Diagrammatic treatment of few-photon scattering from a Rydberg-blockaded atomic ensemble in a cavity

In a previous Letter [A. Grankin et al., Phys. Rev. Lett. 117, 253602 (2016)] we studied the giant optical nonlinearities of a Rydberg atomic medium within an optical cavity, in the Schwinger-Keldysh formalism. In particular, we calculated the nonlinear contributions to the spectrum of the light transmitted through the cavity. In this article we spell out the essential details of this calculation and we show how it can be extended to higher input photon numbers and higher-order correlation functions. As a relevant example, we calculate and discuss the three-photon correlation function of the transmitted light and discuss its physical significance in terms of the polariton energy levels of the Rydberg medium within the optical cavity.