Abstract
We demonstrate that the Wigner function is suitable to describe single-electron tunneling in ultrasmall tunnel junctions. For a current-biased junction, we derive a master equation for the Wigner function which describes charging effects on Cooper-pair tunneling, quasiparticle-pair tunneling, and normal single-electron tunneling in an unified way. We obtain the current-voltage I-V characteristic for an ultrasmall Josephson junction that recovers those for the Josephson junction with no charging energy in the limit of e/C\ensuremath{\rightarrow}0. It is predicted that, due to charging effect in the ultrasmall Josephson, the position of the Riedel peak and the gap of the quasiparticle-pair interference term are changed from 2\ensuremath{\Delta}/e to 2\ensuremath{\Delta}/e-e/2C, while the quasiparticle current's onset changes from 2\ensuremath{\Delta}/e to 2\ensuremath{\Delta}/e+e/2C. \textcopyright{} 1996 The American Physical Society.
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