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.
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