Time Dependent $\mathcal{PT}$-Symmetric Quantum Mechanics

TL;DR

This paper develops a framework for analyzing time-dependent PT-symmetric quantum mechanics, extending the theory beyond spectral properties to include dynamics and geometric phases, with potential links to conventional quantum mechanics.

Contribution

It proposes an axiom for time-dependent PTQM, introduces a mapping to standard quantum mechanics, and explores Berry phase phenomena in PT-symmetric systems.

Findings

Proper mapping relates PTQM to conventional quantum mechanics.

Berry phase in PTQM can be interpreted as flux of a tunable magnetic monopole.

Time-dependent PTQM offers a rich structure for quantum dynamics.

Abstract

The parity-time-reversal- ($\mathcal{PT}$) symmetric quantum mechanics (PTQM) has developed into a noteworthy area of research. However, to date most known studies of PTQM focused on the spectral properties of non-Hermitian Hamiltonian operators. In this work, we propose an axiom in PTQM in order to study general time-dependent problems in PTQM, e.g., those with a time-dependent $\mathcal{PT}$-symmetric Hamiltonian and with a time-dependent metric. We illuminate our proposal by examining a proper mapping from a time-dependent Schr\"odinger-like equation of motion for PTQM to the familiar time-dependent Schr\"odinger equation in conventional quantum mechanics. The rich structure of the proper mapping hints that time-dependent PTQM can be a fruitful extension of conventional quantum mechanics. Under our proposed framework, we further study in detail the Berry phase generation in a class of $\mathcal{PT}$-symmetric two-level systems. It is found that a closed path in the parameter space of PTQM is often associated with an open path in a properly mapped problem in conventional quantum mechanics. In one interesting case we further interpret the Berry phase as the flux of a continuously tunable fictitious magnetic monopole, thus highlighting the difference between PTQM and conventional quantum mechanics despite the existence of a proper mapping between them.