Exceptional and regular spectra of a generalized Rabi model

We study the spectrum of a generalized Rabi model in which co- and counter-rotating terms have different coupling strengths. It is also equivalent to the model of a two-dimensional electron gas in a magnetic field with Rashba and Dresselhaus spin-orbit couplings. Like in the case of the Rabi model, the spectrum of our generalized Rabi model consists of the regular and the exceptional parts. The latter is represented by the energy levels which cross at certain parameter values which we determine explicitly. The wave functions of these exceptional states are given by finite order polynomials in the Bargmann representation. The roots of these polynomials satisfy a Bethe ansatz equation of the Gaudin type. At the exceptional points the model is therefore quasiexactly solvable. An analytical approximation is derived for the regular part of the spectrum in the weak- and strong-coupling limits. In particular, in the strong-coupling limit the spectrum consists of two ladders of equidistant levels.