Spectral resolution of the Liouvillian of the Lindblad master equation for a harmonic oscillator

TL;DR

This paper provides a formal solution to the Lindblad master equation for a harmonic oscillator, revealing the spectral structure of the Liouvillian and its implications for quantum open systems.

Contribution

It offers the explicit spectral resolution of the Liouvillian, including eigenvalues, eigenprojections, and ladder operators, enhancing understanding of open quantum system dynamics.

Findings

Spectral resolution of the Liouvillian obtained

Eigenvalues and eigenprojections explicitly derived

Ladder operators constructed to clarify spectral structure

Abstract

A Lindblad master equation for a harmonic oscillator, which describes the dynamics of an open system, is formally solved. The solution yields the spectral resolution of the Liouvillian, that is, all eigenvalues and eigenprojections are obtained. This spectral resolution is discussed in depth in the context of the biorthogonal system and the rigged Hilbert space, and the contribution of each eigenprojection to expectation values of physical quantities is revealed. We also construct the ladder operators of the Liouvillian, which clarify the structure of the spectral resolution.