  Geometric phases in quantum mechanics play an extraordinary role in
broadening our understanding of fundamental significance of geometry in nature.
One of the best known examples is the Berry phase (M.V. Berry (1984), Proc.
Royal. Soc. London A, 392:45) which naturally emerges in quantum adiabatic
evolution. So far the applicability and measurements of the Berry phase were
mostly limited to systems of weakly interacting quasi-particles, where
interference experiments are feasible. Here we show how one can go beyond this
limitation and observe the Berry curvature and hence the Berry phase in generic
systems as a non-adiabatic response of physical observables to the rate of
change of an external parameter. These results can be interpreted as a
dynamical quantum Hall effect in a parameter space. The conventional quantum
Hall effect is a particular example of the general relation if one views the
electric field as a rate of change of the vector potential. We illustrate our
findings by analyzing the response of interacting spin chains to a rotating
magnetic field. We observe the quantization of this response, which term the
rotational quantum Hall effect.
