Quantum Smoluchowski equation for driven systems

TL;DR

This paper derives a quantum Smoluchowski equation for a driven quantum harmonic oscillator strongly coupled to a heat bath, extending classical stochastic descriptions into the quantum regime with a focus on high friction conditions.

Contribution

It introduces a derivation of the quantum Smoluchowski equation for driven systems using a Green's function approach and analyzes its validity range.

Findings

Derived the quantum Smoluchowski equation for driven systems.

Analyzed the validity range of the equation.

Discussed the special case of a Brownian parametric oscillator.

Abstract

We consider a driven quantum harmonic oscillator strongly coupled to a heat bath. Starting from the exact quantum Langevin equation, we use a Green's function approach to determine the corresponding semiclassical equation for the Wigner phase space distribution. In the limit of high friction, we apply Brinkman's method to derive the quantum Smoluchowski equation for the probability distribution in position space. We further determine the range of validity of the equation and discuss the special case of a Brownian parametric oscillator.