Quasiparticle poisoning and Josephson current fluctuations induced by Kondo impurities

We introduce a toy model that allows us to study the physical properties of a spin impurity coupled to the electrons in the superconducting island. We show that when the coupling of the spin is of the order of the superconducting gap two almost degenerate subgap states are formed. By computing the Berry phase that is associated with the superconducting phase rotations in this model, we prove that these subgap states are characterized by a different charge and demonstrate that the switching between these states has the same effect as quasiparticle poisoning (unpoisoning) of the island. We also show that an impurity coupled to both the island and the lead generates Josepshon current fluctuations.