  The motion of two distant trapped particles or mechanical oscillators can be
strongly coupled by light modes in a high finesse optical resonator. In a two
mode ring cavity geometry, trapping, cooling and coupling is implemented by the
same modes. While the cosine mode provides for trapping, the sine mode
facilitates ground state cooling and mediates non-local interactions. For
classical point particles the centre-of-mass mode is strongly damped and the
individual momenta get anti-correlated. Surprisingly, quantum fluctuations
induce the opposite effect of positively-correlated particle motion, which
close to zero temperature generates entanglement. The non-classical
correlations and entanglement are dissipation-induced and particularly strong
after detection of a scattered photon in the sine mode. This allows for
heralded entanglement by post-selection. Entanglement is concurrent with
squeezing of the particle distance and relative momenta, while the
centre-of-mass observables acquire larger uncertainties.
