  We theoretically examine the driven-dissipative Bose-Hubbard dimer in the
optical bistable regime. Various approximation schemes based on a Gutzwiller
mean field decoupling are applied and compared. Depending on the system
parameters we show that a decoupling with respect to the real space or to the
reciprocal space can be more accurate. The Gutzwiller decoupling is applied
both at the level of the density matrix and for the wavefunction during a
quantum trajectory simulation. The latter is shown to be a more accurate
approximation. A Gaussian approximation for the non-homogeneous anti-bonding
mode is also explored. We also show that entanglement in this system is
witnessed by squeezing in reciprocal space.
