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Lightshift of a state due to the applied laser in an atomic system vanishesat the tune-out wavelengths ($\lambda_T$s). Similarly, differential light shiftof a transition vanishes at the magic wavelengths ($\lambda_{magic}$s). In manyof the earlier studies, values of the electric dipole (E1) matrix elements wereinferred precisely by combining measurements of $\lambda_{magic}$ with thecalculated their values. Similarly, the $\lambda_T$ values of an atomic statecan be used to infer the E1 matrix element as it involves dynamic electricdipole ($\alpha$) values of only one state whereas the $\lambda_{magic}$ valuesare dealt with $\alpha$ values of two states. However, both the $\lambda_T$ and$\lambda_{magic}$ values depend on angular momenta and their magneticcomponents ($M$) of states. Here, we report the $\lambda_T$ and$\lambda_{magic}$ values of many $S_{1/2}$ and $D_{3/2,5/2}$ states, andtransitions among these states of the Mg$^{+}$, Ca$^{+}$, Sr$^{+}$ and Ba$^{+}$ions that are independent of $M$- values. Measuring these wavelengths in aspecial set-up as discussed in the paper, it could be possible to infer a largenumber of E1 matrix elements of the above ions accurately.

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