  Ultracold atoms in optical lattices have emerged as powerful quantum
simulators of translationally invariant systems with many applications in e.g.
strongly-correlated and topological systems. However, the ability to locally
tune all Hamiltonian parameters remains an outstanding goal that would enable
the simulation of a wider range of quantum phenomena. Motivated by recent
advances in quantum gas microscopes and optical tweezers, we here show
theoretically how local control over individual tunnelling links in an optical
lattice can be achieved by incorporating local time-periodic potentials. We
propose to periodically modulate the on-site energy of individual lattice sites
and employ Floquet theory to demonstrate how this can result in full individual
control over the tunnelling amplitudes in one dimension. We provide various
example configurations realising interesting topological models such as
extended Su-Schrieffer-Heeger models that would be challenging to realize by
other means. Extending to two dimensions, we present that local periodic
driving in a three-site plaquette allows for full simultaneous control over the
relative tunnelling amplitudes and the gauge-invariant flux piercing the
plaquette, providing a clear stepping stone in building a fully programmable 2D
tight-binding model. This local modulation scheme is applicable to many
different lattice geometries.
