Dissipative quantum phase transition in a single electron transistor

We study the transport properties of a single electron transistor (SET) with highly resistive gate electrodes, and show that the SET displays a quantum phase transition analogous to the famous dissipative phase transition studied by Leggett. At temperature $T=0$, the charge on the central island of a conventional SET changes smoothly as a function of gate voltage, dueto quantum fluctuations. However, sufficiently-strong dissipation, $R_g>R_C$, can freeze out charge fluctuations on the island even at the degeneracy point, causing the charge on the island to change in sharp steps as a function of gate voltage. For $R_g