Quantum electrodynamics in absorbing nonlinear media

An effective Hamiltonian for the second-order nonlinear process of parametric downconversion in the presence of absorption is derived. The electromagnetic field is quantized using the Green function method. Heisenberg's equations of motion for the interaction of a single atom with a driving electric field are then solved to second order in perturbation theory in the rotating-wave approximation. The real-cavity model is used to embed the interacting atom in a bulk medium and correct for local field effects of the material. The resulting Hamiltonian is found to be trilinear in the electric and noise polarization fields and reduces to the phenomenological nonlinear Hamiltonian for the cases where absorption vanishes.