Nonlinear dynamics of the two-photon Dicke model

We derive an effective two-photon Hamiltonian for a collection of two-level atoms interacting with a quantized field in the regime of two-photon resonance. This Hamiltonian naturally includes the dynamical Stark shift terms. Considering, in detail, the case of an initially strong quantum field, we show that the effective Hamiltonian acquires nonlinear terms in the su (2) algebra generators. For this case we find the evolution operator and prove the wavefunction factorization for special initial states. We show that collective effects modify the dynamics of the atomic inversion leading to a modulation of the collapse-revival structure.