Analytic solutions to various dissipation models of the simple and driven quantum harmonic oscillator

TL;DR

This paper derives analytical solutions for various dissipation models of the quantum harmonic oscillator, including driven and open systems, using the Wigner function Fourier transform approach to simplify the dynamics analysis.

Contribution

It introduces a straightforward method to solve the open quantum harmonic oscillator dynamics across multiple dissipation models using the Wigner Fourier transform.

Findings

Analytic solutions for zero and finite temperature baths.

Unified approach simplifies dynamics of damped oscillators.

Application to driven open quantum harmonic oscillators.

Abstract

We obtain analytic solutions to various models of dissipation of the quantum harmonic oscillator, employing a simple method in the Wigner function Fourier transform description of the system; and study as an exemplification, the driven open quantum harmonic oscillator. The environmental models we use are based on optical master equations for the zero and finite temperature bath and whose open dynamics are described by a Lindblad master equation, and also we use the Caldeira-Leggett model for the high temperature limit, in the the under damped an the over damped case. Under the Wigner Fourier transform or chord function as it has been called, it becomes particularly simple to solve the dynamics of the open oscillator in the sense that the dynamics of the system are reduced to the application of an evolution matrix related to the damped motion of the oscillator.