  We demonstrate a cryogenic optomechanical system comprising a flexible Si3N4
membrane placed at the center of a free-space optical cavity in a 400 mK
cryogenic environment. We observe a mechanical quality factor Q > 4 x 10^6 for
the 261-kHz fundamental drum-head mode of the membrane, and a cavity resonance
halfwidth of 60 kHz. The optomechanical system therefore operates in the
resolved sideband limit. We monitor the membrane's thermal motion using a
heterodyne optical circuit capable of simultaneously measuring both of the
mechanical sidebands, and find that the observed optical spring and damping
quantitatively agree with theory. The mechanical sidebands exhibit a Fano
lineshape, and to explain this we develop a theory describing heterodyne
measurements in the presence of correlated classical laser noise. Finally, we
discuss the use of a passive filter cavity to remove classical laser noise, and
consider the future requirements for laser cooling this relatively large and
low-frequency mechanical element to very near its quantum mechanical ground
state.
