cooled to a low temperature and confined to a region in space with dimensions less than an optical wavelength. It thus constitutes a pure and well isolated quantum system that lends itself to applications where control and isolation from the environment are called for: Narrow optical transitions in trapped ions can be used to realize optical frequency standards [@Madej2001a; @Margolis2009a], and ion traps are also promising systems for the implementation of a scalable quantum computer [@Leibfried2003a; @Wineland2011a].

We study even isotopes of the alkaline-earth-metal ions that have no nuclear spin and thus no hyperfine structure, most notably $^{24}$Mg$^+$, $^{40}$Ca$^+$, $^{88}$Sr$^+$, and $^{138}$Ba$^+$, but also including less abundant isotopes of these elements. Numerical calculations and a comparison with experimental results are performed for $^{88}\mathrm{Sr}^+$, the lowest-lying energy levels of which are shown in Fig. \[fig:SrII\]. The $^2S_{1/2} - \,^2P_{1/2}$ transition is used to Doppler-cool the ion. The narrow $^2S_{1/2} - \,^2D_{5/2}$ electric quadrupole transition is used as the clock transition in optical clocks, for sub-Doppler cooling using resolved sideband cooling [@Wineland1975a], and as a qubit candidate for quantum computing [@Nagerl2000a]. The occurrence of a transition to the $^2D_{5/2}$ state is detected using Dehmelt’s electron shelving technique [@Dehmelt1982a; @Dehmelt1975a], i.e., as a 