Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion

We perform a quantum theoretical calculation of the noise power spectrum for a phase measurement of the light output from a coherently driven optical cavity with a freely moving rear mirror. We examine how the noise resulting from the quantum back action appears among the various contributions from other noise sources. We do not assume an ideal (homodyne) phase measurement, but rather consider phase modulation detection, which we show has a different shot noise level. We also take into account the effects of thermal damping of the mirror, losses within the cavity, and classical laser noise. We relate our theoretical results to experimental parameters, so as to make direct comparisons with current experiments simple. We also show that in this situation, the standard Brownian motion master equation is inadequate for describing the thermal damping of the mirror, as it produces a spurious term in the steady-state phase fluctuation spectrum. The corrected Brownian motion master equation [L. Diosi, Europhys. Lett. {\bf 22}, 1 (1993)] rectifies this inadequacy.