  Developing theoretical models for nonequilibrium quantum systems poses
significant challenges. Here we develop and study a multimode model of a
driven-dissipative Josephson junction chain of atomic Bose-Einstein
condensates, as realised in the experiment of Labouvie et al. [Phys. Rev. Lett.
116, 235302 (2016)]. The model is based on c-field theory, a beyond-mean-field
approach to Bose-Einstein condensates that incorporates fluctuations due to
finite temperature and dissipation. We find the c-field model is capable of
capturing all key features of the nonequilibrium phase diagram, including
bistability and a critical slowing down in the lower branch of the bistable
region. Our model is closely related to the so-called Lugiato-Lefever equation,
and thus establishes new connections between nonequilibrium dynamics of
ultracold atoms with nonlinear optics, exciton-polariton superfluids, and
driven damped sine-Gordon systems.
