  Artificial gauge fields open new possibilities to realize quantum many-body
systems with ultracold atoms, by engineering Hamiltonians usually associated
with electronic systems. In the presence of a periodic potential, artificial
gauge fields may bring ultracold atoms closer to the quantum Hall regime. Here,
we describe a one-dimensional lattice derived purely from effective
Zeeman-shifts resulting from a combination of Raman coupling and radiofrequency
magnetic fields. In this lattice, the tunneling matrix element is generally
complex. We control both the amplitude and the phase of this tunneling
parameter, experimentally realizing the Peierls substitution for ultracold
neutral atoms.
