[Paper Review] PT phase transition in open quantum systems with Lindblad dynamics

Yuma Nakanishi, Tomohiro Sasamoto|arXiv (Cornell University)|Apr 15, 2021

Quantum Mechanics and Non-Hermitian Physics17 references4 citations

TL;DR

This paper investigates PT phase transitions in open quantum systems with Lindblad dynamics, using third quantization to analyze an open 2-spin model with balanced gain and loss. It validates a recent criterion for Liouvillian PT symmetry, showing that eigenvalue structure changes from purely imaginary (PT unbroken) to real (PT broken) at the transition point, leading to a dynamical shift from oscillatory to overdamped behavior in the long-time limit.

ABSTRACT

We investigate parity-time reversal (PT) phase transitions in open quantum systems and discuss a criterion of Liouvillian PT symmetry proposed recently by J. Huber, P. Kirton, S. Rotter and P. Rabl. 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 2-spin model with exactly balanced gain and loss if the total spin is large. Specially, 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 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 dynamics of the physical quantities for quadratic bosonic systems. Our results support validity of the proposed criterion of Liouvillian PT symmetry.

Motivation & Objective

• To investigate PT phase transitions in open quantum systems governed by Lindblad dynamics.
• To test the recently proposed criterion for Liouvillian PT symmetry by Huber et al.
• To analyze the dynamical consequences of PT symmetry breaking in a 2-spin model with balanced gain and loss.
• To establish a direct link between the PT symmetry criterion and observable physical dynamics in quadratic bosonic systems.

Proposed method

• Employing the third quantization method to solve the Lindblad equation for open quadratic systems.
• Applying the proposed criterion of Liouvillian PT symmetry to a 2-spin model with exactly balanced gain and loss.
• Analyzing the eigenvalue spectrum of the Liouvillian operator to identify PT unbroken and broken phases.
• Tracking the long-time dynamics of the system to observe transitions from oscillatory to overdamped behavior.
• Establishing a direct correspondence between the PT symmetry criterion and the time evolution of physical observables in quadratic bosonic systems.
• Using analytical techniques to demonstrate the structural change in the Liouvillian spectrum at the phase transition point.

Experimental results

Research questions

• RQ1 Does the proposed criterion for Liouvillian PT symmetry correctly predict the phase transition in open quantum systems with balanced gain and loss?
• RQ2 How does the eigenvalue spectrum of the Liouvillian change across the PT phase transition in a 2-spin model?
• RQ3 What dynamical behavior emerges in the long-time limit when PT symmetry is broken?
• RQ4 Is there a direct connection between the Liouvillian PT symmetry criterion and the physical dynamics of quadratic bosonic systems?
• RQ5 How does the total spin magnitude influence the nature of the PT phase transition in the model?

Key findings

• In the PT unbroken phase, the Liouvillian eigenvalues are purely imaginary, indicating oscillatory dynamics.
• In the PT broken phase, all Liouvillian eigenvalues become real, signaling overdamped relaxation.
• The transition between these phases coincides with a qualitative change in long-time dynamics—from oscillatory to overdamped behavior.
• The proposed criterion of Huber et al. for Liouvillian PT symmetry is validated through analytical evidence from the eigenvalue spectrum.
• A direct correspondence is established between the PT symmetry criterion and the time evolution of physical observables in quadratic bosonic systems.
• The results confirm that the PT symmetry breaking point marks a critical change in the system's dissipative dynamics.