  We show that for the very same reason that excitons are bright, i.e. coupled
to photons, they have a higher energy than dark excitons, even for electrons
spatially separated from holes, such as in a double quantum well. Indeed, the
same channel which produces the finite electron-hole effective overlap
responsible for the absorption and emission of photon allows for Coulomb
interband exchange processes, which are nothing but a sequence of virtual
recombination and creation of one electron-hole pair. Consequently, this
additional repulsive electron-hole Coulomb exchange interaction exists for
bright excitons, but not for dark excitons. If we now remember that dark
excitons with spins $\pm 2$ are formed in a natural way through carrier
exchange between two opposite spin bright excitons, we are led to predict that
in a double quantum well sample with one trap -- a configuration appropriate to
get high density -- exciton Bose-Einstein condensation should appear when
cooling down the sample as a dark spot made of $(\pm 2)$ excitons at the center
of the trap.
