  We report on cooling the center-of-mass motion of a nanoparticle due to a
purely quadratic coupling between its motion and the optical field of a high
finesse cavity. The resulting interaction gives rise to a Van der Pol nonlinear
damping, which is analogous to conventional parametric feedback where the
cavity provides passive feedback without measurement. We show experimentally
that like feedback cooling the resulting energy distribution is strongly
nonthermal and can be controlled by the nonlinear damping of the cavity. As
quadratic coupling has a prominent role in proposed protocols to generate
deeply nonclassical states, our work represents a first step for producing such
states in a levitated system.
