Wigner Crystallization of Single Photons in Cold Rydberg Ensemble

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The coupling of weak light fields to Rydberg states of atoms under conditions of electromagnetically induced transparency (EIT) leads to the formation of Rydberg polaritons which are quasi-particles with tunable effective mass and long-range interactions. Confined to one spatial dimension their low energy physics is that of a moving-frame Luttinger liquid which due to the long-range character of the repulsive interaction can form a Wigner crystal. We calculate the Luttinger K parameter using density-matrix renormalization group (DMRG) simulations and find that under typical slow-light conditions kinetic energy contributions are too strong for crystal formation. However, adiabatically increasing the polariton mass by turning a light pulse into stationary spin excitations allows to generate true crystalline order over a finite length. The dynamics of this process and asymptotic correlations are analyzed in terms of a time-dependent Luttinger theory.