Abstract
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 electromotive force $E$ is much smaller than the level spacing, $\ensuremath{\Delta}$. Electron states exponentially localized in energy are found for irrational values of the ratio $A\ensuremath{\equiv}\ensuremath{\Delta}/2eE$. Because of relaxation localization does not develop if $A$ is close to a rational number. As a result the accumulated energy as a function of $A$ contains sharp peaks at rational values (fractional pumping).
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