$\mathcal{PT}$ phase transition in open quantum systems with Lindblad dynamics

TL;DR

This paper explores $ ext{PT}$ phase transitions in open quantum systems using the third quantization method, demonstrating eigenvalue structure changes at the $ ext{PT}$ symmetry breaking point and linking symmetry criteria to system dynamics.

Contribution

It provides an analytical demonstration of eigenvalue changes at the $ ext{PT}$ transition in open quadratic systems and validates a recent $ ext{PT}$ symmetry criterion for Liouvillian dynamics.

Findings

Eigenvalues are purely imaginary in the unbroken phase.

Eigenvalues become real in the broken phase.

System dynamics shift from oscillatory to overdamped at the transition.

Abstract

We investigate parity-time ($\mathcal{PT}$) phase transitions in open quantum systems and discuss a criterion of Liouvillian $\mathcal{PT}$ symmetry proposed recently by Huber \textit{et al}. [J. Huber \textit{et al}., SciPost Phys. $\textbf{9}$, 52 (2020)]. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of $\mathcal{PT}$ symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the $\mathcal{PT}$ symmetry breaking point for an open 2-spin model with exactly balanced gain and loss if the total spin is large. In particular, in a $\mathcal{PT}$ unbroken phase, some eigenvalues are pure imaginary numbers while in a $\mathcal{PT}$ broken phase, all the eigenvalues are real. From this result, it is analytically shown for an open quantum system including quantum jumps that the dynamics in the long time limit changes from an oscillatory to an overdamped behavior at the proposed $\mathcal{PT}$ symmetry breaking point. Furthermore, we show a direct relation between the criterion of Huber \textit{et al}. of Liouvillian $\mathcal{PT}$ symmetry and the dynamics of the physical quantities for quadratic bosonic systems. Our results support the validity of the proposed criterion of Liouvillian $\mathcal{PT}$ symmetry.