  We consider the open two-site Bose-Hubbard dimer, a well-known quantum
mechanical model that has been realised recently for photons in two coupled
photonic crystal nanocavities. The system is described by a Lindblad master
equation which, for large numbers of photons, gives rise to a limiting
semiclassical model in the form of a four-dimensional vector field. From the
situation where both sites trap the same amount of photons under symmetric
pumping, one encounters a transition that involves symmetry breaking, the
creation of periodic oscillations and multistability as the pump strength is
increased. We show that the associated one-parameter bifurcation diagram of the
semiclassical model captures the essence of statistical properties of computed
quantum trajectories as the pump strength is increased. Even for small numbers
of photons, the fingerprint of the semiclassical bifurcations can be recognised
reliably in observables of quantum trajectories.
