  This article presents a full operator analytical method for studying the
quadratic nonlinear interactions in quantum optomechanics. The method is based
on the application of higher-order operators, using a six-dimensional basis of
second order operators which constitute an exactly closed commutators. We
consider both types of standard position-field and the recently predicted
non-standard momentum-field quadratic interactions, which is significant when
the ratio of mechanical frequency to optical frequency is not negligible. This
unexplored regime of large mechanical frequency can be investigated in few
platforms including the superconducting electromechanics and simulating quantum
cavity electrodynamic circuits. It has been shown that the existence of
non-standard quadratic interaction could be observable under appropriate
conditions.
