  The ability to engineer both linear and non-linear coupling with a mechanical
resonator is an important goal for the preparation and investigation of
macroscopic mechanical quantum behavior. In this work, a measurement based
scheme is presented where linear or square mechanical displacement coupling can
be achieved using the optomechanical interaction linearly proportional to the
mechanical position. The resulting square displacement measurement strength is
compared to that attainable in the dispersive case using the direct interaction
to the mechanical displacement squared. An experimental protocol and parameter
set are discussed for the generation and observation of non-Gaussian states of
motion of the mechanical element.
