Precision tests of nonadiabatic perturbation theory with measurements on the DT molecule

K.-F. Lai,1 P. Czachorowski,2 M. Schlösser,3 M. Puchalski,4 J. Komasa,4 K. Pachucki,2 W. Ubachs,1 and E. J. Salumbides 1,* 1Department of Physics and Astronomy, LaserLaB, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, Netherlands 2Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland 3Tritium Laboratory Karlsruhe, Institute of Technical Physics, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany 4Faculty of Chemistry, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznań, Poland


I. INTRODUCTION
Seventy years after its development, quantum electrodynamics (QED) has emerged to be the best-tested theory in physics. While QED is most accurately tested from the measurement of the anomalous magnetic moment of a free electron [1] and atomic recoil measurements [2,3], tests in the bound atomic hydrogen system are at present limited by effects of the proton structure [4,5]. Precision tests in the hydrogen atom rely ultimately on the narrowest transition involving the long-lived 2S quantum state with a natural lifetime of 0.12 s [6]. In contrast, the additional rotational and vibrational degrees of freedom in H 2 give rise to a multitude of states in the ground electronic manifold with extremely long lifetimes on the order of 10 5 -10 6 s [7]. In addition, access to all six isotopic variants of molecular hydrogen enables a robust validation of nonadiabatic perturbation theory.
The additional complexity of the four-body molecular hydrogen system presents formidable challenges in firstprinciples calculations. The last decade has shown great improvements in calculation of the nonrelativistic energies using a perturbative approach [8,9], or a recent nonperturbative treatment [10]. Concurrent developments in the calculation of relativistic [11], QED [12], and associated recoil corrections have led to sub-MHz accuracies in level energies of the stable molecular hydrogen species H 2 , HD, and D 2 [13], but have not been applied to tritiated isotopologues until now.
Parallel progress in experiments has resulted in an accurate measurement of the dissociation energy of the H 2 molecule [14,15], the measurement of its fundamental vibrational splitting [16], as well as very weak quadrupole overtone transitions [17][18][19]. Similar spectroscopic studies have also been applied to the other stable D 2 (cf. [20][21][22]) and the mixed isotopologue HD (cf. [23][24][25][26]). On the other hand, very few precision studies have been undertaken on the tritium-bearing species [27][28][29], on account of difficulties in handling radioactive tritium. If these practical challenges were overcome, access to tritium-bearing species T 2 , HT, and DT would double the number of molecular hydrogen test systems.
Here we present highly accurate calculations of the rovibrational transitions for all tritiated molecular hydrogen that are two orders of magnitude more accurate than previous studies [30]. This improvement is obtained by applying a recently developed nonadiabatic perturbation theory approach to obtain accurate nonrelativistic energies [9], as well as by systematic treatment of leading-order, higher-order, and recoil relativistic and QED corrections based on nonrelativistic wave functions. The calculations are benchmarked by accurate measurements of DT transition energies, which enables precision tests in tritiated species that are now sensitive to QED effects.

II. EXPERIMENT
As for the experimental study, we use the nonlinear frequency-mixing scheme of coherent anti-Stokes Raman spectroscopy (CARS), illustrated in the inset of Fig. 1. An anti-Stokes coherent beam is produced for signal detection, at frequency ω AS = 2ω P − ω S corresponding to λ AS ∼ 464 nm, whenever the frequency difference between the pump (ω P ) and Stokes (ω S ) frequencies is in resonance with a vibrational mode (ω R ) in the molecule. The high-resolution CARS setup has been described in a previous work on T 2 [31]. An injection-seeded and frequency-doubled Nd:YAG laser provides the pump beam (λ P ∼ 532 nm), while the Stokes radiation (λ S ∼ 623 nm) originates from a narrow-band pulsed dye amplifier (PDA) system [32], which is seeded by a continuous-wave (cw) ring dye laser. DT at a partial pressure of 4 mbar is contained in a 4-cm 3 gas cell, prepared from a 4:1 mixture of D 2 and T 2 at the Tritium Laboratory Karlsruhe and transported to LaserLaB Amsterdam for the spectroscopic measurements [33].
Recordings of the Q(1) Raman transition at different intensities are shown in Fig. 1, manifesting ac-Stark broadening and shifting. The cw-seed frequency for the ω S radiation is calibrated using a HeNe-stabilized etalon in combination with an absolute frequency reference from saturation I 2 spectroscopy [34]. The cw-pulse frequency offset induced by frequency chirp effects in the pulsed-dye amplification [32,35,36] is measured and corrected for [37]. The frequency of the ω P pulse is monitored online using a highresolution wavemeter (High Finesse Ångstrom WSU-30), which is periodically calibrated against several absolute frequency standards at different operating wavelengths, including calibrations against a Cs clock with the aid of an optical frequency comb. This wavemeter also measures the correct frequency that includes any chirp-induced frequency offset as verified when using a narrow-band titanium sapphire pulsed laser source with an adjustable cw-pulse frequency offset used in Ref. [14]. The Raman shift, ω R = ω P − ω S , is derived from the simultaneous frequency calibrations of both pump ω P and Stokes ω S laser frequencies.
The ac-Stark shift for the Q(1) transition is plotted in Fig. 2, panel (a), as a function of the pump I P and Stokes beam I S intensities, and in panel (c) as a function of the total intensity, I P + I S , of both pump and Stokes beams. It was established that the extrapolation to the unperturbed zero-intensity frequency by fitting a plane in (a) or a line in (c) yields the same value within 2 MHz. This linear dependence is expected since the polarizabilities at λ P and λ S for both the v = 0 and v = 1 levels of molecular hydrogen are very similar [38]. In this manner, the ac-Stark dependence on the total intensity for all other Q(J = 0, 2-5) transitions was treated by linear extrapolation and found to be accurate to 6 MHz. The full widths at half maximum are plotted in Fig. 2(b), which extrapolates to the Doppler width at zero intensity. Collisional shifts on the DT transitions are conservatively estimated to be ∼1 MHz (for DT:D 2 :T 2 partial pressures of 4:8:0.5 mbar), based on investigations of the stable molecular hydrogen species [39][40][41]. Since we cannot vary the pressure of DT, we have verified the estimates by pressure-dependent measurements of D 2 in an identical gas cell. The hyperfine structure of DT is expected to be similar to that of HD with the hyperfine splittings spanning within ∼1 MHz, and is not observed in our Doppler-limited linewidths. Possible shifts in the hyperfine center of gravity of the transitions are expected to be well below a MHz and are neglected. Table I shows the uncertainty contributions, where a final uncertainty of 12 MHz or 4×10 −4 cm −1 is estimated for the Q(J = 0, 2-5) lines. The Q(1) transition has a slightly smaller uncertainty due to more measurements collected for the assessment of systematic shifts. The reproducibility of  measurements performed on multiple days is indicated in the statistics entry in Table I.
Measurements of the Q(0), Q(1), and Q(2) transitions of the fundamental band of D 2 with 8-mbar partial pressure inside the same DT cell were also performed. High-accuracy D 2 measurements using molecular beams have been performed with a completely different spectroscopic approach [16,42], allowing for the in situ assessment of any other systematic effects. D 2 comparisons with Ref. [42] yield an average deviation, shown in Fig. 3, that is consistent with and validates the independently estimated uncertainty of the present CARS study. These experimental results are compared with the theoretical results presented in the following discussion.

III. THEORY
In order to calculate molecular rovibrational levels and transition energies accurately, we use a variant of nonrelativistic quantum electrodynamics (NRQED)-an effective theory approximating QED at low-energy scales [12]. It assumes an expansion of the binding energy in powers of the fine-structure constant α, where E (i) is a contribution of order α i m (with the electron mass m) and may include powers of ln α. Each E (i) can be expressed as an expectation value of some effective Hamiltonian with the nonrelativistic wave function. The E (i) terms can be calculated directly (see, e.g., [10,[43][44][45]) or expanded further in the m/μ n mass ratio [with μ n = M A M B /(M A + M B )the nuclear reduced mass] in the spirit of the nonadiabatic perturbation theory (NAPT) [9,46]. This yields the wellknown components of the nonrelativistic energy-the Born-Oppenheimer (BO) energy E (2,0) , the adiabatic correction E (2,1) , and the nonadiabatic correction E (2,2) for a given rovibronic state. Similarly, NAPT enables the relativistic E (4) and QED E (5) corrections to be evaluated as a sum of the leading (infinite nuclear mass) and the recoil (finite mass) components, although finite mass QED corrections have not yet been incorporated.
The Schrödinger equation for a hydrogen molecule isotopologue, written in a center-of-mass frame, with the origin in the geometric center of the nuclei, is where and where the 1, 2 indices denote electrons, A, B denote nuclei, R = R A − R B , and ∇ el = ( ∇ 1 + ∇ 2 )/2. The last term in H n is present in heteronuclear isotopologues and is a source of the gerade/ungerade mixing effects, of relevance to the DT species investigated here. Within NAPT, the wave function is represented as where one assumes δ |ψ el = 0 (integration over electronic coordinates only), Y ( n) is a spherical harmonic, and n = R/R. Here ψ ( r 1 , r 2 ) is an eigenfunction of the electronic Schrödinger equation, with the eigenvalue dependent on the internuclear distance R. In the NAPT leading order, the function χ satisfies the following nuclear equation, where J is the rotational quantum number. E (2,0) (R) from the electronic Schrödinger equation (7) serves as a potential for the movement of the nuclei, present in the nuclear equation (8). We solve Eq. (8) with a discrete variable representation method [47,48] for χ , which is then used in a perturbative manner to calculate the rovibrational energy contributions χ |E (i,k) (R)|χ , where E (i,k) (R) denotes a correction of the order α i m(m/μ n ) k to the electronic potential. Regarding E (2) , its BO approximation E (2,0) is surely not accurate enough for our purposes. At the same time, direct calculation of E (2,1) and E (2,2) is not very convenient. This is why the nuclear Schrödinger equation is solved again instead-with the NAPT-corrected Hamiltonian, including the nonadiabatic effects up to the (m/μ n ) 2 level 033124- 3  TABLE II. Calculated contributions to the Q(1) transition energy (in units of cm −1 ) in the fundamental band of tritium-bearing molecular hydrogen. E FS is the finite nuclear size correction with r p = 0.841 4 (19) fm [52], r d = 2.127 99(74) fm [52], and r t = 1.759 1(363) fm [53], for the proton, deuteron, and triton sizes, respectively. The fine-structure constant α, Rydberg constant R ∞ , and Bohr radius a 0 are taken from [52], as are the respective proton, deuteron, and triton electron mass ratios: m p /m = 1 836.152 673 43 (11) All the functions μ (R), μ ⊥ (R), W (R), and Y (R) in the above are defined and provided as analytic fits in Refs. [9,46,48]. Note that Y (R) incorporates both the adiabatic and nonadiabatic effects, and the g/u-mixing term from Eq. (5) in particular.
Total theoretical uncertainties of all the transitions reported in this paper are dominated by the missing E (2,3) in the nonrelativistic contribution. The calculated contributions to the Q(1) transition energy in the fundamental vibrational splitting (v = 0 → 1) for all tritiated species DT, HT, and T 2 are listed in Table II. The theoretical results obtained here are in agreement with Ref. [30], with the present accuracy of 0.000 11 cm −1 representing more than a 100-fold improvement. The 0.02 cm −1 uncertainty in Ref. [30] is dominated by the uncertainty in the nonadiabatic nonrelativistic correction (E (2,2) ), and only includes leading-order relativistic (E (4,0) ) and QED corrections using a Bethe logarithm that is 20% off from the modern value, or a deviation of >0.001 cm −1 in E (5) .

IV. RESULTS, DISCUSSIONS AND CONCLUSION
The experimental and theoretical values for Q(J = 0-5) transition energies in DT are listed in Table III. The present measurements with 0.0004 cm −1 accuracy are consistent with, but are two orders of magnitude more precise than, the previous investigation in Ref. [28]. The comparison of the present experimental and theoretical values demonstrates excellent agreement as listed in Table III and shown graphically in Fig. 3, with the differences (indicated by data points) falling well within 1σ of the combined measurement and calculation uncertainty (represented by the error bars). In summary, we have determined Q(J = 0-5) transition energies of the fundamental band of DT with a 250-fold improvement over all other previous measurements. Highly accurate calculations are also presented with similar improvements of uncertainty. Studies using the heavier tritiated species are useful in disentangling various mass-dependent effects that have been frequently overlooked in the literature [54]. In view of a present 2.7σ discrepancy in experiment and calculations in the dissociation energy of HD [44] while there is perfect agreement for H 2 and D 2 , investigations on heteronuclear species such as DT may be helpful in the resolution of the HD discrepancy.
Since the present experimental and theoretical values are in very good agreement, these can be used to constrain hypothetical long-range fifth forces between hadrons [55]. In fifthforce investigations on diatomic molecules, the hypothetical interaction is parametrized as a Yukawa potential, where α 5 and λ 5 are the interaction strength and length parameters, respectively, while R is the distance between the two nuclei of nuclear numbers N 1 and N 2 . The most stringent constraint for these hypothetical fifth forces in the range λ 5 ∼ 1 Å is derived from HD + [56] yielding α 5 < 8×10 −10 α, where α is the electromagnetic coupling strength. This tight bound is largely due to the sub-MHz uncertainties obtained in the HD + measurements. Applying the method in Ref. [55] on the DT Q(1) transition, the strength of a fifth force for an interaction range of ∼1 Å is constrained at α 5 < 2×10 −8 α, which is more than an order of magnitude weaker than the HD + derived bound. With the sixfold enhanced sensitivity of DT with respect to the lightest species H 2 as seen from Eq. (15), the present DT results yield a limit that is slightly more stringent than that of H 2 . This is despite the H 2 (v = 0 → 1) Q lines [16,42] having 2.7 times better accuracy than the present DT study. With the inherent sensitivity of the heavier tritiated species (with N t = 3) this limit can be further tightened when an accuracy in the kHz level is reached, for example by using techniques recently applied to HD [25].