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Quantum microscopy allows microscopic properties of matter and quantum particles to be measured and imaged. Various types of microscopy use quantum principles. The first microscope to do so was the scanning tunneling microscope , which paved the way for development of the photoionization microscope and the quantum entanglement microscope.
The scanning tunneling microscope (STM) uses the concept of quantum tunneling to directly image atoms. A STM can be used to study the three-dimensional structure of a sample, by scanning the surface with a sharp, metal, conductive tip close to the sample. Such an environment is conducive to quantum tunneling: a quantum mechanical effect that occurs when electrons move through a barrier due to their wave-like properties. Tunneling depends on the thickness of the barrier; the Schrödinger equation gives the probability that a particle will be detected on the far side and, for a sufficiently thin barrier, predicts some electrons will cross it. This creates a current across the tunnel. The number of electrons that tunnel is dependent on the thickness of the barrier, therefore the current through the barrier also depends on this thickness. The distance between the tip and the sample affects the current measured by the tip. The tip is formed by a single atom that slowly moves across the surface at a distance of one atomic diameter. By observing the current, the distance can be kept fairly constant, allowing the tip to move up and down according to the structure of the sample.
The STM works best with conducting materials in order to create a current. However, since its creation, various implementations allow for a larger variety of samples, such as spin polarized scanning tunneling microscopy (SPSTM), and atomic force microscopy (AFM).
The wave function is central to quantum mechanics. It contains the maximum information that can be known about a single particle's quantum state . The square of the wave function is the probability of a particle's location at any given moment. Direct imaging of a wave function used to be considered only a gedanken experiment , but became routine. [ 1 ] An image of an atom's exact position or the movement of its electrons is almost impossible to measure because any direct observation of an atom disturbs its quantum coherence. As such, observing an atom's wave function and getting an image of its full quantum state requires many measurements to be made, which are then statistically averaged. The photoionization microscope directly visualizes atomic structure and quantum states. [ 2 ]
A photoionization microscope employs photoionization, along with quantum properties and principles, to measure atomic properties. The principle is to study the spatial distribution of electrons ejected from an atom in a situation in which the De Broglie wavelength becomes large enough to be observed on a macroscopic scale. An atom in an electric field is ionized by a focused laser. The electron is drawn toward a position-sensitive detector, and the current is measured as a function of position. The application of an electric field during photoionization allows confining the electron flux along one dimension. [ 3 ] [ 4 ]
Multiple classical paths lead from the atom to any point in the classically allowed region on the detector, and waves travelling along these paths produce an interference pattern. An infinite set of trajectory families lead to a complicated interference pattern on the detector. As such, photoionization microscopy relies on the existence of interference between various trajectories by which the electron moves from the atom to the plane of observation, for example, of a hydrogen atom in parallel electric and magnetic fields. [ 5 ] [ 6 ] [ 7 ]
The idea stemmed from an experiment proposed by Demkov and colleagues in the early 1980s. [ 8 ] The researchers suggested that electron waves could be imaged when interacting with a static electric field as long as the de Broglie Wavelength of these electrons was large enough. [ 8 ] It was not until 1996 that anything resembling these ideas bore fruit. [ 1 ] In 1996 a team of French researchers developed the first photodetachment microscope. It allowed for direct observation of the oscillatory structure of a wave function. [ 1 ] Photodetachment is the removal of electrons from an atom using interactions with photons or other particles. [ 9 ] Photodetachment microscopy made it possible to image the spatial distribution of the ejected electron. The microscope developed in 1996 was the first to image photodetachment rings of a negative Bromine ion. [ 10 ] These images revealed interference between two electron waves on their way to the detector.
The first attempts to use photoionization microscopy were performed on atoms of Xenon by a team of Dutch researchers in 2001. [ 1 ] The differences between direct and indirect ionization create different trajectories for the outbound electron. Direct ionization corresponds to electrons ejected down-field towards the bottleneck in the Coulomb + dc electric field potential, whereas indirect ionization corresponds to electrons ejected away from the bottleneck in the Coulomb + dc electric field and only ionize upon further Coulomb interactions. [ 1 ] These trajectories produce a distinct pattern that can be detected by a two-dimensional flux detector and subsequently imaged. [ 11 ] The images exhibit an outer ring that correspond to the indirect ionization process and an inner ring, which correspond to the direct ionization process. This oscillatory pattern can be interpreted as interference among the trajectories of the electrons moving from the atom to the detector. [ 1 ]
The next group to attempt photoionization microscopy used the excitation of Lithium atoms in the presence of a static electric field. [ 8 ] This experiment was the first to reveal evidence of quasibound states. [ 8 ] A quasibound state is a "state having a connectedness to true bound state through the variation of some physical parameter". [ 12 ] This was done by photoionizing the Lithium atoms in the presence of a ≈1 kV/cm static electric field. This experiment was an important precursor to the imaging of the hydrogen wave function because, contrary to the experiments done with Xenon, Lithium wave function microscopy images are sensitive to the presence of resonances. [ 8 ] Therefore, the quasibound states were directly revealed.
In 2013, Aneta Stodolna and colleagues imaged the hydrogen atom's wave function by measuring an interference pattern on a 2D detector. [ 4 ] [ 13 ] The electrons are excited to their Rydberg state . In this state, the electron orbital is far from the centre nucleus. The Rydberg electron is in a dc field, which causes it to be above the classical ionization threshold, but below the field-free ionization energy. The electron wave ends up producing an interference pattern because the portion of the wave directed towards the 2D detector interferes with the portion directed away from the detector. This interference pattern shows a number of nodes that is consistent with the nodal structure of the Hydrogen atom orbital [ 4 ]
The same team of researchers that imaged the hydrogen electron's wave function are attempting to image helium. They report considerable differences, since helium has two electrons, which may enable them to 'see' entanglement. [ 1 ]
Quantum metrology makes precise measurements that cannot be achieved classically. Typically, entanglement of N particles is used to measure a phase with precision ∆φ = 1/N, called the Heisenberg limit . This exceeds the ∆φ = 1/ √ N precision limit possible with N non-entangled particles, called the standard quantum limit (SQL). The signal-to-noise ratio (SNR) for a given light intensity is limited by SQL, which is critical for measurements where the probe light intensity is limited in order to avoid damaging the sample. The SQL can be tackled using entangled particles.
The microscope first imaged a relief pattern of a glass plate. In one test, the pattern was 17 nanometers higher than the plate. [ 14 ] [ 15 ]
Quantum entanglement microscopes are a form of confocal-type differential interference contrast microscope . Entangled photon pairs and more generally, NOON states are the illumination source. Two beams of photons are beamed at adjacent spots on a flat sample. The interference pattern of the beams are measured after they are reflected. When the two beams hit the flat surface, they both travel the same length and produce a corresponding interference pattern. This interference pattern changes when the beams hit regions of different heights. The patterns can be resolved by analysing the interference pattern and phase difference. A standard optical microscope would be unlikely to detect something so small. The image is precise when measured with entangled photons, as each entangled photon gives information about the other. Therefore, they provide more information than independent photons, creating sharper images. [ 14 ] [ 16 ]
Entanglement-enhancement principles can be used to improve the image. Researchers are thereby able to overcome the Rayleigh criterion . This is ideal for studying biological tissues and opaque materials. However, the light intensity must be lowered to avoid damaging the sample. [ 14 ] [ 15 ]
Entangled-photon microscopy can avoid the phototoxicity and photobleaching that comes with two-photon scanning fluorescence microscopy. In addition, since the interaction region within entangled microscopy is controlled by two beams, image site selection is flexible, which provides enhanced axial and lateral resolution [ 17 ]
In addition to biological tissues, high-precision optical phase measurements have applications such as gravitational wave detection, measurement of materials properties, as well as medical and biological sensing. [ 14 ] [ 15 ]
Researchers have developed quantum light microscopes based on squeezed states of light . [ 18 ] [ 19 ] [ 20 ] Squeezed states of light have noise characteristics that are reduced beneath the shot noise level in one quadrature (such as amplitude or phase) at the expense of increased noise in the orthogonal quadrature. This reduced noise can be used to improve signal-to-noise ratio. Squeezed states have been shown to allow a signal-to-noise ratio improvement of as much as a factor of thirty. [ 20 ]
The first biological quantum light microscope used squeezed light in an optical tweezer to probe the interior of a living yeast cell. [ 18 ] In experiments it was shown that squeezed light allowed more precise tracking [ compared to? ] of lipid granules that naturally occur within the cell, and that this provided a more accurate measurement [ compared to? ] of the local viscosity of the cell. Viscosity is an important property of cells that is connected to their health, structural properties and local function. Later, the same microscope was employed as a photonic force microscope, tracking a granule as it diffused spatially. [ 19 ] This allowed quantum enhanced resolution to be demonstrated, and for this to be achieved in a far-sub-diffraction limited microscope.
Squeezed light has also been used to improve nonlinear microscopy. [ 20 ] Nonlinear microscopes use intense laser illumination, close to the levels at which biological damage can occur. This damage is a key barrier to improving their performance, preventing the intensity from being increased and therefore putting a hard limit on SNR. By using squeezed light in such a microscope, researchers have shown that this limit can be broken - that SNR beyond that achievable beneath photo-damage limits of regular microscopy can be achieved. [ 20 ]
In a fluorescence microscope , images of objects that contain fluorescent particles are recorded. Each such particle can emit not more than one photon at a time, a quantum-mechanical effect known as photon antibunching . Recording anti-bunching in a fluorescence image provides additional information that can be used to enhance the microscope's resolution beyond the diffraction limit , [ 21 ] and was demonstrated for several types of fluorescent particles. [ 22 ] [ 23 ] [ 24 ]
Intuitively, antibunching can be thought of as detection of ‘missing’ events of two photons emitted from every particle that cannot simultaneously emit two photons. [ contradictory ] It is therefore used to produce an image that would have been produced using photons with half the wavelength of the detected photons. [ clarification needed ] By detecting N-photon events, the resolution can be improved by up to a factor of N over the diffraction limit.
In conventional fluorescence microscopes, antibunching information is ignored, as simultaneous detection of multiple photon emission requires temporal resolution higher than that of most commonly available cameras. [ clarification needed ] However, improved detector technology enabled demonstrations of quantum enhanced super-resolution using fast detector arrays, such as single-photon avalanche diode arrays. [ 25 ]
Quantum correlations offer an SNR beyond the photo-damage limit (the amount of energy that can be delivered without damage to the sample) of conventional microscopy. A coherent Raman microscope offers sub-wavelength resolution and incorporates bright quantum correlated illumination. Molecular bonds within a cell can be imaged with a 35 per cent improved SNR compared with conventional microscopy, corresponding to a 14% concentration sensitivity improvement. [ 20 ] | https://en.wikipedia.org/wiki/Quantum_microscopy |
The quantum mind or quantum consciousness is a group of hypotheses proposing that local physical laws and interactions from classical mechanics or connections between neurons alone cannot explain consciousness , [ 1 ] positing instead that quantum-mechanical phenomena, such as entanglement and superposition that cause nonlocalized quantum effects, interacting in smaller features of the brain than cells, may play an important part in the brain's function and could explain critical aspects of consciousness. These scientific hypotheses are as yet unvalidated, and they can overlap with quantum mysticism .
Eugene Wigner developed the idea that quantum mechanics has something to do with the workings of the mind. [ 2 ] He proposed that the wave function collapses due to its interaction with consciousness. Freeman Dyson argued that "mind, as manifested by the capacity to make choices, is to some extent inherent in every electron". [ 3 ]
Other contemporary physicists and philosophers considered these arguments unconvincing. [ 4 ] Victor Stenger characterized quantum consciousness as a "myth" having "no scientific basis" that "should take its place along with gods, unicorns and dragons". [ 5 ]
David Chalmers argues against quantum consciousness. He instead discusses how quantum mechanics may relate to dualistic consciousness . [ 6 ] Chalmers is skeptical that any new physics can resolve the hard problem of consciousness . [ 7 ] [ 8 ] [ 9 ] He argues that quantum theories of consciousness suffer from the same weakness as more conventional theories. Just as he argues that there is no particular reason why particular macroscopic physical features in the brain should give rise to consciousness, he also thinks that there is no particular reason why a particular quantum feature, such as the EM field in the brain, should give rise to consciousness either. [ 9 ]
David Bohm viewed quantum theory and relativity as contradictory, which implied a more fundamental level in the universe. [ 10 ] He claimed that both quantum theory and relativity pointed to this deeper theory, a quantum field theory . This more fundamental level was proposed to represent an undivided wholeness and an implicate order , from which arises the explicate order of the universe as we experience it. [ 10 ]
Bohm's proposed order applies both to matter and consciousness. He suggested that it could explain the relationship between them. He saw mind and matter as projections into our explicate order from the underlying implicate order. Bohm claimed that when we look at matter, we see nothing that helps us to understand consciousness. [ 11 ]
Bohm never proposed a specific means by which his proposal could be falsified, nor a neural mechanism through which his "implicate order" could emerge in a way relevant to consciousness. [ 10 ] He later collaborated on Karl Pribram 's holonomic brain theory as a model of quantum consciousness. [ 12 ]
David Bohm also collaborated with Basil Hiley on work that claimed mind and matter both emerge from an "implicate order" . [ 13 ] Hiley in turn worked with philosopher Paavo Pylkkänen . [ 14 ] According to Pylkkänen, Bohm's suggestion "leads naturally to the assumption that the physical correlate of the logical thinking process is at the classically describable level of the brain, while the basic thinking process is at the quantum-theoretically describable level". [ 15 ]
Theoretical physicist Roger Penrose and anaesthesiologist Stuart Hameroff collaborated to produce the theory known as " orchestrated objective reduction " (Orch-OR). Penrose and Hameroff initially developed their ideas separately and later collaborated to produce Orch-OR in the early 1990s. They reviewed and updated their theory in 2013. [ 16 ] [ 17 ]
Penrose's argument stemmed from Gödel's incompleteness theorems . In his first book on consciousness, The Emperor's New Mind (1989), [ 18 ] he argued that while a formal system cannot prove its own consistency, Gödel's unprovable results are provable by human mathematicians. [ 19 ] Penrose took this to mean that human mathematicians are not formal proof systems and not running a computable algorithm. According to Bringsjord and Xiao, this line of reasoning is based on fallacious equivocation on the meaning of computation. [ 20 ] In the same book, Penrose wrote: "One might speculate, however, that somewhere deep in the brain, cells are to be found of single quantum sensitivity. If this proves to be the case, then quantum mechanics will be significantly involved in brain activity." [ 18 ] : 400
Penrose determined that wave function collapse was the only possible physical basis for a non-computable process. Dissatisfied with its randomness, he proposed a new form of wave function collapse that occurs in isolation and called it objective reduction . He suggested each quantum superposition has its own piece of spacetime curvature and that when these become separated by more than one Planck length , they become unstable and collapse. [ 21 ] Penrose suggested that objective reduction represents neither randomness nor algorithmic processing but instead a non-computable influence in spacetime geometry from which mathematical understanding and, by later extension, consciousness derives. [ 21 ]
Hameroff provided a hypothesis that microtubules would be suitable hosts for quantum behavior. [ 22 ] Microtubules are composed of tubulin protein dimer subunits. The dimers each have hydrophobic pockets that are 8 nm apart and may contain delocalized π electrons . Tubulins have other smaller non-polar regions that contain π-electron-rich indole rings separated by about 2 nm. Hameroff proposed that these electrons are close enough to become entangled. [ 23 ] He originally suggested that the tubulin-subunit electrons would form a Bose–Einstein condensate , but this was discredited. [ 24 ] He then proposed a Frohlich condensate, a hypothetical coherent oscillation of dipolar molecules, but this too was experimentally discredited. [ 25 ]
In other words, there is a missing link between physics and neuroscience. [ 26 ] For instance, the proposed predominance of A-lattice microtubules, more suitable for information processing, was falsified by Kikkawa et al. , [ 27 ] [ 28 ] who showed that all in vivo microtubules have a B lattice and a seam. The proposed existence of gap junctions between neurons and glial cells was also falsified. [ 29 ] Orch-OR predicted that microtubule coherence reaches the synapses through dendritic lamellar bodies (DLBs), but De Zeeuw et al. proved this impossible [ 30 ] by showing that DLBs are micrometers away from gap junctions. [ 31 ]
In 2014, Hameroff and Penrose claimed that the discovery of quantum vibrations in microtubules by Anirban Bandyopadhyay of the National Institute for Materials Science in Japan in March 2013 [ 32 ] corroborates Orch-OR theory. [ 17 ] [ 33 ] Experiments that showed that anaesthetic drugs reduce how long microtubules can sustain suspected quantum excitations appear to support the quantum theory of consciousness. [ 34 ]
In April 2022, the results of two related experiments at the University of Alberta and Princeton University were announced at The Science of Consciousness conference, providing further evidence to support quantum processes operating within microtubules. In a study Stuart Hameroff was part of, Jack Tuszyński of the University of Alberta demonstrated that anesthetics hasten the duration of a process called delayed luminescence, in which microtubules and tubulins re-emit trapped light. Tuszyński suspects that the phenomenon has a quantum origin, with superradiance being investigated as one possibility. In the second experiment, Gregory D. Scholes and Aarat Kalra of Princeton University used lasers to excite molecules within tubulins, causing a prolonged excitation to diffuse through microtubules further than expected, which did not occur when repeated under anesthesia. [ 35 ] [ 36 ] However, diffusion results have to be interpreted carefully, since even classical diffusion can be very complex due to the wide range of length scales in the fluid filled extracellular space. [ 37 ] Nevertheless, University of Oxford quantum physicist Vlatko Vedral told that this connection with consciousness is a really long shot. [ citation needed ]
Also in 2022, a group of Italian physicists conducted several experiments that failed to provide evidence in support of a gravity-related quantum collapse model of consciousness, weakening the possibility of a quantum explanation for consciousness. [ 38 ] [ 39 ]
Although these theories are stated in a scientific framework, it is difficult to separate them from scientists' personal opinions. The opinions are often based on intuition or subjective ideas about the nature of consciousness. For example, Penrose wrote: [ 40 ]
[M]y own point of view asserts that you can't even simulate conscious activity. What's going on in conscious thinking is something you couldn't properly imitate at all by computer.... If something behaves as though it's conscious, do you say it is conscious? People argue endlessly about that. Some people would say, "Well, you've got to take the operational viewpoint; we don't know what consciousness is. How do you judge whether a person is conscious or not? Only by the way they act. You apply the same criterion to a computer or a computer-controlled robot." Other people would say, "No, you can't say it feels something merely because it behaves as though it feels something." My view is different from both those views. The robot wouldn't even behave convincingly as though it was conscious unless it really was—which I say it couldn't be, if it's entirely computationally controlled.
Penrose continues: [ 41 ]
A lot of what the brain does you could do on a computer. I'm not saying that all the brain's action is completely different from what you do on a computer. I am claiming that the actions of consciousness are something different. I'm not saying that consciousness is beyond physics, either—although I'm saying that it's beyond the physics we know now.... My claim is that there has to be something in physics that we don't yet understand, which is very important, and which is of a noncomputational character. It's not specific to our brains; it's out there, in the physical world. But it usually plays a totally insignificant role. It would have to be in the bridge between quantum and classical levels of behavior—that is, where quantum measurement comes in.
Hiroomi Umezawa and collaborators proposed a quantum field theory of memory storage. [ 42 ] [ 43 ] Giuseppe Vitiello and Walter Freeman proposed a dialog model of the mind. This dialog takes place between the classical and the quantum parts of the brain. [ 44 ] [ 45 ] [ 46 ] Their quantum field theory models of brain dynamics are fundamentally different from the Penrose–Hameroff theory. [ citation needed ]
As described by Harald Atmanspacher, "Since quantum theory is the most fundamental theory of matter that is currently available, it is a legitimate question to ask whether quantum theory can help us to understand consciousness."
The original motivation in the early 20th century for relating quantum theory to consciousness was essentially philosophical. It is fairly plausible that conscious free decisions (“free will”) are problematic in a perfectly deterministic world , so quantum randomness might indeed open up novel possibilities for free will. (On the other hand, randomness is problematic for goal-directed volition!) [ 47 ]
Ricciardi and Umezawa proposed in 1967 a general theory of quanta of long-range coherent waves within and between brain cells, and showed a possible mechanism of memory storage and retrieval in terms of Nambu–Goldstone bosons . [ 48 ] Mari Jibu and Kunio Yasue later popularized these results under the name "quantum brain dynamics" (QBD) as the hypothesis to explain the function of the brain within the framework of quantum field theory with implications on consciousness. [ 49 ] [ 50 ] [ 51 ]
Karl Pribram 's holonomic brain theory (quantum holography) invoked quantum field theory to explain higher-order processing of memory in the brain. [ 52 ] [ 53 ] He argued that his holonomic model solved the binding problem . [ 54 ] Pribram collaborated with Bohm in his work on quantum approaches to the thought process. [ 55 ] Pribram suggested much of the processing in the brain was done in distributed fashion. [ 56 ] He proposed that the fine fibered, felt-like dendritic fields might follow the principles of quantum field theory when storing and retrieving long term memory. [ 57 ]
Henry Stapp proposed that quantum waves are reduced only when they interact with consciousness. He argues from the orthodox quantum mechanics of John von Neumann [ clarify ] that the quantum state collapses when the observer selects one among the alternative quantum possibilities as a basis for future action. The collapse, therefore, takes place in the expectation that the observer associated with the state. Stapp's work drew criticism from scientists such as David Bourget and Danko Georgiev. [ 58 ] [ 59 ] [ 60 ] [ 61 ]
CNET is a hypothesized neural signaling mechanism in catecholaminergic neurons that would use quantum mechanical electron transport. [ 62 ] [ 63 ] The hypothesis is based in part on the observation by many independent researchers that electron tunneling occurs in ferritin, an iron storage protein that is prevalent in those neurons, at room temperature and ambient conditions. [ 64 ] [ 65 ] [ 66 ] [ 67 ] The hypothesized function of this mechanism is to assist in action selection, but the mechanism itself would be capable of integrating millions of cognitive and sensory neural signals using a physical mechanism associated with strong electron-electron interactions. [ 68 ] [ 69 ] [ 70 ] Each tunneling event would involve a collapse of an electron wave function, but the collapse would be incidental to the physical effect created by strong electron-electron interactions. [ citation needed ]
CNET predicted a number of physical properties of these neurons that have been subsequently observed experimentally, such as electron tunneling in substantia nigra pars compacta (SNc) tissue and the presence of disordered arrays of ferritin in SNc tissue. [ 71 ] [ 72 ] [ 73 ] [ 74 ] The hypothesis also predicted that disordered ferritin arrays like those found in SNc tissue should be capable of supporting long-range electron transport and providing a switching or routing function, both of which have also been subsequently observed. [ 75 ] [ 76 ] [ 77 ]
Another prediction of CNET was that the largest SNc neurons should mediate action selection. This prediction was contrary to earlier proposals about the function of those neurons at that time, which were based on predictive reward dopamine signaling. [ 78 ] [ 79 ] A team led by Dr. Pascal Kaeser of Harvard Medical School subsequently demonstrated that those neurons do in fact code movement, consistent with the earlier predictions of CNET. [ 80 ] While the CNET mechanism has not yet been directly observed, it may be possible to do so using quantum dot fluorophores tagged to ferritin or other methods for detecting electron tunneling. [ 81 ]
CNET is applicable to a number of different consciousness models as a binding or action selection mechanism, such as Integrated Information Theory (IIT) and Sensorimotor Theory (SMT). [ 82 ] It is noted that many existing models of consciousness fail to specifically address action selection or binding. For example, O’Regan and Noë call binding a “pseudo problem,” but also state that “the fact that object attributes seem perceptually to be part of a single object does not require them to be ‘represented’ in any unified kind of way, for example, at a single location in the brain, or by a single process. They may be so represented, but there is no logical necessity for this.” [ 83 ] Simply because there is no “logical necessity” for a physical phenomenon does not mean that it does not exist, or that once it is identified that it can be ignored. Likewise, global workspace theory (GWT) models appear to treat dopamine as modulatory, [ 84 ] based on the prior understanding of those neurons from predictive reward dopamine signaling research, but GWT models could be adapted to include modeling of moment-by-moment activity in the striatum to mediate action selection, as observed by Kaiser. CNET is applicable to those neurons as a selection mechanism for that function, as otherwise that function could result in seizures from simultaneous actuation of competing sets of neurons. While CNET by itself is not a model of consciousness, it is able to integrate different models of consciousness through neural binding and action selection. However, a more complete understanding of how CNET might relate to consciousness would require a better understanding of strong electron-electron interactions in ferritin arrays, which implicates the many-body problem .
These hypotheses of the quantum mind remain hypothetical speculation, as Penrose admits in his discussions. Until they make a prediction that is tested by experimentation, the hypotheses are not based on empirical evidence. In 2010, Lawrence Krauss was guarded in criticising Penrose's ideas. He said: "Roger Penrose has given lots of new-age crackpots ammunition... Many people are dubious that Penrose's suggestions are reasonable, because the brain is not an isolated quantum-mechanical system. To some extent it could be, because memories are stored at the molecular level, and at a molecular level quantum mechanics is significant." [ 85 ] According to Krauss, "It is true that quantum mechanics is extremely strange, and on extremely small scales for short times, all sorts of weird things happen. And in fact, we can make weird quantum phenomena happen. But what quantum mechanics doesn't change about the universe is, if you want to change things, you still have to do something. You can't change the world by thinking about it." [ 85 ]
The process of testing the hypotheses with experiments is fraught with conceptual/theoretical, practical, and ethical problems.
The idea that a quantum effect is necessary for consciousness to function is still in the realm of philosophy. Penrose proposes that it is necessary, but other theories of consciousness do not indicate that it is needed. For example, Daniel Dennett proposed a theory called multiple drafts model , which doesn't indicate that quantum effects are needed, in his 1991 book Consciousness Explained . [ 86 ] A philosophical argument on either side is not a scientific proof, although philosophical analysis can indicate key differences in the types of models and show what type of experimental differences might be observed. But since there is no clear consensus among philosophers, there is no conceptual support that a quantum mind theory is needed. [ 87 ]
A possible conceptual approach is to use quantum mechanics as an analogy to understand a different field of study like consciousness, without expecting that the laws of quantum physics will apply. An example of this approach is the idea of Schrödinger's cat . Erwin Schrödinger described how one could, in principle, create entanglement of a large-scale system by making it dependent on an elementary particle in a superposition. He proposed a scenario with a cat in a locked steel chamber, wherein the cat's survival depended on the state of a radioactive atom—whether it had decayed and emitted radiation. According to Schrödinger, the Copenhagen interpretation implies that the cat is both alive and dead until the state has been observed. Schrödinger did not wish to promote the idea of dead-and-alive cats as a serious possibility; he intended the example to illustrate the absurdity of the existing view of quantum mechanics. [ 88 ] But since Schrödinger's time, physicists have given other interpretations of the mathematics of quantum mechanics , some of which regard the "alive and dead" cat superposition as quite real. [ 89 ] [ 90 ] Schrödinger's famous thought experiment poses the question of when a system stops existing as a quantum superposition of states. In the same way, one can ask whether the act of making a decision is analogous to having a superposition of states of two decision outcomes, so that making a decision means "opening the box" to reduce the brain from a combination of states to one state. This analogy of decision-making uses a formalism derived from quantum mechanics, but does not indicate the actual mechanism by which the decision is made.
In this way, the idea is similar to quantum cognition . This field clearly distinguishes itself from the quantum mind, as it is not reliant on the hypothesis that there is something micro-physical quantum-mechanical about the brain. Quantum cognition is based on the quantum-like paradigm, [ 91 ] [ 92 ] generalized quantum paradigm, [ 93 ] or quantum structure paradigm [ 94 ] that information processing by complex systems such as the brain can be mathematically described in the framework of quantum information and quantum probability theory. This model uses quantum mechanics only as an analogy and does not propose that quantum mechanics is the physical mechanism by which it operates. For example, quantum cognition proposes that some decisions can be analyzed as if there is interference between two alternatives, but it is not a physical quantum interference effect. [ 95 ]
The main theoretical argument against the quantum-mind hypothesis is the assertion that quantum states in the brain would lose coherency before they reached a scale where they could be useful for neural processing. This supposition was elaborated by Max Tegmark . His calculations indicate that quantum systems in the brain decohere at sub-picosecond timescales. [ 96 ] [ 97 ] No response by a brain has shown computational results or reactions on this fast of a timescale. Typical reactions are on the order of milliseconds, trillions of times longer than sub-picosecond timescales. [ 98 ]
Daniel Dennett uses an experimental result in support of his multiple drafts model of an optical illusion that happens on a timescale of less than a second or so. In this experiment, two different-colored lights, with an angular separation of a few degrees at the eye, are flashed in succession. If the interval between the flashes is less than a second or so, the first light that is flashed appears to move across to the position of the second light. Furthermore, the light seems to change color as it moves across the visual field. A green light will appear to turn red as it seems to move across to the position of a red light. Dennett asks how we could see the light change color before the second light is observed. [ 86 ] Velmans argues that the cutaneous rabbit illusion , another illusion that happens in about a second, demonstrates that there is a delay while modelling occurs in the brain and that this delay was discovered by Libet . [ 99 ] These slow illusions that happen at times of less than a second do not support a proposal that the brain functions on the picosecond timescale. [ citation needed ]
Penrose says: [ 41 ]
The problem with trying to use quantum mechanics in the action of the brain is that if it were a matter of quantum nerve signals, these nerve signals would disturb the rest of the material in the brain, to the extent that the quantum coherence would get lost very quickly. You couldn't even attempt to build a quantum computer out of ordinary nerve signals, because they're just too big and in an environment that's too disorganized. Ordinary nerve signals have to be treated classically. But if you go down to the level of the microtubules, then there's an extremely good chance that you can get quantum-level activity inside them.
For my picture, I need this quantum-level activity in the microtubules; the activity has to be a large-scale thing that goes not just from one microtubule to the next but from one nerve cell to the next, across large areas of the brain. We need some kind of coherent activity of a quantum nature which is weakly coupled to the computational activity that Hameroff argues is taking place along the microtubules. [ citation needed ]
There are various avenues of attack. One is directly on the physics, on quantum theory, and there are certain experiments that people are beginning to perform, and various schemes for a modification of quantum mechanics. I don't think the experiments are sensitive enough yet to test many of these specific ideas. One could imagine experiments that might test these things, but they'd be very hard to perform.
Penrose also said in an interview:
...whatever consciousness is, it must be beyond computable physics.... It's not that consciousness depends on quantum mechanics, it's that it depends on where our current theories of quantum mechanics go wrong. It's to do with a theory that we don't know yet. [ 100 ]
A demonstration of a quantum effect in the brain has to explain this problem or explain why it is not relevant, or that the brain somehow circumvents the problem of the loss of quantum coherency at body temperature. As Penrose proposes, it may require a new type of physical theory, something "we don't know yet." [ 100 ]
Deepak Chopra has referred a "quantum soul" existing "apart from the body", [ 101 ] human "access to a field of infinite possibilities", [ 102 ] and other quantum mysticism topics such as quantum healing or quantum effects of consciousness. Seeing the human body as being undergirded by a "quantum-mechanical body" composed not of matter but of energy and information, he believes that "human aging is fluid and changeable; it can speed up, slow down, stop for a time, and even reverse itself", as determined by one's state of mind. [ 103 ] Robert Carroll states that Chopra attempts to integrate Ayurveda with quantum mechanics to justify his teachings. [ 104 ] Chopra argues that what he calls "quantum healing" cures any manner of ailments, including cancer, through effects that he claims are based on the same principles as quantum mechanics. [ 105 ] This has led physicists to object to his use of the term quantum in reference to medical conditions and the human body. [ 105 ] Chopra said: "I think quantum theory has a lot of things to say about the observer effect , about non-locality, about correlations. So I think there’s a school of physicists who believe that consciousness has to be equated, or at least brought into the equation, in understanding quantum mechanics." [ 106 ] On the other hand, he also claims that quantum effects are "just a metaphor. Just like an electron or a photon is an indivisible unit of information and energy, a thought is an indivisible unit of consciousness." [ 106 ] In his book Quantum Healing , Chopra stated the conclusion that quantum entanglement links everything in the Universe, and therefore it must create consciousness. [ 107 ]
According to Daniel Dennett, "On this topic, Everybody's an expert... but they think that they have a particular personal authority about the nature of their own conscious experiences that can trump any hypothesis they find unacceptable." [ 108 ]
While quantum effects are significant in the physiology of the brain, critics of quantum mind hypotheses challenge whether the effects of known or speculated quantum phenomena in biology scale up to have significance in neuronal computation, much less the emergence of consciousness as phenomenon. Daniel Dennett said, "Quantum effects are there in your car, your watch, and your computer. But most things—most macroscopic objects—are, as it were, oblivious to quantum effects. They don't amplify them; they don't hinge on them." [ 41 ] | https://en.wikipedia.org/wiki/Quantum_mind |
Quantum natural language processing (QNLP) is the application of quantum computing to natural language processing (NLP). It computes word embeddings as parameterised quantum circuits that can solve NLP tasks faster than any classical computer. [ 1 ] [ 2 ] [ 3 ] It is inspired by categorical quantum mechanics and the DisCoCat framework, making use of string diagrams to translate from grammatical structure to quantum processes. [ 4 ] [ 5 ] [ 6 ]
The first quantum algorithm for natural language processing used the DisCoCat framework and Grover's algorithm to show a quadratic quantum speedup for a text classification task . [ 1 ] It was later shown that quantum language processing is BQP-Complete , [ 2 ] i.e. quantum language models are more expressive than their classical counterpart, unless quantum mechanics can be efficiently simulated by classical computers. [ 7 ]
These two theoretical results assume fault-tolerant quantum computation and a QRAM , i.e. an efficient way to load classical data on a quantum computer. Thus, they are not applicable to the noisy intermediate-scale quantum (NISQ) computers available today.
The algorithm of Zeng and Coecke [ 1 ] was adapted to the constraints of NISQ computers and implemented on IBM quantum computers to solve binary classification tasks. [ 8 ] [ 9 ] Instead of loading classical word vectors onto a quantum memory , the word vectors are computed directly as the parameters of quantum circuits . These parameters are optimised using methods from quantum machine learning to solve data-driven tasks such as question answering , [ 8 ] machine translation [ 10 ] and even algorithmic music composition . [ 11 ] | https://en.wikipedia.org/wiki/Quantum_natural_language_processing |
Quantum non-equilibrium is a concept within stochastic formulations of the De Broglie–Bohm theory of quantum physics .
In quantum mechanics, the Born rule states that the probability density of finding a system in a given state, when measured, is proportional to the square of the amplitude of the system's wavefunction at that state, and it constitutes one of the fundamental axioms of the theory.
This is not the case for the De Broglie–Bohm theory, where the Born rule is not a basic law. Rather, in this theory the link between the probability density and the wave function has the status of a hypothesis, called the quantum equilibrium hypothesis , which is additional to the basic principles governing the wave function, the dynamics of the quantum particles and the Schrödinger equation . (For mathematical details, refer to the derivation by Peter R. Holland.)
Accordingly, quantum non-equilibrium describes a state of affairs where the Born rule is not fulfilled; that is, the probability to find the particle in the differential volume d 3 x {\displaystyle d^{3}x} at time t is unequal to | ψ ( x , t ) | 2 . {\displaystyle |\psi (\mathbf {x} ,t)|^{2}.}
Recent advances in investigations into properties of quantum non-equilibrium states have been performed mainly by theoretical physicist Antony Valentini , and earlier steps in this direction were undertaken by David Bohm , Jean-Pierre Vigier , Basil Hiley and Peter R. Holland . The existence of quantum non-equilibrium states has not been verified experimentally; quantum non-equilibrium is so far a theoretical construct. The relevance of quantum non-equilibrium states to physics lies in the fact that they can lead to different predictions for results of experiments, depending on whether the De Broglie–Bohm theory in its stochastic form or the Copenhagen interpretation is assumed to describe reality. (The Copenhagen interpretation, which stipulates the Born rule a priori , does not foresee the existence of quantum non-equilibrium states at all.) That is, properties of quantum non-equilibrium can make certain classes of Bohmian theories falsifiable according to the criterion of Karl Popper .
In practice, when performing Bohmian mechanics computations in quantum chemistry , the quantum equilibrium hypothesis is simply considered to be fulfilled, in order to predict system behaviour and the outcome of measurements.
The causal interpretation of quantum mechanics has been set up by de Broglie and Bohm as a causal, deterministic model, and it was extended later by Bohm, Vigier, Hiley, Valentini and others to include stochastic properties.
Bohm and other physicists, including Valentini, view the Born rule linking R {\displaystyle R} to the probability density function ρ = R 2 {\displaystyle \rho =R^{2}} as representing not a basic law, but rather as constituting a result of a system having reached quantum equilibrium during the course of the time development under the Schrödinger equation . It can be shown that, once an equilibrium has been reached, the system remains in such equilibrium over the course of its further evolution: this follows from the continuity equation associated with the Schrödinger evolution of ψ . {\displaystyle \psi .} [ 2 ] However, it is less straightforward to demonstrate whether and how such an equilibrium is reached in the first place.
In 1991, Valentini provided indications for deriving the quantum equilibrium hypothesis which states that ρ ( X , t ) = | ψ ( X , t ) | 2 {\displaystyle \rho (X,t)=|\psi (X,t)|^{2}} in the framework of the pilot wave theory . (Here, X {\displaystyle X} stands for the collective coordinates of the system in configuration space ). Valentini showed that the relaxation ρ ( X , t ) → | ψ ( X , t ) | 2 {\displaystyle \rho (X,t)\to |\psi (X,t)|^{2}} may be accounted for by an H-theorem constructed in analogy to the Boltzmann H-theorem of statistical mechanics. [ 3 ] [ 4 ]
Valentini's derivation of the quantum equilibrium hypothesis was criticized by Detlef Dürr and co-workers in 1992, and the derivation of the quantum equilibrium hypothesis has remained a topic of active investigation. [ 5 ]
Numerical simulations demonstrate a tendency for Born rule distributions to arise spontaneously at short time scales. [ 6 ]
Valentini showed that his expansion of the De Broglie–Bohm theory would allow “signal nonlocality ” for non-equilibrium cases in which ρ ( x , y , z , t ) ≠ | ψ ( x , y , z , t ) | 2 , {\displaystyle \rho (x,y,z,t)\neq |\psi (x,y,z,t)|^{2},} [ 3 ] [ 4 ] thereby violating the assumption that signals cannot travel faster than the speed of light .
Valentini furthermore showed that an ensemble of particles with known wave function and known nonequilibrium distribution could be used to perform, on another system, measurements that violate the uncertainty principle . [ 7 ]
These predictions differ from predictions that would result from approaching the same physical situation by means of the standard axioms of quantum mechanics and therefore would in principle make the predictions of this theory accessible to experimental study. As it is unknown whether or how quantum non-equilibrium states can be produced, it is difficult or impossible to perform such experiments.
However, also the hypothesis of quantum non-equilibrium Big Bang gives rise to quantitative predictions for nonequilibrium deviations from quantum theory which appear to be more easily accessible to observation. [ 8 ] | https://en.wikipedia.org/wiki/Quantum_non-equilibrium |
In quantum physics and chemistry , quantum numbers are quantities that characterize the possible states of the system.
To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantum numbers includes the principal , azimuthal , magnetic , and spin quantum numbers. To describe other systems, different quantum numbers are required. For subatomic particles, one needs to introduce new quantum numbers, such as the flavour of quarks , which have no classical correspondence.
Quantum numbers are closely related to eigenvalues of observables . When the corresponding observable commutes with the Hamiltonian of the system, the quantum number is said to be " good ", and acts as a constant of motion in the quantum dynamics.
In the era of the old quantum theory , starting from Max Planck 's proposal of quanta in his model of blackbody radiation (1900) and Albert Einstein 's adaptation of the concept to explain the photoelectric effect (1905), and until Erwin Schrödinger published his eigenfunction equation in 1926, [ 1 ] the concept behind quantum numbers developed based on atomic spectroscopy and theories from classical mechanics with extra ad hoc constraints. [ 2 ] : 106 Many results from atomic spectroscopy had been summarized in the Rydberg formula involving differences between two series of energies related by integer steps. The model of the atom , first proposed by Niels Bohr in 1913, relied on a single quantum number. Together with Bohr's constraint that radiation absorption is not classical, it was able to explain the Balmer series portion of Rydberg's atomic spectrum formula. [ 3 ]
As Bohr notes in his subsequent Nobel lecture, the next step was taken by Arnold Sommerfeld in 1915. [ 4 ] Sommerfeld's atomic model added a second quantum number and the concept of quantized phase integrals to justify them. [ 5 ] : 207 Sommerfeld's model was still essentially two dimensional, modeling the electron as orbiting in a plane; in 1919 he extended his work to three dimensions using 'space quantization' in place of the quantized phase integrals. [ 6 ] : 152 Karl Schwarzschild and Sommerfeld's student, Paul Epstein , independently showed that adding third quantum number gave a complete account for the Stark effect results.
A consequence of space quantization was that the electron's orbital interaction with an external magnetic field would be quantized. This seemed to be confirmed when the results of the Stern-Gerlach experiment reported quantized results for silver atoms in an inhomogeneous magnetic field. The confirmation would turn out to be premature: more quantum numbers would be needed. [ 7 ]
The fourth and fifth quantum numbers of the atomic era arose from attempts to understand the Zeeman effect . Like the Stern-Gerlach experiment, the Zeeman effect reflects the interaction of atoms with a magnetic field; in a weak field the experimental results were called "anomalous", they diverged from any theory at the time. Wolfgang Pauli 's solution to this issue was to introduce another quantum number taking only two possible values, ± ℏ / 2 {\displaystyle \pm \hbar /2} . [ 8 ] This would ultimately become the quantized values of the projection of spin , an intrinsic angular momentum quantum of the electron. In 1927 Ronald Fraser demonstrated that the quantization in the Stern-Gerlach experiment was due to the magnetic moment associated with the electron spin rather than its orbital angular momentum. [ 7 ] Pauli's success in developing the arguments for a spin quantum number without relying on classical models set the stage for the development of quantum numbers for elementary particles in the remainder of the 20th century. [ 8 ]
Bohr, with his Aufbau or "building up" principle, and Pauli with his exclusion principle connected the atom's electronic quantum numbers in to a framework for predicting the properties of atoms. [ 9 ] When Schrödinger published his wave equation and calculated the energy levels of hydrogen, these two principles carried over to become the basis of atomic physics.
With successful models of the atom, the attention of physics turned to models of the nucleus. Beginning with Heisenberg's initial model of proton-neutron binding in 1932, Eugene Wigner introduced isospin in 1937, the first 'internal' quantum number unrelated to a symmetry in real space-time. [ 10 ] : 45
As quantum mechanics developed, abstraction increased and models based on symmetry and invariance played increasing roles. Two years before his work on the quantum wave equation, Schrödinger applied the symmetry ideas originated by Emmy Noether and Hermann Weyl to the electromagnetic field. [ 11 ] : 198 As quantum electrodynamics developed in the 1930s and 1940s, group theory became an important tool. By 1953 Chen Ning Yang had become obsessed with the idea that group theory could be applied to connect the conserved quantum numbers of nuclear collisions to symmetries in a field theory of nucleons. [ 11 ] : 202 With Robert Mills , Yang developed a non-abelian gauge theory based on the conservation of the nuclear isospin quantum numbers.
Good quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian , quantities that can be known with precision at the same time as the system's energy. Specifically, observables that commute with the Hamiltonian are simultaneously diagonalizable with it and so the eigenvalues a {\displaystyle a} and the energy (eigenvalues of the Hamiltonian) are not limited by an uncertainty relation arising from non-commutativity. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together. Many observables have discrete spectra (sets of eigenvalues) in quantum mechanics, so the quantities can only be measured in discrete values. In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers ; although they could approach infinity in some cases.
The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a quantum operator in the form of a Hamiltonian , H . There is one quantum number of the system corresponding to the system's energy; i.e., one of the eigenvalues of the Hamiltonian. There is also one quantum number for each linearly independent operator O that commutes with the Hamiltonian. A complete set of commuting observables (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different basis that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations.
Four quantum numbers can describe an electron energy level in a hydrogen-like atom completely:
These quantum numbers are also used in the classical description of nuclear particle states (e.g. protons and neutrons). [ citation needed ] A quantum description of molecular orbitals requires other quantum numbers, because the symmetries of the molecular system are different.
The principal quantum number describes the electron shell of an electron. The value of n ranges from 1 to the shell containing the outermost electron of that atom, that is [ 12 ] n = 1 , 2 , … {\displaystyle n=1,2,\ldots }
For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6. The average distance between the electron and the nucleus increases with n .
The azimuthal quantum number, also known as the orbital angular momentum quantum number , describes the subshell , and gives the magnitude of the orbital angular momentum through the relation
L 2 = ℏ 2 ℓ ( ℓ + 1 ) . {\displaystyle L^{2}=\hbar ^{2}\ell (\ell +1).}
In chemistry and spectroscopy, ℓ = 0 is called s orbital, ℓ = 1 , p orbital, ℓ = 2 , d orbital, and ℓ = 3 , f orbital.
The value of ℓ ranges from 0 to n − 1 , so the first p orbital ( ℓ = 1 ) appears in the second electron shell ( n = 2 ), the first d orbital ( ℓ = 2 ) appears in the third shell ( n = 3 ), and so on: [ 13 ]
ℓ = 0 , 1 , 2 , … , n − 1 {\displaystyle \ell =0,1,2,\ldots ,n-1}
A quantum number beginning in n = 3, ℓ = 0 , describes an electron in the s orbital of the third electron shell of an atom. In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles . The azimuthal quantum number can also denote the number of angular nodes present in an orbital. For example, for p orbitals, ℓ = 1 and thus the amount of angular nodes in a p orbital is 1.
The magnetic quantum number describes the specific orbital within the subshell, and yields the projection of the orbital angular momentum along a specified axis :
L z = m ℓ ℏ {\displaystyle L_{z}=m_{\ell }\hbar }
The values of m ℓ range from − ℓ to ℓ , with integer intervals. [ 14 ] [ page needed ]
The s subshell ( ℓ = 0 ) contains only one orbital, and therefore the m ℓ of an electron in an s orbital will always be 0. The p subshell ( ℓ = 1 ) contains three orbitals, so the m ℓ of an electron in a p orbital will be −1, 0, or 1. The d subshell ( ℓ = 2 ) contains five orbitals, with m ℓ values of −2, −1, 0, 1, and 2.
The spin magnetic quantum number describes the intrinsic spin angular momentum of the electron within each orbital and gives the projection of the spin angular momentum S along the specified axis:
S z = m s ℏ {\displaystyle S_{z}=m_{s}\hbar }
In general, the values of m s range from − s to s , where s is the spin quantum number, associated with the magnitude of particle's intrinsic spin angular momentum: [ 15 ]
m s = − s , − s + 1 , − s + 2 , ⋯ , s − 2 , s − 1 , s {\displaystyle m_{s}=-s,-s+1,-s+2,\cdots ,s-2,s-1,s}
An electron state has spin number s = 1 / 2 , consequently m s will be + 1 / 2 ("spin up") or − 1 / 2 "spin down" states. Since electron are fermions they obey the Pauli exclusion principle : each electron state must have different quantum numbers. Therefore, every orbital will be occupied with at most two electrons, one for each spin state.
A multi-electron atom can be modeled qualitatively as a hydrogen like atom with higher nuclear charge and correspondingly more electrons. The occupation of the electron states in such an atom can be predicted by the Aufbau principle and Hund's empirical rules for the quantum numbers. The Aufbau principle fills orbitals based on their principal and azimuthal quantum numbers (lowest n + l first, with lowest n breaking ties; Hund's rule favors unpaired electrons in the outermost orbital). These rules are empirical but they can be related to electron physics. [ 16 ] : 10 [ 17 ] : 260
When one takes the spin–orbit interaction into consideration, the L and S operators no longer commute with the Hamiltonian , and the eigenstates of the system no longer have well-defined orbital angular momentum and spin. Thus another set of quantum numbers should be used. This set includes [ 18 ] [ 19 ]
For example, consider the following 8 states, defined by their quantum numbers:
The quantum states in the system can be described as linear combination of these 8 states. However, in the presence of spin–orbit interaction , if one wants to describe the same system by 8 states that are eigenvectors of the Hamiltonian (i.e. each represents a state that does not mix with others over time), we should consider the following 8 states:
In nuclei , the entire assembly of protons and neutrons ( nucleons ) has a resultant angular momentum due to the angular momenta of each nucleon, usually denoted I . If the total angular momentum of a neutron is j n = ℓ + s and for a proton is j p = ℓ + s (where s for protons and neutrons happens to be 1 / 2 again ( see note )), then the nuclear angular momentum quantum numbers I are given by:
I = | j n − j p | , | j n − j p | + 1 , | j n − j p | + 2 , ⋯ , ( j n + j p ) − 2 , ( j n + j p ) − 1 , ( j n + j p ) {\displaystyle I=|j_{n}-j_{p}|,|j_{n}-j_{p}|+1,|j_{n}-j_{p}|+2,\cdots ,(j_{n}+j_{p})-2,(j_{n}+j_{p})-1,(j_{n}+j_{p})}
Note: The orbital angular momenta of the nuclear (and atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and proton are half-integer multiples. It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, I , of any odd-A nucleus and integer values for any even-A nucleus.
Parity with the number I is used to label nuclear angular momentum states, examples for some isotopes of hydrogen (H), carbon (C), and sodium (Na) are; [ 20 ]
The reason for the unusual fluctuations in I , even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of NMR spectroscopy in organic chemistry , [ 19 ] and MRI in nuclear medicine , [ 20 ] due to the nuclear magnetic moment interacting with an external magnetic field .
Elementary particles contain many quantum numbers which are usually said to be intrinsic to them. However, it should be understood that the elementary particles are quantum states of the standard model of particle physics , and hence the quantum numbers of these particles bear the same relation to the Hamiltonian of this model as the quantum numbers of the Bohr atom does to its Hamiltonian . In other words, each quantum number denotes a symmetry of the problem. It is more useful in quantum field theory to distinguish between spacetime and internal symmetries.
Typical quantum numbers related to spacetime symmetries are spin (related to rotational symmetry), the parity , C-parity and T-parity (related to the Poincaré symmetry of spacetime ). Typical internal symmetries [ clarification needed ] are lepton number and baryon number or the electric charge . (For a full list of quantum numbers of this kind see the article on flavour .)
Most conserved quantum numbers are additive, so in an elementary particle reaction, the sum of the quantum numbers should be the same before and after the reaction. However, some, usually called a parity , are multiplicative; i.e., their product is conserved. All multiplicative quantum numbers belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing ( involution ). | https://en.wikipedia.org/wiki/Quantum_number |
In quantum mechanics , a quantum operation (also known as quantum dynamical map or quantum process ) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan . [ 1 ] The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation , a quantum operation is called a quantum channel .
Note that some authors use the term "quantum operation" to refer specifically to completely positive (CP) and non-trace-increasing maps on the space of density matrices, and the term " quantum channel " to refer to the subset of those that are strictly trace-preserving. [ 2 ]
Quantum operations are formulated in terms of the density operator description of a quantum mechanical system. Rigorously, a quantum operation is a linear , completely positive map from the set of density operators into itself. In the context of quantum information, one often imposes the further restriction that a quantum operation E {\displaystyle {\mathcal {E}}} must be physical , [ 3 ] that is, satisfy 0 ≤ Tr [ E ( ρ ) ] ≤ 1 {\displaystyle 0\leq \operatorname {Tr} [{\mathcal {E}}(\rho )]\leq 1} for any state ρ {\displaystyle \rho } .
Some quantum processes cannot be captured within the quantum operation formalism; [ 4 ] in principle, the density matrix of a quantum system can undergo completely arbitrary time evolution. Quantum operations are generalized by quantum instruments , which capture the classical information obtained during measurements, in addition to the quantum information .
The Schrödinger picture provides a satisfactory account of time evolution of state for a quantum mechanical system under certain assumptions. These assumptions include
The Schrödinger picture for time evolution has several mathematically equivalent formulations. One such formulation expresses the time rate of change of the state via the Schrödinger equation . A more suitable formulation for this exposition is expressed as follows:
This means that if the system is in a state corresponding to v ∈ H at an instant of time s , then the state after t units of time will be U t v . For relativistic systems, there is no universal time parameter, but we can still formulate the effect of certain reversible transformations on the quantum mechanical system. For instance, state transformations relating observers in different frames of reference are given by unitary transformations. In any case, these state transformations carry pure states into pure states; this is often formulated by saying that in this idealized framework, there is no decoherence .
For interacting (or open) systems, such as those undergoing measurement, the situation is entirely different. To begin with, the state changes experienced by such systems cannot be accounted for exclusively by a transformation on the set of pure states (that is, those associated to vectors of norm 1 in H ). After such an interaction, a system in a pure state φ may no longer be in the pure state φ. In general it will be in a statistical mix of a sequence of pure states φ 1 , ..., φ k with respective probabilities λ 1 , ..., λ k . The transition from a pure state to a mixed state is known as decoherence.
Numerous mathematical formalisms have been established to handle the case of an interacting system. The quantum operation formalism emerged around 1983 from work of Karl Kraus , who relied on the earlier mathematical work of Man-Duen Choi . It has the advantage that it expresses operations such as measurement as a mapping from density states to density states. In particular, the effect of quantum operations stays within the set of density states.
Recall that a density operator is a non-negative operator on a Hilbert space with unit trace.
Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that
Note that, by the first condition, quantum operations may not preserve the normalization property of statistical ensembles. In probabilistic terms, quantum operations may be sub-Markovian . In order that a quantum operation preserve the set of density matrices, we need the additional assumption that it is trace-preserving.
In the context of quantum information , the quantum operations defined here, i.e. completely positive maps that do not increase the trace, are also called quantum channels or stochastic maps . The formulation here is confined to channels between quantum states; however, it can be extended to include classical states as well, therefore allowing quantum and classical information to be handled simultaneously.
Kraus ' theorem (named after Karl Kraus ) characterizes completely positive maps , which model quantum operations between quantum states. Informally, the theorem ensures that the action of any such quantum operation Φ {\displaystyle \Phi } on a state ρ {\displaystyle \rho } can always be written as Φ ( ρ ) = ∑ k B k ρ B k ∗ {\textstyle \Phi (\rho )=\sum _{k}B_{k}\rho B_{k}^{*}} , for some set of operators { B k } k {\displaystyle \{B_{k}\}_{k}} satisfying ∑ k B k ∗ B k ≤ 1 {\textstyle \sum _{k}B_{k}^{*}B_{k}\leq \mathbf {1} } , where 1 {\displaystyle \mathbf {1} } is the identity operator.
Theorem . [ 5 ] Let H {\displaystyle {\mathcal {H}}} and G {\displaystyle {\mathcal {G}}} be Hilbert spaces of dimension n {\displaystyle n} and m {\displaystyle m} respectively, and Φ {\displaystyle \Phi } be a quantum operation between H {\displaystyle {\mathcal {H}}} and G {\displaystyle {\mathcal {G}}} . Then, there are matrices { B i } 1 ≤ i ≤ n m {\displaystyle \{B_{i}\}_{1\leq i\leq nm}} mapping H {\displaystyle {\mathcal {H}}} to G {\displaystyle {\mathcal {G}}} such that, for any state ρ {\displaystyle \rho } , Φ ( ρ ) = ∑ i B i ρ B i ∗ . {\displaystyle \Phi (\rho )=\sum _{i}B_{i}\rho B_{i}^{*}.} Conversely, any map Φ {\displaystyle \Phi } of this form is a quantum operation provided ∑ k B k ∗ B k ≤ 1 {\textstyle \sum _{k}B_{k}^{*}B_{k}\leq \mathbf {1} } .
The matrices { B i } {\displaystyle \{B_{i}\}} are called Kraus operators . (Sometimes they are known as noise operators or error operators , especially in the context of quantum information processing , where the quantum operation represents the noisy, error-producing effects of the environment.) The Stinespring factorization theorem extends the above result to arbitrary separable Hilbert spaces H and G . There, S is replaced by a trace class operator and { B i } {\displaystyle \{B_{i}\}} by a sequence of bounded operators.
Kraus matrices are not uniquely determined by the quantum operation Φ {\displaystyle \Phi } in general. For example, different Cholesky factorizations of the Choi matrix might give different sets of Kraus operators. The following theorem states that all systems of Kraus matrices representing the same quantum operation are related by a unitary transformation:
Theorem . Let Φ {\displaystyle \Phi } be a (not necessarily trace-preserving) quantum operation on a finite-dimensional Hilbert space H with two representing sequences of Kraus matrices { B i } i ≤ N {\displaystyle \{B_{i}\}_{i\leq N}} and { C i } i ≤ N {\displaystyle \{C_{i}\}_{i\leq N}} . Then there is a unitary operator matrix ( u i j ) i j {\displaystyle (u_{ij})_{ij}} such that C i = ∑ j u i j B j . {\displaystyle C_{i}=\sum _{j}u_{ij}B_{j}.}
In the infinite-dimensional case, this generalizes to a relationship between two minimal Stinespring representations .
It is a consequence of Stinespring's theorem that all quantum operations can be implemented by unitary evolution after coupling a suitable ancilla to the original system.
These results can be also derived from Choi's theorem on completely positive maps , characterizing a completely positive finite-dimensional map by a unique Hermitian-positive density operator ( Choi matrix ) with respect to the trace. Among all possible Kraus representations of a given channel , there exists a canonical form distinguished by the orthogonality relation of Kraus operators, Tr A i † A j ∼ δ i j {\displaystyle \operatorname {Tr} A_{i}^{\dagger }A_{j}\sim \delta _{ij}} . Such canonical set of orthogonal Kraus operators can be obtained by diagonalising the corresponding Choi matrix and reshaping its eigenvectors into square matrices.
There also exists an infinite-dimensional algebraic generalization of Choi's theorem, known as "Belavkin's Radon-Nikodym theorem for completely positive maps", which defines a density operator as a "Radon–Nikodym derivative" of a quantum channel with respect to a dominating completely positive map (reference channel). It is used for defining the relative fidelities and mutual informations for quantum channels.
For a non-relativistic quantum mechanical system, its time evolution is described by a one-parameter group of automorphisms {α t } t of Q . This can be narrowed to unitary transformations: under certain weak technical conditions (see the article on quantum logic and the Varadarajan reference), there is a strongly continuous one-parameter group { U t } t of unitary transformations of the underlying Hilbert space such that the elements E of Q evolve according to the formula
The system time evolution can also be regarded dually as time evolution of the statistical state space. The evolution of the statistical state is given by a family of operators {β t } t such that Tr ( β t ( S ) E ) = Tr ( S α − t ( E ) ) = Tr ( S U t E U t ∗ ) = Tr ( U t ∗ S U t E ) . {\displaystyle \operatorname {Tr} (\beta _{t}(S)E)=\operatorname {Tr} (S\alpha _{-t}(E))=\operatorname {Tr} (SU_{t}EU_{t}^{*})=\operatorname {Tr} (U_{t}^{*}SU_{t}E).}
Clearly, for each value of t , S → U * t S U t is a quantum operation. Moreover, this operation is reversible .
This can be easily generalized: If G is a connected Lie group of symmetries of Q satisfying the same weak continuity conditions, then the action of any element g of G is given by a unitary operator U : g ⋅ E = U g E U g ∗ . {\displaystyle g\cdot E=U_{g}EU_{g}^{*}.} This mapping g → U g is known as a projective representation of G . The mappings S → U * g S U g are reversible quantum operations.
Quantum operations can be used to describe the process of quantum measurement . The presentation below describes measurement in terms of self-adjoint projections on a separable complex Hilbert space H , that is, in terms of a PVM ( Projection-valued measure ). In the general case, measurements can be made using non-orthogonal operators, via the notions of POVM . The non-orthogonal case is interesting, as it can improve the overall efficiency of the quantum instrument .
Quantum systems may be measured by applying a series of yes–no questions . This set of questions can be understood to be chosen from an orthocomplemented lattice Q of propositions in quantum logic . The lattice is equivalent to the space of self-adjoint projections on a separable complex Hilbert space H .
Consider a system in some state S , with the goal of determining whether it has some property E , where E is an element of the lattice of quantum yes-no questions. Measurement, in this context, means submitting the system to some procedure to determine whether the state satisfies the property. The reference to system state, in this discussion, can be given an operational meaning by considering a statistical ensemble of systems. Each measurement yields some definite value 0 or 1; moreover application of the measurement process to the ensemble results in a predictable change of the statistical state. This transformation of the statistical state is given by the quantum operation S ↦ E S E + ( I − E ) S ( I − E ) . {\displaystyle S\mapsto ESE+(I-E)S(I-E).} Here E can be understood to be a projection operator .
In the general case, measurements are made on observables taking on more than two values.
When an observable A has a pure point spectrum , it can be written in terms of an orthonormal basis of eigenvectors. That is, A has a spectral decomposition A = ∑ λ λ E A ( λ ) {\displaystyle A=\sum _{\lambda }\lambda \operatorname {E} _{A}(\lambda )} where E A (λ) is a family of pairwise orthogonal projections , each onto the respective eigenspace of A associated with the measurement value λ.
Measurement of the observable A yields an eigenvalue of A . Repeated measurements, made on a statistical ensemble S of systems, results in a probability distribution over the eigenvalue spectrum of A . It is a discrete probability distribution , and is given by Pr ( λ ) = Tr ( S E A ( λ ) ) . {\displaystyle \operatorname {Pr} (\lambda )=\operatorname {Tr} (S\operatorname {E} _{A}(\lambda )).}
Measurement of the statistical state S is given by the map S ↦ ∑ λ E A ( λ ) S E A ( λ ) . {\displaystyle S\mapsto \sum _{\lambda }\operatorname {E} _{A}(\lambda )S\operatorname {E} _{A}(\lambda )\ .} That is, immediately after measurement, the statistical state is a classical distribution over the eigenspaces associated with the possible values λ of the observable: S is a mixed state .
Shaji and Sudarshan argued in a Physical Review Letters paper that, upon close examination, complete positivity is not a requirement for a good representation of open quantum evolution. Their calculations show that, when starting with some fixed initial correlations between the observed system and the environment, the map restricted to the system itself is not necessarily even positive. However, it is not positive only for those states that do not satisfy the assumption about the form of initial correlations. Thus, they show that to get a full understanding of quantum evolution, non completely-positive maps should be considered as well. [ 4 ] [ 6 ] [ 7 ] | https://en.wikipedia.org/wiki/Quantum_operation |
In condensed matter physics , quantum oscillations describes a series of related experimental techniques used to map the Fermi surface of a metal in the presence of a strong magnetic field . [ 1 ] These techniques are based on the principle of Landau quantization of Fermions moving in a magnetic field. [ 2 ] For a gas of free fermions in a strong magnetic field, the energy levels are quantized into bands, called the Landau levels , whose separation is proportional to the strength of the magnetic field. In a quantum oscillation experiment, the external magnetic field is varied, which causes the Landau levels to pass over the Fermi surface, which in turn results in oscillations of the electronic density of states at the Fermi level ; this produces oscillations in the many material properties which depend on this, including resistance (the Shubnikov–de Haas effect ), Hall resistance , [ 2 ] and magnetic susceptibility (the de Haas–van Alphen effect ). Observation of quantum oscillations in a material is considered a signature of Fermi liquid behaviour. [ 3 ]
Quantum oscillations have been used to study high temperature superconducting materials such as cuprates and pnictides . [ 1 ] Studies using these experiments have shown that the ground state of underdoped cuprates behave similar to a Fermi liquid , and display characteristics such as Landau quasiparticles . [ 4 ]
In 2021 this technique has been used to observe a predicted state called "electron–phonon fluid", [ 5 ] [ 6 ] a similar particle-quasiparticle state already known is the exciton–polariton fluid .
When a magnetic field is applied to a system of free charged fermions , their energy states are quantized into the so-called Landau levels, given by [ 7 ]
ε l = e B m ∗ ( ℓ + 1 2 ) {\displaystyle \varepsilon _{l}={\frac {eB}{m^{*}}}\left(\ell +{\frac {1}{2}}\right)}
for integer-valued ℓ {\displaystyle \ell } , where B {\displaystyle B} is the external magnetic field and e , m ∗ {\displaystyle e,m^{*}} are the fermion charge and effective mass respectively.
When the external magnetic field B {\displaystyle B} is increased in an isolated system, the Landau levels expand, and eventually "fall off" the Fermi surface. This leads to oscillations in the observed energy of the highest occupied level, and hence in many physical properties (including Hall conductivity, resistivity, and susceptibility). The periodicity of these oscillations can be measured, and in turn can be used to determine the cross-sectional area of the Fermi surface. [ 8 ] If the axis of the magnetic field is varied at constant magnitude, similar oscillations are observed. The oscillations occur whenever the Landau orbits touch the Fermi surface. In this way, the complete geometry of the Fermi sphere can be mapped. [ 8 ]
Studies of underdoped cuprate compounds such as YBa 2 Cu 3 O 6+ x through probes such as ARPES have indicated that these phases show characteristics of non-Fermi liquids , [ 9 ] and in particular, the absence of well-defined Landau quasiparticles . [ 10 ] However, quantum oscillations have been observed in these materials at low temperatures, if their superconductivity is suppressed by a sufficiently high magnetic field, [ 2 ] which is evidence for the presence of well-defined quasiparticles with fermionic statistics . These experimental results thus disagree with those from ARPES and other probes. [ 7 ] | https://en.wikipedia.org/wiki/Quantum_oscillations |
Quantum paraelectricity [ 1 ] is a type of incipient ferroelectricity where the onset of ferroelectric order is suppressed by quantum fluctuations. [ 2 ] From the soft mode theory of ferroelectricity, [ 3 ] this occurs when a ferroelectric instability is stabilized by quantum fluctuations . In this case the soft-mode frequency never becomes unstable (Fig. 1a) as opposed to a regular ferroelectric.
Experimentally this is associated with an anomalous behaviour of the dielectric susceptibility, for example in SrTiO 3 . [ 4 ] In a normal ferroelectric, close to the onset of the phase transition the dielectric susceptibility diverges as the temperature approaches the Curie temperature . However, in the case of a quantum paraelectric the dielectric susceptibility diverges until it reaches a temperature low enough for quantum effects to cancel out the ferroelectricity (Fig. 1b). In the case of SrTiO 3 this is around 4K.
Other known quantum paraelectrics are KTaO 3 and potentially CaTiO 3 . [ 5 ] | https://en.wikipedia.org/wiki/Quantum_paraelectricity |
In physics , a quantum phase transition ( QPT ) is a phase transition between different quantum phases ( phases of matter at zero temperature ). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physical parameter—such as magnetic field or pressure—at absolute zero temperature. The transition describes an abrupt change in the ground state of a many-body system due to its quantum fluctuations. Such a quantum phase transition can be a second-order phase transition . [ 1 ] Quantum phase transitions can also be represented by the topological fermion condensation quantum phase transition, see e.g. strongly correlated quantum spin liquid . In case of three dimensional Fermi liquid , this transition transforms the Fermi surface into a Fermi volume. Such a transition can be a first-order phase transition , for it transforms two dimensional structure ( Fermi surface ) into three dimensional . As a result, the topological charge of Fermi liquid changes abruptly, since it takes only one of a discrete set of values.
To understand quantum phase transitions, it is useful to contrast them to classical phase transitions (CPT) (also called thermal phase transitions). [ 2 ] A CPT describes a cusp in the thermodynamic properties of a system. It signals a reorganization of the particles; A typical example is the freezing transition of water describing the transition between liquid and solid. The classical phase transitions are driven by a competition between the energy of a system and the entropy of its thermal fluctuations. A classical system does not have entropy at zero temperature and therefore no phase transition can occur. Their order is determined by the first discontinuous derivative of a thermodynamic potential.
A phase transition from water to ice, for example, involves latent heat (a discontinuity of the internal energy U {\displaystyle U} ) and is of first order. A phase transition from a ferromagnet to a paramagnet is continuous and is of second order. (See phase transition for Ehrenfest's classification of phase transitions by the derivative of free energy which is discontinuous at the transition). These continuous transitions from an ordered to a disordered phase are described by an order parameter, which is zero in the disordered and nonzero in the ordered phase. For the aforementioned ferromagnetic transition, the order parameter would represent the total magnetization of the system.
Although the thermodynamic average of the order parameter is zero in the disordered state, its fluctuations can be nonzero and become long-ranged in the vicinity of the critical point, where their typical length scale ξ (correlation length) and typical fluctuation decay time scale τ c (correlation time) diverge:
where
is defined as the relative deviation from the critical temperature T c . We call ν the ( correlation length ) critical exponent and z the dynamical critical exponent . Critical behavior of nonzero temperature phase transitions is fully described by classical thermodynamics ; quantum mechanics does not play any role even if the actual phases require a quantum mechanical description (e.g. superconductivity ).
Talking about quantum phase transitions means talking about transitions at T = 0: by tuning a non-temperature parameter like pressure, chemical composition or magnetic field, one could suppress e.g. some transition temperature like the Curie or Néel temperature to 0 K.
As a system in equilibrium at zero temperature is always in its lowest-energy state (or an equally weighted superposition if the lowest-energy is degenerate), a QPT cannot be explained by thermal fluctuations . Instead, quantum fluctuations , arising from Heisenberg's uncertainty principle , drive the loss of order characteristic of a QPT. The QPT occurs at the quantum critical point (QCP), where quantum fluctuations driving the transition diverge and become scale invariant in space and time.
Although absolute zero is not physically realizable, characteristics of the transition can be detected in the system's low-temperature behavior near the critical point. At nonzero temperatures, classical fluctuations with an energy scale of k B T compete with the quantum fluctuations of energy scale ħω. Here ω is the characteristic frequency of the quantum oscillation and is inversely proportional to the correlation time. Quantum fluctuations dominate the system's behavior in the region where ħω > k B T , known as the quantum critical region. This quantum critical behavior manifests itself in unconventional and unexpected physical behavior like novel non Fermi liquid phases. From a theoretical point of view, a phase diagram like the one shown on the right is expected: the QPT separates an ordered from a disordered phase (often, the low temperature disordered phase is referred to as 'quantum' disordered).
At high enough temperatures, the system is disordered and purely classical. Around the classical phase transition, the system is governed by classical thermal fluctuations (light blue area). This region becomes narrower with decreasing energies and converges towards the quantum critical point (QCP). Experimentally, the 'quantum critical' phase, which is still governed by quantum fluctuations, is the most interesting one. | https://en.wikipedia.org/wiki/Quantum_phase_transition |
Quantum phases are quantum states of matter at zero temperature. Even at zero temperature a quantum-mechanical system has quantum fluctuations and therefore can still support phase transitions. As a physical parameter is varied, quantum fluctuations can drive a phase transition into a different phase of matter. An example of a canonical quantum phase transition is the well-studied Superconductor Insulator Transition in disordered thin films which separates two quantum phases having different symmetries. Quantum magnets provide another example of QPT. The discovery of new quantum phases is a pursuit of many scientists. These phases of matter exhibit properties and symmetries which can potentially be exploited for technological purposes and the benefit of mankind.
The difference between these states and classical states of matter is that classically, materials exhibit different phases which ultimately depends on the change in temperature and/or density or some other macroscopic property of the material whereas quantum phases can change in response to a change in a different type of order parameter (which is instead a parameter in the Hamiltonian of the system, unlike the classical case) of the system at zero temperature – temperature does not have to change. The order parameter plays a role in quantum phases analogous to its role in classical phases. Some quantum phases are the result of a superposition of many other quantum phases.
This condensed matter physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_phases |
Quantum photoelectrochemistry is the investigation of the quantum mechanical nature of photoelectrochemistry , the subfield of study within physical chemistry concerned with the interaction of light with electrochemical systems, typically through the application of quantum chemical calculations . [ 1 ] Quantum photoelectrochemistry provides an expansion of quantum electrochemistry to processes involving also the interaction with light ( photons ). It therefore also includes essential elements of photochemistry . Key aspects of quantum photoelectrochemistry are calculations of optical excitations, photoinduced electron and energy transfer processes, excited state evolution, as well as interfacial charge separation and charge transport in nanoscale energy conversion systems. [ 2 ]
Quantum photoelectrochemistry in particular provides fundamental insight into basic light-harvesting and photoinduced electro-optical processes in several emerging solar energy conversion technologies for generation of both electricity ( photovoltaics ) and solar fuels. [ 3 ] Examples of such applications where quantum photoelectrochemistry provides insight into fundamental processes include photoelectrochemical cells , [ 4 ] [ 5 ] semiconductor photochemistry, [ 6 ] as well as light-driven electrocatalysis in general, and artificial photosynthesis in particular. [ 7 ]
Quantum photoelectrochemistry constitutes an active line of current research, with several publications appearing in recent years that relate to several different types of materials and processes, including light-harvesting complexes , [ 8 ] light-harvesting polymers , [ 9 ] as well as nanocrystalline semiconductor materials. [ 10 ] [ 11 ] | https://en.wikipedia.org/wiki/Quantum_photoelectrochemistry |
A quantum point contact ( QPC ) is a narrow constriction between two wide electrically conducting regions, of a width comparable to the electronic wavelength (nano- to micrometer). [ 1 ]
The importance of QPC lies in the fact that they prove quantisation of ballistic conductance in mesoscopic systems. The conductance of a QPC is quantized in units of 2 e 2 / h {\displaystyle 2e^{2}/h} , the so-called conductance quantum .
Quantum point contacts were first reported in 1988 by a Dutch team from Delft University of Technology and Philips Research [ 2 ] and, independently, by a British team from the Cavendish Laboratory . [ 3 ] They are based on earlier work by the British group which showed how split gates could be used to convert a two-dimensional electron gas into one-dimension, first in silicon [ 4 ] and then in gallium arsenide . [ 5 ] [ 6 ]
This quantisation is reminiscent of the quantisation of the Hall conductance , but is measured in the absence of a magnetic field. The zero-field conductance quantisation and the smooth transition to the quantum Hall effect on applying a magnetic field are essentially consequences of the equipartition of current among an integer number of propagating modes in the constriction.
There are several different ways of fabricating a quantum point contact. It can be realized in a break-junction by pulling apart a piece of conductor until it breaks. The breaking point forms the point contact. In a more controlled way, quantum point contacts are formed in a two-dimensional electron gas (2DEG), e.g. in GaAs / AlGaAs heterostructures . By applying a voltage to suitably shaped gate electrodes, the electron gas can be locally depleted and many different types of conducting regions can be created in the plane of the 2DEG, among them quantum dots and quantum point contacts. Another means of creating a QPC is by positioning the tip of a scanning tunneling microscope close to the surface of a conductor.
Geometrically, a quantum point contact is a constriction in the transverse direction which presents a resistance to the motion of electrons . Applying a voltage V {\displaystyle V} across the point contact induces a current to flow, the magnitude of this current is given by I = G V {\displaystyle I=GV} , where G {\displaystyle G} is the conductance of the contact. This formula resembles Ohm's law for macroscopic resistors. However, there is a fundamental difference here resulting from the small system size which requires a quantum mechanical analysis. [ 7 ]
It is most common to study QPC in two dimensional electron gases. This way the geometric constriction of the point contact turns the conductance through the opening to a one dimensional system. Moreover, it requires a quantum mechanical description of the system that results in the quantisation of conductance. Quantum mechanically, the current through the point contact is equipartitioned among the 1D subands, or transverse modes, in the constriction.
It is important to state that the previous discussion does not take into account possible transitions among modes. The Landauer formula can actually be generalized to express this possible transitions
G = 2 e 2 h ∑ n , m | T n , m | 2 {\displaystyle G={2e^{2} \over h}\sum _{n,m}|T_{n,m}|^{2}} ,
where T n , m {\displaystyle T_{n,m}} is the transition matrix which incorporates non-zero probabilities of transmission from mode n to m .
At low temperatures and voltages, unscattered and untrapped electrons contributing to the current have a certain energy/momentum/wavelength called Fermi energy /momentum/wavelength. Much like in a waveguide , the transverse confinement in the quantum point contact results in a "quantization" of the transverse motion—the transverse motion cannot vary continuously, but has to be one of a series of discrete modes. The waveguide analogy is applicable as long as coherence is not lost through scattering, e.g., by a defect or trapping site. The electron wave can only pass through the constriction if it interferes constructively, which for a given width of constriction, only happens for a certain number of modes N {\displaystyle N} . The current carried by such a quantum state is the product of the velocity times the electron density. These two quantities by themselves differ from one mode to the other, but their product is mode independent. As a consequence, each state contributes the same amount of e 2 / h {\displaystyle e^{2}/h} per spin direction to the total conductance G = N G 0 {\displaystyle G=NG_{0}} .
This is a fundamental result; the conductance does not take on arbitrary values but is quantized in multiples of the conductance quantum G 0 = 2 e 2 / h {\displaystyle G_{0}=2e^{2}/h} , which is expressed through the electron charge e {\displaystyle e} and the Planck constant h {\displaystyle h} . The integer number N {\displaystyle N} is determined by the width of the point contact and roughly equals the width divided by half the electron wavelength . As a function of the width of the point contact (or gate voltage in the case of GaAs/AlGaAs heterostructure devices), the conductance shows a staircase behavior as more and more modes (or channels) contribute to the electron transport. The step-height is given by G Q {\displaystyle G_{Q}} .
On increasing the temperature, one finds experimentally that the plateaux acquire a finite slope until they are no longer resolved. This is a consequence of the thermal smearing of the Fermi-Dirac distribution . The conductance steps should disappear for T ≲ Δ E / 4 k B ≈ 4 K {\displaystyle T\lesssim \Delta E/4k_{\rm {B}}\approx 4\;\mathrm {K} } (here ∆ E is the subband splitting at the Fermi level ). This is confirmed both by experiment and by numerical calculations. [ 9 ]
An external magnetic field applied to the quantum point contact lifts the spin degeneracy and leads to half-integer steps in the conductance. In addition, the number N {\displaystyle N} of modes that contribute becomes smaller. For large magnetic fields, N {\displaystyle N} is independent of the width of the constriction, given by the theory of the quantum Hall effect .
Anomalous features on the quantized conductance steps are often observed in transport measurements of quantum point contacts. A notable example is the plateau at 0.7 G Q {\displaystyle 0.7G_{Q}} , the so-called 0.7-structure, arising due to enhanced electron-electron interactions arising from a smeared van Hove singularity in the local 1D density of states in the vicinity of the charge constriction. [ 10 ] Unlike the conductance steps, the 0.7-structure becomes more pronounced at higher temperature. 0.7-structure analogues are sometimes observed on higher conductance steps. Quasi-bound states arising from impurities, charge traps, and reflections within the constriction may also result in conductance structure close to the 1D limit.
Apart from studying fundamentals of charge transport in mesoscopic conductors, quantum point contacts can be used as extremely sensitive charge detectors. Since the conductance through the contact strongly depends on the size of the constriction, any potential fluctuation (for instance, created by other electrons) in the vicinity will influence the current through the QPC. It is possible to detect single electrons with such a scheme. In view of quantum computation in solid-state systems, QPCs can be used as readout devices for the state of a quantum bit (qubit). [ 11 ] [ 12 ] [ 13 ] [ 14 ] In device physics, the configuration of QPCs is used for demonstrating a fully ballistic field-effect transistor. [ 15 ] Another application of the device is its use as a switch. A nickel wire is brought close enough to a gold surface and then, by the use of a piezoelectric actuator, the distance between the wire and the surface can be changed and thus, the transport characteristics of the device change between electron tunneling and ballistic. [ 16 ] | https://en.wikipedia.org/wiki/Quantum_point_contact |
The quantum potential or quantum potentiality is a central concept of the de Broglie–Bohm formulation of quantum mechanics , introduced by David Bohm in 1952.
Initially presented under the name quantum-mechanical potential , subsequently quantum potential , it was later elaborated upon by Bohm and Basil Hiley in its interpretation as an information potential which acts on a quantum particle. It is also referred to as quantum potential energy , Bohm potential , quantum Bohm potential or Bohm quantum potential .
In the framework of the de Broglie–Bohm theory, the quantum potential is a term within the Schrödinger equation which acts to guide the movement of quantum particles. The quantum potential approach introduced by Bohm [ 1 ] [ 2 ] provides a physically less fundamental exposition of the idea presented by Louis de Broglie : de Broglie had postulated in 1925 that the relativistic wave function defined on spacetime represents a pilot wave which guides a quantum particle, represented as an oscillating peak in the wave field, but he had subsequently abandoned his approach because he was unable to derive the guidance equation for the particle from a non-linear wave equation. The seminal articles of Bohm in 1952 introduced the quantum potential and included answers to the objections which had been raised against the pilot wave theory.
The Bohm quantum potential is closely linked with the results of other approaches, in particular relating to works of Erwin Madelung in 1927 [ 3 ] and Carl Friedrich von Weizsäcker in 1935. [ 4 ]
Building on the interpretation of the quantum theory introduced by Bohm in 1952, David Bohm and Basil Hiley in 1975 presented how the concept of a quantum potential leads to the notion of an "unbroken wholeness of the entire universe", proposing that the fundamental new quality introduced by quantum physics is nonlocality . [ 5 ]
The Schrödinger equation
is re-written using the polar form for the wave function ψ = R exp ( i S / ℏ ) {\displaystyle \psi =R\exp(iS/\hbar )} with real-valued functions R {\displaystyle R} and S {\displaystyle S} , where R {\displaystyle R} is the amplitude ( absolute value ) of the wave function ψ {\displaystyle \psi } , and S / ℏ {\displaystyle S/\hbar } its phase. This yields two equations: from the imaginary and real part of the Schrödinger equation follow the continuity equation and the quantum Hamilton–Jacobi equation respectively. [ 1 ] [ 6 ]
The imaginary part of the Schrödinger equation in polar form yields
which, provided ρ = R 2 {\displaystyle \rho =R^{2}} , can be interpreted as the continuity equation ∂ ρ / ∂ t + ∇ ⋅ ( ρ v ) = 0 {\displaystyle \partial \rho /\partial t+\nabla \cdot (\rho v)=0} for the probability density ρ {\displaystyle \rho } and the velocity field v = 1 m ∇ S {\displaystyle v={\frac {1}{m}}\nabla S}
The real part of the Schrödinger equation in polar form yields a modified Hamilton–Jacobi equation
− ∂ S ∂ t = ‖ ∇ S ‖ 2 2 m + V + Q {\displaystyle -{\frac {\partial S}{\partial t}}={\frac {\|\nabla S\|^{2}}{2m}}+V+Q}
also referred to as quantum Hamilton–Jacobi equation . [ 7 ] It differs from the classical Hamilton–Jacobi equation only by the term
Q = − ℏ 2 2 m ∇ 2 R R . {\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}.}
This term Q {\displaystyle Q} , called quantum potential , thus depends on the curvature of the amplitude of the wave function. [ 8 ] [ 9 ]
In the limit ℏ → 0 {\displaystyle \hbar \to 0} , the function S {\displaystyle S} is a solution of the (classical) Hamilton–Jacobi equation; [ 1 ] therefore, the function S {\displaystyle S} is also called the Hamilton–Jacobi function, or action , extended to quantum physics.
Hiley emphasised several aspects [ 10 ] that regard the quantum potential of a quantum particle:
In 1979, Hiley and his co-workers Philippidis and Dewdney presented a full calculation on the explanation of the two-slit experiment in terms of Bohmian trajectories that arise for each particle moving under the influence of the quantum potential, resulting in the well-known interference patterns . [ 13 ]
Also the shift of the interference pattern which occurs in presence of a magnetic field in the Aharonov–Bohm effect could be explained as arising from the quantum potential. [ 14 ]
The collapse of the wave function of the Copenhagen interpretation of quantum theory is explained in the quantum potential approach by the demonstration that, after a measurement, "all the packets of the multi-dimensional wave function that do not correspond to the actual result of measurement have no effect on the particle" from then on. [ 15 ] Bohm and Hiley pointed out that
...the quantum potential can develop unstable bifurcation points, which separate classes of particle trajectories according to the "channels" into which they eventually enter and within which they stay. This explains how measurement is possible without "collapse" of the wave function, and how all sorts of quantum processes, such as transitions between states, fusion of two states into one and fission of one system into two, are able to take place without the need for a human observer. [ 16 ]
Measurement then "involves a participatory transformation in which both the system under observation and the observing apparatus undergo a mutual participation so that the trajectories behave in a correlated manner, becoming correlated and separated into different, non-overlapping sets (which we call 'channels')". [ 17 ]
The Schrödinger wave function of a many-particle quantum system cannot be represented in ordinary three-dimensional space . Rather, it is represented in configuration space , with three dimensions per particle. A single point in configuration space thus represents the configuration of the entire n-particle system as a whole.
A two-particle wave function ψ ( r 1 , r 2 , t ) {\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)} of identical particles of mass m {\displaystyle m} has the quantum potential [ 18 ]
where ∇ 1 2 {\displaystyle \nabla _{1}^{2}} and ∇ 2 2 {\displaystyle \nabla _{2}^{2}} refer to particle 1 and particle 2 respectively. This expression generalizes in straightforward manner to n {\displaystyle n} particles:
In case the wave function of two or more particles is separable, then the system's total quantum potential becomes the sum of the quantum potentials of the two particles. Exact separability is extremely unphysical given that interactions between the system and its environment destroy the factorization; however, a wave function that is a superposition of several wave functions of approximately disjoint support will factorize approximately. [ 19 ]
That the wave function is separable means that ψ {\displaystyle \psi } factorizes in the form ψ ( r 1 , r 2 , t ) = ψ A ( r 1 , t ) ψ B ( r 2 , t ) {\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)} . Then it follows that also R {\displaystyle R} factorizes, and the system's total quantum potential becomes the sum of the quantum potentials of the two particles. [ 20 ]
In case the wave function is separable, that is, if ψ {\displaystyle \psi } factorizes in the form ψ ( r 1 , r 2 , t ) = ψ A ( r 1 , t ) ψ B ( r 2 , t ) {\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)} , the two one-particle systems behave independently. More generally, the quantum potential of an n {\displaystyle n} -particle system with separable wave function is the sum of n {\displaystyle n} quantum potentials, separating the system into n {\displaystyle n} independent one-particle systems. [ 21 ]
Bohm, as well as other physicists after him, have sought to provide evidence that the Born rule linking R {\displaystyle R} to the probability density function
can be understood, in a pilot wave formulation, as not representing a basic law, but rather a theorem (called quantum equilibrium hypothesis ) which applies when a quantum equilibrium is reached during the course of the time development under the Schrödinger equation. With Born's rule, and straightforward application of the chain and product rules
the quantum potential, expressed in terms of the probability density function, becomes: [ 22 ]
The quantum force F Q = − ∇ Q {\displaystyle F_{Q}=-\nabla Q} , expressed in terms of the probability distribution, amounts to: [ 23 ]
M. R. Brown and B. Hiley showed that, as alternative to its formulation terms of configuration space ( x {\displaystyle x} -space), the quantum potential can also be formulated in terms of momentum space ( p {\displaystyle p} -space). [ 24 ] [ 25 ]
In line with David Bohm's approach, Basil Hiley and mathematician Maurice de Gosson showed that the quantum potential can be seen as a consequence of a projection of an underlying structure, more specifically of a non-commutative algebraic structure, onto a subspace such as ordinary space ( x {\displaystyle x} -space). In algebraic terms, the quantum potential can be seen as arising from the relation between implicate and explicate orders : if a non-commutative algebra is employed to describe the non-commutative structure of the quantum formalism, it turns out that it is impossible to define an underlying space, but that rather " shadow spaces " (homomorphic spaces) can be constructed and that in so doing the quantum potential appears. [ 25 ] [ 26 ] [ 27 ] [ 28 ] [ 29 ] The quantum potential approach can be seen as a way to construct the shadow spaces. [ 27 ] The quantum potential thus results as a distortion due to the projection of the underlying space into x {\displaystyle x} -space, in similar manner as a Mercator projection inevitably results in a distortion in a geographical map. [ 30 ] [ 31 ] There exists complete symmetry between the x {\displaystyle x} -representation, and the quantum potential as it appears in configuration space can be seen as arising from the dispersion of the momentum p {\displaystyle p} -representation. [ 32 ]
The approach has been applied to extended phase space , [ 32 ] [ 33 ] also in terms of a Duffin–Kemmer–Petiau algebra approach. [ 34 ] [ 35 ]
It can be shown [ 36 ] that the mean value of the quantum potential Q = − ℏ 2 ∇ 2 ρ / ( 2 m ρ ) {\displaystyle Q=-\hbar ^{2}\nabla ^{2}{\sqrt {\rho }}/(2m{\sqrt {\rho }})} is proportional to the probability density's Fisher information about the observable x ^ {\displaystyle {\hat {x}}}
Using this definition for the Fisher information, we can write: [ 37 ]
Giovanni Salesi, Erasmo Recami and co-workers showed in 1998 that, in agreement with the König's theorem , the quantum potential can be identified with the kinetic energy of the internal motion (" zitterbewegung ") associated with the spin of a spin-1/2 particle observed in a center-of-mass frame. More specifically, they showed that the internal zitterbewegung velocity for a spinning, non-relativistic particle of constant spin with no precession, and in absence of an external field, has the squared value: [ 38 ]
from which the second term is shown to be of negligible size; then with | s | = ℏ / 2 {\displaystyle |\mathbf {s} |=\hbar /2} it follows that
Salesi gave further details on this work in 2009. [ 39 ]
In 1999, Salvatore Esposito generalized their result from spin-1/2 particles to particles of arbitrary spin, confirming the interpretation of the quantum potential as a kinetic energy for an internal motion. Esposito showed that (using the notation ℏ {\displaystyle \hbar } =1) the quantum potential can be written as: [ 40 ]
and that the causal interpretation of quantum mechanics can be reformulated in terms of a particle velocity
where the "drift velocity" is
and the "relative velocity" is v S × s {\displaystyle \mathbf {v} _{S}\times \mathbf {s} } , with
and s {\displaystyle \mathbf {s} } representing the spin direction of the particle. In this formulation, according to Esposito, quantum mechanics must necessarily be interpreted in probabilistic terms, for the reason that a system's initial motion condition cannot be exactly determined. [ 40 ] Esposito explained that "the quantum effects present in the Schrödinger equation are due to the presence of a peculiar spatial direction associated with the particle that, assuming the isotropy of space, can be identified with the spin of the particle itself". [ 41 ] Esposito generalized it from matter particles to gauge particles , in particular photons , for which he showed that, if modelled as ψ = ( E − i B ) / 2 {\displaystyle \psi =(\mathbf {E} -i\mathbf {B} )/{\sqrt {2}}} , with probability function ψ ∗ ⋅ ψ = ( E 2 + B 2 ) / 2 {\displaystyle \psi ^{*}\cdot \psi =(\mathbf {E} ^{2}+\mathbf {B} ^{2})/2} , they can be understood in a quantum potential approach. [ 42 ]
James R. Bogan, in 2002, published the derivation of a reciprocal transformation from the Hamilton-Jacobi equation of classical mechanics to the time-dependent Schrödinger equation of quantum mechanics which arises from a gauge transformation representing spin, under the simple requirement of conservation of probability . This spin-dependent transformation is a function of the quantum potential. [ 43 ]
B. Hiley and R. E. Callaghan re-interpret the role of the Bohm model and its notion of quantum potential in the framework of Clifford algebra , taking account of recent advances that include the work of David Hestenes on spacetime algebra . They show how, within a nested hierarchy of Clifford algebras C ℓ i , j {\displaystyle C\ell _{i,j}} , for each Clifford algebra an element of a minimal left ideal Φ L ( r , t ) {\displaystyle \Phi _{L}(\mathbf {r} ,t)} and an element of a right ideal representing its Clifford conjugation Φ R ( r , t ) = Φ ~ L ( r , t ) {\displaystyle \Phi _{R}(\mathbf {r} ,t)={\tilde {\Phi }}_{L}(\mathbf {r} ,t)} can be constructed, and from it the Clifford density element (CDE) ρ c ( r , t ) = Φ L ( r , t ) Φ ~ L ( r , t ) {\displaystyle \rho _{c}(\mathbf {r} ,t)=\Phi _{L}(\mathbf {r} ,t){\tilde {\Phi }}_{L}(\mathbf {r} ,t)} , an element of the Clifford algebra which is isomorphic to the standard density matrix but independent of any specific representation. [ 44 ] On this basis, bilinear invariants can be formed which represent properties of the system. Hiley and Callaghan distinguish bilinear invariants of a first kind, of which each stands for the expectation value of an element B {\displaystyle B} of the algebra which can be formed as T r B ρ c {\displaystyle {\rm {Tr}}B\rho _{c}} , and bilinear invariants of a second kind which are constructed with derivatives and represent momentum and energy. Using these terms, they reconstruct the results of quantum mechanics without depending on a particular representation in terms of a wave function nor requiring reference to an external Hilbert space. Consistent with earlier results, the quantum potential of a non-relativistic particle with spin ( Pauli particle ) is shown to have an additional spin-dependent term, and the momentum of a relativistic particle with spin ( Dirac particle ) is shown to consist in a linear motion and a rotational part. [ 45 ] The two dynamical equations governing the time evolution are re-interpreted as conservation equations. One of them stands for the conservation of energy ; the other stands for the conservation of probability and of spin . [ 46 ] The quantum potential plays the role of an internal energy [ 47 ] which ensures the conservation of total energy. [ 46 ]
Bohm and Hiley demonstrated that the non-locality of quantum theory can be understood as limit case of a purely local theory, provided the transmission of active information is allowed to be greater than the speed of light, and that this limit case yields approximations to both quantum theory and relativity. [ 48 ]
The quantum potential approach was extended by Hiley and co-workers to quantum field theory in Minkowski spacetime [ 49 ] [ 50 ] [ 51 ] [ 52 ] and to curved spacetime. [ 53 ]
Carlo Castro and Jorge Mahecha derived the Schrödinger equation from the Hamilton-Jacobi equation in conjunction with the continuity equation, and showed that the properties of the relativistic Bohm quantum potential in terms of the ensemble density can be described by the Weyl properties of space. In Riemann flat space, the Bohm potential is shown to equal the Weyl curvature . According to Castro and Mahecha, in the relativistic case , the quantum potential (using the d'Alembert operator ◻ {\displaystyle \scriptstyle \Box } and in the notation ℏ = 1 {\displaystyle \hbar =1} ) takes the form
and the quantum force exerted by the relativistic quantum potential is shown to depend on the Weyl gauge potential and its derivatives. Furthermore, the relationship among Bohm's potential and the Weyl curvature in flat spacetime corresponds to a similar relationship among Fisher Information and Weyl geometry after introduction of a complex momentum. [ 54 ]
Diego L. Rapoport, on the other hand, associates the relativistic quantum potential with the metric scalar curvature (Riemann curvature). [ 55 ]
In relation to the Klein–Gordon equation for a particle with mass and charge, Peter R. Holland spoke in his book of 1993 of a "quantum potential-like term" that is proportional ◻ R / R {\displaystyle \Box R/R} . He emphasized however that to give the Klein–Gordon theory a single-particle interpretation in terms of trajectories, as can be done for nonrelativistic Schrödinger quantum mechanics, would lead to unacceptable inconsistencies. For instance, wave functions ψ ( x , t ) {\displaystyle \psi (\mathbf {x} ,t)} that are solutions to the Klein–Gordon or the Dirac equation cannot be interpreted as the probability amplitude for a particle to be found in a given volume d 3 x {\displaystyle d^{3}x} at time t {\displaystyle t} in accordance with the usual axioms of quantum mechanics, and similarly in the causal interpretation it cannot be interpreted as the probability for the particle to be in that volume at that time. Holland pointed out that, while efforts have been made to determine a Hermitian position operator that would allow an interpretation of configuration space quantum field theory, in particular using the Newton–Wigner localization approach, but that no connection with possibilities for an empirical determination of position in terms of a relativistic measurement theory or for a trajectory interpretation has so far been established. Yet according to Holland this does not mean that the trajectory concept is to be discarded from considerations of relativistic quantum mechanics. [ 56 ]
Hrvoje Nikolić derived Q = − ( 1 / 2 m ) ◻ R / R {\displaystyle Q=-(1/2m)\,\Box R/R} as expression for the quantum potential, and he proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wave functions. [ 57 ] He also developed a generalized relativistic-invariant probabilistic interpretation of quantum theory, [ 58 ] [ 59 ] [ 60 ] in which | ψ | 2 {\displaystyle |\psi |^{2}} is no longer a probability density in space but a probability density in space-time. [ 61 ] [ 62 ]
Starting from the space representation of the field coordinate, a causal interpretation of the Schrödinger picture of relativistic quantum theory has been constructed. The Schrödinger picture for a neutral, spin 0, massless field Ψ [ ψ ( x , t ) ] = R [ ψ ( x , t ) ] e S [ ψ ( x , t ) ] {\displaystyle \Psi \left[\psi (\mathbf {x} ,t)\right]=R\left[\psi (\mathbf {x} ,t)\right]e^{S\left[\psi (\mathbf {x} ,t)\right]}} , with R [ ψ ( x , t ) ] , S [ ψ ( x , t ) ] {\displaystyle R\left[\psi (\mathbf {x} ,t)\right],S\left[\psi (\mathbf {x} ,t)\right]} real-valued functionals , can be shown [ 63 ] to lead to
This has been called the superquantum potential by Bohm and his co-workers. [ 64 ]
Basil Hiley showed that the energy–momentum-relations in the Bohm model can be obtained directly from the energy–momentum tensor of quantum field theory and that the quantum potential is an energy term that is required for local energy–momentum conservation. [ 65 ] He has also hinted that for particle with energies equal to or higher than the pair creation threshold, Bohm's model constitutes a many-particle theory that describes also pair creation and annihilation processes. [ 66 ]
In his article of 1952, providing an alternative interpretation of quantum mechanics , Bohm already spoke of a "quantum-mechanical" potential. [ 67 ]
Bohm and Basil Hiley also called the quantum potential an information potential , given that it influences the form of processes and is itself shaped by the environment. [ 12 ] Bohm indicated "The ship or aeroplane (with its automatic Pilot) is a self-active system, i.e. it has its own energy. But the form of its activity is determined by the information content concerning its environment that is carried by the radar waves. This is independent of the intensity of the waves. We can similarly regard the quantum potential as containing active information . It is potentially active everywhere, but actually active only where and when there is a particle." (italics in original). [ 68 ]
Hiley refers to the quantum potential as internal energy [ 27 ] and as "a new quality of energy only playing a role in quantum processes". [ 69 ] He explains that the quantum potential is a further energy term aside the well-known kinetic energy and the (classical) potential energy and that it is a nonlocal energy term that arises necessarily in view of the requirement of energy conservation; he added that much of the physics community's resistance against the notion of the quantum potential may have been due to scientists' expectations that energy should be local. [ 70 ]
Hiley has emphasized that the quantum potential, for Bohm, was "a key element in gaining insights into what could underlie the quantum formalism. Bohm was convinced by his deeper analysis of this aspect of the approach that the theory could not be mechanical. Rather, it is organic in the sense of Whitehead . Namely, that it was the whole that determined the properties of the individual particles and their relationship, not the other way round." [ 71 ] [ 72 ]
Peter R. Holland , in his comprehensive textbook, also refers to it as quantum potential energy . [ 73 ] The quantum potential is also referred to in association with Bohm's name as Bohm potential , quantum Bohm potential or Bohm quantum potential .
The quantum potential approach can be used to model quantum effects without requiring the Schrödinger equation to be explicitly solved, and it can be integrated in simulations, such as Monte Carlo simulations using the hydrodynamic and drift diffusion equations . [ 74 ] This is done in form of a "hydrodynamic" calculation of trajectories: starting from the density at each "fluid element", the acceleration of each "fluid element" is computed from the gradient of V {\displaystyle V} and Q {\displaystyle Q} , and the resulting divergence of the velocity field determines the change to the density. [ 75 ]
The approach using Bohmian trajectories and the quantum potential is used for calculating properties of quantum systems which cannot be solved exactly, which are often approximated using semi-classical approaches. Whereas in mean field approaches the potential for the classical motion results from an average over wave functions, this approach does not require the computation of an integral over wave functions. [ 76 ]
The expression for the quantum force has been used, together with Bayesian statistical analysis and expectation-maximisation methods, for computing ensembles of trajectories that arise under the influence of classical and quantum forces. [ 23 ] | https://en.wikipedia.org/wiki/Quantum_potential |
In quantum mechanics , a quantum process is a somewhat ambiguous term which usually refers to the time evolution of an ( open ) quantum system. Under very general assumptions, a quantum process is described by the quantum operation formalism (also known as a quantum dynamical map ), which is a linear , trace-preserving, and completely positive map from the set of density matrices to itself.
For instance, in quantum process tomography , the unknown quantum process is assumed to be a quantum operation.
However, not all quantum processes can be captured within the quantum operation formalism; [ 1 ] [ 2 ] in principle, the density matrix of a quantum system can undergo completely arbitrary time evolution.
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_process |
Quantum pseudo-telepathy describes the use of quantum entanglement to eliminate the need for classical communications. [ 1 ] [ 2 ] A nonlocal game is said to display quantum pseudo-telepathy if players who can use entanglement can win it with certainty while players without it can not. The prefix pseudo refers to the fact that quantum pseudo-telepathy does not involve the exchange of information between any parties. Instead, quantum pseudo-telepathy removes the need for parties to exchange information in some circumstances.
Quantum pseudo-telepathy is generally used as a thought experiment to demonstrate the non-local characteristics of quantum mechanics . However, quantum pseudo-telepathy is a real-world phenomenon which can be verified experimentally. It is thus an especially striking example of an experimental confirmation of Bell inequality violations.
A simple magic square game demonstrating nonclassical correlations was introduced by P. K. Aravind [ 3 ] based on a series of papers by N. David Mermin [ 4 ] [ 5 ] and Asher Peres [ 6 ] and Adán Cabello [ es ] [ 7 ] [ 8 ] that developed simplifying demonstrations of Bell's theorem . The game has been reformulated to demonstrate quantum pseudo-telepathy. [ 9 ]
This is a cooperative game featuring two players, Alice and Bob , and a referee. The referee asks Alice to fill in one row, and Bob one column, of a 3×3 table with plus and minus signs. Their answers must respect the following constraints: Alice's row must contain an even number of minus signs, Bob's column must contain an odd number of minus signs, and they both must assign the same sign to the cell where the row and column intersects. If they manage they win, otherwise they lose.
Alice and Bob are allowed to elaborate a strategy together, but crucially are not allowed to communicate after they know which row and column they will need to fill in (as otherwise the game would be trivial).
It is easy to see that if Alice and Bob can come up with a classical strategy where they always win, they can represent it as a 3×3 table encoding their answers. But this is not possible, as the number of minus signs in this hypothetical table would need to be even and odd at the same time: every row must contain an even number of minus signs, making the total number of minus signs even, and every column must contain an odd number of minus signs, making the total number of minus signs odd.
With a bit further analysis one can see that the best possible classical strategy can be represented by a table where each cell now contains both Alice and Bob's answers, that may differ. It is possible to make their answers equal in 8 out of 9 cells, while respecting the parity of Alice's rows and Bob's columns. This implies that if the referee asks for a row and column whose intersection is one of the cells where their answers match they win, and otherwise they lose. Under the usual assumption that the referee asks for them uniformly at random, the best classical winning probability is 8/9.
Use of quantum pseudo-telepathy would enable Alice and Bob to win the game 100% of the time without any communication once the game has begun.
This requires Alice and Bob to possess two pairs of particles with entangled states. These particles must have been prepared before the start of the game. One particle of each pair is held by Alice and the other by Bob, so they each have two particles. When Alice and Bob learn which column and row they must fill, each uses that information to select which measurements they should make to their particles. The result of the measurements will appear to each of them to be random (and the observed partial probability distribution of either particle will be independent of the measurement performed by the other party), so no real "communication" takes place. [ citation needed ]
However, the process of measuring the particles imposes sufficient structure on the joint probability distribution of the results of the measurement such that if Alice and Bob choose their actions based on the results of their measurement, then there will exist a set of strategies and measurements allowing the game to be won with probability 1.
Note that Alice and Bob could be light years apart from one another, and the entangled particles will still enable them to coordinate their actions sufficiently well to win the game with certainty.
Each round of this game uses up one entangled state. Playing N rounds requires that N entangled states (2N independent Bell pairs, see below) be shared in advance. This is because each round needs 2-bits of information to be measured (the third entry is determined by the first two, so measuring it isn't necessary), which destroys the entanglement. There is no way to reuse old measurements from earlier games.
The trick is for Alice and Bob to share an entangled quantum state and to use specific measurements on their components of the entangled state to derive the table entries. A suitable correlated state consists of a pair of entangled Bell states :
here | + ⟩ {\displaystyle \left|+\right\rangle } and | − ⟩ {\displaystyle \left|-\right\rangle } are eigenstates of the Pauli operator S x with eigenvalues +1 and −1, respectively, whilst the subscripts a, b, c, and d identify the components of each Bell state, with a and c going to Alice, and b and d going to Bob. The symbol ⊗ {\displaystyle \otimes } represents a tensor product .
Observables for these components can be written as products of the Pauli matrices :
Products of these Pauli spin operators can be used to fill the 3×3 table such that each row and each column contains a mutually commuting set of observables with eigenvalues +1 and −1, and with the product of the observables in each row being the identity operator, and the product of observables in each column equating to minus the identity operator. This is a so-called Mermin–Peres magic square. It is shown in below table.
Effectively, while it is not possible to construct a 3×3 table with entries +1 and −1 such that the product of the elements in each row equals +1 and the product of elements in each column equals −1, it is possible to do so with the richer algebraic structure based on spin matrices.
The play proceeds by having each player make one measurement on their part of the entangled state per round of play. Each of Alice's measurements will give her the values for a row, and each of Bob's measurements will give him the values for a column. It is possible to do that because all observables in a given row or column commute, so there exists a basis in which they can be measured simultaneously. For Alice's first row she needs to measure both her particles in the Z {\displaystyle Z} basis, for the second row she needs to measure them in the X {\displaystyle X} basis, and for the third row she needs to measure them in an entangled basis. For Bob's first column he needs to measure his first particle in the X {\displaystyle X} basis and the second in the Z {\displaystyle Z} basis, for second column he needs to measure his first particle in the Z {\displaystyle Z} basis and the second in the X {\displaystyle X} basis, and for his third column he needs to measure both his particles in a different entangled basis, the Bell basis . As long as the table above is used, the measurement results are guaranteed to always multiply out to +1 for Alice along her row, and −1 for Bob down his column. Of course, each completely new round requires a new entangled state, as different rows and columns are not compatible with each other.
It has been demonstrated that the above-described game is the simplest two-player game of its type in which quantum pseudo-telepathy allows a win with probability one. [ 10 ] Other games in which quantum pseudo-telepathy occurs have been studied, including larger magic square games, [ 11 ] graph colouring games [ 12 ] giving rise to the notion of quantum chromatic number , [ 13 ] and multiplayer games involving more than two participants. [ 14 ]
In July 2022 a study reported the experimental demonstration of quantum pseudotelepathy via playing the nonlocal version of Mermin-Peres magic square game. [ 15 ]
The Greenberger–Horne–Zeilinger (GHZ) game [ 16 ] [ 17 ] is another example of quantum pseudo-telepathy. Classically, the game has 0.75 winning probability. However, with a quantum strategy, the players can achieve a winning probability of 1, meaning they always win. [ 18 ]
In the game there are three players, Alice, Bob, and Carol playing against a referee. The referee poses a binary question to each player (either 0 {\displaystyle 0} or 1 {\displaystyle 1} ). The three players each respond with an answer again in the form of either 0 {\displaystyle 0} or 1 {\displaystyle 1} . Therefore, when the game is played the three questions of the referee x, y, z are drawn from the 4 options { ( 0 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 1 , 0 , 1 ) , ( 0 , 1 , 1 ) } {\displaystyle \{(0,0,0),(1,1,0),(1,0,1),(0,1,1)\}} . For example, if question triple ( 0 , 1 , 1 ) {\displaystyle (0,1,1)} is chosen, then Alice receives bit 0, Bob receives bit 1, and Carol receives bit 1 from the referee. Based on the question bit received, Alice, Bob, and Carol each respond with an answer a, b, c , also in the form of 0 or 1. The players can formulate a strategy together prior to the start of the game. However, no communication is allowed during the game itself.
The players win if m o d 2 ( a + b + c ) = x ∨ y ∨ z {\displaystyle \mathrm {mod} _{2}(a+b+c)=x\lor y\lor z} , where ∨ {\displaystyle \lor } indicates OR condition and m o d 2 {\displaystyle mod_{2}} indicates summation of answers modulo 2. In other words, the sum of three answers has to be even if x = y = z = 0 {\displaystyle x=y=z=0} . Otherwise, the sum of answers has to be odd.
Classically, Alice, Bob, and Carol can employ a deterministic strategy that always end up with odd sum (e.g. Alice always output 1. Bob and Carol always output 0). The players win 75% of the time and only lose if the questions are ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} .
This is the best classical strategy: only 3 out of 4 winning conditions can be satisfied simultaneously. Let a 0 , a 1 {\displaystyle a_{0},a_{1}} be Alice's response to question 0 and 1 respectively, b 0 , b 1 {\displaystyle b_{0},b_{1}} be Bob's response to question 0, 1, and c 0 , c 1 {\displaystyle c_{0},c_{1}} be Carol's response to question 0, 1. We can write all constraints that satisfy winning conditions as a 0 + b 0 + c 0 = 0 mod 2 a 1 + b 1 + c 0 = 1 mod 2 a 1 + b 0 + c 1 = 1 mod 2 a 0 + b 1 + c 1 = 1 mod 2 {\displaystyle {\begin{aligned}&a_{0}+b_{0}+c_{0}=0\mod 2\\&a_{1}+b_{1}+c_{0}=1\mod 2\\&a_{1}+b_{0}+c_{1}=1\mod 2\\&a_{0}+b_{1}+c_{1}=1\mod 2\end{aligned}}}
Suppose that there is a classical strategy that satisfies all four winning conditions, all four conditions hold true. Through observation, each term appears twice on the left hand side. Hence, the left side sum = 0 mod 2. However, the right side sum = 1 mod 2. The contradiction shows that all four winning conditions cannot be simultaneously satisfied.
When Alice, Bob, and Carol decide to adopt a quantum strategy they share a tripartite entangled state | ψ ⟩ = 1 2 ( | 000 ⟩ + | 111 ⟩ ) {\textstyle |{\psi }\rangle ={\frac {1}{\sqrt {2}}}(|000\rangle +|111\rangle )} , known as the GHZ state .
If question 0 is received, the player makes a measurement in the X basis { | + ⟩ , | − ⟩ } {\textstyle \{|+\rangle ,|-\rangle \}} . If question 1 is received, the player makes a measurement in the Y basis { 1 2 ( | 0 ⟩ + i | 1 ⟩ ) , 1 2 ( | 0 ⟩ − i | 1 ⟩ ) } {\textstyle \left\{{\frac {1}{\sqrt {2}}}(|0\rangle +i|1\rangle ),{\frac {1}{\sqrt {2}}}(|0\rangle -i|1\rangle )\right\}} . In both cases, the players give answer 0 if the result of the measurement is the first state of the pair, and answer 1 if the result is the second state of the pair. With this strategy the players win the game with probability 1. | https://en.wikipedia.org/wiki/Quantum_pseudo-telepathy |
A quantum reference frame is a reference frame which is treated quantum theoretically. It, like any reference frame , is an abstract coordinate system which defines physical quantities, such as time , position, momentum , spin , and so on. Because it is treated within the formalism of quantum theory , it has some interesting properties which do not exist in a normal classical reference frame.
Consider a simple physics problem: a car is moving such that it covers a distance of 1 mile in every 2 minutes, what is its velocity in metres per second? With some conversion and calculation, one can come up with the answer "13.41m/s"; on the other hand, one can instead answer "0, relative to itself". The first answer is correct because it recognises a reference frame is implied in the problem. The second one, albeit pedantic, is also correct because it exploits the fact that there is not a particular reference frame specified by the problem. This simple problem illustrates the importance of a reference frame: a reference frame is quintessential in a clear description of a system, whether it is included implicitly or explicitly.
When speaking of a car moving towards east, one is referring to a particular point on the surface of the Earth; moreover, as the Earth is rotating, the car is actually moving towards a changing direction, with respect to the Sun. In fact, this is the best one can do: describing a system in relation to some reference frame. Describing a system with respect to an absolute space does not make much sense because an absolute space, if it exists, is unobservable. Hence, it is impossible to describe the path of the car in the above example with respect to some absolute space. This notion of absolute space troubled a lot of physicists over the centuries, including Newton. Indeed, Newton was fully aware of this stated that all inertial frames are observationally equivalent to each other. Simply put, relative motions of a system of bodies do not depend on the inertial motion of the whole system. [ 1 ]
An inertial reference frame (or inertial frame in short) is a frame in which all the physical laws hold. For instance, in a rotating reference frame , Newton's laws have to be modified because there is an extra Coriolis force (such frame is an example of non-inertial frame). Here, "rotating" means "rotating with respect to some inertial frame". Therefore, although it is true that a reference frame can always be chosen to be any physical system for convenience, any system has to be eventually described by an inertial frame, directly or indirectly. Finally, one may ask how an inertial frame can be found, and the answer lies in the Newton's laws , at least in Newtonian mechanics : the first law guarantees the existence of an inertial frame while the second and third law are used to examine whether a given reference frame is an inertial one or not.
It may appear an inertial frame can now be easily found given the Newton's laws as empirical tests are accessible. Quite the contrary; an absolutely inertial frame is not and will most likely never be known. Instead, inertial frame is approximated. As long as the error of the approximation is undetectable by measurements, the approximately inertial frame (or simply "effective frame") is reasonably close to an absolutely inertial frame. With the effective frame and assuming the physical laws are valid in such frame, descriptions of systems will ends up as good as if the absolutely inertial frame was used. As a digression, the effective frame Astronomers use is a system called " International Celestial Reference Frame " (ICRF), defined by 212 radio sources and with an accuracy of about 10 − 5 {\displaystyle 10^{-5}} radians. However, it is likely that a better one will be needed when a more accurate approximation is required.
Reconsidering the problem at the very beginning, one can certainly find a flaw of ambiguity in it, but it is generally understood that a standard reference frame is implicitly used in the problem. In fact, when a reference frame is classical, whether or not including it in the physical description of a system is irrelevant. One will get the same prediction by treating the reference frame internally or externally.
To illustrate the point further, a simple system with a ball bouncing off a wall is used. In this system, the wall can be treated either as an external potential or as a dynamical system interacting with the ball. The former involves putting the external potential in the equations of motions of the ball while the latter treats the position of the wall as a dynamical degree of freedom . Both treatments provide the same prediction, and neither is particularly preferred over the other. However, as it will be discussed below, such freedom of choice cease to exist when the system is quantum mechanical.
A reference frame can be treated in the formalism of quantum theory, and, in this case, such is referred as a quantum reference frame. Despite different name and treatment, a quantum reference frame still shares much of the notions with a reference frame in classical mechanics . It is associated to some physical system, and it is relational .
For example, if a spin-1/2 particle is said to be in the state | ↑ z ⟩ {\displaystyle \left|\uparrow z\right\rangle } , a reference frame is implied, and it can be understood to be some reference frame with respect to an apparatus in a lab. It is obvious that the description of the particle does not place it in an absolute space, and doing so would make no sense at all because, as mentioned above, absolute space is empirically unobservable. On the other hand, if a magnetic field along y-axis is said to be given, the behaviour of the particle in such field can then be described. In this sense, y and z are just relative directions. They do not and need not have absolute meaning.
One can observe that a z direction used in a laboratory in Berlin is generally totally different from a z direction used in a laboratory in Melbourne. Two laboratories trying to establish a single shared reference frame will face important issues involving alignment. The study of this sort of communication and coordination is a major topic in quantum information theory .
Just as in this spin-1/2 particle example, quantum reference frames are almost always treated implicitly in the definition of quantum states, and the process of including the reference frame in a quantum state is called quantisation/internalisation of reference frame while the process of excluding the reference frame from a quantum state is called dequantisation [ citation needed ] /externalisation of reference frame. Unlike the classical case, in which treating a reference internally or externally is purely an aesthetic choice, internalising and externalising a reference frame does make a difference in quantum theory. [ 2 ]
One final remark may be made on the existence of a quantum reference frame. After all, a reference frame, by definition, has a well-defined position and momentum, while quantum theory, namely uncertainty principle , states that one cannot describe any quantum system with well-defined position and momentum simultaneously, so it seems there is some contradiction between the two. It turns out, an effective frame, in this case a classical one, is used as a reference frame, just as in Newtonian mechanics a nearly inertial frame is used, and physical laws are assumed to be valid in this effective frame. In other words, whether motion in the chosen reference frame is inertial or not is irrelevant.
The following treatment of a hydrogen atom motivated by Aharanov and Kaufherr can shed light on the matter. [ 3 ] Supposing a hydrogen atom is given in a well-defined state of motion, how can one describe the position of the electron? The answer is not to describe the electron's position relative to the same coordinates in which the atom is in motion, because doing so would violate uncertainty principle, but to describe its position relative to the nucleus. As a result, more can be said about the general case from this: in general, it is permissible, even in quantum theory, to have a system with well-defined position in one reference frame and well-defined motion in some other reference frame.
Consider a hydrogen atom. Coulomb potential depends on the distance between the proton and electron only:
With this symmetry, the problem is reduced to that of a particle in a central potential:
Using separation of variables , the solutions of the equation can be written into radial and angular parts:
where l , {\displaystyle l,} m {\displaystyle m} , and n {\displaystyle n} are the orbital angular momentum, magnetic, and energy quantum numbers, respectively.
Now consider the Schrödinger equation for the proton and the electron:
A change of variables to relational and centre-of-mass coordinates yields
where M {\displaystyle M} is the total mass and μ {\displaystyle \mu } is the reduced mass. A final change to spherical coordinates followed by a separation of variables will yield the equation for Φ ( r , θ , ϕ ) {\displaystyle \Phi (r,\theta ,\phi )} from above.
However, if the change of variables done early is now to be reversed, centre-of-mass needs to be put back into the equation for Φ ( r , θ , ϕ ) {\displaystyle \Phi (r,\theta ,\phi )} :
The importance of this result is that it shows the wavefunction for the compound system is entangled , contrary to what one would normally think in a classical standpoint. More importantly, it shows the energy of the hydrogen atom is not only associated with the electron but also with the proton, and the total state is not decomposable into a state for the electron and one for the proton separately. [ 1 ]
Superselection rules, in short, are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originally introduced to impose additional restriction to quantum theory beyond those of selection rules . As an example, superselection rules for electric charges disallow the preparation of a coherent superposition of different charge eigenstates.
As it turns out, the lack of a reference frame is mathematically equivalent to superselection rules. This is a powerful statement because superselection rules have long been thought to have axiomatic nature, and now its fundamental standing and even its necessity are questioned. Nevertheless, it has been shown that it is, in principle, always possible (though not always easy) to lift all superselection rules on a quantum system.
During a measurement, whenever the relations between the system and the reference frame used is inquired, there is inevitably a disturbance to both of them, which is known as measurement back action . As this process is repeated, it decreases the accuracy of the measurement outcomes, and such reduction of the usability of a reference frame is referred to as the degradation of a quantum reference frame. [ 4 ] [ 5 ] A way to gauge the degradation of a reference frame is to quantify the longevity, namely, the number of measurements that can be made against the reference frame until certain error tolerance is exceeded.
For example, for a spin- j {\displaystyle j} system, the maximum number of measurements that can be made before the error tolerance, ϵ {\displaystyle \epsilon } , is exceeded is given by n m a x ≃ ϵ j 2 {\displaystyle n_{max}\simeq \epsilon j^{2}} . So the longevity and the size of the reference frame are of quadratic relation in this particular case. [ 6 ]
In this spin- j {\displaystyle j} system, the degradation is due to the loss of purity of the reference frame state. On the other hand, degradation can also be caused by misalignment of background reference. It has been shown, in such case, the longevity has a linear relation with the size of the reference frame. [ 4 ] | https://en.wikipedia.org/wiki/Quantum_reference_frame |
In quantum mechanics , the quantum revival [ 1 ] is a periodic recurrence of the quantum wave function from its original form during the time evolution either many times in space as the multiple scaled fractions
in the form of the initial wave function (fractional revival) or approximately or exactly to its original
form from the beginning (full revival). The quantum wave function periodic in time exhibits therefore the full revival
every period . The phenomenon of revivals is most readily observable for the wave functions being well localized wave packets at the beginning of the time evolution for example in the hydrogen atom. For Hydrogen, the fractional revivals show up
as multiple angular Gaussian bumps around the circle drawn by the radial maximum of leading circular state component (that with the highest amplitude in the eigenstate expansion) of the
original localized state and the full revival as the original Gaussian. [ 2 ] The full revivals are exact for the infinite quantum well , harmonic oscillator or the hydrogen atom , while for shorter times are approximate
for the hydrogen atom and a lot of quantum systems. [ 3 ]
The plot of collapses and revivals of quantum oscillations of the JCM atomic inversion. [ 4 ]
Consider a quantum system with the energies E i {\displaystyle E_{i}} and the eigenstates ψ i {\displaystyle \psi _{i}}
and let the energies be the rational fractions of some constant C {\displaystyle C}
(for example for hydrogen atom M i = 1 {\displaystyle M_{i}=1} , N i = i 2 {\displaystyle N_{i}=i^{2}} , C = − 13.6 e V {\displaystyle C=-13.6eV} .
Then the truncated (till N m a x {\displaystyle \mathbb {N} _{max}} of states) solution of the time dependent Schrödinger equation is
.
Let L c m {\displaystyle L_{cm}} be to lowest common multiple of all N i {\displaystyle N_{i}} and L c d {\displaystyle L_{cd}} greatest common divisor of all M i {\displaystyle M_{i}} then for each N i {\displaystyle N_{i}} the L c m / N i {\displaystyle {L_{cm}}/N_{i}} is an integer, for each M i {\displaystyle M_{i}} the M i / L c d {\displaystyle {M_{i}}/L_{cd}} is an integer, 2 π M i L c m / ( N i L c d ) {\displaystyle 2\pi M_{i}{L_{cm}}/(N_{i}L_{cd})} is the full multiple of 2 π {\displaystyle 2\pi } angle and
after the full revival time time
For the quantum system as small as Hydrogen and N m a x {\displaystyle \mathbb {N} _{max}} as small as 100 it may take quadrillions of years till it will fully revive. Especially once created by fields the Trojan wave packet in a
hydrogen atom exists without any external fields stroboscopically and eternally repeating itself
after sweeping almost the whole hypercube of quantum phases exactly every full revival time.
The striking consequence is that no finite-bit computer can propagate the numerical wave function accurately for the arbitrarily long
time. If the processor number is n- bit long floating point number then the number can be stored by the computer only with the finite accuracy after the comma and the energy is (up to 8 digits after the comma) for example 2.34576893 = 234576893/100000000 and as the finite fraction it
is exactly rational and the full revival occurs for any wave function of any quantum system after the time t / 2 π = 100000000 {\displaystyle t/2\pi =100000000} which is its maximum exponent and so on that may not be true for all quantum systems or all stationary quantum systems undergo the full and exact revival numerically.
In the system with the rational energies i.e. where the quantum exact full revival exists its existence immediately proves the quantum Poincaré recurrence theorem and the time of the full quantum revival equals to the Poincaré recurrence time.
While the rational numbers are dense in real numbers and the arbitrary function of
the quantum number can be approximated arbitrarily exactly with Padé approximants with the
coefficients of arbitrary decimal precision for the arbitrarily long time each quantum system therefore revives
almost exactly. It also means that the Poincaré recurrence and the full revival is mathematically the same thing [ 5 ] and it is
commonly accepted that the recurrence is called the full revival if it occurs after the reasonable and physically measurable time
that is possible to be detected by the realistic apparatus and this happens due to a very special energy spectrum having a large basic energy
spacing gap of which the energies are arbitrary (not necessarily harmonic) multiples. | https://en.wikipedia.org/wiki/Quantum_revival |
In quantum mechanics , quantum scarring is a phenomenon where the eigenstates of a classically chaotic quantum system have enhanced probability density around the paths of unstable classical periodic orbits. [ 2 ] [ 3 ] The instability of the periodic orbit is a decisive point that differentiates quantum scars from the more trivial observation that the probability density is enhanced in the neighborhood of stable periodic orbits. The latter can be understood as a purely classical phenomenon, a manifestation of the Bohr correspondence principle , whereas in the former, quantum interference is essential. As such, scarring is both a visual example of quantum-classical correspondence, and simultaneously an example of a (local) quantum suppression of chaos.
A classically chaotic system is also ergodic , and therefore (almost) all of its trajectories eventually explore evenly the entire accessible phase space. Thus, it would be natural to expect that the eigenstates of the quantum counterpart would fill the quantum phase space in the uniform manner up to random fluctuations in the semiclassical limit. However, scars are a significant correction to this assumption. Scars can therefore be considered as an eigenstate counterpart of how short periodic orbits provide corrections to the universal spectral statistics of the random matrix theory . There are rigorous mathematical theorems on quantum nature of ergodicity, [ 4 ] [ 5 ] [ 6 ] proving that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. Nonetheless, the quantum ergodicity theorems do not exclude scarring if the quantum phase space volume of the scars gradually vanishes in the semiclassical limit.
On the classical side, there is no direct analogue of scars. On the quantum side, they can be interpreted as an eigenstate analogy to how short periodic orbits correct the universal random matrix theory eigenvalue statistics. Scars correspond to nonergodic states which are permitted by the quantum ergodicity theorems. In particular, scarred states provide a striking visual counterexample to the assumption that the eigenstates of a classically chaotic system would be without structure. In addition to conventional quantum scars, the field of quantum scarring has undergone its renaissance period, sparked by the discoveries of perturbation-induced scars and many-body scars that have subsequently paved the way towards emerging concepts within the field, such as antiscarring and quantum birthmarks .
The existence of scarred states is rather unexpected based on the Gutzwiller trace formula , [ 7 ] [ 8 ] which connects the quantum mechanical density of states to the periodic orbits in the corresponding classical system. According to the trace formula, a quantum spectrum is not a result of a trace over all the positions, but it is determined by a trace over all the periodic orbits only. Furthermore, every periodic orbit contributes to an eigenvalue, although not exactly equally. It is even more unlikely that a particular periodic orbit would stand out in contributing to a particular eigenstate in a fully chaotic system, since altogether periodic orbits occupy a zero-volume portion of the total phase space volume. Hence, nothing seems to imply that any particular periodic orbit for a given eigenvalue could have a significant role compared to other periodic orbits. Nonetheless, quantum scarring proves this assumption to be wrong. The scarring was first seen in 1983 by S. W. McDonald in his thesis on the stadium billiard as an interesting numerical observation. [ 9 ] They did not show up well in his figure because they were fairly crude "waterfall" plots. This finding was not thoroughly reported in the article discussion about the wave functions and nearest-neighbor level spacing spectra for the stadium billiard. [ 10 ] A year later, Eric J. Heller published the first examples of scarred eigenfunctions together with a theoretical explanation for their existence. [ 2 ] The results revealed large footprints of individual periodic orbits influencing some eigenstates of the classically chaotic Bunimovich stadium, named as scars by Heller.
A wave packet analysis was a key in proving the existence of the scars, and it is still a valuable tool to understand them. In the original work of Heller, [ 2 ] the quantum spectrum is extracted by propagating a Gaussian wave packet along a periodic orbit. Nowadays, this seminal idea is known as the linear theory of scarring. [ 2 ] [ 3 ] [ 11 ] [ 12 ] Scars stand out to the eye in some eigenstates of classically chaotic systems, but are quantified by projection of the eigenstates onto certain test states, often Gaussians, having both average position and average momentum along the periodic orbit. These test states give a provably structured spectrum that reveals the necessity of scars. [ 13 ] However, there is no universal measure on scarring; the exact relationship of the stability exponent χ {\displaystyle \chi } to the scarring strength is a matter of definition. Nonetheless, there is a rule of thumb: [ 3 ] [ 7 ] quantum scarring is significant when χ < 2 π {\displaystyle \chi <2\pi } , and the strength scales as χ − 1 {\displaystyle \chi ^{-1}} . Thus, strong quantum scars are, in general, associated with periodic orbits that are moderately unstable and relatively short. The theory predicts the scar enhancement along a classical periodic orbit, but it cannot precisely pinpoint which particular states are scarred and how much. Rather, it can be only stated that some states are scarred within certain energy zones, and by at least by a certain degree.
The linear scarring theory outlined above has been later extended to include nonlinear effects taking place after the wave packet departs the linear dynamics domain around the periodic orbit. [ 12 ] At long times, the nonlinear effect can assist the scarring. This stems from nonlinear recurrences associated with homoclinic orbits. A further insight on scarring was acquired with a real-space approach by E. B. Bogomolny [ 14 ] and a phase-space alternative by Michael V. Berry [ 15 ] complementing the wave-packet and Hussimi space methods utilized by Heller and L. Kaplan. [ 2 ] [ 3 ] [ 12 ]
As well as there being no universal measure for the level of scarring, there is also no generally accepted definition of it. Originally, it was stated [ 2 ] that certain unstable periodic orbits are shown to permanently scar some quantum eigenfunctions as ℏ → 0 {\displaystyle \hbar \rightarrow 0} , in the sense that extra density surrounds the region of the periodic orbit. However, a more formal definition for scarring would be the following: [ 3 ] A quantum eigenstate of a classically chaotic system is scarred by a periodic if its density on the classical invariant manifolds near and all along that periodic is systematically enhanced above the classical, statistically expected density along that orbit.
Most of the research on quantum scars has been restricted to non-relativistic quantum systems described by the Schrödinger equation , where the dependence of the particle energy on momentum is quadratic. However, scarring can occur in a relativistic quantum systems described by the Dirac equation, where the energy-momentum relation is linear instead. [ 16 ] [ 17 ] [ 18 ] Heuristically, these relativistic scars are a consequence of the fact that both spinor components satisfy the Helmholtz equation, in analogue to the time-independent Schrödinger equation. Therefore, relativistic scars have the same origin as the conventional scarring [ 2 ] introduced by E. J. Heller. Nevertheless, there is a difference in terms of the recurrence with respect to energy variation. Furthermore, it was shown that the scarred states can lead to strong conductance fluctuations in the corresponding open quantum dots via the mechanism of resonant transmission. [ 16 ]
The first experimental confirmations of scars were obtained in microwave billiards in the early 1990s. [ 19 ] [ 20 ] Further experimental evidence for scarring has later been delivered by observations in, e.g., quantum wells, [ 21 ] [ 22 ] [ 23 ] optical cavities [ 24 ] [ 25 ] and the hydrogen atom. [ 26 ] In the early 2000s, the first observations were achieved in an elliptical billiard. [ 27 ] Many classical trajectories converge in this system and lead to pronounced scarring at the foci, commonly called as quantum mirages. [ 28 ] In addition, recent numerical results indicated the existence of quantum scars in ultracold atomic gases. [ 29 ] Aside from these analog scars in classical wave experiments, the verification of this phenomenon had remained elusive in true quantum systems — until the recent achievement of using scanning tunneling microscopy to directly visualize (relativistic) quantum scars in a stadium-shaped, graphene-based quantum dot. [ 30 ]
In addition to scarring described above, there are several similar phenomena, either connected by theory or appearance. First of all, when scars are visually identified, some of the states may remind of classical ''bouncing-ball'' motion, excluded from quantum scars into its own category. For example, a stadium billiard supports these highly nonergodic eigenstates, which reflect trapped bouncing motion between the straight walls. [ 3 ] It has been shown that the bouncing states persist at the limit ℏ → 0 {\displaystyle \hbar \rightarrow 0} , but at the same time this result suggests a diminishing percentage of all the states in the agreement with the quantum ergodicity theorems of Alexander Schnirelman, Yves Colin de Verdière , and Steven Zelditch . [ 4 ] [ 5 ] [ 6 ] Secondly, scarring should not be confused with statistical fluctuations. Similar structures of an enhanced probability density occur even as random superpositions of plane waves, [ 31 ] in the sense of the Berry conjecture. [ 32 ] [ 33 ] Furthermore, there is a genre of scars, not caused by actual periodic orbits, but their remnants, known as ghosts . They refer to periodic orbits that are found in a nearby system in the sense of some tunable, external system parameter. [ 34 ] [ 35 ] Scarring of this kind has been associated to almost-periodic orbits. [ 36 ] Another subclass of ghosts stems from complex periodic orbits which exist in the vicinity of bifurcation points. [ 37 ] [ 38 ]
A new class of quantum scars was discovered in disordered two-dimensional nanostructures. [ 39 ] [ 40 ] [ 1 ] [ 41 ] [ 42 ] Even though similar in appearance to ordinary quantum scars described earlier, these scars have a fundamentally different origin. In the case, the disorder arising from small perturbations (see red dots in the figure) is sufficient to destroy classical long-time stability. Hence, there is no moderately unstable periodic in the classical counterpart to which a scar would corresponds in the ordinary scar theory. Instead, scars are formed around periodic orbits of the corresponding unperturbed system. Ordinary scar theory is further excluded by the behavior of the scars as a function of the disorder strength. When the potential bumps are made stronger while keeping they otherwise unchanged, the scars grow stronger and then fade away without changing their orientation. In contrary, a scar caused by conventional theory should become rapidly weaker due to the increase of the stability exponent of a periodic orbit with increasing disorder. Furthermore, comparing scars at different energies reveals that they occur in only a few distinct orientations This too contradicts predictions of ordinary scar theory.
The area of quantum many-body scars is a subject of active research. [ 43 ] [ 44 ]
Scars have occurred in investigations for potential applications of Rydberg states to quantum computing , specifically acting as qubits for quantum simulation . [ 45 ] [ 46 ] The particles of the system in an alternating ground state -Rydberg state configuration continually entangled and disentangled rather than remaining entangled and undergoing thermalization . [ 45 ] [ 46 ] [ 47 ] Systems of the same atoms prepared with other initial states did thermalize as expected. [ 46 ] [ 47 ] The researchers dubbed the phenomenon "quantum many-body scarring". [ 48 ] [ 49 ]
The causes of quantum scarring are not well understood. [ 45 ] One possible proposed explanation is that quantum scars represent integrable systems , or nearly do so, and this could prevent thermalization from ever occurring. [ 50 ] This has drawn criticisms arguing that a non-integrable Hamiltonian underlies the theory. [ 51 ] Recently, a series of works [ 52 ] [ 53 ] has related the existence of quantum scarring to an algebraic structure known as dynamical symmetries . [ 54 ] [ 55 ]
Fault-tolerant quantum computers are desired, as any perturbations to qubit states can cause the states to thermalize, leading to loss of quantum information . [ 45 ] Scarring of qubit states is seen as a potential way to protect qubit states from outside disturbances leading to decoherence and information loss.
A fascinating consequence of quantum scarring is its dual partner—antiscarring, [ 3 ] which refers to a systematic depression of the probability density in quantum states along the path of the scar-generating periodic orbit. The existence of antiscarring is confirmed by a general stacking theorem: [ 56 ] the cumulative probability density of the eigenstates becomes uniform when the energy window of the summed eigenstates is larger than the energy scale associated with the shortest periodic orbit in the system. Since there may be strongly scarred states among the eigenstates, the necessity for a uniform average over a large number of states requires the existence of antiscarred states with low probability in the region of ''regular'' scars. This effect has been demonstrated in the context of variational scarring, where it is promoted by the strength and similar orientation of the scars within a moderate energy window. [ 56 ] Furthermore, it has been realized [ 57 ] that some decay processes have antiscarred states with anomalously long escape times.
A hallmark of classical ergodicity is the complete loss of memory of initial conditions, resulting from the eventual uniform exploration of phase space. In a quantum system, however, classical ergodic behavior can break down, as exemplified by the presence of quantum scars. The concept of a quantum birthmark [ 58 ] bridges the short-term effects, such as due to scarring, and the long-term predictions of random matrix theory. By extending beyond quantum scarring, quantum birthmarks offer a new paradigm for understanding the elusive quantum nature of ergodicity.
Figure depicts two birthmarks unveiled within the time-averaged probability density of a wavepacket launched under different initial conditions (indicated by the black arrows) in the stadium. The upper plot shows the result for a wavepacket released vertically from the center along a bouncing-ball orbit; whereas the lower plot depicts an wapacket prepared off-center at an arbitrary angle, corresponding to a generic initial state. Notably, quantum birthmarks in each case respect the two reflection symmetries of the stadium. These two cases clearly demonstrate that a quantum system can violate the classical ergodicity assumption in the sense that the probability density becomes uniform even at infinite time. Therefore, quantum scars presents a new form of weak ergodicity breaking, beyond quantum scarring taking place at the eigenstate level. While any initial state and its short-term behavior will be memorized by a quantum system, [ 58 ] the strength of the corresponding birthmark depends on the dynamical details of the birthplace, particularly this quantum memory effect is boosted in the presence of scarring. | https://en.wikipedia.org/wiki/Quantum_scar |
Quantum simulators permit the study of a quantum system in a programmable fashion. In this instance, simulators are special purpose devices designed to provide insight about specific physics problems. [ 1 ] [ 2 ] [ 3 ] Quantum simulators may be contrasted with generally programmable "digital" quantum computers , which would be capable of solving a wider class of quantum problems.
A universal quantum simulator is a quantum computer proposed by Yuri Manin in 1980 [ 4 ] and Richard Feynman in 1982. [ 5 ]
A quantum system may be simulated by either a Turing machine or a quantum Turing machine , as a classical Turing machine is able to simulate a universal quantum computer (and therefore any simpler quantum simulator), meaning they are equivalent from the point of view of computability theory . The simulation of quantum physics by a classical computer has been shown to be inefficient. [ 6 ] In other words, quantum computers provide no additional power over classical computers in terms of computability, but it is suspected that they can solve certain problems faster than classical computers, meaning they may be in different complexity classes , which is why quantum Turing machines are useful for simulating quantum systems. This is known as quantum supremacy , the idea that there are problems only quantum Turing machines can solve in any feasible amount of time.
A quantum system of many particles could be simulated by a quantum computer using a number of quantum bits similar to the number of particles in the original system. [ 5 ] This has been extended to much larger classes of quantum systems. [ 7 ] [ 8 ] [ 9 ] [ 10 ]
Quantum simulators have been realized on a number of experimental platforms, including systems of ultracold quantum gases , polar molecules, trapped ions, photonic systems, quantum dots, and superconducting circuits. [ 11 ]
Many important problems in physics, especially low-temperature physics and many-body physics , remain poorly understood because the underlying quantum mechanics is vastly complex. Conventional computers, including supercomputers, are inadequate for simulating quantum systems with as few as 30 particles because the dimension of the Hilbert space grows exponentially with particle number. [ 12 ] Better computational tools are needed to understand and rationally design materials whose properties are believed to depend on the collective quantum behavior of hundreds of particles. [ 2 ] [ 3 ] Quantum simulators provide an alternative route to understanding the properties of these systems. These simulators create clean realizations of specific systems of interest, which allows precise realizations of their properties. Precise control over and broad tunability of parameters of the system allows the influence of various parameters to be cleanly disentangled.
Quantum simulators can solve problems which are difficult to simulate on classical computers because they directly exploit quantum properties of real particles. In particular, they exploit a property of quantum mechanics called superposition , wherein a quantum particle is made to be in two distinct states at the same time, for example, aligned and anti-aligned with an external magnetic field. Crucially, simulators also take advantage of a second quantum property called entanglement , allowing the behavior of even physically well separated particles to be correlated. [ 2 ] [ 3 ] [ 13 ]
Recently quantum simulators have been used to obtain time crystals [ 14 ] [ 15 ] and quantum spin liquids . [ 16 ] [ 17 ]
Ion trap based system forms an ideal setting for simulating interactions in quantum spin models. [ 18 ] A trapped-ion simulator, built by a team that included the NIST can engineer and control interactions among hundreds of quantum bits (qubits). [ 19 ] Previous endeavors were unable to go beyond 30 quantum bits. The capability of this simulator is 10 times more than previous devices. It has passed a series of important benchmarking tests that indicate a capability to solve problems in material science that are impossible to model on conventional computers.
The trapped-ion simulator consists of a tiny, single-plane crystal of hundreds of beryllium ions , less than 1 millimeter in diameter, hovering inside a device called a Penning trap . The outermost electron of each ion acts as a tiny quantum magnet and is used as a qubit, the quantum equivalent of a “1” or a “0” in a conventional computer. In the benchmarking experiment, physicists used laser beams to cool the ions to near absolute zero. Carefully timed microwave and laser pulses then caused the qubits to interact, mimicking the quantum behavior of materials otherwise very difficult to study in the laboratory. Although the two systems may outwardly appear dissimilar, their behavior is engineered to be mathematically identical. In this way, simulators allow researchers to vary parameters that could not be changed in natural solids, such as atomic lattice spacing and geometry.
Friedenauer et al., adiabatically manipulated 2 spins, showing their separation into ferromagnetic and antiferromagnetic states. [ 20 ] Kim et al., extended the trapped ion quantum simulator to 3 spins, with global antiferromagnetic Ising interactions featuring frustration and showing the link between frustration and entanglement [ 21 ] and Islam et al., used adiabatic quantum simulation to demonstrate the sharpening of a phase transition between paramagnetic and ferromagnetic ordering as the number of spins increased from 2 to 9. [ 22 ] Barreiro et al. created a digital quantum simulator of interacting spins with up to 5 trapped ions by coupling to an open reservoir [ 23 ] and
Lanyon et al. demonstrated digital quantum simulation with up to 6 ions. [ 24 ] Islam, et al., demonstrated adiabatic quantum simulation of the transverse Ising model with variable (long) range interactions with up to 18 trapped ion spins, showing control of the level of spin frustration by adjusting the antiferromagnetic interaction range. [ 25 ] Britton, et al. from NIST has experimentally benchmarked Ising interactions in a system of hundreds of qubits for studies of quantum magnetism. [ 19 ] Pagano, et al., reported a new cryogenic ion trapping system designed for long time storage of large ion chains demonstrating coherent one and two-qubit operations for chains of up to 44 ions. [ 26 ] Joshi, et al., probed the quantum dynamics of 51 individually controlled ions, realizing a long-range interacting spin chain. [ 27 ]
Many ultracold atom experiments are examples of quantum simulators. These include experiments studying bosons or fermions in optical lattices , the unitary Fermi gas, Rydberg atom arrays in optical tweezers . A common thread for these experiments is the capability of realizing generic Hamiltonians, such as the Hubbard or transverse-field Ising Hamiltonian. Major aims of these experiments include identifying low-temperature phases or tracking out-of-equilibrium dynamics for various models, problems which are theoretically and numerically intractable. [ 28 ] [ 29 ] Other experiments have realized condensed matter models in regimes which are difficult or impossible to realize with conventional materials, such as the Haldane model and the Harper-Hofstadter model . [ 30 ] [ 31 ] [ 32 ] [ 33 ] [ 34 ]
Quantum simulators using superconducting qubits fall into two main categories. First, so called quantum annealers determine ground states of certain Hamiltonians after an adiabatic ramp. This approach is sometimes called adiabatic quantum computing . Second, many systems emulate specific Hamiltonians and study their ground state properties, quantum phase transitions , or time dynamics. [ 35 ] Several important recent results include the realization of a Mott insulator in a driven-dissipative Bose-Hubbard system and studies of phase transitions in lattices of superconducting resonators coupled to qubits. [ 36 ] [ 37 ] | https://en.wikipedia.org/wiki/Quantum_simulator |
Quantum singular value transformation is a framework for designing quantum algorithms . It encompasses a variety of quantum algorithms for problems that can be solved with linear algebra , including Hamiltonian simulation , search problems , and linear system solving . [ 1 ] [ 2 ] [ 3 ] It was introduced in 2018 by András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, generalizing algorithms for Hamiltonian simulation of Guang Hao Low and Isaac Chuang inspired by signal processing. [ 4 ]
The basic primitive of quantum singular value transformation is the block-encoding. A quantum circuit is a block-encoding of a matrix A if it implements a unitary matrix U such that U contains A in a specified sub-matrix. For example, if ( ⟨ 0 | ⊗ I ) U ( | 0 ⟩ | ϕ ⟩ ) = A | ϕ ⟩ {\displaystyle (\langle 0|\otimes I)U(|0\rangle |\phi \rangle )=A|\phi \rangle } , then U is a block-encoding of A .
The fundamental algorithm of QSVT is one that converts a block-encoding of A to a block-encoding of p ( A , A † ) {\displaystyle p(A,A^{\dagger })} , where p is a polynomial of degree d and A † {\displaystyle A^{\dagger }} denotes the conjugate transpose , with only d applications of the circuit and one ancilla qubit. This can be done for a large class of polynomials p which correspond to applying a polynomial to the singular values of A , giving a "singular value transformation".
A variant of this algorithm can also be performed when A is Hermitian , corresponding to an "eigenvalue transformation". That is, given a block-encoding of A with eigendecomposition of a matrix A = ∑ λ i u i u i † {\displaystyle A=\sum \lambda _{i}u_{i}u_{i}^{\dagger }} , one can get a block-encoding for ∑ p ( λ i ) u i u i † {\displaystyle \sum p(\lambda _{i})u_{i}u_{i}^{\dagger }} , provided p is bounded. [ 4 ]
[ 2 ]
This algorithms or data structures -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_singular_value_transformation |
A quantum solvent is essentially a superfluid (aka a quantum liquid ) used to dissolve another chemical species . Any superfluid can theoretically act as a quantum solvent, but in practice the only viable superfluid medium that can currently be used is helium-4 , and it has been successfully accomplished in controlled conditions. Such solvents are currently under investigation for use in spectroscopic techniques in the field of analytical chemistry , due to their superior kinetic properties. [ 1 ]
Any matter dissolved (or otherwise suspended) in the superfluid will tend to aggregate together in clumps, encapsulated by a 'quantum solvation shell '. Due to the totally frictionless nature of the superfluid medium, the entire object then proceeds to act very much like a nanoscopic ball bearing, allowing effectively complete rotational freedom of the solvated chemical species . A quantum solvation shell consists of a region of non-superfluid helium-4 atoms that surround the molecule(s) and exhibit adiabatic following around the centre of gravity of the solute. As such, the kinetics of an effectively gaseous molecule can be studied without the need to use an actual gas (which can be impractical or impossible). It is necessary to make a small alteration to the rotational constant of the chemical species being examined, in order to compensate for the higher mass entailed by the quantum solvation shell.
Quantum solvation has so far been achieved with a number of organic, inorganic and organometallic compounds, and it has been speculated that as well as the obvious use in the field of spectroscopy , quantum solvents could be used as tools in nanoscale chemical engineering, perhaps to manufacture components for use in nanotechnology .
This nanotechnology-related article is a stub . You can help Wikipedia by expanding it .
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_solvent |
In quantum mechanics , a quantum speed limit ( QSL ) is a limitation on the minimum time for a quantum system to evolve between two distinguishable (orthogonal) states. [ 1 ] QSL theorems are closely related to time-energy uncertainty relations. In 1945, Leonid Mandelstam and Igor Tamm derived a time-energy uncertainty relation that bounds the speed of evolution in terms of the energy dispersion. [ 2 ] Over half a century later, Norman Margolus and Lev Levitin showed that the speed of evolution cannot exceed the mean energy, [ 3 ] a result known as the Margolus–Levitin theorem. Realistic physical systems in contact with an environment are known as open quantum systems and their evolution is also subject to QSL. [ 4 ] [ 5 ] Quite remarkably it was shown that environmental effects, such as non-Markovian dynamics can speed up quantum processes, [ 6 ] which was verified in a cavity QED experiment. [ 7 ]
QSL have been used to explore the limits of computation [ 8 ] [ 9 ] and complexity. In 2017, QSLs were studied in a quantum oscillator at high temperature. [ 10 ] In 2018, it was shown that QSL are not restricted to the quantum domain and that similar bounds hold in classical systems. [ 11 ] [ 12 ] In 2021, both the Mandelstam-Tamm and the Margolus–Levitin QSL bounds were concurrently tested in a single experiment [ 13 ] which indicated there are "two different regimes: one where the Mandelstam-Tamm limit constrains the evolution at all times, and a second where a crossover to the Margolus-Levitin limit occurs at longer times."
In quantum sensing , QSLs impose fundamental constraints on the maximum achievable time resolution of quantum sensors. These limits stem from the requirement that quantum states must evolve to orthogonal states to extract precise information. For example, in applications like Ramsey interferometry , the QSL determines the minimum time required for phase accumulation during control sequences, directly impacting the sensor's temporal resolution and sensitivity. [ 14 ]
The speed limit theorems can be stated for pure states , and for mixed states ; they take a simpler form for pure states. An arbitrary pure state can be written as a linear combination of energy eigenstates:
The task is to provide a lower bound for the time interval t ⊥ {\displaystyle t_{\perp }} required for the initial state | ψ ⟩ {\displaystyle |\psi \rangle } to evolve into a state orthogonal to | ψ ⟩ {\displaystyle |\psi \rangle } . The time evolution of a pure state is given by the Schrödinger equation :
Orthogonality is obtained when
and the minimum time interval t = t ⊥ {\displaystyle t=t_{\perp }} required to achieve this condition is called the orthogonalization interval [ 2 ] or orthogonalization time. [ 15 ]
For pure states, the Mandelstam–Tamm theorem states that the minimum time t ⊥ {\displaystyle t_{\perp }} required for a state to evolve into an orthogonal state is bounded below:
where
is the variance of the system's energy and H {\displaystyle H} is the Hamiltonian operator . The quantum evolution is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space ; the distance along this curve is measured by the Fubini–Study metric . [ 16 ] This is sometimes called the quantum angle , as it can be understood as the arccos of the inner product of the initial and final states.
The Mandelstam–Tamm limit can also be stated for mixed states and for time-varying Hamiltonians. In this case, the Bures metric must be employed in place of the Fubini–Study metric. A mixed state can be understood as a sum over pure states, weighted by classical probabilities ; likewise, the Bures metric is a weighted sum of the Fubini–Study metric. For a time-varying Hamiltonian H t {\displaystyle H_{t}} and time-varying density matrix ρ t , {\displaystyle \rho _{t},} the variance of the energy is given by
The Mandelstam–Tamm limit then takes the form
where D B {\displaystyle D_{B}} is the Bures distance between the starting and ending states. The Bures distance is geodesic , giving the shortest possible distance of any continuous curve connecting two points, with σ H ( t ) {\displaystyle \sigma _{H}(t)} understood as an infinitessimal path length along a curve parametrized by t . {\displaystyle t.} Equivalently, the time τ {\displaystyle \tau } taken to evolve from ρ {\displaystyle \rho } to ρ ′ {\displaystyle \rho '} is bounded as
where
is the time-averaged uncertainty in energy. For a pure state evolving under a time-varying Hamiltonian, the time τ {\displaystyle \tau } taken to evolve from one pure state to another pure state orthogonal to it is bounded as [ 17 ]
This follows, as for a pure state, one has the density matrix ρ t = | ψ t ⟩ ⟨ ψ t | . {\displaystyle \rho _{t}=|\psi _{t}\rangle \langle \psi _{t}|.} The quantum angle (Fubini–Study distance) is then D B ( ρ 0 , ρ t ) = arccos | ⟨ ψ 0 | ψ t ⟩ | {\displaystyle D_{B}(\rho _{0},\rho _{t})=\arccos |\langle \psi _{0}|\psi _{t}\rangle |} and so one concludes D B = arccos 0 = π / 2 {\displaystyle D_{B}=\arccos 0=\pi /2} when the initial and final states are orthogonal.
For the case of a pure state, Margolus and Levitin [ 3 ] obtain a different limit, that
where ⟨ E ⟩ {\displaystyle \langle E\rangle } is the average energy, ⟨ E ⟩ = E avg = ⟨ ψ | H | ψ ⟩ = ∑ n | c n | 2 E n . {\displaystyle \langle E\rangle =E_{\text{avg}}=\langle \psi |H|\psi \rangle =\sum _{n}|c_{n}|^{2}E_{n}.} This form applies when the Hamiltonian is not time-dependent, and the ground-state energy is defined to be zero.
The Margolus–Levitin theorem can also be generalized to the case where the Hamiltonian varies with time, and the system is described by a mixed state . [ 17 ] In this form, it is given by
with the ground-state defined so that it has energy zero at all times.
This provides a result for time varying states. Although it also provides a bound for mixed states, the bound (for mixed states) can be so loose as to be uninformative. [ 18 ] The Margolus–Levitin theorem has not yet been experimentally established in time-dependent quantum systems, whose Hamiltonians H t {\displaystyle H_{t}} are driven by arbitrary time-dependent parameters, except for the adiabatic case. [ 19 ]
In addition to the original Margolus–Levitin limit, a dual bound exists for quantum systems with a bounded energy spectrum. This dual bound, also known as the Ness–Alberti–Sagi limit or the Ness limit, depends on the difference between the state's mean energy and the energy of the highest occupied eigenstate. In bounded systems, the minimum time τ ⊥ {\displaystyle \tau _{\perp }} required for a state to evolve to an orthogonal state is bounded by
τ ⊥ ≥ h 4 ( E max − ⟨ E ⟩ ) , {\displaystyle \tau _{\perp }\geq {\frac {h}{4(E_{\max }-\langle E\rangle )}},}
where E max {\displaystyle E_{\max }} is the energy of the highest occupied eigenstate and ⟨ E ⟩ {\displaystyle \langle E\rangle } is the mean energy of the state. This bound complements the original Margolus–Levitin limit and the Mandelstam–Tamm limit, forming a trio of constraints on quantum evolution speed. [ 20 ]
A 2009 result by Lev B. Levitin and Tommaso Toffoli states that the precise bound for the Mandelstam–Tamm theorem is attained only for a qubit state. [ 15 ] This is a two-level state in an equal superposition
for energy eigenstates E 0 = 0 {\displaystyle E_{0}=0} and E 1 = ± π ℏ / Δ t {\displaystyle E_{1}=\pm \pi \hbar /\Delta t} . The states | E 0 ⟩ {\displaystyle \left|E_{0}\right\rangle } and | E 1 ⟩ {\displaystyle \left|E_{1}\right\rangle } are unique up to degeneracy of the energy level E 1 {\displaystyle E_{1}} and an arbitrary phase factor φ . {\displaystyle \varphi .} This result is sharp, in that this state also satisfies the Margolus–Levitin bound, in that E avg = δ E {\displaystyle E_{\text{avg}}=\delta E} and so t ⊥ = ℏ π / 2 E avg = ℏ π / 2 δ E . {\displaystyle t_{\perp }=\hbar \pi /2E_{\text{avg}}=\hbar \pi /2\delta E.} This result establishes that the combined limits are strict:
Levitin and Toffoli also provide a bound for the average energy in terms of the maximum. For any pure state | ψ ⟩ , {\displaystyle \left|\psi \right\rangle ,} the average energy is bounded as
where E max {\displaystyle E_{\text{max}}} is the maximum energy eigenvalue appearing in | ψ ⟩ . {\displaystyle \left|\psi \right\rangle .} (This is the quarter-pinched sphere theorem in disguise, transported to complex projective space .) Thus, one has the bound
The strict lower bound E max t ⊥ = π ℏ {\displaystyle E_{\text{max}}t_{\perp }=\pi \hbar } is again attained for the qubit state | ψ q ⟩ {\displaystyle \left|\psi _{q}\right\rangle } with E max = E 1 {\displaystyle E_{\text{max}}=E_{1}} .
The quantum speed limit bounds establish an upper bound at which computation can be performed. Computational machinery is constructed out of physical matter that follows quantum mechanics, and each operation, if it is to be unambiguous, must be a transition of the system from one state to an orthogonal state. Suppose the computing machinery is a physical system evolving under Hamiltonian that does not change with time. Then, according to the Margolus–Levitin theorem, the number of operations per unit time per unit energy is bounded above by
This establishes a strict upper limit on the number of calculations that can be performed by physical matter. The processing rate of all forms of computation cannot be higher than about 6 × 10 33 operations per second per joule of energy. This is including "classical" computers, since even classical computers are still made of matter that follows quantum mechanics. [ 21 ] [ 22 ]
This bound is not merely a fanciful limit: it has practical ramifications for quantum-resistant cryptography . Imagining a computer operating at this limit, a brute-force search to break a 128-bit encryption key requires only modest resources. Brute-forcing a 256-bit key requires planetary-scale computers, while a brute-force search of 512-bit keys is effectively unattainable within the lifetime of the universe, even if galactic-sized computers were applied to the problem.
The Bekenstein bound limits the amount of information that can be stored within a volume of space. The maximal rate of change of information within that volume of space is given by the quantum speed limit. This product of limits is sometimes called the Bremermann–Bekenstein limit ; it is saturated by Hawking radiation . [ 1 ] That is, Hawking radiation is emitted at the maximal allowed rate set by these bounds. | https://en.wikipedia.org/wiki/Quantum_speed_limit |
The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional semiconductors that have a quantized spin-Hall conductance and a vanishing charge-Hall conductance. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, and that does not require the application of a large magnetic field. The quantum spin Hall state does not break charge conservation symmetry and spin- S z {\displaystyle S_{z}} conservation symmetry (in order to have well defined Hall conductances).
The first proposal for the existence of a quantum spin Hall state was developed by Charles Kane and Gene Mele [ 1 ] who adapted an earlier model for graphene by F. Duncan M. Haldane [ 2 ] which exhibits an integer quantum Hall effect. The Kane and Mele model is two copies of the Haldane model such that the spin up electron exhibits a chiral integer quantum Hall Effect while the spin down electron exhibits an anti-chiral integer quantum Hall effect. A relativistic version of the quantum spin Hall effect was introduced in the 1990s for the numerical simulation of chiral gauge theories; [ 3 ] [ 4 ] the simplest example consisting of a parity and time reversal symmetric U(1) gauge theory with bulk fermions of opposite sign mass, a massless Dirac surface mode, and bulk currents that carry chirality but not charge (the spin Hall current analogue). Overall the Kane-Mele model has a charge-Hall conductance of exactly zero but a spin-Hall conductance of exactly σ x y spin = 2 {\displaystyle \sigma _{xy}^{\text{spin}}=2} (in units of e 4 π {\displaystyle {\frac {e}{4\pi }}} ). Independently, a quantum spin Hall model was proposed by Andrei Bernevig and Shoucheng Zhang [ 5 ] in an intricate strain architecture which engineers, due to spin-orbit coupling, a magnetic field pointing upwards for spin-up electrons and a magnetic field pointing downwards for spin-down electrons. The main ingredient is the existence of spin–orbit coupling , which can be understood as a momentum-dependent magnetic field coupling to the spin of the electron.
Real experimental systems, however, are far from the idealized picture presented above in which spin-up and spin-down electrons are not coupled. A very important achievement was the realization that the quantum spin Hall state remains to be non-trivial even after the introduction of spin-up spin-down scattering, [ 6 ] which destroys the quantum spin Hall effect. In a separate paper, Kane and Mele introduced a topological Z 2 {\displaystyle \mathbb {Z} _{2}} invariant which characterizes a state as trivial or non-trivial band insulator (regardless if the state exhibits or does not exhibit a quantum spin Hall effect). Further stability studies of the edge liquid through which conduction takes place in the quantum spin Hall state proved, both analytically and numerically that the non-trivial state is robust to both interactions and extra spin-orbit coupling terms that mix spin-up and spin-down electrons. Such a non-trivial state (exhibiting or not exhibiting a quantum spin Hall effect) is called a topological insulator , which is an example of symmetry-protected topological order protected by charge conservation symmetry and time reversal symmetry. (Note that the quantum spin Hall state is also a symmetry-protected topological state protected by charge conservation symmetry and spin- S z {\displaystyle S_{z}} conservation symmetry. We do not need time reversal symmetry to protect quantum spin Hall state. Topological insulator and quantum spin Hall state are different symmetry-protected topological states. So topological insulator and quantum spin Hall state are different states of matter.)
Since graphene has extremely weak spin-orbit coupling, it is very unlikely to support a quantum spin Hall state at temperatures achievable with today's technologies. Two-dimensional topological insulators (also known as the quantum spin Hall insulators) with one-dimensional helical edge states were predicted in 2006 by Bernevig, Hughes and Zhang to occur in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride, [ 7 ] and were observed in 2007. [ 8 ]
Different quantum wells of varying HgTe thickness can be built. When the sheet of HgTe in between the CdTe is thin, the system behaves like an ordinary insulator and does not conduct when the Fermi level resides in the band-gap. When the sheet of HgTe is varied and made thicker (this requires the fabrication of separate quantum wells), an interesting phenomenon happens. Due to the inverted band structure of HgTe, at some critical HgTe thickness, a Lifshitz transition occurs in which the system closes the bulk band gap to become a semi-metal, and then re-opens it to become a quantum spin Hall insulator.
In the gap closing and re-opening process, two edge states are brought out from the bulk and cross the bulk-gap. As such, when the Fermi level resides in the bulk gap, the conduction is dominated by the edge channels that cross the gap. The two-terminal conductance is G x x = 2 e 2 h {\displaystyle G_{xx}=2{\frac {e^{2}}{h}}} in the quantum spin Hall state and zero in the normal insulating state. As the conduction is dominated by the edge channels, the value of the conductance should be insensitive to how wide the sample is. A magnetic field should destroy the quantum spin Hall state by breaking time-reversal invariance and allowing spin-up spin-down electron scattering processes at the edge. All these predictions have been experimentally verified in an experiment [ 9 ] performed in the Molenkamp labs at Universität Würzburg in Germany. | https://en.wikipedia.org/wiki/Quantum_spin_Hall_effect |
In condensed matter physics , a quantum spin liquid is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglement , fractionalized excitations , and absence of ordinary magnetic order . [ 1 ]
The quantum spin liquid state was first proposed by physicist Phil Anderson in 1973 as the ground state for a system of spins on a triangular lattice that interact antiferromagnetically with their nearest neighbors, i.e. neighboring spins seek to be aligned in opposite directions. [ 2 ] Quantum spin liquids generated further interest when in 1987 Anderson proposed a theory that described high-temperature superconductivity in terms of a disordered spin-liquid state. [ 3 ] [ 4 ]
The simplest kind of magnetic phase is a paramagnet , where each individual spin behaves independently of the rest, just like atoms in an ideal gas . This highly disordered phase is the generic state of magnets at high temperatures, where thermal fluctuations dominate. Upon cooling, the spins will often enter a ferromagnet (or antiferromagnet ) phase. In this phase, interactions between the spins cause them to align into large-scale patterns, such as domains , stripes, or checkerboards. These long-range patterns are referred to as "magnetic order," and are analogous to the regular crystal structure formed by many solids. [ 5 ]
Quantum spin liquids offer a dramatic alternative to this typical behavior. One intuitive description of this state is as a "liquid" of disordered spins, in comparison to a ferromagnetic spin state, [ 6 ] much in the way liquid water is in a disordered state compared to crystalline ice. However, unlike other disordered states, a quantum spin liquid state preserves its disorder to very low temperatures. [ 7 ] A more modern characterization of quantum spin liquids involves their topological order , [ 8 ] long-range quantum entanglement properties, [ 1 ] and anyon excitations. [ 9 ]
Several physical models have a disordered ground state that can be described as a quantum spin liquid.
Localized spins are frustrated if there exist competing exchange interactions that can not all be satisfied at the same time, leading to a large degeneracy of the system's ground state. A triangle of Ising spins (meaning that the only possible orientation of the spins are either "up" or "down"), which interact antiferromagnetically, is a simple example for frustration. In the ground state, two of the spins can be antiparallel but the third one cannot. This leads to an increase of possible orientations (six in this case) of the spins in the ground state, enhancing fluctuations and thus suppressing magnetic ordering.
To build a ground state without magnetic moment, valence bond states can be used, where two electron spins form a spin 0 singlet due to the antiferromagnetic interaction. If every spin in the system is bound like this, the state of the system as a whole has spin 0 too and is non-magnetic. The two spins forming the bond are maximally entangled , while not being entangled with the other spins. If all spins are distributed to certain localized static bonds, this is called a valence bond solid (VBS).
There are two things that still distinguish a VBS from a spin liquid: First, by ordering the bonds in a certain way, the lattice symmetry is usually broken, which is not the case for a spin liquid. Second, this ground state lacks long-range entanglement. To achieve this, quantum mechanical fluctuations of the valence bonds must be allowed, leading to a ground state consisting of a superposition of many different partitionings of spins into valence bonds. If the partitionings are equally distributed (with the same quantum amplitude), there is no preference for any specific partitioning ("valence bond liquid"). This kind of ground state wavefunction was proposed by P. W. Anderson in 1973 as the ground state of spin liquids [ 2 ] and is called a resonating valence bond (RVB) state. These states are of great theoretical interest as they are proposed to play a key role in high-temperature superconductor physics. [ 4 ]
The valence bonds do not have to be formed by nearest neighbors only and their distributions may vary in different materials. Ground states with large contributions of long range valence bonds have more low-energy spin excitations, as those valence bonds are easier to break up. On breaking, they form two free spins. Other excitations rearrange the valence bonds, leading to low-energy excitations even for short-range bonds. Something very special about spin liquids is that they support exotic excitations , meaning excitations with fractional quantum numbers. A prominent example is the excitation of spinons which are neutral in charge and carry spin S = 1 / 2 {\displaystyle S=1/2} . In spin liquids, a spinon is created if one spin is not paired in a valence bond. It can move by rearranging nearby valence bonds at low energy cost.
The first discussion of the RVB state on square lattice using the RVB picture [ 10 ] only consider nearest neighbour bonds that connect different sub-lattices. The constructed RVB state is an equal amplitude superposition of all the nearest-neighbour bond configurations. Such a RVB state is believed to contain emergent gapless U ( 1 ) {\displaystyle U(1)} gauge field which may confine the spinons etc. So the equal-amplitude nearest-neighbour RVB state on square lattice is unstable and does not corresponds to a quantum spin phase. It may describe a critical phase transition point between two stable phases. A version of RVB state which is stable and contains deconfined spinons is the chiral spin state. [ 11 ] [ 12 ] Later, another version of stable RVB state with deconfined spinons, the Z2 spin liquid, is proposed, [ 13 ] [ 14 ] which realizes the simplest topological order – Z2 topological order . Both chiral spin state and Z2 spin liquid state have long RVB bonds that connect the same sub-lattice. In chiral spin state, different bond configurations can have complex amplitudes, while in Z2 spin liquid state, different bond configurations only have real amplitudes. The RVB state on triangle lattice also realizes the Z2 spin liquid, [ 15 ] where different bond configurations only have real amplitudes. The toric code model is yet another realization of Z2 spin liquid (and Z2 topological order ) that explicitly breaks the spin rotation symmetry and is exactly solvable. [ 16 ]
Since there is no single experimental feature which identifies a material as a spin liquid, several experiments have to be conducted to gain information on different properties which characterize a spin liquid. [ 17 ]
In a high-temperature, classical paramagnet phase, the magnetic susceptibility is given by the Curie–Weiss law χ ∼ C T − Θ C W {\displaystyle \chi \sim {\frac {C}{T-\Theta _{CW}}}}
Fitting experimental data to this equation determines a phenomenological Curie–Weiss temperature, Θ C W {\displaystyle \Theta _{CW}} . There is a second temperature, T c {\displaystyle T_{c}} , where magnetic order in the material begins to develop, as evidenced by a non-analytic feature in χ ( T ) {\displaystyle \chi (T)} . The ratio of these is called the frustration parameter f = | Θ c w | T c {\displaystyle f={\frac {|\Theta _{cw}|}{T_{c}}}}
In a classic antiferromagnet, the two temperatures should coincide and give f = 1 {\displaystyle f=1} . An ideal quantum spin liquid would not develop magnetic order at any temperature ( T c = 0 ) {\displaystyle (T_{c}=0)} and so would have a diverging frustration parameter f → ∞ {\displaystyle f\to \infty } . [ 18 ] A large value f > 100 {\displaystyle f>100} is therefore a good indication of a possible spin liquid phase. Some frustrated materials with different lattice structures and their Curie–Weiss temperature are listed in the table below. [ 7 ] All of them are proposed spin liquid candidates.
One of the most direct evidence for absence of magnetic ordering give NMR or μSR experiments. If there is a local magnetic field present, the nuclear or muon spin would be affected which can be measured. 1 H- NMR measurements [ 22 ] on κ-(BEDT-TTF) 2 Cu 2 (CN) 3 have shown no sign of magnetic ordering down to 32 mK, which is four orders of magnitude smaller than the coupling constant J≈250 K [ 23 ] between neighboring spins in this compound. Further investigations include:
Neutron scattering measurements of cesium chlorocuprate Cs 2 CuCl 4 , a spin-1/2 antiferromagnet on a triangular lattice, displayed diffuse scattering. This was attributed to spinons arising from a 2D RVB state. [ 25 ] Later theoretical work challenged this picture, arguing that all experimental results were instead consequences of 1D spinons confined to individual chains. [ 26 ]
Afterwards, it was observed in an organic Mott insulator (κ-(BEDT-TTF) 2 Cu 2 (CN) 3 ) by Kanoda's group in 2003. [ 22 ] It may correspond to a gapless spin liquid with spinon Fermi surface (the so-called uniform RVB state). [ 2 ] The peculiar phase diagram of this organic quantum spin liquid compound was first thoroughly mapped using muon spin spectroscopy . [ 27 ]
Herbertsmithite is one of the most extensively studied QSL candidate materials. [ 18 ] It is a mineral with chemical composition ZnCu 3 (OH) 6 Cl 2 and a rhombohedral crystal structure. Notably, the copper ions within this structure form stacked two-dimensional layers of kagome lattices . Additionally, superexchange over the oxygen bonds creates a strong antiferromagnetic interaction between the S = 1 / 2 {\displaystyle S=1/2} copper spins within a single layer, whereas coupling between layers is negligible. [ 18 ] Therefore, it is a good realization of the antiferromagnetic spin-1/2 Heisenberg model on the kagome lattice, which is a prototypical theoretical example of a quantum spin liquid. [ 28 ] [ 29 ]
Synthetic, polycrystalline herbertsmithite powder was first reported in 2005, and initial magnetic susceptibility studies showed no signs of magnetic order down to 2K. [ 30 ] In a subsequent study, the absence of magnetic order was verified down to 50 mK, inelastic neutron scattering measurements revealed a broad spectrum of low energy spin excitations, and low-temperature specific heat measurements had power law scaling. This gave compelling evidence for a spin liquid state with gapless S = 1 / 2 {\displaystyle S=1/2} spinon excitations. [ 31 ] A broad array of additional experiments, including 17 O NMR , [ 32 ] and neutron spectroscopy of the dynamic magnetic structure factor , [ 33 ] reinforced the identification of herbertsmithite as a gapless spin liquid material, although the exact characterization remained unclear as of 2010. [ 34 ]
Large (millimeter size) single crystals of herbertsmithite were grown and characterized in 2011. [ 35 ] These enabled more precise measurements of possible spin liquid properties. In particular, momentum-resolved inelastic neutron scattering experiments showed a broad continuum of excitations. This was interpreted as evidence for gapless, fractionalized spinons. [ 36 ] Follow-up experiments (using 17 O NMR and high-resolution, low-energy neutron scattering) refined this picture and determined there was actually a small spinon excitation gap of 0.07–0.09 meV. [ 37 ] [ 38 ]
Some measurements were suggestive of quantum critical behavior. [ 39 ] [ 40 ] Magnetic response of this material displays scaling relation in both the bulk ac susceptibility and the low energy dynamic susceptibility, with the low temperature heat capacity strongly depending on magnetic field. [ 41 ] [ 42 ] This scaling is seen in certain quantum antiferromagnets , heavy-fermion metals , and two-dimensional 3 He as a signature of proximity to a quantum critical point. [ 43 ]
In 2020, monodisperse single-crystal nanoparticles of herbertsmithite (~10 nm) were synthesized at room temperature, using gas-diffusion electrocrystallization , showing that their spin liquid nature persists at such small dimensions. [ 44 ]
It may realize a U(1)-Dirac spin liquid. [ 47 ]
Another evidence of quantum spin liquid was observed in a 2-dimensional material in August 2015. The researchers of Oak Ridge National Laboratory , collaborating with physicists from the University of Cambridge, and the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, measured the first signatures of these fractional particles, known as Majorana fermions , in a two-dimensional material with a structure similar to graphene . Their experimental results successfully matched with one of the main theoretical models for a quantum spin liquid, known as a Kitaev honeycomb model . [ 48 ] [ 49 ]
The strongly correlated quantum spin liquid ( SCQSL ) is a specific realization of a possible quantum spin liquid (QSL) [ 7 ] [ 39 ] representing a new type of strongly correlated electrical insulator (SCI) that possesses properties of heavy fermion metals with one exception: it resists the flow of electric charge . [ 46 ] [ 50 ] At low temperatures T the specific heat of this type of insulator is proportional to T n , with n less or equal 1 rather than n =3, as it should be in the case of a conventional insulator whose heat capacity is proportional to T 3 . When a magnetic field B is applied to SCI the specific heat depends strongly on B , contrary to conventional insulators. There are a few candidates of SCI; the most promising among them is Herbertsmithite , [ 50 ] a mineral with chemical structure ZnCu 3 (OH) 6 Cl 2 .
Ca 10 Cr 7 O 28 is a frustrated kagome bilayer magnet , which does not develop long-range order even below 1 K, and has a diffuse spectrum of gapless excitations.
In December 2021, the first direct measurement of a quantum spin liquid of the toric code type was reported, [ 51 ] [ 52 ] it was achieved by two teams: one exploring ground state and anyonic excitations on a quantum processor [ 53 ] and the other implementing a theoretical blueprint [ 54 ] of atoms on a ruby lattice held with optical tweezers on a quantum simulator . [ 55 ]
The experimental facts collected on heavy fermion (HF) metals and two dimensional Helium-3 demonstrate that the quasiparticle effective mass M * is very large, or even diverges. Topological fermion condensation quantum phase transition (FCQPT) preserves quasiparticles , and forms flat energy band at the Fermi level . The emergence of FCQPT is directly related to the unlimited growth of the effective mass M *. [ 43 ] Near FCQPT, M* starts to depend on temperature T , number density x , magnetic field B and other external parameters such as pressure P , etc. In contrast to the Landau paradigm based on the assumption that the effective mass is approximately constant, in the FCQPT theory the effective mass of new quasiparticles strongly depends on T , x , B etc. Therefore, to agree/explain with the numerous experimental facts, extended quasiparticles paradigm based on FCQPT has to be introduced. The main point here is that the well-defined quasiparticles determine the thermodynamic , relaxation , scaling and transport properties of strongly correlated Fermi systems and M* becomes a function of T , x , B , P , etc.
The data collected for very different strongly correlated Fermi systems demonstrate universal scaling behavior; in other words distinct materials with strongly correlated fermions unexpectedly turn out to be uniform, thus forming a new state of matter that consists of HF metals , quasicrystals , quantum spin liquid, two-dimensional Helium-3 , and compounds exhibiting high-temperature superconductivity . [ 39 ] [ 43 ]
Materials supporting quantum spin liquid states may have applications in data storage and memory. [ 56 ] In particular, it is possible to realize topological quantum computation by means of spin-liquid states. [ 57 ] Developments in quantum spin liquids may also help in the understanding of high temperature superconductivity . [ 58 ] | https://en.wikipedia.org/wiki/Quantum_spin_liquid |
A spin model is a mathematical model used in physics primarily to explain magnetism . Spin models may either be classical or quantum mechanical in nature. Spin models have been studied in quantum field theory as examples of integrable models . Spin models are also used in quantum information theory and computability theory in theoretical computer science . The theory of spin models is a far reaching and unifying topic that cuts across many fields.
In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. Ferromagnetic materials below their Curie temperature , however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.
The study of the behavior of such "spin models" is a thriving area of research in condensed matter physics . For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. In certain magnets, the magnetic dipoles are only free to rotate in a 2D plane, a system which can be adequately described by the so-called xy-model .
The lack of a unified theory of magnetism [ 1 ] forces scientist to model magnetic systems theoretically with one, or a combination of these spin models in order to understand the intricate behavior of atomic magnetic interactions. Numerical implementation of these models has led to several interesting results, such as quantitative research in the theory of phase transitions .
A quantum spin model is a quantum Hamiltonian model that describes a system which consists of spins either interacting or not and are an active area of research in the fields of strongly correlated electron systems, quantum information theory , and quantum computing . [ 2 ] The physical observables in these quantum models are actually operators in a Hilbert space acting on state vectors as opposed to the physical observables in the corresponding classical spin models - like the Ising model - which are commutative variables. | https://en.wikipedia.org/wiki/Quantum_spin_model |
Quantum spin tunneling , or quantum tunneling of magnetization, is a physical phenomenon by which the quantum mechanical state that describes the collective magnetization of a nanomagnet is a linear superposition of two states with well defined and opposite magnetization. Classically, the magnetic anisotropy favors neither of the two states with opposite magnetization, so that the system has two equivalent ground states.
Because of the quantum spin tunneling, an energy splitting between the bonding and anti-bonding linear combination of states with opposite magnetization classical ground states arises, giving rise to a unique ground state [ 1 ] separated by the first excited state by an energy difference known as quantum spin tunneling splitting . The quantum spin tunneling splitting also occurs for pairs of excited states with opposite magnetization.
As a consequence of quantum spin tunneling, the magnetization of a system can switch between states with opposite magnetization that are separated by an energy barrier much larger than thermal energy. Thus, quantum spin tunneling provides a pathway to magnetization switching forbidden in classical physics.
Whereas quantum spin tunneling shares some properties with quantum tunneling in other two level systems such as a single electron in a double quantum well or in a diatomic molecule, it is a multi-electron phenomenon, since more than one electron is required to have magnetic anisotropy. The multi-electron character is also revealed by an important feature, absent in single-electron tunneling: zero field quantum spin tunneling splitting is only possible for integer spins, and is certainly absent for half-integer spins, as ensured by Kramers degeneracy theorem . In real systems containing Kramers ions, like crystalline samples of single ion magnets, the degeneracy of the ground states is frequently lifted through dipolar interactions with neighboring spins, and as such quantum spin tunneling is frequently observed even in the absence of an applied external field for these systems. [ citation needed ]
Initially discussed in the context of magnetization dynamics [ 2 ] [ 3 ] [ 4 ] [ 5 ] of magnetic nanoparticles , the concept was known as macroscopic quantum tunneling , a term that highlights both the difference with single electron tunneling and connects this phenomenon with other macroscopic quantum phenomena . In this sense, the problem of quantum spin tunneling lies in the boundary between the quantum and classical descriptions of reality.
A simple single spin Hamiltonian that describes quantum spin tunneling for a spin S {\displaystyle S} is given by:
H ^ = D S ^ z 2 + E ( S ^ x 2 − S ^ y 2 ) {\textstyle {\hat {H}}=D{\hat {S}}_{z}^{2}+E({\hat {S}}_{x}^{2}-{\hat {S}}_{y}^{2})} [1]
where D and E are parameters that determine the magnetic anisotropy, and S ^ x , S ^ y , S ^ z {\displaystyle {\hat {S}}_{x},{\hat {S}}_{y},{\hat {S}}_{z}} are spin matrices of dimension 2 S + 1 {\displaystyle 2S+1} . It is customary to take z as the easy axis so that D<0 and |D|>> E . For E=0, this Hamiltonian commutes with S ^ z {\displaystyle {\hat {S}}_{z}} , so that we can write the eigenvalues as E ( S z ) = D S z 2 {\displaystyle E(S_{z})=DS_{z}^{2}} , where S z {\displaystyle S_{z}} takes the 2S+1 values in the list (S, S-1, ...., -S)
describes a set of doublets, with E = 0 {\displaystyle E=0} and D < 0 {\displaystyle D<0} . In the case of integer spins the second term of the Hamiltonian results in the splitting of the otherwise degenerate ground state doublet. In this case, the zero field quantum spin tunneling splitting is given by:
Δ ∝ E ( E D ) ( S − 1 ) {\displaystyle \Delta \propto E\left({\frac {E}{D}}\right)^{(S-1)}}
From this result, it is apparent that, given that E/D is much smaller than 1 by construction, the quantum spin tunnelling splitting becomes suppressed in the limit of large spin S, i.e., as we move from the atomic scale towards the macroscopic world. The magnitude of the quantum spin tunnelling splitting can be modulated by application of a magnetic field along the transverse hard axis direction (in the case of Hamiltonian [1], with D<0 and E>0, the x axis). The modulation of the quantum spin tunnelling splitting results in oscillations of its magnitude, [ 6 ] including specific values of the transverse field at which the splitting vanishes. This accidental degeneracies are known as diabolic points .
Quantum tunneling of the magnetization was reported in 1996 for a crystal of Mn 12 ac molecules with S=10. [ 7 ] Quoting Thomas and coworkers, [ 7 ] " in an applied magnetic field, the magnetization shows hysteresis loops with a distinct 'staircase' structure: the steps occur at values of the applied field where the energies of different collective spin states of the manganese clusters coincide. At these special values of the field, relaxation from one spin state to another is enhanced above the thermally activated rate by the action of resonant quantum-mechanical tunneling ". Quantum tunneling of the magnetization was reported in ferritin [ 8 ] present in horse spleen proteins [ 9 ]
A direct measurement of the quantum spin tunneling splitting energy can be achieved using single spin scanning tunneling inelastic spectroscopy, that permits to measure the spin excitations of individual atoms on surfaces. [ 10 ] Using this technique, the spin excitation spectrum of an individual integer spin was obtained by Hirjibehedin et al. [ 11 ] for a S=2 single Fe atom on a surface of Cu 2 N/Cu(100), that made it possible to measure a quantum spin tunneling splitting of 0.2 meV. Using the same technique other groups measured the spin excitations of S=1 Fe phthalocyanine molecule on a copper surface [ 12 ] and a S=1 Fe atom on InSb, [ 13 ] both of which had a quantum spin tunneling splitting of the S z = ± 1 {\displaystyle S_{z}=\pm 1} doublet larger than 1 meV.
In the case of molecular magnets with large S and small E/D ratio, indirect measurement techniques are required to infer the value of the quantum spin tunneling splitting. For instance, modeling time dependent magnetization measurements of a crystal of Fe 8 molecular magnets with the Landau-Zener formula, Wernsdorfer and Sessoli [ 14 ] inferred tunneling splittings in the range of 10 −7 Kelvin. | https://en.wikipedia.org/wiki/Quantum_spin_tunneling |
The term quantum state discrimination collectively refers to quantum-informatics techniques, with the help of which, by performing a small number of measurements on a physical system, its specific quantum state can be identified . And this is provided that the set of states in which the system can be is known in advance, and we only need to determine which one it is. This assumption distinguishes such techniques from quantum tomography, which does not impose additional requirements on the state of the system, but requires many times more measurements.
If the set of states in which the investigated system can be is represented by orthogonal vectors, the situation is particularly simple. To unambiguously determine the state of the system, it is enough to perform a quantum measurement in the basis formed by these vectors. The given quantum state can then be flawlessly identified from the measured value. Moreover, it can be easily shown that if the individual states are not orthogonal to each other, there is no way to tell them apart with certainty. Therefore, in such a case, it is always necessary to take into account the possibility of incorrect or inconclusive determination of the state of the system. However, there are techniques that try to alleviate this deficiency. With exceptions, these techniques can be divided into two groups, namely those based on error minimization and then those that allow the state to be determined unambiguously in exchange for lower efficiency.
The first group of techniques is based on the works of Carl W. Helstrom from the 60s and 70s of the 20th century [ 1 ] and in its basic form consists in the implementation of projective quantum measurement, where the measurement operators are projective representations. The second group is based on the conclusions of a scientific article published by ID Ivanovich in 1987 [ 2 ] and requires the use of generalized measurement, in which the elements of the POVM set are taken as measurement operators. Both groups of techniques are currently the subject of active, primarily theoretical, research, and apart from a number of special cases, there is no general solution that would allow choosing measurement operators in the form of expressible analytical formula.
More precisely, in its standard formulation, the problem involves performing some POVM ( E i ) i {\displaystyle (E_{i})_{i}} on a given unknown state ρ {\displaystyle \rho } , under the promise that the state received is an element of a collection of states { σ i } i {\displaystyle \{\sigma _{i}\}_{i}} , with σ i {\displaystyle \sigma _{i}} occurring with probability p i {\displaystyle p_{i}} , that is, ρ = ∑ i p i σ i {\displaystyle \rho =\sum _{i}p_{i}\sigma _{i}} . The task is then to find the probability of the POVM ( E i ) i {\displaystyle (E_{i})_{i}} correctly guessing which state was received. Since the probability of the POVM returning the i {\displaystyle i} -th outcome when the given state was σ j {\displaystyle \sigma _{j}} has the form Prob ( i | j ) = tr ( E i σ j ) {\displaystyle {\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})} , it follows that the probability of successfully determining the correct state is P s u c c e s s = ∑ i p i tr ( σ i E i ) {\displaystyle P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})} . [ 3 ] [ 4 ]
The discrimination of two states can be solved optimally using the Helstrom measurement . [ 5 ] With two states { σ 0 , σ 1 } {\displaystyle \{\sigma _{0},\sigma _{1}\}} comes two probabilities { p 0 , p 1 } {\displaystyle \{p_{0},p_{1}\}} and POVMs { E 0 , E 1 } {\displaystyle \{E_{0},E_{1}\}} . Since ∑ i E i = I {\displaystyle \sum _{i}E_{i}=I} for all POVMs, E 1 = I − E 0 {\displaystyle E_{1}=I-E_{0}} . So the probability of success is:
To maximize the probability of success, the trace needs to be maximized. That's accomplished when E 0 {\displaystyle E_{0}} is a projector on the positive eigenspace of p 0 σ 0 − p 1 σ 1 {\displaystyle p_{0}\sigma _{0}-p_{1}\sigma _{1}} , [ 5 ] and the maximal probability of success is given by
where ‖ ⋅ ‖ 1 {\displaystyle \|\cdot \|_{1}} denotes the trace norm .
If the task is to discriminate between more than two quantum states, there is no general formula for the optimal POVM and success probability. Nonetheless, the optimal success probability, for the task of discriminating between the elements of a given ensemble { ( p i , σ i ) } i = 1 N {\displaystyle \{(p_{i},\sigma _{i})\}_{i=1}^{N}} , can always be written as [ 4 ] P s u c c e s s = max { E i } ∑ i p i tr ( E i σ i ) . {\displaystyle P_{\rm {success}}=\max _{\{E_{i}\}}\sum _{i}p_{i}\operatorname {tr} (E_{i}\sigma _{i}).} This is obtained observing that p i {\displaystyle p_{i}} is the a priori probability of getting the i {\displaystyle i} -th state, and tr ( E i σ i ) {\displaystyle \operatorname {tr} (E_{i}\sigma _{i})} is the probability of (correctly) guessing the input to be σ i {\displaystyle \sigma _{i}} , conditioned to having indeed received the state σ i {\displaystyle \sigma _{i}} .
While this expression cannot be given an explicit form in the general case, it can be solved numerically via Semidefinite programming . [ 4 ] An alternative approach to discriminate between a given ensemble of states is to the use the so-called Pretty Good Measurement (PGM), also known as the square root measurement . This is an alternative discrimination strategy that is not in general optimal, but can still be shown to work pretty well. [ 6 ] | https://en.wikipedia.org/wiki/Quantum_state_discrimination |
In quantum information theory , quantum state purification refers to the process of representing a mixed state as a pure quantum state of higher-dimensional Hilbert space . The purification allows the original mixed state to be recovered by taking the partial trace over the additional degrees of freedom. The purification is not unique, the different purifications that can lead to the same mixed states are limited by the Schrödinger–HJW theorem .
Purification is used in algorithms such as entanglement distillation , magic state distillation and algorithmic cooling .
Let H S {\displaystyle {\mathcal {H}}_{S}} be a finite-dimensional complex Hilbert space , and consider a generic (possibly mixed ) quantum state ρ {\displaystyle \rho } defined on H S {\displaystyle {\mathcal {H}}_{S}} and admitting a decomposition of the form ρ = ∑ i p i | ϕ i ⟩ ⟨ ϕ i | {\displaystyle \rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|} for a collection of (not necessarily mutually orthogonal) states | ϕ i ⟩ ∈ H S {\displaystyle |\phi _{i}\rangle \in {\mathcal {H}}_{S}} and coefficients p i ≥ 0 {\displaystyle p_{i}\geq 0} such that ∑ i p i = 1 {\textstyle \sum _{i}p_{i}=1} . Note that any quantum state can be written in such a way for some { | ϕ i ⟩ } i {\displaystyle \{|\phi _{i}\rangle \}_{i}} and { p i } i {\displaystyle \{p_{i}\}_{i}} . [ 1 ]
Any such ρ {\displaystyle \rho } can be purified , that is, represented as the partial trace of a pure state defined in a larger Hilbert space. More precisely, it is always possible to find a (finite-dimensional) Hilbert space H A {\displaystyle {\mathcal {H}}_{A}} and a pure state | Ψ S A ⟩ ∈ H S ⊗ H A {\displaystyle |\Psi _{SA}\rangle \in {\mathcal {H}}_{S}\otimes {\mathcal {H}}_{A}} such that ρ = Tr A ( | Ψ S A ⟩ ⟨ Ψ S A | ) {\displaystyle \rho =\operatorname {Tr} _{A}{\big (}|\Psi _{SA}\rangle \langle \Psi _{SA}|{\big )}} . Furthermore, the states | Ψ S A ⟩ {\displaystyle |\Psi _{SA}\rangle } satisfying this are all and only those of the form | Ψ S A ⟩ = ∑ i p i | ϕ i ⟩ ⊗ | a i ⟩ {\displaystyle |\Psi _{SA}\rangle =\sum _{i}{\sqrt {p_{i}}}|\phi _{i}\rangle \otimes |a_{i}\rangle } for some orthonormal basis { | a i ⟩ } i ⊂ H A {\displaystyle \{|a_{i}\rangle \}_{i}\subset {\mathcal {H}}_{A}} . The state | Ψ S A ⟩ {\displaystyle |\Psi _{SA}\rangle } is then referred to as the "purification of ρ {\displaystyle \rho } ". Since the auxiliary space and the basis can be chosen arbitrarily, the purification of a mixed state is not unique; in fact, there are infinitely many purifications of a given mixed state. [ 2 ] Because all of them admit a decomposition in the form given above, given any pair of purifications | Ψ ⟩ , | Ψ ′ ⟩ ∈ H S ⊗ H A {\displaystyle |\Psi \rangle ,|\Psi '\rangle \in {\mathcal {H}}_{S}\otimes {\mathcal {H}}_{A}} , there is always some unitary operation U : H A → H A {\displaystyle U:{\mathcal {H}}_{A}\to {\mathcal {H}}_{A}} such that | Ψ ′ ⟩ = ( I ⊗ U ) | Ψ ⟩ . {\displaystyle |\Psi '\rangle =(I\otimes U)|\Psi \rangle .}
The Schrödinger–HJW theorem is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications of the density operators . The theorem is named after Erwin Schrödinger who proved it in 1936, [ 3 ] and after Lane P. Hughston , Richard Jozsa and William Wootters who rediscovered in 1993. [ 4 ] The result was also found independently (albeit partially) by Nicolas Gisin in 1989, [ 5 ] and by Nicolas Hadjisavvas building upon work by E. T. Jaynes of 1957, [ 6 ] [ 7 ] while a significant part of it was likewise independently discovered by N. David Mermin in 1999 who discovered the link with Schrödinger's work. [ 8 ] Thanks to its complicated history, it is also known by various other names such as the GHJW theorem , [ 9 ] the HJW theorem , and the purification theorem .
Consider a mixed quantum state ρ {\displaystyle \rho } with two different realizations as ensemble of pure states as ρ = ∑ i p i | ϕ i ⟩ ⟨ ϕ i | {\textstyle \rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|} and ρ = ∑ j q j | φ j ⟩ ⟨ φ j | {\textstyle \rho =\sum _{j}q_{j}|\varphi _{j}\rangle \langle \varphi _{j}|} . Here both | ϕ i ⟩ {\displaystyle |\phi _{i}\rangle } and | φ j ⟩ {\displaystyle |\varphi _{j}\rangle } are not assumed to be mutually orthogonal. There will be two corresponding purifications of the mixed state ρ {\displaystyle \rho } reading as follows:
The sets { | a i ⟩ } {\displaystyle \{|a_{i}\rangle \}} and { | b j ⟩ } {\displaystyle \{|b_{j}\rangle \}} are two collections of orthonormal bases of the respective auxiliary spaces. These two purifications only differ by a unitary transformation acting on the auxiliary space, namely, there exists a unitary matrix U A {\displaystyle U_{A}} such that | Ψ S A 1 ⟩ = ( I ⊗ U A ) | Ψ S A 2 ⟩ {\displaystyle |\Psi _{SA}^{1}\rangle =(I\otimes U_{A})|\Psi _{SA}^{2}\rangle } . [ 10 ] Therefore, | Ψ S A 1 ⟩ = ∑ j q j | φ j ⟩ ⊗ U A | b j ⟩ {\textstyle |\Psi _{SA}^{1}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle \otimes U_{A}|b_{j}\rangle } , which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system. | https://en.wikipedia.org/wiki/Quantum_state_purification |
In physics , a quantum state space is an abstract space in which different "positions" represent not literal locations, but rather quantum states of some physical system . It is the quantum analog of the phase space of classical mechanics .
In quantum mechanics a state space is a separable complex Hilbert space . The dimension of this Hilbert space depends on the system we choose to describe. [ 1 ] [ 2 ] The different states that could come out of any particular measurement form an orthonormal basis , so any state vector in the state space can be written as a linear combination of these basis vectors. Having a nonzero component along multiple dimensions is called a superposition . In the formalism of quantum mechanics these state vectors are often written using Dirac's compact bra–ket notation . [ 3 ] : 165
The spin state of a silver atom in the Stern–Gerlach experiment can be represented in a two state space. The spin can be aligned with a measuring apparatus (arbitrarily called 'up') or oppositely ('down'). [ 4 ] In Dirac's notation these two states can be written as | u ⟩ , | d ⟩ {\displaystyle |u\rangle ,|d\rangle } . The space of a two spin system has four states, | u u ⟩ , | u d ⟩ , | d u ⟩ , | d d ⟩ {\displaystyle |uu\rangle ,|ud\rangle ,|du\rangle ,|dd\rangle } .
The spin state is a discrete degree of freedom ; quantum state spaces can have continuous degrees of freedom. For example, a particle in one space dimension has one degree of freedom ranging from − ∞ {\displaystyle -\infty } to ∞ {\displaystyle \infty } . In Dirac notation, the states in this space might be written as | q ⟩ {\displaystyle |q\rangle } or | ψ ⟩ {\displaystyle |\psi \rangle } . [ 5 ] : 302
Even in the early days of quantum mechanics, the state space (or configurations as they were called at first) was understood to be essential for understanding simple quantum-mechanical problems. In 1929, Nevill Mott showed that "tendency to picture the wave as existing in ordinary three dimensional space, whereas we are really dealing with wave functions in multispace" makes analysis of simple interaction problems more difficult. [ 6 ] Mott analyzes α {\displaystyle \alpha } -particle emission in a cloud chamber . The emission process is isotropic, a spherical wave in quantum mechanics, but the tracks observed are linear.
As Mott says, "it is a little difficult to picture how it is that an
outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space". This issue became known at the Mott problem . Mott then derives the straight track by considering correlations between the positions of the source and two representative atoms, showing that consecutive ionization results from just that state in which all three positions are co-linear. [ 7 ]
Classical mechanics for multiple objects describes their motion in terms of a list or vector of every object's coordinates and velocity. As the objects move, the values in the vector change; the set of all possible values is called a phase space . [ 8 ] : 88 In quantum mechanics a state space is similar, however in the state space two vectors which are scalar multiples of each other represent the same state. Furthermore, the character of values in the quantum state differ from the classical values: in the quantum case the values can only be measured statistically (by repetition over many examples) and thus do not have well defined values at every instant of time. [ 5 ] : 294 | https://en.wikipedia.org/wiki/Quantum_state_space |
In physics , in the area of quantum information theory and quantum computation , quantum steering is a special kind of nonlocal correlation, which is intermediate between Bell nonlocality and quantum entanglement . A state exhibiting Bell nonlocality must also exhibit quantum steering, a state exhibiting quantum steering must also exhibit quantum entanglement. But for mixed quantum states, there exist examples which lie between these different quantum correlation sets. The notion was initially proposed by Erwin Schrödinger , [ 1 ] [ 2 ] and later made popular by Howard M. Wiseman , S. J. Jones, and A. C. Doherty. [ 3 ]
In the usual formulation of quantum steering, two distant parties, Alice and Bob , are considered, they share an unknown quantum state ρ {\displaystyle \rho } with induced states ρ A {\displaystyle \rho _{A}} and ρ B {\displaystyle \rho _{B}} for Alice and Bob respectively. Alice and Bob can both perform local measurements on their own subsystems, for instance, Alice and Bob measure x {\displaystyle x} and y {\displaystyle y} and obtain the outcome a {\displaystyle a} and b {\displaystyle b} . After running the experiment many times, they will obtain measurement statistics p ( a , b | x , y ) {\displaystyle p(a,b|x,y)} , this is just the symmetric scenario for nonlocal correlation. Quantum steering introduces some asymmetry between two parties, viz., Bob's measurement devices are trusted, he knows what measurement his device carried out, and thus can perform a tomographically complete measurement. Meanwhile, Alice's devices are untrusted, she doesn't know what she measures but can still record each choice of measurement and outcome. Bob's goal is to determine if Alice influences his states in a quantum mechanical way or just using some of her prior knowledge of his partial states and some classical means. The classical way for Alice to influence Bob's states is known as the scenario having a local hidden state model which is, in some sense, a generalisation of the local hidden variable model for Bell nonlocality and also a restriction of the separable state model for quantum entanglement.
Mathematically, consider Alice having some finite number of measurements { M x } {\displaystyle \{M^{x}\}} indexed by x {\displaystyle x} , where for each x {\displaystyle x} , we have that M x = { M 1 x , M 2 x , … , M n x } {\displaystyle M^{x}=\{M_{1}^{x},M_{2}^{x},\ldots ,M_{n}^{x}\}} is a POVM with outcomes { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} , or a {\displaystyle a} in general. The assemblage between Alice and Bob is then a set of unnormalised quantum states on Bob's side indexed by the measurement choices and outcomes of Alice:
where p ( a | x ) := T r ( ρ A B M a x ⊗ I B ) = T r ( ρ A M a x ) {\displaystyle p(a|x):=\mathrm {Tr} (\rho _{AB}M_{a}^{x}\otimes I_{B})=\mathrm {Tr} (\rho _{A}M_{a}^{x})} and ρ a | x := T r A ( ρ A B M a x ⊗ I B ) / p ( a | x ) {\displaystyle \rho _{a|x}:=\mathrm {Tr} _{A}(\rho _{AB}M_{a}^{x}\otimes I_{B})/p(a|x)} , this latter trace denoting a partial trace over Alice's system.
An unsteerable assemblage is one that can be described by a so-called local hidden state model: a probability distribution p ( λ ) {\displaystyle p(\lambda )} over an exogenous variable λ {\displaystyle \lambda } , together with a set of quantum states { σ λ } {\displaystyle \{\sigma _{\lambda }\}} on Bob's side and a probability distribution p ( a | x , λ ) {\displaystyle p(a|x,\lambda )} on Alice's side. We say that an assemblage is unsteerable if for all elements of the assemblage:
If an assemblage is not unsteerable it is called steerable. A state is called unsteerable (steerable) if there exists measurements such that an unsteerable (steerable) assemblage can be created from it.
Let us do some comparison among Bell nonlocality, quantum steering, and quantum entanglement. By definition, a Bell nonlocal which does not admit a local hidden variable model for some measurement setting, a quantum steering state is a state which does not admit a local hidden state model for some measurement assemblage and state assemblage, and quantum entangled state is a state which is not separable. They share a great similarity. | https://en.wikipedia.org/wiki/Quantum_steering |
A pump is an alternating current -driven device that generates a direct current (DC). In the simplest configuration a pump has two leads connected to two reservoirs. In such open geometry, the pump takes particles from one reservoir and emits them into the other. Accordingly, a current is produced even if the reservoirs have the same temperature and chemical potential.
Stirring is the operation of inducing a circulating current with a non-vanishing DC component in a closed system. The simplest geometry is obtained by integrating a pump in a closed circuit. More generally one can consider any type of stirring mechanism such as moving a spoon in a cup of coffee.
Pumping and stirring effects in quantum physics have counterparts in purely classical stochastic and dissipative processes. [ 1 ] The studies of quantum pumping [ 2 ] [ 3 ] and of quantum stirring [ 4 ] emphasize the role of quantum interference in the analysis of the induced current. A major objective is to calculate the amount Q {\displaystyle Q} of transported particles per a driving cycle. There are circumstances in which Q {\displaystyle Q} is an integer number due to the topology of parameter space. [ 5 ] More generally Q {\displaystyle Q} is affected by inter-particle interactions , disorder, chaos, noise and dissipation.
Electric stirring explicitly breaks time-reversal symmetry. This property can be used to induce spin polarization in conventional semiconductors by purely electric means. [ 6 ] Strictly speaking, stirring is a non-linear effect, because in linear response theory (LRT) an AC driving induces an AC current with the same frequency. Still an adaptation of the LRT Kubo formalism allows the analysis of stirring. The quantum pumping problem (where we have an open geometry) can be regarded as a special limit of the quantum stirring problem (where we have a closed geometry). Optionally the latter can be analyzed within the framework of scattering theory . Pumping and Stirring devices are close relatives of ratchet systems. [ 7 ] The latter are defined in this context as AC driven spatially periodic arrays, where DC current is induced.
It is possible to induce a DC current by applying a bias, or if the particles are charged then by applying an electro-motive-force. In contrast to that a quantum pumping mechanism produces a DC current in response to a cyclic deformation of the confining potential. In order to have a DC current from an AC driving, time reversal symmetry (TRS) should be broken. In the absence of magnetic field and dissipation it is the driving itself that can break TRS. Accordingly, an adiabatic pump operation is based on varying more than one parameter, while for non-adiabatic pumps [ 8 ] [ 9 ] [ 10 ] modulation of a single parameter may suffice for DC current generation. The best known example is the peristaltic mechanism that combines a cyclic squeezing operation with on/off switching of entrance/exit valves.
Adiabatic quantum pumping is closely related to a class of current-driven nanomotors named Adiabatic quantum motor . While in a quantum pump, the periodic movement of some classical parameters pumps quantum particles from one reservoir to another, in a quantum motor a DC current of quantum particles induce the cyclic motion of the classical device. Said relation is due to the Onsager reciprocal relations between electric currents I {\displaystyle I} and current-induced forces F {\displaystyle F} , taken as generalized fluxes on one hand, and the chemical potentials biases δ μ {\displaystyle \delta \mu } and the velocity of the control parameters X ˙ {\displaystyle {\dot {X}}} , taken as generalized forces on the other hand., [ 4 ] [ 11 ]
where j {\displaystyle j} and α {\displaystyle \alpha } are indexes over the mechanical degrees of freedom and the leads respectively, and
the subindex " e q {\displaystyle eq} " implies that the quantities should be evaluated at equilibrium, i.e. X ˙ = 0 {\displaystyle {\dot {X}}=0} and δ μ = 0 {\displaystyle \delta \mu =0} . Integrating the above equation for a system with two leads yields the well known relation between the pumped charge per cycle Q {\displaystyle Q} , the work done by the motor W {\displaystyle W} , and the voltage bias V {\displaystyle V} , [ 11 ]
Consider a closed system which is described by a Hamiltonian H ( X ) {\displaystyle {\mathcal {H}}(X)} that depends on some control parameters X = ( X 1 , X 2 , X 3 ) {\displaystyle X=(X_{1},X_{2},X_{3})} . If X 3 {\displaystyle X_{3}} is an Aharonov Bohm magnetic flux through the ring, then by Faraday law − X 3 ˙ {\displaystyle -{\dot {X_{3}}}} is the electro motive force. If linear response theory applies we have the proportionality I = − G 33 X ˙ 3 {\displaystyle I=-G^{33}{\dot {X}}_{3}} , where G 33 {\displaystyle G^{33}} is the called the Ohmic conductance. In complete analogy if we change X 1 {\displaystyle X_{1}} the current is I = − G 31 X ˙ 1 {\displaystyle I=-G^{31}{\dot {X}}_{1}} , and if we change X 2 {\displaystyle X_{2}} the current is I = − G 32 X ˙ 2 {\displaystyle I=-G^{32}{\dot {X}}_{2}} , where G 31 {\displaystyle G^{31}} and G 32 {\displaystyle G^{32}} are elements of a conductance matrix. Accordingly, for a full pumping cycle:
The conductance can be calculated and analyzed using the Kubo formula approach to quantum pumping, [ 12 ] which is based on the theory of adiabatic processes. [ 5 ] Here we write the expression that applies in the case of low frequency "quasi static" driving process (the popular terms "DC driving" and "adiabatic driving" turn out to be misleading so we do not use them):
where I {\displaystyle {\mathcal {I}}} is the current operator, and F j = − ∂ H / ∂ X j {\displaystyle {\mathcal {F}}^{j}=-\partial {\mathcal {H}}/\partial X_{j}} is the generalized force that is associated with the control parameter X j {\displaystyle X_{j}} . Though this formula is written using quantum mechanical notations it holds also classically if the commutator is replaced by Poisson brackets. In general G {\displaystyle G} can be written as a sum of two terms: one has to do with dissipation, while the other, denoted as B {\displaystyle B} has to do with geometry. The dissipative part vanishes in the strict quantum adiabatic limit, while the geometrical part B {\displaystyle B} might be non-zero. It turns out that in the strict adiabatic limit B {\displaystyle B} is the " Berry curvature " (mathematically known as ``two-form"). Using the notations B 1 = − G 32 {\displaystyle B_{1}=-G^{32}} and B 2 = G 31 {\displaystyle B_{2}=G^{31}} we can rewrite the formula for the amount of pumped particles as
where we define the normal vector d s → = ( d X 2 , − d X 1 ) {\displaystyle {\vec {ds}}=(dX_{2},-dX_{1})} as illustrated. The advantage of this point of view is in the intuition that it gives for the result: Q {\displaystyle Q} is related to the flux of a field B → {\displaystyle {\vec {B}}} which is created (so to say) by "magnetic charges" in X {\displaystyle X} space. In practice the calculation of B → {\displaystyle {\vec {B}}} is done using the following formula:
This formula can be regarded as the quantum adiabatic limit of the Kubo formula. The eigenstates of the system are labeled by the index n {\displaystyle n} . These are in general many body states, and the energies are in general many body energies. At finite temperatures a thermal average over n 0 {\displaystyle n_{0}} is implicit. The field B {\displaystyle B} can be regarded as the rotor of "vector potential" A {\displaystyle A} (mathematically known as the "one-form"). Namely, B → = ∇ ∧ A → {\displaystyle {\vec {B}}=\nabla \wedge {\vec {A}}} . The `` Berry phase " which is acquired by a wavefunction at the end of a closed cycle is
Accordingly, one can argue that the "magnetic charge" that generates (so to say) the B {\displaystyle B} field consists of quantized "Dirac monopoles". It follows from gauge invariance that the degeneracies of the system are arranged as vertical Dirac chains. The "Dirac monopoles" are situated at X {\displaystyle X} points where n 0 {\displaystyle n_{0}} has a degeneracy with another level. The Dirac monopoles picture [ 13 ] is useful for charge transport analysis: the amount of transported charge is determined by the number of the Dirac chains encircled by the pumping cycle. Optionally it is possible to evaluate the transported charge per pumping cycle from the Berry phase by differentiating it with respect to the Aharonov–Bohm flux through the device. [ 14 ]
The Ohmic conductance of a mesoscopic device that is connected by leads to reservoirs is given by the Landauer formula: in dimensionless units the Ohmic conductance of an open channel equals its transmission. The extension of this scattering point of view in the context of quantum pumping leads to the Brouwer-Buttiker-Pretre-Thomas (BPT) formula [ 2 ] which relates the geometric conductance to the S {\displaystyle S} matrix of the pump. In the low temperature limit it yields
Here P A {\displaystyle P_{A}} is a projector that restrict the trace operations to the open channels of the lead where the current is measured. This BPT formula has been originally derived using a scattering approach, [ 15 ] but later its relation to the Kubo formula has been worked out. [ 16 ]
A very recent work considers the role of interactions in the stirring of Bose condensed particles. [ 17 ] Otherwise the rest of the literature concerns primarily electronic devices. [ 18 ] Typically the pump is modeled as a quantum dot. The effect of electron–electron interactions within the dot region is taken into account in the Coulomb blockade regime or in the Kondo regime. In the former case charge transport is quantized even in the case of small backscattering. Deviation from the exact quantized value is related to dissipation. In the Kondo regime, as the temperature is lowered, the pumping effect is modified. There are also works that consider interactions over the whole system (including the leads) using the Luttinger liquid model.
A quantum pump, when coupled to classical mechanical degrees of freedom, may also induce cyclic variations of the mechanical degrees of freedom coupled to it. In such a configuration, the pump works similarly to an Adiabatic quantum motor . A paradigmatic example of this class of systems is a quantum pump coupled to an elastically deformable quantum dot. [ 19 ] The mentioned paradigm has been generalized to include non-linear effects and stochastic fluctuations. [ 20 ] [ 21 ] | https://en.wikipedia.org/wiki/Quantum_stirring,_ratchets,_and_pumping |
Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system.
An example is a qubit used in quantum information processing . A qubit state is most generally a superposition of the basis states | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } :
where | Ψ ⟩ {\displaystyle |\Psi \rangle } is the quantum state of the qubit, and | 0 ⟩ {\displaystyle |0\rangle } , | 1 ⟩ {\displaystyle |1\rangle } denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes c 0 {\displaystyle c_{0}} and c 1 {\displaystyle c_{1}} that both are complex numbers. Here | 0 ⟩ {\displaystyle |0\rangle } corresponds to the classical 0 bit , and | 1 ⟩ {\displaystyle |1\rangle } to the classical 1 bit. The probabilities of measuring the system in the | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } state are given by | c 0 | 2 {\displaystyle |c_{0}|^{2}} and | c 1 | 2 {\displaystyle |c_{1}|^{2}} respectively (see the Born rule ). Before the measurement occurs the qubit is in a superposition of both states.
The interference fringes in the double-slit experiment provide another example of the superposition principle.
The theory of quantum mechanics postulates that a wave equation completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be linear and homogeneous . These conditions mean that for any two solutions of the wave equation, Ψ 1 {\displaystyle \Psi _{1}} and Ψ 2 {\displaystyle \Psi _{2}} , a linear combination of those solutions also solve the wave equation: Ψ = c 1 Ψ 1 + c 2 Ψ 2 {\displaystyle \Psi =c_{1}\Psi _{1}+c_{2}\Psi _{2}} for arbitrary complex coefficients c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} . [ 1 ] : 61 If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions.
The quantum wave equation can be solved using functions of position, Ψ ( r → ) {\displaystyle \Psi ({\vec {r}})} , or using functions of momentum, Φ ( p → ) {\displaystyle \Phi ({\vec {p}})} and consequently the superposition of momentum functions are also solutions: Φ ( p → ) = d 1 Φ 1 ( p → ) + d 2 Φ 2 ( p → ) {\displaystyle \Phi ({\vec {p}})=d_{1}\Phi _{1}({\vec {p}})+d_{2}\Phi _{2}({\vec {p}})} The position and momentum solutions are related by a linear transformation , a Fourier transformation . This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves. [ 1 ] : 244
Other transformations express a quantum solution as a superposition of eigenvectors , each corresponding to a possible result of a measurement on the quantum system. An eigenvector ψ i {\displaystyle \psi _{i}} for a mathematical operator, A ^ {\displaystyle {\hat {A}}} , has the equation A ^ ψ i = λ i ψ i {\displaystyle {\hat {A}}\psi _{i}=\lambda _{i}\psi _{i}} where λ i {\displaystyle \lambda _{i}} is one possible measured quantum value for the observable A {\displaystyle A} . A superposition of these eigenvectors can represent any solution: Ψ = ∑ n a i ψ i . {\displaystyle \Psi =\sum _{n}a_{i}\psi _{i}.} The states like ψ i {\displaystyle \psi _{i}} are called basis states.
Important mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the Dirac bra-ket notation : [ 1 ] : 245 | v ⟩ = d 1 | 1 ⟩ + d 2 | 2 ⟩ {\displaystyle |v\rangle =d_{1}|1\rangle +d_{2}|2\rangle } This approach is especially effect for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics and superposition of basis states is a fundamental tool in quantum mechanics.
Paul Dirac described the superposition principle as follows:
The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B , such that there exists an observation which, when made on the system in state A , is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b , according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e., either a or b ]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states. [ 2 ]
Anton Zeilinger , referring to the prototypical example of the double-slit experiment , has elaborated regarding the creation and destruction of quantum superposition:
"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear. [ 3 ]
Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:
where | n ⟩ {\displaystyle |n\rangle } are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, | x ⟩ {\displaystyle |x\rangle } :
where ϕ α ( x ) = ⟨ x | α ⟩ {\displaystyle \phi _{\alpha }(x)=\langle x|\alpha \rangle } is the projection of the state into the | x ⟩ {\displaystyle |x\rangle } basis and is called the wave function of the particle. In both instances we notice that | α ⟩ {\displaystyle |\alpha \rangle } can be expanded as a superposition of an infinite number of basis states.
Given the Schrödinger equation
where | n ⟩ {\displaystyle |n\rangle } indexes the set of eigenstates of the Hamiltonian with energy eigenvalues E n , {\displaystyle E_{n},} we see immediately that
where
is a solution of the Schrödinger equation but is not generally an eigenstate because E n {\displaystyle E_{n}} and E n ′ {\displaystyle E_{n'}} are not generally equal. We say that | Ψ ⟩ {\displaystyle |\Psi \rangle } is made up of a superposition of energy eigenstates. Now consider the more concrete case of an electron that has either spin up or down. We now index the eigenstates with the spinors in the z ^ {\displaystyle {\hat {z}}} basis:
where | ↑ ⟩ {\displaystyle |{\uparrow }\rangle } and | ↓ ⟩ {\displaystyle |{\downarrow }\rangle } denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state:
where the probability of finding the particle with either spin up or down is normalized to 1. Notice that c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} are complex numbers, so that
is an example of an allowed state. We now get
If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:
where we have a general state Ψ {\displaystyle \Psi } is the sum of the tensor products of the position space wave functions and spinors.
Successful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed.
In quantum computers , a qubit is the analog of the classical information bit and qubits can be superposed. [ 11 ] : 13 Unlike classical bits, a superposition of qubits represents information about two states in parallel. [ 11 ] : 31 Controlling the superposition of qubits is a central challenge in quantum computation. Qubit systems like nuclear spins with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results. [ 11 ] : 278 | https://en.wikipedia.org/wiki/Quantum_superposition |
In quantum computing , quantum supremacy or quantum advantage is the goal of demonstrating that a programmable quantum computer can solve a problem that no classical computer can solve in any feasible amount of time, irrespective of the usefulness of the problem. [ 1 ] [ 2 ] [ 3 ] The term was coined by John Preskill in 2011, [ 1 ] [ 4 ] but the concept dates to Yuri Manin 's 1980 [ 5 ] and Richard Feynman 's 1981 [ 6 ] proposals of quantum computing.
Conceptually, quantum supremacy involves both the engineering task of building a powerful quantum computer and the computational-complexity-theoretic task of finding a problem that can be solved by that quantum computer and has a superpolynomial speedup over the best known or possible classical algorithm for that task. [ 7 ] [ 8 ]
Examples of proposals to demonstrate quantum supremacy include the boson sampling proposal of Aaronson and Arkhipov, [ 9 ] and sampling the output of random quantum circuits . [ 10 ] [ 11 ] The output distributions that are obtained by making measurements in boson sampling or quantum random circuit sampling are flat, but structured in a way so that one cannot classically efficiently sample from a distribution that is close to the distribution generated by the quantum experiment . For this conclusion to be valid, only very mild assumptions in the theory of computational complexity have to be invoked. In this sense, quantum random sampling schemes can have the potential to show quantum supremacy. [ 12 ]
A notable property of quantum supremacy is that it can be feasibly achieved by near-term quantum computers, [ 4 ] since it does not require a quantum computer to perform any useful task [ 13 ] or use high-quality quantum error correction , [ 14 ] both of which are long-term goals. [ 2 ] Consequently, researchers view quantum supremacy as primarily a scientific goal, with relatively little immediate bearing on the future commercial viability of quantum computing. [ 2 ] Due to unpredictable possible improvements in classical computers and algorithms, quantum supremacy may be temporary or unstable, placing possible achievements under significant scrutiny. [ 15 ] [ 16 ]
In 1936, Alan Turing published his paper, “On Computable Numbers”, [ 17 ] in response to the 1900 Hilbert Problems . Turing's paper described what he called a “universal computing machine”, which later became known as a Turing machine . In 1980, Paul Benioff used Turing's paper to propose the theoretical feasibility of Quantum Computing. His paper, “The Computer as a Physical System: A Microscopic Quantum Mechanical Hamiltonian Model of Computers as Represented by Turing Machines“, [ 18 ] was the first to demonstrate that it is possible to show the reversible nature of quantum computing as long as the energy dissipated is arbitrarily small. In 1981, Richard Feynman showed that quantum mechanics could not be efficiently simulated on classical devices. [ 19 ] During a lecture, he delivered the famous quote, “Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy.” [ 19 ] Soon after this, David Deutsch produced a description for a quantum Turing machine and designed an algorithm created to run on a quantum computer. [ 20 ]
In 1994, further progress toward quantum supremacy was made when Peter Shor formulated Shor's algorithm , streamlining a method for factoring integers in polynomial time. [ 21 ] In 1995, Christopher Monroe and David Wineland published their paper, “Demonstration of a Fundamental Quantum Logic Gate”, [ 22 ] marking the first demonstration of a quantum logic gate , specifically the two-bit " controlled-NOT ". In 1996, Lov Grover put into motion an interest in fabricating a quantum computer after publishing his algorithm, Grover's Algorithm , in his paper, “A fast quantum mechanical algorithm for database search”. [ 23 ] In 1998, Jonathan A. Jones and Michele Mosca published “Implementation of a Quantum Algorithm to Solve Deutsch's Problem on a Nuclear Magnetic Resonance Quantum Computer”, [ 24 ] marking the first demonstration of a quantum algorithm.
Vast progress toward quantum supremacy was made in the 2000s from the first 5-qubit nuclear magnetic resonance computer (2000), the demonstration of Shor's theorem (2001), and the implementation of Deutsch's algorithm in a clustered quantum computer (2007). [ 25 ] In 2011, D-Wave Systems of Burnaby, British Columbia, Canada became the first company to sell a quantum computer commercially. [ 26 ] In 2012, physicist Nanyang Xu landed a milestone accomplishment by using an improved adiabatic factoring algorithm to factor 143. However, the methods used by Xu were met with objections. [ 27 ] Not long after this accomplishment, Google purchased its first quantum computer. [ 28 ]
Google had announced plans to demonstrate quantum supremacy before the end of 2017 with an array of 49 superconducting qubits . [ 29 ] In early January 2018, Intel announced a similar hardware program. [ 30 ] In October 2017, IBM demonstrated the simulation of 56 qubits on a classical supercomputer , thereby increasing the computational power needed to establish quantum supremacy. [ 31 ] In November 2018, Google announced a partnership with NASA that would "analyze results from quantum circuits run on Google quantum processors, and ... provide comparisons with classical simulation to both support Google in validating its hardware and establish a baseline for quantum supremacy." [ 32 ] Theoretical work published in 2018 suggested that quantum supremacy should be possible with a "two-dimensional lattice of 7×7 qubits and around 40 clock cycles" if error rates can be pushed low enough. [ 33 ] The scheme discussed was a variant of a quantum random sampling scheme in which qubits undergo random quantum circuits featuring quantum gates drawn from a universal gate set, followed by measurements in the computational basis.
On June 18, 2019, Quanta Magazine suggested that quantum supremacy could happen in 2019, according to Neven's law . [ 34 ] On September 20, 2019, the Financial Times reported that "Google claims to have reached quantum supremacy with an array of 54 qubits out of which 53 were functional, which were used to perform a series of operations in 200 seconds that would take a supercomputer about 10,000 years to complete". [ 35 ] [ 36 ] On October 23, Google officially confirmed the claims. [ 37 ] [ 38 ] [ 39 ] IBM responded by suggesting some of the claims were excessive and suggested that it could take 2.5 days instead of 10,000 years, listing techniques that a classical supercomputer may use to maximize computing speed. IBM's response is relevant as the most powerful supercomputer at the time, Summit , was made by IBM. [ 40 ] [ 15 ] [ 41 ] Researchers have since developed better algorithms for the sampling problem used to claim quantum supremacy, giving substantial reductions to the gap between Google's Sycamore processor and classical supercomputers [ 42 ] [ 43 ] [ 44 ] and even beating it. [ 45 ] [ 46 ] [ 47 ]
In December 2020, a group based in the University of Science and Technology of China (USTC) led by Pan Jianwei reached quantum supremacy by implementing gaussian boson sampling on 76 photons with their photonic quantum computer Jiuzhang . [ 48 ] [ 49 ] [ 50 ] The paper states that to generate the number of samples the quantum computer generates in 200 seconds, a classical supercomputer would require 2.5 billion years of computation. [ 3 ]
In October 2021, teams from USTC again reported quantum primacy by building two supercomputers called Jiuzhang 2.0 and Zuchongzhi. The light-based Jiuzhang 2.0 implemented gaussian boson sampling to detect 113 photons from a 144-mode optical interferometer and a sampling rate speed up of 10 24 – a difference of 37 photons and 10 orders of magnitude over the previous Jiuzhang. [ 51 ] [ 52 ] Zuchongzhi is a programmable superconducting quantum computer that needs to be kept at extremely low temperatures to work efficiently and uses random circuit sampling to obtain 56 qubits from a tunable coupling architecture of 66 transmons —an improvement over Google's Sycamore 2019 achievement by 3 qubits, meaning a greater computational cost of classical simulation of 2 to 3 orders of magnitude. [ 53 ] [ 54 ] [ 55 ] A third study reported that Zuchongzhi 2.1 completed a sampling task that "is about 6 orders of magnitude more difficult than that of Sycamore" "in the classic simulation". [ 56 ]
In June 2022, Xanadu reported a boson sampling experiment summing to those of Google and USTC. Their setup used loops of optical fiber and multiplexing to replace the network of beam splitters by a single one which made it also more easily reconfigurable. They detected a mean of 125 up to 219 photons from 216 squeezed modes (squeezed light follows a photon number distribution so they can contain more than one photon per mode) and claim to have obtained a speedup 50 million times more than previous experiments. [ 57 ] [ 58 ]
In March of 2024, D-Wave Systems reported on an experiment using a quantum annealing based processor that out-performed classical methods including tensor networks and neural networks. They argued that no known classical approach could yield the same results as the quantum simulation within a reasonable time-frame and claimed quantum supremacy. The task performed was the simulation of the non-equilibrium dynamics of a magnetic spin system quenched through a quantum phase transition. [ 59 ]
Complexity arguments concern how the amount of some resource needed to solve a problem (generally time or memory ) scales with the size of the input. In this setting, a problem consists of an inputted problem instance (a binary string) and returned solution (corresponding output string), while resources refers to designated elementary operations, memory usage, or communication. A collection of local operations allows for the computer to generate the output string. A circuit model and its corresponding operations are useful in describing both classical and quantum problems; the classical circuit model consists of basic operations such as AND gates , OR gates , and NOT gates while the quantum model consists of classical circuits and the application of unitary operations. Unlike the finite set of classical gates, there are an infinite amount of quantum gates due to the continuous nature of unitary operations. In both classical and quantum cases, complexity swells with increasing problem size. [ 60 ] As an extension of classical computational complexity theory , quantum complexity theory considers what a theoretical universal quantum computer could accomplish without accounting for the difficulty of building a physical quantum computer or dealing with decoherence and noise. [ 61 ] Since quantum information is a generalization of classical information, quantum computers can simulate any classical algorithm . [ 61 ]
Quantum complexity classes are sets of problems that share a common quantum computational model, with each model containing specified resource constraints. Circuit models are useful in describing quantum complexity classes. [ 62 ] The most useful quantum complexity class is BQP (bounded-error quantum polynomial time), the class of decision problems that can be solved in polynomial time by a universal quantum computer . Questions about BQP still remain, such as the connection between BQP and the polynomial-time hierarchy, whether or not BQP contains NP-complete problems, and the exact lower and upper bounds of the BQP class. Not only would answers to these questions reveal the nature of BQP, but they would also answer difficult classical complexity theory questions. One strategy for better understanding BQP is by defining related classes, ordering them into a conventional class hierarchy, and then looking for properties that are revealed by their relation to BQP. [ 63 ] There are several other quantum complexity classes, such as QMA (quantum Merlin Arthur) and QIP (quantum interactive polynomial time). [ 62 ]
The difficulty of proving what cannot be done with classical computing is a common problem in definitively demonstrating quantum supremacy. Contrary to decision problems that require yes or no answers, sampling problems ask for samples from probability distributions . [ 64 ] If there is a classical algorithm that can efficiently sample from the output of an arbitrary quantum circuit , the polynomial hierarchy would collapse to the third level, which is generally considered to be very unlikely. [ 10 ] [ 11 ] Boson sampling is a more specific proposal, the classical hardness of which depends upon the intractability of calculating the permanent of a large matrix with complex entries, which is a #P-complete problem. [ 65 ] The arguments used to reach this conclusion have been extended to IQP Sampling, [ 66 ] where only the conjecture that the average- and worst-case complexities of the problem are the same is needed, [ 64 ] as well as to Random Circuit Sampling, [ 11 ] which is the task replicated by the Google [ 38 ] and USTC research groups. [ 48 ]
The following are proposals for demonstrating quantum computational supremacy using current technology, often called NISQ devices . [ 2 ] Such proposals include (1) a well-defined computational problem, (2) a quantum algorithm to solve this problem, (3) a comparison best-case classical algorithm to solve the problem, and (4) a complexity-theoretic argument that, under a reasonable assumption, no classical algorithm can perform significantly better than current algorithms (so the quantum algorithm still provides a superpolynomial speedup). [ 7 ] [ 67 ]
This algorithm finds the prime factorization of an n -bit integer in O ~ ( n 3 ) {\displaystyle {\tilde {O}}(n^{3})} time [ 68 ] whereas the best known classical algorithm requires 2 O ( n 1 / 3 ) {\displaystyle 2^{O(n^{1/3})}} time and the best upper bound for the complexity of this problem is O ( 2 n / 3 + o ( 1 ) ) {\displaystyle O(2^{n/3+o(1)})} . [ 69 ] It can also provide a speedup for any problem that reduces to integer factoring , including the membership problem for matrix groups over fields of odd order. [ 70 ]
This algorithm is important both practically and historically for quantum computing . It was the first polynomial-time quantum algorithm proposed for a real-world problem that is believed to be hard for classical computers. [ 68 ] Namely, it gives a superpolynomial speedup under the reasonable assumption that RSA , a well-established cryptosystem , is secure. [ 71 ]
Factoring has some benefit over other supremacy proposals because factoring can be checked quickly with a classical computer just by multiplying integers, even for large instances where factoring algorithms are intractably slow. However, implementing Shor's algorithm for large numbers is infeasible with current technology, [ 72 ] [ 73 ] so it is not being pursued as a strategy for demonstrating supremacy.
This computing paradigm based upon sending identical photons through a linear-optical network can solve certain sampling and search problems that, assuming a few complexity-theoretical conjectures (that calculating the permanent of Gaussian matrices is #P-Hard and that the polynomial hierarchy does not collapse) are intractable for classical computers. [ 9 ] However, it has been shown that boson sampling in a system with large enough loss and noise can be simulated efficiently. [ 74 ]
The largest experimental implementation of boson sampling to date had 6 modes so could handle up to 6 photons at a time. [ 75 ] The best proposed classical algorithm for simulating boson sampling runs in time O ( n 2 n + m n 2 ) {\displaystyle O(n2^{n}+mn^{2})} for a system with n photons and m output modes. [ 76 ] [ 77 ] The algorithm leads to an estimate of 50 photons required to demonstrate quantum supremacy with boson sampling. [ 76 ] [ 77 ]
The best known algorithm for simulating an arbitrary random quantum circuit requires an amount of time that scales exponentially with the number of qubits , leading one group to estimate that around 50 qubits could be enough to demonstrate quantum supremacy. [ 33 ] Bouland, Fefferman, Nirkhe and Vazirani [ 11 ] gave, in 2018, theoretical evidence that efficiently simulating a random quantum circuit would require a collapse of the computational polynomial hierarchy . Google had announced its intention to demonstrate quantum supremacy by the end of 2017 by constructing and running a 49-qubit chip that would be able to sample distributions inaccessible to any current classical computers in a reasonable amount of time. [ 29 ] The largest universal quantum circuit simulator running on classical supercomputers at the time was able to simulate 48 qubits. [ 78 ] But for particular kinds of circuits, larger quantum circuit simulations with 56 qubits are possible. [ 79 ] This may require increasing the number of qubits to demonstrate quantum supremacy. [ 31 ] On October 23, 2019, Google published the results of this quantum supremacy experiment in the Nature article, “Quantum Supremacy Using a Programmable Superconducting Processor” in which they developed a new 53-qubit processor, named “Sycamore”, that is capable of fast, high-fidelity quantum logic gates , in order to perform the benchmark testing. Google claims that their machine performed the target computation in 200 seconds, and estimated that their classical algorithm would take 10,000 years in the world's fastest supercomputer to solve the same problem. [ 80 ] IBM disputed this claim, saying that an improved classical algorithm should be able to solve that problem in two and a half days on that same supercomputer. [ 81 ] [ 82 ] [ 83 ]
Quantum computers are much more susceptible to errors than classical computers due to decoherence and noise . [ 84 ] The threshold theorem states that a noisy quantum computer can use quantum error-correcting codes [ 85 ] [ 86 ] to simulate a noiseless quantum computer, assuming the error introduced in each computer cycle is less than some number. [ 87 ] Numerical simulations suggest that that number may be as high as 3%. [ 88 ] However, it is not yet definitively known how the resources needed for error correction will scale with the number of qubits . [ 89 ] Skeptics point to the unknown behavior of noise in scaled-up quantum systems as a potential roadblock for successfully implementing quantum computing and demonstrating quantum supremacy. [ 84 ] [ 90 ]
Some researchers have suggested that the term "quantum supremacy" should not be used, arguing that the word "supremacy" evokes distasteful comparisons to the racist belief of white supremacy . A controversial [ 91 ] [ 92 ] commentary article in the journal Nature signed by thirteen researchers asserts that the alternative phrase "quantum advantage" should be used instead. [ 93 ] John Preskill , the professor of theoretical physics at the California Institute of Technology who coined the term, has since clarified that the term was proposed to explicitly describe the moment that a quantum computer gains the ability to perform a task that a classical computer never could. He further explained that he specifically rejected the term "quantum advantage" as it did not fully encapsulate the meaning of his new term: the word "advantage" would imply that a computer with quantum supremacy would have a slight edge over a classical computer while the word "supremacy" better conveys complete ascendancy over any classical computer. [ 4 ] Nature's Philip Ball wrote in December 2020 that the term "quantum advantage" has "largely replaced" the term "quantum supremacy". [ 94 ] | https://en.wikipedia.org/wiki/Quantum_supremacy |
A quantum t-design is a probability distribution over either pure quantum states or unitary operators which can duplicate properties of the probability distribution over the Haar measure for polynomials of degree t or less. Specifically, the average of any polynomial function of degree t over the design is exactly the same as the average over Haar measure. Here the Haar measure is a uniform probability distribution over all quantum states or over all unitary operators. Quantum t-designs are so called because they are analogous to t-designs in classical statistics, which arose historically in connection with the problem of design of experiments . Two particularly important types of t-designs in quantum mechanics are projective and unitary t-designs. [ 1 ]
A spherical design is a collection of points on the unit sphere for which polynomials of bounded degree can be averaged over to obtain the same value that integrating over surface measure on the sphere gives. Spherical and projective t-designs derive their names from the works of Delsarte, Goethals, and Seidel in the late 1970s, but these objects played earlier roles in several branches of mathematics, including numerical integration and number theory. Particular examples of these objects have found uses in quantum information theory , [ 2 ] quantum cryptography , and other related fields.
Unitary t-designs are analogous to spherical designs in that they reproduce the entire unitary group via a finite collection of unitary matrices . [ 1 ] The theory of unitary 2-designs was developed in 2006 [ 1 ] specifically to achieve a practical means of efficient and scalable randomized benchmarking [ 3 ] to assess the errors in quantum computing operations, called gates. Since then unitary t-designs have been found useful in other areas of quantum computing and more broadly in quantum information theory and applied to problems as far reaching as the black hole information paradox. [ 4 ] Unitary t-designs are especially relevant to randomization tasks in quantum computing since ideal operations are usually represented by unitary operators.
In a d-dimensional Hilbert space when averaging over all quantum pure states the natural group is SU(d), the special unitary group of dimension d. [ citation needed ] The Haar measure is, by definition, the unique group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries.
A particularly widely used example of this is the spin 1 2 {\displaystyle {\tfrac {1}{2}}} system. For this system the relevant group is SU(2) which is the group of all 2x2 unitary operators with determinant 1. Since every operator in SU(2) is a rotation of the Bloch sphere , the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere. This implies that the Haar measure is the rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere.
An important class of complex projective t-designs, are symmetric informationally complete positive operator-valued measures ( POVMs ), which are complex projective 2-design. Since such 2-designs must have at least d 2 {\displaystyle d^{2}} elements, a SIC-POVM is a minimal sized complex projective 2-designs. [ 5 ]
Complex projective t-designs have been studied in quantum information theory as quantum t-designs. [ 6 ] These are closely related to spherical 2t-designs of vectors in the unit sphere in R d {\displaystyle \mathbb {R} ^{d}} which when naturally embedded in C d {\displaystyle \mathbb {C} ^{d}} give rise to complex projective t-designs. [ 7 ]
Formally, we define a probability distribution over quantum states ( p i , | ϕ i ⟩ ) {\displaystyle (p_{i},|\phi _{i}\rangle )} to be a [ 6 ] complex projective t-design if
∑ i p i ( | ϕ i ⟩ ⟨ ϕ i | ) ⊗ t = ∫ ψ ( | ψ ⟩ ⟨ ψ | ) ⊗ t d ψ {\displaystyle \sum _{i}p_{i}(|\phi _{i}\rangle \langle \phi _{i}|)^{\otimes t}=\int _{\psi }(|\psi \rangle \langle \psi |)^{\otimes t}d\psi }
Here, the integral over states is taken over the Haar measure on the unit sphere in C d {\displaystyle \mathbb {C} ^{d}}
Exact t-designs over quantum states cannot be distinguished from the uniform probability distribution over all states when using t copies of a state from the probability distribution. However, in practice even t-designs may be difficult to compute. For this reason approximate t-designs are useful.
Approximate t-designs are most useful due to their ability to be efficiently implemented. i.e. it is possible to generate a quantum state | ϕ ⟩ {\displaystyle |\phi \rangle } distributed according to the probability distribution p i | ϕ i ⟩ {\displaystyle p_{i}|\phi _{i}\rangle } in O ( log c d ) {\displaystyle O(\log ^{c}d)} time.
This efficient construction also implies that the POVM of the operators N p i | ϕ i ⟩ ⟨ ϕ i | {\displaystyle Np_{i}|\phi _{i}\rangle \langle \phi _{i}|} can be implemented in O ( log c d ) {\displaystyle O(\log ^{c}d)} time.
The technical definition of an approximate t-design is:
If ∑ i p i | ϕ i ⟩ ⟨ ϕ i | = ∫ ψ | ψ ⟩ ⟨ ψ | d ψ {\displaystyle \sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|=\int _{\psi }|\psi \rangle \langle \psi |d\psi }
and ( 1 − ϵ ) ∫ ψ ( | ψ ⟩ ⟨ ψ | ) ⊗ t d ψ ≤ ∑ i p i ( | ϕ i ⟩ ⟨ ϕ i | ) ⊗ t ≤ ( 1 + ϵ ) ∫ ψ ( | ψ ⟩ ⟨ ψ | ) ⊗ t d ψ {\displaystyle (1-\epsilon )\int _{\psi }(|\psi \rangle \langle \psi |)^{\otimes t}d\psi \leq \sum _{i}p_{i}(|\phi _{i}\rangle \langle \phi _{i}|)^{\otimes t}\leq (1+\epsilon )\int _{\psi }(|\psi \rangle \langle \psi |)^{\otimes t}d\psi }
then ( p i , | ϕ i ⟩ ) {\displaystyle (p_{i},|\phi _{i}\rangle )} is an ϵ {\displaystyle \epsilon } -approximate t-design.
It is possible, though perhaps inefficient, to find an ϵ {\displaystyle \epsilon } -approximate t-design consisting of quantum pure states for a fixed t.
For convenience d is assumed to be a power of 2.
Using the fact that for any d there exists a set of N d {\displaystyle N^{d}} functions {0,...,d-1} → {\displaystyle \rightarrow } {0,...,d-1} such that for any distinct k 1 , . . . , k N ∈ {\displaystyle k_{1},...,k_{N}\in } {0,...,d-1} the image under f, where f is chosen at random from S, is exactly the uniform distribution over tuples of N elements of {0,...,d-1}.
Let | ψ ⟩ = ∑ i = 1 d α i | i ⟩ {\displaystyle |\psi \rangle =\sum _{i=1}^{d}\alpha _{i}|i\rangle } be drawn from the Haar measure. Let P d {\displaystyle P_{d}} be the probability distribution of α 1 {\displaystyle \alpha _{1}} and let P = lim d → ∞ d P d {\displaystyle P=\lim _{d\rightarrow \infty }{\sqrt {d}}P_{d}} . Finally let α {\displaystyle \alpha } be drawn from P. If we define X = | α | {\displaystyle X=|\alpha |} with probability 1 2 {\displaystyle {\tfrac {1}{2}}} and X = − | α | {\displaystyle X=-|\alpha |} with probability 1 2 {\displaystyle {\tfrac {1}{2}}} then: E [ X j ] = 0 {\displaystyle E[X^{j}]=0} for odd j and E [ X j ] = ( j 2 ) ! {\displaystyle E[X^{j}]=({\tfrac {j}{2}})!} for even j.
Using this and Gaussian quadrature we can construct p f , g = ∑ i = 1 d a f , i 2 | S 1 | | S 2 | {\displaystyle p_{f,g}={\frac {\sum _{i=1}^{d}a_{f,i}^{2}}{|S_{1}||S_{2}|}}} so that p f , g | ψ f , g ⟩ {\displaystyle p_{f,g}|\psi _{f,g}\rangle } is an approximate t-design.
Unitary t-designs are analogous to spherical designs in that they reproduce the entire unitary group via a finite collection of unitary matrices . [ 1 ] The theory of unitary 2-designs was developed in 2006 [ 1 ] specifically to achieve a practical means of efficient and scalable randomized benchmarking [ 3 ] to assess the errors in quantum computing operations, called gates. Since then unitary t-designs have been found useful in other areas of quantum computing and more broadly in quantum information theory and in fields as far reaching as black hole physics. [ 4 ] Unitary t-designs are especially relevant to randomization tasks in quantum computing since ideal operations are usually represented by unitary operators.
Elements of a unitary t-design are elements of the unitary group, U(d), the group of d × d {\displaystyle d\times d} unitary matrices. A t-design of unitary operators will generate a t-design of states.
Suppose U k {\displaystyle {U_{k}}} is a unitary t-design (i.e. a set of unitary operators). Then for any pure state | ψ ⟩ {\displaystyle |\psi \rangle } let | ψ k ⟩ = U k | ψ ⟩ {\displaystyle |\psi _{k}\rangle =U_{k}|\psi \rangle } . Then | ψ k ⟩ {\displaystyle {|\psi _{k}\rangle }} will always be a t-design for states.
Formally define a unitary t-design , X, if
1 | X | ∑ U ∈ X U ⊗ t ⊗ ( U ∗ ) ⊗ t = ∫ U ( d ) U ⊗ t ⊗ ( U ∗ ) ⊗ t d U {\displaystyle {\frac {1}{|X|}}\sum _{U\in X}U^{\otimes t}\otimes (U^{*})^{\otimes t}=\int _{U(d)}U^{\otimes t}\otimes (U^{*})^{\otimes t}dU}
Observe that the space linearly spanned by the matrices U ⊗ r ⊗ ( U ∗ ) ⊗ s d U {\displaystyle U^{\otimes r}\otimes (U^{*})^{\otimes s}dU} over all choices of U is identical to the restriction U ∈ X {\displaystyle U\in X} and r + s = t {\displaystyle r+s=t} This observation leads to a conclusion about the duality between unitary designs and unitary codes.
Using the permutation maps it is possible [ 6 ] to verify directly that a set of unitary matrices forms a t-design. [ 8 ]
One direct result of this is that for any finite X ⊆ U ( d ) {\displaystyle X\subseteq U(d)}
1 | X | 2 ∑ U , V ∈ X | tr ( U ∗ V ) | 2 t ≥ ∫ U ( d ) | tr ( U ∗ V ) | 2 t d U {\displaystyle {\frac {1}{|X|^{2}}}\sum _{U,V\in X}|\operatorname {tr} (U*V)|^{2t}\geq \int _{U(d)}|\operatorname {tr} (U*V)|^{2t}dU}
With equality if and only if X is a t-design.
1 and 2-designs have been examined in some detail and absolute bounds for the dimension of X, |X|, have been derived. [ 9 ]
Define Hom ( U ( d ) , t , t ) {\displaystyle \operatorname {Hom} (U(d),t,t)} as the set of functions homogeneous of degree t in U {\displaystyle U} and homogeneous of degree t in U ∗ {\displaystyle U^{*}} , then if for every f ∈ Hom ( U ( d ) , t , t ) {\displaystyle f\in \operatorname {Hom} (U(d),t,t)} :
1 | X | ∑ U ∈ X f ( U ) = ∫ U ( d ) f ( U ) d U {\displaystyle {\frac {1}{|X|}}\sum _{U\in X}f(U)=\int _{U(d)}f(U)dU}
then X is a unitary t-design.
We further define the inner product for functions f {\displaystyle f} and g {\displaystyle g} on U ( d ) {\displaystyle U(d)} as the average value of f ¯ g {\displaystyle {\bar {f}}g} as:
⟨ f , g ⟩ := ∫ U ( d ) f ( U ) ¯ g ( U ) d X {\displaystyle \langle f,g\rangle :=\int _{U(d)}{\bar {f(U)}}g(U)dX}
and ⟨ f , g ⟩ X {\displaystyle \langle f,g\rangle _{X}} as the average value of f ¯ g {\displaystyle {\bar {f}}g} over any finite subset X ⊂ U ( d ) {\displaystyle X\subset U(d)} .
It follows that X is a unitary t-design if and only if ⟨ 1 , f ⟩ X = ⟨ 1 , f ⟩ ∀ f {\displaystyle \langle 1,f\rangle _{X}=\langle 1,f\rangle \quad \forall f} .
From the above it is demonstrable that if X is a t-design then | X | ≥ dim ( Hom ( U ( d ) , ⌈ t 2 ⌉ , ⌊ t 2 ⌋ ) ) {\displaystyle |X|\geq \dim(\operatorname {Hom} (U(d),\left\lceil {\tfrac {t}{2}}\right\rceil ,\left\lfloor {\tfrac {t}{2}}\right\rfloor ))} is an absolute bound for the design. This imposes an upper bound on the size of a unitary design. This bound is absolute meaning it depends only on the strength of the design or the degree of the code, and not the distances in the subset, X. [ 10 ]
A unitary code is a finite subset of the unitary group in which a few inner product values occur between elements. Specifically, a unitary code is defined as a finite subset X ⊂ U ( d ) {\displaystyle X\subset U(d)} if for all U ≠ M {\displaystyle U\neq M} in X | tr ( U ∗ M ) | 2 {\displaystyle |\operatorname {tr} (U^{*}M)|^{2}} takes only distinct values.
It follows that | X | ≤ dim ( Hom ( U ( d ) , s , s ) ) {\displaystyle |X|\leq \dim(\operatorname {Hom} (U(d),s,s))} and if U and M are orthogonal: | X | ≤ dim ( Hom ( U ( d ) , s , s − 1 ) ) {\displaystyle |X|\leq \dim(\operatorname {Hom} (U(d),s,s-1))} | https://en.wikipedia.org/wiki/Quantum_t-design |
Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. The sender does not have to know the particular quantum state being transferred. Moreover, the location of the recipient can be unknown, but to complete the quantum teleportation, classical information needs to be sent from sender to receiver. Because classical information needs to be sent, quantum teleportation cannot occur faster than the speed of light.
One of the first scientific articles to investigate quantum teleportation is "Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels" [ 1 ] published by C. H. Bennett , G. Brassard , C. Crépeau , R. Jozsa , A. Peres , and W. K. Wootters in 1993, in which they proposed using dual communication methods to send/receive quantum information. It was experimentally realized in 1997 by two research groups, led by Sandu Popescu and Anton Zeilinger , respectively. [ 2 ] [ 3 ]
Experimental determinations [ 4 ] [ 5 ] of quantum teleportation have been made in information content – including photons, atoms, electrons, and superconducting circuits – as well as distance, with 1,400 km (870 mi) being the longest distance of successful teleportation by Jian-Wei Pan 's team using the Micius satellite for space-based quantum teleportation. [ 6 ]
In matters relating to quantum information theory , it is convenient to work with the simplest possible unit of information: the two-state system of the qubit . The qubit functions as the quantum analog of the classic computational part, the bit , as it can have a measurement value of both a 0 and a 1, whereas the classical bit can only be measured as a 0 or a 1. The quantum two-state system seeks to transfer quantum information from one location to another location without losing the information and preserving the quality of this information. This process involves moving the information between carriers and not movement of the actual carriers , similar to the traditional process of communications, as two parties remain stationary while the information (digital media, voice, text, etc.) is being transferred, contrary to the implications of the word "teleport". The main components needed for teleportation include a sender, the information (a qubit), a traditional channel, a quantum channel, and a receiver. The sender does not need to know the exact contents of the information being sent. The measurement postulate of quantum mechanics – when a measurement is made upon a quantum state, any subsequent measurements will "collapse" or that the observed state will be lost – creates an imposition within teleportation: if a sender measures their information, the state could collapse when the receiver obtains the data since the state had changed from when the sender made the initial measurement and in so making it different.
For actual teleportation, it is required that an entangled quantum state be created for the qubit to be transferred. Entanglement imposes statistical correlations between otherwise distinct physical systems by creating or placing two or more separate particles into a single, shared quantum state. This intermediate state contains two particles whose quantum states are related to each other: measuring one particle's state provides information about the measurement of the other particle's state. These correlations hold even when measurements are chosen and performed independently, out of causal contact from one another, as verified in Bell test experiments . Thus, an observation resulting from a measurement choice made at one point in spacetime seems to instantaneously affect outcomes in another region, even though light hasn't yet had time to travel the distance, a conclusion seemingly at odds with special relativity . This is known as the EPR paradox . However, such correlations can never be used to transmit any information faster than the speed of light, a statement encapsulated in the no-communication theorem . Thus, teleportation as a whole can never be superluminal , as a qubit cannot be reconstructed until the accompanying classical information arrives.
The sender will combine the particle, whose information is teleported, with one of the entangled particles, causing a change of the overall entangled quantum state. Of this changed state, the particles in the receiver's possession are then sent to an analyzer that will measure the change of the entangled state. The "change" measurement will allow the receiver to recreate the original information that the sender had, resulting in the information being teleported or carried between two people that have different locations. Since the initial quantum information is "destroyed" as it becomes part of the entangled state, the no-cloning theorem is maintained as the information is recreated from the entangled state and not copied during teleportation.
The quantum channel is the communication mechanism that is used for all quantum information transmission and is the channel used for teleportation (relationship of quantum channel to traditional communication channel is akin to the qubit being the quantum analog of the classical bit). However, in addition to the quantum channel, a traditional channel must also be used to accompany a qubit to "preserve" the quantum information. When the change measurement between the original qubit and the entangled particle is made, the measurement result must be carried by a traditional channel so that the quantum information can be reconstructed and the receiver can get the original information. Because of this need for the traditional channel, the speed of teleportation can be no faster than the speed of light (hence the no-communication theorem is not violated). The main advantage with this is that Bell states can be shared using photons from lasers , making teleportation achievable through open space, as there is no need to send information through physical cables or optical fibers.
Quantum states can be encoded in various degrees of freedom of atoms. For example, qubits can be encoded in the degrees of freedom of electrons surrounding the atomic nucleus or in the degrees of freedom of the nucleus itself. Thus, performing this kind of teleportation requires a stock of atoms at the receiving site, available for having qubits imprinted on them. [ 8 ]
As of 2015, [update] the quantum states of single photons, photon modes, single atoms, atomic ensembles, defect centers in solids, single electrons, and superconducting circuits have been employed as information bearers. [ 9 ]
Understanding quantum teleportation requires a good grounding in finite-dimensional linear algebra , Hilbert spaces and projection matrices . A qubit is described using a two-dimensional complex number -valued vector space (a Hilbert space), which are the primary basis for the formal manipulations given below. A working knowledge of quantum mechanics is not absolutely required to understand the mathematics of quantum teleportation, although without such acquaintance, the deeper meaning of the equations may remain quite mysterious.
The resources required for quantum teleportation are a communication channel capable of transmitting two classical bits, a means of generating an entangled Bell state of qubits and distributing to two different locations, performing a Bell measurement on one of the Bell state qubits, and manipulating the quantum state of the other qubit from the pair. Of course, there must also be some input qubit (in the quantum state | ϕ ⟩ {\displaystyle |\phi \rangle } ) to be teleported. The protocol is then as follows:
It is worth noticing that the above protocol assumes that the qubits are individually addressable, meaning that the qubits are distinguishable and physically labeled. However, there can be situations where two identical qubits are indistinguishable due to the spatial overlap of their wave functions. Under this condition, the qubits cannot be individually controlled or measured. Nevertheless, a teleportation protocol analogous to that described above can still be (conditionally) implemented by exploiting two independently prepared qubits, with no need of an initial Bell state. This can be made by addressing the internal degrees of freedom of the qubits (e.g., spins or polarisations) by spatially localized measurements performed in separated regions A and B where the two spatially overlapping, indistinguishable qubits can be found. [ 10 ] This theoretical prediction has been then verified experimentally via polarized photons in a quantum optical setup. [ 11 ]
Work in 1998 verified the initial predictions, [ 2 ] and the distance of teleportation was increased in August 2004 to 600 meters, using optical fiber . [ 12 ] Subsequently, the record distance for quantum teleportation has been gradually increased to 16 kilometres (9.9 mi), [ 13 ] then to 97 km (60 mi), [ 14 ] and is now 143 km (89 mi), set in open air experiments in the Canary Islands , done between the two astronomical observatories of the Instituto de Astrofísica de Canarias . [ 14 ] There has been a recent record set (as of September 2015 [update] ) using superconducting nanowire detectors that reached the distance of 102 km (63 mi) over optical fiber. [ 15 ] For material systems, the record distance is 21 metres (69 ft). [ 16 ]
A variant of teleportation called "open-destination" teleportation, with receivers located at multiple locations, was demonstrated in 2004 using five-photon entanglement. [ 17 ] Teleportation of a composite state of two single qubits has also been realized. [ 18 ] In April 2011, experimenters reported that they had demonstrated teleportation of wave packets of light up to a bandwidth of 10 MHz while preserving strongly nonclassical superposition states. [ 19 ] [ 20 ] In August 2013, the achievement of "fully deterministic" quantum teleportation, using a hybrid technique, was reported. [ 21 ] On 29 May 2014, scientists announced a reliable way of transferring data by quantum teleportation. Quantum teleportation of data had been done before but with highly unreliable methods. [ 22 ] [ 23 ] On 26 February 2015, scientists at the University of Science and Technology of China in Hefei, led by Chao-yang Lu and Jian-Wei Pan carried out the first experiment teleporting multiple degrees of freedom of a quantum particle. They managed to teleport the quantum information from ensemble of rubidium atoms to another ensemble of rubidium atoms over a distance of 150 metres (490 ft) using entangled photons. [ 24 ] [ 25 ] [ 26 ] In 2016, researchers demonstrated quantum teleportation with two independent sources which are separated by 6.5 km (4.0 mi) in Hefei optical fiber network. [ 27 ] In September 2016, researchers at the University of Calgary demonstrated quantum teleportation over the Calgary metropolitan fiber network over a distance of 6.2 km (3.9 mi). [ 28 ] In December 2020, as part of the INQNET collaboration, researchers achieved quantum teleportation over a total distance of 44 km (27.3 mi) with fidelities exceeding 90%. [ 29 ] [ 30 ]
Researchers have also successfully used quantum teleportation to transmit information between clouds of gas atoms, notable because the clouds of gas are macroscopic atomic ensembles. [ 31 ] [ 32 ]
It is also possible to teleport logical operations , see quantum gate teleportation . In 2018, physicists at Yale demonstrated a deterministic teleported CNOT operation between logically encoded qubits. [ 33 ]
First proposed theoretically in 1993, quantum teleportation has since been demonstrated in many different guises. It has been carried out using two-level states of a single photon, a single atom and a trapped ion – among other quantum objects – and also using two photons. In 1997, two groups experimentally achieved quantum teleportation. The first group, led by Sandu Popescu , was based in Italy. An experimental group led by Anton Zeilinger followed a few months later.
The results obtained from experiments done by Popescu's group concluded that classical channels alone could not replicate the teleportation of linearly polarized state and an elliptically polarized state. The Bell state measurement distinguished between the four Bell states, which can allow for a 100% success rate of teleportation, in an ideal representation. [ 2 ]
Zeilinger's group produced a pair of entangled photons by implementing the process of parametric down-conversion. In order to ensure that the two photons cannot be distinguished by their arrival times, the photons were generated using a pulsed pump beam. The photons were then sent through narrow-bandwidth filters to produce a coherence time that is much longer than the length of the pump pulse. They then used a two-photon interferometry for analyzing the entanglement so that the quantum property could be recognized when it is transferred from one photon to the other. [ 3 ]
Photon 1 was polarized at 45° in the first experiment conducted by Zeilinger's group. Quantum teleportation is verified when both photons are detected in the | Ψ − ⟩ 12 {\displaystyle |\Psi ^{-}\rangle _{12}} state, which has a probability of 25%. Two detectors, f1 and f2, are placed behind the beam splitter, and recording the coincidence will identify the | Ψ − ⟩ 12 {\displaystyle |\Psi ^{-}\rangle _{12}} state. If there is a coincidence between detectors f1 and f2, then photon 3 is predicted to be polarized at a 45° angle. Photon 3 is passed through a polarizing beam splitter that selects +45° and −45° polarization. If quantum teleportation has happened, only detector d2, which is at the +45° output, will register a detection. Detector d1, located at the −45° output, will not detect a photon. If there is a coincidence between d2f1f2, with the 45° analysis, and a lack of a d1f1f2 coincidence, with −45° analysis, it is proof that the information from the polarized photon 1 has been teleported to photon 3 using quantum teleportation. [ 3 ]
Zeilinger's group developed an experiment using active feed-forward in real time and two free-space optical links, quantum and classical, between the Canary Islands of La Palma and Tenerife, a distance of over 143 kilometers. The results were published in 2012. In order to achieve teleportation, a frequency-uncorrelated polarization-entangled photon pair source, ultra-low-noise single-photon detectors and entanglement assisted clock synchronization were implemented. The two locations were entangled to share the auxiliary state: [ 14 ]
La Palma and Tenerife can be compared to the quantum characters Alice and Bob. Alice and Bob share the entangled state above, with photon 2 being with Alice and photon 3 being with Bob. A third party, Charlie, provides photon 1 (the input photon) which will be teleported to Alice in the generalized polarization state:
where the complex numbers α {\displaystyle \alpha } and β {\displaystyle \beta } are unknown to Alice or Bob.
Alice will perform a Bell-state measurement (BSM) that randomly projects the two photons onto one of the four Bell states with each one having a probability of 25%. Photon 3 will be projected onto | ϕ ⟩ {\displaystyle |\phi \rangle } , the input state. Alice transmits the outcome of the BSM to Bob, via the classical channel, where Bob is able to apply the corresponding unitary operation to obtain photon 3 in the initial state of photon 1. Bob will not have to do anything if he detects the | ψ − ⟩ 12 {\displaystyle |\psi ^{-}\rangle _{12}} state. Bob will need to apply a π {\displaystyle \pi } phase shift to photon 3 between the horizontal and vertical component if the | ψ + ⟩ 12 {\displaystyle |\psi ^{+}\rangle _{12}} state is detected. [ 14 ]
The results of Zeilinger's group concluded that the average fidelity (overlap of the ideal teleported state with the measured density matrix) was 0.863 with a standard deviation of 0.038. The link attenuation during their experiments varied between 28.1 dB and 39.0 dB, which was a result of strong winds and rapid temperature changes. Despite the high loss in the quantum free-space channel, the average fidelity surpassed the classical limit of 2/3. Therefore, Zeilinger's group successfully demonstrated quantum teleportation over a distance of 143 km. [ 14 ]
In 2004, a quantum teleportation experiment was conducted across the Danube River in Vienna, a total of 600 meters. An 800-meter-long optical fiber wire was installed in a public sewer system underneath the Danube River, and it was exposed to temperature changes and other environmental influences. Alice must perform a joint Bell state measurement (BSM) on photon b, the input photon, and photon c, her part of the entangled photon pair (photons c and d). Photon d, Bob's receiver photon, will contain all of the information on the input photon b, except for a phase rotation that depends on the state that Alice observed. This experiment implemented an active feed-forward system that sends Alice's measurement results via a classical microwave channel with a fast electro-optical modulator in order to exactly replicate Alice's input photon. The teleportation fidelity obtained from the linear polarization state at 45° varied between 0.84 and 0.90, which is well above the classical fidelity limit of 0.66. [ 12 ]
Three qubits are required for this process: the source qubit from the sender, the ancillary qubit, and the receiver's target qubit, which is maximally entangled with the ancillary qubit. For this experiment, Ca + 40 {\displaystyle {\ce {^{40}Ca+}}} ions were used as the qubits. Ions 2 and 3 are prepared in the Bell state | ψ + ⟩ 23 = 1 2 ( | 0 ⟩ 2 | 1 ⟩ 3 + | 1 ⟩ 2 | 0 ⟩ 3 ) {\displaystyle |\psi ^{+}\rangle _{23}={\frac {1}{\sqrt {2}}}(|0\rangle _{2}|1\rangle _{3}+|1\rangle _{2}|0\rangle _{3})} . The state of ion 1 is prepared arbitrarily. The quantum states of ions 1 and 2 are measured by illuminating them with light at a specific wavelength. The obtained fidelities for this experiment ranged between 73% and 76%. This is larger than the maximum possible average fidelity of 66.7% that can be obtained using completely classical resources. [ 34 ]
The quantum state being teleported in this experiment is | χ ⟩ 1 = α | H ⟩ 1 + β | V ⟩ 1 {\displaystyle |\chi \rangle _{1}=\alpha |H\rangle _{1}+\beta |V\rangle _{1}} , where α {\displaystyle \alpha } and β {\displaystyle \beta } are unknown complex numbers, | H ⟩ {\displaystyle |H\rangle } represents the horizontal polarization state, and | V ⟩ {\displaystyle |V\rangle } represents the vertical polarization state. The qubit prepared in this state is generated in a laboratory in Ngari, Tibet. The goal was to teleport the quantum information of the qubit to the Micius satellite that was launched on August 16, 2016, at an altitude of around 500 km. When a Bell state measurement is conducted on photons 1 and 2 and the resulting state is | ϕ + ⟩ 12 = 1 2 ( | H ⟩ 1 | H ⟩ 2 + | V ⟩ 1 | V ⟩ 2 ) ) {\displaystyle |\phi ^{+}\rangle _{12}={\frac {1}{\sqrt {2}}}(|H\rangle _{1}|H\rangle _{2}+|V\rangle _{1}|V\rangle _{2}))} , photon 3 carries this desired state. If the Bell state detected is | ϕ − ⟩ 12 = 1 2 ( | H ⟩ 1 | H ⟩ 2 − | V ⟩ 1 | V ⟩ 2 ) {\displaystyle |\phi ^{-}\rangle _{12}={\frac {1}{\sqrt {2}}}(|H\rangle _{1}|H\rangle _{2}-|V\rangle _{1}|V\rangle _{2})} , then a phase shift of π {\displaystyle \pi } is applied to the state to get the desired quantum state. The distance between the ground station and the satellite changes from as little as 500 km to as large as 1,400 km. Because of the changing distance, the channel loss of the uplink varies between 41 dB and 52 dB. The average fidelity obtained from this experiment was 0.80 with a standard deviation of 0.01. Therefore, this experiment successfully established a ground-to-satellite uplink over a distance of 500–1,400 km using quantum teleportation. This is an essential step towards creating a global-scale quantum internet. [ 6 ]
Quantum teleportation has been demonstrated over fiber optic cables simultaneously carrying regular telecommunications traffic. This eliminates the need for separate, dedicated infrastructure for quantum networking and shows that quantum teleportation and classical communications can coexist on the same fiber optic cables. A less crowded wavelength of light was used for the quantum signal and special filters were required to reduce noise from other traffic. [ 35 ] [ 36 ] [ unreliable source? ]
In April 2025, researchers at the University of Illinois Urbana-Champaign achieved quantum teleportation with 94% fidelity using a nanophotonic indium - gallium - phosphide platform to perform nonlinear sum frequency generation (SFG). This method mitigated multiphoton noise and boosted teleportation efficiency by a factor of 10,000 compared to prior SFG-based systems. [ 37 ]
There are a variety of ways in which the teleportation protocol can be written mathematically. Some are very compact but abstract, and some are verbose but straightforward and concrete. The presentation below is of the latter form: verbose, but has the benefit of showing each quantum state simply and directly. Later sections review more compact notations.
The teleportation protocol begins with a quantum state or qubit | ψ ⟩ {\displaystyle |\psi \rangle } , in Alice's possession, that she wants to convey to Bob. This qubit can be written generally, in bra–ket notation , as:
The subscript C above is used only to distinguish this state from A and B , below.
Next, the protocol requires that Alice and Bob share a maximally entangled state. This state is fixed in advance, by mutual agreement between Alice and Bob, and can be any one of the four Bell states shown. It does not matter which one.
In the following, assume that Alice and Bob share the state | Φ + ⟩ A B . {\displaystyle |\Phi ^{+}\rangle _{AB}.} Alice obtains one of the particles in the pair, with the other going to Bob. (This is implemented by preparing the particles together and shooting them to Alice and Bob from a common source.) The subscripts A and B in the entangled state refer to Alice's or Bob's particle.
At this point, Alice has two particles ( C , the one she wants to teleport, and A , one of the entangled pair), and Bob has one particle, B . In the total system, the state of these three particles is given by
Alice will then make a local measurement in the Bell basis (i.e. the four Bell states) on the two particles in her possession. To make the result of her measurement clear, it is best to write the state of Alice's two qubits as superpositions of the Bell basis. This is done by using the following general identities, which are easily verified:
and
After expanding the expression for | ψ ⟩ C ⊗ | Φ + ⟩ A B {\textstyle {\begin{aligned}|&\psi \rangle _{C}\otimes \ |\Phi ^{+}\rangle _{AB}\end{aligned}}} , one applies these identities to the qubits with A and C subscripts. In particular, α 1 2 | 0 ⟩ C ⊗ | 0 ⟩ A ⊗ | 0 ⟩ B = α 1 2 ( | Φ + ⟩ C A + | Φ − ⟩ C A ) ⊗ | 0 ⟩ B , {\displaystyle \alpha {\frac {1}{\sqrt {2}}}|0\rangle _{C}\otimes |0\rangle _{A}\otimes |0\rangle _{B}=\alpha {\frac {1}{2}}(|\Phi ^{+}\rangle _{CA}+|\Phi ^{-}\rangle _{CA})\otimes |0\rangle _{B},} and the other terms follow similarly. Combining similar terms, the total three particle state of A , B and C together becomes the following four-term superposition: [ 38 ]
Note that all three particles are still in the same total state since no operations have been performed. Rather, the above is just a change of basis on Alice's part of the system. This change has moved the entanglement from particles A and B to particles C and A. The actual teleportation occurs when Alice measures her two qubits (C and A) in the Bell basis
Equivalently, the measurement may be done in the computational basis, { | 0 ⟩ , | 1 ⟩ } {\displaystyle \{|0\rangle ,|1\rangle \}} , by mapping each Bell state uniquely to one of { | 0 ⟩ ⊗ | 0 ⟩ , | 0 ⟩ ⊗ | 1 ⟩ , | 1 ⟩ ⊗ | 0 ⟩ , | 1 ⟩ ⊗ | 1 ⟩ } {\displaystyle \{|0\rangle \otimes |0\rangle ,|0\rangle \otimes |1\rangle ,|1\rangle \otimes |0\rangle ,|1\rangle \otimes |1\rangle \}} with the quantum circuit in the figure to the right.
The result of Alice's (local) measurement is a collection of two classical bits (00, 01, 10 or 11) related to one of the following four states (with equal probability of 1/4), after the three-particle state has collapsed into one of the states:
Alice's two particles are now entangled to each other, in one of the four Bell states , and the entanglement originally shared between Alice's and Bob's particles is now broken. Bob's particle takes on one of the four superposition states shown above. Note how Bob's qubit is now in a state that resembles the state to be teleported. The four possible states for Bob's qubit are unitary images of the state to be teleported.
The result of Alice's Bell measurement tells her which of the above four states the system is in. She can now send her result to Bob through a classical channel. Two classical bits can communicate which of the four results she obtained. After Bob receives the message from Alice, he will know which of the four states his particle is in. Using this information, he performs a unitary operation on his particle to transform it to the desired state α | 0 ⟩ B + β | 1 ⟩ B {\displaystyle \alpha |0\rangle _{B}+\beta |1\rangle _{B}} :
to recover the state.
to his qubit.
Teleportation is thus achieved. The above-mentioned three gates correspond to rotations of π radians (180°) about appropriate axes (X, Y and Z) in the Bloch sphere picture of a qubit.
Some remarks:
When implementing the quantum teleportation protocol, different experimental noises may arise affecting the state transference. [ 39 ] The usual way to benchmark a particular teleportation procedure is by using the average fidelity : Given an arbitrary teleportation protocol producing output states ρ i {\displaystyle \rho _{i}} with probability p i {\displaystyle p_{i}} for an initial state ρ = | ψ ⟩ ⟨ ψ | {\displaystyle \rho =|\psi \rangle \langle \psi |} , the average fidelity is defined as: [ 40 ]
⟨ F ¯ ⟩ = ∫ ∑ i p i F ( ρ , ρ i ) d ψ {\displaystyle \langle {\overline {F}}\rangle =\int \sum _{i}p_{i}F(\rho ,\rho _{i})d\psi }
where the integration is performed over the Haar measure defined by assuming maximal uncertainty over the initial quantum states | ψ ⟩ {\displaystyle |\psi \rangle } , and F ( ρ , ρ i ) = ( Tr ρ ρ i ρ ) 2 {\displaystyle F(\rho ,\rho _{i})=\left({\text{Tr}}{\sqrt {{\sqrt {\rho }}\rho _{i}{\sqrt {\rho }}}}\right)^{2}} is the Uhlmann-Jozsa fidelity .
The widely known classical threshold is obtained by optimizing the average fidelity over all classical protocols (i.e. when the sender Alice and the receiver Bob can use just a classical channel to communicate with each other). When teleportation involves qubit states, the maximal classical average fidelity is 2 / 3 {\displaystyle 2/3} . [ 41 ] [ 42 ] In this way, a particular protocol with average fidelity ⟨ F ¯ ⟩ {\displaystyle \langle {\overline {F}}\rangle } is certified as useful if ⟨ F ¯ ⟩ ≥ 2 / 3 {\displaystyle \langle {\overline {F}}\rangle \geq 2/3} . [ 6 ] [ 12 ] [ 14 ]
However, using the Uhlmann-Jozsa fidelity as the unique distance measure for benchmarking teleportation is not justified, and one may choose different distinguishability measures. [ 43 ] For example, there may exist reasons depending on the context in which other measures might be more suitable than fidelity. [ 44 ] In this way, the average distance of teleportation is defined as: [ 45 ]
⟨ D ¯ ⟩ = ∫ ∑ i p i D ( ρ , ρ i ) d ψ {\displaystyle \langle {\overline {D}}\rangle =\int \sum _{i}p_{i}D(\rho ,\rho _{i})d\psi }
being D ( ρ , σ ) {\displaystyle D(\rho ,\sigma )} a well-behaved (i.e. satisfying identity of indiscernibles and unitary invariance) distinguishability measure between quantum states. Consequently, different classical thresholds exist, depending on the considered distance measure (classical thresholds for Trace distance , quantum Jensen–Shannon divergence , transmission distance, Bures distance , wootters distance, and quantum Hellinger distance, among others, were obtained in Ref. [ 45 ] ). This points out a particular issue when certifying quantum teleportation: Given a teleportation protocol, its certification is not a universal fact in the sense that depends on the distance used. Then, a particular protocol might be certified as useful for a set of distance quantifiers, and non-useful for other distinguishability measures. [ 45 ]
There are a variety of different notations in use that describe the teleportation protocol. One common one is by using the notation of quantum gates .
In the above derivation, the unitary transformation that is the change of basis (from the standard product basis into the Bell basis) can be written using quantum gates. Direct calculation shows that this gate is given by
where H is the one qubit Walsh-Hadamard gate and CNOT {\displaystyle \operatorname {CNOT} } is the Controlled NOT gate.
Teleportation can be applied not just to pure states, but also mixed states , that can be regarded as the state of a single subsystem of an entangled pair. The so-called entanglement swapping is a simple and illustrative example.
If Alice and Bob share an entangled pair, and Bob teleports his particle to Carol, then Alice's particle is now entangled with Carol's particle. This situation can also be viewed symmetrically as follows:
Alice and Bob share an entangled pair, and Bob and Carol share a different entangled pair. Now let Bob perform a projective measurement on his two particles in the Bell basis and communicate the result to Carol. These actions are precisely the teleportation protocol described above with Bob's first particle, the one entangled with Alice's particle, as the state to be teleported. When Carol finishes the protocol she now has a particle with the teleported state, that is an entangled state with Alice's particle. Thus, although Alice and Carol never interacted with each other, their particles are now entangled.
A detailed diagrammatic derivation of entanglement swapping has been given by Bob Coecke , [ 50 ] presented in terms of categorical quantum mechanics .
An important application of entanglement swapping is distributing Bell states for use in entanglement distributed quantum networks . A technical description of the entanglement swapping protocol is given here for pure Bell states.
The basic teleportation protocol for a qubit described above has been generalized in several directions, in particular regarding the dimension of the system teleported and the number of parties involved (either as sender, controller, or receiver).
A generalization to d {\displaystyle d} -level systems (so-called qudits ) is straight forward and was already discussed in the original paper by Bennett et al. : [ 1 ] the maximally entangled state of two qubits has to be replaced by a maximally entangled state of two qudits and the Bell measurement by a measurement defined by a maximally entangled orthonormal basis. All possible such generalizations were discussed by Werner in 2001. [ 51 ]
The generalization to infinite-dimensional so-called continuous-variable systems was proposed by Braunstein and Kimble [ 52 ] and led to the first teleportation experiment that worked unconditionally. [ 53 ]
The use of multipartite entangled states instead of a bipartite maximally entangled state allows for several new features: either the sender can teleport information to several receivers either sending the same state to all of them (which allows to reduce the amount of entanglement needed for the process) [ 54 ] or teleporting multipartite states [ 55 ] or sending a single state in such a way that the receiving parties need to cooperate to extract the information. [ 56 ] A different way of viewing the latter setting is that some of the parties can control whether the others can teleport.
In general, mixed states ρ may be transported, and a linear transformation ω applied during teleportation, thus allowing data processing of quantum information. This is one of the foundational building blocks of quantum information processing. This is demonstrated below.
A general teleportation scheme can be described as follows. Three quantum systems are involved. System 1 is the (unknown) state ρ to be teleported by Alice. Systems 2 and 3 are in a maximally entangled state ω that are distributed to Alice and Bob, respectively. The total system is then in the state
A successful teleportation process is a LOCC quantum channel Φ that satisfies
where Tr 12 is the partial trace operation with respect systems 1 and 2, and ∘ {\displaystyle \circ } denotes the composition of maps. This describes the channel in the Schrödinger picture.
Taking adjoint maps in the Heisenberg picture, the success condition becomes
for all observable O on Bob's system. The tensor factor in I ⊗ O {\displaystyle I\otimes O} is 12 ⊗ 3 {\displaystyle 12\otimes 3} while that of ρ ⊗ ω {\displaystyle \rho \otimes \omega } is 1 ⊗ 23 {\displaystyle 1\otimes 23} .
The proposed channel Φ can be described more explicitly. To begin teleportation, Alice performs a local measurement on the two subsystems (1 and 2) in her possession. Assume the local measurement have effects
If the measurement registers the i -th outcome, the overall state collapses to
The tensor factor in ( M i ⊗ I ) {\displaystyle (M_{i}\otimes I)} is 12 ⊗ 3 {\displaystyle 12\otimes 3} while that of ρ ⊗ ω {\displaystyle \rho \otimes \omega } is 1 ⊗ 23 {\displaystyle 1\otimes 23} . Bob then applies a corresponding local operation Ψ i on system 3. On the combined system, this is described by
where Id is the identity map on the composite system 1 ⊗ 2 {\displaystyle 1\otimes 2} .
Therefore, the channel Φ is defined by
Notice Φ satisfies the definition of LOCC . As stated above, the teleportation is said to be successful if, for all observable O on Bob's system, the equality
holds. The left hand side of the equation is:
where Ψ i * is the adjoint of Ψ i in the Heisenberg picture. Assuming all objects are finite dimensional, this becomes
The success criterion for teleportation has the expression
A local explanation of quantum teleportation is put forward by David Deutsch and Patrick Hayden , with respect to the many-worlds interpretation of quantum mechanics. Their paper asserts that the two bits that Alice sends Bob contain "locally inaccessible information" resulting in the teleportation of the quantum state. "The ability of quantum information to flow through a classical channel [...], surviving decoherence, is [...] the basis of quantum teleportation." [ 57 ]
While quantum teleportation is in an infancy stage, there are many aspects pertaining to teleportation that scientists are working to better understand or improve the process that include:
Quantum teleportation can improve the errors associated with fault tolerant quantum computation via an arrangement of logic gates. Experiments by D. Gottesman and I. L. Chuang have determined that a "Clifford hierarchy" [ 58 ] gate arrangement which acts to enhance protection against environmental errors. Overall, a higher threshold of error is allowed with the Clifford hierarchy as the sequence of gates requires less resources that are needed for computation. While the more gates that are used in a quantum computer create more noise, the gates arrangement and use of teleportation in logic transfer can reduce this noise as it calls for less "traffic" that is compiled in these quantum networks. [ 59 ] The more qubits used for a quantum computer, the more levels are added to a gate arrangement, with the diagonalization of gate arrangement varying in degree. Higher dimension analysis involves the higher level gate arrangement of the Clifford hierarchy. [ 60 ]
Considering the previously mentioned requirement of an intermediate entangled state for quantum teleportation, there needs to be consideration of the purity of this state for information quality. A protection that has been developed involves the use of continuous variable information (rather than a typical discrete variable) creating a superimposed coherent intermediate state. This involves making a phase shift in the received information and then adding a mixing step upon reception using a preferred state, which could be an odd or even coherent state, that will be "conditioned to the classical information of the sender", creating a two mode state that contains the originally sent information. [ 61 ]
There have also been developments with teleporting information between systems that already have quantum information in them. Experiments done by Feng, Xu, Zhou et al. have demonstrated that teleportation of a qubit to a photon that already has a qubit's worth of information is possible due to using an optical qubit-ququart entangling gate. [ 4 ] This quality can increase computation possibilities as calculations can be done based on previously stored information, allowing for improvements on past calculations. | https://en.wikipedia.org/wiki/Quantum_teleportation |
Quantum thermodynamics [ 1 ] [ 2 ] is the study of the relations between two independent physical theories: thermodynamics and quantum mechanics . The two independent theories address the physical phenomena of light and matter.
In 1905, Albert Einstein argued that the requirement of consistency between thermodynamics and electromagnetism [ 3 ] leads to the conclusion that light is quantized, obtaining the relation E = h ν {\displaystyle E=h\nu } . This paper is the dawn of quantum theory. In a few decades quantum theory became established with an independent set of rules. [ 4 ] Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs from quantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. In addition, there is a quest for the theory to be relevant for a single individual quantum system.
There is an intimate connection of quantum thermodynamics with the theory of open quantum systems . [ 5 ] Quantum mechanics inserts dynamics into thermodynamics, giving a sound foundation to finite-time-thermodynamics. The main assumption is that the entire world is a large closed system, and therefore, time evolution is governed by a unitary transformation generated by a global Hamiltonian . For the combined system
bath scenario, the global Hamiltonian can be decomposed into: H = H S + H B + H SB {\displaystyle H=H_{\text{S}}+H_{\text{B}}+H_{\text{SB}}} where H S {\displaystyle H_{\text{S}}} is the system Hamiltonian, H B {\displaystyle H_{\text{B}}} is the bath Hamiltonian and H SB {\displaystyle H_{\text{SB}}} is the system-bath interaction.
The state of the system is obtained from a partial trace over the combined system and bath:
ρ S ( t ) = Tr B ( ρ SB ( t ) ) . {\displaystyle \rho _{\text{S}}(t)=\operatorname {Tr} _{\text{B}}(\rho _{\text{SB}}(t)).}
Reduced dynamics is an equivalent description of the system dynamics utilizing only system operators.
Assuming Markov property for the dynamics the basic equation of motion for an open quantum system is the Lindblad equation (GKLS): [ 6 ] [ 7 ] ρ ˙ S = − i ℏ [ H S , ρ S ] + L D ( ρ S ) {\displaystyle {\dot {\rho }}_{\text{S}}=-{i \over \hbar }[H_{\text{S}},\rho _{\text{S}}]+L_{\text{D}}(\rho _{\text{S}})} H S {\displaystyle H_{\text{S}}} is a ( Hermitian ) Hamiltonian part and L D {\displaystyle L_{\text{D}}} : L D ( ρ S ) = ∑ n [ V n ρ S V n † − 1 2 ( ρ S V n † V n + V n † V n ρ S ) ] {\displaystyle L_{\text{D}}(\rho _{\text{S}})=\sum _{n}\left[V_{n}\rho _{\text{S}}V_{n}^{\dagger }-{\tfrac {1}{2}}\left(\rho _{\text{S}}V_{n}^{\dagger }V_{n}+V_{n}^{\dagger }V_{n}\rho _{\text{S}}\right)\right]} is the dissipative part describing implicitly through system operators V n {\displaystyle V_{n}} the influence of the bath on the system.
The Markov property imposes that the system and bath are uncorrelated at all times ρ SB = ρ s ⊗ ρ B {\displaystyle \rho _{\text{SB}}=\rho _{s}\otimes \rho _{\text{B}}} . The L-GKS equation is unidirectional and leads any initial state ρ S {\displaystyle \rho _{\text{S}}} to a steady state solution which is an invariant of the equation of motion ρ ˙ S ( t → ∞ ) = 0 {\displaystyle {\dot {\rho }}_{\text{S}}(t\to \infty )=0} . [ 5 ]
The Heisenberg picture supplies a direct link to quantum thermodynamic observables. The dynamics of a system observable represented by the operator, O {\displaystyle O} , has the form: d O d t = i ℏ [ H S , O ] + L D ∗ ( O ) + ∂ O ∂ t {\displaystyle {\frac {dO}{dt}}={\frac {i}{\hbar }}[H_{\text{S}},O]+L_{\text{D}}^{*}(O)+{\frac {\partial O}{\partial t}}} where the possibility that the operator, O {\displaystyle O} is explicitly time-dependent, is included.
When O = H S {\displaystyle O=H_{\text{S}}} the first law of thermodynamics emerges: d E d t = ⟨ ∂ H S ∂ t ⟩ + ⟨ L D ∗ ( H S ) ⟩ {\displaystyle {\frac {dE}{dt}}=\left\langle {\frac {\partial H_{\text{S}}}{\partial t}}\right\rangle +\langle L_{\text{D}}^{*}(H_{\text{S}})\rangle } where power is interpreted as
P = ⟨ ∂ H S ∂ t ⟩ {\displaystyle P=\left\langle {\frac {\partial H_{\text{S}}}{\partial t}}\right\rangle }
and the heat current [ 8 ] [ 9 ] [ 10 ]
J = ⟨ L D ∗ ( H S ) ⟩ . {\displaystyle J=\left\langle L_{\text{D}}^{*}(H_{\text{S}})\right\rangle .}
Additional conditions have to be imposed on the dissipator L D {\displaystyle L_{\text{D}}} to be consistent with thermodynamics.
First the invariant ρ S ( ∞ ) {\displaystyle \rho _{\text{S}}(\infty )} should become an equilibrium Gibbs state . This implies that the dissipator L D {\displaystyle L_{\text{D}}} should commute with the unitary part generated by H S {\displaystyle H_{\text{S}}} . [ 5 ] In addition an equilibrium state is stationary and stable. This assumption is used to derive the Kubo-Martin-Schwinger stability criterion for thermal equilibrium i.e. KMS state .
A unique and consistent approach is obtained by deriving the generator, L D {\displaystyle L_{\text{D}}} , in the weak system bath
coupling limit. [ 11 ] In this limit, the interaction energy can be neglected. This approach represents a thermodynamic idealization: it allows energy transfer, while keeping a tensor product separation
between the system and bath, i.e., a quantum version of an isothermal partition.
Markovian behavior involves a rather complicated cooperation between system and bath dynamics. This means that in
phenomenological treatments, one cannot combine arbitrary system Hamiltonians, H S {\displaystyle H_{\text{S}}} , with a given L-GKS generator. This observation is particularly important in the context of quantum thermodynamics, where it is tempting to study Markovian dynamics with an arbitrary control Hamiltonian. Erroneous derivations of the quantum master equation can easily lead to a violation of the laws of thermodynamics.
An external perturbation modifying the Hamiltonian of the system will also modify the heat flow. As a result, the L-GKS generator has to be renormalized. For a slow change, one can adopt the adiabatic approach and use the instantaneous system’s Hamiltonian to derive L D {\displaystyle L_{\text{D}}} . An important class of problems in quantum thermodynamics is periodically driven systems. Periodic quantum heat engines and power-driven refrigerators fall into this class.
A reexamination of the time-dependent heat current expression using quantum transport techniques has been proposed. [ 12 ]
A derivation of consistent dynamics beyond the weak coupling limit has been suggested. [ 13 ]
Phenomenological formulations of irreversible quantum dynamics consistent with the second law and implementing the geometric idea of "steepest entropy ascent" or "gradient flow" have been suggested to model relaxation and strong coupling. [ 14 ] [ 15 ]
The second law of thermodynamics is a statement on the irreversibility of dynamics or, the breakup of time reversal symmetry ( T-symmetry ). This should be consistent with the empirical direct definition: heat will flow spontaneously from a hot source to a cold sink.
From a static viewpoint, for a closed quantum system, the 2nd law of thermodynamics is a consequence of the unitary evolution. [ 16 ] In this approach, one accounts for the entropy change before and after a change in the entire system. A dynamical viewpoint is based on local accounting for the entropy changes in the subsystems and the entropy generated in the baths.
In thermodynamics, entropy is related to the amount of energy of a system that can be converted into mechanical work in a concrete process. [ 17 ] In quantum mechanics, this translates to the ability to measure and manipulate the system based on the information gathered by measurement. An example is the case of Maxwell’s demon , which has been resolved by Leó Szilárd . [ 18 ] [ 19 ] [ 20 ]
The entropy of an observable is associated with the complete projective measurement of an observable, ⟨ A ⟩ {\displaystyle \langle A\rangle } , where the operator A {\displaystyle A} has a spectral decomposition: A = ∑ j α j P j , {\displaystyle A=\sum _{j}\alpha _{j}P_{j},} where P j {\displaystyle P_{j}} are the projection operators of the eigenvalue α j . {\displaystyle \alpha _{j}.} The probability of outcome j {\displaystyle j} is p j = Tr ( ρ P j ) . {\displaystyle p_{j}=\operatorname {Tr} (\rho P_{j}).} The entropy associated with the observable ⟨ A ⟩ {\displaystyle \langle A\rangle } is the Shannon entropy with respect to the possible outcomes: S A = − ∑ j p j ln p j {\displaystyle S_{A}=-\sum _{j}p_{j}\ln p_{j}}
The most significant observable in thermodynamics is the energy represented by the Hamiltonian operator H , {\displaystyle H,} and its associated energy entropy, S E . {\displaystyle S_{E}.} [ 21 ]
John von Neumann suggested to single out the most informative observable to characterize the entropy of the system. This invariant is obtained by minimizing the entropy with respect to all possible observables. The most informative observable operator commutes with the state of the system. The
entropy of this observable is termed the Von Neumann entropy and is equal to S vn = − Tr ( ρ ln ρ ) . {\displaystyle S_{\text{vn}}=-\operatorname {Tr} (\rho \ln \rho ).} As a consequence, S A ≥ S vn {\displaystyle S_{A}\geq S_{\text{vn}}} for all observables. At thermal equilibrium the energy entropy is equal to the von Neumann entropy : S E = S vn . {\displaystyle S_{E}=S_{\text{vn}}.}
S vn {\displaystyle S_{\text{vn}}} is invariant to a unitary transformation changing the state. The Von Neumann entropy S vn {\displaystyle S_{\text{vn}}} is additive only for a system state that is composed of a tensor product of its subsystems: ρ = Π j ⊗ ρ j {\displaystyle \rho =\Pi _{j}\otimes \rho _{j}}
No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.
This statement for N-coupled heat baths in steady state becomes ∑ n J n T n ≥ 0 {\displaystyle \sum _{n}{\frac {J_{n}}{T_{n}}}\geq 0}
A dynamical version of the II-law can be proven, based on Spohn 's inequality: [ 8 ] Tr ( L D ρ [ ln ρ ( ∞ ) − ln ρ ] ) ≥ 0 , {\displaystyle \operatorname {Tr} \left(L_{\text{D}}\rho \left[\ln \rho (\infty )-\ln \rho \right]\right)\geq 0,} which is valid for any L-GKS generator, with a stationary state, ρ ( ∞ ) {\displaystyle \rho (\infty )} . [ 5 ]
Consistency with thermodynamics can be employed to verify quantum dynamical models of transport. For example, local models for networks where local L-GKS equations are connected through weak links have been thought to violate the second law of thermodynamics . [ 22 ] In 2018 has been shown that, by correctly taking into account all work and energy contributions in the full system, local master equations are fully coherent with the second law of thermodynamics . [ 23 ]
Thermodynamic adiabatic processes have no entropy change. Typically, an external control modifies
the state. A quantum version of an adiabatic process can be modeled by an externally controlled time dependent
Hamiltonian H ( t ) {\displaystyle H(t)} . If the system is isolated, the dynamics are unitary, and therefore, S vn {\displaystyle S_{\text{vn}}} is a constant. A quantum adiabatic process is defined by the energy entropy S E {\displaystyle S_{E}} being constant. The quantum adiabatic condition is therefore equivalent to no net change in the population of the instantaneous energy levels. This implies that the Hamiltonian should commute with itself at different times: [ H ( t ) , H ( t ′ ) ] = 0 {\displaystyle [H(t),H(t')]=0} .
When the adiabatic conditions are not fulfilled, additional work is required to reach the final control value. For an isolated system, this work is recoverable, since the dynamics is unitary and can be reversed. In this case, quantum friction can be suppressed using shortcuts to adiabaticity as demonstrated in the laboratory using a unitary Fermi gas in a time-dependent trap. [ 24 ]
The coherence stored in the off-diagonal elements of the density operator carry the required information to recover the extra energy cost and reverse the dynamics. Typically, this energy is not recoverable, due to interaction with a bath that causes energy dephasing. The bath, in this case, acts like a measuring apparatus of energy. This lost energy is the quantum version of friction. [ 25 ] [ 26 ]
There are seemingly two independent formulations of the third law of thermodynamics . Both were originally stated by Walther Nernst . The first formulation is known as the Nernst heat theorem , and can be phrased as:
The second formulation is dynamical, known as the unattainability principle [ 27 ]
At steady state the second law of thermodynamics implies that the total entropy production is non-negative.
When the cold bath approaches the absolute zero temperature,
it is necessary to eliminate the entropy production divergence at the cold side
when T c → 0 {\displaystyle T_{\text{c}}\rightarrow 0} , therefore S ˙ c ∝ − T c α , α ≥ 0 . {\displaystyle {\dot {S}}_{\text{c}}\propto -T_{\text{c}}^{\alpha }~~~,~~~~\alpha \geq 0~~.} For α = 0 {\displaystyle \alpha =0} the fulfillment of the second law depends on the entropy production of the other baths, which should compensate for the negative entropy production of the cold bath.
The first formulation of the third law modifies this restriction. Instead of α ≥ 0 {\displaystyle \alpha \geq 0} the third law imposes α > 0 {\displaystyle \alpha >0} , guaranteeing that at absolute zero the entropy production at the cold bath is zero: S ˙ c = 0 {\displaystyle {\dot {S}}_{\text{c}}=0} . This requirement leads to the scaling condition of the heat current J c ∝ T c α + 1 {\displaystyle {J}_{\text{c}}\propto T_{\text{c}}^{\alpha +1}} .
The second formulation, known as the unattainability principle can be rephrased as; [ 28 ]
The dynamics of the cooling process is governed by the equation: J c ( T c ( t ) ) = − c V ( T c ( t ) ) d T c ( t ) d t . {\displaystyle {J}_{\text{c}}(T_{\text{c}}(t))=-c_{V}(T_{\text{c}}(t)){\frac {dT_{\text{c}}(t)}{dt}}~~.} where c V ( T c ) {\displaystyle c_{V}(T_{\text{c}})} is the heat capacity of the bath. Taking J c ∝ T c α + 1 {\displaystyle {J}_{\text{c}}\propto T_{\text{c}}^{\alpha +1}} and c V ∼ T c η {\displaystyle c_{V}\sim T_{\text{c}}^{\eta }} with η ≥ 0 {\displaystyle {\eta }\geq 0} , we can quantify this formulation by evaluating the characteristic exponent ζ {\displaystyle \zeta } of the cooling process, d T c ( t ) d t ∝ − T c ζ , T c → 0 , ζ = α − η + 1 {\displaystyle {\frac {dT_{\text{c}}(t)}{dt}}\propto -T_{\text{c}}^{\zeta },~~~~~T_{\text{c}}\to 0,\;\;\quad {\zeta =\alpha -\eta +1}}
This equation introduces the relation between the characteristic exponents ζ {\displaystyle \zeta } and α {\displaystyle \alpha } . When ζ < 0 {\displaystyle \zeta <0} then the bath is cooled to zero temperature in a finite time, which implies a violation of the third law. It is apparent from the last equation, that the unattainability principle is more restrictive than the Nernst heat theorem .
The basic idea of quantum typicality is that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time. This is meant to apply to Schrödinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average. [ 29 ]
Quantum ergodic theorem originated by John von Neumann is a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of termed normal typicality, i.e. the statement that, for typical large systems, every initial wave function ψ 0 {\displaystyle \psi _{0}} from an energy shell is ‘normal’: it evolves in such a way that ψ t {\displaystyle \psi _{t}} for most t, is macroscopically equivalent to the micro-canonical density matrix. [ 30 ]
The second law of thermodynamics can be interpreted as quantifying state transformations which are statistically unlikely so that they become effectively forbidden. The second law typically applies to systems composed of many particles interacting; Quantum thermodynamics resource theory is a formulation of thermodynamics in the regime where it can be applied to a small number of particles interacting with a heat bath. For processes which are cyclic or very close to cyclic, the second law for microscopic systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on what state transformations are possible, but an entire family of constraints. These second laws are not only relevant for small systems, but also apply to individual macroscopic systems interacting via long-range interactions, which only satisfy the ordinary second law on average. By making precise the definition of thermal operations, the laws of thermodynamics take on a form with the first law defining the class of thermal operations, the zeroth law emerging as a unique condition ensuring the theory is nontrivial, and the remaining laws being a monotonicity property of generalised free energies. [ 31 ] [ 32 ]
Thermodynamic systems typically conserve quantities—known as charges—such as energy and particle number. These charges are often implicitly assumed to commute. This assumption underlies, for example, the derivation of thermal state forms, the Eigenstate Thermalization Hypothesis , and transport coefficients . However, key quantum phenomena, including uncertainty relations, arise precisely from the noncommutation of observables. How does this noncommutation affect thermodynamic behaviour? [ 33 ]
The noncommutation of conserved charges has been shown to challenge standard assumptions: it can invalidate conventional derivations of the thermal state, [ 34 ] increase entanglement, [ 35 ] induce critical dynamics, [ 36 ] alter entropy production, [ 37 ] and conflict with the eigenstate thermalization hypothesis, [ 38 ] among other effects.
A central open question remains: evidence suggests that noncommuting charges can both hinder and enhance thermalization, revealing a conceptual tension at the heart of this growing field. [ 39 ]
Nanoscale allows for the preparation of quantum systems in physical states without classical analogs. There, complex out-of-equilibrium scenarios may be produced by the initial preparation of either the working substance or the reservoirs of quantum particles, the latter dubbed as "engineered reservoirs".
There are different forms of engineered reservoirs. Some of them involve subtle quantum coherence or correlation effects, [ 40 ] [ 41 ] [ 42 ] while others rely solely on nonthermal classical probability distribution functions. [ 43 ] [ 44 ] [ 45 ] [ 46 ] Interesting phenomena may emerge from the use of engineered reservoirs such as efficiencies greater than the Otto limit, [ 42 ] violations of Clausius inequalities, [ 47 ] or simultaneous extraction of heat and work from the reservoirs. [ 41 ] | https://en.wikipedia.org/wiki/Quantum_thermodynamics |
Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. [ 1 ] The source of these states may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general mixed states . To be able to uniquely identify the state, the measurements must be tomographically complete . That is, the measured operators must form an operator basis on the Hilbert space of the system, providing all the information about the state. Such a set of observations is sometimes called a quorum . The term tomography was first used in the quantum physics literature in a 1993 paper introducing experimental optical homodyne tomography. [ 2 ]
In quantum process tomography on the other hand, known quantum states are used to probe a quantum process to find out how the process can be described. Similarly, quantum measurement tomography works to find out what measurement is being performed. Whereas, randomized benchmarking scalably obtains a figure of merit of the overlap between the error prone physical quantum process and its ideal counterpart.
The general principle behind quantum state tomography is that by repeatedly performing many different measurements on quantum systems described by identical density matrices, frequency counts can be used to infer probabilities , and these probabilities are combined with Born's rule to determine a density matrix which fits the best with the observations.
This can be easily understood by making a classical analogy. Consider a harmonic oscillator (e.g. a pendulum). The position and momentum of the oscillator at any given point can be measured and therefore the motion can be completely described by the phase space . This is shown in figure 1. By performing this measurement for a large number of identical oscillators we get a probability distribution in the phase space (figure 2). This distribution can be normalized (the oscillator at a given time has to be somewhere) and the distribution must be non-negative. So we have retrieved a function W ( x , p ) {\displaystyle W(x,p)} which gives a description of the chance of finding the particle at a given point with a given momentum.
For quantum mechanical particles the same can be done. The only difference is that the Heisenberg's uncertainty principle mustn't be violated, meaning that we cannot measure the particle's momentum and position at the same time. The particle's momentum and its position are called quadratures (see Optical phase space for more information) in quantum related states. By measuring one of the quadratures of a large number of identical quantum states will give us a probability density corresponding to that particular quadrature. This is called the marginal distribution , p r ( X ) {\displaystyle \mathrm {pr} (X)} or p r ( P ) {\displaystyle \mathrm {pr} (P)} (see figure 3). In the following text we will see that this probability density is needed to characterize the particle's quantum state, which is the whole point of quantum tomography.
Quantum tomography is applied on a source of systems, to determine the quantum state of the output of that source. Unlike a measurement on a single system, which determines the system's current state after the measurement (in general, the act of making a measurement alters the quantum state), quantum tomography works to determine the state(s) prior to the measurements.
Quantum tomography can be used for characterizing optical signals, including measuring the signal gain and loss of optical devices, [ 3 ] as well as in quantum computing and quantum information theory to reliably determine the actual states of the qubits . [ 4 ] [ 5 ] One can imagine a situation in which a person Bob prepares many identical objects (particles or fields) in the same quantum states and then gives them to Alice to measure. Not confident with Bob's description of the state, Alice may wish to do quantum tomography to classify the state herself.
Using Born's rule , one can derive the simplest form of quantum tomography. Generally, being in a pure state is not known in advance, and a state may be mixed. In this case, many different types of measurements will have to be performed, many times each. To fully reconstruct the density matrix for a mixed state in a finite-dimensional Hilbert space , the following technique may be used.
Born's rule states P ( E i | ρ ) = T r a c e ( E i ρ ) {\displaystyle \mathrm {P} (E_{i}|\rho )=\mathrm {Trace} (E_{i}\rho )} , where E i {\displaystyle E_{i}} is a particular measurement outcome projector and ρ {\displaystyle \rho } is the density matrix of the system.
Given a histogram of observations for each measurement, one has an approximation p i {\displaystyle p_{i}} to P ( E i | ρ ) {\displaystyle \mathrm {P} (E_{i}|\rho )} for each E i {\displaystyle E_{i}} .
Given linear operators S {\displaystyle S} and T {\displaystyle T} , define the inner product
where T → {\displaystyle {\vec {T}}} is representation of the T {\displaystyle T} operator as a column vector and S → † {\displaystyle {\vec {S}}^{\dagger }} a row vector such that S → † T → {\displaystyle {\vec {S}}^{\dagger }{\vec {T}}} is the inner product in C d {\displaystyle \mathbb {C} ^{d}} of the two.
Define the matrix A {\displaystyle A} as
Here E i is some fixed list of individual measurements (with binary outcomes), and A does all the measurements at once.
Then applying this to ρ → {\displaystyle {\vec {\rho }}} yields the probabilities :
Linear inversion corresponds to inverting this system using the observed relative frequencies p → {\displaystyle {\vec {p}}} to derive ρ → {\displaystyle {\vec {\rho }}} (which is isomorphic to ρ {\displaystyle \displaystyle \rho } ).
This system is not going to be square in general, as for each measurement being made there will generally be multiple measurement outcome projectors E i {\displaystyle E_{i}} . For example, in a 2-D Hilbert space with 3 measurements σ x , σ y , σ z {\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}} , each measurement has 2 outcomes, each of which has a projector E i , for 6 projectors, whereas the real dimension of the space of density matrices is (2⋅2 2 )/2=4, leaving A {\displaystyle A} to be 6 x 4. To solve the system, multiply on the left by A T {\displaystyle A^{T}} :
Now solving for ρ → {\displaystyle {\vec {\rho }}} yields the pseudoinverse :
This works in general only if the measurement list E i is tomographically complete. Otherwise, the matrix A T A {\displaystyle A^{T}A} will not be invertible .
In infinite dimensional Hilbert spaces , e.g. in measurements of continuous variables such as position, the methodology is somewhat more complex. One notable example is in the tomography of light , known as optical homodyne tomography . Using balanced homodyne measurements, one can derive the Wigner function and a density matrix for the state of the light . [ 6 ]
One approach involves measurements along different rotated directions in phase space . For each direction θ {\displaystyle \theta } , one can find a probability distribution w ( q , θ ) {\displaystyle w(q,\theta )} for the probability density of measurements in the θ {\displaystyle \theta } direction of phase space yielding the value q {\displaystyle q} . Using an inverse Radon transformation (the filtered back projection) on w ( q , θ ) {\displaystyle w(q,\theta )} leads to the Wigner function , W ( x , p ) {\displaystyle \mathrm {W} (x,p)} , [ 7 ] which can be converted by an inverse Fourier transform into the density matrix for the state in any basis. [ 5 ] A similar technique is often used in medical tomography .
The density matrix of a single qubit can be expressed in terms of its Bloch vector r → {\displaystyle {\vec {r}}} and the Pauli vector σ → {\displaystyle {\vec {\sigma }}} :
The single-qubit state tomography can be performed by means of single-qubit Pauli measurements: [ 8 ]
This algorithm is the foundation for qubit tomography and is used in some quantum programming routines, like that of Qiskit . [ 9 ] [ 10 ]
Electromagnetic field amplitudes (quadratures) can be measured with high efficiency using photodetectors together with temporal mode selectivity. Balanced homodyne tomography is a reliable technique of reconstructing quantum states in the optical domain. This technique combines the advantages of the high efficiencies of photodiodes in measuring the intensity or photon number of light, together with measuring the quantum features of light by a clever set-up called the homodyne tomography detector.
Quantum homodyne tomography is understood by the following example.
A laser is directed onto a 50-50% beamsplitter , splitting the laser beam into two beams. One is used as a local oscillator (LO) and the other is used to generate photons with a particular quantum state . The generation of quantum states can be realized, e.g. by directing the laser beam through a frequency doubling crystal [ 11 ] and then onto a parametric down-conversion crystal. This crystal generates two photons in a certain quantum state. One of the photons is used as a trigger signal used to trigger (start) the readout event of the homodyne tomography detector. The other photon is directed into the homodyne tomography detector, in order to reconstruct its quantum state. Since the trigger and signal photons are entangled (this is explained by the spontaneous parametric down-conversion article), it is important to realize that the optical mode of the signal state is created nonlocal only when the trigger photon impinges the photodetector (of the trigger event readout module) and is actually measured. More simply said, it is only when the trigger photon is measured, that the signal photon can be measured by the homodyne detector.
Now consider the homodyne tomography detector as depicted in figure 4 (figure missing). The signal photon (this is the quantum state we want to reconstruct) interferes with the local oscillator , when they are directed onto a 50-50% beamsplitter . Since the two beams originate from the same so called master laser , they have the same fixed phase relation. The local oscillator must be intense, compared to the signal so it provides a precise phase reference. The local oscillator is so intense, that we can treat it classically (a = α) and neglect the quantum fluctuations.
The signal field is spatially and temporally controlled by the local oscillator, which has a controlled shape. Where the local oscillator is zero, the signal is rejected. Therefore, we have temporal-spatial mode selectivity of the signal.
The beamsplitter redirects the two beams to two photodetectors. The photodetectors generate an electric current proportional to the photon number . The two detector currents are subtracted and the resulting current is proportional to the electric field operator in the signal mode, depended on relative optical phase of signal and local oscillator.
Since the electric field amplitude of the local oscillator is much higher than that of the signal the intensity or fluctuations in the signal field can be seen. The homodyne tomography system functions as an amplifier . The system can be seen as an interferometer with such a high intensity reference beam (the local oscillator) that unbalancing the interference by a single photon in the signal is measurable. This amplification is well above the photodetectors noise floor .
The measurement is reproduced a large number of times. Then the phase difference between the signal and local oscillator is changed in order to ‘scan’ a different angle in the phase space . This can be seen from figure 4. The measurement is repeated again a large number of times and a marginal distribution is retrieved from the current difference. The marginal distribution can be transformed into the density matrix and/or the Wigner function . Since the density matrix and the Wigner function give information about the quantum state of the photon, we have reconstructed the quantum state of the photon.
The advantage of this balanced detection method is that this arrangement is insensitive to fluctuations in the intensity of the laser .
The quantum computations for retrieving the quadrature component from the current difference are performed as follows.
The photon number operator for the beams striking the photodetectors after the beamsplitter is given by:
where i is 1 and 2, for respectively beam one and two.
The mode operators of the field emerging the beamsplitters are given by:
The a ^ {\displaystyle {\hat {a}}} denotes the annihilation operator of the signal and alpha the complex amplitude of the local oscillator.
The number of photon difference is eventually proportional to the quadrature and given by:
Rewriting this with the relation:
Results in the following relation:
where we see clear relation between the photon number difference and the quadrature component q ^ θ {\displaystyle {\hat {q}}_{\theta }} . By keeping track of the sum current, one can recover information about the local oscillator's intensity, since this is usually an unknown quantity, but an important quantity for calculating the quadrature component q ^ θ {\displaystyle {\hat {q}}_{\theta }} .
One of the primary problems with using linear inversion to solve for the density matrix is that in general the computed solution will not be a valid density matrix. For example, it could give negative probabilities or probabilities greater than 1 to certain measurement outcomes. This is particularly an issue when fewer measurements are made.
Another issue is that in infinite dimensional Hilbert spaces , an infinite number of measurement outcomes would be required. Making assumptions about the structure and using a finite measurement basis leads to artifacts in the phase space density. [ 5 ]
Maximum likelihood estimation (also known as MLE or MaxLik) is a popular technique for dealing with the problems of linear inversion. By restricting the domain of density matrices to the proper space, and searching for the density matrix which maximizes the likelihood of giving the experimental results, it guarantees the state to be theoretically valid while giving a close fit to the data. The likelihood of a state is the probability that would be assigned to the observed results had the system been in that state.
Suppose the measurements { | y j ⟩ ⟨ y j | } {\displaystyle \{|y_{j}\rangle \langle y_{j}|\}} have been observed with frequencies f j {\displaystyle f_{j}} . Then the likelihood associated with a state ρ ^ {\displaystyle {\hat {\rho }}} is
where ⟨ y j | ρ ^ | y j ⟩ {\displaystyle \langle y_{j}|{\hat {\rho }}|y_{j}\rangle } is the probability of outcome y j {\displaystyle y_{j}} for the state ρ ^ {\displaystyle {\hat {\rho }}} .
Finding the maximum of this function is non-trivial and generally involves iterative methods. [ 12 ] [ 13 ] The methods are an active topic of research.
Maximum likelihood estimation suffers from some less obvious problems than linear inversion. One problem is that it makes predictions about probabilities that cannot be justified by the data. This is seen most easily by looking at the problem of zero eigenvalues . The computed solution using MLE often contains eigenvalues which are 0, i.e. it is rank deficient . In these cases, the solution then lies on the boundary of the n-dimensional Bloch sphere . This can be seen as related to linear inversion giving states which lie outside the valid space (the Bloch sphere). MLE in these cases picks a nearby point that is valid, and the nearest points are generally on the boundary. [ 4 ]
This is not physically a problem, the real state might have zero eigenvalues . However, since no value may be less than 0, an estimate of an eigenvalue being 0 implies that the estimator is certain the value is 0, otherwise they would have estimated some ϵ {\displaystyle \epsilon } greater than 0 with a small degree of uncertainty as the best estimate. This is where the problem arises, in that it is not logical to conclude with absolute certainty after a finite number of measurements that any eigenvalue (that is, the probability of a particular outcome) is 0. For example, if a coin is flipped 5 times and each time heads was observed, it does not mean there is 0 probability of getting tails, despite that being the most likely description of the coin. [ 4 ]
Bayesian mean estimation (BME) is a relatively new approach which addresses the problems of maximum likelihood estimation . It focuses on finding optimal solutions which are also honest in that they include error bars in the estimate. The general idea is to start with a likelihood function and a function describing the experimenter's prior knowledge (which might be a constant function), then integrate over all density matrices using the product of the likelihood function and prior knowledge function as a weight.
Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional Bloch sphere . In the case of a coin flipped N times to get N heads described above, with a constant prior knowledge function, BME would assign 1 N + 2 {\displaystyle \scriptstyle {\frac {1}{N+2}}} as the probability for tails. [ 4 ]
BME provides a high degree of accuracy in that it minimizes the operational divergences of the estimate from the actual state. [ 4 ]
The number of measurements needed for a full quantum state tomography for a multi-particle system scales exponentially with the number of particles, which
makes such a procedure impossible even for modest system sizes. Hence, several methods have been developed to
realize quantum tomography with fewer measurements.
The concept of matrix completion and compressed sensing have been applied to reconstruct density matrices from an incomplete set of measurements (that is, a set of measurements which is not a quorum). In general, this is impossible, but under assumptions (for example, if the density matrix is a pure state, or a combination of just a few pure states) then the density matrix has fewer degrees of freedom, and it may be possible to reconstruct the state from the incomplete measurements. [ 14 ]
Permutationally Invariant Quantum Tomography [ 15 ] is a procedure that has been developed mostly for states that are close to being
permutationally symmetric, which is typical in nowadays experiments. For two-state particles, the number of measurements needed scales only quadratically with the number of particles. [ 16 ] Besides the modest measurement effort, the processing of the measured data can also be done efficiently:
It is possible to carry out the fitting of a physical density matrix on the measured data even for large systems. [ 17 ] Permutationally Invariant Quantum Tomography has been combined with compressed sensing in a six-qubit
photonic experiment. [ 18 ]
One can imagine a situation in which an apparatus performs some measurement on quantum systems, and determining what particular measurement is desired. The strategy is to send in systems of various known states, and use these states to estimate the outcomes of the unknown measurement. Also known as "quantum estimation", tomography techniques are increasingly important including those for quantum measurement tomography and the very similar quantum state tomography. Since a measurement can always be characterized by a set of POVM 's, the goal is to reconstruct the characterizing POVM 's Π l {\displaystyle \Pi _{l}} . The simplest approach is linear inversion. As in quantum state observation, use
Exploiting linearity as above, this can be inverted to solve for the Π l {\displaystyle \Pi _{l}} .
Not surprisingly, this suffers from the same pitfalls as in quantum state tomography: namely, non-physical results, in particular negative probabilities. Here the Π l {\displaystyle \Pi _{l}} will not be valid POVM 's, as they will not be positive. Bayesian methods as well as Maximum likelihood estimation of the density matrix can be used to restrict the operators to valid physical results. [ 19 ]
Quantum process tomography (QPT) deals with identifying an unknown quantum dynamical process. The first approach, introduced in 1996 and sometimes known as standard quantum process tomography (SQPT) involves preparing an ensemble of quantum states and sending them through the process, then using quantum state tomography to identify the resultant states. [ 20 ] Other techniques include ancilla-assisted process tomography (AAPT) and entanglement-assisted process tomography (EAPT) which require an extra copy of the system. [ 21 ]
Each of the techniques listed above are known as indirect methods for characterization of quantum dynamics, since they require the use of quantum state tomography to reconstruct the process. In contrast, there are direct methods such as direct characterization of quantum dynamics (DCQD) which provide a full characterization of quantum systems without any state tomography. [ 22 ]
The number of experimental configurations (state preparations and measurements) required for full quantum process tomography grows exponentially with the number of constituent particles of a system. Consequently, in general, QPT is an impossible task for large-scale quantum systems. However, under weak decoherence assumption, a quantum dynamical map can find a sparse representation. The method of compressed quantum process tomography (CQPT) uses the compressed sensing technique and applies the sparsity assumption to reconstruct a quantum dynamical map from an incomplete set of measurements or test state preparations. [ 23 ]
A quantum process, also known as a quantum dynamical map, E ( ρ ) {\displaystyle {\mathcal {E}}(\rho )} , can be described by a completely positive map
where ρ ∈ B ( H ) {\displaystyle \rho \in {\mathcal {B(H)}}} , the bounded operators on Hilbert space ; with operation elements A i {\displaystyle \displaystyle A_{i}} satisfying ∑ i A i † A i ≤ I {\displaystyle \textstyle \sum _{i}A_{i}^{\dagger }A_{i}\leq I} so that T r [ E ( ρ ) ] ≤ 1 {\displaystyle \mathrm {Tr} [{\mathcal {E}}(\rho )]\leq 1} .
Let { E i } {\displaystyle \displaystyle \{E_{i}\}} be an orthogonal basis for B ( H ) {\displaystyle {\mathcal {B(H)}}} . Write the A i {\displaystyle \displaystyle A_{i}} operators in this basis
This leads to
where χ m n = ∑ i a i m a i n ∗ {\displaystyle \chi _{mn}=\sum _{i}a_{im}a_{in}^{*}} .
The goal is then to solve for χ {\displaystyle \displaystyle \chi } , which is a positive superoperator and completely characterizes E {\displaystyle {\mathcal {E}}} with respect to the { E i } {\displaystyle \displaystyle \{E_{i}\}} basis. [ 21 ] [ 22 ]
SQPT approaches this using d 2 {\displaystyle d^{2}} linearly independent inputs ρ j {\displaystyle \rho _{j}} , where d {\displaystyle d} is the dimension of the Hilbert space H {\displaystyle {\mathcal {H}}} . For each of these input states ρ j {\displaystyle \rho _{j}} , sending it through the process gives an output state E ( ρ ) {\displaystyle {\mathcal {E}}(\rho )} which can be written as a linear combination of the ρ k {\displaystyle \rho _{k}} , i.e. E ( ρ j ) = ∑ k c j k ρ k {\displaystyle \textstyle {\mathcal {E}}(\rho _{j})=\sum _{k}c_{jk}\rho _{k}} . By sending each ρ j {\displaystyle \rho _{j}} through many times, quantum state tomography can be used to determine the coefficients c j k {\displaystyle c_{jk}} experimentally.
Write
where B {\displaystyle B} is a matrix of coefficients.
Then
Since ρ k {\displaystyle \rho _{k}} form a linearly independent basis,
Inverting B {\displaystyle B} gives χ {\displaystyle \chi } : | https://en.wikipedia.org/wiki/Quantum_tomography |
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology .
Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with topological spaces and the related embedding of one space within another such as knots and links in three-dimensional space. This bra–ket notation of kets and bras can be generalised, becoming maps of vector spaces associated with topological spaces that allow tensor products . [ 1 ]
Topological entanglement involving linking and braiding can be intuitively related to quantum entanglement . [ 1 ]
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_topology |
In a quantum field theory , charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as quantum triviality . Strong evidence supports the idea that a field theory involving only a scalar Higgs boson is trivial in four spacetime dimensions, [ 1 ] [ 2 ] but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the Standard Model of particle physics , the question of triviality in Higgs models is of great importance.
This Higgs triviality is similar to the Landau pole problem in quantum electrodynamics , where this quantum theory may be inconsistent at very high momentum scales unless the renormalized charge is set to zero, i.e., unless the field theory has no interactions. The Landau pole question is generally considered to be of minor academic interest for quantum electrodynamics because of the inaccessibly large momentum scale at which the inconsistency appears. This is not however the case in theories that involve the elementary scalar Higgs boson, as the momentum scale at which a "trivial" theory exhibits inconsistencies may be accessible to present experimental efforts such as at the Large Hadron Collider (LHC) at CERN . In these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses of the W and Z bosons , as well as lepton masses like those of the electron and muon . If realistic models of particle physics such as the Standard Model suffer from triviality issues, the idea of an elementary scalar Higgs particle may have to be modified or abandoned.
The situation becomes more complex in theories that involve other particles however. In fact, the addition of other particles can turn a trivial theory into a nontrivial one, at the cost of introducing constraints. Depending on the details of the theory, the Higgs mass can be bounded or even calculable. [ 2 ] These quantum triviality constraints are in sharp contrast to the picture one derives at the classical level, where the Higgs mass is a free parameter. Quantum triviality can also lead to a calculable Higgs mass in asymptotic safety scenarios. [ 2 ]
Modern considerations of triviality are usually formulated in terms of the real-space renormalization group , largely developed by Kenneth Wilson and others. Investigations of triviality are usually performed in the context of lattice gauge theory . A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional renormalizable theories, came from condensed matter physics. Leo P. Kadanoff 's paper in 1966 proposed the "block-spin" renormalization group. [ 3 ] The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances.
This approach covered the conceptual point and was given full computational substance [ 4 ] in Wilson's extensive important contributions. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem , in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971 [ citation needed ] . He was awarded the Nobel prize for these decisive contributions in 1982.
In more technical terms, let us assume that we have a theory described by a certain function Z {\displaystyle Z} of the state variables { s i } {\displaystyle \{s_{i}\}} and a certain set of coupling constants { J k } {\displaystyle \{J_{k}\}} . This function may be a partition function , an action , a Hamiltonian , etc. It must contain the
whole description of the physics of the system.
Now we consider a certain blocking transformation of the state variables { s i } → { s ~ i } {\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}} ,
the number of s ~ i {\displaystyle {\tilde {s}}_{i}} must be lower than the number of s i {\displaystyle s_{i}} . Now let us try to rewrite the Z {\displaystyle Z} function only in terms of the s ~ i {\displaystyle {\tilde {s}}_{i}} . If this is achievable by a certain change in the parameters, { J k } → { J ~ k } {\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}} , then the theory is said to be renormalizable . The most important information in the RG flow are its fixed points . The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be trivial . Numerous fixed points appear in the study of lattice Higgs theories , but the nature of the quantum field theories associated with these remains an open question. [ 2 ]
The first evidence of possible triviality of quantum field theories was obtained by Landau, Abrikosov, and Khalatnikov [ 5 ] [ 6 ] [ 7 ] by finding the following relation of the observable charge g obs with the "bare" charge g 0 ,
where m is the mass of the particle, and Λ is the momentum cut-off. If g 0 is finite, then g obs tends to zero in the limit of infinite cut-off Λ .
In fact, the proper interpretation of Eq.1 consists in its inversion, so that g 0 (related to the length scale 1/Λ ) is chosen to give a correct value of g obs ,
The growth of g 0 with Λ invalidates Eqs. ( 1 ) and ( 2 ) in the region g 0 ≈ 1 (since they were obtained for g 0 ≪ 1 ) and the existence of the "Landau pole" in Eq.2 has no physical meaning.
The actual behavior of the charge g ( μ ) as a function of the momentum scale μ is determined by the full Gell-Mann–Low equation
which gives Eqs.( 1 ),( 2 ) if it is integrated under conditions g ( μ ) = g obs for μ = m and g ( μ ) = g 0 for μ = Λ , when only the term with β 2 {\displaystyle \beta _{2}} is retained in the right hand side.
The general behavior of g ( μ ) {\displaystyle g(\mu )} relies on the appearance of the function β ( g ) . According to the classification by Bogoliubov and Shirkov, [ 8 ] there are three qualitatively different situations:
The latter case corresponds to the quantum triviality in the full theory (beyond its perturbation context), as can be seen by reductio ad absurdum . Indeed, if g obs is finite, the theory is internally inconsistent. The only way to avoid it, is to tend μ 0 {\displaystyle \mu _{0}} to infinity, which is possible only for g obs → 0 .
As a result, the question of whether the Standard Model of particle physics is nontrivial remains a serious unresolved question. Theoretical proofs of triviality of the pure scalar field theory exist, but the situation for the full standard model is unknown. The implied constraints on the standard model have been discussed. [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ] [ 14 ] | https://en.wikipedia.org/wiki/Quantum_triviality |
The quantum tunneling of water occurs when water molecules in nanochannels exhibit quantum tunneling behavior that smears out the positions of the hydrogen atoms into a pair of correlated rings. [ 1 ] In that state, the water molecules become delocalized around a ring and assume an unusual double top-like shape. At low temperatures, the phenomenon showcases the quantum motion of water through the separating potential walls, which is forbidden in classical mechanics , but allowed in quantum mechanics . [ 2 ]
The quantum tunneling of water occurs under ultraconfinement in rocks, soil and cell walls. [ 2 ] The phenomenon is predicted to help scientists better understand the thermodynamic properties and behavior of water in confined environments such as water diffusion , transport in the channels of cell membranes and in carbon nanotubes .
Quantum tunneling in water was reported as early as 1992. At that time it was known that motions can destroy and regenerate the weak hydrogen bond by internal rotations of the substituent water monomers . [ 3 ]
On 18 March 2016, it was reported that the hydrogen bond can be broken by quantum tunneling in the water hexamer . Unlike previously reported tunneling motions in water, this involved the concerted breaking of two hydrogen bonds. [ 4 ]
On 22 April 2016, the journal Physical Review Letters reported the quantum tunneling of water molecules as demonstrated at the Spallation Neutron Source and Rutherford Appleton Laboratory . First indications of this phenomenon were seen by scientists from Russia and Germany in 2013 [ 5 ] based on the splitting of terahertz absorption lines of a water molecule captured in five- ångström channels in beryl . Subsequently it was directly observed using neutron scattering and analyzed by ab initio simulations. [ 6 ] In a beryl channel, the water molecule can occupy six symmetrical orientations, in agreement with the known crystal structure . [ 1 ] A single orientation has the oxygen atom approximately in the center of the channel, with the two hydrogens pointing to the same side toward one of the channel’s six hexagonal faces. Other orientations point to other faces, but are separated from each other by energy barriers of around 50 meV . [ 1 ] These barriers, however, do not stop the hydrogens from tunneling among the six orientations and thus split the ground state energy into multiple levels. [ 1 ] | https://en.wikipedia.org/wiki/Quantum_tunneling_of_water |
In physics, quantum tunnelling , barrier penetration , or simply tunnelling is a quantum mechanical phenomenon in which an object such as an electron or atom passes through a potential energy barrier that, according to classical mechanics , should not be passable due to the object not having sufficient energy to pass or surmount the barrier.
Tunneling is a consequence of the wave nature of matter , where the quantum wave function describes the state of a particle or other physical system , and wave equations such as the Schrödinger equation describe their behavior. The probability of transmission of a wave packet through a barrier decreases exponentially with the barrier height, the barrier width, and the tunneling particle's mass, so tunneling is seen most prominently in low-mass particles such as electrons or protons tunneling through microscopically narrow barriers. Tunneling is readily detectable with barriers of thickness about 1–3 nm or smaller for electrons, and about 0.1 nm or smaller for heavier particles such as protons or hydrogen atoms. [ 1 ] Some sources describe the mere penetration of a wave function into the barrier, without transmission on the other side, as a tunneling effect, such as in tunneling into the walls of a finite potential well . [ 2 ] [ 3 ]
Tunneling plays an essential role in physical phenomena such as nuclear fusion [ 4 ] and alpha radioactive decay of atomic nuclei. Tunneling applications include the tunnel diode , [ 5 ] quantum computing , flash memory , and the scanning tunneling microscope . Tunneling limits the minimum size of devices used in microelectronics because electrons tunnel readily through insulating layers and transistors that are thinner than about 1 nm. [ 6 ]
The effect was predicted in the early 20th century. Its acceptance as a general physical phenomenon came mid-century. [ 7 ]
Quantum tunnelling falls under the domain of quantum mechanics . To understand the phenomenon , particles attempting to travel across a potential barrier can be compared to a ball trying to roll over a hill. Quantum mechanics and classical mechanics differ in their treatment of this scenario.
Classical mechanics predicts that particles that do not have enough energy to classically surmount a barrier cannot reach the other side. Thus, a ball without sufficient energy to surmount the hill would roll back down. In quantum mechanics, a particle can, with a small probability, tunnel to the other side, thus crossing the barrier. The reason for this difference comes from treating matter as having properties of waves and particles .
The wave function of a physical system of particles specifies everything that can be known about the system. [ 8 ] Therefore, problems in quantum mechanics analyze the system's wave function. Using mathematical formulations, such as the Schrödinger equation , the time evolution of a known wave function can be deduced. The square of the absolute value of this wave function is directly related to the probability distribution of the particle positions, which describes the probability that the particles would be measured at those positions.
As shown in the animation, a wave packet impinges on the barrier, most of it is reflected and some is transmitted through the barrier. The wave packet becomes more de-localized: it is now on both sides of the barrier and lower in maximum amplitude, but equal in integrated square-magnitude, meaning that the probability the particle is somewhere remains unity. The wider the barrier and the higher the barrier energy, the lower the probability of tunneling.
Some models of a tunneling barrier, such as the rectangular barriers shown, can be analysed and solved algebraically. [ 9 ] : 96 Most problems do not have an algebraic solution, so numerical solutions are used. " Semiclassical methods " offer approximate solutions that are easier to compute, such as the WKB approximation .
The Schrödinger equation was published in 1926. The first person to apply the Schrödinger equation to a problem that involved tunneling between two classically allowed regions through a potential barrier was Friedrich Hund in a series of articles published in 1927. He studied the solutions of a double-well potential and discussed molecular spectra . [ 10 ] Leonid Mandelstam and Mikhail Leontovich discovered tunneling independently and published their results in 1928. [ 11 ]
In 1927, Lothar Nordheim , assisted by Ralph Fowler , published a paper that discussed thermionic emission and reflection of electrons from metals. He assumed a surface potential barrier that confines the electrons within the metal and showed that the electrons have a finite probability of tunneling through or reflecting from the surface barrier when their energies are close to the barrier energy. Classically, the electron would either transmit or reflect with 100% certainty, depending on its energy. In 1928 J. Robert Oppenheimer published two papers on field emission , i.e. the emission of electrons induced by strong electric fields. Nordheim and Fowler simplified Oppenheimer's derivation and found values for the emitted currents and work functions that agreed with experiments. [ 10 ]
A great success of the tunnelling theory was the mathematical explanation for alpha decay , which was developed in 1928 by George Gamow and independently by Ronald Gurney and Edward Condon . [ 12 ] [ 13 ] [ 14 ] [ 15 ] The latter researchers simultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between the half-life of the particle and the energy of emission that depended directly on the mathematical probability of tunneling. All three researchers were familiar with the works on field emission, [ 10 ] and Gamow was aware of Mandelstam and Leontovich's findings. [ 16 ]
In the early days of quantum theory, the term tunnel effect was not used, and the effect was instead referred to as penetration of, or leaking through, a barrier. The German term wellenmechanische Tunneleffekt was used in 1931 by Walter Schottky. [ 10 ] The English term tunnel effect entered the language in 1932 when it was used by Yakov Frenkel in his textbook. [ 10 ]
In 1957 Leo Esaki demonstrated tunneling of electrons over a few nanometer wide barrier in a semiconductor structure and developed a diode based on tunnel effect. [ 17 ] In 1960, following Esaki's work, Ivar Giaever showed experimentally that tunnelling also took place in superconductors . The tunnelling spectrum gave direct evidence of the superconducting energy gap . In 1962, Brian Josephson predicted the tunneling of superconducting Cooper pairs . Esaki, Giaever and Josephson shared the 1973 Nobel Prize in Physics for their works on quantum tunneling in solids. [ 18 ] [ 7 ]
In 1981, Gerd Binnig and Heinrich Rohrer developed a new type of microscope, called scanning tunneling microscope , which is based on tunnelling and is used for imaging surfaces at the atomic level. Binnig and Rohrer were awarded the Nobel Prize in Physics in 1986 for their discovery. [ 19 ]
Tunnelling is the cause of some important macroscopic physical phenomena.
Tunnelling is a source of current leakage in very-large-scale integration (VLSI) electronics and results in a substantial power drain and heating effects that plague such devices. It is considered the lower limit on how microelectronic device elements can be made. [ 20 ] Tunnelling is a fundamental technique used to program the floating gates of flash memory .
Cold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionic emission , where electrons randomly jump from the surface of a metal to follow a voltage bias because they statistically end up with more energy than the barrier, through random collisions with other particles. When the electric field is very large, the barrier becomes thin enough for electrons to tunnel out of the atomic state, leading to a current that varies approximately exponentially with the electric field. [ 21 ] These materials are important for flash memory, vacuum tubes, and some electron microscopes.
A simple barrier can be created by separating two conductors with a very thin insulator . These are tunnel junctions, the study of which requires understanding quantum tunnelling. [ 22 ] Josephson junctions take advantage of quantum tunnelling and superconductivity to create the Josephson effect . This has applications in precision measurements of voltages and magnetic fields , [ 21 ] as well as the multijunction solar cell .
Diodes are electrical semiconductor devices that allow electric current flow in one direction more than the other. The device depends on a depletion layer between N-type and P-type semiconductors to serve its purpose. When these are heavily doped the depletion layer can be thin enough for tunnelling. When a small forward bias is applied, the current due to tunnelling is significant. This has a maximum at the point where the voltage bias is such that the energy level of the p and n conduction bands are the same. As the voltage bias is increased, the two conduction bands no longer line up and the diode acts typically. [ 23 ]
Because the tunnelling current drops off rapidly, tunnel diodes can be created that have a range of voltages for which current decreases as voltage increases. This peculiar property is used in some applications, such as high speed devices where the characteristic tunnelling probability changes as rapidly as the bias voltage. [ 23 ]
The resonant tunnelling diode makes use of quantum tunnelling in a very different manner to achieve a similar result. This diode has a resonant voltage for which a current favors a particular voltage, achieved by placing two thin layers with a high energy conductance band near each other. This creates a quantum potential well that has a discrete lowest energy level . When this energy level is higher than that of the electrons, no tunnelling occurs and the diode is in reverse bias. Once the two voltage energies align, the electrons flow like an open wire. As the voltage further increases, tunnelling becomes improbable and the diode acts like a normal diode again before a second energy level becomes noticeable. [ 24 ]
A European research project demonstrated field effect transistors in which the gate (channel) is controlled via quantum tunnelling rather than by thermal injection, reducing gate voltage from ≈1 volt to 0.2 volts and reducing power consumption by up to 100×. If these transistors can be scaled up into VLSI chips , they would improve the performance per power of integrated circuits . [ 25 ] [ 26 ]
While the Drude-Lorentz model of electrical conductivity makes excellent predictions about the nature of electrons conducting in metals, it can be furthered by using quantum tunnelling to explain the nature of the electron's collisions. [ 21 ] When a free electron wave packet encounters a long array of uniformly spaced barriers , the reflected part of the wave packet interferes uniformly with the transmitted one between all barriers so that 100% transmission becomes possible. The theory predicts that if positively charged nuclei form a perfectly rectangular array, electrons will tunnel through the metal as free electrons, leading to extremely high conductance , and that impurities in the metal will disrupt it. [ 21 ]
The scanning tunnelling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer , may allow imaging of individual atoms on the surface of a material. [ 21 ] It operates by taking advantage of the relationship between quantum tunnelling with distance. When the tip of the STM's needle is brought close to a conduction surface that has a voltage bias, measuring the current of electrons that are tunnelling between the needle and the surface reveals the distance between the needle and the surface. By using piezoelectric rods that change in size when voltage is applied, the height of the tip can be adjusted to keep the tunnelling current constant. The time-varying voltages that are applied to these rods can be recorded and used to image the surface of the conductor. [ 21 ] STMs are accurate to 0.001 nm, or about 1% of atomic diameter. [ 24 ]
Quantum tunnelling is an essential phenomenon for nuclear fusion. The temperature in stellar cores is generally insufficient to allow atomic nuclei to overcome the Coulomb barrier and achieve thermonuclear fusion . Quantum tunnelling increases the probability of penetrating this barrier. Though this probability is still low, the extremely large number of nuclei in the core of a star is sufficient to sustain a steady fusion reaction. [ 27 ]
Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the tunnelling of a particle out of the nucleus (an electron tunneling into the nucleus is electron capture ). This was the first application of quantum tunnelling. Radioactive decay is a relevant issue for astrobiology as this consequence of quantum tunnelling creates a constant energy source over a large time interval for environments outside the circumstellar habitable zone where insolation would not be possible ( subsurface oceans ) or effective. [ 27 ]
Quantum tunnelling may be one of the mechanisms of hypothetical proton decay . [ 28 ] [ 29 ]
Chemical reactions in the interstellar medium occur at extremely low energies. Probably the most fundamental ion-molecule reaction involves hydrogen ions with hydrogen molecules.
The quantum mechanical tunnelling rate for the same reaction using the hydrogen isotope deuterium , D − + H 2 → H − + HD, has been measured experimentally in an ion trap.
The deuterium was placed in an ion trap and cooled. The trap was then filled with hydrogen. At the temperatures used in the experiment, the energy barrier for reaction would not allow the reaction to succeed with classical dynamics alone. Quantum tunneling allowed reactions to happen in rare collisions. It was calculated from the experimental data that collisions happened one in every hundred billion. [ 30 ]
In chemical kinetics , the substitution of a light isotope of an element with a heavier one typically results in a slower reaction rate. This is generally attributed to differences in the zero-point vibrational energies for chemical bonds containing the lighter and heavier isotopes and is generally modeled using transition state theory . However, in certain cases, large isotopic effects are observed that cannot be accounted for by a semi-classical treatment, and quantum tunnelling is required. R. P. Bell developed a modified treatment of Arrhenius kinetics that is commonly used to model this phenomenon. [ 31 ]
By including quantum tunnelling, the astrochemical syntheses of various molecules in interstellar clouds can be explained, such as the synthesis of molecular hydrogen , water ( ice ) and the prebiotic important formaldehyde . [ 27 ] Tunnelling of molecular hydrogen has been observed in the lab. [ 32 ]
Quantum tunnelling is among the central non-trivial quantum effects in quantum biology . [ 33 ] Here it is important both as electron tunnelling and proton tunnelling . Electron tunnelling is a key factor in many biochemical redox reactions ( photosynthesis , cellular respiration ) as well as enzymatic catalysis. Proton tunnelling is a key factor in spontaneous DNA mutation. [ 27 ]
Spontaneous mutation occurs when normal DNA replication takes place after a particularly significant proton has tunnelled. [ 34 ] A hydrogen bond joins DNA base pairs. A double well potential along a hydrogen bond separates a potential energy barrier. It is believed that the double well potential is asymmetric, with one well deeper than the other such that the proton normally rests in the deeper well. For a mutation to occur, the proton must have tunnelled into the shallower well. The proton's movement from its regular position is called a tautomeric transition . If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardised, causing a mutation. [ 35 ] Per-Olov Lowdin was the first to develop this theory of spontaneous mutation within the double helix . Other instances of quantum tunnelling-induced mutations in biology are believed to be a cause of ageing and cancer. [ 36 ]
The time-independent Schrödinger equation for one particle in one dimension can be written as − ℏ 2 2 m d 2 d x 2 Ψ ( x ) + V ( x ) Ψ ( x ) = E Ψ ( x ) {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=E\Psi (x)} or d 2 d x 2 Ψ ( x ) = 2 m ℏ 2 ( V ( x ) − E ) Ψ ( x ) ≡ 2 m ℏ 2 M ( x ) Ψ ( x ) , {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)\Psi (x)\equiv {\frac {2m}{\hbar ^{2}}}M(x)\Psi (x),} where
The solutions of the Schrödinger equation take different forms for different values of x , depending on whether M ( x ) is positive or negative. When M ( x ) is constant and negative, then the Schrödinger equation can be written in the form d 2 d x 2 Ψ ( x ) = 2 m ℏ 2 M ( x ) Ψ ( x ) = − k 2 Ψ ( x ) , where k 2 = − 2 m ℏ 2 M . {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)=-k^{2}\Psi (x),\qquad {\text{where}}\quad k^{2}=-{\frac {2m}{\hbar ^{2}}}M.}
The solutions of this equation represent travelling waves, with phase-constant + k or − k . Alternatively, if M ( x ) is constant and positive, then the Schrödinger equation can be written in the form d 2 d x 2 Ψ ( x ) = 2 m ℏ 2 M ( x ) Ψ ( x ) = κ 2 Ψ ( x ) , where κ 2 = 2 m ℏ 2 M . {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)={\kappa }^{2}\Psi (x),\qquad {\text{where}}\quad {\kappa }^{2}={\frac {2m}{\hbar ^{2}}}M.}
The solutions of this equation are rising and falling exponentials in the form of evanescent waves . When M ( x ) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It follows that the sign of M ( x ) determines the nature of the medium, with negative M ( x ) corresponding to medium A and positive M ( x ) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive M ( x ) is sandwiched between two regions of negative M ( x ), hence creating a potential barrier.
The mathematics of dealing with the situation where M ( x ) varies with x is difficult, except in special cases that usually do not correspond to physical reality. A full mathematical treatment appears in the 1965 monograph by Fröman and Fröman. Their ideas have not been incorporated into physics textbooks, but their corrections have little quantitative effect.
The wave function is expressed as the exponential of a function: Ψ ( x ) = e Φ ( x ) , {\displaystyle \Psi (x)=e^{\Phi (x)},} where Φ ″ ( x ) + Φ ′ ( x ) 2 = 2 m ℏ 2 ( V ( x ) − E ) . {\displaystyle \Phi ''(x)+\Phi '(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right).} Φ ′ ( x ) {\displaystyle \Phi '(x)} is then separated into real and imaginary parts: Φ ′ ( x ) = A ( x ) + i B ( x ) , {\displaystyle \Phi '(x)=A(x)+iB(x),} where A ( x ) and B ( x ) are real-valued functions.
Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:
A ′ ( x ) + A ( x ) 2 − B ( x ) 2 = 2 m ℏ 2 ( V ( x ) − E ) . {\displaystyle A'(x)+A(x)^{2}-B(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right).}
To solve this equation using the semiclassical approximation, each function must be expanded as a power series in ℏ {\displaystyle \hbar } . From the equations, the power series must start with at least an order of ℏ − 1 {\displaystyle \hbar ^{-1}} to satisfy the real part of the equation; for a good classical limit starting with the highest power of the Planck constant possible is preferable, which leads to A ( x ) = 1 ℏ ∑ k = 0 ∞ ℏ k A k ( x ) {\displaystyle A(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}A_{k}(x)} and B ( x ) = 1 ℏ ∑ k = 0 ∞ ℏ k B k ( x ) , {\displaystyle B(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}B_{k}(x),} with the following constraints on the lowest order terms, A 0 ( x ) 2 − B 0 ( x ) 2 = 2 m ( V ( x ) − E ) {\displaystyle A_{0}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-E\right)} and A 0 ( x ) B 0 ( x ) = 0. {\displaystyle A_{0}(x)B_{0}(x)=0.}
At this point two extreme cases can be considered.
Case 1
If the amplitude varies slowly as compared to the phase A 0 ( x ) = 0 {\displaystyle A_{0}(x)=0} and B 0 ( x ) = ± 2 m ( E − V ( x ) ) {\displaystyle B_{0}(x)=\pm {\sqrt {2m\left(E-V(x)\right)}}} which corresponds to classical motion. Resolving the next order of expansion yields Ψ ( x ) ≈ C e i ∫ d x 2 m ℏ 2 ( E − V ( x ) ) + θ 2 m ℏ 2 ( E − V ( x ) ) 4 {\displaystyle \Psi (x)\approx C{\frac {e^{i\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}+\theta }}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}}}
Case 2
If the phase varies slowly as compared to the amplitude, B 0 ( x ) = 0 {\displaystyle B_{0}(x)=0} and A 0 ( x ) = ± 2 m ( V ( x ) − E ) {\displaystyle A_{0}(x)=\pm {\sqrt {2m\left(V(x)-E\right)}}} which corresponds to tunneling. Resolving the next order of the expansion yields Ψ ( x ) ≈ C + e + ∫ d x 2 m ℏ 2 ( V ( x ) − E ) + C − e − ∫ d x 2 m ℏ 2 ( V ( x ) − E ) 2 m ℏ 2 ( V ( x ) − E ) 4 {\displaystyle \Psi (x)\approx {\frac {C_{+}e^{+\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}+C_{-}e^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}}
In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points E = V ( x ) {\displaystyle E=V(x)} . Away from the potential hill, the particle acts similar to a free and oscillating wave; beneath the potential hill, the particle undergoes exponential changes in amplitude. By considering the behaviour at these limits and classical turning points a global solution can be made.
To start, a classical turning point, x 1 {\displaystyle x_{1}} is chosen and 2 m ℏ 2 ( V ( x ) − E ) {\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)} is expanded in a power series about x 1 {\displaystyle x_{1}} : 2 m ℏ 2 ( V ( x ) − E ) = v 1 ( x − x 1 ) + v 2 ( x − x 1 ) 2 + ⋯ {\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=v_{1}(x-x_{1})+v_{2}(x-x_{1})^{2}+\cdots }
Keeping only the first order term ensures linearity: 2 m ℏ 2 ( V ( x ) − E ) = v 1 ( x − x 1 ) . {\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=v_{1}(x-x_{1}).}
Using this approximation, the equation near x 1 {\displaystyle x_{1}} becomes a differential equation : d 2 d x 2 Ψ ( x ) = v 1 ( x − x 1 ) Ψ ( x ) . {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)=v_{1}(x-x_{1})\Psi (x).}
This can be solved using Airy functions as solutions. Ψ ( x ) = C A A i ( v 1 3 ( x − x 1 ) ) + C B B i ( v 1 3 ( x − x 1 ) ) {\displaystyle \Psi (x)=C_{A}Ai\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)+C_{B}Bi\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)}
Taking these solutions for all classical turning points, a global solution can be formed that links the limiting solutions. Given the two coefficients on one side of a classical turning point, the two coefficients on the other side of a classical turning point can be determined by using this local solution to connect them.
Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships between C , θ {\displaystyle C,\theta } and C + , C − {\displaystyle C_{+},C_{-}} are C + = 1 2 C cos ( θ − π 4 ) {\displaystyle C_{+}={\frac {1}{2}}C\cos {\left(\theta -{\frac {\pi }{4}}\right)}} and
With the coefficients found, the global solution can be found. Therefore, the transmission coefficient for a particle tunneling through a single potential barrier is T ( E ) = e − 2 ∫ x 1 x 2 d x 2 m ℏ 2 [ V ( x ) − E ] , {\displaystyle T(E)=e^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left[V(x)-E\right]}}},} where x 1 , x 2 {\displaystyle x_{1},x_{2}} are the two classical turning points for the potential barrier.
For a rectangular barrier, this expression simplifies to: T ( E ) = e − 2 2 m ℏ 2 ( V 0 − E ) ( x 2 − x 1 ) . {\displaystyle T(E)=e^{-2{\sqrt {{\frac {2m}{\hbar ^{2}}}(V_{0}-E)}}(x_{2}-x_{1})}.}
Some physicists have claimed that it is possible for spin-zero particles to travel faster than the speed of light when tunnelling. [ 7 ] This appears to violate the principle of causality , since a frame of reference then exists in which the particle arrives before it has left. In 1998, Francis E. Low reviewed briefly the phenomenon of zero-time tunnelling. [ 37 ] More recently, experimental tunnelling time data of phonons , photons , and electrons was published by Günter Nimtz . [ 38 ] Another experiment overseen by A. M. Steinberg , seems to indicate that particles could tunnel at apparent speeds faster than light. [ 39 ] [ 40 ]
Other physicists, such as Herbert Winful , [ 41 ] disputed these claims. Winful argued that the wave packet of a tunnelling particle propagates locally, so a particle can't tunnel through the barrier non-locally. Winful also argued that the experiments that are purported to show non-local propagation have been misinterpreted. In particular, the group velocity of a wave packet does not measure its speed, but is related to the amount of time the wave packet is stored in the barrier. Moreover, if quantum tunneling is modeled with the relativistic Dirac equation , well established mathematical theorems imply that the process is completely subluminal. [ 42 ] [ 43 ]
The concept of quantum tunneling can be extended to situations where there exists a quantum transport between regions that are classically not connected even if there is no associated potential barrier. This phenomenon is known as dynamical tunnelling. [ 44 ] [ 45 ]
The concept of dynamical tunnelling is particularly suited to address the problem of quantum tunnelling in high dimensions (d>1). In the case of an integrable system , where bounded classical trajectories are confined onto tori in phase space , tunnelling can be understood as the quantum transport between semi-classical states built on two distinct but symmetric tori. [ 46 ]
In real life, most systems are not integrable and display various degrees of chaos. Classical dynamics is then said to be mixed and the system phase space is typically composed of islands of regular orbits surrounded by a large sea of chaotic orbits. The existence of the chaotic sea, where transport is classically allowed, between the two symmetric tori then assists the quantum tunnelling between them. This phenomenon is referred as chaos-assisted tunnelling. [ 47 ] and is characterized by sharp resonances of the tunnelling rate when varying any system parameter.
When ℏ {\displaystyle \hbar } is small in front of the size of the regular islands, the fine structure of the classical phase space plays a key role in tunnelling. In particular the two symmetric tori are coupled "via a succession of classically forbidden transitions across nonlinear resonances" surrounding the two islands. [ 48 ]
Several phenomena have the same behavior as quantum tunnelling. Two examples are evanescent wave coupling [ 49 ] (the application of Maxwell's wave-equation to light ) and the application of the non-dispersive wave-equation from acoustics applied to "waves on strings" . [ citation needed ]
These effects are modeled similarly to the rectangular potential barrier . In these cases, one transmission medium through which the wave propagates that is the same or nearly the same throughout, and a second medium through which the wave travels differently. This can be described as a thin region of medium B between two regions of medium A. The analysis of a rectangular barrier by means of the Schrödinger equation can be adapted to these other effects provided that the wave equation has travelling wave solutions in medium A but real exponential solutions in medium B.
In optics , medium A is a vacuum while medium B is glass. In acoustics, medium A may be a liquid or gas and medium B a solid. For both cases, medium A is a region of space where the particle's total energy is greater than its potential energy and medium B is the potential barrier. These have an incoming wave and resultant waves in both directions. There can be more mediums and barriers, and the barriers need not be discrete. Approximations are useful in this case.
A classical wave-particle association was originally analyzed as analogous to quantum tunneling, [ 50 ] but subsequent analysis found a fluid dynamics cause related to the vertical momentum imparted to particles near the barrier. [ 51 ] | https://en.wikipedia.org/wiki/Quantum_tunnelling |
In physics , a quantum vortex represents a quantized flux circulation of some physical quantity . In most cases, quantum vortices are a type of topological defect exhibited in superfluids and superconductors . The existence of quantum vortices was first predicted by Lars Onsager in 1949 in connection with superfluid helium. [ 2 ] Onsager reasoned that quantisation of vorticity is a direct consequence of the existence of a superfluid order parameter as a spatially continuous wavefunction . Onsager also pointed out that quantum vortices describe the circulation of superfluid and conjectured that their excitations are responsible for superfluid phase transitions . These ideas of Onsager were further developed by Richard Feynman in 1955 [ 3 ] and in 1957 were applied to describe the magnetic phase diagram of type-II superconductors by Alexei Alexeyevich Abrikosov . [ 4 ] In 1935 Fritz London published a very closely related work on magnetic flux quantization in superconductors. London's fluxoid can also be viewed as a quantum vortex.
Quantum vortices are observed experimentally in type-II superconductors (the Abrikosov vortex ), liquid helium , and atomic gases [ 5 ] (see Bose–Einstein condensate ), as well as in photon fields ( optical vortex ) and exciton-polariton superfluids .
In a superfluid, a quantum vortex "carries" quantized orbital angular momentum , thus allowing the superfluid to rotate; in a superconductor, the vortex carries quantized magnetic flux .
The term "quantum vortex" is also used in the study of few body problems. [ 6 ] [ 7 ] Under the de Broglie–Bohm theory , it is possible to derive a "velocity field" from the wave function. In this context, quantum vortices are zeros on the wave function, around which this velocity field has a solenoidal shape, similar to that of irrotational vortex on potential flows of traditional fluid dynamics.
In a superfluid, a quantum vortex is a hole with the superfluid circulating around the vortex axis; the inside of the vortex may contain excited particles, air, vacuum, etc. The thickness of the vortex depends on a variety of factors; in liquid helium , the thickness is of the order of a few Angstroms .
A superfluid has the special property of having phase, given by the wavefunction , and the velocity of the superfluid is proportional to the gradient of the phase (in the parabolic mass approximation). The circulation around any closed loop in the superfluid is zero if the region enclosed is simply connected . The superfluid is deemed irrotational ; however, if the enclosed region actually contains a smaller region with an absence of superfluid, for example a rod through the superfluid or a vortex, then the circulation is:
where ℏ {\displaystyle \hbar } is the Planck constant divided by 2 π {\displaystyle 2\pi } , m is the mass of the superfluid particle, and Δ tot ϕ v {\displaystyle \Delta ^{\text{tot}}\phi _{v}} is the total phase difference around the vortex. Because the wave-function must return to its same value after an integer number of turns around the vortex (similar to what is described in the Bohr model ), then Δ tot ϕ v = 2 π n {\displaystyle \Delta ^{\text{tot}}\phi _{v}=2\pi n} , where n is an integer . Thus, the circulation is quantized:
A principal property of superconductors is that they expel magnetic fields ; this is called the Meissner effect . If the magnetic field becomes sufficiently strong it will, in some cases, "quench" the superconductive state by inducing a phase transition. In other cases, however, it will be energetically favorable for the superconductor to form a lattice of quantum vortices, which carry quantized magnetic flux through the superconductor. A superconductor that is capable of supporting vortex lattices is called a type-II superconductor, vortex-quantization in superconductors is general.
Over some enclosed area S, the magnetic flux is
Substituting a result of London's equation : j s = − n s e s 2 m A + n s e s ℏ m ∇ ϕ {\displaystyle \mathbf {j} _{s}=-{\frac {n_{s}e_{s}^{2}}{m}}\mathbf {A} +{\frac {n_{s}e_{s}\hbar }{m}}{\boldsymbol {\nabla }}\phi } , we find (with B = c u r l A {\displaystyle \mathbf {B} =\mathrm {curl} \,\,\mathbf {A} } ):
where n s , m , and e s are, respectively, number density, mass, and charge of the Cooper pairs .
If the region, S , is large enough so that j s = 0 {\displaystyle \mathbf {j} _{s}=0} along ∂ S {\displaystyle \partial S} , then
The flow of current can cause vortices in a superconductor to move, causing the electric field due to the phenomenon of electromagnetic induction . This leads to energy dissipation and causes the material to display a small amount of electrical resistance while in the superconducting state. [ 8 ]
The vortex states in ferromagnetic or antiferromagnetic material are also important, mainly for information technology. [ 9 ] They are exceptional, since in contrast to superfluids or superconducting material one has a more subtle mathematics: instead of the usual equation of the type curl v → ( x , y , z , t ) ∝ Ω → ( r , t ) ⋅ δ ( x , y ) , {\displaystyle \operatorname {curl} \ {\vec {v}}(x,y,z,t)\propto {\vec {\Omega }}(\mathrm {r} ,t)\cdot \delta (x,y),} where Ω → ( r , t ) {\displaystyle {\vec {\Omega }}(\mathrm {r} ,t)} is the vorticity at the spatial and temporal coordinates, and where δ ( x , y ) {\displaystyle \delta (x,y)} is the Dirac function , one has:
where now at any point and at any time there is the constraint m x 2 ( r , t ) + m y 2 ( r , t ) + m z 2 ( r , t ) ≡ M 0 2 {\displaystyle m_{x}^{2}(\mathrm {r} ,t)+m_{y}^{2}(\mathrm {r} ,t)+m_{z}^{2}(\mathrm {r} ,t)\equiv M_{0}^{2}} . Here M 0 {\displaystyle M_{0}} is constant, the constant magnitude of the non-constant magnetization vector m → ( x , y , z , t ) {\displaystyle {\vec {m}}(x,y,z,t)} . As a consequence the vector m → {\displaystyle {\vec {m}}} in eqn. (*) has been modified to a more complex entity m → e f f {\displaystyle {\vec {m}}_{\mathrm {eff} }} . This leads, among other points, to the following fact:
In ferromagnetic or antiferromagnetic material a vortex can be moved to generate bits for information storage and recognition, corresponding, e.g., to changes of the quantum number n . [ 9 ] But although the magnetization has the usual azimuthal direction, and although one has vorticity quantization as in superfluids, as long as the circular integration lines surround the central axis at far enough perpendicular distance, this apparent vortex magnetization will change with the distance from an azimuthal direction to an upward or downward one, as soon as the vortex center is approached.
Thus, for each directional element d φ d ϑ {\displaystyle \mathrm {d} \varphi \,\mathrm {d} \vartheta } there are now not two, but four bits to be stored by a change of vorticity: The first two bits concern the sense of rotation, clockwise or counterclockwise; the remaining bits three and four concern the polarization of the central singular line, which may be polarized up- or downwards. The change of rotation and/or polarization involves subtle topology . [ 10 ]
As first discussed by Onsager and Feynman, if the temperature in a superfluid or a superconductor is raised, the vortex loops undergo a second-order phase transition . This happens when the configurational entropy overcomes the Boltzmann factor , which suppresses the thermal or heat generation of vortex lines. The lines form a condensate. Since the centre of the lines, the vortex cores , are normal liquid or normal conductors, respectively, the condensation transforms the superfluid or superconductor into the normal state. The ensembles of vortex lines and their phase transitions can be described efficiently by a gauge theory .
In 1949 Onsager analysed a toy model consisting of a neutral system of point vortices confined to a finite area. [ 2 ] He was able to show that, due to the properties of two-dimensional point vortices the bounded area (and consequently, bounded phase space), allows the system to exhibit negative temperatures . Onsager provided the first prediction that some isolated systems can exhibit negative Boltzmann temperature. Onsager's prediction was confirmed experimentally for a system of quantum vortices in a Bose-Einstein condensate in 2019. [ 11 ] [ 12 ]
In a nonlinear quantum fluid, the dynamics and configurations of the vortex cores can be studied in terms of effective vortex–vortex pair interactions. The effective intervortex potential is predicted to affect quantum phase transitions and giving rise to different few-vortex molecules and many-body vortex patterns. [ 13 ] [ 14 ] Preliminary experiments in the specific system of exciton-polaritons fluids showed an effective attractive–repulsive intervortex dynamics between two cowinding vortices, whose attractive component can be modulated by the nonlinearity amount in the fluid. [ 15 ]
Quantum vortices can form via the Kibble–Zurek mechanism . As a condensate forms by quench cooling, separate protocondensates form with independent phases. As these phase domains merge quantum vortices can be trapped in the emerging condensate order parameter. Spontaneous quantum vortices were observed in atomic Bose–Einstein condensates in 2008. [ 16 ] | https://en.wikipedia.org/wiki/Quantum_vortex |
In particle physics , the quantum yield (denoted Φ ) of a radiation -induced process is the number of times a specific event occurs per photon absorbed by the system . [ 1 ]
Φ ( λ ) = number of events number of photons absorbed {\displaystyle \Phi (\lambda )={\frac {\text{ number of events }}{\text{ number of photons absorbed }}}}
The fluorescence quantum yield is defined as the ratio of the number of photons emitted to the number of photons absorbed. [ 2 ]
Φ = # p h o t o n s e m i t t e d # p h o t o n s a b s o r b e d {\displaystyle \Phi ={\frac {\rm {\#\ photons\ emitted}}{\rm {\#\ photons\ absorbed}}}}
Fluorescence quantum yield is measured on a scale from 0 to 1.0, but is often represented as a percentage . A quantum yield of 1.0 (100%) describes a process where each photon absorbed results in a photon emitted. Substances with the largest quantum yields, such as rhodamines , display the brightest emissions; however, compounds with quantum yields of 0.10 are still considered quite fluorescent.
Quantum yield is defined by the fraction of excited state fluorophores that decay through fluorescence:
Φ f = k f k f + ∑ k n r {\displaystyle \Phi _{f}={\frac {k_{f}}{k_{f}+\sum k_{\mathrm {nr} }}}}
where
Non-radiative processes are excited state decay mechanisms other than photon emission, which include: Förster resonance energy transfer , internal conversion , external conversion, and intersystem crossing . Thus, the fluorescence quantum yield is affected if the rate of any non-radiative pathway changes. The quantum yield can be close to unity if the non-radiative decay rate is much smaller than the rate of radiative decay, that is k f > k nr . [ 2 ]
Fluorescence quantum yields are measured by comparison to a standard of known quantum yield. [ 2 ] The quinine salt quinine sulfate in a sulfuric acid solution was regarded as the most common fluorescence standard, [ 3 ] however, a recent study revealed that the fluorescence quantum yield of this solution is strongly affected by the temperature, and should no longer be used as the standard solution. The quinine in 0.1M perchloric acid ( Φ = 0.60) shows no temperature dependence up to 45 °C, therefore it can be considered as a reliable standard solution. [ 4 ]
Experimentally, relative fluorescence quantum yields can be determined by measuring fluorescence of a fluorophore of known quantum yield with the same experimental parameters (excitation wavelength , slit widths, photomultiplier voltage etc.) as the substance in question. The quantum yield is then calculated by:
Φ = Φ R × I n t I n t R × 1 − 10 − A R 1 − 10 − A × n 2 n R 2 {\displaystyle \Phi =\Phi _{\mathrm {R} }\times {\frac {\mathit {Int}}{{\mathit {Int}}_{\mathrm {R} }}}\times {\frac {1-10^{-A_{\mathrm {R} }}}{1-10^{-A}}}\times {\frac {{n}^{2}}{{n_{\mathrm {R} }}^{2}}}}
where
The subscript R denotes the respective values of the reference substance. [ 5 ] [ 6 ] The determination of fluorescence quantum yields in scattering media requires additional considerations and corrections. [ 7 ]
Förster resonance energy transfer efficiency ( E ) is the quantum yield of the energy-transfer transition, i.e. the probability of the energy-transfer event occurring per donor excitation event:
E = Φ F R E T = k E T k E T + k f + ∑ k n r {\displaystyle E=\Phi _{\mathrm {FRET} }={\frac {k_{\mathrm {ET} }}{k_{\mathrm {ET} }+k_{f}+\sum k_{\mathrm {nr} }}}}
where
A fluorophore's environment can impact quantum yield, usually resulting from changes in the rates of non-radiative decay. [ 2 ] Many fluorophores used to label macromolecules are sensitive to solvent polarity. The class of 8-anilinonaphthalene-1-sulfonic acid (ANS) probe molecules are essentially non-fluorescent when in aqueous solution, but become highly fluorescent in nonpolar solvents or when bound to proteins and membranes. The quantum yield of ANS is ~0.002 in aqueous buffer, but near 0.4 when bound to serum albumin .
The quantum yield of a photochemical reaction describes the number of molecules undergoing a photochemical event per absorbed photon: [ 1 ]
Φ = # m o l e c u l e s u n d e r g o i n g r e a c t i o n o f i n t e r e s t # p h o t o n s a b s o r b e d b y p h o t o r e a c t i v e s u b s t a n c e {\displaystyle \Phi ={\frac {\rm {\#\ molecules\ undergoing\ reaction\ of\ interest}}{\rm {\#\ photons\ absorbed\ by\ photoreactive\ substance}}}}
In a chemical photodegradation process, when a molecule dissociates after absorbing a light quantum , the quantum yield is the number of destroyed molecules divided by the number of photons absorbed by the system. Since not all photons are absorbed productively, the typical quantum yield will be less than 1.
Φ = # m o l e c u l e s d e c o m p o s e d # p h o t o n s a b s o r b e d {\displaystyle \Phi ={\frac {\rm {\#\ molecules\ decomposed}}{\rm {\#\ photons\ absorbed}}}}
Quantum yields greater than 1 are possible for photo-induced or radiation-induced chain reactions , in which a single photon may trigger a long chain of transformations . One example is the reaction of hydrogen with chlorine , in which as many as 10 6 molecules of hydrogen chloride can be formed per quantum of blue light absorbed. [ 10 ]
Quantum yields of photochemical reactions can be highly dependent on the structure, proximity and concentration of the reactive chromophores, the type of solvent environment as well as the wavelength of the incident light. Such effects can be studied with wavelength-tunable lasers and the resulting quantum yield data can help predict conversion and selectivity of photochemical reactions. [ 11 ]
In optical spectroscopy , the quantum yield is the probability that a given quantum state is formed from the system initially prepared in some other quantum state. For example, a singlet to triplet transition quantum yield is the fraction of molecules that, after being photoexcited into a singlet state, cross over to the triplet state.
Quantum yield is used in modeling photosynthesis : [ 12 ]
Φ = μ m o l C O 2 f i x e d μ m o l p h o t o n s a b s o r b e d {\displaystyle \Phi ={\frac {\rm {\mu mol\ CO_{2}\ fixed}}{\rm {\mu mol\ photons\ absorbed}}}} | https://en.wikipedia.org/wiki/Quantum_yield |
The quaquaversal tiling is a nonperiodic tiling of Euclidean 3-space introduced by John Conway and Charles Radin . It is analogous to the pinwheel tiling in 2 dimensions having tile orientations that are dense in SO(3) . The basic solid tiles are 30-60-90 triangular prisms arranged in a pattern such that some copies are rotated by π/3, and some are rotated by π/2 in a perpendicular direction. [ 1 ]
They construct the group G( p , q ) given by a rotation of 2π/ p and a perpendicular rotation by 2π/ q ; the orientations in the quaquaversal tiling are given by G(6,4). G( p ,1) are cyclic groups , G( p ,2) are dihedral groups , G(4,4) is the octahedral group , and all other G( p , q ) are infinite and dense in SO(3); if p and q are odd and ≥3, then G( p , q ) is a free group . [ 1 ]
Radin and Lorenzo Sadun constructed similar honeycombs based on a tiling related to the Penrose tilings and the pinwheel tiling; the former has orientations in G(10,4), and the latter has orientations in G( p ,4) with the irrational rotation 2π/ p = arctan(1/2) . They show that G( p ,4) is dense in SO(3) for the aforementioned value of p , and whenever cos(2π/ p ) is transcendental . [ 2 ]
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quaquaversal_tiling |
In particle physics , the quark model is a classification scheme for hadrons in terms of their valence quarks —the quarks and antiquarks that give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)" , or the Eightfold Way , the successful classification scheme organizing the large number of lighter hadrons that were being discovered starting in the 1950s and continuing through the 1960s. It received experimental verification beginning in the late 1960s and is a valid and effective classification of them to date. The model was independently proposed by physicists Murray Gell-Mann , [ 1 ] who dubbed them "quarks" in a concise paper, and George Zweig , [ 2 ] [ 3 ] who suggested "aces" in a longer manuscript. André Petermann also touched upon the central ideas from 1963 to 1965, without as much quantitative substantiation. [ 4 ] [ 5 ] Today, the model has essentially been absorbed as a component of the established quantum field theory of strong and electroweak particle interactions, dubbed the Standard Model .
Hadrons are not really "elementary", and can be regarded as bound states of their "valence quarks" and antiquarks, which give rise to the quantum numbers of the hadrons. These quantum numbers are labels identifying the hadrons, and are of two kinds. One set comes from the Poincaré symmetry — J PC , where J , P and C stand for the total angular momentum , P-symmetry , and C-symmetry , respectively.
The other set is the flavor quantum numbers such as the isospin , strangeness , charm , and so on. The strong interactions binding the quarks together are insensitive to these quantum numbers, so variation of them leads to systematic mass and coupling relationships among the hadrons in the same flavor multiplet.
All quarks are assigned a baryon number of 1 / 3 . Up , charm and top quarks have an electric charge of + 2 / 3 , while the down , strange , and bottom quarks have an electric charge of − 1 / 3 . Antiquarks have the opposite quantum numbers. Quarks are spin- 1 / 2 particles, and thus fermions . Each quark or antiquark obeys the Gell-Mann–Nishijima formula individually, so any additive assembly of them will as well.
Mesons are made of a valence quark–antiquark pair (thus have a baryon number of 0), while baryons are made of three quarks (thus have a baryon number of 1). This article discusses the quark model for the up, down, and strange flavors of quark (which form an approximate flavor SU(3) symmetry ). There are generalizations to larger number of flavors.
Developing classification schemes for hadrons became a timely question after new experimental techniques uncovered so many of them that it became clear that they could not all be elementary. These discoveries led Wolfgang Pauli to exclaim "Had I foreseen that, I would have gone into botany." and Enrico Fermi to advise his student Leon Lederman : "Young man, if I could remember the names of these particles, I would have been a botanist." These new schemes earned Nobel prizes for experimental particle physicists, including Luis Alvarez , who was at the forefront of many of these developments. Constructing hadrons as bound states of fewer constituents would thus organize the "zoo" at hand. Several early proposals, such as the ones by Enrico Fermi and Chen-Ning Yang (1949), and the Sakata model (1956), ended up satisfactorily covering the mesons, but failed with baryons, and so were unable to explain all the data.
The Gell-Mann–Nishijima formula , developed by Murray Gell-Mann and Kazuhiko Nishijima , led to the Eightfold Way classification, invented by Gell-Mann, with important independent contributions from Yuval Ne'eman , in 1961. The hadrons were organized into SU(3) representation multiplets, octets and decuplets, of roughly the same mass, due to the strong interactions; and smaller mass differences linked to the flavor quantum numbers, invisible to the strong interactions. The Gell-Mann–Okubo mass formula systematized the quantification of these small mass differences among members of a hadronic multiplet, controlled by the explicit symmetry breaking of SU(3).
The spin- 3 / 2 Ω − baryon , a member of the ground-state decuplet, was a crucial prediction of that classification. After it was discovered in an experiment at Brookhaven National Laboratory , Gell-Mann received a Nobel Prize in Physics for his work on the Eightfold Way, in 1969.
Finally, in 1964, Gell-Mann and George Zweig , discerned independently what the Eightfold Way picture encodes: They posited three elementary fermionic constituents—the " up ", " down ", and " strange " quarks—which are unobserved, and possibly unobservable in a free form. Simple pairwise or triplet combinations of these three constituents and their antiparticles underlie and elegantly encode the Eightfold Way classification, in an economical, tight structure, resulting in further simplicity. Hadronic mass differences were now linked to the different masses of the constituent quarks.
It would take about a decade for the unexpected nature—and physical reality—of these quarks to be appreciated more fully (See Quarks ). Counter-intuitively, they cannot ever be observed in isolation ( color confinement ), but instead always combine with other quarks to form full hadrons, which then furnish ample indirect information on the trapped quarks themselves. Conversely, the quarks serve in the definition of quantum chromodynamics , the fundamental theory fully describing the strong interactions; and the Eightfold Way is now understood to be a consequence of the flavor symmetry structure of the lightest three of them.
The Eightfold Way classification is named after the following fact: If we take three flavors of quarks, then the quarks lie in the fundamental representation , 3 (called the triplet) of flavor SU(3) . The antiquarks lie in the complex conjugate representation 3 . The nine states (nonet) made out of a pair can be decomposed into the trivial representation , 1 (called the singlet), and the adjoint representation , 8 (called the octet). The notation for this decomposition is
Figure 1 shows the application of this decomposition to the mesons. If the flavor symmetry were exact (as in the limit that only the strong interactions operate, but the electroweak interactions are notionally switched off), then all nine mesons would have the same mass. However, the physical content of the full theory [ clarification needed ] includes consideration of the symmetry breaking induced by the quark mass differences, and considerations of mixing between various multiplets (such as the octet and the singlet).
N.B. Nevertheless, the mass splitting between the η and the η′ is larger than the quark model can accommodate, and this " η – η′ puzzle " has its origin in topological peculiarities of the strong interaction vacuum, such as instanton configurations.
Mesons are hadrons with zero baryon number . If the quark–antiquark pair are in an orbital angular momentum L state, and have spin S , then
If P = (−1) J , then it follows that S = 1, thus PC = 1. States with these quantum numbers are called natural parity states ; while all other quantum numbers are thus called exotic (for example, the state J PC = 0 −− ).
Since quarks are fermions , the spin–statistics theorem implies that the wavefunction of a baryon must be antisymmetric under the exchange of any two quarks. This antisymmetric wavefunction is obtained by making it fully antisymmetric in color, discussed below, and symmetric in flavor, spin and space put together. With three flavors, the decomposition in flavor is 3 ⊗ 3 ⊗ 3 = 10 S ⊕ 8 M ⊕ 8 M ⊕ 1 A . {\displaystyle \mathbf {3} \otimes \mathbf {3} \otimes \mathbf {3} =\mathbf {10} _{S}\oplus \mathbf {8} _{M}\oplus \mathbf {8} _{M}\oplus \mathbf {1} _{A}~.} The decuplet is symmetric in flavor, the singlet antisymmetric and the two octets have mixed symmetry. The space and spin parts of the states are thereby fixed once the orbital angular momentum is given.
It is sometimes useful to think of the basis states of quarks as the six states of three flavors and two spins per flavor. This approximate symmetry is called spin-flavor SU(6) . In terms of this, the decomposition is 6 ⊗ 6 ⊗ 6 = 56 S ⊕ 70 M ⊕ 70 M ⊕ 20 A . {\displaystyle \mathbf {6} \otimes \mathbf {6} \otimes \mathbf {6} =\mathbf {56} _{S}\oplus \mathbf {70} _{M}\oplus \mathbf {70} _{M}\oplus \mathbf {20} _{A}~.}
The 56 states with symmetric combination of spin and flavour decompose under flavor SU(3) into 56 = 10 3 2 ⊕ 8 1 2 , {\displaystyle \mathbf {56} =\mathbf {10} ^{\frac {3}{2}}\oplus \mathbf {8} ^{\frac {1}{2}}~,} where the superscript denotes the spin, S , of the baryon. Since these states are symmetric in spin and flavor, they should also be symmetric in space—a condition that is easily satisfied by making the orbital angular momentum L = 0 . These are the ground-state baryons.
The S = 1 / 2 octet baryons are the two nucleons ( p + , n 0 ), the three Sigmas ( Σ + , Σ 0 , Σ − ), the two Xis ( Ξ 0 , Ξ − ), and the Lambda ( Λ 0 ). The S = 3 / 2 decuplet baryons are the four Deltas ( Δ ++ , Δ + , Δ 0 , Δ − ), three Sigmas ( Σ ∗+ , Σ ∗0 , Σ ∗− ), two Xis ( Ξ ∗0 , Ξ ∗− ), and the Omega ( Ω − ).
For example, the constituent quark model wavefunction for the proton is | p ↑ ⟩ = 1 18 [ 2 | u ↑ d ↓ u ↑ ⟩ + 2 | u ↑ u ↑ d ↓ ⟩ + 2 | d ↓ u ↑ u ↑ ⟩ − | u ↑ u ↓ d ↑ ⟩ − | u ↑ d ↑ u ↓ ⟩ − | u ↓ d ↑ u ↑ ⟩ − | d ↑ u ↓ u ↑ ⟩ − | d ↑ u ↑ u ↓ ⟩ − | u ↓ u ↑ d ↑ ⟩ ] . {\displaystyle |{\text{p}}_{\uparrow }\rangle ={\frac {1}{\sqrt {18}}}[2|{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }\rangle +2|{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }\rangle +2|{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }\rangle -|{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }\rangle -|{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }\rangle -|{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }\rangle -|{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }\rangle -|{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }\rangle -|{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }\rangle ]~.}
Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other quantities that the model predicts successfully.
The group theory approach described above assumes that the quarks are eight components of a single particle, so the anti-symmetrization applies to all the quarks. A simpler approach is to consider the eight flavored quarks as eight separate, distinguishable, non-identical particles. Then the anti-symmetrization applies only to two identical quarks (like uu, for instance). [ 6 ]
Then, the proton wavefunction can be written in a simpler form:
and the
If quark–quark interactions are limited to two-body interactions, then all the successful quark model predictions, including sum rules for baryon masses and magnetic moments, can be derived.
Color quantum numbers are the characteristic charges of the strong force, and are completely uninvolved in electroweak interactions. They were discovered as a consequence of the quark model classification, when it was appreciated that the spin S = 3 / 2 baryon, the Δ ++ , required three up quarks with parallel spins and vanishing orbital angular momentum. Therefore, it could not have an antisymmetric wavefunction, (required by the Pauli exclusion principle ). Oscar Greenberg noted this problem in 1964, suggesting that quarks should be para-fermions . [ 7 ]
Instead, six months later, Moo-Young Han and Yoichiro Nambu suggested the existence of a hidden degree of freedom, they labeled as the group SU(3)' (but later called 'color). This led to three triplets of quarks whose wavefunction was anti-symmetric in the color degree of freedom.
Flavor and color were intertwined in that model: they did not commute. [ 8 ]
The modern concept of color completely commuting with all other charges and providing the strong force charge was articulated in 1973, by William Bardeen , Harald Fritzsch , and Murray Gell-Mann . [ 9 ] [ 10 ]
While the quark model is derivable from the theory of quantum chromodynamics , the structure of hadrons is more complicated than this model allows. The full quantum mechanical wavefunction of any hadron must include virtual quark pairs as well as virtual gluons , and allows for a variety of mixings. There may be hadrons which lie outside the quark model. Among these are the glueballs (which contain only valence gluons), hybrids (which contain valence quarks as well as gluons) and exotic hadrons (such as tetraquarks or pentaquarks ). | https://en.wikipedia.org/wiki/Quark_model |
A quark star is a hypothetical type of compact , exotic star , where extremely high core temperature and pressure have forced nuclear particles to form quark matter , a continuous state of matter consisting of free quarks . [ 1 ]
Some massive stars collapse to form neutron stars at the end of their life cycle , as has been both observed and explained theoretically. Under the extreme temperatures and pressures inside neutron stars, the neutrons are normally kept apart by a degeneracy pressure , stabilizing the star and hindering further gravitational collapse. [ 2 ] However, it is hypothesized that under even more extreme temperature and pressure, the degeneracy pressure of the neutrons is overcome, and the neutrons are forced to merge and dissolve into their constituent quarks, creating an ultra-dense phase of quark matter based on densely packed quarks. In this state, a new equilibrium is supposed to emerge, as a new degeneracy pressure between the quarks, as well as repulsive electromagnetic forces , will occur and hinder total gravitational collapse .
If these ideas are correct, quark stars might occur, and be observable, somewhere in the universe. Such a scenario is seen as scientifically plausible, but has not been proven observationally or experimentally; the very extreme conditions needed for stabilizing quark matter cannot be created in any laboratory and has not been observed directly in nature. The stability of quark matter, and hence the existence of quark stars, is for these reasons among the unsolved problems in physics .
If quark stars can form, then the most likely place to find quark star matter would be inside neutron stars that exceed the internal pressure needed for quark degeneracy – the point at which neutrons break down into a form of dense quark matter. They could also form if a massive star collapses at the end of its life, provided that it is possible for a star to be large enough to collapse beyond a neutron star but not large enough to form a black hole .
If they exist, quark stars would resemble and be easily mistaken for neutron stars: they would form in the death of a massive star in a Type II supernova , be extremely dense and small, and possess a very high gravitational field. They would also lack some features of neutron stars, unless they also contained a shell of neutron matter, because free quarks are not expected to have properties matching degenerate neutron matter. For example, they might be radio-silent, or have atypical sizes, electromagnetic fields, or surface temperatures, compared to neutron stars.
The analysis about quark stars was first proposed in 1965 by Soviet physicists D. D. Ivanenko and D. F. Kurdgelaidze . [ 3 ] [ 4 ] Their existence has not been confirmed.
The equation of state of quark matter is uncertain, as is the transition point between neutron-degenerate matter and quark matter. [ 5 ] Theoretical uncertainties have precluded making predictions from first principles . Experimentally, the behaviour of quark matter is being actively studied with particle colliders, but this can only produce very hot (above 10 12 K ) quark–gluon plasma blobs the size of atomic nuclei, which decay immediately after formation. The conditions inside compact stars with extremely high densities and temperatures well below 10 12 K cannot be recreated artificially, as there are no known methods to produce, store or study "cold" quark matter directly as it would be found inside quark stars. The theory predicts quark matter to possess some peculiar characteristics under these conditions. [ citation needed ]
It is hypothesized that when the neutron-degenerate matter , which makes up neutron stars , is put under sufficient pressure from the star's own gravity or the initial supernova creating it, the individual neutrons break down into their constituent quarks ( up quarks and down quarks ), forming what is known as quark matter. This conversion may be confined to the neutron star's center or it might transform the entire star, depending on the physical circumstances. Such a star is known as a quark star. [ 7 ] [ 8 ]
Ordinary quark matter consisting of up and down quarks has a very high Fermi energy compared to ordinary atomic matter and is stable only under extreme temperatures and/or pressures. This suggests that the only stable quark stars will be neutron stars with a quark matter core, while quark stars consisting entirely of ordinary quark matter will be highly unstable and re-arrange spontaneously. [ 9 ] [ 10 ]
It has been shown that the high Fermi energy making ordinary quark matter unstable at low temperatures and pressures can be lowered substantially by the transformation of a sufficient number of up and down quarks into strange quarks , as strange quarks are, relatively speaking, a very heavy type of quark particle. [ 9 ] This kind of quark matter is known specifically as strange quark matter and it is speculated and subject to current scientific investigation whether it might in fact be stable under the conditions of interstellar space (i.e. near zero external pressure and temperature). If this is the case (known as the Bodmer– Witten assumption), quark stars made entirely of quark matter would be stable if they quickly transform into strange quark matter. [ 11 ]
Stars made of strange quark matter are known as strange stars. These form a distinct subtype of quark stars. [ 11 ]
Theoretical investigations have revealed that quark stars might not only be produced from neutron stars and powerful supernovas, they could also be created in the early cosmic phase separations following the Big Bang . [ 9 ] If these primordial quark stars transform into strange quark matter before the external temperature and pressure conditions of the early Universe makes them unstable, they might turn out stable, if the Bodmer–Witten assumption holds true. Such primordial strange stars could survive to this day. [ 9 ]
Quark stars have some special characteristics that separate them from ordinary neutron stars. Under the physical conditions found inside neutron stars, with extremely high densities but temperatures well below 10 12 K, quark matter is predicted to exhibit some peculiar characteristics. It is expected to behave as a Fermi liquid and enter a so-called color-flavor-locked (CFL) phase of color superconductivity , where "color" refers to the six "charges" exhibited in the strong interaction , instead of the two charges (positive and negative) in electromagnetism . At slightly lower densities, corresponding to higher layers closer to the surface of the compact star, the quark matter will behave as a non-CFL quark liquid, a phase that is even more mysterious than CFL and might include color conductivity and/or several additional yet undiscovered phases. None of these extreme conditions can currently be recreated in laboratories so nothing can be inferred about these phases from direct experiments. [ 12 ]
At least under the assumptions mentioned above, the probability of a given neutron star being a quark star is low, [ citation needed ] so in the Milky Way there would only be a small population of quark stars. If it is correct, however, that overdense neutron stars can turn into quark stars, that makes the possible number of quark stars higher than was originally thought, as observers would be looking for the wrong type of star. [ citation needed ]
A neutron star without deconfinement to quarks and higher densities cannot have a rotational period shorter than a millisecond; even with the unimaginable gravity of such a condensed object the centripetal force of faster rotation would eject matter from the surface, so detection of a pulsar of millisecond or less period would be strong evidence of a quark star.
Observations released by the Chandra X-ray Observatory on April 10, 2002, detected two possible quark stars, designated RX J1856.5−3754 and 3C 58 , which had previously been thought to be neutron stars. Based on the known laws of physics, the former appeared much smaller and the latter much colder than it should be, suggesting that they are composed of material denser than neutron-degenerate matter . However, these observations are met with skepticism by researchers who say the results were not conclusive; [ 13 ] and since the late 2000s, the possibility that RX J1856 is a quark star has been excluded.
Another star, XTE J1739-285 , [ 14 ] has been observed by a team led by Philip Kaaret of the University of Iowa and reported as a possible quark star candidate.
In 2006, You-Ling Yue et al., from Peking University , suggested that PSR B0943+10 may in fact be a low-mass quark star. [ 15 ]
It was reported in 2008 that observations of supernovae SN 2006gy , SN 2005gj and SN 2005ap also suggest the existence of quark stars. [ 16 ] It has been suggested that the collapsed core of supernova SN 1987A may be a quark star. [ 17 ] [ 18 ]
In 2015, Zi-Gao Dai et al. from Nanjing University suggested that Supernova ASASSN-15lh is a newborn strange quark star. [ 19 ]
In 2022 it was suggested that GW190425, which likely formed as a merger between two neutron stars giving off gravitational waves in the process, could be a quark star. [ 20 ]
Apart from ordinary quark matter and strange quark matter, other types of quark-gluon plasma might hypothetically occur or be formed inside neutron stars and quark stars. This includes the following, some of which has been observed and studied in laboratories: | https://en.wikipedia.org/wiki/Quark_star |
Quark–gluon plasma ( QGP or quark soup ) is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word plasma signals that free color charges are allowed. In a 1987 summary, Léon Van Hove pointed out the equivalence of the three terms: quark gluon plasma, quark matter and a new state of matter. [ 2 ] Since the temperature is above the Hagedorn temperature —and thus above the scale of light u,d-quark mass—the pressure exhibits the relativistic Stefan–Boltzmann format governed by temperature to the fourth power ( T 4 {\displaystyle T^{4}} ) and many practically massless quark and gluon constituents. It can be said that QGP emerges to be the new phase of strongly interacting matter which manifests its physical properties in terms of nearly free dynamics of practically massless gluons and quarks. Both quarks and gluons must be present in conditions near chemical (yield) equilibrium with their color charge open for a new state of matter to be referred to as QGP.
In the Big Bang theory, quark–gluon plasma filled the entire Universe before matter as we know it was created. Theories predicting the existence of quark–gluon plasma were developed in the late 1970s and early 1980s. [ 3 ] Discussions around heavy ion experimentation followed suit, [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] and the first experiment proposals were put forward at CERN [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ] [ 14 ] and BNL [ 15 ] [ 16 ] in the following years. Quark–gluon plasma [ 17 ] [ 18 ] was detected for the first time in the laboratory at CERN in the year 2000. [ 19 ] [ 20 ] [ 21 ]
Quark–gluon plasma is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities . [ 22 ] These particles are the quarks and gluons that compose baryonic matter. [ 23 ] In normal matter quarks are confined ; in the QGP quarks are deconfined . In classical quantum chromodynamics (QCD), quarks are the fermionic components of hadrons ( mesons and baryons) while the gluons are considered the bosonic components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts.
Quark–gluon plasma is studied to recreate and understand the high energy density conditions prevailing in the Universe when matter formed from elementary degrees of freedom (quarks, gluons) at about 20 μs after the Big Bang . Experimental groups are probing over a 'large' distance the (de)confining quantum vacuum structure, which determines prevailing form of matter and laws of nature. The experiments give insight to the origin of matter and mass: the matter and antimatter is created when the quark–gluon plasma 'hadronizes' and the mass of matter originates in the confining vacuum structure. [ 19 ]
QCD is one part of the modern theory of particle physics called the Standard Model . Other parts of this theory deal with electroweak interactions and neutrinos . The theory of electrodynamics has been tested and found correct to a few parts in a billion. The theory of weak interactions has been tested and found correct to a few parts in a thousand. Perturbative forms of QCD have been tested to a few percent. [ 24 ] Perturbative models assume relatively small changes from the ground state, i.e. relatively low temperatures and densities, which simplifies calculations at the cost of generality. In contrast, non-perturbative forms of QCD have barely been tested. The study of the QGP, which has both a high temperature and density, is part of this effort to consolidate the grand theory of particle physics.
The study of the QGP is also a testing ground for finite temperature field theory , a branch of theoretical physics which seeks to understand particle physics under conditions of high temperature. Such studies are important to understand the early evolution of our universe: the first hundred microseconds or so. It is crucial to the physics goals of a new generation of observations of the universe ( WMAP and its successors). It is also of relevance to Grand Unification Theories which seek to unify the three fundamental forces of nature (excluding gravity).
The generally accepted model of the formation of the Universe states that it happened as the result of the Big Bang . In this model, in the time interval of 10 −10 –10 −6 s after the Big Bang, matter existed in the form of a quark–gluon plasma. It is possible to reproduce the density and temperature of matter existing of that time in laboratory conditions to study the characteristics of the very early Universe. So far, the only possibility is the collision of two heavy atomic nuclei accelerated to energies of more than a hundred GeV. Using the result of a head-on collision in the volume approximately equal to the volume of the atomic nucleus, it is possible to model the density and temperature that existed in the first instants of the life of the Universe.
A plasma is matter in which charges are screened due to the presence of other mobile charges. For example: Coulomb's Law is suppressed by the screening to yield a distance-dependent charge, Q → Q e − r / α {\displaystyle Q\rightarrow Qe^{-r/\alpha }} , i.e., the charge Q is reduced exponentially with the distance divided by a screening length α. In a QGP, the color charge of the quarks and gluons is screened. The QGP has other analogies with a normal plasma. There are also dissimilarities because the color charge is non-abelian , whereas the electric charge is abelian. Outside a finite volume of QGP the color-electric field is not screened, so that a volume of QGP must still be color-neutral. It will therefore, like a nucleus, have integer electric charge.
Because of the extremely high energies involved, quark-antiquark pairs are produced by pair production and thus QGP is a roughly equal mixture of quarks and antiquarks of various flavors, with only a slight excess of quarks. This property is not a general feature of conventional plasmas, which may be too cool for pair production (see however pair instability supernova ).
One consequence of this difference is that the color charge is too large for perturbative computations which are the mainstay of QED. As a result, the main theoretical tools to explore the theory of the QGP is lattice gauge theory . [ 25 ] [ 26 ] The transition temperature (approximately 175 MeV ) was first predicted by lattice gauge theory. Since then lattice gauge theory has been used to predict many other properties of this kind of matter. The AdS/CFT correspondence conjecture may provide insights in QGP, moreover the ultimate goal of the fluid/gravity correspondence is to understand QGP. The QGP is believed to be a phase of QCD which is completely locally thermalized and thus suitable for an effective fluid dynamic description.
Production of QGP in the laboratory is achieved by colliding heavy atomic nuclei (called heavy ions as in an accelerator atoms are ionized) at relativistic energy in which matter is heated well above the Hagedorn temperature T H = 150 MeV per particle, which amounts to a temperature exceeding 1.66×10 12 K . This can be accomplished by colliding two large nuclei at high energy (note that 175 MeV is not the energy of the colliding beam). Lead and gold nuclei have been used for such collisions at CERN SPS and BNL RHIC , respectively. The nuclei are accelerated to ultrarelativistic speeds ( contracting their length ) and directed towards each other, creating a "fireball", in the rare event of a collision. Hydrodynamic simulation predicts this fireball will expand under its own pressure , and cool while expanding. By carefully studying the spherical and elliptic flow , experimentalists put the theory to test.
There is overwhelming evidence for production of quark–gluon plasma in relativistic heavy ion collisions. [ 27 ] [ 28 ] [ 29 ] [ 30 ] [ 31 ]
The important classes of experimental observations are
The cross-over temperature from the normal hadronic to the QGP phase is about 156 MeV . [ 32 ] This "crossover" may actually not be only a qualitative feature, but instead one may have to do with a true (second order) phase transition , e.g. of the universality class of the three-dimensional Ising model . The phenomena involved correspond to an energy density of a little less than 1 GeV /fm 3 . For relativistic matter, pressure and temperature are not independent variables, so the equation of state is a relation between the energy density and the pressure. This has been found through lattice computations, and compared to both perturbation theory and string theory . This is still a matter of active research. Response functions such as the specific heat and various quark number susceptibilities are currently being computed.
The discovery of the perfect liquid was a turning point in physics. Experiments at RHIC have revealed a wealth of information about this remarkable substance, which we now know to be a QGP. [ 33 ] Nuclear matter at "room temperature" is known to behave like a superfluid . When heated the nuclear fluid evaporates and turns into a dilute gas of nucleons and, upon further heating, a gas of baryons and mesons (hadrons). At the critical temperature, T H , the hadrons melt and the gas turns back into a liquid. RHIC experiments have shown that this is the most perfect liquid ever observed in any laboratory experiment at any scale. The new phase of matter, consisting of dissolved hadrons, exhibits less resistance to flow than any other known substance. The experiments at RHIC have, already in 2005, shown that the Universe at its beginning was uniformly filled with this type of material—a super-liquid—which once the Universe cooled below T H evaporated into a gas of hadrons. Detailed measurements show that this liquid is a quark–gluon plasma where quarks, antiquarks and gluons flow independently. [ 34 ]
In short, a quark–gluon plasma flows like a splat of liquid, and because it is not "transparent" with respect to quarks, it can attenuate jets emitted by collisions. Furthermore, once formed, a ball of quark–gluon plasma, like any hot object, transfers heat internally by radiation. However, unlike in everyday objects, there is enough energy available so that gluons (particles mediating the strong force ) collide and produce an excess of the heavy (i.e., high-energy ) strange quarks . Whereas, if the QGP did not exist and there was a pure collision, the same energy would be converted into a non-equilibrium mixture containing even heavier quarks such as charm quarks or bottom quarks . [ 35 ] [ 36 ]
The equation of state is an important input into the flow equations. The speed of sound (speed of QGP-density oscillations) is currently under investigation in lattice computations. [ 37 ] [ 38 ] [ 39 ] The mean free path of quarks and gluons has been computed using perturbation theory as well as string theory . Lattice computations have been slower here, although the first computations of transport coefficients have been concluded. [ 40 ] [ 41 ] These indicate that the mean free time of quarks and gluons in the QGP may be comparable to the average interparticle spacing: hence the QGP is a liquid as far as its flow properties go. This is very much an active field of research, and these conclusions may evolve rapidly. The incorporation of dissipative phenomena into hydrodynamics is another active research area. [ 42 ] [ 43 ] [ 44 ]
Detailed predictions were made in the late 1970s for the production of jets at the CERN Super Proton–Antiproton Synchrotron . [ 45 ] [ 46 ] [ 47 ] [ 48 ] UA2 observed the first evidence for jet production in hadron collisions in 1981, [ 49 ] which shortly after was confirmed by UA1 . [ 50 ]
The subject was later revived at RHIC. One of the most striking physical effects obtained at RHIC energies is the effect of quenching jets. [ 51 ] [ 52 ] [ 53 ] At the first stage of interaction of colliding relativistic nuclei, partons of the colliding nuclei give rise to the secondary partons with a large transverse impulse ≥ 3–6 GeV/s. Passing through a highly heated compressed plasma, partons lose energy. The magnitude of the energy loss by the parton depends on the properties of the quark–gluon plasma (temperature, density). In addition, it is also necessary to take into account the fact that colored quarks and gluons are the elementary objects of the plasma, which differs from the energy loss by a parton in a medium consisting of colorless hadrons. Under the conditions of a quark–gluon plasma, the energy losses resulting from the RHIC energies by partons are estimated as d E d x = 1 GeV/fm {\displaystyle {\frac {dE}{dx}}=1~{\text{GeV/fm}}} . This conclusion is confirmed by comparing the relative yield of hadrons with a large transverse impulse in nucleon-nucleon and nucleus-nucleus collisions at the same collision energy. The energy loss by partons with a large transverse impulse in nucleon-nucleon collisions is much smaller than in nucleus-nucleus collisions, which leads to a decrease in the yield of high-energy hadrons in nucleus-nucleus collisions. This result suggests that nuclear collisions cannot be regarded as a simple superposition of nucleon-nucleon collisions. For a short time, ~1 μs, and in the final volume, quarks and gluons form some ideal liquid. The collective properties of this fluid are manifested during its movement as a whole. Therefore, when moving partons in this medium, it is necessary to take into account some collective properties of this quark–gluon liquid. Energy losses depend on the properties of the quark–gluon medium, on the parton density in the resulting fireball, and on the dynamics of its expansion. Losses of energy by light and heavy quarks during the passage of a fireball turn out to be approximately the same. [ 54 ]
In November 2010, CERN announced the first direct observation of jet quenching, based on experiments with heavy-ion collisions. [ 55 ] [ 56 ] [ 57 ] [ 58 ]
Direct photons and dileptons are arguably most penetrating tools to study relativistic heavy ion collisions. They are produced, by various mechanisms spanning the space-time evolution of the strongly interacting fireball. They provide in principle a snapshot on the initial stage as well. They are hard to decipher and interpret as most of the signal is originating from hadron decays long after the QGP fireball has disintegrated. [ 59 ] [ 60 ] [ 61 ]
Since 2008, there is a discussion about a hypothetical precursor state of the quark–gluon plasma, the so-called "Glasma", where the dressed particles are condensed into some kind of glassy (or amorphous) state, below the genuine transition between the confined state and the plasma liquid. [ 62 ] This would be analogous to the formation of metallic glasses, or amorphous alloys of them, below the genuine onset of the liquid metallic state.
Although the experimental high temperatures and densities predicted as producing a quark–gluon plasma have been realized in the laboratory, the resulting matter does not behave as a quasi-ideal state of free quarks and gluons, but, rather, as an almost perfect dense fluid. [ 63 ] Actually, the fact that the quark–gluon plasma will not yet be "free" at temperatures realized at present accelerators was predicted in 1984, as a consequence of the remnant effects of confinement. [ 64 ] [ 65 ]
It has been hypothesized that the core of some massive neutron stars may be a quark–gluon plasma. [ 66 ]
A quark–gluon plasma (QGP) [ 67 ] or quark soup [ 68 ] [ 69 ] is a state of matter in quantum chromodynamics (QCD) which exists at extremely high temperature and/or density . This state is thought to consist of asymptotically free strong-interacting quarks and gluons, which are ordinarily confined by color confinement inside atomic nuclei or other hadrons. This is in analogy with the conventional plasma where nuclei and electrons, confined inside atoms by electrostatic forces at ambient conditions, can move freely. Experiments to create artificial quark matter started at CERN in 1986/87, resulting in first claims that were published in 1991. [ 70 ] [ 71 ] It took several years before the idea became accepted in the community of particle and nuclear physicists. Formation of a new state of matter in Pb–Pb collisions was officially announced at CERN in view of the convincing experimental results presented by the CERN SPS WA97 experiment in 1999, [ 72 ] [ 31 ] [ 73 ] and later elaborated by Brookhaven National Laboratory's Relativistic Heavy Ion Collider . [ 74 ] [ 75 ] [ 30 ] Quark matter can only be produced in minute quantities and is unstable and impossible to contain, and will radioactively decay within a fraction of a second into stable particles through hadronization ; the produced hadrons or their decay products and gamma rays can then be detected. In the quark matter phase diagram, QGP is placed in the high-temperature, high-density regime, whereas ordinary matter is a cold and rarefied mixture of nuclei and vacuum, and the hypothetical quark stars would consist of relatively cold, but dense quark matter. It is believed that up to a few microseconds (10 −12 to 10 −6 seconds) after the Big Bang, known as the quark epoch , the Universe was in a quark–gluon plasma state.
The strength of the color force means that unlike the gas-like plasma, quark–gluon plasma behaves as a near-ideal Fermi liquid , although research on flow characteristics is ongoing. [ 76 ] Liquid or even near-perfect liquid flow with almost no frictional resistance or viscosity was claimed by research teams at RHIC [ 77 ] and LHC's Compact Muon Solenoid detector. [ 78 ] QGP differs from a "free" collision event by several features; for example, its particle content is indicative of a temporary chemical equilibrium producing an excess of middle-energy strange quarks vs. a nonequilibrium distribution mixing light and heavy quarks ("strangeness production"), and it does not allow particle jets to pass through ("jet quenching").
Experiments at CERN's Super Proton Synchrotron (SPS) begun experiments to create QGP in the 1980s and 1990s: the results led CERN to announce evidence for a "new state of matter" [ 79 ] in 2000. [ 80 ] Scientists at Brookhaven National Laboratory's Relativistic Heavy Ion Collider announced they had created quark–gluon plasma by colliding gold ions at nearly the speed of light, reaching temperatures of 4 trillion degrees Celsius. [ 81 ] Current experiments (2017) at the Brookhaven National Laboratory 's Relativistic Heavy Ion Collider (RHIC) on Long Island (New York, USA) and at CERN's recent Large Hadron Collider near Geneva (Switzerland) are continuing this effort, [ 82 ] [ 83 ] by colliding relativistically accelerated gold and other ion species (at RHIC) or lead (at LHC) with each other or with protons. [ 83 ] Three experiments running on CERN's Large Hadron Collider (LHC), on the spectrometers ALICE , [ 84 ] ATLAS and CMS , have continued studying the properties of QGP. CERN temporarily ceased colliding protons , and began colliding lead ions for the ALICE experiment in 2011, in order to create a QGP. [ 85 ] A new record breaking temperature was set by ALICE: A Large Ion Collider Experiment at CERN in August 2012 in the ranges of 5.5 trillion ( 5.5 × 10 12 ) kelvin as claimed in their Nature PR. [ 86 ]
The formation of a quark–gluon plasma occurs as a result of a strong interaction between the partons (quarks, gluons) that make up the nucleons of the colliding heavy nuclei called heavy ions. Therefore, experiments are referred to as relativistic heavy ion collision experiments. Theoretical and experimental works show that the formation of a quark–gluon plasma occurs at the temperature of T ≈ 150–160 MeV, the Hagedorn temperature, and an energy density of ≈ 0.4–1 GeV / fm 3 . While at first a phase transition was expected, present day theoretical interpretations propose a phase transformation similar to the process of ionisation of normal matter into ionic and electron plasma. [ 87 ] [ 88 ] [ 89 ] [ 90 ] [ 30 ]
The central issue of the formation of a quark–gluon plasma is the research for the onset of deconfinement . From the beginning of the research on formation of QGP, the issue was whether energy density can be achieved in nucleus-nucleus collisions. This depends on how much energy each nucleon loses. An influential reaction picture was the scaling solution presented by Bjorken . [ 91 ] This model applies to ultra-high energy collisions. In experiments carried out at CERN SPS and BNL RHIC more complex situation arose, usually divided into three stages: [ 92 ]
More and more experimental evidence points to the strength of QGP formation mechanisms—operating even in LHC-energy scale proton-proton collisions. [ 28 ] | https://en.wikipedia.org/wiki/Quark–gluon_plasma |
A tub or quarry tub is a type of railway or tramway wagon used in quarries and other industrial locations for the transport of minerals (such as coal , sand , ore , clay and stone ) from a quarry or mine face to processing plants or between various parts of an industrial site. [ 1 ] This type of wagon may be small enough for one person to push, or designed for haulage by a horse, or for connection in a train hauled by a locomotive . The tubs are designed for ease of emptying, usually by a side-tipping action. This type of rail vehicle is now mainly obsolete, its function having been mostly replaced by conveyor belts .
This article about mining is a stub . You can help Wikipedia by expanding it .
This rail-transport related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quarry_tub |
In computing , a nibble , [ 1 ] or spelled nybble to match byte , is a unit of information that is an aggregation of four- bits ; half of a byte/ octet . [ 1 ] [ 2 ] [ 3 ] The unit is alternatively called nyble , nybl , half-byte [ 4 ] or tetrade . [ 5 ] [ 6 ] In networking or telecommunications , the unit is often called a semi-octet , [ 7 ] quadbit , [ 8 ] or quartet . [ 9 ] [ 10 ]
As a nibble can represent sixteen ( 2 4 ) possible values, a nibble value is often shown as a hexadecimal digit (hex digit). [ 11 ] A byte is two nibbles, and therefore, a value can be shown as two hex digits.
Four-bit computers use nibble-sized data for storage and operations; as the word unit. Such computers were used in early microprocessors , pocket calculators and pocket computers . They continue to be used in some microcontrollers . In this context, 4-bit groups were sometimes also called characters [ 12 ] rather than nibbles. [ 1 ]
The term nibble originates from its representing "half a byte", with byte a homophone of the English word bite . [ 4 ] In 2014, David B. Benson, a professor emeritus at Washington State University , remembered that he playfully used (and may have possibly coined) the term nibble as "half a byte" and unit of storage required to hold a binary-coded decimal (BCD) digit around 1958, when talking to a programmer from Los Alamos Scientific Laboratory . The alternative spelling nybble reflects the spelling of byte , as noted in editorials of Kilobaud and Byte in the early 1980s. Another early recorded use of the term nybble was in 1977 within the consumer-banking technology group at Citibank. It created a pre- ISO 8583 standard for transactional messages between cash machines and Citibank's data centers that used the basic data unit 'nabble'.
Nibble is used to describe the amount of memory used to store a digit of a number stored in packed decimal format (BCD) within an IBM mainframe. This technique is used to make computations faster and debugging easier. An 8-bit byte is split in half and each nibble is used to store one decimal digit. The last (rightmost) nibble of the variable is reserved for the sign. Thus a variable which can store up to nine digits would be "packed" into 5 bytes. Ease of debugging resulted from the numbers' being readable in a hex dump where two hex numbers are used to represent the value of a byte, as 16×16 = 2 8 . For example, a five-byte BCD value of 31 41 59 26 5C represents a decimal value of +314159265.
Historically, there are cases where nybble was used for a group of bits greater than 4. On the Apple II , much of the disk drive control and group-coded recording was implemented in software. Writing data to a disk was done by converting 256-byte pages into sets of 5-bit (later, 6-bit ) nibbles and loading disk data required the reverse. [ 13 ] [ 14 ] [ 15 ] Moreover, 1982 documentation for the Integrated Woz Machine refers consistently to an "8 bit nibble". [ 16 ] The term byte once had the same ambiguity and meant a set of bits but not necessarily 8, hence the distinction of bytes and octets or of nibbles and quartets (or quadbits ). Today, the terms byte and nibble almost always refer to 8-bit and 4-bit collections respectively and are very rarely used to express any other sizes.
A nibble-sized value can be represented in different numeric bases:
The low and high nibbles of a byte are its two halves that are the less and the more significant bits within the byte, respectively. In a graphical representation of bits within a byte, the leftmost bit could represent the most significant bit ( MSB ), corresponding to ordinary decimal notation in which the digit at the left of a number is the most significant. In such an illustration, the four bits on the left end of the byte form the high nibble, and the remaining four bits form the low nibble. [ 17 ] For example,
the high nibble is 0110 2 ( 6 16 ), and the low nibble is 0001 2 ( 1 16 ). The total value is high-nibble × 16 10 + low-nibble ( 6 × 16 + 1 = 97 10 ). | https://en.wikipedia.org/wiki/Quartet_(computing) |
In mathematics , a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is
where a ≠ 0.
The quartic is the highest order polynomial equation that can be solved by radicals in the general case.
Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately. [ 1 ] The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book Ars Magna (1545).
The proof that this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois before his death in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result. [ 2 ]
Consider a quartic equation expressed in the form a 0 x 4 + a 1 x 3 + a 2 x 2 + a 3 x + a 4 = 0 {\displaystyle a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{3}x+a_{4}=0} :
There exists a general formula for finding the roots to quartic equations, provided the coefficient of the leading term is non-zero. However, since the general method is quite complex and susceptible to errors in execution, it is better to apply one of the special cases listed below if possible.
If the constant term a 4 = 0, then one of the roots is x = 0, and the other roots can be found by dividing by x , and solving the resulting cubic equation ,
Call our quartic polynomial Q ( x ) . Since 1 raised to any power is 1,
Thus if a 0 + a 1 + a 2 + a 3 + a 4 = 0 , {\displaystyle \ a_{0}+a_{1}+a_{2}+a_{3}+a_{4}=0\ ,} Q (1) = 0 and so x = 1 is a root of Q ( x ) . It can similarly be shown that if a 0 + a 2 + a 4 = a 1 + a 3 , {\displaystyle \ a_{0}+a_{2}+a_{4}=a_{1}+a_{3}\ ,} x = −1 is a root.
In either case the full quartic can then be divided by the factor ( x − 1) or ( x + 1) respectively yielding a new cubic polynomial , which can be solved to find the quartic's other roots.
If a 1 = a 0 k , {\displaystyle \ a_{1}=a_{0}k\ ,} a 2 = 0 {\displaystyle \ a_{2}=0\ } and a 4 = a 3 k , {\displaystyle \ a_{4}=a_{3}k\ ,} then x = − k {\displaystyle \ x=-k\ } is a root of the equation. The full quartic can then be factorized this way:
Alternatively, if a 1 = a 0 k , {\displaystyle \ a_{1}=a_{0}k\ ,} a 3 = a 2 k , {\displaystyle \ a_{3}=a_{2}k\ ,} and a 4 = 0 , {\displaystyle \ a_{4}=0\ ,} then x = 0 and x = − k become two known roots. Q ( x ) divided by x ( x + k ) is a quadratic polynomial.
A quartic equation where a 3 and a 1 are equal to 0 takes the form
and thus is a biquadratic equation , which is easy to solve: let z = x 2 {\displaystyle z=x^{2}} , so our equation becomes
which is a simple quadratic equation, whose solutions are easily found using the quadratic formula:
When we've solved it (i.e. found these two z values), we can extract x from them
If either of the z solutions were negative or complex numbers, then some of the x solutions are complex numbers.
Steps:
This leads to:
If the quartic has a double root , it can be found by taking the polynomial greatest common divisor with its derivative. Then they can be divided out and the resulting quadratic equation solved.
In general, there exist only four possible cases of quartic equations with multiple roots, which are listed below: [ 3 ]
So, if the three non-monic coefficients of the depressed quartic equation, x 4 + p x 2 + q x + r = 0 {\displaystyle x^{4}+px^{2}+qx+r=0} , in terms of the five coefficients of the general quartic equation are given as follows: p = 8 a c − 3 b 2 8 a 2 {\displaystyle p={\frac {8ac-3b^{2}}{8a^{2}}}} , q = b 3 − 4 a b c + 8 a 2 d 8 a 3 {\displaystyle q={\frac {b^{3}-4abc+8a^{2}d}{8a^{3}}}} and r = 16 a b 2 c − 64 a 2 b d − 3 b 4 + 256 a 3 e 256 a 4 {\displaystyle r={\frac {16ab^{2}c-64a^{2}bd-3b^{4}+256a^{3}e}{256a^{4}}}} , then the criteria to identify a priori each case of quartic equations with multiple roots and their respective solutions are shown below.
To begin, the quartic must first be converted to a depressed quartic .
Let
be the general quartic equation which it is desired to solve. Divide both sides by A ,
The first step, if B is not already zero, should be to eliminate the x 3 term. To do this, change variables from x to u , such that
Then
Expanding the powers of the binomials produces
Collecting the same powers of u yields
Now rename the coefficients of u . Let
The resulting equation is
which is a depressed quartic equation .
If b = 0 {\displaystyle \ b=0\ } then we have the special case of a biquadratic equation , which is easily solved, as explained above. Note that the general solution, given below, will not work for the special case b = 0 . {\displaystyle \ b=0\ .} The equation must be solved as a biquadratic.
In either case, once the depressed quartic is solved for u , substituting those values into
produces the values for x that solve the original quartic.
After converting to a depressed quartic equation
and excluding the special case b = 0, which is solved as a biquadratic, we assume from here on that b ≠ 0 .
We will separate the terms left and right as
and add in terms to both sides which make them both into perfect squares .
Let y be any solution of this cubic equation :
Then (since b ≠ 0)
so we may divide by it, giving
Then
Subtracting, we get the difference of two squares which is the product of the sum and difference of their roots
which can be solved by applying the quadratic formula to each of the two factors. So the possible values of u are:
Using another y from among the three roots of the cubic simply causes these same four values of u to appear in a different order. The solutions of the cubic are:
using any one of the three possible cube roots. A wise strategy is to choose the sign of the square-root that makes the absolute value of w as large as possible.
Otherwise, the depressed quartic can be solved by means of a method discovered by Lodovico Ferrari . Once the depressed quartic has been obtained, the next step is to add the valid identity
to equation ( 1 ), yielding
The effect has been to fold up the u 4 term into a perfect square : ( u 2 + a) 2 . The second term, au 2 did not disappear, but its sign has changed and it has been moved to the right side.
The next step is to insert a variable y into the perfect square on the left side of equation ( 2 ), and a corresponding 2 y into the coefficient of u 2 in the right side. To accomplish these insertions, the following valid formulas will be added to equation ( 2 ),
and
These two formulas, added together, produce
which added to equation ( 2 ) produces
This is equivalent to
The objective now is to choose a value for y such that the right side of equation ( 3 ) becomes a perfect square. This can be done by letting the discriminant of the quadratic function become zero. To explain this, first expand a perfect square so that it equals a quadratic function:
The quadratic function on the right side has three coefficients. It can be verified that squaring the second coefficient and then subtracting four times the product of the first and third coefficients yields zero:
Therefore to make the right side of equation ( 3 ) into a perfect square, the following equation must be solved:
Multiply the binomial with the polynomial,
Divide both sides by −4, and move the − b 2 /4 to the right,
Divide both sides by 2,
This is a cubic equation in y . Solve for y using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula ). Any of the three possible roots will do.
With the value for y so selected, it is now known that the right side of equation ( 3 ) is a perfect square of the form
so that it can be folded:
Therefore equation ( 3 ) becomes
Equation ( 5 ) has a pair of folded perfect squares, one on each side of the equation. The two perfect squares balance each other.
If two squares are equal, then the sides of the two squares are also equal, as shown by:
Collecting like powers of u produces
Equation ( 6 ) is a quadratic equation for u . Its solution is
Simplifying, one gets
This is the solution of the depressed quartic, therefore the solutions of the original quartic equation are
Given the quartic equation
its solution can be found by means of the following calculations:
If b = 0 , {\displaystyle \,b=0,} then
Otherwise, continue with
(either sign of the square root will do)
(there are 3 complex roots, any one of them will do)
Ferrari was the first to discover one of these labyrinthine solutions [ citation needed ] . The equation which he solved was
which was already in depressed form. It has a pair of solutions which can be found with the set of formulas shown above.
If the coefficients of the quartic equation are real then the nested depressed cubic equation ( 5 ) also has real coefficients, thus it has at least one real root.
Furthermore the cubic function
where P and Q are given by ( 5 ) has the properties that
lim v → ∞ C ( v ) = ∞ , {\displaystyle \lim _{v\to \infty }C(v)=\infty ,} where a and b are given by ( 1 ).
This means that ( 5 ) has a real root greater than a 3 {\displaystyle a \over 3} ,
and therefore that ( 4 ) has a real root greater than − a 2 {\displaystyle -a \over 2} .
Using this root the term a + 2 y {\displaystyle {\sqrt {a+2y}}} in ( 6 ) is always real,
which ensures that the two quadratic equations ( 6 ) have real coefficients. [ 5 ]
It could happen that one only obtained one solution through the formulae above, because not all four sign patterns are tried for four solutions, and the solution obtained is complex . It may also be the case that one is only looking for a real solution. Let x 1 denote the complex solution. If all the original coefficients A , B , C , D and E are real—which should be the case when one desires only real solutions – then there is another complex solution x 2 which is the complex conjugate of x 1 . If the other two roots are denoted as x 3 and x 4 then the quartic equation can be expressed as
but this quartic equation is equivalent to the product of two quadratic equations:
and
Since
then
Let
so that equation ( 9 ) becomes
Also let there be (unknown) variables w and v such that equation ( 10 ) becomes
Multiplying equations ( 11 ) and ( 12 ) produces
Comparing equation ( 13 ) to the original quartic equation, it can be seen that
and
Therefore
Equation ( 12 ) can be solved for x yielding
One of these two solutions should be the desired real solution.
Most textbook solutions of the quartic equation require a substitution that is hard to memorize. Here is an approach that makes it easy to understand. The job is done if we can factor the quartic equation into a product of two quadratics . Let
By equating coefficients, this results in the following set of simultaneous equations:
This is harder to solve than it looks, but if we start again with a depressed quartic where b = 0 {\displaystyle b=0} , which can be obtained by substituting ( x − b / 4 ) {\displaystyle (x-b/4)} for x {\displaystyle x} , then r = − p {\displaystyle r=-p} , and:
It's now easy to eliminate both s {\displaystyle s} and q {\displaystyle q} by doing the following:
If we set P = p 2 {\displaystyle P=p^{2}} , then this equation turns into the cubic equation :
which is solved elsewhere. Once you have p {\displaystyle p} , then:
The symmetries in this solution are easy to see. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of p {\displaystyle p} for the square root of P {\displaystyle P} merely exchanges the two quadratics with one another.
A suitably chosen Möbius transformation can transform a quartic equation into a quadratic equation in the new variable squared. This is a known method. [ 6 ]
Finding such a Möbius transformation involves solving a cubic equation and so simplifies the problem. For example, start with the depressed quartic equation with unity leading coefficient and with neither a 1 {\displaystyle a_{1}} nor a 0 {\displaystyle a_{0}} equal to zero:
x 4 + a 2 x 2 + a 1 x + a 0 = 0 {\displaystyle x^{4}+a_{2}x^{2}+a_{1}x+a_{0}=0}
and do the Möbius transformation:
x = A + B y 1 + y {\displaystyle x={\frac {A+By}{1+y}}}
Set the first and third order coefficients of the resulting quartic equation in y {\displaystyle y} to zero. After some algebra, one finds A + B {\displaystyle A+B} is to be obtained from the cubic equation
a 1 ( A + B ) 3 + ( 4 a 0 − 2 a 1 a 2 − a 2 2 ) ( A + B ) 2 − 2 a 1 a 2 ( A + B ) − a 1 2 = 0 {\displaystyle a_{1}(A+B)^{3}+(4a_{0}-2a_{1}a_{2}-{a_{2}}^{2})(A+B)^{2}-2a_{1}a_{2}(A+B)-{a_{1}}^{2}=0}
and, regarding A + B {\displaystyle A+B} as known, A {\displaystyle A} is to be obtained from the quadratic equation
2 ( A + B ) A 2 − 2 ( A + B ) 2 A − a 2 ( A + B ) − a 1 = 0 {\displaystyle 2(A+B)A^{2}-2(A+B)^{2}A-a_{2}(A+B)-a_{1}=0} Solving the resulting quadratic equation for y 2 {\displaystyle y^{2}} gives two values for y 2 {\displaystyle y^{2}} and each square root of y 2 {\displaystyle y^{2}} has two values, giving a total of four solutions, as expected.
The cubic equation in A + B {\displaystyle {\textbf {A}}+{\textbf {B}}} given earlier is the same as P 2 − Q ( A + B ) 2 = 0 {\displaystyle P^{2}-Q(A+B)^{2}=0} , where P ≡ b 1 − b 3 2 ( A − B ) = 2 A B ( A + B ) + a 2 ( A + B ) + a 1 {\displaystyle P\equiv {\frac {b_{1}-b_{3}}{2(A-B)}}=2\,A\,B\,(A+B)+a_{2}(A+B)+a_{1}} Q ≡ B b 1 − A b 3 A − B = 4 A 2 B 2 − a 1 ( A + B ) − 4 a 0 = 0 {\displaystyle Q\equiv {\frac {B\,b_{1}-A\,b_{3}}{A-B}}=4A^{2}B^{2}-a_{1}(A+B)-4a_{0}=0} Here b i are the coefficients of the quartic polynomial in y. This shows how this equation was obtained.
The symmetric group S 4 on four elements has the Klein four-group as a normal subgroup . This suggests using a resolvent whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform of the roots.
Suppose r i for i from 0 to 3 are roots of
If we now set
then since the transformation is an involution , we may express the roots in terms of the four s i in exactly the same way. Since we know the value s 0 = − b /2, we really only need the values for s 1 , s 2 and s 3 . These we may find by expanding the polynomial
which if we make the simplifying assumption that b = 0, is equal to
This polynomial is of degree six, but only of degree three in z 2 , and so the corresponding equation is solvable. By trial we can determine which three roots are the correct ones, and hence find the solutions of the quartic.
We can remove any requirement for trial by using a root of the same resolvent polynomial for factoring; if w is any root of (3), and if
then
We therefore can solve the quartic by solving for w and then solving for the roots of the two factors using the quadratic formula.
The methods described above are, in principle, exact root-finding methods. It is also possible to use successive approximation methods which iteratively converge towards the roots, such as the Durand–Kerner method . Iterative methods are the only ones available for quintic and higher-order equations, beyond trivial or special cases. | https://en.wikipedia.org/wiki/Quartic_equation |
In algebra , a quartic function is a function of the form
where a is nonzero,
which is defined by a polynomial of degree four, called a quartic polynomial .
A quartic equation , or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form
where a ≠ 0 . [ 1 ] The derivative of a quartic function is a cubic function .
Sometimes the term biquadratic is used instead of quartic , but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form
Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity . If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum . Likewise, if a is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.
The degree four ( quartic case) is the highest degree such that every polynomial equation can be solved by radicals , according to the Abel–Ruffini theorem .
Lodovico Ferrari is credited with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately. [ 2 ] The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book Ars Magna . [ 3 ]
The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois prior to dying in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result. [ 4 ]
Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus . It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics , computer-aided design , computer-aided manufacturing and optics . Here are examples of other geometric problems whose solution involves solving a quartic equation.
In computer-aided manufacturing , the torus is a shape that is commonly associated with the endmill cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the z -axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated. [ 5 ]
A quartic equation arises also in the process of solving the crossed ladders problem , in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found. [ 6 ]
In optics, Alhazen's problem is " Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer. " This leads to a quartic equation. [ 7 ] [ 8 ] [ 9 ]
Finding the distance of closest approach of two ellipses involves solving a quartic equation.
The eigenvalues of a 4×4 matrix are the roots of a quartic polynomial which is the characteristic polynomial of the matrix.
The characteristic equation of a fourth-order linear difference equation or differential equation is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending. [ 10 ]
Intersections between spheres, cylinders, or other quadrics can be found using quartic equations.
Letting F and G be the distinct inflection points of the graph of a quartic function, and letting H be the intersection of the inflection secant line FG and the quartic, nearer to G than to F , then G divides FH into the golden section : [ 11 ]
Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area.
Given the general quartic equation
with real coefficients and a ≠ 0 the nature of its roots is mainly determined by the sign of its discriminant
This may be refined by considering the signs of four other polynomials:
such that P / 8 a 2 is the second degree coefficient of the associated depressed quartic (see below );
such that R / 8 a 3 is the first degree coefficient of the associated depressed quartic;
which is 0 if the quartic has a triple root; and
which is 0 if the quartic has two double roots.
The possible cases for the nature of the roots are as follows: [ 12 ]
There are some cases that do not seem to be covered, but in fact they cannot occur. For example, ∆ 0 > 0 , P = 0 and D ≤ 0 is not one of the cases. In fact, if ∆ 0 > 0 and P = 0 then D > 0, since 16 a 2 Δ 0 = 3 D + P 2 ; {\displaystyle 16a^{2}\Delta _{0}=3D+P^{2};} so this combination is not possible.
The four roots x 1 , x 2 , x 3 , and x 4 for the general quartic equation
with a ≠ 0 are given in the following formula, which is deduced from the one in the section on Ferrari's method by back changing the variables (see § Converting to a depressed quartic ) and using the formulas for the quadratic and cubic equations .
where p and q are the coefficients of the second and of the first degree respectively in the associated depressed quartic
and where
(if S = 0 or Q = 0 , see § Special cases of the formula , below)
with
and
Consider the general quartic
It is reducible if Q ( x ) = R ( x )× S ( x ) , where R ( x ) and S ( x ) are non-constant polynomials with rational coefficients (or more generally with coefficients in the same field as the coefficients of Q ( x ) ). Such a factorization will take one of two forms:
or
In either case, the roots of Q ( x ) are the roots of the factors, which may be computed using the formulas for the roots of a quadratic function or cubic function .
Detecting the existence of such factorizations can be done using the resolvent cubic of Q ( x ) . It turns out that:
In fact, several methods of solving quartic equations ( Ferrari's method , Descartes' method , and, to a lesser extent, Euler's method ) are based upon finding such factorizations.
If a 3 = a 1 = 0 then the function
is called a biquadratic function ; equating it to zero defines a biquadratic equation , which is easy to solve as follows
Let the auxiliary variable z = x 2 .
Then Q ( x ) becomes a quadratic q in z : q ( z ) = a 4 z 2 + a 2 z + a 0 . Let z + and z − be the roots of q ( z ) . Then the roots of the quartic Q ( x ) are
The polynomial
is almost palindromic , as P ( mx ) = x 4 / m 2 P ( m / x ) (it is palindromic if m = 1 ). The change of variables z = x + m / x in P ( x ) / x 2 = 0 produces the quadratic equation a 0 z 2 + a 1 z + a 2 − 2 ma 0 = 0 . Since x 2 − xz + m = 0 , the quartic equation P ( x ) = 0 may be solved by applying the quadratic formula twice.
For solving purposes, it is generally better to convert the quartic into a depressed quartic by the following simple change of variable. All formulas are simpler and some methods work only in this case. The roots of the original quartic are easily recovered from that of the depressed quartic by the reverse change of variable.
Let
be the general quartic equation we want to solve.
Dividing by a 4 , provides the equivalent equation x 4 + bx 3 + cx 2 + dx + e = 0 , with b = a 3 / a 4 , c = a 2 / a 4 , d = a 1 / a 4 , and e = a 0 / a 4 .
Substituting y − b / 4 for x gives, after regrouping the terms, the equation y 4 + py 2 + qy + r = 0 ,
where
If y 0 is a root of this depressed quartic, then y 0 − b / 4 (that is y 0 − a 3 / 4 a 4 ) is a root of the original quartic and every root of the original quartic can be obtained by this process.
As explained in the preceding section, we may start with the depressed quartic equation
This depressed quartic can be solved by means of a method discovered by Lodovico Ferrari . The depressed equation may be rewritten (this is easily verified by expanding the square and regrouping all terms in the left-hand side) as
Then, we introduce a variable m into the factor on the left-hand side by adding 2 y 2 m + pm + m 2 to both sides. After regrouping the coefficients of the power of y on the right-hand side, this gives the equation
which is equivalent to the original equation, whichever value is given to m .
As the value of m may be arbitrarily chosen, we will choose it in order to complete the square on the right-hand side. This implies that the discriminant in y of this quadratic equation is zero, that is m is a root of the equation
which may be rewritten as
This is the resolvent cubic of the quartic equation. The value of m may thus be obtained from Cardano's formula . When m is a root of this equation, the right-hand side of equation ( 1 ) is the square
However, this induces a division by zero if m = 0 . This implies q = 0 , and thus that the depressed equation is bi-quadratic, and may be solved by an easier method (see above). This was not a problem at the time of Ferrari, when one solved only explicitly given equations with numeric coefficients. For a general formula that is always true, one thus needs to choose a root of the cubic equation such that m ≠ 0 . This is always possible except for the depressed equation y 4 = 0 .
Now, if m is a root of the cubic equation such that m ≠ 0 , equation ( 1 ) becomes
This equation is of the form M 2 = N 2 , which can be rearranged as M 2 − N 2 = 0 or ( M + N )( M − N ) = 0 . Therefore, equation ( 1 ) may be rewritten as
This equation is easily solved by applying to each factor the quadratic formula . Solving them we may write the four roots as
where ± 1 and ± 2 denote either + or − . As the two occurrences of ± 1 must denote the same sign, this leaves four possibilities, one for each root.
Therefore, the solutions of the original quartic equation are
A comparison with the general formula above shows that √ 2 m = 2 S .
Descartes [ 14 ] introduced in 1637 the method of finding the roots of a quartic polynomial by factoring it into two quadratic ones. Let
By equating coefficients , this results in the following system of equations:
This can be simplified by starting again with the depressed quartic y 4 + py 2 + qy + r , which can be obtained by substituting y − b /4 for x . Since the coefficient of y 3 is 0 , we get s = − u , and:
One can now eliminate both t and v by doing the following:
If we set U = u 2 , then solving this equation becomes finding the roots of the resolvent cubic
which is done elsewhere . This resolvent cubic is equivalent to the resolvent cubic given above (equation (1a)), as can be seen by substituting U = 2m.
If u is a square root of a non-zero root of this resolvent (such a non-zero root exists except for the quartic x 4 , which is trivially factored),
The symmetries in this solution are as follows. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of u for the square root of U merely exchanges the two quadratics with one another.
The above solution shows that a quartic polynomial with rational coefficients and a zero coefficient on the cubic term is factorable into quadratics with rational coefficients if and only if either the resolvent cubic ( 2 ) has a non-zero root which is the square of a rational, or p 2 − 4 r is the square of rational and q = 0 ; this can readily be checked using the rational root test . [ 15 ]
A variant of the previous method is due to Euler . [ 16 ] [ 17 ] Unlike the previous methods, both of which use some root of the resolvent cubic, Euler's method uses all of them. Consider a depressed quartic x 4 + px 2 + qx + r . Observe that, if
then
Therefore, ( r 1 + r 2 )( r 3 + r 4 ) = − s 2 . In other words, −( r 1 + r 2 )( r 3 + r 4 ) is one of the roots of the resolvent cubic ( 2 ) and this suggests that the roots of that cubic are equal to −( r 1 + r 2 )( r 3 + r 4 ) , −( r 1 + r 3 )( r 2 + r 4 ) , and −( r 1 + r 4 )( r 2 + r 3 ) . This is indeed true and it follows from Vieta's formulas . It also follows from Vieta's formulas, together with the fact that we are working with a depressed quartic, that r 1 + r 2 + r 3 + r 4 = 0 . (Of course, this also follows from the fact that r 1 + r 2 + r 3 + r 4 = − s + s .) Therefore, if α , β , and γ are the roots of the resolvent cubic, then the numbers r 1 , r 2 , r 3 , and r 4 are such that
It is a consequence of the first two equations that r 1 + r 2 is a square root of α and that r 3 + r 4 is the other square root of α . For the same reason,
Therefore, the numbers r 1 , r 2 , r 3 , and r 4 are such that
the sign of the square roots will be dealt with below. The only solution of this system is:
Since, in general, there are two choices for each square root, it might look as if this provides 8 (= 2 3 ) choices for the set { r 1 , r 2 , r 3 , r 4 }, but, in fact, it provides no more than 2 such choices, because the consequence of replacing one of the square roots by the symmetric one is that the set { r 1 , r 2 , r 3 , r 4 } becomes the set {− r 1 , − r 2 , − r 3 , − r 4 }.
In order to determine the right sign of the square roots, one simply chooses some square root for each of the numbers α , β , and γ and uses them to compute the numbers r 1 , r 2 , r 3 , and r 4 from the previous equalities. Then, one computes the number √ α √ β √ γ . Since α , β , and γ are the roots of ( 2 ), it is a consequence of Vieta's formulas that their product is equal to q 2 and therefore that √ α √ β √ γ = ± q . But a straightforward computation shows that
If this number is − q , then the choice of the square roots was a good one (again, by Vieta's formulas); otherwise, the roots of the polynomial will be − r 1 , − r 2 , − r 3 , and − r 4 , which are the numbers obtained if one of the square roots is replaced by the symmetric one (or, what amounts to the same thing, if each of the three square roots is replaced by the symmetric one).
This argument suggests another way of choosing the square roots:
Of course, this will make no sense if α or β is equal to 0 , but 0 is a root of ( 2 ) only when q = 0 , that is, only when we are dealing with a biquadratic equation , in which case there is a much simpler approach.
The symmetric group S 4 on four elements has the Klein four-group as a normal subgroup . This suggests using a resolvent cubic whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform of the roots; see Lagrange resolvents for the general method. Denote by x i , for i from 0 to 3 , the four roots of x 4 + bx 3 + cx 2 + dx + e . If we set
then since the transformation is an involution we may express the roots in terms of the four s i in exactly the same way. Since we know the value s 0 = − b / 2 , we only need the values for s 1 , s 2 and s 3 . These are the roots of the polynomial
Substituting the s i by their values in term of the x i , this polynomial may be expanded in a polynomial in s whose coefficients are symmetric polynomials in the x i . By the fundamental theorem of symmetric polynomials , these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If, for simplification, we suppose that the quartic is depressed, that is b = 0 , this results in the polynomial
This polynomial is of degree six, but only of degree three in s 2 , and so the corresponding equation is solvable by the method described in the article about cubic function . By substituting the roots in the expression of the x i in terms of the s i , we obtain expression for the roots. In fact we obtain, apparently, several expressions, depending on the numbering of the roots of the cubic polynomial and of the signs given to their square roots. All these different expressions may be deduced from one of them by simply changing the numbering of the x i .
These expressions are unnecessarily complicated, involving the cubic roots of unity , which can be avoided as follows. If s is any non-zero root of ( 3 ), and if we set
then
We therefore can solve the quartic by solving for s and then solving for the roots of the two factors using the quadratic formula .
This gives exactly the same formula for the roots as the one provided by Descartes' method .
There is an alternative solution using algebraic geometry [ 18 ] In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three reducible quadratic curves (pairs of lines) that pass through these points (this corresponds to the resolvent cubic, the pairs of lines being the Lagrange resolvents), and then use these linear equations to solve the quadratic.
The four roots of the depressed quartic x 4 + px 2 + qx + r = 0 may also be expressed as the x coordinates of the intersections of the two quadratic equations y 2 + py + qx + r = 0 and y − x 2 = 0 i.e., using the substitution y = x 2 that two quadratics intersect in four points is an instance of Bézout's theorem . Explicitly, the four points are P i ≔ ( x i , x i 2 ) for the four roots x i of the quartic.
These four points are not collinear because they lie on the irreducible quadratic y = x 2 and thus there is a 1-parameter family of quadratics (a pencil of curves ) passing through these points. Writing the projectivization of the two quadratics as quadratic forms in three variables:
the pencil is given by the forms λF 1 + μF 2 for any point [ λ , μ ] in the projective line — in other words, where λ and μ are not both zero, and multiplying a quadratic form by a constant does not change its quadratic curve of zeros.
This pencil contains three reducible quadratics, each corresponding to a pair of lines, each passing through two of the four points, which can be done ( 4 2 ) {\displaystyle \textstyle {\binom {4}{2}}} = 6 different ways. Denote these Q 1 = L 12 + L 34 , Q 2 = L 13 + L 24 , and Q 3 = L 14 + L 23 . Given any two of these, their intersection has exactly the four points.
The reducible quadratics, in turn, may be determined by expressing the quadratic form λF 1 + μF 2 as a 3×3 matrix: reducible quadratics correspond to this matrix being singular, which is equivalent to its determinant being zero, and the determinant is a homogeneous degree three polynomial in λ and μ and corresponds to the resolvent cubic. | https://en.wikipedia.org/wiki/Quartic_function |
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 4 ≡ p (mod q ) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x 4 ≡ p (mod q ) to that of x 4 ≡ q (mod p ).
Euler made the first conjectures about biquadratic reciprocity. [ 1 ] Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He said [ 2 ] that a third monograph would be forthcoming with the proof of the general theorem, but it never appeared. Jacobi presented proofs in his Königsberg lectures of 1836–37. [ 3 ] The first published proofs were by Eisenstein. [ 4 ] [ 5 ] [ 6 ] [ 7 ]
Since then a number of other proofs of the classical (Gaussian) version have been found, [ 8 ] as well as alternate statements. Lemmermeyer states that there has been an explosion of interest in the rational reciprocity laws since the 1970s. [A] [ 9 ]
A quartic or biquadratic residue (mod p ) is any number congruent to the fourth power of an integer (mod p ). If x 4 ≡ a (mod p ) does not have an integer solution, a is a quartic or biquadratic nonresidue (mod p ). [ 10 ]
As is often the case in number theory, it is easiest to work modulo prime numbers, so in this section all moduli p , q , etc., are assumed to positive, odd primes. [ 10 ]
The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 3 (mod 4) then a residue r is a quadratic residue (mod q ) if and only if it is a biquadratic residue (mod q ). Indeed, the first supplement of quadratic reciprocity states that −1 is a quadratic nonresidue (mod q ), so that for any integer x , one of x and − x is a quadratic residue and the other one is a nonresidue. Thus, if r ≡ a 2 (mod q ) is a quadratic residue, then if a ≡ b 2 is a residue, r ≡ a 2 ≡ b 4 (mod q ) is a biquadratic residue, and if a is a nonresidue, − a is a residue, − a ≡ b 2 , and again, r ≡ (− a ) 2 ≡ b 4 (mod q ) is a biquadratic residue. [ 11 ]
Therefore, the only interesting case is when the modulus p ≡ 1 (mod 4).
Gauss proved [ 12 ] that if p ≡ 1 (mod 4) then the nonzero residue classes (mod p ) can be divided into four sets, each containing ( p −1)/4 numbers. Let e be a quadratic nonresidue. The first set is the quartic residues; the second one is e times the numbers in the first set, the third is e 2 times the numbers in the first set, and the fourth one is e 3 times the numbers in the first set. Another way to describe this division is to let g be a primitive root (mod p ); then the first set is all the numbers whose indices with respect to this root are ≡ 0 (mod 4), the second set is all those whose indices are ≡ 1 (mod 4), etc. [ 13 ] In the vocabulary of group theory , the first set is a subgroup of index 4 (of the multiplicative group Z /p Z × ), and the other three are its cosets.
The first set is the biquadratic residues, the third set is the quadratic residues that are not quartic residues, and the second and fourth sets are the quadratic nonresidues. Gauss proved that −1 is a biquadratic residue if p ≡ 1 (mod 8) and a quadratic, but not biquadratic, residue, when p ≡ 5 (mod 8). [ 14 ]
2 is a quadratic residue mod p if and only if p ≡ ±1 (mod 8). Since p is also ≡ 1 (mod 4), this means p ≡ 1 (mod 8). Every such prime is the sum of a square and twice a square. [ 15 ]
Gauss proved [ 14 ]
Let q = a 2 + 2 b 2 ≡ 1 (mod 8) be a prime number. Then
Every prime p ≡ 1 (mod 4) is the sum of two squares. [ 16 ] If p = a 2 + b 2 where a is odd and b is even, Gauss proved [ 17 ] that
2 belongs to the first (respectively second, third, or fourth) class defined above if and only if b ≡ 0 (resp. 2, 4, or 6) (mod 8). The first case of this is one of Euler's conjectures:
For an odd prime number p and a quadratic residue a (mod p ), Euler's criterion states that a p − 1 2 ≡ 1 ( mod p ) , {\displaystyle a^{\frac {p-1}{2}}\equiv 1{\pmod {p}},} so if p ≡ 1 (mod 4), a p − 1 4 ≡ ± 1 ( mod p ) . {\displaystyle a^{\frac {p-1}{4}}\equiv \pm 1{\pmod {p}}.}
Define the rational quartic residue symbol for prime p ≡ 1 (mod 4) and quadratic residue a (mod p ) as ( a p ) 4 = ± 1 ≡ a p − 1 4 ( mod p ) . {\displaystyle {\Bigg (}{\frac {a}{p}}{\Bigg )}_{4}=\pm 1\equiv a^{\frac {p-1}{4}}{\pmod {p}}.} It is easy to prove that a is a biquadratic residue (mod p ) if and only if ( a p ) 4 = 1. {\displaystyle {\Bigg (}{\frac {a}{p}}{\Bigg )}_{4}=1.}
Dirichlet [ 18 ] simplified Gauss's proof of the biquadratic character of 2 (his proof only requires quadratic reciprocity for the integers) and put the result in the following form:
Let p = a 2 + b 2 ≡ 1 (mod 4) be prime, and let i ≡ b / a (mod p ). Then
In fact, [ 19 ] let p = a 2 + b 2 = c 2 + 2 d 2 = e 2 − 2 f 2 ≡ 1 (mod 8) be prime, and assume a is odd. Then
Going beyond the character of 2, let the prime p = a 2 + b 2 where b is even, and let q be a prime such that ( p q ) = 1. {\displaystyle ({\tfrac {p}{q}})=1.} Quadratic reciprocity says that ( q ∗ p ) = 1 , {\displaystyle ({\tfrac {q^{*}}{p}})=1,} where q ∗ = ( − 1 ) q − 1 2 q . {\displaystyle q^{*}=(-1)^{\frac {q-1}{2}}q.} Let σ 2 ≡ p (mod q ). Then [ 20 ]
The first few examples are: [ 22 ]
Euler had conjectured the rules for 2, −3 and 5, but did not prove any of them.
Dirichlet [ 23 ] also proved that if p ≡ 1 (mod 4) is prime and ( 17 p ) = 1 {\displaystyle ({\tfrac {17}{p}})=1} then
This has been extended from 17 to 17, 73, 97, and 193 by Brown and Lehmer. [ 24 ]
There are a number of equivalent ways of stating Burde's rational biquadratic reciprocity law.
They all assume that p = a 2 + b 2 and q = c 2 + d 2 are primes where b and d are even, and that ( p q ) = 1. {\displaystyle ({\tfrac {p}{q}})=1.}
Gosset's version is [ 9 ]
Letting i 2 ≡ −1 (mod p ) and j 2 ≡ −1 (mod q ), Frölich's law is [ 25 ]
Burde stated his in the form: [ 26 ] [ 27 ] [ 28 ]
Note that [ 29 ]
Let p ≡ q ≡ 1 (mod 4) be primes and assume ( p q ) = 1 {\displaystyle ({\tfrac {p}{q}})=1} . Then e 2 = p f 2 + q g 2 has non-trivial integer solutions, and [ 30 ]
Let p ≡ q ≡ 1 (mod 4) be primes and assume p = r 2 + q s 2 . Then [ 31 ]
Let p = 1 + 4 x 2 be prime, let a be any odd number that divides x , and let a ∗ = ( − 1 ) a − 1 2 a . {\displaystyle a^{*}=\left(-1\right)^{\frac {a-1}{2}}a.} Then [ 32 ] a * is a biquadratic residue (mod p ).
Let p = a 2 + 4 b 2 = c 2 + 2 d 2 ≡ 1 (mod 8) be prime. Then [ 33 ] all the divisors of c 4 − p a 2 are biquadratic residues (mod p ). The same is true for all the divisors of d 4 − p b 2 .
In his second monograph on biquadratic reciprocity Gauss displays some examples and makes conjectures that imply the theorems listed above for the biquadratic character of small primes. He makes some general remarks, and admits there is no obvious general rule at work. He goes on to say
The theorems on biquadratic residues gleam with the greatest simplicity and genuine beauty only when the field of arithmetic is extended to imaginary numbers, so that without restriction, the numbers of the form a + bi constitute the object of study ... we call such numbers integral complex numbers . [ 34 ] [bold in the original]
These numbers are now called the ring of Gaussian integers , denoted by Z [ i ]. Note that i is a fourth root of 1.
In a footnote he adds
The theory of cubic residues must be based in a similar way on a consideration of numbers of the form a + bh where h is an imaginary root of the equation h 3 = 1 ... and similarly the theory of residues of higher powers leads to the introduction of other imaginary quantities. [ 35 ]
The numbers built up from a cube root of unity are now called the ring of Eisenstein integers . The "other imaginary quantities" needed for the "theory of residues of higher powers" are the rings of integers of the cyclotomic number fields ; the Gaussian and Eisenstein integers are the simplest examples of these.
Gauss develops the arithmetic theory of the "integral complex numbers" and shows that it is quite similar to the arithmetic of ordinary integers. [ 36 ] This is where the terms unit, associate, norm, and primary were introduced into mathematics.
The units are the numbers that divide 1. [ 37 ] They are 1, i , −1, and − i . They are similar to 1 and −1 in the ordinary integers, in that they divide every number. The units are the powers of i .
Given a number λ = a + bi , its conjugate is a − bi and its associates are the four numbers [ 37 ]
If λ = a + bi , the norm of λ, written Nλ, is the number a 2 + b 2 . If λ and μ are two Gaussian integers, Nλμ = Nλ Nμ; in other words, the norm is multiplicative. [ 37 ] The norm of zero is zero, the norm of any other number is a positive integer. ε is a unit if and only if Nε = 1. The square root of the norm of λ, a nonnegative real number which may not be a Gaussian integer, is the absolute value of lambda.
Gauss proves that Z [ i ] is a unique factorization domain and shows that the primes fall into three classes: [ 38 ]
Thus, inert primes are 3, 7, 11, 19, ... and a factorization of the split primes is
The associates and conjugate of a prime are also primes.
Note that the norm of an inert prime q is N q = q 2 ≡ 1 (mod 4); thus the norm of all primes other than 1 + i and its associates is ≡ 1 (mod 4).
Gauss calls a number in Z [ i ] odd if its norm is an odd integer. [ 39 ] Thus all primes except 1 + i and its associates are odd. The product of two odd numbers is odd and the conjugate and associates of an odd number are odd.
In order to state the unique factorization theorem, it is necessary to have a way of distinguishing one of the associates of a number. Gauss defines [ 40 ] an odd number to be primary if it is ≡ 1 (mod (1 + i ) 3 ). It is straightforward to show that every odd number has exactly one primary associate. An odd number λ = a + bi is primary if a + b ≡ a − b ≡ 1 (mod 4); i.e., a ≡ 1 and b ≡ 0, or a ≡ 3 and b ≡ 2 (mod 4). [ 41 ] The product of two primary numbers is primary and the conjugate of a primary number is also primary.
The unique factorization theorem [ 42 ] for Z [ i ] is: if λ ≠ 0, then
where 0 ≤ μ ≤ 3, ν ≥ 0, the π i s are primary primes and the α i s ≥ 1, and this representation is unique, up to the order of the factors.
The notions of congruence [ 43 ] and greatest common divisor [ 44 ] are defined the same way in Z [ i ] as they are for the ordinary integers Z . Because the units divide all numbers, a congruence (mod λ) is also true modulo any associate of λ, and any associate of a GCD is also a GCD.
Gauss proves the analogue of Fermat's theorem : if α is not divisible by an odd prime π, then [ 45 ]
Since Nπ ≡ 1 (mod 4), α N π − 1 4 {\displaystyle \alpha ^{\frac {N\pi -1}{4}}} makes sense, and α N π − 1 4 ≡ i k ( mod π ) {\displaystyle \alpha ^{\frac {N\pi -1}{4}}\equiv i^{k}{\pmod {\pi }}} for a unique unit i k .
This unit is called the quartic or biquadratic residue character of α (mod π) and is denoted by [ 46 ] [ 47 ]
It has formal properties similar to those of the Legendre symbol . [ 48 ]
The biquadratic character can be extended to odd composite numbers in the "denominator" in the same way the Legendre symbol is generalized into the Jacobi symbol . As in that case, if the "denominator" is composite, the symbol can equal one without the congruence being solvable:
Gauss stated the law of biquadratic reciprocity in this form: [ 2 ] [ 51 ]
Let π and θ be distinct primary primes of Z [ i ]. Then
Just as the quadratic reciprocity law for the Legendre symbol is also true for the Jacobi symbol, the requirement that the numbers be prime is not needed; it suffices that they be odd relatively prime nonunits. [ 52 ] Probably the most well-known statement is:
Let π and θ be primary relatively prime nonunits. Then [ 53 ]
There are supplementary theorems [ 54 ] [ 55 ] for the units and the half-even prime 1 + i .
if π = a + bi is a primary prime, then
and thus
Also, if π = a + bi is a primary prime, and b ≠ 0 then [ 56 ]
Jacobi defined π = a + bi to be primary if a ≡ 1 (mod 4). With this normalization, the law takes the form [ 57 ]
Let α = a + bi and β = c + di where a ≡ c ≡ 1 (mod 4) and b and d are even be relatively prime nonunits. Then
The following version was found in Gauss's unpublished manuscripts. [ 58 ]
Let α = a + 2 bi and β = c + 2 di where a and c are odd be relatively prime nonunits. Then
The law can be stated without using the concept of primary:
If λ is odd, let ε(λ) be the unique unit congruent to λ (mod (1 + i ) 3 ); i.e., ε(λ) = i k ≡ λ (mod 2 + 2 i ), where 0 ≤ k ≤ 3. Then [ 59 ] for odd and relatively prime α and β, neither one a unit,
For odd λ, let λ ∗ = ( − 1 ) N λ − 1 4 λ . {\displaystyle \lambda ^{*}=(-1)^{\frac {N\lambda -1}{4}}\lambda .} Then if λ and μ are relatively prime nonunits, Eisenstein proved [ 60 ]
The references to the original papers of Euler, Dirichlet, and Eisenstein were copied from the bibliographies in Lemmermeyer and Cox, and were not used in the preparation of this article.
This was actually written 1748–1750, but was only published posthumously; It is in Vol V, pp. 182–283 of
The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § n ". Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n ".
These are in Gauss's Werke , Vol II, pp. 65–92 and 93–148
German translations are in pp. 511–533 and 534–586 of the following, which also has the Disquisitiones Arithmeticae and Gauss's other papers on number theory.
These papers are all in Vol I of his Werke .
both of these are in Vol I of his Werke .
These two papers by Franz Lemmermeyer contain proofs of Burde's law and related results: | https://en.wikipedia.org/wiki/Quartic_reciprocity |
Quartz dolerite or quartz diabase is an intrusive rock similar to dolerite (also called diabase), but with an excess of quartz . Dolerite is similar in composition to basalt , which is volcanic , and gabbro , which is plutonic . The differing crystal sizes are due to the different rate of cooling, basalt cools quickly and has a very fine structure, while gabbro cools very slowly, at great depth, and large crystals develop. Dolerite is intermediate. [ citation needed ]
Quartz dolerite is very common in central Scotland , in intrusive formations, sills and dykes , and is widely quarried for roadstone. [ citation needed ] It was used with some success for making millstones at one time, the Millstone Grit part of the carboniferous strata not being present in Scotland, but it is no longer used for this purpose, and would probably be illegal now due to the formation of small quartz and other silicate particles, which could cause the serious respiratory disease silicosis .
In Scotland, quartz dolerite is commonly known as whin or whinstone . [ citation needed ]
Quartz dolerite contains many cooling fractures and weathers readily, becoming unstable. It is not uncommon for large boulders to break loose, and significant rockfalls are not uncommon. It is regarded as dangerous as far as climbing is concerned. [ citation needed ]
This article related to petrology is a stub . You can help Wikipedia by expanding it .
This geochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quartz-dolerite |
Within surface science , a quartz crystal microbalance with dissipation monitoring ( QCM-D ) is a type of quartz crystal microbalance (QCM) based on the ring-down technique . It is used in interfacial acoustic sensing . Its most common application is the determination of a film thickness in a liquid environment (such as the thickness of an adsorbed protein layer). It can be used to investigate further properties of the sample, most notably the layer's softness.
Ring-down as a method to interrogate acoustic resonators was established in 1954. [ 1 ] In the context of the QCM, it was described by Hirao et al. [ 2 ] and Rodahl et al. [ 3 ] The active component of a QCM is a thin quartz crystal disk sandwiched between a pair of electrodes. [ 4 ] The application of an AC voltage over the electrodes causes the crystal to oscillate at its acoustic resonance frequency . When the AC voltage is turned off, the oscillation decays exponentially ("rings down"). This decay is recorded and the resonance frequency (f) and the energy dissipation factor (D) are extracted. D is defined as the loss of energy per oscillation period divided by the total energy stored in the system. D is equal to the resonance bandwidth divided by the resonance frequency. Other QCM instruments determine the bandwidth from the conductance spectra. Being a QCM, the QCM-D works in real-time, does not need labeling, and is surface-sensitive. Current QCM-D equipment enables measuring of more than 200 data points per second.
Changes in the resonance frequency (Δf) are primarily related to mass uptake or release at the sensor surface. When employed as a mass sensor, the instrument has a sensitivity of about 0.5ng/cm 2 according to the manufacturer. Changes in the dissipation factor (ΔD) are primarily related to the viscoelasticity (softness). [ 5 ] The softness, in turn, often is related to structural changes of the film adhering at the sensor surface.
When operated as a mass sensor , the QCM-D is often used to study molecular adsorption /desorption and binding kinetics to various types of surfaces. In contrast to optical techniques such as surface plasmon resonance (SPR) spectroscopy, ellipsometry , or dual polarisation interferometry , the QCM determines the mass of the adsorbed film including trapped solvent. Comparison of the "acoustic thickness" as determined with the QCM and the "optical thickness" as determined by any of the optical techniques therefore allows to estimate the degree of swelling of the film in the ambient liquid. [ 6 ] The difference in dry and wet mass measured by QCM-D and MP-SPR is more significant in highly hydrated layers as can be seen in. [ 7 ] [ 8 ] [ 9 ]
Since the softness of the sample is affected by a large variety of parameters, the QCM-D is useful for studying molecular interactions with surfaces as well as interactions between molecules. The QCM-D is commonly used in the fields of biomaterials , cell adhesion , drug discovery, materials science, and biophysics . Other typical applications are characterizing viscoelastic films, conformational changes of deposited macromolecules, build-up of polyelectrolyte multilayers, and degradation or corrosion of films and coatings. | https://en.wikipedia.org/wiki/Quartz_crystal_microbalance_with_dissipation_monitoring |
A quasar ( / ˈ k w eɪ z ɑːr / KWAY -zar ) is an extremely luminous active galactic nucleus (AGN). It is sometimes known as a quasi-stellar object , abbreviated QSO . The emission from an AGN is powered by accretion onto a supermassive black hole with a mass ranging from millions to tens of billions of solar masses , surrounded by a gaseous accretion disc . Gas in the disc falling towards the black hole heats up and releases energy in the form of electromagnetic radiation . The radiant energy of quasars is enormous; the most powerful quasars have luminosities thousands of times greater than that of a galaxy such as the Milky Way . [ 2 ] [ 3 ] Quasars are usually categorized as a subclass of the more general category of AGN. The redshifts of quasars are of cosmological origin . [ 4 ]
The term quasar originated as a contraction of "quasi-stellar [star-like] radio source"—because they were first identified during the 1950s as sources of radio-wave emission of unknown physical origin—and when identified in photographic images at visible wavelengths, they resembled faint, star-like points of light. High-resolution images of quasars, particularly from the Hubble Space Telescope , have shown that quasars occur in the centers of galaxies , and that some host galaxies are strongly interacting or merging galaxies. [ 5 ] As with other categories of AGN, the observed properties of a quasar depend on many factors, including the mass of the black hole, the rate of gas accretion, the orientation of the accretion disc relative to the observer, the presence or absence of a jet , and the degree of obscuration by gas and dust within the host galaxy.
About a million quasars have been identified with reliable spectroscopic redshifts, [ 6 ] and between 2-3 million identified in photometric catalogs. [ 7 ] [ 8 ] The nearest known quasar is about 600 million light-years from Earth, while the record for the most distant known AGN is at a redshift of 10.1, corresponding to a comoving distance of 31.6 billion light-years, or a look-back time of 13.2 billion years. [ 9 ] [ 10 ]
Quasar discovery surveys have shown that quasar activity was more common in the distant past; the peak epoch was approximately 10 billion years ago. [ 11 ] Concentrations of multiple quasars are known as large quasar groups and may constitute some of the largest known structures in the universe if the observed groups are good tracers of mass distribution.
The term quasar was first used in an article by astrophysicist Hong-Yee Chiu in May 1964, in Physics Today , to describe certain astronomically puzzling objects: [ 12 ]
So far, the clumsily long name "quasi-stellar radio sources" is used to describe these objects. Because the nature of these objects is entirely unknown, it is hard to prepare a short, appropriate nomenclature for them so that their essential properties are obvious from their name. For convenience, the abbreviated form "quasar" will be used throughout this paper.
Between 1917 and 1922, it became clear from work by Heber Doust Curtis , Ernst Öpik and others that some objects (" nebulae ") seen by astronomers were in fact distant galaxies like the Milky Way. But when radio astronomy began in the 1950s, astronomers detected, among the galaxies, a small number of anomalous objects with properties that defied explanation.
The objects emitted large amounts of radiation of many frequencies, but no source could be located optically, or in some cases only a faint and point-like object somewhat like a distant star . The spectral lines of these objects, which identify the chemical elements of which the object is composed, were also extremely strange and defied explanation. Some of them changed their luminosity very rapidly in the optical range and even more rapidly in the X-ray range, suggesting an upper limit on their size, perhaps no larger than the Solar System . [ 13 ] This implies an extremely high power density . [ 14 ] Considerable discussion took place over what these objects might be. They were described as "quasi-stellar [meaning: star-like] radio sources" , or "quasi-stellar objects" (QSOs), a name which reflected their unknown nature, and this became shortened to "quasar".
The first quasars ( 3C 48 and 3C 273 ) were discovered in the late 1950s, as radio sources in all-sky radio surveys. [ 15 ] [ 16 ] [ 17 ] [ 18 ] They were first noted as radio sources with no corresponding visible object. Using small telescopes and the Lovell Telescope as an interferometer , they were shown to have a very small angular size. [ 19 ] By 1960, hundreds of these objects had been recorded and published in the Third Cambridge Catalogue while astronomers scanned the skies for their optical counterparts. In 1963, a definite identification of the radio source 3C 48 with an optical object was published by Allan Sandage and Thomas A. Matthews . Astronomers had detected what appeared to be a faint blue star at the location of the radio source and obtained its spectrum, which contained many unknown broad emission lines. The anomalous spectrum defied interpretation.
British-Australian astronomer John Bolton made many early observations of quasars, including a breakthrough in 1962. Another radio source, 3C 273 , was predicted to undergo five occultations by the Moon . Measurements taken by Cyril Hazard and John Bolton during one of the occultations using the Parkes Radio Telescope allowed Maarten Schmidt to find a visible counterpart to the radio source and obtain an optical spectrum using the 200-inch (5.1 m) Hale Telescope on Mount Palomar . This spectrum revealed the same strange emission lines. Schmidt was able to demonstrate that these were likely to be the ordinary spectral lines of hydrogen redshifted by 15.8%, at the time, a high redshift (with only a handful of much fainter galaxies known with higher redshift). If this was due to the physical motion of the "star", then 3C 273 was receding at an enormous velocity, around 47,000 km/s , far beyond the speed of any known star and defying any obvious explanation. [ 20 ] Nor would an extreme velocity help to explain 3C 273's huge radio emissions. If the redshift was cosmological (now known to be correct), the large distance implied that 3C 273 was far more luminous than any galaxy, but much more compact. Also, 3C 273 was bright enough to detect on archival photographs dating back to the 1900s; it was found to be variable on yearly timescales, implying that a substantial fraction of the light was emitted from a region less than 1 light-year in size, tiny compared to a galaxy.
Although it raised many questions, Schmidt's discovery quickly revolutionized quasar observation. The strange spectrum of 3C 48 was quickly identified by Schmidt, Greenstein and Oke as hydrogen and magnesium redshifted by 37%. Shortly afterwards, two more quasar spectra in 1964 and five more in 1965 were also confirmed as ordinary light that had been redshifted to an extreme degree. [ 21 ] While the observations and redshifts themselves were not doubted, their correct interpretation was heavily debated, and Bolton's suggestion that the radiation detected from quasars were ordinary spectral lines from distant highly redshifted sources with extreme velocity was not widely accepted at the time.
An extreme redshift could imply great distance and velocity but could also be due to extreme mass or perhaps some other unknown laws of nature. Extreme velocity and distance would also imply immense power output, which lacked explanation. The small sizes were confirmed by interferometry and by observing the speed with which the quasar as a whole varied in output, and by their inability to be seen in even the most powerful visible-light telescopes as anything more than faint starlike points of light. But if they were small and far away in space, their power output would have to be immense and difficult to explain. Equally, if they were very small and much closer to this galaxy, it would be easy to explain their apparent power output, but less easy to explain their redshifts and lack of detectable movement against the background of the universe.
Schmidt noted that redshift is also associated with the expansion of the universe, as codified in Hubble's law . If the measured redshift was due to expansion, then this would support an interpretation of very distant objects with extraordinarily high luminosity and power output, far beyond any object seen to date. This extreme luminosity would also explain the large radio signal. Schmidt concluded that 3C 273 could either be an individual star around 10 km wide within (or near to) this galaxy, or a distant active galactic nucleus. He stated that a distant and extremely powerful object seemed more likely to be correct. [ 20 ]
Schmidt's explanation for the high redshift was not widely accepted at the time. A major concern was the enormous amount of energy these objects would have to be radiating, if they were distant. In the 1960s no commonly accepted mechanism could account for this. The currently accepted explanation, that it is due to matter in an accretion disc falling into a supermassive black hole , was only suggested in 1964 by Edwin E. Salpeter and Yakov Zeldovich , [ 22 ] and even then it was rejected by many astronomers, as at this time the existence of black holes at all was widely seen as theoretical.
Various explanations were proposed during the 1960s and 1970s, each with their own problems. It was suggested that quasars were nearby objects, and that their redshift was not due to the expansion of space but rather to light escaping a deep gravitational well . This would require a massive object, which would also explain the high luminosities. However, a star of sufficient mass to produce the measured redshift would be unstable and in excess of the Hayashi limit . [ 23 ] Quasars also show forbidden spectral emission lines, previously only seen in hot gaseous nebulae of low density, which would be too diffuse to both generate the observed power and fit within a deep gravitational well. [ 24 ] There were also serious concerns regarding the idea of cosmologically distant quasars. One strong argument against them was that they implied energies that were far in excess of known energy conversion processes, including nuclear fusion . There were suggestions that quasars were made of some hitherto unknown stable form of antimatter in similarly unknown types of region of space, and that this might account for their brightness. [ 25 ] Others speculated that quasars were a white hole end of a wormhole , [ 26 ] [ 27 ] or a chain reaction of numerous supernovae . [ 28 ]
Eventually, starting from about the 1970s, many lines of evidence (including the first X-ray space observatories , knowledge of black holes and modern models of cosmology ) gradually demonstrated that the quasar redshifts are genuine and due to the expansion of space , that quasars are in fact as powerful and as distant as Schmidt and some other astronomers had suggested, and that their energy source is matter from an accretion disc falling onto a supermassive black hole. [ 29 ] This included crucial evidence from optical and X-ray viewing of quasar host galaxies, finding of "intervening" absorption lines, which explained various spectral anomalies, observations from gravitational lensing , Gunn 's 1971 finding that galaxies containing quasars showed the same redshift as the quasars, [ 30 ] and Kristian 's 1973 finding that the "fuzzy" surrounding of many quasars was consistent with a less luminous host galaxy. [ 31 ]
This model also fits well with other observations suggesting that many or even most galaxies have a massive central black hole. It would also explain why quasars are more common in the early universe: as a quasar draws matter from its accretion disc, there comes a point when there is less matter nearby, and energy production falls off or ceases, as the quasar becomes a more ordinary type of galaxy.
The accretion-disc energy-production mechanism was finally modeled in the 1970s, and black holes were also directly detected (including evidence showing that supermassive black holes could be found at the centers of this and many other galaxies), which resolved the concern that quasars were too luminous to be a result of very distant objects or that a suitable mechanism could not be confirmed to exist in nature. By 1987 it was "well accepted" that this was the correct explanation for quasars, [ 32 ] and the cosmological distance and energy output of quasars was accepted by almost all researchers.
Later it was found that not all quasars have strong radio emission; in fact only about 10% are "radio-loud". Hence the name "QSO" (quasi-stellar object) is used (in addition to "quasar") to refer to these objects, further categorized into the "radio-loud" and the "radio-quiet" classes. The discovery of the quasar had large implications for the field of astronomy in the 1960s, including drawing physics and astronomy closer together. [ 34 ]
In 1979, the gravitational lens effect predicted by Albert Einstein 's general theory of relativity was confirmed observationally for the first time with images of the double quasar 0957+561. [ 35 ]
A study published in February 2021 showed that there are more quasars in one direction (towards Hydra ) than in the opposite direction, seemingly indicating that the Earth is moving in that direction. But the direction of this dipole is about 28° away from the direction of the Earth's motion relative to the cosmic microwave background radiation. [ 36 ]
In March 2021, a collaboration of scientists, related to the Event Horizon Telescope , presented, for the first time, a polarized-based image of a black hole , specifically the black hole at the center of Messier 87 , an elliptical galaxy approximately 55 million light-years away in the constellation Virgo , revealing the forces giving rise to quasars. [ 37 ]
It is now known that quasars are distant but extremely luminous objects, so any light that reaches the Earth is redshifted due to the expansion of the universe . [ 38 ]
Quasars inhabit the centers of active galaxies and are among the most luminous, powerful, and energetic objects known in the universe, emitting up to a thousand times the energy output of the Milky Way , which contains 200–400 billion stars. This radiation is emitted across the electromagnetic spectrum almost uniformly, from X-rays to the far infrared with a peak in the ultraviolet optical bands, with some quasars also being strong sources of radio emission and of gamma-rays. With high-resolution imaging from ground-based telescopes and the Hubble Space Telescope , the "host galaxies" surrounding the quasars have been detected in some cases. [ 39 ] These galaxies are normally too dim to be seen against the glare of the quasar, except with special techniques. Most quasars, with the exception of 3C 273 , whose average apparent magnitude is 12.9, cannot be seen with small telescopes.
Quasars are believed—and in many cases confirmed—to be powered by accretion of material into supermassive black holes in the nuclei of distant galaxies, as suggested in 1964 by Edwin Salpeter and Yakov Zeldovich . [ 15 ] Light and other radiation cannot escape from within the event horizon of a black hole. The energy produced by a quasar is generated outside the black hole, by gravitational stresses and immense friction within the material nearest to the black hole, as it orbits and falls inward. [ 32 ] The huge luminosity of quasars results from the accretion discs of central supermassive black holes, which can convert between 5.7% and 32% of the mass of an object into energy , [ 40 ] compared to just 0.7% for the p–p chain nuclear fusion process that dominates the energy production in Sun-like stars. Central masses of 10 5 to 10 9 solar masses have been measured in quasars by using reverberation mapping . Several dozen nearby large galaxies, including the Milky Way galaxy, that do not have an active center and do not show any activity similar to a quasar, are confirmed to contain a similar supermassive black hole in their nuclei (galactic center) . Thus it is now thought that all large galaxies have a black hole of this kind, but only a small fraction have sufficient matter in the right kind of orbit at their center to become active and power radiation in such a way as to be seen as quasars. [ 41 ]
This also explains why quasars were more common in the early universe, as this energy production ends when the supermassive black hole consumes all of the gas and dust near it. This means that it is possible that most galaxies, including the Milky Way, have gone through an active stage, appearing as a quasar or some other class of active galaxy that depended on the black-hole mass and the accretion rate, and are now quiescent because they lack a supply of matter to feed into their central black holes to generate radiation. [ 41 ]
The matter accreting onto the black hole is unlikely to fall directly in, but will have some angular momentum around the black hole, which will cause the matter to collect into an accretion disc . Quasars may also be ignited or re-ignited when normal galaxies merge and the black hole is infused with a fresh source of matter. [ 43 ] In fact, it has been suggested that a quasar could form when the Andromeda Galaxy collides with the Milky Way galaxy in approximately 3–5 billion years. [ 32 ] [ 44 ] [ 45 ] [ 46 ]
In the 1980s, unified models were developed in which quasars were classified as a particular kind of active galaxy , and a consensus emerged that in many cases it is simply the viewing angle that distinguishes them from other active galaxies, such as blazars and radio galaxies . [ 47 ]
The highest-redshift quasar known (as of August 2024 [update] ) is UHZ1 , with a redshift of approximately 10.1, [ 48 ] which corresponds to a comoving distance of approximately 31.7 billion light-years from Earth (these distances are much larger than the distance light could travel in the universe's 13.8-billion-year history because the universe is expanding).
More than 900,000 quasars have been found (as of July 2023), [ 6 ] most from the Sloan Digital Sky Survey . All observed quasar spectra have redshifts between 0.056 and 10.1 (as of 2024), which means they range between 600 million and 30 billion light-years away from Earth . Because of the great distances to the farthest quasars and the finite velocity of light, they and their surrounding space appear as they existed in the very early universe.
The power of quasars originates from supermassive black holes that are believed to exist at the core of most galaxies. The Doppler shifts of stars near the cores of galaxies indicate that they are revolving around tremendous masses with very steep gravity gradients, suggesting black holes.
Although quasars appear faint when viewed from Earth, they are visible from extreme distances, being the most luminous objects in the known universe. The brightest quasar in the sky is 3C 273 in the constellation of Virgo . It has an average apparent magnitude of 12.8 (bright enough to be seen through a medium-size amateur telescope ), but it has an absolute magnitude of −26.7. [ 49 ] From a distance of about 33 light-years, this object would shine in the sky about as brightly as the Sun . This quasar's luminosity is, therefore, about 4 trillion (4 × 10 12 ) times that of the Sun, or about 100 times that of the total light of giant galaxies like the Milky Way . [ 49 ] This assumes that the quasar is radiating energy in all directions, but the active galactic nucleus is believed to be radiating preferentially in the direction of its jet. In a universe containing hundreds of billions of galaxies, most of which had active nuclei billions of years ago but only seen today, it is statistically certain that thousands of energy jets should be pointed toward the Earth, some more directly than others. In many cases it is likely that the brighter the quasar, the more directly its jet is aimed at the Earth. Such quasars are called blazars .
The hyperluminous quasar APM 08279+5255 was, when discovered in 1998, given an absolute magnitude of −32.2. High-resolution imaging with the Hubble Space Telescope and the 10 m Keck Telescope revealed that this system is gravitationally lensed . A study of the gravitational lensing of this system suggests that the light emitted has been magnified by a factor of ~10. It is still substantially more luminous than nearby quasars such as 3C 273.
Quasars were much more common in the early universe than they are today. This discovery by Maarten Schmidt in 1967 was early strong evidence against steady-state cosmology and in favor of the Big Bang cosmology. Quasars show the locations where supermassive black holes are growing rapidly (by accretion ). Detailed simulations reported in 2021 showed that galaxy structures, such as spiral arms, use gravitational forces to 'put the brakes on' gas that would otherwise orbit galaxy centers forever; instead the braking mechanism enabled the gas to fall into the supermassive black holes, releasing enormous radiant energies. [ 50 ] [ 51 ] These black holes co-evolve with the mass of stars in their host galaxy in a way not fully understood at present. One idea is that jets, radiation and winds created by the quasars shut down the formation of new stars in the host galaxy, a process called "feedback". The jets that produce strong radio emission in some quasars at the centers of clusters of galaxies are known to have enough power to prevent the hot gas in those clusters from cooling and falling on to the central galaxy.
Quasars' luminosities are variable, with time scales that range from months to hours. This means that quasars generate and emit their energy from a very small region, since each part of the quasar would have to be in contact with other parts on such a time scale as to allow the coordination of the luminosity variations. This would mean that a quasar varying on a time scale of a few weeks cannot be larger than a few light-weeks across. The emission of large amounts of power from a small region requires a power source far more efficient than the nuclear fusion that powers stars. The conversion of gravitational potential energy to radiation by infalling to a black hole converts between 6% and 32% of the mass to energy, compared to 0.7% for the conversion of mass to energy in a star like the Sun. [ 40 ] It is the only process known that can produce such high power over a very long term. (Stellar explosions such as supernovas and gamma-ray bursts , and direct matter – antimatter annihilation, can also produce very high power output, but supernovae only last for days, and the universe does not appear to have had large amounts of antimatter at the relevant times.)
Since quasars exhibit all the properties common to other active galaxies such as Seyfert galaxies , the emission from quasars can be readily compared to those of smaller active galaxies powered by smaller supermassive black holes. To create a luminosity of 10 40 watts (the typical brightness of a quasar), a supermassive black hole would have to consume the material equivalent of 10 solar masses per year. The brightest known quasars devour 1,000 solar masses of material every year (equivalent to 10 Earths per second). Quasar luminosities can vary considerably over time, depending on their surroundings. Since it is difficult to fuel quasars for many billions of years, after a quasar finishes accreting the surrounding gas and dust, it becomes an ordinary galaxy.
Radiation from quasars is partially "nonthermal" (i.e., not due to black-body radiation ), and approximately 10% are observed to also have jets and lobes like those of radio galaxies that also carry significant (but poorly understood) amounts of energy in the form of particles moving at relativistic speeds . Extremely high energies might be explained by several mechanisms (see Fermi acceleration and Centrifugal mechanism of acceleration ). Quasars can be detected over the entire observable electromagnetic spectrum , including radio , infrared , visible light , ultraviolet , X-ray and even gamma rays . Most quasars are brightest in their rest-frame ultraviolet wavelength of 121.6 nm Lyman-alpha emission line of hydrogen, but due to the tremendous redshifts of these sources, that peak luminosity has been observed as far to the red as 900.0 nm, in the near infrared. A minority of quasars show strong radio emission, which is generated by jets of matter moving close to the speed of light. When viewed downward, these appear as blazars and often have regions that seem to move away from the center faster than the speed of light ( superluminal expansion). This is an optical illusion due to the properties of special relativity .
Quasar redshifts are measured from the strong spectral lines that dominate their visible and ultraviolet emission spectra. These lines are brighter than the continuous spectrum. They exhibit Doppler broadening corresponding to mean speed of several percent of the speed of light. Fast motions strongly indicate a large mass. Emission lines of hydrogen (mainly of the Lyman series and Balmer series ), helium, carbon, magnesium, iron and oxygen are the brightest lines. The atoms emitting these lines range from neutral to highly ionized, leaving it highly charged. This wide range of ionization shows that the gas is highly irradiated by the quasar, not merely hot, and not by stars, which cannot produce such a wide range of ionization.
Like all (unobscured) active galaxies, quasars can be strong X-ray sources. Radio-loud quasars can also produce X-rays and gamma rays by inverse Compton scattering of lower-energy photons by the radio-emitting electrons in the jet. [ 52 ]
Iron quasars show strong emission lines resulting from low-ionization iron (Fe II ), such as IRAS 18508-7815.
Quasars also provide some clues as to the end of the Big Bang 's reionization . The oldest known quasars ( z = 6) [ needs update ] display a Gunn–Peterson trough and have absorption regions in front of them indicating that the intergalactic medium at that time was neutral gas . More recent quasars show no absorption region, but rather their spectra contain a spiky area known as the Lyman-alpha forest ; this indicates that the intergalactic medium has undergone reionization into plasma , and that neutral gas exists only in small clouds.
The intense production of ionizing ultraviolet radiation is also significant, as it would provide a mechanism for reionization to occur as galaxies form. Despite this, current theories suggest that quasars were not the primary source of reionization; the primary causes of reionization were probably the earliest generations of stars , known as Population III stars (possibly 70%), and dwarf galaxies (very early small high-energy galaxies) (possibly 30%). [ 55 ] [ 56 ] [ 57 ] [ 58 ] [ 59 ] [ 60 ]
Quasars show evidence of elements heavier than helium , indicating that galaxies underwent a massive phase of star formation , creating population III stars between the time of the Big Bang and the first observed quasars. Light from these stars may have been observed in 2005 using NASA 's Spitzer Space Telescope , [ 61 ] although this observation remains to be confirmed.
The taxonomy of quasars includes various subtypes representing subsets of the quasar population having distinct properties.
Because quasars are extremely distant, bright, and small in apparent size, they are useful reference points in establishing a measurement grid on the sky. [ 67 ] The International Celestial Reference System (ICRS) is based on hundreds of extra-galactic radio sources, mostly quasars, distributed around the entire sky. Because they are so distant, they are apparently stationary to current technology, yet their positions can be measured with the utmost accuracy by very-long-baseline interferometry (VLBI). The positions of most are known to 0.001 arcsecond or better, which is orders of magnitude more precise than the best optical measurements.
A grouping of two or more quasars on the sky can result from a chance alignment, where the quasars are not physically associated, from actual physical proximity, or from the effects of gravity bending the light of a single quasar into two or more images by gravitational lensing .
When two quasars appear to be very close to each other as seen from Earth (separated by a few arcseconds or less), they are commonly referred to as a "double quasar". When the two are also close together in space (i.e. observed to have similar redshifts), they are termed a "quasar pair", or as a "binary quasar" if they are close enough that their host galaxies are likely to be physically interacting. [ 68 ]
As quasars are overall rare objects in the universe, the probability of three or more separate quasars being found near the same physical location is very low, and determining whether the system is closely separated physically requires significant observational effort. The first true triple quasar was found in 2007 by observations at the W. M. Keck Observatory in Mauna Kea , Hawaii . [ 69 ] LBQS 1429-008 (or QQQ J1432-0106) was first observed in 1989 and at the time was found to be a double quasar. When astronomers discovered the third member, they confirmed that the sources were separate and not the result of gravitational lensing. This triple quasar has a redshift of z = 2.076. [ 70 ] The components are separated by an estimated 30–50 kiloparsecs (roughly 97,000–160,000 light-years), which is typical for interacting galaxies. [ 71 ] In 2013, the second true triplet of quasars, QQQ J1519+0627, was found with a redshift z = 1.51, the whole system fitting within a physical separation of 25 kpc (about 80,000 light-years). [ 72 ] [ 73 ]
The first true quadruple quasar system was discovered in 2015 at a redshift z = 2.0412 and has an overall physical scale of about 200 kpc (roughly 650,000 light-years). [ 74 ]
A multiple-image quasar is a quasar whose light undergoes gravitational lensing , resulting in double, triple or quadruple images of the same quasar. The first such gravitational lens to be discovered was the double-imaged quasar Q0957+561 (or Twin Quasar) in 1979. [ 75 ] An example of a triply lensed quasar is PG1115+08. [ 76 ] Several quadruple-image quasars are known, including the Einstein Cross and the Cloverleaf Quasar , with the first such discoveries happening in the mid-1980s. | https://en.wikipedia.org/wiki/Quasar |
The Quasar Equatorial Survey Team ( QUEST ) is a joint venture between Yale University , Indiana University , and Centro de Investigaciones de Astronomia (CIDA) to photographically survey the sky using a digital camera , an array of 112 charge-coupled devices . Since 2009, it has used the 1 m ESO Schmidt Telescope in Chile. From 2003–2007, it used the 48 inch (1.22 m) Samuel Oschin telescope at the Palomar Observatory . Before that, it had used the 1.0-metre Schmidt telescope at the Llano del Hato National Astronomical Observatory in Venezuela .
As of 08/09/2017 all the following links are broken.
This article about an organization or institute connected with astronomy is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasar_Equatorial_Survey_Team |
In mathematics , a quasi-Lie algebra in abstract algebra is just like a Lie algebra , but with the usual axiom
replaced by
In characteristic other than 2, these are equivalent (in the presence of bilinearity ), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering Lie algebras over the integers.
In a quasi-Lie algebra,
Therefore, the bracket of any element with itself is 2-torsion, if it does not actually vanish.
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasi-Lie_algebra |
In mathematics , the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function . All continuous functions are quasi-continuous but the converse is not true in general.
Let X {\displaystyle X} be a topological space . A real-valued function f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is quasi-continuous at a point x ∈ X {\displaystyle x\in X} if for any ϵ > 0 {\displaystyle \epsilon >0} and any open neighborhood U {\displaystyle U} of x {\displaystyle x} there is a non-empty open set G ⊂ U {\displaystyle G\subset U} such that
Note that in the above definition, it is not necessary that x ∈ G {\displaystyle x\in G} .
Consider the function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } defined by f ( x ) = 0 {\displaystyle f(x)=0} whenever x ≤ 0 {\displaystyle x\leq 0} and f ( x ) = 1 {\displaystyle f(x)=1} whenever x > 0 {\displaystyle x>0} . Clearly f is continuous everywhere except at x=0, thus quasi-continuous everywhere except (at most) at x=0. At x=0, take any open neighborhood U of x. Then there exists an open set G ⊂ U {\displaystyle G\subset U} such that y < 0 ∀ y ∈ G {\displaystyle y<0\;\forall y\in G} . Clearly this yields | f ( 0 ) − f ( y ) | = 0 ∀ y ∈ G {\displaystyle |f(0)-f(y)|=0\;\forall y\in G} thus f is quasi-continuous.
In contrast, the function g : R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } defined by g ( x ) = 0 {\displaystyle g(x)=0} whenever x {\displaystyle x} is a rational number and g ( x ) = 1 {\displaystyle g(x)=1} whenever x {\displaystyle x} is an irrational number is nowhere quasi-continuous, since every nonempty open set G {\displaystyle G} contains some y 1 , y 2 {\displaystyle y_{1},y_{2}} with | g ( y 1 ) − g ( y 2 ) | = 1 {\displaystyle |g(y_{1})-g(y_{2})|=1} . | https://en.wikipedia.org/wiki/Quasi-continuous_function |
Quasi-crystals are supramolecular aggregates exhibiting both crystalline (solid) properties as well as amorphous, liquid-like properties.
Self-organized structures termed "quasi-crystals" were originally described in 1978 by the Israeli scientist Valeri A. Krongauz of the Weizmann Institute of Science , in the Nature paper, Quasi-crystals from irradiated photochromic dyes in an applied electric field . [ 1 ]
In his 1978 paper Krongauz coined the term “Quasi-Crystals” for the new self-organized colloidal particles . The Quasi-crystals are supramolecular aggregates manifesting both crystalline properties e.g. Bragg scattering , as well as amorphous, liquid-like properties i.e. drop-like shapes, fluidity, extensibility and elasticity in electric field. The supramolecular Quasi-crystals are produced in photochemical reaction by exposing solutions of photochromic spiropyran molecules to UV radiation. The ultraviolet light induces the conversion of the spiropyrans to merocyanine molecules that manifest electric dipole moments . (see Scheme 1). The quasi-crystals have external shape of submicron globules and their internal structure consists of crystals enveloped by an amorphous matter (see Fig. 1). The crystals are formed by self-assembled stacks of the merocyanine molecular dipoles aligning themselves in a parallel manner, while amorphous envelopes consist of the same merocyanine dipoles aligned in an anti-parallel manner (Fig. 1, Scheme 2). [ 2 ] [ 3 ] [ 4 ] In an applied electrostatic field, quasi-crystals form macroscopic threads that show linear optical dichroism . [ 1 ] [ 5 ]
Later Krongauz described unusual phase transitions of molecules composed of mesogenic and spiropyran moieties, which he named "quasi-liquid crystals." A micrograph of their mesophase appeared on the cover of Nature in a 1984 paper, “Quasi-Liquid Crystals.” [ 6 ] The investigation of spiropyran-merocyanine self-organized systems, including macromolecules (see, for example, Fig. 2), has continued over the years. [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ]
These studies have resulted in discoveries of unusual and practically significant phenomena. Thus, in the electrostatic field, quasi-crystals and quasi-liquid crystals have exhibited 2nd order non-linear optical properties. [ 12 ] [ 13 ] [ 14 ]
Potential applications of these fascinating materials have been described and patented. [ 15 ] [ 16 ] [ 17 ]
Work on spiropyran-merocyanine self-assemblies currently continues in several laboratories. [ 18 ] | https://en.wikipedia.org/wiki/Quasi-crystals_(supramolecular) |
Quasi-empirical methods are scientific methods used to gain knowledge in situations where empirical evidence cannot be gathered through experimentation, or experience cannot falsify the ideas involved. Quasi-empirical methods aim to be as closely analogous to empirical methods as possible. [ 1 ]
Empirical research relies on, and its empirical methods involve experimentation and disclosure of apparatus for reproducibility , by which scientific findings are validated by other scientists. Empirical methods are studied extensively in the philosophy of science , but they cannot be used directly in fields whose hypotheses cannot be falsified by real experiment (for example, mathematics , philosophy , theology , and ideology ). Because of such limits, the scientific method must rely not only on empirical methods but sometimes also on quasi-empirical ones. The prefix quasi- came to denote methods that are "almost" or "socially approximate" an ideal of truly empirical methods.
Quasi-empirical method usually refers to a means of choosing problems to focus on (or ignore), selecting prior work on which to build an argument or proof, notations for informal claims, peer review and acceptance, and incentives to discover, ignore, or correct errors. To disprove a theory logically, it is unnecessary to find all counterexamples to a theory; all that is required is one counterexample. The converse does not prove a theory; Bayesian inference simply makes a theory more likely, by weight of evidence. Since it is not possible to find all counter-examples to a theory, it is also possible to argue that no science is strictly empirical, but this is not the usual meaning of "quasi-empirical".
Albert Einstein 's discovery of the general relativity theory relied upon thought experiments and mathematics . Empirical methods only became relevant when confirmation was sought. Furthermore, some empirical confirmation was found only some time after the general acceptance of the theory.
Thought experiments are almost standard procedure in philosophy , where a conjecture is tested out in the imagination for possible effects on experience; when these are thought to be implausible, unlikely to occur, or not actually occurring, then the conjecture may be either rejected or amended. Logical positivism was a perhaps extreme version of this practice, though this claim is open to debate.
Quasi-empiricism in mathematics is an important topic in post-20th-century philosophy of mathematics , especially as reflected in the actual mathematical practice of working mathematicians.
This philosophy of science -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasi-empirical_method |
Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice , in particular, relations with physics , social sciences , and computational mathematics , rather than solely to issues in the foundations of mathematics . Of concern to this discussion are several topics: the relationship of empiricism (see Penelope Maddy ) with mathematics , issues related to realism , the importance of culture , necessity of application , etc.
A primary argument with respect to quasi-empiricism is that whilst mathematics and physics are frequently considered to be closely linked fields of study, this may reflect human cognitive bias . It is claimed that, despite rigorous application of appropriate empirical methods or mathematical practice in either field, this would nonetheless be insufficient to disprove alternate approaches.
Eugene Wigner (1960) [ 1 ] noted that this culture need not be restricted to mathematics, physics, or even humans. He stated further that "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning." Wigner used several examples to demonstrate why 'bafflement' is an appropriate description, such as showing how mathematics adds to situational knowledge in ways that are either not possible otherwise or are so outside normal thought to be of little notice. The predictive ability, in the sense of describing potential phenomena prior to observation of such, which can be supported by a mathematical system would be another example.
Following up on Wigner , Richard Hamming (1980) [ 2 ] wrote about applications of mathematics as a central theme to this topic and suggested that successful use can sometimes trump proof, in the following sense: where a theorem has evident veracity through applicability, later evidence that shows the theorem's proof to be problematic would result more in trying to firm up the theorem rather than in trying to redo the applications or to deny results obtained to date. Hamming had four explanations for the 'effectiveness' that we see with mathematics and definitely saw this topic as worthy of discussion and study.
For Willard Van Orman Quine (1960), [ 3 ] existence is only existence in a structure. This position is relevant to quasi-empiricism because Quine believes that the same evidence that supports theorizing about the structure of the world is the same as the evidence supporting theorizing about mathematical structures. [ 4 ]
Hilary Putnam (1975) [ 5 ] stated that mathematics had accepted informal proofs and proof by authority, and had made and corrected errors all through its history. Also, he stated that Euclid 's system of proving geometry theorems was unique to the classical Greeks and did not evolve similarly in other mathematical cultures in China , India , and Arabia . This and other evidence led many mathematicians to reject the label of Platonists , along with Plato's ontology – which, along with the methods and epistemology of Aristotle , had served as a foundation ontology for the Western world since its beginnings. A truly international culture of mathematics would, Putnam and others (1983) [ 6 ] argued, necessarily be at least 'quasi'-empirical (embracing 'the scientific method' for consensus if not experiment).
Imre Lakatos (1976), [ 7 ] who did his original work on this topic for his dissertation (1961, Cambridge ), argued for ' research programs ' as a means to support a basis for mathematics and considered thought experiments as appropriate to mathematical discovery. Lakatos may have been the first to use 'quasi-empiricism' in the context of this subject.
Several recent works pertain to this topic. Gregory Chaitin 's and Stephen Wolfram 's work, though their positions may be considered controversial, apply. Chaitin (1997/2003) [ 8 ] suggests an underlying randomness to mathematics and Wolfram ( A New Kind of Science , 2002) [ 9 ] argues that undecidability may have practical relevance, that is, be more than an abstraction.
Another relevant addition would be the discussions concerning interactive computation , especially those related to the meaning and use of Turing 's model ( Church-Turing thesis , Turing machines , etc.).
These works are heavily computational and raise another set of issues. To quote Chaitin (1997/2003):
Now everything has gone topsy-turvy. It's gone topsy-turvy, not because of any philosophical argument, not because of Gödel 's results or Turing 's results or my own incompleteness results. It's gone topsy-turvy for a very simple reason—the computer! [ 8 ] : 96
The collection of "Undecidables" in Wolfram ( A New Kind of Science , 2002) [ 9 ] is another example.
Wegner's 2006 paper "Principles of Problem Solving" [ 10 ] suggests that interactive computation can help mathematics form a more appropriate framework ( empirical ) than can be founded with rationalism alone. Related to this argument is that the function (even recursively related ad infinitum) is too simple a construct to handle the reality of entities that resolve (via computation or some type of analog) n-dimensional (general sense of the word) systems. | https://en.wikipedia.org/wiki/Quasi-empiricism_in_mathematics |
A linear differential operator L is called quasi-exactly-solvable ( QES ) if it has a finite-dimensional invariant subspace of functions { V } n {\displaystyle \{{\mathcal {V}}\}_{n}} such that L : { V } n → { V } n , {\displaystyle L:\{{\mathcal {V}}\}_{n}\rightarrow \{{\mathcal {V}}\}_{n},} where n is a dimension of { V } n {\displaystyle \{{\mathcal {V}}\}_{n}} . There are two important cases:
The most studied cases are one-dimensional s l ( 2 ) {\displaystyle sl(2)} -Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian
{ H } = − d 2 d x 2 + a 2 x 6 + 2 a b x 4 + [ b 2 − ( 4 n + 3 + 2 p ) a ] x 2 , a ≥ 0 , n ∈ N , p = { 0 , 1 } , {\displaystyle \{{\mathcal {H}}\}=-{\frac {d^{2}}{dx^{2}}}+a^{2}x^{6}+2abx^{4}+[b^{2}-(4n+3+2p)a]x^{2},\ a\geq 0\ ,\ n\in \mathbb {N} \ ,\ p=\{0,1\},}
where ( n+1 ) eigenstates of positive (negative) parity can be found algebraically . Their eigenfunctions are of the form
Ψ ( x ) = x p P n ( x 2 ) e − a x 4 4 − b x 2 2 , {\displaystyle \Psi (x)\ =\ x^{p}P_{n}(x^{2})e^{-{\frac {ax^{4}}{4}}-{\frac {bx^{2}}{2}}}\ ,}
where P n ( x 2 ) {\displaystyle P_{n}(x^{2})} is a polynomial of degree n and (energies) eigenvalues are roots of an algebraic equation of degree ( n+1 ). In general, twelve families of one-dimensional QES problems are known, two of them characterized by elliptic potentials. | https://en.wikipedia.org/wiki/Quasi-exact_solvability |
Quasi-extinction refers to the state in which a species or population has declined to critically low numbers, making its recovery highly unlikely, even though a small number of individuals may still persist. This concept is often used in conservation biology to identify species at extreme risk of extinction and to guide management strategies aimed at preventing complete extinction . Quasi-extinction is typically characterized by an inability of the population to sustain itself due to genetic , demographic , or environmental factors. [ 1 ] [ 2 ] [ 3 ] [ 4 ]
The quasi-extinction threshold , or sometimes called the quasi-extinction risk is the population size below which a species is considered to be at extreme risk of quasi-extinction. [ 5 ] This threshold varies by species and is influenced by several factors, including reproductive rates, habitat requirements, and genetic diversity. It is often used in population viability analyses (PVA) to model the likelihood of a species declining to levels where recovery becomes nearly impossible. [ 6 ] [ 7 ] | https://en.wikipedia.org/wiki/Quasi-extinction |
The quasi-harmonic approximation is a phonon -based model of solid-state physics used to describe volume-dependent thermal effects, such as the thermal expansion . It is based on the assumption that the harmonic approximation holds for every value of the lattice constant , which is to be viewed as an adjustable parameter.
The quasi-harmonic approximation expands upon the harmonic phonon model of lattice dynamics. The harmonic phonon model states that all interatomic forces are purely harmonic , but such a model is inadequate to explain thermal expansion , as the equilibrium distance between atoms in such a model is independent of temperature.
Thus in the quasi-harmonic model, from a phonon point of view, phonon frequencies become volume-dependent in the quasi-harmonic approximation, such that for each volume, the harmonic approximation holds.
For a lattice, the Helmholtz free energy F in the quasi-harmonic approximation is
F ( T , V ) = E l a t ( V ) + U v i b ( T , V ) − T S ( T , V ) {\displaystyle F(T,V)=E_{\rm {lat}}(V)+U_{\rm {vib}}(T,V)-TS(T,V)}
where E lat is the static internal lattice energy , U vib is the internal vibrational energy of the lattice, or the energy of the phonon system, T is the absolute temperature, V is the volume and S is the entropy due to the vibrational degrees of freedom.
The vibrational energy equals
U v i b ( T , V ) = 1 N ∑ k , i 1 2 ℏ ω k , i ( V ) + 1 N ∑ k , i ℏ ω k , i ( V ) exp ( Θ k , i ( V ) / T ) − 1 = 1 N ∑ k , i [ 1 2 + n k , i ( T , V ) ] ℏ ω k , i ( V ) {\displaystyle U_{\rm {vib}}(T,V)={\frac {1}{N}}\sum _{\mathbf {k} ,i}{\frac {1}{2}}\hbar \omega _{\mathbf {k} ,i}(V)+{\frac {1}{N}}\sum _{\mathbf {k} ,i}{\frac {\hbar \omega _{\mathbf {k} ,i}(V)}{\exp(\Theta _{\mathbf {k} ,i}(V)/T)-1}}={\frac {1}{N}}\sum _{\mathbf {k} ,i}[{\frac {1}{2}}+n_{\mathbf {k} ,i}(T,V)]\hbar \omega _{\mathbf {k} ,i}(V)}
where N is the number of terms in the sum, Θ k , i ( V ) = ℏ ω k , i ( V ) / k B {\textstyle \Theta _{\mathbf {k} ,i}(V)=\hbar \omega _{\mathbf {k} ,i}(V)/k_{B}} is introduced as the characteristic temperature for a phonon with wave vector k in the i -th band at volume V and n k , i ( T , V ) {\textstyle n_{\mathbf {k} ,i}(T,V)} is shorthand for the number of ( k , i )-phonons at temperature T and volume V . As is conventional, ℏ {\textstyle \hbar } is the reduced Planck constant and k B is the Boltzmann constant . The first term in U vib is the zero-point energy of the phonon system and contributes to the thermal expansion as a zero-point thermal pressure.
The Helmholtz free energy F is given by
F = E l a t ( V ) + 1 N ∑ k , i 1 2 ℏ ω k , i ( V ) + 1 N ∑ k , i k B T ln [ 1 − exp ( − Θ k , i ( V ) / T ) ] {\displaystyle F=E_{\rm {lat}}(V)+{\frac {1}{N}}\sum _{\mathbf {k} ,i}{\frac {1}{2}}\hbar \omega _{\mathbf {k} ,i}(V)+{\frac {1}{N}}\sum _{\mathbf {k} ,i}k_{B}T\ln \left[1-\exp(-\Theta _{\mathbf {k} ,i}(V)/T)\right]}
and the entropy term equals
S = − ( ∂ F ∂ T ) V = − 1 N ∑ k , i k B ln [ 1 − exp ( − Θ k , i ( V ) / T ) ] + 1 N T ∑ k , i ℏ ω k , i ( V ) exp ( Θ k , i ( V ) / T ) − 1 {\displaystyle S=-\left({\frac {\partial F}{\partial T}}\right)_{V}=-{\frac {1}{N}}\sum _{\mathbf {k} ,i}k_{B}\ln \left[1-\exp(-\Theta _{\mathbf {k} ,i}(V)/T)\right]+{\frac {1}{NT}}\sum _{\mathbf {k} ,i}{\frac {\hbar \omega _{\mathbf {k} ,i}(V)}{\exp(\Theta _{\mathbf {k} ,i}(V)/T)-1}}} ,
from which F = U - TS is easily verified.
The frequency ω as a function of k is the dispersion relation . Note that for a constant value of V , these equations corresponds to that of the harmonic approximation.
By applying a Legendre transformation , it is possible to obtain the Gibbs free energy G of the system as a function of temperature and pressure.
G ( T , P ) = min V [ E l a t ( V ) + U v i b ( V , T ) − T S ( T , V ) + P V ] {\displaystyle G(T,P)=\min _{V}\left[E_{\rm {lat}}(V)+U_{\rm {vib}}(V,T)-TS(T,V)+PV\right]}
Where P is the pressure. The minimal value for G is found at the equilibrium volume for a given T and P .
Once the Gibbs free energy is known, many thermodynamic quantities can be determined as first- or second-order derivatives. Below are a few which cannot be determined through the harmonic approximation alone.
V ( P , T ) is determined as a function of pressure and temperature by minimizing the Gibbs free energy.
The volumetric thermal expansion α V can be derived from V ( P , T ) as
α V = 1 V ( ∂ V ∂ T ) P {\displaystyle \alpha _{V}={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}}
The Grüneisen parameter γ is defined for every phonon mode as
γ i = − ∂ ln ω i ∂ ln V {\displaystyle \gamma _{i}=-{\frac {\partial \ln \omega _{i}}{\partial \ln V}}}
where i indicates a phonon mode. The total Grüneisen parameter is the sum of all γ i s. It is a measure of the anharmonicity of the system and closely related to the thermal expansion. | https://en.wikipedia.org/wiki/Quasi-harmonic_approximation |
In universal algebra , a quasi-identity is an implication of the form
where s 1 , ..., s n , t 1 , ..., t n , s , and t are terms built up from variables using the operation symbols of the specified signature .
A quasi-identity amounts to a conditional equation for which the conditions themselves are equations. Alternatively, it can be seen as a disjunction of inequations and one equation s 1 ≠ t 1 ∨ ... ∨ s n ≠ t n ∨ s = t —that is, as a definite Horn clause . A quasi-identity with n = 0 is an ordinary identity or equation, so quasi-identities are a generalization of identities.
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasi-identity |
In mathematics , a quasi-invariant measure μ with respect to a transformation T , from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T . An important class of examples occurs when X is a smooth manifold M , T is a diffeomorphism of M , and μ is any measure that locally is a measure with base the Lebesgue measure on Euclidean space . Then the effect of T on μ is locally expressible as multiplication by the Jacobian determinant of the derivative ( pushforward ) of T .
To express this idea more formally in measure theory terms, the idea is that the Radon–Nikodym derivative of the transformed measure μ′ with respect to μ should exist everywhere; or that the two measures should be equivalent (i.e. mutually absolutely continuous ):
That means, in other words, that T preserves the concept of a set of measure zero . Considering the whole equivalence class of measures ν , equivalent to μ , it is also the same to say that T preserves the class as a whole, mapping any such measure to another such. Therefore, the concept of quasi-invariant measure is the same as invariant measure class .
In general, the 'freedom' of moving within a measure class by multiplication gives rise to cocycles , when transformations are composed.
As an example, Gaussian measure on Euclidean space R n is not invariant under translation (like Lebesgue measure is), but is quasi-invariant under all translations.
It can be shown that if E is a separable Banach space and μ is a locally finite Borel measure on E that is quasi-invariant under all translations by elements of E , then either dim( E ) < +∞ or μ is the trivial measure μ ≡ 0. | https://en.wikipedia.org/wiki/Quasi-invariant_measure |
Quasi-linkage equilibrium (QLE) is a mathematical approximation used in solving population genetics problems. Motoo Kimura introduced the notion to simplify a model of Fisher's fundamental theorem . QLE greatly simplifies population genetic equations whilst making the assumption of weak selection and weak epistasis . [ 1 ] Selection under these conditions rapidly changes allele frequencies to a state where they evolve as if in linkage equilibrium . Kimura originally provided the sufficient conditions for QLE in two-locus systems, but recently several researchers have shown how QLE occurs in general multilocus systems. [ 2 ] QLE allows theorists to approximate linkage disequilibria by simple expressions, often simple functions of allele or genotype frequencies, thereby providing solutions to highly complex problems involving selection on multiple loci or polygenic traits. [ 3 ] QLE also plays an important role in justifying approximations in the derivation of quantitative genetic equations from mendelian principles.
Let X {\displaystyle X} , Y {\displaystyle Y} , Z {\displaystyle Z} and U {\displaystyle U} represent the frequencies of the four possible genotypes in a haploid two-locus-two-allele model. Kimura's original model [ 1 ] showed that
R = X U Y Z {\displaystyle R={\frac {XU}{YZ}}}
approaches a stable state R ^ {\displaystyle {\hat {R}}} rapidly if epistatic effects are small relative to recombination. Deviations from R ^ {\displaystyle {\hat {R}}} will be reduced by the recombination fraction every generation.
This evolution -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasi-linkage_equilibrium |
The concept of a quasi-median network is a generalization of the concept of a median network that was introduced to represent multistate characters. Note that, unlike median networks, quasi-median networks are not split networks . A quasi-median network is defined as a phylogenetic network , the node set of which is given by the quasi-median closure of the condensed version of M (let M be a multiple sequence alignment of DNA sequences on X) and in which any two nodes are joined by an edge if and only if the sequences associated with the nodes differ in exactly one position. The quasi-median closure is defined as the set of all sequences that can be obtained by repeatedly taking the quasi-median of any three sequences in the set and then adding the result to the set. [ 1 ] [ 2 ]
This bioinformatics-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasi-median_networks |
Quasi-opportunistic supercomputing is a computational paradigm for supercomputing on a large number of geographically disperse computers . [ 3 ] Quasi-opportunistic supercomputing aims to provide a higher quality of service than opportunistic resource sharing . [ 4 ]
The quasi-opportunistic approach coordinates computers which are often under different ownerships to achieve reliable and fault-tolerant high performance with more control than opportunistic computer grids in which computational resources are used whenever they may become available. [ 3 ]
While the "opportunistic match-making" approach to task scheduling on computer grids is simpler in that it merely matches tasks to whatever resources may be available at a given time, demanding supercomputer applications such as weather simulations or computational fluid dynamics have remained out of reach, partly due to the barriers in reliable sub-assignment of a large number of tasks as well as the reliable availability of resources at a given time. [ 5 ] [ 6 ]
The quasi-opportunistic approach enables the execution of demanding applications within computer grids by establishing grid-wise resource allocation agreements; and fault tolerant message passing to abstractly shield against the failures of the underlying resources, thus maintaining some opportunism, while allowing a higher level of control. [ 3 ]
The general principle of grid computing is to use distributed computing resources from diverse administrative domains to solve a single task, by using resources as they become available. Traditionally, most grid systems have approached the task scheduling challenge by using an "opportunistic match-making" approach in which tasks are matched to whatever resources may be available at a given time. [ 5 ]
BOINC , developed at the University of California, Berkeley is an example of a volunteer-based , opportunistic grid computing system. [ 2 ] The applications based on the BOINC grid have reached multi-petaflop levels by using close to half a million computers connected on the internet, whenever volunteer resources become available. [ 7 ] Another system, Folding@home , which is not based on BOINC, computes protein folding , has reached 8.8 petaflops by using clients that include GPU and PlayStation 3 systems. [ 8 ] [ 9 ] [ 2 ] However, these results are not applicable to the TOP500 ratings because they do not run the general purpose Linpack benchmark.
A key strategy for grid computing is the use of middleware that partitions pieces of a program among the different computers on the network. [ 10 ] Although general grid computing has had success in parallel task execution, demanding supercomputer applications such as weather simulations or computational fluid dynamics have remained out of reach, partly due to the barriers in reliable sub-assignment of a large number of tasks as well as the reliable availability of resources at a given time. [ 2 ] [ 10 ] [ 9 ]
The opportunistic Internet PrimeNet Server supports GIMPS , one of the earliest grid computing projects since 1997, researching Mersenne prime numbers. As of May 2011 [update] , GIMPS's distributed research currently achieves about 60 teraflops as an volunteer-based computing project. [ 11 ] The use of computing resources on " volunteer grids " such as GIMPS is usually purely opportunistic: geographically disperse distributively owned computers are contributing whenever they become available, with no preset commitments that any resources will be available at any given time. Hence, hypothetically, if many of the volunteers unwittingly decide to switch their computers off on a certain day, grid resources will become significantly reduced. [ 12 ] [ 2 ] [ 9 ] Furthermore, users will find it exceedingly costly to organize a very large number of opportunistic computing resources in a manner that can achieve reasonable high performance computing . [ 12 ] [ 13 ]
An example of a more structured grid for high performance computing is DEISA , a supercomputer project organized by the European Community which uses computers in seven European countries. [ 14 ] Although different parts of a program executing within DEISA may be running on computers located in different countries under different ownerships and administrations, there is more control and coordination than with a purely opportunistic approach. DEISA has a two level integration scheme: the "inner level" consists of a number of strongly connected high performance computer clusters that share similar operating systems and scheduling mechanisms and provide a homogeneous computing environment; while the "outer level" consists of heterogeneous systems that have supercomputing capabilities. [ 15 ] Thus DEISA can provide somewhat controlled, yet dispersed high performance computing services to users. [ 15 ] [ 16 ]
The quasi-opportunistic paradigm aims to overcome this by achieving more control over the assignment of tasks to distributed resources and the use of pre-negotiated scenarios for the availability of systems within the network. Quasi-opportunistic distributed execution of demanding parallel computing software in grids focuses on the implementation of grid-wise allocation agreements, co-allocation subsystems, communication topology-aware allocation mechanisms, fault tolerant message passing libraries and data pre-conditioning. [ 17 ] In this approach, fault tolerant message passing is essential to abstractly shield against the failures of the underlying resources. [ 3 ]
The quasi-opportunistic approach goes beyond volunteer computing on a highly distributed systems such as BOINC , or general grid computing on a system such as Globus by allowing the middleware to provide almost seamless access to many computing clusters so that existing programs in languages such as Fortran or C can be distributed among multiple computing resources. [ 3 ]
A key component of the quasi-opportunistic approach, as in the Qoscos Grid , is an economic-based resource allocation model in which resources are provided based on agreements among specific supercomputer administration sites. Unlike volunteer systems that rely on altruism, specific contractual terms are stipulated for the performance of specific types of tasks. However, "tit-for-tat" paradigms in which computations are paid back via future computations is not suitable for supercomputing applications, and is avoided. [ 18 ]
The other key component of the quasi-opportunistic approach is a reliable message passing system to provide distributed checkpoint restart mechanisms when computer hardware or networks inevitably experience failures. [ 18 ] In this way, if some part of a large computation fails, the entire run need not be abandoned, but can restart from the last saved checkpoint. [ 18 ] | https://en.wikipedia.org/wiki/Quasi-opportunistic_supercomputing |
A quasi-peak detector is a type of electronic detector or rectifier . Quasi-peak detectors for specific purposes have usually been standardized with mathematically precisely defined dynamic characteristics of attack time , integration time, and decay time or fall-back time.
Quasi-peak detectors play an important role in electromagnetic compatibility (EMC) testing of electronic equipment, where allowed levels of electromagnetic interference (EMI), also called radio frequency interference (RFI), are given with reference to measurement by a specified quasi-peak detector. This was originally done because the quasi-peak detector was believed to better indicate the subjective annoyance level experienced by a listener hearing impulsive interference to an AM radio station. [ 1 ] Over time standards incorporating quasi-peak detectors as the measurement device were extended to frequencies up to 1 GHz , [ 2 ] although there may not be any justification beyond previous practice for using the quasi-peak detector to measure interference to signals other than AM radio. [ 1 ] The quasi-peak detector parameters to be used for EMC testing vary with frequency. [ 3 ] Both CISPR and the U.S. Federal Communications Commission (FCC) limit EMI at frequencies above 1 GHz with reference to an average-power detector, rather than quasi-peak detector. [ 4 ]
Conceptually, a quasi-peak detector for EMC testing works like a peak detector followed by a lossy integrator. A voltage impulse entering a narrow-band receiver produces a short-duration burst oscillating at the receiver centre frequency. The peak detector is a rectifier followed by a low-pass filter to extract a baseband signal consisting of the slowly (relative to the receiver centre frequency) time-varying amplitude of the impulsive oscillation. The following lossy integrator has a rapid rise time and longer fall time, so the measured output for a sequence of impulses is higher when the pulse repetition rate is higher. The quasi-peak detector is calibrated to produce the same output level as a peak-power detector when the input is a continuous wave tone. [ 5 ]
The CISPR quasi-peak detector is used in EMC testing and is defined in Publication 16 of the International Special Committee on Radio Interference (CISPR) of the International Electrotechnical Commission (IEC). The CISPR quasi-peak detector applied to most conducted emissions measurements (0.15–30 MHz) is a detector with an attack time of 1 ms, a decay time of 160 ms and an IF filter setting of 9 kHz. The quasi-peak detector applied to most radiated emissions measurements (30–1000 MHz) has an attack time of 1 ms, a decay time 550 ms and an IF filter bandwidth of 120 kHz. [ citation needed ]
In audio quality measurement , quasi-peak rectifiers are specified in several standards. For example ITU-R 468 noise weighting uses a special rectifier incorporating two cascaded charging time constants . The PPM or peak programme meter used to measure programme levels is actually a quasi-peak reading meter, again with precisely defined dynamics. Flutter measurement also involves a standardised quasi-peak reading meter. In every case the dynamics are chosen to reflect the sensitivity of human hearing to brief sounds, ignoring those so brief that we do not perceive them, and weighting those of intermediate duration according to audibility. | https://en.wikipedia.org/wiki/Quasi-peak_detector |
In mathematics , a quasi-polynomial ( pseudo-polynomial ) is a generalization of polynomials . While the coefficients of a polynomial come from a ring , the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for various objects.
A quasi-polynomial can be written as q ( k ) = c d ( k ) k d + c d − 1 ( k ) k d − 1 + ⋯ + c 0 ( k ) {\displaystyle q(k)=c_{d}(k)k^{d}+c_{d-1}(k)k^{d-1}+\cdots +c_{0}(k)} , where c i ( k ) {\displaystyle c_{i}(k)} is a periodic function with integral period. If c d ( k ) {\displaystyle c_{d}(k)} is not identically zero, then the degree of q {\displaystyle q} is d {\displaystyle d} . Equivalently, a function f : N → N {\displaystyle f\colon \mathbb {N} \to \mathbb {N} } is a quasi-polynomial if there exist polynomials p 0 , … , p s − 1 {\displaystyle p_{0},\dots ,p_{s-1}} such that f ( n ) = p i ( n ) {\displaystyle f(n)=p_{i}(n)} when i ≡ n mod s {\displaystyle i\equiv n{\bmod {s}}} . The polynomials p i {\displaystyle p_{i}} are called the constituents of f {\displaystyle f} .
This combinatorics -related article is a stub . You can help Wikipedia by expanding it .
This polynomial -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasi-polynomial |
In topology , a branch of mathematics, the quasi-relative interior of a subset of a vector space is a refinement of the concept of the interior . Formally, if X {\displaystyle X} is a linear space then the quasi-relative interior of A ⊆ X {\displaystyle A\subseteq X} is qri ( A ) := { x ∈ A : c o n e ¯ ( A − x ) is a linear subspace } {\displaystyle \operatorname {qri} (A):=\left\{x\in A:\operatorname {\overline {cone}} (A-x){\text{ is a linear subspace}}\right\}} where c o n e ¯ ( ⋅ ) {\displaystyle \operatorname {\overline {cone}} (\cdot )} denotes the closure of the conic hull . [ 1 ]
Let X {\displaystyle X} be a normed vector space. If C ⊆ X {\displaystyle C\subseteq X} is a convex finite-dimensional set then qri ( C ) = ri ( C ) {\displaystyle \operatorname {qri} (C)=\operatorname {ri} (C)} such that ri {\displaystyle \operatorname {ri} } is the relative interior . [ 2 ]
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasi-relative_interior |
Quasi-solid , false-solid , or partial-solid are terms for a substance which is not clearly a solid or a liquid . While similar to solids in some respects, such as having the ability to support their own weight and hold their shapes, a quasi-solid also shares some properties of liquids, such as conforming in shape to something applying pressure to it and the ability to flow under pressure. [ citation needed ] The words quasi-solid, partial-solid, and partial-liquid are used interchangeably. The term "semi-solid" is sometimes used interchangeably with these terms but is not a correct term, as "semi" means two equal halves.
Quasi-solids and partial-solids are sometimes described as amorphous because at the microscopic scale they have a disordered structure unlike crystalline solids. They should not be confused with amorphous solids as they are not solids and exhibit properties such as flow which solids do not.
This physical chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasi-solid |
In mathematics , a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by Pfluger (1961) and Tienari (1962) , in the older literature (in German) they were referred to as quasiconformal curves , a terminology which also applied to arcs . [ 1 ] [ 2 ] In complex analysis and geometric function theory , quasicircles play a fundamental role in the description of the universal Teichmüller space , through quasisymmetric homeomorphisms of the circle. Quasicircles also play an important role in complex dynamical systems .
A quasicircle is defined as the image of a circle under a quasiconformal mapping of the extended complex plane . It is called a K -quasicircle if the quasiconformal mapping has dilatation K . The definition of quasicircle generalizes the characterization of a Jordan curve as the image of a circle under a homeomorphism of the plane. In particular a quasicircle is a Jordan curve. The interior of a quasicircle is called a quasidisk . [ 3 ]
As shown in Lehto & Virtanen (1973) , where the older term "quasiconformal curve" is used, if a Jordan curve is the image of a circle under a quasiconformal map in a neighbourhood of the curve, then it is also the image of a circle under a quasiconformal mapping of the extended plane and thus a quasicircle. The same is true for "quasiconformal arcs" which can be defined as quasiconformal images of a circular arc either in an open set or equivalently in the extended plane. [ 4 ]
Ahlfors (1963) gave a geometric characterization of quasicircles as those Jordan curves for which the absolute value of the cross-ratio of any four points, taken in cyclic order, is bounded below by a positive constant.
Ahlfors also proved that quasicircles can be characterized in terms of a reverse triangle inequality for three points: there should be a constant C such that if two points z 1 and z 2 are chosen on the curve and z 3 lies on the shorter of the resulting arcs, then [ 5 ]
This property is also called bounded turning [ 6 ] or the arc condition . [ 7 ]
For Jordan curves in the extended plane passing through ∞, Ahlfors (1966) gave a simpler necessary and sufficient condition to be a quasicircle. [ 8 ] [ 9 ] There is a constant C > 0 such that if z 1 , z 2 are any points on the curve and z 3 lies on the segment between them, then
These metric characterizations imply that an arc or closed curve is quasiconformal whenever it arises as the image of an interval or the circle under a bi-Lipschitz map f , i.e. satisfying
for positive constants C i . [ 10 ]
If φ is a quasisymmetric homeomorphism of the circle, then there are conformal maps f of [ z | < 1 and g of | z |>1 into disjoint regions such that the complement of the images of f and g is a Jordan curve. The maps f and g extend continuously to the circle | z | = 1 and the sewing equation
holds. The image of the circle is a quasicircle.
Conversely, using the Riemann mapping theorem , the conformal maps f and g uniformizing the outside of a quasicircle give rise to a quasisymmetric homeomorphism through the above equation.
The quotient space of the group of quasisymmetric homeomorphisms by the subgroup of Möbius transformations provides a model of universal Teichmüller space . The above correspondence shows that the space of quasicircles can also be taken as a model. [ 11 ]
A quasiconformal reflection in a Jordan curve is an orientation-reversing quasiconformal map of period 2 which switches the inside and the outside of the curve fixing points on the curve. Since the map
provides such a reflection for the unit circle, any quasicircle admits a quasiconformal reflection. Ahlfors (1963) proved that this property characterizes quasicircles.
Ahlfors noted that this result can be applied to uniformly bounded holomorphic univalent functions f ( z ) on the unit disk D . Let Ω = f ( D ). As Carathéodory had proved using his theory of prime ends , f extends continuously to the unit circle if and only if ∂Ω is locally connected, i.e. admits a covering by finitely many compact connected sets of arbitrarily small diameter. The extension to the circle is 1-1 if and only if ∂Ω has no cut points, i.e. points which when removed from ∂Ω yield a disconnected set. Carathéodory's theorem shows that a locally set without cut points is just a Jordan curve and that in precisely this case is the extension of f to the closed unit disk a homeomorphism. [ 12 ] If f extends to a quasiconformal mapping of the extended complex plane then ∂Ω is by definition a quasicircle. Conversely Ahlfors (1963) observed that if ∂Ω is a quasicircle and R 1 denotes the quasiconformal reflection in ∂Ω then the assignment
for | z | > 1 defines a quasiconformal extension of f to the extended complex plane.
Quasicircles were known to arise as the Julia sets of rational maps R ( z ). Sullivan (1985) proved that if the Fatou set of R has two components and the action of R on the Julia set is "hyperbolic", i.e. there are constants c > 0 and A > 1 such that
on the Julia set, then the Julia set is a quasicircle. [ 5 ]
There are many examples: [ 13 ] [ 14 ]
Quasi-Fuchsian groups are obtained as quasiconformal deformations of Fuchsian groups . By definition their limit sets are quasicircles. [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ]
Let Γ be a Fuchsian group of the first kind: a discrete subgroup of the Möbius group preserving the unit circle. acting properly discontinuously on the unit disk D and with limit set the unit circle.
Let μ( z ) be a measurable function on D with
such that μ is Γ-invariant, i.e.
for every g in Γ. (μ is thus a "Beltrami differential" on the Riemann surface D / Γ.)
Extend μ to a function on C by setting μ( z ) = 0 off D .
The Beltrami equation
admits a solution unique up to composition with a Möbius transformation.
It is a quasiconformal homeomorphism of the extended complex plane.
If g is an element of Γ, then f ( g ( z )) gives another solution of the Beltrami equation, so that
is a Möbius transformation.
The group α(Γ) is a quasi-Fuchsian group with limit set the quasicircle given by the image of the unit circle under f .
It is known that there are quasicircles for which no segment has finite length. [ 21 ] The Hausdorff dimension of quasicircles was first investigated by Gehring & Väisälä (1973) , who proved that it can take all values in the interval [1,2). [ 22 ] Astala (1993) , using the new technique of "holomorphic motions" was able to estimate the change in the Hausdorff dimension of any planar set under a quasiconformal map with dilatation K . For quasicircles C , there was a crude estimate for the Hausdorff dimension [ 23 ]
where
On the other hand, the Hausdorff dimension for the Julia sets J c of the iterates of the rational maps
had been estimated as result of the work of Rufus Bowen and David Ruelle , who showed that
Since these are quasicircles corresponding to a dilatation
where
this led Becker & Pommerenke (1987) to show that for k small
Having improved the lower bound following calculations for the Koch snowflake with Steffen Rohde and Oded Schramm , Astala (1994) conjectured that
This conjecture was proved by Smirnov (2010) ; a complete account of his proof, prior to publication, was already given in Astala, Iwaniec & Martin (2009) .
For a quasi-Fuchsian group Bowen (1979) and Sullivan (1982) showed that the Hausdorff dimension d of the limit set is always greater than 1. When d < 2, the quantity
is the lowest eigenvalue of the Laplacian of the corresponding hyperbolic 3-manifold . [ 24 ] [ 25 ] | https://en.wikipedia.org/wiki/Quasicircle |
A quasiperiodic crystal , or quasicrystal , is a structure that is ordered but not periodic . A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry . [ 2 ] While crystals, according to the classical crystallographic restriction theorem , can possess only two-, three-, four-, and six-fold rotational symmetries , the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders—for instance, five-fold. [ 3 ]
Aperiodic tilings were discovered by mathematicians in the early 1960s, and some twenty years later, they were found to apply to the study of natural quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the field of crystallography . In crystallography, the quasicrystals were predicted in 1981 by a five-fold symmetry study of Alan Lindsay Mackay , [ 4 ] —that also brought in 1982, with the crystallographic Fourier transform of a Penrose tiling , [ 5 ] the possibility of identifying quasiperiodic order in a material through diffraction.
Quasicrystals had been investigated and observed earlier, [ 6 ] but, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter. In 2009, after a dedicated search, a mineralogical finding, icosahedrite , offered evidence for the existence of natural quasicrystals. [ 7 ]
Roughly, an ordering is non-periodic if it lacks translational symmetry , which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions. Symmetrical diffraction patterns result from the existence of an indefinitely large number of elements with regular spacing, a property loosely described as long-range order . Experimentally, the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than two, three, four, or six.
In 1982, materials scientist Dan Shechtman observed that certain aluminium – manganese alloys produced unusual diffractograms, which today are seen as revelatory of quasicrystal structures. Due to fear of the scientific community's reaction, it took him two years to publish the results. [ 8 ] [ 9 ] Shechtman's discovery challenged the long-held belief that all crystals are periodic. Observed in a rapidly solidified Al-Mn alloy, quasicrystals exhibited icosahedral symmetry , which was previously thought impossible in crystallography. [ 10 ] This breakthrough, supported by theoretical models and experimental evidence, led to a paradigm shift in the understanding of solid-state matter. Despite initial skepticism, the discovery gained widespread acceptance, prompting the International Union of Crystallography to redefine the term " crystal ." [ 11 ] The work ultimately earned Shechtman the 2011 Nobel Prize in Chemistry [ 12 ] and inspired significant advancements in materials science and mathematics.
On 25 October 2018, Luca Bindi and Paul Steinhardt were awarded the Aspen Institute 2018 Prize for collaboration and scientific research between Italy and the United States after discovering icosahedrite , the first quasicrystal known to occur naturally.
The first representations of perfect quasicrystalline patterns can be found in several early Islamic works of art and architecture such as the Gunbad-i-Kabud tomb tower, the Darb-e Imam shrine and the Al-Attarine Madrasa . [ 14 ] [ 15 ] On July 16, 1945, in Alamogordo, New Mexico, the Trinity nuclear bomb test produced icosahedral quasicrystals. They went unnoticed at the time of the test but were later identified in samples of red trinitite , a glass-like substance formed from fused sand and copper transmission lines. Identified in 2021, they are the oldest known anthropogenic quasicrystals. [ 16 ] [ 17 ]
In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student Robert Berger constructed a set of some 20,000 square tiles (now called Wang tiles ) that can tile the plane but not in a periodic fashion. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1974 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles , that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later Alan Mackay showed theoretically that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp ' delta ' peaks arranged in a fivefold symmetric pattern. [ 18 ] Around the same time, Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry.
In 1972, R. M. de Wolf and W. van Aalst [ 19 ] reported that the diffraction pattern produced by a crystal of sodium carbonate cannot be labeled with three indices but needed one more, which implied that the underlying structure had four dimensions in reciprocal space . Other puzzling cases have been reported, [ 20 ] but until the concept of quasicrystal came to be established, they were explained away or denied. [ 21 ] [ 22 ]
Dan Shechtman first observed ten-fold electron diffraction patterns in 1982, while conducting a routine study of an aluminium – manganese alloy, Al 6 Mn, at the US National Bureau of Standards (later NIST). [ 23 ] Shechtman related his observation to Ilan Blech, who responded that such diffractions had been seen before. [ 24 ] [ 25 ] [ 26 ] Around that time, Shechtman also related his finding to John W. Cahn of the NIST, who did not offer any explanation and challenged him to solve the observation. Shechtman quoted Cahn as saying: "Danny, this material is telling us something, and I challenge you to find out what it is". [ 27 ]
The observation of the ten-fold diffraction pattern lay unexplained for two years until the spring of 1984, when Blech asked Shechtman to show him his results again. A quick study of Shechtman's results showed that the common explanation for a ten-fold symmetrical diffraction pattern, a type of crystal twinning , was ruled out by his experiments. Therefore, Blech looked for a new structure containing cells connected to each other by defined angles and distances but without translational periodicity. He decided to use a computer simulation to calculate the diffraction intensity from a cluster of such a material, which he termed as "multiple polyhedral ", and found a ten-fold structure similar to what was observed. The multiple polyhedral structure was termed later by many researchers as icosahedral glass. [ 28 ]
Shechtman accepted Blech's discovery of a new type of material and chose to publish his observation in a paper entitled "The Microstructure of Rapidly Solidified Al 6 Mn", which was written around June 1984 and published in a 1985 edition of Metallurgical Transactions A . [ 29 ] Meanwhile, on seeing the draft of the paper, John Cahn suggested that Shechtman's experimental results merit a fast publication in a more appropriate scientific journal. Shechtman agreed and, in hindsight, called this fast publication "a winning move". This paper, published in the Physical Review Letters , [ 9 ] repeated Shechtman's observation and used the same illustrations as the original paper.
Originally, the new form of matter was dubbed "Shechtmanite". [ 30 ] The term "quasicrystal" was first used in print by Paul Steinhardt and Dov Levine [ 2 ] shortly after Shechtman's paper was published.
Also in 1985, T. Ishimasa et al. reported twelvefold symmetry in Ni-Cr particles. [ 31 ] Soon, eightfold diffraction patterns were recorded in V-Ni-Si and Cr-Ni-Si alloys. [ 32 ] Over the years, hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were thermodynamically unstable: when heated, they formed regular crystals. However, in 1987, the first of many stable quasicrystals were discovered, making it possible to produce large samples for study and applications. [ 33 ]
In 1992, the International Union of Crystallography altered its definition of a crystal, reducing it to the ability to produce a clear-cut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic. [ 8 ] [ 34 ]
In 2001, Steinhardt hypothesized that quasicrystals could exist in nature and developed a method of recognition, inviting all the mineralogical collections of the world to identify any badly cataloged crystals. In 2007 Steinhardt received a reply by Luca Bindi , who found a quasicrystalline specimen from Khatyrka in the University of Florence Mineralogical Collection. The crystal samples were sent to Princeton University for other tests, and in late 2009, Steinhardt confirmed its quasicrystalline character. This quasicrystal, with a composition of Al 63 Cu 24 Fe 13 , was named icosahedrite and it was approved by the International Mineralogical Association in 2010. Analysis indicates it may be meteoritic in origin, possibly delivered from a carbonaceous chondrite asteroid. In 2011, Bindi, Steinhardt, and a team of specialists found more icosahedrite samples from Khatyrka. [ 36 ] A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al 71 Ni 24 Fe 5 . This quasicrystal is stable in a narrow temperature range, from 1120 to 1200 K at ambient pressure, which suggests that natural quasicrystals are formed by rapid quenching of a meteorite heated during an impact-induced shock. [ 35 ]
Shechtman was awarded the Nobel Prize in Chemistry in 2011 for his work on quasicrystals. "His discovery of quasicrystals revealed a new principle for packing of atoms and molecules," stated the Nobel Committee and pointed that "this led to a paradigm shift within chemistry." [ 8 ] [ 37 ] In 2014, Post of Israel issued a stamp dedicated to quasicrystals and the 2011 Nobel Prize. [ 38 ]
While the first quasicrystals discovered were made out of intermetallic components, later on quasicrystals were also discovered in soft-matter and molecular systems. Soft quasicrystal structures have been found in supramolecular dendrimer liquids [ 39 ] and ABC Star Polymers [ 40 ] in 2004 and 2007. In 2009, it was found that thin-film quasicrystals can be formed by self-assembly of uniformly shaped, nano-sized molecular units at an air-liquid interface. [ 41 ] It was demonstrated that these units can be both inorganic and organic. [ 42 ] Additionally in the 2010s, two-dimensional molecular quasicrystals were discovered, driven by intermolecular interactions [ 43 ] and interface-interactions. [ 44 ]
In 2018, chemists from Brown University announced the successful creation of a self-constructing lattice structure based on a strangely shaped quantum dot. While single-component quasicrystal lattices have been previously predicted mathematically and in computer simulations, [ 45 ] they had not been demonstrated prior to this. [ 46 ]
There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr (mathematician brother of Niels Bohr ). The concept of an almost periodic function (also called a quasiperiodic function) was studied by Bohr, including work of Bohl and Escanglon. [ 47 ] He introduced the notion of a superspace. Bohr showed that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice (an intersection with one or more hyperplanes ), and discussed their Fourier point spectrum. These functions are not exactly periodic, but they are arbitrarily close in some sense, as well as being a projection of an exactly periodic function.
In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures; [ 48 ] similarly, icosahedral quasicrystals in three dimensions are projected from a six-dimensional hypercubic lattice, as first described by Peter Kramer and Roberto Neri in 1984. [ 49 ] Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors , which are the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice. [ 50 ]
Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group . Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasilattices must be used. Instead of groups, groupoids , the mathematical generalization of groups in category theory , is the appropriate tool for studying quasicrystals. [ 51 ]
Using mathematics for construction and analysis of quasicrystal structures is a difficult task. Computer modeling, based on the existing theories of quasicrystals, however, greatly facilitated this task. Advanced programs have been developed [ 52 ] allowing one to construct, visualize and analyze quasicrystal structures and their diffraction patterns. The aperiodic nature of quasicrystals can also make theoretical studies of physical properties, such as electronic structure, difficult due to the inapplicability of Bloch's theorem . However, spectra of quasicrystals can still be computed with error control. [ 53 ]
Study of quasicrystals may shed light on the most basic notions related to the quantum critical point observed in heavy fermion metals. Experimental measurements on an Au –Al– Yb quasicrystal have revealed a quantum critical point defining the divergence of the magnetic susceptibility as temperature tends to zero. [ 54 ] It is suggested that the electronic system of some quasicrystals is located at a quantum critical point without tuning, while quasicrystals exhibit the typical scaling behaviour of their thermodynamic properties and belong to the well-known family of heavy fermion metals.
Since the original discovery by Dan Shechtman , hundreds of quasicrystals have been reported and confirmed. Quasicrystals are found most often in aluminium alloys (Al–Li–Cu, Al–Mn–Si, Al–Ni–Co, Al–Pd–Mn, Al–Cu–Fe, Al–Cu–V, etc.), but numerous other compositions are also known (Cd–Yb, Ti–Zr–Ni, Zn–Mg–Ho, Zn–Mg–Sc, In–Ag–Yb, Pd–U–Si, etc.). [ 55 ]
Two types of quasicrystals are known. [ 52 ] The first type, polygonal (dihedral) quasicrystals, have an axis of 8-, 10-, or 12-fold local symmetry (octagonal, decagonal, or dodecagonal quasicrystals, respectively). They are periodic along this axis and quasiperiodic in planes normal to it. The second type, icosahedral quasicrystals, are aperiodic in all directions. Icosahedral quasicrystals have a three dimensional quasiperiodic structure and possess fifteen 2-fold, ten 3-fold and six 5-fold axes in accordance with their icosahedral symmetry. [ 56 ]
Quasicrystals fall into three groups of different thermal stability: [ 57 ]
Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by X-ray and electron diffraction revealing peak widths as sharp as those of perfect crystals such as Si. Diffraction patterns exhibit fivefold, threefold, and twofold symmetries, and reflections are arranged quasiperiodically in three dimensions.
The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystal-forming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails in the undercooled liquid state for the metastable quasicrystals.
A nanoscale icosahedral phase was formed in Zr-, Cu- and Hf-based bulk metallic glasses alloyed with noble metals. [ 58 ]
Most quasicrystals have ceramic-like properties including high thermal and electrical resistance, hardness and brittleness, resistance to corrosion, and non-stick
properties. [ 59 ] Many metallic quasicrystalline substances are impractical for most applications due to their thermal instability ; the Al–Cu–Fe ternary system and the Al–Cu–Fe–Cr and Al–Co–Fe–Cr quaternary systems, thermally stable up to 700 °C, are notable exceptions.
The quasi-ordered droplet crystals could be formed under Dipolar forces in the Bose Einstein condensate. [ 60 ] While the softcore Rydberg dressing interaction has forms triangular droplet-crystals, [ 61 ] adding a Gaussian peak to the plateau type interaction would form multiple roton unstable points in the Bogoliubov spectrum. Therefore, the excitation around the roton instabilities would grow exponentially and form multiple allowed lattice constants leading to quasi-ordered periodic droplet crystals. [ 60 ]
Quasicrystalline substances have potential applications in several forms.
Metallic quasicrystalline coatings can be applied by thermal spraying or magnetron sputtering . A problem that must be resolved is the tendency for cracking due to the materials' extreme brittleness. [ 59 ] The cracking could be suppressed by reducing sample dimensions or coating thickness. [ 62 ] Recent studies show typically brittle quasicrystals can exhibit remarkable ductility of over 50% strains at room temperature and sub-micrometer scales (<500 nm). [ 62 ]
An application was the use of low-friction Al–Cu–Fe–Cr quasicrystals [ 63 ] as a coating for frying pans . Food did not stick to it as much as to stainless steel making the pan moderately non-stick and easy to clean; heat transfer and durability were better than PTFE non-stick cookware and the pan was free from perfluorooctanoic acid (PFOA); the surface was very hard, claimed to be ten times harder than stainless steel, and not harmed by metal utensils or cleaning in a dishwasher ; and the pan could withstand temperatures of 1,000 °C (1,800 °F) without harm. However, after an initial introduction the pans were a chrome steel, probably because of the difficulty of controlling thin films of the quasicrystal. [ 64 ]
The Nobel citation said that quasicrystals, while brittle, could reinforce steel "like armor". When Shechtman was asked about potential applications of quasicrystals he said that a precipitation-hardened stainless steel is produced that is strengthened by small quasicrystalline particles. It does not corrode and is extremely strong, suitable for razor blades and surgery instruments. The small quasicrystalline particles impede the motion of dislocation in the material. [ 65 ]
Quasicrystals were also being used to develop heat insulation, LEDs , diesel engines, and new materials that convert heat to electricity. Shechtman suggested new applications taking advantage of the low coefficient of friction and the hardness of some quasicrystalline materials, for example embedding particles in plastic to make strong, hard-wearing, low-friction plastic gears. The low heat conductivity of some quasicrystals makes them good for heat insulating coatings. [ 65 ] One of the special properties of quasicrystals is their smooth surface, which despite the irregular atomic
structure, the surface of quasicrystals can be smooth and flat. [ 66 ]
Other potential applications include selective solar absorbers for power conversion, broad-wavelength reflectors, and bone repair and prostheses applications where biocompatibility, low friction and corrosion resistance are required. Magnetron sputtering can be readily applied to other stable quasicrystalline alloys such as Al–Pd–Mn. [ 59 ]
Applications in macroscopic engineering have been suggested, building quasi-crystal-like large scale engineering structures, which could have interesting physical properties. Also, aperiodic tiling lattice structures may be used instead of isogrid or honeycomb patterns . None of these seem to have been put to use in practice. [ 67 ] | https://en.wikipedia.org/wiki/Quasicrystal |
In physics, quasielastic scattering designates a limiting case of inelastic scattering , characterized by energy transfers being small compared to the incident energy of the scattered particles.
The term was originally coined in nuclear physics. [ 1 ]
It was applied to thermal neutron scattering by Leon van Hove [ 2 ] and Pierre Gilles de Gennes [ 3 ] ( quasielastic neutron scattering , QENS).
Finally, it is sometimes used for dynamic light scattering (also known by the more expressive term photon correlation spectroscopy). | https://en.wikipedia.org/wiki/Quasielastic_scattering |
In mathematics, quasilinearization is a technique which replaces a nonlinear differential equation or operator equation (or system of such equations) with a sequence of linear problems, which are presumed to be easier, and whose solutions approximate the solution of the original nonlinear problem with increasing accuracy. It is a generalization of Newton's method ; the word "quasilinearization" is commonly used when the differential equation is a boundary value problem . [ 1 ] [ 2 ]
Quasilinearization replaces a given nonlinear operator N with a certain linear operator which, being simpler, can be used in an iterative fashion to approximately solve equations containing the original nonlinear operator. This is typically performed when trying to solve an equation such as N(y) = 0 together with certain boundary conditions B for which the equation has a solution y . This solution is sometimes called the "reference solution". For quasilinearization to work, the reference solution needs to exist uniquely (at least locally). The process starts with an initial approximation y 0 that satisfies the boundary conditions and is "sufficiently close" to the reference solution y in a sense to be defined more precisely later. The first step is to take the Fréchet derivative of the nonlinear operator N at that initial approximation, in order to find the linear operator L(y 0 ) which best approximates N(y)-N(y 0 ) locally. The nonlinear equation may then be approximated as N ( y ) = N(y k ) + L(y k )( y - y k ) + O( y-y k ) 2 , taking k=0 . Setting this equation to zero and imposing zero boundary conditions and ignoring higher-order terms gives the linear equation L(y k )( y - y k ) = - N(y k ) . The solution of this linear equation (with zero boundary conditions) might be called y k+1 . Computation of y k for k =1, 2, 3, ... by solving these linear equations in sequence is analogous to Newton's iteration for a single equation, and requires recomputation of the Fréchet derivative at each y k . The process can converge quadratically to the reference solution, under the right conditions. Just as with Newton's method for nonlinear algebraic equations, however, difficulties may arise: for instance, the original nonlinear equation may have no solution, or more than one solution, or a multiple solution, in which cases the iteration may converge only very slowly, may not converge at all, or may converge instead to the wrong solution.
The practical test of the meaning of the phrase "sufficiently close" earlier is precisely that the iteration converges to the correct solution. Just as in the case of Newton iteration, there are theorems stating conditions under which one can know ahead of time when the initial approximation is "sufficiently close".
One could instead discretize the original nonlinear operator and generate a (typically large) set of nonlinear algebraic equations for the unknowns, and then use Newton's method proper on this system of equations. Generally speaking, the convergence behavior is similar: a similarly good initial approximation will produce similarly good approximate discrete solutions. However, the quasilinearization approach (linearizing the operator equation instead of the discretized equations) seems to be simpler to think about, and has allowed such techniques as adaptive spatial meshes to be used as the iteration proceeds. [ 3 ]
As an example to illustrate the process of quasilinearization, we can approximately solve the two-point boundary value problem for the nonlinear node d 2 d x 2 y ( x ) = y 2 ( x ) , {\displaystyle {\frac {d^{2}}{dx^{2}}}y(x)=y^{2}(x),} where the boundary conditions are y ( − 1 ) = 1 {\displaystyle y(-1)=1} and y ( 1 ) = 1 {\displaystyle y(1)=1} . The exact solution of the differential equation can be expressed using the Weierstrass elliptic function ℘, like so: y ( x ) = 6 ℘ ( x − α | 0 , β ) {\displaystyle y(x)=6\wp (x-\alpha |0,\beta )} where the vertical bar notation means that the invariants are g 2 = 0 {\displaystyle g_{2}=0} and g 3 = β {\displaystyle g_{3}=\beta } . Finding the values of α {\displaystyle \alpha } and β {\displaystyle \beta } so that the boundary conditions are satisfied requires solving two simultaneous nonlinear equations for the two unknowns α {\displaystyle \alpha } and β {\displaystyle \beta } , namely 6 ℘ ( − 1 − α | 0 , β ) = 1 {\displaystyle 6\wp (-1-\alpha |0,\beta )=1} and 6 ℘ ( 1 − α | 0 , β ) = 1 {\displaystyle 6\wp (1-\alpha |0,\beta )=1} . This can be done, in an environment where ℘ and its derivatives are available, for instance by Newton's method. [ a ]
Applying the technique of quasilinearization instead, one finds by taking the Fréchet derivative at an unknown approximation y k ( x ) {\displaystyle y_{k}(x)} that the linear operator is L ( ε ) = d 2 d x 2 ε ( x ) − 2 y k ( x ) ε ( x ) . {\displaystyle L(\varepsilon )={\frac {d^{2}}{dx^{2}}}\varepsilon (x)-2y_{k}(x)\varepsilon (x).} If the initial approximation is y 0 ( x ) = 1 {\displaystyle y_{0}(x)=1} identically on the interval − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} , then the first iteration (at least) can be solved exactly, but is already somewhat complicated. A numerical solution instead, for instance by a Chebyshev spectral method using n = 21 {\displaystyle n=21} Chebyshev—Lobatto points x k = cos ( π ( n − 1 − k ) / ( n − 1 ) ) {\displaystyle x_{k}=\cos(\pi (n-1-k)/(n-1))} for k = 0 , 1 , ⋯ , n − 1 {\displaystyle k=0,1,\cdots ,n-1} gives a solution with residual less than 5 ⋅ 10 − 9 {\displaystyle 5\cdot 10^{-9}} after three iterations; that is, y 3 ( x ) {\displaystyle y_{3}(x)} is the exact solution to d 2 d x 2 y ( x ) − y 2 ( x ) = 5 ⋅ 10 − 9 v ( x ) {\textstyle {\frac {d^{2}}{dx^{2}}}y(x)-y^{2}(x)=5\cdot 10^{-9}v(x)} , where the maximum value of | v ( x ) | {\displaystyle |v(x)|} is less than 1 on the interval − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} . This approximate solution (call it u 1 {\displaystyle u_{1}} ) agrees with the exact solution 6 ⋅ ℘ ( x − α | 0 , β ) {\displaystyle 6\cdot \wp (x-\alpha |0,\beta )} with { α ≈ 3.524459420 , β ≈ 0.006691372637 } . {\displaystyle \{\alpha \approx 3.524459420,\beta \approx 0.006691372637\}.}
Other values of α {\displaystyle \alpha } and β {\displaystyle \beta } give other continuous solutions to this nonlinear two-point boundary-value problem for ODE, such as { α ≈ 2.55347391110 , β ≈ − 1.24923895273 } . {\displaystyle \{\alpha \approx 2.55347391110,\beta \approx -1.24923895273\}.} The solution corresponding to these values plotted in the figure is called u 2 {\displaystyle u_{2}} . Yet other values of the parameters can give discontinuous solutions because ℘ has a double pole at zero and so y ( x ) {\displaystyle y(x)} has a double pole at x = α {\displaystyle x=\alpha } . Finding other continuous solutions by quasilinearization requires different initial approximations to the ones used here. The initial approximation y 0 = 5 x 2 − 4 {\displaystyle y_{0}=5x^{2}-4} approximates the exact solution u 2 {\displaystyle u_{2}} and can be used to generate a sequence of approximations converging to u 2 {\displaystyle u_{2}} . Both approximations are plotted in the accompanying figure. | https://en.wikipedia.org/wiki/Quasilinearization |
In linear algebra , functional analysis and related areas of mathematics , a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by ‖ x + y ‖ ≤ K ( ‖ x ‖ + ‖ y ‖ ) {\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)} for some K > 1. {\displaystyle K>1.}
A quasi-seminorm [ 1 ] on a vector space X {\displaystyle X} is a real-valued map p {\displaystyle p} on X {\displaystyle X} that satisfies the following conditions:
A quasinorm [ 1 ] is a quasi-seminorm that also satisfies:
A pair ( X , p ) {\displaystyle (X,p)} consisting of a vector space X {\displaystyle X} and an associated quasi-seminorm p {\displaystyle p} is called a quasi-seminormed vector space .
If the quasi-seminorm is a quasinorm then it is also called a quasinormed vector space .
Multiplier
The infimum of all values of k {\displaystyle k} that satisfy condition (3) is called the multiplier of p . {\displaystyle p.} The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition.
The term k {\displaystyle k} -quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to k . {\displaystyle k.}
A norm (respectively, a seminorm ) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is 1. {\displaystyle 1.} Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).
If p {\displaystyle p} is a quasinorm on X {\displaystyle X} then p {\displaystyle p} induces a vector topology on X {\displaystyle X} whose neighborhood basis at the origin is given by the sets: [ 2 ] { x ∈ X : p ( x ) < 1 / n } {\displaystyle \{x\in X:p(x)<1/n\}} as n {\displaystyle n} ranges over the positive integers.
A topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space .
Every quasinormed topological vector space is pseudometrizable .
A complete quasinormed space is called a quasi-Banach space . Every Banach space is a quasi-Banach space, although not conversely.
A quasinormed space ( A , ‖ ⋅ ‖ ) {\displaystyle (A,\|\,\cdot \,\|)} is called a quasinormed algebra if the vector space A {\displaystyle A} is an algebra and there is a constant K > 0 {\displaystyle K>0} such that ‖ x y ‖ ≤ K ‖ x ‖ ⋅ ‖ y ‖ {\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|} for all x , y ∈ A . {\displaystyle x,y\in A.}
A complete quasinormed algebra is called a quasi-Banach algebra .
A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin. [ 2 ]
Since every norm is a quasinorm, every normed space is also a quasinormed space.
L p {\displaystyle L^{p}} spaces with 0 < p < 1 {\displaystyle 0<p<1}
The L p {\displaystyle L^{p}} spaces for 0 < p < 1 {\displaystyle 0<p<1} are quasinormed spaces (indeed, they are even F-spaces ) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology).
For 0 < p < 1 , {\displaystyle 0<p<1,} the Lebesgue space L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} is a complete metrizable TVS (an F-space ) that is not locally convex (in fact, its only convex open subsets are itself L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} and the empty set) and the only continuous linear functional on L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} is the constant 0 {\displaystyle 0} function ( Rudin 1991 , §1.47).
In particular, the Hahn-Banach theorem does not hold for L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} when 0 < p < 1. {\displaystyle 0<p<1.} | https://en.wikipedia.org/wiki/Quasinorm |
Quasinormal modes ( QNM ) are the modes of energy dissipation of a perturbed object or field, i.e. they describe perturbations of a field that decay in time.
A familiar example is the perturbation (gentle tap) of a wine glass with a knife: the glass begins to ring, it rings with a set, or superposition, of its natural frequencies — its modes of sonic energy dissipation. One could call these modes normal if the glass went on ringing forever. Here the amplitude of oscillation decays in time, so we call its modes quasi-normal . To a high degree of accuracy, quasinormal ringing can be approximated by
where ψ ( t ) {\displaystyle \psi \left(t\right)} is the amplitude of oscillation, ω ′ {\displaystyle \omega ^{\prime }} is the frequency, and ω ′ ′ {\displaystyle \omega ^{\prime \prime }} is the decay rate. The quasinormal
frequency is described by two numbers,
or, more compactly
Here, ω {\displaystyle \mathbf {\omega } } is what is commonly referred to as the quasinormal mode frequency . It is a complex number with two pieces of information: real part is the temporal oscillation; imaginary part is the temporal, exponential decay .
In certain cases the amplitude of the wave decays quickly, to follow the decay for a longer time one may plot log | ψ ( t ) | {\displaystyle \log \left|\psi (t)\right|}
In theoretical physics , a quasinormal mode is a formal solution of linearized differential equations (such as the linearized equations of general relativity constraining perturbations around a black hole solution) with a complex eigenvalue ( frequency ). [ 1 ] [ 2 ]
Black holes have many quasinormal modes (or ringing modes) that describe the exponential decrease of asymmetry of the black hole in time as it evolves towards the perfect spherical shape.
Recently, the properties of quasinormal modes have been tested in the context of the AdS/CFT correspondence . Also, the asymptotic behavior of quasinormal modes was proposed to be related to the Immirzi parameter in loop quantum gravity , but convincing arguments have not been found yet.
There are essentially two types of resonators in optics. In the first type, a high- Q factor optical microcavity is achieved with lossless dielectric optical materials, with mode volumes of the order of a cubic wavelength, essentially limited by the diffraction limit. Famous examples of high-Q microcavities are micropillar cavities, microtoroid resonators, photonic-crystal cavities. In the second type of resonators, the characteristic size is well below the diffraction limit, routinely by 2-3 orders of magnitude. In such small volumes, energies are stored for a small period of time. A plasmonic nanoantenna supporting a localized surface plasmon quasinormal mode essentially behaves as a poor antenna that radiates energy rather than stores it. Thus, as the optical mode becomes deeply sub-wavelength in all three dimensions, independent of its shape, the Q-factor is limited to about 10 or less.
Formally, the resonances (i.e., the quasinormal mode) of an open (non-Hermitian) electromagnetic micro or nanoresonators are all found by solving the time-harmonic source-free Maxwell’s equations with a complex frequency , the real part being the resonance frequency and the imaginary part the damping rate. The damping is due to energy loses via leakage (the resonator is coupled to the open space surrounding it) and/or material absorption. Quasinormal-mode solvers exist to efficiently compute and normalize all kinds of modes of plasmonic nanoresonators and photonic microcavities. The proper normalisation of the mode leads to the important concept of mode volume of non-Hermitian (open and lossy) systems. The mode volume directly impact the physics of the interaction of light and electrons with optical resonance, e.g. the local density of electromagnetic states, Purcell effect , cavity perturbation theory , strong interaction with quantum emitters, superradiance . [ 3 ]
In computational biophysics, quasinormal modes, also called quasiharmonic modes, are derived from diagonalizing the matrix of equal-time correlations of atomic fluctuations. | https://en.wikipedia.org/wiki/Quasinormal_mode |
In condensed matter physics , a quasiparticle is a concept used to describe a collective behavior of a group of particles that can be treated as if they were a single particle. Formally, quasiparticles and collective excitations are closely related phenomena that arise when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum .
For example, as an electron travels through a semiconductor , its motion is disturbed in a complex way by its interactions with other electrons and with atomic nuclei . The electron behaves as though it has a different effective mass travelling unperturbed in vacuum. Such an electron is called an electron quasiparticle . [ 1 ] In another example, the aggregate motion of electrons in the valence band of a semiconductor or a hole band in a metal [ 2 ] behave as though the material instead contained positively charged quasiparticles called electron holes . Other quasiparticles or collective excitations include the phonon , a quasiparticle derived from the vibrations of atoms in a solid, and the plasmon , a particle derived from plasma oscillation .
These phenomena are typically called quasiparticles if they are related to fermions , and called collective excitations if they are related to bosons , [ 1 ] although the precise distinction is not universally agreed upon. [ 3 ] Thus, electrons and electron holes (fermions) are typically called quasiparticles , while phonons and plasmons (bosons) are typically called collective excitations .
The quasiparticle concept is important in condensed matter physics because it can simplify the many-body problem in quantum mechanics . The theory of quasiparticles was started by the Soviet physicist Lev Landau in the 1930s. [ 4 ] [ 5 ]
Solids are made of only three kinds of particles : electrons , protons , and neutrons . None of these are quasiparticles; instead a quasiparticle is an emergent phenomenon that occurs inside the solid. Therefore, while it is quite possible to have a single particle (electron, proton, or neutron) floating in space, a quasiparticle can only exist inside interacting many-particle systems such as solids.
Motion in a solid is extremely complicated: Each electron and proton is pushed and pulled (by Coulomb's law ) by all the other electrons and protons in the solid (which may themselves be in motion). It is these strong interactions that make it very difficult to predict and understand the behavior of solids (see many-body problem ). On the other hand, the motion of a non-interacting classical particle is relatively simple; it would move in a straight line at constant velocity. This is the motivation for the concept of quasiparticles: The complicated motion of the real particles in a solid can be mathematically transformed into the much simpler motion of imagined quasiparticles, which behave more like non-interacting particles.
In summary, quasiparticles are a mathematical tool for simplifying the description of solids.
The principal motivation for quasiparticles is that it is almost impossible to directly describe every particle in a macroscopic system. For example, a barely-visible (0.1mm) grain of sand contains around 10 17 nuclei and 10 18 electrons. Each of these attracts or repels every other by Coulomb's law . In principle, the Schrödinger equation predicts exactly how this system will behave. But the Schrödinger equation in this case is a partial differential equation (PDE) on a 3×10 18 -dimensional vector space—one dimension for each coordinate (x, y, z) of each particle. Directly and straightforwardly trying to solve such a PDE is impossible in practice. Solving a PDE on a 2-dimensional space is typically much harder than solving a PDE on a 1-dimensional space (whether analytically or numerically); solving a PDE on a 3-dimensional space is significantly harder still; and thus solving a PDE on a 3×10 18 -dimensional space is quite impossible by straightforward methods.
One simplifying factor is that the system as a whole, like any quantum system, has a ground state and various excited states with higher and higher energy above the ground state. In many contexts, only the "low-lying" excited states, with energy reasonably close to the ground state, are relevant. This occurs because of the Boltzmann distribution , which implies that very-high-energy thermal fluctuations are unlikely to occur at any given temperature.
Quasiparticles and collective excitations are a type of low-lying excited state. For example, a crystal at absolute zero is in the ground state , but if one phonon is added to the crystal (in other words, if the crystal is made to vibrate slightly at a particular frequency) then the crystal is now in a low-lying excited state. The single phonon is called an elementary excitation . More generally, low-lying excited states may contain any number of elementary excitations (for example, many phonons, along with other quasiparticles and collective excitations). [ 6 ]
When the material is characterized as having "several elementary excitations", this statement presupposes that the different excitations can be combined. In other words, it presupposes that the excitations can coexist simultaneously and independently. This is never exactly true. For example, a solid with two identical phonons does not have exactly twice the excitation energy of a solid with just one phonon, because the crystal vibration is slightly anharmonic . However, in many materials, the elementary excitations are very close to being independent. Therefore, as a starting point , they are treated as free, independent entities, and then corrections are included via interactions between the elementary excitations, such as "phonon- phonon scattering ".
Therefore, using quasiparticles / collective excitations, instead of analyzing 10 18 particles, one needs to deal with only a handful of somewhat-independent elementary excitations. It is, therefore, an effective approach to simplify the many-body problem in quantum mechanics. This approach is not useful for all systems, however. For example, in strongly correlated materials , the elementary excitations are so far from being independent that it is not even useful as a starting point to treat them as independent.
Usually, an elementary excitation is called a "quasiparticle" if it is a fermion and a "collective excitation" if it is a boson . [ 1 ] However, the precise distinction is not universally agreed upon. [ 3 ]
There is a difference in the way that quasiparticles and collective excitations are intuitively envisioned. [ 3 ] A quasiparticle is usually thought of as being like a dressed particle : it is built around a real particle at its "core", but the behavior of the particle is affected by the environment. A standard example is the "electron quasiparticle": an electron in a crystal behaves as if it had an effective mass which differs from its real mass. On the other hand, a collective excitation is usually imagined to be a reflection of the aggregate behavior of the system, with no single real particle at its "core". A standard example is the phonon , which characterizes the vibrational motion of every atom in the crystal.
However, these two visualizations leave some ambiguity. For example, a magnon in a ferromagnet can be considered in one of two perfectly equivalent ways: (a) as a mobile defect (a misdirected spin) in a perfect alignment of magnetic moments or (b) as a quantum of a collective spin wave that involves the precession of many spins. In the first case, the magnon is envisioned as a quasiparticle, in the second case, as a collective excitation. However, both (a) and (b) are equivalent and correct descriptions. As this example shows, the intuitive distinction between a quasiparticle and a collective excitation is not particularly important or fundamental.
The problems arising from the collective nature of quasiparticles have also been discussed within the philosophy of science, notably in relation to the identity conditions of quasiparticles and whether they should be considered "real" by the standards of, for example, entity realism . [ 7 ] [ 8 ]
By investigating the properties of individual quasiparticles, it is possible to obtain a great deal of information about low-energy systems, including the flow properties and heat capacity .
In the heat capacity example, a crystal can store energy by forming phonons , and/or forming excitons , and/or forming plasmons , etc. Each of these is a separate contribution to the overall heat capacity.
The idea of quasiparticles originated in Lev Landau's theory of Fermi liquids , which was originally invented for studying liquid helium-3 . For these systems a strong similarity exists between the notion of quasiparticle and dressed particles in quantum field theory . The dynamics of Landau's theory is defined by a kinetic equation of the mean-field type . A similar equation, the Vlasov equation , is valid for a plasma in the so-called plasma approximation . In the plasma approximation, charged particles are considered to be moving in the electromagnetic field collectively generated by all other particles, and hard collisions between the charged particles are neglected. When a kinetic equation of the mean-field type is a valid first-order description of a system, second-order corrections determine the entropy production , and generally take the form of a Boltzmann -type collision term, in which figure only "far collisions" between virtual particles . In other words, every type of mean-field kinetic equation, and in fact every mean-field theory , involves a quasiparticle concept.
This section contains most common examples of quasiparticles and collective excitations. | https://en.wikipedia.org/wiki/Quasiparticle |
In mathematics and theoretical physics , quasiperiodic motion is motion on a torus that never comes back to the same point. This behavior can also be called quasiperiodic evolution, dynamics, or flow . The torus may be a generalized torus so that the neighborhood of any point is more than two-dimensional. At each point of the torus there is a direction of motion that remains on the torus. Once a flow on a torus is defined or fixed, it determines trajectories. If the trajectories come back to a given point after a certain time then the motion is periodic with that period, otherwise it is quasiperiodic.
The quasiperiodic motion is characterized by a finite set of frequencies which can be thought of as the frequencies at which the motion goes around the torus in different directions. For instance, if the torus is the surface of a doughnut, then there is the frequency at which the motion goes around the doughnut and the frequency at which it goes inside and out. But the set of frequencies is not unique – by redefining the way position on the torus is parametrized another set of the same size can be generated. These frequencies will be integer combinations of the former frequencies (in such a way that the backward transformation is also an integer combination). To be quasiperiodic, the ratios of the frequencies must be irrational numbers. [ 1 ] [ 2 ] [ 3 ] [ 4 ]
In Hamiltonian mechanics with n position variables and associated rates of change it is sometimes possible to find a set of n conserved quantities. This is called the fully integrable case. One then has new position variables called action-angle coordinates , one for each conserved quantity, and these action angles simply increase linearly with time. This gives motion on " level sets " of the conserved quantities, resulting in a torus that is an n -manifold – locally having the topology of n -dimensional space. [ 5 ] The concept is closely connected to the basic facts about linear flow on the torus . These essentially linear systems and their behaviour under perturbation play a significant role in the general theory of non-linear dynamic systems. [ 6 ] Quasiperiodic motion does not exhibit the butterfly effect characteristic of chaotic systems . In other words, starting from a slightly different initial point on the torus results in a trajectory that is always just slightly different from the original trajectory, rather than the deviation becoming large. [ 4 ]
Rectilinear motion along a line in a Euclidean space gives rise to a quasiperiodic motion if the space is turned into a torus (a compact space ) by making every point equivalent to any other point situated in the same way with respect to the integer lattice (the points with integer coordinates), so long as the direction cosines of the rectilinear motion form irrational ratios. When the dimension is 2, this means the direction cosines are incommensurable . In higher dimensions it means the direction cosines must be linearly independent over the field of rational numbers . [ 5 ]
If we imagine that the phase space is modelled by a torus T (that is, the variables are periodic, like angles), the trajectory of the quasiperiodic system is modelled by a curve on T that wraps around the torus without ever exactly coming back on itself. Assuming the dimension of T is at least two, these can be thought of as one-parameter subgroups of the torus given group structure (by specifying a certain point as the identity element ).
A quasiperiodic motion can be expressed as a function of time whose value is a vector of " quasiperiodic functions ".
A quasiperiodic function f on the real line is a function obtained from a function F on a standard torus T (defined by n angles), by means of a trajectory in the torus in which each angle increases at a constant rate. [ 7 ] There are n "internal frequencies", being the rates at which the n angles progress, but as mentioned above the set is not uniquely determined. In many cases the function in the torus can be expressed as a multiple Fourier series . For n equal to 2 this is:
If the trajectory is
then the quasiperiodic function is:
This shows that there may be an infinite number of frequencies in the expansion, not multiples of a finite number of frequencies. Depending on which coefficients C j k {\displaystyle C_{jk}} are non-zero the "internal frequencies" ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} themselves may not contribute terms in this expansion, even if one uses an alternative set of internal frequencies such as ω 1 {\displaystyle \omega _{1}} and ω 1 + ω 2 . {\displaystyle \omega _{1}+\omega _{2}.} [ 8 ] If the C j k {\displaystyle C_{jk}} are non-zero only when the ratio i / j {\displaystyle i/j} is some specific constant, then the function is actually periodic rather than quasiperiodic.
See Kronecker's theorem for the geometric and Fourier theory attached to the number of modes. The closure of (the image of) any one-parameter subgroup in T is a subtorus of some dimension d . In that subtorus the result of Kronecker applies: there are d real numbers, linearly independent over the rational numbers, that are the corresponding frequencies.
In the quasiperiodic case, where the image is dense, a result can be proved on the ergodicity of the motion: for any measurable subset A of T (for the usual probability measure), the average proportion of time spent by the motion in A is equal to the measure of A . [ 9 ]
The theory of almost periodic functions is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions. The early discussion of quasi-periodic functions, by Ernest Esclangon following the work of Piers Bohl , in fact led to a definition of almost-periodic function, the terminology of Harald Bohr . [ 10 ] Ian Stewart wrote that the default position of classical celestial mechanics , at this period, was that motions that could be described as quasiperiodic were the most complex that occurred. [ 11 ] For the Solar System , that would apparently be the case if the gravitational attractions of the planets to each other could be neglected: but that assumption turned out to be the starting point of complex mathematics. [ 12 ] The research direction begun by Andrei Kolmogorov in the 1950s led to the understanding that quasiperiodic flow on phase space tori could survive perturbation. [ 13 ]
NB: The concept of quasiperiodic function , for example the sense in which theta functions and the Weierstrass zeta function in complex analysis are said to have quasi-periods with respect to a period lattice , is something distinct from this topic. | https://en.wikipedia.org/wiki/Quasiperiodic_motion |
Quasiperiodicity is the property of a system that displays irregular periodicity . Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". [ 1 ] Quasiperiodic behavior is almost but not quite periodic. [ 2 ] The term used to denote oscillations that appear to follow a regular pattern but which do not have a fixed period. The term thus used does not have a precise definition and should not be confused with more strictly defined mathematical concepts such as an almost periodic function or a quasiperiodic function .
Climate oscillations that appear to follow a regular pattern but which do not have a fixed period are called quasiperiodic . [ 3 ] [ 4 ]
Within a dynamical system such as the ocean-atmosphere system, oscillations may occur regularly when they are forced by a regular external forcing: for example, the familiar winter-summer cycle is forced by variations in sunlight from the (very close to perfectly) periodic motion of the Earth around the Sun. Or, like the recent ice age cycles, they may be less regular but still locked by external forcing. However, when the system contains the potential for an oscillation, but there is no strong external forcing to be phase-locked to, the "period" is likely to be irregular.
The canonical example of quasiperiodicity in climatology is El Niño–Southern Oscillation (ENSO). [ 5 ] ENSO is highly consequential for wheat cultivation in Australia. [ 5 ] Models to predict and thereby assist adaptation to ENSO have a large potential benefit to Australian wheat farmers. [ 5 ] In the modern era, it has a "period" somewhere between four and twelve years and a peak spectral density around five years. [ citation needed ]
This article about atmospheric science is a stub . You can help Wikipedia by expanding it .
This systems -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasiperiodicity |
In mathematics , a quasirandom group is a group that does not contain a large product-free [ clarification needed ] subset . Such groups are precisely those without a small non-trivial irreducible representation . The namesake of these groups stems from their connection to graph theory : bipartite Cayley graphs over any subset of a quasirandom group are always bipartite quasirandom graphs .
The notion of quasirandom groups arises when considering subsets of groups for which no two elements in the subset have a product in the subset; such subsets are termed product-free . László Babai and Vera Sós asked about the existence of a constant c {\displaystyle c} for which every finite group G {\displaystyle G} with order n {\displaystyle n} has a product-free subset with size at least c n {\displaystyle cn} . [ 1 ] A well-known result of Paul Erdős about sum-free sets of integers can be used to prove that c = 1 3 {\textstyle c={\frac {1}{3}}} suffices for abelian groups , but it turns out that such a constant does not exist for non-abelian groups . [ 2 ]
Both non-trivial lower and upper bounds are now known for the size of the largest product-free subset of a group with order n {\displaystyle n} . A lower bound of c n 11 14 {\textstyle cn^{\frac {11}{14}}} can be proved by taking a large subset of a union of sufficiently many cosets , [ 3 ] and an upper bound of c n 8 9 {\textstyle cn^{\frac {8}{9}}} is given by considering the projective special linear group PSL ( 2 , p ) {\displaystyle \operatorname {PSL} (2,p)} for any prime p {\displaystyle p} . [ 4 ] In the process of proving the upper bound, Timothy Gowers defined the notion of a quasirandom group to encapsulate the product-free condition and proved equivalences involving quasirandomness in graph theory.
Formally, it does not make sense to talk about whether or not a single group is quasirandom. The strict definition of quasirandomness will apply to sequences of groups, but first bipartite graph quasirandomness must be defined. The motivation for considering sequences of groups stems from its connections with graphons , which are defined as limits of graphs in a certain sense.
Fix a real number p ∈ [ 0 , 1 ] . {\displaystyle p\in [0,1].} A sequence of bipartite graphs ( G n ) {\displaystyle (G_{n})} (here n {\displaystyle n} is allowed to skip integers as long as n {\displaystyle n} tends to infinity) with G n {\displaystyle G_{n}} having n {\displaystyle n} vertices, vertex parts A n {\displaystyle A_{n}} and B n {\displaystyle B_{n}} , and ( p + o ( 1 ) ) | A n | | B n | {\displaystyle (p+o(1))|A_{n}||B_{n}|} edges is quasirandom if any of the following equivalent conditions hold:
It is a result of Chung–Graham–Wilson that each of the above conditions is equivalent. [ 5 ] Such graphs are termed quasirandom because each condition asserts that the quantity being considered is approximately what one would expect if the bipartite graph was generated according to the Erdős–Rényi random graph model ; that is, generated by including each possible edge between A n {\displaystyle A_{n}} and B n {\displaystyle B_{n}} independently with probability p . {\displaystyle p.}
Though quasirandomness can only be defined for sequences of graphs, a notion of c {\displaystyle c} -quasirandomness can be defined for a specific graph by allowing an error tolerance in any of the above definitions of graph quasirandomness. To be specific, given any of the equivalent definitions of quasirandomness, the o ( 1 ) {\displaystyle o(1)} term can be replaced by a small constant c > 0 {\displaystyle c>0} , and any graph satisfying that particular modified condition can be termed c {\displaystyle c} -quasirandom. It turns out that c {\displaystyle c} -quasirandomness under any condition is equivalent to c k {\displaystyle c^{k}} -quasirandomness under any other condition for some absolute constant k ≥ 1. {\displaystyle k\geq 1.}
The next step for defining group quasirandomness is the Cayley graph. Bipartite Cayley graphs give a way from translating quasirandomness in the graph-theoretic setting to the group-theoretic setting.
Given a finite group Γ {\displaystyle \Gamma } and a subset S ⊆ Γ {\displaystyle S\subseteq \Gamma } , the bipartite Cayley graph BiCay ( Γ , S ) {\displaystyle \operatorname {BiCay} (\Gamma ,S)} is the bipartite graph with vertex sets A {\displaystyle A} and B {\displaystyle B} , each labeled by elements of G {\displaystyle G} , whose edges a ∼ b {\displaystyle a\sim b} are between vertices whose ratio a b − 1 {\displaystyle ab^{-1}} is an element of S . {\displaystyle S.}
With the tools defined above, one can now define group quasirandomness. A sequence of groups ( Γ n ) {\displaystyle (\Gamma _{n})} with | Γ n | = n {\displaystyle |\Gamma _{n}|=n} (again, n {\displaystyle n} is allowed to skip integers) is quasirandom if for every real number p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} and choice of subsets S n ⊆ Γ n {\displaystyle S_{n}\subseteq \Gamma _{n}} with | S n | = ( p + o ( 1 ) ) | Γ n | {\displaystyle |S_{n}|=(p+o(1))|\Gamma _{n}|} , the sequence of graphs BiCay ( Γ n , S n ) {\displaystyle \operatorname {BiCay} (\Gamma _{n},S_{n})} is quasirandom. [ 4 ]
Though quasirandomness can only be defined for sequences of groups, the concept of c {\displaystyle c} -quasirandomness for specific groups can be extended to groups using the definition of c {\displaystyle c} -quasirandomness for specific graphs.
As proved by Gowers, group quasirandomness turns out to be equivalent to a number of different conditions.
To be precise, given a sequence of groups ( Γ n ) {\displaystyle (\Gamma _{n})} , the following are equivalent:
Cayley graphs generated from pseudorandom groups have strong mixing properties ; that is, BiCay ( Γ n , S ) {\displaystyle \operatorname {BiCay} (\Gamma _{n},S)} is a bipartite ( n , d , λ ) {\displaystyle (n,d,\lambda )} -graph for some λ {\displaystyle \lambda } tending to zero as n {\displaystyle n} tends to infinity. (Recall that an ( n , d , λ ) {\displaystyle (n,d,\lambda )} graph is a graph with n {\displaystyle n} vertices, each with degree d {\displaystyle d} , whose adjacency matrix has a second largest eigenvalue of at most λ . {\displaystyle \lambda .} )
In fact, it can be shown that for any c {\displaystyle c} -quasirandom group Γ {\displaystyle \Gamma } , the number of solutions to x y = z {\displaystyle xy=z} with x ∈ X {\displaystyle x\in X} , y ∈ Y {\displaystyle y\in Y} , and z ∈ Z {\displaystyle z\in Z} is approximately equal to what one might expect if S {\displaystyle S} was chosen randomly; that is, approximately equal to | X | | Y | | Z | | Γ | . {\displaystyle {\tfrac {|X||Y||Z|}{|\Gamma |}}.} This result follows from a direct application of the expander mixing lemma .
There are several notable families of quasirandom groups. In each case, the quasirandomness properties are most easily verified by checking the dimension of its smallest non-trivial representation. | https://en.wikipedia.org/wiki/Quasirandom_group |
In the mathematical field of analysis , quasiregular maps are a class of continuous maps between Euclidean spaces R n of the same dimension or, more generally, between Riemannian manifolds of the same dimension, which share some of the basic properties with holomorphic functions of one complex variable.
The theory of holomorphic (= analytic ) functions of one complex variable is one of the most beautiful and most useful parts of the whole mathematics.
One drawback of this theory is that it deals only with maps between two-dimensional spaces ( Riemann surfaces ). The theory of functions
of several complex variables has a different character, mainly because analytic functions of several variables are not conformal . Conformal maps can be defined between Euclidean spaces of arbitrary dimension, but when the dimension is greater than 2, this class of maps is very small: it consists of Möbius transformations only.
This is a theorem of Joseph Liouville ; relaxing the smoothness assumptions does not help, as proved by Yurii Reshetnyak . [ 1 ]
This suggests the search of a generalization of the property of conformality which would give a rich and interesting class of maps in higher dimension.
A differentiable map f of a region D in R n to R n is called K -quasiregular if the following inequality holds at all points in D :
Here K ≥ 1 is a constant, J f is the Jacobian determinant , Df is the derivative, that is the linear map defined by the Jacobi matrix , and ||·|| is the usual (Euclidean) norm of the matrix.
The development of the theory of such maps showed that it is unreasonable to restrict oneself to differentiable maps in the classical sense, and that the "correct" class of maps consists of continuous maps in the Sobolev space W 1, n loc whose partial derivatives in the sense of distributions have locally summable n -th power, and such that the above inequality is satisfied almost everywhere . This is a formal definition of a K -quasiregular map. A map is called quasiregular if it is K -quasiregular with some K . Constant maps are excluded from the class of quasiregular maps.
The fundamental theorem about quasiregular maps was proved by Reshetnyak: [ 2 ]
This means that the images of open sets are open and that preimages of points consist of isolated points. In dimension 2, these two properties give a topological characterization of the class of non-constant analytic functions:
every continuous open and discrete map of a plane domain to the plane can be pre-composed with a homeomorphism , so that the result is an analytic function. This is a theorem of Simion Stoilov .
Reshetnyak's theorem implies that all pure topological results about analytic functions (such that the Maximum Modulus Principle , Rouché's theorem etc.) extend to quasiregular maps.
Injective quasiregular maps are called quasiconformal . A simple example of non-injective quasiregular map is given in cylindrical coordinates in 3-space by the formula
This map is 2-quasiregular. It is smooth everywhere except the z -axis. A remarkable fact is that all smooth quasiregular maps are local homeomorphisms. Even more remarkable is that every quasiregular local homeomorphism R n → R n , where n ≥ 3, is a homeomorphism (this is a theorem of Vladimir Zorich [ 2 ] ).
This explains why in the definition of quasiregular maps it is not reasonable to restrict oneself to smooth maps: all smooth quasiregular maps of R n to itself are quasiconformal.
Many theorems about geometric properties of holomorphic functions of one complex variable have been extended to quasiregular maps. These extensions are usually highly non-trivial.
Perhaps the most famous result of this sort is the extension of Picard's theorem which is due to Seppo Rickman: [ 3 ]
When n = 2, this omitted set can contain at most one point (this is a simple extension of Picard's theorem). But when n > 2, the omitted set can contain more than one point, and its cardinality can be estimated from above in terms of n and K . In fact, any finite set
can be omitted, as shown by David Drasin and Pekka Pankka. [ 4 ]
If f is an analytic function, then log | f | is subharmonic , and harmonic away from the zeros of f . The corresponding fact for quasiregular maps is that log | f | satisfies a certain non-linear partial differential equation of elliptic type .
This discovery of Reshetnyak stimulated the development of non-linear potential theory , which treats this kind of equations
as the usual potential theory treats harmonic and subharmonic functions. | https://en.wikipedia.org/wiki/Quasiregular_map |
The quasispecies model is a description of the process of the Darwinian evolution of certain self-replicating entities within the framework of physical chemistry . A quasispecies is a large group or " cloud " of related genotypes that exist in an environment of high mutation rate (at stationary state [ 1 ] ), where a large fraction of offspring are expected to contain one or more mutations relative to the parent. This is in contrast to a species , which from an evolutionary perspective is a more-or-less stable single genotype, most of the offspring of which will be genetically accurate copies. [ 2 ]
It is useful mainly in providing a qualitative understanding of the evolutionary processes of self-replicating macromolecules such as RNA or DNA or simple asexual organisms such as bacteria or viruses (see also viral quasispecies ), and is helpful in explaining something of the early stages of the origin of life . Quantitative predictions based on this model are difficult because the parameters that serve as its input are impossible to obtain from actual biological systems. The quasispecies model was put forward by Manfred Eigen and Peter Schuster [ 3 ] based on initial work done by Eigen. [ 4 ]
When evolutionary biologists describe competition between species, they generally assume that each species is a single genotype whose descendants are mostly accurate copies. (Such genotypes are said to have a high reproductive fidelity .) In evolutionary terms, we are interested in the behavior and fitness of that one species or genotype over time. [ 5 ]
Some organisms or genotypes, however, may exist in circumstances of low fidelity, where most descendants contain one or more mutations. A group of such genotypes is constantly changing, so discussions of which single genotype is the most fit become meaningless. Importantly, if many closely related genotypes are only one mutation away from each other, then genotypes in the group can mutate back and forth into each other. For example, with one mutation per generation, a child of the sequence AGGT could be AGTT, and a grandchild could be AGGT again. Thus we can envision a " cloud " of related genotypes that is rapidly mutating, with sequences going back and forth among different points in the cloud. Though the proper definition is mathematical, that cloud, roughly speaking, is a quasispecies. [ citation needed ] [ 6 ]
Quasispecies behavior exists for large numbers of individuals existing at a certain (high) range of mutation rates. [ 7 ]
In a species, though reproduction may be mostly accurate, periodic mutations will give rise to one or more competing genotypes. If a mutation results in greater replication and survival, the mutant genotype may out-compete the parent genotype and come to dominate the species. Thus, the individual genotypes (or species) may be seen as the units on which selection acts and biologists will often speak of a single genotype's fitness . [ 8 ]
In a quasispecies, however, mutations are ubiquitous and so the fitness of an individual genotype becomes meaningless: if one particular mutation generates a boost in reproductive success, it can't amount to much because that genotype's offspring are unlikely to be accurate copies with the same properties. Instead, what matters is the connectedness of the cloud. For example, the sequence AGGT has 12 (3+3+3+3) possible single point mutants AGGA, AGGG, and so on. If 10 of those mutants are viable genotypes that may reproduce (and some of whose offspring or grandchildren may mutate back into AGGT again), we would consider that sequence a well-connected node in the cloud. If instead only two of those mutants are viable, the rest being lethal mutations, then that sequence is poorly connected and most of its descendants will not reproduce. The analog of fitness for a quasispecies is the tendency of nearby relatives within the cloud to be well-connected, meaning that more of the mutant descendants will be viable and give rise to further descendants within the cloud. [ 9 ]
When the fitness of a single genotype becomes meaningless because of the high rate of mutations, the cloud as a whole or quasispecies becomes the natural unit of selection.
Quasispecies represents the evolution of high-mutation-rate viruses such as HIV and sometimes single genes or molecules within the genomes of other organisms. [ 10 ] [ 11 ] [ 12 ] Quasispecies models have also been proposed by Jose Fontanari and Emmanuel David Tannenbaum to model the evolution of sexual reproduction. [ 13 ] Quasispecies was also shown in compositional replicators (based on the Gard model for abiogenesis ) [ 14 ] and was also suggested to be applicable to describe cell's replication, which amongst other things requires the maintenance and evolution of the internal composition of the parent and bud.
The model rests on four assumptions: [ 15 ]
In the quasispecies model, mutations occur through errors made in the process of copying already existing sequences. Further, selection arises because different types of sequences tend to replicate at different rates, which leads to the suppression of sequences that replicate more slowly in favor of sequences that replicate faster. However, the quasispecies model does not predict the ultimate extinction of all but the fastest replicating sequence. Although the sequences that replicate more slowly cannot sustain their abundance level by themselves, they are constantly replenished as sequences that replicate faster mutate into them. At equilibrium, removal of slowly replicating sequences due to decay or outflow is balanced by replenishing, so that even relatively slowly replicating sequences can remain present in finite abundance. [ 16 ]
Due to the ongoing production of mutant sequences, selection does not act on single sequences, but on mutational "clouds" of closely related sequences, referred to as quasispecies . In other words, the evolutionary success of a particular sequence depends not only on its own replication rate, but also on the replication rates of the mutant sequences it produces, and on the replication rates of the sequences of which it is a mutant. As a consequence, the sequence that replicates fastest may even disappear completely in selection-mutation equilibrium, in favor of more slowly replicating sequences that are part of a quasispecies with a higher average growth rate. [ 17 ] Mutational clouds as predicted by the quasispecies model have been observed in RNA viruses and in in vitro RNA replication. [ 18 ] [ 19 ]
The mutation rate and the general fitness of the molecular sequences and their neighbors is crucial to the formation of a quasispecies. If the mutation rate is zero, there is no exchange by mutation, and each sequence is its own species. If the mutation rate is too high, exceeding what is known as the error threshold , the quasispecies will break down and be dispersed over the entire range of available sequences. [ 20 ]
A simple mathematical model for a quasispecies is as follows: [ 1 ] let there be S {\displaystyle S} possible sequences and let there be n i {\displaystyle n_{i}} organisms with sequence i . Let's say that each of these organisms asexually gives rise to A i {\displaystyle A_{i}} offspring. Some are duplicates of their parent, having sequence i , but some are mutant and have some other sequence. Let the mutation rate q i j {\displaystyle q_{ij}} correspond to the probability that a j type parent will produce an i type organism. Then the expected fraction of offspring generated by j type organisms that would be i type organisms is w i j = A j q i j {\displaystyle w_{ij}=A_{j}q_{ij}} ,
where ∑ i q i j = 1 {\displaystyle \sum _{i}q_{ij}=1} .
Then the total number of i -type organisms after the first round of reproduction, given as n i ′ {\displaystyle n'_{i}} , is
Sometimes a death rate term D i {\displaystyle D_{i}} is included so that:
where δ i j {\displaystyle \delta _{ij}} is equal to 1 when i=j and is zero otherwise. Note that the n-th generation can be found by just taking the n-th power of W substituting it in place of W in the above formula.
This is just a system of linear equations . The usual way to solve such a system is to first diagonalize the W matrix. Its diagonal entries will be eigenvalues corresponding to certain linear combinations of certain subsets of sequences which will be eigenvectors of the W matrix. These subsets of sequences are the quasispecies. Assuming that the matrix W is a primitive matrix ( irreducible and aperiodic ), then after very many generations only the eigenvector with the largest eigenvalue will prevail, and it is this quasispecies that will eventually dominate. The components of this eigenvector give the relative abundance of each sequence at equilibrium. [ 21 ]
W being primitive means that for some integer n > 0 {\displaystyle n>0} , that the n t h {\displaystyle n^{th}} power of W is > 0, i.e. all the entries are positive. If W is primitive then each type can, through a sequence of mutations (i.e. powers of W ) mutate into all the other types after some number of generations. W is not primitive if it is periodic, where the population can perpetually cycle through different disjoint sets of compositions, or if it is reducible, where the dominant species (or quasispecies) that develops can depend on the initial population, as is the case in the simple example given below. [ citation needed ]
The quasispecies formulae may be expressed as a set of linear differential equations. If we consider the difference between the new state n i ′ {\displaystyle n'_{i}} and the old state n i {\displaystyle n_{i}} to be the state change over one moment of time, then we can state that the time derivative of n i {\displaystyle n_{i}} is given by this difference, n ˙ i = n i ′ − n i {\displaystyle {\dot {n}}_{i}=n'_{i}-n_{i}} we can write:
The quasispecies equations are usually expressed in terms of concentrations x i {\displaystyle x_{i}} where
The above equations for the quasispecies then become for the discrete version:
or, for the continuum version:
The quasispecies concept can be illustrated by a simple system consisting of 4 sequences. Sequences [0,0], [0,1], [1,0], and [1,1] are numbered 1, 2, 3, and 4, respectively. Let's say the [0,0] sequence never mutates and always produces a single offspring. Let's say the other 3 sequences all produce, on average, 1 − k {\displaystyle 1-k} replicas of themselves, and k {\displaystyle k} of each of the other two types, where 0 ≤ k ≤ 1 {\displaystyle 0\leq k\leq 1} . The W matrix is then:
The diagonalized matrix is:
And the eigenvectors corresponding to these eigenvalues are:
Only the eigenvalue 1 + k {\displaystyle 1+k} is more than unity. For the n-th generation, the corresponding eigenvalue will be ( 1 + k ) n {\displaystyle (1+k)^{n}} and so will increase without bound as time goes by. This eigenvalue corresponds to the eigenvector [0,1,1,1], which represents the quasispecies consisting of sequences 2, 3, and 4, which will be present in equal numbers after a very long time. Since all population numbers must be positive, the first two quasispecies are not legitimate. The third quasispecies consists of only the non-mutating sequence 1. It's seen that even though sequence 1 is the most fit in the sense that it reproduces more of itself than any other sequence, the quasispecies consisting of the other three sequences will eventually dominate (assuming that the initial population was not homogeneous of the sequence 1 type). [ citation needed ] | https://en.wikipedia.org/wiki/Quasispecies_model |
Quasistatic approximation(s) refers to different domains and different meanings. In the most common acceptance, quasistatic approximation refers to equations that keep a static form (do not involve time derivatives ) even if some quantities are allowed to vary slowly with time. In electromagnetism it refers to mathematical models that can be used to describe devices that do not produce significant amounts of electromagnetic waves. For instance the capacitor and the coil in electrical networks .
The quasistatic approximation can be understood through the idea that the sources in the problem change sufficiently slowly that the system can be taken to be in equilibrium at all times. This approximation can then be applied to areas such as classical electromagnetism, fluid mechanics, magnetohydrodynamics, thermodynamics, and more generally systems described by hyperbolic partial differential equations involving both spatial and time derivatives . In simple cases, the quasistatic approximation is allowed when the typical spatial scale divided by the typical temporal scale is much smaller than the characteristic velocity with which information is propagated. [ 1 ] The problem gets more complicated when several length and time scales are involved. In the strict acceptance of the term the quasistatic case corresponds to a situation where all time derivatives can be neglected. However some equations can be considered as quasistatic while others are not, leading to a system still being dynamic. There is no general consensus in such cases.
In fluid dynamics , only quasi- hydrostatics (where no time derivative term is present) is considered as a quasi-static approximation. Flows are usually considered as dynamic as well as acoustic waves propagation.
In thermodynamics , a distinction between quasistatic regimes and dynamic ones is usually made in terms of equilibrium thermodynamics versus non-equilibrium thermodynamics . As in electromagnetism some intermediate situations also exist; see for instance local equilibrium thermodynamics .
In classical electromagnetism , there are at least two consistent quasistatic approximations of Maxwell equations: quasi- electrostatics and quasi- magnetostatics depending on the relative importance of the two dynamic coupling terms. [ 2 ] These approximations can be obtained using time constants evaluations or can be shown to be Galilean limits of electromagnetism . [ 3 ]
In magnetostatics equations such as Ampère's Law or the more general Biot–Savart law allow one to solve for the magnetic fields produced by steady electrical currents. Often, however, one may want to calculate the magnetic field due to time varying currents (accelerating charge) or other forms of moving charge. Strictly speaking, in these cases the aforementioned equations are invalid, as the field measured at the observer must incorporate distances measured at the retarded time , that is the observation time minus the time it took for the field (traveling at the speed of light ) to reach the observer. The retarded time is different for every point to be considered, hence the resulting equations are quite complicated; often it is easier to formulate the problem in terms of potentials; see retarded potential and Jefimenko's equations .
In this point of view the quasistatic approximation is obtained by using time instead of retarded time or equivalently to assume that the speed of light is infinite. To first order, the mistake of using only Biot–Savart's law rather than both terms of Jefimenko's magnetic field equation fortuitously cancel. [ 4 ] | https://en.wikipedia.org/wiki/Quasistatic_approximation |
In thermodynamics , a quasi-static process , also known as a quasi-equilibrium process (from Latin quasi , meaning ‘as if’ [ 1 ] ), is a thermodynamic process that happens slowly enough for the system to remain in internal physical (but not necessarily chemical) thermodynamic equilibrium . An example of this is quasi-static expansion of a mixture of hydrogen and oxygen gas, where the volume of the system changes so slowly that the pressure remains uniform throughout the system at each instant of time during the process. [ 2 ] Such an idealized process is a succession of physical equilibrium states, characterized by infinite slowness. [ 3 ]
Only in a quasi-static thermodynamic process can we exactly define intensive quantities (such as pressure, temperature , specific volume , specific entropy ) of the system at any instant during the whole process; otherwise, since no internal equilibrium is established, different parts of the system would have different values of these quantities, so a single value per quantity may not be sufficient to represent the whole system. In other words, when an equation for a change in a state function contains P or T , it implies a quasi-static process.
While all reversible processes are quasi-static, most authors do not require a general quasi-static process to maintain equilibrium between system and surroundings and avoid dissipation, [ 4 ] which are defining characteristics of a reversible process. For example, quasi-static compression of a system by a piston subject to friction is irreversible; although the system is always in internal thermal equilibrium, the friction ensures the generation of dissipative entropy, which goes against the definition of reversibility. Any engineer would remember to include friction when calculating the dissipative entropy generation.
An example of a quasi-static process that is not idealizable as reversible is slow heat transfer between two bodies on two finitely different temperatures, where the heat transfer rate is controlled by a poorly conductive partition between the two bodies. In this case, no matter how slowly the process takes place, the state of the composite system consisting of the two bodies is far from equilibrium, since thermal equilibrium for this composite system requires that the two bodies be at the same temperature. Nevertheless, the entropy change for each body can be calculated using the Clausius equality for reversible heat transfer. | https://en.wikipedia.org/wiki/Quasistatic_process |
In algebra and in particular in algebraic combinatorics , a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions . This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables (but its elements are neither polynomials nor functions).
The ring of quasisymmetric functions , denoted QSym, can be defined over any commutative ring R such as the integers .
Quasisymmetric
functions are power series of bounded degree in variables x 1 , x 2 , x 3 , … {\displaystyle x_{1},x_{2},x_{3},\dots } with coefficients in R , which are shift invariant in the sense that the coefficient of the monomial x 1 α 1 x 2 α 2 ⋯ x k α k {\displaystyle x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{k}^{\alpha _{k}}} is equal to the coefficient of the monomial x i 1 α 1 x i 2 α 2 ⋯ x i k α k {\displaystyle x_{i_{1}}^{\alpha _{1}}x_{i_{2}}^{\alpha _{2}}\cdots x_{i_{k}}^{\alpha _{k}}} for any strictly increasing sequence of positive integers i 1 < i 2 < ⋯ < i k {\displaystyle i_{1}<i_{2}<\cdots <i_{k}} indexing the variables and any positive integer sequence ( α 1 , α 2 , … , α k ) {\displaystyle (\alpha _{1},\alpha _{2},\ldots ,\alpha _{k})} of exponents. [ 1 ] Much of the study of quasisymmetric functions is based on that of symmetric functions .
A quasisymmetric function in finitely many variables is a quasisymmetric polynomial .
Both symmetric and quasisymmetric polynomials may be characterized in terms of actions of the symmetric group S n {\displaystyle S_{n}} on a polynomial ring in n {\displaystyle n} variables x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} .
One such action of S n {\displaystyle S_{n}} permutes variables,
changing a polynomial p ( x 1 , … , x n ) {\displaystyle p(x_{1},\dots ,x_{n})} by iteratively swapping pairs ( x i , x i + 1 ) {\displaystyle (x_{i},x_{i+1})} of variables having consecutive indices.
Those polynomials unchanged by all such swaps
form the subring of symmetric polynomials.
A second action of S n {\displaystyle S_{n}} conditionally permutes variables,
changing a polynomial p ( x 1 , … , x n ) {\displaystyle p(x_{1},\ldots ,x_{n})} by swapping pairs ( x i , x i + 1 ) {\displaystyle (x_{i},x_{i+1})} of variables except in monomials containing both variables. [ 2 ] [ 3 ] Those polynomials unchanged by all such conditional swaps form
the subring of quasisymmetric polynomials. One quasisymmetric polynomial in four variables x 1 , x 2 , x 3 , x 4 {\displaystyle x_{1},x_{2},x_{3},x_{4}} is the polynomial
The simplest symmetric polynomial containing these monomials is
QSym is a graded R - algebra , decomposing as
where QSym n {\displaystyle \operatorname {QSym} _{n}} is the R {\displaystyle R} - span of all quasisymmetric functions that are homogeneous of degree n {\displaystyle n} . Two natural bases for QSym n {\displaystyle \operatorname {QSym} _{n}} are the monomial basis { M α } {\displaystyle \{M_{\alpha }\}} and the fundamental basis { F α } {\displaystyle \{F_{\alpha }\}} indexed by compositions α = ( α 1 , α 2 , … , α k ) {\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{k})} of n {\displaystyle n} , denoted α ⊨ n {\displaystyle \alpha \vDash n} . The monomial basis consists of M 0 = 1 {\displaystyle M_{0}=1} and all formal power series
The fundamental basis consists F 0 = 1 {\displaystyle F_{0}=1} and all formal power series
where α ⪰ β {\displaystyle \alpha \succeq \beta } means we can obtain α {\displaystyle \alpha } by adding together adjacent parts of β {\displaystyle \beta } , for example, (3,2,4,2) ⪰ {\displaystyle \succeq } (3,1,1,1,2,1,2). Thus, when the ring R {\displaystyle R} is the ring of rational numbers , one has
Then one can define the algebra of symmetric functions Λ = Λ 0 ⊕ Λ 1 ⊕ ⋯ {\displaystyle \Lambda =\Lambda _{0}\oplus \Lambda _{1}\oplus \cdots } as the subalgebra of QSym spanned by the monomial symmetric functions m 0 = 1 {\displaystyle m_{0}=1} and all formal power series m λ = ∑ M α , {\displaystyle m_{\lambda }=\sum M_{\alpha },} where the sum is over all compositions α {\displaystyle \alpha } which rearrange to the integer partition λ {\displaystyle \lambda } . Moreover, we have Λ n = Λ ∩ QSym n {\displaystyle \Lambda _{n}=\Lambda \cap \operatorname {QSym} _{n}} . For example, F ( 1 , 2 ) = M ( 1 , 2 ) + M ( 1 , 1 , 1 ) {\displaystyle F_{(1,2)}=M_{(1,2)}+M_{(1,1,1)}} and m ( 2 , 1 ) = M ( 2 , 1 ) + M ( 1 , 2 ) . {\displaystyle m_{(2,1)}=M_{(2,1)}+M_{(1,2)}.}
Other important bases for quasisymmetric functions include the basis of quasisymmetric Schur functions, [ 4 ] the "type I" and "type II" quasisymmetric power sums, [ 5 ] and bases related to enumeration in matroids. [ 6 ] [ 7 ]
Quasisymmetric functions have been applied in enumerative combinatorics , symmetric function theory, representation theory, and number theory. Applications of
quasisymmetric functions include enumeration of P-partitions, [ 8 ] [ 9 ] permutations, [ 10 ] [ 11 ] [ 12 ] [ 13 ] tableaux, [ 14 ] chains of posets, [ 14 ] [ 15 ] reduced decompositions in finite Coxeter groups (via Stanley symmetric functions ), [ 14 ] and parking functions. [ 16 ] In symmetric function theory and representation theory, applications include the study of Schubert polynomials , [ 17 ] [ 18 ] Macdonald polynomials , [ 19 ] Hecke algebras, [ 20 ] and Kazhdan–Lusztig polynomials. [ 21 ] Often quasisymmetric functions provide a powerful bridge between combinatorial structures and symmetric functions.
As a graded Hopf algebra , the dual of the ring of quasisymmetric functions is the ring of noncommutative symmetric functions.
Every symmetric function is also a quasisymmetric function, and hence the ring of symmetric functions is a subalgebra of the ring of quasisymmetric functions.
The ring of quasisymmetric functions is the terminal object in category of graded Hopf algebras with a single character. [ 22 ] Hence any such Hopf algebra has a morphism to the ring of quasisymmetric functions.
One example of this is the peak algebra . [ 23 ]
The Malvenuto–Reutenauer algebra [ 24 ] is a Hopf algebra based on permutations that relates the rings of symmetric functions, quasisymmetric functions, and noncommutative symmetric functions , (denoted Sym, QSym, and NSym respectively), as depicted the following commutative diagram . The duality between QSym and NSym mentioned above is reflected in the main diagonal of this diagram.
Many related Hopf algebras were constructed from Hopf monoids in the category of species by Aguiar and Majahan. [ 25 ]
One can also construct the ring of quasisymmetric functions in noncommuting variables. [ 3 ] [ 26 ] | https://en.wikipedia.org/wiki/Quasisymmetric_function |
In mathematics , a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent. [ 1 ]
Let ( X , d X ) and ( Y , d Y ) be two metric spaces . A homeomorphism f : X → Y is said to be η-quasisymmetric if there is an increasing function η : [0, ∞) → [0, ∞) such that for any triple x , y , z of distinct points in X , we have
A map f:X→Y is said to be H-weakly-quasisymmetric for some H > 0 {\displaystyle H>0} if for all triples of distinct points x , y , z {\displaystyle x,y,z} in X {\displaystyle X} , then
Not all weakly quasisymmetric maps are quasisymmetric. However, if X {\displaystyle X} is connected and X {\displaystyle X} and Y {\displaystyle Y} are doubling , then all weakly quasisymmetric maps are quasisymmetric. The appeal of this result is that proving weak-quasisymmetry is much easier than proving quasisymmetry directly, and in many natural settings the two notions are equivalent.
A monotone map f : H → H on a Hilbert space H is δ-monotone if for all x and y in H ,
To grasp what this condition means geometrically, suppose f (0) = 0 and consider the above estimate when y = 0. Then it implies that the angle between the vector x and its image f ( x ) stays between 0 and arccos δ < π /2.
These maps are quasisymmetric, although they are a much narrower subclass of quasisymmetric maps. For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ -monotone will always map the real line to a rotated graph of a Lipschitz function L :ℝ → ℝ. [ 2 ]
Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives. [ 3 ] An increasing homeomorphism f :ℝ → ℝ is quasisymmetric if and only if there is a constant C > 0 and a doubling measure μ on the real line such that
An analogous result holds in Euclidean space. Suppose C = 0 and we rewrite the above equation for f as
Writing it this way, we can attempt to define a map using this same integral, but instead integrate (what is now a vector valued integrand) over ℝ n : if μ is a doubling measure on ℝ n and
then the map
is quasisymmetric (in fact, it is δ -monotone for some δ depending on the measure μ ). [ 4 ]
Let Ω {\displaystyle \Omega } and Ω ′ {\displaystyle \Omega '} be open subsets of ℝ n . If f : Ω → Ω´ is η -quasisymmetric, then it is also K - quasiconformal , where K > 0 {\displaystyle K>0} is a constant depending on η {\displaystyle \eta } .
Conversely, if f : Ω → Ω´ is K -quasiconformal and B ( x , 2 r ) {\displaystyle B(x,2r)} is contained in Ω {\displaystyle \Omega } , then f {\displaystyle f} is η -quasisymmetric on B ( x , 2 r ) {\displaystyle B(x,2r)} , where η {\displaystyle \eta } depends only on K {\displaystyle K} .
A related but weaker condition is the notion of quasi-Möbius maps where instead of the ratio only the cross-ratio is considered: [ 5 ]
Let ( X , d X ) and ( Y , d Y ) be two metric spaces and let η : [0, ∞) → [0, ∞) be an increasing function. An η -quasi-Möbius homeomorphism f : X → Y is a homeomorphism for which for every quadruple x , y , z , t of distinct points in X , we have | https://en.wikipedia.org/wiki/Quasisymmetric_map |
In magnetic confinement fusion , quasisymmetry (sometimes hyphenated as quasi-symmetry ) is a type of continuous symmetry in the magnetic field strength of a stellarator . [ 1 ] Quasisymmetry is desired, as Noether's theorem implies that there exists a conserved quantity in such cases. This conserved quantity ensures that particles stick to the flux surface , resulting in better confinement and neoclassical transport .
It is currently unknown if it is mathematically possible to construct a quasi-symmetric magnetic field which upholds magnetohydrodynamic force balance, which is required for stability. There are stellarator designs which are very close to being quasisymmetric, [ 2 ] and it is possible to find solutions by generalizing the magnetohydrodynamic force balance equation. [ 3 ] Quasisymmetric systems are a subset of omnigenous systems. The Helically Symmetric eXperiment and the National Compact Stellarator Experiment are designed to be quasisymmetric. [ citation needed ] | https://en.wikipedia.org/wiki/Quasisymmetry |
In mathematics , a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space .
They were introduced by Spanier, who showed that there is a natural quasi-topology on the space of continuous maps from one space to another.
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quasitopological_space |
In mathematics , a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry . A smooth 2 n {\displaystyle 2n} -dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an n {\displaystyle n} -dimensional torus , with orbit space an n {\displaystyle n} -dimensional simple convex polytope .
Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz, [ 1 ] who called them "toric manifolds". However, the term "quasitoric manifold" was eventually adopted to avoid confusion with the class of compact smooth toric varieties , which are known to algebraic geometers as toric manifolds . [ 2 ]
Quasitoric manifolds are studied in a variety of contexts in algebraic topology , such as complex cobordism theory , and the other oriented cohomology theories . [ 3 ]
Denote the i {\displaystyle i} -th subcircle of the n {\displaystyle n} -torus T n {\displaystyle T^{n}} by T i {\displaystyle T_{i}} so that T 1 × … × T n = T n {\displaystyle T_{1}\times \ldots \times T_{n}=T^{n}} . Then coordinate-wise multiplication of T n {\displaystyle T^{n}} on C n {\displaystyle \mathbb {C} ^{n}} is called the standard representation .
Given open sets X {\displaystyle X} in M 2 n {\displaystyle M^{2n}} and Y {\displaystyle Y} in C n {\displaystyle \mathbb {C} ^{n}} , that are closed under the action of T n {\displaystyle T^{n}} , a T n {\displaystyle T^{n}} -action on M 2 n {\displaystyle M^{2n}} is defined to be locally isomorphic to the standard representation if h ( t x ) = α ( t ) h ( x ) {\displaystyle h(tx)=\alpha (t)h(x)} , for all t {\displaystyle t} in T n {\displaystyle T^{n}} , x {\displaystyle x} in X {\displaystyle X} , where h {\displaystyle h} is a homeomorphism X → Y {\displaystyle X\rightarrow Y} , and α {\displaystyle \alpha } is an automorphism of T n {\displaystyle T^{n}} .
Given a simple convex polytope P n {\displaystyle P^{n}} with m {\displaystyle m} facets , a T n {\displaystyle T^{n}} -manifold M 2 n {\displaystyle M^{2n}} is a quasitoric manifold over P n {\displaystyle P^{n}} if,
The definition implies that the fixed points of M 2 n {\displaystyle M^{2n}} under the T n {\displaystyle T^{n}} -action are mapped to the vertices of P n {\displaystyle P^{n}} by π {\displaystyle \pi } , while points where the action is free project to the interior of the polytope.
A quasitoric manifold can be described in terms of a dicharacteristic function and an associated dicharacteristic matrix . In this setting it is useful to assume that the facets F 1 , … , F m {\displaystyle F_{1},\dots ,F_{m}} of P n {\displaystyle P^{n}} are ordered so that the intersection F 1 ∩ ⋯ ∩ F n {\displaystyle F_{1}\cap \dots \cap F_{n}} is a vertex v {\displaystyle v} of P n {\displaystyle P^{n}} , called the initial vertex .
A dicharacteristic function is a homomorphism λ : T m → T n {\displaystyle \lambda :T^{m}\rightarrow T^{n}} , such that if F i 1 ∩ ⋯ ∩ F i k {\displaystyle F_{i_{1}}\cap \dots \cap F_{i_{k}}} is a codimension - k {\displaystyle k} face of P n {\displaystyle P^{n}} , then λ {\displaystyle \lambda } is a monomorphism on restriction to the subtorus T i 1 × ⋯ × T i k {\displaystyle T_{i_{1}}\times \dots \times T_{i_{k}}} in T m {\displaystyle T^{m}} .
The restriction of λ to the subtorus T 1 × … × T n {\displaystyle T_{1}\times \ldots \times T_{n}} corresponding to the initial vertex v {\displaystyle v} is an isomorphism, and so λ ( T 1 ) , … , λ ( T n ) {\displaystyle \lambda (T_{1}),\ldots ,\lambda (T_{n})} can be taken to be a basis for the Lie algebra of T n {\displaystyle T^{n}} . The epimorphism of Lie algebras associated to λ may be described as a linear transformation Z m → Z n {\displaystyle \mathbb {Z} ^{m}\rightarrow \mathbb {Z} ^{n}} , represented by the n × m {\displaystyle n\times m} dicharacteristic matrix Λ {\displaystyle \Lambda } given by
The i {\displaystyle i} th column of Λ {\displaystyle \Lambda } is a primitive vector λ i = ( λ 1 , i , … , λ n , i ) {\displaystyle \lambda _{i}=(\lambda _{1,i},\dots ,\lambda _{n,i})} in Z n {\displaystyle \mathbb {Z} ^{n}} , called the facet vector . As each facet vector is primitive, whenever the facets F i 1 ∩ ⋯ ∩ F i n {\displaystyle F_{i_{1}}\cap \dots \cap F_{i_{n}}} meet in a vertex, the corresponding columns λ i 1 , … λ i n {\displaystyle \lambda _{i_{1}},\dots \lambda _{i_{n}}} form a basis of Z n {\displaystyle \mathbb {Z} ^{n}} , with determinant equal to ± 1 {\displaystyle \pm 1} . The isotropy subgroup associated to each facet F i {\displaystyle F_{i}} is described by
for some θ {\displaystyle \theta } in R {\displaystyle \mathbb {R} } .
In their original treatment of quasitoric manifolds, Davis and Januskiewicz [ 1 ] introduced the notion of a characteristic function that mapped each facet of the polytope to a vector determining the isotropy subgroup of the facet, but this is only defined up to sign. In more recent studies of quasitoric manifolds, this ambiguity has been removed by the introduction of the dicharacteristic function and its insistence that each circle λ ( T i ) {\displaystyle \lambda (T_{i})} be oriented, forcing a choice of sign for each vector λ i {\displaystyle \lambda _{i}} . The notion of the dicharacteristic function was originally introduced V. Buchstaber and N. Ray [ 4 ] to enable the study of quasitoric manifolds in complex cobordism theory. This was further refined by introducing the ordering of the facets of the polytope to define the initial vertex, which eventually leads to the above neat representation of the dicharacteristic matrix Λ {\displaystyle \Lambda } as ( I n ∣ S ) {\displaystyle (I_{n}\mid S)} , where I n {\displaystyle I_{n}} is the identity matrix and S {\displaystyle S} is an n × ( m − n ) {\displaystyle n\times (m-n)} submatrix. [ 5 ]
The kernel K ( λ ) {\displaystyle K(\lambda )} of the dicharacteristic function acts freely on the moment angle complex Z P n {\displaystyle Z_{P^{n}}} , and so defines a principal K ( λ ) {\displaystyle K(\lambda )} -bundle Z P n → M 2 n {\displaystyle Z_{P^{n}}\rightarrow M^{2n}} over the resulting quotient space M 2 n {\displaystyle M^{2n}} . This quotient space can be viewed as
where pairs ( t 1 , p 1 ) {\displaystyle (t_{1},p_{1})} , ( t 2 , p 2 ) {\displaystyle (t_{2},p_{2})} of T n × P n {\displaystyle T^{n}\times P^{n}} are identified if and only if p 1 = p 2 {\displaystyle p_{1}=p_{2}} and t 1 − 1 t 2 {\displaystyle t_{1}^{-1}t_{2}} is in the image of λ {\displaystyle \lambda } on restriction to the subtorus T i 1 × ⋯ × T i k {\displaystyle T_{i_{1}}\times \dots \times T_{i_{k}}} that corresponds to the unique face F i 1 ∩ ⋯ ∩ F i k {\displaystyle F_{i_{1}}\cap \dots \cap F_{i_{k}}} of P n {\displaystyle P^{n}} containing the point p 1 {\displaystyle p_{1}} , for some 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} .
It can be shown that any quasitoric manifold M 2 n {\displaystyle M^{2n}} over P n {\displaystyle P^{n}} is equivariently diffeomorphic to a quasitoric manifold of the form of the quotient space above. [ 6 ]
The moment angle complex Z Δ n {\displaystyle Z_{\Delta ^{n}}} is the ( 2 n + 1 ) {\displaystyle (2n+1)} -sphere S 2 n + 1 {\displaystyle S^{2n+1}} , the kernel K ( λ ) {\displaystyle K(\lambda )} is the diagonal subgroup { ( t , … , t ) } < T n + 1 {\displaystyle \{(t,\dots ,t)\}<T^{n+1}} , so the quotient of Z Δ n {\displaystyle Z_{\Delta ^{n}}} under the action of K ( λ ) {\displaystyle K(\lambda )} is C P n {\displaystyle \mathbb {C} P^{n}} . [ 7 ]
for integers a ( i , j ) {\displaystyle a(i,j)} .
The moment angle complex Z I n {\displaystyle Z_{I^{n}}} is a product of n {\displaystyle n} copies of 3-sphere embedded in C 2 n {\displaystyle \mathbb {C} ^{2n}} , the kernel K ( λ ) {\displaystyle K(\lambda )} is given by
so that the quotient of Z I n {\displaystyle Z_{I^{n}}} under the action of K ( λ ) {\displaystyle K(\lambda )} is the n {\displaystyle n} -th stage of a Bott tower. [ 8 ] The integer values a ( i , j ) {\displaystyle a(i,j)} are the tensor powers of the line bundles whose product is used in the iterated sphere-bundle construction of the Bott tower. [ 9 ]
Canonical complex line bundles ρ i {\displaystyle \rho _{i}} over M 2 n {\displaystyle M^{2n}} given by
can be associated with each facet F i {\displaystyle F_{i}} of P n {\displaystyle P^{n}} , for 1 ≤ i ≤ m {\displaystyle 1\leq i\leq m} , where K ( λ ) {\displaystyle K(\lambda )} acts on C i {\displaystyle \mathbb {C} _{i}} , by the restriction of K ( λ ) {\displaystyle K(\lambda )} to the i {\displaystyle i} -th subcircle of T m {\displaystyle T^{m}} embedded in C {\displaystyle \mathbb {C} } . These bundles are known as the facial bundles associated to the quasitoric manifold. By the definition of M 2 n {\displaystyle M^{2n}} , the preimage of a facet π − 1 ( F i ) {\displaystyle \pi ^{-1}(F_{i})} is a 2 ( n − 1 ) {\displaystyle 2(n-1)} -dimensional quasitoric facial submanifold M i {\displaystyle M_{i}} over F i {\displaystyle F_{i}} , whose isotropy subgroup is the restriction of λ {\displaystyle \lambda } on the subcircle T i {\displaystyle T_{i}} of T m {\displaystyle T^{m}} . Restriction of ρ i {\displaystyle \rho _{i}} to M i {\displaystyle M_{i}} gives the normal 2-plane bundle of the embedding of M i {\displaystyle M_{i}} in M 2 n {\displaystyle M^{2n}} .
Let x i {\displaystyle x_{i}} in H 2 ( M 2 n ; Z ) {\displaystyle H^{2}(M^{2n};\mathbb {Z} )} denote the first Chern class of ρ i {\displaystyle \rho _{i}} . The integral cohomology ring H ∗ ( M 2 n ; Z ) {\displaystyle H^{*}(M^{2n};\mathbb {Z} )} is generated by x i {\displaystyle x_{i}} , for 1 ≤ i ≤ m {\displaystyle 1\leq i\leq m} , subject to two sets of relations. The first are the relations generated by the Stanley–Reisner ideal of P n {\displaystyle P^{n}} ; linear relations determined by the dicharacterstic function comprise the second set:
Therefore only x n + 1 , … , x m {\displaystyle x_{n+1},\dots ,x_{m}} are required to generate H ∗ ( M 2 n ; Z ) {\displaystyle H^{*}(M^{2n};\mathbb {Z} )} multiplicatively. [ 1 ] | https://en.wikipedia.org/wiki/Quasitoric_manifold |
Quaternary is a term used in organic chemistry to classify various types of compounds (e. g. amines and ammonium salts ). [ 1 ]
Quaternary central atoms compared with primary , secondary and tertiary central atoms. | https://en.wikipedia.org/wiki/Quaternary_(chemistry) |
A quaternary carbon is a carbon atom bound to four other carbon atoms. [ 1 ] For this reason, quaternary carbon atoms are found only in hydrocarbons having at least five carbon atoms. Quaternary carbon atoms can occur in branched alkanes , but not in linear alkanes. [ 2 ]
The formation of chiral quaternary carbon centers has been a synthetic challenge. Chemists have developed asymmetric Diels–Alder reactions , [ 3 ] Heck reaction , Enyne cyclization , cycloaddition reactions , [ 4 ] C–H activation , Allylic substitution , [ 5 ] Pauson–Khand reaction , [ 6 ] etc. to construct asymmetric quaternary carbon atoms.
One of the most industrially important compounds containing a quaternary carbon is bis-phenol A (BPA). The central atom is a quaternary carbon. Retrosynthetically, that carbon is the central atom of an acetone molecule before condensation with two equivalents of phenol - BPA Production Process | https://en.wikipedia.org/wiki/Quaternary_carbon |
In materials chemistry , a quaternary phase is a chemical compound containing four elements . Some compounds can be molecular or ionic, examples being chlorodifluoromethane ( CHClF 2 ) and sodium bicarbonate ( NaCO 3 H ). More typically quaternary phase refers to extended solids. A famous example are the yttrium barium copper oxide superconductors. [ 1 ]
This article about chemical compounds is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quaternary_phase |
In differential geometry , a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold , is a Riemannian symmetric space . Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected , and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups .
For any compact simple Lie group G , there is a unique G / H obtained as a quotient of G by a subgroup
Here, Sp(1) is the compact form of the SL(2)-triple associated with the highest root of G , and K its centralizer in G . These are classified as follows.
The twistor spaces of quaternion-Kähler symmetric spaces are the homogeneous holomorphic contact manifolds , classified by Boothby: they are the adjoint varieties of the complex semisimple Lie groups .
These spaces can be obtained by taking a projectivization of
a minimal nilpotent orbit of the respective complex Lie group.
The holomorphic contact structure is apparent, because
the nilpotent orbits of semisimple Lie groups
are equipped with the Kirillov-Kostant holomorphic symplectic form. This argument also explains how one
can associate a unique Wolf space to each of the simple
complex Lie groups. | https://en.wikipedia.org/wiki/Quaternion-Kähler_symmetric_space |
In mathematics , quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
As with complex and real analysis , it is possible to study the concepts of analyticity , holomorphy , harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals , the four notions do not coincide.
The projections of a quaternion onto its scalar part or onto its vector part, as well as the modulus and versor functions, are examples that are basic to understanding quaternion structure.
An important example of a function of a quaternion variable is
which rotates the vector part of q by twice the angle represented by the versor u .
The quaternion multiplicative inverse f 2 ( q ) = q − 1 {\displaystyle f_{2}(q)=q^{-1}} is another fundamental function, but as with other number systems, f 2 ( 0 ) {\displaystyle f_{2}(0)} and related problems are generally excluded due to the nature of dividing by zero .
Affine transformations of quaternions have the form
Linear fractional transformations of quaternions can be represented by elements of the matrix ring M 2 ( H ) {\displaystyle M_{2}(\mathbb {H} )} operating on the projective line over H {\displaystyle \mathbb {H} } . For instance, the mappings q ↦ u q v , {\displaystyle q\mapsto uqv,} where u {\displaystyle u} and v {\displaystyle v} are fixed versors serve to produce the motions of elliptic space .
Quaternion variable theory differs in some respects from complex variable theory. For example: The complex conjugate mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, non-analytic operation. Indeed, conjugation changes the orientation of plane figures, something that arithmetic functions do not change.
In contrast to the complex conjugate , the quaternion conjugation can be expressed arithmetically, as f 4 ( q ) = − 1 2 ( q + i q i + j q j + k q k ) {\displaystyle f_{4}(q)=-{\tfrac {1}{2}}(q+iqi+jqj+kqk)}
This equation can be proven, starting with the basis {1, i, j, k}:
Consequently, since f 4 {\displaystyle f_{4}} is linear ,
The success of complex analysis in providing a rich family of holomorphic functions for scientific work has engaged some workers in efforts to extend the planar theory, based on complex numbers, to a 4-space study with functions of a quaternion variable. [ 1 ] These efforts were summarized in Deavours (1973) . [ a ]
Though H {\displaystyle \mathbb {H} } appears as a union of complex planes , the following proposition shows that extending complex functions requires special care:
Let f 5 ( z ) = u ( x , y ) + i v ( x , y ) {\displaystyle f_{5}(z)=u(x,y)+iv(x,y)} be a function of a complex variable, z = x + i y {\displaystyle z=x+iy} . Suppose also that u {\displaystyle u} is an even function of y {\displaystyle y} and that v {\displaystyle v} is an odd function of y {\displaystyle y} . Then f 5 ( q ) = u ( x , y ) + r v ( x , y ) {\displaystyle f_{5}(q)=u(x,y)+rv(x,y)} is an extension of f 5 {\displaystyle f_{5}} to a quaternion variable q = x + y r {\displaystyle q=x+yr} where r 2 = − 1 {\displaystyle r^{2}=-1} and r ∈ H {\displaystyle r\in \mathbb {H} } .
Then, let r ∗ {\displaystyle r^{*}} represent the conjugate of r {\displaystyle r} , so that q = x − y r ∗ {\displaystyle q=x-yr^{*}} . The extension to H {\displaystyle \mathbb {H} } will be complete when it is shown that f 5 ( q ) = f 5 ( x − y r ∗ ) {\displaystyle f_{5}(q)=f_{5}(x-yr^{*})} . Indeed, by hypothesis
In the following, colons and square brackets are used to denote homogeneous vectors .
The rotation about axis r is a classical application of quaternions to space mapping. [ 2 ] In terms of a homography , the rotation is expressed
where u = exp ( θ r ) = cos θ + r sin θ {\displaystyle u=\exp(\theta r)=\cos \theta +r\sin \theta } is a versor . If p * = − p , then the translation q ↦ q + p {\displaystyle q\mapsto q+p} is expressed by
Rotation and translation xr along the axis of rotation is given by
Such a mapping is called a screw displacement . In classical kinematics , Chasles' theorem states that any rigid body motion can be displayed as a screw displacement. Just as the representation of a Euclidean plane isometry as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and the screw axis required, is a matter of quaternion arithmetic with homographies: Let s be a right versor, or square root of minus one, perpendicular to r , with t = rs .
Consider the axis passing through s and parallel to r . Rotation about it is expressed [ 3 ] by the homography composition
where z = u s − s u = sin θ ( r s − s r ) = 2 t sin θ . {\displaystyle z=us-su=\sin \theta (rs-sr)=2t\sin \theta .}
Now in the ( s,t )-plane the parameter θ traces out a circle u − 1 z = u − 1 ( 2 t sin θ ) = 2 sin θ ( t cos θ − s sin θ ) {\displaystyle u^{-1}z=u^{-1}(2t\sin \theta )=2\sin \theta (t\cos \theta -s\sin \theta )} in the half-plane { w t + x s : x > 0 } . {\displaystyle \lbrace wt+xs:x>0\rbrace .}
Any p in this half-plane lies on a ray from the origin through the circle { u − 1 z : 0 < θ < π } {\displaystyle \lbrace u^{-1}z:0<\theta <\pi \rbrace } and can be written p = a u − 1 z , a > 0. {\displaystyle p=au^{-1}z,\ \ a>0.}
Then up = az , with ( u 0 a z u ) {\displaystyle {\begin{pmatrix}u&0\\az&u\end{pmatrix}}} as the homography expressing conjugation of a rotation by a translation p.
Since the time of Hamilton, it has been realized that requiring the independence of the derivative from the path that a differential follows toward zero is too restrictive: it excludes even f ( q ) = q 2 {\displaystyle \ f(q)=q^{2}\ } from differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable. [ 4 ] [ 5 ] Considering the increment of polynomial function of quaternionic argument shows that the increment is a linear map of increment of the argument. [ dubious – discuss ] From this, a definition can be made:
A continuous function f : H → H {\displaystyle \ f:\mathbb {H} \rightarrow \mathbb {H} \ } is called differentiable on the set U ⊂ H , {\displaystyle \ U\subset \mathbb {H} \ ,} if at every point x ∈ U , {\displaystyle \ x\in U\ ,} an increment of the function f {\displaystyle \ f\ } corresponding to a quaternion increment h {\displaystyle \ h\ } of its argument, can be represented as
where
is linear map of quaternion algebra H , {\displaystyle \ \mathbb {H} \ ,} and o : H → H {\displaystyle \ o:\mathbb {H} \rightarrow \mathbb {H} \ } represents some continuous map such that
and the notation ∘ h {\displaystyle \ \circ h\ } denotes ... [ further explanation needed ]
The linear map d f ( x ) d x {\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}} is called the derivative of the map f . {\displaystyle \ f~.}
On the quaternions, the derivative may be expressed as
Therefore, the differential of the map f {\displaystyle \ f\ } may be expressed as follows, with brackets on either side.
The number of terms in the sum will depend on the function f . {\displaystyle \ f~.} The expressions d s p d f ( x ) d x f o r p = 0 , 1 {\displaystyle ~~{\frac {\operatorname {d} _{sp}\operatorname {d} f(x)}{\operatorname {d} x}}~~{\mathsf {\ for\ }}~~p=0,1~~} are called
components of derivative.
The derivative of a quaternionic function is defined by the expression
where the variable t {\displaystyle \ t\ } is a real scalar.
The following equations then hold:
For the function f ( x ) = a x b , {\displaystyle \ f(x)=a\ x\ b\ ,} where a {\displaystyle \ a\ } and b {\displaystyle \ b\ } are constant quaternions, the derivative is
and so the components are:
Similarly, for the function f ( x ) = x 2 , {\displaystyle \ f(x)=x^{2}\ ,} the derivative is
and the components are:
Finally, for the function f ( x ) = x − 1 , {\displaystyle \ f(x)=x^{-1}\ ,} the derivative is
and the components are: | https://en.wikipedia.org/wiki/Quaternionic_analysis |
A quaternionic matrix is a matrix whose elements are quaternions .
The quaternions form a noncommutative ring , and therefore addition and multiplication can be defined for quaternionic matrices as for matrices over any ring.
Addition . The sum of two quaternionic matrices A and B is defined in the usual way by element-wise addition:
Multiplication . The product of two quaternionic matrices A and B also follows the usual definition for matrix multiplication. For it to be defined, the number of columns of A must equal the number of rows of B . Then the entry in the i th row and j th column of the product is the dot product of the i th row of the first matrix with the j th column of the second matrix. Specifically:
For example, for
the product is
Since quaternionic multiplication is noncommutative, care must be taken to preserve the order of the factors when computing the product of matrices.
The identity for this multiplication is, as expected, the diagonal matrix I = diag(1, 1, ... , 1). Multiplication follows the usual laws of associativity and distributivity . The trace of a matrix is defined as the sum of the diagonal elements, but in general
Left scalar multiplication, and right scalar multiplication are defined by
Again, since multiplication is not commutative some care must be taken in the order of the factors. [ 1 ]
There is no natural way to define a determinant for (square) quaternionic matrices so that the values of the determinant are quaternions. [ 2 ] Complex valued determinants can be defined however. [ 3 ] The quaternion a + bi + cj + dk can be represented as the 2×2 complex matrix
This defines a map Ψ mn from the m by n quaternionic matrices to the 2 m by 2 n complex matrices by replacing each entry in the quaternionic matrix by its 2 by 2 complex representation. The complex valued determinant of a square quaternionic matrix A is then defined as det(Ψ( A )). Many of the usual laws for determinants hold; in particular, an n by n matrix is invertible if and only if its determinant is nonzero.
Quaternionic matrices are used in quantum mechanics [ 4 ] and in the treatment of multibody problems . [ 5 ] | https://en.wikipedia.org/wiki/Quaternionic_matrix |
In geometry , a quaternionic polytope is a generalization of a polytope in real space to an analogous structure in a quaternionic module , where each real dimension is accompanied by three imaginary ones. Similarly to complex polytopes , points are not ordered and there is no sense of "between", and thus a quaternionic polytope may be understood as an arrangement of connected points, lines, planes and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Likewise, each line must contain multiple points, each plane multiple lines, and so on. Since the quaternions are non- commutative , a convention must be made for the multiplication of vectors by scalars, which is usually in favour of left-multiplication. [ 1 ]
As is the case for the complex polytopes, the only quaternionic polytopes to have been systematically studied are the regular ones. Like the real and complex regular polytopes, their symmetry groups may be described as reflection groups. For example, the regular quaternionic lines are in a one-to-one correspondence with the finite subgroups of U 1 ( H ): the binary cyclic groups , binary dihedral groups , binary tetrahedral group , binary octahedral group , and binary icosahedral group . [ 2 ]
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quaternionic_polytope |
In mathematics , quaternionic projective space is an extension of the ideas of real projective space and complex projective space , to the case where coordinates lie in the ring of quaternions H . {\displaystyle \mathbb {H} .} Quaternionic projective space of dimension n is usually denoted by
and is a closed manifold of (real) dimension 4 n . It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line H P 1 {\displaystyle \mathbb {HP} ^{1}} is homeomorphic to the 4-sphere.
Its direct construction is as a special case of the projective space over a division algebra . The homogeneous coordinates of a point can be written
where the q i {\displaystyle q_{i}} are quaternions, not all zero. Two sets of coordinates represent the same point if they are 'proportional' by a left multiplication by a non-zero quaternion c ; that is, we identify all the
In the language of group actions , H P n {\displaystyle \mathbb {HP} ^{n}} is the orbit space of H n + 1 ∖ { ( 0 , … , 0 ) } {\displaystyle \mathbb {H} ^{n+1}\setminus \{(0,\ldots ,0)\}} by the action of H × {\displaystyle \mathbb {H} ^{\times }} , the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside H n + 1 {\displaystyle \mathbb {H} ^{n+1}} one may also regard H P n {\displaystyle \mathbb {HP} ^{n}} as the orbit space of S 4 n + 3 {\displaystyle S^{4n+3}} by the action of Sp ( 1 ) {\displaystyle {\text{Sp}}(1)} , the group of unit quaternions. [ 1 ] The sphere S 4 n + 3 {\displaystyle S^{4n+3}} then becomes a principal Sp(1)-bundle over H P n {\displaystyle \mathbb {HP} ^{n}} :
This bundle is sometimes called a (generalized) Hopf fibration .
There is also a construction of H P n {\displaystyle \mathbb {HP} ^{n}} by means of two-dimensional complex subspaces of H 2 n {\displaystyle \mathbb {H} ^{2n}} , meaning that H P n {\displaystyle \mathbb {HP} ^{n}} lies inside a complex Grassmannian .
The space H P ∞ {\displaystyle \mathbb {HP} ^{\infty }} , defined as the union of all finite H P n {\displaystyle \mathbb {HP} ^{n}} 's under inclusion, is the classifying space BS 3 . The homotopy groups of H P ∞ {\displaystyle \mathbb {HP} ^{\infty }} are given by π i ( H P ∞ ) = π i ( B S 3 ) ≅ π i − 1 ( S 3 ) . {\displaystyle \pi _{i}(\mathbb {HP} ^{\infty })=\pi _{i}(BS^{3})\cong \pi _{i-1}(S^{3}).} These groups are known to be very complex and in particular they are non-zero for infinitely many values of i {\displaystyle i} . However, we do have that
It follows that rationally, i.e. after localisation of a space , H P ∞ {\displaystyle \mathbb {HP} ^{\infty }} is an Eilenberg–Maclane space K ( Q , 4 ) {\displaystyle K(\mathbb {Q} ,4)} . That is H P Q ∞ ≃ K ( Z , 4 ) Q . {\displaystyle \mathbb {HP} _{\mathbb {Q} }^{\infty }\simeq K(\mathbb {Z} ,4)_{\mathbb {Q} }.} (cf. the example K(Z,2) ). See rational homotopy theory .
In general, H P n {\displaystyle \mathbb {HP} ^{n}} has a cell structure with one cell in each dimension which is a multiple of 4, up to 4 n {\displaystyle 4n} . Accordingly, its cohomology ring is Z [ v ] / v n + 1 {\displaystyle \mathbb {Z} [v]/v^{n+1}} , where v {\displaystyle v} is a 4-dimensional generator. This is analogous to complex projective space. It also follows from rational homotopy theory that H P n {\displaystyle \mathbb {HP} ^{n}} has infinite homotopy groups only in dimensions 4 and 4 n + 3 {\displaystyle 4n+3} .
H P n {\displaystyle \mathbb {HP} ^{n}} carries a natural Riemannian metric analogous to the Fubini-Study metric on C P n {\displaystyle \mathbb {CP} ^{n}} , with respect to which it is a compact quaternion-Kähler symmetric space with positive curvature.
Quaternionic projective space can be represented as the coset space
where Sp ( n ) {\displaystyle \operatorname {Sp} (n)} is the compact symplectic group .
Since H P 1 = S 4 {\displaystyle \mathbb {HP} ^{1}=S^{4}} , its tangent bundle is stably trivial. The tangent bundles of the rest have nontrivial Stiefel–Whitney and Pontryagin classes . The total classes are given by the following formulas:
where v {\displaystyle v} is the generator of H 4 ( H P n ; Z ) {\displaystyle H^{4}(\mathbb {HP} ^{n};\mathbb {Z} )} and u {\displaystyle u} is its reduction mod 2. [ 2 ]
The one-dimensional projective space over H {\displaystyle \mathbb {H} } is called the "projective line" in generalization of the complex projective line . For example, it was used (implicitly) in 1947 by P. G. Gormley to extend the Möbius group to the quaternion context with linear fractional transformations . For the linear fractional transformations of an associative ring with 1, see projective line over a ring and the homography group GL(2, A ).
From the topological point of view the quaternionic projective line is the 4-sphere, and in fact these are diffeomorphic manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a Hopf fibration .
Explicit expressions for coordinates for the 4-sphere can be found in the article on the Fubini–Study metric .
The 8-dimensional H P 2 {\displaystyle \mathbb {HP} ^{2}} has a circle action , by the group of complex scalars of absolute value 1 acting on the other side (so on the right, as the convention for the action of c above is on the left). Therefore, the quotient manifold
may be taken, writing U(1) for the circle group . It has been shown that this quotient is the 7- sphere , a result of Vladimir Arnold from 1996, later rediscovered by Edward Witten and Michael Atiyah . | https://en.wikipedia.org/wiki/Quaternionic_projective_space |
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