Solvable two dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime

TL;DR

This paper presents exact solutions for a two-dimensional time-dependent non-Hermitian quantum system, showing that it maintains real energy expectation values despite being in the broken PT-regime, and compares solution methods.

Contribution

It introduces a novel approach using invariants and pseudo-Hermiticity to solve time-dependent non-Hermitian systems more efficiently than direct methods.

Findings

Time-dependent non-Hermitian system has real energy expectations.

Invariant-based method simplifies solving Dyson equations.

Pseudo-Hermiticity relates invariants of non-Hermitian and Hermitian systems.

Abstract

We provide exact analytical solutions for a two dimensional explicitly time-dependent non-Hermitian quantum system. While the time-independent variant of the model studied is in the broken PT-symmetric phase for the entire range of the model parameters, and has therefore a partially complex energy eigenspectrum, its time-dependent version has real energy expectation values at all times. In our solution procedure we compare the two equivalent approaches of directly solving the time-dependent Dyson equation with one employing the Lewis-Riesenfeld method of invariants. We conclude that the latter approach simplifies the solution procedure due to the fact that the invariants of the non-Hermitian and Hermitian system are related to each other in a pseudo-Hermitian fashion, which in turn does not hold for their corresponding time-dependent Hamiltonians. Thus constructing invariants and subsequently using the pseudo-Hermiticity relation between them allows to compute the Dyson map and to solve the Dyson equation indirectly. In this way one can bypass to solve nonlinear differential equations, such as the dissipative Ermakov-Pinney equation emerging in our and many other systems.