Bosons hopping across sites and interacting on-site are the essence of theBose-Hubbard model (BHM). Inspired by the success of BHM simulators with atomsin optical lattices, proposals for implementing the BHM with photons in couplednonlinear cavities have emerged. Two coupled semiconductor microcavitiesconstitute a model system where the hopping, interaction, and decay of excitonpolaritons --- mixed light-matter quasiparticles --- can be engineered incombination with site-selective coherent driving to implement thedriven-dissipative two-site optical BHM. Here we explore the interplay ofinterference and nonlinearity in this system, in a regime where three distinctdensity profiles can be observed under identical driving conditions. Wedemonstrate how the phase acquired by polaritons hopping between cavities canbe controlled through effective polariton-polariton interactions. Our resultsopen new perspectives for synthesizing density-dependent gauge fields forpolaritons in two-dimensional multicavity systems.
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