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Physical Review Letters

Quantum-Classical Correspondence in Many-Dimensional Quantum Chaos

Satoshi Adachi, Mikito Toda, Kensuke S. Ikeda

Published Aug 8, 1988• 82 citations
Publisher

Abstract

Quantum-classical correspondence in many-dimensional quantum chaos is investigated by use of a coupled quantum kicked-rotors model. Even when the number of rotors is only two, results obtained are drastically different from those for a single-rotor system; that is, in the semiclassical limit the coupled system restores the essential features of classical chaos under appropriate conditions. In particular, the time-reversal experiment reveals that the classical chaotic mixing is recovered almost entirely; however, the recovered mixing is "conditional" in the sense that there exists a threshold for the recovery.

DOI
10.1103/physrevlett.61.659

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Quantum-classical correspondence in many-dimensional quantum chaos is investigated by use of a coupled quantum kicked-rotors model. Even when the number of rotors is only two, results obtained are drastically different from those for a single-rotor system; that is, in the semiclassical limit the coupled system restores the essential features of classical chaos under appropriate conditions. In particular, the time-reversal experiment reveals that the classical chaotic mixing is recovered almost entirely; however, the recovered mixing is "conditional" in the sense that there exists a threshold for the recovery.
A study of time reversibility reveals that a quantum chaotic system manifests the same potential for mixing as its classical counterpart under the influence of externally applied noise. The quantum-chaotic system, however, reveals its own nature below a quite small "classicalization" threshold of noise intensity: In the case of the kicked rotor there exists a quite wide regime for which the mixing property is understood neither from classical dynamics nor from quantum perturbation theory. The relationship between such a curious behavior and the pure quantum evolution without the noise is elucidated.
In this thesis, a comparison between quantum and classical mechanics was made for two systems whose classical behaviour can be chaotic.The general correspondence between classical mechanics and quantum mechanics in the classical regime has always been assumed but never proved.Recent research in the field of quantum chaos has brought this assumption into question.For some systems, such as the kicked rotor in the classically chaotic regime, quantum mechanics does not display the classical behaviour of unbounded momentum.It has been suggested that quantum mechanics will fail to produce classical chaos for a broad range of simple systems.A comparison of quantum and classical mechanics was made using numerical simulations, without recourse to taking limits such as 7i going to zero.The two systems studied were a driven pendulum and a pair of pendula joined by a spring.The quantum and classical expectation values for a number of observables were computed, as were the probability distributions in phase space.For periodic motion of the driven pendulum it was found that the theories agreed well for times up to 150 driving periods.Similar agreement was found for initial conditions whose classical behaviour was chaotic, but only for times up to a few tens of driving periods.For longer times, it was found that the quantum system displayed a boundedness in position and/or momentum that was absent in the classical system.A possible explanation was given in terms of the quasienergy expansion.In the coupled-pendula system no such disagreements between the theories were found.I would like to thank Dr. Ballentine for his continued support during this project and his help in editing and revising this thesis.I would like to thank the members of my committee, Drs.Bechhoefer, Boa1 and Loss for their many suggestions.For many helpful discussions, I would like to thank Dominic Mimnagh and Jim Zibin.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.
The relation between the degree of entanglement and time scale of time-irreversible behavior is investigated for classically chaotic quantum coupled kicked rotors by comparing the entanglement entropy (EE) and the lifetime of correspondence with classical decay of correlation, which was recently introduced. Both increase on average drastically with a strong correlation when the strength of coupling between the kicked rotors exceeds a certain threshold. The EE shows an anomalously large fluctuation resembling a critical fluctuation around the threshold value of coupling strength where the entanglement sharply increases toward full entanglement. In this regime it can be shown that, although the correlation is hidden, EE and the lifetime of individual eigenfunctions also have a positive correlation that can be seen via another measure.
The kicked rotor provides a simple yet powerful model for introducing many of the central concepts of classical and quantum chaos. Despite its apparent simplicity, it exhibits rich dynamical behavior and has found applications across a wide range of fields, including atomic and optical physics, condensed matter physics, and emerging quantum technologies. This chapter begins by exploring foundational ideas using the kicked rotor as a unifying framework. We first discuss the transition from regular to chaotic motion in the classical system, and then introduce key quantum phenomena such as dynamical localization and quantum resonances. Special attention is devoted to the emergence of characteristic time scales and their role in the quantum-classical correspondence. To make these ideas more concrete, we also provide a brief overview of experimental realizations of the kicked rotor and its variants, illustrating how theoretical concepts are implemented in practice. In the second part of the chapter, we guide the reader toward more recent and advanced developments. Topics include near-resonant dynamics, topological features of kicked systems, the emergence of quantum dynamical phases inferred from classical transport properties, and extensions to non-Hermitian physics. We conclude with a discussion of open problems and future perspectives, outlining directions in which the kicked rotor continues to offer valuable insights.