Momentum statistics of tunneling electrons in nanoelectromechanical systems

When a mesoscopic conductor is used to measure the position of a nanomechanical oscillator, electrons in the conductor exert a fluctuating back-action force on the oscillator. What is the statistical distribution of the momentum transferred to the oscillator by this force? Motivated by recent experiments that studied a mechanical oscillator coupled to a single tunnel junction \footnote{N. E. Flowers-Jacobs {\it et al.}, Phys. Rev. Lett. {\bf 98}, 096804 (2007).} or a quantum point contact \footnote{M. Poggio {\it et al.}, Nat. Phys. {\bf 4}, 635 (2008).}, we investigate theoretically the statistics of back-action force in these systems as well as correlations between the force and the current. Our approach is based on a scattering matrix that depends parametrically on the oscillator position, allowing us to go beyond weak tunneling and study conductors with arbitrary transmission. We identify two mechanisms of momentum transfer: one involves forces exerted in the scattering region and dominates in the limit of weak tunneling; the other is associated with transferred electron momentum and dominates in the limit of perfect transmission. We also discuss the effects of a spatially asymmetric conductor on the force noise and on the quantum limit of position detection.