Formation Of A Cold Antihydrogen Beam in AEGIS For Gravity Measurements

Abstract

The formation of the antihydrogen beam in the AEGIS experiment through the use of inhomogeneous electric fields is discussed and simulation results including the geometry of the apparatus and realistic hypothesis about the antihydrogen initial conditions are shown. The resulting velocity distribution matches the requirements of the gravity experiment. In particular it is shown that the inhomogeneous electric fields provide radial cooling of the beam during the acceleration.

Full text

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Formation Of A Cold Antihydrogen Beam in
AEGIS For Gravity Measurements
G. Testeraa, A.S. Belovb , G. Bonomic, I. Boscolod, N. Brambillad, R. S.
Brusae, V.M. Byakovf, L. Cabaretg, C. Canalih, C. Carraroa, F. Castellid, S.
Cialdid, M. de Combarieui, D. Comparatg, G. Consolatij, N. Djourelovl, M.
Doserm, G. Drobychevw, A. Dupasquierj, D. Fabrisv, R. Ferragutj, G.
Ferrario, A. Fischerh, A. Fontanac, P. Forgetg, L. Formarod, M. Lunardonv,
A. Gervasinid, M.G. Giammarchid, S.N. Gninenkob, G. Gribakinp, R.
Heyneh, S.D. Hoganq, A. Kellerbauerh, D. Krasnickyu, V. Lagomarsinoa,
G. Manuzioa, S. Mariazzie, V.A. Matveevb, F. Merktq, S. Morettov, C.
Morhardh, G. Nebbiav, P. Nedelecn, M.K. Oberthalerr, P. Parii, V.
Petraceku, M. Prevedellio, I. Y. Al-Qaradawiz , F. Quassoj, O. Rohnet, S.
Pesentev, A. Rotondic, S. Stapnest, D. Silloun, S.V. Stepanovg, H. H.
Strokes, G. Tino o, A. Vairod, G. Viestiv , H. Waltersp, U. Warringh, S.
Zavatarellia , A. Zenonic, D.S. Zvezhinskijg (AEGIS Proto-Collaboration)

a Istituto Nazionale di Fisica Nucleare and Department of Physics, University of Genoa, via
Dodecaneso 33, 16146 Genova, Italy
bInstitute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia
cIstituto Nazionale di Fisica Nucleare and Department of Physics, University of Pavia, via Bassi 6,
27100 Pavia, Italy
dIstituto Nazionale di Fisica Nucleare and Department of Physics, University of Milan, via Celoria 16,
20133 Milano, Italy
eDepartment of Physics, University of Trento, via Sommarive 14, 38100 Povo, Italy
f Institute for Theoretical and Experimental Physics, Moscow 117218, Russia
g Laboratoire Aimé Cotton, CNRS, Univ Paris-Sud, Bât. 505, 91405 Orsay, France
hMax Planck Institute for Nuclear Physics, Postfach 103980, 69029 Heidelberg, Germany
iLaboratoire de Basse Tempe´rature du SPEC/DRECAM/DSM, CEA Saclay, 91191 Gif-sur-Yvette
Cedex-France
lInstitute for Nuclear Research and Nuclear Energy, 72 Tzarigradsko Chaussee, 1784 Sofia, Bulgaria
mDepartment of Physics, CERN, 1211 Gene`ve 23, Switzerland
n Lab. d’Annecy-le-Vieux de Phys. des Particules, 9 Chemin de Bellevue, B.p. 110, 74941 Annecy Cedex,
France
jDepartment of Physics, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy.
oIstituto Nazionale di Fisica Nucleare and Department of Physics, University of Florence and LENS,
CNR-INFM, via Sansone 1, 50019 Firenze, Italy
pDepartment of Applied Mathematics and Theoretical Physics, Queen’s University, University Road,
Belfast BT7 1NN, United Kingdom
qLaboratorium für Physikalische Chemie, ETH Zürich, Zürich 8093, Switzerland
rKirchhoff Institute of Physics, University of Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg,
Germany
sDepartment of Physics, New York University, 4 Washington Place, New York, NY 10003, USA

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tUniversity of Oslo, Department of Physics, Oslo, Norway
u Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering
vIstituto Nazionale di Fisica Nucleare and Department of Physics, University of Padova ,
Via Marzolo 8, 35131 Padova - Italy
z Qatar University, Doha, Qatar P. O. Box 2713, Doha, Qatar
wInstitute for Nuclear Problems of the Belarus State University, 11, Bobruiskaya Str., Minsk 220030,
Belarus


Abstract. The formation of the antihydrogen beam in the AEGIS experiment through the use of
inhomogeneous electric fields is discussed and simulation results including the geometry of the
apparatus and realistic hypothesis about the antihydrogen initial conditions are shown. The
resulting velocity distribution matches the requirements of the gravity experiment. In particular
it is shown that the inhomogeneous electric fields provide radial cooling of the beam during the
acceleration.
Keywords: Antihydrogen, Equivalence Principle, Rydberg atoms.
PACS: 04.80.Cc, 32.80.Ee, 34.70.+e, 36.10.-k, 37.10.Ty

INTRODUCTION
The equivalence principle is a foundation of General Relativity and a large
experimental effort is placed in testing its consequences in all possible fields: this
research activity includes tests about the equality of the inertial and gravitational mass,
the universality of the free fall, the search for non Newtonian corrections to the
gravitational law, the measurement of the gravitational red shift, the search for time
variation of the fundamental constants. Measurements about the equality of the
inertial and gravitational mass of different bodies began well before the general
relativity (they started with Galileo and Newton) and then continued over the years
with larger and larger precisions: sensitivity of the order of 10-15 and 10-18 are
expected from modern space experiments [1] while laboratory torsion balance setups
have already reached an accuracy of 1 part in 1013 [2].
While all these tests concern macroscopic bodies, a sensitivity to the gravitational
acceleration g of the order of 10-10 has been obtained with cold atoms in atomic
fountains [3].
The important point is that all these measurements (with macroscopic bodies or cold
atoms) are performed on matter system: there are no direct measurements about the
validity of the principle of equivalence for antimatter. The validity of the equivalence
principle for antimatter is extrapolated from the matter results or it is inferred using
indirect arguments (several of them are controversial and model dependent [4]).
Particularly interesting is that some quantum gravity models leave room for possible
violations of the equivalence principle for antimatter [4].
The AEGIS experiment has been proposed [5,6] at the CERN Antiproton Decelerator
to directly measure the Earths gravitational acceleration g on a beam of cold
antihydrogen. The basic setup of the experiment should allow one to reach an
accuracy of 1% and the experiment is designed to allow higher precision
measurements through radial cooling of the beam.

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The gravitational acceleration g will be obtained by detecting the vertical deflection of
the antihydrogen beam flying horizontally with a velocity of a few 100 m/s for a path
length of about 1 meter. The small deflection (few tens of µm) will be measured using
two material gratings coupled to a position sensitive detector working as Moiré
deflectometer in the classical regime. A device of this type with an additional grating
in place of the position sensitive detector has been already operated with atoms [7] by
members of the collaboration.
While details about the antihydrogen formation in AEGIS and the Moiré device are
reported elsewhere [5], here we will focus on the formation of the antihydrogen beam.

PULSED COLD ANTIHYDROGEN FORMATION
The antihydrogen beam formation will take places in AEGIS by a two step process:
the first one is the production of very cold antihydrogen in Rydberg states and the
second one is the acceleration in one direction (the horizontal one) of the Rydberg
antihydrogen using inhomogeneous electric fields. A key point is that the
antihydrogen formation will happen within a short time (of the order of 1 µs). This
“pulsed production scheme” strongly differs from the existing schemes where the
antihydrogen is produced during long time intervals (seconds in the ATHENA
experiment) [8].
Cold antihydrogen in Rydberg states will be produced by the charge exchange reaction
[5] [9]

→+
p Ps
between Rydberg positronium Ps* and cold (100 mK or below) antiprotons. Rydberg
positronium (with principal quantum number n of the order of 20-30) will be formed
by laser exciting the ground state positronium emitted by a porous material hit by a
positron bunch (108 positrons with a time length of 10-20 ns).
The production of antihydrogen happens when the Rydberg positronium traverses the
cold antiproton cloud. Taking into account that the velocity of the Rydberg
positronium must be of the order of 104-105 m/s to optimize the cross section for the
reaction of Eq. 1 and that the antiproton cloud dimensions are of the order of a few
mm, therefore the production time is defined within about 1 µs.
This pulsed production scheme of antihydrogen offers the possibility to easily
measure the resulting antihydrogen temperature (by time of flight); it also provides a
t=0 time for gravity measurement and it allows to form the antihydrogen beam as
described below.
The trapped antiproton temperature is the main factor determining the resulting
antihydrogen temperature provided that it is greater than about 100 mK: below this
value kinematics effects in the charge exchange reaction are not completely negligible.
FIGURE 1 (not to scale) shows the scheme of the region of the experiment where the
antihydrogen will be formed, accelerated and sent through the grating system. Not
shown in the drawing are the regions of the apparatus devoted to catch and cool
antiprotons from the CERN Antiproton Decelerator and to accumulate positrons.

−
+
eH
**
(1)

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FIGURE 1. The figure (not to scale) shows the two parallel Penning-Malmberg traps that will be used
in AEGIS to manipulate the antiprotons and positrons and to form and accelerate antihydrogen. They
will be mounted inside a 100 mK cryostat in 1 Tesla magnetic field. The upper trap is devoted to
antiprotons. Cold antiprotons wait for positronium in the black region. The lower trap is devoted to
positrons: they will be sent to to the porous target mounted in front of the antiproton trap to produce
positronium. Laser pulses will excite the positronium to selected Rydberg states having n in the range
20-30. The yellow region of the upper trap shows the bunch of antihydrogen after the acceleration (as
described in the text). Finally the two material gratings followed by a position sensitive detector for the
gravity measurement are shown. Ls is 30 cm, L=40 cm. The trap radius is 8 mm. The gratings and
detector dimensions are 20x20 cm2.

THE PENNING TRAP RYDBERG ACCELARATOR
The use of inhomogeneous electric fields to accelerate and manipulate Rydberg atoms
has been proposed long time ago [10] but only recently this idea has become an
experimental reality. In particular experimental demonstration of the acceleration,
deceleration and even trapping has been achieved by members of AEGIS with
hydrogen atoms in Rydberg states with quantum numbers very similar to those to be
used in AEGIS [11].
To first approximation the energy levels E of the (anti)hydrogen atom in an external,
homogeneous electric field of magnitude F are given, in atomic units, by
3
2

nkF
n
E
2
1
2+−=
(2)
Gratings for the g measurement
e+ trap
Target for positronium
production
Antiproton trap
Position sensitive
detector

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where k is a quantum number which runs from −(n − 1 − |m|) to (n−1−|m|) in steps of
two and m is the azimuthal quantum number. Here 1 a.u. = 27.211 eV in the case of
the energy and 5.14 × 109 Vcm−1 for the electric field strength.
If the excited atoms are moving in a region where the amplitude of the electric field is
changing then their internal energy changes according to Eq. 2 and thus, to conserve
the total energy, they are accelerated or decelerated; the change of kinetic energy is
3/2 nk ∆F. This is described as a force acting on the atom by taking the space
derivative of Eq. 1.

The experiments with hydrogen have shown that using time varying electric fields
changes of velocity of the several hundreds m/s can be achieved in a few mm space
using electric fields only limited by field ionization.
The formation of the beam of antihydrogen in AEGIS will be obtained by switching
the voltages applied to the antiproton trap electrodes (immediately after the
antihydrogen formation) from the usual Penning trap configuration to a new
configuration that we call the “Rydberg accelerator”.
To achieve this it is required to generate an electric field having an amplitude
decreasing (or increasing) with z. Antihydrogen with positive (negative) k will be
accelerated toward the system of gratings . The accelerating electric field will stay on
for a selected time interval.
The gravity measurement procedure sets the requirements about the beam. The
gratings of the Moiré device select specific trajectories of the atoms that can arrive on
the detector. The distribution of the number of atoms arriving on the detector as a
function of the vertical coordinate shows a periodical pattern due to the gratings. The
gravity force causes a vertical shift of this pattern which depends on the time of flight
T between the two gratings. Then to get g it is necessary to reconstruct, in addition to
the vertical position on the detector, the horizontal velocity of the particles while they
travel through the gratings. For this reason, it is important to have a start time well
defined and a limited axial spread of the beam at the time when the accelerating
electric field is switched off.
The transverse velocity of the beam does not influence the measurement result but it
affects of course the number of particles arriving on the detector and the size of the
gratings and detector itself. It is then mandatory to implement an acceleration
procedure that does not heat the beam in a significant way in the radial direction.
We have simulated the expected velocity distribution in AEGIS taking into account
the electric field obtained with the true geometry of the trap electrodes and realistic
hypothesis about the initial antihydrogen spatial extension ( few mm radius and about
1 cm axial length ) and the expected state population.
The antihydrogen state population has been calculated by a dedicated CTMC code
simulating the charge-exchange process in the AEGIS regime [5]. The Rydberg
antihydrogen atoms will be produced not in a single quantum state but they will
populate a distribution of states. The mean value of the principal quantum number nH
of antihydrogen is related by the positronium one by
having a rms deviation of few units is expected.
*
2
Ps
H
nn ≈
but a distribution

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As a reference the results here reported refer to a Gaussian distribution of nH having
mean value 30 and a rms deviation equal to 4. For every n all the possible k have been
included. In these conditions the product nk ranges from zero to about 1000.
The main consequence is that the various atoms will acquire different velocities in the
same electric field and a beam with a large velocity spread will be obtained. To
partially reduce the velocity spread it is important to use a time-varying electric field
as shown in FIGURE 2.







FIGURE 2. The figure shows the amplitude of the accelerating electric field on the trap axis versus z
after 10, 30, 60 µs from the beginning of the acceleration process. The electric field stays on for 80 µsec
in these simulations. It is obtained by applying voltages to radial sectors of the cylindrical trap
electrodes. The accelerating electrodes are constituted by those of the antiproton trap shown in figure 1
together with some neighboring electrodes. These electrodes are split into four sectors (in the radial
plane) with 135 degrees and 45 degrees angular extension. A voltage V0 is applied to one of the big
sectors while a voltage –V0 is applied to the one opposite to it. The smaller sectors are grounded. The
required distribution of electric field strength along the propagation direction of the beam is generated
by applying a time-dependent potential to the relevant electrodes. The resulting electric field direction is
almost perpendicular to the magnetic field. The antihydrogen cloud after production is centered at z=0.



FIGURE 3 shows the axial velocity before and after the acceleration for a cloud of
antihydrogen produced with antiprotons cooled to 100 mK and distributed uniformly
within 3 mm radius (that is within a large fraction of the 8 mm trap radius) and 8 mm
length. Only 10% of the particles are lost on the trap electrodes. The corresponding
distribution of the axial positions immediately after the accelerating electric field is
switched off is shown in Fig 4: its spread is consistent with the requirements about the
gravity measurement.




t=60 µsec
t=30 µsec
t=10 µsec

Electric field strength (V/cm)
Horizontal position z (cm)

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FIGURE 3. The red continuous curve is the distribution dN/dvz of the horizontal velocity at the time
when the accelerating electric field is switched off . For comparison the dotted black is the initial
velocity distribution (Maxwell-Boltzmann distribution at a temperature of 100 mK ). The mean value
of the red curve can be tuned by playing with the field intensity.








FIGURE 4. Distribution of the horizontal position of antihydrogen after acceleration. The plot refers
to the same conditions as figure 3.


Horizontal velocity vz (m/s)
Horizontal position z (mm)
(m/s)
Number of antiH per velocity units ( 1/ms-1)
Number of antiH per unit length ( mm-1)

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TRANSVERSE VELOCITY DISTRIBUTION OF THE
ANTIHYDROGEN BEAM
The amplitude of the radial electric field increases with the distance from the trap
center thus providing an average deceleration force resulting in a reduction of the
mean radial velocity. This is particular relevant for antihydrogen produced within 1-2
mm from the trap center. In the conditions of figure 2 (100 mK antihydrogen
distributed within 3 mm radius) the distributions of the transverse velocity before and
after the acceleration are practically identical. This is already a positive result because
it indicates that, on average, the acceleration procedure is not destroying the efforts
placed in producing ultracold antihydrogen. Particularly interesting if the comparison
between the transverse velocity before and after the acceleration for antihydrogen
initially produced within a smaller radius: 1.5 mm. In this case the net effect is radial
cooling as shown in FIGURE 5.









a) 100 mK
antihydrogen

b) 400 mK
antihydrogen
Number of antiH per velocity units ( 1/ms-1)
Transverse velocity vr (m/s)
Transverse velocity vr (m/s)

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FIGURE 5. These two figures compare the transverse velocity distribution dN/dvr as a function of vr
of antihydrogen before (black dotted plot) and after (red continuous plot) the acceleration. It is assumed
that the antihydrogen is initially produced within a cylinder having 1.5 mm radius and 8 mm length.
The initial velocity, before the acceleration, corresponds to a Maxwell distribution with 100 mK in
figure a) and 400 mK in figure b). The cooling effect is evident in both cases.

EFFECT OF THE MAGNETIC FIELD
The Stark acceleration of antihydrogen in AEGIS will happen in presence of the
uniform magnetic field of the Penning-Malmberg traps. Under completely general
conditions, the dynamics of Rydberg hydrogen atoms in electric and magnetic fields
with arbitrary mutual orientation are rich and complex [12]. Several regimes are
possible depending on the range of parameters (values of the electric and magnetic
fields and principal quantum number values). The magnetic field in the antihydrogen
formation region is chosen following a compromise between the requirement of the
charged particle trap (that demands high magnetic fields) and the need to softly
perturb the Rydberg antihydrogen. The choice of B=1 Tesla realizes this compromise
for Rydberg antihydrogen having n not higher than about n=35. In this regime n is still
a good quantum number and the diamagnetic correction is small (but not completely
negligible) compared to the Zeeman one in the Hamiltonian and the classical
dynamics is regular [13]. For larger n and higher magnetic field the classical dynamics
could become chaotic and the quantum levels can show large avoided crossing [11].
These effects could influence the efficiency of the acceleration procedure. Detailed
quantum calculations to exactly quantify these effects are in progress. The presence of
the magnetic field is here accounted modifying Eq. 2 with the one giving the energy
levels to the first order in the perturbation theory for perpendicular electric and
magnetic fields [14]
1
n
n
Here γ is the magnetic field in atomic units and n’ is a quantum number assuming the



2
2
2
Figure 6 compares the simulated horizontal velocity distribution of figure 3 with the
one obtained using the same electric field and the same initial conditions of the
antihydrogen cloud but including the magnetic field. The effect is a reduction of the
final velocity values which is tolerable. The radial cooling effect discussed in the
previous paragraph is maintained in presence of magnetic field.



222'
2
9
2
FnE
++−=
γ
. (3)
values



−




−
−




−
−
1
,.....
3
,
1nnn
.

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FIGURE 6. Simulated distributions of the horizontal antihydrogen velocity with (filled green plot) 1
Tesla magnetic field and without (black curve). The black curve is the one of figure 3. The initial
antihydrogen cloud are the ones used to obtain figure 3 (a cloud of 100 mK temperature with 3 mm
radial size and 8 mm length and having a mean value of n equal 30 with rms=4). For every n all
possible values of n’ with uniform distribution are considered.





CONCLUSIONS
The recent developments about the manipulation of Rydberg hydrogen by using
inhomogeneous electric fields together with the experience gained at the CERN AD
to form antihydrogen are merged together in the design of AEGIS to form a beam of
cold antihydrogen. The beam will be used to perform the first gravity measurement on
antimatter. In the long term physics program of the collaboration several
improvements are planned: here we mention the use of negative ions to better cool the
antiprotons [15] and of quasi-continuous Lyman-α laser for cooling the formed
antihydrogen beam [5,16]. Finally the possibility of accelerating (and decelerating)
and thus transporting antihydrogen atoms opens the way to investigate novel
antihydrogen trapping strategies in which the production and the trapping regions are
spatially separated [5].


Axial velocity vz (m/s)
Number of antiH per velocity units ( 1/ms-1)

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