Synchronization in one-dimensional arrays of photon condensates

We investigate the synchronization of coupled photonic Bose-Einstein condensates in dye-filled microcavity arrays, subject to a potential gradient. We show that in the absence of noise the desynchronisation threshold is the result of a dynamical instability. For a two-site system, we analytically derive the maximal potential difference that allows for a synchronized solution as a function of the system parameters and find that it crucially depends on the thermalization rate. For systems with $N$ sites, we find a $N^{-3}$ power law scaling for the critical potential difference. When quantum fluctuations are included, stochastic phase slips broaden the sharp transitions, yet the phase velocity and first order spatial coherence still reliably identify the instability threshold.