  Recent experiments show that periodic modulations of cold atoms in optical
lattices may be used to engineer and explore interesting models. We show that
double modulation, combining lattice shaking and modulated interactions allows
for the engineering of a much broader class of lattice models with correlated
hopping, which we study for the particular case of one-dimensional systems. We
show, in particular, that by using this double modulation it is possible to
study Hubbard models with asymmetric hopping, which, contrary to the standard
Hubbard model, present insulating phases with both parity and string order.
Moreover, double modulation allows for the simulation of lattice models in
unconventional parameter regimes, as we illustrate for the case of the
spin-$1/2$ Fermi-Hubbard model with correlated hopping, a relevant model for
cuprate superconductors.

Quantum Gases