  We propose a general method by which experiments on ultracold gases can be
used to determine the topological properties of the energy bands of optical
lattices, as represented by the map of the Berry curvature across the Brillouin
zone. The Berry curvature modifies the semiclassical dynamics and hence the
trajectory of a wave packet undergoing Bloch oscillations. However, in two
dimensions these trajectories may be complicated Lissajous-like figures, making
it difficult to extract the effects of Berry curvature in general. We propose
how this can be done using a "time-reversal" protocol. This compares the
velocity of a wave packet under positive and negative external force, and
allows a clean measurement of the Berry curvature over the Brillouin zone. We
discuss how this protocol may be implemented and explore the semiclassical
dynamics for three specific systems: the asymmetric hexagonal lattice, and two
"optical flux" lattices in which the Chern number is nonzero. Finally, we
discuss general experimental considerations for observing Berry curvature
effects in ultracold gases.
