Sub-Doppler frequency metrology in HD for test of fundamental physics
F.M.J. Cozijn, P. Dupre, E.J. Salumbides, K.S.E. Eikema, W. Ubachs
SSub-Doppler frequency metrology in HD for test of fundamental physics
F. M. J. Cozijn, P. Dupr´e, E. J. Salumbides, K. S. E. Eikema, and W. Ubachs Department of Physics and Astronomy, LaserLaB, Vrije Universiteit Amsterdam,de Boelelaan 1081, 1081 HV Amsterdam, The Netherlands Laboratoire de Physico-Chimie de l’Atmosph`ere, Universit´e du Littoral Cˆote d’Opale,189A Avenue Maurice Schumann, 59140 Dunkerque, France (Dated: December 25, 2017)Weak transitions in the (2,0) overtone band of the HD molecule at λ = 1 . µ m were measuredin saturated absorption using the technique of noise-immune cavity-enhanced optical heterodynemolecular spectroscopy. Narrow Doppler-free lines were interrogated with a spectroscopy laserlocked to a frequency comb laser referenced to an atomic clock to yield transition frequencies [R(1)= 217 105 181 895 (20) kHz; R(2) = 219 042 856 621 (28) kHz; R(3) = 220 704 304 951 (28) kHz] atthree orders of magnitude improved accuracy. These benchmark values provide a test of QED inthe smallest neutral molecule, and open up an avenue to resolve the proton radius puzzle, as wellas constrain putative fifth forces and extra dimensions. Molecular hydrogen, the smallest neutral molecule, hasevolved into a benchmark quantum test system for funda-mental physics now that highly accurate measurementschallenge the most accurate theoretical calculations in-cluding relativity and quantum electrodynamics (QED)[1, 2], even to high orders in the fine structure constant(up to mα ) [3]. The measurement of the H dissociationenergy [4] was a step in a history of mutually stimulat-ing advancement in both theory and experiment witness-ing an improvement over seven orders of magnitude sincethe advent of quantum mechanics [5]. Accurate resultson the fundamental vibrational splitting in hydrogen iso-topologues [6], with excellent agreement between experi-ment and theory, have been exploited to put constraintson the strengths of putative fifth forces in nature [7] andon the compactification of extra dimensions [8].A straightforward strategy to obtain accurate rovibra-tional level splittings in the hydrogen molecule is to mea-sure weak quadrupole transitions, as was done for H inthe first [9] and second overtone band [10, 11], as wellas in the fundamental [12] and overtone [13, 14] bandsof D . In the heteronuclear isotopologue HD, exhibitinga charge asymmetry and a weak dipole moment [15], asomewhat more intense electric dipole spectrum occurs,first measured by Herzberg [16]. The dipole moment ofthe (2-0) band is calculated at 20 µ D [17], in reasonableagreement with experiment [18, 19]. Accurate Doppler-broadened spectral lines in the HD (2-0) band were re-ported using sensitive cavity ring down techniques [19].These lines exhibit a width in excess of 1 GHz at roomtemperature, which challenges the determination of cen-tre frequencies in view of various speed-dependent colli-sional broadening and shifting phenomena [20]. Carefulline shape analysis has led to accuracies of ∼
30 MHz, inaccordance with the ab initio calculated values [21].Here we report on the implementation of an absorp-tion technique that combines the advantages of fre-quency modulation spectroscopy for noise reduction andcavity-enhanced spectroscopy for increasing the interac-
FIG. 1: (Color online) Experimental setup. The spec-troscopy laser (ECDL) is sent through a modulator (EOM)to impose both f PDH and f FSR modulations. f PDH is usedto stabilize the laser (carrier) frequency to the optical cavity(also the HD absorption cell) and f FSR to generate sidebandfrequencies that are resonant to adjacent cavity modes. Thespectroscopy laser is locked to a Cs atomic clock via an opticalfrequency comb laser for long-term stabilization. Additionalcavity-length dither modulation f dith is applied for lock-indetection of the HD saturated absorption signals. tion length between the light beam and the sample. Thisextremely sensitive technique, known as Noise-ImmuneCavity-Enhanced Optical Heterodyne Molecular Spec-troscopy (NICE-OHMS) [22–25], was applied to molecu-lar frequency standards [26] and to precision measure-ments on molecules of astrophysical interest [27]. Inthe present study, weak electric dipole transitions in HDhave been saturated, allowing for a reduction of linewidthdown to 150 kHz (FWHM), some four orders of magni-tude narrower than the Doppler-broadened lines previ-ously reported [19]. The experimental scheme is depictedin Fig. 1, where the spectroscopy laser is simultaneously a r X i v : . [ phy s i c s . a t o m - ph ] D ec - 4 0 0 - 2 0 0 0 2 0 0 4 0 024681 0 Relative absorption (10-12 cm-1) f - 2 1 7 1 0 5 1 8 1 8 9 5 ( k H z ) 2 . 0 P a R ( 1 )
FIG. 2: (Color online) Recordings of the HD (2,0) R(1) linefor three different pressure conditions averaging 5 scans for 2.0Pa, 7 scans for 1.0 Pa, and 4 scans for 0.5 Pa. The solid (red)lines are fits using a line shape function based on a derivativeof dispersion [23] while allowing for a baseline slope. Thecurves have been shifted in the vertical direction for clarity. locked to the stable optical cavity, and also to a Cs-clock-referenced frequency comb laser to provide an absolutefrequency scale during the measurements.The 48 . ∼
130 000) cav-ity comprises a pair of curved high-reflectors (Layertec, 1-m radius of curvature), with one of the mirrors mountedon a piezoelectric actuator. This stabilized optical cavityalso provides short-term frequency stability to our spec-troscopy laser, and transfers the absolute accuracy of thefrequency standard. The setup provides an intracavitypower in the order of 100 W that is sufficient for satura-tion of HD, while the equivalent absorption path lengthamounts to ∼
40 km. The cavity itself is enclosed withina vacuum chamber, which can be pumped and filled withthe HD gas sample.The laser source (ECDL, Toptica DL Pro) operatingaround 1 . µ m is mode-matched and phase-locked to theoptical cavity . The laser beam is fiber coupled and split,with one part for the frequency calibration and metrol-ogy, while the main part is phase-modulated through afiber-coupled EOM (Jenoptik PM1310), allowing for thesimultaneous modulation of two frequencies f PDH ∼ f FSR ∼
310 MHz. The reflected beam of thecavity is collected onto an amplified photoreceiver, thesignal of which is used for locking both the laser frequency f opt through the Pound-Drever-Hall (PDH) scheme [28]and the cavity free spectral range frequency f FSR withthe DeVoe-Brewer scheme [29]. The beam transmittedthrough the cavity is collected with another high-speedphotoreceiver, with the amplified signal demodulated by f FSR in a double-balanced mixer. The resulting disper-sive NICE-OHMS signal is sent to a lock-in amplifier toextract the 1 f signal component at the dither frequency - 4 0 0 - 2 0 0 0 2 0 0 4 0 012345 - 4 0 0 - 2 0 0 0 2 0 0 4 0 0 R ( 3 )
Relative absorption (10-12 cm-1) f - 2 1 9 0 4 2 8 5 6 6 2 1 k H z R ( 2 ) f - 2 2 0 7 0 4 3 0 4 9 5 1 k H z FIG. 3: (Color online) Saturation spectra of the R(2) andR(3) transitions of the HD (2,0) overtone band at 1 Pa pres-sure. [R(2): 12-scan average; R(3): 5-scan average] f dith ∼
430 Hz. The noise equivalent absorption for thesetup is estimated to be 3 × − / (cm √ Hz).The long term frequency stability and accuracy of thesystem is obtained by beating the spectroscopy laserwith a frequency comb (Menlo Systems FC1500-250-WG)stabilized to a Cs clock frequency standard (MicrosemiCSIII Model 4301B). The acquired beatnote frequency f beat is counted using an RF counter, and is also used togenerate the steering signal for locking the cavity length,thereby tuning the laser frequency f opt , which is deter-mined via: f opt = f ceo + n × f rep + f beat , (1)where f ceo = 20 MHz is the carrier-envelope frequencyoffset of the frequency comb laser, f rep ∼
250 MHz is itsrepetition rate, and n ∼ . × is the mode number.The absolute frequency of f opt is determined with anaccuracy better than 1 kHz.R(1) transitions, recorded at different pressures, areplotted in Fig. 2, where each curve is an average of 4 to7 measurements. A typical scan takes about 12 minutes,with frequency intervals of 12.5 kHz, and with each datapoint averaged over 6 seconds. Fig. 3 displays weakerresonances, where the R(2) spectrum is an average of 12scans and that of R(3) an average of 5 scans.The assessment of systematic effects was performedprimarily on the R(1) transition, where the signal-to-noise ratio is the highest. The R(1) transition frequencywas measured at different pressures in the range 0 . − . −
450 kHz. This allowed the determination of apressure-dependent shift coefficient at − obtained FWHM width (kHz) f - 217 105 181 820 kHz ( a ) P r e s s u r e ( P a ) ( b )
FIG. 4: (Color online) Pressure-dependent frequency shift (a)and broadening (b) of the R(1) transition in the 0 . from studies (e.g. [10]) involving pressures higher thankPa. For R(2) and R(3) transitions measured at 1 Pa, apressure shift correction of − [11] andD [13], where it was shown that the collisional shiftparameters only slightly depend on rotational quantumnumber.As seen in Fig. 2, there is an increase in line shapeasymmetry with increasing pressure to which several ef-fects, associated with line broadening (see below), cancontribute. In addition, we observe that water vapor ab-sorption in the vicinity of the HD resonances is a likelycause of asymmetry. This asymmetry ultimately lim-its the present determination of the transition center toan accuracy of ∼ / µ m,the recoil shift is 34 kHz, that results in a doublet split-ting of 68 kHz but does not produce a shift. At half theintracavity laser power, no significant shift of the linecenter is observed, and we estimate an upper limit of10 kHz for the power-dependent or ac-Stark shift. Thesecond-order Doppler shift is calculated to be 1 . TABLE I: List of corrections ∆ f and uncertainty estimates σ f in units of kHz for the transition frequencies.R(1) R(2), R(3)Contribution ∆ f σ f ∆ f σ f line fitting 0 15 0 20pressure shift a − < < − − a R(1) has been extrapolated to zero pressure, while for R(2) andR(3) a correction is applied based on pressure-shift coefficient ofR(1). issues see discussion below.The collisional or pressure broadening, plotted inFig. 4(b) for the R(1) line, also follows a linear behaviorwith a slope of 70(7) kHz/Pa. It is remarkable that thelinear trend extends even to the lowest pressure of 0 . . µ m [31, 32]. Simi-lar observations of strongly reduced linewidths below thetransit-time limit have been shown in methane [33, 34]and acetylene [23], where it was attributed to the domi-nant contribution of slow-moving molecules in the satura-tion signal. Even if the entire width of 150 kHz would beattributed to transit-time broadening an effective kinetictemperature of 185 K would result, but in view of otherlinewidth contributions the temperature must be signif-icantly lower. From this we also deduce that hyperfinestructure only can have a minor contribution to the linebroadening, even though hyperfine splittings between F -components in the HD ( v = 0 , J = 1) level span about220 kHz [35]. This may be explained by the hyperfinecomponents in the near-infrared transition overlappingin view of hyperfine level splittings in v = 2 to be similaras those in v = 0.Table I lists the error budget of the present study.The statistics entry demonstrates the reproducibility ofmeasurements performed on different days, with the beststatistics at 10 kHz obtained for R(1). We estimate atotal uncertainty, including systematics, of σ f = 20 kHzfor the R(1) transition frequency, and σ f = 28 kHz for TABLE II: Comparison of R-branch transition frequencies in the HD (2,0) band obtained from the present study with previousexperimental determination ∆ exp [19], and with most accurate ab initio calculations ∆ calc [21]. Values are given in MHz withuncertainties in units of the last digit indicated in between parentheses. See text for a discussion of the theoretical uncertainty.Line This study Ref. [19] ∆ exp
Theory [21] ∆ calc
R(1) 217 105 181 .
895 (20) 217 105 192 (30) −
10 217 105 180 2R(2) 219 042 856 .
621 (28) 219 042 877 (30) −
20 219 042 856 1R(3) 220 704 304 .
951 (28) 220 704 321 (30) −
16 220 704 303 2 the R(2) and R(3) resonances. Resulting transition fre-quencies of the R(1), R(2), and R(3) lines are listed inTable II. These values are compared to results of the pre-vious experimental determination by Kassi and Campar-gue [19] obtained under Doppler-broadened conditions,showing good agreement, with the present results repre-senting a three order of magnitude improvement in accu-racy. Theoretical level energy calculations by Pachuckiand Komasa [21] were claimed to be accurate to 30 MHz,but values were provided to 3 MHz (10 − cm − ) accu-racy. Since we compare with the energy splittings be-tween v = 0 →
2, the theoretical transition frequenciesin Table II should be more accurate due cancellations invarious energy contributions. This assessment of the cal-culation uncertainty is supported by the excellent agree-ment between our measurements and the theoretical val-ues that is better than 2 MHz.The 30-kHz absolute accuracy (10 − relative accu-racy) achieved in this study constitutes a thousand-foldimprovement over previous work and demonstrates thefirst sub-Doppler determination of pure ground statetransitions in HD, and in fact in any molecular hydrogenisotopologue. The experimental results challenge currentactivities in first principles relativistic and QED calcula-tions of the benchmark hydrogen molecules [2, 3, 21, 36].When such calculations reach the same accuracy level asthe experiment there is a potential to constrain theoriesof physics beyond the Standard Model, as was shown pre-viously [7, 8]. The finite size of the proton contributes ∼
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