Coulomb blockade for an oscillating tunnel junction

We consider a tunnel junction formed between a fixed electrode and an oscillating one. Accumulation of the charge on the junction capacitor induces a force on the nanomechanical oscillator. The junction is voltage biased and connected in series with an impedance Z(omega). We discuss how the picture of Coulomb blockade is modified by the presence of the oscillator. Quantum fluctuations of the mechanical oscillator modify the I-V characteristics particularly in the strong Coulomb blockade limit. We show that the oscillator can be taken into account by a simple modification of the effective impedance of the circuit. We discuss in some details the case of a single inductance Z(omega)=iL omega and of a constant resistance Z(omega)=R. With little modifications the theory applies also to incoherent transport in Josephson junctions in the tunneling limit.