## Introduction

Vibrational and rotational molecular spectra are the target of several fundamental Physics experiments relying on ultra-precise frequency measurements1. These include refined Quantum Electro-Dynamics (QED) tests2,3,4, detection of parity-violating effects5,6,7, investigation of hypothetical extra forces and dimensions8, assessment of the time stability of fundamental constants9,10, and identification of symmetry-violating electromagnetic moments11.

Towards such ambitious goals, in addition to the most sophisticated probe laser sources and interrogation techniques, a powerful tool is now represented by the buffer-gas-cooling (BGC) technology for producing gas molecular samples at cryogenic temperatures, down to a few kelvins12,13. Indeed, according to the Boltzmann statistics, cooling of rotational temperatures reduces the number of populated internal states, thus suppressing spectral congestion and eventually increasing the total number of detected molecules (if the investigated transition starts from the ground state or from one energetically close to it). Simultaneously, cooling the external degrees of freedom restricts the distribution of velocities in the sample, resulting in narrower line-shapes, and raising the matter-radiation interaction time. Moreover, when using specimens in form of beams, spectra are also freed from collisional broadening effects.

Actually, up to now, these favorable properties paid off most in the case of large-size species or even complex chemical mixtures, where the drastic simplification of the spectra plays the most important role, allowing probing molecular structure and dynamics with unprecedented spectral resolution and specificity14,15. In this respect, a bright example is given by the observation of quantum state-resolved ro-vibrational transitions in C60 fullerene, based on the combination of BGC with cavity-enhanced direct frequency comb spectroscopy16. Equally noteworthy is the achievement of sub-Hz-precision differential spectroscopy at frequencies near 15 GHz (9.7 × 10−11 in fractional terms) between buffer-gas-cooled enantiopure samples of (R)- and (S)- 1,2-propanediol, by means of a tunable microwave Fabry–Perot cavity17.

Concerning diatomic or simple polyatomic molecules, several extensive spectroscopic works exist in the literature, with particular relevance to laser cooling and trapping. Here, the main purpose is the determination of vibronic branching ratios to the 10−5 level, which is essentially accomplished by application of laser-induced fluorescence (LIF) on cryogenic buffer gas beams18. For example, this produced accurate characterization of the B2Σ+ state of BaH19, and of the X2Σ1/2 and A2Π1/2 states in CaOH and SrOH20. Application of LIF or absorption spectroscopy to cryogenic buffer-gas cooled samples has also resulted in accurate transition frequency measurements. For instance, hyperfine-resolved spectra were obtained for the $${b}^{3}{{{\Sigma }}}^{+},\nu {\prime}=0\leftarrow {X}^{1}{{{\Sigma }}}^{+},\nu {\prime}{\prime}=1$$ and $${b}^{3}{{{\Sigma }}}^{+},\nu {\prime}=0\leftarrow {a}^{3}{{\Pi }},\nu {\prime}{\prime}=0$$ bands of AlF21 and for the $${\tilde{A}}^{2}{{{\Pi }}}_{1/2}(0,0,0)-{\tilde{X}}^{2}{{{\Sigma }}}^{+}(0,0,0)$$ band of 171,173YbOH22, yielding fine- and hyperfine structure parameters with a precision of a few MHz. Similarly, the [557] − X2Σ+(ν = 0, 1, 2, 3) and the [561] − X2Σ+(ν = 2) visible electronic transitions of YbF were recorded at a near-natural linewidth spectral resolution; particularly, the 174YbF isotopologue spectral features were analyzed to produce a new set of spectroscopic parameters for the X2Σ+(ν = 2), X2Σ+(ν = 3), [557], and [561] states23.

Notwithstanding the numerous applications discussed above, the enormous potential offered by buffer-gas-cooled samples for absolute frequency metrology of ro-vibrational spectra has not been fully exploited yet, mainly due to the difficulty in entirely transferring highly developed spectroscopic methods to complex apparatuses involving cryostats. In particular, implementation of effective cavity-enhanced interrogation schemes is restrained by the vibrations originating from the pulse-tube cryocooler which, via the vacuum chamber, couple to the resonator and deteriorate its mechanical stability24. In a previous pioneering experiment, we partially overcame this drawback by using specially contrived damping mounts for the high-reflectivity (HR) cavity mirrors, thus succeeding in extending to the cold-temperature range the early proven association of cavity-ring-down (CRD) spectroscopy with sub-Doppler saturation measurements25,26. In that case, a statistical uncertainty of 12 kHz (6 × 10−11 in fractional terms) was achieved for the frequency measurement of the target transition27.

In the present work, we improve this result by one order of magnitude, through an original opto-mechanical design that allows the BGC source to be efficiently coupled to a cavity with finesse as high as 95,000 (HFC). On this basis, we realize an optical-frequency-comb(OFC)-assisted, Pound-Drever-Hall(PDH)-locked CRD spectrometer28, additionally making use of two orthogonally polarized laser beams, one for PDH-locking and the other for probing the molecular sample29,30,31. This offers the great advantage of a fast acquisition of cavity-decay events without interrupting the PDH locking, in turn enabling the implementation of a profitable Lamb-dip saturated-absorption cavity ring-down (SCAR) scheme, just recently demonstrated at room temperature by our group32. In this configuration, by way of example, we address the acetylene (ν1 + ν3) R(1)e ro-vibrational line at 6561.0941 cm−1 in a 20-K cell specimen, demonstrating absolute line-center frequency measurements with an overall (statistical + systematic) uncertainty as low as 1.2 kHz (6 × 10−12 in fractional terms). In fact, by ranking with the highest accuracy levels obtained so far with deep-rooted methodologies involving room-temperature cell samples, either in the Doppler-limited33,34 or Doppler-free regime35,36,37,38,39,40, our achievement finally opens a metrological perspective to the spectroscopy of buffer-gas-cooled molecules.

## Results

### Experimental apparatus

The experimental apparatus as a whole is shown in Fig. 1. Described in detail in a previous paper41, the heart of the BGC source consists of a two-stage pulse tube (PT) cryocooler (Cryomech, PT415) housed in a stainless-steel (S-S) vacuum vessel. In particular, the second stage yields a temperature around 5 K provided that its heat load is kept below 1.5 W; for this purpose, each PT plate is enclosed in a gold-plated shield (equipped with accesses for the passage of the laser beam) in order to suppress black-body radiation. A single S-S pipe, thermally insulated from both the cryocooler stages, is used to introduce both acetylene and helium (contained in room-temperature bottles) into the BGC cell. The injected flux for each gas (fAc and fHe), here expressed in Standard Cubic Centimeters per Minute (1 SCCM = 4.5  1017 particle/s), is regulated independently upstream by a 1-%-accuracy flow controller. The BGC cell is a copper cube of side length L = 40 mm, in contact with the second PT plate, equipped with two facing circular holes (5 mm diameter) aligned along the axis of the high-finesse spectroscopic cavity. In this configuration, C2H2 molecules are cooled down to 18 K through multiple collisions with the He buffer gas.

We now come to the delicate point of how to minimize the deleterious effects of the vibrations from the cryocooler on the constituent elements of the high-finesse cavity (HFC), when this is integrated into the BGC source, going to form what from here on we will call the BGC-HFC. The mechanical noise is essentially dominated by the low-frequency components driven by the compression cycle, but high-order harmonics are also present due to distortion modes of the PT tubes and smaller parts of the system with higher resonant frequencies. To address this issue, we act on two fronts.

On one side, there is a careful mechanical arrangement which can be schematically described as follows. The two BGC-HFC mirrors (3 m radius of curvature) are housed in the aforementioned damping mounts at a distance d = 98 cm from each other, corresponding to a free spectral range (FSR) of  150 MHz. Additionally, the mirror mounts are mechanically isolated from the cryostat dewar by a system of edge-welded bellows forming the vacuum connection, and are incorporated in as many massive S-S bars, each resting on its own optical bench by means of damping material (Vulcuren) feet. Finally, two S-S rods, girdling the cryostat dewar, connect the two bars, thus fixing the gross cavity length. Additional Vulcuren damping cylinders are placed in the points of the apparatus most stressed by vibrations.

On the other side, a sophisticated SCAR interrogation scheme is implemented. The radiation source is a cw single-mode external-cavity diode laser (ECDL) emitting around 30 mW in the 1470–1560 nm wavelength interval. The main fraction of the ECDL output beam is divided into the s- and p-polarized radiation, which are separately directed to the BGC-HFC for the PDH locking and the SCAR measurements, respectively. After being phase-modulated, the s-polarized beam is sent to the cavity via an optical circulator. In this way, the light reflected off the cavity is conveyed to an InGaAs photodetector (PD2) in order to generate the PDH error signal, which is used to continuously lock the laser frequency to a resonance of the cavity. The p-polarized component (SCAR beam) double-passes through an acousto-optic modulator (AOM) in cat’s eye configuration42, driven at νAOM = 150 MHz, emerging shifted by 300 MHz (i.e., twice the cavity FSR) with respect to the PDH beam. This ensures the cavity resonance condition for the SCAR beam as well, while avoiding interference with the PDH radiation. Then, the PDH (120 μW) and SCAR beam combine together through a polarizing beam splitter, before entering the BGC-HFC. Emerging from the latter, the transmitted PDH and SCAR beam are again separated, and eventually collected by two identical photo-detectors (PD3 and PD4, respectively). The actual extinction factor is below 10−3, as determined by the ratio of the voltages measured by PD3 and PD4, keeping only the SCAR beam on. Moreover, in order to control the BGC-HFC length, while simultaneously providing absolute frequency calibration of the spectra, the beat note between the ECDL and the N-th tooth of an OFC is phase-locked to a local oscillator (νLO = 30 MHz). The free-running and in-loop phase noise of the overall lock chain (ECDL → BGC-HFC → OFCS → Rb/GPS) are shown in Fig. 2, both at room and cryogenic (6 K) temperature, pointing out a total bandwidth of around 9 kHz. In the end, the residual laser linewidth actually entering the final spectroscopic measurement is substantially set by the width of the comb tooth (less than 100 kHz).

In the above configuration, the absolute frequency of the SCAR beam interacting with the cold sample reads as: νSCAR = 2 νAOM + νCEO + NνRR + νLO, where νCEO (20 MHz) and νRR (250 MHz) are the OFC carrier-envelope offset and mode spacing, respectively. Thus, tuning across the target molecular transition (centered at ν0) is accomplished by varying νRR in discrete steps. For each νSCAR value, under the continuous PDH locking, the SCAR beam is abruptly switched on/off by the AOM (automatically triggered by a digital delay generator), and the resulting ring-down events are recorded by a 18-bit (vertical resolution) oscilloscope (NI, PXI-5922). Then, based on the theoretical model developed specifically for the Lamb-dip regime in a previous work32, the average of 600 acquisitions (each lasting about four times τ0) is fitted by the function

$${{{{{{{\mathcal{W}}}}}}}}(t)={{{{{{{\mathcal{B}}}}}}}}+{{{{{{{\mathcal{A}}}}}}}}\,{e}^{-{\gamma }_{c}t}f[t;{\gamma }_{c},{\gamma }_{g},{U}_{g}({\nu }_{{{{{{{{\rm{SCAR}}}}}}}}}-{\nu }_{0})],$$
(1)

with the equation

$$f=-{\gamma }_{g}\,\frac{2}{1+\sqrt{1+{e}^{-{\gamma }_{c}t}f(t)\,{U}_{g}({\nu }_{{{{{{{{\rm{SCAR}}}}}}}}}-{\nu }_{0})}}\,f(t)$$
(2)

being numerically integrated in the fitting routine (f(0) = 1). In the above equations, $${{{{{{{\mathcal{A}}}}}}}}$$ and $${{{{{{{\mathcal{B}}}}}}}}$$ are the signal amplitude and background, respectively, γc ≡ 1/τ0 is the empty-cavity decay rate, γg denotes the decay rate due to the linear gas absorption, and $${U}_{g}({\nu }_{{{{{{{{\rm{SCAR}}}}}}}}}-{\nu }_{0})=S\,[1+{{{{{{{\mathcal{D}}}}}}}}({\nu }_{{{{{{{{\rm{SCAR}}}}}}}}}-{\nu }_{0})]$$ represents the dip saturation profile, with S = Icav,0/Isat (i.e., the ratio of the intra-cavity radiation intensity to the saturation intensity of the investigated transition) and $${{{{{{{\mathcal{D}}}}}}}}$$ being a lineshape function that is equal to 1 in the center and decreases to zero as the detuning increases in either direction. A typical scan across the Lamb-dip profile consists of 56 points spaced on average 180 kHz apart, and takes approximately 1 min (the maximum acquisition rate for ring-down events is 800 Hz). The ultimate spectral feature is the average over 50 single Lamb-dip profiles.

### Spectroscopic data and analysis

With the ultimate aim of minimizing the uncertainty on ν0, Lamb-dip spectra (with corresponding underlying Doppler-limited profiles) are initially acquired as a function of fHe, for the lowest He flux and SCAR beam power (fAc = 3 SCCM and PSCAR 40 μW). For a coupling factor C 0.9 and a laser beam waist radius w0 730 μm, this corresponds to an intra-cavity radiation intensity as high as $${I}_{{{{{{{{\rm{cav}}}}}}}},0}=(C{P}_{{{{{{{{\rm{SCAR}}}}}}}}}{{{{{{{\mathcal{F}}}}}}}})/({\pi }^{2}{w}_{0}^{2})\simeq 60$$ W/cm2. The values extracted for ν0 and the full width at half maximum (ΓL) by a Lorentzian fit to the dip saturation profiles are plotted in Fig. 3. In particular, the slope of a linear fit to the ΓL(fHe) data points provides the (low-temperature) foreign collisional-broadening coefficient, γHe = (28 ± 3) kHz/SCCM; on the contrary, the collisional shift coefficient is not significantly estimated: δHe = (−0.1 ± 0.3) kHz/SCCM. In order to provide the above coefficients in more commonly used units, we denote by Nin (Nout) the number of helium atoms injected (emerging) into (from) the BGC cell. Then, in stationary conditions, we have $${f}_{{{{{{{{\rm{He}}}}}}}}}\equiv d{N}_{{{{{{{{\rm{in}}}}}}}}}/dt=d{N}_{{{{{{{{\rm{out}}}}}}}}}/dt=({n}_{He}\,{\overline{v}}_{{{{{{{{\rm{He}}}}}}}}}\,{A}_{{{{{{{{\rm{ape}}}}}}}}})/4$$, where nHe and $${\overline{v}}_{{{{{{{{\rm{He}}}}}}}}}=\sqrt{8{k}_{B}T/(\pi {m}_{{{{{{{{\rm{He}}}}}}}}})}$$ (with mHe being the helium mass) are the steady-state density and the mean thermal velocity of the gas atoms inside the cell, respectively, and Aape is the total (for both exit holes) aperture area43. Finally, the conversion factor $${n}_{{{{{{{{\rm{He}}}}}}}}}/{f}_{{{{{{{{\rm{He}}}}}}}}}=4/({\overline{v}}_{{{{{{{{\rm{He}}}}}}}}}{A}_{{{{{{{{\rm{ape}}}}}}}}})$$ is found, which allows to rewrite γHe = (2.4 ± 0.3) cm−1/atm and δHe = (−0.01 ± 0.03) cm−1/atm. A similar analysis is repeated as a function of fAc (fHe = 3 SCCM, PSCAR 40 μW), in order to determine the self collisional-broadening coefficient, γAc = (30 ± 5) kHz/SCCM, and the corresponding shift coefficient, δAc = (−0.2 ± 0.4) kHz/SCCM. In this case, however, a real conversion between SCCM and atm units cannot be done. Indeed, contrary to the case of helium, many of the C2H2 molecules injected into the cell freeze upon impact on the walls (whose temperature is well below the acetylene melting point). Therefore, the acetylene density inside the cell depends not only on the injected C2H2 flux, but also on the helium density, as the random walk experienced by the molecules in the He bath increases the diffusion time (i.e., the lapse between entering the cell and sticking on its walls). Ultimately, even with the same fAc value, very different molecular densities can occur, depending on the amount of helium present.

Finally, as shown in Fig. 4, we investigate the behavior of the line-center frequency as a function of PSCAR (fAc = fHe = 3 SCCM), finding a power shift coefficient of δP = (0.01 ± 0.03) kHz/μW; as illustrated in the corresponding inset, the SCAR analysis leaves saturation broadening effects out of the widths in the Ug profiles32.

As displayed in Fig. 5, the choice fAc = fHe = 3 SCCM with PSCAR = 40 μW corresponds to the highest SNR (90) and narrowest linewidth, while simultaneously minimizing the molecular sample temperature: T = (20 ± 2) K, as measured by the width $$\sigma=({\nu }_{0}/c)\sqrt{8\ln 2{M}^{-1}{k}_{B}T}$$ of the Gaussian profile in which the dip is dug (with M being the acetylene mass). The observed width, Γ3,3 = (370 ± 5) kHz, is consistent with the combination of the three main broadening contributions: the collisional term γcoll = (170 ± 30) kHz, the residual laser emission linewidth γlas 100 kHz, and the transit-time $${\gamma }_{{{{{{{{\rm{tt}}}}}}}}}=\sqrt{16\ln 2/{\pi }^{3}}\,{w}_{0}^{-1}\sqrt{({k}_{B}T)/M}\simeq 70$$ kHz (referring to a single molecule traveling with the mean thermal speed of the ensemble). Specifically, after adding the first two Lorentzian contributions (both of the homogeneous type), and then taking the Voigt convolution with the remaining (inhomogeneous-type) transit-time term, one gets the expected final dip width: $$0.5346\cdot ({\gamma }_{{{{{{{{\rm{coll}}}}}}}}}+{\gamma }_{{{{{{{{\rm{las}}}}}}}}})+\sqrt{{\gamma }_{{{{{{{{\rm{tt}}}}}}}}}^{2}+0.2166\cdot {({\gamma }_{{{{{{{{\rm{coll}}}}}}}}}+{\gamma }_{{{{{{{{\rm{las}}}}}}}}})}^{2}}\simeq 320$$ kHz (Olivero’s formula44).

Concerning the determination of the absolute line-center frequency, a statistical uncertainty as low as 1.1 kHz is found from the Lorentzian lineshape fit, while the relevant sources of systematic uncertainty are listed in Table 1. On the time scale typical of our spectroscopic acquisitions, the stability of the GPS-based reference chain contributes with an uncertainty of 0.5 kHz (the contribution related to the AOM driving frequency is only at the Hz-level). The accuracy of the flow meters (1%) translates into an overall collisional shift uncertainty of 0.03 SCCM × (0.3 + 0.4) kHz/SCCM = 0.02 kHz. Moreover, while the recoil shift is canceled out in Lamb-dip spectroscopy, the second-order Doppler shift is estimated as ν0kBT/(Mc2) = 0.02 kHz. Also, a 15% uncertainty on the SCAR beam power gives a contribution of (40  0.15) μW × 0.03 kHz/μW = 0.18 kHz. Finally, by allowing the static width of the Lorentzian lineshape to vary sigmoidally, an uncertainty around 70 Hz is assigned to asymmetries in the Lamb-dip fit. By adding in quadrature all the terms, the overall (type A + type B) 1σ uncertainty amounts to 1.2 kHz (6 × 10−12 in fractional terms): ν0 = (196 696 652 914.3 ± 1.2) kHz. Our value is consistent with the most accurate (room-temperature) determination so far: (196 696 652 918 ± 2) kHz, by ref. 45.

## Discussion

In conclusion, by applying the Lamb-dip SCAR technique to a buffer-gas-cooled molecular sample, we have measured the absolute frequency of a given ro-vibrational transition with an overall relative uncertainty of 6 × 10−12. Except for selected lines of exclusive molecules (SF6, OsO4, CH4) which, thanks to a long course of spectroscopic research on secondary frequency standards, are known with accuracy below 10−1346,47,48,49, our achievement is in line with the state of the art on room-temperature cell samples. Therefore, being straightforwardly extendable to many species in different spectral regions, our approach establishes a complementary frequency-metrology tool, especially useful for those molecules (even simple ones) whose room-temperature spectrum is congested. In this regard, it should be emphasized that acetylene exhibits pretty isolated lines already at room temperature, especially in the Lamb-dip regime. Thus, the effects of the expected BGC-induced spectral decongestion are not directly demonstrated here, and a franker experimental proof implies moving to a more complex molecular species. Indeed, we are planning a high-resolution spectroscopy experiment on buffer-gas-cooled fluoroform (CF3H), in order to excite a predicted two-photon transition at around 8.6 μm wavelength50, so far not identified due to the presence of numerous concurrent lines51.

Moving to the field of ultracold molecular gases, our result also fairly compares to the records in the low 10−12 range so far attained with trapped ions4,52 or photo-associated dimers (e.g., Sr2) in optical lattices53, in spite of the vastly different linewidth observed for the spectroscopic feature: kHz/sub-KHz in the former case, a few tens of Hz in the latter case. Therefore, already at the current level, our method represents a valuable option for frequency metrology of cold spectra in a different low-temperature range, on a much wider class of molecules.

At the same time, our novel approach still has ample room for improvement, laying the groundwork for approaching the 10−13 accuracy range. Relevant upgrades could include the prestabilization of the probe laser against an ultra-low-expansion (ULE) reference cavity, as well as the implementation of even more sophisticated spectroscopic techniques, such as 2-photon cavity-ring-down spectroscopy54 or noise-immune cavity-enhanced optical-heterodyne molecular spectroscopy (NICE-OHMS)2,55,56. Accordingly, the Rb-GPS standard should be replaced with the clock laser delivered from Istituto Nazionale di Ricerca Metrologica (INRIM) through the National Optical Fiber Link (already fully operational in our lab and routinely referenced to the primary Cs fountain57). Definitely, a big step forward will be moving from a static cryogenic molecular sample to a buffer-gas-cooled beam (high brilliance, slow laboratory-frame velocity, and collision-free environment), until reaching the observation of two-photon Ramsey fringes in the spatial domain9.

Ultimately, this will trigger a new generation of low-temperature precision spectroscopy studies, of relevance both for the aforementioned fundamental Physics researches and for the identification of more efficient laser cooling schemes toward the achievement of quantum degeneracy in molecular systems58.

## Methods

### Optical setup

The BGC-HFC input mirror (M1) has a 0.25-inch diameter (with a thickness of 4 mm) and reflectivity > 99.97%, while the second mirror (M2) has a 1-inch diameter and reflectivity > 99.995%. The measured empty-cavity decay time is τ0 42μs, corresponding to a finesse $${{{{{{{\mathcal{F}}}}}}}}=(\pi c{\tau }_{0})/d\simeq 95000$$. M1 and M2 are mounted on annular PZTs, PZT1 (HPSt150/20-15/12, Piezomechanik) and PZT2 (PD120.31, Round PICMA), respectively. The s-polarized beam emerging from PBS1 first passes through a Glan-Taylor calcite polarizer (GTP), with a vertical transmission axis, providing an extinction ratio of 100 000. Afterward, the PDH beam is frequency modulated by a temperature-controlled electro-optic phase modulator (EOM) forming sidebands at 30 MHz relative to the carrier, and then enters an optical circulator, consisting of PBS3 and a Faraday rotator (FR). The generated PDH error signal is processed by a proportional-integral-derivative (PID) servo (Toptica Photonics, FALC 110), which provides external correction to the ECDL frequency, acting both on the piezo-electric transducer (PZT) of the external cavity (100 Hz bandwidth) and on the laser current driver (1 MHz bandwidth).

The beat note between the ECDL and the OFC (Menlo Systems, FC-1500-250-WG) is detected by a PIN-InGaAs fast photodiode (PID1) and, after proper band-pass filtering (to minimize noise contribution from other comb lines) and amplification, processed by an electronic servo (Toptica Photonics, mFALC110), whose correction is carried out via a slow feedback to PZT1, plus a fast one to PZT2. In turn, both νCEO and νRR are stabilized against a Rb/GPS clock, providing an accuracy of 10−13 and a fractional stability (Allan deviation) between 4 × 10−13 and 8 × 10−13 for an integration time between 10 and 1000 s.