Spectroscopy and Dynamics of Laser-Cooled Ca+ Ions in a Penning Trap Richard James Hendricks October 2006 Thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy of the University of London and for the Diploma of Imperial College Abstract I report on work geared towards improving the efficiency of laser cooling in the Penning trap by manipulating the dynamics of the trapped ions. The wider aim of this project is to measure heating and decoherence rates in order to de- termine the feasibility of quantum information processing using ions trapped in a Penning trap. In this thesis I introduce the basic concepts of quantum information processing and its implementation in radiofrequency trapped-ion systems. I report on the difficulties currently being faced and how many of these obstacles can be circumvented by using Penning traps for ion confine- ment. Laser cooling in the Penning trap is usually rather inefficient because of the instability of the magnetron radial mode of motion. By extending an analytical model for laser cooling in the Penning trap I show that a technique called ‘axialisation’ can overcome this inefficiency by coupling the magnetron mode to another, stable radial mode. I present the results of an experiment to implement the axialisation technique and measure the improvement in laser cooling rates of calcium ions, and I assess the extent to which these results agree with those derived from our model. Finally I document our efforts to trap and cool single calcium ions in the Penning trap and discuss modifications to the experimental procedure that may make this possible in the near future. 2 Contents 1 Introduction 11 2 Quantum information processing 15 2.1 Properties of quantum systems .................. 17 2.1.1 Quantum bits ........................ 17 2.1.2 Multi-qubit systems .................... 19 2.1.3 Decoherence ......................... 20 2.2 Quantum algorithms ........................ 23 2.2.1 Universal computers .................... 23 2.2.2 Quantum gates ....................... 24 2.2.3 Shor’s quantum factoring algorithm ........... 25 2.2.4 Other quantum algorithms ................ 27 2.3 Quantum error correction ..................... 28 2.3.1 Classical error correction ................. 29 2.3.2 Principles of QEC ..................... 31 2.3.3 Application of QEC .................... 32 2.3.4 Requirements of QEC ................... 34 2.4 Criteria for the implementation of QIP ............. 35 2.4.1 System characterisation and scalability .......... 36 2.4.2 System initialisation .................... 37 2.4.3 Long decoherence times .................. 38 2.4.4 A universal set of quantum logic gates .......... 38 2.4.5 Qubit measurement .................... 39 2.5 Conclusions ............................. 40 3 Quantum information processing in ion traps 42 3.1 Quantum computation with trapped ions ............ 43 3 CONTENTS 3.1.1 Ion cooling in the linear RF trap ............. 43 3.1.2 Electronic states as qubits ................. 45 3.1.3 Quantum gates for trapped ions ............. 48 3.2 Quantum computing potential of trapped ions .......... 51 3.3 Realisation of ion trap quantum information processing . 53 3.3.1 Previous developments in ion trap QIP ......... 53 3.3.2 Current issues in ion trap QIP .............. 61 3.3.3 Future directions ...................... 64 3.4 Penning trap quantum information processing .......... 70 3.5 Conclusions ............................. 72 4 Trapping and laser cooling in the Penning trap 74 4.1 Electrode configuration ...................... 75 4.2 Ion motion in the Penning trap .................. 77 4.3 Laser cooling in the Penning trap ................. 81 4.3.1 The laboratory frame ................... 81 4.3.2 The rotating frame ..................... 89 4.4 Axialisation ............................. 91 4.4.1 A quantum picture of axialisation ............ 92 4.4.2 A classical analogy for axialisation ............ 93 4.4.3 A model for laser cooling in the presence of axialisation 97 4.4.4 Amplitude and phase response to a driving field . 109 4.5 Ion Clouds and the Effects of Space Charge ........... 117 5 Experimental setup 120 5.1 Ca+ electronic structure ......................120 5.2 Experimental overview .......................123 5.3 The split-ring trap .........................125 5.3.1 Trap structure .......................125 5.3.2 Ion generation .......................127 5.3.3 Vacuum system .......................127 5.3.4 Magnetic field generation ................. 129 5.3.5 Operation as Penning trap ................ 129 5.3.6 Operation as RF trap ................... 130 5.4 Doppler cooling lasers .......................132 5.5 Repumper lasers ..........................134 4 CONTENTS 5.6 Laser diagnostics ..........................137 5.7 Laser locking ............................142 5.7.1 Optical reference cavities ................. 142 5.7.2 Locking setup ........................144 5.7.3 Locking procedure .....................146 5.7.4 Cavity stability .......................147 5.8 Fluorescence detection .......................149 5.9 General procedure .........................151 6 Axialisation 155 6.1 Experimental methods .......................156 6.2 Results ................................165 6.2.1 iCCD images ........................165 6.2.2 Cooling rate measurements ................ 170 6.2.3 Classical avoided crossing ................. 177 6.3 Conclusions .............................183 7 Towards single-ion work in the Penning trap 185 7.1 Loading and detection of small numbers of ions ......... 186 7.2 Single ions in the RF trap .....................189 7.3 Small clouds of ions in the Penning trap ............. 194 7.4 Contaminant ions ..........................203 7.5 Coherent population trapping ................... 212 7.6 Laser and magnetic field instability ................ 229 8 Discussion 231 8.1 Discussion of results ........................231 8.2 Future directions ..........................232 8.2.1 Improving magnetic field stability ............ 232 8.2.2 Improving laser cavity stability .............. 234 8.2.3 Sideband cooling and beyond ............... 235 Bibliography 237 5 List of Figures 2.1 The Bloch sphere .......................... 18 3.1 Linear RF trap structure ..................... 44 3.2 Sideband cooling .......................... 46 3.3 Partial energy level diagram of Ca+ ............... 47 3.4 Cirac and Zoller CNOT gate ................... 50 3.5 Magnetic field insensitive qubit states .............. 63 3.6 Microfabricated planar RF trap .................. 66 3.7 Microfabricated array of RF traps ................ 66 3.8 Heating rate measurements in RF traps ............. 68 3.9 Variation of heating rate with endcap separation in a RF trap 69 4.1 Electrode structure of an ‘ideal’ Penning trap .......... 76 4.2 Ion motion in the Penning trap .................. 80 4.3 Schematic of the model for laser cooling in the Penning trap . 82 4.4 Linear approximations used in the laser cooling model ..... 85 4.5 Damping rates derived from the laser cooling model ...... 90 4.6 Energy level diagram showing Penning trap motional states . 93 4.7 Ion motion in the rotating frame ................. 95 4.8 Motion of an axialised ion in the laboratory frame ....... 96 4.9 Motional frequencies and damping rates when (/ω )2 M 2 . 105 1 2 2 4.10 Motional frequencies and damping rates when (/ω1) = M . 106 4.11 Motional frequencies and damping rates when (/ω )2 M 2 . 108 1 4.12 Avoided crossing measured with a FT-ICR technique . 110 4.13 Response of a weakly axialised ion to a dipole drive ....... 114 4.14 Response of a strongly axialised ion to a dipole drive . 115 4.15 Argand plots showing ion response to a dipole drive . 116 6 LIST OF FIGURES 5.1 Partial energy level diagram of calcium at 0.98 tesla . 121 5.2 Zeeman splittings of relevant transitions at 0.98 tesla . 122 5.3 Experimental setup .........................124 5.4 Scale drawings of the split-ring trap ............... 126 5.5 Ultra-high vacuum apparatus ................... 128 5.6 Application of the quadrupolar and dipolar drives ....... 130 5.7 Setup for operating as a RF trap ................. 131 5.8 The Littrow configuration extended cavity diode laser . 133 5.9 Spectrum analysis of the 866nm laser sidebands ......... 136 5.10 Optical layout of the wavemeter .................. 139 5.11 Spread in successive wavemeter measurements .......... 141 5.12 Basic design of the optical reference cavities ........... 144 5.13 Optical setup for laser locking ................... 145 5.14 Block diagram of the laser locking electronics .......... 146 5.15 Plots of optical cavity frequency drift ............... 148 5.16 Diagram of the fluorescence detection systems .......... 150 5.17 Fluorescence trace from a large cloud of ions .......... 153 5.18 Fluorescence from ions during and after loading ......... 154 6.1 Harmonic oscillator phase relative to a driving force . 159 6.2 Experimental setup for RF–photon correlation ......... 161 6.3 Example RF–photon correlation traces .............. 163 6.4 Example motional phase measurements of an ion cloud . 164 6.5 iCCD images of ion clouds with and without axialisation . 167 6.6 Fitting of Gaussian functions to iCCD images ......... 168 6.7 Variation of ion cloud aspect ratio with axialisation amplitude 169 6.8 Fluorescence rates as a function of axialisation amplitude . 171 6.9 Motional phase measurements showing two resonances . 172 6.10 Variation of magnetron cooling rate with axialisation amplitude 174 6.11 Variation