Nanomechanical pumping of Cooper pairs through NS junction

We consider a system consisting of a single-level quantum dot that performs mechanical periodic oscillations between spatially distant normal and superconducting electrodes, approaching them at a distance that allows the exchange of electrons through the vacuum tunnel barrier. Considering that the distance between the electrodes is much greater than the tunneling length, we show that charge pumping occurs in such a nanosystem even when the electrochemical potentials of the electrodes coincide. In this case, the direction of the electron flow is determined by the position of the quantum dot level relative to the electrochemical potential in bulk electrodes. The latter can be controlled by applying a voltage between the ground and the electrodes. It is also shown that the value of the average current is critically sensitive to the strength of the tunnel coupling between the quantum dot and the superconducting electrode, which, in turn, is controlled by the amplitude of mechanical oscillations.