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Cavity cooling of leptons for increased antihydrogen production at ALPHA Evetts, Nathan 2015

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Cavity Cooling of Leptons for IncreasedAntihydrogen Production at ALPHAbyNathan EvettsB.Sc., The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2015c© Nathan Evetts 2015AbstractPrecise spectroscopic measurements of anti-hydrogen at the ALPHA experi-ment are hindered by small numbers of cold anti-atoms. This thesis describesa cooling technique for positron plasmas which can be used to increase thenumber of trappable anti-hydrogen atoms. The technique builds on previouswork which allows control of spontaneous emission via the Purcell Effect.Our implementation incorporates a novel microwave resonator into an exist-ing Penning trap to enhance spontaneous emission. Preliminary data sug-gests that temperatures and cooling rates for these plasmas can be improvedby at least a factor of 10. Eventually this work could result in an order ofmagnitude increase in anti-hydrogen production at ALPHAiiPrefaceChapter 7 details measurements of electron plasma temperatures and cool-ing rates at resonances within the cavity presented in chapter 3. The datapresented in chapter 7 was taken primarily by Eric Hunter at Berkeley. Theplasma apparatus (with the exception of the cavity) was built and main-tained by Alex Povilus at Berkeley.The cavity was conceptualized at the University of British Columbiaby Walter Hardy. Later I designed, simulated and characterized the cavitypresented in this thesis.The electro-deposition (described in section 3.4) of a nichrome-like alloyonto the cavity was conducted by Isaac Martens.The nuclear magnetic resonance measurements of chapter 6 were per-formed by Carl Michal.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Antimatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 ALPHA apparatus . . . . . . . . . . . . . . . . . . . . . . . . 41.4 The Penning trap . . . . . . . . . . . . . . . . . . . . . . . . 52 Overview of the Purcell Effect . . . . . . . . . . . . . . . . . . 82.1 Theory and history . . . . . . . . . . . . . . . . . . . . . . . 82.2 Some experiments . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Rydberg atoms . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Solid state systems . . . . . . . . . . . . . . . . . . . 132.2.3 Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 A null result at ALPHA . . . . . . . . . . . . . . . . . . . . . 172.4 Circuit formulation of the Purcell Effect for a cyclotron oscil-lator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.1 Cooling in free space . . . . . . . . . . . . . . . . . . 202.4.2 Cyclotron cooling of a particle inside a resonant cavity 212.4.3 Decay rate for an oscillator inside a cavity with non-uniform fields . . . . . . . . . . . . . . . . . . . . . . 232.4.4 Comparison with the quantum result . . . . . . . . . 26ivTable of Contents3 The Bulge Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Resonances of a cylindrical cavity . . . . . . . . . . . . . . . 283.2 Bulge design . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Simulations of resonant fields and frequencies . . . . . . . . . 353.4 Lowering the cavity Q . . . . . . . . . . . . . . . . . . . . . . 353.4.1 Anti-static coatings . . . . . . . . . . . . . . . . . . . 393.4.2 Magnetism . . . . . . . . . . . . . . . . . . . . . . . . 403.5 Cavity fill factor χ . . . . . . . . . . . . . . . . . . . . . . . . 423.6 Chokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Simulations of N Coupled Resonators: Cavity and Electrons 464.1 The circuit model . . . . . . . . . . . . . . . . . . . . . . . . 464.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 Overloading analysis . . . . . . . . . . . . . . . . . . . 524.2 Averaging theory and the rotating frame . . . . . . . . . . . 554.2.1 General formulation . . . . . . . . . . . . . . . . . . . 554.2.2 Simple example: a single damped harmonic oscillator 564.2.3 N coupled harmonic oscillators . . . . . . . . . . . . . 575 Laser Pass - Microwave Stop Tubes . . . . . . . . . . . . . . 625.1 Theory of operation . . . . . . . . . . . . . . . . . . . . . . . 635.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Microwave attenuation . . . . . . . . . . . . . . . . . . . . . 685.4 Thermal conductivity measurement . . . . . . . . . . . . . . 685.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 726 Magnetic Electrodes . . . . . . . . . . . . . . . . . . . . . . . . 746.1 Measurement technique . . . . . . . . . . . . . . . . . . . . . 746.2 Model for a thin magnetized tube . . . . . . . . . . . . . . . 756.3 Approximating the thickness of the nickel strike . . . . . . . 776.4 Extrapolating to many electrodes . . . . . . . . . . . . . . . 776.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 Preliminary Observation of Enhanced Cooling of Electronsat Cavity Resonances . . . . . . . . . . . . . . . . . . . . . . . 797.0.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 83Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85vTable of ContentsAppendicesA Detailed Drawings of a Bulge Resonator . . . . . . . . . . . 91B A Model for Effective Sheet Resistance of Multiple ThinConducting Layers . . . . . . . . . . . . . . . . . . . . . . . . . 93B.1 Complex poynting theorem . . . . . . . . . . . . . . . . . . . 93B.2 Resistance and reactance . . . . . . . . . . . . . . . . . . . . 94B.3 Sheet resistance for a metallic plane . . . . . . . . . . . . . . 95B.4 Sheet resistance for layered thin metals . . . . . . . . . . . . 96B.4.1 Observation . . . . . . . . . . . . . . . . . . . . . . . 97B.4.2 Thin gold on thick nichrome . . . . . . . . . . . . . . 98B.4.3 Thin nichrome on thick copper . . . . . . . . . . . . . 102B.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 103viList of Tables1.1 A summary of matter-antimatter measurements. . . . . . . . 23.1 Microwave resonances of the bulge cavity. . . . . . . . . . . . 363.2 Measured sheet resistance of Aerodag G at room-temperatureand at 4 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Typical simulation parameters for the N-electron problem. . . 504.2 Cavity resonance overload predictions. . . . . . . . . . . . . . 544.3 Simplifying limits for the rotating frame system. . . . . . . . 595.1 A summary of the thermal conductance measurements on thelaser pass tubes. . . . . . . . . . . . . . . . . . . . . . . . . . 737.1 Cavity cooling enhancement summary. . . . . . . . . . . . . . 83viiList of Figures1.1 ALPHA Penning and Octupole Trap. . . . . . . . . . . . . . . 41.2 Penning Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Cyclotron motion . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Typical environments which suppress or enhance spontaneousemission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Example density of states for a rectangular waveguide . . . . 112.3 A typical experiment exhibiting spontaneous emission controlof Rydberg atoms. . . . . . . . . . . . . . . . . . . . . . . . . 122.4 A semi-conductor sample is sandwiched between two distribu-tive Bragg reflectors to form a cavity. . . . . . . . . . . . . . . 142.5 A Penning trap with hyperbolic electrodes. . . . . . . . . . . 162.6 The plasma quadrupole mode. . . . . . . . . . . . . . . . . . . 182.7 A failed temperature scan across cavity modes at ALPHA. . . 192.8 An electron in circular orbit in a capacitor . . . . . . . . . . . 212.9 The capacitor of figure 2.8 in a resonant circuit. . . . . . . . . 222.10 Capacitively coupled LC resonators. . . . . . . . . . . . . . . 243.1 A cylindrical cavity. . . . . . . . . . . . . . . . . . . . . . . . 283.2 Electric fields for transverse electric resonances of a cylindricalcavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Electric fields for transverse magnetic resonances of a cylin-drical cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Axial dependence of TE and TM modes. . . . . . . . . . . . . 333.5 The bulge resonator. . . . . . . . . . . . . . . . . . . . . . . . 343.6 TE131 fields in the bulge resonator . . . . . . . . . . . . . . . 363.7 Axial field depenence inside the bulge resonator . . . . . . . . 373.8 The layered inner surface of the bulge cavity. . . . . . . . . . 403.9 A Hall probe measurement of the magnetic field from theBulge nichrome layer. . . . . . . . . . . . . . . . . . . . . . . 413.10 On axis magnetic fields from nichrome plated bulge cavity. . . 41viiiList of Figures3.11 A cavity pertubation apparatus used to measure our fill fac-tor, χ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.12 Perturbed cavity resonance data from which our fill factor,χ, is infered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.13 A transmission line schematic. . . . . . . . . . . . . . . . . . . 443.14 The cavity choke with λ/4 transforms labeled. . . . . . . . . . 454.1 N resonator model . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Electron cooling times as a function of number. . . . . . . . . 504.3 Overload decay time prediction from single particle case. . . . 534.4 Overload number prediction compared to simulation result. . 544.5 Overload number versus resonator Q. . . . . . . . . . . . . . . 554.6 Single Particle in the rotating frame . . . . . . . . . . . . . . 614.7 A few electrons in the rotating frame . . . . . . . . . . . . . . 615.1 Exploded view of a laser-pass tube. . . . . . . . . . . . . . . . 625.2 An assembled cross-section of a laser-pass tube. . . . . . . . . 635.3 A λ/4 trick for absorbing plane waves. . . . . . . . . . . . . . 635.4 A cross section of the laser pass tube. The λ/4 requirementwrapped onto a cylindrical geometry. . . . . . . . . . . . . . 645.5 Schematic nichrome deposition apparatus for laser pass tubes. 655.6 Web cam sensor data monitoring nichrome vapour depostionfor laser pass tubes. . . . . . . . . . . . . . . . . . . . . . . . . 675.7 Laser-pass microwave measurement apparatus. . . . . . . . . 695.8 Transmitted microwave power of a laser pass tube . . . . . . . 705.9 Laser-pass thermal conductivity measurement apparatus. . . . 715.10 Temperature data expressing thermal conductivity of laserpass tubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.1 Set up for ALPHA electrode magnetism measurment. . . . . . 756.2 Magnetic Fields from the ALPHA stack . . . . . . . . . . . . 767.1 A sketch of the Berkeley experiment. . . . . . . . . . . . . . . 797.2 Plasma temperature at a cavity resonance . . . . . . . . . . . 807.3 The plasma cooling rate estimated from temperature data. . . 817.4 Plasma cooling rate at a cavity resonance . . . . . . . . . . . 82A.1 Detailed bulge cavity drawings . . . . . . . . . . . . . . . . . 91A.2 Detailed drawings of single electrode within bulge cavity. . . . 92ixList of FiguresB.1 General circuit element used to derive resistance / reactancevalues from field quantities. . . . . . . . . . . . . . . . . . . . 94B.2 A thin metallic layer on top of a different metal. . . . . . . . 96B.3 A coaxial resonator for measuring surface resistance of layeredmetallic surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . 98B.4 A thin layer of gold on semi-infinite nichrome. . . . . . . . . . 99B.5 Effective surface resistance of thin gold on a nichrome substrate.100B.6 Effective conductivity of thin gold on a nichrome substrate . . 100B.7 Q's for coaxial resonator with a thin electroplated layer ofgold on nichrome. . . . . . . . . . . . . . . . . . . . . . . . . . 101B.8 Q's for coaxial resonator with a thin electroplated layer ofnichrome on copper. . . . . . . . . . . . . . . . . . . . . . . . 102xAcknowledgementsA special thanks to my supervisor, Prof. Walter Hardy. I learned a greatdeal from Walter. In particular I am grateful for his willingness to get hishands dirty working on new, possibly crazy ideas. Without his supportnone of this work would be possible.Thanks also to Makoto Fujiwara for consistent support (both moral andfinancial). Makoto displayed a quiet confidence in this project and in mewhich called out for success.Finally I'd like to acknowledge the Berkeley crowd: Alex Povilus, EricHunter, and Joel Fajans have all worked long and hard on this collaboration.Without them, we would not have a plasma experiment!xiChapter 1Introduction1.1 AntimatterAt the time of the big bang when the universe was very hot and very dense,matter-antimatter particle pairs are believed to have been created in abun-dance and to have eventually cooled to produce today's observable universe.This paradigm has the fundamental problem of predicting a universe pop-ulated with equal parts matter and antimatter. The postulate remains instark contrast with our matter dominated universe and has posed a mysterywhich cries out for investigation.Formally the equivalence of matter and antimatter is expressed via theCPT theorem. CPT symmetry states that any local Lorentz-invariant quan-tum field theory is invariant under combined operations of charge-conjugation(C), parity (P), and time-reversal (T) [38]. Antimatter is what results whenthese operations are performed on matter. Every particle, therefore, hasan antimatter counter particle with identical mass but opposite charge andspin. The invariant nature of the symmetry would demand that the physicsof a pure antimatter system be identical to that of its matter analogue. Inessence, the goal of the ALPHA experiment is to observe a violation of thissymmetry. That is, we are attempting to detect a difference between anti-matter and matter, or at least to place constraints on possible differences.Our chosen antimatter system is anti-hydrogen: the bound state of apositron (anti-electron) to an anti-proton. Multiple measurements on otherantimatter systems have already been made and place upper bounds on thepossible discrepancies between certain quantities (for examples, see table1.1).1Hydrogen, however, has the advantage that it is a very simple, neutralsystem. As a result spectroscopic properties of hydrogen have been measured1Signals of CPT violation have also been observed in the cosmic microwave background(CMB) at 2σ [23] and by the BaBar collaboration using Bo−Bo oscillations at 2.2σ [15].11.2. Thesis overviewto extremely high precision. The most promising measurements are the 1S- 2S transition (which has been done to a relative precision of 4.2 × 10−15[45]), and the ground state hyperfine splitting (with a relative precision of10−12 [31]).Quantity RelativeprecisionAbsoluteEnergyDifferenceReferenceKo −Ko mass 4× 10−18 300 kHz [3]e+-e mass 8× 10−9 0.9 THz [22]charge 4× 10−8 - [32]gyromagnetic moment 2.1× 10−12 - [54]p¯− p mass 7× 10−10 159 THz [16]charge 7× 10−10 - [16]Table 1.1: A summary of matter-antimatter measurements. Where appli-cable I have included absolute energy difference (in frequency units) since ithas been argued to be the relevant figure of merit [6] rather than the relativeprecision.ALPHA is not likely to reach these precisions any-time soon, however afirst step towards hyperfine spectroscopy has been taken in 2012 when thespin of antihydrogen atoms was likely flipped [18]. Since demonstrating anability to trap ground state antihydrogen atoms for long periods of time (upto 15 minutes) [20, 19], the ALPHA apparatus has been modified to incorpo-rate laser windows for optical spectroscopy. In particular 1S-2S, 1S-2P andhyperfine spectroscopy measurements are being pursued. All of these mea-surements, however, are greatly hindered by the small number of antiatomswhich can be trapped. Typically only one or two atoms are contained at anygiven time!The goal of this work is to demonstrate a cooling technique which canalleviate this problem by increasing the number of trappable antihydrogenatoms at ALPHA.1.2 Thesis overviewThis chapter is intended to provide background, motivation, and context forthe work of this thesis. The backdrop is, of course, antimatter measure-21.2. Thesis overviewments, CPT violation and the ALPHA experiment. Section 1.1 provides aterse summary of antimatter mysteries and measurements. Section 1.3 willgive a very brief overview of the ALPHA apparatus and the antihydrogenproduction procedure.Chapter 2 is an overview of the Purcell effect. At heart a method forresonantly enhancing a spontaneous emission rate, this effect is the founda-tion on which our proposed cooling technique is built. The original theoret-ical framework as presented by Purcell is given and many of the original ex-periments are discussed. This chapter also characterizes a failed 2011 searchfor spontaneous emission enhancement at ALPHA. Finally, our own classi-cal model of the Purcell effect, specific to the case of a positron in cyclotronmotion, is developed and predictions about possible cooling enhancementsare made.Chapter 3 describes a novel microwave resonator designed for com-patibility with a Penning trap. Resonator design considerations arepresented along with cavity resonance characterization both via numericalsimulation and observation. Considerations for enhancing the cooling powerof this resonator with unique surface preparations are detailed.Chapter 4 models many body effects unique to this refrigerativeimplementation of the Purcell effect. Namely, if there are too manyparticles radiating energy into our cavity, its refrigeration powers will bediminished. This is a many body problem in the vicinity of a resonance.Numerical results are shown from systems of a few particles up to a fewhundred particles. The results are extrapolated in order to make predictionsabout the maximum number of particles our technique is capable of cooling.Chapter 5 describes a laser pass - microwave filter device wehave constructed. This filter is designed with a tubular geometry in-tended to allow laser access to antihydrogen trapping regions while attenuat-ing unwanted microwave radiation which might cause heating of the positronplasma.Chapter 6 describes measurements which characterize the mag-netic properties of the ALPHA Penning trap electrodes.31.3. ALPHA apparatusChapter 7 presents preliminary data which confirms our assertionthat positrons can be cooled via the Purcell effect.1.3 ALPHA apparatusThe ALPHA apparatus is, at its heart, a Penning trap superimposed witha magnetic minimum trap. The Penning trap is an apparatus for control-ling charged particles: positrons and antiprotons. Once these are mixedand (neutral) antihydrogen is formed, a magnetic minimum trap is used tocontain the antiatom(s).Figure 1.1: The Penning trap of the ALPHA apparatus is shown surroundedby Octupole and mirror coils which impose a magnetic minimum trap for antiatoms. Also shown is the silicon detector used to reconstruct annihilationevents. Figure adapted from [2].Antiprotons are created by colliding high energy protons on an iridiumtarget. The antiprotons produced by these collisions are decelerated to en-ergies of ' 5.3 MeV by a facility called the Antiprotron Decelerator (AD)[39]. These low energy antiprotons are shared between a number of antimat-ter and antihydrogen experiments which have developed around this uniquefacility. At ALPHA we direct antiprotons from the AD into a dedicated41.4. The Penning trapPenning trap (called the "catching trap") where they are cooled, both sym-pathetically by an electron plasma and, afterwards by evaporation. Oncecooled, antiprotons are transferred to the "mixing trap" shown in figure 1.1.Positrons are emitted via beta decay from a radioactive sodium-22 source.A Surko-type positron accumulator [43] stores these positrons for injectioninto the mixing trap.Once both antiprotons and positrons are in the same central Penningtrap, the two species are mixed using a technique called auto-resonant mix-ing [17]. The technique is tuned to mix the particles while heating themminimally. During mixing antihydrogen forms, likely through three bodycollisions [49] between two positrons and one antiproton.Most of the antiatoms which are created in this manner likely annihilateon the inner wall of the Penning trap. However, those in either of the uppertwo spin states, with a kinetic energy less than about 0.5 K can be trapped.(The trap depth is set by the strength of the octupole magnetic field at thePenning trap wall: Antiatoms with enough energy to reach higher fields willannihilate here and be lost.)The ALPHA apparatus in its entirety is an extremely complex deviceand many important aspects have been left out of this description. Some ofthe omitted subsystems include: particle detectors (silicon, caesium-iodide,plastic scintillators); plasma diagnostics (faraday cup, micro-channel plates,phosphor screens); vacuum and cryogenics; microwave equipment, lasers andoptics. Greater detail can be found in [2, 41].1.4 The Penning trapBecause of its central importance to both the ALPHA experiment and thepresent work, a more detailed overview of the Penning trap is provided here.The Penning trap is merely a segmented hollow metal tube which trapscharged particles using static electric and magnetic fields. Each segment,or electrode, of the tube is electrically isolated from all the others. Thisisolation allows each electrode to be held at a different voltage. To trappositrons, for example, a central electrode (or group of electrodes) is heldat some low voltage (ground in figure 1.2) while outer electrodes are held athigh voltage. If the potential energy hill provided by the end electrodes ishigh enough, particles will be axially confined.51.4. The Penning trapFigure 1.2: A sketch of a Penning trap which contains positrons axially viaelectrostatic forces. Figure adapted from [10].To obtain 3-dimensional confinement, a large axial magnetic field is ap-plied. This field imposes a centripetal Lorentz force:F = q~v × ~B (1.1)which causes the charged particles to oscillate in a circular motion (cyclotronmotion) as in figure 1.3.2Because these charged particles are accelerating, they are also radiating.This radiation is the primary cooling mechanism for positron plasmas at AL-PHA. Until very recently3the positron temperatures were typically 50 - 100K. Speculation is that radiative heating from objects in poor thermal contactwith the cryostat, electronic noise, and/or magnetic field inhomogeneitiescould prevent positrons from reaching temperatures set by the Penning trapwall at ' 7− 10 K.2A magnetron motion, where the plasma as a whole, spins about its axis, is alsopresent in trapped plasmas, though not discussed in this thesis.3Near the end of 2014, ALPHA saw positrons with temperatures as low as 30 K.61.4. The Penning trapFigure 1.3: An axial magnetic field imposes a centripetal force which causescircular motion. Radiation is emitted at the cyclotron frequency.A competition between these heating mechanisms with cooling due toradiation sets the positron temperature. In this light, Chapter 3 will pointout that a cavity can reduce the equilibrium positron temperature on twofronts. First the cooling rate will be greatly enhanced. Second, radiativeheating can be reduced since photons from warm external objects tend notto couple into the cavity. In the absence of any other heating mechanisms,the positrons would cool to the temperature of the Penning trap electrodes.7Chapter 2Overview of the Purcell Effect2.1 Theory and historyThe historical beginnings of this work lie in a 1946 abstract for a paperpublished by Purcell [1]. Purcell noted that nuclear magnetic resonance(NMR) transitions could be enhanced by a large factor if the sample undermeasurement is coupled to a resonant circuit4. In that work Purcell pointedout that the decay rate of some excited state |i > to some lower state |f > ismodified under this resonant coupling. This decay rate is expressed accordingto Fermi's Golden rule asΓ = 2piρ(E)| < f |H|i > |2h¯(2.1)where H is a perturbing Hamiltonian; ρ(E) is the density of states at energyEf − Ei = hν, for frequency ν; and where 2pih¯ is Plank's constant.If an excited state of some system decays in a free space environment, thedensity of states for the emitted photon is 8piν2/c3. In the case of couplingto a resonator, however, there now exists only one state in the frequencybandwidth of the resonator ∆ω. The enhancement factor Purcell derived is3Qλ34pi2V(2.2)where λ is the resonant wavelength of radiation emitted by NMR transitions,Q is the resonator quality factor and V is the resonator volume.Decades later, Kleppner [37] would elucidate a physical interpretation forthis effect: that the resonator has enhanced vacuum field fluctuations at theresonant frequency (ν) which "stimulate" spontaneous emission. Indeed thestate transition frequency must be matched to a resonance of the environ-ment for enhancement to occur. If the emission frequency is off resonance thedensity of states approaches zero and spontaneous emission can be greatlyinhibited.4In NMR this effect is usually termed "radiation damping" [4].82.1. Theory and historyEnvironments inhibiting and enhancing spontaneous emission are sketchedqualitatively in figure 2.1 alongside a free space environment. Both inhibitedand enhanced spontaneous emission have been observed across a number ofdiverse systems throughout the 1980's. These experiments (to be discussedfurther in the next section) involve electron plasmas, Rydberg atoms, semi-conductor devices and dye molecules.Figure 2.1: Three separate environments are sketched. a) shows a metallictwo dimensional box which constrains the density of states for photons emit-ted from a radiating dipole (red arrow). If the electromagnetic resonancefrequency of the box is tuned to the characteristic emission frequency of thedipole, spontaneous emission is enhanced. b) Shows the same dipole in a boxsignificantly smaller than the resonant wavelength. No states exist for pho-tons at this frequency and the dipole will not emit. c) Free space boundaryconditions allow the dipole to emit "normally".Although not the focus of this thesis, it would be imprudent to omittwo other main features of this theoretical framework. Firstly, in additionto affecting the spontaneous emission rate of a multi-state system, a cavityalso affects the energy levels of that system. That is, the energies of states|i > and |f > when coupled to a cavity are different than in free space.Fortunately (for this work) the energy shift is a higher order effect whichcan be ignored when one is solely interested in the decay rate, Γ.Secondly, and more importantly, all these cavity effects fall under thelarger branch of what is now known as Cavity Quantum Electrodynamics,or Cavity QED. Cavity QED represents a large and interesting field in itsown right. While emission rate and energy level shifts typically occur un-der weak cavity coupling, the strong coupling regime offers an even richervariety of physics. Under strong coupling, cavity photons can become en-tangled with a matter system. These systems offer opportunities for tests ofquantum mechanics, quantum non-demolition measurements, and potential92.2. Some experimentsapplications in quantum information [56]. Most of cavity QED, however, liesbeyond the scope of this thesis and will not be considered further.Finally, applications of the Purcell effect are also widespread in solid statesystems where device performance is often limited by spontaneous emission.Much work been done to improve these limits using cavities or cavity-likestructures [62].2.2 Some experimentsAlthough the first experiment was actually performed with dye molecules[13], experimental Cavity QED has largely resided within the realms ofatomic and solid state physics. This section will therefore provide a briefreview of experiments from these fields which demonstrate control of spon-taneous emission before moving on to the context of lepton plasmas withwhich we are primarily concerned. For a more comprehensive review see[56].2.2.1 Rydberg atomsIn 1981 Kleppner first proposed that Rydberg atoms would make good candi-dates for observations of spontaneous emission control [37]. Rydberg atomswere attractive candidates since transitions between closely spaced energylevels would produce resonance frequencies of hundreds of GHz. This meantthat the wavelength of the emitted radiation would be on the scale of µm ormm and that cavities could therefore be easily constructed to constrain or en-hance this radiation. The environment Kleppner proposed was a waveguidein the vicinity of cut-off.Given that the mode structure of waveguides is well known [48], Kleppnerwas able to analytically calculate the density of states (see figure 2.2). Thehigh frequency, long wavelength limit approaches free space conditions butlarge departures from this limit appear near waveguide cut-off frequencieswhere the density of states for photons is very high.With this geometry in mind, prototypical Rydberg atom experimentswere conceived as in figure 2.3. Generally some atomic Rydberg state iscreated in an atomic beam by use of appropriate laser excitation. Next thebeam is directed into a cavity consisting of two metallic plates or mirrorswhich constrain the density of states for photons in that region. After sometransit time the states of atoms in the beam are analysed usually by a com-bination of laser induced transitions and ionization. A signal is produced inthe detection region by tuning the laser or ionizing electric field such that102.2. Some experimentsFigure 2.2: A reproduction of Kleppner's 1981 calculation for the photonicdensity of states in a rectangular waveguide. The result is compared withthe free space case112.2. Some experimentsatoms are ionized according to their state. By comparing the transit timefor atoms to cross the cavity to the spontaneous emission rate in free space,inhibition or enhancement can be observed.Figure 2.3: A typical experiment exhibiting spontaneous emission control ofRydberg atoms.In [33] inhibited spontaneous emission was inferred when Caesium atomsin state |n, |m| >= |22, 21 > were observed to transit a cavity without de-caying. The cavity transit time being about equal to the free space decaytime meant that more |22, 21 > atoms arrived on the other side of the cavitythan should have. The transition resonance was tuned using the Stark effect:by applying an electric field to the Rydberg atom, the resonant transitionfrequency could be shifted above and below the cut-off of the cavity.A similar experiment was performed in [35] using Caesium atoms in the5D5/2 state. Although multiple atomic decay channels from that state exist,spontaneous emission inhibition was observed when some of those channelsare blocked. Decay channels with ∆m = ±1 emit photons polarized differ-ently than those with∆m = 0, and coupling to these decay channels could betuned by varying a background magnetic field such that this field directionwas either parallel or perpendicular to the cavity plates.The first observation of enhanced spontaneous emission [26] followed thearchetypal set-up of figure 2.3 except that the cavity transit time was shorterthan the free space decay time. Sodium atoms prepared in the 23S state wereobserved in the detection region to have decayed to states 22P1/2 and 22P3/2,much faster than predicted by free space spontaneous emission. An addi-122.2. Some experimentstional difference between this and previous work includes the low temper-ature nature of the apparatus. Superconducting niobium plates in contactwith a liquid helium bath were used to form the cavity. The superconduct-ing plates set a cavity Q ' 106 . Additionally, the atomic production regionwas separated from cavity and state detection region by thermal shields andmicrowave absorbers. This cold environment had the advantage that back-ground (blackbody) radiation induced emission from the Rydberg atoms wasnegligible. The radiation temperature was set by the temperature of the cav-ity plates (about 7 K). Tuning the cavity on or off the transition resonancewas achieved by moving one plate with a fine tuning screw (with the greatestenhancement observed on resonance).Later, transitions at optical frequencies in ytterbium atoms were observedin a Fabry-Perot cavity [30]. For some time these resonators were thoughtto be impractical since the enhancement factor (λ3/V from equation 2.2) issmall at optical frequencies. To counteract this effect the experiment usesa large number of degenerate (transverse mode) resonances. They observecontrol of spontaneous emission by measuring florescence emitted from atomsin the resonator.In a similar set up, this group was able to observe energy level shiftsin Barium atoms resulting from cavity interactions [29]. Energy levels aremapped by weakly coupling a laser into the Fabry-Perot resonator, and scan-ning both the cavity length and laser frequency. When the cavity comes intoresonance with the atomic transition, many excited atoms suddenly decayand a lower intensity of florescence is measured.2.2.2 Solid state systemsSoon after the initial results in atomic and plasma physics Yablonovitch [61]noted that it might be beneficial to control spontaneous emission in electron-hole pairs in semiconductors. These electron-hole pairs form dipoles withenergy levels and radiation patterns not dissimilar to atomic systems. Thepractical devices (lasers, transistors, solar cells etc) based on these materialsoften exhibit performance limited by spontaneous emission.Enhancing spontaneous emission rates can provide lasers with largerbandwidth (when tunability is desired), lower lasing thresholds, and in-creased quantum efficiency [62]. Inhibited spontaneous emission rates canbe used to sharpen spectral features and enhance operation speeds of lightemitting diodes (LEDs). In the strong coupling regime where Cavity QEDeffects produce entanglement of emitters and photons, quantum cryptogra-phy applications result [57].132.2. Some experimentsYablonovitch proposed that a Fabry-Perot type resonator with distribu-tive Bragg reflectors (DBR) in place of mirrors surrounding a semi-conductorfiled cavity (see figure 2.4) would provide the sought after control.5Figure 2.4: A distributive Bragg reflector is constructed of alternating layersof high and low dielectric materials. Each layer has a thickness λ/4 whichenforces an interference pattern demanding reflection. If two such structureswere placed on either side of a semi conducting slab, a Fabry Perot cavity isformed which constrains spontaneous emission.Demonstration of spontaneous emission control came soon afterwards[60] with a different experiment than proposed above. These authors placedthin semi-conducting films on a variety of different substrates in an effortto change the local electric field which induces spontaneous emission. Asmotivation, the authors note that the electric field creation operator ( E+ )within Fermi's Golden rule (expressed as Γ = 2piρ(E) |<xE+>|2h¯ , with x thedipole operator) is also modified by the environment.65Yablonovitch thought of the DBR as having a "photonic band gap" - a range offrequencies for which electromagnetic wave propagation was forbidden. This thoughtforeshadows the field of photonic crystals where 3-dimensional periodic structures exhibitphotonic band gaps with widespread applications, among them control of spontaneousemission [44].6We also hope to capitalize on this type of enhancement. See section 2.4 for a discussion142.2. Some experimentsAs an example, if a semi-conducting sphere is surrounded by some ex-ternal material with a different index of refraction η (rather than an indexmatched material), then the field needed to be rescaled according to standardresults from electrostatics:E+int =3η3int/η3ext + 2E+ext (2.3)By varying the substrate which supports the semi-conductor sample, radia-tive recombination was inhibited by a factor of about 5.2.2.3 PlasmasIn 1985 Gabrielse observed the microwave cavity formed by a set of Penningtrap electrodes to inhibit spontaneous emission [25].Those electrodes took theform of a metallic box with hyperbolic walls7as in figure 2.5. The experimentwas conducted in a magnetic field of 6 T which sets the cyclotron wavelengthto be about 2 mm. Due to the hyperbolic shape of the electrodes, the exactmicrowave mode structure was not known. It was, however, inferred thatcavity effects must play a role since the distance between the electron andmetal wall (' 6.7 mm) was comparable to the cyclotron wavelength.This work was carried out in the context of precision measurements ofthe electron g-factor. Measurements of g required precision measurementsof the cyclotron frequency (ωc). As with Rydberg atoms, the cavity signif-icantly perturbed the energy levels of the system (which, in this case, setthe cyclotron frequency). Cavity effects had to be accounted for [9, 46] andmuch experimental and theoretical work was devoted to this end.To make theoretical predictions about this effect, the authors includedelectric forces resulting from microwave resonances in the Penning trap in theequations of motion for their trapped electron [9, 8] (equation 2.4). Otherterms in equation 2.4 result from the usual trapping forces; namely, theapplied magnetic and electrostatic field.~˙v − ~ωc × ~v + (e/m)∇V (~r) +12γc~v = (e/m) ~E(~r) (2.4)By changing to a circular cylindrical Penning trap geometry the microwaveresonances of the system became analytically tractable. Solving for~E(~r)with the method of images the authors were able to predict shifts in ωc andof our fill factor.7This choice of electrode geometry was selected to obtain a quadrupole electrostaticpotential.152.2. Some experimentsFigure 2.5: Hyperbolic electrodes were used to create a harmonic axial po-tential. The electrodes are circularly symmetric about the vertical axis.162.3. A null result at ALPHAγc resulting from interaction with microwave resonances of the trap. In par-ticular those resonances with strong transverse electric fields at the locationof the electron were shown to strongly perturb the cyclotron frequency andenhance the spontaneous emission rate.Methods for detecting cavity modes in-situ were developed [53, 50, 51,27]. These methods made use of equivalent circuit analyses [12, 59], and theperturbative coupling of the axial plasma oscillation to the cyclotron and spindegrees of freedom via magnetic nickel strips. Voltage measurements acrosselectrodes at the axial plasma frequency, ωz (typically in the MHz regime),allow inference of the cyclotron spontaneous decay rate and a mapping ofthe cavity resonance frequencies.2.3 A null result at ALPHAThe first attempt to realize the Purcell Effect as a cooling technique atALPHA occurred in 2011.By matching the cyclotron frequency of positron plasmas to a microwaveresonance of the ALPHA Penning trap (measured in [21]) we hoped to ob-serve plasma temperatures drop to a new equilibrium temperature set by thePenning trap wall. The drop should occur at a much enhanced (relative tofree space) rate set by the Purcell Effect.By slowly incrementing the axial magnetic field we were able to scan thecyclotron frequency over a region thought to contain microwave resonances.We simultaneously monitor the temperature via a non-destructive technique[14] sketched in figure 2.6.The technique excites and measures the quadrupole mode of electrostaticoscillation in the plasma. Detection is achieved through voltages induced bythe plasma oscillation on a nearby electrode.The quadrupole mode frequency, f2, depends on the plasma temperature(T ) according to(f ′2)2 − (f2)2 = 5(3−α22f2p(f c2)2∂2g(α)∂α2)kB∆Tmpi2L2(2.5)where α = L/2r is the plasma aspect ratio, f c2 is the quadrupole fre-quency in the cold fluid limit, kB is the Boltzmann constant, and g(α) =2Q1(α/√α2 − 1)/(α2− 1) with Q1 the first order Legendre function of thesecond kind. The plasma frequency is fp = 12pi√ne2/mo, where n is theplasma number density.172.3. A null result at ALPHAIn the limitf ′2−f2f2 1, changes in the plasma quadrupole oscillationfrequency are proportional to changes in the plasma temperature.∆f2 ' β∆T (2.6)By measuring changes in this oscillation frequency, the relative plasma tem-perature can be monitored.Figure 2.6: The plasma quadrupole mode inside a Penning trap is sketched. A transmission line attached to a nearby electrode detects this oscillationsignal via image charges induced in the electrode.Figure 2.7 shows the plasma temperature and quadrupole frequency ver-sus the cyclotron frequency. The plot reveals complicated structure notconsistent with the resonant frequencies of the trap. Additionally the tem-perature shows no great drop at any frequency. Multiple cross checks weremade on this measurement involving alternate temperature measurementmethods, magnetic field calibrations, reduction of positron number withinthe plasma, and simple repetition, none of which revealed an obvious result.A more detailed description of that test can be found in [24].Two candidates for potential disruption of the cavity cooling effect atALPHA have been identified. Firstly, these experiments were conductedwith N ' 105 - 106 positrons, a number much greater than other experiments182.3. A null result at ALPHAFigure 2.7: The quadrupole mode is monitored while scanning the cyclotronfrequency across the measured frequencies of three cavity modes. No obviousmode structure revealed itself in the positron temperature measurements.Figure adapted from [24].192.4. Circuit formulation of the Purcell Effect for a cyclotron oscillatormentioned above. If the cooling effect is distributed over too many electronsthen it could feasibly become ineffective and negligible. This many bodyeffect is discussed in Chapter 4 and, using different techniques , in [47] . Ourchosen solution is outlined in section 3.4.Secondly, the properties of microwave resonances in the replica ALPHAstack may vary significantly depending on the environment. This seemslikely since microwaves were found to leak through the gaps between elec-trodes allowing environmental factors to influence cavity resonances. Forexample, a number of striplines run axially down the outside of the ALPHAstack at CERN which were not present for the measurements in [21]. Thesestriplines could act as antennas, carrying energy away and reducing boththe Q and the positron-cavity coupling (the fill factor, χ, of section 2.4). Tomitigate possible microwave leakage at gaps between electrodes and desen-sitize the cavity to its surrounding environment, choke structures have beenincorporated into the cavity presented in chapter 3.2.4 Circuit formulation of the Purcell Effect for acyclotron oscillatorThis section presents our own formulation of the Purcell effect for the case ofa single charged particle in a strong magnetic field. For comparison's sake,we first derive the free space decay rate. Both models are purely classical.2.4.1 Cooling in free spaceThe cyclotron cooling rate of a particle in a field B, due to radiation intofree space is given by1/τc =PEKE(2.7)where P is the power radiated by a charged particle in a circular orbit andEKE is the kinetic energy of the particle. Combining the Larmor formulafor the radiation of an accelerated particleP =e2a26pioc3(2.8)with EKE = 12mω2r2, one obtains the well known formula202.4. Circuit formulation of the Purcell Effect for a cyclotron oscillator1/τc =e2ω23pioc3m=e4B23pioc3m3. (2.9)For a plasma sufficiently dense that the (two) cyclotron degrees of freedomeasily equilibrate with the (third) axial degree of freedom, this cooling ratehas to be multiplied by a factor 2/3 to give:1/τnet =2e4B29pioc3m3. (2.10)For electrons and positrons in a 1 Tesla field this gives τc = 3.87 sec, to becompared to measured values of about 4 sec in the ALPHA apparatus.Note that this result is derived for a single charged particle. It is appli-cable to a dilute plasma under the assumption that the individual phasesof the cyclotron orbits of the particles are randomly distributed. Also, theformula is only valid in the classical limit, where h¯ω is much less than theaverage excitation energy of the cyclotron levels. For electrons and positronsin a 1 Tesla field, h¯ω ' 1.5 Kelvin, so that we expect the results to apply tothe ALPHA lepton plasmas.2.4.2 Cyclotron cooling of a particle inside a resonantcavityA cavity with conducting walls can strongly perturb the spectrum of thefinal states into which a particle can decay. We use a very simple model of acharged particle in a B-field parallel to the plates of a capacitor (figure 2.8,and then generalize the result to an arbitrarily shaped cavity.Figure 2.8: An electron in circular orbit in a capacitorThe oscillating component of the charge on the capacitor is 2er cos(ωt)/dso that the open circuit voltage across the capacitor is212.4. Circuit formulation of the Purcell Effect for a cyclotron oscillatorv =qC=2er cos(ωt)d× (oA/d)=er cos(ωt)oA(2.11)where ω is the cyclotron frequency eB/m, and d is the gap between capacitiveplates of area A.Figure 2.9: The capacitor of figure 2.8 in a resonant circuit.If we now put the capacitor in a series resonant circuit (see figure 2.9)such that ωo = 1/√LC coincides with the cyclotron frequency, then theaverage power delivered to the resistance R isP =< v2 >R=4e2r222oA2R(2.12)Therefore the damping rate of the cyclotron motion is1/τc =PEKE=4e2oA2mω2R. (2.13)Using R = 1/(ωCQ) we obtain1/τc =4e2QomωVc. (2.14)where Vc = Ad is the volume of the capacitor.Using ω = eB/m and including the factor of 2/3 to account for the axialdegree of freedom one obtains1/τnet =8eQ3oBVc. (2.15)222.4. Circuit formulation of the Purcell Effect for a cyclotron oscillatorFor a cavity with non-uniform values of the electric field~E, this generalizesto1/τnet =8eQ3oBVcE2o< E2 >(2.16)or more formally1/τnet =8eQ3oBE2o´E2 dV(2.17)Here, for a particular cavity mode, Eo is the value of the electric field normalto the cyclotron orbit when the cavity is excited and, at the same excitationlevel, < E2 > is the mean square electric field in the cavity. For a simplecapacitor with d√A, ~E is uniform and χ = 1/V . For the bulge resonatorproposed in section 3 (with V = 26cm3) the fill factor is seen to vary byfactors of 10 depending on mode type. Simulation and measurements of χyieldχ =E2o´E2 dV' 1061m3. (2.18)for the TE1lm modes.As an example, a cavity with the above fill factor, a resonant frequencyat 28 GHz with Q = 4000 predicts a cooling time of1/τnet ' 5ms (2.19)representing an improvement over the free space case by a factor of about1000.2.4.3 Decay rate for an oscillator inside a cavity withnon-uniform fieldsConsider the case of a physically small LC circuit inside the capacitor ofa physically large LRC circuit as in figure 2.10. The mutual capacitancewill be used to calculate the voltage induced on the latter resonator by theformer. The cooling rate will be derived in this context in order to motivatethe fill factor, χ.The decay rate is now232.4. Circuit formulation of the Purcell Effect for a cyclotron oscillatorFigure 2.10: A small loss-less LC resonator is placed near the capacitiveedge of a large, lossy LRC resonator.242.4. Circuit formulation of the Purcell Effect for a cyclotron oscillator1/τ =PU1(2.20)where P =< V 22 > /R2 is the power dissipated in the large resonator andU1 =q212C1is the energy stored in the small resonator.The voltages across these capacitors (for respective charges q1 and q2) areV1 =q1C1+q2C21V2 =q2C2+q1C12Neglecting the self-capacitive terms we haveP =q212C212R2(2.21)so that the decay rate (with the definition for U1) is1/τ =C1C212R2(2.22)In order to find C12 we make use of the reciprocity of mutual capacitanceC12 = C21 (2.23)and writeV1 'q2C21V1 =E1d1V1 =E2(r)d1to obtainC12 = C21 =q2E2(r)d1(2.24)together with C1 = oA1/d1 the decay rate is1/τ =oA1d21E22(r)d1q22R2(2.25)252.4. Circuit formulation of the Purcell Effect for a cyclotron oscillatorThe energy in the second capacitor U2 =q222C2= o2´E2 dV allows us toremove q2 from equation 2.25.1/τ =V1C2R2E22(r)´E2 dV(2.26)Finally we use Q2ωo = 1/R2C2 to arrive at1/τ = V1ωoQ2χ (2.27)This can be seen to match the result of the last section if one allows somephenomenological or effective volumeV1 =4e2omω2o(2.28)2.4.4 Comparison with the quantum resultBeginning with Fermi's Golden Rule (equation 2.1) we attempt to derive thecavity enhanced cooling rate.Γ = 2piρ(E)| < f |H|i > |2h¯(2.29)The perturbing Hamiltonian results from a dipole interaction:H = exE(x) (2.30)with e the electron charge, x the charge's position and E(x) the electricfield at that location. Expressing the position operator as a sum of raisingand lowering operators (x =√h¯/2mωo[a+ + a−]), allows us to compute theemission and absorption rates:Γemission =piρe2E2(x)mωo(l + 1)Γabsorption =piρe2E2(x)mωolwhere l is the quantum number for the harmonic oscillator excitation andwe have used a−|l >=√l|l − 1 > and a+|l >=√l + 1|l + 1 >.262.4. Circuit formulation of the Purcell Effect for a cyclotron oscillatorThe cooling rate isΓ =Γemission − Γabsorption=piρe2E2(x)2mωo=pie2E2(x)2mωoh¯∆ω=piQe2E2(x)2mω2o h¯where we have used the density of states appropriate for a cavity resonance.ρ =1h¯∆ω=Qh¯ωoFollowing [7] we normalize the electric field according to the number of pho-tons in the cavity. We perform the rescaling E2 → E2/n withnh¯ωo =o2ˆE2 dV. (2.31)The cooling rate becomesΓ =piQe2E2(x)h¯mω2o2h¯ωoo´E2 dV=2piQe2omωoχwhich matches the classical expression up to a factor of pi/2.27Chapter 3The Bulge CavityThis chapter of the thesis describes a microwave cavity which has been shownto cool electrons at a collaborative plasma experiment at Berkeley. A de-scription of the Berkeley apparatus can be found in [47].The Penning trap at Berkeley is designed in the style of that used at AL-PHA and often used as a testing ground for new plasma techniques. Absentfrom the experiment at Berkeley are many of the strict design requirementsnecessary for a full antihydrogen experiment. This relaxation allows us to de-sign a so called bulge resonator compatible with the open tubular geometryof this style of Penning trap. The resonator is shown in figure 3.5.3.1 Resonances of a cylindrical cavityFigure 3.1: The right circular cylinder represents the inner space of ametallic container.The resonator described in this section strongly resembles a simple cylin-drical cavity (figure 3.1). Therefore, throughout our study we make use ofthe nomenclature surrounding this geometry. Additionally, we often revertto thinking about the right circular cylinder when testing simple ideas ordeveloping intuition about the behaviour of the more complicated cavityresonances to be presented in later sections. Accordingly, a short section isnow devoted to electromagnetic resonances inside a conducting cylinder.283.1. Resonances of a cylindrical cavityBeginning from the Helmholtz equation ( ∇2 ~E + k2 ~E = 0 ) for electricand magnetic fields, one may split the Laplacian operator into parts as∇2 → ∇2T +∂2∂z2(3.1)where ∇2T represents the transverse Laplacian in the cylindrical coordinatesr and φ. Assuming that these electromagnetic waves propagate axially, thez dependence satisfies∂2 ~E(r, φ, z)∂z2= −β2 ~E(r, φ, z) (3.2)and the Helmholtz equation becomes∇2T ~E = k2c~E (3.3)where the cut-off wavevector has been defined k2c = k2 − β2.By using Maxwell's equations ( ∇ × ~E = −iω ~H and ∇ × ~H = iω ~E )together with tedious algebra, one may show that all field components canbe expressed in terms of the axial fields only.Er = −ik2c(β∂Ez∂r+ωµr∂Hz∂φ)Eφ =ik2c(−βr∂Ez∂φ+ ωµ∂Hz∂r)Hr =ik2c(ωr∂Ez∂φ+ β∂Hz∂r)Hφ = −ik2c(ω∂Ez∂r+βr∂Hz∂φ)(3.4)Therefore, a solution to∇2TEz = k2cEz (3.5)together with the correct boundary conditions for the cylinder will yieldcomplete field solutions to cavity resonances.These solutions generally fall into two categories: transverse electric (TE)waves and transverse magnetic (TM) waves. The former have Ez = 0 andthe latter have Hz = 0. Separation of variables shows that, for the TE waves,293.1. Resonances of a cylindrical cavityHz = (AJn(kcr) +BNn(kcr)) (C cos(nφ) +D sin(nφ)) (3.6)where Jn and Nn are the nth order Bessel functions of the first and secondkind respectively. Applying boundary conditions allows us to reduce thisexpression considerably. B = 0 since fields must be finite at the origin(r = 0) and D = 0 by a judicious choice of the origin for φ.Figure 3.2: Electric fields for transverse electric resonances of a cylindricalcavity: TE01 , TE11 ,TE21, TE02, TE12, TE22. The above field contours aretaken for sin(βz) = 1.The conducting boundary conditions on the radius of the cylinder (Eφ =0 at r = a) giveskc =p′nlawhere p′nl is the zero of of the relevant Bessel function derivative (J′(p′nl) =0). Imposing the axial boundary conditions on equations 3.4303.1. Resonances of a cylindrical cavityHz = AJn(kcr) cos(nφ) (AL exp(−iβz) +AR exp(+iβz))Er ∝ Eφ ∝ i (AL exp(−iβz) +AR exp(+iβz))Hr ∝ Hφ ∝ i (−βAL exp(−iβz) + βAR exp(+iβz)) ,gives AR = −1/2 and AL = 1/2. As a result Er ∝ Eφ ∝ sin(βz) with an"axial wavevector" β = mpiL .Finally, the solutions take the formHz = AJn(kcr) cos(nφ) sin(mpiz/L)Hφ =nβk2crAJn(kcr) sin(nφ) cos(mpiz/L)Hr =βkcAJ ′n(kcr) cos(nφ) cos(mpiz/L)Er = −iωµnk2crAJn(kcr) sin(nφ) sin(mpiz/L)Eφ = −iωµkcAJ ′n(kcr) cos(nφ) sin(mpiz/L)where the factor of i in the electric field expressions represents a pi/2 phaseshift (in time) relative to the magnetic fields. The fields of a specific cav-ity resonance are completely determined by the three wavenumbers: n, l,and m. It is common, therefore, to denote cavity resonances with the no-tation TEnlm. These field patterns are shown in figure 3.2 for a handful ofresonances.The frequency of such a resonance can be writtenflnm = c√(p′nl2pia)2+(m2L)2The corresponding solutions for TM modes (shown in figure 3.3) are313.1. Resonances of a cylindrical cavityHφ = −iωkcAJ ′n(kcr) cos(nφ) cos(mpiz/L)Hr = −inωk2crAJn(kcr) sin(nφ) cos(mpiz/L)Ez = AJn(kcr) cos(nφ) cos(mpiz/L)Er =βkcAJ ′n(kcr) cos(nφ) sin(mpiz/L)Eφ =βnk2crAJn(kcr) sin(nφ) sin(mpiz/L)withfnlm = c√( pnl2pia)2+(m2L)2Figure 3.3: Electric fields of TMnlm resonances of a cylindrical cavity. :TM01 , TM11 ,TM21, TM02, TM12, TM22. The fields point axially out ofthe page. The above field contours are taken for sin(βz) = 1.The axial variation for a few of these modes is shown in figure 3.4.323.2. Bulge designFigure 3.4: Axial dependence of TE and TM modes.3.2 Bulge designOur bulge resonator maintains the cylindrical symmetry of the right circu-lar cylinder (above), but removes the traditional axial boundary conditions.Instead we create some localized region where a microwave mode can propa-gate, surrounded by regions where propagation is forbidden. To achieve this,the inner radius swells across three electrodes from small end regions to alarger central region. In the central region a mode can propagate, while inthe end regions this same mode is beyond cut off and must decay exponen-tially. Both the outer surfaces of the electrodes and the segmented natureof the cavity were designed to be compatible with the existing Penning trapat Berkeley.The inner bulge surface is parametrized asr =− 1.25mm cos(t) + 11.25mmz =L2pit+ bwhere L = 75.6 mm, b is a constant that places the maximum radius inthe middle of the central electrode and t ∈ (0, 2pi). Detailed drawings areincluded in appendix A.This geometry was chosen to produce a high Q resonance near the elec-tron cyclotron frequency at 1 Tesla (28 GHz). A high Q dictates a highemission rate according to the Purcell effect. In order to couple to the cy-clotron motion the resonance must also have strong transverse electric fields333.2. Bulge designFigure 3.5: Profiles of the bulge resonator are shown. a) shows an isometricouter view of three electrodes; b) the same view in cross-section with elec-trically isolating ceramics labelled; c) shows another cross-section labelledaccording to regions which trap microwaves versus regions in which propa-gation of these modes is forbidden. The inset of (c) shows the choke structurewhich prevents microwave leakage through the gaps between electrodes.343.3. Simulations of resonant fields and frequenciesat the location of the plasma. For on-axis plasmas this limits useful reso-nances to the set of TE1lm and TM1lm modes. Additionally, higher order(higher l) TE modes are generally preferred since they produce the mostfavourable fill factors (χ, equation 2.17 ). We also note that an appropri-ately smooth evolution of the inner radius is required. Sharp steps in theradius may case so-called mode conversion, leakage of microwave energy fromthe cavity, and consequent lowering of the resonance Q.3.3 Simulations of resonant fields and frequenciesAlthough we use the right circular cylinder to guide our intuition about mi-crowave resonances in this structure, the exact boundary conditions for thisgeometry are significantly more complicated and make analytical solutionsfor the resonant electric and magnetic fields impossible. In order to deter-mine resonant frequencies of this cavity, a commercial microwave simulationprogram (HFSS) is employed. Using finite element analysis multiple trappedmodes of the bulge are identified. As expected, resonances resemble thoseof the right circular cylinder. Figure 3.6 shows field patterns of the TE131mode identified via simulation. The axial dependence of the magnitude ofthe electric field is shown in figure 3.7 where one sees that the mode is welllocalized within the bulge region.Gaps between electrodes (as well as the choke structure of section 3.6) arenot shown but were included and found to have little effect on the resonances.Opened axial faces of end electrodes are approximated to have free spaceboundary conditions (Z = 377 Ω). The radial face is modelled as a generalconductor having the resistivity of our alloy (see section 3.4 and appendixB). A summary of the simulation results for cavity modes observed to coolelectrons is in table 3.1.3.4 Lowering the cavity QAn attempt to lower the Qmay seem contrary to our desire for strong coolingvia a high spontaneous emission rate according to equation 2.17. However,this cooling rate prediction was made for a single particle interacting withthe cavity. To be useful for anti-hydrogen production, many positrons mustbe cooled. Indeed a typical experiment either at Berkeley or ALPHA involvesmillions or hundreds of millions of leptons.In light of the many-particle reality, it seems possible that the cavitywould be receiving energy from leptons faster than it can dissipate this en-353.4. Lowering the cavity QFigure 3.6: Simulated resonant electric fields resembling a TE131 modeare shown. The frequency and Q of this mode were near 33 GHz and 4000respectively.Cavity Mode fo (GHz) Q χ (106) m−3 τ (ms) (equation 2.17)TE121 21.8 1580 1.66 6.2TE123 25.3 1090 0.99 17.4TE131 34.3 2660 3.20 2.9TE132 36.5 2200 2.46 5.2TE133 38.4 1440 2.10 9.2TE134 40.1 1300 1.63 14.0Table 3.1: A summary of the simulated cavity resonances. Only the TE131was characterized experimentally. Its measured frequency and Q were fo =33.80 GHz and Q ' 2500. χ = E2o/´E2 dV is the fill factor (equation 2.18).363.4. Lowering the cavity QFigure 3.7: |E(r = 0)|2, the simulated on-axis electric field magnitude isshown as a function of the axial coordinate z.373.4. Lowering the cavity Qergy. A deeper analysis of this effect can be found in chapter 4 and by Povilusin [47], both of which make predictions for the maximum number of leptonsa cavity can cool. Both of these analyses conclude that, in order to cool alarge number of electrons, the cavity Q needs to be appropriately lowered soas to remove energy from the lepton-cavity coupled system faster.Energy loss from our cavity is set by two primary mechanisms: leakageout the axial end faces, and resistive losses in the metallic cavity walls.Ideally, radiation should not be able to couple into or out of our cavity.Such coupling would allow room temperature blackbody radiation from re-moved parts of the experiment to affect the equilibrium temperature of theplasmas. One expects a copper cavity of our size, with no leakage, to have aQ ' 104. That the measured (and simulated) Q of this cavity is only ' 103indicates that microwave leakage is significant. Unfortunately, reducing lossby this mechanism would require a redesign of the cavity geometry.The second energy loss mechanism, resistive losses in the cavity, can beaffected by coating the inner face of the copper cavity with a resistive alloy8.Isaac Martens of the Bizzotto Electro-chemistry group at UBC intervenedhere to electroplate the inner face of the resonator with a Nichrome-likealloy. Nichrome (nickel-chromium) is a metallic alloy with a temperatureindependent conductivity a factor of about 100 lower than room temperaturecopper.If resistive losses dominated the resonatorQ, we would expect a reductionin Q by a factor√σNichromeσcopper' 10 (3.7)However, since microwave leakage is not negligible, we see some intermediatereduction and achieve Q ' 2200 in both simulation and experiment. (Thisrepresents reduction by a factor of ' 2 from the case of a bare copper cavitywall: Q = 4500 ).We note that it was not obvious beforehand that this method for loweringthe Q would work. Discussion is in appendix B8Attempts to lower the cavity Q by use of a coupling loop (see, for example, figure3.11) terminated by a cold load, were attempted but not used. Simulations showed thatthis kind of local perturbation caused severe reduction of the fill factor due to generationof unwanted waveguide modes.383.4. Lowering the cavity Q3.4.1 Anti-static coatingsNichrome, like copper, will oxidize. The oxide will be poorly conducting andmay hold a static charge which could compromise the cylindrical symmetryof the trap. This has the potential to induce diocotron instabilities [58]which threaten our control of the plasma. Therefore, as is common with anyPenning trap apparatus, the surfaces exposed to the plasmas must be coatedin some non-oxidizing material. Usually gold is electroplated onto the trapelectrodes.GoldCoating the inner surface of our microwave cavity with gold would also raisethe cavity Q thus counteracting the effect of the nichrome. We investigatedwhether or not a thin layer of gold could be used as an anti-static coatingwithout shielding cavity modes from resistive losses which are desired in theunderlying nichrome layer. Calculations and experimental confirmation areleft to an appendix, but our conclusion is that a thin gold layer (thinner thanthe skin depth) would be acceptable. At 30 GHz the skin depth of gold is '450 nm and results (see figure B.4) show that a gold layer of 100 nm wouldnot change the surface resistance unbearably.GraphiteAnother solution for anti-static coatings exists: colloidal graphite. Graphiteis preferable since it does not require difficult electroplating processes. Also,the shielding effect which occurs with a gold anti-static layer is negligible.Manufacturer specifications [52] indicate a graphite (Acheson Aerodag G)resistivity ofρg ' 3× 10−2 Ω ·m (3.8)which implies a skin depth at 30 GHz of ' 0.5 mm. As a test of the graphite'slow temperature resistivity, we applied about 1 µm of graphite to a glass tubeand measured the sheet resistance in a liquid helium bath. The results areshown in table 3.2.300 K 4 KRs kΩ/square 24 89Table 3.2: Measured sheet resistance of Aerodag G at room-temperatureand at 4 K.393.4. Lowering the cavity QAn increase in the resistivity of a factor of about 3.5 is seen at lowtemperatures. This is consistent with literature results [36]. Since the anti-static layer is much thinner than the skin depth, its effect on the microwavelosses in the cavity will be negligible.For this cavity we choose the graphite anti-static coating. The final innersurface of the bulge cavity is shown in figure 3.8 with nichrome and graphitelayered on top of copper.Figure 3.8: The layered inner surface of the bulge cavity is shown. Anichrome like alloy was electroplated to the bulk copper in an effort to lowerthe cavity Q. Colloidal graphite was added on the outermost layer as anelectrostatic shield.3.4.2 MagnetismThe nichrome-like alloy displays considerably higher magnetic susceptibilitythan standard 80/20 nichrome. This is not too surprising in light of anelemental analysis conducted by Isaac Martens which revealed that the alloyis principally nickel (and < 1% chromium). Once plated to the inner cavitywall we magnetize the alloy with an ion pump magnet (B ' 1200 Gauss).After the magnet is removed we used a Hall probe to measure the on-axismagnetic field resulting from the alloy (see set-up in figure 3.9)Figure 3.10 shows the magnetic field fit to the model discussed in chapter6.Equation 6.9 provides an estimate for the alloy thickness:t ' 7µm (3.9)403.4. Lowering the cavity QFigure 3.9: The set up for measurements of the magnetic field resulting froma nichrome-like alloy plated to the inner surface of the cavity.Figure 3.10: Hall probe measurements of the on-axis magnetic field areplotted with a fit to the model of chapter 6.413.5. Cavity fill factor χ3.5 Cavity fill factor χWe measure the cavity fill factor by inserting a small dielectric bead insidethe cavity (see figure 3.11) and employing cavity perturbation analysis.Figure 3.11: A small teflon bead is inserted into the cavity. The changesin frequency of two resonances are monitored via inductive coupling to anetwork analyzer. The frequency shifts allow us to map the electric field ofthese modes.Following Waldron [55] we have∂ωωo=−3(− 1)V12(+ 2)E2o´E2 dV(3.10)where ∂ω is the shift in resonance frequency due to the presence of a smalldielectric sphere, ωo is the unperturbed resonance frequency, V1 is the di-electric volume, and  is the relative permittivity of the perturbing teflon.This, combined with the definition for the fill factorE2o´E2 dV (section 2.4)gives423.5. Cavity fill factor χχ = −∂ωωo2(+ 2)3(− 1)V1(3.11)Figure 3.12: The frequency shift of two orthogonal cavity resonances as afunction of the radial position of a small perturbing teflon bead.Using the frequency shift at r = 0 we obtainχ = 3.2× 106 m−3 (3.12)in good agreement with the simulated result from table 3.1.433.6. Chokes3.6 ChokesChokes are λ/4 structures conventionally used in microwave waveguides toprevent radiation leakage at joints where two guides have been connected.At such a joint electrical contact is typically poor and this structure servesto fake good contact. We employ the structure to prevent microwaves fromleaking radially at the gaps between electrodes: see the inset in figure 3.5 (c)Transmission line theory provides some intuition for the working mech-anism of a choke. The input impedance (Zin) of a lossless transmission lineof characteristic impedance Z0, terminated with a load impedance ZL as infigure 3.13, isZin = Z0ZL + iZ0 tan(2pil/λ)Z0 + iZL tan(2pil/λ)(3.13)Figure 3.13: A transmission line schematic with input impedance Zin,characteristic impedance Z0, and load impedance ZL labelled. If the length(l) is tuned correctly, ZL can be transformed to desired values at Zin.In the limit l = λ/4, we have Zin → Z20/ZL. If the load is a short circuit(ZL = 0) the input impedance is infinite. On the other hand if the load isan open circuit (ZL → ∞) then the input impedance is zero. We thereforesee that λ/4 is a special length for a transmission line. A short circuit atthis distance is transformed into a open circuit . Similarly, an open circuit istransformed into a short. Transforms of this type (see figure 3.14 ) simulateelectrical contact between electrodes for a range of microwave frequencies(centred on c/λ).443.6. ChokesFigure 3.14: The metal bottom of the λ/4 groove acts as a short whichis transformed twice to the edge of the cavity where it serves to simulateelectrical contact.45Chapter 4Simulations of N CoupledResonators: Cavity andElectronsThe purpose of this chapter is to investigate the cooling of many electrons bya single cavity. In the case of a single electron the description of section 2.4is valid. To be useful as a cooling technique at ALPHA, the method mustbe able to cool large numbers of positrons simultaneously. However, thereexists a certain number of particles above which the cooling effectiveness ofthe cavity begins to diminish. This occurs when the rate of energy enteringthe cavity (due to the combined cyclotron radiation from all leptons) startsto perturb the equilibrium thermal excitation of the relevant cavity mode.Here we develop a classical circuit model to describe the many bodycooling process. This simplified model neglects much of plasma physics: inparticular the z motion, as well as the magnetron motion of the particles.In spite of this, we predict that our cavity can effectively cool 104- 105electrons.4.1 The circuit modelWe model N electrons in cyclotron motion coupled to a dissipative microwavecavity resonance as N RLC circuits inductively coupled to one lossy RLC res-onator as in figure 4.1. The requirement of low loss in our electron resonatorsrelative to the cavity can be expressed as Ri/R0  1 or Qi/Q0  1 whereQ is the circuit quality factor.For electrons directly coupled to the cavity (but not to each other) wecan write the circuit equations as0 =q0C0+ q˙0R0 + q¨0L0 +N∑i=1Miq¨i (4.1)and464.1. The circuit modelFigure 4.1: N RLC circuits (model electrons) with low Ri, inductivelycoupled to one lossy (high R0) RLC circuit (a microwave cavity resonator).0 =qiCi+ q˙iRi + q¨iLi +Miq¨0 (4.2)where Mi is the mutual inductance between the ith electron-circuit andthe "cavity" circuit, and qi is the charge on its capacitor. q0 represents thecharge on the cavity-circuit capacitor.Dividing through by the inductances and usingω =1√LCQ =ωLR(4.3)we can rewrite equations 4.1 and 4.2 as474.1. The circuit model0 = ω20q0 +ω0Q0q˙0 + q¨0 +N∑i=1kiq¨i (4.4)0 = ω2i qi +ωiQiq˙i + q¨i + kiq¨0 (4.5)with a coupling parameter ki = Mi/Li.Substituting 4.5 into 4.4 to eliminate q¨i0 = ω20q0 +ω0Q0q˙0 + q¨0 −N∑i=1ki(ω2i qi +ωiQiq˙i + kiq¨0)= ω20q0 +ω0Q0q˙0 + (1−Nk2i )q¨0 −N∑i=1ki(ω2i qi +ωiQiq˙i)(4.6)gives us a solution for q¨0. Setting all coupling constants equal to each other,ki = k, we haveq¨0 =−ω20q0 −ω0Q0q˙0 +∑Ni=1 k(ω2i qi +ωiQiq˙i)1−Nk2(4.7)substituting this into 4.5 we get an analogous equation for q¨iq¨i = −ω2i qi −ωiQiq˙i − kq¨0q¨i = −ω2i qi −ωiQiq˙i − k−ω20q0 −ω0Q0q˙0 +∑Nj=1 k(ω2j qj +ωjQjq˙j)1−Nk2q¨i =kω01−Nk2q0 −∑i 6=jk2ω2j1−Nk2qj −(k2ω2i1−Nk2+ ω2i)qi +kω0(1−Nk2)Q0q˙0−∑i 6=jk2ωj(1−Nk2)Qjq˙j −(kωi(1−Nk2)Qi+ωiQi)q˙i (4.8)We may now solve the system of coupled differential equations as an eigen-value matrix problem.Defining a vector484.1. The circuit model~q =q0q1q2...qN−1qNq˙0q˙1q˙2...q˙N−1q˙N(4.9)and assuming that the time dependence of each charge follows qi = eαt, thenthere are 2(N + 1) eigenvalues (α) which are the solution to~˙q = α~q = A~q (4.10)A is a matrix whose components are found as coefficients in equations 4.74.8. The decay rates and frequencies are then Re(α) and Im(α) respectively.Once the eigenvalues are known the general solution isqp(t) =N∑l=0clvp,leαlt(4.11)where vp,l is the pth component of the lth eigenvector, αl is the correspondingeigenvalue, and cl is a constant set by the initial conditions.4.1.1 ResultsWe define a cavity decay (αc) rate as the average of the two fastest decayrates (Re(α)) and a electron decay rate (αe) as the average of the remainingRe(α). The decay times areτc =1αcτe =1αe494.1. The circuit modelω0 2pi × 30GHzQ0 1000Qi 109Table 4.1: Typical simulation parameters.An example simulation result for these decay times as a function of thenumber of electrons coupled to the cavity is shown in figure 4.2. We choseresonant frequencies and quality factors (table 4.1) to resemble the physicalcase of electrons in a 1 Tesla magnetic field.The resonant frequencies of the electrons were chosen from a normaldistribution centred around ω0 and with standard deviation σ ≈ω010Q0. Thecoupling parameter k was chosen such that, for the single electron case (N =1),τe ≈ 100τc. (4.12)This condition is satisfied for k between 10−4 and 10−5. This choice was madesince, to first order, we expect overloading when τe = τcN , and since N =100 requires a comfortable amount of computational power for a reasonabledesktop computer.Figure 4.2: Decay times for N electrons coupled to a microwave resonanceas a function of N. The result is normalized to the single particle decay timefor the cavity.504.1. The circuit modelThe result in figure 4.2 shows overload to occur just below N = 100.514.1. The circuit model4.1.2 Overloading analysisIn terms of energy conservation, overloading will occur when the power dis-sipated in the cavity is less than that dissipated by the electronsPc < NPeEτc/2< NEτe/2τc >τeN(4.13)as in the last section. This gives an overload number NoNo =τeτc(4.14)However since τc and τe are both functions of N (as is obvious when lookingat figure 4.2), the overload number must be determined self-consistently:No =τe(No)τc(No)(4.15)In addition to this complication, τe and τc may also depend on all the otherparameters in the problem k, σ, ω0, Q0, and Qi. It would therefore beeasier if we knew No in terms of the single particle decay times which canbe calculated from theory according to equation 2.17.We may relate the single electron decay time to that at overload throughempirical trends present in the simulations.The electron decay time at overload is approximately half that as for thesingle electron case9, while the cavity decay time at overload is about twicethat as for the single electron case (This trend is shown for various couplings,k, in figure 4.3).τe(No) =τe,12τc(No) =2τc,1We may write the single electron cavity decay time using Qc =ω02αcτc,1 =2Q0ω0(4.16)9We note that this result is dubious: Why should the cavity be not be able to cool asingle particle as well as some intermediate number?524.1. The circuit modelFigure 4.3: Decay times at overload versus single particle decay timesTherefore the overload number isNo =τe,14τc,1No =τe,1ω08Q0(4.17)This prediction is plotted versus the overload number read off from simula-tion results in figure 4.4.Next we use the cavity cooling equation for a single electron ( equation2.17 with a factor of 2 inserted to account for the doubly degenerate reso-nance of this cavity)τe,1 =3oB16eQ0χτe,1 =3omω016e2Q0χ(4.18)for fill factor χ and magnetic field B = mω0/e.The overload number isNo =3omω20128e2Q20χ(4.19)534.1. The circuit modelFigure 4.4: Predicted overload number versus overload result from simula-tion.This overload number is plotted as a function of Q in figure 4.5The overload number is calculated in table 4.2 for cavity resonances de-scribed in chapter 3.Cavity Mode No(104)TE121 3.4TE123 15.9TE131 1.4TE132 3.6TE133 9.4TE134 17.0Table 4.2: Cavity resonance overload predictions.544.2. Averaging theory and the rotating frameFigure 4.5: Overloading numbers for the ALPHA cavity and the Berkeleycavity as a function of cavity Q. For the Berkeley Cavity, ω0 = 2pi× 34GHzand χ = 3.2× 106 m−3 was used. For the ALPHA cavity the parameters areω0 = 2pi × 25GHz and χ = 5.1× 104 m−3. The ALPHA Q was 6000, givingNo = 490,000, while a Berkeley Q = 2660 gives No = 14,000.4.2 Averaging theory and the rotating frame4.2.1 General formulationIn non-linear dynamics, averaging theory is a well known and powerful toolfor solving complex systems. The general systemx¨+ x+ δh(x, x˙) = 0 (4.20)can be recast in a "rotating frame" with the transformationsx = ρ(t) cos(t+ φ(t))y = x˙ = −ρ(t) sin(t+ φ(t))(4.21)where the amplitude and phase (ρ and φ) are both functions of time. At firstglance this seems incorrect. With the second transform defining y, did we notneglect time derivatives of ρ and φ? Actually Equation 4.21 is a definitionand we will shortly return to impose the correct differential equation on ournew variables.554.2. Averaging theory and the rotating frameWe imposey˙ = −x− δh(x, y)y = x˙(4.22)Then,ρ2 = x2 + y2ρρ˙ = x˙x+ y˙yρρ˙ = yx− y(x+ δh(x, y))ρρ˙ = −δh(x, y)y(4.23)Now substitute the rotating frame transform for yρρ˙ = δh(x, y)ρ sin(t+ φ)(4.24)to obtain the equation of motion for ρρ˙ = δh(x, y) sin(t+ φ).(4.25)We are now in a position to employ averaging theory on this rotating framevariable ρ4.2.2 Simple example: a single damped harmonic oscillatorFirst, a simple example is shown to illustrate the elegance of this technique.A damped harmonic oscillator obeysx¨+ x+ δx˙ = 0 (4.26)for which h(x, y) = yThe equation of motion is now564.2. Averaging theory and the rotating frameρ˙ = −δρ sin2(t+ φ)(4.27)and by averaging over one cycle ( < sin2(t+φ) >= 1/2 ) we obtain a familiarresultρ˙ = −δ2ρ(4.28)The solution for ρ is an exponentialρ ∝ e−δ2 t(4.29)so that the amplitude or envelope of the oscillator decays at a rate δ/2 aswe know that it should using simpler methods.4.2.3 N coupled harmonic oscillatorsWe now move on to the system of interest: many coupled oscillators.As before we create new variables from the rotating frame transformation:x = q0 = ρ(t) cos(ω0t+ φ(t))y = q˙0 = −ω0ρ(t) sin(ω0t+ φ(t))a = qi = ri(t) cos(ωit+ θ(t))b = q˙i = −ωir(t) sin(ωit+ θ(t))(4.30)Next we apply the differential equations of the system:y = x˙b = a˙(4.31)along with equations 4.7 and 4.8574.2. Averaging theory and the rotating frametoρ2 = x2 + y2r2i = a2i + b2iω0t+ φ = tan−1(−yω0x)ωit+ θi = tan−1(−biωiai)(4.32)Below are the equations of motion in the rotating frame for a singleelectronρ˙ = −ω0ρ2Q0(1− k2)+kω211− k2r[s0s1Q1− s0c1]r˙ = −ωir2Q1(1− k2)+kω20ω1(1− k2)ρ[s0s1Q0− c0s1]φ˙ =ω0k22(1− k2)+kω21ω0(1− k2)rρ[c0s1Q1− c0c1]θ˙ =ω1k22(1− k2)+ω20kω1(1− k2)ρr[s0c1Q0− c0c1](4.33)and the averaging has been lumped into the factorss0si =< sin(ω0t+ φ) sin(ωit+ θi) >=ω02piˆ t+pi/ω0t−pi/ω0sin(ω0t+ φ) sin(ωit+ θi) dts0ci =< sin(ω0t+ φ) cos(ωit+ θi) >=ω02piˆ t+pi/ω0t−pi/ω0sin(ω0t+ φ) cos(ωit+ θi) dtc0si =< cos(ω0t+ φ) sin(ωit+ θi) >=ω02piˆ t+pi/ω0t−pi/ω0cos(ω0t+ φ) sin(ωit+ θi) dtc0ci =< cos(ω0t+ φ) cos(ωit+ θi) >=ω02piˆ t+pi/ω0t−pi/ω0cos(ω0t+ φ) cos(ωit+ θi) dt(4.34)584.2. Averaging theory and the rotating frameWeak coupling k  1loss-less electrons Q1  1On resonance ω0 = ω1 = ωicold cavity, hot electrons ρ rArbitrary initial cavity phase φ = 0Table 4.3: Simplifying limits for the rotating frame system.Next I will impose the limits from table 4.3 in order to simplify the modeland draw insights about the system.Mathematically, the phase dynamics for this single particle limiting caseobeysθ˙ ' 0φ˙ ∝ −kωorρc0c1,so that φ˙ = 0 when c0c1 = 0. Therefore, the oscillator phases will evolve toan equilibrium atφ− θ → pi/2, 3pi/2, 5pi/2 . . . (4.35)The single particle amplitude equations are reduced toρ˙ '−ρω02Q0(1− k2)±krω2i2(1− k2)ω0ρ˙ '−ρω02Q0(1− k2)andr˙ '−rωi2Q1(1− k2)∓kρω202(1− k2)ωir˙ '∓kρω202(1− k2)ωiFor the N electron problem, the equations of motion are594.2. Averaging theory and the rotating frameρ˙ = −ρω02Q0(1−Nk2)+kω0(1−Nk2)N∑i=1ω2i ri[s0siQi− s0ci]r˙i = −riωi2Qi(1 +k21−Nk2)+kω20ωi(1−Nk2)ρ[s0siQ0− c0si]+k2ωi(1−Nk2)N∑j 6=iω2j rj[−sjsiQj+ sicj]φ˙ =ω0Nk22(1−Nk2)+k1−Nk2N∑i=1ω2i riω0ρ[c0siQi− c0ci]θ˙i =ωik22(1−Nk2)+kω20ωi(1−Nk2)ρri[s0ciQ0− c0ci]+k21−Nk2N∑j 6=iω2j rjωiri[−sjciQj+ cicj](4.36)By integrating these equations numerically, we confirm the phase predictionsof equation 4.35. Figure 4.6 shows the phase and amplitude behaviour for thesingle particle case. Figure 4.7 shows the same result for N = 5 electrons.For the latter case, the phase dynamics is unclear. In the limit of largeelectron numbers however, equation 4.36 suggests that the cavity phase willevolve such thatN∑i=1c0ci → 0 (4.37)Unfortunately solving for large numbers of electrons in the time domain isnot computationally feasible. Inferences about the large N electron coolingtime and/or phase dynamics have therefore not been attempted.604.2. Averaging theory and the rotating frameFigure 4.6: The phases and amplitudes of one electron (red, left scale)coupled to a cavity (blue).Figure 4.7: The phases and amplitudes of five electrons (red, left scale)coupled to a cavity (blue). The electrons were given uniformly random initialphases (between 0 and 2pi)61Chapter 5Laser Pass - Microwave StopTubesThe ALPHA apparatus has been upgraded so as to include apertures whichallow lasers access to the anti-hydrogen region 5.1. These 14.3 mm ID coppertubes act as waveguides and direct the microwave band of room temperatureblack body radiation towards the cryogenic region of the experiment. Inparticular this radiation will cause resonant heating of positrons which couldresult in a devastating reduction in anti-hydrogen production. This chapterdescribes a design which allows laser light through but which attenuates theunwanted microwave radiation.Figure 5.1: An exploded view of a laser pass tube. Optical spectroscopynecessitates use of windows and apertures to allow laser light to interact withtrapped anti-hydrogen. These windows allow room temperature blackbodyradiation to propagate into the experiment and causes unwanted heating ofpositron plasmas.625.1. Theory of operationFigure 5.2: An assembled view of a laser pass tube.5.1 Theory of operationThe design for the laser-pass tubes was inspired by a well known result fromclassical electromagnetism (see, for example [48]). As in figure 5.3, if a thinresistive sheet is placed a distance of λ/4 from a conducting plane then anormally incident plane wave will be 100% absorbed if the sheet impedanceis matched to free space (Zsheet = Zo).Figure 5.3: A plane wave incident on a good conductor can be perfectlyabsorbed if a thin sheet with surface impedance (Zsheet) matched to freespace (Zo) is placed a distance λ/4 from the conductor.An analogy with transmission line theory explains the mechanism for thisabsorption. We have already seen in section 3.6 that λ/4 is a special length635.1. Theory of operationfor a transmission line. At that length a short (like the conducting planein figure 5.3) is transformed into an open circuit. Any object placed at thisdistance is therefore in parallel with an open circuit, or infinite impedance.The plane wave will therefore not use this branch of the circuit and seeonly the impedance presented by the thin resistive sheet.If the sheet has the impedance of free space (Zo) then no reflections occur:S11 = Ereflected/Eincident =Zsheet − ZoZsheet + Zo(5.1)The above geometry achieves 100% absorption only for normal incidenceand only at one frequency. However, by modifying to a tubular geometry(figure 5.4 we allow nearly all the radiation to be incident at a glancingangle. The angle of incidence modifies the λ/4 requirement and complicatesthe attenuation analysis. A distribution of incident angles will produce adistribution of resonantly absorbed frequencies. Perhaps because of this, weobserve that the filter is effective over a wider range of frequencies ratherthan just one.Figure 5.4: The λ/4 requirement wrapped onto a cylindrical geometry. Thegap between the alumina tube is exaggerated to allow visualization of theberyllium-copper springs used for heat contact. The assembled gap is 0.5mm.645.2. Fabrication5.2 FabricationThin nichrome films are deposited on the inner face of an alumina tube viaa thermal evaporation process.A tungsten filament tightly wrapped in nichrome wire is threaded throughthe alumina tube. This apparatus is placed in a moderate vacuum of about10−5Torr and high a current is run through the filament heating the nichrome.Hot atoms evaporate from the nichrome wire and are deposited on the coolerpolished alumina surface.Figure 5.5: A cartoon of the apparatus used to deposit a thin nichromelayer onto the inner surface of a long, narrow alumina tube. A high cur-rent is passed through a tungsten filament which evaporates atoms from thenichrome wire.The narrowness of the alumina tube (inner diameter 11 mm) means thatthe hot tungsten filament is necessarily very close to the target surface duringdeposition. This presented a number of challenges to be overcome.Firstly, the nichrome generally deposited unevenly on the alumina. Nor-mally during vapour deposition, the filament is placed far from the targetsurface such that the particle flux at the target location is nearly uniform.To mitigate this effect great care was taken to place the filament at the cen-ter of the tube: even a small displacement from center results in an uneven655.2. Fabricationdeposition. However, even if carefully centred, the filament will sag or distortits shape and position once heated. To reduce this sagging, the filament wasplaced under tension using a spring such that once hot and deformable, thefilament stretched and roughly maintained its orientation.Secondly, the alumina tube reflected outgoing thermal radiation back to-wards the filament creating a furnace effect. This caused the temperatureinside the tube to be very sensitive to currents applied to the filament. As aresult the filament would often overheat and melt.10Lastly, a quartz crystal oscillator is commonly used as a thickness monitor.The crystal oscillation frequency will change as material is deposited on itssurface. If this oscillator is placed near the target sample an accurate mea-sure of the deposition thickness can be obtained. In our geometry however,the alumina target shields the thickness monitor from the nichrome sourceessentially blinding the sensor.These last two problems were solved by implementing a common web-camas a sensor. The sensor is capable of detecting relative deposition rates aswell as changes in temperature. Figure 5.6 shows the red, green and bluepixel intensities recorded in our camera during a typical nichrome deposition.Abrupt jumps in this signal correspond to increases in current. The subse-quent decay in transmitted light intensity indicates nichrome deposition onthe inner surface of the alumina tube. As expected, higher decay rates inthe transmission signal are associated with higher currents which producehotter filaments with faster deposition rates.The absolute thickness of deposited nichrome for these filters is not known.However, the technique could be calibrated to produce such a measurement.The thickness can be inferred from measurements of the nichrome surfaceresistance and correlated with the deposition time as well as the exponentialdecay rate of the transmitted light.10Molten nickel is known to attack tungsten [5] quickly, so we suspect that once thenichrome has melted, a nickel tungsten alloy forms. This alloy could have a melting pointlower than either nichrome or tungsten. Formation of such an alloy would explain thesudden liquefaction of the filament.665.2. FabricationFigure 5.6: CCD pixel values for red, green, and blue are recorded during athermal evaporation of nichrome onto the inner surface of an alumina tube.675.3. Microwave attenuation5.3 Microwave attenuationMicrowave transmission through a cylindrical waveguide which incorporatesa laser-pass tube is characterized using the apparatus shown in figure 5.7.Two large cavities serve as input/output ports for microwave power from anetwork analyser. The cavities are connected with a cylindrical waveguidewhich matches laser aperture dimensions at ALPHA and which can house ourmicrowave filtering tube. The cavities are designed to approximate free-spaceand irregular conductors are placed inside to help randomize the microwavemode structure thereby approximating microwave thermal radiation. Thetransmitted microwave power is measured both with the microwave filteringtube in place and absent from the apparatus. To determine the effect of thefilter, the former scenario is normalized to the latter. The power transmittedthrough the filter (relative to an empty waveguide) is shown in figure 5.8Additionally, the surface resistance of the nichrome film deposited onthe inner face of one of the alumina tubes was measured using a Ohm-meter. The nichrome surface deposited in 5.6 had a surface resistance ofRs = 200± 40Ω/square. Using standard resistivity values for nichrome andthe relationship ρ = Rst one can approximate the thickness of the nichromedeposition at t ' 7 nm.5.4 Thermal conductivity measurementAbsorbing thermal microwaves will cause heating of the ceramic tubes. Wemust therefore test the thermal conductance of the beryllium copper springswhich connect the alumina tube to the surrounding copper tube at 4 K.We silver epoxy the springs to a mock laser tube made from aluminium.The aluminium tube is inserted into a copper tube with the same innerdiameter as at ALPHA. The copper tube is placed in thermal contact witha liquid helium bath at about 4 K. The void between the aluminium tubeand copper is pumped to a vacuum pressure of about 10−5Torr. The set upcan be viewed in figure 5.9.Two resistors are fixed with good thermal contact to the aluminium tube.R1 is a "Heater" resistor and R2 is a thermometer. The heater is a thin metalfilm type resistor and its resistance does not change much with temperature.R1 is a ceramic core resistor whose resistance depends strongly on tempera-ture thus allowing inference of temperature from a resistance measurement.685.4. Thermal conductivity measurementFigure 5.7: The apparatus used to measure microwave transmission througha laser-pass tube. Two large cavities serve to randomize the microwavemode structure in an attempt to approximate free-space thermal radiation.Transmission through the laser pass tube is normalized to that of the emptywaveguide.695.4. Thermal conductivity measurementFigure 5.8: The microwave power transmitted through the laser pass tube.705.4. Thermal conductivity measurementFigure 5.9: Thermal contact experiment set-up.715.4. Thermal conductivity measurement5.4.1 ResultsBy applying different voltages to the heater resistor, we vary the heat load tobe conducted to the surrounding liquid helium bath via the beryllium coppersprings. When a heat load is applied a new, higher, equilibrium temperatureis reached. Once in equilibrium we abruptly remove the heat load and thepart is allowed to cool to near 4 K. An exponential fit to this temperaturedecay is performed with the fit functionT − 4.2K = ∆Toe−t/τ(5.2)where T is temperature, t is the time since the heat load is removed, ∆Tois the temperature change of the tube at t = 0, and τ is the characteristiccooling time of the part.Figure 5.10: Different voltages (heat loads) are applied to the tubes andexponential cooling times are measured from the decay (red dots) whichresults when the load is turned off and the part comes back to equilibriumwith the surrounding liquid helium.725.4. Thermal conductivity measurementFigure 5.10 shows temperature data for the part while multiple differentheat loads are applied and then removed. The results are summarized intable 5.1Scenario ∆To (K) τ (s) Heat load (mW) Thermal Conductance (mW/K)a) 0.33 6.2 1.1 3.32b) 0.62 6.9 2.12 3.42c) 1.2 7.0 4.33 3.60d) 3.0 9.1 13.26 4.42e) 4.9 11.8 27.06 5.52f) 7.9 16.0 54.25 6.87g) 12.2 23.5 108.20 8.87Table 5.1: A summary of the thermal conductance measurements on thelaser pass tubes.These numbers can be compared to the total power incident on the laseraperture according to the Stephan-Boltzmann lawP = σT 4Ωpipir2 (5.3)where σ = 5.67× 10−8 Wm2K4 is the Stephan-Boltzmann constant, T = 300 K(room temperature), Ω is the solid angle of room temperature surface seen bythe laser aperture (conservatively estimated at pi/2), and r = 11 mm is theradius of the aperture. The calculation results in P ' 87 mW. Comparingwith the results of table 5.1 we see that the part may heat as much as 10 Kif the thermal radiation can be characterized as 300 K radiation. However,given the construction of the ALPHA apparatus, the effective microwavetemperature in the region of the tubes is likely to be lower than this.73Chapter 6Magnetic ElectrodesGreat care is taken at ALPHA to remove magnetic materials from the ap-paratus. Any such material has the potential to become magnetized in someunpredictable way by the many coils regularly turned on or off during a typi-cal antihydrogen production sequence. Field impurities from these materialshave the potential to disrupt plasma control, affect antihydrogen dynamicsand degrade the magnetic field homogeneity necessary for precision spec-troscopy.This chapter quantifies the magnetic properties of the ALPHA catchingtrap electrodes. These electrodes are constructed from aluminium and anantistatic coating is achieved with a gold plated surface. Unfortunately goldwill not stick well when plated directly onto aluminium, and it is necessaryto first plate a intermediate adhesion layer which will stick to both metals.We have discovered that this adhesion layer is magnetic, likely containingnickel.6.1 Measurement techniqueThe ALPHA electrodes are placed in a home-built nuclear magnetic reso-nance (NMR) spectrometer operating at proton resonance frequency of 360MHz (' 8.5 T) [40]. A 1 mm diameter water NMR probe is placed on theaxis of the electrodes and the precession frequency is monitored as a mea-sure of the magnetic field inhomogeneity introduced by the electrode. Thespectra were acquired by Carl Michal of the UBC Solid State NMR groupusing a simple Bloch decay pulse sequence.By moving the electrodes relative to the probe and monitoring the pre-cession frequency we can map the net magnetic field as a function of positionf =γ2piB(z) (6.1)With γ = eg2m where g is the gyromagnetic ratio, e the elementary charge,and m the proton mass. For the water probe, the frequency to magneticfield conversion is746.2. Model for a thin magnetized tube∆B =f − fo42.58× 106 HzT(6.2)where fo is the measured precession frequency without electrodes presentand ∆B is the field caused by the presence of the electrode.A sketch of the apparatus is shown in figure 6.1Figure 6.1: A 1 mm diameter water NMR probe is placed on the axis of ourelectrodes. The precession frequency is used to monitor the magnetic field.6.2 Model for a thin magnetized tubeA model for the magnetic field of a thin magnetized tube is derived.The on-axis field for a uniformly magnetized, solid rod isB =µoM2(cosβ − cosα) (6.3)B =µoM2(z + L√(z + L)2 +R2−z − L√(z − L)2 +R2)(6.4)For a thin tube, to first order756.2. Model for a thin magnetized tubeBtube = t∂B∂R= −tµoMR2(z + L[(z + L)2 +R2]3/2−z − L[(z − L)2 +R2]3/2)(6.5)For R = 2.3 cm and L = 1.0 cm L = 2.0 cm (for the case of one and twoelectrodes respectively) and employing a fit to the two electrode case (thereis not enough data for a good fit in the one electrode case) we can plot Btubeon top of the dataFigure 6.2: The measured extra magnetic field as a function of the axialposition of the water probe is fit to the model in equation 6.6.In figure 6.2 the fit function isBtube = BotR(z + L[(z + L)2 +R2]3/2−z − L[(z − L)2 +R2]3/2)(6.6)and the result is BotR = 2.4± 0.4 Gauss·cm2.766.3. Approximating the thickness of the nickel strike6.3 Approximating the thickness of the nickelstrikeAssuming that this field is the result of a thin nickel "strike" layer applied tothe aluminium electrodes as a part of the gold plating process, we can deter-mine the thickness of this nickel layer. We take the saturation magnetizationfor nickel to beBsat = µoMsat = 0.57 Tesla (6.7)and use the z = 0 field value from equation 6.5Btube = −tµoMsatRL(L2 +R2)3/2. (6.8)Rearrangingt = −BtubeBsat(L2 +R2)3/22LR. (6.9)where the extra factor of 1/2 has been added to account for a strike on eachside of the tube (since both sides are gold plated).From figure 6.4, for two electrodes, we see that Btube ' −0.35 × 10−4Tesla , givingt ' 1.9µm (6.10)6.4 Extrapolating to many electrodesUsing equation 6.6 with z = 0 and varying L, we can estimate the field insidethe magnetized Penning trap at ALPHA. Considering 13 central electrodeswith a total length of 27.4 cm (L = 13.7 cm) we findBtube = (2.4± 0.4)× 10−2Gauss (6.11)The field as a function of tube length is shown in figure 6.4.6.5 ConclusionAlthough the nickel layer seems excessively thick (∼ 2µm) for the purposesof an adhesion layer between gold and aluminium, the length of the ALPHAPenning trap helps to mitigate the effect by about an order of magnitude.776.5. ConclusionField vs length of Penning trap.The expected magnetic field added by such an adhesion layer at ALPHA is∼ 0.05 G. This is to be compared with a recent spin-flip measurement at AL-PHA with uncertainties governed by magnetic field inconsistencies [18]. Thatexperiment achieved a spin-flip frequency uncertainty of 100 MHz which re-sults from a ∼ 3000 G magnetic field anomaly due to pinning of magneticflux in nearby superconducting wire. Further, the next most limiting fieldinhomogeneity is ∼ 350 G and results from variance in the field of the mag-netic minimum trap. Indeed, a 0.05 G magnetic field is small enough togo unnoticed by a precision hyperfine spectroscopy experiment in the styleof [28]. A simple solution to this minor problem may be to use colloidalgraphite as an anti-static shielding layer (as in chapter 3) in favour of gold.78Chapter 7Preliminary Observation ofEnhanced Cooling of Electronsat Cavity ResonancesFigure 7.1: A sketch of the Berkeley Experiment with the UBC bulgecavity installed is shown. The broken connection indicated left one of thecavity electrodes floating at some unknown, but seemingly stable voltage.This uncertainty means that the location of the plasma in the cavity is notprecisely known.An experiment is conducted at Berkeley which demonstrates the coolingpower of the cavity from chapter 3.Large plasmas are contained in two storage regions within the Penningtrap at Berkeley (see figure 7.1). These reservoirs allow rapid temperaturemeasurements of different plasmas within the bulge cavity. We are thereforeable to slowly scan the plasma cyclotron frequency while monitoring thetemperature of plasmas which are continually loaded into and ejected fromthe trap.A typical sequence goes as follows: A plasma of about one million elec-trons is loaded from a reservoir into the cavity where cooling occurs. Aftera specified time ( usually 0.5 seconds) this same plasma is ejected onto a79Chapter 7. Preliminary Observation of Enhanced Cooling of Electrons at Cavity ResonancesFigure 7.2: Electron plasma temperature is measured as a function of thecyclotron frequency ωc. When ωc matches a cavity mode, the temperatureis reduced by about an order of magnitude. This particular peak matchesthe frequency of the TE133 mode to better than 1 %. Different field scans(shown in different colours) show that only the temperature drop near theTE133 is reproducible.80Chapter 7. Preliminary Observation of Enhanced Cooling of Electrons at Cavity Resonancesmicro-channel plate (MCP). Secondary electrons leaving the MCP are inci-dent on a phosphor screen which converts these electrons to photons. Theemitted light is directed onto a photodiode by Fresnel lenses. From the pho-todiode signal a plasma temperature can be inferred11. The process is thenrepeated at different cyclotron frequencies.We observe that, near a cavity resonance, cooling occurs (see figure 7.2).Figure 7.3: Electron plasma are held in the cavity for various times beforea temperature measurement is performed. An exponential fit (red) of theform T (t) = Toe−Γt + Tf allows us to infer the cooling rate.Once a cavity resonance is identified, we leave the cyclotron frequencyconstant at the resonance frequency, but vary the amount of time the plasma11The photodiode signal is modelled as resulting from the hottest electrons (which leavethe Penning trap first as the electrode voltage is lowered) from a Maxwell-Boltzmanndistributed electron plasma. A fit to the photodiode current and the electrode voltageyields the plasma temperature. [47]81Chapter 7. Preliminary Observation of Enhanced Cooling of Electrons at Cavity ResonancesFigure 7.4: Electron plasma cooling rate is measured as a function of thecyclotron frequency ωc. When ωc matches a cavity mode, the rate is en-hanced. This particular peak matches the frequency of the TE133 mode tobetter than 1 %.82Chapter 7. Preliminary Observation of Enhanced Cooling of Electrons at Cavity Resonancesis allowed in the cavity before ejecting to the MCP and performing a temper-ature measurement. Figure 7.3 shows the plasma temperature versus timespent in the cavity. An exponential fit allows us to infer the plasma coolingrate. Performing this procedure at various cyclotron frequencies in the vicin-ity of the cavity resonance shows the rate enhancement due to the Purcelleffect (figure 7.4).By scanning the cyclotron frequency and measuring the plasma temper-ature we were able to identify six cavity resonances with enhanced coolingability. Figures 7.2, 7.3, and 7.4 focus on the TE133 resonance since it wasfound to have the strongest cooling power. Cooling ability of the remainingdetected modes are summarized in table 7.1.More details on the plasma apparatus and experiment can be found in[47].Cavity Mode B (Tesla) fc = eB2pim (GHz) Relativefrequencydiscrepancy%CoolingRate En-hancementTE121 0.775 21.70 -0.18 1.4TE123 0.91 25.48 -0.12 2TE131 1.209 33.84 0.21 2TE132 1.309 36.65 -0.20 4TE133 1.380 38.64 -0.05 10TE134 1.443 40.40 -0.35 2.3Table 7.1: A summary of the observed cooling power of different cavitymodes. The relative frequency discrepancy represents the fractional differ-ence between the cyclotron frequency and the measured or simulated cavityresonance frequency (see table 3.1) after a correction for the thermal con-traction has been applied.7.0.1 OutlookThe observed spontaneous emission rate enhancements of order 1 -10 fromtable 7.1 may seem rather disappointing given that section 2.4 predictedan enhancement factor of ' 1000. However, one must remember that thisprediction was made for the case of a single particle coupled to the cavity.These experiments involved about 106particles and section 4.1.2 predictedthat the cavity could not cool more than 104- 105electrons (depending onthe mode).83Chapter 7. Preliminary Observation of Enhanced Cooling of Electrons at Cavity ResonancesThese results come with another flaw: the broken connection to a cavityelectrode (see figure 7.1). Since this electrode was floating at some unknown(but seemingly stable) voltage, the exact axial position of any plasma in thisexperiment is also unknown. This means that the fill factor of section 2.4needs to be computed as an average weighted by the plasma density. 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Journal of Physics B: Atomic,Molecular and Optical Physics, 41(19):192001, 2008.[50] J. Tan and G. Gabrielse. Synchronization of parametrically pumpedelectron oscillators with phase bistability. Phys. Rev. Lett., 67:30903093, Nov 1991.[51] J. Tan and G. Gabrielse. Parametrically pumped electron oscillators.Phys. Rev. A, 48:31053122, Oct 1993.89[52] Ted Pella Inc. Graphite Aerosol, 4 2015.[53] R. S. Van Dyck, F. L. Moore, D. L. Farnham, P. B. Schwinberg, andH. G. Dehmelt. Microwave-cavity modes directly observed in a Penningtrap. Phys. Rev. A, 36:34553456, Oct 1987.[54] Robert S. Van Dyck, Paul B. Schwinberg, and Hans G. Dehmelt. Newhigh-precision comparison of electron and positron g factors. Phys. Rev.Lett., 59:2629, Jul 1987.[55] R.A. Waldron. The Theory of Waveguides and Cavities. Maclaren, 1967.[56] Herbert Walther, Benjamin T. H. Varcoe, Berthold-Georg Englert, andThomas Becker. Cavity quantum electrodynamics. Reports on Progressin Physics, 69(5):1325, 2006.[57] C. Weisbuch, H. Benisty, and R. Houdre. Overview of fundamentals andapplications of electrons, excitons and photons in confined structures.Journal of Luminescence, 85(4):271  293, 2000.[58] W. D. White, J. H. Malmberg, and C. F. Driscoll. Resistive-wall desta-bilization of diocotron waves. Phys. Rev. Lett., 49:18221826, Dec 1982.[59] D. J. Wineland and H. G. Dehmelt. Principles of the stored ion calorime-ter. Journal of Applied Physics, 46(2):919930, 1975.[60] E. Yablonovitch, T. J. Gmitter, and R. Bhat. Inhibited and enhancedspontaneous emission from optically thin AlGaAs/GaAs double het-erostructures. Phys. Rev. Lett., 61:25462549, Nov 1988.[61] Eli Yablonovitch. Inhibited spontaneous emission in solid-state physicsand electronics. Phys. Rev. Lett., 58:20592062, May 1987.[62] Yoshihisa Yamamoto, Susumu Machida, and G. Bjork. Micro-cavitysemiconductor lasers with controlled spontaneous emission. Optical andQuantum Electronics, 24(2):S215S243, 1992.90Appendix ADetailed Drawings of a BulgeResonatorDetailed drawings of the bulge resonator are presented.A A 18.891  18.891 ESECTION A-A SCALE 1 : 1.5Electrode 1Electrode 2Electrode 3This is an equation driven curve. The radius is r= -1.25mm*cos(t) + 11.25 mm , so that r=10mm at minimum (near the edge of Electrodes 1,3)  and 12.5 at maximum (in the middle of Electrode 2)t = 2pi line 2  2.170  1  0.635  2.540  3.175  0.635  1.270  3.810  0.635  0.635 DETAIL E SCALE 4 : 1Circular ceramic spacers go between electrodes for electrical isolation. These already exist and do not need to be made.t = 0 lineDO NOT SCALE DRAWING3ElectrodeBulgeSHEET 1 OF 2UNLESS OTHERWISE SPECIFIED:SCALE: 1:2 WEIGHT: REVDWG.  NO.ASIZETITLE: Electrode Stack assemblyNAME DATECOMMENTS:Q.A.MFG APPR.ENG APPR.CHECKEDDRAWNOFHC copperFINISHMATERIALINTERPRET GEOMETRICTOLERANCING PER:DIMENSIONS ARE IN mmTOLERANCES: 25 micronFRACTIONALANGULAR: 0.5 degreesTWO PLACE DECIMAL    THREE PLACE DECIMAL  APPLICATIONUSED ONNEXT ASSYPROPRIETARY AND CONFIDENTIALTHE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>.  ANY REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS PROHIBITED.5 4 3 2 1Figure A.1: Detailed bulge cavity drawings91Appendix A. Detailed Drawings of a Bulge Resonator6X  1.067  2.540 (#58 Drill) 60.00° A A 2.540  17.780  48.260  3.175  1.905  6.350  50.800  24.765 10X  0.711  2.540 (#70 Drill) 43.180  41.910  2.540  3.175  3.810  3.810  0.635 SECTION A-A2X  1.191  3.810 (#0-80 Tap) 22.143  17.780  7.637  3.555 DO NOT SCALE DRAWING5ElectrodeMk2VN34SHEET 1 OF 2UNLESS OTHERWISE SPECIFIED:SCALE: 1:1 WEIGHT: REVDWG.  NO.ASIZETITLE: Electrode 2NAME DATECOMMENTS: Hole alignment is critical and tolerances should be higher than normal. About 12 micron (Or 0.0005 inches)Q.A.MFG APPR.ENG APPR.CHECKEDDRAWNOFHC copperFINISHMATERIALINTERPRET GEOMETRICTOLERANCING PER:DIMENSIONS ARE IN mmTOLERANCES: 25 micronFRACTIONALANGULAR: 0.5 degreesTWO PLACE DECIMAL    THREE PLACE DECIMAL  APPLICATIONUSED ONNEXT ASSYPROPRIETARY AND CONFIDENTIALTHE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>.  ANY REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS PROHIBITED.5 4 3 2 1Figure A.2: Detailed drawings of single electrode within bulge cavity.92Appendix BA Model for Effective SheetResistance of Multiple ThinConducting LayersB.1 Complex poynting theoremThe time averaged work done by fields on sources (each with assumed timedependence eiωtis12ˆ~J∗ · ~E d3x (B.1)using the complex Maxwell Equations for harmonic fields we have∇× ~E = iω ~B (B.2)∇× ~H + iω ~D = ~J (B.3)together with the vector identity∇ · (E ×H) = H · (∇× ~E)− E · (∇× ~H) (B.4)Equation B.1 becomes12ˆ~J∗ · ~E d3x =12ˆ~E · (∇× ~H∗ − iω ~D∗) d3x=ˆ−∇ ·( ~E × ~H∗)2−iω2( ~E · ~D∗ − ~B · ~H∗) d3x=ˆ−∇ · ~S − i2ω(we − wm) d3x(B.5)where the electric and magnetic energy densities are93B.2. Resistance and reactancewe =14~E · ~D∗ (B.6)wm =14~B · ~H∗ (B.7)Conservation of energy is then expressed via the complex Poynting the-orem:12ˆ~J∗ · ~E d3x− i2ωˆwe − wm d3x+˛~S · ~nda = 0 (B.8)B.2 Resistance and reactanceFor a coaxial input (with current I and voltage V ) to a general two terminal,linear, passive electromagnetic system (with impedance Z) as in figure B.1,the power dissipation isFigure B.1: An arbitrary surface S surrounding a general two terminalelectromagnetic structure. ~n is the unit normal vector outwards from thesurface, and Si is an input surface occupied by the coaxial line.12I∗V =12ˆ~J∗ · ~E d3x− i2ωˆwe − wm d3x+˛S−Si~S · ~nda. (B.9)With V = ZI and Z = R− iX, the real and imaginary parts of EquationB.9 give us the resistance and reactance (R, X) in terms of field quantities.94B.3. Sheet resistance for a metallic planeR =1|I|2Re[ˆ~J∗ · ~E d3x− i4ωˆwe − wm d3x+ 2˛S−Si~S · ~nda](B.10)andX =1|I|2Im[ˆ~J∗ · ~E d3x− i4ωˆwe − wm d3x+ 2˛S−Si~S · ~nda](B.11)The last term in Equations B.9, B.10 and B.11 relates to escaping radia-tion and can usually be neglected in the low frequency limit. However, for thecase of our bulge cavity, this term represents the dominate loss mechanism:leakage out the open ends of the resonator.Disregarding this for the moment, the expressions for R and X can beapproximated in the low frequency limitR =1|I|2ˆσ|E|2 d3x (B.12)X =4ω|I|2ˆ(we − wm) d3x (B.13)As a side note, dropping the second term in equation B.10 will always bevalid at a cavity resonance since the resonance condition can be expressedas we = wm [42].Up to now the treatment has followed Jackson Section 6.9 [34]. Wenow look to combine this expression for R with the known solutions forelectromagnetic fields near a conducting surface.B.3 Sheet resistance for a metallic planeFor some tangential magnetic field (Ho) outside a conducting semi-infiniteplane occupying z ≥ 0 the magnetic field inside the metal isHc = Hoe−z/δeiz/δ (B.14)where δ =√2ωµσ is the skin depth and σ is the conductivity. Neglectingthe displacement current in the conductor as well as x and y derivativeswithin the ∇ operator, the electric field is found according toσ ~Ec = ∇× ~Hc (B.15)95B.4. Sheet resistance for layered thin metals~Ec =√µω2σ(1− i)(~n× ~Ho)e−z/δeiz/δ (B.16)|Ec|2 =2σ2δ2(~n× ~Ho)2e−2z/δ (B.17)Inserting equation B.17 into equation B.12 and integrating from zero toinfinity we obtain a comforting resultdRdA= Rs =(~n× ~Ho)2|I|2ˆ ∞02σe−2z/δσ2δ2dz=2σδ2[e0 − e−∞2/δ]=1σδ=ρδ(B.18)B.4 Sheet resistance for layered thin metalsFigure B.2: A thin layer of material 1 on top of a thick (semi-infinite) layerof material 2.We now apply expression B.12 to the case of Figure B.2 : one thin metal(of thickness ξ, conductivity σ1 and skin depth δ1) on top of a thick (semi-infinite) metal plane (of conductivity σ2 and skin depth δ2). We may write96B.4. Sheet resistance for layered thin metalsRs =(~n× ~Ho)2|I|2[ˆ ξ02e−2z/δ1σ1δ21dz +ˆ ∞ξ2e−2z/δ2σ2δ22dz]=1− e−2ξ/δ1σ1δ1+1σ2δ2(e−2ξ/δ2 − e−∞)=1− e−2ξ/δ1σ1δ1+e−2ξ/δ2σ2δ2= Rs1(1− e−2ξ/δ1) + e−2ξ/δ2Rs2 (B.19)The expression satisfies our expectations for limiting cases. Namely,limξ→∞Rs = Rs1limξ→0Rs = Rs2 (B.20)An effective conductivity can be derived from equation B.19 by settingRs =1σeδe=√ωµ2σe(B.21)and solving for σe. After some algebra we arrive atσe =σ1σ2(√σ2(1− e−2ξ/δ1) +√σ1e−2ξ/δ2)2(B.22)Like the surface resistance Rs the effective conductivity still depends onfrequency via the skin depths δ1 and δ2.Two cases of primary interest to the experiment will now be explored viadirect observation.B.4.1 ObservationThis model is tested using a coaxial resonator as in figure B.3. The centerconductor of the resonator is electroplated with thin layers of a second metaland the resonance Q allows us to infer the surface resistance. The Q of sucha resonator is [55, 11]Q =ωoµoLln(a/b)Rsa La +RsbLb(B.23)97B.4. Sheet resistance for layered thin metalsFigure B.3: A short center conductor placed in a long metallic tube will actas a coaxial resonator. The electric fields of the first harmonic are sketchedon the center conductor. An inductive loop coupler allows us to measure thefrequency and Q of the resonance using a network analyser. Knowledge ofthe Q allows us to infer the surface resistance of the inner conductor.where L is the length of the center conductor, a is its radius, Rsa is itssurface resistance, b is the radius of the outer conductor, Rsb is its surfaceresistance, and ωo/2pi is the mode frequency.Using B.19 for Rsa we can make a prediction for the Q of these modesas a function of the thickness of the electro-deposited top metallic layer onthe center conductor. The result, for gold electroplated onto nichrome is infigure B.7. For a nichrome-like alloy plated onto copper, see figure B.8.B.4.2 Thin gold on thick nichromeFirstly we inquire about the sheet resistance of a thick layer of nichromeunderneath a thin layer of gold (figure B.4). Our aim here is to test thepossibility of lowering the Q of a resonant mode (at about 30 GHz) insideour bulge cavity while maintaining good electrostatic shielding for plasmas.The thin gold layer will shield the plasmas from oxide layers on the nichromethat may acquire static charges. However, if the gold is thick enough, it willalso shield the relevant microwave mode from excess losses occurring in the98B.4. Sheet resistance for layered thin metalsFigure B.4: A thin layer of gold on top of a thick (semi-infinite) layer ofNichrome.nichrome and thus prevent the lowering of the mode Q. Figures B.5 and B.6show the predicted effective surface resistance and conductivity for a layerof gold (of thickness z) over top of a thick layer of nichrome.Figure B.7 shows the measured Q of four different resonance from thecavity of figure B.3 as a function of the thickness of electroplated gold. A fitto the conductivity of gold was allowed since the deposition is likely porousand uneven. The fit conductivity for gold was always 60% - 80% of theaccepted value.99B.4. Sheet resistance for layered thin metalsFigure B.5: Effective surface resistance for a layer of gold (thickness z)over top of a thick layer of Nichrome. The result is normalized to the sheetresistance of Nichrome at 30 GHz (Rs,nichrom = 422mΩ ) and approachesthe sheet resistance of gold (Rs,gold = 54mΩ ) for large z. The skin depthsof gold and nichrome at 30 GHz are 450 nm and 3.5 µm respectively.Figure B.6: Effective conductivity for a layer of gold (thickness z) over topof a thick layer of nichrome.100B.4. Sheet resistance for layered thin metalsFigure B.7: The measured Q's (blue) for various cavity resonances as afunction of the thickness of an electroplated gold layer on the inner conductor(see figure B.3). A fit for the conductivity of gold was allowed since thedeposition is likely porous and uneven. The fit conductivity for gold wasalways 60% - 80% of the accepted value. The best fit models are in red.101B.4. Sheet resistance for layered thin metalsB.4.3 Thin nichrome on thick copperEstimate of alloy conductivityFigure B.8: The measured Q's for various cavity resonances as a functionof the thickness of an electroplated nichrome layer on a copper inner con-ductor (see figure B.3). The fit is poor for small z suggesting the presenceof a heavy oxide layer on the copper outer conductor.After the fact it was discovered that our electroplated nichrome actuallycontained very little chromium (<5 %). However, a measurement of a cavityQ at both room-temperature and liquid nitrogen temperatures gave identicalvalues to 5% . This leads us to believe that the electroplated metal is actingas an electrical alloy.To estimate the conductivity of this alloy the coaxial resonator of figureB.3 is utilized, this time with a copper center conductor onto which we plateour nichrome-like alloy.Again the cavity Q's are measured as a function of the thickness of de-posited alloy. This data is fit to the theoretical Q (equation B.23) using theour model for effective surface resistance of metallic layers (equation B.19).The best fit conductivity for this alloy isσalloy ' 4.5× 105S/m, (B.24)slightly lower than, but more or less in agreement with the accepted valuefor nichrome: ' 6.6 × 105S/m. The data along with the best fit model is102B.4. Sheet resistance for layered thin metalsshown in figure B.8. The fit is poor for small z indicating that our value forthe conductivity of copper (σc = 5.8 × 107S/m) may be too low, possiblydue to a heavy oxide layer on the outer conductor.Both model and data show that a nichrome layer of 5 - 6 µm would beenough to minimize a resonator Q by this method.Sheet resistance at 10 kHzWe want to check that diocotron waves [58] are not excited as a result ofintroducing resistive materials to the Penning trap. We model the effectivesheet resistance for a surface of nichrome (thickness z) over a thick copperlayer. The result is plotted in Figure B.4.3. We see immediately that even avery thick nichrome layer will not change the surface resistance much at 10kHz.Effective surface resistance for a layer of nichrome(thickness z) over top ofa thick layer of Copper. The result is normalized to the sheet resistance ofcopper at 10 kHz (Rs,copper = 26µΩ). The skin depths of nichrome andcopper at 10 kHz are 6 mm and 650 µm respectively.B.4.4 ConclusionsTo achieve an increased surface resistance at 30 GHz we electroplate about7µm of the nichrome-like alloy to the inner face of the bulge cavity. Themodel and data presented in this appendix show that a nichrome layer of 5103B.4. Sheet resistance for layered thin metals- 6 µm is enough to minimize the resonator Q. Q measurements performedat liquid nitrogen temperatures show that the alloy has a temperature inde-pendent conductivity about the same as nichrome despite there being verylittle chromium present in the alloy.To provide an anti-static shielding layer we utilize colloidal graphite.However, this appendix has investigated the practicality of a gold electro-static shielding layer. We find that this gold layer must have a thicknessless than about 100 nm to maintain resistivity introduced by the underlyingalloy.Meanwhile the surface resistance at 10 kHz would not appreciate consid-erably due to the presence of the nichrome unless the nichrome layer had athickness comparable to or greater than a millimetre.104

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