  We describe a method to generate a synthetic gauge potential for ultracold
atoms held in an optical lattice. Our approach uses a time-periodic driving
potential based on two quickly alternating signals to engineer the appropriate
Aharonov-Bohm phases, and permits the simulation of a uniform tunable magnetic
field. We explicitly demonstrate that our split driving scheme reproduces the
behavior of a charged quantum particle in a magnetic field over the complete
range of field strengths, and obtain the Hofstadter butterfly band-structure
for the Floquet quasienergies at high fluxes.
