⇦Chinese Physics Letters, 2021, Vol. 38, No. 11, Article code 113101 Effect of Electron Correlation and Breit Interaction on Energies, Oscillator Strengths, and Transition Rates for Low-Lying States of Helium Qing Liu (刘青)1, Jiguang Li (李冀光)2, Jianguo Wang (王建国)2, and Yizhi Qu (屈一至)1* Affiliations 1School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China 2Data Center for High Energy Density Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Received 28 July 2021; accepted 27 September 2021; published online 13 October 2021 Supported by the National Key Research and Development Program of China (Grant No. 2017YFA0402300), and the National Natural Science Foundation of China (Grant Nos. 11774344 and 11474033).
*Corresponding author. Email: yzqu@ucas.ac.cn Citation Text: Liu Q, Li J G, Wang J G, and Qu Y Z 2021 Chin. Phys. Lett. 38 113101    Abstract The transition energies, E1 transitional oscillator strengths of the spin-allowed as well as the spin-forbidden and the corresponding transition rates, and complete M1, E2, M2 forbidden transition rates for 1$s^{2}$, 1$s$2$s$, and 1$s2p$ states of He I, are investigated using the multi-configuration Dirac–Hartree–Fock method. In the subsequent relativistic configuration interaction computations, the Breit interaction and the QED effect are considered as perturbation, separately. Our transition energies, oscillator strengths, and transition rates are in good agreement with the experimental and other theoretical results. As a result, the QED effect is not important for helium atoms, however, the effect of the Breit interaction plays a significant role in the transition energies, the oscillator strengths and transition rates. DOI:10.1088/0256-307X/38/11/113101 © 2021 Chinese Physics Society Article Text In the stellar corona and nebula environment, the line intensity ratios are sensitive to the radiation and collision processes, which are conformed as tools to diagnose the density.[1,2] The abundance of helium, oxygen and neon can determine the formation of bright planetary nebulae. Helium, as the simplest multi-electron atom and the second most abundant element in the universe, has always been one of the most concerning elements in plasma physics, atom physics, and astrophysics.[3–15] For instance, its energy level interval, transition probability, and oscillator strength are of great application to the manipulation of atoms or ions in quantum information, the extraction of proton radius, and the determination of fine structure constant.[7,9,16] Despite the great progress has been made in experimental and theoretical research, there are still unknown parameters such as the “proton radius puzzle”,[17,18] which makes the precise measurement of atoms particularly significant. In the field of precision spectroscopy, the study of fine structure and transition process can be used to determine the radius of the nuclear charge and to test quantum electrodynamics (QED) theory.[11,19–23] So far, there are many works on the fine structure, transition probability, and oscillator strength among 1$s^{2}$, 1$s2s$, and 1$s2p$ states of the helium atom, such as the research of transition frequencies,[10,20,24,25] Shannon entropy[26] and transition rates,[9,27–30] which have improved the measurement of the fine structure constant[31,32] and the nuclear charge radius.[33] In 2008, Bo et al. merely obtained the splitting of fine energy levels $2 ^{3}\!P_{1}^{\rm o}$–$2^{3}\!P_{0}^{\rm o}$ and $2 ^{3}\!P_{2}^{\rm o}$–$2 ^{3}\!P_{0}^{\rm o}$, using the multi-configuration Dirac–Fock (MCDF) method.[34] A few years later, the fine structure and transition including the relativistic effect and the influence of finite nuclear mass were further studied by Morton et al.[27,29] using the relativistic many-body perturbation theory (RMBPT). In 2020, the electric dipole (E1) oscillator strengths of 1$s2p$ states were investigated by Głowacki[30] using the configuration interaction Dirac–Fock (CIDF) approach. Although there are a lot of works[4,6,29] about magnetic quadrupole (M2) transition, it is still meaningful to study systematically the electric quadrupole (E2) and M2 transitions. Furthermore, the forbidden magnetic dipole (M1) transition 2$^{3}\!S_{1}$–$1^{1}\!S_{0}$ was considered and its rate was measured to be $1.10 \times 10^{-4}$ s$^{-1}$ in the experiment of Woodworth et al.[3] It differs from the result $1.73 \times 10^{-4}$ s$^{-1}$ acquired by Lin et al.[35] using the relativistic random-phase approximation (RRPA) method. Indelicato,[36] Johnson et al.[4] and Łach et al.[6] improved the accuracy of transition rate by considering sufficient electronic correlation effect using the MCDF, RMBPT, and QED theory, their results were $1.279 \times 10^{-4}$, $1.266 \times 10^{-4}$ and $1.272426 \times 10^{-4}$ s$^{-1}$, respectively, and had small differences compared with the experimental result.[3] Another extremely weak M1 2$^{1}\!S_{0}$–$2^{3}\!S_{1}$ transition has also attracted much attention. In 1977, Lin et al. studied the radiative transitions of the He I isoelectronic sequence with RRPA. In their work the transition 2$^{1}\!S_{0}$–$2^{3}\!S_{1}$ rate was predicted to be $1.51 \times 10^{-7}$ s$^{-1}$.[35] However, Baklanov and Denisov calculated this forbidden transition rate as $6.11 \times 10^{-8}$ s$^{-1}$ (Ref. [5]) in 1997 by using the correlated nonrelativistic wave functions discussed the detail in Ref. [37]. Until 2011, Rooij et al. obtained the transition rate as $9.1 \times 10^{-8}$ s$^{-1}$ using the 1557 nm laser to directly measure the forbidden transition 2$^{1}\!S_{0}$–$2^{3}\!S_{1}$ of He I.[8] The experimental result is different from the earlier theoretical results,[5,35] which arouses the interest to further calculate the transition rate. In the present investigation, the transitions energies, E1 transitional oscillator strengths of the spin-allowed as well as the spin-forbidden and the corresponding transition rates, and complete M1, E2, M2 forbidden transition rates for He I 1$s^{2}$, 1$s2s$ and 1$s2p$ states are investigated using the GRASP2K program[38–40] based on the multi-configuration Dirac–Hartree–Fock (MCDHF) method. Particularly, the influences of Breit interaction and QED effects are discussed in detail for each result. The results show that the effect of Breit interaction on the oscillator strengths and transition rates for E1 transition of spin-forbidden are greater than that of the spin-allowed E1 transition. In addition, its effect also plays a significant role in the M1, E2 transitions of 1$s2p$ ($^{3}\!P_{2,1,0,}$ and $^{1}\!P_{1}$) states. Theoretical Method. In the MCDHF method, the atomic energy level and wave functions are obtained by solving the Dirac equation. The Dirac–Coulomb (DC) Hamiltonian is $$ H^{\rm DC}=\sum\limits_{i=1}^N {\Big[{c\boldsymbol{\alpha }_{i} \cdot \boldsymbol{p}_{i}+({\beta_{i}-1})c^{2}+V_{\rm nuc}({r_{i}})}\Big]} +\sum\limits_{i>j} {\frac{1}{r_{ij} }},~~ \tag {1} $$ where $V_{\rm nuc}({r_{i}})$ represents the interaction between the nucleus and electrons, $\boldsymbol{\alpha }_{i}$ and $\beta_{i}$ are the Dirac matrices. The wave function of the atom is denoted by an atomic state wave function (ASF) $|{\varGamma PJM} \rangle$, which is obtained by the linear combination of the configuration wave functions (CSFs) $|{\gamma_{i} PJM}\rangle$ with the same parity $P$ and angular quantum numbers $J, M$: $$ |{\varGamma PJM}\rangle=\sum\limits_{i=1}^{n_{c}}{c_{_{\scriptstyle i\varGamma}}|{\gamma_{i} PJM}\rangle }.~~ \tag {2} $$ Here $c_{_{\scriptstyle i\varGamma}}$ indicates the configuration mixing coefficients meaning the correlation effect of the CSFs. The CSF is antisymmetric functions obtained by the product of all one-electron orbitals. To minimize the energy eigenvalue, the radial parts of the orbitals and the mixing coefficients are optimized at the same time.[41–43] In the relativistic configuration interaction (RCI), the electronic correlation effects are further considered. Moreover, the Breit interaction in the low-frequency approximation reads $$ H_{ij}^{\rm B} =-\frac{1}{2r_{ij} }\Big[{{\boldsymbol \alpha }_{{\boldsymbol i}} \cdot {\boldsymbol \alpha }_{{\boldsymbol j}} +\frac{({\boldsymbol \alpha }_{{\boldsymbol i}} \cdot {\boldsymbol r}_{{\boldsymbol ij}})({\boldsymbol \alpha }_{{\boldsymbol i}} \cdot {\boldsymbol r}_{{\boldsymbol ij}})}{r_{ij}^{2} }}\Big],~~ \tag {3} $$ and low-order QED effects (self-energy and vacuum polarization) are considered as the perturbation.[38,40] Therefore, they do not influence the calculated orbit, only affect the mixing coefficient of different states. In addition, the first term of Breit interaction represents the magnetic interaction between electrons, the second term indicates the retarded effect of exchanging virtual photons in Eq. (3). After determining the wave function, the atomic parameters can be calculated by the corresponding reduced matrix elements $$ \Big\langle \psi (f)\Big\|T^{(k)}\Big\|\psi (i)\Big\rangle,~~ \tag {4} $$ where ${T}^{{\rm (k)}}$ is the multipole radiation interaction tensor operator.[44–47] The standard Racah algebra requires that the wave functions of the initial and final states correspond to the same orbital set.[48] In actual calculations, this restriction can be relaxed through the biorthogonal transform technology.[49] Therefore, the reduced matrix elements can be calculated by using independently optimized orbital sets. According to the reduced matrix elements, the line strength is given by $$ S=\Big|{\Big\langle\psi (f)\Big\|T^{(k)}\Big\|\psi(i)\Big\rangle}\Big|^{2},~~ \tag {5} $$ where $k = 0$ and $k = 1$ indicate the magnetic and electric multiples, respectively.[50–52] The transition rate ($A_{ji}$) and absorption oscillator strength ($f_{ij}$) from the initial $|\psi(i)\rangle$ to final states $|\psi(f)\rangle$ are related to the line strength $S$,[51,52] and the relation is shown by the following equation. For the E1 transitions, the transition rates and absorption oscillator strength can be expressed as $$\begin{align} &{A}_{ji} =2.0261\times 10^{18}\frac{1}{\lambda_{ji}^{3}[J_{i} ]}S^{\rm E1},\\ &f_{ij} =303.75\frac{1}{\lambda_{ji} [J_{i} ]}S^{\rm E1} ,~~ \tag {6} \end{align} $$ then the M1 and E2 transitions are expressed, respectively, as $$\begin{align} &{A}_{ji} =2.6974\times 10^{13}\frac{1}{\lambda_{ji}^{3}[J_{i} ]}S^{\rm M1},\\ &f_{ij} =4.044\times 10^{-3}\frac{1}{\lambda_{ji} [J_{i} ]}S^{\rm M1} ,~~ \tag {7} \end{align} $$ $$\begin{align} &{A}_{ji} =1.1199\times 10^{18}\frac{1}{\lambda_{ji}^{5}[J_{i} ]}S^{\rm E2},\\ &f_{ij} =167.89\frac{1}{\lambda_{ji}^{3}[J_{i} ]}S^{\rm E2} .~~ \tag {8} \end{align} $$ For the M2 transitions $$\begin{align} &{A}_{ji} =1.4910\times 10^{13}\frac{1}{\lambda_{ji}^{5}[J_{i} ]}S^{\rm M2},\\ &f_{ij} =2.236\times 10^{-3}\frac{1}{\lambda_{ji}^{3}[J_{i} ]}S^{\rm M2} .~~ \tag {9} \end{align} $$ Here, $[{J_{a}}]=\frac{1}{2J_{a} +1}$ is the statistical weights of $J_{a}$ levels, $\lambda_{ij}$ is transition energy in Å.[45] Results and Discussion—Transition Energies. In the MCDHF calculation, the even parity states 1$s^{2}$ ($^{1}\!S_{0}$),1$s2s$ ($^{3}\!S_{1}$,$^{1}\!S_{0}$) and odd parity states 1$s2p$ ($^{3}\!P_{2,1,0}$ and $^{1}\!P_{1}$) are optimized, respectively, to obtain occupied orbitals. Based on the active space approach, CSF expansions are obtained by single and double (SD) excitation to an active set of orbitals with principal quantum number $n = 7$ and orbital angular momenta $l=i$, followed by SD excitation to orbital sets with higher $n < 10$ ($l < {g}$), from multi-reference 1$s^{2}$,1$s2s$ for the even states and 1$s2p$ for the odd states. In order to ensure the convergence of the calculation, the active orbital sets were systematically added layer by layer, and only the outermost layers of orbitals were optimized by using relativistic self-consistent field procedure each time.[46,47] For the RCI calculation, based on the obtained orbitals, the Breit interaction and QED effects (self-energy and vacuum polarization) are considered using the perturbation theory. Before calculating the transition rates, the wave function generated above is biorthonormal using the BIOTRA component of the GRASP2K package.[38–40]
Table 1. The transition energies (cm$^{-1}$) for 1$s2s$ ($^{3}\!S_{1}$,$^{1}\!S_{0}$) and 1$s2p$ ($^{3}\!P_{2,1,0}$ and $^{1}\!P_{1}$) of He I with the different correlation models.
Model $2^{1}\!S_{0}$–$2^{3}\!S_{1}$ $2^{1}\!P_{1}^{\rm o} $–$2^{3}\!P_{0}^{\rm o}$ $2^{3}\!P_{0}^{\rm o} $–$2^{3}\!P_{1}^{\rm o}$ $2^{3}\!P_{1}^{\rm o} $–$2^{3}\!P_{2}^{\rm o}$
DF 9010.58 2277.25 0.17 0.35
DF + QED 9010.66 2277.25 0.17 0.35
DF + Breit 9012.88 2275.54 0.91 0.08
DF + QED + Breit 9012.96 2275.53 0.91 0.07
3SD 7049.40 2070.63 0.19 0.38
3SD + QED 7049.48 2070.63 0.19 0.38
3SD + Breit 7050.31 2068.79 0.96 0.09
3SD + QED + Breit 7050.39 2068.79 0.96 0.09
4SD 6539.27 2048.69 0.20 0.40
4SD + QED 6539.40 2048.69 0.20 0.40
4SD + Breit 6540.01 2046.77 1.00 0.09
4SD + QED + Breit 6540.14 2046.77 1.00 0.09
5SD 6481.86 2051.54 0.21 0.41
5SD + QED 6481.98 2051.54 0.21 0.41
5SD + Breit 6482.52 2049.63 0.99 0.08
5SD + QED + Breit 6482.64 2049.63 0.99 0.08
6SD 6430.69 2049.65 0.21 0.41
6SD + QED 6430.83 2049.65 0.21 0.41
6SD + Breit 6431.32 2047.74 0.99 0.08
6SD + QED + Breit 6431.46 2047.74 0.99 0.08
7SD 6426.33 2048.17 0.20 0.41
7SD + QED 6426.46 2048.17 0.20 0.41
7SD + Breit 6426.93 2046.25 0.99 0.08
7SD + QED + Breit 6427.07 2046.25 0.99 0.08
8$f$SD 6426.32 2048.03 0.20 0.41
8$f$SD + QED 6426.46 2048.03 0.20 0.41
8$f$SD + Breit 6426.93 2046.12 0.99 0.08
8$f$SD + QED + Breit 6427.06 2046.12 0.99 0.08
9$f$SD 6426.32 2047.96 0.20 0.41
9$f$SD + QED 6426.45 2047.96 0.20 0.41
9$f$SD + Breit 6426.93 2046.04 0.99 0.08
9$f$SD + QED + Breit 6427.06 2046.04 0.99 0.08
NIST[53] 6421.47 2047.07 0.99 0.08
The transition energies of 1$s2s$ ($^{3}\!S_{1}$,$^{1}\!S_{0}$) and 1$s2p$ ($^{3}\!P_{2,1,0}$ and $^{1}\!P_{1}$) states with the different correlation models are listed in Table 1, compared with the experimental results.[53] The different models show that the $n$ and $l$ represent the maximum principal and orbital angular momentum quantum numbers in the active orbital space for the $nl$SD model, respectively. The Breit interaction and QED effects (marked as + Breit, + QED or + QED + Breit) are also considered. The results of the DF model represent the calculations of the single configuration, which differ from that of other models. Taking the results of the DF + QED + Breit model as an example, the difference in transition energies of $2^{1}\!S_{0} $–$2^{3}\!S_{1}$ and $2^{1}\!P_{1}^{\rm o} $–$2^{3}\!P_{0}^{\rm o}$ are greater than 2000 cm$^{-1}$ and 200 cm$^{-1}$ compared with other models, respectively. Therefore, the consideration of electronic correlation effects is important. The differences of transition energies for 1$s2s$ ($^{3}\!S_{1}$,$^{1}\!S_{0}$) and 1$s2p$ ($^{3}\!P_{2,1,0}$ and $^{1}\!P_{1}$) are reduced with the expanding active orbital space, which indicates that the good convergence has been reached in our calculations. For example, the transition energies hardly change from the 7SD to the 9$f$SD model, and the largest difference of excitation energy is only 0.21 cm$^{-1}$ corresponding to the transition $2^{1}\!P_{1}^{\rm o} $–$2^{3}\!P_{0}^{\rm o}$, which shows the convergence of the calculated results. The Breit interaction plays an important role in the transitions among 1$s2p$ ($^{3}\!P_{2,1,0}$ and $^{1}\!P_{1}$) states of He I. It should be noted that the contribution of QED effect to transition energies are offset, but its contribution to the energy level cannot be ignored.[54] The comparison of excitation energy for principal quantum numbers $n = 8$ and $n = 9$ is made. We give that the present calculations for energy levels are converged less than 0.10 cm$^{-1}$ for 1$s2s$ ($^{3}\!S_{1}$,$^{1}\!S_{0}$) and 1$s2p$ ($^{3}\!P_{2,1,0}$ and $^{1}\!P_{1}$). For the convergent 9$f$SD model, the influence of the Breit interaction is around 7.10 cm$^{-1}$, while the greatest contribution of QED effect is 3.37 cm$^{-1}$ for 1$s2s$ ($^{3}\!S_{1}$) state. In addition, the maximum difference of excitation energy is 21.18 cm$^{-1}$ between ours and the experimental results[53] of the 1$s2s$ ($^{3}\!S_{1}$) state. Although the results of the MCDHF method for few-electron atoms are not as accurate as that of the Hylleraas method,[55–57] it should be mentioned that the MCDHF method is used not only to calculate the atomic structure but also to obtain high-precision transition probability and oscillator strength. In addition, the MCDHF method is still effective for complex atomic systems.[58] For example, it can be used to calculate excitation energies and oscillator strengths of Sm I ($Z = 62$).[46] E1 Transition. As discussed above, the 9$f$SD models are adopted in the subsequent transition calculation. From Tables 2–5, the oscillator strengths $gf$ in Babushkin, as well as Coulomb gauges and the corresponding transition rates $A$ are listed for E1 transitions of 1$s^{2}$, 1$s2s$ and 1$s2p$ states. These two gauges correspond to the length and velocity form of the non-relativistic transition amplitude, respectively, in which the consistency of oscillator strengths $gf$ further verify the calculated model. For the initial and final states with the spin-allowed transition, the oscillator strengths $gf$ of the two kinds of gauges are in good agreement with the other theoretical results,[4,6,7,27,29,30] while the spin-forbidden transitions are inconsistent between two gauges even with the above energy converged model (9$f$SD). The similar problems have also appeared in the calculated 2$s2p$ $^{3}\!P_{1}^{\rm o} $–$2 s^{2}{}^{1}\!S_{0}$ transition of Be-like C ion by Ynnerman et al.[59] and Jö nsson et al.[60] Furthermore, Chen et al.[61] pointed that this phenomenon is caused by the neglect of negative energy states, which significantly influences the results of the Coulomb gauge. Therefore, it is considered that the differences of the oscillator strengths $gf$ between Babushkin and Coulomb gauges for those transitions may be due to the same reason. Since the Babushkin gauge is hardly affected by the negative energy state, its result is usually recommended. As shown in Tables 2–5, the contribution from the Breit interaction to the oscillator strengths is indeed significant through comparison of the 9$f$SD and 9$f$SD + Breit models. In addition, by comparing the 9$f$SD and 9$f$SD + QED models, although the QED effect has a greater impact on the E1 transitions of spin-forbidden than the E1 transitions of the spin-allowed, the largest impact on oscillator strengths $gf$ is only 0.93%.
Table 2. The oscillator strengths $gf$ in (B) the Babushkin gauge and (C) the Coulomb gauge, and the transition rates $A$ (s$^{-1}$) in (B) the Babushkin gauge for the He I $1s2p (^{3,1}P_{1})$–$1s^{2}(^{1}\!S_{0}$) E1 transitions.
Model $2^{3}\!P_{1}^{\rm o}$–$1^{1}\!S_{0}$ $2^{1}\!P_{1}^{\rm o}$–$1^{1}\!S_{0}$
$gf$ ($\times 10^{-8}$) $A$ ($\times 10^{2}$) $gf$ ($\times 10^{-1}$) $A$ ($\times 10^{9}$)
B C B B C B
9$f$SD 0.6420 0.6766 0.4081 2.7629 2.7620 1.7989
9$f$SD + QED 0.6480 0.6730 0.4278 2.7629 2.7621 1.7989
9$f$SD + Breit 2.7803 3.2656 1.7670 2.7631 2.7624 1.7989
9$f$SD + QED + Breit 2.7928 3.2578 1.7750 2.7632 2.7624 1.7989
Głowacki et al.[30] 2.789 2.761
Morton et al.[29] 2.794 1.776
Morton et al.[27] 2.7925 1.7758 2.7616
Johnson et al.[4] 2.774 1.757 2.762 1.798
Drake et al.[7] 2.7935 2.7616
Łach et al.[6] 1.7758
By considering Breit correlation and QED effects, the complete results of E1 transitions are obtained in Tables 2–5, which agree with other theoretical calculations[4,6,7,27,29,30] within 1%. For $2^{3}\!P_{1}^{\rm o}$–$1^{1}\!S_{0}$ transition, the present oscillator strength differs from the results of Głowacki et al.,[30] Morton et al.,[27,29] Johnson et al.[4] and Drake et al.[7] about 0.14%, 0.04%, 0.01%, 0.68%, and 0.03%, respectively. The result $2.774 \times 10^{-8}$ was calculated by Johnson et al. in Ref. [4] for this transition using RMBPT theory, which differs slightly from other results. As pointed out by Łach et al.,[6] the difference may be due to the strong numerical cancelation and inadequate electronic correlation. Moreover, the source of difference for Ref. [30] is the use of non-relativistic electron-electron interaction in the calculation using the CIDF method compared with other results. In addition, our result is in better agreement with the result obtained by Morton et al.[27,29] and Drake et al.[7] using the Hylleraas theory.
Table 3. The oscillator strengths $gf$ in (B) the Babushkin gauge and (C) the Coulomb gauge, and the transition rates $A$ (s$^{-1}$) in (B) the Babushkin gauge for the He I 1$s2p (^{3,1}P_{1})$–$1s2s(^{1}\!S_{0}$) E1 transitions.
Model $2^{3}\!P_{1}^{\rm o}$–$2^{1}\!S_{0}$ $2^{1}\!P_{1}^{\rm o}$–$2^{1}\!S_{0}$
$gf$ ($\times 10^{-8}$) $A$ ($\times 10^{-2}$) $gf$ ($\times 10^{-1}$) $A$ ($\times 10^{6}$)
B C B B C B
9$f$SD 0.3253 1.5487 0.5692 3.7516 3.7497 1.9647
9$f$SD + QED 0.3266 1.5566 0.5715 3.7517 3.7517 1.9648
9$f$SD + Breit 1.5066 4.2385 2.6363 3.7511 3.7489 1.9637
9$f$SD + QED + Breit 1.5094 4.2516 2.6415 3.7511 3.7489 1.9639
Morton et al.[29] 1.5294 2.683
Morton et al.[27] 1.5305 3.7644
Drake et al.[7] 1.6899
Table 4. The oscillator strengths $gf$ in (B) the Babushkin gauge and (C) the Coulomb gauge, and the transition rates $A$ (s$^{-1}$) in (B) the Babushkin gauge for the He I 1$s2p (^{3}\!P_{0}$,$_{1})$–$1s2s(^{3}\!S_{1}$) E1 transitions.
Model $2^{3}\!P_{0}^{\rm o}$–$2^{3}\!S_{1}$ $2^{3}\!P_{1}^{\rm o}$–$2^{3}\!S_{1}$
$gf$ ($\times 10^{-1}$) $A$ ($\times 10^{7}$) $gf$ ($\times 10^{-1}$) $A$ ($\times 10^{7}$)
B C B B C B
9$f$SD 1.7972 1.7969 1.0216 5.3917 5.3906 1.0217
9$f$SD + QED 1.7972 1.7968 1.0217 5.3919 5.3904 1.0218
9$f$SD + Breit 1.7976 1.7972 1.0222 5.3922 5.3911 1.0219
9$f$SD + QED + Breit 1.7977 1.7972 1.0223 5.3923 5.3909 1.0220
Głowacki et al.[30] 1.796 5.988
Drake et al.[7] 1.7969 5.989
Table 5. The oscillator strengths $gf$ in (B) the Babushkin gauge and (C) the Coulomb gauge, and the transition rates $A$ (s$^{-1}$) in (B) the Babushkin gauge for the He I 1$s2p (^{1}\!P_{1}$, $^{3}\!P_{2})$–$1s2s (^{3}\!S_{1}$) E1 transitions.
Model $2^{1}\!P_{1}^{\rm o}$–$2^{3}\!S_{1}$ $2^{3}\!P_{2}^{\rm o}$–$2^{3}\!S_{1}$
$gf$ ($\times 10^{-8}$) $A$ ($\times 10^{-1}$) $gf$ ($\times 10^{-1}$) $A$ ($\times 10^{7}$)
B C B B C B
9$f$SD 0.4096 0.2395 3.4762 2.9956 2.9947 1.0218
9$f$SD + QED 0.4111 0.2398 3.4888 2.9957 2.9946 1.0219
9$f$SD + Breit 1.8349 1.1704 1.5571 2.9956 2.9947 1.0219
9$f$SD + QED + Breit 1.8380 1.1711 1.5598 2.9957 2.9946 1.0220
Głowacki et al.[30] 1.817 2.994
Morton et al.[29] 1.8262 1.5493
Morton et al.[27] 1.8257 1.5489
Drake et al.[7] 2.9948
Łach et al.[6] 1.5489
M1, E2, M2 Transitions. The forbidden M1, E2, M2 transition rates for 1$s^{2}$, 1$s2s$, and 1$s2p (^{3}\!P_{0,1,2}$ and $^{1}\!P_{1}$) states are listed from Tables 6–9. In Table 6, the results show that the contributions of Breit correlation and QED effects are all less than 0.01% for 2$^{3}\!S_{1}$–$1^{1}\!S_{0}$ and 2$^{3}\!S_{1}$–$2^{1}\!S_{0}$ M1 transitions. However, as shown in Table 7, the Breit interaction has a great influence on the M1 transition rates of the 1$s2p (^{3}\!P_{0,1,2}$ and $^{1}\!P_{1}$) states. Similar to the M1 transition, the Breit interaction also plays an important role in E2 transitions in Table 8. As can be seen from Table 6, the 2$^{3}\!S_{1} - 1^{1}\!S_{0}$ transition was measured by Woodworth and Moos with the results of $1.10(33) \times 10^{-4}$ s$^{-1}$.[3] The theoretical result $1.73 \times 10^{-4}$ s$^{-1}$ was obtained by Lin et al. using the RRPA method,[35] which is different from the results of experiment[3] because of the inadequate description for electronic correlations. Later, the rate $1.266 \times 10^{-4}$ s$^{-1}$ was obtained by Johnson et al. using the RMBPT theory.[4] However, Łach et al. pointed that the electron correlations were not well considered by Johnson et al.,[4] and they obtained the rate $1.272426 \times 10^{-4}$ s$^{-1}$ using the QED theory.[6] Moreover, the role of the negative continuum effects and electron-electron correlation are considered by Indelicato using the MCDF method for this M1 transition,[36] his result $0.782 \times 10^{-4}$ s$^{-1}$ is more consistent with our result $1.0641 \times 10^{-4}$ s$^{-1}$. For the 2$^{3}\!S_{1}$–$2^{1}\!S_{0}$ M1 transition, our result $8.1268 \times 10^{-8}$ s$^{-1}$ is better in agreement with the experimental result $9.1 \times 10^{-8}$ s$^{-1}$ (Ref.[8]) than the theoretical results $15.1 \times 10^{-8}$ s$^{-1}$ of Lin et al.[35] and $6.1 \times 10^{-8}$ s$^{-1}$ of Baklanov et al.[5] The result of Baklanov et al.[5] differs from the results of experiment,[8] because the correlated nonrelativistic wave functions were used. The M2 transition rates are presented in Table 9, which are greatly consistent with the results of Morton et al.,[29] Johnson et al.,[4] and Łach et al.[6] By comparing the calculated models, the contributions of Breit correlation and QED effects are extremely small, ranging from 0.003% to 0.07%. As shown in Table 9, the $2^{3}\!P_{2}^{\rm o} $–$1 {}^{1}\!S_{0}$ M2 transition rate of $3.2684 \times 10^{-1}$ s$^{-1}$ is much larger than that of other M2 transitions, and it differs from other theoretical values of 0.26%,[29] 0.08%[4] and 0.05%,[6] respectively. In addition, for $2^{3}\!P_{2}^{\rm o}$–$2^{1}\!S_{0}$ and $2^{1}\!P_{1}^{\rm o}$–$2^{3}\!S_{1}^{\rm o}$ M2 transitions, the differences between our calculated rates and the results of Morton et al.[29] are all less than 1%. Moreover, the M2 transition rates of $2^{3}\!P_{1}^{\rm o} $–$2^{3}\!S_{1}$ and $2^{3}\!P_{2}^{\rm o} $–$2^{3}\!S_{1}$ are also obtained. It should be mentioned that the comparison of line strengths for principal quantum numbers $n = 8$ and $n = 9$ is made. We give that the present calculations for line strengths of the E1, M1, E2, M2 transitions are all converged less than $3 \times 10^{-4}$ a.u.
Table 6. The transition rates $A$ (s$^{-1}$) of the 1$s2s$ ($^{3}\!S_{1}$)–1$s^{2}$($^{1}\!S_{0}$) and 1$s2s$ ($^{1}\!S_{0}$)–1$s2s(^{3}\!S_{1}$) M1 transitions for He I.
Model $2^{3}\!S_{1}$–$1^{1}\!S_{0}$ $2^{1}\!S_{0}$–$2^{3}\!S_{1}$
$A$ ($\times 10^{-4}$) $A$ ($\times 10^{-8}$)
9$f$SD 1.0643 8.1241
9$f$SD + QED 1.0642 8.1246
9$f$SD + Breit 1.0642 8.1263
9$f$SD + QED + Breit 1.0641 8.1268
Woodworth et al.[3] (Expt.) 1.10
Johnson et al.[4] 1.266
Indelicato[36] 0.782
Łach et al.[6] 1.272426
Rooij et al.[8] (Expt.) 9.1
Lin et al.[35] 1.73 15.1
Baklanov et al.[5] 6.11
Table 7. The transition rates $A$ (s$^{-1}$) of the 1$s2p$ ($^{3}\!P_{0,1,2}$ and $^{1}\!P_{1}$) M1 transitions for He I.
Model $2^{1}\!P_{1}^{\rm o}$–$2^{3}\!P_{0}^{\rm o}$ $2^{3}\!P_{0}^{\rm o}$–$2^{3}\!P_{1}^{\rm o}$ $2^{1}\!P_{1}^{\rm o} $–$2^{3}\!P_{1}^{\rm o}$ $2^{3}\!P_{1}^{\rm o} $–$2^{3}\!P_{2}^{\rm o}$ $2^{1}\!P_{1}^{\rm o}$–$2^{3}\!P_{2}^{\rm o}$
$A$ ($\times 10^{-9}$) $A$ ($\times 10^{-13}$) $A$ ($\times 10^{-9}$) $A$ ($\times 10^{-13}$) $A$ ($\times 10^{-10}$)
9$f$SD 4.7238 1.5326 5.8323 9.2004 2.7932
9$f$SD + QED 4.7386 1.5440 5.8466 9.2715 2.8334
9$f$SD + Breit 16.017 519.674 16.009 10.744 66.454
9$f$SD + QED + Breit 16.044 518.839 16.033 10.325 66.649
Table 8. The transition rates $A$ (s$^{-1}$) of the 1$s2p (^{3}\!P_{0,1,2}$ and $^{1}\!P_{1}$) E2 transitions for He I.
Model $2^{1}\!P_{1}^{\rm o}$–$2^{3}\!P_{1}^{\rm o}$ $2^{3}\!P_{0}^{\rm o}$–$2^{3}\!P_{2}^{\rm o}$ $2^{3}\!P_{1}^{\rm o}$–$2^{3}\!P_{2}^{\rm o}$ $2^{1}\!P_{1}^{\rm o}$–$2^{3}\!P_{2}^{\rm o}$
$A$
9$f$SD $4.8549 \times 10^{-11}$ $5.1060 \times 10^{-22}$ $1.5134 \times 10^{-22}$ $1.6540 \times 10^{-11}$
9$f$SD + QED $4.8711 \times 10^{-11}$ $5.1711 \times 10^{-22}$ $1.5329 \times 10^{-22}$ $1.6584 \times 10^{-11}$
9$f$SD + Breit $2.2182 \times 10^{-10}$ $4.0613 \times 10^{-20}$ $6.4715 \times 10^{-26}$ $7.3366 \times 10^{-11}$
9$f$SD + QED + Breit $2.2217 \times 10^{-10}$ $4.0317 \times 10^{-20}$ $6.0565 \times 10^{-26}$ $7.3458 \times 10^{-11}$
Table 9. The transition rates $A$ (s$^{-1}$) for the M2 transitions among 1$s2p$, 1$s2s,$ and 1$s^{2}$ states for He I.
Model $2^{3}\!P_{2}^{\rm o}$–$1^{1}\!S_{0}$ $2^{3}\!P_{2}^{\rm o}$–$2^{1}\!S_{0}$ $2^{3}\!P_{1}^{\rm o}$–$2^{3}\!S_{1}$ $2^{1}\!P_{1}^{\rm o}$–$2^{3}\!S_{1}$ $2^{3}\!P_{2}^{\rm o}$–$2^{3}\!S_{1}$
$A$ ($\times 10^{-1}$) $A$ ($\times 10^{-8}$) $A$ ($\times 10^{-6}$) $A$ ($\times 10^{-5}$) $A$ ($\times 10^{-6}$)
9$f$SD 3.2690 1.1811 3.2025 1.9047 5.7681
9$f$SD + QED 3.2688 1.1814 3.2030 1.9050 5.7689
9$f$SD + Breit 3.2686 1.1800 3.2021 1.9050 5.7683
9$f$SD + QED + Breit 3.2684 1.1803 3.2026 1.9052 5.7691
Morton et al.[29] 3.2768 1.1927 1.9072
Johnson et al.[4] 3.271
Łach et al.[6] 3.270
In summary, the transition energies, the E1 transitional oscillator strengths of the spin-allowed as well as spin-forbidden and the corresponding transition rates, and forbidden M1, E2, M2 transition rates for 1$s^{2}$, 1$s2s$, and 1$s2p$ states of He I, are investigated using the MCDHF method. Based on the active space approach, the electronic correlation effects are considered systematically. In subsequent RCI computations, the Breit interaction and the QED effects are considered separately. Our results are in good agreement with other theoretical[4,6,7,27,29,30] and experimental results.[3,8] For E1 transition, the contribution from the Breit interaction to the oscillator strengths is indeed significant. It should be pointed out that the Babushkin gauge is insensitive to negative energy state, therefore its result is usually recommended, especially the spin-forbidden E1 transitions. For forbidden 2$^{3}\!S_{1}$–$1^{1}\!S_{0}$ and $2^{3}\!S_{1}$–$2^{1}\!S_{0}$ M1 transitions, the results show that the contributions of Breit correlation are all less than 0.01%. However, its effects have a great influence on the M1 transition rates of the 1$s2p (^{3}\!P_{0,1,2}$ and $^{1}\!P_{1}$) states. In addition, our result $8.1268 \times 10^{-8}$ s$^{-1}$ is in better agreement with the experimental result $9.1 \times 10^{-8}$ s$^{-1}$ (Ref. [8]) than the theoretical results $15.1 \times 10^{-8}$ s$^{-1}$ of Lin et al.[35] and $6.1 \times 10^{-8}$ s$^{-1}$ of Baklanov et al.[5] for 2$^{3}\!S_{1}$–$1^{1}\!S_{0}$ M1 transition. Similar to the M1 transition of 1$s2p (^{3}\!P_{0,1,2}$ and $^{1}\!P_{1}$) states, the Breit interaction also plays an important role in E2 transitions. For the M2 transition, the complete transition rates are obtained, and it is found that the contributions of Breit correlation are extremely small, ranging from 0.003% to 0.07%. It is worth noting that the contribution of QED effect to the fine structure of the 1$s2p$ state is offset, so its effect cannot be ignored.[54] References E1-forbidden transition rates in ions of astrophysical interestSpace-Resolved Diagnosis for Electron Temperature of Laser-Produced Aluminum PlasmaExperimental determination of the single-photon transition rate between the 2 S 1 3 and 1 S 0 1 states of He iAdvances In Atomic, Molecular, and Optical PhysicsGeneration of the fundamental tone and first overtone radiations in a pulsed H 2 — F 2 laserForbidden transitions in the helium atomA Multiplet Table for Neutral Helium ( 4 He i ) with Transition RatesFrequency Metrology in Quantum Degenerate Helium: Direct Measurement of the 2 3S1 -> 2 1S0 TransitionRelativistic configuration interaction calculations for some two- and three-electron systems with screened hydrogenic spin orbitalsMeasurement of the Frequency of the 2   S 3 − 2   P 3 Transition of He 4 Direct Measurement of the Forbidden 2 S 3 1 → 3 S 3 1 Atomic Transition in HeliumExperiments on the 2 P 3 State of Helium. II. Measurements of the Zeeman EffectCalculation of the Decay Rate for 2 S 1 3 → 1 S 0 1 + One Photon in HeliumNeutral Helium Triplet Spectroscopy of Quiescent Coronal Rain with Sensitivity Estimates for Spectropolarimetric Magnetic Field DiagnosticsInfluence of Debye Plasma on the KLL Dielectronic Recombination of H-Like Helium IonsUniversality of the Dynamic Characteristic Relationship of Electron Correlation in the Two-Photon Double Ionization Process of a Helium-Like SystemProton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic HydrogenA measurement of the atomic hydrogen Lamb shift and the proton charge radiusMetastable Helium: A New Determination of the Longest Atomic Excited-State LifetimeImproved Theory of Helium Fine StructureComplete α 7 m Lamb shift of helium triplet statesTune-out wavelength around 413 nm for the helium 2 3 S 1 state including relativistic and finite-nuclear-mass correctionsApplication of the Hylleraas- B -spline basis set: Static dipole polarizabilities of heliumReexamination of the helium fine structureFine Structure of Heliumlike Ions and Determination of the Fine Structure ConstantShannon Entropy as a Measurement of the Information in a Multiconfiguration Dirac—Fock WavefunctionRelativistic corrections to He I transition ratesThis paper was presented at the International Conference on Precision Physics of Simple Atomic Systems, held at École de Physique, les Houches, France, 30 May – 4 June, 2010.Tune-out wavelengths for metastable heliumOscillator strengths and radiative decay rates for spin-changing S–P transitions in helium: finite nuclear mass effectsRelativistic configuration interaction calculations of transitions for low-lying states in the helium isoelectronic sequenceDetermination of the Fine Structure Constant Using Helium Fine StructureUltrahigh-Precision Measurement of the n = 2 Triplet P Fine Structure of Atomic Helium Using Frequency-Offset Separated Oscillatory FieldsHigh-Precision Spectroscopy of the Forbidden 2   S 1 3 → 2   P 1 1 Transition in Quantum Degenerate Metastable HeliumTheoretical Study of Interesting Fine-Structure Splittings for 2 3 P 0,1,2 States along Helium Isoelectronic SequenceRadiative decays of the n = 2 states of He-like ionsCorrelation and Negative Continuum Effects for the Relativistic M 1 Transition in Two-Electron Ions using the Multiconfiguration Dirac-Fock MethodTheory of Relativistic Magnetic Dipole Transitions: Lifetime of the Metastable 2 S 3 State of the Heliumlike IonsThe grasp2K relativistic atomic structure packageGRASP2018—A Fortran 95 version of the General Relativistic Atomic Structure PackageNew version: Grasp2K relativistic atomic structure packageMCDHF calculations and study of plasma parameters for Li-like ionsWeak- and hyperfine-interaction-induced 1s2s 1 S 0 → 1s 2 1 S 0 E1 transition rates of He-like ionsRelativistic configuration interaction calculations of energy levels, isotope shifts, hyperfine structures, and transition rates in the 2s 2 2p 2 –2s2p 3 transition array for the carbon-like sequenceE1 Transitions between States with n = 1–6 in Helium‐like Carbon, Nitrogen, Oxygen, Neon, Silicon, and ArgonEnergy levels, radiative rates and electron impact excitation rates for transitions in He-like Li II, Be III, B IV and C VMulticonfiguration Dirac-Hartree-Fock calculations of excitation energies, oscillator strengths, and hyperfine structure constants for low-lying levels of Sm IEffects of the electron correlation and Breit and hyperfine interactions on the lifetime of the 2 p 5 3 s states in neutral neonInteraction between Configurations with Several Open ShellsTransition probability calculations for atoms using nonorthogonal orbitalsAtomic data from the Iron ProjectAtomic Transition Probabilities of Sodium and Magnesium. A Critical CompilationRadiative transition properties of singly charged magnesium, calcium, strontium and barium ionsQED Corrections of O ( m c 2 α 7 ln α ) to the Fine Structure Splittings of Helium and He-like IonsHylleraas-configuration-interaction nonrelativistic energies for the 1 S ground states of the beryllium isoelectronic sequenceHigh-precision calculation of the geometric quantities of two-electron atoms based on the Hylleraas configuration-interaction basisComplex coordinate rotation method based on gradient optimizationForbidden transition properties of fine-structure 2p 3 4 S 3/2 −2p 3 2 D 3/2,5/2 for nitrogen-like ionsMulticonfigurational-Dirac-Fock calculation of the 2 s 2 1 S 0 –2 s 2 p 3 P 1 spin-forbidden transition for the Be-like isoelectronic sequenceMulticonfiguration Dirac-Fock calculations of the 2 s 21 S 0 – 2 s 2 p 3 P 1 intercombination transition in C IIILarge-scale relativistic configuration-interaction calculation of the 2s^2 ^1S_0–2s2p ^3P_1 intercombination transition in C III
[1] Träbert E 2014 Phys. Scr. 89 114003
[2] Zhao Y, Deng B, Xiong G, Hu Z M, Wei M X, Zhu T, Shang W L, Li J, Yang G H, Zhang J Y, and Yang J M 2011 Chin. Phys. Lett. 28 065201
[3] Woodworth J R and Moos H W 1975 Phys. Rev. A 12 2455
[4] Johnson W R, Plante D R, and Sapirstein J 1995 Adv. At. Mol. Opt. Phys. 35 255
[5] Baklanov E V and Denisov A 1997 Quantum Electron. 27 465
[6] Lach G and Pachucki K 2001 Phys. Rev. A 64 042510
[7] Drake G W F and Morton D C 2007 Astrophys. J. Suppl. Ser. 170 251
[8] Rooij R V, Borbely J S, Simonet J, Hoogerland M D, Eikema K S E, Rozendaal R A, and Vassen W 2011 Science 333 196
[9] Głowacki L and Migdałek J 2014 Phys. Rev. A 89 042503
[10] Zheng X, Sun Y R, Chen J J, Jiang W, Pachucki K, and Hu S M 2017 Phys. Rev. Lett. 119 263002
[11] Thomas K F, Ross J A, Henson B M, Shin D K, Baldwin K G H, Hodgman S S, and Truscott A G 2020 Phys. Rev. Lett. 125 013002
[12] Lewis S A, Pichanick F M J, and Hughes V W 1970 Phys. Rev. A 2 86
[13] Feinberg G and Sucher J 1971 Phys. Rev. Lett. 26 681
[14] Schad T A 2018 Astrophys. J. 865 31
[15] Zhang D H, Xie L Y, Jiang J, and Dong C Z 2019 Chin. Phys. Lett. 36 083401
[16] Li F, Yang Y J, Chen J, Liu X J, Wei Z Y, and Wang B B 2020 Chin. Phys. Lett. 37 113201
[17] Antognin A, Nez F, Schuhmann K, and Amaro F D 2013 Science 339 417
[18] Bezginov N, Valdez T, Horbatsch M, Marsman A, Vutha A C, and Hessels E A 2019 Science 365 1007
[19] Hodgman S S, Dall R G, Byron L J, Baldwin K G, Buckman S J, and Truscott A G 2009 Phys. Rev. Lett. 103 053002
[20] Pachucki K 2006 Phys. Rev. Lett. 97 013002
[21] Patkóš V, Yerokhin V A, and Pachucki K 2021 Phys. Rev. A 103 042809
[22] Zhang Y H, Tang L Y, Zhang X Z, and Shi T Y 2016 Phys. Rev. A 93 052516
[23] Yang S J, Mei X S, Shi T Y, and Qiao H X 2017 Phys. Rev. A 95 062505
[24] Pachucki K and Yerokhin V A 2009 Phys. Rev. A 79 062516
[25] Pachucki K and Yerokhin V A 2010 Phys. Rev. Lett. 104 070403
[26] Wan J J 2015 Chin. Phys. Lett. 32 023102
[27] Morton D C, Moffatt P, and Drake G W F 2011 Can. J. Phys. 89 129
[28] Mitroy J and Tang L Y 2013 Phys. Rev. A 88 052515
[29] Morton D C, Schulhoff E E, and Drake G W F 2015 J. Phys. B 48 235001
[30] Głowacki L 2020 At. Data Nucl. Data Tables 133 101344
[31] Smiciklas M and Shiner D 2010 Phys. Rev. Lett. 105 123001
[32] Kato K, Skinner T D G, and Hessels E A 2018 Phys. Rev. Lett. 121 143002
[33] Notermans R P and Vassen W 2014 Phys. Rev. Lett. 112 253002
[34] Qing B, Chen S H, Gao X, and Li J M 2008 Chin. Phys. Lett. 25 2448
[35] Lin C D, Johnson W R, and Dalgarno A 1977 Phys. Rev. A 15 154
[36] Indelicato P 1996 Phys. Rev. Lett. 77 3323
[37] Drake G W F 1971 Phys. Rev. A 3 908
[38] Jönsson P, He X, Froese F C, and Grant I P 2007 Comput. Phys. Commun. 177 597
[39] Froese F C, Gaigalas G, Jönsson P, and Bieroń J 2019 Comput. Phys. Commun. 237 184
[40] Jönsson P, Gaigalas G, Bieroń J, Fischer C F, and Grant I P 2013 Comput. Phys. Commun. 184 2197
[41] Khatri I, Goyal A, Aggarwal S, Singh A K, and Man M 2016 Radiat. Phys. Chem. 123 46
[42] Radžiūtė L, Gaidamauskas E, Gaigalas G, Li J G, Dong C Z, and Per J 2015 Chin. Phys. B 24 043103
[43] Jönsson P and Bieroń J 2010 J. Phys. B 43 074023
[44] Johnson W R, Savukov I M, Safronova U I, and Dalgarno A 2002 Astrophys. J. Suppl. Ser. 141 543
[45] Aggarwal K M, Kato T, Keenan F P, and Murakami I 2011 Phys. Scr. 83 015302
[46] Zhou F, Qu Y, Li J, and Wang J 2015 Phys. Rev. A 92 052505
[47] Li J, Jönsson P, Godefroid M, Dong C, and Gaigalas G 2012 Phys. Rev. A 86 052523
[48] Fano U 1965 Phys. Rev. 140 A67
[49] Olsen J, Godefroid M R, Jonsson P, Malmqvist P, and Fischer C F 1995 Phys. Rev. E 52 4499
[50] Nahar S N, Eissner W, Chen G X, and Pradhan A K 2003 Astron. & Astrophys. 408 789
[51] Kelleher D E and Podobedova L I 2008 J. Phys. Chem. Ref. Data 37 267
[52] Kaur M, Dar D F, Sahoo B K, and Arora B 2021 At. Data Nucl. Data Tables 137 101381
[53]Kramida A, Ralchenko Y, Reader J, and Team N A 2020 NIST Atomic Spectra Database (ver. 5.8) (Gaithersburg, MD: National Institute of Standards and Technology)
[54] Zhang T, Yan Z C, and Drake G W F 1996 Phys. Rev. Lett. 77 1715
[55] Sims J S and Hagstrom S A 2014 J. Chem. Phys. 140 224312
[56] Jiao L G, Zan L R, Zhu L, Zhang Y Z, and Ho Y K 2019 Phys. Rev. A 100 022509
[57] Bai Z D, Zhong Z X, Yan Z C, and Shi T Y 2021 Chin. Phys. B 30 023101
[58] He X K, Liu J P, Zhang X, Shen Y, and Zou H X 2018 Chin. Phys. B 27 83102
[59] Ynnerman A and Fischer C F 1995 Phys. Rev. A 51 2020
[60] Jönsson P and Fischer C F 1998 Phys. Rev. A 57 4967
[61] Chen M, Cheng K, and Johnson W 2001 Phys. Rev. A 64 042507