  We investigate the intrinsic noise of a micro-mechanical displacement
detector based on the radio-frequency single-electron transistor (rf-SET).
Using the noise analysis of a SET by Korotkov as our starting point, we
determine the spectral density of the displacement noise due to the tunneling
current shot noise. The resulting mechanical displacement noise decreases in
inverse proportion to the increasing gate voltage. In contrast, the
displacement noise due to the fluctuating SET island charge increases
approximately linearly with increasing gate voltage. Taking into account both
of these noise sources results in an optimum gate voltage value for the lowest
displacement noise and hence best sensitivity. We show that a displacement
sensitivity of about 10^-4 Angstroms and a force sensitivity of about 10^-16 N
are predicted for a micron-sized cantilever with a realizable resonant
frequency 100 MHz and quality factor Q~10^4. Such sensitivities would allow the
detection of quantum squeezing in the mechanical motion of the micro-mechanical
cantilever and the detection of single-spin magnetic resonance in magnetic
resonance force microscopy (MRFM).
