We have cooled a two-ion-crystal to the ground state of its collective modes
of motion. Laser cooling, more specific resolved sideband cooling is performed
sympathetically by illuminating only one of the two 40Ca+ ions in the
crystal. The heating rates of the motional modes of the crystal in our linear
trap have been measured, and we found them considerably smaller than those
previously reported by Q. Turchette {\em et. al.} Phys. Rev. A 61, 063418
(2000) in the case of trapped 9Be+ ions. After the ground state is
prepared, coherent quantum state manipulation of the atomic population can be
performed. Within the coherence time, up to 12 Rabi oscillations are observed,
showing that many coherent manipulations can be achieved. Coherent excitation
of each ion individually and ground state cooling are important tools for the
realization of quantum information processing in ion traps