A stochastic golden rule and quantum Langevin equation for the low density limit

TL;DR

This paper rigorously derives a quantum Langevin equation from microscopic dynamics in the low density limit, using a stochastic golden rule, and shows it has a Lindblad form for the generator of the master equation.

Contribution

It introduces a mathematical procedure called the stochastic golden rule to derive the quantum Langevin equation in the low density limit from microscopic models.

Findings

Derivation of the quantum Langevin equation from microscopic dynamics.

The generator of the master equation has the Lindblad form.

The approach applies to a system coupled with a Bose gas reservoir.

Abstract

A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.