Tune-out and magic wavelengths of ${\mathrm{Ba}}^{+}$ ions

The static and dynamic polarizabilities of the ground state and low-lying excited states of ${\mathrm{Ba}}^{+}$ ions are calculated by using a relativistic semiempirical method: the relativistic configuration interaction plus core polarization (RCICP) method. The tune-out wavelengths of the ground state and magic wavelengths of the $6s_{1/2}\to 6p_{1/2,3/2},5d_{3/2,5/2}$ transitions are determined. One visible tune-out wavelength, 480.658(18) nm, is found. High-precision measurements on this tune-out wavelength could be used to determine the ratio of the oscillator strengths of the $6s_{1/2}\to 6p_{1/2,3/2}$ transitions. In addition, we suggest that the measurements on the magic wavelengths corresponding to the $6s_{1/2}\to 5d_{3/2,5/2}$ transitions could be used to determine the oscillator strengths of the $5d_{5/2}\to 4f_{7/2}$ and $5d_{3/2}\to 4f_{5/2}$ transitions, while the measurements of the magic wavelengths near 416 nm for the $6s_{1/2}\to 6p_{3/2}$ transition could be used to determine the ratio of the oscillator strengths of the $6p_{3/2}\to 6d_{3/2,5/2}$ transitions.

Figure 1

Dynamic polarizabilities (a.u.) of the ground state of ${\mathrm{Ba}}^{+}$ ions. Three tune-out wavelengths are identified with arrows. The vertical dotted lines indicate the positions of the resonant $6s_{1/2}\to n{p}_{1/2,3/2}$ transitions.

Figure 2

Dynamic polarizabilities (a.u.) of the $6s_{1/2}$ and $5d_{5/2}$ states of ${\mathrm{Ba}}^{+}$ ions. The magic wavelengths are identified by arrows. The vertical dotted lines indicate the positions and wavelengths of the resonant transitions. The upper panel shows the scalar dynamic polarizabilities of the $6s_{1/2}$ and $5d_{5/2}$ states, while the lower panel shows the total dynamic polarizabilities of the $6s_{1/2}$ state and the magnetic sublevels $m=5/2$, 3/2, and 1/2 of the $5d_{5/2}$ state.

Figure 3

Same as Fig. 2 but for the $6s_{1/2}$ and $5d_{3/2}$ states of ${\mathrm{Ba}}^{+}$ ions.

Figure 4

Same as Fig. 2 but for the $6s_{1/2}$ and $6p_{3/2}$ states of ${\mathrm{Ba}}^{+}$ ions.

Figure 5

Dynamic polarizabilities (a.u.) for the $6s_{1/2}$ and $6p_{1/2}$ states of ${\mathrm{Ba}}^{+}$ ions. The magic wavelengths are identified by arrows. The vertical dotted lines identify the positions and wavelengths of the resonant transitions.