  We show that a multicolor modulation of the depth of an optical lattice
allows for a flexible independent control of correlated hopping,
occupation-dependent gauge fields, effective on-site interactions without
Feshbach resonances, and nearest-neighbor interactions. As a result, the
lattice-depth modulation opens the possibility of engineering with minimal
experimental complexity a broad class of lattice models in current experiments
with ultra-cold atoms, including Hubbard models with correlated hopping,
peculiar extended models, and two-component anyon-Hubbard models.
