Quantum fluctuations in a single electron box

Recent experiments have demonstrated that the numbern of additional electrons on a small metallic island is a staircase function of a continuous external chargen _x for temperaturesT small compared to the single electron charging energyU. We show that the finite conductanceg of the tunnel barrier connecting the island to the external gate gives rise to quantum fluctuations inn which lead to a smearing of the staircase even at zero temperature. In the experimentally relevant case of wide junctions and in the limit of small conductanceg1 the slope <n>/n _x at the turning point between two plateaus saturates at a finite value of order 1/g asT0 instead of diverging likeU/T as predicted with thermal fluctuations only. The experimentally observed broadening however is still much larger which is probably due to extrinsic effects.