We develop quantum models for the combined external and internal motion of atoms in a strongly coupled driven cavity mode including the transverse degrees of freedom. Using a simplified Gaussian mode function we determine the parameter regimes and prospects of 3D cooling and confinement of one or two atoms in the cavity field. Analysing the field dynamics for slow atoms traversing the cavity, we show that the spectrum of the transmitted and spontaneously scattered light contains ample information on the motional dynamics of the atom and can be nicely used to investigate the cooling properties of the system. Including several atoms in the dynamics we show how motional correlations build up by the common interaction with the cavity field. This can be looked upon as collisions at far distance and can be monitored via the transmitted field dynamics.
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