  Using a field-theoretic approach within the Schwinger-Keldysh formalism, we
study a Bose-Hubbard model in the presence of a driving field and dissipation
due to one-body losses. We recover the bistability diagram from the
Gross-Pitaevski equation and analyze the different phases with respect to their
elementary excitations and correlations. We find the low-density solution to be
subdivided into a dynamically instable, a gapped, and a gapless regime. The
correlations decay exponentially, but a substantial increase of correlation
length marks the regime of gapless excitations.
