  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 $eV/\hbar \Omega$, 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 $eV/\hbar \Omega\gg 1$. 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 $eV$,
$\hbar \Omega$, 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.
