Unitary quantum evolution for time-dependent quasi-Hermitian systems with non-observable Hamiltonians

TL;DR

This paper shows that unitary evolution can be maintained in time-dependent non-Hermitian quantum systems by solving the Dyson and quasi-Hermiticity equations, but at the expense of making the Hamiltonian non-observable.

Contribution

It provides a consistent method to handle time-dependent non-Hermitian systems with a dynamic metric, resolving previous incompatibility issues.

Findings

Time-dependent Dyson and quasi-Hermiticity equations can be solved consistently.

The non-Hermitian Hamiltonian becomes non-observable when the metric is time-dependent.

Unitary evolution is achievable despite the non-observability of the Hamiltonian.

Abstract

It has been argued that it is incompatible to maintain unitary time-evolution for time-dependent non-Hermitian Hamiltonians when the metric operator is explicitly time-dependent. We demonstrate here that the time-dependent Dyson equation and the time-dependent quasi-Hermiticity relation can be solved consistently in such a scenario for a time-dependent Dyson map and time-dependent metric operator, respectively. These solutions are obtained at the cost of rendering the non-Hermitian Hamiltonian to be a non-observable operator as it ceases to be quasi-Hermitian when the metric becomes time-dependent.