Absolute measurement of the S(0) and S(1) lines in the electric quadrupole fundamental band of $D2$ around $3μm$

Emerging trends in fundamental chemistry and physics, such as exact first-principles calculations of molecular properties based on quantum computing,¹ cooling and trapping of molecules,² as well as the search for space-time variation of fundamental constants,^3,4 have, in recent years, aroused renewed interest in precision spectroscopy of $H2$ isotopomers.^5–10 In particular, it has been pointed out that, due to their extremely long lifetimes, the weak quadrupole transitions represent, in principle, the ideal tool to measure with unprecedented accuracy levels the molecular hydrogen line-center frequencies.¹¹ However, a number of experimental issues, related to the availability of ultrastable laser sources and high-sensitivity interrogation techniques, have so far prevented the practical achievement of such results. In this framework, the use of modern optical frequency comb synthesizers^12,13 and the perspectives for production of cold/trapped molecules offer considerable scope for improvement.

The $H2$ electric quadrupole rovibrational spectrum was first observed in 1949 by Herzberg¹⁴ and then extensively investigated, both experimentally and theoretically, in the following two decades. The corresponding quadrupole features in $D2$ were also explored in a series of pioneering experiments making use of different spectroscopic/interferometric techniques usually involving high gas pressures (up to some tens of atmospheres).^15–19 Such early studies proved the ability to measure line-center frequencies with fractional uncertainties ranging from some $10−6$ to a few hundred $10−9$⁠. More recently, an improved redetermination of the $v=2←0$ band was carried out in the near infrared via off-axis integrated-cavity-output spectroscopy by comparison with spectra of well-characterized gases.⁷ Concerning the fundamental band, the most accurate study dates back to 1978.²⁰ In that work, a difference-frequency-generation (DFG) source was used in conjunction with a multiple reflection cell to measure positions of several transitions in the O, Q, and S branches against known lines of various reference molecules. An accuracy between 120 and 150 MHz was achieved for the line-center frequencies, ultimately limited by the adopted standards. Since all the spectra were recorded at only one pressure (1 atm), neither the zero-pressure frequency nor the strength of the observed lines could be measured.

Here, we revisit the fundamental rovibrational band of $D2$ by exploiting the most advanced tools of high-sensitivity spectroscopy and absolute frequency metrology. At a glance, the weak quadrupole transitions are detected by cavity ring-down spectroscopy (CRDS) using a DFG source referenced to the Cs-clock primary standard via an optical frequency comb synthesizer (OFCS). The effectiveness of such measurement scheme is evaluated on the S(0) and S(1) transitions, resulting in an accuracy of $6×10−8$ for the line position (at zero-pressure) and $10−2$ for the line strength.

The experimental setup, shown in Fig. 1, was described in detail in a previous paper.²¹ Briefly, coherent radiation around $3μm$ is produced by difference-frequency generation between a $1μm$ pump and a $1.5μm$ signal beam. The latter originates from an external-cavity diode laser (ECDL) power boosted by an Er-doped fiber amplifier. The pump radiation also comes from an ECDL seeding an Yb-doped fiber amplifier and passes through an acousto-optic modulator (AOM). The first-order diffracted beam is superimposed with the signal radiation and then focused into a periodically poled $LiNbO3$ crystal. The resulting idler beam has a power of a few milliwatts and an emission linewidth of a few hundred kilohertz. Its absolute frequency is determined by referencing both the ECDLs to a near infrared OFCS covering the $1–2μm$ octave (MenloSystems, FC1500-250). For this purpose, each beat note at frequency $fsbeat$ $(fpbeat)$ between signal (pump) and the $Ns$ $(Np)th$ comb tooth is sent to a corresponding electronic servo. The two servos lock these beat notes to given local oscillator values by feeding back proper corrections both to the external-cavity piezos and the laser current drivers. In this way, the DFG emission frequency reads¹³ 

where $M=Np−Ns$⁠, $fr$ is the comb mode spacing (≈250 MHz) and $fAOM$ (91.8 MHz) the frequency of the signal driving the AOM. The link to the Cs-clock standard is established by stabilizing $fr$ against a high-quality 10 MHz BVA (Boîter à Vieillissement Amélioré Enclosure with Improved Aging) quartz oscillator (OscilloQuartz, model 8627) which is disciplined, in turn, by a Rb/GPS (Global Positioning System) clock (TimingSolutions, 4410A); the same reference chain is used to lock the time base of the frequency synthesizers generating the signals at $fpbeat$⁠, $fsbeat$⁠, and $fAOM$⁠, respectively. Finally, the integers $Np$ and $Ns$⁠, and hence $M$ in Eq. (1), are determined by means of a near infrared wavemeter having an accuracy much better than $fr/2$⁠. In this way, the absolute frequency of the DFG radiation can be monitored by simultaneously measuring the terms of Eq. (1). Coarse tuning of $vDFG$ across the molecular resonance is accomplished by changing $M$⁠, while fine steps are realized by slightly changing $fr$ at a given $M$⁠.

For each $vDFG$ value, the $D2$ absorption is recorded by means of a continuous-wave CRDS scheme.²² The $3μm$ wavelength beam is coupled to an enhancement optical resonator consisting of a vacuum-proof stainless-steel tube (length $L=45cm$⁠) equipped with two spherical (1 m radius of curvature) high-reflectivity $(Rm=99.97%)$ mirrors at its ends. The corresponding cavity finesse is $F=πRm/(1−Rm)≈10500$ which gives an effective absorption pathlength $(L⋅F/π)$ of about 1.5 km. A diaphragm-molecular-drag pump is used to evacuate the resonator, while gas samples are let in from a 99.8% $D2$ cylinder; an absolute capacitance manometer is used to measure the gas pressure with 0.25% accuracy. The resonator length is continuously dithered by an annular piezoelectric actuator mounted on the input mirror. As a resonance builds up, a threshold detector triggers a large shift in the AOM frequency that rapidly brings the DFG radiation out of resonance. The subsequent ring-down decay is detected by a thermoelectrically cooled InAs detector (1 MHz electrical bandwidth). The average of 200 acquisitions, recorded by the oscilloscope, is then used to extract $τ$ by means of a LABVIEW least-squares fitting routine; finally, several hundred determinations of $τ$ yield the average cavity decay time $τ¯$ together with its standard error $Δτ$ (typically between 0.005 and 0.01 times $τ¯$⁠). According to this, in the presence of a molecular resonance, the absorption coefficient $α(v)$ is recovered through the well-known relation²³ 

where $c$ is the speed of light and $τ0$ is the empty-cavity decay constant $(≈5μs)$⁠.

As already mentioned, the above method was applied to the investigation of the first two transitions in the S branch. Figure 2 shows the corresponding absorption spectra [Eq. (2)] recorded for different pressure values across an absolute frequency scale [Eq. (1)]. A standard Voigt profile was fitted to the experimental data points to extract, in particular, the line-center frequency $v0$ and the integrated absorbance (IA) (i.e., the area under the molecular absorption profile). We remark that, in order to account for the Dicke narrowing effect, in the Voigt function the Gaussian width was left as a free parameter too. Actually, the measured spectra were also fitted with a Galatry profile, but neither advantage in terms of residuals was found, nor reliable information on the collisional physics (i.e., narrowing coefficient and collision width per unit pressure) could be extracted. The search for a more suitable fitting function will be the object of further investigation.²⁴ 

Line-center frequencies, measured as a function of deuterium pressure, are shown in Fig. 3. The intercept of a linear fit to the data points gives the absolute position of the transition [3166.3620(2) and $3278.5220(2)cm−1$⁠, respectively], while the slope provides the pressure shift [$−2.5(3)×10−3$ and $−2.0(2)×10−3cm−1/atm$]. The latter values are very close to that measured by Henesian and co-workers¹⁷ for the Q(2) line in the fundamental band $(−2.2×10−3cm−1/atm)$ and to those reported in Ref. 7 for the first two S transitions in the $v=2←0$ band [$−2.17(3)×10−3$ and $−1.98(9)×10−3cm−1/atm$].

Concerning the measurement of the transition line strength $S$ (expressed in cm/molecule), it is by definition related to the absorption coefficient by the following expression:

where the last equality is valid for an ideal gas (density $n$⁠, pressure $P$⁠, and temperature $T$⁠) and $g(v,T)$⁠, (having units of centimeter), is the normalized lineshape $(∫−∞+∞g(v,T)dv=1)$⁠. Therefore, by integrating Eq. (3) over the frequency, one gets

indicating that, for a given temperature, the slope of a linear fit to the IA values measured against $P$ directly yields $S(T)$⁠. With all measurements carried out at $T=296(1)K$⁠, the result is $S=3.77(4)×10−27cm/molecule$ for the S(0) line and $S=2.73(5)×10−27cm/molecule$ for the S(1) one (see Fig. 4). Since there are no line strength data in literature for these transitions, the congruence of our values can be assessed only through a comparison with the quadrupole moment matrix elements $⟨vJ|Q|v′J′⟩$ first calculated in Ref. 25 and then confirmed by a thorough theoretical study.²⁶ To this aim, we use the following relationship (in CGS units)²⁷ 

where $h$ is the Planck constant, $fv$$(fJ)$ is the vibrational (rotational) partition function, and $(2J′+1)$$(J2J′000)2$ is a Clebsch–Gordan coefficient which, for the S branch $(J′=J+2)$⁠, simplifies to $3(J+1)(J+2)/[2(2J+1)(2J+3)]$⁠. By assuming from the literature the approximate values $ωe≈2994cm−1$ and $Be≈30cm−1$ for the equilibrium vibrational and rotational constant, respectively, and using our experimental values for $SvJv′J′(T)$ and $v0$⁠, from Eq. (5) we obtain $|⟨00|Q|12⟩|=0.066(1)a.u.$ and $|⟨01|Q|13⟩|=0.064(1)a.u.$⁠, which are in good agreement with the theoretical predictions (0.068 33 and 0.064 65 a.u.). Within our uncertainty, basically determined by that of the line strength, such values are not affected by the exact choice of the molecular constants. The results obtained in this work are summarized in Table I.

A metrological approach for high-precision spectroscopy of molecular hydrogen isotopomers has been demonstrated by use of an OFCS-referenced highly sensitive $3μm$ spectrometer. The S(0) and S(1) transitions in the $D2$ fundamental band have been recalibrated in an absolute way in terms of position, pressure shift, and line strength. In particular, the accuracy in the line-center frequency is improved by more than one order of magnitude, while a 1% uncertainty is reported for the first determination of the line strength. Extensive use of this approach is planned in the next future for a number of transitions belonging to all the three branches. This would yield an absolute, ultra-accurate set of spectroscopic parameters (molecular constants and line strengths) for comparison with new quantum mechanical theories beyond the Born–Oppenheimer approximation,²⁸ as well as with possible future astrophysical observations.