  Detection of nano-mechanical displacement by transport techniques has reached
high level of sensitivity and versatility. In order to detect the amplitude of
oscillation of nano-mechanical oscillator a widely used technique consists to
couple this motion capacitively to a single-electron transistor and to detect
the high-frequency modulation of the current through the non-linear mixing with
an electric signal at a slighltly detuned frequency. The method known as
current-mixing technique is employed in particular for the detection of
suspended carbon nanotubes. In this paper we study theoretically the limiting
conditions on the sensitivity of this method. The sensitivity is increased by
increasing the response function to the signal, but also by reducing the noise.
For these reasons we study systematically the response function, the effect of
current-and displacement-fluctuations, and finally the case where the
tunnelling rate of the electrons are of the same order or larger of the
resonating frequency. We find thus upper bounds to the sensitivity of the
detection technique.
