  Photons strongly coupled to material systems constitute a novel system for
studying the dynamics of non-equilibrium quantum many-body systems. We give a
fully analytical description of the dynamics of photons coupled to a
one-dimensional array of two-level atoms. The description incorporates input,
scattering inside the medium, and the ejection of photons from the array. We
show that inelastic collisions, previously identified in small systems, also
occur in infinite systems and elucidate the physics behind this effect. The
developed theory constitutes an effective field theory for the dynamics, which
can be used as a starting point for studies of many-body dynamics. We discuss
different parameter regimes and touch upon possible applications and the limit
of many excitations.
