  Second-order time correlation measurements with a temporal resolution better
than 3 ps were performed on a CdTe microcavity where spontaneous Bose-Einstein
condensation is observed. After the laser pulse, the nonresonantly excited
thermal polariton population relaxes into a coherent polariton condensate.
Photon statistics of the light emitted by the microcavity evidences a clear
phase transition from the thermal state to a coherent state, which occurs
within 3.2 ps after the onset of stimulated scattering. Following this very
fast transition, we show that the emission possesses a very high coherence that
persists for more than 100 ps after the build-up of the condensate.
