  Extended Hubbard models have proven to bear novel quantum states, but their
experimental realization remains challenging. In this work we propose to use
bosonic quantum gases dressed with molecular bound states in Rydberg
interaction potentials for the observation of these quantum states. We study
the molecular Rabi coupling with respect to principal quantum number and
trapping frequency of the ground state atoms for various molecular potentials
of Rubidium and Potassium, and the hereby resulting dressed interaction
strength. Additionally, we propose a two-color excitation scheme which
significantly increases the dressed interaction and cancels AC Stark shifts
limiting the atomic motion in the itinerant regime. We study the various
equilibrium phases of the corresponding extended Bose-Hubbard model by means of
the Cluster Gutzwiller approach and perform time evolution simulations via the
Lindblad master equation. We find a supersolid phase by slowly ramping the
molecular Rabi coupling of an initially prepared superfluid and discuss the
role of dissipation.
