The fine-structure splitting of the n = 2 states of hydrogen is the separation of the 2P 3/2 and 2P 1/2 levels at zero magnetic field. This splitting, predicted by the Dirac theory of relativistic quantum mechanics11, originates from the spin–orbit interaction between the non-zero orbital angular momentum (L = 1) and the electron spin. The ‘classic’ Lamb shift is defined as the splitting between the 2S 1/2 and 2P 1/2 states at zero field12, and is a manifestation of the interaction of the electron with the quantum fluctuations of the vacuum electromagnetic field, an effect explained by quantum electrodynamics (QED)12,13,14. Today, it is understood that the classic Lamb shift in hydrogen is dominated by the QED effects on the 2S energy level, and that the 1S level receives even stronger QED corrections than the 2S level12,13. Although QED corrections in levels n ≠ 2 are now also sometimes referred to as Lamb shifts, in this Article we restrict our definition of the Lamb shift to be the classic n = 2 shift.

In a magnetic field, the Zeeman effect causes the 2P 3/2 state to also split into four sublevels (labelled 2P a , 2P b , 2P c and 2P d ), whereas the 2S 1/2 and 2P 1/2 states each split into two (2S ab and 2S cd ; 2P e and 2P f ). These fine-structure levels further split into two hyperfine states owing to the proton spin (see Fig. 1 for the expected energy levels for the case of antihydrogen, where the spin orientations are reversed with respect to those of hydrogen.)

Fig. 1: Expected antihydrogen energy levels. Calculated energies of the fine structure and the hyperfine sublevels of the 1S 1/2 , 2S 1/2 , 2P 3/2 and 2P 1/2 states are shown as functions of magnetic-field strength. The spin orientations for antihydrogen are shown; they are reversed for hydrogen. The centroid energy difference, E 1S–2S = 2.4661 × 1015 Hz, has been suppressed on the vertical axis. Details of the energy levels relevant to this work at a magnetic field of B = 1.0329 T are shown on the right. Each state is labelled using conventional notation. For the 1S and 2S states, the hyperfine states are labelled with subscripts a–d in order of increasing energy (see, for example, ref. 7); namely, \({{\rm{S}}}_{{\rm{a}}}=|\uparrow \Uparrow \rangle \), \({{\rm{S}}}_{{\rm{b}}}=|\uparrow \Downarrow \rangle \), \({{\rm{S}}}_{{\rm{c}}}=|\downarrow \Uparrow \rangle \) and \({{\rm{S}}}_{{\rm{d}}}=|\downarrow \Downarrow \rangle \), where the ket notation represents the positron spin (left; ↓ or ↑) and antiproton spin (right; ⇓ or ⇑) states in the high-field limit. The labels S ab and S cd are used when the antiproton spins are unpolarized. For the 2P states, the fine-structure splittings are labelled with subscripts a–f in order of decreasing energy at low magnetic fields, whereas the hyperfine splitting due to the antiproton spin is specified by subscripts + and − for spin parallel (⇑) and anti-parallel (⇓) to the magnetic field in the high-field limit, respectively. The symbol (↓,↑) in the figure indicates that the positron spin states are mixed for the 2P c and 2P f states. The vertical solid arrows indicate the one-photon laser transitions probed here: 1S d → 2P f− (bold red), 1S c → 2P f+ (thin red), 1S d → 2P c− (bold blue) and 1S c → 2P c+ (thin blue). The dashed red and blue arrows indicate relaxation to the same trappable level, which is not detectable in the present experiment, and the dashed black arrows indicate relaxation to untrappable levels, which is detectable via annihilation signals (see text). The bold black arrow shows the microwave transition used to eliminate 1S c state atoms to prepare a doubly spin-polarized antihydrogen sample. Full size image

Lamb’s original work used the then newly developed techniques of an excited-state atomic hydrogen beam and resonant microwave spectroscopy to study direct transitions between the n = 2 fine-structure states in various magnetic fields. The Lamb shift was then determined to 10% precision by extrapolating frequency measurements to zero field1. Here, we report the observation of the splitting between the 2P c and 2P f states in antihydrogen in a field of 1 T, by studying laser-induced transitions from the ground state. Assuming the validity of the Zeeman and hyperfine interactions, and using the value of the previously measured 1S–2S transition frequency7, we infer from our results the values of the zero-field fine-structure splitting and the classic Lamb shift in antihydrogen. Such studies have become possible owing to the combination of several recent advances: the accumulation15 of hundreds of anti-atoms in each run, their confinement for many hours16, control of the hyperfine polarization of the antihydrogen samples17 and the development of a narrow-line, pulsed, Lyman-α laser6,18.

Details of the production, trapping and control of antihydrogen in the ALPHA experiment have been provided elsewhere6,7,15,16,17,18,19,20,21,22,23,24,25, so the following description is brief. The ALPHA-2 apparatus (Fig. 2) incorporates a cylindrical magnetic trapping volume (about 400 cm3) for neutral anti-atoms; the magnetic-field minimum at the centre of the trap was set to 1.0329 ± 0.0004 T for this work. (All uncertainties given herein are 1σ.) By combining 90,000 trapped antiprotons from the CERN Antiproton Decelerator23 and three million positrons from a positron accumulator24,25, about 10–30 cold (below 0.54 K) anti-atoms are confined in the magnetic trap in a 4-min cycle. Under normal conditions, the storage lifetime16 of the trapped antihydrogen is greater than 60 h, which permits loading from repeated cycles15 to obtain hundreds of antihydrogen atoms in a few hours.

Fig. 2: The ALPHA-2 central apparatus. A cylindrical trapping volume for neutral antimatter with a diameter of 44.35 mm and an axial length of 280 mm is located inside several Penning trap electrodes and surrounded by an octupole coil, five mirror coils and two solenoids, all superconducting. The three-layer silicon vertex annihilation detector is shown schematically in green. Laser light (purple line) enters from the positron (e+) side (right) and is transmitted to the antiproton (\(\bar{p}\)) side (left) through vacuum-ultraviolet-grade MgF 2 ultrahigh-vacuum windows. The laser beam crosses the trap axis at an angle of 2.3°. The transmitted 121.6-nm pulses are detected by a solar-blind photomultiplier tube (PMT) at the antiproton side. Microwaves used to prepare the doubly spin-polarized samples are introduced from the positron side through a waveguide, shown in blue. The external solenoid magnet for the Penning traps is not shown here. THG, third-harmonic generation. Full size image

Two types of antihydrogen samples were used in these studies. The positron spin of an antihydrogen atom confined in the ALPHA-2 trap is necessarily polarized, because only the 1S c and 1S d states can be magnetically trapped (Fig. 1). The antiproton spin, on the other hand, is unpolarized a priori, with both orientations equally likely. Thus, the initial samples are singly spin-polarized. On the other hand, doubly spin-polarized samples, in which both the positron and antiproton spins are polarized, can be prepared by injecting microwaves to resonantly drive the 1S c atoms to the untrappable 1S b state (Fig. 1), effectively depopulating the 1S c state from the trap17.

Spectroscopy in the vacuum ultraviolet range is challenging even for ordinary atoms, owing in part to the lack of convenient laser sources and optical components26,27,28. Our pulsed, coherent 121.6-nm radiation was produced by generating the third harmonic of 365-nm pulses in a Kr/Ar gas mixture at a repetition rate of 10 Hz (ref. 18). The typical pulse width at 121.6 nm was 12 ns, and the bandwidth was estimated from the Fourier transform of the temporal pulse shape to be 65 MHz (full-width at half-maximum, FWHM). The 121.6-nm light was linearly polarized because of the three-photon mixing of linearly polarized 365-nm light. In the antihydrogen trap, the polarization vector was nearly perpendicular to the direction of the axial magnetic field. The laser beam had a radius of 3.6 mm and was roughly collimated across the trapping region (Fig. 2). The average pulse energies in the antihydrogen trapping volume ranged from 0.44 nJ to 0.72 nJ over different runs, as evaluated from the pulse waveforms recorded with a calibrated, solar-blind photomultiplier detector.

In this experiment, single-photon transitions from the 1S c (1S d ) states to the 2P c+ (2P c− ) and 2P f+ (2P f− ) states are driven by the 121.6-nm light (red and blue arrows in Fig. 1). When antihydrogen is excited to the 2P c± or 2P f± state, it decays to the ground-state manifold within a few nanoseconds by emitting a photon at 121.6 nm. The mixed nature of the positron spin states in the 2P c+ (2P c− ) and 2P f+ (2P f− ) states implies that these states can decay to the 1S b (1S a ) states via a positron spin flip (black dashed arrows in Fig. 1). Atoms in these final states are expelled from the trap and are annihilated on the trap walls. Annihilation products (charged pions) are in turn detected by a silicon vertex detector29 with an efficiency greater than 80%.

Table 1 summarizes our data. In total, four series of measurements were performed using either singly or doubly spin-polarized samples. The Series 1 data, previously reported in ref. 6, have been reanalysed. Each series consisted of two or four runs, and in each run about 500 antihydrogen atoms were accumulated over approximately two hours, typically involving over 30 production cycles. The trapped anti-atoms were then irradiated for about two hours by a total of 72,000 laser pulses at twelve different frequencies (that is, 6,000 pulses per frequency point for each run) spanning the range −3.10 GHz to +2.12 GHz relative to the expected (hydrogen) transition frequencies. The laser frequency was changed every 20 s in a non-monotonic fashion to minimize effects related to the depletion of the sample of antihydrogen. After the laser exposure, the remaining antihydrogen atoms were released by shutting down the trap magnets, typically in 15 s, and counted via detection of their annihilation events. 40–60% of the trapped antihydrogen atoms experienced resonant, laser-induced spin flips, and their annihilations were detected during the two-hour laser irradiation period.

Table 1 Experimental parameters and number of detected events Full size table

A combination of time-gated antihydrogen detection (enabled by the use of a pulsed laser), the accumulation of a large number of anti-atoms and the use of supervised machine-learning analysis29 (based on a boosted decision-tree classifier) suppressed the background to a negligible level (less than 2 counts per 2-h irradiation period).

The measured spectra, obtained from counting the laser-induced spin-flip events, are shown in Fig. 3 for both singly and doubly spin-polarized antihydrogen samples. For each run, the probability at each frequency point is determined from dividing the number of annihilation events recorded at that frequency by the total number of trapped atoms in that run, and further dividing by the ratio of the average laser energy to a standard value of 0.5 nJ. The normalization to the standard laser energy is to account for the expected linear dependence of the transition probability on the laser power in our regime. The data plotted in Fig. 3 are spectrum-averaged over the runs for each series. For the singly polarized sample (Fig. 3a), each transition shows a linewidth of about 1.5 GHz (FWHM). This is consistent with the expected Doppler broadening in our trapping condition (1 GHz FWHM) and the hyperfine splitting of the 1S–2P f and 1S–2P c transitions (0.71 GHz for both transitions). The hyperfine structure cannot be resolved in these singly polarized samples owing to the Doppler broadening.

Fig. 3: 1S–2P fine-structure spectrum of antihydrogen. a, b, Experimental data (filled circles) and fitted lineshapes for singly spin-polarized (a) and doubly spin-polarized (b) antihydrogen samples. The data points were obtained from the detected spin-flip events, normalized to the total number of trapped antihydrogen atoms, for a laser pulse energy of 0.5 nJ. The error bars are 1σ counting uncertainties. The frequency is offset by 2,466,036.3 GHz. We note that no data were taken between the two peaks (~2–12 GHz). The red fit curves were obtained via our standard fitting procedure (Model 1), and the blue curves were derived from an alternative fitting model (Model 2), illustrating the sensitivity of our results to the fitting procedure. See text and Methods for detailed discussion. Full size image

Figure 3b shows the spectra obtained from doubly spin-polarized antihydrogen samples. For these data, microwave radiation of ~28 GHz (power ~0.4 W, measured at the trap entrance) was applied before the start of optical spectroscopy, in the form of a 9-MHz sweep, covering the 1S c –1S b transition in the magnetic-field minimum17. As shown in Table 1, about half of the total trapped antihydrogen atoms underwent a positron spin-flip and annihilated during microwave irradiation. This is consistent with our experience from earlier studies, in which 1S c -state atoms were removed with about 95% efficiency7,17. The spectral lines of the 1S–2P transitions in doubly spin-polarized antihydrogen (Fig. 3b) are narrower than those in the singly spin-polarized samples (Fig. 3a) because the former involves only one hyperfine state in the ground state. The peaks are red-shifted because the frequencies of the transition from the 1S d state to the 2P f and 2P c states are expected to be about 700 MHz lower than those from the 1S c state. The observed width of ~1 GHz FWHM of these lines is in agreement with the Doppler width expected for our trapping conditions.

The procedure used to extract the frequencies of the fine-structure transitions and to evaluate their associated uncertainties is described in Methods. We summarize the results of this analysis in Table 2. A simulation was used to model the motion of trapped antihydrogen atoms in the ALPHA-2 trap and their interaction with pulsed laser radiation. The resonance transition frequencies were obtained by comparing simulated and experimental lineshapes. Extensive investigations were performed to evaluate systematic uncertainties in our measurement (Table 3). The validity of our analysis procedure was tested by using different lineshape-fitting models. Two representative curve fits are shown in Fig. 3. The fit of Model 1 uses a function constrained to fit the simulation shape, whereas in Model 2 the shape parameters of this function are allowed to vary to best fit the experimental data; see Methods for details. The sensitivity of the results to the experimental and simulation parameters was tested by repeating the analysis procedure for a number of simulations with varied input. These included the initial antihydrogen conditions (such as the initial temperature, the quantum state, and the cloud diameter of antihydrogen at formation) and laser properties (such as linewidth, beam waist size and beam position); see Methods and Extended Data Fig. 1. Other sources of systematic uncertainties include the calibration accuracy and a possible frequency drift of the wavemeter, frequency shifts of the 730-nm amplification laser cavity, and possible incomplete clearing of the 1S c state in the preparation of the doubly spin-polarized samples (Table 3 and Methods).

Table 2 1S–2P transition frequencies Full size table

Table 3 Summary of uncertainties Full size table

Within the uncertainties, the measured transition frequencies agree with theoretical expectations for hydrogen for all four series (Table 2, Fig. 4). The fact that the four measurements are consistent, despite having different systematics, increases the confidence in our overall results. The results can be combined to give a test of charge–parity–time (CPT) invariance in the 1S–2P transitions at the level of 16 parts per billion (Fig. 4).

Fig. 4: Comparison of antihydrogen and hydrogen transition frequencies. The experimentally measured frequencies for the 1S–2P transitions in antihydrogen f res (exp) are compared with those theoretically expected for hydrogen f res (th) (Table 2). All four measurements are consistent with hydrogen, and their average gives a combined test of CPT invariance at 16 parts per billion (ppb). The error bars are 1σ, and the calculation of the error bar for the average takes into account correlated uncertainties (Methods). Full size image

Fundamental physical quantities of antihydrogen can be extracted from our optical measurements of the 1S–2P transitions by combining them with our earlier measurement of the 1S–2S transition in the same magnetic trapping field7. From the weighted average of the results between the singly polarized and doubly polarized measurements (Table 1), we obtain a 2P c− –2P f− splitting of 14.945 ± 0.075 GHz, a 2S d –2P c− splitting of 9.832 ± 0.049 GHz and a 2S d –2P f− splitting of 24.778 ± 0.060 GHz at 1.0329 T (Methods). Only two of these three splittings are independent, and they all agree with the values predicted for hydrogen in the same field.

In interpreting our data, we categorize features in the spectrum based on the order of the fine-structure constant α in a perturbative series expansion in quantum field theory (which is assumed to be valid for the purpose of our categorization). Those features that can be described by the Dirac theory (the Zeeman, hyperfine and fine-structure effects) are referred to as ‘tree-level effects’ and follow from the lower-order terms (up to order ~α2Ry, where Ry is the Rydberg constant). On the other hand, the Lamb shift originates from the so-called ‘loop effects’ (order ~α3Ry), the calculation of which requires the concept of renormalization to avoid infinities12,13,14. Each of the measured splittings has different sensitivity to different terms. At the level of our precision, the 2P c –2P f splitting is sensitive to the tree-level terms with negligible QED effects, whereas the 2S–2P f and 2S–2P c splittings are sensitive to the field-independent Lamb shift, in addition to the tree-level terms (we note that the Lamb shift is predicted to have negligible dependence on the magnetic field14). The agreement between our measurement and the Dirac prediction for the 2P c− –2P f− splitting supports the consistency of the tree-level theory in describing the Zeeman, hyperfine and fine-structure interactions in the 2P states of antihydrogen. If we hence assume that we can correctly account for the tree-level effects in our measurements, we can infer from our measured splittings the values of the zero-field fine-structure splitting in antihydrogen to be 10.88 ± 0.19 GHz. By combining the current result with the much more precisely measured 1S–2S transition frequency in antihydrogen7, we obtain a classic Lamb shift of 0.99 ± 0.11 GHz (Methods). If we use the theoretical value of the fine-structure splitting from the Dirac prediction (rather than treat it as a parameter), we can derive a tighter constraint on the Lamb shift, 1.046 ± 0.035 GHz.

When considering the first measurements on an exotic system such as antihydrogen, it is necessary to adopt a framework within which it is possible to compare the results to the expectations of well established models for normal matter. The choice of which effects can be assumed to be true in interpreting the data are, of necessity, somewhat arbitrary. The approach illustrated here is based on the order of perturbation in the coupling constant α; we have assumed (lower-order) tree-level effects in order to extract (higher-order) renormalizable loop effects. Other approaches are possible in interpreting our data. We note that if the standard theory for the hydrogen atom applies to antihydrogen, most of the expected QED effect is on the 2S level, rather than on the 2P level. Furthermore, the 1S level receives approximately n3 = 8 times larger QED corrections than the 2S level; hence, our earlier accurate determination of the antihydrogen 1S–2S level difference7 gives strong constraints on new interactions within the QED framework. However, it is possible that a new effect could show up in the antihydrogen classic Lamb shift while satisfying the 1S–2S constraint. See ref. 8 for an example in a Lorentz-violating effective-field theory framework.

We have investigated the fine structure of the antihydrogen atom in the n = 2 states. The splitting between the 2P c and 2P f states, two of the 2P Zeeman sublevels belonging to the J = 3/2 and J = 1/2 manifolds (J, total angular momentum), has been observed in a magnetic field of 1 T. The energy levels of the 1S–2P transitions agree with the Dirac theory predictions for hydrogen at 1 T to 16 parts per billion, and their difference to 0.5%. By assuming the standard Zeeman and hyperfine effects, and by combining our results with the earlier result of 1S–2S spectroscopy7, we have inferred the zero-field fine-structure splitting and the classic Lamb shift in the n = 2 level.

These observations expand the horizons of antihydrogen studies, providing opportunities for precision measurements of the fine structure and the Lamb shift—both of which are longstanding goals in the field. Prospects exist for considerable improvements in the precision beyond this initial determination. With the advent of the ELENA ring in 2021, an upgrade to the Antiproton Decelerator with an anticipated increase in the antiproton flux, the statistical uncertainties are expected to be dramatically reduced. The development of laser cooling30 would reduce the Doppler width to a level comparable to the natural linewidth, which in turn would improve the precision of the frequency determination. It would also permit direct experimental determination of the hyperfine splitting in the 2P states, for which theoretical values were assumed in this study.

Such measurements will provide tests of CPT invariance that are complementary to other precision measurements in antihydrogen, such as the 1S–2S frequency and the ground-state hyperfine splitting. Furthermore, a precise value of the classic Lamb shift, combined with that of the 1S–2S interval, will permit an antimatter-only determination of the antiproton charge radius9,10, without referring to matter measurements—that is, independent of the proton charge radius puzzle31,32,33. These examples signify the importance of broad and complementary measurements in testing fundamental symmetries. In the absence of compelling theoretical arguments to guide the way to possible asymmetries, it is essential to address the antihydrogen spectrum as comprehensively as is practical. Finally, the results reported here demonstrate our capability to precisely and reproducibly drive vacuum ultraviolet transitions on a few anti-atoms, and indicate our readiness for laser cooling of antihydrogen30, an eagerly anticipated development in antimatter studies with far-reaching implications for both spectroscopic and gravitational studies34.