^{†}Corresponding author. Email: zyan@unb.ca
^{‡}Corresponding author. Email: tyshi@wipm.ac.cn
^{*}
In this paper, we overview recent advances in highprecision structure calculations of the hydrogen molecular ions (
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 selfconsistency 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 protontoelectron 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 ^{12}C^{5+ } 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 protontoelectron 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 subHz, 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 highorder 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 protontoelectron 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 selfconsistency among the proton mass m_{p}, the electron mass m_{e}, and the ratio m_{p}/m_{e}. The selfconsistency 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 selfconsistency may indicate the existence of new physics.
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
Compared to p̄ He^{+ }, rovibrational states of hydrogen molecular ions (
This article is organized as follows. The background of highprecision spectroscopy is introduced in Section 1.2 for hydrogen molecular ions, which includes theory and experiment. In Section 2, theoretical formulation for solving a threebody boundstate problem is presented, including the nonrelativistic energy eigenvalue problem and the leading and higherorder 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.
The hydrogen molecular ion, the simplest molecule in nature, can be considered as a threebody quantum mechanical system consisting of one electron and two nuclei. Some of its rovibrational states have lifetimes longer than 10 μ s, and the probability of making a dipole transition can be strong enough to allow for a precision measurement of its spectroscopy.^{[2, 27, 33– 35]} On the other hand, the transition frequency between two specific rovibrational states can be calculated to sufficiently high precision using the threebody QED theory.^{[30, 31]} Since the theoretical frequency of a rovibrational transition in the hydrogen molecular ion contains effects due to the protontoelectron mass ratio, a comparison between theory and experiment of its spectroscopy holds a potential of extracting a precise value of m_{p}/m_{e}.^{[2, 3]}
Theoretically, a transition frequency in the hydrogen molecular ion can be expanded in powers of the fine structure constant α (≈ 1/137) according to the theory of nonrelativistic QED (NRQED).^{[36, 37]} The structure of an energy level contains the contributions from the nonrelativistic term, the leadingorder relativistic and QED corrections of order R_{∞ }α ^{2} and R_{∞ }α ^{3} respectively, where R_{∞ } is the Rydberg constant, and the correction due to the finite nuclear size. Higherorder relativistic and QED corrections can be added in when they become necessary.
In order to determine a transition frequency, the nonrelativistic energy eigenfunction and the eigenvalue can be obtained by solving the threebody Schrö dinger equation variationally. In previous work, the rovibrational states of the hydrogen molecular ions were calculated at different levels of approximation, such as the Born– Oppenheimer approximation, the adiabatic approximation, and the nonadiabatic approximation.^{[38– 40]} In 1985, Bishop and Solunac^{[41, 42]} broke down the Born– Oppenheimer and adiabatic approximation and calculated the energy levels of the hydrogen molecular ions precisely. From then on, the energy levels of
In 2006, Korobov^{[30]} not only evaluated the R_{∞ }α ^{2} and R_{∞ }α ^{3} order corrections to high precision but also considered for the first time the higherorder terms of R_{∞ }α ^{4} and R_{∞ }α ^{5}, beyond the adiabatic approximation. The largest part of the uncertainty in his calculation was due to the theoretical approximation of the R_{∞ }α ^{5} term. The uncertainty in the transition frequency (v, L) : (0, 0) → (1, 0) for
Spectroscopic measurements of rovibrational and hyperfine structures of hydrogen molecular ions have been carried out since 1960s. Figure 3 shows a time evolution of some important measurements. During the early stages, the measured spectra were much less precise than their neutral counterparts. Highorder relativistic and QED effects of spin– spin and spin– orbital interactions have never been tested on these ions until recently. With developments of new experimental techniques, such as laser cooling and trapping, and sympathetic cooling, ^{[85]} together with the emergence of new laser systems, ^{[86]} precision measurements of cold molecules and molecular ions have received considerable attention. Cold and trapped ions have a range of advantages for spectroscopic applications. First, experiment on single trapped ions can avoid inhomogeneous spectral line broadening resulting from ensemble averaging. Second, the low translational velocities of the ions can minimize Doppler broadening. Third, a long storage time allows a measurement to go over an extended period of time. Finally, an ultrahigh vacuum apparatus provides a welldefined experimental environment where interactions with surroundings are minimized or well controlled. All these can make enhancement of spectroscopic resolution possible.
Schiller’ s group at Dü sseldorf^{[23, 33]} pioneered an experimental measurement of a rovibrational transition in sympathetically cooled HD^{+ } ions. In their experiments, the HD^{+ } ions were sympathetically cooled into a Be^{+ } Coulomb crystal. Highresolution infrared spectroscopy on the HD^{+ } ions was recorded for the rovibrational transition of (v, L) : (0, 2) → (4, 3) at a wavelength around 1.4 μ m. Though the spectral resolution of about 40 MHz was limited by the residual Doppler broadening, the spectral line position was determined with a relative accuracy of 2 ppb. Combining with the Korobov’ s theoretical prediction, the protontoelectron mass ratio m_{p}/m_{e} was determined at the level of 5 ppb. In 2012, the same group made further progress on the measurement of the rovibrational transition (v, L) : (0, 0) → (1, 1) with the accuracy of 1.1 ppb.^{[26]} The improvement of the experimental accuracy was achieved by using the optical pumping method to increase the rovibrational ground state population. The population transfer efficiency was achieved at 60%– 70% in their experiments.^{[87]} There are other two groups currently working on the precision spectroscopy of the hydrogen molecular ions HD^{+ } and
According to the theory of NRQED, ^{[36, 37]} the total energy of a state in
where E_{nr} is the nonrelativistic energy, E_{nuc} is the correction due to the finite nuclear size, E_{hfs} is the correction from the hyperfine structure, and E^{(n)} is the correction of order R_{∞ }α ^{n}. The effect of hyperfine structure in an experimental measurement can be removed according to theoretical calculation of hyperfine splittings, and the transitional frequency is then obtained. E_{hfs} has been well calculated in series of works for HD^{+ }^{[69]} and
We begin with the nonrelativistic Hamiltonian of a threebody 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 H_{0}, is
For the purpose of isolating the centerofmass motion, we make the following coordinate transformations:
where
where r_{12} =  r_{1} − r_{2} and μ _{i} = m_{i}m_{0}/(m_{i} + m_{0}). Since X is cyclic, we can simply neglect X. Therefore in the centerofmass 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
The Schrö dinger equation for the system is therefore expressed as H_{0}ψ _{0} = E_{0}ψ _{0}, where E_{0} is the energy eigenvalue and ψ _{0} is the corresponding eigenfunction. For solving the eigenvalue problem for H_{0} variationally, we seek the common eigenstates of the commuting observables {H_{0}, L^{2}, L_{z}, Π }, where Π is the parity operator, using the following basis set in Hylleraas coordinates:
In the above,
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 Gaussianlike 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
If the two nuclei are identical, such as
Each configuration (l_{1}, l_{2}) 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} subblocks, 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 r_{2}:
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 r_{2}, the rate of convergence, along with the numerical stability in the wavefunction, can be further enhanced.^{[56]} Hence another smallsize block with j_{0} = l_{2} 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 _{2}F_{1}(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.
The nonrelativistic energies of the hydrogen molecular ions
The leadingorder 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 fourcomponent wavefunction of the Dirac equation to a twocomponent 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 socalled 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
which is rooted from the dependence of the relativistic mass on velocity:
and the Darwin term for HD^{+ } is
The complete contribution of the leadingorder relativistic correction is thus
For the case of
The numerical values for the above discussed operators are presented in Table 2 for
The leadingorder 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 oneloop selfenergy 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 oneloop 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
The Bethe logarithm β (v, L), the most difficult to calculate, is responsible for the nonrelativistic part of the selfenergy 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 E_{0}; J = ∑ _{a}q_{a}P_{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 (H_{0} − E_{0} + k)^{− 1} is the resolvent operator associated with the nonrelativistic Hamiltonian H_{0}. 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
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 selfenergy and the vacuum polarization corrections at this order are respectively
and
Note that in Ref. [31], the above term
Finaly the contribution due to the secondorder 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 twocenter 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 r_{1} and r_{2} 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
where
which can be evaluated numerically. The numerical values of
The next order of corrections that we should consider is R_{∞ }α ^{5}, which contain the contributions of the selfenergy, the vacuum polarization, the Wichman– Kroll term, the complete twoloop diagrams, and the threeloop 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 twocenter 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 selfenergy 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 twocenter problem. The details of the derivation can be found in Ref. [84]. The final result is
where A_{62} = − 1, and the expressions of A_{60} and A_{61} 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 twocenter problem. For the vacuum polarization in a strong external field, the leadingorder 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 V_{60}(nS) is taken from Refs.[121]– [124] and V_{61}(nS) from Ref. [125]. The Wichman– Kroll term^{[126]} can be expressed in the form
The next contribution is the complete twoloop Feynman diagrams^{[127– 129]}
where B_{50} = − 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 threeloop 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 secondorder perturbation with two oneloop selfenergy 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 sstate hydrogen or deuterium. According to Refs. [114] and [115], the leadingorder correction of the finite nuclear size is given by
where R_{0} and R_{2} are the rootmeansquare 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 nextorder correction is given by (see Ref. [6] and references therein)
where
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
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 higherorder 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.
The motioninduced 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 firstorder 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 lasercooled atomic ions.^{[138– 140]} Since the molecular ions can exchange kinetic energy with the lasercooled atomic species by Coulomb interactions, it is possible to reduce the molecular ions’ kinetic energy. Thus, the atomicion species act as a coolant for the molecular ions resulting in the formation of bicomponent (or molecular) Coulomb crystals.
The hydrogen molecular ions can be sympathetically cooled either using lasercooled 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 highefficiency 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 lasercooled and sympathetically cooled ions and their translational energies. We simulated 150 lasercooled Mg^{+ } ions to sympathetically cool ten HD^{+ } ions and obtained an image of bicomponent 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
Translationally cold molecular ions are conventionally produced from “ warm” samples by sympathetic cooling. Because lowenergy 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 rotationalstate 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 stateselected molecular ions in a linear Paul ion trap.^{[143, 144]} The technique relies on the generation of molecular ions in a welldefined rovibrational quantum state using photoionization followed by sympathetic cooling of the translational motion with lasercooled atomic ions.
Such a technique can be applied to prepare a single rovibrational quantum state in HD^{+ } by resonanceenhanced 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.
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
Highresolution infrared spectroscopy on sympathetically cooled HD^{+ } ions can be performed by using resonanceenhanced 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 timeresolved image change of the bicomponent 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 minutelong 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.
The past few years have witnessed impressive progress in the field of sympathetic cooling of molecular ions. New experimental methods have been developed, which allow for an effective control of molecular internal quantum sates. Such developments have opened up a variety of opportunities for spectroscopic measurements of molecular ions with high precision. In our project/proposal, the hydrogen molecular ions HD^{+ } can be generated to the rovibrational ground state by RETPI, and can be sympathetically cooled to millikelvin by lasercooled atomic ions. With this accurate control of internal and external degrees of freedom, the overtone transition frequency from the rovibrational ground state to the 6th rovibrational excited state can be determined by the REMPD method. The relative accuracy of the measured transition frequency around 991 nm is expected to reach 0.4 ppb.
On the theoretical side, the contributions up to R_{∞ }α ^{5} have been taken into account in the calculations of the transition frequencies of
We would like to thank ShuMin Zhao, LiYan Tang, PeiPei Zhang, and QuanLong Tian for their participation and contributions. We also thank Kelin Gao and Hua Guan for valuable discussion, and V. I. Korobov of JINR for his expertise and encouragement.
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