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 H-eff having the structure A + (PD-1P dagger) where D-1 is the propagator of the intermediate states and A, P two operators given by Rabi hamiltonian H-R. The adiabatic elimination of Ken et al. [Phys Rev. A 92, 023842 (2015)] is recovered as time independent limit where D-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 D-1 like B-1 - B-2 i partial derivative/partial derivative t and leads to a refined expression of the effective H-eff. Others features are also studied.