Comparison of quantization of charge transport in periodic and open pumps

We compare the charges transported in two systems, a spatially periodic and an open quantum pump, both depending periodically and adiabatically on time. The charge transported in a cycle was computed by Thouless [Phys. Rev. B 27, 6083 (1983)] and by Buttiker [Z. Phys. B: Condens. Matter 94, 133 (1994)] in the two cases. We show that the results agree in the limit where the two physical situations become the same, i.e., that of a large open pump.