Pumping of energy into a mesoscopic ring: exactly solvable model

We consider the energy stored in a one-dimensional ballistic ring with a barrier subjected to a linearly time-dependent magnetic flux. An exact analytical solution for the quantum dynamics of electrons in the ring is found for the case when the electromotive force multiplied by the electron charge, eε, is much smaller than the inter level spacing, Δ. Electron states exponentially localized in energy space are found for irradiational values of the dimensionless ratio Α equivalent Δ/2 eε. Relaxation limits the dynamic evolution and the localization does not develop if Α is sufficiently close to a rational number. As a result the accumulated energy becomes a regular function of Α containing a set of sharp peaks at rational values with small enough denominators (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