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Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Quantum Information".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 69832

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Guest Editor
CAMTP-Center for Applied Mathematics and Theoretical Physics, University of Maribor, SI-2000 Maribor, Slovenia
Interests: quantum chaos; classical chaos; generic/mixed-type systems; chaotic diffusion; dynamical localization; structure of chaotic eigenstates; statistical properties of phase space localized chaotic eigenstates; statistical properties of energy spectra; semiclassical approximations; WKB theory; time-dependent classical and quantum Hamilton systems

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to collect original research articles on the most recent research in quantum chaos, as well as comprehensive review articles covering quantum chaos from either a theoretical or experimental viewpoint. A review can focus on either a wide context or the recent research contributions of the author(s) and related works of other researchers on the same topic. For example, reviews of results obtained so far on the out-of-time-order correlator or on dynamical localization, etc., are welcome. Topics can include any subject in the area of quantum or wave chaos, as well as subjects in classical chaos in a manner that is relevant to quantum chaos.

Moreover, we welcome articles focusing on the mathematical methods and tools used to study quantum chaos, such as random matrix theories—including supersymmetric methods, semiclassical methods and theoretical results, and sophisticated advanced numerical methods. Needless to say, papers can address phenomena both in the time evolution of systems as well as their stationary properties, such as the structure and properties of the eigenstates of chaotic systems.

Research focusing on applications is welcome as well, such as research in microwave and optical systems; in acoustics; in mesoscopic and nano systems; and in atomic, molecular, and solid-state physics. In particular, works on quantum many-body systems are welcome, as well as their classical counterparts, if they exist.

Professor Giulio Casati has had an admirable impact on physics, especially in classical and quantum chaos, where his work over more than five decades has laid down the foundations, but also in many other areas of theoretical physics and applications.

Moreover, he has been and still is an active organizer of science, both in creating and leading elite academic and research institutions, as well as in organizing many world-top level conferences, workshops, and schools. In particular, he has been supporting young researchers all over the world in an important way.

In recognition of his creative and influential life opus and of his plentiful contributions for the good of our scientific community, this Special Issue is dedicated to him on the occasion of his 80th birthday (9 December 2022).

Prof. Dr. Marko Robnik
Guest Editor

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • theoretical and experimental quantum or wave chaos
  • time evolution of quantum systems
  • structure of eigenstates and energy spectra
  • random matrix theories
  • supersymmetric methods
  • semiclassical methods and results
  • advanced numerical methods
  • applications in wave systems
  • mesoscopic systems
  • nano systems
  • atomic, molecular and solid-state systems
  • quantum many body systems

Published Papers (40 papers)

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Editorial

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2 pages, 156 KiB  
Editorial
Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday
by Marko Robnik
Entropy 2023, 25(9), 1279; https://doi.org/10.3390/e25091279 - 31 Aug 2023
Viewed by 870
Abstract
Quantum chaos is the study of phenomena in the quantum domain which correspond to classical chaos [...] Full article

Research

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17 pages, 1058 KiB  
Article
Chaos Detection by Fast Dynamic Indicators in Reflecting Billiards
by Gabriele Gradoni, Giorgio Turchetti and Federico Panichi
Entropy 2023, 25(9), 1251; https://doi.org/10.3390/e25091251 - 23 Aug 2023
Viewed by 890
Abstract
The propagation of electromagnetic waves in a closed domain with a reflecting boundary amounts, in the eikonal approximation, to the propagation of rays in a billiard. If the inner medium is uniform, then the symplectic reflection map provides the polygonal rays’ paths. The [...] Read more.
The propagation of electromagnetic waves in a closed domain with a reflecting boundary amounts, in the eikonal approximation, to the propagation of rays in a billiard. If the inner medium is uniform, then the symplectic reflection map provides the polygonal rays’ paths. The linear response theory is used to analyze the stability of any trajectory. The Lyapunov and reversibility error invariant indicators provide an estimate of the sensitivity to a small initial random deviation and to a small random deviation at any reflection, respectively. A family of chaotic billiards is considered to test the chaos detection effectiveness of the above indicators. Full article
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20 pages, 2464 KiB  
Article
Quantum Entropies and Decoherence for the Multiparticle Quantum Arnol’d Cat
by Giorgio Mantica
Entropy 2023, 25(7), 1004; https://doi.org/10.3390/e25071004 - 29 Jun 2023
Viewed by 892
Abstract
I study the scaling behavior in the physical parameters of dynamical entropies, classical and quantum, in a specifically devised model of collision-induced decoherence in a chaotic system. The treatment is fully canonical and no approximations are involved or infinite limits taken. I present [...] Read more.
I study the scaling behavior in the physical parameters of dynamical entropies, classical and quantum, in a specifically devised model of collision-induced decoherence in a chaotic system. The treatment is fully canonical and no approximations are involved or infinite limits taken. I present this model in a detailed way, in order to clarify my views in the debate about the nature, definition, and relevance of quantum chaos. Full article
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19 pages, 26273 KiB  
Article
Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles
by Barbara Dietz
Entropy 2023, 25(5), 762; https://doi.org/10.3390/e25050762 - 6 May 2023
Cited by 1 | Viewed by 1844
Abstract
Rectangular billiards have two mirror symmetries with respect to perpendicular axes and a twofold (fourfold) rotational symmetry for differing (equal) side lengths. The eigenstates of rectangular neutrino billiards (NBs), which consist of a spin-1/2 particle confined through boundary conditions to a planar domain, [...] Read more.
Rectangular billiards have two mirror symmetries with respect to perpendicular axes and a twofold (fourfold) rotational symmetry for differing (equal) side lengths. The eigenstates of rectangular neutrino billiards (NBs), which consist of a spin-1/2 particle confined through boundary conditions to a planar domain, can be classified according to their transformation properties under rotation by π (π/2) but not under reflection at mirror-symmetry axes. We analyze the properties of these symmetry-projected eigenstates and of the corresponding symmetry-reduced NBs which are obtained by cutting them along their diagonal, yielding right-triangle NBs. Independently of the ratio of their side lengths, the spectral properties of the symmetry-projected eigenstates of the rectangular NBs follow semi-Poisson statistics, whereas those of the complete eigenvalue sequence exhibit Poissonian statistics. Thus, in distinction to their nonrelativistic counterpart, they behave like typical quantum systems with an integrable classical limit whose eigenstates are non-degenerate and have alternating symmetry properties with increasing state number. In addition, we found out that for right triangles which exhibit semi-Poisson statistics in the nonrelativistic limit, the spectral properties of the corresponding ultrarelativistic NB follow quarter-Poisson statistics. Furthermore, we analyzed wave-function properties and discovered for the right-triangle NBs the same scarred wave functions as for the nonrelativistic ones. Full article
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14 pages, 608 KiB  
Article
Density of Avoided Crossings and Diabatic Representation
by Anatoly E. Obzhirov and Eric J. Heller
Entropy 2023, 25(5), 751; https://doi.org/10.3390/e25050751 - 4 May 2023
Cited by 1 | Viewed by 1512
Abstract
Electronic structure theory describes the properties of solids using Bloch states that correspond to highly symmetrical nuclear configurations. However, nuclear thermal motion destroys translation symmetry. Here, we describe two approaches relevant to the time evolution of electronic states in the presence of thermal [...] Read more.
Electronic structure theory describes the properties of solids using Bloch states that correspond to highly symmetrical nuclear configurations. However, nuclear thermal motion destroys translation symmetry. Here, we describe two approaches relevant to the time evolution of electronic states in the presence of thermal fluctuations. On the one hand, the direct solution of the time-dependent Schrodinger equation for a tight-binding model reveals the diabatic nature of time evolution. On the other hand, because of random nuclear configurations, the electronic Hamiltonian falls into the class of random matrices, which have universal features in their energy spectra. In the end, we discuss combining two approaches to obtain new insights into the influence of thermal fluctuations on electronic states. Full article
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13 pages, 2488 KiB  
Article
On Two Non-Ergodic Reversible Cellular Automata, One Classical, the Other Quantum
by Tomaž Prosen
Entropy 2023, 25(5), 739; https://doi.org/10.3390/e25050739 - 30 Apr 2023
Cited by 1 | Viewed by 1185
Abstract
We propose and discuss two variants of kinetic particle models—cellular automata in 1 + 1 dimensions—that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing [...] Read more.
We propose and discuss two variants of kinetic particle models—cellular automata in 1 + 1 dimensions—that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing two species of quasiparticles: stable massless matter particles moving with velocity ±1 and unstable standing (zero velocity) field particles. We discuss two distinct continuity equations for three conserved charges of the model. While the first two charges and the corresponding currents have support of three lattice sites and represent a lattice analogue of the conserved energy–momentum tensor, we find an additional conserved charge and current with support of nine sites, implying non-ergodic behaviour and potentially signalling integrability of the model with a highly nested R-matrix structure. The second model represents a quantum (or stochastic) deformation of a recently introduced and studied charged hardpoint lattice gas, where particles of different binary charge (±1) and binary velocity (±1) can nontrivially mix upon elastic collisional scattering. We show that while the unitary evolution rule of this model does not satisfy the full Yang–Baxter equation, it still satisfies an intriguing related identity which gives birth to an infinite set of local conserved operators, the so-called glider operators. Full article
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13 pages, 1670 KiB  
Article
Quantization of Integrable and Chaotic Three-Particle Fermi–Pasta–Ulam–Tsingou Models
by Alio Issoufou Arzika, Andrea Solfanelli, Harald Schmid and Stefano Ruffo
Entropy 2023, 25(3), 538; https://doi.org/10.3390/e25030538 - 21 Mar 2023
Cited by 1 | Viewed by 1399
Abstract
We study the transition from integrability to chaos for the three-particle Fermi–Pasta–Ulam–Tsingou (FPUT) model. We can show that both the quartic β-FPUT model (α=0) and the cubic one (β=0) are integrable by introducing an [...] Read more.
We study the transition from integrability to chaos for the three-particle Fermi–Pasta–Ulam–Tsingou (FPUT) model. We can show that both the quartic β-FPUT model (α=0) and the cubic one (β=0) are integrable by introducing an appropriate Fourier representation to express the nonlinear terms of the Hamiltonian. For generic values of α and β, the model is non-integrable and displays a mixed phase space with both chaotic and regular trajectories. In the classical case, chaos is diagnosed by the investigation of Poincaré sections. In the quantum case, the level spacing statistics in the energy basis belongs to the Gaussian orthogonal ensemble in the chaotic regime, and crosses over to Poissonian behavior in the quasi-integrable low-energy limit. In the chaotic part of the spectrum, two generic observables obey the eigenstate thermalization hypothesis. Full article
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10 pages, 4198 KiB  
Article
Spectral Form Factor and Dynamical Localization
by Črt Lozej
Entropy 2023, 25(3), 451; https://doi.org/10.3390/e25030451 - 4 Mar 2023
Viewed by 1550
Abstract
Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is [...] Read more.
Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is typically computed from the corresponding classical dynamics. In this paper, we present an alternative approach based purely on the study of spectral fluctuations of the quantum system. The information about the transport times is encoded in the spectral form factor, which is the Fourier transform of the two-point spectral autocorrelation function. We compute large samples of the energy spectra (of the order of 106 levels) and spectral form factors of 22 stadium billiards with parameter values across the transition between the localized and extended eigenstate regimes. The transport time is obtained from the point when the spectral form factor transitions from the non-universal to the universal regime predicted by random matrix theory. We study the dependence of the transport time on the parameter value and show the level repulsion exponents, which are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way. Full article
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12 pages, 1348 KiB  
Article
The Effect of On-Site Potentials on Supratransmission in One-Dimensional Hamiltonian Lattices
by Tassos Bountis and Jorge E. Macías-Díaz
Entropy 2023, 25(3), 423; https://doi.org/10.3390/e25030423 - 26 Feb 2023
Cited by 2 | Viewed by 1086
Abstract
We investigated a class of one-dimensional (1D) Hamiltonian N-particle lattices whose binary interactions are quadratic and/or quartic in the potential. We also included on-site potential terms, frequently considered in connection with localization phenomena, in this class. Applying a sinusoidal perturbation at one [...] Read more.
We investigated a class of one-dimensional (1D) Hamiltonian N-particle lattices whose binary interactions are quadratic and/or quartic in the potential. We also included on-site potential terms, frequently considered in connection with localization phenomena, in this class. Applying a sinusoidal perturbation at one end of the lattice and an absorbing boundary on the other, we studied the phenomenon of supratransmission and its dependence on two ranges of interactions, 0<α< and 0<β<, as the effect of the on-site potential terms of the Hamiltonian varied. In previous works, we studied the critical amplitude As(α,Ω) at which supratransmission occurs, for one range parameter α, and showed that there was a sharp threshold above which energy was transmitted in the form of large-amplitude nonlinear modes, as long as the driving frequency Ω lay in the forbidden band-gap of the system. In the absence of on-site potentials, it is known that As(α,Ω) increases monotonically the longer the range of interactions is (i.e., as α0). However, when on-site potential terms are taken into account, As(α,Ω) reaches a maximum at a low value of α that depends on Ω, below which supratransmission thresholds decrease sharply to lower values. In this work, we studied this phenomenon further, as the contribution of the on-site potential terms varied, and we explored in detail their effect on the supratransmission thresholds. Full article
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16 pages, 2518 KiB  
Article
Towards the Resolution of a Quantized Chaotic Phase-Space: The Interplay of Dynamics with Noise
by Domenico Lippolis and Akira Shudo
Entropy 2023, 25(3), 411; https://doi.org/10.3390/e25030411 - 24 Feb 2023
Cited by 1 | Viewed by 997
Abstract
We outline formal and physical similarities between the quantum dynamics of open systems and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, which, for suitable timescales, becomes analogous to the [...] Read more.
We outline formal and physical similarities between the quantum dynamics of open systems and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, which, for suitable timescales, becomes analogous to the Fokker–Planck equation describing classical advection and diffusion. This correspondence allows, in principle, to surmise a finite resolution, other than the Planck scale, for the quantized state space of the open system, particularly meaningful when the latter underlies chaotic classical dynamics. We provide representative examples of the quantum-stochastic parallel with noisy Hopf cycles and Van der Pol-type oscillators. Full article
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14 pages, 459 KiB  
Article
Statistical Topology—Distribution and Density Correlations of Winding Numbers in Chiral Systems
by Thomas Guhr
Entropy 2023, 25(2), 383; https://doi.org/10.3390/e25020383 - 20 Feb 2023
Cited by 1 | Viewed by 1711
Abstract
Statistical Topology emerged as topological aspects continue to gain importance in many areas of physics. It is most desirable to study topological invariants and their statistics in schematic models that facilitate the identification of universalities. Here, the statistics of winding numbers and of [...] Read more.
Statistical Topology emerged as topological aspects continue to gain importance in many areas of physics. It is most desirable to study topological invariants and their statistics in schematic models that facilitate the identification of universalities. Here, the statistics of winding numbers and of winding number densities are addressed. An introduction is given for readers with little background knowledge. Results that my collaborators and I obtained in two recent works on proper random matrix models for the chiral unitary and symplectic cases are reviewed, avoiding a technically detailed discussion. There is a special focus on the mapping of topological problems to spectral ones as well as on the first glimpse of universality. Full article
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17 pages, 573 KiB  
Article
A Physical Measure for Characterizing Crossover from Integrable to Chaotic Quantum Systems
by Chenguang Y. Lyu and Wen-Ge Wang
Entropy 2023, 25(2), 366; https://doi.org/10.3390/e25020366 - 17 Feb 2023
Viewed by 1412
Abstract
In this paper, a quantity that describes a response of a system’s eigenstates to a very small perturbation of physical relevance is studied as a measure for characterizing crossover from integrable to chaotic quantum systems. It is computed from the distribution of very [...] Read more.
In this paper, a quantity that describes a response of a system’s eigenstates to a very small perturbation of physical relevance is studied as a measure for characterizing crossover from integrable to chaotic quantum systems. It is computed from the distribution of very small, rescaled components of perturbed eigenfunctions on the unperturbed basis. Physically, it gives a relative measure to prohibition of level transitions induced by the perturbation. Making use of this measure, numerical simulations in the so-called Lipkin-Meshkov-Glick model show in a clear way that the whole integrability-chaos transition region is divided into three subregions: a nearly integrable regime, a nearly chaotic regime, and a crossover regime. Full article
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17 pages, 10097 KiB  
Article
The Metastable State of Fermi–Pasta–Ulam–Tsingou Models
by Kevin A. Reiss and David K. Campbell
Entropy 2023, 25(2), 300; https://doi.org/10.3390/e25020300 - 6 Feb 2023
Cited by 1 | Viewed by 1628
Abstract
Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For [...] Read more.
Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan–Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a “metastable” state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi–Pasta–Ulam–Tsingou (FPUT) models. We treat both the α-FPUT and β-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure—the spectral entropy (η)—and show that this measure has the power to quantify the distance from equipartition. For the α-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the α-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1-Q1 plane. Applying this procedure gives us a power-law scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα20. We examine the energy spectrum E(k) over time in the α-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the “wave turbulence” theory. We next apply a similar approach to the β-FPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating tm in the β-FPUT model, a very different task than for the α-FPUT model, because the β-FPUT model is not a truncation of an integrable nonlinear model. Full article
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14 pages, 3263 KiB  
Article
Records and Occupation Time Statistics for Area-Preserving Maps
by Roberto Artuso, Tulio M. de Oliveira and Cesar Manchein
Entropy 2023, 25(2), 269; https://doi.org/10.3390/e25020269 - 1 Feb 2023
Viewed by 1296
Abstract
A relevant problem in dynamics is to characterize how deterministic systems may exhibit features typically associated with stochastic processes. A widely studied example is the study of (normal or anomalous) transport properties for deterministic systems on non-compact phase space. We consider here two [...] Read more.
A relevant problem in dynamics is to characterize how deterministic systems may exhibit features typically associated with stochastic processes. A widely studied example is the study of (normal or anomalous) transport properties for deterministic systems on non-compact phase space. We consider here two examples of area-preserving maps: the Chirikov–Taylor standard map and the Casati–Prosen triangle map, and we investigate transport properties, records statistics, and occupation time statistics. Our results confirm and expand known results for the standard map: when a chaotic sea is present, transport is diffusive, and records statistics and the fraction of occupation time in the positive half-axis reproduce the laws for simple symmetric random walks. In the case of the triangle map, we retrieve the previously observed anomalous transport, and we show that records statistics exhibit similar anomalies. When we investigate occupation time statistics and persistence probabilities, our numerical experiments are compatible with a generalized arcsine law and transient behavior of the dynamics. Full article
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13 pages, 1425 KiB  
Article
Quantum Bounds on the Generalized Lyapunov Exponents
by Silvia Pappalardi and Jorge Kurchan
Entropy 2023, 25(2), 246; https://doi.org/10.3390/e25020246 - 30 Jan 2023
Cited by 10 | Viewed by 1860
Abstract
We discuss the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of [...] Read more.
We discuss the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents Lq via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation–dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos. Full article
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12 pages, 784 KiB  
Article
Generalized Survival Probability
by David A. Zarate-Herrada, Lea F. Santos and E. Jonathan Torres-Herrera
Entropy 2023, 25(2), 205; https://doi.org/10.3390/e25020205 - 20 Jan 2023
Cited by 3 | Viewed by 1514
Abstract
Survival probability measures the probability that a system taken out of equilibrium has not yet transitioned from its initial state. Inspired by the generalized entropies used to analyze nonergodic states, we introduce a generalized version of the survival probability and discuss how it [...] Read more.
Survival probability measures the probability that a system taken out of equilibrium has not yet transitioned from its initial state. Inspired by the generalized entropies used to analyze nonergodic states, we introduce a generalized version of the survival probability and discuss how it can assist in studies of the structure of eigenstates and ergodicity. Full article
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22 pages, 2693 KiB  
Article
Scarring in Rough Rectangular Billiards
by Felix M. Izrailev, German A. Luna-Acosta and J. A. Mendez-Bermudez
Entropy 2023, 25(2), 189; https://doi.org/10.3390/e25020189 - 18 Jan 2023
Viewed by 1364
Abstract
We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is [...] Read more.
We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is related to the bouncing ball trajectories in the configuration space of the corresponding classical billiard. A second set of scar-like states emerges in the momentum space, which originated from the plane-wave states of the unperturbed flat billiard. In the case of billiards with one rough surface, the numerical data demonstrate the repulsion of eigenstates from this surface. When two horizontal rough surfaces are considered, the repulsion effect is either enhanced or canceled depending on whether the rough profiles are symmetric or antisymmetric. The effect of repulsion is quite strong and influences the structure of all eigenstates, indicating that the symmetric properties of the rough profiles are important for the problem of scattering of electromagnetic (or electron) waves through quasi-one-dimensional waveguides. Our approach is based on the reduction of the model of one particle in the billiard with corrugated surfaces to a model of two artificial particles in the billiard with flat surfaces, however, with an effective interaction between these particles. As a result, the analysis is conducted in terms of a two-particle basis, and the roughness of the billiard boundaries is absorbed by a quite complicated potential. Full article
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12 pages, 1209 KiB  
Article
Ray-Stretching Statistics and Hot-Spot Formation in Weak Random Disorder
by Sicong Chen and Lev Kaplan
Entropy 2023, 25(1), 161; https://doi.org/10.3390/e25010161 - 13 Jan 2023
Viewed by 1353
Abstract
Weak scattering in a random disordered medium and the associated extreme-event statistics are of great interest in various physical contexts. Here, in the context of non-relativistic particle motion through a weakly correlated random potential, we show how extreme events in particle densities are [...] Read more.
Weak scattering in a random disordered medium and the associated extreme-event statistics are of great interest in various physical contexts. Here, in the context of non-relativistic particle motion through a weakly correlated random potential, we show how extreme events in particle densities are strongly related to the stretching exponents, where the ’hot spots’ in the intensity profile correspond to minima in the stretching exponents. This strong connection is expected to be valid for different random potential distributions, as long as the disorder is correlated and weak, and is also expected to apply to other physical contexts, such as deep ocean waves. Full article
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6 pages, 1193 KiB  
Article
Quantum Manifestation of the Classical Bifurcation in the Driven Dissipative Bose–Hubbard Dimer
by Pavel Muraev, Dmitrii Maksimov and Andrey Kolovsky
Entropy 2023, 25(1), 117; https://doi.org/10.3390/e25010117 - 5 Jan 2023
Cited by 2 | Viewed by 1060
Abstract
We analyze the classical and quantum dynamics of the driven dissipative Bose–Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency [...] Read more.
We analyze the classical and quantum dynamics of the driven dissipative Bose–Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency unrelated to the driving frequency. This bifurcation is shown to lead to a peculiarity in the stationary single-particle density matrix of the quantum system. The case of the Bose–Hubbard trimer, where the discussed limit cycle bifurcates into a chaotic attractor, is briefly discussed. Full article
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15 pages, 832 KiB  
Article
Implications of Spectral Interlacing for Quantum Graphs
by Junjie Lu, Tobias Hofmann, Ulrich Kuhl and Hans-Jürgen Stöckmann
Entropy 2023, 25(1), 109; https://doi.org/10.3390/e25010109 - 4 Jan 2023
Cited by 1 | Viewed by 1505
Abstract
Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the [...] Read more.
Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed. Full article
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10 pages, 660 KiB  
Article
Entanglement Dynamics and Classical Complexity
by Jiaozi Wang, Barbara Dietz, Dario Rosa and Giuliano Benenti
Entropy 2023, 25(1), 97; https://doi.org/10.3390/e25010097 - 3 Jan 2023
Viewed by 1256
Abstract
We study the dynamical generation of entanglement for a two-body interacting system, starting from a separable coherent state. We show analytically that in the quasiclassical regime the entanglement growth rate can be simply computed by means of the underlying classical dynamics. Furthermore, this [...] Read more.
We study the dynamical generation of entanglement for a two-body interacting system, starting from a separable coherent state. We show analytically that in the quasiclassical regime the entanglement growth rate can be simply computed by means of the underlying classical dynamics. Furthermore, this rate is given by the Kolmogorov–Sinai entropy, which characterizes the dynamical complexity of classical motion. Our results, illustrated by numerical simulations on a model of coupled rotators, establish in the quasiclassical regime a link between the generation of entanglement, a purely quantum phenomenon, and classical complexity. Full article
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10 pages, 1217 KiB  
Article
Universality and beyond in Optical Microcavity Billiards with Source-Induced Dynamics
by Lukas Seemann and Martina Hentschel
Entropy 2023, 25(1), 95; https://doi.org/10.3390/e25010095 - 3 Jan 2023
Cited by 1 | Viewed by 1399
Abstract
Optical microcavity billiards are a paradigm of a mesoscopic model system for quantum chaos. We demonstrate the action and origin of ray-wave correspondence in real and phase space using far-field emission characteristics and Husimi functions. Whereas universality induced by the invariant-measure dominated far-field [...] Read more.
Optical microcavity billiards are a paradigm of a mesoscopic model system for quantum chaos. We demonstrate the action and origin of ray-wave correspondence in real and phase space using far-field emission characteristics and Husimi functions. Whereas universality induced by the invariant-measure dominated far-field emission is known to be a feature shaping the properties of many lasing optical microcavities, the situation changes in the presence of sources that we discuss here. We investigate the source-induced dynamics and the resulting limits of universality while we find ray-picture results to remain a useful tool in order to understand the wave behaviour of optical microcavities with sources. We demonstrate the source-induced dynamics in phase space from the source ignition until a stationary regime is reached comparing results from ray, ray-with-phase, and wave simulations and explore ray–wave correspondence. Full article
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31 pages, 1844 KiB  
Article
Extreme Eigenvalues and the Emerging Outlier in Rank-One Non-Hermitian Deformations of the Gaussian Unitary Ensemble
by Yan V. Fyodorov, Boris A. Khoruzhenko and Mihail Poplavskyi
Entropy 2023, 25(1), 74; https://doi.org/10.3390/e25010074 - 30 Dec 2022
Cited by 4 | Viewed by 1511
Abstract
Complex eigenvalues of random matrices J=GUE+iγdiag(1,0,,0) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is [...] Read more.
Complex eigenvalues of random matrices J=GUE+iγdiag(1,0,,0) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions N1 the eigenvalue density of J undergoes an abrupt restructuring at γ=1, the critical threshold beyond which a single eigenvalue outlier (“broad resonance”) appears. We provide a detailed description of this restructuring transition, including the scaling with N of the width of the critical region about the outlier threshold γ=1 and the associated scaling for the real parts (“resonance positions”) and imaginary parts (“resonance widths”) of the eigenvalues which are farthest away from the real axis. In the critical regime we determine the density of such extreme eigenvalues, and show how the outlier gradually separates itself from the rest of the extreme eigenvalues. Finally, we describe the fluctuations in the height of the eigenvalue outlier for large but finite N in terms of the associated large deviation function. Full article
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11 pages, 539 KiB  
Article
Relaxation Exponents of OTOCs and Overlap with Local Hamiltonians
by Vinitha Balachandran and Dario Poletti
Entropy 2023, 25(1), 59; https://doi.org/10.3390/e25010059 - 28 Dec 2022
Cited by 1 | Viewed by 1323
Abstract
OTOC has been used to characterize the information scrambling in quantum systems. Recent studies have shown that local conserved quantities play a crucial role in governing the relaxation dynamics of OTOC in non-integrable systems. In particular, the slow scrambling of OTOC is seen [...] Read more.
OTOC has been used to characterize the information scrambling in quantum systems. Recent studies have shown that local conserved quantities play a crucial role in governing the relaxation dynamics of OTOC in non-integrable systems. In particular, the slow scrambling of OTOC is seen for observables that have an overlap with local conserved quantities. However, an observable may not overlap with the Hamiltonian but instead with the Hamiltonian elevated to an exponent larger than one. Here, we show that higher exponents correspond to faster relaxation, although still algebraic, and such exponents can increase indefinitely. Our analytical results are supported by numerical experiments. Full article
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21 pages, 5934 KiB  
Article
Chaos and Thermalization in the Spin-Boson Dicke Model
by David Villaseñor, Saúl Pilatowsky-Cameo, Miguel A. Bastarrachea-Magnani, Sergio Lerma-Hernández, Lea F. Santos and Jorge G. Hirsch
Entropy 2023, 25(1), 8; https://doi.org/10.3390/e25010008 - 21 Dec 2022
Cited by 14 | Viewed by 1941
Abstract
We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of [...] Read more.
We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called “efficient basis” over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis. Full article
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13 pages, 6993 KiB  
Article
On the Quantization of AB Phase in Nonlinear Systems
by Xi Liu, Qing-Hai Wang and Jiangbin Gong
Entropy 2022, 24(12), 1835; https://doi.org/10.3390/e24121835 - 16 Dec 2022
Viewed by 1351
Abstract
Self-intersecting energy band structures in momentum space can be induced by nonlinearity at the mean-field level, with the so-called nonlinear Dirac cones as one intriguing consequence. Using the Qi-Wu-Zhang model plus power law nonlinearity, we systematically study in this paper the Aharonov–Bohm (AB) [...] Read more.
Self-intersecting energy band structures in momentum space can be induced by nonlinearity at the mean-field level, with the so-called nonlinear Dirac cones as one intriguing consequence. Using the Qi-Wu-Zhang model plus power law nonlinearity, we systematically study in this paper the Aharonov–Bohm (AB) phase associated with an adiabatic process in the momentum space, with two adiabatic paths circling around one nonlinear Dirac cone. Interestingly, for and only for Kerr nonlinearity, the AB phase experiences a jump of π at the critical nonlinearity at which the Dirac cone appears and disappears (thus yielding π-quantization of the AB phase so long as the nonlinear Dirac cone exists), whereas for all other powers of nonlinearity, the AB phase always changes continuously with the nonlinear strength. Our results may be useful for experimental measurement of power-law nonlinearity and shall motivate further fundamental interest in aspects of geometric phase and adiabatic following in nonlinear systems. Full article
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25 pages, 2133 KiB  
Article
Canonical Density Matrices from Eigenstates of Mixed Systems
by Mahdi Kourehpaz, Stefan Donsa, Fabian Lackner, Joachim Burgdörfer and Iva Březinová
Entropy 2022, 24(12), 1740; https://doi.org/10.3390/e24121740 - 29 Nov 2022
Cited by 3 | Viewed by 5956
Abstract
One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N) limit of large quantum many-body systems, canonical density matrices [...] Read more.
One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N) limit of large quantum many-body systems, canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates, it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum-chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos, we find a continuous and universal (i.e., size-independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy. Full article
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17 pages, 3674 KiB  
Article
Universal Single-Mode Lasing in Fully Chaotic Billiard Lasers
by Mengyu You, Daisuke Sakakibara, Kota Makino, Yonosuke Morishita, Kazutoshi Matsumura, Yuta Kawashima, Manao Yoshikawa, Mahiro Tonosaki, Kazutaka Kanno, Atsushi Uchida, Satoshi Sunada, Susumu Shinohara and Takahisa Harayama
Entropy 2022, 24(11), 1648; https://doi.org/10.3390/e24111648 - 14 Nov 2022
Cited by 4 | Viewed by 1860
Abstract
By numerical simulations and experiments of fully chaotic billiard lasers, we show that single-mode lasing states are stable, whereas multi-mode lasing states are unstable when the size of the billiard is much larger than the wavelength and the external pumping power is sufficiently [...] Read more.
By numerical simulations and experiments of fully chaotic billiard lasers, we show that single-mode lasing states are stable, whereas multi-mode lasing states are unstable when the size of the billiard is much larger than the wavelength and the external pumping power is sufficiently large. On the other hand, for integrable billiard lasers, it is shown that multi-mode lasing states are stable, whereas single-mode lasing states are unstable. These phenomena arise from the combination of two different nonlinear effects of mode-interaction due to the active lasing medium and deformation of the billiard shape. Investigations of billiard lasers with various shapes revealed that single-mode lasing is a universal phenomenon for fully chaotic billiard lasers. Full article
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17 pages, 27772 KiB  
Article
Ray–Wave Correspondence in Microstar Cavities
by Julius Kullig and Jan Wiersig
Entropy 2022, 24(11), 1614; https://doi.org/10.3390/e24111614 - 5 Nov 2022
Cited by 1 | Viewed by 1570
Abstract
In a previous work published by the authors in 2020, a novel concept of light confinement in a microcavity was introduced which is based on successive perfect transmissions at Brewster’s angle. Hence, a new class of open billiards was designed with star-shaped microcavities [...] Read more.
In a previous work published by the authors in 2020, a novel concept of light confinement in a microcavity was introduced which is based on successive perfect transmissions at Brewster’s angle. Hence, a new class of open billiards was designed with star-shaped microcavities where rays propagate on orbits that leave and re-enter the cavity. In this article, we investigate the ray–wave correspondence in microstar cavities. An unintuitive difference between clockwise and counterclockwise propagation is revealed which is traced back to nonlinear resonance chains in phase space. Full article
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16 pages, 400 KiB  
Article
Mather β-Function for Ellipses and Rigidity
by Michael Bialy
Entropy 2022, 24(11), 1600; https://doi.org/10.3390/e24111600 - 3 Nov 2022
Cited by 1 | Viewed by 1360
Abstract
The goal of the first part of this note is to get an explicit formula for rotation number and Mather β-function for ellipse. This is done here with the help of non-standard generating function of billiard problem. In this way the derivation [...] Read more.
The goal of the first part of this note is to get an explicit formula for rotation number and Mather β-function for ellipse. This is done here with the help of non-standard generating function of billiard problem. In this way the derivation is especially simple. In the second part we discuss application of Mather β-function to rigidity problem. Full article
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13 pages, 3534 KiB  
Article
Quantum Chaos in the Extended Dicke Model
by Qian Wang
Entropy 2022, 24(10), 1415; https://doi.org/10.3390/e24101415 - 4 Oct 2022
Cited by 8 | Viewed by 1931
Abstract
We systematically study the chaotic signatures in a quantum many-body system consisting of an ensemble of interacting two-level atoms coupled to a single-mode bosonic field, the so-called extended Dicke model. The presence of the atom–atom interaction also leads us to explore how the [...] Read more.
We systematically study the chaotic signatures in a quantum many-body system consisting of an ensemble of interacting two-level atoms coupled to a single-mode bosonic field, the so-called extended Dicke model. The presence of the atom–atom interaction also leads us to explore how the atomic interaction affects the chaotic characters of the model. By analyzing the energy spectral statistics and the structure of eigenstates, we reveal the quantum signatures of chaos in the model and discuss the effect of the atomic interaction. We also investigate the dependence of the boundary of chaos extracted from both eigenvalue-based and eigenstate-based indicators on the atomic interaction. We show that the impact of the atomic interaction on the spectral statistics is stronger than on the structure of eigenstates. Qualitatively, the integrablity-to-chaos transition found in the Dicke model is amplified when the interatomic interaction in the extended Dicke model is switched on. Full article
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9 pages, 410 KiB  
Article
Elliptic Flowers: New Types of Dynamics to Study Classical and Quantum Chaos
by Hassan Attarchi and Leonid A. Bunimovich
Entropy 2022, 24(9), 1223; https://doi.org/10.3390/e24091223 - 1 Sep 2022
Viewed by 1418
Abstract
We construct examples of billiards where two chaotic flows are moving in opposite directions around a non-chaotic core or vice versa; the dynamics in the core are chaotic but flows that are moving in opposite directions around it are non-chaotic. These examples belong [...] Read more.
We construct examples of billiards where two chaotic flows are moving in opposite directions around a non-chaotic core or vice versa; the dynamics in the core are chaotic but flows that are moving in opposite directions around it are non-chaotic. These examples belong to a new class of dynamical systems called elliptic flowers billiards. Such systems demonstrate a variety of new behaviors which have never been observed or predicted to exist. Elliptic flowers billiards, where a chaotic (non-chaotic) core coexists with the same (chaotic/non-chaotic) type of dynamics in flows were recently constructed. Therefore, all four possible types of coexisting dynamics in the core and tracks are detected. However, it is just the beginning of studies of elliptic flowers billiards, which have already extended the imagination of what may happen in phase spaces of nonlinear systems. We outline some further directions of investigation of elliptic flowers billiards, which may bring new insights into our understanding of classical and quantum dynamics in nonlinear systems. Full article
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16 pages, 5801 KiB  
Article
Pseudoclassical Dynamics of the Kicked Top
by Zhixing Zou and Jiao Wang
Entropy 2022, 24(8), 1092; https://doi.org/10.3390/e24081092 - 9 Aug 2022
Cited by 3 | Viewed by 1690
Abstract
The kicked rotor and the kicked top are two paradigms of quantum chaos. The notions of quantum resonance and the pseudoclassical limit, developed in the study of the kicked rotor, have revealed an intriguing and unconventional aspect of classical–quantum correspondence. Here, we show [...] Read more.
The kicked rotor and the kicked top are two paradigms of quantum chaos. The notions of quantum resonance and the pseudoclassical limit, developed in the study of the kicked rotor, have revealed an intriguing and unconventional aspect of classical–quantum correspondence. Here, we show that, by extending these notions to the kicked top, its rich dynamical behavior can be appreciated more thoroughly; of special interest is the entanglement entropy. In particular, the periodic synchronization between systems subject to different kicking strength can be conveniently understood and elaborated from the pseudoclassical perspective. The applicability of the suggested general pseudoclassical theory to the kicked rotor is also discussed. Full article
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9 pages, 697 KiB  
Article
Localization Detection Based on Quantum Dynamics
by Kazue Kudo
Entropy 2022, 24(8), 1085; https://doi.org/10.3390/e24081085 - 5 Aug 2022
Cited by 2 | Viewed by 1571 | Correction
Abstract
Detecting many-body localization (MBL) typically requires the calculation of high-energy eigenstates using numerical approaches. This study investigates methods that assume the use of a quantum device to detect disorder-induced localization. Numerical simulations for small systems demonstrate how the magnetization and twist overlap, which [...] Read more.
Detecting many-body localization (MBL) typically requires the calculation of high-energy eigenstates using numerical approaches. This study investigates methods that assume the use of a quantum device to detect disorder-induced localization. Numerical simulations for small systems demonstrate how the magnetization and twist overlap, which can be easily obtained from the measurement of qubits in a quantum device, changing from the thermal phase to the localized phase. The twist overlap evaluated using the wave function at the end of the time evolution behaves similarly to the one evaluated with eigenstates in the middle of the energy spectrum under a specific condition. The twist overlap evaluated using the wave function after time evolution for many disorder realizations is a promising probe for detecting MBL in quantum computing approaches. Full article
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11 pages, 282 KiB  
Article
Quantum Chaos, Random Matrices, and Irreversibility in Interacting Many-Body Quantum Systems
by Hans A. Weidenmüller
Entropy 2022, 24(7), 959; https://doi.org/10.3390/e24070959 - 11 Jul 2022
Viewed by 1503
Abstract
The Pauli master equation describes the statistical equilibration of a closed quantum system. Simplifying and generalizing an approach developed in two previous papers, we present a derivation of that equation using concepts developed in quantum chaos and random-matrix theory. We assume that the [...] Read more.
The Pauli master equation describes the statistical equilibration of a closed quantum system. Simplifying and generalizing an approach developed in two previous papers, we present a derivation of that equation using concepts developed in quantum chaos and random-matrix theory. We assume that the system consists of subsystems with strong internal mixing. We can then model the system as an ensemble of random matrices. Equilibration results from averaging over the ensemble. The direction of the arrow of time is determined by an (ever-so-small) coupling to the outside world. The master equation holds for sufficiently large times if the average level densities in all subsystems are sufficiently smooth. These conditions are quantified in the text, and leading-order correction terms are given. Full article
28 pages, 1093 KiB  
Article
Effective Field Theory of Random Quantum Circuits
by Yunxiang Liao and Victor Galitski
Entropy 2022, 24(6), 823; https://doi.org/10.3390/e24060823 - 13 Jun 2022
Cited by 2 | Viewed by 2317
Abstract
Quantum circuits have been widely used as a platform to simulate generic quantum many-body systems. In particular, random quantum circuits provide a means to probe universal features of many-body quantum chaos and ergodicity. Some such features have already been experimentally demonstrated in noisy [...] Read more.
Quantum circuits have been widely used as a platform to simulate generic quantum many-body systems. In particular, random quantum circuits provide a means to probe universal features of many-body quantum chaos and ergodicity. Some such features have already been experimentally demonstrated in noisy intermediate-scale quantum (NISQ) devices. On the theory side, properties of random quantum circuits have been studied on a case-by-case basis and for certain specific systems, and a hallmark of quantum chaos—universal Wigner–Dyson level statistics—has been derived. This work develops an effective field theory for a large class of random quantum circuits. The theory has the form of a replica sigma model and is similar to the low-energy approach to diffusion in disordered systems. The method is used to explicitly derive the universal random matrix behavior of a large family of random circuits. In particular, we rederive the Wigner–Dyson spectral statistics of the brickwork circuit model by Chan, De Luca, and Chalker [Phys. Rev. X 8, 041019 (2018)] and show within the same calculation that its various permutations and higher-dimensional generalizations preserve the universal level statistics. Finally, we use the replica sigma model framework to rederive the Weingarten calculus, which is a method of evaluating integrals of polynomials of matrix elements with respect to the Haar measure over compact groups and has many applications in the study of quantum circuits. The effective field theory derived here provides both a method to quantitatively characterize the quantum dynamics of random Floquet systems (e.g., calculating operator and entanglement spreading) and a path to understanding the general fundamental mechanism behind quantum chaos and thermalization in these systems. Full article
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