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Quantum theory of Kerr nonlinearity with Rydberg slow light polaritons

We study the propagation of Rydberg slow light polaritons through an atomic medium for intermediate interactions. Then, the dispersion relation for the polaritons is well described by the slow light velocity alone, which allows for an analytical solution for arbitrary shape of the atomic cloud. We demonstrate the connection of Rydberg polaritons to the behavior of a conventional Kerr nonlinearity for weak interactions, and determine the leading quantum corrections for increasing interactions. We propose an experimental setup which allows one to measure the effective two-body interaction potential between slow light polaritons as well as higher body interactions.