  Optical measurements of a nanoscale silicon optomechanical crystal cavity
with a mechanical resonance frequency of 3.6GHz are performed at sub-kelvin
temperatures. We infer optical-absorption-induced heating and damping of the
mechanical resonator from measurements of phonon occupancy and motional
sideband asymmetry. At the lowest probe power and lowest fridge temperature
(10mK), the localized mechanical resonance is found to couple at a rate of
400Hz (Q=9x10^6) to a thermal bath of temperature 270mK. These measurements
indicate that silicon optomechanical crystals cooled to millikelvin
temperatures should be suitable for a variety of experiments involving coherent
coupling between photons and phonons at the single quanta level.
