  In order to study the temporal coherence of a single-mode dye-cavity photon
condensate, a model is developed for the dynamics which treats the condensate
mode on a quantum-mechanical level. The effects of driving-dissipation and Kerr
interactions on the number fluctuations are studied analytically and
numerically, including the finding of a long-$\tau$ antibunching effect.
Depending on the interaction strength, we quantitatively observe an exponential
Schawlow-Townes-like decay or Gaussian Henry-like decay of phase correlations.
The adequacy of a heuristic phasor model originating from laser physics in
describing number and phase dynamics is validated within the experimentally
relevant parameter regime. The ratio of the first and second order coherence
times is shown to be inversely proportional to the number fluctuations, with a
prefactor that varies smoothly throughout the crossover between canonical and
grandcanonical statistics.
