Cooling the Collective Motion of Trapped Ions to Initialize a Quantum Register

We report preparation in the ground state of collective modes of motion of two trapped ${}^9{\mathrm{Be}}^{+}$ ions. This is a crucial step towards realizing quantum logic gates which can entangle the ions' internal electronic states. We find that heating of the modes of relative ion motion is substantially suppressed relative to that of the center-of-mass modes, suggesting the importance of these modes in future experiments.