Theoretical calculation of dynamic polarizability of 4s2 1S0—4s4p 3P0 transition for Ga+ ion

LOU Zongshuai; WANG Yuefei; KANG Boyi; LI Rui; ZHANG Wenjun; WEI Yuanfei; BU Minglu; CAI Yiyu

doi:10.7498/aps.74.20250125

Abstract

The transition of Ga⁺ ions from 4s² ¹S₀ to 4s4p ³P₀ has advantages such as a high quality factor and a small motional frequency shift, making it suitable as a reference for precision measurement experiments like optical clocks. Calculating the dynamic polarizability of 4s² ¹S₀—4s4p ³P₀ transition for Ga⁺ ion is of great significance for exploring the potential applications of the Ga⁺ ion in the field of quantum precision measurement and for testing atomic and molecular structure theories. In this paper, the dynamic polarizability of the Ga⁺ ion 4s² ¹S₀—4s4p ³P₀ transition is theoretically calculated using the relativistic configuration interaction plus many-body perturbation (RCI+MBPT) method. The “tune-out” wavelengths for the 4s² ¹S₀ state and the 4s4p ³P₀ state, as well as the “magic” wavelength of the 4s² ¹S₀—4s4p ³P₀ transition, are also computed. It is observed that the resonant lines situated near a certain “turn-out” and “magic” wavelength can make dominant contributions to the polarizability, while the remaining resonant lines generally contribute the least. These “tune-out” and “magic” wavelengths provide theoretical guidance for precise measurements, which is important for studying the atomic structure of Ga⁺ ions. The accurate determination of the difference in static polarizability between the 4s² ¹S₀ and 4s4p ³P₀ states is of significant importance. Additionally, based on the “polarizability scaling” method, this work also discusses how the theoretical calculation errors in static polarizability measurements vary with wavelength, which provides theoretical guidance for further determining the static polarizability of the 4s² ¹S₀ and 4s4p ³P₀ states with high precision. This is crucial for minimizing the uncertainty of the blackbody radiation (BBR) frequency shift in Ga⁺ optical clock and suppressing the systematic uncertainty.

Keywords:

Authors and contacts

1.
School of Physics and Electronic Information, Weifang University, Weifang 261061, China
2.
School of Physical Science, Nankai University, Tianjin 300071, China
3.
School of Physics, Beihang University, Beijing 100191, China
4.
Macao Institute of Materials Science and Engineering (MIMSE), Faculty of Innovation Engineering, Macau University of Science and Technology, Macau 999078, China
5.
Macau University of Science and Technology Zhuhai MUST Science and Technology Research Institute, Zhuhai 519031, China
6.
University of Chinese Academy of Sciences, Beijing 100049, China

Corresponding author: ZHANG Wenjun, 20210018@wfu.edu.cn ; WEI Yuanfei, weiyuanfei@must.edu.mo ; CAI Yiyu, yycai@must.edu.mo

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12304151, 12404421), the Science and Technology Development Fund, Macao SAR (Grant No. 0024/2024/RIB1), the Natural Science Foundation of Shandong Province, China (Grant No. ZR2022QA085), and the Innovation and Entrepreneurship Training Program for College Students of Weifang University, China (Grant No. S202411067021).

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References

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Cited By

图 1 Ga⁺离子的4s² ¹S₀态和4s4p ³P₀态的动态极化率曲线

Figure 1. Dynamic polarizabilities α(ω) of the 4s² ¹S₀ and 4s4p ³P₀ states in Ga⁺ ion.

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图 2 Ga⁺离子的4s² ¹S₀态和4s4p ³P₀态静态极化率之差的理论计算误差随波长变化曲线

Figure 2. Theoretical calculation error of the difference in static polarizability between the 4s² ¹S₀ state and the 4s4p ³P₀ state of Ga⁺ ion as a function of wavelength.

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表 1 Ga⁺离子的最低的23个能级的能量值 (cm^–1), 第1行数值为基态的绝对能量值, 其他行数值为激发态相对于基态的激发能

Table 1. Energies of 23 lowest energy levels of Ga⁺ ion (cm^–1), the value in the first row represents the absolute energy of the ground state, while the values in the other rows represent the excitation energies of the excited states relative to the ground state.

State	RCI	RCI+MBPT	NIST	Diff./%	Refs.
4s² ¹S₀	396252.81	412181.94	413285.38	–0.27	413285.41^CICP^[5]
4s4p ³P₀	43174.90	47338.76	47367.55	–0.060	47367.57 ^{CICP [5]}; 47032 ^{MCDHF [35]}; 47368^Expt^[36]
4s4p ³P₁	43584.84	47792.83	47814.114	–0.044	47469^{MCDHF [35]}; 47814 ^Expt^[36]
4s4p ¹P₁	68389.17	70709.25	70701.427	0.011	70701.42 ^{CICP [5]}; 70455^MCDHF^[35]; 70701^Expt^[36]
4s5s ³S₁	96653.91	102623.02	102944.595	–0.31	100749.90 ^CICP^[5]; 102665^{MCDHF [35]}; 102945^Expt^[36]
4s4d ³D₁	106905.35	113471.29	113815.885	–0.30	113815.87 ^CICP^[5]; 113305^MCDHF^[35]; 113816^Expt^[36]
4p² ³P₁	109300.03	115272.94	115224.47	–0.042	115224.49 ^{CICP [5]}; 114590^{MCDHF [35]}; 115224^Expt^[36]
4s5p ³P₁	111880.88	118110.59	118518.461	–0.34	118236^MCDHF^[35]; 118518^Expt^[36]
4s5p ¹P₁	114324.78	120211.72	120550.431	–0.28	120715.81 ^CICP^[5]; 120322 ^MCDHF^[35]; 120550 ^{Expt [36]}
4s6s ¹S₀	126011.01	132559.65	133010.30	–0.34	130793.68 ^CICP^[5]; 133517^MCDHF^[35]; 133741^Expt^[36]
4s5d ³D₁	129989.22	136706.55	137157.524	–0.33	137155.79 ^{CICP [5]}; 136759^{MCDHF [35]}; 137157^Expt^[36]
4s6p ³P₁	132039.60	138646.98	—	—	—
4s6p ¹P₁	132671.60	139209.74	—	—	—
4s7s ¹S₀	138282.16	145005.12	145494.205	–0.34	145176^MCDHF^[35]; 145494^Expt^[36]
4s6d ³D₁	140241.35	147033.96	147520.34	–0.33	—
4s8s ¹S₀	144640.75	151383.73	151923.93	–0.36	—
4s7d ³D₁	145733.76	152563.77	153064.92	–0.33	—
4s7p ³P₁	141291.01	148033.38	—	—	—
4s7p ¹P₁	141504.08	148239.62	—	—	—
4s8p ³P₁	146349.69	153163.32	—	—	—
4s8p ¹P₁	146429.38	153251.11	—	—	—
4s9s ¹S₀	148859.67	154959.21	—	—	—
4s8d ³D₁	149030.92	155885.88	156386.7	–0.32	—

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表 2 使用RCI+MBPT方法得到的Ga⁺离子 4s² ¹S₀态和4s4p ³P₀态的电偶极约化跃迁矩阵元 (a.u.)

Table 2. Reduced matrix elements of E1 transition for the 4s² ¹S₀ and 4s4p ³P₀ of Ga⁺, obtained by using the RCI+MBPT methods (a.u.).

Method	RCI		RCI+MBPT		Recommend	Refs.
Guage	Length	Velocity	Length	Velocity	Recommend	Refs.
4s² ¹S₀—4s4p ³P₁	0.055752	0.059065	0.064832	0.072400	0.065 (17)	0.0744 ^[34]; 0.0895 ^{MCDHF [35]}; 0.0802^{RRPA [38]}
4s² ¹S₀—4s4p ¹P₁	3.0918	3.0507	2.8480	3.0361	2.84 (24)	2.69 ^CICP^[5]; 2.87 ^[34]; 2.68^{MCDHF [35]}; 2.81^{MP [37]}; 2.79^{RRPA [38]}; 2.71^{MCHF [39]}; 2.78 (11)^{Expt [40]}
4s² ¹S₀—4s5p ³P₁	0.000595	0.000455	0.00594	0.000400	0.006 (6)	—
4s² ¹S₀—4s5p ¹P₁	0.28458	0.27302	0.15426	0.23982	0.15 (13)	0.138 ^[34]
4s² ¹S₀—4s6p ³P₁	0.00229	0.00237	0.00815	0.00272	0.0082 (59)	—
4s² ¹S₀—4s6p ¹P₁	0.0741	0.0684	0.0868	0.0594	0.087 (27)	—
4s² ¹S₀—4s7p ¹P₁	0.0264	0.0229	0.0143	0.0205	0.014 (12)	—
4s² ¹S₀—4s7p ³P₁	0.00232	0.00249	0.00803	0.00298	0.008 (6)	—
4s² ¹S₀—4s8p ¹P₁	0.00986	0.00753	0.0183	0.00768	0.02 (1)	—
4s² ¹S₀—4s8p ³P₁	0.00202	0.00233	0.00787	0.003104	0.008 (6)	—
4s4p ³P₀—4s5s ³S₁	0.93304	0.92359	0.92029	0.90827	0.920 (25)	0.974^CICP^[5]; 1.00^{MCDHF [35]}; 0.982^{MP [37]}
4s4p ³P₀—4s4d ³D₁	2.1286	2.0871	2.0181	2.0670	2.02 (11)	2.00^CICP^[5]; 2.08^[34]; 2.02^{MCDHF [35]}; 2.05^{MP [37]}
4s4p ³P₀—4p² ³P₁	1.8133	1.7818	1.6470	1.7695	1.65 (17)	1.64^CICP^[5]; 1.64^{MCDHF [35]}; 1.72^{MP [37]}
4s4p ³P₀—4s6s ³S₁	0.26761	0.26392	0.26890	0.26353	0.269 (5)	0.214^CICP^[5]; 0.205^{MCDHF [35]}; 0.217^{MP [37]}
4s4p ³P₀—4s5d ³D₁	0.66449	0.64536	0.62443	0.65590	0.62 (4)	0.461^CICP^[5]; 0.442^{MCDHF [35]}; 0.479^{MP [37]}
4s4p ³P₀—4s7s ³S₁	0.14979	0.14750	0.15084	0.14798	0.151 (3)	—
4s4p ³P₀—4s6d ³D₁	0.36574	0.35358	0.33986	0.36272	0.340 (26)	—
4s4p ³P₀—4s8s ³S₁	0.10206	0.10042	0.10065	0.098613	0.1021 (34)	—
4s4p ³P₀—4s7d ³D₁	0.24440	0.23567	0.22522	0.24281	0.225 (19)	—
4s4p ³P₀—4s9s ³S₁	0.088839	0.087378	0.078599	0.078310	0.079 (11)	—
4s4p ³P₀—4s8d ³D₁	0.18107	0.17433	0.16616	0.18060	0.181 (14)	—

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表 3 相关跃迁对态4s² ¹S₀和态4s4p ³P₀的静态极化率α(0)的贡献

Table 3. Contributions of individual transitions to the static polarizabilities α(0) for 4s² ¹S₀ and 4s4p ³P₀.

Transition	Contributions	Refs.
$ \alpha \left(0\right)( $4s² ¹S₀$ ) $	—	—
4s² ¹S₀—4s4p ³P₁	0.013 (7)	—
4s² ¹S₀—4s4p ¹P₁	16.69 (2.82)	16.601^[5]
4s² ¹S₀—4s5p ³P₁	4.4 (4.4)×10^–5	—
4s² ¹S₀—4s5p ¹P₁	0.027 (27)	0.016^[5]
4s² ¹S₀—4s6p ³P₁	7.1 (7.1)×10^–5	—
4s² ¹S₀-—4s6p ¹P₁	0.008 (5)	—
4s² ¹S₀—4snp ³P₁, n = 7—8	0.00012 (12)	—
4s² ¹S₀—4snp ¹P₁, n = 7—9	0.0006 (4)	—
Core	1.24 (1)^[5]	1.24 (1)^[5]
Total	17.98 (2.82)	17.95 (34)^[5]
$ \alpha \left(0\right)( $4s4p ³P₀$ ) $	—	—
4s4p ³P₀—4s5s ³S₁	2.23 (12)	2.257^[5]
4s4p ³P₀—4s4d ³D₁	8.98 (98)	8.668^[5]
4s4p ³P₀—4p² ³P₁	5.87 (1.21)	5.945^[5]
4s4p ³P₀—4s6s ³S₁	0.124 (5)	—
4s4p ³P₀—4s5d ³D₁	0.62 (8)	—
4s4p ³P₀—4sns ³S₁, n = 7—9	0.057 (13)	—
4s4p ³P₀—4snd ³D₁, n = 6—8	0.283 (29)	—
Core	1.24 (1)^[5]	1.24 (1)^[5]
Total	19.41 (1.56)	19.58 (38)^[5]
$ {{\Delta }}\alpha \left(0\right) $	1.43 (3.2)	1.63 (72)^[5]

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表 4 “幻零”波长和“魔幻”波长的不确定度评估表, 占主导贡献的部分在表中加重

Table 4. The uncertainty evaluation table for the ‘Tune-out’ wavelength and the ‘Magic’ wavelength, the dominant contribution is emphasized in the table.

Transition	‘Tune-out’ wavelengths					‘Magic’ wavelengths
Transition	209.101	176.42	148.61	117.197	113.09	209.286	168.1	148.27	116.38	106.7
4s² ¹S₀—4s4p ³P₁	0.025					0.085	<0.1	<0.001	<0.001	<0.1
4s² ¹S₀—4s4p ¹P₁	0.0076					0.049	2.7	0.049	0.16	2.5
4s² ¹S₀—4s5p ¹P₁	<0.001					<0.01	<0.1	<0.001	<0.01	0.11
4s² ¹S₀—4s6p ¹P₁	<0.001					<0.01	<0.1	<0.001	<0.01	<0.1
4s4p ³P₀—4s5s ³S₁		0.17	<0.001	0.0017	0.0073	<0.01	0.27	<0.001	<0.01	<0.1
4s4p ³P₀—4s4d ³D₁		0.19	0.077	0.034	0.0095	0.026	1.6	0.047	0.036	0.68
4s4p ³P₀—4p² ³P₁		0.22	0.15	0.048	0.13	0.030	1.7	0.16	0.047	0.96
4s4p ³P₀—4s6s ³S₁		<0.01	<0.001	<0.001	0.0065	<0.01	<0.1	<0.001	<0.01	<0.1
4s4p ³P₀—4s5d ³D₁		<0.01	<0.001	<0.001	<0.001	<0.01	<0.1	<0.001	<0.01	<0.1
Others	<0.001	<0.01	<0.01	<0.001	<0.01	<0.01	<0.1	<0.01	<0.01	<0.1
Total	0.026	0.34	0.16	0.059	0.16	0.11	3.6	0.17	0.17	2.8

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表 5 “幻零”波长和“魔幻”波长处相关跃迁对两个光钟态4s² ¹S₀和4s4p ³P₀的动力学极化率的贡献, 占绝对贡献的部分在表中加重

Table 5. Breakdown of the contributions of individual transitions to the dynamic polarizabilities at the “tune-out” wavelengths and “Magic” wavelengths for the 4s² ¹S₀ and 4s4p ³P₀ clock states of Ga⁺, the dominant contribution is emphasized in the table.

Transition	‘Tune-out’ wavelengths					‘Magic’ wavelengths
Transition	209.101	176.42	148.61	117.197	113.09	209.286	168.1	148.27	116.38	106.7
4s² ¹S₀—4s4p ³P₁	–32.06	–0.03	–0.01	–0.01	–0.01	9.51	–0.02	–0.01	–0.01	0.00
4s² ¹S₀—4s4p ¹P₁	30.77	46.73	177.29	–36.57	–29.58	30.72	57.11	184.64	–34.99	–22.00
4s² ¹S₀—4s5p ¹P₁	0.03	0.03	0.04	0.05	0.06	0.03	0.04	0.04	0.06	0.06
4s² ¹S₀—4s6p ¹P₁	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01
Others	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01
Core	1.24	1.24	1.24	1.24	1.24	1.24	1.24	1.24	1.24	1.24
Total	0.00	47.99	178.58	–35.27	–28.27	41.52	58.39	185.93	–33.68	–20.68
4s4p ³P₀—4s5s ³S₁	8.59	–55.40	–4.78	–1.64	–1.46	8.54	–15.34	–4.71	–1.60	–1.21
4s4p ³P₀—4s4d ³D₁	18.64	33.00	–352.26	–13.85	–11.66	18.61	45.19	–297.43	–13.37	–9.07
4s4p ³P₀—4p² ³P₁	11.66	19.42	353.37	–10.10	–8.41	11.64	25.33	484.39	–9.73	–6.46
4s4p ³P₀—4s6s ³S₁	0.18	0.22	0.32	16.76	–1.87	0.18	0.24	0.33	–18.78	–0.62
4s4p ³P₀—4s5d ³D₁	0.87	1.0	1.43	6.46	20.77	0.87	1.12	1.44	7.43	–6.93
Others	0.44	0.52	0.68	1.13	1.39	0.44	0.61	0.68	1.13	2.37
Core	1.24	1.24	1.24	1.24	1.24	1.24	1.24	1.24	1.24	1.24
Total	41.62	0.00	0.00	0.00	0.00	41.52	58.39	185.93	–33.68	–20.68
Diff.						0.00	0.00	0.00	0.0	0.0

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Theoretical calculation of dynamic polarizability of 4s^{2 1}S₀—4s4p³P₀ transition for Ga⁺ ion

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Corresponding author: ZHANG Wenjun, 20210018@wfu.edu.cn ; WEI Yuanfei, weiyuanfei@must.edu.mo ; CAI Yiyu, yycai@must.edu.mo

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