Bose-Hubbard model: Relation between driven-dissipative steady states and equilibrium quantum phases

Abstract

We present analytical solutions for the mean-field master equation of the driven-dissipative Bose-Hubbard model for cavity photons, in the limit of both weak pumping and weak dissipation. Instead of pure Mott-insulator states, we find statistical mixtures with the same second-order coherence ${g}^{(2)}(0)$ as a Fock state with $n$ photons, but a mean photon number of $n/2.$ These mixed states occur when $n$ pump photons have the same energy as $n$ interacting photons inside the nonlinear cavity and survive up to a critical tunneling coupling strength, above which a crossover to a classical coherent state takes place. We also explain the origin of both antibunching and superbunching predicted by P-representation mean-field theory at higher pumping and dissipation. In particular, we show that the strongly correlated region of the associated phase diagram cannot be described within the semiclassical Gross-Pitaevskii approach.