arXiv:2403.05898 · Physical Review A

Dynamic polarizabilities and triply magic trapping conditions for 5s2S01→5s5p3P0,2 transitions of Cd atoms

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

Published Mar 27, 2024• 7 citations

Abstract

The dynamic electric dipole polarizabilities of the $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}$, $5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$, and $5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{2}$ 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.

DOI: 10.1103/physreva.109.032821

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arXiv:2403.05898 · Physical Review A

Dynamic polarizabilities and triply magic trapping conditions for 5s2S01→5s5p3P0,2 transitions of Cd atoms

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

Published Mar 27, 2024• 7 citations

arXiv HTML PDF

Abstract

The dynamic electric dipole polarizabilities of the

5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}

5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}

, and

5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{2}

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.

DOI: 10.1103/physreva.109.032821

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Magic wavelengths and triple magic trapping conditions for $5s^2~^1\!S_0$ and $5s5p~^3\!P_{0,2}$ states of Sr atomsMagic 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 +4 more

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.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.

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arXiv:2003.13855Physical Review AJul 20, 202016 citations

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arXiv:1804.01218Physical Review AOct 5, 201817 citations

Magic wavelengths for optical cooling and trapping of potassium

Marianna S. Safronova, Marianna S. Safronova, Charles W. Clark

We carry out a systematic study of the static and dynamic polarizabilities of the potassium atom using a first-principles high-precision relativistic all-order method in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values and uncertainties are provided for the relevant electric-dipole matrix elements. Polarizabilities of the $4s$ , $4{p}_{j}$ , $5s$ , $5{p}_{j}$ , and $3{d}_{j}$ states are compared with other theoretical and experimental values when possible. We identify magic wavelengths for the $4s\ensuremath{-}np$ transitions for $n=4,5$ , i.e., those wavelengths for which the two levels have the same ac Stark shifts. The magic wavelengths for the $4s\ensuremath{-}5p$ transitions are of particular interest for attaining a quantum gas of potassium at high phase-space density. We find 20 such wavelengths in the technically interesting region of $1050--1130$ nm.We carry out a systematic study of the static and dynamic polarizabilities of the potassium atom using a first-principles high-precision relativistic all-order method in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values and uncertainties are provided for the relevant electric-dipole matrix elements. Polarizabilities of the $4s$, $4{p}_{j}$, $5s$, $5{p}_{j}$, and $3{d}_{j}$ states are compared with other theoretical and experimental values when possible. We identify magic wavelengths for the $4s\ensuremath{-}np$ transitions for $n=4,5$, i.e., those wavelengths for which the two levels have the same ac Stark shifts. The magic wavelengths for the $4s\ensuremath{-}5p$ transitions are of particular interest for attaining a quantum gas of potassium at high phase-space density. We find 20 such wavelengths in the technically interesting region of $1050--1130$ nm.

Physical Review AMay 6, 201329 citations

Dynamic dipole polarizabilities for the low-lying triplet states of helium

Yong-Hui Zhang, Li-Yan Tang, Xian Zhou Zhang +1 more

The dynamic dipole polarizabilities for the four lowest triplet states ( $2{\phantom{\rule{0.16em}{0ex}}}^{3}S, 3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}P$ , and $3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ ) of helium are calculated using the $B$ -spline configuration interaction method. Present values of the static dipole polarizabilities in the length, velocity, and acceleration gauges are in good agreement with the best Hylleraas results. Also the tune-out wavelengths in the range 400 nm--4.2 $\ensuremath{\mu}\mathrm{m}$ for the four lowest triplet states are identified, and the magic wavelengths in the range 460 nm--3.5 $\ensuremath{\mu}\mathrm{m}$ for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P$, and $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transitions are determined. We show that the tune-out wavelength of $2{\phantom{\rule{0.16em}{0ex}}}^{3}S$ state is $413.038\phantom{\rule{0.16em}{0ex}}28(3)$ nm, which corroborates the value of Mitroy and Tang [Phys. Rev. A 88, 052515 (2013)], and the magic wavelength around 1066 nm for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transition can be expected for precision measurement to determine the ratio of transition matrix elements $(2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P)/(3{\phantom{\rule{0.16em}{0ex}}}^{3}P\ensuremath{\rightarrow}6{\phantom{\rule{0.16em}{0ex}}}^{3}S)$.The dynamic dipole polarizabilities for the four lowest triplet states ($2{\phantom{\rule{0.16em}{0ex}}}^{3}S, 3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}P$, and $3{\phantom{\rule{0.16em}{0ex}}}^{3}P$) of helium are calculated using the $B$-spline configuration interaction method. Present values of the static dipole polarizabilities in the length, velocity, and acceleration gauges are in good agreement with the best Hylleraas results. Also the tune-out wavelengths in the range 400 nm--4.2 $\ensuremath{\mu}\mathrm{m}$ for the four lowest triplet states are identified, and the magic wavelengths in the range 460 nm--3.5 $\ensuremath{\mu}\mathrm{m}$ for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P$, and $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transitions are determined. We show that the tune-out wavelength of $2{\phantom{\rule{0.16em}{0ex}}}^{3}S$ state is $413.038\phantom{\rule{0.16em}{0ex}}28(3)$ nm, which corroborates the value of Mitroy and Tang [Phys. Rev. A 88, 052515 (2013)], and the magic wavelength around 1066 nm for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transition can be expected for precision measurement to determine the ratio of transition matrix elements $(2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P)/(3{\phantom{\rule{0.16em}{0ex}}}^{3}P\ensuremath{\rightarrow}6{\phantom{\rule{0.16em}{0ex}}}^{3}S)$.

arXiv:1503.07300Physical Review AJul 24, 201528 citations

The dynamic electric dipole polarizabilities of the

5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}

5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{0}

, and

5s5p\phantom{\rule{0.16em}{0ex}}^{3}P_{2}

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}

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.

Magic wavelengths and triple magic trapping conditions for

5s^2~^1\!S_0

and

5s5p~^3\!P_{0,2}

states of Sr atoms

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.

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.

5{s}^{2}

$^{1}{S}_{0}\ensuremath{-}5s5p$

^{3}{P}_{0}^{o}

clock transition. We carried out calculations of the magnetic dipole and electric quadrupole polarizabilities and linear and circular hyperpolarizabilities of the

5{s}^{2}

^{1}{S}_{0}

and

5s5p

^{3}{P}_{0}^{o}

clock states at the magic wavelength and estimated uncertainties of these quantities. We also evaluated the second-order Zeeman clock transition frequency shift.

A large-scale full-configuration-interaction calculation based on the Dirac-Coulomb-Breit (DCB) Hamiltonian is performed for the

2{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}

and

2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}

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

n

(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

^{4}\mathrm{He}

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

^{3}\mathrm{He}

would provide theoretical support for experimentally designing an optical dipole trap to precisely determine the nuclear charge radius of helium in the future.

4s

4{p}_{j}

5s

5{p}_{j}

, and

3{d}_{j}

states are compared with other theoretical and experimental values when possible. We identify magic wavelengths for the $4s\ensuremath{-}np$ transitions for

n=4,5

, i.e., those wavelengths for which the two levels have the same ac Stark shifts. The magic wavelengths for the $4s\ensuremath{-}5p$ transitions are of particular interest for attaining a quantum gas of potassium at high phase-space density. We find 20 such wavelengths in the technically interesting region of

1050--1130

nm.

The dynamic dipole polarizabilities for the four lowest triplet states (

2{\phantom{\rule{0.16em}{0ex}}}^{3}S, 3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}P

, and

3{\phantom{\rule{0.16em}{0ex}}}^{3}P

) of helium are calculated using the

B

-spline configuration interaction method. Present values of the static dipole polarizabilities in the length, velocity, and acceleration gauges are in good agreement with the best Hylleraas results. Also the tune-out wavelengths in the range 400 nm--4.2 $\ensuremath{\mu}\mathrm{m}$ for the four lowest triplet states are identified, and the magic wavelengths in the range 460 nm--3.5 $\ensuremath{\mu}\mathrm{m}$ for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P$, and $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transitions are determined. We show that the tune-out wavelength of

2{\phantom{\rule{0.16em}{0ex}}}^{3}S

state is

413.038\phantom{\rule{0.16em}{0ex}}28(3)

nm, which corroborates the value of Mitroy and Tang [Phys. Rev. A 88, 052515 (2013)], and the magic wavelength around 1066 nm for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transition can be expected for precision measurement to determine the ratio of transition matrix elements $(2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P)/(3{\phantom{\rule{0.16em}{0ex}}}^{3}P\ensuremath{\rightarrow}6{\phantom{\rule{0.16em}{0ex}}}^{3}S)$.