  Accessing distinctly quantum aspects of the interaction between light and the
position of a mechanical object has been an outstanding challenge to
cavity-optomechanical systems. Only cold-atom implementations of cavity
optomechanics have indicated effects of the quantum fluctuations in the optical
radiation pressure force. Here we use such a system, in which quantum
photon-number fluctuations significantly drive the center of mass of an atomic
ensemble inside a Fabry-Perot cavity. We show that the optomechanical response
both amplifies and ponderomotively squeezes the quantum light field. We also
demonstrate that classical optical fluctuations can be attenuated by 26 dB or
amplified by 20 dB with a weak input pump power of < 40 pW, and characterize
the optomechanical amplifier's frequency-dependent gain and phase response in
both the amplitude and phase-modulation quadratures.
