  Optical lattice systems provide exceptional platforms for quantum simulation
of many-body systems. We focus on the doubly modulated Bose-Hubbard model
driven by both time-dependent on-site energy and interaction, and predict the
emergence of the nearest neighbour interaction and density-assisted tunnelling.
By specifically designing a bi-chromatic driving pattern for a one dimensional
lattice, we demonstrate that the doubly modulated fields can be tuned to
realize desired quantum phases, e.g. the Mott insulator phase with selective
defects, and density wave phase.
