  We study the quantum measurement of a cantilever using a
parametrically-coupled electromagnetic cavity which is driven at the two
sidebands corresponding to the mechanical motion. This scheme, originally due
to Braginsky et al. [V. Braginsky, Y. I. Vorontsov, and K. P. Thorne, Science
209, 547 (1980)], allows a back-action free measurement of one quadrature of
the cantilever's motion, and hence the possibility of generating a squeezed
state. We present a complete quantum theory of this system, and derive simple
conditions on when the quantum limit on the added noise can be surpassed. We
also study the conditional dynamics of the measurement, and discuss how such a
scheme (when coupled with feedback) can be used to generate and detect squeezed
states of the oscillator. Our results are relevant to experiments in
optomechanics, and to experiments in quantum electromechanics employing
stripline resonators coupled to mechanical resonators.
