Correlation between the magic wavelengths and the polarization direction of the linearly polarized laser in the Ca⁺ optical clock

The magic wavelengths for different Zeeman components are measured based on the ⁴⁰Ca⁺ optical clock. The dynamic dipole polarizability of a non-zero angular moment level has correlation with the polarization direction of the linearly polarized laser beam, and we show that the four hyperfine structure levels of 4s_{1/2, m = ± 1/2} and 3d_{5/2, m = ± 1/2} for ⁴⁰Ca⁺ have the same dynamic dipole polarizability at the magic wavelength and a certain polarization direction. In addition, the existence of a specific direction of polarization may provide a new idea for improving the precision of magic wavelength measurement in experiment.