Transport of Trapped-Ion Qubits within a Scalable Quantum Processor by R. Bradford Blakestad B.S., California Institute of Technology, 2002 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics 2010 This thesis entitled: Transport of Trapped-Ion Qubits within a Scalable Quantum Processor written by R. Bradford Blakestad has been approved for the Department of Physics Dr. David J. Wineland Prof. Murray J. Holland Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Blakestad, R. Bradford (Ph.D., Physics) Transport of Trapped-Ion Qubits within a Scalable Quantum Processor Thesis directed by Dr. David J. Wineland Key requirements for efficient large-scale quantum computing (QC) include reliable transport of information throughout the processor and the ability to perform gates between arbitrarily selected qubits. Trapped ions are a useful system for studying the elements of QC and can potentially satisfy these requirements. The internal atomic state of the ion can be used as a qubit. Quantum gate operations can be performed with electromagnetic fields and entangling operations are achieved by using those fields to map information between the qubits and a shared motional mode of the trapped ions. One approach to extending this to large-scale QC would be to distribute ions over separate zones in an array, where information would be shared between zones by physically transporting the ions. Multidimensional arrays incorporating junctions would enable ions selected from arbitrary locations to be grouped together for multi-qubit gates. However, excitation of the harmonic motion of the ions during transport can negatively impact computational fidelity. Here I report a novel ion-trap design including an `X' junction. Reliable transport of 9Be+ ions through this array has been demonstrated with motional excitation corresponding to less than one quantum in the local confining potential. The process of designing and building the trap, as well as producing the control potentials that shuttle the ions will be described. In addition, sources of motional excitation will be discussed. This work is an important step in demonstrating the scalability of this particular approach to trapped-ion QC. Such techniques should provide a path toward transfer of information in a larger- scale quantum processor and the potential to employ an increased number of qubits in quantum algorithm experiments. Dedication To my parents and grandparents, for showing me what is important and for all the support over the years. v Acknowledgements Throughout my dissertation work, I have had the honor and privilege of working with and learning from a lot of talented colleagues. First and foremost, I thank my adviser, Dave Wineland for his guidance. He has unparalleled knowledge and wisdom on everything related to trapped ions. Dave was quick to spot problems in my work and point me in the correct direction. In addition, Dave does an amazing job of managing one of the preeminent atomic physics groups in the world and I learned a lot as part of this efficient research machine. Didi Leibfried also deserves recognition for his assistance. He was always there to answer those silly graduate-student questions and has an uncanny knack for pinpointing what is going wrong in the experiment and fixing it. Without him, this dissertation would have dragged on many more years. I had the benefit of working closely with a number of excellent postdocs. Chirstian Ospelkaus has been a great friend and colleague. He made significant contributions to the experimental setup and was always willing to help with those details that were so important to the experiment's success. Aaron VanDevender was a big help, both with the lasers and in taking the actual data. And Jason Amini's provided invaluable assistance with the computer systems. I also appreciate Janus Wesenberg's involvement in the trap modeling and waveform software. I enjoyed working with Signe Seidelin while I fabricated the X-junction trap, and I am sure that she shares my contempt for electroplating. Joe Britton, my office mate for the majority of my graduate career, had a significant impact on my project. He was always willing to help work through a problem or explain whatever it was vi that I did not understand. Joe has also been great for bouncing ideas off of, no matter how crazy the ideas were. In addition to those singled out above, I had the opportunity to learn from a number of other great colleagues in the NIST Ion Storage Group: Jim Bergquist, Sarah Bickman, Mike Biercuk, John Bollinger, Ryan Bowler, Kenton Brown, Anders Brusch, John Chiaverini, James Chou, Yves Colombe, Bob Drullinger, Ryan Epstein, David Hanneke, Jonathan Home, David Hume, Wayne Itano, Brana Jelenkovic, John Jost, Manny Knill, Jeroen Koelemeij, Chris Langer, Robert Maiwald, Roee Ozeri, Rainer Reichle, Till Rosenband, Piet Schmidt, Nobu Shiga, Mike Thorpe, Hermann Uys, and Ulirch Warring. It has truly been great to work with such a wonderful group of people, who have made my graduate-school years enjoyable and successful. Finally, I would like to acknowledge my dissertation committee: Dave Wineland, Murray Holland (my titular adviser), Frank Barnes, Manny Knill, Scott Parker, and Jun Ye. vii Contents Chapter 1 Introduction 1 2 9Be+ qubits - atom/laser interactions 12 2.1 9Be+ atomic structure . 12 2.2 State detection . 15 2.3 State initialization . 17 2.4 Doppler cooling . 20 2.5 Single-ion coherent operations . 23 2.5.1 Single-photon operations . 23 2.5.2 Two-photon stimulated-Raman operations . 29 2.5.3 Ground state cooling . 33 2.5.4 Shelving . 34 2.6 Two-ion coherent operations . 35 2.7 24Mg+ ions . 37 3 Linear rf Paul ion traps 38 3.1 Static confining potential . 39 3.2 rf Paul trap . 41 3.2.1 Adiabatic approximation . 43 3.2.2 Linear Paul trap . 45 viii 3.3 Mathieu equation . 48 3.4 Inhomogenous Mathieu equation . 54 3.4.1 Mathieu equation with monopole and dipole terms . 55 3.4.2 Mathieu equation with nonlinear terms . 58 3.5 Real trap geometries . 63 3.5.1 Common linear trap architectures . 66 3.5.2 Comparison of multipole terms in different trap geometries . 71 4 Junction architectures and the X-junction array 77 4.1 Junction architectures . 78 4.1.1 Two-layer junctions . 78 4.1.2 Three-layer junctions . 82 4.2 X-junction array . 84 4.2.1 X-junction . 87 4.2.2 Experiment zone . 87 4.2.3 Load zone . 89 5 Ion transport 92 5.1 Transport introduction . 93 5.2 Modeling of the trap potentials . 96 5.3 Solving for a single harmonic potential . 98 5.3.1 Singular value decomposition and the pseudoinverse . 99 5.3.2 Constraints . 107 5.3.3 Issues . 113 5.4 Solving for transport waveforms . 117 5.4.1 Problems encountered solving for waveforms . 120 5.4.2 Linear least-squares solution method . 122 5.5 Other types of waveforms . 125 ix 5.6 Waveforms for the X-junction array . 126 6 Apparatus 136 6.1 Ion trap . 136 6.1.1 Trap fabrication . 138 6.1.2 Assembly . 143 6.2 Trapping potential . 146 6.2.1 rf potential . 146 6.2.2 Control potentials . 150 6.3 Vacuum system . 156 6.4 Loading . 160 6.5 Lasers and optics . 163 6.5.1 Doppler lasers . 165 6.5.2 Raman lasers . 168 6.5.3 Light detection . 171 6.6 Controlling the experiment . 172 7 Demonstration of transport through the X-junction 173 7.1 Experiment procedure . 173 7.2 Transport success rates . 175 7.3 Motional excitation rates . 178 7.4 Qubit coherence . 180 7.5 Transporting 24Mg+ ....................................181 7.6 Mechanism that excite the secular motion . 183 7.6.1 rf-noise heating mechanism . 183 7.6.2 DAC update noise . 186 7.6.3 Other mechanisms . 188 7.7 Possible improvements . 190 x 8 Conclusion 191 Bibliography 193 Appendix A Glossary of acronyms and symbols 203 B Methods for modeling trap potentials 206 C Possible linear constraints 212 D Maximum waveform step size 217 E Measuring motional excitation 223 xi Figures Figure 1.1 Ions in a linear array. 6 1.2 Possible multiplexed ion-trap array. 8 2.1 Energy level diagram for 9Be+............................... 14 2.2 Detection Histograms. 16 2.3 Energy level diagram with resonant lasers. 19 2.4 Rabi flopping of a single ion between the bright and dark qubit states. 27 2.5 Qubit energy level diagram that includes the Raman beam energies. 30 2.6 Raman beam geometry. 31 3.1 Diagram of a linear rf Paul trap. 42 3.2 Solutions to the Mathieu equation. 51 3.3 Two-wafer trap. ..