  The radiation pressure of light can act to damp and cool the vibrational
motion of a mechanical resonator. In understanding the quantum limits of this
cooling, one must consider the effect of shot noise fluctuations on the final
thermal occupation. In optomechanical sideband cooling in a cavity, the finite
Stokes Raman scattering defined by the cavity linewidth combined with shot
noise fluctuations dictates a quantum backaction limit, analogous to the
Doppler limit of atomic laser cooling. In our work we sideband cool to the
quantum backaction limit by using a micromechanical membrane precooled in a
dilution refrigerator. Monitoring the optical sidebands allows us to directly
observe the mechanical object come to thermal equilibrium with the optical
bath.
