Quantized adiabatic charge pumping and resonant transmission

Adiabatically pumped charge, carried by non interacting electrons through a quantum dot in a turnstile geometry, is studied as a function of the strength of the two modulating potentials (related to the conductances of the two point contacts to the leads) and of the phase shift between them. It is shown that the magnitude and sign of the pumped charge are determined by the relative position and orientation of the closed contour traversed by the system in the parameter plane, and the transmission peaks (or resonances) in that plane. Integer values (in units of the electronic charge e) of the pumped charge (per modulation period) are achieved when a transmission peak falls inside the pumping contour. The integer value is given by the winding number of the pumping contour: double winding in the same direction gives a charge of 2, while winding around two opposite branches of the transmission peaks or winding in opposite directions can give a charge close to zero.