An experimental limit on the charge of antihydrogen

Abstract

The properties of antihydrogen are expected to be identical to those of hydrogen, and any differences would constitute a profound challenge to the fundamental theories of physics. The most commonly discussed antiatom-based tests of these theories are searches for antihydrogen-hydrogen spectral differences (tests of CPT (charge-parity-time) invariance) or gravitational differences (tests of the weak equivalence principle). Here we, the ALPHA Collaboration, report a different and somewhat unusual test of CPT and of quantum anomaly cancellation. A retrospective analysis of the influence of electric fields on antihydrogen atoms released from the ALPHA trap finds a mean axial deflection of 4.1±3.4 mm for an average axial electric field of 0.51 V mm^-1 . Combined with extensive numerical modelling, this measurement leads to a bound on the charge Qe of antihydrogen of Q=(-1.3±1.1±0.4) × 10^-8 . Here, e is the unit charge, and the errors are from statistics and systematic effects.

ARTICLE

Received 12 Jan 2014 | Accepted 24 Apr 2014 | Published 3 Jun 2014

DOI: 10.1038/ncomms4955 OPEN

An experimental limit on the charge of antihydrogen

C. Amole1, M.D. Ashkezari2, M. Baquero-Ruiz3, W. Bertsche4,5, E. Butler6,7, A. Capra1, C.L. Cesar8, M. Charlton9,S. Eriksson9, J. Fajans3,10, T. Friesen11, M.C. Fujiwara12, D.R. Gill12, A. Gutierrez13, J.S. Hangst7,14, W.N. Hardy13,15, M.E. Hayden2, C.A. Isaac9, S. Jonsell16, L. Kurchaninov12, A. Little3, N. Madsen9, J.T.K. McKenna17, S. Menary1, S.C. Napoli9, P. Nolan17, K. Olchanski12, A. Olin12, A. Povilus3, P. Pusa17, C.. Rasmussen14, F. Robicheaux18,E. Sarid19, D.M. Silveira8, C. So3, T.D. Tharp3, R.I. Thompson11, D.P. van der Werf9, Z. Vendeiro3, J.S. Wurtele3,10, A.I. Zhmoginov3,10 & A.E. Charman3

The properties of antihydrogen are expected to be identical to those of hydrogen, and any differences would constitute a profound challenge to the fundamental theories of physics. The most commonly discussed antiatom-based tests of these theories are searches for anti-hydrogen-hydrogen spectral differences (tests of CPT (charge-parity-time) invariance) or gravitational differences (tests of the weak equivalence principle). Here we, the ALPHA Collaboration, report a different and somewhat unusual test of CPT and of quantum anomaly cancellation. A retrospective analysis of the inuence of electric elds on antihydrogen atoms released from the ALPHA trap nds a mean axial deection of 4.13.4 mm for an average axial electric eld of 0.51 Vmm 1. Combined with extensive numerical modelling, this measurement leads to a bound on the charge Qe of antihydrogen of Q ( 1.31.10.4)

10 8. Here, e is the unit charge, and the errors are from statistics and systematic effects.

1 Department of Physics and Astronomy, York University, Toronto, Ontario, Canada M3J 1P3. 2 Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6. 3 Department of Physics, University of California at Berkeley, Berkeley, California 94720-7300, USA. 4 School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK. 5 The Cockcroft Institute, Daresbury Laboratory, Warrington WA4 4AD, UK. 6 Centre for Cold Matter, Imperial College, London SW7 2BW, UK. 7 Physics Department, CERN, CH-1211 Geneva 23, Switzerland. 8 Instituto de Fsica, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-972, Brazil. 9 Department of Physics, College of Science, Swansea University, Swansea SA2 8PP, UK.

10 Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA. 11 Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada T2N 1N4. 12 TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A3. 13 Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1. 14 Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark.

15 Canadian Institute of Advanced Research, Toronto, Ontario, Canada M5G 1ZA. 16 Department of Physics, Stockholm University, SE-10691 Stockholm, Sweden. 17 Department of Physics, University of Liverpool, Liverpool L69 7ZE, UK. 18 Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA. 19 Department of Physics, NRCN-Nuclear Research Center Negev, Beer Sheva IL-84190, Israel. Correspondence and requests for materials should be addressed to J.F. (email: mailto:[email protected]

Web End [email protected] ) or to J.S.H. (email: mailto:[email protected]

Web End [email protected] ) or to J.S.W. (email: mailto:[email protected]

Web End [email protected] ).

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The recent creation13 and trapping47 of antihydrogen has opened new opportunities to explore natures fundamental symmetries. The two most often proposed symmetry tests

explore CPT (charge-parity-time) invariance by comparing the spectra of antihydrogen with that of hydrogen and probe the weak equivalence principle by measuring the gravitational behaviour of antihydrogen. Recently, the ALPHA collaboration reported initial experimental results in both these areas8,9; future experiments expect to obtain much more precise results915.

Here, we report on a retrospective search for an electric eld-induced deection of antihydrogen atoms as they are released from the ALPHA trap. This search tests for the charge neutrality of these antiatoms, as the antiatoms would only be deected if they possessed a charge. Theoretically, charge neutrality of antimatter atoms, as well as of matter atoms, is expected from the condition for quantum anomaly cancellation, which is required for theoretical consistency in quantum eld theory16. Furthermore, experiments show that normal matter atoms and molecules are charge neutral17 to about 10 21e for diverse species such as He, H2 and SF6, where e is the elementary charge. (There do not appear to be precision measurements for H.) As CPT invariance requires that antihydrogen and hydrogen have opposite charge, it predicts that antimatter atoms are charge neutral to a similar level. However, the methods17 used to test for charge neutrality in normal-matter studies are unsuitable for antihydrogen as they require macroscopic quantities of atoms or molecules; to date, only B500 antihydrogen atoms have been trapped and detected, and there are no prospects for trapping macroscopic quantities. Thus, an electric eld-based deection measurement of antihydrogens charge constitutes a novel test of the fundamental consistencies of quantum eld theory and of CPT invariance.

We observe a mean axial deection of 4.13.4 mm for antiatoms subjected to an average axial electric eld of0.51 V mm 1. By using extensive numeric simulations to estimate the mean deection of putatively charged antihydrogen atoms by these electric elds, as well as those deections that might be caused by systematic effects, we deduce a bound on the charge of antihydrogen; we nd that antihydrogen is charge neutral to Q ( 1.31.10.4) 10 8 (1s condence level),

where Qe is the antihydrogen charge, and where the rst error is from statistics and the second from known systematic effects. (We note that at the 90% condence level, the statistical error is 1.8 10 8 and covers zero.) This bound on the charge of

antihydrogen is B106 times lower than the best previous experimental bound18.

ResultsApparatus and procedures. ALPHA traps antihydrogen atoms by producing and capturing them in a minimum-B trap19. The trap connes those antiatoms whose magnetic moment l H is aligned such that they are attracted to the minimum of the trap magnetic eld B, and whose kinetic energy is below the trap well depth, m

H B

j jWall B

j jCentre

Mirror coils

V(V)B(T)

2.0

1.0

0

100

Count

(A.U.)

0

150 50 0 50 100 150

100

Axial position z (mm)

. In ALPHA (see Fig. 1a), this magnetic minimum is created by an octupole magnet, which produces transverse elds of magnitude 1.54 T at the trap wall (RWall 22.3 mm), and two mirror coils, which produce axial

elds of 1 T at their centres. The mirror coil centres are located at distances z 137 mm from the trap centre at z 0 (see

Fig. 1b). These elds are superimposed on a uniform axial eld of 1 T produced by an external solenoid20,21. Taken together, these elds result in an antiatom trap depth of 540 mK, where kelvin is used as an energy unit.

The general methods by which antiatoms are produced from antiprotons and positrons and then captured in our trap are

described in refs 46,22,23. In this Article, we concentrate only on the last phase of the experiments, during which antiatoms are released from the minimum-B trap by turning off the octupole and mirror elds after at least 0.4 s of connement. The magnet turnoff time constants of B9 ms result in all antiatoms escaping within 30 ms. The escaping antiatoms are then detected when they annihilate on the trap wall; a silicon-based annihilation vertex imaging detector24 records the times and locations of the pions that result from the antiproton components of these annihilations. The locations have an axial uncertainty FWHM of5.6 mm.

An electric eld, (see Fig. 1c for the corresponding potential), is

present in the trap when the antiatoms are released. This electric eld would discriminate between antihydrogen atoms and any antiprotons inadvertently left in the trap, as the negatively charged antiprotons would be swept out of the axial ends of the trap4,5,25. As expected25, we did not nd evidence for any such antiprotons; the application of these electric elds was precautionary. It is these electric elds, however, that allow us to place a bound on the charge of antihydrogen. Two electric eld congurations were employed: a Bias-Right (R) conguration, which would sweep antiprotons to the right (positive axial position z), and a Bias-Left (L) conguration, which would sweep antiprotons to the left (negative z).

Observations. In the B1,300 trapping trials analysed here, 386 of the events recorded by our detector were identied as antiatoms that passed all our selection criteria; these selection cuts were determined in blind analysis, that is, without reference to the

Octupole Electrodes

Figure 1 | Experimental summary. (a) A schematic of the antihydrogen production and trapping region of the ALPHA apparatus, showing the cryogenically cooled cylindrical PenningMalmberg trap electrodes, and the mirror and octupole magnet coils. Our positron source (not shown) is towards the right, and the antiproton decelerator (not shown) is towards the left. (b) The on-axis magnetic eld B as a function of z. (c) The on-axis electrostatic potentials V used to establish the Bias-Right (red dashed line) and Bias-Left (blue solid line) congurations. (d) Normalized histograms of the experimental z positions of the annihilations in the Bias-Right (red dashed line) and Bias-Left (blue solid line) congurations. The error bars show the expected deviation of the distribution based on the number of observed antiatoms in each bin.

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experimental data, and are described in the Methods section below. Of these, 241 were detected in the Bias-Right conguration, and 145 were detected in the Bias-Left conguration. Histograms of their z positions are shown in Fig. 1d, and their average positions are /zSR 7.94.2 mm, and /zSL 0.25.3 mm. (The errors associated with these quantities are standard errors of the mean.) These yield sum and difference positions of /zS 0.5[/zSR /zSL] 3.93.4 mm and

/zSD 0.5[/zSR /zSL] 4.13.4 mm, where the errors are

found by taking half the quadrature sum of the errors of /zSR and /zSL. The value of the sum /zS is determined by systematic errors discussed in the Methods. A nonzero value for the difference /zSD, beyond deviations that could be caused be sampling errors, could only result from a nonzero antiatom charge as there is no other property of the orbital dynamics of the antiatoms that depends on an electric eld (aside from a negligible contribution from the polarization of ground state anti-hydrogen: see Methods).

Charge calculation. Including the presence of a putative, nonzero charge Q, the on-axis potential energy of an antiatom in the trap is given by

U z

m

Q = 0

Normalized counts (a.u.)

Q = +4108bias-right Q = +4108 bias-left

0 150 100 50 0 50 100 150Axial position z (mm)

Figure 2 | Simulated annihilation z-distributions. Three simulated annihilation z-distributions, for antiatoms with Q 0 (black solid line) and

Q 4 10 8 under Bias-Right (red dashed line) and Bias-Left (blue

dotted line) conditions. The vertical dashed lines indicate the locations of the cuts at z 136 mm.

QeEkB z; 1

where U(z) is given using kelvin as an energy unit, m

H

0:67 K T 1 is the normalized antihydrogen magnetic moment, veried experimentally in ref. 8 to the accuracy necessary here and kB is the Boltzmann constant. We approximate the electric eld inside the trap as the constant E. When Q 0, this potential

has a minimum at z 0. Consequently, as B is reduced during the

magnet shutdown, the antihydrogen annihilations will be centred around z 0. A nonzero Q will shift the potential minimum, and,

hence, the annihilation centre.

From equation (1), we can solve for the approximate shift in the minimum of the potential U(z) that results from a nonzero Q, or, alternately, the Q required to produce a given size shift /zSD, given the average Bias-Right and Bias-Left electric elds, ER and EL, evaluated near the trap centre:

Q

4m

HbkB e ER EL

Simulation z(mm)

15

5

0

5

15

10

z R

z L

z

10

4 2 0 2 4

Q (108)

Figure 3 | Simulated dependence of the axial shifts on Q. The average annihilation locations /zS in the Bias-Right (red diamonds) and Bias-Left (blue squares) congurations, as well as the resulting /zSD (black dots), as found in the simulations with detector efciency and resolution corrections.

The statistical error of 0.1 mm in the calculated values of /zSR and /zSL, and 0.07 mm for /zSD, is too small to be shown with clarity. The black solid line is the least-squares linear t that best describes the variation of /zSD with Q. The t is constrained to pass through /zSD 0

at Q 0, consistent with the simulations to within statistical uncertainty

and with our expectation that the bias electric elds have no effect on particles for which Q 0.

h iD: 2

Here, we have approximated the quasi-parabolic central magnetic eld at the start of the magnet shutdown as B(z)

B0 bz2. For our experimental parameters, b 1.6 10 5

T mm 2, ER 0.50 V mm 1 and EL 0.52 V mm 1; thus,

equation (2) predicts Q 3.7 10 9(mm 1)/zSD. Using

the measured /zSD yields the approximate Q ( 1.51.3) 10 8, where the uncertainty comes from the statistical uncertainty in /zSD.

While equation (2) allows us to estimate Q, the antiatom orbits in the three-dimensional, time-dependent, elds E and B are sufciently complicated that numerical simulations13,25 are required to determine the true relation between Q and /zSD. Simulations were performed for the Bias-Right and

Bias-Left electric elds of Fig. 1c. Typical results of the simulations are shown in Fig. 2 and are further discussed in the Methods. A compilation of the results of many simulations, shown in Fig. 3, yields an approximately linear relation, Q s/zSD, between Q and /zSD, where the sensitivity

s (3.310.04) 10 9 mm 1. From this we nd Q

( 1.31.1) 10 8; the error here is one s.d. and derives

from the statistical uncertainty in /zSD, and does not yet include systematic effects or the 0.04 10 9 mm 1 error in the

sensitivity.

Principle data set. The annihilation location data used here were collected during experiments primarily intended to trap4 and hold5 antihydrogen atoms, and to eventually measure8 the microwave spectral properties of these atoms. Thus, this analysis is a retrospective analysis of data taken for other purposes, and the data were not collected in the optimal manner for this analysis. In particular, the Bias-Right and Bias-Left conditions were not well alternated, and the antiatom synthesis sequences were varied and improved. All of the Bias-Left data were collected in 2010, and the Bias-Right data were split between 2010 (27 events) and 2011 (214 events). As our analysis relies on subtracting the means of the Bias-Right and Bias-Left data, it is critical that the experiment has not drifted in any way that could affect these means. Because of the retrospective nature of the

z

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analysis, we were limited to employing numerical simulations to analyse the potential drift mechanisms. From an extensive set of these simulations, described in the Methods, we conclude that the known systematic errors contribute an uncertainty to Q of 0.4 10 8.

DiscussionBy searching for a deection of antiatoms in an electric eld, we have determined that the charge Qe of the antihydrogen atom is Q ( 1.31.10.4) 10 8, with a statistical condence level

of 1s, and close to the prediction of zero from CPT and from quantum anomaly cancellation. The only previous direct bound on the antihydrogen charge, |Q|o0.02, was deduced18 from the observation that Qv B forces were insufcient to deect an

energetic (momentum 1.94 GeV c 1) beam26 of antihydrogen atoms away from a detector. Thus, our bound is 106 times more precise than the only previous direct bound.

An indirect bound on the antihydrogen charge can be inferred by comparing the experimentally measured charges of the antiproton and positron. The charge anomaly of the antiproton, q p

e

=e, is known2729 to be o7 10 10 with a condence

level of 90% by measurements28 on

pHe . The charge anomaly of the positron27,31 is less well known: |(qe e)/e|o2.5 10 8

(no condence level given explicitly, but assumed here to be 1s), determined by measurements of the positron cyclotron frequency and the positronium Rydberg constant32. From these limits on the individual charge anomalies, we can then infer a bound on the charge of antihydrogen of |Q|o2.5 10 8. (This inference relies

on the assumption that the positron and antiproton charges add exactly to form the charge of the antihydrogen atom. A similar inference was used to bound the charge anomaly of the antimuon to ( 1.12.1) 10 9 by measurements of the muonium

1s 2s energy interval33.) Thus, our direct bound is B2 times

lower than the best inferred bound.

It should be possible to laser cool the trapped antihydrogen atoms34 to 20 mK. This would allow us to decrease the conning magnetic elds to make a well that is 30 mK deep, much less than the current depth of 540 mK, while still retaining the majority of the trapped antihydrogen atoms. The b in equation (3) for a 30 mK well would be lower than the current b by a factor of B10, and the resulting bound on Q would improve by a similar factor. It would still require, however, a substantial data set. A different technique, based on stochastic acceleration35,36, could be used to bound |Q| at the 10 12 level with only a few tens of antiatoms.

Stochastic acceleration, as applied here, uses many cycles of a randomly oscillating electric eld to cause antiatoms to diffuse out of the trap were they to possess a charge (see ref. 35 for details). A few cycles of oscillating electric elds were incidentally applied to the current data set. Stochastic acceleration analysis of the effect of these elds yields a bound of |Q|o2 10 7.

This is roughly 10 times less precise than the deection-based bound reported here, but more cycles of stronger electric elds applied to a cooled antihydrogen distribution could achieve the aforementioned lower bound.

We note that a charge on antihydrogen could cause a signicant systematic error in a gravity measurement37. The conducting tubes in which gravitational experiments are likely to occur all exhibit anomalous patch electric elds. There is considerable variation38 in the magnitude of these patch elds depending on the geometry, surface material and surface preparation of the tubes, but elds on the order of 10 3 V mm 1 have been suggested37 at the centre of a 10 mm tube, with elds thousands of times higher near the tube wall.

These electric elds would deect an antihydrogen atom if it had a charge; for elds of 10 3 V mm 1, the gravitational force

would only exceed the electrical force if the charge was less than approximately 10 7e; near the wall, the charge would have to be proportionally less. Thus, our measurement of the anti-hydrogen charge appears to be sufcient to dismiss this systematic error for an up or down gravity measurement. For a precision measurement, the tolerable charge would depend on experimental details and would likely require a more accurate measurement than we report here, but probably not more accurate than can be achieved with stochastic acceleration.

Methods

Systematic effects. The discussion in this section is directed towards understanding the systematic and statistical errors that affect our determination of Q. Quantitative values for the systematic errors that we analysed are listed in Table 1. The systematic effects can be classied into one of four categories in order of descending importance (see also Table 2).

Category A: systematic effects that cause the sensitivity s, the relation between Q and /zSD, to differ from that predicted by our nominal simulations. As an example, inaccurate knowledge of the mirror coil currents would cause this type of error. These effects are assumed to be static in time.

Category B: systematic effects that change with time. These effects typically change /zSD. A z-dependent change in the performance of the detector with time would, for example, cause this type of error. While these effects could occur continuously, we estimate their size by assuming a discontinuous change between the 2010 and 2011 antiproton seasons; thus, we assume that all of the 2010 data (all of the Bias-Left data and 11% of the Bias-Right data) were collected with one value of the parameter in question, and all of the 2011 data (the remaining 89% of the Bias-Right data) were collected with a different value.

Category C: similar to Category A effects, but with the sensitivity assumed to jump discontinuously between 2010 and 2011. As these errors are necessarily less than half of the Category A errors (see Table 2), and the Category A errors are already small, these errors will not be further discussed.

Category D: systematic effects that cause an error in /zS, but not in /zSD. An inaccurate calculation of the detector efciency as a function of z would cause this type of error. As these errors do not affect /zSD, they have no effect on the bounds we set on Q.

All of the systematic effects described in Table 1 and in the Methods are classied into one of these four categories. Many of these effects were investigated by comparing simulations with and without the purported systematic effect. Often, the differences between the two simulations were within the sampling errors that derive from the large, but nite, number of antihydrogen trajectories modelled in the simulations. In these cases, we deem the effect to be statistically insignicant as we do not know whether the systematic effect has a real effect on the experiment. Fortunately, all these insignicant systematic effects would have only a small effect on Q even if they were real. Here, as elsewhere where we discuss systematic errors, we dene small to mean small relative to the statistical sampling errors engendered by the limited number of antihydrogen atoms that we observed.

Simulation details. The simulations model the antiatom equation of motion,

M d2rdt2 r l

B r; t

Qe E r; t

; 3 where r is the centre of mass position of the antiatom, and E(r, t) and B(r, t) are the position and time-dependent electric and magnetic elds. For the low eld seeking antiatoms modelled here, the magnetic moment, l

H and B are antialigned. The simulations employ a symplectic propagator. The electric eld in the simulation is determined by rst using COMSOL and a precise model of the trap to solve Laplaces equation for the electrostatic potential. The potential is calculated on a dense grid that covers the trapping volume, and interpolation coefcients are found for all positions. Sets of these coefcients are found for every electrode bias voltage condition employed in the experiment. The time evolution of the potential was modelled by implementing transitions between the different bias condition coefcient sets, taking into account the programmed ramp times and the measured response times of the real electrodes. The time-dependent electric eld is found from the gradient of this potential. The magnetic eld in the simulation is calculated from an analytic approximation accurate to better than 2% over the trap volume25. The magnetic eld model was checked against experiments with antiprotons25. Detailed descriptions of similar simulations, various benchmarking tests, and the analytic magnetic eld model have been given in previous publications9,25.

The simulations track each antiatom for 1 s, followed by four applications of antiproton clearing electric elds25, the Bias-Right or Bias-Left electric elds, and nally by the magnetic eld turn off. The total simulation time is 1.19 s. This sequence closely mimics the complete time history of B71% of the antihydrogen atoms, but does not include some of the electric eld manipulations used to diagnose the positron plasmas from which the antiatoms are synthesized. As these manipulations involve electric elds weaker than the bias and clearing elds, their omission has little effect. The remaining 29% of the antihydrogen atoms (36% of

_

r B r; t

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Table 1 | Systematic errors.

Error source Normalized variation s Category A(10 9/mm)

/zSD Category B (mm)

Time invariant /zS Category D (mm)

Q error net (10 8)

10 s Hold time. 0.04*

Improved B(r,t), dB/dt terms, adaptive Runge-Kutta propagator.

0.01 [0.01]* 0.09 [0.03]w 0.20w 0.03w

Degraded E(r,t). 0.01 [0.01]* 0.10 [0.03]w 0.24w 0.03w

|c4 Space m

H included. 0.01 [0.01]* 0.07* 0.01*

1% Differential mirror drift. 0.13 [0.04]z 0.29z 0.05z

1% Common mode mirror drift. 0.01 [0.01]z 0.02 [0.00]z 0.03z 0.02z

2 mrad Solenoid tilt. 0.01 [0.02]* 0.07 [0.02]* 0.15* 0.03* 1% Octupole drift. 0.01 [0.00]* 0.03* 0.01*

External magnet. 0.06 [0.02]z 0.11z 0.02z

3 Radius, 2 length. 0.01 [0.02]* 0.07 [0.02]* 0.15* 0.03*

Initial energy distribution. 0.22 [0.29]z 0.04 [0.01]z 0.05z 0.31z

Anisotropic initial distribution. 0.08 [0.11]w 0.17 [0.06]w 0.23w 0.12w

Detector z-centre relative to electrode zcentre.

0.17 [0.06]z 0.2z 0.06z

Mirror z-centre relative to electrode z-centre. 0.5z

Detector efciency. 0.03 [0.04]z 0.02 [0.01]z 3.6z 0.04z

Long-term detector drift. 0.04 [0.01]z 0.09z 0.01z

Cosmic background. 0.25 [0.08]y 0.08

y

Antiproton background. 0.13 [0.04]z 0.04z Entries in [] are the induced Q error associated with the adjacent entries, scaled by 10 8. (These entries do not always sum to the corresponding entries in the Q Error Net column because of rounding and because the entries in the sum are sometimes known to have opposite sign.) The normalized variation in the sensitivity s dQ/d/zS is dened to be s/s 1, where s is the sensitivity

evaluated with nominal parameters. Entries that are zero to two digits are designated with a .

*Effects for which the 1s sampling errors in the simulations used to study the effects are larger than the size of the effect predicted by the simulation; thus, we cannot determine whether these effects would cause an actual change in the experimental observations. More precisely, the simulations generally predict an effect of size ab. For these entries, |a|ob; to give an estimate of the worst case errors possible for these effects, we report the value of b, not that of a. For all other entries, we report a.

wComparisons between simulations where |a|4b.

zThe potential effect caused by some independently measurable parameter cd is real, that is, the simulations or calculations predict |a(c)|4b(c), but the 1s errors in the measurement of the relevant parameter are compatible with zero, that is, |c|od. Consequently, we do not know whether these error sources cause an actual change in our data.yAn effect based on an independently measured parameter that is not compatible with zero; this effect is likely to have caused a change to our data.

Table 2 | Systematic error formulas.

Category Effect Ond/zSL d/zSR d/zSD dsL dsR dQ

A ses ses ses/zSD

B (N2011/NR)ez 0.5(N2011/NR)ez 0.5sez(N2011/NR)

C (N2011/NR)ses 0.5(N2011/NR)ses|/zSD|

D ez ez 0 0

The change induced in the average Bias-Left z position (d/zS ), the average Bias-Right z position (d/zS ), the difference in these positions (d/zS ), the sensitivity for the Bias-Left data (ds ), the sensitivity for the Bias-Right data (ds ) and the resulting charge (dQ) for systematic errors that produce errors in z of size e and errors in the relative sensitivity e . Temporal changes in e and e are assumed to occur between 2010 and 2011; N /N is the fraction of Bias-Right events that occurred in 2011. Unaffected quantities are blank; the zeros in the Category D row emphasize that these errors have no effect on Q.

the Bias-Right and 17% of the Bias-Left anti-atoms) were held for times longer than B1 s. Simulations of atoms for 10.19 s show no signicant differences with respect to those held for 1.19 s.

Approximately 430,000 antiatoms were simulated for each value of Q. To mimic detector resolution effects on the data, the antiatom annihilation locations were individually shifted in z by a random amount consistent with the known24 detector resolution function.

We cross-checked the primary (symplectic) simulations against fourth order, adaptive step RungeKutta simulations. The latter employed a somewhat more accurate magnetic eld model (up to a factor of three times more accurate in some regions of the trap) and also included the electric elds induced by the changing magnetic elds; these effects were not employed in the primary simulations for reasons of computational efciency. The differences between the results predicted by the two types of simulations are small. We also cross-checked the primary simulations against symplectic simulations employing a simplied electric eld structure; in this third simulation, all trap electrodes were assumed to have the same radius. (There is a radially inwards step in the trap radius to RWall 16.8 mm

at the trap boundaries at z 137.6 mm.) This causes signicant electric eld

errors at large z, for example, about 5% at z 120 mm, but again results in only

small differences in Q.In all nominal simulations, m

H is assumed to be constant. Special simulations that allow it to vary with B, as it does slightly for one of the trapped

antihydrogen states (the |c4 state in ref. 8), show that such variations have insignicant effects.

Magnetic eld errors. Mirror coil current errors would cause various systematic errors. The coils are driven by individual power supplies rated accurate to 0.5%. Continuous, low precision, monitoring sets upper limits of about 1% for deviations in the mirror currents. There are small Category A, B and D effects from both differential and common mode current deviations.

The maximum tilt between the axial magnetic eld and the trap axis allowed by mechanical constraints, 2 mrad, produces insignicant errors. Errors or drifts in the octupole current are likewise insignicant. External magnets are a potential source of Category B and D errors, but their effect is very small. Consider an extreme example: a 1 T solenoid, 1 m long, with a bore diameter of 0.4 m. Such a solenoid would produce a shift in /zSD on the order of 0.06 mm if it were located 2 m away from our trap during 2011. No such external magnets are even remotely this close to our experiment.

Electric eld errors. Experimentally, the electric eld is generated by multiple high voltage ampliers driven by high precision digital to analogue converters (DACs). Each amplier channel is accurate to 1% and controls one trap

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electrode. The electric eld error from one channel is anticorrelated in adjacent electrodes. Thus, the average electric eld error is much smaller the error from one individual channel. The resulting error in the sensitivity s is much less than 1% and will not be further discussed.

Initial conditions for the simulations. Experimentally, antihydrogen atoms are created by collisions between antiprotons and positrons. Consequently the antiatoms are born in the volume occupied by the positron plasmas. The simulations are initiated with antiatoms born in this same volume. As a check, we also ran simulations with the antiatoms born in a volume with a radius three times greater and a length two times greater. This resulted in small, insignicant Category A, B and D errors.

Previous studies have established that trapped antiatoms in the ALPHA experiment have a distribution in centre-of-mass energy E that scales like E

p dE

below the trapping threshold5,25: that is, close to a cutoff Maxwellian distribution. These studies are based on examining the time history of escaping antiatoms and show that the experimental data are much better t by this Maxwellian distribution than by bracketing linear or uniform energy distributions. Separate examinations of the 2010 and 2011 time history data strongly suggest that the antiatom distribution did not change between these 2 years (see Fig. 4). Nonetheless, we can estimate the worst case effects of deviations from Maxwellian by evaluating the consequences of using a linear or uniform distribution. The resulting errors (principally Category A), while still much smaller than the statistical error, have the largest effect on Q of all of the systematic effects studied.

We have also studied anisotropic initial conditions E8 E> and E8 0.25E>,

where both are still consistent with the total energy distribution E

p dE. (The

isotropic normal condition is E8 0.5E>.) These anisotropies have a modest effect

on Q. There are no obvious mechanisms that could create such anisotropies. Even if the non-cutoff distribution function were anisotropic, (a bi-Maxwellian, for instance), the cutoff, low energy portion of the distribution, from which our trapped antiatoms are sampled, would reduces to an isotropic distribution.

An anisotropy would have to originate after the antiatom formation.

Location errors. The trap electrodes and the detector have some freedom to move relative to each other along z. We used the annihilation locations of antiprotons on background gas under poor vacuum conditions to measure this axial displacement and found that the centre of the trap is displaced from the centre of the detector by24.00.2 mm. This displacement is used to shift the detector-reported locations of the antihydrogen annihilations to the electrode reference frame; thus, the 24.0 mm displacement causes no errors in our analysis (although the uncertainty of0.2 mm does cause a Category D error.) This displacement changed by less than 0.34 mm between 2010 and 2011, a Category B error.

The trap electrode centre could also be displaced from the magnet system centre. By employing electron cyclotron heating39, we have measured this displacement to be 0.50.5 mm. Mechanical constraints make this

displacement time invariant.

Detector errors. The efciency with which we can reconstruct antihydrogen annihilations is z dependent, peaking near z 0 and diminishing towards both

ends. The corresponding efciency function, Eff(z), has been found by detailed Monte-Carlo simulations of the detector using GEANT3. It is incorporated into the simulation results by stochastically dropping simulated annihilations in a manner consistent with the efciency function. Because of the known (and time-invariant) failure of one of the detector modules, z-asymmetries in scattering materials, and the 24.0 mm displacement between the detector and the trap centres, the efciency is not mirror symmetric around z 0. This leads to a bias in the expected average

annihilation position of /zS 0.13.6 mm. Fortunately, this is largely an error of

Category D, which has little effect on Q. However, this also causes a small Category A error.

A time-dependent variation in Eff(z) would produce a Category B error. We have performed careful studies of the detector performance as a function of time using cosmic ray particles (typically muons) as a source; no antiparticles were present in the trap at the time of these studies. In each study, the number of hits, and hence the occupancy, was determined for each of the 60 detector modules. (A hit is dened by the coincident detection of a signal in the orthogonally oriented detector strips on both sides of a single module, thereby allowing the locationof the event that caused the signals to be determined. The occupancy is the number of particle track-forming hits per module, normalized to the number of cosmic ray events detected. The tracks were found using the standard ALPHA reconstruction methodology24.) From the individual occupancies, we determined the average occupancy for the modules in the right and left halves of the detector. Between 2010 and 2011 occupancies, the right occupancy changed by

0.20.2% and the left occupancy by 0.20.2%. These measurements are compatible with no change in the detector performance between the 2 years. However, we explored the consequences of these occupancies, which would lead to a real change in the detector efciency function Eff(z) of 0.2% on the right

side and of 0.2% on the left side. As expected, this produces a small Category B

error of /zSD 0.010.04 mm.

We also searched for a drift in the average z-position of annihilations created during the mixing, or antihydrogen synthesis, phase of our experiment46,22. During this phase, antihydrogen atoms too energetic to be trapped, as well as reionized antiprotons, annihilate on the trap wall. The small observed drift is loosely equivalent to a Category B error of 0.350.39 mm and is statistically consistent

with no detector drift. As the resolution of this technique is much poorer than that of the cosmic technique, and the results are consistent with the cosmic technique, we use the cosmic technique as our estimate of the detector error. Even if we were to use the mixing technique, however, it would be equivalent to an error in Q of only (0.130.15) 10 8, which is signicantly smaller than the sampling error.

Antiproton and cosmic background. Because our particle detector cannot distinguish antiprotons from antihydrogen atoms, we designed our experiment to minimize the chance that antiprotons are trapped. Measurements, comparisons with simulations and calculations have established that few, if any, antiprotons are trapped4,6,25; nonetheless, it is important to estimate the magnitude of the false /zSD signal (a Category B error) that antiprotons might engender. Previous theoretical and experimental studies of the behaviour of deliberately trapped antiprotons25 show that all but B5% annihilate in an elliptical region in z-t space centred on z 130 mm and t 5 ms, and with axial half width 20 mm and

temporal half width 4 ms. Here, t 0 marks the beginning of the magnet shutdown

phase of the experiment, and the sign of the z-centre is positive for Bias Right conditions and negative for Bias Left conditions (see Figures 6a and 9d in ref. 25). Consequently, we reduce the possible contamination from antiprotons by cutting annihilation events located within these two elliptical regions from our analysis set. To maintain symmetry, we cut with both ellipses from data taken under either Bias condition, although only one ellipse, on the appropriate side, is necessary to eliminate antiprotons from each Bias condition individually. We also cut all experimental annihilations that occur at |z|4136 mm because of the mirror maxima and the radial step in the trap wall that occurs near these locations.

In addition to the z cuts, we also cut all annihilations that occur 30 ms or more after the magnet shutdown because of the increasing chance of cosmic contamination for such late events9. As discussed earlier, 386 events pass all cuts. (In the B1,300 trapping trials used to obtain our event set, we would expect approximately two cosmic rays to be misclassied as antihydrogen atoms, and engender a modest error. For categorical simplicity, we list this error (and the error that comes from antiprotons) as a Category B error as these cause shifts in /zSD;

we do not know in which year (perhaps both) the cosmic rays might have been misclassied.)

A total of nine antiproton candidates are cut by the elliptical exclusion regions. It is not likely that all nine candidates are antiprotons, however, because Q 0

simulations predict that we should have observed B7.7 antihydrogen atoms in the appropriate-side exclusion regions. Since the probability, as predicted by Poisson statistics, of observing 0 antihydrogen atoms when 7.7 are expected is P 0.0005,

the likelihood that all nine candidates are antiprotons is very low. Indeed, itis not unlikely that all nine candidates are legitimate antihydrogen atoms. However, if we assume that all nine candidates are antiprotons, we would predict that an additional 9 0.05 0.45 antiprotons annihilate outside the exclusion regions,

creating a false signal. Simulations predict that the z-averaged annihilation location

1.0

0.8

0.6

0.4

0.2

0.0

f(E) (a.u.)

1.0

0.5

0.00.0 0.2 0.4 0.6 Energy (K)

Time-reversed CDF

Uniform

Linear

Maxwellian

0 5 10 20

15 30

25

Time (ms)

Figure 4 | Measured and simulated antihydrogen cumulative distribution function. The simulated time-reversed cumulative distribution function (CDF) of the time of annihilation for the nominal, Maxwellian distribution (solid red), uniform (short dashed green) distribution and linear (long dashed orange) distribution. The inset gure depicts the candidate energy distributions f(E). The Maxwellian distribution is a much better match to both the 2010 (solid dark blue) experimental data and the 2011 (dashed light blue) experimental data than either the Uniform or Linear distributions. The error bars show the expected deviation of the CDF based on the number of observed antiatoms used to compute the CDF at each time.

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of the few antiprotons outside the exclusion regions is approximately 110 mm. Thus, for our sample size of 386 antiatoms outside of all exclusion regions, we would expect an error in |/zSD| of B0.13 mm under the very pessimistic assumption that all nine candidates are antiprotons.

Simulated z-distributions and z cuts. Simulated z-distributions of antiproton annihilations are shown in Fig. 2 for antiatoms with Q 0 and for antiatoms with

Q 4 10 8 under Bias-Right and Bias-Left conditions. Detector smearing

and efciency are taken into account. However, no z-cuts (that is, neither the |z|o136 mm cut nor the elliptical cut) are employed. Beyond the shifts in average z for nonzero Q, which are used to calculate /zSD, the most remarkable feature of the data is the sharp fall-off in annihilations outside of 136 mmozo136 mm.

For Q 0, only 3.8% of the simulated particles annihilate in the outside region.

(Experimentally 4.9% annihilated in the outside region. The experimental value matches the simulated value with a w2 P-value of 0.22.) The paucity of annihilations outside of 136 mm is due to the location of the magnetic mirror maxima at z

137 mm (see Fig. 1b) and the relative size of the axial and radial well depths. At all times after the initiation of the magnet shutdown, the axial well depth exceeds the radial well depth: for instance, at t 0 ms, it is 80 mK deeper; at t 10 ms, it is

140 mK deeper; at t 20 ms, it is 70 mK deeper; and, at t 30 ms, it is 20 mK

deeper. Thus, it is always favourable for antiatoms to leave the trap radially, and few antiatoms surmount the axial well maxima. (The poor coupling13 between the transverse and axial motions does allow a few antiatoms to escape axially.) Antiatoms with |Q|o4 10 8 are also bound by the mirror maximum and largely

annihilate within 136 mm. Thus, our cut at 136 mm is loose compared with the data, and has only a small effect on the sensitivity s.

Stark forces. One source of systematic error that is negligible, but deserves attention, is the Stark force that comes from polarization of the antihydrogen atoms (which we here assume is identical to that of hydrogen). The antiatoms analysed here were all held for 0.4 s or longer; this is long enough to allow for at least 99.5% of them to decay to the ground state5. The polarizability of ground state antihydrogen is small; given that the largest electric elds in the trap are on the order of10 V mm 1, and such large elds exist only in the gap between the electrodes, a value of Q greater than about 10 14 will dominate over polarization effects. Since our bound on Q is well above this value, we can safely ignore such effects.

Systematic error summary. Adding all the systematic errors in Table 1 in quadrature yields a net systematic error in Q of 0.4 10 8. As this is less than

the 1s counting statistic error, 1.1 10 8, the precision of our determination of

Q is dominated by our sample size. Systematics also engender a net bias and error in /zS, namely 0.13.7 mm. Experimentally, we nd an average position,/zS 3.93.4 mm. We stress that this bias does not enter /zSD, and, hence, Q;

rather it provides a check on the completeness of our systematics study.

Statistical methods and blind analysis. The statistical techniques used here were developed using pseudo-data generated by the simulations. We investigated more complex statistical techniques than the simple averages we chose to employ, but these techniques were not found to be more powerful. We did not compute the experimental values of the averages /zSR, /zSL, and hence, /zSD and Q until all of the statistical procedures were xed and all cuts determined.

The statistical Q condence interval is based on the s.e. of the mean of the 386 event experimental data set, s/zS 3.390.14 mm. We also computed the s.e. of

the mean of the simulation data, assuming a 386 event subset: sS 3.22 mm. The

agreement between these two alternatives further validates the simulations and the detector efciency curves. In principle, the sampling error of the s.e., 0.14 mm, creates some small uncertainty in the Q condence region range, but we do not propagate this uncertainty into the nal answer. To show that our result covers zero at the 90% condence level, we use the factor 1.64 to scale from a 1s condence; Students t-test shows that the error in this factor due to our use of the sample rather than the population mean is negligible.

Expanded data set. As described above, the data analysed in this article are limited to those antiatoms that met several cut criterion in z and t, as well as to those antiatoms that were held for times greater than or equal to 0.4 s. Some likely antiatom candidates annihilated in the elliptical cut regions (designed to eliminate the possibility of including mirror-trapped antiprotons in the analysis). Others annihilated outside of |z|o136 mm. Still others were held for times between 0.17 s, the minimum hold time ever employed by ALPHA, and 0.4 s. These last antiatoms

a

20

z (mm)

10

10

20

30

0

b

Q (108 )

10

5

0

5

10 0 100 200 300 400 500 600

Number of anti-atoms

Figure 5 | Data selection. The /zSD and Q plotted as a function of the number of antiatoms included in the analysis for the data in Table 3. The principle case is the blue square point. Note that the data are generally, but not always, cumulative with increasing number of antiatoms. Thus, the points are not generally independent. Also note that the sensitivity, s, used to scale from /zSD to Q varies from 3.31 10 9mm 1 (Principle data

set) to 0.224 10 9mm 1 (no z cut data sets). In a, the error bars

show the s.e. values of the mean in /zSD, and in b, these errors scaled by

the sensitivity s.

Table 3 | Data selection.

Condition Selection 2010 only No z cut NR NL /zSR(mm) /zSL(mm) /zSD(mm) No bias Disjoint O 39 1.710

Well alternated Disjoint O 11 13 22.322.3 1.820.3 10.215.1

Well alternated Disjoint O O 14 14 13.928.8 17.524.5 1.818.9

Hold timesr0.4 s Disjoint O 11 70 22.322.3 1.68.5 10.311.9

All hold times Sub/Super O 38 220 18.310.3 0.14.6 9.15.6

All hold times Sub/Super O O 42 229 12.012.5 1.45.0 5.36.7

Hold timesZ10 s Sub 94 25 9.86.8 5.111.5 7.56.7

Principle 241 145 7.94.2 0.25.3 4.13.4

Hold times40.4 s Super 279 150 5.54.1 0.75.5 2.43.4 All hold times Super 290 220 4.44.0 0.14.6 2.23.1

All hold times Super O 310 229 2.84.2 1.45.0 2.13.2

Different data sets as described in the condition column; the principle row is the data set principally analysed in this article. It is the only row to include the antiproton-suppressing elliptical cut. The No Bias row reports data taken with the bias electric elds off. The Well alternated rows come from trapping trials in which the Bias-Right and Bias-Left conditions were strictly alternated. As not every trial trapped antiatoms, the data are not strictly alternated. The selection column indicates if the rows data set is a subset, superset or disjoint set relative to the principle data set. Some of the rows include both a subset of the principle data and some additional data; these rows are designated Sub/Super. All rows include the |z|o136 mm cut unless otherwise noted.

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were likely to have been in the ground state, but cannot be guaranteed to have been so. Further, some antiatoms passed all cuts and were detected during trapping trials in which the Bias-Right and Bias-Left conditions were strictly alternated. As discussed earlier, most of our data were not collected in this optimal manner. In Table 3, we compare various disjoint sets, subsets and supersets of our complete data set to our Principle data set. The Table reports /zSR, /zSL and /zSD for these different data selections. The comparisons, which are summarized in Fig. 5, show that the /zSD are mutually consistent and are consistent with the average z annihilation position when no electric elds are applied.

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Acknowledgements

This work was supported by: CNPq, FINEP/RENAFAE (Brazil); ISF (Israel); FNU (Denmark); VR (Sweden); NSERC, NRC/TRIUMF, AITF, FQRNT (Canada); DOE, NSF, LBNL-LDRD (USA); and EPSRC, the Royal Society and the Leverhulme Trust (UK). We are grateful for the efforts of the CERN AD team, without which these experiments could not have taken place.

Author contributions

This retrospective analysis was based on data collected in 2010 and 2011 by the entire ALPHA collaboration, using the antihydrogen trapping apparatus and methods developed by the collaboration. The electrostatic deection methodology was rst suggested by M.C.F. and investigated by M.B.-R., A.E.C., J.F., J.S.W. and A.I.Z. with help from F.R. The detector drifts were analysed by A.C., J.T.K.Mc.K. and A.O., and with help from S.M. and P.P. The stochastic acceleration methodology was proposed by J.F. and investigated by M.B.-R., J.F., F.R. and J.S.W. This article was written by M.B.-R., J.F. and J.S.W., with help from E.B., W.B., A.E.C., M.C., S.E., M.E.H., N.M., J.T.K.Mc.K., S.M., A.O. and P.P., and then improved and approved by all the authors. This article constitutes part of the Ph.D work of M.B.-R. All authors except A.E.C. contributed to this work as members of the ALPHA antihdyrogen collaboration.

Additional information

Competing nancial interests The authors declare no competing nancial interests.

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