Quantum Langevin equation

TL;DR

This paper introduces a quantum Langevin equation for bounded particles that incorporates both fluctuation and dissipation forces, leading to a quantum Fokker-Planck-Kramers equation ensuring the correct thermal equilibrium.

Contribution

It presents a novel formulation of the quantum Langevin equation with a non-Hermitian dissipation force, extending classical stochastic methods to quantum systems with bounded particles.

Findings

The fluctuation force is modeled as white noise proportional to temperature.

The dissipation force is not limited to velocity dependence and ensures Gibbs equilibrium.

Applied to the harmonic oscillator, the dissipation force is non-Hermitian and depends on position and velocity.

Abstract

We propose a Langevin equation to describe the quantum Brownian motion of bounded particles based on a distinctive formulation concerning both the fluctuation and dissipation forces. The fluctuation force is similar to that employed in the classical case. It is a white noise with a variance proportional to the temperature. The dissipation force is not restrict to be proportional to the velocity and is determined in a way as to guarantee that the stationary state is given by a density operator of the Gibbs canonical type. To this end we derived an equation that gives the time evolution of the density operator, which turns out to be a quantum Fokker-Planck-Kramers equation. The approach is applied to the harmonic oscillator in which case the dissipation force is found to be non Hermitian and proportional to the velocity and position.