Master equation for quantum Brownian motion derived by stochastic methods

TL;DR

This paper derives the master equation for quantum Brownian motion using a stochastic approach, providing an alternative to existing methods based on path integrals and Wigner function evolution.

Contribution

It introduces a stochastic derivation of the master equation for open quantum systems, offering a new perspective compared to traditional techniques.

Findings

Derivation of the master equation as a Fokker-Planck equation

Demonstration that the reduced Wigner function matches a stochastic distribution

Provides an alternative method for analyzing quantum Brownian motion

Abstract

The master equation for a linear open quantum system in a general environment is derived using a stochastic approach. This is an alternative derivation to that of Hu, Paz and Zhang, which was based on the direct computation of path integrals, or to that of Halliwell and Yu, based on the evolution of the Wigner function for a linear closed quantum system. We first show by using the influence functinal formalism that the reduced Wigner function for the open system coincides with a distribution function resulting from averaging both over the initial conditions and the stochastic source of a formal Langevin equation. The master equation for the reduced Wigner function can then be deduced as a Fokker-Planck equation obtained from the formal Langevin equation.