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We study the nonequilibrium steady state of the driven-dissipativeBose-Hubbard model with Kerr nonlinearity. Employing a mean-field decouplingfor the intercavity hopping $J$, we find that the steep crossover between lowand high photon-density states inherited from the single cavity transforms intoa gas$-$liquid bistability at large cavity-coupling $J$. We formulate a van derWaals like gas$-$liquid phenomenology for this nonequilibrium situation anddetermine the relevant phase diagrams, including a new type of diagram where alobe-shaped boundary separates smooth crossovers from sharp, hysteretictransitions. Calculating quantum trajectories for a one-dimensional system, weprovide insights into the microscopic origin of the bistability.

Nonequilibrium gas-liquid transition in the driven-dissipative photonic lattice