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Physical Review A

Magic-zero and magic wavelengths of the Ca atom

Ru-Kui Zhang, Jun Jiang, Chenzhong Dong, Yong-Bo Tang

Published Jul 7, 2023• 7 citations
Publisher

Abstract

The relativistic configuration interaction plus many-body perturbation (RCI $+$ MBPT) method is utilized to calculate the wave functions, energy levels, and reduced $E1$ transition matrix elements of Ca atoms in detail. The static and dynamic polarizabilities of the $4{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}, 4s4p\phantom{\rule{0.16em}{0ex}}^{1}P_{1}$, and $4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{0,1,2}$ states are determined by the sum-over-states method. A number of magic-zero wavelengths for the ground and $4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ 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 $4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ 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.

DOI
10.1103/physreva.108.012803

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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 4s21S0\ensuremath→4s4p3P04{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}4s21S0​\ensuremath→4s4p3P0​ and 4s21S0\ensuremath→4s4p3P14{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{1}4s21S0​\ensuremath→4s4p3P1​ 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 4s21S0\ensuremath→4s4p3P14{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}\ensuremath{\rightarrow}4s4p\phantom{\rule{0.16em}{0ex}}^{3}P_{1}4s21S0​\ensuremath→4s4p3P1​ transition are provided, which are of particular interest for optical frequency measurements, Bose-Einstein condensate generation, and superradiant laser technology.
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.
A large-scale full-configuration-interaction calculation based on the Dirac-Coulomb-Breit (DCB) Hamiltonian is performed for the 21S02{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}21S0​ and 23S12{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}23S1​ states of helium. The operators of the normal and specific mass shifts are directly included in the DCB framework to take the finite nuclear mass correction into account. High-accuracy energies and matrix elements involved nnn (the main quantum number) up to 13 are obtained from one diagonalization of Hamiltonian. The dynamic dipole polarizabilities are calculated by using the sum rule of intermediate states, and a series of magic wavelengths with QED and hyperfine effects included for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}$ transition of helium are identified. In addition, the high-order ac Stark shift determined by the dynamic hyperpolarizabilities at the magic wavelengths are also evaluated. Since the most promising magic wavelength for application in experiment is 319.8 nm, the high-accuracy magic wavelength of 319.815 3(6) nm of 4He^{4}\mathrm{He}4He is in good agreement with the recent measurement value of 319.815 92(15) nm [R. J. Rengelink et al., arXiv:1804.06693], and the present magic wavelength of 319.830 2(7) nm for 3He^{3}\mathrm{He}3He would provide theoretical support for experimentally designing an optical dipole trap to precisely determine the nuclear charge radius of helium in the future.
The dynamic polarizabilities of the atomic states with angular momentum j>1/2j>1/2j>1/2 are sensitive to the angle between the quantization axis ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{e}}_{z}$ and the polarization vector $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\epsilon}}$ owing to the contribution of anisotropic tensor polarizabilities. The magic wavelength at which the differential Stark shift of an atomic transition nullifies depends on this angle. We identified the magic wavelengths for the $4{s}_{1/2}\ensuremath{\rightarrow}3{d}_{3/2,5/2}$ transitions of Ca+{\mathrm{Ca}}^{+}Ca+ ions at different angles between ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{e}}_{z}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\epsilon}}$ in the case of linearly polarized light. The magic wavelengths near 395.79 nm, which lie between the $4{s}_{1/2}\ensuremath{\rightarrow}4{p}_{1/2}$ and $4{s}_{1/2}\ensuremath{\rightarrow}4{p}_{3/2}$ transition wavelengths, remain insensitive to the angle, while the longest magic wavelength, which is around 1000 nm, for each of the magnetic sublevel transitions is very sensitive to the angle. We suggest that accurate measurements on the longest magic wavelengths for the $4{s}_{1/2}\ensuremath{\rightarrow}3{d}_{5/2}$ and $4{s}_{1/2}\ensuremath{\rightarrow}3{d}_{3/2}$ transitions can be used to determine the oscillator strengths for the $4{s}_{1/2}\ensuremath{\rightarrow}4{p}_{1/2,3/2}, 3{d}_{5/2}\ensuremath{\rightarrow}4{p}_{3/2}$, and $3{d}_{3/2}\ensuremath{\rightarrow}4{p}_{1/2,3/2}$ transitions, and the difference of the static polarizabilities of the 4s1/24{s}_{1/2}4s1/2​ and 3d5/23{d}_{5/2}3d5/2​ states, and the tensor polarizability of the 3d5/23{d}_{5/2}3d5/2​ state.
We have measured the ac Stark shift of the 4s21S0−−4s4p3P14{s}^{2}\phantom{\rule{0.2em}{0ex}}^{1}S_{0}--4s4p\phantom{\rule{0.2em}{0ex}}^{3}P_{1}4s21S0​−−4s4p3P1​ line in 40Ca^{40}\mathrm{Ca}40Ca for perturbing laser wavelengths between 780nm780\phantom{\rule{0.3em}{0ex}}\mathrm{nm}780nm and 1064nm1064\phantom{\rule{0.3em}{0ex}}\mathrm{nm}1064nm with a time domain Ramsey-Bord\'e atom interferometer. We found a zero crossing of the shift for the mS=0−−mP=0{m}_{S}=0--{m}_{P}=0mS​=0−−mP​=0 transition and $\ensuremath{\sigma}$ polarized perturbation at 800.8(22)nm800.8(22)\phantom{\rule{0.3em}{0ex}}\mathrm{nm}800.8(22)nm. The data was analyzed by a model deriving the energy shift from known transition wavelengths and strengths. To fit our data, we adjusted the Einstein AAA coefficients of the 4s3d3D−−4s4p3P4s3d\phantom{\rule{0.2em}{0ex}}^{3}D--4s4p\phantom{\rule{0.2em}{0ex}}^{3}P4s3d3D−−4s4p3P and 4s5s3S−−4s4p3P4s5s\phantom{\rule{0.2em}{0ex}}^{3}S--4s4p\phantom{\rule{0.2em}{0ex}}^{3}P4s5s3S−−4s4p3P fine structure multiplets. With these we can predict vanishing ac Stark shifts for the 1S0m=0−−3P1m=1^{1}S_{0}\phantom{\rule{0.2em}{0ex}}m=0--^{3}P_{1}\phantom{\rule{0.2em}{0ex}}m=11S0​m=0−−3P1​m=1 transition and ${\ensuremath{\sigma}}^{\ensuremath{-}}$ light at 983(12)nm983(12)\phantom{\rule{0.3em}{0ex}}\mathrm{nm}983(12)nm and at 735.5(20)nm735.5(20)\phantom{\rule{0.3em}{0ex}}\mathrm{nm}735.5(20)nm for the transition to the 3P0^{3}P_{0}3P0​ level.
The wave functions, energy levels and matrix elements of Yb+ ions are calculated using the relativistic configuration interaction plus core polarization (RCICP) method. The static and dynamic electric dipole polarizabilities of the ground state and low-lying excited states are determined. Then, the magic wavelengths of the magnetic sublevel 6s1/2,m = 1/2 → 5d3/2,m = ±3/2,±1/2 and 6s1/2,m = 1/2 → 5d5/2,m = ±5/2,±3/2,±1/2 transitions in the linearly, right-handed, and left-handed polarized light are further determined. The dependence of the magic wavelengths upon the angle between the direction of magnetic field and the direction of laser polarization is analyzed.