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In this paper we propose a model of polariton condensation with Kerr-type nonlinear photons. We introduce a generalized Dicke Hamiltonian to describe our system. By constructing the partition function as a path integral, the analytical and numerical solutions are presented. On the mean-field level, it is shown that the polariton condensation can occur and the Kerr nonlinearity affects the character of the polariton condensate. As the nonlinear coefficient increases, the condensate evolves from more photon-like to more exciton-like. Although the photon nonlinearity gives rise to a chemical potential greater than the photon energy, the quasiparticle excitation spectrum is still fully gapped. For the condensate collective excitations, the nonlinearity destroys the Goldstone modes and mixes the phase modes with the amplitude modes, resulting four non-zero-frequency collective modes. In addition, the influence of the photon-exciton detuning on the polariton condensate is also discussed.

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Topological phase transitions of a two-dimensional topologically nontrivial polaritonic system are studied. A generic model of semiconductor excitons strongly coupled with tailored photonic modes is considered. We introduce a pseudospin operator, measuring the polariton polarization between photonic-like and excitonic-like. The associated pseudospin spectrum and pseudospin Chern numbers are calculated. It is shown that the pseudospin Chern number phase diagram exhibits certain features resembling the topological phase of quantum-spin-Hall-like. Moreover, a series of topological phase transitions may occur with the closing of the bulk energy gap or the pseudospin spectrum gap. In a tight-binding form, the edge-mode simulation is done numerically to confirm the analytically results.

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