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arXiv:2512.23326

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

Yan-Min Wang, Qing-Yi Liu, Yong-Bo Tang, Lei Wu, Deng-Hong Zhang, Chen-Zhong Dong, Jun Jiang

Published Dec 29, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2512.23326

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Magic wavelengths and triple magic trapping conditions for 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​ and 5s5p 3 ⁣P0,25s5p~^3\!P_{0,2}5s5p 3P0,2​ states of Sr atoms
The static and dynamic electric dipole polarizabilities of the 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​ and 5s5p 3 ⁣P0,25s5p~^3\!P_{0,2}5s5p 3P0,2​ 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 5s2 1 ⁣S0→5s5p 3 ⁣P05s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{0}5s2 1S0​→5s5p 3P0​, 5s2 1 ⁣S0→5s5p 3 ⁣P25s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{2}5s2 1S0​→5s5p 3P2​, and 5s5p 3 ⁣P0→5s5p 3 ⁣P25s5p~^3\!P_0\rightarrow 5s5p~^3\!P_{2}5s5p 3P0​→5s5p 3P2​. 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 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​, 5s5p 3 ⁣P05s5p~^3\!P_{0}5s5p 3P0​ and 5s5p 3 ⁣P25s5p~^3\!P_{2}5s5p 3P2​ states are investigated. In the case of linearly polarized light, when the angle (θpθ_pθp​) between the laser polarization direction and the magnetic field is 79.1(0.7)∘79.1(0.7)^\circ79.1(0.7)∘, triple magic trapping for the 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​, 5s5p 3 ⁣P05s5p~^3\!P_{0}5s5p 3P0​, and 5s5p 3 ⁣P2 M=05s5p~^3\!P_{2}~M=05s5p 3P2​ 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 ⁣P2 M=25s5p~^3\!P_{2}~M=25s5p 3P2​ M=2 state can be achieved when θp=37.4(0.3)∘θ_p= 37.4(0.3)^\circθp​=37.4(0.3)∘. The conditions for achieving triple magic trapping with circularly and arbitrarily elliptically polarized light are also presented.
The dynamic electric dipole polarizabilities of the 5s21S05{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}5s21S0​, 5s5p3P05s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}5s5p3P0​, and 5s5p3P25s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{2}5s5p3P2​ states for Cd atoms are calculated using the relativistic configuration interaction plus many-body perturbation theory method. The magic wavelengths for the $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ and $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{2}$ transitions within a range of 300--500 nm are identified. The possibility of achieving triple magic trapping for the transitions $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ and $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{2}$ is investigated. It is found that no common magic wavelength could be identified for achieving triple magic trapping with the linearly polarized light. However, if the degree of ellipticity is between 0.358 and 1, the triple magic trapping can be achieved at 419.88 nm for the $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{2}$ $({M}_{i}=\ifmmode\pm\else\textpm\fi{}2)$ and $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ transitions.
The relativistic configuration interaction plus many-body perturbation (RCI +++ MBPT) method is utilized to calculate the wave functions, energy levels, and reduced E1E1E1 transition matrix elements of Ca atoms in detail. The static and dynamic polarizabilities of the 4s21S0,4s4p1P14{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}, 4s4p\phantom{\rule{0.16em}{0ex}}^{1}P_{1}4s21S0​,4s4p1P1​, and 4s4p3P0,1,24s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{0,1,2}4s4p3P0,1,2​ states are determined by the sum-over-states method. A number of magic-zero wavelengths for the ground and 4s4p3P04s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}4s4p3P0​ states, as well as magic wavelengths for the $4{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ and $4{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ transitions, are obtained. We suggest that the measurements of the magic-zero wavelengths of the 4s4p3P04s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}4s4p3P0​ states can be used to determine oscillator strengths with high precision. In addition, the angle-dependent magic wavelengths of each magnetic sublevel transition of the $4{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ transition are provided, which are of particular interest for optical frequency measurements, Bose-Einstein condensate generation, and superradiant laser technology.
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We demonstrate magic wavelengths for the 6s 2S1/2−6p 2P3/2,1/26s ~ {^2}S_{1/2}-6p ~ {^2}P_{3/2,1/2}6s 2S1/2​−6p 2P3/2,1/2​ transitions in the Cs atom due to the circularly polarized light. In addition, we also obtain magic wavelengths using the linearly polarized light in order to verify the previously reported values and make a comparative study with the values obtained for the circularly polarized light. Most of these wavelengths are found to be in the optical region and could be of immense interest to the experimentalists for carrying out high precision measurements. To obtain these wavelengths, we have calculated dynamic dipole polarizabilities of the ground, 6p 2P1/26p ~{^2}P_{1/2}6p 2P1/2​ and 6p 2P3/26p ~{^2}P_{3/2}6p 2P3/2​ states. We use the available precise values of the electric dipole (E1) matrix elements for the most dominantly contributing transitions from the lifetime measurements of the above excited states and obtain other relevant important E1 matrix elements by employing a relativistic coupled-cluster method approximating at the singles and doubles level. Accuracies in these quantities are substantiated by comparing our evaluated scalar polarizability values with the corresponding experimental results.
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Magic wavelengths (${\ensuremath{\lambda}}_{\mathrm{magic}}$) for the $7{S}_{1/2}\ensuremath{-}7{P}_{1/2,3/2}$ transitions (D lines) in Fr were reported by Dammalapati et al. [U. Dammalapati, K. Harada, and Y. Sakemi, Phys. Rev. A 93, 043407 (2016)]. These ${\ensuremath{\lambda}}_{\mathrm{magic}}$ were determined by plotting dynamic polarizabilities ($\ensuremath{\alpha}$) of the involved states with the above transitions against a desired range of wavelengths. Electric dipole (E1) matrix elements listed in [J. E. Sansonetti, J. Phys. Chem. Ref. Data 36, 497 (2007)], from the measured lifetimes of the 7P1/2,3/27{P}_{1/2,3/2}7P1/2,3/2​ states and from the calculations considering core-polarization effects in the relativistic Hartree-Fock (HFR) method, were used to determine $\ensuremath{\alpha}$. However, contributions from core correlation effects and from the E1 matrix elements of the $7P\ensuremath{-}7S, 7P\ensuremath{-}8S$, and $7P\ensuremath{-}6D$ transitions to $\ensuremath{\alpha}$ of the 7P7P7P states were ignored. In this work, we demonstrate importance of these contributions and improve accuracies of $\ensuremath{\alpha}$ further by replacing the E1 matrix elements taken from the HFR method by the values obtained employing relativistic coupled-cluster theory. Our static $\ensuremath{\alpha}$ are found to be in excellent agreement with the other available theoretical results, whereas substituting the E1 matrix elements used by Dammalapati et al. gives very small $\ensuremath{\alpha}$ values for the 7P7P7P states. Owing to this, we find disagreement in ${\ensuremath{\lambda}}_{\mathrm{magic}}$ reported by Dammalapati et al. for linearly polarized light, especially at wavelengths close to the D lines and in the infrared region. As a consequence, a ${\ensuremath{\lambda}}_{\mathrm{magic}}$ reported at 797.75 nm which was seen supporting a blue detuned trap in their work is now estimated at 771.03 nm and is supporting a red detuned trap. Also, none of our results match with the earlier results for circularly polarized light. Moreover, our static values of $\ensuremath{\alpha}$ will be very useful for guiding experiments to carry out their measurements.