Quantum Shuttle Phenomena in a Nanoelectromechanical Single-Electron Transistor

An analytical analysis of quantum shuttle phenomena in a nanoelectromechanical single-electron transistor has been performed in the realistic case, when the electron tunnelling length is much greater than the amplitude of the zero point oscillations of the central island. It is shown that when the dissipation is below a certain threshold value, the vibrational ground state of the central island is unstable. The steady-state into which this instability develops is studied. It is found that if the electric field ${\cal E}$ between the leads is much greater than a characteristic value ${\cal E}_q$, the quasiclassical shuttle picture is recovered, while if ${\cal E}\ll{\cal E}_q$ a new quantum regime of shuttle vibrations occurs. We show that in the latter regime small quantum fluctuations result in large (i.e. finite in the limit $\hbar \to 0$) shuttle vibrations.