  Bilayer materials may support interlayer excitons comprised of electrons in
one layer and holes in the other. In experiments, a non-zero exciton density is
typically sustained by a bias chemical potential, implemented either by optical
pumping or by electrical contacts connected to the two layers. We show that if
charge can tunnel between the layers, the chemical potential bias means that an
exciton condensate is in the dynamical regime of ac Josephson effect. It has
physical consequences such as tunneling currents and the ability to tune a
condensate from bright (emitting coherent photons) to dark by experimental
controlling knobs. If the system is placed in an optical cavity, coupling with
cavity photons favors different dynamical states depending on the bias,
realizing superradiant phases.
