  We introduce a scheme that combines photon-assisted tunneling by a moving
optical lattice with strong Hubbard interactions, and allows for the quantum
simulation of paradigmatic quantum many-body models. We show that, in a certain
regime, this quantum simulator yields an effective Hubbard Hamiltonian with
tunable bond-charge interactions, a model studied in the context of
strongly-correlated electrons. In a different regime, we show how to exploit a
correlated destruction of tunneling to explore Nagaoka ferromagnetism at finite
Hubbard repulsion. By changing the photon-assisted tunneling parameters, we can
also obtain a $t$-$J$ model with independently controllable tunneling $t$,
super-exchange interaction $J$, and even a Heisenberg-Ising anisotropy. Hence,
the full phase diagram of this paradigmatic model becomes accessible to
cold-atom experiments, departing from the region $t\gg J$ allowed by standard
single-band Hubbard Hamiltonians in the strong-repulsion limit. We finally show
that, by generalizing the photon-assisted tunneling scheme, the quantum
simulator yields models of dynamical Gauge fields, where atoms of a given
electronic state dress the tunneling of the atoms with a different internal
state, leading to Peierls phases that mimic a dynamical magnetic field.
