  We demonstrate a density-dependent gauge field, induced by atomic
interactions, for quantum gases. The gauge field results from the synchronous
coupling between the interactions and micromotion of the atoms in a modulated
two-dimensional optical lattice. As a first step, we show that a coherent
shaking of the lattice in two directions can couple the momentum and
interactions of atoms and break the four-fold symmetry of the lattice. We then
create a full interaction-induced gauge field by modulating the interaction
strength in synchrony with the lattice shaking. When a condensate is loaded
into this shaken lattice, the gauge field acts to preferentially prepare the
system in different quasimomentum ground states depending on the modulation
phase. We envision that these interaction-induced fields, created by fine
control of micromotion, will provide a stepping stone to model new quantum
phenomena within and beyond condensed matter physics.
