  Cooling the center-of-mass motion of levitated nanoparticles provides a route
to quantum experiments at mesoscopic scales. Here we demonstrate
three-dimensional sympathetic cooling and detection of the center-of-mass
motion of a levitated silica nanoparticle. The nanoparticle is
electrostatically coupled to a feedback-cooled particle while both particles
are trapped in the same Paul trap. We identify two regimes, based on the
strength of the cooling: in the first regime, the sympathetically cooled
particle thermalizes with the directly cooled one, while in the second regime,
the sympathetically cooled particle reaches a minimum temperature. This result
provides a route to efficiently cool and detect particles that cannot be
illuminated with strong laser light, such as absorptive particles, and paves
the way for controlling the motion of arrays of several trapped nanoparticles.
