Recently, the combination of slow light polaritons with the strong interactions between Rydberg atoms has emerged as a promising system for inducing a strong interaction between photons. Potential applications range from the implementation of phase gate for photons, to single photon sources, as well as the generation of strongly correlated states of photons. From theoretical point of view, the many-body setup of propagating Rydberg polaritons is hard to describe, especially when polaritons are strongly interacting . W show that in the regime of intermediate interaction strengths the many-body problem has an exact analytical solution for an arbitrary atomic density. Using developed theoretical framework , we discuss the solution and its regimes of validity. For weak nonlinearities, we establish the relation between the polariton setup and classical behaviour of a Kerr nonlinearity. Furthermore, we provide quantum correction to the classical Kerr nonlinearity. This theoretical platform is applicable to ongoing experiments and can be used as a benchmark for other theoretical methods.
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