Fractional pumping of energy into a ballistic quantum ring

TL;DR

This paper analyzes how energy is pumped into a ballistic quantum ring with a barrier under a slowly changing magnetic flux, revealing fractional energy localization at rational ratios of system parameters.

Contribution

It provides an exact analytic solution for the quantum dynamics in such a system and uncovers fractional energy pumping phenomena related to rational parameter ratios.

Findings

Energy localization occurs at irrational ratios of system parameters.

Energy peaks sharply at rational ratios, indicating fractional pumping.

Backscattering and relaxation influence the shape and spacing of energy peaks.

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 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.