  We show that pairs of atoms optically excited to the Rydberg states can
strongly interact with each other via effective long-range dipole-dipole or van
der Waals interactions mediated by their non-resonant coupling to a common
microwave field mode of a superconducting coplanar waveguide cavity. These
cavity mediated interactions can be employed to generate single photons and to
realize in a scalable configuration a universal phase gate between pairs of
single photon pulses propagating or stored in atomic ensembles in the regime of
electromagnetically induced transparency.
