  Phase transitions between different states of matter can profoundly modify
the order in physical systems, with the emergence of ferromagnetic or
topological order constituting important examples. Correlations allow to
quantify the degree of order and classify different phases. Here we report
measurements of first-order spatial correlations in a harmonically trapped
two-dimensional photon gas below, at, and above the critical particle number
for Bose-Einstein condensation, using interferometric measurements of the
emission of a dye-filled optical microcavity. For the uncondensed gas, the
transverse coherence decays on a length scale determined by the thermal de
Broglie wavelength of the photons, which shows the expected scaling with
temperature. At the onset of Bose-Einstein condensation true long-range order
emerges, and we observe quantum statistical effects as the thermal wave packets
overlap. The excellent agreement with equilibrium Bose gas theory prompts
microcavity photons as promising candidates for studies of critical scaling and
universality in optical quantum gases.
