D Mondelain
email: didier.mondelain@univ-grenoble-alpes.fr

S Kassi

A Campargue

Transition frequencies in the (2-0) band of D 2 with MHz accuracy

Keywords: Deuterium, hydrogen, D2, transition frequencies, CRDS, frequency comb

Series of spectra of the S(0) and Q(1)-Q(4) transitions of the (2-0) band of D 2 near 5860 cm -1 are recorded by comb-referenced cavity ring down spectroscopy at four pressures up to 200 Torr. After averaging, high resolution (30 MHz step) and low noise (noise equivalent absorption,  min ≈ 1.5×10 -11 cm -1 ) spectra are obtained for each pressure value. This leads to signal to noise ratios of several hundred for the considered very weak electric quadrupole transitions with intensities ranging between 4×10 -29 and 4×10 -28 cm/molecule. The Q(3) and Q(4) transitions are newly detected.

A multi-spectrum treatment of the D 2 line profiles at different pressures performed using a quadratic speed-dependent Nelkin-Ghatak profile leads to residuals close to the noise level. Considering statistical fit errors and possible biases, total uncertainties on the frequencies extrapolated at zero pressure are estimated between 0.30 MHz and 1.83 MHz.

The derived experimental frequencies and intensities of (2-0) transitions are gathered with literature values and compared to ab initio values reported in J. Komasa, M. Puchalski, P. Czachorowski, G. Łach, K. Pachucki, Phys. Rev. A 100, 032519 (2019). An agreement within the combined uncertainties is achieved for the S(0), Q(1), Q(2) and Q(3) transitions. Line intensities are also discussed in relation with literature values.

Introduction

The hydrogen molecule (H 2 ) and its isotopologues (HD, D 2 ) are the most simple neutral molecular systems, allowing thus for sophisticated ab initio calculations of their electronic ground state rovibrational energies [START_REF] Pachucki | Nonadiabatic corrections to rovibrational levels of H 2[END_REF][START_REF] Piszczatowski | Theoretical Determination of the Dissociation Energy of Molecular Hydrogen[END_REF][START_REF] Komasa | Quantum Electrodynamics Effects in Rovibrational Spectra of Molecular Hydrogen[END_REF]. Very recently, updated calculations have been made available including relativistic and quantum electrodynamic (QED) corrections and leading to accuracy on the transition frequencies between 10 -3 and 10 -7 cm -1 [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF]. The accuracy achieved in the calculations of the hydrogen spectrum gives an exceptional opportunity for a constructive interplay between theory and experiment. This matter of fact motivated recent experimental efforts to determine the absolute frequency of different absorption and Raman transitions for H 2 [START_REF] Dickenson | Fundamental Vibration of Molecular Hydrogen[END_REF][START_REF] Salumbides | QED Effects in Molecules: Test on Rotational Quantum States of H 2[END_REF][START_REF] Campargue | The absorption spectrum of H 2 : CRDS measurements of the (2-0) band, review of the literature data and accurate ab initio line list up to 35 000 cm 1[END_REF][START_REF] Kassi | Electric quadrupole transitions and collision-induced absorption in the region of the first overtone band of H 2 near 1.25 µm[END_REF][START_REF] Cheng | Electric-quadrupole transition of H 2 determined to 10 9 precision[END_REF][START_REF] Tan | Cavity ring-down spectroscopy of the electric quadrupole transitions of H 2 in the 784-852 nm region[END_REF][START_REF] Niu | Test of quantum chemistry in vibrationally hot hydrogen molecules[END_REF], HD [START_REF] Dickenson | Fundamental Vibration of Molecular Hydrogen[END_REF][START_REF] Kassi | Electric quadrupole and dipole transitions of the first overtone band of HD by CRDS between 1[END_REF][START_REF] Tao | Toward a determination of the protonelectron mass ratio from the lamb-dip measurement of HD[END_REF][START_REF] Cozijn | Sub-doppler frequency metrology in hd for tests of fundamental physics[END_REF][START_REF] Diouf | Lamb-dips and lamb-peaks in the saturation spectrum of HD[END_REF] and D 2 [START_REF] Dickenson | Fundamental Vibration of Molecular Hydrogen[END_REF][START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF][START_REF] Maddaloni | Absolute measurement of the S(0) and S(1) lines in the electric quadrupole fundamental band of D 2 around 3 μm[END_REF][START_REF] Fakhr-Eslam | Anomalous broadening and shifting in D 2 and D 2 -He mixtures[END_REF][START_REF] Borysow | NIR Raman spectrometer for monitoring protonation reactions in gaseous hydrogen[END_REF][START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF][START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF][START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF][START_REF] Martínez | Accurate wavenumber measurements for the S 0 (0), S 0 (1), and S 0 (2) pure rotational Raman lines of D 2[END_REF].

Transition frequencies reported with an accuracy down to a 10 -9 relative uncertainty provide crucial tests of quantum electrodynamic theory and interesting insights for physics beyond the Standard Model [START_REF] Ubachs | Physics beyond the Standard Model from hydrogen spectroscopy[END_REF].

In the case of the symmetric species (H 2 , D 2 ), the absorption spectrum consists of weak electric quadrupole (E2) vibrational bands. In the case of the HD isotopologue, the isotopic substitution leads to the appearance of a small electric dipole and the HD spectrum includes both electric dipole (E1) and E2 transitions. As a result, the HD spectrum is dominated by E1 bands which provide the most stringent tests for ab initio calculations. In particular, the accuracy of the line centre determination in HD transitions has been considerably improved in

Ref. [START_REF] Tao | Toward a determination of the protonelectron mass ratio from the lamb-dip measurement of HD[END_REF] and Ref. [START_REF] Cozijn | Sub-doppler frequency metrology in hd for tests of fundamental physics[END_REF] using saturation spectroscopy. Accuracies of 80 kHz and 20 kHz have been reported for the frequency of the R(1) transition in the (2-0) overtone band, respectively.

Nevertheless, the two values show a difference of ~1 MHz, largely beyond their combined accuracies even if the latter uncertainty was recently reconsidered and increased to 50 kHz because of a complex hyperfine structure [START_REF] Diouf | Lamb-dips and lamb-peaks in the saturation spectrum of HD[END_REF].

In the case of D 2 , the intensity of the E2 lines are several orders of magnitude weaker and saturation spectroscopy is hardly achievable. An accurate determination of the line centre in Doppler regime requires thus both a very high sensitivity and a high accuracy of the frequency axis. Highly sensitive techniques, such as CRDS, allow recordings at low pressures, minimizing the error due to extrapolation of the measured positions to zero pressure which is required for comparison to theory. Nowadays, the accurate calibration of the frequency axis of the recorded spectra can be achieved by linking the experimental setup to an optical frequency comb (OFC) referenced to an atomic clock [START_REF] Margolis | Spectroscopic applications of femtosecond optical frequency combs[END_REF]. This method has been recently applied to the determination of the S(2) transition frequency of the (2-0) band of D 2 [START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF][START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF] near 6204 cm -1 in Doppler regime. Line centers reported with small uncertainties of 500 kHz and 390 kHz, respectively, show a difference of 540 kHz, thus within the combined error bars. These experimental values were in disagreement with the best ab initio value provided at that time with a difference (~4 MHz) close to the 3σ uncertainty on the calculated value [START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF]. Nevertheless, the latest calculations [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF] show a better agreement (1 MHz difference instead of 4 MHz [START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF]) with these combined experimental values and a reduced uncertainty (0.75 MHz [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF] instead of 1.25 MHz [START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF]). The Torun group has recently remeasured the same transition with an improved accuracy of 160 kHz with a frequencystabilized cavity ring-down spectrometer in the frequency agile, rapid scanning spectroscopy (FARS) mode [START_REF] Zaborowski | Ultra-high finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules[END_REF]. This most recent value shows a larger difference of 1590 kHz (~5.3×10 -5 cm -1 ) compared to the theoretical value of Ref. [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF] reported with a 1σ uncertainty of 750 kHz.

The aim of the present contribution is to measure frequencies of other transitions of the same (2-0) band of D 2 with an accuracy at the MHz level or below in order to provide further validation tests of the latest ab initio calculations available [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF]. An overview of the (2-0) band as predicted by ab initio calculations is presented in Fig. 1. If we except the first observation of the Q(1)-Q(3) lines at high pressure (>35 bar) [START_REF] Brannon | Electric field induced spectra of molecular hydrogen, deuterium and deuterium hydride[END_REF], previous absorption studies of transitions are limited to (i) the S(0)-S(3) transitions reported by Gupta et al. by off-axis ICOS with position uncertainties between 3 MHz and 300 MHz retrieved from a standard Voigt profile analysis, [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF] (ii) our previous CRDS study of the Q(1), Q [START_REF] Piszczatowski | Theoretical Determination of the Dissociation Energy of Molecular Hydrogen[END_REF] and S(0)-S(8) transitions including the extremely weak S(8) line among the weakest absorption lines ever reported (line intensity of 1.8×10 -31 cm/molecule), which were analyzed using a Galatry line profile [START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] and (iii) the above mentioned studies dedicated to the high accuracy determination of the S(2)

line center [START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF][START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF][START_REF] Zaborowski | Ultra-high finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules[END_REF].

In the present work, the Q(1)-Q(4) and the S(0) absorption lines are measured (see Fig. 1). The very weak Q(3) and Q(4) lines are detected for the first time. The positions of the S(0), Q(1) and Q(2) transitions were previously reported by CRDS in [START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] with an accuracy limited to 30 MHz due to the absence of an absolute reference standard to calibrate the frequency axis. In the present experiment, an important gain on the frequency accuracy is achieved by linking the CRDS setup to a frequency comb referenced to a GPS referenced 10

MHz rubidium clock. In the following, the setup and spectra recordings are described (Part 2).

Then, the data analysis, line profile fitting procedures together with the evaluation of the uncertainties on the measured transition frequencies will be presented in Part 3 before comparison to ab initio calculations (Part 4). [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF]. The lines studied in Ref. [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF], Ref. [START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] and in this work are marked with red dots, blue stars and green circles, respectively. The S( 2) transition (square) was studied in Refs [START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF][START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF] and very recently in [START_REF] Zaborowski | Ultra-high finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules[END_REF].

Experimental setup and spectra recordings

The frequency comb referenced CRDS method and setup used for the recordings have been described in details in Refs. [START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by comb-assisted cavity ring down spectroscopy[END_REF][START_REF] Mondelain | Comb-Assisted Cavity Ring Down Spectroscopy of 17 O enriched waterbetween 7443 and 7921 cm -1[END_REF]. The accurate frequency values associated "on the fly" to each ring-down value allows not only for an absolute calibration of the frequency axis but also for a more accurate characterization of the line profile due to a reduced noise amplitude in particular on the sharp slopes of the line profile [START_REF] Tran | High sensitivity spectroscopy of the O 2 band at 1.27 μm: (II) air-broadened line profile parameters[END_REF][START_REF] Konefał | High sensitivity spectroscopy of the O 2 band at 1.27 μm: (I) pure O 2 line parameters above 7920 cm 1[END_REF][START_REF] Vasilchenko | Accurate absorption spectroscopy of water vapor near 1.64 μm in support of the MEthane Remote LIdar missioN (MERLIN)[END_REF].

Here, four distributed feedback (DFB) laser diodes were employed to measure the S(0), Q(1)-Q( 4) transitions. The optical cavity was fitted with two high reflectivity mirrors optimized between 5435 cm -1 and 5780 cm -1 leading to finesse between 130 000 for the Q( 4)

transition around 5826 cm -1 and 50 000 for the S(0) transition around 6035 cm -1 . Typical beat note widths of 700 kHz at the 1 ms time scale were measured between the different laser diodes and a tooth of a self-referenced frequency comb (from Menlo Systems). This provides the absolute frequency of the laser light, f exp , for each ring down (RD) event from:

f exp = f DFB + f AOM = f CEO + nf rep ± f BN + f AOM Eq. ( 1
)
The 250 MHz repetition rate, f rep , and the carrier-envelop offset, f CEO , (here equal to -20

MHz) of the comb were referenced to a GPS referenced 10 MHz rubidium clock. The beat note frequency, f BN , was determined at a 1 kHz repetition rate by Fourier transforming, with a 7 kHz resolution, the photodiode signal digitalized with a fast ADC (250 MHz sampling rate).

The frequency of the sinusoidal wave, f AOM , applied to the acousto-optic modulator, used to obtain the RD events by switching off the laser light in the cavity, was delivered by a direct digital synthesizer (DDS). Both the fast ADC clock and DDS were referenced to the 10 MHz reference.

Series of spectra for pure D 2 (Aldrich, chemical purity of 99.999% with atomic deuterium isotopic abundance of 99.96%) were recorded in static regime for each transition at four different pressures between 50 and 200 Torr (see Table 1). For a single scan, the noise equivalent absorption (NEA) evaluated as the rms of the baseline fluctuations varies between 6.5×10 11 and 8×10 11 cm -1 for the Q(1)-Q( 4) transitions and is around 2×10 10 cm 1 for the S(0) transition because of the much shorter RD time of ~80 µs instead of ~200 µs at lower wavenumbers. The gas pressure in the optical cavity was continuously monitored using a 1000 mbar pressure gauge (Model 622 Barocel pressure sensor from Edwards, 0.15% accuracy of the reading) and the temperature of the cell was measured with a temperature sensor (TSic 501, IST-AG, ±0.1 K accuracy) fixed on the cell surface, covered by an external blanket for thermal isolation. During all the recordings the mean temperature was 295 K with an excursion of ±0.5 K. The number of spectra (between 14 and 45) recorded for each transition and each pressure value is included in Table 1 leading to a total of more than 500 spectra with a typical duration of 10 min each. Each spectrum (typically 0.3 cm -1 to 0.5 cm -1 wide) is composed of several ten thousands of RD events, each of them associated to an absolute frequency calculated from Eq. (1). In order to reduce the amount of data, an averaging procedure was applied to each series of spectra corresponding to a given pressure value. It consists in gathering the CRDS measurement points within 30 MHz wide bins, removing the outliers, and averaging. As a result, typical NEA of 1.5×10 11 cm 1 was achieved after averaging. Figure 2 shows the pressure dependence of the Q(4) lines. Signal to noise ratios between 280 and 580 are achieved for the maximum absorption coefficients of few 10 more than 1000 for the highest pressure.

Transition S(0) Q(1) Q(2) Q(3) Q(4)

Data analysis and uncertainty budget

The first step of the spectra analysis was to remove the contribution of some weak impurity lines overlapping with the D the recordings, lines due water vapor and methane ppm and a few tens ppb, respectively, spectrum, corresponding to a species was simulated with the HITRAN subtracted (see Figure 3). profile [START_REF] Ngo | An isolated line-shape model to go beyond the Voigt profile in spectroscopic databases and radiative transfer codes[END_REF][START_REF] Tran | Efficient computation of some speed-dependent isolated line profiles[END_REF][START_REF] Ngo | Erratum to "An isolated line-shape model to go beyond the Voigt profile in spectroscopic databases and radiative transfer codes[END_REF][START_REF] Tran | Efficient computation of some speed-dependent isolated line profiles[END_REF]. For each spectrum, the baseline (including the contribution of the collision induced absorption (CIA) [START_REF] Frommhold | Collision-Induced Absorption in Gases[END_REF]) was assumed to be a quadratic polynomial of the wavenumber and adjusted by the fit. The multi-spectrum fitting procedure constrained the pressure broadening coefficient (γ 0 ) and the ν VC , γ 2 , δ 2 parameters to be identical for the different pressure values but the positions and line areas were fitted independently for each spectrum, δ 0 , being fixed to zero. Figure 2 shows the quality of the profile reproduction for the Q(1) and Q(4) transitions using a qSDNG profile. The residuals of the fit displayed on the lower panel of Fig. 2 are close to the noise level (1.8×10 11 cm 1 and 1.3×10 11 cm 1 for Q [START_REF] Pachucki | Nonadiabatic corrections to rovibrational levels of H 2[END_REF] and Q(4), respectively). The derived line parameters are listed in Table 2. The position at zero pressure, ν 0 and the pressure shift coefficient were retrieved from a weighted linear fit of the fitted positions versus the pressure. Each data point was weighted according to the inverse of its squared uncertainty (evaluated below). Similarly, the line intensity, S(T), was obtained from the proportionality factor of the areas versus the pressure. An inaccurate determination of the spectrum baseline or of the concentration of the interfering species contributes to the error budget of the position (and intensity). To evaluate this possible bias, we fitted each averaged spectrum alone because each spectrum can be affected differently. Hence, we investigated the influence of a change by ±25% of the CH 4 and H 2 O concentrations, this variation corresponding to a mismatch of the concentration clearly visible in the residuals. Similarly, we estimated the impact of the baseline by varying the spectral range over which the fit was performed. These type-B uncertainties [START_REF]Evaluation of measurement data -Guide to the expression of uncertainty in measurement[END_REF] were used to weight the positions and areas in the above described fit of the zero-pressure positions, line intensities and self-pressure shift parameters. The obtained fitted values of  0 , S(296 K) and δ 0 are included in Table 2 with their corresponding statistical uncertainty provided by the fit. The final uncertainty on the transition frequency range between 0.30 MHz and 1.83 MHz. Note that the signal-to-noise ratio of our spectra allowed us to evidence and take into account the asymmetry of the line shape which has an impact on the obtained line center.

Table 2. Line shape parameters obtained from multi-spectrum treatment of the CRDS spectra recorded for a series of pressure values. The δ 0 , γ 0 , ν VC , γ 2 , and δ 2 values are given in 10 -3 cm -1 atm -1 Unit. For a pressure series, the γ 0 , ν VC , γ 2 , and δ 2 values were constrained to be identical but the line centers and line areas were fitted independently for each spectrum, and used afterwards to derive the zero-pressure position ( 0 ,), line intensity, S(T) and self-pressure shift (δ 0 ). Line intensities were converted to the usual reference temperature (296 K). The δ 0 , γ 0 , ν VC , γ 2 , and δ 2 parameter values are plotted on Figure 6 versus m, where m= J" and J"+1 for the Q and S branch, respectively. A clear numerical correlation between γ 0 , ν VC and γ 2 parameters is observed γ 0 and γ 2 evolving similarly with m while ν VC is anticorrelated to these parameters (see discussion in Ref. [START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF]). Note that due to these numerical correlations, the physical meaning of the fitted line-shape parameters is subject to caution.

Comparison with ab initio calculations

In Table 3, we have gathered the positions, extrapolated to null pressure, and intensities and their final uncertainties, obtained in this work with literature values relative to the (2-0) band of D 2 [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF][START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF][START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF][START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF][START_REF] Zaborowski | Ultra-high finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules[END_REF]. The table includes the ab initio position values and uncertainties provided by the computer code of Ref. [START_REF]H2SPECTRE ver. 7.0 Fortran source code[END_REF] supported by calculations of Ref. [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF]. The experimental and theoretical uncertainties and the difference between the experimental and calculated values are plotted in Figure 7. For the S(0) and Q(1)-Q(4) transitions studied here, the uncertainty on the calculated values is experimental error bars ranging between 1.

between the calculated and measured

times the combined uncertainties

Our previous CRDS determination of the reported with uncertainties larger than at that time relied on some water reference line center known with such accuracy.

previous measured frequencies show a reasonable experimental error bar. This is not the cas axis ICOS technique which show deviations with an claimed error bar of 10

The S(2) line center is the one which was determined with the highest accuracy both in our group [START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF] (510 kHz) and in the Torun's group [ error bars is observed between the newly reported [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF][START_REF]H2SPECTRE ver. 7.0 Fortran source code[END_REF] and the combined experimental value (6241.127647 [START_REF] Niu | Test of quantum chemistry in vibrationally hot hydrogen molecules[END_REF] 6) transitions [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF][START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF][START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF][START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF][START_REF] Zaborowski | Ultra-high finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules[END_REF] calculations of Ref. [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF]. The grey rectangles on the right panel [START_REF]H2SPECTRE ver. 7.0 Fortran source code[END_REF]. All the error bars are given at 1σ.

The difference between the combination difference relations of the wavenumbers of these transitions. An excellent agreement (~5×10 the uncertainty on the calculated values is about 2×10 -5 cm -1 , on the same order experimental error bars ranging between 1.0×10 -5 cm -1 and 6.2×10 -5 cm and measured positions is satisfactory with maximum differences times the combined uncertainties.

Our previous CRDS determination of the Q(1)-Q [START_REF] Piszczatowski | Theoretical Determination of the Dissociation Energy of Molecular Hydrogen[END_REF] and S(0)-S( 8) line centers uncertainties larger than 10 -3 cm -1 (30 MHz) because the frequency calibration some water reference line center known with such accuracy.

previous measured frequencies show a reasonable agreement with ab initio experimental error bar. This is not the case for the S(2) position value measured by the off which show deviations of more than 6×10 -4 cm -1 while it was reported 10 -4 cm -1 (Figure 7).

(2) line center is the one which was determined with the highest accuracy both in and in the Torun's group [START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF] (390 kHz). An agreement within the observed between the newly reported ab initio value (6241.127614( 25 [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF][START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] for the Q(1)-Q( 4) and S(0)-S( 6) transitions and the calculated values [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF]. All the error bars are given at 1σ. Note the break on the y-axis scale.

Conclusion

Thanks to series of spectra recorded at different pressures using a CRDS spectrometer Trans.

Line position Line intensity at 296 K Reference c

Calc. (cm -1 ) [START_REF]H2SPECTRE ver. 7.0 Fortran source code[END_REF] Meas. (cm -1 ) δ 0 a (10 -3 cm -1 atm -1 )

Calc. [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF] (10 -28 cm/molecule) Meas.

(10 Notes a Literature values of δ 0 obtained with asymmetric line profiles are not reported in this table. The δ 0 values of our measurements differ from those listed in Table 2. For comparison purpose, they have been obtained with δ 2 fixed to zero (see Text).

b (Meas.-Calc.)/Calc.×100 c Previous works: Gupta2007 [START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF], Kassi2012 [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF], Mondelain2016 [START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF], Wcislo2018 [START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF] and Zaborowski2020 [START_REF] Zaborowski | Ultra-high finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules[END_REF].

Figure 1 .

 1 Figure 1. Overview of the calculated spectrum of the (2-0) band of D 2[START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF]. The lines studied in Ref.[START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF], Ref.[START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] and in this work are marked with red dots, blue stars and green circles, respectively. The S(2) transition (square) was studied in Refs[START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF][START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF] and very recently in[START_REF] Zaborowski | Ultra-high finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules[END_REF].

Figure 2 .

 2 Figure 2. Averaged spectra of the Q(1) (left panel) and Q(4) (right panel) transitions overtone band of D 2 , at four pressure values. The lower panels show the residuals multi-spectrum treatment using a qSDNG profile and right-hand panels.

  Averaged spectra of the Q(1) (left panel) and Q(4) (right panel) transitions at four pressure values. The lower panels show the residuals a qSDNG profile (see Text). Note the different scale between the leftshows the pressure dependence of the averaged line profile of the Signal to noise ratios between 280 and 580 are achieved for the ficients of few 10 -9 cm -1 . For the Q(1) line, it reaches a value of more than 1000 for the highest pressure.and uncertainty budgetthe spectra analysis was to remove the contribution of some weak overlapping with the D 2 lines of interest. As a result of the high sensitivity of water vapor and methane present with relative abundance of ~10, respectively, were observed in the spectra. For each averaged corresponding to a given pressure value, the contribution of these interfering simulated with the HITRAN2016 database[START_REF] Gordon | The HITRAN2016 Molecular Spectroscopic Database[END_REF] and a Voigt profile and then 7 Averaged spectra of the Q(1) (left panel) and Q(4) (right panel) transitions in the first at four pressure values. The lower panels show the residuals obtained from a . Note the different scale between the leftline profile of the Q(1) and Signal to noise ratios between 280 and 580 are achieved for the Q(4) line despite (1) line, it reaches a value of the spectra analysis was to remove the contribution of some weak As a result of the high sensitivity of present with relative abundance of ~10 were observed in the spectra. For each averaged contribution of these interfering and a Voigt profile and then

Figure 3 .

 3 Figure 3. Spectrum of the (2-0) Q(1) transition of D 2 recorded at 50 Torr after averaging (black line) and subtracting the contribution of methane present as an impurity with a few tens ppb relative concentration (red line). The CH 4 transitions provided by the HITRAN2016 database (blue circles) and the simulated methane spectrum (cyan solid line) are also shown.The averaged spectra free of impurity lines were used for a line profile fit of the D 2 transitions. As illustrated in Figure4, the standard Voigt profile is not sufficient to account for the observation. Dicke effect, characterized by the frequency of the velocity-changing collisions parameter, ν VC , in the hard-collision model, and speed dependence of the broadening and shift (γ 2 and δ 2 , respectively), modeled with a quadratic law here, lead to the narrowing and asymmetry of the line profile. A home-made multi-spectrum fitting program was used to fit the parameters of the quadratic speed-dependence Nelkin-Ghatak (qSDNG)

Figure 4 .

 4 Figure 4. Averaged spectrum of the Q(2) transition at 150 Torr (upper panel). The residuals obtained after subtraction of the spectrum fitted with Voigt (VP), Nelkin-Ghatak (NGP) and quadratic speeddependent Nelkin-Ghatak (qSDNGP) profiles are shown on the lower panels. Note the ×10 scaling between the residuals for the Voigt profile and for the other profiles.

Figure 5

 5 Figure5illustrates the linear behavior of the fitted positions versus the pressure. Except for the Q(4) transition, the rms deviation of the linear fit is smaller than the plotted uncertainties due to interfering lines and base line determination which are estimated below at each pressure.

Figure 5 .

 5 Figure 5. Pressure dependence of the line centers and corresponding linear fit. The zero-pressure values determined from the fit have been substracted from the measured values. For the sake of clarity, the values relative to the Q(1), Q(2), Q(3) and Q(4) lines have been vertically shifted by -1, -2, -3 and -4 (in 10 -4 cm -1 unit).

Figure 6 .

 6 Figure 6. Overview of the line profile parameters of the S(0) and Q(1)-Q(4) transitions of the (2-0) band of D 2 versus m where m=J" and J"+1 for the Q and S branch, respectively.

  cm measurements of Refs. [21,22 measured with an improved accuracy of 160 kHz FARS mode leading to a value of of 1590 kHz (~5.3×10 -5 cm uncertainties.

Figure 7 .

 7 Figure 7. Differences between the wavenumbers obtained experimentally for the Q(1) S(6) transitions[START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF][START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF][START_REF] Mondelain | Sub-MHz accuracy measurement of the S(2) 2-0 transition frequency of D 2 by Comb-Assisted Cavity Ring Down spectroscopy[END_REF][START_REF] Wcisło | Accurate deuterium spectroscopy for fundamental studies[END_REF][START_REF] Zaborowski | Ultra-high finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules[END_REF] calculations of Ref.[START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF]. The grey rectangles on the right panel[START_REF]H2SPECTRE ver. 7.0 Fortran source code[END_REF]. All the error bars are given at 1σ.

  ) cm ] and the combined experimental value (6241.127647[START_REF] Niu | Test of quantum chemistry in vibrationally hot hydrogen molecules[END_REF] cm 22]. The line center of this transition has recently an improved accuracy of 160 kHz with a frequency-stabilized leading to a value of 6241.1276670(54) cm -1 [26]. This value cm -1 ) with the theoretical value i. e. 2.1 times the combined between the wavenumbers obtained experimentally for the Q(1) 26] and the calculated values from computer code [ ]. The grey rectangles on the right panel represent the theoretical uncertainties ]. All the error bars are given at 1σ.The difference between the J= 0 and J= 2 ground state levels can be obtained using combination difference relations of the Q(2) and S(0) experimental and wavenumbers of these transitions. An excellent agreement (~5×10 -6 cm , on the same order than the cm -1 . The agreement with maximum differences of line centers[START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF] was the frequency calibration some water reference line center known with such accuracy. These ab initio values within the measured by the offwhile it was reported (2) line center is the one which was determined with the highest accuracy both in An agreement within the value (6241.127614(25) cm -1 ) ] and the combined experimental value (6241.127647(11) cm -1 ) derived from has recently been restabilized CRDS in the . This value shows a difference 2.1 times the combined between the wavenumbers obtained experimentally for the Q(1)-Q(4) and S(0)and the calculated values from computer code[START_REF]H2SPECTRE ver. 7.0 Fortran source code[END_REF] supported by represent the theoretical uncertainties n be obtained using (0) experimental and ab initio[START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF] cm -1 or 150 kHz) is

Figure 8 .

 8 Figure 8. Ratios between the intensities obtained in this work and in Refs[START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF][START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] for the Q(1)-Q(4) and S(0)-S(6) transitions and the calculated values[START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF]. All the error bars are given at 1σ. Note the break on the y-axis scale.

  linked to a rubidium clock referenced frequency comb, accurate transition wavenumbers were determined for the very weak Q(1)-Q(4) and S(0) transitions for the (2-0) band of D 2 . (line intensities between 4×10 -29 cm/molecule and 3.7×10 -28 cm/molecule). Accuracies between 330 kHz and 1860 kHz have been achieved corresponding to relative uncertainties from 1.8×10 -9 to 1.0×10 -8 . Two of these very weak transitions, Q(4) and Q(3), are measured for the first time. Comparison with the latest ab initio frequencies[START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF][START_REF]H2SPECTRE ver. 7.0 Fortran source code[END_REF] shows a good agreement within the combined uncertainties for S(0), Q(1), Q(2) and Q(3) transitions confirming the validity of the high order QED corrections of the calculations.

Table 1 .

 1

	Pressure. (Torr) 75/100/150/200 50/75/100/150 50/75/100/150 50/75/100/150 75/100/125/150
	Nb of spectra	27/28/16/14	26/19/29/14	33/33/24/19	37/27/28/18	45/23/28/17

Recording pressures and corresponding number of spectra acquired for each absorption line of the (2-0) band of D 2 .

Table 3 .

 3 Centre, intensity and pressure shift coefficient for the transitions of the (2-0) band of D 2 . Comparison with calculated values.

  -28 cm/molecule)

						Relative
						Difference(%) b
	Q(4)	5826.479170(22) 5826.479099(37)	-2.53(20)	0.428	0.420(8)	-1.9(19)	This work
	Q(3)	5843.084689(22) 5843.084584(61)	-2.54(34)	0.537	0.524(12)	-2.4(23)	This work
	Q(2)	5855.583348(22) 5855.583(1)	-2.16(18)	1.949	1.93(8)	-1.0(40)	Kassi2012
		5855.583358(14)	-2.99(8)		1.930(20)	-1.0(10)	This work
	Q(1)	5863.937095(22) 5863.938(1)	-2.67(12)	1.458	1.43(4)	-1.9(28)	Kassi2012
		5863.937052(29)	-2.85(18)		1.463(49)	+0.3(33)	This work
	S(0)	6034.650446(24) 6034.650(1)	-2.33(25)	3.780	3.76(7)	-0.5(20)	Kassi2012
		6034.6498(10)	-2.169(26)		3.47(1)	-8.2(3)	Gupta2007
		6034.650463(10)	-2.63(6)		3.692(31)	-2.3(8)	This work
	S(1)	6140.618429(24) 6140.620 (1)	1.7(4)	2.774	2.75(6)	+0.8(20)	Kassi2012
		6140.6205(100)	1.978(92)		3.35(2)	+20.8(6)	Gupta2007
	S(2)	6241.127614(25) 6241.129 (1)	2.5(5)	4.782	4.78(9)	-0.0(20)	Kassi2012
		6241.1270(1)	1.536(28)		4.36(2)	-8.8(5)	Gupta2007
		6241.127637(17)					Mondelain2016
		6241.127655(13)	a				Wcislo2018
		6241.127667(5)	a				Zaborowski2020
	S(3)	6335.717250(25) 6335.718 (1)		1.388	1.38(3)	+0.6(20)	Kassi2012
		6335.7168(10)	1.805(22)		1.32(1)	+4.9(8)	Gupta2007
	S(4)	6423.966767(26) 6423.968 (1)	1.859(76)	1.138	1.145(22)	+0.6(20)	Kassi2012
	S(5)	6505.501516(26) 6505.502 (1)	1.8(5)	0.1686	0.177(4)	+5.0(23)	Kassi2012
	S(6)	6579.997182(27) 6580.002 (1)	2.6	0.07349	0.0732	+0.4(20)	Kassi2012

obtained between the calculated and measured values as reflected by the very close (exp.-ab initio) deviations of the Q(2) and S(0) transitions plotted in Figure 7 (to be compared to an absolute uncertainty on the calculated ground state energy levels around 1.5×10 -4 cm -1 [START_REF] Komasa | Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory[END_REF][START_REF]H2SPECTRE ver. 7.0 Fortran source code[END_REF]).

Only three sources of experimental intensities are available for comparison with ab initio calculations (Table 3 and Figure 8). The intensities measured in this work with error bars ranging between 0.8 and 3.3% show differences of a few percent compared to the calculated values reported with relative uncertainties below 0.1% [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF]. The largest difference concerns the S(0) transition for which a deviation close to 3σ is observed while the previous determination of Ref. [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF] was much closer to the ab initio value. For the Q(1), Q [START_REF] Piszczatowski | Theoretical Determination of the Dissociation Energy of Molecular Hydrogen[END_REF] and S(0) transitions, our intensity values agree with the previous CRDS values [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF] within the error bars. Again, the error bars reported by Gupta et al. [START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] seem largely underestimated. Deviations of +20.8 and -8.8 % are noticed for the S(0) and S(0) intensities reported with uncertainties of only 0.45% and 0.3%, respectively, which seems unrealistic considering the simple Voigt profile used in that reference. We note that overall our present intensity values have the tendency to be smaller than calculated values but the differences remain close to the experimental error bars. Unfortunately, the Torun group limited their study of the S(2) line to a sophisticated profile analysis and did not report the corresponding intensity value which would have been particularly valuable for comparison.

Pressure-shift coefficients given in Refs. [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF][START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] are included in Table 3 for comparison purpose. Note that the listed δ 0 values relative to our work differ from those given in Table 2:

as most of the literature values did not consider an asymmetric profile in their analysis, we performed a new multi-spectrum analysis constraining δ 2 to be zero. An agreement within 1σ is observed with [START_REF] Kassi | The absorption spectrum of D 2 : Ultrasensitive cavity ring down spectroscopy of the (2-0) band near 1.7 µm and accurate ab initio line list up to 24 000 cm -1[END_REF] for the S(0) and Q(1) transitions but a discrepancy larger than 3σ is noticed for the Q(2) transition. A large disagreement with [START_REF] Gupta | Quantitative determination of the (2,0) band of deuterium in the near infrared via off-axis ICOS[END_REF] is also observed for the S(0) transition.