Raman Sideband Cooling and Coherent Manipulation of Trapped Ions A thesis submitted for the degree of Doctor of Philosophy 0.7 0.6 0.5 Fraction of ions shelved 0.4 0.3 0.2 −1000 −800 −600 −400 −200 0 200 400 600 800 1000 Raman detuning /kHz Simon Webster Hilary Term St. John’s College 2005 Oxford Abstract This thesis describes experiments primarily concerned with the manipulation of the spin and quantum motional state of single trapped ions and pairs of ions. A two stage photoionization system is described. This allows the loading of ions of a specific isotope of calcium into our trap. We measure the absolute cross section for 1 photoionization from the 4s4p P1 state of neutral Ca. We demonstrate the loading and cooling of all the naturally occurring isotopes of calcium, including of pure crystals of 43Ca+. A magnetic resonance is used to determine accurately the Zeeman splitting of the ground state of a 40Ca+ ion. The coherent manipulation of the spin state of a 40Ca+ ion using this magnetic resonance is shown. We observe high quality Rabi oscillations between the two spin states with π times of order 100 µs and with decay times of order milliseconds. By using multiple pulses of this magnetic resonance in a Ramsey type experiment we are able to measure the net light shift induced on the two spin states by a circularly polarised far-off-resonant beam, and thus deduce its intensity precisely, and also ensure that such a beam is linearly polarised to high accuracy. We demonstrate a laser-driven rotation of the qubit with a π time of order a few microseconds. We also implement Rabi flopping by Raman carrier transitions, with a similar π time. We investigate three Raman sideband cooling methods applied to a single 40Ca+ ion. Two of these provide continuous cooling, with the lasers which drive the Raman transitions also providing the repumping. A simple theoretical model for this continuous cooling process is presented. The third method is pulsed sideband cooling, using pulses of far-off- resonant Raman beams to drive the first red motional sideband of the ion, and then use a near-resonant beam to act as a repumper. The mean vibrational excitation of the ion is measured by comparing sideband heights, for these three cooling schemes. We achieve maximum ground state occupation probabilities of over 85%. We also demonstrate sub-Doppler cooling of a pair of trapped ions. Using a continuous cooling method followed by pulsed sideband cooling of the two axial motional modes we demonstrate cooling of both modes into the Lamb-Dicke regime. We present a rate equations program developed to study 43Ca+, which we intend to use to embody our qubit in future experiments. We use our program to fit data taken from a trapped 43Ca+ ion during photoionization experiments and to study the effect of the hyperfine structure of the ion on producing a high scattering rate during Doppler cooling of the ion. i Acknowledgements My first thanks go to my supervisor Prof. Andrew Steane for all his help and guidance which has made this work possible, and whose enthusiasm to try new things in the lab is unmatched. My thanks also go to Prof. Derek Stacey whose enthusiasm to get things working correctly in the lab is unmatched. Dr. David Lucas has been a great help in getting the work presented here done. His knowledge and mastery of the experiment is awe inspiring at times and my thanks go to him. I must thank all the other ion trappers for their friendship, hard work on the experiment and time spent drinking tea in the Clarendon common room: Dr. Matthew McDonnell, Dr. J.-P. Stacey, Dr. Angel Ramas, Dr. Marek Saˇsura,ˇ Jonathan Home, Ben Keitch, Nick Thomas and Greg Imreh. Out of all these I would particularly like to thank Matt and J.-P. for their friendship when I first joined the group. Thanks also to Graham Quelch, George Matthews and Rob Harris for their practical help and advice in the workshop. Outside of the lab thanks to all the friends I’ve made in Oxford, both as an under- graduate and a graduate, and who have made my time here so enjoyable. My parents have made all of this possible. While they may not understand why I want to be a physicist, their love and support have helped me to become one. Finally I thank Katrina, for her love and for being there for me over the past five years. ii Contents Abstract i Acknowledgements ii 1 Introduction 1 1.1 Quantumcomputinginaniontrap. 1 1.1.1 Quantumgates .............................. 3 1.2 Summaryofthethesis .............................. 7 2 Apparatus 9 2.1 Buildingblocks .................................. 9 2.1.1 Calcium.................................. 9 2.1.2 Paultrap ................................. 10 2.1.3 Lasersystems............................... 14 2.1.4 Magnetic resonance coil . 18 2.1.5 Imagingsystem.............................. 18 2.2 Opticallayout................................... 19 2.2.1 Coolingbeams .............................. 20 2.2.2 Ramanbeams............................... 21 2.3 Controloftheexperiment . 23 3 Photoionization 26 3.1 Geometricfactor ................................. 26 3.2 Photoionization cross section . ..... 27 4 Single qubit Rabi flopping experiments 29 4.1 Fluorescence, Doppler cooling and qubit initialization ............ 30 4.2 QubitReadout .................................. 30 4.3 Basic concepts in magnetic resonance . ...... 33 4.3.1 Rabiflopping ............................... 33 4.3.2 RamseyandSpinecho. .. .. .. .. .. .. .. 34 4.4 Magnetic resonance experiments . 35 4.4.1 Determining the qubit splitting . 35 4.4.2 Rabifloppingandcoherencetime. 38 4.5 Raman transitions and coupling to the motion . ...... 39 4.5.1 Characterisation of the Raman beams . 41 4.5.2 Sidebandspectrumofasingleion . 44 iii CONTENTS iv 5 Cooling 47 5.1 Notes on Cooling and Diagnostics . 47 5.1.1 CoolingProcesses ............................ 47 5.1.2 Temperature Diagnostics . 49 5.2 ContinuousRamanSidebandCooling . 51 5.3 RedDetunedContinuousCooling. 55 5.3.1 Fluorescencespectrum. 56 5.3.2 Temperaturemeasurements . 64 5.4 ‘EIT’cooling ................................... 68 5.5 Pulsedsidebandcooling . 70 5.5.1 Conclusions................................ 72 6 Cooling two ions 74 6.1 Dynamicsoftwotrappedions. 74 6.1.1 Normalmodes .............................. 74 6.1.2 Quantisation of the motion . 75 6.1.3 Determiningthequbitstate . 76 6.1.4 Temperaturediagnostics. 77 6.2 Experiments.................................... 80 6.2.1 ContinuousSidebandCooling . 80 6.2.2 Pulsedsidebandcooling . 83 6.3 Conclusions .................................... 84 7 43Ca+ 85 7.1 Whychangeisotope?............................... 85 7.2 Modelling 43Ca+ ................................. 86 7.2.1 Rateequations .............................. 89 7.3 Programresults.................................. 90 7.3.1 Coolingwiththe393laser. 90 7.3.2 Cooling with multiple 397 frequencies . 91 7.3.3 Cooling using circularly polarised 397 light . ....... 93 8 Conclusion 96 A Photoionization publication 100 B Rate equations code 101 Chapter 1 Introduction The implementation of quantum information processing (QIP) is a subject of great inter- est in many areas of physics. Techniques developed in atomic physics for high resolution spectroscopy and metrology applications are almost uniquely suited for the task of imple- menting quantum information processing. The world of metrology and atomic clocks has advanced by developing extremely precise control over quantum systems, and indeed the Ramsey split pulse method can be recast as the manipulation of single qubit systems. The use of laser cooled trapped ions for metrology has advanced rapidly over the last 25 years. Frequency standards based on optical transitions in trapped ions (rather than microwave transitions in neutral atoms) are becoming increasingly accurate, and are poised to overtake the cesium standard[1]. Meanwhile, ion trap quantum computing is arguably the most successful of the raft of physical systems under investigation for QIP potential, with on demand production of entangled states[2] and teleportation of quantum information[3, 4] achieved as well as a demonstration of basic error-correction[5, 6]. This introductory chapter briefly outlines the properties of trapped ions which make them suitable for QIP and describes quantum gates, including the one we are working towards implementing, before giving a summary of the work presented in this thesis. 1.1 Quantum computing in an ion trap The network model of a quantum computer, based on a register of qubits and a fixed finite set of gate operations, was proposed by Deutsch[7]. To implement a quantum algorithm, such as Shor’s factoring algorithm[8], we start with an initialized quantum register. This is then fed through a network of quantum gates, which act on both individual and collections of two or more qubits. In addition to these gates there are also measurements of individual qubits, both within the network for the purposes of error correction, where the results influence subsequent gate operations, and at the end of the network, where the result of performing the algorithm is obtained. DiVincenzo criteria The requirements on a physical system and our control over it which must be met for it to be used as described above were first outlined by Deutsch. A more extensive discussion is summarised by the DiVincenzo criteria[9] which are listed below (in italics). We explain how they are met by our system. 1 CHAPTER 1. INTRODUCTION 2 1. A scaleable physical system with well-characterised qubits The first of these criteria calls for well-characterised qubits. These qubits are provided by the trapped ions, each ion embodying a qubit in its internal state. The majority of experiments in this thesis are concerned with 40Ca+, and the qubit basis states are the two Zeeman sublevels of the ground state. The ion trap group at the University of Innsbruck use one of these sublevels as a qubit basis state, the other being a sublevel in the metastable 3D5/2 level. The ion trap can be scaled up in two ways. For a small number of qubits, holding all the ions in a single harmonic potential is possible.