Schrodinger-equation formalism for a dissipative quantum system

TL;DR

This paper develops a formalism using a Schrödinger-equation approach to model dissipative quantum systems by coupling a quantum oscillator to a classical string, enabling explicit solutions and applications to decay and resonance phenomena.

Contribution

It introduces a mean-field based formalism that couples quantum and classical parts, deriving an effective dissipative Schrödinger equation with explicit solutions.

Findings

Explicit solutions for classical subsystem equations of motion

Derivation of an effective dissipative Schrödinger equation

Application to decay of quasi-stationary states and nonlinear resonance

Abstract

We consider a model dissipative quantum-mechanical system realized by coupling a quantum oscillator to a semi-infinite classical string which serves as a means of energy transfer from the oscillator to the infinity and thus plays the role of a dissipative element. The coupling between the two -- quantum and classical -- parts of the compound system is treated in the spirit of the mean-field approximation and justification of the validity of such an approach is given. The equations of motion of the classical subsystem are solved explicitly and an effective dissipative Schrodinger equation for the quantum subsystem is obtained. The proposed formalism is illustrated by its application to two basic problems: the decay of the quasi-stationary state and the calculation of the nonlinear resonance line shape.