Effective nonlinear Hamiltonians in dielectric media

We derive an effective Hamiltonian for the nonlinear process of parametric downconversion in the presence of absorption. Based upon the Green-function method for quantizing the electromagnetic field, we first set up Heisenberg's equations of motion for a single atom driven by an external electric field and in the presence of an absorbing dielectric material. The equations of motion are then solved to second order in perturbation theory which, in rotating-wave approximation, yields the standard effective interaction Hamiltonian known from free-space nonlinear optics. In a second step, we derive the local-field-corrected Hamiltonian for an atom embedded in a dielectric host medium, that is, a nonlinear crystal. Here we show that the resulting effective Hamiltonian is found to be trilinear in the electric and noise-polarization fields and is thus capable of describing nonlinear noise processes. Furthermore, it reduces to the phenomenological nonlinear Hamiltonian for the cases where absorption, and hence the noise-polarization field, vanishes.