  Cavity-mediated cooling of the center--of--mass motion of a transversally,
coherently pumped atom along the axis of a high--Q cavity is studied. The
internal dynamics of the atomic dipole strongly coupled to the cavity field is
treated by a non-perturbative quantum mechanical model, while the effect of the
cavity on the external motion is described classically in terms of the
analytically obtained linear friction and diffusion coefficients. Efficient
cavity-induced damping is found which leads to steady-state temperatures
well-below the Doppler limit. We reveal a mathematical symmetry between the
results here and for a similar system where, instead of the atom, the cavity
field is pumped. The cooling process is strongly enhanced in a degenerate
multimode cavity. Both the temperature and the number of scattered photons
during the characteristic cooling time exhibits a significant reduction with
increasing number of modes involved in the dynamics. The residual number of
spontaneous emissions in a cooling time for large mode degeneracy can reach and
even drop below the limit of a single photon.
