Dynamics and Current Fluctuations in AC driven Charge Shuttle

The behavior of a charge shuttle under a pure AC field has been recently considered theoretically and experimentally. If the system presents an asymmetry in the tunneling amplitudes the device acts as a nano-electromechanical rectifier, transforming a pure AC voltage field into a direct curren. In this paper we first review the model and the appearance of the rectifying effect for bias voltages below the threshold of self-oscillation. We discuss in some details the dynamics of the central island that, like the current, presents strong dependence on the forcing AC field frequency. In presence of both a constant and a small oscillating bias voltage we analyze the transition from the static to self-oscillating solution. We then consider current fluctuations (full counting statistics) for periodic motion of the grain. We explicitly evaluate the current noise numerically and we find that it shows clear signatures of correlated transport at certain locking frequencies. In the adiabatic limit we obtain a simple expression for the full-counting statistics and calculate explicitly the first four moments.