  We demonstrate application of the method of higher-order operators to
nonlinear standard optomechanics. It is shown that a symmetry breaking in
frequency shifts exists, corresponding to inequivalency of red and blue
side-bands. This arises from nonlinear higher-order processes leading to
inequal detunings. Similarly, a higher-order resonance shift exists appearing
as changes in both of the optical and mechanical resonances. We provide the
first known method to explicitly estimate the population of coherent phonons.
We also calculate corrections to spring effect due to higher-order interactions
and coherent phonons, and show that these corrections can be quite significant
in measurement of single-photon optomechanical interaction rate. It is shown
that there exists non-unique and various choices for the higher-order operators
to solve the optomechanical interaction with different multiplicative noise
terms, among which a minimal basis offers exactly linear Langevin equations,
while decoupling one Langevin equation and thus leaving the whole standard
optomechanical problem exactly solvable by explicit expressions. We finally
present a detailed treatment of multiplicative noise as well as nonlinear
dynamic stability phases by the method of higher-order operators. Similar
approach can be used outside the domain of standard optomechanics to quadratic
and all other types of nonlinear interactions in quantum physics.
