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We introduce a scheme that combines photon-assisted tunneling by a movingoptical lattice with strong Hubbard interactions, and allows for the quantumsimulation of paradigmatic quantum many-body models. We show that, in a certainregime, this quantum simulator yields an effective Hubbard Hamiltonian withtunable bond-charge interactions, a model studied in the context ofstrongly-correlated electrons. In a different regime, we show how to exploit acorrelated destruction of tunneling to explore Nagaoka ferromagnetism at finiteHubbard repulsion. By changing the photon-assisted tunneling parameters, we canalso 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 tocold-atom experiments, departing from the region $t\gg J$ allowed by standardsingle-band Hubbard Hamiltonians in the strong-repulsion limit. We finally showthat, by generalizing the photon-assisted tunneling scheme, the quantumsimulator yields models of dynamical Gauge fields, where atoms of a givenelectronic state dress the tunneling of the atoms with a different internalstate, leading to Peierls phases that mimic a dynamical magnetic field.

Interaction-Dependent Photon-Assisted Tunneling in Optical Lattices: A Quantum Simulator of Strongly-Correlated Electrons and Dynamical Gauge Fields