Data acquisition
Here we give further relevant details on the experimental scheme and apparatus (both described in detail in ref. 23). The run time of the cryogenic atomic beam is limited to freezing cycles of approximately 2 h by the accumulation of frozen (molecular) hydrogen inside the nozzle, which is removed by heating the nozzle to room temperature. The vacuum (2 × 10−7 mbar, dominated by molecular hydrogen) inside the 2S–6P spectroscopy region is maintained by differential pumping with a cryopump to minimize pressure shifts50,51,52, with the temperature of the apparatus allowed to equilibrate on each measurement day before collecting data.
The power of the linearly polarized, 243-nm 1S–2S preparation laser53,54 is resonantly enhanced in an in-vacuum, standing-wave cavity to P 1S–2S ≈ 1 W per direction (297 μm 1/e2 intensity waist radius). The observed 1S–2S transition linewidth is approximately 3 kHz (FWHM; atomic detuning, as opposed to laser detuning), limited by single-photon ionization of the 2S level55. Therefore, the \({2{\rm{S}}}_{1/2}^{F=1}\) levels are only populated by Doppler-sensitive two-photon excitation, leading to a population of approximately 7 × 10−7 in each sublevel relative to the population in the \({2{\rm{S}}}_{1/2}^{F=0}\) level. The detuning of the preparation laser is set several times per freezing cycle by observing the 1S–2S transition. An equal-slit-width optical chopper running at 160 Hz periodically blocks the preparation laser (which sets delay time τ = 0 μs) to enable the velocity-resolved detection. The channel electron multipliers are switched off with a fast high-voltage switch while the preparation laser is unblocked to prevent saturation from scattered 243-nm light.
By using a linearly polarized 2S–6P spectroscopy laser53,54 and the \({2{\rm{S}}}_{1/2}^{F=0}\) level as the initial level, only transitions to the \({6{\rm{P}}}_{1/2}^{F=1}\) hyperfine level (2S–6P 1/2 transition) and the \({6{\rm{P}}}_{3/2}^{F=1}\) hyperfine level (2S–6P 3/2 transition) are dipole-allowed, whereas the excitation of the \({6{\rm{P}}}_{1/2}^{F=0}\) and \({6{\rm{P}}}_{3/2}^{F=2}\) levels is prevented by angular momentum conservation (Fig. 2a). The line strengths (∝μ2, in which μ is the dipole moment) of the 2S–6P 1/2 and 2S–6P 3/2 transitions have a 1:2 ratio. We use spectroscopy laser powers P 2S–6P with a ratio of 2:1 for the two transitions to keep their Rabi frequencies \({\varOmega }_{0}\propto \mu \sqrt{{P}_{2{\rm{S}}-6{\rm{P}}}}\) identical (the peak Rabi frequency is (2π × 126) krad s−1 at our highest spectroscopy laser powers of 30 and 15 μW, respectively). Most 6P decays (branching ratio Γ e–1S /Γ = 88.2%) are Lyman decays to the 1S manifold, with the most energetic, direct Lyman-ε decay (Γ det /Γ = 80.5%) dominating, whereas the remaining γ e–2S /Γ = 11.8% are Balmer decays to the 2S manifold, of which in turn a fraction γ ei /Γ leads back to the initial \({2{\rm{S}}}_{1/2}^{F=0}\) level (see Section 1.2 in the Supplementary Methods). The metastable 2S levels are treated as stable here, as their natural lifetime (122 ms) is much longer than the time the atoms spend in the atomic beam (see Section 2.6 in the Supplementary Methods for the 2S decay contribution to the signal background).
The channel electron multipliers count the fluorescence photons from the 6P decays, either by detecting the photoelectrons emitted by the photons from the detector cylinder walls or, to a lesser extent, directly detecting the photons. Because the photoelectron yield strongly increases with photon energy (for both colloidal graphite and oxidized aluminium, the surface materials of the Faraday cage and the detector cylinder, respectively), fluorescence from Lyman-ε decay (13.2 eV photon energy) constitutes approximately 97% of the signal detected by the channel electron multipliers. The counts are binned into the 16 velocity groups by their delay time τ, with the bins chosen to cover a wide range of mean speeds \(\overline{v}\) while exhibiting a sufficient signal-to-noise ratio (bin width 50–550 μs; Extended Data Table 2). For each line scan, we accumulate counts over 160 chopper cycles at each spectroscopy laser detuning. Intermittently, excess scatter and spiking was observed for the bottom detector and its signal was subsequently discarded (≈11% of line scans; Extended Data Table 3). We attribute this to the bottom detector being cooled down to close to, and possibly below, its lower operating temperature limit because of its vicinity to the cryopump.
At least once per measurement day, the nozzle and the collimating aperture are centred on the preparation laser. At the start of each freezing cycle, the atomic beam offset angle α 0 , which is controlled by a linear motor equipped with a position sensor, is aligned to zero with a 1 mrad alignment uncertainty. This is achieved by blocking the returning beam of the spectroscopy laser (using an in-vacuum shutter in front of the high-reflectivity mirror of the AFR25,26), determining the (now unsuppressed) Doppler slope κ for several angles and moving to the angle at which κ is zero, which is set as α 0 = 0 mrad. To record line scans at a non-zero angle ±α 0 , we first move to either +α 0 or −α 0 (chosen randomly) and then to the opposite sign, recording typically 5–10 scans at each angle, and repeating this procedure several times per freezing cycle. The fibre–collimator distance of the AFR is optimized25,26 at least once per freezing cycle.
We use fixed sets of 30 symmetric (15 unique) detunings Δ of the spectroscopy laser frequency to sample the 2S–6P fluorescence line shape, with different sets used for α 0 = 0, ±7.5 and ±12 mrad to account for the different line shapes (see Section 5.1.3 and Table 5.2 of ref. 23), all of which have ±50 MHz as the largest detuning. The detunings were chosen to minimize the statistical uncertainty of the Doppler-free transition frequency ν e , whereas the number of detunings and the acquisition time (1 s) at each detuning were chosen to balance between sufficient line sampling, signal-to-noise ratio and number of line scans per freezing cycle. For each line scan, the order of the detunings is randomized to minimize the influence of drifts in the signal. At the beginning of each freezing cycle, the centre laser frequency (to which the detunings are added) was chosen randomly from a normal distribution. This distribution was centred on the 2S–6P transition frequency expected from the muonic measurement of the proton radius9, with a standard deviation of 12 kHz to cover the transition frequency expected from the CODATA 2014 value of the proton radius16. The laser frequencies are referenced to the caesium frequency standard using an optical frequency comb56,57 and a global navigation satellite system (GNSS)-referenced hydrogen maser (see Section 2.7 in the Supplementary Methods).
We switched between examining the 2S–6P 1/2 and 2S–6P 3/2 transitions several times during the measurement, except during measurement run C, for which only the 2S–6P 1/2 transition was examined for several values of α 0 (Extended Data Table 3).
Voigt and Voigt doublet line shapes
A Voigt line shape23,58 is the convolution of a Lorentzian line shape (with FWHM linewidth Γ L ) and a Gaussian line shape (with FWHM linewidth Γ G ). It has a combined FWHM linewidth \({\varGamma }_{{\rm{F}}}\approx 0.5346{\varGamma }_{{\rm{L}}}\,+\) \(\sqrt{0.2166{{\varGamma }_{{\rm{L}}}}^{2}+{{\varGamma }_{{\rm{G}}}}^{2}}\) (ref. 59) and amplitude A and is centred on the resonance frequency ν 0 . A constant offset y 0 is added to account for the experimentally observed signal background, resulting in five free parameters.
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