  We provide a quantum analysis of a DC SQUID mechanical displacement detector
within the sub-critical Josephson current regime. A segment of the SQUID loop
forms the mechanical resonator and motion of the latter is transduced
inductively through changes in the flux threading the loop. Expressions are
derived for the detector signal response and noise, which are used to evaluate
the position and force detection sensitivity. We also investigate cooling of
the mechanical resonator due to back reaction noise from the detector.
