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Cold antihydrogen experiments and radial compression of antiproton clouds in the ALPHA apparatus at CERN Gutierrez, Andrea 2016

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Cold Antihydrogen Experiments and RadialCompression of Antiproton Clouds in theALPHA Apparatus at CERNbyAndrea GutierrezB.Sc., Université de Montréal, 2008M.Sc., Université de Montréal, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2016c© Andrea Gutierrez 2015AbstractAntihydrogen is the simplest neutral antimatter atom. Precision comparisons between hydrogen andantihydrogen would provide stringent tests of CPT (charge conjugation/parity transformation/timereversal) invariance and the weak equivalence principle. In the last few years, the ALPHA collabo-ration has produced, and trapped antihydrogen [1, 2]. Most recently, this collaboration has probedantihydrogen’s internal structure by inducing hyperfine transitions in ground state atoms [3]. In thisthesis, many details of the cold antihydrogen formation, trapping and measurements of antihydrogenperformed in the ALPHA apparatus are presented, with a focus on antiproton cloud compression.Such compression is an important tool for the formation and trapping of cold antihydrogen, sinceit allows control of the radial size and density of the antiproton cloud. Compression of non-neutralplasmas can be achieved using a rotating time-varying azimuthal electric field, which has beencalled rotating wall technique.In this work, we have observed a new mechanism for compression of a non-neutral plasma, specifi-cally where antiprotons embedded in an electron plasma are compressed by a rotating wall drive at afrequency close to the sum of the axial bounce and rotation frequencies (in a frequency range of 50– 750 kHz). The radius of the antiproton cloud is reduced by up to a factor of 20 with the smallestradius measured to be ∼ 0.2 mm. We have studied antiproton cloud compression as a function ofthe rotating wall frequency, the duration of compression, the rotating wall amplitude, the numbersof electrons and antiprotons, the magnetic field and the shape of the potential well.The frequency range over which compression is evident is compared to the sum of the antiprotonbounce frequency and the system’s rotation frequency. It is suggested that bounce resonant transportis a likely explanation for the compression of antiproton clouds in this regime.iiPrefaceThe research presented in this thesis was carried out as a part of the ALPHA collaboration, a groupof about 40 physicists from 16 different institutions around the world. The ALPHA experiment isbased at CERN near Geneva, Switzerland and as a member of the ALPHA collaboration, I spenta total of nineteen months at CERN. Ten months (2010/2011) were devoted to the experimentalmeasurements of trapped antihydrogen and the spin-flip of antihydrogen.During the remaining nine months (2012), I worked on the commissioning of the ALPHA-2 appa-ratus and experiments on antiproton cloud compression.My most significant contributions are:Antihydrogen experiments in the ALPHA-1 apparatus:• I specifically played a large part in preparing data, validating data runs and identifying can-didate antihydrogen annihilations used as the dataset in the publications Resonant quantumtransitions in trapped antihydrogen atoms [3], Confinement of antihydrogen for 1,000 sec-onds [2], Discriminating between antihydrogen and mirror-trapped antiprotons in a minimum-B trap [4], Description and first application of a new technique to measure the gravitationalmass of antihydrogen [5] and An experimental limit on the charge of antihydrogen [6].• As a part of a team, I contributed to the running and maintenance of the ALPHA-1 appara-tus, collecting data and performing the online and offline antihydrogen annihilation detectionanalysis.• I participated in day-to-day operations such as the maintenance of cryogenic systems (liquidhelium and nitrogen), vacuum systems, shift preparation and routine shift work.iiiPreface• I participated in the optimization of techniques for trapped charged particles for the produc-tion of antihydrogen, such as radial compression (antiproton, electrons, positrons), evapora-tive cooling (antiprotons, electrons, positrons) and autoresonant excitation (antiprotons) (seechapters 3 and 5).• I presented Trapped antihydrogen and Resonant quantum transitions in trapped antihydrogenatoms at the 23rd International Conference on Atomic Physics (ICAP) in Paris (2012).ALPHA-2 apparatus commissioning:• One of my major contributions was the commissioning and the assembly of the antiprotoncapture trap, which is intended to be used as an antiproton accumulator that provides antipro-tons to the atom trap for antihydrogen studies (see section 2.3.3).• I was in charge of the cabling and connections of the electrodes of the Penning-Malmbergtrap of the antiproton trap and the atom trap (appendix B).• I coordinated the production of new filter boards for the electrodes.• I contributed to the setup for the high-voltage electrodes and hardware work with NIM mod-ules for coincidence/logic/triggers to trap antiprotons.• I participated in the wiring of the superconducting magnets for the ALPHA-2 antihydrogentrap.• I was in charge of providing hardware pieces for the silicon detector (in collaboration withthe ALPHA/TRIUMF group).• I participated in the assembly of the silicon modules of the ALPHA-2 silicon detector us-ing the facilities at the Liverpool Semiconductor Detector Centre with collaborators at theUniversity of Liverpool.Antiproton cloud compression:ivPreface• I was the Run Coordinator for a six-week period of beam-time and led a program to devise amethod to improve the antiproton cloud properties.• During this period of beam-time, we demonstrated cooling of the antiprotons with secondaryelectrons, which are produced when an antiproton deposits energy into the degrader layer.Usually, cooling is performed using electrons from an electron gun. The electron plasmacools the antiproton cloud by Coulomb collisions while the electrons cool through emissionof cyclotron radiation. This new technique improved the cooling efficiency from 60% to 90%,which we believe is a result of the electron and the antiproton plasma having an almost perfectoverlap inside the trap, chapter 5.• I also identified a new regime of compression where the antiproton cloud is radially com-pressed in a frequency range close to the sum of the axial bounce and rotation frequencies.The antiprotons are embedded in an electron plasma, providing cooling to the antiprotons. Icarried out many measurements of the process under different conditions to understand thephysics behind the compression and was able to compress the antiproton cloud by up to afactor of 20 (see chapter 6).• I numerically calculated the distributions of the antiproton axial bounce and rotational fre-quencies and showed that the compression occurs when the rotating wall frequency is closeto the sum of those frequencies, chapter 7.• I worked with experts in non-neutral plasma theory from UC Berkeley, who developed an abinitio theory to explain my experimental observations.• I presented my work on antiproton cloud compression at the 6th international conference onTrapped Charged Particles and Fundamental Physics (TCP), where I was awarded best studentpresentation. I also presented my work at the Winter Nuclear Particle Physics Conference(WNPPC) and in a seminar at the non-neutral plasma physics group at the University ofCalifornia, San Diego (UCSD)• A peer-reviewed proceeding of TCP, titled Antiproton cloud compression in the ALPHA ap-vPrefaceparatus at CERN, has been accepted for publication in the journal Hyperfine Interactions(Springer). For this publication I will be the first and corresponding author.• I am preparing a peer-reviewed paper reporting the antiproton cloud compression measure-ments.Other projects:• I performed a simulation for a veto detector prototype. My work consisted of optimizingthe geometry using three constraints: time of flight of the particles, scintillator surface anddistance from the trap, see appendix C.• I contributed to inclusion of the motion of a dipole in an inhomogeneous magnetic field (as fortrapped antihydrogen and ultra cold neutrons), in collaboration with Dr. Peter Gumplinger ofthe GEANT4 collaboration. This physics is now included in GEANT4. I performed prelim-inary simulations of trapped antihydrogen and gravity interaction for a prototype apparatus,see appendix D.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiI ALPHA apparatus and cold antihydrogen experiments . . . . . . . . . . . . . 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Antimatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Antihydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Why study antihydrogen? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 ALPHA Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 The Antiproton Decelerator experimental hall . . . . . . . . . . . . . . . . . . . . 92.2 ALPHA-1 and ALPHA-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Penning-Malmberg traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Charged particle motion in a Penning trap/Malmberg-Penning trap . . . . . 14viiTABLE OF CONTENTS2.3.2 ALPHA-1 Penning-Malmberg trap . . . . . . . . . . . . . . . . . . . . . . 172.3.3 ALPHA-2 Penning-Malmberg traps . . . . . . . . . . . . . . . . . . . . . 202.4 Antiproton production, capture and cooling . . . . . . . . . . . . . . . . . . . . . 212.4.1 The Antiproton Decelerator (AD) . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Antiproton capture and cooling . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Positron Accumulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.6 Movable stick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.7 Charged particle detection and diagnostic devices . . . . . . . . . . . . . . . . . . 322.7.1 Faraday Cup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.7.2 MCP/phosphor/CCD detector assembly . . . . . . . . . . . . . . . . . . . 332.7.3 Scintillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.7.4 Plasma temperature diagnostics . . . . . . . . . . . . . . . . . . . . . . . 352.8 Atom traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8.1 Antihydrogen motion in a magnetic field minimum trap . . . . . . . . . . . 382.8.2 ALPHA-1 atom trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.8.3 ALPHA-2 atom trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.9 Laser access in ALPHA-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.10 Antihydrogen detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.10.1 Antiproton annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.10.2 ALPHA-1 silicon tracking detector . . . . . . . . . . . . . . . . . . . . . 482.10.3 ALPHA-2 silicon tracking detector . . . . . . . . . . . . . . . . . . . . . 542.11 Cryostat and vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Trapped antihydrogen experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1 Antihydrogen formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2 Antihydrogen production in ALPHA . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.1 Evaporative cooling technique . . . . . . . . . . . . . . . . . . . . . . . . 603.2.2 Autoresonant injection technique . . . . . . . . . . . . . . . . . . . . . . . 62viiiTABLE OF CONTENTS3.2.3 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Antihydrogen trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3.1 Plasma preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.2 Mirror trapped antiproton background . . . . . . . . . . . . . . . . . . . . 673.3.3 38 trapped antihydrogen atoms . . . . . . . . . . . . . . . . . . . . . . . . 693.3.4 Antihydrogen confinement for 1,000 s . . . . . . . . . . . . . . . . . . . . 713.3.5 Summary of trapped antihydrogen measurements . . . . . . . . . . . . . . 733.4 Resonant quantum transitions in antihydrogen . . . . . . . . . . . . . . . . . . . . 743.4.1 Hyperfine structure of the ground state . . . . . . . . . . . . . . . . . . . . 743.4.2 Microwave injection and transition probability . . . . . . . . . . . . . . . 803.4.3 Static magnetic field measured with the electron cyclotron resonance . . . . 823.4.4 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.4.5 Description of measurements . . . . . . . . . . . . . . . . . . . . . . . . 863.4.6 Results of disappearance mode analysis . . . . . . . . . . . . . . . . . . . 883.4.7 Results of appearance mode analysis . . . . . . . . . . . . . . . . . . . . . 893.5 Future experiments on antihydrogen . . . . . . . . . . . . . . . . . . . . . . . . . 90II Antiproton cloud radial compression . . . . . . . . . . . . . . . . . . . . . . . . 964 Non-neutral plasma confinement and radial compression in a Penning-Malmberg trap 974.1 Motivation for the compression of antiproton clouds . . . . . . . . . . . . . . . . . 974.2 Non-neutral plasma confinement in a Penning-Malmberg trap . . . . . . . . . . . . 994.2.1 Non-neutral plasma fundamental properties . . . . . . . . . . . . . . . . . 994.2.2 Plasma rotation frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.2.3 Theory of confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2.4 Global thermal equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2.5 Antiproton-electron plasmas . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3 Rotating wall mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107ixTABLE OF CONTENTS4.4 Sympathetic compression of antiproton clouds . . . . . . . . . . . . . . . . . . . . 1124.4.1 Antiproton cloud compression by the ASACUSA collaboration . . . . . . 1164.4.2 Antiproton cloud compression by the ATHENA collaboration . . . . . . . 1175 Experimental setup for the compression of antiproton clouds in ALPHA-2 . . . . . 1185.1 Capturing antiprotons and secondary electrons . . . . . . . . . . . . . . . . . . . . 1185.1.1 Antiproton capture in ALPHA-2 . . . . . . . . . . . . . . . . . . . . . . . 1185.1.2 Secondary electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2 Antiproton cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.3 Pulsed electron ejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.4 Rotating wall application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.5 Detection and analysis of antiproton clouds . . . . . . . . . . . . . . . . . . . . . 1316 Antiproton cloud radial compression measurements . . . . . . . . . . . . . . . . . . 1376.1 History of the measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.2 Observation of a new regime of antiproton cloud compression . . . . . . . . . . . 1386.2.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.2.2 Antiproton cloud radial compression for various frequencies and electronnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.2.3 Electron plasma compression without antiprotons . . . . . . . . . . . . . . 1456.2.4 Compression of antiproton cloud without electrons . . . . . . . . . . . . . 1466.2.5 Compression at higher frequencies (ωRW/2pi > 750 kHz) . . . . . . . . . . 1476.2.6 Observation of a new mechanism? . . . . . . . . . . . . . . . . . . . . . 1496.3 Other measurements of the antiproton cloud compression . . . . . . . . . . . . . . 1506.3.1 Compression as a function of duration of the rotating wall . . . . . . . . . 1506.3.2 Compression when changing the potential well . . . . . . . . . . . . . . . 1536.3.3 Compression as a function of the applied rotating wall amplitude . . . . . . 1566.3.4 Compression for various numbers of antiprotons and electrons . . . . . . . 1566.3.5 Expansion after the rotating wall finishes . . . . . . . . . . . . . . . . . . 159xTABLE OF CONTENTS6.3.6 Plasma temperature before and after compression . . . . . . . . . . . . . . 1606.3.7 Compression as a function of magnetic field . . . . . . . . . . . . . . . . . 1616.4 Summary of observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.5 Comparison of the direct compression of antiproton clouds versus sympathetic com-pression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657 Compression mechanisms and calculation of the plasma frequencies associated withthe bounce resonant transport of antiprotons . . . . . . . . . . . . . . . . . . . . . . 1697.1 Radial compression mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.1.1 Plasma frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.1.2 Diocotron mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.1.3 Compression by Trivelpiece-Gould excitation . . . . . . . . . . . . . . . . 1727.1.4 Strong drive regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.1.5 Sympathetic compression mechanism . . . . . . . . . . . . . . . . . . . . 1727.1.6 Compression by magnetron sideband cooling . . . . . . . . . . . . . . . . 1737.1.7 Compression by bounce resonance transport . . . . . . . . . . . . . . . . . 1747.2 Investigation of the bounce resonant transport of antiprotons . . . . . . . . . . . . 1757.3 Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot) . . . . 1767.3.1 f (ωb + ωrot) of antiprotons cooled by 4 × 106 electrons . . . . . . . . . . . 1817.4 f (ωb + ωrot) of antiprotons cooled by 20 × 106 electrons . . . . . . . . . . . . . . 1847.4.1 f (ωb + ωrot) distribution at the initial conditions . . . . . . . . . . . . . . 1847.4.2 f (ωb + ωrot) as a function of the partial compression of the electron plasma. 1877.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195xiTABLE OF CONTENTSAppendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A Antiproton energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207B Electrodes cabling and connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210C Veto detector simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213D Derivation of the magnetic dipole force in GEANT4 . . . . . . . . . . . . . . . . . . 218D.1 Magnetic dipole introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218D.2 GEANT4 implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219xiiList of Tables1.1 Measurements that test CPT invariance. . . . . . . . . . . . . . . . . . . . . . . . 62.1 Frequencies of charged particles in the ALPHA-2 antiproton trap in a typical poten-tial well in a 3 T magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 ALPHA-1 and ALPHA-2 different module configuration and radii of each layer. . . 543.1 The value of the constant C as a function of the magnetic field. . . . . . . . . . . . 603.2 Number of annihilation events for antihydrogen trapping and heated positrons (an-tihydrogen formation suppression) experiments for different bias electric fields. . . 713.3 Number of annihilation events for different times of confinement. . . . . . . . . . . 723.4 Table showing a total of six combinations of experimental parameters. . . . . . . . 843.5 Results of disappearance mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.6 Summary of flip-spin transition measurements. . . . . . . . . . . . . . . . . . . . 894.1 Electron plasma parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1 A summary of degrader material and thickness. . . . . . . . . . . . . . . . . . . . 1196.1 Number of particles and densities of the plasmas radially compressing at a specificrotating wall frequency (measurements performed with 1×105 antiprotons and 20×106 electrons). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.2 Voltages applied to the electrodes producing the potential well during application ofthe rotating wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.3 Characteristics of the beam for different beam percentages. . . . . . . . . . . . . . 158xiiiLIST OF TABLES6.4 Temperature of 20 × 106 electrons after applying the rotating wall for 100 s at dif-ferent frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.1 Electron plasma parameters. λD is calculated from equation 4.3 and ωp/2pi is cal-culated from equation 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.2 Peak density and rotation frequency of a fraction of the electron plasma (3 × 106electrons) at 500 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190A.1 Range for protons and antiprotons with kinetic energy of 5.3 MeV for aluminiumand beryllium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209xivList of Figures2.1 Schematic of the AD hall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 ALPHA-1 trapping region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Photograph of the ALPHA-1 apparatus. . . . . . . . . . . . . . . . . . . . . . . . 132.4 Photograph of the ALPHA-2 apparatus. . . . . . . . . . . . . . . . . . . . . . . . 132.5 The charged particle motion in a Penning trap. . . . . . . . . . . . . . . . . . . . . 172.6 Diagram of the ALPHA-1 electrode stack. . . . . . . . . . . . . . . . . . . . . . . 192.7 Diagram of the ALPHA-2 electrode stack of the antiproton capture trap. . . . . . . 212.8 Diagram of the ALPHA-2 electrode stack of the mixing capture trap. . . . . . . . . 222.9 AD cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.10 Schematic illustration of the antiproton capture. . . . . . . . . . . . . . . . . . . . 252.11 Schematic illustration of antiproton cooling. . . . . . . . . . . . . . . . . . . . . . 262.12 Antiproton cooling efficiency as a function of the cooling time. . . . . . . . . . . . 272.13 Antiproton cloud radius as a function of the radius of the electron plasma and an-tiproton cooling efficiency as a function of the radius of the electron plasma. . . . . 282.14 Positron accumulator schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.15 Photograph of the ALPHA-2 antiproton capture trap and movable stick. . . . . . . 322.16 Example of the FC traces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.17 Schematic of the MCP/phosphor/CCD assembly. . . . . . . . . . . . . . . . . . . 342.18 Picture of the ALPHA-2 apparatus and scintillators. . . . . . . . . . . . . . . . . . 362.19 Integrated number of antiproton loss as a function of the well depth. . . . . . . . . 382.20 On-axis magnetic field produced by the two mirror coils of the ALPHA-1 atom trap,superimposed on 1 T external solenoid. Image from [67]. . . . . . . . . . . . . . . 40xvLIST OF FIGURES2.21 Transverse magnetic field as a function of radius for a quadrupole, a sextuple andan octupole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.22 Photograph of the first layer of the octupole windings for ALPHA-1. . . . . . . . . 422.23 The magnetic field strength of the ALPHA-1 antihydrogen trap. . . . . . . . . . . 432.24 Current decay of the octupole and mirror coils as a function of time. . . . . . . . . 452.25 Example of the simplest picture to understand antiproton annihilation. . . . . . . . 472.26 Histogram of the distribution of the number of pions (neutral and charged) perproton-antiproton annihilation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.27 ALPHA-2 silicon tracking detector assembly at the Liverpool semiconductor detec-tor centre at the University of Liverpool in July 2012 [105]. . . . . . . . . . . . . . 492.28 Schematic of the cross section at the axial centre of the ALPHA-1 apparatus. . . . 502.29 Schematic cross section of the three layer silicon detector. . . . . . . . . . . . . . . 512.30 Measured antiproton annihilation signal and background distributions for the dis-criminating variables before and after applying the cuts. . . . . . . . . . . . . . . . 532.31 Transverse section of ALPHA-1 and ALPHA-2 detector. . . . . . . . . . . . . . . 552.32 Cross sectional schematic of ALPHA-2. 1) Antiproton capture trap, 2) transferline with two external solenoid to transfer antiprotons to the antihydrogen trap, 3)antihydrogen trap and 4) cryostat tower with the liquid helium inlet and transfer linewith one external solenoid to transfer positrons from the positron accumulator (notshown) to the antihydrogen trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1 Example of potential wells used during evaporative cooling where the most ener-getic antiprotons escape to the left. . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 Results of evaporative cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3 Autoresonant injection schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4 Axial energy distribution of 15,000 antiprotons driven to different final frequencies. 643.5 Antiproton annihilation vertices measured during antihydrogen production. . . . . 653.6 Simulated t and z coordinates of released mirror trapped antiprotons. . . . . . . . . 69xviLIST OF FIGURES3.7 Simulated t and z coordinates of released antihydrogen after the trap shutdown. . . 703.8 Simulated t and z coordinates of released antihydrogen after the trap shutdown. . . 733.9 Distributions of vertex variables for trapped antihydrogen. . . . . . . . . . . . . . 753.10 The Breit-Rabi diagram showing the hyperfine structure of the energy levels of the(anti)hydrogen atom in an external magnetic field. . . . . . . . . . . . . . . . . . . 793.11 Schematic of the ALPHA-1 apparatus. . . . . . . . . . . . . . . . . . . . . . . . . 813.12 Transition probability as a function of the frequency in the ALPHA-1 trap. . . . . . 813.13 Quadrupole mode frequency as a function of time . . . . . . . . . . . . . . . . . . 833.14 Example of calibration of the external solenoid using the cyclotron frequency. . . . 833.15 Microwave injection cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.16 The Series 1 probability transition as a function of the frequency when the minimumon-axis magnetic field is Bmin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.17 Series 2 and 4. Probability as a function of the frequency when the minimum on-axis magnetic field is Bmin+3.6 mT. . . . . . . . . . . . . . . . . . . . . . . . . . . 883.18 Antihydrogen event counts as a function of time. . . . . . . . . . . . . . . . . . . 913.19 Antihydrogen event counts as a function of the axial position for 0 < t < 30 s. . . . 924.1 Experimental density and rotation frequency evolving over time to global thermalequilibrium as a function of radius. . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.2 Calculated density as a function of radius for 20 × 106 electrons for different peakdensities n0 and different temperatures. . . . . . . . . . . . . . . . . . . . . . . . . 1064.3 Example of centrifugal separation of an antiproton-electron plasma. . . . . . . . . 1074.4 Measured compression rate of an electron plasma as a function of the rotating wallfrequency fs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.5 Schematic representing a transverse section of the rotating wall. . . . . . . . . . . 1114.6 Rotating wall radial potential when applying a 1 V sinusoidal potential to the 6segmented electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.7 Example of the radial compression of an antiproton-electron plasma. . . . . . . . . 114xviiLIST OF FIGURES4.8 Electron plasma radius as a function of compression time for slow and fast com-pression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.9 Electron plasma radius as a function of antiproton cloud radius. . . . . . . . . . . . 1164.10 Transport efficiency εexp as a function of the rotating wall frequency. . . . . . . . . 1175.1 Measurement of the antiproton capture efficiency as a function of the thickness ofthe tuneable degrader. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2 Antiproton capture efficiency as a function of the closing time. . . . . . . . . . . . 1215.3 Antiproton capture efficiency as a function of E01 voltage. . . . . . . . . . . . . . 1215.4 Schematic illustration of the antiproton and secondary electron capture. . . . . . . 1235.5 The number of secondary electrons, normalized to the number of incident antipro-tons as a function of the closing time. . . . . . . . . . . . . . . . . . . . . . . . . 1245.6 Schematic illustration of the antiproton cooling process. . . . . . . . . . . . . . . . 1265.7 Antiproton cooling efficiency as a function of closing time. . . . . . . . . . . . . . 1275.8 Schematic illustration of electron ejection while holding antiprotons. . . . . . . . . 1285.9 The boxes on the top represent the electrodes and show the voltage applied to them. 1295.10 Voltage applied to the electrodes and the resulting on axis potential for differentnumbers of electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.11 Schematic illustration of the particle dump to the MCP. . . . . . . . . . . . . . . . 1325.12 Magnetic field as a function of the axial position. . . . . . . . . . . . . . . . . . . 1335.13 MCP image of an antiproton-electron plasma after applying the rotating wall. . . . 1345.14 MCP image of an antiproton-electron plasma after applying the rotating wall. . . . 1356.1 MCP image of an antiproton-electron plasma before applying the rotating wall. Thescale bar is calculated from equation 5.2, so that it corresponds to the size in amagnetic field of 3 T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.2 MCP images of an antiproton-electron plasmas after applying the rotating wall for100 s with an amplitude of 1 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141xviiiLIST OF FIGURES6.3 MCP images of antiproton-electron plasmas after applying the rotating wall for100 s with an amplitude of 1 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.4 Central density after applying the rotating wall for 100 s, at 1 V and at a chosenfrequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.5 MCP image of the electron plasma before applying the rotating wall. . . . . . . . . 1456.6 Central density of an electron plasma of about 20 × 106 electrons as a function ofthe rotating wall frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.7 MCP image of the electron plasma after the rotating wall is applied for 100 s withan amplitude of 1 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.8 MCP images of antiproton clouds after applying the rotating wall for 100 s and 1 Vfor different frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.9 MCP image of an antiproton-electron plasma radially compressed by the rotatingwall at 600 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.10 Central density of the antiproton cloud after applying the rotating wall at 1 V atdifferent frequencies as a function of the rotating wall time when cooled by 20×106electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.11 MCP images after compression at 150 kHz for 1.5 × 105 antiprotons and 20 × 106electrons at different times of compression. . . . . . . . . . . . . . . . . . . . . . 1526.12 Central density of the antiproton cloud after applying a 1 Volt rotating wall at severalfrequencies as a function of the rotating wall duration. An electron plasma of 4×106electrons was used to cool the antiprotons. . . . . . . . . . . . . . . . . . . . . . . 1536.13 MCP images after compression at 140 kHz for 1.5 × 105 antiprotons and 4 × 106electrons at different times of compression. . . . . . . . . . . . . . . . . . . . . . 1546.14 Voltages applied to the electrodes and the resulting potential well for different num-ber of electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.15 Antiproton cloud central density as a function of the rotating wall frequency whenusing different potential wells and electron numbers for cooling. . . . . . . . . . . 156xixLIST OF FIGURES6.16 Antiproton cloud central density as a function of the rotating wall time for severalvoltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.17 MCP images of antiproton-electron plasmas after applying the rotating wall for100 s with an amplitude of 1 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586.18 Central density of the antiproton cloud as a function of the rotating wall frequencyfor different numbers of electrons and antiprotons. . . . . . . . . . . . . . . . . . . 1596.19 Antiproton cloud central density as a function of time after the rotating wall wasapplied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.20 MCP images after stopping a 100 s rotating wall, for 1.5 × 105 antiprotons and20 × 106 electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.21 MCP images of the antiproton-electron plasma after applying the rotating wall at500 kHz, with an amplitude of 1 V and for 100 s. . . . . . . . . . . . . . . . . . . 1626.22 Antiproton cloud central density as a function of the rotating wall frequency at 3 Tand 4 T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.1 Calculated potential as a function of the axial position for 4 × 106 electrons, withTe = 300 K and re = 4 mm (n0 = 10 × 106 cm−3). . . . . . . . . . . . . . . . . . . 1777.2 Calculated density as a function of radius and axial position for 4 × 106 electronswith Te = 300 K and re = 4 mm (n0 = 10 × 106 cm−3). . . . . . . . . . . . . . . . 1777.3 Maxwell-Boltzmann energy distribution for 300 K, 500 K and 700 K, normalized tothe peak value of 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.4 ωb/2pi as a function of radius and energy. . . . . . . . . . . . . . . . . . . . . . . 1807.5 Energy distribution for 300 K and also the antiprotons’ bounce frequency as a func-tion of the energy at the radial positions r = 0 mm and r = 4 mm, with re = 4mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.6 Self-consistent on-axis potential (left plot) and radial density (right plot) for differ-ent plasma radii at 300 K for 4 × 106 electrons. . . . . . . . . . . . . . . . . . . . 181xxLIST OF FIGURES7.7 Self-consistent on-axis potential (left plot) and radial density (right plot) for differ-ent temperatures for re = 3.2 mm for a plasma with 4 × 106 electrons. . . . . . . . 1827.8 Bounce frequency distribution of antiprotons as a function of the antiproton bouncefrequency for different temperatures and electron densities for 4 × 106 electrons. . . 1837.9 Rotation frequency of the electron plasma as a function of radius for several plasmaradii and temperatures of a 4 × 106 electron plasma. . . . . . . . . . . . . . . . . . 1847.10 Antiprotons f (ωb + ωrot) as a function of the temperature and plasma radii whencooled by 4 × 106 electrons. Black solid dots are the measured antiproton centraldensity (arbitrary units) as a function of the applied rotating wall frequency. . . . . 1857.11 f (ωb+ωrot) of antiprotons when cooled by 20×106 electrons at the initial conditionsfor different temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.12 MCP images after 100 s of compression for various frequencies for 1× 105 antipro-tons and 20 × 106 electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.13 MCP images for various compression times at 550 kHz for 1 × 105 antiprotons and20 × 106 electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.14 Antiproton cloud central density as a function of the rotating wall application timewhen cooled by 20 × 106 electrons at ωRW = 350 kHz and ωRW = 550 kHz. . . . . 1907.15 The antiprotons’ f (ωb + ωrot) distribution when cooled by an electron plasma withelectron densities 5 × 109 cm−3 and 9 × 109 cm−3, which correspond to the fastestcompression when applying the rotating wall at 350 kHz and 550 kHz, respectively. 191A.1 Stopping power as a function of the particle energy for antiprotons and protons inaluminium and beryllium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209B.1 Stack of electrodes of the antiproton capture trap. . . . . . . . . . . . . . . . . . . 211B.2 Micro-D connector and PCB connected in the flange between the UHV and OVCregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212C.1 Schematic of the cosmic veto detector. . . . . . . . . . . . . . . . . . . . . . . . . 214C.2 Distribution of ∆t of cosmic rays and antiproton annihilations for dtop = 150 cm. . . 216xxiLIST OF FIGURESC.3 Distribution of ∆t of cosmic rays and antiproton annihilations for dtop = 200 cm. . . 217xxiiAcknowledgementsFirstly, I would like to thank my supervisor, Makoto Fujiwara, for his endless guidance and supportduring my PhD, and of course for giving me the opportunity to be part of this incredible journeyon antihydrogen research. I would also like to thank my co-supervisor Walter Hardy for sharing hisknowledge and experience. I am also grateful to the other members of ALPHA Canada, especiallyArt Olin and Dave Gill, who were always supportive and helped me a lot to improve my researchskills and oral presentations.I would like to thank all the members of the ALPHA Collaboration, it was amazing to spend timeat CERN, work at the experiment and be part of ALPHA-1 and ALPHA-2. I am especially thankfulto the people that spent night shifts with me taking data for the antiproton cloud compression:Eoin, Andrey, Phil, Ryan, Chuckman, Eli and Marcelo. I would also like to thank everybody whowas interested in the antiproton cloud compression project, it was very motivating. Thanks to Joel,Jonathan, Andrey, Mike, Dirk, and Aled for the helpful discussions about the mechanism of rotatingwall compression.I would also like to thank my family for all their support and my husband, for all his help and love.xxiiiPart IALPHA apparatus and coldantihydrogen experiments1Chapter 1Introduction1.1 AntimatterPhysicist and mathematician Paul Dirac first predicted the existence of the antielectron in 1931 [7],while developing a relativistic quantum theory for the electron. The quantum theory for slow par-ticles had already been established by Schrödinger without taking into account relativistic effects.Dirac obtained four solutions to his equation. Two corresponded to the electron (spin up and spindown), while the other two predicted the existence of a negative energy particle. The origin of suchsolutions can be seen at a glance in the equation for relativistic energy that carries a square root andallows a positive and a negative solutions:E = ±√(m2c4 + ~p2c2), (1.1)where E is the energy, m is the particle’s mass, c is the speed of light and ~p is the particle’s momen-tum.The obvious first guess was that the proton (the only positive particle known at that time) was theother particle in the solution of Dirac’s equation. But the mass difference between the electronand the proton constituted a major problem, since the Dirac equation imposed an exact symmetrybetween the particles. The candidacy of the proton as a possible solution was soon ruled out inde-pendently by Robert Oppenheimmer and Igor Tamm [8]. They argued that it was not in agreementwith the stability of the hydrogen atom since the Dirac equation also predicted the annihilation ofthe two particles when they were close to each other. After this rejection, Dirac introduced the new21.1. Antimatterconcept of a “sea of electrons” to explain how a particle with negative energy in the sea becomes aparticle with positive energy leaving a “hole” in the sea. Matter will be stable as long as the “sea ofelectrons” is not perturbed. This idea was replaced by the Feynman-Stueckelberg interpretation [9]in which, instead of using the concept of negative energy, antiparticles have positive energy runningbackwards in time. Dirac finally predicted the existence of the antielectron in a paper published in1931 [7]:“A hole, if there were one, would be a new kind of particle, unknown to experimen-tal physics, having the same mass and opposite charge to an electron. We may call sucha particle an antielectron.”It did not take too long for the detection of such an antiparticle. One year later, Carl Andersondetected the antielectron for the first time, while he was doing research on cosmic rays [10]. Theantielectron was the first and the only antiparticle to receive its own name [10] :“From an examination of the energy-loss and ionization produced it is concludedthat the charge is less than twice and, and it is probably exactly equal to, that of theproton. [...] These particles will be called positrons.”After the discovery of the positron, the existence of a matter-antimatter system composed of anelectron and a positron, analogous to the hydrogen atom, was predicted in 1936 by Stjepan Mo-horovicˇic´ and was called “electrum” [11]. This system was experimentally observed by MartinDeutsch in 1951 and became known as “positronium” [12].The first observation of antiprotons occurred in 1955 [13]. Between the discovery of the positronand the antiproton, other particles were discovered. Among them are: the neutron (1932) [14], themuon (1937) [15], the charged pion (1947) [16, 17] and the neutral pion (1950) [18].The solutions of Dirac’s equation for particles with spin 12 predicted that the proton should haveits matching antiparticle, called the antiproton. But to produce antiprotons, very highly energeticprotons were needed.31.2. AntihydrogenThe proposal to construct a proton accelerator needed for antiproton production was presented in1946 and accepted in 1948 [19]. The Bevatron (Billions of eV Synchrotron) was developed and builtat Lawrence Berkeley National Laboratory over several years and finally began operation in January1954. Protons were accelerated up to energies of 6.5 GeV and the reaction during the collision witha stationary target to produce antiprotons is:p + p→ p + p + p¯ + p (1.2)The Bevatron was used for both the production of antiprotons and antineutrons [21].In 1965, the antideuteron (antiproton and antineutron nucleus) was observed at the the Proton Syn-chrotron at CERN [22] and at the Alternating Gradient Synchrotron (AGS) at Brookhaven NationalLaboratory [23].The observation of the antiproton was of great importance, since it demonstrated that Dirac equationalso applies to other particles of spin 12 . Furthermore, the observation of the antideuteron revealedthat the nuclear force is also valid for antimatter.1.2 AntihydrogenIn 1995, antihydrogen was produced for the first time at CERN [25]. Ten atoms of antihydrogenwere produced in flight from the interaction between high energy antiprotons from the Low EnergyAntiproton Ring (LEAR) and a xenon jet gas target. When the antiprotons passed near the nucleus,antihydrogen was formed predominantly through the two-photon mechanism [25, 26]:p + Z → p + 2γ + Z → p + e+ + e− + Z → H + e− + Z, (1.3)where the photon-photon interaction produced an electron-positron pair and the positron could bebound to the antiproton, resulting in antihydrogen formation [27].41.3. Why study antihydrogen?Later, in 1997, 101 atoms of antihydrogen were observed at Fermilab [28]. Anithydrogen was pro-duced by the interaction of a circulating antiproton beam with a jet of molecular hydrogen gas [29].Nevertheless, the antihydrogen atoms created in 1995 and 1997 were too energetic to be capturedand studied.In 2002, the ATHENA collaboration produced for the first time about 50,000 low-energy (a few eV)atoms of antihydrogen at CERN [30], followed by the ATRAP collaboration, who produced around170,000 atoms of antihydrogen [31, 32]. For these measurements, antiprotons and positrons werecaptured in Penning traps and were made to interact to form antihydrogen.In 2004, the ATRAP collaboration produced for the first time antihydrogen through laser controlledcharge-exchange collisions [33].In 2010, the ALPHA collaboration produced and trapped 38 cold antihydrogen atoms with energiesless than ∼ 42 µeV [1], and trapped hundreds of them later in 2010 and in 2011. The ATRAPcollaboration trapped 105 atoms of antihydrogen in 2012 [34].In 2011, ALPHA published a paper where the atoms were confined for at least 1,000 s [2]. In2012, state transitions of the antihydrogen spin were induced by causing the atoms to interact withmicrowave radiation [3].While the ALPHA and ATRAP experiments produce and confine antihydrogen atoms, the ASACUSAcollaboration synthetized cold antihydrogen in a cusp trap in 2010 for the production of a spin po-larized antihydrogen beam for precision spectroscopy of the ground state hyperfine splitting ofantihydrogen [35]. In 2014, an antihydrogen beam was created by ASACUSA, where 80 atoms ofantihydrogen were detected [36].1.3 Why study antihydrogen?Neutral antimatter, in the form of antihydrogen, is a promising testbed for tests of the CPT (chargeconjugation/parity transformation/time reversal) invariance and the weak equivalence principle [37].51.3. Why study antihydrogen?In the 1950’s, explicit proofs of the CPT theorem based on the Lagrangian quantum field theorywhere derived by G. Lüders, W. Pauli and J.S. Bell [38–40]. The CPT theorem states that undersimultaneous transformation of the charge conjugation (C), space reversal or parity (P) and timereversal (T), there is an exact symmetry of any interaction. This means that for each particle, thereis an antiparticle with:• the opposite charge,• the opposite internal quantum numbers• the opposite magnetic moment• the same lifetime• the same (inertial) massCPT invariance has been tested in many sectors, by the comparison between matter and antimatterproperties. Among some of the CPT invariance tests are shown in table 1.1. Until now, all themeasurements performed are consistent with CPT symmetry.Type of measurement Quantity measured Value measurede− and e+ mass [41] |me+ − me− |/maverage < 8 × 10−9e− and e+ charge [42] |qe+ + qe− |/e < 4 × 10−8e− and e+ gyromagnetic ratio [43] (ge+ − ge−)/gaverage (−0.5 ± 2.1) × 10−12p and p magnetic moment [44] (µp − µp)/µaverage (0 ± 5) × 10−6p and p mass [45, 46] |mp − mp)|maverage < 7 × 10−10p and p charge [45, 46] |qp − qp|/e < 7 × 10−10K0 and K0mass [47] |mK0 − mK0 |/maverage < 6 × 10−19K0 and K0lifetime [47] (ΓK0 − ΓK0)/Γaverage (8 ± 8) × 10−18Table 1.1: Measurements that test CPT invariance. For a more extensive list, see reference [48].Antihydrogen is the simplest antimatter atomic system, analogous to the hydrogen atom, but com-posed of an antiproton with an orbiting positron. Its matter partner, hydrogen, is the best understoodatom in physics and has been studied to a very high precision. CPT symmetry implies that theeigenenergies of antihydrogen and hydrogen are the same. The measurements of the electronic(positronic) transitions of antihydrogen is a promising measure that would test the CPT invariance.61.3. Why study antihydrogen?The most precise measurements performed on hydrogen that are suitable for a comparison betweenhydrogen and antihydrogen are:• The 1S–2S electronic transition is one of the best candidate for a high precision measurementin antihydrogen since its natural width is very narrow (1.3 Hz) and it has been measured to4.2 parts in 1015 in hydrogen [49]. Future experiments of antihydrogen, including 1S–2Sspectroscopy, are discussed in section 3.5.• The ground state of the hyperfine transition is known to 1.4 parts in 1012 in hydrogen [50].Measurements of the hyperfine transition of antihydrogen are discussed in section 3.4.The Standard Model is a very successful Quantum Field theory that explains the experimental obser-vations of Particles Physics on a microscopy scale. On the other hand, it seems that it is incompletesince it fails to explain the matter-antimatter inbalance, the nature of dark matter and the workingof gravity on a microscopic scale. The CPT theorem states that for all Quantum Field theoriesthat incorporate Lorentz invariance, locality and unitarity (of which the Standard Model is one),CPT invariance holds exactly [38, 51, 52]. A detection of CPT violation, even the tiniest amount,would imply that one or more of these conditions do not hold, which in turn would mean that ourunderstanding of the Nature must be reformulated at the most fundamental level.The CPT theorem does not provide any information about the gravitational acceleration of antihy-drogen (antimatter) on the Earth (matter). By CPT invariance, the free fall of antihydrogen on anantimatter Earth (“anti-Earth”), should be the same as hydrogen on the Earth. Also, Einstein’s weakequivalence principle is a foundation stone of general relativity and testing it is of great interest.According to the weak equivalence principle, any object (matter or antimatter) should fall to theEarth at the same rate. Even though antimatter was discovered decades ago, a test with neutralantimatter is more suitable for a direct measurement on the gravity interaction between matter andantimatter. The reason is that the motion of charged (antimatter) particles can be easily distorted byelectric fields, among other disruptions [53]. The ALPHA collaboration recently reported a limit onthe ratio of the gravitational mass to the inertial mass of antihydrogen [5].71.4. Thesis overview1.4 Thesis overviewDuring my PhD, I was part of the first experiments on trapped antihydrogen and was also involvedin the transition between the ALPHA-1 and the ALPHA-2 apparatus. The ALPHA-1 apparatus wasused to trap and make the first measurements on antihydrogen by flipping the spin of the positron.In 2012, the ALPHA collaboration disassembled ALPHA-1 and the new ALPHA-2 apparatus wasinstalled. This thesis is separated into two parts: part I is about cold antihydrogen experiments inALPHA-1, while part II is about antiproton cloud compression in the antiproton trap of ALPHA-2.During part I, in chapter 2, I describe the ALPHA-1 and ALPHA-2 apparatus, along with the sim-ilarities and differences between them. Furthermore, I explain how charged particles and neutralparticles can be trapped in the apparatus and how they are detected. Chapter 3 is dedicated to ex-periments with antihydrogen. Here, I describe the techniques used by ALPHA to produce and trapantihydrogen. I also present the experimental measurements on trapped antihydrogen, its confine-ment for 1,000 s and the first experimental measurements on the internal structure of antihydrogen.During part II, in chapter 4, I first discuss the motivation for compressing antiproton clouds in AL-PHA. Then, I give a theoretical overview of the confinement of non-neutral plasmas in a Penningtrap and its properties. I describe the rotating wall mechanism, which is used to radially com-press non-neutral plasmas. This includes an overview of the theoretical knowledge, which it mustbe stressed, is as yet incomplete. Then I discuss the way compression is routinely performed inALPHA for the production of antihydrogen. In chapter 5, I present the experimental set up andprocedure used for the measurements of antiproton cloud compression in ALPHA-2. The resultsfor these measurements are presented in chapter 6. The compression process was studied under dif-ferent conditions (time, frequency, voltage, magnetic field, particle number, etc..) to understand thephysics behind the compression. In chapter 7, I discuss possible compression mechanisms that canbe responsible for the antiproton cloud compression. Bounce resonant transport seems to be a goodcandidate and I performed numerical calculations of the axial bounce frequency of the antiprotonsand the rotation frequency of the system to compare the sum of these frequencies to the range offrequencies where compression was observed.8Chapter 2ALPHA Apparatus2.1 The Antiproton Decelerator experimental hallThe ALPHA apparatus is located at CERN in the Antiproton Decelerator (AD) hall. The AD cur-rently supplies antiprotons to six experiments aiming to study the properties of antimatter. Theseexperiments are:• ACE (Antiproton Cell Experiment): The goal of the experiment is to study the effectivenessand suitability of an antiproton beam for treating cancer [54].• AEGIS (Antihydrogen Experiment: Gravity, Interferometry, Spectroscopy): The goal of theexperiment is to produce an antihydrogen beam to make a direct measurement of the Earth’sgravitational acceleration with the use of a moiré deflectometer [55].• ALPHA (Antihydrogen Laser PHysics Apparatus) : The goal of the experiment is to produce,trap and study the properties of antihydrogen such as the 1S-2S transition [56]. ALPHA is asuccessor of the ATHENA experiment.• ASACUSA (Atomic Spectroscopy And Collisions Using Slow Antiprotons): The goals of theexperiment are to produce an antihydrogen beam to study the hyperfine structure of the atom,produce and study antiprotonic helium (where one electron of the helium atom is replaced byan antiproton) and finally, to study the interaction of antiproton beams with matter [57].• ATRAP (Antihydrogen TRAP): The goal of the experiment is to produce, trap and studyproperties of antihydrogen [58].92.2. ALPHA-1 and ALPHA-2• BASE (Baryon Antibaryon Symmetry Experiment): The goal of the experiment is to trapantiprotons and measure their magnetic moment [59].The six experiments are located in the AD hall as shown in figure 2.1. Note that GBAR experiment(Gravitational Behaviour of Antihydrogen at Rest) is under construction and it is not shown infigure 2.1.Figure 2.1: Schematic of the AD hall. Image modified from [60]. Antiproton deceleration andcooling is explained in section 2.4.1.2.2 ALPHA-1 and ALPHA-2The ALPHA experiment is a successor of the ATHENA experiment [61], which produced low en-ergy antihydrogen in 2002, but without trapping it [30]. The ALPHA-1 apparatus was constructedin 2005 and replaced in late 2011 by the ALPHA-2 apparatus. Figure 2.2 illustrates the key com-ponents of the ALPHA-1 apparatus, a charged particle trap and an antihydrogen trap, which are102.2. ALPHA-1 and ALPHA-2surrounded by the three-layer silicon detector.ElectrodesMirror CoilsOctupoleAnnihilationDetectorX YZFigure 2.2: ALPHA-1 trapping region. The annihilation detector is the silicon tracking detectorwhich surrounds the magnetic trap and the Penning-Malmberg trap. The 1 Tesla external solenoidis not shown here. Image from [62].ALPHA-1 and ALPHA-2 have similar characteristics:• They include Penning-Malmberg traps, which are composed of cylindrical electrodes for axialconfinement and an uniform, axial magnetic field radial confinement of charged particles(antiprotons, electrons and positrons).• The Penning-Malmberg traps are placed inside of an ultra-high vacuum space with a pressureas low as 10−13 mbar. The ultra-high vacuum space is surrounded by a liquid helium cryostatso that the walls of the trap can reach temperatures near 4 K.• Both apparatuses also possess a magnetic atom trap superimposed on the Penning-Malmbergtrap. The trap is composed of a transverse octupole and several axial mirror coils. Groundstate antihydrogen with energies up to ∼43 µeV (0.5 K) can be confined in the magnetic trap.• Antihydrogen annihilation is detected by a three-layer cylindrical silicon detector surroundingthe apparatus. The vertex of the annihilation can be determined in time and in space.112.2. ALPHA-1 and ALPHA-2• ALPHA-2 inherited the ALPHA-1 positron accumulator, which provides positrons to produceantihydrogen. The positron accumulator is briefly described in section 2.5.During the operation of the ALPHA-1 apparatus, the ALPHA collaboration mastered new tech-niques to optimize antiproton and positron plasma parameters to produce trappable antihydrogen.Among those techniques are:• The rotating wall mechanism, which compresses antiproton clouds and positron plasmas re-sulting in denser groups of particles [63]. For more details, see section 4.3.• Evaporative cooling, which is used to cool particles to temperatures below those otherwiseachievable [64]. For more details, see section 3.2.1.• The autoresonant injection of antiprotons, used to inject antiprotons into the positron plasmato create antihydrogen [65]. For more details, see section 3.2.2.Antihydrogen was successfully trapped by ALPHA in the ALPHA-1 apparatus in 2010 [1] and thefirst experiments on the internal structure of antihydrogen were performed in 2011 [3].To achieve the ultimate goal of the ALPHA collaboration, precision laser spectroscopy on anti-hydrogen, laser access is needed. Therefore, despite the fruitful performance of the ALPHA-1apparatus, it was decided to build a new apparatus (ALPHA-2) to allow laser access, as well otherimprovements.The commissioning of the ALPHA-2 apparatus began in 2012. Partial operation initiated in June2012 and first antihydrogen trapping measurements were performed during 2014. I personallyworked on the antiproton capture trap of ALPHA-2 (section 2.3.3 and performed the measurementson antiproton cloud compression for this thesis in late 2012 (chapter 6).Figure 2.3 and figure 2.4 are photographs of the ALPHA-1 and the ALPHA-2 apparatuses, respec-tively. In this chapter, we will present the ALPHA-1 and the ALPHA-2 apparatus and how theywork, including the details of their similarities and differences.122.2. ALPHA-1 and ALPHA-2Figure 2.3: Photograph of the ALPHA-1 apparatus. The grey cylindrical volume is the 1 T externalsolenoid. The silicon detector and the particle/atom traps are inside it. The antiprotons from the ADare injected from the left . A part of the positron accumulator can be seen on the right hand side ofthe picture.Figure 2.4: Photograph of the ALPHA-2 apparatus. The "Carlsberg" grey cylindrical volume isthe 1 T external solenoid, the silicon detector and the atom trap with laser access are placed insideit. The green cylindrical volume at the left hand side is the 3 T external solenoid of the antiprotoncapture trap. The antiprotons arrive from the left side and the positron accumulator is on the righthand side of the apparatus (not shown here). The light blue rectangular pads are the antiprotonscintillators.132.3. Penning-Malmberg traps2.3 Penning-Malmberg trapsBoth Penning traps and Penning-Malmberg traps (sometimes called cylindrical Penning traps) con-sist of a uniform, axial magnetic field for the radial confinement of charged particles and an electricfield for the axial confinement. The difference between them is that the original Penning trap hadhyperbolic electrodes, creating a quadratic electric potential, while the Penning-Malmberg trap hashollow cylindrical electrodes creating an electric potential with anharmonic contributions. The mo-tion of charged particles in a Penning trap can be analytically described, but the trap presents somelimitations since it does not allow an easy access to the trap volume. The Penning-Malmberg trapis much more versatile, since it is possible to stack the hollow cylindrical electrodes. As a result,differently charged particles can be held in the trap, segmented electrodes can provide variable ax-ial electric fields, and there is access at the end of the electrode stack to place diagnostic devices.Even though Penning-Malmberg traps have non-harmonic contributions, the theoretical motion ofcharged particles in a Penning trap can be considered to be a first approximation of the equations ofmotion in a Penning-Malmberg trap.2.3.1 Charged particle motion in a Penning trap/Malmberg-Penning trapIn this section, we discuss the motion of single particles. See section 4.2, for plasmas, for whichspace charge and collective effects are important.The classic Penning trap consists of an axial, uniform magnetic field:~B = Bzzˆ, (2.1)and a quadratic electric potential:φ(x, y, z) =V02d2(z2 − 12x2 − 12y2), (2.2)142.3. Penning-Malmberg trapswhere V0 is the applied potential and d is a characteristic trap length [66].The motion of charged particles in such a trap can be derived from the Lorentz force equation:md2~rdt2= q(−~∇φ + d~rdt× Bzzˆ), (2.3)where ~r = (x, y, z). In Cartesian coordinates, the equations of motion can be written as [66]:d2xdt2− ωc dydt −12ω2z x = 0, (2.4)d2ydt2+ ωcdxdt− 12ω2z y = 0, (2.5)andd2zdt2+ ω2z z = 0, (2.6)whereωc =qBzm(2.7)is the cyclotron frequency andωz =√qV0md2(2.8)is the axial frequency. From the zˆ equation of motion, we can observe that the charged particleundergoes simple harmonic motion. The xˆ and yˆ equations of motion are coupled, and by making acomplex substitution u = x + iy, we obtain [66]:d2udt2+ iωcdudt− 12ω2z u = 0. (2.9)Equation 2.9 has a solution of the form u = exp(−iω±t) withω± =12(ωc ±√ω2c − 2ω2z). (2.10)152.3. Penning-Malmberg trapsParticle ωc/2pi ωz/2pi ωm/2piElectron/positron 84 GHz 12 MHz 1 kHzAntiproton 46 MHz 300 kHz 1 kHzTable 2.1: Frequencies of charged particles in the ALPHA-2 antiproton trap in a typical potentialwell in a 3 T magnetic field.ω2c > 2ω2z is required for a confined oscillatory motion in the xˆ and yˆ directions. The high frequencysolution is the modified cyclotron frequency and it is usually known as ω′c with ω′c ≡ ω+ = ωc−ωm,where ωm ≡ ω−. The low frequency solution is the magnetron frequency and can be expressed asωm ≡ ω− = ω2z/2ω′c. The three oscillation frequencies of charged particles in a Penning trap are:ω′c ' ωc =qBzm, (2.11)ωz =√qV0md2, (2.12)andωm =ω2z2ω′c. (2.13)These oscillation frequencies follow the hierarchy:ωc  ωz  ωm, (2.14)and figure 2.5 shows an example of the motion of a charged particle in a Penning trap. Typical oscil-lation frequencies of antiprotons, electrons and positrons in the ALPHA experiment are calculatedin table 2.1. The magnetron frequency is the same for all these particles since it does not depend onthe mass of the particles.162.3. Penning-Malmberg trapsCyclotron MotionMagnetron MotionAxial MotionFigure 2.5: The charged particle motion in a Penning trap is the superposition of the three oscillatorymodes: cyclotron, axial and magnetron oscillation. Image from [67].2.3.2 ALPHA-1 Penning-Malmberg trapThe ALPHA-1 Penning-Malmberg trap is composed of a stack of thirty-five electrodes immersedin a 1 T solenoidal magnetic field directed along the trap axis. The stack of electrodes is illustratedin figure 2.6. The cylindrical electrodes are made from aluminium and are gold-plated to avoidoxidization. The electrodes have different radii and lengths for specific purposes and they are iso-lated from each other by synthetic ruby spheres or ceramic spacers (in the case of high voltageelectrodes). By applying appropriate voltages to the electrodes, it is possible to create an electricpotential that axially confines the charged particles.The ALPHA-1 trap can be divided into 3 sections:• Antiproton trap: Left hand side section, where the antiprotons are captured from the ADusing high voltage electrodes and sympathetically cooled by an electron plasma (see sec-tion 2.4). A six-segmented electrode provides the rotating wall field to radially compress theantiproton cloud (see section 4.3). An additional internal solenoid surrounding this regioncan be energized to give a total magnetic field of 3 T.• Positron trap: The right hand side section is used to prepare positrons plasmas after capturingthem from the positron accumulator (see section 2.5). One segmented electrode is used to172.3. Penning-Malmberg trapsradially compress the positron plasmas.• Mixing trap: The central section is composed of thirteen electrodes where four of them arespecially filtered using low noise amplifiers. Antihydrogen production is performed in thissection of the trap. The electrodes of the mixing trap are designed to have a larger innerdiameter and very small thickness to allow antihydrogen to have the maximum radial motioninside the magnetic trap without annihilating (see section 2.8.1).The inner diameter of the electrodes of the antiproton trap and the positron trap is 36.6 mm andthe inner diameter of the electrodes in the mixing trap is 44.6 mm. Voltages are applied to theelectrodes through a circuit board with a passive RC-filter for each electrode. The filter has a low-pass connection for DC voltages, which is the normal operation of the electrodes, and a high-passconnection for fast pulses, which is used for the segmented electrodes (rotating wall electrodes) orelectrodes where fast pulses are required (e.g. electrodes with pulses for electron ejection). Normalamplifiers can sustain voltages in a ± 140 V range to a precision of 4 mV, while the specially filteredamplifiers have low-noise voltages in a ± 72 V range within 2 mV. High-voltages (up to ∼5 kV) areapplied directly to the electrodes without filters.182.3.Penning-Malmbergtrapscatching trap mixing trap positron trapE02 E03 E05E06E07 E08 E10 E11E04HVAE01HVBE09E12 E14 E16 E17 E19 E20 E21 E22 E24E18E15 E23E13E25 E29 E30E28 E29E26E27 E28 E33HVCE34 E35 E36/Flapper1010 mmE31 E32Normal ElectrodesFiltered ElectrodesSegmented ElectrodesHigh-Voltage ElectrodesFigure 2.6: Diagram of the ALPHA-1 electrode stack. Image from [67]192.3. Penning-Malmberg traps2.3.3 ALPHA-2 Penning-Malmberg trapsThe ALPHA-2 apparatus possesses two disconnected Penning-Malmberg traps. The antiprotoncapture trap is separated from the mixing trap, so that independent operation can be carried out. Theantiproton capture trap is intended to be used as an accumulator of antiprotons, which will makehigher numbers of antiprotons available to the mixing trap. Another advantage for the separation ofthe antiproton capture trap and the mixing trap is that the antiproton degrader will not obstruct thepath for laser access.ALPHA-2 antiproton capture trapThe ALPHA-2 Penning-Malmberg antiproton capture trap is composed of a stack of twenty elec-trodes immersed in a 3 T solenoidal magnetic field, directed along the trap axis. Figure 2.7 illustratesthe electrode stack. Two high-voltage electrodes are used to trap antiprotons from the AD (see sec-tion 2.4.2) and two segmented electrodes near each end can be used to compress the antiprotonclouds. The ideal use of the catching trap is to constantly capture antiprotons and accumulate themat the end of the trap. Then, optimal amounts of antiprotons can be transferred to the mixing trapto mix them with positrons to produce antihydrogen. Therefore, antiprotons would be available ‘ondemand’.The inner diameter of each electrode is 29.6 mm. The electrodes filters and connections are thesame as for ALPHA-1 (section 2.3.2). I personally contributed to the cabling and connection of theelectrodes. For more details about this work, see appendix B.ALPHA-2 mixing trapThe ALPHA-2 Penning-Malmberg mixing trap is composed of a stack of twenty-seven electrodesimmersed in a 1 T solenoidal magnetic field directed along the trap axis. Figure 2.8 illustrates theelectrode stack. The trap is similar to the one of ALPHA-1, with three different sections:202.4. Antiproton production, capture and coolingHVRW RW395.86 mm29.6 mmHVDegraderEFigure 2.7: Diagram of the ALPHA-2 electrode stack of the antiproton capture trap. The highvoltage (HV) electrodes are in red and the six-segmented rotating wall (RW) electrodes are in blue.• Antiproton trap: Left hand section for antiproton capture, where antiprotons are re-capturedfrom the antiproton trap. It is composed of a stack of seven electrodes with an inner diameterof 29.6 mm, including a rotating wall electrode. An internal solenoid can be energized toprovide a maximum axial field of 2 T. This trap section can have an overall axial magneticfield of 3 T.• Positron trap: Placed at the right hand side, this trap is similar to the antiproton trap at theopposite end. It is used to capture positrons from the positron accumulator and can also havean overall axial magnetic field of 3 T.• Mixing trap: It is the core of the trap, with a total of thirteen electrodes. The inner diameteris larger (44.6 mm), the same as in the ALPHA-1 apparatus.2.4 Antiproton production, capture and coolingAs discussed in section 1.1, the antiproton was not observed until 1955, more than twenty yearsafter the discovery of the positron. The reason for such a long wait was that a proton beam ofvery high energy was needed to produce an antiproton by creating a proton-antiproton pair. As athreshold, the incident proton must carry at least 6 times its mass in energy. The production and thedetection of the first antiproton was carried out in the Bevatron at the Lawrence Berkeley National212.4. Antiproton production, capture and coolingRWantiproton trap mixing trapRWpositron trap503.13 mm29.6 mm44.6 mmFigure 2.8: Diagram of the ALPHA-2 electrode stack of the mixing capture trap. Six-segmentedrotating wall (RW) electrodes are in blue.Laboratory, where protons were accelerated up to energies of 6.5 GeV [13]. The experiment wasmainly undertaken by Owen Chamberlain and Emilio Segrè, who shared the Nobel Prize in 1959for this discovery.Collision experiments between a proton and an antiproton beam were performed from the 80’s,when the W and Z bosons were discovered at CERN [68, 69] and the top quark was discovered inthe Tevatron at Fermilab [70].Nowadays, antiproton beams are decelerated and stored to make low energy measurements of theproperties of antimatter.2.4.1 The Antiproton Decelerator (AD)The Antiproton Decelerator (AD) is a unique machine designed to decelerate and cool antiprotonsfrom the giga-electronvolt to the mega-electronvolt energy range [71] and is a successor of LEAR(Low Energy Antiproton Ring) [72]. A pulse of 1013 protons with a momentum of 26 GeV/c isreceived from the Proton Synchrotron (PS) at CERN. Before entering the AD, the pulse of protonscollides with a target consisting of a thin iridium rod embedded in graphite to create antiprotons [73].Antiprotons are produced through a proton-proton collision reaction where a proton-antiproton pairis created:p + p→ p + p + p + p¯. (2.15)222.4. Antiproton production, capture and coolingAbout 5 × 107 antiprotons are created through this reaction and about 3 × 107 antiprotons with amomentum of 3.5 GeV/c are injected into the AD after being focussed and selected with a magnetichorn [74].The AD decelerates and cools antiprotons using a cycle of ∼ 100 s, where it alternates decelerationand cooling of the beam [74]. Figure 2.9 shows a typical AD cycle. Radio-frequency (rf) decelera-tion is applied when the antiproton pulse passes through a rf cavity. Deceleration causes spreadingof the beam momentum and an increase in divergence (in accelerator physics terms, an increaseof the emittance of the beam). To counteract this effect, two techniques of cooling are employed:stochastic cooling and electrons cooling.0.10.323.570 12 35 54 71 77 85Antiprotonmomentum[Gev/c]Time [s]17 s6.6 s16 s8 sstochastic coolingelectron coolingp¯ injectionp¯ extractionFigure 2.9: AD cycle during deceleration and cooling [74].Stochastic cooling is applied to antiprotons with momentum of 3.5 GeV/c and 2 GeV/c. Stochasticcooling is a feedback system that corrects the orbital motion of the particles [75]. A "pick-up"detects the motion of the particles and produces a correction derived from the deviation of the232.4. Antiproton production, capture and coolingparticles from the ideal trajectories. After a short time, the particles pass through the kicker andan electric field is applied to correct the position and momentum of the particle. The beam needsto circulate many times to concentrate the particles around the chosen orbit. Simon van der Meershared the 1984 Nobel prize with Carlo Rubbia. Simon van der Meer contributed to the discovery ofthe W and Z bosons by inventing the technique of stochastic cooling to accumulate intense beams ofantiprotons, while Carlo Rubbia led the UA1 experiment that detected the W and Z bosons [68, 69].Electron cooling is performed by injecting a dense electron beam along the antiproton beam withthe same average velocity [76]. The antiproton beam is cooled by the electron beam by Coulombcollisions on a short section of the ring. The electrons are then extracted from the AD ring.After cooling and deceleration, the antiproton beam is extracted to the experiment. The beam is a∼200 ns long bunch, has about 3 × 107 particles and a kinetic energy of 5.3 MeV (momentum of100 MeV/c). The cycle is repeated and the experiments receive a bunch of antiprotons every 100 s.2.4.2 Antiproton capture and coolingAntiprotons are extracted from the AD into the experiment with a kinetic energy of 5.3 MeV (seesection 2.4.1). Since we are only able to trap antiprotons having an energy less than ∼ 5 keV or less,very thin layers of material (typically aluminium) are placed before the trap entrance to cause theantiprotons to lose energy in the material before the capture. This process is called degrading andthe layers are called a “degrader”. About 3 × 107 antiprotons are extracted from the AD, but onlyabout 0.5% survive the degrading process with an energy less than 5 keV.In principle, the antiproton capture and cooling mechanism is the same in ALPHA-1 and ALPHA-2but there are some differences such as the thickness and the degrader material used. In ALPHA-1, the degrader is a 12.5 µm thick stainless steel foil and a 218 µm thick aluminum foil, while inALPHA-2, there are layers of aluminium and beryllium. A part from the thickness, the degradermaterial must be compatible with the vacuum requirements and be strong enough to hold at least 1atm of pressure. More details about antiproton capture in ALPHA-2 is found in section 5.1.1.242.4. Antiproton production, capture and coolingAfter antiprotons traverse the degrader, they are captured in the Penning-Malmberg trap. Figure 2.10illustrates the antiproton capture procedure. The capture is performed by erecting a high voltage (5keV) electrode (E13) a few seconds before the extraction of the antiprotons from the AD. E13 is23 cm away from the degrader and only antiprotons with an energy less than 5 keV are reflectedback towards the degrader. A second high voltage electrode (E01), which is placed next to thedegrader, is triggered to trap the antiprotons between E01 and E13. The electrode E01 is triggeredat a time called the “closing time”, which is the time between the beam extraction signal from theAD and switching the voltage.-6000-30000Potential[V]a)-6000-30000Potential[V]b)-6000-300000 100 200 300Potential[V]z [mm]c)E01 E13DegraderFigure 2.10: Schematic illustration of the antiproton capture. The black curve is the resulting po-tential of the voltage applied to the electrodes. The stack of electrodes in the capture region areshown at the top of the drawing. The two high-voltage electrodes are in yellow and the rotating wallelectrode is in indigo. The stack of degraders is represented as one layer next to the first electrode(E01). a) The antiprotons (in red) travel from the AD to be degraded before entering into the cap-ture region while the voltage in E13 is already raised. b) A fraction of the antiprotons traverse thematerial and only the particles with an energy below 5 keV are reflected by the potential at E13.c) E01 is erected to trap the remaining antiprotons.252.4. Antiproton production, capture and coolingWe typically load electrons from an electron gun (see section 2.6) before the antiproton captureprocedure. The electrons are used to sympathetically cool the antiprotons through Coulomb col-lisions [77], since the electrons are quickly cooled in a strong magnetic field through cyclotronradiation. The electrons cool exponentially with a time constant of ∼ 0.43 s in a 3 T magneticfield [63, 78]. Figure 2.11 shows the antiproton cooling schematically. The antiprotons and elec-trons are usually allowed to interact for 80 s. A large fraction of antiprotons are cooled and migrateto the low energy well. After the interaction time, one high voltage electrode is changed to a lowervoltage so hot antiprotons are able to escape.0 100 200 300 400z [mm]a)b)c)90 VFigure 2.11: Schematic illustration of antiproton cooling. The black curve is the confining potential.a) The electrons (in blue) are loaded in advance and the antiprotons (in red) are captured by the highvoltage potential as already shown in figure 2.10. b) The antiprotons and electrons are allowed tointeract (usually about 80 s) and as a result, a large fraction of the antiprotons are cooled. c) The highvoltage electrode on the left hand side is switched to a lower voltage so remaining hot antiprotonsare allowed to escape (red arrow).The antiproton cooling is a non-exponential process which accelerates as the antiprotons cool [77].The electron plasma is limited in how fast it can dissipate the energy from the antiprotons, whichdepends on the ratio of the numbers of electrons to antiprotons. A low number of electrons per262.4. Antiproton production, capture and coolingantiprotons limits the cooling process by an increase of the electron plasma temperature, accordingto the demonstration in Ref. [77]. We have observed that the number of cooled antiprotons alsodepends on the interaction time. The efficiency of antiproton cooling as a function of the interactiontime is shown in figure 2.12. The efficiency of antiproton cooling is the ratio of the number ofcooled antiprotons (antiprotons that do not escape after high voltage electrode is switched to a lowervoltage) to the number of captured antiprotons. We observed that at least several tens of seconds isneeded to obtain a cooling efficiency higher than 50%.00.20.40.60.810 20 40 60 80 100 120 140 160 180 200p¯coolingefficiencyCooling time [s]Figure 2.12: Antiproton cooling efficiency as a function of the cooling time. The cooling time is thetime for which antiprotons and electrons are allowed to interact.The antiproton cooling efficiency depends on the cooling time but also on the radial overlap betweenthe antiproton cloud and the electron plasma. If the electron plasma is radially smaller than theantiproton cloud, the cooling is less efficient because a fraction of the antiprotons can not interactwith the electrons. It is possible to use the rotating wall technique to compress or expand the radiusof the electron plasma to match the radius of the antiproton cloud (see the rotating wall techniquein section 4.3). Figure 2.13 shows the effect on the antiproton cooling efficiency, when varying theelectron plasma radius before the antiproton capture, as well as the resulting antiproton radius. Thecooling efficiency increases as the electron plasmas radius is increased. It is estimated that the initialradius of the antiproton cloud is ∼ 4 mm [63].272.4. Antiproton production, capture and coolingFigure 2.13: Top: Antiproton cloud radius as a function of the radius of the electron plasma usedto cool antiprotons. Symbols correspond to trials with different total number of electrons (1×108 –1.65 × 108 electrons). Open symbols represent radii measured directly from MCP images and radiifrom red solid symbols are calculated from central intensity from the image on the MCP. For theseplasmas, it is not possible to accurately measure the radius so it is assumed that the electron plasmaradial profiles are self similar and infer the plasma radius from the peak density. The inset figureshows this approach. Bottom: Antiproton cooling efficiency as a function of the electron plasmaradius. The cooling time is 30 s. Image from [63].282.5. Positron AccumulatorIn ALPHA-2, instead of using electrons from the electron gun, we used the secondary electronsproduced when the antiprotons pass through the degrader. Section 5.1.2 has more details about thisprocedure.2.5 Positron AccumulatorThe ALPHA-2 positron accumulator is the same as the one used in ALPHA-1 and ATHENA. Anew 1.42 GBq 22Na radioactive source was installed in 2013 and the source region was rebuilt, butthe core remains the same.Positrons are spontaneously generated in β+ radioactive decay, in which a positron is emitted. AL-PHA uses a 22Na source which has a half-life of 2.6 years and has a positron yield of 90.4%. The22Na decay reactions in which the positrons are produced are:2211Na→2211Ne∗ + e+ + νe2211Ne∗ →2211 Ne + γ(2.16)Shortly after the 22Na decay, a 1.27 MeV γ is released from the 22Ne as it relaxes to the groundstate.The positron accumulator is a Surko-type positron accumulator and the techniques used to accu-mulate positrons are pioneered by the positron research group at the University of California, SanDiego [79]. See Ref. [80] for a comprehensive review of Surko-type positron accumulators andtheir applications. Figure 2.14 shows a schematic of the positron accumulator.The positron accumulation is described in detail in Ref. [81], and here we summarize its stages:1. The emitted positrons are directed into a cold neon layer (called the “moderator”), where thepositrons thermalize and diffuse through the material.2. About 0.4 % of the positrons escape the moderator with a kinetic energy of about 50 eVand are guided by a magnetic field into the positron trapping region. There, a 0.15 T axial,292.5. Positron AccumulatorFigure 2.14: On top, positron accumulator schematic. The source is positioned at the left and thepositrons are accumulated as a plasma in the right side (green ellipse). The positrons are eventuallyextracted to the right, where they are trapped in the mixing trap for antihydrogen production. Onthe bottom, on-axis trapping electric potential as a function of the position, showing the pressure. Itshows the cooling stages for the positrons. Image adapted from [67].uniform magnetic field provides radial confinement and a Penning-Malmberg trap providesaxial confinement.3. The positrons lose energy by electronic excitation of a nitrogen buffer gas. The gas inlet isplaced at the source side of the accumulator, where the electrodes have the smallest diameter.The diameter increases the further from the source, creating a pressure gradient. The positronsare cooled and trapped in the region with the lowest pressure. The rotating wall technique isapplied to radially confine the positron plasma. This mode of accumulation lasts for about200 s and about 1 × 108 positrons are accumulated.4. After the accumulation, two high-rate cryo-pumps are used to remove the nitrogen gas andimprove the vacuum from 1 × 10−5 to 1 × 10−9 mbar in about 40 s. Then, a valve betweenthe accumulator and the atom trap is opened and the positrons are transferred and recapturedin the Penning-Malmberg trap of the atom trap. Up to 80% of the positrons survive thisprocedure.302.6. Movable stick2.6 Movable stickThe stick is composed of different devices that can be used according to requirements of the exper-iment. The components are:• Electron source: Electrons are used in ALPHA to cool antiprotons because of their shortcooling time (about 0.4 s in 3 T) in a strong magnetic field through cyclotron radiation [78].Electrons are produced by thermionic emission from a barium-oxide filament, which is placedinside an electron gun [82]. The resulting electron beam is guided by the magnetic field tothe antiproton capture trap, where electrons are trapped by the axial potentials.• Micro-Channel Plate (MCP) detector system: This consists of an assembly of a phosphorscreen, an MCP and a mirror, which are used to record a transverse image of the particlesclouds/plasmas. For more details, refer to section 2.7.2.• Microwave horn: In ALPHA-1, the horn antenna injects microwave radiation to the centreof the trap. It was used to perform microwave experiments on antihydrogen. For more detailson the measurements, refer to section 3.4.• Microwave mirror: Another way to inject microwaves is to place a horn outside the appara-tus in front of a window and to direct the microwaves inside the trap through the microwavemirror. This technique was used to measure the magnetic field by determining the cyclotronfrequency of an electron plasma.• Pass-through: A cylindrical electrode allowing the passage of particles to the atom trap.The movable stick is the same in ALPHA-1 and in ALPHA-2 antiproton capture trap. Figure 2.15shows the exterior of the movable stick in the ALPHA-2 antiproton capture trap. The stick is placedat one side of the antiproton capture trap and can be moved along its vertical axis. The atom traphas another movable stick, but it is not shown here.312.7. Charged particle detection and diagnostic devicesFigure 2.15: On the left, a photograph of the ALPHA-2 antiproton capture trap and the movablestick, which is about 130 cm from the antiproton capture trap. On the right, an illustration of themovable stick with its components.2.7 Charged particle detection and diagnostic devicesALPHA uses various types of charged particle detectors. They can measure the number of particlesand parameters of the plasmas such as the density and the transverse size. Some can also be used tomeasure the temperature of charged particles.2.7.1 Faraday CupThe Faraday cup (FC) consists of a thin layer of conducting material. When the electrons orpositrons are dumped to the FC, the charge of the particles induces a voltage and with a suitableamplifier, the voltage can be measured. Since the capacitance of the conductor can be measured, theamount of charge collected can be calculated, giving the number of particles. At least a few hundredthousand particles is needed to get a measurement above the noise level.Figure 2.16 shows an example of the resulting traces from the FC. A background, probably comingfrom an electrical coupling with the other electrodes, is subtracted. The resulting peak gives theelectron number.322.7. Charged particle detection and diagnostic devicesFigure 2.16: Example of the FC traces. The measured FC trace is in yellow and the backgroundtrace is in green. The trace giving the electron number is in red (∼ 5 × 106 electrons).The antiproton capture trap in ALPHA-2 has a Faraday cup placed next to the first electrode. ThisFC is also used as a degrader (discussed in section 5.1.1). The FC is a 165 µm thick beryllium layer.The antiproton number is not measured with the FC because antiprotons ionize and annihilate inthe conductor producing a large number of secondary particles, which carry away charge. Hencethe measurement is not reliable and plastic scintillators are used instead to detect radiation fromantiproton annihilations (see section 2.7.3).2.7.2 MCP/phosphor/CCD detector assemblyThe microchannel plate (MCP) detector system is a two dimensional sensor that amplifies the de-tected signal with high efficiency and high speed [83]. It consists of a plate of a semiconductingmaterial with an hexagonal array of miniature electron multipliers placed parallel to each other.Since the channels’ axes have a small tilt (usually about 8 degrees) with the direction of incidentparticle, when the particle strikes into the front surface of the MCP, it impacts the inner surface ofone of the microchannels. Such impact creates a cascade of secondary electrons which are acceler-ated to the rear surface of the MCP. The cascade is proportional to the incident number of particles332.7. Charged particle detection and diagnostic devicesand is directed onto a phosphor screen. The cascade of electrons excites the phosphor atoms, whichdeexcites by emitting light. The light is redirected into a charged coupled device (CCD) camera bya 45◦ mirror [85], which produces an image of the charge distribution. A schematic is shown infigure 2.17.MCP100 V 600 V 5 kVInicidentparticlePhosphor screenMirrore-PhotonsCCDFigure 2.17: Schematic of the MCP, phosphor screen, 45◦ mirror and CCD assembly. The incidentparticle can be an antiproton, electron or positron. Objects are not drawn to scale.The MCP/phosphor/CCD assembly is used in ALPHA to obtain information about the density andthe radial size of the antiproton, positron and electron plasmas. In ALPHA, this detector assemblyis placed in the movable stick (see 2.6).The MCP in ALPHA is a E050VP47 device manufactured by El-Mul Technologies [84]. The diam-eter of each channel is 10 µm with 12 µm centre-to-centre spacing. The active diameter of the MCPis 43.5 mm. A metallic coating deposited on the front and at the rear surfaces of the MCP allowselectrical contact, where the particles are accelerated by the potential difference (typically hundreds342.7. Charged particle detection and diagnostic devicesof volts). A minimum gain of 1 × 104 is achieved at a bias voltage of 1.2 kV.2.7.3 ScintillatorsAntiprotons are dumped into a material (typically the degrader layer) so their annihilation can bedetected. Each annihilation produces an average of three charged pions [86], which can be detectedwith plastic scintillators. When charged particles pass through plastic scintillators, they electroni-cally excite the atoms/molecules, which release light when they relax. The scintillators are coupledto a photomultiplier tube (PMT). The light is guided, then collected by a photocathode, where aprimary electron is emitted due to the photoelectric effect. The primary electron is focused into theelectron multiplier section and accelerated trough a series of dynodes creating secondary electronemission. The final cascade is converted into an electronic signal.In ALPHA, 40 cm wide by 60 cm high scintillator pads are placed vertically, normal to the floor,which minimizes their sensitivity to the cosmic rays. They are assembled by pairs and there is onepair on each side of the trap, about 60 cm from the trap and centred at the axial position of thedegrader. Figure 2.18 shows a photograph of the setup. If both of the scintillators in a pair detect asignal passing a voltage threshold in a defined time window (i.e. in “coincidence”), it is considereda “count”. Since the scintillators cover a relatively small solid angle, a GEANT4 simulation isused to translate the “counts” into the total number of dumped antiprotons. Taking the geometryinto account, the detection efficiency for antiproton annihilation on the degrader is ∼ 20%. Thebackground rate from cosmic rays is 40 s−1.2.7.4 Plasma temperature diagnosticsThe temperatures of the antiproton and positron plasmas are very important since they affect theantihydrogen production and trapping rate.Since the trapped plasma/cloud is assumed to be in thermal equilibrium, the kinetic energy distribu-tion parallel to the axial magnetic field is expected to follow a one-dimensional Maxwell-Boltzmann352.7. Charged particle detection and diagnostic devicesFigure 2.18: Picture of the ALPHA-2 apparatus. Red lines indicates the position of a pair of scintil-lators/PMT assemblies at one side of the antiproton capture trap. The scintillator assembly is placedat about 60 cm from the trap. Another pair of detectors is symmetrically placed at the other side ofthe apparatus (not shown in the picture). The annihilation of the antiprotons occurs at the degrader,placed inside the trap (approximated position is indicated by the yellow rectangle).362.8. Atom trapsdistribution [87]:f (E‖) ∝ exp(− E‖kBT), (2.17)where E‖ is the parallel energy, kB is Boltzmann’s constant and T is the temperature. By slowlylowering the confining axial well potential at one side, it is possible to measure the parallel energydistribution of the particles. As particles escape from the well, they are detected. By knowingthe relation between time and the escaping potential, it is possible to build up the parallel energydistribution f (E‖) and to extract the temperature using equation 2.17:dln f (E‖)dE‖' − 1kBT. (2.18)In ALPHA we are able to measure the temperature of the antiprotons using the scintillators (seesection 2.7.3) and the temperature of the positrons using the MCP/Phosphor assembly (see sec-tion 2.7.2).Figure 2.19 shows examples of temperature measurements for antiprotons. The temperature isextracted by fitting a straight line on a semi-log plot. The non-exponential behaviour of low energyparticles escaping from the well is due to the space charge of the plasma that changes the height ofthe potential. Corrections for this effect are discussed in Ref. [88].2.8 Atom trapsIn ALPHA, Penning-Malmberg traps are used to confine antiprotons, electrons and positrons. Whenneutral antihydrogen is synthesized, a neutral atom trap is superimposed on the Penning-Malmbergtrap and is used to confine the antihydrogen atoms.In this section, we will discuss how antihydrogen is confined in a magnetic trap. We will alsopresent the ALPHA-1 atom trap, where antihydrogen was trapped for the first time, as well as theALPHA-2 atom trap, where laser spectroscopy of antihydrogen will be performed.372.8. Atom trapsFigure 2.19: Integrated number of antiproton loss as a function of the well depth. The well depthis ramped down, so time flows from right to left. Each set of points represents one measurement.The calculated temperatures are: 1040 K (A), 325 K (B), 57 K (C), 23 K (D), 19 K (E) and 9 K (F).Image from [64].2.8.1 Antihydrogen motion in a magnetic field minimum trapAntihydrogen atoms can be trapped by using the interaction of the atom’s magnetic dipole momentwith an inhomogenous magnetic field. The antihydrogen atom has an intrinsic magnetic dipolemoment ~µ due to the spins of the antiproton and the positron and the orbital motion of the positron.Since the magnetic dipole moment is, to first approximation, inversely proportional to the massand me/mp = 5.4 × 10−4, the antihydrogen magnetic dipole moment can be approximated as thepositron’s dipole magnetic moment. If the antihydrogen atom is in its ground state, the orbitalangular momentum is zero, hence only the positron’s spin angular momentum contributes to theatom’s magnetic dipole moment.The magnetic potential energy of the antihydrogen magnetic dipole in an inhomogenous magneticfield can be written asU = −~µ.~B, (2.19)382.8. Atom trapswhere ~µ is the magnetic moment of antihydrogen and ~B is the magnetic field.Assuming that the rate of change of the direction of the magnetic field at the particle’s position isslow compared to the Larmor frequency (precession of the magnetic moment of the atom aroundthe external magnetic field), the spin follows adiabatically the direction of ~B. In a strong magneticfield, the spins of the antiproton and the positron are essentially uncoupled and, there are two stableconfigurations, where ~µ and ~B are parallel or antiparallel to each other. For an atom in the groundstate, the two possible magnetic potential energies areU = ±µBB, (2.20)where µB is the Bohr magnetron. These two magnetic potential energies correspond to two cases:• Low field seeking atom: The magnetic dipole moment of the particle is antiparallel to themagnetic field. Such particles are attracted to regions with a low magnetic field strength.• High field seeking atom: The magnetic dipole moment of the particle is parallel to the mag-netic field. These particles are attracted to regions with a high magnetic field strength.To create a static trap, an extremum (maximum or minimum) magnetic field is required. It is onlypossible to trap low field seeking particles by constructing a three dimensional static magnetic fieldminimum. High field seeking particles cannot be trapped because it is impossible to create a staticmagnetic field maximum in free space [89].For stable trapping, the kinetic energy of the atom must be lower than the depth of the magneticpotential well and, as already mentioned, the magnetic dipole moment must move adiabatically inthe magnetic field. In a region where the magnetic field is too small, the spin cannot follow thechanging direction of the magnetic field and can flip its orientation relative to the magnetic field,escaping from the trap. This loss of particles from the trap is called Majorana losses [90]. Magnetictraps are usually constructed so that the minimum magnetic field is large enough to prevent suchlosses.392.8. Atom trapsThe well depth is proportional to the difference between the minimum magnetic field magnitudeand the maximum at the trap boundary and can be expressed as a potential energy,kBT = µ(B − B0), (2.21)where for ground state of antihydrogen, µ = µB.2.8.2 ALPHA-1 atom trapThe ALPHA-1 atom trap consists of three superconducting magnets and one external solenoid.Two mirror coils provide axial confinement, while an octupole coil provides radial confinement ofantihydrogen. The external solenoid (1 T) provides radial confinement for antiprotons and positronsbefore synthesizing antihydrogen and provides the minimum magnetic field of the atom trap.The two mirror coils are co-axially positioned at each side of the trap at about 28 cm apart. When thetwo mirror coils are energized, they create an axial magnetic field minimum, as shown in figure 2.20.The coils operate at a current of 600 A, each producing a maximum longitudinal field of 1.2 T.00.511.522.5-0.2 -0.1 0 0.1 0.2MagneticField[T]z [m]Figure 2.20: On-axis magnetic field produced by the two mirror coils of the ALPHA-1 atom trap,superimposed on 1 T external solenoid. Image from [67].A multipolar field produces the radial confinement well to trap antihydrogen. The simplest magnetic402.8. Atom trapstrap is the Ioffe-Pritchard, which uses a quadrupole magnetic field [91]. In this experiment, wherethe antiproton and positron plasmas must remain radially confined, the order of the multipolar fieldwas carefully chosen because a transverse magnetic field breaks the Penning-Malmberg trap cylin-drical symmetry [92]. The octupole coil was chosen because its magnetic field close to the axis islower than a quadrupole or a sextupole magnetic field, as shown in figure 2.21. This characteristicminimizes the transverse-field effects on charged particles, and storage of antiproton and positronsplasmas is possible without significant loss of the particles [93]. When trapping antihydrogen, itis very important to maximize the well depth so antihydrogen atoms do not annihilate on the innersurface of the electrode. Higher order multipoles have a steep gradient near the trap boundary, sothat a significant amount of trap depth can be lost in the material of the vacuum system and theelectrodes. For this reason, and the difficulties in manufacture, multipoles of higher order than anoctupole were not used. To increase the well depth, the electrodes in this region of the apparatus (asshown in fig. 2.6) have larger inner radius (22.2 mm) and are very thin (0.5 mm).010 0.2 0.4 0.6 0.8 1B⊥/Bmaxr/RMagQuadrupoleSextupoleOctupoleFigure 2.21: Transverse magnetic field as a function of radius for a quadrupole, a sextuple and anoctupole. We can observe by looking at the vertical line that the octupole produces the shallowesttrap depth. The vertical line represents the radius of the inner electrode. Image from [67].Figure 2.22 shows a picture of the first layer of the octupole windings. There is a total of eight412.8. Atom trapslayers, which are needed to generate a strong magnetic field. A serpentine pattern was used to windthe octupole, rather than a racetrack pattern, because this cancels out the axial fields by azimuthallystaggering each layer by 45◦ with respect to each other [94]. The octupole is operated with a currentof 900 A producing 1.55 T at the inner radius of the electrodes. The mirror coils and the octupoleare wound directly onto the vacuum vessel wall as shown in figure 2.22, where they are in directcontact with a liquid helium bath at a temperature of 4.2 K. The contact of the vacuum vessel wallwith the liquid helium also acts as a cyopump, which allows the volume in the trap chamber to reachvery low pressures (see section 2.11).Figure 2.22: Photograph of the first layer of the octupole windings for ALPHA-1. Image from [56]The minimum well depth is given by the difference between the radial magnetic field magnitudeat the electrode wall and minimum magnetic field magnitude at the centre of the well (since ourmirror coils can provide stronger fields than the octupole). The former is the sum in quadratureof the octupole’s magnetic field at the electrode wall Bw and the axial solenoidal field Bz, whichare orthogonal to each other. The latter is just the axial solenoidal field and the mirror coils. The422.8. Atom trapsdifference between the magnetic field magnitude is written as∆B =√B2z + B2w − Bz. (2.22)To maximize the probability of trapping antihydrogen, the well depth needs to be as large as possi-ble. For that reason, antihydrogen production takes place in the 1 T region of the trap.The trap depth is usually given in units of kelvin, which can be converted to kinetic energy bymultiplying by the Boltzmann constant kB. For the currents given above, the trap depth is about0.5 K, so that antihydrogen with a kinetic energy equal to or less than about 0.5 K× kB = 44 µeV istrapped, while antihydrogen with higher energies will eventually escape from the trap.Figure 2.23 shows the total 3-D magnetic field minimum antihydrogen trap of ALPHA-1, where themirror coil, the octupole magnet and the external solenoid have been superimposed.r [mm]z [mm]1.52B [T]05101520-200-100010020011.52Figure 2.23: The magnetic field strength of the ALPHA-1 antihydrogen trap. The octupole, twoaxial mirror coils and external solenoid are superimposed.432.8. Atom trapsQuench protectionThe magnets are made from niobium-titanium (NbTi) wires embedded in a copper matrix. NbTi isa type II superconductor alloy, with a critical temperature of 9.2 K [95]. The copper matrix givesmechanical stability and in the event of a quench (that is, loss of superconducting state), the copperprovides a path for the large currents. If the superconducting wires alone carry the current duringa quench, Joule heating (heat generated by the passage of an electric current through a conductor)would lead to too high temperatures, damaging the coils.A quench protection system (QPS) is required to protect the magnets. The QPS safely extracts thecurrent when a quench is identified. A quench occurs when a region of the superconducting ma-terial becomes resistive and heats other parts of the superconductor above the critical temperature,creating a chain reaction. This situation can be avoided by monitoring the voltage across severalregions of the magnet using a system controlled by a Field Programmable Gate Array (FPGA). Ifan abnormally high voltage, indicating a quench, is detected, a high current insulated-gate bipolartransistor (IGBT) is used to safely extract the current through a resistor network, where the energyis dissipated as heat.This system was designed so it could quickly shutdown the magnetic trap to detect antihydrogen.A quick shutdown reduces the background due to noise counts and cosmic rays. The magnetshave extremely low inductances, which allows the current to be removed in a very short time. Themagnetic fields decay with a time constant of 9 ms, as shown in figure 2.24. Antihydrogen detectionis performed in a 30 ms time window from the beginning of the shutdown.2.8.3 ALPHA-2 atom trapThe ALPHA-2 atom trap is composed of nine superconducting magnets. The octupole and themirror coils have the same design as the magnets in ALPHA-1, and they operate at the same currentsand produce the same magnetic fields.442.8. Atom traps11010010000 10 20 30 40 50Current[A]Time [ms]OctupoleUpstream MirrorDownstream MirrorFigure 2.24: Current decay of the octupole and mirror coils as a function of time. 0 ms is the timewhen the magnets begin to be ramped down. Image from [67].• One octuple: As in ALPHA-1, an octupole magnetic field is used to provide radial confine-ment of antihydrogen atoms. The octupole produces a magnetic field of 1.55 T at a current of900 A.• Five mirror coils provide the axial confinement of antihydrogen atoms. The mirror coils canbe independently energized, so different axial lengths and well depths can be obtained duringthe experiments. Each mirror coil can be energized up to 600 A, producing a maximum axialfield of 1.2 T.• External solenoid: Provides the minimum magnetic field of the trap, while keeping thecharged particles (antiprotons and positrons) radially confined during antihydrogen produc-tion. The nominal magnetic field is 1 T. The solenoid is new for ALPHA-2 and is designed tobe capable of quickly changing its field.• Two solenoids at each end of the trap: These solenoids are energized at a current of 250 Ato give a field of 2 T during the manipulation of charged particles, to improve the radialconfinement, and reduce the cooling time. They are de-energized before synthesizing antihy-drogen.452.9. Laser access in ALPHA-2The ALPHA-2 magnetic trap is more versatile, allowing the length of the well to be changed and thefield profile to be fine-tuned. The magnetic trap has the same quench protection system, providing asafe mechanism to de-energize the superconducting magnets during a quench, as already describedin section 2.8.2.2.9 Laser access in ALPHA-2ALPHA-2 has 8 windows for optical access to the antihydrogen trap. There are 4 on each side of thetrap and have a direct path to the centre of the trap. The laser accesses will be used for experimentson 1S–2S two photon spectroscopy, Lyman-α spectroscopy among other experiments. A cavity toenhance the laser power is built inside the apparatus. More details about future experiments onantihydrogen are described in section 3.5.2.10 Antihydrogen detectionIn this section, we present the ALPHA-1 silicon tracking detector which was used during the ex-periments with trapped antihydrogen during 2010 and 2011. The detection principle remains thesame for the ALPHA-1 and ALPHA-2 silicon tracking detectors but the geometry of the ALPHA-2detector is slightly different, occupying a larger volume and including more silicon modules. Wefirst discuss antiproton annihilation, which is the main process to be detected when studying anti-hydrogen, and then we discuss how to discriminate antihydrogen from cosmic rays.2.10.1 Antiproton annihilationAntiprotons are slowed in matter just like protons, following the Bethe-Block formula but witha smaller energy transfer to electrons (more details in appendix A). When an antiproton stops inmatter and comes close to a nucleus, it annihilates. The simplest way to understand the annihilationof an antiproton with a proton or a neutron is the rearrangement of the quarks. Figure 2.25 shows462.10. Antihydrogen detectionan example of the quarks’ rearrangement into three pions when the antiproton annihilates with aproton.puududpi+uupi0dupi−puudFigure 2.25: Example of the simplest picture to understand antiproton annihilation.Proton-antiproton annihilation (at rest), usually results in the production of a combination of pions(pi+, pi−, pi0), ranging from two to eight in number, with an average of five pions. An average ofthree charged pions and two neutral pions are produced and the multiplicity of pions is shown infigure 2.26 [100–102].The pions are created directly or through the decay of mesonic resonances. When five pions areproduced, each one has a kinetic energy of about 236 MeV [96]. For an isolated proton-antiprotonannihilation at rest, these pions will directly escape at different angles set by momentum conserva-tion. If the annihilation occurs in a nucleus, a pion could enter the nucleus and be absorbed [97],have a charge-exchange process [98] or cause the fragmentation of the nucleus itself [99]. Becauseof these different possible reactions, proton-antiproton and neutron-antiproton annihilation are noteasy to distinguish. Pions, able to escape from the nucleus, will fly away from the annihilation point.The lifetime of pions is short. The lifetime of charged pions is 2.6 × 10−8 s, while the lifetime ofneutral pions is 8.4×10−17 s [103]. The lifetime of charged pions is long enough to pass through thedetector, but neutral pions will decay into gamma rays, which most often produce electron-positronpairs [102].472.10. Antihydrogen detection00.10.20.30.40.52 3 4 5 6 7 8FrequencyNumber of pions per eventFigure 2.26: Histogram of the distribution of the number of pions (neutral and charged) per proton-antiproton annihilation. Data from [100].In ALPHA, when antihydrogen is released from the magnetic trap, it annihilates on the surface ofthe electrodes. It could also annihilate with background gases inside the trap. In antihydrogen anni-hilation, the positron can annihilate with an electron by producing two or three photons, while theantiproton annihilation is more complicated. The electrodes are made from gold-plated aluminium,so an antiproton annihilation with a heavy nucleus must be considered, as well as the fact that theelectrodes may be covered by frozen gas molecules.2.10.2 ALPHA-1 silicon tracking detectorThe detector is composed of 60 double sided silicon microstrip modules arranged in three concentriclayers co-axially placed around the antihydrogen trapping region, between the cryostat and theexternal solenoid.Each module, called a hybrid, consists of a semiconductor microstrip detector and the front endelectronics. One module is composed of two 6 cm× 23 cm silicon wafers, one on the front and oneon the rear of a Printed Circuit-Board (PCB). Each side reads out 256 signal strips, which are placed482.10. Antihydrogen detectionorthogonally to the other side, giving positional information when a particle deposits energy.The detector is symmetrically divided into two axial sections, as shown in figure 2.27. Each sectionconsists of 30 modules, arranged into three concentric layers around the trap. The total axial lengthof the detector is 46 cm, providing a solid angle coverage of ∼ 72% of tracks originated at the axialcentre of the trap, travelling in a straight line and interacting at least one with the active area of eachlayer [104].Figure 2.27: ALPHA-2 silicon tracking detector assembly at the Liverpool semiconductor detectorcentre at the University of Liverpool in July 2012 [105].When a charged particle passes through the silicon area, the signals from the orthogonally positionedstrips give information about the position of the particle. If the signal is larger than a threshold,the point of intersection of the two signals is called a "hit". As already mentioned, an antiprotonannihilation produces on average about three charged pions and two neutral pions. Between thesurface of the electrodes and the detector layers, there are several materials that could scatter andabsorb the particles. The pions resulting from the annihilation are so-called minimum ionizingparticles (MIP), meaning that the mean energy loss rate is close to the minimum and most of theMIPs can at least travel through the apparatus before being stopped.The passage of the charged particles produced from the annihilation triggers a hit on each layerof the detector. Given this information and the geometry of the detector inside the apparatus, theannihilation vertex is reconstructed. The detector is surrounded by a 1 T external solenoid that bendsthe path of the charged pions. For this reason, the trajectories of the particles are reconstructed using492.10. Antihydrogen detectionhelices instead of straight lines. Figure 2.28 illustrates an example of an antiproton annihilationinside the apparatus. For a detailed description of hardware and software used in antihydrogendetection, see Ref. [102]. Using Monte Carlos simulations, the resolution of the reconstructedvertex position is estimated to be (0.67± 0.04) cm in the axial direction, (0.68± 0.04) cm in theradial direction and (0.82± 0.04) cm in the azimuthal direction at the trap wall.Figure 2.28: Schematic of the cross section at the axial centre of the ALPHA-1 apparatus. Theobjects are labelled: a) electrodes, b) trap vacuum wall, c) octupole magnet winding, d) liquidhelium reservoir, e) inner isolation vacuum wall, f) outer isolation vacuum wall, g) silicon detectorand h) external solenoid magnet. The yellow star is the reconstructed vertex of an antiproton thatannihilated into three pions (two charged and one neutral). The red curves are the reconstructedtrajectories of the particles that pass through the detector producing hits (red points on the siliconlayers). The neutral pion quickly decayed into two photons, one of them is absorbed on the octupolemagnet winding and the other produced an electron-positron pair. Image from [110].In addition to silicon modules, the ATHENA experiment used a detector with CsI crystals for the502.10. Antihydrogen detectiondetection of photons from positron annihilation [30, 106]. Because of the material between thetrap and the detector, the total efficiency for the detection of two simultaneous photons was about5% [107]. The ALPHA experiment has more material between the trap and the detector (e.g. super-conducting coils and cryostat) and it was deemed impractical to try to detect the photons, due thegamma absorption in the materials.Figure 2.29: Schematic cross section of the three layer silicon detector. Red tracks are the re-constructed trajectories of the particles from the hits in the detector (red dots). The blue diamondis the reconstructed vertex and the black circle at the center is the electrode wall. a) Antiprotonannihilation and b) cosmic ray. Image from [1].The main background when detecting antihydrogen comes from cosmic rays. Fortunately the topol-ogy of a cosmic ray’s passage through the detector is usually very different from that of an antiprotonannihilation. Figure 2.29 shows an example of track reconstruction for an antiproton annihilationand a cosmic ray.To discriminate annihilations from cosmic rays, three variables are used:1. Number of charged particle tracks: The majority of the cosmic ray events register twotracks from the passage of the charged particle through the detector, one on the upper half ofthe detector and the other on the bottom side of the detector. About 46% of the reconstructedantiproton annihilation events have at least three tracks from three charged pions (in additionto some neutral pions).512.10. Antihydrogen detection2. Combined linear fit residual: Because of their high energies, cosmic rays are expected tofollow straight line trajectories, but are sometimes bent by the magnetic fields or scatteredon materials. However, antiproton annihilations are not expected to have so many co-lineartracks. For this reason, the combined linear fit residualδ = min ∑i∈Nhits d2⊥,i , (2.23)where d⊥,i is the residual between the fitted line and the i − th hit in the set of Nhits is calcu-lated. δ is calculated for every pair of tracks and the combination providing the smallest valuefor is chosen. Cosmic rays trajectories will have a δ close to zero, compared to antiprotonannihilation trajectories.3. Vertex radius: When released from the trap, antihydrogen will almost certainly annihilateon the wall of the electrodes. The reconstructed vertex is then constrained to be inside agiven volume, chosen to be slightly larger than the electrode radius to allow for the finitereconstruction resolution.The discriminating variables are optimized from two sets of data. One set is the background cosmicray data when the magnetic trap is energized and there are no particles in the apparatus. The sec-ond set of data is from the annihilation signal of antihydrogen when antiprotons and positrons aremixed. The optimization was performed using a blind analysis approach, so the data from trappedantihydrogen was not used for this propose. This eliminates the possibility of the algorithm beingimproperly biased. Figure 2.30 shows the results of the optimization for the two sets of data.After the optimization, the ALPHA-1 detector background rate is (47 ± 2)×10−3 events/s and (64.4± 0.1)% of the antiproton annihilation events are accepted. The overall annihilation event efficiencyis (58 ± 7)%, when combined with the (90 ± 10)% trigger efficiency [110].522.10. Antihydrogen detectionFigure 2.30: Measured antiproton annihilation signal and background distributions for the discrimi-nating variables before and after applying the cuts. a) Distribution of the number of charged particletracks, b) vertex radius of the reconstructed events, and the combined linear residuals for c) eventswith two charged tracks and d) events with three tracks and more. Note that in d), the backgrounddistribution is multiplied by a factor of 20 to allow comparison. The shaded areas represent theregions rejected by the cuts. Image from [110].532.11. Cryostat and vacuum2.10.3 ALPHA-2 silicon tracking detectorThe ALPHA-2 silicon tracking detector is an upgrade of the ALPHA-1 detector. The 60 doublesided hybrid modules from ALPHA-1 were reused and reassembled with the addition of 12 extradouble sided hybrid modules. There is then a total of 72 modules. Additionally, the radii of thelayers were enlarged to accommodate the trap and cryostat of ALPHA-2. Table 2.2 shows thenumber of modules and radius of each layer, while figure 2.31 compares the ALPHA-1 and theALPHA-2 module configuration. The silicon modules of the ALPHA-2 detector are staggered,which improves the solid angle by 5% [104].Layer ALPHA-1 ALPHA-2 ALPHA-1 ALPHA-2position number of modules number of modules radius [cm] radius [cm]Inner16 20 7.5 8.99.45Middle20 24 9.55 10.811.35Outer24 28 10.8 12.711.4 13.25Table 2.2: ALPHA-1 and ALPHA-2 different module configuration and radii of each layer.2.11 Cryostat and vacuumWhen filled with liquid helium, the cryostat keeps the apparatus at cryogenic temperatures, and themagnets of the atom trap in the superconducting state. The cryostat has a volume filled with liq-uid helium, located between the ultra-high vacuum (UHV) volume and the outer vacuum chamber(OVC) of the apparatus. Superconducting magnets are placed inside this volume and are wound di-rectly to the outer surface of the UHV vessel wall containing the electrode stack (see section 2.8.2).Liquid helium (temperature of 4.2 K) covers the magnets. The electrodes are adjacent to the cryostatvacuum pipe but are in high vacuum. They are cooled to about 8 K. The pressure inside the trapis estimated, from the antiproton annihilation rate, to be in the range of 10−13 mbar to 10−14 mbar[67]. Such low pressures are only attainable at cryogenic temperatures where the cold surfaces ofthe UHV volume act as a cryopump, freezing background gases. Low pressure is very important542.11. Cryostat and vacuumFigure 2.31: On the top, transverse section of the ALPHA-1 silicon detector. On the bottom, trans-verse section of the ALPHA-2 silicon detector. Image from [104].552.11. Cryostat and vacuumwhen manipulating antimatter, since collisions with any background atoms or molecules will causeannihilation. A heat shield and the OVC insulate the cryostat from room temperature. The sili-con detector is placed around the OVC, at room temperature. Figure 2.32 shows a cross sectionalschematic of the ALPHA-2 apparatus.Figure 2.32: Cross sectional schematic of ALPHA-2. 1) Antiproton capture trap, 2) transfer linewith two external solenoid to transfer antiprotons to the antihydrogen trap, 3) antihydrogen trap and4) cryostat tower with the liquid helium inlet and transfer line with one external solenoid to transferpositrons from the positron accumulator (not shown) to the antihydrogen trap.56Chapter 3Trapped antihydrogen experimentsOne of the ultimate goals of the ALPHA collaboration is to perform precision laser spectroscopyof antihydrogen and to compare the resulting spectrum with that of hydrogen. ALPHA has mademajor technological advances to tailor antiproton and positron plasmas, and to merge them togetherto produce trapped antihydrogen. The ALPHA collaboration successfully trapped antihydrogenduring 2010 – 2011 in the ALPHA-1 apparatus.In this chapter, we first discuss how antihydrogen can be produced. We discuss evaporative cool-ing, a technique that is used by ALPHA to reduce the plasma temperatures to produce trappableantihydrogen. We also describe the autoresonant injection, a technique used for controlled injectionof antiprotons into the positron plasma. The rotating wall technique, used to achieve small plasmaradii, will be presented in the second part of this thesis. Each of these techniques is important foroptimizing the plasma parameters prior to synthesizing antihydrogen.Following this, we discuss how antihydrogen trapping experiments are performed and we will showthe results of the measurements from 2010 and 2011. We also present measurements of the antihy-drogen lifetime inside the magnetic trap and the first demonstration of changing the spin of groundstate antihydrogen, which was induced by resonant microwave radiation.At the end of this chapter, we discuss future experiments on antihydrogen that may be carried outin the ALPHA-2 apparatus.573.1. Antihydrogen formation3.1 Antihydrogen formationAntihydrogen is the bound state of an antiproton and a positron and the antimatter analogue of thehydrogen atom.As already mentioned in section 1.2, relativistic antihydrogen was produced by the interaction ofantiprotons with a jet gas target. In ALPHA, the formation of low energy antihydrogen is requiredsince it needs to be confined in a shallow magnetic trap, as discussed in section 2.8. The formationof low energy antihydrogen can be achieved via several different mechanisms, which are detailedbelow. More detailed descriptions can be found in Ref. [111].Spontaneous radiative recombinationThis is the simplest process, in which a photon carries away the excess energy required to form abound state:p + e+ → H + hν, (3.1)where h is Planck’s constant and ν is the frequency of the photon [112]. This process is the inverseof photo-ionization.Laser-stimulated recombinationThe rate of the above interaction can be enhanced by stimulating the capture by irradiating thesystem with k photons of energy hνp + e+ + khν→ H + (k + 1)hν. (3.2)This process could potentially enhance production into a particular quantum state [113], but it is yetto be experimentally demonstrated [114].Three-body recombinationThis is expected to be the most efficient process in experiments such as ALPHA, that use dense583.2. Antihydrogen production in ALPHApositron plasmas (∼ 107 cm−3) at low temperature [115]. Experimental evidence, including mea-surements of the state distribution [31], the lack of stimulated recombination [114] and the highproduction rate strongly support this.The reaction isp + e+ + e+ → H + e+. (3.3)Essentially, two positrons scatter near the antiproton, and one loses enough energy to become boundto the antiproton, while the other carries away the binding energy.Charge exchangeCharge exchange consists of the collision between a positronium (Ps), the bound state of an electronand a positron, with an antiproton:p + Ps→ H + e−. (3.4)Here the antiproton exchanges with the electron of the positronium. One of the advantages of thisprocess is that the final state of the antihydrogen can be selected to some extent by using positroniumin defined atomic states [116]. The ATRAP experiment [117], the AEGIS collaboration [118] andthe GBAR collaboration [119] will make use of this process to form antihydrogen.Pulsed field recombinationIn this proposal, the Coulomb potential of the antiproton is modified to trap a positron in a boundstate by using a pulsed electric field [120]. This has not been demonstrated experimentally.3.2 Antihydrogen production in ALPHAAs already mentioned, antihydrogen formation in ALPHA is thought to occur via three body re-combination (TBR). The evidence for this includes measurements that states in which the atoms areformed are weakly bound [31] and the rate of formation is much higher than the rate that is expectedfor radiative recombination at the measured positron temperature [122]. The rate of antihydrogenformation depends strongly on the positron plasma temperature, density and the magnetic field. In593.2. Antihydrogen production in ALPHAthe case of a infinitely large steady-state plasma, the rate is [37]ΓT BR = 8 × 10−12C(4.2 KT)9/2 ( necm−3)2s−1, (3.5)where ne is the positron plasma density, and C is a numerical constant, which depends on themagnetic field (see table 3.1). The rate also varies with temperature as T−9/2. The scaling withT has been studied both theoretically [121] and experimentally [122]. In Ref. [122], a scaling ofT−1.1±0.5 was obtained, which is in strong disagreement with equation 3.5. However, in experiments,the conditions of an infinite plasma in steady state conditions do not hold, which is expected to affectthe scaling. The recombination process is still not entirely understood, and it is still being studied.For a recent review, see Ref. [111].Magnetic field CB→ 0 ∼ 0.76B = 1 T ∼ 0.1B→ ∞ ∼ 0.07Table 3.1: The value of the constant C as a function of the magnetic field, which decreases as themagnetic field increases. Values extracted from [123–125].However, it is clear that to optimize for the highest rate of low energy antihydrogen formation, itis very important to have cold and dense positron plasmas. For this purpose, evaporative coolingis used to cool the positron plasmas (see section 3.2.1) and the rotating wall technique is used toradially compress them (see section 4.3).3.2.1 Evaporative cooling techniqueEvaporative cooling was first applied to neutral atoms in 1986 [126] and famously used to formBose-Einstein condensates of dilute alkali gases in 1995 [127, 128]. It consists of lowering thedepth of the trapping potential, and sometimes using a radio frequency to remove the most energeticatoms [126]. The remaining atoms in the trap can re-thermalize to a lower average temperature, andthe process continues until the desired temperature is reached. Using this technique, temperaturesas low as 450 pK have been reached [129]. Evaporative cooling was also used on highly charged603.2. Antihydrogen production in ALPHAions in an electron beam ion trap at high temperatures [130]. The ALPHA collaboration applied theevaporative cooling to cold clouds of antiprotons [64] and electron and positron plasmas [1] for thefirst time.Since charged particles in a Penning-Malmberg trap are strongly bound to the magnetic field, evap-orative cooling relies on letting the most energetic particles escape in the axial direction by reducingthe depth of the confining potential. Figure 3.1 shows an example of the electric potential used toachieve evaporative cooling in Ref. [64]. The left hand side of the potential is lowered from 1.5 Vto 10 mV relative to the potential minimum. The escaping antiprotons followed the magnetic fieldlines and annihilate on the degrader. The antiprotons remaining inside the well thermalize for 10 sand afterwards the temperature was inferred from measuring the Maxwell-Boltzmann distributionof the particles escaping from the trap (see section 2.7.4).Figure 3.1: Example of potential wells used during evaporative cooling where the most energeticantiprotons escape to the left. Image from [64].The results of these experiments are shown in figure 3.2a, where temperatures as low as (9± 4) K(10 mV well depth) were reached. The limitation of this procedure is that the technique relieson losing the particles; for example, at the lowest temperature, only ∼ 6 % of the initial 45,000antiprotons remain. This is shown in figure 3.2b). At the lowest temperature, the radius of the cloud613.2. Antihydrogen production in ALPHAhas expanded from 0.6 mm before evaporative cooling to 3 mm afterwards.Figure 3.2: a) Temperature as a function of the on-axis well depth after performing the evaporativecooling technique. b) Fraction of antiprotons remaining as a function of the on-axis well depth.Images from [64].3.2.2 Autoresonant injection techniqueAutoresonant excitation is used in ALPHA to inject antiprotons into positron plasmas to form an-tihydrogen [65]. It was one of the key techniques that led to the observation of trapped antihydro-gen [1].A trapped antiproton plasma can be thought of as an oscillator and autoresonant injection is based onthe excitation of the motion by a swept-frequency drive. In the relevant experimental parameters,the system behaves as a non-linear oscillator, with frequency of small-amplitude oscillation ω0and with a monotonic relation between its amplitude and frequency. When the perturbation ischirped downwards in frequency, the energy of the system increases so that the oscillation frequencymatches the driving frequency, then the oscillator follows the drive and becomes “locked”. To injectthe antiprotons, the frequency chirp is continued until the antiprotons have just enough energy toenter the positron plasma. There, there is an abrupt change in the frequency of the oscillation andthe antiprotons lose resonance with the drive. Therefore, the antiprotons have close to the minimalenergy required to enter the positron plasma. This is shown in figure 3.3.Figure 3.4a shows the axial energy distribution for several final drive frequencies measured in a623.2. Antihydrogen production in ALPHA- 3 V 18 V 0 V 18 V - 3 Vpbar in resonance pbar not in resonancee+ plasmaFigure 3.3: On the top, the voltages applied to the electrodes to create the nested well for positronsand antiprotons. On the bottom, the black curve is the on-axis potential (nested well). The blueshading represents the approximate space charge of the positron plasma, which is nested withina surrounding well with antiprotons. Before the autoresonant drive is applied, the antiprotons arelocated at the top of the left peak of the nested well. The red curve illustrates the axial motion of theantiprotons when applying the autoresonant chirp generator. Antiprotons lose autoresonance aftertraversing the positron well.633.2. Antihydrogen production in ALPHAseries of experiments. One can see how the energy of the antiprotons can be chosen by selectingthe final drive frequency. There is an inverse relationship between the amplitude and the oscillatorfrequency, shown in figure 3.4b.Figure 3.4: a) Axial energy distribution of 15,000 antiprotons driven to different final frequencies.Frequencies in a) are normalized to ω0/2pi = 410 kHz. b) Measurements of the axial energy ofantiprotons as a function of the final drive frequency (open squares). Calculations of the axialenergy as a function of the drive frequency is shown for the vacuum potential (solid blue line), fora potential with 15,000 antiprotons (green dashed line) and for a potential with 50,000 antiprotons(red dotted-dashed line). Image from [65].Further studies and simulations were performed to improve the rate of trapped antihydrogen [131].Once the antiprotons are in the positron plasma, formation of antihydrogen can take place, a processthat is called mixing and is presented in the following section.The autoresonant injection of antiprotons has been proven to be very effective when producingtrappable antihydrogen since antiprotons are injected into the positron plasma with little excesslongitudinal energy. Other techniques to inject antiprotons into the positron plasmas use antiprotonsat higher energies (several eV). The antiprotons are released into the nested well from a higherpotential so they can interact with the positrons [30, 31].643.2. Antihydrogen production in ALPHA3.2.3 MixingThe superconducting magnets of the atom trap are energized, before mixing the antiprotons andpositrons. The particles are confined in a nested well, where the positrons are nested in a centralwell within a surrounding well with antiprotons, as already shown in figure 3.3 [132].The antiprotons are injected into the positron plasma with a frequency sweep of 350 to 200 kHz over1 ms. After injection, the particles are allowed to interact for 1 s to form antihydrogen through three-body recombination (see section 3.2). During mixing, hot antihydrogen, not trapped in the magnetictrap, annihilates on the walls of the electrodes and is detected by the silicon tracking detector.Figure 3.5 shows the x − y and z − φ projection vertex distribution of the annihilations recordedduring this time. The annihilation vertices form a ring centred on the radius of the electrodes,uniform in φ, and centred axially on the position of the positron plasma.-4 -2 0 2 4x [cm]-4-2024y[cm]-10 -5 0 5 10z [cm]−pi−pi20pi2piφFigure 3.5: Colour-density maps of antiproton annihilation vertices measured during antihydrogenformation. On the left is the x−y projection and the white circle is the inner surface of the electrode.On the right is the z − φ projection. Image from [67].653.3. Antihydrogen trapping3.3 Antihydrogen trappingExperiments on antihydrogen trapping were performed during 2010 and 2011. The first measure-ments were reported in Nature in 2010, where 38 antihydrogen atoms were detected after beingtrapped for 172 ms in the ALPHA-1 apparatus [1]. This data set consisted of 335 attempts. In eachattempt, the antiprotons and the positrons were prepared and mixed to form antihydrogen. The stepsthat were followed in each attempt to trap antihydrogen were:1. Antiprotons and positrons were captured from the antiproton beam and accumulator, respec-tively. The rotating wall is applied to achieved the desired plasma densities (see section 3.3.1).2. Antiprotons and positrons were placed at the centre of the magnetic atom trap in a nestedwell.3. The superconducting magnets of the magnetic atom trap were energized to their maximumfield over 25 s.4. EVC was performed to positrons and antiprotons.5. Antiprotons were injected into the positron plasma by applying an autoresonant drive (sec-tion 3.2.2).6. Antiprotons and positrons were allowed to interact for 1 s (section 3.2.3).7. The electrodes were grounded, allowing remaining charged particles to axially escape fromthe Penning-Malmberg trap.8. Four axial electric field pluses were applied to remove any “mirror-trapped” antiprotons (sec-tion 3.3.2).9. 172 ms after mixing, a static electric field was raised to deflect any remaining antiprotons, andthe superconducting magnets of the atom trap were quickly de-energized.10. Any trapped antihydrogen released annihilated on the electrodes of the apparatus. Antihydro-gen annihilations were detected during a time window 30 ms long.663.3. Antihydrogen trappingThe interpretation of trapped antihydrogen results will be presented in section 3.3.3. The summaryof all of the measurements during 2010/2011 will be reported, as well as experiments where anti-hydrogen was confined for longer times. At the end of this section, we will show the experimentson the resonant quantum transitions of antihydrogen and present a discussion of possible futureexperiments.3.3.1 Plasma preparationEach trapping attempt begins by catching and cooling antiprotons from the AD (section 2.4). Foreach attempt about 3 × 104 antiprotons are caught and cooled. The antiproton cloud is radiallycompressed using the rotating wall (see section 4.3) to a radius of about 0.8 mm and a density ofabout 6.5 × 106 cm−3. Evaporative cooling (EVC) is used on antiprotons to achieve a temperatureof about 200 K (see section 3.2.1).A positron plasma containing about 2 × 106 positrons is transferred to the mixing trap from thepositron accumulator (section 2.5). The rotating wall technique and the EVC technique are per-formed and the resulting positron plasma has a radius of about 0.9 mm, a density of about 5.5 ×107cm−3 and a temperature of about 40 K.3.3.2 Mirror trapped antiproton backgroundAs discussed in section 2.8, antihydrogen can be trapped in an inhomogeneous magnetic field.However, bare antiprotons can be also "mirror trapped" by the magnetic atom trap. Mirror trappedantiprotons have been extensively studied since they produce one of the main backgrounds whentrapping and detecting antihydrogen [4] . When antihydrogen is released from the trap to annihilateand be detected, the silicon detector is sensitive to the charged particles produced in the antiprotonannihilation but not to the gamma rays produced in the positron annihilation. For this reason, it iscrucial to have a way to discriminate between the annihilation of a bare antiproton from antihydro-gen.673.3. Antihydrogen trappingCharged particles can be mirror trapped because they possess a magnetic moment that originates intheir motion. From the first adiabatic invariant [133], we can state that the magnetic dipole momentof a gyrating particle is constant. The magnetic moment is given byµ =12 mv2⊥B= constant, (3.6)where µ is the magnetic moment, B is the external magnetic field and 12 mv2⊥ is kinetic energy ofthe particle perpendicular to B. If the magnetic field changes spatially or temporally, the perpendic-ular component of the kinetic energy of the particle must change to maintain a constant magneticmoment. Additionally, the total energy of the particle isEtot =12mv2‖ +12mv2⊥ + qφ, (3.7)where qφ is the electrostatic potential energy in the trap. The total energy of the particle is con-served. In the absence of an electric field, when the particle moves from a low magnetic field toa higher magnetic field, v⊥ must also increase to compensate the change in the magnetic field. Asconsequence, v‖ must decrease. If the magnetic field is high enough, v‖ goes to zero and the particleis then reflected from the region of high magnetic field.To avoid a mirror-trapped antiproton background during the trapping experiments, four pulses ofaxial electric field (up to 500 V/m) are applied so that mirror trapped antiprotons can escape fromthe trap. We can always postulate that there could be antiprotons with higher magnetic momentsthat would remain trapped despite the pulses.Fortunately, information about the different dynamics of antihydrogen and antiprotons can be usedto distinguish them. After they are released from the magnetic trap, the silicon detector providesthe position and timing of the annihilations. During the magnet shutdown, a bias electric field isapplied, which will deflect the trajectory of antiprotons, depending on the direction of the electricfield, while the trajectory of neutral antihydrogen remains unchanged. This technique is intended tohelp discriminate mirror trapped antiprotons from antihydrogen. Additionally, we can compare the683.3. Antihydrogen trappingspatial and temporal distributions to simulations.In order to validate this technique, mirror trapped antiprotons were deliberately created and detectedduring the magnetic trap shutdown for bias electric fields of about 500 V/m in strength that pushedantiprotons either to the right side of the apparatus (right bias) or to the left side of the apparatus(left bias). Figure 3.6 shows the experimental data and simulations for mirror trapped antiprotonswhen using no bias, left bias and right bias. The simulations match the experimental data well:we observe that when a bias is applied, trapped antiprotons are well localized around a given axialposition.0102030-300 -200 -100 0 100 200 300t[ms]z [mm]t[ms]No biasLeft biasRight biasFigure 3.6: Simulated t and z coordinates of released mirror trapped antiprotons for no bias (greenscattered dots), left bias (blue scattered dots) and right bias (red scattered dots). Solid points are theexperimental data of mirror trapped antiprotons. t = 0 is the time when the magnetic trap is shutdown.3.3.3 38 trapped antihydrogen atomsThe report of the first trapping of antihydrogen atoms was published in Nature by the ALPHAcollaboration in 2010 [1]. 38 antihydrogen atoms were detected after being trapped for 172 ms.After trapping the antihydrogen atoms, they are released from the trap, and the annihilation vertex isreconstructed from the data gathered by the silicon detector. Figure 3.7 shows the annihilation vertexevents as a function of time and axial position, z, where 0 ms is the time when the superconductingmagnets of the atom trap are shutdown. Simulations are also shown, for antihydrogen (a) and formirror trapped antiprotons (b). One can see that the data match the simulations of antihydrogen well693.3. Antihydrogen trappingand not the one for mirror trapped antiprotons. Additionally, the data taken with different electricfields are all similar, indicating that the events are not due to charged particles.Figure 3.7: a) Simulated t and z coordinates of released antihydrogen after the trap shutdown (greyscattered dots). b) Simulated t and z coordinates of released mirror trapped antiprotons for no bias(green scattered dots), left bias (blue scattered dots) and right bias (red scattered dots). Annihilationevents of antihydrogen trapping experiments are plotted for no bias (green circles), left bias (bluetriangles) and right bias (red triangles). One annihilation event was also detected during the heatedpositrons experiments (violet star).Furthermore, control experiments were performed where antihydrogen formation was suppressedby heating the positrons to a temperature of about 1,100 K. The goal of these experiments was torule out any other source of background during antihydrogen trapping. The control experiments areexpected to suppress antihydrogen formation, so that no antihydrogen should be detected. Table 3.2shows the results for the experiments with trapped antihydrogen and for the experiment with heatedpositrons. Only one event was detected during the heated positron experiments in 246 attempts(a rate of 0.0041 events per attempt) and 38 events in 335 attempts were detected during trappedantihydrogen experiments (rate of 0.11 events per attempt).703.3. Antihydrogen trappingType of Bias Number of Annihilation Estimatedmeasurement electric field attempts events backgroundAntihydrogen trapping None 137 15 0.19Antihydrogen trapping Left 101 11 0.14Antihydrogen trapping Right 97 12 0.14Heated positrons None 132 1 0.19Heated positrons Left 60 0 0.08Heated positrons Right 54 0 0.08Table 3.2: Number of annihilation events for antihydrogen trapping and heated positrons (antihy-drogen formation suppression) experiments for different bias electric fields.3.3.4 Antihydrogen confinement for 1,000 sThe confinement time of 172 ms was the shortest time possible to hold antihydrogen in our magnetictrap and still perform the manipulations to clear the charged particles. The short time assures thehighest probability to detect antihydrogen atoms before they might be lost through annihilation withbackground gases, collisional energy transfer or any other loss mechanisms. One of the motivationsto study the confinement time is to investigate the specific capabilities of the ALPHA trap to holdantihydrogen for a long time. It is known that magnetically trapped atoms can be confined for upto 1 s in room temperature traps [134] and up to 10 – 30 min in cryogenic temperature traps [135–137]. Another motivation to confine antihydrogen for long times is to allow formation groundstate antihydrogen, since ground state antihydrogen will be necessary for precision spectroscopy.As already mentioned before, antihydrogen is created through three-body recombination, whichresults in antihydrogen formed in excited states. The de-excitation of antihydrogen to its groundstate takes place via radiative and collisional processes. It was calculated that after 0.5 s, 99% oftrapped antihydrogen will be in the ground state [2]. Finally, the transition rate for spectroscopicmeasurements is expected to be low, so a long interaction time will be needed to obtain a signal.Table 3.3 shows the results of the confinement measurements. Confinement measurements wereperformed up to 2,000 s, where only one antihydrogen annihilation was detected in three attempts.We observed trapping times longer than 1 s, implying that a sample of ground-state antihydrogenwas obtained for the first time [2].713.3. Antihydrogen trappingHolding Number of Annihilation Estimated Statisticaltime [s] attempts events background significance (σ)0.4 119 76 0.17  2010.4 6 6 0.01 850.4 13 4 0.02 5.7180 32 14 0.05 11600 12 4 0.02 5.81,000 16 7 0.02 82,000 3 1 0.004 2.6Table 3.3: Number of annihilation events for different times of confinement and the respectiveestimated background.The rate of events per attempt appears to decrease from 0.64± 0.07 (0.4 s holding time) to 0.44± 0.16(1,000 s holding time). However, we do not have sufficient statistics to extract quantitative informa-tion on the trapping lifetime of antihydrogen.The trapping rate at short times has been improved from the one in section 3.3.3 after evaporativecooling and autoresonant injection were further optimized. Nonetheless, there are several possiblemechanisms which could lead to loss of trapped antihydrogen over time. These are mostly due tothe presence of background gases (expected to be composed of He and H2) inside the trap [138].Here is an overview of the possible mechanisms for antihydrogen loss [2]:• Destruction: Collisions of antihydrogen with neutral atoms can cause the antihydrogen toannihilate. This happens when the antiproton gets close to a nucleus, or when exotic boundstates are created with background gas atoms.• Heating: During an elastic collision between antihydrogen (< 0.5 K) and background gases(∼ 10 K), the antihydrogen can gain energy and be expelled from the trap.• Quasi-trapped orbits: This mechanism is well known for magnetically trapped neutrons [139].Antihydrogen atoms with a kinetic energy higher that the trap depth can be temporarily con-fined since the depth of the magnetic trap is anisotropic. The mirror coils can produce adeeper axial trap than the octupole produces in the transverse direction.723.3. Antihydrogen trapping3.3.5 Summary of trapped antihydrogen measurementsDuring the 2010 experimental run, we observed 320 annihilation events compatible with antihy-drogen. Those events were gathered in experiments performed for different bias electric fields andfor different holding times ranging from 172 ms up to 2000 s. Each experimental attempt was per-formed as already described in section 3.3 but with small variations in the plasma preparation thatresulted in more efficient antihydrogen trapping.During 2011, we detected an additional 275 atoms: 65 of these events were associated with mi-crowave experiments (see section 3.4). A summary t − z plot of the detected antihydrogen events ispresented in figure 3.8.0102030t[ms]t[ms]t[ms]0102030-200 -100 0 100 200t[ms]z [mm]t[ms]t[ms]Figure 3.8: Top panel: simulated t and z coordinates of released antihydrogen after the trap shut-down (grey scattered dots). Bottom panel: simulated t and z coordinates of released mirror trappedantiprotons for no bias (green scattered dots), left bias (blue scattered dots) and right bias (red scat-tered dots). The black solid points are the combined trapped antihydrogen data sets from 2010 and2011. t = 0 is the time when the magnetic trap is shut down.733.4. Resonant quantum transitions in antihydrogenFigure 3.9 shows distributions of vertex variables of the 595 trapped antihydrogen events. Weobserve that the majority of trapped antihydrogen has either 2 or 3 tracks. The vertex radius distri-bution (figure 3.9b) has a peak at 2 cm < R < 2.5 cm, which is compatible with the electrodes’ innerradius (2.2 cm). The linear residual distribution (figure 3.9c) are compatible with the results of theblind analysis shown in figure 2.30. We observe in figure 3.9d that the vertices are concentratedat the centre of the trap between -15 cm < z < 15 cm, the length of the trap as defined by the mirrorcoils. Figure 3.9e shows the distribution of the time of detection. The vast majority of events havet < 30 ms, as predicted by the simulations (figure 3.8). Figure 3.9f shows the distribution of thevertex azimuthal angle, which should not have a direction preference, as we observe.3.4 Resonant quantum transitions in antihydrogen3.4.1 Hyperfine structure of the ground stateIn this section, we discuss the hyperfine structure of (anti)hydrogen atoms, with or without anapplied magnetic field. The antihydrogen atom is composed of a positron orbiting a antiproton. Thepositron and the antiproton each have spin one-half and can be oriented either "up" or "down". Tostudy the ground state of the atom, we will use more convenient and simpler basis states, where thesingle arrow refers to the positron spin and the double arrow refers to the antiproton spin:positron spin "up" and antiproton spin "up": |↑⇑〉 (3.8a)positron spin "up" and antiproton spin "down": |↑⇓〉 (3.8b)positron spin "down" and antiproton spin "up": |↓⇑〉 (3.8c)743.4. Resonant quantum transitions in antihydrogen0501001502002502 3 4 5 6 7 8 9 10NumberofeventsNumber of tracksa)0204060801001200 1 2 3 4 5 6 7 8NumberofeventsVertex radius, R [cm]b)0204060801001200.1 1 10 100NumberofeventsLinear residual, δ [cm2]c)020406080100120140-20 -10 0 10 20NumberofeventsVertex axial position, z [cm]d)0501001502002500 10 20 30 40 50NumberofeventsTime of detection, t [ms]e)01020304050-180 -90 0 90 180Numberofeventsφ [deg]f)Figure 3.9: Distributions of vertex variables for trapped antihydrogen: a) distribution of number oftracks, b) distribution of vertex radius, c) distribution of linear residual, d) distribution of vertex axialposition, e) distribution of time of detection, at t = 0 s the magnetic trap is turned off, f) distributionof vertex azimuthal angle. For detailed information on the cuts performed, see section 2.10.2; for acomparison with blind analysis, see figure 2.30.753.4. Resonant quantum transitions in antihydrogenpositron spin "down" and antiproton spin "down": |↓⇓〉 (3.8d)The spin-spin coupling between the positron and the antiproton is responsible for the“hyperfinestructure” in the energy levels [140]. Spin-orbit coupling causes “fine structure”, but does not applyto the ground state, where the orbital angular momentum of the positron should be zero. In theabsence of an external magnetic field, there are four ground states composed of a triplet:|↑⇑〉1√2|↑⇓〉 + |↓⇑〉|↓⇓〉with total spin 1, (3.9)and a singlet1√2|↑⇓〉 − |↓⇑〉 , with total spin 0. (3.10)For matter hydrogen, the energy difference between the triplet and singlet state is∆E ' 5.88 × 10−6eV, (3.11)which corresponds to the zero-field hyperfine splitting frequency [141, 142]:fH = 1, 420, 405, 751.7667 ± 0.001 Hz, (3.12)which is known to very high precision. This frequency is in the microwave range and correspondsto the famous "21-centimeter line" of hydrogen, which is of great importance in astronomy andcosmology, because of the ubiquity of hydrogen in the universe [143].In this presentation, we use the well established formalism for hydrogen to describe the types ofmeasurements that will be made. The object of the ALPHA experiments is to see to what accuracythis formalism describes antihydrogen.763.4. Resonant quantum transitions in antihydrogenIn the presence of a magnetic field, we need to consider the interaction of the positron and theantiproton spins with the magnetic field. The change in the energy of the state due to an externalmagnetic field is known as "Zeeman effect" [140]. The hyperfine states in a magnetic field are [144]:|a〉 = cos(θn) |↑⇓〉 − sin(θn) |↓⇑〉, (3.13)|b〉 = |↑⇑〉, (3.14)|c〉 = cos(θn) |↓⇑〉 + sin(θn) |↑⇓〉, (3.15)|d〉 = |↓⇓〉, (3.16)where n is the principal quantum number and the mixing angles are [37]tan(2θn) ' 51 mTn3B . (3.17)The corresponding energies of the hyperfine states for the ground state of (anti)hydrogen are writtenasEa = −A1 + 2√1 +(µe − µp)2B2/4A2 , (3.18)Eb = A −(µe + µp)B, (3.19)Ec = A−1 + 2√1 +(µe − µp)2B2/4A2 , (3.20)Ed = A +(µe + µp)B, (3.21)where µe and µp are the (anti)electron and the (anti)proton magnetic moments respectively, A=h fH/4 and B is the external magnetic field [140].For a magnetic field of∼ 1 T, which is the case applicable to ALPHA, and for antihydrogen in ground773.4. Resonant quantum transitions in antihydrogenstate, cos(θ1)∼ 1, while sin(θ1)∼ 0. The spins of the positron and the antiproton are essentiallyuncoupled and behave as if they were alone in the external magnetic field. For antihydrogen, thestates are approximated to:|a〉 = |↑⇓〉, (3.22)|b〉 = |↑⇑〉, (3.23)|c〉 = |↓⇑〉, (3.24)|d〉 = |↓⇓〉, (3.25)where the single arrow is the positron spin and the double arrow is the antiproton spin.As already discussed in section 2.8.1, an antihydrogen atom confined in the magnetic trap if itsmagnetic dipole moment is antiparallel to the magnetic field. Since µe/µp 'mp/me, so µe  µp. Forthis reason, states |c〉 and |d〉 correspond to the "low-field seeking" states (trappable states). On theother hand, antihydrogen atoms in "high-field seeking" states (|a〉 and |b〉 ) are expelled from thetrap. Figure 3.10 shows the Zeeman splitting of the ground state of (anti)hydrogen as a function ofthe external magnetic field. This diagram is known as the Breit-Rabi diagram.By applying resonant microwaves to the atoms, it is possible to induce a transition between thehyperfine splitting states. The radiation is resonant if the photon energy equals the energy differencebetween the states. Two vertical arrows representing the transitions for spin flip from trappable statesto untrappable states have been drawn in figure 3.10. These transitions involve flipping the spin ofthe positron, and can be represented as:|d〉 = |↓⇓〉 → |a〉 = |↑⇓〉, (3.26)|c〉 = |↓⇑〉 → |b〉 = |↑⇑〉. (3.27)783.4. Resonant quantum transitions in antihydrogen-20-15-10-5051015200 0.2 0.4 0.6 0.8 1 1.2 1.4Relativeenergyinfrequencyunits[GHz]External magnetic field [T]|a〉|b〉|c〉|d〉TrappablestatesUntrappable statesFigure 3.10: The Breit-Rabi diagram showing the hyperfine structure of the energy levels of the(anti)hydrogen atom in an external magnetic field. The vertical dashed line intercepts the 1 T mag-netic field that is used during the experiment in ALPHA-1. The two black arrows join the statesused during the experiment in section 3.4.4. The arrows have been offset horizontally for clarity.793.4. Resonant quantum transitions in antihydrogenThe approximate corresponding frequencies at the minimum on-axis magnetic field in ALPHA-1(Bmin) arefad = 28.72 GHz (3.28)andfbc = 27.30 GHz. (3.29)If fad and fbc are precisely measured under the same conditions, the difference between the twofrequencies fad − fbc gives the zero-field hyperfine splitting frequency fH .3.4.2 Microwave injection and transition probabilityThe spin-flip measurements relies on the fact that resonant microwaves can induce a transition froma trappable state into an untrappable state. An antihydrogen atom in untrappable state is expelledfrom the trap and the annihilation can be detected by the silicon detector.During the antiproton beam time of 2011, ALPHA performed measurements of the spin flip tran-sition. The time-varying magnetic field was produced by an Agilent 8257D PSG synthesizer andwith a maximum output power of about 700 mW. The radiation entered the vacuum system via awaveguide and was injected into the trap by a horn antenna, which was placed on the movable stickof the apparatus, described in section 2.6 and illustrated in figure 3.11.The spin-flip transition as a function of frequency in the ALPHA-1 trap was calculated using MonteCarlo simulations [3, 145], the results of which are shown in figure 3.12. The shape of the line hasa sharp peak with a long tail on the high-frequency side. The abrupt low-frequency edges are dueto the minimum magnetic field in the center of trap. The long high-frequency tails are associatedwith the inhomogeneity of the magnetic field of the antihydrogen trap. One can see that transition|c〉 → |b〉 also has a small probability to occur at the same frequencies as the transition |d〉 → |a〉 .803.4. Resonant quantum transitions in antihydrogenFigure 3.11: Schematic of the ALPHA-1 apparatus. A microwave horn at the right hand side of theapparatus illustrates the microwave injection. Image from [3].Figure 3.12: Transition probability as a function of the frequency in the ALPHA-1 trap. The fre-quency difference between the two transitions is the zero-field hyperfine splitting frequency. Iffrequencies are applied near the |d〉 → |a〉 transitions, there is a small probability that |c〉 → |b〉 isalso induced, since they overlap. Image from [3].813.4. Resonant quantum transitions in antihydrogen3.4.3 Static magnetic field measured with the electron cyclotron resonanceThe static magnetic field in the ALPHA-1 trap was measured using a novel method, where thecyclotron motion of an electron plasma was excited using microwave radiation [146, 147]. Theresulting temperature increase was detected by monitoring the frequency of the plasma’s quadrupolemode.To carry out a measurement, a series of 4 µs long microwave pulses are injected into the trap, wherethe electron plasma is confined. The microwave frequency is changed for each pulse so that ascan near the cyclotron frequency of the plasma is made. When the microwave frequency matchesthe cyclotron frequency, the electron plasma temperature increases. The change of temperatureis determined by measuring the frequency of the plasma’s quadrupole mode, which depends onthe plasma’s aspect ratio and temperature [149]. Figure 3.13a shows the quadrupole frequencymeasurement as a function of time for a single scan. The multiple peaks correspond to the excitationof the cyclotron motion, when applying the microwave pulse, which results on the increase of theplasma temperature and, therefore in the quadrupole mode frequency of the plasma. The plasma isthen allowed to cool through cyclotron radiation over 30 s. The cyclotron frequency of the plasmais determined by examining the height of the step in the quadrupole frequency as a function of themicrowave frequency (figure 3.13b). The microwaves are on resonance with the cyclotron frequencyat the maximum of this curve.After identifying the cyclotron frequency of the electron plasma, the static magnetic field can bededuced from the equation:B =2pim fcq, (3.30)where fc is the cyclotron frequency, q is the elementary charge, and m is the mass of the electron.This technique was used to calibrate the magnetic fields used during the spin-flip measurements.Figure 3.14 shows an example of calibration.823.4. Resonant quantum transitions in antihydrogenFigure 3.13: a) Quadrupole mode frequency as a function of time. A microwave pulse of 4 µs isapplied every 30 s. The microwave frequency is scanned near the cyclotron frequency. b) Changein the quadrupole mode frequency as a function of the applied microwave frequency gives the cy-clotron frequency of the plasma. Image from [147].Figure 3.14: Example of calibration of the external solenoid using the cyclotron frequency. Imagefrom [150].833.4. Resonant quantum transitions in antihydrogen3.4.4 Experimental procedureThe microwave radiation is reflected from metallic structures in the apparatus. The amount of powerthat is transmitted to the trapping volume depends on the microwave frequency in a very complicatedway because of multiple reflections of the microwaves within the electrode stack. It is importantto ensure that as much microwave power as possible is delivered to the atoms. This dependancewas determined by measuring the reflected power as a function of frequency. Two frequenciesthat were separated by the ground state hyperfine splitting (equation 3.12), and that also have goodtransmission were selected as resonant frequencies to be used in the experiments. The magneticfield that should tune the atoms’ transitions can be calculated by inverting equations 3.18, 3.19, 3.20and 3.21. The magnetic field was set to this value by varying the current in the external solenoid andmeasuring the cyclotron frequency. This field is referred to as Bmin, and the corresponding selectedresonant frequencies fres(ad, bc).To avoid changing the power delivered to the atoms because of the frequency-dependent trans-mission, the magnetic field was instead changed by 3.6 mT to change between on-resonance andoff-resonance. This corresponds to a 100 MHz detuning of the transitions. In further experimentswe decided to change the microwave frequency by 100 MHz to bring the radiations back into reso-nance with atoms. This gives a total of six combinations of experimental parameters, as shown intable 3.4.Series Microwave Magnetic Type ofnumber frequency field measurementSeries 1 fres(ad, bc) Bmin On-resonanceSeries 2 fres(ad, bc) Bmin + 3.6 mT Off-resonanceSeries 3 fres(ad, bc) + 100 MHz Bmin + 3.6 mT On-resonanceSeries 4 fres(ad, bc) Bmin + 3.6 mT Off-resonanceSeries 5 Power off Bmin No-microwavesSeries 6 Power off Bmin + 3.6 mT No-microwavesTable 3.4: Table showing a total of six combinations of experimental parameters.All the measurements have the same experimental procedure:843.4. Resonant quantum transitions in antihydrogen• Antihydrogen productionThe antiproton and positron plasma preparation and antihydrogen formation (1 s mixing) isthe same as the trapped antihydrogen measurements discussed at the beginning of this chapter(section 3.3).• 60 s of atom confinementIn this period of time, the magnetic field could be changed from Bmin to Bmin + 3.6 mT, tochoose between on-resonance and off-resonance. The magnetic field was then allowed tostabilize.This period of time was also used as a background and as we discussed in section 3.3.4, toensure antihydrogen had de-excited to the ground state.• 180 s of microwave irradiationDuring this period of time microwaves were injected.We do not know a priori in which hyperfine state trapped antihydrogen is; it could be either in|d〉 or |c〉 . For this reason, we apply microwaves resonant with both of the positron spin fliptransitions |d〉 → |a〉 and |c〉 → |b〉 , one at a time. A frequency sweep from ( fbc − 5 MHz)to ( fbc + 10 MHz) is applied for 15 s, then a frequency sweep from ( fad − 5 MHz) to ( fad +10 MHz) is applied for another 15 s. This cycle is repeated six times, for a total time of 180 s.Figure 3.15 shows the cycle of microwave injection. The frequencies are resonant with theminimum on-axis magnetic field Bmin or Bmin + 3.6 mT.Since the magnetic field is highly inhomogenous away from the central axis, a transitionwill likely occur only when the antihydrogen passes through the magnetic field minimum.180 s should be enough time for antihydrogen to pass through the center of the trap. Duringthe 180 s time window, we expect to see antihydrogen annihilations when an atom passesthrough either of the transitions |d〉 → |a〉 or |c〉 → |b〉 .As will be discussed later in section 3.4.7, during this detection time window, we use theappearance mode analysis.853.4. Resonant quantum transitions in antihydrogen• The magnetic trap shut downThe magnetic trap is turned off in the same way as other experiments described in 3.3. In thistime window, we use disappearance mode analysis (see section 3.4.6), and detect antihydro-gen that remained in the trap after the microwave irradiation.0 15 30 45 60 75 90 105 120 135 150 165 180Time [s]fbc - 5 MHzfbc + 10 MHzTrappablestatesUntrappable statesfbc - 5 MHzfbc + 10 MHzfad - 5 MHzfad + 10 MHzFigure 3.15: Microwave injection cycle. During the first 15 s, microwaves resonant to fbc areapplied, then from 15 < t < 30 s , microwaves resonant to fad are applied. This 30 s cycle isrepeated six times for a total time of 180 s.3.4.5 Description of measurementsAs we mentioned before, we performed 6 different types of measurements (see table 3.4).Figure 3.16 shows the injected frequency range along with the transition probability as a functionof frequency for the two different on-resonance measurements. These measurements are calledSeries 1 (when using Bmin) and Series 3 (when using Bmin+3.6 mT).Off-resonance measurements were performed in alternation with every attempt of Series 1 and Se-ries 3. Series 2 was alternated with every attempt of Series 1. Likewise, off-resonance measure-ments, Series 4, was alternated with every attempt of Series 3. Series 2 and Series 4 are identical.Figure 3.17 illustrates the off-resonance measurements.863.4. Resonant quantum transitions in antihydrogen15MHz15MHzfAbcFrequencyfAadTransitionProbability(arbitraryunits)b |c>→ |b>|d>→ |a>Series1//1420.4MHz|c>→ |b>|d>→ |a>d Series315MHz15MHz//1420.4MHzfBbc=fAbc+100MHz fBad=fAad+100MHzFrequencyTransitionProbability(arbitraryunits)Figure 3.16: Top: the Series 1 probability transition as a function of the frequency when the min-imum on-axis magnetic field is Bmin. Bottom: the Series 3 probability transition as a function ofthe frequency when the minimum on-axis magnetic field is Bmin+3.6 mT (resonance shifted by 100MHz compared to series 1). Both series are on-resonance measurements. The two yellow bins ineach plot represent the frequency ranges that are applied. Image from [3].873.4. Resonant quantum transitions in antihydrogenSeries2& 4c |c>→ |b>|d>→ |a>15MHz15MHz100MHz100MHzfBbc=fAbc+100MHz fBad=fAad+100MHzFrequencyTransitionProbability(arbitraryunits)Figure 3.17: Series 2 and 4. Probability as a function of the frequency when the minimum on-axismagnetic field is Bmin+3.6 mT. Yellow bins represent the ranges of the frequencies applied, whichare 100 MHz lower than the transition resonance frequency. We observe that the upper frequencysweep overlaps with the tail of the |c〉 → |b〉 transition, meaning that there is a small but non-zeroprobability to induce this transition.3.4.6 Results of disappearance mode analysisIn the disappearance mode analysis, we detect any antihydrogen atoms that remain after the mi-crowave irradiation. The detection occurs during the 30 ms time window, using the standard an-tihydrogen event analysis (see section 2.10). Table 3.5 shows the number of antihydrogen eventsdetected for each series, along with the rate per run. The different experiments are grouped in ta-ble 3.6. The rates of detection in the on-resonance measurements are clearly smaller than the ratesof the off-resonance measurements. This indicates that during microwave injection, antihydrogenatoms are transferred to untrappable states and have escaped from the trap before the disappearancemode analysis.If we compare the off-resonance rates with the no-microwaves measurements, we observe that off-resonance microwaves appear to reduce the rate of detection. There are two possible explanationsfor this. Recall that there is a small probability to induce |c〉 → |b〉 when injecting off-resonant883.4. Resonant quantum transitions in antihydrogenmicrowaves, as illustrated in figure 3.17. Additionally, the microwaves induce currents in the elec-trodes, resulting in heating. Cryogenically absorbed gas can evaporate, resulting in a poorer vac-uum, and a reduced antihydrogen lifetime. During the microwave injection, our temperature sensorsplaced on the electrodes outside the trapping region, measured temperatures changed as high as 3 K.Series Microwave Magnetic Type of Number Number Ratenumber frequency field measurement of attempts of events (events/attempt)1 fres(ad, bc) Bmin On-resonance 79 1 0.01 ± 0.012 fres(ad, bc) Bmin Off-resonance 88 16 0.18 ± 0.05+3.6 mT3 fres(ad, bc) Bmin On-resonance 24 1 0.04 ± 0.04+ 100 MHz +3.6 mT4 fres(ad, bc) Bmin Off-resonance 22 7 0.32 ± 0.12+3.6 mT5 Power off Bmin No-microwaves 52 17 0.33 ± 0.086 Power off Bmin No-microwaves 48 23 0.48 ± 0.10+3.6 mTTable 3.5: Results of disappearance mode for on-resonance, off-resonance and no-microwave mea-surements. Results are from [3].Type of Number of Number of Ratemeasurement attempts events (events/attempt)On-resonance (Series 1+3) 103 2 0.02 ± 0.01Off-resonance (Series 2+4) 110 23 0.21 ± 0.04No-microwaves (Series 5+6) 100 40 0.40 ± 0.06Table 3.6: Summary of flip-spin transition measurements.3.4.7 Results of appearance mode analysisIn the appearance mode analysis, we look for events during the 180 s long microwave injectionwindow, during which antihydrogen atoms could be ejected from the trap. Because of the longobservation time, the cosmic background becomes higher than the expected signal. We expect toget about 8 cosmic events in 180 s compared to the expected antihydrogen event number (about 0.5893.5. Future experiments on antihydrogendetected event per attempt). To improve the signal-to-noise ration, an alternative criteria for annihi-lation event identification was introduced, using a technique known as a bagged decision tree clas-sifier, in the random forest approach [3, 151–153]. This reduces the cosmic rate to (1.7 ± 0.3) mHzfrom (47 ± 2) mHz. This method is ten times more effective in the rejection of cosmic rays, whileaccepting 75% of antihydrogen events compared to the standard analysis.Figure 3.18 shows the number of events selected from the alternative criteria as a function oftime in the 180 s appearance mode window. The expected cosmic background per bin per runis 0.026 ± 0.005 events.Clearly, the on-resonance measurements have a higher number of events than the off-resonance andno-microwave measurements. The events are mostly detected during the first microwave sweep0 < t < 30 s, implying that the first sweep has enough time and power to induce the transitions. Infact, 14 out of 37 events in the first 30 s occur in the first second of one of the frequency sweeps.After 30 s, the on-resonance measurements look similar to off-resonance and no-microwave mea-surements. In the off-resonance measurements, there is an excess of events during 15 < t < 30 s,indicating that |c〉 → |b〉 transitions might be induced when applying the sweep around fad.Figure 3.19 shows the antihydrogen event counts as a function of the axial position. One can seethat the on-resonance events occur for |z| < 6 cm, similar to the events produced in the numericalsimulations. Antihydrogen annihilations with background gases are more scattered in space, sinceannihilations are not concentrated at the magnetic field minimum.Taking these pieces of evidence together, we can conclude that we have observed quantum tran-sitions in trapped antihydrogen atoms. We have localized the transition frequencies at the trapminimum to approximately 100 MHz, which corresponds to a relative precision of 4 × 10−3.3.5 Future experiments on antihydrogenWith the ALPHA-2 apparatus fully commissioned and ready to take measurements, a new era ofantihydrogen research is around the corner. Soon, the ALPHA collaboration will be able to make903.5. Future experiments on antihydrogenTime, t (s)-60 -30 0 30 60 90 120 150 180Events/(15s)0510152025on res. - 103 runsoff res. - 110 runsw - 100 runsµnoaFigure 3.18: Antihydrogen event counts as a function of time. Microwave radiation is injected att = 0 s for 180 s (see section 3.4.4). The errors bars are due to statistics. Image from [3].913.5. Future experiments on antihydrogenAxial position, z (cm)-20 -10 0 10 20Events/(1.6cm)051015on res. - 103 runsoff res. - 110 runsw - 100 runsµnobFigure 3.19: Antihydrogen event counts as a function of the axial position for 0 < t < 30 s. The greyhistogram is the result of numerical simulations, where spin-flip transitions are induced. The dashedblack curve histogram is the result of numerical simulations, where antihydrogen annihilates withbackground gases. Image from [3].923.5. Future experiments on antihydrogenlaser spectroscopy measurements. Here we will discuss several future measurements that can beperformed in antihydrogen:• 1S–2S two photon spectroscopyThe 1S–2S electronic transition is known to a precision of 4.2×10−15 in hydrogen, one of themost precise measurements in physics [49]. This spectroscopic measurement in antihydrogenwill allow a comparison between hydrogen and antihydrogen. The transition is forbidden forsingle photons, so two photons each with wavelength 243 nm must be absorbed to induce thetransition. Using two counter-propagating photons allows the doppler effect to be cancelledto the first order. Also, the fact that single-photon decay is forbidden means that the transitionhas a natural line width of only 1.3 Hz, which is why such high-precision measurements canbe made. However, the line will be broadened by the Zeeman effect and time of flight broad-ening [154]. One of the biggest challenges is to perform spectroscopy in a large trap volumewith very few antihydrogen atoms (about 1 atom per attempt). The ALPHA-2 apparatus isdesigned to allow laser access and the trap has an optical resonant cavity that increases thelaser power in the trapping volume.• 1S–2P one photon spectroscopy and coolingAlso called Lyman-α transition, this transition has a line width of 99.7 MHz [156], and issuitable for laser cooling of trapped antihydrogen in the ground state [156]. This transitionrequires radiation at a wavelength of 121.56 nm, which is in the vacuum ultraviolet (VUV)and its production is very challenging. Doppler cooling of magnetically trapped hydrogen wasfirst reported in 1993 [155] and no other similar experiments have been reported since then.Within the ALPHA collaboration, a pulsed Lyman-α laser has been developed [157, 158].The VUV light was generated using a four-wave mixing process in a mixture of krypton andargon, where two wavelengths are mixed to obtain the Lyman-α wavelength [158]. Based oncalculations it has been estimated that antihydrogen can be cooled to temperatures as low as20 mK, using the available power levels [159].933.5. Future experiments on antihydrogen• Positrons cooled sympathetically with laser cooled atomsOne way to increase the rate of trapped antihydrogen is to further cool the constituent par-ticles. In particular, the temperature of the positron plasma greatly influences both the rateof antihydrogen production and its final temperature (see section 3.2). Sympathetic cool-ing of positrons using laser cooled 9Be+ ions was demonstrated in 2002 [160, 161]. Thistechnique has been studied computationally in the context of antihydrogen experiments and,it was found that the positron temperature could be reduced by about one order of magni-tude [162]. It was also found that antihydrogen formation in the presence of 9Be+ ions is notaffected [162].• Adiabatic expansion cooling of antihydrogenAdiabatic expansion cooling consists of adiabatically changing the shape of the magneticwell, which will perform work on the atoms and cool them. With the new ALPHA-2 atomtrap, the length of the magnetic trap can be effectively expanded in the axial direction [163].• Nuclear magnetic resonance (NMR) transitionsWhen antihydrogen is in a trappable state (|c〉or |d〉 ), the spin of the antiprotons can also beflipped. Such transitions are called NMR transitions. The transition frequency passes througha broad maximum at a magnetic field of 0.65 T, which makes the NMR transition much lesssensitive to the magnetic field inhomogenity [164]. At 0.65 T, the frequency corresponding tothe |c〉 → |d〉 transition is about 655 MHz with a corresponding wavelength of about 46 cm.Since the wavelength is larger than the diameter of the electrodes, the radiation will not prop-agate down the electrode stack and a coaxial transmission line and a built-in resonator will berequired [165].• Electric charge of antihydrogenUsing the experimental data from 2010 and 2011, ALPHA derived an experimental limiton the charge of antihydrogen. It was found that the charge of antihydrogen is Q/e =(1.3 ± 1.1 ± 0.4) × 10−8, where e is the unit charge and the errors are from statistical andsystematic effects [6]. It has been proposed that by applying randomly oscillating electric943.5. Future experiments on antihydrogenfields to a sample of trapped antihydrogen, it should be possible to increase the precision ofthe measurement of the antihydrogen charge by several orders of magnitude [166, 167].• GravityUsing again the experimental data from 2010 and 2011, ALPHA reported directly measuredlimits on the ratio of the gravitational mass Mg to the inertial mass of antihydrogen M, F =Mg/M. Ratios above F = 75 (statistics alone) and F = 110 (including worst-case systematiceffects for gravity) and F = -65 (combined statistical and systematic effects) were ruled out [5].It was found with simulations that, by cooling antihydrogen to ∼ 30 mK, and by increasingthe magnetic trap shut down time to ∼ 300 ms, it could be possible to measure F to the levelF =±1 [5].95Part IIAntiproton cloud radial compression96Chapter 4Non-neutral plasma confinement andradial compression in aPenning-Malmberg trapIn section 4.1, we discuss the motivation for antiproton cloud compression in antihydrogen ex-periments. Later, we give a theoretical overview of the confinement of non-neutral plasmas in aPenning-Malmberg trap (section 4.2. We also discuss the rotating wall mechanism, which consistsof a rotating, time-varying, azimuthal, electric field and is used to radially compress non-neutralplasmas (section 4.3). ALPHA has been routinely using the rotating wall mechanism to compressantiprotons clouds and electron and positrons plasmas since 2008 [63]. In section 4.4, we describehow ALPHA sympathetically compresses the antiproton clouds with an electron plasma, whichdiffers from the measurements found in this thesis, where the electron plasma is not directly com-pressed (chapter 6).4.1 Motivation for the compression of antiproton cloudsWhen working with charged particles in Penning-Malmberg traps, it is necessary to counteract theslow expansion and loss of the plasma due to the imperfections in static fields and the presence ofbackground gases (more details in section 4.3). Radial compression with the rotating wall mecha-nism has proven to be a very efficient way to increase the lifetime of particles inside a trap [168].974.1. Motivation for the compression of antiproton cloudsDuring antihydrogen experiments, it is important to be able to control the radial size of the antipro-ton clouds and positron plasmas, and to avoid expansion and loss during preparation of the plasmafor mixing.As already mentioned in section 2.8.2, the presence of the octupole field, which is needed to trapthe neutral antihydrogen, breaks the cylindrical symmetry of the Penning-Malmberg trap. Suchdisturbance causes the diffusion and expansion of confined non-neutral plasmas, but which can becounteracted with the rotating wall. Furthermore, the presence of the transverse magnetic field willcause the loss of particles beyond a critical radius since charged particles closely follow the magneticfield lines [169]. Radially small clouds of antiprotons are less sensitive to this perturbation.Yet another reason to reduce the radial size of antiprotons clouds is to achieve good overlap betweenthe antiproton cloud and the positron plasma during the formation of antihydrogen. As already men-tioned in section 3.2, the rate of antihydrogen formation strongly depends on the positron plasmadensity. For this reason, the positron plasmas are typically small (∼ 0.9 mm) and when mixed withthe antiprotons, a similar radial size for the antiproton cloud is needed to increase the probability ofinteraction between the two species.A low kinetic energy is a requirement for trapping antihydrogen, since only antihydrogen atomswith energies less than 44 µeV (0.5 K) are confined in the trap, as already seen in section 2.8.2.Perhaps most importantly, the kinetic energy of antihydrogen formed from antiprotons and positronsdepends on the azimuthal velocity of the former, vθ, which is expressed as:vθ = ωrotr, (4.1)where ωrot is the rotation frequency and r is the radius of the cloud. If the antiproton cloud iscompressed, the radius is reduced, and therefore the azimuthal velocity too. Since the electric fieldproduced by the antiprotons is negligible, the azimuthal velocity is determined by the radial positionof the plasma, and smaller antiproton clouds are almost always better.984.2. Non-neutral plasma confinement in a Penning-Malmberg trap4.2 Non-neutral plasma confinement in a Penning-Malmberg trapA non-neutral plasma is a many-body collection of charged particles, in which the total chargeis noticeably different from zero. The name, non-neutral plasma, was first used in the publica-tion of Davidson’s monograph, Theory of Nonneutral plasmas (1974) [170]. Non-neutral plasmasshare some of their properties with electrically neutral plasmas, such as plasma waves, instabili-ties, and Debye shielding [171]. On the other hand, such non-neutral plasmas are characterizedby intense self-electric fields that play an important, and sometimes dominant, role in the plasmadynamics [172]. Non-neutral plasmas can be confined in Penning traps by static electric and mag-netic fields and also be in a state of global thermal equilibrium, unlike neutral and quasi-neutralplasmas [172]. Examples of such plasmas are pure electron plasmas, positron plasmas, pure ionplasmas of one or more species and in our case, antiproton-electron plasmas.4.2.1 Non-neutral plasma fundamental propertiesThe non-neutral plasma properties discussed in this section follow that of Ref. [133, 171]. Eventhough non-neutral plasmas are not charge neutral, they can be called plasmas because they havemany collective properties common to neutral plasmas, as discussed above.The plasma frequency is the most fundamental time-scale in plasma physics:ωp =√ne20m, (4.2)where n is the number density, e is the elementary charge, m the mass of the particle and 0 isthe vacuum permittivity. For a given density, electron plasmas have the fastest plasma frequenciesbecause of the dependance of ωp on the mass of the particle. The plasma frequency corresponds tothe typical oscillation frequency in response to the displacement of a small charge in the plasma.The plasma period τp = 1/ωp is also of great importance, since plasma oscillation can only beclearly observed over time periods longer than τp. Similarly, plasmas can only be observed over994.2. Non-neutral plasma confinement in a Penning-Malmberg trapa length longer than the distance traveled by the plasma particle τpv, where v is the speed of theparticle. This distance is called the Debye length:λD =√0kBTne2, (4.3)where T is the temperature, kB is Boltzmann constant and n is the density of the plasma. λD isindependent of mass, so is generally comparable for all plasma species.To be considered a plasma, the Debye length must be smaller than the overall dimensions of thesystem L:λDL 1. (4.4)A single charged particle inside a non-neutral plasma will attract other particles with oppositecharge, or repel other particles with same charge. Plasma particles act collectively to cancel orshield the field of an extra charge introduced into the plasma. This phenomena is called Debyeshielding [171]. Particles are shielded from external view and the particle’s Coulomb field falls offexponentially as e−r/λD/r, instead of 1/r2, as for an isolated charge.4.2.2 Plasma rotation frequencyThe rotation frequency of the plasma is analogous to the magnetron frequency of a single chargedparticle (see section 2.3.1). Here, the non-neutral plasma self-electric field produced by the ‘self-charge’ or ‘space charge’ creates an ~E × ~B circular drift about the symmetry axis of the trap.Many of the plasmas used in ALPHA are typically long and have small radii, and we can approx-imate them as infinite cylinders. Using the Lorentz force equations of motion, and Gauss’ law tofind the radial electric field inside a infinite cylinder, we can calculate the rotation frequency [173].Since ~E is perpendicular to ~B (i.e. ~B = Bzˆ and ~E = Erˆ), only the perpendicular equation of motion1004.2. Non-neutral plasma confinement in a Penning-Malmberg trapis modified:md~v(t)dt= q(~E + ~v × ~B). (4.5)A constant azimuthal velocity,~vθ =~E × ~BB2(4.6)is a solution of equation 4.5, where ~vθ = ωrotrθˆ, .From Gauss’ law, one can find that inside an infinite charged cylinder, the radial electric field isEr =enr20. (4.7)It will be shown in section 4.2.4 that the density is almost constant. The magnitude of the plasmaazimuthal velocity can then be written as:vθ =enr20Bz= ωrotr. (4.8)The plasma rotation frequency ωrot is independent of the species’ charge, mass and radial coordi-nate, so that non-neutral plasmas are characterized by global rigid-rotation [174].4.2.3 Theory of confinementConfinement in a Penning-Malmberg trap is underpinned by the conservation of the total canonicalangular momentum of the plasma [172]. As already mentioned in section 2.3, axial confinementis provided by the electrostatic field and radial confinement by the axial magnetic field. It canbe shown that due to the (~E × ~B) forces, the plasma revolves about the magnetic field axis. The1014.2. Non-neutral plasma confinement in a Penning-Malmberg trapcanonical angular momentum is expressed as [172]:Pθ =N∑j=1mvθ jr j −N∑j=1|e|Aθ(r j)r j, (4.9)where m is the mass of the particles, vθ j is the azimuthal velocity, r j is the distance from the axisof symmetry (radius) of the jth particle, N is the total number of particles, |e| is the absolute valueof the elementary electric charge and Aθ is the azimuthal component of the vector potential. Thefirst right-hand term of equation 4.9 is the total mechanical angular momentum and the secondright-hand term comes from the rotation of the charged particles in the magnetic field. The vectorpotential of a uniform, axial magnetic field is Aθ(r j) =Br j2 , where ~B = Bzˆ and ~A = Aθ(r)θˆ. Here(r, θ, z) denote a cylindrical coordinate system with the z axis coincident with the axis of symmetryof the trap. Equation 4.9 becomes:Pθ =N∑j=1mvθ jr j −N∑j=1|e|B2r2j . (4.10)For a strong magnetic field, the second term dominates, and the canonical angular momentum canbe approximated asPθ ≈ −N∑j=1|e|B2r2j . (4.11)Since we are dealing with particles carrying a single sign of charge, the charge and the other con-stants can be excluded from the sum:Pθ ≈ −|e|B2N∑j=1r2j ≈ −|e|B2< r2 > N, (4.12)where < r2 > is the mean-square radius of the plasma.Given the conservation of the total canonical angular momentum, we conclude that the mean-square1024.2. Non-neutral plasma confinement in a Penning-Malmberg trapradius of the plasma must be approximately constant, hence the particles are largely confined. Thisis an important feature of non-neutral plasmas, distinct from neutral plasmas whose confinementhas proven very difficult. In section 4.3 we show how increasing the canonical angular momentumresults on plasma radial compression.4.2.4 Global thermal equilibriumElectron plasmas can evolve to a state of global thermal equilibrium after a long confinementtime [175]. Plasma particles reach such global thermal equilibrium by Coulomb collisions witheach other and, assuming conservation of the total energy and total canonical angular momentum,the thermal distribution of the plasma can be described by a one-particle distribution:f (h, pθ) = n0(m2pikBT)3/2exp[− 1kBT(h + ωpθ)], (4.13)where h is the one-particle energy and pθ is the one-particle canonical angular momentum. Hereh =12mv2 + eφ(r, z), (4.14)pθ = mvθr +eB2r2 (4.15)and φ(r, z) is the electric potential.By substituting equations 4.14 and 4.15 into equation 4.13, the thermal distribution can be expressedas:f (r,v) = n(r, z)(m2pikBT)3/2exp(− m2kBT[v + ωrθˆ]2), (4.16)wheren(r, z) = n0exp(− 1kBT[eφR(r, z) + eφp(r, z)]). (4.17)φR(r, z) is the effective trap potential in the rotating frame and φp(r, z) is the plasma space-charge1034.2. Non-neutral plasma confinement in a Penning-Malmberg trappotential. Equation 4.16 is a Maxwellian velocity distribution rotating as a rigid body at a uniformrotation frequency ω (also called ωrot). For simplicity, we use Φ(r, z) = φR(r, z) + φp(r, z). TheMaxwell-Boltzman distribution becomesn(r, z) = n0exp[eΦ(r, z)kBT]. (4.18)The plasma density largely depends on the plasma’s space charge potential, so it is possible to findthese two quantities by self-consistently solving Poisson’s equation:∇2Φ(r, z) = −en(r, z)0. (4.19)Figure 4.1 shows measurements of the density and the rotating frequency over time as the plasmasreaches a global thermal equilibrium state.Figure 4.1: Experimental density and rotation frequency evolving over time to global thermal equi-librium as a function of radius. After 10 s, global thermal equilibrium is reached and the rotationfrequency becomes almost constant as a function of radius, as for rotation of a rigid body. Imagefrom [175].1044.2. Non-neutral plasma confinement in a Penning-Malmberg trapGlobal thermal equilibrium solverA global thermal equilibrium solver [176], developed by Prof. F. Robicheaux of Purdue Universityand extensively used by the ALPHA collaboration, has been used for the analysis of the experimen-tal results of this thesis.On providing the number of particles, peak intensity, temperature, electrode configuration and volt-ages, the algorithm outputs the charge density and potential in the trap. The calculation is initializedusing an initial guess of the charge density to solve Poisson’s equation. After the electric potentialis calculated, it is used to find a new charge density. The algorithm is repeated until both solu-tions (density and potential) converge to stable values. Figure 4.2 shows examples of the densityas a function of radius calculated by self-consistently solving Poisson’s equation and the Maxwell-Boltzmann distribution and table 4.1 shows their plasma parameters. It has been demonstrated that,provided λD  L, the plasma density (equation 4.17) is nearly constant until the radius is within afew λD of the edge and then falls off to zero. From figure 4.2, one can see that colder plasmas havea flatter density and almost abruptly falls off to zero, while for hotter plasmas, the density falls offmore gradually.Number Peak density Temperature λdebye Lz ωrotof electrons [cm−3] [K] [mm] [mm] [kHz]20×106 5×106 100 ∼0.3 ∼19 ∼220×106 1×108 100 ∼0.1 ∼37 ∼4820×106 5×106 2000 ∼1 ∼19 ∼220×106 1×108 2000 ∼0.3 ∼35 ∼47Table 4.1: Electron plasma parameters as calculated by a global thermal equilibrium solver [176].Number of electrons and plasma temperature are input parameters, while the remaining parametersare output from the solver.It is possible to force the solver to use a specific radial profile density. After each iteration, thedensity solution is normalized to the desired radial profile. In this case, the solution obtained willnot in general correspond to global thermal equilibrium.1054.2. Non-neutral plasma confinement in a Penning-Malmberg trap00.510 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16n(r)/n0r [mm]5 × 106 cm−3, 100 K1 × 108 cm−3, 100 K5 × 106 cm−3, 2000 K1 × 108 cm−3, 2000 KFigure 4.2: Calculated density as a function of radius for 20 × 106 electrons for different peakdensities n0 and different temperatures. Densities are normalized by their peak.4.2.5 Antiproton-electron plasmasUnderstanding antiproton-electron plasmas is very important when synthesizing antihydrogen. Elec-trons provide a source of cooling for the antiprotons while applying the rotating wall, and while per-forming manipulations before the production of antihydrogen (see section 2.4.2). For these reasons,electrons are important for the antiproton experiments performed in ALPHA.In ALPHA, we usually refer to trapped electrons as an electron plasma. We typically use about20 × 106 electrons in a plasma having ∼ 30 mm length and λD ∼ 0.3 mm. The self-electric field(space charge) of the electron plasma is large enough to create an ~E × ~B rotation, as explained insection 4.2.2. If a second species (antiprotons) is added to the electron plasma, the antiprotons willrotate under the influence of the electron plasma’s space charge. The number of antiprotons is smallcompared to the number of electrons (about 1.5 × 105 antiprotons) and thus has a negligible spacecharge. Also, the Debye length is large for a pure antiproton cloud. For these reasons, we refer totrapped antiprotons as an antiproton cloud, rather than a plasma.1064.3. Rotating wall mechanismCentrifugal separationCentrifugal separation can be important for multi-species plasmas in a global thermal equilibriumstate at low temperatures [177] and has been observed for species having different charge-to-massratios, for example, for laser cooled Be+-ion [178–180] and Be+-positron [181] systems.ALPHA observed centrifugal separation between antiprotons and electrons in 2010 [182]. Fig-ure 4.3 shows an example of centrifugal separation at ∼ 100 K, for different magnetic fields. Cen-trifugal separation was also reported by the ATRAP experiment [183].Figure 4.3: Example of centrifugal separation of an antiproton-electron plasma in a (a) 1 T and (b)3 Tesla magnetic field. Image from [182]Even though the centrifugal separation dynamics can be important when using antiproton-electronplasmas and synthesizing antihydrogen, it will not be discussed here since we do not observe it in themeasurements of this thesis. The reason for this is that the temperatures of the antiproton-electronplasmas, about 500 K, are not low enough.4.3 Rotating wall mechanismIn a perfect trap, plasmas can be confined for an indefinite time via conservation of the canonicalmomentum of the plasma [184]. However, in a real trap, asymmetries in the static fields and the1074.3. Rotating wall mechanismpresence of background gases exert a drag on the plasma [184] and, as a consequence, there is aslow expansion and loss of particles [185–187]. Fortunately, non-neutral plasmas can be radiallycompressed using a time-varying rotating electric field, called a “rotating wall”.The rotating wall was first developed by the non-neutral plasma group at the University of Californiain San Diego in 1997 to radially compress magnesium ion plasmas [168], see also work at TRIUMFin Ref. [188]. Later, compression of electron plasmas was achieved [189]. Heating when using therotating wall was observed and to counteract these effects, cooling gases such as SF6, CF4 or CO2were added to positron plasmas [190].A large amount of experimental and theoretical work has improved our understanding of the mech-anism of the rotating wall. However, a complete microscopic theory remains a future goal. For-tunately, a macroscopic, phenomenological description provides a useful explanation of the mostimportant features. In this theory, the rotating wall balances or exceeds the drag on the plasmaby applying positive torque to the plasma [184]. From the equations of Theory of Confinement(section 4.2.3), we can see how applying a positive torque can reduce the radial size of the plasma.Equation 4.12 shows the relation between the canonical angular momentum and the mean-squareradius of the plasma. If the rotating wall applies a positive torque to the plasma:τ =dPθdt> 0 (4.20)then,τ ≈ −|e|B2d〈r2〉dtN > 0, (4.21)which implies a decrease in the radius of the plasma:d〈r2〉dt< 0. (4.22)A positive torque will increase the canonical angular momentum while decreasing its magnitude(the canonical angular momentum is negative). Therefore 〈r2〉 will decrease and the plasma will1084.3. Rotating wall mechanismcompress.One of the mechanisms to apply torque to the plasma is via excitation of plasma modes. It has beenstudied that it is possible to achieve plasma compression by exciting Trivelpiece-Gould modes (TGmodes) [191–193]. TG modes are surface waves that rotate in the azimuthal direction. When theydamp, the angular momentum associated with the wave is transferred to the bulk of the plasma.If the rotating wall continuously excites the TG modes, this results in a torque. Modes that rotatefaster than the plasma are found to provide a positive torque, resulting in the compression of theplasma. On the other hand, modes rotating slower than the plasma provide a negative torque andexpansion of the plasma is observed [193]. The degree of compression depends on the efficiency ofcoupling of the external torque to the plasma [194]. It was experimentally observed that an electronplasma compresses near the predicted and observed TG mode frequencies of the electron plasma(see figure 4.4) [191].Figure 4.4: Measured compression rate of an electron plasma as a function of the rotating wallfrequency fs. The calculated and measured frequencies of different modes (labelled by (mθ,mz,mr))are shown on the bottom. Image from [193].1094.3. Rotating wall mechanismCompression has been observed in several different regimes and there is not a single explanationthat covers them all. Here is an overview of several compression regimes:• Weak drive: For weak rotating wall excitations, it was shown that electric fields transfertorque to the plasma by exciting Trivelpiece-Gould plasma modes [191]. The Trivelpiece-Gould modes carry angular momentum that can be transferred to the particles by a wave-particle coupling [189] as discussed above, also see section 7.1.3.• Strong drive: When a larger amplitude rotating electric field coupled with a source of coolingis used, compression over a broad range of frequencies, without the need to couple the plasmato a specific mode, is observed [195–197].• Sideband cooling: The sideband cooling mechanism in the single particle regime relies on thecoupling of the cyclotron or axial motion with the magnetron motion of the particle producedby the rotating electric field. For example, the excitation of the cyclotron mode leads to adecrease of the radius of the magnetron mode, which results in the “axialization" of particles(i.e. the particle radially moves towards the axis), see section 7.1.6.• Bounce resonant transport: The resonance on the bounce transport (axial and rotationmotion) of particles (plasma or single particle regime) can result in compression (see sec-tion 7.1.7) .• Sympathetic compression: With two species of particles, compressing one species with therotating wall can cause the other species to follow via Coulomb collisions [63]. ALPHA per-forms this type of compression, by applying a rotating wall to an antiproton-electron plasma,to compress the electron plasma that sympathetically compresses the antiproton cloud (seesection 4.4).The rotating electric field is produced by an electrode divided into azimuthally isolated segmentsand by applying to each segment a sinusoidal potential Vω j of frequency ωRW and phase θ j = 2pi j/k,1104.3. Rotating wall mechanismwhere k is the number of segments. The potential is then:Vω j = Aωcos[mθ(θ j − ωRW t)] (4.23)where mθ = 1 corresponds to the dipole mode and mθ = 2 to the quadrupole mode. Figure 4.5illustrates a six-segmented rotating wall setup, where a sinusoidal waveform voltage generator isconnected to each segmented electrode with a phase shift of 60◦.Figure 4.5: Schematic representing a transverse section of the rotating wall. A sinusoidal potentialis applied to a six-segmented electrode with a phase difference of 60◦ between each segment. Thered circle illustrates the plasma, and the blue arrow the direction of rotation.In ALPHA, the amplitude of the rotating wall can vary between 0 V to 10 V. Figure 4.6 shows therotating wall potential for various phases. In ALPHA, the generator is AC coupled to the electrode,so when applying the rotating wall, the potential of the segmented electrode is added to an overallbias potential.1114.4. Sympathetic compression of antiproton cloudsa) b)-1-0.5 0 0.5 1c)d) e)-1-0.5 0 0.5 1f)Figure 4.6: Rotating wall radial potential when applying a 1 V sinusoidal potential to the 6 seg-mented electrode. Each image illustrates the potential at different phases: a) 0 ◦, b) 60 ◦, c) 120 ◦,d) 180 ◦, e) 240 ◦ and f) 300 ◦.4.4 Sympathetic compression of antiproton cloudsIn 2008, ALPHA published a paper with the first detailed measurement of radial compression ofantiproton clouds in the ALPHA1 apparatus [63]. This paper summarizes the procedure for the an-tiproton cloud radial compression that was used in the experiments to trap antihydrogen. A rotatingwall drive with a frequency in the 10 MHz range is applied to the antiproton-electron plasma. Sucha drive radially compresses the electrons plasma that is co-located with the antiproton cloud. Whenthe electrons move towards the centre of the trap, it is thought that the electrons transfer angularmomentum to the antiprotons by Coulomb collisions and, as a consequence, the antiproton cloudalso becomes radially smaller. Such compression is called "sympathetic compression". Antiprotonclouds were compressed to a radius as small as 0.29 mm in a 3 T magnetic field when applying therotating wall at ωRW/2pi= 20 – 25 MHz [63].Sympathetic compression differs from the direct compression studied in this thesis (chapter 6) bythe fact that for the sympathetic compression, the rotating wall acts directly on the electron plasma(in the 10 MHz range). This was demonstrated by applying the rotating wall drive in the absence of1124.4. Sympathetic compression of antiproton cloudsantiprotons and observing that the electron plasma compresses.The electron plasma used during sympathetic compression typically has a radius of about 0.8 mmand its rotation frequency and axial bounce frequency is in the 10 MHz range. As stated in Ref. [63],the electron plasma is unchanged when applying the rotating wall at ωRW/2pi= 400 kHz. This fre-quency is in the same range as the antiproton axial bounce frequency.Figure 4.7 shows some typical examples of compression after applying the rotating wall for 1 s,20 s and 60 s. The bare antiproton MCP images are obtained by ejecting the electrons just beforedumping the particles to the MCP. One can see that the electron plasma clearly compresses whenapplying the rotating wall and the antiproton clouds’ radial sizes follow that of the electron plasma.There are about 11,000 antiprotons captured and cooled and the smallest radius measured at thisfrequency is 0.42 mm.Figure 4.8 shows the electron plasma radius and the antiproton cloud density as a function of therotating wall compression time. These measurements were performed for both slow and fast electronplasma compression. Fast compression is achieved by increasing the rotating wall voltage by afactor of 5 with respect to the slow compression. One can observe in figure 4.8a that during fastcompression, the electron plasma takes as little as 5 s to reach a minimum size, compared to the slowcompression, which takes at least 45 s. However, the antiproton cloud only compresses when theslow electron plasma compression is used (see fig. 4.8b). It is thought that during fast compression,the antiprotons are “left behind” in a region of low electron density, because the antiproton-electroncollision rate is too low to keep the two species coupled.Figure 4.9 shows the electron plasma radius as a function of the antiproton cloud radius for differentcompression times. The measurements were performed using slow compression. The radii of bothspecies get proportionally smaller as the compression time increases, showing that the electronplasma radius and the antiproton cloud radius compress at similar rates.The smallest antiproton cloud radius achieved was 0.29 mm, using frequencies of about 20 – 25 MHz.These measurements use frequencies that directly compress the electron plasma, which is a differentapproach from the one in chapter 6, where for the frequencies applied, the rotating wall does not act1134.4. Sympathetic compression of antiproton cloudsFigure 4.7: Example of the radial compression of an antiproton-electron plasma forωRW/2pi∼ 10 MHz. The center figures are MCP images at different times. It is observed that elec-tron plasma is radially compressed. The outer columns show their resulting radial profiles, wherethe red lines are generalized Gaussians [i.e. exp(−| rr0 |k) where k ≈ 2] fitting the radial profiles.Image from [63].1144.4. Sympathetic compression of antiproton cloudsFigure 4.8: (a) Electron plasma radius as a function of compression time for slow and fast com-pression. The electron plasmas is radially compressed for both kinds of compression to a similarradius. (b) Antiproton cloud density as a function of time of compression for slow and fast com-pression. It is observed that the antiproton cloud is not radially compressed in fast compression.Image from [63].1154.4. Sympathetic compression of antiproton cloudsFigure 4.9: Electron plasma radius as a function of antiproton cloud radius. Measurements areperformed for different times of compression (slow compression) and it is suggested that both radiiare proportionally compressed. Image from [63].directly on the electron plasmas.4.4.1 Antiproton cloud compression by the ASACUSA collaborationIn the same year (2008), the ASACUSA collaboration published a paper in which antiproton cloudswere radially compressed, but without the use of an electron plasma or any other source of cool-ing [201]. The antiproton cloud compression was achieved at frequencies between 200 kHz and1000 kHz. Figure 4.10 shows the observed compression of the antiproton clouds as a function ofthe rotating wall frequency.The ASACUSA collaboration concluded that there are no resonant structures in their range of fre-quencies, and that the compression is phenomenologically similar to the strong drive compression.At this point, there is no full understanding of the compression mechanism.Even though the ASACUSA collaboration used the same frequencies that we use in the measure-ments of this thesis, the inclusion of electrons drastically changes the potential due to the spacecharge. Also, the cooling provided by the electrons could potentially change the behaviour of theantiprotons.1164.4. Sympathetic compression of antiproton cloudsFigure 4.10: Transport efficiency εexp as a function of the rotating wall frequency. εexp is the ratioof the number of antiprotons detected to the number of the trapped antiproton, which increases asthe radius gets smaller. Figure a) shows a wide range of rotating wall frequencies, while figure b)shows the results of the compression around 247 kHz (sideband frequency). Image from [201].4.4.2 Antiproton cloud compression by the ATHENA collaborationIn 2006, the ATHENA collaboration reported evidence of sideband cooling of antiprotons in anelectron gas [202]. It consisted of applying a quadrupolar excitation at the cyclotron frequency(∼ 45 MHz), which converts the magnetron motion of all or some of the antiprotons to cyclotronmotion. The experiment was performed using thousands of antiprotons and between 1 and 3×106electrons. In order to find the exact cyclotron frequency, the frequency of the quadrupolar excita-tion was scanned around the expected cyclotron frequency. It was found that about 20 – 40 % ofantiprotons appeared to be affected by the excitation. It was assumed that the final magnetron radiusshould be of the same order as the cyclotron radius, but no direct measurements were made thatcould support this.117Chapter 5Experimental setup for the compressionof antiproton clouds in ALPHA-2In this chapter, we describe the experimental set-up used in the measurements on antiproton cloudcompression in ALPHA-2. This includes: 1) antiproton capture in the Penning-Malmberg trap, 2)antiproton cooling by the electron plasma, 3) manipulation of the potential to move the particlesaxially, 4) ejection of the electrons from the trap, 5) application of the rotating wall and 6) detectionof the particles.5.1 Capturing antiprotons and secondary electronsIn this section, we first focus on the capture of antiprotons. Then, we introduce the production andcapture of secondary electrons, which are used in the measurements of antiproton cloud compres-sion.5.1.1 Antiproton capture in ALPHA-2The degrader in ALPHA-2 consists of four layers: two beryllium (Be) and two aluminium (Al)foils. Table 5.1 is a summary of the degrader layers’ thicknesses. The degrader is divided intofour layers because each layer also has a secondary use. After entering the experiment from theAD, the antiprotons first encounter the tuneable Al degrader. It is possible to have access to thislayer to change the thickness without dismantling most of the apparatus. This allows the thickness1185.1. Capturing antiprotons and secondary electronsto be optimized for the maximum number of antiprotons trapped. Nevertheless, this still requiresopening the vacuum system and was performed only 5 times during 2012. The second layer, witha thickness of 10 µm, is part of the heat shield between the 4 K section and the room temperatureenvironment, and is cooled to ∼ 40 K. The third layer that the antiprotons encounter is the 50 µm Bevacuum window that separates the isolation vacuum and the ultra-high quality trap vacuum. Thelast layer is the Faraday cup (FC), which is made of Be with a thickness of 165 µm. The FC collectscharge and can be used to measure the number of electrons that are incident on it. The precision ofthe thickness of the Be layers is quoted by the manufacturer to be between 10% to 15%. The Allayers were measured during commissioning of the trap and have an uncertainty less than 5%. Dueto these uncertainties, it was necessary to experimentally optimize the thickness by measuring theantiproton capture efficiency for different thicknesses.Layer Material Thickness [µm] Distance from valve [mm]Tuneable degrader aluminium 24 - 35 (±5%) 1004Heat shield aluminium 10 (±5%) 1022Vacuum window beryllium 50 (±15%) 1051Faraday cup beryllium 165 (±10%) 1072Table 5.1: A summary of degrader material and thickness. The valve separates the ALPHA experi-ment from the AD and is used as a convenient reference point. The thickness of tuneable degraderwas changed between 24 µm and 35 µm.More information about the energy loss and the range of the antiprotons is presented in appendix A.Figure 5.1 shows the results of an experiment to measure of the efficiency of capturing antiprotonsas a function of the tuneable degrader thickness. The antiproton capture efficiency is the percentageof antiprotons captured with the 5 keV electrodes, measured by annihilations on the scintillators,as a function of the initial number of antiprotons, measured by a current transformer in beam line(section 2.4.2). The best efficiency was achieved when the tuneable degrader had a thickness of(26± 1.3) µm.The “closing time”, the time between the beam extraction signal from the AD and switching theE01 voltage to trap antiprotons (see figure 2.10 in section 2.4.2), is also experimentally tuned andfigure 5.2 shows the antiproton capture efficiency as a function of the closing time. Note that there is1195.1. Capturing antiprotons and secondary electrons00.20.40.60.820 25 30 35 40p¯captureefficiency[%]Tuneable degrader thickness [µm]Figure 5.1: Measurement of the antiproton capture efficiency as a function of the thickness of thetuneable degrader. The efficiency was calculated from the incident number of antiprotons. Theoptimal thickness is (26± 1.3) µm.a constant offset, so a closing time = 0 does not mean the time of the antiproton arrival. The optimaltime is 430 ns, where the highest fraction of antiprotons are captured. If E01 is raised too late, thefastest antiprotons from the distribution reflected by the potential of E13 escape back towards theAD. If E01 is raised too fast, the voltage prevents a fraction of the antiprotons from even enteringthe capture region.The 5 kV voltage applied to E01 corresponds to the highest possible voltage that can be applied tothe electrodes and cabling without risking sparking or discharging. Figure 5.3 shows the measure-ments of the antiproton capture efficiency as a function of E01 voltage. As expected, we can seethat the number of captured antiprotons increases monotonically with the voltage applied.5.1.2 Secondary electronsElectrons provide cooling to the antiprotons and, in ALPHA, were produced by thermionic emissionfrom a filament and loaded into the trap before capturing the antiprotons (see section 2.4.2). Forthe measurements discussed in chapter 6, we used a new technique, where secondary electrons arecreated during the passage of the antiprotons during the degrading process. Antiprotons lose energy1205.1. Capturing antiprotons and secondary electrons00.20.40.6500 1000 1500 2000p¯captureefficiency[%]Closing time [ns]Figure 5.2: Antiproton capture efficiency as a function of the closing time. The optimal time is430 ns.00.20.40.60.80 1 2 3 4 5p¯captureefficiency[%]E01 voltage [kV]Figure 5.3: Antiproton capture efficiency as a function of E01 voltage.1215.1. Capturing antiprotons and secondary electronsin the degrader via excitation and ionization of the atoms. When an antiproton deposits energy thatexceeds the binding energy of an electron in an atom of degrader material, the electron is ejectedfrom the atom. The electron can escape the material into free space. In some cases, these secondaryelectrons have enough energy to produce further ionization [203].If the energy transferred by the antiproton to an atom is lower than the binding energy, electronsin low energy levels can be excited. When they dexcite, the energy is released in the form ofelectromagnetic radiation or Auger electrons [204]. Finally, if an antiproton annihilates, high energypions are produced which can also liberate electrons from the material.In summary, there are many ways that electrons can be ejected from the atoms to become part ofthe particle beam.Previously, production of secondary electrons was considered to be an undesirable effect. This isthe first time they have been used to cool antiprotons. This technique can be useful when using theantiproton capture trap as an accumulator, since electrons do not need to be loaded while antiprotonsare already present. Additionally, it can be useful as a backup if the electron gun is fails.Figure 5.4 shows an schematic of the antiproton and secondary electron capture, similar to fig-ure 2.10, which also shows the production and capture of secondary electrons.Figure 5.5a shows the ratio of the number of secondary electrons captured to the number of an-tiprotons extracted from the AD, Ne−/Np, as a function of the closing time. The time at which themaximum occurs is shorter than that for antiprotons alone (figure 5.2) because for the same energy,electrons have a higher velocity than antiprotons. Figure 5.5b shows Ne−/Np as a function of theE01 voltage, which only changes for voltages below about 2 kV.To measure the energy of the captured secondary electrons, a voltage was applied to E02 during theextraction of the antiprotons, to block the secondary electrons with the axial energy below a certainlevel. Ne−/Np as a function of the blocking voltage is shown in figure 5.5c. Almost all electronsare blocked at voltages above 10 V. Clearly, there is a contradiction between 5.5b and 5.5c. Almostall the secondary electrons are blocked at about 10 V, so it is not understood why only half of the1225.1. Capturing antiprotons and secondary electrons0 100 200 300 400 500z [mm]p¯p¯ + e−p¯ + e−p¯ + e−a)b)c)d)5 kVFigure 5.4: Schematic illustration of the antiproton and secondary electron capture. The blackcurve is the electric potential, the red dots are the antiprotons and the blue dots are the electrons.a) Antiproton beam before passing though the degrader (grey layer). b) After the antiprotons passthrough degrader, secondary electrons are created. c) Particles with an energy less than 5 keV arereflected by the high voltage electrode E13. d) The voltage in electrode E01 is then raised, and boththe antiprotons and secondary electrons are trapped.1235.1. Capturing antiprotons and secondary electrons00.511.52500 1000 1500 2000Ne−/Np¯Closing time [ns]00.20.40.60.810 1 2 3 4 5Ne−/Np¯E01 voltage [kV]00.20.40.60 10 20 30Ne−/Np¯Blocking voltage [V]a)b)c)Figure 5.5: a) The number of secondary electrons, normalized to the number of incident antiprotonsas a function of the closing time. The optimal time is 200 ns. b) The number of secondary electronsnormalized to the number of incident antiprotons as function of E01 at a closing time of 430 ns. c)The number of secondary electrons normalized to the number of incident antiprotons as a functionof the blocking voltage.1245.2. Antiproton coolingelectrons are trapped when E01 is at 0.5 keV.5.2 Antiproton coolingAfter capturing the antiprotons and the secondary electrons, we perform the cooling process. Ourgoal is to cool a large number of captured antiprotons to energies below a few electron-volts. Apotential well is constructed so that the antiprotons and electrons can accumulate in the minimumof the well. The electrons exchange energy with the antiprotons through Coulomb collisions andlose this energy via cyclotron radiation emission. The cooling efficiency depends on the time forwhich the particles are allowed to interact, the number of electrons, and the radial overlap betweenthe antiprotons and electrons.After the cooling has finished and a large number of antiprotons have been cooled, the high voltageon E01 is turned off, and any uncooled “hot” antiprotons escape to annihilate on the degrader. Thenumber of hot antiprotons is measured by detecting the particles resulting from the annihilation withthe scintillators. Figure 5.6 illustrates the cooling process with secondary electrons.We also varied the closing time to study how the antiproton cooling efficiency changes. The re-sults are shown in figure 5.2. For loaded electrons separately, the number of electrons is constant.However, for secondary electrons the number changes with the closing time, as can be seen in fig-ure 5.5. For loaded electrons, the antiproton cooling efficiency seems to be the same (40 %) at allvalues of closing time. For secondary electrons, the maximum cooling efficiency is around 90 %,but decreases for smaller numbers of secondary electrons (i.e. longer closing time). It is apparentthat the radial size of the plasma plays an important role in the antiproton cooling. We can infer thatthe high antiproton cooling efficiencies are possible because the electrons occupy the same volumeas the antiprotons. We estimated that the radial size of the secondary electron plasma is ∼ 4 mm(see section ). When electrons are loaded from the filament, there is no guarantee that the elec-tron plasma has the same radial size as the antiprotons, which can lead to lower antiproton coolingefficiency. Typical sizes of the loaded electron plasmas are ∼ 0.5 mm.1255.2. Antiproton cooling-200-1000100200Potential[V]a)-200-1000100200Potential[V]b)-200-10001002000 50 100 150 200 250 300Potential[V]z [mm]c)E01 E13hotFigure 5.6: Schematic illustration of the antiproton cooling process. a) Well potential produced bythe electrodes after the capture. A 100 V voltage on two electrodes is used to create the cooling well.The electrodes are illustrated at the top of the image. b) Antiprotons and electrons after the coolingtime. Cool antiprotons and electrons fall into the cooling potential well, but a few antiprotons remainhot. c) The voltage on E01 is turned off to release the remaining hot antiprotons, which strike thedegrader and their annihilation is detected.1265.3. Pulsed electron ejectionAll the measurements of antiproton cloud compression reported in chapter 6 of this thesis wereperformed with secondary electron cooling. About 1.5 × 105 antiprotons are cooled with about20 × 106 electrons participating in the cooling.0204060801001200 500 1000 1500 2000Coolingefficiency[%]Closing time [ns]a)0120 500 1000 1500 2000Numberofcoldantiprotons×105Closing time [ns]b)Secondary e−Loaded e−Secondary e−Loaded e−Figure 5.7: a) Antiproton cooling efficiency as a function of closing time, and b) number of coldantiprotons as a function of closing time for secondary electrons and loaded electrons.5.3 Pulsed electron ejectionBy taking advantage of the difference between the masses of antiprotons and electrons, we canremove a fraction, or all, of the electrons from the antiproton-electron plasma. Figure 5.8 shows aschematic of the electron ejection method. The process works by applying a short pulse (∼ 100 ns)to the voltage of one of the electrodes that confines the electrons. The well opens and a fractionof the electron plasma escapes. On the other hand, the antiprotons remain trapped since they areheavier and move much more slowly than the electrons. The short voltage pulse can be appliedseveral times, depending on whether a partial or total ejection of the electrons is needed.During the measurements of the antiproton cloud compression, we used the electron ejection methodto change the number of electrons after antiproton cooling while keeping the number of antiprotonsconstant. We used 20 × 106, 12 × 106, 7 × 106 and 4 × 106 electrons.1275.3. Pulsed electron ejection-60-300Potential[V]a)-60-300Potential[V]b)-60-300100 150 200Potential[V]z [mm]c)E01E13e−Figure 5.8: Schematic illustration of electron ejection while holding antiprotons. a) Potential wellproduced by the electrodes before the electron ejection. The electrodes are illustrated at the topof the figure. b) E03 (hatched pattern on the top) is pulsed. A fraction of the electron plasma hasenough time to escape from the well, while the antiprotons move too slowly to escape. c) Thepotential well is restored after the pulse. Most of the antiprotons are kept, while only a fraction ofthe electrons remain trapped.1285.4. Rotating wall application5.4 Rotating wall applicationFollowing removal of some of the electrons, the antiproton-electron plasma is moved to a positionnear the rotating wall electrode. The typical voltages applied to the electrodes and the resultingpotential well in the absence of the plasmas are shown in figure 5.9. The bare potential was chosento be harmonic near the centre of the potential. The harmonic approximation is shown as the redline. However, we will see later that, due to the electron space charge, the potential experienced byan individual particle is far from being harmonic.010203040-40 -20 0 20 40Potential[V]z [mm]E01 0.0 V 18.7 V 25.8 V 30.0 V 25.8 V 18.7 V 0.0 VE13DegraderFigure 5.9: The boxes on the top represent the electrodes and show the voltage applied to them. Theblack curve is the resulting potential and the red curve is a quadratic function that approximates thepotential in the central region, where it was intended to be harmonic.Note that the rotating wall electrode (in purple) is at one side of the centre of the potential well. Ex-perimentally, we have found that compression works better in this configuration than if the electrodeis centred on the cloud.Figure 5.10 shows the potential well including the space charge of the electrons, which is verydifferent from the vacuum potential (the potential without space charge).We typically apply the rotating wall for 100 s and the amplitude can be varied up to 10 V. At low1295.4. Rotating wall application2223242526272829-10 0 10Potential[V]z [mm]4 × 106 e−7 × 106 e−12 × 106 e−20 × 106 e−No space chargeE01 0.0 V25.8 V 30.0 V 25.8 V0.0 VFigure 5.10: Voltage applied to the electrodes and the resulting on axis potential for different num-bers of electrons. The potential is calculated by simultaneously solving the Poisson and Boltzmannequations. The purple electrode and hatched region illustrates the position of the rotating wall elec-trode. The black curve is the vacuum potential.1305.5. Detection and analysis of antiproton cloudsamplitudes, the generator itself produces low-quality waveforms. A 20 dB attenuator is used toprovide weaker drives, while keeping the generator amplitude high. These amplitudes refer to thevoltages at the input of the passive RC-filter. The signal then passes through the passive RC-filterand the electrode cabling, which attenuates the amplitude by an unknown factor.The rotating wall frequency is defined by a sweep with initial and final frequencies separated by0.2 kHz. These cannot be identical for technical reasons. During the measurements we usuallysweep from high to low frequencies. Reverse sweeps were also applied and the results found to beidentical. The width of the frequency sweep was chosen to be very small, so we can consider thefrequency to be constant to a precision of 0.1 kHz.5.5 Detection and analysis of antiproton cloudsAfter applying the rotating wall, the antiproton-electron plasma is “dumped” (ejected from the well)onto a micro-channel plate (MCP). For more information about the MCP/phosphor/CCD detectorassembly, refer to section 2.7.2. The dump is performed by slowly raising the potential (i.e. reducingthe depth of the well) on the right side of the well and letting the particles escape. Figure 5.11illustrates how the potentials change during the dump of the antiproton-electron plasma onto theMCP.When the particles are released, in the adiabatic limit the antiprotons and the electrons follow theaxial magnetic field lines, which are generated by the 3 T solenoid and a transfer solenoid placedbetween the trap and the MCP. Figure 5.12 shows the magnetic field along z. The magnetic fieldat the MCP is 0.042 T, so the particles move from a high to a low magnetic field region. Sincethe magnetic field lines diverge, the sizes of the particles’ orbits increase. From the third adiabaticinvariant, the total magnetic flux φ passing through a drift surface S is conserved for adiabaticprocesses [133]:φ =∫SB.ds = constant. (5.1)1315.5. Detection and analysis of antiproton clouds-200-1000100200Potential[V]a)-200-1000100200Potential[V]b)-200-10001002000 50 100 150 200 250 300Potential[V]z [mm]c)E01 E13p and e− to the MCPFigure 5.11: Schematic illustration of the particle dump to the MCP. a) Potential well before theprocedure. b) The potential is lowered before the ejection of the particles. c) The voltage on theelectrode holding the particles is raised to allow the particles escape. The particles are released tothe right, where they strike the MCP.1325.5. Detection and analysis of antiproton cloudsWe can deduce that the radius of the plasma changes as:rtrap = rMCP√BMCPBtrap, (5.2)where rtrap is the radius of the plasma in the trap region when the magnetic field is Btrap. rMCP isthe radius of the plasma in the MCP region when the magnetic field is BMCP. When showing MCPimages in this thesis, the scale bars are the size of the plasma inside the trap (3 T), calculated fromequation 5.2.130.0010.010.1-1 -0.5 0 0.5 1 1.5 2Magneticfield[T]z [m]MCPFigure 5.12: Magnetic field as a function of the axial position. The vertical red line is the MCPposition. z = 0 is the centre of the antiproton trap. A small solenoid is placed at z = 1.2 m to guidethe particles.We use this radius conversion for electron plasmas and antiproton clouds, even though it is not aperfect scaling since the adiabatic requirement may not always be fulfilled. This depends on themass of the particles. The electrons are tightly bound to the field lines and follow them very closely,contrary to the antiprotons, which are not so strongly bound to the field lines and experience acentrifugal drift in the region of low magnetic field [85]. Also, the magnetic field axis and theelectrode axis are not perfectly aligned, causing an off-set between the positions of the two particles.We always observe that the antiprotons are on the left side of the MCP image and electrons on the1335.5. Detection and analysis of antiproton cloudsright side, as shown in figure 5.13. If the alignment was perfect, the two species should overlap.Figure 5.13: MCP image of an antiproton-electron plasma after applying the rotating wall. Thelarge outer circle in the background of the image is due to a mechanical aperture. The antiprotoncloud is at the left hand side of the image, with an ellipse-like shape. Next to the antiproton cloud, atthe right hand side, the electron plasma has a circular shape. The scale bar is the size of the plasmainside the trap (3 T), calculated from equation 5.2.To extract information from the image, we performed a two dimensional Gaussian fit. This isimplemented in the ROOT Data Analysis Framework [205]. Since the antiproton cloud has anelliptical shape and has tilted axes, a fit of the following form was performed:B + np¯ exp(−(a (x − x0)2 + 2b(x − x0) (y − y0) + c (y − y0)2))(5.3)where B is the background on the MCP, np¯ is the height of the peak (central density) and (x0, y0) isthe centre of the cloud. The coefficients a, b and c are:a =cos2θ2σ2x+sin2θ2σ2y, (5.4)b = −sin2θ4σ2x+sin2θ4σ2y, (5.5)andc =sin2θ2σ2x+cos2θ2σ2y(5.6)where θ is the angle between the ellipse axis and the vertical axis and σx and σy are the semi-axial1345.5. Detection and analysis of antiproton cloudslengths of the cloud. There are a total of 7 free parameters.For electrons, the distribution detected is nearly circular and is fit well by a generalized Gaussian:B + ne−exp−12√( x − x0σ)2+√(y − y0σ)2k (5.7)where ne− is the central density of the electron plasma and k∼ 2. When an antiproton-electronplasma is detected, the fit with the sum of equation 5.3 and equation 5.7 is implemented. Figure 5.14shows an example of the resulting fit for the data across one line of the MCP image.Figure 5.14: a) MCP image of an antiproton-electron plasma after applying the rotating wall. Thelarge circle in the background of the image is due to a mechanical aperture. The antiproton cloudis on the left hand side of the image, with an ellipse-like shape. Next to the antiproton cloud, onthe right hand side, the electron plasma has a circular shape. b) Dots are the data from the radialprofile across the arrow shown in a) and the (red) curve is the respective fit. Position and sizes ofthe plasma are calculated from equation 5.2, so that they correspond to the size in a magnetic fieldof 3 T.To allow comparison of the images, the same voltages were always applied to the MCP, so the gainwas fixed and low enough to avoid saturation of the signal. It is known that the brightness on theMCP is linearly proportional to the charge [85]. We calibrate the MCP brightness using indepen-dent measurements of the number of particles. The electron number is measured with the FC (seesection 2.7.1) and the antiproton number is measured with the scintillators/PMTs (see section 2.7.3).For all measurements of the antiproton cloud compression we use the central density np¯, not the1355.5. Detection and analysis of antiproton cloudsradius, as a measure of the degree of compression. This is because for MCP images, when there isno significant compression and the plasma is larger than the mechanical aperture, the radius has tobe determined indirectly, while the central density can always be observed.However, even when the plasma is larger than the MCP mechanical aperture, it is necessary toestimate the radius of the plasma. For these cases, we use np¯ and the number of particles fromindependent measurements, to calculate the radius of the plasma.136Chapter 6Antiproton cloud radial compressionmeasurementsIn this chapter, we present data showing evidence of direct antiproton cloud compression.The rotating wall is applied to a mixed antiproton-electron plasma and the degree of compressionof the antiproton cloud is measured by imaging it on an MCP. As we will see, we can achieve goodcompression at frequencies in the range 100 kHz – 750 kHz.We will examine different aspects of the compression. We study the performance of the compres-sion as we vary the rotating wall frequency, the electron and antiproton number, the compressionduration, and the rotating wall amplitude, among others.6.1 History of the measurementsAs already discussed in section 4.4, in ALPHA, we typically use sympathetic compression whenreducing the radial size of the antiproton clouds. During the periods of antiproton beam, fromaround June to November each year, the ALPHA experiment receives about 8 hours of antiprotonbeam per day. During the beam time, one of our goals is to optimize the antiproton clouds forantihydrogen formation and trapping. Once this is successfully done, the beam time is dedicated toperform measurements on antihydrogen.Since the antiproton beam is scarce, the baseline and optimization of the positron plasma is per-formed a few hours before the antiproton beam time. Similarly, a study of the electron plasma1376.2. Observation of a new regime of antiproton cloud compressioncompression is performed, so that we can use optimal parameters when the antiproton beam arrives.The parameters do not change very much from one day to another.An advantage of sympathetic compression is that optimal parameters can be determined before thebeam time. One must do this carefully because sympathetic compression involves requirements,such as the compression speed, which need to be slow enough to allow the antiprotons to follow theelectrons. Some antiproton beam time can be used to check this.Since this compression technique worked well for ALPHA’s purposes, measurements of the antipro-ton cloud compression at frequencies where the electron plasma does not compress had not beencarried out.However, during the ALPHA-2 apparatus commissioning in November 2012, we had an exceptionalopportunity to study antiproton cloud compression in the antiproton capture trap, while the atom trapwas being commissioned. Inspired by the ALPHA collaborators Isaac et al. paper “Compression ofpositron clouds in the independent particle regime” [206], where positron clouds were compressedby applying the rotating wall close to the axial bounce frequency, we decided to attempt compressionof the antiproton clouds near the axial bounce frequency of the antiprotons (100 kHz range). Wenoted that the electron plasma does not compress in that frequency range.During this time, for technical reasons, the loaded electron number and size were fluctuating on aday-to-day basis. The use of secondary electrons had proven to be more stable, thus we used themduring the measurements of the antiproton cloud compression.Once the atom trap was commissioned, the use of the beam time was fully used to trap antiprotonsin the atom trap. Due to the scarcity of the antiproton beam, the data we were able to collect isrelatively limited compared to experiments that do not depend on an accelerator beam.6.2 Observation of a new regime of antiproton cloud compressionIn this section, we give quantitative evidence of a new regime of antiproton cloud compression.We first present the initial conditions, and then results of the compression of the antiproton clouds1386.2. Observation of a new regime of antiproton cloud compressionas a function of the rotating wall frequency. We performed these measurements when coolingwith different numbers of electrons. We also report on experiments using only electrons or onlyantiprotons.6.2.1 Initial conditionsFigure 6.1 shows a typical MCP image of the antiproton-electron plasma before applying the rotat-ing wall. The intensity of the image is low and the edges of the plasma are not visible (the circlein figure 6.1 is the result of a mechanical aperture), meaning that the radius of the plasma is largerthan the mechanical aperture. The aperture has a radius of about 1 mm in the trap.Figure 6.1: MCP image of an antiproton-electron plasma before applying the rotating wall. Thescale bar is calculated from equation 5.2, so that it corresponds to the size in a magnetic field of 3 T.We indirectly estimated the radius in the initial conditions from MCP images by using the centraldensity, and the number of particles measured by the Faraday cup. By inverting equation 5.7, wedetermined the radius to be 3.2 ± 1.5 mm. The error bar is the standard deviation of shot-to-shot fluctuations. Simulations of the antiproton degrading process predict that the radius is about4 mm [63].1396.2. Observation of a new regime of antiproton cloud compression6.2.2 Antiproton cloud radial compression for various frequencies and electronnumbersThe measurements were performed with 1.5 × 105 antiprotons and with different numbers of sec-ondary electrons: 4 × 106, 7 × 106, 12 × 106 and 20 × 106 electrons. The number of electrons isselected by executing the electron ejection procedure described in section 5.3, and by varying thenumber or strength of the pulses applied.Examples of MCP images are presented in figure 6.2, for different numbers of electrons. Eachimage shows the best antiproton compression observed when varying the rotating wall frequency,for different number of electrons. Comparing the images, one can observe that compression of theantiproton cloud varies with the number of electrons. The smallest clouds are those cooled by a largenumber of electrons (12 × 106 and 20 × 106 electrons). A striking difference between the imagesis the evident compression of the electron plasma when using high numbers of electrons (12 × 106and 20 × 106 electrons). About 15% of the electron plasma is being compressed in these images,with the rest appearing as a diffuse background. Such partial compression of the electron plasma isa qualitatively new behaviour since, as will be seen in section 6.2.3, a pure electron plasma does notcompress in the frequency ranges used. We deduce that this new behaviour is due to the presenceof the antiprotons.After showing how compression changes as a function of the electron number, we will discusscompression as a function of frequency. Figure 6.3 shows MCP images of antiproton compressionat different frequencies for different numbers of electrons. We observed, for different numbers ofelectrons, that at low frequencies (∼ 100 kHz), the compressions look very similar. As the frequencyincreases, compressions for 4 × 106 and 7 × 106 electrons seem similar (note that the frequenciesare different), while when using 12 × 106 and 20 × 106 electrons, the compression process clearlyproduces a compressed electron plasma next to the antiproton cloud. We can conclude that partialcompression of the electron plasma becomes evident around 300 kHz.A summary of antiproton cloud compression as a function of the rotating wall frequency and elec-tron number is shown in figure 6.4. Each point represents an independent measurement where we1406.2. Observation of a new regime of antiproton cloud compressionFigure 6.2: MCP images of an antiproton-electron plasmas after applying the rotating wall for 100 swith an amplitude of 1 V. Each image has about 1.5 × 105 antiprotons and different numbers ofelectrons providing cooling. The antiproton cloud has an ellipse-like shape at the left side of theMCP and the electron plasma has a round shape to the right, near the centre of the MCP. a) 4 × 106electrons at 140 kHz, b) 7 × 106 electrons at 300 kHz, c) 12 × 106 electrons at 700 kHz and d)20 × 106 electrons at 600 kHz. The scale bar is calculated from equation 5.2, so that it correspondsto the size in a magnetic field of 3 T.1416.2. Observation of a new regime of antiproton cloud compressionFigure 6.3: MCP images of antiproton-electron plasmas after applying the rotating wall for 100 swith an amplitude of 1 V. Each image has about 1.5×105 antiprotons with different numbers of elec-trons providing cooling. Row a) antiproton-electron plasma with 4 × 106 electrons when applyingthe rotating wall at different frequencies. Row b) antiproton-electron plasma with 7 × 106 electronswhen applying the rotating wall at different frequencies. Row c) antiproton-electron plasma with12 × 106 electrons when applying the rotating wall at different frequencies. Row d) antiproton-electron plasma with 20 × 106 electrons when applying the rotating wall at different frequencies.1426.2. Observation of a new regime of antiproton cloud compressionapply the rotating wall for 100 s, with an amplitude of 1 V at the appropriate frequency, and with awidth of 0.2 kHz. The central density is extracted from the fit, as described in section 5.5. The cen-tral density is a quantitative measure of the degree of compression of the cloud provided the numberof particles remains constant. In all of these trials, we never observe the rotating wall to inducea loss of antiprotons, as monitored by the scintillators. Therefore, if the central density increases,the cloud radius must become smaller. In figure 6.4, we show the central density as a function offrequency for several electron numbers, with fixed antiproton number. The error bars on the centraldensity are too small to be seen. The antiproton maximum central density increases with the elec-tron number. When cooling with 4 × 106 electrons, the antiproton maximum central density occursat ∼ 140 kHz, when cooling with 7 × 106 electrons the maximum is at ∼ 400 kHz and when coolingwith 12 × 106 and 20 × 106 electrons, the maximum is at ∼ 600 kHz.Furthermore, increasing the electron number allows better compression and over a broader rangeof frequencies. When cooling with 4 × 106 and 7 × 106 electrons, the antiproton cloud compressesover the range 50 – 200 kHz and 50 – 600 kHz respectively. When cooling with 12 × 106 and 20 ×106 electrons, very similar compression is produced and the antiproton cloud compresses between50 – 800 kHz. One might think that a higher number of electrons enhances the antiproton cloudcompression because it provides a better source of cooling. However, we cannot ignore the fact thata fraction of the electron plasma is itself compressed for trials with large numbers of electrons (seefigure 6.2c and 6.2d) and that this might also explain the different antiproton cloud compressionbehaviour.At the bottom of figure 6.4, we plot the central density of the electron plasma while compressingthe antiprotons. We do not plot the central density for 4 × 106 electron plasma, since there is eitherno electron plasma compressing or if it is compressing, it is too diffuse to be seen in the antiprotoncloud and the analysis cannot be performed.During the measurement period of 5 weeks, we performed a series of baseline measurements ondaily basis, where we checked the antiproton number, the secondary electron number and the com-pression of the antiproton cloud when applying the rotating wall at 500 kHz for 40 s, when cooled1436.2. Observation of a new regime of antiproton cloud compression03006009001200150018000 200 400 600 800 1000p¯centraldensity[a.u]01002003000 200 400 600 800 1000e−centraldensity[a.u]Rotating wall frequency [kHz]4 × 106 e−7 × 106 e−12 × 106 e−20 × 106 e−a)b)Figure 6.4: Central density after applying the rotating wall for 100 s, at 1 V and at a chosen fre-quency. a) Antiproton central density when cooled by different number of electrons. b) Electroncentral density from the same measurements in a). No points are shown for 4 × 106 electrons, be-cause the electron plasma was not visible and a fit could not be performed. The arbitrary units (a.u.)are the same for antiprotons and electrons.1446.2. Observation of a new regime of antiproton cloud compressionby 20 × 106 electrons. Compression baselines were always satisfactory and MCP images showingcompression were reproducible. We observed that the secondary electron number may increase ifthe background pressure of the trap increases. This problem was solved by thermally cycling thetrap (allowing the trap to warm to ∼ 100 K and cooling again).6.2.3 Electron plasma compression without antiprotonsHaving observed compression of mixed antiproton-electron plasmas, it was important to study theradial behaviour of the electron plasma when applying the rotating wall, particularly when the elec-tron plasma was alone, without antiprotons. This measurement was key to deciding if the antiprotoncloud was sympathetically compressed by the electron plasma or not.The measurement procedure was almost the same as the one with an antiproton-electron plasma,but instead of cooling for 80 s, the particles were cooled for only 5 s. After 5 s, almost all of theantiprotons were still hot, thus they were released by turning off E01 and only the electrons kept inthe trap. It is important to note that the cyclotron cooling time for the electrons is ∼ 0.4 s.Figure 6.5 shows examples of MCP images of the electron plasma before and after compression.Before compression, the radial size of the plasma is estimated to be3.2± 1.5 mm (see section 6.2.1)and after compression at 1.5 MHz, is 0.14± 0.01 mm.Figure 6.5: a) MCP image of the electron plasma before applying the rotating wall. b) MCP imageof the electron plasma after applying the rotating wall for 100 s, with an amplitude of 1 V and at1.5 MHz.1456.2. Observation of a new regime of antiproton cloud compressionThe peak density of the electron plasma as a function of the rotating wall frequency is shown infigure 6.6. We observe that below ∼ 750 kHz, there is no radial compression of the electron plasma,but above ∼ 750 kHz, the plasma begins to compress.02004006000 200 400 600 800 1000 1200 1400 1600e−Centraldensity[a.u]Rotating Wall Frequency [kHz]Figure 6.6: Central density of an electron plasma of about 20 × 106 electrons as a function of therotating wall frequency. The rotating wall was applied for 100 s with an amplitude of 1 V. Theplasma radially compresses above 750 kHz.To closely observe the initial compression of the electron plasma, we show the MCP images of theelectrons after compressing at 700 kHz, 800 kHz and 900 kHz (see figure 6.7). At 700 kHz, there isno sign of compression, but at 800 kHz, we observe that the plasma is a little more dense and thecompression continues for higher frequencies.6.2.4 Compression of antiproton cloud without electronsWe also performed measurements of the antiproton cloud compression without electrons. The elec-trons were ejected from the trap before applying the rotating wall. Figure 6.8 shows the MCPimages after applying the rotating wall at different frequencies. There is no evident compression ofthe antiproton cloud. Furthermore, we detected an abnormal loss of about 30% of the antiprotonswhile applying the rotating wall. We think they might be lost due to heating by the electric fieldperturbation, but with no source of cooling to counteract this effect.1466.2. Observation of a new regime of antiproton cloud compressionFigure 6.7: MCP image of the electron plasma after the rotating wall for 100 s with an amplitude of1 V at a) 700 kHz, b) 800 kHz and c) 900 kHz. The scale bars give the size the plasma would havein a 3 T magnetic field.6.2.5 Compression at higher frequencies (ωRW/2pi > 750 kHz)As already seen in section 6.2.3, a large fraction of the plasma (∼ 8 × 106 electrons) radially com-presses for a rotating wall frequency above ∼ 750 kHz. This drastically changes the potential welland dominates the compression of the antiproton cloud. A comparison between the MCP imagesat 600 kHz and 1 MHz shows two different behaviours of the antiproton-electron plasma (see im-ages 6.9). We can see at 1 MHz how the electron plasma compresses to a higher density and howthe antiproton cloud follows with a lower density. At this frequency, we think that the electrons aresympathetically compressing the antiprotons. The poor sympathetic compression of the antiprotoncloud may be because the electrons compress so quickly that they do not efficiently compress theantiprotons.Table 6.1 shows the densities and the numbers of particles that have been concentrated near thecentre at 600 kHz and 1 MHz. Only about 10% of the antiproton cloud is radially compressed at 1MHz. Also, a larger fraction of the electron plasma radially compresses at 1 MHz.1476.2. Observation of a new regime of antiproton cloud compressionFigure 6.8: MCP images of antiproton clouds after applying the rotating wall for 100 s and 1 V fordifferent frequencies.ωRW/2pi p¯ density p¯ number e− density e− number600 kHz ∼ 1 × 108 cm −3 ∼ 1.5 × 105 ∼ 3 × 109 cm−3 ∼ 3 × 1061 MHz ∼ 6 × 106 cm −3 ∼ 1.5 × 105 ∼ 3 × 109 cm−3 ∼ 8 × 106Table 6.1: Number of particles and densities of the plasmas radially compressing at a specific rotat-ing wall frequency (measurements performed with 1 × 105 antiprotons and 20 × 106 electrons).1486.2. Observation of a new regime of antiproton cloud compressionFigure 6.9: a) MCP image of an antiproton-electron plasma radially compressed by the rotating wallat 600 kHz and b) at 1 MHz. Both measurements were performed with ∼ 1.5 × 105 antiprotons and∼ 20 × 106 electrons. In a), the antiproton cloud is at the left hand side of the MCP image and nextto the antiproton cloud, the electron plasma is at the right hand side near the centre of the MCP. Inb) the antiproton cloud is not very visible at left hand side of the image. The electron plasma has acircular shape near the centre of the MCP image.6.2.6 Observation of a new mechanism?We have observed compression of antiproton clouds when applying the rotating wall in the fre-quency range of 50 – 750 kHz. The radius of the antiproton cloud was reduced by up to a factor 20and the smallest radius measured was ∼ 0.2 mm (central density of 1 × 108 cm−3).We have also observed that a pure electron plasma does not compress in the frequency range ofinterest and as a consequence it is very unlikely to induce sympathetic compression of antiprotonclouds. We have also observed that the use of an electron plasma is necessary for the compressionto be effective.When compressing the antiproton clouds, it is evident that a fraction of the electron plasma (about15%) also compresses when using 12 × 106 and 20 × 106 electrons. Such an effect is not evidentwhen using fewer electrons (4 × 106 and 7 × 106 electrons).We believe that the rotating wall is acting directly on the antiproton clouds and that we have observeda compression mechanism that is different from sympathetic compression [63] (see section 4.4)and also different from the compression reported by the ASACUSA Collaboration [57] (see sec-tion 4.4.1). Sympathetic compression is unlikely to be the mechanism of compression, since a pure1496.3. Other measurements of the antiproton cloud compressionelectron plasma does not compress when using the frequency range of interest. On the other hand,the compression observed here is distinguished from that reported by the ASACUSA Collaboration,since we used electrons as a source of cooling and observed a new behaviour in the compressionof the antiproton clouds that depends on the number of electrons. In contrast, ASACUSA reportedthat they did not have any electrons present while the rotating wall was applied.In chapter 7, we will discuss possible compression mechanisms that could explain the observedantiproton cloud compression. In the next section, we further investigate the antiproton cloud com-pression as a function of, for example, compression time and the potential well used.6.3 Other measurements of the antiproton cloud compression6.3.1 Compression as a function of duration of the rotating wallHere we vary duration of the rotating wall time, with the frequency and amplitude fixed. Thesemeasurements allow us to observe how the antiproton cloud compression evolves over time. Fig-ure 6.10 shows the antiproton central density when being cooled with 20 × 106 electrons at severalfrequencies.When cooling with 20 × 106 electrons, we can see that the antiproton cloud compresses slowly atfirst, but after ∼ 10 s, speeds up until levelling out between 20 – 40 s. Also, the antiproton cloudcompresses to higher central densities at the higher frequencies (e.g. 350 kHz, 450 kHz and 550kHz). This can also be seen in figure 6.4.Figure 6.11a shows the MCP images at different times at 150 kHz, while figure 6.11b shows theMCP images at 450 kHz. We observe for both frequencies that during the first few seconds there isvery slow compression. We also notice that the antiproton cloud reaches higher densities at 450 kHzand that the electron behaviour is different. We can barely see the electron plasma compressing atthe right hand side of the antiproton cloud at 30 s and 40 s when applying the rotating wall at 150kHz (figure 6.11a). In contrast, at 450 kHz (figure 6.11b), electron plasma compression is clearly1506.3. Other measurements of the antiproton cloud compression0300600900120015000 10 20 30 40p¯centraldensity[a.u]Rotating wall time duration [s]150 kHz200 kHz350 kHz450 kHz550 kHzFigure 6.10: Central density of the antiproton cloud after applying the rotating wall at 1 V at differentfrequencies as a function of the rotating wall time when cooled by 20 × 106 electrons.observable after 17.5 s. From the fit to the compressed electron plasma, we conclude that there areabout 3×106 electrons. It seems that the rest of the electron plasma remains in a larger, more diffusedistribution.In figure 6.12, we show the central density as a function of time when cooling with 4×106 electrons.It takes a longer time (∼ 20 s) before a noticeable compression begins. Also, the antiproton clouddoes not compress as much as in the previous case. Some of the points at 140 kHz do not follow thetrend (the points at 70 s and 100 s), which may indicate that the compression is unstable. We didnot have enough time to investigate this fully.Figure 6.13 shows the MCP images of the antiproton-electron plasma when applying the rotatingwall at 140 kHz (with 4 × 106 electrons). There is no evident compression of the electron plasma,but we can not rule out that there could be a small number of electrons next to the antiproton cloud.We can see at 90 s, that the compressed antiproton cloud is in a different position, which we attributeto a diochotron instability [80]. We observed such position changes when using 4× 106 and 7× 106electrons (figure 6.3), but not in other data sets. Diochotron motion has been often observed in1516.3. Other measurements of the antiproton cloud compressionFigure 6.11: a) MCP images after compression at 150 kHz for 1.5 × 105 antiprotons and 20 × 106electrons at different times of compression. b) Same as in a) but at 450 kHz.1526.3. Other measurements of the antiproton cloud compression02004006000 10 20 30 40 50 60 70 80 90 100p¯centraldensity[a.u]Rotating wall time duration [s]100 kHz140 kHz150 kHz170 kHz200 kHzFigure 6.12: Central density of the antiproton cloud after applying a 1 Volt rotating wall at severalfrequencies as a function of the rotating wall duration. An electron plasma of 4 × 106 electrons wasused to cool the antiprotons.ALPHA, often induced by rapid changes in the trap such as when the plasma is dumped to theMCP.From these observations, we conclude that the rate of compression of the antiproton cloud dependson the electron number. Increasing the number of electrons causes the antiproton cloud to compressmore quickly and to higher densities.6.3.2 Compression when changing the potential wellAntiproton cloud compression was also performed using two other potential wells. Table 6.2 showsthe electrode voltages used during the measurements. Potential A is the typical potential with aharmonic shape near the minimum with ωb/2pi ∼ 270 kHz, potential B is a non-harmonic potentialand potential C is a steeper harmonic potential with a bounce frequency of ∼ 370 kHz. The objectiveof these measurements was to determine whether compression could be achieved using other kindsof potentials and to check whether there is anything special with potential A, or not.1536.3. Other measurements of the antiproton cloud compressionFigure 6.13: MCP images after compression at 140 kHz for 1.5 × 105 antiprotons and 4 × 106electrons at different times of compression.Potential E04 E05 E06 (RW) E07 E08 E09 E10A 0 V 18.7 V 25.8 V 30 V 25.8 V 18.7 V 0 VB 0 V 15 V 30 V 30 V 30 V 15 V 0 VC 0 V 37.5 V 51.6 V 60 V 51.6 V 37.5 V 0 VTable 6.2: Voltages applied to the electrodes producing the potential well during application of therotating wall. Potential A is the typical potential used during the measurements, potential B is anon-harmonic potential (even without charge), and potential C is a deeper potential.1546.3. Other measurements of the antiproton cloud compressionFigure 6.14 shows the resulting potential for case A, B and C. The shapes of the potentials with andwithout space charge are shown.-10-5051015-10 0 10Potential[V]z [mm]Potential A: No space chargePotential B: No space chargePotential C: No space chargePotential A: 4 × 106 e−Potential B: 4 × 106 e−Potential C: 20 × 106 e−E06 (RW) E07 E08Figure 6.14: Voltages applied to the electrodes and the resulting potential well for different numberof electrons. The hatched region illustrates the rotating wall position. Potential A is the typicalpotential, potential B is the non-harmonic potential and potential C is a deeper well. The potentialwells are offset so that the potential maxima without space charge are set at 0 V. For the voltagesapplied to the electrodes, see table 6.2.Figure 6.15 shows the antiproton central density as a function of the rotating wall frequency whenusing different potential wells. One can observe that antiproton cloud is compressed regardless ofthe potential well used. When using 4 × 106 electrons with potential A and B, the antiproton cloudcompression looks the same. We can deduce that potential A and B are not different enough to makea change in the compression. For an unknown reason, when using 20×106 electrons, the antiprotoncloud does not compress at frequencies lower than ∼ 200 kHz for potentials B and C. Because of thelack of the antiproton beam time, the set of measurements of compression of the antiproton cloudswith potential B is incomplete. When using potential C, the compression of the antiproton cloudsseems shifted to higher frequencies by ∼ 100 kHz.We have observed that compression is achieved regardless of the potential shape. Compression ina steeper potential well (higher axial bounce frequency) is slightly different as a function of the1556.3. Other measurements of the antiproton cloud compression030060090012001500180021000 200 400 600 800 1000p¯centraldensity[a.u]Rotating wall frequency [kHz]Potential A with 4 × 106 e−Potential B with 4 × 106 e−Potential A with 20 × 106 e−Potential B with 20 × 106 e−Potential C with 20 × 106 e−Figure 6.15: Antiproton cloud central density as a function of the rotating wall frequency whenusing different potential wells and electron numbers for cooling. For more information about thepotentials, see table 6.2.rotating wall frequency.6.3.3 Compression as a function of the applied rotating wall amplitudeThe compression of the antiproton cloud was also measured using different rotating wall amplitudes.Figure 6.16 shows the results when cooling the antiprotons with 4 × 106 electrons. Higher voltagesspeed up the compression. For amplitudes used in the preceding sections of 1 V, compressionbecame visible after 20 s. When the amplitude was increased to 1.5 V, the compression sped up andbecame visible after 10 s. When the amplitude was reduced, it took up to 50 s to see a compressionat 0.5 V.6.3.4 Compression for various numbers of antiprotons and electronsWe wanted to know if the number of antiprotons changes the compression, and moreover, if thereis a minimum number of antiprotons needed to achieve compression. For these measurements, we1566.3. Other measurements of the antiproton cloud compression02004006008000 20 40 60 80 100p¯centraldensity[a.u]Rotating wall time [s]0.50.71 V1.5 VFigure 6.16: Antiproton cloud central density as a function of the rotating wall time for severalvoltages. The measurement was performed when cooling with 4 × 106 electrons and at a frequencyof 140 kHz.changed the number of antiprotons in the beam. This required of changing the intensity of the protonbeam in the Proton Synchrotron, so the production of antiprotons is reduced. As a consequence, thenumber of secondary electrons is also reduced. In these measurements, we are changing both thenumber of cooled antiprotons, Np, and the number of secondary electrons, Ne− . We used roughly25%, 50% and 100% of the usual number of antiprotons. We also obtained 200% by accumulatingtwo pulses of the antiproton beam.Figure 6.17 shows examples of MCP images of the antiproton-electron plasma after compression forfrequencies where the central density is highest. Table 6.3 gives more information on the parametersused during these measurements. We use the ratio Ne−/Np to provide information on the numbersof the electrons providing cooling versus the number of antiprotons.Figure 6.18 shows the antiproton central density for different numbers of antiprotons and electrons.As expected, one can see that the antiproton central density increases at higher numbers of antipro-tons. However, the electron number also changes so this may show the importance of the electrons’role as a coolant.1576.3. Other measurements of the antiproton cloud compressionFigure 6.17: MCP images of antiproton-electron plasmas after applying the rotating wall for 100 swith an amplitude of 1 V for a) 6 × 103 antiprotons and 7 × 106 electrons at 600 kHz, b) 9 × 104antiprotons and 12 × 106 electrons at 600 kHz, c) 1.5 × 105 antiprotons and 20 × 106 electrons at600 kHz and d) 3.3 × 105 antiprotons and 30 × 106 electrons at 500 kHz.Beam percentage p¯ Secondary e− Ne−/Np25% 6 × 103 7 × 106 11750% 9 × 104 12 × 106 133100 % 1.5 × 105 20 × 106 133200 % 3.3 × 105 30 × 106 91Table 6.3: Characteristics of the beam when using different beam percentages. 100% correspondsto 1 stack of the beam, and 200 % corresponds to a stack of 2 beam pulses. Ne−/Np is the ratiobetween the electron and the antiproton number.1586.3. Other measurements of the antiproton cloud compression0300600900120015001800210024000 200 400 600 800p¯centraldensity[a.u]Rotating wall frequency [kHz]Ne−/Np = 117,Np = 6 × 103Ne−/Np = 133,Np = 9 × 104Ne−/Np = 133,Np = 1.5 × 105Ne−/Np = 91,Np = 3.3 × 105Figure 6.18: Central density of the antiproton cloud as a function of the rotating wall frequency fordifferent numbers of electrons and antiprotons. For more information about the particle number, seetable 6.3.6.3.5 Expansion after the rotating wall finishesIt is important to study how the plasma expands after we stop the rotating wall. Ideally, for furtheruse in the experiment, one might hope that the plasma would maintain a small radius and expandslowly. If the plasma expands too quickly, it would be difficult to perform the further manipulationsneeded to produce antihydrogen. It is required that the antiproton cloud remains compressed withlittle expansion for at least 10 s.Figure 6.19 shows how the antiproton cloud expands when cooling with 4 × 106 and 20 × 106electrons after applying the rotating wall for 40 s or 100 s (see fig. 6.19). One can see that byabout 10 s after the rotating wall is stopped, the compression has already been undone. When using20 × 106 electrons, it does not matter for how long the rotating wall is applied (40 s or 100 s) sincethe central density has already reached a maximum, as seen in section 6.3.1. However, it seems thatapplying the rotating wall for 100 s helps to reduce the rate of expansion after stopping the rotatingwall. When cooling with 4 × 106 electrons, the rate of expansion looks about the same, regardless1596.3. Other measurements of the antiproton cloud compressionof whether it was compressed for 40 s or 100 s.02004006008001000120014000 5 10 15p¯centraldensity[a.u]Waiting time after rotating wall [s]20 × 106 e−, RW for 40 s20 × 106 e−, RW for 100 s4 × 106 e−, RW for 40 s4 × 106 e−, RW for 100 sFigure 6.19: Antiproton cloud central density as a function of the time after the rotating wall wasapplied. The rotating wall was applied either for 40 s or for 100 s. The antiprotons are cooled by4 × 106 or by 20 × 106 electrons.These measurements were performed without ejecting the electrons after application of the rotatingwall, so that expansion is observed for both species. Figure 6.20 shows the MCP images of theexpansion of the antiproton-electron plasma after applying the rotating wall. We observe that bothspecies expand. The fraction of the electron plasma that is not compressed (∼ 85% of the electrons)has a very large radius, and we think that after the rotating wall is applied, the antiproton cloud hasa tendency to redistribute towards the outside of this distribution.In future manipulations, it would seem more appropriate to eject the electron plasma after applyingthe rotating wall, so that the antiproton cloud compression will not be influenced by the electronplasma.6.3.6 Plasma temperature before and after compressionIt was not possible to perform more than few measurements of plasmas temperatures due to therestricted availability of the antiproton beam time. We measured the electron temperature after1606.3. Other measurements of the antiproton cloud compressionFigure 6.20: MCP images after stopping a 100 s rotating wall, for 1.5×105 antiprotons and 20×106electrons.applying the rotating wall to an antiproton-electron plasma. However, the temperature diagnosticof the antiprotons was not performed since the space charge of the electrons changes the shape ofthe potential well and the measurements are not easily interpreted (unless of course the electronsare ejected before the temperature measurements, but this can in turn affect the temperature). Weperformed measurements of the electron temperature when using 20 × 106 electrons and 1.5 ×105 antiprotons after applying the rotating wall for 100 s. Table 6.4 shows the temperature ofthe electrons after applying the rotating wall. We observe that for frequencies in the range 100 –500 kHz, the temperature is ∼ 500 K and at 700 – 900 kHz the temperature increases to ∼ 700 K.The temperature before applying the rotating wall is ∼ 300 K.6.3.7 Compression as a function of magnetic fieldSince the plasma rotation frequency depends inversely on the magnetic field (equation 4.8), wethought it could be useful to study how the compression of the antiproton clouds changes whenincreasing the magnetic field.Figure 6.21 shows example MCP images of the antiproton-electron plasma after compression when1616.3. Other measurements of the antiproton cloud compressionωRW/2pi [kHz] Temperature [K]100 557300 539450 580500 561700 758900 727Table 6.4: Temperature of 20 × 106 electrons after applying the rotating wall for 100 s at differentfrequencies.using 3 T and 4 T magnetic fields ( 3 T is the typical magnetic field used in the measurements so fardiscussed). The antiproton central density as a function of the rotating wall frequency is shown infigure 6.22. Unfortunately, we can not directly compare the two measurements because the captureand the initial conditions of the antiproton-electron plasmas are different. At 4 T, there is about1.9 × 105 antiprotons and 20 × 106 electrons, meaning that Ne−/Np is different and the resultingcompression could be influenced by the electron cooling, as previously observed. Also, since themagnetic field at the MCP position is slightly higher, the extraction properties of the particles isdifferent.Figure 6.21: MCP images of the antiproton-electron plasma after applying the rotating wall at500 kHz, with an amplitude of 1 V and for 100 s. The magnetic field is a) 3 T and b) 4 T.We conclude that there is still compression at 4 T, but there are too many changes of the initial1626.4. Summary of observations03006009001200150018000 200 400 600 800p¯centraldensity[a.u]Rotating wall frequency [kHz]3 T4 TFigure 6.22: Antiproton cloud central density as a function of the rotating wall frequency at 3 T and4 T. The antiprotons are cooled by 20 × 106 electrons.conditions of the plasma (particle number, size, rotation frequency) to make a useful quantitativecomparison.6.4 Summary of observationsWe will now present a list of key observations of the antiproton cloud compression:• Antiproton cloud compression is achieved when cooled by 4 × 106, 7 × 106, 12 × 106 and20×106 electrons in the range of hundreds of kHz. The rotating wall is applied for 100 s withan amplitude of 1 V.• The antiproton cloud central density increases when the number of electrons is increased.• The antiproton cloud radius before compression is estimated to be ∼ 4 mm, and measured tobe ∼ 0.2 mm after compression.• When increasing the electron number, the antiproton cloud is compressed over a wider range1636.4. Summary of observationsof frequencies.• When using 12 × 106 and 20 × 106 electrons, antiproton cloud compression is observed overa range of 50 – 750 kHz and the compressions are very similar. An evident partial electronplasma compression is observed on the MCP after the rotating wall is applied. About 3× 106of the electrons are compressed, while the rest seem to retain their initial conditions.• When using a smaller number of electrons, 7×106 electrons, the antiproton cloud compressesover a range of frequencies 50 – 500 kHz. The partial compression of the electron plasma isnot so evident, but there is a shadow next to the antiproton cloud that resembles the electronplasma.• When using 4 × 106 electrons, the antiproton cloud compresses at 50 – 200 kHz. In this case,the electron plasma seems to retain their initial conditions. We cannot rule out that someelectrons may be compressing, but they are not dense enough to be seen.• The temperature after compression for 20 × 106 electrons is ∼ 500 K at 100 – 500 kHz, in-creasing to ∼ 700 K at 700 – 900 kHz, where the antiproton cloud is no longer compressing.The temperature before applying the rotating wall is ∼ 300 K.• The electron plasma (without antiprotons) compresses when applying the rotating wall at afrequency above ∼ 750 kHz, but not at the lower frequencies where antiproton compressionis observed.• No compression is observed when applying the rotating wall to the antiprotons alone.• When using 20 × 106 electrons, the antiproton cloud compresses very slowly during the firstfew seconds ( 0 – 10 s), and then the rate of compression increases until saturating after ∼ 40 s.• When using 4 × 106 electrons, the antiproton cloud compresses very slowly during the first20 s.• When using 4×106 electrons, the compression is sped up when using a higher amplitude (1.5V) and slowed down when using lower amplitude (0.7 and 0.5 V).1646.5. Comparison of the direct compression of antiproton clouds versus sympathetic compression• Compression is achieved independent of the shape of the potential.• The antiproton cloud compression is undone within 10 s after stopping the rotating wall, atleast when the electrons are present.• When the antiproton number is increased, the central density is higher.• Increase of the magnetic field from 3 T to 4 T changed the compression behaviour at rotatingwall frequencies above about 500 kHz.6.5 Comparison of the direct compression of antiproton cloudsversus sympathetic compressionIn this section we discuss the advantages and disadvantages of using direct compression of an-tiproton clouds and secondary electrons in an experiment such as ALPHA. First, we compare thecompression method, regardless of the origin of the electrons, then we compare the use of secondaryelectrons and electrons from a source.Advantages and disadvantages of direct compression of antiproton clouds (versussympathetic compression)Advantages• In the sympathetic compression scheme, the antiproton clouds are compressed only indirectlyand therefore compression relies critically on the coupling between the electrons and theantiprotons. As such, it occasionally exhibits instabilities, due possibly to the electron initialconditions and the speed of the electron plasma compression. For example, if the electronplasma is compressed too rapidly, the antiproton cloud can be left uncompressed. Also, itwas occasionally observed that the antiproton clouds are not fully compressed, sometimesforming a halo around a partially compressed antiproton cloud.1656.5. Comparison of the direct compression of antiproton clouds versus sympathetic compressionDirect compression, on the other hand, does not rely on the coupling between the electronsand the antiprotons (at least with small electron numbers). Therefore it is expected to be morerobust, regardless of the origin of the electrons. In fact, during the work reported here, wehave observed that direct compression of antiproton clouds have been always reproducible atthe same frequencies.• While direct compression uses rotating wall frequencies in the range of a few 100 kHz, sym-pathetic compression uses higher rotating wall frequencies (a few MHz), which are moresusceptible to phase shifts and attenuation that varies between the channels depending ontechnical variations in the electrical circuits. In ALPHA, the circuits were only guaranteed topass frequencies below around 10 MHz with good fidelity. This could lead to defects in therotating field, reducing the compression speed or the limit.Disadvantages• Electron ejection after direct compression can be difficult since the strength of the electronpulse has a radial dependence and the electron plasma is radially large. However, this couldin principle be fixed by quickly compressing the electron plasma before the ejection. Electronejection is of great importance because without ejection, the antiproton cloud will expand veryfast to the size of the electron plasma. In this thesis we did not have the chance to performmeasurements with electron ejection.• If secondary electrons are not available (such as in the atom trap where there is not a degrader),an electron source can be used, but the radial size and density of the electron plasma must betuned to similar initial conditions as when using secondary electrons.1666.5. Comparison of the direct compression of antiproton clouds versus sympathetic compressionAdvantages and disadvantages of secondary electrons (versus electrons from asource)Advantages• One of the advantages of using secondary electrons is that the plasma has a big radius and thusa higher antiproton cooling efficiency can be achieved. Initial antiproton clouds have a radiussimilar to secondary electrons plasma radius, whereas when cooling with electrons from thesource (described in section 2.6), the cooling efficiency is limited by the poor radial overlap.In principle, this characteristic can be reproduced with electrons loaded from the electronsource by using the rotating wall to expand the plasma, but the plasma size and electronnumber is not always stable and the electron’s trajectory can be blocked by a mechanicalaperture.• In some cases, the use of secondary electrons can be advantageous because they can be cap-tured and trapped in one end of the trap, while positrons or antiprotons are already trapped inthe other end of the trap. This would not be possible with electrons from a source since theymust go though the entire trap.• The secondary electron plasma has a low density, which appears to be necessary for directcompression of the antiproton clouds.Disadvantages• Electrons from a source could have provide additional flexibility to investigate compressionprocess. With secondary electrons it is not possible, for example, to increase the number ofelectrons.• The use of secondary electrons could be inconvenient when the antiproton beam is fluctuatingsince the secondary electron number also changes.• The number of secondary electrons increases when the vacuum in the apparatus is poor.1676.5. Comparison of the direct compression of antiproton clouds versus sympathetic compressionFor antihydrogen experiments like ALPHA, where the goal for the antiproton capture procedure isto cool the highest number of antiprotons possible and to compress the cloud to be small enoughto transfer to the atom trap, using a sequence that is as robust and as free of complicated steps aspossible, direct compression has the potential to be the method of choice.168Chapter 7Compression mechanisms andcalculation of the plasma frequenciesassociated with the bounce resonanttransport of antiprotonsIn this chapter, we will discuss the possible mechanisms that could potentially explain the antiprotoncloud compression. First of all, we will calculate the plasma parameters to determine if the antipro-tons and electrons are in the plasma regime. As already discussed, compression can be achievedfor plasmas, as well as for particles in the single particle regime. We will also investigate if dio-cotron modes can be excited in the kHz range, since there are indications we have observed thisphenomenon for plasmas with 4 × 106 and 7 × 106 electrons.We first discuss the possibility that the antiproton cloud compression is due to a resonance withthe electron plasma. We will determine the Trivelpiece-Gould frequency and then discuss othermechanisms of compression, such as sympathetic compression and the strong drive regime. Later,we will discuss magnetron side band cooling and bounce resonant transport on the electrons and onthe antiprotons.After discussing possible compression mechanisms, we concentrate on the hypothesis that the an-tiproton cloud compresses by bounce resonant transport since it seems that to be the most likelymechanism of compression. To study the bounce resonant transport of antiprotons, we numerically1697.1. Radial compression mechanismscalculate the axial bounce frequency distribution of the antiprotons and the rotation frequency of theantiproton-electron plasma. The sum of these two frequencies gives a frequency distribution of thesupposed resonance, which we can compare to the frequencies where compression was observed.We study two cases. The first is when the antiprotons are cooled by 4 × 106 electrons. The axialbounce and rotation frequency can be calculated straightforwardly because there is no evidence ofcompression of the electron plasma, so we assume that the electron plasma does not change whilethe rotating wall is applied. We then consider cooling with 20 × 106 electrons. This is a morecomplex case because the electron plasma partially compresses over time and in a way that varieswith the rotating wall frequency. The partial compression of the electron plasma changes both theelectrostatic potential and the rotation frequency of the system.7.1 Radial compression mechanismsCompression of single component plasmas has been extensively studied both theoretically [172,184] and experimentally [168, 189, 190, 195, 206].In this chapter, we intend to explain compression of an antiproton cloud embedded in an electronplasma. We believe that the rotating wall acts directly on the antiprotons. The challenge lies inthe fact that we need to demonstrate that the electron plasma is not excited during the antiprotoncloud compression, even though the electron plasma space charge dominates the rotational motionand distorts the potential well. Systems with two species have been studied, but in the independentparticle regime [206, 211, 212] and never in the current situation, where the antiproton cloud motionis affected by the space charge of the electrons.7.1.1 Plasma frequencyAs already seen in section 4.2.1, a collection of charges is a plasma if it fulfills the condition :λDL 1, (7.1)1707.1. Radial compression mechanismsNumber Temperature Density Plasma radius Plasma length λD ωp/2pielectrons [K] [cm−3] [mm] [mm] [mm] [MHz]4 × 106 500 1 × 107 4 12 0.5 2820 × 106 500 3 × 107 4 30 0.3 49Table 7.1: Electron plasma parameters. λD is calculated from equation 4.3 and ωp/2pi is calculatedfrom equation 4.2.where λD is the Debye length and L is the length of the plasma. Here we have 20 × 106 electronswith a radius rp ∼ 4 mm, and a length L∼ 30 mm giving a density of ∼ 3 × 107 cm−3. Thus for T =500 K, λD ∼ 0.3 mm. Therefore, L ∼ 100λD and rp ∼ 13λD. λD is much smaller than the length andthe radius so it can be considered as a plasma.Table 7.1 shows some parameters of the electron plasma.7.1.2 Diocotron modeThe diocotron mode is not intentionally used for the radial compression of plasmas when applyingthe rotating wall, but it is helpful to be aware if diocotron effects can be excited during the compres-sion. The diocotron modes are generally used for diagnostic purposes [207, 208] and for plasmacontrol and manipulations [209, 210]. The diocotron motion occurs when the plasma moves offthe axis of the cylindrical electrodes and the electric image charge in the conductive wall of theelectrodes causes the plasma to move in an orbit around the magnetic field axis. The motion is ananalogue to the ~E × ~B motion, but instead of the self electric field of the plasma, ~E becomes ~Ei, theimage electric field. The diocotron frequency is expressed as [80]:ωmθ = ωrotmθ − 1 + ( rpRW)2mθ , (7.2)where ωrot is the rotation frequency of the plasma, mθ is the mode order, rp is the plasma radius andRW is the electrode radius. For the plasma used here, ωmθ=1/2pi ∼ 1 kHz and ωmθ=2/2pi ∼ 40 kHz.Thus, there is a possibility that high order diocotron modes can be excited.1717.1. Radial compression mechanisms7.1.3 Compression by Trivelpiece-Gould excitationWhen the rotating wall frequency coincides with the Trivelpiece-Gould (TG) modes, angular mo-mentum is injected into the system leading to radial compression of the plasma. The frequency ofthe first order TG mode is [80]:ωTG =kzωpk, (7.3)where ωp is the plasma frequency, k and kz are typical values from a standing wave, kz = mzpi/L,k2 = k2z + k2⊥ and k⊥ = 1rp(2ln(RW/rp))1/2.The plasma studied here has a first order TG frequency of ωTG/2pi ∼ 15 MHz and the subse-quent modes can be excited at higher frequencies. The TG modes are excited at frequencies in the10 MHz and higher range, thus this mechanism is not consistent with the compression observed inthe 100 kHz range.7.1.4 Strong drive regimeThe strong drive regime has been observed in single component plasmas and can be achieved overa broad range of frequencies without tuning to plasma modes [195]. During compression, the max-imum density occurs when the rotation frequency of the plasma is equal to the rotating wall fre-quency [195]. From the measurements of compression of a pure electron plasma (section 6.2.3),we have observed that compression occurs from ∼ 800 kHz, which is probably in the strong driveregime. As for the antiproton cloud, we observe a resonant structures over a relatively narrow rangeof frequencies which implies that this is probably due to another kind of compression mechanism.7.1.5 Sympathetic compression mechanismFor sympathetic compression, where the antiprotons follow the compressing electron plasma, weneed the electron plasma to be able to compress in the frequencies of interest. We have observedthat a pure electron plasma does not compress in the 50 – 700 kHz frequency range. This indicates1727.1. Radial compression mechanismsthat sympathetic compression is not the observed mechanism. However, sympathetic compressionis observed above about 800 kHz, where the electron plasma compresses.7.1.6 Compression by magnetron sideband coolingThe magnetron motion can be cooled or heated by exciting either the cyclotron motion or the axialmotion in the single particle regime [66]. The magnetron motion can be coupled to the axial motion,or to the cyclotron motion, by introducing oscillating electric fields into the trap, for example, byapplying the rotating wall.By exciting the cyclotron motion with an oscillating potential, the magnetron radius will decrease.Hence, the particles will move towards the center [202]. In our experiment, in a magnetic field of 3T, ωc/2pi ∼ 84 GHz for electrons and ∼ 46 MHz for antiprotons, therefore we are clearly not excitingthe cyclotron motion when applying the rotating wall.On the other hand, the coupling of the axial motion with the magnetron motion can also result ina decrease of the magnetron radius [66]. Theory says that the magnetron motion can be cooledwhen exciting the system at the frequency ωb + ωM, and heated when applying the frequency ωb −ωM. Here, ωM is the magnetron frequency and ωM/2pi ∼ 1 kHz. The axial bounce frequencyωb/2pi ∼ 12 MHz for the electrons and ∼ 270 kHz for the antiprotons. These frequencies arecalculated by considering a very accurate harmonic potential. Recently, compression in a slightlyanharmonic potential, for positrons in the single particle regime was achieved [211]. Sidebandcooling is an appealing candidate because compression occurs aroundωb. However, in our case, thismechanism becomes ineffective, since the space charge of the electrons greatly distorts the potentialand it becomes highly anharmonic. Furthermore, if we consider the axial motion of the antiprotons,we can see that the upper and the lower band are very close together and we should be able to observeboth compression and expansion of the cloud. Recently, a theory of a new form of sideband coolingwas developed, where only one side band is active for positrons in the single particle regime [206].However, since we are not in the single particle regime and the potential is highly anharmonic, weconclude that the antiproton cloud is not being compressed by this mechanism.1737.1. Radial compression mechanisms7.1.7 Compression by bounce resonance transportIt has been previously shown that particles with an axial bounce frequency and rotation frequency inresonance with an asymmetric time-varying potential (rotating wall), undergo radial inward or out-ward movement [198]. This mechanism has been studied theoretically and experimentally for pureelectron plasmas [199, 213, 214] and has also been observed in positron plasmas [212]. Moreover,this phenomenon was observed for positrons in the single particle regime in a harmonic well [198],where it was hypothesized that the same bounce resonant transport mechanism applies, using themagnetron frequency as the rotation frequency. Generally, the axial bounce frequency is muchhigher than the magnetron frequency, so the resonance frequency is approximated as the axialbounce frequency.We now present an overview of bounce resonant transport when using a rotating electric field forsingle particles and the plasma regime [198]:A plasma is resonant with the asymmetry ifωres = ωb + mθωrot (7.4)is equal to the rotating wall frequency ωRW . Here, ωb is the axial bounce frequency, ωrot is therotation frequency of the plasma and mθ is the azimuthal wavenumber of the asymmetry, wheremθ = 1 corresponds to the dipole mode and mθ = 2 to the quadrupole mode of the rotating wall.Recall that for the experiments in chapter 6, we used the dipole mode. In this case equation 7.4becomes:ωres = ωb + ωrot, (7.5)which can be generalized to:ωres = nωb + lωrot, (7.6)where l and n are the azimuthal and axial wavenumbers, and correspond to higher harmonics.1747.2. Investigation of the bounce resonant transport of antiprotonsIn non-neutral plasma physics, the resonance condition has also been developed as [214]:ωRW − lωrot − npivzL = 0, (7.7)where vz is the axial velocity, L is the length of the plasma, l and n are the azimuthal and axialwavenumbers. The resonance condition is the same as the one in equation 7.6.Instead of all the particles having the same axial bounce frequency, the particles will have a dis-tribution of axial bounce frequencies due to the non-harmonic potential, and this may explain whycompression is observed over such a wide range of frequencies. From the measurement we knowthat ωrot/2pi ∼ 10 – 40 kHz, the antiproton’s ωb is in the 100 kHz range and the electron’s ωb isin the 10 MHz range. From this information, we assess that the bounce resonance transport of theantiprotons is the more promising explanation.7.2 Investigation of the bounce resonant transport of antiprotonsTo further discuss if bounce resonant transport can be the compression of mechanism observed, wewill calculate the resonant condition, so that we can compare it to the frequencies where compres-sion is observed.In our case, we have an antiproton cloud co-located with an electron plasma. The antiproton cloudrotates at the same frequency as the electron plasma due to the electron plasma’s self-electric field,where ωrot = (~E × ~B)/(B2r). As a result of the space charge of the electrons, the potential well isnot harmonic and there is not an unique axial bounce frequency. Instead, the bounce frequency ofany single antiproton depends on several variables:• Total electric field due to the electron plasma and the trap ~Etot, which depends on the elec-tron peak density (n0) or electron plasma radius (re), electron temperature (Te) and electronnumber (Ne).• Antiproton energy E p¯.1757.3. Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot)• Radial position of the antiproton rp¯.Averaging over the entire antiproton population yields a distribution of antiproton bounce frequen-cies f (ωb), which can be numerically calculated taking all these factors into account.The distribution is written as:f (ωres) = f (ωb + ωrot), (7.8)7.3 Numerical calculation of the antiproton distribution f (ωb) andf (ωb + ωrot)We will first calculate the antiproton distribution f (ωb) and then f (ωb + ωrot). The latter is thefrequency distribution, corresponding to particles in resonance if compression is due the bounceresonant transport mechanism on antiprotons. The potential and the electron density are calcu-lated by self-consistently solving the Maxwell-Boltzmann distribution and Poisson’s equation, asdescribed in section 4.2.4. The electron plasma is considered to be in global thermal equilibriumand the solution is sensitive to three parameters: Te, re and Ne. The temperature of the electronplasma is ∼ 300 K before the rotating wall and ∼ 500 K after the rotating wall. The values 300 K,500 K and 700 K are used to study the sensitivity of the bounce frequency to temperature. Theinitial size of the plasma was estimated to be re ∼ 3.2± 1.5 mm (see section 6.2.1). We study thedependance on radius for three values: 3.2 mm, 4.0 mm and 4.7 mm, which correspond to densitiesof 8×106 cm−3, 10×106 cm−3 and 12×106 cm−3 respectively, for a plasma of 4×106 electrons. Westudy large radii since the MCP image shows that the cloud initially has a radius larger than 3 mm.Figure 7.1 shows an example of the potential at various radial positions for Ne = 4×106 electrons, re= 4 mm and Te = 300 K. One can see that at small radii (where the electron plasma is concentrated),the potential is highly anharmonic due to the space charge and, as the radius increases, the effect ofthe space charge vanishes. Figure 7.2 shows the corresponding solved density.1767.3. Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot)23242526-10 -5 0 5 10Potential[V]z [mm]r = 0 mmr = 2 mmr = 4 mmr = 6 mmno e−, r = 0 mmFigure 7.1: Calculated potential as a function of the axial position for 4 × 106 electrons, withTe = 300 K and re = 4 mm (n0 = 10 × 106 cm−3). The potential is shown for several radialpositions. The black solid curve is the calculated potential without space charge.024681012-10 -5 0 5 10Density[×106cm−3]z [mm]r = 0 mmr = 2 mmr = 4 mmr = 6 mmFigure 7.2: Calculated density as a function of radius and axial position for 4 × 106 electrons withTe = 300 K and re = 4 mm (n0 = 10 × 106 cm−3).1777.3. Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot)We assume the particles are in equilibrium and since we are interested in the axial energy of the par-ticles, we have assumed that the energy of the particles follows a Maxwell-Boltzmann distributionwith one degree of freedom:f (E) =√EpikBTexp(− EkBT). (7.9)From figure 7.3, we can see the Maxwell-Boltzmann energy distributions for different temperaturesthat are used for the antiproton thermal distribution (300 K, 500 K and 700 K).00.20.40.60.810 0.1 0.2 0.3 0.4f(E)[a.u]Energy [eV]300 K500 K700 KFigure 7.3: Maxwell-Boltzmann energy distribution for 300 K, 500 K and 700 K, normalized to thepeak value of 1.After generating the potential, the density and the energy distribution, we numerically calculate theaxial bounce frequency for several thousand antiprotons. For each antiproton, a random energy ispicked from the Maxwell-Boltzmann distribution. This energy is used as the initial potential energyof the particle, and we place the particle at an initial axial position when the particle is at rest andthe kinetic energy is zero. Since the potential changes with radius, a random radial position isalso selected from the density distribution. We assume that the antiprotons follow the same densitydistribution as the electrons.1787.3. Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot)These two parameters (E p¯ and r p¯) define a position where the antiproton can be released from restto calculate its trajectory.We perform a fourth-order Runge-Kutta integration of the motion [215]. By definition:Fz = md2zdt2= eEz (7.10)Where Ez = −dΦ(r, z)/dz, so:d2zdt2= − emdΦ(r, z)dz(7.11)A single antiproton moves in the potential well and the calculation is stopped when the velocitychanges sign (i.e. it is at its turning point). The time taken for the antiproton to reach this point isT/2, where T is the period. The frequency is calculated:ωb/2pi =1T(7.12)Figure 7.4 shows the calculated antiproton axial bounce frequencies as function of the antipro-ton’s energy and radial position for one set of parameters. The axial bounce frequency rangesfrom ∼ 10 kHz to ∼ 400 kHz. The axial bounce frequency increases with energy and radius becauseof the shape of the potential.Figure 7.5 shows the energy distribution and the calculated axial bounce frequency as a function ofthe energy for two radial positions: r = 0 mm and r = 4 mm. One can see that for the same energy,the axial bounce frequency is slightly higher for larger radii. At the peak of the energy distribution,ωb/2pi is between ∼ 100 kHz and ∼ 150 kHz.The rotation frequency of the plasma is calculated as a function of the radial position from:ωrot/2pi = −dΦ(r, z)dr12piBr(7.13)Where dΦ(r, z)/dr is the radial electric field and B is the magnetic field.1797.3. Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Energy [eV]012345Radius[mm]0 kHz100 kHz200 kHz300 kHz400 kHzFigure 7.4: ωb/2pi as a function of radius and energy. The bounce frequency of antiprotons wascalculated for 300 K and re = 4 mm (n0 = 10 × 106 cm−3).00.510 0.05 0.1 0.15 0.2050100150200250300f(E)[a.u]ωb/2pi[kHz]Energy [eV]Energy distributionr = 0 mmr = 4 mmFigure 7.5: Energy distribution for 300 K and also the antiprotons’ bounce frequency as a functionof the energy at the radial positions r = 0 mm and r = 4 mm, with re = 4 mm.1807.3. Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot)7.3.1 f (ωb + ωrot) of antiprotons cooled by 4 × 106 electronsThis is the simplest case since the measurements do not show a measurable change in the electrondensity (at least as observed via the MCP image), as seen in section 6.2.2. We therefore assumethat any change in the electron density is too small to affect the potential or the rotation frequencyduring the 100 s of compression. The distributions of bounce frequency and rotation frequency arethen calculated from the initial conditions.Figure 7.6 shows the calculated thermal equilibrium potential and the density for different electronradii at 300 K. One can see that as the density increases, the corresponding radius gets smaller andthe space charge becomes larger.2525.52626.5-10 -5 0 5 10Potential[V]z [mm]00.510 1 2 3 4 5 6 7Normalizeddensityr [mm]3.2 mm 4.0 mm 4.7 mmFigure 7.6: Self-consistent on-axis potential (left plot) and radial density (right plot) for differentplasma radii at 300 K for 4 × 106 electrons.If we fix the plasma radius to re = 3.2 mm (n0 = 8 × 106 cm−3) and we change the temperature(see figure 7.7), we observe a less pronounced change in the height of the potential, but the shape isslightly different, since for colder temperatures, the density is flatter than for warmer temperatures.The antiproton axial bounce frequency (ωb) was calculated for several thousand antiprotons. Theindividual values of ωb were then binned to create the distribution, f (ωb). Figure 7.8 shows theantiprotons’ f (ωb) for different electron densities and temperatures. To allow comparison, the dis-tributions are normalized to their maximum value. We also plot the data (solid black dots), whichshows the antiproton cloud central density on the MCP as a function of the applied rotating wall fre-1817.3. Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot)2626.5-10 -5 0 5 10Potential[V]z [mm]00.510 1 2 3 4 5 6 7Normalizeddensityr [mm]300 K 500 K 700 KFigure 7.7: Self-consistent on-axis potential (left plot) and radial density (right plot) for differenttemperatures for re = 3.2 mm for a plasma with 4 × 106 electrons.quency. It is important to note that we are not intending to directly compare the two quantities, weonly want to compare the frequency range of the distribution and the frequencies where compres-sion is achieved. At first glance, we can see that the bounce frequencies cover a similar frequencyrange as the rotating wall frequencies where the compression occurs. However, we have not yettaken the rotation frequency of the plasma into account.We calculate the rotation frequency as a function of radius using equation 7.13. Figure 7.9 showsthe resulting rotation frequency for different temperatures and plasma radii. As we can see fromequation 4.8, ωrot depends on the density of the plasma. For the peak densities that we are studying,the peak rotation frequency varies between 4 kHz and 6 kHz and it is mostly constant over theplasma. When the temperature increases, the rotation frequency becomes less flat, but this effect issmall.Figure 7.10 shows f (ωb + ωrot) for different temperatures and plasma radii. We remark that ωrot istypically negligible (4 kHz < ωrot/2pi < 6 kHz), compared to ωb, whose peak frequency ranges from100 kHz to 200 kHz. There is a very small shift to the right at higher electron density due to thecontribution of the rotation frequency, but is almost negligible. f (ωb +ωrot) is more sensitive to theplasma temperature than to the radius (for the studied values). There is little change when varyingthe radius, but when the plasma is warmer, there is a shift in the distribution to higher frequencies.Overall, the calculated antiproton distribution f (ωb +ωrot) has a range of frequencies similar to the1827.3. Numerical calculation of the antiproton distribution f (ωb) and f (ωb + ωrot)00.510200400600f(ωb)[a.u]Centraldensity[a.u]a) 300 K00.510200400600f(ωb)[a.u]Centraldensity[a.u]b) 500 K00.510 100 200 3000200400600f(ωb)[a.u]Centraldensity[a.u]ω/2pi [kHz]c) 700 K3.2 mm4.0 mm4.7 mmDataFigure 7.8: Bounce frequency distribution of antiprotons as a function of the antiproton bouncefrequency for different temperatures and electron densities for 4 × 106 electrons. Black solid dotsare the measured antiproton central density (arbitrary units) as a function of the applied rotatingwall frequency.1837.4. f (ωb + ωrot) of antiprotons cooled by 20 × 106 electrons02468100 1 2 3 4 5 6ωrot/2pi[kHz]Radius [mm]a) 300 K0 1 2 3 4 5 6Radius [mm]b) 500 K0 1 2 3 4 5 6Radius [mm]c) 700 K3.2 mm 4.0 mm 4.7 mmFigure 7.9: Rotation frequency of the electron plasma as a function of radius for several plasmaradii and temperatures of a 4 × 106 electron plasma.frequency where compression was observed. This suggests that a resonance between the motion ofthe antiprotons and the rotating wall drive might be the explanation of compression.7.4 f (ωb + ωrot) of antiprotons cooled by 20 × 106 electronsIn this section we study the case where the antiprotons are cooled by 20×106 electrons. This case isdifferent from the previous case (when the antiprotons are cooled by 4 × 106 electrons) because theelectron plasma partially compresses, which means that the system behaves differently. Moreover,the measurements show that the antiproton cloud can be compressed using rotating wall frequencyas high as ∼ 700 kHz.7.4.1 f (ωb + ωrot) distribution at the initial conditionsAs for the case of 4 × 106 electrons, we numerically calculate the antiprotons’ f (ωb + ωrot) at theinitial conditions. Figure 7.11 shows f (ωb + ωrot) for re ∼ 4 mm (n0 = 28 × 106 cm−3) and atdifferent temperatures (300 K, 500 K and 700 K). All the distributions have a rotation frequencyωrot/2pi ∼13 kHz. The peak of f (ωb + ωrot) is at ∼150 kHz and the resonant frequencies range1847.4. f (ωb + ωrot) of antiprotons cooled by 20 × 106 electrons00.510200400600f(ωb+ωrot)[a.u]Centraldensity[a.u]a) 300 K00.510200400600f(ωb+ωrot)[a.u]Centraldensity[a.u]b) 500 K00.510 100 200 3000200400600f(ωb+ωrot)[a.u]Centraldensity[a.u]ω/2pi [kHz]c) 700 K3.2 mm4.0 mm4.7 mmDataFigure 7.10: Antiprotons f (ωb +ωrot) as a function of the temperature and plasma radii when cooledby 4 × 106 electrons. Black solid dots are the measured antiproton central density (arbitrary units)as a function of the applied rotating wall frequency.1857.4. f (ωb + ωrot) of antiprotons cooled by 20 × 106 electronsbetween 10 kHz and 250 kHz. These distributions can only explain the antiproton compression atlow frequency (up to ∼ 250 kHz), but not at higher frequencies.00.20.40.60.811.20 100 200 300 400 500 600 700 800 900 1000020040060080010001200140016001800f(ωb+ωrot)[a.u]AntiprotonCentralDensity[a.u]Frequency [kHz]300 K500 K700 KDataFigure 7.11: f (ωb + ωrot) of antiprotons when cooled by 20 × 106 electrons at the initial conditionsfor different temperatures.The observation of compression at higher frequencies (ωRW > 250 kHz) by the bounce resonanttransport implies that there should be a resonance in that range of frequencies. Furthermore, thereshould be a resonance with the initial conditions of the plasma to at least start the compression. Forexample, if we apply the rotating wall at 550 kHz, why is there compression, even though the initialresonance is between 50 kHz and 250 kHz (see figure 7.11). To resolve this, we call on the azimuthaland axial wavenumbers of the resonant bounce transport resonance condition (equation 7.7). Itis possible that the antiprotons bounce axially several times for each rotation. As well as beingresonant at a frequency of ∼150 kHz, we infer that the antiprotons should also be in resonance atinteger multiples of this frequency (i.e. ∼ 300 kHz, ∼ 450 kHz, ∼ 600 kHz). Note that we did notobserve this effect when using 4 × 106 electrons. It may be that compression via a resonance athigher harmonics requires a better source of cooling and 4 × 106 electrons do not provide enoughcooling. 7×106 electrons provide more cooling and perhaps for this reason, it is possible to achievecompression to frequencies up to ∼ 500 kHz. If this explanation holds, it indicates that above a1867.4. f (ωb + ωrot) of antiprotons cooled by 20 × 106 electronscertain electron number, one can get the same cooling effects (as for 12×106 and 20×106 electrons),where the antiproton cloud has almost the same compression behaviour. However this effect onlyindicates an initial resonance but does not explain why a better antiproton cloud compression isachieved at higher frequencies. As we observed in chapter 6, there is a partial compression of theelectrons, which clearly modifies the potential and the rotation frequency of the system. The effectof such partial compression will be studied in the next section.7.4.2 f (ωb + ωrot) as a function of the partial compression of the electron plasma.Figure 7.12 shows MCP images of the antiproton-electron plasma (with 20×106 electrons) after ap-plying the rotating wall for 100 s at different frequencies. Recall that the electrons do not compresswhen the antiprotons are absent, as seen in section 6.2.3. There is an evident partial compression ofthe electron plasma after applying the rotating wall at 200 kHz and higher frequencies. The electronplasma compresses when applying the rotating wall from 200 kHz to 800 kHz, and is most intenseat 500 kHz, 600 kHz and 700 kHz. At these frequencies, a fraction of the electron plasma (about3 × 106 electrons) becomes denser and the rotation frequency of the system increases.Figure 7.13 shows the MCP images when applying the rotating wall at 550 kHz for different timesto ∼ 1 × 105 antiprotons and 20 × 106 electrons. We can see that from 1 s to 10 s, the antiprotoncloud is still large, revealing that the compression is very slow. Here we speculate that compressionis due to a resonance with higher harmonics.Faster compression of the antiproton cloud can be observed from ∼ 15 s to ∼ 20 s. In those images,one can see on the right hand side, next to the antiprotons, a fraction of the electron plasma com-pressing. After ∼ 30 s, one can see that the density of the antiproton cloud and the electron plasmareaches a maximum.Figure 7.14 shows the data of the antiproton cloud central density as a function of the rotatingwall application time for ωRW = 350 kHz and ωRW = 550 kHz. Note that the antiproton cloudcompression speeds up earlier when ωRW = 350 kHz than when ωRW = 550 kHz. The antiprotoncloud compresses fastest at the point of steepest slope: in the ωRW = 550 kHz data, this happens1877.4. f (ωb + ωrot) of antiprotons cooled by 20 × 106 electronsFigure 7.12: MCP images after 100 s of compression for various frequencies for 1×105 antiprotonsand 20 × 106 electrons. About 15% of the electron plasma compresses.1887.4. f (ωb + ωrot) of antiprotons cooled by 20 × 106 electronsFigure 7.13: MCP images for various compression times at 550 kHz for 1 × 105 antiprotons and20 × 106 electrons.1897.4. f (ωb + ωrot) of antiprotons cooled by 20 × 106 electronsωRW/2pi [kHz] Time [s] Density [cm−3] ωrot/2pi [kHz]550 20 9 × 108 440350 15 5 × 108 220Table 7.2: Peak density and rotation frequency of a fraction of the electron plasma (3×106 electrons)at 500 K.at ∼ 20 s, and for ωRW = 350 kHz at ∼ 15 s. By analyzing the corresponding MCP image, we canfind the peak density and the rotation frequency of the electron plasma, which is shown in table 7.2.The electron plasma is denser and ωrot is not negligible anymore, but is close to the rotating wallfrequency.03006009001200150018000 10 20 30 40 50AntiprotonCentralDensity[a.u]Rotating wall time duration [s]350 kHz550 kHzFigure 7.14: Antiproton cloud central density as a function of the rotating wall application timewhen cooled by 20 × 106 electrons at ωRW = 350 kHz and ωRW = 550 kHz.We plot the calculated distributions f (ωb + ωrot) for these two points in figure 7.15. The verticallines show the applied rotating wall frequencies. The applied rotating wall frequency is close tothe peak of f (ωb + ωrot) at the moment when the rate of compression is fastest. This indicatesthat this occurs when the largest number of antiprotons are in resonance. We conclude that, as thefraction of the electron plasma compresses, ωrot increases and brings a larger number of particlesinto resonance, which speeds up the compression.The calculations of f (ωb + ωrot) allow us to make an interpretation of the antiproton cloud com-1907.4. f (ωb + ωrot) of antiprotons cooled by 20 × 106 electrons00.510 100 200 300 400 500 600 700 800 900 1000f(ωb+ωrot)[a.u](ωb + ωrot)/2pi [kHz]350 kHz550 kHzFigure 7.15: The antiprotons’ f (ωb + ωrot) distribution when cooled by an electron plasma withelectron densities 5 × 109 cm−3 and 9 × 109 cm−3, which correspond to the fastest compressionwhen applying the rotating wall at 350 kHz and 550 kHz, respectively. Vertical lines show theapplied rotating wall frequency.pression. However, another question remains: why there is a partial compression of the electronplasma? This is not yet understood. As observed in section 6.2.3, for a pure the electron plasma (noantiprotons), there is no compression at all. We speculate that it could be an inverse sympatheticmechanism: the antiproton cloud compresses during the first seconds, which sympathetically com-presses a fraction of the electron plasma. We cannot rule out that it could also be an effect of themixed antiproton-electron plasma. As already seen in section 7.1, the electron plasma has plasmamodes in the 10 MHz region and the electron bounce frequency is at ∼ 12 MHz, which indicates thatthe compression of the antiprotons cannot be linked to a resonance with the electron plasma, butwith a resonance with the antiprotons themselves. It seems that the partial compression of the elec-tron plasma is a secondary effect, which in turn, after a certain time, enhances the compression ofthe antiprotons at high frequencies (e.g. 500 kHz and 600 kHz) by providing cooling and increasingthe rotation frequency of the system and consequently f (ωb + ωrot) to higher frequencies.1917.5. Conclusion7.5 ConclusionWe have discussed possible compression mechanisms. We have found that bounce resonant trans-port is the most likely explanation for the compression of the antiproton clouds. For this mechanism,there is compression if the particles’ ωb + ωrot is equal to ωRW .To further investigate this mechanism, we calculate the frequency distribution f (ωb + ωrot), whichshould indicate if the motion of the antiprotons is in resonance with the rotating wall drive. We haveobserved that the antiprotons’ f (ωb+ωrot) is highly dependant on the electron plasma characteristicssince the space charge makes significant changes to the potential well and the rotation frequency.When the antiprotons are cooled by 4 × 106 electrons, the electron plasma remains in conditionssimilar to the initial conditions, and there is no major change in the potential well or the rotationfrequency over time. The simulation shows that the antiprotons’ frequency distribution f (ωb +ωrot)is very close to the frequencies where the compression of antiproton cloud occurs. This resultindicates that the antiproton cloud may be compressed by a resonance between ωb + ωrot and ωRW .If the number of electrons is increased, the same underlying mechanism of compression seems toapply. However, a fraction of the electron plasma compresses, which complicates the analysis. Wethink that the antiproton cloud causes a partial compression of the electron plasma. As a conse-quence, the potential well changes and thus f (ωb) and f (ωb + ωrot) of the antiprotons change overtime. We specifically studied the case when the antiprotons are cooled by 20 × 106 electrons. Here,there is an evident compression of a fraction of the electron plasma, which changes the potentialwell and consequently, the rotation frequency of the system increases. By analyzing the electronplasma on the MCP image, we calculated the antiprotons’ f (ωb +ωrot) in these new conditions. Wefound that f (ωb +ωrot) is shifted to higher frequencies, and the fastest compression occurs when itspeak is close to the rotating wall frequency.192Chapter 8Summary and conclusionAntihydrogen holds the promise of a stringent test of CPT invariance. In the first part of this thesiswe have discussed the first measurements on antihydrogen trapping and spin-flip transitions in theALPHA apparatus. In the second part, we have observed a new mechanism of compression ofthe antiproton clouds. Compression of antiproton clouds is an important tool for antihydrogenformation and trapping.In 2010, the ALPHA collaboration trapped 38 atoms for the first time. Antihydrogen trapping mea-surements continued until 2011 and a total of 595 antihydrogen atoms were trapped. In 2011, theALPHA collaboration induced spin-flip transitions of antihydrogen with microwave radiation. Thetransition frequencies at the minimum trap were localized to a relative precision of 4 × 10−3. Theseexperiments have demonstrated that antihydrogen can be trapped for enough time to be interrogatedand enable other experiments.In 2012, the ALPHA apparatus was upgraded and renamed ALPHA-2. The ALPHA-2 apparatushas laser access and a new era of antihydrogen research is around the corner. Future experiments onantihydrogen include 1S–2S two photon spectroscopy, 1S–2P one photon spectroscopy and cooling,and gravity experiments among other (for a detailed list, see section 3.5).The second part of this thesis focuses on antiproton cloud radial compression using the rotating walltechnique and performed in the antiproton capture trap of the ALPHA-2 apparatus. Compression ofan antiproton cloud (∼ 1.5× 105 antiprotons) was observed over a frequency range of 50 – 750 kHz,when cooled by 4 × 106, 7 × 106, 12 × 106 and 20 × 106 electrons. The radius of the antiprotoncloud was decreased by up to a factor of 20, with the smallest radius being ∼ 0.2 mm. Different193Chapter 8. Summary and conclusioncompression behaviours were observed that depended on the electron number. A higher number ofelectrons (12× 106 and 20× 106 electrons) enhances and allows antiproton cloud compression overa wider range of frequencies. For those electron numbers, we also observed that about 15% of theelectron plasma compresses.After discussing several mechanisms of compression, we concluded that bounce resonant transportis the most likely mechanism to explain compression of the antiproton clouds at these frequencies.To further study this mechanism, we calculated the axial bounce and rotation frequency distributionsof the antiprotons for a comparison with the frequency range where compression was observed. Wehave found that both ranges of frequencies are similar when cooling the antiprotons with 4 × 106electrons, but not when using higher numbers of electrons. However, the partial compression of theelectron plasma increases the rotation frequency of the system and seems to be the reason behindcompression at slightly higher frequencies. This is the first time that bounce resonant transport hasbeen observed on antiprotons. The explanation of compression was challenging because we havea two species plasma in which the electron plasmas space charge dominates the rotation frequencyand the potential shape.Nevertheless, compression of the antiproton clouds could be further studied by measuring the tem-perature during compression and at higher rotating wall amplitudes. Another remaining experimentis to perform ejection of the electron plasma to study the antiproton cloud expansion after compres-sion. 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The energy loss per length unit (−dE/dx) for charged heavy particles is also called thestopping power. The energy loss is mainly due to inelastic collisions where target atoms are ionizedor excited (electronic stopping power). At low energy, elastic collisions become important since theprobability of the atomic target to recoil becomes higher and contributes to the energy loss of theincident particle (nuclear stopping power).The stopping power is accurately described by the Bethe formula for high energies (see equa-tion A.1) [109].−〈dEdx〉=( e24pi0)2 4piz2mec2β2NAZAL0, (A.1)whereE: particle energy z: charge of incident particlex: the distance travelled Z: target atomic numbere: elementary charge A: atomic mass of the target0: vacuum permittivity NA: Avogadro’s numberme: mass of the electron L0: Bethe stopping functionc: speed of light β: particle velocity (v/c).The Bethe stopping function is [216]L0 =12ln2mec2β2γ2TmaxI2− β2 − δ(βγ)2, (A.2)207Appendix A. Antiproton energy losswhere γ is the Lorentz factor 1/√1 − β2, I is the mean excitation energy, δ(βγ) is the density effectcorrection and Tmax is the maximum kinetic energy that can be transferred to an electron in a singlecollision. Tmax can be written asTmax =2mec2β2γ21 + 2γme/M + (me/M)2, (A.3)where M is the mass of the incident particle.Note that the Bethe stopping function is independent of z. The Bethe stopping function can begeneralized and higher-order z terms are introduced:L(β) = L0(β) + zL1(β) + z2L2(β) + ..., (A.4)where the zL1(β) term is known as the Barkas correction and the z2L2(β) term is called the Blochcorrection.The Barkas correction is responsible for the difference in the stopping power between protons andantiprotons. Antiprotons are negatively charged, hence the Barkas term will decrease the magnitudeof the stopping power. Figure A.1 shows the stopping power for protons and antiprotons as afunction of energy in aluminium and beryllium. For antiprotons, the maximum stopping poweris about 35% smaller. This difference can be explained by the polarization of the target by theprojectile.When the antiproton is slow enough, it repels electrons along its path, so the energy loss is smallerthan for protons (figure A.1) [216].The mean particle range is the distance traveled by the particle before depositing all its kineticenergy in the material and can be directly calculated from the stopping power:R(E0) =∫ 0E0dx =∫ 0E0dEdxdE=∫ E00dE−dE/dx , (A.5)where E0 is the initial energy of the incident particle. The range for protons and antiprotons in Al208Appendix A. Antiproton energy loss05010015010−4 10−3 10−2 10−1 100-dE/dx[MeV/mm]Energy [MeV]Antiprotons in AlAntiprotons in BeProtons in AlProtons in BeFigure A.1: Stopping power as a function of the particle energy for antiprotons and protons inaluminium and beryllium. Data extracted from GEANT4 [217].and Be are shown in table A.1.Particle Material Range [µm]antiproton aluminium 227beryllium 269proton aluminium 211beryllium 260Table A.1: Range for protons and antiprotons with kinetic energy of 5.3 MeV for aluminium andberyllium. Data extracted from GEANT4 [217].209Appendix BElectrodes cabling and connectionsThe antiproton capture Penning-Malmberg trap is composed of 20 cylindrical electrodes, wheretwo of them are high voltage electrodes and two are six-segmented electrodes (rotating wall), asdescribed in section 2.3.3. Figure B.1a is a picture of the electrode stack. The electrodes are in-dependently connected to electric feedthrough using a designed flexible (flex) circuit, with 35 µmthick copper conducting layers and with kapton insulating layers. The layer with the traces is sand-wiched between two ground layers to improve the insolation between the traces and the rejection ofexternal noise. The flex circuit is connected with screws to the electrodes as shown in figure B.1b.High voltages that are supplied to high voltage electrodes cannot be applied through the flex, and soare directly connected with 50 coaxial cable (KAP50) [218], as shown in figure B.1c.The end of the flex has a serpentine shape to increase the path of the circuit and by consequence,to decrease the heat flow. This is necessary to allow the electrodes to cool to temperatures close to∼ 4 K. The ground layer of the flex circuit is directly clamped to different regions of the apparatus.In figure B.1, on the left hand side, the red circle shows where the flex circuit is clamped to the 40 Kregion. On the right hand side, the blue circle shows where the flex is clamped to the 4 K region.Micro-D connectors are soldered to the flex and connected to the 40 K flange that separates theUHV region and the OVC region. In the OVC region, small PCBs (printed circuit boards) connectthe micro-D connectors and coaxial wires as shown in figure B.2a. The coaxial cables (KAP3) [218]have a copper conductor with a diameter of 0.25 mm and a length of ∼ 1.5 m. The end of the cablesare soldered to small PCBs with subminiature-D (sub-D) connectors. The sub-D are connected tothe flange between the OVC and the room temperature area, as shown in figure B.2b. The room210Appendix B. Electrodes cabling and connectionsFigure B.1: a) Stack of electrodes of the antiproton capture trap. The green double arrow shows thelength of the stack. b) A close up picture of the connection of the electrode with a circuit flex. c)Picture of the high voltage electrode connected to a coaxial wire. The white ceramic spacer insulatesthe high voltage electrode. d) Picture of the end of the flex. The blue circle on the left represents thearea that is clamped to the 4 K region. The red circle on the right represents the area that is clampedto the 40 K region.211Appendix B. Electrodes cabling and connectionsFigure B.2: a) Micro-D connector and PCB connected in the flange between the UHV and OVCregion. b) Sub-D connectors and PCB connected in the flange between the OVC and the roomtemperature region.temperature sub-D are connected to a circuit board with a passive RC-filter (one for each electrode),which are connected to the respective amplifiers (see section 2.3.2).212Appendix CVeto detector simulationsIn 2010, ALPHA studied the possibility to implement a cosmic veto detector around the apparatus.I performed Monte Carlo simulations to study if a cosmic veto detector was suitable to discriminatebetween antiproton/antihydrogen annihilations and cosmic rays.The prototype cosmic veto detector consisted of two parallel scintillator pads, one on the top ofthe apparatus and one on the bottom of the apparatus (on the floor). Using the time of detectionof the scintillator on the top (ttop) and the scintillator on the bottom (tbottom), we can calculate thetime difference ∆t = tbottom − ttop. Because of the different origin of the cosmic rays and antiprotonannihilations, ∆t is different for each case. Cosmic rays passing through the apparatus are expectedto pass first through the scintillator on the top and then, after ∆tcosmic, through the scintillator onthe bottom. Antiproton/antihydrogen annihilations originate close to the center of the apparatus,thus the resulting particles from the annihilation are detected by the scintillators on the top and onthe bottom almost simultaneously, with ∆tannihilation < ∆tcosmic. Figure C.1 shows a schematic ofthe geometry of the prototype detector. The challenge of the cosmic veto detector resided in thedetector time resolution needed and on the space and geometry available in the experiment.Monte Carlo simulations were performed using GEANT3 package as described in Ref. [102]. Twoscintillator pads were implemented. The distance between the scintillators was varied, as well as theresolution time of the detector. The distance from the centre to the bottom is fixed to dbottom = 110 cmsince it is limited by the floor. Because of the space available in the experimental zone, the bottompad dimensions are also limited to 160 cm× 110 cm. The distance from the centre to the top scintil-lator is not fixed but has to be greater than dtop > 150 cm. Figure C.2 shows the distribution of ∆t for213Appendix C. Veto detector simulationszyxScintillator on the topScintillator on the bottomFigure C.1: Schematic of the cosmic veto detector. The silicon detector is represented in yellow. Anexample of a cosmic ray path is represented in green and the path from an antiproton annihilation isrepresented in red.214Appendix C. Veto detector simulationscosmic rays and antiproton annihilations for a distance of dtop = 150 cm (centre to top scintillator)and for various time resolutions. For this distance, ∆t for cosmic rays has a distribution with a meanaround ∆tcosmic ∼ 10 ns, while the distribution for annihilations has a mean around ∆tannihilation ∼ -1 ns. Even though the two distributions have means about 10 ns apart, both distributions are wideand a time resolution of 2 ns is not enough to detangle both signals.Figure C.3 shows the distribution of ∆t for cosmic rays and antiproton annihilations for a distanceof dtop = 200 cm (centre to top scintillator) and for various time resolutions. If the top scintillatoris placed further from the centre, the signal of the cosmic rays and antiproton annihilations almostcompletely separated when using a time resolution of 1 ns. However, when the top scintillator ismoved farther up, the size must be increased to cover sufficient solid angle.The project of a cosmic veto detector was delayed since scintillators with so large area and shorttime resolution were not found. A new project is been studied, where a structure with various smallscintillator pads are placed around the external magnet [219].215Appendix C. Veto detector simulationsFigure C.2: Distribution of ∆t of cosmic rays and antiproton annihilations for dtop = 150 cm. On thetop: time resolution of 1 ns. On the bottom: time resolution of 2 ns.216Appendix C. Veto detector simulationsFigure C.3: Distribution of ∆t of cosmic rays and antiproton annihilations for dtop = 200 cm. On thetop: time resolution of 1 ns. On the bottom: time resolution of 2 ns.217Appendix DDerivation of the magnetic dipole forcein GEANT4D.1 Magnetic dipole introductionThe magnetic potential energy of a magnetic dipole in an inhomogenous magnetic field is:U = −~µ · ~B, (D.1)where ~µ is the magnetic dipole moment and ~B is the magnetic field.There are two stable configurations ( ~µ and ~B parallel or antiparallel). A particle can be trapped ina magnetic field minimum if its magnetic moment is antiparallel to the magnetic field (~µ·~B = −µ|~B|).The force acting on such a "low field seeking" particle is:~F =d~Pdt= −µ∇|~B|. (D.2)Note that equation D.2 only applies when the rate of change of the direction of the magnetic field218D.2. GEANT4 implementationat the particle’s position is slow compared to the Larmor frequency ( ωl):1Bd~B(~r)dt ωl, (D.3)where ωl = γB, γ being the gyromagnetic ratio of the system. This requirement is easily fulfilledfor most experiments.D.2 GEANT4 implementationThe force acting on a magnetic dipole can be described as the differential equation:Fi =dPidt= −µ∂|~B|∂xi, (D.4)where i is 0, 1 or 2 corresponding to x, y or z.However, GEANT4 uses the derivative along the curved trajectory dPids to calculate the trajecto-ries of particles, so:dPids=dPidtdtds= −µ∂|~B|∂xidtds. (D.5)We can replace dtds by1|~v| . And since E = γmc2 and P = γmv, we get:dtds=E|~P|c2 . (D.6)219D.2. GEANT4 implementationIn reality, GEANT4 deals with the quantity Pc (in MeV) instead of P, so equation D.4 becomes:d(Pic)dt= −µ∂|~B|∂xic, (D.7)and the units are:[MeVns]=[MeVTesla] [Teslamm] [mmns]. (D.8)Equation D.6 is the same:dtds=E(|~P|c)c , (D.9)with units:[ nsmm]=[MeVMeV mmns]. (D.10)In GEANT4, equation D.5 becomes:d(Pic)ds= −µ∂|~B|∂xicE(|~P|c)c, (D.11)= −µ∂|~B|∂xiE(|~P|c). (D.12)The quantities d(Pic)ds in GEANT4 are dydx[3], dydx[4] and dydx[5], E =√(pc)2 + (mc2)2 and(|~P|c) =√(pxc)2 + (pyc)2 + (pzc)2 where pic are y[3], y[4] and y[5].220

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