Quantum and Nonlinear Optics in Strongly Interacting Atomic Ensembles

Coupling light to ensembles of strongly interacting particles has emerged as a promising route toward achieving few photon nonlinearities. One specific way to implement this kind of nonlinearity is to interface light with highly excited atomic Rydberg states by means of electromagnetically induced transparency, an approach which allows freely propagating photons to acquire synthetic interactions of hitherto unprecedented strength. Here, we present an overview of this rapidly developing field, from classical effects to quantum manifestations of the nonlocal nonlinearities emerging in such systems. With an emphasis on underlying theoretical concepts, we describe the many experimental breakthroughs so far demonstrated and discuss potential applications looming on the horizon.