Time-dependent $\mathcal{C}$-operators as Lewis-Riesenfeld invariants in non-Hermitian theories

Highlights

• Defined time-dependent $\mathcal{C}$-operator.

• Established link between $\mathcal{C}$-operator and Lewis-Riesenfeld invariants.

• Presented a worked out example of a two-level system in all three $\mathcal{PT}$-regimes.

Abstract

$\mathcal{C}$-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 $\mathcal{C}(t)$-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 $\mathcal{C}$-operator and the Hamiltonian in the time-independent case is replaced by the Lewis-Riesenfeld equation in the time-dependent scenario. Thus, $\mathcal{C}(t)$-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 $\mathcal{C}(t)$ and the time-dependent metric operator may be found that hold in all three $\mathcal{PT}$-regimes, i.e., the $\mathcal{PT}$-regime, the spontaneously broken $\mathcal{PT}$-regime and at the exceptional point.