Derivation of exact master equation with stochastic description: Dissipative harmonic oscillator

TL;DR

This paper presents a systematic stochastic approach to derive the exact master equation for a dissipative harmonic oscillator, demonstrating its equivalence to the Hu-Paz-Zhang equation and extending to time-dependent fields.

Contribution

It introduces a detailed stochastic derivation method for the master equation of a dissipative harmonic oscillator, highlighting conditions for closed-form solutions and their relation to existing formalisms.

Findings

Derived the exact Lindblad-form master equation for the harmonic oscillator

Showed the equivalence to the Hu-Paz-Zhang equation

Extended the derivation to systems with time-dependent fields

Abstract

A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function. For a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled and the master equation naturally comes out. Such an equation possesses the Lindblad form in which time dependent coefficients are determined by a set of integral equations. It is proved that the obtained master equation is equivalent to the well-known Hu-Paz-Zhang equation based on the path integral technique. The procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time-dependent fields.