# Time-dependent $\mathcal{PT}$-symmetric quantum mechanics in generic non-Hermitian systems

## TL;DR

This paper introduces a framework for time-dependent $ ext{PT}$-symmetric quantum mechanics using a dynamic metric operator, enabling consistent interpretation of non-Hermitian system dynamics with real eigenvalues and unitary evolution.

## Contribution

It proposes a novel approach employing a time-dependent metric operator to treat the dynamics of non-Hermitian systems within $ ext{PT}$-symmetry, restoring unitarity and real spectra.

## Key findings

- Enables interpretation of non-Hermitian dynamics with real eigenvalues
- Restores unitarity in time evolution of non-Hermitian systems
- Provides applications in quantum thermodynamics

## Abstract

$\mathcal{PT}$-symmetric quantum mechanics has been considered an important theoretical framework for understanding physical phenomena in $\mathcal{PT}$-symmetric systems, with a number of $\mathcal{PT}$-symmetry related applications. This line of research was made possible by the introduction of a time-independent metric operator to redefine the inner product of a Hilbert space. To treat the dynamics of generic non-Hermitian systems under equal footing, we advocate in this work the use of a time-dependent metric operator for the inner-product between time-evolving states. This treatment makes it possible to always interpret the dynamics of arbitrary (finite-dimensional) non-Hermitian systems in the framework of time-dependent $\mathcal{PT}$-symmetric quantum mechanics, with unitary time evolution, real eigenvalues of an energy observable, and quantum measurement postulate all restored. Our work sheds new lights on generic non-Hermitian systems and spontaneous $\mathcal{PT}$-symmetry breaking in particular. We also illustrate possible applications of our formulation with well-known examples in quantum thermodynamics.