[@2008_Nature_Thompson_Harris], in which a flexible dielectric membrane is placed inside a free-space optical cavity. The typical dimensions of free-space optical cavities lead to the requirement that the membrane have a lateral dimension $\sim$ 1 mm to avoid clipping losses at the beam waist. This leads to a fundamental drum-head mode with a resonance frequency $\sim 10^5$ Hz, requiring laser cooling to $\sim$ 1 K in order to reach the ground state. Despite this low temperature, this type of optomechanical system is appealing for a number of reasons. The [Si$_3$N$_4$]{}  membranes used here exhibit exceptionally high quality factors $Q$ (even when they are patterned into more complex shapes [@KimblePersonal]), low optical absorption [@Sankey2010Strong], and compatibility with monolithic, fiber-based optical cavities [@FlowersJacobs2012FiberCavityBased]. Furthermore, the membrane-in-the-middle geometry provides access to different types of optomechanical coupling that may serve as useful tools for addressing quantum vibrations [@2008_Nature_Thompson_Harris; @Sankey2010Strong; @2008_NJP_Jayich_Harris].

At a cryogenic base temperature of 400 mK, we observe a mechanical quality factor $Q > 4\times 10^6$ for the 261-kHz fundamental membrane mode, and a cavity resonance halfwidth of 60 kHz, meaning the system operates in the resolved sideband limit. We monitor the membrane’s thermal motion using a heterodyne optical circuit capable of simultaneously 