  Methods for controlling the motion of single particles, optically levitated
in vacuum, have developed rapidly in recent years. The technique of cold
damping makes use of feedback-controlled, electrostatic forces to increase
dissipation without introducing additional thermal fluctuations. This process
has been instrumental in the ground-state cooling of individual electrically
charged nanoparticles. Here we show that the same method can be applied to a
pair of nanoparticles, coupled by optical binding forces. These optical binding
forces are about three orders of magnitude stronger than typical Coulombic
inter-particle force and result in a coupled motion of both nanoparticles
characterized by a pair of normal modes. We demonstrate cold damping of these
normal modes, either independently or simultaneously, to sub-Kelvin
temperatures at pressures of 5x10^{-3} mbar. Experimental observations are
captured by a theoretical model which we use to survey the parameter space more
widely and to quantify the limits imposed by measurement noise and time delays.
Our work paves the way for the study of quantum interactions between meso-scale
particles and the exploration of multiparticle entanglement in levitated
optomechanical systems.
