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Physical Review A

Destruction of localization due to coupling to a bath in the kicked-rotator problem

Doron Cohen

Published Jan 1, 1991• 39 citations
Publisher

Abstract

The quantum kicked rotator is linearly coupled to a heat bath via its momentum coordinate. A momentum-space formulation of the Feynman-Vernon formalism is introduced to find the reduced propagator. The Wigner picture of the dynamics is adopted. It is found that it is a good approximation to ignore the ``friction'' effect and to replace the bath by a correlated noise source. For certain values of the friction coefficient, this replacement is exact. Destruction of coherence at zero temperature is qualitatively different for an undriven rotator.

DOI
10.1103/physreva.43.639

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The quantum kicked rotator is linearly coupled to a heat bath via its momentum coordinate. A momentum-space formulation of the Feynman-Vernon formalism is introduced to find the reduced propagator. The Wigner picture of the dynamics is adopted. It is found that it is a good approximation to ignore the ``friction'' effect and to replace the bath by a correlated noise source. For certain values of the friction coefficient, this replacement is exact. Destruction of coherence at zero temperature is qualitatively different for an undriven rotator.
Localization in the quantum kicked rotator (QKR) problem leads to nontrivial dynamical correlations that are absent in the classical limit. These correlations are related to the spectral properties of the model. The effect of noise on coherence is studied. Also, its effect on the dynamical correlations is investigated, and thus one is able to determine analytically the diffusion that is induced. If the noise is white, then coherence time is the same as that for the undriven system. For colored noise it is essential to take into account the nature of the dynamical correlations. It follows that a Markovian treatment of the dynamics for a system that is coupled to a low-temperature heat bath is not valid even if the system is classically chaotic. The formalism applies also to the problem of QKR with kicks that are not strictly periodic in time, and also to the quantum kicked particle problem, where a nonperturbative mechanism for destruction of coherence has been found.
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.
A test system is assumed to interact with a heat bath consisting of harmonic oscillators or an equivalent bath with a proper frequency spectrum producing a Gaussian-Markoffian random perturbation. The effect of reaction of the test system to the bath is considered in the high temperature approximation. Elimination of the bath using the influence functional method of Feynman and Vernon yields a continuous fraction expression for the reduced density matrix of the test system. The result affords a basis to clarify the relationship between the stochastic and the dynamical approaches to treat the problem of partial destruction of quantum coherence of a system interacting with its environment.
We analyze the irreversible behavior of a quantum harmonic oscillator immersed in an arbitrary thermal reservoir. For the fully coupled (FC) model, i.e., the coupling between subsystems linear in the coordinate of the oscillator, we construct a generalized master equation for the reduced density operator of the oscillator. We apply the rotating-wave approximation (RWA) and investigate its connection with the FC master equation. By means of the Wigner distribution, we obtain the corresponding Fokker-Planck equations in the semiclassical representation. We find that, when the RWA is not imposed, the correct classical limit is obtained (Kramer's equation). Then, we establish the conditions for the equivalence between the FC Fokker-Planck equation and that obtained in the RWA. Quantum maps for a kicked oscillator immersed in a heat bath are obtained in both cases. These maps are studied in the classical and semiclassical limits, and it is shown that they coincide for low kicking frequency compared to the damping rate.