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

Abstract

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.