corso di laurea specialistica in scienze fisiche 5.12 time of ﬂight spectrum: the red triangles...

UNIVERSITA DEGLI STUDI DI PAVIA

FACOLTA DI SCIENZE MM. FF. NN.

CORSO DI LAUREA SPECIALISTICA IN

SCIENZE FISICHE

Geant4 simulation of a moiredeflectometer for gravity

measurements with antihydrogen atAEgIS

Simulazione con Geant4 di undeflettometro moire per misure digravita con antiidrogeno in AEgIS

Tesi sperimentale per la Laurea Specialistica diAndrea Capra

RelatoreDr. Andrea Fontana

INFN sez. Pavia

Anno Accademico 2009/2010

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ALL men by nature desire to know.[...] they philosophized in order to

escape from ignorance, evidently theywere pursuing science in order to know,

and not for any utilitarian end.Aristotle’s Metaphysics

Contents

1 Theory for antimatter studies 11.1 CPT theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . 31.3 CPT, gravity and antimatter . . . . . . . . . . . . . . . . . . . 51.4 Test of CPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Test of EEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Atom interferometry 152.1 Atom diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 The Ramsey fringes pattern . . . . . . . . . . . . . . . 172.1.2 Diffraction gratings for atoms . . . . . . . . . . . . . . 20

2.2 Interference fringe pattern . . . . . . . . . . . . . . . . . . . . 222.2.1 Building an interferometer . . . . . . . . . . . . . . . . 232.2.2 Three gratings interferometers . . . . . . . . . . . . . . 252.2.3 Atomic fountain . . . . . . . . . . . . . . . . . . . . . . 272.2.4 Phase shift in the Mach-Zehnder interferometer . . . . 28

2.3 Classical shadow pattern . . . . . . . . . . . . . . . . . . . . . 302.3.1 The moire deflectometer . . . . . . . . . . . . . . . . . 312.3.2 Phase shift in the moire deflectometer . . . . . . . . . 332.3.3 Classical Deflectometer vs. Quantum Interferometer . . 37

3 Gravity experiments on antimatter 393.1 Gravity measurements on

charged antiparticles . . . . . . . . . . . . . . . . . . . . . . . 403.1.1 Gravitational field . . . . . . . . . . . . . . . . . . . . 413.1.2 Electric field . . . . . . . . . . . . . . . . . . . . . . . . 423.1.3 Electric field gradients . . . . . . . . . . . . . . . . . . 423.1.4 Magnetic field . . . . . . . . . . . . . . . . . . . . . . . 433.1.5 Magnetic field gradients . . . . . . . . . . . . . . . . . 433.1.6 Interaction with radiation . . . . . . . . . . . . . . . . 453.1.7 Residual gas scattering . . . . . . . . . . . . . . . . . . 45

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vi CONTENTS

3.1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 473.2 AD experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Production and detection of antihydrogen . . . . . . . 483.3 The AEgIS experiment . . . . . . . . . . . . . . . . . . . . . . 53

3.3.1 General design . . . . . . . . . . . . . . . . . . . . . . 543.3.2 Charge Exchange Reaction . . . . . . . . . . . . . . . . 643.3.3 Stark acceleration . . . . . . . . . . . . . . . . . . . . . 653.3.4 The moire deflectometer . . . . . . . . . . . . . . . . . 67

4 Simulation of the deflectometer 694.1 Detector Construction . . . . . . . . . . . . . . . . . . . . . . 704.2 Primary Generator . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Gravitational field classes . . . . . . . . . . . . . . . . . . . . 76

4.3.1 Uniform Gravitational Field . . . . . . . . . . . . . . . 774.3.2 Equation of Motion . . . . . . . . . . . . . . . . . . . . 784.3.3 Principles of tracking in a field . . . . . . . . . . . . . . 80

4.4 Sensitive detector classes . . . . . . . . . . . . . . . . . . . . . 82

5 Analysis of the simulated data 855.1 Analytical solution for the

moire deflectometer . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Analysis procedure . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Geant4 parameters and output . . . . . . . . . . . . . . . . . . 945.4 Discussion of the results . . . . . . . . . . . . . . . . . . . . . 101

6 Outlook 105

A CHiral Invariant Phase Space 107

B Solving ODE 111B.1 Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . 112B.2 The Euler Method . . . . . . . . . . . . . . . . . . . . . . . . 114B.3 The Runge-Kutta Method . . . . . . . . . . . . . . . . . . . . 115B.4 Adaptive Stepsize Control . . . . . . . . . . . . . . . . . . . . 117

List of Figures

1.1 Sketch of the gedanken experiment. . . . . . . . . . . . . . . . 41.2 Comparison between the accuracy of CPT tests on leptons and

hadrons. Also show the desirable accuracy for the atomic test. 81.3 The (anti)hydrogen energy levels. The circles indicates the

spectroscopic goals of the ATRAP, ALPHA and ASACUSAexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Selected tests for WEP. Taken from [1]. Blue band shows thecurrent bounds on η from lunar laser ranging [2]. . . . . . . . 11

1.5 Selected tests for LLI. Taken from [1]. . . . . . . . . . . . . . . 121.6 Selected tests for LPI. Taken from [1]. . . . . . . . . . . . . . 14

2.1 Transition probability |c2(t)|2 as function of the frequency de-tuning ω0 − ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 The Ramsey fringes. From [3]. . . . . . . . . . . . . . . . . . . 202.3 Scheme of the level involved in the Raman transition. The

detuning ∆ prevents the level 2 to became populate, avoidingspontaneous emission. Taken from [4]. . . . . . . . . . . . . . 21

2.4 Top: Mach-Zehnder light interferometer. Bottom: Mach-Zehnder atom interferometer. The mirrors in the light in-terferometer are replaced by the second grating in the atominterferometer while the wave is splitted and recombined bythe first and the third grating, respectively. Taken from[4]. . . 25

2.5 Scheme for a Raman interferometer. Taken from [4]. . . . . . . 262.6 Left: sketch of the atomic fountain, taken from [4]. T is the

time of flight. Right: atomic fountain apparatus, taken from [5]. 272.7 Scheme of the Mach-Zehnder interferometer. Thin dashed

lines are the two gratings (G1 and G2), thin line is the de-tector and thick lines are the matter waves wavevectors. . . . . 28

2.8 a) Moire pattern formed by two identical superposed gratings,mutually rotated. b) Enlarged view. Taken from [6] . . . . . . 31

2.9 Atoms’ trajectories in the deflectometer. Taken from [7]. . . . 32

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viii LIST OF FIGURES

2.10 Scheme of the moire deflectometer. Thin dashed lines are thegratings, thin solid line is the detector and thick solid lines arethe propagating rays. Dotted-dashed lines are useful for thegeometric construction exposed in the text. . . . . . . . . . . . 34

3.1 Schematic view of the drift tube apparatus used by the Stan-ford group in their free-fall experiments with electrons. Takenfrom [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 A scheme of the antimatter factory AD . . . . . . . . . . . . . 483.3 Typical AD cycle, taken from [9] . . . . . . . . . . . . . . . . 493.4 ATHENA apparatus . . . . . . . . . . . . . . . . . . . . . . . 503.5 Shape of the trapping potential plotted against length along

the trap. The dashed line is the potential immediately be-fore antiproton transfer. The solid line is the potential duringmixing. Image taken from [10] . . . . . . . . . . . . . . . . . . 51

3.6 Antihydrogen detector . . . . . . . . . . . . . . . . . . . . . . 523.7 Reconstructed vertexes of the antihydrogen annihilation . . . . 523.8 Layout of the AD zones . . . . . . . . . . . . . . . . . . . . . 553.9 Layout of the AEgIS apparatus . . . . . . . . . . . . . . . . . 553.10 On the left, antihydrogen beam formation. On the right, moire

deflectometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.11 Magnetic field profile and inhomogeneity (see eq. (3.34)) in

AEgIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.12 Sketch of the p (bottom) and e+ (top) traps . . . . . . . . . . 603.13 Sketch of the p (bottom) and e+ (top) traps. The Ps converter

is pictured, too. . . . . . . . . . . . . . . . . . . . . . . . . . . 603.14 Mechanism of Ps production in a porous film. . . . . . . . . . 613.15 Energy level involved in the excitation process (not to scale). . 623.16 Stark structure of the n = 30 ml = 0 state of atomic hydrogen

as a function of electric field strength. The vertical axis in-dicates the detuning from the energy position of the field-freeRydberg state (1 cm−1 ≈ 0.12 · 103− eV) . . . . . . . . . . . . . 66

4.1 A schematic view of the grating. The blue rectangles are theelementary blocks projected in the y-z plane. Here a representsthe period, while w is the gap among blocks. . . . . . . . . . . 71

4.2 The grids are pictured in blue and the silicon detector in red.The moire texture is highly visible on the grids. . . . . . . . . 73

4.3 Distribution of two dimensional vector modulo, whose compo-nents are Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4 Antiproton annihilation on the first grating. . . . . . . . . . . 84

4.5 Antiproton annihilation on the second grating. . . . . . . . . . 844.6 Antiproton annihilation on the detector, along with several

annihilation on the gratings. . . . . . . . . . . . . . . . . . . . 84

5.1 Shadow pattern. Top: g = 0. Bottom: g = 9.81 m s−2. . . . . . 895.2 Folded shadow pattern. Top: g = 0. Bottom: g = 9.81 m s−2.

The shift due to gravity is highly visible. . . . . . . . . . . . . 905.3 Red curve: oscillation in the number of detected antiatoms.

Black curve: fit function. . . . . . . . . . . . . . . . . . . . . . 925.4 Detected atoms as function of the shift of the third grating

for antihydrogen longitudinal velocities in m/s: 300, 350, 400,450, 500, 550, 600. . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5 Phase shift vs time of flight . . . . . . . . . . . . . . . . . . . 945.6 Comparison between Runge-Kutta fourth order stepper with

different miss distance. . . . . . . . . . . . . . . . . . . . . . . 955.7 σx = 0, r = 0, vL = 400 m/s, σL = 0, σT = 29 m/s, Ls =

10 cm. Left: g = 0. Right:g = 9.81 m/s2. . . . . . . . . . . . . 975.8 σx = 0, r = 1 cm, vL = 400 m/s, σL = 0, σT = 0, Ls = 10 cm.

Left: g = 0. Right:g = 9.81 m/s2. . . . . . . . . . . . . . . . . 975.9 σx = 0, r = 1 cm, vL = 400 m/s, σL = 0, σT = 29 m/s,

Ls = 10 cm. Left: g = 0. Right:g = 9.81 m/s2. . . . . . . . . . 975.10 N(∆y) for antihydrogen Gaussian velocities vL centred at:

300, 350, 450, 500, 550, 600, 650 m/s. Top: g = 0. Bottom:g =9.81 m/s2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.11 Zoom of N(∆y). The red curve (300 m/s) is affected by nu-merical errors arising from the approximate solution of theequation of motion. . . . . . . . . . . . . . . . . . . . . . . . . 99

5.12 Time of flight spectrum: the red triangles indicates the peaks. 1005.13 Antihydrogen velocity spectrum: thin lines refer to different

runs with different velocities, dashed refers to their convolution.1005.14 Phase shift vs time of flight: fraction = 0.385, half-width of

restricted interval = 0.21 π . . . . . . . . . . . . . . . . . . . . 1015.15 Further output of the Geant 4 simulation and of the analysis

procedure with g = 1.6 m/s2 (left) and g = 20 m/s2 (right) . . 102

B.1 Scheme for RK4 steps. The black dots are the starting andthe final points, the white dots are the trial evaluations of thederivatives: twice at midpoints and once at endpoint. . . . . . 116

B.2 Initial conditions are xs = ys = 0 v = 40 mm/ms θ =0 rad . Stepsize and number of step are τ = 10−3 ms ν = 104 . 117

Summary

Modern physical theories rely upon symmetries, among which Lorentz co-variance represents certainly one of the most important. It is the claim thata physical law or the outcome of an experiment do not depend on the ori-entation or on the speed of the laboratory. As this principle, that is thefoundation of Special Relativity, is embodied into Quantum Mechanics, theexistence of antiparticles is a mandatory consequence. Indeed, soon after itstheoretical prediction, the antiparticle of the electron, the positron, was dis-covered in the cosmic rays, while several other antiparticles were producedin laboratory by colliding particles.In addition to the proper orthochronous Lorentz group, that is the back-bone of the Special Relativity, the microscopic systems obey also to the CPTinvariance, which means that the equations ruling those system remain un-changed under the combined transformation of charge conjugation C, parityreflection P and time reversal T. This symmetry is seen to be exact in everyexperiment ever performed and constitutes one of the successes in theoreti-cal and experimental Particle Physics. Indeed, the antiparticles are includedinto the Quantum Field Theories, which are based on the gauge symmetries,that give rise to the so-called Standard Model of elementary particles andfundamental interactions: the CPT theorem is a cornerstone of this model.In this picture of the fundamental interactions, gravitation is not yet for-mulated in terms of a renormalizable gauge field theory, differently from theother forces that are described by the Standard Model. Einstein’s GeneralRelativity, that is the modern Theory of Gravitation, does not explicitly takeinto account antiparticles, though it is founded on the Equivalence Principlethat contains the statement, called Weak Equivalence Principle, that the freefall of a body does not depend on the body’s material. Hence, the EquivalencePrinciple, that is a principle of symmetry, requires the inertial behaviour ofthe antiparticles to be indistinguishable from that of the particles.Any violation of the fundamental symmetries principles listed above leadsto the formulation of new physical theories. Moreover, recent high energyexperiments, as well as, neutrinos experiments, signal the necessity of an

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extension of the Standard Model. Indeed, the fundamental incompatibilitybetween Quantum Field Theory and General Relativity is a further hint ofnew physics beyond the Standard Model, where antimatter is a valid candi-date to test new physical theories.

Antimatter, as a composite system of antiparticles, seems absent fromthe Earth and, even, from our Galaxy. Therefore, it is necessary to produceantimatter in laboratory in order to test its properties. Particular attentionhas been devoted to antihydrogen, that is the simplest antiatomic system,since it offers the opportunity to test CPT invariance through spectroscopicmeasurements that have reached very high precision in the realm of AtomicPhysics, due to the astonishing development of trapping and cooling tech-niques. ATHENA and ATRAP are the first collaborations that succeeded inproducing a large amount of cold antiatoms, whereas ALPHA and ASACUSAdeveloped pioneering techniques to trap antihydrogen for spectroscopic pur-poses.The other relevant aspect concerning antimatter is its gravitational behaviour,which is very interesting especially from a cosmological point of view, as wellas, in the light of an extension of the Standard Model with a new type of inter-action. Indeed, a direct test of the Weak Equivalence Principle on antimatterhas never been performed: the AEgIS experiment (Antimatter Experiment:gravity, Interferometry, Spectroscopy) is devised exactly for this purpose andwill be performed, as the experiments mentioned above, at the AntiprotonDecelerator facility at CERN. AEgIS makes use of interferometric tools, bor-rowed from Atomic Physics, to measure the gravitational acceleration on theEarth’s surface of a beam of antihydrogen atoms. In particular, the deviceadopted is a moire deflectometer, that is the classical analogous of the morecommon Mach-Zehnder interferometer, since the working conditions neededby the latter instrument are unmatched at the present state of the antimattertechnology. However, the antiatomic fringes pattern displayed by the moiredeflectometer experiences a vertical shift due to gravity which can be easilymeasured through the moire effect.It is worth noting that the cited breakthrough in trapping and cooling atomshas allowed the rapid development of atom interferometry which representsan important tool for testing the modern physical theories with unprece-dented accuracy.

The present thesis studies the AEgIS moire deflectometer adopting aMonte Carlo approach: the geometric setup and the antihydrogen tracking

in the Earth’s gravitational field are simulated with the Geant4 toolkit. It isimportant to highlight that the method followed in this thesis is very peculiarsince it represents one of the first attempts to simulate gravity in a ParticlePhysics experiment. Moreover, Geant4 has never been used to simulate anatom interferometer and the software developed in this thesis can be consid-ered as the first attempt to perform this task.The first chapter expands the notions given at the beginning of this intro-duction and exhibits an updated collection of experimental results on CPTinvariance and Weak Equivalence Principle violations from recent data. Thesecond chapter deals with the basic concepts of atom interferometry, payingparticular attention to the properties of the moire deflectometer and to theexpected theoretical fringes shift due to gravity. The third chapter reviewsthe known facts about cold antihydrogen production, together with a detaileddescription of the AEgIS experiment. The fourth chapter describes the codeof the simulation and explains the method adopted to simulate gravity. Fi-nally, the fifth chapter shows the results of the simulation and presents ananalysis procedure that could be used when the experimental data will beavailable.

Sommario

Le moderne teorie fisiche si basano su princıpi di simmetria, dei quali la co-varianza di Lorentz rappresenta sicuramente uno dei piu importanti. Essoafferma che una legge fisica o il risultato di un esperimento non dipendonodall’orientamento o dalla velocita del laboratorio. Quando questo principio,che e il fondamento della Relativita Speciale, viene incluso nella MeccanicaQuantistica, si arriva in modo naturale a postulare l’esistenza delle antiparti-celle. Infatti, poco dopo le previsioni teoriche, l’antiparticella dell’elettrone,il positrone, e stata scoperta nei raggi cosmici, mentre diverse altre antipar-ticelle sono state successivamente prodotte in laboratorio.In aggiunta al gruppo proprio ortocrono di Lorentz, che e il fondamento dellaRelativita Speciale, i sistemi microscopici obbediscono anche all’invarianzaCPT, il che significa che le equazioni che governano quei sistemi rimangonoinvarianti per la trasformazione combinata di coniugazione di carica C, paritaP ed inversione del tempo T. Questa simmetria risulta essere rispettata inogni esperimento finora effettuato e costituisce uno dei successi teorici e spe-rimentali in Fisica delle Particelle. Infatti, le antiparticelle sono incluse nelleTeorie Quantistiche di Campo, che si basano sulle simmetrie di gauge, chedanno luogo al cosiddetto Modello Standard delle particelle elementari e delleinterazioni fondamentali: il teorema CPT e una pietra miliare di questo mo-dello.In questo quadro delle interazioni fondamentali, la gravitazione non e ancorastata formulata in termini di una teoria di gauge rinormalizzabile, a differenzadi altre forze che sono descritte dal Modello Standard. La Relativita Gene-rale di Einstein, che e la moderna Teoria della Gravitazione, non tiene contoesplicitamente delle antiparticelle, anche se e fondata sul Principio di Equi-valenza che contiene l’asserto, chiamato Principio di Equivalenza Debole, chela caduta libera di un corpo non dipenda dalla sua composizione materiale.Quindi, il Principio di Equivalenza, che e un principio di simmetria, richiedeche il comportamento inerziale delle antiparticelle sia indistinguibile da quellodelle particelle.Ogni violazione di questi princıpi fondamentali di simmetria porta alla for-

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mulazione di nuove teorie fisiche. Inoltre, recenti esperimenti di alta energia,come pure, esperimenti con neutrini, segnalano la necessita di un’estensionedel Modello Standard. Infatti, l’incompatibilita di fondo tra Teoria Quan-tistica dei Campi e Relativita Generale e un ulteriore indizio di nuova fisicaoltre il Modello Standard: in questo contesto l’antimateria e un valido can-didato per testare nuove teorie fisiche.

L’antimateria, intesa come sistema composito di antiparticelle, sembraassente dalla Terra e, anche, dalla nostra Galassia. Pertanto, e necessarioprodurre l’antimateria in laboratorio per testarne le proprieta. Particolareattenzione e stata dedicata all’ antiidrogeno, che e il sistema antiatomico piusemplice, poiche offre l’opportunita di testare l’invarianza CPT attraversomisure spettroscopiche che hanno raggiunto una precisione molto elevata nelcampo della Fisica Atomica, grazie al sorprendente sviluppo delle tecnichedi cattura e di raffreddamento. ATHENA e ATRAP sono le prime collabo-razioni che sono riuscite a produrre una grande quantita di antiatomi freddi,mentre ALPHA e ASACUSA hanno sviluppato tecniche d’avanguardia perl’intrappolamento dell’antiidrogeno a scopi spettroscopici.Un altro aspetto rilevante dell’antimateria e il suo comportamento gravi-tazionale, che e molto interessante soprattutto dal punto di vista cosmologico,cosı come, alla luce di una estensione del Modello Standard, con un nuovotipo di interazione. Infatti, una verifica diretta del Principio di EquivalenzaDebole su antimateria non e mai stata eseguita: l’esperimento AEgIS (An-timatter Experiment: gravity, Interferometry , Spectroscopy) e stato ideatoproprio per questo scopo e sara effettuato, come gli altri esperimenti citati,presso l’Antiproton Decelerator del CERN. AEgIS utilizza strumenti interfe-rometrici, presi in prestito dalla Fisica Atomica, per misurare l’accelerazionegravitazionale sulla superficie terrestre di un fascio di atomi di antiidrogeno.In particolare, il dispositivo adottato e un deflettometro moire, che e l’analogoclassico del piu comune interferometro Mach-Zehnder, dato che i requisiti ne-cessari per il funzionamento di quest’ultimo strumento non sono soddisfattidall’attuale tecnologia dell’antimateria. Comunque, le frange prodotte dagliantiatomi subiscono uno spostamento verticale causato della gravita che puoessere facilmente misurato attraverso l’effetto moire.Vale la pena sottolineare che i progressi sopra citati nella cattura e nel raf-freddamento di atomi hanno permesso il rapido sviluppo dell’interferometriaatomica, che rappresenta un importante strumento per testare le moderneteorie fisiche con una precisione senza precedenti.

Questa tesi studia il deflettometro moire di AEgIS adottando un approc-cio Monte Carlo: la geometria dell’apparato e il tracciamento dell’antiidrogenonel campo gravitazionale della Terra sono simulati con Geant4. Il metodoseguito in questa tesi e particolarmente originale in quanto rappresenta unodei primi tentativi di simulare la gravita in un esperimento di Fisica delleParticelle. Inoltre, Geant4 non e mai stato usato per simulare un interfero-metro atomico e il software sviluppato in questa tesi puo essere consideratoil primo tentativo in questa direzione.Il primo capitolo espande le nozioni date all’inizio di questa introduzioneed espone una raccolta aggiornata di risultati sperimentali sull’invarianzaCPT e sulla violazione del Principio di Equivalenza Debole, a partire da datirecenti. Il secondo capitolo riguarda i concetti base di interferometria ato-mica, con particolare attenzione alle proprieta del deflettometro moire e allospostamento teorico atteso delle frange, causato dalla gravita. Il terzo capi-tolo esamina quanto gia noto in letteratura sulla produzione di antiidrogenofreddo, unitamente ad una descrizione dettagliata dell’esperimento AEgIS. Ilquarto capitolo descrive il codice della simulazione e spiega il metodo adot-tato per simulare la gravita. Infine, il quinto capitolo mostra i risultati dellasimulazione e presenta una procedura di analisi che potrebbe essere utilizzataquando i dati sperimentali saranno disponibili.

Chapter 1

Theoretical motivations forstudies with atomic antimatter

The path leading to the current knowledge about particles and antiparti-cles starts with the Dirac equation, written for the first time in 1928, andpasses through several experimental verifications, such as the discovery ofthe positron, made by Anderson in 1932, of the antiproton, made by Segreand Chamberlain in 1955, of the antineutron in 1956 and of the antideuteronwithin a decade (1965). After producing the elementary particles, there wereattempts to produce bound system of antiparticles, i.e. antiatoms: in recentyears, the production of large quantities of antihydrogen atoms, which arebound states of an antiproton with a positron, becomes a matter of fact.This chapter presents some theoretical ideas which have been stimulating theresearch on antihydrogen. This system represents a fundamental testbenchfor the modern physical theories which are based on symmetry principles.The planned measurements on antihydrogen extend the phenomenologicalbasis of the Quantum Field Theories and of the General Relativity sincethey have not been ever tested on the atomic antimatter.The concept of antimatter is reviewed in the first section and along withits relation with the CPT invariance of the Quantum Field Theories. Thissymmetry is of a central importance in the Elementary Particle Theory. Inthe subsequent sections, the Equivalence Principle of the General Relativityis presented, paying particular attention to its role in Newtonian mechanics.Successively, the relation between antimatter and the Theory of Gravitationis discussed. A quantum theory of gravitation is far from being establishedand, if antimatter fails to obey the Equivalence Principle, several theoreticalscenarios are open. The fourth section collects some experimental boundson quantities that can reveal violation of the CPT symmetry. These ex-periments measures possible CPT violation by seeking modifications in the

1

2 CHAPTER 1. THEORY FOR ANTIMATTER STUDIES

values of constants related to leptons and antileptons and to hadrons andantihadrons, hence atomic spectroscopic techniques can be successfully ap-plied to determine differences in the antihydrogen spectra with respect tothe hydrogen ones. Finally, some tests on the Equivalence Principle are ex-plored by testing its different statements: the Weak Equivalence Principle isof particular relevance for the present thesis.

1.1 CPT theorem

The efforts spent to build a model which merges Quantum Mechanics withSpecial Relativity has led to the Quantum Field Theory (QFT). In this frame-work, antiparticles come as a consequence by demanding adherence to thecausality principle, as pointed out by Feynman [11]: indeed, according to theFeynman-Stueckelberg interpretation of the QFT, a particle moving forwardin time can be seen by an inertial observer as an antiparticle moving back-ward in time.An antiparticle state is obtained from a particle state by applying to it adiscrete Lorentz transformation: the charge conjugation operator C, whichinduces a unitary transformation in the Hilbert space of the states1. Thisoperator changes the internal quantum numbers of a state, as for exampleelectric charge or flavour numbers, to the opposite ones. The two other dis-crete transformations relevant to the present context are the time reversaloperator T and the parity operator P . The former inverts the sign of thetime coordinate, while the latter causes the reflection of the spatial coordi-nates. The intimate connection between antiparticles and the symmetries ofthe QFT results in the CPT theorem. This theorem claims that a QFT is in-variant under the composition of the operators time reversal T , parity P andcharge conjugation C. In a flat spacetime with coordinates x = xµ = (ct,x),the action of the antiunitary CPT operator, Θ, on a conserved vector currentis given by

ΘJµ(x)Θ−1 = −Jµ(−x) (1.1)

and on the energy-momentum tensor

ΘT µν(x)Θ−1 = T µν(−x) . (1.2)

The final formalisation of the statement in the ’50s by Luders [13] requiresthat the QFT is built from fields belonging to finite dimensional representa-tions of the Lorentz group, undertaking local interactions invariant with re-spect to the proper Lorentz group and described by an hermitian Lagrangian.

1Neutrinos behave in a slight different manner which is not addressed here, see [12].

1.2. EQUIVALENCE PRINCIPLE 3

The CPT theorem is a cornerstone of the QFT upon which rests the standardmodel (SM) of the known fundamental (anti)particles and their interactions.Such quantum description includes the strong and the electroweak forces. Re-markably, it has been demonstrated that the weak sector of the SM violatesP , C, CP and even T , instead the CPT symmetry has been verified in everyexperiment ever done. Thus, CPT invariance turns out to be the minimalrequirement for the existence of antiparticles within a QFT. In other words,the theorem assures that every particle has an antiparticle with oppositeinternal quantum numbers and the same inertial mass.

1.2 Equivalence Principle

As regards the remaining force, gravitation, which does not fit in the frameof the SM, it is described by a classical theory, the General Relativity (GR).It is firmly based on the Einstein’s Equivalence Principle (EEP) which, inEinstein’s own words [14], sounds like: ”The laws of physics must be of suchnature that they apply to systems of reference in any kind of motion.” Whatis implicit in this statement is that the reference frames occupy only a smallregion of spacetime, hence they are called local inertial frames (LIFs). Thesmallness of the reference frame is dictated by the gravitational field strengthwhich has to vary little over the extention of the reference frame.At a closer look, the EEP states that [15]:

1. The trajectory of a freely falling body is independent of its internalstructure and composition, known as the Weak Equivalence Principle(WEP).

2. The outcome of any local non-gravitational experiment is independentof the velocity of the freely falling reference frame in which it is per-formed, known as the Local Lorentz Invariance (LLI).

3. The outcome of any local non-gravitational experiment is independentof where and when in the universe it is performed, known as the LocalPosition Invariance (LPI).

A non-gravitational experiment could be the measurement of an electric fieldexperienced by a particle, but obviously not the Cavendish experiment. Ithas also been speculated by Schiff that WEP implies LLI and LPI: there isnot a rigorous proof but some plausible arguments can be formulated [1].Einstein was able to envisage the EEP by looking at the Eotvos experiment,which verifies with an accuracy of one part per twenty milion, what the WEP


claims: the gravitational mass mG is equal to the inertial mass mI

mG = mI . (1.3)

The former mass is the one entering in the Newton’s law of gravity

F = −G mGm′G

r2(1.4)

where G is the Newton’s constant and r is the distance between the bodieswith gravitational masses mG and m

′G, while the latter is the one entering in

the Newton’s second law

F = mIa . (1.5)

The WEP has been known since Galilei who found that different bodies,dropped from the top of the leaning Tower of Pisa, fall with the same ac-celeration. An interesting consequence of the WEP can be figured out bythinking at a non-gravitational experiment performed inside the cabin of afree falling elevator in a uniform gravitational field2 g. In this gedanken ex-periment, see figure 1.1, a voltage difference is applied between the floor andthe ceiling of the cabin while a bunch of charged particles travels with awell-defined initial velocity between its walls.

Figure 1.1: Sketch of the gedanken experiment.

2The elevator is a non-rotating cabin falling along the field lines which are parallel toeach others.

1.3. CPT, GRAVITY AND ANTIMATTER 5

An inertial observer outside the elevator attributes the motion of a par-ticle to a force F = mIa which is the sum of an electric force

FE = mIα (1.6)

where α is the acceleration in such an external field, with the force ofgravity

FG = mGg (1.7)

where g is the gravitational acceleration. Since the Newton’s secondprinciple holds, the total force on the particle is

mIa = mIα+mGg . (1.8)

The observer inside the cabin measures the acceleration of the particlegiven by

a′= a− g (1.9)

as follows from ”classical” accelerations addition. The force on theparticle in this reference frame is given by

mI(a− g) = mIα+mGg −mIg (1.10)

as follows by substituting the first term in the LHS with its value in(1.8). If the WEP is valid, then the non-gravitational part of force act-ing on the particle FE = F−FG is equal to the external force measuredby the outer observer. Therefore, the Newton’s second principle holdseven in the reference frame of the cabin

mIa′= mIα . (1.11)

Though this example does not take into account Lorentz invariance, it showshow the WEP works: the free falling elevator is mechanically equivalent toan inertial reference frame without gravity [16].

1.3 CPT, gravity and antimatter

Being GR a classical theory, it does not explicitly take into account antimat-ter. Hence it has been speculated that antimatter may fall in the Earth’sgravitational field in a different manner with respect to the ordinary mat-ter. The CPT theorem assures that the inertial masses of a particle and itsantiparticle are equal

mI = mI (1.12)


but it does not imply that

mG = mI = mI = mG (1.13)

since the last equality has not yet tested directly. This means that a WEPviolation does not necessarily results in a violation of the CPT symmetry.The reasoning is as follows: the CPT theorem grants that an antiapple fallson an antiearth in the same way as an apple does on the Earth, but it isimpossible to say what happens to an antiapple falling on the Earth.A suggestive situation comes from the possibility the antimatter might ex-perience ”antigravity”, namely that an antiparticle ”falls up” in the Earth’sgravitational field. Nevertheless, this hypothesis has to be discarded on thebasis of the arguments due to Schiff, Morrison and Good, which are exten-sively discussed in [17]. For instance, Schiff looked at the role of the EEP inQFT. He ruled out antigravity by considering the contribution of positrons,through the vacuum polarisation diagrams, to the mass of the atoms, con-cluding that if there were antigravity, the effect would be so huge that itshould become apparent in the Eotvos experiment.From a theoretical field viewpoint, GR is a tensor theory, in the sense thatthe gravitational force is mediated by a spin two boson. Furthermore, it isunderstood that every even-integer spin bosons mediate force that are al-ways attractive, instead odd-integer spin bosons give rise to field that can beattractive or repulsive depending on the sign of the charges involved. Thevalidity of this result is based on the CPT theorem, therefore the possibilitythat matter-antimatter repels each other can be ruled out.A question related to this line of thought comes from the observation of theso-called baryon-lepton asymmetry in our part of universe. At present, thereis no evidence of an equal abundance of matter and antimatter, though afterthe Big Bang the production of the same amount of both of them shouldhave taken place. In the present day, universe antimatter seems to havedisappeared: such domination of matter on antimatter can be seen as somekind of broken symmetry, for instance of gravitational origin.However, the possible difference between matter and antimatter as regardsthe gravitational interaction is really rising the more general question whetherdifferent forms of energy gravitate differently. This conjecture relies upon theincompatibility of the QFT and GR as matter of principle. Indeed, while theformer is taking a many-path point of view [18, 19], the latter deals withgeodesic motion in spacetime [16, 20]. Thus, the required knowledge of pre-cise information about the position and momentum coordinates in GR isinconsistent with the quantum approach. Moreover, the CPT theorem hasnever been demonstrated in curved spacetime.Currently, several theoretical scenarios are open since various attempts have

1.4. TEST OF CPT 7

been made to find out a theory which is not in contrast with the fundamentalprinciples of QFT and GR. As an example of the difficulties concerning theformulation of a well-motivated and self-consistent theory which allows anasymmetry between matter and antimatter, it has been proved that the vi-olation of the CPT symmetry necessarily imply the violation of the Lorentzinvariance [21].On the other hand, a quantum gravity model, in addition to the spin-twoboson mentioned above, called graviton, can introduce two new partners: aspin-zero boson, called graviscalar, and a spin-one boson, called gravivector.The simplest linearised static potential for the exchange of only one gra-vivector and one graviscalar, with ranges v and s, respectively, between twopointlike masses m and m

′at distance r, can be given in the form

V = −Gmm′(1∓ ae−r/v + be−r/s

r

)(1.14)

where a and b are the coupling strenghts relative to the Newton’s constant G.The sign in front of the vector-mediated interaction is crucial for the antimat-ter behaviour. Such a model shows composition dependent couplings. Thesenew interactions have not yet been detected due to a notable weakness of thestrenghts involved, several orders of magnitude less than the one of gravity.Otherwise, there are some mechanism inducing a cancellation between theseforces at some level of precision[17]. On the chances that the latter scenariois practicable, a matter-antimatter asymmetry becomes measureable.A better comprehension of the phenomena involved with the gravitation ofthe antimatter definitely comes from the experiments devoted to test its in-ertial properties, even though it is possible to establish some bounds to theasymmetry [22].

1.4 Test of CPT

Antihydrogen offers the possibility to extend the metrological basis of thelaws of Physics. The measurement of fundamental constants provides thebest tests for the basis of the current physical knowledge. Precise tests ofthe CPT symmetry strengthen the physical theory of the elementary parti-cles, since it is not well founded as the energy-momentum conservation, forexample.The validity of the CPT theorem has been tested for the masses of the neutralkaons K0 and K0 to a very good level of accuracy∣∣∣∣m(K0)−m(K0)

m(K0)

∣∣∣∣ ≤ 5 · 10−18 . (1.15)


However this result is indirect and model-dependent, since it depends on theconnection between the decay amplitude and the kaon mass matrix. Directtests are the ”equality” of the electron e− and positron e+ gyromagneticratios ∣∣∣∣ g(e+)− g(e−)

[g(e+) + g(e−)]/2

∣∣∣∣ = (−0.5± 2.1) · 10−12 (1.16)

and the ”equality” of the proton p and antiproton p cyclotron frequenciesωc = eB/m in the same magnetic field B∣∣∣∣ωc(p)− ωc(p)

ωc(p)

∣∣∣∣ ≤ 4 · 10−8 . (1.17)

Figure 1.2: Comparison between the accuracy of CPT tests on leptons andhadrons. Also show the desirable accuracy for the atomic test.

The arrangement of the energy levels in the hydrogen atom should be thesame as in the antihydrogen atom, according to the CPT symmetry of theQuantum Electrodynamic, which is the simplest among the Quantum fieldtheories. The structures present in the spectra comprise the Bohr levels ofthe gross structure, the Dirac fine structure with the Lamb shifts and thehyperfine splitting. The 1S−2S two photon transition, depicted in figure 1.3,can be used to compare the two spectra with high precision. In particular,the 2S metastable level has a lifetime of 1/8 s, which indicates an attainableprecision of 10−15 to 10−18 [23]. Such transition allows to determine theRydberg constant for the antihydrogen, which can be expressed in terms ofthe p and e+ charges, ep, ee+ , respectively, and masses, mp, me+ , respectively,

RH =mpme+

mp +me+

epee+

8ε0ch3(1.18)

1.5. TEST OF EEP 9

where ε0 is the dielectric permeability, c is the speed of light and h isthe Planck’s constant. Assuming that the measurement with antihydrogenreaches the same accuracy obtained for the hydrogen, this CPT test yields aresult comparable to the one involving the neutral kaons masses.A further example of spectroscopic measurement of the CPT symmetry in-volves the hyperfine splitting of the (anti)hydrogen ground state 1S. Thesublevels emerge from the interaction between the (positron) electron spinSe and the (anti)proton spin Sp that form the total angular momentumF = Se + Sp by angular momentum addition. The transition, depicted infigure 1.3, involves the F = 0 and F = 1 states and it is measured for thehydrogen atom with an accuracy of the order of 10−14.A detailed analysis of the effects of some CPT violating models is presentedin [24].

Figure 1.3: The (anti)hydrogen energy levels. The circles indicates the spec-troscopic goals of the ATRAP, ALPHA and ASACUSA experiments.

1.5 Test of EEP

In order to detect EEP violations, the three statements listed at the beginningof section 1.2, which constitute the bulk of this fundamental principle, canbe tested individually.A body, whose inertial mass mI is no longer equal to its gravitational mass


mG, experiences an acceleration a given by

a =mG

mI

g (1.19)

due to a gravitational field g. Since the internal energy of a typical laboratorybody is made up of several contributes, such as rest energy, kinetic energy,electromagnetic energy etc., a WEP violation results if one of these forms ofenergy contributes to mG differently with respect to mI . This condition canbe written as

mG = mI +∑

i

ηAEA

c2(1.20)

where EA is the internal energy of the body due to the interaction A andηA is a dimensionless parameter that accounts for the WEP violation of thatinteraction. The Eotvos ratio gives the differential acceleration between twobodies

η(1, 2) = 2|a1 − a2||a1 + a2|

=∑

A

ηA

(EA

1

m1c2− EA

2

m2c2

)(1.21)

where the subscript I in the mass is dropped, hence experimental constraintson η result in limit on the ηA [1].Modern experiments with the torsion balance reach accuracies that are greaterthan the classical Eotvos experiment by several order of magnitude (see fig.1.4). The typical setup consists of two objects of different composition con-nected by a rod or placed on a tray and suspended by a thin wire. Then, ifthe gravitational acceleration of the bodies differs, the suspended wire feelsa torque related to the mutual orientation between the wire and the grav-itational field. For instance, the Eot-Wash experiments [25] performed atthe University of Washington compare the acceleration of various materialstoward local topography on Earth, movable laboratory masses, the Sun andthe galaxy. The Eot-Wash experiments, which have the goal of searchingfor a new force, have led to confirmation of the WEP validity, since theyobtained a null result (green band in fig. 1.4). Their results expressed asEotvos ratio are

η(Be,Al) = (−0.2± 2.8) · 10−12 (1.22)

η(Be,Cu) = (−1.9± 2.5) · 10−12 (1.23)

and are printed in [25], where it is explained how the source of systematicerrors were careful eliminated. By considering the fractional contributions ofthe forces between the nucleons to the inertial mass of the Cu and Be testbodies, it is possible to set limits on the WEP violations ηS and ηEM occur-ring, respectively, in the strong interactions and electromagnetic. Treating

1.5. TEST OF EEP 11

Figure 1.4: Selected tests for WEP. Taken from [1]. Blue band shows thecurrent bounds on η from lunar laser ranging [2].

the nucleons as elementary particle, the limit set by (1.23) on the strongforce is given by

ηS =η(Be,Cu)

fSBe − fS

Cu

= (−7.7± 10.1) · 10−10 (1.24)

where fSBe = −6.685 · 10−3 and fS

Cu = −9.158 · 10−3 are the fractional con-tribution of the nuclear binding energy to the inertial mass of the Be andCu, respectively. As regards the electromagnetic energy, the contribution tothe internucleon force are fEM

Be = 2.48 · 10−3 and fEMCu = 4.95 · 10−4, which

results in

ηEM =η(Be,Cu)

fEMBe − fEM

Cu

= (9.6± 12.6) · 10−10 . (1.25)

The second piece of the EEP, the LLI, can be tested by searching for spaceanisotropy. A useful way to deal with this violation is to suppose that theelectromagnetic interaction suffers a Lorentz invariance violation, through achange in the velocity of propagation of the electromagnetic disturbances c


relative to the limiting speed of a material test particle c0 = 1, by a suitablechoice of units. Thus, c 6= 1 selects a preferred rest frame. Such a Lorentzviolating interaction produce shifts in the atomic and nuclear energy levelsthat depend on the quantum numbers of the states and on the orientation oftheir quantisation axis relative to the selected frame. If the states are affecteddifferently by the violation, an energy level appears shifted with respect toone another. The Drever experiment [26] measures with high precision theground state of the 7Li nucleus placed in a magnetic field through magneticresonance techniques. Considering the ordinary electromagnetic interaction,the ground state is splitted in four equally-spaced sublevels due to the cou-pling between its total angular momentum J = 3/2 and the magnetic field.An anomalous energy shift can be described by the parameter δ = c−2 − 1that quantifies the violation of LLI. Recent experiments (see fig. 1.5) havedetermined more accurately the upper bound of δ through atomic measu-rements carried out by applying the atom manipulation techniques, such aslaser cooling and trapping.A test on the LPI consists in the measurement of the gravitational redshift

Figure 1.5: Selected tests for LLI. Taken from [1].

1.5. TEST OF EEP 13

occurring in the frequency of the photons exchanged by two sources placedat different heights in the Earth’s gravitational field. The two sources actas ideal clocks with equal frequencies at rest in a gravitational potential andthe shift can be estimated with the parameter

Z =∆ν

ν= −∆λ

λ(1.26)

In the case of Newtonian gravitational potential ∆U between the receiverand the emitter, the associated redshift is

Z =∆U

c2(1.27)

and a deviation from this value can be taken into account by introducing theparameter α via

Z = (1 + α)∆U

c2(1.28)

that depends upon the nature of the clock whose shift is being measured. Avery precise measurement of α, discussed in [27], was performed by compar-ing the wave signals generated from a hydrogen maser located in a spacecraft,launched nearly vertically upward to 104 km, and at an Earth station. Dueto the masers’ frequency stability and the sophisticated acquisition system,the data analysis yields |α| < 2 · 10−4. On the contrary, the Pound-Rebkaexperiment [28] is less accurate than the previous (see fig. 1.6). The gravita-tional shift in the frequency of the gamma photons emitted by 57Fe nucleusis determined by means of the Mossbauer effect [29], that is related to thevibrational properties of the crystals where nuclei absorb or emit gammaphotons without recoiling.

Each of the cited experiments tests the EEP in the matter realm, hencethe tasks of performing gravitational measurements on antimatter is notsubsidiary, in order to extend the phenomenological basis of a Theory ofGravitation. The AEgIS experiment, which is the subject of this thesis, isdevoted to search for WEP violation on antihydrogen with an accuracy of1%.


Figure 1.6: Selected tests for LPI. Taken from [1].

Chapter 2

Atom interferometry

As shown in the chapter 1, the relevance of accurate tests on the basics ofGeneral Relativity is outstanding. Atom interferometry represents a power-ful tool which can be set more stringent bounds on putative violations of theEquivalence Principle (see sec. 1.5).This chapter is devoted to develop some basic concepts about interference ofthe atomic matter waves and related phenomena, paying particular attentionto the far-field limit, where Fraunhofer diffraction takes place, and to the casein which the (anti)atom diffraction is negligible: this is the classical regimewhere the interference fringes are substituted by a shadow pattern. In thefollowing, it is shown that devices, based on the quantum behaviour of thematter (i.e. interference) or on the classical deflection of the (anti)atoms, arevery sensitive tools that could make precise measurements of quantities rele-vant to fundamental physics. It is also shown that a particular setup, calledMach-Zehnder interferometer, has the same response function wheather itworks in the quantum regime or it does not. In the latter case, it is calledmoire deflectometer.The first two sections deal with atom optics which is a discipline that pro-vides methods to manipulate atoms in such a way that the relative phase ofthe matter waves is under the control of the experimenter, i.e. preserving thecoherence. This collection of techniques is so named because it borrows sev-eral ideas from classical light optics as, for example, mirrors and diffractiongratings. The coherent atoms manipulation, introduced in the first section,finds its main application in the constructions of interferometers that is thesubject of the second section. The last section discusses the properties of themoire deflectometer. Among the capabilities of such a device, there is thehigh inertial sensitivity that enables to accurately measure the accelerationexperienced by the apparatus due to gravity or rotation, as reported in theexperiment described in [7].

15

16 CHAPTER 2. ATOM INTERFEROMETRY

The AEgIS experiment, which is the subject of this thesis, has adopted amoire deflectometer as the instrument for the measurement of the accelera-tion of neutral antimatter in the Earth’s gravitational field, which is the goalof the AEgIS collaboration.

2.1 Atom diffraction

The wave character of matter was established by Davisson and Germer in1927, when they fired a beam of electrons through a nickel crystal, findingthat these particles form a fringe pattern very similar to the one formed bylight wave diffracting on a grating. The equations governing the electromag-netic radiation, the Maxwell’s equations, are very different from the equationsgoverning non-relativistic matter, the Schrodinger’s equation, nevertheless,both give rise to a quite similar wave-like behaviour. The wave-function, so-lution to the Schrodinger’s equation, is currently understood as a probabilityamplitude and it is the modern version of the hypothesis formulated by deBroglie in 1924, according to which a wave is associated to the motion ofevery particle. If a particle has momentum p, the wave has a wavelengthexpressed by the de Broglie relation

λdB =h

p(2.1)

where h is the Planck’s constant.Matter exhibits a wave-like behaviour when its de Broglie wavelength is com-parable to the relevant length scales in the system under study. Hence amatter wave can be diffracted by a slit of the appropriate dimension or by aperiodical grating: diffraction is a distinctive feature of the wave behaviour.When the matter waves add-up coherently, they interfere forming a regu-lar pattern of fringes, that is an alternating sequence of local maxima andminima of the atom density, stemming, respectively, for constructive anddestructive interference. Two waves are coherent if there is a well-definedrelationship between their phases, because it is this relative phase which mo-dulates the fringe pattern. Coherence is a fundamental property because aill-defined relative phase causes a random superposition of the matter waves,washing out the interference fringe pattern.An interesting feature of the atom optics is the interplay between light andmatter. Since 1930 it has been proposed by Kapitza and Dirac that a standingwave of light can diffract electrons, but only recently it has been proved exper-imentally that such light gratings produce Fresnel and Fraunhofer diffractionof matter waves in much the same way as material gratings do for laser light.

2.1. ATOM DIFFRACTION 17

A very general and unified approach, that treats either the material and thelight gratings on the same foot, is given by C. J. Borde in his contributionin [30]. In such a theoretical framework, the diffraction comes from a scat-tering potential with spatial periodicity. In the following, an example of thereversed role of light and matter is presented.

2.1.1 The Ramsey fringes pattern

The following calculation is suited for a two-level atom (ground state plus anexcited state) and makes use of the time-dependent perturbation theory. Theproblem can be stated as follows: What happens to the levels’ populationwhen the quantum system under study interacts with a classical radiationfield? This semiclassical treatment (i.e. the radiation is considered a classicalelectric field while the atom obeys to the quantum mechanical laws) givesthe response of a two-level atom subjected to two pulses of radiation.The time-dependent Schrodinger’s equation is

ı~∂Ψ

∂t= HΨ (2.2)

where the Hamiltonian H has two parts

H = H0 +HI(t) . (2.3)

The part depending on the time, HI(t), describes the interaction with anoscillating electric field which perturbs the eigenstates of H0. Setting thewavefunction for the level with energy En = ~ωn as

Ψn(r, t) = ψn(r)e−ıωnt , (2.4)

the spatial wavefunctions ψn(r) satisfy

H0ψ1(r) = E1ψ1(r)

H0ψ2(r) = E2ψ2(r) . (2.5)

These equations amount to say that ψn(r) are the unperturbed eigenfunctionsof H0 and En are the relative unperturbed eigenvalues. Since these wavefunc-tions are not stationary states with respect to the full Hamiltonian H, thesolution to (2.2) at any instant of time can be expressed as a superpositionof them

Ψ(r, t) = c1(t)ψ1(r)e−ıω1t + c2(t)ψ2(r)e

−ıω2t . (2.6)

Besides, the usual normalisation requirements impose

|c1(t)|2 + |c2(t)|2 = 1 . (2.7)


The interaction Hamiltonian HI(t) corresponds to the energy of an electricdipole d placed in the electric field E(t) = E0 cos(ωt)

HI(t) = er · E0 cos(ωt) (2.8)

where d is thought as arising from a single electron of charge −e and position,with respect to atom’s centre of mass, r. Furthermore, it is useful to considerthe electric field amplitude |E0| uniform over the atomic size. This dipoleapproximation holds when λ a0, where λ is the radiation wavelength anda0 is Bohr’s radius.The substitution of equation (2.6) in (2.2) leads to the coupled differentialequations for the coefficients c1(t) and c1(t)

ıdc1(t)

dt= Ω cos(ωt)e−ıω0tc2(t) (2.9)

ıdc2(t)

dt= Ω∗ cos(ωt)eıω0tc1(t) (2.10)

where ω0 = (E2−E1)/~ is the atom’s resonance frequency and Ω is the Rabifrequency. This fundamental parameter gives the strength of the interactionand it is defined, for radiation linearly polarised along the x-axis E(t) =|E0 |i cos(ωt), by

Ω =e

~|E0|

∫d3rψ∗1(r)xψ2(r) . (2.11)

The initial conditions c1(0) = 1 and c2(0) = 0 allow to integrate (2.9) and(2.10), yielding

c1(t) = 1 (2.12)

c2(t) =Ω∗

2

1− exp[ı(ω0 + ω)t]

ω0 + ω+

1− exp[ı(ω0 − ω)t]

ω0 − ω

. (2.13)

A further approximation, called rotating-wave, makes use of the fact that theradiation usually has a frequency close to the atomic resonance frequency,hence ω0 + ω ≈ 2ω0 and the first term inside the curly brackets can beneglected. The probability of finding the atom at the upper level at the timet is given by the squared module of (2.13)

|c2(t)|2 =1

4|Ω|2t2 sin2 ξ

ξ2(2.14)

where ξ = (ω0−ω)t/2. The functional form of (2.14), shown in figure 2.1, isthe same as the Fraunhofer diffraction of light passing through a slit.


Figure 2.1: Transition probability |c2(t)|2 as function of the frequency detun-ing ω0 − ω.

The previous formula gives the response of a two-level system to a singlesquare pulse of radiation, i.e. the amplitude of the oscillating electric field isconstant from the time t = 0 to time τ and zero otherwise. Now, the goalis to figure out the transition probability when the atom interacts with twopulses of this type, once from t = 0 to τ and again from t = T to T + τ . Inthe same approximation as above (|ω0 − ω| ω), the integration of (2.10),with the same initial condition, gives

c2(t) =Ω∗

2

1− exp[ı(ω0 − ω)τ ]

ω0 − ω+ exp[ı(ω0 − ω)T ]

1− exp[ı(ω0 − ω)τ ]

ω0 − ω

.

(2.15)This formula is a sum of two terms: the first arises from the interactionwith the first pulse, while the second one arises in a similar way from thesecond pulse but it is weighted by a phase factor exp[ı(ω0−ω)T ]. Of course,each pulse produces the same effects on the system when acts independently.Nevertheless, when both pulses are present, the amplitudes interfere yieldinga transition probability

|c2(t)|2 =

∣∣∣∣Ωτ2∣∣∣∣2[sin(∆τ/2)

∆τ/2

]2

cos2

(∆T

2

)(2.16)

where ∆ = ω0 − ω is the frequency detuning. The overall envelope is mod-ulated by sin2ξ/ξ2 while the cosine function determines the positions of themaxima. Figure 2.2 shows this fringes pattern, called Ramsey fringes. Theyshare the functional form with the fringes seen in the classical Young’s double-slit experiment with light.The reversed role of light and matter in this quantum version of the Young’s

experiment plays a central role in the development of atom interferometry.Moreover, it is an amazing expression of the wave-particle duality.


Figure 2.2: The Ramsey fringes. From [3].

2.1.2 Diffraction gratings for atoms

The starting point to build an interferometer is a coherent beam splitter. Thistask is usually carried out by diffracting the incident atom beam through adiffraction grating, which can be a nanofabricated structure or a standingwave of light. A grating of period a generates multiple momentum compo-nents for the scattered wave, transferring to the nth component an amountequal to

δpn =nh

a= n~G (2.17)

where h is the Planck’s constant and

G =2π

a(2.18)

is the reciprocal lattice vector of the grating [31]. For a beam centeredaround p in momentum space and for normal incidence, the diffraction angleis approximated by

θn ≈δpn

p=nλdB

a‘ (2.19)

where λdB is the de Broglie wavelength given by (2.1).The current material gratings achieves typical periodicity of 100 nm. Theystop a fraction of the atomic beam depending on the slits width, but transmitthe remaining almost unperturbed. Several studies [32] [33] have establishedthat the van der Waals interaction, between the atoms and the surfaces ofthe grating, plays a central role in such a beam splitter, preventing missingorders.


Figure 2.3: Scheme of the level involved in the Raman transition. The de-tuning ∆ prevents the level 2 to became populate, avoiding spontaneousemission. Taken from [4].

The construction of material gratings relies upon nanotechnology which al-lows to build single slits, double slits, diffraction gratings and structures witharbitrary pattern. These nanofabricated devices can diffract every atomicand molecular species and can be produced with a finer periodicity with re-spect to the standing wave of light. Nevertheless, the disadvantages comefrom the reduced transmitted flux and from the material clogging up.The composite nature of the atoms (and the molecules) opens the road tomanipulations of the matter waves that has no counter-part in classical lightoptics. As shown in 2.1.1, the stimulated absorption of atoms producesdiffraction phenomena. Generally speaking, atoms, that undergo a change inthe internal state under the control of the experimenter, behave coherently.For example, the absorption of a photon causes the atoms to recoil withvelocity ~k/m, where λ = 2π/k is the radiation wavelength, and creates asuperposition of states whose populations depend on the Rabi frequency, thedetuning from the atom resonance and the interaction time trough a relationsimilar to (2.14).The cited examples are useless since spontaneous emission destroys the co-herence, unless long-lived metastable states are involved. Random decays ofthe atoms can be avoided with a second radiation source making the beamsplitting mechanism a two photon process based on stimulated transitions.The best known of such processes is the Raman transition (see figure 2.3)which gives rise to the same transition probability as (2.14) but with a gener-alised Rabi frequency. On the whole, the atom gains a momentum ~(k1−k2)where the two wavevectors characterised, respectively, the first and the se-cond laser beam. It is customary to use as second radiation source the firstlaser beam reflected backwards by, for example, a common optical mirror.


This counter-propagating beams creates a standing wave which changes theatom momentum by units of 2~k, since k1 ∼ −k2. This lattice has a periodequal to a = λ/2, where λ is the radiation wavelength.It is worth noting that the standing waves can also work as mirrors. Indeed,a light pulse can transfer all the atoms from their ground state to an excitedone, if it lasts for a sufficiently long time. Such a pulse, devised with theappropriate frequency and intensity, causes the total depletion of the groundlevel, while the other is filled coherently, and imparts the same amount ofmomentum to each atom. This interaction that reverses the population iscalled a π pulse. Obviously, when the radiation causes a superposition ofstate, i.e. beam splitting, it is called a π/2 pulse. The application of thesetechniques to interferometry will be shown in the following.The periodic structure generated by standing waves is generally called opti-cal lattice and it is described by a reciprocal the lattice vector G = k1 − k2.It can be classified according to the thickness and to the relative strengthbetween the energy scale of the grating (∼ ~G2/2m) and the atom’s recoilenergy. A good explanation of the physics involved in these distinctions isgiven in [34].

p atom momentum vectorλdB = ~

pde Broglie wavelenght

λ radiation wavelenghta grating periodG = 2π

au grating reciprocal vector

u unit vector ortoghonal to pk = 2π

λphoton wavector

~G = 2~k atom recoil momentum (two photon transition)a = λ/2 standing wave period

Table 2.1: Relevant quantities in the text

2.2 Interference fringe pattern

Atom interferometry is a rapidly growing field that develops several theore-tical and experimental tools in order to gain a deeper comprehension of thequantum models and to improve the accuracy of some fundamental physicalconstants. It collects a variety of techniques that are applied to the coherentmanipulation of the translational motion of the atoms. Some fundamentalstudies concern the Bose-Einstein condensate, the decoherence phenomenaand the related limits on the size of the bodies having a quantum behaviour.

2.2. INTERFERENCE FRINGE PATTERN 23

On the other hand, several quantities relevant to Physics, such as the finestructure constant or the Newton’s constant or the molecular/atomic pola-rizability, can be measured with better accuracy than one attainable withthe best optical interferometers [31]. The superior accuracy obtainable withmatter wave interferometers is due to the shorter (de Broglie) wavelengthof the atoms with respect to lasers wavelength and the atom mass allows toexplore gravitational interactions. Moreover, different atoms can be used tomake measurements over a wide range of variations of the atomic properties.The present work especially focuses on the inertial sensitivity of the atominterferometers, in particular on their use as gravimeter.The matter wave interference arises from the phase difference accumulatedby the splitted waves along the interferometer arms. It results in a oscilla-tion of the detected atomic population whose periodicity is determined bythe beam splitting mechanism. This modulation corresponds to a sinusoidaldependence of the recorded intensity on the phase difference. For exam-ple, two matter waves, travelling along different arms of the interferometer,change their relative phase as soon as a different interaction is applied to thearms. By varying the interaction strength, the consequent phase shift allowsto study the interaction of the atoms with such an external field.

2.2.1 Building an interferometer

There are several types of atom and molecule interferometers that have beenbuilt since 1991, when the first one was realised. They results from the devel-opment of the atom optics required to localize atoms and to transfer atomsbetween two different localitions in a coherent manner. These techniquesconcern the manipulation of either the internal atomic states or the positionand momentum degrees of freedom, therefore interferometers can be dividedin two broad categories:

• atomic state interferometers where the beam splitter produces a super-position of internal states,

• de Broglie wave interferometers where diffraction creates distinct pathin the position space and does not change the internal state of theatoms.

Every interferometric measurement, even the ones with light, goes troughthe following five steps:

1. initial state preparation,

2. coherent splitting,


3. free propagation,

4. coherent recombination,

5. detection.

In the initial step, the atoms (or the molecules) are usually prepared inmomentum space, since localisation in position space is hindered by the un-certainty principle and most beam splitting mechanism are momentum de-pendent. Therefore a coherent source is produced reducing the momentump of an ensemble of atoms and its spread ∆p throughout techniques that arereferred as slowing and cooling, respectively. The coherent length ∆x is themaximum distance over which all the points in the wave have a well-definedrelative phase, and it is related to the momentum spread [4] by

∆x∆p ∼ ~ , (2.20)

hence a narrower momentum distribution means a larger coherence length.The momentum selection can be as simple as two collimating slits that re-duce the transverse momentum. More preferably, laser cooling and magnetictrapping can be used to select the atoms’ phase space.The second and the fourth step are accomplished by employing materialdiffraction gratings or laser pulses, as described in 2.1.2. During the prop-agation of the atoms inside the interferometer, the associated matter wavesaccumulate a relative phase due to different path and/or different appliedinteraction.After the recombination, the information carried by the phase of the super-position is translated into the population of the states. Then the fringes areobserved as oscillations, function of the phase difference, between momentumstates or in the position space by either moire filtering (see sec. 2.3) or byimaging the atoms (holography). A very general way to detect atom consistsin producing countable ions by laser excitation or by electron bombardment.A considerable difference with respect to light interferometers is the smallerwavelengths involved in the atomic interferometry (∼ 10 pm for thermalatoms and up to ∼ 1µm for ultracold atoms) and also the smaller coher-ence length (∼ 100 pm for thermal atoms and up to ∼ 10µm for ultracoldatoms). This requires that the position and the periodicity of the interferencefringes pattern is not dependent on the de Broglie wavelength. An interfe-rometer having such a property is called white light interferometer and itstypical layout corresponds to the three gratings Mach-Zehnder setup.Another difference concerns the strong interaction of atoms with each other,making atom interferometers often non linear. Moreover, the possibility oftrapping atoms leads to a new class of interferometers for confined particles.


2.2.2 Three gratings interferometers

The three gratings setup represents the atomic analogous of the Mach-Zehnderinterferometer for light, as can be seen in figure 2.4. The gratings are sepa-reted by a fixed distance and have the periodicity conceived in such mannerthat Fraunhofer diffraction occurs, that is the so-called far-field limit. Threegratings interferometers working in the near-field limit are called Talbot-Lauinterferometer [31].

Figure 2.4: Top: Mach-Zehnder light interferometer. Bottom: Mach-Zehnderatom interferometer. The mirrors in the light interferometer are replaced bythe second grating in the atom interferometer while the wave is splitted andrecombined by the first and the third grating, respectively. Taken from[4].

According to the classification made before, de Broglie wave interferome-ters display separated atomic path. If the diffraction gratings impart enoughtransverse momentum to the atoms such that the spread of the diffractedmatter wave exceeds the initial spread, they can be considered as movingalong different trajectories. Since the fringes appear in the position space,it is possible to replace the third grating by a position sensitive detector.In the following it is shown how the fringes shift arises by simple quantummechanical considerations.As regards the atomic state interferometry, the Ramsey-Borde interferometeris a remarkable example of device based on the interaction of a electromag-netic field with the internal degrees of freedom of the atoms. The atomic


internal states are manipulated by four laser fields which serve as splitter,mirror and recombiner. The description of the processes underlying this tech-nique is envisaged in 2.1.1 where the Ramsey fringes are shown. From anhistorical point of view, the ability of change the atoms’ internal state witha rf is due to Rabi, while Ramsey in 1949 creates a long-lived coherent su-perposition of quantum states. In 1989 Borde recognised such manipulationof the atomic states, intended for spectroscopic purpose, as a Mach-Zehnderinterferometer.Raman interferometry is closely related to the previous one. Indeed, Ramantransitions between the ground state |1〉 and the excited state |3〉, depictedin figure 2.3, are induced by two laser beams which differ in frequency by anamount equal to the resonance transition. Such interaction of atoms withradiation takes place via a virtual level (dashed line in the figure 2.3) whichis above the intermediate level |2〉 by a quantity ∆.In a Raman interferometer, schematically depicted in figure 2.5, the source

Figure 2.5: Scheme for a Raman interferometer. Taken from [4].

is prepared by selecting the atoms in the state |1〉. As long as the matterwave passes the first interaction region, part of the population of the groundlevel is transferred to the excited level, leaving the atoms in a superpositionof states labelled by the momentum associated with the internal states andother internal quantum numbers. This splitting mechanism is called π/2pulse. The name is due to the product of the Rabi frequency of the system(see eq.(2.11)) with the interaction time which, in the case of superpositionof states with equal amplitude, is π/2. Doubling the interaction time inthe second interaction region causes the complete transfer from |3〉 to |1〉or viceversa, hence such π pulse acts as mirror since each atom is deflectedequally. A π/2−π−π/2 sequence of pulses certainly represents a three grat-ing interferometer and the resulting interference pattern is an oscillation in


the population of the different internal states, which is measured by means ofa state-sensitive detector placed downstream to the recombing laser pulser.

2.2.3 Atomic fountain

Two photons Raman transitions, together with laser cooling and trappingof atoms, are successfully used in high precision gravity measurements. Theexperiments, carried out at the University of Stanford [35, 36, 5], make useof a atomic fountain to determine the gravitational acceleration of caesiumatoms with an absolute uncertainty of ∆g/g ≈ 3 · 10−10. This interferometeruses the π/2− π − π/2 pulses scheme described above on the two magnetic-field-insensitive hyperfine ground states of caesium launched upward.

Figure 2.6: Left: sketch of the atomic fountain, taken from [4]. T is the timeof flight. Right: atomic fountain apparatus, taken from [5].

The apparatus, shown in figure 2.6, is the atomic fountain used by Peterset al.. About 5 · 108 atoms are loaded in the magneto-optic trap (MOT) in600ms where they are cooled through the optical molasses technique [37].When the temperature reaches the minimum, about ∼ 1.5µK, the atoms arelaunched using moving polarisation gradient optical molasses (this techniqueis described in the cited references). At the start the atoms have a velocity ofabout 3 m/s and achieves an height of 0.46 m following a ballistic trajectory.


The interference region is shielded from magnetic fields and isolated from vi-bration: this conditions allows to apply the Raman pulses every T = 160 ms.One minute of integration time yields a precision of 3 · 10−9g and after twodays of integration time the precision reaches 1 · 10−10g.This measurement is one of the best test of the Equivalence Principle (see 1.5)between a quantum and macroscopic object. Other tests of the EquivalencePrinciple are reported in [38, 39, 40]. The atom interferometric approach inGeneral Relativity is deeply discussed in [41, 42].

2.2.4 Phase shift in the Mach-Zehnder interferometer

The Mach-Zehnder interferometer is constituted by a first grating G1, thatacts as a beam splitter, a second grating G2, that acts as a mirror and adetector which recombines the beams. The distance between G1 and G2 issame as the distance between G2 and the detector and it is supposed to beL. The magnitude of the reciprocal lattice vector of the gratings is given byequation (2.18). This vector lies in the plane of the gratings and it is parallelto the slits width.The following calculation yields the formula for the shift of the fringe pattern,formed by a matter wave crossing the Mach-Zehnder apparatus, when a(classical) force is applied. The matter wave is associated to an atom ofmass m and speed v, and it is assumed that it propagates in the directionorthogonal to the plane containing the gratings. The magnitude of the matterwavevector is mv/~ and therefore it is orthogonal to the reciprocal latticevector of the gratings.

Figure 2.7: Scheme of the Mach-Zehnder interferometer. Thin dashed linesare the two gratings (G1 and G2), thin line is the detector and thick linesare the matter waves wavevectors.


A matter wave, described by the amplitude Ψ, is split by G1 into theamplitudes Ψ′ and Ψ′′ which represent, respectively, the 0 and -1 diffractionorders [31]. Hence Ψ′ and Ψ′′ differ in momentum by one unit of ~G (see eq.(2.17)) and are given by

Ψ′ = A0

Ψ′′ = A−1 exp(−ıG · r1) (2.21)

where r1 is the position vector of G1 and the amplitudes An, with n = 0,−1,take into account the atom-grating interaction. The above formula can bestated as follows: the running wave Ψ′′ has acquired a phase with respect toΨ′ due to diffraction.G2 diffracts the two matter waves again, after a time τ = L/v. Consideringonly the -1 and 1 diffraction orders of Ψ′ and Ψ′′, respectively, these matterwaves are now given by

Ψ′ = A0−1 exp(−ıG · r2)

Ψ′′ = A−1+1Ψ exp(ıG · r1) exp(−ıG · r2) (2.22)

where r2 is the position vector of G2 and Anm, with m = −1,+1, take intoaccount the atom-grating interaction.As shown in figure 2.7, the matter waves superimpose to a point in thedetector after another time interval τ . The sum of their amplitudes is

Ψ′ + Ψ′′ = exp(−ıG · r2)[A0−1 + A−1+1 exp(ıG · (2r2 − r1))] (2.23)

Defining the (mean) intensity as

I = |A0−1|2 + |A−1+1|2 (2.24)

and the contrast as

C =A∗0−1A−1+1 + A∗−1+1A0−1

|A0−1|2 + |A−1+1|2(2.25)

the squared module is

|Ψ′ + Ψ′′|2 = I(1 + C cos ΦI) (2.26)

whereΦI = G · (2r2 − r1) (2.27)

is the phase of the standing wave formed by the interfering matter waves.Since the freely propagating atoms in the interferometer form the fringe pat-tern with respect to an inertial reference frame, a force acting on the in-terferometer causes the shift of the fringes. More generally, the fringe are


shifted when the gratings move with respect to the inertial frame reference.For the sake of concreteness, the gravitational force on the Earth’s surfaceis considered as acting in the direction of reciprocal grating vector. By tak-ing the origin of the time coordinate when the matter waves impinge on thedetector, the position vectors of the gratings, when they are crossed by thematter waves, are

r1(−2τ) =1

2g(−2τ)2 = 2gτ 2

r2(−τ) =1

2g(−τ)2 =

1

2gτ 2 (2.28)

where g is the Earth’s gravitational field (i.e. the acceleration vector due togravity). Substituting equation (2.28) in (2.27) yields

ΦI = G · (2r2(−τ)− r1(−2τ)) = Ggτ 2 (2.29)

where the acceleration vector magnitude is indicated by g.

2.3 Classical shadow pattern

Besides atom interferometry and the quantum description so far illustrated,a purely classical approach can be used to determine the deflection of theatoms in a classical field of forces. A device based on the geometric prop-agation of the atoms takes advantage of the imaging properties of a seriesof identical periodic structures. An atomic beam passing a three gratingssetup forms a shadow image, very similar to the fringes obtained with aninterferometer, which can be measured by using the moire effect.The moire phenomenon occurs when repetitive structures are superposed orviewed against each other. It is a new pattern of bright and dark zones thatis clearly observable at the superposition, although it does not appear in anyof the original structures [6]. The term comes from the French, where it re-ferred to a type of textile, traditionally of silk, with a wavy appearance. Themoire effect is due to an ”interaction” between the overlaid structures: whendark areas of the original structures fall on top of each other, they appearbrighter than areas where dark elements fill the space between each other(see figure 2.8).Since the moire effect is sensible to the slightest displacements, variations or

distortions in the superposed structures, it finds its application in a variety offields, such as optical alignment, mechanical strain analysis or measurementof very small angles. On the other hand, the moire phenomenon comes asundesired nuisance in graphic arts, where moire patterns may appear in the

2.3. CLASSICAL SHADOW PATTERN 31

Figure 2.8: a) Moire pattern formed by two identical superposed gratings,mutually rotated. b) Enlarged view. Taken from [6]

reproduction of colour images.In the present work, the moire effect is used to detected the small displace-ment of atomic fringes, due to the gravitational force on the Earth’s surface,with respect to an inertial frame of reference. The apparatus needed forsuch a challenging measurement was firstly conceived by Oberthaler and hisresearch group in the middle of ’90s. They developed a moire deflectometerwith high inertial sensitivity and measured the acceleration and the rotationon a test system. The results of their studies, summarised in the following,are published in [7] and in the second chapter of [30].

2.3.1 The moire deflectometer

This device is constituted by an atomic source, an atoms detector and threegratings equally spaced and aligned parallel to each other. The atoms emergefrom the source as a broad beam and travel along the apparatus through clas-sical paths which can be depicted as undirected rays. The first two gratingsselect the directions of the rays in such a manner that a modulation in thetransmitted atom density is created at the position of the third grating. Bylooking at figure 2.9, the characteristic self-focusing of this two grating setupis evident. The image, formed by the intersection of the atomic trajecto-ries at distances from the second grating that are integer multiples of thedistance between the first two gratings, displays a sequence of fringes withthe gratings periodicity. Such atomic fringes represent the oscillation in theatom density which is detected through the moire effect by superimposingto it a grating with the same period. The third grating is displaced alongthe direction of the reciprocal grating vector, that is orthogonal to the prop-


agation direction of the beam. The detector records the transmitted atomicintensity as function of the grating displacement, exhibiting the oscillatorybehaviour of the moire phenomenon.The moire deflectometer shares the same geometry with the interferome-

Figure 2.9: Atoms’ trajectories in the deflectometer. Taken from [7].

ters described in 2.2.4, but it works with different length scales, making thediffraction negligible. This is the case when the gratings period a and thegratings distance L are dimensioned so that the atoms are diffracted by anangle θx which is large enough in order to redirect the atomic beam alongits path by one grating period [7]. The angle θx is the angle formed by twocomponents of the atomic beam, which have crossed the first grating, and itcan be approximated by

θx ≈x

L(2.30)

where x is the separation of the beam components at the position of thesecond grating. The first diffraction order of a matter wave, with wavelengthλdB, is at an angle

θdB ≈λdB

a(2.31)

with respect to the incident beam. Hence, the previous condition on theredirection of the beam can be stated as follows

θx θdB (2.32)

and, since the critical value of x is a, it can be recast in the form

a√λdBL . (2.33)


The last formula is a natural condition which makes the ”three-gratings”setup a device working in the classical regime. Indeed, the deflectometerselects only the atom trajectories that overlap at the position of the detector(the third grating), forming a shadow pattern. On the other hand, if thedisequality changes sign

a√λdBL . (2.34)

the quantum mechanical behaviour is restored and the device works as aninterferometer. Hence, the deflectometer can be view as the classical versionof the Mach-Zehnder interferometer corresponding to an infinitely long deBroglie wavelength.The experimental apparatus consists in a metastable Argon gas source, thethree gratings and a channeltron detector. Argon atoms in states with life-times of the order of ∼ 10 s, emerge from the source with an average velocityof ∼ 102 m/s and a velocity spread of the same order of magnitude. The grat-ings are made up of gold with a grating period of 10µm and a slit width of3µm within each period. They are mounted on an optical bench, separatedby 27 cm and aligned parallel to each other within a tolerance of 300µrad.The gratings size is (3× 3) mm2. The third grating can be moved linearly inthe direction perpendicular to the bars with an accuracy better than 1µm.The transmitted metastable atoms successively impact on the surface of thedetector emitting Auger electrons, which are detected trough a single channelelectron multiplier (commonly known as channeltron).The relation between the period and the slit width w determines the trans-mission of the gratings and the contrast

C =Imax − Imin

Imax + Imin

(2.35)

where Imax and Imin are, respectively, the maximum and the minimum recordedintensity. The choice for the ratio w/a ∼ 30% takes into account the need ofhigh contrast fringes (here around 80%) against the transmission efficiency(about 2.7%). With the parameter listed above and a total time elapsed inthis experiment of 105 minutes, the value of the gravitational accelerationhas been determined as (9.86± 0.07) m/s2.

2.3.2 Phase shift in the moire deflectometer

According to [7] and [30], the moire deflectometer is an inertial sensor, suitedfor the precise measurement of the acceleration experienced by the apparatuswhen a classical force acts upon it. The acceleration of the apparatus resultsin a fringes shift. Before going on the dynamical case, it is interesting to


digress on some geometrical features of the classical rays propagating insidethe deflectometer.

Figure 2.10: Scheme of the moire deflectometer. Thin dashed lines are thegratings, thin solid line is the detector and thick solid lines are the propa-gating rays. Dotted-dashed lines are useful for the geometric constructionexposed in the text.

With reference to the figure 2.10, two rays, starting from a common point,impinge on the detector at a distance AD that is equal to twice their distanceat the second grating OC ′ minus their distance at the first one A′A′′, or moreconcisely

AD = 2OC ′ − A′A′′ . (2.36)

This equality is equivalent to

AD = 2AC − AB (2.37)

as follows from the figure and it can be proved assuming that OA′ = OA = L.Defining AE = 2AC, (2.37) can be recast newly as

AD = AE − AB . (2.38)

On the other hand, AC = AB +BC hence

AE = 2AB + 2BC . (2.39)

By hypothesis and by elementary geometric considerations, it follows that thetriangle of vertexes A′′B′C ′ is congruent with which one of vertexes C ′CD,thereby

BC = CD (2.40)


where BC is the projection of the cathetus B′C ′ on the detector plane. SinceAD = AB +BC + CD, the equality (2.40) gives

2BC = AD − AB . (2.41)

Inserting (2.41) in (2.39) gives the identity

AE = AB + AD (2.42)

that proves formula (2.38).Now, if the deflectometer is accelerating, the gratings position vectors dependon the time ri(t), with i = 1, 2, 3. The atoms move with speed v in thelaboratory frame, spending the time τ = L/v between each pair of gratings.By taking the origin of the time coordinate when the atoms arrive on thedetector, the position vector of the atom fringes is rA(0). The atoms crossthe first grating at the time −2τ and the second one at the time −τ . Then,the final displacement of the atom fringes with respect to the position of thedetector r3(0) is

R = rA(0)− r3(0) . (2.43)

A new perspective is gained if the point A in figure 2.10 is considered as fixedin a inertial frame reference (the fringes) and the point D anchored to theaccelerating detector. Indeed, it allows to translate the previous geometricalquantities as follows

AB = A′A′′ = r1(−2τ)

AC = OC ′ = r2(−τ) (2.44)

AD = R

where A′A′′ and OC ′ are the displacements with respect to the inertial frameof, respectively, the first grating at the time −2τ and the second grating atthe time −τ . This point of view makes the fringe position fixed in spaceand refers all positions to it, as implied by (2.44). Then the fringe patternappears shifted if the gratings accelerate. Hence, the formula (2.37) turnsout to be the result given in [7]

R = 2r2(−τ)− r1(−2τ) . (2.45)

The fringe shift can be obtained by formal dot product of R with the reci-procal grating vector G, as given by equation (2.18),

ΦM = G · (2r2(−τ)− r1(−2τ)) (2.46)


discovering that it recovers the formula (2.27) for the fringe shift in the Mach-Zehnder interferometer.As in the previous case, a (gravitational) force acting in the direction of thereciprocal grating vector produces, according to equations (2.28) and (2.46),a fringes shift given by

ΦM = Ggτ 2 (2.47)

where G and g are the magnitude of the reciprocal grating vector and of theacceleration vector, respectively.The fringes shift due to an acceleration of the gratings (in this example,gravity) can be obtained with a classical mechanics calculation. Provided theformula (2.45), it is possible to determine the phase shift ΦM by consideringthe analytical expression for the law of motion of the gratings. Working inthe laboratory reference frame, the direction of propagation of the atoms istaken as x axis, while the y axis is taken parallel to the force and the z axisis irrelevant. Obviously, these choices imply that the force is orthogonal tothe beam. The origin of the coordinate system is the position of the detectorwhen the atoms impinge on it at the time t = 0, which means r3(0) = (0, 0),r2(0) = (−L, 0) and r1(0) = (−2L, 0). The gratings position vectors at thetime t are given by

r1(t) = (−2L− vt,−1

2gt2)

r2(t) = (−L− vt,−1

2gt2) (2.48)

r3(t) = (−vt,−1

2gt2) .

Then the fringes appear shifted by δy due to the gratings motion, therebytheir position vector at the time t = 0 is

rA(0) = (0, δy) . (2.49)

Putting equations (2.48) and (2.49), evaluated at the appropriate time, in(2.43) and (2.45), the final displacement is

R = (0, δy) = (0, gτ 2) (2.50)

which means that the fringes shift is

δy = gτ 2 . (2.51)

In order to reproduce (2.47), the previous expression must be converted inradians. The conversion is possible through the relation

δy

a=

ΦM

2π(2.52)


where a is the grating period and ΦM is the angular displacement of thefringes.

2.3.3 Classical Deflectometer vs. Quantum Interfero-meter

The equality of the formulae (2.27) and (2.46) is basically due to the ge-ometrical setup shared by the moire deflectometer and the Mach-Zehnderinterferometer: both employ three grantings and exhibit white light fringes(see 2.2.1). The deflectometer forms a shadow pattern at the third gratingby selecting the classical rays that superimpose right there. On the otherhand, the Mach-Zehnder splits the matter wave at the first grating and re-combines it with the second grating at the position of the third. The thirdgrating in both cases serves as ruler to measure the fringes shift, through theoscillation of the transmitted intensity. The interferometer produces an in-tensity modulation perfectly sinusoidal while the moire pattern results fromthe convolution of the shadow image with the third grating. Indeed, thephase shifts (2.29) and (2.47), calculated for the concrete examples, dependonly on geometric quantities and on atom velocity.A deeper comprehension of the analogy between these devices relies upon theidentification of the classical force F, which causes deflection, with a poten-tial gradient F = −∇U . The latter can be explained as the phase gradientneeded to account for the tilting of the matter wavefront that yields the shiftin the de Broglie phase. Indeed, the matter wave experiences a position de-pendent phase gradient which can be understood in terms of path integralquantum mechanics (see [19] for a more formal treatment). According tothis formulation, the matter waves accumulate a phase difference along thedifferent path in the interferometer expressed by

Φ =1

~(S1 − S2) (2.53)

where

Si =

∫Γi

dt L[x(t), p(t)] i = 1, 2 (2.54)

is the classical action for the two path Γ1 and Γ2 depicted in figure 2.7.The Lagrangian L for a one-dimensional system with a position dependentpotential is given by

L =p2

2m− U(x) (2.55)

where x is the position and p is the momentum. Since this phase differencedetermines the phase shift of the interference pattern, it is helpful to express


it in a nicer fashion through equation (2.55)

Si =

∫Γi

dt

[pdx

dt−

(p2

2m+ U(x)

)]=

∫Γi

(dx p− dtH) (2.56)

where H = T + U is the Hamiltonian governing the classical motion of theparticle. For an Hamiltonian independent from the time, the time integra-tions cancel each out in the difference (2.53), leaving

Φ =

∫Γ1

dx k1(x)−∫

Γ2

dx k2(x) (2.57)

where

k(x) =1

~√

2m(E − U(x)) (2.58)

is the local wavevector of the particle with total energy E.Moreover, the phase shift is decomposable in a sum of two terms Φ = Φpath +∆Φ thanks to the action that is stationary with respect to small perturbationof the path. The first term depends on the path through the interferometer

Φpath =2π

a(x1 − 2x2 + x3) (2.59)

where a is the grating period and xi are the transverse position of the grating.This relation results from the integration of (2.57) by considering that thegratings change the de Broglie wavevector by multiple of G. The second termis given by the interaction with a potential and at first order is

∆Φ =

∫Γ0

1

dt L[x(t), p(t)]−∫

Γ02

dt L[x(t), p(t)] (2.60)

where Γ01 and Γ0

2 denote the classical path through the interferometer withno applied interaction. It is worth noting that ∆Φ is zero, unless the twointerferometer arms experience a different potential.The fringe shift arises from the presence of a potential gradient and is thesame as the envelope shift due to a classical force. The equation (2.59) isexactly of the same form of (2.27) and (2.46), giving a unified view to theorigins of phase shift. Anyway, there are phase shift with no counterpart inthe envelope shift, for example an interaction phase shift due to the applica-tion of a uniform potential to only one arm of the interferometer, or a purelytopological phase shift, as the one involved in the Aharonov-Bohm effect.

Chapter 3

Gravity experiments onantimatter

This chapter deals with some experiments on antimatter systems. It is shownthat charged antiparticles seriously limit the precision attainable in gravityexperiment due to the presence of stray electromagnetic field and that neu-tral antimatter systems naturally emerge. In addition, the progress madein controlling the translational motion of the atoms, as well as charged andneutral particle cooling and trapping techniques, focuses the attention ofthe researcher on atomic antimatter. The simplest antiatom is the antihy-drogen, which is a bound state of an antiproton and a positron. For thepurpose of fundamental Physics measurements with high accuracy, the avail-ability of very cold antihydrogen is of great value: spectroscopy, trappingstudies and gravitational tests require antiatoms at very low temperates, ofthe order of few Kelvin. The lower temperature, the more accurate is themeasurement, as can be seen by considering the thermal Doppler broaden-ing in spectroscopy, or the velocity dependent deflection of a particle in thegravitational field. Since the critical step in producing low-temperature anti-hydrogen is the availability of low-energy antiprotons, the importance of theAntiproton Decelerator (AD) facility at CERN is highlighted. Subsequently,some details about ATHENA, the first experiment succeeded in 2002 in pro-ducing cold antihydrogen, are given in order to provide useful introductionto antihydrogen studies. A further step in the Antihydrogen Physics is repre-sented by the experiments devoted to carry out measurements on antimatter:AEgIS, the central topic of this thesis, aims to measure the gravitational ac-celeration of antihydrogen on the Earth’s surface. It is described in greatdetails in the last part of this chapter.

39

40 CHAPTER 3. GRAVITY EXPERIMENTS ON ANTIMATTER

3.1 Gravity measurements on

charged antiparticles

General Relativity is a classical theory of gravitation that does not explicitlytake into account antimatter. Its fundamental axiom, the Einstein’s Equiv-alence Principle (EEP), can be stated as follows: ”a body behaves locallyin such a manner that is impossible to say whether it feels a gravitationalforce or it does not”, where ”locally” refers to a set of inertial frames usedto describe the motion. This statement is an extension of an assumption,already made by Galilei, that is empirically known to be valid in classicalmechanics and has the following mathematical expression

mI = mG = m (3.1)

where mI is the inertial mass, the one entering in Newton’s second law, mG

is the gravitational mass, the one entering in Newton’s law of gravity, and mis simply called the mass of the body. Due to its limited domain of validity, itis called Weak Equivalence Principle (WEP). If antimatter is seen to violatethis principle, it fails to satisfy also the EEP and new theories of gravitationmust come into play.Testing WEP for antimatter requires that antiparticles propagate freely inspace.A possible approach, that has been adopted since the first experimental mea-surements performed at Stanford the mid of ’60s of the last century, makesuse of an air evacuated metal drift tube (see fig. 3.1) cooled to few Kelvin,where the (anti)particles are free to fall. When the antiparticle’s initial ve-locity is known, the measurement of vertical time of flight determines the netforce on it.

The simplest probes are positrons and antiprotons, but they carry elec-tric charge and this fact poses serious problems. In fact, the gravitationalpotential energy difference mgh for an antiproton over a 1 m long verticalpath is only of order 10−7 eV and for positrons is of order 10−10 eV; theseenergies are much lower than the electromagnetic energies experimented bythe particles from the environment. Due to the weakness of gravitationalinteraction with respect to electromagnetic one, the antiparticles must becarefully shielded from external electricmagnetic fields. More generally, acharged (anti)particle interacts with the environment via:

1) gravitational field through its mass m,

2) electric field through its charge q,

3.1. GRAVITY MEASUREMENTS ON CHARGED ANTIPARTICLES 41

Figure 3.1: Schematic view of the drift tube apparatus used by the Stanfordgroup in their free-fall experiments with electrons. Taken from [8]

3) electric field gradient through its polarizability αe,

4) magnetic field through its charge and velocity V,

5) magnetic field gradient through its magnetic momentum µ and diama-gnetic polarizability αm,

6) pressure radiation through the photon interaction cross-section σr,

7) scattering on residual gas atoms through the collisional cross-sectionσg.

This list should be exhaustive [8], apart for the annihilation process that willbe treated separately. In this section, each of these couplings will be analysedalong with some other effects that occur in drift-tube experiments [43] [44].

3.1.1 Gravitational field

In this context, gravitational refers to all kind of non-electromagnetic inter-action. The gravitational force on a non-relativistic particle is given by

Fg = mg. (3.2)

where g is the gravitational field. In terms of gravitational potential φ, g isgiven by

g = −∇φ (3.3)


3.1.2 Electric field

The main issues in free-falling experiments are unwanted electrical interac-tions. The magnitude of electric field that equals gravity is given by

E =mg

e(3.4)

which yields 0.56×10−10 V/m for positrons and 1.02×10−7 V/m for antipro-tons, so that a very weak electromagnetic interference is enough to perturbsignificantly a gravitational measurement.Although metal drift tube provides shielding against electrostatic fields, acharged particle is attracted by its image charge. In a perfect and infinitelylong cylinder a particle experiences no force on the axis. However it is unsta-ble against radial attraction and must be maintained on axis by a longitudinalmagnetic field. The finite length of the cylinder allows external fields to pen-etrate inside tube, reducing the effectively shielded length from its physicallength, L, by about the tube’s diameter, 2R, at either end for L R. Typ-ical lengths are L ≈ 1 m and R ≈ 2 cm.In drift-tube experiments, a particle may feel an axial force, if the tube’sinterior diameter is not uniform, and a radial force, if guiding magnetic fieldis not aligned with the tube’s axis. In general, the tolerance for antiprotonis wider than positron due to larger mass of the former.

3.1.3 Electric field gradients

Additional forces arise in case of polarizable systems. Negative hydrogenions, that have been used to compare their time of flight with those obtainedwith antiprotons [43], have electric polarizability and then they experience aforce

Fα = αeE · ∇E (3.5)

The polarizability αe is approximately given by

αe =ea0

∆ε(3.6)

where a0 is the Bohr radius of hydrogen and ∆ε is the energy required toexcite the lowest opposite parity state. Taking ∆ε = 0.1 eV, the ratio∣∣∣Fα

Fg

∣∣∣ 1 (3.7)

even with the very large magnitudes E = 10 V/m and ∇E = 103 V/m2.Hence, electric field gradient forces are negligible.


3.1.4 Magnetic field

For a given particle, the image-charge force is negligible if a magnetic field Bmaintains the particle on the drift tube’s axis. Although solenoids with highdegree of homogeneity are commercially available, without careful magneticshielding, ambient transverse magnetic field variations of order B′ ≈ 10 mGcan be expected. The net magnetic field may guide the particle far off axis,hence B must be large enough to overwhelm B′.The motion of charged particles in magnetic field is helical. Classically, theradius of the helix is

r =mVt

eB(3.8)

which, for a typical slow transverse velocity Vt ≈ 10 m/s, gives r(B = 20 G) ≈28 nm for positrons and r(B = 50 kG) ≈ 21 nm for antiprotons. Thus thecharged particles follow the magnetic field lines very closely by comparisonwith the scale of the typical apparatus.

3.1.5 Magnetic field gradients

Taking a system of reference where the z axis is parallel to g, a magneticfield, almost parallel to g, can be expressed in cylindrical coordinates by

B = Bzz +Bρρ (3.9)

where the parallelism condition is satisfied when |Bz| |Bρ|. Two differenteffects are possible: paramagnetism and diamagnetism. Electrons, positrons,protons and antiprotons possess a orbital magnetic moment due to quantizedorbital momentum in a magnetic field and a intrinsic magnetic moment dueto spin. This results in a attraction to regions of higher magnetic field den-sity (paramagnetism). However the H− system cited above, shows also adiamagnetic behaviour, i.e. a repulsion from regions of high magnetic energydensity, due to the anticorrelated electrons in its orbitals.

Paramagnetism The force exerted by a magnetic field gradient in the zdirection on a particle with magnetic moment µ, that results from thesum of its orbital and intrinsic angular momenta, is given by

Fz = µz∂Bz

∂z(3.10)

where the magnetic moment is

µz = 2µi

(l +

1

2± 1

2γ)

l = 0, 1, 2, . . . (3.11)


and

µi =e~2mi

i = e, p (3.12)

is the Bohr magneton for positron and the nuclear magneton for an-tiproton. The γ factor is 1.001159 for positrons and 2.792847 for an-tiprotons. The orbital quantum number l assumes non-negative integervalues and for an ensemble of particles with kinetic temperature T , hasa statistical average given by

l =kBT

~ωc

(3.13)

where ωc is the cyclotron frequency, given by

ωc =e|Bz|m

. (3.14)

The ratio of the magnetic force to that of gravity for electrons orpositrons is given by∣∣∣Fz

Fg

∣∣∣ =(2.1× 106 m

T

)(l +

1

2± γ

2

)∂Bz

∂z(3.15)

and for protons or antiprotons is given by∣∣∣Fz

Fg

∣∣∣ =(0.62

m

T

)(l +

1

2± γ

2

)∂Bz

∂z(3.16)

In all cases, for all practicable values of Bz, it can be shown that l 1.For electrons and positrons, this means that to have a negligible effectthe field uniformity should be unrealistically high: this makes verydifficult to shield magnetic field gradients. For protons and antiprotonshowever, by using very high magnetic fields (Bz ≈ 50 kG) and verylow temperatures (T < 10 K), the perturbations of this effect can bereduced.

Diamagnetism The diamagnetic polarizability of the H− is expressed by

αm = −4π

3a3

0α2( %

a0

)2

(3.17)

where α is the fine structure constant and % is a root-mean-squareradius of order a0. This charge radius determines a force, that repelsthe H− from regions of high magnetic energy density, given by

Fz =αm

µ0

Bz∂Bz

∂z(3.18)


where µ0 is the permeability of vacuum. The ratio∣∣∣Fz

Fg

∣∣∣ =(1.6× 10−3 m

T2

)Bz∂Bz

∂z(3.19)

shows that this force is negligible in comparison with that due to µ forall practicable condition. In this way it is proved the previous statementabout the similar behaviour of H− and p [45].

3.1.6 Interaction with radiation

It is possible to estimate the influence of radiation on the motion of chargedparticles, recalling that photons exert pressure, the so-called radiation pres-sure. Assuming the validity of the Stefan-Boltzmann law, the radiation forceon a particle of radiation cross-section σr is given by

Fr =S

cσr (3.20)

with S = σT 4 where σ is the Stefan constant.Considering the scattering of a photon of energy ~ω mc2 from a free pointparticle of charge e and mass m, the cross-section σr can be approximatedby the Thomson formula

σr ≈ σ0 =8π

3r2c (3.21)

where

rc =e2

4πεmc2(3.22)

is the classical radius of the particle. It is straightforward to show that∣∣∣Fr

Fg

∣∣∣ = (7.53× 10−24 K−4)T 4, (3.23)

hence the radiation pressure is negligible for all acceptable temperature val-ues.It can be shown that also in the case of a bound electron, where the Rayleighresonant cross-section has to be used instead of the Thomson one, the effectof radiation pressure can be neglected.

3.1.7 Residual gas scattering

The drifting particle’s time of flight can be significantly affected by collisionswith residual gas in the drift tube. The dominant component expected in


the residual gas is given by Helium and only its interaction with the beamof charged (anti)particles will be considered.An estimation of the mean time between these collisions is given by

τ =1

nσgVg

(3.24)

where Vg is the velocity of residual gas atoms, n is the number density of gasatoms given by the equation of state P = nkBT and σg is the cross-sectionfor the particle to be scattered and can be defined as

σ(∆V ) = πr20 (3.25)

where r0 is the distance of closest approach. The velocity of the atoms isassumed to be distributed as a Maxwellian, then

Vg =

√8kBT

πMg

(3.26)

where Mg is the mass of the atom.A point particle of charge q interacts with the neutral background gas atomthrough induced electric dipole moment that gives rise to an attractive force

F = 2αeq2

(4πε0)2r5(3.27)

where αe is the atomic polarizability. When an atom of the residual gaspasses with distance of closest approach r0, it imparts to the point particlea change in velocity[8]

∆V =3π

8

1

mVgr40

2αeq2

(4πε0)2(3.28)

and a variation of kinetic energy

∆E =m

2∆V 2. (3.29)

By substituting the numerical values, the estimation for the mean time be-tween collisions is

τ = (6.0× 104)T 3/4(m∆E)1/4P−1 s. (3.30)

For electrons and positrons, T = 4.2 K, P = 10−12 Torr and ∆E = 10−11 eV,the value obtained is τ ≈ 1.5 s, that is somehow larger than the time requiredto cross the apparatus. For protons and antiprotons, with the same valuesof pressure and temperature, but ∆E = 10−9 eV, that value is τ ≈ 30 s.

3.2. AD EXPERIMENTS 47

3.1.8 Conclusions

As it can be seen in the previous paragraphs, it is not always possible toperfectly shield charged (anti)particles from any electromagnetic interaction.These stray fields pose serious limits on the attainable precision in gravitymeasurement.Besides, unwanted electromagnetic interactions are particularly harmful tointerferometric measurement, where they produce a decoherence in the inter-fering waves that in turn results in the vanishing of the interference fringes.A reasonable way to overcome these difficulties makes use of neutral systems,such as atoms, antiatoms, exotic atoms (e.g. positronium and muonium) andneutrons.Taking advantage of the recent developments in atom optics, especially inthe precise measurement of fundamental physical constants, a gravitationalexperiment on antimatter could be performed with antiatoms. The simplestone is the antihydrogen and in the next section it will be shown that measu-rements on neutral antimatter are feasible.

3.2 AD experiments

As seen in the previous section, the measurement of the acceleration of grav-ity for charged antiparticles has serious problems due to stray electric andmagnetic fields that have to be shielded to the appropriate degree. Instead,the use of neutral antimatter vastly reduces the shielding requirements, butintroduces the problem of making and controlling it.Some examples of simplest suitable systems are the positronium Ps, the an-tineutron n and the antihydrogen H.The force of gravity on positronium, which is made of the same amount ofmatter and antimatter, needs careful analysis in order to take into accountthe force on the electron. Moreover, the system must be prepared in a highlyexcited state in order to survive long enough to be measurably influenced bygravity.Antineutrons are copiously produced along with antiprotons at the antipro-ton sources at both Fermilab and CERN, but they are hard to manage.Cold antihydrogen allows a deep insight in the physics of antimatter andits presumed symmetry with matter in connection with CPT and WEP (seechapter 1), makes it a very interesting system to study.The place where pioneering studies about H production have been conductedon a broad and systematic scale is the Antiproton Decelerator (AD) facilityat CERN.


Figure 3.2: A scheme of the antimatter factory AD

The first antiatoms were formed in 1995 by a team of Italian and Germanphysicists (experiment PS210) at the Low Energy Antiproton Ring (LEAR).Soon after the LEAR was closed and the construction of the AD started. TheAD ring, approximately 188 m long, constitutes a self-contained low-energyantiprotons factory (fig.3.2).The antiprotons are produced by striking an Iridium fixed target with protonsfrom the CERN PS, then they are cooled, slowed-down and delivered to theexperiments by AD. The basic AD cycle is sketched in figure 3.3, where theessential steps are depicted. In particular, the p beam undergoes stochasticand electron coolings, which reduce the momentum p of the particles downto 0.1 GeV/c and the momentum spread ∆p/p down to 0.01%. Typically,AD delivers about 2× 107 antiprotons in pulses lasting 200 ns every 100 s.

In 2002 the goal of producting and detecting a large number of H wasachieved by the ATHENA [10] and the ATRAP [46] experiments. The formerwas closed in 2005 and will be reviewed [47] in the next section. Threeexperiments are currently taking data: ATRAP, ASACUSA and ALPHA.AEgIS, which is the subject of this thesis, is in preparation and is scheduledto take data at the end of 2011.

3.2.1 Production and detection of antihydrogen

The process through which a p captures a e+, forming a bound state (theantiatom), takes place via the following mechanisms:

a) spontaneous radiative recombination: p+ e+ → H + γ,


Figure 3.3: Typical AD cycle, taken from [9]

b) three-body recombination: p+ e+ + e+ → H + e+.

The cross-sections of these processes, and consequently the H formation rate,depend on the temperature and on the number density of positrons. The a)reaction results predominantly in atomic levels with low principal quantumnumbers and the H production rate is given by

Γrad = 3× 10−11(4.2

T

)1/2

n2e [s−1] (3.31)

where ne is the positron plasma density. Instead, b) mainly populates atomiclevels with high principal quantum numbers and the H production rate inmagnetic field is given by [48]

Γ3−body = 6× 10−13(4.2

T

)9/2

n2e [s−1] (3.32)

This rate takes into account the e+ motion in the magnetic field, but there aresome uncertainties due to other assumptions required for the calculation. Forexample, the calculated rate is for highly excited antihydrogen, which maynot be stable enough to live through field ionization as it exits the plasma.There are other proposed alternatives to produce antihydrogen:

c) laser stimulated recombination: p+ e+ + nγ → H + (n+ 1)γ,

d) charge exchange reaction: p+ Ps→ H + e−.


ATHENA studied reaction a) and b) and also made an attempt employingreaction c) [49]. The process labelled as d) takes place also with positroniumin highly excited states, Ps∗, yielding Rydberg antihydrogen H∗.The design of the ATHENA apparatus (fig.3.4) was conceived to maximizethe H formation and aimed to have a very high e+ density ne and a lowtemperature T : during routine operations, it allowed to keep in contact 104

antiprotons and 7×107 positrons for sufficiently long time to make antihydro-gens at a temperature of 4K. Four main modules constitute this apparatus:

• the p catching trap,

• the e+ accumulator,

• the p e+ mixing trap,

• the H annihilation detector.

Also the following elements are essential for the H production:

• a 3 T solenoid, coaxial with traps,

• a cryogenic system to cool the traps to few Kelvin,

• the vacuum system.

Figure 3.4: ATHENA apparatus

All traps are variations of the Penning trap and are placed adjacent to eachother. The 3 T axial magnetic field transversely confines the charged parti-cles while a series of cylindrical electrodes, having 1.25 cm inner radius, trapsthem axially with a particularly shaped potential well. The p with kineticenergy 5.3 MeV are slowed by a thin foil and captured by a pulsed electricfield. The cooling of p is provided by 3 × 108 electrons through Coulombinteraction.


The positron accumulator has its own axial magnetic field of magnitude0.14 T. The positron source is the 22Na isotope whose activity was 1.4×109 Bqat the beginning of the experiment. The accumulation lasts 300 s and yields1.5 × 108 e+, half of which are successfully transferred to the mixing trapwhere they cool by synchrotron radiation. The spheroidal cloud of e+, whosetypical dimensions are radius of 2− 2.5 mm and length of 32 mm, is charac-terised in ATHENA by exciting and detecting axial plasma oscillations. Thecloud’s maximum density is 2.5× 108 cm3.Electrode’s configuration in the mixing trap confines p and e+ in the sameregion, the so-called nested trap (see fig. 3.5), keeping them in contact for190 s.

Figure 3.5: Shape of the trapping potential plotted against length along thetrap. The dashed line is the potential immediately before antiproton transfer.The solid line is the potential during mixing. Image taken from [10]

The H formation is detected through a large solid angle array of position-sensitive silicon microstrip and CsI crystals, surrounding the mixing trap.The detector is sketched in figure 3.6.

Summaryzing, the standard mixing cycle passes through the followingsteps [50]:

1) e− loading in the catching trap,

2) p loading in the cathing trap and electron cooling,

3) e+ transfer in the mixing trap,

4) e− and p transfer in the mixing trap (separated from e+),


Figure 3.6: Antihydrogen detector

5) electron kick-out,

6) p transfer in the nested trap and beginning of the mixing.

After dumping all the potential, a new cycle begins.

Figure 3.7: Reconstructed vertexes of the antihydrogen annihilation

When an antiatom is produced, because it has no electric charge, it isfree to move inside the trap and annihilates on its walls. The signature ofthis annihilation is given by two photons, emitted back-to-back, in spatialand temporal coincidence with few pions and neutral pions. The goal of the

3.3. THE AEGIS EXPERIMENT 53

detector is to provide a unambiguous evidence for antihydrogen productionand to reconstruct the position of annihilation (vertex ). The charged pions,passing through both microstrip layers, determine approximately a straightline. The real trajectory is a helix due to the magnetic field, but the errorcommitted with such an approximation is estimated with Montecarlo meth-ods. The uncertainty in vertex determination is approximately 4 mm (1σ)and is dominated by the unmeasured curvature. Moreover, the positron an-nihilation yields two back-to-back photons, which are detected by the CsIcrystal in the appropriate energy range. A plot of the experimental H an-nihilation vertices is shown in figure 3.7, taken from [10]. Immediately, onerecognise that the vertexes lie on a circumference of radius given by the innerradius of the mixing trap.

3.3 The AEgIS experiment

The theoretical framework exposed in the chapter 1 represents the physicalmotivations which have lead the AEgIS collaboration to plan the measure-ment of the acceleration of antihydrogen atoms due to the Earth’s gravi-tational field g [51]. Furthermore, the AD facility at CERN is devised toprovide a beam of antiprotons with moderate energy to the experiments in-volved in the production of antihydrogen. The possibility of producting largequantities of antiatoms has been demonstrated by the experiments ATHENAand ATRAP installed in the AD facility (see 3.2) in the first decade of thiscentury. More recently, the new experiments ALPHA and ASACUSA demon-strated the possibility of antihydrogen trapping [52] and beam formation [53]for at rest and in flight spectroscopy. AEgIS, which is an experiment of se-cond generation as the last two mentioned, is designed to make measurementson the antihydrogen atom and not only to study its production. Indeed, inaddition to the experimental methods developed by the cited experiments,AEgIS relies upon the formation of an antihydrogen beam which has to befired across a moire deflectometer. This device is an inertial sensor basedon the classical propagation of the antiatoms through a system of materialgratings and it is described in the chapter 2 within the more general contextof atom interferometry.The production of antihydrogen and the measurement of its gravitationalinteraction is carried out according to the following conditions:

• antihydrogen H formation through the charge exchange reaction be-tween Rydberg positronium Ps∗ and cold antiprotons p

P s∗ + p→ H∗ + e− (3.33)


which produces Rydberg antihydrogen,

• Stark acceleration of the H atoms, which takes advantage of the sen-sitivity of highly excited atoms to electric field gradients, in order toproduce a beam with a longitudinal velocity of hundreds of m/s and aradial velocity of tens of m/s,

• H propagation inside the moire deflectometer coupled to a positionsensitive detector. The vertical position and the arrival time of eachantihydrogen atom are recorded.

The present section deals with the methods proposed in [54] by the AEgIScollaboration in order to make a 1% accuracy measurement of the gravi-tational acceleration of antihydrogen. Such experimental techniques mergeaspects of Atomic and high energy Particle Physics. Besides the gravitationalacceleration, H and Ps spectroscopy, as well as search for new physics in theexotic decays of Ps, represent additional goals of the experiment.

3.3.1 General design

The previous conditions are matched by an apparatus which is at presentunder construction in the so called AD DEM zone (see fig. 3.8). Its maincomponents are:

• 5 MeV p beam pipe from AD,

• a Surko-type e+ accumulator,

• a superconducting magnet divided in two sections of 5 T and 1 T re-spectively,

• a dilution refrigerator cryostat,

• several electromagnetic traps for charged particles,

• a Ps converter,

• the gratings system for the gravity measurement,

• various types of detectors and

• many laser systems.


Figure 3.8: Layout of the AD zones

Figure 3.9: Layout of the AEgIS apparatus

The coils of the magnets wrap the cryostat which provide the environmentneeded to handle, cool and trap the e+ and the p. Indeed the trap electrodes,


the target for the Ps and the Stark accelerator electrodes, as well as the moiredeflectometer with the H position detector, are placed inside the cryostat.The cryogenic environment is conceived in such a way that in the centralregion, where the H is formed, the temperature is ∼ 0.1 K, while elsewherein the apparatus is ∼ 4 K. The e+ are injected in the main magnet from thesame side of AD pipe since the other side of the apparatus has to be free forthe gravity measurements. Figure 3.9 shows an overall view of the apparatusas it will look.When operational, the standard sequence of events in AEgIS goes throughthe following experimental steps:

1. About 107 p, with a bunch length of ∼ 10 ns, are delivered by AD every100 s and they are trapped in the catching trap (Malmberg-Penningtype) mounted inside the cryostat in the bore of a 5 T magnetic field.In this region, which is at a temperature of 4 K, electron cooling takesplace and the p reach sub-eV energies.

2. The p are transferred, troughout the transfer trap, to the low field regionof 1 T in a colder environment at 100 mK, which is the H formationtrap. Since cooling in the formation trap is carried out in parallel withcatching and trapping new p coming from AD, an average of 105 p areready for recombination by stacking several AD shots.

3. About 108 e+ are accumulated by a Surko type device in 200 − 300 sand are transferred to a dedicated trap, the UHV positron trap, hostednearby the p catching trap. Here the bunch is compressed in space andtime with standard non-neutral plasma techniques. Then the e+ aretransported across the transfer trap to the region close to porous targetfor Ps production.

4. After e+ acceleration toward the porous material, ground state ortho-Ps is produced with velocity of about 104 m/s.

5. The emerging Ps is excited by two laser pulses into a selected Rydbergstate with quantum number n ranging from 18 to about 30. It isestimated to obtain about 5 · 106 Ps∗.

6. The p cloud at 100 mK is trapped near the Ps production target. ColdRydberg antihydrogen H∗ is formed by charge exchange reaction whilePs∗ flights inside the p cloud. It is expected to produce 102 − 103 H∗

per cycle.

7. Antihydrogen formation is monitored by the central detector which iscurrently under study.


8. p, which have not recombined, are transferred in the trap adjacent tothe formation trap in order to reuse them.

9. Stark acceleration of H∗ is carried out by a suitable setting of thetrapping electrods and those nearby. The electric field gradient is largein the direction parallel to the magnetic field lines and small in theradial direction.

10. Radial cooling of the Rydberg antihydrogen beam using pulsed (quasiCW) Lyman-α laser light can be implemented to focus the beam ontothe gratings.

Figure 3.10: On the left, antihydrogen beam formation. On the right, moiredeflectometer.

It is worth noting that the preparation of the cold p and the Ps∗ lasts ∼100 s, while the recombination occurs in less than 1µs. Hence the AEgISantihydrogen source is pulsed on the contrary to ATHENA and the otherexperiments employing a nested trap configuration (see fig. 3.5), where theH are produced continuously over a time of order of tens of seconds. Thepulsed production allows to define a production time and consequently toturn on the electric field necessary to the beam formation. Moreover, thedifference between the production time and the time of annihilation on thetrap walls determines the H velocity immediately after the formation. Thedefinition of the start time for the Stark acceleration is important also in themeasurement of the H time of flight along the moire deflectometer.

Positron accumulator and transfer section

The e+ source is the 22Na radionuclide with a nominal activity of 40 mCi.The emitted particles are moderated by solid neon which yields a slow beam


containing 8 · 106 e+/s. The accumulator traps this continuous beam witha Malmberg-Penning trap and cools it through electronic excitation of thenitrogen gas. Such a transition is favoured in nitrogen compared to Ps for-mation, which is the only other major inelastic channel open in the kineticenergy range of interest. A differential gas pressure is maintained along thetrap axis, with the lowest values located at the minimum of the potentialwell. A magnetic field of 0.15 T is usually employed. Standard non-neutralplasma techniques are used to radially compress the e+ cloud, increasing thestorage time and the accumulated particles.The injection of the e+ plasma into the main magnet, in the trap locatedin the 4 K region, is carried out by the transfer section through an appro-priate electrostatic guiding system together with a pulsed magnet. Beforetransferring the e+, the nitrogen gas is pumped out of the accumulator andsuccessively the valve, which separates its vacuum chamber from one of thecryogenic region, is opened. This procedure is conceived in order to limit theflux of gas to the ultrahigh vacuum inside the main magnet. Depending onthe exact transfer procedure, the e+ arrive in the high magnetic field regionwith a kinetic energy of several tens or hundreds of eV. Nevertheless, suchlight particles cool by synchrotron radiation thanks to the high magneticfield, with a cooling time constant of about 1 s, reaching the thermal equilib-rium at a temperature of the order of 4 K. The e+ cloud behaves as a coldnon-neutral plasma.

Magnetic field

High magnetic field is required to trap and cool the p while lower values areneeded in the region where the Stark acceleration of the Rydberg H takesplace. Indeed, the main apparatus consists of two magnets, the first one at5 T in the catching region in order to optimise the trapping efficiency for the pdelivered by AD, and the second one at 1 T in the H formation region wherereaching low-temperature demands high homogeneity. The inhomogeneityI(%, z) is defined in cylindrical coordinated by

I(%, z) =∣∣∣Bz(%, z)−B0

B0

∣∣∣ (3.34)

where B0 is the field value at the centre of the given volume and Bz(%, z)is the z-component of the magnetic field. The behaviour of the magneticfield in AEgIS as function of the axial coordinate is shown in figure 3.11.Moreover, particular attention is devoted to reduce the fringing field in thedirection of the grating system.The dilution refrigerator cryostat, able to reach 100 mK in the region of


Figure 3.11: Magnetic field profile and inhomogeneity (see eq. (3.34)) inAEgIS

H formation, is independent of the one housing the magnet’s coils and isinserted inside the bore. The design of the magnets allows access from thetwo sides (p and e+ injection on the upstream side and H beam extractionon the downstream side) and in between the two magnets for cabling anddiagnostics.

Traps

A set of foils, called beam degrader, placed before the catching trap entrancereduce the energy of the p coming from AD with a kinetic energy of 5 MeV.A fraction of p is caught in flight by appropriately setting the voltages onthe trap electrods and is cooled by electrons, loaded in the inner region ofthe trap before the arrival of the p bunch.The traps have cylindrical shape with a typical radius of 1 cm and are consti-tuted by two parallel series of electrodes mounted close to each other insidethe cryostat, called trap cryostat. One of them is used to handle the p whilethe second one is devoted to e+.

The p catching trap and the e+ UHV trap are located in the high field(5 T) region and share the two outermost electrods which serve to capturethe p bunch from AD (see figure 3.12). The central electrods of the catchingtrap keep the p in contact with the e− and provide an harmonic potential(Penning trap). The UHV trap stores and shrinks the e+ bunch from theSurko-type accumulator, which cools by synchrotron radiation emission down


Figure 3.12: Sketch of the p (bottom) and e+ (top) traps

to the cryogenic temperature of 4 K. Also in this region, the possibility offorming an harmonic potential enables the monitoring of the shape of thepositron cloud through detection of the plasma oscillatory modes. As men-tioned, it is important to limit the gas flow from the Surko-type device duringthe e+ transfer. Sufficiently long p storage time is obtained with a residualgas pressure of 10−12 mbar.The approximately 20 electrodes located in the region where the magnetic eldchanges from 5 to 1 T are designed to transfer the p and the e+ in the ultracold environment at 100 mK. While the magnetic field changes by a factor 3,the cloud radius increases by

√3. By applying suitable electric fields, the e+

bunch is accelerated towards a porous target in order to produce Ps. In thefinal p trap a semitransparent electrode has to be used in order to allow thepassage of the Ps∗ (see figure 3.13). The porous converter and the formationtrap are placed inside the dilution refrigerator cryostat at 100 mK.

Figure 3.13: Sketch of the p (bottom) and e+ (top) traps. The Ps converteris pictured, too.


Positronium formation and excitation

Ps production in vacuum proceeds via implantation of e+, with a kineticenergy of order of several hundreds of eV or few keV, into a solid materialtarget, called converter. Annihilation process takes more time comparedto the slowing down to thermal energies, so the e+ can pick up e− beforeescaping from the converter in the form of Ps. The energy distributionand the efficiency depend on the nature of the converter material and, fora specic material, on the implantation depth and on the temperature of thetarget. The emitted Ps is in the ground state which can be a singlet (para-positronium, p−Ps) or a triplet (ortho-positronium, o−Ps). The lifetime ofthe former configuration is 125 ps and the self-annihilation mainly producetwo 511 keV photons. The annihilation of the o−Ps, instead, occurs throughthe emission of 3 photons, whose total energy sums to twice the electronmass, hence the lifetime is 142 ns, much longer than the previous one.

Figure 3.14: Mechanism of Ps production in a porous film.

AEgIS is interested only in the o−Ps, since it can flight with an approxi-mate velocity of 104 m/s for some tens of ns in order to reach a distance fromthe converter surface of few mm where laser excitation easily takes place.The AEgIS design has adopted a geometry where the incoming e+ and theoutgoing Ps flow from the same side of the converter (see fig. 3.13). It isimportant also to remark that the target is mounted in the cryogenic envi-ronment at 100 mK and in the low field region 1 T. In order to select theconverter suited for AEgIS, particular attention has been devoted to porousmaterials. If the pores are connected to the surface, a Ps, formed in the bulk


Figure 3.15: Energy level involved in the excitation process (not to scale).

of the target, can diffuse into a pore, colliding with its wall, and then escapetowards the vacuum. Such a mechanism is schematically pictured in figure3.14. It has been demonstrated that the Ps cools during the collisions andthat the energy spectrum can be tailored to match the required values.The laser system excites the Ps atoms to Rydberg states with principalquantum number n ranging from 20 to 40. The optimal value is determinedby experimental tests. The Rydberg state is split in several sublevels by themotional Stark effect and by linear and quadratic Zeeman effect. Such effectsresult in a Rydberg level-band which is broader (3 · 10−4 eV at n = 35) thanthe Doppler width of the reference Rydberg level (1.6 · 10−4 eV) by about afactor two.Since the difference in energy between the ground state and one of those

Rydberg level is greater than 6 eV and since there are no commercial lasersfor transition of this energy, the laser system consists in a two stage pro-cess. The power and the spectral bandwidth of the two laser pulses takeinto account the geometry, the Rydberg levelband and the timing of the Psexpanding cloud. Indeed, the power of the lasers has to be sufficiently high inorder to excite the whole Ps cloud within few ns and their linewidth must beof the same order as that of the Rydberg level-band of the Ps atoms in orderto maximize the excitation efficiency due to resonant interaction. The first


stage provides the energy necessary to reach an allowed n = 3 sublevel andthe second one drives the transition from this state to the required Rydberglevel-band. The excitation scheme is depicted in figure 3.15. This approachavoids the issue of the rapid decay of the n = 2 level, whose lifetime is about3 ns in contrast to the 10.5 ns of the n = 3 level, and reduces the leakagedue to ionization, which is the process in competition with the desired tran-sitions.The present design allows to insert in the space between the converter andthe p trap a suitable set of electrodes which focus the Ps cloud onto the p,maximizing the H yield.Equipment suitable for these studies is currently installed in Trento and inMilano, whose research groups are member of the AEgIS collaboration.

Detectors

Several diagnostic tools are devised to control each stage in the experiment.They are Particle Physics detectors, as well as detectors common to Atomicand non-neutral cold Plasma Physics. The former are grouped into three in-dependent sets: monitor detectors, antihydrogen detector and position sen-sitive g-measurement detector. The first two are devoted to control thedifferent steps needed to produce the H beam, while the third measures thedeviation of the beam (shift of the moire pattern) due to the gravitationalforce.The monitor detectors are used to check the intensity and alignment of thep beam and to follow the movements of the p inside the AEgIS apparatus.They are a beam counter and several plastic scintillator detectors. The beamcounter is an active silicon layer divided in pads with thickness of only 70µm.This device is very useful to diagnose the beam and to trigger the p catchingtrap since it is located in front of the trap. A set of two scintillators, coupledto Hybrid Photo Diodes (HPD), are positioned close to the beam degraderso that the beam intensity is evaluated detecting the p annihilation on thefoils. Scintillator detectors equipped with photomultipliers and placed allalong the apparatus, are used to record the annihilation on the traps wallsin order to monitor each phase of the H beam formation.Since one of the AEgIS goal consists in producing H in a stable and con-trolled way, it is of central importance to conceive an efficient method totrack back the H formation, as mentioned in 3.2.1. Moreover, the presenceof the central detector, together with the pulsed production of H, allows tomeasure the H velocity after its formation. The details of the detector arecurrently being defined by Monte Carlo (MC) simulation and R&D tests. Itconsists of:


• a set of scintillating fibers layers to measure the z position of the Hannihilation and its velocity,

• a tracker based on position sensitive solid state detectors (Si strips,CCDs, etc.) to reconstruct the annihilation vertex in the r-φ plane.

The position sensitive detector coupled to the moire deflectometer measuresthe H vertical coordinate and its arrival time. Its resolution must be of theorder of 10µm and its size about 20× 20 cm2.Several other diagnostic tools monitor the performance of the e+ accumulatorand of the transfer section, such as imaging systems, constituted by phosphorscreens with a suitable read-out (CCD and/or MCP), Faraday cups, whichcount the particle impinging on them, and CsI crystals for annihilation pho-tons detection.Contrary to the previous destructive diagnostic methods, there are non-destructive techniques, such as the Plasma Modes detection. Charged par-ticles inside the AEgIS apparatus are confined in a Malmberg-Penning trapat low temperature. At thermal equilibrium, they behave as rigidly rotatingspheroidal plasmas with sharp boundaries, whose characteristics are obtainedthrough measurements of the first two axial electrostatic mode frequencieswhich depend on the plasma size, density, and temperature. The dipole andquadrupole mode can be excited by applying sinusoidally time-varying po-tentials to one trap electrode, while acquiring the induced current due to theplasma response on the another electrode. The technique has been devel-oped by the AEgIS Genova group, successfully demonstrated by ATHENAand used again by AEgIS.

3.3.2 Charge Exchange Reaction

The charge exchange reaction, displayed in equation (3.33), was proposedsome years ago and later experimentally demonstrated by the ATRAP col-laboration. However, the method devised by AEgIS for H production signif-icantly differs from the one of ATRAP.Charge exchange reactions between Rydberg atoms and ions are largely stud-ied in Atomic Physics, for example in [55] it was demonstrated that hydrogenatoms can be formed in the interaction of protons with positronium, whichcan be viewed as the charge conjugate of the reaction under study [56]. Themain advantages in this reaction are:

• the large cross-section

σ ∝ nPsa0 (3.35)


where a0 ≈ 0.05 nm is the Bohr’s radius and nPs is the principal quan-tum number of the Rydberg Ps,

• the Rydberg antiatoms are produced in a reasonably narrow range offinal states, whose population is strictly related to that of Ps∗,

• the possibility of forming very cold H by a suitable tuning of the ex-perimental parameters.

Each of these features works together in optimizing the H production andmaximizing the efficiency in forming the beam and its quality. Predictionsabout H production rate, temperature and angular distribution as functionof the Ps kinetic energy are the subject of MC studies [57].The difference between the binding energy of the H∗ and that of Ps∗ is

Q = RH

( 1

nH

− 1

2nPs

)(3.36)

where RH = 13.6 eV is the Rydberg’s constant and nH is the principal quan-tum number of the Rydberg H∗. From the previous equation, assumingthat the p and the e+ recombine at rest, the H∗ has a null velocity only ifnH =

√2nPs. Even if this is the most probable H∗ quantum number, there

are states, differing in n by some units, that are also populated with highprobability, hence the H∗ final velocity distribution results from the contri-bution of ∼ 20 m/s, if nPs = 20, and ∼ 2 m/s, if nPs = 30. As lower is nPs,as such contribution becomes non-negligible.Additionally, the Ps∗ velocity enhances the H∗ final velocity by a term oforder of vH ≈

√2vPsme/mp. With a Ps velocity of order of 104 m/s, which is

an optimal value as regard the maximization of the reaction cross-section un-der the AEgIS experimental conditions, the contribution to the H∗ velocityis about 15− 20 m/s.

3.3.3 Stark acceleration

The manipulation of Rydberg atoms by means of inhomogeneous electric fieldwas proposed in [58]. Since Rydberg atoms exhibit a large dipole moment, itcan be shown that inhomogeneous fields exert force on them. The change inkinetic energy comes from the Stark shift of Rydberg states. The feasibilityof decelerating neutral atoms was experimentally demonstrated, for example,by a group at ETH Zurich, which is member of the AEgIS collaboration.The induced electric dipole moment in Rydberg states scales with the squareof the principal quantum number and for the hydrogen atom at n = 30reaches the value of 1300 Debye. The linear Stark shift of the energy levels in


the hydrogen atom in a homogeneous electric field of magnitude E is given,in atomic units, by

E = − 1

2n2+

3

2nkE (3.37)

where k is a quantum number ranging from −(n−1−|ml|) to n−1−|ml| insteps of two and ml is the azimuthal quantum number. The manifold of suchStark shifted Rydberg state is schematically sketched in figure 3.16. Then,if the H (or H) atom is excited to the k = +29 state, for example at thepoint labelled A, which is shifted at higher energy by the electric field, itsinternal energy decreases while it moves out of the electric field. Hence theatom accelerate and, similarly, if it is excited to the k = −29 state under thesame conditions it decelerates as it moves out of the field. The corresponding

Figure 3.16: Stark structure of the n = 30 ml = 0 state of atomic hydrogen asa function of electric field strength. The vertical axis indicates the detuningfrom the energy position of the field-free Rydberg state (1 cm−1 ≈ 0.12 ·103− eV)

force experienced by the atom is

F = −3

2nk∇E . (3.38)


Experimental results have shown that acceleration up to 2 · 108m/s2 over aflight distance of only 3 mm in a time of less than 5µm. Because of the largedipole moments of Rydberg Stark states, fields of only a few kV/cm, whichare easily produced, are required to achieve accelerations of this magnitude.In AEgIS the ring electrodes forming the Malmberg-Penning trap which ini-tially confines the p are split, so as to produce a field gradient along thecommon axis of the trap and the superconducting magnet at the time whenthe H∗ are to be Stark accelerated. Here, the difference with the previouslymentioned experiment is the presence of the magnetic field. The magneticfield in the H formation region is chosen following a compromise between therequirement of the charged particle trap (that demands high magnetic fields)and the need to reduce the perturbation on the H∗ atoms. The behaviourof hydrogen atoms in electric and magnetic field with arbitrary mutual ori-entation is a complex matter and several regimes are possible depending onthe range of parameters. Quantum calculations with the AEgIS parametersare very useful to drive the experimental choices. Experimental tests withhydrogen are planned at ETH, as long as numerical calculation (ClassicalTrajectories Monte Carlo).

3.3.4 The moire deflectometer

Despite the severe care used in minimizing the H beam radius, the temper-ature is still sufficiently high to cause an appreciable broadening, after theflight path of about 1 m. The deflection of the H with a longitudinal velocityvL ≈ 400 m/s under the influence of the force of gravity for a path of lengthL ≈ 1.1 m is

h =g

2

( LvL

)2

≈ 45µm (3.39)

where g is the gravitational acceleration. By comparing this value with thesize of the beam of about 1 cm, obtained by multiplying the time of flightt ≈ 3 ms with the H radial velocity v⊥ ≈ 30 m/s, it is evident that the fallof the antiatoms cannot be detected in this way.The gravitational acceleration of H is determined by measuring the phaseshift due to gravity in the periodic shadow pattern produced by the moiredeflectometer. The origin of the moire effect and its inertial sensitivity isdiscussed in 2.3. Here it is pointed out that the condition (2.34), required tomake an actual interferometric measurement (see 2.2), is unmatched at thepresent state of antimatter technologies. Indeed, the temperatures requiredby an atomic fountain (see 2.2.3) are of the order of 1µK. Instead, theparameters of the H beam, whose de Broglie wavelength at the velocitiesof interest is of the order of magnitude of ∼ 1 nm, impose the utilisation of


a classical moire deflectometer, which works well even for an uncollimatedbeam [7].

v[m/s] λdB[m] xc[µm] p[eV/c]300 1.32e-09 22.98 938.9400 9.90e-10 19.90 1251.9500 7.92e-10 17.80 1564.9600 6.60e-10 16.25 1877.8700 5.66e-10 15.05 2190.8800 4.95e-10 14.07 2503.8

Table 3.1: Antihydrogen longitudinal velocities, de Broglie wavelength, crit-ical period xc =

√LλdB for L = 40 cm (see 2.3.1), linear momentum.

The ideas related to an inertial sensitive device based on the moire ef-fect were proposed in the middle of ’90s by a group of Heidelberg, whichis member of AEgIS. An upgraded version of the device presented in [7]consists in substituting the third movable gratings with a position sensitivedetector. This solution, adopted in AEgIS, grants an increase in the recordedantiatoms. Currently, the solid state detector, which is planned to be used todetermine the vertical coordinate of the H annihilation, is under developmentby a group of Bergen.

Chapter 4

Simulation of the gravityinterferometric measurement inGeant 4

The simulation of the moire deflectometer has been developed in the Geant4 environment [59]. Although this framework is designed for High EnergyPhysics and medical applications, it is suited to investigate also the featuresof the AEgIS gravity detector, in particular both the tracking of a neutralsystem in a external field and the annihilation process of the p. Indeed, itallows to easily generate the geometry of the apparatus and efficiently trackthe particles in a external user-defined field. Besides, it manages a varietyof physics processes, in particular in the low-energy range that interests us,part of which are discussed in appendix A.The Geant 4 simulation toolkit (in the following G4) is written in C++ andeach of the following sections describes a C++ class that deserves attentiondue to its importance in the simulation. The running of the simulation, aC++ program, requires the definition of three mandatory classes that deter-mine the geometry, the physical processes and the generation of the primaryevent. Conventionally, a run is made of events, that is a predetermined num-ber of particles, with a user-defined kinematics, from which the G4 enginestarts the simulation, hence the tracking, the hit recording and the outputstorage and visualization.A run starts when the G4RunManager receives the command /run/beamOn

from the command-line or from a macro. The main program (Detg.cc) cre-ates the G4RunManager and initialises it by invoking the setters methods, towhich the mandatory classes are passed as argument.The definition of the gravitational field is demanded to the constructor ofthe geometry. Given that there are no electromagnetic fields and that the

69

70 CHAPTER 4. SIMULATION OF THE DEFLECTOMETER

positron mass affects less than 0.2% of the antihydrogen mass, the primaryparticle used is antiproton in order to simplify the tracking in the externalfield. Nevertheless the definition of a new particle (an antiatom) is currentlyunder development for future studies.In addition to the mandatory UserAction classes, it is possible to defineclasses which retrieve and store information from the simulation at variouslevels (run, event, track and step). In particular, they record data on ROOTfiles that are subsequently used in ROOT macros that analyse the simulateddata. The information stored are usually event number, particle ID, particlePDG code [60], position, global time and similar quantities.Currently, the most important data yielded by the present simulation con-cern the annihilation point of the antiproton in the detector. Such a siliconsensitive detector is represented by DetgSilSD which ”fill” the collection ofDetgSilHit.

4.1 Detector Construction

The building block of the geometry is a parallelepiped whose copies are re-peatedly positioned (along the y-direction), forming, in such a way, two peri-odical gratings at a given distance (along the x-direction, which is the beamdirection). The silicon detector is a sensitive volume and is momentarilyrepresented by a parallelepiped with the same dimensions of the grids, butthicker (x-direction). The DetgDetectorConstruction class has six privatemembers that parametrise the dimensions and the positions of the volumes.

class DetgDetectorConstruction : public G4VUserDetectorConstruction...private:G4int ngrid;G4int ngap;G4double distance;G4double thick;G4double period;G4double gap;

;

They are the number of grids and of periods (in each grid), the distancebetween the grids (that must be the same also between the second gridand silicon detector) and the thickness of the elementary block. The lasttwo parameters determine the apparatus opening-fraction, of , through thegrating period, a, and the height of gap, w, in between two solid blocks (see

4.1. DETECTOR CONSTRUCTION 71

fig. 4.1)

of =w

a. (4.1)

Then the height of the block, h, is obtained by subtraction

h = a− w . (4.2)

In the present work of = 30%, as found in [7], where this value is justifiedconsidering the transmitted flux of the (anti)atoms and the fringe constrastas function of the open fraction. The number of blocks equals the number ofperiods (ngap) plus one. The default constructor initialises these parametersand invokes the DefineField() method that is reported below and will bedescribed later.

Figure 4.1: A schematic view of the grating. The blue rectangles are theelementary blocks projected in the y-z plane. Here a represents the period,while w is the gap among blocks.

In the Construct() method, that is pure virtual in the G4VUserDetectorConstruction abstract class and it is therefore overloaded by the user class,the grid’s height (y-direction) and width (z-direction) are computed and theyare assumed to be equal and identified with a variable named lato. The cave(world volume) y-z half-dimensions are computed as multiple of this variable,while the half-length is a multiple of distance.

G4VPhysicalVolume* DetgDetectorConstruction::Construct()G4int nblocks = ngap + 1;G4double lato = period * nblocks - gap;


G4double half_cave_x = 5.0 * distance;G4double half_cave_y = lato;G4double half_cave_z = lato;

G4double grid_x = thick;G4double grid_y = period - gap;G4double grid_z = lato;

...

In this setup, the blocks are made of gold (density 19.3 g/cm3) and thesensitive volume (the detector) is made of silicon (density 2.6 g/cm3). Thecave is filled with vacuum, whose apparently contradictory definition as ma-terial is needed since G4 does not know the ideal vacuum. Instead, a low-density (10−30 g/cm3) hydrogen gas is used, characterised by the thermody-namic variable temperature (2.73 K) and pressure (3 · 10−18 Pa).The desired setup of the gratings is achieved through two for-cycles thatphysically place ngrid × nblocks copies of the elementary box made ofgold. Each placed block is identified through a copy-number. The position ofeach volume is assigned through a Cartesian coordinate system whose originis placed in the cave’s geometric centre, corresponding to the centre of grav-ity of the second grid, that is which of the median block in this grid. Sincethe grids must be rigorously aligned, the z coordinate of the centre of grav-ity of each block is fixed to 0, while the x and the y position are computedaccording to the following algorithm:

G4int j = 0;G4double half_ngap = 0.5*ngap;for(G4int i=0;i<ngrid;i++)G4double gridPos_x = - ( 1 - i ) * distance;for(G4int k=0;k<nblocks;k++)

G4double gridPos_y = ( k - half_ngap) * period;new G4PVPlacement(0, G4ThreeVector(gridPos_x, gridPos_y,

gridPos_z),grid_log, "grid", cave_log, false, j);

++j;

Here j represents the copy-number.The center of the silicon detector is placed at distance from the second

4.1. DETECTOR CONSTRUCTION 73

grid in the positive x direction. In the present code, the silicon detectorhas a thickness of 380µm and a infinite spatial resolution. The storage ofthe information about the particle in the detector volume is accomplishedby the DetgSilSD class, whose description is postponed. Indeed, the lasttask performed by Construct() consists in making the silicon silicon volumesensitive through the setter methods of the G4SDManager class. The completegeometric setup is shown in figure 4.2.The DefineField() method, invoked at the end of the constructor, takes

Figure 4.2: The grids are pictured in blue and the silicon detector in red.The moire texture is highly visible on the grids.

care of registering the user-defined field to the G4FieldManager, in otherwords it tells to the G4 engine that the particles must be tracked accordingto the presence of an external field.

void DetgDetectorConstruction::DefineField()// create gravitational fieldDetgUniformGravField* gField = new DetgUniformGravField();

// equation of motionDetgEqGravField* EquationOfMotion= new DetgEqGravField(gField);

// stepper for equation of motionG4MagIntegratorStepper* stepper = new G4ClassicalRK4(

EquationOfMotion, 8);

// driverG4MagInt_Driver* pIntgrDriver = new G4MagInt_Driver(0.001*mm,

stepper,


stepper->GetNumberOfVariables());

// chordfinderG4ChordFinder* chord = new G4ChordFinder(pIntgrDriver);

// field managerG4FieldManager* fieldMgr =G4TransportationManager::GetTransportationManager()

->GetFieldManager();fieldMgr->SetDetectorField(gField);fieldMgr->SetChordFinder(chord);

The work of this method is more understandable when the classes repre-senting the gravitational field are explained in the section 4.3.

4.2 Primary Generator

The instantiation of the primary particle and the generation of its kine-matics is deferred to AegisGenerator class. Its default constructor as-signs to the private member aegisParticleDefinition the outcome ofthe AntiProtonDefinition() public method belonging to G4AntiProtonclass, and sets the private variable aegisTime to zero. Every event be-gins with the generation of the primary particle kinematics, which consistsof a point in the space and the three Cartesian components of momentum.The GeneratePrimaryVertex(G4Event *evt) method is devoted to pick outthese values according to the experimental conditions.Six private members encode the primary particle initial conditions, which aredetermined by employing statistical algorithms in order to reproduce the anti-hydrogen (antiproton) beam spatial dispersion and the momenta distributionat finite temperature. In the system of reference previously highlightened,the x coordinate is uniformly distributed around a central value, smaller thanthe x position of the first grid. Instead, in the transverse plane, the particlesare supposed to be uniformly distributed inside a circumference with givenradius.

void AegisGenerator::GeneratePrimaryVertex(G4Event *evt)G4double mx = -70.;G4double sx = 0.02;vx = CLHEP::RandFlat::shoot( mx - sx , mx + sx )*cm;

4.2. PRIMARY GENERATOR 75

G4double csi = CLHEP::RandFlat::shoot();G4double phi = 2.0*M_PI*CLHEP::RandFlat::shoot();G4double r = 3.*cm;vy = r*sqrt(csi)*cos(phi);vz = r*sqrt(csi)*sin(phi);

...

The momentum components have Gaussian distribution with differentmeans and standard deviations, so that the (squared) radial velocity is ex-tracted from a Maxwellian, and the mean axial velocity can be set to whatevervalue, fixed or not, depending on the standard deviation of the x component.This procedure is conceived in order to simulate the particle beam after Starkacceleration (see 3.3.3), that reduces the momentum spread and imparts anaxial velocity of order of hundreds of meter per seconds. Setting the standarddeviation of the x component to zero makes possible to extract a single valuefrom the velocity spectrum in each run, allowing a simpler analysis of thegravitational shift.The expected temperature is T ≈ 0.1 K, resulting in a transverse momentumCartesian components distributed as a normal with mean equal to zero andstandard deviation, in SI units,

σ =√mkBT (4.3)

where m is the mass of the particle. A reasonable value is σ ≈ 90 eV/c. Thetransverse velocity distribution is plotted in figure 4.3.

The three spatial components, together with aegisTime, are passed asarguments to the instance of a G4PrimaryVertex. Then, a setter method ofthis class assigns the particle type and the momentum components.

...px = CLHEP::RandGauss::shoot(1252.5 , 0.)*eV; // 400 m/spy = CLHEP::RandGauss::shoot( 0., 90.)*eV;pz = CLHEP::RandGauss::shoot( 0., 90.)*eV;

G4PrimaryVertex *vt = new G4PrimaryVertex(vx, vy, vz, aegisTime);G4PrimaryParticle *particle = new G4PrimaryParticle(

aegisParticleDefinition,px, py, pz);

vt->SetPrimary(particle);evt->AddPrimaryVertex(vt);

Lastly, the primary vertex is added to a G4Event and G4 is ready to trackparticles from it.


Figure 4.3: Distribution of two dimensional vector modulo, whose compo-nents are Gaussian.

4.3 Gravitational field classes

The capability of G4 of describing a large variety of fields and propagatingparticles inside them is a very attractive feature and it is almost unique. Inaddition to the main configurations of electric, magnetic and electromagneticfields, the user can specify any type of external field. Coupling distinct fieldswith distinct (logical) volumes is another income of G4, but it is not takeninto account in the present work, since the gravity is a space-time property(curved space-time).A new type of field can be created inheriting from G4VField and associatingto it the equations of motion class (inheriting from G4EquationOfMotion).Since the particle trajectory in a non-uniform field may be not analyticallyknown, the equations of motion, i.e. a system of second order ordinarydifferential equations (ODE), are numerically integrated using a steppingalgorithm. In spite of this time-consuming tracking method, the versatilityand the accuracy reached by G4 are unmatched.In the literature there is an application of these tools to the simulation ofthe ultracold neutron (UCN) experiment [61]. Neutrons with kinetic energiesless than about 300 neV, stored in material bottles, are strongly affected by

4.3. GRAVITATIONAL FIELD CLASSES 77

gravity. Comparing experimental observables (as neutrons storage time) withmaterial parameters (as loss probability per wall collision) needs a model,which can be built through Montecarlo calculations but the simulation musttake into account the gravitational field effect on the neutrons trajectories.In the code presented in [61] the gravity is introduced as an electric field andthe same approach has been adopted in the present work.In order to implement in G4 a new external field, the following objects mustbe created:

1. the field,

2. the equation of motion,

3. the stepper, giving to it the equation,

4. the driver, passing to it the stepper,

5. the chord finder, passing to it the driver.

Each of these objects is instantiated in the DefineField() method alreadymentioned. The user only needs to implement the first two classes, which inthe present case are:

• DetgUniformGravField,

• DetgEqGravField.

The former fixes the value of the gravitational acceleration through its con-structor method, while the latter points out which equations describe the(classical) motion of the particle in the field, i.e. it determines which forceacts on the particle at each step during the tracking. The description of thelast three classes, which are provided by G4, is postponed to the section 4.3.3.

4.3.1 Uniform Gravitational Field

The class DetgUniformGravField is a concrete implementation that inheritsfrom G4ElectricField, in much the same way as G4UniformElectricFielddoes. This means that the Earth’s gravitational field is implemented as anuniform electric field: the different behaviour is determined by the equationsof motion through the different force acting on the particle. The Newtonforce and the Coulomb force are both conservative, since they are the gra-dient of some scalar field (the potential energy), and they share a similar


mathematical expression, for example a charge q in a parallel-plate capacitorexperiences a force given by

Fe = qE (4.4)

where the uniform electric field E, with a suitable choice of reference frame,has only one non-zero component, given by the voltage across the platesdivided by their distance. In the same way, the gravitational field g affectsthe particle motion through the force

Fg = mg (4.5)

and this coupling is determined by the particle mass m, which is the privatemember fMass of the DetgEqGravField class.The instantiation of the field is left to the class constructor DetgUniformGravField(), which assigns the field components to a three dimensional array.

DetgUniformGravField::DetgUniformGravField(const G4double gy)

fFieldComponents[0] = 0.0;fFieldComponents[1] = gy;fFieldComponents[2] = 0.0;

...

The default value of gy is the earth’s gravitational acceleration (9.81 m/s2)divided by the speed of light, in order to conform to the G4 units system.The implementation of GetFieldValue(const G4double[3], G4double *G)

makes the DetgUniformGravField class concrete and establishes how manycomponents the field has and which values they take at each point in thespace. Since the earth’s gravitational field is uniform, the point, representedby a three dimensional array passed as first argument to this method, doesnot matter, instead the three field components are assigned to its secondargument.

4.3.2 Equation of Motion

Once the new field is created, it needs to be supplied by the object that repre-sents the equations of motion and this goal is achieved by the DetgEqGravFieldclass. It inherits from the G4EquationOfMotion class which has two pure vir-tual methods:

• SetChargeMomentumMass(G4double, G4double,

G4double particleMass)


• EvaluateRhsGivenB(const G4double y[],

const G4double G[], G4double dydx[])

The first one is a setter that assigns to the private member fMass its thirdparameter, while the first two are omitted since the gravitational force isindependent from electric charge and momentum of the particle.Owing to G4 conventions, the first argument of the second method is anarray of six components: the position vector (x, y, z) and the momentumvector (px, py, pz) of the particle. The second argument is given by the fieldcomponents, hence the method’s name ”given B”, in reminiscence of themagnetic field in the Right-Hand-Side of the differential equation. The thirdargument is their derivatives with respect to the path length s. When thisfunction is expressed as function of time, it can be written

s(t) =√x(t)2 + y(t)2 + z(t)2, (4.6)

where x(t), y(t), z(t) are precisely the components 0, 1, 2 of the arrayy, kept dependent of time. Therefore, the first three components of thederivatives array are

dx

ds=

dt

ds

dx

dt=vx

vdy

ds=

dt

ds

dy

dt=vy

v(4.7)

dz

ds=

dt

ds

dz

dt=vz

v

where v is the magnitude of the velocity vector. The subsequent array com-ponents are given by

dpx

ds=

dt

ds

dpx

dt=Fx

vdpy

ds=

dt

ds

dpy

dt=Fy

v(4.8)

dpz

ds=

dt

ds

dpz

dt=Fz

v.

The expression for the force (see eq. 4.5) is given by

F = mg =E

c2g (4.9)

and the particle energy is given by the usual relativistic expression

E

c=

√m2

0c2 + p2 (4.10)


where m0 is the particle rest mass and p2 = p2x + p2

y + p2z. The introduction

of the energy in the expression for the gravity force is a subtlety that is rela-tivistically correct but, in the case under examination, there is no differencein magnitude among the relativistic and rest particle mass (p/m0 ≈ 10−6).Recalling that the velocity vector magnitude is v = p/m0, the expression forthe components 3, 4, 5 of the dydx[] array follows from equations 4.8 and4.9

Fx

v=m0

p

E

c

gx

cFy

v=m0

p

E

c

gy

c(4.11)

Fz

v=m0

p

E

c

gz

c.

(4.12)

In the selected reference frame gx = gz = 0.

4.3.3 Principles of tracking in a field

After the creation of the field and of the equations of motion, the user needsto instantiate four objects in the DefineField() method. Each one belongsto the G4 kernel and is conceived to propagate the particles according tothe geometry, the field configuration and within a predetermined accuracy.Before explaining what these classes represent, some notions about particlestracking in G4 are given in the following paragraphs.In G4 the equations of motion are integrated by a numerical algorithm. Thedefault method is the classical fourth order Runge-Kutta, that is reviewedin section B.3 because it is also adopted in the code shown in this thesis.The integration of the equations of motion results in a curved path that isbroken up into linear chord segments. Both end-points of these segmentsbelong to the real trajectory of the particle, hence the chords well approxi-mate the path as long as their number is large enough. The set of chords isclose to the trajectory with an accuracy determined by a parameter calledmiss distance, or chord distance or delta chord. Its values is an upper boundfor the sagitta which is the distance between the real curved path and theapproximate linear chord.The chords are used to ask to the geometry navigator (G4Navigator class)whether the track has crossed a volume boundary. This is a fundamentaltask since the particle may undergo interactions with different cross-sectionin this new material or the field may even cease to be defined. Each volume


interrogation is made according to an accuracy within the miss distance,which is a private member of the G4ChordFinder class and its default andoptimal value is set to 10−6 mm by the constructor.The first of the four objects which have to be instantiated in order to set upthe new field is the stepper algorithm. It provides the ODE integration andG4 offers several methods to perform this task. Some examples of steppersare implicit and explicit Euler, simple Runge, classical fourth order Runge-Kutta, Kash-Carp Runge-Kutta-Fehlberg and many others specifically de-signed for magnetic fields. Each of the classes, representing a numericalintegration method, inherits from G4MagIntegratorStepper and it shouldbe chosen according to the smoothness of the field. As usual the tag ”Mag”in the class name reminds the use for magnetic fields.In the present thesis, the choice falls on G4ClassicalRK4 class. It performsa dumb step, in the sense that it does not know about errors but it advancesa solution over an interval of the independent variable (the time) and returnsthe incremented dependent variables (the position and velocity vectors). Itsconstructor needs two arguments: the equations of motion and the numberof variables, which are the six components of two vector mentioned above,plus the time and the energy.Taking care of the step accuracy is left to the G4MagInt Driver class, thatis a driver for ODE integrator: it talks to the stepper and insures that theerror is within an acceptable bounds through the so-called adaptive stepsizecontrol (see sec. B.4). Indeed, its constructor receives as second argumentthe stepper, while the first one is the minimum allowed step (in the presentcase set to 10−3 mm) and the third one is number of variables which is de-termined by the getter method of the stepper class.The third required object is instantiated by passing the driver to the oneparameter constructor of the G4ChordFinder class. It provides the ODEintegration, maintaining the accuracy within the limit set by the miss dis-tance. This is achieved through the AdvanceChordLimited(...) methodthat makes use of the driver to figure out the chord endpoint satisfying thecriterion

d < ∆ (4.13)

where d is the computed distance between the chord and the curve and ∆ isthe miss distance. The instantiation of this object is also required by standardmagnetic fields, but it is usually constructed in a slight different mannerinvoking the CreateChordFinder(G4MagneticField *MagField) method ofthe G4FieldManager class and passing to it a concrete field class. Indeed,this method makes use of a G4ChordFinder constructor that creates theequations of motion, the stepper and the driver.


The last DefineField() instructions retrieve the global field manager fromG4TransportationManager and tell to the G4FieldManager which are theG4ChordFinder and the G4Field through its setter methods.

4.4 Sensitive detector classes

A hit is the fundamental piece of information in a Montecarlo simulationand it often is referred as MC truth. When a volume is sensitive, i.e. it is adetector, it can record a collection of hits. In the G4 language, a hit containsthe information associated to a step performed by a particle in the sensitivevolume. Here a hit is constituted by

• the PDG particle code,

• the track identification number,

• its Cartesian position vector,

• the global time.

Each of them is a private member of the DetgSilHit class. The first is asigned integer which uniquely identifies a particle [60]. The second is an in-teger which is equal to 1 if it is associated to the primary particle and biggerthan 1 if it is a secondary particle. The position vector gives the Cartesiancoordinates of the point on the particle track with respect to the global frameof reference. The global time is the total elapsed time from the generationof the event.The setter methods of the DetgSilHit class are invoked in the ProcessHits(G4Step *aStep) method of the DetgSilSD class which retrieves the dynami-cal information, contained in G4Step, from the simulation. The solely privatemember of the DetgSilSD class is DetgSilHitsCollection which is a con-tainer for the hits registered in the current event. The instructions are asfollowing:

G4bool DetgSilSD::ProcessHits(G4Step *aStep,G4TouchableHistory*)...DetgSilHit* aHit = new DetgSilHit();aHit->SetPDGc (aStep->GetTrack()->GetDefinition()

->GetPDGEncoding());aHit->SetID (aStep->GetTrack()->GetTrackID());aHit->SetPos (aStep->GetPreStepPoint()->GetPosition());aHit->SetGTime(aStep->GetTrack()->GetGlobalTime());silCollection->insert( aHit );

4.4. SENSITIVE DETECTOR CLASSES 83

...

In order to analyse the simulation output, the data representing the MCtruth are stored in a TNtuple which belongs to the ROOT data analysispackage. The constructor of this object is invoked in the BeginOfRunActionmethod of the DetgRunAction

TNtuple("nt","nt","ev:pdgc:id:pos:gt")

. The first two arguments are the name and the title of the n-tuple, whilethe third argument represents the names of the various n-tuple branches,separated by colons. Each branch is exactly one of the DetgSilHit privatemember plus the event number. At the end of each G4Event, the n-tuple isfilled with each member of every hits stored in the collection through

void DetgEventAction::EndOfEventAction(const G4Event* evt)...G4HCofThisEvent * HCE = evt->GetHCofThisEvent();SHC = (DetgSilHitsCollection*)(HCE->GetHC(silCollID));

...int n_hit = SHC->entries();G4int pdg;G4int hID;G4ThreeVector P;G4double time;for(int i=0;i<n_hit;i++)

pdg = (*SHC)[i] ->GetPDGc();hID = (*SHC)[i] ->GetID();P = (*SHC)[i] ->GetPos();time = ((*SHC)[i]->GetGTime())/ms;nt->Fill( nevt, pdg, hID, P.getY(), time );

It is worth noting that the collection is identified by silCollID, whichis obtained by a getter method of the sensitive detectors mananger classG4SDManager, and that only the y component of the position vector is rele-vant since the gravitational force acts along this direction.The hit NTuple produced by the simulation is stored in ROOT file and isready for the analysis, which it is shown in the next chapter. It is also possi-ble to visualize the geometry and the tracks (primaries and secondaries) byusing the standard Geant 4 graphical libraries, as shown in figure 4.4, 4.5and 4.6.


Figure 4.4: Antiproton annihilation on the first grating.

Figure 4.5: Antiproton annihilation on the second grating.

Figure 4.6: Antiproton annihilation on the detector, along with several an-nihilation on the gratings.

Chapter 5

Analysis of the simulated data

This chapter is devoted to the analysis of data produced with the simulationof a moire deflectometer, of the type used in the AEgIS experiment to mea-sure the gravitational acceleration of the antihydrogen. The code, developedwithin the Geant 4 toolkit and called Detg, is discussed in 4. When an anti-hydrogen impinges on the detector, a hit is recorded and the information isstored in a ROOT TNtuple [62, 63]. The aim of this chapter is to conceive adata analysis procedure which will be also applied to the experimental datawhen they will become available.Before discussing the procedure, it is useful to introduce a simplified sim-ulation program, which does not rely upon Geant 4, in order to calibratethe parameters of the analysis and to remove some subtlety related to theantihydrogen source and to the tracking in the gravitational field. This pro-gram, called moire and presented in the first section, takes advantage of thewell-known analytical solution of the problem under exam. The second sec-tion describes step-by-step the procedure conceived to analyse the simulationdata in order to extract the gravitational phase shift and, hence, the gravi-tational acceleration. Successively, the output of the Geant 4 simulation isshown along with a detailed analysis. The last part of this chapter discussesthe results so far obtained.

5.1 Analytical solution for the

moire deflectometer

The solution of the equations of motion for a test particle in the Earth’sgravitational field is well-known. Adopting a system of Cartesian coordinates,

85

86 CHAPTER 5. ANALYSIS OF THE SIMULATED DATA

with the y axis parallel to the gravitational field

g = (0,−g, 0) , (5.1)

the trajectory of the particle, released from a point of coordinates (x0, y0, z0),with initial velocity (vx, vy, vz), can be projected out on the x-y plane (vx vy, vz), obtaining a parabola given by

y(x) = −g2

(x− x0

vx

)2

+ vyx− x0

vx

+ y0 . (5.2)

The lateral displacement along the z axis is not relevant in the context of theanalysis presented here, as can be seen by the geometrical setup discussed inthe following.The goal of the C++ program moire is to produce a TNtuple identical tothe one produced by Detg (see 4.4). This is accomplished by the followingsteps:

• definition of the gratings,

• computation of the particle initial conditions,

• calculation of the particle trajectory through (5.2).

The gratings of the deflectometer are defined by the function hit(y) thatreturns 1 if the y of the particle, passed to the function as argument, liesinto one of the intervals covered by the blocks in the y direction, and 0 if theparticle crosses the grating. The block has an height (y direction) related tothe gratings period and open fraction (see 4.1), while the x and z dimensionare idealised since it is assumed that the former is zero and the latter isinfinite.Since the grating is a periodical structure, it is helpful to focus on the outcomeof the function fmod(p,q), provided by the cmath functions belonging to thestandard C++ library [64]: it computes the remainder r of the divisionbetween the first p and the second q argument according to the formula

r = p− nq (5.3)

where n is the quotient of the division p/q rounded towards zero to an integer.The fmod(p,q) function allows to take into account only the blocks at thepositive and negative side of the y axis. Moreover, particular attention hasbeen devoted to the alignment of the gratings, in such a manner that thecentre of the gap between the two blocks coincides with the x axis, i.e. theequation of the straight line joining the centres of the gratings is given byy = 0. Hence, the code is as follows

5.1. ANALYTICAL SOLUTION FOR THEMOIRE DEFLECTOMETER87

int hit(double y)double a=0.008;double w=0.0024;double res=fmod(y+0.5*w,a);...

The first two lines declare the gratings periodicity a and the gap widthbetween the blocks w in centimeters, while the third line computes the valueneeded to decide whether the particle has hit the grating, taking into accountthe gratings alignment by shifting the y by half the gap width.In order to identify the intervals mentioned above, two cases must be distin-guished according to the sign of the variable res returned by fmod()

...if(res>0. && res>w) return 1;else if(res<0. && res>w-a) return 1;else return 0;

If res is positive and greater than the height of the gap, then the particlehas hit the grating. If res is negative but greater than the extension ofthe block in the negative y direction, then the particle again has hit thegrating. In both cases the function returns 1, indicating that the positionof the particle lies in the interval occupied by the block. If the previousconditions are not fulfilled, the function returns 0, meaning that the particlehas not been stopped by the grating.The main of the moire program performs the remaining two tasks for every”generated” particle through a for-cycle. At the beginning of the loop, theinitial conditions for a particle with mass m, emitted from a thermal sourcewith temperature T , are computed. The particle has a longitudinal boost(x direction) v1 of the order of hundreds of m/s, according to 3.3.3. Thetransverse velocities, v2 and v3, are chosen in such a manner that theirsquared modules are distributed as Maxwellian functions with temperatureT . The initial position of the particle (xprim,yprim,zprim) is extracted froma Gaussian distribution, in order to mimic the finite size of the source. Therandom generator used gRandom is the one provided by ROOT. The relevantlines of the code are shown in the following.

int main(int argc, char *argv[])...double tim,pc=-2212.,pID=1.;for(int i=0;i<1000000;i++)


tim=0.;v1=gRandom->Gaus(500.,0.);v2=gRandom->Gaus(0.,sqrt(k*T/m));v3=gRandom->Gaus(0.,sqrt(k*T/m));xprim=gRandom->Gaus(0.,0.);yprim=gRandom->Gaus(0.,0.01);zprim=gRandom->Gaus(0.,0.01);

...

The variable declared before the loop are the laboratory time tim, setequal to zero when the particle is generated, the particle PDG code [60] pcand the track identification number pID. The latter quantities, pc and pID,are relevent only in the Geant 4 simulation and kept here for analogy and tosimplify the analysis. The k in the code is the Boltzmann’s constant.The evaluation of the positions of the particle at the locations of the grat-ings and of the detector is carried out according to the equation (5.2). Thedistance between the source and the first grating is 30 cm and the distance be-tween the gratings and between the second grating and the detector is 40 cm.Hence, the laboratory time of the particle tim, travelling in the x direction,is computed according to the law of uniform motion. The obtained value isinserted in (5.2), which gives the vertical position of the particle ycurr atthe time it reaches one of the grating or the detector. In the former case, thehit(ycurr) function decides whether the particle is stopped by the gratingsand, through a if-statement, the loop continues with the next particle. Inthe latter case, the particle impinges on the detector and a TNtuple nt isfilled with the computed value for ycurr, in the same way as it is done forthe Geant 4 simulation.

...tim = (30. - xprim)/v1;ycurr=yprim+v2*tim-0.5*g*tim*tim;if(hit(ycurr)) continue;

tim = (70. - xprim)/v1;ycurr=yprim+v2*tim-0.5*g*tim*tim;if(hit(ycurr)) continue;

tim = (110. - xprim)/v1;ycurr=yprim+v2*tim-0.5*g*tim*tim;

...nt->Fill(i,pc,pID,ycurr*10.,tim*1.e3);

...

5.2. ANALYSIS PROCEDURE 89

5.2 Analysis procedure

The typical output of the previous code is shown in figure 5.1, where thehistogram represents the interference fringes pattern. The abscissa is thecoordinate of the antihydrogen annihilation point, along the gravitationalfield direction. In order to measure the fringes shift due to the gravitationalfield, the information about the phase (see 2.3.2) must be extracted with asuitable procedure.

Figure 5.1: Shadow pattern. Top: g = 0. Bottom: g = 9.81 m s−2.

Since the shadow pattern is periodic, it is possible to reduce the problemby considering only one period, without loss of information. This is accom-plished by performing the operation of folding of the fringes pattern, thatmakes use of the function fmod(p,q), described in previous section. The his-togram h1, plotted in figure 5.2, is obtained by calculating the remainder ofthe integer division between the annihilation coordinate and the period, thatis the task of fmod(p,q), then the value returned by this function is divided


again by the period in order to obtain a dimensionless quantity ranging inthe interval [0, 1]. If y is the position of the antihydrogen annihilation, thenew folded variable y is given by

y =fmod(y, a)

a(5.4)

where a is the grating period. Hence, the number of atoms arrived on thedetector is described by the function N(y).

Figure 5.2: Folded shadow pattern. Top: g = 0. Bottom: g = 9.81 m s−2.The shift due to gravity is highly visible.

To analyse the folded pattern, it is possible to the introduce a maskfunction, displayed in the following, playing the role of the third grating inthe Mach-Zehnder interferometer (see 2.2.2).

double ladder(double x, double deltax)


double f;double fraction=0.3;if(fmod(x+deltax,1.)<fraction) f=1.;else f=0.;return f;

The goal of the third grating is to detect the modulation of the atomicdensity on the detector by scanning the fringes in the vertical direction. Suchfiltering procedure is carried out by moving the grating by a fraction ∆y of thegrating period, yielding the number of atoms as a periodic function N(∆y)of ∆y with wavenumber G = 2π/a. This function can be obtained by takingthe convolution of N(y) with m(y,∆y), which numerically corresponds to theresponse function of the third grating shifted vertically by ∆y

N(∆y) =

∫N(y)m(y,∆y)dy . (5.5)

Indeed, the ladder(x,deltax) function is the mathematical expression fora grating identical to the first two, which have an open fraction equal to0.3, but shifted by deltax where x ranges in the interval [0, 1].The algorithm devised to compute the integral (5.5) is as follows

for(Float_t deltax = 0.; deltax <= 1.; deltax += step)

nbin=0;for(double x = 0; x<=1.; x += step)

factor = ladder(x,deltax);h2->SetBinContent(nbin,(h1->GetBinContent(nbin))*factor);nbin++;

vdeltax[ndx] = TMath::TwoPi()*deltax;vmask[ndx] = h2->Integral();ndx++;

TGraph *g1=new TGraph(bins, vdeltax, vmask);

The outer loop reproduces the motion of the third grating by increasingdeltax by a small step and the inner loop fills a temporary histogram h2

with the product of the mask function with the bin content of the foldedhistogram h1. The arrays vdeltax[] and vmask[] contains, respectively,the grating shift in radians and the integral of h2 for a fixed deltax, thatis N(∆y) at discrete points. The number of these points is given by the


inverse of the number of step, called bins, as can be seen by looking at theconstructor of the TGraph g1.The phase shift due to gravity is computed through a best fit of g1 withthe least square method [62, 63] in a restriced interval, centred around theminimum, with the function

f(∆y) = α cos

(2π∆y

a+ φ

)+ β (5.6)

where α, β and φ are the parameters of the fit. Hence, the phase inducedby the gravitational field is given by the minimum of the fit function. Theresults of the procedure described in this paragraph are plotted in figure 5.3.The restricted interval can be obtained by fitting the whole curve with a high

Figure 5.3: Red curve: oscillation in the number of detected antiatoms. Blackcurve: fit function.

degree polynomial, for instance of five degree, then adding and subtractingto its minimum a fixed quantity, for instance π/4.In order to determine the gravitational acceleration, the previous procedure is


repeated for different antihydrogen longitudinal velocities, as shown in figure5.4, and the extracted phases are plotted against the time of flight betweentwo gratings

τ =L

2L+ Ls

t (5.7)

where L is the distance between the gratings, Ls is the distance between theantihydrogen source and the first grating and t is the global time spent bythe antiatoms crossing the entire apparatus. The resulting points are plotted

Figure 5.4: Detected atoms as function of the shift of the third grating forantihydrogen longitudinal velocities in m/s: 300, 350, 400, 450, 500, 550,600.

in figure 5.5 where they are best fitted with a parabola

Φ(τ) = Aτ 2 +Bτ + C , (5.8)

hence the gravitational acceleration is given by

g =aA

2πτ 2. (5.9)

The uncertainty associated with the phase shift Φ is computed though thetheoretical variance [30]

σ2Φ =

1

C2N(5.10)


where C is the fringes contrast (see 2.3.1) and N is the number of antiatomsrecorded by the detector. Instead, the relative uncertainty on the time offlight τ is assumed of 1%.

Figure 5.5: Phase shift vs time of flight

The value obtained with the analytical solution for the moire deflectome-ter with a statistics of 106 particles is g = 9.76± 0.06.

5.3 Geant4 parameters and output

In the previous section the data produced by the moire program were pro-cessed through an algorithm that takes as input several ROOT TNtuple andgives the estimation of the gravity acceleration of the antihydrogen. Here,the same scheme is applied to the TNtuple produced by the Geant 4 sim-ulation program Detg. It is worth noting that the Geant 4 program takesinto account the physical interaction undertaken by the antiprotons, that aretracked as if they were antihydrogens, such as annihilation on nuclei (seeappendix A): since only the gravitational field is simulated the results areidentical. Moreover, as shown in 4.3.3, the particle trajectory is broken upinto segments according to a numerical integration algorithm.The Detg program makes use of the Adaptive Rung-Kutta fourth order method

5.3. GEANT4 PARAMETERS AND OUTPUT 95

(see appendix B) in order to integrate the equation of motion and the preci-sion according to which the curved trajectory is replaced by a set of chordsis determined by the parameter called miss distance. Figure 5.6 shows thefunction (5.2) and two numerical solution, obtained with the Runge-Kuttaintegrator, with different miss distance, nonetheless the value used for thesimulation is kept fixed at 1 nm.

Figure 5.6: Comparison between Runge-Kutta fourth order stepper withdifferent miss distance.

There are two other parameters, related to the geometry (see sec. 4.1),that are not present in the moire program but editable in Detg:

• the grating thickness,

• the grating size.

In order to maintain some similarities with the analytical program, Detg setsthe grating thickness at 10µm and computes the size by considering:

• the period a,

• the number of gaps,

• their width w,


in such a manner that the grating sides are about 20 cm. The ratio w/adetermines also the grating open fraction, upon which depends the numberof transmitted antiatoms and the fringes contrast. The value adopted for theperiod is a = 80µm and for the gap width is w = 24µm, yielding an openfraction of 30% [7]. The geometric properties of the moire deflectometer alsoinclude:

• the distance between the gratings L,

• the distance between the source and the first grating Ls.

The former quantity is fixed at L = 40 cm.The parameters of the antihydrogen source (see sec. 4.2) are:

• the longitudinal length of the cylindrical antiatom cloud σx,

• its radius r,

• the antiatom longitudinal velocity vL,

• its spread σL,

• the spread in the radial velocity σT .

Figure 5.7 shows the folded pattern produced by a pointlike source placedat Ls = 10 cm from the first grating. The longitudinal velocity is fixed atvL = 400 m/s by imposing that its spread is zero σL = 0. The spread in theradial velocity is given by

σT =

√kBT

m≈ 29 m/s (5.11)

where kB is the Boltzmann’s constant, T ≈ 100 mK is the temperature of thesource and m = 1.67262158 · 10−27 kg is the (anti)proton’s mass. Figure 5.8is obtained by assuming that the antiatoms are uniformly distributed intoa disc of radius r = 1 cm but neglecting the spread in the radial velocity.Figure 5.9 represents the standard conditions under which the extrapolationof the gravitational acceleration is simulated. The source is idealised as aslice of the antihydrogen cloud, hence a disc projected onto the transverseplane, with a fixed temperature that affects only the transverse componentsof the antiatom velocity.

By comparing figures 5.7, 5.8 and 5.9, it is found that the moire deflec-tometer shows a fringes shift even with an uncollimated beam. The poordefinition of the minima is due to the low statistics, since the particles gene-rated in each of these runs is 105.


Figure 5.7: σx = 0, r = 0, vL = 400 m/s, σL = 0, σT = 29 m/s, Ls = 10 cm.Left: g = 0. Right:g = 9.81 m/s2.

Figure 5.8: σx = 0, r = 1 cm, vL = 400 m/s, σL = 0, σT = 0, Ls = 10 cm.Left: g = 0. Right:g = 9.81 m/s2.

Figure 5.9: σx = 0, r = 1 cm, vL = 400 m/s, σL = 0, σT = 29 m/s, Ls =10 cm. Left: g = 0. Right:g = 9.81 m/s2.

As seen in 5.2, the phase shift due to gravity is figured out by superimpos-ing onto the folded pattern a mask which is a periodic structure similar tothe gratings but with a changeable open fraction. This parameter is veryimportant since it determines the moire effect (see 2.3) that is a oscillating


shadow image, plotted in figure 5.10, arising by the superimposition of twoperiodic patterns. Moreover, the mask scans the folded pattern with smallsteps whose number matches the bins in the folding histogram.

Figure 5.10: N(∆y) for antihydrogen Gaussian velocities vL centred at: 300,350, 450, 500, 550, 600, 650 m/s. Top: g = 0. Bottom:g = 9.81 m/s2.

The shift of the fringes pattern is enlarged in figure 5.11 and the minimumof each curve can be computed through a best fit in a restricted interval withthe function (5.6). The plotted curves are not cosines since they are obtainedthrough the convolution of the (anti)atomic fringes with a mask function,hence the extrema of the interval are very sensitive parameters and several


efforts are made to devise the best method to compute them. In 5.2 it isshown how to determine the best interval by fitting the curves in the wholerange with a five degree polynomial. The shift of the minima due to thevelocity-dependent deflection of the antihydrogen in the gravitational field isclearly visible in figure 5.11.

Figure 5.11: Zoom of N(∆y). The red curve (300 m/s) is affected by numer-ical errors arising from the approximate solution of the equation of motion.

If the spread in the longitudinal velocity is different from zero, for instanceσL = 29 m/s, the antihydrogen time of flight displays a finite width as shownin figure 5.12 where three peaks are present. The first and the second peakare the time employed by the antiatom to travel to the first and to the secondgrating, respectively. The third peak is the one of interest since it is the timeat which the particles annihilate on the detector. The time of flight betweentwo gratings is computed through (5.7), where the global time t is the meanof the Gaussian distribution which best fit the third peak (see fig. 5.12).The calculation of the gravitational acceleration is performed by applying

the analysis procedure to several runs with different longitudinal velocitiesextracted from a Gaussian distribution with mean values ranging between300 and 800 m/s, as shown in figure 5.13 for some of them. The uncertaintyassociated to the phase shift and to the time of flight are as in 5.2. In figure5.14 the points, obtained in 20 different Geant 4 runs, are best fitted witha parabola and also a preliminary value of g is displayed with the relativeuncertainty of 3%. For each of these runs 106 primary particles are generated,while 103 arrives on the detector. A further test on the correctness of the


Figure 5.12: Time of flight spectrum: the red triangles indicates the peaks.

Figure 5.13: Antihydrogen velocity spectrum: thin lines refer to differentruns with different velocities, dashed refers to their convolution.

simulation code and of the analysis procedure is shown in figure 5.15. Theoutput of two simulations with an fictious gravitational acceleration of g =1.6 m/s2 and g = 20 m/s2 are analysed through the procedure describedabove, giving results in agreement with those expected.

5.4. DISCUSSION OF THE RESULTS 101

Figure 5.14: Phase shift vs time of flight: fraction = 0.385, half-width ofrestricted interval = 0.21 π

5.4 Discussion of the results

The location of the minima in the folded patterns, shown in figure 5.2 and5.9, depends on the gratings alignment with respect to the centre of theantihydrogen beam. For both plots the centre of the antihydrogen sourcelies on the straight line that links the centres of the gap within the medianperiod in each grating. The analytical solution in absence of gravity (see fig.5.2 left) displays the minimum at the midpoint of the interval [0, 1], that isone period, but the Geant 4 data in the same condition (see fig. 5.9 left)appear to be shifted toward smaller values: this is due to truncation errorsin the numerical integration of the equations of motion and is an effect thatcan be minimised by reducing the miss distance parameter of the Geant 4stepping (see sec. 4.3.3). This difference explains why the minima of thecurves shown in figure 5.4 are shifted with respect to the one 5.10 at thesame particle velocity.


Figure 5.15: Further output of the Geant 4 simulation and of the analysisprocedure with g = 1.6 m/s2 (left) and g = 20 m/s2 (right)

Besides, an overall phase occurs in both groups of curves shown in figure 5.4and 5.10, as can be seen by inspecting the parabola, which best fit the pointsin figure 5.5 and 5.14. These functions have a constant term that is notzero along with the first order term, indicating that the quadratic relationbetween the phase shift and the time of flight, characteristic of a field force,is complicated by a global effect which is independent from the gravitationalacceleration. Nevertheless, the phase shift highlighted in figure 5.11 can bereconducted to shift induced by the gravitational field on the moire fringesthrough equation (5.9).Another feature of the curves present in figure 5.10 can be seen by lookingat the range of the ordinate which shows the large variability in the numberof detected atoms, on the contrary to 5.4 that displays about the statisticsfor different velocities. A probable cause for this issue is due to the Physicsclasses of Geant 4 that work in the present simulation in the extreme limit ofvery low energy, while the simpler analytical solution is valid for a pointlikeparticle and without any interaction with the materials.The fringes contrast in each set of data is compatible with the expected valueof 80%.Obviously, the relative error on the gravitational acceleration, computed bypropagating the uncertainty from the fit on A in equation (5.8), becomessmaller by increasing the number of generated particles. The extrapolationof the acceleration of gravity becomes more accurate if it is carried out withseveral different simulation at different velocities, taking into account the

5.4. DISCUSSION OF THE RESULTS 103

spread in the radial velocity and the finite extension of the antihydrogencloud in the longitudinal direction, which produce a broad time of flightspectra.This approach requires an improvement of the analysis procedure, currentlyunder study, consisting in the following steps:

• from the simulated data, a distribution of the annihilation events as afunction of the time of flight squared is built;

• the events of the distribution are grouped in classes, by collecting to-gether a given number of channels. For each class the average time offlight and the phase shift is computed as shown in 5.2;

• the fringes shift obtained is plotted against the time of flight and bestfitted with a parabolic function;

• by studying the variation of the χ2 of the fit by changing some of theanalysis parameters (for instance the fraction and the width of therestricted interval in the mask function best fit), the optimal value forg can be determined;

• possible sources of errors can be identified and the requirements on theapparatus can be optimised.

This improved procedure and also other approaches to the analysis are cur-rently under investigation and will be studied by the AEgIS collaborationusing the tools developed for this thesis.

Chapter 6

Outlook

This thesis has presented the challenging topic of a gravitational measure-ment on neutral antimatter, following in particular the approach of the AEgISexperiment: its goal is to determine the gravitational acceleration g on theEarth’s surface of the antihydrogen atom with a balistic measurement in anatom interferometer. Such measurement would constitute the first direct testof the Weak Equivalence Principle in the realm of antimatter. The deviceadopted by the AEgIS collaboration to perform this test is a moire deflec-tometer, which is a very special type of atom interferometer working in theclassical regime. Thus, the interference fringes pattern is substituted by ashadow image that is produced by the self-focusing of the two gratings setupand that is analysed by superimposing a third (virtual) grating, giving riseto the moire effect.The moire deflectometer is studied in the present work through Monte Carlomethods using the tools provided by the Geant4 simulation toolkit. The simu-lation output is analysed through the CERN ROOT data analysis framework.The results, obtained by processing the Geant4 data with ROOT macros,agree, within the statistical uncertainty, very well with the input value ofthe Earth’s gravitational acceleration, thus successfully demonstrating thevalidity of the Monte Carlo approach in this context.The procedure devised in this thesis to extract the gravitational accelerationfrom the phase shift of the fringes is based on several best fit of the oscillat-ing antiatomic patterns, hence the data analysis can be more precise if thesignificance level of these fit is studied as function of the analysis parameters.

The actual experimental conditions require some improvements of thesimulation in order to take into account the magnetic field gradient from thefringing field of the 1 T solenoid, as well as, the finite resolution of the silicon

105

106 CHAPTER 6. OUTLOOK

position-sensitive detector. The former issue needs the definition in Geant4 of a new particle, the antihydrogen atom, with its physical properties,such as the magnetic moment, and, consequently, the implementation ofa suitable Physics list that describes the physical processes undertaken bythe antiatoms. The latter question is more trivial since it can be treatedby smearing the antihydrogen annihilation coordinate sampling it from aGaussian distribution centered at such point and with the standard deviationgiven by the resolution of the detector.A further study concerns the products of the antihydrogen annihilation on thegratings and on the walls of the ultra high-vacuum apparatus: the chargedand neutral pions created by the antiproton annihilation and the gamma fromthe positron annihilation constitute noise for the detector. The estimationof such background can be of great help to increase the accuracy of themeasurement.All these topics can be studied and optimized by using an extension of thecode developed here.

Although several improvements in the simulation and in the analysis soft-wares are needed, the results given in this thesis are very encouraging fortesting the fundamental physical principles on neutral antimatter systems.

Appendix A

CHiral Invariant Phase Space

Geant4 models the physical interaction between the particles through a setof processes, represented by specific classes, that are collected in a PhysicsList whose implementation is mandatory. Besides transportation, which isthe default process for all particles, Geant4 distinguishes among three kindof processes:

• electromagnetic interactions,

• hadronic interactions,

• particles decay.

Geant4 provides numerous prefabricated Physics Lists and, among them,there is the CHiral Invariant Phase Space, called CHIPS [65] [66], that isused in the present simulation of the moire deflectometer in AEgIS. Thesimulation takes into account electromagnetic interactions, as well as decays,by combining standard Geant4 processes, and the hadronic interaction aredescribed by the CHIPS model.CHIPS is a computer code designed to generate the fragmentation of hadronicsystems into hadrons [67]. It is non-perturbative and employs a 3D quark-level SU(3) approach, hence the phase space refers to massless partons andconsiders only light (u, d, s) quarks. Nevertheless, c, b, and t quarks arecreated in the model as a result of the gluon-gluon or photo-gluon fusion,even though they are not implemented directly.The CHIPS model sets simple rules which determine the microscopic quark-level behaviour in order to model the macroscopic hadronic systems with alarge number of degrees of freedom. Indeed, this generator works in sucha manner that simple kinematic mechanisms account for the hadronizationof quarks into hadrons, yielding a good description of the Physics at high

107

108 APPENDIX A. CHIRAL INVARIANT PHASE SPACE

energies, for instance in nuclear and nucleons excitation, as well as, at lowenergies, despite its quark nature.The fundamental concepts underlying this model are:

• the critical temperature Tc,

• the quasmon,

• the quark fusion.

The critical temperature, determined as Tc ≈ 200 MeV by the comparison ofsimulation with experimental data, plays the role of the ΛQCD in QuantumChromoDynamics and defines the number of 3D partons, n, in the hadronicsystem with total energy W through the formula

W 2 = 4T 2c (n− 1)n (A.1)

assuming that the masses of all partons is zero. Qualitatively, the exis-tence of a critical temperature means that the addition of more energy tothe quark-gluon hadronic system increases only the number of constituentquark-partons while the temperature remains constant. This is the only nonkinematic concept of the model and represents its main parameter.The hadronic or nuclear interaction results in the creation of a quasmon thatrepresents the excited hadronic matter. It is an intermediate state which ra-diates energy (decay) through the quark fusion mechanism, that is the onlypossibility in vacuum, or the quark exchange with surrounding nucleons orclusters of nucleons.The quasmon is constituted by an ensemble of quark-partons that are homo-geneously distributed over the invariant phase space. Commonly, an hadronis characterised by its quantum numbers and its mass, with a finite or infinitelifetime, letting aside the quark content. Instead, the quasmon is defined byits mass and its quark content: indeed, it can be considered, for a fixed massand quark content, as a superposition of traditional hadrons.The quark fusion mechanism is responsible for the production of the finalstate hadrons. Fusion occurs when a quark-parton joins with another quark-parton of the same quasmon or of a neighboring nucleon (or nuclear cluster),forming a white hadron that can be radiated. Such hadron is produced onits mass shell and has an energy spectrum that reflects the momentum dis-tribution of the quarks in the system.The CHIPS models constitutes a generalisation of the hadronic phase spacedistribution approach, presented in [68], but also gives the multiplicity distri-butions for different kinds of hadrons. Moreover, it fits the mass of hadrons

109

better than the bag model and is a successful competitor of the cascade mod-els [67], the so-called Bertini models. For all these motivations, CHIPS iscurrently used for simulations also in other experiments in different physicalregimes (LHC, AD, hadrontherapy, etc.).

110 APPENDIX A. CHIRAL INVARIANT PHASE SPACE

Appendix B

Solving Ordinary DifferentialEquations

The problem under study is the determination of the gravitational acceler-ation of the antihydrogen in the Earth’s field through a deflectometer (seesec. 2.3), hence a simulation of this apparatus must take into account howthe particle trajectory is computed. The classical nature of the experimentimposes a calculation based on the Newton’s second law.In order to determine the path of a particle obeying the Newton’s second law,it is necessary to solve a system of second order ordinary differential equa-tions (in the following ODE). In the simplest case of motion of a particle ina gravitational field, the analytical solution exist. In general, it is necessaryto include effects of other forces during the motion, as electric stray field ormagnetic gradients, and in this case it is hard to find analytical solutionsfor such coupled and non-linear ODE. Nevertheless, numerical solutions arealways available.Though they are only approximate, algorithms developed for modern com-puters can reduce the error to an acceptable value. Moreover, analyticalsolutions can be superfluous and can be replaced by tabulated values if oneneeds to evaluate the unknown function only at specified points. This is ourcase, where the particle trajectory is constrained by interaction points.Arbitrary order ODE can be always traced back to a set of first order ODE.For definiteness, a second order ODE of the type

d2y

dx2+ a(x)

dy

dx= b(x) (B.1)

111

112 APPENDIX B. SOLVING ODE

can be rewritten as two first order equations

dy

dx= z(x) (B.2)

dz

dx= b(x)− a(x)z(x)

where z is a new variable. Hence a generic ODE problem has the generalform

dyi(x)

dx= fi(x, y0, . . . , yN−1) i = 0, . . . , N − 1 (B.3)

where the functions fi on the right-hand-side (RHS) are known.A ODE problem is not completely specified unless the boundary conditionsare given. These algebraic equations are crucial in determining how to attacka problem numerically and they divide into two broad categories:

• initial value problem, where all the yi are given at some starting valuexs and it is desired to find the yi’s at some final point xf ,

• two-point boundary value problems, where boundary conditions are spec-ified at more than one x.

The present section deals with two methods that integrate the ODE initialvalue problems. As introduction to these topics some notion about numeri-cal differentiation is given. In the last part, the adaptive stepsize control isshown: it is a tool that automatically controls the error, changing the fun-damental stepsize. Obviously, this computation slows down the integrationbut the reward is not negligible.

B.1 Numerical Differentiation

Everyone knows the exact derivative formula

dy

dx≡ y′(x) = lim

h→0

y(x+ h)− y(x)

h(B.4)

but for a ”small enough” value of h, one can figure out a ”good enough”answer

y′(x) =y(x+ h)− y(x)

h+ error term. (B.5)

On computers, there are several issues associated with (B.5), such as theallowed range of h (e.g., for single precision h > 10−37) and the, more serious,round-off error in the calculation of x + h and in the subtraction in the

B.1. NUMERICAL DIFFERENTIATION 113

numerator (see the first chapter of [69]). The idea is to do as well as possiblegiven these constraints [70].The first step is to obtain an explicit expression for the error term. TheTaylor expansion of y can be expressed as

y(x+ h) = y(x) + hy′(x) +1

2h2y′′(x) + . . . (B.6)

where higher order terms are replaced by dots. An equivalent form of theTaylor series used in numerical analysis is

y(x+ h) = y(x) + hy′(x) +1

2h2y′′(ξ) (B.7)

where ξ ∈ [x, x+h]. There are no dropped terms, this expansion has a finitenumber of terms: Taylor’s theorem guarantees that exists some values ξ forwhich (B.7) is true.The previous equation gives the so-called right derivative formula indeed,solving for the first order derivative

y′(x) =y(x+ h)− y(x)

h− 1

2hy′′(ξ) (B.8)

where ξ ∈ [x, x+h] and the last term on the right is known as truncation error.This kind of error depends on the approximations used in the algorithm, whilethe round-off error, mentioned above, depends on the hardware. Droppingthe error term, (B.8) gives an approximation for the first derivative of y(x).Sometimes the truncation error term is replaced by O(h), specifying its orderin h. It is essential to note that the error introduced by the truncation of theTaylor series in (B.8) is linear in h. Anyway, it is possible to improve thisformula, writing the derivative as

y′(x) = limh→0

y(x+ h)− y(x− h)

2h(B.9)

namely, centred at x. Again, using Taylor expansion

y(x+ h)− y(x− h) = 2hy′(x) +2

3h3y′′′(ξ) (B.10)

where ξ ∈ [x− h, x+ h], the centred derivative is

y′(x) =y(x+ h)− y(x− h)

2h− 1

3h2y′′′(ξ) (B.11)

and the truncation error is now quadratic in h.


B.2 The Euler Method

Coming back to the equation (B.3) with initial value problem, the straightfor-ward procedure to compute its solution consists in replacing the infinitesimalincrements dy and dx by finite steps ∆y and ∆x and multiplying the alge-braic relation by ∆x. From now on, the quantity h = ∆x is referred asstepsize.A more formal proof comes when the results of the previous section are used.In particular the right derivative formula

dy

dx=y(x+ h)− y(x)

h−O(h) (B.12)

gives the left-hand-side (LHS) of (B.3). For the sake of simplicity, the idependence is ruled out and the equation takes the form

y(x+ h) = y(x) + hf(x, y) +O(h2) (B.13)

where O(h2) = hO(h). This numerical scheme takes the name of Eulermethod. Introducing the notation

yn ≡ y(xn) xn+1 ≡ xn + h n = 0, 1, . . . (B.14)

the previous equation can be cast in a suitable form for numerical calculus

yn+1 = yn + hf(xn, yn) . (B.15)

Summeryzing the procedure outlined above:

1. specify the initial condition x0 = xs y0 = y(xs),

2. choose the stepsize h,

3. calculate f given the current value of x and y,

4. use the Euler step to compute the new value of y,

5. increment x and go to step 3 until x = xf .

This Eulerian scheme is referred as first order method, since a kth-ordermethod has a truncation error of the type O(hk+1).The Euler method is conceptually important but it is of little practical use,since it is not very accurate when compared to other methods running at theequivalent stepsize and neither it is very stable [69]. However, all practicalmethods come down to the same idea: add small increments to the functions

B.3. THE RUNGE-KUTTA METHOD 115

corresponding to the derivatives (RHS of the ODE) multiplied by the step-size.Before ending this topic, it is worth noting the truncation error that appearsin equation (B.13). It is called local error, since it is made in a single step.Instead, the global truncation error ε can be estimated multiplying the num-ber of steps ν = xf/h by the local error O(h2). Expressing this relation in amore general form, namely for a local error O(hn)

ε ∝ νO(hn) =xf

hO(hn) = xfO(hn−1) (B.16)

it is possible to judge the accuracy of the chosen method. Hence the Eulermethod has a global truncation error linear in the stepsize. This analysisis only an estimation, since the actual global error depends strongly on theproblem under study, in particular it is not a priori known if the local errorscancel each other or accumulate.

B.3 The Runge-Kutta Method

The Runge-Kutta methods combine several Euler steps, each involving anevaluation of the RHS, to get an answer matching the Taylor expansion upsome higher order.For example, the formula (B.15) can be used to compute a trial step to themidpoint of the interval [x, x+ h]

k1 = hf(xn, yn) (B.17)

k2 = hf(xn +1

2h, yn +

1

2k1)

then the values of x and y obtained are used to compute the real step

yn+1 = yn + k2 +O(h3) . (B.18)

As indicated, the first order truncation error cancels out in this symmetri-sation, making this scheme second order, that is called second order Runge-Kutta.The basic idea is to add up the right combination of f(x, y), weighted by suit-able coefficients, so that the error terms are eliminated order by order. Thereare many ways to apply this idea by evaluating the RHS many times alongthe step. By far, the most popular scheme is the fourth order Runge-Kutta


which implies four evaluation of the RHS per step

k1 = hf(xn, yn) (B.19)

k2 = hf(xn +h

2, yn +

k1

2)

k3 = hf(xn +h

2, yn +

k2

2)

k4 = hf(xn + h, yn + k3) (B.20)

yn+1 = yn +k1

6+k2

3+k3

3+k4

6+O(h5)

A picture of the Runge-Kutta scheme is shown in figure B.1. Higher orderschemes require more evaluation of f than the order itself, although not morethan the order plus two [69].

Let’s compare the Euler and the fourth order Runge-Kutta in a simple

Figure B.1: Scheme for RK4 steps. The black dots are the starting and thefinal points, the white dots are the trial evaluations of the derivatives: twiceat midpoints and once at endpoint.

case to verify these formulas and the associated errors: for concreteness,the trajectory of a particle with a mass of 1 g, freely falling in the Earth’sgravitational field, is computed by the C++ program gravity. It allows toset up the initial condition by specifying the coordinate in the x-y planeof the starting point, the velocity vector magnitude and its angle with thex-axis. It also lets decide to the user the length of the time step and theirnumber. The force of gravity acts in the negative y direction.Two generic algorithm are implemented in this program and the user canswitch between them at running time. One is the fourth order Runge-Kuttascheme through the RK4 function [70], while the other one is the Euler methodthrough the euler function. Each one needs to know the RHS of the ODE,which is furnished by EqOfMotion. The result of the RK4 integration is shown

B.4. ADAPTIVE STEPSIZE CONTROL 117

in figure B.1 together with the parameters used.Since the problem can be solved analytically, it is easy to figure out the

Figure B.2: Initial conditions are xs = ys = 0 v = 40 mm/ms θ = 0 rad .Stepsize and number of step are τ = 10−3 ms ν = 104 .

accuracy for both methods. Paying attention to the coordinate affected bythe force of gravity, the global error is

ε = |ytrue − ycalc| . (B.21)

Hence, with the parameters of the figure, their values are

Runge-Kutta 4th ε = 7.8 · 10−15 mm,

Euler ε = 4.9 · 10−5 mm.

As expected, the accuracy of the fourth order Runge-Kutta method is over-whelming.

B.4 Adaptive Stepsize Control

The numerical schemes presented above, which integrate the ODE, are gen-erally called stepper algorithms. The RK4 and euler are two examples ofalgorithm routines. They perform a dumb step that does not knows about


its error. Monitoring the truncation error and adjusting the stepsize, alongwith the calls to the routines, are left to the stepper routine which has the fun-damental task to take the largest possible step consistent with the specifiedaccuracy and performance. Instead, the driver routine talks to the stepperand stops the integration acting as an interface with the user or with themain program. In the code studied in section 4.3.3 these last two routinesare written in the same file.As regards the Runge-Kutta stepping algorithm of the fourth order, the moststraightforward technique to estimate the truncation error is the step dou-bling. Each step is taken twice, ybig as a full step of length 2h, and ysmall,independently, as two half steps each of length h. Then the difference ofthese numerical estimations

∆ ≡ ysmall − ybig (B.22)

gives an indicator of the truncation error since the true solution and the twonumerical approximations are related by Taylor expansion

y(x+ 2h) = ybig + (2h)5φ (B.23)

y(x+ 2h) = ysmall + 2(h)5φ

where φ ≈ y5/5! is constant over the step.As long as the truncation error is approximately known, the stepper candecide whether retry the present failed step, reducing the stepsize, or computethe next step with a larger stepsize. This operation can be safely performedby determining the relation among the stepsize h and the accuracy ∆. If astep of length h1 is taken with an error ∆1, then the step of length h0 producean error ∆0 according to

h0 = h1

∣∣∣∆0

∆1

∣∣∣0.2

(B.24)

since, from equation (B.23), ∆ is of order of h5. Therefore, denoting ∆0

the desired accuracy, the last equation determines how much to decrease thestepsize if ∆0 < ∆1, and how much to increase it for the next step if ∆0 > ∆1.

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