  Frequency shifts of red- and blue-scattered (Stokes/anti-Stokes) side-bands
in quantum optomechanics are shown to be counter-intuitively inequal, resulting
in an unexpected symmetry breaking. This difference is referred to as Side-band
Inequivalenve (SI), which normally leans towards red, and being a nonlinear
effect it depends on optical power or intracavity photon number. Also there
exists a maximum attainable SI at an optimal operation point. The mathematical
method employed here is a combination of operator algebra equipped with
harmonic balance, which allows a clear understanding of the associated
nonlinear process. This reveals the existence of three distinct operation
regimes in terms of pump power, two of which have immeasurably small SI.
Compelling evidence from various experiments sharing similar interaction
Hamiltonians, including quantum optomechanics, ion/Paul traps, electrooptic
modulation, Brillouin scattering, and Raman scattering unambiguously confirm
existence of a previously unnoticed SI.
