A thorium-229 optical nuclear clock with feedback loop

Over the last 70 years, atomic clocks based on microwave and optical electronic transitions have successfully served as frequency standards based on their high levels of stability and accuracy^2,8,9,10. To elevate these atomic timekeeping devices to the next generation, in 2003, Peik and Tamm¹ proposed a nuclear clock based on the extraordinarily low-energy isomeric state of the thorium-229 (²²⁹Th) isotope at 8.4 eV. With a lifetime approaching 1 h for a bare nucleus in vacuum, this transition offers a potential resonance quality factor of the order of 10¹⁹ (ref. ¹¹). Additionally, the nucleus couples only weakly to perturbative fields¹², which allows the construction of an optical clock implemented in a solid material at room temperature¹³. By doping microgram amounts of ²²⁹Th into a crystal host, an experimentally simple and robust clock can be created, offering a large gain in signal-to-noise ratio (SNR) compared with approaches based on a single or a few isolated nuclei, while only moderately degrading the resonance quality factor^1,4,11,14.

A clock based on the ²²⁹Th nuclear transition is of great interest for testing theories beyond the Standard Model of particle physics that indicate that the fundamental constants vary over time⁵. The low energy of the ²²⁹Th isomer transition originates from a coincidental near-cancellation of the mega-electronvolt-scale Coulomb and nuclear contributions to the binding energies of the ²²⁹Th ground and excited states, making the transition frequency highly sensitive to fluctuations in the coupling constants of the interactions involved^5,6,15. As a result, a solid-state nuclear clock in its development phase would already be competitive with or may even surpass the sensitivity of advanced comparable atomic clocks in the search for dark matter.

The realization of a nuclear clock builds upon half a century of scientific discoveries. See the reviews in refs. ^12,16. In a research stream initiated by Kroger and Reich in 1976 (ref. ¹⁷), the energy of the ²²⁹Th nuclear transition was narrowed down from an originally conjectured value of below 100 eV to −1(4) eV (ref. ¹⁸), 3.5(1.0) eV (ref. ¹⁹) and then 7.8(5) eV (refs. ^20,21) using gamma spectroscopy. Using the internal conversion decay channel, the existence of the ²²⁹Th isomer was proven in 2016 (ref. ²²) and the energy value refined to 8.28(17) eV (ref. ²³). An optical detection of the ²²⁹Th radiative nuclear decay was realized in 2022, which determined the isomer energy to be 8.338(24) eV (ref. ²⁴) in the vacuum-ultraviolet (VUV) range. Thereafter, resonant laser excitation in Th:CaF₂, Th:LiSrAlF₆ crystals and ThF₄ and ThO₂ thin films was reported in 2024 and 2025 (refs. ^25,26,27,28), quickly followed by the resolution of the nuclear quadrupole structure in Th:CaF₂ (refs. ^13,29).

The development of 148-nm continuous-wave laser systems in 2025 (refs. ^30,31), in combination with highly doped ²²⁹Th:CaF₂ crystals^32,33, has enabled the measurement of the nuclear transition with absorption spectroscopy⁷. With this technique, it became possible to probe the ²²⁹Th nuclear transition continuously, without being limited by the long isomeric state lifetime of approximately 10 min. Moreover, absorption provides a higher detection efficiency, yielding three orders of magnitude more signal photons per second compared with fluorescence.

An optical clock is based on an atomic transition with a narrow linewidth, which is probed by a clock laser with a long coherence time. By stabilizing this laser to the atomic resonance, it can inherit the desired stability and reproducibility from the atom. Aiming for millihertz resolution and accuracy, the feedback times in this process typically extend to several seconds, requiring a pre-stabilization of the laser for higher-frequency noise to an optical cavity that has a suitable short-term stability. With the combination of these two control loops, very stable laser frequencies can be generated and then converted into a time signal through a frequency comb acting as an optical clockwork³⁴.

Here, we transfer this concept from an atomic to a nuclear reference transition. Although previous studies^7,13,14 demonstrated that it is possible to excite the ²²⁹Th nucleus using VUV lasers stabilized to external frequency standards, it has not yet been possible to construct a system in which the laser interrogating the ²²⁹Th nuclei is steered by the nuclear transition itself. Our system is a nuclear clock that operates as a stand-alone device.

The nuclear clock is composed of a cavity-stabilized laser for short-term stability (referred to as the clock laser) and a ²²⁹Th interrogation system providing long-term stability (Fig. 1a). We use a continuous-wave VUV laser source to interrogate the ²²⁹Th nuclei^7,30. This laser source comprises a 1,187-nm frequency-quadrupled commercial laser (the seed laser in Fig. 1) and a single-pass second-harmonic-generation stage using a randomly quasi-phase-matched strontium tetraborate (SBO) crystal. The frequency and linewidth of the 1,187-nm interrogation laser are determined by an offset phase lock to the clock laser. We modulate the offset lock frequency to perform absorption measurements. For the clock laser, we use a high-finesse cavity-stabilized external-cavity diode laser operating at 1,187 nm. The stabilization of the clock laser to the cavity uses a Pound–Drever–Hall scheme on a sideband generated by an electro-optical modulator, allowing us to shift the frequency of the clock laser independently of the cavity resonances.

The 148-nm laser enters a vacuum chamber containing the Th:CaF₂ crystal and the detector. We use a segment of the X2 crystal; other pieces of the same ingot were used in refs. ^{13,14,25,29,35,36,37,38,39}. The temperature of the crystal is monitored throughout all measurements; without active stabilization it remains within an interval of plus or minus 0.5 K around 294.7 K. The detection is based on a photomultiplier tube (PMT) with a CsI photocathode, operating in photon-counting mode, placed behind the crystal for the absorption spectroscopy of the nuclear transitions⁷.

The beat frequency f_b for clock comparison and diagnostics is measured between the 1,187-nm output of the clock laser and the nearest mode of a stabilized infrared-frequency comb (Fig. 1b). The repetition rate of the comb is stabilized by locking the respective nearest comb mode to a high-finesse cavity-stabilized external-cavity laser at a wavelength of 1,542 nm. The absolute frequency of the comb is calibrated with radiation from an external-cavity laser at 1,542 nm, provided through a Doppler-compensated fibre link from the Austrian Federal Office of Metrology and Surveying (Bundesamt für Eich- und Vermessungswesen or BEV), where it is referenced to a Yb⁺ single-ion clock. We use this for comparison measurements between the Yb⁺ ion clock and the ²²⁹Th nuclear clock. Using the beat signal between the clock laser and the comb, we characterize the clock laser cavity drift and compensate for the dominant linear component of approximately 200 mHz s⁻¹.

In clock operation, the clock laser is stabilized to the interrogated nuclear transition frequency with the help of a feedback loop that corrects for residual instability or drift of the cavity. To obtain the error signal of the ²²⁹Th absorption line, we modulate the interrogation frequency between two frequencies separated by \({f}_{{\rm{FWHM}}}/\sqrt{3}\), with f_FWHM the full-width at half-maximum (FWHM) of the measured absorption peak, and we record the difference between the signals. Scanning is then done by moving the centre frequency over the scan range. The resulting curve has a zero crossing at the absorption peak and a near-linear slope around it. For a detailed description and an example of the error signal shape obtained, see Methods.

We can neglect absorption saturation effects as we are operating in a regime where the number of ²²⁹Th nuclei in the excited state is much lower than the ground-state population at any time. Therefore, the fractional absorption amplitude A is constant in time. The \({\rm{SNR}}\,\approx \frac{A}{\sqrt{1-A}}\sqrt{\phi }\), where ϕ is the number of photons detected per second. With 65 pW of VUV laser power reaching the PMT, which has a detection efficiency of 10%, and 0.75% absorption on the ²²⁹Th 5/2 → 3/2 transition, this equation yields SNR ≈ 17 after 1 s of measurement.

In each clock cycle, we perform a measurement at the centre frequency determined in the preceding cycle for a given integration time T. The result is a point on the previously measured error signal function, which we then invert to get the actual frequency deviation from the zero crossing. This deviation, expressed as an infrared frequency, is then added to the current driving frequency of the electro-optical modulator in a single adjustment step to shift the laser back to the ²²⁹Th resonance. The actuation of each adjustment step takes about 1 s. The size of the step depends on the integration time T and the stability of the high-finesse cavity used in the clock laser.

If T is short, the noise in the detected PMT counts will result in a high uncertainty in the estimated line centre. If T is long, the residual drift of the cavity during the measurement also increases this uncertainty. We can choose T such that the SNR of the error signal is high enough to avoid degrading the short-term stability of the clock laser inherited from the cavity. For T = 30 min, the feedback signal of the nuclear resonance barely impacts the fractional frequency instability at 1 s, which is then comparable with the measured cavity instability of approximately 10⁻¹⁵. In contrast to fluorescence measurements, absorption spectroscopy also enables us to use feedback times much shorter than the isomer lifetime. With T = 20 s, the short-term stability deteriorates more strongly from the level obtained with the cavity alone, but the laser is locked to the ²²⁹Th frequency on a shorter timescale. This enables faster comparison of the ²²⁹Th and Yb⁺ transition frequencies, which extends the accessible mass range for ultralight dark matter searches.

The standard metric for characterizing the stability of a clock is the Allan deviation, the square root of the two-sample variance of consecutive measurements of the normalized clock output frequency. In Fig. 2 we show the overlapping Allan deviation σ_y (ref. ⁴⁰) plotted versus the averaging time τ. For τ > T, the instability is determined by the shot-noise-limited interrogation of the ²²⁹Th nuclear transition \({\sigma }_{y}(\tau )\approx (\varGamma /{f}_{0})(1/\mathrm{SNR}){(\tau /{\rm{s}})}^{-1/2}\). Here f₀ = 2.0204 × 10¹⁵ Hz is the nuclear transition frequency⁷ and Γ is the measured linewidth of approximately 100 kHz.

The current state of the power stability of the VUV laser system allows us to operate the nuclear clock without any intervention for 1 day. The clock operation is not affected by fluctuations of the crystal temperature of approximately 0.1 K, consistent with the temperature dependence of the 5/2 → 3/2 transition (1.8 kHz K⁻¹ at 295 K) reported in ref. ³⁷.

Although the clock reaches fractional frequency instabilities in the low 10⁻¹⁴ range within a continuous clock run, we observed that the reproducibility between runs on different days was limited to approximately 5 × 10⁻¹³. Before each clock operation run, it is necessary to realign the laser system. In our current set-up, this necessarily leads to a small change in the axis and location at which the VUV laser traverses the crystal. Therefore, each clock operation probes slightly different local regions of the X2 crystal, as the laser beam has a diameter of approximately 0.5 mm and the crystal has a diameter of 3.1 mm. To test whether this explains the reduced reproducibility, we used an x–y translation stage to perform measurements on four different crystal positions.

We determined the respective line centres through T = 20 s measurements over a total duration of t_tot = 20 min on each of the probed crystal positions. As visible in Fig. 3, the deviations δf_i of the line centres from the frequency average of all five measurements differ by up to 1.7 kHz between datasets, which is well above the 0.13-kHz spread expected from statistical fluctuations in our measurement but within the uncertainty reported in previous studies of the line-centre reproducibility¹⁴. Remeasuring at the same crystal position (scans 2 and 4 in Fig. 3) yields a reproducible line-centre frequency. We conjecture that local strains induced by structural inhomogeneity causes these frequency shifts. In future implementations of the Th:CaF₂-based clock, constraining the optical path through the crystal to a reproducible volume or improving doping homogeneity and minimizing strain through optimized growth conditions may enhance reproducibility.

To address some of the most pressing open questions in fundamental physics, such as the origin and properties of dark matter, a variety of theories predict the existence of new ultralight scalar bosons⁴¹. Even if these bosons only have feeble interactions with the known matter particles, their influence might still be observable in high-precision experiments. For example, couplings to the electromagnetic force, strong force or quarks would lead to an apparent oscillation of the fine structure constant α, the quantum chromodynamics (QCD) scale parameter Λ_QCD and quark masses m_q, respectively, with a frequency corresponding to the mass of the new boson. Because the transition energies in atoms and nuclei depend on these fundamental constants, their values would oscillate too⁴². Here the ²²⁹Th nuclear transition offers a big advantage: the extremely low energy of the isomer state is a result of a coincidental near-cancellation of mega-electronvolt-scale electromagnetic and strong nuclear force effects. This makes the transition very sensitive to changes in α, Λ_QCD and m_q. Owing to unknown nuclear parameters, precise values of these sensitivity factors cannot at present be calculated^15,43, but it is predicted that they will exceed those of the most sensitive atomic clock transitions by several orders of magnitude^5,6,43,44. Thus, even if the ²²⁹Th nuclear clock does not yet reach the same level of frequency instability as the best atomic clocks, it already yields competitive results in searches for fundamental physics.

We searched for periodic signals in the ²²⁹Th transition frequency in the data from the T = 20 s clock operation shown in Fig. 2 (red curve). We related the variation in the infrared beat frequency f_b to the variation in the ratio between the Yb⁺ and ²²⁹Th transitions \({f}_{{\rm{Th}}}/{f}_{{{\rm{Yb}}}^{+}}\) through \({\partial }_{t}\,\log ({f}_{\mathrm{Th}}/{f}_{{\mathrm{Yb}}^{+}})=8({\partial }_{t}\,{f}_{{\rm{b}}})/{f}_{0}\). To search for possible oscillations of this ratio, we compute the best-fitting amplitudes \({{\mathcal{A}}}_{{\rm{f}}}=\sqrt{4{P}_{{\rm{f}}}/{N}_{\mathrm{tot}}}/{f}_{0}\) using the Lomb–Scargle method (Methods). Here P_f is the power spectrum of the beat frequencies converted into the VUV, and N_tot is the total number of statistically independent measurements. This result is sensitive to oscillations with frequencies between 1/t_tot and 1/T, where t_tot is the total measurement time of approximately 23 h.

To identify the statistical significance of the peaks in the Lomb–Scargle periodogram, we used Monte Carlo simulations of the clock feedback loop to calculate a 5% detection threshold. As no amplitudes exceeded this detection threshold, we established an upper limit for the amplitude of oscillations in the investigated frequency range (Fig. 4a and Methods). The upper limit can further be used to restrict possible couplings to scalar dark matter. The size of the coupling constant d_e for the interaction between a scalar boson ϕ and photons relates to the measured power spectrum P_f as follows (Methods):

The first factor contains the ²²⁹Th-specific enhancement through the large sensitivity factor \({k}_{{\rm{Th}}}^{\alpha }\) and transition frequency. The second factor indicates how the uncertainty decreases with reduced noise and higher statistics. The final factor leads to the decreasing sensitivities for higher boson masses m_ϕ and contains a suppression factor that is the same for any clock measurement, where c is the speed of light. For variations of the fine structure constant α, which corresponds to a scalar coupling to photons, the nuclear clock presented in this work imposes limits like those of the best comparable atomic clocks (Fig. 4). The transition frequencies of atomic clocks are less dependent on nuclear properties, even for those clocks that are based on hyperfine transitions, like the Cs clock. Thus, their sensitivities to variations of Λ_QCD and m_q are lower. The nuclear clock presented in this work reached a factor of 10 to 1,000 deeper into the parameter space than earlier experiments with atomic clocks. Compared with ref. ⁴⁴, in which measurements of the ²²⁹Th line position and shape over 10 months were analysed, our measurement offers an improvement of around one order of magnitude in the accessible frequency range. The fast feedback of the absorption scheme allowed us to directly probe masses that could previously only be restricted through a line-shape analysis^44,45. In future clock measurements, further enhancements in sensitivity can be expected. The frequency limits of the Lomb–Scargle periodogram are given by 1/t_tot and 1/T. Therefore, lower frequencies could be reached by operating the clock for longer, whereas higher frequencies are accessible by reducing the cycle time. The sensitivity over the whole frequency range scales with the square root of the number of statistically independent measurements N_tot.

Apart from searches for oscillations, we analysed the frequency ratio \({f}_{{\rm{Th}}}/{f}_{{{\rm{Yb}}}^{+}}\) data for possible slow drifts. A linear fit of the T = 20 s and t_tot ≈ 23 h data gave a slope of (2 ± 4) × 10⁻¹⁴ d⁻¹, consistent with 0. Using the same sensitivity constants introduced above, this corresponds to linear drifts of \(\log \alpha \), \(\log \,{\varLambda }_{\mathrm{QCD}}\) and \(\log \,{m}_{q}\) of (2 ± 4)/k_Th × 10⁻¹⁴ d⁻¹. The aforementioned line-centre reproducibility will be a limiting factor when extending this search to months or years.

The performance of the ²²⁹Th clock could be straightforwardly improved by increasing the VUV laser power. The power limitation of the laser source used in our experiment is defined by the spontaneous domain-inversion structure of the SBO crystal. Only a fraction of the domains contributes to the frequency upconversion of the 297-nm radiation.

One order of magnitude gain in the VUV power could be achieved by using the presented randomly quasi-phase-matched SBO in an enhancement cavity, which is limited due to optical losses at 297 nm. Further enhancement of the VUV power to the hundreds of nanowatts level could be reached by using commercially available 3-W ultraviolet lasers.

Higher VUV power levels could be obtained using four-wave mixing in Cd vapour³¹, for which 300 nW has already been reported, or by fabricating patterned SBO⁴⁶ and BaMgF₄ crystals⁴⁷. For a first-order periodically poled nonlinear crystal, a VUV power of at least 1 μW can be expected. However, this has not yet been demonstrated due to technical challenges in the submicrometre periodic poling.

The shot-noise-limited clock instability scales with \({\sigma }_{y}(\tau )\approx (\varGamma /{f}_{0})\) \((1/\mathrm{SNR}){(\tau /{\rm{s}})}^{-1/2}\). Beyond increasing the laser power, there are two other ways to improve the clock performance: increasing the absorption and decreasing the linewidth Γ. We operate in a low-saturation regime, where the fractional absorption amplitude A does not change with time. This will be the case as long as the number of photons that are absorbed during the lifetime of the isomeric state is much lower than the number of ²²⁹Th nuclei in the beam path. In the current experiment, approximately 3 × 10⁸ photons were absorbed in 10 min, although there were approximately 10^{16 229}Th nuclei in the beam path. Therefore, absorption measurements are expected to operate far from saturation, even if the available laser power increases significantly with further laser developments.

In ref. ¹⁴, it was found that Γ increases with the ²²⁹Th density ρ_Th. The absorption A also depends on ρ_Th and increases as \(A=1-\exp (-\bar{\sigma }L{\rho }_{{\rm{Th}}}\,\chi )\). Here L is the length of the light path through the crystal, χ quantifies the percentage of Th atoms that are in a D centre (for X2, 58.6%; ref. ²⁹) and \(\bar{\sigma }\) is the absorption cross section, which itself scales with Γ⁻¹. These scalings imply that an optimal ρ_Th exists for which σ_y is minimized. The shot noise limit resulting from this theoretically optimal doping density and χ = 1 is only a factor of 2 lower than that of our current crystal, X2. Therefore, no large improvements in σ_y can be expected by modifying ρ_Th in future Th:CaF₂ crystals. It should be possible with longer crystals and VUV cavities to increase the path length L by a factor of 10, resulting in an order of magnitude higher SNR. With this, and a laser power of 1 μW, ²²⁹Th-doped CaF₂ crystals could reach a fractional frequency instability of approximately \(1{0}^{-15}/\sqrt{\tau /{\rm{s}}}\).

Going beyond Th:CaF₂, there are other possible host crystals. We expect crystals containing Th as part of the stoichiometric crystal structure, such as ThF₄, to have reduced linewidths. In such crystals, ²²⁹Th is not a dopant and, thus, does not distort the crystal structure, presumably reducing the inhomogeneous broadening of the nuclear transition. This is also expected to improve the line-centre reproducibility.

Another relevant contribution to the linewidth is the magnetic interaction with neighbouring F-19 nuclei, which causes broadenings in the range 0.1–1 kHz (refs. ^4,27). Even smaller linewidths might become accessible with spinless solids⁴⁸. To benefit from these improvements, the linewidth of future VUV lasers needs to be narrowed down equivalently. Using a conservative linewidth estimate of Γ = 1 kHz, a crystal with similar doping density and dimensions as the one presently used would yield an optical depth \(\bar{\sigma }L{\rho }_{{\rm{Th}}}\,\chi \approx 1\). With this, and using a 100-pW laser with a high stability cavity, a fractional frequency instability of approximately \(1{0}^{-16}/\sqrt{\tau /{\rm{s}}}\) could be reached. This would bring the frequency instability of the nuclear clock to the same level as state-of-the-art optical atomic clocks^49,50,51 while still maintaining the form factor and simplicity of a solid-state clock. Compared with the nuclear clock introduced in this work, this would enable a four orders of magnitude increase in sensitivity to the variation of the fundamental constants.

Laser set-up

In the ²²⁹Th interrogation system, a commercial laser (FHG-TA Pro, TOPTICA) provides an output power of approximately 500 mW at a wavelength of 296.8 nm, frequency-quadrupled in two doubling stages and one amplification stage from a diode seed laser at 1,187 nm. The final second-harmonic-generation process is based on nonlinear frequency conversion in an SBO crystal kept under a high-purity N₂ atmosphere (purity 5.0). Fundamental radiation still present in the beam after the final second-harmonic-generation step is separated using three dielectric-coated mirrors. For the absorption measurements, we used a head-on type PMT (R6835, Hamamatsu) mounted inside the vacuum chamber and operated at 2.5 kV. We used the same segment of the X2 sample⁶⁰ as in ref. ⁷. The segment in use has a cylindrical geometry with a diameter of 3.1(1) mm and a length of 4.2(1) mm and was oriented such that the beam traversed along the centre line of the piece. The measured average concentration of ²²⁹Th in the segment was 6.6(5) × 10¹⁵ mm⁻³.

The seed laser of the FHG-TA was locked to the clock laser (CLS, TOPTICA) through an offset frequency phase lock. A fast photodiode with a bandwidth of 10 GHz measured the beat frequency between the two lasers, which was then mixed with a reference frequency. The fast loop of this lock actuated on the seed laser diode current, and the slow loop was used to steer the grating of the laser diode. The laser was scanned by modulating the reference frequency.

Like the actuators of the offset lock, the Pound–Drever–Hall lock of the clock laser to the cavity was also configured such that the fast feedback loop actuated on the laser diode current, whereas the slow feedback signal of the Pound–Drever–Hall scheme controlled the position of the grating for optical feedback to the laser diode.

For clock comparison measurements and drift rate measurements of the cavity, the beat frequency f_b of the clock laser with the frequency comb (FC1500-250-ULN, Menlo) was recorded using a dead-time-free frequency counter (K+K FXE) operated in Π-type counting mode^61,62 at a gate time of 1 s. Our method of comparing the thorium clock at TU Wien with the Yb⁺ clock (TOPTICLOCK, TOPTICA)⁶³ at BEV follows the scheme described in ref. ⁷. The Yb⁺ clock was based on the 435.5-nm E2 transition of a single ¹⁷¹Yb ion.

All frequency synthesizers and counters in the ²²⁹Th clock system and the frequency comb were referenced to the 10-MHz signal of a commercial Rb clock (FS725, SRS).

To perform an absorption measurement at a specific frequency, we modulated (square wave) the offset lock frequency between the seed laser diode and the clock laser with 10 Hz between the target frequency and an off-resonance frequency. Each modulation cycle was also reflected in a synchronization signal connected to a time-resolved pulse counter (TimeTagger Ultra, Swabian Instruments). The modulation ensured that power fluctuations in the laser output did not affect the measurement. The detection set-up consists of a PMT, a radiofrequency amplifier and the pulse counter, which binned the arriving pulses from the PMT relative to the last synchronization edge. The absorption was then calculated by subtracting the on-resonance counts from the off-resonance counts and dividing by the off-resonance counts.

Clock operation

To operate the set-up as a clock, we first acquired a single absorption spectrum, as shown in Extended Data Fig. 2a. We observed that a Lorentzian line shape fits our absorption measurement data well. We then measured the expected error signal (Extended Data Fig. 2b). To measure an error signal, we modulated the offset lock frequency with 10 Hz between two frequencies f_low and f_high. The difference Δf = f_high − f_low was kept constant during the measurement, and only the centre was swept across the resonance. In our measurements, we used a frequency deviation Δf of \({f}_{{\rm{FWHM}}}/\sqrt{3}\), with f_FWHM describing the FWHM of the measured absorption peak (Extended Data Fig. 2a). We fitted this signal with the function

where A_L is the amplitude and γ is the half width at half maximum of the Lorentzian. For the clock operation, the laser frequency was again switched between two frequencies while we adjusted the centre position with the feedback. For both frequency positions, the counts c_low,i and c_high,i were recorded, where i is the measurement index for averaging the signal. The applied error signal E can be calculated as

with the number of cycles L and the inverse fit function \({y}_{{\rm{fit}}}^{-1}(c)\). Extended Data Fig. 2c illustrates an adjustment step inferred by the calculation of E. With the integration times set in the clock operation, the beat was shifted after every interrogation cycle by the calculated value E, as shown in Extended Data Fig. 1.

Variation of the fundamental constants

The variation of α(t) relates to the variation of the ratio of the ²²⁹Th and ytterbium clock frequencies through: \(({k}_{\mathrm{Th}}^{\alpha }-{k}_{\mathrm{Yb}}^{\alpha })\times {\partial }_{t}\,\log \,\alpha (t)\,=\) \({\partial }_{t}\,\log ({f}_{\mathrm{Th}}/{f}_{\mathrm{Yb}})\). Here \({k}_{{\rm{Th}}}^{\alpha }\) and \({k}_{{\rm{Yb}}}^{\alpha }\) are sensitivity factors that describe how much the respective transition frequency depends on α. The log-derivative directly corresponds to the measured infrared beat frequency f_b through \({\partial }_{t}\log ({f}_{\mathrm{Th}}/{f}_{\mathrm{Yb}})=8({\partial }_{t}\,{f}_{{\rm{b}}})/{f}_{0}\), where the factor of 8 originates from the three frequency-doubling steps⁶⁴.

The anomalously low energy of the ²²⁹Th nuclear transition arises from a coincidental near-cancellation of the mega-electronvolt-scale strong force and Coulomb contributions to the binding energies of the nuclear states involved¹⁵. We, therefore, expect that \({k}_{\mathrm{Th}}^{\alpha }\approx {k}_{\mathrm{Th}}^{{\rm{g}}}\approx {k}_{\mathrm{Th}}^{{m}_{q}}\), where \({k}_{{\rm{Th}}}^{g}\) and \({k}_{{\rm{Th}}}^{{m}_{q}}\) are the sensitivity constants that relate between the log-derivatives of the frequency ratio and Λ_QCD and m_q, respectively. They by far dominate the sensitivity of the Yb⁺ reference for which the corresponding sensitivities are approximately 1 (ref. ⁵⁹). We determined the amplitude of possible oscillations in the experimental data by computing the Lomb–Scargle power spectrum P_f of the beat signal translated into the VUV. It corresponds to an oscillation amplitude A_f through \({A}_{{\rm{f}}}=\sqrt{4{P}_{{\rm{f}}}/{N}_{\mathrm{tot}}}/{f}_{0}\), where N_tot is the number of data points⁶⁵.

Various theories suggest that dark matter could consist of yet undiscovered ultralight scalar bosons⁴¹. Such bosons would interact only very weakly with other particles and their behaviour can be approximately described by a free field: \(\phi =\frac{\sqrt{2{(\hbar c)}^{3}{\rho }_{\mathrm{DM}}}}{{m}_{\phi }{c}^{2}}\,\cos \left(\frac{{m}_{\phi }{c}^{2}}{\hbar }t+{\delta }\right)\). Here ħ is the reduced Planck constant, ρ_DM = 0.4 GeV cm⁻³ is the observed dark matter density in the Milky Way⁴¹, m_ϕ is the unknown mass of the boson and δ is some unknown phase. The coupling of such particles to matter can be expressed through the following Lagrangian density^42,56,66:

Here we included only the terms that are relevant for this study. ϕ is the scalar field, F_μν the electromagnetic field, \({G}_{\mu \nu }^{a}\) the gluon field and ψ_q the quark field. \(\kappa =\frac{\sqrt{4{\rm{\pi }}}}{{M}_{\mathrm{Pl}}{c}^{2}}\), where M_Pl is the (unreduced) Planck mass. μ₀ is the vacuum permeability and β₃ is the QCD beta function that describes the running of the coupling constant g₃. \({\gamma }_{{m}_{q}}{d}_{{\rm{g}}}\) describes the anomalous and m_q the bare contribution to the quark mass. The first term leads to a modification of the electromagnetic coupling strength α → α + ακd_eϕ (ref. ⁶⁶). The amplitude of this oscillation is given by \({{\mathcal{A}}}_{{\rm{f}}}=\frac{\alpha \kappa {d}_{{\rm{e}}}\sqrt{2{(\hbar c)}^{3}{\rho }_{\mathrm{DM}}}}{{m}_{\phi }{c}^{2}}\). Finally, using the Lomb–Scargle power spectrum P_f of the VUV equivalent of the measured beat signal, we arrived at the following expression for the scalar-photon coupling strength:

Here we included a factor of 3 to account for possible stochastic fluctuations in the dark matter density⁶⁷. The corresponding equations for Λ_QCD and m_q follow completely analogously and differ only by the respective sensitivity factor.

To evaluate the statistical significance of the amplitudes in the Lomb–Scargle power spectrum, we performed Monte Carlo simulations of the clock feedback loop. For this, we took N = T + 1 random values from the Poissonian distribution to simulate the actual experimental cycle. where T is the integration time in seconds. This distribution follows from the SNR of the Lorentzian fit of the measured data at positions \({f}_{\mathrm{est}}\pm \mathrm{FWHM}/2\sqrt{3}\). We then added white noise to these values with an amplitude that matched the noise we observed on the beat frequency. We then averaged these N values and used the error function to calculate a new estimate for f_est, with which we repeated the procedure. We assumed a decrease in SNR for each subsequent random value to model the linear decrease in laser power that we experienced during the data taking. Extended Data Fig. 1a shows that the simulated beat frequency has the same characteristics as the measured one. We further saw that the power spectrum and Allan deviation of the simulated data fitted well to the experiment values.

For the experimental measurement acquired between 2 April 2026 (08:30:15 UTC) and 3 April 2026 (07:04:15 UTC), we extracted one beat-frequency value in each cycle to construct a sample of independent frequency ratios. The Lomb–Scargle periodograms for both the experimental data and simulations were computed using the Astropy Python package⁶⁸ with the setting ‘normalization = psd’.

To assess the statistical significance of peaks in the experimental periodogram and account for the look-elsewhere effect, we estimated a 5% detection threshold. When finding a peak above this threshold, the probability of it being a false detection is less than p₀ = 5%. Adopting the method of refs. ^52,65, we performed 1,000 Monte Carlo simulations of the clock loop and computed the corresponding Lomb–Scargle periodograms. Then, we fitted the histogram of the cumulative power at each frequency with an exponential cumulative distribution function of the form: \(1-\exp (-a({P}_{{\rm{f}}}-{P}_{0}))\). The fitting parameters, a and P₀, allowed us to estimate the detection threshold P_f,th and the corresponding amplitude according to \({P}_{{\rm{f}},\mathrm{th}}={P}_{0}-\frac{1}{a}{\rm{ln}}[1-{(1-{p}_{0})}^{1/{n}_{\mathrm{ind}}}]\), where \({n}_{\mathrm{ind}}\approx {t}_{\mathrm{tot}}/T\approx \) 3,800 is the number of independent frequencies. Here the estimated threshold fits well to the analytical solution for an exponential distribution, which is expected for white noise. We found no amplitudes exceeding the detection threshold, indicating the absence of DM oscillations in our measurements.

The local 95% confidence level was determined using 1,000 Monte Carlo simulations. For each simulation, a random offset was added to the experimental beat frequency at each time step. The offset followed a Gaussian white noise distribution, with a standard deviation σ = 1.34 kHz, derived from the experimental beat frequency. The local 95% confidence level was then defined as the 95th percentile of the amplitude distribution at each frequency, allowing us to exclude, with 95% certainty, the presence of any oscillations with amplitudes exceeding this limit.

The data supporting the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.22657608⁶⁹.

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Author notes

1. These authors contributed equally: L. Toscani De Col, T. Riebner

Authors and Affiliations

1. Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Vienna, Austria

   L. Toscani De Col, T. Riebner, I. Morawetz, F. Schneider, N. Sempelmann, J. Schlachet-Lépinay, F. Schaden, M. Bartokos, G. A. Kazakov, K. Beeks, B. Gerstenecker, M. Pimon, S. Lahs & T. Schumm

2. Bundesamt für Eich- und Vermessungswesen (BEV), Vienna, Austria

   T. Riebner, A. Hellerschmied, T. Lercher, J. Premper, A. Niessner & M. Matus

3. Wolfgang Pauli Institut, Vienna, Austria

   G. A. Kazakov

4. Institut für Erdmessung, Leibniz Universität Hannover, Hanover, Germany

   H. Denker

5. Institute of Scientific Instruments, CAS v. v. i., Brno, Czech Republic

   M. Čížek & O. Číp

6. Physikalisch-Technische Bundesanstalt (PTB), Braunschweig, Germany

   V. Lal, G. Zitzer, J. Tiedau, M. V. Okhapkin & E. Peik

7. Max-Born-Institute for Nonlinear Optics and Ultrafast Spectroscopy, Berlin, Germany

   V. Petrov

Contributions

T.S. and E.P. led the conceptual development of the experiment. L.T.D.C., T.R., I.M., F. Schneider, N.S., F. Schaden, M.B., S.L., J.S.-L. and K.B. performed the measurements and analysed the data. G.A.K., J.S.-L., M.P. and S.L. performed the simulations and numerical modelling. T.R., B.G., A.H., T.L., J.P., A.N., M.M. and H.D. established and maintained the optical reference at BEV. T.R., B.G., M.Č. and O.Č. established and maintained the fibre links used in this study. V.L., M.V.O., J.T., G.Z. and V.P. developed the continuous-wave laser system operating at 148 nm and performed the installation and optical alignment. J.T., M.V.O. and V.L. assisted with the data acquisition and laser stabilization. All authors assisted in interpreting the data and writing the Article.

Corresponding authors

Correspondence to E. Peik or T. Schumm.

Competing interests

The authors declare no competing interests.

Peer review information

Nature thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.

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