Controlled Dynamics of Laser-Cooled Ions in a Penning Trap Eoin Seymour Gwyn Phillips December 2004 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 a scheme for improved control over the motion of small numbers of laser-cooled ions in a Penning trap. This work is part of a larger project to assess the suitability of the Penning trap for quantum information processing (QIP). Laser cooling of ions in this type of trap is complicated by large Zeeman shifts of the cooling- and repumping-transition wavelengths from those in zero mag- netic field. Lasers can not, therefore, be tuned to the unshifted atomic spectral lines using an external reference such as a hollow-cathode lamp. In order to address this difficulty I built a wavemeter with a precision of one part in 107 for direct tuning of the lasers to the transition wavelengths. It was also necessary to improve the laser-frequency stabilities by building reference cavities to which they could be locked. In evaluating the suitability of the Penning trap for QIP, one must address the poor localisation of the ions due to the difficulty in cool- ing one of the radial motions: the magnetron motion. I have implemented a technique called axialisation that reduces this motion's amplitude by coupling it to the other radial motion: the modified cyclotron motion, which is efficiently laser-cooled. This was done with small clouds of magnesium ions and later with small clouds of calcium ions. The damping rate of both radial motions was measured using near-resonant motional excitation and an RF–photon correla- tion technique. The coupling rate could be inferred from a shift of the motional frequency in a classical analogue of a quantum avoided crossing. Increases in damping rates of the magnetron motion of more than an order of magnitude were observed for both calcium and magnesium ions and coupling rates of or- der 1 kHz were measured. 2 Do mo thuismitheoirı´ 3 Contents 1 Introduction 15 2 Ion trapping and laser-cooling 19 2.1 RF traps . 20 2.2 The Penning trap . 23 2.3 The combined trap . 27 2.4 Buffer-gas cooling . 27 2.5 Resistive-cooling . 28 2.6 Light pressure and Doppler-cooling . 29 2.7 Doppler-cooling in the Penning trap . 31 2.8 Resolved-sideband cooling . 35 2.8.1 The Lamb–Dicke parameter . 36 2.9 Raman- and EIT-cooling . 38 3 Quantum Information Processing 40 3.1 Information theory . 40 3.1.1 Classical information . 41 3.1.2 Quantum information . 42 3.2 Quantum algorithms . 43 3.2.1 The c-NOT gate . 44 3.3 Quantum error correction . 45 3.4 The DiVincenzo criteria . 46 3.5 A proposal for a quantum gate: the Cirac–Zoller scheme . 47 3.6 QIP in a magnetic field . 51 3.6.1 Spin-spin coupling . 51 3.6.2 QIP with electrons in a Penning trap . 53 3.7 Experimental realisations of QIP . 53 4 CONTENTS 3.8 Technical limitations . 55 3.9 Discussion . 56 4 Experimental set-up 57 4.1 Trap . 57 4.1.1 Atomic-beam ovens . 59 4.1.2 Vacuum system . 62 4.1.3 Magnet . 65 4.2 Laser-cooling of Ca+ . 65 4.2.1 The cooling cycle . 65 4.3 Lasers . 67 4.3.1 Extended-cavity diode lasers . 67 4.3.2 Cooling lasers . 69 4.3.3 Re-pumping lasers . 70 4.3.4 Quadrupole-transition laser . 72 4.4 Locking cavities . 73 4.4.1 Temperature stabilisation . 76 4.4.2 Mode matching . 76 4.4.3 Locking the lasers . 79 4.5 Laser tuning . 81 4.6 Fluorescence Detection . 82 4.7 Loading the trap . 84 5 Wavemeter 86 5.1 Introduction . 86 5.2 Basic wavemeter design . 87 5.3 Resolution . 89 5.4 Errors . 91 5.5 Optical and Mechanical Design . 93 5.5.1 Stabilised reference laser . 95 5.5.2 Cart . 96 5.6 Electronics . 98 5.7 Alignment . 99 5.8 Calibration . 100 5.9 Discussion . 102 5 CONTENTS 6 Axialisation of ions in a Penning trap 104 6.1 Plasma confinement . 104 6.2 FT-ICR spectroscopy . 105 6.3 Excitation frequencies and geometries . 107 6.3.1 Dipolar excitation . 107 6.3.2 Parametric excitation . 107 6.3.3 Azimuthal quadrupolar excitation . 108 6.4 Coupling motional modes . 110 6.4.1 Coupling frequencies . 110 6.4.2 Coupling to the axial mode . 111 6.5 Coupling the magnetron and modified cyclotron motions . 112 6.5.1 The rotating frame . 113 6.5.2 The azimuthal quadrupole field in the rotating frame . 115 6.5.3 A quantum mechanical picture . 118 6.6 Measuring the damping rate of the motions . 120 6.6.1 Uncoupled motion . 120 6.6.2 Coupled oscillators . 122 6.7 Coupling and avoided crossings . 124 6.7.1 Dressed states . 125 6.8 Axialisation in the presence of laser-cooling . 128 7 Axialisation of magnesium ions 130 7.1 Introduction . 130 7.2 Doppler-cooling magnesium . 130 7.2.1 Energy level structure . 131 7.2.2 Laser system . 131 7.3 Axialisation . 136 7.3.1 Loading technique . 137 7.3.2 Turning on the drive . 137 7.4 RF–photon correlation . 138 7.5 Measured cooling rates . 140 8 Axialisation of calcium ions 144 8.1 Axialisation-drive circuit . 145 8.2 Imaging of clouds and qualitative results . 145 8.2.1 Laser-cooling rate . 149 6 CONTENTS 8.2.2 Axialisation-drive amplitude . 149 8.2.3 Cloud size . 151 8.2.4 Trap bias . 151 8.2.5 Observation of motional excitation on the MCS . 151 8.3 Quantitative results . 155 8.3.1 Magnetron-motion damping rate . 155 8.3.2 Modified-cyclotron-motion damping rate . 158 8.3.3 Shifts in motional frequencies with axialisation . 159 8.3.4 Coupling rate of the motions . 164 8.4 Loading with axialisation . 168 8.5 Discussion . 168 8.5.1 Future directions . 169 9 Cavity QED experiments with trapped ions 172 9.1 Introduction . 172 9.2 Single-photon sources . 173 9.3 Quantum communication and state transfer . 174 9.3.1 Cirac proposal . 175 9.3.2 Experimental progress . 177 9.4 Laser and cavity locking . 178 9.4.1 Sources of noise . 179 9.4.2 Measuring frequency fluctuations . 180 9.4.3 Locking to the side of a fringe . 182 9.4.4 The Pound–Drever–Hall lock . 183 9.4.5 Cavity Stabilisation . 187 9.5 Experimental Setup . 188 9.5.1 Ion-trap cavity . 188 9.6 Ti:sapphire laser lock . 192 9.7 Future Directions . 196 10 Discussion 198 10.1 Future Directions . 200 A Octave code for cavity mode matching 203 B Split-ring trap 207 7 CONTENTS Bibliography 210 8 List of Figures 2.1 The Paul trap . 21 2.2 Paul trap potential . 22 2.3 Paul trap stability diagram . 23 2.4 Penning trap motional frequencies . 26 2.5 Penning trap radial motion . 27 2.6 Penning trap cooling-beam offset . 32 2.7 Influence of the motional-frequency difference on the cooling rate 33 2.8 The effect of laser offsets on cooling rate (low laser power) . 34 2.9 The effect of laser offsets on cooling rate (high laser power) . 34 2.10 Motional sidebands . ..