On the localized quantum oscillators in a common heat bath

TL;DR

This paper develops a quantum model for two localized oscillators in a shared heat bath, addressing previous limitations by using scalar fields and canonical quantization, and derives their dynamics, interactions, and response functions.

Contribution

It introduces a novel modeling scheme for the medium using scalar fields and derives the quantum dynamics of coupled oscillators in a common heat bath.

Findings

Derived equations of motion and fluctuation-dissipation relations.

Identified an induced force between oscillators from medium fluctuations.

Analyzed oscillator positions in large time and weak coupling regimes.

Abstract

To remedy the failure of minimal coupling method in describing the quantum dynamics of two localized Brownian oscillators interacting with a common medium, a scheme is introduced to modeling the medium by a continuum of complex scalar fields or equivalently two independent real scalar fields. The starting point is a Lagrangian of the total system and quantization is achieved in the framework of canonical quantization. The equations of motion, memory or response functions and fluctuation-dissipation relations are obtained. An induced force between oscillators is obtained originating from the fluctuations of the medium. Ohmic regime and Drude regularization is discussed and the positions of oscillators are obtained approximately in large time limit and weak coupling regime.