  We report on the observation of quantum coherence of Bose-Einstein condensed
photons in an optically-pumped, dye-filled microcavity. We find that coherence
is long-range in space and time above condensation threshold, but short-range
below threshold, compatible with thermal-equilibrium theory. Far above
threshold, the condensate is no longer at thermal equilibrium and is fragmented
over non-degenerate, spatially overlapping modes. A microscopic theory
including cavity loss, molecular structure and relaxation shows that this
multimode condensation is similar to multimode lasing induced by imperfect gain
clamping.
