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arXiv:quant-ph/9807062

Quantum Brownian motion

Fabián H. Gaioli, Alvarez, Edgardo T. Garcia, Javier Guevara

Published Jul 21, 1998• 2 citations
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Abstract

We study the behavior of a subsystem (harmonic oscillator) in contact with a thermal reservoir (finite set of uncoupled harmonic oscillators). We exactly solve the eigenvalue problem and obtain the temporal evolution of the dynamical variables of interest. We show how the subsystem goes to equilibrium and give quantitative estimates of the Poincaré recurrence times. We study the behavior of the subsystem mean ocuppation number in the limit of a dense bath and compare it with the expected exponential decay law.

DOI
10.48550/arxiv.quant-ph/9807062

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We study the behavior of a subsystem (harmonic oscillator) in contact with a thermal reservoir (finite set of uncoupled harmonic oscillators). We exactly solve the eigenvalue problem and obtain the temporal evolution of the dynamical variables of interest. We show how the subsystem goes to equilibrium and give quantitative estimates of the Poincaré recurrence times. We study the behavior of the subsystem mean ocuppation number in the limit of a dense bath and compare it with the expected exponential decay law.
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The effects of system-environment correlations on the dynamics of an open quantum system are investigated for the standard model of a set of quantum harmonic oscillators interacting with a heat bath of oscillators. By definition, a subdynamics is described by transformations of the open system observables. It is shown that such a construction can reproduce the observable properties of the exact dynamics only when the states of system and environment are uncorrelated, while for classical systems there is always a subdynamics. A quantum subdynamics cannot have the properties we associate with thermal fluctuations; the KMS relation at a finite temperature for the open system implies that the system must be closed. The conditions for having a subdynamics as a good approximation to the exact closed dynamics are investigated, and so are the similar but stronger conditions for a Markovian dynamics. It is also shown that a subdynamics defines the response of the open system to some types of time dependent external forces.
A finite number of harmonic oscillators coupled to infinitely many environment oscillators is fundamental to the problem of understanding quantum dissipation of a small system immersed in a large environment. Exact operator solution as a function of time is given to this problem, by using diagonalized dynamical variable of the entire system, the small system plus the environment. The decay law of prepared initial configuration is worked out in greatest detail. A clear separation of the exponential- and the power-law decay period is made possible by our method. Behavior of physical quantities at asymptotically late times can be understood in terms of the overlap probability of the system variable with the diagonal variable of the entire system.
The solution to a quantum Markovian master equation of a harmonic oscillator weakly coupled to a thermal reservoir is investigated as a non-Hermitian eigenvalue problem in space coordinates. In terms of a pair of quantum action-angle variables, the equation becomes separable and a complete set of biorthogonal eigenfunctions can be constructed. Properties of quantum states, such as the change in the quantum coherence length, damping in the motion, and disappearance of the spatial interference pattern, can then be described as the decay of the nonequilibrium modes in the eigenbasis expansion. It is found that the process of gaining quantum coherence from the environment takes a longer time than the opposite process of losing quantum coherence to the environment. An estimate of the time scales of these processes is obtained.
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.