  While cavity cooling of a single trapped emitter was demonstrated, cooling of
many particles in an array of harmonic traps needs investigation and poses a
question of scalability. This work investigates the cooling of a one
dimensional atomic array to the ground state of motion via the interaction with
the single mode field of a high-finesse cavity. The key factor ensuring the
cooling is found to be the mechanical inhomogeneity of the traps. Furthermore
it is shown that the pumped cavity mode does not only mediate the cooling but
also provides the necessary inhomogeneity if its periodicity differs from the
one of the array. This configuration results in the ground state cooling of
several tens of atoms within a few milliseconds, a timescale compatible with
current experimental conditions. Moreover, the cooling rate scaling with the
atom number reveals a drastic change of the dynamics with the size of the
array: atoms are either cooled independently, or via collective modes. In the
latter case the cavity mediated atom interaction destructively slows down the
cooling as well as increases the mean occupation number, quadratically with the
atom number. Finally, an order of magnitude speed up of the cooling is
predicted as an outcome the optimization scheme based on the adjustment of the
array versus the cavity mode periodicity.
