Quantum Integrable Multiatom Matter-Radiation Models With and Without the Rotating-Wave Approximation

New integrable multiatom matter-radiation models with and without the rotating-wave approximation (RWA) are constructed and exactly solved via the algebraic Bethe ansatz. The models with the RWA are generated using the ancestor model approach in a unified way. The rational case yields the standard type of matter-radiation models, while the trigonometric case corresponds to their q-deformations. The models without the RWA are obtained from the elliptic case in the limit of the Gaudin model and high spin.