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We study the ground state phase diagram of a nonlinear two-photonRabi-Hubbard (RH) model in one dimension using quantum Monte Carlo (QMC)simulations and density matrix renormalization group (DMRG) calculations. Ourmodel includes a nonlinear photon-photon interaction term. Absent this term,the RH model has only one phase, the normal disordered phase, and suffers fromspectral collapse at larger values of the photon-qubit interaction orinter-cavity photon hopping. The photon-photon interaction, no matter howsmall, stabilizes the system which now exhibits {\it two} quantum phasetransitions: Normal phase to {\it photon pair} superfluid (PSF) transition andPSF to single particle superfluid (SPSF). The discrete $Z_4$ symmetry of theHamiltonian spontaneously breaks in two stages: First it breaks partially asthe system enters the PSF and then completely breaks when the system finallyenters the SPSF phase. We show detailed numerical results supporting this, andmap out the ground state phase diagram.

Nonlinear two-photon Rabi-Hubbard model: superradiance and photon/photon-pair Bose-Einstein condensate