[en] 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^† 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 H_R. A new limit, named as linear approximation, is also obtained; it corresponds to approximating 𝒟^-1 like ℬ^-1 − ℬ^-2 (i∂/∂t) and leads to a refined expression of the effective H_eff. Others features are also studied. Graphical abstract: .