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

Using the technique of classical r-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 detail two subclasses of this model and the corresponding examples associated with concrete classical r-matrices with spectral parameters. We show that the so-called 'diagonal' in the root basis r-matrices produces integrable Jaynes-Cummings- Dicke-type models with the 'rotating wave approximation', while all other types of classical r-matrices produce integrable JCD models without the rotating wave approximation. We construct an explicit example of an integrable Hermitian Jaynes-Cummings-Dicke-type Hamiltonian containing both 'rotating' and 'counter-rotating' terms. In the case of the diagonal r-matrices we find the spectrum of the corresponding quantum Hamiltonians using the algebraic Bethe ansatz. © 2011 IOP Publishing Ltd and SISSA.