  Microcavity electron-hole-photon systems in two-dimensions are long
anticipated to exhibit a crossover from Bose-Einstein condensate (BEC) to
Bardeen-Cooper-Schrieffer (BCS) superfluid, when carrier density is tuned to
reach the Mott transition density. Yet, theoretical understanding of such a
BEC-BCS crossover largely relies on the mean-field framework and the nature of
the carriers at the crossover remains unclear to some extent. Here, motivated
by the recent demonstration of a BCS polariton laser {[}Hu \textit{et al.},
arXiv:1902.00142{]} and based on a simplified short-range description of the
electron-hole attraction, we examine the role of quantum fluctuations in an
exciton-polariton condensate at thermal equilibrium and determine the number of
different type carriers at the crossover beyond mean-field. Near Mott density
and with ultra-strong light-matter coupling, we find an unexpectedly large
phase window for a strongly correlated BCS polariton condensate, where both
fermionic Bogoliubov quasi-particles and bosonic excitons are significantly
populated and strongly couple to photons. We predict its photoluminescence
spectra and show that the upper polariton energy gets notably renormalized,
giving rise to a high-energy side-peak at large carrier density, as observed in
recent experiments.
