Random-matrix model for quantum Brownian motion

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Abstract

We use a random band-matrix model for the system–bath interaction to derive a Markovian master equation for the time evolution of one-dimensional quantum systems weakly coupled to a heat bath. We study in detail the damped harmonic oscillator and discuss the fluctuation-dissipation relation. In the large-bandwidth, high-temperature limit, our master equation coincides with the equations derived by Agarwal and Caldeira–Leggett. This shows that the Markovian master equations for quantum Brownian motion are independent of model assumptions used in their derivation and, thus, universal.

Introduction

The description of the interaction of an open system with its environment is an important issue both in classical and in quantum physics. If the environment is modeled as a heat bath, the interaction will lead to relaxation and dissipation processes in the system, and to an irreversible approach towards equilibrium [1]. The standard model for quantum dissipation – the Caldeira–Leggett model [2] – consists of a quantum system coupled to an infinite set of harmonic oscillators. Caldeira and Leggett have shown that it is always possible to treat the bath as an ensemble of independent oscillators provided the system–bath coupling is weak. These authors also assumed that the coupling is linear in both the position coordinate of the quantum system and bath variables. For the quantum system, this choice follows from the requirement that, in the classical limit, the friction force in the Langevin equation should be linear in the velocity. For the bath, the choice was primarily made for computational convenience. To the best of our knowledge, there is no compelling argument for choosing the interaction term linear in the bath coordinates.

It is expected, of course, that the relaxation process described by a master equation should be insensitive to the detailed form of the system–bath interaction. The motivation of this work aims at proving this statement. We do so with the help of an alternative model for the interaction. We use an ensemble of random matrices [3]. The ensemble encompasses all forms of system–bath interaction which are linear in the position coordinate of the quantum system, and which respect fundamental symmetries of the problem like time-reversal invariance. The ensemble is characterized by a few parameters which establish the relevant time scales. The Markovian master equations derived in this fashion are then valid for all possible forms of the interaction between quantum system and heat bath. We find that in the high-temperature limit, the Markovian master equations derived by Caldeira and Leggett and others are independent of both the specific structure of the bath and of the specific form of the system–bath interaction. This establishes the universality of the Markovian master equation for quantum Brownian motion.

The outline of the paper is as follows. In Section 2, we present the model for system plus bath with emphasis on the statistical properties of the interaction, a random band-matrix. In Section 3, we derive the Markovian master equation for the averaged density operator of the system. In Section 4, we apply our results to the damped harmonic oscillator. We show that in the large-bandwidth high-temperature limit, we recover the Agarwal and Caldeira–Leggett master equations. We finally conclude in Section 5.

Section snippets

The model

We consider a simple quantum system interacting with a large environment considered as a heat bath. We write the Hamiltonian of the composite system in the formH=HS1B+1S⊗HB+Q⊗V=H0+W,where HS and HB describe the system S and the bath B, respectively, and W=QV is the system–bath interaction. In the present approach, in contrast to the Caldeira–Leggett model, the actual form of HB is not specified. In particular, there are no bath oscillators. We denote by |m〉(|a〉) the eigenstates of the system

The master equation

In this section, we derive a Markovian master equation for ρ̂S(t). We start from the evolution equations , . Because of the presence of a stochastic quantity in the Hamiltonian, the time-evolution operator and, consequently, the density operator are themselves random variables. We, therefore, have to determine their average over the random-matrix ensemble. The averaging procedure consists in [5] (i) expanding the operators in powers of the random interaction W (Born series), then (ii) averaging

Damped harmonic oscillator

We illustrate our results for the case where the system S is a harmonic oscillator with mass M and frequency ω. We introduce the usual creation and annihilation operators a and a. We assume a coupling linear in the position of the system S and, accordingly, set Q=x=(2Mω)−1/2(a+a). The elements of the matrix Wabmn vanish unless |mn|=1. The generalized transition probabilities Eqs. (13) are then easily evaluated using , . We find, for instance,Wnn−1n−1n(1)=A0πe−ω2/2Δ2neβω/2.Proceeding

Conclusion

Using a random band-matrix model for the system–bath interaction, we have derived a Markovian master equation for the averaged reduced density operator of a one–dimensional quantum system. The equation is valid in a domain of parameter values specified by inequalities , of Section 3. We have applied the master equation to the damped harmonic oscillator assuming that the system–bath interaction is linear in the position coordinate of the quantum system. We have obtained the same equations as

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