Theoretical calculation of dynamic polarizability of 4s<sup>2 1</sup>S<sub>0</sub>—4s4p<sup> 3</sup>P<sub>0</sub> transition for Ga<sup>+</sup> ion

The transition of Ga<sup>+</sup> ions from 4s<sup>2</sup> <sup>1</sup>S<sub>0</sub> to 4s4p <sup>3</sup>P<sub>0</sub> has advantages such as a high quality factor and a small motional frequency shift, making it suitable as a reference for precision measurement experiments like optical clocks. Calculating the dynamic polarizability of 4s<sup>2</sup> <sup>1</sup>S<sub>0</sub>—4s4p <sup>3</sup>P<sub>0</sub> transition for Ga<sup>+</sup> ion is of great significance for exploring the potential applications of the Ga<sup>+</sup> ion in the field of quantum precision measurement and for testing atomic and molecular structure theories. In this paper, the dynamic polarizability of the Ga<sup>+</sup> ion 4s<sup>2</sup> <sup>1</sup>S<sub>0</sub>—4s4p <sup>3</sup>P<sub>0</sub> transition is theoretically calculated using the relativistic configuration interaction plus many-body perturbation (RCI+MBPT) method. The “tune-out” wavelengths for the 4s<sup>2</sup> <sup>1</sup>S<sub>0</sub> state and the 4s4p <sup>3</sup>P<sub>0</sub> state, as well as the “magic” wavelength of the 4s<sup>2</sup> <sup>1</sup>S<sub>0</sub>—4s4p <sup>3</sup>P<sub>0</sub> transition, are also computed. It is observed that the resonant lines situated near a certain “turn-out” and “magic” wavelength can make dominant contributions to the polarizability, while the remaining resonant lines generally contribute the least. These “tune-out” and “magic” wavelengths provide theoretical guidance for precise measurements, which is important for studying the atomic structure of Ga<sup>+</sup> ions. The accurate determination of the difference in static polarizability between the 4s<sup>2</sup> <sup>1</sup>S<sub>0</sub> and 4s4p <sup>3</sup>P<sub>0</sub> states is of significant importance. Additionally, based on the “polarizability scaling” method, this work also discusses how the theoretical calculation errors in static polarizability measurements vary with wavelength, which provides theoretical guidance for further determining the static polarizability of the 4s<sup>2</sup> <sup>1</sup>S<sub>0</sub> and 4s4p <sup>3</sup>P<sub>0</sub> states with high precision. This is crucial for minimizing the uncertainty of the blackbody radiation (BBR) frequency shift in Ga<sup>+</sup> optical clock and suppressing the systematic uncertainty.