  We investigate signatures of a self-trapping transition in the
driven-dissipative Bose Hubbard dimer, in presence of incoherent pump and
single-particle losses. For fully symmetric couplings the stationary state
density matrix is independent of any Hamiltonian parameter, and cannot
therefore capture the competition between hopping-induced delocalization and
the interaction-dominated self-trapping regime. We focus instead on the exact
quantum dynamics of the particle imbalance after the system is prepared in a
variety of initial states, and on the frequency-resolved spectral properties of
the steady state, as encoded in the single-particle Green's functions. We find
clear signatures of a localization-delocalization crossover as a function of
hopping to interaction ratio. We further show that a finite a pump-loss
asymmetry restores a delocalization crossover in the steady-state imbalance and
leads to a finite intra-dimer dissipation.
