Polariton dynamics in one-dimensional arrays of atoms coupled to waveguides

Photons strongly coupled to material systems constitute a novel system for realizing non-linear optics at the level of individual photons and studying the dynamics of non-equilibrium quantum many-body system. We give a simple physical polariton-picture of the dynamics of photons coupled to a one-dimensional array of two-level atoms. This picture allows a fully analytical description of the dynamics in terms of polariton scattering inside the medium and reflections of the polaritons from the edge of the array. We show that inelastic collisions, previously identified in small systems, also occur in infinite systems and are related to the existence of multiple bands in the dispersion relation. The developed theory constitutes an effective field theory for the dynamics, which can be used for studies of non-linear optics and many-body dynamics. As a specific example we map the system to the Lieb-Liniger model and show that a so-called Tonks-Girardeau gas of photons is a stable eigenstate of the system in the limit of many emitters.