Jannes B. Wübbena,1 Sana Amairi,1 Olaf Mandel,1 and Piet O. Schmidt1, 2
1 QUEST Institute for Experimental Quantum Metrology,
Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany
2 Institut für  Quantenoptik, Leibniz Universität Hannover, 30167 Hannover, Germany

Sympathetic cooling of trapped ions has become an indispensable tool for quantum information processing and precision spectroscopy. In the simplest situation a single Doppler-cooled ion sympathetically cools another ion which typically has a different mass. Trap-induced heating of the ion crystal prevents achieving the Doppler cooling limit of modes for which the cooling ion has a small oscillation amplitude. In this publication we analytically investigate the effect of the mass ratio of such an ion crystal on the achievable temperature limit in the presence of external heating. We apply our findings on the cooling of a single Al⁺ ion with five different cooling partners (Be⁺, Mg⁺, Ca⁺, Sr⁺ and Yb⁺). Figure 1 shows the second order Doppler shift of the Al⁺ clock transition at the Doppler cooling limit for varying heating rates (expressed as electric field spectral densities at the trap frequencies in multiples of S_E0=1.26x10^-9 V² / m² Hz). Optimal cooling intensities and trap frequencies were chosen for each point. Since the x-axis interval is chosen such that the heating rates of most macroscopic ion traps lie well within its boundaries, this shows that Ca⁺, Be⁺ and Mg⁺ will perform comparably as Al⁺ cooling ion species whereas the heavier ions will significantly increase the Doppler cooling limit of the Al⁺ ion.

Reference: Phys. Rev. A 85, 043412 (2012)