Abstract
Antihydrogen, the lightest atom consisting purely of antimatter, is an ideal laboratory to study the CPT symmetry by comparison with hydrogen. With respect to absolute precision, transitions within the groundstate hyperfine structure (GSHFS) are most appealing by virtue of their small energy separation. ASACUSA proposed employing a beam of cold antihydrogen atoms in a Rabitype experiment, to determine the GSHFS in a fieldfree region. Here we present a measurement of the zerofield hydrogen GSHFS using the spectroscopy apparatus of ASACUSA’s antihydrogen experiment. The measured value of ν_{HF}=1,420,405,748.4(3.4) (1.6) Hz with a relative precision of 2.7 × 10^{−9} constitutes the most precise determination of this quantity in a beam and verifies the developed spectroscopy methods for the antihydrogen HFS experiment to the p.p.b. level. Together with the recently presented observation of antihydrogen atoms 2.7 m downstream of the production region, the prerequisites for a measurement with antihydrogen are now available within the ASACUSA collaboration.
Introduction
Investigations of the hydrogen atom have been a driving force for the discovery of more profound theories^{1} and contribute to the basis of physics through their prominent influence on the definition of fundamental constants^{2}. Most notable from a precision point of view are the recent measurement of the 1S–2S transition via twophoton spectroscopy^{3} and the determination of the hyperfine splitting in hydrogen maser experiments in the early 1970s (refs 4, 5, 6, 7, 8, 9). The achieved absolute (relative) precisions are 10 Hz (4 × 10^{−15}) and 2 mHz (1.4 × 10^{−12}), respectively. A revival of the interest in hydrogen is founded on prospects of antihydrogen research^{10,11}. The structure of the simplest antiatom consisting of a positron bound to an antiproton is predicted to be identical to that of hydrogen, if the combined symmetry of charge conjugation, parity and time reversal (CPT) is conserved. Hence, antihydrogen spectroscopy promises precise tests of the CPT symmetry, which is a cornerstone of the Standard Model of particle physics. A vivid physics programme is currently underway at the Antiproton Decelerator of CERN aiming at spectroscopic^{12,13,14,15,16,17} and gravity tests^{18,19} along with other CPT tests such as the neutrality of antihydrogen^{20,21}, as well as measurements of the chargetomass ratio^{22} and magnetic moment^{23,24} of the antiproton.
Among the spectroscopic tests of CPT, the comparison of the groundstate hyperfine structure (GSHFS) of hydrogen and antihydrogen has the potential to reach the highest sensitivity on an absolute energy scale^{25,26,27}. However, the aforementioned most precise measurement of this quantity for hydrogen was made using a maser^{8}. Such a technique is not applicable to antimatter, which would annihilate with the confining matter enclosure. The measurement proposed by the ASACUSA (Atomic Spectroscopy And Collisions Using Slow Antiprotons) collaboration at the Antiproton Decelerator of CERN therefore makes use of a beam of cold antihydrogen atoms^{28,29}. In addition to avoiding wall interaction, the actual measurement takes place in a fieldfree region, ultimately allowing for higher precision compared with the observation of resonant quantum transitions between the hyperfine states in trapped antihydrogen in a highfield environment^{30}.
In the present experiment, the Zeemanshifted hyperfine transitions at various external magnetic field strengths were determined for subsequent extraction of the zerofield value and resulted in
The numbers in brackets are the 1 s.d. (1σ) statistical and systematic uncertainties. Added in quadrature the total uncertainty of 3.8 Hz constitutes an improvement by more than an order of magnitude in comparison with the previously achieved best precision by Rabitype spectroscopy of 50 Hz^{31,32}. Our result is in agreement within 1 s.d. with the literature value of ν_{lit}=1,420,405,751.768 (2) Hz, which relies on the more precise hydrogen maser measurements^{4,5}. In view of the initial goal for antihydrogen GSHFS of ≲1 p.p.m. relative precision^{33}, our hydrogen measurement shows that at this level, systematic uncertainties will be well under control. Our estimate suggests that at least 8,000 antihydrogen atoms of the usually assumed properties will be needed to determine hyperfine transition frequency of antihydrogen with 1 p.p.m. precision.
Results
Spectroscopy principle
Rabitype magnetic resonance spectroscopy^{34,35} applies rotating (or oscillating) magnetic fields to induce quantum transitions and exploits the force of magnetic field gradients on the statedependent magnetic moment of atoms (or molecules), to spatially separate the atoms in a beam with respect to their quantum states (Stern–Gerlach separation). Typically, magnetic sextupole fields are employed to focus atoms in lowfieldseeking states (lfs) and defocus highfield seekers (hfs). In the case of groundstate hydrogen, the hyperfine structure consists of a lowerlying singlet state with total angular momentum quantum number F=0 (being proton and electron spin, respectively) and a triplet state F=1. As illustrated by the Breit–Rabi diagram in Fig. 1, the triplet state degeneracy is lifted in the presence of a magnetic field. The singlet state and the triplet state with magnetic quantum number M_{F}=−1 are hfs, whereas the other two states (F=1, M_{F}=0, 1) are lfs. In the present work, the σ_{1}transition from (F=1, M_{F}=0) to (F=0, M_{F}=0) has been studied^{36,37}.
Experimental setup
The main components of the experiment are a source of cooled and polarized atomic hydrogen, the hyperfine spectrometer of the hyperfine splitting (HFS) setup (that is, a microwave cavity and a superconducting sextupole magnet) and a hydrogen detector (cf. Fig. 2). The atomic hydrogen source maintains a microwavedriven plasma in a pyrex cylinder to dissociate molecular hydrogen (H_{2}→H+H)^{38}. Hydrogen atoms are allowed into the first vacuum chamber through a polythetrafluorethylen (PTFE) tubing, which is kept under cryogenic temperatures, to cool the hydrogen atoms and hence reduce their velocity^{39}. Two tubing configurations are used in which the plasmacontaining pyrex cylinder is either mounted perpendicular to or on axis with the beam. In the first case, a 90° bent tubing assures an efficient and complete interaction of the hydrogen atoms with the cold PTFE surface. In the latter case, a straight tubing keeps the recombinations caused by wall interactions down to a minimum. The cooled atomic hydrogen beam is directed onto a skimmer of 1 mm in diameter and reaches the second, differentially pumped chamber, which houses two permanent sextupole magnets with a pole field of ∼1.3 T at a radius of 5 mm over a mechanical length of 65 mm each^{40}. In addition to providing the initial spinpolarization, those sextupole magnets are moveable and feature a midway aperture (aperture 1) to allow for the adjustable selection of a narrow velocity range. As the focusing length depends on the beam velocity, only a certain velocity component is focused onto the aperture and can pass, whereas the offaxis portions of all other components are blocked. The variable distance to the aperture located at half the distance between the sextuple magnets (d_{s}) therefore selects a velocity component. The resulting velocity distribution is much narrower than a Maxwell–Boltzmann distribution and roughly of Gaussian shape. The spinpolarized and velocityselected hydrogen beam passes another aperture (aperture 2) and is then modulated by a tuning fork chopper in the next differentially pumped section. The modulation adds timeofflight measurements to the beam diagnostic tools, as well as suppression of background originating from residual hydrogen via lockin amplification. Downstream of the chopper, apertures of different diameters (aperture 3) can be installed, to produce different beam sizes at the entrance of the microwave cavity.
The HFS spectrometer has been designed with an open diameter of 100 mm, as a large acceptance is crucial in view of small production rates. The amplitude of the oscillating magnetic field B_{osc} has to be sufficiently uniform over the large open diameter, to guarantee a trajectoryindependent stateconversion probability. This requirement is best met by a cavity of the socalled stripline geometry^{41,42}. Two highly transparent meshes confine the microwaves at the entrance and exit of the stateconversion cavity, which are separated by half a wavelength of the hyperfine splitting transition (L_{cav}∼λ_{HF}/2∼105.5 mm). A standing wave forms between them and, as a consequence, B_{osc} is not constant along the beam propagation direction, causing a doubledip resonance line shape. The origin of this structure is outlined below and explained in detail in the Methods. The cavity length and the beam velocity V_{H} define the interaction time of the hydrogen atoms with the microwave field T_{int}=L_{cav}/V_{H} and restrict the achievable resonance line width to . A synthesizer coupled to an external rubidium clock for frequency stabilization produces microwaves, which are fed radially to the cavity via an antenna after amplification. On the opposite side of the cavity, another antenna is used for pickup and monitoring of the microwave power (P_{MW} ∝ B_{osc}^{2}) using a spectrum analyser. Helmholtz coils are mounted onto the cavity to generate a homogeneous external magnetostatic field B_{stat}, parallel to B_{osc}, and of several Gauss in magnitude at the interaction region for fine control of the Zeeman splitting. A current source with a relative stability of 20 p.p.m. supplies the Helmholtz coils’ current I_{HC}, which is independently monitored by an amperemeter. I_{HC} is directly proportional to B_{stat} and turned out to be a better proxy for the magnetic field inside the cavity than a dedicated external magnetic field measurement. The microwave cavity and the Helmholtz coils are surrounded by a twolayer cuboidal Mumetal shielding to block the Earth’s magnetic field, as well as the fringe field of the closely succeeding superconducting sextupole magnet. Owing to the pole strength of up to 3.5 T, this magnet generates sizeable magnetic field gradients despite the large open diameter of 100 mm. The integrated gradient amounts to 150 Tm^{−1} and ensures refocusing of 50 K lfs atoms within a distance of ∼1 m.
The detection of hydrogen suffers from a large background rate and small efficiencies. A crossedbeam quadrupole mass spectrometer (QMS) with a 3 mm opening ionizes beam atoms and residual gas by electron impact and selectively guides protons to a channeltron for efficient single mass=1 ion counting. The small sensitive area and hydrogen ionization efficiency result in detected beam rates of only a few kHz in spite of typical H_{2} flowrates of 1.8 × 10^{17} s^{−1}. The background rate is kept at a level of only few tens of kHz by combining twostage turbomolecular pumping and nonevaporablegetter pumps, thereby maintaining ultrahigh vacuum conditions (≲5 × 10^{−10} mbar) in the detection chamber. Furthermore, the QMS can be moved twodimensionally in the plane perpendicular to the beam for optimizing count rates and investigating beam profiles.
Measurement procedure
The dissociation plasma was operated under stable standard conditions. Before starting frequency scans, the microwave power P_{MW} supplied to the cavity was adjusted to yield the largest stateconversion probability by observing a Rabi oscillation. A single measurement cycle was obtained by scanning the frequency once in a random sequence across the desired range. Typically, this included N∼39 frequency points distributed over ∼40 kHz. At each frequency point, the channeltron events of the QMS were summed several times for typical intervals of 5–60 s from which an average count rate was retrieved. Such cycles over the frequency range were repeated on average five times with changing random sequences to result in a complete scan at a given I_{HC}. This was repeated at different values and polarity of I_{HC} to yield a set of scans suitable for determination of the fieldfree hyperfine splitting. The number of I_{HC} values per set ranged from 6 to 16. In total, ten such sets have been recorded, which differ in various of the experimental settings and arrangements (cf. Table 1).
Raw data corrections
Initially, a fit as described below was applied to the detected count rates. Two systematic effects were identified in the residuals and corrected for. The first correction compensates slow time drifts. The second correction concerns a type of memory effect, which became evident in an increased likelihood of observing positive or negative residuals if the previous data point was taken at higher or lower count rate, respectively. This indicated that the settling of the hydrogen rate in the detection chamber following a change of the excitation frequency had a nonnegligible time constant when compared with the measurement time at each frequency step. These two effects were corrected for at the raw data level and led to an improvement of the fit quality without affecting the extracted ν_{HF} values. The application of a random sequence of frequencies in the cycles seemed to suppress systematic impacts of the drift and the memory effect below the statistical sensitivity.
Analysis
The central frequency ν_{c} was extracted from every cycle by a fit to the spectrum as illustrated in Fig. 3a, where the excitation frequency ν is given as the difference to ν_{lit}. The doubledip line shape originates from the sinusoidal dependence of B_{osc} along the beam axis, which follows half a cosine period. At the actual transition frequency, the highest count rate between the two dips is observed. The theoretical line shape for a monoenergetic beam is well understood and accurately described within the framework of the twolevel system with the interaction Hamiltonian
where is the magnetic moment operator as defined in equation (9). The time dependence of the magnetic field includes a cos(πt/T_{int})term in addition to the microwave oscillations. The resulting equations were solved numerically to obtain the stateconversion probability as a function of the frequency ν and amplitude B_{osc} of the driving field for a monoenergetic beam. A realistic fit function for the measured stateconversion probabilities was obtained by convolution of the shape for monoenergetic beams with a velocity distribution as described in the Methods. Consequently, the fit function could extract the physical parameters B_{osc}, the mean velocity of the polarized atomic hydrogen beam and the width of the velocity distribution σ_{V} in addition to ν_{c} of the transition. Two further fit parameters of less relevant physical content scaled the stateconversion probability to the count rate and correspond to the count rate baseline R_{0} and the count rate drop ΔR for complete state conversion. In the final analysis, only ν_{c} was extracted from every cycle individually. For B_{osc}, a relation to the monitored microwave power was established based on the complete available data. This enabled individual fixation of this parameter for every set and avoided nonconverging fits due to a strong correlation of B_{osc} with ΔR. For and σ_{V}, a common fit value for a complete set was used, as all settings of direct impact on the beam velocity remained unchanged during data collection of a set.
As illustrated in Fig. 3a, the line shape thus obtained resulted in good fits to the observed count rates at all I_{HC} settings with reduced χ^{2}values close to unity as summarized in Table 1. The reliability of the fit function was important, as ν_{c} could be extracted with typical statistical uncertainties on the order of tens of Hz, whereas the width of the doubledip structure is on the order of tens of kHz. In Fig. 3c, the extracted ν_{c} value of each cycle of set 8 are plotted against the Helmholtz coils’ current I_{HC} at which it was recorded. The Zeemanshifted frequency of the σ_{1}transition ν_{σ}(B_{stat}) has only a secondorder dependence on the static external magnetic field B_{stat} as apparent from the Breit–Rabi diagram (Fig. 1) and described by the Breit–Rabi formula^{43}
with μ_{B}=5.7883818012 × 10^{−5} eVT^{−1} and μ_{N}=3.1524512550 × 10^{−8} eVT^{−1} being the Bohr and nuclear magneton, respectively, g_{e}=−2.00231930436182 and g_{p}=5.585694702 (ref. 2) being the gfactors of the electron and proton, respectively, and h=2πħ the Planck constant. To extract the zerofield hyperfine transition frequency ν_{HF}, a fit function was required, which used I_{HC} as a variable. A factor k converting I_{HC} to a magnetic field and a residual field B_{res} at I_{HC}=0 added two further fit parameters and established a linear relation to B_{stat}, which enters the Breit–Rabi formula
The notation for the fit function separates the variable from the parameters by a semicolon. The zerofield values ν_{HF} as obtained via this Breit–Rabi fit are plotted in Fig. 3d as the deviation from ν_{lit}.
Systematic tests
The following experimental arrangements and conditions have undergone changes for the ten sets (summarized in the top part of Table 1). The beam velocity varied due to different settings of d_{s} and the temperature of the PTFE tubing. The first three sets operated with the straight PTFE tubing, then the bent tubing was used. The need for an improved monitoring of I_{HC} and the advantage of a faster data acquisition scheme based on the total count rate instead of the lockin amplifier signal became evident in a preliminary evaluation of the first three sets and motivated the additional changes at that stage. Two opening diameters for aperture 3, resulting in different beam sizes at the entrance of the cavity, were also investigated. This is of special interest as an even larger beam diameter is expected for the HFS spectroscopy. In addition, the last four sets were performed with a second cavity of the same but slightly upgraded design. Three aspects were only changed for individual sets. For set 7 only one instead of two layers of magnetic shielding were used, for set 3 the superconducting sextupole was operated with a larger magnetic field strength leading to a shorter focal length and for set 4 the direction of the static magnetic field (Helmholtz coils) was not reversed.
The obtained results for ν_{HF} of the ten sets by first fitting all cycles in a set using the fit function (12) and second the Breit–Rabi fit (4) are presented in Fig. 3d. In addition, the average reduced χ^{2} of all fits to cycles within a set and the reduced χ^{2} of the Breit–Rabi fit are given in Table 1. On the level of the achieved statistical precision, no significant dependence of the ten results on any of the changed experimental conditions could be found. This justified to combine the ten individual results into one weighted mean value. Our final result deviates from the literature value by ν_{HF}−ν_{lit}=−3.4 Hz with a total uncertainty of σ_{tot}=3.8 Hz, which corresponds to a relative precision of 2.7 p.p.b. The mean value is shown in Fig. 3d as the dashed red line and the total 1σ uncertainties as the greyshaded area.
The fit parameters B_{osc}, and σ_{V}, which were fixed to a common average value for each set, were varied to assess the potential systematic uncertainties originating from the fit procedure. The complete analysis was repeated six times with setting each of the three parameters individually to its lower and upper 1σ boundary. The observed shifts of ν_{HF} for each parameter are listed in Table 2. However, those three values added in quadrature yielded 0.06 Hz and present a negligible systematic uncertainty. The rubidium clock, which served as frequency standard, supplied a 10 MHz reference signal to the microwave synthesizer. A calibration was performed several months after the measurement campaign and revealed a shift of 11.4 mHz or equivalently 1.14 p.p.b. This corresponds to 1.6 Hz for ν_{HF}, which is less than half the total statistical error. Given the timespan between the measurement and the calibration and the unknown evolution in time of the shift, it was not corrected for but instead conservatively added as a 1σ systematic uncertainty. A correction would have brought the central value closer to the literature value by roughly half a s.d. Table 2 summarizes the error budget.
Discussion
In the antihydrogen Rabi spectroscopy proposal^{33}, the firststage precision goal is ≲1 p.p.m. According to the Standard Model Extension framework^{25}, the absolute precision is more decisive, to quantify and compare the sensitvity level of CPT tests. A p.p.m. measurement of the GSHFS of antihydrogen corresponds to kHz frequencies or peV energies. This would already be several orders of magnitude better than the ∼2 neV precision of the kaon–antikaon comparison^{44} and competitive with the best achievable test using the 1S2S transition^{17}. In addition, at the level of ∼40 p.p.m. the antiproton structure becomes relevant in the calculation of the hyperfine structure of antihydrogen through the Zemach and nuclear polarizability corrections^{45}. Therefore, together with independent measurements of the antiproton magnetic moment^{23,24}, a p.p.m. measurement of the GSHFS of antihydrogen would give access to the electric and magnetic form factors of the antiproton. The result presented here on hydrogen shows that systematic uncertainties can be controlled much beyond the p.p.m. level. The data can also be used to assess the prospects for antihydrogen hyperfine spectroscopy. As first documented in ref. 46, three main terms quantitatively describe the precision with which a fit parameter of a resonance spectrum line shape can be determined. They relate to the signaltonoise ratio, the number of data samples per line width and a lineshapedependent factor. If the fit parameter of interest is the central frequency, the last term primarily expresses the resolution of the method. We adapt this formula to our case:
where the signal is identified with the count rate drop for complete state conversion ΔR and the noise with the average error bar σ_{R} of a data point. Instead of the data sample density, the dimensionless number of frequency data points N enters as the inverse square root. The interaction time T_{int} appears explicitly, as it is inversely proportional to the line width, which is a measure for the resolution. The dimensionless constant C should be unity and is inserted to test the relation. All remaining quantities are collected in a dimensionless lineshapedependent factor ɛ, which for instance accounts for the effects of different velocity distributions. Formula (5) is verified using the present hydrogen data. Apart from a consistent underestimation by a factor of C∼1.07, the precision is predicted reliably. More details can be found in the Methods.
The prospects of an antihydrogen GSHFS measurement using the σ_{1}transition can now be assessed by formula (5) making assumptions for yet unknown beam properties. Under Poisson statistics, σ_{R} will be proportional to the square root of the total number of events registered at the annihilation detector , which includes lfs of groundstate antihydrogen, as well as excited antihydrogen and false identifications from cosmic radiation or upstream annihilations. The number of excited states will be limited in the antihydrogen experiment by field ionization of Rydberg atoms down to the main quantum numbers n=12 (ref. 47). Given the typical decay times of states with n<12, most atoms should reach the cavity in the ground state. Conservatively, we assume that half of the atoms remain in excited or metastable states. According to statistical weights, another half of the lfs will be in the state (F=1, M_{F}=1). For a completely polarized beam, this remaining quarter of the total detected events would correspond to ΔR. We assume a polarization effect by the cusp magnetic field gradients of p=1/3 (refs 48, 49), which leads to a further reduction of ΔR proportional to 2p/(p+1). The annihilation detector, which consists of a central calorimeter^{50} and a double layer hodoscope for pion tracking^{51}, suppresses background from false identifications to negligible levels. For the last term of formula (5), we find
A signaltonoise ratio of 2 or 3 will require 250–600 events for each data point. The ASACUSA collaboration aims at reaching an antihydrogen temperature below 50 K. We assume a Maxwell–Boltzmann velocity distribution of the emerging beam as the worst case scenario, as other selection mechanisms are expected to lead to a smaller velocity spread. The most probable velocity of antihydrogen with this temperature is 909 ms^{−1}; hence, the estimated interaction time is 116 μs. The correction coefficient ɛ can reach values of ∼1.2 by concentrating on the central peak of the resonance instead of resolving the full doubledip structure as discussed in the Methods. In addition, the central peak can be encompassed with a low number of frequency points of N∼8. Inserting all numbers into formula (5) result in a statistical precision of δν_{c}=1.38 kHz or ≲1 p.p.m. with 2,000 detected events (and δν_{c}=0.89 kHz with 4,800 events). For a zerofield determination as demonstrated in the present measurement on hydrogen, a minimum of four resonances will be required as the Breit–Rabi fit (4) has three parameters. Consequently, an estimated minimum of 8,000 events will be required to determine with a precision of 1 p.p.m.
Methods
Resonance line shape
The σ_{1}transition in groundstate hydrogen is driven by an external microwave field, which is generated in a stripline cavity and takes the form
where e_{B} is the unit vector pointing in the direction of the magnetic field (z axis in the frame of the atoms, x axis in the coordinate system of the experiment) and ν=ω/2π is the applied microwave frequency. The term cos(ω_{cav}t) describes the changing amplitude of the magnetic field in the cavity along the beam propagation direction. T_{int} is the interaction time, which in turn follows from the hydrogen beam velocity V_{H} and the length of the cavity L_{cav}.
The small external magnetic field is aligned parallel to the oscillating magnetic field, which only for the σ_{1}transition leads to nonvanishing matrix elements. In addition, the Zeeman shift separates the groundstate hydrogen sublevels by more than the observed resonance width. Therefore, the transition dynamics is well described within the framework of the twolevel system
To obtain the time evolution of this system under the influence of the oscillating magnetic field, the Hamiltonian needs to be extended by the interaction with
for hydrogen. Here, are the spin operators acting on the electron or proton spinor as indicated by the superscript. An analytical solution can be found for conventional Rabi experiments, where the oscillating (or rotating) magnetic field has a constant amplitude B_{osc} and does not include the term cos(ω_{cav}t). If the system is initially prepared purely in state , then the conversion probability of finding it after a given interaction time T_{int} in the second state depends on the strength of B_{osc} and the detuning Ω_{D}=ω−ω_{12} with ħω_{12}=E_{2}−E_{1}:
where Ω_{R} is the Rabi frequency, which is proportional to the amplitude of the oscillating magnetic field. The relation for the σ_{1}transition is
Including the term cos(ω_{cav}t) requires numerical methods to determine the stateconversion probability. Figure 4 shows a comparison of as a function of the detuning Ω_{D} and the driving strength B_{osc} of conventional Rabi spectroscopy and the stripline cavity designed for the antihydrogen experiment. The latter case features the distinct doubledip structure with vanishing effects at the actual transition frequency. For a given interaction time T_{int}, the best precision is achieved with the first fullstate conversion in both situations. For the conventional case, this corresponds to a socalled πpulse, indicating that the condition Ω_{R}·T_{int}=π is satisfied or alternatively . The doubledip resonance reaches the first fullstate conversion when applying a somewhat stronger oscillating magnetic field .
From a twodimensional (2D) map as shown in Fig. 4, fit functions of the stateconversion probabilities for a monoenergetic beam can be derived with ν_{c}, the strength of B_{osc} and the hydrogen beam velocity V_{H} as fit parameters. This was realized by constructing a 2D spline interpolation to the numerically generated stateconversion probabilities at discrete points. A more realistic resonance line shape is then obtained by including the effect of the velocity distribution of the hydrogen beam, which translates to a not sharply defined interaction time T_{int}. It is noteworthy that both axis of the 2D maps are normalized to . Therefore, on an absolute scale for Ω_{D} and B_{osc}, a change of T_{int} is equivalent to a 2D zooming of the stateconversion probability map. The roughly Gaussian velocity distribution of the hydrogen beam after passage of the polarizing and velocityselecting permanent sextupole magnets is approximated by binomial coefficients for a discrete numerical realization of the convolution
The result of a convolution with such a velocity distribution is illustrated in Fig. 5 and compared with a measured map. The present analysis used M=6, because choices of M>6 did not change nor improve the fit results. For completeness, two more fit parameters were needed. To scale the stateconversion probability, which is a number between 0 and 1, to the observed count rates, a count rate baseline R_{0} and a count rate drop for complete state conversion ΔR were introduced
Investigations on the precision
The precision with which a parameter x_{i} of a line shape can be determined by minimum least square fitting is related to the noise σ of the resonance spectrum by
The elements of the matrix H are given by
where the summation is taken over the N data points^{46}. Equation (5) can be derived from the relations above and yields the following expression for the lineshapedependent correction factor
where 2·ΔΩ_{D} gives the covered scan range in units of and the sum has been simplified to a symmetric integral.
Figure 6 visualizes the test of equation (5) using the present data by plotting the precision extracted by the fit against the predicted precision with and without inclusion of the correction factor ɛ. The data set covers all 545 cycles of the 10 recorded sets. The correction for lineshapedependent effects by ɛ improves the reliability of the predicted precision and plays a more important role when making projections for antihydrogen spectroscopy, where different beam properties have to be expected. The slope of a line fit through the origin yields the dimensionless constant C.
Figure 7 compares the line shapes and correction factors ɛ of a monoenergetic beam to Gaussian and Maxwell–Boltzmann distributed beams. The crosses mark the correction factors determined for the ten sets, which depend on the chosen scan range, on the amplitude of the oscillating magnetic field B_{osc} and on the relative width of a Gaussian velocity distribution , as determined by the fit. For the resonance line shape of a Maxwell–Boltzmann distributed beam, the correction factors are generally smaller, which indicates reduced precision. The curves for the Maxwell–Boltzmann distributed beam show the cases, when B_{osc} is optimized for either the most probable or rootmeansquare velocity. For the antihydrogen measurement with limited count rates, the scan range dependence of the correction factor reveals that it will be beneficial to restrict the resonance scan to the central peak.
Data availability
The data sets generated and analysed during the current study are available from the corresponding author on reasonable request.
Additional information
How to cite this article: Diermaier, M. et al. Inbeam measurement of the hydrogen hyperfine splitting and prospects for antihydrogen spectroscopy. Nat. Commun. 8, 15749 doi: 10.1038/ncomms15749 (2017).
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Acknowledgements
We want to thank H. Kundsen, H.P.E. Kristiansen, F. Caspers, T. Kroyer, S. Federmann, P. Caradonna, M. Wolf, M. Heil, F. Pipper, C. Klaushofer, S. Friedreich and B. Wünschek for their contributions. We acknowledge technical support by the CERN Cryolab and Instrumentation group TECRGCI, as well as the CERN Magnet Normal Conducting group TEMSCMNC. This work has been supported by the European Research Council under European Union’s Seventh Framework Programme (FP7/20072013)/ERC Grant agreement (291242), the Austrian Ministry of Science and Research, and the Austrian Science Fund (FWF): W1252N27.
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M.D. prepared and performed the experiment, carried out the data analysis and wrote parts of the initial manuscript. C.B.J. performed the experiment and carried out parts of the data analysis. B.K. and C.S. performed simulations. C.M. and M.C.S. prepared and performed the experiment, guided and carried out parts of the data analysis, and wrote the manuscript. O.M. and J.Z. prepared the experiment. E.W. proposed and prepared the experiment, guided the data analysis and wrote the manuscript.
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Diermaier, M., Jepsen, C., Kolbinger, B. et al. Inbeam measurement of the hydrogen hyperfine splitting and prospects for antihydrogen spectroscopy. Nat Commun 8, 15749 (2017). https://doi.org/10.1038/ncomms15749
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