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Physica Scripta

Quantum effects in the long time behavior of the dissipative standard map

Thomas Dittrich, R. Graham

Published Sep 1, 1989• 4 citations
Publisher

Abstract

A dissipative quantum system exhibiting chaos in its classical limit is constructed by coupling the quantum kicked rotator to a reservoir with exchange of action. A version of the quantized standard map that incorporates dissipation is derived from this model and iterated numerically to study the long-time behavior in various regions of the damping rate. It is found that even weak damping is capable of disrupting dynamical localization which suppresses chaotic motion in the conservative standard map, and thus restores diffusion in the action variable. On the time scale of classical relaxation, a steady state is reached which is the quantum analogue of a classical strange attractor. For strong dissipation, observable deviations from classical behavior both in the transients and in the steady state are due to quantum noise. They are reproduced by a classical stochastic map which is approached by the dissipative quantum map as its semi-classical limit.

DOI
10.1088/0031-8949/40/3/025

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A dissipative quantum system exhibiting chaos in its classical limit is constructed by coupling the quantum kicked rotator to a reservoir with exchange of action. A version of the quantized standard map that incorporates dissipation is derived from this model and iterated numerically to study the long-time behavior in various regions of the damping rate. It is found that even weak damping is capable of disrupting dynamical localization which suppresses chaotic motion in the conservative standard map, and thus restores diffusion in the action variable. On the time scale of classical relaxation, a steady state is reached which is the quantum analogue of a classical strange attractor. For strong dissipation, observable deviations from classical behavior both in the transients and in the steady state are due to quantum noise. They are reproduced by a classical stochastic map which is approached by the dissipative quantum map as its semi-classical limit.
The time evolution of the density matrix generated by a quantum map which corresponds to the classical dissipative standard map is studied numerically. The system approaches a steady state which is the quantum analogue of a classical ensemble on an attractor. The quantum-statistical steady state is represented both as Wigner quasi-probability density and as the diagonal elements of the density matrix in coherent states. It is compared with the corresponding results for a stochastic classical map that is equivalent to the semi-classical limit of the quantum map.
We apply the QSD to the quantum version of Duffing oscillator and discuss the QCC and dissipative quantum chaos in this system. We calculate the constant phase map and the pseudo-Lyapunov exponent, varying Planck constant effectively and show there is a certain clear critical stage in which a crossover from classical to quantum behavior occurs. Furthermore, we discuss the effect of dissipation in the dissipative quantum chaos by means of a degrees of localization introduced in the QSD formalism[5]. Unfortunately, the pseudo-Lyapunov exponent is not a quantity characterized a dissipative system. Therefore, it will be necessary to discuss the connection between the classical trajectory (strange attractor) and the quantum observable which characterize a dissipative quantum system.
Two models for the investigation of the quantum damped kicked-rotator problem are introduced and analysed in a unified fashion. For the first model we follow the Caldeira-Leggett approach while the second constitutes a simplification of the Dittrich-Graham model. These models enable one to investigate the effects of noise and dissipation for systems that exhibit chaos in the classical limit and quantum localization otherwise. The rotator is coupled via its angle coordinate to a heat bath that is held at an arbitrary temperature. Noise time-autocorrelations which may arise from such coupling and the validity of the Markovian approximation are discussed.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.
The quantized standard map (kicked rotor) is coupled to macroscopic systems acting as measuring devices for the probability distribution of the action variable or some of its moments. As a result of such measurements, which are continuous in time but have a limited time resolution, localization of the action variable is destroyed in sufficiently long time scales and replaced by diffusion. The diffusion constant, in general, differs from that of the classical chaotic diffusion and depends on the measurement performed.