The Schr\"odinger-Langevin equation with and without thermal fluctuations

TL;DR

This paper investigates the Schr"odinger-Langevin equation's effectiveness in modeling open quantum systems interacting with thermal baths, focusing on state damping and thermal relaxation through analytical and numerical methods.

Contribution

It clarifies the impact of dissipation and stochastic terms on quantum state evolution and assesses the SL equation's capability to reach thermal equilibrium.

Findings

Excited states experience non-zero damping due to proper wave function transformation.

The SL equation can model thermal relaxation under certain assumptions.

Limitations of the SL formalism are discussed.

Abstract

The Schr\"odinger-Langevin (SL) equation is considered as an effective open quantum system formalism suitable for phenomenological applications involving a quantum subsystem interacting with a thermal bath. We focus on two open issues relative to its solutions: the stationarity of the excited states of the non-interacting subsystem when one considers the dissipation only and the thermal relaxation toward asymptotic distributions with the additional stochastic term. We first show that a proper application of the Madelung/polar transformation of the wave function leads to a non zero damping of the excited states of the quantum subsystem. We then study analytically and numerically the SL equation ability to bring a quantum subsystem to the thermal equilibrium of statistical mechanics. To do so, concepts about statistical mixed states and quantum noises are discussed and a detailed analysis is carried with two kinds of noise and potential. We show that within our assumptions the use of the SL equation as an effective open quantum system formalism is possible and discuss some of its limitations.