Momentum and position detection in nanoelectromechanical systems beyond 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 $eV/\hslash \Omega$, where V is the applied bias of the tunnel junction and $\Omega$ 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 $eV/\hslash \Omega \gg 1$. Furthermore, we also go beyond the Born approximation because we calculate 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 non-stationary oscillator, we find a complex finite frequency noise of the tunnel junction. By analyzing the real and the imaginary part of this noise separately, we conclude that a simple tunnel junction detector can probe both position- and momentum-based observables of the non-stationary oscillator. On the other hand, for a stationary oscillator, a more complicated setup based on 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 for the first time what happens if the energy scales $eV$, $\hslash \Omega$, and $k_{B}T$ take arbitrary values with respect to each other where T is the temperature of an external heat bath. Under these circumstances, we show that it is possible to uniquely identify the quantum state of the oscillator by a finite frequency noise measurement.

15 pages, 10 figures

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