Lewis-Riesenfeld invariants for PT-symmetrically coupled oscillators from two dimensional point transformations and Lie algebraic expansions

TL;DR

This paper develops a systematic method using Lie algebra and point transformations to construct Lewis-Riesenfeld invariants for PT-symmetric coupled oscillators, facilitating analysis of their time-dependent quantum behavior.

Contribution

It introduces a novel approach combining Lie algebraic expansions and point transformations to derive invariants for non-Hermitian, PT-symmetric oscillators.

Findings

Constructed invariants using symplectic $sp(4)$ algebra

Solved coupled differential equations via time-ordered exponentials

Demonstrated direct construction of Dyson maps from point transformations

Abstract

We construct Lewis-Riesenfeld invariants from two dimensional point transformations for two oscillators that are coupled to each other in space in a PT-symmetrical and time-dependent fashion. The non-Hermitian Hamiltonian of the model is conveniently expressed in terms of generators of the symplectic $sp(4)$ Lie algebra. This allows for an alternative systematic approach to find Lewis-Riesenfeld invariants leading to a set of coupled differential equations that we solve by using time-ordered exponentials. We also demonstrate that point transformations may be utilized to directly construct time-dependent Dyson maps from their respective time-independent counterparts in the reference system.