Metric versus observable operator representation, higher spin models

TL;DR

This paper explores the metric representation in non-Hermitian quantum systems, demonstrating how to solve time-dependent spin models using the Dyson relation and simplifying the equations to linear and nonlinear forms.

Contribution

It introduces a method to solve time-dependent Hermitian and non-Hermitian spin models by decoupling the Dyson and quasi-Hermiticity equations into simpler differential equations.

Findings

Solutions for spin 1/2, 1, and 3/2 models with time-dependent metrics.

Decoupling of coupled equations into linear differential equations.

Reduction to the nonlinear Ermakov-Pinney equation.

Abstract

We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schroedinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the nonlinear Ermakov-Pinney equation.