General integrable two-level, one-mode Jaynes–Cummings–Dicke models and classicalr-matrices with spectral parameters

Abstract

Using the technique of classicalr-matrices and quantum Lax operators we construct the most general form of the quantum integrable‘two-level, one-mode’ spin-boson Jaynes–Cummings–Dicke-type model. We consider in detailtwo subclasses of this model and the corresponding examples associated with concrete classicalr-matrices with spectral parameters. We show that the so-called ‘diagonal’ in the root basisr-matrices produces integrable Jaynes–Cummings–Dicke-type models withthe ‘rotating wave approximation’, while all other types of classicalr-matrices produce integrable JCD models without the rotating wave approximation. Weconstruct an explicit example of an integrable Hermitian Jaynes–Cummings–Dicke-typeHamiltonian containing both ‘rotating’ and ‘counter-rotating’ terms. In the case of the diagonalr-matrices we find the spectrum of the corresponding quantum Hamiltonians using thealgebraic Bethe ansatz.