Coupled harmonic oscillator in a multimode harmonic oscillator bath: Derivation of quantum propagator and master equation using white noise analysis

TL;DR

This paper applies white noise functional analysis to derive the quantum propagator and master equation for a coupled harmonic oscillator system interacting with a multimode harmonic bath, offering a rigorous and user-friendly method.

Contribution

It introduces a novel application of white noise analysis to derive the quantum propagator and master equation for a complex open quantum system with coupled oscillators and a multimode bath.

Findings

Derived the quantum propagator as a product of three simpler propagators.

Obtained the time evolution equation for the reduced density matrix.

Demonstrated the method's potential for analyzing open quantum systems.

Abstract

This paper presents an application of white noise functional approach to derive the quantum propagator and the evolution of the reduced density matrix for an open quantum system consisting of coupled harmonic oscillators which are coupled to a bath consisting of a multimode harmonic oscillator. It is shown that the full quantum propagator is a product of three individual propagators. These propagators were obtained after two successive transformations of the coordinates for the system, and the coordinates for the coupling between the bath and one of the two harmonic oscillators in the system. The obtained propagator is then used to derive the time evolution equation for the density matrix describing the system. The method used to derive the propagator of the open quantum system considered in this paper shows promise for analyzing open quantum systems, due to its ease of use and mathematical rigor.