Path-integral description of quantum nonlinear optics in arbitrary media

We present a method, based on Feynman path integrals, to describe the propagation and properties of the quantized electromagnetic field in an arbitrary, nonlinear medium. We provide a general theory, valid for any order of optical nonlinearity, and we then specialize the case of second-order nonlinear processes. In particular, we show that second-order nonlinear processes in arbitrary media, under the undepleted pump approximation, can be described by an effective free electromagnetic field, propagating in a vacuum, dressed by the medium itself. Moreover, we show that the probability of such processes to occur is related to the biphoton propagator, which contains information about the structure of the medium, its nonlinear properties, and the structure of the pump beam.