Introduction

The invention of optical frequency combs has enabled, in the past two decades, accurate measurements of the energy values of multiple atomic and molecular transitions across a large part of the electromagnetic spectrum1,2,3. One of the areas where precise absolute frequency calibration plays a pivotal role for fundamental physics is the spectroscopic investigation of molecular hydrogen and its isotopologs, whose transition frequencies can be accurately predicted by theory4. The comparison between theory and experiments represents a testbed for molecular quantum electrodynamics5 (QED) as well as a compelling approach to explore physics beyond the standard model6,7,8 or to determine fundamental quantities such as the nucleon-electron mass ratios9.

A stumbling block to the accuracy of experimental determinations of transition frequencies is the intrinsic weakness of quadrupole rovibrational transitions, the only ones allowed in homonuclear species such as H2 and D2. Many experiments circumvented this weakness using high-finesse optical cavities providing enhancement of the effective absorption length up to several kilometers10. Thanks to the comb referencing of the frequency of the cavity-injected laser, transition frequencies have been measured with uncertainties of few parts per billion (ppb)11,12,13,14,15,16, at the same level of ab initio calculations4. The main limiting factors are the signal-to-noise ratio at low pressures and the nontrivial extrapolation of the line center to zero pressure because of the complexity of absorption lineshapes17. Cavity-enhanced measurements have produced so far accurate determinations only on overtone bands in the near-infrared. Their extension to fundamental rovibrational transitions which fall in the mid-infrared is a complex task, due the decreased quality of lasers, mirrors, detectors, and modulators in this region of the spectrum.

An alternative solution for determining the frequency of fundamental transitions is resonantly-enhanced multi-photon ionization (REMPI) applied to a molecular beam: it involves state-selective ionization of the excited molecule through a pulsed ultraviolet laser and subsequent mass-selective detection of the generated ion. In 2015, Ubachs and co-workers successfully obtained by REMPI an uncertainty of 2 × 10−4 cm−1 on the Q(0), Q(1) and Q(2) lines of the main hydrogen isotopologs18. For D2, this benchmark was improved recently, down to 6 × 10−7 cm−1 (0.2 ppb at a frequency of 3167 cm−1) on the S(0) 1-0 transition19, primarily thanks to an efficient excitation of the molecules in the upper vibrational state and to an accurate reduction of systematic errors related to the fine structure of the transition. The excitation was fostered by a strong static electric field that induces a transition dipole moment in the molecule and thus enhances the absorption of an intense comb-referenced mid-infrared laser. The overall setup remains quite complex and challenging to scale to purely rotational transitions, that would require far-infrared lasers20.

Here, we tackle the challenge of precision spectroscopy of infrared-inactive vibrational transitions by revitalizing and adding metrological quality to the stimulated Raman scattering (SRS) technique21. SRS is a third-order nonlinear spectroscopic process that makes use of two laser fields with proper frequency detuning, the so-called pump and Stokes fields, to excite a given vibration. It has been the approach of election in the past century to address transitions of dipole-inactive species like H222,23, but it has never benefited from the enhanced precision afforded by frequency combs. Here we introduce comb-assisted SRS metrology to study with high accuracy fundamental infrared-inactive transitions. The approach has the potential to cover a two-decade-spanning range of frequencies, from 50 to 5000 cm−1, by changing the wavelength of the pump laser in the near infrared, leaving all other parts of the apparatus mostly unaltered. We apply the spectrometer to the most studied stretching mode of H2 at ≈ 4155 cm−1, corresponding to the Q(1) line of the 1-0 band. We improve the state of the art for H2, obtained through REMPI, by 20 times18, achieving a combined uncertainty of 1.0 × 10−5 cm−1 (310 kHz) that challenges the theoretical benchmark4. This required the implementation of a multi-spectra fitting procedure for the extrapolation to zero pressure of the line center, with a set of fitting parameters fixed to ab initio values inferred from quantum-scattering calculations on H2–H2 collisions24.

Results

Comb-calibrated coherent Raman spectrometer

SRS requires the use of two detuned lasers (pump field at frequency \({\nu }_{{{{{{\rm{p}}}}}}}\) and Stokes field at frequency \({\nu }_{{{{{{\rm{S}}}}}}} < {\nu }_{{{{{{\rm{p}}}}}}}\)) to probe a rovibrational transition at frequency \({\nu }_{{{{{{\rm{p}}}}}}}-{\nu }_{{{{{{\rm{S}}}}}}}\). An energy transfer from the pump to the Stokes beam occurs when their frequency detuning is resonant with a Raman-active transition, in our case the Q(1) transition of the fundamental rovibrational band of H2 around 4155 cm−1. The measured quantity is the so-called Stimulated Raman Loss (SRL), which is the normalized change \(\Delta {I}_{{{{{{\rm{p}}}}}}}/{I}_{{{{{{\rm{p}}}}}}}\) of the pump intensity \({I}_{{{{{{\rm{p}}}}}}}\) induced by the Raman interaction. It is proportional, without any limitation given by phase matching, to the interaction length (\(L\)) and to the Stokes field intensity (\({I}_{{{{{{\rm{S}}}}}}}\))25:

$${{{{{\rm{SRL}}}}}}=\frac{\Delta {I}_{{{{{{\rm{p}}}}}}}}{{I}_{{{{{{\rm{p}}}}}}}}\propto {{{{{\rm{Im}}}}}}\big\{{\chi }^{\left(3\right)}({\nu }_{{{{{{\rm{p}}}}}}}-{\nu }_{{{{{{\rm{S}}}}}}})\big\}L\,{I}_{{{{{{\rm{S}}}}}}}$$

The resonant third-order nonlinear susceptibility \({\chi }^{(3)}\) of the medium entails the dependence of the SRL on the pump-Stokes frequency detuning, which reflects in H2 a complex collisional physics. To maximize the SRL signal, we used a multi-pass cell and a high-power Stokes laser.

The spectrometer (depicted in Fig. 1) employs a pair of single-frequency lasers for the pump (2 mW at 740 nm, <300 kHz linewidth) and Stokes (5 W at 1064 nm, <100 kHz linewidth) fields, with optical frequencies calibrated on a primary frequency standard through an optical frequency comb (see “Methods” for details). The two beams are superimposed by an active alignment system (see “Methods” for details) and mode matched to a multi-pass cell filled with H2. The cell can contain gas in a pressure range from 10−3 to 5 bar and is equipped with broadband dielectric mirrors that guarantee a total transmission around 50% from 700 to 1100 nm upon 70 reflections inside the cell and a total optical path length of 30 m. Gas temperature and pressure are actively stabilized to user-set values (see Methods for details). At the cell output, the pump is separated from the Stokes and is sent to a photodiode to measure the SRL signal, which is at the 10−4 level at 1 bar of H2 pressure in our experimental conditions. The signal-to-noise ratio (SNR) of the measurement is brought to the shot-noise limit (corresponding to root-mean square fluctuations of 3.3 × 10−8 for 1 s measurement time, see Supplementary Note 2 for details) by a modulation transfer technique that exploits high-frequency (9.7 MHz) electro-optical modulation of the Stokes intensity and synchronous detection of the SRL with a lock-in amplifier. Spectral acquisitions are carried out by measuring the SRL while repeatedly scanning the Stokes frequency over about 12 GHz around the center of the transition. The absolute frequency calibration comes from offset-locking the pump laser frequency to the nearest mode of a frequency comb and by real-time tracking of the Stokes frequency through high-speed acquisition and processing of its beat note against the same comb. A single spectral scan takes from 0.5 to 5 s depending on the gas pressure. Spectral points acquired over many subsequent scans with stable thermodynamic conditions are binned to obtain a single averaged spectrum with 1 MHz point spacing.

Fig. 1: Comb-assisted stimulated Raman scattering.
figure 1

EOM electro-optic modulator, M mirror, BC beam combiner, BS beam sampler, DM dichroic mirror, BD beam dumper, PD photodiode, OFC optical frequency comb, SHG second-harmonic generation. Comb referencing of pump and Stokes lasers is obtained through second-harmonic generation of selected spectral portions of an octave-spanning continuum generated by an Er:fiber comb (see “Methods” for details).

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Spectral measurements

Spectral acquisitions were made at 9 pressure values in the range 0.05–4 bar and at constant temperature T = 303.1 K. The exploration of such a large pressure range allows one to follow the complex collisional phenomena taking place in the ensemble of gas molecules and enables a robust determination of the transition frequency, as described in the following section and in Supplementary Note 4. A typical set of spectra is shown in Fig. 2a in an absolute SRL scale. The shape and the width of the line change considerably in the explored pressure range (see Fig. 2b for a zoom on lower pressures). Qualitatively, the most evident effect is that at high pressure the spectra are narrower and tend to exhibit a Lorentzian lineshape, whereas at lower pressure—see Fig. 2b—the lineshape tends to be broader and Gaussian. This is due to the Dicke effect26, which is the narrowing of the velocity distribution of the gas molecules and thus of the Doppler profile at increasing collision rates: this effect is prominent in H2 around 2 bar, where it leads to an almost complete extinction of Doppler broadening and to a minimum spectral width of 260 MHz. As a result, the SNR of high-pressure spectra is high even with short measurement times, reaching around 8000 over 10 min with a 1 MHz sampling. Another evident qualitative effect is a pronounced shift of the line peak with pressure, which has to be properly accounted for to extrapolate an accurate line center to zero pressure and compare it with that calculated on the isolated molecule. This calls for a refined modeling of the collisional physics of H2 and for the two-step analysis described below.

Fig. 2: Stimulated Raman scattering metrology of H2.
figure 2

a Set of spectra at different pressures plotted in an absolute SRL scale. Legend applies to all panels. b Zoom of a showing a detail of the spectra acquired at pressures equal to 0.05 and 0.1 bar. c Residuals (data—model) normalized to the corresponding spectral maxima, obtained from a global fitting with 5 free parameters and 3 fixed to ab initio value. \({\chi }_{r}^{2}\) represents the reduced chi square of the fit. d Residuals as above, obtained from a global fitting with 2 free (\({\nu }_{0}\), \({\delta }_{0}\)) and 6 fixed parameters.

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Multi-pressure global fitting

As a first step for data analysis we modeled the experimental spectra with the \(\beta\)-corrected Hartmann–Tran profile (\(\beta\)HTP), which is a spectral line profile optimized for H2 isotopologs27. It accounts for all main collisional phenomena at play, including velocity-changing collisions and speed-dependent effects, while exhibiting an analytical representation that can be efficiently integrated into least-square fitting procedures. To reduce the correlations among the 8 collisional parameters that define the \(\beta\)HTP profile24, we fixed a subset of them to ab initio values inferred from quantum-scattering calculations of H2–H2 collisions (described in Supplementary Note 3). For every fit we left as free parameters the unperturbed (zero-pressure) transition frequency \({\nu }_{0}\), which is the quantity of main interest here, and the linear pressure shift of the line center \(\delta_0\), which is hard to be accurately predicted by theory and can be better constrained by measurements. Figure 2c, d show the residuals of two types of fitting conducted in a global form over the entire multi-pressure dataset, using just 5 (Fig. 2c) or 2 (\({\nu }_{0}\), \({\delta }_{0}\), Fig. 2d) free parameters, respectively (details given in Supplementary Note 4). In both cases we observe structured residuals at a level of 1% of the line maximum, with similar reduced chi-square values (1.3 and 1.2) and fitted \({\delta }_{0}\) parameters (−2.7 and −2.69 cm−1). This is an indication of consistency between experimental spectra and lineshape model with ab initio collisional calculations. On the other hand, the two fits converge to different values of the line center \({\nu }_{0}\), with a discrepancy of almost 2.5 MHz. This is due to the concurrence of two effects, namely the difficulty of \(\beta\)HTP to fully model the collisional physics of H2 in such a very large pressure range24 and the inclusion of high-pressure spectra in the fit. While being of key importance for the retrieval of robust collisional parameters, they introduce a strong leveraging effect that amplifies the impact on the line center of any imperfect assessment of collisional parameters.

Extrapolation of the zero-pressure line center

The second step in our data analysis was to develop a procedure to extrapolate the line center overcoming the limitations of the model while reducing the impact of high-pressure spectra. To this purpose, we first investigated how the inferred \({\nu }_{0}\) changes upon restricting the fit to experimental data at lower and lower pressures (i.e., progressively discarding high-pressure spectra), down to a value pmax = 0.2 bar, to include in the fit at least three pressures. Figure 3a reports \({\nu }_{0}\) as a function of pmax, taking all collisional parameters except \({\delta }_{0}\) fixed to ab initio values. Three trajectories are retrieved, corresponding to three different fixed values of \({\delta }_{0}\), from −2.7 to −2.36 cm−1 atm−1, which define the limits of our confidence interval for this parameter (see Supplementary Note 4 for details on such an interval). They showcase a strong dependence of \({\nu }_{0}\) on both pmax and \({\delta }_{0}\), but at the same time also a clear convergence to one another while reducing pmax: this suggests an extrapolation of these trajectories to pmax = 0 as a viable approach to infer the line center frequency of the isolated molecule, minimizing the errors introduced by the model and by the uncertainty on \({\delta }_{0}\). This extrapolation is shown in Fig. 3b using a quadratic fit over the last four points of the trajectories.

Fig. 3: Determination of the frequency of the Q(1) 1-0 line of H2.
figure 3

a Inferred \({\nu }_{0}\) as a function of the maximum pressure pmax considered for the global fitting. The collisional parameters are fixed to ab initio parameters except \({\delta }_{0}\), which is fixed to the three different values specified in the legend, generating the three trajectories shown here. b Zoomed-in view of the low-pressure range together with parabolic extrapolations (lines) to pmax = 0. The vertical axis reports the difference between the fitted \({\nu }_{0}\) and the ab initio value \({\nu }_{{{{{{\rm{ref}}}}}}}\) calculated by Komasa et al.4.

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Uncertainty budget

The line center uncertainty due to \({\delta }_{0}\) can be quantified with the help of the inset in Fig. 3, which provides a zoomed-in view of the three trajectories and their quadratic extrapolations to pmax = 0. The half peak-to-peak distance between the two outer extrapolations amounts to 80 kHz, which was taken as an estimate for this systematic uncertainty term. As a second source of systematic uncertainty, we evaluated the impact on \({\nu }_{0}\) of the uncertainty on the ab initio parameters kept fixed in the fitting. For every such parameter varied in its confidence interval (see Supplementary Notes 4 and 5 for details) we calculated trajectories similar to those reported in Fig. 3 and considered the half peak-to-peak distance of the outer extrapolations as the corresponding uncertainty contribution to \({\nu }_{0}\), to be summed up in quadrature to the other contributions. This procedure leads to an additional term of systematic uncertainty of 220 kHz. To quantify the statistical uncertainty, we applied a bootstrap procedure taking advantage of the many spectral datasets acquired, running many fits on randomly assembled spectral datasets obtained by combinations of spectra acquired in different days over several months. This allows to account not only for white noise sources on the vertical and horizontal scales of the measurement, but also for contributions due to long-term drifts in the spectrometer or to uneven thermodynamic conditions. Differently from the systematic uncertainties above, we used here the root-mean square deviation of the different fits as an estimator for a final value of 200 kHz. The combined uncertainty budget for the extrapolated transition frequency, obtained by quadrature sum of the three contributions above, amounts to 310 kHz (10−5 cm−1), thus half that of the theoretical benchmark for the transition4 and 20 times better than the experimental benchmark18.

Discussion

The measured transition frequency is 4155.253790(10) cm−1. It is compared in Table 1 with the best experimental18 and theoretical4 determinations reported so far. For both cases, the agreement with our determination is within 1\(\sigma\), using the combined uncertainty of the two compared frequencies as an estimator for \(\sigma\).

Table 1 Comparison between the frequency of the Q(1) 1-0 transition obtained in this work with the best available experimental18 and theoretical4 data.
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To put this result in a broader context, we report in Fig. 4 the deviations between theory and experiments for all transition frequencies accurately measured so far on neutral molecular hydrogen and its isotopologs. Experimental determinations are in excess of calculated values in all cases, independently of the isotopolog, of the rovibrational branch, of the experimental approach and of whether the transition is dipole or quadrupole allowed, with discrepancies from 0.4 up to 2.3\(\sigma\). Focusing on quadrupole lines, by far the most accurate experimental determinations were obtained on D2, with an uncertainty down to 161 kHz for lines of the 2-0 band using cavity-ring-down spectroscopy11, and of 17 kHz for lines of the 1-0 band on a molecular beam19, in both cases well below the theoretical uncertainty. For H2, the center frequency here reported is the most accurate obtained so far and the only value exhibiting an accuracy comparable with theory, thus making it of significance for QED tests. The discrepancy with theory is of the same sign and of the same order of magnitude (\(1\sigma\)) encountered in several measurements on D2, and is dominated by the theoretical uncertainty. Overall, experimental accuracy has grown up at a faster pace than theory in recent years, reaching and even surpassing theory in some cases, yet not at point to challenge our current QED modeling of molecules. This challenge requires further efforts, over both axes.

Fig. 4: Comparison of experiment and theory for various rovibrational transitions of H2 isotopologs.
figure 4

Data points represent the discrepancy on the transition frequency (experiment–theory), with error bars given by the experimental uncertainty (1\(\sigma\)) and the height of the red shaded area at the bottom representing the theoretical uncertainty. Line names are reported next to data points, while the rovibrational band is identified through the point color (see legend). The data point corresponding to this work summarizes 10 measurement series acquired over 5 months.

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In conclusion, our work brings SRS metrology in the arena of available techniques for tests of fundamental physics in molecular systems. For the Q(1) 1-0 line of H2 we demonstrated an improvement by more than one order of magnitude with respect to the most accurate experimental determination, and surpassed the uncertainty of state-of-the-art theoretical calculations by a factor of 2. The strength of our setup lies in the ability to target a great variety of rovibrational transitions, in the mid or far infrared, by simple tuning of a near-infrared cw laser. This metrological tool enables probing other fundamental rovibrational transitions of H2 and its isotopologs. It also opens up for addressing Raman-active transitions of heavier molecules, as well as extending comb-calibration to so far unaddressed purely rotational bands.

Methods

Absolute frequency calibration of Raman spectra

The frequency detuning between the pump and Stokes laser fields is calibrated against a frequency comb generated by an amplified Er:fiber femtosecond oscillator (Menlo C-comb) emitting pulses at a repetition frequency \({f}_{{{{{{\rm{rep}}}}}}}=100\,{{{{{\rm{MHz}}}}}}\), stabilized on a GPS-disciplined Rb clock (Timing Solutions 4410 A). The experimental configuration is described in detail in Supplementary Note 1. Briefly, we first broaden the comb inside a highly nonlinear fiber, then frequency double (through second harmonic generation in periodically poled nonlinear crystals) spectral regions around 1480 nm and 2128 nm to obtain narrow frequency combs around 740 and 1064 nm. These are mixed with the pump (Toptica DL Pro) and Stokes (NKT Photonics Boostik) beams, respectively, to obtain beat notes between the comb and each cw laser. If we write the frequency of the n-th tooth of the broadened comb as \({\nu }_{n}=n{f}_{{{{{{\rm{rep}}}}}}}+{f}_{{{{{{\rm{ceo}}}}}}}\), the frequencies of the pump and Stokes lasers can be expressed as \({\nu }_{{{{{{\rm{p}}}}}}/{{{{{\rm{s}}}}}}}={n}_{{{{{{\rm{p}}}}}}/{{{{{\rm{s}}}}}}}{f}_{{{{{{\rm{rep}}}}}}}+2{f}_{{{{{{\rm{ceo}}}}}}}\pm {f}_{{{{{{\rm{beat}}}}}},{{{{{\rm{p}}}}}}/{{{{{\rm{s}}}}}}}\), where the carrier-envelope frequency \({f}_{{{{{{\rm{ceo}}}}}}}\) is doubled by second harmonic generation and the beat note \({f}_{{{{{{\rm{beat}}}}}},{{{{{\rm{p}}}}}}/{{{{{\rm{s}}}}}}}\) is added or subtracted depending on the relative position of the pump/Stokes frequencies with respect to their closest comb tooth of index \({n}_{{{{{{\rm{p}}}}}}/{{{{{\rm{s}}}}}}}\). The detuning between the two cw lasers can thus be unambiguously determined from the relationship \({\nu }_{{{{{{\rm{p}}}}}}}-{\nu }_{{{{{{\rm{s}}}}}}}=\left({n}_{{{{{{\rm{p}}}}}}}-{n}_{{{{{{\rm{s}}}}}}}\right){f}_{{{{{{\rm{rep}}}}}}}\pm ({f}_{{{{{{\rm{beat}}}}}},{{{{{\rm{p}}}}}}}-{f}_{{{{{{\rm{beat}}}}}},{{{{{\rm{s}}}}}}})\) once the pump and Stokes beat notes are measured and the correct comb-index detuning \(\triangle n=({n}_{{{{{{\rm{p}}}}}}}-{n}_{{{{{{\rm{s}}}}}}})\) is identified, with no contribution from \({f}_{{{{{{\rm{ceo}}}}}}}\). In our case the pump beat note is stabilized at 10 MHz while that of the Stokes beat note is measured in real time while scanning the frequency, as described in the next paragraph. The integer number \(\triangle n\) is determined by minimizing the distance between the experimental line center and its reference value from literature.

Spectra acquisition and processing

During spectral acquisitions the pump frequency is kept constant while the Stokes frequency is repeatedly scanned over about 12 GHz with a rate from 1 Hz at high pressure to 0.1 Hz at low pressure28. To ensure the proper calibration of spectra, we use a single digital acquisition board to synchronously measure the SRL signal and the beat note of the Stokes field. The Stokes beat note is digitized at 14 bit and 100 MSamples s−1 and its frequency calculated in real-time by FFT applied at every 1024 samples via a field-programmable gate array available for on-board processing (FPGA, PXIe-7961 FPGA board and NI-5781 add-on, National Instruments). The demodulated SRL signal (from a Zurich Instruments H2FLI lock-in amplifier) is simultaneously acquired and averaged over the same 1024 samples. The absolute frequency is reconstructed in post-processing by unwrapping the measured beat note barycentre28,29. Spectral acquisition times typically vary from 10 to 30 min depending on the pressure, but longer times could be used to further enhance the SNR. The spectral points pertaining to subsequent scans are binned to produce the final averaged spectra with frequency spacing of 1 MHz. Fitted spectra and residuals are further binned with bin width of 10 MHz to help identification of non-flat residuals (see example in Fig. 2).

Active alignment of beams

Any angular misalignment between pump and Stokes beams during the Raman interaction was found to originate a shift of the measured frequency detuning28. Manual alignment is accurate to about 300 μrad in our setup, corresponding to fluctuations of the measured center frequency by about 1–2 MHz from measurement to measurement. To ensure optimal superposition of the laser beams, we built an active alignment system based on the imaging of the near and far fields of both beams on a camera to track any displacement and correct it in a servo loop that acts on piezo-tilt mirror holders. To this purpose, the pump and Stokes beams are sampled right after the beam combiner (BC in Fig. 2), then sent through a beam splitter to generate two replicas. The first replica impinges on the sensor of a color CMOS camera (Thorlabs CS235CU) after passing through a lens that provides imaging of the BC plane onto the sensor; we refer to this plane as the near field (NF). The second replica impinges on the sensor after a total propagation distance equal to that from the beam combiner to the center of the cell, that we refer to as the far field (FF). The differential sensitivity of the color pixels of the camera at pump and Stokes wavelengths allow disentangling the two colors and control their relative displacements in both the NF and FF. This takes place in real time thanks to a LabView software that, upon calculation of the pump and Stokes centroids from the fitting of their intensity profiles, optimizes the overlap of the Stokes beam onto the pump beam through a 4-channels proportional-integral-derivative (PID) controller followed by 4 piezoelectric tip-tilt mirror adjusters (see Supplementary Note 6 for a photograph of the apparatus). The system provides an angular root mean square (RMS) stability of 6 μrad (corresponding to RMS fluctuations of the measured frequency detuning by 65 kHz) over few hours.

Active stabilization of thermodynamic gas parameters

Uniform and constant temperature of the gas inside the multipass cell is obtained by encasing the cell inside a box made with thick Styrofoam. A pair of rubber heater stripes (Minco) is placed onto the cell central cylinder, and two fans circulate the air inside the Styrofoam box to ensure temperature homogeneity better than 100 mK on the outer surface of the cell. The temperature is measured through a calibrated temperature sensor (a Pt100 probe paired to a 6 ½ digit multimeter, Keysight 34461A) with an accuracy of 50 mK, whose output is used by a software PID controller (implemented in LabView) to regulate the current passing through the heater stripes and stabilize the temperature to 303.1 K. To compensate for small leaks of the cell, a constant flow of about 10−2 L min−1 is established in the cell using two flow controllers (Bronkhorst F-201CV-1K0-RAD-33-V), the first one installed between the H2 cylinder and the cell, the second one between the cell and the vacuum pump. The pressure inside the cell is measured via a calibrated pressure sensor (Honeywell Sensotec TJE read by a Burster 9206 interface) with relative uncertainty better than 10−3. Through a software PID control loop, the output flow is regulated to maintain a constant pressure inside the cell.