Eigenvalues of the Liouvillians of Quantum Master Equation for a Harmonic Oscillator

TL;DR

This paper characterizes the eigenvalues of Liouvillians in quantum harmonic oscillator systems, showing their generic form and establishing a relation to known Liouvillian spectra, with implications for understanding quantum dissipative dynamics.

Contribution

It proves that the eigenvalues of quadratic Liouvillians for harmonic oscillators have a universal form by relating them to the Kossakowski--Lindblad equation.

Findings

Eigenvalues have a generic form for quadratic Liouvillians.

Left and right eigenfunctions form a complete biorthogonal set.

Examples of related eigenfunctions are provided.

Abstract

The eigenvalues of the Liouvillians of Markovian master equation for a harmonic oscillator have a generic form. The Liouvillians considered are quadratic in the position coordinates or creation and annihilation operators, as well as having positive renormalized frequencies. We prove this by showing that a generic Liouvillian of this form can be similarly related to the Liouvillian of the Kossakowski--Lindblad equation, whose eigenvalues are already known. The left and right eigenfunctions of the generic Liouvillian also form a complete and biorthogonal set. Examples of similarly related right eigenfunctions are given.