  We propose and analyse analogs of optical cavities for atoms using two-well
Bose-Hubbard models with pumping and losses. With one well pumped, we find that
both the mean-field dynamics and the quantum statistics show a quantitative
dependence on the choice of damped well. Both the systems we analyse remain far
from equilibrium, preserving good coherence between the wells in the
steady-state. We find a small degree of quadrature squeezing and mode
entanglement for some parameter regimes. Due to recent experimental advances,
it should be possible to demonstrate the effects we investigate and predict.
