  We propose and analyse analogs of optical cavities for atoms using three-well
inline Bose-Hubbard models with pumping and losses. With one well pumped and
one damped, we find that both the mean-field dynamics and the quantum
statistics show a qualitative dependence on the choice of damped well. The
systems we analyse remain far from equilibrium, although most do enter a
steady-state regime. We find quadrature squeezing, bipartite and tripartite
inseparability and entanglement, and states exhibiting the EPR paradox,
depending on the parameter regimes. We also discover situations where the
mean-field solutions of our models are noticeably different from the quantum
solutions for the mean fields. Due to recent experimental advances, it should
be possible to demonstrate the effects we predict and investigate in this
article.
