  Recent experiments have illustrated that long range two-body interactions can
be induced by laser coupling atoms to highly excited Rydberg states. Stimulated
by this achievement, we study supersolidity of lattice bosons in an
experimentally relevant situation. In our setup, we consider two-component
atoms on a square lattice, where one species is weakly dressed to an
electronically high-lying (Rydberg) state, generating a tunable, soft-core
shape long-range interaction. Interactions between atoms of the second species
and between the two species are characterized by local inter- and intra-species
interactions. Using a dynamical mean-field calculation, we find that
interspecies onsite interactions can stabilize a pronounced region of
supersolid phases. This is characterized by two distinctive types of
supersolids, where the bare species forms supersolid phases that are immersed
in strongly correlated quantum phases, i.e. a crystalline solid or supersolid
of the dressed atoms. We show that the interspecies interaction leads to a
roton-like instability in the bare species and therefore is crucially important
to the supersolid formation. We provide a detailed calculation of the
interaction potential to show how our results can be explored under current
experimental conditions.
