Fundamental physical constants appear in mathematical formulas describing fundamental interactions in physical theories. Comparison of the same fundamental physical constant measured in different areas of physics can test the universality of the constant as well as the self-consistency of physical laws. The precision of a fundamental physical constant determines the accuracy of a theoretical prediction. Moreover, a space– time variation of a fundamental physical constant can reveal the domain of validity of a theory and provides further evidence for constructing a more general theory beyond the “ standard model of particle physics” .

The proton-to-electron mass ratio m_p/m_e is one of the fundamental physical constants, which is relevant to the quantum chromodynamic scale.^[1] Its high precision value is of great importance for testing the standard model of particle physics, and for exploring new physics beyond the standard model.^{[2– 5]} Currently, the Committee on Data for Science and Technology (CODATA) recommended value for m_p/m_e is 1836.15267245 with the relative standard uncertainty of 4.1 × 10^{− 10}.^[6] This value is determined indirectly by measuring the masses of the proton and the electron separately. Since the relative uncertainties of m_p and m_e in the unified atomic mass unit (u) are 8.9 × 10^{− 11} and 4.0 × 10^{− 10[6]} respectively, the uncertainty in m_p/m_e is mainly due to the uncertainty of the electron mass. In fact, direct experimental measurement of m_e is challenging. One way to extract the electron mass is to compare the experimentally measured g factor of the electron in a hydrogenlike ion with the corresponding theoretical calculation based on quantum electrodynamic (QED) theory. This approach has been the most accurate method in the last decade.^[7] Recently, Max Planck Institute for Nuclear Physics^[8] in Germany has made a breakthrough on measuring the g factor of the electron in the hydrogenlike ¹²C⁵⁺ and has determined the electron mass with the relative precision of 3 × 10^{− 11}, a factor of 13 improvement over the CODATA recommended value.

In addition to indirect measurements, the proton-to-electron mass ratio can also be measured directly. Figure  1 shows the precision of m_p/m_e measured directly over the past 40 years. In earlier times, m_p/m_e was determined by measuring the cyclotron frequencies of a free proton and a free electron in the same magnetic field, ^{[9– 16]} where the energy scales involved were in the high energy range of GeV to TeV. In the last decade, due to rapid development of new techniques, such as femtosecond optical comb and laser cooling and trapping, atomic and molecular spectroscopy can be measured to an unprecedented high precision at the level of Hz or sub-Hz, which has created many opportunities to test physical laws in low energy eV region.^{[17, 18]} With an increase of precision, small shifts in atomic and molecular energy levels can reveal small but subtle physical effects, such as high-order relativistic and QED (radiative) corrections, contributions from unknown particles beyond the standard model of particle physics, and effects due to symmetry breakings.^{[19, 20]} A comparison between spectroscopic measurements and theoretical calculations can be used to determine fundamental physical constants involved, such as the proton-to-electron mass ratio m_p/m_e (or m_p̄ /m_e)^{[21– 23]} (see the last three measurements in Fig.  1). Direct and indirect measurements of m_p/m_e can not only provide diversified methods but also check self-consistency among the proton mass m_p, the electron mass m_e, and the ratio m_p/m_e. The self-consistency can ensure that these three constants are fundamental; furthermore a precise mass ratio m_p/m_e can impose a strict constraint on the individual values of m_p and m_e. A violation of this self-consistency may indicate the existence of new physics.

Fig.  1.

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Fig.  1. Relative uncertainty of mp/me measured directly. These measurements were carried out at Johannes Gutenberg-Universitä t Mainz (JGU), [9, 10] University of Washington (UWash), [11– 13] National Institute of Standards and Technology (NIST), [14] Harvard University (Harvard), [16] the European Organization for Nuclear Research (CERN), [21, 22] and Heinrich-Heine-Universitä t Dü sseldorf (UDuss).[23]

So far, several experiments have been performed to measure m_p/m_e (or m_p̄ /m_e) directly by studying spectroscopic data. These high precision measurements focus on the antiprotonic helium and the hydrogen molecular ions which can be used to directly determinate m_p̄ /m_e and m_p/m_e, respectively. Moreover, a comparison between m_p̄ /m_e and m_p/m_e can test the CPT (charge, parity, and time reversal) symmetry variation.

Since its establishment, the atomic spectroscopy and collisions using slow antiprotons (ASACUSA) collaboration has been working on precision spectroscopies of atoms containing the antiproton, ^[20] such as the antihydrogen atom and the antiprotonic helium atom (p̄ He⁺ ), at the European organization for nuclear research (CERN). In 2006, ASACUSA measured 12 transition frequencies of the antiprotonic helium at the fractional precision of 9– 16  ppb (1  ppb ≡ 10^{− 9}), yielding the value of the antiproton-to-electron mass ratio m_p̄ /m_e = 1836.152674(5), ^[21] in combination with the three-body QED calculations. In 2011, the two-photon spectroscopy of the antiprotonic helium containing 3 transition frequencies was reported with the achieved fractional precision of 2.3– 5  ppb, which leads to the determination of m_p̄ /m_e = 1836.1526736(23).^[22] The p̄ He⁺ atom can be generated by slowing down antiprotons in low temperature helium. Soon after its generation, the p̄ He⁺ atom is then cooled to superfluid temperature T ∼ 1.5  K for spectroscopic measurements within its short-lived 1-μ s-scale lifetime (see Ref.  [24] and references therein). Currently, the ASACUSA of CERN is the only group that is performing the measurements of p̄ He⁺ transitional frequencies.

Compared to p̄ He⁺ , rovibrational states of hydrogen molecular ions ( and HD⁺ ) have longer lifetimes (ms to days) and can be sympathetically cooled to lower temperature of about 10  mK.^[25] In 2007, Schiller and coworkers measured the transition frequency (v, L) : (0, 2) → (4, 3) in HD⁺ to a relative precision of 2  ppb, which is consistent with the value of QED calculation to within 5  ppb.^[23] In the above, v is the vibrational quantum number and L is the rotational quantum number. Later, they measured another HD⁺ transition frequency (v, L) : (0, 0) → (1, 1) with a relative uncertainty being 1.1  ppb.^[26] It should be pointed out that the theoretical calculations of the hydrogen molecular ions and the antiprotonic helium used in the above determinations are mainly due to the work by Korobov.^{[28– 32]} Although at this moment the direct determination of m_p/m_e is expected to be still less precise than the indirect determination, it will provide an important reference to the corresponding CODATA recommended value.

This article is organized as follows. The background of high-precision spectroscopy is introduced in Section  1.2 for hydrogen molecular ions, which includes theory and experiment. In Section  2, theoretical formulation for solving a three-body bound-state problem is presented, including the nonrelativistic energy eigenvalue problem and the leading- and higher-order relativistic and QED corrections. In Section  3, an experimental scheme is described for measuring the rovibrational transition frequency v : 0 → 6 in HD⁺ , which is currently underway at the Wuhan Institute of Physics and Mathematics. In Section  4, a summary is given.

We begin with the nonrelativistic Hamiltonian of a three-body Coulombic system,

where m_i, q_i, and R_i are the mass, charge, and position vector of the ith particle, respectively. The total angular momentum operator, which commutes with H₀, is

For the purpose of isolating the center-of-mass motion, we make the following coordinate transformations:

where is the total mass of the system and X is the position vector of the center of mass of the system. In terms of the new coordinates r₁, r₂, and X, H₀ and L can be written in the form of

where r₁₂ = | r₁ − r₂| and μ _i = m_im₀/(m_i + m₀). Since X is cyclic, we can simply neglect X. Therefore in the center-of-mass frame,

This Hamiltonian can be thought of as the one describing two “ effective particles” moving relative to particle 0 sitting at the origin. It is noted that one has freedom to choose any particle as particle 0. For the hydrogen molecular ions and HD⁺ , we choose the nucleus of larger mass as particle 0, the electron as particle 1, and the other nucleus as particle 2.

The Schrö dinger equation for the system is therefore expressed as H₀ψ ₀ = E₀ψ ₀, where E₀ is the energy eigenvalue and ψ ₀ is the corresponding eigenfunction. For solving the eigenvalue problem for H₀ variationally, we seek the common eigenstates of the commuting observables {H₀, L², L_z, Π }, where Π is the parity operator, using the following basis set in Hylleraas coordinates:

In the above, is the vector coupled product of spherical harmonics for particles 1 and 2 forming the common eigenstate of L², L_z, and Π with the corresponding eigenvalues of L(L + 1), M_L, and (− 1)^{l₁+ l₂}, respectively

The vibrational degrees of freedom between two the nuclei should be sufficiently represented in the basis set, according to the picture of the Born– Oppenheimer approximation, as pointed out by Bhatia and Drachman.^[89] They suggested to use Gaussian-like function ρ ^N e^{− bρ} to simulate the vibrational modes, where ρ is the distance between the two nuclei, and N and b are two big numbers satisfying b ≈ N/2. Since we choose one nucleus as particle 0 and the other as particle 2, the vibrational part in our basis can be described by . For low rovibrational states, the lowest power of r₂ is set to be j₀ = 35 and 45 for and HD⁺ , respectively. Also i, j, and k in Eq.  (9) are non-negative integers. Furthermore, the nonlinear parameters are initially chosen to be α ≈ 1 and β ≈ 15– 20.

If the two nuclei are identical, such as , the symmetry is preserved in the Hamiltonian for the permutation of the two nuclei. We can thus choose a basis which has the permutational symmetry explicitly adapted. On the other hand, we are also free to choose a basis by not explicitly imposing the permutational symmetry.^[53] We denote the permutation operation between the two identical nuclei by P_nuc, which implies that the eigenvalues of P_nuc are (− 1)^S_p with S_p = 0, 1, corresponding to the singlet and triplet states, respectively. Furthermore, L is also invariant under P_nuc. It is thus possible to find the common eigenstates for the commuting observables {H₀, L², L_z, Π , P_nuc}. For a given total angular momentum L, for a state of natural parity, the independent angular configurations (l₁, l₂) in a wavefunction are those satisfying the condition l₁ + l₂ = L

Each configuration (l₁, l₂) has its own nonlinear parameters α and β . Since the nuclei carry most of the angular momentum, the leading configuration (0, L) is divided into N_L sub-blocks, each having its own nonlinear parameters α and β , according to the following scheme:^[53] the nth block in the configuration (0, L) contains all the terms satisfying the following relations for the power j_n of r₂:

where

int[x] stands for the integer part of x, and Ω is an integer controlling the size of each block. The basis set is generated by including all terms such that

The basis set constructed this way can sufficiently represent the vibrational modes in the wavefunction. Moreover, if we include some terms with lower powers of r₂, the rate of convergence, along with the numerical stability in the wavefunction, can be further enhanced.^[56] Hence another small-size block with j₀ = l₂ is added to our basis set. A complete optimization is performed with respect to those nonlinear parameters by minimizing the energy eigenvalue. This involves the calculation of the energy derivative ∂ E/∂ α with respect to each nonlinear parameters.^[55]

The basic type of integrals that appear in our variational calculation are of the form

which can be evaluated analytically in terms of the hypergeometric function ₂F₁(a, b; c; x).^[90] The procedure for handling singular integrals appearing in the evaluation of the Breit operators can be found in Ref.  [91] where the singular parts are exactly canceled out with each other. Another method to deal with singular integrals was developed by Harris and coworkers^[92] where the singular parts are simply dropped out. It should be pointed out that these two methods are equivalent.

Table 1.

Table 1.

Table 1. Nonrelativistic energies for various rovibrational states in and HD+ . The mass ratios of mp/me = 1836.15267245 and md/me = 3670.4829652 are taken from the 2010 CODATA set.[6] (v, L) HD+ (0, 0) − 0.597139063079144251108031025395(3) − 0.597897968608892577899511752764(6) (0, 1) − 0.59687373878444226891335830(3) − 0.59769812819205956807874305(7) (0, 2) − 0.5963452054890752846020373(2) − 0.597299643351606083357910(2) (0, 3) − 0.595557638979926416186902(4) − 0.59670488276168354929401(3) (0, 4) − 0.5945171692384930253637(2) − 0.5959173422125540486957(3) (1, 0) − 0.5871556790955326878290194007(1) − 0.5891818295565784691823214177(4) (1, 1) − 0.5869043209185137007114739(5) − 0.588991111991646700668530(2) (1, 2) − 0.586403631527974188229395(6) − 0.58861082938937351713882(3) (1, 3) − 0.5856576118768781977862(2) − 0.5880432641623919793933(4) (1, 4) − 0.58467213422805030304(6) − 0.58729178437373152151(3) (2, 0) − 0.577751904414067229657555467(2) − 0.580903700217773751175193655(2) (2, 1) − 0.57751403405641093579582(1) − 0.5807218281203265529408(2) (2, 2) − 0.5770402371619496762028(3) − 0.5803591951992570177865(6) (2, 3) − 0.57633435021849557018(6) − 0.57981800202721379983(2) (2, 4) − 0.5754020032976371412(2) − 0.5791014955647967925(2) (3, 0) − 0.568908498729535152478035082(7) − 0.5730505465510793119298908(2) (3, 1) − 0.568683708258842165049(1) − 0.572877277093417222702(2) (3, 2) − 0.568235992970197563839(7) − 0.57253181032515666322(5) (3, 3) − 0.567569034832072494(5) − 0.5720162692316722661(2) (3, 4) − 0.56668823662821325(2) − 0.57133378604512134(2) (4, 0) − 0.5606092208480679590637575(2) − 0.565611042075852914783012(1) (4, 1) − 0.56039717139867397930(2) − 0.5654461662766091807(2) (4, 2) − 0.55997486481840974852(5) − 0.5651174497627543881(5) (4, 3) − 0.55934583822658360(2) − 0.564626942061051144(2) (4, 4) − 0.5585152816241481(3) − 0.5639776668665789(5)

Table 1. Nonrelativistic energies for various rovibrational states in and HD+ . The mass ratios of mp/me = 1836.15267245 and md/me = 3670.4829652 are taken from the 2010 CODATA set.[6]

The nonrelativistic energies of the hydrogen molecular ions and HD⁺ for various rovibrational states, together with the corresponding uncertainties, are presented in Table  1. The energy eigenvalues are converged to about 30 significant digits for the ground states of and HD⁺ . For excited rovibrational states, however, we can still obtain at least 16 significant digits, which is sufficient for our calculations of relativistic and radiative corrections.

The leading-order relativistic corrections due to the Breit operators are R_∞ α ². For a general atomic or molecular system, these corrections can be obtained by reducing the relativistic four-component wavefunction of the Dirac equation to a two-component one in its nonrelativistic form.^{[93– 96]} The major part of the R_∞ α ² corrections is due to the relativistic bound electron which, for the hydrogen molecular ion, has the form of

There also exist the so-called recoil corrections of R_∞ α ²(m_e/m_nuc) and R_∞ α ²(m_e/m_nuc)², which are caused by the finite nuclear masses. The first of this type is the retardation terms coming from the exchange of transverse photon

where

The correction can be understood as the interaction between the magnetic dipole moments of the particles involved, arising from the orbital motion of the charges (also called the orbit– orbit interaction). The next part is the relativistic kinetic energy correction for the two nuclei

which is rooted from the dependence of the relativistic mass on velocity: . Furthermore, there is the so-called Darwin term that arises from the nuclear spin-dependent recoil correction for a particle of spin, such as the proton. The Darwin term vanishes for the case of spin-0 or spin-1 nucleus, such as the ⁴He nucleus or the deuteron.^[97] Therefore, the Darwin term for is

and the Darwin term for HD⁺ is

The complete contribution of the leading-order relativistic correction is thus

For the case of , the permutation symmetry of the two protons implies that

The numerical values for the above discussed operators are presented in Table  2 for and in Table  3 and Table  4 for HD⁺ .

Table 2.

Table 2.

Table 2. Numerical values of the Breit operators 〈 δ (r1)〉 , , , R01, and R02 for , where v = 0− 4 and L = 0− 4. (v, L) 〈 δ (r1)〉 R01 R02 (0, 0) 0.206736476288664004(4) 6.2856600593074270(7) 79.79764930578(5) 1.170117625032930(1) 4.60193431229276(3) (0, 1) 0.2064913201586455(1) 6.27803903738084(1) 85.0504556068(1) 1.168818663818102(7) 4.83433647345744(1) (0, 2) 0.20600454282539647(2) 6.26290995769249(3) 96.91090513728(1) 1.1662390068467653(1) 5.293427387814051(1) (0, 3) 0.205283077054385(3) 6.240494331036(2) 117.976351526(8) 1.16241450178322(5) 5.96806152239636(1) (0, 4) 0.204336990211821(1) 6.21111373390(1) 151.874397816(6) 1.15739726546214(3) 6.8421981480130571(2) (1, 0) 0.2013106647017149(4) 6.12451980773668(3) 334.89830189(5) 1.14080522712808(8) 12.89614649422(1) (1, 1) 0.201081173073537(2) 6.11739730750(2) 347.54872540(2) 1.139597149439(4) 13.0952920268(3) (1, 2) 0.200625548551648(1) 6.10325962914552(6) 373.997618687(2) 1.137198302449808(6) 13.488286549835(3) (1, 3) 0.19995037648835(4) 6.0823171614(3) 416.4349042(3) 1.1336426477395(5) 14.06480755353873(5) (1, 4) 0.19906521542714(5) 6.05487519861(4) 477.89237015(7) 1.1289795422565(4) 14.810022086899788(2) (2, 0) 0.1962945882927300(1) 5.976228559885276(9) 762.803695574(1) 1.11407722548336(1) 19.878981012746(1) (2, 1) 0.196079942033917(5) 5.9695793169698(7) 780.9216157(2) 1.112955681491(6) 20.0477078807(2) (2, 2) 0.1956538427886285(7) 5.956382859214(3) 818.12463906(1) 1.1107290113251(1) 20.380248727985(3) (2, 3) 0.19502254990888(7) 5.9368391478(2) 876.2466269(3) 1.107429422937(1) 20.86703767611787(1) (2, 4) 0.1941951450844(1) 5.9112382321(2) 957.7993482(5) 1.103103710671(6) 21.494350160804(2) (3, 0) 0.191662497250660(4) 5.8400118488205(1) 1304.20928519(5) 1.08980160860771(4) 25.661621333295(5) (3, 1) 0.191461981700093(3) 5.8338140022328(6) 1326.12198978(1) 1.08876292594733(1) 25.80234568898(3) (3, 2) 0.19106398832511(1) 5.821515365437(5) 1370.7553662(2) 1.086701150024(5) 26.07923566010(5) (3, 3) 0.1904744702670(2) 5.803306116(3) 1439.630107(2) 1.083646859106(4) 26.48342297917(8) (3, 4) 0.1897020619107(7) 5.7794620161(6) 1534.8012755(8) 1.079644469587(5) 27.00220303997(3) (4, 0) 0.1873918466646(3) 5.71519849517(8) 1908.37170(2) 1.0678682734(1) 30.33829428(1) (4, 1) 0.187204844125(2) 5.70943335888(2) 1932.6198429(5) 1.0669094254(3) 30.45305164(3) (4, 2) 0.1868337304667(1) 5.697995472387(8) 1981.782888(1) 1.065006550320(2) 30.67833544895(8) (4, 3) 0.1862841710550(7) 5.681065831(1) 2057.103974(2) 1.06218871317(1) 31.0059220217(1) (4, 4) 0.185564380519(5) 5.65890681(2) 2160.234041(5) 1.0584981285(1) 31.424047350261(4)

Table 2. Numerical values of the Breit operators 〈 δ (r1)〉 , , , R01, and R02 for , where v = 0− 4 and L = 0− 4.

Table 3.

Table 3.

Table 3. Numerical values of the Breit operators 〈 δ (r12)〉 , 〈 δ (r1)〉 , , , and 〈 (∇ r1 + ∇ r2)4〉 for HD+ , where v = 0− 4 and L = 0− 4. (v, L) 〈 δ (r12)〉 〈 δ (r1)〉 〈 (∇ r1 + ∇ r2)4〉 (0, 0) 0.2070425994774719004(2) 0.2073481417802970666(2) 6.3001999476785933(6) 104.371713686093(1) 104.443848901053(2) (0, 1) 0.20685769957275256(1) 0.207163241677423572(7) 6.294450746055923(5) 110.38493981778(2) 110.4613313632(1) (0, 2) 0.20648990097608551(7) 0.2067954443349440(4) 6.2830163299151(5) 123.79001412764(6) 123.8756587418(6) (0, 3) 0.20594314768281(1) 0.2062486970164612(4) 6.266022789768(7) 147.25876381(2) 147.36009295(4) (0, 4) 0.20522321312906(3) 0.205528777980949(6) 6.24365466610(2) 184.59247475(2) 184.7179604(1) (1, 0) 0.20228886473978993(2) 0.20260117861162289(2) 6.15902235229201(4) 449.45675946471(2) 449.7391462670(1) (1, 1) 0.20211421220190804(1) 0.20242655716340313(3) 6.15359974575108(2) 464.36193814804(1) 464.6533872074(1) (1, 2) 0.2017668215015289(1) 0.2020792303568918(5) 6.142815739370(1) 495.3679016186(8) 495.678116390(6) (1, 3) 0.2012504647824(3) 0.2015629737358(2) 6.1267909226(3) 544.7845009(2) 545.1244240(5) (1, 4) 0.2005706622275(5) 0.20088331251301(2) 6.1057017310(3) 615.8801839(4) 616.262509(1) (2, 0) 0.19784583746893604(2) 0.1981667951256018414(8) 6.02761432267228(1) 1042.81683129264(3) 1043.44331653088(5) (2, 1) 0.1976809559785677(9) 0.198001981206342(1) 6.0225036774882(2) 1064.624132727(1) 1065.263406622(4) (2, 2) 0.197353025118044(5) 0.1976741873189976(5) 6.012340962862(5) 1109.271531087(7) 1109.93693530(3) (2, 3) 0.19686565432911(8) 0.19718702654295(2) 5.9972415652(1) 1178.74702791(2) 1179.4529688(1) (2, 4) 0.1962241242204(1) 0.1965457848848(2) 5.97737426240(4) 1275.84569327(1) 1276.6080713(2) (3, 0) 0.1936960139249636(3) 0.1940278413804545(3) 5.9054534324885(7) 1812.997565364(1) 1814.063236354(2) (3, 1) 0.1935404783766321(6) 0.193872416785920(1) 5.9006418093939(4) 1839.951687858(5) 1841.032921943(3) (3, 2) 0.193231161405204(1) 0.193563323743184(3) 5.89107464757(3) 1894.7472155(5) 1895.8600500(8) (3, 3) 0.192771518257(4) 0.19310402152252(7) 5.876862410(2) 1979.084658(2) 1980.246050(8) (3, 4) 0.192166602681(1) 0.192499569868(1) 5.8581666305(8) 2095.33663918(2) 2096.564775(2) (4, 0) 0.189823702852585(7) 0.19016909089076(3) 5.7920773795339(5) 2697.669274(1) 2699.2354157(9) (4, 1) 0.18967713560504(1) 0.1900226872284(1) 5.787553428565(6) 2728.2144158(1) 2729.7980517(2) (4, 2) 0.18938568121580(3) 0.18973156232654(8) 5.77855925079(2) 2790.0612310(3) 2791.6802590(4) (4, 3) 0.18895264869(6) 0.18929902980(2) 5.7652006889(6) 2884.65201(1) 2886.32533(6) (4, 4) 0.188382876738(6) 0.188729935330(3) 5.747632304(5) 3013.980528(6) 3015.72739(3)

Table 3. Numerical values of the Breit operators 〈 δ (r12)〉 , 〈 δ (r1)〉 , , , and 〈 (∇ r1 + ∇ r2)4〉 for HD+ , where v = 0− 4 and L = 0− 4.

Table 4.

Table 4.

Table 4. Numerical values of the Breit operators R02, R01, and R12 for HD+ , where v = 0− 4 and L = 0− 4. (v, L) R02 R01 R12 (0, 0) 5.35463051901711943(7) 1.174487825932605556(3) 1.170770145051139727(2) (0, 1) 5.589562255342723(1) 1.1735785001855910(1) 1.1697193799366195(4) (0, 2) 6.055104818249343733(4) 1.171769108405540(2) 1.16762928124835329(3) (0, 3) 6.74277630617217739(6) 1.16907791520748(2) 1.1645223841766(2) (0, 4) 7.640242669140628(6) 1.16553169532956(5) 1.1604316913454(5) (1, 0) 15.14482588936280077(6) 1.1504812637673131(2) 1.1433664050045(2) (1, 1) 15.350575279863064(1) 1.1496258585220391(3) 1.14238073961236855(7) (1, 2) 15.75802763752410(4) 1.147923888699637(7) 1.140420315240445(2) (1, 3) 16.35924480624(3) 1.145392790044(2) 1.13750662824(1) (1, 4) 17.1426952788899(3) 1.1420581168039(4) 1.133671151272(3) (2, 0) 23.59899696623698(2) 1.1282362008927494(6) 1.1181235693136(2) (2, 1) 23.777762089355404(4) 1.1274325102467439(1) 1.11720015102388(1) (2, 2) 24.13150123361982(4) 1.12583357622946(1) 1.11536372709830(3) (2, 3) 24.6527816700128(9) 1.1234560568468(8) 1.112634817630(2) (2, 4) 25.3308170266259(7) 1.120324362118(2) 1.109043458742(1) (3, 0) 30.817653008622(1) 1.10767720643593(3) 1.0949375427689(2) (3, 1) 30.97134959374(1) 1.1069233223353(1) 1.09407389213715(9) (3, 2) 31.275189712792(3) 1.1054236322514(2) 1.0923565391832(2) (3, 3) 31.72221152212(6) 1.103194064780(2) 1.0898050864(1) (3, 4) 32.30232200847(1) 1.100257957806(8) 1.086448221081(5) (4, 0) 36.88807887511(1) 1.088739929649(8) 1.073718403917(3) (4, 1) 37.0183653341(6) 1.08803423109(2) 1.07291239499(1) (4, 2) 37.275607599566(2) 1.086630566009(1) 1.07130988884988(7) (4, 3) 37.65328339(4) 1.0845441798(5) 1.068929626(1) (4, 4) 38.14194609(1) 1.0817974074(2) 1.0657990275(3)

Table 4. Numerical values of the Breit operators R02, R01, and R12 for HD+ , where v = 0− 4 and L = 0− 4.

The leading-order radiative corrections of R_∞ α ³ is the largest part of the QED shift. For an atomic or molecular bound state, the current theoretical formalism for evaluating the QED corrections is based on the NRQED theory.^{[36, 98– 100]} The one-loop self-energy correction is

where β (v, L) is the Bethe logarithm. The correction of the anomalous magnetic moment of order R_∞ α ³ is

which may be integrated into Eq.  (26) as a contribution from the form factor of the electron.^[101] The Coulomb interaction between the bound electron and an external field can be represented by exchanging Coulomb photons. According to QED, a Coulomb photon generates virtual electron– positron pairs which are then annihilated into photons. As a result, the energy of the Coulomb photon has a shift. The correction of one-loop vacuum polarization, which has one electron– positron pair, is

The interaction that the bound electron exchanges a Coulomb photon and a transverse photon with nucleus contributes to the energy shift. Thus, the order R_∞ α ³(m_e/m_nuc) correction of one transverse photon exchange is

where Q(r) is the Q term introduced by Araki^[102] and Sucher.^[103] Summarizing all the contributions of order R_∞ α ³, one obtains

In the basis set of Eq.  (9), the Q terms can be expressed as^[104]

where γ _E is Euler’ s constant, ε is the radius of a sphere about r = 0 to be excluded from the integration for Q(r). Numerical values of these terms can be found in Table  5 and Table  6 for and HD⁺ , respectively.

Table 5.

Table 5.

Table 5. Numerical values of Q(r1) for , where v = 0− 4 and L = 0− 4. L\v 0 1 2 3 4 0 − 0.13442622791325256(8) − 0.131286375375801(2) − 0.128392575302857(4) − 0.12573150011197(2) − 0.12329168120902(4) 1 − 0.1342668168626330(4) − 0.131136925403617(2) − 0.12825265604664(3) − 0.12560074245121(4) − 0.123169774407(2) 2 − 0.133950459810467(2) − 0.13084038002659(3) − 0.12797506716106(2) − 0.125341377055(1) − 0.122928016028(1) 3 − 0.133481987355(3) − 0.13040135187(2) − 0.12756421444(3) − 0.12495761350(4) − 0.1225704297(3) 4 − 0.132868404802(1) − 0.12982652672(1) − 0.1270264810(1) − 0.12445554797(5) − 0.122102840(2)

Table 5. Numerical values of Q(r1) for , where v = 0− 4 and L = 0− 4.

Table 6.

Table 6.

Table 6. Numerical values of Q(r1) and Q(r12) for HD+ , where v = 0− 4 and L = 0− 4. For each (v, L), the first entry is for Q(r1) and the second is for Q(r12). L\v 0 1 2 3 4 0 − 0.1348622766081903(2) − 0.132113355129(1) − 0.129551987571271(1) − 0.1271692904455(1) − 0.12495749557739(3) − 0.1345911955685105(1) − 0.131838977797(3) − 0.129272929986259(7) − 0.1268839283180(2) − 0.12466388687278(6) 1 − 0.134742001057392(5) − 0.13199961143967(2) − 0.1294445278685(4) − 0.1270678977803(2) − 0.1248619827842(2) − 0.13447097875759(1) − 0.131725268104580(5) − 0.1291654760689(3) − 0.1267825085601(1) − 0.1245683078029(8) 2 − 0.13450284686085(8) − 0.1317734666211(3) − 0.1292308990657(3) − 0.126866353373(3) − 0.1246721506275(1) − 0.13423194023120(3) − 0.1314991894516(1) − 0.1289518569532(7) − 0.126580907975(1) − 0.124378340936(7) 3 − 0.1341475653900(1) − 0.131437564457(6) − 0.12891364214(1) − 0.12656709998(4) − 0.124390349(3) − 0.133876827784(1) − 0.13116338201(2) − 0.12863460989(3) − 0.1262815656(3) − 0.12409633(1) 4 − 0.1336801802328(6) − 0.130995769843(4) − 0.128496470584(6) − 0.12617370653(3) − 0.1240200070(4) − 0.133409659744(2) − 0.13072170536(5) − 0.12821744286(1) − 0.1258880438(2) − 0.12372570179(5)

Table 6. Numerical values of Q(r1) and Q(r12) for HD+ , where v = 0− 4 and L = 0− 4. For each (v, L), the first entry is for Q(r1) and the second is for Q(r12).

The Bethe logarithm β (v, L), the most difficult to calculate, is responsible for the nonrelativistic part of the self-energy of a charged particle in a quantum state interacting with transverse photons. The Bethe logarithm can either be represented by a sum over intermediate states^{[95, 105]} or by an integral over intermediate photon momentum k, ^[106] according respectively to

where | ψ ₀〉 is the energy eigenstate of interest with the corresponding energy eigenvalue E₀; J = ∑ _aq_aP_a/m_a is the electric current density operator of the system with q_a, P_a, and m_a being respectively the charge, the momentum operator relative to a laboratory coordinate system, and the mass of particle a; the summation over i covers all virtual intermediate states connected to | ψ ₀〉 by the operator J; and (H₀ − E₀ + k)^{− 1} is the resolvent operator associated with the nonrelativistic Hamiltonian H₀. Korobov^[107] applied the approach of Schwartz^[106] to evaluate β (v, L) for the hydrogen molecular ions according to Eq.  (34) and obtained about 5– 6 significant digits for HD^{+ [108]} and .^[109] In 2012, a further development on the Schwartz approach was made by Korobov^{[110, 111]} to deal with a general three-body system, such as helium, , and HD⁺ , in which the dipole matrix elements are expressed in the velocity gauge. After that, this method was applied to and HD⁺ by Korobov and Zhong^[81] for the rovibrational states of v = 0– 4 and L = 0– 4. Besides the Schwartz approach, Drake and Goldman^[112] developed a different method for the calculation of the Bethe logarithm for the case of atomic systems. Their method only requires a single matrix diagonalization of the Hamiltonian in a basis set containing a wide range of distance scales. The Drake– Goldman method was successfully applied to by Zhong  et al.^[113] for v = 0– 1 and L = 0. The precision achieved for these two states is a few parts in 10⁹, which demonstrates the feasibility of the Drake– Goldman method for simple molecular systems. The details of the calculations of β (v, L) can be found in Ref.  [110] for the Schwartz method and in Ref.  [113] for the Drake– Goldman method. The latest numerical values of β (v, L) can be found in Refs.  [81] and [113] that are quoted in Table  7 and Table  8.

Table 7.

Table 7.

Table 7. Bethe logarithms for low-lying rovibrational states of , see Table  4 of Ref.  [81]. L\v 0 1 2 3 4 0 3.012230335(1) 3.012547548(3) 3.01267873(2) 3.01262269(4) 3.01237775(6) 3.012230329(7)  a 3.012547533(3)  a 1 3.01220132(1) 3.01251393(2) 3.01264054(3) 3.01258051(4) 3.0123316(1) 2 3.01214395(1) 3.01244742(2) 3.01256542(3) 3.01249674(4) 3.0122395(1) 3 3.01205949(2) 3.01234936(3) 3.01245429(4) 3.01237302(5) 3.0121036(1) 4 3.01194983(3) 3.01222182(3) 3.01230955(5) 3.01221169(6) 3.0119263(2) a Ref.  [113]

Table 7. Bethe logarithms for low-lying rovibrational states of , see Table  4 of Ref.  [81].

Table 8.

Table 8.

Table 8. Bethe logarithms for low-lying rovibrational states of HD+ , see Table  5 of Ref.  [81]. L\v 0 1 2 3 4 0 3.01233626(2) 3.01263268(3) 3.01278948(4) 3.01280622(6) 3.0126822(1) 1 3.01231470(2) 3.01260814(3) 3.01276198(5) 3.0127760(1) 3.0126490(1) 2 3.01227206(2) 3.01255942(3) 3.01270766(5) 3.0127160(1) 3.0125836(2) 3 3.01220877(3) 3.01248727(4) 3.01262691(6) 3.0126269(1) 3.0124865(2) 4 3.01212616(4) 3.01239292(4) 3.0125211(1) 3.0125102(1) 3.0123593(2)

Table 8. Bethe logarithms for low-lying rovibrational states of HD+ , see Table  5 of Ref.  [81].

The corrections of order R_∞ α ⁴ are also needed to be taken into consideration.^{[31, 114, 115]} However, the recoil corrections of order R_∞ α ⁴(m_e/m_nuc) and higher are small and can be neglected at the precision of ∼ 0.01– 0.1  ppb. The self-energy and the vacuum polarization corrections at this order are respectively

and

Note that in Ref.  [31], the above term of the vacuum polarization is incorporated into Eq.  (38) below. The contribution from the anomalous magnetic moment of order R_∞ α ⁴ is

Finaly the contribution due to the second-order slope of the Dirac form factor is

Summing up all the contributions, the total radiative correction of order R_∞ α ⁴ in the external field approximation can be expressed as

Currently, the order R_∞ α ⁴ relativistic corrections are calculated in the two-center approximation.^{[31, 116]} In this approximation, the Hamiltonian of the nonrelativistic Schrö dinger equation is expressed as

where R is the distance between the two nuclei, and r₁ and r₂ are the distances from the electron to nuclei 1 and 2, respectively. The total relativistic correction to the energy of a bound electron at order R_∞ α ⁴ is calculated according to

where Q = 1 − | ψ ₀〉 〈 ψ ₀| is the projection operator and H_B is the Breit– Pauli Hamiltonian defined by

The effective Hamiltonian represents the interaction of the bound electron with the external field generated by the two nuclei and has the form of

where and ρ = δ (r₁) + δ (r₂). It should be mentioned that both terms in Eq.  (41) are divergent and the details for a complete removal of the divergence between these two terms can be found in Ref.  [116]. The final finite result is

which can be evaluated numerically. The numerical values of are listed in Table  9 and Table  10 for and HD⁺ , respectively.^[31] Finally the total correction of this order is

Table 9.

Table 9.

Table 9. Relativistic corrections of order R∞ α 4 for . Units are α 4  a.u. L\v 0 1 2 3 4 0 − 0.042097 − 0.042908 − 0.043786 − 0.044732 − 0.045729 1 − 0.042100 − 0.042912 − 0.043792 − 0.044740 − 0.045738 2 − 0.042107 − 0.042922 − 0.043805 − 0.044757 − 0.045756 3 − 0.042117 − 0.042938 − 0.043825 − 0.044782 − 0.045783 4 − 0.042133 − 0.042959 − 0.043854 − 0.044818 − 0.045820

Table 9. Relativistic corrections of order R∞ α 4 for . Units are α 4  a.u.

Table 10.

Table 10.

Table 10. Relativistic corrections of order R∞ α 4 for HD+ . Units are α 4  a.u. L\v 0 1 2 3 4 0 − 0.042043 − 0.042738 − 0.043483 − 0.044278 − 0.045126 1 − 0.042045 − 0.042741 − 0.043487 − 0.044284 − 0.045132 2 − 0.042050 − 0.042748 − 0.043496 − 0.044295 − 0.045146 3 − 0.042058 − 0.042759 − 0.043510 − 0.044312 − 0.045167 4 − 0.042069 − 0.042773 − 0.043529 − 0.044336 − 0.045195

Table 10. Relativistic corrections of order R∞ α 4 for HD+ . Units are α 4  a.u.

The next order of corrections that we should consider is R_∞ α ⁵, which contain the contributions of the self-energy, the vacuum polarization, the Wichman– Kroll term, the complete two-loop diagrams, and the three-loop diagrams. The R_∞ α ⁵ order corrections for hydrogenlike ions can be found in Refs.  [114], [115], and [117]. In early times, these corrections were calculated in the hydrogen approximation where the parameters of hydrogen were adopted.^{[30, 31]} In 2014, the R_∞ α ⁵ order corrections were extended by Korobov et al.^{[83, 84]} to a system of two Coulomb centers. Thus the two-center approximation used for the R_∞ α ⁴ order relativistic corrections will also be used in the following equations.

The main part of this shift is the self-energy contribution. The general results obtained in Refs.  [118]– [120] for a bound electron in an external field were taken as the starting point in the work of Korobov et al. Since in Refs.  [118]– [120] the form of field is generic, it is possible to extend their approach to the two-center problem. The details of the derivation can be found in Ref.  [84]. The final result is

where A₆₂ = − 1, and the expressions of A₆₀ and A₆₁ are

In the above, 〈   〉 _fin means that the matrix elements are redefined in finite forms, and the other notations can be found in Section  2.4. The most difficult part to calculate is the relativistic Bethe logarithm 𝓛 (R), which has been evaluated by Korobov and coworkers^[82] for the two-center problem. For the vacuum polarization in a strong external field, the leading-order correction is the Uehling term,

which contributes at the level of 1  kHz and can thus be estimated in the hydrogen approximation. The coefficients for an S state of the atomic hydrogen are

where V₆₀(nS) is taken from Refs.[121]– [124] and V₆₁(nS) from Ref.  [125]. The Wichman– Kroll term^[126] can be expressed in the form

The next contribution is the complete two-loop Feynman diagrams^{[127– 129]}

where B₅₀ = − 21.55447(12). This expression is valid for a bound electron in an arbitrary configuration of a few pointlike Coulomb sources. The contribution from the three-loop Feynman diagrams at this order is^{[130– 133]}

which may be negligible. The total contribution of order R_∞ α ⁵ is therefore

In addition to these corrections discussed above, the dominant part of R_∞ α ⁶ is also considered, which represents the second-order perturbation with two one-loop self-energy operators

The electron in the hydrogen molecular ion is settled in the 1σ _g state with its principal quantum number n = 1 and the angular momentum quantum number l = 0. In the hydrogen approximation, it can be simplified as a 1s state. Thus, we can calculate the corrections of the finite nuclei size using the formula for the s-state hydrogen or deuterium. According to Refs.  [114] and [115], the leading-order correction of the finite nuclear size is given by

where R₀ and R₂ are the root-mean-square radii of the two nuclear charge distributions. It is R_p = 0.8775(51)  fm for the proton and R_d = 2.1424(21)  fm for the deuteron.^[6] The next-order correction is given by (see Ref.  [6] and references therein)

where are constants that depend on the charge distributions in the nuclei with for the proton and for the deuteron. The leading-order correction E_{nuc− 1} has been evaluated in earlier works, ^{[30, 79, 80]} while E_{nuc− 2} has not been considered before. We found that E_{nuc− 2} can have a noticeable contribution to a transition frequency at the current required precision. The remaining corrections of the finite nuclear size are at the level of Hz or smaller and can thus be neglected. Finally, the total correction due to the finite nuclear size is

The total transition frequency is obtained by summing up all the contributions

where E_i and E_f are the energy levels of the initial and final states, respectively. Table  11 lists individual contributions up to R_∞ α ⁶ to the fundamental transition (v, L) : (0, 0) → (1, 0) in and HD⁺ . The nonrelativistic energies and the leading-order relativistic corrections were calculated by using the fundamental physical constants of the 2010 CODATA set.^[6] The uncertainty of Δ E_nr comes from the uncertainty of the Hartree energy E_h.^[6] It should be pointed out that the uncertainty in Δ E⁽²⁾ is entirely due to the fundamental physical constants used. Also listed in Table  11 is the earlier value of Δ E⁽²⁾ for ^[79] calculated using the 2006 CODATA recommended values, ^[134] which has a slight discrepancy of about 10  Hz. The computational uncertainty in Δ E⁽²⁾ can be ignored due to the high precision numerical values of the energy levels in Table  1. The remaining uncertainty in Δ E⁽²⁾ is from the uncertainty of the Rydberg constant R_∞ due to the unit conversion, which is less than the total uncertainty in the transitional frequency and will not affect the m_p/m_e determination at the present targeted precision. The current uncertainty of Δ E⁽³⁾ is mainly due to the numerical uncertainties of the Bethe logarithm. For HD⁺ , the uncertainty of Δ E⁽³⁾ was reduced from 40  Hz^[81] to 8  Hz because the high precision values of the Q term were adopted. The uncertainty in Δ E⁽⁴⁾ is due to , which was estimated to be 10^{− 5}α ⁴  a.u. The uncertainty in the Δ E⁽⁵⁾ term was reduced from 20  kHz to 2  kHz because of the improved calculation of the one-loop self-energy.^{[83, 84]} Finally the uncertainty in Δ E⁽⁶⁾ was estimated by the two one-loop self-energy corrections and its magnitude is smaller than the uncertainty of the Δ E⁽⁵⁾ term. The finite nuclear size effect has a significant contribution to the transition frequency. In early works, only the leading-order finite-nuclear size correction was calculated. We estimated that the next order correction has a contribution beyond the uncertainty of the Δ E⁽⁵⁾ correction and it should be taken into account. As a result, the present value of Δ E_nuc is different from the previous one. This in turn leads to a difference in the total frequency by about 20 to 30  kHz from the latest calculations of Korobov and coworkers.^{[83, 84]} It is, however, that the R_∞ α ⁵ order correction remains the source of uncertainty in the total transition frequency.

Table 11.

Table 11.

Table 11. Summary of contributions to the transition frequency (v, L) : (0, 0) → (1, 0) of and HD+ . Units are MHz. HD+ Δ Enr 65687511.071 42(17) 57349439.97334(15) Δ E(2) 1 091.081365(7) 958.276694(8)[80] 1 091.081355(2)[79] 1 091.081(03)a 958.277(01)a Δ E(3) – 276.545038(12)b – 242.126263(8) – 276.545049(4)[81] – 242.12626(4)[81] – 276.544(02)[30] – 242.125(02)[30] Δ E(4) – 1.99730(26)c – 1.74837(26)c Δ E(5) 0.138(2)[83, 84] 0.120(2)[83, 84] 0.120(23)[31] 0.105(19)[31] Δ E(6) 0.001(1)[83, 84] 0.001(1)[83, 84] Δ Enuc – 0.03899(50) – 0.0924(16) – 0.0410(6)[79] – 0.1252(17)[80] – 0.0410(3)a – 0.125(2)a Δ Etotal 65 688323.710 5(28) 57350154.404 0(28) [83, 84] 65688323.708(2) 57350154.371(2) [80] 57350154.355(22) [31] 65688323.688(25) 57350154.355(21) [28] 65688323.745(80) 57350154.412(70) a Obtained using the data of Table  V in Ref.  [30]. b Obtained using the values of the Bethe logarithm from Ref.  [80]. c Obtained using the values of the R∞ α 4 corrections from Ref.  [31].

Table 11. Summary of contributions to the transition frequency (v, L) : (0, 0) → (1, 0) of and HD+ . Units are MHz.

Table  12 lists some important transition frequencies in HD⁺ , together with the available experimental measurements. For the transition (v, L) : (0, 0) → (1, 1), there is a discrepancy of about 150  kHz between theory and experiment. At first glance, this seems to be introduced by the higher-order correction of the finite nuclear size effect that is ignored in previous works. However, this correction to (v, L) : (0, 0) → (1, 1) is estimated to be only + 34  kHz, indicating that the resulting discrepancy is now increased to 180  kHz. The reason for this discrepancy is unknown.

Table 12.

Table 12.

Table 12. Theoretical and experimental transition frequencies in HD+ . The last two measurements are currently underway.[27, 135] Units are MHz. (v, L) → (v′ , L′ ) Experiment Theory (0, 2) → (4, 3) 214978560.6(5)[23] 214978560.948(8)[83, 84] (0, 0) → (1, 1) 58605052.00(6)[26] 58605052.156(2)[83, 84] (0, 0) → (0, 1) [135] 1314925.7523(1)[84] (0, 2) → (8, 3) [27] 383407177.150(15)[84]

Table 12. Theoretical and experimental transition frequencies in HD+ . The last two measurements are currently underway.[27, 135] Units are MHz.

The motion-induced Doppler broadening is the main factor affecting the precision and resolution of rovibrational spectroscopy in the hydrogen molecular ions. Reducing the translational energy of the ions and confining them in the Lamb– Dicke region can minimize the first-order Doppler shift and broadening to the greatest extent.^[25]

The complexity of the molecular energy level structure usually prevents the implementation of closed optical cycles so that laser cooling is generally not applicable to molecules or molecular ions. Over the past 15 years, sympathetic cooling^{[136, 137]} has emerged as a versatile method for cooling a broad range of molecular ions to millikelvin secular temperatures with laser-cooled atomic ions.^{[138– 140]} Since the molecular ions can exchange kinetic energy with the laser-cooled atomic species by Coulomb interactions, it is possible to reduce the molecular ions’ kinetic energy. Thus, the atomic-ion species act as a coolant for the molecular ions resulting in the formation of bi-component (or molecular) Coulomb crystals.

The hydrogen molecular ions can be sympathetically cooled either using laser-cooled beryllium ions Be⁺ or magnesium ions Mg⁺ . Due to the lower mass ratio between Be⁺ and HD⁺ , using Be⁺ as a coolant can provide higher cooling efficiency and lower secular temperature than using Mg⁺ . However, making a radiation source for laser cooled Be⁺ is technologically demanding; on the other hand, using laser cooled Mg⁺ as a coolant for sympathetically cooling HD⁺ has advantages in terms of experimental complexity and cost.

Early theoretical work^[141] gives the critical mass ratio for using heavy ions to sympathetically cool light ions, but this theoretical threshold has been overcome experimentally.^[142] The more reliable method to determine the feasibility of performing high-efficiency sympathetic cooling between different ions is the use of molecular dynamics (MD) to simulate such a cooling process. An MD simulation can also provide the characteristic information of molecular Coulomb crystal, e.g., the number of the laser-cooled and sympathetically cooled ions and their translational energies. We simulated 150 laser-cooled Mg⁺ ions to sympathetically cool ten HD⁺ ions and obtained an image of bi-component Coulomb crystal (see Fig.  4). The result shows that HD⁺ can be sympathetically cooled into the Mg⁺ Coulomb crystal and form a chain of almost stationary (T ≤ 10  mK) molecular ions. At this temperature, the resolution of rovibrational spectroscopy of HD⁺ can be obtained in about 10  MHz range. As we know, the spectral linewidth resulting from the Doppler broadening is proportional to ; thus, the spectral precision will be similar using either laser-cooled Mg⁺ or Be⁺ species as a coolant.

Fig.  4.

	Figure Option ViewDownloadNew Window
	Fig.  4. Molecular dynamics simulation of a fluorescence image of an Mg+ /HD+ bi-component Coulomb crystal. The spatial distribution of the non-fluorescing molecular HD+ ions has been made visible in blue in the simulated image.

Translationally cold molecular ions are conventionally produced from “ warm” samples by sympathetic cooling. Because low-energy collisions between ions are dominated by the Coulomb interaction which does not couple to the internal degrees of freedom, sympathetically cooled ions exhibit a broad distribution of the rotational-state population.^{[143, 144]} In such translationally cold but internally warm samples, the population can be accumulated in the rotational ground state using optical pumping scheme as demonstrated by Drewsen et al.^[139] and Schneider et al.^[87] Although optical pumping can increase the population of the rovibrational ground state by an order of magnitude higher than the corresponding thermal population at room temperature, the state preparation is not complete. A novel method has been developed to prepare internally state-selected molecular ions in a linear Paul ion trap.^{[143, 144]} The technique relies on the generation of molecular ions in a well-defined rovibrational quantum state using photoionization followed by sympathetic cooling of the translational motion with laser-cooled atomic ions.

Such a technique can be applied to prepare a single rovibrational quantum state in HD⁺ by resonance-enhanced threshold photoionization (RETPI) method.^[145] The experimental process includes the following three steps, as depicted in Fig.  5. First, neutral HD molecules are accumulated at low rovibrational states in a molecular beam due to the adiabatic expansion; second, an excitation laser of λ ₁ ≈ 327  nm is applied to populate HD to an exited electronic state. Finally, by setting a photoionization photon energy of λ ₂ ≈ 314  nm slightly above the desired ionic state accessible by the selection rules, HD⁺ ions can be generated in a single rovibrational level.

Fig.  5.

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	Fig.  5. State selective [3+ 1′ ] resonance-enhanced threshold-photoionization scheme used to generate HD+ ions in the rovibrational ground state.

Since the RETPI method can produce molecular ions selectively due to the propensity rule of the excitation and ionization, it avoids generating undesired high abundant molecular species, such as , MgH⁺ , and MgD⁺ . Furthermore, the relative spectral intensities need to be precisely measured for the determination of spectral line positions. The RETPI method ensures that all HD⁺ ions are prepared in a single rovibrational quantum state and a transition from such a single state would offer the highest possible detecting sensitivity and the most accurate relative intensity. Previous measurements on HD⁺ rovibrational transitions have improved the accuracy from 2  ppb to 1.1  ppb by using rotational cooling to increase the ground-state population.^[84] We expect that the accuracy of the line position could further be improved to 0.4  ppb if a pure ionic species and a single rovibrational state could be obtained by using the RETPI method.

High-resolution infrared spectroscopy on sympathetically cooled HD⁺ ions can be performed by using resonance-enhanced multiphoton dissociation (REMPD) spectroscopy.^{[23, 146]} The rovibrational transition (v, L) : (0, 0) → (6, 1) can be measured by using a single photon excitation at 991  nm. The absorption of an additional 355  nm photon from the vibrationally excited state results in photofragmentation of the ions. By comparing with an MD simulation, the rate of fragmentation can be derived from the time-resolved image change of the bi-component Coulomb crystal.

The probability of vibrational overtone transitions is much lower than the fundamental transitions. For example, the probabilities of transitions from v : 0 → 4 and v : 0 → 8 are 4 and 7 orders of magnitude lower than the fundamental transition of v : 0 → 1 respectively. However, the sympathetically cooled molecular ions are spatially stationary (T ∼   mK, trapping time ∼ hours), which can provide a minute-long interaction time between photons and ions. This unprecedentedly long interaction time ensures successful excitation of the vibrational overtone transitions.

In 2007, Schiller’ s group observed the overtone transitions of v : 0 → 4 in HD⁺ at the rate of 22  s^{− 1} using a laser with 170  μ W in power and 200  μ m of beam waist.^[23] The Einstein coefficient B of the v : 0 → 6 transition is proximately 1/400 of the transition v : 0 → 4. We estimate the rate of transition by employing a laser of 100  kHz linewidth, 100  mW in power, and 200  μ m beam waist to be achieved at 64  s^{− 1}, which is faster than the other observed overtone transitions of v : 0 → 8.^[27] The hyperfine transition linewidth in sympathetically cooled HD⁺ ions is around 10  MHz, which is dominated by the Doppler broadening of the residual thermal energy. The radiation intensity of the above proposed laser system would not bring an additional broadening of the measured transition linewidth. The uncertainty of the induced AC Stark shift is on the order of 10^{− 12}, which will evidently not affect the targeted accuracy.