  Optomechanical systems explore and exploit the coupling between light and the
mechanical motion of matter. A nonlinear coupling offers access to rich new
physics, in both the quantum and classical regimes. We investigate a dynamic,
as opposed to the usually studied static, nonlinear optomechanical system,
comprising of a nanosphere levitated and cooled in a hybrid electro-optical
trap. An optical cavity offers readout of both linear-in-position and
quadratic-in-position (nonlinear) light-matter coupling, whilst simultaneously
cooling the nanosphere to millikelvin temperatures for indefinite periods of
time in high vacuum. We observe cooling of the linear and non-linear motion,
leading to a $10^5$ fold reduction in phonon number $n_p$, attaining final
occupancies of $n_p = 100-1000$. This work puts cavity cooling of a levitated
object to the quantum ground-state firmly within reach.
