We have developed a path-integral based framework, for describing the propagation of the quantised electromagnetic field in an arbitrarily shaped, nonlinear medium. In the undeleted pump approximation, quantum nonlinear effects can be described by Feynman diagrams in an effective vacuum, dressed by the material and pump properties.
Check out the details in our preprint:
Mosbah Difallah, Alexander Szameit, and Marco Ornigotti, “A path integral description of quantum nonlinear optics in arbitrary media“, arXiv:1904.02548 (2019).