Improved position measurement of nanoelectromechanical systems using cross correlations

We consider position measurements using the cross-correlated output of two tunnel-junction position detectors. Using a fully quantum treatment, we calculate the equation of motion for the density matrix of the coupled detector-detector-mechanical-oscillator system. After discussing the presence of a bound on the peak-to-background ratio in a position measurement using a single detector, we show how one can use detector cross correlations to overcome this bound. We analyze two different possible experimental realizations of the cross-correlation measurement and show that in both cases, the maximum cross-correlated output is obtained when using twin detectors and applying equal bias to each tunnel junction. Furthermore, we show how the double-detector setup can be exploited to drastically reduce the added displacement noise of the oscillator.