions $\\rho(\\nu)$,\n$\\wt\\rho(\\nu)$\n\\bea\n\\Phi(\\bbox{x})=\\oint\\!\\!\\oint (\\psi(\\lambda,\\mu;\\bbox{x}))\n\\rho(\\lambda)\n\\wt\\rho(\\mu) d\\lambda\\,d\\mu.\n\\nn\n\\eea\n\\section{M\\\"obius Symmetry}\nIn this section we will concentrate on M\\\"obius symmetry of\nKPSM equation (\\ref{KPSM}), using only equations of generalized\nhierarchy and connections between them, without explicit use of\nbilinear technique underlying the construction. We will demonstrate\nthat M\\\"obius symmetry on the level of KPSM hierarchy generates\nbinary B\\\"acklund transformations on the level of the basic KP\nhierarchy.\n\nCharacteristic feature of singular manifold equation\n(\\ref{KPSM}) is its invariance under M\\\"obius transformation\n$$\n\\Phi\\rightarrow {a\\Phi+ b\\over c\\Phi+d},\n$$\nwhich can be easily checked. Now we are going to find the symmetry\nof the basic KP hierarchy (\\ref{KPa1}), which corresponds to the\nM\\\"obius transformation on the level of the singular manifold\nequation. To do that, we define the transformations of the\nwave functions $\\Psi$, $\\wt \\Psi$\nusing the equations (\\ref{Phi1}),\nand then we substitute the wave functions into linear equations\n(\\ref{KPbasea13}) and (\\ref{KPbaseb13})\nto find the transformation of the potential $u$. Generic\nM\\\"obius transformation can be represented as composition of\ntranslation, scaling and inversion. Translation and scaling of $\\Phi$\ndo not change the potential $u$ (translation doesn't change wave\nfunctions, and scaling of wave functions doesn't change the\npotential), and so in principle our problem is to find the\ntransformation of potential $u$ corresponding to inversion\n$\n\\Phi\\rightarrow {\\Phi}^{-1}\n$.\nTransformations of the wave functions, according to equations\n(\\ref{Phi1}), look like\n$\n\\wt\\Psi\\rightarrow -\\Phi^{-1}\\wt \\Psi$, $\\Psi\\rightarrow\n\\Phi^{-1}\\Psi\n$,\nand, substituting them into linear equations (\\ref{KPbasea13})\nand (\\ref{KPbaseb13}),\nwe get the formula for the transformation of\npotential $u$,\n\\be\nu(\\Phi^{-1})=u(\\Phi)- \\Psi\\Phi^{-1}\\wt\\Psi.\n\\label{utrans}\n\\ee\nTaking into account equation\n\\be\n\\partial_x\\Phi=\\wt\\Psi\\Psi,\n\\label{derivative}\n\\ee\narising in the zeroth order of expansion of equation (\\ref{Phi1}),\nit is possible to rewrite formula (\\ref{utrans}) as\n\\be\nu({\\Phi}^{-1})=u(\\Phi)-\\partial_x\\ln \\Phi,\n\\label{utrans1}\n\\ee\nwhich is a well-known binary B\\\"acklund transformation.\nThus we have shown that inversion on the level\nof KPSM equation hierarchy leads to binary B\\\"acklund transformation\non the level of the basic KP hierarchy. The connection\nbetween M\\\"obius transformation and binary B\\\"acklund transformation\nwas discovered in the framework of Painleve analysis \\cite{Weiss}.\n\nLet us consider a one-parametric subgroup of the M\\\"obius group\n\\be\n\\Phi(\\Theta)={\\Phi_0\\over 1-\\Theta \\Phi_0}\n\\label{Phicont}\n\\ee\ncharacterized by the equation\n\\be\n\\partial_\\Theta \\Phi=\\Phi^2.\n\\label{Phicont1}\n\\ee\nUsing equation (\\ref{utrans1}), it is easy to find continuous\nsymmetry of KP hierarchy corresponding to this subgroup.\nFirst, directly from (\\ref{utrans1}) we get a formula\n\\be\nu(\\Theta):=u\\left({\\Phi_0\\over 1-\\Theta \\Phi_0}\\right)=\nu\\left({1-\\Theta \\Phi_0\\over\\Phi_0}\\right)-\\partial_x\\ln\n\\left({1-\\Theta \\Phi_0\\over\\Phi_0}\\right).\n\\label{utrans11}\n\\ee\nTaking into account that\n$$\nu\\left({1-\\Theta \\Phi_0\\over\\Phi_0}\\right)=u(\\Phi_0^{-1}),\n$$\nand transforming $u(\\Phi_0^{-1})$ using formula\n(\\ref{utrans1}), we finally find symmetry transformation\nof potential $u$ depending on continuous parameter $\\Theta$,\n\\be\nu(\\Theta)=u_0-\\partial_x\\ln(1-\\Theta\\Phi_0).\n\\label{trans}\n\\ee\nPotential $u$ satisfies differential relation\n\\be\n\\partial_\\Theta u=\\partial_x\\Phi,\n\\label{difftrans}\n\\ee\nor, taking into account formula (\\ref{derivative}),\n\\be\n\\partial_\\Theta u=\\Psi\\wt\\Psi.\n\\label{difftrans1}\n\\ee\nExpression $\\Psi\\wt\\Psi$ represents infinitesimal\n(in general non-isospectral) symmetry of KP hierarchy \\cite{Orlov1},\nand the formula (\\ref{trans}) defines a one-parametric group of\ntransformations connected with this symmetry (specified by\nextra relation (\\ref{Phicont1})). Exactly this form of\nsymmetry generator is used to define constrained KP hierarchy\n\\cite{KS0}, \\cite{KS},\nwhich will be one of the objects of our study.\n\nWe will also consider more general symmetry transformations\nof KPSM hierarchy (\\ref{KPSM}), which we call multicomponent\nM\\\"obius-type transformations. We have used an arbitrary pair of wave\nfunctions $\\Psi$, $\\wt\\Psi$ to define the function $\\Phi$ through the set\nof equations (\\ref{Phi1}). Let us fix a set of wave functions and dual\nwave functions $\\Psi_k$, $\\wt\\Psi_k$, $1\\leq k\\leq \\infty$\n(in terms of Hirota bilinear identity\nwe should fix a set of weight functions $\\rho_k(\\nu)$,\n$\\wt\\rho_k(\\nu)$).\nThen equations (\\ref{Phi1}) define a matrix of solutions\nof equation (\\ref{KPSM}) $\|\\Phi\|$ connected with the same solution\n$u$ of KP hierarchy (\\ref{KPa1}).\nMatrix\nentries $\\Phi_{kp}$ satisfy the equations\n\\be\n{\\Delta_i\\over a_i}\\Phi_{kp}=\\wt\\Psi_k T_i \\Psi_p,\n\\ee\nor, in matrix form,\n\\be\n{\\Delta_i\\over a_i}\|\\Phi\|=\|\\wt\\Psi\\rangle T_i \\langle\\Psi\|.\n\\label{transmulti}\n\\ee\nIt is easy to check that matrix inversion\n$\n\|\\Phi\|\\rightarrow\|\\Phi\|^{-1}\n$\nleads to the same equation (\\ref{transmulti})\nwith transformed $\|\\wt\\Psi\\rangle$, $\\langle\\Psi\|$,\n$\n\|\\wt\\Psi\\rangle\\rightarrow \|\\Phi\|^{-1}\|\\wt\\Psi\\rangle$,\n$\\langle\\Psi\|\\rightarrow \\langle\\Psi\|\|\\Phi\|^{-1}$.\nSubstituting transformed vectors of wave functions\ninto linear equations (\\ref{KPbasea13}), (\\ref{KPbaseb13}),\nwe come to the conclusion that all components of the wave functions\ngive the same transformed potential\n\\be\nu\\rightarrow u -\\langle\\Psi\|\\Phi^{-1}\|\\wt\\Psi\\rangle.\n\\label{transmulti1}\n\\ee\nTaking into account that equation\n$\n\\partial_x\|\\Phi\|=\|\\wt\\Psi\\rangle\\langle\\Psi\|\n$\nimply the identity\n$$\n\\langle\\Psi\|\\Phi^{-1}\|\\wt\\Psi\\rangle=\\partial_x\\ln\\det \|\\Phi\|,\n$$\nwe get another form of transformation of potential\ncorresponding to multicomponent M\\\"obius-type transformation,\n\\be\nu(\|\\Phi\|^{-1})=u(\|\\Phi\|)-\\partial_x\\ln\\det \|\\Phi\|,\n\\label{transmulti2}\n\\ee\nwhich represents a composition formula for several binary B\\\"acklund\ntransformations.\n\nMulticomponent continuous M\\\"obius-type symmetry\n$$\n\|\\Phi(\\Theta)\|=\|\\Phi_0\|(I-\\Theta\|\\Phi_0\|)^{-1},\\qquad\n\\partial_\\Theta \|\\Phi\|=\|\\Phi\|\|\\Phi\|,\n$$\nleads to continuous symmetry for the potential\n\\bea\nu(\\Theta)=u_0-\\partial_x\\ln\\det(I-\\Theta\|\\Phi_0\|),\n\\label{transmulti3}\n\\\\\n\\partial_\\Theta u=\\partial_x \\text{tr}\|\\Phi\|=\\sum_{i=1}^N \\Psi_i\\wt\\Psi_i.\n\\label{diff2}\n\\eea\n\\section{Symmetry Constraints and Calogero-Moser System}\nThe concept of generalized hierarchy is rather effective tool\nin the study\nof symmetry constraints.\nA standard symmetry constraint for KP hierarchy is\n\\cite{KS0}, \\cite{KS}\n\\be\nu_x=\\Psi\\wt\\Psi,\n\\label{constraint}\n\\ee\nand it was shown in \\cite{KS0} that it leads\nto AKNS hierarchy for the wave functions. It is possible also to derive\ntwo-dimensional equation for one function (either $u$ or $\\Phi$).\nIndeed, it was shown above that\n$\n\\Phi_x=\\Psi\\wt\\Psi\n$,\nso for the constrained hierarchy\n$$\nu_x=\\Phi_x.\n$$\nThus $u$ and $\\Phi$ represent almost the same object, and the meaning\nof the constraint is that it glues the first and the third level of\ngeneralized hierarchy. We know that $u$ satisfies KP equation\n(\\ref{KP00}), and\n$\\Phi$ satisfies KPSM equation, but, using the constraint, we can\nwrite two equations for both of these functions. Combining these\nequations, it is easy to eliminate the terms containing partial\nderivative over $t$ and get two-dimensional differential relation,\nfor $\\Phi$\nit looks like (see also \\cite{KS0})\n%\\begin{equation}\n%\\partial_x^2\\left(\\Phi_t-{1\\over 4}\\Phi_{xxx}+\n%{3\\over2}(\\Phi_x)^2\\right)={3\\over4}\n%\\Phi_{xyy}\\,,\n%\\label{KP01}\n%\\end{equation}\n%\\begin{eqnarray}\n%&&\\Phi_t=\\mbox{${1\\over4}$}\\Phi_{xxx}+\\mbox{${3\\over8}$} {\\frac{\n%\\Phi_y^2-\\Phi_{xx}^2}{\\Phi_x}}+ \\mbox{${3\\over4}$}\\Phi_x W_y ,\n%\\quad W_x={\n%\\frac{\\Phi_y}{\\Phi_x}}\\,.\n%\\label{singman2}\n%\\end{eqnarray}\n%Applying operator $\\partial_x^2$ to the second equation and taking\n%difference of two equation, we get differential relation for $\\phi$\n%containing only partial derivatives over $x$ and $y$,\n\\bea\n\\partial_x\\left(\n\\partial_x\\bigl(\\mbox{${3\\over2}$}(\\Phi_x)^2-\n\\mbox{${3\\over8}$} {\\frac{\n\\Phi_y^2-\\Phi_{xx}^2}{\\Phi_x}}\\bigr)+\n\\mbox{${3\\over4}$}{\\Phi_y\\Phi_{xy}\\over\\Phi_x}-\n\\mbox{${3\\over4}$}\\Phi_{xx}W_y\\right)=0,\n%\\nn\\\\\n\\quad W_x={\n\\frac{\\Phi_y}{\\Phi_x}}.\n\\nn\n\\eea\n\nLet us consider one-parametric group of symmetry transformations\nof potential $u$ defined by the formula (\\ref{transmulti3});\nwe have shown that $u$ satisfies differential relations\n\\bea\n\\partial_\\Theta u=\\partial_x \\text{tr}\|\\Phi\|=\\sum_{i=1}^N \\Psi_i\\wt\\Psi_i.\n%\\label{diff2}\n\\nn\n\\eea\nAccording to these relations, standard constrains of the type\n(\\ref{constraint}) can be interpreted as an equation\n\\be\nu_\\Theta=u_x,\\quad \\Theta=0.\n\\label{constraint2}\n\\ee\n\nThere is stronger symmetry constraint, for which\nrelation (\\ref{constraint2}) is required\nto be satisfied {\\em for all}\n$\\Theta$, not only at the origin. The dependence of $u$ on extra\ntime $\\Theta$ is rational, so the constraints of this type impose\nrational dependence of $u$ on $x$. In this way we come to\nrational Calogero-Moser system. Indeed, let us make a simple\ntransformation of the\nformula (\\ref{transmulti3}) using relation (\\ref{transmulti2}),\n\\bea\nu(\\Theta)=u(\|\\Phi_0\|^{-1})-\\partial_x\\ln\\det(\|\\Phi_0\|^{-1}-\\Theta),\n\\label{transmulti4}\n\\eea\nand substitute the result to the equation (\\ref{constraint2}).\nComparing the singularities, we come to the conclusion that\n\\be\nv=-2u_x=\\sum_{i=1}^N{-2\\over(\\phi_i(y,\\cdots)-x-\\Theta)^2},\n\\label{Calogero}\n\\ee\nwhere $\\phi_i$ are eigenvalues of the matrix\n$\|\\Phi(x=0,y,t,\\cdots)\|^{-1}$. Due to the constraint the eigenvalues of\nthis matrix should depend linearly on $x$,\n$\n\\phi_i(x)=\\phi_i(0)-x\n$, and also $\\partial_x u(\|\\Phi_0\|^{-1})=0$.\nThe substitution for the potential (\\ref{Calogero})\ncharacterizes Calogero-Moser integrable system of particles\non the line \\cite{Krichever}. Thus we have demonstrated that this system can\nbe obtained through the symmetry constraint of KP hierarchy.\n\\section*{Acknowledgments}\nThe first author (LB) is grateful to the\nDipartimento di Fisica dell' Universit\\`a\nand Sezione INFN, Lecce, for hospitality and support;\n(LB) also acknowledges partial support from the\nRussian Foundation for Basic Research under grants\nNo 98-01-00525 and 96-15-96093 (scientific schools).\n\\begin{thebibliography}{99}\n\\bibitem{AB1}\nL.V. Bogdanov and B.G. Konopelchenko,\n\\Journal{\\JMP}{39(9)}{4683}{1998}.\n\\bibitem{AB2}\nL.V. Bogdanov and B.G. Konopelchenko,\n\\Journal{\\JMP}{39(9)}{4701}{1998}.\n\\bibitem{dbar}\nV.E. Zakharov and S.V. Manakov,\n\\Journal{\\LOMI}\n{133}{77}{1984} [in Russian].\n\\\\\nV.E. Zakharov and S.V. Manakov,\n\\Journal{\\FAP}{19(2)}{11}{1985}\n[in Russian].\n\\bibitem{NLS}\nL.V. Bogdanov, \\Journal{\\PHD}{87(1-4)}{58}{1995}.\n\n\\bibitem{Sato}\nM. Sato,\n\\Journal{RIMS, Kokyuroku, Kyoto Univ.}{439}{30}{1981},\n\\\\\n\\Proc{M. Sato and Y. Sato}{}\n{Nonlinear partial differential equations in applied science,\neds. {H. Fujita et al}}\n{259}\n{North-Holland, Amsterdam-New York}{1983}.\n\\bibitem{Wilson}\nG. Segal and G. Wilson,\n\\Journal{Inst. Hautes \\'Etudes Sci. Publ. Math.}\n{61}{5}{1985},\\\\\n\\Book{G. Segal and A. Pressley}{Loop groups}\n{Clarendon, Oxford}{1986}.\n\\bibitem{Orlov}\nA.Yu. Orlov and E.I. Shulman,\n\\Journal{\\LMP}{12}{171}{1986}.\n\\bibitem{Date}\n\\Proc{E. Date, M. Kashiwara, M. Jimbo, T. Miwa}{}\n{Nonlinear integrable systems---classical theory and quantum theory,\neds. {M. Jimbo and T. Miwa}}\n{39}{World Scientific Publishing,\nSingapore}\n{1983}.\n\\bibitem{Weiss}\nJ. Weiss, \\Journal{\\JMP}{24}{1405}{1983}.\n\n\\bibitem{Moeb}\nL.V. Bogdanov and B.G. Konopelchenko, \\Journal{\\PLA}{256}{39}{1999}.\n\\bibitem{KS0}\nB.Konopelchenko and W.Strampp,\n\\Journal{Inverse Problems}{7}{L17}{1991}.\n\\bibitem{KS}\nB.Konopelchenko and W.Strampp,\n\\Journal{\\JMP}{33}{3676}{1992}.\n\\bibitem{Orlov1}\n\\Proc{A. Yu. Orlov}{}\n{Plasma Theory and Nonlinear and Turbulent Processes in Physics,\ned. {Baryakhtar}}\n{116}{World Scientific Publishing,\nSingapore}\n{1988}.\n\n\\bibitem{Krichever} I.M. Krichever,\n\\Journal{Funct. Anal. Appl.}{12}{59}{1978}.\n\\end{thebibliography}\n\n\\end{document}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% End of sprocl.tex\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n\n" } ]	[ { "name": "solv-int9912005.extracted_bib", "string": "{AB1 L.V. Bogdanov and B.G. Konopelchenko, \\Journal{\\JMP{39(9){4683{1998." }, { "name": "solv-int9912005.extracted_bib", "string": "{AB2 L.V. Bogdanov and B.G. Konopelchenko, \\Journal{\\JMP{39(9){4701{1998." }, { "name": "solv-int9912005.extracted_bib", "string": "{dbar V.E. Zakharov and S.V. Manakov, \\Journal{\\LOMI {133{77{1984 [in Russian]. \\\\ V.E. Zakharov and S.V. Manakov, \\Journal{\\FAP{19(2){11{1985 [in Russian]." }, { "name": "solv-int9912005.extracted_bib", "string": "{NLS L.V. Bogdanov, \\Journal{\\PHD{87(1-4){58{1995." }, { "name": "solv-int9912005.extracted_bib", "string": "{Sato M. Sato, \\Journal{RIMS, Kokyuroku, Kyoto Univ.{439{30{1981, \\\\ \\Proc{M. Sato and Y. Sato{ {Nonlinear partial differential equations in applied science, eds. {H. Fujita et al {259 {North-Holland, Amsterdam-New York{1983." }, { "name": "solv-int9912005.extracted_bib", "string": "{Wilson G. Segal and G. Wilson, \\Journal{Inst. Hautes \\'Etudes Sci. Publ. Math. {61{5{1985,\\\\ \\Book{G. Segal and A. Pressley{Loop groups {Clarendon, Oxford{1986." }, { "name": "solv-int9912005.extracted_bib", "string": "{Orlov A.Yu. Orlov and E.I. Shulman, \\Journal{\\LMP{12{171{1986." }, { "name": "solv-int9912005.extracted_bib", "string": "{Date \\Proc{E. Date, M. Kashiwara, M. Jimbo, T. Miwa{ {Nonlinear integrable systems---classical theory and quantum theory, eds. {M. Jimbo and T. Miwa {39{World Scientific Publishing, Singapore {1983." }, { "name": "solv-int9912005.extracted_bib", "string": "{Weiss J. Weiss, \\Journal{\\JMP{24{1405{1983." }, { "name": "solv-int9912005.extracted_bib", "string": "{Moeb L.V. Bogdanov and B.G. Konopelchenko, \\Journal{\\PLA{256{39{1999." }, { "name": "solv-int9912005.extracted_bib", "string": "{KS0 B.Konopelchenko and W.Strampp, \\Journal{Inverse Problems{7{L17{1991." }, { "name": "solv-int9912005.extracted_bib", "string": "{KS B.Konopelchenko and W.Strampp, \\Journal{\\JMP{33{3676{1992." }, { "name": "solv-int9912005.extracted_bib", "string": "{Orlov1 \\Proc{A. Yu. Orlov{ {Plasma Theory and Nonlinear and Turbulent Processes in Physics, ed. {Baryakhtar {116{World Scientific Publishing, Singapore {1988." }, { "name": "solv-int9912005.extracted_bib", "string": "{Krichever I.M. Krichever, \\Journal{Funct. Anal. Appl.{12{59{1978." } ]
solv-int9912006		[]		[ { "name": "lame.tex", "string": "\\documentclass[12pt]{article}\n\\usepackage{rotating}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\setlength{\\oddsidemargin}{0.0cm}\n\\setlength{\\topmargin}{-0.5cm}\n\\setlength{\\textwidth}{16.cm}\n\\setlength{\\textheight}{23.0cm}\n\\itemsep=0pt\n%\\input{tcilatex} \n\n\\begin{document}\n\n\n\\vspace{2.5cm}\n\n\\begin{center}\n{\\Large {\\sc Group Theoretical Properties and Band Structure of the Lam\\'{e}\nHamiltonian}}\n\n\\vspace{1.8cm}\n\nHui LI\\footnote{%\nemail: huili@nst4.physics.yale.edu}, Dimitri KUSNEZOV\\footnote{%\nemail: dimitri@nst4.physics.yale.edu}, Francesco IACHELLO\n\n{\\sl Center for Theoretical Physics, Sloane Physics Laboratory,\\\\[0pt]\nYale University, New Haven, CT 06520-8120}\n\n\\vskip 1.2 cm\n\n{\\it December 1999}\n\n\\vspace{1.2cm}\n\n\\parbox{13.0cm}\n{\\begin{center}\\large\\sc ABSTRACT \\end{center}\n{\\hspace{0.3cm}\nWe study the group theoretical properties of the Lam\\'e equation\nand its relation to $su(1,1)$ and $su(2)$. We compute the band\nstructure, dispersion relation and transfer matrix and discuss the \ndynamical symmetry limits.}}\n\\end{center}\n\n\\vspace{3mm}\n\n\\noindent PACs numbers: 03.65.Fd, 02.20.-a, 02.20.Sv, 11.30.-j\\newline\n\n\\noindent keywords: representation theory, dynamical symmetry, exactly\nsolvable models, periodic potentials, band structure.\n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\setcounter{page}{2} \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Introduction}\n\n\nThe use of group theory to study bound and scattering states has found wide\napplication in physics\\cite{intro}. A common starting point is to identify a\nspectrum generating algebra or SGA for the problem of interest. This is\npossible when the Hamiltonian can be expressed in terms of generators of\nsome algebra $G$. Representation theory then can be used to identify exactly\nsolvable limits of the theory, which can then be translated to explicit\nforms of the Hamiltonian\\cite{prldk}. Connections between bound state or\nscattering state problems and representations of compact and non-compact\ngroups are well understood. The remaining category, having to do with band\nstructure and periodic potentials, was suggested initially in\n\\cite{gursey}, but remained an open problem until recently. In \\cite{prllk}\nit was shown that dynamical symmetry techniques and representation\ntheory can be used to solve band structure problems. In\nparticular, for the Scarf Hamiltonian, \nband structure arises when one uses the complementary\nseries of the projective representations of $su(1,1)$ (and $so(2,2)$ for the\nextended Scarf potential). Further, these dynamical symmetries allowed\nsimple evaluation of the transfer matrix and dispersion relations. In the\npresent study, we consider the band structure problem associated with the\nLam\\'e equation, which is not a dynamical symmetry situation, but more\ngenerally that of a SGA. We discuss several realizations of the Lam\\'e\nequation and its relation to $su(2)$ and $su(1,1)$, including a discussion\nof dynamical symmetry limits, generators, dispersion relation and transfer\nmatrix.\n\nWhile the Lam\\'e equation at first might seem to be an obscure differential\nequation, it does find surprisingly wide application in physics. For\ninstance, it has been shown that if one uses the periodic potentials in\nsuper--symmetric quantum mechanics, the super partners of the Lam\\'{e}\npotentials are distinctly different from the original except for $n=1$ which\nis not self-isospectral. This provides new solvable periodic potentials\\cite\n{a}. The Lam\\'{e} equation also appears in the topics ranging from solitons\nto exactly-solvable models. This includes associations with solutions of the\nperiodic KdV equation\\cite{novikov}, BPS monopoles\\cite{sutcliffe},\nsphaleron solutions of the (1+1)-dimensional abelian Higgs model\\cite\n{brihaye}, sine-Gordon solitons\\cite{Liang}, as well as relations to\nCalogero-Moser systems\\cite{enolskii}. The analysis of this equation has\nmany group theoretical aspects as well. For instance, the Lam\\'{e} equation\ncan be written in terms of the composition of two first-order matrix\noperators, where the coefficients of which satisfy so(3) Nahm's equation\\cite\n{b}. It also arises naturally in the context of $su(2)$ when one tries to\nseparate Laplace's equation in certain coordinate systems\\cite{miller,patera}\nas well as in the general classification of Lie algebraic potentials\\cite\n{karman} and in quasi-exactly solvable $sl(2)$ models\\cite{turbiner}. The\ndiscussion of band structure in the context of group theory is more recent\nhowever\\cite{gursey}, although the system was not solved and the transfer\nmatrix was not determined.\n\nThe Jacobian form of the Lam\\'e equation is\n\\begin{equation}\n-\\frac{d^{2}}{dx^{2}}+\\kappa ^{2}\\ell (\\ell +1){\\rm sn}^{2}x={\\cal E}.\n\\end{equation}\n%\nWe will view this as a Schr\\\"odinger equation with mass $M=1/2$ when $x\n$ is real valued. This equation was first studied group\ntheoretically in \\cite{patera}. The elliptic functions sn$\\alpha =$sn$(\\alpha \|\\kappa )$,\ncn$\\alpha ={\\rm cn(}\\alpha \|\\kappa )$, and ${\\rm dn}\\alpha ={\\rm dn(}\\alpha\n\|\\kappa )$ are doubly-periodic functions in the complex plane, of modulus $\n\\kappa $, where $0\\leq \\kappa \\leq 1$. We will omit the modulus except in\nfunctions where it differs from $\\kappa $. The complementary modulus is\ndefined as $\\kappa ^{\\prime }=(1-\\kappa ^{2})^{1/2}$. The periods of the\nJacobi elliptic functions are related to the complete elliptic integrals $\nK=(\\pi /2)F(1/2,1/2,1;\\kappa ^{2})$ and $K^{\\prime }=(\\pi\n/2)F(1/2,1/2,1;\\kappa ^{\\prime 2})$, where $F$ is the hypergeometric\nfunction, by:\n\n\\begin{equation}\n\\begin{array}[t]{ll}\n{\\rm sn}\\,(\\alpha +\\tau )={\\rm sn}\\,\\alpha \\qquad & \\tau =2iK^{\\prime\n},\\quad 4K+4iK^{\\prime },\\quad 4K \\\\ \n{\\rm cn}\\,(\\alpha +\\tau )={\\rm cn}\\,\\alpha & \\tau =4iK^{\\prime },\\quad\n2K+2iK^{\\prime },\\quad 4K \\\\ \n{\\rm dn}\\,(\\alpha +\\tau )={\\rm dn}\\,\\alpha & \\tau =4iK^{\\prime },\\quad\n4K+4iK^{\\prime },\\quad 2K\n\\end{array}\n\\end{equation}\nIn the limit $\\kappa =1$, the real period related to $K$ becomes infinite,\nand the functions are no longer periodic on the real axis. \\ Along the real\naxis, the potential ${\\rm sn}^{2}x$ is periodic and bounded, while along the\nimaginary axis, it has periodic singularities of the type $1/x^{2}.$\nExamples are shown in Fig. 1 for $\\kappa ^{2}=1/2$ for potentials $V(x)=\n{\\rm sn}^{2}(ax+b)$, with several choices of $a$ and $b$.\n\nThe Lam\\'e equation has also been related to the band structure problem\nassociated with the P\\\"oschl-Teller potential, $V_{pt}(x)\\sim 1/\\cosh ^{2}x$\n\\cite{sutherland,prllk}, by using the expansions for the elliptic\nfunctions in terms of trigonometric functions found in the exercises of\nWhittaker and Watson\\cite{whitwat}. Specifically, one can start with the 1-d\nband structure problem \n\\begin{equation}\nH=-\\frac{d^{2}}{dx^{2}}+\\sum_{k=-\\infty }^{\\infty }\\frac{g}{\\cosh\n^{2}((x-ka)/x_{0})}.\n\\end{equation}\nIf we write the coupling constant as \n\\begin{equation}\ng=-\\frac{\\ell (\\ell +1)}{x_{0}^{2}}\n\\end{equation}\nwe can make the coordinate transformation \n\\begin{equation}\nx=i\\left( \\frac{\\pi x_{0}}{2K}\\right) z-\\left( \\frac{a+i\\pi x_{0}}{2}\\right) \n\\end{equation}\nwhich transforms the Hamiltonian to the Lam\\'e equation. In this article, we\nfirst discuss the relation of $su(2)$ Hamiltonians to the Lam\\'e equation,\nfollowed by a discussion of band edges. Next we consider $su(1,1)$ and the\nrelation to scattering states in the dynamical symmetry limits. \nFinally we derive the transfer matrix and dispersion relation for the Lam\\'e\nequation followed by the relation to the group theoretical SGA\\ Hamiltonians.\n\n\\section{$su(2)$ Realizations of the Lam\\'e Equation}\n\n\\subsection{Sphero-conal Coordinates for $su(2)$}\n\n\\bigskip\n\nIn order to realize the Lam\\'e equation from $su(2)$\\ or $su(1,1)$, we will\nuse coordinate systems defined in terms of Jacobi elliptic functions. These\nare often refered to as sphero-conal\\cite{Arscott} or conical\\cite{miller}\ncoordinates. One can find variations that have been considered in the past,\nand not all provide a good description for all values of the modulus $\\kappa\n.$ Consider first the mapping: \n\\begin{equation}\nx=\\kappa \\,{\\rm sn}\\,\\alpha \\,{\\rm sn}\\,\\beta ,\\,y=i\\frac{\\kappa }{\\kappa\n^{\\prime }}\\,{\\rm cn}\\,\\alpha \\,{\\rm cn}\\,\\beta ,\\,z=\\frac{1}{\\kappa\n^{\\prime }}\\,{\\rm dn}\\,\\alpha \\,{\\rm dn}\\,\\beta ,\n\\end{equation}\nThe ranges of the angles $\\alpha ,\\beta $ are defined in terms of the\nelliptic integrals $K$ and $K^{\\prime }$, so that they depend on the value\nof $\\kappa $. Specifically $-2K<\\alpha <2K$ and $\\,K\\leq \\beta\n<K+2iK^{^{\\prime }}$ where $\\kappa ^{\\prime 2}=1-\\kappa ^{2}$, and $\nx^{2}+y^{2}+z^{2}=1.$ The contours of constant $\\alpha $ and $\\beta $ cover\nthe sphere as illustrated in Fig. 2(a). For $\\kappa =1/2$, four typical\ncontours (for $\\alpha =\\pm K/2$ and $\\beta =K+iK^{\\prime }/2,K+3iK^{\\prime\n}/2$)\\ are shown in\\ Fig. 2(b). This parametrization can be used to\nconstruct the generators of $su(2)$. A direct computation of the generators\nyields the realization (I) given in Table 1. These\ngenerators satisfy the commutation relations in the form $\\left[ L_{a},L_{b}\n\\right] =\\epsilon _{abc}L_{c}$ with Casimir invariant\n\\begin{eqnarray}\nC_{2} &=&L_{x}^{2}+L_{y}^{2}+L_{z}^{2} \\nonumber \\\\\n&=&\\frac{1}{\\kappa ^{2}({\\rm sn}^{2}\\,\\alpha -{\\rm sn}\\,^{2}\\beta )}\\left( \n\\frac{\\partial ^{2}}{\\partial \\alpha ^{2}}-\\frac{\\partial ^{2}}{\\partial\n\\beta ^{2}}\\right) .\n\\end{eqnarray}\nThe $su(2)$ algebras we discuss here have Casimir invariants which have\nexpectation value $-\\ell (\\ell +1)$. We now consider which types of\nquadratic Hamiltonians of the form \n\\begin{equation}\nH=\\sum_{j}\\eta _{j}L_{j}^{2}\n\\end{equation}\ncan be constructed which result in Schr\\\"odinger equations in $\\alpha $ and $\n\\beta .$ Specifically, when the coordinate representation of $H$ contains\ndifferential opertors only of the type $\\partial ^{2}/\\partial \\alpha ^{2}$\nand $\\partial ^{2}/\\partial \\beta ^{2}$, one can use $H$ together with $C_{2}\n$ to perform a separation of variables resulting in two independent Lam\\'e\nequations. We find the three forms in the top right of Table 1, which are\nsimply related through the Casimir operator. To demonstrate the separation,\nconsider $H_{3}$, \n\\begin{eqnarray}\nH_{3} &=&L_{z}^{2}+\\kappa ^{2}L_{y}^{2} \\nonumber \\\\\n&=&\\frac{1}{({\\rm sn}^{2}\\,\\beta -{\\rm sn}\\,^{2}\\alpha )}\\left( {\\rm sn}\n^{2}\\,\\beta \\frac{\\partial ^{2}}{\\partial \\alpha ^{2}}-{\\rm sn}^{2}\\,\\alpha \n\\frac{\\partial ^{2}}{\\partial \\beta ^{2}}\\right) , \\label{ham1}\n\\end{eqnarray}\nand require it to be a constant of the motion with eigenvalue $H_{3}={\\cal E}\n$. Using basis states denoted by the direct product, $\\Psi (\\alpha ,\\beta\n)=\\psi (\\alpha )\\phi (\\beta )$, we arrive at two identical Lam\\'{e}\nequations with identical eigenvalue: \n\\begin{eqnarray}\n\\ \\left[ -\\frac{d^{2}}{d\\alpha ^{2}}+\\kappa ^{2}\\ell (\\ell +1)\\,{\\rm sn}\n^{2}\\alpha \\right] \\psi (\\alpha ) &=&{\\cal E}\\psi (\\alpha ), \\\\\n\\ \\left[ -\\frac{d^{2}}{d\\beta ^{2}}+\\kappa ^{2}\\ell (\\ell +1)\\,{\\rm sn}\n^{2}\\beta \\right] \\phi (\\beta ) &=&{\\cal E}\\phi (\\beta ). \\nonumber\n\\end{eqnarray}\nThus solution of the eigenvalue problem for $H_{3}=L_{z}^{2}+\\kappa\n^{2}L_{y}^{2}$ yields the single valued solutions to these equations, which\nwill correspond to the band edges. One should keep in mind that while $\n\\alpha $ is defined on the real axis, $\\beta $ is complex. Also, while the\nHamiltonians $H_{k}$ are well defined in their algebraic form, the\ngenerators and coordinates have singularities in both $\\kappa =0$ and $\n\\kappa =1$ limits. Hence we consider a different realization below.\n\nTo reduce these Hamiltonians to more conventional Schr\\\"odinger\nequations in $\\alpha$ and $\\beta$, we would\nlike the coordinates to be defined along the real axis. To do so, we make a\nshift (i) $\\beta \\rightarrow \\beta +K+iK^{^{\\prime }}$ followed by (ii) $\n\\beta \\rightarrow -i\\beta $. This results in the transformations: \n\\begin{eqnarray}\n{\\rm sn}(\\beta \|\\kappa ) &\\rightarrow &\\frac{1}{\\kappa }{\\rm dn}(\\beta\n\|\\kappa ^{\\prime }), \\nonumber \\\\\n{\\rm cn}(\\beta \|\\kappa ) &\\rightarrow &\\frac{\\kappa ^{\\prime }}{i\\kappa }\n{\\rm cn}(\\beta \|\\kappa ^{\\prime }), \\\\\n{\\rm dn}(\\beta \|\\kappa ) &\\rightarrow &\\kappa ^{\\prime }{\\rm sn}(\\beta\n\|\\kappa ^{\\prime }). \\nonumber\n\\end{eqnarray}\nThe new coordinate system is no longer singular, and is parameterized as \n\\begin{equation}\nx={\\rm sn}\\,\\alpha \\,{\\rm dn}(\\beta \|\\kappa ^{\\prime }),\\,y={\\rm cn}\\,\\alpha\n\\,{\\rm cn}(\\beta \|\\kappa ^{\\prime }),\\,z={\\rm dn}\\,\\alpha \\,{\\rm sn}(\\beta\n\|\\kappa ^{\\prime }),\n\\end{equation}\nwhere $-2K<\\alpha <2K,\\,-K^{^{\\prime }}\\leq \\beta <K^{^{\\prime }}$ are now\nboth real valued. Contours of constant $\\alpha $ and $\\beta $ are shown in\nFig. 2(c) for $\\alpha =\\pm K/2$ and $\\beta =\\pm K^{\\prime }/2$. If we\nconstruct the generators of $su(2)$ in this basis, we obtain the realization\n(II)\\ given in Table 1. The Casimir invariant now has the form \n\\[\nC_{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=\\frac{-1}{\\kappa ^{2}{\\rm cn}^{2}\\alpha\n+\\kappa ^{\\prime 2}{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime })}\\left( \\frac{\n\\partial ^{2}}{\\partial \\alpha ^{2}}+{\\rm \\ }\\frac{\\partial ^{2}}{\\partial\n\\beta ^{2}}\\right) \n\\]\nThe quadratic Hamiltonians which give rise to Schr\\\"odinger equations when\ncombined with $C_{2}$ are given in the lower right of Table 1. Consider\nagain $H_{3}=L_{z}^{2}+\\kappa ^{2}L_{y}^{2}$. If we set $H_{3}={\\cal E}$ and \n$C_{2}=-\\ell (\\ell +1)$, we obtain \n\\begin{eqnarray}\n \\left[ -\\frac{d^{2}}{d\\alpha ^{2}}+\\kappa ^{2}\\ell (\\ell +1){\\rm sn}\n^{2}\\alpha \\right] \\psi (\\alpha ) &=&{\\cal E}\\psi (\\alpha ), \\\\\n\\ \\left[ -\\frac{d^{2}}{d\\beta ^{2}}+\\kappa ^{\\prime 2}\\ell (\\ell +1){\\rm sn}\n^{2}(\\beta \|\\kappa ^{\\prime })\\right] \\phi (\\beta ) &=&(\\ell (\\ell +1)-{\\cal \nE})\\phi (\\beta ).\n\\end{eqnarray}\nIn this case the coordinates are both real valued. For integer $\\ell $, and\nboth potentials are non-negative, \\ we see that $0\\leq {\\cal E}\\leq \\ell\n(\\ell +1).$ Further the ground state of one Hamiltonian corresponds the the\nhighest energy state of the other, and vice-versa. (Similar equations\nare obtained if we use $H_{2}$ or $H_{1}.$) Before we discuss some of the\ngeneral properties of these equations, we consider first the dynamical\nsymmetry limits which are when $\\kappa =0$ or $1$.\n\n\\subsection{$\\kappa =0$ Limit}\n\nIn the limit $\\kappa =0$, and $H_{3}=L_{z}^{2}=$ $m^{2}$, so that ${\\cal E}\n=m^{2}$, and the coordinate Hamiltonians are: \n\\begin{eqnarray}\n\\ -\\frac{d^{2}}{d\\alpha ^{2}}\\psi (\\alpha ) &=&{\\cal E}\\psi (\\alpha ), \\\\\n\\ \\left[ -\\frac{d^{2}}{d\\beta ^{2}}-\\frac{\\ell (\\ell +1)}{\\cosh ^{2}\\beta }\n\\right] \\phi (\\beta ) &=&-{\\cal E}\\phi (\\beta ).\n\\end{eqnarray}\nThe generators in this limit have the simpler form: \n\\begin{eqnarray}\nI_{\\pm } &=&\\ e^{\\pm i\\alpha }\\left[ \\mp \\cosh \\beta \\ \\frac{\\partial }{\n\\partial \\beta }+i\\sinh \\beta \\frac{\\partial }{\\partial \\alpha }\\right] \n\\nonumber \\\\\nI_{3} &=&\\ -i\\frac{\\partial }{\\partial \\alpha }\n\\end{eqnarray}\nThe first equation is free motion, while the second corresponds to bound\nstates of the P\\\"oschl-Teller potential. For the discrete representations of $\nsu(2)$, one obtains the bound state spectrum $\\ -{\\cal E}=-m^{2}.$ The\nwavefunctions are of the form: \n\\begin{equation}\n\\Psi (\\alpha ,\\beta ;\\kappa =0)\\sim P_{\\ell }^{m}(\\tanh \\beta )\\exp (\\pm\nim\\alpha ),\\qquad m=-\\ell ,...,\\ell ;\\quad \\ell =0,1,2,...\n\\end{equation}\nwith $-\\pi <\\alpha <\\pi ,\\,-\\infty <\\beta <\\infty .$\n\n\\subsection{$\\kappa =1$ Limit}\n\nIn the limit $\\kappa =1$, $H_{3}=L^{2}-L_{x}^{2},$ the equations reduce to \n\\begin{eqnarray}\n\\ \\left[ -\\frac{d^{2}}{d\\alpha ^{2}}-\\frac{\\ell (\\ell +1)}{\\cosh ^{2}\\alpha }\n\\right] \\ \\psi (\\alpha ) &=&[{\\cal E}-\\ell (\\ell +1)]\\psi (\\alpha )={\\cal E}\n^{\\prime }\\psi (\\alpha ) \\\\\n-\\frac{d^{2}}{d\\beta ^{2}}\\phi (\\beta ) &=&[\\ell (\\ell +1)-{\\cal E}]\\phi\n(\\beta )=-{\\cal E}^{\\prime }\\phi (\\beta )\n\\end{eqnarray}\nThe generators are the same as in the $\\kappa =0$ limit, with $\\alpha $ and $\n\\beta $ interchanged. The second equation is now free motion, while the\nfirst corresponds to bound states of the P\\\"oschl-Teller potential with a\nshifted eigenvalue. For the discrete representations of $su(2)$, one obtains\nthe bound state spectrum ${\\cal E}^{\\prime }=-m^{2}.$ The wavefunctions are\nof the form: \n\\begin{equation}\n\\Psi (\\alpha ,\\beta ;\\kappa =1)\\sim P_{\\ell }^{m}(\\tanh \\alpha )\\exp (\\pm\nim\\beta ),\\qquad m=-\\ell ,...,\\ell ;\\quad \\ell =0,1,2,...\n\\end{equation}\nwith $-\\infty <\\alpha <\\infty ,\\,-\\pi <\\beta <\\pi .$ \\ \n\n\\subsection{ Band Edges and $su(2)$}\n\nFor general values of $\\kappa $, the Lam\\'e Hamiltonians are periodic (as seen\nin Fig. 1), and will have band structure. As the discrete representations of \n$su(2)$ correspond to single-valued wavefunctions, the discrete\nrepresentations can at most describe the band edges. Since we have an\nalgebraic realization of the Hamiltonians whose spectrum corresponds to the\nLam\\'e equations, we can obtain the eigenvalues of the Hamiltonian $H_{k}$\n(and hence the band edges) by a direct diagonalization in the spherical\nharmonic basis $\\left\| \\ell m\\right\\rangle $, which can then be related to\nthe results for the elliptic basis through the coordinate transformations.\nThe resulting functions are \ndoubly periodic solutions of Lam\\'e's equation known as Lam\\'e\npolynomials\\cite{Arscott}. These polynomials are of the form\nsn$^a x$cn$^b x$dn$^c x F_p($sn$^2 x )$, where $a,b,c=0,1$ and\n$a+b+c+2p=\\ell$. Here $F_p(z)$ is a polynomial in $z$ of order\n$p$.\nA discussion of these functions can be found in \\cite{bateman,whitwat,Arscott}.\nFor $\\ell =1$, there are three eigenstates of $H_{3}$ given by \n\\begin{eqnarray}\n\\Psi _{1}(\\alpha ,\\beta ;\\kappa ) &=&\\left\| 10\\right\\rangle \\sim {\\rm sn}\n\\,\\alpha {\\rm dn}(\\beta \|\\kappa ^{\\prime }) \\nonumber \\\\\n\\Psi _{2}(\\alpha ,\\beta ;\\kappa ) &=&\\frac{1}{\\sqrt{2}}\\left[ \\left\|\n11\\right\\rangle +\\left\| 1-1\\right\\rangle \\right] \\sim {\\rm \\ cn}\\,\\alpha \n{\\rm cn}(\\beta \|\\kappa ^{\\prime }) \\\\\n\\Psi _{3}(\\alpha ,\\beta ;\\kappa ) &=&\\frac{1}{\\sqrt{2}}\\left[ \\left\|\n11\\right\\rangle -\\left\| 1-1\\right\\rangle \\right] \\sim {\\rm \\ dn}\\,\\alpha \n{\\rm sn}(\\beta \|\\kappa ^{\\prime }) \\nonumber\n\\end{eqnarray}\nwith eigenvalues $E_{1}=1+\\kappa ^{2}$, $E_{2}=1$, $E_{3}=\\kappa ^{2}$. In\nthe dynamical symmetry limits, these three states reduce to those discussed\nabove.\n\nFor $\\ell =2$, we have \n\\begin{eqnarray}\n\\Psi _{1}(\\alpha ,\\beta ;\\kappa ) &=&\\frac{1}{\\sqrt{2}}\\left[ \\left\|\n22\\right\\rangle -\\left\| 2-2\\right\\rangle \\right] \\sim {\\rm cn}\\,\\alpha {\\rm \ndn}\\,\\alpha {\\rm sn}(\\beta \|\\kappa ^{\\prime }){\\rm cn}(\\beta \|\\kappa\n^{\\prime }) \\nonumber \\\\\n\\Psi _{2}(\\alpha ,\\beta ;\\kappa ) &=&\\frac{1}{\\sqrt{2}}\\left[ \\left\|\n21\\right\\rangle +\\left\| 2-1\\right\\rangle \\right] \\sim {\\rm sn}\\,\\alpha {\\rm \ncn}\\,\\alpha {\\rm cn}(\\beta \|\\kappa ^{\\prime }){\\rm dn}(\\beta \|\\kappa\n^{\\prime }) \\nonumber \\\\\n\\Psi _{3}(\\alpha ,\\beta ;\\kappa ) &=&\\frac{1}{\\sqrt{2}}\\left[ \\left\|\n21\\right\\rangle -\\left\| 2-1\\right\\rangle \\right] \\sim {\\rm sn}\\,\\alpha {\\rm \ndn}\\,\\alpha {\\rm sn}(\\beta \|\\kappa ^{\\prime }){\\rm dn}(\\beta \|\\kappa\n^{\\prime }) \\\\\n\\Psi _{4}(\\alpha ,\\beta ;\\kappa ) &\\sim &\\left\| 22\\right\\rangle +\\left\|\n2-2\\right\\rangle +\\sqrt{\\frac{2}{3}}\\frac{2f_{+}(\\kappa )-1-\\kappa ^{2}}{\n1-\\kappa ^{2}}\\left\| 20\\right\\rangle \\nonumber \\\\\n&\\sim &(1-f_{+}(\\kappa ){\\rm sn}\\,^{2}\\alpha )(1-f_{-}(\\kappa ^{\\prime })\n{\\rm sn}\\,^{2}(\\beta \|\\kappa ^{\\prime })) \\nonumber \\\\\n\\Psi _{5}(\\alpha ,\\beta ;\\kappa ) &\\sim &\\left\| 22\\right\\rangle +\\left\|\n2-2\\right\\rangle +\\sqrt{\\frac{2}{3}}\\frac{2f_{-}(\\kappa )-1-\\kappa ^{2}}{\n1-\\kappa ^{2}}\\left\| 20\\right\\rangle \\nonumber \\\\\n&\\sim &(1-f_{-}(\\kappa ){\\rm sn}\\,^{2}\\alpha )(1-f_{+}(\\kappa ^{\\prime })\n{\\rm sn}\\,^{2}(\\beta \|\\kappa ^{\\prime })) \\nonumber\n\\end{eqnarray}\nwith eigenvalues \n\\begin{eqnarray}\nE_{1} &=&1+\\kappa ^{2},\\qquad E_{2}=4+\\kappa ^{2},\\qquad E_{3}=1+4\\kappa\n^{2},\\qquad \\\\\nE_{4} &=&2f_{+}(\\kappa ),\\qquad E_{5}=\\ 2f_{-}(\\kappa ),\\qquad \n\\end{eqnarray}\nwhere $f_{\\pm }(\\kappa )=\\left( 1+\\kappa ^{2}\\pm \\sqrt{1-\\kappa\n ^{2}\\kappa ^{\\prime 2}}\\right) $. \nIn general there will\nbe a relation between the eigenstates $\\left\| \\ell\n m\\right\\rangle $ of the $2\\ell+1$ band edges and\nproducts of Lam\\'e polynomials. The band edges for $\\ell =1,2$ are shown in\nFig. 3 as a function of $\\kappa ^{2}$. The bands are indicated by the shaded\nregions. The dashed line indicates the height of the potential. One can see\nthat the bands merge at $\\kappa =0$, and pass to the bound and scattering\nstates of the P\\\"oschl-Teller potential for $\\kappa =1$.\n\n\\section{su(1,1) Realizations of the Lam\\'e Equation}\n\n\\subsection{Elliptic Parametrization of su(1,1)}\n\nIn order to discuss the eigenstates in the band, and develop the dispersion\nrelation, we require more general representations. In the spirit of recent\nwork on the Scarf potential where it was shown that one can use a $su(1,1)$\ndynamical symmetry to analytically solve for the dispersion relation\\cite\n{prllk}, states and transfer matrix, we consider transforming our generators\nto $su(1,1)$. Consider the transformation of our previous coordinates\nassociated with $x\\rightarrow -ix,\\,y\\rightarrow -iy:$ \n\\begin{equation}\nx=-i\\kappa \\,{\\rm sn}\\,\\alpha \\,{\\rm sn}\\,\\beta ,\\,y=\\frac{\\kappa }{\\kappa\n^{\\prime }}\\,{\\rm cn}\\,\\alpha \\,{\\rm cn}\\,\\beta ,\\,z=\\frac{1}{\\kappa\n^{\\prime }}\\,{\\rm dn}\\,\\alpha \\,{\\rm dn}\\,\\beta ,\n\\end{equation}\nwhere $0\\leq \\alpha <4K,\\,0\\leq \\beta <iK^{^{\\prime }}$\\cite{patera}.\nThis now parametrizes the hyperbolic surface $z^{2}-x^{2}-y^{2}=1$\nillustrated in Fig. 4(a) for $\\kappa ^{2}=1/2$. Contours are shown for\nselected values of $\\alpha $ and $\\beta $ in Fig. 4(b). The generators of $\nsu(1,1)$ algebra are given in Table 2 as realization (I), and satisfy the\ncommutation relations $\\left[ L_{x},L_{y}\\right] =-L_{z}$, $\\left[\nL_{y},L_{z}\\right] =L_{x}$, and $\\left[ L_{z},L_{x}\\right]\n=L_{y}.$ The\nCasimir invariant now has the form \n\\begin{equation}\nC_{2}=L_{z}^{2}-L_{x}^{2}-L_{y}^{2}=\\frac{-1}{\\kappa ^{2}({\\rm sn}\n^{2}\\,\\alpha -{\\rm sn}^{2}\\,\\beta )}\\left( \\frac{\\partial ^{2}}{\\partial\n\\alpha ^{2}}-\\frac{\\partial ^{2}}{\\partial \\beta ^{2}}\\right) \n\\end{equation}\nAs in the $su(2)$ case, there are three forms of bilinear Hamiltonians which\nlead to Schr\\\"odinger equations in $\\alpha $ and $\\beta $. These are given in\nthe top right of Table 2. If we choose \\ $H_{3}=L_{z}^{2}-\\kappa\n^{2}L_{x}^{2}={\\cal E}$, together with the Casimir invariant $C_{2}=-\\ell\n(\\ell +1)$, we obtain the decoupled Lam\\'e equations:\n\\begin{eqnarray}\n\\ \\left[ -\\frac{d^{2}}{d\\alpha ^{2}}+\\kappa ^{2}\\ell (\\ell +1){\\rm sn}\n^{2}\\alpha \\right] \\psi (\\alpha ) &=&-{\\cal E}\\psi (\\alpha ), \\label{sua} \\\\\n\\ \\left[ -\\frac{d^{2}}{d\\beta ^{2}}+\\kappa ^{2}\\ell (\\ell +1){\\rm sn}\n^{2}\\beta \\right] \\phi (\\beta ) &=&-{\\cal E}\\phi (\\beta ). \\label{sub}\n\\end{eqnarray}\nAgain similar results are obtained if we use $H_{1}$ or $H_{2}$, the only\ndifference arising in the definition of the eigenvalue. Note that the range\nof $\\beta $ is along the imaginary axis. This realization is\nproblematic in the $\\kappa =0$ and $\\kappa =1$ limits since the\ngenerators become singular. Consequently we consider a slightly different realization of\n$su(1,1)$ below.\n\n\\subsection{Another su(1,1) Realization}\n\nWhile we do not have a coordinate system which avoids the singularities at $\n\\kappa =1$, it is possible to at least allow a study of the scattering\nstates when $\\kappa =0.$ To do this we make the transformations (i) $\\beta\n\\rightarrow \\beta +K+iK^{^{\\prime }}$ followed by (ii) $\\beta \\rightarrow\n-i\\beta $. As a result: \n\\begin{eqnarray}\n{\\rm sn}(\\beta \|\\kappa ) &\\rightarrow &\\frac{1}{\\kappa }{\\rm dn}(\\beta\n\|\\kappa ^{\\prime }), \\nonumber \\\\\n{\\rm cn}(\\beta \|\\kappa ) &\\rightarrow &\\frac{\\kappa ^{\\prime }}{i\\kappa }\n{\\rm cn}(\\beta \|\\kappa ^{\\prime }), \\nonumber \\\\\n{\\rm dn}(\\beta \|\\kappa ) &\\rightarrow &\\kappa ^{\\prime }{\\rm sn}(\\beta\n\|\\kappa ^{\\prime }).\n\\end{eqnarray}\nThis results in coordinates\\cite{patera} \n\\begin{equation}\nx=-i\\,{\\rm sn}\\,\\alpha \\,{\\rm dn}(\\beta \|\\kappa ^{\\prime }),\\,y=-i\\,{\\rm cn}\n\\,\\alpha \\,{\\rm cn}(\\beta \|\\kappa ^{\\prime }),\\,z={\\rm dn}\\,\\alpha \\,{\\rm sn}\n(\\beta \|\\kappa ^{\\prime }),\n\\end{equation}\nwhere $0\\leq \\alpha <4K,\\,-iK\\leq \\beta <-iK+K^{^{\\prime }}$, also\nsatisfying $z^{2}-x^{2}-y^{2}=1$. Typical contours for this parametrization\nare shown in Fig. 4(c). The generators are given in Table 2 as realization\n(II) and satisfy $\\left[ L_{x},L_{y}\\right] =-L_{z}$, $\\left[ L_{y},L_{z}\n\\right] =L_{x}$, and $\\left[ L_{z},L_{x}\\right] =L_{y}$ with Casimir\ninvariant \n\\begin{equation}\nC=L_{z}^{2}-L_{x}^{2}-L_{y}^{2}=\\ \\frac{1}{\\kappa ^{2}{\\rm cn}^{2}\\alpha\n+\\kappa ^{\\prime 2}{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime })}\\left( \\frac{\n\\partial ^{2}}{\\partial \\alpha ^{2}}+\\frac{\\partial ^{2}}{\\partial \\beta ^{2}\n}\\right)\n\\end{equation}\n\\qquad The Hamiltonians which result in seperable Hamiltonians are given in\nthe bottom right of Table 2. Using $H_{1}=L_{x}^{2}+\\kappa ^{\\prime\n2}L_{y}^{2}={\\cal E}\\ $and $C_{2}=-\\ell (\\ell +1)$ leads to the two Lam\\'{e}\nequations: \n\\begin{eqnarray}\n\\ \\left[ -\\frac{d^{2}}{d\\alpha ^{2}}+\\kappa ^{2}\\ell (\\ell +1){\\rm sn}\n^{2}\\alpha \\right] \\psi (\\alpha ) &=&{\\cal E}\\psi (\\alpha ), \\nonumber \\\\\n\\ \\left[ -\\frac{d^{2}}{d\\beta ^{2}}+\\kappa ^{\\prime 2}\\ell (\\ell +1){\\rm sn}\n^{2}(\\beta \|\\kappa ^{\\prime })\\right] \\phi (\\beta ) &=&(\\ell (\\ell +1)-{\\cal \nE})\\phi (\\beta ).\n\\end{eqnarray}\nIt should be kept in mind that $\\beta $ is still complex valued. Never the\nless this realization allows an analysis of the $\\kappa =0$ dynamical\nsymmetry.\n\n\\subsection{$\\kappa =0$ Limit}\n\n\\bigskip\n\nThis realization is not singular in the $\\kappa =0$ limit (although it is in\nthe $\\kappa =1$ case). Taking $\\kappa =0$, and shifting $\\beta $ to be on\nthe real axis by $\\beta =\\theta -i\\pi /2$, we find \n\\begin{eqnarray}\nL_{\\pm } &=&\\pm e^{\\pm i\\alpha }\\left( \\mp \\sinh \\theta \\frac{\\partial }{\n\\partial \\theta }+i\\cosh \\theta \\frac{\\partial }{\\partial \\alpha }\\right) \n\\nonumber \\\\\nL_{z} &=&-\\ \\frac{\\partial }{\\partial \\alpha }.\n\\end{eqnarray}\nTo recover the usual $su(1,1)$ commutation relations $[I_{z},I_{\\pm }]=\\pm\nI_{\\pm }$, $[I_{+},I_{-}]=-2I_{z}$, we then identify $I_{\\pm }=\\pm L_{\\pm }$\nand $I_{z}=iL_{z}$. It is convient to perform the\ntransformations: $\\theta \\rightarrow \\theta -i\\pi /2$, $\\tanh \\theta\n\\rightarrow \\cos \\theta $, followed by a similarity transformation $f(\\theta\n)=\\sqrt{\\sin \\theta }$, and $\\theta \\rightarrow i\\theta $. Then we have the\nform: \n\\begin{eqnarray}\n\\ -\\frac{d^{2}}{d\\alpha ^{2}}\\ \\psi (\\alpha ) &=&m^{2}\\psi (\\alpha ), \n\\nonumber \\\\\n\\ \\left[ -\\frac{d^{2}}{d\\theta ^{2}}+\\frac{m^{2}-1/4}{\\sinh ^{2}\\theta }\n\\right] \\phi (\\theta ) &=&-(\\ell +\\frac{1}{2})^{2}\\phi (\\theta ).\n\\end{eqnarray}\nThe principal series $\\ell =-1/2+i\\rho $ of the projective representations\nof $su(1,1)$ now describe these scattering states, and the eigenfunctions\nare of the form: \n\\begin{equation}\n\\Psi _{\\ell }^{m}\\sim \\sqrt{i\\sinh \\theta }P_{\\ell }^{m}(\\cosh \\theta\n)e^{\\pm im\\alpha }\\quad \\qquad m\\in \\Re ,\\ \\ell =-1/2+i\\rho ,\\ \\rho >0.\n\\end{equation}\n\n\\section{Band Structure of the Lam\\'{e} Hamiltonian}\n\nWe now focus on the properties of the Lam\\'e equation in the form of Eq. (1).\nAs we are interested in band structure, \\ the eigenstates must satisfy\nBloch's theorem. For a potential which is periodic with period $a$, $\nV(x+a)=V(x)$, the wavefunctions must be of the form \n\\begin{equation}\n\\Psi _{k}(x)=u_{k}(x)\\exp [-ikx],\n\\end{equation}\nwhere $k$ is the wavenumber and $u_{k}(x)$ has the periodicity of the\nlattice: $u_{k}(x+a)=u_{k}(x)$. As the eigenstates $\\Psi _{k}(x)$ are not\nperiodic, the doubly periodic solutions of the\nLam\\'e equation do not play a role for energies in the band. Rather, we look\nto the more general class of solutions expressed in terms of Jacobi theta\nfunctions\\cite{whitwat}. Starting with the Hamiltonian \n\\begin{equation}\nH\\psi =\\left[ -\\frac{d^{2}}{dx^{2}}+\\kappa ^{2}\\ell (\\ell +1){\\rm sn}\n^{2}(x\|\\kappa )\\right] \\psi (x)={\\cal E}\\psi (x), \\label{Lameham}\n\\end{equation}\nthe solutions for positive integer $\\ell $ are given parametrically by\n\n\\begin{equation}\n\\psi (x)=\\prod_{n=1}^{\\ell }\\left[ \\frac{{\\cal H}(x+\\alpha _{n})}{\\theta (x)}\ne^{-xZ(\\alpha _{n})}\\right]\n\\end{equation}\nwhere ${\\cal H}$ and $\\theta$ are theta functions, and\n$\\alpha _{1},\\alpha _{2},...\\alpha _{\\ell }$ are constants\ndetermined by the constraints: \n\\begin{eqnarray}\n{\\cal E} &=&\\sum_{n=1}^{\\ell }\\frac{1}{{\\rm sn}^{2}\\alpha _{n}}-\\left[ \\sum_{n=1}^{\\ell }\n{\\rm cn}\\alpha _{n}{\\rm dn}\\alpha _{n}/{\\rm sn}\\alpha _{n}\\right] ^{2}\n\\label{eq:const} \\\\\n0 &=&\\sum_{p=1}^{\\ell }\\frac{{\\rm sn}\\alpha _{p}{\\rm cn}\\alpha _{p}{\\rm dn}\n\\alpha _{p}+{\\rm sn}\\alpha _{n}{\\rm cn}\\alpha _{n}{\\rm dn}\\alpha _{n}}{{\\rm \nsn}^{2}\\alpha _{p}-{\\rm sn}^{2}\\alpha _{n}}\\quad (p\\neq n)\n\\end{eqnarray}\nIf this solution is not doubly periodic, a second solution is\n\n\\begin{equation}\n\\psi ^{^{\\prime }}(x)=\\prod_{n=1}^{\\ell }\\left[ \\frac{{\\cal H}(x-\\alpha _{n})}{\n\\theta (x)}e^{xZ(\\alpha _{n})}\\right].\n\\end{equation}\nWe can then identify the dispersion relation by putting these\nwavefunctions in Bloch form and by using the periodicity of the\ntheta functions to extract $u_k(x)$. We find\n\n\\begin{equation}\nk({\\cal E})=-i\\sum_{n=1}^{\\ell }Z(\\alpha _{n}\|\\kappa ^{2})+\\frac{\\ell \\pi }{2K}.\n\\label{eq:disp}\n\\end{equation}\nWe will start with the case of $\\ell =1$, where simple anlytic\nresults can be obtained. We then derive the transfer matrix for\nthe general case of integer $\\ell .$\n\n\\subsection{$ \\ell =1$ Results}\n\nSince there is only one parameter $\\alpha ,$ the constraint equation is\nsimply\n\n\\begin{equation}\n{\\rm dn}^{2}\\alpha ={\\cal E}-\\kappa ^{2}.\n\\end{equation}\nThe condition that the dispersion relation is real,\nRe$Z(\\alpha\|\\kappa^2)=0$, results in two energy bands given by \n\\begin{equation}\n\\kappa ^{2}\\leq {\\cal E}\\leq 1,\\quad 1+\\kappa ^{2}\\leq {\\cal E}. \n\\end{equation}\nThese are shown in Fig. 3 (top). In the lower band, $\\alpha $ has the form $\n\\alpha =K+i\\eta $, where $\\eta $ ranges from $K^{\\prime }$ at ${\\cal E}\n=\\kappa ^{2}$, to $0$ at ${\\cal E}=1.$ In the upper band, $\\alpha =i\\eta $,\nwhere $\\eta $ ranges from $0$ at ${\\cal E}=1+\\kappa ^{2}$, to $K^{\\prime }$\nas ${\\cal E}\\rightarrow \\infty $. This path traced out by the\nparameter $\\alpha$ as a function energy ${\\cal E}$ is shown schematically in Fig. 5. The upper\nand lower sides correspond to band gaps while the right and left\nedges are the energy bands.\nUsing the specific forms of $\\alpha $ for each band, the dispersion relation\nbecomes \n\\begin{equation}\nk({\\cal E})=\\left\\{ \n\\begin{array}{ll}\n-Z(\\eta \|\\kappa ^{\\prime 2})+\\frac{\\pi }{2K}(1-\\frac{\\eta }{K^{\\prime }})+\n\\sqrt{\\frac{({\\cal E}-\\kappa ^{2})(1-{\\cal E)}}{1+\\kappa ^{2}-{\\cal E}}} & \n\\kappa ^{2}\\leq {\\cal E}\\leq 1, \\\\ \n-Z(\\eta \|\\kappa ^{\\prime 2})+\\frac{\\pi }{2K}(1-\\frac{\\eta }{K^{\\prime }})+\n\\sqrt{\\frac{({\\cal E}-\\kappa ^{2}-1)({\\cal E}-\\kappa ^{2})}{{\\cal E}-1}} & \n1+\\kappa ^{2}\\leq {\\cal E}\n\\end{array}\n\\right.\n\\end{equation}\n%\nThis is plotted in Fig. 6 (a) for the case of $\\ell=1$. The\nmomentum $k$ is plotted up to the edge of the Brillouin zone,\nwhich is $k=\\pi/2K$. In the figure we use $\\kappa^2=1/2$, so\nthat the band edges are ${\\cal E}=1/2$, 1 and 3/2. (The\nanalogous behavior for $\\ell=2$ is indicated in Fig. 6(b).)\nThe solution of the Lam\\'e equation in Bloch form is now \n%\n\\begin{equation}\n\\psi _{k}(x)=\\left\\{ \n\\begin{array}{ll}\n\\left[ \\frac{H_{1}(x+i\\eta )}{\\Theta (x)}\\exp (i\\pi x/2K)\\right] \\exp (-ikx)\n& \\kappa ^{2}\\leq {\\cal E}\\leq 1, \\\\ \n\\left[ \\frac{H(x+i\\eta )}{\\Theta (x)}\\exp (i\\pi x/2K)\\right] \\exp (-ikx) & \n1+\\kappa ^{2}\\leq {\\cal E}\n\\end{array}\n\\right. .\n\\end{equation}\nThe component of the wavefunction in square brackets can be checked to be\nperiodic with the periodicity of the direct lattice: $x\\rightarrow x+2K.$\n\nThe dispersion relation displays the desired limits. One can see that $k(\n{\\cal E})\\rightarrow 0$ as ${\\cal E}\\rightarrow \\kappa ^{2}$, and $k({\\cal E}\n)\\rightarrow \\pi /2K$ as ${\\cal E}\\rightarrow 1.$ Further, as the modulus of\nthe elliptic function vanishes, $\\kappa ^{2}\\rightarrow 0$, the Hamiltonian\nbecomes that of a free system, and we find $k({\\cal E})\\rightarrow \\sqrt{\n{\\cal E}}$ as desired. In the P\\\"oschl-Teller limit, $\\kappa ^{2}\\rightarrow 1$\n, the band vanishes, and we find $k({\\cal E})\\rightarrow 0$. Similar\nresults hold for the upper band as well.\n\nFrom the dispersion relation we can compute the group velocity and effective\nmass. To do so, we use the relation between the zeta function and the\nelliptic integrals of the first and second kind\n\n\\begin{equation}\nZ(\\alpha )=E(\\alpha )-\\frac{E(\\kappa ^{2})}{K}\\alpha\n\\end{equation}\nwhere $E(\\kappa ^{2})=\\frac{\\pi }{2}F(-1/2,1/2;1;\\kappa ^{2})$ and $K=\\frac{\n\\pi }{2}F(1/2,1/2;1;\\kappa ^{2})$ are the complete elliptic integrals and $\nE(\\alpha )$ is incomplete. Then, the group velocity is given by:\n\n\\begin{equation}\n\\frac{1}{\\nu }=\\frac{dk({\\cal E})}{d{\\cal E}}\n\\end{equation}\nor\n\n\\begin{equation}\n\\nu ({\\cal E})=\\frac{\\ 2\\sqrt{(1-{\\cal E})({\\cal E}-\\kappa ^{2})(1+\\kappa\n^{2}-{\\cal E})}}{\\kappa ^{2}+\\frac{E(\\kappa ^{2})}{K}-{\\cal E}}.\n\\end{equation}\nWe plot $\\nu ^{2}$ as a function of energy in Fig. 7 for several values of $\n\\kappa .$ As $\\kappa$ approaches zero, the energy gaps vanish,\nand the group velocity approaches the free particle limit $E=M\\nu^2/2=\\nu^2/4$\n(dot-dashed line). As $\\kappa$ approaches unity, the lower band\nvanishes becoming a bound state, and the group velocity is only\nnon-vanishing for the continuum states of the P\\\"oschl-Teller\npotential with ${\\cal E}\\ge 2$. \n\nThe effective mass $M^{\\ast }$ is determined by\n\n\\begin{eqnarray}\n\\frac{1}{M^{\\ast }} &=&\\frac{d\\nu }{dk} \\nonumber \\\\\n&=&-2\\frac{({\\cal E}-\\kappa ^{2})(1+\\kappa ^{2}-{\\cal E})+({\\cal E}\n-1)(1+\\kappa ^{2}-{\\cal E})-({\\cal E}-1)({\\cal E}-\\kappa ^{2})}{({\\cal E}\n-\\kappa ^{2}-\\frac{E(\\kappa ^{2})}{K})^{2}} \\nonumber \\\\\n&&+4\\frac{({\\cal E}-1)({\\cal E}-\\kappa ^{2})(1+\\kappa ^{2}-{\\cal E})}{({\\cal \nE}-\\kappa ^{2}-\\frac{E(\\kappa ^{2})}{K})^{3}}\n\\end{eqnarray}\nWe plot $1/M^$ in Fig. 8 for selected values of $\\kappa .$ One can see that\nas $\\kappa ^{2}\\rightarrow 0$, the gaps vanish and $M^{\\ast }\\rightarrow M=1/2$ as expected for\nthe free particle.\n\n\\subsection{Transfer Matrix for the Lam\\'e Hamiltonian}\n\nThe general form of the transfer matrix is computed using the\ndefinitions in the Appendix. Using the wavefunctions and Eq. (A1), we have \n\\begin{equation}\nr=-\\sum_{n=1}^{\\ell }[Z(\\alpha _{n})-\\frac{{\\rm sn}\\,\\alpha _{n}\\,{\\rm dn}\n\\,\\alpha _{n}}{{\\rm cn}\\,\\alpha _{n}}]-i\\frac{\\ell \\pi }{2k}.\n\\end{equation}\nConsequently, we see that \n\\begin{equation}\nr+ik({\\cal E})=\\sum_{n=1}^{\\ell }\\frac{{\\rm sn}\\,\\alpha _{n}\\,{\\rm dn}\n\\,\\alpha _{n}}{{\\rm cn}\\,\\alpha _{n}}.\n\\end{equation}\nThe transfer matrix then has the form \n\\begin{equation}\nT\\,=\\,\\left( \n\\begin{array}{cc}\n\\cos 2k({\\cal E})K & i\\left( \\sum_{n=1}^{\\ell }\\frac{{\\rm sn}\\,\\alpha _{n}\\,\n{\\rm dn}\\,\\alpha _{n}}{{\\rm cn}\\,\\alpha _{n}}\\right) ^{-1}\\sin 2k({\\cal E})K\n\\\\ \ni(\\sum_{n=1}^{\\ell }\\frac{{\\rm sn}\\,\\alpha _{n}\\,{\\rm dn}\\,\\alpha _{n}}{{\\rm \ncn}\\,\\alpha _{n}})\\sin 2k({\\cal E})K & \\cos 2k({\\cal E})K\n\\end{array}\n\\right) .\n\\end{equation}\n%\n(Note that this is in the form of Eq. (A.3) rather than (A.5).)\nWhile this expression requires knowledge of the parameters $\\alpha _{n}$,\none can obtain various limits of this for the free particle and\nP\\\"oschl-Teller potentials.\n\n\\subsection{The $\\kappa =0$ Free Particle Limit}\n\nWe start first with the $\\ell =1$ case. Taking $\\kappa =0$ in our transfer\nmatrix, we obtain for the upper and lower bands\n\\begin{equation}\nT\\,=\\,\\left( \n\\begin{array}{cc}\n\\cosh 2k({\\cal E})K & i\\frac{{\\rm cn}\\,\\alpha }{{\\rm sn}\\,\\alpha \\,{\\rm dn}\n\\,\\alpha }\\ \\sinh 2k({\\cal E})K \\\\ \ni\\frac{{\\rm sn}\\,\\alpha \\,{\\rm dn}\\,\\alpha }{{\\rm cn}\\,\\alpha }\\sinh 2k(\n{\\cal E})K & \\cosh 2k({\\cal E})K\\ \n\\end{array}\n\\right) .\n\\end{equation}\n$\\ $According to Eq. (\\ref{eq:const}), \n\\begin{equation}\n\\frac{{\\rm sn}\\,\\alpha \\,{\\rm dn}\\,\\alpha }{{\\rm cn}\\,\\alpha }=\\left[ \\frac{\n(1+\\kappa ^{2}-{\\cal E})({\\cal E}-\\kappa ^{2})}{({\\cal E}-1)}\\right] ^{1/2}.\n\\label{eq:ratio}\n\\end{equation}\nTaking $\\kappa =0$, we have $r\\rightarrow 0$, so that \n\\begin{equation}\nk({\\cal E})=-i\\frac{{\\rm sn}\\,\\alpha \\,{\\rm dn}\\,\\alpha }{{\\rm cn}\\,\\alpha }=\n\\sqrt{{\\cal E}}.\n\\end{equation}\nThe transfer matrix becomes that for a free particle, given by: \n\\begin{equation}\nT\\,=\\,\\left( \n\\begin{array}{cc}\n\\cos \\pi \\sqrt{{\\cal E}} & {\\cal E}^{-1/2}\\sin \\pi \\sqrt{{\\cal E}} \\\\ \n-{\\cal E}^{1/2}\\sin \\pi \\sqrt{{\\cal E}}\\ & \\ \\cos \\pi \\sqrt{{\\cal E}}\n\\end{array}\n\\right) . \\label{eq:freetrans}\n\\end{equation}\n For the general case of integer $\\ell $, we start with\nthe constraint equations (\\ref{eq:const}) and take the limit $\\kappa\n\\rightarrow 0.$ We first must show that \n\\begin{equation}\nr+ik({\\cal E})=\\sum_{n=1}^{\\ell }\\tan \\alpha _{n}\\rightarrow i\\sqrt{{\\cal E}}\n. \\label{eq:sumtan}\n\\end{equation}\nThen, from the definition of $r$ in the appendix, it must vanish for free\nmotion, so that $r\\rightarrow 0$ implies $k({\\cal E})\\rightarrow \\sqrt{{\\cal \nE}}$. Then we recover the free particle transfer matrix. While we can show\nthat the sum in Eq. (\\ref{eq:sumtan}) tends to $i\\sqrt{{\\cal E}}$ for small $\n\\ell $ on a case by case basis, we do not yet have a general proof. However,\nsince the transfer matrix must be that of a free particle in this limit, it\nis clear that Eq. (\\ref{eq:sumtan}) must hold, and we can use this instead\nto provide an additional relation among the parameters $\\alpha _{n}$.\n\n\\subsection{The $\\kappa =1$ P\\\"oschl-Teller Limit}\n\n\\bigskip\n\nIn the limit $\\kappa =1$, the Hamiltonian $H$ becomes the P\\\"oschl-Teller\nHamiltonian, and our transfer matrix should reduce to that case. We start\nfirst with $\\ell =1$ and examine the upper band.(The lower band becomes\ndegenerate at $\\kappa =1$). In this limit, $K\\rightarrow \\infty $ and $\nK^{\\prime }\\rightarrow \\pi /2$. For the upper band, \n\\begin{equation}\nk({\\cal E})\\rightarrow \\sqrt{{\\cal E}-2}+\\frac{\\pi +2i\\alpha }{2K},\n\\end{equation}\nFrom Eq. (\\ref{eq:ratio}), we have in the $\\kappa =1$ limit: \n\\begin{equation}\n\\frac{{\\rm sn}\\,\\alpha \\,{\\rm dn}\\,\\alpha }{{\\rm cn}\\,\\alpha }=\\tanh \\alpha\n=i\\sqrt{{\\cal E}-2}. \\label{leq:ratioa}\n\\end{equation}\nThe asymptotic form of the transfer matrix (as $K\\rightarrow \\infty $)\nbecomes: \n\\begin{equation}\nT\\,=\\,\\left( \n\\begin{array}{cc}\n\\cos (2K\\sqrt{{\\cal E}-2}+2i\\alpha ) & ({\\cal E}-2)^{-1/2}\\sin (2K\\sqrt{\n{\\cal E}-2}+2i\\alpha ) \\\\ \n-\\ ({\\cal E}-2)^{1/2}\\sin (2K\\sqrt{{\\cal E}-2}+2i\\alpha ) & \\cos (2K\\sqrt{\n{\\cal E}-2}+2i\\alpha )\\ \n\\end{array}\n\\right) .\n\\end{equation}\nWe must now change this form of the transfer matrix, defined for periodic\npotentials $T$, to the form used in the P\\\"oschl-Teller case which is not\nperiodic, ${\\cal T}\\,$(see Appendix). Using $k({\\cal E})=\\sqrt{{\\cal E}-2}$\n, we find: \n\\begin{equation}\n{\\cal T}\\,_{\\kappa =1}=\\,\\left( \n\\begin{array}{cc}\n\\frac{\\Gamma (ik)\\Gamma (1+ik)}{\\Gamma (2+ik)\\Gamma (ik-1)} & \\ 0 \\\\ \n0 & \\ \\frac{\\Gamma (-ik)\\Gamma (1-ik)}{\\Gamma (2-ik)\\Gamma (-ik-1)}\n\\end{array}\n\\right) .\n\\end{equation}\n\nFor the case of arbitrary $\\ell $, we have the more general relations: \n\\begin{equation}\n\\sum_{n=1}^{\\ell }\\frac{\\,{\\rm sn}\\alpha _{n}{\\rm cn}\\alpha _{n}}{{\\rm dn}\n\\alpha _{n}}\\rightarrow \\sum_{n=1}^{\\ell }\\tanh \\alpha _{n}=i\\sqrt{{\\cal E}\n-\\ell (\\ell +1)}.\n\\end{equation}\nThe dispersion relation becomes \n\\begin{equation}\nk({\\cal E})\\rightarrow -i\\sum_{n=1}^{\\ell }(\\tanh \\alpha _{n}-\\frac{\\alpha\n_{n}}{K})+\\ell \\frac{\\pi }{2K}=\\sqrt{{\\cal E}-\\ell (\\ell +1)}+\\ell \\frac{\\pi \n}{2K}+i\\frac{1}{K}\\sum_{n=1}^{\\ell }\\alpha _{n}.\n\\end{equation}\nTo compute the form of the transfer matrix for the P\\\"oschl-Teller potential,\nwe use the notation\n\n\\begin{equation}\n{\\cal T}\\,=\\,\\left( \n\\begin{array}{cc}\nF & G\\ \\\\ \nG^{\\ast } & F^{\\ast }\n\\end{array}\n\\right) .\n\\end{equation}\nwhere, from the appendix, we have: \n\\begin{equation}\nF=(-)^{\\ell }\\exp \\left[ -2\\sum_{n=1}^{\\ell }\\alpha _{n}\\right]\n=\\prod_{n=1}^{\\ell }\\frac{\\tanh \\alpha _{n}-1}{\\tanh \\alpha _{n}+1}.\n\\end{equation}\nThis can be further simplified using the limiting form of the constraint\nequations (\\ref{eq:const}): \n\\begin{eqnarray}\n{\\cal E} &=&-\\left[ \\sum_{n=1}^{\\ell }\\frac{1-\\tanh ^{2}\\alpha _{n}}{\\tanh\n\\alpha _{n}}\\right] ^{2}+\\sum_{n=1}^{\\ell }\\coth ^{2}\\alpha _{n}\\ , \n\\nonumber \\\\\n0 &=&\\sum_{p=1(p\\neq n)}^{\\ell }\\frac{(1-\\tanh ^{2}\\alpha _{n})\\tanh \\alpha\n_{n}+(1-\\tanh ^{2}\\alpha _{p})\\tanh \\alpha _{p}}{\\tanh ^{2}\\alpha _{p}-\\tanh\n^{2}\\alpha _{n}}\\ .\n\\end{eqnarray}\nFor small $\\ell $ ($\\ell =1,2$) we can solve these equations to show that \n\\begin{equation}\nF=\\frac{\\Gamma (ik)\\Gamma (1+ik)}{\\Gamma (1+ik+\\ell )\\Gamma (ik-\\ell )}\n,\\qquad G=0.\n\\end{equation}\nThis reduces to the correct form of the transfer matrix. For arbitrary $\\ell \n$, we have not been able to solve these equations, but the result must still\nhold, since it is governed by the form of the Hamiltonian. Again, we can use\nthese relations to provide additional relations between the parameters $\n\\alpha _{n}$.\n\n\\section{Conclusions}\n\nWe have presented a group theoretical analysis of the Lam\\'e equation, which\nis an example of a SGA band structure problem for $su(2)$ and $su(1,1)$. We\nhave computed the dispersion relation and transfer matrix, and discussed the\nlimiting dynamical symmetry limits of these results which correspond to the\nP\\\"oschl-Teller and free particle Hamiltonians. Because the general\nHamiltonian is not of the dynamical symmetry the spectrum cannot be obtained\nin closed form. Never the less, a diagonalization of Hamiltonians which are\nbilinear in the angular momentum generators will provide the general\nsolution. There are still many open questions associtated with the group\ntheoretical treatment of the Lam\\'e equation. It would be nice to develop a $\nsu(1,1)$ parametrization which is non-singular for all values of $\\kappa .$\n In addition, the\nScarf and Mathieu equation limits would be interesting to realize more expliticly in\nthe transfer matrix and dispersion relations. Finally, the diagonalization\nof the algebraic Hamiltonians in the continuum $su(1,1)$ bases would be\ninteresting to study. It is clear that the result of the diagonalization\nmust yield the same transcendental equations for the parameters $\\alpha _{n}$\n, but their origin would be different.\n\n\\vspace{2cm}\n\n\\section{\\bf Appendix: Form of the transfer matrix for periodic potentials}\n\\setcounter{equation}{1}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\nFor symmetric periodic potentials with period $\\tau $, we can derive a\ngeneral formular of the transfer matrix. Suppose for a specific energy E, we\nhave two bloch solutions $u_{+}(x)e^{ikx}$, $u_{-}(x)e^{-ikx}$, where $x$ is\nthe coordinate, $k=k({\\cal E})$ is the dispersion relation. Since the\npotential is symmetric, we can define\\cite{James} \n\\begin{equation}\nr=\\frac{u_{+}^{^{\\prime }}(\\tau /2)}{u_{+}(\\tau /2)}=-\\frac{u_{-}^{^{\\prime\n}}(\\tau /2)}{u_{-}(\\tau /2)}.\n\\end{equation}\nThen the transfer matrix is \n\\begin{equation}\nT\\,=\\,\\left( \n\\begin{array}{cc}\n\\cos k\\tau & \\frac{i}{ik+r}\\sin k\\tau \\\\ \ni(ik+r)\\sin k\\tau & \\cos k\\tau \n\\end{array}\n\\right) .\n\\end{equation}\nThe forms of transfer matrices used for periodic and non-periodic potentials\nis different. In the discussion of the Lam\\'e equation, the $\\kappa\n\\rightarrow 1$ limit takes a periodic potential to a non-periodic one, so\nthat we require the transformations that take us from one standard form to\nthe other. If we express the transfer matrix $T$ for the periodic potential\nas \n\\begin{equation}\nT=\\left( \n\\begin{array}{cc}\n{\\rm Re}(F\\exp (ik\\tau )+G) & \\frac{1}{k}{\\rm Im}(F\\exp (ik\\tau )+G) \\\\ \nk{\\rm Im}(F\\exp (ik\\tau )-G)\\ & {\\rm Re}(F\\exp (ik\\tau )-G)\\ \n\\end{array}\n\\right) ,\n\\end{equation}\nthen that of the non-periodic limit will have the form \n\\begin{equation}\n{\\cal T}=\\left( \n\\begin{array}{cc}\nF & G \\\\ \nG^{\\ast }\\ & F^{\\ast }\n\\end{array}\n\\right) .\n\\end{equation}\nIn this notation, $k=k({\\cal E})$ is the dispersion relation, and $\\tau $ is\nthe period of the periodic potential which tends to $\\infty .$\n\n\\newpage\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{intro} For a survey, see for example: F. Iachello, {\\sl Rev. Nuovo\n Cimento} {\\bf 19} 1 (1996), and references there\n in; {\\sl Dynamical Groups and\n Spectrum Generating Algebras}, Eds. A.Barut, A. Bohm and Y.Ne'eman (World\n Scientific, Singapore, 1987);\n F. Iachello and R. Levine, {\\it Algebraic Theory of Molecules} \n (Oxford Press, Oxford, 1995); F. Iachello and\n A. Arima, {\\sl The Interacting Boson\n Model}.\n\n\n\\bibitem{prldk} D. Kusnezov, {\\sl Phys. Rev. Lett.} {\\bf 79} 537 (1997).\n\n\\bibitem{gursey} Y. Alhassid, F. G\\\"ursey and F. Iachello, \n {\\sl Phys. Rev. Lett.} {\\bf 50},\n 873 (1983); F. G\\\"ursey, in Group Theoretical Methods in Physics XI \n (Springer-Verlag, Berlin, 1983) p.106.\n\n\\bibitem{prllk} H. Li and D. Kusnezov, {\\sl Phys. Rev. Lett.}\n {\\bf 83} 1283 (1999); H. Li and D. Kusnezov, preprint (1999).\n\n\n\\bibitem{a} A. Khare and U. Sukhatme, quant-ph/9906044;\n G. Dunne and J. Mannix, {\\sl Phys. Lett.}{\\bf B428},\n 115 (1998);\n G. Dunne and J. Feinberg, {\\sl Phys. Rev.}{\\bf D57},\n 1271 (1998).\n\\bibitem{novikov} S. Novikov, S.V. Manakov, L.P. Pitaevskii and\n V.E. Zakharov, {\\it Theory of Solitons}, (New York, Plenum, 1984);\n\n\\bibitem{sutcliffe} P.M. Sutcliffe,{\\sl J. Phys.}{\\bf A29}, 5187 (1996).\n\n\n\n\\bibitem{brihaye} Y. Brihaye, S. Giller, P. Kosinski and J. Kunz,\n {\\sl Phys. Lett.}{\\bf B293}, 383 (1992).\n\n\\bibitem{Liang} J. Liang, H.J.W. Muller-Kirsten and D.H. Tchrakian, \n {\\sl Phys. Lett.}{\\bf B282}, 105 (1992).\n\n\\bibitem{enolskii} V.I. Enolskii and J.C. Eilbeck, {\\sl J. Phys.} {\\bf A28},\n 1069(1995); \n J. Dittrich and V.I. Inozemtsev, {\\sl J. Phys.} {\\bf A26},\n L753 (1993).\n\n\n\\bibitem{b} R. S. Ward, {\\sl J. Phys.} {\\bf A20}, 2679 (1987).\n\n\n\\bibitem{miller} W. Miller, {\\it Symmetry and Separation of Variables},\n (Addison-Wesley, Reading, MA, 1977).\n\n\\bibitem{patera} J. Patera and P. Winternitz,\n {\\sl J. Math. Phys.} {\\bf 14}, 1130 (1973);\n N.W. Macfadyen and P. Winternitz, \n {\\sl J. Math. Phys.} {\\bf 12}, 281(1971);\n E. Kalnins and W. Miller Jr., {\\sl J. Math. Phys.} {\\bf 15}, 1263 (1974).\n\n\\bibitem{karman} N. Kamran and P.J. Olver,\n {\\sl J. Math. Anal. Appl.} {\\bf 145}, 342 (1990).\n\n\\bibitem{turbiner} A. V. Turbiner, {\\sl J. Phys.} {\\bf A22},\n L1 (1989).\n\n\n\\bibitem{sutherland} B. Sutherland, {\\sl Phys. Rev. } {\\bf A 8},\n 2514 (1973).\n\n\\bibitem{whitwat} E.T. Whittaker and G.N. Watson, {\\it A Course of Modern\n Analysis}, (Cambridge Univ. Press, Cambridge, 1980).\n\n\\bibitem{Arscott} F. M. Arscott, {\\it Periodic Differential\n Equations}, (Pergamon, Oxford, 1981).\n\n\\bibitem{bateman} A. Erd\\'elyi {\\sl et al}, {\\it Higher Transcedental\n Functions}, Vol. 3, (McGraw-Hill, New York, 1953).\n\n\\bibitem{scatt1} Y. Alhassid, F. G\\\"ursey, F. Iachello, {\\sl Ann. Phys. \n(NY)} {\\bf 148}, 346 (1983); {\\sl Ann. Phys. (NY)} {\\bf 167}, 181 (1983).\n\n\\bibitem{James} H.M. James, {\\sl Phys. Rev.} {\\bf 76}, 1602 (1949).\n\\end{thebibliography}\n \n\\newpage\n\n\n\\begin{center}\nFigure Captions\n\\end{center}\n\n\\begin{enumerate}\n\\item[Figure 1.] Various forms of the Lam\\'e potential which are real valued.\nWe plot $V(x)$ for modulus $\\kappa ^{2}=1/2$ given by {\\it (i)} ${\\rm sn}\n^{2}(x\|\\kappa )$ (solid), {\\it (ii)}$\\ $ ${\\rm sn}^{2}(x+iK^{\\prime }\|\\kappa\n)$ (dots), {\\it (iii)}$\\ $ ${\\rm sn}^{2}(x+K+iK^{\\prime }\|\\kappa )$\n(dashes), {\\it (iv)}$\\ {\\rm sn}^{2}(ix\|\\kappa )$ (dot-dashes). For other\nvalues of $x$, the potential is periodic, but generally complex.\n\n\\item[Figure 2.] Coordinate systems for $su(2)$. (a) Parametrization of the\nsphere. (b) Contours of constant $\\alpha$ and $\\beta$. (c) Same as (b) but\nfor the second parametrization.\n\n\\item[Figure 3.] Band edges from the $su(2)$ realization of the Lam\\'e\nequation as a function of $\\kappa ^{2}$ for $\\ell =1$ (top) and $\\ell =2$\n(bottom). The eigenvalues are indicated. The bands are given by the shaded\nregions. The dashed line indicates the value of the energy which is at the\nmaximum of the potential $V(x)=\\kappa ^{2}\\ell (\\ell +1){\\rm sn}^{2}x$\n\n\\item[Figure 4.] Coordinate systems for $su(1,1)$. (a) Parametrization of\nthe hyperboloid. (b) Contours of constant $\\alpha$ and $\\beta$. (c) Same as\n(b) but for the second parametrization.\n\n\\item[Figure 5.] Evolution of the parameter $\\alpha $ for the\n $\\ell =1$ Lam\\'{e} equation. The lower energy band has\n parameter $\\alpha =K+i\\eta $ where \n$\\eta $ ranges from $K^{\\prime }$ at the lower end to $0$ at the upper end.\nThe valence band starts with $\\alpha =0$ and grows to $\\alpha =iK^{\\prime }$\nas the energy $E\\rightarrow \\infty .$\n\n\\item[Figure 6.] (a) Dispersion relation ${\\cal E}(k)$ for the $\\ell=1$ Lam\\'e\n equation with $\\kappa^2=1/2$. The momentum is plotted in units\n of $\\pi/2K$. (b) Analogous behavior for the $\\ell=2$ case.\n\n\\item[Figure 7.] Group velocity for the $\\ell =1$ Lam\\'{e} equation for\nseveral values of the modulus $\\kappa $. We show $v^{2}$ as a function of\nenergy ${\\cal E}$. When $\\kappa =0$, the Hamiltonian is that of a free\nparticle so ${\\cal E}\\propto v^{2}$ (dot-dashes). As $\\kappa $ increases to\nunity, the lower energy band shrinks to a single bound state of multiplity 2\ncorresponding to the P\\\"oschl-Teller potential, and the group velocity is only\nnon-zero in the continuum, ${\\cal E}>2$.\n\n\\item[Figure 8.] Behavior of $1/M^{\\ast }$ as a function of energy ${\\cal E}$\nfor $\\ell =1$ Lam\\'e equation, where $M^{\\ast }$ is the effective mass. As $\n\\kappa \\rightarrow 0$, the gap closes, and $M^{\\ast }\\rightarrow M=1/2$ for\nthis Hamiltonian.\n\\end{enumerate}\n\\newpage\n\n\\sideways\n\n\\begin{tabular}{ll\|ll}\n\\multicolumn{4}{l}{Table 1. Realizations of $SU(2)$ with\n $[L_i,L_j]=\\epsilon_{ijk}L_k$ and corresponding\nHamiltonians which lead to the Lam\\'e equation. }\\\\\n\\multicolumn{4}{l}{ }\\\\\n\\multicolumn{4}{l}{ }\\\\\n&$SU(2)$ Realization & & Hamiltonians \\\\ \\hline\\hline\n& & & \\\\\n(I): && & \\\\\n& & &\\\\\n$L_{x}$& $ =i\\left[\\kappa ({\\rm sn}^{2}\\alpha \n-{\\rm sn}^{2}\\beta)\\right]^{-1}$ & $H_1$ & $= L_{x}^{2}+\\kappa\n^{\\prime 2}L_{y}^{2}$\\\\ \n& $\\times\n\\left[ {\\rm sn}\\,\\alpha \\,{\\rm cn}\\,\\beta \\,\n{\\rm dn}\\,\\beta \\partial_\\alpha -\\,{\\rm cn}\n\\,\\alpha \\,{\\rm dn}\\,\\alpha \\,{\\rm sn}\\,\\beta \n\\partial_\\beta \\right] $ & &$ = \n \\left[\\kappa^2({\\rm sn}^{2}\\,\\alpha -{\\rm sn}\\,^{2}\\beta) \\right]^{-1}\n\\left( {\\rm dn}^{2}\\,\\beta \\partial_\\alpha\n^{2} -{\\rm dn}^{2}\\,\\alpha \\partial_\\beta ^{2}\\right)$\\\\\n$L_{y}$ & $ =\\left[\\kappa \\kappa ^{\\prime }({\\rm sn}^{2}\\alpha -{\\rm \nsn}^{2}\\beta) \\right]^{-1}$ & $H_2$ & $ = \\kappa^2 L_x^2 -\n\\kappa^{\\prime 2} L_z^2$\\\\\n& $\\times \\left[ {\\rm cn}\\,\\alpha \\,{\\rm sn}\\,\\beta\n\\,{\\rm dn}\\,\\beta \\partial_\\alpha -\\,{\\rm \n{\\rm sn}}\\,\\alpha \\,{\\rm dn}\\,\\alpha \\,{\\rm cn}\n\\,\\beta \\partial_\\beta \\right]$ & & $ = \\left[{\\rm\n sn}^{2}\\,\\beta -{\\rm sn}\\,^{2}\\alpha\\right]^{-1}\n \\left( {\\rm cn}^{2}\\,\\beta \\partial_\\alpha\n ^{2} -{\\rm cn}^{2}\\,\\alpha \\partial_\\beta ^{2}\\right)$\\\\\n$L_{z}$ & $ = i\\left[\\kappa ^{\\prime }({\\rm sn}^{2}\\alpha -{\\rm sn}\n^{2}\\beta )\\right]^{-1}$ & $H_3$ & $= L_{z}^{2}+\\kappa ^{2}L_{y}^{2} $\\\\\n& $\\times \\left[ {\\rm dn}\\,\\alpha \\,{\\rm sn}\\,\\beta \\,\n{\\rm cn}\\,\\beta \\partial_\\alpha -\\,{\\rm {\\rm \nsn}}\\,\\alpha \\,{\\rm cn}\\,\\alpha \\,{\\rm dn}\\,\\beta\\partial_\\beta\n\\right] $ & & \n $ = \\left[{\\rm sn}^{2}\\,\\beta -{\\rm sn}\\,^{2}\\alpha \\right]^{-1}\n \\left( {\\rm sn}^{2}\\,\\beta \\partial_\\alpha^{2} \n -{\\rm sn}^{2}\\,\\alpha \\partial_\\beta ^{2}\\right)$ \\\\\n & &&\\\\ \n$C_2$ & $ =L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=\\left[\\kappa ^{2}({\\rm sn}^{2}\\,\\beta -{\\rm sn}\n\\,^{2}\\alpha )\\right]^{-1}\\left( \\partial_\\alpha ^{2}-\\partial_\\beta ^{2}\\right)$ & &\\\\\n & &&\\\\\\hline\\hline\n&&&\\\\\n(II): &&& \\\\\n& & &\\\\\n$L_{x}$ & $ =\\left[\\kappa ^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}\n{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1} $ & $H_1$ & $\n =L_{x}^{2}+\\kappa ^{\\prime 2}L_{y}^{2}$\\\\\n& $\\times\\left[ \\kappa ^{\\prime 2}\\,{\\rm sn}\\,\\alpha {\\rm sn}\n(\\beta \|\\kappa ^{\\prime }){\\rm cn}(\\beta \|\\kappa ^{\\prime\n })\\partial_\\alpha \n +\\,{\\rm cn}\\,\\alpha \\,{\\rm {\\rm dn}\n}\\,\\alpha \\,{\\rm dn}(\\beta \|\\kappa ^{\\prime })\n\\partial_\\beta \\right] $ & & $ =\\left[\\kappa ^{2}{\\rm\ncn}^{2}\\alpha +\\kappa ^{\\prime 2}{\\rm cn}^{2}(\\beta\n\|\\kappa ^{\\prime })\\right]^{-1}\\left( \\kappa ^{\\prime 2}{\\rm sn}^{2}\\,(\\beta\n\|\\kappa ^{\\prime })\\partial_\\alpha ^{2}+{\\rm \ndn}\\,^{2}\\alpha \\partial_\\beta^{2}\\right) $\\\\ \n$L_{y}$ & $=\\left[\\kappa ^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}\n{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1}$ & $H_2$ & $=\\kappa\n^{2}L_{x}^{2}-\\kappa ^{\\prime 2}L_{z}^{2}$\\\\\n& $\\times\\left[ -{\\rm cn}\\,\\alpha {\\rm sn}(\\beta \|\\kappa\n^{\\prime }){\\rm dn}(\\beta \|\\kappa ^{\\prime })\n\\partial_\\alpha +\\,{\\rm sn}\\,\\alpha \\,{\\rm dn}\\,\\alpha\n\\,{\\rm cn}(\\beta \|\\kappa ^{\\prime })\\partial\n\\beta \\right]$ & & $=-\\left[\\kappa\n^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}{\\rm cn}\n^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1}\\left( \\kappa ^{\\prime 2}{\\rm cn}\n^{2}\\,(\\beta \|\\kappa ^{\\prime })\\partial_\\alpha ^{2}\n-\\kappa ^{2}{\\rm cn}\\,^{2}\\alpha\\partial_\\beta^{2}\\right)$\\\\\n$L_{z}$ & $=\\left[\\kappa ^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}\n{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1}$ & $H_3$ &\n $=L_{z}^{2}+\\kappa ^{2}L_{y}^{2}$\\\\\n&$\\times\\left[ -\\,{\\rm dn}\\,\\alpha \\,{\\rm cn}(\\beta \|\\kappa\n^{\\prime }){\\rm dn}(\\beta \|\\kappa ^{\\prime })\n\\partial_\\alpha -\\kappa ^{2}\\,{\\rm sn}\\,\\alpha \\,{\\rm {\\rm cn}\n}\\,\\alpha \\,{\\rm sn}(\\beta \|\\kappa ^{\\prime })\n\\partial_\\beta \\right]$ & & $=\\left[\\kappa\n^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}{\\rm cn}\n^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1}\\left({\\rm dn}\n^{2}\\,(\\beta \|\\kappa ^{\\prime })\\partial_\\alpha ^{2}\n+\\kappa ^{2}{\\rm sn}\\,^{2}\\alpha\\partial_\\beta^{2}\\right)$\\\\\n\n& & &\\\\\n$C_2$ & $=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}$ & &\\\\\n& $=\\left[\\kappa ^{2}{\\rm cn}\n^{2}\\alpha +\\kappa ^{\\prime 2}{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime\n})\\right]^{-1}\\left( \\partial_\\alpha ^{2}+{\\rm \\ }\\partial_\\beta ^{2}\\right) $ & &\\\\\n& & &\\\\\n \\hline\\hline\n\\end{tabular}\n\\endsideways\n\n\n\\newpage\n\n\n\\sideways\n\n\\begin{tabular}{ll\|ll}\n\\multicolumn{4}{l}{Table 2. Realizations of $SU(1,1)$ and corresponding\nHamiltonians which lead to the Lam\\'e equation.}\\\\\n\\multicolumn{4}{l}{ }\\\\\n\\multicolumn{4}{l}{ }\\\\\n&$SU(1,1)$ Realization & & Hamiltonians \\\\ \\hline\\hline\n& & & \\\\\n(I): && & \\\\\n& & &\\\\\n$L_{x}$ & $=i\\left(\\kappa \\kappa ^{\\prime }({\\rm sn}^{2}\\beta -{\\rm\n sn}^{2}\\alpha )\\right)^{-1}$ & $H_1$ & $=\\kappa\n ^{2}L_{y}^{2}+\\kappa ^{\\prime 2}L_{z}^{2}$\\\\\n & $\\times \\left[ {\\rm cn}\\,\\alpha \\,{\\rm sn}\\,\\beta\n\\,{\\rm dn}\\,\\beta \\partial_\\alpha -\\,{\\rm \n{\\rm sn}}\\,\\alpha \\,{\\rm dn}\\,\\alpha \\,{\\rm cn}\n\\,\\beta \\partial_\\beta \\right]$ & & $= \\left[{\\rm \n{\\rm sn}}^{2}\\,\\alpha -{\\rm sn}^{2}\\,\\beta \\right]^{-1}\\left( {\\rm \n{\\rm cn}}^{2}\\beta \\partial ^{2}_\\alpha-{\\rm \n{\\rm cn}}^{2}\\alpha \\partial ^{2}_\\beta\\right)$\\\\\n$L_y$ & $=\\left(\\kappa ({\\rm sn}^{2}\\beta -{\\rm sn}^{2}\\alpha\n )\\right)^{-1}$ & $H_2$ & $=L_{y}^{2}+\\kappa ^{\\prime\n 2}L_{x}^{2}$ \\\\\n & $\\times \\left[ {\\rm sn}\\,\\alpha \\,{\\rm cn}\\,\\beta \\,{\\rm \n{\\rm dn}}\\,\\beta\\partial_\\alpha -\\,{\\rm cn}\n\\,\\alpha \\,{\\rm dn}\\,\\alpha \\,{\\rm sn}\\,\\beta \n\\partial_\\beta \\right]$ & & $=\\left[\\kappa ^{2}({\\rm \n{\\rm sn}}^{2}\\,\\alpha -{\\rm sn}^{2}\\,\\beta )\\right]^{-1}\\left( {\\rm \n{\\rm dn}}^{2}\\beta \\partial_\\alpha ^{2}-\\ {\\rm \n{\\rm dn}}\\,^{2}\\alpha \\partial_\\beta^{2}\\right) $\\\\\n$L_z$ & $=i\\left[\\kappa ^{\\prime }({\\rm sn}^{2}\\beta -{\\rm sn}\n^{2}\\alpha )\\right]^{-1}$ & $H_3$ & $= L_z^2-\\kappa^2 L_x^2$\\\\\n & $\\times\\left[ {\\rm dn}\\,\\alpha \\,{\\rm sn}\\,\\beta \\,\n{\\rm cn}\\,\\beta \\partial_\\alpha -\\,{\\rm {\\rm \nsn}}\\,\\alpha \\,{\\rm cn}\\,\\alpha \\,{\\rm dn}\\,\\beta \n\\partial_\\beta \\right]$ & & $= -\\left[{\\rm sn}\n^{2}\\,\\alpha -{\\rm sn}^{2}\\,\\beta \\right]^{-1}\\left( {\\rm \\ {\\rm sn}}\n^{2}\\beta \\partial_\\alpha^{2}-\\ {\\rm sn}\n^{2}\\alpha \\partial_\\beta^{2}\\right)$\\\\\n& & &\\\\\n$C_2$ & $=L_{z}^{2}-L_{x}^{2}-L_{y}^{2}=-\\left[\\kappa ^{2}({\\rm sn}\n^{2}\\,\\alpha -{\\rm sn}^{2}\\,\\beta )\\right]^{-1}\\left(\n\\partial_\\alpha ^{2} - \\partial_\\beta\n^{2}\\right)$ & &\\\\\n & &&\\\\\\hline\\hline\n&&&\\\\\n(II): &&& \\\\\n& & &\\\\\n$L_{x}$ & $=i\\left[\\kappa ^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}\n{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1}$ & $H_1$ & \n $=L_{x}^{2}+\\kappa ^{\\prime 2}L_{y}^{2}$\\\\\n & $\\times\\left[ -\\kappa ^{\\prime 2}\\,{\\rm sn}\\,\\alpha \\,{\\rm sn}\n(\\beta \|\\kappa ^{\\prime }){\\rm cn}(\\beta \|\\kappa ^{\\prime\n })\\partial_\\alpha -\\,{\\rm cn}\\,\\alpha \\,{\\rm {\\rm dn}\n}\\,\\alpha \\,{\\rm dn}(\\beta \|\\kappa ^{\\prime })\n\\partial_\\beta \\right]$ & & $=-\\left[\\kappa ^{2}{\\rm \n{\\rm cn}}^{2}\\alpha +\\kappa ^{\\prime 2}{\\rm cn}^{2}(\\beta\n\|\\kappa ^{\\prime })\\right]^{-1}\\left( \\kappa ^{\\prime 2}{\\rm sn}^{2}(\\beta\n\|\\kappa ^{\\prime }){\\rm \\ }\\partial_\\alpha^{2}+{\\rm \\ \n{\\rm dn}}\\,^{2}\\alpha\\partial_\\beta\n^{2}\\right) $\\\\\n$L_y$ & $=i\\left[\\kappa ^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}\n{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1}$ & $H_2$ &\n$=\\kappa ^{2}L_{y}^{2}-L_{z}^{2}$\\\\\n& $\\times\\left[ {\\rm cn}\\,\\alpha \\,{\\rm sn}(\\beta \|\\kappa\n^{\\prime }){\\rm dn}(\\beta \|\\kappa ^{\\prime })\n\\partial_\\alpha -\\,{\\rm sn}\\,\\alpha \\,{\\rm dn}\\,\\alpha \n{\\rm cn}(\\beta \|\\kappa ^{\\prime })\\partial_\\beta \n\\right]$ & & $=-\\left[\\kappa ^{2}{\\rm {\\rm cn\n}}^{2}\\alpha +\\kappa ^{\\prime 2}{\\rm cn}^{2}(\\beta \|\\kappa\n^{\\prime })\\right]^{-1}\\left( {\\rm dn}^{2}(\\beta \|\\kappa\n^{\\prime })\\partial_\\alpha^{2}\n +\\kappa ^{2}{\\rm sn}\n^{2}\\alpha \\partial_\\beta^{2}\\right)$\\\\\n$L_z$ & $=\\left[\\kappa ^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}\n{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1}$ & $H_3$ &\n$=\\kappa ^{2}L_{x}^{2}+\\kappa ^{\\prime 2}L_{z}^{2}$\\\\ \n & $\\times\\left[ -{\\rm dn}\\,\\alpha \\,{\\rm cn}(\\beta \|\\kappa\n^{\\prime }){\\rm dn}(\\beta \|\\kappa ^{\\prime })\n\\partial_\\alpha -\\kappa ^{2}\\,{\\rm sn}\\,\\alpha \\,{\\rm {\\rm cn}\n}\\,\\alpha \\,{\\rm sn}(\\beta \|\\kappa ^{\\prime })\n\\partial_\\beta \\right]$ & & $=\\left[\\kappa\n^{2}{\\rm cn}^{2}\\alpha +\\kappa ^{\\prime 2}{\\rm cn}\n^{2}(\\beta \|\\kappa ^{\\prime })\\right]^{-1}\\left( \\kappa ^{\\prime 2}{\\rm cn}\n^{2}(\\beta \|\\kappa ^{\\prime })\\;\\partial_\\alpha ^{2}\n-\\kappa ^{2}{\\rm cn}^{2}\\alpha\\; \\partial_\\beta^{2}\\right) $\\\\\n& & &\\\\\n$C_2$ & $= L_{z}^{2}-L_{x}^{2}-L_{y}^{2}$ & & \\\\\n& $= \\left[\\kappa ^{2}{\\rm cn}\n^{2}\\alpha +\\kappa ^{\\prime 2}{\\rm cn}^{2}(\\beta \|\\kappa ^{\\prime\n})\\right]^{-1}\\left( \\partial_\\alpha ^{2}+\n\\partial_\\beta ^{2}\\right) $ & & \\\\\n& & &\\\\\n\\hline\\hline\n\\end{tabular}\n\\endsideways\n\\end{document}\n\n\n\\end{document}\n" } ]	[ { "name": "solv-int9912006.extracted_bib", "string": "{intro For a survey, see for example: F. Iachello, {\\sl Rev. Nuovo Cimento {19 1 (1996), and references there in; {\\sl Dynamical Groups and Spectrum Generating Algebras, Eds. A.Barut, A. Bohm and Y.Ne'eman (World Scientific, Singapore, 1987); F. Iachello and R. Levine, {Algebraic Theory of Molecules (Oxford Press, Oxford, 1995); F. Iachello and A. Arima, {\\sl The Interacting Boson Model." }, { "name": "solv-int9912006.extracted_bib", "string": "{prldk D. Kusnezov, {\\sl Phys. Rev. Lett. {79 537 (1997)." }, { "name": "solv-int9912006.extracted_bib", "string": "{gursey Y. Alhassid, F. G\\\"ursey and F. Iachello, {\\sl Phys. Rev. Lett. {50, 873 (1983); F. G\\\"ursey, in Group Theoretical Methods in Physics XI (Springer-Verlag, Berlin, 1983) p.106." }, { "name": "solv-int9912006.extracted_bib", "string": "{prllk H. Li and D. Kusnezov, {\\sl Phys. Rev. Lett. {83 1283 (1999); H. Li and D. Kusnezov, preprint (1999)." }, { "name": "solv-int9912006.extracted_bib", "string": "{a A. Khare and U. Sukhatme, quant-ph/9906044; G. Dunne and J. Mannix, {\\sl Phys. Lett.{B428, 115 (1998); G. Dunne and J. Feinberg, {\\sl Phys. Rev.{D57, 1271 (1998)." }, { "name": "solv-int9912006.extracted_bib", "string": "{novikov S. Novikov, S.V. Manakov, L.P. Pitaevskii and V.E. Zakharov, {Theory of Solitons, (New York, Plenum, 1984);" }, { "name": "solv-int9912006.extracted_bib", "string": "{sutcliffe P.M. Sutcliffe,{\\sl J. Phys.{A29, 5187 (1996)." }, { "name": "solv-int9912006.extracted_bib", "string": "{brihaye Y. Brihaye, S. Giller, P. Kosinski and J. Kunz, {\\sl Phys. Lett.{B293, 383 (1992)." }, { "name": "solv-int9912006.extracted_bib", "string": "{Liang J. Liang, H.J.W. Muller-Kirsten and D.H. Tchrakian, {\\sl Phys. Lett.{B282, 105 (1992)." }, { "name": "solv-int9912006.extracted_bib", "string": "{enolskii V.I. Enolskii and J.C. Eilbeck, {\\sl J. Phys. {A28, 1069(1995); J. Dittrich and V.I. Inozemtsev, {\\sl J. Phys. {A26, L753 (1993)." }, { "name": "solv-int9912006.extracted_bib", "string": "{b R. S. Ward, {\\sl J. Phys. {A20, 2679 (1987)." }, { "name": "solv-int9912006.extracted_bib", "string": "{miller W. Miller, {Symmetry and Separation of Variables, (Addison-Wesley, Reading, MA, 1977)." }, { "name": "solv-int9912006.extracted_bib", "string": "{patera J. Patera and P. Winternitz, {\\sl J. Math. Phys. {14, 1130 (1973); N.W. Macfadyen and P. Winternitz, {\\sl J. Math. Phys. {12, 281(1971); E. Kalnins and W. Miller Jr., {\\sl J. Math. Phys. {15, 1263 (1974)." }, { "name": "solv-int9912006.extracted_bib", "string": "{karman N. Kamran and P.J. Olver, {\\sl J. Math. Anal. Appl. {145, 342 (1990)." }, { "name": "solv-int9912006.extracted_bib", "string": "{turbiner A. V. Turbiner, {\\sl J. Phys. {A22, L1 (1989)." }, { "name": "solv-int9912006.extracted_bib", "string": "{sutherland B. Sutherland, {\\sl Phys. Rev. {A 8, 2514 (1973)." }, { "name": "solv-int9912006.extracted_bib", "string": "{whitwat E.T. Whittaker and G.N. Watson, {A Course of Modern Analysis, (Cambridge Univ. Press, Cambridge, 1980)." }, { "name": "solv-int9912006.extracted_bib", "string": "{Arscott F. M. Arscott, {Periodic Differential Equations, (Pergamon, Oxford, 1981)." }, { "name": "solv-int9912006.extracted_bib", "string": "{bateman A. Erd\\'elyi {\\sl et al, {Higher Transcedental Functions, Vol. 3, (McGraw-Hill, New York, 1953)." }, { "name": "solv-int9912006.extracted_bib", "string": "{scatt1 Y. Alhassid, F. G\\\"ursey, F. Iachello, {\\sl Ann. Phys. (NY) {148, 346 (1983); {\\sl Ann. Phys. (NY) {167, 181 (1983)." }, { "name": "solv-int9912006.extracted_bib", "string": "{James H.M. James, {\\sl Phys. Rev. {76, 1602 (1949)." } ]
solv-int9912007		[]		[ { "name": "scarf_pre.tex", "string": "%%\n%% Note: To include figures in text, just uncomment the lines\n%% that start with %*** and latex the file.\n%%\n\n\\documentstyle[12pt]{article}\n\\setlength{\\oddsidemargin}{0.0cm}\n\\setlength{\\topmargin}{-0.5cm}\n\\setlength{\\textwidth}{16.cm}\n\\setlength{\\textheight}{23.0cm}\n\\itemsep=0pt\n%\n\\def\\tdfullfigure #1 #2 #3 #4 #5 #6\n {\\begin{figure}\\vspace{#3 cm}\\vspace{12.3cm}\n \\tdpsinput #1 #6 #5 #3\n \\vspace{-4.0cm}\\vspace{#4 cm}\n \\caption[]{#2}\\end{figure}\n \\typeout{TDINPUT: FIGURE \\thefigure. produced with file #1}}\n\\def\\tdpsinput#1 #2 #3 #4 {\\special{ps::\n/beginexecute{\n /level0 save def\n /showpage{}def\n /jobname exch def\n} bind def\n/endexecute{\n level0 restore\n} bind def\n(topdrawer_input) beginexecute}\n \\special{ps::\n-1 1 scale\n-90 rotate\n-1700 -2342 translate\n#2 #2 scale\n#4 -118 mul #3 -118 mul translate\n3.5 3.5 scale\n }\n \\special{ps: plotfile #1 asis}\n \\special{ps:: endexecute\n }}\n%%%%%%%%%%%%%%%%%%%%\n\\setlength{\\baselineskip}{13pt}\n\n\\textheight 23.5cm\n\\textwidth 16cm\n\\parskip 1ex\n\\jot = .5ex\n\n\\begin{document}\n\n%\\vspace{1mm} {\\raggedleft YCTP-N-99\\\\} \\mbox{} \\vspace{1cm}\n\\vspace{2.5cm}\n\n\\begin{center}\n{\\Large {\\sc Dynamical Symmetry Approach to Periodic\n Hamiltonians}}\n\n\\vspace{1.8cm}\n\n\nHui LI\\footnote{\nemail: huili@nst4.physics.yale.edu} {\\rm and} Dimitri KUSNEZOV\\footnote{\nemail: dimitri@nst4.physics.yale.edu}\\\\\n\n\\vspace{1cm}\n\n{\\sl Center for Theoretical Physics, Sloane Physics Laboratory,\\\\Yale\nUniversity, New Haven, CT 06520-8120}\n\n\\vskip 1.2 cm\n\n{\\it April 1999}\n\n\\vspace{1.2cm}\n\n\\parbox{13.0cm}\n{\\begin{center}\\large\\sc ABSTRACT \\end{center}\n{\\hspace{0.3cm}\nWe show that dynamical symmetry methods can be applied to\nHamiltonians with periodic potentials. We construct dynamical\nsymmetry Hamiltonians for the Scarf\npotential and its extensions using representations of $su(1,1)$\nand $so(2,2)$. Energy bands and gaps are readily understood \nin terms of representation theory. We compute the transfer \nmatrices and dispersion relations for these systems, and find\nthat the complementary series plays a central role as well as\nnon-unitary representations.}}\n\\end{center}\n\n\\vspace{3mm}\n\n\\noindent PACs numbers: 03.65.Fd, 02.20.-a, 02.20.Sv, 11.30.-j\\\\\n\n\\noindent keywords: representation theory, dynamical symmetry, exactly\nsolvable models, periodic potentials, band structure.\n\n\\newpage\n \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\setcounter{page}{2} \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{center}\n{\\large I. Introduction}\n\\end{center}\n\\vspace{7mm}\n\nLie-algebraic techniques have found wide application to\nphysical systems and generally provide descriptions of bound\nstates or scattering states\\cite{1,2,AGI}. Once an algebraic\nstructure is identified, such as a spectrum generating algebra, \nexactly solvable limits of the theory, or dynamical symmetries,\ncan be constructed\\cite{fi1}. Here representation\ntheory provides a full classification of states and often\ntransitions\\cite{dk}. These dynamical symmetry limits can be \nintuitive guides to the more general structure and behavior of\nsolutions of the problem. Quantum systems can be\ncharacterized by three types of spectra: discrete (bound\nstates), continuous (scattering states) and bands (periodic\npotentials). The third case corresponds to spectra with \nenergy bands and gaps.\nUp to now, however, dynamical symmetry treatments have\nfocused only on the first two, leaving the case of band\nstructure and its connection to representation theory unclear. \n\n\nIn this article, we extend the dynamical symmetry \napproach to quantum systems by showing that Lie\nalgebras and representation theory can also be used to treat Hamiltonians\nwith periodic potentials, allowing the calculation of dispersion\nrelations and transfer matrices\\cite{li}.\nWe will focus our attention here on the Scarf\npotential\\cite{scarf} and its generalizations and\nshow how representations of $so(2,1)$ and\n$so(2,2)$ can be used to explain energy bands and gaps. \nThe representations which will be necessary are the projective\nrepresentations of \n $su(1,1)\\sim so(2,1)$. These have three\nfamilies, known as the discrete, principal, and complementary\nseries. The discrete and principal series have found\nmuch application in physics. For instance, the P\\\"oschl-Teller\nHamiltonian, $H=-d/dx^2 + g/\\cosh^2 x$, can be expressed as an\n$su(1,1)$ dynamical symmetry\\cite{fi2}, with the discrete\nand principal series describing the bound and scattering states.\nThe complementary series, however, with $-1/2<j<0$, has found\nlittle application in physics and is considered \n to be more of a curiosity. We\nwill see that this series is precisely what is needed to\ndescribe band structure in certain periodic potentials, and\nfurther, that the unitary representations correspond to the\nenergy gaps, rather than the bands. \n\n\n\\vspace{1cm}\n\\begin{center}\n{\\large II. Scarf Potential}\n\\end{center}\n\\vspace{7mm}\n\nThe Scarf potential\\cite{scarf} provides a convenient starting\npoint for the dynamical symmetry analysis of periodic\nsystems. It was originally introduced as an example of an\nexactly solvable crystal model. The starting point is the\nHamiltonian\n%\n\\begin{equation}\n H_{sc} = -\\frac{d^2}{dx^2} + \\frac{g}{\\sin^2 x}.\n\\end{equation}\n%\nThe potential is shown in Fig. 1. (We choose units with mass\n$M=1/2$ and $\\hbar=1$.)\nThe strength of the potential $g$ is usually expressed as\n$g=s^2-1/4$ since for $g \\leq -1/4$, one can no longer define a\nHilbert space for which the Hamiltonian is\nself-adjoint\\cite{gesztesy1}. The dispersion relation for this\nHamiltonian was found to be\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%% FIG 1 %%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%\\tdfullfigure fig1_jmp.eps {\\small Scarf potential\n%* $V(x)=g/\\sin^2 x$ for $g=-0.1$. } 4 -4. 8. 1.2\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%\n\\begin{equation}\n E(k) = \\frac{1}{\\pi^2}\\left[ \\cos^{-1} \\left( \\sin\\pi s\\cos\n k\\pi\\right)\\right]^2\n\\end{equation}\n%\nwith the band edges for the $n-$th band:\n%\n\\begin{equation}\nE_n^\\pm= (n+\\frac{1}{2} \\pm s)^2.\n\\end{equation}\n%\nThe bands become degenerate as $s\\rightarrow 0$. For $s=1/2$, the motion is\nthat of a free particle with $E(k)=k^2$. While Scarf originally showed\nthat the potential admits band structure for $0< s \\leq 1/2$, it\nwas demonstrated more recently that the Hamiltonian has\nbands for $1/2\\leq s<1$\\cite{gesztesy1}. In our\nanalysis, we will see that the entire range of $0< s < 1$\narises naturally from representation theory.\n\n\nIn order to realize the Scarf problem as a dynamical symmetry,\nwe consider the Lie algebras isomorphic to $so(3)$. We will see\nthat while different constructions are possible, not all are fruitful.\n\n\n\\vspace{7mm}\n\\noindent{\\sc A. $so(3)$ realization}\n\\vspace{7mm}\n\nThe relationship of the Scarf Hamiltonian to $so(3)$ was noted\nsome time ago by G\\\"ursey\\cite{gursey}. Consider the \nrealization of $so(3)$ given by the generators:\n%\n\\begin{eqnarray}\nI_{\\pm} &=& e^{\\pm i \\phi} [ \\pm\\frac{\\partial}{\\partial \\theta} +\n\\cot\\theta (\\mp\\frac{1}{2} + i \\frac{\\partial}{\\partial \\phi})] \\\\\nI_{3} &=& - i \\frac{\\partial}{\\partial \\phi} \\\\\nI^{2} &=& I_+I_- + I_3^2 - I_3 \\nonumber \\\\\n&=& - \\frac{\\partial^{2}}{\\partial \\theta^{2}} -\n\\frac{1}{\\sin^2\\theta} \n(\\frac{\\partial^{2}}{\\partial \\phi^{2}} + \\frac{1}{4}) - \\frac{1}{4}\n\\end{eqnarray}\nwhich satisfy the usual commutation relations:\n%\n\\begin{equation}\n\\lbrack I_{3},I_{+}]=I_{+},\\,\\ [I_{3},I_{-}]=-I_{-},\\,\\ [I_{+},I_{-}]=2I_{3}.\n\\end{equation}\n%\nThen, using the basis $\\psi _{j}^{m}=\\sqrt{\\sin \\theta }\\,P_{j}^{m}(\\cos\n\\theta )$, with the unitary representations of $so(3)$ labeled by\n$(j,m)$, the Casimir invariant $I^2$ can be rewritten as the\nSchr\\\"odinger equation:\n\n\\begin{equation}\n\\lbrack -\\frac{d^{2}}{d \\theta ^{2}}+\\frac{m^{2}-\\frac{1}{4}}{%\n\\sin ^{2}\\theta }]\\,\\psi _{j}^{m}(\\theta )=(j+\\frac{1}{2})^{2}\\,\\psi\n_{j}^{m}(\\theta ).\n\\end{equation}\n%\nWhile this is Scarf's Hamiltonian with $g=m^2-1/2$\n(similar to $g=s^2-1/2$ in (1)), it is not a useful\nrealization for several reasons.\nFor instance, one cannot\nobtain any band structure from the discrete representations of\n$so(3)$. Here the spectrum is labeled by $(j+1/2)$, which\nidentifies only bound states. Further, the strength of the\npotential, $m^2-1/4$, is only negative for $m=0$. In this case\n$g=-1/4$ and the Hamiltonian is no longer self-adjoint. Finally,\nsince $m$ appears in the strength $g$ of the potential, a given\nrepresentation $j$ would correspond to different forms of the\nHamiltonian, rather than the spectrum of a single Hamiltonian. For this\nreason, the previous realizations of $H_{Sc}$ are not useful for\nthe discussion of band structure.\n\n\n\\vspace{7mm}\n\\noindent{\\sc B. $so(2,1)$ realization}\n\\vspace{7mm}\n\nA more suitable realization of the Scarf Hamiltonian can be\nfound using $so(2,1)\\sim su(1,1)$. To obtain this form, we\nperform the following transformations of the $so(3)$ algebra:\n{\\it (i)} scaling the wavefunction by $\\frac{1}{\n\\sqrt{\\sin \\theta }}$ , {\\it (ii)} changing $\\cos \\theta \\rightarrow $ $\\tanh\n\\theta $ and {\\it (iii)} taking $\\theta\\rightarrow\ni\\theta$. The result is the $so(2,1)$ realization\n%\n\\begin{eqnarray}\nI_{\\pm } &=&e^{\\pm i\\phi }(\\mp \\sin \\theta \\frac{\\partial }{\\partial \\theta }\n+i\\cos \\theta \\frac{\\partial }{\\partial \\phi }) \\\\\nI_{3} &=&-i\\frac{\\partial }{\\partial \\phi } \\\\\nI^{2} &=&-I_{+}I_{-}+I_{3}^{2}-I_{3} \\nonumber \\\\\n&=&\\sin ^{2}\\theta (\\frac{\\partial ^{2}}{\\partial \\theta ^{2}}-\\frac{%\n\\partial ^{2}}{\\partial \\phi ^{2}})\n\\end{eqnarray}\nwhich satisfies the commutation relations\n%\n\\begin{equation}\n\\lbrack I_{3},I_{+}]=I_{+},\\,\\ [I_{3},I_{-}]=-I_{-},\\,\\\n[I_{+},I_{-}]=-2I_{3}.\n\\end{equation}\n%\nThe Casimir operator, using the basis states \n$\\psi _{j}^{m}(\\theta )=P_{j}^{m}(i\\cot \\theta ),0<\\theta\n<\\frac{\\pi }{2}$, reduces to Scarf's Hamiltonian in the\ndynamical symmetry form:\n\n\\begin{equation} \\label{eq:sds}\n\\lbrack -\\frac{d^{2}}{d\\theta ^{2}}+\\frac{j(j+1)}{\\sin\n^{2}\\theta }]\\,\\psi _{j}^{m}(\\theta )=m^{2}\\,\\psi\n_{j}^{m}(\\theta ) .\n\\end{equation}\n%\n\nWhile this Hamiltonian is more pleasing than Eq. (8) in the sense that a\nsingle representation $j$ will account for the spectral\nproperties, given by $m^2$, the standard unitary representations\n(given in Appendix A) are not yet sufficient to describe the bands. \nThese come in three series. The principal series with\n$j=-1/2+i\\rho$, $\\rho>0$, the discrete series $D_j^\\pm$ where\n$j=-n/2$ for $n=1,2,...$, and the complementary series,\n$-1/2<j<0$. \n\nIn order to realize band structure as a dynamical symmetry, it\nis clear that we must consider slightly more general representations. For\nHamiltonians with periodic potentials, $V(x+a)=V(x)$, Bloch's\ntheorem requires the form of the wavefunctions to be\\cite{kittel}\n%\n\\begin{equation}\\label{eq:bloch}\n \\Psi_k(x) = e^{ikx} u_k(x),\\qquad u_k(x+a)=u_k(x),\n\\end{equation}\n%\nso that $\\Psi_k(x+a)=\\exp(ika)\\Psi_k(x)$ is not single\nvalued. To obtain multi-valued functions, we pass\nto the projective unitary representations of $su(1,1)\\sim\nso(2,1)$\\cite{bargmann,pukanszky}.\nIn contrast to the more familiar representations of $so(3)$\nwhich are related to the orthogonal symmetries in the vector\nspace ${\\cal R}^3$, the\nprojective representations are associated with equivalence classes of\nvectors defined up to a phase ( as in Eq. \\ref{eq:bloch}). The\naction of a group on the projective space (rather than a vector\nspace), defined by this\nequivalence class of states, leads to the projective representations.\nWhile these are multi-valued representations of the group, they\nare proper representations of the algebra and are hence\nsuitable. Consequently, \nthe single-valued representations of this covering group of\n$su(1,1)$ are infinitely many-valued representations of\n$su(1,1)$. Such representations have been used to describe bound\nand scattering states in the P\\\"oschl-Teller\npotential\\cite{fi2}.\nThey fall into the same three series as the usual\nunitary representations of $su(1,1)$ discussed above (see \nAppendix A). We will see that for our Scarf dynamical symmetry (\\ref{eq:sds}),\nthe discrete series corresponds to the band edges, the\ncomplementary series provides the bands and gaps, while the\nprincipal series is unphysical, corresponding to the regime\nwhere the Hamiltonian is not self-adjoint.\n\nConsider first the complementary series of the projective unitary\nrepresentations of $so(2,1)$. Here we must have \n\\begin{equation}\n -\\frac{1}{2}<j<0,\\qquad {\\rm or}\\qquad -\\frac{1}{4}<j(j+1)<0.\n\\end{equation}\nThis is precisely the range of $g=j(j+1)$ studied initially by\nScarf in Eq. (1). The states are labeled by two quantum numbers\n$j,m$, with unitary representations given by the range of\nquantum numbers:\n\n% \n\\begin{equation}\n m=m_{0}\\pm n\\,(n=0,1,\\cdots ),\n \\qquad 0\\leq m_{0}<1,\\qquad m_{0}(1-m_{0})<-j(j+1)<\\frac{1}{4}.\n\\end{equation}\n%\nThe last condition provides the range:\n%\n\\begin{equation}\n 0<m_{0}<-j,\\qquad {\\rm and}\\qquad 1+j<m_{0}<1,\n\\end{equation}\n%\nwhich is illustrated in Fig. 2. For a given value of $j$,\n$j(j+1)$ (dots) separates unitary from non-unitary\nrepresentations. The unitary representations are given by values\nof $m$ for which the periodically continued parabola (dashes and\nsolid) are above $j(j+1)$. One can now see that these unitary\nrepresentations correspond to the band gaps rather than the\nbands by taking $j\\rightarrow 0$. In this case the Hamiltonian\n(13) is that of a free particle, so that the spectrum is\n$E=m^2\\ge 0$. From Eqs. (16)-(17) and Fig. 2, we see that as \n$j\\rightarrow 0$ the allowed values of $m$ become restricted \nto $m=0,\\pm1,\\pm2,...$. Therefore, for a specific \n$j$, $E=m^{2}$ has band structure, with the range of $m$ from\nunitary projective representations giving the energy gaps.\nThe {\\sl non-unitary} projective representations of the\ncomplementary series give the energy bands\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%% FIG 2 %%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%*\\tdfullfigure fig2_jmp.ps {\\small Complementary series of projective\n%* representations of $so(2,1)$. Unitary representations are\n%* labeled by $j,m$ where $-1/2<j<0$ (so that $-1/4<j(j+1)<0$) and\n%* $m$ given by the following construction: $m=m_0\\pm n$, where\n%* $0\\leq m_0<1$, and $m_0(m_0-1)>j(j+1)\\geq -1/4$. In the\n%* figure, we plot $j(j+1)$ for some $j$ (dots), the\n%* parabola $m_0(m_0-1)$ (solid), and its periodic extension\n%* $(m\\pm n)(m\\pm n-1)$ (dashes). The unitary representations\n%* $(j,m)$ are the values of $m$ for which the solid and dashed \n%* lines are above $j(j+1)$. These correspond to the\n%* energy gaps of the Scarf potential, while the non-unitary\n%* regions correspond to the energy bands.} 4. -5. 10. 1.5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n\\begin{equation}\n (-j+n)^{2}<E<(1+j+n)^{2},\\qquad n-j<m<1+j+n.\n\\end{equation}\n\nThe band edges are not contained in the complementary\nseries. In contrast to the states in the band, the edge states\nare periodic. They form a discrete set of states which are\nassociated with the discrete series. These series $D_{j}^{\\pm }$ \nhave the representations $j<0$ with $m$ given by\n%\n\\begin{eqnarray}\n D_j^+ \\;& : & m=-j,1-j,2-j,...\\\\\n D_j^- \\;& : & m=j,j-1,j-2,... \n\\end{eqnarray}\n%\nWhen we restrict to the range of physical interest,\n$-\\frac{1}{2}<j<0$, this series provides the upper and lower\nband edges (compare to Eq. (18)):\n%\n\\begin{eqnarray}\n D_j^\\pm ({\\rm lower}) &: & E=(n-j)^2\\\\\n D_{-j-1}^\\pm ({\\rm upper}) &: & E=(n+j+1)^2.\n\\end{eqnarray}\n%\nEq. (22) arises from the invariance of our Hamiltonian (13)\nunder $j\\rightarrow -1-j$, allowing both discrete series\n$D^\\pm_j$ and $D^\\pm_{-1-j}$.\nOther discrete representations with $j<-1$ are not useful for band\nstructure. The band spectrum of the Scarf potential which\nincludes both the discrete and complementary series is shown in\nFig. 3. The shaded region corresponds to the bands (non-unitary) and the\nunshaded to the gaps (unitary).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%% FIG 3 %%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%*\\tdfullfigure fig3_jmp.ps {\\small Band structure of the Scarf potential\n%* $j(j+1)/sin^2x$. The dynamical\n%* symmetry spectrum $E=m^2$ is plotted as a function of $j$.\n%* The energy gaps (unshaded)\n%* for each $j$ correspond to the unitary representations of\n%* Fig. 2. One can see that the complementary series describes\n%* the spectral properties of this potential. The discrete\n%* representations $D_j^\\pm$ ($D_{-1-j}^\\pm$) are seen to explain\n%*** the lower (upper) band edges. } 8. -8. 8. 1.7\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n \nThe remaining representations, the principal series, has\n$j=-\\frac{1}{2}+i\\rho$ $(\\rho >0)$. This gives a potential with strength\n$g=j(j+1)<-\\frac{1}{4}$, for which the Hamiltonian is no longer\nself-adjoint and is of no physical interest. \n\n\nNote that we have explained the band structure for strengths of\nthe potential $-1/4 < g=j(j+1)<0$ and found agreement with\nScarf\\cite{scarf}. More recently it was noted that for\n$0 \\le g < 3/4$, there is also band\nstructure \\cite{gesztesy1}. In this range the potential is\nstrictly positive. (The origin of the band structure here is that\nthe matching conditions on the wavefunctions around the\nsingularity in the potential, needed to have a self-adjoint\nHamiltonian, in a sense `dilute' the infinite potential at these\npoints and allow bands.) While our $so(2,1)$ realization above cannot\naccount for this range of $g$, we will see in Section III, that a\nlimiting case of an $so(2,2)$ dynamical symmetry will account\nfor this range using the same complementary series. For \n$g \\ge 3/4$, there is no band structure and the discrete projective\nrepresentation then describe the bound state spectrum.\n\n\\vspace{7mm}\n\\noindent{\\sc C. Transfer matrix}\n\\vspace{7mm}\n\nThe transfer matrix T for the period $x\\in (-\\frac{\\pi }{2},\\frac{\\pi\n}{2})$ can be computed directly from wave functions. However,\nthe quadratic singularity of the potential\nrequires some care. There are two approaches one can consider, but both are\nequivalent\\cite{scarf,gesztesy1,james}. In the first, we compute\nthe transfer matrix at $x=\\pm\\varepsilon$. We then match the\ntransfer matrices on both sides of the singularity as\n$\\varepsilon\\rightarrow 0$, which results in matching conditions\non the wavefunctions. This procedure is not equivalent to an analytical\ncontinuation around the origin. The second arises in the\nconstruction of the\nHilbert space of functions for which $H$ is self-adjoint. This\ngives rise to equivalent matching conditions around the origin\\cite{gesztesy1}.\nThe matrix elements of the transfer matrix are related to the values of the\neven and odd solutions and their first derivatives at \n$\\frac{\\pi}{2}$ (see Appendix B). We find\n\n\\begin{equation}\nT \\, = \\, \\left(\n\\begin{array}{cc}\n\\alpha & \\beta \\\\\n\\beta^{} & \\alpha^{}\n\\end{array}\n\\right)\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are determined by the representations\nof the complementary and discrete series $j,m$ as:\n\n\\begin{eqnarray}\n\\alpha &=& e^{- i m \\pi} \\, \\left[ \\frac{\\cos\\pi m}{\\sin\\pi(j+\\frac{1}{2})}\n \\right.\\nonumber \\\\\n & & \\left. + \\, i \\left(-\\frac{2}{m} \\frac{ \n \\Gamma (\\frac{1-j+m}{2}) \\Gamma (\\frac{1 -j-m}{2} )}{\n \\Gamma (- \\frac{j-m}{2}) \\Gamma ( - \\frac{j+m}{2} )} + \\frac{m}{2}\n \\frac{ \\Gamma ( \\frac{1+j+m}{2}) \n \\Gamma ( \\frac {1+j - m}{2})}{ \\Gamma (\\frac{2+j+m}{2} ) \n \\Gamma ( \\frac{2 +j -m}{2})}\\right)\\frac{\\cos\\pi\n \\frac{j+m}{2} \\; \\cos\\pi\n \\frac{j-m}{2}}{\\sin\\pi(j+\\frac{1}{2})} \\right]\\\\\n\\beta &=& i \\, e^{i m \\pi} \\, \\frac{\\cos\\pi \\frac{j+m}{2} \\/\n \\cos\\pi \\frac{j-m}{2}}{\\sin\\pi(j+\\frac{1}{2})} \\nonumber \\\\\n& & \\left[\\frac{2}{m} \\frac{ \\Gamma\n (\\frac{1-j+m}{2})\\Gamma(\\frac{1-j-m}{2})}\n {\\Gamma ( -\\frac{j-m}{2}) \\Gamma ( -\n \\frac{j+m}{2})} + \\frac{m}{2} \\frac{\\Gamma (\n \\frac{1+j+m}{2}) \n \\Gamma (\\frac{1+j-m}{2})}{ \\Gamma (\n \\frac{2+j+m}{2}) \\Gamma ( \\frac{2+j-m}{2})}\\right] .\n\\end{eqnarray}\n%\nAlthough the Scarf Hamiltonian can be obtained from the\nP\\\"oschl-Teller potential $V(x)=g/\\cosh^2x$ through a\ntransformation, the above transfer matrix is not related to that\nof the P\\\"oschl-Teller in any simple manner.\n\nThe Bloch form of the $so(2,1)$ wave functions for the $n-$th\nperiod, $(n-\\frac{1}{2})\\pi < x \\leq (n+\\frac{1}{2})\\pi$, \nof the Scarf Hamiltonian are readily found to be\n%\n\\begin{equation}\n\\Psi_k (x) = f_k (x-n\\pi) e^{ikx}\n\\end{equation}\n%\nwhere:\n%\n\\begin{equation}\nf_k(z) = e^{-ik(z+\\frac{\\pi}{2})}[a P_j^m (i\\cot z) +\n b P_j^{-m}(i\\cot z)]\n\\end{equation}\nand:\n\\begin{eqnarray}\na &=& (-)^{-j/2}\\frac{\\sqrt{\\pi}2^{-m}}{\\sin m\\pi} \\left[\\cos\\frac{k\\pi}{2}\n\\frac{\\Gamma(\\frac{1-j-m}{2})}{\\Gamma(\\frac{-j+m}{2})} \n - \\sin\\frac{k\\pi}{2}\n\\frac{\\Gamma(\\frac{2+j-m}{2})}{\\Gamma(\\frac{1+j+m}{2})} (-)^j\\right] \\\\\nb &=& -(-)^{-j/2}\\frac{\\sqrt{\\pi}2^m}{\\sin m\\pi} \\left[\\cos\\frac{k\\pi}{2}\n\\frac{\\Gamma(\\frac{1-j+m}{2})}{\\Gamma(\\frac{-j-m}{2})} \n - \\sin\\frac{k\\pi}{2}\n\\frac{\\Gamma(\\frac{2+j+m}{2})}{\\Gamma(\\frac{1+j-m}{2})} (-)^j \\right].\n\\end{eqnarray}\n%\nSince $-\\frac{\\pi}{2} < z\\leq \\frac{\\pi}{2}$, $f_k(z)$ is made\nperiodic, and $\\Psi_k(x)$ satisfies Bloch's theorem.\n\n\n\\vspace{7mm}\n\\noindent{\\sc D. Dispersion relation}\n\\vspace{7mm}\n\nOnce we have the transfer matrix, the dispersion relation is \nobtained from $\\alpha$ by the condition\\cite{kittel,cohen}:\n%\n\\begin{equation}\n\\cos \\pi k \\, = \\, Re(\\alpha e^{im\\pi}) \\, = \\, \\frac{\\cos \\pi m}{\\sin \\pi\n(j + \\frac{1}{2})} \\, .\n\\end{equation}\n%\nSolving for the energy $E=m^2$, we find:\n%\n\\begin{equation}\nE(k) = m^2 = \\frac{1}{\\pi^2}[\\cos^{-1}(\\sin\\pi(j+\\frac{1}{2})\\cos\\pi k)]^2 .\n\\end{equation}\n%\nThis is precisely the result (2) obtained by Scarf. Again, the\nvalues of $j$ and $m$ are determined from the representations\ngiven in (18) and (21)-(22). From the dispersion\nrelation, we can also compute the group velocity ${\\cal V}$ and\nthe effective mass $M^$. These will depend only upon the\nrepresentation labels $j$ and $m$. We have:\n%\n\\begin{equation}\n{\\cal V}(j,m) = \\frac{\\partial E}{\\partial\n k} = 2 m \\frac{\\sqrt{\\cos^2 \\pi j -\\cos^2\\pi m}}{\\sin \\pi m}\n\\end{equation}\n%\nThis is plotted in Fig. 4(a) for selected values of $j$. ${\\cal\n V}(j,m)$ vanishes on the band edges. For\n$j=0$, the Hamiltonian (13) describes free motion, and we expect\n${\\cal V}=\\pm k/M=\\pm 2k$ (dots), while for $j\\rightarrow-1/2$, we\n have degenerate bands,\nand ${\\cal V}\\rightarrow 0$ at half-integer values of $m$. \nFor the effective mass:\n%\n\\begin{eqnarray}\n\\frac{1}{M^(j,m)} &=& \\frac{\\partial^2 E}{\\partial\n k^2}\\nonumber \\\\\n &=& 2\\left[\\frac{\\cos^2j\\pi}{\\sin^2 m\\pi} - \\cot^2 m\\pi +\n m\\pi\\frac{\\sin^2j\\pi \\cot m\\pi}{\\sin^2 m\\pi}\\right].\n\\end{eqnarray}\n%\n(Note that this differs slightly from the result derived in \\cite{scarf}.)\nIn Fig. 4(b), $1/M^(j,m)$ is shown for selected values of $j$. For\n$j= 0$, $M^= M=1/2$, while for\n$j\\rightarrow -1/2$, $1/M^\\rightarrow 0$.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%% FIG 4 %%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%\\tdfullfigure fig4_jmp.ps {\\small(a) The group velocity of the\n%* Scarf potential ${\\cal V}(j,m)$ shown as a function of $m$ \n%* for several strengths of\n%* the potential. For $j= 0$, the motion becomes free and one has\n%* ${\\cal V}=\\pm k/M=\\pm 2k$. For $j\\rightarrow -\\frac{1}{2}$, the\n%* bands become degenerate and ${\\cal V}\\rightarrow 0$ at the\n%* values of $m=\\pm (2\\ell +1)/2$, $\\ell=0,1,2,...$.\n%* (b) Effective mass $M^$ of the Scarf potential. We shown\n%** $1/M^$ as a function of $m$ for several values of $j$. For\n% $j\\rightarrow -1/2$, the bands become degenerate, and the\n%* effective mass diverges at $m=\\pm (2\\ell +1)/2$,\n%*** $\\ell=0,1,2,...$. For $j\\rightarrow 0$, $M^\\rightarrow\n% M=1/2$. } 4. -4.7 9. 1.1\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n\n\n\n\n\\vspace{7mm}\n\\noindent{\\sc E. Variation of Scarf potential}\n\\vspace{7mm}\n\nIn the next section we will present a dynamical symmetry\nHamiltonian for a variation of the Scarf potential using\n$so(2,2)$. This potential will have several limits where the\nHamiltonian reduces to the Scarf case, including the\n$1/cos^2x$ potential. In order to compare the transfer matrix in this\nlimit to the Scarf result, we consider the Scarf Hamiltonian\ntranslated by $\\frac{\\pi }{2}$,\n%\n\\begin{equation}\n\\lbrack -\\frac{d ^{2}}{d \\theta ^{2}}+\\frac{j(j+1)}{\\cos\n^{2}\\theta }]\\,\\psi _{j}^{m}(\\theta )=m^{2}\\,\\psi\n_{j}^{m}(\\theta ).\n\\end{equation}\n%\nThe dispersion relation $E(k)$ and the energy band structure\nwill remain the same as before. The\ntransfer matrix for $(-\\frac{\\pi }{2},\\frac{\\pi }{2})$, on the\nother hand, will change. The new transfer matrix \ncan be calculated easily from a translation of the solutions of Scarf case:\n\n\\begin{eqnarray}\n\\alpha &=&e^{-im\\pi }\\, \\left[\\frac{\\cos \\pi m}{\\sin \\pi (j+\\frac{1}{2})}\n+\\,i\\left(-m \\frac{\\Gamma (j+\\frac{1}{2})\\Gamma (\\frac{1-j+m}{2})\\Gamma\n(\\frac{ 1-j-m}{2})}{\\Gamma (-j-\\frac{1}{2})\\Gamma\n(\\frac{2+j+m}{2})\\Gamma (\\frac{ 2+j-m}{2})}\\right.\\right.\n\\nonumber \\\\\n & & \\left.\\left.+\\frac{1}{m}\\frac{\\Gamma (-j-\\frac{1}{2})\\Gamma\n(\\frac{1+j+m}{2} )\\Gamma (\\frac{1+j-m}{2})}{\\Gamma\n(j+\\frac{1}{2})\\Gamma (-\\frac{j-m}{2} )\\Gamma (-\\frac{j+m}{2})}\\right)\n\\frac{\\cos \\pi \\frac{j+m}{2}\\/\\cos \\pi\n\\frac{j-m}{2}}{\\sin \\pi (j+\\frac{1}{ 2})}\\right]\\\\\n\\beta &=& -i\\,e^{im\\pi }\\,\\frac{\\cos \\pi \\frac{j+m}{2}\\/\\cos \\pi \\frac{j-m}{2}\n}{\\sin \\pi (j+\\frac{1}{2})} \\left[m\\frac{\\Gamma (j+\\frac{1}{2})\\Gamma\n(\\frac{1-j+m}{2})\\Gamma (\\frac{1-j-m}{ 2})}{\\Gamma\n(-j-\\frac{1}{2})\\Gamma (\\frac{2+j+m}{2})\\Gamma (\\frac{2+j-m}{2})}\\right.\n\\nonumber \\\\\n && \\left. +\\frac{1}{m}\\frac{\\Gamma (-j-\\frac{1}{2})\\Gamma\n(\\frac{1+j+m}{2})\\Gamma (\\frac{1+j-m}{2})}{\\Gamma\n(j+\\frac{1}{2})\\Gamma (-\\frac{j-m}{2})\\Gamma (-\n\\frac{j+m}{2})}\\right]\n\\end{eqnarray}\n%\n\n\n\\vspace{1cm}\n\\begin{center}\n{\\large III. Generalized Scarf Potential}\n\\end{center}\n\\vspace{7mm}\n\nWe have now shown that band structure can arise naturally as a\ndynamical symmetry. We would like to build on the analysis of\nthe Scarf problem and study a different class of\nperiodic potentials. Consider an extension of the Scarf potential\ngiven by a generalized P\\\"oschl-Teller Hamiltonian\\cite{poschl}\n%\n\\begin{equation}\n[-\\frac{d^{2}}{d x^{2}} + \\frac{g_1}{\\sin^2 x} +\n\\frac{g_2}{\\cos^2 x} ] \\, \\Psi(x) \\, = \\, E \\, \\Psi(x),\n\\qquad (g_1,g_2 > -\\frac{1}{4})\n\\end{equation}\n%\nWhile this Hamiltonian is exactly solvable, we would like to see\nhow band structure can be obtained from representation theory\nusing dynamical symmetry considerations.\nWe will relate this Hamiltonian to the $so(4)$ and\n$so(2,2)$ algebras and develop the band structure from the\ncomplementary series. We plot some forms of this potential in\nFig. 5 for several values of $g_1$ and $g_2$. Our study will be\nrestricted to the range $-1/4< g_1,g_2\\leq 0$. \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%% FIG 5 %%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%\\tdfullfigure fig5_jmp.ps {\\small Forms of the generalized Scarf potential\n% $V(x)=g_1/sin^2x +g_2/cos^2x$ using (a)\n%* $g_1=g_2=-0.1$, (b) $g_1=-0.25$, $g_2=-0.001$, and (c)\n%* $g_1=-0.01$, $g_2=-0.25$.\n%* Situation (a) is one of the three limiting cases where $V(x)$\n%* becomes the Scarf potential. } 4. -4.7 8. 1.1\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n\n\\vspace{7mm}\n\\noindent{\\sc A. $so(4)$ realization}\n\\vspace{7mm}\n\nWe start with the realization of the $so(4)$ algebra:\n\n\\begin{eqnarray}\nA_{\\pm } &=&\\frac{1}{2}e^{\\pm i(\\phi +\\alpha )} \\left[\\pm\n \\frac{\\partial }{\\partial \\theta }+\\cot 2\\theta\n \\left(i\\frac{\\partial }{\\partial \\phi }+ i\\frac{\\partial\n }{\\partial \\alpha }\n \\mp 1\\right) -\\frac{i}{\\sin\n 2\\theta }\\left(\\frac{\\partial }{\\partial \\phi }-\\frac{\\partial\n }{\\partial \\alpha }\\right)\\right] \\nonumber \\\\\nA_{3} &=&-\\frac{i}{2}\\left(\\frac{\\partial }{\\partial \\phi }+\\frac{\\partial }{%\n \\partial \\alpha }\\right) \\\\\nB_{\\pm } &=&\\frac{1}{2}e^{\\pm i(\\phi -\\alpha )}\n\\left[\\pm\\frac{\\partial}{\\partial\\theta}\n +\\cot 2\\theta \\left(i\\frac{\\partial }{\\partial \\phi\n }-i\\frac{\\partial }{\\partial \\alpha }\\mp\n 1\\right)\n -\\frac{i}{\\sin2\\theta } \\left(\\frac{\\partial }{\\partial \\phi }+\\frac{\\partial\n }{\\partial \\alpha }\\right) \\right]\\nonumber\\\\\nB_{3} &=&-\\frac{i}{2}\\left(\\frac{\\partial }{\\partial \\phi\n }-\\frac{\\partial }{\\partial \\alpha }\\right)\\nonumber\n\\end{eqnarray}\nwhich have the commutation relations:\n\\begin{equation}\n\\begin{array}{cccc}\n{\\lbrack A_{3},A_{+}]}=A_{+},& {[A_{3},A_{-}]}=-A_{-},\n&{[A_{+},A_{-}]}=2A_{3}, &\\\\\n{\\lbrack B_{3},B_{+}]}=B_{+},& {[B_{3},B_{-}]}=-B_{-}, &\n{[B_{+},B_{-}]}=2B_{3}, & {[A,B]}=0.\\end{array}\n\\end{equation}\n%\nSince this is the direct product of two $so(3)$ algebras, the quadratic Casimir\ninvariant has the form:\n\n\\begin{eqnarray}\nC_{2} &=&2(A^{2}+B^{2}) \\nonumber \\\\\n&=&2(A_{+}A_{-}+A_{3}^{2}-A_{3}+B_{+}B_{-}+B_{3}^{2}-B_{3}) \\\\\n&=&-\\frac{\\partial ^{2}}{\\partial \\theta ^{2}}+\\frac{1}{\\cos ^{2}\\theta }\\left[\n-\\frac{\\partial ^{2}}{\\partial \\phi ^{2}}-\\frac{1}{4}\\right]\n+\\frac{1}{\\sin ^{2}\\theta }\\left[-\\frac{\\partial ^{2}\n}{\\partial \\alpha ^{2}}-\\frac{1}{4}\\right]-1 \\nonumber\n\\end{eqnarray}\n\nThe representations of $so(4)$ can be labeled by\n($j_1,m$; $j_2,c$), where $j_1,j_2,m,c$ are non-negative\nintegers or half\nintegers and $-j_1 \\le m \\le j_1$, $-j_2 \\le c \\le j_2$. It is easy\nto check that, as differential operators, $A^2=B^2$. So for this\nrealization, we only need to consider symmetric representations\nwith $j_1=j_2=j$. Hence, $C_2=4j(j+1)$. The resulting Schr\\\"odinger\nequation is\n\n\\begin{equation}\n\\lbrack -\\frac{d^{2}}{d \\theta ^{2}}+\\frac{(m+c)^{2}-\\frac{1}{%\n4}}{\\cos ^{2}\\theta }+\\frac{(m-c)^{2}-\\frac{1}{4}}{\\sin ^{2}\\theta }]\\,\\psi\n_{j}^{m,c}(\\theta )=(2j+1)^{2}\\,\\psi _{j}^{m,c}(\\theta )\n\\end{equation}\n%\nWhile this is suitable for bound states, the discrete representations of\n$so(4)$ do not explain band structure, and the strength of the\npotential is not in the range of physical interest.\n\n\\vspace{7mm}\n\\noindent{\\sc B. $so(2,2)$ realization}\n\\vspace{7mm}\n\nWe can derive a more suitable realization by passing to\n$so(2,2)$. Starting with the above generators, we {\\it (i)}\nscale the wavefunctions by\n$\\frac{1}{\\sqrt{\\sin\\theta}}$ , {\\it (ii)} transform \n$\\cos\\theta \\rightarrow \\tanh \\theta$ and {\\it (iii)} take\n$\\theta\\rightarrow i\\theta$. This results in the $so(2,2)$ realization:\n\n\\begin{eqnarray}\nA_{\\pm} &=& \\frac{1}{2} e^{\\pm i (\\phi + \\alpha)} \\, \\left[ \\pm \\cos \\theta \\frac{\n \\partial}{\\partial \\theta} + \\, i \\frac{1 +\n \\sin^2\\theta}{2\\sin\\theta}\n \\left(\\frac{\\partial}{\\partial\\phi} \n + \\frac{\\partial}{\\partial \\alpha}\\right)\\right. \\nonumber \\\\\n& & \\left.+ i \\frac {\\cos^2\\theta}{2\\sin\\theta} \n \\left(\\frac{\\partial}{\\partial \\phi} - \\frac{\\partial\n }{\\partial \\alpha}\\right) \\mp \\frac{1}{2\\sin\\theta} \\right] \\nonumber\\\\\nA_{3} &=& - \\frac{i}{2} \\left(\\frac{\\partial}{\\partial \\phi} + \\frac{\\partial}{\n \\partial \\alpha}\\right) \\\\\nB_{\\pm} &=& \\frac{1}{2} e^{\\pm i (\\phi - \\alpha)} \\, \\left[ \\pm \n \\cos\\theta\\frac{\n \\partial}{\\partial \\theta} + \\, i \\frac{1 +\n \\sin^2\\theta}{2\\sin\\theta} \\left(\\frac{\\partial}{\\partial\\phi} \n -\\frac{\\partial}{\\partial \\alpha}\\right)\\right. \\nonumber \\\\\n& & \\left.+ i \\frac\n{\\cos^2\\theta}{2\\sin\\theta} \\left(\\frac{\\partial}{\\partial \\phi} + \\frac{\\partial\n}{\\partial \\alpha}\\right) \\mp \\frac{1}{2\\sin\\theta} \\right] \\nonumber\\\\\nB_{3} &=& - \\frac{i}{2} \\left(\\frac{\\partial}{\\partial \\phi} - \\frac{\\partial}{\n\\partial \\alpha}\\right)\\nonumber\n\\end{eqnarray}\nwith the commutation relations\n%\n\\begin{equation}\n\\begin{array}{cccc}\n{\\lbrack A_{3},A_{+}]}=A_{+},& {[A_{3},A_{-}]}=-A_{-},\n&{[A_{+},A_{-}]}=-2A_{3}, &\\\\\n{\\lbrack B_{3},B_{+}]}=B_{+},& {[B_{3},B_{-}]}=-B_{-}, &\n{[B_{+},B_{-}]}=-2B_{3}, & {[A,B]}=0.\\end{array}\n\\end{equation}\n%\nThe quadratic Casimir invariant now has the form\n%\n\\begin{eqnarray}\nC_{2} &=&2(A^{2}+B^{2}) \\nonumber \\\\\n&=&2(-A_{+}A_{-}+A_{3}^{2}-A_{3}-B_{+}B_{-}+B_{3}^{2}-B_{3}) \\\\\n&=&\\cos ^{2}\\theta \\frac{\\partial ^{2}}{\\partial \\theta ^{2}}-\\cos\n^{2}\\theta \\frac{\\partial ^{2}}{\\partial \\alpha ^{2}}+\\frac{\\cos ^{2}\\theta\n}{\\sin ^{2}\\theta }(\\frac{\\partial ^{2}}{\\partial \\phi ^{2}}+\\frac{1}{4})-%\n\\frac{3}{4} \\nonumber \n\\end{eqnarray}\n\nThe states of the representations of $so(2,2)$ can be labeled by\na direct product of representations of $so(2,1)$, denoted \n($j_1,m$; $j_2,c$). Again, as\ndifferential operators, $ A^2 = B^2$ so that $j_1 = j_2 = j$.\nReplacing $\\theta$ by $x$, this leads to the Schr\\\"odinger equation:\n\n\\begin{equation}\n\\lbrack -\\frac{d^{2}}{d x^{2}}+\\frac{(m+c)^{2}-\\frac{1}{%\n4}}{\\sin ^{2}x }+\\frac{(2j+1)^{2}-\\frac{1}{4}}{\\cos ^{2}x }]\\,\\psi\n_{j}^{m,c}(x )=(m-c)^{2}\\,\\psi _{j}^{m,c}(x )\n\\end{equation}\n%\nTwo independent solutions \\cite{gesztesy2,miller} in the\nregion $0< x <\\frac{\\pi}{2}$ are:\n\n\\begin{eqnarray}\n\\psi_1(x) &=& (\\sin^2 x)^{\\frac{1}{4}-\\frac{m+c}{2}}\n(\\cos^2 x)^{-j-\\frac{1}{4}} {}_2F_1(-c-j,-m-j;1-m-c; \\sin^2 x) \\\\\n\\psi_2(x) &=& (\\sin^2 x)^{\\frac{1}{4}+\\frac{m+c}{2}}\n (\\cos^2 x)^{-j-\\frac{1}{4}} {}_2F_1(m-j,c-j;1+m+c; \\sin^2 x) \\nonumber \n\\end{eqnarray}\n\nIn order to develop the band structure of this Schr\\\"odinger\nequation, we must construct the complementary series of the\nprojective representations of $so(2,2)\\sim su(1,1)\\oplus\nsu(1,1)$. This direct product structure allows us to simply use\nthe results discussed in the Scarf dynamical symmetry.\n\nThe complementary series, labeled by $(j,m,c)$, is constructed as\nfollows. For ranges of $m$ and $c$ which correspond to unitary\nrepresentations of (projective) complementary series $su(1,1)$, \nthe resulting $so(2,2)$ representation is also\nunitary. For ranges of $m$ and $c$ which are both non-unitary,\nthe resulting direct product becomes unitary in the strip\nof physical interest, $0<\|m+c\|\\leq 1/2$ . The\nremaining cases when $m$ is unitary and $c$ is non-unitary and\nthe case with $m$ and $c$ interchanged, result in non-unitary\nrepresentations of the complementary series of $so(2,2)$. These\nnon-unitary representations correspond to the energy bands of the extended\nScarf potential, which can be seen by taking limiting cases\nwhere (i) the potential reduces to the Scarf case (see below)\nand (ii) the potential vanishes and the spectrum is continuous.\n\nSince the eigenvalue of our Hamiltonian is $E=(m-c)^2$, and the\nstrength of the potential is labeled by $j$ and $m+c$, it is\nconvenient to plot the resulting unitary and non-unitary\nrepresentations of $so(2,2)$ versus $m+c$ for selected\nvalues of $j$. This is done in Fig. 6. Here the energy gaps\ncorrespond to the shaded regions and the bands to the unshaded\nregions. Three values of $j$ are chosen: (a) $j=-0.45$, (b)\n$j=-0.35$ and (c) $j=-0.25$. Case (c) corresponds to the Scarf\npotential limit. As $j\\rightarrow -1/2$ or $\|m+c\|\\rightarrow 0$,\nthe bands become degenerate. On the other hand, when\n$j\\rightarrow -1/4$ and\n $\|m+c\|\\rightarrow 1/2$, the spectrum\nbecomes continuous. For the band edges, one takes the direct\nproduct of $su(1,1)$ discrete projective representations.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%% FIG 6 %%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%\\tdfullfigure fig6_jmp.ps {\\small Complementary series of the projective\n% representation\n%* of $so(2,2)$. The spectrum of $m-c$ is shown for $0<\|m+c\|\\le\n%* 1/2$ for (a) $j=-0.45$, (b) $j=-0.35$ and (c) $j=-0.25$. The\n%* dashed lines are the spectra for particular strengths of\n%* the potential for which the group velocity is computed in\n%* Fig. 7. The unshaded regions are non-unitary representations,\n%* the striped (checkered) regions are unitary representations which arise\n%* from direct products of two $su(1,1)$ unitary (non-unitary) \n%*** representations. } 4. -4.5 9. 1.5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n\n\nThe bands $E=(m-c)^2$ are given by the following ranges of\nquantum numbers in the $(m,c)$ plane:\n\n\\begin{eqnarray}\n2n -(m_o+c_o)-2j &\\leq & m-c\\leq 2n+1-\\mid 2j+1-m_o-c_o\\mid,\\\\\n2n+1+\\mid 2j+1-m_o-c_o\\mid &\\leq & m-c\\leq 2n+2+2j+m_o+c_o,\\nonumber\n\\end{eqnarray}\n%\nwhere $n=0,1,2,...$ and\n%\n\\begin{equation}\n 0< \\mid m+c\\mid \\leq \\frac{1}{2},\\qquad 0< 2j+1\\leq \\frac{1}{2}.\n\\end{equation}\n\n\n\\vspace{7mm}\n\\noindent{\\sc C. Transfer matrix}\n\\vspace{7mm}\n\nDue to the strong singularity structure of the potential, one\nagain must introduce boundary conditions for the solutions at\nsingularities such that the Schr\\\"odinger operator can be made\nself-adjoint. Such an analysis has been undertaken in Refs. \n\\cite {gesztesy1,gesztesy2}. We can then easily\ncompute the transfer matrix for the interval $x\\in\n(-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\nusing the boundary values and first derivatives at $\\frac{\\pi}{2}$.\nThe transfer matrix is:\n\n\\begin{equation}\nT\\,=\\,\\left(\n\\begin{array}{cc}\n\\alpha & \\beta \\\\\n\\beta ^{\\ast } & \\alpha ^{\\ast }\n\\end{array}\n\\right)\n\\end{equation}\n%\nwhere\n\n\\begin{eqnarray}\n\\alpha &=& e^{- i (m-c) \\pi} \\, \\left[ \\frac{\\cos\\pi (m-c) + \n \\cos \\pi (2j+1) \\cos\\pi (m+c)}{\\sin\\pi(2j+1)\n \\sin\\pi(m+c)}\\right. \\nonumber \\\\\n& & \\: + \\, i \\left( \\frac{1}{m-c} \\frac{ \\Gamma(-2j-1)\\Gamma (1+j-m) \\Gamma (1\n+j -c)}{ \\Gamma (2j+1) \\Gamma (-j-m)\\Gamma(-j-c)}\\right. \\nonumber \\\\\n& & \\left.- \\, (m-c) \\frac{ \\Gamma (2j+1 ) \\Gamma (-j+m)\\Gamma(-j+c)}{\n\\Gamma(-2j-1)\\Gamma (1+j+m) \\Gamma (2 +j +c)}\\right) \\nonumber \\\\\n& & \\left.\\frac{\\sin\\pi(j-m)\\sin\\pi(j-c)}{\\sin\\pi(2j+1)\\sin\\pi(m+c)}\\right]\\\\\n\\beta &=& -i \\; e^{i (m-c) \\pi}\\frac{\\sin\\pi(j-m)\\sin\\pi(j-c)}{%\n\\sin\\pi(2j+1)\\sin\\pi(m+c)} \\nonumber \\\\\n& & \\left[\\frac{1}{m-c} \\frac{ \\Gamma(-2j-1)\\Gamma (1+j-m) \\Gamma (1 +j -c)}{\n\\Gamma (2j+1) \\Gamma (-j-m)\\Gamma(-j-c)} \\right. \\nonumber \\\\\n& & + \\, \\left.(m-c) \\frac{ \\Gamma (2j+1 ) \\Gamma (-j+m)\\Gamma(-j+c)}{\n\\Gamma(-2j-1)\\Gamma (1+j+m) \\Gamma (2 +j +c)}\\right]\n\\end{eqnarray}\n\n\n\\vspace{7mm}\n\\noindent{\\sc D. Dispersion relation}\n\\vspace{7mm}\n\nThe dispersion relation is computed as before, using $\\cos \\pi k\n= Re(\\alpha e^{i(m-c)\\pi})$:\n\n\\begin{equation}\n\\cos\\pi k = \\frac{\\cos\\pi (m-c) + \\cos \\pi (2j+1) \\cos \\pi (m+c)}{%\n\\sin\\pi(2j+1)\\sin\\pi(m+c)} .\n\\end{equation}\n%\nIf we denote\n\\begin{equation}\n m_+ = m+c,\\qquad m_-=m-c,\n\\end{equation}\n%\nthen $E=(m-c)^2=m_-^2$, and we find:\n\n\\begin{eqnarray}\nE(k) &=& m_-^2 \\nonumber \\\\\n&=& \\frac{1}{\\pi^2}\\left[\\cos^{-1}(\\cos\\pi k \\sin\\pi(2j+1)\\sin\\pi m_+\n -\\cos\\pi(2j+1)\\cos\\pi m_+)\\right]^2\n\\end{eqnarray}\n\nThe band structure could be explained through the projective representations\nof $so(2,2)$ when $0<\|m+c\|\\leq \\frac{1}{2}$ and $-\\frac{1}{2} < j \\leq\n-\\frac{1}{4}$. Again, non-unitary\nrepresentations give the energy bands while unitary representations\ncorrespond to energy gaps.\n\nThe group velocity for this potential is \n%\n\\begin{eqnarray} \n{\\cal V}(j,m,c) &=& \\frac{\\partial E}{\\partial\n k}\\nonumber\\\\\n & =& \\frac{2 m_-}{\\sin \\pi m_-} \\left[\\sin^2(2\\pi j) - \\cos^2\n \\pi m_+ -\\cos^2 \\pi m_-\\right.\\\\\n & & \\left.+ 2\\cos\\pi m_-\\cos\\pi m_+\\cos 2\\pi j\\right]^{1/2}\\nonumber\n\\end{eqnarray}\n%\nThe behavior is shown in Fig. 7 for selected values of $j$ and\n$m+c$ given by the dashed lines in Fig. 6.\nThe effective mass $M^(j,m,c)$ is given by:\n%\n\\begin{eqnarray}\n\\frac{1}{M^} &=& \\frac{\\partial^2 E}{\\partial k^2}\\nonumber\\\\\n & =& 2 m_-\\pi \\left[ \\cot\\pi m_- - \\cos 2\\pi j\\cos\n \\pi m_+\\csc\\pi m_-\\right]\\nonumber \\\\\n & & + 2\\csc^2\\pi m_-\\left(1- m_-\\pi\\cot\\pi\n m_-\\right)\\left(\\sin^22\\pi j - \\cos^2\\pi m_+ \\right.\\\\\n & & \\left. - \\cos^2\\pi m_- +\n 2\\cos\\pi m_-\\cos\\pi m_+\\cos 2\\pi j\\right) \\nonumber\n\\end{eqnarray}\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%% FIG 7 %%%%%%%%%%%%%%%%%%5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n%*\\tdfullfigure fig7_jmp.ps {\\small Group velocity for the three cases shown\n%* in Fig. 6. For $j=-1/4$ and $\|m+c\|=1/2$, one has free motion.\n%*** } 4. -4.5 9. 1.3\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n\n\n\\vspace{7mm}\n\\noindent{\\sc E. Limiting cases}\n\\vspace{7mm}\n\nThere are three cases where the extended Scarf potential reduces\nto the Scarf case: {\\it (i)} When $2j+1 = \\frac{1}{2}$, the potential\nbecomes the Scarf potential and the transfer matrix is\nequivalent to Eqs. (24)-(25). {\\it (ii)} When $m+c=\\frac{1}{2}$,\nEq. (38) reduces to the\npotential\n\\begin{equation}\n \\frac{ (2j+1)^2 - \\frac{1}{4}}{\\cos^2\\theta}\n\\end{equation}\n% \nand the transfer matrix is consistent with the results of Sec. II.E.\n{\\it (iii)} When $\|m+c\|=2j+1$, the Hamiltonian\nreduces to the Scarf potential with twice the period. \n\nOf the three limiting cases, it is case {\\it (ii)} which\nprovides something new. To compare to the Scarf results, we let\n$2j+1=\\tilde j+\\frac{1}{2}$, so that the potential (57) becomes\n$\\tilde j(\\tilde j+1)/\\sin^2x$. For the full complementary\nseries $-1/2< j\\leq 0$, we have $-1/2< \\tilde j\\leq 1/2$ which\ncorresponds to potentials $g/\\sin^2 x$ with $-1/4< g <\n3/4$. From (47) we find that the energy bands are given by\n\n\n\\begin{eqnarray}\n2n-\\tilde j &\\leq & m-c\\leq 2n+1+\\tilde j,\\\\\n2n+1-\\tilde j &\\leq & m-c\\leq 2n+2 + \\tilde j.\\nonumber\n\\end{eqnarray}\n%\nThis is precisely the band structure obtained in Eq. (18),\n(21)-(22), but now extended to positive coupling constants,\n$0\\leq g < 3/4$, while using the same complementary series. This\nagrees with the more recent observation that the Scarf potential\nadmits band structure for ranges of the strength which are \npositive\\cite{gesztesy1}. It also exemplifies the fact that a\ndynamical symmetry does not necessarily exhaust all\npossible regimes of band structure, and that other realizations\nmight provide additional regions. In principle we can\nextend our analysis of the generalized Scarf potential to\n$g_1,g_2 >0$ as well, but we do not do so here.\n\n\n\\vspace{1cm}\n\\begin{center}\n{\\large IV. Conclusions}\n\\end{center}\n\\vspace{7mm}\n\n\nWe have shown that dynamical symmetry techniques can be applied to\nHamiltonians with periodic potentials, and band structure can arise\nnaturally from representation theory. This fills a long-standing gap in the\nalgebraic approach to quantum systems. We have constructed\ndynamical symmetry Hamiltonians in $so(2,1)$ and $so(2,2)$ which\ncan be expressed as Schr\\\"odinger operators with periodic\npotentials. Using projective representations motivated by Bloch's theorem,\nwe have seen that the complementary series of $so(2,1)$ and $so(2,2)$\n(and their {\\it non-unitary} representations) are needed to \nexplain band structure, while the discrete representations are\nimportant for band edges. As far as we know, this is the first\napplication of the $su(1,1)$ complementary series to a physical\nproblem. It now seems reasonable to loosely associate the\nthree series of projective representations, discrete, principal\nand complementary, with the quantum problems of bound states,\nscattering states and energy bands.\n\n Using our dynamical symmetries, Hamiltonians such as Scarf's\nand its extension can be reduced to quadratic forms of the\nCartan subalgebra generators, such as $H=J_z^2$,\nwhich are readily solved. We are then able to derive not only the band\nstructure, but the dispersion relation and transfer matrix as\nwell. It would be interesting to develop\nhigher dimensional periodic Hamiltonians connected to\nrepresentations of $u(n,m)$ or $so(n,m)$. In this case, the\ninclusion of additional discrete symmetries using point groups\nwould be possible, and extensions to non-dynamical symmetry\nproblems could be pursued.\n\n\\vspace{7mm}\n\\noindent{\\bf ACKNOWLEDGMENTS}\n\\vspace{7mm}\n\nWe would like to thank F. Iachello for many useful discussions. This work\nwas supported by DOE grant DE-FG02-91ER40608.\n\n\\newpage\n\n\\appendix\n\\renewcommand{\\thesection}{{\\bf APPENDIX} \\Alph{section}.}\n\\setcounter{equation}{0}\n\\setcounter{section}{1}\n\\renewcommand{\\theequation}{\\Alph{section}\\arabic{equation}}\n\n\\begin{center}\n{\\large Appendix A: Representations of $so(2,1)$}\n\\end{center}\n\\vspace{7mm}\n\nFirst, let us recall the presentation of $so(3)$. The algebra can\nbe realized as differential operators on the sphere $x^2 + y^2 +\nz^2 = 1$. The representations are labeled by $(j,m)$ where $j$ is any\nnon-negative half integer and $-j \\le m \\le j$.\n\nThe $so(2,1)$ algebra can be realized as differential operators on a\nhyperboloid $- x^2 - y^2 + z^2 = 1$. The unitary representations are \\cite\n{bargmann}:\n\n\\begin{itemize}\n\\item The principal series $j=-\\frac{1}{2}+i\\rho ,\\,\\rho >0,\\,m=0,\\pm\n1,\\ldots $ or $m=\\pm \\frac{1}{2},\\pm \\frac{3}{2},\\ldots $.\n\n\\item The complementary series $-\\frac{1}{2}<j<0,\\,m=0,\\pm 1,\\ldots $.\n\n\\item The discrete series $D_{j}^{+}$, where $j$ is a negative integer or\nhalf integer and $m=-j,-j+1,\\ldots $.\n\n\\item The discrete series $D_{j}^{-}$, where $j$ is a negative integer or\nhalf integer and $m=j,j-1,\\ldots $.\n\\end{itemize}\n\nA more general form of the representations of the algebra are\n the projective representations. The projective unitary\nrepresentations of $so(2,1)$ are \\cite{pukanszky}:\n\n\\begin{itemize}\n\\item The principal series $j=-\\frac{1}{2}+i\\rho ,\\,\\rho >0,\\,0\\leq\nm_{0}<1,\\,m=m_{0}\\pm n,\\,n=0,1,2,\\ldots $\n\n\\item The complementary series $-\\frac{1}{2}<j<0,\\,0\\leq m_{0}<1,\\\n,m_{0}(m_{0}-1)>j(j+1)\\geq -\\frac{1}{4},\\,m=m_{0}\\pm n,n=0,1,\\ldots $.\n\n\\item The discrete series $D_{j}^{+},j<0$, $m=-j, -j+1,...$\n\n\\item The discrete series $D_{j}^{-},j<0$, $m=j, j-1,...$\n\\end{itemize}\n\nSince we find that the non-unitary representations are important\nfor the bands, we review their origin\\cite{pukanszky}.\nAssuming $I_3 f = m_0 f$, with $0\\leq m_0< 1$, and using the \ncommutation relations for $ so(2,1) $, we have\n\\begin{eqnarray}\nI_{3}I_{+}f &=&(m_0 +1)I_{+}f \\\\\nI_{3}I_{-}f &=&(m_0 -1)I_{-}f \\\\\nI_{-}I_{+}f &=&[-j(j+1)+m_0 (m_0 +1)]f \\\\\nI_{+}I_{-}f &=&[-j(j+1)+m_0 (m_0 -1)]f\n\\end{eqnarray}\nwhere $I^{2}=j(j+1)$ is the Casimir, a constant for a specific\nrepresentation.\nReplacing $f$ by $I_{+}^{n-1}f$ and $I_{-}^{n-1}f\\,(n=1,2,\\dots )$ in\nthe last two equations, we get\n\n\\begin{eqnarray}\nI_{-}I_{+}^{n}f &=&\\alpha _{n}I_{+}^{n-1}f \\\\\nI_{+}I_{-}^{n}f &=&\\beta _{n}I_{-}^{n-1}f\n\\end{eqnarray}\nwhere $\\alpha _{n}=-j(j+1)+(m_0 +n-1)(m_0 +n)$ and \n$\\beta _{n}=-j(j+1)+(m_0-n)(m_0 -n+1)$. The above relations imply\n\\begin{eqnarray}\n\|\|I_{+}^{n+1}f\|\|^{2} &=&(I_{+}^{n+1}f,I_{+}^{n+1}f)=\\alpha\n_{n+1}\|\|I_{+}^{n}f\|\|^{2} \\\\\n\|\|I_{-}^{n+1}f\|\|^{2} &=&(I_{-}^{n+1}f,I_{-}^{n+1}f)=\\beta\n_{n+1}\|\|I_{-}^{n}f\|\|^{2}\n\\end{eqnarray}\n%\n\nStarting with the initial state $f$, we can generate the\ncoefficients $\\alpha_k$ and $\\beta_k$ ($k>0$). Of these\ncoefficients, only $\\beta_1$ can be positive or negative.\nThis distinguishes the unitary and non-unitary\nrepresentations. For instance $\\beta_1>0$ when\n$m_0(m_0-1)>j(j+1)$, which gives the complementary series. When\nwe are in the region $-j< m_0< 1+j$, $\\beta_1< 0$. So\nif we start with a state $f$ labeled by $(j,m_0)$ with $-j<\nm_0<1+j$, we find that all states obtained by operating with $I_+$\nwill have norms of the same sign. These are related to all the\nstates $I_-^n f$ by a sign change in the norm. \nConsequently, the states of the non-unitary representation can\nbe divided into two families. In each family, the states have norms of the\nsame sign, while the two families are related by a change in\nsign in the norm. \n\n\n\\vspace{1cm}\n\\begin{center}\n{\\large Appendix B: A Formula for the Transfer Matrix}\n\\end{center}\n\\vspace{7mm}\n\\setcounter{equation}{0}\n\\setcounter{section}{2}\n\nWhen the potential is symmetric about the center of each period, it\nis convenient to consider even and odd solutions $g(E,x)$, $u(E,x)$\nsuch that\n\\begin{eqnarray}\ng(E,0)=1,& g'(E,0)=0, \\\\\nu(E,0)=0,& u'(E,0)=1.\n\\end{eqnarray}\n\nLet us define \\cite{james}\n\n\\begin{eqnarray}\ng(E,-\\frac{a}{2}) = g(E, \\frac{a}{2}) = g_0 (E); \\\\\ng'(E,-\\frac{a}{2}) = -g'(E, \\frac{a}{2}) = g'_0 (E); \\\\\nu(E,-\\frac{a}{2}) = -u(E, \\frac{a}{2}) = u_0 (E); \\\\\nu'(E,-\\frac{a}{2}) = u'(E, \\frac{a}{2}) = u'_0 (E);\n\\end{eqnarray}\n\nAccording to the definition of transfer matrix \\cite{cohen}, we can\nderive a formula as follows:\n\n\\begin{equation}\nT \\, = \\, \\left(\n\\begin{array}{cc}\n\\alpha & \\beta \\\\\n\\beta^{} & \\alpha^{}\n\\end{array}\n\\right)\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\alpha &=& e^{-ika} [(g_0 u'_0 + g'_0 u_0) + i(u'_0 g'_0 /k- u_0 g_0 k)] \\\\\n\\beta &=& -i e^{ika} (u'_0 g'_0 /k + u_0 g_0 k)\n\\end{eqnarray}\nand $k=\\sqrt E$, $a$ is the period.\n\n\\newpage\n\n\\begin{thebibliography}{99}\n\\bibitem{1} For a survey, see for example: {\\sl Dynamical Groups and\n Spectrum Generating Algebras}, Eds. A.Barut, A. Bohm and Y.Ne'eman (World\n Scientific, Singapore, 1987).\n\n\\bibitem{2} F. Iachello, {\\it Chem. Phys. Lett.} {\\bf 78} 581\n (1981); F. Iachello and R. D. Levine, {\\it\n J. Chem. Phys.} {\\bf 77} 3046 (1982);\n F. Iachello and R. Levine, {\\it Algebraic Theory of Molecules} \n (Oxford Press, Oxford, 1995); F. Iachello and\n A. Arima, {\\sl The Interacting Boson Model}\n (Cambridge Press, Cambridge, 1987).\n \n\n\\bibitem{AGI} Y. Alhassid, F. G\\\"{u}rsey and F. Iachello, {\\it\n Phys. Rev. Lett.} {\\bf 50} 873 (1983).\n\n\\bibitem{fi1} See for example, F. Iachello, {\\sl Rev. Nuovo\n Cimento} {\\bf 19} 1 (1996), and references there in.\n\n\\bibitem{dk} D. Kusnezov, {\\sl Phys. Rev. Lett.} {\\bf 79} 537 (1997).\n\n\\bibitem{li} H. Li and D. Kusnezov, Yale Univ. preprint (1999);\n {\\it ibid}, in {\\it Group 22: International Colloquium on Group\n Theoretical Methods in Physics}, Eds. S.P. Corney,\n R. Delbourgo and P.D. Jarvis, (International, Cambridge, MA,\n 1999), p. 310.\n\n\\bibitem{scarf} F. L. Scarf, {\\it Phys. Rev.} {\\bf 112} 1137 (1958).\n\n\\bibitem{fi2} Y. Alhassid, F. G\\\"ursey and F. Iachello,\n {\\it Ann. Phys. (NY)} {\\bf 167} 181 (1983); A. Frank and K. B. Wolf,\n {\\it Phys. Rev. Lett.} {\\bf 52} 1737 (1984).\n\n\\bibitem{gesztesy1} F. Gesztesy and W. Kirsch, {\\it Journal \n f\\\"{u}r Mathematik} {\\bf 362} 28 (1984).\n\n\n\\bibitem{gursey} F. G\\\"{u}rsey, in {\\it Group Theoretical\n Methods in Physics XI},\n (Springer-Verlag, Berlin, 1983) p.106.\n\n\\bibitem{kittel} C. Kittel, {\\it Quantum Theory of Solids},\n (Wiley, New York, 1963).\n\n\\bibitem{bargmann} V. Bargmann, {\\it Ann. Math.} {\\bf 48} 568 (1947).\n\n\\bibitem{pukanszky} L. Puk\\'{a}nszky, {\\it Math. Annalen} {\\bf\n 156} 96 (1964); A. O. Barut and C. Fronsdal, {\\it\n Proc. Roy. Soc. London} {\\bf A287}, 532 (1965).\n\n\\bibitem{james} H. M. James, {\\it Phys. Rev.} {\\bf 76} 1602 (1949).\n\n\\bibitem{cohen} C. Cohen-Tannoudji, B. Diu and F. Laloe,\n {\\it Quantum Mechanics} (Wiley, New York, 1977) Vol.1 .\n\n\\bibitem{poschl} G. P\\\"oschl and E. Teller, {\\it Z. Phys.} {\\bf\n 83} 143 (1933); L.D. Salem and R. Montemayor, {\\it\n Phys. Rev.} {\\bf A47} 105 (1993).\n\\bibitem{gesztesy2} F. Gesztesy, C. Macdeo and L. Streit, {\\it\n J. Phys. A: Math. Gen.}{\\bf 18} L503 (1985).\n\n\\bibitem{miller} W. Miller Jr., {\\it Lie Theory and Special\n Functions}, (Academic Press, New York, 1968).\n\n\\end{thebibliography}\n\n\\end{document}\n" } ]	[ { "name": "solv-int9912007.extracted_bib", "string": "{1 For a survey, see for example: {\\sl Dynamical Groups and Spectrum Generating Algebras, Eds. A.Barut, A. Bohm and Y.Ne'eman (World Scientific, Singapore, 1987)." }, { "name": "solv-int9912007.extracted_bib", "string": "{2 F. Iachello, {Chem. Phys. Lett. {78 581 (1981); F. Iachello and R. D. Levine, {J. Chem. Phys. {77 3046 (1982); F. Iachello and R. Levine, {Algebraic Theory of Molecules (Oxford Press, Oxford, 1995); F. Iachello and A. Arima, {\\sl The Interacting Boson Model (Cambridge Press, Cambridge, 1987)." }, { "name": "solv-int9912007.extracted_bib", "string": "{AGI Y. Alhassid, F. G\\\"{ursey and F. Iachello, {Phys. Rev. Lett. {50 873 (1983)." }, { "name": "solv-int9912007.extracted_bib", "string": "{fi1 See for example, F. Iachello, {\\sl Rev. Nuovo Cimento {19 1 (1996), and references there in." }, { "name": "solv-int9912007.extracted_bib", "string": "{dk D. Kusnezov, {\\sl Phys. Rev. Lett. {79 537 (1997)." }, { "name": "solv-int9912007.extracted_bib", "string": "{li H. Li and D. Kusnezov, Yale Univ. preprint (1999); {ibid, in {Group 22: International Colloquium on Group Theoretical Methods in Physics, Eds. S.P. Corney, R. Delbourgo and P.D. Jarvis, (International, Cambridge, MA, 1999), p. 310." }, { "name": "solv-int9912007.extracted_bib", "string": "{scarf F. L. Scarf, {Phys. Rev. {112 1137 (1958)." }, { "name": "solv-int9912007.extracted_bib", "string": "{fi2 Y. Alhassid, F. G\\\"ursey and F. Iachello, {Ann. Phys. (NY) {167 181 (1983); A. Frank and K. B. Wolf, {Phys. Rev. Lett. {52 1737 (1984)." }, { "name": "solv-int9912007.extracted_bib", "string": "{gesztesy1 F. Gesztesy and W. Kirsch, {Journal f\\\"{ur Mathematik {362 28 (1984)." }, { "name": "solv-int9912007.extracted_bib", "string": "{gursey F. G\\\"{ursey, in {Group Theoretical Methods in Physics XI, (Springer-Verlag, Berlin, 1983) p.106." }, { "name": "solv-int9912007.extracted_bib", "string": "{kittel C. Kittel, {Quantum Theory of Solids, (Wiley, New York, 1963)." }, { "name": "solv-int9912007.extracted_bib", "string": "{bargmann V. Bargmann, {Ann. Math. {48 568 (1947)." }, { "name": "solv-int9912007.extracted_bib", "string": "{pukanszky L. Puk\\'{anszky, {Math. Annalen {156 96 (1964); A. O. Barut and C. Fronsdal, {Proc. Roy. Soc. London {A287, 532 (1965)." }, { "name": "solv-int9912007.extracted_bib", "string": "{james H. M. James, {Phys. Rev. {76 1602 (1949)." }, { "name": "solv-int9912007.extracted_bib", "string": "{cohen C. Cohen-Tannoudji, B. Diu and F. Laloe, {Quantum Mechanics (Wiley, New York, 1977) Vol.1 ." }, { "name": "solv-int9912007.extracted_bib", "string": "{poschl G. P\\\"oschl and E. Teller, {Z. Phys. {83 143 (1933); L.D. Salem and R. Montemayor, {Phys. Rev. {A47 105 (1993)." }, { "name": "solv-int9912007.extracted_bib", "string": "{gesztesy2 F. Gesztesy, C. Macdeo and L. Streit, {J. Phys. A: Math. Gen.{18 L503 (1985)." }, { "name": "solv-int9912007.extracted_bib", "string": "{miller W. Miller Jr., {Lie Theory and Special Functions, (Academic Press, New York, 1968)." } ]
solv-int9912008	The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff	[ { "author": "M. Adler\\thanks{Department of Mathematics" }, { "author": "Brandeis University" }, { "author": "Waltham" }, { "author": "Mass 02454" }, { "author": "USA. E-mail: adler@math.brandeis.edu. The support of a National Science Foundation grant \\# DMS-98-4-50790 is gratefully acknowledged." } ]		[ { "name": "solv-int9912008.tex", "string": "\\documentclass[12pt]{article}\n\n\n\\title{The Pfaff lattice, Matrix integrals and a map from\n Toda to Pfaff}\n\n\\author{M. Adler\\thanks{Department of Mathematics,\nBrandeis University, Waltham, Mass 02454, USA. E-mail:\nadler@math.brandeis.edu. The support of a National Science\nFoundation grant \\# DMS-98-4-50790 is gratefully\nacknowledged.}~~~~~~P. van Moerbeke\\thanks{Department of\nMathematics, Universit\\'e de Louvain, 1348 Louvain-la-Neuve,\nBelgium and Brandeis University, Waltham, Mass 02454, USA. E-mail:\nvanmoerbeke@geom.ucl.ac.be and @math.brandeis.edu. The support of\na National Science Foundation grant \\# DMS-98-4-50790, a Nato, a\nFNRS and a Francqui Foundation grant is gratefully acknowledged.}}\n\n\n%\\input{ucl}\n\\date{March 8, 1999}\n\n\\newcommand{\\MAT}[1]{\\left(\\begin{array}{#1c}}\n\\newcommand{\\mat}{\\end{array}\\right)}\n\\newcommand{\\qed}\n{%\n\\mbox{}%\n\\nolinebreak%\n\\hfill%\n\\rule{2mm}{2mm}%\n\\medbreak%\n\\par%\n}\n\n\n\n\n\\newcommand{\\sumbis}[2]%\n{%\n\\renewcommand{\\arraystretch}{0.5}\n\\begin{array}[t]{c}\n\\sum\\\\\n{\\scriptstyle #1}\\\\\n{\\scriptstyle #2}\n\\end{array}\n\\renewcommand{\\arraystretch}{1}\n}\n\n\n\\def\\boxed#1{\\vbox{\\hrule\\hbox{\\vrule$\\displaystyle #1$\n\\vrule}\\hrule}}\n\n\\newcommand{\\rg}{\\rightarrow}\n\\newcommand{\\lrg}{\\longrightarrow}\n\\newcommand{\\Rg}{\\Rightarrow}\n\\newcommand{\\DF}{\\Longleftrightarrow}\n\\newcommand{\\pp}{\\ldots}\n\\newcommand{\\TT}{\\tilde\\tau}\n\\newcommand{\\DR}{{\\cal D}}\n\\newcommand{\\LR}{{\\cal L}}\n\\newcommand{\\AR}{{\\cal A}}\n\\newcommand{\\BF}{{\\Bbb F}}\n\\newcommand{\\BC}{{\\Bbb C}}\n\\newcommand{\\BD}{{\\Bbb D}}\n\\newcommand{\\BX}{{\\Bbb X}}\n\\newcommand{\\BY}{{\\Bbb Y}}\n\\newcommand{\\BZ}{{\\Bbb Z}}\n\\newcommand{\\Bk}{{\\Bbb k}}\n\\newcommand{\\Bn}{{\\Bbb n}}\n\\newcommand{\\Bu}{{\\Bbb u}}\n\\newcommand{\\Sg}{\\Sigma}\n\\newcommand{\\iy}{\\infty}\n\\newcommand{\\pl}{\\partial}\n\\newcommand{\\al}{\\alpha}\n\\newcommand{\\proof}{\\underline{\\sl Proof}: }\n\\newcommand{\\remark}{\\underline{\\sl Remark}: }\n\\newcommand{\\example}{\\underline{\\sl Example}: }\n\\newcommand{\\om}{\\omega}\n\\newcommand{\\HR}{{\\cal H}}\n\\newcommand{\\JR}{{\\cal J}}\n\\newcommand{\\VR}{{\\cal V}}\n\\newcommand{\\UR}{{\\cal U}}\n\\newcommand{\\SR}{{\\cal S}}\n\\newcommand{\\TR}{{\\cal T}}\n\\newcommand{\\MR}{{\\cal M}}\n\\newcommand{\\CR}{{\\cal C}}\n\\newcommand{\\CB}{{\\cal B}}\n\\newcommand{\\PR}{{\\cal P}}\n\\newcommand{\\NR}{{\\cal N}}\n\\newcommand{\\WR}{{\\cal W}}\n\\newcommand{\\FR}{{\\cal F}}\n\\newcommand{\\GR}{{\\cal G}}\n\\newcommand{\\vp}{\\varphi}\n\\newcommand{\\la}{\\langle}\n\\newcommand{\\ra}{\\rangle}\n\\newcommand{\\ga}{\\gamma}\n\\newcommand{\\Ga}{\\Gamma}\n\\newcommand{\\dt}{\\delta}\n\\newcommand{\\Dt}{\\Delta}\n\\newcommand{\\vr}{\\varepsilon}\n\\newcommand{\\sg}{\\sigma}\n\\newcommand{\\BR}{{\\Bbb R}}\n\\newcommand{\\lb}{\\lambda}\n\\newcommand{\\Lb}{\\Lambda}\n\\newcommand{\\tr}{\\mbox{tr}}\n\n\\newcommand{\\pk}{\\pi_{{\\bf k}}}\n \\newcommand{\\PK}{\\pi_{{\\bf K}}}\n \\newcommand{\\ps}{\\pi_{{\\bf s}}}\n \\newcommand{\\PS}{\\pi_{{\\bf S}}}\n\n\n\\newcommand{\\BJ}{{\\Bbb J}}\n%\\def\\span{\\mathop{\\rm span}}\n\\def\\diag{\\mathop{\\rm diag}}\n\\def\\Res{\\mathop{\\rm Res}}\n\n\n\\def\\be#1\\ee{\\begin{equation}#1\\end{equation}}\n\\def\\bea#1\\eea{\\begin{eqnarray}#1\\end{eqnarray}}\n\\def\\bean#1\\eean{\\begin{eqnarray}#1\\end{eqnarray}}\n\n\\ifx\\undefined\\Bbb\n \\let\\Bbb\\bf\n\\fi\n\n\n\\catcode`\\@=11\n\\def\\ps@X{\\let\\@mkboth\\@gobbletwo\n \\def\\@oddhead{\\tt A %&\n v M\n :%\n ~~Pfaff/matrix integrals\\hfil March 8, 1999\\ \\#1\\hfil\\S\\thesection,\np.\\thepage\n }\n \\def\\@oddfoot{\\rm\\hfil\\thepage\\hfil}\n \\let\\@evenhead\\@oddhead\n \\let\\@evenfoot\\@oddfoot}\n\\catcode`@=12\n\\pagestyle{X}\n\n\n\n%%% NUMEROTATION DES EQUATIONS\n\n%%% Le type de numerotation qeu tu souhaites\n%%% n'est prevu de maniere standard que par LaTeX 2e.\n%%% Voici une redefinition repondant a tes souhaits\n%%% et valable en general.\n\n% Redefinition du compteur \"equation\"\n% pour que la numerotation soit subordonne a celle des sous-sections\n\n%%% NOUVELLES COMMANDES\n\n% Redefinition du compteur \"equation\"\n% pour que la numerotation soit subordonne a celle des sous-sections\n\n\\catcode`\\@=11\n\\let\\c@equation=\\relax\n\\newcounter{equation}[subsection]\n\\def\\theequation{\\thesubsection.\\arabic{equation}}\n\\catcode`\\@=12\n\n\n\\newtheorem{definition}{Definition}[%sub\nsection]\n\n\n\\newtheorem{theorem}[definition]{Theorem}\n\n\n\\newtheorem{lemma}[definition]{Lemma}\n\\newtheorem{corollary}[definition]{Corollary}\n\\newtheorem{proposition}[definition]{Proposition}\n\n\n\\catcode`\\@=11\n\\let\\c@equation=\\relax\n\\newcounter{equation}[section]\n\\def\\theequation{\\thesection.\\arabic{equation}}\n\\catcode`\\@=12\n\n\n\n\n\\begin{document}\n\\maketitle\n\n\n\n%\\abstract{We study the Pfaff lattice, introduced by us\n%in the context of a Lie algebra splitting of gl(infinity) into\n%sp(infinity) and lower-triangular matrices. We establish a set of\n%bilinear identities, which we show to be equivalent to the Pfaff\n%Lattice. In the semi-infinite case, the tau-functions are\n%Pfaffians; interesting examples are the matrix integrals over\n%symmetric matrices (symmetric matrix integrals) and matrix\n%integrals over self-dual quaternionic Hermitean matrices\n%(symplectic matrix integrals).\n\n%There is a striking parallel of the Pfaff lattice with the Toda\n%lattice, and more so, there is a map from one to the other. In\n%particular, we exhibit two maps, dual to each other,\n\n\n%(i) from the the Hermitean matrix integrals to the symmetric matrix\n%integrals, and\n\n%(ii) from the Hermitean matrix integrals to the symplectic matrix\n%integrals.\n\n%The map is given by the skew-Borel decomposition of a\n%skew-symmetric operator. We give explicit examples for the classical weights. }\n\n\\tableofcontents\n\\setcounter{section}{-1}\n\n\\setcounter{equation}{0}\n\n\n\n\\newpage\n\\section{Introduction}\nConsider a weight on $\\BR$, depending on $t=(t_1,t_2,\\ldots)\n\\in\n\\BC^{\\iy}$,\n\\be\\rho_t(z)dz=e^{\\sum t_i\nz^i}\\rho(z)dz=e^{-V(z)+\\sum_1^{\\iy}t_i\nz^i}dz,~~\\mbox{with}~-\\frac{\\rho'(z)}{\\rho(z)}=V'(z)=\\frac{g(z)}{f(z)}.\\ee\n\n\n\n{\\bf Hermitean matrix integrals} ({\\sl revisited}) This weight\nleads to a $t$-dependent moment matrix\n $$\n m_n(t)=(\\mu_{k+\\ell}(t))_{0\n \\leq k,\\ell\\leq n-1}\n =\\left( \\int_{\\BR}z^{k+\\ell}\\rho_t(z) dz\n \\right)_{0\n \\leq k,\\ell\\leq n-1},\n$$\nwith the semi-infinite moment matrix $m_{\\iy}$, satisfying the\ncommuting equations\n\\be\\pl m_{\\iy} / \\pl t_k=\\Lb^k m_{\\iy}= m_{\\iy}\\Lb^k,\\ee\nwhere $\\Lb$ is the customary shift matrix. Considering the lower-\nand an upper-triangular matrix Borel decomposition\n\\be\nm_{\\iy}=S^{-1} S^{\\top -1},\n\\ee\nwhich is determined by the following $t$-dependent matrix\n integrals\\footnote{We set vol$( \\UR(n))=1$ for all $n$.} ($n\\geq 0$)\n\\be\n\\tau_n(t):=\\int_{\\HR_n}e^{Tr (- V(X)+\\sum t_i X^i)}dX\n=\\det m_n, ~~\\mbox{and}~~\\tau_0=1,\n\\ee\nwith Haar measure $dX$ on the ensemble\n$\n{\\cal H}_n=\\{n\\times n\\mbox{\\,\\,Hermitean matrices}\\}.\n$\nAs is well known, the integral (0.4) is a solution to the following\ntwo systems,\n\\newline\\noindent{\\em (i) the KP-hierarchy}\n \\be\n\\left(p_{k+4}(\\tilde\\pl)-\\frac{1}{2}\\frac{\\pl^2}{\\pl\nt_1\\pl t_{k+3}}\\right)\\tau_{n}\\circ\\tau_{n}=0,\n ~~\\mbox{for}~~ k,n=0,1,2,\\dots.\n\\ee\n\\newline\\noindent{\\em (ii) the Toda lattice}; i.e., the tridiagonal\nmatrix\n \\be\nL(t):= S\\Lb S^{ -1}=\\pmatrix{\n \\frac{\\pl}{\\pl t_1}\\log \\frac{\\tau_1}{\\tau_0} &\n \\left(\\frac{\\tau_{0}\\tau_{2}}{\\tau_{1}^2}\\right)^{1/2}& 0 \\cr\n\\left(\\frac{\\tau_{0}\\tau_{2}}{\\tau_{1}^2}\\right)^{1/2}&\n\\frac{\\pl}{\\pl t_1}\\log \\frac{\\tau_2}{\\tau_1} &\n\\left(\\frac{\\tau_{1}\\tau_{3}}{\\tau_{2}^2}\\right)^{1/2}\\cr\n0&\\left(\\frac{\\tau_{1}\\tau_{3}}{\\tau_{2}^2}\\right)^{1/2}&\n \\frac{\\pl}{\\pl t_1}\\log \\frac{\\tau_3}{\\tau_2} & \\ddots \\cr\n& ~~~~\\ddots &~~~~~~~~~\\ddots }\\ee satisfies the following\ncommuting Toda equations\n$$\n\\frac{\\pl L}{\\pl t_n}=\\left[ \\frac{1}{2}(L^n)_{sk},L \\right],\n$$\nwhere $(A)_{sk}$ denotes the skew-part of the matrix $A$ for the\nLie algebra splitting into skew and lower-triangular matrices.\nMoreover, the following $t$-dependent polynomials in $z$,\n are defined\nby the $S$-matrix obtained from the Borel decomposition (0.3);\n it is also given, on the one hand, by a classic\n determinantal formula,\nand on the other hand, in terms of the functions $\\tau_n(t)$:\n\\bean\np_n(t,z):&=&(S(t)\\chi(z))_n=\\frac{1}{\\sqrt{\\tau_n\\tau_{n+1}}}\\det\n\\left(\\begin{array}{ccc\|c}\n & & &1\\\\\n & & &z \\\\\n &m_n(t)& &\\vdots\\\\\n & & & \\\\\n\\hline%{1-3}\n\\mu_{n,0}(t)&\\pp&\\mu_{n,n-1}(t)&z^n\n\\end{array}\\right)\\\\\n&=& z^n \\frac{\\tau_n(t-[z^{-1}])}{\\sqrt{\\tau_n \\tau_{n+1}}}\n.\\eean\nThe $p_n$'s are orthonormal with respect to the (symmetric)\ninner-product $\\la,\\ra_{sy}$, defined by $\\la\nz^i,z^j\\ra_{sy}=\\mu_{ij}$, which is a restatement of the Borel\ndecomposition (0.3). The vector $p(t,z)=(p_n(t,z))_{n\\geq 0}$ is an\neigenvector of the matrix $L(t)$ in (0.6):\n$$L(t)p(t,z)=zp(t,z).$$\n\n\n\n\n\n\\noindent{\\bf Symmetric and symplectic matrix integrals.}\n~ Instead consider the following skew-symmetric matrix\n$\n m_{\\iy}=(\\mu_{ij})_{i,j\\geq 0}$ of moments\\footnote{$\\vr(x)=1$, for\n$x\\geq 0$ and $=-1$, for $x<0$ and $\\{f,g\\}=f'g-fg'$.}\n\\be\\mu^{(1)}_{ij}(t)=\\int\\!\\int_{\\BR^2}x^iy^j\\vr(x-y)\\rho_t(x)\\rho_t(y)dx dy\n~~~\\mbox{or}~~~\n\\mu^{(2)}_{ij}(t)=\\int_{\\BR}\n \\{y^i,y^j\\}\\rho_t(y)^2 dy,\n \\ee\nboth satisfying the equations\n\\be\n\\frac{\\pl m_{\\iy}}{\\pl t_i}=\\Lb^im_{\\iy}+m_{\\iy}\\Lb^{\\top i}.\n\\ee\nIn this paper, we consider symmetric matrix\n integrals\\footnote{where again we set the volume of the\n orthogonal and symplectic groups equal to $1$.}\n \\be\n\\tau^{(1)}_{2n}(t):= \\frac{1}{(2n)!}\\int_{{\\cal S}_{2n}}\n e^{Tr~(- V(X)+\\sum_1^{\\iy} t_i X^i )} dX=pf(m^{(1)}_{2n}),\n\\ee\nand symplectic matrix integrals\n\\be\n \\tau^{(1)}_{2n}(t):= \\frac{1}{n!}\\int_{{\\cal T}_{2n}}\n e^{2Tr~(- V(X)+\\sum_1^{\\iy} t_i X^i )} dX=pf(m^{(2)}_{2n}),\n \\ee\nboth expressed in terms of the Pfaffian of the ``moment\"\n matrix $m^{(i)}_{\\iy},$\nwhere\n\\newline (1) in the first case, $dX$ denotes Haar measure\non the space $S_{2n}$ of symmetric matrices and,\n\\newline (2) in the\nsecond case, $dX$ denotes Haar measure\n on the $2n\n\\times 2n$ matrix realization $\n\\TR_{2n}$ of the space of self-dual $n\\times n$ Hermitean matrices, with quaternionic\nentries.\n\\newline Since $m_{\\iy}$ is skew-symmetric, the Borel decomposition of\n $m_{\\iy}$ will require the interjection of a skew-symmetric\n matrix $J,$ used throughout this paper,\n\\be\nJ=\\left(\n\\begin{array}{cc@{}c@{}cc}\n\\ddots &&&& \\\\\n &\\boxed{\\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array}} &&0& \\\\\n && \\boxed{\\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array}} &&\\\\\n &0&& \\boxed{\\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array}} & \\\\\n &&&& \\ddots\n \\end{array}\n \\right)~~\\mbox{with}~J^2=-I\n \\ee\n and the order 2 involution on the space $\\DR$ of\n infinite matrices\n \\be\n \\JR:\\DR\\longrightarrow\n\\DR: a\\longmapsto\\JR(a):=Ja^{\\top}J.\n\\ee\nThe skew-Borel decomposition\n \\be\nm_{\\iy}(t)=Q^{-1}(t)JQ^{-1\n\\top}(t)\n,\\ee\n can entirely be expressed in terms of the integrals $\\tau_{2n}(t)$,\n (0.9) and (0.10) corresponding respectively to the first and second\n moment matrix (0.7). They satisfy both\n\\newline\\noindent{\\em (i) the Pfaffian KP-hierarchy} for\n$k,n=0,1,2,\\dots$\\,,\n\\be\n\\left(p_{k+4}(\\tilde\\pl)-\\frac{1}{2}\\frac{\\pl^2}{\\pl\nt_1\\pl t_{k+3}}\\right)\\tau_{2n}\\circ\\tau_{2n}=p_k(\\tilde\n\\pl)~\\tau_{2n+2}\\circ\\tau_{2n-2},\n\\ee\n\\newline\\noindent{\\em (ii) the Pfaff lattice}; i.e., the matrix,\nconstructed by dressing up $\\Lb$ with $Q$ and which this time is\nfull below the main diagonal,\n$$ L =Q\\Lb Q^{-1}=\n h^{-1/2} \\pmatrix{\n \\hat L_{00}&\\hat L_{01}&0&0&\\cr\n\\hat L_{10}&\\hat L_{11}&\\hat L_{12}&0&\\cr &\\hat L_{21} &\\hat\nL_{22}&\\hat L_{23}&\\cr\n &\n&\\hat L_{32}&\\hat L_{33}&\\dots\\cr & & &\\vdots& &\n\\cr} h^{1/2},\n$$\nsatisfies the Hamiltonian commuting equations\n\\be\n\\frac{\\pl L}{\\pl\nt_i}=\n\\left[\\left((L^i)_++\\JR ((L^i)_+)\\right)+\\frac{1}{2}\n\\left((L^i)_0+ \\JR ((L^i)_0) \\right),L\\right],\n\\ee\nwith the entries $\\hat L_{ij}$ and the entries of $h$, being\n$2\\times 2$ matrices\n$$\nh=\\mbox{diag}(h_0I_2,h_2I_2,h_4I_2,\\dots),\n~h_{2n}=\\tau_{2n+2}/\\tau_{2n},\n$$\nand ($ {}^.=\\frac{\\pl}{\\pl t_1}$)\n$$\n\\hat L_{nn}:=\\pmatrix{-(\\log \\tau_{2n})^. & &1 \\cr \\cr\n -\\frac{S_2(\\tilde \\pl)\\tau_{2n}}{\\tau_{2n}}\n-\\frac{S_2(-\\tilde \\pl)\\tau_{2n+2}}{\\tau_{2n+2}} & &\n (\\log \\tau_{2n+2})^.\\cr}~~\n ~~~~~~\\hat L_{n,n+1}:=\\pmatrix{0& 0 \\cr\n 1 & 0\\cr}\n$$\n\n\\vspace{0.4cm}\n\n\\be\n\\hat L_{n+1,n}:=\\pmatrix{&(\\log \\tau_{2n+2})^{..}& \\cr\n &\\cr}.\n \\ee\nThe following $t$-dependent polynomials $q_n(t,z)=(S\\chi(z))_n$ in\n$z$, defined by the $S$ matrix of the skew-Borel\n decomposition (0.13), have determinantal and\n Pfaffian $\\tau$-function expression, in analogy with\n the Hermitean case:\n\\bean\nq_{2n}(t,z)&=&\\frac{1}{\\sqrt{\\tau_{2n}(t)\\tau_{2n+2}(t)}}pf\n\\left(\\begin{array}{ccc\|c}\n & & &1\\\\\n & & &z \\\\\n &m_{2n+1}(t)& &\\vdots\\\\\n & & &z^{2n} \\\\\n\\hline%{1-3}\n-1&\\pp&-z^{2n}&0\n\\end{array}\\right)\\\\\n&=& z^{2n} \\frac{\\tau_{2n}(t-[z^{-1}])}\n {\\sqrt{\\tau_{2n}(t) \\tau_{2n+2}(t)}}\n\\eean\nand\n\\bea\nq_{2n+1}(t,z)&=&\\frac{1}{\\sqrt{\\tau_{2n}\\tau_{2n+2}}}pf\n\\left(\\begin{array}{ccc\|cc}\n & & & 1&\\mu_{0,2n+1}\\\\\n& & & z &\\mu_{1,2n+1} \\\\\n &m_{2n}(t) & & \\vdots&\\vdots\\\\\n& & &z^{2n} &\\mu_{2n,2n+1} \\\\\n\\hline%{1-3}\n-1&-z&\\pp&0 &-z^{2n+1}\\\\\n-\\mu_{0,2n+1}&-\\mu_{1,2n+1} &\\pp&z^{2n+1}&0\n\\end{array}\\right)\\nonumber\\\\\n&=&\\frac{1}{\\sqrt{\\tau_{2n}(t)\\tau_{2n+2}(t)}}\n\\left( z+\\frac{\\pl}{\\pl t_1} \\right)\n\\sqrt{\\tau_{2n}(t)\\tau_{2n+2}(t)}~q_{2n}(t,z)\\nonumber\\\\\n&=& z^{2n} \\frac{ (z+\\frac{\\pl}{\\pl t_1})\\tau_{2n}(t-[z^{-1}])}\n {\\sqrt{\\tau_{2n}(t) \\tau_{2n+2}(t)}}\n\\eea\nform skew-orthonormal sequences with respect to the skew\ninner-product\n $\\la,\\ra_{sk}$, defined by $\\la y^i,z^j\\ra_{sk}=\\mu_{ij}$,\n namely, we have the following restatement of the\n skew-Borel dcomposition (0.13):\n\\be\n(\\la q_i, q_j\\ra)_{0\\leq i,j<\\iy}=J.\n\\ee\nFinally, the vector $q(z)=(q_n(z))_{n\\geq 0}$ forms a eigenvector\nfor the matrix L:\n\\be\nL(t)q(t,z)=zq(t,z).\n\\ee\n\nIn section 2, we show how a general skew-symmetric infinite matrix\n flowing via (0.8) and its skew-Borel decomposition\n (0.13), lead to wave vectors $\\Psi$,\n satisfying bilinear relations and differential equations.\n Section 3 deals with the existence, in the above general\n setting, of a so-called Pfaffian\n $\\tau$-function, satisfying bilinear equations and a KP-\n type hierarchy. In \\cite{ASV}, these results were obtained,\n by embedding the system in 2-Toda theory, while in this paper,\n they are obtained in an intrinsic fashion.\n\nFor $k=0$, the KP-like equation (0.14) has already appeared in the\ncontext of the charged BKP hierarchy, studied by V. Kac and van de\nLeur \\cite{KvdL}; the precise relationship between the charged BKP\nhierarchy of Kac and van de Leur and the Pfaff Lattice, introduced\nhere, deserves further investigation. See the recent paper of\nvan de Leur \\cite{vdL}.\n\n\\vspace{1cm}\n\n\n\\noindent {\\bf A remarkable map from Toda to\n Pfaff lattice}:\nRemembering the notation (0.1), we act with\n the $z$-operator,\n\\be\n\\Bn_t:=\\sqrt{\\frac{f}{\\rho_t}}\\frac{d}{dz}\n\\sqrt{f\\rho_t}\n=e^{-\\frac{1}{2}\\sum t_k\nz^k}\\left(\\frac{d}{dz}f(z)-\\frac{f'+g}{2}(z)\\right)\ne^{\\frac{1}{2}\\sum t_k z^k}\n\\ee\non the $t$-dependent orthonormal\n polynomials $p_n(t,z)$ in $z$; in \\cite{AvM2}, we\n showed that the matrix $\\NR$ defined by\n \\be\n \\Bn_t p(t,z)= \\left(f(L)M-\\frac{f'+g}{2}(L)\\right)p(t,z)\n =: \\NR p(t,z)\n \\ee\nis skew-symmetric. The $t$-dependent matrix $\\NR$ is expressed in\nterms of $L$ and a new matrix $M$, defined by\n \\be\n zp=Lp~~~\\mbox{and}~~~\n e^{-\\frac{1}{2}\\sum t_k z^k}\\frac{d}{dz}\n e^{\\frac{1}{2}\\sum t_k z^k}p=Mp.\n \\ee\nConsider now the skew-Borel decomposition of $\\NR(2t)$ and its\ninverse\\footnote{See the appendix, section 9.} $\\NR(2t)^{- 1}$, in\nterms of lower-triangular matrices $O_{(+) }(t)$ and $O_{(-)}(t)$\nrespectively\\footnote{The upper-signs (respectively, lower-signs)\ncorrespond to each other throughout this section.}\n:\n\\be\n\\NR(2t)^{\\pm 1}=-O_{(\\pm )}^{-1}(t)JO_{(\\pm) }^{\\top -1}(t).\n\\ee\nThen, the lower-triangular matrices $O_{(\\pm)}(t)$ map {\\em\northonormal} into {\\em skew-orthonormal} polynomials, and the\ntridiagonal $L$-matrix into an $\\tilde L$-matrix:\n\\bea\np_n(t,z)~~~~~~&\\longmapsto& q^{(\\pm)}_n(t,z)=O_{(\\pm)}(t)p_n(t,z)\n \\nonumber\\\\\nL(t)~~~~~~~~ ~&\\longmapsto& \\tilde\nL(t)=O_{(\\pm)}(t)L(2t)O_{(\\pm)}(t)^{-1} .\\\\\n\\mbox{({\\bf Toda Lattice})}& & ~~~\\mbox{({\\bf Pfaff Lattice})}\n \\nonumber\n\\eea\nIt also maps the weight into a new weight\n$$\\rho(z)=e^{-V(z)}\\longmapsto \\tilde\n \\rho_{\\pm}(z)=e^{-\\tilde V(z)}:=e^{-\\frac{1}{2}(V(z)\\mp\\log\nf(z))},$$\n and the corresponding string of $\\tau$-functions into\n a new string of pfaffian $\\tau$-functions: (remember $V_t(z)=V(z)-\\sum_1^{\\iy} t_i z^i$)\n$$\n\\tau_k(t)=\\int_{{\\cal H}_{k}}\n e^{Tr~(- V_t(X))} dX\\longmapsto \\left\\{\\begin{array}{l}\n \\displaystyle\n\\tau^{(+)}_{2n}(t):= \\int_{{\\cal T}_{2n}}\n e^{Tr~2(- \\tilde V_t(X))} dX ,~( \\beta=4)\n \\\\[10pt]\n \\displaystyle\n \\tau^{(-)}_{2n}(t):= \\int_{{\\cal S}_{2n}}\n e^{Tr~(- \\tilde V_t(X) )} dX~ ~( \\beta=1) .\n%\\\\\n%$p_n(t,z) ~~\\mbox{orthonormal polynomials in $z$}$\n\\end{array}\n\\right.\n$$\nFor the {\\bf classical orthogonal polynomials} $p_n(z)$, we have\nshown in\n \\cite{AvM2}, that $\\NR(0)$ is not only skew-symmetric, but also\n tridiagonal; i.e.,\n\\be\nL=\\left[\\begin{array}{ccccc}\nb_0&a_0& & & \\\\\na_0&b_1&a_1& & \\\\\n &a_1&b_2&\\ddots&\\\\\n & &\\ddots&\\ddots&\n\\end{array}\n\\right],\\quad -\\NR=\\left[\\begin{array}{ccccc}\n0&c_0& & & \\\\\n-c_0&0&c_1& & \\\\\n &-c_1&0&\\ddots&\\\\\n & &\\ddots& &\n\\end{array}\\right] .\n\\ee\nIn section 6 and 7, we show that the maps $O_{(-)}$ and $O_{(+)}$\nonly involves three steps, in the following sense:\n\\bea\n q^{(-)}_{2n}(0,z)&=&\\sqrt{\\frac{c_{2n}}{a_{2n}}}p_{2n}(0,z)\\nonumber\\\\\n q^{(-)}_{2n+1}(0,z)&=&\\sqrt{\\frac{a_{2n}}{c_{2n}}}\\nonumber\\\\\n&&\n\\left(-c_{2n-1}p_{2n-1}(0,z)+\\frac{c_{2n}}{a_{2n}}(\\sum_0^{2n}b_i)\n p_{2n}(0,z)+c_{2n}p_{2n+1}(0,z)\\right)\\nonumber\\\\\n&&\\hspace{8cm} (\\beta=1)\n\\eea\n\\bea\n p_{2n}(0,z)&=&\n-c_{2n-1}\\sqrt{\\frac{a_{2n-2}}{c_{2n-2}}}q^{(+)}_{2n-2}(0,z)\n+\\sqrt{a_{2n}c_{2n}}~q^{(+)}_{2n}(0,z)\\nonumber\\\\\n p_{2n+1}(0,z)&=&\n -c_{2n}\\sqrt{\\frac{a_{2n-2}}{c_{2n-2}}}\n q^{(+)}_{2n-2}(0,z)-(\\sum_0^{2n} b_i)\\sqrt{\\frac{c_{2n}}{a_{2n}}}\n q^{(+)}_{2n}(0,z)+\\sqrt{\\frac{c_{2n}}{a_{2n}}}q^{(+)}_{2n+1}(0,z),\\nonumber\\\\\n &&\\hspace{8cm} (\\beta=4)\n \\eea\n\nThe abstract map $O_{(-)}$ for $t=0$ appears already in the\nwork of E. Br\\'ezin and H. Neuberger \\cite{BN}. This\nhas been applied in \\cite{AFNV} to a problem\nin the theory of random matrices.\n\n\n\\section{Splitting theorems, as applied to the Toda\nand Pfaff Lattices}\n\nIn this section, we show how each of the equations\n\\be\n\\frac{\\pl m_{\\iy}}{\\pl t_i}=\\Lb^im_{\\iy}~~~\\mbox{and}~~~\n\\frac{\\pl m_{\\iy}}{\\pl t_i}=\\Lb^im_{\\iy}+m_{\\iy}\\Lb^{\\top i},\n\\ee\nlead to commuting Hamiltonian vector fields related to a Lie\nalgebra splitting. First recall the splitting theorem, due to\nAdler-Kostant-Symes \\cite{AvM1} and the R-version to\n Reiman and\nSemenov-Tian-Shansky \\cite{RS}. The R-version allows for more\ngeneral initial conditions.\n\n \\begin{proposition} Let ${\\bf g} = {\\bf k} + {\\bf n}$ be a\n (vector space) direct sum of a Lie algebra ${\\bf g}$ in terms of Lie\n subalgebras ${\\bf k}$ and ${\\bf n}$, with ${\\bf g}$ paired with\n itself via a non-degenerate {\\rm ad}-invariant inner\n product\\footnote{$\\la{\\rm Ad}_gX;Y\\ra =\\la X,{\\rm Ad}_{g-1}Y\\ra$,\n $g\\in G$, and thus $\\la [z,x],y\\ra =\\la x,-[z,y]\\ra.$} $\\la\\,\n ,\\,\\ra$; this in turn induces a decomposition ${\\bf g} = {\\bf\n k}^{\\bot} + {\\bf n}^{\\bot}$ and isomorphisms ${\\bf g}\\simeq {\\bf\n g}^$, ${\\bf k}^{\\bot}\\simeq {\\bf n}^$, ${\\bf n}^{\\bot}\\simeq {\\bf\n k}^$. $\\pi_{{\\bf k}}$ and $\\pi_{{\\bf n}}$ are\n projective onto ${\\bf k}$ and ${\\bf n}$ respectively. Let $\\GR, ~\\GR_{\\Bk}$\n and $ \\GR_{\\Bn}$ be the groups associated with the Lie algebras ${\\bf g},\n {\\bf k}$ and ${\\bf n}$. Let ${\\cal I}({\\bf g})$\n be the {\\rm Ad}$^* \\simeq$ {\\rm Ad}-invariant functions on ${\\bf\n g}^* \\simeq {\\bf g}$.\n\n {\\bf (i)} Then, given an element\n $$\n \\vr \\in {\\bf g} :\n [\\vr,{\\bf k}] \\subset {\\bf k}^{\\bot} \\mbox{\\,\\, and\\, \\,} [\\vr,{\\bf\n n}] \\subset {\\bf n}^{\\bot},\n $$\n the functions\n \\be\n \\vp(\\vr +\\xi')\|_{{\\bf k}^{\\bot}}\\mbox{\\, with \\,}\n \\vp\\in {\\cal I}({\\bf g})~ \\mbox{and}~ \\xi' \\in\n {\\bf k}^{\\bot}, \\ee\n respectively Poisson commute for the\n respective Kostant-Kirillov symplectic structures of $n^* \\simeq\n {\\bf k}^{\\bot}$; the associated\n Hamiltonian flows are expressed in terms of the Lax\n pairs\\footnote{$\\nabla\\vp$ is defined as the element in ${\\bf g}^$\n such that $d\\vp(\\xi)=\\la\\nabla\\vp,d\\xi\\ra$, $\\xi\\in{\\bf g}$.}\n \\be\n \\dot\\xi = [-\\pi_{{\\bf k}}\\nabla\\vp(\\xi),\\xi]=[\\pi_{{\\bf\n n}}\\nabla\\vp(\\xi),\\xi]\\mbox{\\,for\\,}\n \\xi \\equiv\\vr+\\xi',\\xi'\\in{\\bf k}^{\\bot}\n \\ee\n\n{\\bf (ii)} The splitting also leads to a second Lie algebra ${\\bf\ng}_R$, derived\n from ${\\bf g}$, such that ${\\bf g}^_R\\simeq {\\bf g}_R$, namely:\n \\be {\\bf g}_R : [x,y]_R = \\frac{1}{2}[Rx,y] +\n \\frac{1}{2}[x,Ry] = [\\pi_{{\\bf k}}x,\\pi_{{\\bf k}}y] - [\\pi_{{\\bf\n n}}x,\\pi_{{\\bf n}}y], \\ee\n with $R = \\pi_{{\\bf\n k}}-\\pi_{{\\bf n}}$. The functions\n$$\\vp(\\xi)\|_{{\\bf g}_R}\n \\mbox{\\, with \\,}\\vp\\in {\\cal I}({\\bf g})~ \\mbox{and}~\n \\xi \\in {\\bf g}_R $$ respectively Poisson commute for the\n respective Kostant-Kirillov symplectic structures\n of ${\\bf g}^_R\\simeq{\\bf g}_R$, with the same associated\n (Hamiltonian) Lax pairs\n \\be\n \\dot\\xi = [-\\pi_{{\\bf k}}\\nabla\\vp(\\xi),\\xi]=[\\pi_{{\\bf\n n}}\\nabla\\vp(\\xi),\\xi]~~\\mbox{\\,for\\,}~~\n \\xi\\in{\\bf g}_R.\n \\ee\n Each of the equations (1.3) and (1.5) has the same solution expressible in two different\nways\\footnote{naively written Ad$_{K(t)}\\xi_0=K(t)\\xi_0K(t)^{-1}$,\nAd$_{S^{-1}}\\xi_0=S^{-1}(t)\\xi_0S(t)$.}:\n\\be\n\\xi(t)={\\rm Ad}_{K(t)}\\xi_0={\\rm Ad}_{S^{-1}(t)}\\xi_0,\n\\ee\nwith\\footnote{with regard to the group factorization\n$A=\\pi_{\\GR_{\\Bk}}A~\n\\pi_{\\GR_{\\Bn}}A$.}\n$$\nK(t) = \\pi_{\\GR_{\\Bk}} e^{t \\nabla\\vp(\\xi_0)},\\quad\n \\mbox{and}\\quad\nS(t)\n=\n \\pi_{\\GR_{\\Bn}} e^{t \\nabla\\vp(\\xi_0)}.\n$$\n\\end{proposition}\n\n\n\n\n\\noindent {\\sl \\underline{Example 1}}:\n{\\bf The standard Toda lattice and the equations\n$\\displaystyle{\\frac{\\pl m}{\\pl t_i}}=\\Lambda^im $ for the H\\\"ankel\nmatrix $m_{\\iy}$}. Since, in particular, the matrix $m_{\\iy}$ is\nsymmetric,\n% (i.e.,$\\Lb m_{\\iy}=m_{\\iy}\\Lb^{\\top}$)\n the Borel decomposition into\nlower- times upper-triangular matrix must be done with the same\nlower-triangular matrix $S$:\n\\be\nm_{\\iy}=S^{-1} S^{\\top -1}.\n\\ee\nIn turn, the matrix $S$ defines a wave vector $\\Psi$, and\n operators\\footnote{In the formulas below $\\chi(z)=(z^0,z,z^2,\\dots)$ and\n $\\pl$ is the matrix such that $\\frac{d}{dz}\\chi(z)\n =\\pl \\chi(z)$. } $L$ and $M$, the same as the\n ones defined in (0.22),\n\\be\n\\Psi(t,z):=e^{\\frac{1}{2} \\sum_1^{\\iy}t_i z^i}S\\chi,\n~~~L:=S\\Lb S^{-1},~~~ M:=S(\\pl +\\frac{1}{2}\\sum_1^{\\iy}it_i\n\\Lb^{i-1})S^{-1}, \\ee\n satisfying the following well-known equations\\footnote\n {where the $()_{sk}$ and $()_{bo}$ refers to the\n skew-part and the lower-triangular (Borel) part respectively;\n i.e., projection onto $\\Bk$ and $\\Bn$ respectively.}:\n\\bea\nL\\Psi=z\\Psi~~&\\mbox{ }&~~M\\Psi=\\frac{\\pl}{\\pl z}\\Psi,\\quad\n\\mbox{with}\\quad [L,M]=1,\\nonumber\\\\\n\\frac{\\pl S}{\\pl\nt_n}=-\\frac{1}{2}{(L^n)}_{bo} S ~~&\\mbox{ }&~~\\frac{\\pl\n\\Psi}{\\pl t_n}=\\frac{1}{2}{(L^n)}_{sk} \\Psi\n\\nonumber\\\\\n \\frac{\\pl L}{\\pl t_n}=\\frac{1}{2}[(L^n)_{sk},L]~~\n &\\mbox{ }&~~\n \\frac{\\pl M}{\\pl t_n}=\\frac{1}{2}[(L^n)_{sk},M].\n\\eea\n The wave vector $\\Psi$ can then be expressed in terms of a sequence\n of $\\tau$-functions $\\tau_n(t)=\\det m_n(t)$, but also\n has the simple expression in terms of orthonormal\n polynomials, with respect to the moment matrix\n $m_{\\iy}$:\n \\bea\n \\Psi(t,z)&=& e^{\\frac{1}{2}\\sum t_iz^i}\n\\left(z^n\\frac{\\tau_n(t-[z^{-1}])}\n{\\sqrt{\\tau_n(t)\\tau_{n+1}(t)}}\\right)\n_{n\\geq 0}\n\\nonumber\\\\\n &=& e^{\\frac{1}{2}\\sum t_iz^i}\\left(p_n(t,z)\\right)_{n\\geq 0}\n.\n \\eea\n\n\nThe vector fields (1.9) on $L$ are commuting Hamiltonian vector\nfields, in view of the Adler-Kostant-Symes splitting theorem\n(version (i)),\n\\be\n\\frac{\\pl L}{\\pl t_i}=[-\\pi_{\\Bk}\\nabla \\HR_i,L]=\n[\\pi_{\\Bn}\\nabla \\HR_i,L],~~~\\HR_i=\\frac{tr L^{i+1}}{i+1}, ~~~\n\\nabla \\HR_i=L^i,\n\\ee\nwith\n\\be\nL=\\Lb^{\\top} a+b +a \\Lb, ~~\\mbox{ $a$ and $b$ diagonal matrices}\n\\ee\n for the splitting of the Lie algebra of semi-infinite\n matrices\n\\bea\n\\DR=gl_{\\iy}&=&\\Bk+\\Bn:=\\{\\mbox {skew-symmetric }\\}+\n\\{ \\mbox {lower-triangular} \\} \\nonumber\\\\\n&=&\\Bk^{\\bot}+\\Bn^{\\bot}:=\\{\\mbox {symmetric }\\}+\n\\{ \\mbox {strictly upper-triangular} \\} , \\nonumber\\\\\n\\eea\nwith the form (1.12) of $L$ being preserved in time.\n Note that the solution (1.6) to (1.5) in the AKS theorem is\n nothing but the factorization of $m_{\\iy}$ followed\n by the dressing up of $\\Lb$.\n\n\\noindent {\\sl \\underline{Example 2}}:\n{\\bf The Pfaff lattice and the equations $\\displaystyle{\\frac{\\pl\nm}{\\pl t_i}}=\\Lambda^im+m\\Lambda^{\\top^i}.$}\n\n\nThroughout this paper the Lie algebra $\\DR = gl_{\\iy}$ of\nsemi-infinite matrices is viewed as composed of $2\\times 2$ blocks.\nIt admits the natural decomposition into subalgebras:\n\\be\n\\DR=\\DR_-\\oplus\\DR_0\\oplus\\DR_+=\\DR_-\\oplus\\DR_0^-\\oplus\\DR_0^+\\oplus\\DR_+\n\\ee\nwhere $\\DR_0$ has $2\\times 2$ blocks along the diagonal with zeroes\neverywhere else and where $\\DR_+$ (resp. $\\DR_-$) is the subalgebra\nof upper-triangular (resp. lower-triangular) matrices with $2\n\\times 2$ zero matrices along $\\DR_0$ and\nzero below (resp. above). As pointed out in (1.14), $\\DR_0$ can\nfurther be decomposed into two Lie subalgebras:\n\\bea\n\\DR_0^-&=&\\{\\mbox{all $2\\times 2$ blocks $\\in \\DR_0$\nare proportional to Id} \\}\\nonumber\\\\\n\\DR_0^+&=&\\{\\mbox{all $2\\times 2$ blocks $\\in \\DR_0$\nhave trace $0$ } \\}.\n\\eea\nRemember from (0.10) and (0.11) in the introduction, the matrix $J$\nand the associated Lie algebra order 2 involution $\\JR$. The\nsplitting into two Lie subalgebras\\footnote{Note $\\Bn$ is the fixed\npoint set of $\\JR$.}\n\\be\n\\DR= \\Bk+\\Bn,\n\\ee\nwith\n \\bea \\Bk&=&\\DR_-+\\DR_0^-\\nonumber\\\\\n &=&~\\mbox{algebra of}~\n\\left(\n\\begin{array}{c@{}c@{}cc}\n\\ddots &&&0 \\\\\n %\\boxed{\\begin{array}{cc} Q_{2n,2n} & 0 \\\\ 0 & Q_{2n+1,2n+1} \\end{array}} &&0& \\\\\n & \\boxed{\\begin{array}{cc}\n Q_{2n,2n} & 0 \\\\ 0 & Q_{2n,2n} \\end{array}} &&\\\\\n && \\boxed{\\begin{array}{cc}\n Q_{2n+2,2n+2} & 0 \\\\ 0 & Q_{2n+2,2n+2} \\end{array}} & \\\\\n &&& \\ddots\n \\end{array}\n \\right)\n ~~\\nonumber\n\\\\\n{\\Bbb n}&=& \\{a\\in \\DR, ~\\mbox{such that}~ \\JR a=a\\}=\\{b+\\JR b,~b\n\\in \\DR \\}=\\mbox{sp}(\\iy),\n\\eea\nwith corresponding Lie groups\\footnote{$\\GR_{\\Bk}$ is the group of\ninvertible elements in $\\Bk$, i.e., invertible lower-triangular\nmatrices, with non-zero $2\\times 2$ blocks proportional to Id along\nthe diagonal.} $\\GR_{\\Bk}$ and $\\GR_{\\Bn}=Sp(\\iy)$, will play a\ncrucial role here. Let $\\pi_{\\Bk}$ and $\\pi_{\\Bn}$ be the\nprojections onto $\\Bk$ and $\\Bn$. Notice that $\\Bn=$sp$(\\iy)$ and\n$\\GR_{\\Bn}=Sp(\\iy)$ stand for the infinite rank affine symplectic\nalgebra and group; e.g. see \\cite{Kac}. Any element $a\\in\\DR$\ndecomposes uniquely into its projections onto $\\Bk$ and $\\Bn$, as\nfollows:\n\\bea\na&=&\\pi_{\\Bk}a+\\pi_{\\Bn}a\\nonumber\\\\ &=&\\left\\{\\left(a_--\\JR\na_+\\right)+\\frac{1}{2}\\left(a_0-\\JR a_0\n\\right)\\right\\}+\\left\\{\\left(a_++\\JR a_+\\right)+\\frac{1}{2}\\left(a_0+\\JR\na_0\n\\right)\\right\\}.\\nonumber\\\\\n\\eea\nThe following splitting, with\n$$\n\\Bk_+=\\DR_++\\DR_0^-~~\\mbox{and}~~\\Bn_+=\\Bn,\n$$\n will also be used in section 2; the projections take on\n the following form,\n\\bea\na&=&\\pi_{\\Bk_+}a+\\pi_{\\Bn_+}a\\nonumber\\\\\n &=&\\left\\{\\left(a_+-\\JR\na_-\\right)+\\frac{1}{2}\\left(a_0-\\JR a_0\n\\right)\\right\\}+\\left\\{\\left(a_-+\\JR a_-\\right)+\n\\frac{1}{2}\\left(a_0+\\JR a_0\n\\right)\\right\\}.\\nonumber\\\\\n\\eea\nNote $\\JR$ intertwines $\\pi_{\\Bk}$ and $\\pi_{\\Bk_+}$:\n\\be\n\\JR \\pi_{\\Bk}=\\pi_{\\Bk_+} \\JR.\n\\ee\n\n\\medbreak\n\nFor a skew-symmetric semi-infinite matrix $m_{\\iy}$, the skew-Borel\ndecomposition \\be m_{\\iy}:=Q^{-1}J Q^{-1\\top}\\mbox{ with }Q\\in\n\\GR_{\\Bk},\\ee is unique, as was shown in \\cite{AHV}. Here we may assume\n $m_{\\iy}$ to be bi-infinite, as long as the factorization\n (1.21)\n is unique, upon imposing a suitable normalization. Then we use $Q$ to dress up $\\Lb$:\n $$\n L=Q\\Lb Q^{-1}.\n $$\nThen letting $m_{\\iy}$ run according to the equations\n$\n\\pl\nm / \\pl t_i=\\Lambda^i m+m\\Lambda^{\\top^i},\n$\nwe show in the next proposition and corollary that $L$ evolves\naccording to a system of commuting equations, which by virtue of\nthe AKS theorem are Hamiltonian vector fields; for details, see\n\\cite{AHV}.\n\n\\begin{proposition} For the matrices\n$$\nm_{\\iy}:=Q^{-1}J Q^{-1\\top} \\quad\\mbox{and}\\quad L:=Q\\Lb\nQ^{-1}\\quad ,\n\\mbox{ with }Q\\in \\GR_{\\Bk},\n$$\nthe following three statements are equivalent\\medbreak\\indent (i) $\\displaystyle{\\frac{\\pl Q}{\\pl\nt_i}Q^{-1}=-\\pi_{\\Bk}L^i}$\n\\medbreak\\indent (ii) $L^i+\\displaystyle{\\frac{\\pl Q}{\\pl\nt_i}}Q^{-1}\\in\\Bn$\\medbreak\n \\indent (iii) $\\displaystyle{\\frac{\\pl m}{\\pl\nt_i}}=\\Lambda^im+m\\Lambda^{\\top^i}.$\n\\medbreak\\noindent Whenever the vector fields on $Q$ or $m$\nsatisfy (i), (ii) or (iii), then the matrix $L=Q\\Lb Q^{-1}$ is a\nsolution of the AKS-Lax pair\n$$\n\\frac{\\pl L}{\\pl t_i}=[-\\pi_{\\Bk}L^i,L]=\n \\left[\\pi_{{\\bf n}} L^i,L \\right].\n$$\n\n\\end{proposition}\n\n\\proof Written out and using (1.18), proposition 1.2 amounts to showing the\nequivalence of the three formulas:\n\\medbreak\n\\noindent(I) $\\displaystyle{\\frac{\\pl Q}{\\pl t_i}Q^{-1}+\\left((L^i)_--J(L^i_+)^{\\top}J\\right)+\\frac{1}{2}\n\\left((L^i)_0-J((L^i)_0)^{\\top}J\\right)=0}$\n\n\\medbreak\n\n\\noindent(II) $\\displaystyle{\\left(L^i+\\frac{\\pl Q}{\\pl t_i}Q^{-1}\\right)-J\n\\left(L^i+\\frac{\\pl Q}{\\pl t_i}Q^{-1}\\right)^{\\top}J=0}$\n\n\\medbreak\n\n\\noindent(III) $\\Lambda^im+m\\Lambda^{\\top}-\\displaystyle{\\frac{\\pl m}{\\pl t_i}}=0$.\n\n\\medbreak\n\n\\noindent The point is to show that\n$$\n(\\mbox{I})_+=0,~~~~(\\mbox{I})_-=(\\mbox{II})_-\n=-J~(\\mbox{II}_+)^{\\top} ~J,~~~~\n(\\mbox{I})_0=\\frac{1}{2}\\left(\\mbox{II}\\right)_0,\n$$\n\\be\nQ^{-1}(\\mbox{II})JQ^{-1^{\\top}}\n=(\\mbox{III}).\n\\ee\nThe reader will find the details of this proof in \\cite{AHV}.\n\\qed\n\n\n\n\\section{Wave functions and their bilinear equations for\nthe Pfaff Lattice}\n\n\\setcounter{equation}{-1}\n\nConsider the commuting vector fields \\be \\pl m_{\\iy}/ \\pl\nt_i=\\Lb^im_{\\iy}+m_{\\iy}\\Lb^{\\top i} \\ee on the skew-symmetric\nmatrix $m_{\\iy}(t)$ and the skew-Borel decomposition\n\\be\nm_{\\iy}(t)=Q^{-1}(t)J\\,Q^{\\top -1}(t),\\quad\\quad Q(t)\\in\n\\GR_{\\Bk};\n\\ee\nremember from (1.17), $Q(t)\\in\n\\GR_{\\Bk}$ means: $Q(t)$ is lower-triangular, with along the ``diagonal\" 2$\\times$2\nmatrices $c_{2n}I$, with $c_{2n}\\neq 0$.\n\n In this section, we give the properties of the wave vectors and\ntheir bilinear relations. Upon setting\n\\be\nQ_1=Q(t)\\mbox{\\,\\,and\\,\\,}Q_2=JQ^{\\top -1}(t),\n\\ee\nthe matrix $Q(t)$ defines wave operators\n\\be\nW_1(t)=Q_1(t)e^{\\sum_1^{\\iy}t_i\\Lb^i},\\quad W_2(t)=Q_2(t)e^{-\\sum_1^{\\iy}\nt_i\\Lb^{\\top i}}=JW_1^{-1\\top}(t),\n\\ee\n$L$-matrices\n\\be\nL:=L_1:=Q_1\\Lb Q^{-1}_1,\\quad L_2:=-\\JR(L_1)=Q_2\\Lb^{\\top}Q_2^{-1},\n\\ee\nand wave and dual wave vectors\n\\be\n\\begin{array}{lll}\n\\Psi_1(t,z)=W_1(t)\\chi(z)&\n&\\Psi_1^{\\ast}(t,z)=W_1^{-1}(t)^{\\top}\\chi(z^{-1})\n =-J\\Psi_2(t,z^{-1})\\\\[4pt]\n\\Psi_2(t,z)=W_2(t)\\chi(z)&\n&\\Psi_2^{\\ast}(t,z)=W_2^{-1}(t)^{\\top}\\chi(z^{-1})=J\\Psi_1(t,z^{-1}).\n\\end{array}\n\\ee\nFrom the definition, it follows that the wave functions $\\Psi_1$ have the\nfollowing asymptotics\n$$\n\\left\\{\\begin{array}{ll}\n\\Psi_{1,2n}(t,z):=e^{\\sum t_kz^k}z^{2n}\nc_{2n}(t)\\psi_{1,2n}(t,z),& \\psi_{1,2n}=1+O(z^{-1})\\\\[4pt]\n\\Psi_{1,2n+1}(t,z)=e^{\\sum\nt_kz^k}z^{2n+1}c_{2n}(t)\\psi_{1,2n+1}(t,z),&\n\\psi_{1,2n+1}=1+O(z^{-2})\n\\end{array}\\right.\n$$\n\n\\be\n\\left\\{\\begin{array}{ll}\n\\Psi_{2,2n}(t,z)=\ne^{-\\sum t_kz^{-k}}z^{2n+1}c^{-1}_{2n}(t)\n\\psi_{2,2n}(t,z),&\n\\psi_{2,2n}=1+O(z)\\\\[4pt]\n\\Psi_{2,2n+1}(t,z)=e^{-\\sum\nt_kz^{-k}}z^{2n}(-c_{2n}^{-1}(t))\\psi_{2,2n+1}(t,z),&\n\\psi_{2,2n+1}=1+O(z^2),\n\\end{array}\\right.\n\\ee\nwhere the $c_i$ are the elements of the diagonal part of $Q$.\n\n\\begin{theorem} The $Q_i$, $L_i$ and $\\Psi_i$ satisfy the equations\n\\be\n\\frac{\\pl Q_1}{\\pl t_i}=-(\\pi_{\\Bk}L_1^i)Q_1\\quad\\quad\\frac{\\pl Q_2}{\\pl\nt_i}=-(\\JR(\\pi_{\\Bk}L_1^i))Q_2=(\\pi_{\\Bk_{+}}L_2^i)Q_2\n\\ee\n\n\\be\n\\frac{\\pl L_1}{\\pl t_i}=[-\\pi_{\\Bk}L_1^i,L_1]\\quad\\quad\\frac{\\pl L_2}{\\pl t_i}\n=[\\pi_{\\Bk_+}L_2^i,L_2]\n\\ee\n\\be\nL_1\\Psi_1=z\\Psi_1\\quad \\quad L_2\\Psi_2=z^{-1}\\Psi_2,\n\\ee\n\\be\n\\frac{\\pl\\Psi_1}{\\pl t_i}=(\\pi_{\\Bn}L_1^i)\\Psi_1\\quad\\quad\\frac{\\pl\\Psi_2}{\\pl\nt_i}=-(L_2^i-\\pi_{\\Bk_+}L_2^i)\\Psi_2=-(\\pi_{\\Bn_+}L_2^i)\\Psi_2,\n\\ee\n with\n$\\Psi_i$ satisfying the following bilinear identity for all\n$n,m\\in\\BZ$,\n\\be\n\\oint_{\\iy}\\Psi_{1,n}(t,z)\\Psi_{2,m}(t',z^{-1})\\frac{dz}{2\\pi\niz}+\\oint_0\\Psi_{2,n}(t,z)\\Psi_{1,m}(t',z^{-1})\\frac{dz}{2\\pi iz}=0.\n\\ee\n\\end{theorem}\n\n\nFor later use, we shall also consider the ``monic\" wave functions, with\nthe factors $c_{2n}(t)$ removed, i.e.,\n\\be\n\\hat \\Psi_1(t,z):=Q_0^{-1}\\Psi_1\\mbox{\\,\\,and\\,\\,}\\hat\\Psi_2(t,z):=Q_0\\Psi_2\n\\ee\nand the matrix $\\hat L_1$, normalized so as to have 1's above the\nmain diagonal, with $\\hat Q:=Q_0^{-1}Q$,\n\\medbreak\n\\bea\\hat L_1&=&Q_0^{-1}L_1Q_0=\n(Q_0^{-1}Q)\\Lb(Q_0^{-1}Q)^{-1}=\\hat Q \\Lb \\hat Q^{-1}, \\nonumber\\\\\n\\hat L_2&=&Q_0L_2Q_0^{-1} =-Q_0\\JR(L_1)Q_0^{-1}=-\\JR(\\hat L_1)\n\\eea\nThen, in terms of the elements $\\hat q_{ij}$ of the matrix $\\hat\nQ:=Q_0^{-1}Q$, one easily computes by conjugation, that $\\hat L_1$\nhas the following block structure:\n\\medbreak\n$$\\hat L_1=Q_0^{-1}L_1Q_0= (Q_0^{-1}Q)\\Lb(Q_0^{-1}Q)^{-1}\n=\n \\pmatrix{&\\vdots& \\cr\n ...&\\hat L_{00}&\\hat L_{01}&0&0&\\cr\n&\\hat L_{10}&\\hat L_{11}&\\hat L_{12}&0&\\cr &&\\hat L_{21} &\\hat\nL_{22}&\\hat L_{23}&\\cr\n &&\n&\\hat L_{32}&\\hat L_{33}&...\\cr & & & &\\vdots& &\n\\cr},\n$$\nwith\n$$\n\\hat L_{ii}:=\\pmatrix{\\hat q_{2i,2i-1}& 1 \\cr\n \\hat q_{2i+1,2i-1}-\\hat q_{2i+2,2i} & -\\hat q_{2i+2,2i+1}\\cr},~~\n \\hat L_{i,i+1}:=\\pmatrix{0& 0 \\cr\n 1 & 0\\cr}\n$$\n\\be\n\\hat L_{i+1,i}:=\\pmatrix{&-\\hat q_{2i+2,2i+1}^2-\\hat q_{2i+3,2i+1}+\\hat\nq_{2i+2,2i}& \\cr\n * &\\cr}.\n \\ee\n\n\n\\begin{theorem} $\\hat L_i,~\\hat Q,~\\hat \\Psi_1,~\\hat \\Psi_2$ satisfy the\n following equations:\n\\be\n\\frac{\\pl\\hat Q}{\\pl t_n}=-\\left((\\hat L_1^n)_--Q_0^{-2}J((\\hat L_1^n)_+)\n^{\\top}JQ_0^2\\right)\\hat Q,\n\\ee\nand\n\\be\n\\hat L_1 \\hat\\Psi_1=z\\hat\\Psi_1\\quad \\quad \\hat L_2 \\hat\\Psi_2\n=z^{-1} \\hat\\Psi_2,\n\\ee\nwith\n$$\n\\frac{\\pl}{\\pl t_n}\\hat \\Psi_1(t,z)=\\left((\\hat L_1^n)_++(\\hat\nL_1^n)_0+Q_0^{-2}\\JR((\\hat L_1^n)_+)Q_0^2\\right)\\hat \\Psi_1(t,z),\n$$\n\\bea\n\\frac{\\pl}{\\pl t_n}\\hat \\Psi_2(t,z)&=&\\JR\\left((\\hat L_1^n)_++(\\hat\nL_1^n)_0+Q_0^{-2}\\JR((\\hat\nL_1^n)_+)Q_0^2\\right)\\hat\\Psi_2(t,z)\\nonumber\\\\ &=&-\\left((\\hat\nL^n_2)_-+(\\hat L^n_2)_0+Q^2_0\\JR((\\hat\nL_2^n)_-)Q_0^{-2}\\right)\\hat\\Psi_2(t,z).\\nonumber\n\\eea\n\\end{theorem}\n\nThe proof of Theorem 2.1 hinges on the following matrix version of\nthe bilinear identities:\n\n\n\\begin{lemma} The matrices $W_1(t)$ and $W_2(t)$,\ndefined in (2.3), satisfy\n\\be\nW_1(t)W_1(t')^{-1}=W_2(t)W_2(t')^{-1}.\n \\ee\n\\end{lemma}\n\n\\proof\nThe solution to the equation (2.0) is given by\n$$\nm_{\\iy}(t)=e^{\\sum t_k\\Lb^k}m_{\\iy}(0)e^{\\sum t_k\\Lb^{\\top k}}.\n$$\nTherefore skew-Borel decomposing $m_{\\iy}(t)$ and $m_{\\iy}(0)$, we\nfind\n\\be\nQ^{-1}(0)JQ^{\\top\n-1}(0)=e^{-\\sum t_i\\Lb^i}Q^{-1}(t)JQ^{\\top -1}(t)e^{-\\sum t_i\\Lb^{\\top\ni}}\n\\ee\nand so, from the definition of $W_1$ and $W_2$,\n\\bea\n W_1^{-1}(0)W_2(0)&=&Q^{-1}(0)JQ^{\\top -1}(0)\\nonumber\\\\\n &=&\\left(Q(t)e^{\\sum_1^{\\iy}t_i\\Lb^i}\\right)^{-1}J\\left(Q(t)e^{\\sum\nt_i\\Lb^i}\\right)^{\\top -1}~\\mbox{using (2.18)}\\nonumber\\\\\n &=&W_1(t)^{-1}JW_1(t)^{\\top\n-1}\\nonumber\\\\ &=&W_1(t)^{-1}W_2(t),\n\\eea\nimplying the independence in $t$ of the right hand\n side of (2.19).\nTherefore, we have\n$$\nW_1(t)^{-1}W_2(t)=W_1(t')^{-1}W_2(t'),\\quad\\mbox{for\nall\\,\\,}t,t'\\in\\BC^{\\iy},\n$$\nand so\n$$\nW_1(t)W_1^{-1}(t')=W_2(t)W_2^{-1}(t').\n$$\n\\qed\n\n\\underline{\\sl Proof of Theorem 2.1 }: The proof of equation\n(2.7) for $Q_1$, namely\n$$\\frac{\\pl Q_1}{\\pl t_i}=-(\\pi_{\\Bk}L_1^i)Q_1,\n$$ follows at once from Proposition 1.2.\n\nThe proof of (2.7) for $Q_2=JQ_1^{\\top -1}$ is based on the\nidentity $\\JR\\pi_{\\Bk}a=\\pi_{\\Bk_+}\\JR a$. Indeed, we compute\n\\bea\n\\frac{\\pl Q_2}{\\pl t_i}Q_2^{-1}&=&-JQ_1^{\\top -1}\\frac{\\pl Q_1^{\\top}}{\\pl\nt_i}Q_1^{\\top -1}Q_2^{-1}\\nonumber\\\\ &=&-JQ_1^{\\top -1}Q_1^{\\top\n}(\\pi_{\\Bk}L_1^i)^{\\top}Q_1^{\\top -1}Q_1^{\\top }J\\nonumber\\\\\n&=&-J(\\pi_{\\Bk}L_1^i)^{\\top}J\\nonumber\\\\\n&=&-\\JR(\\pi_{\\Bk}L_1^i)\\nonumber\\\\ &=&-\\pi_{\\Bk_+}\\JR\nL_1^i\\nonumber\\\\ &=&-\\pi_{\\Bk_+}\\JR(-\\JR L_2)^i, ~~\\mbox{using\n(2.4)},\\nonumber\\\\ &=&-\\pi_{\\Bk_+}\\JR(-1)^i(\\JR L_2)^i\\nonumber\\\\\n&=&-\\pi_{\\Bk_+}\\JR(-1)^i(-1)^{i-1}\\JR L_2^i\\nonumber\\\\\n&=&\\pi_{\\Bk_+}L_2^i.\\nonumber\n\\eea\nEquations (2.8) and (2.10) for $L_1,~L_2$ and $\\Psi_1,~\\Psi_2$ are\nthen straightforward.\n\nFinally, the proof of the bilinear identity (2.11) proceeds as\nfollows: By a well-known lemma (see \\cite{DJKM}) ,\n$$W_{{1}\\atop{2}}W_{{1}\\atop{2}}\n(t)W_{{1}\\atop{2}}(t')^{-1}=\n\\oint_{{\\iy}\\atop{0}}\\Psi_{{1}\\atop{2}}(t,z)\\otimes\n\\Psi^_{{1}\\atop{2}}(t',z)\\frac{dz}{2\\pi iz}\n$$\nand so the statement of Lemma 2.3 yields\n$$\n\\oint_{\\iy}\\Psi_1(t,z)\\otimes\\Psi_1^(t',z)\\frac{dz}{2\\pi\niz}=\\oint\\Psi_2(t,z)\\otimes\\Psi_2^(t',z)\\frac{dz}{2\\pi iz},\n$$\nwhose $(m,n)$th component is\n$$\n\\oint_{\\iy}\\Psi_{1,n}(t,z)\\Psi^_{1,m}(t',z)\n\\frac{dz}{2\\pi\niz}-\\oint_0\\Psi_{2,n}(t,z)\\Psi^_{2,m}(t',z)\n\\frac{dz}{2\\pi iz}=0.\n$$\n Next we use the relations\n$\\Psi^_1(t,z)=-J\\Psi_2(t,z^{-1})$\nand $\\Psi^_2(t,z)=J\\Psi_1(t,z^{-1})$, to yield\n$$\n\\oint_{\\iy}\\Psi_1(t,z)\\otimes J\\Psi_2(t',z^{-1})\\frac{dz}{2\\pi\niz}+\\oint_0\\Psi_2(t,z)\\otimes J\\Psi_1(t',z^{-1})\\frac{dz}{2\\pi iz}=0,\n$$\nwhich again componentwise leads to (2.11).\\qed\n\n\\vspace{0.5cm}\n\n\\underline{\\sl Proof of Theorem 2.2 }: To prove (2.15), remember from Theorem 2.1,\n$$\n\\frac{\\pl Q}{\\pl t_n}Q^{-1}=-\\pi_{\\Bk}L^n=-((L^n)_-+J(L^n_+)^{\\top}J)-\\frac{1}{2}\n((L^n)_0-J((L^n)_0)^{\\top}J);\n$$\nhence, taking the $(\\,)_0$-part of this expression, yields\n$$\n\\frac{\\pl \\log Q_0}{\\pl t_n}=\\left(\\frac{\\pl Q}{\\pl\nt_n}Q^{-1}\\right)_0=-\\pi_{\\Bk}(L^n)_0=-\\frac{1}{2}\n(L^n)_0+\\frac{1}{2}J(L^n)_0^{\\top}J.\n$$\nUsing the fact that $Q_0,Q_0^{-1},\\dot Q_0\\in G_k\\cap\\DR_0$ commute\namong themselves and commute with $J$ and the fact that\n$\\DR_0\\DR_{\\pm}$, $\\DR_{\\pm}\\DR_0\\subset\\DR_{\\pm}$, we compute for\n$\\hat Q=Q_0^{-1} Q,~ \\hat L_1= Q_0^{-1} L_1 Q_0,$ (see (2.13))\n%\\medbreak\n%\\noindent $\\displaystyle{\\frac{\\pl}{\\pl t_n}(Q_0^{-1}Q)(Q_0^{-1}Q)^{-1}}$\n\\bea\n\\frac{\\pl \\hat Q}{\\pl t_n}\\hat Q^{-1}&=&-Q_0^{-1}\\dot Q_0Q_0^{-1}QQ^{-1}Q_0+Q_0^{-1}\\dot QQ^{-1}Q_0\\nonumber\\\\\n&=&-Q_0^{-1}\\dot Q_0+Q_0^{-1}\\dot QQ^{-1}Q_0\\nonumber\\\\\n&=&Q_0^{-1}\\left(-\\dot Q_0Q_0^{-1}+\\dot\nQQ^{-1}\\right)Q_0\\nonumber\\\\\n&=&Q_0^{-1}\\left(-(L_1^n)_-+J(L^n_{1+})^{\\top}J\\right)Q_0\\nonumber\\\\\n&=&-(Q_0^{-1}L_1^nQ_0)_-+Q_0^{-1}J\n\\left(Q_0(Q_0^{-1}L_1^nQ_0)_+Q_0^{-1}\\right)^{\\top}\nJQ_0\\nonumber\\\\ &=&-(\\hat L_1^n)_-+Q_0^{-2}J((\\hat\nL_1^n)_+)^{\\top}JQ_0^2.\\nonumber\n\\eea\nUsing this result and $\\hat L_1\\hat \\Psi_1(t,z)=z \\hat\n\\Psi_1(t,z)$, we find\n\\bea\n\\lefteqn{\n\\frac{\\pl \\hat \\Psi_1(t,z)}{\\pl t_n}\n}\\nonumber\\\\\n&=&\\frac{\\pl}{\\pl t_n}e^{\\sum t_iz^i}\\hat Q\\chi(z)\\nonumber\\\\\n&=&z^ne^{\\sum_1^{\\iy}t_iz^i}\\hat\nQ\\chi(z)+e^{\\sum_1^{\\iy}t_iz^i}\\left(-(\\hat L_1^n)_-\n+Q_0^{-2}J((\\hat L_1^n)_+)^{\\top}JQ_0^2\\right)\\hat\nQ\\chi(z)\\nonumber\\\\ &=&(\\hat L_1^n-(\\hat L_1^n)_-+Q_0^{-2}J((\\hat\nL_1^n)_+)^{\\top}JQ_0^2)\\hat\\Psi_1(t,z)\\nonumber\\\\ &=&((\\hat\nL_1^n)_++(\\hat L_1^n)_0+Q_0^{-2}(\\JR(\\hat\nL_1^n)_+)Q^2_0)\\hat\\Psi_1(t,z).\n\\eea\nBut, we also have that $\\hat \\Psi_1=Q_0^{-1}\\Psi_1(t,z)$ and $\\hat\n\\Psi_2=Q_0\\Psi_2(t,z)$ satisfy, using $W_2=JW_1^{-1 \\top}$,\n\\bea\n\\frac{\\pl \\hat\\Psi_1(t,z)}{\\pl t_n}=(Q_0^{-1}W_1)^{.}\\chi(z)&=&\n (Q_0^{-1}W_1)^{.} (Q_0^{-1}W_1)^{-1} \\hat\\Psi_1(t,z)\\\\\n\\frac{\\pl \\hat\\Psi_2(t,z)}{\\pl t_n}=(Q_0W_2)^{.}\\chi(z)&=&(Q_0W_2)\n^{.}(Q_0W_2)^{-1}\n(Q_0\\Psi_2)\\nonumber\\\\ &=&(\\dot Q_0W_2+Q_0\\dot\nW_2)W_2^{-1}Q^{-1}_0(Q_0\\Psi_2)\\nonumber\\\\ &=&(\\dot\nQ_0Q^{-1}_0+Q_0\\dot W_2W_2^{-1}Q^{-1}_0)Q_0\\Psi_2\\nonumber\\\\\n&=&(\\dot Q_0Q_0^{-1}+Q_0\\JR(\\dot\nW_1W_1^{-1})Q_0^{-1})Q_0\\Psi_2\\nonumber\\\\ &=&(\\dot\nQ_0Q_0^{-1}+Q_0J(\\dot\nW_1W_1^{-1})^{\\top}JQ_0^{-1})Q_0\\Psi_2\\nonumber\\\\ &=&(-J\\dot\nQ_0Q_0^{-1}+J(Q^{-1}_0\\dot\nW_1W_1^{-1}Q_0)^{\\top})JQ_0\\Psi_2\\nonumber\\\\ &=&J(-Q_0^{-1}\\dot\nQ_0+Q_0^{-1}\\dot W_1W_1^{-1}Q_0)^{\\top}J Q_0\\Psi_2\\nonumber\\\\\n&=&\\JR\\left((Q^{-1}_0W_1)^{.}(Q_0^{-1}W_1)^{-1}\\right)(Q_0\\Psi_2).\n \\eea\nComparing (2.21), (2.22) and (2.23), and using\n$$\n-\\JR(\\hat L_1^n)=\\hat L_2^n,\n$$\nand so, in particular,\n$$\n-\\JR((\\hat L_{1 }^n)_{\\pm})=(\\hat L_{2 }^n)_{\\mp}~~\\mbox{and}~~\n-\\JR((\\hat L_{1}^n)_{0})=(\\hat L_{2 }^n)_0,\n$$\n$$\n\\frac{\\pl \\hat\\Psi_2(t,z)}{\\pl t_n}=-((\\hat L_2^n)_-+(\\hat\nL_2^n)_0+Q^2_0(\\JR(L^n_2)_-)Q_0^{-2})\\hat\\Psi_2(t,z),\n$$ which establishes theorem 2.2.\n\\qed\n\n\n\n\n\n\\section{Existence of the Pfaff $\\tau$-function}\n\nThe point of this section is to show that the solution of the Pfaff\nLattice can be expressed in terms of a sequence of functions\n$\\tau$, which are not $\\tau$-functions in the usual sense, but\nenjoys a different set of bilinear identities and partial\ndifferential equations.\n\n\\begin{proposition}There exists functions $\\tau_{2n}(t)$\n such that\n\\be\n \\psi_{1,2n}(t,z)=\\frac{\\tau_{2n}(t-[z^{-1}])}{\\tau_{2n}(t)}~\n \\mbox{ and }~\n\\psi_{2,2n}(t,z)=\\frac{\\tau_{2n+2}(t+[z])}{\\tau_{2n+2}(t)}.\n\\ee\n\\end{proposition}\n\n\\bigbreak\n\nThe proof of proposition 3.1 will be postponed until later. For\nfuture use, we define the diagonal matrix\n\\be h=\\mbox{diag}(...,h_{-2},h_{-2}, h_0, h_0, h_2,\n h_2,...)\\in \\DR_0^-,~~\\mbox{with}~~ h_{2n}=\n \\frac{ \\tau_{2n+2}}{ \\tau_{2n}}.\n\\ee\n\n\\begin{theorem}\n\n\\bea\n\\Psi_{1,2n}(t,z)&=&e^{\\sum t_i z^i}z^{2n}\n\\frac{\\tau_{2n}(t-[z^{-1}])}\n{\\sqrt{\\tau_{2n}(t)\\tau_{2n+2}(t)}}\n\\nonumber\\\\\n \\Psi_{1,2n+1}(t,z)&=&e^{\\sum t_i z^i}z^{2n}\n\\frac{(z+\\pl/ \\pl t_1)\\tau_{2n}(t-[z^{-1}])}\n{\\sqrt{\\tau_{2n}(t)\\tau_{2n+2}(t)}}\n\\nonumber\\\\\n \\Psi_{2,2n}(t,z)&=&e^{-\\sum t_i z^{-i}}z^{2n+1}\n\\frac{\\tau_{2n+2}(t+[z])}\n{\\sqrt{\\tau_{2n}(t)\\tau_{2n+2}(t)}}\n\\nonumber\\\\\n \\Psi_{2,2n+1}(t,z)&=&e^{-\\sum t_i z^{-i}}z^{2n+1}\n\\frac{(z^{-1}-\\pl/ \\pl t_1)\\tau_{2n+2}(t+[z])}{\\sqrt{\\tau_{2n}(t)\\tau_{2n+2}(t)}}\n,\\nonumber\\\\\n \\eea\nwith the $\\tau_{2n}(t)$ satisfying the following bilinear identity\n\\bea\n& &\\oint_{z=\\iy}\\tau_{2n}(t-[z^{-1}])\\tau_{2m+2}(t'+[z^{-1}])\ne^{\\sum(t_i-t'_i)z^i} z^{2n-2m-2}\\frac{dz}{2\\pi i}\\nonumber\\\\\n&&\\quad+\\oint_{z=0}\\tau_{2n+2}(t+[z])\\tau_{2m}(t'-[z])\ne^{\\sum(t'_i-t_i)z^{-i}}z^{2n-2m}\\frac{dz}{2\\pi i}=0.\n\\eea\nThen $L$ has the following representation in terms of the Pfaffian\n$\\tau$-functions:\n\\bigbreak\n$$ h^{1/2} L h^{-1/2}=\n \\pmatrix{&\\vdots& \\cr\n ...&\\hat L_{00}&\\hat L_{01}&0&0&\\cr\n&\\hat L_{10}&\\hat L_{11}&\\hat L_{12}&0&\\cr &&\\hat L_{21} &\\hat\nL_{22}&\\hat L_{23}&\\cr\n &&\n&\\hat L_{32}&\\hat L_{33}&...\\cr & & & &\\vdots& &\n\\cr},\n$$\nwith $({}^.=\\frac{\\pl}{\\pl t_1})$\n$$\n\\hat L_{nn}:=\\pmatrix{-(\\log \\tau_{2n})^. & &1 \\cr \\cr\n -\\frac{S_2(\\tilde \\pl)\\tau_{2n}}{\\tau_{2n}}\n-\\frac{S_2(-\\tilde \\pl)\\tau_{2n+2}}{\\tau_{2n+2}} & &\n (\\log \\tau_{2n+2})^.\\cr}~~\n ~~~~~~\\hat L_{n,n+1}:=\\pmatrix{0& 0 \\cr\n 1 & 0\\cr}\n$$\n\n\\vspace{0.5cm}\n\n\\be\n\\hat L_{n+1,n}:=\\pmatrix{&(\\log \\tau_{2n+2})^{..}& \\cr\n * &\\cr}.\n \\ee\n\n\n\n\n\\end{theorem}\n\nThe following bilinear relations are due to \\cite{ASV}:\n\n\\begin{corollary}\nThe functions $\\tau_{2n}(t)$ satisfy the following ``differential\nFay identity\"\\footnote{$\\{f,g\\}=f'g-fg'$, where $'=\\pl/\\pl t_1$.}\n\n\\bigbreak\n\n$\n\\{\\tau_{2n}(t-[u]),\\tau_{2n}(t-[v])\\}\n$\\hfill\n\\bea\n& & \\hspace{2cm}+(u^{-1}-v^{-1})(\\tau_{2n}(t-[u])\\tau_{2n}(t-[v])\n-\\tau_{2n}(t)\\tau_{2n}(t-[u]-[v]))\\nonumber\\\\\n& & \\nonumber\\\\\n & &~~~=uv(u-v)\\tau_{2n-2}(t-[u]-[v])\\tau_{2n+2}(t),\n\\eea\nand Hirota type bilinear equations, always involving nearest\nneighbours:\n\\be\n\\left(p_{k+4}(\\tilde\\pl)-\\frac{1}{2}\\frac{\\pl^2}{\\pl\nt_1\\pl t_{k+3}}\\right)\\tau_{2n}\\circ\\tau_{2n}=p_k(\\tilde\n\\pl)~\\tau_{2n+2}\\circ\\tau_{2n-2}\n\\ee\n\\hfill $k,n=0,1,2,...~.$\n\\end{corollary}\n\n\\begin{lemma} Consider an arbitrary function $\\vp(t,z)$ depending on\n$t\\in\\BC^{\\iy}$, $z\\in\\BC$, having the asymptotics\n$\\vp(t,z)=1+O(\\frac{1}{z})$ for $z\\nearrow\\iy$ and\n satisfying the functional\nrelation\n$$\n\\frac{\\vp(t-[z^{-1}_2],z_1)}{\\vp(t,z_1)}=\\frac{\\vp(t-[z^{-1}_1],z_2)}\n{\\vp(t,z_2)}, \\quad t\\in\\BC^{\\iy},z\\in\\BC.\n$$\nThen there exists a function $\\tau(t)$ such that\n$$\\vp(t,z)=\\frac{\\tau(t-[z^{-1}])}{\\tau(t)}.\n$$\n\\end{lemma}\n\n\\proof See appendix (Section 10)\n\n\n\\begin{lemma} The following holds for the\nPfaffian wave function $\\Psi_1$ and $\\Psi_2$, as in (2.6),\n\\be\n\\frac{\\psi_{1,2n}(t-[z^{-1}_2],z_1)}{\\psi_{1,2n}(t,z_1)}=\n\\frac{\\psi_{1,2n}(t-[z^{-1}_1],z_2)}{\\psi_{1,2n}(t,z_2)}\n\\ee\nand\n\\be\n\\psi_{2,2n-2}(t-[z^{-1}],z^{-1})\\psi_{1,2n}(t,z)=1.\n\\ee\n\\end{lemma}\n\n\\proof Setting (2.6) in the bilinear equation (2.11), with $n\\mapsto 2n,~m\\mapsto 2n-2$, yields\n$$\n\\frac{c_{2n}(t)}{c_{2n-2}(t)}\\oint_{\\iy}e^{\\sum\n(t_i-t'_i)z^i}\\psi_{1,2n}(t,z)\\psi_{2,2n-2}(t',z^{-1})\\frac{dz}{2\\pi i}\n$$\n$$\n+\\frac{c_{2n-2}(t)}{c_{2n}(t)}\\oint_0 e^{\\sum\n(t'_i-t_i)z^i}\\psi_{2,2n}(t,z)\\psi_{1,2n-2}(t',z^{-1})\\frac{z^2dz}{2\\pi\ni}=0.\n$$\nSetting\n$$\nt-t'=[z^{-1}_1]+[z^{-1}_2]\n$$\nin the above and using $e^{\\sum_1^{\\iy}x^i/i}=1/(1-x)$ yields\n$$\n\\frac{c_{2n}}{c_{2n-2}}\\oint_{\\iy}\\frac{\\psi_{1,2n}(t,z)\\psi_{2,2n-2}(t',z^{-1})}\n{\\left(1-\\frac{z}{z_1}\\right)\\left(1-\\frac{z}{z_2}\\right)}\n\\frac{dz}{2\\pi i}\\hspace{6cm}\n$$\n$$\n\\hspace{1cm}=-\\frac{c_{2n-2}}\n{c_{2n}}\\oint_{z=0}z^2\\left(1-\\frac{z}{z_1}\\right)\\left(1-\\frac{z}{z_2}\\right)\n\\psi_{2,2n}(t,z)\\psi_{1,2n-2}(t',z^{-1})\\frac{dz}{2\\pi i}.$$\n\n\\medbreak\n\\noindent Since the integrand on the right hand side is holomorphic, it suffices to\nevaluate the integral on the left hand side, which can be viewed as an\nintegral along a contour encompassing $\\iy$ and the points $z_1$ and $z_2$,\nthus leading to\n\\be\n\\psi_{1,2n}(t,z_1)\\psi_{2,2n-2}(t-[z_1^{-1}]-[z_2^{-1}],z_1^{-1})=\\psi_{1,2n}(t,z_2)\\psi_{2,2n-2}(t-\n[z_1^{-1}]-[z_2^{-1}],z_2^{-1})\n\\ee\nwith\n$$\n\\psi_{1,2n}(t,z)=1+O\\left(z^{-1}\\right),~~~\n\\psi_{2,2n-2}(t-[z_1^{-1}]-[z_2^{-1}],z^{-1})\n=1+O(z^{-2}).\n$$\nTherefore, letting $z_2\\nearrow\\iy$, one finds\n\\be\n\\psi_{1,2n}(t,z_1)\\psi_{2,2n-2}(t-[z_1^{-1}],z_1^{-1})=1\n,\\ee\n yielding (3.9), and so, upon shifting $t\\mapsto t-[z_2^{-1}]$,\n$$\n\\psi_{2,2n-2}(t-[z_1^{-1}]-[z_2^{-1}],z_1^{-1})=\\frac{1}{\\psi_{1,2n}(t-[z_2^{-1}],z_1)};\n$$\nsimilarly\n\\be\n\\psi_{2,2n-2}(t-[z_1^{-1}]-[z_2^{-1}],z_2^{-1})=\\frac{1}{\\psi_{1,2n}(t-[z_1^{-1}],z_2)}.\n\\ee\nSetting the two expressions (3.12) in (3.10) yields\n$$\n\\frac{\\psi_{1,2n}(t-[z_2^{-1}],z_1)}{\\psi_{1,2n}(t,z_1)}=\\frac{\\psi_{1,2n}(t-[z_1^{-1}],z_2)}\n{\\psi_{1,2n}(t,z_2)}.\n$$\n\\qed\n\n\n\\underline{\\sl Proof of Proposition 3.1 }: From Lemmas 3.4 and 3.5, there exists,\nfor each $2n$, a function $\\tau_{2n}$ such that the first relation of (3.1)\nis satisfied, i.e.,\n$$\n\\psi_{1,2n}(t,z)=\\frac{\\tau_{2n}(t-[z^{-1}])}{\\tau_{2n}(t)},\n$$\nand so from (3.9)\n$$\n\\psi_{2,2n-2}(t-[z^{-1}],z^{-1})=\\frac{1}{\\psi_{1,2n}(t,z)}=\\frac{\\tau_{2n}(t)}{\\tau_{2n}(t-[z^{-1}])},\n$$\nthus leading to\n$$\n\\psi_{2,2n-2}(t,z)=\\frac{\\tau_{2n}(t+[z])}{\\tau_{2n}(t)},\n$$\nwhich is the second relation of (3.1).\\qed\n\n\n\\underline{\\sl Proof of Theorem 3.2 }: At first, remembering that\n$\\hat Q=Q_0^{-1}Q$, observe that\n\\bea\ne^{\\sum t_iz^i}((\\hat\nQ)\\chi(z))_{2n}&=&(Q^{-1}_0\\Psi_1(t,z))_{2n}\\nonumber\\\\ &=&e^{\\sum\nt_iz^i}z^{2n}\\psi_{1,2n}(t,z)\\nonumber\\\\ &=&e^{\\sum\nt_iz^i}z^{2n}\\frac{\\tau_{2n}(t-[z^{-1}])}{\\tau_{2n}(t)}\\nonumber\\\\\n&=&e^{\\sum\nt_iz^i}z^{2n}\\left(1+\\sum_{n=1}^{\\iy}\\frac{S_k(-\\tilde\\pl)\\tau_{2n}(t)}{\\tau_{2n}(t)}\\right),\\nonumber\n\\eea\nshowing that a few subdiagonals of the matrix $\\hat Q$ are\n given by\n$$\n\\hat Q=\\left(\\begin{array}{cc@{}c@{}ccc}\n \\ddots& &&&&\\\\\n &\\boxed{\\begin{array}{cc}~~~ 1 ~~~~~~&~~~~~~~~ 0~~~~ \\\\\n~~~ 0~~~~~~&~~~~~~~~1~~~~ \\end{array}}& & & &\\\\\n &\\boxed{\\begin{array}{cc} \\hat q_{2n,2n-2}&\\hat q_{2n,2n-1} \\\\\n \\hat q_{2n+1,2n-2}&\\hat q_{2n+1,2n-1} \\end{array}}&\n \\boxed{\\begin{array}{cc} ~~~~ 1 ~~~~~~&~~~~~~~~ 0~~~~ \\\\\n~~~~ 0~~~~~~&~~~~~~~~1~~~~ \\end{array}} &&&\\\\\n & &&& \\ddots\n \\end{array}\n \\right)\n $$\nwith\n\\be\n\\hat q_{2n,2n-1}=-\\frac{\\pl}{\\pl t_1}\\log\\tau_{2n},\\quad\\hat\nq_{2n,2n-2}=\\frac{S_2(-\\tilde\\pl)\\tau_{2n}}{\\tau_{2n}}.\n\\ee\nRemembering that\n\\be\n\\left\\{\\begin{tabular}{ll}\n$\\hat\\Psi_{1,2n}(t,z)=e^{\\sum t_kz^k}z^{2n}\n\\psi_{1,2n}(t,z),$& $\\psi_{1,2n}=1+O(z^{-1})$\\\\\n$\\hat \\Psi_{1,2n+1}(t,z)=e^{\\sum\nt_kz^k}z^{2n+1}\\psi_{1,2n+1}(t,z),$& $\\psi_{1,2n+1}=1+O(z^{-2})$\n\\end{tabular}\\right.\n\\ee\n\\bea\n\\left\\{\\begin{tabular}{ll}\n$\\hat \\Psi_{2,2n}(t,z)= e^{-\\sum t_kz^{-k}}z^{2n+1}\n\\psi_{2,2n}(t,z),$&\n$\\psi_{2,2n}=1+O(z)$\\nonumber\\\\\n $\\hat \\Psi_{2,2n+1}(t,z)=-e^{-\\sum\nt_kz^{-k}}z^{2n}\\psi_{2,2n+1}(t,z),\n $& $\\psi_{2,2n+1}=1+O(z^2),$\n\\end{tabular}\\right.\\nonumber\\\\\n\\eea\nwe now compute, using theorem 2.2,\n\\bea\n\\lefteqn{\ne^{\\sum t_iz^i}\\left(\\frac{\\pl}{\\pl\nt_1}+z\\right)z^{2n}\\psi_{1,2n}(t,z)\n}\n\\nonumber\\\\\n&=&\\left(\\frac{\\pl}{\\pl t_1}\\hat\n\\Psi_1(t,z)\\right)_{2n}\\nonumber\\\\\n&=&\\left(\\left((\\hat\nL_1)_++(\\hat L_1)_0+Q_0^{-2}J(\\hat\nL_{1+})^{\\top}JQ_0^2\\right)\\hat\\Psi_1(t,z)\\right)_{2n}%\\nonumber\n\\eea\nand\n\\bea\n-\\lefteqn{\ne^{-\\sum t_iz^{-i}}\\left(\\frac{\\pl}{\\pl\nt_1}-\\frac{1}{z}\\right)z^{2n+1}\\psi_{2,2n}(t,z)\n}\n\\nonumber\\\\\n&=&\\frac{\\pl}{\\pl t_1}(\\hat\\Psi_2(t,z))_{2n}\\nonumber\\\\\n&=&\\left(\\left(\\JR((\\hat L_1)_++(\\hat L_1)_0 +Q_0^{-2}J(\\hat\nL_{1+})^{\\top}JQ_0^2)\\right)\\hat \\Psi_2(t,z)\\right)_{2n}.%\\nonumber\n\\eea\nIn this expression, the matrix equals, according to (2.13),\n\n\n\\medbreak\n\\bean\n\\lefteqn{(\\hat L_1)_++(\\hat L_1)_0+Q_0^{-2}J(\\hat\nL_{1+})^{\\top}JQ_0^2=}\n\\\\\n&&\n\\pmatrix{&\\vdots& \\cr\n ...&\\hat q_{0,-1}&1&\\vline&0&0&\\vline&0&0\\cr\n&\\hat q_{1,-1}-\\hat q_{20}&-\\hat\nq_{21}&\\vline&1&0&\\vline&0&0\\cr\\hline\n &0 & 0\n&\\vline&\\hat q_{21}&1&\\vline&0&0\\cr &{c_0^2}/{c_2^2}&0&\\vline&\\hat\nq_{31}-\\hat q_{42}&-\\hat q_{43}&\\vline&1&0\\cr\\hline\n &0&0&\\vline&0&0&\\vline&\\hat q_{43}&1\\cr\n &0&0&\\vline&{c_2^2}/{c_4^2} &0&\\vline&\\hat q_{53}-\\hat\nq_{64}&-\\hat q_{65}&...\\cr\n & & & & & & & &\\vdots\n\\cr},\n\\eean\nand, acting with $\\JR$ on this matrix,\n\n\\medbreak\n\\noindent$\\displaystyle{\\JR\\left((\\hat L_1)_++(\\hat L_1)_0+Q_0^{-2}J(\\hat\nL_{1+})^{\\top}JQ_0^2\\right)=}$\n\n$$\n\\pmatrix{&\\vdots& \\cr\n ...&-\\hat q_{21}&1&\\vline&0&0&\\vline&0&0\\cr\n&\\hat q_{1,-1}-\\hat q_{20}&\\hat\n q_{0,-1}&\\vline&c^2_0/c^2_2&0&\\vline&0&0\\cr\\hline\n &0 & 0&\\vline&-\\hat q_{43}&1&\\vline&0&0\\cr\n &1&0&\\vline&\\hat q_{31}-\\hat\nq_{42}&\\hat q_{21}&\\vline&c_2^2/c^2_4&0\\cr\\hline\n &0&0&\\vline&0&0&\\vline&-\\hat\nq_{65}&1\\cr\n &0 &0&\\vline&1&0&\\vline&\\hat q_{53}-\\hat q_{64}&\\hat\nq_{43}&...\\cr & & & & & & & & \\vdots \\cr},\n$$\nusing the fact that\n$$\n\\JR\\pmatrix{0&0\\cr\n1&0\\cr}=\\pmatrix{0&0\\cr\n1&0\\cr}.\n$$\nTherefore the $2n$th rows of both matrices respectively have the\nform\n$$(0,\\dots,0,\\begin{array}[t]{@{}c@{}}\n\\hat q_{2n,2n-1}(t)\\\\\\stackrel{\\uparrow}{2n}\\end{array},1,0,0,\\dots)$$\n$$(0,\\dots,0,\\begin{array}[t]{@{}c@{}}\n-\\hat q_{2n+2,2n+1}(t)\\\\\\stackrel{\\uparrow}{2n}\\end{array},1,0,0,\\dots)$$\nand thus from (3.16) and (3.17), and the expansions (3.14) and\n(3.15), we have\n\\bea\n\\left(\\frac{\\pl}{\\pl t_1}+z\\right)\\psi_{1,2n}(t,z)&=&\\hat q_{2n,2n-1}(t)\\psi_{1,2n}+z\\psi_{1,2n+1}\n\\nonumber\\\\\n\\left(\\frac{\\pl}{\\pl t_1}-z^{-1}\\right)\\psi_{2,2n}(t,z)&=&\\hat q_{2n+2,2n+1}(t)\n\\psi_{2,2n}+z^{-1}\\psi_{2,2n+1}\n\\eea\nand so, using the expression (3.13) for $\\hat q_{2n,2n-1}$ and the\nfirst expression (3.1),\n\n\\medbreak\n\n\\noindent\n$\\displaystyle{z^{2n+1}\\psi_{1,2n+1}(t,z)}$\n\\bea\n&=&\\left(z+\\frac{\\pl}{\\pl t_1}\\right)z^{2n}\\psi_{1,2n}(t,z)-\\hat\nq_{2n,2n-1}(t)z^{2n}\\psi_{1,2n}(t,z)\\nonumber\\\\\n&=&\\left(z+\\frac{\\pl}{\\pl\nt_1}\\right)z^{2n}\\psi_{1,2n}(t,z)+\\left(\\frac{\\pl}{\\pl\nt_1}\\log\\tau_{2n}(t)\\right)z^{2n}\\psi_{1,2n}(t,z)\\nonumber\\\\\n&=&\\left(z+\\frac{\\pl}{\\pl\nt_1}\\right)z^{2n}\\frac{\\tau_{2n}(t-[z^{-1}])}{\\tau_{2n}(t)}+\\left(\\frac{\\pl}{\\pl\nt_1}\\tau_{2n}(t)\\right)z^{2n}\\frac{\\tau_{2n}(t-[z^{-1}])}{\\tau^2_{2n}(t)}\\nonumber\\\\\n&=&z^{2n}\\frac{\\left(z+\\frac{\\pl}{\\pl\nt_1}\\right)\\tau_{2n}(t-[z^{-1}])}{\\tau_{2n}(t)}%\\nonumber\n\\eea\nand similarly, using the relation (3.17),\n\\be\nz^{2n}\\psi_{2,2n+1}(t,z)=z^{2n+1}\\frac{\\left(-z^{-1}+\\frac{\\pl}{\\pl\nt_1}\\right)\\tau_{2n+2}(t+[z])}{\\tau_{2n+2}(t)}.\n\\ee\nTherefore, we also have\n\\bea\n\\psi_{1,2n+1}(t,z)&=&\\frac{1}{z}\\frac{1}{\\tau_{2n}(t)}\\left(z+\\frac{\\pl}{\\pl\nt_1}\\right)\\left(\\tau_{2n}(t)-\\frac{\\pl\\tau_{2n}}{\\pl\nt_1}z^{-1}+S_2(-\\tilde\\pl)\\tau_{2n}z^{-2}+\\cdots\\right)\\nonumber\\\\\n&=&1+\\frac{1}{\\tau_{2n}(t)}\\left(-\\frac{\\pl^2}{\\pl\nt_1^2}+S_2(-\\tilde\\pl)\\right)\\tau_{2n}z^{-2}+O(z^{-3})\\nonumber\n\\eea\nthus\n\\be\n\\hat q_{2n+1,2n}=0,\\quad\n\\hat q_{2n+1,2n-1}=\\frac{1}{\\tau_{2n}}\\left(S_2(-\\tilde\\pl)-\\frac{\\pl^2}\n{\\pl\nt_1^2}\\right)\\tau_{2n}=\\frac{-S_2(\\tilde\\pl)\\tau_{2n}}{\\tau_{2n}}.\n\\ee\n\nSetting $n\\mapsto 2n$ and $m\\mapsto 2n$ in the bilinear relation\n(2.11) and substituting, using (2.6) and the expressions for\n$\\psi_{1,2n}(t,z)$ and $\\psi_{2,2n}(t,z)$ in the proof of\nproposition 3.1,\n$$\n\\Psi_{1,2n}(t,z)=e^{\\sum\nt_kz^k}z^{2n}c_{2n}(t)\\frac{\\tau_{2n}(t-[z^{-1}])}{\\tau_{2n}(t)}\n$$\nand\n$$\n\\Psi_{2,2n}(t',z)=e^{-\\sum\nt_kz^{-k}}z^{2n+1}c_{2n}^{-1}(t')\\frac{\\tau_{2n+2}(t+[z])}{\\tau_{2n+2}(t)}\n$$\ninto\n$$\n\\oint_{\\iy}\\Psi_{1,2n}(t,z)\\Psi_{2,2n}(t',z^{-1})\\frac{dz}{2\\pi iz}+\n\\oint_0\\Psi_{2,2n}(t,z)\\Psi_{1,2n}(t',z^{-1})\\frac{dz}{2\\pi iz}=0\n$$\nyields\n$$\n\\frac{c_{2n}(t)}{c_{2n}(t')}\\oint_{\\iy}e^{\\sum(t_k-t'_k)z^k}\n\\frac{\\tau_{2n}(t-[z^{-1}])\\tau_{2n+2}(t'+[z^{-1}])}{\\tau_{2n}(t)\n\\tau_{2n+2}(t')}\\frac{dz}{2\\pi iz^2}\n$$\n$$\n+\\frac{c_{2n}(t')}{c_{2n}(t)}\\oint_0e^{\\sum(t'_k-t_k)z^{-k}}\n\\frac{\\tau_{2n+2}(t+[z])\\tau_{2n}(t'-[z])}{\\tau_{2n+2}(t)\n\\tau_{2n}(t')}\\frac{dz}{2\\pi i}.\n$$\nSetting $t'=t+[\\al]$ amounts to replacing the exponential:\n$$\ne^{\\sum(t_k-t'_k)z^k}=1-\\al z,~\\quad\ne^{\\sum(t'_k-t_k)z^{-k}}=\\frac{1}{1-\\al/z},\n$$\nso that the first integral has a simple pole at $z=\\iy$ and the second\nintegral one at $z=\\al$. Evaluating the integrals yield\n$$\n-\\al\\frac{c^2_{2n}(t)\\frac{\\tau_{2n+2}(t)}\n {\\tau_{2n}(t)}}{c^2_{2n}(t')\n\\frac{\\tau_{2n+2}(t')}{\\tau_{2n}(t')}}+\\al=0;\n$$\ni.e.,\n$$\n\\left(e^{\\sum\\frac{\\al^i}{i}\\frac{\\pl}{\\pl t_i}}-1\\right)\nc_n^2(t)\\frac{\\tau_{2n+2}(t)}{\\tau_{2n}(t)}=0\n$$\nyielding the following relation, which involves a constant $c_n$,\nindependent of time,\n\\be\nc_{2n}^2(t)=c_n\\frac{\\tau_{2n}(t)}{\\tau_{2n+2}(t)}=c_n\\cdot\nh^{-1}_{2n}(t).\n\\ee\nRescaling $\\tau_{2n}\\mapsto \\tau_{2n}/(c_1 c_2\\cdots c_{n-1}) $\nyields (3.3). Using the expressions for $\\psi_{1,2n}(t,z)$ and\n$\\psi_{2,2n}(t,z)$ (see the proof of proposition\n 3.1), (3.19), (3.20), (3.21), (2.6) and substituting\n (3.3) into (2.11) yields (3.4).\n\nFinally to derive the form (3.5) of the matrix $L$, set (3.13) and\n(3.20) in the elements just below the main diagonal of matrix\n(2.14), to yield ($^.=\\pl/\\pl t_1$)\n\\bea\n\\lefteqn{- \\hat q_{2n,2n-1}^2- \\hat q_{2n+1,2n-1}+\n\\hat q_{2n,2n-2}}\n\\nonumber\\\\\n&=&-\\left(\\frac{\\dot{\n \\tau}_{2n}}{ \\tau_{2n}}\\right)^2-\n \\frac{(S_2(-\\tilde\\pl)-\n\\frac{\\pl^2}{\\pl t_1^2})\\tau_{2n}}{\\tau_{2n}}+\n\\frac{S_2(-\\tilde\\pl)\\tau_{2n}}{\\tau_{2n}}\\nonumber\\\\\n&=&\\frac{\\ddot\\tau_{2n}}{\\tau_{2n}}-\\left(\\frac{\\dot\\tau_{2n}}{\\tau_{2n}}\\right)^2\\nonumber\\\\\n&=&(\\log\\tau_{2n})^{..}\\nonumber\n\\eea\nand\n\\bea\n \\hat q_{2n+1,2n-1}-\\hat q_{2n+2,2n}&=&\\frac{(S_2(-\\tilde\\pl)-\n\\frac{\\pl^2}{\\pl\nt_1^2})\\tau_{2n}}{\\tau_{2n}}-\\frac{S_2(-\\tilde\\pl)\\tau_{2n+2}}{\\tau_{2n+2}}\\nonumber\\\\\n&=&-\\frac{S_2(\\tilde\\pl)\\tau_{2n}}{\\tau_{2n}}-\\frac{S_2(-\\tilde\\pl)\n\\tau_{2n+2}}{\\tau_{2n+2}},\\nonumber\n\\eea\nconcluding the proof of theorem 3.2, upon substituting these\nrelations into (2.14).\\qed\n\n%\\newpage\n\n\\section{Semi-infinite matrices $m_{\\iy}$,\n(skew-)orthogonal polynomials and matrix integrals}\n\n\n\n\\subsection{$\\pl m / \\pl t_k=\\Lb^k m $, orthogonal polynomials and\nHermitean matrix integrals. }\n\n\n For the sake of completeness and analogy, we add this\n subsection, which summarizes some of \\cite{AvM2}. Consider a $t$-dependent weight\n$\\rho_t(dz):=e^{-V_t(z)}dz:=e^{-V(z)+\\sum t_i z^i}dz=e^{\\sum t_i\nz^i}\\rho(dz)$ on $\\BR$, as in (0.0) and the induced $t$-dependent\nmeasure\n\\be\ne^{Tr (- V(X)+\\sum t_i X^i)}dX,\n\\ee\non the ensemble\n$\n{\\cal H}_n$ of Hermitean matrices, with Haar measure $dX$; the\nlatter can be decomposed into a spectral part (radial part) and an\nangular part:\n\\be\ndX:=\\displaystyle{\\prod_1^n dX_{ii}\\prod_{1\\leq i<j\\leq n}}(d\\Re\nX_{ij}\\,d\\Im X_{ij})=\\Delta^2(z)dz_1\\cdots dz_n~dU,\n\\ee\nwhere $\\Delta(z)=\\displaystyle{\\prod_{1\\leq i<j\\leq n}}(z_i-z_j)$\nis the Vandermonde determinant. Here we form the following matrix\nintegral\n\\be\n\\int_{\\HR_n}e^{Tr (- V(X)+\\sum t_i X^i)}dX=c_n\n\\int_{\\BR^n}\\Delta^2(z)\\prod^n_1\\rho_t(dz_k).\n\\ee\nThe weight $\\rho_t(dz)$ defines a (symmetric) inner product\n$$\n\\la f,g\\ra^{sy}=\\int f(z)g(z)\\rho_t(dz)\n$$\nand so, the moments\n$$\n\\mu_{ij}(t):=\\la\nz^i,z^j\\ra^{sy}=\n\\int_{\\BR} z^{k+\\ell}e^{\\sum t_iz^i}\\rho(dz)=\n\\mu_{i+\\ell,j}(t)\n$$\nsatisfy\n$$\n\\frac{\\pl\\mu_{ij}}{\\pl t_{\\ell}}=\\int_{\\BR} z^{i+j+\\ell}\\,e^{\\sum t_k z^k}\\rho(dz)\n.$$\nTherefore the semi-infinite moment matrix\n$m_{\\iy}=(\\mu_{ij})_{i,j\\geq 0}$ satisfies\n\\be\n\\frac{\\pl m_{\\iy}}{\\pl t_i}=\\Lb^i m_{\\iy}=m_{\\iy}\n \\Lb^{\\top ^i}.\n\\ee\nThe point now is that the following integral can be expressed as a\ndeterminant of moments, namely\n\\bean\nn! \\tau_n(t)=\\int_{\\HR_n}e^{-Tr\\,V_t(X)}dX\n &=&\\int_{\\BR^n}\\Delta^2(z)\\prod^n_{k=1}\\rho_t(dz_k)\\\\\n &=&\\int_{\\BR^n}\\sum_{\\sg\\in S_n}\\det\n \\left(z^{\\ell-1}_{\\sg(k)}z^{k-1}_{\\sg(k)}\\right)_{1\\leq\\ell,k\\leq\nn}\\prod^n_{k=1}\\rho_t(dz_k)\\\\\n &=&\\int_{\\BR^n}\\sum_{\\sg\\in\nS_n}\\det\\left(z_{\\sg(k)}^{\\ell+k-2}\\right)_{1\\leq\\ell,k\\leq\nn}\\prod^n_{k=1}\n\\rho_t(dz_{\\sg(k)})\\\\\n&=&\\sum_{\\sg\\in S_n}\\det\\left(\\int_{\\BR}\nz_{\\sg(k)}^{\\ell+k-2}\\rho_t(dz_{\\sg(k)})\n\\right)_{1\\leq\\ell,k\\leq n}\\\\\n&=&n!\\det\\left(\\int_{\\BR}\nz^{\\ell+k-2}\\rho_t(dz)\\right)_{1\\leq\\ell,k\\leq n}\\\\\n&=&n!\\det(\\mu_{ij})_{0\\leq i,j\\leq n-1}\n\\eean\nis a $\\tau$-function for the KP-equation;\n also in view of (4.4) and the upper-lower Borel decomposition\n (0.3) of $m_{\\iy}$,\n the integrals form a vector of $\\tau$-functions for the Toda\n lattice.\n\n\\subsection{$\\pl m / \\pl t_k=\\Lb^k m +m\n\\Lb^{\\top k} $, skew-orthogonal polynomials and\nsymmetric and symplectic matrix integrals. }\n\n\\setcounter{equation}{4}\n\n\nConsider a skew-symmetric semi-infinite matrix $$ m_{\\iy}(t)\n=(\\mu_{ij}(t))_{i,j\\geq 0},~~\\mbox{with}~~m_n(t)=\n (\\mu_{ij}(t))_{0\\leq i,j\\leq n-1},$$\nsatisfying \\be\n\\pl m_{\\iy} / \\pl t_k=\\Lb^k m_{\\iy} +m_{\\iy}\n\\Lb^{\\top n} .\\ee\nThen we have shown in previous sections that, upon skew-Borel\ndecomposing $m_{\\iy}$, these equations ultimately imply the\nexistence of functions $\\tau(t)$ satisfying the bilinear equations\n(3.4). Remember also\n$$ h(t)=\\mbox{diag}(\\ldots,h_{-2},h_{-2}, h_0, h_0, h_2,\nh_2,\\ldots)\\in\n\\DR_0^-,~~\\mbox{with}~~ h_{2n}(t)=\n\\frac{ \\tau_{2n+2}(t)}{ \\tau_{2n}(t)}.$$\n Here, we need the Pfaffian $ pf (A)$ of a\n skew-symmetric matrix\n$\nA=(a_{ij})_{0\\leq i,j\\leq n-1}$ for\\footnote{In the formula below\n$\n (i_0,i_1,\\ldots\n,i_{n-2},i_{n-1})=\\sigma (0,1,\\ldots,n-1)$, where $\\sigma$ is a\npermutation and $\\vr(\\sigma) $ its parity.} even $n$:\n\\begin{eqnarray}\n\\lefteqn{pf(A)dx_0\\wedge\\cdots\\wedge dx_{n-1}}\\nonumber\\\\\n&=&\\frac{1}{n!}\\left(\n\\sum_{0\\leq i<j\\leq n-1}\na_{ij}dx_i\\wedge dx_j\\right)^n\\nonumber\\\\\n&=&\\frac{1}{2^{n/2}(n/2)!}\\left(\\sum_{\\sg}\\vr(\\sg)a_{i_0,i_1}a_{i_2,i_3}\n\\cdots a_{i_{n-2},i_{n-1}}\n\\right)dx_0\\wedge \\cdots\\wedge dx_{n-1},\\nonumber\\\\\n\\end{eqnarray}\nso that $pf(A)^2=\\det A$. We now state the following theorem due to\nAdler-Horozov-van Moerbeke\\cite{AHV}.\n\n\\begin{theorem}\nConsider a semi-infinite skew-symmetric matrix $m_{\\iy}$, evolving\naccording to (4.5); setting\n\\be\n\\tau_{2n}(t)=pf (m_{2n}(t)),~\\mbox{and}~~ h_{2n}=\n\\frac{pf (m_{2n+2}(t))}{pf (m_{2n}(t))},\n\\ee\n then the wave vector $\\Psi_1$, defined by (3.3) is a sequence of polynomials, except for the\nexponential,\n\\be\n\\Psi_{1,k}(t,z)=e^{\\sum t_i z^i} q_k(t,z),\n\\ee\nwhere the $ q_k$'s are skew-orthonormal polynomials of the form\n(0.17) satisfying\n\\be\n(\\la q_i, q_j\\ra^{sk})_{0\\leq i,j<\\iy}\n =J, ~~\\mbox{with}~~\\la y^i, z^j\\ra^{sk}:=\\mu_{ij}.\n\\ee\nThe matrix $Q$ defined by $q(z)=Q\\chi(z)$ is the unique solution\n(modulo signs)\n to the skew-Borel decomposition of $m_{\\iy}$:\n\\be\nm_{\\iy}(t)=Q^{-1}JQ^{\\top -1}, ~~\\mbox{with}~~Q\\in \\Bk. \\ee The\nmatrix $L=Q\\Lb Q^{-1}$, also defined by $$z q(t,z)=L q(t,z),$$\n and the diagonal matrix $ h$ satisfy the equations\n\\be\n\\frac{\\pl L}{\\pl t_i}=\\left[-\\pi_{\\Bk}L^i, L \\right].\n~~\\mbox{and}~~\n h^{-1}\\frac{\\pl h}{\\pl t_i}\n=2\\pi_{\\Bk} (L^i)_0 .\n\\ee\n\\end{theorem}\n\n\\underline{\\sl Sketch of proof}: at first note that looking\n for skew-orthogonal polynomials is tantamount to the\n skew-Borel decomposition of $m_{\\iy}$, so that\n (4.9) and (4.10) are equivalent. The skew-orthogonality of\n the polynomials (0.17) follows from expanding the determinants\n explicitly in terms of $z$-columns, upon using the expression\n for the pfaffian in terms of a column\n $$\n\\sum_{0\\leq k\\leq\\ell-1}(-1)^k a_{ ki}\npf(0,\\dots,\\hat k,\\dots,\\ell-1)=pf(0,\\dots,\\ell-1,i).\n$$\nFor details, see \\cite{AHV}. On the other hand, Theorem 3.2 gives\n$\\Psi(t,z)$ and hence $Q$ in terms of $\\tau_n(t)$ of\n (4.7). By the uniqueness of the decomposition (4.10),\n the two ways of arriving at $Q$, (0.16) and (3,3) must coincide.\\qed\n\n\\vspace{1.2cm}\n\n\\noindent \\underline{\\sl Important remark}: The polynomials\n (0.16) provide an explicit algorithm to perform the skew-Borel\n decomposition of the skew-symmetric matrix $m_{\\iy}$. Namely,\n the coefficients of the polynomials $q_i$ provide the entries\n of the matrix $Q$. This fact will be used later in the examples.\n\n\\vspace{2cm}\n\n\n\\noindent{\\bf Symmetric matrix integrals}\nHere we shall focus on integrals of the type\n \\be\n \\int_{{\\cal S}_{2n}}\n e^{Tr~(- V(X)+\\sum_1^{\\iy} t_i X^i )} dX,\n \\ee\nwhere $dX$ denotes Haar measure\n\\be\ndX:=\\displaystyle{\\prod_{1\\leq i\\leq j\\leq n}}d\\Re X_{ij}=\|\\Delta\n(z)\|dz_1\\cdots dz_n~dU,\n\\ee\nover the space $S_{2n}$ of symmetric matrices. As will appear\nbelow, the integral (4.12) leads to\n%The weight $\\rho(dz):=e^{-V(z)}dz$\n%on $\\BR$ defines\na skew-inner-product with weight\n$\\rho_t(z)dz:=e^{-V_t(z)}dz:=e^{-V(z)+\\sum t_i z^i}dz=e^{\\sum t_i\nz^i}\\rho(z)dz$ on an interval $\\subseteq\\BR$, as in (0.1),\n\\be\n\\la f(x),g(y)\\ra:= \\int\\!\\int_{\\BR^2}f(x)g(y) \\vr(x-y) \\rho_t(dx)\\rho_t(dy)\n\\ee\nand therefore skew-symmetric moments\\footnote{$\\vr(x)=1$, for\n$x\\geq 0$ and $=-1$, for $x<0$.}\n\\bea\n\\mu_{ij}(t)&=&\\int\\!\\int_{\\BR^2}x^iy^j\\vr(x-y)\n\\rho_t(x)\\rho_t(y)dxdy\n\\nonumber\\\\\n&=&\\int\\!\\int_{x\\geq y}(x^iy^j-x^jy^i)\\rho_t(x)\\rho_t(y)dxdy\n\\nonumber\\\\ &=&\\int_{\\BR}\\left(F_j(x)G_i(x)-F_i(x)G_j(x)\\right)dx.\n\\eea\nwhere\n$$ F_i(x):=\\int^x_{-\\iy}y^i\\rho_t(y)dy\n\\quad\\mbox{and}\n\\quad G_i(x):=F_i'(x)=x^i\\rho_t(x).\n$$\nBy simple inspection, the moments\n $\\mu_{k\\ell}(t)$ satisfy\n\\bean\n\\frac{\\pl \\mu_{k\\ell}}{\\pl\nt_i}&=&\\int\\!\\int_{\\BR^2}(x^{k+i}y^{\\ell}+x^ky^{\\ell\n+i})\\vr(x-y)\\rho_t(x)\\rho_t(y)dx dy\\\\\n&=&\\mu_{k+i,\\ell}+\\mu_{k,\\ell+i},\n\\eean\nand so $m_{\\iy}$ satisfies (4.5).\n%$$\n%\\frac{\\pl m_{\\iy}}{\\pl t_i}=\\Lb^im_{\\iy}+m_{\\iy}\\Lb^{\\top i}.\n%$$\n\n\n\n\nAccording to Mehta \\cite{M}, the symmetric matrix integral can now\nbe expressed in terms of the pfaffian, as follows, taking into\naccount a constant $c_{2n}$, coming from integrating the orthogonal\ngroup:\n\n\\bean\n\\lefteqn{\\frac{1}{(2n)!}\\int_{{\\cal S}_{2n}(E)}\ne^{Tr~( -V(X)+\\sum t_i X^i) } dX}\n\\\\\n&=&\\frac{1}{(2n)!}\\int_{\\BR^{2n}}\|\\Delta_{2n}(z)\|\n\\prod_{i=1}^{2n}\n\\rho_t(z_i)dz_i\\\\\n&=&\\int_{-\\iy<z_1<z_2<\\cdots<z_{2n}<\\iy}\\det\\left(z_{j+1}^i\n\\rho_t(z_{j+1})\\right)_{0\\leq i,j\\leq 2n-1}\\prod_{i=1}^{2n}\ndz_i,\\\\ &=&\\int_{-\\iy<z_2<z_4<\\cdots<z_{2n}<\\iy}\n\\prod^n_{k=1}\\rho_t(z_{2k})dz_{2k} \\\\ & &\n\\det\\left(\\int_{-\\iy}^{z_2}z_1^i\\rho_t(z_1)dz_1~,~z_2^i~,\\dots,~\n\\int_{z_{2n-2}}^{z_{2n}}z_{2n-1}^i\\rho_t(z_{2n-1})dz_{2n-1}~,\n~z_{2n}^i\n\\right)_{0\\leq i\\leq 2n-1}\\\\\n&=&\\int_{-\\iy<z_2<z_4<\\cdots<z_{2n}<\\iy}\n\\prod^n_{k=1}\\rho_t(z_{2k})dz_{2k}\\\\ & &\n\\det\\left(F_i(z_2)~,~z_2^i~,~F_i(z_4)-F_i(z_2)~,~\nz_4^i~,~\\dots~,~F_i(z_{2n})-F_i(z_{2n-2})~,~z^i_{2n}\\right)_{0\\leq\ni\\leq 2n-1}\\\\\n &=&\\int_{-\\iy<z_2<z_4<\\cdots<z_{2n}<\\iy}\\prod^n_1dz_i~~\n\\det\\Bigl(F_i(z_2)~,~G_i(z_2)~,~\\dots,~F_i(z_{2n})~,~\nG_i(z_{2n})\\Bigr)_{0\\leq i\\leq 2n-1},\\\\\n &=&\\frac{1}{n!}\\int_{\\BR^n}\\prod^n_1dy_i~~\n\\det\\Bigl(F_i(y_1)~,~G_i(y_1)~,~\\dots,~F_i(y_n)~,~\nG_i(y_n)\\Bigr)_{0\\leq i\\leq 2n-1},\\\\\n&=&{\\det}^{1/2}\\left(\\int_{\\BR}\n (G_i(y)F_j(y)-F_i(y) G_j(y))dy\\right)_{0\\leq\ni,j\\leq 2n-1}\\\\\n & &\\mbox{\\hspace{5cm} using de\nBruijn's Lemma \\cite{M},p.446},\\\\ &=& pf\n \\left(\\mu_{ij}\\right)_{0\\leq\ni,j\\leq 2n-1} \\\\ &=& \\tau_{2n}(t),\n\\eean\nwhich is a Pfaffian $\\tau$-function.\n\n\n\\noindent{\\bf Symplectic matrix integrals}: Here we shall concentrate on integrals of the type\n \\be\n \\int_{{\\cal T}_{2n}}\n e^{2Tr~(- V(X)+\\sum_1^{\\iy} t_i X^i )} dX,\n \\ee\nwhere $dX$ denotes Haar measure\\footnote{$\\bar X$ means the usual complex\nconjugate. The condition on the $2\\times 2 $ matrices $X_{k \\ell}$\nimplies that $X_{kk}=X_k I$, with $X_k\\in\n\\BR$ and $I$ the identity.}\n$$\ndX=\\prod^N_1 dX_k \\prod_{k \\leq \\ell}\n dX_{k \\ell}^{(0)}\\bar {dX_{k \\ell}^{(0)}}\n dX_{k \\ell}^{(1)} \\bar {dX_{k \\ell}^{(1)}},\n $$\n on the space $\n\\TR_{2N}$ of self-dual $N\\times N$ Hermitean matrices, with quaternionic\nentries; the latter can be realized as the space of $2N\n\\times 2N$ matrices with entries $X^{(i)}_{\n\\ell k}\\in \\BC$\n$$\n\\TR_{2N}=\\left\\{ X=(X_{k \\ell})_{1\\leq k ,\\ell \\leq N} ,\nX_{k \\ell}=\\pmatrix{X^{(0)}_{k \\ell}&X^{(1)}_{k \\ell}\\\\ \\cr\n -\\bar X^{(1)}_{k \\ell}&\\bar X^{(0)}_{k \\ell} }\n \\mbox{ with }~X_{ \\ell k}=\\bar X_{k \\ell}^{\\top}\n\\right\\},\n$$\n\n\nA more exotic\n skew-symmetric matrix $m_{\\iy}$ satisfying (4.5) is given by the\n moments, with $V(y,t)=e^{2 (-V(y)+\\sum t_{\\al}y^{\\al})}$,\n\\begin{eqnarray}\n \\mu_{ij}(t)&=&\\int_{\\BR}\n \\{y^i,y^j\\}e^{2 (-V(y)+\\sum t_{\\al}y^{\\al})}I_E(y)dy\\nonumber\\\\\n &=& \\int_{\\BR}\n \\{y^i e^{ V(y,t)},y^j e^{ V(y,t)}\\} I_E(y)dy\\nonumber\\\\\n &=& \\int_{\\BR}\n (G_i(y)F_j(y)-F_i(y) G_j(y))dy,\n \\end{eqnarray}\nupon setting\n$$\n F_j(x)=x^j e^{V(x,t)} ~~\\mbox{and}~~\n G_j(x):=F'_j(x)=\\left(x^j e^{V(x,t)}\\right)' .\n$$\nThat $m_{\\iy}$ satisfies (4.5) follows at once from the\n first expression (4.17) above.\n\n\\bean\n\\mu_{k\\ell}(t)&=&\\int\\{y^k,y^{\\ell}\\}\\rho_t(y)^2 dy\\\\\n&=&\\int(k-\\ell)y^{k+\\ell-1}\\rho_t(y)^2 dy\\\\\n \\frac{\\pl\\mu_{k\\ell}}{\\pl t_i}&=&2\\int\\{y^k,y^{\\ell}\\}\n y^ie^{2(-V(y)+\\sum t_iy^i)}dy\\\\\n &=&\\int\\left((k+i-\\ell)y^{k+i+\\ell\n-1}+(k-\\ell-i)y^{k+i+\\ell -1}\\right) \\rho_t(y)^2 dy\\\\\n&=&\\mu_{k+i,\\ell}+\\mu_{k,\\ell +i},\n\\eean\nthus leading to (4.5). Using the relation\n$$\n\\prod_{1\\leq i,j\\leq n}(x_i-x_j)^4=\\det\\left(x_1^i~~~(x_1^i)'~~~x_2^i~~~(x_2^i)'~\\dots~\n~x_n^i~~~(x_n^i)'\\right)_{0\\leq i\\leq 2n-1},\n$$\none computes, using again de Bruijn's Lemma,\n\n\n\\noindent $\\displaystyle{\\frac{1}{(n)!}\n\\int_{{\\cal T}_{2n}}e^{2~Tr\n (-V(X)+\\sum t_i X^i )} dX}$\n \\bean\n&=&\\frac{1}{n!}\\int_{\\BR^n}\\prod_{1\\leq i,j\\leq n}(x_i-x_j)^4\n \\prod_{i=1}^{n}\\left(e^{-2V(x,t)} dx_i \\right)\\\\\n &=&\\frac{1}{n!}\\int_{\\BR^n}\n\\prod^n_{k=1}\\left(dx_{k} e^{-2V(x_{k},t)}dx_k\n \\right)\\\\ & &\n\\hspace{3cm}\\det\\left(x_1^i~~~(x_1^i)'~~~x_2^i~~~(x_2^i)'~\\dots~\n~x_n^i~~~(x_n^i)'\\right)_{0\\leq i\\leq 2n-1}\\\\\n&=&\\frac{1}{n!}\\int_{\\BR^n}\\prod^n_1dy_i~~\n\\det\\Bigl(F_i(y_1)~~G_i(y_1)~~\\dots~~F_i(y_n)~~\nG_i(y_n)\\Bigr)_{0\\leq i\\leq 2n-1},\\\\\n %& &\\mbox{\\hspace{5cm} using de\n%Bruijn's Lemma \\cite{M1},p.446},\\\\\n&=& {\\det}^{1/2} \\left(\\int_{\\BR}\n (G_i(y)F_j(y)-F_i(y) G_j(y))dy\\right)_{0\\leq\ni,j\\leq 2n-1}\\\\ &=&pf \\left(\\mu_{ij}\\right)_{0\\leq i,j\\leq 2n-1}\\\\\n\\\\ &=& \\tau_{2n(t)},\\eean\nwhich is a Pfaffian $\\tau$-function as well.\n\n\n\n\n\n\n\\section{A map from the Toda to the Pfaff lattice}\n\nRemember from (0.1), the notations $\\rho_t(z)=\\rho (z)\n e^{\\sum t_k z^k}$,\nand $\\rho'/\\rho =-g/f$.\n Assuming, in addition, $f(z)\\rho(z)$ vanishes at the endpoints\n of the interval under consideration (which could be finite, infinite\n or semi-infinite), one checks the the $t$-dependent\n operator in $z$,\n\\bea\n\\Bn_t:&=&\\sqrt{\\frac{f}{\\rho_t}}\\frac{d}{dz}\n\\sqrt{f\\rho_t}\\nonumber\\\\\n&=&e^{-\\frac{1}{2}\\sum t_k\nz^k}\\left(\\frac{d}{dz}f(z)-\\frac{f'+g}{2}(z)\\right)\ne^{\\frac{1}{2}\\sum t_k z^k}\\nonumber\\\\ &=&\n \\frac{d}{dz}f(z)-\\frac{f'+g_t}{2}(z),~~\\mbox{with}~\n g_t(z)=g(z)-f(z)\\sum_1^{\\iy}kt_k z^{k-1},\\nonumber\\\\\n\\eea\nmaintains $\\HR_+=\\{1,z,z^2,...\\} $ and is skew-symmetric with\nrespect to the\n $t$-dependent inner-product $\\la ~,\\ra_t^{sy}$, defined by\n the weight $\\rho_t(z)dz$,\n $$\n \\la \\Bn_t\\varphi,\\psi\\ra_t^{sy}\n =\\int_E (\\Bn_t\\varphi )(z)\\psi(z)\\rho_t(z) dz=\n -\\int_E \\varphi (\\Bn_t\\psi)\\rho_t dz\n =-\\la \\varphi, \\Bn_t\\psi\\ra_t^{sy}.\n $$\n The orthonormality of the $t$-dependent\n polynomials $p_n(t,z)$ in $z$ imply\n$$\\la p_n(t,z), p_m(t,z)\\ra_t^{sy}=\\dt_{mn}.$$\n The matrices $L$ and $M$ are defined by\n $$\n zp=Lp~~~\\mbox{and}~~~\n e^{-\\frac{1}{2}\\sum t_k z^k}\\frac{d}{dz}\n e^{\\frac{1}{2}\\sum t_k z^k}p=Mp.\n $$\n The skewness of $\\Bn_t$ implies the skew-symmetry\n of the matrix\n \\be\n\\NR(t)=f(L)M-\\frac{f'+g}{2}(L),~~\\mbox{such that} ~~\\Bn_t p(t,z)\n=\\NR p(t,z);\n\\ee\nso $\\NR(t)$ can be viewed as the operator $\\Bn_t$, expressed in the\npolynomial basis $(p_0(t,z),p_1(t,z),...)$.\n\nIn the next theorem, we shall\n consider functions $F$ of two (non-commutative)\n variables $z$ and $\\Bn_t$ so that the (pseudo)-differential\n operator in $z$ and the matrix\n$$\\Bu_t:=F(z,\\Bn_t)~~\\mbox{and}~~\\UR:=F(L,\\NR), $$\nrelated by\\footnote{with the understanding that $F(L,\\NR)$\n reverses the order of $z, \\Bu$ in $F(z,\\Bu)$.}\n$$F(z,\\Bn_t)p(t,z)= F(L,\\NR)p(t,z),$$ are skew-symmetric as well. Examples of\n$F$'s are\\footnote{$\\{A,B\\}^{\\dag}=AB+BA.$}\n$$F(z,\\Bn_t):=\\Bn_t,~\\Bn_t^{-1} ~\\mbox{or}~\n\\{z^{\\ell},\\Bn_t^{2k+1}\\}^{\\dag},\n$$ corresponding to $$F(L,\\NR)=\n\\NR,~~\\NR^{-1}~\\mbox{or}~~\n\\{\\NR^{2k+1},L^{\\ell}\\}^{\\dag}.$$\n\n\n\\begin{theorem}\nAny H\\\"ankel matrix $m_{\\iy}$ evolving according to the vector\nfields\n$$\n\\frac{\\pl m_{\\iy}(t)}{\\pl t_k}=\\Lb^k m_{\\iy}\n$$\nleads to matrices $L$ and $M$, evolving according to the Toda\nlattice (1.9). Consider a function $F$ of two variables,\n such that the operator $\\Bu_t:=F(z,\\Bn_t)$ is skew-symmetric\n with respect to $\\la ~,\\ra_t^{sy}$ and so the matrix\n $$\\UR(t)=F(L(t),\\NR(t)),~~\n \\mbox{defined by}~~\\Bu_t p(t,z)=\\UR p(t,z)\n $$ is skew-symmetric. This induces a natural\n lower-triangular matrix $O(t)$, mapping the Toda\n lattice into the Pfaff lattice:\n\n\\vspace{.5cm}\n$$\n\\mbox{Toda lattice}\\left\\{\\begin{tabular}{l}\n$p_n(t,z)=\\left(S(t)\\chi(z)\\right)_n ~~\n \\mbox{orthonormal with respect to} $\\\\\n $\\hspace{4cm}~m_{\\iy}(t)=\\left( \\la z^i,z^j\\ra^{sy}_t\n \\right)_{0\\leq i,j \\leq \\iy}=S^{-1}S^{\\top -1}$\n \\\\\n \\\\\n $L(t)=S\\Lb S^{-1} ~~\\mbox{satisfies}~\\quad\\displaystyle{\\frac{\\pl L}{\\pl t_j}\n =\\left[-\\frac{1}{2}\\pi_{bo} L^j,\nL\\right],~j=1,2,...}$\\\\\n\\end{tabular}\n\\right.\n$$\n$$\\quad\\left\\downarrow\n\\vbox to1.1in{\\vss}\\right.\n\\mbox{ map $O(2t)$ such that }\n\\left\\{\n\\begin{array}{l}\n -\\UR(2t)= O^{-1}(2t)JO^{\\top -1}(2t)\\\\[6pt]\n O(2t) \\mbox{ is lower-triangular } \\\\[6pt]\n O(2t)S(2t) \\in \\GR_{\\Bk}\n\\end{array}\n\\right.\n.\n$$\n$$\n\\mbox{Pfaff lattice}\\left\\{\\begin{tabular}{l}\n $q_n(t,z)=%\\left(\\tilde Q(t)\\chi(z)\\right)_n=\n \\left(O(2t)p(2t,z)\\right)_n,\n \\mbox{skew-orthonormal with regard to }$\\\\\n\\\\\n \\hspace{1.4cm}$\n \\tilde m_{\\iy}(t):=-S^{-1}(2t)\\UR(2t) S^{\\top\n-1}(2t)=Q^{-1}(t)JQ^{\\top -1}(t)$\\\\ \\\\\n$ \\hspace{2.8cm}=\\left( \\la z^i, z^j\\ra^{sk}_t \\right)\n_{0\\leq i,j \\leq \\iy} $\\\\ \\\\\n \\hspace{2.7cm} $=\\left( \\la z^i, \\Bu_{2t}z^j\\ra^{sy}_{2t} \\right)\n_{0\\leq i,j \\leq \\iy}\n$\\\\\n \\\\\n \\\\\n $\\tilde L(t):=O(2t)L(2t)O(2t)^{-1}~~\\mbox{satisfies} ~\n\\displaystyle{\\frac{\\pl \\tilde L}{\\pl t_j}}=[-\\pi_{\\Bk}\\tilde\nL^j,\\tilde L]~,j=1,...$\\\\\n\\end{tabular}\n\\right.\n$$\n\\end{theorem}\n\n\n\n\n\n\n\n\n\\proof Since $\\UR(t)$ is skew-symmetric, it admits a\n skew-Borel decomposition\n\\be\n-\\UR(t)=O^{-1}(t)JO^{\\top -1}(t),~~\\mbox{with lower-triangular }O(t)\n.\\ee\nBut the new matrix, defined by\n\\be\n\\tilde m_{\\iy}(t):=-S^{-1}(2t)\\UR(2t)S^{\\top -1}(2t),\n\\ee\nis skew-symmetric and thus admits a unique skew-Borel decomposition\n\\be\n\\tilde m_{\\iy}(t)=\\tilde Q^{-1}(t)J\\tilde\nQ(t)^{\\top -1}~~\\mbox{with}~~\\tilde Q(t)\\in \\GR_{\\Bk}.\n\\ee\nComparing (5.3), (5.4) and (5.5) leads to a unique choice of matrix\n$O(t)$, skew-Borel decomposing $-\\UR(2t)$, as in (5.3), such that\n\\be\nO(2t)S(2t)=\\tilde Q(t) \\in \\GR_{\\Bk}.\n\\ee\nUsing\n$$\n\\frac{\\pl \\UR}{\\pl t_k}(2t)=[\\pi_{sy}L^k(2t),\\UR(2t)]\n$$\nand\n$$\n\\frac{\\pl S}{\\pl t_k}(2t)=-(\\pi_{bo} L^k(2t))S(2t),\n$$\nwe compute\n\n\\medbreak\n\n\\noindent $\\displaystyle{\\frac{\\pl\\tilde m_{\\iy}}{\\pl t_k}(t)}$\n\\begin{eqnarray}\n&=&S^{-1}\\frac{\\pl S}{\\pl t_k}(2t)S^{-1}\\UR(2t)S^{\\top\n-1}(2t)-S^{-1}(2t)\\left(\\frac{\\pl}{\\pl t_k}\\UR(2t)\\right)S^{\\top\n-1}(2t)\\\\ & &\\hspace{2cm}+S^{-1}(2t)\\UR(2t)S^{\\top -1}\\frac{\\pl\nS^{\\top}}{\\pl t_k} (2t)S^{\\top -1}\\\\ &=&-S^{-1}(\\pi_{bo} L^k(2t))\n\\UR S^{\\top\n-1}-S^{-1}[\\pi_{sy}L^k,\\UR]S^{\\top -1}-\nS^{-1}\\UR(\\pi_{bo} L^k)^{\\top}S^{\\top -1}\\\\\n &=&-S^{\n-1}(\\pi_{bo}L^k+\\pi_{sy} L^k)\\UR S^{\\top\n-1}-S^{-1}\\UR ((\\pi_{bo}L^k)^{\\top}-\\pi_{sy}L^k)S^{\\top -1}\\\\\n&=&-S^{ -1}L^k \\UR S^{\\top -1}-S^{ -1}\\UR L^{\\top k}S^{\\top\n-1},\\,\\,\\mbox{using (5.6) below}\\\\ &=&-\\Lb^kS^{ -1}\\UR S^{\\top -1}-S^{\n-1}\\UR S^{\\top -1}\\Lb^{\\top k}S^{\\top}S^{\\top\n-1},\\,\\,\\mbox{using\\,\\,}L^k=S\\Lb^kS^{ -1},\\\\\n&=&\\Lb^k\\tilde m_{\\iy}(t)+\\tilde m_{\\iy}(t)\\Lb^{\\top k}.\n\\end{eqnarray}\n\nFor an arbitrary matrix $A$, we have\n\\be A=A^{\\top}\\Longleftrightarrow\nA=(A_{bo})^{\\top}-A_{sy}.\n\\ee\nIndeed, remembering that\\footnote{$A_{\\pm}$ means the usual\nstrictly upper(lower)-triangular part and $A_0$ the diagonal part\nin the common sense.}$A_{bo}=2A_-+A_0$ and $A_{sy}=A_+-A_-$, one\nchecks\n$$\n(A_{bo})^{\\top}-A_{sy}-A=\n2(A_-)^{\\top}+A_0-(A_+-A_-)-A_--A_+-A_0=-2(A_+-(A_-)^{\\top}).\n$$\nso that the left hand side vanishes, if the right hand side does;\nthe latter means A is symmetric.\n\nWe now define $\\tilde L(t)$ by conjugation of $L(2t)$ by $O(2t)$:\n$$\n\\tilde L(t):=O(2t)L(2t)O(2t)^{-1}=O(2t)S(2t)\\Lb\nS^{-1}(2t)O(2t)^{-1}=\\tilde Q(t)\\Lb\\tilde Q^{-1}(t).\n$$\nTherefore the sequence of polynomials\n$$\nq(t,z):=O(2t)p(2t,z)=O(2t)S(2t)\\chi(z)=\\tilde Q(t)\\chi(z)\n$$\nis skew-orthonormal\n$$\n\\la q_i(t,z),q_j(t,z)\\ra^{sk}=J_{ij}\n$$\nwith regard to the skew inner-product specified by the matrix $\\tilde m_{\\iy}$:\n$$\n\\la z^i,z^j\\ra_t^{sk}=\\tilde\\mu_{ij}(t).\n$$\nIn the last step, we show that\n $\\la\\vp,\\psi\\ra^{sk}=\\la\\vp,\\Bu\\psi\\ra^{sy}$. Since\n\\begin{eqnarray}\n\\UR(2t)&=&-O^{-1}(2t)JO^{\\top\n-1}(2t) ,\n%&=&-S(2t)\\tilde Q^{-1}(t)J\\tilde Q(t)^{\\top\n%-1}S^{\\top}(2t)\\nonumber\\\\\n%&=&-S(2t)\\tilde m_{\\iy}(t)S^{\\top}(2t),\\nonumber\\\\\n\\end{eqnarray}\nwe compute\n\\begin{eqnarray}\n \\la q_i(t,z),(\\Bu_{2t} q)_j(t,z)\\ra_{2t}^{sy}&=&\\left\\la (Op)_i(2t),\n (\\Bu Op)_j(2t)\\right\\ra_{2t}^{sy}\\nonumber\\\\\n &=&\\la (Op)_i(2t),(O\\Bu p)_j(2t)\\ra_{2t}^{sy}\\nonumber\\\\\n &=&\\la (Op)_i(2t),(O \\UR p)_j(2t)\\ra_{2t}^{sy}\\nonumber\n \\\\ &=&(O(2t)\\left\\la p_k(2t),p_{\\ell}(2t)\\right\\ra^{sy}_{k,\\ell\\geq\n 0}(O \\UR )^{\\top}(2t))_{ij}\\nonumber\n \\\\ &=&(O(2t)I(O \\UR )^{\\top}(2t))_{ij}\\nonumber\\\\\n &=&(O(2t)\\UR^{\\top}(2t)O^{\\top}(2t))_{ij}\\nonumber\\\\\n &=&-(O(2t)\\UR (2t)O^{\\top}(2t))_{ij}\\nonumber\\\\ &=&J_{ij}\n ,~\\mbox{using} ~(5.8).\n\\end{eqnarray}\nTherefore, defining a new skew inner-product $\\la \\, ,\\,\\ra^{\n sk^{\\prime}}$\n$$\n\\la\\vp,\\psi\\ra^{sk^{\\prime}}:=\\la\\vp,\\Bu \\psi\\ra^{sy},\n$$\nwe have shown\n$$\n\\la q_i,q_j\\ra_t^{sk^{\\prime}}=\\la q_i,q_j\\ra_t^{sk}=J_{ij},\n$$\nand so by completeness of the basis $q_i$, we have\n$$\n\\la ~,~\\ra_t^{sk^{\\prime}}=\\la ~,~\\ra_t^{sk},\n$$\nthus ending the proof of Theorem 4.1.\n\\qed\n\n\n\\medbreak\n\n\\section{Example 1: From Hermitean to symmetric matrix integrals}\n\nStriking examples are given by using the map $O(t)$ obtained from\nskew-borel decomposing $\\NR^{-1}(t)$ and $\\NR(t)$; see (5.2). This\nsection deals with $\\NR^{-1}(t)$, whereas the next will deal with\n$\\NR(t)$.\n\n\n\\begin{proposition}\n The special transformation\n$$\\UR(t)=\\NR^{-1}(t)=\\left(f(L)M-\\frac{f'+g}{2}(L)\\right)^{-1}(2t)$$\n maps the Toda lattice $\\tau$-functions with initial weight\\footnote{Remember\n $\\rho'/\\rho=-V'=-g/f$.}\n$\\rho=e^{-V},~V'=-g/f$ (Hermitean matrix integral) to the Pfaff\nlattice $\\tau$-functions (symmetric matrix integral), with initial\nweight\n%\\newpage\n $$\\tilde \\rho_t(z):=\\left( \\frac{\\rho_{2t}(z)}{f(z)} \\right)^{\\frac{1}{2}}\n =e^{-\\frac{1}{2}(V(z)+\\log\nf(z)-2\\sum_1^{\\iy} t_i z^i)}=:e^{-\\tilde V(z)+\\sum_1^{\\iy} t_i\nz^i}.$$ To be precise:\n\n\n$$\n\\mbox{Toda lattice}\\left\\{\\begin{tabular}{l}\n$p_n(t,z) ~~\\mbox{orthonormal polynomials in $z$\n for the inner-product\n}$ \\\\\\hspace{4cm} $\\la \\varphi,\\psi\\ra_t^{sy}=\n=\\displaystyle{\\int \\varphi(z)\\psi(z)\n \\rho_t(z)dz} $,\\\\\n$\\mu_{ij}(t)=\\la z^i , z^j \\ra^{sy}_t $ and\n %\\displaystyle{\\int z^{i+j}\n %e^{\\sum^{\\iy}_1 t_kz^k}\\rho(z)dz,\\quad\n $ m_n=(\\mu_{ij})_{0\\leq i,j\\leq n-1},$\\\\ $\\tau_n(t)=\\det\nm_n=\\displaystyle{\\frac{1}{n!}\\int_{\\HR_n}\ne^{Tr(-V(X)+\\sum_1^{\\iy}t_iX^i)}dX}$\n\\\\\n\n\\end{tabular}\n\\right.\n$$\n$$\\quad\\left\\downarrow\\vbox to1.1in{\\vss}\\right.\n\\mbox{map $O(2t)$ such that}\\\n\\left\\{\n\\begin{array}{l}\n -\\NR^{-1}(2t)= O^{-1}(2t)JO^{\\top -1}(2t)\\\\[6pt]\n O(2t) \\mbox{ is lower-triangular } \\\\[6pt]\n O(2t)S(2t) \\in \\GR_{\\Bk}\n\\end{array}\n\\right.\n$$\n$$\n\\mbox{Pfaff lattice}\\left\\{\\begin{tabular}{l}\n $q_n(t,z)=O (2t) p_n(2t,z) ~~\\mbox{skew-orthonormal\n polynomials}$\\\\\n\\hspace{3.0cm}$\\mbox{ in\n$z$ for the skew-inner-product (weight $\\tilde \\rho$}$),\n \\\\\n\n\\\\\n \\hspace{1cm}$\n \\la \\varphi,\\psi \\ra_t^{sk}\n :=\\la \\varphi,\\Bn_{2t}^{-1}\\psi \\ra_{2t}^{sy}$\\\\ \\\\\n$ \\hspace{2.6cm}=\\displaystyle{\\frac{1}{2}\n \\int\\!\\int_{\\BR^2}\\varphi(x) \\psi(y)\n \\vr(x-y) \\tilde\\rho_t(x) \\tilde\\rho_t(y)dx\\,dy}\n $\\\\ \\\\\n\n\n $\\tilde\n\\mu_{ij}(t)=\\la x^i, y^j\\ra ^{sk}_t$ and\n $\\tilde m_n=(\\tilde\\mu_{ij})_{0\\leq i,j\\leq n-1}$\n \\\\\n \\\\\n$\\tilde\\tau_{2n}(t)=pf(\\tilde\nm_{2n})=\\displaystyle{\\frac{1}{(2n)!}\\int_{{\\cal\nS}_{2n}}e^{Tr(-\\tilde V(X)+\\sum_1^{\\iy}t_iX^i)}dX}.$\n\n\\end{tabular}\n\\right.\n$$\n\\end{proposition}\n\n\\bigbreak\n\nIn the first integral defining $\\tau_n(t)$, $dX$ denotes Haar\nmeasure on Hermitean matrices (see section 4.1), whereas the second\nintegral $\\tilde\\tau_{2n}(t)$ involves Haar measure on symmetric\nmatrices (see section 4.2)\n\n\n\\proof At first, check that\n\\be\n\\left(\\frac{d}{dx}\\right)^{-1}\\vp(x)=\\frac{1}{2}\\int\\vr(x-y)\\vp(y)dy.\n\\ee\nIndeed,\n\\begin{eqnarray}\n\\frac{d}{dx}\\left(\\frac{d}{dx}\\right)^{-1}\\vp(x)&=&\\int\\frac{1}{2}\\frac{\\pl}{\\pl x}\n\\vr(x-y)\\vp(y)dy\\\\\n&=&\\int\\delta(x-y)\\vp(y)dy~~\\mbox{using}~~\\frac{\\pl}{\\pl x}\\vr(x)=2\n\\dt(x)\\\\ &=&\\vp(x).\n\\end{eqnarray}\nConsider now the operator\n$$\n\\Bu_t=\\Bn_t^{-1}=\\left(\\sqrt{\\frac{f}{\\rho_t}}\\frac{d}{dz}\n \\sqrt{ f \\rho_t}\\right)^{-1},~\\mbox{so that }~\n \\Bu_tp=\\Bn_t^{-1}p=\\NR^{-1}p ,\n$$\naccording to (5.2). Let it act on a function $\\vp(x)$:\n\\begin{eqnarray}\n\\Bn_t^{-1}\\vp (x)%&=&\\left(\\frac{d}{dx}f(x)-\\frac{f'+g}{2}\\right)^{-1}\\vp(x)\\\\\n%&=&\\left(\\sqrt{\\frac{f(x)}{\\rho(x)}}\\frac{d}{dx}\\sqrt{f(x)\\rho(x)}\\right)^{-1}\\vp(x)\\\\\n&=&\\left(\\frac{1}{\\sqrt{f(x)\\rho_t(x)}}\\left(\\frac{d}{dx}\\right)^{-1}\n\\sqrt{\\frac{\\rho_t(x)}{f(x)}}\\right)\\vp(x)\\\\\n&=&\\int_{\\BR}\\frac{1}{\\sqrt{f(x)\\rho_t(x)}}\\frac{\\vr(x-y)}{2}\n\\sqrt{\\frac{\\rho_t(y)}{f(y)}}\\vp(y)dy,\n\\,\\,\\mbox{using (6.1)}.\n\\end{eqnarray}\nOne computes\n\\begin{eqnarray}\n\\la\\vp,\\psi\\ra_t^{sk}\n &=&\\la\\vp,\\Bu_{2t}\\psi\\ra_{2t}^{sy}\\\\\n &=&\\la\\vp,\\Bn^{-1}_{2t}\\psi\\ra_{2t}^{sy}\\\\\n&=&\\frac{1}{2}\\int\\!\\int_{\\BR^2}\\sqrt{\\frac{\\rho_{2t}(x)}{f(x)}}\n \\vr(x-y) \\sqrt{\\frac{\\rho_{2t}(y)}{f(y)}}\\vp(x)\\psi(y)dx\\,dy\\\\\n&=&\\frac{1}{2}\\int\\!\\int_{\\BR^2}\\tilde\\rho(x) \\tilde\\rho(y)\n e^{\\sum_1^{\\iy} t_k(x^k+y^k)}\\vr(x-y)\\vp(x)\\psi(y)dx\\,dy.\n\\end{eqnarray}\nSo, finally setting $\\tilde V(x)=\\frac{1}{2}(V(x)+\\log f(x))$\nyields\n$$\n\\tilde\\tau_{2n}(t)=pf(\\tilde m_{2n})=\\frac{1}{(2n)!}\\int_{{\\cal S}_{2n}}e^{Tr(-\\tilde\nV(X)+\\sum_1^{\\iy}t_iX^i)}dX.\n$$\n\n\\bigbreak\n\n\\noindent{\\bf The map $O$ for the\nclassical orthogonal polynomials at $t=0$}: {\\em Then, the matrix\n$O$, mapping orthonormal $p_k$ into skew-orthonormal polynomials\n$q_k$, is given by a lower-triangular three-step relation:\n\\bea\n q_{2n}(0,z)&=&\\sqrt{\\frac{c_{2n}}{a_{2n}}}p_{2n}(0,z)\\nonumber\\\\\n q_{2n+1}(0,z)&=&\\sqrt{\\frac{a_{2n}}{c_{2n}}}\\nonumber\\\\\n&&\n\\left(-c_{2n-1}p_{2n-1}(0,z)+\\frac{c_{2n}}{a_{2n}}(\\sum_0^{2n}b_i)\n p_{2n}(0,z)+c_{2n}p_{2n+1}(0,z)\\right)\\nonumber\\\\\n\\eea\nwhere the $a_i$ and $b_i$ are the entries in the tridiagonal matrix\ndefining the orthonormal polynomials, and the $c_i$'s are the\nentries of the skew-symmetric matrix $\\NR$.}\n\n\nIn \\cite{AvM2}, we showed that then $\\NR$ is tridiagonal, at the\nsame time as $L$, (see Appendix 2)\n\\be\nL=\\left[\\begin{array}{ccccc}\nb_0&a_0& & & \\\\\na_0&b_1&a_1& & \\\\\n &a_1&b_2&\\ddots&\\\\\n & &\\ddots&\\ddots&\n\\end{array}\n\\right],\\quad -\\NR=\\left[\\begin{array}{ccccc}\n0&c_0& & & \\\\\n-c_0&0&c_1& & \\\\\n &-c_1&0&\\ddots&\\\\\n & &\\ddots& &\n\\end{array}\\right]\\,,\n\\ee\n with the following precise entries:\n\n\\noindent \\underline{Hermite}: $\\rho(z)=e^{-z^2}$,\n$a_{n-1}=\\sqrt{\\frac{n}{2}}$, $b_n=0$, $c_n=a_n$\n\n\\noindent \\underline{Laguerre}: $\\rho(z)=e^{-z}z^{\\al}I_{[0,\\iy)}(z)$, $a_{n-1}=\\sqrt{n(n+\\al)}$,\n$b_n=2n+\\al+1$, $c_n=a_n /2$\n\n\\vspace{0.4cm}\n\n\\noindent \\underline{Jacobi}: $\\rho(z)=(1-z)^{\\al}(1+z)^{\\rho}I_{[-1,1]}(z)$\n\n\\bea\na_{n-1}&=&\\left(\\frac{4n(n+\\al+\\beta)(n+\\al)(n+\\beta)}{(2n+\\al+\\beta)^2(\n2n+\\al+\\beta+1)(2n+\\al+\\beta-1)}\\right)^{1/2}\\nonumber\\\\\nb_n&=&\\frac{\\al^2-\\beta^2}{(2n+\\al+\\beta)(2n+\\al+\\beta-2)}\\nonumber\\\\\n c_n&=&a_n\\left( \\frac{\\alpha+\\beta}{2}+n+1 \\right)\\nonumber\n \\eea\n If the skew-symmetric matrix $\\NR$ has the\ntridiagonal form above, then one checks its inverse has the\nfollowing form:\n\\be\n-\\NR^{-1}=\\left(\\begin{array}{ccccccccc}\n0&-\\frac{1}{c_0}&0&\\frac{-c_1}{c_0c_2}&0&\n\\frac{-c_1c_3}{c_0c_2c_4}&0&\n\\frac{-c_1c_3c_5}{c_0c_2c_4c_6}&\\\\\n\\frac{1}{c_0}&0&0&0&0&0&0&0&\\\\\n0&0&0&-\\frac{1}{c_2}&0&\\frac{-c_3}{c_2c_4}&0&\\frac{-c_3c_5}{c_2c_4c_6}\\\\\n\\frac{c_1}{c_0c_2}&0&\\frac{1}{c_2}&0&0&0&0& & \\\\\n0&0&0&0&0&-\\frac{1}{c_4}&0&\\frac{-c_5}{c_4c_6} &\\\\\n\\frac{c_1c_3}{c_0c_2c_4}&0&\\frac{c_3}{c_2c_4}&0&\\frac{1}{c_4}&0&0&0\\\\\n0&0&0&0&0&0&0&-\\frac{1}{c_6}& \\\\\n\\frac{c_1c_3c_5}{c_0c_2c_4c_6}&0&\\frac{c_3c_5}{c_2c_4c_6}\n&0&\\frac{c_5}{c_4c_6}&0&\\frac{1}{c_6}&0 &\\\\ &&&&&&&&\\ddots\n\\end{array}\n\\right).\n\\ee\nIn order to find the matrix $O$, we must perform the skew-Borel\ndecomposition of the matrix $-\\UR$\n$$-\\UR=-\\NR^{-1}=O^{-1}JO^{\\top -1}.\n$$\nThe recipe for doing so was given in theorem 4.1 (see also the\nimportant remark, following that theorem). It suffices to form the\npfaffians (0.17), by appropriately bordering the matrix\n$-\\NR^{-1}$, as in (0.17), with rows and columns of powers of $z$,\nyielding monic skew-orthogonal polynomials; we choose to call them\n$r$'s, instead of the $q$'s of theorem 4.1,\n with $O\\chi(z)=r(z)$.\nThey turn out to be the following simple polynomials, with\n$1/\\tilde{\\tilde\\tau}_{2n}=c_0c_2c_4\\cdots c_{2n-2}$\n\\bea\n r_{2n}(z)&=&\\frac{1}{\\sqrt{\\tilde{\\tilde\\tau}_{2n}\\tilde{\\tilde\\tau}_{2n+2}}}\n \\frac{c_{2n}z^{2n}}{c_0c_2\\cdots c_{2n}}=\n \\frac{1}{\\sqrt{c_{2n}}}c_{2n}z^{2n}\\nonumber\\\\\n r_{2n+1}(z)&=&\n\\frac{1}{\\sqrt{\\tilde{\\tilde\\tau}_{2n}\\tilde{\\tilde\\tau}_{2n+2}}}\n\\frac{c_{2n}z^{2n+1}-c_{2n-1}z^{2n-1}}{c_0c_2\\cdots c_{2n}}=\n\\frac{1}{\\sqrt{c_{2n}}}(c_{2n}z^{2n+1}-c_{2n-1}z^{2n-1}).\\nonumber\n\\eea\nThen, also from appendix 1, in order to get\n $O \\rightarrow \\hat O$ in the correct form,\nwe compute the skew-orthonormal polynomials $\\hat r_k$,\n with $\\hat O \\chi(z)=\\hat r(z)$:\n\\bea\n\\hat r_{2n}(z)&=&\\frac{1}{\\sqrt{a_{2n}}}~r_{2n}(z)=\\sqrt{\\frac{c_{2n}}{a_{2n}}}\nz^{2n}\\nonumber\\\\\n\\hat r_{2n+1}(z)&=&\\frac{\\sum_0^{2n}b_i}{\\sqrt{a_{2n}}}~r_{2n}(z)\n+\\sqrt{a_{2n}} r_{2n+1}(z)\\nonumber\\\\\n &=&\\sqrt{\\frac{a_{2n}}{c_{2n}}}%\\nonumber\\\\\n %&&\n\\left(-c_{2n-1}z^{2n-1}+\\frac{c_{2n}}{a_{2n}}(\\sum_0^{2n}b_i)\n z^{2n}+c_{2n}z^{2n+1}\\right).\n\\nonumber\\\\\n\\eea\nFrom the coefficients of the polynomial $\\hat r_k$, one reads off\nthe transformation matrix from orthonormal to skew-orthonormal\npolynomials; it is given by the matrix $\\hat O $, such that\n $\\hat O\\chi(z)=\\hat r(z)$. Therefore\n $q(t,z)=\\hat O(2t)p(2t,z)$ yields, after setting $t=0$,\n\\bea\n q_{2n}(0,z)&=&\\sqrt{\\frac{c_{2n}}{a_{2n}}}p_{2n}(0,z)\\nonumber\\\\\n q_{2n+1}(0,z)&=&\\sqrt{\\frac{a_{2n}}{c_{2n}}}\\nonumber\\\\\n&&\n\\left(-c_{2n-1}p_{2n-1}(0,z)+\\frac{c_{2n}}{a_{2n}}(\\sum_0^{2n}b_i)\n p_{2n}(0,z)+c_{2n}p_{2n+1}(0,z)\\right),\\nonumber\\\\\n\\eea\nconfirming (6.2).\n\n\n\\medbreak\n\n\n\\section{Example 2: From Hermitean to symplectic matrix integrals}\n\n\\begin{proposition}\nThe matrix transformation\n$$\n\\NR=f(L)M-\\frac{f'+g}{2}(L),\n$$\n maps the Toda lattice $\\tau$-functions with $t$-dependent\n weight\n $$\\rho_t(z)=e^{-V(z)+\\sum_1^{\\iy}t_iz^i},~V'=g/f $$\n (Hermitean matrix\nintegral) to the Pfaff lattice $\\tau$-functions (Symplectic matrix\nintegral), with $t$-dependent weight\n $$\\tilde \\rho_t(z):=(\\rho_{2t}(z)f(z))^{\\frac{1}{2}}\n =e^{-\\frac{1}{2}(V(z)-\\log\nf(z)-2\\sum_1^{\\iy}t_i z^i)}=:e^{-\\tilde V(z)+\\sum t_i z^i}.\n $$\n To be precise:\n\n%\\newpage\n\n$$\n\\mbox{Toda lattice}\\left\\{\\begin{tabular}{l}\n$p_n(t,z) ~~\\mbox{orthonormal polynomials in\n $z$ for the inner-product }$\\\\\n\n\\hspace{3.0cm}$\\la\\varphi,\\psi\\ra^{sy}_t=\\int\n \\varphi (z)\\psi (z) \\rho_t(z)dz $ \\\\ \\\\\n\n$\\mu_{ij}(t)=\\la z^i,z^j\\ra^{sy}_t,$ and $ m_n=\n (\\mu_{ij})_{0\\leq i,j\\leq n-1},$\\\\\n$\\tau_n(t)=\\det\nm_n(t)=\\displaystyle{\\frac{1}{n!}\\int_{\\HR_n}e^{Tr(-V(X)+\\sum\nt_iX^i)}dX}$\n\\\\\n\\end{tabular}\n\\right.\n$$\n\n$$\\quad\\left\\downarrow\\vbox to1.1in{\\vss}\\right.\n\\mbox{map $O(2t)$ such that}\\\n\\left\\{\n\\begin{array}{l}\n -\\NR(2t)= O^{-1}(2t)JO^{\\top -1}(2t)\\\\[6pt]\n O(2t) \\mbox{ is lower-triangular } \\\\[6pt]\n O(2t)S(2t) \\in \\GR_{\\Bk}\n\\end{array}\n\\right.\n$$\n\n\n$$\n\\mbox{Pfaff lattice}\\left\\{\\begin{tabular}{l}\n$q_n(t,z)=O (2t) p_n(2t,z) ~~\\mbox{skew-orthonormal\n polynomials}$\\\\\n\\hspace{3.0cm}$\\mbox{ in\n$z$ for the skew-inner-product\n (weight $\\tilde \\rho_t$}$),\n \\\\\n\n\\\\\n \\hspace{1cm}$\n \\la \\varphi,\\psi \\ra_t^{sk}\n :=\\la \\varphi,\\Bn_{2t}\\psi \\ra_{2t}^{sy}$\\\\ \\\\\n$ \\hspace{2.6cm}=\\displaystyle{-\\frac{1}{2}\n \\int\\!\\int_{\\BR^2}\\{\\varphi(z), \\psi(z)\\}\n \\tilde\\rho^2_t(z) dz}\n $\\\\ \\\\\n\n$\\tilde\n\\mu_{ij}(t)=\\la z^i,z^j\\ra^{sk}_t$, and\n$\\tilde m_n=\\det(\\tilde\\mu_{ij})_{0\\leq i,j\\leq n-1}$\\\\\n$\\tilde\\tau_{2n}(t)=pf(\\tilde\nm_{2n}(t))=\\displaystyle{\\frac{1}{(-2)^n n!}\\int_{{\\cal\nT}_{2n}}e^{2Tr(-\\tilde V(X)+\\sum t_iX^i)}dX}.$\n\\\\\n\\end{tabular}\n\\right.\n$$\n\\end{proposition}\n\n\n\n\\bigbreak\n\n\\proof\nRepresenting $d/dx$ as an integral operator\n$$\n\\frac{d}{dx}\\vp(x)=\\int_{\\BR}\\delta(x-y)\\vp'(y)dy=-\\int_{\\BR}\\frac{\\pl}{\\pl\ny}\\delta(x-y)\\vp(y)dy=\\int_{\\BR}\\delta'(x-y)\\vp(y)dy,\n$$\ncompute\n$$\n\\Bu_t=\\Bn_t= \\sqrt{\\frac{f}{\\rho_t}}\\frac{d}{dz}\n \\sqrt{ f \\rho_t} ,\n~~\\mbox{so that ~$\\Bn_tp(t,z)=\\NR p(t,z) $;}\n$$ remember $\\NR$ from (5.2). Let it act on a function $\\vp(x)$:\n\\begin{eqnarray}\n\\Bu_t\\vp(x)\n %&=&\\left(\\frac{d}{dx}f(x)+\\frac{f'+g}{2}(x)\\right)\\vp(x)\\\\\n&=&\\left(\\sqrt{\\frac{f}{\\rho_t}}\\frac{d}{dx}\\sqrt{f\\rho_t}\\right)\\vp(x)\\\\\n&=&\\int_{\\BR}\\sqrt{\\frac{f(x)}{\\rho_t(x)}}\\delta'(x-y)\n\\sqrt{f(y)\\rho_t(y)}\\vp(y)dy.\n\\end{eqnarray}\nThen\n\\begin{eqnarray}\n\\la\\vp,\\psi\\ra_t^{sk}&=&\\la\\vp,\\Bu_{2t}\\psi\\ra_{2t}^{sy}\n =\\la\\vp,\\Bn_{2t}\\psi\\ra_{2t}^{sy}\\\\\n&=&\\int\\!\\int_{\\BR^2}\\rho_{2t}(x)\\vp(x)\\sqrt{\\frac{f(x)}\n{\\rho_{2t}(x)}}\\delta'(x-y)\\sqrt{f(y)\\rho_{2t}(y)}\n\\psi(y)dx\\,dy\\\\\n&=&\\int\\!\\int_{\\BR^2}\\sqrt{f(x)\\rho_{2t}(x)}\\vp(x)\n\\delta'(x-y)\\sqrt{f(y)\\rho_{2t}(y)}\\psi(y)dx\\,dy\\\\\n&=&-\\int\\!\\int_{\\BR^2}\\left(\\frac{\\pl}{\\pl\nx}\\sqrt{f(x)\\rho_{2t}(x)}\\vp(x)\\right)\\delta(x-y)\n\\sqrt{f(y)\\rho_{2t}(y)}\\psi(y)dx\\,dy\\\\\n&=&-\\int_{\\BR}\\left(\\frac{\\pl}{\\pl\nx}\\sqrt{f(x)\\rho_{2t}(x)}\\vp(x)\\right)\\sqrt{f(x)\\rho_{2t}(x)}\\psi(x)dx\\\\\n&=&-\\frac{1}{2}\\int_{\\BR}\\left(\\frac{\\pl}{\\pl\nx}\\sqrt{f(x)\\rho_{2t}(x)}\\vp(x)\\right)\\sqrt{f(x)\\rho_{2t}(x)}\\psi(x)dx\\\\\n& &\\quad\n+\\frac{1}{2}\\int_{\\BR}\\sqrt{f(x)\\rho_{2t}(x)}\\vp(x)\n \\left(\\frac{\\pl}{\\pl x}\\sqrt{f(x)\\rho_{2t}(x)}\\psi(x)\\right)dx\\\\\n&=&-\\frac{1}{2}\\int_{\\BR}\\{\\sqrt{f(x)\\rho_{2t}(x)}\\vp(x),\n \\sqrt{f(x)\\rho_{2t}(x)}\\psi(x)\\}dx\\\\\n&=&-\\frac{1}{2}\\int_{\\BR}\\{\\vp(x),\\psi(x)\\}\\tilde\\rho_0^2(x)\n e^{2\\sum_1^{\\iy}t_i x^i}dx,\n\\end{eqnarray}\nusing the notation in the statement of this proposition. Setting\n$\\tilde \\rho(x)=e^{-\\tilde V(x)}$, with $\\tilde\nV(x)=\\frac{1}{2}(V(x)-\\log f)$\n\\begin{eqnarray}\n\\la x^i,x^j\\ra^{sk}&=&-\\frac{1}{2}\\int_{\\BR}\\{x^i,x^j\\}\n\\tilde\\rho^2(x)\n e^{2\\sum_1^{\\iy}t_i x^i}dx\\\\\n&=&-\\frac{1}{2}\\int_{\\BR}\\{x^i,x^j\\} e^{-2(\\tilde V(x)-\\sum\nt_ix^i)}dx,\n\\end{eqnarray}\nand so\n$$\n\\tau_{2n}(t)=pf (\\tilde m_{2n}(t))\n=\\frac{1}{(-2)^n n!}\\int_{\\TR_{2n}}\ne^{2Tr(-\\tilde V(x)+\\sum t_ix^i)}dx.\n$$\n\n\\bigbreak\n\n\\noindent{\\bf The map $O^{-1}$ for the\nclassical orthogonal polynomials at $t=0$}: {\\em Then, the matrix\n$O$, mapping orthonormal $p_k$ into skew-orthonormal polynomials\n$q_k$, is given by a lower-triangular three-step relation :\n\\bea\np_{2n}(0,z)&=&\n-c_{2n-1}\\sqrt{\\frac{a_{2n-2}}{c_{2n-2}}}q_{2n-2}(0,z)\n+\\sqrt{a_{2n}c_{2n}}~q_{2n}(0,z)\\nonumber\\\\\n p_{2n+1}(0,z)&=& -c_{2n}\\sqrt{\\frac{a_{2n-2}}{c_{2n-2}}}\n q_{2n-2}(0,z)-(\\sum_0^{2n} b_i)\\sqrt{\\frac{c_{2n}}{a_{2n}}}\n q_{2n}(0,z)+\\sqrt{\\frac{c_{2n}}{a_{2n}}}q_{2n+1}(0,z),\\nonumber\\\\\n \\eea\nwhere the $a_i$ and $b_i$ are the entries in the tridiagonal matrix\ndefining the orthonormal polynomials, and the $c_i$ the entries in\nthe skew-symmetric matrix.}\n\n\n\nIn this case, we need to perform the following skew-Borel\ndecomposition at $t=0$,\n$$-\\UR=-\\NR=O^{-1}JO^{\\top -1},\n$$\nwhere $\\NR$ is the matrix (6.3). Here again, in order to find $O$,\nwe use the recipe given in theorem 4.1, namely writing down the\ncorresponding skew-orthogonal polynomials (0.17), but where the\n $\\mu_{ij}$ are the entries of $-\\UR=-\\NR$:\nconsider the pfaffians of the bordered matrices (0.17); they have\nleading term\n$$\\tilde{\\tilde\\tau}_{2n}= \\prod_0^{n-1} c_{2j}.$$\n Then one computes\n\\bea\nr_{2n}&=&\\frac{1}{\\sqrt{\\tilde{\\tilde\\tau}_{2n}\\tilde{\\tilde\\tau}_{2n+2}}}\\sum_{i=0}^n\n z^{2n-2i}\\left( \\prod_0^{n-i-1} c_{2j} \\right)\n \\left( \\prod_0^{i-1} c_{2n-2j-1} \\right)\\nonumber\\\\\nr_{2n+1}&=&\\frac{1}{\\sqrt{\\tilde{\\tilde\\tau}_{2n}\\tilde{\\tilde\\tau}_{2n+2}}}\\left(z^{2n+1}\n\\prod_0^{n-1} c_{2j}+\\sum_{i=1}^n\n z^{2n-2i}\\left( \\prod_0^{n-i-1} c_{2j} \\right)\n \\left( \\prod_0^{i-1} c_{2n-2j-1} \\right)\\right)\\nonumber\\\\\n \\eea\n with\n $$\\sqrt{\\tilde{\\tilde\\tau}_{2n}\\tilde{\\tilde\\tau}_{2n+2}}=\nc_0 c_2...c_{2n-2}\\sqrt{c_{2n}}\n~~~\n \\sqrt{\\tilde{\\tilde\\tau}_{0}\\tilde{\\tilde\\tau}_{2}}=\\sqrt{c_{0}}\n.$$\nSetting\n$$\nD:=\\diag\n(\\sqrt{\\tilde{\\tilde\\tau}_0\\tilde{\\tilde\\tau}_2},\\sqrt{\\tilde{\\tilde\n\\tau}_0\\tilde{\\tilde\\tau}_2},\\sqrt{\\tilde{\\tilde\\tau}_2\\tilde{\\tilde\n\\tau}_4},\n\\sqrt{\\tilde{\\tilde\\tau}_2\\tilde{\\tilde\\tau}_4},...)\n,$$ the matrix $O$ is the set of coefficients of the polynomials\n above, i.e.,\n\n\\bea O&=&D^{-1}\\pmatrix{1&0&0&0&0&0&0&0\\cr 0&1&0&0&0&0&0&0\\cr c_1&0&%\n {\\rm c_0}&0&0&0&0&0\\cr {\\rm c_2}&0&0&{\\rm c_0}&0&0&0&0\\cr {\\rm %\n c_1}\\,{\\rm c_3}&0&{\\rm c_0}\\,{\\rm c_3}&0&c_0 c_2&0&0&0%\n \\cr {\\rm c_1}\\,{\\rm c_4}&0&{\\rm c_0}\\,{\\rm c_4}&0&0&{\\rm c_0}\n \\,{\\rm c_2}&0%\n &0\\cr {\\rm c_1}\\,{\\rm c_3}\\,{\\rm c_5}&0&{\\rm c_0}\\,{\\rm c_3}\n \\,{\\rm c_5%\n }&0&{\\rm c_0}\\,{\\rm c_2}\\,{\\rm c_5}&0&{\\rm c_0}\\,{\\rm c_2}\\,{\\rm c_4}\n &0\\cr {\\rm c_1}\\,{\\rm c_3}\\,{\\rm c_6}&0&{\\rm c_0}\\,{\\rm c_3}\\,{\\rm c_6}&0&%\n {\\rm c_0}\\,{\\rm c_2}\\,{\\rm c_6}&0&0&{\\rm c_0}\\,{\\rm c_2}\\,{\\rm c_4}%\n \\cr &&&&&&&&\\ddots\n }\\nonumber\\\\\n &=:&D^{-1}R\n \\eea\nAs before, in order to get the skew-symmetric polynomials in the\nright form, from the orthogonal ones, one needs to multiply to the\nleft with the matrix $E$, defined in (8.2) in the appendix:\n \\be\n\\hat O=EO=ED^{-1}R,\n\\ee\nand so,\n \\be\n\\hat O^{-1}=R^{-1}DE^{-1};\n\\ee\nit turns out the matrix $\\hat O$ is complicated, but its inverse is\nsimple. Namely, compute\n$$R^{-1}= \\pmatrix{1&0&0&0&0&0&0&0\\cr 0&1&0&0&0&0&0&0\\cr\n -{{{ c_1}%\n }\\over{{ c_0}}}&0&{{1}\\over{{\\rm c_0}}}&0&0&0&0&0\\cr\n -{{{\\rm c_2%\n }}\\over{{\\rm c_0}}}&0&0&{{1}\\over{{\\rm c_0}}}&0&0&0&0\\cr\n 0&0&-{{%\n {\\rm c_3}}\\over{{\\rm c_0}\\,{\\rm c_2}}}&0&{{1}\\over{{\\rm c_0}\\,\n {\\rm c_2}}}%\n &0&0&0\\cr 0&0&-{{{\\rm c_4}}\\over{{\\rm c_0}\\,{\\rm c_2}}}&0&0&{{1%\n }\\over{{\\rm c_0}\\,{\\rm c_2}}}&0&0\\cr 0&0&0&0&-{{{\\rm c_5}}\\over{%\n {\\rm c_0}\\,{\\rm c_2}\\,{\\rm c_4}}}&0&{{1}\\over{{\\rm c_0}\\,{\\rm c_2}\\,{\\rm %\n c_4}}}&0\\cr 0&0&0&0&-{{{\\rm c_6}}\\over{{\\rm c_0}\\,{\\rm c_2}\\,{\\rm c_4%\n }}}&0&0&{{1}\\over{{\\rm c_0}\\,{\\rm c_2}\\,{\\rm c_4}}}\\cr\n \\cr&&&&&&&&\\ddots } ,$$\n and\n\\be\nE^{-1}\n=\\left(\n\\begin{array}{c@{}c@{}cc}\n \\boxed{\\begin{array}{cc} \\al_0 & 0 \\\\ -\\beta_0 & \\frac{1}{\\al_0}\n \\end{array}} &&&0 \\\\\n & \\boxed{\\begin{array}{cc} \\al_2 & 0 \\\\ -\\beta_2&\\frac{1}{\\al_2} \\end{array}} &&\\\\\n && \\boxed{\\begin{array}{cc} \\al_4&0 \\\\ -\\beta_4&\\frac{1}{\\al_4} \\end{array}} & \\\\\n 0& && \\ddots\n \\end{array}\n \\right),\n\\ee\nwith $\\al_{2n}$ and $\\beta_{2n}$ as in (8.5). Carrying out the\nmultiplication (7.5) leads to the matrix $\\hat O^{-1}$, with a few\nnon-zero bands, yielding the map (7.1).\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix 1: Free parameter in the skew-Borel decomposition}\n\n\nIf the Borel decomposition of $-H=O^{-1}JO^{\\top -1}$ is given by a\nmatrix $O \\in \\GR_{\\Bk}$, with the diagonal part of $O$ being\n\\be\n(O)_0=\\left(\n\\begin{array}{c@{}c@{}cc}\n \\boxed{\\begin{array}{cc} \\sg_0&0 \\\\ 0&\\sg_0 \\end{array}} && &0 \\\\\n & \\boxed{\\begin{array}{cc} \\sg_2&0 \\\\ 0&\\sg_2 \\end{array}} &&\\\\\n && \\boxed{\\begin{array}{cc} \\sg_4&0 \\\\ 0&\\sg_4 \\end{array}} & \\\\\n 0&&& \\ddots\n \\end{array}\n \\right),\n\\ee\nthen the new matrix\n\\be\n\\hat O:\n=\\left(\n\\begin{array}{c@{}c@{}cc}\n \\boxed{\\begin{array}{cc} 1/\\al_0 & 0 \\\\ \\beta_0 & \\al_0\n \\end{array}} &&&0 \\\\\n & \\boxed{\\begin{array}{cc} 1/\\al_2 & 0 \\\\ \\beta_2&\\al_2 \\end{array}} &&\\\\\n && \\boxed{\\begin{array}{cc} 1/\\al_4&0 \\\\ \\beta_4&\\al_4 \\end{array}} & \\\\\n 0& && \\ddots\n \\end{array}\n \\right)O=:EO,\n\\ee\nwith free parameters $\\al_{2n},~\\beta_{2n}$, is a solution of the\nBorel decomposition $-H=\\hat O^{-1}J\\hat O^{\\top\n-1}$, as well. The diagonal part of $\\hat O$ consists of $2\\times 2$ blocks\n$$\n\\pmatrix{1/\\al_{2n}&0\\cr\n \\beta_{2n} & \\al_{2n}}\n\\pmatrix{\\sg_{2n}&0\\cr\n 0 & \\sg_{2n}}=\n\\pmatrix{\\sg_{2n}/\\al_{2n}&0\\cr\n \\beta_{2n}\\sg_{2n} & \\al_{2n}\\sg_{2n}}.\n$$\nImposing the condition that\n$$\nq(z)=\\hat O p(z), ~\\mbox{with}~ p_k(z)=\\sum^k_{i=0}p_{ki}z^i\n$$\nhas the required form, i.e., the same leading term for $q_{2n}$ and\n$q_{2n+1}$ and no $z^{2n}$-term in $q_{2n+1}$,\n\\bea\nq_{2n}(z)&=&q_{2n,2n}z^{2n}+\\cdots\\nonumber\\\\\n q_{2n+1}(z)&=&q_{2n,2n}z^{2n+1}+q_{2n,2n-1}z^{2n-1}+\\cdots\n\\eea\nimplies\n$$\n\\frac{\\sg_{2n}}{\\al_{2n}}p_{2n,2n}=\\sg_{2n} \\al_{2n}p_{2n+1,2n+1}\n$$\n$$\n \\sg_{2n}\\beta_{2n}p_{2n,2n}+\\sg_{2n} \\al_{2n}p_{2n+1,2n}=0\n$$\nyielding, upon using the explicit form of the coefficients\n$p_{k\\ell}$ of the polynomials $p_k$, associated with three step\nrelations (see next lemma),\n\\bea\n\\al_{2n}^2&=&\\frac{p_{2n,2n}}{p_{2n+1,2n+1}}=a_{2n}\\nonumber\\\\\n \\frac{\\beta_{2n}}{\\al_{2n}}&=&\n -\\frac{p_{2n+1,2n}}{p_{2n,2n}}=\\frac{\\sum_0^{2n} b_i}{a_{2n}}.\n\\eea\nHence\n\\be\n \\al_{2n}=\\sqrt{a_{2n}}~~~\\mbox{and}~~~\\beta_{2n}=\n \\frac{1}{\\sqrt{a_{2n}}}\n \\sum_0^{2n} b_i.\n \\ee\n\nSo, if\n$$\nr(z)=O\\chi(z),\n$$\nthen\n$$\n\\hat r (z):=\\hat O\\chi(z)=\\left(\n\\begin{array}{c@{}c@{}cc}\n \\boxed{\\begin{array}{cc} 1/\\al_0 & 0 \\\\ \\beta_0 & \\al_0\n \\end{array}} &&&0 \\\\\n & \\boxed{\\begin{array}{cc} 1/\\al_2 & 0 \\\\ \\beta_2&\\al_2 \\end{array}} &&\\\\\n && \\boxed{\\begin{array}{cc} 1/\\al_4&0 \\\\ \\beta_4&\\al_4 \\end{array}} & \\\\\n 0& && \\ddots\n \\end{array}\n \\right) r(z)=E r(z),\n $$\n and thus\n \\bea\n \\hat r_{2n}(z)&=&\\frac{1}{\\sqrt{a_{2n}}}r_{2n}(z)\\\\\n \\hat r_{2n+1}(z)&=&\\frac{\\sum_0^{2n} b_i}{\\sqrt{a_{2n}}}\nr_{2n}+\\sqrt{a_{2n}}r_{2n+1}(z)\n\\eea\n\n\n\n\\begin{lemma} A sequence of polynomials $p_n(z)=\\sum^n_{i=0}p_{ni}z^i$ of\ndegree $n$ satisfying three-step recursion relation\\footnote{with\n$a_{-1}=0$.}\n\\be\nzp_n=a_{n-1}p_{n-1}+b_np_n+a_np_{n+1},\\quad n=0,1,\\dots,\n\\ee\nhas the form\n$$\np_{n+1}(z)=\\frac{p_{n,n}}{a_n}\\left(z^{n+1}-\n(\\sum_0^{n}b_i)z^n+\\cdots\\right).\n$$\n\\end{lemma}\n\n\\proof Equating the $z^{n+1}$ and $z^n$ coefficients of (8.8) divided by\n$p_{n,n}$ yields\n$$\n\\frac{p_{n+1,n+1}}{p_{n,n}}=\\frac{1}{a_n}\n$$\nand\n$$\n\\frac{p_{n,n-1}}{p_{n,n}}=a_n\\frac{p_{n+1,n}}{p_{n,n}}+b_n.\n$$\nCombining both equations leads to\n$$\na_n\\frac{p_{n+1,n}}{p_{n,n}}-a_{n-1}\\frac{p_{n,n-1}}{p_{n-1,n-1}}=b_n,\n$$\nyielding\n$$\na_n\\frac{p_{n+1,n}}{p_{n,n}}=-\\sum_0^nb_i,\\quad \\mbox{using $a_{-1}=0$}.\n$$\n\\qed\n\n\\section{Appendix 2: Simultaneous (skew) - symmetrization of $L$ and $\\NR$.}\n\n{\\em For the classical polynomials, the matrices $L$ and $\\NR$ can\nbe simultaneously symmetrized and skew-symmetrized}.\n\nWe sketch the proof of this statement, which has been established\nby us in \\cite{AvM2}. Given the monic orthogonal polynomials\n$\\tilde p_n$ with respect to the weight $\\rho$, with\n$\\rho'/\\rho=-g/f$, we have that the operators $z$ and\n $$\n \\Bn=\\sqrt{\\frac{f}{\\rho}} \\frac{d}{dz} \\sqrt{f\\rho}\n =f \\frac{d}{dz} +\\frac{f'-g}{2}.\n $$\nacting on the polynomials $\\tilde p_n$'s have the following form:\n\\bea\nz\\tilde p_n&=&a_{n-1}^2\\tilde p_{n-1}+b_n\\tilde p_n+\\tilde p_{n+1}\n \\nonumber\\\\\n \\Bn\\tilde p_n&=&\n ...-\\gamma_n \\tilde p_{n+1},\n \\eea\n in view of the fact that for the classical orthogonal\n polynomials\\footnote{with respective weights $\\rho=e^{-z^2},~\n \\rho=e^{-z}z^{\\al},~\n\\rho=(1-z)^{\\al}(1+z)^{\\beta}$.},\n\n$\\left\\{\n\\noindent\\begin{tabular}{l l }\nHermite: & $\\Bn=\\frac{d}{dz}-z $\\\\ Laguerre: &\n$\n\\Bn=z\\frac{d}{dz}-\n\\frac{1}{2}(z-\\alpha -1)$ \\\\ Jacobi: & $\\Bn=(1-z^2)\\frac{d}{dz}\n-\\frac{1}{2}((\\alpha +\\beta\n+2)z+(\\alpha -\\beta)).$\\\\\n% & & &-\\frac{1}{2}((\\alpha +\\beta\n%+2)z+(\\alpha -\\beta))\n\\end{tabular}\\right.\n$\n\nFor the orthonormal polynomials, the matrices $L$ and $-\\NR$ are\nsymmetric and skew-symmetric respectively. Therefore the right hand\nside of these expressions must have the form:\n\\bean\nz\\tilde p_n&=&a_{n-1}^2\\tilde p_{n-1}+b_n\\tilde p_n+\\tilde p_{n+1}\n \\\\\n \\Bn\\tilde p_n&=&\n a^2_{n-1}\\gamma_{n-1}\\tilde p_{n-1}-\\gamma_n \\tilde p_{n+1}.\n \\eean\nTherefore, upon rescaling the $\\tilde p_n$'s, to make them\northonormal, we have\n\\bean\nz p_n&=&(Lp)_n =a_{n-1} p_{n-1}+b_n p_n+ a_n p_{n+1}\n \\\\\n \\Bn p_n&=& (\\NR p)_n=\n a_{n-1}\\gamma_{n-1}\\tilde p_{n-1}-a_n\\gamma_n \\tilde p_{n+1},\n \\eean\nfrom which it follows that $$-\\NR=\\left[\\begin{array}{ccccc} 0&c_0&\n& & \\\\\n-c_0&0&c_1& & \\\\\n &-c_1&0&\\ddots&\\\\\n & &\\ddots& &\n\\end{array}\\right],~~\n\\mbox{with}~~ c_n=-a_n \\gamma_n~,\n$$\nwhere $-\\gamma_n$ is the leading term in the expression (9.1).\n\n\n\\section{Appendix 3: Proof of Lemma 3.4}\n\nFor future use, consider the first order differential operators\n \\be\n \\eta(t,z)=\\sum^{\\iy}_{j=1}\\frac{z^{-j}}{j}\\frac{\\pl}{\\pl\n t_j}\\quad\\mbox{and}\\quad\n B(z)=-\\frac{\\pl}{\\pl z}+\\sum^{\\iy}_{j=1}z^{-j-1}\\frac{\\pl}{\\pl t_j}\n \\ee\n having the property\n \\be\n B(z)e^{-\\eta(z)}f(t)=B(z)f(t-[z^{-1}])=0.\n \\ee\n\n\n\\begin{lemma} Consider an arbitrary function $\\vp(t,z)$ depending on\n$t\\in\\BC^{\\iy}$, $z\\in\\BC$, having the asymptotics\n$\\vp(t,z)=1+O(\\frac{1}{z})$ for $z\\nearrow\\iy$ and\n satisfying the functional\nrelation\n\\be\n\\frac{\\vp(t-[z^{-1}_2],z_1)}{\\vp(t,z_1)}=\\frac{\\vp(t-[z^{-1}_1],z_2)}\n{\\vp(t,z_2)}, \\quad t\\in\\BC^{\\iy},z\\in\\BC.\n\\ee\nThen there exists a function $\\tau(t)$ such that\n\\be\n \\vp(t,z)=\\frac{\\tau(t-[z^{-1}])}{\\tau(t)}.\n\\ee\n\\end{lemma}\n\n\\proof\nApplying $B_1:=B(z_1)$ to the logarithm of (10.3) and using\n (10.1) and (10.3) yields\n \\begin{eqnarray}\n (e^{-\\eta(z_2)}-1)B_1\\log\\vp(t;z_1)&=&-B_1\\log\\vp(t,z_2)\\\\\n &=&\\sum^{\\iy}_{j=1}z_1^{-j-1}\\frac{\\pl}{\\pl t_j}\\log\\vp(t,z_2),\n \\end{eqnarray}\n which, upon setting\n $$\n f_j(t)=\\mbox{\\,Res}_{z_1=\\iy}z_1^jB_1\\log\\vp(t,z_2)\n, $$\n yields termwise in $z_1$,\n \\be\n (e^{-\\eta(z_2)}-1)f_j(t)=-\\frac{\\pl}{\\pl t_j}\\log\\vp(t,z_2).\n \\ee\n Acting with $\\frac{\\pl }{\\pl t_i}$ on the latter expression\n and with $ \\frac{\\pl }{\\pl t_j}$\n on the same expression with $j$ replaced by $i$, and subtracting\\footnote{It\n is obvious that\n $\\left[\\frac{\\pl}{\\pl t_i},e^{-\\eta(z)}\\right]=0$.}, one finds\n $$\n (e^{-\\eta(z_2)}-1)\\left(\\frac{\\pl f_i}{\\pl t_j}-\n \\frac{\\pl f_j}{\\pl t_i}\\right)=0,\n $$\n yielding\n $$\n \\frac{\\pl f_i}{\\pl t_j}-\\frac{\\pl f_j}{\\pl t_i}=0;\n $$\n the constant vanishes, because $\\frac{\\pl f_i}{\\pl t_j}$ never contains\n constant terms.\n\n Therefore there exists a function\n $\\log\\tau(t_1,t_2,...)$ such that\n $$\n -\\frac{\\pl }{\\pl t_j}\\log\\tau=f_j(t)=\\mbox{\\,Res}_{z=\\iy}z^jB\\log\\vp\n $$\n and hence, using (10.5)\n $$\n \\frac{\\pl }{\\pl t_j}\\log\\vp(t,z)=(e^{-\\eta(z)}-1)\n \\frac{\\pl }{\\pl\n t_j}\\log\\tau\n $$\n or, what is the same,\n $$\n \\frac{\\pl }{\\pl t_j}(\\log\\vp-(e^{-\\eta}-1)\\log\\tau)=0,\n $$\n from which it follows that\n $$\n \\log\\vp-(e^{-\\eta}-1)\\log\\tau=-\\sum_1^{\\iy}\\frac{b_i}{i}z^{-i}\n $$\n is, at worst, a holomorphic series in $z^{-1}$ with constant coefficients,\n which we call $-b_i/i$. Hence\n \\begin{eqnarray}\n \\vp(t,z)&=&\\frac{\\tau(t-[z^{-1}]e^{-\\sum_1^{\\iy}\\frac{b_i}{i}\n z^{-i}}}{\\tau(t)}\\\\\n &=&\\frac{\\tau(t-[z^{-1}])e^{\\sum_1^{\\iy}b_i(t_i-\\frac{z^{-i}}{i})}}\n {\\tau(t)e^{\\sum_1^{\\iy}b_it_i}},\n \\end{eqnarray*}\n i.e.\n $$\n \\vp(t,z)=\\frac{\\tilde\\tau(t-[z^{-1}])}{\\tilde\\tau(t)},\n $$\n where\n $$\n \\tilde\\tau=\\tau(t)e^{\\sum_1^{\\iy}b_it_i}.\n $$\n \\qed\n\n\n\n\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{AvM1} M. Adler and P.~ van Moerbeke: {\\em Completely\nintegrable system, Euclidean Lie algebras and curves}, Advances in\nMath. 38, 267-317 (1980) .\n\n\n\\bibitem{AvM2} M.~ Adler and P.~ van Moerbeke: {\\em Matrix\nintegrals, Toda symmetries, Virasoro constraints and orthogonal\npolynomials} Duke Math.J., {\\bf 80} (3), 863--911 (1995)\n\n\n\\bibitem{AvM3} M. Adler and P. van Moerbeke: {\\em Group factorization,\nmoment matrices and 2-Toda lattices}, Intern. Math. Research Notices,\n{\\bf 12}, 555-572 (1997)\n\n\\bibitem{AvM4} M. Adler and P. van Moerbeke: {\\em The spectrum of symmetric random\nmatrices and the Pfaff Lattice}, Annals of Math.,\n(2000) (to appear).\n\n\n\n\n\n\n\n\n \\bibitem{AHV} M.~Adler, ~E.~Horozov and P.~ van Moerbeke :\n{\\em The Pfaff lattice and skew-orthogonal polynomials},\n Int. Math. Res. Notices, {\\bf 11} 569-588 (1999).\n\n\n \\bibitem{ASV} M.~Adler, T.~Shiota and P.~ van Moerbeke :\n{\\em The Pfaffian $\\tau$-functions } (to appear).\n\n\n\\bibitem{BN} E.~Br\\'ezin, H.~Neuberger : {\\em\nMulticritical points of unoriented random surfaces}, Nuclear\nPhysics {\\bf B 350}, 513-553 (1991).\n\n\\bibitem{Dickey} L. Dickey: Soliton equations and integrable\nsystems, World Scientific (1991).\n\n\n\n\n\n\\bibitem{Date} E. Date, M. Jimbo, M. Kashiwara, T. Miwa: {\\em\nTransformation groups for soliton equations}, In: Proc.\\ RIMS\nSymp.\\ Nonlinear integrable systems --- Classical and quantum\ntheory (Kyoto 1981), pp.\\ 39--119.\\ Singapore : World Scientific\n1983.\n\n\n\n\n\n\n\n\n\n\n\n\\bibitem{For} P. Forrester: Random matrices, Cambridge University press.\n\n\\bibitem{Kac} V.G. Kac: Infinite dimensional Lie algebras, 3rd\nedition, Cambridge University press.\n\n\n\\bibitem{KvdL} V.G. Kac and J. van de Leur: {\\em\n The geometry of spinors and the multicomponent\n BKP and DKP hierarchies} in \"The bispectral problem\n (Montreal PQ, 1997)\", CRM Proc. Lecture notes {\\bf 14},\n AMS, Providence, 159-202 (1998).\n\n\n\n\\bibitem{M} M.L. Mehta: Random matrices, 2nd ed.\\\nBoston: Acad.\\ Press, 1991\n\n\\bibitem{AFNV} M.~Adler, P.J.~Forrester,\nT.~Nagao and P.~van Moerbeke:\n{\\em\nClassical skew orthogonal polynomials and random\nmatrices},\nJournal of Statistical Physics, 2000\n\n\n\n\n\\bibitem{RS} A.G.~Reyman and M.A. Semenov-Tian-Shansky : {\\em\n Reduction of Hamiltonian systems, affine Lie algebras and\n Lax equations }, Inv. Math., {\\bf 54}, 81-100 (1979).\n\n\\bibitem{vdL} J. van de Leur: {\\em\nMatrix Integrals and Geometry of Spinors }\n (solv-int/9909028)\n\n\n\n\n\n\n\\end{thebibliography}\n\n\n\\end{document}\n" } ]	[ { "name": "solv-int9912008.extracted_bib", "string": "{AvM1 M. Adler and P.~ van Moerbeke: {\\em Completely integrable system, Euclidean Lie algebras and curves, Advances in Math. 38, 267-317 (1980) ." }, { "name": "solv-int9912008.extracted_bib", "string": "{AvM2 M.~ Adler and P.~ van Moerbeke: {\\em Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials Duke Math.J., {80 (3), 863--911 (1995)" }, { "name": "solv-int9912008.extracted_bib", "string": "{AvM3 M. Adler and P. van Moerbeke: {\\em Group factorization, moment matrices and 2-Toda lattices, Intern. Math. Research Notices, {12, 555-572 (1997)" }, { "name": "solv-int9912008.extracted_bib", "string": "{AvM4 M. Adler and P. van Moerbeke: {\\em The spectrum of symmetric random matrices and the Pfaff Lattice, Annals of Math., (2000) (to appear)." }, { "name": "solv-int9912008.extracted_bib", "string": "{AHV M.~Adler, ~E.~Horozov and P.~ van Moerbeke : {\\em The Pfaff lattice and skew-orthogonal polynomials, Int. Math. Res. Notices, {11 569-588 (1999)." }, { "name": "solv-int9912008.extracted_bib", "string": "{ASV M.~Adler, T.~Shiota and P.~ van Moerbeke : {\\em The Pfaffian $\\tau$-functions (to appear)." }, { "name": "solv-int9912008.extracted_bib", "string": "{BN E.~Br\\'ezin, H.~Neuberger : {\\em Multicritical points of unoriented random surfaces, Nuclear Physics {B 350, 513-553 (1991)." }, { "name": "solv-int9912008.extracted_bib", "string": "{Dickey L. Dickey: Soliton equations and integrable systems, World Scientific (1991)." }, { "name": "solv-int9912008.extracted_bib", "string": "{Date E. Date, M. Jimbo, M. Kashiwara, T. Miwa: {\\em Transformation groups for soliton equations, In: Proc.\\ RIMS Symp.\\ Nonlinear integrable systems --- Classical and quantum theory (Kyoto 1981), pp.\\ 39--119.\\ Singapore : World Scientific 1983." }, { "name": "solv-int9912008.extracted_bib", "string": "{For P. Forrester: Random matrices, Cambridge University press." }, { "name": "solv-int9912008.extracted_bib", "string": "{Kac V.G. Kac: Infinite dimensional Lie algebras, 3rd edition, Cambridge University press." }, { "name": "solv-int9912008.extracted_bib", "string": "{KvdL V.G. Kac and J. van de Leur: {\\em The geometry of spinors and the multicomponent BKP and DKP hierarchies in \"The bispectral problem (Montreal PQ, 1997)\", CRM Proc. Lecture notes {14, AMS, Providence, 159-202 (1998)." }, { "name": "solv-int9912008.extracted_bib", "string": "{M M.L. Mehta: Random matrices, 2nd ed.\\ Boston: Acad.\\ Press, 1991" }, { "name": "solv-int9912008.extracted_bib", "string": "{AFNV M.~Adler, P.J.~Forrester, T.~Nagao and P.~van Moerbeke: {\\em Classical skew orthogonal polynomials and random matrices, Journal of Statistical Physics, 2000" }, { "name": "solv-int9912008.extracted_bib", "string": "{RS A.G.~Reyman and M.A. Semenov-Tian-Shansky : {\\em Reduction of Hamiltonian systems, affine Lie algebras and Lax equations , Inv. Math., {54, 81-100 (1979)." }, { "name": "solv-int9912008.extracted_bib", "string": "{vdL J. van de Leur: {\\em Matrix Integrals and Geometry of Spinors (solv-int/9909028)" } ]
solv-int9912009		[]	We show explicitly how to construct the quantum Lax pair for systems with open boundary conditions. We demonstrate the method by applying it to the Heisenberg XXZ model with the general integrable boundary terms.	[ { "name": "solv-int9912009.tex", "string": "\n\\documentstyle[12pt]{article}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%TCIDATA{Created=Tue Dec 14 12:09:02 1999}\n%TCIDATA{LastRevised=Tue Dec 14 12:09:02 1999}\n%TCIDATA{Language=American English}\n\n\\topmargin0cm\n\\textwidth155mm\n\\textheight22cm\n\\oddsidemargin0cm\n\\catcode`\\@=11\n\n\\begin{document}\n\n\n\\newlength{\\lno} \\lno1.5cm \\newlength{\\len} \\len=\\textwidth%\n\\addtolength{\\len}{-\\lno}\n\n\\baselineskip7mm \\renewcommand{\\thefootnote}{\\fnsymbol{footnote}} \\newpage\n\n\\begin{titlepage}\n\\vspace{2.0cm}\n\\begin{center}\n{\\Large\\bf Quantum Lax Pair From Yang-Baxter Equations}\\\\\n\\vspace{1.5cm}\n{\\large A. Lima-Santos }\\footnote{e-mail: dals@power.ufscar.br} \\\\\n\\vspace{1.0cm}\n{\\large \\em Universidade Federal de S\\~ao Carlos, Departamento de F\\'{\\i}sica \\\\\nCaixa Postal 676, CEP 13569-905~~S\\~ao Carlos, Brasil}\\\\\n\\end{center}\n\\vspace{3.0cm}\n\n\\begin{abstract}\nWe show explicitly how to construct the quantum Lax pair for systems\nwith open boundary conditions. We demonstrate the method by applying it\nto the Heisenberg XXZ model with the general integrable boundary terms.\n\\end{abstract}\n\\vfill\n\\end{titlepage}\n\n\\baselineskip6mm\n\n\\newpage{}\n\nEver since the introduction of the quantum Lax pair into statistical\nmechanics by McCoy and Wu \\cite{Wu}\\ and of the notion of a one-parameter\nfamily of commuting transfer matrices by Baxter \\cite{Baxter1}, a great deal\nof effort has been expended in the search for spin chains with Hamiltonian $%\n{\\cal H}$ and two-dimensional statistical models with transfer matrix $\\tau $\nwhich have the integrability property.\n\nThe quantum inverse scattering method places the theory of completely\nintegrable quantum system and solvable statistical models in a unified\nframework \\cite{Faddeev}. The basis to apply it to a completely integrable\nsystem is to associate an operator version of an auxiliary linear problem \n\\cite{Wadati}: \n\\begin{equation}\n\\Psi _{n+1}=L_{n}\\Psi _{n}\\ \\quad ,\\quad \\ \\stackrel{.}{\\Psi }_{n}=A_{n}\\Psi\n_{n}, \\label{eq1}\n\\end{equation}\nwhere $L_{n}$ and $A_{n}$ are matrix operators depending on the spectral\nparameter $u$, and a dot signifies a time derivative. The consistency\ncondition for equations (\\ref{eq1}) with $\\stackrel{.}{u}=0$ yields the Lax\npair equation: \n\\begin{equation}\n\\stackrel{.}{L}_{n}=A_{n+1}L_{n}-L_{n}A_{n}. \\label{eq2}\n\\end{equation}\nAll the solved integrable models appear to imply that \\ a model is\ncompletely integrable if we can find a Lax pair $\\left\\{ L_{n},A_{n}\\right\\} \n$ \\ such that the Lax equation (\\ref{eq2}) is equivalent to the equation of\nmotion of the model \\cite{Wadati}.\n\nStarting with a local Lax operator $L_{n}$ (a matrix acting in the auxiliary\nspace $V$ with elements acting in the quantum space $h_{n}$, at site $n$),\nfor most quantum integrable system the product direct of two Lax operators $%\nL_{n}$, with different spectral parameters satisfy a similarity relation \n\\begin{equation}\n{\\cal R}(u-v)L_{n}(u)\\otimes L_{n}(v)=L_{n}(v)\\otimes L_{n}(u){\\cal R}(u-v)\n\\label{eq3}\n\\end{equation}\nwith ${\\cal R}$ a $c$-number matrix acting in $V\\otimes V$. The above\nrelation is referred to as the local Yang-Baxter relation and can be\nrepresented graphically by figure $1$.\n\n\\begin{center}\n\\setlength{\\unitlength}{5947sp}\\begingroup\\makeatletter\\ifx\\SetFigFont%\n\\undefined\\gdef\\SetFigFont#1#2#3#4#5{\\reset@font\\fontsize{#1}{#2pt} %\n\\fontfamily{#3}\\fontseries{#4}\\fontshape{#5} \\selectfont}\\fi\\endgroup%\n\\begin{picture}(3024,1050)(1189,-1723)\n\\thinlines\n\\put(1201,-1411){\\line( 2, 1){1200}}\n\\put(1201,-961){\\line( 2,-1){1200}}\n\\put(2101,-661){\\line( 0,-1){1050}}\n\\put(3001,-811){\\line( 2,-1){1200}}\n\\put(3001,-1561){\\line( 2, 1){1200}}\n\\put(3301,-661){\\line( 0,-1){1050}}\n\\put(1445,-1331){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$\\cal{R}$$(u$-$v)$}}}\n\\put(2051,-1036){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{n}(v)$}}}\n\\put(2051,-1411){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{n}(u)$}}}\n\\put(2976,-1036){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{n}(u)$}}}\n\\put(2976,-1400){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{n}(v)$}}}\n\\put(3576,-1361){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$\\cal{R}$$(u$-$v)$}}}\n\\end{picture}\n\nfigure $1$\n\\end{center}\n\nIn terms of the operators $L_{n}$, the monodromy matrix for a chain with\nlength $N$ is expressed as \n\\begin{equation}\nT(u)=L_{N}(u)L_{N-1}(u)\\cdots L_{2}(u)L_{1}(u) \\label{eq4}\n\\end{equation}\nand if $L_{n}^{\\prime }$s with different $n$ commute, we further have \n\\begin{equation}\n{\\cal R}(u-v)T(u)\\otimes T(v)=T(v)\\otimes T(u){\\cal R}(u-v) \\label{eq5}\n\\end{equation}\ncalled global Yang-Baxter relation.\n\nWe assume the auxiliary space $V$ and the quantum space $h$ are the same and\nregard the elements of $L$ and ${\\cal R}$ as those of the $R$-matrix (${\\cal %\nR}={\\cal P}R$ with ${\\cal P}$ the permutation operator which interchanges\nspaces of $V_{n}$ with $V_{n-1}$ ), in terms of which the local Yang-Baxter\nrelation (\\ref{eq3}) has the form \n\\begin{equation}\nR_{12}(u-v)R_{13}(u)R_{23}(v)=R_{23}(v)R_{13}(u)R_{12}(u-v), \\label{eq5a}\n\\end{equation}\nwhere $R_{ij}$ is a matrix in $V\\times V\\times V$, which acts non-trivially\nin the spaces $V_{i}$ and $V_{j}$ only.\n\nIn two-dimensional statistical mechanics, the elements of $R$ may be\nconsidered as vertices of a vertex model. Baxter first noticed the\nimportance of the relation (\\ref{eq5a}) in that context and regarded it as\nthe solvability condition of the vertex model \\cite{Baxter1}: we introduce\nthe transfer matrix $\\tau (u)$ as a trace of the monodromy matrix $\\tau (u)=%\n{\\rm tr}T(u)$. The relation (\\ref{eq5}) indicates that there exists a family\nof commuting transfer matrices and that $u$-expansion of $\\tau (u)$ gives a\nset of conserved quantities which are involutive.\n\nThe integrability conditions for a system with open boundary condition are\nformulated in order that both the Yang-Baxter equation and the boundary\nYang-Baxter equations (or reflection equations) are satisfied \\cite{Sklyanin}%\n. To a quantum system on a finite interval with independent boundary\nconditions at each end, we have to introduce reflection matrices $K^{\\mp\n}(u) $ to describe such boundary conditions. For a $PT$-invariant $R$%\n-matrix, the fundamental reflection-factorization relations obeyed by $%\nK^{-}(u)$ and $K^{+}(u)$\\ are \\cite{Mezin}: \n\\begin{equation}\nR_{12}(u-v)K_{1}^{-}(u)R_{21}(u+v)K_{2}^{-}(v)=K_{2}^{-}(v)R_{12}(u+v)K_{1}^{-}(u)R_{21}(u-v)\n\\label{eq6}\n\\end{equation}\nwhich is represented graphically by figure $2$\n\n\\begin{center}\n{}\\setlength{\\unitlength}{4947sp}\\begingroup\\makeatletter\\ifx\\SetFigFont%\n\\undefined\\gdef\\SetFigFont#1#2#3#4#5{\\reset@font\\fontsize{#1}{#2pt} %\n\\fontfamily{#3}\\fontseries{#4}\\fontshape{#5} \\selectfont}\\fi\\endgroup%\n\\begin{picture}(2475,1674)(1801,-2023)\n\\thinlines\n\\put(2401,-361){\\line( 0,-1){1650}}\n\\put(2101,-1861){\\line( 2, 5){300}}\n\\put(2401,-1111){\\line(-1, 2){300}}\n\\put(1951,-1711){\\line( 2, 1){450}}\n\\put(2401,-1486){\\line(-2, 1){450}}\n\\put(1951,-1261){\\line( 0, 1){ 0}}\n\\put(1951,-1261){\\line( 0, 1){ 0}}\n\\put(1951,-1261){\\line( 0, 1){ 0}}\n\\put(4201,-361){\\line( 0,-1){1650}}\n\\put(3901,-1861){\\line( 2, 5){300}}\n\\put(4201,-1111){\\line(-1, 2){300}}\n\\put(3751,-1036){\\line( 2, 1){450}}\n\\put(4201,-811){\\line(-2, 1){450}}\n\\put(3751,-586){\\line( 0, 1){ 0}}\n\\put(3751,-586){\\line( 0, 1){ 0}}\n\\put(3751,-586){\\line( 0, 1){ 0}}\n\\put(1826,-1670){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$u$}}}\n\\put(2026,-1861){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$v$}}}\n\\put(3826,-1861){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$v$}}}\n\\put(3626,-976){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$u$}}}\n\\put(2326,-1486){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$K^{\\scriptstyle{-}}_{\\scriptstyle{1}}(u)$}}}\n\\put(2326,-1111){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$K^{\\scriptstyle{-}}_{\\scriptstyle{2}}(v)$}}}\n\\put(4126,-811){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$K^{\\scriptstyle{-}}_{\\scriptstyle{1}}(u)$}}}\n\\put(4126,-1111){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$K^{\\scriptstyle{-}}_{\\scriptstyle{2}}(v)$}}}\n\\put(1801,-1231){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}-$u$}}}\n\\put(1951,-491){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}-$v$}}}\n\\put(3601,-566){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}-$u$}}}\n\\put(3751,-491){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}-$v$}}}\n\\end{picture}\n\nfigure $2$\n\\end{center}\n\nFor the left boundary we have a similar equation: \n\\begin{eqnarray}\n&&R_{12}(-u+v)K_{1}^{+}(u)^{t_{1}}M_{1}^{-1}R_{21}(-u-v-2\\rho\n)M_{1}K_{2}^{+}(v)^{t_{2}} \\nonumber \\\\\n&=&K_{2}^{+}(v)^{t_{2}}M_{1}R_{12}(-u-v-2\\rho\n)M_{1}^{-1}K_{1}^{+}(u)^{t_{1}}R_{21}(-u+v) \\label{eq7}\n\\end{eqnarray}\nHere $t_{i}$ denotes the transposition in the $i^{{\\it th}}$ space; $%\nK_{1}^{\\pm }=K^{\\pm }\\otimes 1,K_{2}^{\\pm }=1\\otimes K^{\\pm }$ , etc. $\\rho $\nis a crossing parameter and $M$ is a crossing matrix, both being specific to\na given $R$-matrix. These relations correspond to the constraint of\nfactorized scattering in the presence of a wall.\n\nThere is an automorphism \\cite{Sklyanin, Mezin} between the $K^{-}$ and $%\nK^{+}$ namely, given a solution $K^{-}(u)$, then $K^{-}(-u-\\rho )^{t}M$ is a\nsolution $K^{+}(u)$.\n\nGiven an $R$ -matrix and $K$-matrices satisfying (\\ref{eq6}) and (\\ref{eq7}%\n), the corresponding open chain transfer matrix $t(u)$ is given by \n\\begin{equation}\nt(u)={\\rm tr}\\left( K^{+}(u)T(u)K^{-}(u)T^{-1}(-u)\\right) \\label{eq8}\n\\end{equation}\nwhich constitutes a one-parameter commutative family.\n\nHamiltonians \\ with boundary terms are obtained from the first derivative of\nthe transfer matrix (\\ref{eq8}) \\cite{Sklyanin}: \n\\begin{equation}\n{\\cal H}=\\sum_{n=2}^{N}H_{n,n-1}+\\frac{1}{2}\\frac{K_{1}^{-\\prime }(0)}{%\nK_{1}^{-}(0)}+\\frac{{\\rm tr}_{a}[K_{a}^{+}(0)H_{Na}]}{{\\rm tr}K^{+}(0)}\n\\label{eq9}\n\\end{equation}\nwhere the local sum is given by \n\\begin{equation}\n\\sum_{n=2}^{N}H_{n,n-1}=\\alpha \\left. \\frac{{\\rm d}}{{\\rm d}u}\\ln \\tau\n(u)\\right\| _{u=0}+{\\rm const\\ }I \\label{eq9a}\n\\end{equation}\nThe index $a$ denotes the auxiliary space and $\\alpha $ is a constant.\n\nThe corresponding Lax pair formulation may be obtained by direct\nconsideration of the equations of motion which can be write in the Lax form\nconsidering the following operator version of an auxiliary linear problem \n\\begin{equation}\n\\Psi _{n+1}=L_{n}\\Psi _{n}\\qquad n=1,2,...,N\\quad ,\\quad \\stackrel{.}{\\Psi }%\n_{n}=A_{n}\\Psi _{n}\\qquad n=2,3,...,N \\label{eq11}\n\\end{equation}\nand \n\\begin{equation}\n\\stackrel{.}{\\Psi }_{1}=Q_{1}\\Psi _{1}\\quad ,\\quad \\stackrel{.}{\\Psi }%\n_{N+1}=Q_{N}\\Psi _{N+1}. \\label{eq12}\n\\end{equation}\nNote that $Q_{1}(u)$ and $Q_{N}(u)$ are responsible for the boundary terms\nin the equations of motion of the system. The consistency conditions for\nthese equations are the following Lax equations: \n\\begin{equation}\n\\stackrel{.}{L}_{n}=A_{n+1}L_{n}-L_{n}A_{n}\\quad {\\rm for\\quad }n=2,...,N-1.\n\\label{eq13}\n\\end{equation}\nand \n\\begin{equation}\n\\stackrel{.}{L}_{1}=A_{2}L_{1}-L_{1}Q_{1}\\quad {\\rm and}\\quad \\stackrel{.}{L}%\n_{N}=Q_{N}L_{N}-L_{N}A_{N} \\label{eq14}\n\\end{equation}\n\nThese equations specify the evolution only of $L_{n}$. \\ It means that $%\nA_{n} $ , $Q_{1}$ and $Q_{N}$ \\ have to be determined in terms of $L_{n}$\nand $H_{n,n-1}$.\n\nIn this note we will show how to find the Lax pair from the boundary\nYang-Baxter equations. In this way we are including the general integrable\nboundary terms in a previous work of Zhang \\cite{Zhang} where the periodic\ncase was considered. We also apply the method \\ to the open {\\small XXZ}\nmodel and compare the result with previous calculations.\n\nTo do this we consider a application of the local Yang-Baxter relation,\nwhere the vertices are being commuted horizontally rather than vertically: \n\\begin{equation}\n{\\cal R}_{n,n-1}(\\epsilon )L_{n}(u+\\epsilon\n)L_{n-1}(u)=L_{n}(u)L_{n-1}(u+\\epsilon ){\\cal R}_{n,n-1}(\\epsilon ).\n\\label{eq21}\n\\end{equation}\nHere the spectral parameters of the local $L$-operators differ only by an\ninfinitesimal amount $\\epsilon $. This relation can be represented\ngraphically by figure $3$\n\n\\begin{center}\n\\setlength{\\unitlength}{2947sp}\\begingroup\\makeatletter\\ifx\\SetFigFont%\n\\undefined\\gdef\\SetFigFont#1#2#3#4#5{\\reset@font\\fontsize{#1}{#2pt} %\n\\fontfamily{#3}\\fontseries{#4}\\fontshape{#5} \\selectfont}\\fi\\endgroup%\n\\begin{picture}(6924,2129)(2089,-3073)\n\\thinlines\n\\put(2101,-2461){\\line( 1, 0){2700}}\n\\put(3901,-961){\\line(-3,-4){1548}}\n\\put(4501,-3061){\\line(-3, 4){1548}}\n\\put(8701,-961){\\line(-3,-4){1548}}\n\\put(8108,-3056){\\line(-3, 4){1548}}\n\\put(6301,-1561){\\line( 1, 0){2700}}\n\\put(6751,-1411){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{n}(u)$}}}\n\\put(8326,-1411){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{n-1}(u$+$\\epsilon)$}}}\n\\put(2001,-2986){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$n$}}}\n\\put(3850,-2986){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$n$-$1$}}}\n\\put(6826,-2986){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$n$}}}\n\\put(8001,-2986){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$n$-$1$}}}\n\\put(2551,-2661){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{n}(u$+$\\epsilon)$}}}\n\\put(4151,-2661){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{n-1}(u)$}}}\n\\put(3351,-1661){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$\\cal{R}$$_{n,n-1}(\\epsilon)$}}}\n\\put(6450,-2461){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$\\cal{R}$$_{n,n-1}(\\epsilon)$}}}\n\\end{picture}\n\nfigure $3$\n\\end{center}\n\nThis is a form of the star-triangle equation used in \\cite{Zhang, Thacker,\nHelen}. This formulation follows also from the non-local quantum Lax pair \n\\cite{BP, ALS}.\n\nUsing the normalization ${\\cal R}(0)=1$ we have the following $\\epsilon $%\n-expansions \n\\begin{eqnarray}\n{\\cal R}(\\epsilon ) &\\sim &1+\\epsilon (\\alpha ^{-1}H+\\beta I)+o(\\epsilon\n^{2}), \\label{eq22a} \\\\\nL_{n}(u+\\epsilon ) &\\sim &L_{n}(u)+\\epsilon L_{n}^{\\prime }(u)+o(\\epsilon\n^{2}), \\label{eq22b}\n\\end{eqnarray}\nwhere a prime stands for $u$-derivative, with $\\alpha $ and $\\beta $ some\nconstants and $H_{n,n-1}$ of the local sum in (\\ref{eq9a}) is the $H$ in (%\n\\ref{eq22a}) acting on the quantum spaces at sites $n$ and $n-1$.\n\nSubstituting (\\ref{eq22a}) and (\\ref{eq22b}) into (\\ref{eq21}) we get as the\nfirst non-trivial consequence the following commutation relation \n\\begin{eqnarray}\n\\alpha ^{-1}[H_{n,n-1},L_{n}(u)L_{n-1}(u)] &=&L_{n}(u)L_{n-1}^{\\prime\n}(u)-L_{n}^{\\prime }(u)L_{n-1}(u) \\nonumber \\\\\nn &=&2,3,...,N \\label{eq23}\n\\end{eqnarray}\nWe multiply out the above equation to arrive at \n\\begin{eqnarray}\n\\lbrack H_{n,n-1},L_{n}] &=&\\alpha (L_{n}L_{n-1}^{\\prime\n}L_{n-1}^{-1}-L_{n}^{\\prime })-L_{n}[H_{n,n-1},L_{n}]L_{n-1}^{-1}\n\\label{eq24} \\\\\n\\lbrack H_{n+1,n},L_{n}] &=&\\alpha (L_{n}^{\\prime\n}-L_{n+1}^{-1}L_{n+1}^{\\prime }L_{n})-L_{n+1}^{-1}[H_{n+1,n},L_{n+1}]L_{n}\n\\label{eq25}\n\\end{eqnarray}\nfor $n=2,3,...,N-1$.\n\nIn order to include the boundary terms into (\\ref{eq24}) and (\\ref{eq25}) we\nalso shall consider a reflection equation where the vertices are being\ncommuted horizontally: \n\\begin{equation}\n\\left[ L_{1}(u)L_{0}(u+2\\epsilon )K_{1}^{-}(\\epsilon )\\right]\nK_{2}^{-}(u+\\epsilon )=\\left[ K_{1}^{-}(\\epsilon )L_{1}(u+2\\epsilon\n)L_{0}(u)\\right] K_{2}^{-}(u+\\epsilon ) \\label{eq26}\n\\end{equation}\nThis equation can be represented graphically by figure $4$\n\n\\begin{center}\n\\setlength{\\unitlength}{2947sp}\\begingroup\\makeatletter\\ifx\\SetFigFont%\n\\undefined\\gdef\\SetFigFont#1#2#3#4#5{\\reset@font\\fontsize{#1}{#2pt} %\n\\fontfamily{#3}\\fontseries{#4}\\fontshape{#5} \\selectfont}\\fi\\endgroup%\n\\begin{picture}(8325,1935)(1351,-3586)\n\\thinlines\n\\put(1501,-3361){\\line( 1, 0){3000}}\n\\put(1501,-2461){\\line( 5,-3){1500}}\n\\put(3001,-3361){\\line( 5, 3){1500}}\n\\put(1501,-1861){\\line( 1,-2){750}}\n\\put(2251,-3361){\\line( 1, 2){750}}\n\\put(6601,-3361){\\line( 1, 0){3000}}\n\\put(8131,-1846){\\line( 1,-2){750}}\n\\put(8881,-3346){\\line( 1, 2){750}}\n\\put(6601,-2461){\\line( 5,-3){1500}}\n\\put(8101,-3361){\\line( 5, 3){1500}}\n\\put(1301,-1711){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$u$+$\\epsilon$}}}\n\\put(2801,-1786){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}-$u$-$\\epsilon$}}}\n\\put(1151,-2461){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$\\epsilon$}}}\n\\put(4376,-2386){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}-$\\epsilon$}}}\n\\put(1776,-3586){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$K^{\\scriptstyle{-}}_{2}(u$+$\\epsilon)$}}}\n\\put(2801,-3586){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$K^{\\scriptstyle{-}}_{1}(\\epsilon)$}}}\n\\put(6326,-2386){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$\\epsilon$}}}\n\\put(7901,-1711){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$u$+$\\epsilon$}}}\n\\put(9401,-1711){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}-$u$-$\\epsilon$}}}\n\\put(9476,-2461){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}-$\\epsilon$}}}\n\\put(7626,-3586){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$K^{\\scriptstyle{-}}_{1}(\\epsilon)$}}}\n\\put(8701,-3586){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$K^{\\scriptstyle{-}}_{2}(u$+$\\epsilon)$}}}\n\\put(1751,-2611){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{1}(u)$}}}\n\\put(2406,-2986){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{0}(u$+$2\\epsilon)$}}}\n\\put(7551,-2961){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{1}(u$+$2\\epsilon)$}}}\n\\put(9076,-2836){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{6}{7.2}{\\rmdefault}{\\mddefault}{\\updefault}$L_{0}(u)$}}}\n\\end{picture}\n\nfigure $4$\n\\end{center}\n\nUsing the the normalization $K^{-}(0)=1$ and \n\\begin{equation}\nK^{-}(\\epsilon )\\sim 1+\\epsilon K^{-^{\\prime }}(0)+o(\\epsilon ^{2}),\n\\label{eq27}\n\\end{equation}\nwe get from (\\ref{eq26}) a new commutation relation \n\\begin{equation}\n\\lbrack \\frac{1}{2}K_{1}^{-\\prime\n}(0),L_{1}(u)L_{0}(u)]=L_{1}(u)L_{0}^{\\prime }(u)-L_{1}^{\\prime }(u)L_{0}(u)\n\\label{eq28}\n\\end{equation}\nWe multiply out the above equation and arrive at \n\\begin{equation}\n\\lbrack \\frac{1}{2}K_{1}^{-\\prime }(0),L_{1}]=(L_{1}L_{0}^{\\prime\n}L_{0}^{-1}-L_{1}^{\\prime })-L_{1}[\\frac{1}{2}K_{1}^{-\\prime\n}(0),L_{1}]L_{0}^{-1} \\label{eq29}\n\\end{equation}\nNote that in these equations we have extended the chain in order to include\nthe site $n=0$ at the right boundary and made use of the non-singular\nproperty of the $L$-operators.\n\nNow we recall the equation (\\ref{eq24}) to see that the identification \n\\begin{equation}\n\\alpha ^{-1}H_{1,0}=\\frac{1}{2}K_{1}^{-\\prime }(0)\\quad \\label{eq210}\n\\end{equation}\nallows its continuation for $n=1$.\n\nBy similar considerations one can derive the left boundary relations. Now\nthe equation (\\ref{eq25}) is continued for $n=N$ with the identification \n\\begin{equation}\n\\alpha ^{-1}H_{N+1,N}=\\frac{{\\rm tr}\\left[ K_{a}^{+}(0)H_{Na}\\right] }{{\\rm %\ntr}K^{+}(0)} \\label{eq211}\n\\end{equation}\n\nThese results tell us that we can keep all considerations made by Zhang for\nthe periodic case\\cite{Zhang}. The main difference consist in remove the\nterm $H_{N,N+1}=H_{N,1}$ from the periodic Hamiltonian and adding two\nboundary terms $H_{1,0}$ and $H_{N+1,N}$ determined by the matrices $K^{\\pm\n}(u)$.\n\nThe equation of motion for $L_{n}$ is the Heisenberg equation \n\\begin{equation}\n\\stackrel{.}{L}_{n}={\\rm i}[{\\cal H},L_{n}]={\\rm i}[H_{n+1,n},L_{n}]+{\\rm i}%\n[H_{n,n-1},L_{n}], \\label{eq212}\n\\end{equation}\nhere we have set $\\hbar =1$. The second equality is because of the locality\nof the ${\\cal H}$ and $L_{n}$. Combining (\\ref{eq24}),(\\ref{eq25}) and (\\ref\n{eq212}), we have \n\\begin{equation}\n\\stackrel{.}{L}_{n}\\!=\\!-{\\rm i}L_{n+1}^{-1}\\!\\left\\{ \\!\\alpha\nL_{n+1}^{\\prime }\\!+\\![H_{n+1,n},L_{n+1}]\\!\\right\\} \\!L_{n}\\!+\\!{\\rm i}%\nL_{n}\\!\\left\\{ \\!\\alpha L_{n-1}^{\\prime }\\!-\\![H_{n,n-1},L_{n-1}]\\!\\right\\}\n\\!L_{n-1}^{-1} \\label{eq213}\n\\end{equation}\nBy comparing with the Lax equation (\\ref{eq13}), we can read off the second\nLax operator \n\\begin{equation}\nA_{n}=-{\\rm i}L_{n}^{-1}\\!\\left\\{ \\alpha L_{n}^{\\prime }\\!+\\left[\nH_{n,n-1},L_{n}\\right] \\!\\!\\right\\} \\! \\label{eq214}\n\\end{equation}\nor \n\\begin{equation}\nA_{n}=-{\\rm i}\\left\\{ \\!\\alpha L_{n-1}^{\\prime\n}\\!-\\![H_{n,n-1},L_{n-1}]\\!\\right\\} \\!L_{n-1}^{-1} \\label{eq215}\n\\end{equation}\nThe compatibility of (\\ref{eq214}) and (\\ref{eq215}) is guaranteed by the\ncommutation relation (\\ref{eq24}).\n\nTherefore, the second Lax operator for completely integrable open chains has\nthe following form: In the bulk it is the same for the corresponding\nperiodic chain \n\\begin{eqnarray}\nA_{n} &=&{\\rm i}H_{n,n-1}-{\\rm i}L_{n}^{-1}H_{n,n-1}L_{n}-{\\rm i}\\alpha\nL_{n}^{-1}\\!L_{n}^{\\prime }\\! \\nonumber \\\\\nn &=&2,3,...,N, \\label{eq216}\n\\end{eqnarray}\nat the right boundary it is given by \n\\begin{equation}\nQ_{1}={\\rm i}\\alpha \\left\\{ \\frac{1}{2}K_{1}^{-\\prime }(0)-\\frac{1}{2}%\nL_{1}^{-1}K_{1}^{-\\prime }(0)L_{1}-L_{1}^{-1}\\!L_{1}^{\\prime }\\right\\}\n\\label{eq217}\n\\end{equation}\nand at the left boundary it is read off from the equation (\\ref{eq215}): \n\\begin{equation}\nQ_{N}={\\rm i}H_{N+1,N}-{\\rm i}L_{N}\\ H_{N+1,N}\\ L_{1}^{-1}-{\\rm i}\\alpha\nL_{N}^{\\prime }\\!\\ L_{N}^{-1}, \\label{eq218}\n\\end{equation}\nwhere $H_{N+1,N}$ is given by (\\ref{eq211}).\n\nFinally, we shall apply this method to a concrete model, the one-dimensional\nHeisenberg {\\small XXZ} open chain with Hamiltonian \\cite{Sklyanin, deVega}: \n\\begin{eqnarray}\n{\\cal H} &=&-\\sum_{k=2}^{N}\\left( \\sigma _{k}^{+}\\sigma _{k-1}^{-}+\\sigma\n_{k}^{-}\\sigma _{k-1}^{+}+\\frac{1}{2}\\cos 2\\eta \\ \\sigma _{k}^{z}\\sigma\n_{k-1}^{z}\\right) \\nonumber \\\\\n&&+\\sin 2\\eta \\left( A_{-}\\sigma _{1}^{z}+B_{-}\\sigma _{1}^{+}+C_{-}\\sigma\n_{1}^{-}+A_{+}\\sigma _{N}^{z}+B_{+}\\sigma _{N}^{+}+C_{+}\\sigma\n_{N}^{-}\\right) \\label{eq31}\n\\end{eqnarray}\nwhere \n\\begin{equation}\nA_{\\mp }=\\frac{1}{2}\\cot (\\xi _{\\mp })\\ ,\\ B_{\\mp }=\\frac{b_{\\mp }}{\\sin \\xi\n_{\\mp }}\\ ,\\ C_{\\mp }=\\frac{c_{\\mp }}{\\sin \\xi _{\\mp }} \\label{eq32}\n\\end{equation}\nHere $\\sigma ^{x},\\sigma ^{y},\\sigma ^{z}$ and $\\sigma ^{\\pm }=\\frac{1}{2}%\n(\\sigma ^{x}\\pm {\\rm i}\\sigma ^{y})$ are the usual Pauli spin-$\\frac{1}{2}$\noperators. $\\eta $ is a parameter associated with the anisotropy, $b_{\\mp }$%\n, $c_{\\mp }$ and $\\xi _{\\mp }$ are some constants describing the boundary\neffects.\n\nIt is not difficult to verify that the equations of motion derived from the\nHamiltonian (\\ref{eq31}) may be cast in the Lax form (\\ref{eq13}) and (\\ref\n{eq14}).\n\nThe first Lax operator $L_{n}$ is identified with the $R$-matrix of the $6$%\n-vertex model \\cite{KS} \n\\begin{eqnarray}\nL_{n} &=&R_{na}=\\left( \n\\begin{array}{ccc}\nw_{4}+w_{3}\\sigma _{n}^{z} & & 2w_{1}\\sigma _{n}^{-} \\\\ \n& & \\\\ \n2w_{1}\\sigma _{n}^{+} & & w_{4}-w_{3}\\sigma _{n}^{z}\n\\end{array}\n\\right) \\nonumber \\\\\n&=&w_{4}+w_{3}\\sigma ^{z}\\sigma _{n}^{z}+2w_{1}(\\sigma ^{-}\\sigma\n_{n}^{+}+\\sigma ^{+}\\sigma _{n}^{-}) \\label{eq33}\n\\end{eqnarray}\nwhere the elements are parametrized by trigonometric functions of $u$%\n\\begin{equation}\nw_{4}+w_{3}=\\sin (u+2\\eta ),\\quad w_{4}-w_{3}=\\sin u,\\quad 2w_{1}=\\sin 2\\eta\n\\label{eq34}\n\\end{equation}\nUsing the $\\epsilon $-expansion (\\ref{eq22a}) we have \n\\begin{equation}\n\\alpha =-\\sin 2\\eta \\quad {\\rm and}\\quad \\beta =-\\frac{1}{2}\\cot 2\\eta\n\\label{eq35}\n\\end{equation}\nThe inverse and the first derivative of $L_{n}$ are well-defined: \n\\begin{eqnarray}\nL_{n}^{-1} &=&v_{4}+v_{3}\\sigma ^{z}\\sigma _{n}^{z}+2v_{1}\\left( \\sigma\n^{-}\\sigma _{n}^{+}+\\sigma ^{+}\\sigma _{n}^{-}\\right) \\nonumber \\\\\nL_{n}^{\\prime } &=&w_{4}^{\\prime }+w_{3}^{\\prime }\\sigma ^{z}\\sigma _{n}^{z}\n\\label{eq36}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nv_{4}+v_{3} &=&\\frac{1}{\\Delta }\\sin (u-2\\eta ),\\quad v_{4}-v_{3}=\\frac{1}{%\n\\Delta }\\sin u,\\quad 2v_{1}=-\\frac{1}{\\Delta }\\sin 2\\eta \\nonumber \\\\\n\\Delta &=&\\sin (u-2\\eta )\\sin (u+2\\eta ) \\label{eq37}\n\\end{eqnarray}\nNow, by direct substitution of these data in (\\ref{eq216}) one can easily\nfind the second Lax operator $A_{n}$ \\ for $n=2,3,...,N$ : \n\\begin{eqnarray}\nA_{n} &=&{\\rm iconst.}I+{\\rm i}d(u)\\sigma _{n}^{z}\\sigma _{n-1}^{z}+{\\rm i}%\nf(u)\\left( \\sigma _{n}^{-}\\sigma _{n-1}^{+}+\\sigma _{n}^{+}\\sigma\n_{n-1}^{-}\\right) \\nonumber \\\\\n&&+{\\rm i}\\sigma ^{z}\\left[ g(u)\\left( \\sigma _{n}^{-}\\sigma\n_{n-1}^{+}-\\sigma _{n}^{+}\\sigma _{n-1}^{-}\\right) -d(u)\\left( \\sigma\n_{n}^{z}+\\sigma _{n-1}^{z}\\right) \\right] \\nonumber \\\\\n&&-{\\rm i}\\sigma ^{+}\\left[ p(u)\\left( \\sigma _{n}^{-}\\sigma\n_{n-1}^{z}-\\sigma _{n}^{z}\\sigma _{n-1}^{-}\\right) +q(u)\\left( \\sigma\n_{n}^{-}+\\sigma _{n-1}^{-}\\right) \\right] \\nonumber \\\\\n&&-{\\rm i}\\sigma ^{-}\\left[ p(u)\\left( \\sigma _{n}^{z}\\sigma\n_{n-1}^{+}-\\sigma _{n}^{+}\\sigma _{n-1}^{z}\\right) +q(u)\\left( \\sigma\n_{n}^{+}+\\sigma _{n-1}^{+}\\right) \\right] \\label{eq38}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nd(u) &=&\\frac{1}{4\\Delta }\\sin 4\\eta \\sin 2\\eta ,\\quad f(u)=\\frac{1}{\\Delta }%\n\\sin \\eta \\left( \\sin \\eta \\cos 2u+\\sin 3u\\right) , \\nonumber \\\\\ng(u) &=&\\frac{1}{2\\Delta }\\sin 2\\eta \\sin 2u,\\quad p(u)=\\frac{1}{2\\Delta }%\n\\sin 4\\eta \\sin u,\\quad q(u)=\\frac{1}{2\\Delta }\\sin 4\\eta \\sin u \\nonumber\n\\\\\n{\\rm const.} &=&\\frac{1}{2\\Delta }\\sin 2\\eta (\\sin 2u-\\sin 2\\eta \\cos 2\\eta )\n\\label{eq39}\n\\end{eqnarray}\nThis solution is the Lax operator for the periodic {\\small XXZ \\ }spin chain\nwhich was obtained by Sogo and Wadati \\cite{SW} in the trigonometric limit\nof the Lax pair operators for the one-dimensional {\\small XYZ} Heisenberg\nspin chain.\n\nNow we have to compute the corresponding boundary operators. By a direct\ncalculation with the operators $L_{1}$ and $L_{1}^{-1}$ given by (\\ref{eq36}%\n) and together with (\\ref{eq210}), one gets the following results for each\nterm of $Q_{1}$: \n\\begin{equation}\n{\\rm i}H_{10}=-{\\rm i}\\alpha \\left( A_{-}\\sigma _{1}^{z}+B_{-}\\sigma\n_{1}^{+}+C_{-}\\sigma _{1}^{-}\\right) \\label{eq310}\n\\end{equation}\n\\begin{eqnarray}\n-{\\rm i}\\alpha L_{1}^{-1}L_{1} &=&-{\\rm i}\\alpha \\left( v_{4}w_{4}^{\\prime\n}+v_{3}w_{3}^{\\prime }\\right) -{\\rm i}\\alpha \\left( v_{4}w_{3}^{\\prime\n}+v_{3}w_{4}^{\\prime }\\right) \\sigma ^{z}\\sigma _{1}^{z} \\nonumber \\\\\n&&-2{\\rm i}v_{1}\\left( w_{4}^{\\prime }-w_{3}^{\\prime }\\right) \\left( \\sigma\n^{-}\\sigma _{1}^{+}+\\sigma ^{+}\\sigma _{1}^{-}\\right) \\label{eq311}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n&&-{\\rm i}L_{1}^{-1}H_{01}L_{1}={\\rm i}\\alpha\nA_{-}(v_{4}w_{4}+v_{3}w_{3}-2v_{1}w_{1})\\sigma _{1}^{z}+{\\rm i}\\alpha\nA_{-}(v_{4}w_{3}+v_{3}w_{4}+2v_{1}w_{1})\\sigma ^{z} \\nonumber \\\\\n&&+2{\\rm i}\\alpha A_{-}\\left[ (v_{4}-v_{3})w_{1}-v_{1}(w_{4}-w_{3})\\right]\n(\\sigma ^{-}\\sigma _{1}^{+}-\\sigma ^{+}\\sigma _{1}^{-}) \\nonumber \\\\\n&&+{\\rm i}\\alpha (v_{4}w_{4}-v_{3}w_{3})(B_{-}\\sigma _{1}^{+}+C_{-}\\sigma\n_{1}^{-})+{\\rm i}\\alpha (v_{3}w_{4}-v_{4}w_{3})\\sigma ^{z}(B_{-}\\sigma\n_{1}^{+}-C_{-}\\sigma _{1}^{-}) \\nonumber \\\\\n&&+{\\rm i}\\alpha \\left[ w_{1}(v_{4}+v_{3})+v_{1}(w_{4}+w_{3})\\right]\n(B_{-}\\sigma ^{+}+C_{-}\\sigma ^{-}) \\nonumber \\\\\n&&+{\\rm i}\\alpha \\left[ w_{1}(v_{4}+v_{3})-v_{1}(w_{4}+w_{3})\\right]\n(B_{-}\\sigma ^{+}-C_{-}\\sigma ^{-})\\sigma _{1}^{z} \\label{eq312}\n\\end{eqnarray}\nWe thus find the following Lax operator \n\\begin{eqnarray}\nQ_{1} &=&{\\rm iconst.}{\\bf 1}+\\frac{{\\rm i}\\sin ^{2}2\\eta }{\\Delta \\sin \\xi\n_{-}}\\times \\nonumber \\\\\n&& \\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\left( \n\\begin{array}{ccc}\n\\begin{array}{c}\n-\\frac{1}{2}\\sin (2\\eta +\\xi _{-})\\sigma _{1}^{z}+\\frac{1}{2}\\cos \\xi\n_{-}\\sin 2\\eta \\\\ \n+b_{-}t(u)\\sigma _{1}^{+}+c_{-}r(u)\\sigma _{1}^{-}\n\\end{array}\n& & \n\\begin{array}{c}\n\\sin (u-\\xi _{-})\\sigma _{1}^{-} \\\\ \n+b_{-}\\sin 2\\eta \\cos u-c_{-}\\sin u\\cos 2\\eta \\sigma _{1}^{z}\n\\end{array}\n\\\\ \n& & \\\\ \n\\begin{array}{c}\n-\\sin (u+\\xi _{-})\\sigma _{1}^{+} \\\\ \n+b_{-}\\sin 2\\eta \\cos u+c_{-}\\sin u\\cos 2\\eta \\sigma _{1}^{z}\n\\end{array}\n& & \n\\begin{array}{c}\n-\\frac{1}{2}\\sin (2\\eta -\\xi _{-})\\sigma _{1}^{z}-\\frac{1}{2}\\cos \\xi\n_{-}\\sin 2\\eta \\\\ \n+b_{-}r(u)\\sigma _{1}^{+}+c_{-}t(u)\\sigma _{1}^{-}\n\\end{array}\n\\end{array}\n\\right) \\nonumber \\\\\n&& \\label{eq314}\n\\end{eqnarray}\nwhere \n\\begin{equation}\nt(u)=\\frac{\\sin u\\cos (u-\\eta )}{\\cos \\eta }-\\sin 2\\eta \\quad {\\rm and}\\quad\nr(u)=\\frac{\\sin u\\cos (u+\\eta )}{\\cos \\eta }-\\sin 2\\eta . \\label{eq315}\n\\end{equation}\nThe left boundary operator $Q_{N}$ is obtained using the equation (\\ref\n{eq218}). This results that $\\ Q_{N}$ \\ is the transposition of $Q_{1}$,\nfollowed by the following substitution: \n\\begin{equation}\n\\xi _{-}\\rightarrow \\xi _{+},\\quad b_{-}\\rightarrow c_{+},\\quad\nc_{-}\\rightarrow b_{+}\\quad {\\rm and}\\quad \\sigma _{1}\\rightarrow \\sigma _{N}\n\\label{eq316}\n\\end{equation}\nThese Lax operators are the trigonometric limit of the Lax pair for the open \n{\\small XYZ} spin chain given in ref.\\cite{Ju}. In particular, the diagonal\ncase ($b_{\\mp }=c_{\\mp }=0$), was first derived in \\cite{Zhou}.\n\nTo summarize: solving the Yang-Baxter equation together with the reflection\nequations, we can read off $L_{n}$ , ${\\cal R}$ and $K^{\\mp }$ operators; if \n$\\ {\\cal H}$ has the form (\\ref{eq9}) with its bulk (\\ref{eq1}) is related\nto ${\\cal R}$ by (\\ref{eq22a}), then the corresponding $A_{n}$ operator is\ngiven by (\\ref{eq216}). Moreover, for every solution $\\ K^{-}$ ($K^{+}$)\\ of\nthe reflection equation (\\ref{eq6}) ((\\ref{eq7})) with $K^{-}(0)\\neq 0$ ($%\n{\\rm Tr}K^{+}(0)\\neq 0$), we can read off the right (left) boundary term of $%\n{\\cal H}$ which has the form (\\ref{eq210})\\ ((\\ref{eq211})), then the\ncorresponding $Q_{1}$ ($Q_{N}$) operator is given by (\\ref{eq217}) ((\\ref\n{eq218})).\n\n\\vspace{0.5cm}{}\n\n{\\bf Acknowledgment:} This work was supported in part by Funda\\c{c}\\~{a}o de\nAmparo \\`{a} Pesquisa do Estado de S\\~{a}o Paulo--FAPESP--Brasil and by\nConselho Nacional de Desenvol\\-{}vimento--CNPq--Brasil.\n\n\\begin{thebibliography}{99}\n\\bibitem{Wu} {\\rm McCoy B M and Wu T T 1968} {\\it Nouvo Cimento} {\\bf B56} \n{\\rm 311}.\n\n\\bibitem{Baxter1} {\\rm Baxter R J 1971} {\\it Phys. Lett.} {\\bf 26} {\\rm 832}%\n; {\\rm 1972} {\\it Ann. Phys.} {\\bf 70} {\\rm 193}.\n\n\\bibitem{Faddeev} {\\rm Faddeev L D, Sklyanin E K and Takhtajan L A 1979} \n{\\it Theor. Math. Phys.} {\\bf 40} {\\rm 194}.\n\n\\bibitem{Wadati} {\\rm Wadati M and Akutsu M 1988} {\\it Prog. Theor. Phys.} \n{\\bf 94} {\\rm 1}.\n\n\\bibitem{Sklyanin} {\\rm Sklyanin E K 1988} {\\it J. Phys. A: Math. Gen.} \n{\\bf 21} {\\rm 2375}.\n\n\\bibitem{Mezin} {\\rm Mezincescu L and Nepomechie R I 1991} {\\it J. Phys. A:\nMath. Gen. }{\\sl \\ }{\\bf 24} {\\rm L17}.\n\n\\bibitem{Zhang} {\\rm Zhang M Q 1991} {\\it Commun. Math. Phys.} {\\bf 141} \n{\\rm 523}.\n\n\\bibitem{Thacker} {\\rm Thacker H B 1981} {\\it Rev. Mod. Phys.} {\\bf 53} \n{\\rm 253}.\n\n\\bibitem{Helen} {\\rm Au-Yang H, McCoy B M, Perk J H H , Tang S and Yan M-L\n1987} {\\it Phys. Lett}{\\sl .} {\\bf A123} {\\rm 219}.\n\n\\bibitem{BP} {\\rm Bashilov Yu A and Pokrovsky S V 1980} {\\it Commun. Math.\nPhys.} {\\bf 76} {\\rm 129}.\n\n\\bibitem{ALS} {\\rm Alcaraz F C and Lima Santos A 1986} {\\it Nucl. Phys.} \n{\\bf B275} {\\rm [FS17] 436}.\n\n\\bibitem{SW} {\\rm Sogo K and Wadati M 1982} {\\it Prog. Theor. Phys.} {\\bf 68%\n} {\\rm 85}.\n\n\\bibitem{deVega} {\\rm de Vega H J and Gonz\\'{a}lez-Ruiz A 1993} {\\it J.\nPhys. A: Math. Gen.} {\\bf 26} {\\rm L519}.\n\n\\bibitem{KS} {\\rm Kulish P P and Sklyanin E K 1979} {\\it Phys. Lett.} {\\bf %\nA70} {\\rm 461}.\n\n\\bibitem{Ju} {\\rm Ju G-xing, Wang S-kun, Wu ke and Xiong Chi 1997} {\\it %\nBoundary K-matrices and the Lax pair for 1D open XYZ spin chain} {\\rm %\nsolv-int/9712011}.\n\n\\bibitem{Zhou} {\\rm Zhou H Q 1966} {\\it J. Phys. A: Math. Gen.} {\\bf 29} \n{\\rm L489}.\n\\end{thebibliography}\n\n\\end{document}\n" } ]	[ { "name": "solv-int9912009.extracted_bib", "string": "{Wu {\\rm McCoy B M and Wu T T 1968 {Nouvo Cimento {B56 {\\rm 311." }, { "name": "solv-int9912009.extracted_bib", "string": "{Baxter1 {\\rm Baxter R J 1971 {Phys. Lett. {26 {\\rm 832% ; {\\rm 1972 {Ann. Phys. {70 {\\rm 193." }, { "name": "solv-int9912009.extracted_bib", "string": "{Faddeev {\\rm Faddeev L D, Sklyanin E K and Takhtajan L A 1979 {Theor. Math. Phys. {40 {\\rm 194." }, { "name": "solv-int9912009.extracted_bib", "string": "{Wadati {\\rm Wadati M and Akutsu M 1988 {Prog. Theor. Phys. {94 {\\rm 1." }, { "name": "solv-int9912009.extracted_bib", "string": "{Sklyanin {\\rm Sklyanin E K 1988 {J. Phys. A: Math. Gen. {21 {\\rm 2375." }, { "name": "solv-int9912009.extracted_bib", "string": "{Mezin {\\rm Mezincescu L and Nepomechie R I 1991 {J. Phys. A: Math. Gen. {\\sl \\ {24 {\\rm L17." }, { "name": "solv-int9912009.extracted_bib", "string": "{Zhang {\\rm Zhang M Q 1991 {Commun. Math. Phys. {141 {\\rm 523." }, { "name": "solv-int9912009.extracted_bib", "string": "{Thacker {\\rm Thacker H B 1981 {Rev. Mod. Phys. {53 {\\rm 253." }, { "name": "solv-int9912009.extracted_bib", "string": "{Helen {\\rm Au-Yang H, McCoy B M, Perk J H H , Tang S and Yan M-L 1987 {Phys. Lett{\\sl . {A123 {\\rm 219." }, { "name": "solv-int9912009.extracted_bib", "string": "{BP {\\rm Bashilov Yu A and Pokrovsky S V 1980 {Commun. Math. Phys. {76 {\\rm 129." }, { "name": "solv-int9912009.extracted_bib", "string": "{ALS {\\rm Alcaraz F C and Lima Santos A 1986 {Nucl. Phys. {B275 {\\rm [FS17] 436." }, { "name": "solv-int9912009.extracted_bib", "string": "{SW {\\rm Sogo K and Wadati M 1982 {Prog. Theor. Phys. {68% {\\rm 85." }, { "name": "solv-int9912009.extracted_bib", "string": "{deVega {\\rm de Vega H J and Gonz\\'{alez-Ruiz A 1993 {J. Phys. A: Math. Gen. {26 {\\rm L519." }, { "name": "solv-int9912009.extracted_bib", "string": "{KS {\\rm Kulish P P and Sklyanin E K 1979 {Phys. Lett. {% A70 {\\rm 461." }, { "name": "solv-int9912009.extracted_bib", "string": "{Ju {\\rm Ju G-xing, Wang S-kun, Wu ke and Xiong Chi 1997 {% Boundary K-matrices and the Lax pair for 1D open XYZ spin chain {\\rm % solv-int/9712011." }, { "name": "solv-int9912009.extracted_bib", "string": "{Zhou {\\rm Zhou H Q 1966 {J. Phys. A: Math. Gen. {29 {\\rm L489." } ]
solv-int9912010		[ { "author": "O. LECHTENFELD~$^\\dag$ and A. SORIN~$^\\ddag$" } ]	\noindent Three nonequivalent real forms of the complex twisted N=2 supersymmetric Toda chain hierarchy (solv-int/9907021) in real N=1 superspace are presented. It is demonstrated that they possess a global twisted N=2 supersymmetry. We discuss a new superfield basis in which the supersymmetry transformations are local. Furthermore, a representation of this hierarchy is given in terms of two twisted chiral N=2 superfields. The relations to the s-Toda hierarchy by H. Aratyn, E. Nissimov and S. Pacheva (solv-int/9801021) as well as to the modified and derivative NLS hierarchies are established.	[ { "name": "74lett3.tex", "string": "\\documentclass{jnmp01b}\n\n\\usepackage{amsmath}\n%\\usepackage{graphicx}\n%\\usepackage{latexsym}\n\n\n\\setcounter{page}{433}\n\n\\numberwithin{equation}{section}\n\n\n%%%%% DOCUMENT SPECIFIC DEFINITIONS \n\n\\allowdisplaybreaks\n\n\\newcommand{\\p}[1]{(\\ref{#1})}\n\n% Fix for old versions of AMS-LaTeX to accept \\over\n%\n\\makeatletter\n\\DeclareRobustCommand{\\primfrac}[1]{%\n \\PackageWarning{amsmath}{%\nForeign command \\@backslashchar#1; %\n\\protect\\frac\\space or \\protect\\genfrac\\space should be used instead%\n }\n \\global\\@xp\\let\\csname#1\\@xp\\endcsname\\csname @@#1\\endcsname\n \\csname#1\\endcsname\n}\n\\makeatother\n\n%%%%% END DOCUMENT SPECIFIC DEFINITIONS\n\n\\begin{document}\n\n\\renewcommand{\\evenhead}{O. Lechtenfeld and A. Sorin}\n\\renewcommand{\\oddhead}{Real Forms of \nSupersymmetric Toda Chain Hierarchy}\n\n% Title\n\n\\thispagestyle{empty}\n\n\\begin{flushleft}\n\\footnotesize \\sf\nJournal of Nonlinear Mathematical Physics \\qquad 2000, V.7, N~4,\n\\pageref{firstpage}--\\pageref{lastpage}.\n\\hfill {\\sc Letter}\n\\end{flushleft}\n\n\\vspace{-5mm}\n\n\\copyrightnote{2000}{O. Lechtenfeld and A. Sorin}\n\n\\Name{Real Forms of the Complex Twisted N=2 Supersymmetric Toda Chain \nHierarchy in Real N=1 and Twisted N=2 Superspaces}\n\n\\label{firstpage}\n\n\\Author{O. LECHTENFELD~$^\\dag$ and A. SORIN~$^\\ddag$}\n\n\\Adress{$^\\dag$ Institut f\\\"ur Theoretische Physik, Universit\\\"at Hannover,\\\\\n~~Appelstra{\\ss}e 2, D-30167 Hannover, Germany \\\\\n~~lechtenf@itp.uni-hannover.de\\\\[10pt]\n$^\\ddag$ Bogoliubov Laboratory of Theoretical Physics, JINR,\\\\\n~~141980 Dubna, Moscow Region, Russia \\\\\n~~sorin@thsun1.jinr.ru}\n\n\\Date{Received March 1, 2000; Revised March 29, 2000; \nAccepted May 22, 2000}\n\n\n\n\\begin{abstract}\n\\noindent\nThree nonequivalent real forms of the complex twisted N=2 supersymmetric\nToda chain hierarchy (solv-int/9907021) in real N=1 superspace are presented. \nIt is demonstrated that they possess a global twisted N=2 supersymmetry. \nWe discuss a new superfield basis in which the supersymmetry transformations \nare local. Furthermore, a representation of this hierarchy is given in terms \nof two twisted chiral N=2 superfields. \nThe relations to the s-Toda hierarchy by H. Aratyn, E. Nissimov and \nS. Pacheva (solv-int/9801021) as well as to the modified and derivative NLS \nhierarchies are established.\n\\end{abstract}\n\n\n\n% The article\n\n\n\n\\section{Introduction}\n\nRecently an $N{=}(1\|1)$ supersymmetric generalization of the \ntwo-dimensional Darboux transformation was proposed in \\cite{ls}\nin terms of $N=(1\|1)$ superfields, and an infinite class of bosonic and\nfermionic solutions of its symmetry equation was constructed in \\cite{ls}\nand \\cite{ols}, respectively. These solutions generate bosonic and\nfermionic flows of the complex $N=(1\|1)$ supersymmetric Toda lattice\nhierarchy\\footnote{A wide class of the complex Toda\nlattices connected with Lie superalgebras was first\nintroduced in the pioneering papers \\cite{olshanetsky,leites,andreev}\n(see also recent papers \\cite{evans} and references therein).}\n which actually possesses a more rich symmetry, namely \ncomplex $N=(2\|2)$ supersymmetry. Its one-dimensional reduction possessing\ncomplex $N=4$ supersymmetry ---the complex $N=4$ Toda chain\nhierarchy--- was discussed in \\cite{dgs}. There, the Lax\npair representations of the bosonic and fermionic flows, the corresponding\nlocal and nonlocal Hamiltonians, finite and infinite discrete symmetries,\nthe first two Hamiltonian structures and the recursion operator were\nconstructed. Furthermore, its nonequivalent real forms in real $N=2$\nsuperspace were analyzed in \\cite{ds}, where the relation to the complex \n$N=4$ supersymmetric KdV hierarchy \\cite{di} was established. \n\nConsecutively, the reduction of the complex $N=4$ supersymmetric Toda chain\nhierarchy from complex $N=2$ superspace to complex $N=1$ superspace was\nanalyzed in \\cite{ols1}, where also its Lax-pair and Hamiltonian\ndescriptions were developed in detail. Here, we call this\nreduction the {\\it complex twisted N=2 supersymmetric Toda chain\nhierarchy}, due to the common symmetry properties of its three different\nreal forms which will be discussed in what follows (see the paragraph after\neq. \\p{conjn23}). The main goals of the present letter are firstly to\nanalyze real forms of this hierarchy in real $N=1$\nsuperspace with one even and one odd real coordinate, secondly to derive a\nmanifest twisted $N=2$ supersymmetric representation of its simplest\nnon-trivial even flows in twisted $N=2$ superspace, and thirdly to clarify\nits relations (if any) with other\nknown hierarchies (s-Toda \\cite{anp1}, modified NLS and derivative NLS\nhierarchies). \n\nLet us start with a short summary of the results that we shall need\nconcerning the complex twisted $N=2$ supersymmetric Toda chain hierarchy\n(see \\cite{ols1,dgs,ls,ols} for more details).\n\nThe complex twisted $N=2$ supersymmetric Toda chain hierarchy in \ncomplex $N=1$ superspace comprises an infinite set of even and odd\nflows for two complex even $N=1$ superfields\n$u(z,\\theta)$ and $v(z,\\theta)$, where\n$z$ and $\\theta$ are complex even and odd\ncoordinates, respectively. The flows are generated by complex\neven and odd evolution derivatives\n$\\{{\\textstyle{\\partial\\over\\partial t_k}}, ~U_k\\}$ and $\\{ D_k,~Q_k\\}$\n($k \\in \\mathbb{N}$), respectively, with the following length\ndimensions:\n\\begin{equation}\n[{\\textstyle{\\partial\\over\\partial t_k}}]=[U_k]=-k, \\quad\n[D_k]=[Q_k]=-k+\\frac{1}{2},\n\\label{dimtimes}\n\\end{equation}\nwhich are derived by the reduction of the supersymmetric KP\nhierarchy in $N=1$ superspace \\cite{maninradul}, \ncharacterized by the Lax operator \n\\begin{equation}\nL=Q+v D^{-1}u.\n\\label{lax1}\n\\end{equation}\n$D$ and $Q$ are the odd covariant derivative\nand the supersymmetry generator, respectively,\n\\begin{equation}\nD\\equiv \\frac{\\partial}{\\partial {\\theta}} +\n{\\theta} {\\partial}, \\quad\nQ\\equiv \\frac{\\partial}{\\partial {\\theta}} -\n{\\theta} {\\partial}.\n\\label{algDQ0}\n\\end{equation}%\n\\resetfootnoterule%\nThey form the algebra\\footnote{We explicitly present only non-zero\nbrackets in this letter.}\n\\begin{equation}\n\\{ D,D\\} = +2{\\partial}, \\quad \\{ Q,Q\\} = -2{\\partial}.\n\\label{alg00}\n\\end{equation}\nThe first few of these flows are:\n\\begin{gather}\n{\\textstyle{\\partial\\over\\partial t_0}}\n\\left(\\begin{array}{cc} v\\\\ u \\end{array}\\right) =\n\\left(\\begin{array}{cc} +v\\\\ -u \\end{array}\\right), \\quad\n{\\textstyle{\\partial\\over\\partial t_1}}\n\\left(\\begin{array}{cc} v\\\\ u \\end{array}\\right) =\n{\\partial}\\left(\\begin{array}{cc} v\\\\ u \\end{array}\\right),\n\\label{eqs1}\n\\\\[1ex]\n\\begin{split}\n&{\\textstyle{\\partial\\over\\partial t_2}} v =\n+v~'' - 2uv(DQv)+(DQv^2u)+v^2(DQu) -2v(uv)^2,\n\\\\\n&{\\textstyle{\\partial\\over\\partial t_2}} u =\n-u~'' - 2uv(DQu)+(DQu^2v)+u^2(DQv) +2u(uv)^2,\n\\end{split}\n\\label{eqs}\n\\\\[1ex]\n\\begin{split}\n{\\textstyle{\\partial\\over\\partial t_3}} v ={}&\nv~''' +3(Dv)~'(Quv)-3(Qv)~'(Duv)\n+3v~'(Du)(Qv) \\\\\n&-3v~'(Qu)(Dv)+6vv~'(DQu)-6(uv)^2v~', \\\\\n{\\textstyle{\\partial\\over\\partial t_3}} u ={}&\nu~''' +3(Qu)~'(Duv)-3(Du)~'(Quv)\n+3u~'(Qv)(Du) \\\\\n&-3u~'(Dv)(Qu)\n+6uu~'(QDv)-6(uv)^2u~',\n\\end{split}\n\\label{flow3}\n\\\\[1ex]\n\\begin{split}\n&\nD_1 v= -Dv+ 2vQ^{-1}(uv),\n\\quad D_1 u= -Du- 2uQ^{-1}(uv), \\\\\n&\nQ_1 v= -Qv- 2vD^{-1}(uv),\n\\quad Q_1 u= -Qu+ 2uD^{-1}(uv), \n\\end{split}\n\\label{ff2-}\n\\\\[1ex]\nU_0\\left(\\begin{array}{cc} v\\\\ u \\end{array}\\right) =\n{\\theta}D \\left(\\begin{array}{cc} v\\\\ u \\end{array}\\right).\n%\\label{ueqs}\n\\label{qqq1}\n\\end{gather}\nThroughout this letter, we shall use the notation $u'=\\partial \nu={\\partial\\over\\partial z}u$. \nUsing the explicit expressions of the flows \\p{eqs1}--\\p{qqq1}, one\ncan calculate their algebra which has the following nonzero brackets:\n\\begin{gather}\n\\Bigl\\{D_k\\,,\\,D_l\\Bigr\\}=\n-2\\;\\frac{{\\partial}}{{\\partial t_{k+l-1}}}, \\quad\n\\Bigl\\{Q_k\\,,\\,Q_l\\Bigr\\}=\n+2\\;\\frac{{\\partial}}{{\\partial t_{k+l-1}}},\n\\label{alg1}\n\\\\[1ex]\n\\Bigl[U_k\\,,\\,D_l\\Bigr]=Q_{k+l}, \\quad\n~\\Bigl[U_k\\,,\\,Q_l\\Bigr]=D_{k+l}.\n\\label{algqqbar}\n\\end{gather}\nThis algebra produces an affinization of the algebra of global\ncomplex $N=2$ supersymmetry, together with an affinization of its\n$gl(1,\\mathbb{C})$ automorphisms. \nIt is the algebra of symmetries of the nonlinear even flows\n\\p{eqs}--\\p{flow3}. The generators may be realized in the superspace\n$\\{t_k,\\theta_k,\\rho_k, h_k \\}$,\n\\begin{equation}\n\\begin{split}\n& D_k= \\frac{\\partial}{\\partial \\theta_k}- \\sum^{\\infty}_{l=1}\\theta_l\n\\frac{\\partial}{{\\partial t_{k+l-1}}},\\quad\nQ_k=\\frac{\\partial}{\\partial {\\rho}_k}+\n\\sum^{\\infty}_{l=1}{\\rho}_l\n\\frac{\\partial}{{\\partial t_{k+l-1}}}, \\\\\n& \nU_k=\\frac{\\partial}{\\partial h_k}-\n\\sum^{\\infty}_{l=1}({\\theta}_l\n\\frac{\\partial}{{\\partial {\\rho}_{k+l}}}+{\\rho}_l\n\\frac{\\partial}{{\\partial {\\theta}_{k+l}}}),\n\\end{split}\n\\label{covder}\n\\end{equation}\nwhere $t_k, h_k$ ($\\theta_k,\\rho_k$) are bosonic (fermionic) abelian\nevolution times with length dimensions\n\\begin{equation}\n[t_k]=[h_k]=k, \\quad [\\theta_k] =[\\rho_k]=k-\\frac{1}{2}\n\\label{dim}\n\\end{equation}\nwhich are in one-to-one correspondence with the length dimensions\n\\p{dimtimes} of the corresponding evolution derivatives.\n\nThe flows $\\{{\\textstyle{\\partial\\over\\partial t_k}},~D_k,~Q_k\\}$ \ncan be derived from the flows \n$\\{{\\textstyle{\\partial\\over\\partial t_k}},~D^{+}_k,~D^{-}_k\\}$ \nof the complex $N=4$ Toda chain hierarchy \\cite{dgs} by the reduction\nconstraint \n\\begin{equation}\n{\\theta}^{+}=i{\\theta}^{-}\\equiv {\\theta} \n\\label{ccccc}\n\\end{equation}\nwhich leads to the correspondence $D_+\\equiv D$ and $D_-\\equiv iQ$ with the \nfermionic derivatives of the present paper, where $i$ is the imaginary unit \nand ${\\theta}^{\\pm}$ are the Grassmann coordinates of the $N=2$ superspace\nin \\cite{dgs}.\n\n\n\\section{Real forms of the complex twisted N=2 Toda chain\nhierarchy} \n\nIt is well known that different real forms derived from the\nsame complex integrable hierarchy are nonequivalent in general.\nKeeping this in mind it seems important to find as many\ndifferent real forms of the complex twisted $N=2$ Toda chain hierarchy as\npossible.\n\nWith this aim let us discuss various nonequivalent complex\nconjugations of the superfields $u(z,\\theta)$ and\n$v(z,\\theta)$, of the superspace coordinates $\\{z,~\\theta\\}$,\nand of the evolution derivatives\n$\\{{\\textstyle{\\partial\\over\\partial t_k}},~U_k,~D_k, ~Q_k\\}$\nwhich should be consistent with the flows \\p{eqs1}--\\p{qqq1}.\nWe restrict our considerations to the case\nwhen $iz$ and $\\theta$ are coordinates of real $N=1$ superspace\nwhich satisfy the following standard complex conjugation properties:\n\\begin{equation}\n(iz,{{\\theta}})^{}=(iz,{\\theta}).\n\\label{conj}\n\\end{equation}\nWe will also use the standard convention regarding complex conjugation of\nproducts involving odd operators and functions (see, e.g., the books\n\\cite{ggrs}). In particular, if $\\mathbb{D}$ is some even differential\noperator acting on a superfield $F$, we define the complex conjugate of \n$\\mathbb{D}$ by $(\\mathbb{D}F)^=\\mathbb{D}^F^$. Then, in the\ncase under consideration one can derive, for example, the following\nrelations\n\\begin{equation}\n\\begin{split}\n&{\\partial}^=-{\\partial}, \\quad\n{\\epsilon}^{}={\\epsilon}, \\quad\n{\\varepsilon}^{}={\\varepsilon}, \\quad\n({\\epsilon}{\\varepsilon})^{}=-{\\epsilon}{\\varepsilon},\\\\\n&({\\epsilon}D)^{}={\\epsilon}D, \\quad\n({\\varepsilon}Q)^{}={\\varepsilon}Q, \\quad (DQ)^{} = - DQ\n\\end{split}\n\\label{conjrel}\n\\end{equation}\nwhich we use in what follows. Here, ${\\epsilon}$ and ${\\varepsilon}$ \nare constant odd real parameters.\n\nLet us remark that, although most of the flows of the\ncomplex twisted $N=2$ supersymmetric Toda chain hierarchy can be derived by\nreduction \\p{ccccc}, its real forms in $N=1$ superspace \\p{conj} cannot be\nderived in this way from the real forms of the complex $N=4$ Toda chain\nhierarchy in the real $N=2$ superspace \n\\begin{equation}\n(iz,{{\\theta}}^{\\pm})^{}=(iz,{\\theta}^{\\pm})\n\\label{conjn4}\n\\end{equation}\nfound in \\cite{ds}. This conflict arises because the constraint \\p{ccccc} is\ninconsistent with the reality properties \\p{conjn4} of the $N=2$\nsuperspace. \n\nWe would like to underline that the flows \\p{eqs1}--\\p{qqq1} form a\nparticular realization of the algebra \\p{alg1}--\\p{algqqbar} in terms of\nthe $N=1$ superfields $u(z,\\theta)$ and $v(z,\\theta)$. Although\nthe classification of real forms of affine and conformal\nsuperalgebras was given in a series of classical papers\n\\cite{new1,new2} (see also interesting paper \\cite{new3} for\nrecent discussions and references therein) we cannot\nobtain the complex conjugations of the target space superfields\n$\\{u(z,\\theta), v(z,\\theta)\\}$ using only this base. It is\na rather different, non-trivial task to construct the\ncorresponding complex conjugations of various realizations\nof a superalgebra which are relevant in the context of\nintegrable hierarchies. Moreover, different complex\nconjugations of a given (super)algebra realization may\ncorrespond to the same real form of the (super)algebra, while\nsome of its other existing real forms may not be reproducible\non the base of a given particular realization. In what follows\nwe will demonstrate that this is exactly the case for the\nrealization under consideration. We shall see that complex conjugations\nof the target space superfields $\\{u(z,\\theta), v(z,\\theta)\\}$\ncorrespond to the twisted real $N=2$ supersymmetry.\n\nDirect verification shows that the flows \\p{eqs1}--\\p{qqq1}\nadmit the following three nonequivalent complex conjugations\n(meaning that it is not possible to relate them via obvious symmetries):\n\\begin{gather}\n\\begin{split}\n&(v,u)^{}= (v,-u), \\quad\n(iz,{{\\theta}})^{}=(iz,{\\theta}), \\\\\n&(t_p,U_p,{\\epsilon}_p D_p,\n{\\varepsilon}_pQ_p)^{}=(-1)^{p}(t_p,U_p,\n-{\\epsilon}_pD_p, -{\\varepsilon}_pQ_p), \n\\end{split}\n\\label{conj1}\n\\\\[1ex]\n\\begin{split}\n& (v,u)^{\\bullet}= (u,v), \\quad\n(iz,{{\\theta}})^{\\bullet}=(iz,{\\theta}), \\\\\n&(t_p,U_p,{\\epsilon}_p D_p,\n{\\varepsilon}_pQ_p)^{\\bullet}=(-t_p,U_p,\n{\\epsilon}_p D_p,{\\varepsilon}_pQ_p), \n\\end{split}\n\\label{conj2}\n\\\\[1ex]\n\\begin{split}\n& (v,~u)^{\\star}=(~-u(QD\\ln u+uv),~ \\frac{1}{u}~),\n\\quad (iz,{\\theta})^{\\star}=(iz,{\\theta}),\\\\\n& (t_p,U_p,{\\epsilon}_p\nD_p,{\\varepsilon}_pQ_p)^{\\star}=(-t_p,U_p,\n-{\\epsilon}_p D_p, -{\\varepsilon}_pQ_p),\n\\end{split}\n\\label{conj3}\n\\end{gather}\nwhere ${\\epsilon}_p$ and ${\\varepsilon}_p$\nare constant odd real parameters. We would like to underline that\nthe complex conjugations of the evolution derivatives (the second\nlines of eqs. \\p{conj1}--\\p{conj3} ) are defined and fixed\ncompletely by the explicit expressions \\p{eqs1}--\\p{qqq1} for the flows. \nThese complex conjugations extract three different real\nforms of the complex integrable hierarchy we started with,\nwhile all the real forms of the flows algebra \\p{alg1}--\\p{algqqbar}\ncorrespond to the same algebra of a twisted global real\n$N=2$ supersymmetry. \nThis last fact becomes obvious if one uses the $N=2$ basis \nof the algebra with the generators\n\\begin{equation}\n{\\cal D}_1\\equiv \\frac{1}{\\sqrt{2}}(Q_1+D_1), \\quad\n{\\overline {\\cal D}}_{1}\\equiv \\frac{1}{\\sqrt{2}} (Q_1-D_1).\n\\label{su(2)}\n\\end{equation}\nThen, the nonzero algebra brackets \\p{alg1}--\\p{algqqbar} and the\ncomplex conjugation rules \\p{conj1}--\\p{conj3} are the \nstandard ones for the twisted $N=2$ supersymmetry\nalgebra together with its non--compact $o(1,1)$ automorphism,\n\\begin{gather}\n\\Bigl\\{{\\cal D}_1\\,,\\,{\\overline {\\cal D}}_1\\Bigr\\}=\n2 \\;\\frac{{\\partial}}{{\\partial t_{1}}}, \\quad\n\\Bigl[U_0\\,,\\,{\\cal D}_1\\Bigr]=+{\\cal D}_1, \\quad \\\n~~\\Bigl[U_0\\,,\\,{\\overline {\\cal D}}_1\\Bigr]=-{\\overline {\\cal D}}_1,\n\\label{algnn2}\n\\\\[1ex]\n({\\textstyle{\\partial\\over\\partial t_1}},\nU_0,{\\gamma}_1{\\cal D}_1, {\\overline {\\gamma}}_1\n{\\overline {\\cal D}}_1)^{}=\n(-{\\textstyle{\\partial\\over\\partial t_1}},U_0,\n+{\\gamma}_1{\\cal D}_1,\n+{\\overline {\\gamma}}_1 {\\overline {\\cal D}}_1), \\quad\n({\\gamma}_1, {\\overline {\\gamma}}_1)^{}=\n({\\gamma}_1, {\\overline {\\gamma}}_1),\n\\label{conjn21}\n\\\\[1ex]\n({\\textstyle{\\partial\\over\\partial t_1}},\nU_0,{\\gamma}_1{\\cal D}_1, {\\overline {\\gamma}}_1\n{\\overline {\\cal D}}_1)^{\\bullet}=\n(-{\\textstyle{\\partial\\over\\partial t_1}},U_0,\n+{\\gamma}_1{\\cal D}_1,\n+{\\overline {\\gamma}}_1 {\\overline {\\cal D}}_1), \\quad\n({\\gamma}_1, {\\overline {\\gamma}}_1)^{\\bullet}=\n({\\gamma}_1, {\\overline {\\gamma}}_1),\n\\label{conjn22}\n\\\\[1ex]\n({\\textstyle{\\partial\\over\\partial t_1}},\nU_0,{\\gamma}_1{\\cal D}_1, {\\overline {\\gamma}}_1\n{\\overline {\\cal D}}_1)^{\\star}=\n(-{\\textstyle{\\partial\\over\\partial t_1}},U_0,\n-{\\gamma}_1{\\cal D}_1,\n-{\\overline {\\gamma}}_1 {\\overline {\\cal D}}_1), \\quad\n({\\gamma}_1, {\\overline {\\gamma}}_1)^{\\star}=\n({\\gamma}_1, {\\overline {\\gamma}}_1),\n\\label{conjn23}\n\\end{gather}\nwhere ${\\gamma}_1, {\\overline {\\gamma}}_1$ are constant odd\nreal parameters. Therefore, we conclude that the complex twisted $N=2$\nsupersymmetric Toda chain hierarchy with the complex conjugations \n\\p{conj1}--\\p{conj3} possesses twisted real $N=2$ supersymmetry.\nFor this reason we like to call it the ``twisted $N=2$\nsupersymmetric Toda chain hierarchy'' (for the supersymmetric Toda chain\nhierarchy possessing untwisted $N=2$ supersymmetry see \\cite{bs1} \nand references therein).\n\nLet us remark that a combination of the two involutions\n(\\ref{conj3}) and (\\ref{conj2}) generates the infinite-dimensional group\nof discrete Darboux transformations \\cite{ols1}\n\\begin{equation}\n\\begin{split}\n&(v,~u)^{\\star \\bullet}=(~v(QD\\ln v-uv),~ \\frac{1}{v}~), \\quad\n(z,{{\\theta}})^{\\star \\bullet}=(z,{\\theta}),\\\\\n& \n(t_p,U_p,D_p,Q_p)^{\\star \\bullet }=(t_p, U_p,-D_p,-Q_p).\n\\end{split}\n\\label{discrsymm}\n\\end{equation}\nThis way of deriving discrete symmetries was proposed\nin \\cite{s} and applied to the construction of discrete symmetry\ntransformations of the $N=2$ supersymmetric GNLS hierarchies.\n\nTo close this section let us stress once more that we cannot claim\nto have exhausted {\\it all} complex conjugations of the twisted $N=2$ \nToda chain hierarchy by the three examples of complex conjugations \n(eqs. \\p{conj1}--\\p{conj3}) we have constructed. Finding complex\nconjugations for affine (super)algebras themselves is a problem \nsolved by the classification of \\cite{new1} but rather different from\nconstructing complex conjugations for different {\\it realizations} of\naffine (super)algebras. To our knowledge, no algorithm yet exists for\nsolving this rather complicated second problem. Thus, classifying {\\it all}\ncomplex conjugations is out of the scope of the present letter.\nRather, we have constructed these examples in order to use them merely as\ntools to generate the important discrete symmetries \\p{discrsymm} as well\nas to construct a convenient superfield basis and a manifest twisted $N=2$\nsuperfield representation (see Sections 3 and 4), with the aim to clarify\nthe relationships of the hierarchy under consideration to other physical\nhierarchies discussed in the literature (see Section~5). \n\n\n\\section{A KdV-like basis with locally realized\nsupersymmetries.}\n\nThe third complex conjugation \\p{conj3} looks rather complicated\nwhen compared to the first two ones \\p{conj1}--\\p{conj2}.\nHowever, it drastically simplifies in another superfield basis defined as\n\\begin{equation}\nJ\\equiv uv + QD\\ln u, \\quad {\\overline J}\\equiv -uv,\n\\label{basis}\n\\end{equation}\nwhere $J \\equiv J(z,\\theta)$ and $ {\\overline J}\\equiv \n{\\overline J}(z,\\theta)$ ($[J]=[{\\overline J}]=-1$) are \nunconstrained even $N=1$ superfields.\nIn this basis the complex conjugations \\p{conj1}--\\p{conj3} \nand the discrete Darboux transformations \\p{discrsymm} are given by\n\\begin{gather}\n(J,~{\\overline J})^{}=-(J,~{\\overline J}),\n\\label{conj1j}\n\\\\[1ex]\n(J,~{\\overline J})^{\\bullet}=(~J - QD\\ln {\\overline J}, ~{\\overline J}~),\n\\label{conj2j}\n\\\\[1ex]\n(J,~{\\overline J})^{\\star}=({\\overline J},~J),\n\\label{conj3j}\n\\\\[1ex]\n(J,~{\\overline J})^{\\star \\bullet}=(~{\\overline J},\n~J - QD\\ln {\\overline J}~),\n\\label{discrsymm1}\n\\end{gather}\nand the equations \\p{eqs}--\\p{qqq1} become simpler as well,\n\\begin{gather}\n\\begin{split}\n&{\\textstyle{\\partial\\over\\partial t_2}} J =\n(-J~' +2J D^{-1}Q{\\overline J}-J^2)~', \\\\\n&{\\textstyle{\\partial\\over\\partial t_2}}{\\overline J} =\n(+{\\overline J}~' + 2{\\overline J}D^{-1}QJ-{\\overline J}^2)~',\n\\end{split}\n\\label{eqs2j}\n\\\\[1ex]\n\\begin{split}\n&{\\textstyle{\\partial\\over\\partial t_3}} J =\n3~\\Bigl[ ~\\frac{1}{3}J~'' +J~J~'-J~'D^{-1}Q{\\overline J} -\n2J^2D^{-1}Q{\\overline J}-JD^{-1}Q{\\overline J}^2 \n+\\frac{1}{3}J^3 ~\\Bigr]~', \\\\\n&{\\textstyle{\\partial\\over\\partial t_3}} {\\overline J} =\n3~\\Bigl[ ~\\frac{1}{3}{\\overline J}~'' -{\\overline J}~{\\overline J}~'+\n{\\overline J}~'D^{-1}QJ-2{\\overline J}^2D^{-1}QJ\n-{\\overline J}D^{-1}QJ^2+\\frac{1}{3}{\\overline J}^3 ~\\Bigr]~',\n\\end{split}\n\\label{eqs2jt3}\n\\end{gather}\nand then\n\\begin{gather}\nD_1\n\\left(\\begin{array}{cc} J\\\\ {\\overline J} \\end{array}\\right) =\nD\\left(\\begin{array}{cc} +J\\\\ -{\\overline J} \\end{array}\\right),\n\\quad Q_1 \\left(\\begin{array}{cc} J \\\\ {\\overline J} \\end{array}\\right) =\nQ\\left(\\begin{array}{cc} + J \\\\ - {\\overline J} \\end{array}\\right),\n\\label{supersflowsj}\n\\\\[1ex]\nU_0 \\left(\\begin{array}{cc} J \\\\ {\\overline J} \\end{array}\\right) =\n{\\theta}D \\left(\\begin{array}{cc} J \\\\ {\\overline J} \\end{array}\\right).\n\\label{qqqj1}\n\\end{gather}\nNotice that the supersymmetry and $o(1,1)$ \ntransformations \\p{supersflowsj}--\\p{qqqj1} of the superfields\n$J$, $\\bar J$ are local functions of the superfields. \nThe evolution equations \\p{eqs2j}--\\p{eqs2jt3} are also local\nbecause the operator $D^{-1}Q$ is a purely differential one,\n$D^{-1}Q\\equiv [\\theta,D]$.\n\n\n\\section{A manifest twisted N=2 supersymmetric representation}\n\nThe existence of a basis with locally and linearly realized twisted $N=2$\nsupersymmetric flows \\p{supersflowsj} would give evidence in favour\nof a possible description of the hierarchy \nin terms of twisted $N=2$ superfields. It turns out that this is indeed\nthe case. In order to show this, let us introduce a twisted $N=2$ \nsuperspace with even coordinate $z$ and two odd real coordinates $\\eta$\nand $\\overline \\eta$ (${\\eta}^{}={\\eta},~ {\\overline {\\eta}}^{}=\n{\\overline {\\eta}}$), as well as odd covariant derivatives ${\\cal D}$ and\n${\\overline {\\cal D}}$ via\n\\begin{equation}\n{\\cal D} \\equiv \\frac{\\partial}{\\partial {\\eta}}+\n{\\overline \\eta} {\\partial}, \\quad\n{\\overline {\\cal D}}\\equiv \\frac{\\partial}\n{\\partial {\\overline \\eta}}+{\\eta} {\\partial}, \\quad\n\\Bigl\\{{\\cal D}\\,,\\,{\\overline {\\cal D}}\\Bigr\\}=2{\\partial}, \\quad\n{\\cal D}^2={\\overline {\\cal D}}^2=0\n\\label{algnn4}\n\\end{equation}\ntogether with twisted $N=2$ supersymmetry generators \n${\\cal Q}$ and ${\\overline {\\cal Q}}$\n\\begin{equation}\n{\\cal Q} \\equiv \\frac{\\partial}{\\partial {\\eta}}-\n{\\overline \\eta} {\\partial}, \\quad\n{\\overline {\\cal Q}}\\equiv \\frac{\\partial}\n{\\partial {\\overline \\eta}}-{\\eta} {\\partial}, \\quad\n\\Bigl\\{{\\cal Q}\\,,\\,{\\overline {\\cal Q}}\\Bigr\\}=-2{\\partial}, \\quad\n{\\cal Q}^2={\\overline {\\cal Q}}^2=0.\n\\label{algnngen4}\n\\end{equation}\nIn this space, we consider two chiral even twisted\n$N=2$ superfields ${\\{\\cal J}(z,\\eta,\\overline \\eta)\n~{\\overline {\\cal J}}(z,\\eta,\\overline \\eta)\\}$, which obey \n\\begin{equation}\n{\\cal D}{\\cal J}=0, \\quad {\\cal D} {\\overline {\\cal J}}= 0\n\\label{N=2constr}\n\\end{equation}\nand are related to the $N=1$ superfields \n$\\{J(z,\\theta),~ {\\overline J}(z,\\theta)\\}$ \\p{basis}.\nMore concretely,\ntheir independent components are related to those of \n$J$ and ${\\overline J}$ as follows,\n\\begin{equation}\n\\begin{split}\n&{\\cal J}\|_{\\eta=\\overline \\eta=0} = J\|_{\\theta=0}, \\quad\n{\\overline {\\cal D}} ~{\\cal J}\|_{\\eta=\\overline \\eta=0} = +D J\|_{\\theta=0},\n\\\\ \n&{\\overline {\\cal J}}\|_{\\eta=\\overline \\eta=0}\n={\\overline J}\|_{\\theta=0}, \\quad\n\\overline {\\cal D}~{\\overline {\\cal J}}\|_{\\eta=\\overline \\eta=0} =\n-D{\\overline J}\|_{\\theta=0}. \n\\end{split}\n\\label{N=4n2rel}\n\\end{equation}\nThen, in terms of these superfields the equations\n\\p{eqs2j}--\\p{eqs2jt3} become\n\\begin{gather}\n\\begin{split}\n&{\\textstyle{\\partial\\over\\partial t_2}} {\\cal J} =\n(-{\\cal J}~' -2{\\cal J} {\\overline {\\cal J}}-{\\cal J}^2)~', \\\\\n&{\\textstyle{\\partial\\over\\partial t_2}}{\\overline {\\cal J}} =\n(+{\\overline {\\cal J}}~' - 2{\\cal J}{\\overline {\\cal J}}-\n{\\overline {\\cal J}}^2)~',\n\\end{split}\n\\label{eqs2jN=4}\n\\\\[1ex]\n\\begin{split}\n{\\textstyle{\\partial\\over\\partial t_3}} {\\cal J} =\n3~\\Bigl( ~\\frac{1}{3}{\\cal J}~'' +{\\cal J}~{\\cal J}~'+\n{\\overline {\\cal J}} {\\cal J}~'+ 2{\\cal J}^2{\\overline {\\cal J}}+\n{\\cal J}{\\overline {\\cal J}}^2 \n+\\frac{1}{3}{\\cal J}^3 ~\\Bigr)~', \\\\\n{\\textstyle{\\partial\\over\\partial t_3}} {\\overline {\\cal J}} =\n3~\\Bigl( ~\\frac{1}{3}{\\overline {\\cal J}}~'' -\n{\\overline {\\cal J}}~{\\overline {\\cal J}}~'-\n{\\cal J}{\\overline {\\cal J}}~'+2{\\overline {\\cal J}}^2{\\cal J}+\n{\\overline {\\cal J}}~{\\cal J}^2+\\frac{1}{3}{\\overline {\\cal J}}^3~\\Bigr)~',\n\\end{split}\n\\label{eqs2jN=4t3}\n\\end{gather}\nand it is obvious that they and the chirality constraints \\p{N=2constr} are\nmanifestly invariant with respect to the transformations generated by the\ntwisted $N=2$ supersymmetry generators ${\\cal Q}$ and \n${\\overline {\\cal Q}}$ \\p{algnngen4}.\n\nLet us also present a manifestly twisted $N=2$ supersymmetric form of \nthe complex conjugations \\p{conj1j}--\\p{conj3j} and the discrete\nDarboux transformations \\p{discrsymm1} in terms of the superfields \n${\\cal J}(z,\\eta,\\overline \\eta)$ and \n${\\overline {\\cal J}}(z,\\eta,\\overline \\eta)$ \\p{N=4n2rel}: \n\\begin{gather}\n({\\cal J},~{\\overline {\\cal J}})^{}=-({\\cal J},~{\\overline {\\cal J}}),\n\\label{conj1jj}\n\\\\[1ex]\n({\\cal J},~{\\overline {\\cal J}})^{\\bullet}=\n(~{\\cal J}-{\\partial}\\ln{\\overline {\\cal J}}, ~{\\overline {\\cal J}}~),\n\\label{conj2jj}\n\\\\[1ex]\n({\\cal J},~{\\overline {\\cal J}})^{\\star}=({\\overline {\\cal J}},~{\\cal J}),\n\\label{conj3jj}\n\\\\[1ex]\n({\\cal J},~{\\overline {\\cal J}})^{\\star \\bullet}=\n(~{\\overline {\\cal J}},~{\\cal J} -{\\partial}\\ln {\\overline {\\cal J}}~),\n\\label{discrsymm2}\n\\end{gather}\nmodulo the standard automorphism which changes the sign of all\nGrassmann odd objects. \n\n\n\\section{Relation with the s-Toda, modified NLS and derivative NLS\nhierarchies} \n\nIt is well known that there are often hidden relationships\nbetween a priori unrelated hierarchies. Some examples are the $N=2$ NLS\nand $N=2$ ${\\alpha}=4$ KdV \\cite{kst}, the ``quasi'' $N=4$ KdV and \n$N=2$ ${\\alpha}= -2$ Boussinesq \\cite{dgi}, the $N=2$ (1,1)-GNLS\nand $N=4$ KdV \\cite{s,bs}, the $N=4$ Toda and $N=4$ KdV \\cite{ds}. These\nrelationships may lead to a deeper understanding of the hierarchies. They\nmay help to obtain a more complete description and to derive solutions. \n \nThe absence of odd derivatives in the equations \n\\p{eqs2jN=4}--\\p{eqs2jN=4t3}, starting off the twisted $N=2$\nsupersymmetric Toda chain hierarchy, gives additional evidence in favour\nof a hidden relationship with some bosonic hierarchy. It\nturns out that such a relationship indeed exists.\nLet us search it first at the level of the Darboux transformations\n\\p{discrsymm2}, then in the second flow equation \\p{eqs2jN=4}.\n\nFor this purpose, we introduce new $N=1$ superfields \n$\\{\\Phi(z,\\theta), {\\Psi}(z,\\theta)\\}$ via \n\\begin{equation}\n\\begin{split}\n&{\\cal J}\|_{\\eta=\\overline \\eta=0} \\equiv \n(\\Phi \\Psi + \\partial \\ln \\Psi)\|_{\\theta=0}, \\quad\n{\\overline {\\cal D}}~{\\cal J}\|_{\\eta=\\overline \\eta=0} \\equiv \nD(\\Phi \\Psi + \\partial \\ln \\Psi)\|_{\\theta=0}, \\\\\n& {\\overline {\\cal J}}\|_{\\eta=\\overline \\eta=0} \n\\equiv -(\\Phi \\Psi)\|_{\\theta=0}, \\quad \\quad \\quad \\quad \n{\\overline {\\cal D}}~ {\\overline {\\cal J}}\|_{\\eta=\\overline \\eta=0} \n\\equiv -D(\\Phi \\Psi)\|_{\\theta=0}.\n\\end{split}\n\\label{basisn1}\n\\end{equation}\nThe Darboux transformations \\p{discrsymm2}, expressed in terms of those \nnew superfields, exactly reproduce the Darboux-Backlund (s-Toda)\ntransformations \n\\begin{equation}\n(\\Phi,~\\Psi)^{\\star \\bullet}=(~\\Phi(\\partial \\ln \\Phi- \\Phi \\Psi),~\n\\frac{1}{\\Phi}~) \n\\label{discrsymm3}\n\\end{equation}\nproposed in \\cite{anp1} in the context of the reduction\nof the supersymmetric KP hierarchy in $N=1$ superspace characterized \nby the Lax operator \n\\begin{equation}\nL=D-2(D^{-1}\\Phi \\Psi)+\\Phi D^{-1}\\Psi.\n\\label{lax2}\n\\end{equation}\nFor completeness, we also present the corresponding second flow equations,\n\\begin{equation}\n{\\textstyle{\\partial\\over\\partial t_2}} \\Phi =\n+\\Phi~'' - 2\\Phi^2\\Psi~' -2(\\Phi\\Psi)^2\\Phi ,\n\\quad {\\textstyle{\\partial\\over\\partial t_2}} \\Psi =\n-\\Psi~'' - 2\\Psi^2\\Phi~' +2(\\Phi\\Psi)^2\\Psi,\n\\label{eqsar}\n\\end{equation}\nwhich also follow from \\cite{anp1}.\nTherefore, we are led to the conclusion that the two integrable hierarchies \nrelated to the reductions \\p{lax1} and \\p{lax2} are {\\it equivalent}.\nIt would be interesting to establish a relationship (if any) between\nthese two hierarchies in the more general case where $v,u$ and $\\Phi,\\Psi$\nentering the corresponding Lax operators \\p{lax1} and \\p{lax2} \nare rectangular (super)matrix-valued superfields \\cite{ols1}, but\nthis rather complicated question is outside the scope of the present letter. \nIn general, these two families of $N=2$ supersymmetric hierarchies\ncorrespond to a non-trivial supersymmetrization\\footnote{By trivial\nsupersymmetrization of bosonic equations we mean just replacing functions\nby superfunctions. In this case the resulting equations are supersymmetric,\nbut they do not contain fermionic derivatives at all.} of bosonic\nhierarchies, except for the simplest case we consider here. \nIndeed, a simple inspection shows that the\nequations \\p{eqs2jN=4}--\\p{eqs2jN=4t3} do not contain fermionic\nderivatives and belong to the hierarchy which is the trivial $N=2$ \nsupersymmetrization of the bosonic modified NLS or derivative NLS\nhierarchy. This last fact becomes obvious if one introduces yet a new\nsuperfield basis $\\{b(z,\\eta,\\overline \\eta), \n{\\overline b}(z,\\eta,\\overline \\eta)\\}$ through \n\\begin{equation}\n{\\cal J}\\equiv (\\ln {\\overline b})~', \\quad {\\overline {\\cal J}}\\equiv \n-b{\\overline b}, \\quad\n{\\cal D}b={\\cal D} {\\overline b}= 0,\n\\label{mnlsbasis}\n\\end{equation}\nin which the second flow (\\ref{eqs2jN=4}) and the Darboux transformations\n\\p{discrsymm2} become\n\\begin{gather}\n{\\textstyle{\\partial\\over\\partial t_2}} b =\n+b~'' + 2b{\\overline b}b~', \\quad\n{\\textstyle{\\partial\\over\\partial t_2}} {\\overline b}\n =-{\\overline b}~'' + 2{\\overline b}b{\\overline b}~',\n\\label{eqsmnls}\n\\\\\nb^{\\star \\bullet \\star \\bullet}=b~(\\ln b^{\\star \\bullet})~', \\quad \n{\\overline b}^{\\star \\bullet \\star \\bullet} = \\frac{1}{b}, \n\\label{discrsymm4}\n\\end{gather}\nrespectively, and the equation \\p{eqsmnls} reproduces the trivial $N=2$\nsupersymmetrization of the modified NLS equation \\cite{cll}. When passing\nto alternative superfields \n$g(z,\\eta,\\overline \\eta)$ and ${\\overline g}(z,\\eta,\\overline \\eta)$\ndefined by the following invertible transformations\n\\begin{equation}\ng = b~ {\\exp (-{\\partial^{-1}} (b {\\overline b}))}, \\quad\n{\\overline g} = {\\overline b}~{ \\exp (+{\\partial^{-1}}\n(b {\\overline b}))},\n\\label{comp2}\n\\end{equation}\nequation \\p{eqsmnls} becomes\n\\begin{equation}\n\\frac{\\partial}{\\partial t_2}g = (+g~' + 2g{\\overline g}g)~', \\quad\n\\frac{\\partial}{\\partial t_2}{\\overline g} =\n(-{\\overline g}~' + 2{\\overline g}g{\\overline g})~' \n\\label{comp3}\n\\end{equation}\nand coincides with the derivative NLS equation \\cite{kn}. \n\nFinally, we would like to remark that one can produce the {\\it non-trivial}\n$N=2$ supersymmetric modified KdV hierarchy by secondary reduction even\nthough the twisted $N=2$ Toda chain hierarchy is a {\\it trivial} $N=2$\nsupersymmetrization of the modified or derivative NLS hierarchy. \nOne of such reductions was described in\n\\cite{ols1}. In terms of the superfields $J$\nand ${\\overline J}$ \\p{basis}, the reduction constraint is \n\\begin{equation}\nJ+{\\overline J}=0,\n\\label{red1}\n\\end{equation}\nand only half of the flows from the set \\p{dimtimes} are consistent\nwith this reduction, namely\n\\begin{equation}\n\\{~{\\textstyle{\\partial\\over\\partial t_{2k-1}}}\n~U_{2k},~D_{2k},~Q_{2k}~\\}\n\\label{consistfl}\n\\end{equation}\n(for details, see \\cite{ols1}). Substituting the constraint \\p{red1} into\nthe third flow equation \\p{eqs2jt3} of the reduced hierarchy, this flow \nbecomes\n\\begin{equation}\n{\\textstyle{\\partial\\over\\partial t_3}} u =\n(J~'' +3(QJ)(DJ)-2J^3)~'.\n\\label{redflow3}\n\\end{equation}\nNow, one can easily recognize that the equation for the bosonic component\nreproduces the modified KdV equation and does not contain the fermionic\ncomponent at all. Nevertheless, it seems\nthat the supersymmetrization \\p{redflow3} is rather non-trivial, because it \ninvolves the odd operators $D$ and $Q$ but does not admit \nodd flows having length dimension $[D]=[Q]=-1/2$. Hence, it does not\nseem to be possible to avoid a dependence of $D$ and $Q$ in a cleverly chosen \nsuperfield basis. To close this discussion let us mention that the\npossible alternative constraint on the twisted $N=2$ superfields ${\\cal J}$\nand ${\\overline {\\cal J}}$, namely\n\\begin{equation}\n{\\cal J}+{\\overline {\\cal J}}=0,\n\\label{red2}\n\\end{equation}\nleads again to the trivial $N=2$ supersymmetrization of the modified KdV\nhierarchy. \n\n\n\\section{Conclusion}\n\nIn this letter we have described three distinct real forms of the \ntwisted $N=2$ Toda chain hierarchy introduced in \\cite{ols1}. It has been\nshown that the symmetry algebra of these real forms is the twisted \n$N=2$ supersymmetry algebra. We have introduced a set of $N=1$ superfields.\nThey enjoy simple conjugation properties and allowed us to\neliminate all nonlocalities in the flows. All flows and complex\nconjugation rules have been rewritten directly in twisted $N=2$\nsuperspace. As a byproduct, relationships between the twisted $N=2$ Toda\nchain, s-Toda, modified NLS, and derivative NLS hierarchies have been\nestablished. These connections enable us to derive new real forms of the\nlast three hierarchies, possessing a twisted $N=2$ supersymmetry.\n\n\n\\subsection{Acknowledgments}\n\nA.S. would like to thank the Institut f\\\"ur Theoretische Physik,\nUniversit\\\"at Hannover for the hospitality during the course of this work.\nThis work was partially supported by the DFG Grant No. 436 RUS 113/359/0\n(R), RFBR-DFG Grant No. 99-02-04022,\nthe Heisenberg-Landau programme HLP-99-13, PICS Project No. 593, RFBR-CNRS\nGrant No. 98-02-22034, RFBR Grant No. 99-02-18417, Nato Grant No. PST.CLG\n974874 and INTAS Grant INTAS-96-0538.\n\n\\begin{thebibliography}{**}\n\\small\n\n\\bibitem{ls}\nLeznov A.N. and Sorin A.S.,\nTwo-Dimensional Superintegrable Mappings and Integrable\nHierarchies in the $(2\|2)$ Superspace,\n{\\it Phys. Lett.}, 1996, V.B389, 494, hep-th/9608166;\nIntegrable Mappings and Hierarchies in the $(2\|2)$ Superspace,\n{\\it Nucl. Phys.} (Proc. 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Phys.}, 1978, V.19, 798.\n\n\\end{thebibliography}\n\n\\label{lastpage}\n\n\\end{document}\n\n\n\n\n" } ]	[ { "name": "solv-int9912010.extracted_bib", "string": "{ls Leznov A.N. and Sorin A.S., Two-Dimensional Superintegrable Mappings and Integrable Hierarchies in the $(2\|2)$ Superspace, {Phys. Lett., 1996, V.B389, 494, hep-th/9608166; Integrable Mappings and Hierarchies in the $(2\|2)$ Superspace, {Nucl. Phys. (Proc. Suppl.), 1997, V.B56, 258." }, { "name": "solv-int9912010.extracted_bib", "string": "{ols Lechtenfeld O. and Sorin A., Fermionic Flows and Tau Function of the N=$(1\|1)$ Superconformal Toda Lattice Hierarchy, {Nucl. Phys., 1999, V.B557, [PM], 535, solv-int/9810009." }, { "name": "solv-int9912010.extracted_bib", "string": "{olshanetsky Olshanetsky M.A., Supersymmetric Two-Dimensional Toda Lattice, {Commun. Math. Phys., 1983, V.88, 63." }, { "name": "solv-int9912010.extracted_bib", "string": "{leites Leites D.A., Saveliev M.V. and Serganova V.V., Embeddings of ${\\rm osp(1\|2)$ and the Associated Nonlinear Supersymmetric Equations, in Group Theoretical Methods in Physics, Vol. I (Yurmala, 1985), VNU Sci. Press, Utrecht, 1986, 255--297." }, { "name": "solv-int9912010.extracted_bib", "string": "{andreev Andreev V.A., Supersymmetric Generalized Toda Lattice, in Group Theoretical Methods in Physics, Vol. 1 (Yurmala, 1985), VNU Sci. Press, Utrecht, 1986, 315--321; Odd Bases of Lie Superalgebras and Integrable Systems, {Theor. Math. Phys., 1987, V.72, 758." }, { "name": "solv-int9912010.extracted_bib", "string": "{evans Evans J.M. and T. Hollowood, Supersymmatric Toda Field Theories, {Nucl. Phys., 1991, V.B352, 723; Evans J.M. and Madsen J.O., On the Classification of Real Forms of Non-Abelian Toda Theories and W-Algebras, {Nucl. Phys., 1998, V.B536, 657, hep-th/9802201." }, { "name": "solv-int9912010.extracted_bib", "string": "{dgs Delduc F., Gallot L. and Sorin A., $N=2$ Local and $N=4$ Nonlocal Reductions of Supersymmetric KP Hierarchy in $N=2$ Superspace, {Nucl. Phys., 1999, V.B558, 545, solv-int/9907004." }, { "name": "solv-int9912010.extracted_bib", "string": "{ds Delduc F. and Sorin A., A Note on Real Forms of the Complex $N=4$ Supersymmetric Toda Chain Hierarchy in Real $N=2$ and $N=4$ Superspaces, {Nucl. Phys., 2000, V.B577, 461, solv-int/9911005." }, { "name": "solv-int9912010.extracted_bib", "string": "{di Delduc F. and Ivanov E., $N=4$ Super KdV Equation, {Phys. Lett., 1993, V.B309, 312, hep-th/9301024; %" }, { "name": "solv-int9912010.extracted_bib", "string": "{dik Delduc F., Ivanov E. and Krivonos S., N=4 Super KdV Hierarchy in N=4 and N=2 Superspaces, {J. Math. Phys., 1996, V.37, 1356; Erratum-ibid, 1997, V.38, 1224, hep-th/9510033." }, { "name": "solv-int9912010.extracted_bib", "string": "{ols1 Lechtenfeld O. and Sorin A., Supersymmetric KP Hierarchy in $N=1$ Superspace and its $N=2$ Reductions, {Nucl. Phys., 2000, V.B566, [PM], 489, solv-int/9907021." }, { "name": "solv-int9912010.extracted_bib", "string": "{anp1 Aratyn H., Nissimov E. and Pacheva S., Supersymmetric KP Hierarchy: ``Ghost'' Symmetry Structure, Reductions and Darboux-B\\\"acklund Solutions, {J. Math. Phys., 1999, V.40, 2922, solv-int/9801021; Berezinian Construction of Super-Solitons in Supersymmetric Constrained KP Hierarchies, in Topics in Theoretical Physics vol. II, Festschrift for A.H. Zimerman, IFT-S\\~{ao Paulo, SP-1998, pp. 17-24, solv-int/9808004." }, { "name": "solv-int9912010.extracted_bib", "string": "{maninradul Manin Yu.I. and Radul A.O., A Supersymmetric Extension of the Kadomtsev-Petviashvili Hierarchy, {Commun. Math. Phys., 1985, V.98, 65." }, { "name": "solv-int9912010.extracted_bib", "string": "{ggrs Gates S.J. Jr., Grisaru M.T., Ro\\v{cek M. and Siegel W., Superspace or One Thousand and One Lessons in Supersymmetry, Benjamin/Cummings, 1983, 58--59; West P., Introduction to Supersymmetry and Supergravity, extended second edition, World Scientific, 1990, 393--394." }, { "name": "solv-int9912010.extracted_bib", "string": "{new1 Serganova V.V., Outer Automorphisms and Real Forms of Kac-Moody Superalgebras, in Group Theoretical Methods in Physics, Vol. 1--3, (Zvenigorod, 1982), Harwood Academic Publ., Chur, 1985, 639--642; Automorphisms of Lie Superalgebras of String Theories (in Russian), {Funktional. Anal. i Prilozhen., 1985, V.19, 75." }, { "name": "solv-int9912010.extracted_bib", "string": "{new2 Feigin B.L., Leites D.A. and Serganova V.V., Kac-Moody Superalgebras, in Group Theoretical Methods in Physics, Vol. 1--3, (Zvenigorod, 1982), Harwood Academic Publ., Chur, 1985, 631--637." }, { "name": "solv-int9912010.extracted_bib", "string": "{new3 Grozman P., Leites D. and Shchepochkina I., Lie Superalgebras of String Theories, hep-th/9702120." }, { "name": "solv-int9912010.extracted_bib", "string": "{bs1 Bonora L. and Sorin A., The N=2 Supersymmetric Toda Lattice, {Nucl. Phys., 1998, V.B521, 444, hep-th/9710118." }, { "name": "solv-int9912010.extracted_bib", "string": "{s Sorin A., The Discrete Symmetry of the N=2 Supersymmetric Modified NLS Hierarchy, {Phys. Lett., 1997, V.B395, 218, hep-th/9611148; Discrete Symmetries of the N=2 Supersymmetric Generalized Nonlinear Schroedinger Hierarchies, {Phys. Atom. Nucl., 1998, V.61, 1768, solv-int/9701020." }, { "name": "solv-int9912010.extracted_bib", "string": "{kst Krivonos S. and Sorin A., The Minimal $N=2$ Superextension of the NLS Equation, {Phys. Lett., 1995, V.B357, 94, hep-th/9504084; Krivonos S., Sorin A. and Toppan F., On the Super-NLS Equation and its Relation with $N=2$ Super-KdV within Coset Approach, {Phys. Lett., 1995, V.A206, 146, hep-th/9504138." }, { "name": "solv-int9912010.extracted_bib", "string": "{dgi Delduc F., Gallot L. and Ivanov E., New Super KdV System with the N=4 SCA as the Hamiltonian Structure, {Phys. Lett., 1997, V.B396, 122, hep-th/9611033." }, { "name": "solv-int9912010.extracted_bib", "string": "{bs Bonora L. and Sorin A., The Hamiltonian Structure of the N=2 Supersymmetric GNLS Hierarchy, {Phys. Lett., 1997, V.B407, 131, hep-th/9704130." }, { "name": "solv-int9912010.extracted_bib", "string": "{cll Chen H.H., Lee Y.C. and Lin C.S., Integrability of Nonlinear Hamiltonian Systems by Inverse Scattering Method, {Phys. Scripta, 1979, V.20, 490." }, { "name": "solv-int9912010.extracted_bib", "string": "{kn Kaup D.J. and Newell A.C., An Exact Solution for a Derivative Nonlinear Schr\\\"odinger Equation, {J. Math. Phys., 1978, V.19, 798." } ]
solv-int9912011	Liouville equation under perturbation	[ { "author": "L.A. Kalyakin" }, { "author": "Institute of Mathematics" }, { "author": "Ufa Sci. Centre" }, { "author": "of Russian Acad. of Sci." }, { "author": "Institute of Mathematics" }, { "author": "112" }, { "author": "Chernyshevsky str." }, { "author": "Ufa" }, { "author": "450000" }, { "author": "Russia" }, { "author": "E-mail: klen@imat.rb.ru \\thanks {This research has been supported by the Russian Foundation of the Fundamental Research under Grants 99-01-00139" }, { "author": "96-15-96241" } ]	Small perturbation of the Liouville equation under smooth initial data is considered. Asymptotic solution which is available for a long time interval is constructed by the two scale method.	[ { "name": "solv-int9912011.tex", "string": "%%%%----------------------- This is a LaTeX file -----\n\\documentclass{article}\n% --------- Sets size of page and margins\n\\oddsidemargin 10mm\n\\evensidemargin 10mm\n\\topmargin 0pt\n\\headheight 0pt\n\\headsep 0pt\n\\baselineskip = 20pt\n\\hsize = 340pt\n\\vsize = 490pt\n\n\n\\font\\Bbb = msbm10.tmf\n\\def\\const {{\\hbox{const}}}\n\\def\\R {{\\hbox {\\Bbb R}}}\n\\def\\O {{\\cal O}}\n\\def\\o {\\hbox{o}}\n\\def\\U {{\\bf U}}\n\n\\newtheorem{Def}{Definition}\n\\newtheorem{Lemma}{Lemma}[section]\n\\newtheorem{Th}{Theorem}[section]\n\\newtheorem{cons}{Corollare}[section]\n\n\n\\title\n{\\bf Liouville equation under perturbation}\n\\author\n{\\bf L.A. Kalyakin\n\\\\\nInstitute of Mathematics, Ufa Sci. Centre, of Russian Acad. of\nSci.\\\\ Institute of Mathematics, 112,Chernyshevsky str., Ufa,\n\\\\450000, Russia\\\\\nE-mail: klen@imat.rb.ru\n\\thanks\n{This research has been supported by the Russian\nFoundation of the Fundamental Research under Grants\n99-01-00139, 96-15-96241} }\n\\date{December, 16, 1999}\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nSmall perturbation of the Liouville equation under smooth\ninitial data is considered. Asymptotic solution which is\navailable for a long time interval is constructed by the\ntwo scale method.\n\\end{abstract}\n\nThe Cauchy problem for the Liouville equation with a\nsmall perturbation\n$$\n\\partial_t^2u -\\partial_x^2u\n+8\\exp u =\\varepsilon{\\bf F}[u],\n\\quad 0< \\varepsilon \\ll 1,\n\\eqno (0.1)\n$$\n$$\nu \|_{t=0}=\\psi_0(x), \\\n\\partial_tu\|_{t=0}=\\psi_1(x),\\quad x\\in\\R\n\\eqno (0.2)\n$$\nis considered. The problem is not suit under soliton\nperturbation theory because the Liouville equation has\nno any soliton solution. Initial functions are here\narbitrary, ones are smooth and decay rapidly at\ninfinity\n\\ ${\\psi}_0,{\\psi}_1(x)=\\O(x^{-N}),\n\\ \|x\|\\to\\infty,\\ \\forall\\, N$. So we deal with a\nsmooth solution; the case of singular solutions was\nconsidered in [1].\n\nThe perturbation operator is determined by two smooth\nfunctions $F_1,F_2$:\n$$\n{\\bf F}[u]=\\partial_xF_1(\\partial_xu,\\partial_tu)+\n\\partial_tF_2(\\partial_xu,\\partial_tu).\n\\eqno (0.3)\n$$\nThe purpose is to construct an asymptotic approach of the\nsolution $u(x,t;\\varepsilon)$ as $\\varepsilon\\to 0$\nuniformly over long time interval $\\{ x\\in\\R,\\ 0<t\\leq\n\\O(\\varepsilon^{-1})\\}$.\n\n{\\bf Results.} {\\it 1. The solution of the unperturbed\nproblem (as $\\varepsilon=0$) decomposes asymptotically\nat infinity (as $t\\to\\infty$) on two simple waves\nwhich travel on a decreasing background\\\n$u(x,t;0)=-4t+A^0_\\pm(s^\\pm)+\\O((s^\\mp)^{-N}),\\\ns^\\mp\\to\\mp\\infty$. 2. The structure of the asymptotic\nsolution as $\\varepsilon\\to 0$ remains the same\n$u(x,t;\\varepsilon)\\approx\n-4t+A_\\pm(s^\\pm,\\tau)+\\O(\\varepsilon)$\nfor long times $t\\approx\\varepsilon^{-1}$. The\nperturbation affects only a slow deformation of the\nwaves $A_\\pm=A_\\pm(s^\\pm,\\tau)$ on the slow time scale\n$\\tau=\\varepsilon t$. 3. The deformation of the waves\nis described by the first order PDE's\n$$\n\\pm 2\\partial_\\tau A_\\pm=H_\\pm(\\partial_s A_\\pm),\\quad\ns\\in\\R, \\ \\tau >0,\n\\eqno (0.4)\n$$\nwhere\n$\nH_\\pm(B)=F_1(B,-4\\pm B)- F_1(0,-4)\\pm F_2(B,-4\\pm\nB)\\mp F_2(0,-4).\n$\nThe initial data are here taken from the fast time\nasymptotics of the unperturbed solution\n$$\nA_\\pm(s,0)=A^0_\\pm(s)\n\\eqno (0.5)\n$$\n}\n\nThis result is very close to the case of linear wave\nunder perturbation, which slow deformation is described\nby the Hopf equation, [2].\n\nWe use here the two scale method and construct an\nasymptotic solution as a piece of the asymptotic series\n$$\nu\\approx\\sum_{n=0}^\\infty\\varepsilon^n\\stackrel{n}{u}\n(x,t,\\tau),\\ \\tau=\\varepsilon t,\n\\quad \\varepsilon\\to 0.\n\\eqno (0.6)\n$$\n\n{\\bf Remark.} The direct asymptotic expansion\n$\nu\\approx\\sum\\varepsilon^n\\stackrel{n}{u}(x,t)\n$\ndoes not provide approach to the solution over long\ntime interval $t\\approx\\O(\\varepsilon^{-1})$ through\nthe secular terms in corrections.\n\n\\section\n{Unperturbed problem ($\\varepsilon=0$)}\n\nGeneral solution of the nonlinear equation\n$$\n\\partial_t^2\\phi -\\partial_x^2\\phi\n+8\\exp\\phi =0,\n\\ (x,t)\\in R^2\n$$\nis given (Liouville, [3]) by the formula\n$$\n\\phi (x,t)=\\ln {{r_+^{\\prime}(s^+)r_-^{\\prime}(s^-)}\\over\n{r^2(s^+,s^-)}},\\quad r=r_+(s^+)+r_-(s^-), \\\ns^{\\pm}=x\\pm t\n\\eqno (1.1)\n$$\nwith arbitrary functions $r_\\pm$. Initial conditions\n(0.2) give two ODE's for the $r_\\pm$ which may be\nlinearized by change of variable\n$r_\\pm^\\prime=w/\\rho_\\pm^2,\\ (\\forall\\, w=\\const\\neq\n0)$, so that\n$$\n\\rho_\\pm^{\\prime\\prime}-\\Psi_\\pm (x)\\rho_\\pm =0.\n\\eqno (1.2)\n$$\nThe potentials $\\Psi_\\pm (x)$ are determined by the\noriginal initial data, [4]\n$$\n\\Psi_\\pm(x)=\\exp(\\psi_0)+\n\\Big({{\\psi_0^\\prime \\pm\\psi_1}\\over 4}\\Big)^2-{1\\over\n4}(\\psi_0^\\prime\n\\pm\\psi_1)\n^\\prime.\n$$\nSo the Cauchy problem for the Liouville equation is\nintegrable.\n\n\\begin{Lemma}\nLet as\n$\n\\psi_0,\\psi_1(x)=\\O(x^{-N}),\\ \|x\|\\to\\infty,\\ \\forall\\, N,\n$\nand equations (1.2) are not on spectrum. Than the\nsolution of the Cauchy problem for the Liouville equation\nhas an asymptotics\n$$\n\\phi (x,t)=-4t+ A_\\pm (s^\\pm )\n+\\O((s^\\mp )^{-N})+\\O(e^{-4t}),\\ s^\\mp\\to\\mp\\infty,\\\nt\\to\\infty\n$$\nwith the matching property\n$$\nA_\\pm (s)=\\cases{-\\ln a^2 +\\O(s^{-N}),\\\ns\\to\\pm\\infty,\\ (a=\\const\\neq 0),\n\\cr \\O(s^{-N}),\\ s\\to\\mp\\infty .}\n\\eqno (1.3)\n$$\n\\end{Lemma}\n\nFunctions $A_\\pm (s)$ are reading from $r_\\pm (s)$. If\nthe $r_\\pm$ are fixed by the conditions at infinity\n$$\nr_\\pm(x)=(1/a)\\exp(2x)[1+\\O(x^{-N})], \\quad x\\to\n-\\infty,\n\\eqno (1.4)\n$$\n$$\nr_\\pm (x)=a\\exp(2x)[1+\\O(x^{-N})], \\quad x\\to +\\infty\n\\eqno (1.5)\n$$\nthen\n$$ A_+(s^+)=\\ln\\Big[{{\\exp(2s^+)r_+^{\\prime}(s^+)\n}/{ar^2_+(s^+)}}\\Big],\\quad\nA_-(s^-)=\\ln\\Big[{{\\exp(-2s^-)r_-^{\\prime}(s^-)\n}/{a}}\\Big].\n\\eqno (1.6)\n$$\nFunctions $A_\\pm (s)$ do not depend on the choice of\n$r_\\pm(s)$ within the Bianchi transform [5] and may be\nused for parametrization of the general solution.\n\n\\begin{cons}\nGeneral solution of the Liouville equation can be\nparametrized by the pare of functions $A_\\pm(s)$ which\nhave the matching property (1.3) so that\n$$\n\\phi(x,t)=\\Phi[A_+,A_-]\\equiv\n\\ln {{r_+^{\\prime}r_-^{\\prime}}\\over {(r_++r_-)^2}}\n\\eqno (1.7)\n$$\nin view of (1.6).\n\\end{cons}\n\n\\bigskip\n\n\\section\n{ Linearized problem for the correction}\n\n\nCorrections $\\stackrel{n}{u}\\ (n \\geq 1)$ are obtained\nfrom linear equations with corresponding initial\nconditions\n$$\n\\partial_t^2\\stackrel{n}{u}\n-\\partial_x^2\\stackrel{n}{u}+\n8{{r_+^{\\prime}r_-^{\\prime}}\\over\n{r^2}}\\stackrel{n}{u}=\\stackrel{n}{f}(x,t;\\varepsilon ),\n\\quad\n\\stackrel{n}{u}\|_{t=0}=\\stackrel{n}{\\psi}_0(x), \\\n\\partial_t\\stackrel{n}{u}\|_{t=0}=\\stackrel{n}{\\psi}_1(x).\n\\eqno (2.1)\n$$\nThe right sides are here determined by the previous\napproaches. Dependence on the fast variables $x,t$ is\nonly determined from these equations.\n\nGeneral solution of the homogeneous linear equation is\ngiven by the formula\n$$\nu_0(x,t;\\varepsilon\n)={{j_+^{\\prime}}\\over{r_+^{\\prime}}}+\n{{j_-^{\\prime}}\\over{r_-^{\\prime}}}- 2{{j_++j_-}\\over{r}}\n$$\nwhere $j_\\pm=\\stackrel{n}{j}_\\pm(s^\\pm)$ are arbitrary\nfunctions. In context of the Cauchy problem they are\ndetermined by the initial data. In this way pare linear\nODE's are obtained which can be solved in explicit form.\n\nA similar formula with $j_\\pm(s^\\pm,t)$ may be used to\nsolve the nonhomogeneous linear equation (2.1). The\nfunctions $j_\\pm(x,t)$ are defined from ODE's as well,\nso that the solution is represented by the integral\n$$\nu(x,t)=\\int_{s^-}^{s^+}\\int_{s^-}^{\\sigma^+}\nK({s^+},{s^-},\\sigma^+,\\sigma^-)f(\\sigma^+,\\sigma^-)\n\\,d\\sigma^- \\,d\\sigma^+\n$$\ntaken over the characteristic triangle. The kernel $K$\nis here expressed by means of $r_\\pm$\n$$\nK({s^+},{s^-},\\sigma^+,\\sigma^-)=\n{1\\over{2r(s_+,s_-)r(\\sigma_+,\\sigma_-)}}\n$$\n$$\n\\Big\\{\nr_+(s^+)r_-(s^-) + r_+(\\sigma^+)r_-(\\sigma^-)\n +{1\\over 2}\\Big[r_+(s^+)-r_-(s^-)\\Big]\n\\Big[r_+(\\sigma^+)-r_-(\\sigma^-)\\Big]\\Big\\}.\n$$\n\n\\begin{Lemma}\nLet both the right side and the initial functions\ndecay rapidly at infinity. Than the solution of the\nCauchy problem for linearized equation is bounded and\nhas an asymptotics\n$$\nu(x,t)=U_\\pm(s^\\pm)+\n\\O((s^\\mp)^{-N}),\\ s^\\mp\\to\\mp\\infty,\n$$\n$$ U_\\pm(s)=\\cases{U+\\O(s^{-N}),\\ s\\to\\pm\\infty,\\\n(U=\\const),\n\\cr \\O(s^{-N}),\\ s\\to\\mp\\infty .}\n$$\n\\end{Lemma}\n\n\n\\section\n{ Perturbed problem ($\\varepsilon\\neq 0$)}\n\nWe construct a formal asymptotic solution in the form\n(0.6) where the leading order term is taken as a\nsolution of the unperturbed equation. The original\nidea is to use $A_\\pm$ - parametrization of this\nsolution\n$\n\\stackrel{0}{u}=\\Phi[A_+,A_-]\\\n$\nas it was pointed in (1.7). The second idea becomes from\nthe two scale method. It is assumed the functions\n$A_\\pm(s^\\pm,\\tau)$ depend on both fast $s^\\pm=x\\pm t$\nand slow $\\tau=\\varepsilon t$ variables. The initial\nvalues for the $A_\\pm(s^\\pm,\\tau)$ as $\\tau=0$ are taken\nin (0.5) from the unperturbed solution. Ones are\ncalculated per the initial function $\\psi_0,\\psi_1(x)$\nfrom the equations (1.2),(1.4),(1.6).\n\nDependence on the slow variable as $\\tau>0$ is determined\nby the differential equations obtained from the secular\ncondition which means the first order correction is\nsmall:\\ $\\varepsilon\\stackrel{1}{u}=\\o(1),\\\n\\varepsilon\\to 0,\\ \\tau=\\varepsilon t$\\ uniformly over long time interval\n$0<t\\leq\\O(\\varepsilon^{-1})$.\n\nThe right side of the first order equation is given by\nformula\n$$\n\\stackrel{1}{f}(x,t,\\tau)={\\bf F}[\\stackrel{0}{u}]\n-2\\partial_\\tau\\partial_t\\stackrel{0}{u}.\n$$\nOne can see from the lemma 2.1 that the secular condition\ncan be formulated through the right side as follows: The\nright side tends to zero at infinity as $s^\\pm\\to 0$.\nThis requirement gives two equations\n$$\n\\pm 2\\partial_\\tau\\partial_sA_\\pm={\\bf F}[-4t+A_\\pm].\n$$\nIf we integrate these relations taking into account\nboundary conditions\n$$\nA_\\pm(s^\\pm,\\tau)\\to 0 \\quad as \\quad s^\\pm\\to\\mp\\infty\n$$\nthan the first order PDE's (0.4) are obtained.\n\n\\begin{Lemma}\n1. If the perturbation operator has the form (0.3)\nthan the Cauchy problem for deformation equations\n(0.4),(0.5) has the unique smooth solution on some\nfinite interval $0\\leq\\tau\\leq\\tau_0$. 2. Under such\nfunctions $A_\\pm(s^\\pm,\\tau)$ the secular condition is\nhold for the first correction, i.e.\n$\\stackrel{1}{u}(x,t,\\tau)$ is bounded uniformly for\nall $x,t\\in \\R^2,\\\n\\tau\\in [0,\\tau_0]$.\n\\end{Lemma}\n\n\\begin{cons}\nThe function $U_1(x,t,\\varepsilon)=\\stackrel{0}{u}\n(x,t,\\varepsilon\nt)+\\varepsilon\\stackrel{1}{u}(x,t,\\varepsilon t)$\\ under\nsubstitution in the equations (0.1),(0.2) gives a\nremainder of order $\\O(\\varepsilon^2)$ uniformly for all\n$x\\in\\R,\\ 0\\leq t\\leq\\tau_0\\varepsilon^{-1}$.\n\\end{cons}\n\n\\begin{Th}\nLet the perturbation operator has the form (0.3) and\nthe initial functions in (0.2) are such that equations\n(1.2) are not on spectrum (i.e. in general position).\nThan the leading order term of the formal asymptotic\nsolution for the perturbed problem (0.1),(0.2) is\ngiven by the Liouville formula (1.1) in which the\nfunctions $r_\\pm(s^\\pm,\\tau)$ depend on the additional\nslow time $\\tau=\\varepsilon t$. Slow deformation of\nthe $r_\\pm(s^\\pm,\\tau)$ is determined from equations\n(0.4),(0.5),(1.6),(1.4).\n\\end{Th}\n\n\n\n\\begin{thebibliography}{99}\n\\bibitem {1}\nL.A. Kalyakin, Teoret. Matemat. Fisika 118, 3 (1999)\n390-396 (in Russian).\n\n\\bibitem {2}\nL.A. Kalyakin, Math. USSR Sbornik 52, 1 (1985) 91-114\nMatem. Sbornik. 124, 1 (1984) 96-120 (in Russian).\n\n\\bibitem {3}\nJ. Liouville, Journ. math. pure et appl. 18 (1853)\n71-74.\n\n\\bibitem {4}\nG.P.Jeorjadze, A.K. Pogrebkov, M.C. Polivanov, Teoret.\nMatemat. Fisika 40, 2 (1979) 221-234 (in Russian).\n\n\\bibitem {5}\nL. Bianchi, Ann. Sci. Norm. Super. Piza, Ser 1, 2 (1879)\n26.\n\n\n\\end{thebibliography}\n\n\\end{document}\n" } ]	[ { "name": "solv-int9912011.extracted_bib", "string": "{1 L.A. Kalyakin, Teoret. Matemat. Fisika 118, 3 (1999) 390-396 (in Russian)." }, { "name": "solv-int9912011.extracted_bib", "string": "{2 L.A. Kalyakin, Math. USSR Sbornik 52, 1 (1985) 91-114 Matem. Sbornik. 124, 1 (1984) 96-120 (in Russian)." }, { "name": "solv-int9912011.extracted_bib", "string": "{3 J. Liouville, Journ. math. pure et appl. 18 (1853) 71-74." }, { "name": "solv-int9912011.extracted_bib", "string": "{4 G.P.Jeorjadze, A.K. Pogrebkov, M.C. Polivanov, Teoret. Matemat. Fisika 40, 2 (1979) 221-234 (in Russian)." }, { "name": "solv-int9912011.extracted_bib", "string": "{5 L. Bianchi, Ann. Sci. Norm. Super. Piza, Ser 1, 2 (1879) 26." } ]
solv-int9912012	Whitham-Toda hierarchy in the Laplacian growth problem\footnote{Talk given at the Workshop NEEDS 99 (Crete, Greece, June 1999)	[ { "author": "M. Mineev-Weinstein \\thanks{Theoretical Division" }, { "author": "MS-B213" }, { "author": "LANL" }, { "author": "Los Alamos" }, { "author": "NM 87545" }, { "author": "USA" } ]	The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain. The Laplacian growth equation itself is the quasiclassical version of the string equation that selects the solution to the hierarchy.	[ { "name": "solv-int9912012.tex", "string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentstyle[12pt]{article}\n%%%%%%%%%%%%%%%% MACROS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\def\\hybrid{\\topmargin 0pt \\oddsidemargin 0pt\n \\headheight 0pt \\headsep 0pt\n \\voffset=-0.5cm\n% \\textwidth 6.5in % US paper\n% \\textheight 9in % US paper\n \\textwidth 6.25in % A4 paper\n \\textheight 9.5in % A4 paper\n \\marginparwidth 0.0in\n \\parskip 5pt plus 1pt \\jot = 1.5ex}\n%\\def\\baselinestretch{1.5}\n\\catcode`\\@=11\n\\def\\marginnote#1{}\n\n\\newcount\\hour\n\\newcount\\minute\n\\newtoks\\amorpm\n\\hour=\\time\\divide\\hour by60\n\\minute=\\time{\\multiply\\hour by60 \\global\\advance\\minute by-\\hour}\n\\edef\\standardtime{{\\ifnum\\hour<12 \\global\\amorpm={am}%\n \\else\\global\\amorpm={pm}\\advance\\hour by-12 \\fi\n \\ifnum\\hour=0 \\hour=12 \\fi\n \\number\\hour:\\ifnum\\minute<10 0\\fi\\number\\minute\\the\\amorpm}}\n\\edef\\militarytime{\\number\\hour:\\ifnum\\minute<10 0\\fi\\number\\minute}\n\n\\def\\draftlabel#1{{\\@bsphack\\if@filesw {\\let\\thepage\\relax\n \\xdef\\@gtempa{\\write\\@auxout{\\string\n \\newlabel{#1}{{\\@currentlabel}{\\thepage}}}}}\\@gtempa\n \\if@nobreak \\ifvmode\\nobreak\\fi\\fi\\fi\\@esphack}\n \\gdef\\@eqnlabel{#1}}\n\\def\\@eqnlabel{}\n\\def\\@vacuum{}\n\\def\\draftmarginnote#1{\\marginpar{\\raggedright\\scriptsize\\tt#1}}\n\n\\def\\draftlabel#1{{\\@bsphack\\if@filesw {\\let\\thepage\\relax\n \\xdef\\@gtempa{\\write\\@auxout{\\string\n \\newlabel{#1}{{\\@currentlabel}{\\thepage}}}}}\\@gtempa\n \\if@nobreak \\ifvmode\\nobreak\\fi\\fi\\fi\\@esphack}\n \\gdef\\@eqnlabel{#1}}\n\\def\\@eqnlabel{}\n\\def\\@vacuum{}\n\\def\\draftmarginnote#1{\\marginpar{\\raggedright\\scriptsize\\tt#1}}\n\n\\def\\draft{\\oddsidemargin -.5truein\n \\def\\@oddfoot{\\sl preliminary draft \\hfil\n \\rm\\thepage\\hfil\\sl\\today\\quad\\militarytime}\n \\let\\@evenfoot\\@oddfoot \\overfullrule 3pt\n \\let\\label=\\draftlabel\n \\let\\marginnote=\\draftmarginnote\n \\def\\@eqnnum{(\\theequation)\\rlap{\\kern\\marginparsep\\tt\\@eqnlabel}%\n\\global\\let\\@eqnlabel\\@vacuum} }\n\n% This causes equations to be numbered by section\n\n%\\def\\numberbysection{\\@addtoreset{equation}{section}\n% \\def\\theequation{\\thesection.\\arabic{equation}}}\n\n\\def\\underline#1{\\relax\\ifmmode\\@@underline#1\\else\n $\\@@underline{\\hbox{#1}}$\\relax\\fi}\n\n\\def\\titlepage{\\@restonecolfalse\\if@twocolumn\\@restonecoltrue\\onecolumn\n \\else \\newpage \\fi \\thispagestyle{empty}\\c@page\\z@\n \\def\\thefootnote{\\fnsymbol{footnote}} }\n\n\\def\\endtitlepage{\\if@restonecol\\twocolumn \\else \\fi\n \\def\\thefootnote{\\arabic{footnote}}\n \\setcounter{footnote}{0}} %\\c@footnote\\z@ }\n%\\catcode`@=12\n\\relax\n\n%\\draft\n\n%\\numberbysection\n\\hybrid\n\n\\def\\beq{\\begin{equation}}\n\\def\\eeq{\\end{equation}}\n\\def\\p{\\partial}\n\\def\\G{\\Gamma}\n\n\\newtheorem{th}{Theorem}\n\\newtheorem{prop}{Proposition}\n\\newtheorem{cor}{Corollary}\n\\newtheorem{lem}{Lemma}\n\n\n\\def\\square{\\hfill\n{\\vrule height6pt width6pt depth1pt} \\break \\vspace{.01cm}}\n\n\\begin{document}\n\n\\begin{titlepage}\n\n\\title{Whitham-Toda hierarchy in the Laplacian\ngrowth problem\\footnote{Talk given at the Workshop NEEDS 99\n(Crete, Greece, June 1999)}}\n\n\n\\author{M. Mineev-Weinstein \\thanks{Theoretical Division, MS-B213, LANL,\n Los Alamos, NM 87545, USA }\n\\and A. Zabrodin\n\\thanks{Joint Institute of Chemical Physics, Kosygina str. 4, 117334,\nMoscow, Russia and ITEP, 117259, Moscow, Russia}}\n\n\n\\date{October 1999}\n\\maketitle\n\n\\begin{abstract}\n\nThe Laplacian growth problem in the limit of zero\nsurface tension\nis proved to be equivalent to\nfinding a particular solution to the dispersionless Toda\nlattice hierarchy. The hierarchical times are harmonic\nmoments of the growing domain. The Laplacian growth equation\nitself is the quasiclassical version of the\nstring equation that selects the solution to the hierarchy.\n\n\\end{abstract}\n\n\n\\vfill\n\n\\end{titlepage}\n\n\nThe Laplacian growth problem is one of the central\nproblems in the theory of pattern formation.\nIt has many different faces and a lot of important\napplications.\nIn general words, this is about dynamics of\nmoving front (interface)\nbetween two different phases. In many cases the dynamics\nis governed by a scalar field that obeys the Laplace equation;\nthat is why this class of growth problems is called Laplacian.\nHere we shall confine ourselves to the two-dimensional (2D)\ncase only.\nTo be definite, we shall speak about two\nincompressible fluids with different viscosities on the plane.\nIn practice, the 2D geometry is\nrealized in the narrow gap between two plates.\nIn this version, this\nis known as the Saffman-Taylor problem\nor viscous fingering in the\nHele-Shaw cell.\nFor a review, see \\cite{RMP}.\n\nWe shall mostly concentrate on the {\\it external radial\nproblem} for it turns out to be the simplest case\nin the frame of the suggested approach.\nLet the exterior of a simply connected domain on the plane\nbe occupied by a viscous fluid (oil) while the interior\nbe occupied by a fluid with small viscosity (water).\nThe oil/water interface is assumed to be a simple\nanalytic curve. Other versions such as internal radial problem,\nwedge or channel geometry are briefly discussed at the end\nof the paper. Basically, they allow for the same approach.\n\nLet $p(x,y)$ be the pressure, then $p$ is constant in the\nwater domain. We set it equal to zero.\nIn the case of zero surface tension $p$\nis a continuous function across the interface, so\n$p=0$ on the interface. In the oil domain\nthe gradient of $p$ is proportional to local velocity\n$\\vec V=(V_x,V_y)$ of the fluid\n(Darcy's law):\n\\beq\n\\label{lg1}\n\\vec V=-\\kappa \\,\\mbox{grad}\\, p\\,,\n\\eeq\nwhere $\\kappa$ is called the filtration coefficient\\footnote{This\ncoefficient is inversely proportional to the viscosity, so the\nDarcy law is formally valid in the water, too.}.\nIn particular, this law holds on the interface thus\ngoverning its dynamics:\n\\beq\n\\label{lg1a}\nV_n=-\\kappa \\frac{\\p p}{\\p n}\\,.\n\\eeq\nHere $V_n$ is the component of the velocity\nnormal to the interface\nand $\\p p/\\p n$ is normal derivative.\nSince the fluid is incompressible ($\\mbox{div}\\vec V =0$),\nthe Darcy law implies\nthat the potential $\\Phi (x,y)=-\\kappa p(x,y)$\nis a harmonic function in the exterior (oil) domain:\n\\beq\n\\label{lg2}\n\\Delta \\Phi (x,y)=0\\,,\n\\eeq\nwhere $\\Delta =\\p_{x}^{2}+\\p_{y}^{2}$ is the Laplace operator\non the plane. The asymptotic behaviour of the function $\\Phi$\nvery far away from the interface (at infinity) is determined\nby the physical condition that there is a sink with constant\ncapacity $q$ placed at infinity. This means that\n$$\n\\oint_{\\gamma}(\\vec V, \\vec n)dl =q\\,,\n$$\nwhere $\\gamma$ is any closed contour encircling the water\ndomain,\n$\\vec n$ is the unit vector normal to $\\gamma$.\nSo, we require $\\Phi =0$ on the interface and\n$\\Phi=\\frac{q}{2\\pi}\\mbox{log}\\,\|z\|$ as $\|z\|\\to \\infty$.\nThe goal is to describe the interface motion subject\nto the local dynamical law $V_n=\\p \\Phi /\\p n$.\n\nAn effective tool for dealing with this\nproblem is the time dependent conformal\nmapping technique (see e.g. \\cite{RMP}).\nPassing to the complex coordinates $z=x+iy$,\n$\\bar z =x-iy$ on the physical plane,\nwe bring into play a conformal map from\na reference domain on the mathematical plane $w$\nto the growing domain on the physical plane.\nBy the Riemann mapping theorem, such a map does exist and,\nunder some conditions, is unique.\nMore precisely, let $z=\\lambda (w)$ be the\nunivalent conformal map from\nthe exterior of the unit circle to the\nexterior of the interface (i.e., to the oil domain)\nsuch that $\\infty$\nis mapped to $\\infty$ and the derivative $\\lambda '(\\infty)$\nis a positive real number $r$. Under these conditions\nthe map is known to be unique.\nThe Laurent expansion of the $\\lambda (w)$ around $\\infty$\nhas then the following general form:\n\\beq\n\\label{z}\n\\lambda (w)=rw+\\sum_{j=0}^{\\infty}u_jw^{-j}\\,.\n\\eeq\n\nIf the interface moves, the conformal map\n$z(w,t)=\\lambda (w,t)$ becomes time-dependent.\nThe interface itself is the image\nof the unit circle $\|w\|=1$: as $w=e^{i\\phi}$,\n$0\\leq \\phi \\leq 2\\pi$, sweeps\nover the unit circle, $z=\\lambda (e^{i\\phi},t)$ sweeps over the\ninterface at the moment $t$.\n\nHaving defined the conformal map, we immediately see that the\nreal part of the logarithm\nof the inverse conformal map, $w(z)$,\nprovides the solution\nto the Laplace equation in the oil domain with\nthe required asymptotics:\n$\\Phi(x,y)=\\frac{q}{2\\pi}\\mbox{Re}\\,\\mbox{log}\\,w(z)$.\nLet us introduce\nthe complex velocity $V=V_x-iV_y$.\nThe obvious formula\n$$\n\\frac{q}{2\\pi}\\frac{\\p \\,\\mbox{log}\\,w(z,t)}{\\p z}=\n\\frac{\\p \\Phi}{\\p x}-i\\frac{\\p \\Phi}{\\p y}\n$$\nallows one to represent\nthe Darcy law (\\ref{lg1}) as follows:\n\\beq\n\\label{lg3}\nV(z)=\\frac{q}{2\\pi}\\,\n\\frac{\\p \\,\\mbox{log}\\,w}{\\p z}\\,,\n\\eeq\nwhere the derivative is taken\nat constant $t$.\n\nIn terms of the time-dependent conformal map,\nthe Darcy law is equivalent to the following\nrelation referred to as the Laplacian growth equation (LGE):\n\\beq\n\\label{lg6a}\n\\mbox{Im}\\left (\n\\frac{\\p z}{\\p \\phi}\n\\frac{\\p \\bar z}{\\p t}\\right )=\\frac{q}{2\\pi}\\,.\n\\eeq\nIt first appeared in 1945 \\cite{DAN} in the works on\nthe mathematical theory of oil production.\n>From now on we set $q=\\pi$ without loss of generality.\n(This amounts to a proper rescaling of $t$.)\nIntroducing the Poisson bracket notation\n\\beq\n\\label{PB}\n\\{f,g\\}=w\\frac{\\p f}{\\p w}\\frac{\\p g}{\\p t}\n-w\\frac{\\p g}{\\p w}\\frac{\\p f}{\\p t}\\,,\n\\eeq\nfor functions $f=f(w,t)$, $g=g(w,t)$ of $w$, $t$, we rewrite\nthe LGE in the suggestive form\\footnote{Given a Laurent series\n$f(z)=\\sum_j f_j z^j$, we set\n$\\bar f(z)=\\sum_j \\bar f_j z^j$, so $z(w)$ and $\\bar z(w^{-1})$\nare complex conjugate only if $\|w\|=1$.}\n\\beq\n\\label{lg6}\n\\{z(w,t), \\bar z(w^{-1},t)\\}=1\\,.\n\\eeq\nThe LGE thus means that the transformation from\n$\\mbox{log}\\,w,\\, t$ to $z, \\bar z$ is canonical.\n\nFor a technical simplicity,\nwe assume that the point $z=0$ lies in the\nwater domain.\nThe interface dynamics given by the Darcy law (or,\nequivalently, by the LGE) implies that if a point $(x,y)$ is\nin the water (interior) domain at the initial moment,\nthen it remains there for all values of time. In particular,\nour assumption that the point $z=0$ belongs to the water\ndomain means that zeros of the function $\\lambda (w)$ are inside\nthe unit circle for any $t$.\n\nThe Laplacian growth is a particular case of\nthe 2D inverse potential problem. The shape of the\ninterface can be characterized by the\nharmonic moments $C_k$ of the oil domain and the area\n$C_0$ of the water domain:\n\\beq\n\\label{h1}\nC_k=-\\displaystyle{\\int \\!\\!\n\\int_{\\mbox{{\\small exterior}}}}z^{-k}dxdy\\,,\n\\;\\;\\;\\;\\;\\; k\\geq 1 ;\\;\\;\\;\\;\\;C_{0}=\\displaystyle{\\int \\!\\!\n\\int_{\\mbox{{\\small interior}}}}dxdy\\,.\n\\eeq\n(The integrals at $k=1,2$ are assumed to be properly\nregularized.) A remarkable result of Richardson \\cite{Rich}\nshows that\nthe LGE (\\ref{lg6}) implies conservation of the harmonic\nmoments $C_k$ when the\ninterface moves, $dC_k/dt =0$, while area of the water\ndomain grows linearly in time: $C_0=\\pi t$. Therefore, the problem\ncan be posed as follows: to find the shape of the domain\nas a function of its area provided all the harmonic moments\nof the exterior are\nkept fixed. Since the harmonic moments\nare coefficients in the expansion of the Coulomb\npotential created by a homogeneously distributed charge\nin the oil domain, this is just a\nspecification of the inverse potential problem.\nWe remind that to know the shape of the domain is the same as\nto know the coefficients $r$, $u_j$ of the conformal map (\\ref{z}).\n\nA good deal of hints that the LGE has much to do\nwith integrable systems have\nbeen known for quite a long time \\cite{Rich}-\\cite{M1}.\nHowever, its status in the realm of integrability\nwas obscure until very recently.\nThe nature of the LGE and its relation to integrability\nare clarified in the work \\cite{MWZ}.\nThe idea is to treat the LGE {\\it not\nas a dynamical equation} but {\\it as a constraint} in\na bigger integrable hierarchy.\nThe latter turns out to be an infinite\nWhitham hierarchy of the type first\nintroduced in \\cite{kri1}. This hierarchy is a\nmulti-dimensional generalization of integrable hierarchies of\nhydrodinamic type \\cite{hydro}. It naturally\nincorporates the general inverse potential problem as well.\nNamely, the coefficients of the conformal map as functions\nof all the harmonic moments are given by a particular solution\nto the dispersionless Toda\nlattice hierarchy (see \\cite{Tak-Tak} for a detailed\nstudy of the latter). The hierarchical evolution\ntimes are just the harmonic moments and their complex\nconjugate. In the Laplacian growth all of them but the\narea $C_0$ are frozen. Making them alive, one moves over\nthe space of initial data for the LGE, and recovers the\nWhitham hierarchy.\n\nFor a more convenient formulation\nof the result, let us rescale the harmonic\nmoments and introduce the new notation for them:\n\\beq\n\\label{11}\nt\\equiv t_{0}=\\frac{C_0}{\\pi},\\;\\;\\;\\;\\;\nt_k=\\frac{C_k}{\\pi k},\\;\\;\\;\\;\\;\\bar\nt_k=\\frac{\\bar C_k}{\\pi k},\\;\\;\\;\\;\\; k\\geq 1.\n\\eeq\nThe symbols $(f(w))_{\\pm}$\nbelow mean a truncated Laurent series, where\nonly terms with positive (negative)\npowers of $w$ are kept, $(f(w))_{0}$ is a\nconstant part ($w^0$) of the series.\n\\begin{th}\nThe conformal map (\\ref{z}) obeys the following\ndifferential equations with respect to the\nharmonic moments:\n\\beq\n\\label{HJ9a}\n\\frac{\\p \\lambda (w)}{\\p t_j} =\\{H_j, \\lambda (w)\\}\\,,\n\\;\\;\\;\\;\\;\\;\n\\frac{\\p \\lambda (w)}{\\p \\bar t_j} =-\\{\\bar H_j, \\lambda (w)\\}\\,,\n\\eeq\n\\beq\n\\label{HJ9b}\n\\frac{\\p \\bar \\lambda (w^{-1})}{\\p t_j}\n=\\{H_j, \\bar \\lambda (w^{-1})\\}\\,,\n\\;\\;\\;\\;\\;\\;\n\\frac{\\p \\bar \\lambda (w^{-1})}{\\p \\bar t_j} =\n-\\{\\bar H_j, \\bar \\lambda (w^{-1})\\}\\,,\n\\eeq\nwhere the Poisson bracket is defined in (\\ref{PB}), and\n\\beq\n\\label{HJ12}\nH_j(w)=\\Bigl (\\lambda ^j(w)\\Bigr )_{+} +\\frac{1}{2}\n\\Bigl (\\lambda ^j(w) \\Bigr )_{0}\\,,\n\\eeq\n\\beq\n\\label{J4}\n\\bar H_j(w)=\\Bigl (\\bar \\lambda ^j(w^{-1})\\Bigr )_{-} +\\frac{1}{2}\n\\Bigl (\\bar \\lambda ^j(w^{-1}) \\Bigr )_{0}\\,.\n\\eeq\n\\end{th}\nThe proof is sketched in\n\\cite{MWZ} and \\cite{WZ}. These are the Lax-Sato\nequations for the dispersionless Toda lattice hierarchy of\nnon-linear differential equations, with the $\\lambda (w)$\nand $\\bar \\lambda (w^{-1})$ being the Lax functions.\nOn comparing coefficients\nin front of powers of $w$ in (\\ref{HJ9a}), (\\ref{HJ9b}),\none obtains an infinite set of non-linear differential\nequations for the coefficients of the comformal map. Altogether,\nthey form the hierarchy. The particular solution that solves the\ninverse potential problem is selected by the constraint\n(\\ref{lg6}), where $z(w,t)=\\lambda (w,t)$,\n$\\bar z(w,t)=\\bar \\lambda (w^{-1},t)$,\nand all $t_k$ are fixed.\nThis constraint is known as (a quasiclassical version of) the\n{\\it string equation}. Surprisingly, this very constraint\nis the key ingredient of the integrable structures in\n2D gravity coupled with $c=1$ matter \\cite{gravity,matrix}.\nThe mathematical theory of the dispersionless hierarchies\nconstrained by string equations was developed in\n\\cite{kri2,Du1} and\nextended to the Toda case in \\cite{Tak-Tak}.\n\nThe Lax-Sato equations imply \\cite{kri2}\nthe existence of the prepotential function\n$F(t, t_k, \\bar t_k)$. This function\nsolves the inverse potential problem\nin the following sense.\nLet $C_{-k}$, $k\\geq 1$, be the complimentary set of\nharmonic moments,\ni.e., the moments of the interior of the domain:\n\\beq\n\\label{h2}\nC_{-k}=\\displaystyle{\\int \\!\\!\n\\int_{\\mbox{{\\small interior}}}}z^{k}dxdy\\,.\n\\eeq\nWere we able to find $C_{-k}$ from a given set $C_0$,\n$C_k$ (i.e., given $t$, $t_k$), this would yield the complete\nsolution, alternative but equivalent\nto knowing the conformal map (\\ref{z}). Indeed, using\nthe contour integral representation of the harmonic\nmoments, it is easy to see that\nthe generating function of {\\it all} the harmonic moments,\nobtained as an analytic continuation of the Laurent series\n\\beq\n\\label{S}\n\\mu (\\lambda )= \\frac{1}{\\pi}\n\\sum_{k\\in {\\bf Z}}C_k \\lambda ^k =\n\\sum_{k=1}^{\\infty}kt_k \\lambda ^{k}+t+\n\\frac{1}{\\pi}\\sum_{k=1}^{\\infty}C_{-k} \\lambda ^{-k}\\,,\n\\eeq\nwould allow us to\nrestore the interface curve via the equation\n$\|z\|^2 =\\mu (z)$ ($S(z)= \\mu (z)/z$ is what is called\n{\\it Schwarz function} of the curve \\cite{Davis}).\nThe function $F$ does the job.\n\\begin{th}\n(\\cite{MWZ,WZ})\nThere exists a real function $F$ of the (rescaled) harmonic\nmoments $t$, $t_k$, $\\bar t_k$ such that\n\\beq\n\\label{F}\nC_{-k}=\\pi \\,\\frac{\\p F}{\\p t_k}\\,,\n\\;\\;\\;\\;\\;\\;\n\\bar C_{-k}=\\pi \\,\\frac{\\p F}{\\p \\bar t_k}\\,,\n\\;\\;\\;\\;\\; k\\geq 1\\,.\n\\eeq\n\\end{th}\nIn particular, this implies the symmetry of\nderivatives of the inner harmonic\nmoments with respect to the outer ones:\n\\beq\n\\label{sym}\n\\frac{\\p C_{-k}}{\\p t_j}=\n\\frac{\\p C_{-j}}{\\p t_k}\\,,\n\\;\\;\\;\\;\\;\\;\n\\frac{\\p C_{-k}}{\\p \\bar t_j}=\n\\frac{\\p \\bar C_{-j}}{\\p t_k}\\,.\n\\eeq\nFor general Whitham hierarchies, the function $F$\nwas introduced in \\cite{kri2}.\nIn the dispersionless Toda case, it is the\ndispersionless limit of logarithm of the\n$\\tau$-function \\cite{Tak-Tak}.\nThis function obeys the dispersionless limit of the Hirota\nequation (a leading term of the differential Fay identity\n\\cite{GiKo1},\\,\\cite{Tak-Tak}):\n\\begin{equation}\n\\label{Hirota}\n(z\\! -\\! \\zeta)\\,\\exp \\! \\left(\\sum_{n,m\\geq\n1}\\frac{F_{nm}}{nm}z^{-n}\\zeta^{-m}\\right)\\! =\n\\! z\\exp \\! \\left(-\\sum_{k\\geq\n1}\\frac{F_{0k}}{k}z^{-k}\\right)\\! -\\!\\zeta\\exp \\! \\left(-\\sum_{k\\geq\n1}\\frac{F_{0k}}{k}\\zeta^{-k}\\right)\n\\end{equation}\nThis is an infinite set of relations between the second derivatives\n$F_{nm}=\n\\p_{t_n}\\p_{t_m}F,\\;\\;\\;F_{0m}=\n\\p_{t}\\p_{t_m}F$.\nThe equations appear while expanding the\nboth sides of (\\ref{Hirota}) in\npowers of $z$ and $\\zeta$.\n\nTo complete the identification with the objects of the\ntheory of Whitham hierarchies, we just mention that\nthe function $\\mu (\\lambda )$ (\\ref{S}) is the quasiclasical limit\nof the Orlov-Shulman operator \\cite{Or} for the 2D Toda lattice.\nThis function obeys the conditions\n$\\{\\lambda , \\mu \\}=\\lambda$,\n$\\{\\bar \\lambda , \\bar \\mu \\}=-\\bar \\lambda$.\nThe string equation can be then written as a relation\n\\beq\n\\label{string1}\n\\bar \\lambda =\\lambda ^{-1} \\mu\n\\eeq\nbetween the Laurent series, together with the\nreality condition $\\mu (\\lambda )=\\bar \\mu (\\bar \\lambda)$.\nWritten in this form,\nit admits generalizations which are also relevant\nto the Laplacian growth\nproblem, and which we are going to discuss now.\n\nConsider the Laplacian growth problem for domains\nsymmetric under the ${\\bf Z}_n$-\\-trans\\-for\\-ma\\-ti\\-ons\n$z\\to e^{2\\pi i l/n}z$, $l=1,\\ldots , n-1$, where\n$n$ is a positive integer. In this case $C_m=0$\nunless $m=0 \\,(\\mbox{mod}\\,n)$, i.e., only the moments\n$C_{kn}$ are non-zero.\nThe conformal map from the exterior of the unit circle\nto the exterior of such a domain\nis given by\n$z(w)=(\\lambda (w^n))^{1/n}$,\nwhere $\\lambda (w)$ has the same general structure as\nin (\\ref{z}).\nEquivalently, the problem may be\nposed as the Laplacian growth in the wedge (sector) domain\nrestricted by the rays $\\mbox{arg}\\,z =0$ and\n$\\mbox{arg}\\,z =2\\pi /n$ with the periodic condition\non the boundary. In the latter case, the conformal map is\n\\beq\n\\label{conf2}\nz(w)=(\\lambda (w))^{1/n}\\,.\n\\eeq\nThe LGE in terms of the conformal map\n$z(w,t)$ has the same form (\\ref{lg6}) if the time\nis rescaled as $t \\to t/n$.\nIn terms of the $\\lambda (w,t)$, one has\n\\beq\n\\label{lg61}\n\\{\\lambda ,\\, \\bar \\lambda \\}=\nn^2(\\lambda \\bar \\lambda )^{\\frac{n-1}{n}}\\,.\n\\eeq\n\nLet us introduce the following notation for the\nnon-zero harmonic moments:\n\\beq\n\\label{111}\nt=\\frac{C_{0}}{n\\pi },\\;\\;\\;\\;\\;\nt_k=\\frac{C_{kn}}{\\pi k},\\;\\;\\;\\;\\;\n\\bar t_k=\\frac{\\bar C_{kn}}{\\pi k},\\;\\;\\;\\;\\; k\\geq 1\n\\eeq\n(cf.\\,(\\ref{11})).\nThe more general string equation (\\ref{lg61}) amounts to\nthe relation\n\\beq\n\\label{string2}\n\\bar \\lambda =n^n \\lambda ^{-1} \\mu ^n\n\\eeq\nbetween the Lax and Orlov-Shulman functions (note that\nthis means $\|z\|^2 =n\\mu (z)$). This relation is familiar\nfrom \\cite{Tak-Tak,gravity}. It is consistent with the\nhierarchy and selects a solution that, as $n>1$, is\ndifferent from the one selected by (\\ref{string1}).\nThe Lax-Sato equations (\\ref{HJ9a}) -- (\\ref{J4}) for\nthe Lax function $\\lambda$ and the Hirota\nequation (\\ref{Hirota}) hold true as they stand\nprovided the times are redefined as in (\\ref{111}).\nThe solution to the Laplacian growth problem\nin the channel geometry (in the\nHele-Shaw cell) can be obtained\nfrom the above formulas as a somewhat tricky limit\n$n\\to \\infty$.\nIt is unclear, however, if the existing finite-dimensional\nsolutions of the LGE [4-6] will survive after these transformations.\nWe should also mention that\ninclusion of the ``no-flux'' boundary conditions\nat the walls of the wedge, which state that\n$\\partial\\log z / \\partial \\log w$\nis real, imposes an extra symmetry in the solution and\ndoubles the number of\nwedges. We will elucidate these questions in the near future.\n\nAt last we point out that the above formulas make sense\nfor negative integer $n$, too. In particular, the case\n$n=-1$ describes the {\\it internal} radial problem\nwhen oil is inside while water is outside.\nIn this case the internal and external\nharmonic moments are interchanged in their role:\nthe internal moments (\\ref{h2}) together with\nthe $C_0$ become independent\nvariables (cf.\\,(\\ref{111})) while the external\nmoments are found as derivatives of the prepotential function\naccording to (\\ref{F}).\n\n\n\\vspace{3mm}\n\nThese results were reported at the XIII NEEDS Workshop\n(Crete, June 1999).\nWe are very grateful to the organizers\nfor the invitation and for the opportunity\nto share these results in such a fruitful and nice atmosphere.\nWe are indebted to Paul Wiegmann for collaboration\nin \\cite{MWZ,WZ} and many stimulating discussions.\nThe work of A.Z. was partially supported\nby RFBR grant 98-01-00344.\n\n\n\n\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{RMP} D.Bensimon, L.P.Kadanoff, S.Liang, B.I.Shraiman\nand C.Tang, Rev. Mod. Phys. {\\bf 58} (1986) 977-999\n\n\\bibitem{DAN}\nL.A. Galin, Dokl. Akad. Nauk SSSR\n {\\bf 47 } (1945) 250-253;\\\\\nP.Ya.Polubarinova-Kochina, Dokl. Akad. Nauk SSSR,\n {\\bf 47 } (1945) 254-257; \\\\\nP.P. Kufarev, Dokl. Akad. Nauk SSSR\n{\\bf 57 } (1947) 335-348\n\n\\bibitem{Rich}\nS.Richardson, J. Fluid Mech. {\\bf 56} (1972) 609-618\n\n\\bibitem{SB} B.Shraiman and D.Bensimon, Phys. Rev. A {\\bf 30}\n(1984) 2840-2842\n\n\n\\bibitem{How} S.D.Howison, J. Fluid Mech. {\\bf 167} (1986) 439-453\n\n\n\\bibitem{M1} M.Mineev, Physica D {\\bf 43} (1990) 288-292;\\\\\nS.P.Dawson and M.Mineev-Weinstein, Physica D {\\bf 73} (1994) 373-387\n\n\n\\bibitem{MWZ} M.Mineev-Weinstein, P.Wiegmann and A.Zabrodin,\nto appear\n\n\\bibitem{kri1}\nI.M.Krichever, Funct. Anal Appl. {\\bf 22} (1989) 200-213\n\n\\bibitem{hydro} B.A.Dubrovin and S.P.Novikov, Soviet Math. Dokl.\n {\\bf 27} (1983) 665-669;\\\\\nS.P.Tsarev, Soviet Math. Dokl. {\\bf 31} (1985) 488-491\n\n\\bibitem{Tak-Tak} T.Takasaki and T.Takebe,\nRev. in Math. Phys. {\\bf 7} (1995) 743-808\n\n\n\\bibitem{WZ} P.Wiegmann and A.Zabrodin, preprint ESI-760,\nITEP-TH-47/99, hep-th/9909147\n\n\n\n\\bibitem{gravity}\nR.Dijkgraaf, G.Moore and R.Plesser,\nNucl.Phys. {\\bf B394} (1993) 356-382; \\\\\nA Hanany, Y.Oz and R.Plesser, Nucl.Phys. {\\bf B425} (1994) 150-172; \\\\\nK.Takasaki, Commun. Math. Phys. {\\bf 170} (1995) 101-116; \\\\\nT.Eguchi and H.Kanno, Phys.Lett. {\\bf 331B} (1994) 330\n\n\\bibitem{matrix}\nJ.M.Daul, V.A.Kazakov and I.K.Kostov, Nucl.Phys.\n{\\bf B409} (1993) 311-338; \\\\\nL.Bonora and C.S.Xiong,\nPhys. Lett. {\\bf B347} (1995) 41-48\n\n\\bibitem{kri2}\nI.M.Krichever, Comm. Pure. Appl. Math. {\\bf 47} (1992) 437-476\n\n\n\\bibitem{Du1}B.A.Dubrovin,\nCommun. Math. Phys. {\\bf 145} (1992) 195-207\n\n\n\\bibitem{Davis} P.J.Davis, The Schwarz\nfunction and its applications, The\nCarus Math. Monographs, No. 17,\nThe Math. Assotiation of America, Buffalo,\nN.Y., 1974\n\n\\bibitem{GiKo1}J.Gibbons and Y.Kodama, Phys.Lett. {\\bf 135 A} (1989)\n167-170;\\\\\nJ.Gibbons and Y.Kodama, Proceedings of NATO ASI, 'Singular Limits of\nDispersive Waves' ed.\nN.Ercolani, Plenum 1994\n\n\\bibitem{Or}A.Orlov and E.Shulman,\nLett. Math. Phys. {\\bf 12} (1986) 171-179\n\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n\n\n\n\n\n" } ]	[ { "name": "solv-int9912012.extracted_bib", "string": "{RMP D.Bensimon, L.P.Kadanoff, S.Liang, B.I.Shraiman and C.Tang, Rev. Mod. Phys. {58 (1986) 977-999" }, { "name": "solv-int9912012.extracted_bib", "string": "{DAN L.A. Galin, Dokl. Akad. Nauk SSSR {47 (1945) 250-253;\\\\ P.Ya.Polubarinova-Kochina, Dokl. Akad. Nauk SSSR, {47 (1945) 254-257; \\\\ P.P. Kufarev, Dokl. Akad. Nauk SSSR {57 (1947) 335-348" }, { "name": "solv-int9912012.extracted_bib", "string": "{Rich S.Richardson, J. Fluid Mech. {56 (1972) 609-618" }, { "name": "solv-int9912012.extracted_bib", "string": "{SB B.Shraiman and D.Bensimon, Phys. Rev. A {30 (1984) 2840-2842" }, { "name": "solv-int9912012.extracted_bib", "string": "{How S.D.Howison, J. Fluid Mech. {167 (1986) 439-453" }, { "name": "solv-int9912012.extracted_bib", "string": "{M1 M.Mineev, Physica D {43 (1990) 288-292;\\\\ S.P.Dawson and M.Mineev-Weinstein, Physica D {73 (1994) 373-387" }, { "name": "solv-int9912012.extracted_bib", "string": "{MWZ M.Mineev-Weinstein, P.Wiegmann and A.Zabrodin, to appear" }, { "name": "solv-int9912012.extracted_bib", "string": "{kri1 I.M.Krichever, Funct. Anal Appl. {22 (1989) 200-213" }, { "name": "solv-int9912012.extracted_bib", "string": "{hydro B.A.Dubrovin and S.P.Novikov, Soviet Math. Dokl. {27 (1983) 665-669;\\\\ S.P.Tsarev, Soviet Math. Dokl. {31 (1985) 488-491" }, { "name": "solv-int9912012.extracted_bib", "string": "{Tak-Tak T.Takasaki and T.Takebe, Rev. in Math. Phys. {7 (1995) 743-808" }, { "name": "solv-int9912012.extracted_bib", "string": "{WZ P.Wiegmann and A.Zabrodin, preprint ESI-760, ITEP-TH-47/99, hep-th/9909147" }, { "name": "solv-int9912012.extracted_bib", "string": "{gravity R.Dijkgraaf, G.Moore and R.Plesser, Nucl.Phys. {B394 (1993) 356-382; \\\\ A Hanany, Y.Oz and R.Plesser, Nucl.Phys. {B425 (1994) 150-172; \\\\ K.Takasaki, Commun. Math. Phys. {170 (1995) 101-116; \\\\ T.Eguchi and H.Kanno, Phys.Lett. {331B (1994) 330" }, { "name": "solv-int9912012.extracted_bib", "string": "{matrix J.M.Daul, V.A.Kazakov and I.K.Kostov, Nucl.Phys. {B409 (1993) 311-338; \\\\ L.Bonora and C.S.Xiong, Phys. Lett. {B347 (1995) 41-48" }, { "name": "solv-int9912012.extracted_bib", "string": "{kri2 I.M.Krichever, Comm. Pure. Appl. Math. {47 (1992) 437-476" }, { "name": "solv-int9912012.extracted_bib", "string": "{Du1B.A.Dubrovin, Commun. Math. Phys. {145 (1992) 195-207" }, { "name": "solv-int9912012.extracted_bib", "string": "{Davis P.J.Davis, The Schwarz function and its applications, The Carus Math. Monographs, No. 17, The Math. Assotiation of America, Buffalo, N.Y., 1974" }, { "name": "solv-int9912012.extracted_bib", "string": "{GiKo1J.Gibbons and Y.Kodama, Phys.Lett. {135 A (1989) 167-170;\\\\ J.Gibbons and Y.Kodama, Proceedings of NATO ASI, 'Singular Limits of Dispersive Waves' ed. N.Ercolani, Plenum 1994" }, { "name": "solv-int9912012.extracted_bib", "string": "{OrA.Orlov and E.Shulman, Lett. Math. Phys. {12 (1986) 171-179" } ]
solv-int9912013	Singular solution of the Liouville equation under perturbation	[ { "author": "L.A. Kalyakin \\thanks {This research has been supported by the Russian Foundation of the Fundamental Research under Grants 99-01-00139" }, { "author": "96-15-96241" } ]		[ { "name": "solv-int9912013.tex", "string": "%%%%----------------------- This is a LaTeX file -----\n\\documentclass{article}\n% --------- Sets size of page and margins\n\\oddsidemargin 10mm\n\\evensidemargin 10mm\n\\topmargin 0pt\n\\headheight 0pt\n\\headsep 0pt\n\\baselineskip = 20pt\n\\hsize = 340pt\n\\vsize = 490pt\n\n\\def\\const {{\\hbox{const}}}\n\\def\\O {{\\cal O}}\n\\newtheorem{Def}{Definition}\n\n\n\\title\n{ Singular solution of the Liouville equation under\nperturbation}\n\\author\n{ L.A. Kalyakin\n\\thanks\n{This research has been supported by the Russian\nFoundation of the Fundamental Research under Grants\n99-01-00139, 96-15-96241} }\n\\date{\\it Institute of Mathematics RAS, Ufa, Russia}\n\n\\begin{document}\n\\maketitle\n\nThe Cauchy problem for the Liouville equation with a\nsmall perturbation\n$$\n\\partial_t^2u -\\partial_x^2u\n+8\\exp u =\\varepsilon{\\bf F}[u];\n\\quad 0< \\varepsilon \\ll 1,\n\\eqno (0.1)\n$$\n$$\nu \|_{t=0}=\\psi_0(x;\\varepsilon ), \\\n\\partial_tu\|_{t=0}=\\psi_1(x;\\varepsilon ),\\quad x\\in R\n\\eqno (0.2)\n$$\nis considered. We are interesting for asymptotics of the\nperturbed solution $u(x,t;\\varepsilon)$\\ as\n$\\varepsilon\\to 0$.\n\nPerturbation theory for integrable equations remains a\nvery attractive task. As a rule a perturbation of smooth\nsolutions such as a single soliton were usually\nconsidered. We intend here to discuss a perturbation of a\nsingular solution under assumption that the perturbed\nsolution has singularities as well. A simple well known\ninstance of this kind is a chock wave under weak\nperturbation as given by the Hopf equation with a small\nperturbation term\n$\nu_t+uu_x=\\varepsilon f(u),\\ 0< \\varepsilon \\ll 1.\n$\nIn this paper we consider a more complicated problem\nnamely a perturbation of the singular solution of\nLiouville equation. We deal with the singularities\nstudied by Pogrebkov and Polivanov, [1].\n\n\\bigskip\n%\\section\n{\\bf 1. Unperturbed equation} (as $\\varepsilon=0$)\n$\n\\partial_t^2u -\\partial_x^2u\n+8\\exp u =0$ was solved by Liouville [2] and a formula\nof the general solution is well known\n$$\nu(x,t)=\\ln {{r_+^{\\prime}(s^+)r_-^{\\prime}(s^-)}\\over\n{r^2(s^+,s^-)}},\\quad r=r_+(s^+)+r_-(s^-), \\\ns^{\\pm}=x\\pm t.\n\\eqno (1.1)\n$$\nInitial equations (0.2) give two ODE's for the $r_\\pm$\nwhich may be linearized.\n\nThe singularities of the solution occur due to zero of\nthe denominate $ r(x+t,x-t)\\equiv r_+(x+t)+r_-(x-t)=0$\nunder condition $\\ r_\\pm^\\prime >0.$ We consider just\nthis case. The singular solution is unique under\nmatching condition imposed on the singular lines\n$\\Gamma$, [3]. That is continuity (zero jump across\nthe singular lines) of two expressions\n$$\n[1/2(u _t^2+u _x^2)+2\\exp u\n-2u _{xx}]_{\\Gamma}=0,\\quad\n [-u _t\\phi _x+ 2u _{tx}]_{\\Gamma}=0.\n\\eqno (1.2)\n$$\n\n\n\\bigskip\n%\\section\n{\\bf 2. Perturbed problem ($\\varepsilon\\neq 0$)}\n\nAs regards the perturbation operator $F[u]$ we assume\nthat one has no higher order derivatives. We desire\nthe singularities do not disperse so that singular\nlines only deform slowly under perturbation. Note that\nan existence theorem for the perturbed problem is not\nproved up to now. We only give a formal asymptotic\nsolution.\n\nThe main goals are both an asymptotic approximation of\nthe singular lines and an asymptotic approximation of\nthe solution everywhere out of narrow neighborhood of\nthe singular lines.\n\n\\begin{Def}\nThe formal asymptotic solution (FAS) of order $N$ is a\nfunction $U_N(x,t;\\varepsilon )$ satisfied to both the\nequation (0.1) and the initial condition (0.2) up to\norder $\\O(\\varepsilon^N)$ everywhere in \\{$x\\in R, \\\n0\\leq t\\leq T=\\const$\\} out of narrow strips of order\n$\\O(\\varepsilon^N)$. In the strips there are smooth\nlines on which the matching conditions (1.2) are\nsatisfied.\n\\end{Def}\n\n{\\bf Remark.} A direct asymptotic expansion as given by\n$\nu\\approx\\sum_n\\varepsilon^n\\stackrel{n}{u}(x,t),\n\\quad \\varepsilon\\to 0\n$\ndoes not provide an approach to both the singular\nlines and the solution inside of stripes (near limit\nsingular lines) whose width has the order of\n$\\O(\\varepsilon)$. This assertion can be verified on a\nsimple example when $F\\equiv 0$ and exact solution is\ntaken in the explicit form (1.1) with the smooth\nfunctions $r_\\pm(s^\\pm;\\varepsilon)$ depending on the\nparameter $\\varepsilon$. Asymptotic expansion of such\nsolution as $\\varepsilon\\to 0$ has coefficients with\nthe increasing order singularities $\\O(r_0^{-n})$ at\nthe limit singular lines $r_0^{-n}\\equiv\nr_+(s^+;0)+r_-(s^-;0)=0$.\n\n{\\bf Anzatz.} A FAS (for any N) is taken as a finite\npeace of the asymptotic series\n$$\nu(x,t;\\varepsilon )\n\\approx\\sum_{n=0}^\\infty\\varepsilon^n\n\\stackrel{n}{u}(x,t;\\varepsilon ),\\quad \\varepsilon\\to 0.\n\\eqno (2.1)\n$$\nThe leading order term is here taken as an exact solution\nof the unperturbed equation\n$$\n\\stackrel{0}{u}=\n\\ln {{r_+^{\\prime}(s^+;\\varepsilon )\nr_-^{\\prime}(s^-;\\varepsilon )}\\over\n{r^2(s^+,s^-;\\varepsilon )}},\\quad (r=r_++r_-).\n\\eqno (2.2)\n$$\nWe permit dependence on a small parameter in the\nfunctions $r_\\pm(s_\\pm;\\varepsilon)$. For these\nfunctions an asymptotics is constructed in the form\n$$\nr_\\pm (s^\\pm;\\varepsilon )\n\\approx\\sum_{n=0}^\\infty\\varepsilon^n\n\\stackrel{n}{r}_\\pm (s^\\pm ),\\quad \\varepsilon\\to 0.\n\\eqno (2.3)\n$$\nThe formulas (2.1)--(2.3) are usually named Bogolubov\n-- Krylov anzatz. This approach provides an asymptotic approximation\nof the singular lines up to any order from the\nequation\n$\nr_+ (s^\\pm;\\varepsilon )+r_- (s^\\pm;\\varepsilon )=0.\n$\nThus the matter is reduced to define the coefficients\n$\\stackrel{n}{u},\\stackrel{n}{r_\\pm}$.\n\n\\bigskip\n%\\section\n{\\bf 3. Linearized problem for the correction}\n\nCorrections $\\stackrel{n}{u},\\ (n \\geq 1)$ are determined\nfrom the linear equations\n$$\n\\partial_t^2\\stackrel{n}{u}\n-\\partial_x^2\\stackrel{n}{u}+\n8{{r_+^{\\prime}r_-^{\\prime}}\\over\n{r^2}}\\stackrel{n}{u}=\\stackrel{n}{f}(x,t;\\varepsilon ),\n\\quad (x,t)\\in R^2\n$$\nwith the corresponding initial data. The right sides are\nhere calculated from previous steps, for example\n$\n\\stackrel{1}{f}={\\bf F}[\\stackrel{0}{u}],\\quad\n\\stackrel{2}{f}=\\delta\n{\\bf F}[\\stackrel{0}{u}]\\stackrel{1}{u}-\n4{{r_+^{\\prime}r_-^{\\prime}}}(\\stackrel{1}{u})^2/ {r^2}.\n$\n\nGeneral solution of the homogeneous linear equation is\ngiven by the formula\n$\nu(x,t;\\varepsilon )={{j_+^{\\prime}}/{r_+^{\\prime}}}+\n{{j_-^{\\prime}}/{r_-^{\\prime}}}- 2{{(j_++j_-)}/{r}}.\n$\nHere $j_\\pm=\\stackrel{n}{j}_\\pm(s^\\pm;\\varepsilon )$\nare arbitrary functions. In context of the Cauchy\nproblem they are determined by the initial functions\nand an expression similar to the D'Alembert formula\ntakes place. Take into account the singularities of\norder $\\O(1/r)$.\n\nTo solve the nonhomogeneous linear equation the similar\nformula may be used with the functions $j_\\pm(x,t)$,\ndepending on two variables, which are defined from ODE's\nin the explicit form. So a solution of the linear\nequation is determined by the integral\n$$\nu(x,t)=\\int_{s^-}^{s^+}\\int_{s^-}^{\\sigma^+}\nK({s^+},{s^-},\\sigma^+,\\sigma^-)f(\\sigma^+,\\sigma^-)\n\\,d\\sigma^- \\,d\\sigma^+\n$$\ntaken over the characteristic triangle. The kernel $K$\nis here expressed by\n$$\nK({s^+},{s^-},\\sigma^+,\\sigma^-)=\n{1\\over{2r(s_+,s_-)r(\\sigma_+,\\sigma_-)}}\n\\Big\\{\nr_+(s^+)r_-(s^-) + r_+(\\sigma^+)r_-(\\sigma^-)\n$$\n$$\n +{1\\over 2}\\Big[r_+(s^+)-r_-(s^-)\\Big]\n\\Big[r_+(\\sigma^+)-r_-(\\sigma^-)\\Big]\\Big\\}.\n$$\nLet us denote by $\\stackrel{n}{u}_1(x,t;\\varepsilon)$\na part of the correction corresponding to a solution\nof the nonhomogeneous linear equation. One depends on\nboth the $r_\\pm (s^\\pm ;\\varepsilon )$ and the\n$\\stackrel{m}{j}_\\pm (s^\\pm;\\varepsilon ),\\ (0<m< n)$\nand one is determined by means of the quite well\nspecific operator $\\stackrel{n}{u}_1=\\stackrel{n}{\\bf\nU} [r_\\pm,\\stackrel{1}{j_\\pm}\n,...,\\stackrel{n-1}{j_\\pm}]$. It is convenient to\nidentify the singular part in these representation\n$$\n\\stackrel{n}{\\bf U}=\\stackrel{n}{\\bf V} [r_\\pm,\\stackrel{1}{j_\\pm}\n,...,\\stackrel{n-1}{j_\\pm} ]- {(2/r)}\\stackrel{n}{\\bf\nW} [r_\\pm,\\stackrel{1}{j_\\pm}\n,...,\\stackrel{n-1}{j_\\pm} ].\n\\eqno (3.1)\n$$\n\nSingularities play here a role of secular terms. If we\ntake a FAS in the form of the direct expansion, then\nwe find that the singularities became stronger on each\nstep. One can guess that this effect is due to the\npoor approximation of the perturbed singular lines.\nHence the singularities must to be eliminated from the\ncorrections. Just this elimination gives us a good\napproximation of the perturbed singular\nlines\\footnote{ One can think these ideas are suitable\nfor another problems with singularities under\nperturbations.}. Of course a representation (3.1) with\nsmooth functions $\\stackrel{n}{\\bf V},\\stackrel{n}{\\bf\nW}$ is possible just only under some restrictions on\nthe perturbation operator $F[u]$. In fact, the more\ngeneral condition on the $F[u]$ is a representation\n(3.1) with smooth functions $\\stackrel{n}{\\bf\nV},\\stackrel{n}{\\bf W}$ on each step $n$.\n\n\n\\bigskip\n%\\section\n{\\bf 4. Identification of corrections}\n\nThe system of linear equations are based on the\nleading term whose parameters $r_\\pm $ are determined\nby initial data from equation:\n$$\n\\ln {{r_+^{\\prime}r_-^{\\prime}}\\over{r^2}}\n+\\sum_{n=1}^\\infty\\varepsilon^n\n\\Big[{{\\stackrel{n}{j_+}^\\prime}/{r_+^\\prime}}\n+{{\\stackrel{n}{j_-}^\\prime}/{r_-^{\\prime}}} -\n{2\\over{r}}\n\\Big(\\stackrel{n}{j_+}+\\stackrel{n}{j_-}\\Big)\n+\\stackrel{n}{u_1}\\Big]=\\psi_0(x;\\varepsilon ),\n\\eqno (4.1)\n$$\n$$\n{{r_+^{\\prime\\prime}}\\over{r_+^{\\prime}}}-\n{{r_-^{\\prime\\prime}}\\over{r_-^{\\prime}}}\n-2{{r_+^{\\prime}-r_-^{\\prime}}\\over{r}}+\n\\sum_{n=1}^\\infty\\varepsilon^n\n\\Big[({{\\stackrel{n}{j_+}^\\prime}/{r_+^\\prime}})^\\prime\n-({{\\stackrel{n}{j_-}^\\prime}/{r_-^{\\prime}}})^\\prime-\n{2\\over{r}}\n\\Big(\\stackrel{n}{j_+}^\\prime-\\stackrel{n}{j_-}^\\prime\\Big)+\n$$\n$$\n+{2\\over{r^2}}\\Big(\\stackrel{n}{j_+}+\\stackrel{n}{j_-}\\Big)\n({r_+}^\\prime-{r_-}^\\prime)\n+\\partial_t\\stackrel{n}{u_1}\\Big]=\\psi_1(x;\\varepsilon ).\n\\eqno (4.2)\n$$\n Note that there are unknown functions\n $\\stackrel{n}{j}_\\pm$ apart from $r_\\pm$.\n\nA naive approach to these equations with a small\nparameter is as follows. Functions $r_\\pm$ are identified\nwith the leading term $r_\\pm=\\stackrel{0}{r_\\pm}(s^\\pm )$\nwhich is defined from the nonlinear equations as\n$\\varepsilon\n=0$. After that all functions $\\stackrel{n}{j_\\pm}(s^\\pm )$\nare determined from the recurrent system of linear\nequations. In this way a direct asymptotic expansion\nis just obtained, whose coefficients have the\nsingularities of increasing order $\\O(r_0^{-n})$ at\nthe limit singular lines $r_0\\equiv\nr_+(s^+;0)+r_-(s^-;0)=0$. Deformation of the singular\nlines is not determined in this way.\n\nIn our approach the functions $r_\\pm$ are taken in the\nform of asymptotic series (2.3). Additional ambiguities\nin the coefficients $\\stackrel{n}{r_\\pm}(s^\\pm ),\\ n\\geq\n1$ are used to solve the (4.1),(4.2) under additional\nrequirements. We desire to eliminate the singularities of\norder $\\O(r^{-1})$ from the corrections $\\stackrel{n}{u}$\non each step. Elimination of these terms give rise to the\nalgebraic equations\n$$\n\\stackrel{n}{j_+}(s^+;\\varepsilon )\n+\\stackrel{n}{j_-}(s^-;\\varepsilon )+\\stackrel{n} {\\bf\nW} [r_\\pm,\\stackrel{1}{j_\\pm}\n,...,\\stackrel{n-1}{j_\\pm} ]=0\\quad {\\rm as} \\quad\nr(s^+,s^-;\\varepsilon )=0.\n\\eqno (4.3)\n$$\nMoreover an additional condition is used\n$$\n\\stackrel{n}{r_+}(x)+\\stackrel{n}{r_-}(x)=0, \\quad\n\\forall x\\in R, \\quad (n\\geq 1).\n\\eqno (4.4)\n$$\nSo we obtain the systems of equations (4.1)--(4.4) for\nthe four functions\n$\\stackrel{n}{r_\\pm},\\stackrel{n}{j_\\pm}$ on each step\n$n=1,2,...$. However these equations contain a small\nparameter throw $r_\\pm (x;\\varepsilon)$. Hence an\nasymptotic expansion have to be constructed for the\nfunctions $\\stackrel{n}{j_\\pm}$ like (2.3)\n$\n\\stackrel{n}{j_\\pm }(x;\\varepsilon )\n\\approx\\sum_{m}\\varepsilon^m\n\\stackrel{n,m}{j_\\pm }(x).\n$\nCoefficients can be here obtained from the recurrent\nsystem of algebraic equations\n$$\n\\stackrel{n,m}{j_+}(s^+)+\\stackrel{n,m}{j_-}(s^-)\n=\\stackrel{n,m}{W}, \\quad (n\\geq 1,\\ 0<m<n)\\quad {\\rm as} \\ \\stackrel{0}{r}(s^+,s^-)=0.\n\\eqno (4.5)\n$$\n\n\nSo all functions\n$\\stackrel{n}{r_\\pm},\\stackrel{n,0}{j_\\pm}(x),\\ (n\\geq\n1)$\\ and\\ $\\stackrel{n,m}{j_\\pm}\\ (1\\leq m<n$)\\ are\ndefined step--by--step from the algebraic and\ndifferential equations. The main objects are differential\nequations, which can be represented in the form\n$$\n{{\\stackrel{n}{y_+}^{\\prime}}/{\\stackrel{0}{r_+}^{\\prime}(x)}}+\n{{\\stackrel{n}{y_-}^{\\prime}}/{\\stackrel{0}{r_-}^{\\prime}(x)}}-\n2{{\\stackrel{n}{y_+}+\\stackrel{n}{y_-}}\\over{r(x,x)}}\n=\\stackrel{n}{\\Psi}_0(x),\n$$\n$$\n\\Big({{\\stackrel{n}{y_+}^{\\prime}}/{\\stackrel{0}{r_+}^{\\prime}(x)}}\n-{{\\stackrel{n}{y_-}^{\\prime}}/{\\stackrel{0}{r_-}^{\\prime}(x)}}\\Big)^{\\prime}-\n2{{\\stackrel{n}{y_+}^{\\prime}-\\stackrel{n}{y_-}^{\\prime}}\\over{r(x,x)}}\n+2{{(\\stackrel{n}{y_+}+\\stackrel{n}{y_-})\n(\\stackrel{0}{r_+}^{\\prime}(x)-\\stackrel{0}{r_-}^{\\prime}(x))}\\over{r^2(x,x)}}=\n\\stackrel{n}{\\Psi}_1(x)\n$$\nfor the combinations\n$\n\\stackrel{n}{y}_\\pm (x)=\n\\stackrel{n}{r}_\\pm (x)+\\stackrel{n,0}{j}_\\pm (x).\n$\nIf we take into account\n${r}(x,x)=\\stackrel{0}{r_+}(x)+\\stackrel{0}{r_-}(x)$,\nthen a solution is obtained in the explicit form\n$$\ny_{\\pm}={1\\over 2}\\int(\\Psi_0\\pm\ng)\\stackrel{0}{r_{\\pm}}^{\\prime}\\,dx +{1\\over 2}\\int\n\\stackrel{0}{r_{\\pm}}^{\\prime}\n\\int\\Big({\\Psi_0{{{r}} ^{\\prime}}/{r}}+\n{{(\\stackrel{0}{r}_+\n-\\stackrel{0}{r}_-)^{\\prime}} g/\n{{r}}}\\Big)\\,dx\\,dx,\n$$\n$$\ng(x)={r}(x,x)\\int\\Big({{\\Psi_1(x)}\n/{{r}(x,x)}}\n+{{\\Psi_0(x)\\Big(r_+(x)-r_-(x)\n\\Big)^{\\prime} }/{{r}^2(x,x)} } \\Big)\\, dx.\n$$\nAfter that the algebraic equations (4.4)--(4.5) are\nsolved. There are some arbitrariness in the solution\n$\\stackrel{n}r_\\pm,\\stackrel{n,m}j_\\pm$, which however do\nnot effect the obtained approximation for the\n$u(x,t;\\varepsilon)$.\n\n{\\bf Conclusion.} The main result of the paper is the\ngiven above manner of determination of the FAS\n(2.1)--(2.3) in the occurence of singularities.\n\n\\begin{thebibliography}{99}\n\\bibitem {1}\nG.P.Jeorjadze, A.K. Pogrebkov, M.C. Polivanov, Teoret.\nMatemat. Fisika 40, 2 (1979) 221-234 (in Russian).\n\\bibitem {2}\nJ. Liouville, Journ. math. pure et appl. 18 (1853)\n71-74.\n\\bibitem {3}\nA.K. Pogrebkov, Doklady Akad. Nauk USSR, 244 (1979)\n873-876 (in Russian)\n\n\\end{thebibliography}\n\\end{document}\n" } ]	[ { "name": "solv-int9912013.extracted_bib", "string": "{1 G.P.Jeorjadze, A.K. Pogrebkov, M.C. Polivanov, Teoret. Matemat. Fisika 40, 2 (1979) 221-234 (in Russian)." }, { "name": "solv-int9912013.extracted_bib", "string": "{2 J. Liouville, Journ. math. pure et appl. 18 (1853) 71-74." }, { "name": "solv-int9912013.extracted_bib", "string": "{3 A.K. Pogrebkov, Doklady Akad. Nauk USSR, 244 (1979) 873-876 (in Russian)" } ]
solv-int9912014	Vertex operator solutions to the discrete KP-hierarchy\footnote{ The final version appeared in: Comm. Math. Phys., {203	[ { "author": "M. Adler\\thanks{Department of Mathematics" }, { "author": "Brandeis University" }, { "author": "Waltham" }, { "author": "Mass 02454" }, { "author": "USA. E-mail: adler@math.brandeis.edu. The support of a National Science Foundation grant \\# DMS-9503246 is gratefully acknowledged." } ]		[ { "name": "solv-int9912014.tex", "string": "\\documentstyle[12pt]{article}\n%\\documentstyle[amsfonts,12pt,oldlfont]{article}\n\n\\title{Vertex operator solutions to the discrete\nKP-hierarchy\\footnote{ The final version appeared in:\nComm. Math. Phys., {\\bf 203}, 185--210 (1999)}}\n\n\\author{M. Adler\\thanks{Department of Mathematics,\nBrandeis University, Waltham, Mass 02454, USA. E-mail:\nadler@math.brandeis.edu. The support of a National Science\nFoundation grant \\# DMS-9503246 is gratefully acknowledged.}~~~~~P.\nvan Moerbeke\\thanks{Department of Mathematics, Universit\\'e de\nLouvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University,\nWaltham, Mass 02454, USA. E-mail: vanmoerbeke@geom.ucl.ac.be and\n@math.brandeis.edu. The support of a National Science Foundation\ngrant \\# DMS-9503246, a Nato, a FNRS and a Francqui Foundation\ngrant is gratefully acknowledged.}}\n\n%\\input{ucl}\n\\date{August 24, 1998}\n\n\\newcommand{\\MAT}[1]{\\left(\\begin{array}{#1c}}\n\\newcommand{\\mat}{\\end{array}\\right)}\n\\newcommand{\\qed}\n{%\n\\mbox{}%\n\\nolinebreak%\n\\hfill%\n\\rule{2mm}{2mm}%\n\\medbreak%\n\\par%\n}\n\n\n\n\n\\newcommand{\\sumbis}[2]%\n{%\n\\renewcommand{\\arraystretch}{0.5}\n\\begin{array}[t]{c}\n\\sum\\\\\n{\\scriptstyle #1}\\\\\n{\\scriptstyle #2}\n\\end{array}\n\\renewcommand{\\arraystretch}{1}\n}\n\n\n\\newcommand{\\rg}{\\rightarrow}\n\\newcommand{\\lrg}{\\longrightarrow}\n\\newcommand{\\Rg}{\\Rightarrow}\n\\newcommand{\\DF}{\\Longleftrightarrow}\n\\newcommand{\\pp}{\\ldots}\n\\newcommand{\\DR}{{\\cal D}}\n\\newcommand{\\LR}{{\\cal L}}\n\\newcommand{\\AR}{{\\cal A}}\n\\newcommand{\\BF}{{\\Bbb F}}\n\\newcommand{\\BC}{{\\Bbb C}}\n\\newcommand{\\BD}{{\\Bbb D}}\n\\newcommand{\\BX}{{\\Bbb X}}\n\\newcommand{\\BY}{{\\Bbb Y}}\n\\newcommand{\\BZ}{{\\Bbb Z}}\n\\newcommand{\\Sg}{\\Sigma}\n\\newcommand{\\iy}{\\infty}\n\\newcommand{\\pl}{\\partial}\n\\newcommand{\\al}{\\alpha}\n\\newcommand{\\proof}{\\underline{\\sl Proof}: }\n\\newcommand{\\remark}{\\underline{\\sl Remark}: }\n\\newcommand{\\example}{\\underline{\\sl Example}: }\n\\newcommand{\\om}{\\omega}\n\\newcommand{\\HR}{{\\cal H}}\n\\newcommand{\\VR}{{\\cal V}}\n\\newcommand{\\SR}{{\\cal S}}\n\\newcommand{\\MR}{{\\cal M}}\n\\newcommand{\\CR}{{\\cal C}}\n\\newcommand{\\CB}{{\\cal B}}\n\\newcommand{\\PR}{{\\cal P}}\n\\newcommand{\\WR}{{\\cal W}}\n\\newcommand{\\FR}{{\\cal F}}\n\\newcommand{\\vp}{\\varphi}\n\\newcommand{\\la}{\\langle}\n\\newcommand{\\ra}{\\rangle}\n\\newcommand{\\ga}{\\gamma}\n\\newcommand{\\Ga}{\\Gamma}\n\\newcommand{\\dt}{\\delta}\n\\newcommand{\\Dt}{\\Delta}\n\\newcommand{\\vr}{\\varepsilon}\n\\newcommand{\\sg}{\\sigma}\n\\newcommand{\\BR}{{\\Bbb R}}\n\\newcommand{\\lb}{\\lambda}\n\\newcommand{\\Lb}{\\Lambda}\n\n\\newcommand{\\BJ}{{\\Bbb J}}\n\\def\\span{\\mathop{\\rm span}}\n\\def\\diag{\\mathop{\\rm diag}}\n\\def\\Res{\\mathop{\\rm Res}}\n\n\n\\def\\be{\\begin{equation}}\n\\def\\ee{\\end{equation}}\n\\def\\bea{\\begin{eqnarray}}\n\\def\\eea{\\end{eqnarray}}\n\n\\ifx\\undefined\\Bbb\n \\let\\Bbb\\bf\n\\fi\n\n\n\\catcode`\\@=11\n\\def\\ps@X{\\let\\@mkboth\\@gobbletwo\n \\def\\@oddhead{\\tt Adler-van Moerbeke:%\n Discrete KP\\hfil%\\today\n \\date{August 24, 1998}\\ \\ \\hfil\\S\\thesection,\np.\\thepage\n }\n \\def\\@oddfoot{\\rm\\hfil\\thepage\\hfil}\n \\let\\@evenhead\\@oddhead\n \\let\\@evenfoot\\@oddfoot}\n\\catcode`@=12\n\\pagestyle{X}\n\n\n\n%%% NUMEROTATION DES ENONCES\n\n%%% Le type de numerotation que tu souhaites\n%%% est prevu de maniere standard par LaTeX.\n%%% Voici cela sur un exemple.\n\n% Environnement \"definition\"\n% dont la numerotation est subordonnee a celle des sous-sections\n\n\\newtheorem{definition}{Definition}[%sub\nsection]\n\n% Environnement \"theorem\"\n% dont la numerotation est subordonnee a celle des definitions\n\n\\newtheorem{theorem}[definition]{Theorem}\n\n% Environnement \"lemma\"\n% dont la numerotation est subordonnee a celle des definitions\n\n\\newtheorem{lemma}[definition]{Lemma}\n\\newtheorem{corollary}[definition]{Corollary}\n\\newtheorem{proposition}[definition]{Proposition}\n\n% et de meme pour propositions, corollaires, ...\n\n\n%%% NUMEROTATION DES EQUATIONS\n\n%%% Le type de numerotation qeu tu souhaites\n%%% n'est prevu de maniere standard que par LaTeX 2e.\n%%% Voici une redefinition repondant a tes souhaits\n%%% et valable en general.\n\n% Redefinition du compteur \"equation\"\n% pour que la numerotation soit subordonne a celle des sous-sections\n\n%%% NOUVELLES COMMANDES\n\n% Redefinition du compteur \"equation\"\n% pour que la numerotation soit subordonne a celle des sous-sections\n\n\\catcode`\\@=11\n\\let\\c@equation=\\relax\n\\newcounter{equation}[%sub\nsection]\n\\def\\theequation{\\thesection.\\arabic{equation}}\n\\catcode`\\@=12\n\n\n%%% UN EXEMPLE\n\n\\begin{document}\n\\maketitle\n\n\\tableofcontents\n\n%\\abstract{{\\em Vertex operators}, which are disguised\n%Darboux maps, transform solutions of the KP equation\n%into new ones. In this paper, we show that the\n%bi-infinite sequence obtained by Darboux transforming\n%an arbitrary KP solution recursively forward and\n%backwards, yields a solution to the {\\em discrete\n%KP-hierarchy}. The latter is a KP hierarchy where the\n%continuous space $x$-variable gets replaced by a\n%discrete $n$-variable. The fact that these sequences\n%satisfy the discrete KP hierarchy is tantamount to\n%certain bilinear relations connecting the consecutive\n%KP solutions in the sequence. At the Grassmannian\n%level, these relations are equivalent to a very simple\n%fact, which is the nesting of the associated\n%infinite-dimensional planes (flag).\n%\n%It turns out that many new and old systems lead to\n%such discrete (semi-infinite) solutions, like\n%sequences of soliton solutions, with more and more\n%solitons, sequences of Calogero-Moser systems, having\n%more and more particles, band matrices, etc... ; this\n%will be developped in another paper. In this paper, as\n%an other example, we show that the {\\em $q$-KP\n%hierarchy} maps, via a kind of {\\em Fourier\n%transform}, into the discrete KP hierarchy, enabling\n%us to write down a very large class of solutions to\n%the $q$-KP hierarchy.}\n\n\\vspace{1cm}\n\n {\\em Vertex operators}, which are\ndisguised Darboux maps, transform solutions of the KP\nequation into new ones. In this paper, we show that\nthe bi-infinite sequence obtained by Darboux\ntransforming an arbitrary KP solution recursively\nforward and backwards, yields a solution to the {\\em\ndiscrete KP-hierarchy}. The latter is a KP hierarchy\nwhere the continuous space $x$-variable gets replaced\nby a discrete $n$-variable. The fact that these\nsequences satisfy the discrete KP hierarchy is\ntantamount to certain bilinear relations connecting\nthe consecutive KP solutions in the sequence. At the\nGrassmannian level, these relations are equivalent to\na very simple fact, which is the nesting of the\nassociated infinite-dimensional planes (flag). The\ndiscrete KP hierarchy can thus be viewed as a\ncontainer for an entire ensemble of vertex or Darboux\ngenerated KP solutions.\n\nIt turns out that many new and old systems lead to such discrete\n(semi-infinite) solutions, like sequences of soliton solutions,\nwith more and more solitons, sequences of Calogero-Moser systems,\nhaving more and more particles, just to mention a few examples;\nthis is developped in \\cite{AvM4}. In this paper, as an other\nexample, we show that the {\\em $q$-KP hierarchy} maps, via a kind\nof {\\em Fourier transform}, into the discrete KP hierarchy,\nenabling us to write down a very large class of solutions to the\n$q$-KP hierarchy. This was also reported in a brief note with E.\nHorozov\\cite{AHV}.\n\n\\bigbreak\n\n\n\nGiven the shift operator $\\Lb=(\\dt_{i,j-1})_{i,j\\in\\BZ}$, consider\nthe Lie algebra\n%$\\DR$ of shift operators\n\\be\n\\DR=\\left\\{\\sum_{-\\iy<i\\ll\\iy}a_i\\Lb^i,a_i\\mbox{\\,\\,diagonal\noperators}\\right\\}=\\DR_-+\\DR_+\n\\ee\nwith the usual splitting $\\DR=\\DR_-+\\DR_+$, into subalgebras\n\\be\n\\DR_+=\\left\\{\\sum_{0\\leq i\\ll\\iy}a_i\\Lb^i\\in\\DR\\right\\},\\DR_-=\n\\left\\{\\sum_{-\\iy<i<0}a_i\\Lb^i\\in\\DR\\right\\}.\n\\ee\nThe discrete\nKP-hierarchy equations\n\\be\n\\frac{\\pl L}{\\pl t_n}=[(L^n)_+,L],\\quad n=1,2,...\n\\ee\nare deformations of an infinite matrix\n\\be\nL=\\sum_{-\\iy<i\\leq 0}a_i(t)\\Lb^i+\\Lb \\in \\DR,\\quad\\mbox{with\n$t=(t_1,t_2,...)\\in\\BC^{\\iy}$.}\n\\ee\nIf we represent $L$ as a dressing up of $\\Lambda$ by a wave operator\n$S\\in I+\\DR_-$\n\\be\nL=S \\Lb S^{-1}=W\\Lb\nW^{-1},\\quad W=Se^{\\sum_1^{\\iy}t_i \\Lb^i},\n\\ee\nthen the $L$-deformations are induced by $S$-deformations and\n$W$-deformations:\n\\be\n\\frac{\\pl S}{\\pl t_n}=-(L^n)_-S,\\quad\\frac{\\pl W}{\\pl\nt_n}=(L^n)_+W,\\quad n=1,2,...;\n\\ee\nIn terms of vectors\n\\be\n\\chi(z)=(z^n)_{n\\in\\BZ},\\quad\\quad \\chi^(z)=\\chi(z^{-1}),\n\\ee\nsuch that\n$ z\\chi(z)=\\Lb\\chi(z),\\quad\nz\\chi^(z)=\\Lb^{\\top}\\chi^(z), $ let us define wave and adjoint wave\nvectors $\\Psi(t,z)$ and $\\Psi^(t,z)$\n\\be\n\\Psi(t,z)=W\\chi(z)~\\mbox{and}~\n\\Psi^(t,z)=(W^{-1})^{\\top}\\chi^(z).\n\\ee\nWe find, using (0.5), (0.8), (0.6), that\n\\bea\nL\\Psi(t,z)=z\\Psi(t,z) & & L^{\\top}\\Psi^(t,z)=z\\Psi^(t,z),\\nonumber\\\\\n\\frac{\\pl\\Psi}{\\pl t_n}=(L^n)_+\\Psi & & \\frac{\\pl\\Psi^}{\\pl\nt_n}=-((L^n)_+)^{\\top}\\Psi^.\n\\eea\n\n\\begin{theorem}\nIf $L$ satisfies the Toda lattice, then the wave vectors $\\Psi(t,z)$\nand\n$\\Psi^(t,z)$ can be expressed in terms of one sequence of\n$\\tau$-functions\n$\\tau(n,t):=\n\\tau_n(t_1,t_2,\\dots),\\quad n\\in\\BZ$,\nto wit:\n$$\n\\Psi(t,z)=\\left(e^{\\sum^{\\iy}_1 t_iz^i}\n\\psi(t,z) \\right)_{n\\in\\BZ}=\\left(\n \\frac{\\tau_n(t-[z^{-1}])}{\\tau_n(t)}\ne^{\\sum^{\\iy}_1 t_iz^i} z^n\n\\right)_{n\\in\\BZ},\n$$\n\\be\n\\Psi^(t,z)=\\left(e^{-\\sum^{\\iy}_1 t_iz^{i}}\\psi^(t,z)\n\\right)_{n\\in\\BZ}=\\left(\n\\frac{\\tau_{n+1}(t+[z^{-1}])}{\\tau_{n+1}(t)}\ne^{-\\sum^{\\iy}_1 t_iz^{i}}z^{-n}\n\\right)_{n\\in\\BZ},\n\\label{1.4}\n\\ee\nsatisfying the bilinear identity\n\\be\n\\oint_{z=\\iy}\\Psi_n(t,z)\\Psi^_m(t',z)\\frac{dz}{2 \\pi iz}=0\n\\ee\nfor all $n>m$. It follows that\n$$\\Psi=W\\chi(z)=e^{\\sum^{\\iy}_1 t_iz^i} S\n\\chi(z),\n$$\n$$\n\\Psi^=\\left(W^{\\top}\\right)^{-1} \\chi^{\\ast}(z) = e^{-\\sum^{\\iy}_1\nt_iz^i}(S^{-1})^{\\top}\\chi^{\\ast}(z),\n$$\nwith\\footnote{In an expression, like $S=\\sum a^{(n)} \\Lambda^{n}$,\n$a^{(n)}=\\mbox{diag}(a^{(n)}_k)_{k \\in \\BZ}$ and $(\\tilde\n\\Lambda a)_k=a_{k+1} \\Lb^0 $.}\n\\be\nS=\\sum_0^{\\iy}\\frac{p_n(-\\tilde\\pl)\\tau(t)}{\\tau(t)}\n\\Lb^{-n}\\quad\\mbox{and}\\quad\nS^{-1}=\\sum_0^{\\iy}\\Lambda^{-n}~\\tilde \\Lambda\\left(\\frac{\n p_n(\\tilde\\pl)\\tau(t)} {\\tau(t)}\\right).\n\\ee\nThen $L^k$ has the following expression in terms of\n$\\tau$-functions\\footnote{where the $p_{\\ell}$ are elementary Schur\npolynomials and where $p_{\\ell}(\\tilde \\pl)f \\circ g$ refers to the\nusual Hirota operation, to be defined in section 1.},\n\\be\nL^k=\\sum_{\\ell=0}^{\\iy}\\mbox{diag}~\\left(\\frac{p_{\\ell}(\\tilde\\pl)\n\\tau_{n+k-\\ell+1}\\circ\\tau_n}\n{\\tau_{n+k-\\ell+1} \\tau_n}\\right)_{n \\in \\BZ}\\Lb^{k-\\ell}\n\\ee\nwith the $\\tau_n$'s satisfying\n\\be\n\\left(\\frac{\\pl}{\\pl\nt_k}-\\sum^{\\ell-1}_{r=0}(\\ell-r)p_r(-\\tilde\\pl)p_{k-r}(\\tilde\\pl)\\right)\n\\tau_{n}\\circ\\tau_{n-\\ell}=0,~~\\mbox{for}~\\ell,k=1,2,3,...\n\\ee\nand\n$$\n\\left(\\frac{1}{2}\\frac{\\pl^2}{\\pl\nt_1\\pl\nt_k}-p_{k+1}(\\tilde\\pl)\\right)\\tau_n\\circ\\tau_n=0,\n~~\\mbox{for}~k=1,2,3,...\n$$\n\\end{theorem}\n\n\\noindent\\remark Equation (0.14) reads\n\\bea\nL^k&=&\\Lb^k+\\left(\\frac{\\pl}{\\pl\nt_1}\\log\\frac{\\tau_{n+k}}{\\tau_n}\\right)_{n\\in\\BZ}\n\\Lb^{k-1}+...\\nonumber\\\\\n& &+\\,\\left(\\frac{\\pl}{\\pl\nt_k}\\log\\frac{\\tau_{n+1}}{\\tau_n}\\right)_{n\\in\\BZ}\\Lb^0+\n\\left(\n\\frac{\\pl^2}{\\pl t_1\\pl t_k}\\log\\tau_n\n\\right)_{n\\in\\BZ}\\Lb^{-1}+...\\,,\\nonumber\\\\\n& &\n\\eea\n\n\n\\bigbreak\n\nWith each component of the wave vector $\\Psi$, or, what is the same,\nwith each component of the $\\tau$-vector, we associate a sequence of\ninfinite-dimensional planes in the Grassmannian\n$Gr^{(n)}$\n\\bea\n\\WR_n&=&\\mbox{ span}_{\\BC}\\left\\{\\left(\\frac{\\pl}{\\pl t_1}\n\\right)^k\\Psi_n(t,z),~~k=0,1,2,...\\right\\}\\nonumber \\\\\n&=&e^{\\sum_1^{\\iy}t_i z^i}\\mbox{ span}_{\\BC}\\left\\{\\left(\n\\frac{\\pl}{\\pl\nt_1}+z\\right)^k\\psi_n(t,z),~~k=0,1,2,...\\right\\}\\nonumber\\\\\n&=:& e^{\\sum_1^{\\iy}t_i z^i} \\WR_n^t.\n\\eea\n\nNote that the plane $z^{-n}\\WR_n \\in Gr^{(0)}$ has so-called virtual\ngenus zero, in the terminology of \\cite{SW}; in particular,\nthis plane contains an element of order $1+O(z^{-1})$. Setting\n$\\{f,g\\}=f'g-fg'$ for $'=\\pl / \\pl t_1$, we have the following\nstatement:\n\n\n\\bigbreak\n\n\n\n\\begin{theorem} The following six statements are\nequivalent\n\\newline\\noindent(i) The discrete KP-equations (0.3)\n\\newline\\noindent(ii) $\\Psi$ and $\\Psi^$, with the proper\nasymptotic behaviour, given by (0.8),\nsatisfy the bilinear identities for all\n$t,t' \\in \\BC^{\\iy }$\n\\be\n\\oint_{z=\\iy}\\Psi_n(t,z) \\Psi^_m(t',z) \\frac{dz}{2 \\pi iz}=0,\n~~~\\mbox{ for all }~~n>m;\n\\label{7}\\ee\n\\newline\\noindent(iii)\nthe $\\tau$-vector satisfies the following bilinear identities\nfor all $n>m$ and $t,t' \\in \\BC^{\\iy }$:\n\\begin{equation}\n\\oint_{z=\\iy}\\tau_n(t-[z^{-1}])\\tau_{m+1}(t'+[z^{-1}])\ne^{\\sum_1^{\\iy}(t_i-t'_i)z^i}\nz^{n-m-1}dz=0;\n\\label{8}\n\\end{equation}\n\\newline\\noindent(iv) The components $\\tau_n$ of a $\\tau$-vector\ncorrespond to a flag of planes in $Gr$,\n\\begin{equation}\n... \\supset \\WR_{n-1}\\supset\n\\WR_{n}\\supset \\WR_{n+1}\\supset....\n\\end{equation}\n\\newline\\noindent(v) A sequence of KP-$\\tau$-functions $\\tau_n$\nsatisfying the equations\n\\be\\{ \\tau_n\n(t-[z^{-1}]),\\tau_{n+1} (t)\\} + z (\\tau_n (t-[z^{-1}]) \\tau_{n+1}\n(t) - \\tau_{n+1} (t-[z^{-1}])\\tau_n (t)) = 0 \\ee\n\\newline\\noindent(vi) A sequence of KP-$\\tau$-functions $\\tau_n$\nsatisfying equations (0.14) for $\\ell=1$, i.e.,\n\\be\n\\left( \\frac{\\pl}{\\pl t_k} - p_k(\\tilde\\pl) \\right)\\tau_{n+1}\\circ\n\\tau_n = 0 ~~\\mbox{for}~k=2,3,...~\\mbox{and}~n\\in \\BZ.\n\\ee\n\\end{theorem}\n\n\\vspace{0.5cm}\n\n\\noindent\\remark The 2-Toda lattice, studied in \\cite{UT}, amounts to\ntwo coupled 1-Toda lattices or discrete KP-hierarchies, thus\nintroducing two sets of times\n$t_n$'s and $s_n$'s. Actually, every 1-Toda lattice can naturally be\nextended to a 2-Toda lattice; this is the content of Theorem 3.4.\n\n\\vspace{0.5cm}\n\n\\noindent{\\bf How to construct discrete KP-solutions}. A wide class of\nexamples of discrete KP-solutions is given in section 4 by the following\nconstruction, involving the simple vertex operators,\n\\be\nX(t,z):=e^{\\sum_1^{\\iy}t_iz^i}e^{-\\sum_1^{\\iy}\n\\frac{z^{-i}}{i}\\frac{\\pl}{\\pl t_i}},\n\\ee\nwhich are disguised Darboux transformations acting on KP\n$\\tau$-functions. We now state:\n\n\n\n\\begin{theorem} Consider an arbitrary $\\tau$-function for the KP\nequation and a family of weights $...,\\nu_{-1}(z)dz,\\nu_0(z)dz,\n\\nu_1(z)dz,...$ on $\\BR$. The infinite sequence of $\\tau$-functions:\n$\\tau_0=\\tau$ and, for $ n> 0$,\n$$\n\\tau_{ n}: = \\left(\\int X (t,\\lambda) \\nu_{n-1}(\\lb)d\\lb...\\int X\n(t,\\lambda)\\nu_0(\\lambda)d\\lb\\right)\n\\tau (t),\n$$\n$$\n\\tau_{-n} := \\left(\\int X (-t,\\lambda) \\nu_{-n}(\\lb)d\\lb...\\int X\n(-t,\\lambda)\\nu_{-1}(\\lambda)d\\lb\\right)\n\\tau (t),\n$$\nform a discrete KP-$\\tau$-vector, i.e., the bi-infinite matrix\n\\be\nL=\\sum_{\\ell=0}^{\\iy}\\mbox{diag}~\\left(\\frac{p_{\\ell}(\\tilde\\pl)\n\\tau_{n+2-\\ell}\\circ\\tau_n}\n{\\tau_{n+2-\\ell} \\tau_n} \\right)_{n \\in \\BZ}\\Lb^{1-\\ell}\n\\ee\nsatisfies the discrete KP hierarchy (0.3).\n\\end{theorem}\n\n\\noindent As an interesting special case of this situation, we study\nin section 6 the {\\em $q$-KP equation}.\n\n\n\\bigbreak\n\\noindent A wide variety of examples are captured by this\nconstruction, like $q$-approximations to KP, discussed in section\n5, but also soliton formulas, matrix integrals, certain integrals\nleading to band matrices, the Calogero-Moser system and others,\ndiscussed in \\cite{AvM4}.\n\n\n\\noindent \\underline{\\em Remark}: A semi-infinite discrete\nKP-hierarchy with $\\tau_0(t)=1$ is equivalent to a bi-infinite discrete\nKP-hierarchy with\n$\\tau_{-n}(t)=\\tau_n(-t)$ and $\\tau_0(t)=1$; this also amounts to\n$\\WR_{-n}=\\WR^{\\ast}_n$, with $\\WR_{0}=\\HR_+$. In such cases, one only\nkeeps the lower right hand corner of $L$, while the lower left hand\ncorner completely vanishes.\n\n\n\n\n\\section{The KP $\\tau$-functions, Grassmannians and a\nresidue formula}\n\nAs is well known \\cite{DJKM}, the bilinear identity\n\\be\n\\oint_{z=\\iy}\\Psi(t,z)\\Psi^(t,z)dz=0,\n\\ee\ntogether with the asymptotics\n\\be\n\\Psi(t,z)=e^{\\sum_1^{\\iy}t_iz^i}\\left(1+O\\left(\\frac{1}{z}\\right)\n\\right),\\Psi^(t,z)=e^{-\\sum_1^{\\iy}t_iz^i}\\left(1+O\\left(\\frac{1}{z}\\right)\n\\right)\n\\ee\nforce $\\Psi,\\Psi^$ to be expressible in terms of $\\tau$-functions\n$$\n\\Psi(t,z)=e^{\\sum_1^{\\iy}t_iz^i}\\frac{\\tau(t-[z^{-1}])}{\\tau(t)},\n\\Psi^(t,z)=e^{-\\sum_1^{\\iy}t_iz^i}\\frac{\\tau(t+[z^{-1}])}{\\tau(t)};\n$$\nmoreover the KP $\\tau$-functions satisfy the differential Fay\nidentity\\footnote{$\\{ f,g\\}:=\\frac{\\pl f}{\\pl t_1} g-f \\frac{\\pl\ng}{\\pl t_1}$.}, for all $y,z\\in\\BC$, as shown in \\cite{AvM1,vM}:\n\\bea\n& &\\{\\tau(t-[y^{-1}]),\\tau(t-[z^{-1}])\\}\\\\\n&\n&\\hspace{1cm}+\\,(y-z)(\\tau(t-[y^{-1}])\\tau(t-[z^{-1}])-\n\\tau(t)\\tau(t-[y^{-1}]\n-[z^{-1}])=0.\\nonumber\n\\eea\nIn fact this identity characterizes the $\\tau$-function, as shown in\n\\cite{TT}.\n\nFrom (1.1), it follows that\n\\bea\n0&=&\\oint\\tau(t-a-[z^{-1}])\\tau(t+a+[z^{-1}])e^{-2\\sum_1^{\\iy}\na_iz^i}\\frac{dz}{2\\pi i}\\nonumber\\\\\n& &=\\sum^{\\iy}_{k=1}a_k\\left(\\frac{\\pl^2}{\\pl t_1\\pl\nt_k}-2p_{k+1}(\\tilde\\pl)\\right)\\tau\\circ\\tau+O(a^2).\n\\eea\nThe Hirota notation used here is the following: Given a polynomial\n$p\\left(\\frac{\\pl}{\\pl t_1},\\frac{\\pl}{\\pl t_2},...\\right)$ in\n$\\frac{\\pl}{\\pl t_i}$, define the symbol\n\\be\np\\left(\\frac{\\pl}{\\pl t_1},\\frac{\\pl}{\\pl\nt_2},...\\right)f\\circ g(t):=p\\left(\\frac{\\pl}{\\pl\nu_1},\\frac{\\pl}{\\pl u_2},...\\right)f(t+u)g(t-u)\\Biggl\|_{u=0},\n\\ee\nand\n$$\n\\tilde\\pl_t:=\\left(\\frac{\\pl}{\\pl t_1},\\frac{1}{2}\\frac{\\pl}{\\pl\nt_2},\\frac{1}{3}\\frac{\\pl}{\\pl t_3},...\\right).\n$$\n\nFor future use, we state the following proposition shown in\n\\cite{AvM1}:\n\n\\begin{proposition} Consider\n$\\tau$-functions\n$\\tau_1$ and $\\tau_2$, the corresponding wave functions\n\\be\n\\Psi_j = e^{\\sum_{i\\geq 1} t_i z^i} {\\tau_j (t-[z^{-1}])\\over\n\\tau_j (t)}=e^{\\sum_{i\\geq 1} t_i z^i}\\left(1+O(z^{-1})\\right)\n\\ee\nand the associated infinite-dimensional planes, as points in the\nGrassmannian $Gr$,\n$$\\tilde \\WR_i=\\span\\left\\{\\left(\n\\frac{\\pl}{\\pl t_1}\\right)^k \\Psi_i(t,z), \\mbox{ for }\nk=0,1,2,...\\right\\}~~\\mbox{with}~~\\tilde\n\\WR^t_i=\\tilde \\WR_i e^{-\\sum_1^{\\iy}t_kz^k};\n$$\nthen the following statements are equivalent\n\\newline\\noindent(i) $ z \\tilde \\WR_2 \\subset \\tilde \\WR_1 $;\n\\vspace{0.2cm}\\newline\\noindent(ii) $ z \\Psi_2 (t,z) =\n\\frac{\\pl}{\\pl t_1}\n\\Psi_1 (t,z) - \\alpha \\Psi_1 (t,z)$, for some function\n$\\alpha = \\alpha (t)$;\n\\vspace{0.2cm}\\newline\\noindent(iii)\\be\\{ \\tau_1\n(t-[z^{-1}]),\\tau_2 (t)\\} + z (\\tau_1 (t-[z^{-1}]) \\tau_2 (t)\n- \\tau_2 (t-[z^{-1}])\\tau_1 (t)) = 0 .\\ee\n\\newline\\noindent When (i), (ii) or (iii) holds, $\\alpha(t)$ is\ngiven by\n\\be\n\\alpha (t) = {\\pl \\over \\pl t_1} \\log {\\tau_2 \\over \\tau_1}.\n\\ee\n\\end{proposition}\n\n\\proof To prove that (i) $\\Rightarrow$ (ii), the inclusion\n$z\\tilde \\WR_2 \\subset \\tilde \\WR_1$, hence\n$z\\tilde \\WR^t_2 \\subset \\tilde \\WR^t_1$, implies by (0.16) that\n$$\nz \\psi_2 (t,z) = z (1+O(z^{-1})) \\in \\tilde \\WR_1^t\n$$\nmust be a linear combination\\footnote{$\\psi_i$ is the same as\n$\\Psi_i$, but without the exponential.}\n\\be\nz \\psi_2 = {\\pl \\psi_1\\over\n\\pl x} + z \\psi_1 -\\alpha (t) \\psi_1,\\mbox{ and thus }\nz\n\\Psi_2 = {\\pl \\over \\pl t_1} \\Psi_1 - \\alpha (t) \\Psi_1.\n\\ee\nThe expression (1.8) for $\\alpha (t)$ follows from equating the\n$z^0$-coefficient in (ii), upon using the $\\tau$-function\nrepresentation (1.6). To show that (ii) $\\Rightarrow$ (i), note\nthat\n$$\nz \\Psi_2 = {\\pl \\over \\pl t_1} \\Psi_1 - \\alpha \\Psi_1 \\in \\tilde\\WR_1\n$$\nand taking $t_1$-derivatives, we have\n$$\nz \\left( \\frac{\\pl}{\\pl t_1}\\right)^j \\Psi_2 = \\left(\n\\frac{\\pl}{\\pl t_1}\\right)^{j+1} \\Psi_1 + \\beta_1 \\left(\n\\frac{\\pl}{\\pl t_1}\\right)^j \\Psi_1 + \\cdots + \\beta_{j+1}\n\\Psi_1,\n$$\nfor some $\\beta_1,\\cdots,\\beta_{j+1}$ depending on $t$ only;\nthis implies the inclusion (i). The equivalence (ii)\n$\\Longleftrightarrow$ (iii) follows from a straightforward\ncomputation using the $\\tau$-function representation (1.6) of (ii)\nand the expression for $\\alpha (t)$.\\qed\n\n\\begin{lemma} The following integral along a clockwise circle in the\ncomplex plane encompassing $z=\\iy$ and $z=\\al^{-1}$, can be evaluated as\nfollows\n\\begin{eqnarray}\n& &\\oint_{z=\\iy}\nf(t+[\\al]-[z^{-1}])g(t-[\\al]+[z^{-1}])\\frac{z^{m+1}}{(z-\n\\al^{-1})^2}\\frac{dz}{2\\pi iz}\\\\\n&=&\\al^{1-m}\\sum^{\\iy}_{k=1}\\al^k\\left(-\\frac{\\pl}{\\pl\nt_k}+\\sum_{r=0}^{m-1}(m-r)p_r(-\\tilde\\pl)p_{k-r}(+\\tilde\\pl)\\right)\nf\\circ\ng.\n\\end{eqnarray}\n\\end{lemma}\n\n\\proof By the residue theorem, the integral above is the sum of\nresidue at $z=\\iy$ and at $z=\\al^{-1}$:\n\\bea\n&\n&\\oint_{z=\\iy}f(t+[\\al]-[z^{-1}])g(t-[\\al]+[z^{-1}])\\frac{z^{m+1}}{\n(z-\\al^{-1})^2}\n\\frac{dz}{2\\pi iz}\\nonumber\\\\\n&=& \\frac{1}{(m-1)!}\\left(\n\\frac{d}{du}\\right)^{m-1}f(t+[\\al]-[u])g(t-[\\al]+[u])\n\\frac{1}{(1-u\\al^{-1})^2}\\Biggl\|_{u=0}\\nonumber\\\\\n& & \\\\\n&\n&-\\frac{d}{dz}z^mf(t+[\\al]-[z^{-1}])g(t-[\\al]+[z^{-1}])\\Biggl\|_{z=\\al\n^{-1}}.\n\\eea\nEvaluating each of the pieces requires a few steps.\n\n\\medbreak\n\n{\\bf Step 1.}\n$$\n\\frac{1}{k!}\\left(\\frac{d}{du}\\right)^kf(t+[\\al]-[u])g(t-[\\al]+\n[u])\\Biggl\|_{u=0}=\\sum^{\\iy}_{\\ell=0}\\al^{\\ell}p_k(-\\tilde\\pl)p_{\\ell}\n(\\tilde\\pl)f\\circ g.\n$$\nAt first note\n\\be\n\\left(\\frac{d}{du}\\right)^kF([u])\\Biggl\|_{u=0}=k!p_k(\\tilde\\pl_s)F(s)\n\\ee\nand, by (1.5) and (1.12),\n\\bea\n\\frac{1}{k!}\\left(\\frac{d}{du}\\right)^kf(t+[u])g(t-[u])\n\\Biggl\|_{u=0}&=&p_k(\\tilde\\pl)f\\circ g \\nonumber\\\\\n&=&p_k(-\\tilde\\pl)g\\circ f \\nonumber\\\\\n&=& \\sum_{i+j=k} p_i(-\\tilde \\pl)g . p_j(\\tilde \\pl)f.\n\\eea\nIndeed\n\\medbreak\n\n$\\displaystyle{\\frac{1}{k!}\\left(\\frac{d}{du}\\right)^kf(t+[\\al]-[u])\ng(t-[\\al]+[u])\\Biggl\|_{u=0}}$\n\\begin{eqnarray}\n&=&p_k(\\tilde\\pl_s)g(t-[\\al]+s)f(t+[\\al]-s)\\Biggl\|_{s=0},\\quad\n\\mbox{using (1.12)}\\\\\n&=&p_k(\\tilde\\pl_s)\\sum^{\n\\iy}_{\\ell=0}\\al^{\\ell}p_{\\ell}(\\tilde\\pl_t)f(t-s)\\circ\ng(t+s)\\Biggl\|_{s=0},\\quad\\mbox{using (1.13)}\\\\\n&=&\\sum^{\\iy}_{\\ell=0}\\al^{\\ell}p_k(\\tilde\\pl_s)p_{\\ell}(\\tilde\\pl_w)\nf(t+w-s)g(t-w+s)\\Biggl\|_{s=w=0},\\quad\\mbox{expressing Hirota,}\\\\\n&=&\\sum^{\\iy}_{\\ell=0}\\al^{\\ell}p_k(\\tilde\\pl_s)p_{\\ell}(-\\tilde\\pl_w)\nf(t-w-s)g(t+w+s)\\Biggl\|_{s=w=0},\\quad\\mbox{flipping signs,}\\\\\n&=&\\sum^{\\iy}_{\\ell=0}\\al^{\\ell}p_k(\\tilde\\pl_v)p_{\\ell}(-\\tilde\\pl_v)\nf(t-v)g(t+v)\\Biggl\|_{v=0}\\\\\n&=&\\sum^{\\iy}_{\\ell=0}\\al^{\\ell}p_k(-\\tilde\\pl)p_{\\ell}(\\tilde\\pl)\nf\\circ g,~~~~~~\\mbox{using} (1.13).\n\\end{eqnarray}\n\n\\medbreak\n\n{\\bf Step 2.} Residue at $\\iy$.\n\nNote\n\\be\n\\left(\\frac{d}{du}\\right)^{\\ell}\\left(\\frac{1}{1-u\\al^{-1}}\\right)^2\n\\Biggl\|_{u=0}=\\left(\\frac{d}{du}\\right)^{\\ell}\n\\sum^{\\iy}_{i=1}i(u\\al^{-1})^{i-1}\\Biggl\|_{u=0}=(\\ell+1)!\\al^{-\\ell};\n\\ee\nthen we find\n\\medbreak\n\n$\\displaystyle{\\frac{1}{(m-1)!}\\left(\\frac{d}{du}\\right)^{m-1}f(t+\n[\\al]-[u])\ng(t-[\\al]+[u])\\frac{1}{(1-u\\al^{-1})^2}\\Biggl\|_{u=0}}$\n\\bea\n&=&\\frac{1}{(m-1)!}\\sum^{m-1}_{r=0}\\MAT{1}m-1\\\\r\\mat\n\\left(\\frac{d}{du}\\right)^rf(t+[\\al]-[u])g(t-[\\al]+[u])\n\\nonumber\\\\\n&\n&\\hspace{6cm}\\left(\\frac{d}{du}\\right)^{m-1-r}\\frac{1}{(1-u\\al^{-1})^2}\\Biggl\|_{u=0}\\nonumber\\\\\n&=&\\sum^{m-1}_{r=0}(m-r)\\sum_{\\ell=0}^{\\iy}\n\\al^{\\ell-m+r+1}p_r(-\\tilde\\pl)p_{\\ell}(\\tilde\\pl)f\\circ\ng,\\quad\\mbox{using step 1 and (1.14)}\\nonumber\\\\\n&=&m\\al^{1-m}f(t)g(t)+\\al^{1-m}\n\\sum^{\\iy}_{k=1}\\al^k\\sum^m_{r=0}(m-r)p_r(-\\tilde\\pl)\np_{k-r}(\\tilde\\pl)f\\circ g,\\quad\\mbox{using $p_0=1$}.\\nonumber\\\\\n& &\n\\eea\n\n\\medbreak\n\n{\\bf Step 3.} Residue at $z=\\al^{-1}$.\n\n\\medbreak\n\n$\\displaystyle{\\frac{d}{dz}z^mf(t+\n[\\al]-[z^{-1}])\ng(t-[\\al]+[z^{-1}])\\Biggl\|_{z=\\al^{-1}}}$\n\\bea\n&=&-u^2\\frac{d}{du}u^{-m}f(t+[\\al]-[u])g(t-[\\al]+[u])\\Biggl\|_{u=\\al}\\nonumber\\\\\n&=&m\\al^{-m+1}f(t)g(t)-\\al^{2-m}\\frac{d}{du}f(t+[\\al]-[u])\ng(t-[\\al]+[u])\\Biggl\|_{u=\\al}\\nonumber\\\\\n&=&m\\al^{1-m}f(t)g(t)+\n\\sum^{\\iy}_{k=1}\\al^{1-m+k}\\frac{\\pl}{\\pl\nt_k}f\\circ g,\\quad\\mbox{by explicit differentiation.}\\nonumber\\\\\n& &\n\\eea\nFinally, putting step 2 and step 3 in (1.11) yields Lemma 1.2.\\qed\n\n\\begin{lemma} The Hirota symbol acts as follows on functions\n$f(t_1,t_2,...)$ and $g(t_1,t_2,...)$:\n\\begin{equation}\n\\frac{1}{fg}\\frac{\\pl^n}{\\pl t_1...\\pl t_n}f\\circ g ~~=\\mbox{ a\npolynomial $P_n$ in}~\\left\\{\n\\begin{array}{l}\n\\frac{\\pl^k}{\\pl t_{i_1}...\\pl t_{i_k}}\\log \\frac{f}{g}~~~\\mbox{for $k$\nodd}\\\\\n\\\\\n\\frac{\\pl^k}{\\pl t_{i_1}...\\pl t_{i_k}}\\log fg~~~\\mbox{for $k$\neven}\n \\end{array}\n\\right.\n\\end{equation}\nover all subsets $\\{i_1,...,i_k\\} \\subset \\{1,...,n\\}$. Upon\ngranting degree $1$ to each partial in $t_i$, the polynomial $P_n$ is\nhomogeneous of degree $n$.\n\\end{lemma}\n\n\\proof By induction, we assume the statement to be valid for an Hirota\nsymbol, involving $\\ell$ partials, and we prove the statement for a\nsymbol involving $\\ell +1$ partials:\n$$\n\\frac{1}{fg}\\frac{\\pl}{\\pl t_{\\ell +1}}\\frac{\\pl^{\\ell}}{\\pl t_1...\\pl\nt_{\\ell}}f(t)\\circ g(t)\\hspace{7cm}\n$$\n\\bea\n&=&\\frac{1}{fg}\\frac{\\pl}{\\pl\nu_{\\ell+1}}f(t+u)g(t-u)\\frac{\\frac{\\pl^{\\ell}}{\\pl t_1...\\pl\nt_{\\ell}}f(t+u)\\circ g(t-u)}{f(t+u)g(t-u)}\\Big\|_{u=0}\\nonumber\\\\\n&=&\\left( \\frac{\\pl}{\\pl t_{\\ell+1}} \\log \\frac{f}{g}\n\\right)\\frac{1}{fg}\\frac{\\pl^{\\ell}}{\\pl t_1...\\pl\nt_{\\ell}}f(t+u)\\circ g(t-u) \\nonumber\\\\\n& & +\\frac{\\pl}{\\pl u_{\\ell+1}}P\\left(\n...,\\frac{\\pl^m}{\\pl t_{i_1}...t_{i_m}}\\log \\frac{f(t+u)}{g(t-u)},...,\n\\frac{\\pl^n}{\\pl t_{j_1}...\\pl t_{j_n}}\\log f(t+u)g(t-u),...\n\\right)\\Big\|_{u=0}, \\nonumber\\\\\n\\eea\nwhere $m$ is odd and $n$ even. The result follows from the simple\ncomputation:\n\\bea\n\\frac{\\pl}{\\pl u_{\\ell+1}} \\frac{\\pl^m}{\\pl t_{i_1}...\\pl t_{i_m}}\\log\n\\frac{f(t+u)}{g(t-u)}\\Big\|_{u=0}&=&\\frac{\\pl^{m+1}}{\\pl\nt_{i_1}...\\pl t_{i_m}.\\pl t_{\\ell+1}}\\log f(t)g(t)\\nonumber\\\\\n\\frac{\\pl}{\\pl u_{\\ell+1}} \\frac{\\pl^n}{\\pl t_{i_1}...\\pl t_{i_n}}\\log\nf(t+u)g(t-u)\\Big\|_{u=0}&=&\\frac{\\pl^{n+1}}{\\pl\nt_{i_1}...\\pl t_{i_n}.\\pl t_{\\ell+1}}\\log \\frac{f(t)}{g(t)}\\nonumber\\\\\n\\eea\n\\qed\n\n\\remark The induction formula (1.18) can be made into an explicit\nformula for $P_n$, involving partitions of the set $\\{1,2,...,n\\}$.\n\n\n\n\\section{The existence of a $\\tau$-vector and the discrete KP\nbilinear identity}\n\n\nBefore proving Theorem 0.1, we shall need two lemmas, which\nare analogues of basic lemmas in the theory of differential\noperators. So the main purpose of this section is\nthreefold, namely, to prove the bilinear identities for the\nwave and adjoint wave vectors, to prove the existence of a\n$\\tau$-vector and finally to give a closed form for\n$L^k$.\n\n\\begin{lemma} For $z$-independent $U,~V \\in \\DR$,\nthe following matrix identities hold \\footnote{\n $(A\\otimes B)_{ij}=A_iB_j$ and remember\n$\\chi^(z)=\\chi(z^{-1})$. The contour in the integration below runs\nclockwise about $\\iy$; i.e., opposite to the usual orientation.}\n\\be\nUV\n=\\oint_{z=\\iy}U \\chi(z)\\otimes\nV^\\top\\chi^(z)\\frac{dz}{2\\pi iz},\n\\ee\n\\end{lemma}\n\n\\proof Set\n$$\nU=\\sum_\\al u_{\\al}\\Lb^\\al\\quad\\hbox{and}\\quad\nV=\\sum_\\beta\\Lb^\\beta v_{\\beta},\n$$\nwhere $u_{\\al}$ and $v_{\\al}$ are diagonal matrices.\nTo prove (2.1), it suffices to compare the\n$(i,j)$-entries on each side. On the left side of (2.1), we have\n\\begin{eqnarray}\n\\left(UV\\right)_{ij}\n&=&\\Bigl(\\sum_{\\al,\\beta}u_{\\al}\\Lb^{\\al+\\beta}\nv_{\\beta}\\Bigr)_{ij}\n\\\\\n&=&\\sum_{\\al,\\beta}u_{\\al}(i)\n(\\Lb^{\\al+\\beta})_{ij}v_{\\beta}(j)\n\\\\\n&=&\\sum_{{\\al,\\beta}\\atop{\\al+\\beta=j-i}}u_{\\al}(i)\nv_{\\beta}(j).\n\\end{eqnarray}\nOn the right side of (2.1), we have\n\\begin{eqnarray}\n\\lefteqn{\n \\oint_{z=\\iy}\\Bigl(U\\chi(z)\\Bigr)_i\n \\Bigl(V^\\top\\chi(z^{-1})\\Bigr)_{j}\n ~\\frac{dz}{2\\pi iz}\n}\\\\\n&=&\\oint_{z=\\iy}\\Bigl(\\sum_{\\al}u_{\\al}z^{\\al}\\chi(z)\\Bigr)_i\n\\Bigl(\\sum_{\\beta}v_{\\beta}z^{\\beta}\\chi(z^{-1})\\Bigr)_{j}\n\\frac{dz}{2\\pi iz}\\\\\n&=&\\oint_{z=\\iy}\n\\sum_{\\al,\\beta}\nu_{\\al}(i)v_{\\beta}(j)z^{\\al+\\beta+i-j}\\frac{dz}{2\\pi iz}\\\\\n&=&\n\\sum_{{\\al,\\beta}\\atop{\\al+\\beta=j-i}}\nu_{\\al}(i)v_{\\beta}(j),\n\\end{eqnarray}\nestablishing (2.1). \\qed\n\n\\begin{lemma} For $W(t)$ a wave operator of the discrete\nKP-hierarchy,\n\\be\nW(t)W^{-1}(t')\\in\\DR_+,\\quad\\forall t,t'.\n\\ee\n\\end{lemma}\n\n\\proof Setting $h(t,t')=W(t)W^{-1}(t')$, compute from (0.6)\n\\be\n\\frac{\\pl h}{\\pl t_n}=(L^n(t))_+h,\\quad\\frac{\\pl h}{\\pl\nt'_n}=-h(L^n(t'))_+,\n\\ee\nsince $h(t,t)=I\\in\\DR_+$, it follows that $h(t,t')$ evolves in\n$\\DR_+$.\\qed\n\n\\medbreak\n\nConsider the wave function, already defined in the introduction, and\nthe adjoint wave function:\n\\bea\n\\Psi(t,z)&=&W\\chi(z)=e^{\\sum_1^{\\iy}t_iz^i}S\\chi(z)=\ne^{\\sum\nt_iz^i}\\left(z^n+\\sum_{i<n}s_i(n)z^i \\right)_{n\\in\\BZ}\n\\nonumber\\\\\n\\Psi^(t,z)&=&(W^{-1})^{\\top}\\chi^(z)=e^{-\\sum_1^{\\iy}t_iz^i}\n(S^{-1})^{\\top} \\chi^(z)\\nonumber\\\\\n& &\\quad\\quad\\quad =e^{-\\sum\nt_iz^i}\\left(z^{-n}+\\sum_{i<-n}s_i^{\\ast}(n)z^i\n\\right)_{n\\in\\BZ}.\n\\eea\n\n\n{\\underline{\\sl Proof of Theorem 0.1}: }\n\n{\\bf Step 1:} Setting\n$$\nU:=W(t)\\quad\\mbox{and}\\quad V^{\\top}:=(W^{-1}(t'))^{\\top}\n$$\nin formula (2.1) of Lemma 2.1, and using formula (0.8) of $\\Psi$ and\n$\\Psi^$ in terms of $W$, one finds for all $t,t'\\in\\BC^{\\iy}$,\n\\be\nW(t)W(t')^{-1}=\\oint_{z=\\iy}\\Psi(t,z)\\otimes\\Psi^(t',z)\\frac{dz}{2\\pi\niz}.\n\\ee\nBut, according to Lemma 2.2, $W(t)W(t')^{-1}\\in\\DR_+$ and thus\n(2.5) is upper-triangular, yielding\n\\be\n\\oint_{z=\\iy}\\Psi_n(t,z)\\Psi^_m(t',z)\\frac{dz}{2\\pi\niz}=0\\quad\\mbox{for all $n>m$}.\n\\ee\nDefining\n\\begin{eqnarray}\n\\Phi_n(t,z)&:=&z^{-n}\\Psi_n(t,z)=e^{\\sum\nt_iz^i}(1+O(z^{-1}))\\\\\n\\Phi_n^(t,z)&:=&z^{n-1} \\Psi^_{n-1}(t,z)=e^{-\\sum\nt_iz^i}(1+O(z^{-1})),\n\\end{eqnarray}\nupon using the asymptotics (0.8), we have, by setting $m=n-1$ in\n(2.6)\n$$\n\\oint_{z=\\iy}\\Phi_n(t,z)\\Phi^_n(t',z)dz=\n\\oint_{z=\\iy}\\Psi_n(t,z)\\Psi^_{n-1}(t',z)\\frac{dz}{z}=0.\n$$\nFrom the KP-theory, there exists a $\\tau$-function $\\tau_n(t)$ for\neach $n$, such that\n$$\n\\Phi_n(t,z)=e^{\\sum\nt_iz^i}\\frac{\\tau_n(t-[z^{-1}])}{\\tau_n(t)},\\quad\\Phi^_n(t,z)=e^{-\\sum\nt_iz^i}\\frac{\\tau_n(t+[z^{-1}])}{\\tau_n(t)},\n$$\nyielding the $\\tau$-function representation (0.10) for $\\Psi_n$ and\n$\\Psi^_n$.\\qed\n\n\\medbreak\n\n{\\bf Step 2:} The following holds for $n\\in\\BZ$:\n\\be\n\\left(\\frac{1}{2}\\frac{\\pl^2}{\\pl\nt_1\\pl\nt_k}-p_{k+1}(\\tilde\\pl)\\right)\\tau_n\\circ\\tau_n=0,\n~~\\mbox{for}~k=1,2,3,...\n\\ee\n\\be\n\\left(\\frac{\\pl}{\\pl\nt_k}-\\sum^{\\ell-1}_{r=0}(\\ell-r)p_r(-\\tilde\\pl)p_{k-r}(\\tilde\\pl)\\right)\n\\tau_{n}\\circ\\tau_{n-\\ell}=0,~~\\mbox{for}~\\ell,k=1,2,3,...\n\\ee\nIndeed the bilinear identity (2.6), upon setting $m=n-\\ell-1$, shifting\n$t\\mapsto t+[\\al]$, $t'\\mapsto t-[\\al]$, using the\n$\\tau$-function representation (0.10) of $\\Psi$ and $\\Psi^$, and lemma\n1.2 with $m=\\ell$, yield\\footnote{\n$e^{m\\sum_1^{\\iy}(\\al z)^i/i}=(1-\\al z)^{-m}$}\n\\begin{eqnarray}\n0&=&-\\al^2\\oint_{z=\\iy}\\Psi_n(t+[\\al],z)\\Psi^_{n-\\ell-1}\n(t-[\\al],z)\\frac{dz}{2\\pi iz}\\tau_n(t+[\\al])\\tau_{n-\\ell\n}(t-[\\al])\\\\\n&=&-\\oint_{z=\\iy}\\tau_n(t+[\\al]-[z^{-1}])\\tau_{n-\\ell}(t-[\\al]+[z^{-1}])\ne^{2\\sum_1^{\\iy}(\\al z)^i/i}\\al^2 z^{\\ell +1}\\frac{dz}{2\\pi iz}\\\\\n&=&\\al^{1-\\ell}\\sum_{k=1}^{\\iy}\\al^k\\left(\\frac{\\pl}{\\pl\nt_k}-\\sum^{\\ell-1}_{r=0}(\\ell-r)p_r(-\\tilde\\pl)p_{k-r}(\\tilde\\pl)\\right)\n\\tau_{n}\\circ\\tau_{n-\\ell},\n\\end{eqnarray}\nestablishing the second relation of (2.8). As for the first one, set\n$m=n-1$,\n$t\\mapsto t-a$ and $t'\\mapsto t+a$ in the bilinear identity, and use\n(1.4), thus yielding (0.14).\n\n\\medbreak\n\n{\\bf Step 3:} To check the formulas (0.12) for $S$, compute\n\\begin{eqnarray}\ne^{\\sum^{\\iy}_1t_iz^i}S\\chi(z)&=:&\\Psi(t,z)\\\\\n&=&e^{\\sum^{\\iy}_1t_iz^i}\\frac{\\tau(t-\n[z^{-1}])}{\\tau(t)}\\chi(z)\\quad\\mbox{(by (0.10))}\\\\\n&=&e^{\\sum^{\\iy}_1t_iz^i}\\sum^{\\iy}_{n=0}\n\\frac{p_n(-\\tilde\\pl)\\tau(t)}{\\tau(t)}z^{-n}\\chi(z)\\\\\n&=&e^{\\sum^{\\iy}_1t_iz^i}\\sum^{\\iy}_0\n\\frac{p_n(-\\tilde\\pl)\\tau(t)}{\\tau(t)}\\Lb^{-n}\\chi(z).\n\\end{eqnarray}\nSimilarly one checks the formula for $S^{-1}$ using the formulas\nfor $\\Psi^(t,z)$ in terms of $S^{-1}$ and $\\tau(t)$. Finally to\ncheck the formula (0.13) for $L^k$, use the formulas (0.12) for $S$\nand $S^{-1}$ (for $\\tilde \\Lb$, see footnote 1):\n\\begin{eqnarray}\nL^k&=&S\\Lb^kS^{-1}\\\\\n&=&\\sum^{\\iy}_{i,j\\geq\n0}\\frac{p_i(-\\tilde\\pl)\\tau}{\\tau}\\Lb\n^{-i-j+k}\\left(\\tilde \\Lb\\frac{p_j(\\tilde\\pl)\\tau}{\\tau}\\right)\\\\\n&=&\\sum^{\\iy}_{i,j\\geq\n0}\\frac{p_i(-\\tilde\\pl)\\tau}{\\tau}\\left(\\tilde \\Lb^{-i-j+k+1}\n\\frac{p_j(\\tilde\\pl)\\tau}{\\tau}\\right)\\Lb^{-i-j+k}\\\\\n&=&\\sum_{\\ell\\geq 0}\\left(\\sum_{{i,j\\geq\n0}\\atop{i+j=\\ell}}\\frac{p_i(-\\tilde\\pl)\\tau_np_j(\\tilde\\pl)\\tau_{n+k\n-\\ell+1})}{\\tau_n\\tau_{n+k-\\ell+1}}\\right)_{n\\in\\BZ}\n\\Lb^{k-\\ell}\\\\\n&=&\\sum_{\\ell\\geq\n0}\\left(\\frac{p_{\\ell}(\\tilde\\pl)\n\\tau_{n+k-\\ell+1}\\circ\\tau_n}{\\tau_{n+k-\\ell+1}\\tau_n}\n\\right)_{n\\in\\BZ}\\Lb^{k-\\ell}\\mbox{\\,\\,}\\quad\\mbox{(using (1.13))}\n\\end{eqnarray}\nyielding (0.13) and (0.15), upon noting,\n\\begin{eqnarray}\n\\mbox{coef}_{\\Lb^{k-1}}L^k&=&\\left(\\frac{p_1(\\tilde\\pl)\\tau_{n+k}\n\\circ\\tau_n}{\\tau_{n+k}\\tau_n}\\right)_{n\\in\\BZ}=\\left(\\frac{\\pl}{\\pl\nt_1}\\log\\frac{\\tau_{n+k}}{\\tau_n}\\right)_{n\\in\\BZ}\\\\\n\\mbox{coef}_{\\Lb^0}L^k&=&\\left(\\frac{p_k(\\tilde\\pl)\\tau_{n+1}\n\\circ\\tau_n}{\\tau_{n+1}\\tau_n}\\right)_{n\\in\\BZ}=\n\\left(\\frac{\\pl}{\\pl\nt_k}\\log\\frac{\\tau_{n+1}}{\\tau_n}\\right)_{n\\in\\BZ}\\mbox{\\,\\,by (2.8)}\\\\\n\\mbox{coef}_{\\Lb^{-1}}L^k\n&=&\\left(\\frac{p_{k+1}(\\tilde\\pl)\\tau_n\n\\circ\\tau_n}{\\tau_n\\tau_n}\\right)_{n\\in\\BZ} =\\left(\\frac{\\pl^2}{\\pl\nt_1\\pl t_k}\\log\\tau_n\\right)_{n\\in\\BZ},\\quad\\mbox{by (2.7)},\n\\end{eqnarray}\nconcluding the proof of the Theorem 0.1.\\qed\n\n\\begin{corollary} Setting $\\ga(t):=(\\tilde \\Lb\\tau(t)/\\tau(t))$, the\nwave operator $W(t)$ for the discrete\nKP-hierarchy has the following property\n$$\n(W(t)W^{-1}(t'))_-=0,\\quad (W(t)W^{-1}(t'))_0=\\frac{\\ga(t)}{\\ga(t')}.\n$$\n\\end{corollary}\n\n\\proof That $h(t,t')=W(t)W^{-1}(t')\\in\\DR_+$ was shown in Lemma 2.2.\nConcerning its diagonal $h_0$, we deduce from (2.3)\nthat\\footnote{$M_0:=$~diagonal part of $M$.}\n$$\n\\frac{\\pl}{\\pl t_k}\\log h_0=(L^k(t))_0,\\quad\n\\frac{\\pl}{\\pl t_k}\\log h_0=-(L^k(t'))_0,\\quad\\mbox{with\n$h_0(t,t)=I.$}\n$$\nNote that $\\ga(t)/\\ga(t')$ satisfies the same differential equations\nas $h_0(t)$ with the same initial condition, upon using (0.15):\n\\begin{eqnarray}\n\\left(\\frac{\\pl}{\\pl t_k}\\log\\frac{\\ga(t)}{\\ga(t')}\\right)_n&=&\n\\frac{\\pl}{\\pl t_k}\\log\\frac{\\tau_{n+1}(t)}{\\tau_n(t)}=L^k(t)_{nn}\\\\\n\\left(\\frac{\\pl}{\\pl t'_k}\\log\\frac{\\ga(t)}{\\ga(t')}\\right)_n&=&\n-\\frac{\\pl}{\\pl\nt'_k}\\log\\frac{\\tau_{n+1}(t')}{\\tau_n(t')}=-L^k(t')_{nn},\n\\end{eqnarray}\nwith $\\ga(t)/\\ga(t')\\Biggl\|_{t=t'}=I$.\n\n\n\n\\section{Sequences of $\\tau$-functions, flags and the discrete\nKP equation}\n\nIn this section, we prove Theorem 0.2; it will be broken up into three\npropositions: the first one is very similar to the analogous statement\nfor the KP theory (see \\cite{DJKM,vM}). One could make an argument\nunifying both cases, in the context of Lie theory. The second statement\nuses Grassmannian technology.\n\n\\begin{proposition} The following equivalences (i)\n$\\Longleftrightarrow$ (ii) $\\Longleftrightarrow$ (iii) hold.\n\\end{proposition}\n\n\\proof (i) $\\Rg$ (ii) was already shown in Theorem 0.1. Regarding\nthe converse (ii) $\\Rg$ (i), we show vectors $\\Psi(t,z)$ and\n$\\Psi^(t,z)$ having the asymptotics (0.8) and satisfying the bilinear\nidentity (ii) are discrete\nKP-hierarchy vectors.\n\n\\medbreak\n\nThe point of the proof is to show that the matrices $S$ and $T^t\\in\nI+\\DR_-$ defined through\n$$\n\\Psi(t,z)=:e^{\\sum_1^{\\iy}t_iz^i}S\\chi(z),\\quad\n\\Psi^(t,z)=:e^{-\\sum_1^{\\iy}t_iz^i}T\\chi^(z)\n$$\nsatisfy the vector fields (0.6) with $T^t=S^{-1}$.\n\n\\medbreak\n\n\\noindent{\\bf Step 1.} $T^t=S^{-1}$.\n\nAssuming the bilinear identities (assumption (ii) of Theorem 0.2),\n\\begin{eqnarray}\n0&=&\\left(\\oint_{z=\\iy}\\Psi(t,z)\\otimes\\Psi^(t,z)\\frac{dz}{2\\pi\niz}\\right)_-\\\\\n&=&\\left(\\oint_{z=\\iy}e^{\\sum_1^{\\iy}t_iz^i}S \\,\\chi(z)\\otimes\ne^{-\\sum_1^{\\iy}t_iz^i}T\\,\\chi(z^{-1})\\frac{dz}{2\\pi iz}\n\\right)_- \\\\\n&=&(ST^{\\top})_-,\\quad\\mbox{by (2.1)}\n\\end{eqnarray}\nbut since $S,T^t\\in I+\\DR_-$, $ST^t=I$, yielding $T^t=S^{-1}$.\n\n\\medbreak\n\n\\noindent{\\bf Step 2.} $W(t)W^{-1}(t')\\in\\DR_+$, upon defining\n$W(t):=S(t)e^{\\sum t_i\\Lb^i}$.\n\nAccording to the bilinear identity, the left hand side of\n\n$$\\oint_{z=\\iy}\\Psi(t,z)\\otimes\\Psi^(t',z)\\frac{dz}{2\\pi\niz}\\hspace{5cm}$$\n\\begin{eqnarray}\n&=&\\oint_{z=\\iy}e^{\\sum t_iz^i}S \\, \\chi(z)\\otimes\ne^{-\\sum_1^{\\iy}t'_iz^i}(S^{-1})\n^{\\top}\\chi(z^{-1})\\frac{dz}{2\\pi\niz}\\\\\n&=&\\oint_{z=\\iy}S(t)e^{\\sum t_i\\Lb^i}\\,\n\\chi(z)\\otimes (S^{-1}(t'))^{\\top}e^{-\\sum\nt'_i\\Lb^{\\top -i}} \\chi(z^{-1})\\frac{dz}{2\\pi iz}\\\\ &=&S(t)e^{\\sum\nt_i\\Lb^i}e^{-\\sum t'_i\n\\Lb^i}S^{-1}(t'),\\quad\\mbox{using Lemma 2.1}\\\\ &=&W(t)W^{-1}(t');\n\\end{eqnarray}\n belongs to $\\DR_+$, and hence so is the right hand side.\n\n\n\\medbreak\n\n\\noindent{\\bf Step 3.}\n\\begin{eqnarray}\n\\left(\\frac{\\pl}{\\pl t_n}-(L^n)_+\\right)\\Psi(t,z)\n&=&\\left(\\frac{\\pl}{\\pl\nt_n}-(L^n)_+\\right)S\\chi(z)e^{\\sum_1^{\\iy}t_iz^i}\\\\\n&=&\\left(\\frac{\\pl S}{\\pl t_n}-(L^n)_+S+S\\,z^n\\right)\\chi(z)\ne^{\\sum_1^{\\iy}t_iz^i}\\\\\n&=&\\left(\\frac{\\pl S}{\\pl\nt_n}-(L^n)_+S+S\\,\\Lb^n(S^{-1}S)\\right)\n\\chi(z)e^{\\sum_1^{\\iy}t_iz^i}\\\\\n&=&\\left(\\frac{\\pl S}{\\pl\nt_n}-(L^n)_+S+L^nS\\right)\\chi(z)\ne^{\\sum_1^{\\iy}t_iz^i}\\\\\n&=&\\left(\\frac{\\pl S}{\\pl\nt_n}+(L^n)_-S)\\right)\\chi(z)e^{\\sum_1^{\\iy}t_iz^i}.\n\\end{eqnarray}\n\n\\medbreak\n\n\\noindent{\\bf Step 4.} From $W(t)W^{-1}(t')\\in\\DR_+$, since\n$\\DR_+$ is an algebra, deduce\n\\begin{eqnarray}\n\\DR_+&\\ni&\\left(\\left(\\frac{\\pl}{\\pl t_n}-(L^n)_+\\right)W(t)\\right)\nW^{-1}(t')\\Biggl\|_{t'=t}\\\\\n&=&\\oint_{z=\\iy}\\left(\\frac{\\pl}{\\pl t_n}-(L^n)_+\\right)\\Psi(t,z)\n\\otimes\\Psi^(t,z)\\frac{dz}{2\\pi iz},\\quad\\mbox{by step 2}\\\\\n&=&\\oint_{z=\\iy}\\left(\\frac{\\pl S(t)}{\\pl t_n}+(L^n)_-S(t)\\right)\n\\chi(z)e^{\\sum_1^{\\iy}t_iz^i}\\otimes(S^{\\top}(t))^{-1}\n\\chi(z^{-1})e^{-\\sum_1^{\\iy}t_iz^i}\\frac{dz}{2\\pi iz},\\\\\n& &\\hspace{7cm}\\quad\\mbox{by step 3}\\\\\n&=&\\left(\\frac{\\pl S(t)}{\\pl t_n}+(L^n)_-S(t)\\right)S(t)^{-1},\\quad\n\\mbox{by Lemma 2.1}\n\\end{eqnarray}\nand thus, since $S\\in I+\\DR_-$ and $\\DR_-$ is an algebra,\n$$\n\\left(\\frac{\\pl S(t)}{\\pl\nt_n}+(L^n)_-S(t)\\right)S(t)^{-1}\\in\\DR_+\\cap\\DR_-=0;\n$$\ntherefore, we have the discrete\nKP-hierarchy equations on $S$\n$$\n\\frac{\\pl S(t)}{\\pl t_n}+(L^n)_-S=0,\\quad n=1,2,...,\n$$\nand on $L=S\\Lb S^{-1}$,\n$$\n\\frac{\\pl L}{\\pl t_n}=[-(L^n)_-,L],\n$$\nending the proof that (ii) $\\Rg$ (i).\\qed\n\n\\medbreak\n\nFinally (ii) $\\Longleftrightarrow$ (iii) upon using the\nequivalence (i) $\\Longleftrightarrow$ (ii) and the\n$\\tau$-function representation (0.10) of $\\Psi$ and $\\Psi^$, shown in\nTheorem 0.1; this establishes Proposition 3.1.\n\nWith each component of the wave vector $\\Psi$, given in (0.10), or,\nwhat is the same, with each component of the $\\tau$-vector, we\nassociate a sequence of infinite-dimensional planes in the\nGrassmannian $Gr^{(n)}$\n\\bea\n\\WR_n&=&\\mbox{ span}_{\\BC}\\left\\{\\left(\\frac{\\pl}{\\pl t_1}\n\\right)^k\\Psi_n(t,z),~~k=0,1,2,...\\right\\}\\nonumber \\\\\n&=&e^{\\sum_1^{\\iy}t_i z^i}\\mbox{ span}_{\\BC}\\left\\{\\left(\n\\frac{\\pl}{\\pl\nt_1}+z\\right)^k\\psi_n(t,z),~~k=0,1,2,...\\right\\}\\nonumber\\\\\n&=:& e^{\\sum_1^{\\iy}t_i z^i} \\WR_n^t.\n\\eea\nand planes\n\\be\n\\WR^\\ast_n=\\mbox{ span}_{\\BC}\\left\\{\\frac{1}{z}\\left(\\frac{\\pl}{\\pl t_1}\n\\right)^k \\Psi^{\\ast}_{n-1}(t,z),~~k=0,1,2,...\\right\\},\n\\ee which are the orthogonal complements of $\\WR_n$ in $Gr^{(n)}$, by\nthe residue pairing\n\\be\n\\la f,g\\ra_{\\iy} :=\\oint_{z=\\iy} f(z) g(z) \\frac{dz}{2\\pi i}.\n\\ee\n\n\n\\begin{proposition} (ii) $\\Longleftrightarrow$ (iv)\n$\\Longleftrightarrow$ (v) holds.\n\\end{proposition}\n\n\\proof The inclusion $...\\supset \\WR_{n-1}\\supset \\WR_n\\supset\n\\WR_{n+1}\\supset ...$ in (iv) implies that $\\WR_n$,\ngiven by (3.1) and (0.10), is also given by\n$$\n\\WR_n=\\mbox{span}_{\\BC}\\{\\Psi_n(t,z),\\Psi_{n+1}(t,z),...\\}.\n$$\nMoreover the inclusions $...\\supset \\WR_n\\supset\n\\WR_{n+1}\\supset ...$ imply, by orthogonality, the inclusions $...\n\\subset \\WR_n^\\subset \\WR^_{n+1}\\subset ...$, and thus $\\WR^_n$,\ngiven by (3.2) and (0.10) and thus specified by $\\Psi^_{n-1}$ and\n$\\tau_n$, is also given by\n$$\n\\WR^_n=\\{\\frac{\\Psi^_{n-1}(t,z)}{z},\\frac{\\Psi^_{n-2}(t,z)}{z},...\\}.\n$$\nThe formula (0.10) for $\\Psi_n$ and $\\Psi^_{n-1}$\n imply the\nbilinear identities (1.1), since each $\\tau_n$ is a\n$\\tau$-function, yielding $\\WR_n^{\\ast}\n=\\WR_n^{\\bot}$, with respect to the residue pairing and so:\n$$\n\\la \\Psi_n(t,z),\\frac{\\Psi^_{n-1}(t',z)}{z}\\ra_{\\iy}\n=\\oint_{z=\\iy}\\Psi_n(t,z)\\Psi^_{n-1}(t',z)\\frac{dz}{2\\pi iz}=0.\n$$\n%implying that, for each $n\\in\\BZ$, $\\Psi^_{n-1}(t,z)\\in \\WR^_n$.\nSince\n$$\n\\WR_n\\subset \\WR_{m+1}=(\\WR^_{m+1})^,\\quad\\mbox{all $n>m$}\n$$\nwe have the orthogonality $\\WR_n\\perp \\WR^_{m+1}$ for all $n>m$, with\nrespect to the residue pairing; since $\\Psi_n(t,z) \\in \\WR_n,\n~\\frac{\\Psi^_m(t',z)}{z}\\in \\WR^_{m+1}(t',z)$, we have\n\\be\n0=\\la \\Psi_n(t,z),\\frac{\\Psi^_m(t',z)}{z}\\ra_{\\iy}\n=\\oint_{z=\\iy}\\Psi_n(t,z)\\Psi^_m(t',z)\\frac{dz}{2\\pi\niz},\\quad\\mbox{all $n>m$,}\n\\ee\nwhich is (ii).\n\nNow assume (ii); then, for fixed $n>m$, we have\n$$\n0=\\oint_{z=\\iy}\\left(\\frac{\\pl}{\\pl t_1}\\right)^k\n\\Psi_n(t,z)\\left(\\frac{\\pl}{\\pl\nt'_1}\\right)^{\\ell}\\Psi^_m(t',z)\\frac{dz}{2\\pi iz},\\quad n>m\n$$\nand thus by (3.1) and (3.2),\n%$$\n%\\WR^_{m+1}\\subseteq \\WR_n^{\\bot}=\\WR_n^,\\quad\\mbox{for $n>m.$}\n%$$\n%Hence\n$$\n\\WR_n\\subseteq (\\WR^{\\ast}_{m+1})^{\\ast}=\\WR_{m+1},\\quad\\mbox{for\n$n>m$,}\n$$\nwhich implies the flag condition $...\\supset \\WR_{n-1}\\supset\n\\WR_n\\supset \\WR_{n+1}\\supset ...$, stated in (iv).\n\n(iv) $\\Longleftrightarrow$ (v), follows from the equivalence of\n(i) and (iii) in Proposition 1.1, by setting $\\tau_1:=\\tau_{n-1}$,\n$\\tau_2=\\tau_n$,\n$\\tilde \\WR_1=z^{-n+1}\\WR_{n-1}$ and $\\tilde \\WR_2=z^{-n}\\WR_n$ and\nnoting\n$$\nz(z^{-n}\\WR_n)\\subset(z^{-n+1}\\WR_{n-1}),\\quad\\mbox{i.e. $\\WR_n\\subset\n\\WR_{n-1}$},\n$$\nconcluding the proof of the proposition.\\qed\n\n\\begin{proposition} (v)\n$\\Longleftrightarrow$ (vi) holds.\n\\end{proposition}\n\n\\proof\n\n{\\bf\\noindent Step 1.} For a given $n\\in\\BZ$, statement (v), namely\n$$\nR_k^{(n)}:=\\{p_{k-1}(-\\tilde\\pl)\\tau_n,\\tau_{n+1}\\}+\\tau_{n+1}p_k(-\\tilde\n\\pl)\\tau_n-\\tau_np_k(-\\tilde\\pl)\\tau_{n+1}=0,\\quad k\\geq 2\n$$\nimplies\n$$R_k^{(n)'}=\\left(\\frac{\\pl}{\\pl\nt_k}-p_k(\\tilde\\pl)\\right)\\tau_{n+1}\\circ\\tau_n=0,\\quad k\\geq 2.\n$$\nSince $R_k^{(n)}$ are the Taylor coefficients of relation (v) in\nTheorem 0.2, statement (v)$_n$ is equivalent to (iv)$_n$ (i.e.\n$\\WR_n\\supset \\WR_{n+1}$). The latter is equivalent to the bilinear\nidentity (iii)$_n$ (i.e., (0.18) with $n\\rg n+1$ and $m\\rg n-1$).\nAccording to the arguments used in the proof of Theorem 0.1,\n(iii)$_n$ implies $R_k^{(n)'}=0$.\n\n\\medbreak\n\n{\\bf\\noindent Step 2.} The converse holds, because, upon using an\ninductive argument,\n$$\nR_k^{(n)}=\\al R_k^{(n)'}+\\mbox{\\,partials of\n$(R_1^{(n)'},...,R_{k-1}^{(n)'})$};\n$$\nthus the vanishing of the $R_1^{(n)'},...,R_k^{(n)'}$ implies the\nvanishing of $R_k^{(n)}$.\\qed\n\n\\bigbreak\n\n\n\n\\begin{theorem}\nEvery 1-Toda lattice is equivalent to a 2-Toda lattice.\n\\end{theorem}\n\n\\proof The 1-Toda theory implies for $S_1:=S\\in I+{\\cal\nD}_-$,\n$L_1:=L$\n$$\n\\frac{\\pl S_1}{\\pl t_n}=-(L_1^n)_-S_1(t),\\quad\\mbox{where\n$L_1=S_1\\Lb S_1^{-1}.$}\n$$\nThen, in view of the 2-Toda theory, define $S_2(t)\\in{\\cal D}_+$ by\nmeans of the differential equations\n$$\n\\frac{\\pl S_2(t)}{\\pl t_n}=(L_1^n)_+S_2(t),\\quad n=1,2,...,\n$$\nwith initial condition $S_2(0)=$ (an invertible element\n$d_+\\in{\\cal D}_+$). Then define\\footnote{The first index in\n$L_{1,2}$ and $S_{1,2}$ corresponds to the upper-sign.}\n$S_{1,2}(t,s)$ and\n$L_{1,2}=S_{1,2}\\Lb^{\\pm 1}S_{1,2}^{-1}$, flowing according to\nthe commuting differential equations\n\\be\n\\frac{\\pl\nS_{1,2}(t,s)}{\\pl\ns_n}=\\pm(L_2^n(t,s))_{\\mp}S_{1,2}(t,s)\\quad\\mbox{with}\\quad\nS_{1,2}(t,0)=S_{1,2}(t).\n\\ee\n$S_{1,2}(t,s)$ satisfies the $t$-equations of 2-Toda for $s=0$, by\nconstruction; now we must check that this holds for $s\\neq 0$;\ntherefore, set\n\\be\nF_{1,2}^{(n)}(t,s)=\\frac{\\pl\nS_{1,2}}{\\pl t_n}(t,s)\\pm(L^n_1(t,s))_{\\mp}S_{1,2}(t,s),\n\\quad\\mbox{for\n$n=1,2,...$}\n\\ee\nCompute, using (3.5) and $[\\pl/\\pl t_n,\\pl/\\pl s_n]=0$, the\nsystem of two differential equations\n$$\n\\frac{\\pl\nF_{1,2}^{(n)}}{\\pl\ns_k}(t,s)=\\pm[F_{2,1}^{(n)}S_2^{-1},L_2^k]_{\\mp}S_{1,2}\\pm(L_2^k)_{\\mp}\nF_{1,2}^{(n)},\\quad k,n=1,2,...;\n$$\nsince $F_{1,2}^{(n)}(t,0)=0$, we have $F_{1,2}^{(n)}(t,s)=0$ for all\n$s$. Thus, by (3.5) and (3.6), $S_{1,2}(t,s)$ flow according to\n2-Toda.\\qed\n\n\n\n\n\\medbreak\n\n\n\\section{Discrete KP-solutions generated by vertex operators}\n\nAn important construction leading to Toda solutions is contained\nin Theorem 0.3, which is based on the following Lemma:\n\n\n\\begin{lemma}\n Particular solutions to equation\n\\be\\{ \\tau_1\n(t-[z^{-1}]),\\tau_2 (t)\\} + z (\\tau_1 (t-[z^{-1}]) \\tau_2 (t)\n- \\tau_2 (t-[z^{-1}])\\tau_1 (t)) = 0 \\ee\nare given, for arbitrary $\\lambda \\in\n\\BC^\\ast$, by pairs $(\\tau_1,\\tau_2)$, defined by:\n\\be\n\\tau_2 (t) = \\left(\\int X (t,\\lambda)\\nu(\\lb)d\\lb\\right) \\tau_1 (t) =\n\\int e^{\\sum t_i\n\\lambda^i}\n\\tau_1 (t-[\\lambda^{-1}])\\nu(\\lb)d\\lb,\n\\ee\nor\n\\be\n\\tau_1 (t) =\\left(\\int X (-t,\\lambda)\\nu'(\\lb)d\\lb\\right) \\tau_2 (t) =\n\\int e^{-\\sum t_i \\lambda^i}\n\\tau_2 (t+[\\lambda^{-1}])\\nu'(\\lb)d\\lb.\n\\ee\n\\end{lemma}\n\n\\proof Using\n$$e^{-\\sum^\\iy_1 {1 \\over i}({\\lambda \\over z})^i } = 1 -\n{\\lambda \\over z},$$ it suffices to check,before even integrating,\nthat $\\tau_2 (t)=X(t,\\lb)\\tau_1(t)$ satisfies the above equation\n(4.1)\n\\bea\n&&e^{-\\sum t_i \\lambda^i} \\left(\\{\\tau_1\n(t-[z^{-1}]),\\tau_2(t)\\}\n + z (\\tau_1 (t-[z^{-1}])\\tau_2(t) - \\tau_2(t-[z^{-1}]) \\tau_1\n(t))\\right) \\nonumber\\\\\n&&~~~~= e^{-\\sum t_i \\lambda^i}\\{\\tau_1 (t-[z^{-1}]),\ne^{\\sum t_i \\lambda^i} \\tau_1 (t-[\\lambda^{-1}])\\} \\nonumber\n\\\\ &&~~~~~~~~~+ z\n(\\tau_1 (t-[z^{-1}]) \\tau_1 (t-[\\lambda^{-1}]) - (1-{\\lambda\n\\over z}) \\tau_1 (t) \\tau_1 (t-[z^{-1}] -\n\\lambda^{-1}]))\\nonumber \\\\ &&~~~~= \\{\\tau_1\n(t-[z^{-1}]),\\tau_1 (t-[\\lambda^{-1}])\\}\\nonumber\n\\\\ &&~~~~~~~~~+\n(z-\\lambda) (\\tau_1 (t-[z^{-1}])\\tau_1 (t-[\\lambda^{-1}]) -\n\\tau_1 (t) \\tau_1 (t-[z^{-1}]-[\\lambda^{-1}]))\\nonumber \\\\\n&&~~~~= 0,\\nonumber\n\\eea\nusing the differential Fay identity (1.3) for the $\\tau$-function\n$\\tau_1$; a similar proof works for the second solution, given by\n$\\tau_1 (t)=X(-t,\\lb)\\tau_2(t)$. Since equation (4.1) is linear in\n$\\tau_1(t)$, and also in $\\tau_2(t)$, the equation remains valid after\nintegrating with regard to $\\lb$.\n\\qed\n\n\\noindent\\underline{\\em Proof of Theorem 0.3}: Note, from the\ndefinition of $\\tau_{\\pm n}$ in Theorem 3, that each $\\tau_n$ is\ndefined inductively by\n$$\n\\tau_{n+1} =\\int X(t,\\lambda) \\nu_n(\\lb) d\\lb ~\\tau_n\\mbox{ and\n}\\tau_{-n-1}= \\int X(-t,\\lambda) \\nu_{-n-1}(\\lb) d\\lb ~\\tau_{-n};\n$$\nthus by Lemma 4.1, the functions $\\tau_{n+1}$ and\n$\\tau_n$ are a solution of equation (v) of Theorem 0.2. Therefore,\ntheorem 0.2 implies that the\n$\\tau_n$'s form a $\\tau$-vector of the discrete KP hierarchy.\\qed\n\n\n\n\n\n\n\n\\section{Example of vertex generated solutions: the $q$-KP equation}\n\nConsider the class of $q$-pseudo-difference operators,\nwith $y$-dependent coefficients, acting on functions $f(y)$\n$$\n\\DR_q=\\{ \\sum a_i(y) D^i\\},~~\\mbox{with}~~Df(y):=f(qy).\n$$\nand the $q$-derivative $D_q$, defined by\n$$\nD_qf(y):=\\frac{f(qy)-f(y)}{(q-1)y}=-\\lb(y)(D-1)f(y),\n~\\mbox{with}~\\lb(y):=-\\frac{1}{(q-1)y};\n$$\nConsider the following $q$-pseudo-difference operators\n$$\nQ=D+u_0(x)D^0+u_{-1}D^{-1}+...\\mbox{\nand }~~Q_q=D_q+v_0(x)D_q^0+v_{-1}(x)D_q^{-1}+...\n$$\nand the following $q$-deformations, which were proposed respectively by\nE. Frenkel \\cite{F} and Khesin, Lyubashenko and Roger \\cite{KLR}, for\n$n=1,2,...$:\n\\be\n{\\pl Q\\over\\pl t_n}=\\bigl[\\left(Q^n\\right)_+,Q\\bigr]\n~~~~~~~~~~~~~~~ ~~~~~~~\\mbox{{\\em (Frenkel system)}}\n\\ee\n\\be\n{\\pl Q_q\\over\\pl\nt_n}=\\bigl[\\left(Q_q^n\\right)_+,Q_q\\bigr],\n~~~~~~~~~~~~~~~~~~~ \\mbox{{\\em (KLR system)}}\n\\ee\nwhere $(~)_+$ and $(~)_-$ refer to the $q$-difference and\nstrictly $q$-pseudo-differential part of $(~)$. %The two systems\n%are closely related, as will become clear from the isomorphism\n%between $q$-operators and difference operators, explained below.\nDefine\n\\be\n c(x) =\n\\left({(1-q) x \\over 1-q} , {(1-q)^2 x^2 \\over 2(1-q^2)} ,\n{(1-q)^3 x^3 \\over 3 (1-q^3)}, ... \\right)\\in \\BC^{\\iy}\\ \\ \\ \\mbox{and}\n\\\n\\\n\\\n\\lambda^{-1}_n = (1-q) x q^{n-1} ,\n\\ee\none checks for $n \\geq 1$, $D^n \\lb_0(x)=\\lb_n(x)$, and\n\\bea\nD^n c(x)&=&c(x)-\\sum_1^n[\\lb_i^{-1}(x)]\\nonumber\\\\\nD^{-n} c(x)&=&c(x)+\\sum_1^{n}[\\lb_{-i+1}^{-1}(x)]\n\\eea\nDetails about these theorems were reported in a joint work with E.\nHorozov\\cite{AHV}.\n\n\\begin{theorem}\nThere is an algebra isomorphism\n$$\n\\hat {}~\n: \\DR_q \\lrg \\DR ,\n$$\nwhich maps the Frenkel and KLR system into the discrete\nKP-hierarchy\n\\be\n{\\pl L\\over\\pl t_n}=\\bigl[\\left(L^n\\right)_+,L \\bigr],\\quad\nn=1,2,.. .\n\\ee\n\\end{theorem}\n\\bigbreak\n\n\n\\begin{theorem} The matrices\n$$L=\\Lb+\\sum_{-\\iy<\\ell\\leq 0}\\diag\\left(\n\\frac{p_{1-\\ell}(\\tilde\\pl)\\tau_{n+\\ell+1}\\circ\\tau_n}{\\tau_{\nn+\\ell+1}\\tau_n}\\right)_{n\\in\\BZ}\\Lb^{\\ell}\n$$\nand\n$$\n\\tilde L=\\vr L\\vr^{-1}\n$$\nwith $\\vr$ as in (5.11), $\\tau_0=\\tau(c(x)+t)$ and\n\\bea\n\\tau_n&=&X(t,\\lb_n)...X(t,\\lb_1)\\tau(c(x)+t)\\nonumber\\\\\n&=&r_n(\\lb)\\left(\n\\prod^n_{k=1}e^{\\sum^{\\iy}_{i=1}t_i\\lb^i_k}\\right)D^n\\tau(c(x)+t)\n\\eea\n\\bea\n\\tau_{-n}&=&X(-t,\\lb_{-n+1})...X(-t,\\lb_0)\\tau(c(x)+t)\\nonumber\\\\\n&=&r_{-n}(\\lb)\\left(\n\\prod^n_{k=1}e^{-\\sum^{\\iy}_{i=1}t_i\\lb^i_\n{-k+1}}\\right)D^{-n}\\tau(c(x)+t) \\nonumber\n\\eea\ntransform, using the map $\\hat {}$, respectively into solutions to\nthe $q$-KP deformations (5.1) and (5.2) of\n$$\nQ=D+\\sum_{-\\iy<i\\leq 0}a_i(y)D^i\\quad\\mbox{or}\\quad\nQ_q=D_q+\\sum_{-\\iy<i\\leq 0}b_i(y)D^i_q,\n$$\nwhere the $b_i$ are related to the $a_i$ by (5.12)\nand\\footnote{$\\pi(k)=$ parity of $k=1$, when $k$ is even, and\n$=-1$, when $k$ is odd.}\n$$\na_{\\ell}(y)=\\mbox{\\,polynomial in}\\left\\{\n\\begin{tabular}{l}\n$\\displaystyle{\\frac{\\pl^k}{\\pl\nt_{i_1}...\\pl\nt_{i_k}}\\log\\left(\\tau(c(y)+t)^{\\pi(k)}D^{\\ell+1}\\tau(c(y)+t)\\right)}$\nfor\n$k\\geq 2$\\\\\n \\\\\n$\\displaystyle{\\sum^{\\ell+1}_{i=1}\\lb_i^j(y)+\\frac{\\pl}{\\pl\nt_j}\\log\\frac{D^{\\ell+1}\\tau(c(y)+t)}{\\tau(c(y)+t)}}$, for $k=1$\n\\end{tabular}\\right.\n$$\n\\end{theorem}\n\n\n\nThe proofs of these theorems, which rely heavily on the next lemma,\nwill be given later. In an elegant recent paper, Iliev \\cite{I} has\nobtained $q$-bilinear identities and $q$-tau functions, as well,\npurely within the KP theory.\n\nConsider an appropriate space of functions $f(y)$ representable by\n``Fourier\" series\n$$\nf(y)=\\sum_{-\\iy}^{\\iy}f_n\\varphi_n(y)\n$$\nin the basis\\footnote{The $\\delta$-function $\\dt(z):=\\sum_{i \\in \\BZ}z^i$;\nenjoys the property $f(za)\\dt(z)=f(a)\\dt(z)$}\n$\\varphi_n(y):=\\dt(q^{-n}x^{-1}y)$ for fixed $q\\neq 1$, and a\nparameter $x \\in \\BR$. Also, remember\n\\be\n\\lb_i:=D^i\\lb_0=\\lb(xq^i).\n\\ee\n\n\\begin{lemma} Then the Fourier transform,\n$$\nf\\longmapsto \\FR f=(...,f_n,...)_{n\\in \\BZ},\n$$\ninduces an algebra isomorphism $\\hat {}$, mapping $D$-operators\ninto $\\Lambda$-operators\n\\bea\n\\hat {}~ : \\DR_q &\\lrg & \\DR \\nonumber\\\\\n\\sum_i a_i(y)D^i &\\longmapsto &\n\\sum \\hat a_i \\Lb^i:=\\sum_i\\mbox{ diag}\\left(\n...,a_i(xq^n),...\\right)_{n\n\\in \\BZ}\\Lb^i.\n\\eea\nMoreover\n\\bea\n\\sum_{i=0}^n b_i(y) D_q^i=\\sum_{i=0}^n a_i(y)(-\\lb D)^i &\\longmapsto &\n\\vr \\left( \\sum_{i=0}^n \\hat a_i \\Lb^i \\right) \\vr^{-1} ,\n\\eea\nwhere the $\\Lb$-operator in brackets is monic,\nwith\\footnote{with $[j]:=\\frac{1-q^j}{1-q}$ and $\\bigl[ \\stackrel{n}{k}\n\\bigr]:=\\frac{[n]~[n-1]~...[n-k+1]}{[k]~[k-1]~...[1]}$}\n\\be\n\\hat\\lb=(...,\\lb_{-1}(x),\\lb_0(x),\\lb_1(x),...)=(...,D^{-1}\\lb,\\lb,D\\lb,...)\n\\ee\n\\be\n\\vr:=\\mbox{diag}~\\left(...,\\lb_{-2}\\lb_{-1},-\\lb_{-1},1,\n-\\frac{1}{\\lb_0},\\frac{1}{\\lb_0\\lb_1},\n-\\frac{1}{\\lb_0 \\lb_1 \\lb_2},...\\right)~~\\mbox{with}~ \\vr_0=1,\n\\ee\n\\be\na_i(y):=\\sum_{0\\leq k \\leq\nn-i}\\frac{\\bigl[\\stackrel{k+i}{k}\\bigr]}{(-y(q-1)q^i)^k} b_{k+i}(y).\n\\ee\n\\end{lemma}\n\n\\proof The operators $D$ and multiplication by a function\n$a(y)$ act on basis elements, as follows:\n$$\nD\\varphi_n(y)=\\varphi_{n-1}(y)~~\\mbox{ and }~~a(y)\n\\varphi_n(y)=a(xq^n)\\varphi_n(y).\n$$\nTherefore $D^k$ and $a(y)$ act on functions $f(y)$, as\n\\bea\nf(y)=\\sum_{n \\in \\BZ} f_n\\varphi_n(y) \\longmapsto D^kf(y)&=&\\sum_{n\n\\in \\BZ}f_n D^k \\varphi_n(y)\\nonumber\\\\\n&=& \\sum_{n \\in \\BZ}f_n \\varphi_{n-k}(y) \\nonumber\\\\\n&=& \\sum_{n \\in \\BZ}f_{n+k} \\varphi_{n}(y),\n\\eea\nand\n\\bea\nf(y)=\\sum_{n \\in \\BZ} f_n\\varphi_n(y) \\longmapsto a(y)f(y)&=&\\sum_{n\n\\in \\BZ}f_n a(y) \\varphi_n(y) \\nonumber\\\\\n&=&\\sum_{n\n\\in \\BZ}f_n a(xq^n) \\varphi_n(y),\n\\eea\nfrom which it follows that\n\\be\n (D^k)\\hat{} =\\Lb^k\n\\ee\n\\be\n\\hat a(y)=\\mbox{diag}~(...,a(xq^n),...)_{n \\in \\BZ}.\n\\ee\nTo establish the algebra isomorphism (5.8), one checks that\n\n\\bea\n\\left( a(y) D^i \\right)\\hat{} ~\\left( b(y) D^j \\right)\\hat{} &=&\\hat\na(y)\n\\Lb^i ~ \\hat b(y) \\Lb^j \\nonumber\\\\ &=&\\hat a(y) \\left( \\Lb^i \\hat\nb(y) \\Lb^{-i} \\right) \\Lb^{i+j}\n\\nonumber\\\\\n&=&\\mbox{diag}(...,a(xq^n)b(xq^{n+i}),...)_{n \\in \\BZ}\\Lb^{i+j}\n\\nonumber\\\\\n&=& \\left( a(y)b(yq^i) D^{i+j} \\right)\\hat{}\\nonumber\\\\\n&=&\\left( a(y) D^i ~~b(y) D^j \\right)\\hat{}.\n\\eea\nUsing the inductively established identity\n$$\nD_q^n=\\frac{1}{y^n (q-1)^n q^{\\frac{n(n-1)}{2}}}\\sum_{k=0}^n(-1)^k\nq^{k(k-1)/2}\\Bigl[\\stackrel{n}{k}\\Bigr] D^{n-k},\n$$\nthe first identity (5.9) is immediate.\n\nThen, using, by virtue of (5.10) and (5.11), $ \\hat \\lb \\Lb=-\\vr \\Lb\n\\vr^{-1}$ and\n$\\vr\n\\hat a\n\\vr^{-1}=\\hat a$ (since $\\hat a$ is diagonal), one computes, using\n the established isomorphism,\n\\bea\n \\left( a_i(y)( -\\lb(y) D)^i\\right)\\hat{}&=& \\hat a_i \\left(\n-\\hat\\lb\n\\hat D \\right)^i\n\\nonumber\\\\\n&=&\\hat a_i \\left( -\\hat \\lb \\Lb \\right)^i\\nonumber\\\\\n&=& \\hat a_i \\left(\\vr \\Lb \\vr^{-1} \\right)^i \\nonumber\\\\\n&=&\\vr \\left(\\hat a_i \\Lb^i \\right) \\vr^{-1}\n\\eea\nestablishing (5.9). \\qed\n\n\n\\noindent\\underline{\\em Proof of Theorem 5.1}:\nIndeed the Frenkel system (5.1) maps at once into (5.5), whereas,\nusing (5.9), the KLR-system maps into\n\\bea\n{\\pl \\vr L \\vr^{-1} \\over \\pl\nt_n}&=&\\bigl[\\left(\\vr L^n \\vr^{-1}\\right)_+,\\vr L \\vr^{-1} \\bigr] \\\\\n&=& \\vr \\bigl[\\left(L^n\\right)_+,L\\bigr] \\vr^{-1},\n\\eea\nwhich upon conjugation by $\\vr$ leads to (5.5) as well.\\qed\n\n\\bigbreak\n\n\\underline{\\sl Proof of Theorem 5.2}: From Theorem 0.3, it follows that\n$L$ with the\n$\\tau_n$'s defined by (5.6), satisfies the Toda lattice; the second\nequality in (5.6) follows from (5.4). According to Lemma 1.3,\n$$\n\\frac{p_{1-\\ell}(\\tilde\\pl)\\tau_{n+\\ell+1}\\circ\\tau_n}{\\tau_{n+\\ell+1}\n\\tau_n}=\\mbox{\\,a\npolynomial in\n$\\left(\\frac{\\pl^k}{\\pl\nt_{i_1}...\\pl t_{i_k}}\\log(\\tau_{n+\\ell+1}\\tau_n^{\\pi(k)})\\right)$},\n$$\nwhere by (5.6), for $k\\geq 2$,\n\n\\medbreak\\noindent\n$\\displaystyle{\\left(\\frac{\\pl^k}{\\pl\nt_{i_1}...\\pl t_{i_k}}\\log(\\tau_{n+\\ell+1}\\tau_n\n^{\\pi (k)})\\right)_{n\\in\\BZ}}$\n\\begin{eqnarray}\n&=&\\left(D^n\\frac{\\pl^k}{\\pl t_{i_1}...\\pl\nt_{i_k}}\\log\\left(\\tau(c(y)+t)^{\\pi(k)}D^{\\ell+1}\\tau(c(y)+t)\\right)\n\\right)_{n\\in\\BZ}\\\\\n&=&\\left(\\frac{\\pl^k}{\\pl\nt_{i_1}...\\pl\nt_{i_k}}\\log\\tau(c(y)+t)^{\\pi(k)}D^{\\ell+1}\n\\tau(c(y)+t)\\right)^{\\wedge},\n\\end{eqnarray}\nand\n\n$\\displaystyle{\\left(\\frac{\\pl}{\\pl\nt_j}\\log\\frac{\\tau_{n+\\ell+1}}{\\tau_n}\\right)_{n\\in\\BZ}}$\n\\begin{eqnarray}\n&=&\\left(\\frac{\\pl}{\\pl\nt_j}\\log\\frac{\\left(\n\\displaystyle{\\prod^{n+\\ell+1}_{\\al=1}e^{\\sum^{\\iy}_{i=1}}t_i\\lb^i_{\\al}}\n\\right)D^{n+\\ell+1}\\tau(c(y)+t)}{\\left(\\displaystyle{\n\\prod^n_{\\al=1}e^{\\sum^{\\iy}_{i=1}}t_i\\lb^i_{\\al}}\n\\right)D^n\\tau(c(y)+t)}\\right)_{n\\in\\BZ}\\\\\n&=&\\left(\\sum_{\\al=n+1}^{n+\\ell+1}\\lb^j_{\\al}(y)+\n\\frac{\\pl}{\\pl t_j}\n\\log\\frac{D^{n+\\ell+1}\\tau(c(y)+t)}{D^n\\tau(c(y)+t)}\n\\right)_{n\\in\\BZ}\\\\\n&=&\\left(D^n\\left(\\sum_{i=1}^{\\ell+1}\\lb^j_i(y)+\n\\frac{\\pl}{\\pl t_j}\n\\log\\frac{D^{\\ell+1}\\tau(c(y)+t)}{\\tau(c(y)+t)}\n\\right)\\right)_{n\\in\\BZ}\\\\\n&=&\\left(\\sum_{i=1}^{\\ell+1}\\lb^j_i(y)+\n\\frac{\\pl}{\\pl t_j}\n\\log\\frac{D^{\\ell+1}\\tau(c(y)+t)}{\\tau(c(y)+t)}\n\\right)^{\\wedge},\n\\end{eqnarray}\nestablishing Theorem 5.2.\\qed\n\n\n\n\\noindent\\underline{\\em Remark}: Note the $\\vr$-conjugation has no\ncounterpart in $\\DR_q$-world.\n\n\nDefining the simple $q$-vertex operators:\n$$\nX_q(x,t,z):=e_q^{xz}X(t,z)\\quad\\mbox{and}\\quad\\tilde X_q\n(x,t,z):=(e_q^{xz})^{-1}X(-t,z)\n$$\nin terms of the vertex operator (6.1) and the $q$-exponential\n$\\displaystyle{e^x_q=e^{\\sum_1^{\\iy}\\frac{(1-q)^kx^k}{k(1-q^k)}}}$ we\nnow state:\n\n\\begin{corollary}Any K-P $\\tau$-function leads to a\n$q$-K-P $\\tau$-function $\\tau(c(x)+t)$ satisfying\n$q$-bilinear relations below for all $x\\in\\BR$,\n$t,t'\\in\\BC^{\\iy}$ and all $n>m$, which tends to the\nstandard K-P bilinear identity when $q$ goes to 1:\n$$\n\\oint_{z=\\iy}D^n(X_q(x,t,z)\\tau(c(x)+t))D^{m+1}(\\tilde\nX_q(x,t',z)\\tau(c(x)+t')dz=0\n$$\n$$\n\\lrg\\int_{z=\\iy}X(t,z)\\tau(\\bar x+t)X(t',z)\\tau(\\bar x+t')dz=0.\n$$\n\\end{corollary}\n\n\\proof The $\\tau$-functions $\\tau_n$ defined in Theorem 5.2 satisfy\nthe usual bilinear identity (0.18), and so, using the following\nidentity\n\\begin{eqnarray}\n\\frac{z^{n-m-1}}{\\prod^n_{k=m+2}(-\\lb)^k}\\prod^n_{k=m+2}\ne^{-\\sum^{\\iy}_{i=1}\\frac{1}{i}\\left(\\frac{\\lb_k}{z}\\right)^i}&=&\n\\prod^n_{k=m+2}\\left(1-\\frac{z}{\\lb_k}\\right)\\\\\n&=&\\prod^n_{k=m+2}e^{-\\sum^{\\iy}_{i=1}\\frac{1}{i}\\left(\\frac{z}{\\lb_k}\n\\right)^i}\\\\\n&=&D^ne_q^{xz}D^{m+1}(e_q^{xz})^{-1}\n\\end{eqnarray}\nin computing $\\tau_n(t-[z^{-1}])$ in the usual bilinear identity\nyields, up to a multiplicative factor $\\al(\\lb,\\nu)$:\n\\begin{eqnarray}\n& &\\al(\\lb,\\nu)\\oint_{z=\\iy}\\tau_n(t-[z^{-1}])\\tau_{m+1}(t'+\n[z^{-1}])e^{\\sum_1^{\\iy}(t_i-t'_i)z^i}z^{n-m}\\frac{dz}{z}\\\\\n&=&\\oint_{z=\\iy}\\tau(c(x)+t-[z^{-1}]-\\sum^n_1[\\lb_i^{-1}])\\tau(c(x)+t'\n+[z^{-1}]+\\sum_1^{m+1}[\\lb_i^{-1}])\\\\\n& &\\quad\\quad\\quad\\quad\\prod^n_{k=m+2}\\left(1-\\frac{z}{\\lb_k}\n\\right)e^{\\sum^{\\iy}_1(t_i-t'_i)z^i}dz\\\\\n&=&\\oint_{z=\\iy}D^n(X_q(x,t,z)\\tau(c(x)+t))D^{m+1}(\\tilde\nX_q(x,t',z)\\tau(c(x)+t'))dz=0,\n\\end{eqnarray}\nthe latter tending as $q\\rg 1$ to the usual KP bilinear identity,\nupon using (5.3).\\qed\n\n\\begin{corollary}If we take $\\tau_0(t)=\\tau(c(x)+t)$\nin Theorem 5.2, with $\\tau(t)$ a $N$-KdV $\\tau$-function, i.e.,\n$\\displaystyle{\\pl\\tau / \\pl t_{iN}=0}$,\n$i=1,2,...$, then\n\\be\n(L^N)=(L^N)_+\\quad\\mbox{and}\\quad\\tilde L^N=(\\tilde L^N)_+\n\\ee\nyielding the $N$-Frenkel and $N$-KLR system:\n\\be\nQ^N=(Q^N)_+\\quad\\mbox{and}\\quad Q_q^N=(Q^N_q)_+.\n\\ee\nThe $q$-differential operator $Q^N_q$ has the form below and tends\nto the differential operator of the $N$-KdV hierarchy as $q$ goes\nto 1:\n\\bea\nQ^N_q=D^N_q&+&\\frac{\\pl}{\\pl\nt_1}\\log\\frac{\\tau(D^Nc+t)}{\\tau(c+t)}D^{N-1}_q\\nonumber\\\\\n&+&\\left(\\sum_{i=0}^{N-1}\\frac{\\pl^2}{\\pl\nt_1^2}\\log\\tau(D^ic+t)\\right.\\nonumber\\\\\n&-&\\sum_{i=0}^{N-2}\\lb_{i+1}\\left(\\frac{\\pl}{\\pl\nt_1}\\log\\frac{\\tau(D^Nc+t)}{\\tau(D^{N-1}c+t)}-\n\\frac{\\pl}{\\pl\nt_1}\\log\\frac{\\tau(D^{i+1}c+t)}{\\tau(D^ic+t)}\\right)\\nonumber\\\\\n&+&\\left.\\sum_{0\\leq i\\leq j\\leq N-2}\\frac{\\pl}{\\pl\nt_1}\\log\\frac{\\tau(D^{i+1}c+t)}{\\tau(D^ic+t)}\n\\frac{\\pl}{\\pl\nt_1}\\log\\frac{\\tau(D^{j+1}c+t)}{\\tau(D^jc+t)}\\right)D_q^{N-2}+...\n\\nonumber\\\\\n&\\lrg&\\left(\\frac{\\pl}{\\pl x}\\right)^N+N\\frac{\\pl^2}{\\pl\nt^2_1}\\log\\tau(\\bar x+t)\\left(\\frac{\\pl}{\\pl x}\\right)^{N-2}+...\n\\eea\n\\end{corollary}\n\n\\proof Note that for $W\\in Gr^{(0)}$, $z^NW\\subset W$ if and only\nif its tau function is of the form $e^{\\sum_1^{\\iy}\nc_{i}t_{iN}}\\tau(t)$, with $\\pl\\tau(t)/ \\pl t_{iN}=0$, $i=1,2,...$.\nThus by hypothesis, we have for each\n$$\nW_k=\\span\\{\\psi_k(t,z),\\psi_{k+1}(t,z),...\\}\n$$\n$z^NW_k\\subset W_k$ and since $L\\psi=z\\psi$,\n$$\nz^N\\psi_k=\\sum^{N-1}_{j=0}a_j\\psi_{k+j}+\\psi_{k+N}=(L^N\\psi)_k,\n$$\nand so $L^N$ is upper-triangular, yielding (5.21), which by the\nisomorphism $^{\\wedge}$ of Lemma 5.3 yields (5.22). From (0.13) and\nthe relationship between $a_i(y)$ and $b_i(y)$ given in (5.12),\ndeduce (5.23).\n\n\n\n\n\n\n\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{AvM1} M.~ Adler and P.~ van Moerbeke: {\\em Birkhoff\nstrata, B\\\"acklund transformations and limits of isospectral\noperators }, Adv. in Math., {\\bf 108} 140--204 (1994).\n\n\n\\bibitem{AvM2} M.~ Adler and P.~ van Moerbeke: {\\em Matrix\nintegrals, Toda symmetries, Virasoro constraints and orthogonal\npolynomials} Duke Math.J., {\\bf 80} (3), 863--911 (1995)\n\n\n\\bibitem{AvM4} M. ~Adler and P.~ van Moerbeke:\n{\\em Generalized\northogonal polynomials, discrete KP and\nRiemann-Hilbert problems }, Comm. Math. Phys. , {\\bf\n207}(3), 589--620 (1999)\n\n\n\n\\bibitem{AHV} M.~ Adler, E.~Horozov and P.~ van Moerbeke: {\\em\nThe solution to the $q$-KdV equation}, Phys. Letters A, {\\bf 242}\n139--151 (1998).\n\n\n\n\\bibitem{DJKM} E. Date, M. Jimbo, M. Kashiwara,\nT. Miwa: {\\em Transformation groups for soliton equations},\nProc. RIMS Symp. Nonlinear integrable systems, Classical\nand quantum theory (Kyoto 1981), pp. 39-119. Singapore~:\nWorld Scientific 1983.\n\n\n\\bibitem{F} E.~ Frenkel: {\\em Deformations of the KdV hierarchy\nand related soliton equations }, Int. Math. Res. Notices, {\\bf 2}\n55--76 (1996).\n\n\n\\bibitem{G} D.~ Gieseker: {\\em The Toda hierarchy and the KdV\nhierarchy }, preprint, alg-geom/9509006.%, {\\bf ?} ??--?? (1997).\n\n\n\\bibitem{HI} L.~ Haine and P. ~Iliev: {\\em The bispectral\nproperty of a $q$-deformation of the Schur polynomials and the\n$q$-KdV hierarchy }, J. of Phys. A: Math. Gen, {\\bf 30}, 7217-7227\n(1997).\n\n\\bibitem{I} P. ~Iliev: {\\em Tau-function solutions to a\n$q$-deformation of the KP-hierarchy}, preprint (1997).\n\n\n\\bibitem{KLR} B.~ Khesin, V.~ Lyubashenko and C.~Roger: {\\em\nExtensions and contractions of the Lie algebra of\n$q$-Peudodifferential symbols on the circle }, J. of functional\nanalysis, {\\bf 143} 55--97 (1997).\n\n\\bibitem{K} B. A.~ Kupershmidt: {\\em Discrete Lax equations and\ndifferential-difference calculus }, Ast\\'erisque, {\\bf 123}\n(1985).\n\n\n\n\\bibitem{SW} G. ~Segal, G. ~Wilson: {\\em Loop groups and equations\nof KdV type}, Publ. Math. IHES {\\bf 61}, 5--65 (1985).\n\n\n\n\\bibitem{TT} K.~Takasaki, T. ~Takebe: {\\em Integrable\nhierarchies and dispersionless limit}, Reviews in Math. Phys.\n{\\bf 7},743--808 (1995) .\n\n\n\\bibitem{UT} K.~Ueno, K.~Takasaki: {\\em Toda Lattice\nHierarchy}, Adv. Studies in Pure Math. {\\bf 4},1-95 (1984) .\n\n\\bibitem{vM} P.~van Moerbeke: Integrable foundations of string\ntheory, in Lecures on Integrable systems, Proceedings of the\nCIMPA-school, 1991, Ed.: O. Babelon, P. Cartier, Y.\nKosmann-Schwarzbach, World scientific, pp 163--267 (1994).\n\n\n\\end{thebibliography}\n\n\n\n\n\\end{document}\n" } ]	[ { "name": "solv-int9912014.extracted_bib", "string": "{AvM1 M.~ Adler and P.~ van Moerbeke: {\\em Birkhoff strata, B\\\"acklund transformations and limits of isospectral operators , Adv. in Math., {108 140--204 (1994)." }, { "name": "solv-int9912014.extracted_bib", "string": "{AvM2 M.~ Adler and P.~ van Moerbeke: {\\em Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials Duke Math.J., {80 (3), 863--911 (1995)" }, { "name": "solv-int9912014.extracted_bib", "string": "{AvM4 M. ~Adler and P.~ van Moerbeke: {\\em Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems , Comm. Math. Phys. , {207(3), 589--620 (1999)" }, { "name": "solv-int9912014.extracted_bib", "string": "{AHV M.~ Adler, E.~Horozov and P.~ van Moerbeke: {\\em The solution to the $q$-KdV equation, Phys. Letters A, {242 139--151 (1998)." }, { "name": "solv-int9912014.extracted_bib", "string": "{DJKM E. Date, M. Jimbo, M. Kashiwara, T. Miwa: {\\em Transformation groups for soliton equations, Proc. RIMS Symp. Nonlinear integrable systems, Classical and quantum theory (Kyoto 1981), pp. 39-119. Singapore~: World Scientific 1983." }, { "name": "solv-int9912014.extracted_bib", "string": "{F E.~ Frenkel: {\\em Deformations of the KdV hierarchy and related soliton equations , Int. Math. Res. Notices, {2 55--76 (1996)." }, { "name": "solv-int9912014.extracted_bib", "string": "{G D.~ Gieseker: {\\em The Toda hierarchy and the KdV hierarchy , preprint, alg-geom/9509006.%, {? ??--?? (1997)." }, { "name": "solv-int9912014.extracted_bib", "string": "{HI L.~ Haine and P. ~Iliev: {\\em The bispectral property of a $q$-deformation of the Schur polynomials and the $q$-KdV hierarchy , J. of Phys. A: Math. Gen, {30, 7217-7227 (1997)." }, { "name": "solv-int9912014.extracted_bib", "string": "{I P. ~Iliev: {\\em Tau-function solutions to a $q$-deformation of the KP-hierarchy, preprint (1997)." }, { "name": "solv-int9912014.extracted_bib", "string": "{KLR B.~ Khesin, V.~ Lyubashenko and C.~Roger: {\\em Extensions and contractions of the Lie algebra of $q$-Peudodifferential symbols on the circle , J. of functional analysis, {143 55--97 (1997)." }, { "name": "solv-int9912014.extracted_bib", "string": "{K B. A.~ Kupershmidt: {\\em Discrete Lax equations and differential-difference calculus , Ast\\'erisque, {123 (1985)." }, { "name": "solv-int9912014.extracted_bib", "string": "{SW G. ~Segal, G. ~Wilson: {\\em Loop groups and equations of KdV type, Publ. Math. IHES {61, 5--65 (1985)." }, { "name": "solv-int9912014.extracted_bib", "string": "{TT K.~Takasaki, T. ~Takebe: {\\em Integrable hierarchies and dispersionless limit, Reviews in Math. Phys. {7,743--808 (1995) ." }, { "name": "solv-int9912014.extracted_bib", "string": "{UT K.~Ueno, K.~Takasaki: {\\em Toda Lattice Hierarchy, Adv. Studies in Pure Math. {4,1-95 (1984) ." }, { "name": "solv-int9912014.extracted_bib", "string": "{vM P.~van Moerbeke: Integrable foundations of string theory, in Lecures on Integrable systems, Proceedings of the CIMPA-school, 1991, Ed.: O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, World scientific, pp 163--267 (1994)." } ]
solv-int9912015		[]	We discuss some algebraic aspects of the integrable vector non-linear Schr\"{odinger hierarchies (GNLS$_{r$). These are hierarchies of zero-curvature equations constructed from affine Kac-Moody algebras $\hat{sl_{r+1$. Using the dressing transformation method and the tau-function formalism, we construct the N-soliton solutions of the GNLS$_{r$ systems. The explicit matrix elements in the case of GNLS$_{1$ are computed using level one vertex operator representations.	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W_\\infty}}\n\\def\\Win1{{\\sf W_{1+\\infty}}}\n\\def\\hWinf{{\\sf {\\hat W}_{\\infty}}}\n\\def\\Rm#1#2{r(\\vec{#1},\\vec{#2})} % r-operator kernel\n\\def\\OR#1{{\\cal O}(R_{#1})} % R-coadjoint orbit\n\\def\\ORti{{\\cal O}({\\widetilde R})} % R-tilde-coadjoint orbit\n\\def\\AdR#1{Ad_{R_{#1}}} % R-adjoint group action\n\\def\\dAdR#1{Ad_{R_{#1}^{\\ast}}} % R-coadjoint group action\n\\def\\adR#1{ad_{R_{#1}^{\\ast}}} % R-coadjoint algebra action\n\\def\\KP{${\\rm \\, KP\\,}$} %% KP\n\\def\\KPl{${\\rm \\,KP}_{\\ell}\\,$} %% KP_l\n\\def\\KPo{${\\rm \\,KP}_{\\ell = 0}\\,$} %% KP_l=0\n\\def\\mKPa{${\\rm \\,KP}_{\\ell = 1}\\,$} %% modified KP-1\n\\def\\mKPb{${\\rm \\,KP}_{\\ell = 2}\\,$} %% modified KP-2\n%\n% fake blackboard bold macros for reals, complex, etc.\n\\def\\rlx{\\relax\\leavevmode}\n\\def\\inbar{\\vrule height1.5ex width.4pt depth0pt}\n\\def\\IZ{\\rlx\\hbox{\\sf Z\\kern-.4em Z}}\n\\def\\IR{\\rlx\\hbox{\\rm I\\kern-.18em R}}\n\\def\\IC{\\rlx\\hbox{\\,$\\inbar\\kern-.3em{\\rm 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Theorem.}\\quad}\n\\def\\name{\\noindent{\\bf Definition.}\\quad}\n\\def\\exam{\\noindent{\\bf Example.}\\quad}\n\\def\\proof{\\noindent{\\bf Proof.}\\quad}\n%%\n%\n% This defines the journal citations\n%\n\\def\\PRL#1#2#3{{\\sl Phys. Rev. Lett.} {\\bf#1} (#2) #3}\n\\def\\NPB#1#2#3{{\\sl Nucl. Phys.} {\\bf B#1} (#2) #3}\n\\def\\NPBFS#1#2#3#4{{\\sl Nucl. Phys.} {\\bf B#2} [FS#1] (#3) #4}\n\\def\\CMP#1#2#3{{\\sl Commun. Math. Phys.} {\\bf #1} (#2) #3}\n\\def\\PRD#1#2#3{{\\sl Phys. Rev.} {\\bf D#1} (#2) #3}\n\\def\\PRE#1#2#3{{\\sl Phys. Rev.} {\\bf E#1} (#2) #3}\n\\def\\PRv#1#2#3{{\\sl Phys. Rev.} {\\bf #1} (#2) #3}\n\\def\\PLA#1#2#3{{\\sl Phys. Lett.} {\\bf #1A} (#2) #3}\n\\def\\PLB#1#2#3{{\\sl Phys. Lett.} {\\bf #1B} (#2) #3}\n\\def\\JMP#1#2#3{{\\sl J. Math. Phys.} {\\bf #1} (#2) #3}\n\\def\\PTP#1#2#3{{\\sl Prog. Theor. Phys.} {\\bf #1} (#2) #3}\n\\def\\SPTP#1#2#3{{\\sl Suppl. Prog. Theor. Phys.} {\\bf #1} (#2) #3}\n\\def\\AoP#1#2#3{{\\sl Ann. of Phys.} {\\bf #1} (#2) #3}\n\\def\\PNAS#1#2#3{{\\sl Proc. Natl. Acad. Sci. USA} {\\bf #1} (#2) #3}\n\\def\\RMP#1#2#3{{\\sl Rev. Mod. Phys.} {\\bf #1} (#2) #3}\n\\def\\PR#1#2#3{{\\sl Phys. Reports} {\\bf #1} (#2) #3}\n\\def\\AoM#1#2#3{{\\sl Ann. of Math.} {\\bf #1} (#2) #3}\n\\def\\UMN#1#2#3{{\\sl Usp. Mat. Nauk} {\\bf #1} (#2) #3}\n\\def\\FAP#1#2#3{{\\sl Funkt. Anal. Prilozheniya} {\\bf #1} (#2) #3}\n\\def\\FAaIA#1#2#3{{\\sl Functional Analysis and Its Application} {\\bf #1} (#2)\n#3}\n\\def\\BAMS#1#2#3{{\\sl Bull. Am. Math. Soc.} {\\bf #1} (#2) #3}\n\\def\\TAMS#1#2#3{{\\sl Trans. Am. Math. Soc.} {\\bf #1} (#2) #3}\n\\def\\InvM#1#2#3{{\\sl Invent. Math.} {\\bf #1} (#2) #3}\n\\def\\LMP#1#2#3{{\\sl Letters in Math. Phys.} {\\bf #1} (#2) #3}\n\\def\\IJMPA#1#2#3{{\\sl Int. J. Mod. Phys.} {\\bf A#1} (#2) #3}\n\\def\\AdM#1#2#3{{\\sl Advances in Math.} {\\bf #1} (#2) #3}\n\\def\\RMaP#1#2#3{{\\sl Reports on Math. Phys.} {\\bf #1} (#2) #3}\n\\def\\IJM#1#2#3{{\\sl Ill. J. Math.} {\\bf #1} (#2) #3}\n\\def\\APP#1#2#3{{\\sl Acta Phys. Polon.} {\\bf #1} (#2) #3}\n\\def\\TMP#1#2#3{{\\sl Theor. Mat. Phys.} {\\bf #1} (#2) #3}\n\\def\\JPA#1#2#3{{\\sl J. Physics} {\\bf A#1} (#2) #3}\n\\def\\JSM#1#2#3{{\\sl J. Soviet Math.} {\\bf #1} (#2) #3}\n\\def\\MPLA#1#2#3{{\\sl Mod. Phys. Lett.} {\\bf A#1} (#2) #3}\n\\def\\JETP#1#2#3{{\\sl Sov. Phys. JETP} {\\bf #1} (#2) #3}\n\\def\\CAG#1#2#3{{\\sl Commun. Anal\\&Geometry} {\\bf #1} (#2) #3}\n\\def\\JETPL#1#2#3{{\\sl Sov. Phys. JETP Lett.} {\\bf #1} (#2) #3}\n\\def\\PHSA#1#2#3{{\\sl Physica} {\\bf A#1} (#2) #3}\n\\def\\PHSD#1#2#3{{\\sl Physica} {\\bf D#1} (#2) #3}\n\\def\\PJA#1#2#3{{\\sl Proc. Japan. Acad.} {\\bf #1A} (#2) #3}\n\\def\\JPSJ#1#2#3{{\\sl J. Phys. Soc. Japan} {\\bf #1} (#2) #3}\n\\def\\SJPN#1#2#3{{\\sl Sov. J. Part. Nucl.} {\\bf #1} (#2) #3}\n%%%\n\n\\begin{document}\n\\begin{titlepage}\n\\vspace{-1cm}\n\n\n\\vspace{.2in}\n\\begin{center}\n{\\large\\bf Vector NLS hierarchy solitons revisited: dressing transformation and tau function approach}\n\\end{center}\n\n\\vspace{1in}\n\n\\begin{center}\nHarold Blas \n \n\n\\vspace{.5 cm}\n\\small\n\n\n\n\\par \\vskip .1in \\noindent\nInstituto de F\\'\\i sica Te\\'orica - IFT/UNESP\\\\\nRua Pamplona 145\\\\\n01405-900 S\\~ao Paulo-SP, BRAZIL\n\n\n\n\\normalsize\n\\end{center}\n\n\\vspace{1.5in}\n\n\\begin{abstract}\nWe discuss some algebraic aspects of the integrable vector non-linear Schr\\\"{o}dinger hierarchies (GNLS$_{r}$). These are hierarchies of zero-curvature equations constructed from affine Kac-Moody algebras $\\hat{sl}_{r+1}$. Using the dressing transformation method and the tau-function formalism, we construct the N-soliton solutions of the GNLS$_{r}$ systems. The explicit matrix elements in the case of GNLS$_{1}$ are computed using level one vertex operator representations. \n\\end{abstract}\n\\end{titlepage}\n\\section{Introduction}\nIt is well known that the $1+1$-dimensional non-linear Schr\\\"{o}dinger ({\\bf NLS}) equation is integrable and possesses exact soliton solutions \\ct{zakharov}. It has been known that many soliton equations in $1+1$ dimensions have integrable matrix generalizations, or more generally, integrable multi-component generalizations. The most well-known example for the multi-component case is the vector {\\bf NLS} equation, first studied by Manakov \\ct{manakov}. These type of equations and their higher-order generalizations find applications in non-linear optics (for a complete review of the most important references in the field see \\ct{nls}). The multi-soliton type solutions of these hierarchies can be obtained using diverse methods. For example, in \\ct{zen} the vector {\\bf NLS} equation has been studied in the framework of the inverse scattering method, the bright and dark multi-soliton solutions and their collisions have been studied. \n\nOne of the most fascinating applications of the Kac-Moody theory and its affine Lie algebras and their relevant groups is to exhibit hidden symmetries of soliton equations. According to the approach of \\ct{ferreira} a common feature of integrable hierarchies presenting soliton solutions is the existence of some special ``vacuum solutions'' such that the Lax operators evaluated on them lie in some abelian subalgebra of the associated Kac-Moody algebra. The soliton type solutions are constructed out of those ``vacuum solutions'' through the so called dressing transformation procedure. These developments lead to a quite general definition of tau functions associated to the hierarchies, in terms of the so called ``integrable highest weight representations'' of the relevant Kac-Moody algebra.\n \nIn this paper we obtain the multi-soliton solutions of the vector {\\bf NLS} equation using the dressing transformation method. We believe that the group theoretical point of view of finding the analitical results for the general case of $N$-soliton interactions could facilitate the study of their properties; for example, the asymptotic behaviour of trains of $N$ solitonlike pulses with approximately equal amplitudes and velocities, as studied in \\ct{kaup1}. The second point we shoul highlight relies upon the possible relevance of the {\\bf NLS} tau functions to its higher-order generalizations. We believe that the tau functions of the higher-order {\\bf NLS} generalization are related somehow to the basic tau functions of the usual (vector) {\\bf NLS} equations (this fact is observed for example in the case of the coupled {\\bf NLS}$+${\\bf DNLS} system \\ct{blas, liu}, in the second Ref. Hirota's method has been used).\n\nThe plan is as follows. In section $2$ we review the theory of the dressing transformations and the definition of the tau-function vectors. In section 3 we present the construction of the vector {\\bf NLS} equations (GNLS$_{r}$) associated to the homogeneous gradation of the Kac-Moody algebra $\\hat{sl}_{r+1}$, their relevant tau functions are defined and the construction of multiple-soliton solutions uotlined. In section 4 we present a detailed study of the GNLS$_{1}$ case; the first conserved charges are constructed in the context of this formalism and the explicit form of the $N$-soliton solutions are presented. Finally, for the sake of completeness, we have included Kac-Moody algebra notations and conventions, as well as, its well known theory of integrable highest weight representations, and level one homogeneous vertex operator representations (see appendices \\ref{appa}, \\ref{appb}, \\ref{appc}). Some of the details regarding the matrix elements appear in the appendices \\ref{appd}, \\ref{appe} and \\ref{appf}. \n\n\n\\section{Dressing Transformations}\n\nConsider a non linear system which can be formulated in terms of a system of\nfirst-order differential equations \n\\begin{equation}\n{\\cal L}_N{\\bf \\Psi =}0, \\lab{dr0}\n\\end{equation}\nwhere ${\\cal L}_N$ are Lax operators of the form \n\\begin{equation}\n\\qquad {\\cal L}_N=\\frac \\partial {\\partial _{t_N}}-A_N \\lab{dr1}\n\\end{equation}\nand the variables $t_N$ are the various ``times'' of the hierarchy.\n\nThen, the equations of the hierarchy are equivalent to the integrability or\nzero curvature conditions of \\rf{dr0}, \n\\begin{equation}\n\\llbrack {\\cal L}_N\\, ,\\, {\\cal L}_M \\rrbrack = 0. \\lab{dr2}\n\\end{equation}\n\nTherefore the Lax operators are of the form of a pure gauge\n\n\\begin{equation}\nA_N=\\frac{\\partial {\\bf \\Psi }}{\\partial _{t_N}}{\\bf \\Psi }^{-1}. \\lab{dr3}\n\\end{equation}\n\nThe type of integrable hierarchy considered here is based on a Kac-Moody\nalgebra $\\widehat{g}$ furnished with an integer gradation labelled by a\nvector $s=(s_o,s_1,...,s_r)$ of $r+1$ non-negative co-prime integers such\nthat\n\n\\br\n\\widehat{g}=\\bigoplus_{i\\in \\IZ}\\widehat{g}_i(s) \\qquad \\mbox{and}\\qquad \n\\llbrack \\widehat{g}_i(s)\\, ,\\, \\widehat{g}_j(s)\\rrbrack \\subseteq \\widehat{g}_{i+j}(s).\n\\lab{dr4}\n\\er\n\nThe connections we consider are of the form\n\n\\begin{equation}\nA_N=\\sum_{i=0}^lA_{N,i}\\qquad \\mbox{where}\\qquad A_{N,i}\\in \\widehat{g}_i(s).\n\\lab{dr5}\n\\end{equation}\n\nThe ``dressing transformations'' are non local gauge transformations on $%\nA_N $ which maintain their form and gradation \\ct{ferreira, babelon}.\nEach of these gauge transformations is made with the help of two group\nelements $\\Theta _{+}$ and $\\Theta _{-}$ , such that \n\\begin{equation}\nA_N\\rightarrow A_N^h\\equiv \\Theta _{\\pm }A_N\\Theta _{\\pm }^{-1}+\\partial\n_N\\Theta _{\\pm }\\Theta _{\\pm }^{-1} \\lab{dr6}\n\\end{equation}\nor \n\\begin{equation}\nA_N^h=\\frac{\\partial (\\Theta _{\\pm }{\\bf \\Psi )}}{\\partial _{t_N}}(\\Theta\n_{\\pm }{\\bf \\Psi )}^{-1} \\lab{dr7}\n\\end{equation}\n\nWe have a residual gauge transformation in (\\ref{dr7})\n\n\\begin{equation}\n{\\bf \\Psi \\rightarrow \\Psi }h, \\lab{dr8}\n\\end{equation}\nwhere $h$ is a constant group element. Therefore we can impose\n\\begin{equation}\n\\Theta _{+}{\\bf \\Psi }=\\Theta _{-}{\\bf \\Psi }h \\label{dr9}\n\\end{equation}\nor equivalently\n\\begin{equation}\n\\Theta _{-}^{-1}\\Theta _{+}={\\bf \\Psi }h{\\bf \\Psi }^{-1} \\label{dr10}\n\\end{equation}\n\n$\\Theta _{-}\\Psi $ \\,\\, defines a new solution\n\\begin{equation}\n{\\bf \\Psi }^h=\\Theta _{-}{\\bf \\Psi } \\label{dr11}\n\\end{equation}\n\nWe admit a Gauss decomposition with respect to the gradation\n\n\\begin{equation}\n{\\bf \\Psi }h{\\bf \\Psi }^{-1}=\\left( {\\bf \\Psi }h{\\bf \\Psi }^{-1}\\right)\n_{-}\\left( {\\bf \\Psi }h{\\bf \\Psi }^{-1}\\right) _0\\left( {\\bf \\Psi }h{\\bf %\n\\Psi }^{-1}\\right) _{+} \\label{dr12}\n\\end{equation}\nWe choose (see (\\ref{dr10}))\n\n\\begin{equation}\n\\Theta _{-}=(\\left( {\\bf \\Psi }h{\\bf \\Psi }^{-1}\\right) _{-})^{-1}\n\\label{dr13}\n\\end{equation}\nand therefore (\\ref{dr11}) can be written as \n\\begin{equation}\n{\\bf \\Psi }^h=(\\left( {\\bf \\Psi }h{\\bf \\Psi }^{-1}\\right) _{-})^{-1}{\\bf %\n\\Psi =}\\Theta _{+}{\\bf \\Psi }h^{-1}=\\left( {\\bf \\Psi }h{\\bf \\Psi }%\n^{-1}\\right) _0\\left( {\\bf \\Psi }h{\\bf \\Psi }^{-1}\\right) _{+}{\\bf \\Psi }%\nh^{-1} \\label{dr14}\n\\end{equation}\nwhere we used (\\ref{dr9}) and (\\ref{dr10}). $\\Psi ^h$ in (\\ref{dr14}) is also a\nsolution of the linear problem (\\ref{dr1}).\n\nWe shall consider solutions which belong to the orbits of the vacuum\nsolutions.\n\nDefine\n\\br\n\\Theta _{-}^{-1}=\\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left(\nvac\\right) -1}\\right) _{-},\\quad B^{-1}=\\left( {\\bf \\Psi }^{\\left(\nvac\\right) }h{\\bf \\Psi }^{\\left( vac\\right) -1}\\right) _0 \n\\er\n\\begin{equation}\nN=\\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right)\n-1}\\right) _{+}\\quad \\mbox{and}\\quad \\Omega =B^{-1}N. \\label{dr15}\n\\end{equation}\n\nTherefore the dressing transformations generated by $h$ becomes \n\\begin{equation}\n{\\bf \\Psi }^{\\left( vac\\right) }\\rightarrow {\\bf \\Psi }^h=\\Theta _{-}{\\bf %\n\\Psi }^{\\left( vac\\right) }=\\Omega {\\bf \\Psi }^{\\left( vac\\right) }h^{-1}.\n\\label{dr16}\n\\end{equation}\n\nDenote by $\\mid \\lambda _i>$ the state of highest weight of a fundamental\nrepresentation such that $s_i\\neq 0$.\n\nDefine the tau-function vector \\ct{ferreira}\n\\begin{eqnarray}\n\\tau _i(t^{\\pm }) &=&\\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }%\n^{\\left( vac\\right) -1}\\right) \\left\| \\lambda _i\\right\\rangle \\nonumber \\\\\n&=&\\Theta _{-}^{-1}B^{-1}\\left\| \\lambda _i\\right\\rangle ,\\qquad \\qquad\ni=0,1,...,r;\\qquad s_i\\neq 0. \\label{dr17}\n\\end{eqnarray}\n\nNote that $N\\mid \\lambda _i>=\\mid \\lambda _i>$ and \n\\begin{equation}\n\\tau _i^{\\left( 0\\right) }(t^{\\pm })=B^{-1}\\left\| \\lambda _i\\right\\rangle\n=\\left\| \\lambda _i\\right\\rangle \\widehat{\\tau }_i^{\\left( 0\\right) }(t^{\\pm\n}), \\label{dr18}\n\\end{equation}\n{\\bf \\ }\nsince $\\mid \\lambda _i>$ is an eigenstate of the sublagebra $\\widehat{g}\n_o(s) $. Then\n\\begin{equation}\n\\widehat{\\tau }_i^{\\left( 0\\right) }(t^{\\pm })=\\left\\langle \\lambda\n_o\\right\| \\left[ {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left(\nvac\\right) -1}\\right] _{\\left( o\\right) }\\left\| \\lambda _o\\right\\rangle ,\n\\label{dr19}\n\\end{equation}\n\nUsing (\\ref{dr17}) we obtain \n\\begin{equation}\n\\Theta _{-}^{-1}\\left\| \\lambda _i\\right\\rangle =\\frac{\\tau _i(t^{\\pm })}{%\n\\widehat{\\tau }_i^{\\left( 0\\right) }(t^{\\pm })} \\label{dr20}\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The $\\tau $ function and the N soliton solution for GNLS$_r$}\n\n\n\nThe GNLS$_r$ model is constructed for example in \\ct{fordy}, where it was studied in the framework of the Zakharov-Shabat formalism and the context of hermitian symmetric spaces. In \\ct{aratyn} an affine Lie algebraic foundation (loop algebra $g\\otimes {\\mathbf C} [\\lambda,\\lambda^{-1}]$ of $g$) for GNLS$_r$ was given.\nHere instead we will consider the full Kac-Moody algebra $\\widehat{sl}(r+1)$\nwith homogeneous gradation\n\\br\ns=(1,\\stackrel{r-zeros}{\\overbrace{0,0,...,0}}),\n\\er\nand a semisimple element $E^{(l)}$.\n\nThe connections are given by \n\\br\nA_1\\equiv A&=&E^{\\left( 1\\right) }+\\sum_{i=1}^r\\Psi _i^{+}E_{\\beta _i}^{\\left(\n0\\right) }+\\sum_{i=1}^r\\Psi _i^{-}E_{-\\beta _i}^{\\left( 0\\right) }+\\nu\n_1C,\\\\\n\\nonu\nA_2\\equiv B&=&E^{\\left( 2\\right) }+\\sum_{i=1}^r\\Psi _i^{+}E_{\\beta _i}^{\\left(\n1\\right) }+\\sum_{i=1}^r\\Psi _i^{-}E_{-\\beta _i}^{\\left( 1\\right)\n}+\\sum_{i=1}^r\\partial _x\\Psi _i^{+}E_{\\beta _i}^{\\left( 0\\right) }-\\\\\n&&\\sum_{i=1}^r\\partial _x\\Psi _i^{-}E_{-\\beta _i}^{\\left( 0\\right)\n}-\\sum_{i=1}^r\\Psi _i^{+}\\Psi _i^{-}\\beta _i.H^{\\left( 0\\right)\n}-\\sum_{i,j=1}^r\\Psi _i^{+}\\Psi _i^{-}\\epsilon \\left( \\beta _i,\\beta\n_j\\right) E_{\\beta _i-\\beta _j}^{\\left( 0\\right) }+\\nu _2C, \\label{dr21}\n\\er\nwhere $\\Psi _i^{+}$, $\\Psi _i^{-}$, $\\nu _1$ and $\\nu _2$ are the fields of\nthe model.\n\nWe have\n\\begin{equation}\n\\quad E^{\\left( l\\right) }=2\\frac{\\mu _r.H^{\\left( l\\right) }}{\\alpha _r^2}%\n,\\quad \\left[ D,E^{\\left( l\\right) }\\right] =lE^{\\left( l\\right) }\n\\label{dr22}\n\\end{equation}\nwhere $\\mu _r$ is a fundamental weight and $\\alpha _{a}$ are the\nsimple roots of the $sl(r+1)$ algebra. The $\\beta _i$ are the positive roots\ndefined by\n\\begin{equation}\n\\beta _i=\\alpha _i+\\alpha _{i+1}+...+\\alpha _r,\\quad \\mbox{with}\\quad \\alpha\n_i^2=2, \\label{dr23}\n\\end{equation}\nand $H_i$ being the generators of the cartan subalgebra in the Weyl-Cartan\nbasis. We need also some relations in the Chevalley basis \n\\begin{eqnarray}\nE &=&\\frac 1{r+1}\\left( \\sum_{a=1}^ra{\\bf H}_a^{\\left( o\\right) }\\right)\n,\\quad \\mbox{with}\\quad {\\bf H}_a=\\alpha _a.H^{\\left( o\\right) } \\nonumber\n\\\\\n\\mu _r &=&\\frac 1{r+1}\\left( \\sum_{a=1}^ra\\alpha _a\\right) ,\\quad \\beta _a.H=%\n{\\bf H}_a+...+{\\bf H}_r,\\quad a=1,...,r \\label{dr24}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\left[ E^{\\left( m\\right) },{\\bf H}_a^{\\left( n\\right) }\\right] &=&\\frac\nm{r+1}\\sum_{a=1}^ra\\eta _{ab}C\\delta _{m,-n}, \\\\\n\\left[ E^{\\left( m\\right) },E_{\\pm \\beta _i}^{\\left( n\\right) }\\right]\n&=&\\pm E_{\\pm \\beta _i}^{\\left( m+n\\right) }, \\\\\n\\left[ {\\bf H}_a^{\\left( m\\right) },E_{\\pm \\beta _i}^{\\left( n\\right)\n}\\right] &=&\\pm \\left( \\sum_{b=i}^rK_{ba}\\right) E_{\\pm \\beta _i}^{\\left(\nm+n\\right) },\n\\end{eqnarray}\nwhere \n\\br\n\\eta _{ab}=\\frac 2{\\alpha _2^2}K_{ab}=\\eta _{ba}\\qquad \\mbox{and}\\qquad\nK_{ab}=2\\frac{\\alpha _a\\cdot \\alpha _b}{\\alpha _b^2}\\cdot \n\\er\n\nThe potentials are in the subspaces \n\\begin{equation}\nA_1\\,\\in\\, \\widehat{g}_o(s)+\\widehat{g}_1(s),\\quad A_2\\,\\in\\, \\widehat{g}_o(s)+%\n\\widehat{g}_1(s)+\\widehat{g}_2(s) \\label{dr25}\n\\end{equation}\n\nThe zero curvature condition\\, $[\\partial _t$ $-B$ $,\\partial _x$ $-$ $A]=0$\\,\ngives the following system of equations\n\\br\n\\partial _t\\Psi _i^{+} &=&\\partial _x^2\\Psi _i^{+}-2\\left( \\sum_{j=1}^r\\Psi\n_j^{+}\\Psi _j^{-}\\right) \\Psi _i^{+}, \\nonumber \\\\\n\\label{dr26}\n\\partial _t\\Psi _i^{-} &=&-\\partial _x^2\\Psi _i^{-}+2\\left( \\sum_{j=1}^r\\Psi\n_j^{+}\\Psi _j^{-}\\right) \\Psi _i^{-}, \\\\\n\\nonu\n\\qquad \\qquad \\qquad \\partial _t\\nu _1-\\partial _x\\nu _2 &=&0. \n\\er\n\nThe system of equations for the $\\Psi^{\\pm}_{j}$ fields in \\rf{dr26}, supplied with a convenient complexification of the time variable and the fields, are related to the well known integrable vector non-linear Schr\\\"{o}dinger equation (vector {\\bf NLS}) \\ct{manakov, zen, faddeev}.\n\nLet us now study the vacuum solutions and dressing transformations. Let us\nnote that $\\Psi _i^{\\pm }=\\nu _1=$ $\\nu _2=0$ is a solution of\nequations (\\ref{dr26}). Therefore from (\\ref{dr21}) we have the connections \n\\begin{equation}\nA_1^{\\left( vac\\right) }\\equiv A^{\\left( vac\\right) }=E^{\\left( 1\\right)\n},\\quad A_2^{\\left( vac\\right) }\\equiv B^{\\left( vac\\right) }=E^{\\left(\n2\\right) }, \\label{dr27}\n\\end{equation}\n\nThese connections can be obtained from\\, $A_N^{(vac)}=\\partial_{t_N}{\\bf \\Psi } {\\bf \\Psi }^{-1}$\\, from the group element\n\n\\begin{equation}\n{\\bf \\Psi }^{\\left( vac\\right) }=e^{xE^{\\left( 1\\right) }+tE^{\\left(\n2\\right) }+X}, \\label{dr28}\n\\end{equation}\nwhere \n\\begin{equation}\nX=\\sum_{n=3}^{+\\infty }t_nE^{(n)},\\,\\,\\,t_n\\,\\,\\mbox{ are real parameters.}\n\\label{dr29}\n\\end{equation}\n\nThe connections in the vacuum orbit are given by \n\\begin{eqnarray}\nA &=&\\Theta _{-}E^{\\left( 1\\right) }\\Theta _{-}^{-1}+\\partial _x\\Theta\n_{-}\\Theta _{-}^{-1}\\qquad \\label{dr30} \\\\\n&=&M^{-1}NE^{\\left( 1\\right) }N^{-1}M-M^{-1}\\partial _xM+M^{-1}\\partial\n_xNN^{-1}M, \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\nB &=&\\Theta _{-}E^{\\left( 2\\right) }\\Theta _{-}^{-1}+\\partial _t\\Theta\n_{-}\\Theta _{-}^{-1} \\label{dr31} \\\\\n&=&M^{-1}NE^{\\left( 2\\right) }N^{-1}M-M^{-1}\\partial _tM+M^{-1}\\partial\n_tNN^{-1}M, \\nonumber\n\\end{eqnarray}\nwith \n\\br\n\\Theta _{-}=\\exp \\left( \\sum_{n>0}\\sigma _{-n}\\right) ,\\quad {\\em \\ }M={\\em %\n\\exp }\\left( \\sigma _o\\right) ,\\quad N=\\exp \\left( \\sum_{n>0}\\sigma\n_n\\right) \n\\er\n\nIn the present case the gradation operator is\\, $Q_s=D$\\, with\n\\begin{equation}\n\\left[ D,\\sigma _n\\right] =n\\sigma _n. \\label{dr32}\n\\end{equation}\n\nWe can relate the fields $\\Psi _i^{\\pm }$, $\\nu _1$ and $\\nu _2$ with some\nof the components of $\\sigma _n$. We have \n\\begin{equation}\nA=E^{\\left( 1\\right) }+\\left[ \\sigma _{-1},E^{\\left( 1\\right) }\\right] +%\n\\mbox{ terms of negative grade.\\qquad } \\label{dr33}\n\\end{equation}\n\n\\br\n=M^{-1}\\left( E^{\\left( 1\\right) }-\\partial _xM.M^{-1}+\\partial _x\\sigma\n_1\\right) M+\\mbox{ terms of grade}>1 \n\\er\n\n\\begin{equation}\nB=E^{\\left( 2\\right) }+\\left[ \\sigma _{-1},E^{\\left( 2\\right) }\\right]\n+\\left[ \\sigma _{-2},E^{\\left( 2\\right) }\\right] +\\frac 12\\left[ \\sigma\n_{-1},\\left[ \\sigma _{-1},E^{\\left( 2\\right) }\\right] \\right] + \\label{dr33.1}\n\\end{equation}\n\\[\n\\mbox{terms of negative grade} \n\\]\n\\[\n=M^{-1}\\left( E^{\\left( 2\\right) }-\\partial _tM.M^{-1}+\\partial _t\\sigma\n_1+\\partial _t\\sigma _2+\\left[ \\sigma _1,\\partial _t\\sigma _1\\right] \\right)\nM+ \n\\]\n\\[\n\\mbox{terms of grade }>2 \n\\]\n\nIn (\\ref{dr33}) the term of degree $-1$ must vanish and therefore \n\\begin{equation}\n\\partial _x\\sigma _{-1}+\\left[ \\sigma _{-2},E^{\\left( 1\\right) }\\right]\n+\\frac 12\\left[ \\sigma _{-1},\\left[ \\sigma _{-1},E^{\\left( 1\\right) }\\right]\n\\right] =0, \\label{dr34}\n\\end{equation}\n\nDenote (consistently with (\\ref{dr33})) \n\\begin{equation}\n\\sigma _{-1}=-\\sum_{i=1}^r\\Psi _i^{+}E_{\\beta _i}^{\\left( -1\\right)\n}+\\sum_{i=1}^r\\Psi _i^{-}E_{-\\beta _i}^{\\left( -1\\right)\n}+\\sum_{a=1}^r\\sigma _{-1}^a{\\bf H}_a^{\\left( -1\\right) }, \\label{dr35}\n\\end{equation}\n\\[\n\\sigma _{-2}=\\sum_{i=1}^r\\sigma _{-2}^{+i}E_{\\beta _i}^{\\left( -2\\right)\n}+\\sum_{i=1}^r\\sigma _{-2}^{-i}E_{-\\beta _i}^{\\left( -2\\right)\n}+\\sum_{a=1}^r\\sigma _{-2}^a{\\bf H}_a^{\\left( -2\\right) }, \n\\]\nand therefore from (\\ref{dr34}) we obtain \n\\begin{eqnarray}\n\\quad \\quad \\partial _x\\sigma _{-1}^a &=&\\sum_{i=1}^r\\Psi _i^{+}\\Psi\n_i^{-},\\quad a,i=1,...r. \\nonumber \\\\\n\\sigma _{-2}^{+i} &=&-\\partial _x\\Psi _i^{+}+\\frac 12\\sum_{a=1}^r\\sigma\n_{-1}^a\\Psi _i^{+}\\left( \\sum_{b=i}^rK_{ba}\\right) , \\label{dr36} \\\\\n\\sigma _{-2}^{-i} &=&-\\partial _x\\Psi _i^{+}+\\frac 12\\sum_{a=1}^r\\sigma\n_{-1}^a\\Psi _i^{-}\\left( \\sum_{b=i}^rK_{ba}\\right) , \\nonumber\n\\end{eqnarray}\n\nSubstitution of $\\sigma _{-1}$, $\\sigma _{-2}$ in (\\ref{dr33}) and (\\ref{dr33.1}%\n) we obtain \n\\[\n\\nu _1=-\\frac 1{r+1}\\sum_{a,b=1}^ra.\\eta _{ab}\\sigma _{-1}^b, \n\\]\n\\begin{equation}\n\\nu _2=-\\frac 2{r+1}\\sum_{a,b=1}^ra.\\eta _{ab}\\sigma _{-2}^b. \\label{dr37}\n\\end{equation}\n\nThe $\\sigma _{-n}$ 's with the higher gradations are used to cancel out the\nundesired componentes.\n\nOne of the tau-function vectors is given by \n\\begin{equation}\n\\tau _o\\left( x,t\\right) =\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }%\n^{\\left( 0\\right) -1}\\right] \\left\| \\lambda _o\\right\\rangle \\label{dr38}\n\\end{equation}\n\\begin{equation}\n=\\Theta _{-}^{-1}M^{-1}\\left\| \\lambda _o\\right\\rangle , \\label{dr39}\n\\end{equation}\nwhere $h$ is a particular constant element of $\\widehat{sl}(r+1)$.\n\nThen we have \n\\begin{equation}\n\\exp \\left( -\\sum_{n>0}\\sigma _{-n}\\right) {\\em \\exp }\\left( -\\sigma\n_o\\right) \\left\| \\lambda _o\\right\\rangle =\\left[ {\\bf \\Psi }^{\\left(\n0\\right) }h{\\bf \\Psi }^{\\left( 0\\right) -1}\\right] \\left\| \\lambda\n_o\\right\\rangle . \\label{dr40}\n\\end{equation}\n\nWe want to express the fields $\\Psi _i^{\\pm }$ in terms of some tau\nfunctions, which are some matrix elements in a appropriate representation of \n$\\widehat{sl}(r+1)$.\n\nWe can write down \n\\begin{equation}\n\\sigma _o=\\sum_{a=1}^r\\sigma _o^{+a}E_{\\alpha _a}^{\\left( 0\\right)\n}+\\sum_{a=1}^r\\sigma _o^{-a}E_{-\\alpha _a}^{\\left( 0\\right)\n}+\\sum_{a=1}^r\\sigma _o^a{\\bf H}_a^{\\left( 0\\right) }+\\eta C \\label{dr41}\n\\end{equation}\nor\n\\begin{equation}\n\\sigma _o=\\sum_{i=1}^r\\sigma _o^{+i}.e_i+\\sum_{i=1}^r\\sigma\n_o^{-i}.f_i+\\sum_{a=1}^r\\sigma _o^a.h_a+\\eta C, \\label{dr42}\n\\end{equation}\nwhere $e_i$ and $f_i$ ($i=0...r$) are the generators in the Chevalley basis\nand \\{ $h_i$ , $D$\\} generates the Cartan subalgebra.\n\nWe have (see Appendix \\ref{appa})\n\\begin{equation}\nh_i\\left\| \\lambda _o\\right\\rangle =0,\\quad f_i\\left\| \\lambda _o\\right\\rangle\n=0,\\quad e_i\\left\| \\lambda _o\\right\\rangle =0\\quad \\mbox{and \\quad }C\\left\|\n\\lambda _o\\right\\rangle =\\left\| \\lambda _o\\right\\rangle ,\\quad \\quad\ni=1,2,...r \\label{dr43}\n\\end{equation}\n\nTherefore the zero gradation of (\\ref{dr40}) is \n\\begin{equation}\n{\\em \\exp }\\left( -\\sigma _o\\right) \\left\| \\lambda _o\\right\\rangle =\\left[ \n{\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right) -1}\\right]\n_{\\left( o\\right) }\\left\| \\lambda _o\\right\\rangle , \\label{dr44}\n\\end{equation}\nand the left hand side can be written as \n\\begin{equation}\n{\\em \\exp }\\left( -\\sigma _o\\right) \\left\| \\lambda _o\\right\\rangle =\\left\|\n\\lambda _o\\right\\rangle \\widehat{\\tau }^{\\left( o\\right) }\\left( x,t\\right)\n\\label{dr45}\n\\end{equation}\nwith $\\widehat{\\tau }^{(0)}$ a real function given by the matrix element \n\\begin{equation}\n\\widehat{\\tau }^{\\left( o\\right) }\\left( x,t\\right) =\\left\\langle \\lambda\n_o\\right\| \\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] _{\\left( o\\right) }\\left\| \\lambda _o\\right\\rangle . \\label{dr46}\n\\end{equation}\n\nThen the term with grade (-1) in (\\ref{dr40}) is \n\\begin{equation}\n-\\sigma _{-1}\\left\| \\lambda _o\\right\\rangle =\\frac{\\left[ {\\bf \\Psi }%\n^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right) -1}\\right] _{\\left(\n-1\\right) }\\left\| \\lambda _o\\right\\rangle }{\\widehat{\\tau }^{\\left( o\\right)\n}\\left( x,t\\right) } \\label{dr47}\n\\end{equation}\nor \n\\[\n\\left( -\\sum_{i=1}^r\\Psi _i^{+}E_{\\beta _i}^{\\left( -1\\right)\n}+\\sum_{i=1}^r\\Psi _i^{-}E_{-\\beta _i}^{\\left( -1\\right)\n}+\\sum_{a=1}^r\\sigma _{-1}^a{\\bf H}_a^{\\left( -1\\right) }\\right) \\left\|\n\\lambda _o\\right\\rangle = \n\\]\n\\begin{equation}\n-\\frac{\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] _{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle }{\\widehat{%\n\\tau }^{\\left( o\\right) }\\left( x,t\\right) }, \\label{dr48}\n\\end{equation}\n\nUsing the commutation rules for the relevant Kac-Moody algebra elements we\nhave \n\\begin{equation}\n\\Psi _i^{+}=\\frac{\\tau _i^{+}}{\\widehat{\\tau }^{\\left( o\\right) }}\\quad \n\\mbox{and}\\quad \\Psi _i^{-}=-\\frac{\\tau _i^{-}}{\\widehat{\\tau }^{\\left(\no\\right) }}\\quad , \\label{dr49}\n\\end{equation}\nwhere the $tau$ functions are defined by \n\\begin{equation}\n\\tau _i^{+}\\equiv \\left\\langle \\lambda _o\\right\| E_{-\\beta _i}^{\\left(\n1\\right) }\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] _{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle , \\label{dr50}\n\\end{equation}\n\\begin{equation}\n\\tau _i^{-}\\equiv \\left\\langle \\lambda _o\\right\| E_{\\beta _i}^{\\left(\n1\\right) }\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] _{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle , \\label{dr51}\n\\end{equation}\nand \n\\begin{equation}\n\\widehat{\\tau }^{\\left( o\\right) }\\equiv \\left\\langle \\lambda _o\\right\|\n\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] _{\\left( o\\right) }\\left\| \\lambda _o\\right\\rangle .\\qquad \\quad\n\\label{dr52}\n\\end{equation}\n\nIn order to obtain the first non trivial solutions of (\\ref{dr26}) let us\nconsider \n\\begin{equation}\nh=e^{aF_j},\\quad F_j=\\sum_{n=-\\infty }^{+\\infty }\\nu _j^nE_{-\\beta\n_j}^{\\left( -n\\right) },\\quad \\mbox{where }\\nu _j\\mbox{ and }a\\mbox{ are\nreal parameters.} \\label{dr53}\n\\end{equation}\nwith $F_j$ being an eigenvector under the adjoint action of the generator $E^{(n)}$, that is \n\\br\n\\left[ xE^{\\left( 1\\right) }+tE^{\\left( 2\\right) }+X,F_j\\right] =-\\left[ \\nu\n_j\\left( x+\\nu _jt\\right) +\\overline{\\nu }_j\\right] F_j, \\label{dr54}\n\\er\nwhere \n\\[\n\\overline{\\nu }_j=\\sum_{n=3}^{+\\infty }z_n\\nu _j^n; \n\\]\ndenoting \\,$\\varphi _j=\\nu _j\\left( x+\\nu _jt\\right) +\\overline{\\nu }_j$\\, one\ncan write \n\\begin{equation}\n\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] =\\exp \\left( e^{-\\varphi _j}aF_j\\right) \\label{dr55}\n\\end{equation}\n\\[\n\\qquad \\qquad \\qquad =1+e^{-\\varphi _j}aF_j, \n\\]\nwhere we have used the fact that $F_j^n=0,$ for $n\\geq 2$, that is, $F_j$\nare nilpotent (see Appendix \\ref{appb} for more details)\n\nThe tau function $\\widehat{\\tau }^{\\left( o\\right) }$ becomes \n\\[\n\\widehat{\\tau }^{\\left( o\\right) }=\\left\\langle \\lambda _o\\right\| \\left(\n1+e^{-\\varphi _j}aF_j\\right) \\left\| \\lambda _o\\right\\rangle =1,\\quad \n\\]\nsince \n\\[\n\\left\\langle \\lambda _o\\right\| E_{-\\beta _i}^{\\left( o\\right) }\\left\|\n\\lambda _o\\right\\rangle =0. \n\\]\nFor the tau function $\\tau _i^{+}$ we have \n\\begin{eqnarray}\n\\tau _i^{+} &=&\\left\\langle \\lambda _o\\right\| \\left[ E_{-\\beta _i}^{\\left(\n1\\right) }\\exp \\left( e^{-\\varphi _j}.a\\sum_{n=-\\infty }^{+\\infty }\\nu\n_j^nE_{-\\beta _i}^{\\left( -n\\right) }\\right) \\right] _{\\left( o\\right)\n}\\left\| \\lambda _o\\right\\rangle \\\\\n&=&\\left\\langle \\lambda _o\\right\| \\left[ E_{-\\beta _i}^{\\left( 1\\right)\n}e^{-\\varphi _j}.a\\sum_{n=-\\infty }^{+\\infty }\\nu _j^nE_{-\\beta _i}^{\\left(\n-n\\right) }\\right] _{\\left( o\\right) }\\left\| \\lambda _o\\right\\rangle ,\n\\end{eqnarray}\nas $\\left[ E_{-\\beta _i}^{\\left( 1\\right) },E_{-\\beta\n_j}^{\\left( -n\\right) }\\right] =0$ \\quad and $\\quad E_{-\\beta\n_i}^{\\left( 1\\right) }\\left\| \\lambda _o\\right\\rangle =0,$\\,\\, it follows\n\n\\[\n\\tau _i^{+}=0. \n\\]\n\nNow \n\\[\n\\tau _i^{-}=\\left\\langle \\lambda _o\\right\| \\left[ E_{\\beta _i}^{\\left(\n1\\right) }e^{-\\varphi _j}.a\\sum_{n=-\\infty }^{+\\infty }\\nu _j^nE_{-\\beta\n_i}^{\\left( -n\\right) }\\right] _{\\left( o\\right) }\\left\| \\lambda\n_o\\right\\rangle , \n\\]\nif $i\\neq j$ the last expression becomes \n\\[\n\\tau _i^{-}=\\left\\langle \\lambda _o\\right\| E_{\\beta _i-\\beta _j}^{\\left(\n0\\right) }e^{-\\varphi _j}.a\\nu _j\\left\| \\lambda _o\\right\\rangle =0, \n\\]\nthen \n\\[\n\\tau _i^{-}=a\\delta _{ij}e^{-\\varphi _j}.\\nu _j\\left\\langle \\lambda\n_o\\right\| E_{\\beta _i}^{\\left( 1\\right) }E_{-\\beta _i}^{\\left( -1\\right)\n}\\left\| \\lambda _o\\right\\rangle ; \n\\]\nthe matrix element can be written as \n\\[\n\\left\\langle \\lambda _o\\right\| \\left\\{ \\sum_{a=1}^r{\\bf H}_a^{\\left(\n0\\right) }+C+E_{-\\beta _i}^{\\left( -1\\right) }E_{\\beta _i}^{\\left( 1\\right)\n}\\right\\} \\left\| \\lambda _o\\right\\rangle . \n\\]\nThen it follows \n\\[\n\\tau _i^{-}=\\delta _{ij}.a.\\nu _{j.}e^{-\\varphi _j}, \n\\]\nand therefore we obtain the solution \n\\begin{equation}\n\\Psi _i^{+}=0,\\qquad \\Psi _i^{-}=-\\delta _{ij}.a.\\nu _{j.}e^{-\\varphi _j}.\n\\label{dr6.41}\n\\end{equation}\n\nWe can also make the choice \n\\[\nh=e^{b.G_j},\\mbox{ with }G_j=\\sum_{n=-\\infty }^{+\\infty }\\rho _j^nE_{\\beta\n_i}^{\\left( -n\\right) },\\quad \\rho _j\\mbox{ being a real\nparameter.} \n\\]\nwith $G_j$ a new eigenvector of the adjoint action of $E^{(l)}.$\n\\[\n\\left[ xE^{\\left( 1\\right) }+tE^{\\left( 2\\right) }+X\\,,\\,G_j\\right] =\\left( \\rho\n_j\\left( x+\\rho _jt\\right) +\\overline{\\rho }_j\\right) G_j, \n\\]\nwith \n\\[\n\\overline{\\rho }_j=\\sum_{n=3}^{+\\infty }z_n\\rho _j^n. \n\\]\n\nAs in the previous case \n\\[\n\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] =\\exp \\left( e^{\\eta _j}bG_j\\right) \n\\]\n\\begin{equation}\n\\qquad \\qquad \\qquad =1+e^{\\eta _j}bG_j, \\label{dr6.43}\n\\end{equation}\nwhere $G_j^n=0$ for $n\\geq 2$. Denoting\\, $\\eta _j=\\rho _j\\left( x+\\rho\n_jt\\right) +\\overline{\\rho }_j$,\\, and performing a similar calculations as in the previous case, one gets\n\\[\n\\widehat{\\tau }^{\\left( o\\right) }=1,\\qquad \\tau _i^{-}=0\\quad \\mbox{and\n\\quad }\\tau _i^{+}=\\delta _{ij}.b.\\rho _{j.}e^{\\eta _j}, \n\\]\nand therefore we have a new solution of the form \n\\begin{equation}\n\\Psi _i^{-}=0,\\qquad \\Psi _i^{+}=\\delta _{ij}.b.\\rho _{j.}e^{\\eta _j}.\n\\label{dr6.44}\n\\end{equation}\n\nConsider now the product \n\\begin{equation}\nh=e^{a_{j_1}F_{j_1}}e^{b_{j_2}G_{j_2}},\\quad j_1,j_2=1,2,...r, \\label{6.45}\n\\end{equation}\nwhere \n\\[\nF_{j_1}=\\sum_{n=-\\infty }^{+\\infty }\\nu _{j_1}^nE_{-\\beta _{j_1}}^{\\left(\n-n\\right) },\\,\\,\\,\\,G_{j_2}=\\sum_{n=-\\infty }^{+\\infty }\\rho\n_{j_2}^nE_{\\beta _{j_2}}^{\\left( -n\\right) };\\,\\,\\,\\,a_{j_1}, %\nb_{j_2}\\mbox{ real parameters.} \n\\]\n\nLet us note that $F_{j_1}$ and $G_{j_2}$ are associated to positive and\nnegative roots respectively.\n\nTherefore \n\\begin{eqnarray}\n\\nonu\n\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] &=&\\left( 1+e^{-\\varphi _{j_1}}a_{j_1}F_{j_1}\\right) \\left(\n1+e^{\\eta _{j_2}}b_{j_2}G_{j_2}\\right) \\\\\n&=&1+e^{-\\varphi _{j_1}}a_{j_1}F_{j_1}+e^{\\eta\n_{j_2}}b_{j_2}G_{j_2}+a_{j_1}b_{j_2}e^{-\\varphi _{j_1}}e^{\\eta\n_{j_2}}F_{j_1}G_{j_2},\n\\label{dr6.46}\n\\end{eqnarray}\nwith $\\varphi _{j_1}=\\nu_{j_1}\\left( x+\\nu _{j_1}t\\right) +\n\\overline{\\nu }_{j_1}$\\quad and \\quad $\\eta _{j_2}=\\rho _{j_2}\\left(\nx+\\rho _{j_2}t\\right) +\\overline{\\rho }_{j_2}$.\nThe corresponding tau function are \n\\begin{equation}\n\\widehat{\\tau }^{\\left( o\\right) }=1+\\delta\n_{j_1,j_2}a_{j_1}b_{j_2}C_{j_1,j_2}e^{-\\varphi _{j_1}}e^{\\eta _{j_2}},\\qquad\nC_{j_1,j_2}=\\frac{\\nu _{j_1}.\\rho _{j_2}}{\\left( \\nu _{j_1}-\\rho\n_{j_2}\\right) ^2}\\quad , \\label{dr6.47}\n\\end{equation}\n\n\\begin{equation}\n\\tau _i^{+}=\\left\\langle \\lambda _o\\right\| E_{-\\beta _i}^{\\left( 1\\right)\n}b_{j_2}e^{\\eta _{j_2}}G_{j_2}\\left\| \\lambda _o\\right\\rangle \\label{dr6.48}\n\\end{equation}\n\\[\n\\quad =\\delta _{i,j_2}b_{j_2}\\rho _{j_2}e^{\\eta _{j_2}},\\qquad \\qquad \n\\]\nand \n\\begin{equation}\n\\tau _i^{-}=\\delta _{i,j_2}a_{j_1}\\nu _{j_1}e^{-\\varphi _{j_1}}.\\qquad\n\\qquad \\qquad \\label{dr6.49}\n\\end{equation}\n\nIn Appendix \\ref{appd} we outline the form of the corresponding matrix elements. We therefore obtain \n\\begin{equation}\n\\Psi _i^{+}=\\frac{\\delta _{i,j_2}b_{j_2}\\rho _{j_2}e^{\\eta _{j_2}}}{1+\\delta\n_{j_1,j_2}a_{j_1}b_{j_2}C_{j_1,j_2}e^{-\\varphi _{j_1}}e^{\\eta _{j_2}}},\\quad\n\\Psi _i^{-}=-\\frac{\\delta _{i,j_2}a_{j_1}\\nu _{j_1}e^{-\\varphi _{j_1}}}{%\n1+\\delta _{j_1,j_2}a_{j_1}b_{j_2}C_{j_1,j_2}e^{-\\varphi _{j_1}}e^{\\eta\n_{j_2}}}\\ \\cdot \\label{dr6.50}\n\\end{equation}\n\nIf $j_{1\\neq }j_2$ in (\\ref{dr6.47}) we recover the solution (\\ref{dr6.41}) and (%\n\\ref{dr6.44}). Therefore in order to have {\\sl one-soliton} solutions we must have $%\nj_1=j_2=i$ and therefore a solution of the system of equations \\rf{dr26} is \n\\begin{equation}\n\\Psi _i^{+}=\\frac{b_i\\rho _ie^{\\eta _i}}{1+a_ib_iC_{i,i}e^{-\\varphi\n_i}e^{\\eta _i}},\\quad \\Psi _i^{-}=-\\frac{a_i\\nu _ie^{-\\varphi _i}}{%\n1+a_ib_iC_{i,i}e^{-\\varphi _i}e^{\\eta _i}}\\quad \\cdot \\label{dr6.51}\n\\end{equation}\n\nIn order to study the $N$-soliton solution consider the following group\nelement \n\\begin{equation}\nh=e^{a_1F_{i_1}}...e^{a_NF_{i_N}}e^{b_1G_{j_1}}...e^{b_NG_{j_N}},\n\\label{dr6.52}\n\\end{equation}\nwhere \n\\[\nF_{j_l}=\\sum_{n=-\\infty }^{+\\infty }\\nu _l^nE_{-\\beta _{j_l}}^{\\left(\n-n\\right) },\\quad G_{j_l}=\\sum_{n=-\\infty }^{+\\infty }\\rho _l^nE_{\\beta\n_{j_l}}^{\\left( -n\\right) },\\quad l=1,2,...N;\\quad \\,\\,i_l,j_l=1,2,...r; \n\\]\n\n\\[\n\\quad \\,\\,a_l,\\quad b_l,\\quad \\,\\,\\nu _l\\quad \\mbox{and \\quad }\\rho\n_l\\mbox{ \\quad are real parameters.} \n\\]\nwhere\n\\[\n\\varphi _l=\\nu _l\\left( x+\\nu _lt\\right) +\\overline{\\nu }_l\\quad ,\\quad \\eta\n_l=\\rho _l\\left( x+\\rho _lt\\right) +\\overline{\\rho }_l\\quad ,l=1,2,...,N. \n\\]\n\nThen\n\\begin{eqnarray}\n\\left[ {\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\right] &=&\\left( 1+e^{-\\varphi _1}a_1F_{i_1}\\right) ...\\left(\n1+e^{-\\varphi _N}a_NF_{i_N}\\right) \\left( 1+e^{\\eta _1}b_1G_{j_1}\\right) ...\n\\nonumber \\\\\n&&\\left( 1+e^{\\eta _N}b_NG_{j_N}\\right) , \\label{dr6.53}\n\\end{eqnarray}\n\nUsing (\\ref{dr50}), (\\ref{dr51}) and (\\ref{dr52}) we calculate the corresponding tau functions. Denoting \n\\begin{equation}\nA_{i_l}\\equiv a_l\\nu _l.e^{-\\varphi _l},\\quad B_{j_l}\\equiv b_{l_l}\\rho\n_le^{\\eta _l}, \\label{dr6.54}\n\\end{equation}\nwe have \n\\[\n\\widehat{\\tau }^{\\left( o\\right) }=\\left\\langle \\lambda _o\\right\| \\left( \n{\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right) -1}\\right)\n\\left\| \\lambda _o\\right\\rangle \n\\]\n\n\\[\n\\widehat{\\tau }^{\\left( o\\right) }=1+\\qquad \\qquad {\\bf \\qquad \\quad } \n\\]\n\n\\br\n\\nonu\n&&\\left\\langle \\lambda _o\\right\| {\\bf \\{}\\sum_{n=1}^N \\sum_{1\\leq\nl_1<l_2<...<l_N \\leq N}\\left( A_{i_{l_1}}...A_{i_{l_n}}\\right) \\left(\nF_{i_{l_1}}...F_{i_{l_n}}\\right) \\sum_{1\\leq k_1<k_2<...<k_n\\leq N}\\left(\nB_{j_{k_1}}...B_{j_{k_n}}\\right) \\\\\n&&\\left( G_{j_{k_1}}...G_{j_{k_n}}\\right){\\bf \\}} \\left\| \\lambda\n_o\\right\\rangle\n\\er\n\n\\br\n=1+\\sum_{n=1}^N\\, \\sum_{1\\leq l_1<l_2<...<l_n \\leq N,\\,\\, 1\\leq k_1<k_2<...<k_n\\leq N} C_{i_{l_1}...i_{l_n},\nj_{k_1}...j_{k_n}}A_{i_{l_1}}...A_{i_{l_n}}B_{j_{k_1}}...B_{j_{k_n}},\n\\label{dr6.55}\n\\er\nwhere the coefficients are given by the matrix elements \n\\begin{equation}\nC_{i_{l_1}...i_{l_n},j_{k_1}...j_{k_n}}=\\left\\langle \\lambda _o\\right\|\nF_{i_{l_1}}...F_{i_{l_n}}G_{j_{k_1}}...G_{j_{k_n}}\\left\| \\lambda\n_o\\right\\rangle , \\label{dr6.56}\n\\end{equation}\n\nSimilarly \n\\[\n\\tau _i^{\\pm }=\\left\\langle \\lambda _o\\right\| E_{\\mp \\beta _i}^{\\left(\n1\\right) }{\\bf \\Psi }^{\\left( 0\\right) }h{\\bf \\Psi }^{\\left( 0\\right)\n-1}\\left\| \\lambda _o\\right\\rangle \n\\]\n\\br\n&=&\\left\\langle \\lambda _o\\right\| E_{\\mp \\beta _i}^{\\left( 1\\right) }\\left(\n1+e^{-\\varphi _1}a_1F_{i_1}\\right) ...\\left( 1+e^{-\\varphi\n_N}a_NF_{i_N}\\right) \\left( 1+e^{\\eta _1}b_1G_{j_1}\\right) ... \\nonumber \\\\\n&&\\left( 1+e^{\\eta _N}b_NG_{j_N}\\right) \\left\| \\lambda _o\\right\\rangle ,\n\\label{dr6.57}\n\\er\n\nTherefore we obtain \n\\br\n\\tau _i^{+}=\\sum_{n=0}^{N-1}\\,\\,\\sum_{1\\leq l_1<l_2<...<l_n\\leq N,\\,\\, 1\\leq\nk_1<k_2<...<k_{n+1}\\leq N } \nC_{ii_{l_1}...i_{l_n},j_{k_1}...j_{k_{n+1}}}^{+}A_{i_{l_1}}...A_{i_{l_n}}B_{j_{k_1}}...B_{j_{k_{n+1}}},\n\\label{dr6.58}\n\\er\n\\br\n\\tau _i^{-}=\\sum_{n=0}^{N-1}\\,\\,\\sum_{ 1\\leq l_1<l_2<...<l_{n+1}\\leq N,\\,\\,\n1\\leq k_1<k_2<...<k_n \\leq N }\nC_{i_{l_1}...i_{l_{n+1}},ij_{k_1}...j_{k_n}}^{-}A_{i_{l_1}}...A_{i_{l_n}}B_{j_{k_1}}...B_{j_{k_{n+1}}},\n\\label{dr6.59}\n\\er\nwith the matrix elements given by \n\\begin{equation}\nC_{ii_{l_1}...i_{l_n},j_{k_1}...j_{k_{n+1}}}^{+}=\\left\\langle \\lambda\n_o\\right\| E_{-\\beta _i}^{\\left( 1\\right)\n}F_{i_{l_1}}...F_{i_{l_n}}G_{j_{k_1}}...G_{j_{k_{n+1}}}\\left\| \\lambda\n_o\\right\\rangle , \\label{dr6.60}\n\\end{equation}\n\n\\begin{equation}\nC_{i_{l_1}...i_{l_{n+1}},ij_{k_1}...j_{k_n}}^{-}=\\left\\langle \\lambda\n_o\\right\| E_{\\beta _i}^{\\left( 1\\right)\n}F_{i_{l_1}}...F_{i_{l_n}}G_{j_{k_1}}...G_{j_{k_n}}\\left\| \\lambda\n_o\\right\\rangle .\\qquad \\label{dr6.61}\n\\end{equation}\n\nThe calculation of the corresponding matrix elements is outlined in Appendix \\ref{appd}.\n\n\n\n\n\n\n\\section{The example of GNLS$_1$}\n\nThe hierarchy GNLS$_1$ has a Lax operator \n\\begin{equation}\nL=\\partial _x-E^{\\left( 1\\right) }-\\Psi ^{+}E_{+}^{\\left( 0\\right) }-\\Psi\n^{-}E_{-}^{\\left( 0\\right) }-\\nu _1C, \\label{7.1}\n\\end{equation}\nwhere $\\Psi ^{\\pm }$ and $\\nu _1$ are the fields of the model.\n\nThe corresponding $\\widehat{sl}(2)$ algebra in the Weyl Cartan basis is \n\\br\n\\left[ H^{\\left( m\\right) },H^{\\left( n\\right) }\\right] &=&\\frac n2C\\delta\n_{m+n,0}, \\nonumber \\\\\n\\left[ H^{\\left( n\\right) },E_{\\pm }^{\\left( m\\right) }\\right] &=&\\pm E_{\\pm\n}^{\\left( m+n\\right) }, \\label{7.2} \\\\\n\\left[ E_{+}^{\\left( m\\right) },E_{-}^{\\left( n\\right) }\\right]\n&=&2H^{\\left( m+n\\right) }+nC\\delta _{m+n,0}. \\nonumber\n\\er\n\nThe equations of the hierarchy are obtained as follows \n\\[\n\\frac{\\partial L}{\\partial t_N}=\\left[ B_N,L\\right] ,\\qquad N>0 \n\\]\nwhere \n\\[\nB_N=\\left( UH^{\\left( N\\right) }U^{-1}\\right) _{\\geq 0}\\in C^\\infty \\left( \n\\IR,\\,\\widehat{g}_{\\geq o}\\left( s\\right) \\right) , \n\\]\n\n\\[\nB_N\\subset \\bigoplus_{i=0}^N\\widehat{g}_i, \n\\]\nwith $U$\\ being a group element obtained by exponentiating the negative\ndegree elements\n\n\\[\nU=\\exp \\left( \\sum_{n>1}T^{\\left( -n\\right) }\\right) ,\\quad \\left[\nD,T^{\\left( n\\right) }\\right] =nT^{\\left( n\\right) }. \n\\]\nThe first two $B_N$ are \n\\br\nB_1&=&H^{\\left( 1\\right) }+\\Psi ^{+}E_{+}^{\\left( 0\\right) }+\\Psi\n^{-}E_{-}^{\\left( 0\\right) }+\\nu _1C, \\label{7.3}\n\\\\\nB_2 &=&H^{\\left( 2\\right) }+\\Psi ^{+}E_{+}^{\\left( 1\\right) }+\\Psi\n^{-}E_{-}^{\\left( 1\\right) }-2\\Psi ^{+}\\Psi ^{-}H^{\\left( 0\\right)\n}+\\partial _x\\Psi ^{+}E_{+}^{\\left( 0\\right) }- \\nonu \\\\\n&&\\partial _x\\Psi ^{-}E_{-}^{\\left( 0\\right) }+\\nu _2C, \\label{7.5}\n\\er\nand the first equations of the hierarchy are \n\\br\n\\partial _{t_1}L &=&\\llbrack B_1, L\\rrbrack :\\nonu \\\\\n\\partial _{t_1}\\Psi ^{\\pm } &=&\\partial _x\\Psi ^{\\pm }, \\label{7.6}\\\\\n\\partial _{t_1}\\nu _1 &=&\\partial _x\\nu _1. \\nonu\n\\er\nand\n\\br\n\\partial _{t_2}L &=&\\llbrack B_2,L\\rrbrack : \\nonumber \\\\\n\\partial _{t_2}\\Psi ^{\\pm } &=&\\pm \\partial _x^2\\Psi ^{\\pm }\\mp 2.\\left(\n\\Psi ^{+}\\Psi ^{-}\\right) \\Psi ^{\\pm }, \\lab{sch} \\\\\n\\partial _{t_2}\\nu _1 &=&\\partial _x\\nu _2. \\nonu \\label{7.8}\n\\er\n\nThe system of equations for the $\\Psi^{\\pm}$ fields in \\rf{sch}, supplied with a convenient complexification of the time variable and the fields, are related to the well known non-linear Schr\\\"{o}dinger equation ({\\bf NLS}) \\ct{kac, leznov}. \nThe zero curvature condition for $B_1$ and $B_N$ can be written as \n\\begin{equation}\n\\llbrack \\partial _{t_N}-B_N,\\partial _x-B_1\\rrbrack =0,\\quad N=1,2,...\n\\label{7.9}\n\\end{equation}\nwhere $B_N$ has the general form \n\\begin{equation}\nB_N=H^{\\left( N\\right) }+\\sum_{n=0}^{N-1}B_N^{\\left( n\\right) },\\quad \\mbox{%\nwith }B_N^{\\left( n\\right) }\\in C^\\infty \\left( {\\bf R}{\\em ,}\\,\\,%\n\\widehat{g}_n\\left( s_{\\hom }\\right) \\right) . \\label{7.10}\n\\end{equation}\n\nAs $\\Psi ^{\\pm }=\\nu _N=0$ is a solution of each system of equations\nof the hierarchy, we have \n\\begin{equation}\nB_1^{\\left( vac\\right) }=H^{\\left( 1\\right) },\\ \\quad B_N^{\\left( vac\\right)\n}=H^{\\left( N\\right) }, \\label{7.11}\n\\end{equation}\nsuch connections can be obtained with the help of\\, $B_N=\\partial _{t_N}{\\bf %\n\\Psi \\Psi }^{-1}$\\, from the group element \n\\br\n{\\bf \\Psi }^{\\left( vac\\right) }=\\exp \\left( xH^{\\left( 1\\right)\n}+t_NH^{\\left( N\\right) }+\\sum_{ n=2,3,... n\\neq N } t_nH^{\\left(\nn\\right) }\\right) \\equiv \\exp \\left( \\sum_{n=1,2,...}t_nH^{\\left( n\\right)\n}\\right) . \\label{7.12}\n\\er\n\nObserve that according to (\\ref{7.6}) we have identified $t_1=x.$\n\nThe connections in the vacuum orbit are given by \n\\begin{equation}\n\\quad B_1=\\Theta H^{\\left( 1\\right) }\\Theta ^{-1}+\\partial _x\\Theta \\Theta\n^{-1},\\qquad \\qquad \\qquad \\qquad \\label{7.13}\n\\end{equation}\n\n\\[\n\\quad =M^{-1}\\left( {\\em N}H^{\\left( 1\\right) }{\\em N}^{-1}-\\partial\n_xMM^{-1}+\\partial _x{\\em NN}^{-1}\\right) M{\\em ,} \n\\]\n\\begin{equation}\n\\quad \\quad B_N=\\Theta H^{\\left( N\\right) }\\Theta ^{-1}+\\partial\n_{t_N}\\Theta \\Theta ^{-1},\\quad \\qquad \\qquad \\qquad \\qquad \\label{7.14}\n\\end{equation}\n\\[\n\\quad \\quad \\quad =M^{-1}\\left( {\\em N}H^{\\left( N\\right) }{\\em N}%\n^{-1}-\\partial _{t_N}MM^{-1}+\\partial _{t_N}{\\em NN}^{-1}\\right) M. \n\\]\n\nDenote \n\\begin{equation}\n\\Theta =\\exp \\left( \\sum_{n>0}\\sigma _{-n}\\right) ,\\quad M=\\exp \\left(\n\\sigma _o\\right) ,\\quad {\\em N}=\\exp \\left( \\sum_{n>0}\\sigma _n\\right) ,\n\\label{7.15}\n\\end{equation}\n\\[\n\\llbrack D,\\sigma _n\\rrbrack =n\\sigma _n, \n\\]\n\nTherefore we can relate $\\Psi ^{\\pm }$ to some $\\sigma _n$. For instance for \n$N=2$ and denoting $t_2=t$, we have \n\\begin{equation}\nB_1=H^{\\left( 1\\right) }+\\llbrack \\sigma _{-1},H^{\\left( 1\\right) }\\rrbrack +%\n\\mbox{ terms of negative grade,\\qquad \\qquad } \\label{7.16}\n\\end{equation}\n\\[\n=M^{-1}\\left( H^{\\left( 1\\right) }-\\partial _xMM^{-1}+\\partial _x\\sigma\n_1\\right) M\\,\\,+\\mbox{ terms of grade }>1, \n\\]\n\\[\n\\quad B_2=H^{\\left( 2\\right) }+\\llbrack \\sigma _{-1},H^{\\left( 2\\right)\n}\\rrbrack +\\llbrack \\sigma _{-2},H^{\\left( 2\\right) }\\rrbrack +\\frac 12\\llbrack\n\\sigma _{-1},\\llbrack \\sigma _{-1},H^{\\left( 2\\right) }\\rrbrack \\rrbrack + \n\\]\n\\begin{equation}\n+\\mbox{ terms of negative grade.} \\label{7.17}\n\\end{equation}\n\\[\n=M^{-1}\\left( H^{\\left( 2\\right) }-\\partial _tMM^{-1}+\\partial _t\\sigma\n_1+\\partial _t\\sigma _2+\\llbrack \\sigma _1,\\partial _t\\sigma _1 \\rrbrack\\right)\nM+\\mbox{ terms of grade}>2.\n\\]\n\nLet us observe that the next term (with degree $-1$) in (\\ref{7.16})\nvanishes, and therefore we have \n\\begin{equation}\n\\partial _x\\sigma _{-1}+\\llbrack \\sigma _{-2},H^{\\left( 1\\right) }\\rrbrack\n+\\frac 12\\llbrack \\sigma _{-1},\\llbrack \\sigma _{-1},H^{\\left( 1\\right) }\\rrbrack\n\\rrbrack =0. \\label{7.18}\n\\end{equation}\n\nDenoting \n\\br\n\\sigma _{-1}=-\\Psi ^{+}E_{+}^{\\left( -1\\right) }+\\Psi ^{-}E_{-}^{\\left(\n-1\\right) }+\\sigma _{-1}^oH^{\\left( -1\\right) } \\label{7.19}\n\\er\nand \n\\br\n\\sigma _{-2}=-\\sigma _{-2}^{+}E_{+}^{\\left( -2\\right) }+\\sigma\n_{-2}^{-}E_{-}^{\\left( -2\\right) }+\\sigma _{-2}^oH^{\\left( -2\\right) },\n\\label{7.20}\n\\er\nfrom (\\ref{7.18}) we obtain \n\\br\n\\partial _x\\sigma _{-1}^o &=&2.\\Psi ^{+}\\Psi ^{-} \\label{7.21} \\\\\n\\sigma _{-2}^{+} &=&-\\partial _x\\Psi ^{+}+\\frac 12\\sigma _{-1}^o\\Psi ^{+}\n\\label{7.22} \\\\\n\\sigma _{-2}^{-} &=&-\\partial _x\\Psi ^{-}+\\frac 12\\sigma _{-1}^o\\Psi ^{-}.\n\\label{7.23}\n\\er\n\nSubstituting these expressions for $\\sigma _{-1}$ and $\\sigma _{-2}$ in (\\ref\n{7.16}) and (\\ref{7.17} ) we obtain (\\ref{7.8} ) with \n\\br\n\\nu _1=-\\frac{\\sigma _{-1}^o}2,\\quad \\nu _1=-\\sigma _{-2}^o. \\label{7.24}\n\\er\n\nThe $\\sigma _{-n}$ 's with higher grades are to cancel the undesired\ncomponents. The term of gradation $-2$ in (\\ref{7.16}) satisfies \\footnote{we use\\, $\\partial\ne^\\sigma e^{-\\sigma }=\\partial \\sigma +\\frac 1{2!}\\left[ \\sigma ,\\partial\n\\sigma \\right] +\\frac 1{3!}\\left[ \\sigma ,\\left[ \\sigma ,\\partial \\sigma\n\\right] \\right] +\\cdot \\cdot \\cdot $} \n\\begin{equation}\n\\partial _x\\sigma _{-2}+\\llbrack \\sigma _{-3},H^{\\left( 1\\right) }\\rrbrack\n+\\frac 12\\llbrack \\sigma _{-2},\\llbrack \\sigma _{-1},H^{\\left( 1\\right) }\\rrbrack\n\\rrbrack +\\frac 12\\llbrack \\sigma _{-1},\\llbrack \\sigma _{-2},H^{\\left( 1\\right)\n}\\rrbrack \\rrbrack =0, \\label{7.25}\n\\end{equation}\nwhere \n\\[\n\\sigma _{-3}=\\sigma _{-3}^{+}E_{+}^{\\left( -3\\right) }+\\sigma\n_{-3}^{-}E_{-}^{\\left( -3\\right) }+\\sigma _{-3}^oH^{\\left( -3\\right) }. \n\\]\n\nFrom (\\ref{7.25}) we obtain \n\\begin{equation}\n\\partial _x\\sigma _{-2}^o=\\Psi ^{-}\\partial _x\\Psi ^{+}-\\Psi ^{+}\\partial\n_x\\Psi ^{-}. \\label{7.26}\n\\end{equation}\n\nEquating to zero the terms of degree ($-1$) and ($-2$) in (\\ref{7.17}) we\ncan obtain \n\\begin{equation}\n\\partial _t\\sigma _{-1}^o=2\\left( \\Psi ^{-}\\partial _x\\Psi ^{+}-\\Psi\n^{+}\\partial _x\\Psi ^{-}\\right) , \\label{7.27}\n\\end{equation}\nand \n\\begin{equation}\n\\partial _t\\sigma _{-2}^o=\\frac 23\\Psi ^{+}\\Psi ^{-}\\left( \\sigma\n_{-1}^o\\right) ^2-2\\partial _x\\Psi ^{+}\\partial _x\\Psi ^{-}-\\frac 23\\left(\n\\Psi ^{+}\\Psi ^{-}\\right) ^2+\\Psi ^{-}\\partial _x\\Psi ^{+}-\\Psi ^{+}\\partial\n_x\\Psi ^{-}. \\label{7.28}\n\\end{equation}\n\nFrom (\\ref{7.21}) and (\\ref{7.27}) we obtain \n\\begin{equation}\n\\partial _t\\left( \\Psi ^{+}\\Psi ^{-}\\right) =\\partial _xF\\llbrack \\Psi\n^{+},\\Psi ^{-}\\rrbrack , \\label{7.29}\n\\end{equation}\nwhere $F$ is a functional of the fields and ${\\cal H}_1$ is the first\nHamiltonian given by\n\n\\[\n{\\cal H}_1=\\int_{-\\infty }^{+\\infty }dx.\\Psi ^{+}\\Psi ^{-}. \n\\]\n\nIn the same way (\\ref{7.26}) and (\\ref{7.28}) gives \n\\begin{equation}\n\\partial _t\\left( \\Psi ^{-}\\partial _x\\Psi ^{+}-\\Psi ^{+}\\partial _x\\Psi\n^{-}\\right) =\\partial _xG\\llbrack \\Psi ^{+},\\Psi ^{-}\\rrbrack , \\label{7.30}\n\\end{equation}\nwhere $G$ is a functional of the fields and ${\\cal H}_2$ is the second\nHamiltonian of GNLS$_1$ system given by \n\\begin{equation}\n{\\cal H}_2=\\int_{-\\infty }^{+\\infty }dx.\\left( \\Psi ^{-}\\partial _x\\Psi\n^{+}-\\Psi ^{+}\\partial _x\\Psi ^{-}\\right) . \\label{7.31}\n\\end{equation}\n\nIn this way one can construct the remaining Hamiltonians of higher order corresponding to every $\\sigma^{0}_n $ ($n<-2$).\n\nLet us define the tau-function vector\n\\begin{equation}\n\\tau \\left( x,t_2,t_3,...\\right) ={\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi \n}^{\\left( vac\\right) -1}\\left\| \\lambda _o\\right\\rangle , \\label{7.32}\n\\end{equation}\n\\begin{equation}\n=\\Theta ^{-1}M^{-1}\\left\| \\lambda _o\\right\\rangle , \\label{7.33}\n\\end{equation}\nwhere $h$ is a particular element of the group $\\widehat{sl}(2)$ which\ngenerates a dressing transformation.\n\nTherefore we have \n\\begin{equation}\n\\exp \\left( -\\sum_{n>0}\\sigma _{-n}\\right) \\exp \\left( -\\sigma _o\\right)\n\\left\| \\lambda _o\\right\\rangle ={\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }%\n^{\\left( vac\\right) -1}\\left\| \\lambda _o\\right\\rangle , \\label{7.34}\n\\end{equation}\nand then one can write\n\\begin{equation}\n\\sigma _o=\\sigma _o^oH+\\sigma _o^{+}E_{+}^{\\left( o\\right) }+\\sigma\n_o^{-}E_{-}^{\\left( o\\right) }+\\eta C, \\label{7.35}\n\\end{equation}\nor \n\\[\n\\sigma _o=\\sigma _o^oh_1+\\sigma _o^{+}e_1+\\sigma _o^{-}f_1+\\eta C, \n\\]\nwhere we have used\\, $h_1\\left\| \\lambda _o\\right\\rangle =0,\\quad f_1\\left\| \\lambda\n_o\\right\\rangle =0$\\quad and \\quad $C\\left\| \\lambda _o\\right\\rangle =\\left\|\n\\lambda _o\\right\\rangle .$ In this way the zero gradation of expression (\\ref\n{7.34}) is \n\\begin{equation}\n\\exp \\left( -\\sigma _o\\right) \\left\| \\lambda _o\\right\\rangle =\\left( {\\bf %\n\\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right) -1}\\right)\n_{\\left( o\\right) }\\left\| \\lambda _o\\right\\rangle , \\label{7.36}\n\\end{equation}\nwhich can be rewritten as\n\\[\n\\exp \\left( -\\sigma _o\\right) \\left\| \\lambda _o\\right\\rangle =\\left\| \\lambda\n_o\\right\\rangle \\widehat{\\tau }^{\\left( o\\right) }\\left( x,t\\right) \n\\]\nwhere \\,$\\widehat{\\tau }^{\\left( o\\right) }\\left( x,t\\right)$ is a function of \n$x$ and the times $t_n$ given by the following matrix element \n\\begin{equation}\n\\widehat{\\tau }^{\\left( o\\right) }\\left( x,t\\right) =\\left\\langle \\lambda\n_o\\right\| \\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left(\nvac\\right) -1}\\right) _{\\left( o\\right) }\\left\| \\lambda _o\\right\\rangle .\n\\label{7.37}\n\\end{equation}\n\nThe term with degree ($-1$) in (\\ref{7.34}) becomes \n\\[\n\\left( -\\sigma _{-1}\\right) \\left\| \\lambda _o\\right\\rangle =\\frac{\\left( \n{\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right) -1}\\right)\n_{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle }{\\widehat{\\tau }^{\\left(\no\\right) }\\left( x,t\\right) }, \n\\]\nor \n\\begin{equation}\n\\left( -\\Psi ^{+}E_{+}^{\\left( -1\\right) }+\\Psi ^{-}E_{-}^{\\left( -1\\right)\n}+\\sigma _{-1}^oH^{\\left( -1\\right) }\\right) \\left\| \\lambda _o\\right\\rangle\n=-\\frac{\\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left(\nvac\\right) -1}\\right) _{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle }{%\n\\widehat{\\tau }^{\\left( o\\right) }\\left( x,t\\right) }, \\label{7.38}\n\\end{equation}\n\nAs $H^{\\left( 0\\right) }\\left\| \\lambda _o\\right\\rangle =h_1\\left\| \\lambda\n_o\\right\\rangle =0,$ $\\quad E_{\\pm }^{\\left( 1\\right) }\\left\| \\lambda\n_o\\right\\rangle =0$ we can write \n\\begin{equation}\n\\Psi ^{+}=\\frac{\\tau ^{+}}{\\widehat{\\tau }^{\\left( o\\right) }}\\qquad \\mbox{%\nand\\qquad }\\Psi ^{-}=-\\frac{\\tau ^{-}}{\\widehat{\\tau }^{\\left( o\\right) }},\n\\label{7.39}\n\\end{equation}\nwhere \n\\begin{equation}\n\\tau ^{+}\\equiv \\left\\langle \\lambda _o\\right\| E_{-}^{\\left( 1\\right)\n}\\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right)\n-1}\\right) _{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle , \\label{7.40}\n\\end{equation}\n\\begin{equation}\n\\tau ^{-}\\equiv \\left\\langle \\lambda _o\\right\| E_{+}^{\\left( 1\\right)\n}\\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right)\n-1}\\right) _{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle . \\label{7.41}\n\\end{equation}\n\nThe relations \\rf{7.37}, \\rf{7.40} and \\rf{7.41} define the tau-functions of the GNLS$_1$ system of equations \\rf{sch}.\n\nIn order to obtain the first non trivial solution we choose \n\\begin{equation}\nh=e^F,\\qquad \\mbox{with\\qquad }F=\\sum_{n=-\\infty }^{+\\infty }\\nu\n_1^nE_{-}^{\\left( -n\\right) }. \\label{7.42}\n\\end{equation}\n\nSince \n\\begin{equation}\n\\left[ \\sum_{n=1}^{+\\infty }t_nH^{\\left( n\\right) },F\\right] =-\\left(\n\\sum_{n=1}^{+\\infty }t_n\\nu _1^n\\right) F,\\qquad \\mbox{with }\\nu _1\\mbox{ a\nreal parameter,} \\label{7.43}\n\\end{equation}\nwe may obtain \n\\br\n\\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right)\n-1}\\right) &=&\\exp \\left( e^{-\\varphi _1}F\\right) \\\\\n&=&1+e^{-\\varphi _1}F,\n\\er\nwith \n\\[\n\\varphi _1=\\sum_{n=1}^{+\\infty }t_n\\nu _1^n. \n\\]\nWhere we have used the property $F^n=0,$ for $n\\geq 2$. The tau functions\nbecome \n\\[\n\\tau ^{\\left( o\\right) }=\\left\\langle \\lambda _o\\right\| \\left( 1+e^{-\\varphi\n_1}E_{-}^{\\left( o\\right) }\\right) \\left\| \\lambda _o\\right\\rangle =1, \n\\]\n\\[\n\\quad \\quad \\tau ^{+}=\\left\\langle \\lambda _o\\right\| E_{-}^{\\left( 1\\right)\n}e^{-\\varphi _1}\\nu _1E_{-}^{\\left( -1\\right) }\\left\| \\lambda\n_o\\right\\rangle =0, \n\\]\n\n\\br\n\\tau ^{-} &=&\\left\\langle \\lambda _o\\right\| E_{+}^{\\left( 1\\right)\n}e^{-\\varphi _1}\\nu _1E_{-}^{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle\n\\\\\n&=&\\nu _1e^{-\\varphi _1}\\left\\langle \\lambda _o\\right\| \\left( 2H^{\\left(\n0\\right) }+C\\right) \\left\| \\lambda _o\\right\\rangle \\\\\n&=&\\nu _1e^{-\\varphi _1}.\n\\er\n\nUsing (\\ref{7.39}) we obtain the following solution of the Eqs. \\rf{sch} \n\\begin{equation}\n\\Psi ^{+}=0\\qquad \\mbox{e \\quad }\\Psi ^{-}=-\\nu _1e^{-\\varphi _1}.\n\\label{7.44}\n\\end{equation}\nNow let us choose \n\\begin{equation}\nh=e^G,\\qquad G=\\sum_{n=-\\infty }^{+\\infty }\\rho _1^nE_{+}^{\\left( -n\\right)\n},\\,\\,\\rho _1\\,\\mbox{is a real parameter.} \\label{7.45}\n\\end{equation}\nSince \n\\[\n\\llbrack \\sum_{n=1}^{+\\infty }t_nH^{\\left( n\\right) },G\\rrbrack =\\left(\n\\sum_{n=1}^{+\\infty }t_n\\rho _1^n\\right) G, \n\\]\nwe obtain \n\\br\n\\left( {\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right)\n-1}\\right) &=&\\exp \\left( e^{\\eta _1}G\\right) \\\\\n&=&1+e^{\\eta _1}G,\\qquad \\mbox{with\\qquad }\\eta _1=\\sum_{n=1}^{+\\infty\n}t_n\\rho _1^n,\n\\er\nwhere we used $G^n=0,$ $n\\geq 2.$ Therefore the tau functions become\n\\[\n\\widehat{\\tau }^{\\left( o\\right) }=\\left\\langle \\lambda _o\\right\| \\left(\n1+e^{\\eta _1}E_{+}^{\\left( o\\right) }\\right) _{\\left( o\\right) }\\left\|\n\\lambda _o\\right\\rangle =1, \n\\]\n\n\\[\n\\tau ^{-}=\\left\\langle \\lambda _o\\right\| \\left( E_{+}^{\\left( 1\\right)\n}e^{\\eta _1}\\rho _1E_{+}^{\\left( -1\\right) }\\right) _{\\left( o\\right)\n}\\left\| \\lambda _o\\right\\rangle =0, \n\\]\n\\[\n\\tau ^{+}=\\left\\langle \\lambda _o\\right\| E_{-}^{\\left( 1\\right) }e^{\\eta\n_1}\\rho _1E_{+}^{\\left( -1\\right) }\\left\| \\lambda _o\\right\\rangle =\\rho\n_1e^{\\eta _1}, \n\\]\nand the corresponding solutions \n\\begin{equation}\n\\Psi ^{-}=0\\qquad \\mbox{and\\qquad }\\Psi ^{+}=\\rho _1e^{\\eta _1}.\n\\label{7.46}\n\\end{equation}\n\nIn order to obtain {\\sl one-soliton} solutions, let us choose\n\n\\br\nh &=&e^{aF}e^{bG},\\qquad \\mbox{where }F\\mbox{ and }G\\mbox{ are given in (\\ref\n{7.42}) and (\\ref{7.45})} \\label{7.47} \\\\\n&&\\qquad \\qquad \\qquad \\mbox{with }a\\mbox{ and }b\\mbox{ real parameters.} \n\\nonumber\n\\er\nThen \n\\br\n{\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right) -1} &=&\\exp\n\\left( e^{-\\varphi }aF\\right) \\exp \\left( e^\\eta bG\\right) \\label{7.48} \\\\\n&=&1+e^{-\\varphi }aF+e^\\eta bG+e^{-\\varphi }e^\\eta aFbG, \\label{7.49}\n\\er\nwith $\\varphi =\\sum_{n=1}^{+\\infty }t_n\\nu ^n$ \\quad and $\\quad \\eta\n=\\sum_{n=1}^{+\\infty }t_n\\rho ^n.$ $\\widehat{\\tau }^{\\left( o\\right) }.$ Let\nus compute the relevant tau functions. The expression for $\\widehat{\\tau }%\n^{\\left( o\\right) }$ becomes\n\n\\begin{equation}\n\\widehat{\\tau }^{\\left( o\\right) }=1+a\\,b\\,c\\,e^{-\\varphi }e^\\eta , \\label{7.50}\n\\end{equation}\nwhere c is an matrix element of the following form \n\\br\nc &=&\\left\\langle \\lambda _o\\right\| \\left( FG\\right) _{\\left( o\\right)\n}\\left\| \\lambda _o\\right\\rangle \\nonumber \\label{7.2.20} \\\\\n&=&\\left\\langle \\lambda _o\\right\| \\left( \\sum_{n,m>0}^{+\\infty }\\nu\n_1^{-n}\\rho _1^mE_{-}^{\\left( n\\right) }E_{+}^{\\left( -m\\right) }\\right)\n_{\\left( o\\right) }\\left\| \\lambda _o\\right\\rangle \\nonumber \\\\\n&=&\\left\\langle \\lambda _o\\right\| \\sum_{n,m>0}^{+\\infty }\\nu _1^{-n}\\rho\n_1^m\\left( -2H^{\\left( n-m\\right) }+m\\delta _{n-m,o}C\\right) _{\\left(\no\\right) }\\left\| \\lambda _o\\right\\rangle \\nonumber \\\\\n&=&\\sum_{n=0}^{+\\infty }n\\left( \\frac{\\rho _1}{\\nu _1}\\right) ^n, \\nonumber\n\\\\\n&=&\\frac{\\nu _1\\rho _1}{\\left( \\rho _1-\\nu _1\\right) ^2}\\cdot\n\\label{7.51}\n\\er\nwhere, for pedagogical reasons, we have used step by step, the properties of the integrable highest weight representation of the algebra $\\hat{sl}(2)$, see Appendix \\ref{appa} (Eqs. \\rf{a29}-\\rf{a35}). The computation of higher order matrix elements is performed very quickly using the vertex operator formalism, see Appendix \\ref{appe}.\n\nThe remaining tau functions are given by \n\\br\n\\tau ^{+} &=&\\left\\langle \\lambda _o\\right\| E_{-}^{\\left( 1\\right) }\\left(\nbe^\\eta \\rho _1E_{+}^{\\left( -1\\right) }\\right) \\left\| \\lambda\n_o\\right\\rangle \\nonumber \\\\\n&=&be^\\eta \\rho _1\\left\\langle \\lambda _o\\right\| E_{-}^{\\left( 1\\right)\n}\\left( -2H^{\\left( 0\\right) }+C\\right) \\left\| \\lambda _o\\right\\rangle \n\\nonumber \\\\\n&=&b\\rho _1e^\\eta \\label{7.52}\n\\er\n\nand \n\\br\n\\quad \\quad \\tau ^{-} &=&\\left\\langle \\lambda _o\\right\| E_{+}^{\\left(\n1\\right) }\\left( a.e^{-\\varphi }\\nu _1E_{-}^{\\left( -1\\right) }\\right)\n\\left\| \\lambda _o\\right\\rangle \\qquad \\qquad \\nonumber \\\\\n&=&a.\\nu _1e^{-\\varphi }. \\label{7.53}\n\\er\nThus, a {\\sl one-soliton} solution is \n\\begin{equation}\n\\Psi ^{+}=\\frac{b\\rho _1e^\\eta }{1+a\\,b\\,c\\,e^{-\\varphi }e^\\eta }, \\label{7.54}\n\\end{equation}\nand \n\\begin{equation}\n\\Psi ^{-}=-\\frac{a\\,\\nu _1e^{-\\varphi }}{1+a\\,b\\,c\\,e^{-\\varphi }e^\\eta }.\n\\label{7.55}\n\\end{equation}\n\nLet us write down the explicit form of this {\\sl one-soliton} solution. As a\nparticular case we set the following relations\n\n\\br\n\\rho _1=-\\nu _1,\\quad b=-a=-2\\quad \\mbox{and \\quad }t_{2n+1}=0\\quad (n\\geq\n0), \n\\er\nthen the relations (\\ref{7.54}) and (\\ref{7.55}) become \n\\begin{equation}\n\\Psi ^{+}=\\nu _1\\exp \\left( \\nu _1^2t+\\overline{\\nu }_1\\right) \\mbox{sech} \\left(\n\\nu _1x\\right) \\label{7.57}\n\\end{equation}\nand \n\\begin{equation}\n\\quad \\Psi ^{-}=-\\nu _1\\exp \\left( -\\nu _1^2t-\\overline{\\nu }_1\\right) \\mbox{sech} \\left( \\nu _1x\\right) ,\\quad \\label{7.58}\n\\end{equation}\nwhere $\\overline{\\nu }_1=\\sum_{n=2}^{+\\infty }\\nu _1^{2n}t_{2n}$ is a phase\nparameter as far as only the equation \\rf{sch} of the whole hierarchy is considered. The\nsolutions of types \\rf{7.57} and (\\ref{7.58}), are known as an `envelope soliton' or `bright soliton' solutions in the context of the non-linear Schr\\\"{o}dinger equation \\ct{drazin}.\n\nNext let us choose\n\n\\br\nh &=&e^{a_1F_1}\\ e^{a_2F_2}\\,\\,e^{b_1G_1}\\,\\,e^{b_2G_2},\\quad\na_i,b_i,\\nu _i,\\rho _i\\mbox{ real parameters;} \\nonumber \\label{7.4.1} \\\\\n\\mbox{ with }\\,F_i &=&\\sum_{n=-\\infty }^{+\\infty }\\nu _i^nE_{-}^{\\left(\n-n\\right) }\\quad \\mbox{and }G_i=\\sum_{n=-\\infty }^{+\\infty }\\rho\n_i^nE_{+}^{\\left( -n\\right) },\\,\\,i=1,2. \\label{7.59}\n\\er\nThen \n\\br\n&&{\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right)\n-1}=1+e^{-\\varphi _1}a_1F_1+e^{-\\varphi _2}a_2F_2+e^{\\eta _1}b_1G_1+e^{\\eta\n_2}b_2G_2+\\nonu\\\\\n&&e^{\\eta _1}e^{\\eta _2}b_1G_1b_2G_2+e^{-\\varphi _1}e^{-\\varphi\n_2}a_1F_1a_2F_2+e^{-\\varphi _1}e^{\\eta _1}a_1F_1b_1G_1+e^{-\\varphi\n_1}e^{\\eta _2}a_1F_1b_2G_2+\\nonu\\\\\n&&e^{-\\varphi _2}e^{\\eta _2}a_2F_2b_2G_2+e^{\\eta _1}e^{-\\varphi\n_2}b_1G_1a_2F_2+e^{\\eta _1}e^{-\\varphi _2}e^{\\eta _2}b_1G_1a_2F_2b_2G_2+ \\nonu\\\\\n&&e^{-\\varphi _1}e^{-\\varphi _2}e^{\\eta _2}a_1F_1a_2F_2b_2G_2+e^{-\\varphi\n_1}e^{\\eta _1}e^{-\\varphi _2}a_1F_1b_1G_1a_2F_2+\\nonu\\\\\n&&e^{-\\varphi _1}e^{\\eta _1}e^{\\eta _2}a_1F_1b_1G_1b_2G_2+e^{-\\varphi\n_1}e^{\\eta _1}e^{-\\varphi _2}e^{\\eta _2}a_1F_1b_1G_1a_2F_2b_2G_2,\n\\label{7.60}\n\\er\nwith \n\\[\n\\varphi _i=\\sum_{n=1}^{+\\infty }\\nu _i^nt_n,\\quad \\eta\n_i=\\sum_{n=1}^{+\\infty }\\rho _i^nt_n,\\quad i=1,2. \n\\]\nThe corresponding tau functions are \n\\br\n&&\n\\widehat{\\tau }^{(o)}=1+ e^{-\\varphi _1}e^{\\eta\n_1}a_1b_1\\left\\langle \\lambda _o\\right\| F_1G_1\\left\| \\lambda _o\\right\\rangle\n+e^{-\\varphi _1}e^{\\eta _2}a_1b_2\\left\\langle \\lambda _o\\right\| F_1G_2\\left\|\n\\lambda _o\\right\\rangle +\\nonu\\\\\n&& e^{-\\varphi _2}e^{\\eta\n_2}a_2b_2\\left\\langle \\lambda _o\\right\| F_2G_2\\left\| \\lambda _o\\right\\rangle\n+e^{\\eta _1}e^{-\\varphi _2}b_1a_2\\left\\langle \\lambda _o\\right\| G_1F_2\\left\|\n\\lambda _o\\right\\rangle + \\nonu\\\\\n&&e^{-\\varphi _1}e^{\\eta _1}e^{-\\varphi _2}e^{\\eta\n_2}a_1b_1a_2b_2\\left\\langle \\lambda_o\\right\| F_1G_1F_2G_2\\left\| \\lambda\n_o\\right\\rangle , \\label{7.61}\n\\er\n\n\\br\n\\tau ^{+} &=&e^{\\eta _1}b_1\\left\\langle \\lambda _o\\right\| E_{-}^{\\left(\n1\\right) }G_1\\left\| \\lambda _o\\right\\rangle +e^{\\eta _2}b_2\\left\\langle\n\\lambda _o\\right\| E_{-}^{\\left( 1\\right) }G_2\\left\| \\lambda _o\\right\\rangle +\n\\nonumber \\\\\n&&e^{-\\varphi _1}e^{\\eta _1}e^{\\eta _2}a_1b_1b_2\\left\\langle \\lambda\n_o\\right\| E_{-}^{\\left( 1\\right) }F_1G_1G_2\\left\| \\lambda _o\\right\\rangle + \n\\nonumber \\\\\n&&e^{\\eta _1}e^{-\\varphi _2}e^{\\eta _2}b_1a_2b_2\\left\\langle \\lambda\n_o\\right\| E_{-}^{\\left( 1\\right) }G_1F_2G_2\\left\| \\lambda _o\\right\\rangle\n\\label{7.62}\n\\er\nand \n\\br\n\\tau ^{-} &=&e^{-\\varphi _1}a_1\\left\\langle \\lambda _o\\right\| E_{+}^{\\left(\n1\\right) }F_1\\left\| \\lambda _o\\right\\rangle +e^{-\\varphi _2}a_2\\left\\langle\n\\lambda _o\\right\| E_{+}^{\\left( 1\\right) }F_2\\left\| \\lambda _o\\right\\rangle +\n\\nonumber \\\\\n&&e^{-\\varphi _1}e^{-\\varphi _2}e^{\\eta _2}a_1a_2b_2\\left\\langle \\lambda\n_o\\right\| E_{+}^{\\left( 1\\right) }F_1F_2G_2\\left\| \\lambda _o\\right\\rangle + \n\\nonumber \\\\\n&&e^{-\\varphi _1}e^{\\eta _1}e^{-\\varphi _2}a_1b_1a_2\\left\\langle \\lambda\n_o\\right\| E_{+}^{\\left( 1\\right) }F_1G_1F_2\\left\| \\lambda _o\\right\\rangle .\n\\label{7.63}\n\\er\nNotice that only some terms of the expansion ${\\bf \\Psi }^{\\left(\nvac\\right) }h{\\bf \\Psi }^{(vac)-1}$ contribute to the tau functions. For\nexample the terms $\\left\\langle \\lambda _o\\right\| F_i\\left\| \\lambda\n_o\\right\\rangle ,$ $\\left\\langle \\lambda _o\\right\| G_i\\left\| \\lambda\n_o\\right\\rangle ,$ $\\left\\langle \\lambda _o\\right\| F_iG_jF_k\\left\| \\lambda\n_o\\right\\rangle $ and $\\left\\langle \\lambda _o\\right\| G_iF_jG_k\\left\|\n\\lambda _o\\right\\rangle $ do not contribute to the computation of $\\widehat{%\n\\tau }^{\\left( o\\right) }$, since these matrix elements vanish.\n\nIn particular let us consider $\\rho _i=-\\nu _i,$ $b_i=-a_i=-2$ and $t_{2n+1}=0,$ then\n\n\\begin{equation}\n\\tau ^{+}=a_1\\nu _1e^{\\widehat{\\varphi }_1}+a_2\\nu _2e^{\\widehat{\\varphi }%\n_2}+a_1a_2^2\\Delta _1e^{\\widehat{\\varphi }_1}e^{-\\varphi _2}e^{\\widehat{%\n\\varphi }_2}+a_1^2a_2\\Delta _2e^{-\\varphi _1}e^{\\widehat{\\varphi }_1}e^{%\n\\widehat{\\varphi }_2},\\qquad \\qquad \\label{7.65}\n\\end{equation}\n\\begin{equation}\n\\tau ^{-}=a_1\\nu _1e^{-\\varphi _1}+a_2\\nu _2e^{-\\varphi _2}+a_1a_2^2\\Delta\n_1e^{-\\varphi _1}e^{-\\varphi _2}e^{\\widehat{\\varphi }_2}+a_1^2a_2\\Delta\n_2e^{-\\varphi _1}e^{\\widehat{\\varphi }_1}e^{-\\varphi _2},\\quad \\label{7.66}\n\\end{equation}\nwhere \n\\br\n\\widehat{\\varphi }_i &=&-\\nu _i\\left( x-\\nu _it\\right) +\\overline{\\nu }_i, \\\\\n\\varphi _i &=&\\nu _i\\left( x+\\nu _it\\right) +\\overline{\\nu }_i,\n\\er\n\\[\n\\Delta _i=\\frac{\\nu _i}4\\left( \\frac{\\nu _1-\\nu _2}{\\nu _1+\\nu _2}\\right)\n^2,\\quad i=1,2; \n\\]\nnow let us choose $a_1$ and $a_2$ such that \n\\[\n\\nu _i=a_j^2\\Delta _i,\\qquad \\left( i\\neq j\\right) , \n\\]\nthen \n\\[\na_1=a_2\\equiv a=2\\left( \\frac{\\nu _1-\\nu _2}{\\nu _1+\\nu _2}\\right) . \n\\]\nLet us remark that the parameters $\\overline{\\nu }_i$ are some phase\nparameters if only the equations \\rf{sch} of the hierarchy are to be\nconsidered. With this choice of parameters the $\\widehat{\\tau }^{\\left(\no\\right) }$ function turns out to be \n\\br\n\\widehat{\\tau }^{\\left( o\\right) } &=&e^{-\\left( \\nu _1+\\nu _2\\right)\nx}\\{\\frac 4{a^2}(e^{\\left( \\nu _1-\\nu _2\\right) x}+e^{-\\left( \\nu _1-\\nu\n_2\\right) x})+e^{\\left( \\nu _1+\\nu _2\\right) x}+e^{-\\left( \\nu _1+\\nu\n_2\\right) x}+ \\nonumber \\\\\n&&4\\frac{\\nu _1\\nu _2}{\\left( \\nu _1-\\nu _2\\right) ^2}(e^{\\left( \\nu\n_1^2-\\nu _2^2\\right) t+\\left( \\overline{\\nu }_1-\\overline{\\nu }_2\\right)\n}+e^{-\\left( \\nu _1^2-\\nu _2^2\\right) t-\\left( \\overline{\\nu }_1-\\overline{%\n\\nu }_2\\right) }\\}. \\label{7.67}\n\\er\nTherefore the fields $\\Psi ^{+}$and $\\Psi ^{-}$ become \n\\[\n\\Psi ^{\\pm }=\\pm ae^{\\pm \\left[ \\left( \\nu _1^2+\\nu _2^2\\right) t+\\left( \n\\overline{\\nu }_1+\\overline{\\nu }_2\\right) \\right] }\\cdot \n\\]\n\\begin{equation}\n\\frac{e^{\\mp \\left( \\nu _1^2t+\\overline{\\nu }_1\\right) }\\nu _2\\cosh \\left(\n\\nu _1x\\right) +e^{\\mp \\left( \\nu _2^2t+\\overline{\\nu }_2\\right) }\\nu\n_1\\cosh \\left( \\nu _2x\\right) }{\\frac 4{a^2}\\cosh \\left[ \\left( \\nu _1-\\nu\n_2\\right) x\\right] +\\cosh \\left[ \\left( \\nu _1+\\nu _2\\right) x\\right] +4%\n\\frac{\\nu _1\\nu _2}{\\left( \\nu _1-\\nu _2\\right) ^2}\\cosh \\left[ \\left( \\nu\n_1^2-\\nu _2^2\\right) t+\\left( \\overline{\\nu }_1-\\overline{\\nu }_2\\right)\n\\right] }, \\label{7.68}\n\\end{equation}\nwhich are the {\\sl two-soliton} solutions of \\rf{sch} or the {\\sl two-soliton} solutions (after relevant complexification) of the corresponding non-linear Schr\\\"{o}dinger equation \\ct{twosoliton}.\n\nRegarding the solutions of the equations of higher order of the hierarchy (\\ref{7.9}), we may argue that the same solutions, 1-soliton \\rf{7.54}-\\rf{7.55} and 2-soliton constructed with the tau functions \\rf{7.61}, \\rf{7.62}\nand \\rf{7.63} satisfying \\rf{sch} should satisfy the higher order\nequations of the hierarchy GNLS$_1,$ each equation with its corresponding time scale $t_n.$ This behaviour is also observed in the study of the KdV system\nusing the perturvative reduction and multiple time scaling approach \\ct{kraenkel}. \n\n\\subsection{N-soliton solutions}\n\nThe generalization for a N-soliton solution can be made choosing\n\n\\begin{equation}\nh=e^{a_1F_1}\\,\\,...e^{a_NF_N}\\,\\,e^{b_1G_1}\\,\\,%\n...\\,\\,e^{b_NG_N}, \\label{7.69}\n\\end{equation}\nwhere \n\\br\nF_i &=&\\sum_{n=-\\infty }^{+\\infty }\\nu _i^nE_{-}^{\\left( -n\\right) },\\quad\nG_i=\\sum_{n=-\\infty }^{+\\infty }\\rho _i^nE_{+}^{\\left( -n\\right) },\\,\\,i=1,2,...,N \\\\\n&&\\mbox{with }a_i,b_i,\\nu _i\\mbox{ and }\\rho _i\\mbox{ are real parameters .}\n\\er\nThe following expression plays an important role in the construction af the\ntau functions \n\\begin{equation}\n{\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right) -1}=\\left(\n1+a_1e^{-\\varphi _1}F_1\\right) \\left( 1+b_1e^{\\eta _1}G_1\\right) ...\\left(\n1+a_Ne^{-\\varphi _N}F_N\\right) \\left( 1+b_Ne^{\\eta _N}G_N\\right) ,\n\\label{7.70}\n\\end{equation}\nwhere \n\\[\n\\varphi _i=\\sum_{n=1}^{+\\infty }\\nu _i^nt_n,\\quad \\eta\n_i=\\sum_{n=1}^{+\\infty }\\rho _i^nt_n,\\quad i=1,2,...,N; \n\\]\nthe parameters $\\nu _i$ and $\\rho _i$ will characterize each soliton.\n\nIt will be convenient to write the various $F_i$ and $G_i$ in terms of the\nvertex operators (see Appendix \\ref{appc})\n\\begin{equation}\nF_i\\longrightarrow \\nu _i\\Gamma _{-}\\left( \\nu _i\\right) ,\\qquad\nG_i\\longrightarrow \\rho _i\\Gamma _{+}\\left( \\rho _i\\right) , \\label{7.71}\n\\end{equation}\nthen \n\\[\n{\\bf \\Psi }^{\\left( vac\\right) }h{\\bf \\Psi }^{\\left( vac\\right) -1}=\\left(\n1+a_1\\nu _1e^{-\\varphi _1}\\Gamma _{-}\\left( \\nu _1\\right) \\right) \\left(\n1+b_1\\rho _1e^{\\eta _1}\\Gamma _{+}\\left( \\rho _1\\right) \\right) ... \n\\]\n\\[\n\\left( 1+a_N\\nu _Ne^{-\\varphi _N}\\Gamma _{-}\\left( \\nu _N\\right) \\right)\n\\left( 1+b_N\\rho _Ne^{\\eta _N}\\Gamma _{+}\\left( \\rho _N\\right) \\right) . \n\\]\nDenoting \n\\[\nA_n\\equiv a_n\\nu _ne^{-\\varphi _n},\\quad B_n\\equiv b_n\\rho _ne^{\\eta _n}, \n\\]\nwe may compute the relevant tau functions \n\\[\n\\widehat{\\tau }^{\\left( o\\right) }=1+\\left\\langle \\lambda _o\\right\|\n\\sum_{n=1}^N \n\\]\n\\[\n\\sum_{1\\leq i_1<i_2<...<i_n\\leq N}A_{i_1}...A_{i_n}\\Gamma _{-}\\left( \\nu\n_{i_1}\\right) ...\\Gamma _{-}\\left( \\nu _{i_n}\\right) \\sum_{1\\leq\nj_1<j_2<...<j_n\\leq N}B_{j_1}...B_{j_n}\\Gamma _{+}\\left( \\rho _{j_1}\\right)\n...\\Gamma _{-}\\left( \\rho _{j_n}\\right) \\left\| \\lambda _o\\right\\rangle . \n\\]\n\nWe are using some properties of the product of vertex operators acting on $%\n\\left\| \\lambda _o\\right\\rangle $ and its dual $\\left\\langle \\lambda\n_o\\right\| $ (see Appendix \\ref{appb} and \\ref{appc}). Then \n\\br\n\\widehat{\\tau }^{\\left( o\\right) } &=&1+\\sum_{n=1}^N\\,\\,\\sum_{ 1\\leq\ni_1<i_2<...<i_n\\leq N,\\,\\, 1\\leq j_1<j_2<...<j_n\\leq N}\nA_{i_1}...A_{i_n}B_{j_1}...B_{j_n}\\cdot \\\\\n&&\\llbrack \\prod_{1\\leq l<m\\leq n}\\left( \\nu _{i_l}-\\nu _{i_m}\\right) ^2\\left(\n\\rho _{j_l}-\\rho _{j_m}\\right) ^2\\epsilon \\left( \\alpha _{i_l},\\alpha\n_{i_m}\\right) \\epsilon \\left( \\alpha _{j_l},\\alpha _{j_m}\\right) \\rrbrack\n\\cdot \\\\\n&&\\llbrack \\prod_{1\\leq l\\leq m<n}\\left( \\nu _{i_l}-\\rho _{j_m}\\right)\n^2\\epsilon \\left( \\alpha _{i_l},\\alpha _{j_m}\\right) \\rrbrack ^{-1}.\n\\er\n\nDenoting \n\\[\n\\epsilon \\left( \\alpha _{i_l},\\alpha _{i_m}\\right) =\\epsilon \\left( -\\alpha\n,-\\alpha \\right) \\equiv \\epsilon \\left( -,-\\right) , \n\\]\n\\[\n\\epsilon \\left( \\alpha _{j_l},\\alpha _{j_m}\\right) =\\epsilon \\left( \\alpha\n,\\alpha \\right) \\equiv \\epsilon \\left( +,+\\right) , \n\\]\n\\[\n\\epsilon \\left( \\alpha _{i_l},\\alpha _{j_m}\\right) =\\epsilon \\left( -\\alpha\n,\\alpha \\right) \\equiv \\epsilon \\left( -,+\\right) , \n\\]\nand using the properties of the cocycles in the case of $\\widehat{sl}(2)$ \\ct{goddard} \n\\br\n\\epsilon \\left( +,+\\right) &=&\\epsilon \\left( -,-\\right) =-1, \\\\\n\\epsilon \\left( +,-\\right) &=&\\epsilon \\left( -,+\\right) =1,\n\\er\nwe can write \n\\br\n\\widehat{\\tau }^{\\left( o\\right) } &=&1+\\sum_{n=1}^N\\,\\,\\sum_{1\\leq\ni_1<i_2<...<i_n\\leq N,\\,\\, 1\\leq j_1<j_2<...<j_n\\leq N}\nA_{i_1}...A_{i_n}B_{j_1}...B_{j_n}\\cdot \\\\\n&&\\llbrack \\prod_{1\\leq l<m\\leq n}\\left( \\nu _{i_l}-\\nu _{i_m}\\right) ^2\\left(\n\\rho _{j_l}-\\rho _{j_m}\\right) ^2\\rrbrack \\llbrack \\prod_{1\\leq l\\leq m\\leq\nn}\\left( \\nu _{i_l}-\\rho _{j_m}\\right) ^2 \\rrbrack ^{-1}\n\\er\nand \n\\br\n\\tau ^{\\pm } &=&\\frac 1{2\\pi i}\\oint dz.z\\left\\langle \\lambda _o\\right\|\n\\Gamma _{\\mp }\\left( z\\right) \\left( 1+a_1\\nu _1e^{-\\varphi _1}\\Gamma\n_{-}\\left( \\nu _1\\right) \\right) \\left( 1+b_1\\rho _1e^{\\eta _1}\\Gamma\n_{+}\\left( \\rho _1\\right) \\right) \\cdot \\cdot \\cdot \\\\\n&&\\left( 1+a_N\\nu _Ne^{-\\varphi _N}\\Gamma _{-}\\left( \\nu _N\\right) \\right)\n\\left( 1+b_N\\rho _Ne^{\\eta _N}\\Gamma _{+}\\left( \\rho _N\\right) \\right)\n\\left\| \\lambda _o\\right\\rangle .\n\\er\nAccording to Eq \\ref{33}, in order to have non vanishing terms, we must have equal number of operators $\\Gamma _{+}$ and $\\Gamma _{-}$\ninside the states $\\left\\langle \\lambda _o\\right\| $ and $\\left\| \\lambda\n_o\\right\\rangle $, then \n\\br\n\\tau ^{\\pm } &=&\\frac 1{2\\pi i}\\oint\ndz.z\\sum_{n=1}^NA_1^{m_1}...A_N^{m_N}B_1^{n_1}...B_N^{n_N}\\left\\langle\n\\lambda _o\\right\| \\Gamma _{\\mp }\\left( z\\right) \\Gamma _{-}^{m_1}\\left( \\nu\n_1\\right) ...\\Gamma _{-}^{m_N}\\left( \\nu _N\\right) \\cdot \\\\\n&&\\Gamma _{+}^{n_1}\\left( \\rho _1\\right) ...\\Gamma _{+}^{n_N}\\left( \\rho\n_N\\right) \\left\| \\lambda _o\\right\\rangle ,\n\\er\nwhere the exponents satisfy the following relations \n\\br\n\\sum_{i=1}^Nm_i\\pm 1 &=&\\sum_{i=1}^Nn_i=n \\\\\nm_i,n_i &=&0,1.\n\\er\nReordering the operators we may write \n\\br\n\\tau ^{+} &=&\\frac 1{2\\pi i}\\oint d\\nu .\\nu \\sum_{n=0}^{N-1}\\,\\,\\sum_{1\\leq\ni_1<i_2<...<i_n\\leq N,\\,\\, 1\\leq j_1<j_2<...<j_{n+1}\\leq N}\nA_{i_1}...A_{i_n}B_{j_1}...B_{j_{n+1}}\\\\\n&&\\left\\langle \\lambda _o\\right\| \\Gamma _{-}\\left( \\nu \\right) \\Gamma\n_{-}\\left( \\nu _{i_1}\\right) ...\\Gamma _{-}\\left( \\nu _{i_n}\\right) \\Gamma\n_{+}\\left( \\rho _{j_1}\\right) ...\\Gamma _{+}\\left( \\rho _{j_{n+1}}\\right)\n\\left\| \\lambda _o\\right\\rangle ,\n\\er\n\\br\n&=&\\frac 1{2\\pi i}\\sum_{n=0}^{N-1}\\,\\,\\sum_{ 1\\leq i_1<i_2<...<i_n\\leq N\\,\\, \n1\\leq j_1<j_2<...<j_{n+1}\\leq N} \\oint d\\nu .\\nu \\left(\n\\prod_{0<l\\leq n}\\epsilon \\left( -\\alpha ,\\alpha _{i_l}\\right) \\left( \\nu\n-\\nu _{i_l}\\right) ^2\\right) \\\\\n&&\\left( \\prod_{0<m\\leq n+1}\\epsilon \\left( -\\alpha ,\\alpha _{j_m}\\right)\n\\left( \\nu -\\rho _{j_m}\\right) ^2\\right) ^{-1}\nA_{i_1}...A_{i_n}B_{j_1}...B_{j_{n+1}} \\\\\n&&\\left( \\prod_{0\\leq l<m\\leq n}\\epsilon \\left( \\alpha _{i_l},\\alpha\n_{i_m}\\right) \\left( \\nu _{i_l}-\\nu _{i_m}\\right) ^2\\right) \\cdot \\left(\n\\prod_{0\\leq l<m\\leq n+1}\\epsilon \\left( \\alpha _{j_l},\\alpha _{j_m}\\right)\n\\left( \\rho _{j_l}-\\rho _{j_m}\\right) ^2\\right) \\\\\n&&\\left( \\prod_{ 1\\leq l\\leq m\\leq n+1,\\, l\\neq n+1 } \\epsilon\n\\left( \\alpha _{i_l},\\alpha _{j_m}\\right) \\left( \\nu _{i_l}-\\rho\n_{j_m}\\right) ^2\\right) ^{-1}\\cdot\n\\er\n\nIn Appendix \\ref{appf} we show that the contour integration in the variable \n$\\nu $ is equal to $2\\pi i$ for any value of $n$. Then, \n\\begin{eqnarray}\n\\tau ^{+} &=&\\sum_{n=0}^{N-1}\\sum_{ 1\\leq i_1<i_2<...<i_n\\leq N,\\, 1\\leq\nj_1<j_2<...<j_{n+1}\\leq N} A_{i_1}...A_{i_n}B_{j_1}...B_{j_{n+1}}%\n\\left( \\prod_{1\\leq l<m\\leq n}\\epsilon \\left( \\alpha _{i_l},\\alpha\n_{i_m}\\right) \\left( \\nu _{i_l}-\\nu _{i_m}\\right) ^2\\right) \\cdot \\\\\n&&\\left( \\prod_{1\\leq l<m\\leq n+1}\\epsilon \\left( \\alpha _{j_l},\\alpha\n_{j_m}\\right) \\left( \\rho _{j_l}-\\rho _{j_m}\\right) ^2\\right) .\\left(\n\\prod_{0<l\\leq n}\\epsilon \\left( -\\alpha ,\\alpha _{i_l}\\right) \\right) \\cdot\n\\\\\n&&\\left( \\prod_{1\\leq l\\leq m\\leq n+1}\\epsilon \\left( \\alpha _{i_l},\\alpha\n_{j_m}\\right) \\left( \\nu _{i_l}-\\rho _{j_m}\\right) ^2\\right) ^{-1}\\left(\n\\prod_{0<m\\leq n+1}\\epsilon \\left( -\\alpha ,\\alpha _{j_m}\\right) \\right)\n^{-1}\\cdot\n\\end{eqnarray}\n\nLikewise we have \n\\begin{eqnarray}\n\\tau ^{-} &=&\\frac 1{2\\pi i}\\oint d\\rho .\\rho \\sum_{n=0}^{N-1}\\,\\,\\sum_{ 1\\leq\ni_1<i_2<...<i_n\\leq N,\\,\\, 1\\leq j_1<j_2<...<j_{n+1}\\leq N}\nB_{i_1}...B_{i_n}A_{j_1}...A_{j_{n+1}}. \\\\\n&&\\left\\langle \\lambda _o\\right\| \\Gamma _{+}\\left( \\rho \\right) \\Gamma\n_{+}\\left( \\rho _{i_1}\\right) ...\\Gamma _{+}\\left( \\rho _{i_n}\\right) \\Gamma\n_{-}\\left( \\nu _{j_1}\\right) ...\\Gamma _{-}\\left( \\nu _{j_{n+1}}\\right)\n\\left\| \\lambda _o\\right\\rangle\n\\end{eqnarray}\n\\quad \n\\begin{eqnarray}\n&=&\\frac 1{2\\pi i}\\sum_{n=0}^{N-1}\\,\\,\\sum_{1\\leq i_1<i_2<...<i_n\\leq N,\\,\\,\n1\\leq j_1<j_2<...<j_{n+1}\\leq N} \\oint d\\rho .\\rho \\left(\n\\prod_{0<l\\leq n}\\epsilon \\left( \\alpha ,\\alpha _{i_l}\\right) \\left( \\rho\n-\\rho _{i_l}\\right) ^2\\right) \\\\\n&&\\left( \\prod_{0<m\\leq n+1}\\epsilon \\left( \\alpha ,\\alpha _{j_m}\\right)\n\\left( \\rho -\\nu _{j_m}\\right) ^2\\right) ^{-1}\nB_{i_1}...B_{i_n}A_{j_1}...A_{j_{n+1}} \\\\\n&&\\left( \\prod_{1\\leq l<m\\leq n}\\epsilon \\left( \\alpha _{i_l},\\alpha\n_{i_m}\\right) \\left( \\rho _{i_l}-\\rho _{i_m}\\right) ^2\\right) \\cdot \\left(\n\\prod_{1\\leq l<m\\leq n+1}\\epsilon \\left( \\alpha _{j_l},\\alpha _{j_m}\\right)\n\\left( \\nu _{j_l}-\\nu _{j_m}\\right) ^2\\right) \\\\\n&&\\left( \\prod_{ 1\\leq l\\leq m\\leq n+1 ,\\, l\\neq n+1} \\epsilon\n\\left( \\alpha _{i_l},\\alpha _{j_m}\\right) \\left( \\rho _{i_l}-\\nu\n_{j_m}\\right) ^2\\right) ^{-1},\n\\end{eqnarray}\nthe contour integration in $\\rho $ is also equal to $2\\pi i$ ( see Appendix \\ref{appf}) for any value of $n.$ Therefore \n\\begin{eqnarray}\n\\tau ^{-} &=&\\sum_{n=0}^{N-1}\\sum_{ 1\\leq i_1<i_2<...<i_n\\leq N,\\, 1\\leq\nj_1<j_2<...<j_{n+1}\\leq N} B_{i_1}...B_{i_n}A_{j_1}...A_{j_{n+1}}%\n\\left( \\prod_{1\\leq l<m\\leq n}\\epsilon \\left( \\alpha _{i_l},\\alpha\n_{i_m}\\right) \\left( \\rho _{i_l}-\\rho _{i_m}\\right) ^2\\right) \\cdot \\\\\n&&\\left( \\prod_{1\\leq l<m\\leq n+1}\\epsilon \\left( \\alpha _{j_l},\\alpha\n_{j_m}\\right) \\left( \\nu _{j_l}-\\nu _{j_m}\\right) ^2\\right) \\left(\n\\prod_{0<l\\leq n}\\epsilon \\left( \\alpha ,\\alpha _{i_l}\\right) \\right) \\cdot\n\\\\\n&&\\left( \\prod_{1\\leq l\\leq m\\leq n+1}\\epsilon \\left( \\alpha _{i_l},\\alpha\n_{j_m}\\right) \\left( \\rho _{i_l}-\\nu _{j_m}\\right) ^2\\right) ^{-1}\\left(\n\\prod_{0<m\\leq n+1}\\epsilon \\left( \\alpha ,\\alpha _{j_m}\\right) \\right)\n^{-1}\\cdot\n\\end{eqnarray}\n\nSimilar expressions were found by Kac and Wakimoto \\ct{wakimoto} in the context of their generalized Hirota equations approach.\n\n\n\\section{Acknowledgements} I am grateful to Profs. L.A. Ferreira and J.F. Gomes for enlightening discussions, and Prof. A.H. Zimerman for suggesting me to study this problem and valuable discussions. I thank Prof. B.M. Pimentel for his encouragement. This work was supported by FAPESP under grant 96/00212-0. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\appendix\n\n\n\n\n\\section{Appendix: The ``untwisted'' Kac-Moody algebras and their integrable highest-weight representations}\n\\label{appa}\n\nWe present the necessary Kac-Moody algebra notations and conventions used to construct integrable models, as well as, the theory of the so called ``highest weight integrable representations'' which are useful in the construction of their soliton solutions. A complete treatment can be found in \\ct{kac, wan}.\n\nAn ``untwisted'' Kac-Moody algebra $\\widehat{g}$ affine to a finite Lie algebra $g$ can be realized as an extension of the ``loop algebra'' of $g$:\n\n\\begin{equation}\n\\widehat{g}=\\left( g\\otimes {\\mathbf C} \\llbrack z,z^{-1}\\rrbrack \\right) \\oplus \n{\\mathbf C} C\\oplus {\\mathbf C} D, \\lab{a1}\n\\end{equation}\nwhere ${\\mathbf C} \\llbrack z,z^{-1}\\rrbrack$ is the algebra of Laurent Polynomials in $z$ and $\\mathbf{C}$$x$ ($x=C,D$) is a 1 dimensional subspace. Writing an element of the loop algebra as $a_{n\\equiv }$ $\\left( a\\otimes z^m\\right) ,$ where $a\\in g$ and $%\nn\\in \\mathbf{Z,}$ then the algebra can be written as\n\\br\n\\llbrack a_n,b_m\\rrbrack \\,=\\,\\llbrack a,b\\rrbrack _{n+m}+\\delta _{m+n,0}\\left(\na,b\\right) mC,\\nonu\\\\\n\\llbrack D,a_n\\rrbrack\\, =\\, na_n \\lab{algebra} \\\\\n\\llbrack D,D\\rrbrack=\\llbrack C,D\\rrbrack =\\llbrack C,C\\rrbrack =\\llbrack C,a_n\\rrbrack\n=0, \n\\er\nwhere $\\left( a,b\\right) $ is the Killing form of $g$ and $\\left[\na,b\\right] $ is the Lie bracket in $g$. $C$ is the central element of $\\widehat{g}$, and $D$ is a derivative operator which induces a natural integer gradation of $\\widehat{g}$\n\n\\[\n\\widehat{g}=\\bigoplus_{i\\in \\mathbf{Z}}\\widehat{g}_i, \n\\]\n$\\ $where $\\left[ D,\\widehat{g}_i\\right] =i\\widehat{g}_i.$ The operator $D$\ndefines the so called ``homogeneous gradation''.\n\nHere let us point out that there is a natural Heisenberg subalgebra of $\\widehat{g}$. Introducing a triangular decomposition of the finite algebra $g=n_{-}\\oplus h\\oplus n_{+}$, one can define an homogeneous Heisenberg algebra as an algebra composed of the elements \\{$h\\otimes z^n,C$\\} such that \n\n\n\\[\n\\llbrack a_n,b_m\\rrbrack =\\delta _{m+n,0}\\left( a,b\\right) mC. \n\\]\n\nIn the Cartan-Weyl basis the commutation relations are given as\n\n\\br\n\\llbrack H_i^m,H_j^n\\rrbrack =mC\\delta _{ij}\\delta _{m,-n}, \\lab{a2}\n\\er\n\\br\n\\llbrack H_i^m,E_\\alpha ^n\\rrbrack =\\alpha _iE_\\alpha ^{m+n}, \\lab{a3}\n\\er\n\\br\n\\llbrack E_\\alpha ^m,E_\\beta ^n\\rrbrack =\\left\\{ \n\\begin{tabular}{l}\n$\\varepsilon (\\alpha ,\\beta )E_{\\alpha +\\beta }^{m+n}\\quad ,\\mbox{if }\\,\\alpha\n+\\beta\\,\\,\\mbox{is a root}$ \\\\ \n$\\frac 2{\\alpha ^2}\\alpha \\cdot H^{m+n}+Cm\\delta _{m+n,0}, \\quad\n\\quad \\mbox{if }\\alpha +\\beta =0$ \\\\ \n$0\\quad \\quad \\quad \\quad \\quad \\quad ,\\mbox{in other case}$%\n\\end{tabular}\n\\right. \\lab{a4}\n\\er\n\\br\n\\llbrack C,E_\\alpha ^m\\rrbrack =\\llbrack C,H_\\alpha ^m\\rrbrack =0 \\lab{a5}\n\\er\n\\br\n\\llbrack D,E_\\alpha ^n\\rrbrack =nE_\\alpha ^n,\\qquad \\llbrack D,H_i^n\\rrbrack\n=nH_i^n. \\lab{a6}\n\\er\n\nIn this case the Cartan subalgebra is formed by the generators \\{$%\nH_i^{(0)},C,D$\\} and the step operators are:\\\\ \n- $E_\\alpha ^n$ associated to the roots $a=\\left( \\alpha ,0,n\\right) ,$ where $\\alpha $ belongs to the set of roots of $g$ and $n$ are integers,\\\\\n- $H_i^n$ associated to the roots $n\\delta =\\left( 0,0,n\\right) $ with $n\\neq 0.$\n\nThe positive roots are $\\left( \\alpha ,0,n\\right) $ for $n>0$ or \n $n=0$ and $\\alpha >0$, and among them the simple roots being $a_i=\\left(\n\\alpha _i,0,0\\right),\\,\\quad i=1,...r$,\\, and $\\alpha _o=\\left( -\\psi ,0,1\\right)\n $ with $\\psi$ the maximal root of $g$.\n\nThe representation theory of the Kac-Moody algebra in terms of vertex operators usually make use of the Cartan-Weyl basis \\ct{goddard}, see Appendix \\ref{appb}. Instead, to construct the so called ``integrable highest weight representations'' \\ct{kac, wan} of the ``untwisted'' affine Kac-Moody algebra we will need the Chevalley basis commutation relations (the notations and presentation here follow closely the Appendix of \\ct{ferreira1})\n\n\\br\n&&\\llbrack \\emph{H}_a^m,\\emph{H}_b^n\\rrbrack =mC\\eta _{ab}\\delta _{m+n,0} \\lab{a7}\\\\\n&&\\llbrack \\emph{H}_a^m,E_\\alpha ^n\\rrbrack =\\sum_{b=1}^rm_b^\\alpha\nK_{ba}E_\\alpha ^{m+n} \n\\lab{a8}\\\\\n&&\\llbrack E_\\alpha ^m,E_{-\\alpha }^n\\rrbrack\n=\\sum_{a=1}^rl_a^\\alpha \\emph{H}_a^{m+n}+\\frac 2{\\alpha ^2}mC\\delta\n_{m+n,0} \\lab{a9}\n\\\\\n&&\\left[ E_\\alpha ^m,E_\\beta ^n\\right] =\\left( q+1\\right)\n\\varepsilon (\\alpha ,\\beta )E_{\\alpha +\\beta }^{m+n};\\qquad \\mbox{se }\\alpha\n+\\beta \\mbox{ \\'{e} uma raiz\\qquad } \\lab{a10}\n\\\\\n&&\\left[ C,E_\\alpha ^m\\right] =\\left[ C,\\emph{H}_\\alpha ^m\\right]\n=0 \\lab{a11}\n\\\\\n&&\\left[ D,E_\\alpha ^n\\right] =nE_\\alpha ^n,\\qquad \\left[ D,\\emph{H}%\n_a^n\\right] =n\\emph{H}_a^n. \\lab{a12}\n\\er\nwhere $K_{ab}=2\\alpha _a\\cdot \\alpha _b/\\alpha _b^2$\\, is the Cartan matrix of the finite simple Lie algebra $g$ associated to $\\widehat{g}$ and generated by \\{$\\emph{H}_a^0,E_\\alpha ^0$\\}. $\\eta _{ab}=\\frac 2{\\alpha\n_a^2}K_{ab}=\\eta _{ba},$ $q$ is the highest positive integer such that $\\beta\n-q\\alpha $ is a root, $\\varepsilon (\\alpha ,\\beta )$ are $(\\pm)$ signs determined by the Jacobi identities, $l_a^\\alpha $ and $m_a^\\alpha $ are the integers in the expansion $\\alpha /\\alpha\n^2=\\sum_{a=1}^rl_a^\\alpha \\alpha _a/\\alpha _a^2$ and $\\alpha\n=\\sum_{a=1}^rm_a^\\alpha \\alpha _a$ respectively, where $\\alpha _1,...,\\alpha _r$ are the simple roots of $g\\,(r\\equiv \\mbox{rank of}\\, g).$ $\\widehat{g}$ \nhas a symmetric non-degenerate bilinear form which can be normalized as\n\\[\nTr\\left( \\emph{H}_a^m\\emph{H}_b^n\\right) =\\eta _{ab}\\delta _{m+n,0}\n\\]\n\\[\n\\qquad \\qquad Tr\\left( E_\\alpha ^mE_\\beta ^n\\right) =\\frac 2{\\alpha\n^2}\\delta _{\\alpha +\\beta ,0}\\delta _{m+n,0}\\quad \n\\]\n\\[\nTr\\left( CD\\right) =1.\\quad \n\\]\nThe integer gradations of $\\widehat{g}$ \n\\[\n\\widehat{g}=\\bigoplus_{n\\in \\mathbf{Z}}\\widehat{g}_n\n\\]\nhave been presented in \\ct{kac}. The gradation operator $Q_s$ satisfying \n\\begin{equation}\n\\left[ Q_{\\mathbf{s}},\\widehat{g}_n\\right] =n\\widehat{g}_n;\\qquad n\\in \n\\mathbf{Z,} \\lab{a13}\n\\end{equation}\nis defined by\n\\begin{equation}\nQ_{\\mathbf{s}}=H_{\\mathbf{s}}+N_{\\mathbf{s}}D+\\sigma C,\\quad H_{\\mathbf{s}%\n}=\\sum_{a=1}^rs_a\\lambda _a^v\\cdot H^0,\\quad H^0=\\left(\nH_1^0,...H_r^0\\right) \\lab{a14}\n\\end{equation}\nwhere $\\left( s_o,s_1,...,s_r\\right) $ is a $n-$tuple of non-negative co-prime integers, and $\\lambda _a^v\\equiv 2\\lambda _a/\\alpha _a^2$\nwith $\\lambda _a$ and $\\alpha _a$ being the fundamental weights and the simple roots of $g$ respectively. Moreover, \n\\begin{equation}\nN_s=\\sum_{i=0}^rs_im_i^\\psi ,\\quad \\psi =\\sum_{a=1}^rm_a^\\psi \\alpha\n_a,\\quad m_0^\\psi \\equiv 1, \\lab{a15}\n\\end{equation}\nwith $\\psi $ being the maximal root of $g$. The value of $\\sigma $ is arbitrary. Therefore \n\\[\n\\quad \\left[ Q_{\\mathbf{s}},\\emph{H}_a^n\\right] =nN_{\\mathbf{s}}\\emph{H}%\n_a^n\\qquad \\qquad \\qquad \\qquad \n\\]\n\\[\n\\left[ Q_{\\mathbf{s}},E_\\alpha ^n\\right] =\\left( \\sum_{a=1}^rm_a^\\alpha\ns_a+nN_{\\mathbf{s}}\\right) E_\\alpha ^n.\n\\]\nThe positive and negative ``step operators'' of $\\widehat{g}$ associated to the simple roots are: \n\\begin{equation}\ne_a\\equiv E_{\\alpha _a}^0,\\quad e_o\\equiv E_{-\\psi }^1,\\quad f_a\\equiv\nE_{-\\alpha _a}^o\\quad \\mbox{e \\quad }f_o\\equiv E_\\psi ^{-1}, \\lab{a16}\n\\end{equation}\nand its Cartan subalgebra generated by\n\\begin{equation}\nh_a\\equiv \\emph{H}_a^0,\\quad h_0\\equiv -\\sum_{a=1}^rl_a^\\psi \\emph{H}%\n_a^0+\\frac 2{\\psi ^2}C\\mbox{ \\quad e \\quad }D, \\lab{a17}\n\\end{equation}\nwith $l_a^\\psi$ given above; then they satisfy \n\\begin{equation}\n\\left[ Q_{\\mathbf{s}},h_i\\right] =\\left[ Q_{\\mathbf{s}},D\\right] =0;\\quad\n\\left[ Q_{\\mathbf{s}},e_i\\right] =s_ie_i;\\qquad \\left[ Q_{\\mathbf{s}%\n},f_i\\right] =-s_if_i;\\quad i=0,1,...,r. \\lab{a18}\n\\end{equation}\n\nAn important class of representations of the Kac-Moody algebra are the so called ``integrable highest-weight representations'' \\ct{kac}. They are defined in terms of a highest weight $\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle $ labelled by the gradation $\\mathbf{s}$ of $\\widehat{g}.$ That state is annihilated by the positive grade generators \n\\begin{equation}\n\\widehat{g}_{+}\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle =0, \\lab{a19}\n\\end{equation}\nand it is an eigenstate of the generators of the subalgebra $\\widehat{g}_o$ \n\\begin{equation}\nh_i\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle =s_i\\left\| \\lambda _{\\mathbf{s}%\n}\\right\\rangle \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\quad \\lab{a20}\n\\end{equation}\n\\begin{equation}\nf_i\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle =0;\\quad \\quad \\mbox{for any }i\\mbox{ with }s_i=0 \\lab{a21}\n\\end{equation}\n\\begin{equation}\nQ_s\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle =\\eta _s\\left\| \\lambda _{%\n\\mathbf{s}}\\right\\rangle \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \n\\lab{a22}\n\\end{equation}\n\\begin{equation}\n\\quad C\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle =\\frac{\\psi ^2}2\\left(\n\\sum_{i=0}^rl_i^\\psi s_i\\right) \\left\| \\lambda _{\\mathbf{s}}\\right\\rangle\n,\\qquad \\qquad \\qquad \\quad \\lab{a23}\n\\end{equation}\nwhere $l_i^\\psi $ is given by \n\\begin{equation}\n\\frac \\psi {\\psi ^2}=\\sum_{a=1}^rl_a^\\psi \\frac{\\alpha _a}{\\alpha _a^2}%\n;\\quad l_0^\\psi =1. \\lab{a24}\n\\end{equation}\nThe eigenvalue of the central element $C$ in the representation $\\left\|\n\\lambda _{\\mathbf{s}}\\right\\rangle ,$ is known as the level of the representation \n\\begin{equation}\nc=\\frac{\\psi ^2}2\\left( \\sum_{i=0}^rl_i^\\psi s_i\\right) , \\lab{a25}\n\\end{equation}\nin particular, the ``highest-weight integrable representations'' with $c=1$ are known as ``basic representations''.\n\nThe states of highest weight $\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle $ can be realized as \n\n\\begin{equation}\n\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle \\equiv \\bigotimes_{i=0}^r\\left\| \n\\widehat{\\lambda }_i\\right\\rangle ^{\\otimes s_i}, \\lab{a26}\n\\end{equation}\nwhere $\\left\| \\widehat{\\lambda }_i\\right\\rangle $ are the highest states of the fundamental representations of $\\widehat{g},$ and $\n\\widehat{\\lambda }_i$ are the relevant fundamental weights of $\\widehat{g},$ . They are given by \\ct{goddard}\n\\begin{equation}\n\\widehat{\\lambda }_0=\\left( 0,\\frac{\\psi ^2}2,0\\right) \\qquad \\lab{a27}\n\\end{equation}\n\\begin{equation}\n\\widehat{\\lambda }_a=\\left( \\lambda _a,\\frac{l_a^\\psi \\psi ^2}2,0\\right) ,\n\\lab{a28}\n\\end{equation}\nwhere $\\lambda _a$, $a=1,2...,r$ are the fundamental weights of the finite Lie algebra $g$ associated to $\\widehat{g}$, $l_a^\\psi $ is defined in \n\\ref{a24}, and the corresponding components are the eigenvalues of $\\emph{H}_a^0,$ $C$ and $%\nD$ respectively, viz. \n\\begin{equation}\n\\emph{H}_a^0\\left\| \\widehat{\\lambda }_0\\right\\rangle =0;\\quad C\\left\| \n\\widehat{\\lambda }_0\\right\\rangle =\\frac{\\psi ^2}2\\left\| \\widehat{\\lambda }%\n_0\\right\\rangle \\lab{a29}\n\\end{equation}\n\\begin{equation}\n\\emph{H}_b^0\\left\| \\widehat{\\lambda }_a\\right\\rangle =\\delta _{a,b}\\left\| \n\\widehat{\\lambda }_a\\right\\rangle ;\\quad C\\left\| \\widehat{\\lambda }%\n_a\\right\\rangle =\\frac{\\psi ^2}2l_a^\\psi \\left\| \\widehat{\\lambda }%\n_0\\right\\rangle \\lab{a30}\n\\end{equation}\nand\n\\begin{equation}\nD\\left\| \\widehat{\\lambda }_i\\right\\rangle =0. \\lab{a31}\n\\end{equation}\nLet us notice that in each of the $r+1$ fundamental representations of $\\widehat{g}$, the (unique) highest weight state satisfies\n\\begin{equation}\nh_j\\left\| \\widehat{\\lambda }_i\\right\\rangle =\\delta _{ij}\\left\| \\widehat{%\n\\lambda }_i\\right\\rangle \\lab{a32}\n\\end{equation}\n\\begin{equation}\ne_j\\left\| \\widehat{\\lambda }_i\\right\\rangle =0,\\,\\forall \\,j\n\\lab{a33}\n\\end{equation}\n\\begin{equation}\n\\qquad \\qquad f_j\\left\| \\widehat{\\lambda }_i\\right\\rangle =0,\\mbox{ \\quad\nfor }j\\neq i \\lab{a34}\n\\end{equation}\n\\begin{equation}\nf_j^2\\left\| \\widehat{\\lambda }_i\\right\\rangle =0.\\qquad \\lab{a35}\n\\end{equation}\n\nThen the generators $e_i$ and $f_{i}$ are nilpotent when acting on $\\left\| \\lambda _{\\mathbf{s}}\\right\\rangle ,$ and these are indeed, integrable representations.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{\\bf The construction of the homogeneous vertex operators}\n\\label{appb}\nThe construction of vertex operator representations of Kac-Moody algebras can be found in \\ct{kac, vertex}. The construction of the homogeneous vertex operators is based on the homogeneous Heisenberg subalgebra\\,${\\mathbf h}$. Here we follow the construction developed in \\ct{kac}.\n\nConsider\n\n\\begin{equation}\ng={\\bf h}\\bigoplus \\left( \\bigoplus_{\\alpha \\in \\Delta }{\\bf C}E_\\alpha\n\\right) , \\label{1}\n\\end{equation}\n$g$ being a simple finite Lie algebra of type $A_l,D_l$ or $E_l$, whose\ncommutation relations are given by \n\\br\n\\llbrack h,h^{,}\\rrbrack =0,\\qquad \\mbox{ \\qquad \\qquad if }h,\\,h^{,}\\in \n{\\bf h\\qquad } \n\\er\n\\br\n\\quad \\quad \\llbrack h,E_\\alpha \\rrbrack =\\left( h\\mid \\alpha \\right) E_\\alpha ,%\n\\mbox{ \\quad \\quad if }h\\in {\\bf h,}\\quad \\alpha \\in \\Delta \\quad \n\\er\n\\[\n\\llbrack E_\\alpha ,E_{-\\alpha }\\rrbrack =-\\alpha ,\\qquad \\qquad \\qquad \\mbox{if \n}\\alpha \\in \\Delta \\qquad \\qquad \n\\]\n\\[\n\\quad \\quad \\llbrack E_\\alpha ,E_\\beta \\rrbrack =0,\\qquad \\mbox{if }\\alpha\n,\\beta \\in \\Delta ,\\quad \\alpha +\\beta \\notin \\Delta \\cup \\left\\{ 0\\right\\} \n\\]\n\\[\n\\llbrack E_\\alpha ,E_\\beta \\rrbrack =\\varepsilon \\left( \\alpha ,\\beta \\right)\nE_{\\alpha +\\beta },\\qquad \\mbox{if }\\alpha ,\\beta ,\\alpha +\\beta \\in \\Delta\n, \n\\]\nwhere $\\Delta =\\left\\{ \\alpha \\in Q/(\\alpha \\mid \\alpha )=2\\right\\} $ and $Q$\nis the root lattice; $\\left( \\quad \\mid \\quad \\right) $ is the invariant\nsymmetric form of $g,$ normalized as follows$:$ \n\\[\n(h\\mid E_\\alpha )=0,\\mbox{ if }h\\in {\\bf h,}\\quad \\alpha \\in \\Delta ;\\qquad\n(E_\\alpha ,E_\\beta )=-\\delta _{\\alpha ,-\\beta },\\mbox{ if }\\alpha ,\\beta \\in\n\\Delta . \n\\]\nLet\n\n\\begin{equation}\n\\widehat{g}={\\bf C}\\left[ t,t^{-1}\\right] \\otimes _{{\\bf C}}g+{\\bf C}C+{\\bf C%\n}D, \\label{2}\n\\end{equation}\nbe an affine algebra of type $A_n^{\\left( 1\\right) },D_n^{\\left( 1\\right) }$\nor $E_n^{\\left( 1\\right) },$ respectively.\n\nConsider the complex commutative associative algebra \n\\begin{equation}\nV=S\\left( \\bigoplus_{j<0}(t^j\\otimes {\\bf h})\\right) \\otimes _{{\\bf C}}{\\bf C%\n}\\left[ Q\\right] , \\label{3}\n\\end{equation}\nwhere $S$ stands for the symmetric algebra and ${\\bf C}\\left[ Q\\right] $\nstands for the group algebra of the root lattice $Q\\subset {\\bf h}$ of $g$.\nLet $\\alpha $ $\\rightarrow e^\\alpha $ denote the inclusion $Q\\subset {\\bf C}%\n\\left[ Q\\right] $(a base of the vector space ${\\bf C}\\left[ Q\\right] $ is\ngiven by the elements $e^\\alpha ,$ $\\alpha \\in Q,$ and, it is defined the\ntwisted product of the group algebra elements, $e^\\alpha e^\\beta\n=\\varepsilon \\left( \\alpha ,\\beta \\right) $ $e^{\\alpha +\\beta }$ \\ct{vandeleur}); $%\nu^{\\left( n\\right) }$ will stand for $t^n\\otimes u$ ($n\\in {\\bf Z},$ $u\\in g$%\n). For $n>0$, $u\\in {\\bf h,}$ denote by $u\\left( -n\\right) $ the operator on \n$V$ of multiplication by $u^{\\left( -n\\right) }$ $.$ For $n\\geq 0$, $u\\in \n{\\bf h,}$ denote by $u\\left( n\\right) $ the derivation of the algebra $V$\ndefined by the formula\n\n\\begin{equation}\nu(n)\\left( v^{\\left( -m\\right) }\\otimes e^\\alpha \\right) =n\\delta\n_{n,-m}\\left( u\\mid v\\right) \\otimes e^\\alpha +\\delta _{n,0}\\left( \\alpha\n\\mid u\\right) v^{\\left( -m\\right) }\\otimes e^\\alpha \\label{4}\n\\end{equation}\n\nChoosing dual bases $u_i$ and $u^i$ of ${\\bf h}$, define the operator $D_o$\non $V$ by the formula \n\\begin{equation}\nD_o=\\sum_{i=1}^l\\left( \\frac 12u_i(0)u^i(0)+\\sum_{n\\geq\n1}u_i(-n)u^i(n)\\right) . \\label{5}\n\\end{equation}\nFurthermore, for $\\alpha \\in Q,$ define the sign operator $c_\\alpha $: \n\\begin{equation}\nc_\\alpha \\left( f\\otimes e^\\beta \\right) =\\varepsilon (\\alpha ,\\beta\n)f\\otimes e^\\beta . \\label{6}\n\\end{equation}\nFinaly, for $\\alpha \\in \\Delta \\subset Q$ introduce the {\\bf vertex operator}\n\\begin{equation}\n\\Gamma _\\alpha (z)=\\exp \\left( \\sum_{j\\geq 1}\\frac{z^j}j\\alpha (-j)\\right)\n\\exp \\left( -\\sum_{j\\geq 1}\\frac{z^{-j}}j\\alpha (j)\\right) e^\\alpha\nz^{\\alpha (0)}c_\\alpha , \\label{7}\n\\end{equation}\nhere $z$ is viewed as an indeterminate. Expanding in powers of $z$: \n\\begin{equation}\n\\Gamma _\\alpha (z)=\\sum_{j\\in {\\bf Z}}\\Gamma _\\alpha ^{\\left( j\\right)\n}z^{-j-1}, \\label{8}\n\\end{equation}\nwe obtain a sequence of operators $\\Gamma _\\alpha ^{\\left( j\\right) }$ on $V \n$. Now we can state the result.\n\n{\\bf Theorem}: The map $\\sigma :\\widehat{g}\\rightarrow End\\left( V\\right) $\ndefined by \n\\begin{eqnarray}\nC &\\longrightarrow &1, \\nonumber \\\\\nu^{\\left( n\\right) } &\\longrightarrow &u\\left( n\\right) ,\\qquad \\mbox{for }%\nu\\in {\\bf h,\\qquad n}\\in {\\bf Z,} \\nonumber \\\\\nE_\\alpha ^{\\left( n\\right) } &\\longrightarrow &\\Gamma _\\alpha ^{\\left(\nn\\right) },\\qquad \\mbox{for }\\alpha \\in \\Delta {\\bf ,\\qquad n}\\in {\\bf Z,} \n\\nonumber \\\\\nD &\\longrightarrow &-D_o, \\label{9}\n\\end{eqnarray}\ndefines the basic representation of the affine algebra $\\widehat{g}$ on $V$. \n$End\\left( V\\right) $ denotes the space of the linear maps of an vector\nspace $V$ on itself.\n\nThe proof of this theorem is presented in \\ct{kac}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{\\bf Homogeneous vertex operator calculus}\n\\label{appc}\nDefining\n\n\\begin{equation}\n\\Gamma _\\alpha ^{\\pm }(z)=\\exp \\sum_{j\\geq 1}\\frac{\\alpha \\left( \\pm\nj\\right) }{\\mp j}z^{\\mp j},\\quad \\Gamma _\\alpha ^0(z)=e^\\alpha z^{\\alpha\n\\left( 0\\right) }c_\\alpha , \\label{19}\n\\end{equation}\nthe following relations can be obtained: \n\\begin{equation}\n\\Gamma _\\alpha ^{-}(z_1)\\Gamma _\\beta ^{+}(z_2)=\\Gamma _\\beta\n^{+}(z_2)\\Gamma _\\alpha ^{-}(z_1)\\left( 1-\\frac{z_2}{z_1}\\right) ^{\\left(\n\\alpha \\mid \\beta \\right) } \\label{20}\n\\end{equation}\nand \n\\begin{equation}\n\\Gamma _\\alpha ^0(z_1)\\Gamma _\\beta ^0(z_2)=e^{\\alpha +\\beta }z_1^{\\alpha\n\\left( 0\\right) }z_2^{\\beta \\left( 0\\right) }z_1^{\\left( \\alpha \\mid \\beta\n\\right) }c_\\alpha c_\\beta \\varepsilon \\left( \\alpha ,\\beta \\right) ,\\quad\n\\label{21}\n\\end{equation}\nwhere $\\left( 1-\\frac{z_2}{z_1}\\right) ^m,$ $m\\in {\\bf Z,}$ with $\\mid \\frac{%\nz_2}{z_1}\\mid $ $\\leq 1.$\n\nFrom (\\ref{20}) and (\\ref{21}) it follows \n\\begin{eqnarray}\n\\Gamma _\\alpha (z_1)\\Gamma _\\beta (z_2) &=&\\left( 1-\\frac{z_2}{z_1}\\right)\n^{\\left( \\alpha \\mid \\beta \\right) }z_1^{\\left( \\alpha \\mid \\beta \\right)\n}\\varepsilon (\\alpha ,\\beta )\\exp \\left[ \\sum_{j\\geq 1}\\frac 1j\\left(\nz_1^j\\alpha (-j)+z_2^j\\beta (-j)\\right) \\right] \\nonumber \\\\\n&&\\exp \\left[ -\\sum_{j\\geq 1}\\frac 1j\\left( z_1^{-j}\\alpha (j)+z_2^{-j}\\beta\n(j)\\right) \\right] e^{\\alpha +\\beta }z_1^{\\alpha \\left( 0\\right) }z_2^{\\beta\n\\left( 0\\right) }c_\\alpha c_\\beta . \\label{22}\n\\end{eqnarray}\nIf $z\\equiv z_1=z_2\\quad $and$\\quad \\alpha =\\beta ,$ from (\\ref{22}) we can\nobtain\n\\[\n\\Gamma _\\alpha (z)\\Gamma _\\alpha (z)=0, \n\\]\nor \n\\begin{equation}\n\\left[ \\Gamma _\\alpha (z)\\right] ^n=0,\\qquad \\mbox{for\\quad }n\\geq 2.\n\\label{23}\n\\end{equation}\n{\\bf A useful formula} \n\\[\n\\Gamma _{\\alpha _N}(z_N)\\Gamma _{\\alpha _{N-1}}(z_{N-1})...\\Gamma _{\\alpha\n_1}(z_1)\\left( 1\\otimes 1\\right) =\\prod_{1\\leq i<j\\leq N}\\varepsilon (\\alpha\n_i,\\alpha _j)\\left( z_i-z_j\\right) ^{\\left( \\alpha _i\\mid \\alpha _j\\right)\n}\\cdot \n\\]\n\\begin{equation}\n\\left( \\prod_{i=1}^N\\exp \\sum_{j\\in {\\bf N}}\\frac{z_i^j}j\\alpha\n_i(-j)\\right) \\otimes \\exp \\left( \\sum_{i=1}^N\\alpha _i\\right) \\label{24}\n\\end{equation}\n\n{\\bf Proof}.: We will prove by induction\n\nfor $N=1$\n\\begin{equation}\n\\Gamma _\\alpha (z_1)\\left( 1\\otimes 1\\right) =\\exp \\sum_{j\\geq 1}\\frac{z_1^j}%\nj\\alpha (-j)\\exp -\\sum_{j\\geq 1}\\frac{z_1^{-j}}j\\alpha (j)e^\\alpha\nz_1^{\\alpha \\left( 0\\right) }c_\\alpha \\quad ; \\label{25}\n\\end{equation}\nwe have the following relations: \n\\begin{eqnarray}\nc_\\alpha \\left( 1\\otimes 1\\right) &=&\\varepsilon (\\alpha ,0)1\\otimes\n1,\\qquad \\mbox{consider the convention: }\\varepsilon (\\alpha ,0)\\equiv 1 \\nonumber \\\\\n&=&1\\otimes 1 \\label{26}\n\\end{eqnarray}\n\\begin{eqnarray}\nz_1^{\\alpha \\left( 0\\right) }\\left( 1\\otimes 1\\right) &=&1\\otimes \\left(\ne^{\\ln z_1\\alpha \\left( 0\\right) }\\right) 1 \\nonumber \\\\\n&=&1\\otimes 1. \\label{27}\n\\end{eqnarray}\nwhere we used \n\\begin{equation}\ne^\\alpha \\left( 1\\otimes 1\\right) \\equiv e^\\alpha \\left( 1\\otimes e^\\alpha\n\\right) =1\\otimes e^\\alpha \\label{28}\n\\end{equation}\n\nFrom (\\ref{4}) we have \n\\[\nu(n)\\left( v^{\\left( 0\\right) }\\otimes e^\\alpha \\right) \\equiv u(n)\\left(\n1\\otimes e^\\alpha \\right) =n\\delta _{n,0}\\left( u\\mid v\\right) \\otimes\ne^\\alpha +\\delta _{n,0}\\left( \\alpha \\mid u\\right) v^{\\left( 0\\right)\n}\\otimes e^\\alpha \n\\]\nthen for $(n>0)$ \n\\begin{equation}\nu(n)\\left( 1\\otimes 1\\right) =0.\\qquad \\label{29}\n\\end{equation}\n\nTherefore \n\\begin{eqnarray}\n\\exp -\\sum_{j\\geq 1}\\frac{z_1^{-j}}j\\alpha (j)\\left( 1\\otimes e^\\alpha\n\\right) &=&\\left[ \\left( 1-\\sum_{j\\geq 1}\\frac{z_1^{-j}}j\\alpha\n(j)+...\\right) 1\\right] \\otimes e^\\alpha \\nonumber \\\\\n&=&1\\otimes e^\\alpha . \\label{30}\n\\end{eqnarray}\n\nBesides we can write \n\\[\n\\exp \\sum_{j\\geq 1}\\frac{z_1^j}j\\alpha (-j)\\left( 1\\otimes e^\\alpha \\right)\n=\\sum_{j\\geq 1}\\frac{z_1^j}j\\alpha (-j)\\otimes e^\\alpha , \n\\]\nthus \n\\begin{equation}\n\\Gamma _{\\alpha _1}(z_1)\\left( 1\\otimes 1\\right) =\\left( \\exp \\sum_{j\\geq 1}%\n\\frac{z_1^j}j\\alpha _1(-j)\\right) \\otimes e^{\\alpha _1}, \\label{31}\n\\end{equation}\nwhich is equal to the equation (\\ref{24}) for $N=1.$\n\nNow, let us assume that (\\ref{24}) is true for a given $N$, and we will\nwrite (\\ref{24}) as \n\\[\n\\Gamma _{\\alpha _N}(z_N)\\Gamma _{\\alpha _{N-1}}(z_{N-1})...\\Gamma _{\\alpha\n1}(z_1)\\left( 1\\otimes 1\\right) =\\prod_{1\\leq i<j\\leq N}\\varepsilon (\\alpha\n_i,\\alpha _j)\\left( z_i-z_j\\right) ^{\\left( \\alpha _i\\mid \\alpha _j\\right)\n}\\cdot \n\\]\n\\begin{equation}\n\\left( \\prod_{i=1}^N\\exp \\sum_{j\\in {\\bf N}}\\frac{z_i^j}j\\alpha\n_i(-j)\\right) \\exp \\left( \\sum_{i=1}^N\\alpha _i\\right) \\left( 1\\otimes\n1\\right) . \\label{32}\n\\end{equation}\n\nMultiplying (\\ref{32}) by $\\Gamma _{\\alpha _{N+1}}(z_{N+1})$ and using the\nrelations (\\ref{20}), (\\ref{21}) and (\\ref{25})-(\\ref{30}) many times, we\ncan write\n\\[\n\\Gamma _{\\alpha _{N+1}}(z_{N+1})\\Gamma _{\\alpha _N}(z_N)...\\Gamma _{\\alpha\n1}(z_1)\\left( 1\\otimes 1\\right) =\\prod_{1\\leq i<j\\leq N}\\varepsilon (\\alpha\n_i,\\alpha _j)\\left( z_i-z_j\\right) ^{\\left( \\alpha _i\\mid \\alpha _j\\right)\n}\\cdot \n\\]\n\n\\[\n\\prod_{1\\leq k<N+1}\\varepsilon (\\alpha _k,\\alpha _{N+1})\\left(\nz_k-z_{N+1}\\right) ^{\\left( \\alpha _k\\mid \\alpha _{N+1}\\right) }\\cdot \\left(\n\\exp \\sum_{j\\in {\\bf N}}\\frac{z_{N+!}^j}j\\alpha _{N+1}(-j)\\right) e^{\\alpha\n_{N+1}}\\cdot \n\\]\n\\[\n\\left( \\prod_{i=1}^N\\exp \\sum_{j\\in {\\bf N}}\\frac{z_i^j}j\\alpha\n_i(-j)\\right) \\exp \\left( \\sum_{i=1}^N\\alpha _i\\right) \\left( 1\\otimes\n1\\right) \n\\]\n\\begin{eqnarray}\n&=&\\prod_{1\\leq i<j\\leq N+1}\\varepsilon (\\alpha _i,\\alpha _j)\\left(\nz_i-z_j\\right) ^{\\left( \\alpha _i\\mid \\alpha _j\\right) }\\left(\n\\prod_{i=1}^{N+1}\\exp \\sum_{j\\in {\\bf N}}\\frac{z_i^j}j\\alpha _i(-j)\\right)\n\\cdot \\\\\n&&\\exp \\left( \\sum_{i=1}^{N+1}\\alpha _i\\right) \\left( 1\\otimes 1\\right) \\\\\n&=&\\prod_{1\\leq i<j\\leq N+1}\\varepsilon (\\alpha _i,\\alpha _j)\\left(\nz_i-z_j\\right) ^{\\left( \\alpha _i\\mid \\alpha _j\\right) }\\left(\n\\prod_{i=1}^{N+1}\\exp \\sum_{j\\in {\\bf N}}\\frac{z_i^j}j\\alpha _i(-j)\\right)\n\\otimes \\exp \\left( \\sum_{i=1}^{N+1}\\alpha _i\\right) ,\n\\end{eqnarray}\nwhich is exactly the expression (\\ref{24}) written for $N+1$. This way the\nequation (\\ref{24}) is proved.\n\nMoreover, making the correspondence\n\n\\[\n\\left\| \\lambda _o\\right\\rangle \\longleftrightarrow 1\\otimes 1, \n\\]\nwhere $\\left\| \\lambda _o\\right\\rangle $ denotes a highest weight state of a\nfundamental representation, we can write\n\\[\n\\left\\langle \\lambda _o\\right\| \\Gamma _{\\alpha _N}(z_N)\\Gamma _{\\alpha\n_{N-1}}(z_{N-1})...\\Gamma _{\\alpha _1}(z_1)\\left\| \\lambda _o\\right\\rangle = \n\\]\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n0,\\qquad \\qquad \\qquad \\mbox{ \\qquad \\qquad \\qquad \\qquad \\qquad if }%\n\\sum_{i=1}^N\\alpha _i\\neq 0 \\\\ \n\\prod_{1\\leq i<j\\leq N}\\varepsilon (\\alpha _i,\\alpha _j)\\left(\nz_i-z_j\\right) ^{\\left( \\alpha _i\\mid \\alpha _j\\right) }\\qquad \\quad \\mbox{%\nif }\\sum_{i=1}^N\\alpha _i=0\n\\end{array}\n\\right. \\label{33}\n\\end{equation}\nwhere we have used the fact that\n\\begin{eqnarray}\n\\left\\langle \\lambda _o\\right\| \\exp \\sum_{j\\in {\\bf N}}\\frac{z_i^j}j\\alpha\n_i(-j) &=&\\left\\langle \\lambda _o\\right\| \\left( 1+\\sum_{j\\in {\\bf N}}\\frac{%\nz_i^j}j\\alpha _i(-j)+...\\right) \\\\\n&=&\\left\\langle \\lambda _o\\right\| ,\n\\end{eqnarray}\n\nand \n\\[\n\\left\\langle \\lambda _o\\right\| \\exp \\left( \\sum_{i=1}^N\\alpha _i\\right)\n=\\left\\{ \n\\begin{array}{c}\n0,\\mbox{\\qquad \\qquad \\qquad \\qquad if }\\sum_{i=1}^N\\alpha _i\\neq 0 \\\\ \n\\left\\langle \\lambda _o\\right\| ,\\mbox{\\qquad \\qquad \\qquad \\quad if }%\n\\sum_{i=1}^N\\alpha _i=0\n\\end{array}\n\\right. \n\\]\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{\\bf Matrix elements using a vertex operator representation of the\nalgebra }$\\widehat{sl}_{r+1}$}\n\n\\label{appd}\nConsider the correspondence \n\n\\[\nF_i\\longrightarrow \\nu _i\\Gamma _{-\\beta _i}\\left( \\nu _i\\right) , \n\\]\n\n\\[\nG_i\\longrightarrow \\rho _i\\Gamma _{\\beta _i}\\left( \\rho _i\\right) .\\qquad \n\\]\nthen we can write \n\\[\nC_{_{l_1...i_{l_1},}\\quad j_{k_1}...j_{k_n}}=\\left\\langle \\lambda _o\\right\|\nF_{i_{l_1}}...F_{i_{l_n}}G_{j_{k_1}}...G_{j_{k_n}}\\left\| \\lambda\n_o\\right\\rangle \n\\]\n\\[\n=\\nu _{i_{l_1}}...\\nu _{i_{l_n}}\\rho _{j_{k_1}}...\\rho\n_{j_{k_n}}\\left\\langle \\lambda _o\\right\| \\Gamma _{-\\beta _{i_{l_1}}}\\left(\n\\nu _{i_{l_1}}\\right) ...\\Gamma _{-\\beta _{i_{l_n}}}\\left( \\nu\n_{i_{l_n}}\\right) \\Gamma _{\\beta _{j_{k_1}}}\\left( \\rho _{j_{k_1}}\\right)\n...\\Gamma _{\\beta _{j_{k_n}}}\\left( \\rho j_{k_n}\\right) \\left\| \\lambda\n_o\\right\\rangle \n\\]\n\\begin{eqnarray}\n\\quad \\quad &=&{\\bf \\{\\delta }_{\\overrightarrow{0},\\beta _{j_{k_1}}+\\cdot\n\\cdot \\cdot \\beta _{j_{k_n}}-\\beta _{i_{l_1}}-\\cdot \\cdot \\cdot \\beta\n_{i_{l_n}}}\\nu _{i_{l_1}}...\\nu _{i_{l_n}}\\rho _{j_{k_1}}...\\rho\n_{j_{k_n}}\\cdot \\\\\n&&\\prod_{1\\leq a<b\\leq n}\\epsilon \\left( \\beta _{j_{k_a}},\\beta\n_{j_{k_b}}\\right) \\epsilon \\left( -\\beta _{i_{l_a}},-\\beta _{i_{l_b}}\\right)\n\\left( \\rho _{j_{k_a}}-\\rho _{j_{k_b}}\\right) ^{\\left( \\beta _{j_{k_a}}\\mid\n\\beta _{j_{k_b}}\\right) }\\cdot\n\\end{eqnarray}\n\\begin{equation}\n\\left( \\nu _{i_{l_a}}-\\nu _{i_{l_b}}\\right) ^{\\left( \\beta _{i_{l_a}}\\mid\n\\beta _{i_{l_b}}\\right) }{\\bf \\}}/{\\bf \\{}\\prod_{1\\leq a<b\\leq n}\\epsilon\n\\left( -\\beta _{i_{l_a}},\\beta _{j_{k_b}}\\right) \\left( \\nu _{i_{l_a}}-\\rho\n_{j_{k_b}}\\right) ^{\\left( \\beta _{i_{l_a}}\\mid \\beta _{j_{k_b}}\\right) }%\n{\\bf \\},} \\label{34}\n\\end{equation}\nwhere we have used equation (\\ref{33})$.$ \n\\begin{eqnarray}\nC_{ii_{l_1}...i_{l_n},j_{k_1}...j_{k_{n+1}}}^{+} &=&\\left\\langle \\lambda\n_o\\right\| E_{-\\beta _i}^{\\left( 1\\right)\n}F_{i_{l_1}}...F_{i_{l_n}}G_{j_{k_1}}...G_{j_{k_{n+1}}}\\left\| \\lambda\n_o\\right\\rangle \\\\\n&=&\\frac 1{2\\pi i}\\oint d\\nu .\\nu \\nu _{i_{l_1}}...\\nu _{i_{l_n}}\\rho\n_{j_{k_1}}...\\rho _{j_{k_{n+1}}}\\cdot\n\\end{eqnarray}\n\\[\n\\left\\langle \\lambda _o\\right\| \\Gamma _{-\\beta _i}\\left( \\nu \\right) \\Gamma\n_{-\\beta _{i_{l_1}}}\\left( \\nu _{i_{l_1}}\\right) ...\\Gamma _{-\\beta\n_{i_{l_n}}}\\left( \\nu _{i_{l_n}}\\right) \\Gamma _{\\beta _{j_{k_1}}}\\left(\n\\rho _{j_{k_1}}\\right) ...\\Gamma _{\\beta _{j_{k_{n+1}}}}\\left( \\rho\nj_{k_{n+1}}\\right) \\left\| \\lambda _o\\right\\rangle \n\\]\n\\begin{eqnarray}\n&=&\\frac 1{2\\pi i}\\oint d\\nu .\\nu \\nu _{i_{l_1}}...\\nu _{i_{l_n}}\\rho\n_{j_{k_1}}...\\rho _{j_{k_{n+1}}}{\\bf \\delta }_{\\overrightarrow{0},\\beta\n_{j_{k_1}}+\\cdot \\cdot \\cdot \\beta _{j_{k_{n+1}}}-\\beta _i-\\beta\n_{i_{l_1}}-\\cdot \\cdot \\cdot \\beta _{i_{l_n}}}\\cdot \\\\\n&&{\\bf \\{}\\prod_{0<a\\leq n}\\epsilon \\left( -\\beta _i,-\\beta\n_{i_{l_a}}\\right) \\left( \\nu -\\nu _{i_{l_a}}\\right) ^{\\left( \\beta _i\\mid\n\\beta _{i_{l_a}}\\right) }\\prod_{0<b\\leq n+1}\\epsilon \\left( -\\beta _i,\\beta\n_{j_{k_b}}\\right) \\left( \\nu -\\rho _{j_{k_b}}\\right) ^{-\\left( \\beta _i\\mid\n\\beta _{j_{k_b}}\\right) }\\cdot \\\\\n&&\\prod_{0\\leq a<b\\leq n}\\epsilon \\left( -\\beta _{i_{l_a}},-\\beta\n_{i_{l_b}}\\right) \\left( \\nu _{i_{l_a}}-\\nu _{i_{l_b}}\\right) ^{\\left( \\beta\n_{i_{l_a}}\\mid \\beta _{i_{l_b}}\\right) }\\prod_{0\\leq a<b\\leq n+1}\\epsilon\n\\left( \\beta _{j_{k_a}},\\beta _{j_{k_b}}\\right) \\cdot \\\\\n&&\n\\end{eqnarray}\n\\begin{equation}\n\\left( \\rho _{j_{k_a}}-\\rho _{j_{k_b}}\\right) ^{-\\left( \\beta _{j_{k_a}}\\mid\n\\beta _{j_{k_b}}\\right) }{\\bf \\}}/\\{\\prod_{ 0\\leq a\\leq b\\leq n ,\\, a\\neq\nn+1} \\epsilon \\left( -\\beta _{i_{l_a}},\\beta _{j_{k_b}}\\right) \\left(\n\\nu _{i_{l_a}}-\\rho _{j_{k_b}}\\right) ^{\\left( \\beta _{i_{l_a}}\\mid \\beta\n_{j_{k_b}}\\right) }\\}. \\label{35}\n\\end{equation}\nIn the last relation we have written the following contour integration: \n\\begin{equation}\nI_1=\\frac 1{2\\pi i}\\oint d\\nu .\\nu \\frac{\\prod_{0<a\\leq n}\\epsilon \\left(\n-\\beta _i,-\\beta _{i_{l_a}}\\right) \\left( \\nu -\\nu _{i_{l_a}}\\right)\n^{\\left( \\beta _i\\mid \\beta _{i_{l_a}}\\right) }}{\\prod_{0<b\\leq n+1}\\epsilon\n\\left( -\\beta _i,\\beta _{j_{k_b}}\\right) \\left( \\nu -\\rho _{j_{k_b}}\\right)\n^{\\left( \\beta _i\\mid \\beta _{j_{k_b}}\\right) }}\\cdot \\label{36}\n\\end{equation}\n\nLikewise we have \n\\begin{eqnarray}\nC_{_{l_1}...i_{l_{n+1}},ij_{k_1}...j_{k_n}}^{-} &=&\\left\\langle \\lambda\n_o\\right\| E_{\\beta _i}^{\\left( 1\\right)\n}F_{i_{l_1}}...F_{i_{l_{n+1}}}G_{j_{k_1}}...G_{j_{k_n}}\\left\| \\lambda\n_o\\right\\rangle \\\\\n&=&\\frac 1{2\\pi i}\\oint d\\rho .\\rho \\nu _{i_{l_1}}...\\nu _{i_{l_{n+1}}}\\rho\n_{j_{k_1}}...\\rho _{j_{k_n}}\\cdot\n\\end{eqnarray}\n\\[\n\\left\\langle \\lambda _o\\right\| \\Gamma _{\\beta _i}\\left( \\rho \\right) \\Gamma\n_{-\\beta _{i_{l_1}}}\\left( \\nu _{i_{l_1}}\\right) ...\\Gamma _{-\\beta\n_{i_{l_{n+1}}}}\\left( \\nu _{i_{l_{n+1}}}\\right) \\Gamma _{\\beta\n_{j_{k_1}}}\\left( \\rho _{j_{k_1}}\\right) ...\\Gamma _{\\beta _{j_{k_n}}}\\left(\n\\rho j_{k_n}\\right) \\left\| \\lambda _o\\right\\rangle \n\\]\n\\begin{eqnarray}\n&=&\\frac 1{2\\pi i}\\oint d\\rho .\\rho \\nu _{i_{l_1}}...\\nu _{i_{l_{n+1}}}\\rho\n_{j_{k_1}}...\\rho _{j_{k_n}}{\\bf \\delta }_{\\overrightarrow{0},\\beta _i+\\beta\n_{j_{k_1}}+\\cdot \\cdot \\cdot \\beta _{j_{k_n}}-\\beta _{i_{l_1}}-\\cdot \\cdot\n\\cdot \\beta _{i_{l_{n+1}}}}\\cdot \\\\\n&&{\\bf \\{}\\prod_{0<a\\leq n}\\epsilon \\left( \\beta _i,\\beta _{j_{k_a}}\\right)\n\\left( \\rho -\\rho _{j_{k_a}}\\right) ^{\\left( \\beta _i\\mid \\beta\n_{j_{k_a}}\\right) }\\prod_{0<b\\leq n+1}\\epsilon \\left( \\beta _i,\\beta\n_{i_{l_b}}\\right) \\left( \\rho -\\nu _{i_{l_b}}\\right) ^{-\\left( \\beta _i\\mid\n\\beta _{i_{l_b}}\\right) }\\cdot \\\\\n&&\\prod_{0\\leq a<b\\leq n}\\epsilon \\left( \\beta _{j_{k_a}},\\beta\n_{j_{k_b}}\\right) \\left( \\rho _{j_{k_a}}-\\rho _{j_{k_b}}\\right) ^{\\left(\n\\beta _{j_{k_a}}\\mid \\beta _{j_{k_b}}\\right) }\\prod_{0\\leq a<b\\leq\nn+1}\\epsilon \\left( -\\beta _{i_{l_a}},-\\beta _{i_{l_b}}\\right) \\cdot\n\\end{eqnarray}\n\\begin{equation}\n\\left( \\nu _{i_{l_a}}-\\nu _{i_{l_b}}\\right) ^{-\\left( \\beta _{i_{l_a}}\\mid\n\\beta _{i_{l_b}}\\right) }{\\bf \\}}/\\{\\prod_{ 0\\leq a\\leq b\\leq n ,\\, b\\neq\nn+1 } \\epsilon \\left( -\\beta _{i_{l_a}},\\beta _{j_{k_b}}\\right) \\left(\n\\nu _{i_{l_a}}-\\rho _{j_{k_b}}\\right) ^{\\left( \\beta _{i_{l_a}}\\mid \\beta\n_{j_{k_b}}\\right) }\\}. \\label{37}\n\\end{equation}\nThis time the relevant contour integration becomes \n\\begin{equation}\nI_2=\\frac 1{2\\pi i}\\oint d\\rho .\\rho \\frac{\\prod_{0<a\\leq n}\\epsilon \\left(\n\\beta _i,-\\beta _{j_{k_a}}\\right) \\left( \\rho -\\rho _{j_{k_a}}\\right)\n^{\\left( \\beta _i\\mid \\beta _{j_{k_a}}\\right) }}{\\prod_{0<b\\leq n+1}\\epsilon\n\\left( \\beta _i,\\beta _{i_{l_b}}\\right) \\left( \\rho -\\nu _{i_{l_b}}\\right)\n^{\\left( \\beta _i\\mid \\beta _{i_{l_b}}\\right) }}\\cdot \\label{38}\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{\\bf Matrix elements using the homogeneous vertex operator\nrepresentation of the Kac-Moody algebra }$\\widehat{sl}_2$}\n\n\\label{appe}\nIn the case of the affine algebra $\\widehat{sl}_2$, we have the generators\n\n\\[\n\\alpha =\\left( \n\\begin{array}{cc}\n1 & 0 \\\\ \n0 & -1\n\\end{array}\n\\right) ,\\quad e=\\left( \n\\begin{array}{cc}\n0 & 1 \\\\ \n0 & 0\n\\end{array}\n\\right) ,\\quad f=\\left( \n\\begin{array}{cc}\n0 & 0 \\\\ \n1 & 0\n\\end{array}\n\\right) \n\\]\nand let us choose the following dual bases of $\\widehat{sl}_2$%\n\\[\n\\left\\{ t^n\\alpha ,t^ne,t^nf,C,D\\right\\} ,\\quad \\mbox{and \\quad }\\left\\{\n\\frac 12t^{-n}\\alpha ,t^{-n}f,t^{-n}e,D,C\\right\\} . \n\\]\nWe have \n\\begin{eqnarray}\nQ &=&{\\bf \\ Z}\\alpha ,\\quad \\left( \\alpha \\mid \\alpha \\right) =2,\\quad\n\\varepsilon (\\alpha ,\\alpha )=\\varepsilon (-\\alpha ,-\\alpha )=-1,\\quad\n\\varepsilon (\\alpha ,\\alpha )=\\varepsilon (-\\alpha ,-\\alpha )=1\\, \\\\\n&&\\mbox{(we are using the ``gauge fixing'' of \\ct{goddard, frenkel} for }\\varepsilon (\\alpha\n,\\beta)).\n\\end{eqnarray}\n\nConsidering $q=e^\\alpha ,$ we identify ${\\bf C}\\left[ Q\\right] $ with ${\\bf C}%\n\\left[ q,q^{-1}\\right] .$ Thus the homogeneous vertex operator construction can be described as follows \n\\[\nL\\left( \\lambda _o\\right) ={\\bf C}\\left[ x_1,x_2,...;q,q^{-1}\\right] ; \n\\]\n\\begin{eqnarray}\n\\alpha ^{\\left( n\\right) } &\\mapsto &2\\frac \\partial {\\partial x_n}\\mbox{\n\\quad and \\quad }\\alpha ^{\\left( -n\\right) }\\mapsto nx_n\\mbox{ \\quad for\n\\quad }n>0,\\qquad \\alpha ^{\\left( 0\\right) }\\mapsto 2q\\frac \\partial\n{\\partial q}; \\\\\nC &\\mapsto &1,\\qquad D\\mapsto -\\left( q\\frac \\partial {\\partial q}\\right)\n^2-\\sum_{n\\geq 1}nx_n\\frac \\partial {\\partial x_n};\n\\end{eqnarray}\n\\[\nE(z):=\\sum_{n\\in {\\bf Z}}E_{\\pm }^{\\left( n\\right) }z^{-n-1}\\mapsto \\Gamma\n_{\\pm }(z) \n\\]\nwhere \n\\begin{equation}\n\\Gamma _{\\pm }(z)=\\exp \\left( \\pm \\sum_{j\\geq 1}z^jx_j\\right) \\exp \\left(\n\\mp 2\\sum_{j\\geq 1}\\frac{z^{-j}}j\\frac \\partial {\\partial x_j}\\right) q^{\\pm\n1}z^{\\pm 2q\\frac \\partial {\\partial q}}c_{\\pm \\alpha }, \\label{39}\n\\end{equation}\n(note that $z^{\\pm 2q\\frac \\partial {\\partial q}}\\left( q^n\\right) =z^{\\pm\n2n}q^n$).\n\nThen, we can make use of the Eqn.(\\ref{33}) to compute the matrix elements,\n viz.,\n\n\\br\n\\left\\langle \\lambda _o\\right\| \\Gamma _{\\alpha _N}(z_N)\\Gamma _{\\alpha\n_{N-1}}(z_{N-1})...\\Gamma _{\\alpha _1}(z_1)\\left\| \\lambda _o\\right\\rangle = \n\\er\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n0,\\qquad \\qquad \\qquad \\mbox{ \\qquad \\qquad \\qquad \\qquad \\qquad if }%\n\\sum_{i=1}^N\\alpha _i\\neq 0 \\\\ \n\\prod_{1\\leq i<j\\leq N}\\varepsilon (\\alpha _i,\\alpha _j)\\left(\nz_i-z_j\\right) ^{\\left( \\alpha _i\\mid \\alpha _j\\right) }\\qquad \\quad \\mbox{%\nif }\\sum_{i=1}^N\\alpha _i=0\n\\end{array}\n\\right. \\label{40}\n\\end{equation}\n\nIn this way we have two types of operators, $\\Gamma _{\\pm }(z)$, associated to $%\n\\alpha $ and -$\\alpha $ respectively. We should have an even number of $%\n\\Gamma ^{,}s$ in Eqn. (\\ref{33}) in order to have a non zero value for $%\n\\left\\langle \\lambda _o\\right\| \\Gamma _{\\alpha _N}(z_N)\\Gamma _{\\alpha\n_{N-1}}(z_{N-1})...\\Gamma _{\\alpha _1}(z_1)\\left\| \\lambda _o\\right\\rangle $;\nthus we may choose $2N$ operators, such that $N$ operators correspond to $\\alpha \n$ and the remaining $N$ of them correspond to $-\\alpha .$\n\nWe provide some of the matrix components we used in the construction of {\\sl one-soliton} and {\\sl two-soliton} solutions of the system {\\bf NLS}. Defining\n\\br\nG_i=\\sum_{n=-\\infty }^{+\\infty }\\nu _i^{-n}E_{+}^{\\left( n\\right) },\\qquad\nF_i=\\sum_{n=-\\infty }^{+\\infty }\\nu _i^{-n}E_{-}^{\\left( n\\right) }; \n\\er\nwe can make the correspondence \n\\br\nG_i\\longrightarrow \\rho _i\\Gamma _{+}\\left( \\rho _i\\right) ,\\qquad\nF_i\\longrightarrow \\nu _i\\Gamma _{-}\\left( \\nu _i\\right) . \n\\er\nThen\n\\begin{eqnarray}\n\\left\\langle \\lambda _o\\right\| G_iF_j\\left\| \\lambda _o\\right\\rangle\n&=&\\left\\langle \\lambda _o\\right\| F_jG_i\\left\| \\lambda _o\\right\\rangle =\\rho\n_i\\nu _j\\left\\langle \\lambda _o\\right\| \\Gamma _{+}(\\rho _i)\\Gamma _{-}(\\nu\n_j)\\left\| \\lambda _o\\right\\rangle \\\\\n&=&\\rho _i\\nu _j\\frac{\\varepsilon (+,-)}{\\left( \\rho _i-\\nu _j\\right) ^2} \\\\\n&=&\\frac{\\rho _i\\nu _j}{\\left( \\rho _i-\\nu _j\\right) ^2}\\cdot\n\\end{eqnarray}\n\nAs a special case we compute the following expression\n\\begin{eqnarray}\n\\left\\langle \\lambda _o\\right\| E_{-}^1G_i\\left\| \\lambda _o\\right\\rangle\n&=&\\frac 1{2\\pi i}\\oint dz\\,z\\rho _i\\left\\langle \\lambda _o\\right\| \\Gamma\n_{-}(z)\\Gamma _{+}(\\rho _i)\\left\| \\lambda _o\\right\\rangle \\\\\n&=&\\frac 1{2\\pi i}\\rho _i\\oint dz.\\frac z{\\left( z-\\rho _i\\right) ^2} \\\\\n&=&\\rho _i,\n\\end{eqnarray}\nwhere we have used \n\\[\nE_{-}^1=\\oint dz\\,z\\Gamma _{-}(z), \n\\]\nwhere integration is over some curve encircling the origin.\n\nThe same method can be used to compute the following matrix element\n\\begin{eqnarray}\n\\left\\langle \\lambda _o\\right\| E_{-}^1G_1F_2G_2\\left\| \\lambda\n_o\\right\\rangle &=&\\frac 1{2\\pi i}\\oint dz\\,z\\,\\rho _{1}\\,\\nu _{2}\\,\\rho\n_2\\left\\langle \\lambda _o\\right\| \\Gamma _{-}(z)\\Gamma _{+}(\\rho _1)\\Gamma\n_{-}(\\nu _2)\\Gamma _{+}(\\rho _2)\\left\| \\lambda _o\\right\\rangle \\\\\n&=&\\frac 1{2\\pi i}\\oint dz\\,z\\,\\rho _{1}\\,\\nu _{2}\\,\\rho _2\\varepsilon\n(+,-)\\varepsilon (+,+)\\varepsilon (+,-)\\varepsilon (-,+)\\varepsilon\n(-,-)\\varepsilon (+,-) \\\\\n&&\\frac{\\left( \\rho _2-\\rho _1\\right) ^2\\left( \\nu _2-z\\right) ^2}{\\left(\n\\rho _2-\\nu _2\\right) ^2\\left( \\nu _2-\\rho _1\\right) ^2\\left( \\rho\n_2-z\\right) ^2\\left( \\rho _1-z\\right) ^2} \\\\\n&=&\\rho _{1}\\,\\nu _{2}\\,\\rho _2\\frac{\\left( \\rho _2-\\rho _1\\right) ^2}{\\left(\n\\rho _2-\\nu _2\\right) ^2\\left( \\nu _2-\\rho _1\\right) ^2}\\cdot\n\\end{eqnarray}\n\nThe remaining matrix elements can be computed in the same way. We give some\nof them \n\\begin{eqnarray}\n\\left\\langle \\lambda _o\\right\| E_{-}^1G_i\\left\| \\lambda _o\\right\\rangle\n&=&\\frac 1{2\\pi i}\\oint dz.z\\rho _i\\left\\langle \\lambda _o\\right\| \\Gamma\n_{-}(z)\\Gamma _{+}(\\rho _i)\\left\| \\lambda _o\\right\\rangle \\qquad \\\\\n&=&\\frac 1{2\\pi i}\\rho _i\\oint dz\\,\\frac z{\\left( z-\\rho _i\\right) ^2} \\\\\n&=&\\rho _i\\,\\quad \\qquad \\qquad \\qquad\n\\end{eqnarray}\n\n\\br\n\\left\\langle \\lambda _o\\right\| F_iG_iF_jG_j\\left\| \\lambda _o\\right\\rangle =%\n\\frac{\\rho _{i}\\,\\nu _{j}\\,\\rho _i\\rho _j\\left( \\rho _j-\\rho _i\\right)\n^2\\left( \\nu _j-\\nu _i\\right) ^2}{\\left( \\rho _j-\\nu _j\\right) ^2\\left( \\rho\n_j-\\nu _i\\right) ^2\\left( \\rho _i-\\nu _i\\right) ^2\\left( \\rho _i-\\nu\n_j\\right) ^2}, \n\\er\n\n\\br\n\\left\\langle \\lambda _o\\right\| E_{-}^1G_iF_jG_j\\left\| \\lambda\n_o\\right\\rangle =\\frac{\\rho _{i}\\,\\nu _{j}\\,\\rho _j\\left( \\rho _j-\\rho\n_i\\right) ^2}{\\left( \\rho _j-\\nu _j\\right) ^2\\left( \\nu _j-\\rho _i\\right) ^2}%\n,\\qquad \\qquad \\qquad \\qquad \\quad \n\\er\n\n\\br\n\\left\\langle \\lambda _o\\right\| E_{+}^1F_i\\left\| \\lambda _o\\right\\rangle =\\nu\n_i,\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \n\\er\n\n\\br\n\\left\\langle \\lambda _o\\right\| E_{+}^1F_iG_iF_j\\left\| \\lambda\n_o\\right\\rangle =\\frac{\\nu _{i}\\,\\nu _{j}\\,\\rho _i\\left( \\nu _j-\\nu _i\\right)\n^2}{\\left( \\rho _i-\\nu _j\\right) ^2\\left( \\nu _i-\\rho _i\\right) ^2}\\cdot\n\\qquad \\qquad \\qquad \\qquad \\quad \n\\er\n\nThese results suggest that a general matrix element could be expressed in terms of Vandermonde-like determinants. In fact, recently in \\ct{meinel} there was derived a natural relationship with the {\\sl Vandermonde-like determinants}. The resulting framework in our case may be well-suited to achieve a compactness and transparency in $N$-soliton formulas. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{\\bf The contour integration of the matrix elements in the case of }%\n$\\widehat{sl}_2$}\n\\label{appf}\nLet us show that the integral \n\\begin{equation}\nI_n=\\frac 1{2\\pi i}\\oint dz.z\\frac{\\left( z-\\nu _1\\right) ^2...\\left( z-\\nu\n_{n-1}\\right) ^2}{\\left( z-\\rho _1\\right) ^2...\\left( z-\\rho _n\\right) ^2}%\n,\\qquad n\\geq 1 \\label{41}\n\\end{equation}\nis equal to unity. We will prove by induction.\n\nFor $n=1:$\n\n\\begin{eqnarray}\nI_1 &=&\\frac 1{2\\pi i}\\oint dz\\frac z{\\left( z-\\rho _1\\right) ^2} \\\\\n&=&1.\n\\end{eqnarray}\nAssuming that $I_n=1$ for some $n>1$ let us determine the form of $I_{n+1}$.\n\nThe expression for $I_{n+1}$ can be written as \n\\begin{equation}\nI_{n+1}=I_n+C_n, \\label{42}\n\\end{equation}\nwhere \n\\[\nC_n=\\frac 1{2\\pi i}\\frac \\partial {\\partial \\rho _1}\\cdot \\cdot \\cdot \\frac\n\\partial {\\partial \\rho _{n+1}}\\oint dz.z\\frac{\\left( z-\\nu _1\\right)\n^2\\cdot \\cdot \\cdot \\left( z-\\nu _{n-1}\\right) ^2}{\\left( z-\\rho _1\\right)\n\\cdot \\cdot \\cdot \\left( z-\\rho _{n+1}\\right) }\\left( \\rho _{n+1}-\\nu\n_n\\right) ^2 \n\\]\n\\begin{eqnarray}\n&=&\\frac 1{2\\pi i}{\\bf \\{}\\frac \\partial {\\partial \\rho _1}\\cdot \\cdot \\cdot\n\\frac \\partial {\\partial \\rho _{n+1}}\\oint dz\\left( \\rho _2-\\nu _1\\right)\n^2\\left( \\rho _3-\\nu _2\\right) ^2\\cdot \\cdot \\cdot \\left( \\rho _{n+1}-\\nu\n_n\\right) ^2\\cdot \\\\\n&&\\frac z{\\left( z-\\rho _1\\right) \\cdot \\cdot \\cdot \\left( z-\\rho\n_{n+1}\\right) }+ \\\\\n&&{\\bf [}\\frac \\partial {\\partial \\rho _1}\\frac \\partial {\\partial \\rho\n_3}\\cdot \\cdot \\cdot \\frac \\partial {\\partial \\rho _{n+1}}\\oint dz\\left(\n\\rho _3-\\nu _2\\right) ^2\\cdot \\cdot \\cdot \\left( \\rho _{n+1}-\\nu _n\\right)\n^2\\cdot \\\\\n&&\\frac z{\\left( z-\\rho _1\\right) \\left( z-\\rho _3\\right) \\cdot \\cdot \\cdot\n\\left( z-\\rho _{n+1}\\right) }+ \\\\\n&&\\frac \\partial {\\partial \\rho _1}\\frac \\partial {\\partial \\rho _2}\\frac\n\\partial {\\partial \\rho _4}\\cdot \\cdot \\cdot \\frac \\partial {\\partial \\rho\n_{n+1}}\\oint dz\\left( \\rho _2-\\nu _1\\right) ^2\\left( \\rho _4-\\nu _3\\right)\n^2\\cdot \\cdot \\cdot \\left( \\rho _{n+1}-\\nu _n\\right) ^2\\cdot \\\\\n&&\\frac z{\\left( z-\\rho _1\\right) \\left( z-\\rho _2\\right) \\left( z-\\rho\n_4\\right) \\cdot \\cdot \\cdot \\left( z-\\rho _{n+1}\\right) }+\\cdot \\cdot \\cdot +\n\\\\\n&&\\frac \\partial {\\partial \\rho _1}\\ \\cdot \\cdot \\cdot \\frac \\partial\n{\\partial \\rho _{n-1}}\\frac \\partial {\\partial \\rho _{n+1}}\\oint dz\\left(\n\\rho _2-\\nu _1\\right) ^2\\left( \\rho _3-\\nu _2\\right) ^2\\cdot \\cdot \\cdot\n\\left( \\rho _{n-1}-\\nu _{n-2}\\right) ^2\\left( \\rho _{n+1}-\\nu _n\\right)\n^2\\cdot \\\\\n&&\\frac z{\\left( z-\\rho _1\\right) \\cdot \\cdot \\cdot \\left( z-\\rho\n_{n-1}\\right) \\left( z-\\rho _{n+1}\\right) }{\\bf ]}+{\\bf [}\\frac \\partial\n{\\partial \\rho _1}\\frac \\partial {\\partial \\rho _4}\\cdot \\cdot \\cdot \\frac\n\\partial {\\partial \\rho _{n+1}}\\cdot \\\\\n&&\\oint dz\\left( \\rho _4-\\nu _3\\right) ^2\\cdot \\cdot \\cdot \\left( \\rho\n_{n+1}-\\nu _n\\right) ^2\\cdot \\\\\n&&\\frac z{\\left( z-\\rho _1\\right) \\left( z-\\rho _4\\right) \\cdot \\cdot \\cdot\n\\left( z-\\rho _{n+1}\\right) }+\\frac \\partial {\\partial \\rho _1}\\frac\n\\partial {\\partial \\rho _3}\\frac \\partial {\\partial \\rho _5}\\cdot \\cdot\n\\cdot \\frac \\partial {\\partial \\rho _{n+1}}\\cdot \\\\\n&&\\oint dz\\left( \\rho _3-\\nu _2\\right) ^2\\left( \\rho _5-\\nu _4\\right)\n^2\\cdot \\cdot \\cdot \\left( \\rho _{n+1}-\\nu _n\\right) ^2\\frac z{\\left( z-\\rho\n_1\\right) \\left( z-\\rho _3\\right) \\left( z-\\rho _5\\right) \\cdot \\cdot \\cdot\n\\left( z-\\rho _{n+1}\\right) }+ \\\\\n&&\\frac \\partial {\\partial \\rho _1}\\frac \\partial {\\partial \\rho _3}\\cdot\n\\cdot \\cdot \\frac \\partial {\\partial \\rho _{n-1}}\\frac \\partial {\\partial\n\\rho _{n+1}}\\oint dz\\left( \\rho _3-\\nu _2\\right) ^2\\cdot \\cdot \\cdot \\left(\n\\rho _{n-1}-\\nu _{n-2}\\right) ^2\\left( \\rho _{n+1}-\\nu _n\\right) ^2\\cdot\n\\end{eqnarray}\n\\[\n\\cdot \\cdot \\cdot + \n\\]\n\\begin{eqnarray}\n&&\\frac z{\\left( z-\\rho _1\\right) \\left( z-\\rho _3\\right) \\cdot \\cdot \\cdot\n\\left( z-\\rho _{n-1}\\right) \\left( z-\\rho _{n+1}\\right) }+\\frac \\partial\n{\\partial \\rho _1}\\frac \\partial {\\partial \\rho _2}\\frac \\partial {\\partial\n\\rho _5}\\cdot \\cdot \\cdot \\frac \\partial {\\partial \\rho _{n+1}}\\cdot \\\\\n&&\\oint dz\\left( \\rho _2-\\nu _1\\right) ^2\\left( \\rho _5-\\nu _4\\right)\n^2\\cdot \\cdot \\cdot \\left( \\rho _{n+1}-\\nu _n\\right) ^2\\frac z{\\left( z-\\rho\n_1\\right) \\left( z-\\rho _2\\right) \\left( z-\\rho _5\\right) \\cdot \\cdot \\cdot\n\\left( z-\\rho _{n+1}\\right) }+ \\\\\n&&\\cdot \\cdot \\cdot +\\frac \\partial {\\partial \\rho _1}\\frac \\partial\n{\\partial \\rho _2}\\frac \\partial {\\partial \\rho _4}\\cdot \\cdot \\cdot \\frac\n\\partial {\\partial \\rho _{n-1}}\\frac \\partial {\\partial \\rho _{n+1}}\\oint\ndz\\left( \\rho _2-\\nu _1\\right) ^2\\left( \\rho _4-\\nu _3\\right) ^2\\cdot \\cdot\n\\cdot \\left( \\rho _{n-1}-\\nu _{n-2}\\right) ^2\\cdot \\\\\n&&\\left( \\rho _{n+1}-\\nu _n\\right) ^2\\frac z{\\left( z-\\rho _1\\right) \\left(\nz-\\rho _2\\right) \\left( z-\\rho _4\\right) \\cdot \\cdot \\cdot \\left( z-\\rho\n_{n-1}\\right) \\left( z-\\rho _{n+1}\\right) }+\\cdot \\cdot \\cdot + \\\\\n&&\\frac \\partial {\\partial \\rho _1}\\ \\cdot \\cdot \\cdot \\frac \\partial\n{\\partial \\rho _{n-2}}\\frac \\partial {\\partial \\rho _{n+1}}\\oint dz\\left(\n\\rho _2-\\nu _1\\right) ^2\\cdot \\cdot \\cdot \\left( \\rho _{n-2}-\\nu\n_{n-3}\\right) ^2\\left( \\rho _{n+1}-\\nu _n\\right) ^2\\cdot \\\\\n&&\\frac z{\\left( z-\\rho _1\\right) \\cdot \\cdot \\cdot \\left( z-\\rho\n_{n-2}\\right) \\left( z-\\rho _{n+1}\\right) }{\\bf ]}+{\\bf [}\\frac \\partial\n{\\partial \\rho _1}\\frac \\partial {\\partial \\rho _5}\\cdot \\cdot \\cdot \\frac\n\\partial {\\partial \\rho _{n+1}}\\oint dz\\left( \\rho _5-\\nu _4\\right) ^2\\cdot\n\\cdot \\cdot \\\\\n&&\\left( \\rho _{n+1}-\\nu _n\\right) ^2\\frac z{\\left( z-\\rho _1\\right) \\left(\nz-\\rho _5\\right) \\cdot \\cdot \\cdot \\left( z-\\rho _{n+1}\\right) }+\\frac\n\\partial {\\partial \\rho _1}\\frac \\partial {\\partial \\rho _4}\\frac \\partial\n{\\partial \\rho _5}\\cdot \\cdot \\cdot \\frac \\partial {\\partial \\rho\n_{n-1}}\\frac \\partial {\\partial \\rho _{n+1}}\\cdot \\\\\n&&\\oint dz\\left( \\rho _4-\\nu _3\\right) ^2\\left( \\rho _5-\\nu _4\\right)\n^2\\cdot \\cdot \\cdot \\left( \\rho _{n-1}-\\nu _{n-2}\\right) ^2\\left( \\rho\n_{n+1}-\\nu _n\\right) ^2\\cdot \\\\\n&&\\frac z{\\left( z-\\rho _1\\right) \\left( z-\\rho _4\\right) \\left( z-\\rho\n_5\\right) \\cdot \\cdot \\cdot \\left( z-\\rho _{n-1}\\right) \\left( z-\\rho\n_{n+1}\\right) }+\\cdot \\cdot \\cdot +\n\\end{eqnarray}\n\\begin{eqnarray}\n&&\\frac \\partial {\\partial \\rho _1}\\ \\cdot \\cdot \\cdot \\frac \\partial\n{\\partial \\rho _{n-3}}\\frac \\partial {\\partial \\rho _{n+1}}\\oint dz\\left(\n\\rho _2-\\nu _1\\right) ^2\\cdot \\cdot \\cdot \\left( \\rho _{n-3}-\\nu\n_{n-4}\\right) ^2\\left( \\rho _{n+1}-\\nu _n\\right) ^2\\cdot \\nonumber \\\\\n&&\\frac z{\\left( z-\\rho _1\\right) \\cdot \\cdot \\cdot \\left( z-\\rho\n_{n-3}\\right) \\left( z-\\rho _{n+1}\\right) }{\\bf ]+\\cdot \\cdot \\cdot +} \n\\nonumber \\\\\n&&\\frac \\partial {\\partial \\rho _1}\\frac \\partial {\\partial \\rho\n_{n+1}}\\oint dz\\left( \\rho _{n+1}-\\nu _n\\right) ^2\\frac z{\\left( z-\\rho\n_1\\right) \\left( z-\\rho _{n+1}\\right) }{\\bf \\}\\cdot } \\label{43}\n\\end{eqnarray}\nIt is easy to show the following:\n\n{\\bf i)} \n\\begin{equation}\n\\frac 1{2\\pi i}\\oint dz\\frac z{\\left( z-\\rho _1\\right) \\left( z-\\rho\n_2\\right) }=1, \\label{44}\n\\end{equation}\nand\n{\\bf ii)} \n\\begin{equation}\n\\oint dz\\frac z{\\left( z-\\rho _1\\right) \\cdot \\cdot \\cdot \\left( z-\\rho\n_n\\right) }=0\\qquad \\mbox{for\\qquad }n>2\\cdot \\label{45}\n\\end{equation}\nwriting \n\\begin{equation}\n\\frac 1{\\left( z-\\rho _1\\right) \\cdot \\cdot \\cdot \\left( z-\\rho _n\\right)\n}=\\frac 1{\\det \\Delta }\\left[ \\frac{A_{n,1}}{z-\\rho _1}+\\frac{A_{n,2}}{%\nz-\\rho _2}+\\cdot \\cdot \\cdot +\\frac{A_{n,n}}{z-\\rho _n}\\right] , \\label{46}\n\\end{equation}\nwith \n\\br\n\\nonu\n&&\\Delta =\\\\\n\\nonu\n&&\\left( \n\\begin{array}{cccc}\n1 & 1 & ...& 1\\\\ \n-\\sum_{i=1,\\, i\\neq 1}^n\\rho _i & - \\sum_{ i=1 ,\\, i\\neq 2} ^n\\rho _i & \\cdot \\cdot \\cdot & \n- \\sum_{ i=1 , \\, i\\neq n} ^n\\rho _i\\\\\n \\sum_{ 1\\leq i<j\\leq n ,\\, i,j\\neq 1 } ^n\\rho _i\\rho\n_j & \\sum_{ 1\\leq i<j\\leq n ,\\, i,j\\neq 2} ^n\\rho\n_i\\rho _j & \\cdot \\cdot \\cdot & \\sum_{ 1\\leq i<j\\leq n \n,\\, i,j\\neq n} ^n\\rho _i\\rho _j \\\\ \n- \\sum_{ 1\\leq i<j<k\\leq n ,\\, i,j,k\\neq 1} ^n\\rho _i\\rho\n_j\\rho _k & - \\sum_{ 1\\leq i<j<k\\leq n ,\\, i,j,k\\neq 2 \n} ^n\\rho _i\\rho _j\\rho _k & \\cdot \\cdot \\cdot & -\n\\sum_{ 1\\leq i<j<k\\leq n ,\\, i,j,k\\neq n} ^n\\rho _i\\rho _j\\rho_k \\\\ \n \\cdot & \\cdot & & \\cdot \\\\ \n \\cdot & \\cdot & & \\cdot \\\\ \n \\cdot & \\cdot & & \\cdot \\\\ \n (-1)^{n-1}\\rho _2\\rho _3...\\rho _n & \\left( -1\\right)\n^{n-1}\\rho _1\\rho _3\\rho _4...\\rho _n & \\cdot \\cdot \\cdot & \\left(\n-1\\right)^{n-1}\\rho _1\\rho _2...\\rho _{n-1}\n\\end{array}\n\\right)\\\\\n\\label{47}\n\\er\n$A_{i,j}$ denotes the cofactor of the element $\\Delta _{i,j}.$ Let us note\nthat, multiplying by $\\sum_{i=1}^n\\rho _i$ the first row and adding to the\nsecond row of the matrix $\\Delta ,$ the $\\det \\Delta $ and the cofators $%\nA_{n,i}$ do not change$;$ then (\\ref{46}), can be written as\n\n\\begin{equation}\n\\frac 1{\\left( z-\\rho _1\\right) \\cdot \\cdot \\cdot \\left( z-\\rho _n\\right)\n}=\\frac 1{\\det d}\\left[ \\frac{B_{n,1}}{z-\\rho _1}+\\frac{B_{n,2}}{z-\\rho _2}%\n+\\cdot \\cdot \\cdot +\\frac{B_{n,n}}{z-\\rho _n}\\right] \\label{48}\n\\end{equation}\nwhere $\\det \\Delta =\\det d$ and $A_{i,j}=B_{i,j},\\,\\,$ $B_{i,j}$\nare the cofactors of the matrix elements $d_{i,j}.$ It is easy to realize\nthat the matrix $d$ has $d_{2,i}=\\rho _i$ as the elements of the second row.\nThen the contour integration of the relevant terms of (\\ref{48}) is \n\\[\n\\frac 1{2\\pi i}\\oint dz\\frac z{\\left( z-\\rho _1\\right) ...\\left( z-\\rho\n_n\\right) }=\\frac 1{\\det d}\\left[ \\rho _1B_{n,1}+\\rho _2B_{n,2}+\\cdot \\cdot\n\\cdot +\\rho _nB_{n,n}\\right] . \n\\]\n\nWe know that \n\\[\n\\sum_{i=1}^nd_{l,j}B_{k,j}=0,\\qquad l\\neq k \n\\]\nand therefore, we can conclude \n\\begin{equation}\n\\oint dz\\frac z{\\left( z-\\rho _1\\right) \\cdot \\cdot \\cdot \\left( z-\\rho\n_n\\right) }=0,\\qquad \\mbox{for }n>2 \\label{49}\n\\end{equation}\n\nThus, the last term of $C_n$ vanishes due to (\\ref{44}) and the remaining\nterms vanish because of (\\ref{45}). Then $C_n=0$ for $n>1,$ and the Eqn.(\\ref\n{42}) becomes\n\n\\[\nI_{n+1}=I_n,\\qquad n>1, \n\\]\nthis shows that\n\\[\nI_n=1\\qquad n\\geq 1. \n\\]\n\n\n\n\n\n\n\n\n\\begin{thebibliography}{*}\n\\bi{zakharov}\nV.E. Zakharov and A.B. Shabat, \\JETPL{34}{1972}{62}.\n\\bi{manakov}\nS.V. Manakov, \\JETPL{38}{1974}{248}.\n\\bi{nls}\nA.V. Mikhailov, E.A. Kuznetsov, A.C. Newell and V.E. Zakharov (Eds.), Proceedings of the Conference on {\\sl The Nonlinear Schr\\\"{o}dinger Equation}, Chernogolovka, 25 July-3 August 1994. Published in \\PHSD{87}{1995}{1}.\n\\bi{zen}\nF.P. Zen and H.I. Elim, solv-int/9902010 v2.\n\\bi{ferreira}L.A. Ferreira, J. L. Miramontes and J. S\\'anchez Guillen, \\JMP{38}{1997}{882} (hep-th/9606066).\n\\bi{kaup1}\nV.S. Gerdjikov, D.J. Kaup, I.M. Uzunov and E.G. Evstatiev, \\PRE{77}{1996}{3943}.\n\\bi{blas}\nH.S. Blas, unpublished manuscript.\n\\bi{liu}\nShan-liang Liu and Wen-zheng Wang, \\PRE{48}{1993}{3054}\n\\bi{babelon}O. Babelon and D. Bernard, Phys. Lett. 260B (1991) 81; O. Babelon\nand D. Bernard, Commun. Math. Phys. 149 (1992) 279; O. Babelon and D.\nBernard, Int.\\ Jour. of Mod. Phys. A,V.8, No.3(1993) 507.\n\\bi{fordy}\nA.P. Fordy and P.P. Kulish, \\CMP{89}{1983}{427};\\\\\n A.P. Fordy, in {\\sl Soliton Theory: a Survey of Results}, (ed. A.P. Fordy) University Press, Manchester (1990), p. 315.\n\\bi{aratyn}\nH. Aratyn, J.F. Gomes and A.H. Zimerman, \\JMP{36}{1995}{3419}.\n\\bi{faddeev}\nL.D. Faddeev and L.A. Takhtajan, {\\sl Hamiltonian Methods in the Theory of Solitons}, Springer-Verlag, London, p. 288 (1987).\n\\bi{kac}\nV.G. Kac, {\\sl Infinite dimensional Lie algebras}, Third Ed., Cambridge\nUniversity Press, Cambridge, 1990.\n\\bi{leznov}\nA.N. Leznov and A.V. Razumov, \\JMP{35}{1994}{1738}.\n\\bi{drazin}\nP.G. Drazin and R.S. Johnson, {\\sl Solitons: an introduction}, Cambridge, University Press, Cambridge, 1989.\n\\bi{twosoliton}\nH. Konno and P.S. Lomdahl, \\JPSJ{63}{1994}{3967};\\\\\n J.N. Maki and T. Kodama, \\PRL{57}{1986}{2097}.\n\\bi{kraenkel}\nR. A. Kraenkel, J.G. Pereira and M.A. Manna, {The reductive perturbative method and the KDV hierarchy} in ``Proc. of the Conference KdV' 95'', The Netherlands 1995.\n\\bi{goddard}\nP. Goddard and D. Olive, \\IJMPA{1}{1986}{303}. \n\\bi{wakimoto}\nV.G. Kac and M. Wakimoto, {\\sl Exceptional hierarchies of soliton equations} in ``Proceedings of Symposia in Pure Mathematics'' Vol. 49 (1989) 191.\n\\bi{wan}\nWan Zhe-xian, {\\sl Introduction to KAC-MOODY ALGEBRA}, World Scientific, 1991.\n\\bi{ferreira1}\nL.A. Ferreira, J.L. Miramontes and J. S\\'anchez Guillen, \\NPB{449}{1995}{631}.\n\\bi{vertex}\nJ. Lepowsky and R.L. Wilson, \\CMP{62}{1978}{43};\\\\\nV. G. Kac, D.A. Kazhdan, J. Lepowsky and R.L. Wilson, \\AdM{42}{1981}{83};\\\\\nF. ten Kroode and J. van de Leur, \\CMP{137}{1991}{67};\\\\ \nI.B. Frenkel, J. Lepowsky and A. Meurman, in ``Mathematical Aspects of\nString Theory '', Adv. Series in Math. Phys. Vol.1. S.T. Yau (Ed.), World\nScientific, 1987;\\\\\nA.J. Feingold, in ``Vertex Operators in Math. and Phys.'',\nJ. Lepowsky, S. Mandelstam and I.M. Singer (Eds.) Math. Sciences Research\nInst. Publications. Springer-Verlag, 1983.\n\\bi{vandeleur}\nV.G. Kac and J.W. van de Leur, in ''Important Developments in\nSoliton Theory '', A.S. Fokas and V.E. Zakharov (Eds.) Springer-Verlag\nBerlin Heidelberg (1993) p. 302.\n\\bi{frenkel}\nI.B. Frenkel and V.G. Kac, \\InvM{62}{1980}{28}.\n\\bi{meinel}\nH. Steudel, R. Meinel and G. Neugebauer, \\JMP{38}{1997}{4692}.\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n" } ]	[ { "name": "solv-int9912015.extracted_bib", "string": "{#1 \\def\\lskip{\\vskip\\baselineskip\\vskip-\\parskip\\noindent \\relax % common physics symbols \\def\\tr{\\mathop{\\rm tr \\def\\Tr{\\mathop{\\rm Tr \\def\\v{\\vert \\def\\bv{\\bigm\\vert \\def\\Bgv{\\;\\Bigg\\vert \\def\\bgv{\\bigg\\vert \\newcommand\\partder[2]{{{\\partial {#1\\over{\\partial {#2 \\newcommand\\funcder[2]{{{\\delta {#1\\over{\\delta {#2 \\newcommand\\Bil[2]{\\Bigl\\langle {#1 \\Bigg\\vert {#2 \\Bigr\\rangle %% <.\|.> \\newcommand\\bil[2]{\\left\\langle {#1 \\bigg\\vert {#2 \\right\\rangle %% <.\|.> \\newcommand\\me[2]{\\left\\langle {#1\\bv {#2 \\right\\rangle %% <.\|.> \\newcommand\\sbr[2]{\\left\\lbrack\\,{#1\\, ,\\,{#2\\,\\right\\rbrack \\newcommand\\pbr[2]{\\{\\,{#1\\, ,\\,{#2\\,\\ \\newcommand\\pbbr[2]{\\lcurl\\,{#1\\, ,\\,{#2\\,\\rcurl % % math symbols \\def\\a{\\alpha \\def\\at{{\\tilde A^R \\def\\atc{{\\tilde {\\cal A^R \\def\\atcm#1{{\\tilde {\\cal A^{(R,#1) \\def\\b{\\beta \\def\\btil{{\\tilde b \\def\\dc{{\\cal D \\def\\d{\\delta \\def\\D{\\Delta \\def\\eps{\\epsilon \\def\\bareps{{\\bar \\epsilon \\def\\vareps{\\varepsilon \\def\\fptil{{\\tilde F^{+ \\def\\fmtil{{\\tilde F^{- \\def\\gh{{\\hat g \\def\\g{\\gamma \\def\\G{\\Gamma \\def\\grad{\\nabla \\def\\h{{1\\over 2 \\def\\l{\\lambda \\def\\L{\\Lambda \\def\\m{\\mu \\def\\n{\\nu \\def\\o{\\over \\def\\om{\\omega \\def\\O{\\Omega \\def\\p{\\phi \\def\\P{\\Phi \\def\\pa{\\partial \\def\\pr{\\prime \\def\\pt{{\\tilde \\Phi \\def\\qs{Q_{s \\def\\ra{\\rightarrow \\def\\s{\\sigma \\def\\S{\\Sigma \\def\\t{\\tau \\def\\th{\\theta \\def\\Th{\\Theta \\def\\tpp{\\Theta_{+ \\def\\tmm{\\Theta_{- \\def\\tpg{\\Theta_{+^{> \\def\\tms{\\Theta_{-^{< \\def\\tp0{\\Theta_{+^{(0) \\def\\tm0{\\Theta_{-^{(0) \\def\\ti{\\tilde \\def\\wti{\\widetilde \\def\\jc{J^C \\def\\bj{{\\bar J \\def\\sj{{\\jmath \\def\\bsj{{\\bar \\jmath \\def\\bp{{\\bar \\p \\def\\vp{\\varphi \\def\\vt{{\\tilde \\varphi \\def\\faa{Fa\\'a di Bruno~ \\def\\ca{{\\cal A \\def\\cb{{\\cal B \\def\\ce{{\\cal E \\def\\cg{{\\cal G \\def\\cgh{{\\hat {\\cal G \\def\\ch{{\\cal H \\def\\chh{{\\hat {\\cal H \\def\\cl{{\\cal L \\def\\cm{{\\cal M \\def\\cn{{\\cal N \\def\\ns{N_{{s \\newcommand\\sumi[1]{\\sum_{#1^{\\infty %% summation till infinity \\newcommand\\fourmat[4]{\\left(\\begin{array{cc %% 2x2 matrix {#1 & {#2 \\\\ {#3 & {#4" } ]
solv-int9912016	\sc On the Miura map between the dispersionless KP and dispersionless modified KP hierarchies\/	[ { "author": "Jen-Hsu Chang$^1$ and Ming-Hsien Tu$^2$" }, { "author": "$^1$Institute of Mathematics" }, { "author": "Academia Sinica" }, { "author": "Nankang" }, { "author": "Taipei" }, { "author": "Taiwan" }, { "author": "E-mail: changjen@math.sinica.edu.tw" }, { "author": "$^2$Department of Physics" }, { "author": "National Chung Cheng University" }, { "author": "Minghsiung" }, { "author": "Chiayi" }, { "author": "Taiwan" }, { "author": "E-mail: phymhtu@ccunix.ccu.edu.tw" } ]	We investigate the Miura map between the dispersionless KP and dispersionless modified KP hierarchies. We show that the Miura map is canonical with respect to their bi-Hamiltonian structures. Moreover, inspired by the works of Takasaki and Takebe, the twistor construction of solution structure for the dispersionless modified KP hierarchy is given.	[ { "name": "solv-int9912016.tex", "string": "\n\\documentstyle[preprint,eqsecnum,aps]{revtex}\n\\pagestyle{plain}\n \\tightenlines\n%\\usepackage{amssymb, amsmath, amsthm}\n\n%\\theoremstyle{plain}\n\n\\newtheorem{theorem}{Theorem}\n\n\\newtheorem{corollary}[theorem]{Corollary}\n\n\\newtheorem{lemma}[theorem]{Lemma}\n\n%\\theoremstyle{definition}\n\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\bea}{\\begin{eqnarray}}\n\\newcommand{\\eea}{\\end{eqnarray}}\n\\newcommand{\\ba}{\\begin{array}}\n\\newcommand{\\ea}{\\end{array}}\n\\newcommand{\\vp}{\\varphi}\n\\newcommand{\\ep}{\\epsilon}\n\\newcommand{\\Th}{\\Theta}\n%\\newcommand{\\th}{\\theta}\n\\newcommand{\\Ga}{\\Gamma}\n\\newcommand{\\ga}{\\gamma}\n\\newcommand{\\ka}{\\kappa}\n\\newcommand{\\La}{\\Lambda}\n\\newcommand{\\la}{\\lambda}\n\\newcommand{\\om}{\\omega}\n\\newcommand{\\Om}{\\Omega}\n\\newcommand{\\de}{\\delta}\n\\newcommand{\\dex}{\\delta(x-y)}\n\\newcommand{\\pa}{\\partial}\n\\newcommand{\\pax}{\\partial_x}\n\\newcommand{\\pari}{\\partial^{-1}}\n\\newcommand{\\no}{\\nonumber}\n\\newcommand{\\pade}{\\pa_x\\de (x-y)}\n\\newcommand{\\epx}{\\ep (x-y)}\n\\newcommand{\\tr}{\\mbox{tr}}\n\\newcommand{\\res}{\\mbox{res}}\n\\newcommand{\\adj}{\\mbox{ad}}\n\\begin{document}\n \\draft\n\n\\title\n{\\sc On the Miura map between the dispersionless KP and\ndispersionless modified KP hierarchies\\/}\n\\author{Jen-Hsu Chang$^1$\n and Ming-Hsien Tu$^2$\\\\ $^1$Institute of Mathematics, Academia Sinica,\\\\\nNankang, Taipei, Taiwan\\\\ E-mail: changjen@math.sinica.edu.tw\\\\\n$^2$Department of Physics, National Chung Cheng University,\n\\\\ Minghsiung, Chiayi, Taiwan\\\\\nE-mail: phymhtu@ccunix.ccu.edu.tw}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nWe investigate the Miura map between the dispersionless KP and\ndispersionless modified KP hierarchies. We show that the Miura map\nis canonical with respect to their bi-Hamiltonian structures.\nMoreover, inspired by the works of Takasaki and Takebe, the\ntwistor construction of solution structure for the dispersionless\nmodified KP hierarchy is given.\n\\end{abstract}\n% \\pacs{{\\it PACS Classification :\\/} 03.65.Fd}\n\n\\newpage\n\n \\section{Introduction}\n\nThe dispersionless KP hierarchy(dKP)\n\\cite{Car1,Car2,KG,LM,Ta2,Ta1} can be thought as the\nsemi-classical limit of the KP hierarchy \\cite{Di}. There are many\nmathematical and physical problems associated with the dKP\nhierarchy and its various reductions, such as Whitham hierarchy,\ntopological field theory and its connections to string theory and\n2D gravity \\cite{Car1,Ak,Du4,du1,kr1,kr2}. Similarly, the\ndispersionless modified KP(dmKP) hierarchy \\cite{Li} can be\nregarded as the semi-classical limit of the modified KP (mKP)\nhierarchy \\cite{jm,ku1} . However, in contrast to dKP, the\nintegrable structures of dmKP are less investigated. This motives\nus to study the relationships between dKP and dmKP and to gain an\ninsight of dmKP from dKP.\n\nThe Miura map \\cite{Mi} has been playing an important role in the\ndevelopment of soliton theory. It's a transformation between two\nnonlinear equations, which in general cannot be solved easily.\nHowever, knowing the solutions of one of the non-linear systems,\none may obtain the solutions of the other one via an appropriate\nMiura map. A typical example is the Miura map between the KP\nequation and the mKP equation \\cite{ki,KO,ku2,OR,St1,ST}.\nMotivated by the Miura map between the KP equation and the mKP\nequation, we will construct the Miura map between dKP and dmKP.\n(In \\cite{Ku3}, this Miura map is constructed in different way.)\nMoreover, since almost all the known integrable systems are\nHamiltonian, exploring the Hamiltonian nature of these Miura maps\nwill deepen our understanding of these relations between these\nintegrable systems.\n\n Recently, the canonical property of the\nMiura map between the mKP and the KP hierarchy has been\ninvestigated \\cite{ST}. It turns out that the Miura map is a\ncanonical map in the sense that the first and second Hamiltonian\nstructures of the mKP hierarchy \\cite{O,OS} are mapped to the\nfirst and second Hamiltonian structures of the KP hierarchy. Since\nthe bi-Hamiltonian structures of mKP and KP have their own\ncorrespondences in dmKP and dKP, thus we expect that the\nbi-Hamiltonian structures of dKP and dmKP are still preserved\nunder the Miura map. We will show, in section 4, that it is indeed\nthe case.\n\nOn the other hand, the solution structure of dKP is also an\ninteresting subject. To extend the tau-function theory in KP\ntheory to the semi-classical one in dKP hierarchy and\ndispersionless analogue of Virasoro constraints \\cite{kr1},\nTakasaki and Takebe \\cite{Ta2} proposed twistor construction of\nthe dKP hierarchy using the Orlov function, which can be regarded\nas the semi-classical limit of the Orlov operator in KP theory\n\\cite{Or1,VM}. Using the Miura map between dKP and dmKP, we can\nconstruct the Orlov function of the dmKP hierarchy and hence\nestablish the twistor construction for dmKP. \\\\ \\indent Our paper\nis organized as follows: Section II is background materials for\ndKP and dmKP; Section III is the Miura map between dKP and dmKP;\nSection IV proves the canonical property of the Miura map; Section\nV shows the twistor construction of the dmKP hierarchy; Section VI\nlists some unsolved problems.\n\n\\section{background materials}\n\n\\subsection{dKP hierarchy}\n\nLet's start with the KP hierarchy. The Lax operator of the KP\nhierarchy is ($\\partial=\\partial_x$)\n \\begin{eqnarray}\n L=\\partial+ \\sum_{n=1}^{\\infty}u_{n+1}\\partial^{-n} \\no\n\\end{eqnarray}\nand the KP hierarchy is determined by the Lax equations\n($\\partial_n=\\frac{\\partial}{\\partial t_n},t_1=x$)\n\\begin{eqnarray}\n\\partial_n L=[B_n, L], \\label{lax}\n\\end{eqnarray}\nwhere $B_n=(L^n)_+$ is the differential part of $L^n$. The Lax\nequation (\\ref{lax}) is equivalent to the existence of the wave\nfunction $\\Psi_{KP}$ such that\n\\begin{eqnarray}\nL \\Psi_{KP} &=&\\lambda \\Psi_{KP}, \\no \\\\\n\\partial_n \\Psi_{KP} &=& B_n \\Psi_{KP}.\\no \\label{line}\n\\end{eqnarray}\nNow for the dKP hierarchy, one can think of fast and slow\nvariables or averaging procedures, by simply taking $t_n \\to\n\\epsilon t_n=T_n$($t_1=x, \\epsilon x=X$) in the KP equation\n\\be\nu_t=\\frac{1}{4}u_{xxx}+3uu_x+\\frac{3}{4} \\partial_x^{-1}u_{yy},\n\\qquad (y=t_2,t=t_3)\n \\label{kp}\n \\ee\n with $\\partial_n \\to \\epsilon\n\\frac{\\partial}{\\partial T_n}$ and $u(t_n) \\to U(T_n)$ to obtain\n\\begin{eqnarray}\n\\partial_T U=3UU_X+\\frac{3}{4}\\partial_X^{-1}U_{YY} \\label{dkp}\n\\end{eqnarray}\nwhen $\\epsilon \\to 0$ and thus the dispersionless term $u_{xxx}$\nis removed. In terms of hierarchies we write\n\\begin{eqnarray}\nL_{\\epsilon}=\\epsilon \\partial +\n\\sum_{n=1}^{\\infty}u_{n+1}(T/\\epsilon) (\\epsilon \\partial)^{-n}\n\\no\n\\end{eqnarray}\nand think of $u_n(T/\\epsilon)=U_n(T)+O(\\epsilon)$, etc. One then\ntakes a WKB form for the wave function $\\Psi_{KP}$ with the action\n$S_{KP}$\n\\begin{eqnarray}\n\\Psi_{KP}=\\exp[\\frac{1}{\\epsilon}S_{KP}(T,\\lambda)] \\no.\n\\end{eqnarray}\nNow, we replace $\\partial_n $ by $\\epsilon\n\\frac{\\partial}{\\partial T_n}$ and define $P=\\partial_X S_{KP}$.\nThen $\\epsilon^i \\partial^i \\Psi_{KP} \\to P^i \\Psi_{KP}$ as\n$\\epsilon \\to 0$ and the equation $L \\Psi_{KP}= \\lambda\n\\Psi_{KP}$ implies\n\\[\n\\lambda=P+\\sum_{n=1}^{\\infty}U_{n+1}(T)P^{-n}.\n\\]\nWe also note from $\\partial_n \\Psi_{KP}=B_n \\Psi_{KP}$ that one\nobtains $\\frac{\\partial S_{KP}}{\\partial T_n}={{\\mathcal B}}_n\n(P)=(\\lambda^n)_+$, where the subscript $(+)$ now refers to powers\nof $P$. The KP hierarchy goes to\n\\begin{equation}\n\\frac{\\partial P}{\\partial T_n}=\\frac{ \\partial {\\mathcal\nB}_n(P)}{\\partial X}. \\label{kph}\n\\end{equation}\n\nAlso, the Lax equation (\\ref{lax}) goes to\n\\begin{equation}\n\\partial_n \\lambda =\\{{\\mathcal B}_n (P), \\lambda \\}, \\label{zero}\n\\end{equation}\nwhere the Poisson bracket $\\{,\\}$ is defined by\n\n\\begin{equation}\n\\{f(X,P),g(X,P)\\}=\\frac{\\partial f}{\\partial P} \\frac{\\partial\ng}{\\partial X}- \\frac {\\partial f}{\\partial X}\\frac{\\partial\ng}{\\partial P}. \\label{poss}\n\\end{equation}\nNotice that both the equations (\\ref{kph}) and (\\ref{zero}) are\ncompatible respectively, i.e, $\\partial^2 \\lambda / \\partial T_n\n\\partial T_m= \\partial^2 \\lambda / \\partial T_m \\partial T_n$ ,\n$\\partial^2 P / \\partial T_n \\partial T_m= \\partial^2 P / \\partial\nT_m \\partial T_n$, and they both imply the dKP hierarchy\n\\begin{equation}\n\\frac{\\partial {\\mathcal B}_n(P)}{\\partial T_m}-\\frac {\\partial\n{\\mathcal B}_m(P)}{\\partial T_n} + \\{{\\mathcal B}_n(P), {\\mathcal\nB}_m(P) \\}=0.\n\\end{equation}\n\nIn particular,\n \\begin{eqnarray}\n{\\mathcal B}_2(P) &= &P^2+2U_2, \\no \\\\ {\\mathcal B}_3 (P) &=&\nP^3+ 3U_2P +3U_3. \\no\n \\end{eqnarray}\n Then ($T_2=Y, T_3=T$)\n\\[\n\\frac{\\partial {\\mathcal B}_2(P)}{\\partial T}-\\frac {\\partial\n{\\mathcal B}_3(P)}{\\partial Y} + \\{{\\mathcal B}_2(P), {\\mathcal\nB}_3(P) \\}=0\n\\]\nbecomes\n\\begin{eqnarray}\nU_{3X}&=& \\frac{1}{2}U_{2Y}, \\\\ U_{3Y} &= & \\frac{2}{3}\nU_{2T}-2U_2U_{2X}\n\\end{eqnarray}\nand thus\n\\[\n\\frac{1}{2}U_{2YY}=\\frac{2}{3}(U_{2T}-3U_2U_{2X})_X.\n\\]\nThis is the dKP equation (\\ref{dkp}) ($U_2=U$).\n\nIn summary, we define the dKP hierarchy by\n\\begin{eqnarray}\n\\lambda &=& P+\\frac{U_2}{P}+\\frac{U_3}{P^2}+\\cdots, \\label{expr}\n\\\\\n\\partial_n \\lambda &=&\\{{\\mathcal B}_n (P), \\lambda \\}. \\label{zero1}\n\\end{eqnarray}\n Let us define the Hamiltonians $H_k= 1/k \\int \\res\n(\\lambda^k),$ where $\\res$ means the coefficient of $P^{-1}$, then\nthe bi-Hamiltonian structure of dKP (\\ref{zero1}) is given by\n ~\\cite{FR,Li}\n\\[\n\\frac{\\partial \\lambda}{\\partial T_k} = \\{H_k, \\lambda\n\\}=\\Theta^{(2)}(dH_k)=\\Theta^{(1)} (dH_{k+1}), \\qquad k=1,2,\n\\cdots\n\\]\nwhere the Hamiltonian one-form $dH_k$ and the Hamiltonian maps\n$\\Th^{(i)}$ are defined by\n\\begin{eqnarray}\n dH_k &=& \\frac{\\delta H_k}{\\delta U_2}+ \\frac{\\delta\nH_k}{\\delta U_3}P+ \\frac{\\delta H_k}{\\delta U_4}P^2 + \\frac{\\delta\nH_k}{\\delta U_5}P^3+ \\cdots, \\no\\\\\n \\Theta^{(2)}(dH_k) &=& \\lambda \\{\\lambda, dH_k \\}_+-\\{\\lambda\n,(\\lambda dH_k)_+ \\} \\label{ham1}\n \\\\ &&+ \\{\\lambda, \\int^X \\res\n\\{\\lambda, dH_k \\} \\}, \\no \\\\ \\Theta^{(1)}(dH_{k+1}) &=&\n\\{\\lambda, dH_{k+1} \\}_+ -\\{\\lambda ,(dH_{k+1})_+ \\} ,\\no\n\\label{ham2}\n\\end{eqnarray}\n the third term of (\\ref{ham1}) being Dirac reduction for\n$U_1=0$. \\\\\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{dmKP}\n\nThe Lax operator of the mKP hierarchy is defined \\cite{KO} by\n\n \\[\nK=\\partial +v_0 + v_1 \\partial^{-1} +v_2 \\partial^{-2} + \\cdots.\n \\]\nwhich satisfies the Lax equations\n\\begin{equation}\n\\partial_n K =[Q_n, K] \\label{mkp},\n\\end{equation}\nwhere $Q_n=(K^n)_{\\geq 1}$ means the part of order $\\geq 1$ of\n$K^n$. Also, the Lax equation (\\ref{mkp}) is equivalent to the\nexistence of wave function $\\Psi_{mKP}$ such that\n\\begin{eqnarray}\nK \\Psi_{mKP} &=& \\mu \\Psi_{mKP}, \\no \\\\\n\\partial_n \\Psi_{mKP} &=& Q_n \\Psi_{mKP}.\\no \\label{lmkp}\n\\end{eqnarray}\n\nTo obtain the dmKP hierarchy, similarly, one takes $t_n \\to\n\\epsilon t_n =T_n(t_1=x \\to \\epsilon t_1 =X) $ in the mKP equation\n\\be\n\\label{mkpeq}\nv_t=\\frac{1}{4}v_{xxx}-\\frac{3}{2}v^2v_x+\\frac{3}{2}v_x\n\\partial_x^{-1}v_y\n+\\frac{3}{4} \\partial_x^{-1}v_{yy},\n \\ee\n with $\\partial_n \\to\n\\epsilon \\partial / \\partial T_n$ and $v(t_n) \\to V(T_n)$ to get\n\\begin{equation}\nV_T=-\\frac{3}{2}V^2V_X+\\frac{3}{2}V_X \\partial_X^{-1}V_Y\n+\\frac{3}{4} \\partial_X^{-1}V_{YY} \\label{dmkp},\n\\end{equation}\nwhen $ \\epsilon \\to 0$ . Thus, the dispersionless term $v_{xxx}$\nis removed, too. In terms of hierarchies, we write\n\\[\nK_{\\epsilon}=\\epsilon \\partial+ v_{1}(T/\\epsilon)(\\epsilon\n\\partial)^{-1}\n+ v_{2}(T/\\epsilon)(\\epsilon \\partial)^{-2} +\\cdots\n\\]\nand think of $v_n( T/ \\epsilon)=V_{n}(T)+0(\\epsilon)$. One then\ntakes a WKB form for the wave function $\\Psi_{mKP}$ with the\naction $S_{mKP}$:\n\\[\n\\Psi_{mKP}= \\exp (\\frac{1}{\\epsilon} S_{mKP}( T, \\mu)).\n\\]\nNow we replace $\\partial_n$ by $ \\epsilon \\partial / \\partial T_n$\nand define $P=\\partial_X S_{mKP}$. Then $\\epsilon^i \\partial^i_X\n\\Psi_{mKP} \\to P^i \\Psi_{mKP}$ as $\\epsilon \\to 0$ and the\nequation $K \\Psi_{mkP} =\\mu \\Psi_{mKP}$ yields\n\\[\n\\mu =P+\\sum_{n=o}^{\\infty}V_{n}(T)P^{-n}.\n\\]\n\nFrom $\\partial_n \\Psi_{mKP}= Q_n \\Psi_{mKP}$, one obtains\n$\\partial S_{mKP}/ \\partial T_n = {\\mathcal Q}_n(P)= (\n\\mu^n)_{\\geq 1}$, where the subscript $\\geq 1$ refers to powers $\n\\geq 1$ of $P$. The dmKP hierarchy goes to\n\\[\n\\frac{\\partial P}{\\partial T_n}= \\frac{\\partial {\\mathcal\nQ}_n(P)}{\\partial X} .\n\\]\n\nIt also can be written as the following zero-curvature form\n\n\\[\n\\frac{\\partial {\\mathcal Q}_n(P)}{\\partial T_m}-\\frac {\\partial\n{\\mathcal Q}_m(P)}{\\partial T_n} + \\{{\\mathcal Q}_n(P), {\\mathcal\nQ}_m(P) \\}=0,\n\\]\nwhere the Poisson bracket is defined by (\\ref{poss}). In\nparticular,\n\\begin{eqnarray}\n{\\mathcal Q}_2(P)& =& P^2+2PV_0, \\\\ {\\mathcal Q}_3(P) &=& P^3\n+3P^2 V_0 +P(V_1 +3V_0^2).\n\\end{eqnarray}\n\nThen the equation($T_2=Y, T_3=T$)\n\\[\n\\frac{\\partial {\\mathcal Q}_2(P)}{\\partial T}-\\frac {\\partial\n{\\mathcal Q}_3(P)}{\\partial Y} + \\{{\\mathcal Q}_2(P), {\\mathcal\nQ}_3(P) \\}=0,\n\\]\nbecomes\n\\begin{eqnarray}\nV_{1X} &=& \\frac{3}{2} V_{0Y}-\\frac{3}{2}(V_0^2)_X, \\label{mot1}\n\\\\\nV_{1Y} &=& 2 V_{0T}-3V_0V_{0Y}-2V_1V_{0X}. \\no\n\\end{eqnarray}\nwhich implies the dmKP (\\ref{dmkp}) ($V_0=V$).\n\nIn summary, we write the dmKP equation as\n\\begin{eqnarray}\n\\mu &= &P+V_0+\\frac{V_1}{P}+ \\frac{V_2}{P^2}+\\cdots, \\no \\\\\n\\partial_n \\mu &=& \\{ {\\mathcal Q}_n(P), \\mu \\}. \\label{dmkp1}\n\\end{eqnarray}\n\\indent If we define the Hamiltonians as $H_k=\\frac{1}{k} \\int\n\\res(\\mu^k)$, then the bi-Hamiltonian structure of (\\ref{dmkp1})\nis described by ~\\cite{Li}\n\\[\n\\frac{\\partial \\mu}{\\partial T_k}=\\{H_k, \\mu\n\\}=J^{(2)}(dH_k)=J^{(1)}(dH_{k+1})\n\\]\nwhere\n\\begin{eqnarray}\ndH_k &=& \\frac{\\delta H_k}{\\delta V_0} P^{-1} + \\frac{\\delta\nH_k}{\\delta V_1} + \\frac{\\delta H_k}{\\delta V_2} P + \\frac{\\delta\nH_k}{\\delta V_3} P^2+ \\cdots, \\no \\\\ J^{(2)}(dH_k) &=& \\mu \\{\\mu\n, dH_k \\}_{\\geq -1}-\\{\\mu, (\\mu dH_k)_{\\geq 1} \\}, \\label{poss1}\n\\\\ J^{(1)}(dH_{k+1}) &=& \\{\\mu , dH_{k+1} \\}_{\\geq -1}-\\{\\mu,\n(dH_{k+1})_{\\geq 1} \\}. \\label{possi}\n\\end{eqnarray}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Dispersionless Miura Map}\n\nIt has been shown \\cite{ki,KO,ku2,OR,St1,ST} that there exists a\ngauge transformation (Miura map) between the Lax operator $L$ of\nKP and the Lax operator $K$ of mKP, namely,\n\\begin{equation}\nK=\\Phi^{-1}(t)L \\Phi(t) \\label{gauge},\n\\end{equation}\nwhere $\\Phi(t)$ is an eigenfunction of L, i.e.,\n\\begin{equation}\n\\partial_n \\Phi=(L^n)_+ \\Phi. \\label{eigen}\n\\end{equation}\nOne generalizes this result to dispersionless limit case.\n \\\\\n\\indent Let\n\\[\n{\\mathcal L}=P^m+a_{m-1}P^{m-1}+a_{m-2}P^{m-2}+\\cdots+a_0+\n\\frac{a_{-1}}{P} +\\frac{a_{-2}}{P^2}+\\cdots,\n\\]\nwhere $a_{m-1}, a_{m-2}, \\cdots,a_0 ,a_{-1}, a_{-2}, \\cdots$ are\nfunctions of $T=(T_1=X, T_2, T_3, \\cdots)$. Also, we suppose\n$\\phi(T)$(independent of $P$) is any function of $T$. We define\n\\begin{eqnarray}\n\\tilde {{\\mathcal L}} &=& e^{-ad \\phi(T)} {\\mathcal L}, \\\\ & =&\n{\\mathcal L}-\\{\\phi, {\\mathcal L}\\}+\\frac{1}{2}\\{\\phi,\n\\{\\phi,{\\mathcal L}\\}\\} -\\frac{1}{3!} \\{\\phi ,\\{\\phi,\n\\{\\phi,{\\mathcal L}\\}\\}\\}+ \\cdots\n\\end{eqnarray}\nwhere the Poisson bracket is defined by (\\ref{poss}). Since $\\phi$\nis independent of $P$, a simple calculation gets\n\\begin{equation}\n\\tilde {{\\mathcal L}} =\\sum_{n=0}^{\\infty} \\frac{1}{n!} (\\phi_X)^n\n\\partial_P^n {\\mathcal L} \\label{simp}.\n\\end{equation}\n\\begin{lemma} Let $ \\tilde {{\\mathcal L}}$ be defined as above. Then\n\\[\n\\tilde {{\\mathcal L}}_{\\geq 1} = e^{-ad \\phi}({\\mathcal L}_{\\geq\n0})-{\\mathcal L}_{\\geq 0} \\vert_{P=\\phi_X},\n\\]\nwhere\n\\[\n {\\mathcal L}_{\\geq 0} \\vert_{P=\\phi_X}= \\phi_{X}^m +a_{m-1} \\phi_{X}^{m-1}\n+\\cdots +a_1 \\phi_X +a_0.\n\\]\n\\end{lemma}\n{\\it Proof\\/}. From (\\ref{simp}), one knows that $\\tilde {\\mathcal\nL}_{\\geq 0}$ comes from the polynomial part of ${\\mathcal L}$.\n Hence\n\\begin{eqnarray}\n\\tilde {{\\mathcal L}}_{\\geq 1} &=& \\tilde {{\\mathcal L}}_{\\geq\n0}-\\tilde {{\\mathcal L}}_0, \\\\ &=& e^{-ad \\phi}({\\mathcal\nL}_{\\geq 0})-e^{-ad \\phi}({\\mathcal L}_{\\geq 0}) \\vert_{P=0}.\n\\end{eqnarray}\n\nUsing (\\ref{simp}), one knows\n\\begin{eqnarray}\n e^{-ad \\phi}({\\mathcal L}_{\\geq 0}) \\vert_{P=0} &=& \\sum_{n=0}^{\\infty}\n \\frac{1}{n!} \\phi_{X}^n (\\partial_P^n {\\mathcal L}_{\\geq 0} \\vert_{P=0}), \\\\\n&=& \\sum_{n=0}^{\\infty} \\frac{1}{n!} \\phi_{X}^n ( a_n n!), \\\\ &=&\n\\phi_X^m + a_{m-1} \\phi_X^{m-1}+a_{m-2} \\phi_X^{m-2} +\\cdots +a_1\n\\phi_X +a_0, \\\\ &=& {\\mathcal L}_{\\geq 0} \\vert_{P=\\phi_X}.\n\\end{eqnarray}\nThis completes the lemma.\\qquad $\\Box$\n\n\n\\begin{lemma}\n$e^{-ad \\phi} \\{f(T,P), g(T,P) \\}=\\{e^{-ad \\phi} f(T,P), e^{-ad\n\\phi} g(T,P)\\}$.\n\\end{lemma}\n{\\it Proof\\/}.\n\\begin{eqnarray}\nr.h.s. &=& \\sum_{m,n=0}^{\\infty} \\frac{1}{m!n!} \\{\\phi_X^n\n\\partial_P^n f, \\phi_X^m \\partial_P^m g \\}, \\\\\n&=& \\sum_{m,n=0}^{\\infty} \\frac{1}{m!n!} \\phi_X^{m+n}\n\\{\\partial_P^n f, \\partial_P^m g \\}, \\\\ &=& \\sum_{m=0}^{\\infty}\n\\frac{\\phi_X^m}{m!} \\sum_{n=0}^m \\left(\n\\begin{array}{c}\n m \\\\ n\n\\end{array}\n\\right ) \\{\\ \\partial_P^n f, \\partial_P^{m-n} g \\}, \\\\ &=& e^{-ad\n\\phi} \\{f,g \\}=l.h.s.\\qquad \\Box\n\\end{eqnarray}\n\\begin{theorem} Let $ \\tilde {{\\mathcal L}}$ be defined as above. Then\n\\[\n\\tilde{{\\mathcal L}}_{T_q}-\\{(\\tilde{{\\mathcal L}}^q )_{\\geq 1} ,\n\\tilde {{\\mathcal L}} \\}=e^{-ad \\phi}({\\mathcal\nL}_{T_q}-\\{({\\mathcal L}^q )_+ , {\\mathcal L} \\})-\n\\{\\phi_{T_q}-({\\mathcal L}^q )_+ \\vert_{P=\\phi_X}, \\tilde{\\mathcal\nL} \\} ,\n\\]\nwhere the subscript $T_q$ means $\\partial / \\partial T_q$.\n\\end{theorem}\n{\\it Proof\\/}. Using (\\ref{simp}), we have\n\\begin{eqnarray}\n\\frac{\\partial \\tilde{{\\mathcal L}}}{\\partial T_q} & = & e^{-ad\n\\phi} \\frac{\\partial {\\mathcal L}}{\\partial T_q}+\n\\sum_{n=0}^{\\infty} \\frac{1}{n !}[\\frac{\\partial}{\\partial T_q}\n(\\phi_X)^n]\n\\partial_P^n {\\mathcal L}, \\\\\n &=& e^{-ad \\phi} \\frac{\\partial {\\mathcal L}}{\\partial T_q}+\n(\\frac{\\partial^2 \\phi}{\\partial T_q \\partial X})\n\\sum_{n=0}^{\\infty} \\frac{1}{n !} \\phi_X^{n} \\partial_P^{n+1}\n{\\mathcal L}, \\\\ &=& e^{-ad \\phi} \\frac{\\partial {\\mathcal\nL}}{\\partial T_q}-\\{\\phi_{T_q}, e^{-ad \\phi} {\\mathcal L} \\}.\n\\end{eqnarray}\nThen, by Lemmas 1 and 2, we have\n\\begin{eqnarray}\n\\tilde{{\\mathcal L}}_{T_q}-\\{(\\tilde{{\\mathcal L}}^q )_{\\geq 1} ,\n\\tilde {{\\mathcal L}} \\} &=& e^{-ad \\phi}({\\mathcal\nL}_{T_q}-\\{\\phi_{T_q}, {{\\mathcal L}}\\})- \\{e^{-ad \\phi}({\\mathcal\nL}^q)_+-({\\mathcal L}^q)_+ \\vert_{P=\\phi_X},\n e^{-ad \\phi} {\\mathcal L} \\}, \\\\\n &=& e^{-ad \\phi}({\\mathcal L}_{T_q}-\\{({\\mathcal L}^q)_+, {\\mathcal L}\\})\n + \\{ ({\\mathcal L}^q)_+ \\vert_{P=\\phi_X}-\\phi_{T_q}, \\tilde{{\\mathcal L}}\\}.\n\\end{eqnarray}\nThis completes the theorem. $\\Box$\n\n\\begin{corollary} Let\n\\[\n{\\mathcal L} =P+\n\\frac{U_2}{P}+\\frac{U_3}{P^2}+\\frac{U_4}{P^3}+\\cdots\n\\]\nand suppose that $U_i(T)$ satisfy the dKP hierarchy (\\ref{zero})\n($\\lambda = {\\mathcal L}$) and $\\phi (T)$ satisfies the equation\n\\begin{equation}\n\\frac{\\partial \\phi}{\\partial T_n}=({\\mathcal L}^n)_+\n\\vert_{P=\\phi_X}. \\label{deigen}\n\\end{equation}\nThen $\\tilde {{\\mathcal L}}=e^{-ad \\phi} {\\mathcal L}$ will\nsatisfy the dmKP hierarchy (\\ref{dmkp1}) ($\\mu=\\tilde {\\mathcal\nL}$).\n\\end{corollary}\n{\\it Proof\\/}. Obvious. $\\Box$ \\\\ From the corollary, one calls\nthe map\n\\begin{equation}\n{\\mathcal L} \\to e^{-ad \\phi} {\\mathcal L} \\label{dmp}\n\\end{equation}\nthe dispersionless Miura map between dKP and dmKP. It's because\none can think the map (\\ref{dmp}) as the dispersionless limit of\nequation (\\ref{gauge}) and, moreover, the equation (\\ref{deigen})\ncan be regarded as the dispersionless limit of equation\n(\\ref{eigen}). As in the case of KP and mKP, the dispersionless\nMiura map gives rise to a transformation between dKP and dmKP in\nterms of \"dispersionless\" eigenfunction $\\phi(T)$. If one assumes\nthat\n\\[\n\\tilde {\\mathcal\nL}=P+V_0+\\frac{V_{1}}{P}+\\frac{V_{2}}{P^2}+\\frac{V_{3}}{P^3} +\n\\cdots,\n\\]\nthen, after some calculations, one gets\n\\begin{eqnarray}\nV_0 &=& \\phi_X, \\no \\\\ V_{1} &=& U_2, \\label{mot2} \\\\ V_{2} &=&\nU_3 +\\phi_X U_2, \\no \\\\ V_{3} &=& U_4 +2 \\phi_X U_3 + \\phi_X^2\nU_2, \\no \\\\ V_{4} &=& U_5 +3 \\phi_X U_4 + 3\\phi_X^2 U_3+\\phi_X^3\nU_2, \\no \\\\ & \\vdots& \\no \\\\ V_{n} &=& \\sum_{i=0}^{n-1} \\left(\n\\begin{array}{c}\n n-1 \\\\ i\n\\end{array}\n\\right) \\phi_X^i U_{n+1-i},\\,\\ n \\geq 1. \\no\n\\end{eqnarray}\n\\indent Finally, it is well known that Miura-type transformations\nbetween (\\ref{kp}) and (\\ref{mkpeq}) are\n \\bea\n u_1 &=& \\frac{3}{2} (-v^2-v_x+\\partial_x^{-1}v_y), \\no \\\\\n u_2&=&\\frac{3}{2} (-v^2+v_x+\\partial_x^{-1}v_y)\\no \\label{miura}.\n \\eea\n\nIn the dispersionless limit, the term $v_x$ is removed and we\nobtain the only transformation\n\\begin{equation}\nU=\\frac{3}{2}(-V^2+\\partial_X^{-1}V_Y). \\label{dmir}\n\\end{equation}\nNotice that we can also obtain the equation (\\ref{dmir}) from\n(\\ref{mot1}) and (\\ref{mot2}). Furthermore, since the term\ncorresponding to $v_x$ is removed, this would explain why we\ncannot find the auto-B\\\"acklund transformation of the dKP\nhierarchy as one did in the ordinary case \\cite{St1}.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{canonical property of the Miura map}\n\nHaving constructed the dispersionless Miura map between the dKP\nhierarchy and the dmKP hierarchy in the Lax formulation, which\nprovides a connection of solutions associated with dKP and dmKP,\nwe next would like to investigate the canonical property of the\nMiura map. As we have seen that both dKP and dmKP hierarchies\nequip a compatible bi-Hamiltonian structure, thus it is quite\nnatural to ask whether their bi-Hamiltonian structures are still\npreserved under the Miura map.\n\nTo proceed the discussion, it is convenient to rewrite the\ndispersionless Miura map as\n\\begin{equation}\\label{gmap}\n G: \\mu(T,P)\\to \\lambda(T,P)=e^{ad \\phi(T)}\\mu(T,P)\n\\end{equation}\nwhere $\\lambda$ and $\\mu$ are Lax operators of the dKP and dmKP\nhierarchies respectively and the function $\\phi(T)=\\int^XV_0$ is\nindependent of $P$. In the following, the symbols $A$, $B$ and $C$\nwill stand for arbitrary Laurent series without further mention.\n\n\\begin{lemma} $e^{-ad \\phi(T)}e^{ad\\phi(T)}A=A.$\n \\end{lemma}\n\n{\\it Proof\\/}. By definition,\n \\bea\n e^{-ad\\phi(T)}e^{ad\\phi(T)}A\n &=&e^{-ad\\phi(T)}\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{n!}\\phi_X^n\\pa_P^nA,\\no\\\\\n&=&\\sum_{m=0}^{\\infty}\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{m!n!}\n\\phi_X^{m+n}\\pa_P^{m+n}A,\\no\\\\\n&=&\\sum_{m=0}^{\\infty}\\frac{1}{m!}\\phi_X^m\\pa_P^mA\n\\sum_{n=0}^{m}(-1)^n \\left(\n\\begin{array}{c}\nm \\\\ n\n\\end{array}\n\\right ) =A.\\no\\qquad \\Box\n \\eea\n\n \\begin{lemma}\n$e^{-ad \\phi}(AB)=(e^{-ad \\phi} A)(e^{-ad \\phi} B)$.\n\\end{lemma}\n{\\it Proof\\/}.\n\\begin{eqnarray}\ne^{-ad \\phi}(AB) &=& \\sum_{n=0}^{\\infty}\n\\frac{\\phi_X^n}{n!}\\partial_P^n(AB), \\\\ &=& \\sum_{n=0}^{\\infty}\n\\frac{\\phi_X^n}{n!} \\left [\\sum_{m=0}^n \\left(\n\\begin{array}{c}\nn \\\\m\n\\end{array}\n\\right) (\\partial_P^m A) (\\partial_P^{n-m} B) \\right ], \\\\ &=&\n\\sum_{n=0}^{\\infty} \\left [\\sum_{m=0}^n \\frac{\\phi_X^{n-m}\n\\phi_X^m}{m!(n-m)!} (\\partial_P^m A) (\\partial_P^{n-m} B) \\right\n],\\\\\n &=& (\\sum_{m=0}^{\\infty} \\frac{ \\phi_X^m}{m!}\n\\partial_P^m A) (\\sum_{n=0}^{\\infty} \\frac{ \\phi_X^n}{n!}\n\\partial_P^n B), \\\\\n&=& (e^{-ad \\phi} A)(e^{-ad \\phi} B).\\qquad \\Box\n\\end{eqnarray}\n\n \\begin{lemma}\n $\\int \\res(A\\{B,C\\})=\\int \\res(\\{A,B\\}C)$\n \\end{lemma}\n {\\it Proof\\/}.\n \\begin{eqnarray}\n l.h.s.&=&\\int \\res \\left[A\\left(\\frac{\\pa B}{\\pa P}\\frac{\\pa C}{\\pa X}-\n \\frac{\\pa B}{\\pa X}\\frac{\\pa C}{\\pa P}\\right)\\right],\\\\\n &=&\\int \\res\\left[-\\frac{\\pa}{\\pa X}\\left(A\\frac{\\pa B}{\\pa P}\\right)C+\n \\frac{\\pa}{\\pa P}\\left(A\\frac{\\pa B}{\\pa X}\\right)C\\right],\\\\\n &=&r.h.s.\\qquad \\Box\n \\end{eqnarray}\n To investigate the canonical property of the Miura map (\\ref{gmap}) we\n shall first construct the tangential map between the tangent\n spaces (to which $\\de \\lambda$ and $\\de \\mu$ belong) of the corresponding\n phase space manifolds.\n\n\\begin{theorem} For the Miura map $G$, the linearized map $G'$ and its\ntransposed map ${G'}^{\\dagger}$ are given by\n \\begin{eqnarray}\n G'&:& B\\to e^{ad\\phi(T)}B+\\{\\int^Xb_0, \\la\\},\n \\label{lmp} \\\\\n {G'} ^{\\dagger}&:& A\\to e^{-ad\\phi(T)}A+P^{-1}\\int^X\\res\\{A, \\la\\}\n \\label{ltmp}\n \\end{eqnarray}\nwhere $b_0\\equiv (B)_0$ and $\\dagger$ is the transposed operation\ndefined by $ \\int\\res( A G'B)=\\int \\res(({G'}^{\\dagger}A) B)$.\n\\end{theorem}\n\n {\\it Proof\\/}. Let $B=\\de\\mu$ be an infinitesimal deformation of the Lax operator\n $\\mu$, then under the Miura map $G$ we have\n \\begin{eqnarray}\n \\mu+B &\\to& e^{ad(\\phi+\\int^X b_0)}(\\mu+B),\\no\\\\\n &=&e^{ad\\phi}\\mu+e^{ad\\phi}B+\\{\\int^X b_0, \\la\\}+O(B^2).\\no\n \\end{eqnarray}\nwhich implies the linearized map (\\ref{lmp}). On the other hand,\nusing Lemmas 5-7 and the fact $\\res(e^{ad\\phi}A)=\\res(A)$ we have\n \\begin{eqnarray}\n \\int \\res(AG'B)&=&\n \\int \\res(A(e^{ad\\phi}B))+\\int \\res(A\\{\\int^X b_0, \\la\\}),\\no\\\\\n &=&\\int \\res((e^{-ad\\phi}A)B)+\\int b_0\\int^X \\res\\{A, \\la\\},\\no\\\\\n &=&\\int \\res((e^{-ad\\phi}A)B)+\\int \\res((P^{-1}\\int^X \\res\\{A,\n \\la\\})B)\\no\n \\end{eqnarray}\nwhere we have used integration by part and $b_0=\\res(BP^{-1})$ to\nreach the last line. Comparing the last line with $\\int\n\\res(({G'}^{\\dagger}A) B)$ we obtain (\\ref{ltmp}). $\\Box$\\\\\n Now we are in a position to investigate the canonical property of the\nMiura map.\n\\begin{theorem} The Miura map $G$ maps the bi-Hamiltonian structure of the\ndmKP hierarchy given by $J^{(1)}$ and $J^{(2)}$ to the\nbi-Hamiltonian structure of the dKP hierarchy given by $\\Th^{(1)}$\nand $\\Th^{(2)}$ respectively, i.e., they are related by\n \\begin{eqnarray}\n \\Th^{(1)}&=&G' J^{(1)}{G'}^{\\dagger},\n \\label{cano1}\\\\\n \\Th^{(2)}&=&G' J^{(2)}{G'}^{\\dagger}\n \\label{cano2}\n \\end{eqnarray}\nwhere $G'$ and ${G'}^{\\dagger}$ are transformations defined in\nTheorem 8.\n\\end{theorem}\n {\\it Proof\\/}. To prove the first structure, let us act the right hand side\n of (\\ref{cano1}) on an arbitrary Laurent series $A$, then $G'J^{(1)}({G'}^{\\dagger}A)=G'B$\n where\n \\begin{eqnarray}\n B&\\equiv&J^{(1)}({G'}^{\\dagger}A),\\no\\\\\n &=&\\{\\mu, {G'}^{\\dagger}A\\}_{\\ge -1}-\\{\\mu, ({G'}^{\\dagger}A)_{\\ge 1}\\},\\no\\\\\n &=&e^{-ad\\phi}\\left(\\{\\la, A\\}_+-\\{\\la, A_+\\}+\\{\\la,\n (e^{-ad\\phi}A)_0\\}\n \\label{B1}\n \\right),\n \\end{eqnarray}\n and thus\n \\begin{equation}\n \\int^X b_0=\\int^X (B)_0=(e^{-ad\\phi}A)_0.\n \\label{b01}\n \\end{equation}\n Substituting (\\ref{B1}) and (\\ref{b01}) into (\\ref{lmp}) we have\n \\begin{eqnarray}\n G'J^{(1)}({G'}^{\\dagger}A)&=&e^{ad\\phi}B+\\{(e^{-ad\\phi}A)_0,\n \\la\\},\\no\\\\\n &=&\\{\\la, A\\}_+-\\{\\la, A_+\\}=\\Th^{(1)}(A).\\no\n \\end{eqnarray}\n This completes the first part of the proof.\n For the second Hamiltonian structure, using (\\ref{poss1}) and (\\ref{ltmp})\n we have\n \\begin{eqnarray}\n B&\\equiv&J^{(2)}({G'}^{\\dagger}A),\\no\\\\\n &=&\\{\\mu, {G'}^{\\dagger}A\\}_+\\mu-\\{\\mu, (\\mu {G'}^{\\dagger}A)_+\\}+\\{\\mu,\n (\\mu{G'}^{\\dagger}A)_0\\}+\\mu P^{-1}\\res\\{\\mu, {G'}^{\\dagger}A\\}\n \\label{b2}\n \\end{eqnarray}\nwhere each term in (\\ref{b2}) can be calculated as follows:\n \\begin{eqnarray}\n (1)&=& e^{-ad\\phi}(\\{\\la, A\\}_+\\la),\\no\\\\\n (2)&=& -e^{-ad\\phi}\\left(\\{\\la, (A\\la)_+\\}+\\{\\la, \\int^X\\res\\{A, \\la\\}\\}\\right),\\no\\\\\n (3)&=& e^{-ad\\phi}\\left(\\{\\la, (e^{-ad\\phi}(A\\la))_0\\}+\n \\{\\la, \\int^X\\res\\{A, \\la\\}\\}\\right),\\no\\\\\n (4)&=&0.\\no\n \\end{eqnarray}\nThen\n \\begin{eqnarray}\n B&=&(1)+(2)+(3)+(4),\\no\\\\\n &=&e^{-ad\\phi}\\left(\\{\\la, A\\}_+\\la-\\{\\la, (A\\la)_+\\}+\n \\{\\la, (e^{-ad\\phi}(A\\la))_0\\} \\right),\n \\label{B2}\n \\end{eqnarray}\nand\n \\begin{equation}\n\\int^X b_0=\\left(e^{-ad\\phi}(\\la A) \\right)_0+\\int^X\\res\\{A,\n\\la\\}. \\label{b02}\n \\end{equation}\nSubstituting (\\ref{B2}) and (\\ref{b02}) into (\\ref{lmp}) we get\n \\[\n G' J^{(2)}({G'}^{\\dagger}A)=\\la\\{\\la, A\\}_+-\\{\\la, (\\la A)_+\\}+\\{\\la, \\int^X\\res\\{\\la,\n A\\}\\}=\\Th^{(2)}(A).\n \\]\n This completes the theorem.\n $\\Box$\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{solution structure of dispersionless mKP}\n\nIn ~\\cite{Ta2,Ta1}, it is shown that the twistor construction\nexists for the solution structure of dKP hierarchy. Based on the\ndispersionless Miura map described in Section III, we can also\nfind a similar twistor construction for solution structure of\ndmKP. This is the purpose of this section.\n\\\\ \\indent First of all, let's recall the twistor construction of\ndKP in ~\\cite{Ta2,Ta1}. Here we change slightly the symbols used\nin those papers. Let's consider the dKP (\\ref{zero1}). It can be\nshown that there exists a Laurent series $\\psi(T,P)$ (dressing\nfunction) such that\n\\[\n\\lambda=e^{ad \\psi}(P),\n\\]\nwhere $\\psi(T,P)$ has the form\n\\[\n\\psi(T,P)=\\sum_{n=1}^{\\infty}\\psi_n(T)P^{-n}.\n\\]\nSuch Laurent series $\\psi(T,P)$ is not unique up to a constant\nLaurent series $\\sum_{i=0}^{\\infty} c_i P^{-i}$. The Orlov\nfunction of dKP is by definition a formal Laurent series\n\\cite{Ta2,Ta1}\n\\[\n{\\mathcal M}=e^{ad \\psi} (\\sum_{n=1}^{\\infty} n T_n P^{n-1}).\n\\]\n\nIt's convenient to expand ${\\mathcal M}$ into a Laurent series of\n$\\lambda$ as\n\\begin{equation}\n{\\mathcal M}=\\sum_{n=1}^{\\infty} n T_n \\lambda^{n-1}+\n\\sum_{i=1}^{\\infty}h_i(T) \\lambda^{-i}. \\label{orlov}\n\\end{equation}\nIt can be also shown that the series ${\\mathcal M}$ satisfies the\nLax equation\n\\begin{equation}\n\\frac{\\partial {\\mathcal M}}{\\partial T_n}=\\{{\\mathcal B}_{n},\n{\\mathcal M} \\} \\label{orlov1}\n\\end{equation}\nand the canonical Possion relation\n\\begin{equation}\n\\{\\lambda , {\\mathcal M} \\}=1. \\label{poss2}\n\\end{equation}\n\nTo get the solution structure of dKP hierarchy, let's consider a\npair of two functions $(f(P,X), g(P,X))$ such that they are\narbitrary holomorphic functions defined in a neighborhood of\n$P=\\infty$ except at $P=\\infty$ itself. Then we have the following\nfact(twistor construction of dKP hierarchy). \\\\\n {\\bf Fact:(K.Takasaki and T. Takebe, \\cite{Ta2})}\n Suppose \\\\ (i) $\\lambda$\nand ${\\mathcal M}$ has the form (\\ref{expr}) and (\\ref{orlov}). \\\\\n (ii) $f(P,X)$ and $g(P,X)$\ndescribed as above satisfy the canonical relation\n\\begin{equation}\n \\{f(P,X), g(P,X) \\}=1. \\label{one}\n\\end{equation}\nThen the following functional equations(in $P$)\n\\begin{equation}\nf(\\lambda, {\\mathcal M})_{\\leq -1}=0 \\hspace{8mm} g(\\lambda ,\n{\\mathcal M})_{\\leq -1}=0 \\label{twist}\n\\end{equation}\nwill imply equations (\\ref{zero1}), (\\ref{orlov1}) and\n(\\ref{poss2}), i.e, the pair $(\\lambda, {\\mathcal M})$ gives a\nsolution of dKP hierarchy. We call $(f(P,X),g(P,X))$ the twistor\ndata of this solution. $\\parallel$\\\\\n \\indent\n Conversely, each solution of dKP hierarchy possesses a twistor data\ncorresponding to the solution, i.e, if $(\\lambda, {\\mathcal M})$\nis a solution of (\\ref{zero1}), (\\ref{orlov1}) and (\\ref{poss2}),\nthen there exists a pair $(f(P,X),g(P,X))$ which satisfies\n(\\ref{one}) and (\\ref{twist}). In fact, if we let $e^{ad\n\\psi(T,P)}$ be the dressing operator corresponding to $(\\lambda,\n{\\mathcal M})$, then the twistor data $(f,g)$ of this solution\nwill be\n\\begin{eqnarray}\nf(P,X)& = & e^{-ad \\psi_0(X,P)} P, \\no \\\\ g(P,X)& = & e^{-ad\n\\psi_0(X,P)} X, \\label{pair}\n\\end{eqnarray}\nwhere $\\psi_0(X, P)=\\psi(T_1=X, T_2=T_3=T_4= \\cdots =0, P)$. \\\\\n\\indent Next, we consider the dispersionless Miura map (\\ref{dmp})\nfrom dKP to dmKP. Let us define\n\\begin{eqnarray}\n\\mu &=& e^{-ad \\phi(T)} \\lambda, \\no \\\\ \\tilde {{\\mathcal M}} &=&\ne^{-ad \\phi(T)} {{\\mathcal M}}. \\label{me}\n\\end{eqnarray}\nThen $\\mu$ satisfies dmKP hierarchy (Theorem 3) and from Lemma 2\nwe have\n\\begin{equation}\n\\{\\mu, \\tilde {{\\mathcal M}} \\}=1. \\label{poss3}\n\\end{equation}\n\nMorever, a similar argument of Theorem 3 can also show that\n\\begin{equation}\n\\frac{\\partial \\tilde {{\\mathcal M}}}{\\partial T_n}=\\{{\\mathcal\nQ}_n(P), \\tilde {{\\mathcal M}} \\}. \\label{tim}\n\\end{equation}\n\nNow, we want to construct a pair of twistor data $(\\tilde f(P,X),\n\\tilde g(P,X))$ corresponding to $\\mu$ and $\\tilde {{\\mathcal M}}$\ndefined in (\\ref{me}).\n\\begin{theorem} Let $(\\lambda, {\\mathcal\nM})$ be a solution of (\\ref{zero1}), (\\ref{orlov1}) and\n(\\ref{poss2}) and $\\mu, \\tilde {{\\mathcal M}} $ is defined by the\nMiura map (\\ref{me}). If we define\n\\begin{eqnarray}\n\\tilde f (P,X) &=& e^{-ad \\psi_0(X,P)} e^{ad \\phi_0(X)}P, \\\\\n\\tilde g(P,X) &=& e^{-ad \\psi_0(X,P)} e^{ad \\phi_0(X)}X=g(P,X),\n\\end{eqnarray}\nwhere $\\psi_0(X,P)$ is defined in (\\ref{pair}) and\n$\\phi_0(x)=\\phi(T_1=X, T_2=T_3=\\cdots =0)$, (obviously, we have\n$\\{\\tilde f, \\tilde g \\}=1$.) then\n\\begin{eqnarray}\n\\tilde f (\\mu, \\tilde {{\\mathcal M}})_{\\leq 0} &= & 0, \\\\ \\tilde g\n(\\mu, \\tilde {\\mathcal M})_{\\leq -1}&= & 0.\n\\end{eqnarray}\n\\end{theorem}\n{\\it Proof\\/}. For convenience, we let $T=0$ mean\n$T_2=T_3=T_4=\\cdots=0$. Since\n\\begin{eqnarray}\n\\lambda (T=0) &= & e^{ad \\psi_0} P, \\\\ {{\\mathcal M}} (T=0) &=&\ne^{ad \\psi_0} X,\n\\end{eqnarray}\nthen we have\n\\begin{eqnarray}\n\\mu(T=0)&=&e^{-ad \\phi_0} \\lambda (T=0) =e^{-ad \\phi_0}e^{ad\n\\psi_0}P, \\\\ \\tilde {{\\mathcal M}}(T=0)&=&e^{-ad \\phi_0} {\\mathcal\nM} (T=0) = e^{-ad \\phi_0}e^{ad \\psi_0}X .\n\\end{eqnarray}\n\nTherefore, by the Lemma 5 and assumptions, we have\n\\begin{eqnarray}\n\\tilde f(\\mu(T=0), \\tilde {{\\mathcal M}}(T=0)) &=& e^{-ad\n\\phi_0}e^{ad \\psi_0} \\tilde f(P,X), \\no \\\\ &=& e^{-ad \\phi_0}e^{ad\n\\psi_0}(e^{-ad \\psi_0}e^{ad \\phi_0}P)=P, \\label{init1} \\\\ \\tilde\ng(\\mu(T=0), \\tilde {\\mathcal M}(T=0)) &=& e^{-ad \\phi_0}e^{ad\n\\psi_0} \\tilde g(P,X), \\no \\\\ &=& e^{-ad \\phi_0}e^{ad\n\\psi_0}(e^{-ad \\psi_0}e^{ad \\phi_0}X)=X. \\label{init2}\\no\n\\end{eqnarray}\n\nNow, we prove that $\\tilde f(\\mu, \\tilde {\\mathcal M})_{\\leq\n0}=0.$ Since $\\mu$ and $\\tilde {\\mathcal M}$ satisfy equations\n(\\ref{dmkp1}) and (\\ref{tim}) respectively, we have\n\\[\n\\frac{\\partial \\tilde f (\\mu, \\tilde {\\mathcal M})}{\\partial\nT_n}=\\{{\\mathcal Q}_n(P), \\tilde f (\\mu, \\tilde {\\mathcal M}) \\}.\n\\]\nUsing (\\ref{init1}), we see that $\\partial \\tilde f(\\mu, \\tilde\nM)/\n\\partial T_n \\vert_{T=0}$ will only contain powers $\\geq 1$ of\n$P$. In this way, we can prove, by induction, that $(\\partial /\n\\partial T)^{\\alpha} \\tilde f(\\mu, \\tilde {\\mathcal M})\n\\vert_{T=0}$, i.e, coefficients of Taylor expansion at $T=0$, will\nonly contain powers $\\geq 1$ of $P$ for any multi-index $\\alpha$.\nThus, we have proved that $\\tilde f(\\mu, \\tilde {\\mathcal\nM})_{\\leq 0}=0.$ As for $\\tilde g(\\mu, \\tilde {\\mathcal M})_{\\leq\n-1}=0$, we notice that the powers of $P$ of $\\{{\\mathcal Q}_n(P),\nX \\}$ are $\\geq 0$. Then it can be proved in the same way. $\\Box$\n\nThis theorem shows the possibility of twistor construction for the\nsolution structure of dmKP without using dispersionless Miura map.\nIndeed, we have the following main theorem of this section.\n\n\\begin{theorem}\nLet\n\\begin{eqnarray}\n\\mu &=& P+V_0+\\frac{V_1}{P}+\\frac{V_2}{P^2}+\\cdots, \\\\ {\\mathcal\nM}_{dmkp} &=& \\sum_{n=1}^{\\infty} nT_n\n\\mu^{n-1}+\\sum_{i=1}^{\\infty}S_i(T) \\mu^{-i}\n\\end{eqnarray}\n(${\\mathcal M}_{dmkp}$ can be defined as the Orlov function of\ndmKP). Suppose that\n\\begin{equation}\n\\{f(P,X), g(P,X) \\}=1. \\label{uni}\n\\end{equation}\n\nThen the functional equations\n\\begin{eqnarray}\nf(\\mu, {\\mathcal M}_{dmkp})_{\\leq 0} &=& 0, \\no \\\\ g(\\mu,\n{\\mathcal M}_{dmkp})_{\\leq -1} &=& 0 \\label{fun}\n\\end{eqnarray}\ncan get a solution of\n\\begin{eqnarray}\n\\partial_{T_n} \\mu &=& \\{{\\mathcal Q}_{\\geq 1}^n(P), \\mu \\}, \\\\\n\\partial_{T_n} {\\mathcal M}_{dmkp} &=& \\{{\\mathcal Q}_{\\geq 1}^n(P), {\\mathcal M}_{dmkp} \\}, \\\\\n\\{ \\mu, {\\mathcal M}_{dmkp} \\} &= & 1.\n\\end{eqnarray}\n\\end{theorem}\n{\\it Proof\\/}. For convenience, we let\n\\begin{eqnarray}\n\\tilde \\mu &=& f(\\mu, {\\mathcal M}_{dmkp}), \\no \\\\ \\tilde\n{\\mathcal M}_{dmkp} &=& g(\\mu, {\\mathcal M}_{dmkp}). \\label{sub}\n\\end{eqnarray}\n\nWe first derive the canonical Poisson relation. By differentiating\nthe last equations with respect to $P$ and $X$, we have\n \\be\n\\left(\n\\begin{array}{cc}\n\\frac{\\partial f(\\mu, {\\mathcal M}_{dmkp})}{\\partial \\mu} &\n\\frac{\\partial f(\\mu, {\\mathcal M}_{dmkp})}{\\partial {\\mathcal\nM}_{dmkp}} \\\\ \\frac{\\partial g(\\mu, {\\mathcal M}_{dmkp})}{\\partial\n\\mu} & \\frac{\\partial g(\\mu, {\\mathcal M}_{dmkp})}{\\partial\n{\\mathcal M}_{dmkp}}\n\\end{array}\n\\right ) \\left(\n\\begin{array}{cc}\n\\frac{\\partial \\mu }{\\partial P} & \\frac{\\partial \\mu }{\\partial\nX}\\\\ \\frac{\\partial {\\mathcal M}_{dnkp}}{\\partial P} &\n\\frac{\\partial {\\mathcal M}_{dmkp} }{\\partial X}\n\\end{array}\n\\right )\n=\n\\left(\n\\begin{array}{cc}\n\\frac{\\partial \\tilde \\mu }{\\partial P} & \\frac{\\partial \\tilde\n\\mu }{\\partial X}\\\\ \\frac{\\partial \\tilde {\\mathcal\nM}_{dnkp}}{\\partial P} & \\frac{\\partial \\tilde {\\mathcal\nM}_{dmkp} }{\\partial X}\n\\end{array}\n\\right ). \\label{mat1}\n \\ee\n\nSince the determinant of the first matrix on the left hand side is\n$1$ because of (\\ref{uni}), the determinants of both hand sides\ngive\n\\[\n\\{\\mu, {\\mathcal M}_{dmkp} \\}=\\{ \\tilde \\mu, \\tilde {\\mathcal\nM}_{dmkp} \\}.\n\\]\n\nOne can calculate the left hand side as\n\\begin{eqnarray}\n\\{\\mu, {\\mathcal M}_{dmkp} \\} &=& \\frac{\\partial \\mu}{\\partial P}\n\\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial X}-\\frac{\\partial\n{\\mathcal M}_{dmkp}}{\\partial P} \\frac{\\partial \\mu}{\\partial X},\n\\\\\n &=& \\frac{\\partial \\mu}{\\partial P} \\left [\\left (\\frac{\\partial M_{dmkp}}{\\partial \\mu} \\right )_{S_{i}(T)\\hspace{2mm} fixed} \\frac{\\partial \\mu}{\\partial X}+1+\\sum_{i=1}^{\\infty} \\frac{\\partial S_{i}(T)}{\\partial X}\n\\mu^{-i} \\right] \\\\\n &-& \\frac{\\partial \\mu}{\\partial X} \\left (\\frac{\\partial M_{dmkp}}\n {\\partial \\mu} \\right )_{S_{i}(T)\\hspace{2mm} fixed} \\frac{\\partial \\mu}{\\partial\n P},\\\\\n&=&1+(negative \\,\\ powers \\,\\ of \\,\\ P)\n\\end{eqnarray}\nwhere we have used the fact that the terms containing $\\left\n(\\frac{\\partial M_{dmkp}}{\\partial \\mu} \\right )_{S_{i}(T)\n\\hspace{2mm} fixed} $ in the last line cancel.\n Moreover, the Laurent expansions of $\\tilde \\mu $ and $ \\tilde {\\mathcal M}_{dmkp}$\ncontain only non-negative powers of $P$ because of the functional\nequations (\\ref{fun}). Therefore strictly negative powers of $P$\nin the last line should be absent, thus\n\\begin{equation}\n\\{\\mu, {\\mathcal M}_{dmkp} \\}=\\{\\tilde \\mu, \\tilde {\\mathcal\nM}_{dmkp} \\}=1. \\label{can}\n\\end{equation}\n\nThis gives the desired canonical Poisson relation. We now show\nthat the Lax equation for $\\mu$ and ${\\mathcal M}_{dmkp}$ are\nindeed satisfied. Differentiating equations (\\ref{sub}) with\nrespect to $T_n$ gives\n\\begin{equation}\n\\left(\n\\begin{array}{cc}\n\\frac{\\partial f(\\mu, {\\mathcal M}_{dmkp})}{\\partial \\mu} &\n\\frac{\\partial f(\\mu, {\\mathcal M}_{dmkp})}{\\partial {\\mathcal\nM}_{dmkp}} \\\\ \\frac{\\partial g(\\mu, {\\mathcal M}_{dmkp})}{\\partial\n\\mu} & \\frac{\\partial g(\\mu, {\\mathcal M}_{dmkp})}{\\partial\n{\\mathcal M}_{dmkp}}\n\\end{array}\n\\right ) \\left(\n\\begin{array}{c}\n\\frac{\\partial \\mu }{\\partial T_n} \\\\ \\frac{\\partial {\\mathcal\nM}_{dmkp}}{\\partial T_n}\n\\end{array}\n\\right )\n=\n\\left(\n\\begin{array}{c}\n\\frac{\\partial \\tilde \\mu }{\\partial T_n} \\\\ \\frac{\\partial\n\\tilde {\\mathcal M}_{dmkp}}{\\partial T_n}\n\\end{array}\n\\right ). \\label{mat2}\n\\end{equation}\n\nCombining equations (\\ref{mat1}) and (\\ref{mat2}), one can\neliminate the derivative matrix of $(f,g)$ by $(\\mu, {\\mathcal\nM}_{dmkp})$ and obtain the matrix relation\n \\[\n\\left(\n\\begin{array}{cc}\n\\frac{\\partial \\mu }{\\partial P} & \\frac{\\partial \\mu }{\\partial\nX}\\\\ \\frac{\\partial {\\mathcal M}_{dnkp}}{\\partial P} &\n\\frac{\\partial {\\mathcal M}_{dmkp} }{\\partial X}\n\\end{array}\n\\right )^{-1} \\left(\n\\begin{array}{c}\n\\frac{\\partial \\mu }{\\partial T_n} \\\\ \\frac{\\partial {\\mathcal\nM}_{dmkp}}{\\partial T_n}\n\\end{array}\n\\right ) = \\left(\n\\begin{array}{cc}\n\\frac{\\partial \\tilde \\mu }{\\partial P} & \\frac{\\partial \\tilde\n\\mu }{\\partial X}\\\\ \\frac{\\partial \\tilde {\\mathcal\nM}_{dnkp}}{\\partial P} & \\frac{\\partial \\tilde {\\mathcal\nM}_{dmkp} }{\\partial X}\n\\end{array}\n\\right )^{-1} \\left(\n\\begin{array}{c}\n\\frac{\\partial \\tilde \\mu }{\\partial T_n} \\\\ \\frac{\\partial\n\\tilde {\\mathcal M}_{dmkp}}{\\partial T_n}\n\\end{array}\n\\right ). \\label{mat3}\n \\]\n\nSince the the determinants of the $2 \\times 2$ matrices on both\nsides are $1$ because of (\\ref{can}), the inverse can also be\nwritten explicitly. In components, thus, the above matrix relation\ngives\n \\begin{eqnarray}\n \\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial X} \\frac{\\partial\n \\mu}{\\partial T_n} -\\frac{\\partial \\mu}{\\partial X} \\frac{\\partial\n {\\mathcal M}_{dmkp}}{\\partial T_n} &=& \\frac{\\partial \\tilde\n {\\mathcal M}_{dmkp}}{\\partial X} \\frac{\\partial \\tilde \\mu}{\\partial\n T_n} -\\frac{\\partial \\tilde \\mu}{\\partial X} \\frac{\\partial \\tilde\n {\\mathcal M}_{dmkp}}{\\partial T_n}, \\no \\\\\n \\frac{\\partial\n {\\mathcal M}_{dmkp}}{\\partial P} \\frac{\\partial \\mu}{\\partial T_n}\n -\\frac{\\partial \\mu}{\\partial P} \\frac{\\partial {\\mathcal\n M}_{dmkp}}{\\partial T_n} &=& \\frac{\\partial \\tilde {\\mathcal\n M}_{dmkp}}{\\partial P} \\frac{\\partial \\tilde \\mu}{\\partial T_n}\n -\\frac{\\partial \\tilde \\mu}{\\partial P} \\frac{\\partial \\tilde\n {\\mathcal M}_{dmkp}}{\\partial T_n}.\n \\label{equal}\n \\end{eqnarray}\n\nThe left hand side of equation (\\ref{equal}) can be calculated\njust as we have done above for derivatives in $(P, X)$. For the\nfirst equation of (\\ref{equal}),\n\\begin{eqnarray}\n& &\\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial X} \\frac{\\partial\n\\mu}{\\partial T_n} -\\frac{\\partial \\mu}{\\partial X} \\frac{\\partial\n{\\mathcal M}_{dmkp}}{\\partial T_n} \\no \\\\ &=&\\left [\\left\n(\\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial \\mu} \\right\n)_{S_{i}(T)\\hspace{2mm} fixed} \\frac{\\partial \\mu}{\\partial\nX}+1+\\sum_{i=1}^{\\infty} \\frac{\\partial S_{i}(T)}{\\partial X}\n\\mu^{-i} \\right] \\frac{\\partial \\mu}{\\partial T_n} \\no \\\\ &-&\n\\frac{\\partial \\mu}{\\partial X} \\left [\\left (\\frac{\\partial\n{\\mathcal M}_{dmkp}}{\\partial \\mu} \\right )_{S_{i}(T)\\hspace{2mm}\nfixed} \\frac{\\partial \\mu}{\\partial T_n}+ n \\mu^{n-1}\n+\\sum_{i=1}^{\\infty} \\frac{\\partial S_{i}(T)}{\\partial X} \\mu^{-i}\n\\right], \\no\n\\end{eqnarray}\nand terms containing $\\left (\\frac{\\partial M_{dmkp}}{\\partial\n\\mu} \\right )_{S_{i}(T)\\hspace{2mm} fixed} $ cancel. Thus,\n\\[\n\\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial X} \\frac{\\partial\n\\mu}{\\partial T_n} -\\frac{\\partial \\mu}{\\partial X} \\frac{\\partial\n{\\mathcal M}_{dmkp}}{\\partial T_n} = -\\frac{\\partial (\\mu^n)_{\\geq\n1}}{\\partial X} +(powers \\,\\ of \\,\\ P \\leq 0). \\label{rest1}\n\\]\nBy the functional equations (\\ref{fun}), we know that the right\nhand side of the first equation of (\\ref{equal}) has Laurent\nexpansion with only powers of $\\geq 1$. Therefore only powers of\n$P \\geq 1$ should survive. Hence\n\\begin{equation}\n\\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial X} \\frac{\\partial\n\\mu}{\\partial T_n} -\\frac{\\partial \\mu}{\\partial X} \\frac{\\partial\n{\\mathcal M}_{dmkp}}{\\partial T_n} = -\\frac{\\partial (\\mu^n)_{\\geq\n1}}{\\partial X}=-\\frac{\\partial {\\mathcal Q}_n}{\\partial X}.\n\\label{fina1}\n\\end{equation}\nFor the second equation of (\\ref{equal}), we have similarly\n\\begin{eqnarray}\n\\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial P} \\frac{\\partial\n\\mu}{\\partial T_n} -\\frac{\\partial \\mu}{\\partial P} \\frac{\\partial\n{\\mathcal M}_{dmkp}}{\\partial T_n} &=& -\\frac{\\partial\n(\\mu^n)_+}{\\partial P} +(negative \\,\\ powers \\,\\ of \\,\\ P ),\n\\\\ &=& -\\frac{\\partial (\\mu^n)_{\\geq 1}}{\\partial P} +(negative\n\\,\\ powers \\,\\ of \\,\\ P ) .\n\\end{eqnarray}\nBy the functional equations (\\ref{fun}), noticing the partial\nderivative $\\partial / \\partial P $, we see that the right hand\nside of the second equation of (\\ref{equal}) have Laurent\nexpansion with only nonnegative powers of $P$. Hence only\nnonnegative powers of $P$ should survive.\n\nThus\n\\begin{equation}\n\\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial P} \\frac{\\partial\n\\mu}{\\partial T_n} -\\frac{\\partial \\mu}{\\partial P} \\frac{\\partial\n{\\mathcal M}_{dmkp}}{\\partial T_n} = -\\frac{\\partial (\\mu^n)_{\\geq\n1}}{\\partial P}=-\\frac{\\partial {\\mathcal Q}_n}{\\partial P}.\n\\label{fina2}\n\\end{equation}\n\nUsing (\\ref{can}), equations (\\ref{fina1}) and (\\ref{fina2}) can\nbe readily solved:\n\\begin{eqnarray}\n\\frac{\\partial \\mu}{\\partial T_n} &= & -\\frac{\\partial\n\\mu}{\\partial P}\\frac{\\partial {\\mathcal Q}_n}{\\partial X} +\n\\frac{\\partial \\mu}{\\partial X}\\frac{\\partial {\\mathcal\nQ}_n}{\\partial P}=\\{{\\mathcal Q}_n, \\mu \\}, \\\\ \\frac{\\partial\n{\\mathcal M}_{dmkp}}{\\partial T_n} &= & -\\frac{\\partial {\\mathcal\nM}_{dmkp}}{\\partial P}\\frac{\\partial {\\mathcal Q}_n}{\\partial X} +\n\\frac{\\partial {\\mathcal M}_{dmkp}}{\\partial X}\\frac{\\partial\n{\\mathcal Q}_n}{\\partial P}=\\{{\\mathcal Q}_n, {\\mathcal M}_{dmkp}\n\\}.\n\\end{eqnarray}\nThis completes the theorem. $\\Box$\n\\section{concluding remarks}\n\nWe have studied the Miura map between the dKP and dmKP\nhierarchies. We show that the Miura map not only preserves the Lax\nformulation of these two hierarchies but also is a canonical map\nin the sense that the bi-Hamiltonian structure of the dmKP\nhierarchy is mapped to the bi-Hamiltonian structure of the dKP\nhierarchy. We further use the twistor construction developed by\nTakasaki and Takebe to investigate the solution structure of the\ndmKP hierarchy.\n\n In spite of the results obtained in the paper,\n there are some related problems deserve further investigations.\n We list some of them in the following. \\\\\n (1) In \\cite{ch}, it is shown that the second\nHamiltonian structure $\\Theta^{(2)}$ of dKP has\n free field realizations. Since the Miura map is canonical, this suggests the\n possibility of free field realizations of second Hamiltonian structure $J^{(2)}$\n of dmKP \\cite{ct}. \\\\\n (2) In \\cite{du2}, we know that bi-Hamiltonian structure of\nDubrovin-Novikov (DN) type \\cite{dn} has geometric structure of\nFrobenius manifold \\cite{du1}. A natural question is : what's the\ngeometric meaning of the Miura map between bi-Hamiltonian\nstructures of DN type? \\\\\n (3) The dmKP theory should be\ninvestigated without using Miura map. The quasi-classical\n$\\tau$-function for dKP has been established in\n\\cite{Ta2,kr1,kr2}. The basic question for dmKP theory is : Does\nthe quasi-classical $\\tau$-function theory exist ? We notice that\nthe Hirota bilinear equations for KP and mKP are essentially\ndifferent \\cite{jm}. Also, in \\cite{Car2}, the dispersionless\nHirota equation for dKP is obtained. Is there an analogue for dmKP\n?\\\\\n\n{\\bf Acknowledgements\\/}\\\\ We would like to thank Prof. J.C Shaw\nfor useful discussions. JHC thanks for the support of the Academia\nSinica and MHT thanks for the support of the National Science\nCouncil of Taiwan under Grant No. NSC 89-2112-M194-018.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Car1}\nR. Carroll, Jour. Nonlin. Sci. {\\bf 4\\/}, 519 (1994).\n\n\\bibitem{Car2}\nR. Carroll and Y. Kodama, J. Phys. A {\\bf 28\\/}, 6373 (1995).\n\n\\bibitem{KG}\nY. Kodama and J. Gibbons, Integrability of dispersionless KP\nhierarchy, Proceedinds Fourth Workshop on Nonlinear and Turbulent\nProcess in Physics, 166 (World Scientific, 1990).\n\n\\bibitem{LM}\nD. Lebedev and Yu. Manin, Phys. Lett. A {\\bf 74\\/}, 154 (1979).\n\n\\bibitem{Ta2} T. Takasaki and T. Takebe, SDiff(2) KP Hierarchy, Proceedings of RIMS\nResearch Project, {\\it Infinite Analysis\\/}, (World Scientific,\n1991).\n\n\\bibitem{Ta1} K. Takasaki and T. Takebe, Rev. Math. Phys. {\\bf 7\\/}, 743 (1995).\n\n\\bibitem{Di}\nL.A. Dickey, {\\it Soliton Equations and Hamiltonian Systems\\/},\n(World Scientific, 1991).\n\n\\bibitem{Ak}\nS. Aoyama and Y. Kodama, Commun. Math. Phys. {\\bf 182\\/}, 185\n(1996).\n\n\\bibitem{Du4}\nB. Dubrovin, Nucl. Phys. B {\\bf 379\\/}, 627 (1992).\n\n\\bibitem{du1}\nB. Dubrovin, Geometry of 2d topological field theories, {\\it\nIntegrable systems and Quantum Groups\\/}, Lecture Notes in Math.\n{\\bf 1620\\/}, 120 ( Springer, 1996).\n\n\\bibitem{kr1}\nI. Krichever, Commun. Math. Phys. {\\bf 143\\/}, 415 (1992).\n\n\\bibitem{kr2}\nI. Krichever, Commun. Pure Appl. Math. {\\bf 47\\/}, 437 (1994).\n\n\\bibitem{Li}\n Luen-Chau Li, Commun. Math. Phys. {\\bf\n203\\/}, 573 (1999).\n\n\\bibitem{jm}\nM. Jimbo and T. Miwa, Publ. RIMS, Kyoto University, {\\bf 19\\/},\n943 (1983).\n\n\\bibitem{ku1}\nB.A. Kupershmidt, Commun. Math. Phys. {\\bf 99\\/}, 51 (1985).\n\n\\bibitem{Mi}\n R.M. Miura, J. Math. Phys. {\\bf 9\\/}, 1202 (1968).\n\n\\bibitem{ki}\nK. Kiso, Prog. Theor. Phys. {\\bf 83\\/}, 1108 (1990).\n\n\\bibitem{KO}\nB. Konopelchenko and W. Oevel, Publ. RIMS, Kyoto Univ. {\\bf 29\\/},\n581 (1993).\n\n\\bibitem{ku2}\nB.A. Kupershmidt, Commun. Math. Phys. {\\bf 167\\/}, 351 (1995).\n\n\\bibitem{OR}\n W.Oevel and C. Rogers, Rev. Math. Phys. {\\bf 157\\/}, 299 (1993).\n\n\\bibitem{St1}\nJiin-Chang Shaw and Ming-Hsien Tu, J. Math. Phys. {\\bf 38\\/}, 5756\n(1997).\n\n\\bibitem{ST} Jiin-Chang Shaw and Ming-Hsien Tu, J. Phys. A {\\bf 30\\/}, 4825\n(1997).\n\n\\bibitem{Ku3}\nB.A. Kupershmidt, J. Phys. A {\\bf 23\\/}, 871 (1990).\n\n\\bibitem{O}\nW. Oevel, Phys. Lett. A {\\bf 186\\/}, 79 (1994).\n\n\\bibitem{OS}\n W. Oevel and W. Strampp, Comm. Math. Phys. {\\bf 157\\/}, 51 (1993).\n\n\\bibitem{Or1}\nA.Y. Orlov and E.I. Schulman, Lett. Math. Phys. {\\bf 12\\/}, 171\n(1986).\n\n\\bibitem{VM}\nP. Van Moerbeke, Integrable fundations of string theory, {\\it\nLectures on Integrable Systems\\/}, Eds. O. Babelon et. el, (World\nScientific, 1994).\n\n\\bibitem{FR}\nJ.M. Figueroa-O'Farrill and E. Ramos, Phys. Lett. B {\\bf 282\\/},\n357 (1992).\n\n\\bibitem{ch}\nY. Cheng and Z.F. Li, Lett. Math. Phys. {\\bf 42\\/}, 73 (1997).\n\n\\bibitem{ct}\nJen-Hsu Chang and Ming-Hsien Tu, in preparation.\n\n\\bibitem{du2}\nB. Dubrovin, Flat pencils of metrics and Frobenius manifolds, {\\it\nIntegrable systems and algebraic geometry\\/}, Kobe/Kyoto, 1997,\n47-72, (World Scientific, 1998).\n\n\\bibitem{dn}\nB. Dubrovin and S.P. Novikov, Russ. Math. Surv. {\\bf 44\\/}, 35\n(1989).\n\n\n\n\n\n\n\n\n\n\n\n\\end{thebibliography}\n\n\n\n\n\n\n\n\n\n\n\n\\end{document}\n" } ]	[ { "name": "solv-int9912016.extracted_bib", "string": "{Car1 R. Carroll, Jour. Nonlin. Sci. {4\\/, 519 (1994)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Car2 R. Carroll and Y. Kodama, J. Phys. A {28\\/, 6373 (1995)." }, { "name": "solv-int9912016.extracted_bib", "string": "{KG Y. Kodama and J. Gibbons, Integrability of dispersionless KP hierarchy, Proceedinds Fourth Workshop on Nonlinear and Turbulent Process in Physics, 166 (World Scientific, 1990)." }, { "name": "solv-int9912016.extracted_bib", "string": "{LM D. Lebedev and Yu. Manin, Phys. Lett. A {74\\/, 154 (1979)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Ta2 T. Takasaki and T. Takebe, SDiff(2) KP Hierarchy, Proceedings of RIMS Research Project, {Infinite Analysis\\/, (World Scientific, 1991)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Ta1 K. Takasaki and T. Takebe, Rev. Math. Phys. {7\\/, 743 (1995)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Di L.A. Dickey, {Soliton Equations and Hamiltonian Systems\\/, (World Scientific, 1991)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Ak S. Aoyama and Y. Kodama, Commun. Math. Phys. {182\\/, 185 (1996)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Du4 B. Dubrovin, Nucl. Phys. B {379\\/, 627 (1992)." }, { "name": "solv-int9912016.extracted_bib", "string": "{du1 B. Dubrovin, Geometry of 2d topological field theories, {Integrable systems and Quantum Groups\\/, Lecture Notes in Math. {1620\\/, 120 ( Springer, 1996)." }, { "name": "solv-int9912016.extracted_bib", "string": "{kr1 I. Krichever, Commun. Math. Phys. {143\\/, 415 (1992)." }, { "name": "solv-int9912016.extracted_bib", "string": "{kr2 I. Krichever, Commun. Pure Appl. Math. {47\\/, 437 (1994)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Li Luen-Chau Li, Commun. Math. Phys. {203\\/, 573 (1999)." }, { "name": "solv-int9912016.extracted_bib", "string": "{jm M. Jimbo and T. Miwa, Publ. RIMS, Kyoto University, {19\\/, 943 (1983)." }, { "name": "solv-int9912016.extracted_bib", "string": "{ku1 B.A. Kupershmidt, Commun. Math. Phys. {99\\/, 51 (1985)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Mi R.M. Miura, J. Math. Phys. {9\\/, 1202 (1968)." }, { "name": "solv-int9912016.extracted_bib", "string": "{ki K. Kiso, Prog. Theor. Phys. {83\\/, 1108 (1990)." }, { "name": "solv-int9912016.extracted_bib", "string": "{KO B. Konopelchenko and W. Oevel, Publ. RIMS, Kyoto Univ. {29\\/, 581 (1993)." }, { "name": "solv-int9912016.extracted_bib", "string": "{ku2 B.A. Kupershmidt, Commun. Math. Phys. {167\\/, 351 (1995)." }, { "name": "solv-int9912016.extracted_bib", "string": "{OR W.Oevel and C. Rogers, Rev. Math. Phys. {157\\/, 299 (1993)." }, { "name": "solv-int9912016.extracted_bib", "string": "{St1 Jiin-Chang Shaw and Ming-Hsien Tu, J. Math. Phys. {38\\/, 5756 (1997)." }, { "name": "solv-int9912016.extracted_bib", "string": "{ST Jiin-Chang Shaw and Ming-Hsien Tu, J. Phys. A {30\\/, 4825 (1997)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Ku3 B.A. Kupershmidt, J. Phys. A {23\\/, 871 (1990)." }, { "name": "solv-int9912016.extracted_bib", "string": "{O W. Oevel, Phys. Lett. A {186\\/, 79 (1994)." }, { "name": "solv-int9912016.extracted_bib", "string": "{OS W. Oevel and W. Strampp, Comm. Math. Phys. {157\\/, 51 (1993)." }, { "name": "solv-int9912016.extracted_bib", "string": "{Or1 A.Y. Orlov and E.I. Schulman, Lett. Math. Phys. {12\\/, 171 (1986)." }, { "name": "solv-int9912016.extracted_bib", "string": "{VM P. Van Moerbeke, Integrable fundations of string theory, {Lectures on Integrable Systems\\/, Eds. O. Babelon et. el, (World Scientific, 1994)." }, { "name": "solv-int9912016.extracted_bib", "string": "{FR J.M. Figueroa-O'Farrill and E. Ramos, Phys. Lett. B {282\\/, 357 (1992)." }, { "name": "solv-int9912016.extracted_bib", "string": "{ch Y. Cheng and Z.F. Li, Lett. Math. Phys. {42\\/, 73 (1997)." }, { "name": "solv-int9912016.extracted_bib", "string": "{ct Jen-Hsu Chang and Ming-Hsien Tu, in preparation." }, { "name": "solv-int9912016.extracted_bib", "string": "{du2 B. Dubrovin, Flat pencils of metrics and Frobenius manifolds, {Integrable systems and algebraic geometry\\/, Kobe/Kyoto, 1997, 47-72, (World Scientific, 1998)." }, { "name": "solv-int9912016.extracted_bib", "string": "{dn B. Dubrovin and S.P. Novikov, Russ. Math. Surv. {44\\/, 35 (1989)." } ]