We develop an analytical approach for calculating the scattering and bound states of two polaritons in a one-dimensional (1D) infinite array of coupled cavities, with each cavity coupled to a two-level system. We find that in such a system a contact interaction between two polaritons is induced by the non-linearity of the Jaynes-Cummigs Hamiltonian. We then solve the two-polariton problem with zero center-of-mass momentum, and find 1D scattering resonances. Our results are relevant to the transport of two polaritons, and are helpful for the investigation of many-body physics in a dilute gas of polaritons in a 1D cavity array.
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