In the OSTI Collections: Quantum Chaos
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| Article Acknowledgement: Dr. William N. Watson, Physicist DOE Office of Scientific and Technical Information |
Chaos in specific quantum-physical systems
Explorations of general theoretical concepts
Reports available through E-print Network
Any theory whose logical consequences contradict experimentally known facts is evidently missing something significant among its premises. If a theory’s implications bore no resemblance to reality, there’d be no reason to take it seriously. Theories are taken very seriously when they agree with experimental observations over a wide range of phenomena, and continue to predict new phenomena that are actually found to occur. Such theories prove useful for some time as guides to previously unknown aspects of nature and to the design of new technologies. But it has sometimes happened that even theories like these are eventually found to imply that certain real phenomena shouldn’t occur at all. Figuring out how such phenomena do happen requires some revision of the theory’s basic concepts that doesn’t just simply accommodate the new phenomena, but provides a new, possibly deeper explanation for phenomena that were previously known.
Even if the new explanations of already-known phenomena are based on radically different concepts, the new theory of these phenomena will still look very much like the old in one respect: its mathematical expression. Although the math may represent quite different physical concepts in the two theories, their mathematical descriptions of the same phenomena will both be similarly accurate and therefore approximate each other. Only for new phenomena outside the earlier theory’s range of accuracy will the new theory be very different mathematically as well as conceptually. The logical requirement that a proposed new theory correspond in this way to an earlier theory in the earlier theory’s realm of accuracy is referred to in physics as the “correspondence principle”.[Wikipedia]
Such correspondence between newer and earlier theories is exemplified by several conceptual advances. Classical thermodynamics implies that the flow of heat is inevitably from warmer to cooler objects; the later theory of statistical mechanics, according to which heat flow can fluctuate the other way, nonetheless implies that for objects made of very large numbers of particles, these fluctuations are so small as to be practically invisible against the predominant warmer-to-cooler trend. The special theory of relativity, with its concept of interrelated space and time, scarcely implies anything different from Newton’s concepts of absolute space and time for processes involving motion at speeds far below that of light in a vacuum, in which the interrelatedness of space and time is barely evident. The general theory of relativity, which implies that very strong gravitational fields can take the forms of black holes and gravitational waves, also implies that the orbits traced out by objects in weak gravitational fields will look almost the same as those deduced from Newton’s gravitational theory. And in phenomena for which a quantity known as the action is significantly larger than the value of Planck’s constant h[Wikipedia], the discontinuities and interference effects that quantum theory predicts for low-action phenomena become so negligible that classical physics gives accurate results.
This latter correspondence helped guide physicists toward a formulation of quantum theory that accurately described many phenomena, whether the actions involved were a few or many times the size of Planck’s constant. (In fact, it was while quantum theory was being formulated that Niels Bohr[Wikipedia] first articulated the correspondence principle and gave it its name.) But at that time, a less familiar type of phenomenon was scarcely considered. The courses of these phenomena are so sensitive to their starting conditions, that systems which differ only slightly in those starting conditions soon behave in ways that bear no resemblance to each other and exhibit little obvious regularity. This type of phenomenon has in recent decades been found ubiquitous in nature and technology, and the name “chaos” has been given to it.[Wikipedia] Chaotic processes occur in weather, planetary motion in our own solar system and others, seismic phenomena, human physiology, electronic devices, and many other things, and much effort has been made to understand them better.
While many known chaotic systems are easily described in terms of classical theory, classical descriptions of their behavior that don’t account for quantum physics are at best just close approximations to what the systems actually do. But the usual mathematical descriptions of quantum-physical systems don’t involve parameters that vary greatly for small changes in the systems’ starting conditions. Because of this, it hasn’t been immediately obvious how quantum physical laws that apparently govern all physical systems even allow for chaotic phenomena to occur, much less how chaotic behavior in a system is related to its quantum-physical features.[Wikipedia] There has been no lack of effort to figure this out, though; as sometimes happens in the early explorations of a new class of phenomena, many hypotheses about it are proposed and tested, with some of them found at least promising to varying degrees. Often, researchers eventually arrive at a comprehensive theory of the new phenomena that they find fits the known experimental facts and accurately predicts the results of further experiments. Judging from numerous recent publications on chaos in quantum-physical systems, this last stage has not been reached, though some of the ideas being proposed may turn out to form the basis of an accurate theory. What we do see in these reports is a wide variety of approaches taken in efforts to understand the relation of chaos to quantum dynamics.
Classical physical theory represents the way interactions affect objects in terms of the objects’ resulting locations and momenta, so that particle motions are described by trajectories that have definite locations and momenta at each instant of time. Quantum theory recognizes that the more definite an object’s position is, the less definite its momentum can be, and vice versa, so that observable behaviors are described, not by trajectory equations, but equations for wavelike entities. Like a water wave or sound wave, whose height or intensity of vibration varies with position, a quantum-physical wave’s amplitude can vary with its independent variables, whether those variables are position, or momentum, or anything else. The absolute square of the quantum-physical wave’s amplitude for a given position, momentum, &c. represents a probability density: the entity that the wave represents (e.g., an electron, a quark, a photon, a molecule) is likelier to have a position, momentum, &c. where the amplitude is large than where the amplitude is low. How the shape of a wave’s amplitude changes with time is generally different from the way a classical particle’s position or momentum would change with time, even if the same forces were to act in the wave or particle’s environment, though the changes would resemble each other more, the larger the actions associated with them.
In classical theory, physical systems are found to exhibit chaotic sensitivity to their initial conditions only if the equations that describe the systems’ response to interactions lack the property of linearity—the property that sums and multiples of the equation’s solutions are also solutions. [Wikipedia; Wikipedia] Such nonlinearity doesn’t guarantee chaotic behavior but is essential for it. Classical equations for, e.g., the locations and motions of billiard balls bouncing off the edges of a stadium-shaped “billiard table” are nonlinear in this sense.[Wikipedia] One report by researchers at Argentina’s Comisión Nacional de Energía Atómica (CNEA) and France’s Université de Toulouse, UPS considers what would happen when a two-dimensional area, whose stadium-like shape induces chaotic motion in any particles confined to bounce around within it, encloses particles that behave more identically as their temperature lowers and thus exhibit the quantum-physical phenomenon of Bose-Einstein condensation. The researchers accordingly model the condensation with a nonlinear wave equation, and calculate how the degree of nonlinearity in the particle set’s interactions changes the distribution of the particles’ energies, as they report in “Dynamical thermalization of Bose-Einstein condensate in Bunimovich stadium”[E-print Network]. They argue that cold atoms trapped by laser beams could serve as particles confined to a “stadium” to allow experimental study of the phenomenon they calculate.
The Bunimovich stadium just mentioned[Wikipedia] is a system that can confine particles of low-enough energy to its interior. Other billiardlike systems have openings for particles to enter and leave, so that particles entering from one direction may be scattered from it in another direction. The chaotic sensitivity to starting conditions manifests itself by large variations in the times, energies, and directions at which classical particles exit from the system given small variations in their entry times, energies, and directions. A quantum-physical analysis of a system of this type is described in the report “Effect of chiral symmetry on chaotic scattering from Majorana zero modes”[E-print Network] by researchers at England’s Lancaster University and the Netherlands’ Universiteit Leiden. A physical system with “chiral symmetry” looks the same as its mirror image. The “Majorana zero modes” are a superconductor’s zero-energy modes of electron-and-“hole” pairs, each of which consists of an electron paired with a “hole”, or bubblelike empty space within a sea of electrons[Wikipedia], and moves as a single unit.
Figure 1. A billiard ball moving on a stadium-shaped billiard table (a Buminovich stadium). Small differences in the ball’s initial speed or direction very quickly turn into big differences in the way the ball moves—a characteristic of a chaotic system. Quantum chaos studies have included the examination of waves confined to propagate within Buminovich stadia and similar billiardlike systems. (From https://commons.wikimedia.org/wiki/File%3ABunimovichStadium.svg. By George Stamatiou convert to svg (inkscape) the XaosBits's work [CC BY 2.5 (http://creativecommons.org/licenses/by/2.5)], via Wikimedia Commons; license and terms https://creativecommons.org/licenses/by/2.5/deed.en.)
A quantum-mechanical scatterer can be characterized by a spectrum of possible time spans that a particle can take to exit the scatterer after entering. If chaotic scattering were accurately described in all respects by classical theory, each possible scattering process would be associated with a definite entry-to-exit time. But quantum theory implies that definite entry-to-exit times only occur for particles that approach the scatterer at certain particular energies and directions; a particle approaching with a slightly different energy or direction from any of these may exit the scatterer at any one of the corresponding definite times, with each possible time having its own definite probability of occurring. The researchers found that chiral symmetry in a chaotic scatterer strongly modifies its spectra of possible delay times and associated particle energies, and also found that the chiral symmetry is necessary for the delay times and energies to depend significantly on the number of Majorana zero modes.
Finite two-dimensional surfaces need not have boundaries. The relation of chaos to waves confined to spheres and tori
[Wikipedia]is explored in the report “On the Distribution of Local Extrema in Quantum Chaos”[E-print Network]. Its authors, at the Institute of Science and Technology Austria and Yale University, begin by noting that except for special cases, quantum-physical waves of definite energy will seem somewhat random, the exceptions being waves whose classical-trajectory counterparts are nonchaotic (e.g., the regular waves of definite energy that can occupy a two-dimensional surface). The researchers further note that usually, for the exceptions, any small perturbation of the space the system occupies will induce chaotic behavior; accordingly, they approximated the irregular forms of waves confined to a surface when the surface’s connectivity is altered by the addition of small randomly placed wormholes, and presented data on how chaotic the approximations appeared to be.
Chaos in specific quantum-physical systems
While these two investigations provided some new information about the quantum physics of chaotic systems, other chaotic systems with known quantum-physical features have been explored further by using the knowledge of those features as a starting point. “Nuclear Level Density: Shell Model vs Mean Field”[E-print Network], by researchers at City University of New York and Michigan State University, explains a new way to calculate how many states of a given atomic nucleus have definite energies within any specific energy range—an essential sort of information for understanding how any quantum system made of interacting particles will behave, and particularly for how a nucleus behaves in technologically and astrophysically interesting nuclear reactions.
The new calculation method builds upon one of the conventional mathematical models that treat the protons and neutrons in a nucleus as being arranged in a “shell structure”[Wikipedia] similar to that of the electrons in an atom that orbit the nucleus[Wikipedia]. What’s new in the method is its accounting for the protons’ and neutrons’ chaotic dynamics that their interactions induce, and the effect of this chaos on what definite energy states are possible for the nucleus. The researchers discuss limitations of the new method, including the inability of finite computational resources to account for all the influences on the physical system, and note that the method accurately counts definite-energy states only for cases in which the set of omitted influences is insignificant; however, the researchers state that the method’s range of accuracy does include energy ranges of astrophysical nuclear interactions. In contrast, other standard methods of calculating definite-energy states simplify the entire set of forces on each proton or neutron that all the other nuclear particles exert on it as if those forces were just a single, average (“mean”) field of force.[Wikipedia] The researchers find that these “mean-field” methods are generally less accurate, and in some cases “not in good agreement with exact results”, but “in many cases they are nevertheless quite reasonable for practical use”. Their main conclusion from the work they describe is that, when the input to their calculation includes a considerable number of the relevant definite-energy states, the calculated number of definite-energy states within a unit-size range of a given energy varies smoothly with the given energy.
A group of researchers affiliated with France’s Centre National de la Recherche Scientifique (CNRS), at the Université de Lille 1 Sciences et Technologies and the Collège de France, have performed experiments on sets of atoms to gain insight into a rather different-looking solid-state process. According to a classical-theory analysis, particles that start drifting freely along one dimension of space with very similar initial positions and momenta, and then receive kicks at regular intervals from a strong-enough force that varies sinusoidally with the particles’ position and with time of occurrence, will quickly end up moving in ways that don’t resemble each other at all—that is, their motion will be made chaotic by the random kicks. The mathematical description of this system, and of a set of objects that revolve freely around one axis in between brief torques, is exactly the same, so either type of system is known as a kicked rotator or kicked rotor, whether its single dimension of motion is a straight line or a circle. The quantum-mechanical analog of a kicked rotor involves a wave motion rather than a classical particle trajectory, and its mathematical description could also apply to an actual rotor or to a wave moving along a straight line that’s periodically refracted to varying degrees. This latter analog, in the form of atoms whose quantum-mechanical waves were periodically refracted by laser beams, was studied by the CNRS researchers, who reported their findings in “Experimental observation of two-dimensional Anderson localization with the atomic kicked rotor”[E-print Network].
The motivation for their study is indicated by the “two-dimensional Anderson localization” in the title. An electrically conducting three-dimensional solid, by the addition of more and more impurities, can gradually become less conductive and more insulating. In quantum-physical terms, the waves that describe the condition of electrons within the solid change from propagating freely among the atoms of a pure conductor to being “localized” at particular atoms in an insulator, as worked out by Nobel laureate Philip W. Anderson.[Wikipedia; Nobelprize.org] But for electrons confined to moving along only two dimensions of a solid, following a surface or a thin layer, how to accurately observe or calculate the effect of impurities on their motion has been less clear.
As the researchers show, however, electrons in two-dimensional motion are as analogous to periodically refracted atoms as periodically refracted atoms are to kicked rotors, but their corresponding variables are different. First, an atom wave subject to refractions whose strength varies with time as well as position along one dimension is mathematically equivalent to an atom wave whose refractive “kick” strengths don’t vary with time, but do vary along two spatial dimensions. Second, the atom wave’s motion doesn’t get localized to particular positions in two dimensions, but does indicate an association with particular momenta along those dimensions. Since the atom wave’s two momentum-like dimensions are analogs of two spatial dimensions for an electron wave, an experiment that determines whether “momentum localization” occurs in kicked atom waves will indicate whether positional localization occurs in a two-dimensional electron wave. The researchers found experimental evidence of such localization in their atom system, and showed that a particular scheme for calculating the scale of the localization yields values that roughly correspond with their experimental findings.
Chaos is only one phenomenon for which the correct quantum-theory description is unobvious. Gravitation is another. According to Einstein’s theory of gravitation, general relativity[Wikipedia], material objects tend to accelerate toward each other because they follow the straightest possible paths in curved spacetime, whose exact curvature is largely determined by the material it contains. While general relativity theory has been repeatedly found accurate as far as it has been tested, it is a classical-type theory that takes no account of matter’s quantum-physical nature, and finding a quantum-physical version is problematic: common mathematical modifications that transform classical theories of other phenomena into their quantum-physical equivalents require that the phenomena exhibit features of a particular type called “Dirac observables”. Such features are lacking in chaotic phenomena, and there’s “good reason” to believe that gravitational processes, as described by general relativity theory, may also lack them, according to the recent report “Chaos, Dirac observables and constraint quantization”[E-print Network] by researchers with Canada’s Perimeter Institute for Theoretical Physics and University of New Brunswick, the Universidad Nacional Autónoma de Mexico, the African Institute for Mathematical Sciences—Ghana, and the University of Ghana.
While methods of getting around this problem have been suggested, it has been argued that they won’t work in the presence of chaos—and there’s also good reason, according to the report, to believe that general-relativistic gravitation can be chaotic. The report argues that Dirac observables, as usually defined, almost certainly don’t exist for the complete general relativity theory, and so explores how redefining the relationships among different spacetime configurations in a certain way might compensate for this lack and provide a way to translate the classical general relativity theory into a quantum-physical version. The researchers note that this translation technique underlies the proposed theory of loop quantum cosmology[Wikipedia] and one formulation of loop quantum gravity[Wikipedia], according to which separations between different points have a minimum possible nonzero value. They found that, when applied to a few simple model theories that share some features of general relativity theory, the technique led to “an interesting quantum theory”, though “it is of course an open question whether these techniques will also work for full general relativity—and ultimately whether our own universe is actually described by such a modified quantum theory.”
Much as some illumination of quantum-physical processes was provided by even the incomplete early forms of quantum theory, some aspects of quantum physics in gravitational fields have been analyzed in terms of the incomplete understanding already achieved. The incompleteness of this understanding is shown by the fact that different sets of apparently reasonable assumptions lead to mutually contradictory conclusions about whether the information represented by a physical system is preserved or lost if that system falls into a black hole. One recently examined set of premises, reported in “Quantum chaos inside Black Holes” by Andrea Addazi of Italy’s Università degli Studi dell'Aquila and Laboratori Nazionali del Gran Sasso, is presented as a matter for further investigations rather than a definitive solution of black-hole problems. Adazzi concludes that, where a black hole can be described accurately in terms of a semiclassical theory of gravitation, the black hole can be thought of as a particular kind of “frizzy” structure within which information will propagate chaotically.
A different relationship between chaos and quantum physics was found by researchers at Ukraine’s V. E. Lashkaryov Institute of Semiconductor Physics and England’s Lancaster University and University of Warwick for electron waves that travel in semiconducting superlattices made of regularly spaced material units much larger than atomic or molecular size[Wikipedia; OSTI]. When the superlattice is subject to an electric or magnetic field alone, the electron waves’ motion is characterized by a frequency—a back-and-forth oscillation frequency for a pure electric field or an orbital frequency for a pure magnetic field. If both types of field are present, calculations show that the electron wave will travel rather than oscillate in place, with the electron drift velocity being high whenever the electric/magnetic field strength ratio is an integer multiple of a value that depends on the size of the superlattice’s repeating unit.
It was already known that classical physical systems analogous to these would exhibit a particular pattern of regular and chaotic motions: if a “classical electron’s” initial position and momentum were within certain regularly spaced ranges, the electron’s motion would be nonchaotic, while if its initial position and momentum were outside those ranges, it would move chaotically. Conjecture had it that the highest drift velocities of real quantum-physical electrons in superlattices were counterparts to classical-theory motion along the chaotic trajectories, implying that chaotic diffusion could control the enhancement of electron transport in a superlattice. But a careful analysis by the researchers showed this conjecture false: the electron drift enhancement is actually a regular process, not a chaotic one, and chaos actually inhibits it. They report these findings in “Regular rather than chaotic origin of the resonant transport in superlattices”[E-print Network].
Explorations of general theoretical concepts
Four other reports describe different candidate criteria for distinguishing chaotic from nonchaotic quantum-physical systems more generally.
One of these reports, by researchers at the University of British Columbia, the Freie Universität Berlin, and the University of New Mexico, relates their recent findings about a method, called “quantum tomography”[Wikipedia], for determining the state of a quantum-physical system. This technique first involves setting up an ensemble of systems that all share an identical quantum-physical description, then measuring the systems in a particular way, and finally deducing from the measurements the single description that best describes all the systems. According to the researchers’ report (“Characterizing and Quantifying Quantum Chaos with Quantum Tomography”[E-print Network]), how chaotic the set of systems is depends not only on how much information each measurement provides about the systems’ common description (an earlier finding), but on the kind of symmetry the systems have in relation to mirror-image and time-reversed versions of themselves. Furthermore, for fully chaotic systems, the amount of information provided by the sequence of measurements is found to be well described by the long-known “random matrix theory”[Wikipedia; Wikipedia] of systems whose own interactions are incompletely specified. The forces that govern how any quantum-physical system develops are ordinarily represented by a matrix; incompletely specified forces can be represented by sets of random matrices, each of which is consistent with the incomplete force specification.
Figure 2. Pairs of 2 x 2 matrices that each exhibit a different type of symmetry. The product of each pair found by matrix multiplication is the unit matrix 
, so that the matrices in each pair are inverses of each other. The first, second, and third pairs of matrices are known as orthogonal, unitary, and symplectic matrices, respectively. Incompletely specified interactions that govern a quantum-physical system can be represented by ensembles of larger n x n matrices, in which all the matrices are of the same symmetry type corresponding to the symmetry of the interactions, with the variables a, b, c, and/or d of each individual matrix replaced by random numbers such as 0 ≤ a,b,c,d ≤ 1.
The probability that a specific matrix actually characterizes the forces governing a system can be calculated from the information we lack about those forces. But the probability can also be determined from the lack of correlation among observable parameters of the system in its equilibrium state, according to “A derivation of the Gaussian ensembles from the quantum ergodic hierarchy”[E-print Network] by researchers at Argentina’s Universidad Nacional de La Plata and Instituto de Física de Rosario. The “quantum ergodic hierarchy” referred to in this report’s title is a hierarchy of quantum systems that exhibit different degrees of randomness; systems that lack the aforementioned correlation among observable parameters are at the next-to-lowest level in this hierarchy. A corresponding hierarchy for systems accurately described by classical theory[Stanford] is described in the online Stanford Encyclopedia of Philosophy.
In “Tunneling as a Source for Quantum Chaos”[E-print Network], by researchers at Ariel University, Bar-Ilan University, and Tel Aviv University in Israel, the prototypical system is one that would look familiar to any student of quantum physics—a wave confined to a space between two walls in one dimension, in the midst of which is a region where the wave has a higher potential energy than in the rest of the space between the walls. In the corresponding classical theory of a particle bouncing back and forth between the two walls, such a region is where the particle’s kinetic energy is reduced as it’s converted to potential energy, meaning that the particle moves slower while it’s in this region and faster while it’s out. To be able to cross the region, a classical particle after entering the region would have to have some kinetic energy left over; otherwise it couldn’t move to the region’s other side, and the region would constitute a barrier for the particle. But a wave of even less total energy could still cross the barrier, in a process known as quantum-mechanical tunneling.[Wikipedia]
Figure 3. Depiction of a wave “tunneling” through a barrier. The incident wave is partially reflected and partially transmitted through the barrier. This quantum-mechanical process has no counterpart in classical particle mechanics; according to classical theory, a particle encountering a point or region within which its potential energy would have to be greater than its total energy would experience a repelling force there and bounce back from it. (From https://en.wikipedia.org/wiki/Quantum_tunnelling#/media/File:Quantum_Tunnelling_animation.gif. By Yuvalr (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons; license and terms https://creativecommons.org/licenses/by-sa/3.0/deed.en.)
What makes this familiar system interesting for quantum chaos relates to the recognition that a moving wave’s classical-theory counterpart isn’t necessarily a single classical-particle motion. Generally, a wave’s motion would correspond to what classical theory represents as an entire family of possible motions for a particle. A quantum-physical wave pulse may spread out as it moves. For a pulse that spreads slowly, the corresponding classical approximation is generally a set of very similar particle motions, with paths that look very similar for slightly different starting conditions. On the other hand, a chaotic family of classical particle motions, with paths that despite close resemblance at the start soon cease to resemble each other at all, approximates a quantum-physical wave pulse that spreads out very quickly. The converse isn’t necessarily true, however; a fast-spreading wave pulse doesn’t necessarily have a classical-physics counterpart—for example, a wave pulse, encountering a region of high potential energy that its classical-particle counterpart would bounce off of, would only partially reflect from the region while partly tunneling through it as well. Accordingly, the authors of “Tunneling as a Source for Quantum Chaos” examine a wave that spreads rapidly in opposite directions from the high-potential region (and becomes more complex in other ways), and show how it can reasonably be considered a chaotic quantum-physical system.
As mentioned at the beginning of this article, quantum-physical systems are more accurately described by classical theory when the significant actions of the system are large compared to Planck’s constant h: the larger a system’s action/h ratio becomes, the more accurate its classical-theory description gets—provided that the action/h ratio increases in an appropriate way. The appropriate way for a multiparticle system has to take account of how this ratio relates, among other things, to the number of particles in the system. The approximate classical descriptions of systems that have large significant actions per particle, or large particle numbers, or large significant actions and large particle numbers, are all quite different. A method of analyzing systems of the second type, with large numbers of particles but not necessarily large action/h ratios per particle, was adapted to systems of the third type with both parameters large by researchers now at Germany’s Universität Duisburg-Essen and Universität zu Köln. Their adaptations, reported in “Classical foundations of many-particle quantum chaos”[E-print Network], treat a system’s configuration and momenta as varying not only with time, but with the number of particles in the system. Thus the classical-theory depictions of the system are not one-dimensional curves, with different points along the curves’ lengths representing the system’s state at different times, but two-dimensional surfaces whose different points represent the system at different times and with different numbers of particles composing it. Whereas chaos-related features of many-particle quantum-physical systems correspond to particular features of related curves when the systems’ significant action/h ratios per particle are fixed, they correspond to features of related surfaces when these ratios and the numbers of particles both tend toward infinity.
Wikipedia
- Correspondence principle
- Planck constant
- Niels Bohr
- Chaos theory
- Quantum chaos
- Dynamical billiards: Bunimovich stadium
- Linearity: In mathematics; Physics
- Electron hole
- Torus
- Nuclear shell model
- Electron configuration: Shells and subshells
- Mean field theory
- Philip Warren Anderson
- General relativity
- Loop quantum cosmology
- Loop quantum gravity
- Superlattice
- Quantum tomography
- Random matrix: Physics
- Quantum tunneling
- Comisión Nacional de Energía Atómica (CNEA), Argentina [English]
- Laboratoire de Physique Théorique [English] du CNRS (Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes—IRSAMC [English]), Université de Toulouse, UPS [English], France
- Lancaster University, United Kingdom
- Instituut-Lorentz, Universiteit Leiden [English], The Netherlands
- IST Austria [English]
- Yale University
- City University of New York
- Michigan State University
- Centre National de la Recherche Scientifique (CNRS), France[English]
- Université de Lille 1 Sciences et Technologies, CNRS, France [English] [Spanish] [Dutch]
- Collège de France [English] [Chinese]
- Perimeter Institute for Theoretical Physics, Canada [French]
- University of New Brunswick, Canada
- Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de Mexico, Mexico City
- African Institute for Mathematical Sciences—Ghana
- University of Ghana
- Università degli Studi dell'Aquila [English], Italy
- Laboratori Nazionali del Gran Sasso [English], Istituto Nazionale di Fisica Nucleare (INFN) [English], Italy
- V. E. Lashkaryov Institute of Semiconductor Physics, Ukraine [Russian] [English]
- Lancaster University, United Kingdom
- University of Warwick, United Kingdom
- University of British Columbia, Canada
- Freie Universität Berlin [English]
- University of New Mexico
- Universidad Nacional de La Plata, Argentina
- Instituto de Física de Rosario, Argentina [English]
- Ariel University, Israel [English]
- Bar-Ilan University, Israel [Arabic] [Chinese] [English]
- Tel Aviv University, Israel [English]
- (Universität Duisburg-Essen, Germany [English])
- (Universität zu Köln, Germany [English])
Reports available through E-print Network
- “Dynamical thermalization of Bose-Einstein condensate in Bunimovich stadium” [Metadata]
- “Effect of chiral symmetry on chaotic scattering from Majorana zero modes” [Metadata]
- “Local Extrema in Quantum Chaos” [Metadata]
- “Nuclear Level Density: Shell Model vs Mean Field” [Metadata]
- “Experimental observation of two-dimensional Anderson localization with the atomic kicked rotor” [Metadata]
- “Chaos, Dirac observables and constraint quantization” [Metadata]
- “Quantum chaos inside Black Holes” [Metadata]
- “Regular rather than chaotic origin of the resonant transport in superlattices” [Metadata]
- “Characterizing and Quantifying Quantum Chaos with Quantum Tomography” [Metadata]
- “A derivation of the Gaussian ensembles from the quantum ergodic hierarchy” [Metadata]
- “Tunneling as a Source for Quantum Chaos” [Metadata]
- “Classical foundations of many-particle quantum chaos” [Metadata]
- Philip W. Anderson’s Nobel lecture
- In the OSTI Collections: Metamaterials
- Stanford Encyclopedia of Philosophy: The Ergodic Hierarchy (Section 3)
