Exactly solvable time-dependent pseudo-Hermitian $su$(1,1) Hamiltonian models

TL;DR

This paper provides an exact analytical solution for time-dependent pseudo-Hermitian su(1,1) Hamiltonians, revealing new solvable models with applications in classical optics and quantum Rabi scenarios, and explores the spectral transition independent of dynamical regimes.

Contribution

It introduces a class of exactly solvable time-dependent pseudo-Hermitian su(1,1) models with a parameter distinguishing dynamical regimes, applicable to classical and quantum systems.

Findings

Identified new exactly solvable models for su(1,1) Hamiltonians.

Demonstrated applications in guided wave optics.

Analyzed the spectral transition unrelated to dynamical regimes.

Abstract

An exact analytical treatment of the dynamical problem for time-dependent 2x2 pseudo-hermitian su(1,1) Hamiltonians is reported. A class of exactly solvable and physically transparent new scenarios are identified within both classical and quantum contexts. Such a class is spanned by a positive parameter $\nu$ that allows to distinguish two different dynamical regimes. Our results are usefully employed for exactly solving a classical propagation problem in a guided wave optics scenario. The usefulness of our procedure in a quantum context is illustrated by defining and investigating the su(1,1) "Rabi" scenario bringing to light analogies and differences with the standard su(2) Rabi model. Our approach, conjugated with the generalized von Neumann equation describing open quantum systems through non-Hermitian Hamiltonians, succeeds in evidencing that the $\nu$-dependent passage from a real to a complex energy spectrum is generally unrelated to the existence of the two dynamical regimes.