Few-electron molecules are attractive systems for precision spectroscopy because their properties can be calculated with high accuracy by quantum-chemical methods (1-5). The measurements serve to test theoretical predictions, ideally at the level where their accuracy is limited by the uncertainties of the fundamental constants or by unrecognized physical effects. I will report on precision measurements of energy intervals in cold samples of He, H₂ and He₂. In particular, we determine their ionization energies with a precision (∆ν/ν) of 10⁻¹⁰ from high-resolution Rydberg spectra (6-8), approaching the level where uncertainties in the size of the nuclei and the nuclear-to-electron mass ratios would limit the accuracy of otherwise exact calculations and calibration requires the use of primary frequency standards. Comparison will be made to recent theoretical results in the context of more-than-100-year-long experimental and theoretical efforts invested in the accurate determination of the properties of fundamental few-electron atoms and molecules. 1.V. I. Korobov, L. Hilico and J.-P. Karr, Phys. Rev. Lett. 118, 23, 001 (2017). 2.M. Puchalski, J. Komasa, P. Czachorowski, and K. Pachucki, Phys. Rev. Lett. 122, 103003 (2019). 3.D. Ferenc, V. I. Korobov and E. Mátyus, Phys. Rev. Lett. 125, 213001 (2020). 4.V. A. Yerokhin, V. Patkóš, M. Puchalski, and K. Pachucki, Phys. Rev. A 102, 012807 (2020) 5.P. Danev, D. Balakov, V. I. Korobov, S. Schiller, Phys. Rev. A 103, 012805 (2021) 6.M. Beyer, N. Hölsch, J. A. Agner, J. Deiglmayr, H. Schmutz and F. Merkt, Phys. Rev. A 97, 012501 (2018). 7.N. Hölsch, M. Beyer, E. J. Salumbides, K. S. E. Eikema, W. Ubachs, Ch. Jungen, and F. Merkt, Phys. Rev. Lett. 122, 103002 (2019). 8.L. Semeria, P. Jansen, G.-M. Camenisch, F. Mellini, H. Schmutz, and F. Merkt, Phys. Rev. Lett. 124, 213001 (2020).

Precision experiments in the hydrogen atom have a long tradition and extensive studies of transitions between low lying n ≤ 12 states were carried out J. C. Garreau et al., J. Phys. France 51, 2293 (1990).C. G. Parthey et al., Phys. Rev. Lett. 107, 203001 (2011) A. Beyer et al., Science 358, 79 (2017)H. Fleurbaey et al., Phys. Rev. Lett. 120, 183001 (2018) N. Bezginov et al., Science 365, 1007 (2019)A. Grinin et al., Science 370, 1061 (2020). These measurements can be used to determine values of the Rydberg constant and the proton charge radius. We present a new experimental approach to perform measurements of transition frequencies between the metastable 2s ²S_1/2 (F = 0,1) state of H and highly excited Rydberg states with principal quantum number n ≥ 23.
We generate the hydrogen atoms by photodissociation of NH₃ in a capillary mounted at the orifice of a pulsed valve. The hydrogen atoms are entrained in supersonic expansions of a rare gas. The atoms enter a magnetically shielded region in which they are photoexcited to a specific hyperfine level of the metastable 2s ²S_1/2 state by a home-built frequency-tripled Fourier-transform-limited pulsed titanium sapphire laser (pulse length 40 ns). Transitions to np Rydberg states are then induced by a narrow-band frequency-doubled continuous-wave titanium sapphire laser, which is phase locked to an optically stabilized frequency comb and referenced to a Cs primary frequency standard. The highly excited Rydberg states are detected by pulsed-field ionization. We will report progress on our efforts to minimize uncertainties from stray electric fields and Doppler shifts and to obtain spectral lines with a FWHM below 10 MHz.

---

Footnotes:

J. C. Garreau et al., J. Phys. France 51, 2293 (1990).

---

Footnotes:

A. Beyer et al., Science 358, 79 (2017)

---

Footnotes:

N. Bezginov et al., Science 365, 1007 (2019)

---

Footnotes:

Helium, as one of the simplest few-electron atoms, is well suited to test fundamental interactions and QED calculations with a high precision. Its importance is underlined by recent advances in the determination of particle properties such as the α-particle charge radiusJ. Krauth et al., Nature 589, 527531 (2021) Yet, the most accurate theoreticalK. Pachucki, V. Patkó\checks, and V. A. Yerokhin, Physical Review A 95, 062510 (2017)^,G. W. Drake and Z. C. Yan, Canadian Journal of Physics 86, 45 (2008)nd experimentalW. Lichten, D. Shiner, and Z.−X. Zhou, Physical Review A 43, 1663 (1991)^,C. J. Sansonetti and J. D. Gillaspy, Physical Review A 45, R1 (1992)ata on the ionization energies of the low-lying electronic states show a significant discrepancy of more than 2 σ (about 3 MHz). We report on the determination of the ionization energy of the metastable 2 ¹S₀ state of helium through Rydberg-series extrapolation with a relative uncertainty of 5 ×10⁻¹¹ through the determination of 20 different np ←2  ¹S₀ transition frequencies in the range of n from 24 to 102, yielding quantum-defect parameters for the np Rydberg states. A one-photon excitation scheme was employed, using the frequency-doubled output of a narrowband cw laser source (312 nm), calibrated to a frequency comb referenced to a GPS-disciplined Rb clock. The major sources of systematic uncertainties are minimized by (i) carrying out the experiment in a doubly skimmed, pulsed supersonic beam of metastable ⁴He atoms to minimize the Doppler broadening, (ii) compensating electric stray fields to below 0.7 mV/cm, and (iii) cancelling the first-order Doppler shift by monitoring two Doppler components resulting from two counter-propagating laser beams. The effects of residual stray electric fields are included as DC-Stark shifts in the analysis. Our new value of the ionization energy of the 2  ¹S₀ state of ⁴He, with an absolute uncertainty better than 50 kHz, represents an improvement by a factor of about five in precision. It also allows for a purely experimental determination of the ionization energies of several other low-lying electronic states of ⁴He, which were previously only accurately known through combination of measured and theoretical energy intervals.

---

Footnotes:

J. Krauth et al., Nature 589, 527531 (2021). K. Pachucki, V. Patkó\checks, and V. A. Yerokhin, Physical Review A 95, 062510 (2017)\end G. W. Drake and Z. C. Yan, Canadian Journal of Physics 86, 45 (2008)a\end W. Lichten, D. Shiner, and Z.−X. Zhou, Physical Review A 43, 1663 (1991) C. J. Sansonetti and J. D. Gillaspy, Physical Review A 45, R1 (1992)d

Highly excited rovibrational states of H₂ in the X ¹Σ_g⁺ electronic ground state are prepared by two-photon photolysis of H₂S [1]. The highest vibrational levels v=14, J=0−4, with a dissociation energy of less than 100 cm⁻¹, and some long-lived quasi-bound resonances (v,J) = (7,21), (8,19), (9,17), and (10,15) are probed through Doppler-free two-photon spectroscopy of F ¹Σ_g⁺ - X ¹Σ_g⁺ transitions. By compensating AC-Stark shifts, absolute transition frequencies were determined with an accuracy on the order of 90 MHz. From combination differences of the F-X transitions the energy level splittings in the electronic ground state are determined and compared with ab initio calculations, including relativistic and QED effects. This study extends precision test of QED theory for the electronic ground state of the hydrogen molecule to the very highest bound level, and into the quasi-bound region. [1] K.-F. Lai, M. Beyer, E.J.Salumbides, W. Ubachs, Photolysis production and spectroscopic investigation of the highest vibrational states in H2 (X¹Σ_g⁺ v=13,14), J. Phys. Chem. A 125, 1221-1228 (2021)

From the precise measurement of the ionization energy of H₂ its dissociation energy can be determinedN. Hölsch, M. Beyer, E.J. Salumbides, K.S.E. Eikema, W. Ubachs, Ch. Jungen, and F. Merkt, Phys. Rev. Lett., 122(10), 103003 (2019) which serves as a benchmark quantity for QED calculationsM. Puchalski, J. Komasa, P. Czachorowski, and K. Pachucki, Phys. Rev. Lett., 122(10), 103003 (2019) Additionally, measurements of ionization energies corresponding to higher rovibrational levels of the ion enable the determination of rovibrational intervals in H₂⁺C. Haase, M. Beyer, Ch. Jungen, and F. Merkt, J. Chem. Phys. 142(6), 064310 (2015) These intervals can be calculated extremely precisely for one-electron systems, offering the prospect of improving the value of fundamental constantsI.V. Kortunov, S. Alighanbari, M.G. Hansen, G.S. Giri, V.I. Korbov, and S. Schiller, Nat. Phys., 1-5 (2021) Until now, the most precise determinations of the ionization energies of molecular hydrogen have relied on the extrapolation of Rydberg series using multichannel quantum-defect theory (MQDT) and quantum defects extracted from high-precision measurements of Rydberg-Rydberg transitions using millimeter-wave (mmW) spectroscopyD. Sprecher, Ch. Jungen, and F. Merkt, J. Chem. Phys. 140, 104303:1-18 (2014) The uncertainty of these extrapolations are typically larger than 500 kHz, currently representing the dominant source of uncertainty in the determinations of ionization energies. Additionally, autoionization lifetimes impede mmW measurements above the lowest ionization threshold which renders accurate extrapolations to higher rovibrational levels of the ion difficult. We present here a method to determine the binding energies of Rydberg states without having to resort to an extrapolation using MQDT. We carry out precision measurements of the linear Stark manifolds associated with near-degenerate high-l states by mmW spectroscopy in the presence of intentionally applied electric fields. Extrapolating the linear Stark manifold to zero field yields accurate values of the zero-quantum-defect positions, given by −R_H2/n² relative to the ionization thresholds. This method will be illustrated for Rydberg states of H₂ above n=50, circumventing the current bottlenecks and opening the route for order-of-magnitude improved ionization energies.

---

Footnotes:

N. Hölsch, M. Beyer, E.J. Salumbides, K.S.E. Eikema, W. Ubachs, Ch. Jungen, and F. Merkt, Phys. Rev. Lett., 122(10), 103003 (2019), M. Puchalski, J. Komasa, P. Czachorowski, and K. Pachucki, Phys. Rev. Lett., 122(10), 103003 (2019). C. Haase, M. Beyer, Ch. Jungen, and F. Merkt, J. Chem. Phys. 142(6), 064310 (2015). I.V. Kortunov, S. Alighanbari, M.G. Hansen, G.S. Giri, V.I. Korbov, and S. Schiller, Nat. Phys., 1-5 (2021). D. Sprecher, Ch. Jungen, and F. Merkt, J. Chem. Phys. 140, 104303:1-18 (2014).

The hydrogen molecule is an important system for the development of molecular quantum mechanics. In particular, the dissociation energy is a benchmark quantity for ab initio calculations. Latest calculations including nonadiabatic QED corrections can now reach a precision of better than 1 MHz for the dissociation energy of molecular hydrogen (M. Puchalski et. al, Phys. Rev. Lett. 121, 073001 (2018)), opening up the prospect of contributing to the more accurate determination of the proton charge radius, if validated experimentally. We present a spectroscopic determination of the ionization and dissociation energies of D₂ with an uncertainty below 1 MHz by combining separate measurements of the intervals between the X and GK states, and the GK and high-nf Rydberg states with MQDT-assisted extrapolation to the ionization limit. The X-GK interval is measured by Doppler-free two-photon spectroscopy using 178 nm radiation, generated by frequency up-conversion using a KBBF crystal. The laser pulses are generated in a seeded, chirp-compensated, Ti:Sa oscillator-amplifier system where the seed is a Ti:Sa laser at 714 nm locked to a frequency comb. The GK-nf Rydberg state intervals are measured in a resonant three-photon excitation scheme, using pulsed VUV and VIS lasers to reach the intermediate GK state and a single-mode continuous-wave (cw) Ti:Sa laser for transitions to nf Rydberg states. The frequency of the cw laser was calibrated with a frequency comb referenced to an atomic clock. To reach an accuracy below 1 MHz, AC- and DC-Stark effects, as well as first- and second-order Doppler effects are analyzed and compensated.

Precise measurements of vibrational transition frequencies in the isotopes of molecular hydrogen can provide a sensitive probe of fundamental physics. Because these transitions can be predicted with high-level ab-initio theory, comparisons between theory and experiment can be used to test quantum electrodynamics, search for new physics, and determine the nucleon-electron mass ratios more precisely. In this talk, I will present our measurement of the 1−0 R(0) transition frequency in hydrogen deuteride (HD) using infrared-ultraviolet double resonance spectroscopy in a pulsed supersonic molecular beam.A. Fast and S. A. Meek., Phys. Rev. Lett. 125, 023001 (2020)D molecules in the v=0, J=0 state are excited to v=1, J=1 using a tunable infrared laser stabilized to an optical frequency comb, and the excitation efficiency is determined by state-selectively ionizing the vibrationally-excited molecules using a pulsed UV laser. We have determined the absolute frequency of the transition with an uncertainty of 13 kHz (0.12 ppb relative uncertainty), limited primarily by the first order Doppler shift. Improvements in the compensation of this shift should make it possible to reduce the uncertainty by another order of magnitude.

---

Footnotes:

A. Fast and S. A. Meek., Phys. Rev. Lett. 125, 023001 (2020)H

Molecular hydrogen has evolved into a benchmark quantum test system as highly accurate measurements can challenge the most accurate theoretical calculations, which allows for various tests of fundamental physics. Especially the rovibrational splitting serves as an excellent probe to put constraints on the strengths of putative fifth forces in nature. Recent Doppler-free measurements in the (2,0) overtone band of the hydrogen deuteride molecule provided an ongoing stimulating debate on how to interpret the unusual line-shapes. The obtained saturated spectra differ significantly from an ordinary Lamb-dip, or even manifest itself as a ‘Lamb-peak’ for the P(1) line. Here we present an overview of our latest work on the hydrogen deuteride molecule to provide insight in this conundrum. Doppler-free measurements were obtained with our frequency comb referenced NICE-OHMS spectrometer and carefully compared with numerical optical Bloch simulations to extract an accurate prediction of the hyperfineless transition frequency. This shows that levels of accuracy and resolution are reached where the hyperfine contribution cannot be neglected for accurate determination of rovibrational transitions in the hydrogen deuteride molecule and is the origin of the distinctively shaped Lamb-dips and peaks F.M.J. Cozijn et al., Sub-Doppler frequency metrology in HD for tests of fundamental physics, Phys. Rev. Lett. 120, 153002 (2018)^,M.L. Diouf et al., Lamb−dips and Lamb−peaks in the saturation spectrum of HD, Opt. Lett. 44, 4733 (2019)^,M.L. Diouf et al., Lamb-peak spectrum of the HD (2-0) P(1) line, Phys. Rev. Res. 2, 023209 (2020)

---

Footnotes:

F.M.J. Cozijn et al., Sub-Doppler frequency metrology in HD for tests of fundamental physics, Phys. Rev. Lett. 120, 153002 (2018)\end M.L. Diouf et al., Lamb−dips and Lamb−peaks in the saturation spectrum of HD, Opt. Lett. 44, 4733 (2019) M.L. Diouf et al., Lamb-peak spectrum of the HD (2-0) P(1) line, Phys. Rev. Res. 2, 023209 (2020).

Molecular hydrogen is the subject of intensive theoretical and experimental studies because of its relevance to benchmark quantum electrodynamics models. We report an accurate study of the Q(1) 1-0 line of H₂ through a nonlinear spectroscopy setup based on Stimulated Raman Scattering (SRS) and comb calibration of the frequency axis. Systematic uncertainties affecting the apparatus, mainly due to pointing instabilities of pump and Stokes beams and to temperature drifts of the gas cell, have been reduced to about 200 kHz, which is comparable to statistical and systematic errors expected from the global fitting of multi-pressure SRS spectra. We performed measurements over two decades of pressure, from 0.05 to 5 bar, using low-pressure spectra to extrapolate the transition frequency of the isolated molecule, and high-pressure spectra to test collisional lineshape models. Our results are likely to be of relevance for a deeper understanding of H₂ physics and for bringing theory-vs-experiment comparison on the Q(1) 1-0 line-centre frequency below 1 MHz.

Continuous-wave optical parametric oscillators (CW-OPOs) can offer broad tuning with high resolution and high power in the mid-infrared (MIR) and have many important potential applications in precision spectroscopy. The widely tunable cw-OPO system (TOPO) developed by TOPTICA Photonics is capable of covering the wavelength range of 1.45-4.0 μm (with a degenerate gap between 2.07-2.19 μm) without optics or nonlinear optical crystal exchanges. A computer-controlled automation system has been developed to achieve a broad (up to 300 GHz) mode-hop-free (MHF) tuning range of the idler wavelengths and high output power ( > 1 W). Its narrow linewidth is demonstrated in the direct absorption measurements and the Doppler-free saturated absorption spectroscopy of atmospheric molecules. Furthermore, a much-improved frequency accuracy has been achieved by referencing the OPO to a frequency comb and creating a phase-coherent bridge between the NIR and MIR spectral ranges. Frequency noise from both the OPO cavity and the seed laser is eliminated by (i) beating the OPO signal output against the fundamental of the frequency comb and (ii) beating the seed laser against a 1064 nm comb extension. The TOPO system has been incorporated in the MIR cavity ring-down (CRD) system and the CRD-based two-photon spectroscopy system in our lab.

Precision spectroscopy of fundamental bands of molecules in the mid-infrared (MIR) region is of great interest in applications of trace detection and testing fundamental physics, where high-power and narrow-linewidth MIR lasers are needed. By controlling the phase noise of the signal and pump light of a continuous wave optical parametric oscillator (OPO), we established a broadly tunable MIR light source which has an output power of several hundred milliwatts and a linewidth of a few tens kilohertz. The long term frequency drift of the MIR laser was reduced and calibrated utilizing a near-infrared optical frequency comb. The performance of the light source was investigated and the laser system will be used for precision measurement of fundamental vibration of HD.

We have carried out infrared-infrared double-resonance spectroscopy of the 2ν₃ A₁ - ν₃ band of methane. The ν₃ band transitions are pumped using a 90.5 THz difference-frequency-generation (DFG) source frequency-controlled with an optical frequency comb (OFC), and ten tetrahedral components of the Q(1) to Q(4) transitions from the pumped levels are observed with a half-width at half maximum of 0.43 MHz using another 88.5 THz DFG source. The transition frequencies are determined with an uncertainty of 10 kHz using the OFC.

The accurate determination of molecular transition frequencies can provide stringent tests and constraints on fundamental physics questions [1-3]. Here, we present recent work on a comb-locked cavity ring-down spectroscopy system which probes Doppler-broadened spectra in the linear absorption regime. These measurements have relative uncertainties in transition frequency below 10⁻¹², which are among the lowest values reported for optical molecular transition frequencies. We use a probe laser which is phase-locked to a commercial optical frequency comb and subsequently coupled to a high-finesse optical cavity based on that of [4]. We observe stationary measurement statistics for measurements of more than 2000 spectra and find that measurements which are replicated in multiple experiments over several months are normally distributed. Single-spectrum signal-to-noise-ratios can exceed 50,000:1, with resulting line center uncertainties below 5 kHz. Because this method is based on linear absorption, it is an attractive alternative for the measurement of molecular transitions that cannot be probed by saturation-based Doppler-free spectroscopy. Furthermore, we demonstrate results having smaller systematic errors than those provided by more complicated nonlinear spectroscopy methods. [1] F. M. J. Cozijn et. al. Phys. Rev. Lett. 120, 153002 (2018). [2] H. Fleurbaey et. al. Phys. Rev. Lett. 120, 183001 (2018). [3] J. Baron et. al. Science 343, 269-272 (2014). [4] H. Lin et. al. J. Quant. Spectros. and Rad. Trans. 161, 11-20 (2015)

r0pt Figure Double resonance (DR) spectroscopy has been frequently applied in state-selective excitation and to reach energy levels forbidden to single-photon transitions. Due to the low cross section of two-photon transitions, usually high-power pulsed lasers are needed, which prevent high-precision measurements. Here we present a newly developed comb-locked cavity-assisted double resonance (COCA-DR) spectroscopy technique, which simultaneously lock two near-ir diode lasers to a high-finesse cavity and an optical frequency comb. Doppler-free spectra of molecules with selected speeds along the laser beam were observed, and the DR line positions were determined with a frequency accuracy of 3 kHz. The V and Λ type DR measurements of CO overtone transitions near 1.6 μm were used to verify the frequency accuracy of lines measured by COCA-DR. Transitions to the highly-excited (60025) (v_CO=8) "dark" state of the CO₂ molecule were measured using the (30013) vibrational state as the intermediate. Rotational energies of the (60025) state were determined for the first time, and they were compared with the calculated values based on effective operators. The study paves the way to pump-probe measurements of molecules with unprecedented precision.