  Levitated particles are an ideal tool for measuring weak forces and
investigating quantum mechanics in macroscopic objects. Arrays of two or more
of these particles have been suggested for improving force sensitivity and
entangling macropscopic objects. In this article, two charged, silica
nanoparticles, that are coupled through their mutual Coulomb repulsion, are
trapped in a Paul trap, and the individual masses and charges of both particles
are characterised. We demonstrate sympathetic cooling of one nanoparticle
coupled via the Coulomb interaction to the second nanoparticle to which
feedback cooling is directly applied. We also implement sympathetic squeezing
through a similar process showing non-thermal motional states can be
transferred by the Coulomb interaction. This work establishes protocols to cool
and manipulate arrays of nanoparticles for sensing and minimising the effect of
optical heating in future experiments.
