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Progress of Theoretical Physics Supplement

Dynamical Aspects of Quantum-Classical Correspondence in Quantum Chaos

Mikito Toda, Satoshi Adachi, Kensuke S. Ikeda

Published Jan 1, 1989• 40 citations
Publisher

Abstract

In the present paper we review our study on the dynamics of quantum chaos with particular emphasis on the problem of dynamical quantum-classical correspondence.
DOI
10.1143/ptps.98.323

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The influence of quantum chaos on small atomic systems is reviewed. It now seems clear that chaos in the usually understood sense (e.g. exponential sensitivity to perturbations) is not found in isolated quantum systems. However, there are phenomena which appear only when the corresponding classical system is chaotic. The stability of the classical dynamics, in other words whether it is regular or chaotic, has a profound effect on the character of the corresponding quantum spectrum and wavefunctions.
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.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.