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Cavity Control in a Single-Electron Quantum Cyclotron: an Improved Measurement of the Electron Magnetic Moment

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Cavity Control in a Single-Electron Quantum Cyclotron: an Improved Measurement of the Electron Magnetic Moment Cavity Control in a Single-Electron Quantum Cyclotron: An Improved Measurement of the Electron Magnetic Moment A thesis presented by David Andrew Hanneke to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts December 2007 c 2007 - David Andrew Hanneke All rights reserved. Thesis advisor Author Gerald Gabrielse David Andrew Hanneke Cavity Control in a Single-Electron Quantum Cyclotron: An Improved Measurement of the Electron Magnetic Moment Abstract A single electron in a quantum cyclotron yields new measurements of the electron magnetic moment, given by g=2 = 1:001 159 652 180 73 (28) [0:28 ppt], and the fine structure constant, α−1 = 137:035 999 084 (51) [0:37 ppb], both significantly improved from prior results. The static magnetic and electric fields of a Penning trap confine the electron, and a 100 mK dilution refrigerator cools its cyclotron motion to the quantum-mechanical ground state. A quantum nondemolition measurement allows resolution of single cyclotron jumps and spin flips by coupling the cyclotron and spin energies to the frequency of the axial motion, which is self-excited and detected with a cryogenic amplifier. The trap electrodes form a high-Q microwave resonator near the cyclotron fre- quency; coupling between the cyclotron motion and cavity modes can inhibit sponta- neous emission by over 100 times the free-space rate and shift the cyclotron frequency, a systematic effect that dominated the uncertainties of previous g-value measure- ments. A cylindrical trap geometry creates cavity modes with analytically calculable couplings to cyclotron motion. Two independent methods use the cyclotron damping rate of an electron plasma or of the single electron itself as probes of the cavity mode structure and allow the identification of the modes by their geometries and couplings, iv the quantification of an offset between the mode and electrostatic centers, and the reduction of the cavity shift uncertainty to sub-dominant levels. Measuring g at four magnetic fields with cavity shifts spanning thirty times the final g-value uncertainty provides a check on the calculated cavity shifts. Magnetic field fluctuations limit the measurement of g by adding a noise-model dependence to the extraction of the cyclotron and anomaly frequencies from their resonance lines; the relative agreement of two line-splitting methods quantifies a line- shape model uncertainty. New techniques promise to increase field stability, narrow the resonance lines, and accelerate the measurement cycle. The measured g allows tests for physics beyond the Standard Model through searches for its temporal variation and comparisons with a \theoretical" g-value cal- culated from quantum electrodynamics and an independently measured fine structure constant. Contents Title Page . i Abstract . iii Table of Contents . v List of Figures . ix List of Tables . xi Acknowledgments . xii 1 Introduction 1 1.1 The Electron Magnetic Moment . 2 1.1.1 The fine structure constant . 3 1.1.2 QED and the relation between g and α ............. 5 1.1.3 Comparing various measurements of α ............. 9 1.1.4 Comparing precise tests of QED . 11 1.1.5 Limits on extensions to the Standard Model . 16 1.1.6 Magnetic moments of the other charged leptons . 24 1.1.7 The role of α in a redefined SI . 26 1.2 Measuring the g-Value . 29 1.2.1 g-value history . 29 1.2.2 An artificial atom . 30 1.2.3 The Quantum Cyclotron . 32 2 The Quantum Cyclotron 36 2.1 The Penning Trap . 36 2.1.1 Trap frequencies and damping rates . 39 2.1.2 The Brown{Gabrielse invariance theorem . 42 2.2 Cooling to the Cyclotron Ground State . 43 2.3 Interacting with the Electron . 45 2.3.1 Biasing the electrodes . 45 2.3.2 Driving the axial motion . 50 2.3.3 Detecting the axial motion . 51 2.3.4 QND detection of cyclotron and spin states . 56 v Contents vi 2.3.5 Making cyclotron jumps . 58 2.3.6 Flipping the spin . 62 2.3.7 \Cooling" the magnetron motion . 63 2.4 The Single-Particle Self-Excited Oscillator . 64 2.5 Summary . 68 3 Stability 69 3.1 Shielding External Fluctuations . 70 3.2 High-Stability Solenoid Design . 71 3.3 Reducing Motion in an Inhomogeneous Field . 73 3.3.1 Stabilizing room temperature . 75 3.3.2 Reducing vibration . 76 3.4 Care with Magnetic Susceptibilities . 78 3.5 Future Stability Improvements . 79 4 Measuring g 82 4.1 An Experimenter's g ........................... 82 4.2 Expected Cyclotron and Anomaly Lineshape . 85 4.2.1 The lineshape in the low and high axial damping limits . 88 4.2.2 The lineshape for arbitrary axial damping . 89 4.2.3 The cyclotron lineshape for driven axial motion . 91 4.2.4 The saturated lineshape . 92 4.2.5 The lineshape with magnetic field noise . 94 4.3 A Typical Nightly Run . 96 4.3.1 Cyclotron quantum jump spectroscopy . 96 4.3.2 Anomaly quantum jump spectroscopy . 99 4.3.3 Combining the data . 101 4.4 Splitting the Lines . 106 4.4.1 Calculating the weighted mean frequencies . 107 4.4.2 Fitting the lines . 109 4.5 Summary . 113 5 Cavity Control of Lifetimes and Line-Shifts 115 5.1 Electromagnetic Modes of an Ideal Cylindrical Cavity . 117 5.2 Mode Detection with Synchronized Electrons . 122 5.2.1 The parametric resonance . 123 5.2.2 Spontaneous symmetry breaking in an electron cloud . 124 5.2.3 Parametric mode maps . 126 5.2.4 Mode map features . 130 5.3 Coupling to a Single Electron . 136 5.3.1 Single-mode approximation . 137 5.3.2 Renormalized calculation . 139 Contents vii 5.3.3 Single-mode coupling with axial oscillations . 146 5.4 Single-Electron Mode Detection . 147 5.4.1 Measuring the cyclotron damping rate . 149 5.4.2 Fitting the cyclotron lifetime data . 150 5.4.3 Axial and radial (mis)alignment of the electron position . 153 5.5 Cavity-shift Results . 159 5.5.1 2006 cavity shift analysis . 159 5.5.2 Current cavity shift analysis . 161 5.6 Summary . 162 6 Uncertainties and a New Measurement of g 165 6.1 Lineshape Model Uncertainty and Statistics . 166 6.1.1 Cyclotron and anomaly lineshapes with magnetic field noise . 167 6.1.2 The line-splitting procedure . 169 6.1.3 2006 lineshape model analysis . 170 6.1.4 Current lineshape model analysis . 171 6.1.5 Axial temperature changes . 175 6.2 Power Shifts . 177 6.2.1 Anomaly power shifts . 178 6.2.2 Cyclotron power shifts . 180 6.2.3 Experimental searches for power shifts . 181 6.3 Axial Frequency Shifts . 185 6.3.1 Anharmonicity . 185 6.3.2 Interaction with the amplifier . 186 6.3.3 Anomaly-drive-induced shifts . 186 6.4 Applied Corrections . 187 6.4.1 Relativistic shift . 188 6.4.2 Magnetron shift . 189 6.4.3 Cavity shift . 190 6.5 Results . 191 6.5.1 2006 measurement . 191 6.5.2 New measurement . 193 7 Future Improvements 196 7.1 Narrower Lines . 197 7.1.1 Smaller magnetic bottle . 197 7.1.2 Cooling directly or with feedback . 199 7.1.3 Cavity-enhanced sideband cooling . 200 7.2 Better Statistics . 210 7.2.1 π{pulse . 210 7.2.2 Adiabatic fast passage . 212 7.3 Remaining questions . 214 Contents viii 7.4 Summary . 215 8 Limits on Lorentz Violation 217 8.1 The Electron with Lorentz Violation . 217 8.1.1 Energy levels and frequency shifts . 218 8.1.2 Experimental signatures . 220 8.1.3 The celestial equatorial coordinate system . 221 8.2 Data Analysis . 222 8.2.1 Local Mean Sidereal Time . 223 8.2.2 Fitting the data . 225 8.2.3 Comparisons with other experiments . 227 8.3 Summary and outlook . 229 9 Conclusion 231 9.1 A New Measurement of g . 231 9.2 Outlook . 233 9.2.1 The e+ g-value and testing CPT . 233 9.2.2 The proton-to-electron mass ratio . 233 9.2.3 The proton and antiproton magnetic moments . 234 9.2.4 A single electron as a qubit . 235 9.3 Summary . 237 A Derivation of Single-Mode Coupling with Axial Oscillations 238 Bibliography 242 List of Figures 1.1 Electron g-value comparisons . 2 1.2 Sample Feynman diagrams.
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