Abstract
We study the effect of quantum charge fluctuations on the discrete spectrum of charge states of a small superconducting island (Cooper-pair box) connected to a large finite-size superconductor by a tunnel junction. In particular, we calculate the reduction of the even-odd energy difference $\ensuremath{\delta}E$ due to virtual tunneling of electrons across the junction. We show that the renormalization effects are important for understanding the quasiparticle ``poisoning'' effect because $\ensuremath{\delta}E$ determines the activation energy of a trapped quasiparticle in the Cooper-pair box. We find that renormalization of the activation energy depends on the dimensionless normal-state conductance of the junction ${g}_{T}$ and becomes strong at ${g}_{T}⪢1$.
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