Fluorescence-signal analysis

The fluorescence spectra of S1 and TS1 were obtained by correcting for two effects in the measured fluorescence signals. Residual excited-state population from preceding measurements was removed using a procedure adapted from refs. ^14,17, whereas variations in the excitation were corrected using the monitored transmitted VUV power during each excitation interval.

For each spectrum, the fluorescence records from all frequency points i are fitted jointly to

where t is measured from the beginning of the acquisition window, A_i is the fluorescence amplitude at the beginning of that window and B is a common constant background count rate for the corresponding spectrum. The lifetime is fixed to the independently measured value τ_e = 631 s.

Because fluorescence acquisition of each frequency point begins before the residual excited-state population from previous acquisitions has decayed fully, the measured fluorescence records contain contributions from earlier excitations. The net fluorescence amplitude at each frequency point is therefore obtained by subtracting the preceding-point amplitude extrapolated to the start of the current fluorescence acquisition window:

where \({t}_{i,\det }\) and \({t}_{i-1,\det }\) are the start times of the fluorescence acquisition windows of the current and previous frequency points, respectively.

To correct for variations in the excitation arising from VUV power fluctuations, the transmitted VUV power is monitored during each excitation interval. In the weak-excitation regime, the excited-state population generated during an infinitesimal excitation interval of dt is proportional to the instantaneous transmitted VUV power P(t), whereas population excited at an earlier time is reduced by spontaneous decay before the fluorescence acquisition window^48,49. The effective excitation dose for the frequency point i is therefore

where t_i,on and t_i,off denote the start and end of the excitation interval of frequency point i.

The net fluorescence amplitude is then normalized to the effective excitation dose to obtain a corrected fluorescence response:

This normalization makes the corrected fluorescence response insensitive to the frequency point acquisition order and variations in the VUV excitation.

The corrected fluorescence responses F_i constitute the fluorescence spectrum for each spectral component s. The spectrum is fitted to a single-Lorentzian profile

where F_0,s is the constant offset, H_s is the Lorentzian peak height, ν_0,s is the line-centre frequency and Δν_s is the FWHM. Applying this Lorentzian fit to TS1 yields linewidths of Δν_b = 27(1) kHz, Δν_c = 39(2) kHz and Δν_QU = 0.898(5) MHz.

For the narrow QR transitions, the spectra of lines a–d in both S1 and TS1 show evidence of a broad pedestal in addition to the narrow central component. A representative example, line b in TS1, is shown in Extended Data Fig. 1. The pedestal could reflect an inhomogeneously broadened subensemble of QR sites, arising from variations in the local fluoride environment, lattice strain or nearby defects, although its microscopic origin remains to be established.

Absorption signal analysis

To extract transition linewidths and on-resonance fractional absorptions, we fit the demodulated absorption signals to the first-harmonic response of a sinusoidally frequency-modulated Lorentzian line shape.

For each spectral component s ∈ {b, c, QU}, the measured normalized demodulated absorption signal is

where \({V}_{1f,s}^{{\rm{RMS}}}(\bar{\nu })\) is the in-phase first-harmonic voltage recorded by the lock-in amplifier in the root-mean-square units and \({V}_{{\rm{DC}}}(\bar{\nu })\) is the direct current signal of the simultaneously monitored transmitted VUV light.

The normalized demodulated absorption signal is modelled as

where A_s ≡ (P_off − P_on,s)/P_off is the on-resonance fractional absorption of the VUV light, with P_off the off-resonant transmitted power and P_on,s the transmitted power at the centre of spectral component s without frequency modulation. E_s is a fitted constant offset. The function \({{\mathcal{D}}}_{s}(\bar{\nu })\) is the first-harmonic response function for a unit-height Lorentzian absorption profile \({{\mathcal{L}}}_{s}(\nu )\),

where

Here \(\bar{\nu }\) is the mean laser frequency, δ_s is the modulation amplitude, θ is the modulation phase, ν_0,s is the line-centre frequency and Δν_s is the Lorentzian FWHM. Fits of (8) to the measured normalized demodulated signals yield the on-resonance fractional absorptions A_s and transition linewidths Δν_s.

The 1-s absorption SNR in the main text is defined as the peak amplitude of the normalized demodulated absorption signal divided by the s.d. of the signal baseline projected to a 1-s measurement time:

In the photon-shot-noise limit, it reduces to

where \(\mathop{{N}_{e}}\limits^{.}\) is the detected photoelectron rate and \({k}_{s}=\frac{1}{\sqrt{2}}{\max }_{\bar{\nu }}| {{\mathcal{D}}}_{s}(\bar{\nu })| \) is the proportionality factor for the spectral component s. The dependence of the \({{\rm{SNR}}}_{1{\rm{s,SN}}}\) on both A_s and \({\mathop{N}\limits^{.}}_{e}\) motivates the successive improvements to the detector and crystal used in absorption spectroscopy and clock operation.

Resonant steady-state excitation fraction

To quantify the resonant nuclear excitation throughout the crystal, we calculate the depth-dependent steady-state excitation fraction p_s(z) for the laser tuned to the centre of each spectral component, s ∈ {b, c, QU}, following ref. ⁴⁹. In the weak-excitation regime,

where \({\Gamma }_{{\rm{exc}}}^{(s)}(z)\) is the excitation rate and τ_e is the measured nuclear lifetime.

The Rabi frequency for component s is

where I(z) is the local VUV intensity, ω₀ is the angular nuclear-transition frequency, Γ_γ ≃ 2π × 63 μHz is the radiative decay rate calculated from the measured excited-state lifetime and χ_s is the effective line-strength factor. For s = b, c and QU, the effective line-strength factors are, respectively,

These values are obtained by summing the squared Clebsch–Gordan coefficients over the unresolved magnetic-sublevel transitions and averaging the resulting coupling strengths over the orientations of the effective quantization axes relative to the laser polarization⁴⁸.

Because resonant nuclear absorption is negligible compared with the off-resonant propagation loss, the local VUV intensity is approximated as

where P_in is the incident VUV power and S_beam is the effective illuminated area. The Fresnel reflectance and effective off-resonant attenuation coefficient are

where n is the refractive index of CaF₂, L is the crystal length and T_xtal is the measured off-resonant transmission through the crystal. For S1 and TS1, T_xtal is approximately 1% over 9.8 mm and 7% over 4.94 mm, corresponding to equivalent 1-mm internal transmittances of 63% and 60%, respectively.

For a spectral component with measured FWHM Δν_s, the resonant excitation rate in the weak-excitation regime is⁴⁹:

giving

Using an incident VUV power of approximately 5 μW with a Gaussian-beam 1/e²  intensity diameter of approximately 1 mm, together with the measured linewidths and off-resonant transmissions of S1 and TS1, we evaluate p_s(z) for the b, c and QU components throughout each crystal. The resulting depth-dependent excitation fractions are shown in Fig. 2i,j.

For the present experimental parameters, the entrance surface Rabi frequencies are Ω_s < 2π × 1 Hz, more than two orders of magnitude smaller than both the crystal-induced optical-decoherence rates and the magnetic-sublevel-mixing rates, which are of order 2π × 10² Hz^4,48. This hierarchy justifies the excitation rate treatment used above, with the measured ensemble linewidth Δν_s providing the effective linewidth of each spectral component.

Frequency discriminator optimization

To minimize the measurement-noise contribution to the fractional frequency instability of the nuclear clock, we optimize the modulation amplitude to maximize the slope of the frequency discriminator at the centre of line b. Following the standard stability analysis for atomic clocks², the fractional frequency instability arising from statistically independent cycle-to-cycle measurement noise is

where σ_S is the fluctuation of the normalized demodulated absorption signal at the line centre, ν₀ is the nuclear-transition frequency, T_c is the clock-cycle duration, τ is the averaging time and

is the magnitude of the discriminator slope at the line centre. Provided that σ_S is approximately independent of the modulation amplitude over the relevant range, maximizing g_b minimizes the corresponding contribution to the clock instability.

For a Lorentzian line with FWHM Δν_b and peak modulation amplitude δ_b, the discriminator slope is

Maximizing g_b with respect to δ_b gives

For the measured b-line linewidth Δν_b = 27(1)kHz, the modulation amplitude that maximizes the line-centre discriminator slope is δ_b,opt ≃ 9.5 kHz. A slightly larger modulation amplitude provides a wider monotonic frequency discrimination range and hence greater tolerance to frequency excursions during feedback operation. We therefore use a near-optimal δ_b = 12 kHz, which increases the width of the monotonic frequency discrimination range by approximately 13% while retaining 96.4% of the maximum line-centre slope. At this modulation amplitude, the discriminator slope is

Clock instability at photon-shot-noise limit

We estimate the photon-shot-noise-limited fractional frequency instability of the optimized TS1 + phototube configuration from the detected VUV power. The detected photoelectron rate is \({\mathop{N}\limits^{.}}_{e}={\eta }_{\det }{P}_{\det }/(h{\nu }_{0})\), where \({P}_{\det }\) is the detected VUV power and \({\eta }_{\det }\) is the phototube quantum efficiency. For a measurement time T_m, the photon-shot-noise-limited s.d. of the normalized demodulated absorption signal is⁵⁰

Substituting this expression into (20) and using the near-optimized discriminator slope g_b = 1.05A_b/Δν_b gives

For the optimized TS1 + phototube configuration, we use \({P}_{\det }=350\,{\rm{nW}}\), \({\eta }_{\det }\simeq 0.10\), A_b = 2.9 × 10⁻⁴, Δν_b = 27 kHz, T_m = 10.0s and T_c = 11.8s, yielding

The measured instability of \(5\times 1{0}^{-13}/\sqrt{\tau /{\rm{s}}}\) is within a factor of two of this limit, indicating near-photon-shot-noise-limited operation. The remaining difference can be accounted for partly by the gradual decrease in detected VUV power during the measurement.

Absolute-frequency determination

For the absolute-frequency measurements of line b in S1 and TS1 at a crystal temperature of 301.4(1) K, the VUV laser is stabilized to the centre of the corresponding absorption resonance. The locked VUV frequency is determined through the comb-referenced optical chain used for VUV generation, including the comb mode numbers, optical beat notes, AOM shifts and fixed RF offsets. All RF synthesizers and frequency counters are referenced to a common 10-MHz hydrogen maser signal. The reconstructed frequency is corrected for the fractional frequency offset of the hydrogen maser relative to International Atomic Time, using the monthly International Atomic Time clock-rate data for the corresponding measurement periods.

The resulting b-line centre frequencies are

The first set of parentheses gives the 1σ statistical uncertainty, and the second gives the systematic uncertainty. These systematic uncertainties arise from the 0.1 K absolute crystal temperature uncertainty, dominated by the sensor calibration, propagated through the local temperature coefficient of line b reported in ref. ¹⁷.

Inter-laboratory frequency comparison

For comparison with the JILA measurements, we fit the publicly available C10 and C13 source data⁴³ using the quadratic temperature-dependence model reported in ref. ¹⁷. Evaluating the fit at our crystal temperature of 301.4(1) K yields

where the 1σ uncertainty combines in quadrature the 1.21 kHz fit uncertainty and the 0.185 kHz contribution from the uncertainty of our crystal temperature.

The offsets of the S1 and TS1 measurements from the JILA value are

The agreement within the combined standard uncertainties verifies the reproducibility of the b-line frequency across independently fabricated crystals and independent experimental platforms.