In-beam measurement of the hydrogen hyperfine splitting - towards antihydrogenspectroscopy

M. Diermaier,1 C. B. Jepsen,2, ∗ B. Kolbinger,1 C. Malbrunot,2, 1

O. Massiczek,1 C. Sauerzopf,1 M. C. Simon,1 J. Zmeskal,1 and E. Widmann1, †

1Stefan-Meyer-Institut fur Subatomare Physik, Osterreichische Akademie der Wissenschaften, Wien 1090, Austria2CERN, Geneve 1211, Switzerland

Antihydrogen, the lightest atom consisting purely of antimatter, is an ideal laboratory to studythe CPT symmetry by comparison to hydrogen. With respect to absolute precision, transitionswithin the ground-state hyperfine structure (GS-HFS) are most appealing by virtue of their smallenergy separation. ASACUSA proposed employing a beam of cold antihydrogen atoms in a Rabi-type experiment to determine the GS-HFS in a field-free region. Here we present a measurementof the zero-field hydrogen GS-HFS using the spectroscopy apparatus of ASACUSA’s antihydrogenexperiment. The measured value of νHF=1 420 405 748.4(3.4)(1.6) Hz with a relative precision of∆νHF/νHF=2.7 × 10−9 constitutes the most precise determination of this quantity in a beam andverifies the developed spectroscopy methods for the antihydrogen HFS experiment to the ppb level.Together with the recently presented observation of antihydrogen atoms 2.7 m downstream of theproduction region, the prerequisites for a measurement with antihydrogen are now available withinthe ASACUSA collaboration.

Investigations of the hydrogen atom have been adriving force for the discovery of more profound theo-ries [1] and contribute to the basis of physics throughtheir prominent influence on the definition of fun-damental constants [2]. Most notable from a preci-sion point of view are the recent measurement of the1S − 2S transition via two-photon spectroscopy [3]and the determination of the hyperfine splitting inhydrogen maser experiments in the early 1970s [4–9].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 onprospects of antihydrogen (H) research [10, 11]. Thestructure of the simplest antiatom, consisting of apositron bound to an antiproton is predicted to beidentical to that of hydrogen, if the combined sym-metry of charge conjugation, parity, and time rever-sal (CPT ) is conserved. Hence, antihydrogen spec-troscopy promises precise tests of the CPT symmetry,which is a cornerstone of the Standard Model of parti-cle physics. A vivid physics program is currently un-derway at the Antiproton Decelerator of CERN aim-ing at spectroscopic [12–16] and gravity tests [17, 18]along with other CPT tests like the neutrality of an-tihydrogen [19, 20] as well as measurements of thecharge-to-mass ratio [21] and magnetic moment [22]of the antiproton.

Among the spectroscopic tests of CPT , the com-parison of the GS-HFS of hydrogen and antihydrogenhas the potential to reach the highest sensitivity onan absolute energy scale [23–25]. However, the afore-

∗ Present address: Department of PhysicsPrinceton University, New Jersey 08544, USA† eberhard.widmann@oeaw.ac.at

mentioned most precise measurement of this quan-tity for hydrogen was made using a maser [8]. Sucha technique is not applicable to antimatter whichwould annihilate with the confining matter enclosure.The measurement proposed by the ASACUSA col-laboration at the Antiproton Decelerator of CERNtherefore makes use of a beam of cold antihydrogenatoms [26, 27]. In addition to avoiding wall interac-tion, the actual measurement takes place in a field-free region, ultimately allowing for higher precisioncompared to the observation of resonant quantumtransitions between the hyperfine states in trappedantihydrogen in a high-field environment [28].

Rabi-type magnetic resonance spectroscopy [29,30] applies rotating (or oscillating) magnetic fieldsto induce quantum transitions and exploits the forceof magnetic field gradients on the state-dependentmagnetic moment of atoms (or molecules) in orderto spatially separate the atoms in a beam with re-spect to their quantum states (Stern-Gerlach sepa-ration). Typically magnetic sextupole fields are em-ployed to focus atoms in low-field-seeking states (lfs)and defocus high-field-seekers (hfs). In the case ofground-state hydrogen the hyperfine structure con-sists of a lower lying singlet state with total angularmomentum quantum number F=0 (F=Sp+Se withSp and Se being proton and electron spin, respec-tively) and a triplet state F=1. As illustrated bythe Breit-Rabi diagram in Fig. 1 the triplet state de-generacy is lifted in the presence of a magnetic field.The singlet state and the triplet state with magneticquantum number MF=−1 are hfs, while the othertwo states (F=1, MF=0,1) are lfs. In the presentexperiment the σ1-transition from (F=1, MF=0)to (F=0, MF=0) has been studied [31, 32]. TheZeeman-shifted frequencies at various external mag-netic field strengths were determined for subsequent

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extraction of the zero-field value and resulted in

νHF = 1 420 405 748.4(3.4)(1.6) Hz. (1)

The numbers in brackets are the one standarddeviation (1σ) statistical and systematic uncertain-ties. Added in quadrature the total uncertainty of3.8 Hz constitutes an improvement by more thanan order of magnitude in comparison to the pre-viously achieved best precision by Rabi-type spec-troscopy of 50 Hz [33, 34]. Our result is in agree-ment within one standard deviation with the liter-ature value of νlit=1 420 405 751.768 (2) Hz, whichrelies on the more precise hydrogen maser measure-ments [4] (see discussion in §6 of [5]).

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.02 0.04 0.06 0.08 0.1

(F=1,MF=-1)

(F=1,MF=0)(F=1,MF=1)

(F=0,MF=0)

π1 σ1

H Low fieldseekers

High fieldseekers

π2

Hyp

erfin

e sp

littin

g (G

Hz)

External magnetic field (T)

Figure 1. Ground state hyperfine splitting inhydrogen. The Breit-Rabi diagram shows theenergy levels in ground-state hydrogen as afunction of the strength of an external magneticfield. The 4 hyperfine states separate into asinglet state and a triplet state, which exhibitdifferent Zeeman shifts. The states with a positiveor negative slope are named low or high fieldseekers, respectively. Three possible hyperfinetransitions between lfs and hfs are denoted byarrows, the σ1-transition occurs between thestates (F=1,MF=0) and (F=0,MF=0).

I. RESULTS

A. Experimental Setup

The main components of the experiment are asource of cooled and polarized atomic hydrogen, thehyperfine spectrometer of the H hyperfine splitting(HFS) setup (i.e. a microwave cavity and a supercon-ducting sextupole magnet), and a hydrogen detector

(cf. Fig. 2). The atomic hydrogen source maintains amicrowave driven plasma in a pyrex cylinder to disso-ciate molecular hydrogen (H2 → H + H) [35]. Hydro-gen atoms are allowed into the first vacuum cham-ber through a PTFE tubing, which is kept undercryogenic temperatures in order to cool the hydro-gen atoms and hence reduce their velocity [36]. Twotubing configurations are used in which the plasma-containing pyrex cylinder is either mounted perpen-dicular to or on axis with the beam. In the firstcase, a 90 bent tubing assures an efficient and com-plete interaction of the hydrogen atoms with the coldPTFE surface. In the latter case, a straight tubingkeeps the recombinations caused by wall interactionsdown to a minimum. The cooled atomic hydrogenbeam is directed onto a skimmer of 1 mm in diameterand reaches the second, differentially-pumped cham-ber, which houses two permanent sextupole magnetswith a pole field of ∼1.3 T at a radius of 5 mm overa mechanical length of 65 mm each [37]. In additionto providing the initial spin-polarisation, those sex-tupole magnets are moveable and feature a midwayaperture (aperture 1) to allow for the adjustable se-lection of a narrow velocity range. As the focusinglength depends on the beam velocity, only a certainvelocity component is focused onto the aperture andcan pass, while the off-axis portions of all other com-ponents are blocked. The variable distance to theaperture located at half the distance between the sex-tuple magnets (ds) therefore selects a velocity com-ponent. The resulting velocity distribution is muchnarrower than a Maxwell-Boltzmann distribution androughly of Gaussian shape. The spin-polarised andvelocity-selected hydrogen beam passes another aper-ture (aperture 2) and is then modulated by a tuningfork chopper in the next differentially-pumped sec-tion. The modulation adds time-of-flight measure-ments to the beam diagnostic tools as well as sup-pression of background originating from residual hy-drogen via lock-in amplification. Downstream of thechopper, apertures of different diameters (aperture 3)can be installed in order to produce different beamsizes at the entrance of the microwave cavity.

The H HFS spectrometer has been designed withan open diameter of 100 mm since a large acceptanceis crucial in view of small H production rates. Theamplitude of the oscillating magnetic field Bosc hasto be sufficiently uniform over the large open diam-eter in order to guarantee a trajectory-independentstate-conversion probability. This requirement isbest met by a cavity of so-called strip-line geome-try [38, 39]. Two highly transparent meshes confinethe microwaves at the entrance and exit of the state-conversion cavity, which are separated by half a wave-length of the hyperfine splitting transition (Lcav ∼λHF /2 ∼ 105.5 mm). A standing wave forms betweenthem and as a consequence Bosc is not constant along

3

Frequency standard!(rubidium clock)! MW synthesiser!

MW amplifier!

Spectrum analyser!

DAQ !Current supply!

Cou

nt ra

te!

Cry

osta

t !

Skim

mer

!

Atomic hydrogen source! Permanent!sextupole!magnets!

Magnetic!shielding!

Helmholtz!coils!

QMS!

Channeltron!

Ionisation!

Chopper!Ap

ertu

re 2!

Aper

ture

3!

Aper

ture

1!

H HFS!spectrometer!

Detector!

x!z!

y!

Strip-line cavity!

Superconducting!sextupole!magnet!

Figure 2. Atomic hydrogen beam setup. Illustration of the three main components of the Rabi-type experimentalsetup (not to scale). Green panel: the source of cold, polarised, and modulated atomic hydrogen. Orange panel:the hyperfine spectrometer of ASACUSA’s antihydrogen experiment. Blue panel: the detector. The sourceconsists of a microwave driven plasma for dissociation of H2, a cryostat for cooling the atomic hydrogen beam ina PTFE tubing, two permanent sextupole magnets for polarisation and velocity selection, and a tuning forkchopper for beam modulation. The hyperfine spectrometer consists of a state-conversion cavity of strip-linegeometry and Helmholtz coils enclosed in a cuboidal Mu-metal shielding followed by a superconducting sextupolemagnet for spin-state analysis. The detector employs a quadrupole mass spectrometer for selective mass=1 ion(H+) counting after ionisation. The count rate is acquired as a function of the driving frequency supplied to thecavity.

the beam propagation direction, causing a double-dipresonance line-shape. The origin of this structure isoutlined below and explained in detail in the Meth-ods. The cavity length and the beam velocity vH de-fine the interaction time of the hydrogen atoms withthe microwave field Tint = Lcav/vH and restrict theachievable resonance line width to ∼ T−1

int . A synthe-siser coupled to an external rubidium clock for fre-quency stabilisation produces microwaves, which arefed radially to the cavity via an antenna after ampli-fication. On the opposite side of the cavity anotherantenna is used for pick-up and monitoring of the mi-crowave power (PMW ∝ B2

osc) using a spectrum anal-yser. Helmholtz coils are mounted onto the cavity togenerate a homogeneous external magnetostatic fieldBstat, parallel to Bosc, and of several Gauss in mag-nitude at the interaction region for fine control of theZeeman splitting. A current source with a relativestability of 20 ppm supplies the Helmholtz coils’ cur-rent IHC, which is independently monitored by anamperemeter. IHC is directly proportional to Bstatand turned out to be a better proxy for the magneticfield inside the cavity than a dedicated external mag-netic field measurement. The microwave cavity and

the Helmholtz coils are surrounded by a two-layercuboidal Mu-metal shielding to block the Earth’smagnetic field as well as the fringe field of the closelysucceeding superconducting sextupole magnet. Ow-ing to the pole strength of up to 3.5 T, this magnetgenerates sizeable magnetic field gradients despite thelarge open diameter of 100 mm. The integrated gra-dient amounts to 150 T/m and ensures refocusing of50 K lfs atoms within a distance of ∼1 m.

The detection of hydrogen suffers from a largebackground rate and small efficiencies. A crossed-beam quadrupole mass spectrometer (QMS) with a3 mm opening ionises beam atoms and residual gasby electron impact and selectively guides protons to achanneltron for efficient single mass=1 ion counting.The QMS can be moved two-dimensionally in theplane perpendicular to the beam for optimising countrates and investigating beam profiles. Ultrahigh vac-uum conditions (p ≤ 5×10−10 mbar), achieved bycombining two-stage turbo-molecular pumping andnon-evaporable-getter pumps, lead to count rates oftens of kHz at the QMS for a typical H2-flowrate of1.8×1017 s−1.

4

B. Measurement procedure

The dissociation plasma was operated under sta-ble standard conditions. Before starting frequencyscans the microwave power PMW supplied to the cav-ity was adjusted to yield the largest state-conversionprobability by observing a Rabi oscillation. A singlemeasurement cycle was obtained by scanning the fre-quency once in a random sequence across the desiredrange. Typically, this included 39 frequency pointsdistributed over ∼40 kHz. At each frequency pointthe channeltron events of the QMS were summed sev-eral times for typical intervals of 5-60 s from whichan average count rate was retrieved. Such cyclesover the frequency range were repeated on average5 times with changing random sequences to result ina complete scan at a given IHC. This was repeated atdifferent values and polarity of IHC to yield a set ofscans suitable for determination of the field-free hy-perfine splitting. The number of IHC values per setranged from 6 to 16. In total 10 such sets have beenrecorded, which differ in various of the experimentalsettings and arrangements (cf. table II).

C. Raw data corrections

Initially a fit as described below was applied to thedetected count rates. Two systematic effects wereidentified in the residuals and corrected for. Thefirst correction compensates slow time drifts. Thesecond correction concerns a type of memory-effect,which became evident in an increased likelihood ofobserving positive or negative residuals if the previ-ous data point was taken at higher or lower countrate, respectively. This indicated, that the settlingof the hydrogen rate in the detection chamber fol-lowing a change of the excitation frequency had anon-negligible time constant when compared to mea-surement time at each frequency step. These twoeffects were corrected for at the raw data level andled to an improvement of the fit quality without af-fecting the extracted νHF values. The application of arandom sequence of frequencies in the cycles seemedto suppress systematic impacts of the drift and thememory-effect below the statistical sensitivity.

D. Analysis

The central frequency νc was extracted from ev-ery cycle by a fit to the spectrum as illustrated inFig. 3a, where the excitation frequency ν is given asthe difference to νlit. The double-dip line shape orig-inates from the sinusoidal dependence of Bosc alongthe beam axis, which follows half a cosine period. Atthe actual transition frequency the highest count rate

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Excitation frequency i-ilit (kHz)

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oils

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(A)

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A)

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A)

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A)

1 2 3 4 5 6 7 8 9

10

-0.02 0.00 0.02 0.04 0.06

Set n

umbe

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Hyperfine transition frequency iHF-ilit (kHz)

set iHFweighted mean

total 1m error

1 2 3 4 5 6 7 8 9

10

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Set n

umbe

r

Hyperfine transition frequency iHF-ilit (kHz)

1 2 3 4 5 6 7 8 9

10

-0.02 0.00 0.02 0.04 0.06

Set n

umbe

r

Hyperfine transition frequency iHF-ilit (kHz)

Figure 3. Resonance spectrum and zero-field valueextraction. (a) Data of one cycle of set 8 at aHelmholtz coils’ current IHC=400 mA fitted withthe resonance curve (FR, full blue line, seeequation (11) in Methods) to extract the centralfrequency νc, which is Zeeman-shifted tovalues > νHF. A dashed-dotted black line at 0 isdrawn through all the plots to represent νlit. (b)State-conversion probabilities (F) as obtainedfrom the fit (FR) of two other cycles of the sameset, but at different settings of IHC (dotted blueline 180 mA, dashed blue line 550 mA, dataomitted for clarity). (c) νc of all 80 cycles of set8 (16 different values of IHC, 5 cycles each)plotted against IHC for extraction of the zero-fieldhyperfine splitting νHF using the Breit-Rabi fitfunction ν′σ (red line) of equation (4). The insetis a zoom into the group of 5 cycles atIHC=400 mA illustrating the typical size of thefrequency and current standard deviations of eachdata point. (d) The resulting νHF as deviationfrom νlit for the 10 sets (red error bars) and theirweighted mean value (dashed red line) with the 1σtotal uncertainty as grey-shaded area.

5

between the two dips is observed. The theoretical lineshape for a mono-energetic beam is well understoodand accurately described within the framework of thetwo-level system with the interaction Hamiltonian

Hint = −µµµ ·Bosc(t), (2)

where µµµ is the magnetic moment operator as de-fined in equation (7). The time dependence of themagnetic field includes a cos(πt/Tint)-term in addi-tion to the microwave oscillations. The resultingequations were solved numerically to obtain the state-conversion probability as a function of the frequencyν and amplitude Bosc of the driving field for a mono-energetic beam. A realistic fit function F for themeasured state-conversion probabilities was obtainedby convolution of the shape for mono-energetic beamswith a velocity distribution as described in the Meth-ods. Consequently, the fit function could extract thephysical parameters Bosc, the mean velocity of thepolarised atomic hydrogen beam vH , and the widthof the velocity distribution σv in addition to νc of thetransition. Two further fit parameters of less relevantphysical content scaled the state-conversion probabil-ity to the count rate and correspond to the count ratebaseline R0 and the count rate drop ∆R for completestate conversion. In the final analysis only νc wasextracted from every cycle individually. For Bosc arelation to the monitored microwave power was es-tablished based on the complete available data. Thisenabled individual fixation of this parameter for everyset and avoided non-converging fits due to a strongcorrelation of Bosc with ∆R. For vH and σv a com-mon fit value for a complete set was used, as all set-tings of direct impact on the beam velocity remainedunchanged during data collection of a set.

As illustrated in Fig. 3a the line shape thus ob-tained resulted in good fits to the observed countrates at all IHC settings with reduced χ2 values closeto unity as summarized in table II. The reliability ofthe fit function was important since νc could be ex-tracted with typical statistical uncertainties on theorder of tens of Hz while the width of the double-dipstructure is on the order of tens of kHz. In Fig. 3c, theextracted νc value of each cycle of set 8 are plottedagainst the Helmholtz coils’ current IHC at which itwas recorded. The Zeeman-shifted frequency of theσ1-transition νσ(Bstat) has only a second order de-pendence on the static external magnetic field Bstatas apparent from the Breit-Rabi diagram (Fig. 1) anddescribed by the Breit-Rabi formula [40]

νσ(Bstat) =√ν2

HF +(µ+

h

)2B2

stat,

µ+ = |ge|µB + gpµN ,

(3)

with µB=5.788 381 8012×10−5 eVT−1 andµN=3.152 451 2550×10−8 eVT−1 being re-spectively the Bohr and nuclear magneton,ge=-2.002 319 304 361 82 and gp=5.585 694 702 [2]being respectively the g-factors of the electron andproton, and h=2π~ the Planck constant. In order toextract the zero-field hyperfine transition frequencyνHF a fit function ν′σ was required, that used IHC asa variable. A factor k converting IHC to a magneticfield and a residual field Bres at IHC = 0 addedtwo further fit parameters and established a linearrelation to Bstat, which enters the Breit-Rabi formula

ν′σ(IHC; νHF,k,Bres) =

=√ν2

HF +(µ+

h

)2(kIHC +Bres)2

.(4)

The notation for the fit function separates the vari-able from the parameters by a semicolon. The zero-field values νHF as obtained via this Breit-Rabi fit areplotted in Fig. 3d as the deviation from νlit.

E. Systematic tests

The following experimental arrangements and con-ditions have undergone changes for the 10 sets (sum-marised in the top part of table II). The beam ve-locity varied due to different settings of ds and thetemperature of the PTFE tubing. The first threesets operated with the straight PTFE tubing thenthe bent tubing was used. The need for an improvedmonitoring of IHC and the advantage of a faster dataacquisition scheme based on the total count rate in-stead of the lock-in amplifier signal became evidentin a preliminary evaluation of the first three sets andmotivated the additional changes at that stage. Twoopening diameters for aperture 3, resulting in differ-ent beam sizes at the entrance of the cavity, werealso investigated. This is of special interest as aneven larger beam diameter is expected for the H HFSspectroscopy. Additionally, the last 4 sets were per-formed with a second cavity of the same but slightlyupgraded design. Three aspects were only changedfor individual sets. For set 7 only one instead of twolayers of magnetic shielding were used, for set 3 thesuperconducting sextupole worked with a larger mag-netic 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 νHF of the 10 sets by firstlyfitting all cycles in a set using the fit function (10)and secondly the Breit-Rabi fit (4) are presented inFig. 3d. Additionally the average reduced χ2 of allfits to cycles within a set and the reduced χ2 of theBreit-Rabi fit are given in table II. On the level of

6

the achieved statistical precision no significant de-pendence of the 10 results on any of the changedexperimental conditions could be found. This jus-tified to combine the 10 individual results into oneweighted mean value. Our final result deviates fromthe literature value by νHF– νlit=–3.4 Hz with a to-tal uncertainty of σtot=3.8 Hz, which corresponds toa relative precision of 2.7 ppb. The mean value isshown in Fig. 3d as the dashed red line and the total1σ uncertainties as the grey shaded area.

The fit parameters Bosc, vH and σv, which werefixed to a common average value for each set, werevaried in order to assess the potential systematic un-certainties originating from the fit procedure. Thecomplete analysis was repeated 6 times with set-ting each of the three parameters individually to itslower and upper 1σ boundary. The observed shiftsof νHF for each parameter are listed in table I. How-ever, those three values added in quadrature yielded0.06 Hz and present a negligible systematic uncer-tainty. The rubidium clock, which served as fre-quency standard, supplied a 10 MHz reference sig-nal to the microwave synthesiser. A calibration wasperformed and revealed a shift of 11.4 mHz or equiv-alently 1.14 ppb. This corresponds to 1.6 Hz for νHF

and is conservatively used as a 1σ systematic uncer-tainty. Table I summarises the error budget.

Table I. Error budget

contribution 1σ st.dev. (Hz)systematic error

frequency standard 1.62common fit parameters

vH 0.05σv 0.03Bosc 0.02

systematic error total 1.62statistical error 3.43

total error 3.79

II. METHODS

A. Resonance line shape

The σ1-transition in ground-state hydrogen isdriven by an external microwave field, which is gen-erated in a strip-line cavity and takes the form

Bosc(t) = BosceB cos(ωt) cos(ωcavt),

ωcav = π

Tint= πvHLcav

,

0 < t < Tint,

(5)

where eB is the unit vector pointing in the direc-tion of the magnetic field (z-axis in the frame of theatoms, x-axis in the coordinate system of the exper-iment) and ν = ω/2π is the applied microwave fre-quency. The term cos(ωcavt) describes the changingamplitude of the magnetic field in the cavity alongthe beam propagation direction. Tint is the interac-tion time, which in turn follows from the hydrogenbeam velocity vH and the length of the cavity Lcav.

The small external magnetic field is aligned parallelto the oscillating magnetic field, which only for theσ1-transition leads to non-vanishing matrix elements.In addition the Zeeman shift separates the ground-state hydrogen sub levels by more than the observedresonance width. Therefore, the transition dynamicsis well described within the framework of the two-level system

|φ〉 = c1(t)|φ1〉+ c2(t)|φ2〉,

|φ1〉 = |F = 0,MF = 0〉 =√

1/2 (| ↑e↓p〉 − | ↓e↑p〉) ,

|φ2〉 = |F = 1,MF = 0〉 =√

1/2 (| ↑e↓p〉+ | ↓e↑p〉) ,Hatom|φi〉 = Ei|φi〉,|c1(t)|2 + |c2(t)|2 = 1.

(6)

To obtain the time evolution of this system un-der the influence of the oscillating magnetic field theHamiltonian needs to be extended by the interactionHint = −µµµ ·Bosc(t) with

µµµ = −|ge|µB1~

Se + gpµN1~

Sp, (7)

for hydrogen. Here, Si are the spin operators act-ing on the electron or proton spinor as indicated bythe superscript. An analytical solution can be foundfor conventional Rabi experiments, where the oscillat-ing (or rotating) magnetic field has a constant ampli-tude Bosc and doesn’t include the term cos(ωcavt). Ifthe system is initially prepared purely in state |φ1〉,then the conversion probability |c2|2 of finding it aftera given interaction time Tint in the second state |φ2〉depends on the strength of Bosc and the detuningΩD = ω − ω12 with ~ω12 = E2 − E1:

|c2|2 = Ω2R

Ω2D + Ω2

R

sin2(

12

√Ω2

D + Ω2R Tint

), (8)

where ΩR is the Rabi frequency, which is propor-tional to the amplitude of the oscillating magneticfield. The relation for the σ1-transition is

ΩR = (|ge|µB + gpµN )︸ ︷︷ ︸µ+

Bosc

2~ . (9)

7

Table II. Parameters of the data sets. Comparison of the 10 sets. The 4 blocks of rows summarize (i)experimental conditions, (ii) statistics of the data acquisition, (iii) average fit parameter of cycles and theaverage reduced chi-squares from applying fit-formula (11), and finally (iv) the fit parameters and thecorresponding reduced chi-square from applying the Breit-Rabi fit (4).

set 1 2 3 4 5 6 7 8 9 10

PTFE tubing straight straight straight 90 deg. 90 deg. 90 deg. 90 deg. 90 deg. 90 deg. 90 deg.cryostat tem-perature (K)

23 16 100 50 50 50 50 50 50 50

ds (mm) 115 35 91 21 21 16 16 16 16 115cavity # 1 # 1 # 1 # 1 # 1 # 1 # 2 # 2 # 2 # 2precise moni-toring of IHC

no no no yes yes yes yes yes yes yes

supercond.sextupole (A)

350 350 400 350 350 350 350 350 350 350

beam diame-ter (mm)

8 8 8 8 8 8 8 8 22 22

shieldinglayers

2 2 2 2 2 2 1 2 2 2

IHC polarity ± ± ± + ± ± ± ± ± ±number ofscans

8 6 6 10 12 12 16 16 12 12

number of cy-cles

23 46 26 50 60 60 80 80 60 60

frequencydata points

41 21 26 39 39 39 39 39 39 39

acqu. time /data point (s)

60 40 40 5 5 5 5 5 5 5

v(m/s)

1066(1)

962(2)

1152(2)

888(2)

857(3)

883(2)

933(2)

922(1)

1049(1)

1131(1)

σv

(m/s)152(2)

145(3)

156(2)

160(2)

184(2)

139(2)

124(2)

129(2)

183(1)

149(1)

Bosc

(10−7 T)6.86(0.01)

6.49(0.01)

8.14(0.01)

5.73(0.01)

5.81(0.01)

5.78(0.01)

6.70(0.03)

6.28(0.05)

6.54(0.03)

6.93(0.03)

R0(Hz)

27088(232)

24420(576)

26517(234)

26998(458)

20889(237)

26118(84)

23100(194)

24825(225)

56390(2806)

31584(1724)

∆R(Hz)

891(30)

476(43)

1112(29)

1471(71)

795(47)

1484(50)

1126(42)

1401(51)

4499(907)

3284(154)

av. χ2/n.d.f.of res. curves

2.6(0.7)

1.6(0.5)

1.9(0.5)

1.2(0.3)

1.0(0.3)

1.2(0.2)

1.1(0.3)

1.1(0.2)

1.9(0.4)

1.9(0.4)

Bres

(10−7 T)4.0(1.0)

3.9(2.6)

2.3(2.0)

11.9(19.1)

5.4(1.9)

5.7(1.0)

3.1(1.3)

3.5(1.1)

2.5(0.5)

2.7(0.5)

k(10−5 T/A)

45.83(0.02)

45.47(0.16)

45.70(0.14)

45.87(0.37)

45.76(0.06)

45.80(0.03)

45.85(0.04)

45.90(0.03)

45.90(0.02)

45.89(0.02)

χ2/n.d.f. ofBreit-Rabi fit

17.5/21 38.6/44 41.3/24 70.0/47 65.3/57 48.5/57 101.0/77 83.2/77 75.0/57 55.6/57

νHF - νlit (Hz) −12.0(10.6)

22.6(22.2)

20.8(19.4)

19.7(46.7)

5.7(23.9)

−9.2(12.8)

7.1(11.4)

−2.9(9.2)

−8.8(6.9)

−3.5(6.4)

Including the term cos(ωcavt) requires numericalmethods to determine the state-conversion probabil-ity. Figure 4 shows a comparison of |c2|2 as a func-tion of the detuning ΩD and the driving strengthBosc of conventional Rabi spectroscopy and the strip-line cavity designed for the antihydrogen experiment.The latter case features the distinct double-dip struc-ture with vanishing effects at the actual transition

frequency. For a given interaction time Tint the bestprecision is achieved with the first full state conver-sion in both situations. For the conventional casethis corresponds to a so-called π-pulse indicating thatthe condition ΩR · Tint = π is satisfied or alterna-tively Bosc = hµ−1

+ T−1int . The double-dip resonance

reaches the first full state conversion when apply-ing a somewhat stronger oscillating magnetic field

8

Conventional Rabi spectroscopy

2

4

6

8

10

12

B osc

(h µ

m-1

Tin

t-1)

Strip-line cavity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Stat

e-co

nver

sion

pro

babi

lity

0.00.20.40.60.81.0

-6 -4 -2 0 2 4 6

Stat

e-co

nv. p

rob.

Detuning 1D (/ Tint-1)

-6 -4 -2 0 2 4 6Detuning 1D (/ Tint

-1)

Figure 4. State-conversion probability maps for the detuning and driving strength. Comparison of thestate-conversion probabilities as a function of the detuning ΩD (in units of πT−1

int ) and the amplitude of theoscillating magnetic field Bosc (in units of hµ−1

+ T−1int ) for the case of conventional Rabi spectroscopy (left) and

when using a strip-line cavity (right) to drive the transition. Both cases refer to a mono-energetic beam, whichtranslates to a fixed interaction time Tint. The dashed horizontal line indicates the required driving strength toreach the first complete state conversion. The plots below are projections of the state-conversion probabilities atthe dashed lines and show the ideal (i.e. mono-energetic) line shapes observed as count rate drops in the Rabiexperiments.

Bosc ∼ 1.86× hµ−1+ T−1

int .From a 2D-map as shown in Fig. 4 fit func-

tions of the state-conversion probabilities for a mono-energetic beam can be derived with νc, the strengthof Bosc, and the hydrogen beam velocity vH as fitparameters. This was realised by constructing a 2D-spline interpolation S(ν; νc,Bosc,vH) to the numeri-cally generated state-conversion probabilities at dis-crete points. A more realistic resonance line shapeis then obtained by including the effect of the veloc-ity distribution of the hydrogen beam, which trans-lates to a not sharply defined interaction time Tint.Note that both axis of the 2D-maps are normalisedto T−1

int . Therefore, on an absolute scale for ΩD andBosc a change of Tint is equivalent to a 2D-zoomingof the state-conversion probability map. The roughlyGaussian velocity distribution of the hydrogen beamafter passage of the polarising and velocity-selectingpermanent sextupole magnets is approximated by bi-nomial coefficients for a discrete numerical realisationof the convolution

F(ν; νc,Bosc,vH ,σv) =

= 2−NN∑n=0

(N

n

)S(ν; νc, Bosc, v(N,n)),

v(N,n) = vH + (n−N/2)dv,dv ∼ 2σvN−1/2.

(10)

The result of a convolution with such a velocitydistribution is illustrated in Fig. 5 and compared toa measured map. The present analysis used N = 6because choices of N > 6 did not change nor im-prove the fit results. For completeness two more fitparameters were needed. In order to scale the state-conversion probability, which is a number between0 and 1, to the observed count rates, a count ratebaseline R0 and a count rate drop for complete stateconversion ∆R were introduced

FR(ν; νc,Bosc,vH ,σv,R0,∆R) == R0 −∆R · F(ν; νc,Bosc,vH ,σv),

(11)

which are albeit not relevant for the obtained re-sults.

III. ACKNOWLEDGEMENT

We want to thank H. Kundsen, H.-P. E. Kris-tiansen, F. Caspers, T. Kroyer, S. Federmann,P. Caradonna, M. Wolf, M. Heil, F. Pitters,C. Klaushofer, S. Friedreich, and B. Wunschek fortheir contributions. We acknowledge technical sup-port by the CERN Cryolab and Instrumentationgroup TE-CRG-CI as well as the CERN Magnet Nor-mal Conducting group TE-MSC-MNC.

9

Theory (numerical)

0.5

1

1.5

2

2.5

3

3.5

B osc

(µT)

Measurement

0

0.2

0.4

0.6

0.8

1

Stat

e-co

nver

sion

pro

babi

lity

0.00.20.40.60.81.0

-30 -20 -10 0 10 20 30

Stat

e-co

nv. p

rob.

Detuning 1D/2/ (kHz)

-30 -20 -10 0 10 20 30Detuning 1D/2/ (kHz)

Figure 5. State-conversion probability maps convoluted with a velocity distribution. Comparison of theoreticaland measured state-conversion probabilities as a function of the detuning ΩD/2π in units of kHz and theamplitude of the oscillating magnetic field Bosc in units of µT. The theoretical map includes the effect of aGaussian-like velocity distribution with vH=1060 m/s and σv=95 m/s. The measurement was taken setting alarge distance between the permanent sextupole magnets of ds=115 mm as it has been used for the sets 1 and 10.The blue and red dashed horizontal lines indicate the driving strengths, where the first and second full stateconversion would be reached in the case of a mono energetic beam. The plots below are projections of thestate-conversion probabilities at the dashed lines showing good agreement between theory and measurement.Frequency spectra across the narrow double-dips at the first state conversion yield the highest precision.

This work has been supported by the Euro-pean Research Council under European Union’s Sev-enth Framework Programme (FP7/2007-2013)/ ERCGrant agreement (291242), the Austrian Ministry ofScience and Research, the Austrian Science Fund(FWF): W1252-N27.

IV. AUTHOR CONTRIBUTIONS

M.D. prepared and performed the experiment, car-ried out the data analysis, and wrote parts of the ini-

tial manuscript. C.B.J. performed the experimentand 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 andcarried out parts of the data analysis, and wrote themanuscript. O.M. and J.Z. prepared the experiment.E.W. proposed and prepared the experiment, guidedthe data analysis, and wrote the manuscript.

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