University of Oxford Department of Physics High-Fidelity Quantum Logic in Ca+ Christopher J. Ballance −1 10 0.9 Photon scattering Motional error Off−resonant lightshift Spin−dephasing error Total error budget Data −2 10 0.99 Gate error Gate fidelity −3 10 0.999 1 2 3 10 10 10 Gate time (µs) A thesis submitted for the degree of Doctor of Philosophy Hertford College Michaelmas term, 2014 Abstract High-Fidelity Quantum Logic in Ca+ Christopher J. Ballance A thesis submitted for the degree of Doctor of Philosophy Michaelmas term 2014 Hertford College, Oxford Trapped atomic ions are one of the most promising systems for building a quantum computer – all of the fundamental operations needed to build a quan- tum computer have been demonstrated in such systems. The challenge now is to understand and reduce the operation errors to below the ‘fault-tolerant thresh- old’ (the level below which quantum error correction works), and to scale up the current few-qubit experiments to many qubits. This thesis describes experimen- tal work concentrated primarily on the first of these challenges. We demonstrate high-fidelity single-qubit and two-qubit (entangling) gates with errors at or be- low the fault-tolerant threshold. We also implement an entangling gate between two different species of ions, a tool which may be useful for certain scalable architectures. We study the speed/fidelity trade-off for a two-qubit phase gate implemented in 43Ca+ hyperfine trapped-ion qubits. We develop an error model which de- scribes the fundamental and technical imperfections / limitations that contribute to the measured gate error. We characterize and minimise various error sources contributing to the measured fidelity, allowing us to account for errors due to the single-qubit operations and state readout (each at the 0:1% level), and to identify the leading sources of error in the two-qubit entangling operation. We achieve gate fidelities ranging between 97:1(2)% (for a gate time tg = 3:8 µs) and 99:9(1)% (for tg = 100 µs), representing respectively the fastest and lowest-error two-qubit gates reported between trapped-ion qubits by nearly an order of mag- nitude in each case. We also characterise single-qubit gates with average errors below 10−4 per operation, over an order of magnitude better than previously achieved with laser-driven operations. Additionally, we present work on a mixed-species entangling gate. We en- tangle of a single 40Ca+ ion and a single 43Ca+ ion with a fidelity of 99:8(5)%, and perform full tomography of the resulting entangled state. We describe how this mixed-species gate mechanism could be used to entangle 43Ca+ and 88Sr+, a promising combination of ions for future experiments. i Acknowledgements Dr. David Lucas has proved an excellent supervisor – as well as sharing his impressive knowledge of atomic physics (and, for better or for worse, Turbo- Pascal and DOS), he also let me get on with whatever I wanted to do in the lab, signed orders, and provided much needed guidance and critique. Over the past few years most of my time has been spent working with Dr. Thomas Harty, Vera Schafer,¨ and Dr. Norbert Linke. Tom has been a spring- board for many an idea (both good and bad), and has been an exceptional quar- termaster. Vera has been a great addition to the ‘old lab’, overcompensating for her smaller stature with phenomenal torque. I have enjoyed Norbert’s company during our rebuilding of the ‘old lab’, despite our differing opinions on a certain micro-fabricated trap. Thank you also to Sarah Woodrow and Keshav Thirumalai, who are busy building the next generation of the experiment, and to the rest of the trapped- ion gang; Martin Sepiol, Jamie Tarlton, Diana Prado Lopes Aude Craik, and Hugo Janacek, as well as Prof. Andrew Steane and Prof. Derek Stacey, the other pillars of the group. I would also like to thank Dr. David Allcock for the ARSES. I am forever grateful to my parents for raising a child who, from a young age, would take apart everything he could get his hands on (but took a while longer to master the art of putting them back together without any pieces left over). Your care, patience, and support have enabled me to become the keen experimentalist I am today – I hope this scientific endeavour will prove less contentious than my previous ventures. My brothers have both encouraged and endured my scientific interests, listening to (and sometimes participating in) many a long technical discussion. I am grateful also for the encouragement and care of my Aunt, Shirley. I would like to thank Isabel, who has been a constant and unswerving source of distraction over the course of my D. Phil. – without her my life would be far less interesting. I gratefully acknowledge funding by the EPSRC and by Hertford College’s Carreras senior scholarship. I would also like to acknowledge the funding I received throughout my undergraduate studies from the Stuart Barber bursary. iii Contents Contents iv 1 Introduction1 1.1 The Quantum Computer........................ 2 1.2 Quantum Computation with Trapped Ions ............. 3 1.3 Thesis Outline.............................. 5 2 Trapped-Ion Qubits7 2.1 The Calcium Ion as a Qubit ...................... 7 2.1.1 Photo-Ionisation ........................ 7 2.1.2 Doppler Cooling ........................ 8 2.1.3 The Qubit and Coherent Manipulations........... 9 2.1.4 Readout of the Qubit State................... 11 2.2 Linear Paul Traps............................ 12 2.2.1 Micro-Motion Detection and Compensation . 14 2.3 Motion of the Ions............................ 15 2.3.1 Quantization of motion .................... 15 2.3.2 Coupling to the Motion .................... 16 3 Raman Interactions 19 3.1 The Raman Coupling.......................... 19 3.2 Rabi Frequencies and Light-Shifts................... 21 3.3 Photon Scattering............................ 21 3.3.1 Raman Scattering........................ 22 3.3.2 Rayleigh Scattering....................... 22 3.3.3 Motional Decoherence from Scattering ........... 23 3.4 Raman Transitions in 43Ca+ ...................... 23 3.4.1 Stretch Qubit .......................... 24 3.4.2 Low-Field Clock Qubit..................... 25 3.4.3 Scaling of Raman Rates with Detuning ........... 25 3.5 Scattering Experiments......................... 28 3.5.1 Ratio of scattering rate to light-shift ............. 28 3.5.2 Stretch Qubit Rayleigh Dephasing.............. 30 4 Two-Qubit Gate Theory 33 4.1 Introduction............................... 33 iv Contents 4.2 The Geometric Phase Gate....................... 34 4.3 The Light-Shift Gate .......................... 37 4.4 Sources of Error............................. 39 4.4.1 Mis-Set Gate Detuning..................... 41 4.4.2 Mis-Set Rabi Frequency .................... 41 4.4.3 Unequal Illumination ..................... 41 4.4.4 Thermal Errors ......................... 42 4.4.5 Motional Heating........................ 44 4.4.6 Motional Dephasing...................... 45 4.4.7 Amplitude and Phase Noise.................. 46 4.4.8 Motional Kerr Cross-Coupling................ 47 4.4.9 Spin Dephasing from Rayleigh Scattering.......... 48 4.4.10 Raman Photon Scattering................... 48 4.4.11 Off-Resonant Excitation.................... 49 4.5 Error Summary ............................. 54 5 Apparatus 57 5.1 The Ion Trap............................... 58 5.1.1 Trap Geometry ......................... 58 5.1.2 Vacuum System......................... 59 5.1.3 Trap RF Source ......................... 60 5.1.4 Trap DC Source......................... 62 5.2 Magnetic Field Coils .......................... 63 5.3 Imaging System............................. 64 5.4 Laser Systems.............................. 65 5.4.1 Ca+ Lasers............................ 65 5.4.2 Photo-Ionisation Lasers .................... 67 5.4.3 Beam-Paths ........................... 67 5.5 Raman Lasers.............................. 69 5.5.1 Laser Sources.......................... 69 5.5.2 AOM Switch-Yard and Optical Fibres............ 70 5.5.3 Beam-Paths ........................... 72 5.6 Microwave Qubit Drive ........................ 74 5.7 Experimental Control.......................... 75 5.7.1 Coherent DDS.......................... 76 5.7.2 Laser Intensity Stabilisation.................. 76 6 Experiment Characterization 79 6.1 Trap Behaviour ............................. 79 6.1.1 Axial Micro-Motion ...................... 79 6.1.2 Magnetic Field Gradient.................... 82 6.1.3 Recrystallisation ........................ 84 v CONTENTS 6.2 State-Preparation and Readout .................... 86 6.2.1 43Ca+ Optical Qubit ...................... 87 6.2.2 43Ca+ Stretch Qubit ...................... 88 6.2.3 43Ca+ Low-Field Clock Qubit................. 88 6.2.4 40Ca+ Zeeman Qubit...................... 89 6.3 Qubit Dephasing and Frequency Drift................ 90 6.3.1 Magnetic Field Drift and Modulation ............ 90 6.3.2 Stretch Qubit Coherence.................... 91 6.3.3 Clock Qubit Coherence .................... 92 6.4 Raman Lasers.............................. 93 6.4.1 Beam-Pointing and Rabi Frequency Stability . 93 6.4.2 Intensity and Phase Noise................... 96 6.5 Motion.................................. 100 6.5.1 Cooling ............................. 100 6.5.2 Motional Frequency Drifts................... 101 6.5.3 Motional Heating........................ 102 6.5.4 Motional Coherence ...................... 103 7 Single-Qubit Gates 107 7.1 Randomized Benchmarking of a Single Qubit . 107 7.2 Experimental Setup........................... 108 7.3 Error