Perturbative approach for strong and weakly coupled time-dependent non-Hermitian quantum systems

TL;DR

This paper introduces a perturbative method to compute the Dyson map and metric operator for time-dependent non-Hermitian quantum systems, enabling recursive solutions without prior guesses, and explores the physical implications of solution ambiguities.

Contribution

The paper presents a novel perturbative approach for time-dependent non-Hermitian systems, allowing systematic recursive solutions for Dyson maps and metrics, and analyzes solution ambiguities affecting physical observables.

Findings

Recursive solutions for Dyson maps and metrics are achievable without guesswork.

Ambiguities in solutions lead to different physical behaviors.

Method applied to coupled harmonic oscillators and anharmonic potentials.

Abstract

We propose a perturbative approach to determine the time-dependent Dyson map and the metric operator associated with time-dependent non-Hermitian Hamiltonians. We apply the method to a pair of explicitly time-dependent two dimensional harmonic oscillators that are weakly coupled to each other in a PT-symmetric fashion and to the strongly coupled explicitly time-dependent negative quartic anharmonic oscillator potential. We demonstrate that once the perturbative Ansatz is set up the coupled differential equations resulting order by order may be solved recursively in a constructive manner, thus bypassing the need for making any guess for the Dyson map or the metric operator. Exploring the ambiguities in the solutions of the order by order differential equations naturally leads to a whole set of inequivalent solutions for the Dyson maps and metric operators implying different physical behaviour as demonstrated for the expectation values of the time-dependent energy operator.