arXiv:quant-ph/9610041

Dynamical peculiarities of the nonlinear quasiclassical systems

B. Kaulakys

Published Oct 28, 1996• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.quant-ph/9610041

Related papers

We discuss the solutions of quantum problems which are nonlinear in the classical limit. It is shown that in this case it is necessary to solve the corresponding nonlinear classical problem and study its bifurcation properties.

K. N. Yugai

Dynamical chaos in quantum systems (quantum chaos) has been studied by a large number of physicists [1-7]. It has been shown that quantum chaos is possible only in the quasiclassical region, for example in the case of quantum states near the dissociation limit in molecules or near the ionization limit in atoms and molecules. However, the quantum equations of motion are linear and hence they cannot contain dynamical chaos. In the present paper we discuss the solution of quantum problems which are nonlinear in the classical limit. It will be show that in solving the quantum problem in this case it is necessary to use the solution of the nonlinear classical problem forming the classical limit of the quantum problem. i. We note first that here we are not interested in the usual field nonlinearities considered, for example, in quantum optics. This is because field nonlinearities cannot determine the bifurcation properties of the quantum equations of motion since the latter are linear in the wave function ~ or density matrix p. In other words, dynamical chaos, which is intimately connected with the bifurcation properties of the corresponding equations, is not determined by nonlinearities in the fields in quantum equations of motion linear in ~ or 0o

Jan 1, 19930 citations

Quantum-“classical” correspondence in a nonadiabatic transition system

Hiroshi Fujisaki

A nonadiabatic transition system which exhibits "quantum chaotic" behavior [H. Fujisaki and K. Takatsuka, Phys. Rev. E 63, 066221 (2001)] is investigated from quasiclassical aspects. Since such a system does not have a naive classical limit, we take the mapping approach [Stock and Thoss, Phys. Rev. Lett. 78, 578 (1997)] to represent the quasiclassical dynamics of the system. We numerically show that there is a sound correspondence between the quantum chaos and classical chaos for the system.

arXiv:quant-ph/0401136Physical Review EMar 31, 20042 citations

Quantum chaos in open systems: a quantum state diffusion analysis

Todd A Brun, Ian Colin Percival, Rüdiger Schack

Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wavepackets in the neighbourhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics.

arXiv:quant-ph/9509015Journal of Physics A Mathematical and GeneralMay 7, 199683 citations

Quantum dynamics with intermediate measurements in agreement with the classical dynamics

B. Kaulakys

The effect of repetitive measurement for quantum dynamics of driven by an intensive external force of the simple few-level systems as well as of the multilevel systems that exhibit the quantum localisation of classical chaos is investigated. Frequent measurement of the simple system yields to the quantum Zeno effect while that of the suppressed quantum system, which classical counterpart exhibits chaos, results in the delocalisation of the quantum suppression. From the analysis we may conclude that continuously observable quasiclassical system evolves essentially classically-like.

arXiv:quant-ph/9610019Oct 14, 19960 citations

Introduction to classical chaos and quantum chaos.

Faure, Frédéric

This article gives an introduction to chaotic systems in classical mechanics and their quantum mechanical counterparts.The author assumes very few prerequisites and starts with a basic introduction to the mathematical formulation of classical and quantum mechanics and discusses the relation between classical and quantum systems by means of the mathematical procedure of quantization. Then, the article proceeds with discussing the notion of ``chaos'' in classical mechanics and particularly focuses on the concepts of hyperbolicity, ergodicity and mixing. In the last section, the quantum mechanical properties of classically chaotic systems are discussed. Here, the focus lies in particular on quantum ergodicity and the role of Egorov's theorem. Finally, the distribution of eigenvalues, according to Weyl's law, {\it M.C. Gutzwiller}'s trace formula [``Periodic orbits and classical quantization conditions'', J. Math. Phys., 12, No. 3, 343--358 (1971; \url{doi:10.1063/1.1665596})] as well as the random matrix conjecture of {\it O. Bohigas} et al. [``Characterization of chaotic quantum spectra and universality of level fluctuation laws'', Phys. Rev. Lett. 52, No. 1, 1--4 (1984; \url{doi:10.1103/PhysRevLett.52.1})] is also discussed.A particularly nice feature throughout the whole text is that it includes both mathematically rigorous definitions which can be illustrated for toy models such as the cat map, as well as phyiscally relevant sytems such as billiards. For the latter it is then outlined, where the difficulties in a mathematically rigorous description lie.

Jan 1, 20140 citations

Nature of semiclassical spectrum in terms of classical trajectories(4) Quantum chaos and semiclassical theory in molecular science and nuclear theory, Chaos and Nonlinear Dynamics in Quantum-Mechanical and Macroscopic Systems)

Hiroshi Ushiyama, Kazuo Takatsuka

The mechanism of semiclassical quantization is examined in detail. We found an important role of destructive interference in building quantum spectrum. Based on this analysis, we propose an efficient method to calculate the semiclassical spectrum. This method is particularly promising in multidimensional chaotic systems and also very useful to identify which trajectories dominate a quantum spectrum. We numerically examine this method by applying to a two-dimensional chaotic system and show an illustrative example of identifying such a dominant trajectory that corresponds to an eigenstate.

物性研究Jun 20, 20050 citations