  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 (TLS). In particular,
we find that in such a system a contact interaction between two polaritons is
induced by the nonlinearity of the Jaynes-Cummigs Hamiltonian. Using our
approach we solve the two-polariton problem with zero center-of-mass momentum,
and find 1D 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.
