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Many-body physics of Rydberg dark-state polaritons in the strongly interacting regime

Coupling light to Rydberg states of atoms under conditions of electromagnetically induced transparency (EIT) leads to the formation of strongly interacting quasi-particles, termed Rydberg polaritons. We derive a one-dimensional model describing the time evolution of these polaritons under paraxial propagation conditions, which we verify by numerical two-excitation simulations. We determine conditions allowing for a description by an effective Hamiltonian of a single-species polariton, and calculate ground-state correlations by use of the density matrix renormalization group (DMRG). Under typical stationary slow-light EIT conditions it is difficult to reach the strongly interacting regime where the interaction energy dominates the kinetic energy. We show that by employing time dependence of the control field the regime of strong interactions can be reached where the polaritons attain quasi crystalline order. We analyze the dynamics and resulting correlations for a translational invariant system in terms of a time-dependent Luttinger liquid theory and exact few-particle simulations and address the effects of nonadiabatic corrections and initial excitations.