  Driven-dissipative condensates, such as those formed from polaritons, expose
how the coherence of Bose-Einstein condensates evolves far from equilibrium. We
consider the phase and frequency ordering in the steady-states of a
one-dimensional lattice of condensates, described by a coupled oscillator model
with non-odd couplings, and include both time-dependent noise and a static
random potential. We present numerical results for the phase and frequency
distributions, and discuss them in terms of the Kardar-Paraisi-Zhang equation
and the physics of spacetime vortices. We find that the nucleation of spacetime
vortices causes the breakdown of the single-frequency steady-state and produces
a variation in the frequency with position. Such variation would provide an
experimental signature of spacetime vortices. More generally, our results
expose the nature of sychronization in oscillator chains with non-odd
couplings, random frequencies, and noise.
