Magic wavelengths and triple magic trapping conditions for $5s^2~^1\!S_0$ and $5s5p~^3\!P_{0,2}$ states of Sr atoms

TL;DR

This work computes static and dynamic polarizabilities for Sr states and identifies magic wavelengths for transitions among the ground and metastable clock states using a relativistic configuration interaction plus MBPT approach. It analyzes how these magic wavelengths shift with laser polarization and geometry and derives conditions for triple magic trapping at 813.4 nm under linear, circular, and elliptical polarization, including specific angle and polarization settings. The results are in good agreement with existing theory and experimental measurements, and they offer practical guidance for implementing robust Sr-based optical clocks and all-optical qutrits. Overall, the study demonstrates how precise control of light-atom interactions can suppress differential Stark shifts across multiple Sr clock-like states, enabling improved metrology and quantum information processing.

Abstract

The static and dynamic electric dipole polarizabilities of the $5s^2~^1\!S_0$ and $5s5p~^3\!P_{0,2}$ states of Sr atoms are calculated using the relativistic configuration interaction plus the many-body perturbation theory (RCI+MBPT) method. Magic wavelengths are determined for the transitions $5s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{0}$, $5s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{2}$, and $5s5p~^3\!P_0\rightarrow 5s5p~^3\!P_{2}$. A comprehensive study is conducted on the dependence of magic wavelengths on the angle between the laser polarization and the magnetic field. Furthermore, the conditions for realizing triple magic trapping at 813.4~nm for the $5s^2~^1\!S_0$, $5s5p~^3\!P_{0}$ and $5s5p~^3\!P_{2}$ states are investigated. In the case of linearly polarized light, when the angle ($θ_p$) between the laser polarization direction and the magnetic field is $79.1(0.7)^\circ$, triple magic trapping for the $5s^2~^1\!S_0$, $5s5p~^3\!P_{0}$, and $5s5p~^3\!P_{2}~M=0$ states can be achieved. This result agrees well with the recent experimental measurement (78.49(3)$^\circ$)[Phys. Rev. Lett. 135, 143401 (2025)]. Meanwhile, triple magic trapping involving the $5s5p~^3\!P_{2}~M=2$ state can be achieved when $θ_p= 37.4(0.3)^\circ$. The conditions for achieving triple magic trapping with circularly and arbitrarily elliptically polarized light are also presented.

Figures (7)

• Figure 1: The schematic of the geometrical parameters of the electromagnetic plane wave and the 1D optical lattice trap. The elliptical area is swept by the electric field vector in one period. The unit vector $\hat{\varepsilon}_\text{maj}$ ($\hat{\varepsilon}_\text{min}$) is aligned with the semimajor (-minor) axis of the ellipse. $\vec{e}_\text{z}$ is the quantization axis, which determines the direction of the magnetic field in the experiment. $\vec{k}$ is the direction of the wave vector. $\theta_p$ is the angle between the polarization vector $\hat{\varepsilon}$ and the quantization axis $\vec{e}_\text{z}$, $\theta_k$ is the angle between $\vec{e}_\text{z}$ and $\vec{k}$. The$\hat{\varepsilon}_\text{maj}$, $\hat{\varepsilon}_\text{min}$, and $\vec{k}$ are orthogonal to each other. $\theta_\text{maj}$ ($\theta_\text{min}$) is the angle between $\hat{\varepsilon}_\text{maj}$ ($\hat{\varepsilon}_\text{min}$) and $\vec{e}_\text{z}$. $\psi$ is directly related to the degree of circular polarization of the electromagnetic plane wave.
• Figure 2: Dynamical polarizabilities of the $5s^2~^1\!S_{0}$ and $5s5p~^3\!P_0$ states. The dashed vertical lines indicate the positions of the resonant transitions. The magic wavelengths are identified by arrows where the polarizabilities of $5s5p~^3\!P_0$ and $5s^2~^1\!S_{0}$ are equal.
• Figure 3: Dynamic polarizabilities of the $^1\!S_0$ and $^3\!P_2$ states for linearly polarized light with $\theta_p=0^{\circ}$. The dashed vertical line indicates the positions of the resonant transitions. The magic wavelengths are identified by arrows.
• Figure 4: The $\theta_p$-dependent magic wavelengths under the linearly polarized light for the $5s^2~^1\!S_0\rightarrow5s5p~^3\!P_2$ transition.The horizontal solid line indicates the position of 813.4 nm.
• Figure 5: The dynamical polarizabilities under linearly polarized light with $\theta_p = 0^\circ$ and $\theta_p$-dependent magic wavelengths of $5s5p~^3\!P_0$ and $5s5p~^3\!P_2$ states. The vertical dashed lines indicate the positions of the resonant transitions and are given at the top of the figures.