  We study the motion of two atoms trapped at distant positions in the field of
a driven standing wave high-Q optical resonator. Even without any direct
atom-atom interaction the atoms are coupled through their position dependent
influence on the intracavity field. For sufficiently good trapping and low
cavity losses the atomic motion becomes significantly correlated and the two
particles oscillate in their wells preferentially with a 90 degrees relative
phase shift. The onset of correlations seriously limits cavity cooling
efficiency, raising the achievable temperature to the Doppler limit. The
physical origin of the correlation can be traced back to a cavity mediated
cross-friction, i.e. a friction force on one particle depending on the velocity
of the second particle. Choosing appropriate operating conditions allows for
engineering these long range correlations. In addition this cross-friction
effect can provide a basis for sympathetic cooling of distant trapped clouds.
