Precision measurement based on rovibrational spectrum of cold molecular hydrogen ion

Zhang Qian-Yu; Bai Wen-Li; Ao Zhi-Yuan; Ding Yan-Hao; Peng Wen-Cui; He Sheng-Guo; Tong Xin

doi:10.7498/aps.73.20241064

Abstract

A molecular hydrogen ion HD⁺, composed of a proton, a deuteron, and an electron, has a rich set of rovibrational transitions that can be theoretically calculated and experimentally measured precisely. Currently, the relative accuracy of the rovibrational transition frequencies of the HD⁺ molecular ions has reached 10^–12. By comparing experimental measurements with theoretical calculations of the HD⁺ rovibrational spectrum, the precise determination of the proton-electron mass ratio, the testing of quantum electrodynamics(QED) theory, and the exploration of new physics beyond the standard model can be achieved. The experiment on HD⁺ rovibrational spectrum has achieved the highest accuracy (20 ppt, 1 ppt = 10^–12) in measuring proton-electron mass ratio. This ppaper comprehensively introduces the research status of HD⁺ rovibrational spectroscopy, and details the experimental method of the high-precision rovibrational spectroscopic measurement based on the sympathetic cooling of HD⁺ ions by laser-cooled Be⁺ ions. In Section 2, the technologies of generating and trapping both Be⁺ ions and HD⁺ ions are introduced. Three methods of generating ions, including electron impact, laser ablation and photoionization, are also compared. In Section 3, we show the successful control of the kinetic energy of HD⁺ molecular ions through the sympathetic cooling, and the importance of laser frequency stabilization for sympathetic cooling of HD⁺ molecular ions. In Section 4, two methods of preparing internal states of HD⁺ molecular ions, optical pumping and resonance enhanced threshold photoionization, are introduced. Both methods show the significant increase of population in the ground rovibrational state. In Section 5, we introduce two methods of determining the change in the number of HD⁺ molecular ions, i.e. secular excitation and molecular dynamic simulation. Both methods combined with resonance enhanced multiphoton dissociation can detect the rovibrational transitions of HD⁺ molecular ions. In Section 6, the experimental setup and process for the rovibrational spectrum of HD⁺ molecular ions are given and the up-to-date results are shown. Finally, this paper summarizes the techniques used in HD⁺ rovibrational spectroscopic measurements, and presents the prospects of potential spectroscopic technologies for further improving frequency measurement precision and developing the spectroscopic methods of different isotopic hydrogen molecular ions.

Keywords:

Authors and contacts

1.
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
2.
University of Chinese Academy of Sciences, Beijing 100049, China
3.
Wuhan Institute of Quantum Technology, Wuhan 430074, China

Corresponding author: Peng Wen-Cui, wencuipeng@wipm.ac.cn ; He Sheng-Guo, hesg@wipm.ac.cn ;

Funds: Project supported by the National Key R&D Program of China (Grant No. 2021YFA1402103) and the National Natural Science Foundation of China (Grant No. 12393825).

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Cited By

图 1 质子电子质量比常数测量

Figure 1. The measurements of proton-electron mass ratio.

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图 2 H₂分子、HD⁺分子离子、反质子氦光谱实验确定强子与强子相互作用的第五种力的上限^[26]

Figure 2. Spectroscopic measurement of H₂ molecule, HD⁺ molecular ion, and antiprotonic helium constrains on the fifth force of hadron-hadron interaction^[26].

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图 3 激光溅射影响下的离子阱电压变化

Figure 3. The change of voltage on the ion trap under the influence of laser ablation.

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图 4 Be原子和HD分子光电离的相关能级　(a) Be原子光电离的相关能级, 黑色箭头表示双光子非共振电离, 紫色箭头表示[1+1]双光子共振电离, 蓝色箭头表示[2+1]三光子共振电离; (b) HD分子光电离的相关能级, 3个蓝色箭头组合表示[2+1]三光子共振电离, 两个蓝色箭头和一个红色箭头组合表示[2+1']三光子共振电离

Figure 4. The related levels of Be atom and HD molecule photoionization: (a) The relevant energy levels for photoionization of the Be atom, black arrows indicate two-photon non-resonant ionization, purple arrows indicate [1+1] two-photon resonant ionization, and blue arrows indicate [2+1] three-photon resonant ionization; (b) the relevant energy levels for photoionization of the HD molecule, three blue arrows represent [2+1] three-photon resonant ionization, and combination of two blue arrows and a red arrow represent [2+1'] three-photon resonant ionization.

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图 5 双组分库仑晶体径向分离示意图, 该图视角为径向截面图, 其中M₁为内层被协同冷却离子的质量, M₂为外层冷却剂离子的质量, b₂和a₂分别为质量为M₂离子壳层的外径和内径, b₁为内层离子的外径

Figure 5. The schematic diagram of a bi-component Coulomb crystal in the view of a radial cross-section, where M₁ is the mass of the sympathetically cooled ions in the inner shell, M₂ is the mass of the laser-cooled ions in the outer shell, b₂ and a₂ are the radius of the outer and inner surface of the ions with the mass of M₂, respectively, b₁ is the radius of the outer surface of the ions with the mass of M₁.

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图 6 Be⁺离子激光冷却相关能级

Figure 6. Related energy levels of Be⁺ laser cooling.

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图 7 313 nm激光器稳频实验装置示意图^[68]

Figure 7. Schematic of the experimental setup for frequency stabilization of the 313 nm laser^[68].

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图 8 将冷却激光的锁定在ULE腔(a)和波长计(b)上的Be⁺库仑晶体的图像^[68], 图像时间点在激光频率锁定后的2, 40, 80, 120, 160, 200, 240 s

Figure 8. The images of Be⁺ Coulomb crystals with cooling laser locked to ULE cavity (a) or wavelength meter (b)^[68], the image time points are at 2, 40, 80, 120, 160, 200, 240 s after the laser frequency is locked.

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图 9 利用光泵浦方法后HD⁺振动基态的转动态分布^[77], 红色、黑色、蓝色的数据点分别为为使用光泵浦方法后的实验采集的信号、模拟的信号、模拟的态布居数, 灰色数据点为没有使用光泵浦方法实验采集的信号

Figure 9. Rotational-state distribution of the vibrational ground state after applying the optical pumping scheme^[77], the red, black, and blue data points represent the experimental collected signals, simulated signals, and simulated population after using the optical pumping method, respectively, the gray data points represent the experimental collected signals without using the optical pumping method.

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图 10 不同201 nm激光能量下[2+1]和[2+1' ]两种REMPI过程产生的HD⁺离子信号^[48]

Figure 10. HD⁺ ion signals produced by two processes under the different power of 201 nm laser^[48].

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图 11 宏运动激发装置示意图, 蓝色、红色小球分别为激光冷却的离子和宏运动激发加热的暗离子

Figure 11. Schematic diagram of secular excitation, the blue and red balls represent coolant ions and dark ions heated by secular excitation, respectively.

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图 12 HD⁺分子离子宏运动激发扫频信号^[48], 红线、蓝线分别为HD⁺分子离子解离前后的扫频信号

Figure 12. The change of fluorescent signals when sweeping frequency of the secular excitation for HD⁺ molecular ions^[48], the red and blue lines represent the fluorescent signals before and after the dissociation of HD⁺ molecular ions, respectively.

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图 13 通过分子动力学模拟确定离子阱内装载的HD⁺分子离子的数量^[35], 比较实验与模拟图像的晶体结构, 其内部暗核的形状和尺寸与HD⁺离子的数量有关(红框内), 含有(15 ± 1)个HD⁺分子离子的模拟图像与实验图像最为符合

Figure 13. Determination of the number of sympathetically cooled HD⁺ ions by molecular dynamics simulation^[35], comparing the crystal structures in the experimental and simulated images, the shape and size of the internal dark core are related to the number of HD⁺ ions (within the red square), and the simulated image containing (15 ± 1) HD⁺molecular ions is the most consistent with the experimental image.

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图 14 HD⁺分子离子共振增强多光子解离(REMPD)过程　(a)解离前后二维电子概率密度ρ的分布图, 其色度正比于lgρ; (b) REMPD过程的相关能级; (c)为转跃迁(v, L):(0, 0)→(6, 1)相关的超精细结构能级图, 其中的量子数F, S, J是电子自旋s_e、质子自旋I_p、氘核自旋I_d和分子旋转N按耦合强弱通过以下耦合方案形成, J = S+L, 其中S = F+I_d, F = s_e+I_p, 4种不同颜色带箭头的线表示符合ΔF = 0, ΔS = 0选择定则的超精细跃迁

Figure 14. Resonance enhanced multiphoton dissociation (REMPD) process of HD⁺ molecular ions: (a) The distribution of electrons two-dimensional probability density ρ before and after dissociation, and its chromaticity is proportional to log₁₀ρ; (b) the relevant energy levels of the REMPD process; (c) the relevant hyperfine structure levels of the rovibrational transition (v, L):(0, 0)→(6, 1), the quantum numbers refer to the following coupling scheme for the electron spin s_e, proton spin I_p, deuteron spin I_d, and molecular rotation N: J = S+L, where S = F+I_d, F = s_e+I_p. The four strongest hyperfine transitions for ΔF = 0 and ΔS = 0 are represented by four different colored arrows.

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图 15 HD⁺分子离子振转跃迁 (v, L):(0, 0)→(6, 1) 光谱实验装置示意图

Figure 15. Schematic diagram of the experimental setup for the HD⁺ molecular ion rovibrational transition (v, L): (0, 0)→(6, 1) spectrum.

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图 16 HD⁺分子离子振转跃迁 (v, L):(0, 0)→(6, 1) 光谱, 数据点为8次测量的平均值, 垂直误差棒为8次测量的标准差

Figure 16. Spectrum of the (v, N): (0, 0) → (6, 1) HD⁺molecular ion rovibrational transition, data points are the average of 8 measurements, and vertical error bars represent the standard deviations.

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表 1 基本物理常数对HD⁺分子离子振转跃迁频率不确定度的影响^[1]

Table 1. Influences of fundamental physical constants on the uncertainty of the vibrational transition frequencies of HD⁺ molecular ions^[1].

	R_∞	μ_pe	μ_de	r_p	r_d	α
当前物理量的相对不确定度	1.9 ppt	60 ppt	35 ppt	0.002	350 ppm	0.15 ppb
频率值对物理量的敏感系数	~1	~0.1	~0.01	~10^–9	~10^–9	~10^–6
物理量对频率相对不确定度影响	~1 ppt	~10 ppt	~1 ppt	~1 ppt	~0.1 ppt	~0.1 ppq
注: 表中ppm(part per million), ppb(part per billion), ppt(part per trillion), ppq(part per quadrillion)分别表示10^–6, 10^–9, 10^–12, 10^–15.

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表 2 QED理论计算的HD⁺振转跃迁 (v, L):(0, 0)→(6, 1)各项贡献

Table 2. Contribution of QED theory calculation of HD⁺ rovibrational transition (v, L):(0, 0)→(6, 1).

	频率/MHz	贡献项
v_nr	303393178.0114(8)	三体非相对论薛定谔方程能量
v_nuc	–0.096(1)	有限核效应
v_α2	4571.102 59(3)	Breit–Pauli近似中的相对论修正
v_α3	–1 234.8136(3)	辐射修正领头项
v_α4	–8.9607(3)	1圈、2圈辐射修正; 高阶的相对论修正
v_α5	0.537(1)	3圈的辐射修正; Wichmann–Kroll贡献项
v_α6	0.003(5)	高阶的辐射修正
v_tot	303303396505.784(5)

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