Operating a quantum pump in a closed circuit

During an adiabatic pumping cycle a conventional two-barrier quantum device takes an electron from the left lead and ejects it to the right lead. Hence, the pumped charge per cycle is naively expected to be Q ≤ e. This zero-order adiabatic point of view is in fact misleading. For a closed device we can get Q > e and even Q ≫ e. In this paper, a detailed analysis of the quantum pump operation is presented. Using the Kubo formula for the geometric conductance, and applying the Dirac chains picture, we derive practical estimates for Q.