Invariant imbedding theory charge pumping in quantum wires

We have developed a method to study the theory of charge pumping through a continuous quantum wire in the invariant imbedding approach. Using this method, the effects of the barriers width of the time-periodic potential barriers on the pumped current as well as the general properties of quantum charge pumping is investigated. Using the invariant imbedding approach allows us to address directly the complex reflection (??) and transmission (??) matrices across the wire instead of considering the spectrum of the Schro??dinger equation. The pumped current depends on the width of the driven potential barriers and in the case of nonequality of the width of barriers, our study predicts a nonzero charge pumping at phase difference ????0 even if the driving potentials are equal, which is consistent with the recent experimental result. Our results also represents that the pumped current shows a sign reversal due to the signs of driving potentials.