  Modern experimental platforms such as supercoducting-circuit arrays call for
the exploration of bosonic tight-binding models in unconventional situations
with no counterpart in real materials. Here we investigate one of such
situations, in which excitations are driven and damped by pairs, leading to
pattern formation and exotic bosonic states emerged from a non-equilibrium
quantum many-body system. Focusing on a two-dimensional driven-dissipative
Bose-Hubbard model, we find that its steady states are characterized by the
condensation of bosons around momenta lying on a "Bose surface", a bosonic
analogue of the Fermi surface in solid-state systems. The interplay between
instabilities generated by the driving, the nonlinear dissipative
mode-coupling, and the underlaying lattice effect, allows the system to
equilibrate into an exotic superfluid state of bosons condensed on a closed
ring in momentum space instead of discrete points. Such an unconventional state
with a spatially uniform density distribution goes beyond the traditional scope
of pattern formation, and thus has no counterpart in the classical literature.
In addition, it is a state connected to several open problems in modern
condensed-matter physics, and here we provide the means to stabilize it,
opening the way to its experimental study. Moreover, we also provide a concrete
experimental implementation of our model in currently-available
superconducting-circuit arrays. We also investigate the relaxation spectrum
around the condensate, which shows a characteristic purely diffusive behavior.
