  Sub-wavelength arrays of quantum emitters offer an efficient free-space
approach to coherent light-matter interfacing, using ultracold atoms or
two-dimensional solid-state quantum materials. The combination of collectively
suppressed photon-losses and emerging optical nonlinearities due to strong
photon-coupling to mesoscopic numbers of emitters holds promise for generating
nonclassical light and engineering effective interactions between freely
propagating photons. While most studies have thus far relied on numerical
simulations, we describe here a diagrammatic Green's function approach that
permits analytical investigations of nonlinear processes. We illustrate the
method by deriving a simple expression for the scattering matrix that describes
photon-photon interactions in an extended two-dimensional array of quantum
emitters, and reproduces the results of numerical simulations of coherently
driven arrays. The approach yields intuitive insights into the nonlinear
response of the system and offers a promising framework for a systematic
development of a theory for interacting photons and many-body effects on
collective radiance in two-dimensional arrays of quantum emitters.
