Ultraviolet spectroscopy plays a pivotal role in studying electronic transitions in atoms and rovibronic transitions in molecules, essential for tests of fundamental physics and of quantum-electrodynamics theory^1,2, determination of fundamental constants³, precision measurements⁴, optical clocks⁵, high-resolution spectroscopy supporting atmospheric chemistry⁶ and astrophysics^7,8, as well as strong-field physics¹¹.

The emerging dual-comb technique^{9,12,13,14,15,16,17,18,19,20,21,22,23} offers spectroscopy a unique host of features, which would provide a unique tool for vacuum- and extreme-ultraviolet spectroscopy. Dual-comb spectroscopy leverages frequency combs, spectra of evenly spaced phase-coherent laser lines that have revolutionized time and frequency metrology^24,25. Dual-comb spectra span over a broad spectral bandwidth and their frequency scale may be directly calibrated within the accuracy of an atomic clock, whereas, as the transitions are interrogated with narrow laser lines, the well-defined instrumental line shape usually has a negligible contribution compared to that of the atomic or molecular profiles⁹. Dual-comb spectroscopy is also a complex technique that involves recording, over long durations, the time-domain interference between the two frequency combs with slightly different repetition frequencies. Dual-comb spectroscopy at present attracts tremendous interest in the infrared range.

Considerable progress has been achieved for generating frequency combs at short wavelengths and using them with techniques of narrow-band spectroscopy^1,2,3,4,5. So far, high-resolution dual-comb spectroscopy with ultraviolet radiation has not been reported. Previous studies have demonstrated nonlinear spectroscopy in rubidium using two-photon excitation with combs at 384 THz (780 nm)^26,27. Linear-absorption spectroscopy is of substantial interest in atmospheric science, and two proposals have discussed possible implementations based on high-power laser systems for reaching short wavelengths^28,29. At present, several research groups are pursuing different approaches for experimental implementation^30,31,32,33.

Here, we take a significant step towards precise spectroscopy over broad spectral bandwidths in the extreme-ultraviolet spectral region by demonstrating a new approach to dual-comb spectroscopy that is suitable for extremely low light levels, as encountered in frequency-comb photonics at short wavelengths. By using photon-counting technology, we have achieved precise, high-resolution, quantum-noise-limited near-ultraviolet dual-comb spectroscopy that operates at photon fluxes more than 10⁶ times lower than commonly used levels in dual-comb spectroscopy and other techniques of comb-based Fourier transform spectroscopy⁹. We demonstrate comb-line-resolved, high-resolution dual-comb absorption spectroscopy in the near-ultraviolet region. We achieve resolutions of 500 MHz (resolving power 1.5 × 10⁶) and 200 MHz (resolving power 4 × 10⁶) at the centre frequency of 772 THz (388 nm). Our innovative approach based on photon counting enables a quantum-noise-limited signal-to-noise ratio, particularly difficult to achieve in dual-comb spectroscopy. Our robust method for interferometry at low light levels overcomes the challenges posed by the low efficiency of nonlinear frequency conversion and, therefore, it lays a solid foundation for extending dual-comb spectroscopy to even shorter wavelengths.

We counterintuitively conduct dual-comb spectroscopy under extremely low light conditions, where, on average, fewer than one detector count occurs every 20 repetition periods of the comb. It is unlikely that two photons, one from each comb generator, would be present in the interferometer at the same time: Two different quantum paths interfere at the counter and the sum of their probability amplitudes provides the probability amplitude of detecting a count¹⁰. We operate with optical powers that are more than a million times weaker than those commonly used in dual-comb spectroscopy. In ref. ¹⁰, we had proposed the conceptual possibility of dual-comb spectroscopy in the photon-counting regime, however, its practical applicability has remained elusive, due to poor spectral resolution.

Two frequency-comb generators emit, at very low light levels, pulse trains with slightly different repetition frequencies, f_rep and f_rep + δf_rep, respectively (Fig. 1). The two beams of the two combs are combined on a beam splitter and their time-domain interference is measured on a fast photon counter. We record the detector counts as a function of time during a specific duration that we call a scan. A scan is initiated by a trigger signal generated in the instrument. Subsequent trigger signals repeat the same scans. By interferometrically controlling the relative timing and phase fluctuations between two combs, the time scale is made to correspond to a scale of optical delay between pairs of pulses. The detector counts at a given optical delay are added to the already accumulated counts of the same delay. Owing to the low light level, it is essential to accumulate many identical scans to reconstruct the interferograms with sufficient statistical data from photon counts. The sequence is repeated until the desired signal-to-noise ratio is achieved in the interferogram (Fig. 1 and Supplementary Video 1). Direct accumulation is only feasible if precise reproducibility of the interferometric scans is achieved. We achieve this by using well-controlled frequency combs and choosing experimental parameters that result in reproducible interferometric waveforms. In particular, we set δf_ceo = 0 (modulo δf_rep), where δf_ceo is the difference between the carrier-offset frequencies of the two combs. The sampling rate of the time bins should be an integer multiple of the difference in repetition frequencies δf_rep. Here, we demonstrate accumulation times of more than 1 hour with quantum-noise limited sensitivity.

The beam of a very low-light-level comb generator based on nonlinear frequency conversion of a near-infrared comb passes through an absorbing sample. It is superimposed on a beam splitter with the beam of another very low-light-level comb generator of slightly different pulse repetition frequency. One beam-splitter output is counted by a photon-counting detector. Fewer than one detector count occurs every 20 comb pulses. At power levels more than 10⁶ times weaker than usually used in dual-comb spectroscopy, the statistics of the detected counts carry the spectral information about the sample.

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Once the acquisition is completed with sufficient statistics, the interferogram can be processed similarly to dual-comb interferograms obtained at higher power. In the frequency domain, pairs of optical comb lines, one from each comb, produce radio-frequency beat notes on the detector, forming, in the radio-frequency domain, a frequency comb of line spacing δf_rep and with a carrier-envelope offset frequency equal to zero. Optical frequencies are thus down-converted into radio frequencies, mδf_rep, where m is an integer.

We illustrate the potential of near-ultraviolet dual-comb spectroscopy by using nonlinear frequency conversion of near-infrared electro-optic frequency combs (Fig. 1, Extended Data Fig. 1 and Methods). The electro-optic system, with its poor conversion efficiency in the ultraviolet, is well suited to test our approach to photon counting. Two frequency combs with slightly different repetition frequencies are generated from a continuous-wave laser that is intensity- and phase-modulated by electro-optic modulators at a centre frequency of 193 THz (1,550 nm). An acousto-optic modulator offsets the centre frequency of one comb, to measure the dual-comb spectrum without aliasing. The near-infrared combs are sequentially frequency doubled twice, once in a periodically poled lithium-niobate crystal and once in a BiB₃O₆ (BIBO) crystal. The near-ultraviolet combs of about 100 lines have a centre frequency tuneable between 770 and 774 THz, and a repetition frequency freely selectable between 100 kHz and 40 GHz. The two low-power ultraviolet frequency-comb beams are combined on a beam splitter. One output of the beam splitter is detected by a photon counter. The detected photon rate of the combined beam of the two combs is at most 5 × 10⁷ photons s⁻¹. This corresponds to an average optical power per comb incident on the counter of about 50 pW, more than 1 million-fold weaker than that commonly used in dual-comb spectroscopy. The fringe visibility is on the order of 30% (Extended Data Table 1). The counts are sampled by a multiscaler at a rate ranging between 12.5 GHz and 500 MHz. The trigger signal for the data acquisition by the multiscaler is generated by frequency division of the 10 MHz clock signal.

The temporal build-up of an interferogram by accumulation of many scans is shown in Supplementary Video 1. An experimental dual-comb interferogram sampled at a rate of 12.5 GHz (Extended Data Fig. 2) recurs with a period of 625 ns, owing to a difference in repetition frequency δf_rep of 1.6 MHz. A Fourier transform reveals the dual-comb spectrum, as in traditional dual-comb spectroscopy. A dual-comb spectrum centred at 770.73 THz contains more than 100 comb lines with a flat-top intensity distribution and results from an accumulation time of 255.5 s (Fig. 2a). The spectral resolution is equal to the line-spacing f_rep of 500 MHz. The span is 50 GHz. The instrumental line shape is a cardinal sine, as expected in an unapodized spectrum (Fig. 2b). By tuning the centre frequency of the continuous-wave near-infrared laser, a sequence of dual-comb spectra corresponding to 15 acquisitions centred at different frequencies is acquired over a total span of about 3 THz (Fig. 2c).

a, Spectrum recorded at a detected rate of 2.7 × 10⁷ photons s⁻¹ with about 100 comb lines. The y scale is linear. b, Magnified portion of a showing three comb lines with cardinal-sine line shapes. c, Illustration of the frequency agility: a sequential acquisition of 15 spectra spanning overall a bandwidth broader than 3 THz. a.u., arbitrary units.

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Next, we illustrate the potential of our photon-counting dual-comb interferometer for absorption spectroscopy of the weak 6S–8P transitions in atomic caesium vapour. One of the comb beams passes through a heated caesium cell before the two comb beams are combined on a beam splitter. The interferograms are detected at a rate of 4.5 × 10⁷ counts s⁻¹, which corresponds to an average optical power per comb of 45 × 10⁻¹² W incident onto the detector of a quantum efficiency of 25% (Extended Data Table 1 and Methods). With roughly 100 comb lines, the power per comb line is estimated to be 4 × 10⁻¹³ W, simply calculated as the optical power divided by the number of comb lines. Spectra with resolved comb lines show the 6S_1/2–8P_1/2 and 6S_1/2–8P_3/2 transitions at 770.73 and 773.21 THz, respectively (Fig. 3a,c, respectively)³⁴. The transmittance spectrum and the dispersion spectrum of the 6S_1/2–8P_1/2 transitions of a Doppler full-width at half-maximum of 1 GHz result from an accumulation time of 152 s (Fig. 3b). The two resonances are due to the hyperfine splitting in the 6S_1/2 ground state (F = 3,4)_, while the hyperfine structure in the excited state is not resolved due to the Doppler broadening. The signal-to-noise ratio, determined as the inverse standard deviation of the normalized absorption baseline of the amplitude spectrum, is 210. The stronger 6S_1/2–8P_3/2 transitions are measured at a shorter accumulation time (64 s) at a signal-to-noise ratio of 195 (Fig. 3d). Absolute frequency measurements of the position of the strongest lines reach an uncertainty of 6 × 10⁻⁹, dominated by the statistical uncertainty (Methods, Extended Data Fig. 3 and Extended Data Table 2). When the two comb beams pass through the cell, only the transmittance spectrum is revealed (Extended Data Fig. 4).

The y scale is linear. The spectra span over 50 GHz. a,b, 6S_1/2–8P_1/2 resonances. a, Amplitude spectrum with resolved comb lines. b, Transmittance and dispersion spectra, obtained by sampling the complex spectrum at the comb line positions, with a signal-to-noise ratio of 210. c,d, 6S_1/2–8P_3/2 resonances. c, Amplitude spectrum. d, Transmittance and dispersion spectra with a signal-to-noise ratio of 195.

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One may argue that each spectrum does not span over extended regions. This feature of dual-comb spectroscopy with electro-optic modulators has been discussed in the context of infrared spectroscopy²⁰: the moderate spectral span is compensated by the frequency agility of the continuous-wave laser and of the driving synthesizers: the centre frequency and the repetition frequency of the combs can be adjusted by a simple knob, and the small number of comb lines provides a high signal-to-noise ratio in short measurement times, with selectable spectral resolution. This makes the setup attractive for some experimental configurations such as atomic spectroscopy in which the spectra are not too dense and crowded. As an illustration of the adjustable resolution, the transmittance spectrum of the 6S_1/2–8P_1/2 transition at a resolution of 500 MHz is compared with that at a resolution of 200 MHz (Extended Data Fig. 5).

Reducing technical noise sources to reach the quantum-noise limit is not easy in dual-comb spectroscopy. Most experiments have been limited by detector noise or intensity noise of the laser sources. In the experimental conditions of Extended Data Fig. 4a,b, we measure the experimental signal-to-noise ratio of the absorption baseline in the spectra as a function of count rate (Fig. 4a) and accumulation time (Fig. 4b). The experimental signal-to-noise ratio scales as the square root of the detected photon rate (Fig. 4a), which demonstrates that the signal-to-noise ratio is limited by counting statistics (also called quantum-noise limit or shot-noise limit). The experimental signal-to-noise is also in good agreement with a simple model of the quantum-noise-limited signal-to-noise ratio (Methods and Extended Data Table 1). The experimental signal-to-noise also scales with the square root of the accumulation time, showing that the interferometric coherence is maintained (Fig. 4b).

a, The signal-to-noise ratio of the transmittance baseline scales as the square root of the photon rate, showing that the quantum-noise limit is reached. The slope in the linear fit is 0.51 (±0.01). b, The signal-to-noise ratio also scales as the square root of the accumulation time, showing that mutual coherence of the two combs is maintained. The slope in the linear fit is 0.50 (±0.01).

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Vacuum- and extreme-ultraviolet combs have only been generated as harmonics of near-infrared femtosecond mode-locked lasers³⁵, so it is crucial to establish the suitability of such lasers for photon-counting dual-comb spectroscopy. We illustrate that, even with fibre mode-locked lasers attenuated to only ten million counts per second, the spectral information in amplitude and phase, of an absorbing sample can be precisely retrieved from the photon-counting statistics.

A pair of frequency-doubled erbium-doped femtosecond mode-locked fibre lasers generate two trains of optical pulses at a central frequency of 384 THz (Extended Data Fig. 6 and Methods). Their repetition frequencies are 100 MHz and their difference δf_rep is equal to −12.5 kHz. One fibre comb is self-referenced against a radio-frequency clock. A feed-forward dual-comb technique³⁶ establishes mutual coherence between the two frequency combs, effectively stabilizing their relative phase and timing fluctuations. The phase-matching properties of the periodically poled lithium-niobate crystals limit the spectral span to roughly 0.12 THz. The self-referenced comb passes through a Rb vapour cell, after which it is combined on a beam splitter with the second comb. At one of the beam-splitter outputs, the beam is attenuated to a detected count rate of 8.4 × 10⁶ counts s⁻¹, which is less than one count every 20 laser pulses. The calculated power per comb incident on the detector is 1.5 × 10⁻¹² W, 10⁸ times weaker than commonly used in dual-comb spectroscopy. The average power per comb line is estimated to be 1.2 × 10⁻¹⁵ W. The signal is captured by a photon-counting detection module. The second output of the beam splitter produces the trigger signal; the dual-comb interference is detected by a silicon photodiode. The fringe occurring at zero delay acts as a trigger of the multiscaler, which adds up the subsequent scans.

The multiscaler directly accumulates the time scans, each comprising 5 × 10⁵ time bins over a duration of 3.28 ms. Over a total duration of 4,592 s, the interference signal gradually builds up in the counting statistics. The final interferogram time trace (Extended Data Fig. 7) consists of 41 identical bursts occurring periodically with a period 1/δf_rep of 8 × 10⁻⁵ s. The Fourier transform of the interferogram reveals a transmittance spectrum with resolved comb lines (Fig. 5). The spectrum spans 0.12 THz with 1,200 comb lines well above the noise level (Fig. 5a). The Doppler-broadened 5S_1/2–5P_3/2 transitions in ⁸⁵Rb and ⁸⁷Rb, are sampled by the comb lines of 100 MHz spacing (Fig. 5b). The comb lines show a transform-limited width (Fig. 5c). The average signal-to-noise ratio of the absorption baseline is 67, whereas the calculated quantum-noise limited signal-to-noise ratio over 1,200 comb lines is 69. The phase spectrum is simultaneously retrieved. The absorption and dispersion profiles of Rb transitions, sampled by the comb lines at 100 MHz resolution (resolving power 4 × 10⁶), show a full-width at half-maximum of about 600 MHz (Extended Data Fig. 8).

The average power per comb line is 1.2 × 10⁻¹⁵ W. a, Dual-comb spectrum with 1,200 comb lines. b, Magnified portion of a showing the 5S_1/2–5P_3/2 transition of rubidium sampled by the comb lines. c, Magnified view of four comb lines. The comb lines show the expected sinc-function instrumental line shape. The full-width at half-maximum of the comb lines is 365 Hz in the radio-frequency domain, corresponding to the Fourier transform limit.

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We implement high-resolution linear-absorption dual-comb spectroscopy in the ultraviolet spectral range. Our experiments use two distinct experimental setups with different types of frequency-comb generator, and they establish that the full capabilities of dual-comb spectroscopy are extended to starved-light conditions, at power levels more than a million-fold weaker than those commonly used in dual-comb spectroscopy. By repeatedly achieving a quantum-noise-limited signal-to-noise ratio, an optimal use of the light available for the experiments is achieved. Our photon-level interferometer accurately reproduces the statistics of photon counting, as shown by the signal-to-noise ratio at the fundamental limit, the instrumental line shape that follows the theoretical expectation and the direct referencing of the frequency scale to a radio-frequency clock. The prospect of harnessing dual-comb spectroscopy at very low light levels may seem counterintuitive¹⁰. Here we have experimentally realized such a milestone, which will unlock new applications. In this section, we discuss the implications and perspectives of our results.

In recent years, dual-comb spectroscopy has emerged as a powerful technique for precise spectroscopy over broad spectral bandwidths. A technique of Fourier transform spectroscopy, it measures the time-domain interference between two frequency combs of slightly different repetition frequencies and reveals the spectrum through harmonic analysis of this interference pattern. However, as a technique relying on frequency measurements rather than on wavelength determination, it does not suffer from the geometric limitations of state-of-the-art interferential or diffractive spectrometers. It promises an as yet untapped potential for high precision and high accuracy. Whereas dual-comb spectroscopy has been mostly used for linear absorption of Doppler- or collision-broadened transitions of small molecules in the gas phase, action spectroscopy has been successfully demonstrated^22,26,27, for example, with Doppler-free two-photon excitation²⁶. Nevertheless, the technique at present uses intense laser beams of an average power higher than several tens of microwatts at the detector and it interrogates samples of millions of atoms or molecules.

In a broader context, however, measuring spectroscopic signatures at extremely low light levels has become critical in many areas of science and technology. These range from precision spectroscopy, in which few atoms or molecules are observed in controlled conditions in which the systematic effects due to interaction between atoms and between atoms and light field are minimized, to light-damageable strongly scattering bio-medical tissues to environmental sounding of the atmosphere over long distances. Our work opens up the prospect of dual-comb spectroscopy at low light levels in these challenging scenarios, which will leverage the host of specific features of the technique.

An exciting forthcoming application is certainly the development of dual-comb spectroscopy in the short-wavelength range, which would unlock precise vacuum- and extreme-ultraviolet molecular spectroscopy over broad spectral spans. Now, broadband ultraviolet spectroscopy is limited in resolution and accuracy^7,37,38, and at short wavelengths, it relies on unique instrumentation at a unique facility^39,40. Ultraviolet dual-comb spectroscopy is a much sought-after target, but a challenging one for three primary reasons. First, similar to other lasers, frequency-comb sources that emit directly in the ultraviolet region are not readily available. The current approach to generating ultraviolet frequency-comb radiation involves nonlinear frequency conversion through harmonic generation in crystals (or rare gas jets for shorter wavelengths). However, this process leads to low conversion efficiencies and consequently weak powers. Second, during this conversion process, the phase noise of the comb source is multiplied, typically by the harmonic number. Last, dual-comb spectroscopy relies on interferometry. Many interferometric applications require fringes to be defined to within a hundredth of the wavelength. Achieving such a precision in an interferometer operating at short wavelengths is technically challenging and, for dual-comb spectroscopy, it becomes even more demanding due to the adverse effect of phase-noise multiplication resulting from nonlinear frequency conversion.

Efficient solutions for generating low-noise combs in the ultraviolet and for building dual-comb interferometers with long coherence times have been reported in other contexts. Building on these provides a starting point. Conversely, the pressing issue of low-power handling in dual-comb spectroscopy remained unexplored until our work. In ref. ⁴, fluorescence excitation of a rare gas by a single comb line with a power of 10 pW unlocked linear direct frequency-comb spectroscopy at 63 nm. Here, by controlling the mutual coherence of two mode-locked lasers with 1 femtowatt per comb line, we show an optimal build-up of the counting statistics of their interference signal over times exceeding 1 hour. Our present concept at the femtowatt level will allow to extend the pioneering experiment in ref. ⁴ to broadband dual-comb excitation spectroscopy²⁶ in the extreme ultraviolet, to measure at even lower powers, and thus at shorter wavelengths, and potentially to acquire linear-absorption broadband spectra. Furthermore, the generation of frequency combs at frequencies higher than 1,600 THz (wavelengths shorter than 180 nm) has relied on cavity-enhanced high-harmonic generation in a rare gas, a complex process³⁵ whose use in dual-comb spectroscopy is daunting. Our work opens up alternative strategies: single-pass high-harmonic generation in a rare gas⁴¹ or the emerging high-harmonic generation in solids⁴² become conceivable options, even at high repetition frequencies.