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Physical Review Letters

Condition of Correspondence between Quantum and Classical Dynamics for a Chaotic System

Andrey R. Kolovsky

Published Jan 15, 1996• 31 citations
Publisher

Abstract

The paper deals with the dynamics of a semiclassical system under the influence of the environment. The effect of the environment is shown to convert the quantum dynamics of the system into the classical one. The condition of correspondence between quantum and classical dynamics is obtained and checked numerically.
DOI
10.1103/physrevlett.76.340

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The paper deals with dynamics of a quantum chaotic system under influence of an environment. The effect of an environment is known to destroy the quantum coherence and can convert the quantum dynamics of a system to classical. We use a semiclassical technique for studying the process of decoherence. The condition for transition from quantum to classical dynamics is obtained in general form and checked numerically for a particular chaotic system, known as quantum the standard map on a torus. The relevance of the obtained results to the problem of correspondence between quantum and classical mechanics is briefly discussed. (c) 1996 American Institute of Physics.
The effect of dissipation on the recovery of the classical chaotic dynamics is discussed in the semiclassical regime of the quantum nonintegrable systems. In the classically chaotic Hamiltonian systems, the semiclassical approach breaks down within rather short time scale due to the interference of the wave function. It is shown that the dissipation suppresses the interference and confines the quantum fluctuation around the semiclassical trajectory which is described by the semiclassical Langevin equation. We also show that the semiclassical condition obtained by Schmid is equivalent to the condition for the suppression of interference. The discussion is valid regardless whether the classical motion is chaotic or not and therefore dissipation helps the recovery of the classical dynamics in the chaotic systems.
The quantum-classical correspondence for dynamics of the nonlinear classically chaotic systems is analysed. The problem of quantum chaos consists of two parts: the quasiclassical quantisation of the chaotic systems and attempts to understand the classical chaos in terms of quantum mechanics. The first question has been partially solved by the Gutzwiller semiclassical trace formula for the eigenvalues of chaotic systems, while the classical chaos may be derived from quantum equations only introducing the decoherence process due to interaction with system's environment or intermediate frequent measurement. We may conclude that continuously observable quasiclassical system evolves essentially classically-like.
An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.
The quantum-classical correspondence in the presence of dissipation is studied. The semiclassical expression for the linear response function of an anharmonic system is expressed in a series containing classical stability matrix elements, which can diverge due to the chaotic behavior of stochastic trajectories. The presence of dissipation in most cases removes the divergence of higher-order correction terms, thus suppressing quantum effects and making the system more classical. The regime of system-bath coupling, which makes quantum dynamics completely classical, is obtained in terms of friction, temperature, and anharmonicity. Special cases when bath coupling may lead to enhancement of quantum effects are discussed.