  Realization of strong synthetic magnetic fields in driven optical lattices
has enabled implementation of topological bands in cold-atom setups. A
milestone has been reached by a recent measurement of a finite Chern number
based on the dynamics of incoherent bosonic atoms. The measurements of the
quantum Hall effect in semiconductors are related to the Chern-number
measurement in a cold-atom setup, however, the design and complexity of the two
types of measurements are quite different. Motivated by these recent
developments, we investigate the dynamics of weakly interacting incoherent
bosons in a two-dimensional driven optical lattice exposed to an external
force, which provides a direct probe of the Chern number. We consider a
realistic driving protocol in the regime of high driving frequency and focus on
the role of weak repulsive interactions. We find that interactions lead to the
redistribution of atoms over topological bands both through the conversion of
interaction energy into kinetic energy during the expansion of the atomic cloud
and due to an additional heating. Remarkably, we observe that the moderate
atomic repulsion facilitates the measurement by flattening the distribution of
atoms in the quasi-momentum space. Our results also show that weak interactions
can suppress the contribution of some higher-order non-topological terms in
favor of the topological part of the effective model.
