Time-dependent PT -symmetric quantum mechanics

Abstract

The parity-time-reversal (PT )-symmetric quantum mechanics (QM) (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 timedependent PT -symmetric Hamiltonian and with a time-dependent metric. We illuminate our proposal by examining a proper mapping from a timedependent Schrodinger-like equation of motion for TQM to the familiar timedependent Schrodinger equation in conventional QM. The rich structure of the proper mapping hints that time-dependent PTQM can be a fruitful extension of conventional QM. Under our proposed framework, we further study in detail the Berry-phase generation in a class of 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 QM. 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 QM despite he existence of a proper mapping between them. © 2013 IOP Publishing Ltd Printed in the UK and the USA.