Quantization of adiabatic pumped charge in the presence of superconducting lead

We investigate the parametric electron pumping of a double barrier structure in the presence of a superconducting lead. The parametric pumping is facilitated by cyclic variation of the barrier heights x₁ and x₂ of the barriers. In the weak-coupling regime, there exists a resonance line in the parameter space (x₁,x₂) so that the energy of the quasibound state is in line with the incoming Fermi energy. Levinson et al. found recently that the pumped charge for each pumping cycle is quantized with $Q=2e$ for normal structure when the pumping contour encircles the resonance line. In the presence of a superconducting lead, we find that the pumped charge is quantized with the value $2e.$