  We demonstrate that a two-dimensional (2D) atomic array can be used as a
novel platform for quantum optomechanics. Such arrays feature both
nearly-perfect reflectivity and ultra-light mass, leading to
significantly-enhanced optomechanical phenomena. Considering the collective
atom-array motion under continuous laser illumination, we study the nonlinear
optical response of the array. We find that the spectrum of light scattered by
the array develops multiple sidebands, corresponding to collective mechanical
resonances, and exhibits nearly perfect quantum-noise squeezing. Possible
extensions and applications for quantum nonlinear optomechanics are discussed.
