Moire excitons promise a new platform with which to generate and manipulate hybrid quantum phases of light and matter in unprecedented regimes of interaction strength. We explore the properties in this regime, through studies of a Bose-Hubbard model of excitons coupled to cavity photons. We show that the steady states exhibit a rich phase diagram with pronounced bi-stabilities governed by multi-photon resonances reflecting the strong inter-exciton interactions. In the presence of an incoherent pumping of excitons we find that the system can realise one- and multi-photon lasers.
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