Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories

TL;DR

This paper introduces a new framework for time-dependent ${ m C}(t)$-operators in non-Hermitian quantum theories, showing they are always Lewis-Riesenfeld invariants and applying this to a two-level system across different ${ m PT}$-regimes.

Contribution

It defines time-dependent ${ m C}(t)$-operators as Lewis-Riesenfeld invariants, extending their role in non-Hermitian theories to dynamic scenarios and different ${ m PT}$-regimes.

Findings

${ m C}(t)$-operators can be expanded in terms of biorthonormal eigenvectors.

Solutions for ${ m C}(t)$ and the metric operator exist across all ${ m PT}$-regimes.

The framework applies to a non-Hermitian two-level Hamiltonian.

Abstract

${\cal C}$-operators were introduced as involution operators in non-Hermitian theories that commute with the time-independent Hamiltonians and the parity/time-reversal operator. Here we propose a definition for time-dependent ${\cal C}(t)$-operators and demonstrate that for a particular signature they may be expanded in terms of time-dependent biorthonormal left and right eigenvectors of Lewis-Riesenfeld invariants. The vanishing commutation relation between the ${\cal C}$-operator and the Hamiltonian in the time-independent case is replaced by the Lewis-Riesenfeld equation in the time-dependent scenario. Thus, ${\cal C}(t)$-operators are always Lewis-Riesenfeld invariants, whereas the inverse is only true in certain circumstances. We demonstrate the working of the generalities for a non-Hermitian two-level matrix Hamiltonian. We show that solutions for ${\cal C}(t)$ and the time-dependent metric operator may be found that hold in all three ${\cal PT}$-regimes, i.e., the ${\cal PT}$-regime, the spontaneously broken ${\cal PT}$-regime and at the exceptional point.