cases/case25/task.md

How is the quantum state of a driving field encoded in crystal high-harmonic radiation?

Background and central research question

The field driving a strong-field process need not be well represented by one classical waveform. Its coherent-state amplitudes can instead be distributed according to different quantum states, even when those states have the same mean photon number.

Investigate how the quantum state of the driving field is encoded in intraband high-harmonic radiation from a crystal. Determine which spectral and time-domain properties of the generated radiation provide defensible state-dependent evidence, explain their physical origin within the stated model, and delimit the strongest inference supported by the calculation. Do not assume the outcome in advance.

Design the task-specific derivation, numerical calculation, evidence and scientific argument needed to answer this one research question. The form and organization of the evidence are yours.

Physical model and indispensable inputs

Use atomic units unless otherwise stated. Consider a one-dimensional cut of a single conduction band along the field-polarization direction. Its dispersion is

$$E(q)=\sum _{l=0}^5{b}_l\cos (laq),E(q)=\backslash sum\_\{l=0\}^\{5\}\; b\_l\backslash cos(laq),$$

with

The constant $b_0$ does not affect the intraband current and may be set to zero. The band is occupied by ten electrons: the five crystal momenta

$$q_m=m\mathrm{\Delta }q,m=-2,-1,0,1,2,\mathrm{\Delta }q=\frac{2\pi }{10a},q\_m=m\backslash Delta\; q,\backslash qquad\; m=-2,-1,0,1,2,\; \backslash qquad\; \backslash Delta\; q=\backslash frac\{2\backslash pi\}\{10a\},$$

are each occupied in both spin orientations.

For a coherent-state amplitude $\alpha=|\alpha|e^{i\phi}$, take the semiclassical vector potential to be

$$A_{\alpha }(t)=\frac{2g_0}{\sqrt{{\omega }_0}}f(t)\mathrm{\mid }\alpha \mathrm{\mid }\sin ({\omega }_{0}t-\varphi ),A\_\backslash alpha(t)=\backslash frac\{2g\_0\}\{\backslash sqrt\{\backslash omega\_0\}\}\; f(t)|\backslash alpha|\backslash sin(\backslash omega\_0t-\backslash phi),$$

where

$$g_0=4.0\times {10}^{-8},{\omega }_0=\mathrm{0.005.}g\_0=4.0\backslash times10^\{-8\},\backslash qquad\; \backslash omega\_0=0.005.$$

The envelope $f(t)$ has a three-cycle $\sin^2$ turn-on, ten cycles at unit amplitude, and a three-cycle $\sin^2$ turn-off. A shift of the time origin is an equivalent carrier-phase convention when applied consistently.

Compare coherent, thermal, Fock and bright squeezed-vacuum driving states at the common mean photon number

$$\bar{N}=7.35\times {10}^{11}\mathrm{.}\backslash bar\; N=7.35\backslash times10^\{11\}.$$

Use the exact diagonal Glauber--Sudarshan weights

$$P_{\mathrm{c}\mathrm{o}\mathrm{h}}(\alpha )={\delta }^{(2)}(\alpha -{\alpha }_L),\mathrm{\mid }{\alpha }_L{\mathrm{\mid }}^2=\bar{N},P\_\{\backslash rm\; coh\}(\backslash alpha)=\backslash delta^\{(2)\}(\backslash alpha-\backslash alpha\_L),\; \backslash qquad\; |\backslash alpha\_L|^2=\backslash bar\; N,$$

and

$$P_{\mathrm{t}\mathrm{h}}(\alpha )=\frac{1}{\pi \bar{N}}\exp \text{\hspace{0.17em}⁣}(-\frac{\mathrm{\mid }\alpha {\mathrm{\mid }}^2}{\bar{N}})\mathrm{.}P\_\{\backslash rm\; th\}(\backslash alpha)=\backslash frac\{1\}\{\backslash pi\backslash bar\; N\}\; \backslash exp\backslash !\backslash left(-\backslash frac\{|\backslash alpha|^2\}\{\backslash bar\; N\}\backslash right).$$

Choose $\alpha_L>0$ to fix the common time-origin convention. For the Fock state $|n\rangle$, use $n=\bar N$ and the large-photon-number approximative positive-$P$ (APP) diagonal weight

$$P_{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{k}}^{\mathrm{A}\mathrm{P}\mathrm{P}}(\alpha )=\frac{e^{-\mathrm{\mid }\alpha {\mathrm{\mid }}^2}\mathrm{\mid }\alpha {\mathrm{\mid }}^{2n}}{\pi n!}\mathrm{.}P\_\{\backslash rm\; Fock\}^\{\backslash rm\; APP\}(\backslash alpha)=\; \backslash frac\{e^\{-|\backslash alpha|^2\}|\backslash alpha|^\{2n\}\}\{\backslash pi\; n!\}.$$

For the bright squeezed vacuum $\hat S(r)|0\rangle$, take real $r$ with

$${\sinh }^{2}r=\bar{N},\backslash sinh^2\; r=\backslash bar\; N,$$

so that $r\simeq14.35$, and use

$$P_{\mathrm{B}\mathrm{S}\mathrm{V}}^{\mathrm{A}\mathrm{P}\mathrm{P}}(\alpha )=\frac{1}{\pi \cosh r}\exp \text{\hspace{0.17em}⁣}[-\frac{2[\mathrm{Re}\alpha {]}^2}{1+e^{-2r}}-\frac{2[\mathrm{Im}\alpha {]}^2}{1+e^{2r}}]\mathrm{.}P\_\{\backslash rm\; BSV\}^\{\backslash rm\; APP\}(\backslash alpha)=\; \backslash frac\{1\}\{\backslash pi\backslash cosh\; r\}\; \backslash exp\backslash !\backslash left[\; -\backslash frac\{2[\backslash operatorname\{Re\}\backslash alpha]^2\}\{1+e^\{-2r\}\}\; -\backslash frac\{2[\backslash operatorname\{Im\}\backslash alpha]^2\}\{1+e^\{2r\}\}\; \backslash right].$$

The coherent and thermal averages are exact within this single-band model. The Fock and bright-squeezed-vacuum weights define the APP approximation used for this task. APP is prescribed only for the harmonic spectrum and the first two time-domain moments of the generated electric field; stronger emitted- field claims require independent support.

Method boundary and numerical freedom

Obtain the four-state finite-pulse emitted-energy signal and the generated- field expectation value and zero-point-subtracted uncertainty. Derive the task-specific relations needed to define and calculate these observables from the stated model; no final output formula is supplied.

The finite-pulse calculation is the spectrum-generating route. A periodic Floquet treatment, Bessel-function analysis or a semiclassical cutoff estimate may be used as supporting methods, but cannot replace the finite-pulse result.

The quadrature or sampling representation, time and frequency domains, grids, steps, Fourier convention, differentiation method, truncations, tolerances, diagnostics and reliability strategy are student choices. State the conventions needed to reconstruct your observables. Equivalent normalizations are acceptable when unscaled data and the comparison convention are retained; absolute intensity is not graded when it is left arbitrary by the model.

Submission and resources

Submit one self-contained scientific report and an auditable offline calculation package containing producing code, reconstructible numerical data for material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run and environment information. The evidence must be sufficiently reliable to support the claims made, but no particular validation procedure, plot layout or number of figures is prescribed.

Resource limits are supplied exclusively by the accompanying resource_limits.json. Its fixed CPU allocation, candidate-active wall time, network boundary and temporary-storage allowance are binding. No separate aggregate CPU-hour limit applies.

Reference and integrity policy

This examination is conducted without network access or live literature retrieval. You may rely on your existing general scientific knowledge and, if any are explicitly supplied with the examination, on those offline materials or software documentation. Record every such source actually consulted. For formulas, methods, constants, or conventions used from general knowledge rather than a consulted source, state enough of the method basis, definition, value, units, and convention to make the calculation auditable; do not invent a citation. You are not required to identify or prove a publication date or to retrieve literature during the examination.

The hidden target article and its identifying metadata, figures or captions, supplementary material, author code or data, target-specific reproduction code, solution or scoring material, online solutions, and worked reproductions of this case must not be used. All submitted scientific figures and numerical data must be generated by your own implementation of the stated problem.

See submission requirements for the common submission entry points.