Sympathetic cooling in a large ion crystal

We analyze the dynamics and steady state of a linear ion array when some of the ions are continuously laser cooled. We calculate the ions' local temperature measured by its position fluctuation under various trapping and cooling configurations, taking into account background heating due to the noisy environment. For a large system, we demonstrate that by arranging the cooling ions evenly in the array, one can suppress the overall heating considerably. We also investigate the effect of different cooling rates and find that the optimal cooling efficiency is achieved by an intermediate cooling rate. We discuss the relaxation time for the ions to approach the steady state, and show that with periodic arrangement of the cooling ions, the cooling efficiency does not scale down with the system size.