Spectral theory of Liouvillians for dissipative phase transitions

TL;DR

This paper investigates the spectral properties of Liouvillian superoperators in open quantum systems, providing a general framework for understanding dissipative phase transitions and illustrating results with quantum optical models.

Contribution

It introduces a comprehensive theoretical framework for analyzing the spectral features of Liouvillians during dissipative phase transitions, including symmetry-breaking cases.

Findings

Derived the general form of steady-state density matrices at criticality

Characterized the Liouvillian spectral gap and eigenmatrices

Applied theory to quantum optical models exhibiting phase transitions

Abstract

A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.