PT phase transition in open quantum systems with Lindblad dynamics

We investigate parity-time (PT) phase transitions in open quantum systems and discuss a criterion of Liouvillian PT symmetry proposed recently by Huber et al. [SciPost Phys. 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 PT symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the PT-symmetry-breaking point for an open two-spin model with exactly balanced gain and loss if the total spin is large. In particular, in a PT-unbroken phase, some eigenvalues are pure imaginary numbers, while in a 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 PT-symmetry-breaking point. Furthermore, we show a direct relation between the criterion of Huber et al. of Liouvillian PT symmetry and the dynamics of the physical quantities for quadratic bosonic systems. Our results support the validity of the proposed criterion of Liouvillian PT symmetry.