  We measure the full photon-number distribution emitted from a Bose condensate
of microcavity exciton-polaritons confined in a micropillar cavity. The
statistics are acquired by means of a photonnumber resolving transition edge
sensor. We directly observe that the photon-number distribution evolves with
the non-resonant optical excitation power from geometric to quasi-Poissonian
statistics, which is canonical for a transition from a thermal to a coherent
state. Moreover, the photon-number distribution allows evaluating the
higher-order photon correlations, shedding further light on the coherence
formation and phase transition of the polariton condensate. The experimental
data is analyzed in terms of thermal coherent states which allows one to
directly extract the thermal and coherent fraction from the measured
distributions. These results pave the way for a full understanding of the
contribution of interactions in light-matter condensates in the coherence
buildup at threshold.
