  The hardcore extended Bose Hubbard model on a bilayer square lattice with
attractive (repulsive) interplane (intraplane) interactions has been
investigated numerically. Focusing on the strong interplane hopping parameter
regime, triplet valence bonds of dimers are stabilized to form valence bond
checkerboard solid at quarter filling. Increasing the particle number we
confirm, for the first time, the presence of the exotic valence bond supersolid
phase, where the valence bond solid ordering and the boson superfluidity
coexist. Rising further the particle number will lead to a checkerboard solid
of dimer pair at half filling for strong intraplane repulsion, or a valence
bond Mott insulator for weak repulsive interactions. We analyze the rich ground
state phase diagrams of this model, which can be experimentally realized in
optical lattices of cold atoms.
