Cavity polaritons with Rydberg blockade and long-range interactions

We study interactions between polaritons, arising when photons strongly couple to collective excitations in an array of two-level atoms trapped in an optical lattice inside a cavity. We consider two types of interactions between atoms: dipolar forces and atomic saturability, which range from hard-core repulsion to Rydberg blockade. We show that, in spite of the underlying repulsion in the subsystem of atomic excitations, saturability induces a broadband bunching of photons for two-polariton scattering states. We interpret this bunching as a result of interference, and trace it back to the mismatch of the quantization volumes for atomic excitations and photons. We also examine bound bipolaritonic states: these include states created by dipolar forces, as well as a gap bipolariton, which forms solely due to saturability effects in the atomic transition. Both types of bound states exhibit strong bunching in the photonic component. We discuss the dependence of bunching on experimentally relevant parameters.