  In this article, we theoretically study the quantum statistical properties of
the light transmitted through or reflected from an optical cavity, filled by an
atomic medium with strong optical non-linearity induced by Rydberg-Rydberg van
der Waals interactions. Atoms are driven on a two-photon transition from their
ground state to a Rydberg level via an intermediate state by the combination of
a weak signal field and a strong control beam. By using a perturbative
approach, we get analytic results which remain valid in the regime of weak
feeding fields, even when the intermediate state becomes resonant. Therefore
they allow us to investigate quantitatively new features associated with the
resonant behaviour of the system. We also propose an effective non-linear
three-boson model of the system which, in addition to leading to the same
analytic results as the original problem, sheds light on the physical processes
at work in the system.
