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Quantum Chaos and Quantum Randomness-Paradigms of Entropy Production on the Smallest Scales. ENTROPY 2019; 21:e21030286. [PMID: 33267001 PMCID: PMC7514766 DOI: 10.3390/e21030286] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/29/2018] [Revised: 02/04/2019] [Accepted: 03/10/2019] [Indexed: 11/16/2022]
Abstract
Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro- to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase-space, reminiscent of Gibbs' paradox in statistical mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos and how the unitary time evolution in a closed system directly implies the "quantum death" of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, could it result from an exchange of entropy with the macroscopic meter? This possibility is explored, presenting a model for spin measurement in a unitary setting and some preliminary analytical results based on it.
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Quantum-classical correspondence principle for work distributions in a chaotic system. Phys Rev E 2016; 93:062108. [PMID: 27415209 DOI: 10.1103/physreve.93.062108] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2016] [Indexed: 11/07/2022]
Abstract
We numerically study the work distributions in a chaotic system and examine the relationship between quantum work and classical work. Our numerical results suggest that there exists a correspondence principle between quantum and classical work distributions in a chaotic system. This correspondence was proved for one-dimensional integrable systems in a recent work [Jarzynski, Quan, and Rahav, Phys. Rev. X 5, 031038 (2015)1063-651X10.1103/PhysRevX.5.031038]. Our investigation further justifies the definition of quantum work via two-point energy measurements.
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A review of sigma models for quantum chaotic dynamics. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2015; 78:086001. [PMID: 26181515 DOI: 10.1088/0034-4885/78/8/086001] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We review the construction of the supersymmetric sigma model for unitary maps, using the color-flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization.
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Shrimp-shape domains in a dissipative kicked rotator. CHAOS (WOODBURY, N.Y.) 2011; 21:043122. [PMID: 22225359 DOI: 10.1063/1.3657917] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Some dynamical properties for a dissipative kicked rotator are studied. Our results show that when dissipation is taken into account a drastic change happens in the structure of the phase space in the sense that the mixed structure is modified and attracting fixed points and chaotic attractors are observed. A detailed numerical investigation in a two-dimensional parameter space based on the behavior of the Lyapunov exponent is considered. Our results show the existence of infinite self-similar shrimp-shaped structures corresponding to periodic attractors, embedded in a large region corresponding to the chaotic regime.
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Abstract
It has been suggested that appropriate periodic sequences of laser pulses can maintain molecular alignment for arbitrarily long times [J. Ortigoso, Phys. Rev. Lett. 93, 073001 (2004)]. These aligned states are found among the cyclic eigenstates of truncated matrix representations of the one-period time propagator U(T,0). However, long time localization of periodic driven systems depends on the nature of the spectrum of their exact propagator; if it is continuous, eigenstates of finite-basis propagators cease to be cyclic, in the long time limit, under the exact time evolution. We show that, for very weak laser intensities, the evolution operator of the system has a point spectrum for most laser frequencies, but for the laser powers needed to create aligned wave packets it is unknown if U(T,0) has a point spectrum or a singular continuous spectrum. For this regime, we obtain error bounds on the exact time evolution of rotational wave packets that allow us to determine that truncated aligned cyclic states do not lose their alignment for millions of rotational periods when they evolve under the action of the exact time propagator.
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Relativistic kicked rotor. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:016213. [PMID: 16090072 DOI: 10.1103/physreve.72.016213] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2004] [Indexed: 05/03/2023]
Abstract
Transport properties in the relativistic analog of the periodically kicked rotor are contrasted under classically and quantum mechanical dynamics. The quantum rotor is treated by solving the Dirac equation in the presence of the time-periodic delta-function potential resulting in a relativistic quantum mapping describing the evolution of the wave function. The transition from the quantum suppression behavior seen in the nonrelativistic limit to agreement between quantum and classical analyses in the relativistic regime is discussed. The absence of quantum resonances in the relativistic case is also addressed.
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Delocalized and resonant quantum transport in nonlinear generalizations of the kicked rotor model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:036220. [PMID: 15903559 DOI: 10.1103/physreve.71.036220] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2004] [Indexed: 05/02/2023]
Abstract
We analyze the effects of a nonlinear cubic perturbation on the delta-kicked rotor. We consider two different models, in which the nonlinear term acts either in the position or in the momentum representation. We numerically investigate the modifications induced by the nonlinearity in the quantum transport in both localized and resonant regimes and a comparison between the results for the two models is presented. Analyzing the momentum distributions and the increase of the mean square momentum, we find that the quantum resonances asymptotically are very stable with respect to nonlinear perturbation of the rotor's phase evolution. For an intermittent time regime, the nonlinearity even enhances the resonant quantum transport, leading to superballistic motion.
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Chaos in a double driven dissipative nonlinear oscillator. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:046219. [PMID: 11690137 DOI: 10.1103/physreve.64.046219] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/27/2000] [Indexed: 05/23/2023]
Abstract
We propose an anharmonic oscillator driven by two periodic forces of different frequencies as a time-dependent model for investigating quantum dissipative chaos. Our analysis is done in the framework of the statistical ensemble of quantum trajectories in a quantum state diffusion approach. The quantum dynamical manifestations of chaotic behavior, including the emergence of chaos, properties of strange attractors, and quantum entanglement, are studied by numerical simulation of the ensemble averaged Wigner function and von Neumann entropy.
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Shape of analyticity domains of Lindstedt series: the standard map. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:015202. [PMID: 11461318 DOI: 10.1103/physreve.64.015202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2000] [Indexed: 05/23/2023]
Abstract
The analyticity domains of the Lindstedt series for the standard map are studied numerically using Padé approximants to model their natural boundaries. We show that if the rotation number is a Diophantine number close to a rational value p/q, then the radius of convergence of the Lindstedt series becomes smaller than the critical threshold for the corresponding Kol'mogorov-Arnol'd-Moser curve, and the natural boundary on the plane of the complexified perturbative parameter acquires a flower-like shape with 2q petals.
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Quantum resonances and regularity islands in quantum maps. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:5057-72. [PMID: 11031548 DOI: 10.1103/physreve.61.5057] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/1999] [Indexed: 11/07/2022]
Abstract
We study analytically as well as numerically the dynamics of a quantum map near a quantum resonance of an order q. The map is embedded into a continuous unitary transformation generated by a time-independent quasi-Hamiltonian. Such a Hamiltonian generates at the very point of the resonance a local gauge transformation described by the unitary unimodular group SU(q). The resonant energy growth is attributed to the zero Liouville eigenmodes of the generator in the adjoint representation of the group while the nonzero modes yield saturating with time contribution. In a vicinity of a given resonance, the quasi-Hamiltonian is then found in the form of power expansion with respect to the detuning from the resonance. The problem is related in this way to the motion along a circle in a (q2 - 1)-component inhomogeneous "magnetic" field of a quantum particle with q intrinsic degrees of freedom described by the SU(q) group. This motion is in parallel with the classical phase oscillations near a nonlinear resonance. The most important role is played by the resonances with the orders much smaller than the typical localization length q << l. Such resonances master for exponentially long though finite times the motion in some domains around them. Explicit analytical solution is possible for a few lowest and strongest resonances.
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Quantum resonances of the kicked rotor and the SU(q) group. PHYSICAL REVIEW LETTERS 2000; 84:3566-3569. [PMID: 11019147 DOI: 10.1103/physrevlett.84.3566] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/1999] [Indexed: 05/23/2023]
Abstract
The quantum kicked rotor map is embedded into a continuous unitary transformation generated by a time-independent quasi Hamiltonian. In some vicinity of a quantum resonance of order q, we relate the problem to the regular motion along a circle in a (q(2)-1) component inhomogeneous "magnetic" field of a quantum particle with q intrinsic degrees of freedom described by the SU(q) group. This motion is in parallel with the classical phase oscillations near a nonlinear resonance.
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From coherent to noise-induced microwave ionization of Rydberg atoms. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1995; 51:4862-4876. [PMID: 9912177 DOI: 10.1103/physreva.51.4862] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Chaos and the quantum-classical correspondence in the kicked pendulum. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:3683-3696. [PMID: 9961653 DOI: 10.1103/physreve.49.3683] [Citation(s) in RCA: 27] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Chaos in Liouville's equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:4215-4220. [PMID: 9961101 DOI: 10.1103/physreve.48.4215] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Signatures of quantum chaos in Wigner and Husimi representations. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:4552-4555. [PMID: 9960530 DOI: 10.1103/physreve.47.4552] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Standard map at complex rotation numbers: Creation of natural boundaries. PHYSICAL REVIEW LETTERS 1992; 68:1443-1446. [PMID: 10045133 DOI: 10.1103/physrevlett.68.1443] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Abstract
The quantum localization of chaotically diffusive classical motion is reviewed, using the kicked rotator as a simple but instructive example. The specific quantum steady state, which results from statistical relaxation in the discrete spectrum, is described in some detail. A new phenomenological theory of quantum dynamical relaxation is presented and compared with the previously existing theory.
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Abstract
The relation between the spectrum of a generalized quasienergy operator and the stability of quantum systems driven by quasiperiodic time-dependent forces is discussed.
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Continuous quantum measurements and chaos. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:4647-4660. [PMID: 9904573 DOI: 10.1103/physreva.42.4647] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Singular continuous quasienergy spectrum in the kicked rotator with separable perturbation: Possibility of the onset of quantum chaos. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:3213-3220. [PMID: 9904399 DOI: 10.1103/physreva.42.3213] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Localized chaos in one-dimensional hydrogen. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:1592-1600. [PMID: 9904192 DOI: 10.1103/physreva.42.1592] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Quantum chaos in the Fermi-accelerator model. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:2306-2310. [PMID: 9903357 DOI: 10.1103/physreva.41.2306] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Classical chaos in one-dimensional hydrogen in strong dc electric fields. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 40:3727-3735. [PMID: 9902589 DOI: 10.1103/physreva.40.3727] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Quantum suppression of irregularity in the spectral properties of the kicked rotator. PHYSICAL REVIEW LETTERS 1988; 60:3-6. [PMID: 10037852 DOI: 10.1103/physrevlett.60.3] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Simple model for strong-laser-field ionization. PHYSICAL REVIEW. A, GENERAL PHYSICS 1987; 36:4311-4320. [PMID: 9899386 DOI: 10.1103/physreva.36.4311] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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