Charge fluctuation between even and odd states of a superconducting island

We theoretically investigate effects of quantum fluctuation between even and odd charge states on the transport properties of a normal-superconducting-normal single-electron tunneling transistor. The charge fluctuation is discussed beyond the orthodox theory. We find that the charging energy renormalization enhances the parity effect: The Coulomb blockade regime for odd state is reduced and that for even state is widened. We show that the renormalization factor can be obtained by measuring the peak position of the conductance. The renormalization factor reduces with increasing the applied bias voltage.