Sub-ppb measurement of a fundamental band rovibrational transition in HD

We report a direct measurement of the 0-1 R(0) vibrational transition frequency in ground-state hydrogen deuteride (HD) using infrared-ultraviolet double resonance spectroscopy in a molecular beam. Ground-state molecules are vibrationally excited using a frequency comb referenced continuous-wave infrared laser, and the excited molecules are detected via state-selective ionization with a pulsed ultraviolet laser. We determine an absolute transition frequency of 111 448 815 477(13) kHz. The 0.12 parts-per-billion (ppb) uncertainty is limited primarily by the residual first-order Doppler shift.

Precise measurements of vibrational transition frequencies in the isotopologues of molecular hydrogen can provide a sensitive probe of fundamental physics. Because these transitions can be predicted with high precision using ab-initio theory, comparisons between theory and experiment can be used to test quantum electrodynamics, search for new forces beyond the standard model, and determine the proton-electron and deuteron-electron mass ratios more precisely [1][2][3][4]. In recent years, many precise measurements of molecular hydrogen transition frequencies have been published [5][6][7][8][9][10][11], with some recent works reporting fractional uncertainties of less than one part-per-billion (ppb, 10 −9 ) on vibrational overtone and electronic transition frequencies [4,[12][13][14][15]. Many of these recent experiments (with a few notable exceptions [9,12]) detect infrared absorption by hydrogen in a gas cell. In order to determine accurate transition frequencies from such measurements, the data analysis must properly account for the effect of collisions on the line shape and position [11]. If saturation techniques are used to achieve sub-Doppler resolution, additional difficulties can arise due to the complex structure of the saturation features [16].
In this work, we demonstrate a technique that avoids both of these issues by measuring the molecules in the low-density, cold environment of a supersonic molecular beam. Ground-state hydrogen deuteride (HD) molecules in the beam are vibrationally excited using a tunable continuous-wave (cw) narrow-linewidth infrared (IR) laser referenced to an optical frequency comb (OFC) for absolute accuracy. To detect the excitation efficiency, the excited molecules are state-selectively ionized using a pulsed ultraviolet (UV) laser, and the HD + ions are massselectively detected using a time-of-flight mass spectrometer. This detection scheme is both efficient and nearly background free, making it sensitive enough to detect weak transitions in a sparse sample. Based on the measured infrared spectra, we are able to determine the absolute frequency of the 0-1 R(0) transition with an uncertainty of 13 kHz or 0.12 ppb fractional uncertainty.
The infrared spectroscopy laser is produced as the idler of a cw optical parametric oscillator (OPO) based on The main panel shows a side view of the molecular beam. HD molecules traveling upward are first excited by a 2690-nm IR spectroscopy laser and are subsequently ionized in a time-offlight mass spectrometer using a pulsed 209-nm UV laser. To compensate for first-order Doppler shifts, the spectroscopy laser is retroreflected from mirror "M1" and returns through a slot "S". The top left panel ("Top View") shows a perpendicular view of outgoing and returning infrared beams near the OPO. The returning beam is slightly offset so that it can be separated from the outgoing beam using the mirror "M2" and detected with a power meter "PM". The lower left panel shows the IR laser stabilization scheme. the design described by Ricciardi et al. [17]. The OPO uses a periodically-poled lithium niobate (PPLN) crystal in a bowtie cavity to convert a ∼10-W, 1064-nm pump laser into a signal beam at 1762 nm and an idler beam at 2690 nm. To measure and stabilize the idler frequency, we use a Ti:Sapphire-based optically-locked OFC, which has been described in detail elsewhere [18]. This comb is stabilized in such a way that the carrier-envelope offset frequency f 0 is zero and the mode number n p has a fixed 100-MHz offset from a 1064-nm reference laser. The OPO pump beam is generated by amplifying the reference laser and thus has the same frequency, ν p . The frequency of the signal beam ν s is measured by frequency doubling the signal output to 881-nm using an external PPLN crystal and measuring the frequency of its beat note with the OFC, f bn,881 . Using this beat-note frequency, we can arXiv:2002.09333v1 [physics.atom-ph] 21 Feb 2020 determine the absolute frequency of the idler beam, ν i = ν p − ν s , using In the current measurements, the mode number of the pump beat note n p is 281 631, and the mode number of the signal second-harmonic beat note n 2s is 340 364. The comb repetition rate f r ≈ 999 996 455.5 Hz is monitored during each measurement relative to a rubidium oscillator disciplined by a global navigation satellite system (GNSS) receiver. The lower left inset of Figure 1 illustrates this scheme for measuring the idler frequency. The main body of Figure 1 shows the molecular beam apparatus. A mixture of 13% HD in xenon is expanded upward through a piezo-actuated pulsed valve at a repetition rate of 50 Hz. The molecules pass into a second differentially-pumped chamber and are collimated by a series of rings, resulting in a beam with an angular spread of 3.4 mrad. Approximately 625 mm from the valve, the molecules pass through the 8-mm wide infrared spectroscopy laser, and 150 mm further downstream, HD molecules in the X 1 Σ + , v = 1, J = 1 state are ionized at the entrance of a time-of-flight mass spectrometer with a pulsed UV laser using 2+1 resonanceenhanced multiphoton ionization (REMPI) through the The UV laser is a frequency-tripled pulsed dye laser that produces ∼1 mJ per pulse at 209 nm, and the pulse arrives 1.95 ms after the molecules leave the nozzle, selecting HD molecules with a velocity of 400 m/s. The extraction field in the mass spectrometer is switched on during a ±100 µs window around the ionization pulse but is switched off while the molecules are passing through the infrared laser to minimize fields in the spectroscopy region. Spectra are measured by recording the time-resolved ion signal while scanning the frequency of the infrared laser. The upper left panel of Figure 2 shows a typical averaged measurement of the ion signal as a function of time delay after the laser pulse and infrared laser frequency, while the lower panel shows the ion signal versus time delay for a single laser frequency. The Gaussian peak at 3.38 µs corresponds to the HD + mass channel. To determine the total ion intensity in this peak, the peak center and its standard deviation σ are first computed by averaging time traces at all laser frequencies and fitting the trace with the sum of a Gaussian function and a linear background. The ion signal is then computed at each laser frequency by averaging the signal over a ±2σ region around the peak and subtracting a background calculated by averaging over regions covering (−8σ,−4σ) and (+4σ,+8σ) relative to the peak. The black curve in the right panel of Figure 2 shows the normalized ion intensity as a function of laser frequency, while the red curve shows a fit to this data using five overlapping Gaussian peaks. In order to determine the absolute 0-1 R(0) transition frequency from the measured spectra, a number of potential systematic shifts have been considered. The recoil shift hν 2 0 /(2m HD c 2 ) ≈ 9.1 kHz and the second-order Doppler shift −ν 0 v 2 /(2c 2 ) ≈ −0.1 kHz can be computed with high accuracy and are corrected in the reported value. Other effects are found to have a negligible influence on the measured transition frequency. Based on the HD polarizability computed by Ko los et al. [19], the alternating-current (ac) Stark shift is estimated to be less than 1 Hz for the laser intensity used in the experiment. External Helmholtz coils are used to reduce the magnetic field in the spectroscopy region to below 3 µT, resulting in a residual Zeeman shift of less than 100 Hz. The gas density in the spectroscopy region is estimated to be ∼ 3 × 10 17 molecules per cubic meter; applying the −10 kHz/Pa pressure shift reported by Cozijn et al. [13] for the 0-2 R(1) transition would result in an estimated pressure shift of 10 Hz. Errors in the frequency of the rubidium reference are expected to contribute less than 0.5 kHz to the overall uncertainty.
Two systematic effects are not so easily ignored and must be considered in further detail. The first and most significant is the residual shift due to the first-order Doppler effect. Although the infrared laser is nominally aligned so that its propagation direction is perpendicular to the central velocity of the molecular beam, even a small error in this alignment can result in a significant shift of the measured transition frequency. To detect such a shift, the infrared laser is retroreflected after passing through the spectroscopy region and interacts with the molecules a second time. If the retroreflection were perfect in both direction and amplitude, the Doppler shift of the second beam would be equal and opposite to the first and both beams would contribute equally to the vibrational excitation, resulting in no net shift. Unfortunately, an angular deviation between the outgoing and returning laser beams along the molecular beam direction or an imbalance of the amplitudes would result in imperfect cancellation.
To limit the angular deviation between the two beams, a 7-mm wide slot near the OPO (labeled "S" in Figure 1) ∼3 m from the retroreflection mirror ("M1") helps constrain the offset between the outgoing and returning beams along the molecular beam direction. The slot is aligned so that it is centered vertically on the outgoing beam, and the returning beam must pass through the same slot to reach a power meter ("PM" in Figure 1). The returning beam is offset by about 7 mm from the outgoing beam in the direction perpendicular to the molecular beam (see the "Top View" panel of Figure 1) to facilitate the power measurement. Based on the sensitivity of the measured power to changes of the slot height, we estimate an uncertainty of the offset between the outgoing and returning beams along the molecular beam direction of 0.5 mm, which translates to a 12 kHz uncertainty of the infrared transition frequency. This error is included both as a random error that contributes to the uncertainty of the transition frequency extracted from each spectrum and as a potential systematic error that shifts all spectra in the same direction.
The mismatch in amplitude between the two beams (caused by losses in the window and retroreflection mirror) is compensated by measuring each spectrum both with and without the retroreflected beam. If the transition frequency determined with both beams (ν 2 ) is the same as the frequency determined with one beam (ν 1 ), then the laser is perpendicular to the molecular beam, but a difference between the two frequencies indicates that ν 2 has been shifted from its true value by an amount proportional to ν 1 −ν 2 . To apply this concept to the measured data, we fit all twelve measured frequency pairs (ν 1 , ν 2 ) with a linear model ν 2 = aν 1 + b using an orthogonal distance regression fit [20] in order to account for the uncertainties in both coordinates. The Doppler-corrected frequency is determined by finding the crossing point between the linear model and the line ν 1 = ν 2 , which occurs at ν 1 = ν 2 = b/(1 − a). The uncertainty of this crossing point is determined by propagating the errors given by the fit covariance matrix for a and b. Figure 3 illustrates the results of such a fit.
Uncertainties in the relative contributions of the individual hyperfine components in the transition can also contribute an error to the measured transition frequency. The R(0) transition contains nine hyperfine components which, due to the degeneracy between the F = 1/2 and F = 3/2 levels in the ground state, results in five unique transition frequencies spread over ∼300 kHz. Figure 4 shows the positions of these components relative to the transition center. These individual components are not resolved in our experiment but are blended into a single ∼500-kHz wide peak. It is therefore important to accurately predict the relative intensities of these five components in order to correctly determine the transition center from a measured spectrum.
To model the hyperfine substructure, we use an effec-tive Hamiltonian defined bŷ (2) A separate set of parameters is used for each vibrational state. The operatorN is the rotational angular momentum andÎ H andÎ D are the spins of the hydrogen and deuterium nuclei, respectively. Band origins T v and rotational constants B v and D v are determined by fitting the energies calculated by the program H2Spectre [1] for the first three rotational levels in each vibrational state. The hyperfine parameters c H,v , c D,v , eQq 0,v , and S v ("c dip ") are taken from Dupré [21]; the signs of c H,v and c D,v have been inverted to correctly reproduce the results from that work. Eigenenergies and eigenvectors based on these parameters are calculated using the program spcat [22].
We then define P IR as the normalized sum of onephoton transition strengths from any of the degenerate X 1 Σ + , v = 0, J = 0 hyperfine levels to a specific M F level in the X 1 Σ + , v = 1, J = 1 state due to an IR laser polarized along the Z axis.
In general, F 1 (defined byF 1 =N +Î H ) is not a good quantum number; the symbol |v , J , F 1 , F , M F is used here as a shorthand for the eigenstate with the largest contribution from the corresponding basis vector. Because the linewidth of the UV laser is broad enough to cover all hyperfine components, the ionization efficiency from a particular M F level in the X 1 Σ + , v = 1, J = 1 state is modeled as the normalized sum of two-photon transition strengths to any EF 1 Σ + , v = 0, J = 1 level due to a UV laser polarized along the X axis.
The normalization factors A IR and A UV are chosen such that the average values of P IR and P UV are 1.
The amplitudes of the black bars in Figure 4 show the strengths of the hyperfine components of the transition calculated by summing P IR (F 1 , F , M F ) over all M F . If the detection efficiency of the UV laser is taken into account by instead summing P IR (F 1 , F , M F ) × P UV (F 1 , F , M F ) over all M F , it is found that certain transitions are detected less efficiently, as indicated by the red dots. The intensities predicted by the second model (red dots) only hold if there is no saturation of the UV transition and no reorientation of the molecules between the IR excitation and UV ionization  [9] and [23] 111 448 818.5(6.6) −3.0(6.6) Ref. [1] 111 448 814.5(6) 1.0 (6) lasers. If either of these conditions does not hold, the relative intensities of the hyperfine components will be more closely described by the first model (black bars).
To account for this possibility, we analyze the measured spectra using the intensities predicted by both models and report the average of the two results as a best estimate; half of the difference is then included in the error budget.  Table I summarizes the contributions to the measured transition frequency. After correcting for shifts due to recoil and second-order Doppler effects, we conclude an absolute frequency for the HD 0-1 R(0) transition of 111 448 815.477(13) MHz, with the uncertainty dominated by residual first-order Doppler shifts. Table II shows a comparison between this result and previous theoretical and experimental values. The experimental value is determined by combining the 0-1 Q(1) transition frequency reported by Niu et al. [9] with the J = 0 → 1 rotational transition frequency reported by Drouin et al. [23]; the theoretical value is computed using H2Spectre [1]. The present result agrees with both previous values but shows a factor of 500 smaller uncertainty than the experimental result and 50 smaller than the theoretical. Interestingly, we note that five other measurements of hydrogen vibrational transition frequencies [7,10,11,15,16], covering all three stable isotopologues, show fractional deviations from the theoretical predictions from H2Spectre consistent with the 8.7×10 −9 deviation measured here to within experimental uncertainty.
The precision of the current result is limited primarily by residual first-order Doppler shifts and possible shifts due to unresolved hyperfine structure. We anticipate that, by improving the retroreflection quality and characterizing the hyperfine effects by changing the relative polarizations between the two lasers, the uncertainty can be reduced below 1 kHz, or 10 −11 fractional uncertainty. Measurements at this level of precision, combined with accurate theoretical predictions, would result in values for the proton-electron and deuteron-electron mass ratios that are more precise than the 2018 CODATA recommended values [24]. With improved sensitivity, the same techniques used here could also be used to measure quadrupole transitions in the homonuclear isotopologues, making it possible to investigate the properties of the proton and deuteron separately. We gratefully acknowledge S. Kaufmann and K. Papendorf for lending us the UV ionization laser, as well as M. De Rosa for his extensive advice on the design of the OPO.