Abstract
We examine the dynamics of Rydberg polaritons with dipolar interactions that propagate in multiple spatial modes. The dipolar excitation exchange between different Rydberg states mediates an effective exchange between polaritons that enables photons to hop across different spatial channels. Remarkably, the efficiency of this photon exchange process can increase with the channel distance and becomes optimal at a finite rail separation. Based on this mechanism, we design a simple photonic network that realizes a two photon quantum gate with a robust π phase, protected by the symmetries of the underlying photon interaction and the geometry of the network. These capabilities expand the scope of Rydberg electromagnetically induced transparency towards multidimensional geometries for nonlinear optical networks and explorations of photonic many-body physics.
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