The transition of interest here, between the ground state and the first excited state of antihydrogen, has an energy of about 10.2 eV. The frequency of this transition in hydrogen has been measured8 to a few parts in 1015. We previously demonstrated7 the existence of the transition in antihydrogen, localizing the frequency to a few parts in 1010. Here we characterize the spectral line shape of the transition to the limits of precision of our current apparatus.

Matter and antimatter annihilate each other, so antihydrogen must be synthesized and then held in ultrahigh vacuum, in isolation from matter, to be studied. The ALPHA-2 apparatus at CERN (Fig. 1) combines antiprotons from the antiproton decelerator9 with positrons from a positron accumulator10, 11 to produce and trap12 atoms of antihydrogen. Antihydrogen can be trapped in ALPHA-2’s magnetic multipole trap if it is produced with a kinetic energy of less than 0.54 K in temperature units. The techniques that we use to produce antihydrogen that is cold enough to trap are described elsewhere12,13,14. In round numbers, a typical trapping trial in ALPHA-2 involves mixing 90,000 antiprotons with 3,000,000 positrons to produce 50,000 antihydrogen atoms, about 20 of which will be trapped. The anti-atoms are confined by the interaction of their magnetic moments with the inhomogeneous magnetic field. The cylindrical trapping volume for antihydrogen has a diameter of 44.35 mm and a length of 280 mm.

Fig. 1: The ALPHA-2 central apparatus and magnetic field profile. a, b, Penning traps, comprising stacks of cylindrical electrodes immersed in a uniform axial magnetic field generated by an external solenoid (not shown), are used to confine and manipulate antiprotons (\(\bar{p}\)) and positrons (e+) to produce antihydrogen. Cold (less that 0.5 K) anti-atoms can be trapped radially by the octupole field and axially by the magnetic well that is formed by the five mirror coils and plotted in b. The 243-nm laser light is injected from the antiproton side (left in a) and is aligned and position-stabilized on the fixed optical cavity axis. The laser beam crosses the trap axis at an angle of 2.3°. The piezoelectric actuator behind the output coupler is used to modulate the cavity length to lock the cavity to the laser frequency. The axial scale in a and b is the same; the radial extent of the annihilation detector is larger than illustrated. The vacuum window and photo-diode are further to the right (by about 1 m) than illustrated. The brown-shaded electrodes are used to apply blocking potentials during the experimental trials to ensure that antiprotons that result from ionization are confined to annihilate in the active volume of the detector7. Full size image

The key to anti-atomic spectroscopy, as developed so far7, 15, 16, is to illuminate a sample of trapped antihydrogen atoms with electromagnetic radiation (microwaves or laser photons) that causes atoms to be lost from the trap if the radiation is on resonance with the transition of interest. ALPHA-2’s silicon vertex detector17 (Fig. 1) affords us single-atom detection capability for the annihilation events associated with lost antihydrogen atoms or antiprotons that encounter the walls of the apparatus. The silicon vertex detector tracks the charged pions from the antiproton annihilation, and various reconstruction algorithms are used to determine the location (vertex) of each annihilation and to distinguish antiprotons from cosmic-ray background using multivariate analysis18 (Methods).

To excite the 1S–2S transition, we use a cryogenic, in vacuo enhancement cavity (Fig. 1) for continuous-wave light from a 243-nm laser system (Methods) to boost the intensity in the trapping volume. Long interaction times are possible, because the anti-atoms have a storage lifetime of at least 60 h in the trap. Two counter-propagating photons can resonantly excite the ground-state atoms to the 2S state. Absorption of a third photon ionizes the atom, leading to loss of the antiproton from the trap. Atoms that decay from the 2S to the 1S state via coupling to the 2P state may also be lost, owing to a positron spin-flip19.

Referring to the energy-level diagram of hydrogen in Fig. 2, there are two trappable, hyperfine substates of the 1S ground state (labelled ‘c’ and ‘d’). In practice, we find that these states are, on average, equally populated in our trap: N c = N d = N i /2, where N i is the number of ground-state atoms that are initially trapped in an experimental trial. The 2S state has corresponding hyperfine levels, and we refer to the transitions between the two manifolds as d–d (Fig. 2) and c–c (not pictured).

Fig. 2: Hydrogenic energy levels. Calculated energies (E; for hydrogen) of the hyperfine sublevels of the 1 S (bottom) and 2 S (top) states are plotted against magnetic field strength. The centroid energy difference E 1S–2S = 2.4661 × 1015 Hz has been suppressed on the vertical axis. The vertical black arrow indicates the two-photon laser transition probed here (frequency f d–d ); the red arrow illustrates the microwave transition used to remove the 1S c state atoms (frequency f c–b ). Full size image

For each experimental trial, we first accumulate antihydrogen atoms from three mixing cycles or ‘stacks’13 and then remove any leftover charged particles using pulsed electric fields. After a wait of about 10 s to allow any excited atoms to decay to the ground state, the trapped population is exposed to laser radiation at a fixed frequency for 300 s. The frequencies used here were chosen to probe only the d–d transition (Fig. 2). Following the laser exposure, we use microwave radiation to remove the 1S c state atoms by driving a resonant spin-flip15, 16. The microwave frequency is scanned over 9 MHz in 32 s; these parameters and the injected power level (160 mW at the vacuum feed-through) are chosen to eject anti-atoms quickly while minimizing the perturbation of the vacuum and cryogenic environment. The silicon vertex detector is used to detect annihilations of antihydrogen atoms that are lost during the laser and microwave exposures. Finally, the atom-trap magnets are ramped down in 1.5 s, so that any surviving anti-atoms would be released and their annihilations detected. If the microwave removal of 1S c -state atoms is 100% effective, then the surviving particles would be only 1S d -state atoms that were not removed by laser action.

We collected data for nine different laser frequencies in four sets. Each set involved four distinct frequencies and 21 (or 23, see below) trials at each of these frequencies. In each set, two of the frequencies were always the calculated hydrogen on-resonance frequency at zero laser power (zero detuning) and a far-off-resonance frequency (−200 kHz detuning at 243 nm), as used previously7. The other two frequencies in each set were chosen to address various detunings in the neighbourhood of the d–d resonance. The data are summarized in Table 1. The repetition of the points at −200 kHz and zero detuning was intended to address variations in laser power and trapping number between sets. The repetition at + 25 kHz was a check of reproducibility. During the accumulation of data for each set, the four frequencies were interleaved in a varying order and the operators were blinded as to the identity of each frequency setting. The power of the enhancement cavity (about 1 W) was monitored by measuring the transmitted power outside of the vacuum chamber (Fig. 1). Each set was preceded by a thermal cycle of the apparatus to regenerate the cryo-pumping surface.

Table 1 Antihydrogen atom counts Full size table

The background-corrected numbers in Table 1 are calculated from raw detector events using the measured, overall efficiencies of the silicon vertex detector. These efficiencies depend on the particular multivariate analysis algorithm that was used to distinguish antiproton annihilations from cosmic rays (Methods) in the relevant time window. The efficiencies and background rates are listed in Table 2.

Table 2 Annihilation detector efficiencies and background rates Full size table

The number of initially trapped atoms N i for a trial is unknown a priori, but was typically about 60 at the beginning of a measurement set. In Table 1, the total number of atoms for each group of trials is assumed to be the sum L + M + S of the numbers of atoms lost during laser (L) or microwave (M) exposure and the number of surviving atoms (S) (see Table 1). The trapping rate declined slowly but reproducibly during each set (Extended Data Fig. 1). The third set has 23 trials at each frequency because of a hardware failure in an early block of four trials; extra trials were added to compensate for the excluded data.

To examine the general features of the measurement results, we plot (Fig. 3a) the four datasets on one graph by using a simple scaling. The points at zero (on-resonance) and −200-kHz detuning (at which no signal is expected7), repeated for each set, are used for the scaling. For the laser exposure (‘appearance’) data, we define a scaled response at detuning D within each set: r l (D) = L(D)/L(0). Similarly, for the surviving population (‘disappearance’ data), we use r s (D) = [S(−200 kHz) − S(D)]/[S(−200 kHz) − S(0)]. The uncertainties shown are due to Poissonian counting errors only. For comparison, we also plot the results of a simulation19 based on the expected behaviour of hydrogen in our trap for a cavity power of 1 W, scaled to the zero-detuning data point. We see that the peak position and the width of the scaled spectral line are consistent with the calculation for hydrogen and that the experiment generally reproduces the predicted asymmetric line shape. There is also good agreement between the appearance and disappearance data (Fig. 3a).

Fig. 3: Spectral line of antihydrogen. a, The complete dataset, scaled as described in the text. The simulated curve (not a fit, drawn for qualitative comparison only) is for a stored cavity power of 1 W and is scaled to the data at zero detuning. ‘Appearance’ refers to annihilations that are detected during laser irradiation; ‘disappearance’ refers to atoms that are apparently missing from the surviving sample. The error bars are 1-s.d. counting uncertainties. b, Three simulated line shapes (for hydrogen) are depicted for different cavity powers to illustrate the effect of power on the size and the frequency at the peak. The width of the simulated line (FWHM) as a function of laser power is plotted in the inset. Full size image

The simulation involves propagating the trapped atoms in an accurate model of the magnetic trap. When an atom crosses the laser beam, which has a waist of 200 μm at the cavity centre, we calculate the two-photon excitation probability, taking into account transit-time broadening, the a.c. Stark shift and the residual Zeeman effect. The simulation determines whether excited atoms are lost owing to ionization or to a spin-flip event. The variable input parameters for the simulation are the cavity power and the laser frequency. The modelled response is asymmetric in frequency owing to the residual Zeeman effect19. The width of the line, for our experimental parameters, is dominated by transit-time broadening, which contributes about 50 kHz full-width at half-maximum (FWHM) at 243 nm. For 1 W of cavity power, the a.c. Stark shift is about 2.5 kHz to higher frequency and the ionization contributes about 2 kHz to the natural line width.

To make a more quantitative comparison of the experimental results with the expectations for hydrogen, it is necessary to scrutinize differences between the four datasets. The overall response should be linear in the number of atoms addressed, so it is possible to normalize for this. However, the line width depends on the stored power in the cavity, as does the frequency of the peak (Fig. 3b). The cavity power is difficult to measure in our geometry because the amount of transmitted light depends sensitively on the small transmission from the output coupler (about 0.05%) and on absorption in the optical elements through which the transmitted light exits (Fig. 1). We observe that the transmitted power can degrade, owing to accumulated ultraviolet damage to the window and mirror substrate, whereas the finesse of the cavity does not change.

A modelling approach that self-consistently accounts for fluctuations in experimental parameters is a simultaneous fit in which we allow the four sets to have distinct powers (P 1–4 ), but the same frequency shift with respect to the hydrogen calculation (Methods). We require that the average powers for the appearance and disappearance data within a set are the same. We find the parameters that best reproduce the data to be: P 1 = 1135(50) mW, P 2 = 904(30) mW, P 3 = 1123(43) mW, P 4 = 957(31) mW and δf = −0.44 ± 1.9 kHz, where δf is the difference (at 243 nm) between the resonant frequency inferred from the fit and the resonant frequency of hydrogen expected for our system, both at zero power. The uncertainties represent the 68% confidence interval of a least-squares fit and do not take into account systematic uncertainties. The fit uses the five variables identified above, and the individual data points at each frequency are weighted by their Poissonian counting errors. We include an uncertainty of 3.8 kHz (Table 3) in the final resonance frequency to represent statistical and curve-fitting uncertainties.

Table 3 Summary of uncertainties Full size table

Considering systematic effects, the microwave removal procedure for the 1S c -state atoms provides a reproducibility check on the strength of the magnetic field at the centre of the trap. At the beginning of each data-taking shift, the magnetic field of the external solenoid magnet was reset to a standard value using an electron cyclotron resonance technique16. For the complete dataset, we find that the variations in the magnetic field at the minimum field of about 1 T are about 3.2 × 10−5 T (1 s.d.). This corresponds to a resonance frequency shift19 of only about 15 Hz at 243 nm for the d–d transition. (At 1 T, the c–c transition is about 20 times more sensitive to magnetic field shifts, which is why the d–d transition is more attractive here.) The laser frequency was tuned with respect to the minimum of the magnetic well, such that the resonance condition should be met in the centre of the trap for zero detuning in the limit of zero laser power. The accuracy of the magnetic-field determination corresponds to an uncertainty of 300 Hz in the 243-nm laser frequency.

Including all of the statistical and systematic uncertainties that we have identified (Table 3, for 121 nm), our fit of the experimental data to the hydrogen model yields

$${f}_{{\rm{d}}-{\rm{d}}}=\mathrm{2,466,061,103,079.4}(5.4)\,{\rm{k}}{\rm{H}}{\rm{z}}$$

The value (Methods) for hydrogen calculated at the minimum field in our system (1.03285(63) T) is

$${f}_{{\rm{d}}-{\rm{d}}}=\mathrm{2,466,061,103,080.3}(0.6)\,{\rm{k}}{\rm{H}}{\rm{z}}$$

where the uncertainty is determined by the experimental error in measuring the field.

Owing to the motion of the antihydrogen atoms in the inhomogeneous trapping field, this comparison is necessarily model-dependent. We therefore conclude that the measured resonance frequency for this transition in antihydrogen is consistent with the expected hydrogen frequency to a precision of about 2 × 10−12. Although the precision of our measurement is still a few orders of magnitude short of the state of the art with a cold hydrogen beam8, the modern frequency reference permits the accuracy of our experiment to exceed that achieved with trapped hydrogen20 as recently as the mid-1990s. We used a total of about 15,000 antihydrogen atoms to obtain this result, compared to 1012 trapped atoms in the analogous matter experiment. Our dataset was accumulated over a period of ten weeks, illustrating that the antihydrogen trapping procedure is robust and that systematic effects are manageable. ALPHA’s emergent antihydrogen production, storage and detection techniques, together with advances in ultraviolet laser technology and frequency metrology, pioneered by Hänsch and colleagues, enable precision anti-atom spectroscopy.

Precision experiments at the antiproton decelerator have recently constrained the properties of the antiproton through studies in Penning traps21, 22 or with antiprotonic helium23. For example, the antiproton charge-to-mass ratio is known to agree with that of the proton to 69 parts per trillion21, equivalent to an energy sensitivity of 9 × 10−27 GeV. The ratio of the antiproton mass to the electron mass has been shown to agree with its proton counterpart23 to 8 × 10−10, and antihydrogen has been shown to be neutral24 to 0.7 parts per billion. Our measurement of antihydrogen probes different and complementary physics at a precision of a few parts per trillion, or an energy level of 2 × 10−20 GeV. This already exceeds the precision (4 × 10−19 GeV) in the mass difference of neutral kaons and antikaons25, which has long been the standard for particle-physics tests of charge–parity–time invariance.

Near-term improvements in the ALPHA-2 apparatus will include a larger waist size for the radiation in the optical cavity to reduce transit-time broadening, operation at lower magnetic fields and operational improvements to accelerate data acquisition and to reduce statistical uncertainties. Future measurements will require an upgrade to our frequency reference to exceed a fractional precision of 8 × 10−13 (Methods). The rapid progress detailed here confirms that, in principle, there is nothing to prevent the achievement of hydrogen-like precision in antihydrogen and the associated very sensitive test of charge–parity–time symmetry in this system.