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

Abstract

An exact analytical treatment of the dynamical problem for time-dependent 2×2 pseudo-Hermitian su(1,1) Hamiltonians is reported. A class of exactly solvable and physically transparent scenarios are identified within both classical and quantum contexts. The class is spanned by a positive parameter ν that allows for distinguishing 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, applied to the generalized von Neumann equation describing open quantum systems through non-Hermitian Hamiltonians, succeeds in evidencing that the ν-dependent passage from a real to a complex energy spectrum is generally unrelated to the existence of the two dynamical regimes.