<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math> phase transition in open quantum systems with Lindblad dynamics
Yuma Nakanishi, Tomohiro Sasamoto
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 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 $\mathcal{PT}$ symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the $\mathcal{PT}$-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 $\mathcal{PT}$-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 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.