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Journal of Physics A Mathematical and General

Noise, dissipation and the classical limit in the quantum kicked-rotator problem

Doron Cohen

Published Jul 21, 1994• 14 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/27/14/011

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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.
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.
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 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.
The evolution of the density matrix is calculated in the Wigner representation for the periodically kicked particle [H=(1/2p2{\mathit{p}}^{2}p2+1/2) Kx2{\mathit{Kx}}^{2}Kx2${\mathrm{\ensuremath{\Sigma}}}_{\mathit{n}}$\ensuremath{\delta}(t-nT)] in the presence of dissipation. The dissipation is introduced via a linear coupling to a bath of harmonic oscillators. The behavior in various regions of the parameter space for the problem is analyzed in detail and compared with the Markovian approximation. The degree of agreement with the Markovian approximation is classified. In various regions in parameter space the model reduces to various examples that were studied in the past (damped particle, harmonic oscillator). Most regions of the parameter space were not explored in the past and are of potential interest. One of these regions exhibits a limit cycle.
A model that generalizes the study of quantum Brownian motion (BM) is constructed. We consider disordered environment that may be either static (quenched), noisy or dynamical. The Zwanzig-Caldeira-Leggett BM model formally constitutes a special case where the disorder autocorrelation length is taken to be infinite. Alternatively, a localization problem is obtained if the noise autocorrelation time is taken to be infinite. Also the general case of weak nonlinear coupling to a thermal, possibly chaotic bath is handled by the same formalism. A general, Feynman-Vernon type path-integral expression for the propagator is introduced. A Wigner transformed version of this expression is utilized in order to facilitate comparison with the classical limit. It is demonstrated that nonstochastic genuine quantal manifestations are associated with the model. It is clarified that such effects are absent in the standard BM model, either the disorder or the chaotic nature of the bath are essential. Quantal correction to the classical diffusive behavior is found even in the limit of high temperatures. The suppression of interference due to dephasing is discussed, leading to the observation that due to the disorder the decay of coherence is exponential in time, and no longer depends on geometrical considerations. Fascinating non-Markovian effects due to time-correlated (colored) noise are explored. For this, a strategy is developed in order to handle the integration over paths. This strategy is extended in order to demonstrate how localization comes out from the path-integral expression.