Characterization of the 1S–2S transition in antihydrogen

In 1928, Dirac published an equation1 that combined quantum mechanics and special relativity. Negative-energy solutions to this equation, rather than being unphysical as initially thought, represented a class of hitherto unobserved and unimagined particles—antimatter. The existence of particles of antimatter was confirmed with the discovery of the positron2 (or anti-electron) by Anderson in 1932, but it is still unknown why matter, rather than antimatter, survived after the Big Bang. As a result, experimental studies of antimatter3–7, including tests of fundamental symmetries such as charge–parity and charge–parity–time, and searches for evidence of primordial antimatter, such as antihelium nuclei, have high priority in contemporary physics research. The fundamental role of the hydrogen atom in the evolution of the Universe and in the historical development of our understanding of quantum physics makes its antimatter counterpart—the antihydrogen atom—of particular interest. Current standard-model physics requires that hydrogen and antihydrogen have the same energy levels and spectral lines. The laser-driven 1S–2S transition was recently observed8 in antihydrogen. Here we characterize one of the hyperfine components of this transition using magnetically trapped atoms of antihydrogen and compare it to model calculations for hydrogen in our apparatus. We find that the shape of the spectral line agrees very well with that expected for hydrogen and that the resonance frequency agrees with that in hydrogen to about 5 kilohertz out of 2.5 × 1015 hertz. This is consistent with charge–parity–time invariance at a relative precision of 2 × 10−12—two orders of magnitude more precise than the previous determination8—corresponding to an absolute energy sensitivity of 2 × 10−20 GeV.

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 elsewhere [12][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.
The key to anti-atomic spectroscopy, as developed so far 7,15,16 , 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 detector 17 (Fig. 1) affords us singleatom 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 analysis 18 (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-flip 19 .
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).
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 Letter reSeArCH 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-flip 15,16 . The microwave frequency is scanned over 9 MHz in 32 s; these parameters and the injected power level (160 mW at the vacuum feedthrough) 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 previously 7 . 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.
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.
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   Letter reSeArCH 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 expected 7 ), repeated for each set, are used for the scaling.  19 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).
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 effect 19 . 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 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.

Letter reSeArCH
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.
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 technique 16 . 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 shift 19 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 magneticfield 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,  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 beam 8 , the modern frequency reference permits the accuracy of our experiment to exceed that achieved with trapped hydrogen 20 as recently as the mid-1990s. We used a total of about 15,000 antihydrogen atoms to obtain this result, compared to 10 12 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 traps 21,22 or with antiprotonic helium 23 . For example, the antiproton charge-to-mass ratio is known to agree with that of the proton to 69 parts per trillion 21 , 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 counterpart 23 to 8 × 10 −10 , and antihydrogen has been shown to be neutral 24 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 antikaons 25 , 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 chargeparity-time symmetry in this system.

Online content
Any Methods, including any statements of data availability and Nature Research reporting summaries, along with any additional references and Source Data files, are available in the online version of the paper at https://doi.org/10.1038/s41586-018-0017-2. The estimated statistical and systematic errors (at 121 nm) are tabulated.