  A full quantum theory beyond the mean-field regime is developed for an
exciton polariton condensate, to gain a complete understanding of quantum
fluctuations. We find analytical solution for the polariton density matrix,
showing the polariton nonlinearity causing fast relaxation correlated with the
pump so as to yield the condensation at threshold. Increasing the pump
intensity, a nonequilibrium phase transition towards the condensation of lower
polaritons emerges, with a statistics transiting from a thermal, through a
super-Poissonian and to a nonclassical distribution beyond the understanding at
the level of off-diagonal long-range order. The results signify the role of
dark states for polariton fluctuations, and lead to a nonclassical counting
statistics of emitted photons, which elaborates the role of the key parameters,
e.g., pump, detuning and temperature.
