  It has been recently suggested that the Bose-Einstein condensate formed by
excitons in the dilute limit must be dark, i.e., not coupled to photons. Here,
we show that, under a density increase, the dark exciton condensate must
acquire a bright component due to carrier exchange in which dark excitons turn
bright. This however requires a density larger than a threshold which seems to
fall in the forbidden region of the phase separation between a dilute exciton
gas and a dense electron-hole plasma. The BCS-like condensation which is likely
to take place on the dense side, must then have a dark and a bright component -
which makes it "gray". It should be possible to induce an internal Josephson
effect between these two coherent components, with oscillations of the
photoluminescence as a strong proof of the existence for this "gray" BCS-like
exciton condensate.
