Theory of magic optical traps for Zeeman-insensitive clock transitions in alkali-metal atoms

Precision measurements and quantum-information processing with cold atoms may benefit from trapping atoms with specially engineered, “magic” optical fields. At the magic trapping conditions, the relevant atomic properties remain immune to strong perturbations by the trapping fields. Here we develop a theoretical analysis of magic trapping for especially valuable Zeeman-insensitive clock transitions in alkali-metal atoms. The involved mechanism relies on applying a magic bias $B$ field along a circularly polarized trapping laser field. We map out these $B$ fields as a function of trapping laser wavelength for all commonly used alkalis. We also highlight a common error in evaluating Stark shifts of hyperfine manifolds.

Figure 1

Importance of full-scale calculations. Dependence of magic $B$ field (in G) on laser frequency (in atomic units) for ${}^{87}\mathrm{Rb}$. Full-scale calculations (solid blue line) are compared with approximate “experimentalist” computations which neglect the HFI contribution to atomic wave functions (dashed red line).

Figure 2

Dependence of magic $B$ field (in G) on laser frequency (in atomic units) for ${}^{23}\mathrm{Na}$ (dashed green line), ${}^{87}\mathrm{Rb}$ (solid blue line), and ${}^{133}\mathrm{Cs}$ (dot-dashed red line). Magic $B$ fields for other isotopes of the same element may be obtained using the scaling law, Eq. (9).