Magnetic Sublevel Independent Magic and Tune-out Wavelengths of the Alkaline-earth Ions

Abstract

Lightshift of a state due to the applied laser in an atomic system vanishes at the tune-out wavelengths (${\lambda }_T$s). Similarly, differential light shift of a transition vanishes at the magic wavelengths (${\lambda }_{magic}$s). In many of the earlier studies, values of the electric dipole (E1) matrix elements were inferred precisely by combining measurements of ${\lambda }_{magic}$ with the calculated their values. Similarly, the ${\lambda }_T$ values of an atomic state can be used to infer the E1 matrix element as it involves dynamic electric dipole ($\alpha$) values of only one state whereas the ${\lambda }_{magic}$ values are dealt with $\alpha$ values of two states. However, both the ${\lambda }_T$ and ${\lambda }_{magic}$ values depend on angular momenta and their magnetic components ($M$) of states. Here, we report the ${\lambda }_T$ and ${\lambda }_{magic}$ values of many $S_{1/2}$ and $D_{3/2,5/2}$ states, and transitions among these states of the Mg${}^{+}$, Ca${}^{+}$, Sr${}^{+}$ and Ba${}^{+}$ ions that are independent of $M$- values. Measuring these wavelengths in a special set-up as discussed in the paper, it could be possible to infer a large number of E1 matrix elements of the above ions accurately.