  We construct a quantum Ginsburg-Landau theory to study the quantum phases and
transitions in electron hole bilayer system. We propose that in the dilute
limit as distance is increased, there is a first order transition from the
excitonic superfluid (ESF) to the excitonic supersolid (ESS) driven by the
collapsing of a roton minimum, then a 2nd order transition from the ESS to
excitonic normal solid. We show the latter transition is in the same
universality class of superfluid to Mott transition in a rigid lattice. We then
study novel elementary low energy excitations inside the ESS. We find that
there are two "supersolidon" longitudinal modes (one upper branch and one lower
branch) inside the ESS, while the transverse mode in the ESS stays the same as
that inside a ENS. We also work out various experimental signatures of these
novel elementary excitations by evaluating the Debye-Waller factor,
density-density correlation, specific heat and vortex -vertex interactions. For
the meta-stable supersolid generated by photon pumping, we show that the angle
resolved spectrum is dominated by the macroscopic super-radiance from its
superfluid component, even it is just a very small percentage of the the whole
system. This fact can be used to detect the metastable ESS state generated by
photon pumping by a power spectrum experiment easily and without any ambiguity.
