Momentum and position detection in nanoelectromechanical systems beyond the Born and Markov approximations

Abstract

We propose and analyze different schemes to probe the quantum nature of nanoelectromechanical systems (NEMS) by a tunnel junction detector. Using the Keldysh technique, we are able to investigate the dynamics of the combined system for an arbitrary ratio of $\mathit{eV}/\ensuremath{\hbar}\ensuremath{\Omega}$, where $V$ is the applied bias of the tunnel junction and $\ensuremath{\Omega}$ is the eigenfrequency of the oscillator. In this sense, we go beyond the Markov approximation of previous works where these parameters were restricted to the regime $\mathit{eV}/\ensuremath{\hbar}\ensuremath{\Omega}\ensuremath{\gg}1$. Furthermore, we also go beyond the Born approximation by expanding the finite frequency current noise of the tunnel junction up to fourth order in the tunneling amplitudes. Interestingly, we discover different ways to probe both position and momentum properties of NEMS. On the one hand, for a nonstationary oscillator, we find a complex finite frequency noise of the tunnel junction, concluding that a simple tunnel junction detector can probe both position- and momentum-based observables of the nonstationary oscillator. On the other hand, for a stationary oscillator, an Aharonov-Bohm-loop tunnel junction detector is needed. It still allows us to extract position and momentum information of the oscillator. For this type of detector, we analyze what happens if the energy scales $\mathit{eV}$, $\ensuremath{\hbar}\ensuremath{\Omega}$, and ${k}_{B}{T}_{\mathrm{env}}$ take arbitrary values with respect to each other where ${T}_{\mathrm{env}}$ is the temperature of an external heat bath.