  Probing the center-of-mass of an ultracold atomic cloud can be used to
measure Chern numbers, the topological invariants underlying the quantum Hall
effects. In this work, we show how such center-of-mass observables can have a
much richer dependence on topological invariants than previously discussed. In
fact, the response of the center of mass depends not only on the current
density, typically measured in a solid-state system, but also on the particle
density, which itself can be sensitive to the topology of the band structure.
We apply a semiclassical approach, supported by numerical simulations, to
highlight the key differences between center-of-mass responses and more
standard conductivity measurements. We illustrate this by analyzing both the
two- and the four-dimensional quantum Hall effects. These results have
important implications for experiments in engineered topological systems, such
as ultracold gases and photonics.
