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Physics-Uspekhi

Quantum models of relaxation

Alexander B. Brailovsky, В. Л. Вакс, V. V. Mityugov

Published Jul 31, 1996• 6 citations
Publisher

Abstract

Exactly solvable quantum models for an atom or a molecule interacting with a physical environment are considered. The kinetic description of the random walk of an open system is shown to be generally consistent with the phenomenological relaxation equations found use in spectroscopy. Some aspects of the problem of entropy increase are also revised.

DOI
10.1070/pu1996v039n07abeh000159

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Exactly solvable quantum models for an atom or a molecule interacting with a physical environment are considered. The kinetic description of the random walk of an open system is shown to be generally consistent with the phenomenological relaxation equations found use in spectroscopy. Some aspects of the problem of entropy increase are also revised.
This paper is intended to provide the axiomatic study of nonequilibrium quantum statistical mechanics with some simple and rigorously solvable models. The models considered here are obtained as generalizations of the Ising model. They illustrate and allow a rational discussion of the following concepts relevant to the theory of irreversible phenomena: coarse-graining and time-smoothing, ergodicity, recurrences, impossibility of a Markovian description of the approach to equilibrium for some physical systems, justification of the various random phase assumptions, properties of the interaction responsible for the approach to equilibrium, master equations, etc.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.
The Kubo-Anderson model is a stochastic model of phase relaxation of an ensemble of systems in a fluctuating environment. This model is usually studied under the assumption that the system-environment interaction is a Gaussian stochastic process. This assumption only holds if the environment changes a very large number of times on the time scale of the system’s motion. This paper reviews our work on the Kubo-Anderson model for the case where this interaction is a continuous-time random walk. A continuous-time random walk is a simple model for a ‘slowly changing environment’, i.e., one which makes a relatively small number of changes on the time scale of the system’s motion. We present the key results from this model and show how to apply them to common problems in magnetic resonance spectroscopy and and non-linear optical spectroscopy.
We study the dynamical correlations in open quantum systems on an example of an exactly solvable dephasing model. The system of non-Markovian kinetic equations for the generalized coherence and quasi-temperature is derived up to the 2-nd order in the coupling constant using the generalized quantum master equation [20]. Numerical and analytical solutions of the kinetic equations are obtained. The analytical result shows a redistribution of the real and imaginary parts of the coherence as compared to the exact one. We give a physical interpretation to the quasi-temperature, relating it to the nonequilibrium kinetic energy of the environment.