On the invariant method for the time-dependent non-Hermitian Hamiltonians

TL;DR

This paper introduces a method using pseudo-Hermitian invariant operators to ensure real phases and unitary evolution in certain time-dependent non-Hermitian quantum systems, exemplified by a complex harmonic oscillator.

Contribution

It develops a scheme employing pseudo-Hermitian invariants for non-Hermitian Hamiltonians with time dependence, extending the understanding of their unitary evolution and phase properties.

Findings

Invariant operators can be pseudo-Hermitian with respect to a time-dependent metric.

The method guarantees real phases and unitary evolution in specific non-Hermitian systems.

Application to a complex harmonic oscillator demonstrates the approach's effectiveness.

Abstract

We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators $H(t)$ that generate a real phase in their time-evolution. This involves the use of invariant operators $I_{PH}(t)$ that are pseudo-Hermitian with respect to the time-dependent metric operator, which implies that the dynamics is governed by unitary time evolution. Furthermore, $H(t)$ is not generally quasi-Hermitian and does not define an observable of the system but $I_{PH}(t)$ obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.