  The idea of making photons effectively interact has attracted a lot of
interest in recent years, for several reasons. Firstly, since photons do not
naturally interact with each other, it is of fundamental physical interest to
see what kind of medium can mediate interactions between these fundamental and
non-interacting particles, and to what extent. Secondly, photonics is a major
candidate for future quantum technology, due to the easy manipulation, readout,
and transport of photons, which makes them ideal for quantum information
processing. Finally, achieving strong and tunable interactions among photons
would open up an avenue for exploring the many-body physics of a fluid of
light. In this thesis, we will see how a cavity formed of subwavelength
lattices of two-level atoms can confine photons to a nonlinear environment for
a long time, such that emitted photons have accumulated strong correlations
both among their momenta and in their temporal statistics. This speaks of a
strong photon-photon interaction within the cavity. The nonlinearity originates
in the saturability of individual atoms, and the lattice structure results in a
strong and low-loss collective interaction with light. While a single atomic
lattice has a largely linear nature, as the effect of individual atoms washes
out in the collective response, the confining geometry of the cavity means the
photons are exposed to the underlying saturability of the atoms for such a long
time that the nonlinearity is revived. We will analyse this system both using a
standard input-output formalism, where the nonlinear physics of the system is
handled numerically, and a powerful Green's function-based approach that allows
for exact analytical results with no additional approximations. This analytical
description has the potential to lead to an exact study of the many-body
physics of interacting photons in a two-dimensional setting.
