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Optimal quantum pump in the presence of a superconducting lead

We investigate the parametric pumping of a hybrid structure consisting of a normal quantum dot, a normal lead, and a superconducting lead. Using the time-dependent scattering matrix theory, we have derived a general expression for the pumped electric current and heat current. We have also derived the relationship among the instantaneous heat current generated by the pump, electric current, and the noise. This gives a lower bound for the heat current generated in the hybrid system similar to that of the normal case obtained by Avron et al. [Phys. Rev. Lett. 87, 236601 (2001)].