  Precision measurement of non-linear observables is an important goal in all
facets of quantum optics. This allows measurement-based non-classical state
preparation, which has been applied to great success in various physical
systems, and provides a route for quantum information processing with otherwise
linear interactions. In cavity optomechanics much progress has been made using
linear interactions and measurement, but observation of non-linear mechanical
degrees-of-freedom remains outstanding. Here we report the observation of
displacement-squared thermal motion of a micro-mechanical resonator by
exploiting the intrinsic non-linearity of the radiation pressure interaction.
Using this measurement we generate bimodal mechanical states of motion with
separations and feature sizes well below 100~pm. Future improvements to this
approach will allow the preparation of quantum superposition states, which can
be used to experimentally explore collapse models of the wavefunction and the
potential for mechanical-resonator-based quantum information and metrology
applications.
