  We report a spin-orbit coupling induced back-action cooling in an
optomechanical system, composed of a spin-orbit coupled Bose-Einstein
condensate trapped in an optical cavity with one movable end mirror, by
suppressing heating effects of quantum noises. The collective density
excitations of the spin-orbit coupling mediated hyperfine states - serving as
atomic oscillators equally coupled to the cavity field - trigger strongly
driven atomic back-action. We find that the back-action not only revamps
low-temperature dynamics of its own but also provides an opportunity to cool
the mechanical mirror to its quantum mechanical ground state. Further, we
demonstrate that the strength of spin-orbit coupling also superintends dynamic
structure factor and squeezes nonlinear quantum noises, like thermo-mechanical
and photon shot noise, which enhances optomechanical features of hybrid cavity
beyond the previous investigations. Our findings are testable in a realistic
setup and enhance the functionality of cavity-optomechanics with spin-orbit
coupled hyperfine states in the field of quantum optics and quantum
computation.
