Angle-dependent magic wavelengths for the 4s1/2→3d5/2,3/2 transitions of Ca+ ions

Abstract

The dynamic polarizabilities of the atomic states with angular momentum $j>1/2$ are sensitive to the angle between the quantization axis ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{e}}_{z}$ and the polarization vector $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\epsilon}}$ owing to the contribution of anisotropic tensor polarizabilities. The magic wavelength at which the differential Stark shift of an atomic transition nullifies depends on this angle. We identified the magic wavelengths for the $4{s}_{1/2}\ensuremath{\rightarrow}3{d}_{3/2,5/2}$ transitions of ${\mathrm{Ca}}^{+}$ ions at different angles between ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{e}}_{z}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\epsilon}}$ in the case of linearly polarized light. The magic wavelengths near 395.79 nm, which lie between the $4{s}_{1/2}\ensuremath{\rightarrow}4{p}_{1/2}$ and $4{s}_{1/2}\ensuremath{\rightarrow}4{p}_{3/2}$ transition wavelengths, remain insensitive to the angle, while the longest magic wavelength, which is around 1000 nm, for each of the magnetic sublevel transitions is very sensitive to the angle. We suggest that accurate measurements on the longest magic wavelengths for the $4{s}_{1/2}\ensuremath{\rightarrow}3{d}_{5/2}$ and $4{s}_{1/2}\ensuremath{\rightarrow}3{d}_{3/2}$ transitions can be used to determine the oscillator strengths for the $4{s}_{1/2}\ensuremath{\rightarrow}4{p}_{1/2,3/2}, 3{d}_{5/2}\ensuremath{\rightarrow}4{p}_{3/2}$, and $3{d}_{3/2}\ensuremath{\rightarrow}4{p}_{1/2,3/2}$ transitions, and the difference of the static polarizabilities of the $4{s}_{1/2}$ and $3{d}_{5/2}$ states, and the tensor polarizability of the $3{d}_{5/2}$ state.