Theoretical calculation of “tune-out” wavelengths for clock states of Al<sup>+</sup>

Abstract

In quantum optical experiments, the polarizabilities of atomic systems play a very important role, which can be used to describe the interactions of atomic systems with external electromagnetic fields. When subjected to a specific electric field such as a laser field with a particular frequency, the frequency-dependent electric-dipole (E1) dynamic polarizability of an atomic state can reach zero. The wavelength corresponding to such a frequency is referred to as the “turn-out” wavelength. In this work, the “turn-out” wavelengths for the 3s<sup>2</sup> <sup>1</sup>S<sub>0</sub> and 3s3p <sup>3</sup>P<sub>0</sub> clock states of Al<sup>+</sup> are calculated by using the configuration interaction plus many-body perturbation theory (CI+MBPT) method. The values of energy and E1 reduced matrix elements of low-lying states of Al<sup>+</sup> are calculated. By combining these E1 reduced matrix elements with the experimental energy values, the E1 dynamic polarizabilities of the 3s<sup>2</sup> <sup>1</sup>S<sub>0</sub> and 3s3p <sup>3</sup>P<sub>0</sub> clock states are determined in the angular frequency range of (0, 0.42 a.u.). The “turn-out” wavelengths are found at the zero-crossing points of the frequency-dependent dynamic polarizability curves for both the 3s<sup>2</sup> <sup>1</sup>S<sub>0</sub> and 3s3p <sup>3</sup>P<sub>0</sub> states. For the ground state 3s<sup>2</sup> <sup>1</sup>S<sub>0</sub>, a single “turn-out” wavelength at 266.994(1) nm is observed. On the other hand, the excited state 3s3p <sup>3</sup>P<sub>0</sub> exhibits four distinct “turn-out” wavelengths, namely 184.56(1) nm, 174.433(1) nm, 121.52(2) nm, and 119.71(2) nm. The contributions of individual resonant transitions to the dynamic polarizabilities at the “turn-out” wavelengths are examined. It is observed that the resonant lines situated near a certain “turn-out” wavelength can provide dominant contributions to the polarizability, while the remaining resonant lines generally contribute minimally. When analyzing these data, we recommend accurately measuring these “turn-out” wavelengths to accurately determine the oscillator strengths or reduced matrix elements of the relevant transitions. This is crucial for minimizing the uncertainty of the blackbody radiation (BBR) frequency shift in Al<sup>+</sup> optical clock and suppressing the systematic uncertainty. Meanwhile, precisely measuring these “turn-out” wavelengths is also helpful for further exploring the atomic structure of Al<sup>+</sup>.