We have investigated pumping in quantum dots from the perspective of non-Abelian (matrix) Berry phases by solving the time dependent Schr{\"o}dinger equation exactly for adiabatic changes. Our results demonstrate that a pumped charge is related to the presence of a finite matrix Berry phase. When consecutive adiabatic cycles are performed the pumped charge of each cycle is different from the previous ones.
@article{arxiv.0706.0947,
title = {Pumping in quantum dots and non-Abelian matrix Berry phases},
author = {N. Y. Hwang and S. C. Kim and P. S. Park and S. -R. Eric Yang},
journal= {arXiv preprint arXiv:0706.0947},
year = {2009}
}