  We consider the energy stored in a one-dimensional ballistic ring with a
barrier subject to a linearly time-dependent magnetic flux. An exact analytic
solution for the quantum dynamics of electrons in the ring is found for the
case when the electro-motive force ${\cal E}$ is much smaller than the level
spacing, $\Delta$. Electron states exponentially localized in energy are found
for irrational values of the ratio $A\equiv\Delta/2e{\cal E}$. Relaxation
limits the dynamic evolution and localization does not develop if $A$ is
sufficiently close to a rational number. As a result the accumulated energy
becomes a regular function of $A$ containing a set of sharp peaks at rational
values (fractional pumping). The shape of the peaks and the distances between
them are governed by the interplay between the strength of backscattering and
the relaxation rate.
