An exact analytical treatment of the dynamical problem for time-dependent $2\times 2$ pseudo-Hermitian $\text{su}(1,1)$ 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 $\nu$ 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 $\text{su}(1,1)$ “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 $\nu$-dependent passage from a real to a complex energy spectrum is generally unrelated to the existence of the two dynamical regimes.

• Received 9 June 2018

DOI:https://doi.org/10.1103/PhysRevA.98.033835