On ultrastrong coupling in quantum Rabi theory

We develop an effective Hamiltonian model to study resonance properties of multi-photon processes in c-QED. This proposal predicts an effective Heff having the structure + -1† where -1 is the propagator of the intermediate states and , two operators given by Rabi hamiltonian HR. The adiabatic elimination of Ken et al. [Phys Rev. A 92, 023842 (2015)] is recovered as time independent limit where -1 is approached by the inverse of a constant matrix ℬ which is also determined from HR. A new limit, named as linear approximation, is also obtained; it corresponds to approximating -1 like ℬ-1 − ℬ-2 (i∂/∂t) and leads to a refined expression of the effective Heff. Others features are also studied