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ON STRUCTURAL INSTABILITY OF STATES OF CHAOTIC QUANTUM-SYSTEMS

GO XU, QL JIE, XU, GO (reprint author), NANJING UNIV,DEPT PHYS,NANJING 210008,PEOPLES R CHINA.

Published Jan 16, 1995• 0 citations
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Abstract

Based on quantum canonical representations corresponding to the same subgroup chain of the dynamical group and the quantum canonical transformation between them, the structural instability of states of chaotic quantum systems due to a sensitive dependence on the initial condition is discussed. Numerical illustrations are provided with a model system which has been shown to be classically chaotic.

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Based on quantum canonical representations corresponding to the same subgroup chain of the dynamical group and the quantum canonical transformation between them, the structural instability of states of chaotic quantum systems due to a sensitive dependence on the initial condition is discussed. Numerical illustrations are provided with a model system which has been shown to be classically chaotic.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and ℏ−1\hbar^{-1}ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.
We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos.
The nature of classically chaotic quantum dynamics is studied by menas of a numerical time reversal experiment. It turns out that there is a fundamental quantum scale below which the system cannot exhibit classical irreversibility to external perturbations. On the other hand, a paradoxical phenomenon manifesting that quantum irreversibility may exceed its classical counterpart is discovered. These features are explained in terms of semicalssical dynamical theory. Introduction Classically chaotic quantum systems in general can exhibit intrinsinc chaotic behaviour on a quite restricted time scale [1], In particular localization phenomena provide direct evidence exhibiting the suppression of quantum ergodicity [2]. On the other hand, recent studies reveal that a many-dimensional system can mimic some aspects peculiar to chaotic dynamics quite well on an unexpectedly long time scale [3]. A remarkable feature that distinguishes chaotic motion from integrable motion is the sensitive dependence of dynamics on external perturbations. For classical dynamical systems such a sensitivity can be well defined by measuring how two nearby trajectories in the phase space separate in time. However, there has been proposed no systematic way to examine the sensitivity of quantum dynamics to external perturbation. In quantum dynamics, it is not possible to trace a well defined trajectory in phase space, because both quantum uncertainty and chaotic instability make a localized wavepacket spread suddenly over the phase space [4]. There is, however, a simple way in which we may test the quantum sensitivity quantitively. This is the time reversal experiment described below. In this chapter I describe several remarkable characteristics of quantal instability which have been clarified by the time reversal experiment.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.