Electronic transport in nanoelectromechanical systems : noise, back-action, and quantum measurement

The important progress made in nanolitography processes in the last decades has had
a profound impact in our daily lives, by making possible the miniaturization of consumer
electronics. Unbeknownst to most consumers, it is nowadays possible to fabricate
free-standing nanoscale devices, that will naturally vibrate under thermal or
external excitation. Over the last decade, a new subfield of physics devoted to studying
these objects emerged: nanomechanics.
In this thesis, we study electronic transport in such nanostructures where mechanical
degrees of freedom play an important role. More precisely, we calculate the full
transport properties (e.g. average current, frequency-dependent current noise) of different
mesoscopic detectors in the presence of coupling to a nanomechanical oscillator.
The objective of our study is twofold. First, there is a strong interest in understanding
the effect that the coupling to electronic degrees of freedom has on the state
of the mechanical system. We will show that under many conditions the interaction
with the detector can be understood in terms of an effective thermal bath, but also
discuss the limitations of this effective environment model. A second main aspect of
the work presented here is the calculation of the signature of the mechanical object
in the transport properties of the detector. As one of the primary goal in the field
of nanoelectromechanical systems is to use the output of such electrical detectors to
achieve position measurements at the quantum limit, this question obviously is of
great relevance to the field.
This thesis is organized in 3 main parts, each associated with a different electronic
detector. After a short introduction to nanoelectromechanical systems, we focus in
Part II on a system composed a single-electron transistor coupled capacitively to a
classical mechanical oscillator. We present a complete study of the transport properties
of the coupled system, going beyond the usual weak-coupling approximation.
In Part III, we discuss the properties of a system where a tunnel junction is coupled
to the mechanical object. Looking at this system from the point of view of quantum
measurement, we analyze the transport properties of a system composed of two independent
tunnel junctions coupled to the same oscillator and demonstrate how, by
using the cross correlated output of the two detectors, one can improve the sensitivity
of position measurements beyond the usual quantum limit. In this part, we also
demonstrate that the current noise of a system composed of two tunnel junctions (one
with fixed transmission amplitude, the other with position-dependent transmission
amplitude) can contain information about the momentum of the mechanical oscillator.
Lastly, in Part IV we study a system composed of a mechanical oscillator coupled
to a superconducting single-electron transistor. The coupled dynamics of the oscillator
and mesoscopic detector are in this case very complex, and we demonstrate how
a numerical approach based on a solution of the Liouville equation can be used to
validate results obtained from approximate analytical approaches. We also demonstrate,
by looking at the frequency-dependence of the charge fluctuations on the superconducting
single-electron transistor, limitations to the model where the effect of
the detector back-action on the oscillator is modeled as an effective environment.