Angle-dependent magic wavelengths for the $4s_{1/2}\to 3d_{5/2,3/2}$ transitions of ${\mathrm{Ca}}^{+}$ ions

The dynamic polarizabilities of the atomic states with angular momentum $j>1/2$ are sensitive to the angle between the quantization axis ${\widehat{e}}_z$ and the polarization vector $\widehat{\epsilon }$ 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 $4s_{1/2}\to 3d_{3/2,5/2}$ transitions of ${\mathrm{Ca}}^{+}$ ions at different angles between ${\widehat{e}}_z$ and $\widehat{\epsilon }$ in the case of linearly polarized light. The magic wavelengths near 395.79 nm, which lie between the $4s_{1/2}\to 4p_{1/2}$ and $4s_{1/2}\to 4p_{3/2}$ 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 $4s_{1/2}\to 3d_{5/2}$ and $4s_{1/2}\to 3d_{3/2}$ transitions can be used to determine the oscillator strengths for the $4s_{1/2}\to 4p_{1/2,3/2}, 3d_{5/2}\to 4p_{3/2}$, and $3d_{3/2}\to 4p_{1/2,3/2}$ transitions, and the difference of the static polarizabilities of the $4s_{1/2}$ and $3d_{5/2}$ states, and the tensor polarizability of the $3d_{5/2}$ state.

Figure 1

Representation of the electromagnetic plane-wave geometrical parameters. The surface represents the ellipse swept out by the electric field vector in one period. The unit vector ${\widehat{\varepsilon }}_{\mathrm{maj}}\left({\widehat{\varepsilon }}_{\min }\right)$ aligns with the semimajor (-minor) axis of the ellipse. ${\widehat{e}}_z$ is the quantization axis, which selects the direction of the magnetic field in the experiment. $\widehat{k}$ represents the direction of the wave vector. ${\theta }_k$ is the angle between ${\widehat{e}}_z$ and $\widehat{k}$. The parameters ${\widehat{\varepsilon }}_{\mathrm{maj}}, {\widehat{\varepsilon }}_{\min }$, and $\widehat{k}$ are mutually orthogonal. ${\theta }_{\mathrm{maj}}$ (${\theta }_{\min }$) is the angle between ${\widehat{\varepsilon }}_{\mathrm{maj}}$ (${\widehat{\varepsilon }}_{\min }$) and ${\widehat{e}}_z$. $\psi$ is directly related to the degree of circular polarization.

Figure 2

Dynamic polarizabilities (in a.u.) of the $4s$ and $3d_{5/2}(m=1/2,3/2,5/2)$ states in the wavelength range 300–1400 nm. The upper panel plots the dynamic polarizabilities in the case of ${\widehat{e}}_z\perp \widehat{\epsilon }$. The lower panel plots the dynamic polarizabilities in the case of ${\widehat{e}}_z\parallel \widehat{\epsilon }$. The approximate position of the $3d_{5/2}\to 4p_{3/2}$ resonance is indicated by the vertical dotted line.

Figure 3

The dependence of magic wavelengths, which lie between the $4s_{1/2}\to 4p_{1/2}$ transition wavelength (393.37 nm) and $4s_{1/2}\to 4p_{3/2}$ transition wavelength (396.85 nm), of each magnetic sublevel transition of $4s_{1/2}\to 3d_{5/2}$ upon ${\cos }^2{\theta }_p$ in the case of linearly polarized light.

Figure 4

Same as Fig. 3 but for magic wavelengths longer than the $3d_{5/2}\to 4p_{3/2}$ transition wavelength (854.21 nm).

Figure 5

Magic wavelengths of the $4s_{1/2}\to 3d_{3/2}$ transition of the ${\mathrm{Ca}}^{+}$ ions for linearly polarized light. (a) Magic wavelengths longer than the $3d_{3/2}\to 4p_{1/2}$ transition wavelength (866.21 nm). (b) Magic wavelengths which lie between the $3d_{3/2}\to 4p_{3/2}$ transition wavelength (849.80 nm) and the $3d_{3/2}\to 4p_{1/2}$ transition wavelength (866.21 nm). (c) Magic wavelengths which lie between the $4s_{1/2}\to 4p_{1/2}$ transition wavelength (393.37 nm) and the $4s_{1/2}\to 4p_{3/2}$ transition wavelength (396.85 nm).