  This chapter summarizes the recent progress in the theory and analytical
tools of quadratic optomechanical interactions, as one of the prominent domains
of contemporary nonlinear quantum optics. Emphasis has been put here first to
show what types of nonlinear interactions do exist, and what physical
interpretations follow each. The standard quadratic interactions between light
and mechanical motion is expressed as the product of cavity light intensity and
squared mirror position. However, there exists a non-standard quadratic
optomechanical interaction as well, which assumes a mathematically different
form and appears as the squared product of field and mirror momenta. This
non-standard type of quadratic interaction originates from two corrections: the
momentum exchange and conservation among mirror and field, as well as
relativistic corrections due to different mechanisms. Both these types of
non-standard interactions become relevant when the ratio of mechanical to
optical frequency is no longer negligible. Next, we turn to the solution
technique of such interactions, and introduce a formal higher-order operator
method to tackle the nonlinear evolution of quantum systems. This enables one
to accurately study any type of quantum nonlinear interaction using the
analysis tools of linear algebra. In order to employ the analytical power of
higher-order operator method, one first needs to identify a closed Lie algebra,
which should satisfy closedness property under commutation either exactly or
approximately, and is referred to as the basis. Having the basis of
higher-order operators known, one may proceed to construct the corresponding
Langevin equations, which can be now conveniently analyzed using the existing
mathematical toolbox of linear algebra to yield the spectral densities,
moments, and expectation values.
