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Journal of the Physical Society of Japan

Semiquantum Chaos

Jun Ma, Renkuan Yuan

Published Aug 15, 1997• 3 citations
Publisher

Abstract

In this paper it is shown by numerical calculations that chaos exists in a quantum-classical coupled system and behavior of the system depends on the initial condition of the system. It is also found that relatively regular or irregular behavior of one subsystem corresponds to similar behavior of the other. So one way arise naturally in which quantum information about such a coupled system can be obtained only by observing the different kinds of motions of the classical subsystem.

DOI
10.1143/jpsj.66.2302

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In this paper it is shown by numerical calculations that chaos exists in a quantum-classical coupled system and behavior of the system depends on the initial condition of the system. It is also found that relatively regular or irregular behavior of one subsystem corresponds to similar behavior of the other. So one way arise naturally in which quantum information about such a coupled system can be obtained only by observing the different kinds of motions of the classical subsystem.
A semiquantum system, composed of a quantum part coupled to a classical part, can exhibit dynamical chaos in the motion of the quantum state vector. However, there has been disagreement as to whether this mathematical chaos is physically real or merely an artifact of the semiquantum approximation. It is shown, for a model of a quantum spin coupled to an approximately classical nonlinear oscillator, that the semiquantum chaos disappears rapidly as the mass of the oscillator is increased to make it more classical. The time interval during which the semiquantum approximation remains accurate increases with the mass, but is not closely related to the Lyapunov time.
Using the method of adiabatic invariants and the Born-Oppenheimer approximation, we have calculated the ground-state wave function of a pair of coupled oscillators in so-called semiquantum chaos. Some interesting characteristics, e.g., the similarities and differences between the wave functions in the regular and chaotic states have been found. Time-correlation functions of the wave functions and their Fourier spectra in two states have also been investigated. The sensitivity of the wave function in the chaotic state to the initial conditions has been identified.
We have identified ultracold atoms in magneto-optical double-well potentials as a very clean setting in which to study the quantum and classical dynamics of a nonlinear system with multiple degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from nonseparable couplings between the atomic spin and its center of mass motion. The main features of the chaotic dynamics are analyzed using action-angle variables and Poincaré surfaces of section. We show that for the initial state prepared in current experiments [D. J. Haycock et al., Phys. Rev. Lett. 85, 3365 (2000)], classical and quantum expectation values diverge after a finite time, and the observed experimental dynamics is consistent with quantum-mechanical predictions. Furthermore, the motion corresponds to tunneling through a dynamical potential barrier. The coupling between the spin and the motional subsystems, which are very different in nature from one another, leads to interesting questions regarding the transition from regular quantum dynamics to chaotic classical motion.
The dynamics of systems consisting of coupled quantum–classical oscillators is numerically investigated. It is shown that, under certain conditions, the quantum oscillator exhibits chaos. When the mass of the classical oscillator increases, the chaos will be suppressed; if the energy of the system and/or the coupling strength between the two oscillators increases, chaotic behaviour of the system appears. This result will be helpful to understand the probability of the emergence of quantum chaos and may be applied to explain the spectra of complex atoms qualitatively.
We discuss a top undergoing constant precession around a magnetic field and suffering a periodic sequence of impulsive nonlinear kicks. The squared angular momentum being a constant of the motion the quantum dynamics takes place in a finite dimensional Hilbert space. We find a distinction between regular and irregular behavior for times exceeding the quantum mechanical quasiperiod at which classical behavior, whether chaotic or regular, has died out in quantum means. The degree of level repulsion depends on whether or not the top is endowed with a generalized time reversal invariance. Introduction The quantum treatment of systems capable of chaotic motion in the classical limit is interesting for several reasons. First, there is the desire to see how the classical distinguishability between regular and chaotic motion gradually arises as the system is turned more and more classical by changing a suitable parameter. By a controlled increase of quantum mechanical time scales such as wave packet spreading times or inverse level spacings, for instance, one would like to study the growth of the life time of effectively chaotic evolution of suitable observables. Perhaps an even greater incentive for quantum mechanical investigations lies in the question whether quantum chaos can be more than a mere transient mimicry of classical chaos. If so, we need intrinsically quantum mechanical criteria to distinguish regular an irregular behavior; the relation of such quantum criteria to the traditional ones for classical chaos would have to be clarified.