  Cavity cooling of an atom works best on a cyclic optical transition in the
strong coupling regime near resonance, where small cavity photon numbers
suffice for trapping and cooling. Due to the absence of closed transitions a
straightforward application to molecules fails: optical pumping can lead the
particle into uncoupled states. An alternative operation in the far
off-resonant regime generates only very slow cooling due to the reduced
field-molecule coupling. We predict to overcome this by using a strongly driven
ring-cavity operated in the sideband cooling regime. As in the optomechanical
setups one takes advantage of a collectively enhanced field-molecule coupling
strength using a large photon number. A linearized analytical treatment
confirmed by full numerical quantum simulations predicts fast cooling despite
the off-resonant small single molecule - single photon coupling. Even ground
state cooling can be obtained by tuning the cavity field close to the
Anti-stokes sideband for sufficiently high trapping frequency. Numerical
simulations show quantum jumps of the molecules between the lowest two trapping
levels, which can be be directly and continuously monitored via scattered light
intensity detection.
