Photon-photon gate via the interaction between two collective Rydberg excitations

We propose a scheme for a deterministic controlled-PHASE gate between two photons that is based on the strong interaction between two stationary collective Rydberg excitations in an atomic ensemble outside the regime of Rydberg blockade. The distance-dependent character of the interaction causes both a momentum displacement of the collective excitations and unwanted entanglement between them. We show that these effects can be overcome by swapping the collective excitations in space and by optimizing the geometry, resulting in a photon-photon gate with high fidelity and efficiency.