ABSTRACT Title of dissertation: COHERENCE IN DC SQUID PHASE QUBITS Hanhee Paik, Doctor of Philosophy, 2007 Dissertation directed by: Professors Chris Lobb and Fred Wellstood Department of Physics I report measurements of energy relaxation and quantum coherence times in an aluminum dc SQUID phase qubit and a niobium dc SQUID phase qubit at 80 mK. In a dc SQUID phase qubit, the energy levels of one Josephson junction are used as qubit states and the rest of the SQUID forms an inductive network to isolate the qubit junction. Noise current from the SQUID?s current bias leads is flltered by the network, with the amount of flltering depending on the ratio of the loop inductance to the Josephson inductance of the isolation junction. The isolation junction inductance can be tuned by adjusting the current, and this allows the isolation to be varied in situ. I quantify the isolation by the isolation factor rI which is the ratio of the current noise power in the qubit junction to the total noise current power on its bias leads. I measured the energy relaxation time T1, the spectroscopic coherence time T ?2 and the decay time constant T 0 of Rabi oscillations in the Al dc SQUID phase qubit AL1 and the Nb dc SQUID phase qubit NBG, which had a gradiometer loop. In particular, I investigated the dependence of T1 on the isolation rI . T1 from the relaxation measurements did not reveal any dependance on the isolation factor rI . For comparison, I found T1 by fltting to the thermally induced background escape rate and found that it depended on rI . However, further investigation suggests that this apparent dependence may be due to a small-noise induced population in j2i so I cannot draw any flrm conclusion. I also measured the spectroscopic coherence time T ?2 , Rabi oscillations and the decay constant T 0 at signiflcantly difierent isolation factors. Again, I did not observe any dependence of T ?2 and T 0 on rI , suggesting that the main decoherence source in the qubit AL1 was not the noise from the bias current. Similar results were found previously in our group?s Nb devices. I compared T1, T ?2 and T 0 for the qubit AL1 with those for NBG and a niobium dc SQUID phase qubit NB1 and found signiflcant difierences in T ?2 and T 0 among the devices but similar T1 values. If ux noise was dominant, NBG which has a gradiometer loop would have the longest Rabi decay time T 0. However, T 0 for NBG was similar to NB1, a Nb dc SQUID phase qubit without a gradiometer. I found that T 0 = 28 ns for AL1, the Al dc SQUID phase qubit, and this was more than twice as long as in NBG (T 0 ? 15 ns) or NB1 (T 0 ? 15 ns). This suggests that materials played an important role in determining the coherence times of the difierent devices. Finally, I discuss the possibility of using a Cooper pair box to produce variable coupling between phase qubits. I calculated the efiective capacitance of a Cooper pair box as a function of gate voltage. I also calculated the energy levels of a Josephson phase qubit coupled to a Cooper pair box and showed that the energy levels of the phase qubit can be tuned with the coupled Cooper pair box. COHERENCE IN dc SQUID PHASE QUBITS by Hanhee Paik Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulflllment of the requirements for the degree of Doctor of Philosophy 2007 Advisory Committee: Professor Christopher J. Lobb, Co-Chair/Advisor Professor Frederick C. Wellstood, Co-Chair/Advisor Professor Alex J. Dragt Professor Michael S. Fuhrer Professor John Melngailis c Copyright by Hanhee Paik 2007 Acknowledgements I always think graduate students are like rice crops (it can be wheats depending on where you are from). Growing them requires lots of efiort, care and time, especially if you want them to be good ones when you harvest, i.e. when they graduate with PhD degrees. Not to mention that the crops have to endure tough times, overcome unexpected obstacles and work hard to be full-grown crops. Whether I was a good rice crop, I don?t know. One thing I can say for sure is I received incredible amount of supports and helps from many people while I was in the graduate school. I certainly know that the space here in \Acknowledgements" is not enough to show what they did for me during all those years and how much I appreciate them. First I thank my dissertation committee members, Prof. Chris Lobb, Prof. Fred Wellstood, Prof. Alex Dragt, Prof. Michael Fuhrer and Prof. John Melngailis for giving me good advice on my thesis and their good suggestions. I owe great debts to my advisors, Professors Chris Lobb, Fred Wellstood and Bob Anderson. I just cannot imagine myself being in the United States, studying physics without them. They are my academic parents who taught me everything I need to know including how to be a good physicist. Also they showed me how to be a good advisor through themselves as the example. They became the reason that I want to be a good physicist and the motivation that kept me going during the troubled time in the graduate school. I flnd sometimes a little bit of Chrisness or Fredness or Dr. Andersonness inside myself and I feel proud of that. Chris taught me how important it is in physics to think deeply and ask ques- tions. Chris is like a physicist that I always imagine to be when I grow up. When I was about to give up after two failed projects, he didn?t give up on me, believed in me and encouraged me to try again. While I was struggling in communication (probably still I am), he was very patient and always willing to listen to me on any subject as a physicist and mentor. I will never be able to say thanks enough to him and I wish I will be able to become one his students that he is proud of. ii Dr. Anderson always gave me warm advice, was very supportive and most of all, he inspired me in every way through his restless will to learn and try new things as a physicist. He showed me joy of learning, and I was able to discover the fun part of quantum computation through him. I admire his young heart and mind. And I hope I will become like him in the future. Fred is my academic \mom". I won?t exist as a physicist without him. I look up to his incredible strength and momentum as well as his intuition and studiousness. He taught me the joy of studying physics. He always encouraged me and listened to me about anything. He helped me to discover myself that I enjoy research in physics and motivated me to pursue physics as a career. I thank Dr. Alex Dragt for giving me good insight in physics in terms of theory. Also I will never forget his warm encouragement during the group meetings for many years and thesis defense. I thank Dr. Michael Fuhrer for introducing me new physics of carbon, giving me good discussions, encouragement and advice. It would have been impossible for me to flnish Ph.D if Dr. Sudeep K. Dutta wasn?t there. He taught me many things such as cryogenic skills, created MATLAB codes which are essential part in analysis in this thesis, and most of all, taught me how to do experiment right. If I didn?t meet him and didn?t have a chance to work together with him, I would have become a bad student (seriously!). I thank Sudeep also for LaTex thesis templates and sharing his bibliography. Having Sudeep in the subbasement was one of the best parts being in the subbasement for me. Dr. Rupert Lewis is like my academic uncle. I owe him for so many things that to describe those, I have to write a thick book about all those years of my life in the subbasement lab; not only he was always encouraging me and being supportive, but also he gave me priceless advice and discussions. He helped me out all those trouble I?ve had during the experiment and the analysis, and still he will be the flrst one that I would ask if I get stuck in research in the future. He is a good advisor and mentor. I thank Rupert for good discussions and for his teaching in physics, experiment and how to be a good physicist. I thank Dr. Roberto Ramos who taught me the beauty of a dilution fridge; he taught me how to deal with the dilution refrigerator. He taught me how to set up the experiment from the scratch that I did in this thesis. He put lots of work on iii the dilution fridge wiring which I also learned from him, setting up the electronics and measurement. He is also naturally good teacher and good friend of mine who gave me lots of advice in physics as well as in life. I thank Tauno Palomaki, a.k.a. \Face of Angel". He is a working- hard graduate student who I learned a lot about how to be a good student. I enjoyed discussions about decoherence with him which inspired me in my T1 analysis in this thesis. I also thank Ben Coooper, a.k.a. \Cooop" or \Beeeeeen" for sharing his deep thoughts on our research with me and good discussions, and sometimes, theoretical supports. I thank Tony Przybysz, a.k.a. \Yo! Toni!" for good discussions and I enjoyed very much working on the small fridge project together. I thank Hyeokshin Kwon, a.k.a. \Hyeokster" for good discussions and I liked exchanging ideas on physics with him. We have incredible members of experimentalists in the subbasement but I shouldn?t forget that we also have incredible theorists working with those experi- mentalists. I thank Dr. Fred Strauch for his theoretical support, calculations on the coupled Cooper pair box, and helping me out for many other physics questions that I?ve asked whenever I got stuck. I thank Kaushik for his calculation on a dc SQUID phase qubit and many good discussions. I also thank Dr. Phil Johnson for his theoretical support and his good notes on Caldeira-Leggett model. All the glory we have in the subbasement started from our starting members, Huizhing Xu, Andrew Berkley, and Mark Gubrud. The legendary Dr. Huizhong Xu; I thank him, who basically set up the subbasement, for his noise theory and his contributions on my experiment setup. He was a good physicist from whom I learned how to be a good physicist. I thank Dr. Andrew Berkley for helping me setting up the small fridge. I also thank Mark Gubrud for the electronics and fridge setup. I thank our \SQUID neighbors", Gus Vlahacos and Dr. John Matthews for good discussions and conversation. I thank John also for the discussions on how to calculate a dc SQUID current- ux map and I-V curves. I thank Ben Palmer for useful comments and discussions at the group meetings. I used to work on a SET microscope project and I thank Dave Tobias and Matt Kenyon for teaching me the basics of device fabrication and good discussions on the iv SET project. Especially I thank David for many things in my life in graduate school; Dave is one of my best friends in the US, and while talking to him, my English skill improved 300 % (this is true!!). I thank Su-Young Lee, who was like my sister, for helping me out to settle down when I came to the US and many good advice on physics and life. I thank Dr. Zhengli Li and Dr. Hua Xu for good discussions. I enjoyed their company as a fellow graduate student in GLG (great Lobb group). I thank Dr. Paola Barbara and Yanfei Yang at Georgetown University for good discussions. Paola was my role model as a women physicist and Yanfei was one of those few fellow women physicists that I really enjoyed talking to about physics as well as our life. The Center for Superconductivity Research is where I spent seven years of the best part of my life. I thank Dr. Richard Greene for many things, especially the superconductivity seminar. I got a job because I gave a good talk at the conference and this is all because I was able to practice a lot for many years through the superconductivity seminar. Dr. Greene - actually he said I could call him Rick, which I really wanted to try, but I was too shy to do so. However, I?m going to try here now - or Rick was such a cheerful person that it was fun to talk to him. I thank people in Greene group; Dr. Pengcheng Li and Dr. Weiqiang Yu for their help on liquid helium and good discussions. I thank Dr. Steve Anlage, Dr. Dragos Mircea, Dr. Sameer Hemmady, Dr. Mike Ricci and Nate Orlofi for good discussions on microwave physics and their company. I thank Dr. Ichiro Takeuchi and his group members for useful discussions. I?m looking forward to playing the famous annual baseball game with Takeuchi group at the center picnic this year. The Center for Superconductivity Research won?t be able to run at all if we didn?t have Belta Pollard, Cleopatra White, Grace Sewlall and Brian Barnaby, and I thank them for everything. I thank Doug Bensen and Brian Straughn for their incredible help on my experiments. All of them are like my family and I enjoyed very much their company. I have to thank the members of Michael?s Nanotechnology lab, Dr. Michael Fuhrer, Dr. Anthony Ayari, Dr. Todd Brintlinger, Dr. Tobias Du?rkop, Enrique Cobas, Dr. Yung-Fu Chen and others for their help and discussions on nanotube transistors. I learned a lot from them. They are also good friends of mine who v enriched my life in graduate school. In the Physics department, I thank Dr. Nicholas Chant who was a great person and the graduate chair that I owe him for many things in my life at school. I won?t be able to go through the graduate school if I didn?t get help from Jane Hessing. I thank Jane for so much help and encouragement that I received during the graduate school years. I thank Linda O?Hara for her incredible help when I started the graduate school and for her encouragement. I thank Bernie Kozlowski for many things including physics tea. I thank Jesse Anderson, Al Godinez, George Butler and Bob Dahms for, flrst of all, being such a caring person, all those warmth they gave me, as well as their professional help on liquid helium and equipments which are essential part in my research. I thank Dr. Kim who is one of my best friends in physics for nice stories on Korean history and for his advice. I thank Dr. Ho Jung Paik (big Dr. Paik) for his good advice and conversation. I thank Dr. Michael Dreyer for useful discussions and his friendship. I should thank Dr. Kevin D. Osborn for good discussions on fabrications and his patience to wait for me to flnish as well. I have buddies that I went through the graduate school together with; Dr. Yung-Fu Chen, Dr. Micwah Ng and Dr. Narupon \Tor" Chattrapiban. I appreciate their friendship which brought joy in my life in the graduate school and even though we are apart in Illinois, Japan and Thailand and Maryland, I hope our friendship will last forever. I thank my Korean fellow physicists; Dr. Junghwan Kim, Dr. Seok-Hwan Chung, Young Soo Youn, Bora Sul, Young-Noh Youn, Zaeil Kim, Hyeokshin Kwon, Chaun Jang and others for their help and friendship. I especially thank Jonghee Lee for his help and friendship. He saved me from many troubles so many times. I have to thank Dr. Jae Hoon Kim who flrst introduced the world of super- conductivity to me for his support for studying abroad. I thank Dr. Kibum Kim for his help and giving me insight to look into physics problems when I was studying at Yonsei University, in Korea. I thank Dr. Mi-Ock Mun, Young Sup Roh and other members of Yonsei Spectroscopy Lab for their help. I thank the professors at the physics department at Yonsei University for their support and teaching that eventually led me to pursue further steps in studying physics. vi I thank Dr. Seunghee Son and Sejin Han who are my college classmates from Korea for their friendship and support. I also thank my year-94 Yonsei Physics classmates for their friendship and support. I should thank Dr. Kiwoong Kim for useful discussions on NMR. Finally I deeply thank my parents-in-law for their incredible support and love, and my brother-in-law Sangbong Jeong and my brother Tae-jong Paik. My parents, Dr. Sunbok Paik and Jeongja Kim who convinced me to study physics instead of electrical engineering 13 years ago when I went to college, I thank them for their good insight on me as well as, not to mention, their endless support and love. I also thank my husband Jae Kwang Jeong who took care of the household jobs for me, cooking, cleaning, etc during my time writing my thesis, for his support and love. I was able to get my PhD because I had his support and I am honestly looking forward to his time writing thesis (I?m not being sadist!) so that I could be supportive for him like he did for me. vii Table of Contents List of Tables xi List of Figures xiii 1 Introduction 1 1.1 Quantum computers and qubits . . . . . . . . . . . . . . . . . . . . . 1 1.2 What is this thesis about . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Josephson junctions, SQUIDs and superconducting qubits 6 2.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Superconducting wave function and ux quantization . . . . . . . . . 7 2.3 Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 dc and ac Josephson efiects . . . . . . . . . . . . . . . . . . . 10 2.3.2 Properties of Josephson tunnel junctions . . . . . . . . . . . . 12 2.3.3 Equation of motion and Lagrangian . . . . . . . . . . . . . . . 13 2.3.4 Hamiltonian of a Josephson junction . . . . . . . . . . . . . . 16 2.3.5 Solution of the Josephson junction Hamiltonian . . . . . . . . 17 2.3.6 ^ and n^ uncertainty relation . . . . . . . . . . . . . . . . . . . 20 2.4 Classical properties of SQUIDs . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 What is a SQUID? . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2 Flux-phase relation: uxoid quantization rule revisited . . . . 24 2.4.3 SQUID potential energy function . . . . . . . . . . . . . . . . 28 2.4.4 Current- ux map . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5 Application of Josephson junctions to quantum computation . . . . . 36 2.5.1 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . 36 2.5.2 dc SQUID phase qubit: design and basic idea, inductive isolation 38 3 Dynamics of a two-level quantum system 42 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Density matrix formalism for a two-level system . . . . . . . . . . . . 43 3.3 Optical Bloch equations: two-level systems and magnetic spin . . . . 45 3.3.1 Representing the Hamiltonian of a two-level system with Pauli matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.2 Equation of motion for a two-level system . . . . . . . . . . . 46 3.4 Including decoherence and dissipation . . . . . . . . . . . . . . . . . . 52 3.5 Solutions of the density matrix equation with T1 and T2 . . . . . . . 54 3.6 From the density matrix to the Bloch vector . . . . . . . . . . . . . . 57 3.7 Bloch vector of the Josephson junction phase qubit . . . . . . . . . . 58 4 Qubit fabrication, experimental techniques and analysis 63 4.1 Fabrication recipe for aluminum dc SQUID phase qubits . . . . . . . 63 4.1.1 Photolithography: Introduction . . . . . . . . . . . . . . . . . 67 4.1.2 Preparation for photolithography . . . . . . . . . . . . . . . . 67 viii 4.1.3 Spinning and baking photoresist . . . . . . . . . . . . . . . . . 71 4.1.4 Expose and develop photoresist . . . . . . . . . . . . . . . . . 73 4.1.5 Deposition of aluminum . . . . . . . . . . . . . . . . . . . . . 73 4.1.6 Lift-ofi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Fabrication recipe for a Cooper pair box: E-beam lithography . . . . 75 4.2.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.2 Spinning resist . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.3 E-beam writing . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.4 Develop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.5 Ion milling and Al deposition . . . . . . . . . . . . . . . . . . 79 4.2.6 Lift-ofi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Table of dc SQUID phase qubits measured in UMD SQC group . . . 82 4.4 Dilution refrigerator setup . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 Measurements and analysis . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5.1 Initialization of the ux state of the dc SQUID phase qubit . . 92 4.5.2 Biasing the qubit junction . . . . . . . . . . . . . . . . . . . . 94 4.5.3 Measurement of the qubit state via the escape rate . . . . . . 95 4.5.4 Spectroscopy and T?2 . . . . . . . . . . . . . . . . . . . . . . . 97 4.5.5 Measurement of relaxation . . . . . . . . . . . . . . . . . . . . 99 4.5.6 Measurement of Rabi oscillations . . . . . . . . . . . . . . . . 100 5 Efiects of variable isolation on high frequency noise and T1 in the dc SQUID phase qubit 108 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2 Variable current isolation and isolation factor . . . . . . . . . . . . . 108 5.3 Arbitrary dissipation model for the dc SQUID phase qubit . . . . . . 113 5.3.1 Calculation of efiective admittance and T1 . . . . . . . . . . . 114 5.3.2 Current noise power spectrum SI1(f) and noise induced tran- sitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3.3 Determination of T1 using thermal escape rate . . . . . . . . . 123 5.4 Measuring the efiect of isolation on an Al dc SQUID phase qubit . . . 128 5.4.1 dc SQUID phase qubit parameters; flt to spectroscopy and I - ' curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.4.2 Observing noise induced transitions. . . . . . . . . . . . . . . 134 5.4.3 T1 measurements using relaxation . . . . . . . . . . . . . . . . 141 5.4.4 T1 measurements from the thermally induced escape rate . . . 144 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6 Measurements of coherence times in dc SQUID phase qubits 157 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 Current noise, isolation and coherence times in the dc SQUID phase qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3 Efiect of current noise on Rabi oscillations . . . . . . . . . . . . . . . 161 6.4 T0 - comparison with a Nb device . . . . . . . . . . . . . . . . . . . . 176 6.5 Spectroscopic coherence time T?2 : comparison with a Nb device . . . 179 ix 6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7 Comparison of coherence times in dc SQUID phase qubits 185 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.2 dc SQUID phase qubits without and with gradiometer loops . . . . . 186 7.3 Measurement of energy levels of NBG . . . . . . . . . . . . . . . . . . 189 7.4 Measurement of T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.5 Measurement of T?2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.6 Rabi oscillations in gradiometer NBG and comparison with magne- tometers AL1 and NB1 . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8 The Cooper pair box as a coupling component in a quantum computer 206 8.1 Introduction: The Cooper pair box . . . . . . . . . . . . . . . . . . . 206 8.2 Charging energy of a Cooper pair box with two voltage bias sources . 209 8.3 Hamiltonian and energy bands of the Cooper pair box . . . . . . . . . 214 8.4 Calculation of efiective capacitance . . . . . . . . . . . . . . . . . . . 219 8.4.1 Efiective capacitance: deflnition . . . . . . . . . . . . . . . . . 222 8.4.2 Efiective capacitance: simulation . . . . . . . . . . . . . . . . 223 8.5 Cooper pair box coupled to a phase qubit . . . . . . . . . . . . . . . 225 8.5.1 Hamiltonian of the coupled box and junction . . . . . . . . . . 227 8.5.2 Solving the coupled Hamiltonian using the Jaynes-Cummings model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.5.3 Calculating the energy levels of the coupled Hamiltonian using perturbation theory. . . . . . . . . . . . . . . . . . . . . . . . 232 8.5.4 Energy level spacings . . . . . . . . . . . . . . . . . . . . . . . 233 8.5.5 Energy level spacings: degenerate case. . . . . . . . . . . . . . 234 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 9 Conclusions 240 9.1 Current status of superconducting quantum computing and future plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 A MATLAB Code 243 A.1 Solution of the Junction Hamiltonian . . . . . . . . . . . . . . . . . . 243 A.2 Non-stationary Master equation solution . . . . . . . . . . . . . . . . 252 A.3 Stationary Master equation solution . . . . . . . . . . . . . . . . . . . 257 Bibliography 260 x List of Tables 1.1 Types of qubits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1 Comparison of the notations on time constants . . . . . . . . . . . . . 54 4.1 Ar Ion beam etching rate at normal incidence for a beam current density of 1.0 mA/cm2 and 500 V acceleration voltage as given by refs. [72 - 74]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Parameters of devices (dc SQUID phase qubit) . . . . . . . . . . . . . 82 4.3 Commercial electronics used in the experiment. . . . . . . . . . . . . 92 4.4 List of the homemade electronics used in the experiment. . . . . . . . 94 5.1 Parameters of dc SQUID phase qubit AL1 obtained from the current- ux map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 Parameters of dc SQUID phase qubit AL1 . . . . . . . . . . . . . . . 132 5.3 Relaxation fltting parameters. . . . . . . . . . . . . . . . . . . . . . . 146 5.4 T1 estimates for qubit AL1 . . . . . . . . . . . . . . . . . . . . . . . . 156 6.1 Summary of fltting parameters for Rabi oscillations in device AL1 at 80 mK with rI = 1000. The microwave was at 7.0 GHz. . . . . . . . . 167 6.2 Summary of fltting parameters for Rabi oscillations in device AL1 at 80 mK with rI = 200. The microwave power varies from 6 dBm to 11 dBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.3 Summary of fltting parameters for Rabi oscillations in device AL1 at 80 mK with rI = 200. The microwave power varies from 12 dBm to 17 dBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.1 Parameters for SQUIDs NBG, NB1 and AL1. . . . . . . . . . . . . . 189 7.2 Relaxation fltting parameters for NBG. . . . . . . . . . . . . . . . . . 194 7.3 Relaxation fltting parameters for AL1. . . . . . . . . . . . . . . . . . 196 xi 7.4 The decay time constant T 0 of Rabi oscillations in NBG measured at 80 mK for difierent microwave frequencies. . . . . . . . . . . . . . . . 201 7.5 Summary of spectroscopic coherence time T?2, time constant T0 for decay of Rabi oscillation, relaxation time T1 and estimated T1 = 3T 0=4 that would occur if all decoherence was due to dissipation. . . . 205 8.1 Parameters of Cooper pair box for the simulation shown in Fig. 8.3. . 218 8.2 Parameters of the Cooper pair box for simulation shown in Fig. 8.4 . 219 8.3 Parameters for energy level simulation of the Cooper pair box coupled to the phase qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 xii List of Figures 2.1 Schematic diagram of the potential energy of the Josephson junction. 11 2.2 Graph of tilted washboard potential. . . . . . . . . . . . . . . . . . . 15 2.3 Metastable states in a well of the tilted washboard potential. . . . . . 18 2.4 Types of SQUIDs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Schematic of a dc SQUID. . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6 Schematic of a dc SQUID. . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Current- ux curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.8 Cooper-pair box with a single ultra small junction. . . . . . . . . . . 37 2.9 Schematic of a dc SQUID phase qubit. . . . . . . . . . . . . . . . . . 39 3.1 Two-level system represented as a vector on the Bloch sphere. . . . . 48 4.1 Photograph of Al/AlOx/Al dc SQUID phase qubit. . . . . . . . . . . 64 4.2 Schematic of double angle deposition after photolithography. . . . . . 65 4.3 Photograph of qubit junction (left) with a coupled Cooper pair box (right) in device AL1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 A zoomed-in photograph of qubit junction in device AL1 from Fig. 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.5 Photograph of isolation junction in device AL1. . . . . . . . . . . . . 69 4.6 SEM picture of a coupled Cooper pair box and a Josephson phase qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7 SEM picture of a ultra-small junction . . . . . . . . . . . . . . . . . 78 4.8 Oxford Instruments Kelvinox 25 dilution refrigerator and sample mount. 83 4.9 Oxford Instruments Kelvinox 25 dilution refrigerator unit. . . . . . . 84 4.10 300 K ange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.11 Wiring on the Oxford Instruments Kelvinox 25 dilution refrigerator. . 86 xiii 4.12 Photograph of the heat exchanger and still showing where the coaxial lines are thermally anchored. . . . . . . . . . . . . . . . . . . . . . . 87 4.13 Wiring schematic for Oxford Instruments Kelvinox 25 dilution refrig- erator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.14 Copper powder fllters and inside of qubit sample holder box . . . . . 89 4.15 Aluminum sample holder shown without the top. . . . . . . . . . . . 90 4.16 Schematic of the measurement setup. . . . . . . . . . . . . . . . . . . 93 4.17 Metastable states in a well of the tilted washboard potential. . . . . . 101 4.18 Biasing scheme for the dc SQUID phase qubit and microwave se- quence. The time interval between tI that the ramping starts and tF when the qubit junction switching voltage V appears is recorded by the frequency counter SR620. . . . . . . . . . . . . . . . . . . . . . . 102 4.19 Total escape rate vs. current for qubit AL1 at 80 mK with 6.9 GHz microwaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.20 Microwave enhancement of the escape rate for AL1 at 80 mK with 6.9 GHz microwaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.21 Lorentzian flt to the microwave enhancement of the escape rate for AL1 at 80 mK with 6.9 GHz microwaves. . . . . . . . . . . . . . . . . 105 4.22 Observed relaxation in the escape rate. . . . . . . . . . . . . . . . . . 106 4.23 Rabi oscillations in the escape rate ? in device NB1 at 25 mK for rI = 1300 and for rI = 450. . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.1 dc SQUID phase qubit circuit diagram and its efiective circuit. . . . . 109 5.2 Isolation factors vs I2/I02. . . . . . . . . . . . . . . . . . . . . . . . . 112 5.3 Schematic of dc SQUID phase qubit and equivalent circuit for the isolation network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.4 Simulation of Riso, Reff and T1. . . . . . . . . . . . . . . . . . . . . . 118 5.5 Plot of simulated thermal current noise power spectral density SI1(f) at 100 mK for Z0 = 50 ? and R1 = 6 k?. . . . . . . . . . . . . . . . 121 5.6 Total escape rate ? versus current I1 for qubit AL1. . . . . . . . . . . 125 xiv 5.7 Plot of (?ME ? ?SME)/?SME versus current. . . . . . . . . . . . . . . 127 5.8 Escape rate of dc SQUID phase qubit AL1 at 80 mK. . . . . . . . . . 129 5.9 The energy spectrum of dc SQUID phase qubit AL1 . . . . . . . . . . 131 5.10 Current- ux characteristic curve of dc SQUID phase qubit AL1 mea- sured at 80 mK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.11 Escape rate versus current when the qubit is most isolated (rI = 1000).135 5.12 Escape rate versus current when the qubit is poorly isolated (rI = 270).136 5.13 Escape rate versus current when the qubit is more poorly isolated (rI = 220). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.14 3-D false color plot of the noise-induced transition peaks in the back- ground escape rate enhancement G0 in device AL1. . . . . . . . . . . 140 5.15 Energy levels of Josephson junction phase qubit using parameters for AL1 in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.16 Observed relaxation in the escape rate in device AL1 for r = 1000. . . 145 5.17 Observed relaxation in the escape rate for rI = 220. . . . . . . . . . . 146 5.18 Total escape rate ? versus current I1 for qubit AL1. . . . . . . . . . . 147 5.19 ?2 map of the escape rate for parameters T1 and T. . . . . . . . . . . 149 5.20 Experimental ?SME and calculated ? versus current I1 for qubit AL1. 151 5.21 Calculated T1 versus current I1 for qubit AL1. . . . . . . . . . . . . . 152 5.22 Comparison of results for T1 vs I1=I01 with rI = 1000 and rI = 400. . 154 6.1 Schematic of dc SQUID phase qubit. . . . . . . . . . . . . . . . . . . 158 6.2 Escape rates of device AL1 at rI = 1000 and rI = 200. . . . . . . . . 163 6.3 Square of the Rabi frequency vs. microwave power for r = 1000 and for r = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.4 Rabi oscillations measured for r = 1000 with powers of 6 dBm to 17 dBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.5 Rabi oscillations in device AL1 at 80 mK with rI = 1000 and fltting results for P = 6, 7 and 8 dBm. . . . . . . . . . . . . . . . . . . . . . 168 xv 6.6 Rabi oscillations for device AL1 at 80 mK with rI = 1000 and fltting results for P = 9, 10 and 11 dBm. . . . . . . . . . . . . . . . . . . . . 169 6.7 Rabi oscillations measured for the poorly isolated situation, with rI = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.8 Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 200 and fltting results for P = 6, 7 and 8 dBm. . . . . . . . . . . . 172 6.9 Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 200 and fltting results for P = 9, 10 and 11 dBm. . . . . . . . . . . 173 6.10 Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 200 and fltting results for P = 12, 13 and 14 dBm. . . . . . . . . . 174 6.11 Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 200 and fltting results for P = 15, 16 and 17 dBm. . . . . . . . . . 175 6.12 Rabi oscillations in AL1 at 80 mK for rI = 1000 and rI = 200. . . . . 177 6.13 Decay time constant T 0 of Rabi oscillations vs. Rabi frequency ?=2? for qubit AL1 at 80 mK. . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.14 Rabi oscillations in the escape rate ? in device NB1 at 25 mK for rI = 1300 and for rI = 450. . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.15 Spectroscopic coherence time T ?2 versus frequency for rI = 1000 in AL1 at 80 mK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.16 Spectroscopic coherence time T ?2 versus isolation factor rI for AL1 and NB1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.1 Schematic of dc SQUID phase qubit. . . . . . . . . . . . . . . . . . . 187 7.2 Schematic of dc SQUID phase qubit with a gradiometer loop. . . . . 188 7.3 Total escape rate vs. current for device NBG at 80 mK. . . . . . . . . 191 7.4 Resonance frequency vs. current for NBG at 80 mK. . . . . . . . . . 192 7.5 Observed relaxation in the escape rate for NBG at 80 mK. . . . . . . 194 7.6 Total escape rate ? versus current I1 for qubit NBG at 80 mK. . . . . 195 7.7 Spectroscopic coherence time T?2 for NBG at 80 mK. . . . . . . . . . 197 xvi 7.8 Measurements of Rabi oscillations in the escape rates in gradiometer NBG at 80 mK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.9 Square of the Rabi frequency in NBG vs. microwave power. . . . . . 200 7.10 Measurements of Rabi oscillations in AL1 at 80 mK, NB1 at 25 mK and NBG at 80 mK. . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.11 The decay time constant T 0 of Rabi oscillations vs. Rabi frequency ?=2? of AL1 at 80 mK, NB1 at 25 mK, NB2 at 25 mK and NBG at 80 mK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.1 Cooper-pair box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.2 Cooper pair box with a voltage bias source Vb and a gate voltage source Vg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.3 Simulation of energy and average number for a Cooper pair box with two difierent EJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.4 Simulation of energy and average number for a Cooper pair box with two difierent Ec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.5 A Cooper-pair box and an equivalent efiective variable capacitor. . . 221 8.6 Simulation of < N > and the efiective capacitance. . . . . . . . . . . 224 8.7 Circuit schematic of a Cooper pair box coupled to a Josephson junction.226 8.8 Simulated energy level spacings for a Cooper pair box coupled to a Josephson phase qubit for Ib = 0.989 IcQ. . . . . . . . . . . . . . . . . 237 8.9 Simulated energy level spacings for a Cooper pair box coupled to a Josephson phase qubit for Ib = 0.991 IcQ. . . . . . . . . . . . . . . . . 238 xvii Chapter 1 Introduction 1.1 Quantum computers and qubits A quantum computer is a device that employs physical mechanisms described by quantum mechanics to perform computations [1]. The idea of quantum computa- tion was flrst proposed by Richard Feynman in 1982 [2]. He showed that only simple quantum systems could be e?ciently simulated on a classical computer, while one quantum system could, in principle, e?ciently simulate another. In 1985, David Deutsch published his description of a quantum Turing machine, showing how to use \quantum parallelism" [3]. The quantum-bit or qubit is the basic unit of a quantum computer and the term \qubit" was flrst used by Benjamin Schumacher, who developed a coding theorem for quantum information theory [4]. Superposition and entanglement are two key quantum properties that are re- quired for quantum computation. The main difierence between a bit and a qubit is that only the qubit is allowed to be in a superposition of j0i and j1i, enabling it to span a 2-dimensional Hilbert space. In addition, the qubit state can be entangled with other qubits. Entanglement can be used to store, exchange or read out infor- mation. In principle, because of superposition and entanglement, N qubits can be placed into of order 22N states while N classical bits only have available 2N distinct states. Moreover, operations can be done on all registers at the same time in a quantum computer. In this sense, an N-qubit quantum register is like a 2N-classical bit register. One of the main motivations to build a quantum computer is that a quantum computer would be able to break RSA encryption [5]. RSA encryption uses mul- tiplication of two large prime numbers to produce a public key. If the key could 1 be factored, then messages that were encrypted could be deciphered. Enormous in- terest in quantum computation developed after Peter Shor developed an algorithm (Shor?s algorithm) for factoring numbers very e?ciently on a quantum computer [6]. With Shor?s algorithm, the difierence in the speed of computation for a clas- sical computer and a quantum computer gets bigger as the size of the number to be factored grows. For example, to factor an integer with k digits, the classical computer would take on the order of 2k1=3 operations while a quantum computer would take on the order of k3 operations [7]. If both computers could factor a 130- digit number in one month, then the classical computer would require 1010 years to factor a 400-digit number, while the quantum computer would take only three years [8]. At present, however, no quantum computer exists that can factor such large numbers. In 2001, Vandersypen et al. used Shor?s algorithm to factor 15 using a solution of molecules that each had flve 19F and two 13C spin-1/2 nuclei qubits in liquid state NMR at room temperature [7], and even this result has been questioned as to whether it was true quantum computation. To be able to perform useful computation, the qubits and their interconnec- tions in a quantum computer must satisfy the DiVincenzo criteria [9]: 1. Be a scalable physical system with well-deflned qubits 2. Be initializable to a simple flducial state such as j000:::i 3. Have long coherence times 4. Have a universal set of quantum gates 5. Permit high quantum e?ciency, qubit-speciflc measurements The coherence time is the time scale that characterizes how long the qubit can remain in a well-deflned superposition of states. Since the qubit needs to occupy 2 superposition states to perform quantum operations, the coherence time is a measure of the time available for a computation, and long coherence time is important. To obtain a long coherence time, the qubit must be isolated from dissipation and all external disturbance [10, 11]. There are many types of qubits that have been proposed and may one day satisfy the DiVincenzo criteria. Table 1.1 summarizes some of the main types of qubits being built and studied currently. I note that the same physical system can provide various types of qubits. For example, photons can use polarization, number or photon time-bin encoding to construct distinct types of qubits. Similarly, superconducting devices with Josephson junctions can be used to construct three main classes of qubits - charge, phase and ux qubits. Each qubit has advantages and disadvantages. Neutral atom qubits (two atomic states as a qubit) and hyperflne qubits (two hyperflne states of a trapped ion) tend to be well-isolated, and in consequence they can have long coherence times. However, they do not interact strongly, making it challenging to control and couple them together. Superconducting qubits are easily controlled and coupled, but they have been plagued by relatively short coherence times. For the superconducting quantum computing community, flnding and removing the causes of decoherence is a major challenge. Since superconducting qubits consist of large numbers of atoms and electrons, they can easily interact with many other unwanted quantum states. As a result, superconducting qubits require more elaborate isolation scheme than naturally well-isolated qubits such as atoms or photons. The basic approach to iso- lating superconducting qubits involves placing a large impedance between the qubit and the environment or noise, so that the qubit does not interact with external degrees of freedom. 3 Table 1.1: Types of qubits. Name of qubit qubit states j0i j1i polarization horizontal vertical photon number vacuum single photon time of arrival early late coherent light (wave) squeezed quadrature amplitude phase electron spin up down charge 0 e nuclear spin (NMR) spin up down neutral atom atomic spin up down trapped ion hyperflne states atomic energy ground state excited state Cooper pair box charge zero 2e one 2e three junction SQUID ux (current) clockwise counterclockwise phase qubit energy (phase) ground state 1st excited state Quantum dot charge e on left dot e on right dot 4 1.2 What is this thesis about In this thesis, I focus on isolation and coherence in large capacitance Josephson- junctions or phase qubits [12]. In particular, I built Al/AlOx/Al Josephson junction with on-chip inductive isolation networks which act as current noise isolation fllters. Because the inductive isolation network and the Josephson junction phase qubit form a dc SQUID, our group calls this type of qubit the \dc SQUID phase qubit". In Chapters 2 and 3, I review essential background material on superconductivity, superconducting qubits and qubit dynamics. In Chapter 4, I describe the experi- mental setup and device fabrication techniques that I used. Chapters 5 and 6 show my experimental results for the relaxation time and coherence time, and I discuss possible causes of decoherence in my qubits. In Chapter 7, I compare the coherence time of an Al dc SQUID qubit to those of Nb dc SQUID qubits. I also discuss how the design and material choices could in uence the coherence times. Chapter 8 discusses the Cooper pair box and how it can be coupled to a dc SQUID phase qubit. Finally in Chapter 9, I provide a summary of the thesis. 5 Chapter 2 Josephson junctions, SQUIDs and superconducting qubits 2.1 Superconductivity Following his success in liquifying helium, superconductivity was flrst discov- ered in mercury by Kamerlingh Onnes in 1911 [13] (also see the Nobel Lecture by Kamerlingh Onnes [14]). Soon after, Meissner and his colleagues found that superconductors were perfect diamagnets (the Meissner efiect) [13]. However, the microscopic origin of superconductivity wasn?t revealed until the Bardeen-Cooper- Schriefier (BCS) theory was introduced in 1957 [15]. The conventional BCS super- conducting state is a thermodynamic phase in which the electrons in a conducting material form pairs and condense into a state with perfect diamagnetism and zero resistivity below a critical temperature Tc [13]. Before the BCS theory was developed, Ginzburg and Landau proposed a phe- nomenological theory that describes superconductivity as a phase transition from the normal to superconducting state [13]. The Ginzburg-Landau (GL) theory does surprisingly well at deflning and explaining the behavior of important parameters such as the coherence length, the penetration depth, and the condensation energy. Later, Gor?kov proved that the GL theory can be derived from the BCS formalism [13]. In this chapter, I use the GL approach to discuss a few important phenomena related to Josephson junctions [13, 16, 17]. 6 2.2 Superconducting wave function and ux quantization The superconducting state is thermodynamically more ordered than the nor- mal state. In particular, for temperature T less than the critical temparature Tc, the free energy is a minimum in the superconducting state. In the GL theory, the superconductor is described by a complex \order parameter" ? ?(r) = p ns(r) exp[i (r)] (2.1) where j?j2 = ns(r) is the density of the \superconducting electrons" that have condensed into the superconducting state and is a phase factor that depends on position r. The \superconducting electrons" were later revealed by the BCS theory to be pairs of electrons that were attracted to each other by the exchange of phonons; they are called \Cooper pairs" [15]. The order parameter ? satisfles the flrst Ginzburg-Landau equation (?i~r? qA)2 2m? ? + fi? + flj?j 2? = 0: (2.2) where q is the charge of the Cooper pair and m? is its mass. Ginzburg and Landau found this equation by minimizing the Gibb?s free energy with respect to ? [13]. Eq. 2.2 resembles the Schro?dinger equation of a particle with charge q and mass m? except for the nonlinear j?j2? term. Keeping the similarity in mind, we treat ?(r) as a wavefunction for the Cooper pairs. ? can be normalized to the total number N of Cooper pairs: Z ??(r)?(r)dV = N: (2.3) where the integral is taken over the volume V of superconductor. 7 The current density J due to the Cooper pairs is given by J = Re[qhv^i] (2.4) where q = ? 2e = ? 2 ? (1.6 ?10?19 C) is the charge of a Cooper pair. Here v^ = p? qA m? ? = 1m? (?i~r? qA) (2.5) is the velocity operator of a Cooper pair in the superconductor and A is the vector potential of any magnetic fleld that is present. Since r? = i?r (r) + exp[i (r)]r p ns(r); (2.6) the current density in Eq. 2.4 can be written as J = Re[qh?jp? qAm? j?i] (2.7) = qns(r)m? [~r (r)? qA]: (2.8) Some of the most interesting features of Cooper pairs comes from the phase factor (r) and its connection to current. For example, the phase (r) has to produce a single-valued wavefunction. If we consider a superconducting ring, this gives a quantization rule I r (r) ? dl = 2?n (2.9) where dl is a line element and the integration is taken along any path in the super- conductor. Thus in a closed-loop of superconductor, the super current must ow so as to satisfy Eq. 2.9. Taking a line integral over both sides of Eq. 2.8, one flnds: I J ? dl = qnsm? ? ~ I r (r) ? dl? q I A ? dl ? (2.10) 8 where I have assumed ns is constant so that it can be taken out of the integral. Using Eq. 2.9 and Stokes? theorem, Eq. 2.10 becomes I J ? dl = q 2ns m? ? 2?~q n? Z B ? da ? (2.11) = q 2ns m? ?nh q ? Z B ? da ? (2.12) where da is an inflnitesimal area element and B is the total magnetic fleld. The integral over B is done over the area enclosed by the contour for the line integral, and Z B ? da = ' (2.13) is the total magnetic ux in the superconducting ring. The flrst term in the brackets in Eq. 2.12 yields nh q ? nh 2e ? n'0 (2.14) where '0 = h=2e is the ux quantum. Using Eq. 2.13 and 2.14, Eq. 2.12 becomes me 2e2ns(r) I J ? dl+ ' = n'0: (2.15) Here I have taken m? = 2me and me is the mass of an electron. This equation describes uxoid quantization. Deep inside a superconductor, we expect J = 0. In this case, Eq. 2.15 implies that the total ux inside a closed path in a superconductor should be an integer multiple of a ux quantum. An important fact to keep in mind is that although the vector potential A is not unique, any physical quantities that involves A must be gauge invariant. For example, suppose we choose a vector potential that satisfles A0 = A+r? (2.16) 9 instead of A = 0. Since Eq. 2.8 must yield the same J for either choice of A or A0, it is necessary to deflne a new gauge invariant phase difierence 12 by [13] 12 = 1 ? 2 ? 2?'0 Z 2 1 A ? dl: (2.17) 2.3 Josephson junctions Josephson junctions are formed from superconductor-insulator-superconductor structures (SIS), superconductor-normal-superconductor junction (SNS) or even by creating a small constriction between two superconducting banks (weak links). In this section, I discuss the basic properties of SIS Josephson junctions and show how the Josephson junction Hamiltonian is found from the Josephson equations. 2.3.1 dc and ac Josephson efiects In 1962, Brian Josephson predicted that for two superconducting electrodes separated by a very thin insulator (see Fig. 2.1), current can ow via tunneling without any voltage drop [18]. He found that the tunneling current owing from superconductor 1 to superconductor 2 across the junction is given by I = I0 sin( 1 ? 2 ? 2?'0 (?1 ? ?2)) (2.18) = I0 sin (2.19) where is the gauge invariant phase difierence between superconductor 1 and su- perconductor 2 (deflned in Eq. 2.17) and I0 is the critical current. Equation 2.18 is called the \dc Josephson efiect". Josephson also found that if the phase changes with respect to time, then a voltage develops across the junction, given by: V = ~2e d dt : (2.20) 10 E0V(x) |?(x)| x x 1 2 1 2 Figure 2.1: Schematic diagram of the potential energy and the magnitude of the wave function for pairs in a Josephson junction with a thin insulating barrier. The thickness of the barrier is 2a. 11 Equation 2.20 is called the "ac Josephson efiect". The dc and ac Josephson efiects were experimentally conflrmed by Anderson and Rowell in 1963 [19]. For historical background to the discoveries of the Josephson junction, see \Foundations of Ap- plied Superconductivity" by Orlando and Delin [16] and the Nobel lectures by B. Josephson and I. Giaever [20]. The dc and ac Josephson efiects provide the basis for the international voltage standard. The essential physics of the technique is that if a constant voltage V is applied to the Josephson junction, the phase across the junction becomes = Z t 0 d dt dt = 2eV ~ t: (2.21) Substituting from Eq. 2.21 into Eq. 2.18, one flnds an oscillating current, I(t) = I0 sin 2eV ~ t ? (2.22) at frequency f = 2eV=h. In practice, a microwave current that is oscillating at an accurately known frequency f is applied and a voltage step is produced at V = hf/2e. 2.3.2 Properties of Josephson tunnel junctions Cooper pairs owing through a Josephson junction are naturally described with two variables: the superconducting phase and the number N of the Cooper pairs that have passed through the junction. They are connected classically by the ac Josephson equation V = '02? d dt = Q C = 2eN C (2.23) where C is the junction capacitance. In fact, we need both N and to obtain the Hamiltonian of the Cooper pairs involved in Josephson tunneling, but we can choose either one as the independent 12 coordinate (see sec. 2.3.6). If the tunneling is very small (small junction), the tun- neling process can be strongly afiected by the Coulomb energy associated with the junction capacitance. In this small capacitance limit, the tunneling is suppressed unless we apply enough energy for the Cooper pairs to overcome the Coulomb charg- ing energy Q2=2C = 2e2=C. This is called the Coulomb blockade efiect [21]. In this limit the pair number N is the natural choice for the independent variable in the Hamiltonian. However, if the junction has a large area, its capacitance is large and the Coulomb energy stored in the junction capacitor becomes less important. In this limit, the device behavior is dominated by the dc and ac Josephson efiects and is a more convenient choice to use in the Hamiltonian. In this thesis, I mainly focus on large-area Josephson junctions, where the Josephson efiect dominates and the dynamics of Cooper pairs is best described using the phase variable. The exception is in Chapter 9, where I discuss some aspects of the Cooper pair box. 2.3.3 Equation of motion and Lagrangian In a real Josephson junction, the junction electrodes form a capacitor and quasiparticles can tunnel as well as pairs. In addition, there can be other normal resistive shunts across the junction. Therefore, displacement current through the capacitor, quasiparticle current and current associated with any normal shunt will ow as well as the Josephson supercurrent. Taking those into account, the total current that ows through a junction can be written as I = I0 sin + CdVdt + V R: (2.24) Here I is the bias current, C is the junction capacitance and R is the efiective resistance due to any normal shunt and quasiparticle tunneling [13]. Substituting 13 Eq. 2.20 into Eq. 2.24, we obtain the equation of motion for the phase difierence I = I0 sin + '0C2? d2 dt2 + '0 2?R d dt : (2.25) This equation of motion is identical to that of a damped driven pendulum with the angular displacement . In this pendulum analog, the current I becomes a torque, the capacitance term '0C=2? is the moment of inertia of the pendulum, and the term '0=2?R is a damping term [17, 22]. Note in particular that the shunting conductance 1/R is related to dissipation in the Josephson junction, i.e. large R yields small dissipation. The Lagrangian for a Josephson junction can be guessed by comparing Eq. 2.25 to Lagrange?s equation d dt @L @ _ ? @L @ = 0: (2.26) Ignoring the dissipation, the equation of motion becomes I = I0 sin + '0C2? d2 dt2 : (2.27) and comparing Eq. 2.27 to Eq. 2.26, we obtain L = 12 '0 2? ?2 C _ 2 + '02? (I0 cos + I ): (2.28) Notice that I multiplied Eq. 2.27 by a factor '0=2? to make L have dimensions of energy. 14 10 5 0 5 10 10 8 6 4 2 0 2 4 6 8 10 ? 2pi U /I 0 ? 0 I/I0 = 0.5 I/I0 = 0.9 ?U (radians) Figure 2.2: Graph of tilted washboard potential U (normalized by '0I0=2?) versus phase difierence . The solid curve is for I = 0.5I0 and the dotted curve for I = 0.9I0. 15 2.3.4 Hamiltonian of a Josephson junction Choosing the phase as a generalized position coordinate, the conjugate mo- mentum p is then p = @L@ _ = '0 2? ?2 C _ (2.29) Then the Hamiltonian H can be found from H(p; ) = p _ ? L( ; _ ) (2.30) = 12C 2? '0 ?2 p2 ? '0 2? (I0 cos + I ): (2.31) The Josephson junction Hamiltonian given by Eq. 2.31 is analogous to a ball of mass m = C('0=2?)2 moving in a tilted washboard potential U( ) = ?'0? (I0 cos + I ): (2.32) Figure 2.2 shows the tilted washboard potential in space. For small bias, we can approximate cos ? 1 ? 12 2. Thus for small the potential looks harmonic and the phase will oscillate at the minimum of the potential with angular frequency !p0 where !p0 ? r2?I0 '0C : (2.33) Examination of Fig. 2.2 reveals that the potential U has local minima that are separated by a barrier of height ?U . Increasing the bias current causes the barrier height to decrease and one flnds in general [23]: ?U = ?'02?I0 ?s 1? I 2 I20 ? II0 cos ?1 I I0 ! : (2.34) The location of the extrema can be found by setting the flrst derivative of the po- tential to zero. The second derivative of the potential at the potential minimum 16 gives the spring constant k of the efiective harmonic potential. Using a cubic ap- proximation [24], we then obtain the plasma frequency !p = r k m = r2?I0 '0C ? 1? I I0 ?2!1=4 = !p0 ? 1? I I0 ?2!1=4 : (2.35) where !p0 is the frequency of a small oscillation (harmonic approximation) at the bottom of the washboard potential. 2.3.5 Solution of the Josephson junction Hamiltonian With the Hamiltonian given by Eq. 2.31, one can substitute p = ?i~@ and write the Schro?dinger equation H? = E?. This equation can be solved for ? using a numerical method [24] or a WKB approximation [23, 25]. F. W. Strauch?s thesis contains a discussion of several methods to solve Schro?dinger equation for the Josephson junction and the accuracy of the difierent approaches [24]. The Josephson junction simulation I used in this thesis is based on a numerical simulation code written by H. Xu and S. K. Dutta [23]. If we cool a large-area Josephson junction and isolate it enough [11], well- deflned metastable resonant energy states [24] will exist, as shown in Fig. 2.3. Inside a well, the discrete resonant states can be labeled as j0i, j1i, j2i, etc. Energy states also exist above the well and form a continuum. Each metastable state in the well is distinguishable spectroscopically because the level spacings are anharmonic. The anharmonicity in the potential increases as we increase the current bias [24]. For T ? ?E=kB where ?E is the energy level spacing, the system will tend to relax to the ground state. We can control the state by applying microwave current to the Josephson junction. When f = ?E=h where f is microwave frequency, the corresponding energy level resonates with the microwave drive and the system can make transitions to higher levels. 17 ?2?1?0|0> |1>|2> continuum states Figure 2.3: Metastable states in a well of the tilted washboard potential. Close to the top of the barrier, the energy levels form a continuous energy band. The depth of the well is exaggerated. ?0, ?1 and ?2 are the escape rates from the energy level j0i, j1i and j2i, respectively. 18 The occupancy of a state can be measured from the escape rate. As we increase the bias current, the energy barrier gets lower and eventually the Josephson junction phase tunnels through the barrier to a running state that produces a voltage across the junction. This phenomena is called macroscopic quantum tunneling [26, 27] since the tunneling involves macroscopic numbers of electrons (also see sec.2.3.6). The escape rate for tunneling depends on the barrier height. For example, the escape rate at zero temperature is given as [28] ?0 = s 120?7:2?U~!p !p 2?exp ? ?7:2?U~!p 1 + 0:87!pReffCeff + ? ? ? ?? (2.36) where Reff is the efiective resistance, and Ceff is the efiective capacitance, both of which are in parallel to the junction. Equation 2.36 includes the efiect of dissipation in Reff and Ceff . More rigorously, escape rates from each levels can be calculated by solving Schr?odinger?s equation for the Hamiltonian in Eq. 2.31 with a full washboard po- tential and a decaying boundary condition [23, 24]. The?escape rate ?n from level n is given by ?n = (7:2Ns)n+1=2 !p2? exp [?7:2Ns + f n ? (Ns)] (2.37) where Ns ? ?U~!p = 1p2 EJ EC ?1=2 h?1? I2r ?1=4 ? Ir ?1? I2r ??1=4 cos?1 Ir i (2.38) is the number of energy levels in the well obtained from a full tilted washboard potential and fn? is a correction term [23, 24]. Appendix A shows a MATLAB code to solve the Schr?odinger?s equation for a single Josephson junction to obtain the escape rates. In my experiment, I measure the total escape rate from all energy levels. Since escape rates from the various energy levels difier by a factor of ? 500 19 to 1000 we are able to distinguish which levels the Josephson junction phase tunnels from. The procedure to distinguish levels is described in detail in chapter 4. The energy level spacings can also be obtained by solving Schr?odinger?s equa- tion for the Hamiltonian in Eq. 2.31 with a full washboard potential and a decaying boundary condition [23, 24]. The calculated energy level spacing between level jni and jn+ 1i can be written as !n;n+1 = !p fn! (Ns) : (2.39) where fn! is a correction term and Ns is the number of energy levels in the well shown in Eq. 2.38. Appendix A gives the code I used to calculate this factor and Strauch?s thesis [24] contains a detailed discussion. 2.3.6 ^ and n^ uncertainty relation While the underlying physics of the Josephson efiects is quantum mechanical, it was not apparent that the dynamics of the phase difierence would require quantum mechanics as well until the discovery of Macroscopic quantum tunneling (MQT) [26]. Although it is known from the BCS theory that the Cooper pairs are in a coherent state (the condensate), the discovery of MQT was surprising because the Josephson junction itself is a macroscopic object that is directly coupled to the rest of the world through leads to the current bias source. Observation of MQT proved that if a macroscopic object like a Josephson junction is reasonably well-isolated [10, 11], it can show quantum mechanical behavior. In a quantum mechanical treatment, the two conjugate variables and p are conjugate operators similar to x^ and p^ where the relationship is ^ $ x^ and p^ $ p^. 20 Thus one expects the commutation relation of ^ and p^ is [13] [ ^; p^ ] = i~: (2.40) I note that p^ is related to the voltage V across the junction because V is related to the time derivative of ^. From Eq. 2.29, p^ is given by p^ = '0 2? ?2 C _ = '02?CV = '0 2? Q^ (2.41) where Q^ is the charge on one plate of the capacitor C of the Josephson junction. If there are N Cooper pairs on the capacitor, then Q^ = - 2eN^ ; N^ is the number operator for the number of Cooper pairs on the capacitor. Therefore, p^ = '02? Q^ = ? ~ 2e2eN^ = ?~N^ (2.42) and the commutation relation, Eq. 2.40 becomes [ ^; ~N^ ] = ?i~ (2.43) or [ ^; N^ ] = ?i: (2.44) This result implies that ^ and N^ obey an uncertainty relation ? ?N ? 1=2 where ? is the uncertainty in ^ and ?N is the uncertainty in N^ . If we know exactly how many Cooper pairs exist on the Josephson junction, we lose information on and the amount of supercurrent owing through the junction. This phenomena can be interpreted as electrostatic energy causing the phase to delocalize [29]. We can choose either the phase representation or the number (charge) represen- 21 tation for the Hamiltonian. In the phase representation, the Hamiltonian becomes H^ = 12 '0 2? ?2 C _ 2 ? '02?I0 cos (2.45) where I = 0. The Josephson coupling energy is '0 2? cos ^ = '0 2? ei ^ + e?i ^ 2 : (2.46) and e?i ^ is the translation operator which satisfles e?i ^jNi = jN currency1 1i: (2.47) In the same manner as a translation operator for x^ [30], e?i ^ changes a number state by ?1. Thus e?i ^ can be written as e?i ^ = X N e?i ^jNihN j = X N jN currency1 1ihN j: (2.48) Therefore, in number representation, the Hamiltonian becomes H^ = 2e 2 C (N^) 2 ? '04?I0 X N (jN ? 1ihN j+ jN + 1ihN j) : (2.49) Examination of Eq. 2.49 reveals that the Hamiltonian does not commute with either ^ or N^ . The kinetic energy part is associated with charge (N^) and gives the \charging energy". The potential energy is associated with phase ( ^) and is the source of the \Josephson coupling energy". In many cases, either N^ or is much more sharply deflned. Which operator is sharper determines which representation we choose for the Hamiltonian. For the Josephson junction phase qubit, is relatively well-deflned, so it is the natural coordinate (phase representation). The number representation is in Ch. 9, where I discuss the Cooper pair box. 22 (a) (b) (c) Figure 2.4: Types of SQUIDs. (a) Schematic of an rf SQUID, (b) a dc SQUID and (c) a three junction SQUID. 23 2.4 Classical properties of SQUIDs 2.4.1 What is a SQUID? SQUID is an acronym for Superconducting QUantum Interference Device. There are three main types of SQUIDs (see Fig. 2.4). The rf SQUID is formed by placing one Josephson junction in a superconducting loop and uses only a ux bias [see Fig 2.4(a)]. The dc SQUID is formed by placing two Josephson junctions in a loop and uses a current bias and a ux bias [see Fig 2.4(b)]. Three junction SQUIDs are formed by placing three small or ultrasmall junctions in a loop [see Fig 2.4(c)]. The dc SQUID was invented in 1964 by Jaklevic, Lambe, Silver, and Mercereau from Ford Research Labs [31]. A year later, Zimmerman and Silver from Ford Research Labs invented the rf SQUID [32]. As the most sensitive known devices for detecting magnetic ux, SQUIDS have been used as a magnetic fleld detector in many applications [33, 34]. In this thesis I am mainly interested in the dc SQUID since it forms the basis for the dc SQUID phase qubit. Here I review some basic classical properties of the dc SQUID [35, 16]. 2.4.2 Flux-phase relation: uxoid quantization rule revisited In Eq. 2.15, I showed the uxoid quantization rule, which I can write as: me 2e2ns I J ? dl+ Z B ? da = n'0: (2.50) This result can be generalized to describe current and ux in a dc SQUID even though the SQUID is not a full superconducting ring. For a SQUID, the phase difierences across each junction must be taken into account. Consider the diagram of a SQUID shown in Fig. 2.5. The points a, b, c and d indicate points on the 24 SQUID loop. First, from Eq. 2.10 I can write I J ? dl = ?2ensme ? ~ I r (r) ? dl+ 2e I A ? dl ? : (2.51) By integrating along the lower half of the SQUID loop from b to c, we obtain [16] Z c b J ? dl = ?2ensme ? ~ Z c b r (r) ? dl+ 2e Z c b A ? dl ? ; (2.52) and then by integrating along the upper half of the SQUID loop from d to a, we obtain Z a d J ? dl = ?2ensm ? ~ Z a d r (r) ? dl+ 2e Z a d A ? dl ? : (2.53) If the superconductor that forms the SQUID is thick enough, we can choose a path inside the superconductor such that integration over the current density is negligible, that is: Z a d J ? dl ? 0 (2.54) and since 2 ? 1 = Z 2 1 r ? dl; (2.55) Eqs. 2.52 and 2.53 become ? me2e~ns Z c b J ? dl = c ? b + 2?'0 Z c b A ? dl = 0; (2.56) ? me2e~ns Z a d J ? dl = a ? d + 2?'0 Z a d A ? dl = 0 (2.57) where I used 2e=~ = 2?='0. From the deflnition of the gauge invariance phase difierences ij, ij = i ? j ? 2?'0 Z j i A ? dl; (2.58) 25 I I1 I2 Figure 2.5: Schematic of a dc SQUID. 26 the phase difierence of the left junction ( ab) and the right junction ( dc) is deflned by ab = a ? b ? 2?'0 Z b a A ? dl (2.59) dc = d ? c ? 2?'0 Z c d A ? dl: (2.60) where ~=2e = '0=2? and the line integrals are taken through the left and right junction, respectively, from top to bottom in Figure 2.5. The superconducting phase around the SQUID loop must be single-valued, i.e. the sum of the phase difierences around the loop must satisfy I r (r) ? dl = 2?n (2.61) where n is an integer. Substituting Eqs. 2.56, 2.57, 2.59 and 2.60 into Eq. 2.61 yields I r (r) ? dl = Z b a r (r) ? dl+ Z c b r (r) ? dl+ Z d c r (r) ? dl+ Z a d r (r) ? dl = ( b ? a) + ( c ? b) + ( d ? c) + ( a ? d) = ? ab ? 2?'0 Z b a A ? dl? 2?'0 Z c b A ? dl+ dc + 2?'0 Z c d A ? dl? 2?'0 Z a d A ? dl = ? ab ? 2?'0 Z b a A ? dl? 2?'0 Z c b A ? dl+ dc ? 2?'0 Z d c A ? dl? 2?'0 Z a d A ? dl = ?2?'0 I A ? dl+ dc ? ab = 2?n: (2.62) Since I A ? dl = Z B ? da = ' (2.63) 27 where ' is the total ux in the SQUID loop. I can then write Eq. 2.62 as [16]: dc ? ab = 2 ? 1 = 2?n+ 2?''0 : (2.64) I set dc = ? cd = 2 and ab = 1 so that the current through each junction ows from the top to the bottom. Here 1 is the phase difierence of the junction 1 (ab) where the current I1 going through the junction 1 ows from a to b, and 2 is the phase difierence of the junction 2 (dc) where the current I2 going through the junction 2 ows from d to c. The total ux ' includes any external applied ux 'a and any ux generated by the current J circulating around the loop. Including these explicitly gives the ux-phase relation: 2 ? 1 = 2?n+ 2?('a + LJ)'0 (2.65) where L is the loop inductance of the SQUID. 2.4.3 SQUID potential energy function In this section I flnd the equations of motion and derive the potential energy function of the dc SQUID [35, 36]. Figure 2.6 shows a more detailed circuit diagram of the dc SQUID. In this diagram, J1 is junction 1 and J2 is junction 2, and C1 and C2 are the junction capacitances of J1 and J2. J1 and J2 are connected through two inductors L1 and L2 that form the SQUID loop. M is the mutual inductance between the SQUID loop and a ux bias current source If which produces an applied ux 'a. Finally, I is the current bias source. Ignoring any normal shunting paths through the junction from current con- servation, we can write I = I1 + I2 = I01 sin 1 + I02 sin 2 + C1'02? ?1 + C2 '0 2? ?2 (2.66) 28 ?a L1 L2 J2J1 C2C1 I I1 I2 Figure 2.6: Schematic of a dc SQUID. J1 is junction 1 and J2 is junction 2. C1 and C2 are junction capacitances of J1 and J2. J1 and J2 are connected through two inductors on the SQUID loop, L1 and L2 and we will assume the total inductance is L = L1 + L2. 'a is the applied ux. I is the current bias source. 29 where I1 is the current in the left arm of the SQUID and I2 is the current in the right arm which are given by I1 = I01 sin 1 + C1'02? ?1 (2.67) I2 = I02 sin 2 + C2'02? ?2: (2.68) The ux-phase relation, as given by Eq. 2.65, can be written as: 2 ? 1 = 2?`a + 2?L1'0 I1 ? 2?L2 '0 I2: (2.69) `a = 'a='0 is a dimensionless applied ux, and I01 and I02 are the critical currents of junction 1 and junction 2, respectively. 1. Here, the current I and dimensionless ux `a are external control parameters. Substituting Eq. 2.67 into Eq. 2.69 gives 2 ? 1 = 2?`a + 2?L1'0 I1 ? 2?L2 '0 (I ? I1) (2.70) = 2?`a + 2?(L1 + L2)'0 (I01 sin 1 + C1 '0 2? ?1)? 2?L2I '0 : (2.71) and substituting Eq. 2.68 into Eq. 2.69 gives 2 ? 1 = 2?`a + 2?L1'0 (I ? I2)? 2?L2 '0 I2; = 2?`a + 2?L1'0 I ? 2?(L1 + L2) '0 (I02 sin 2 + C2 '0 2? ?2): (2.72) 1I assume that a ux '1 generated from the inductance L1 and current I1, is calculated with respect to the area of the SQUID loop and so as '2 30 Thus the equations of motion for 1 and 2 become '0 2?(L1 + L2)( 2 ? 1) = '0 (L1 + L2)`a + I01 sin 1 + C1 '0 2? ?1 ? L2 L1 + L2 I (2.73) '0 2?(L1 + L2)( 2 ? 1) = '0 (L1 + L2)`a ? I02 sin 2 ? C2 '0 2? ?2 + L1 L1 + L2 I (2.74) Now if we had the Lagrangian L, the equations of motion could be found from, d dt @L @ _ 1 ? @L @ 1 = 0 (2.75) d dt @L @ _ 2 ? @L @ 2 = 0: (2.76) Comparing Eqs. 2.73 and 2.74 to Eqs. 2.75 and 2.76, we flnd the following La- grangian as L = ' 2 0 4?2 ? (C1 _ 12 + C2 _ 22) + 2?(I01 cos 1 + I02 cos 2)'0 ? ( 2 ? 1)2 2L (2.77) ? 2?`a( 2 ? 1)L + 2?I '0 L2 1 + L1 2 L ?? where the total loop inductance L ? L1 + L2. Here a constant '0=2? is multiplied to L to give the Lagrangian the dimension of energy. The potential energy U for the dc SQUID phase qubit is then obtained by inspection from Eq. 2.77 U = '02? ? ?I01 cos 1 ? I02 cos 2 + '04?L( 2 ? 1) 2 + '0L `a( 2 ? 1)? I L2 1 + L1 2 L ?? : (2.78) It is convenient to normalize U with respect to the total critical current I0 and deflne a dimensionless potential: u = 2?U'0I0 = ? I01 I0 cos 1? I02 I0 cos 2+ '0 4?LI0 ( 2? 1+2?`a) 2+4?2`2a? I I0 L2 1 + L1 2 L ? : (2.79) 31 where 2I0 = I01 + I02. The flrst and the second terms in Eq. 2.79 are due to the Josephson coupling energies of junction 1 and 2 respectively (see flg. 2.5). The third and the fourth terms can be combined and yields a quadratic term in 2? 1?2?`a which causes coupling between the two junction phases. This term accounts for the magnetic energy stored in the SQUID inductances. The last term is the energy due to the bias current. 2.4.4 Current- ux map In a dc SQUID, the critical current Ic is the maximum current that can ow through the SQUID loop with zero voltage drop. Many key properties of the SQUID arise from the fact that the critical current changes as a function of the applied magnetic ux (see Fig. 2.6). Moreover, depending on the inductances and the critical currents of each junction, the SQUID can have a single critical current or multiple critical currents at a given applied ux. Multiple critical currents typically occur in our dc SQUID qubit because we choose fl = ?L(I01 + I02)/'0 1 for isolation purposes and this allows the loop to trap a persistent circulating current. The relation between critical current and applied ux is best visualized by plotting the switching current vs. ux; i.e. I versus 'a. In practice, I determine SQUID parameters such as the total loop inductance L, the critical currents of each junction, and the mutual inductance between the SQUID loop and the feedback coil from the measurements of the current- ux map. We can calculate the current- ux map classically from the equations of motion. In the classical model, the critical current is the maximum static current that can ow through the SQUID with constant phase across the junctions. To calculate the critical current as a function of the applied ux, I use an approach described by 32 Figure 2.7: Critical current versus ux curve. Yellow dots show results from the calculation of the critical current using a method of Tsang et al. [37]. Solid curves (which are made from small dashes) are measured switching currents for the dc SQUID phase qubit AL1 at 80 mK. 33 Tsang et al. [37]. They start from the static current conservation equation, I = I01 sin 1 + I02 sin 2 (2.80) and use the ux-phase relation as a constraint: 1 ? 2 = 2?`a + 2?L1'0 I1 ? 2?L2 '0 I2 (2.81) where I1 = I01 sin 1 (2.82) I2 = I02 sin 2 (2.83) are the static currents through junctions 1 and 2, respectively. Here any normal shunts across the junctions are ignored, since no current ows through at V = 0. Also the displacement currents from the capacitances do not contribute because we are dealing with the static, zero-voltage, situation before switching. Our goal is to flnd the maximum of I subject to the constraint given by Eq. 2.81. The Euler-Lagrange equation is a convenient tool to flnd extrema functions that are subject to constraints. The goal is to flnd a function F of 1 and 2 which satisfles @F @ 1 = 0 (2.84) @F @ 2 = 0 (2.85) @F @ 2 = 0 (2.86) when I( 1; 2) is maximized and the constraint given by Eq. 2.81 is also satisfled. 34 The appropriate F is F ( 1(t); 2(t)) = I01 sin 1 + I02 sin 2 + ? 1 ? 2 ? 2?`a ? 2?L1'0 I1 + 2?L2 '0 I2 ? (2.87) where ? is the Lagrange multiplier. From the solutions 1 and 2 of Eqs. 2.84 to 2.86, we flnd the currents I1( 1) and I2( 2) which maximize or minimize I. The equations can not be solved analytically but the numerical solution is straightforward. Figure 2.7 shows a plot of a best flt calculation of the critical currents vs. applied ux (circles) compared to experimental data (solid curves). The data was collected for an Al/AlOx/Al dc SQUID, device AL1. AL1 is an asymmetric dc SQUID withL1 L2. In the current- ux map, this results in switching events from each junction being distinguishable [37, 23]. The section of the I-'a curves with higher slope with respect to the applied ux is due to junction 1 switching flrst, and the section with lower slope with respect to the applied ux is due to junction 2 switching flrst. For the best flt, I flnd good agreement between the data and the simulation. The maximum of the curve gives the sum of the critical currents of J1 and J2 and the minimum of the line with higher slope gives the difierence of the critical currents of J1 and J2, although this was not resolved in the data. Any two adjacent curves in this flgure are separated along the current axis by almost exactly '0=L, so I can get a good estimate for the total loop inductance L. Also the switchings curves are strictly periodic with period of '0=M along the x-axis, so I can also obtain the mutual inductance M between the SQUID loop and the ux bias. In this case, the best flt was for I01 = 21.401 ?A, I02 = 9.445 ?A, L1 = 1.236 nH, L2 = 5 pH, and M = 15 pH. 35 2.5 Application of Josephson junctions to quantum computation This thesis concerns the potential application of Josephson junctions to quan- tum computation. In this section, I review the types of superconducting qubits and summarize progress on them. I also introduce our dc SQUID phase qubit. 2.5.1 Superconducting qubits Superconducting qubits are classifled into three main types: charge qubits, ux qubits and phase qubits. Charge qubits are based on ultra-small Josephson junctions in which the charg- ing energy (Q2=2C) is dominant (see Fig. 2.8). Charge qubits use the two lowest energy states, represented in charge basis, as qubit states. For charge qubit, the Hamiltonian is dominated by the charging energy stored in the junction capacitor. The state can be manipulated by applying a gate voltage. The Cooper pair box is the best-known type of charge qubit [38]. The Hamiltonian of the Cooper pair box can essentially be obtained from Eq. 2.31 by changing from the phase basis into the charge (number) basis. Although the charge (number) states are not the exact eigenstates of the Hamiltonian [29], the energy eigenstates can be found from superpositions of a few number states by treating the Josephson energy term in Eq. 2.31 as a perturbation. An in-depth discussion of the Cooper pair box is given in chapter 9. The flrst experimental demonstration of coherent oscillations in a su- perconducting qubit was performed on a Cooper pair box by Nakamura et al. in 1999 [38]. The longest coherence time in charge qubit has been obtained by the Yale group [39, 40]; using a non-demolition readout [41, 42] they recently reported flnding T2 ? 2 ?s in a hybrid charge/phase qubit called the \transmon" [43]. Flux qubits are SQUIDs with one, two or three junctions. The basis states correspond to difierent amounts of ux in the SQUID loop or superpositions of such 36 Vg Cg CJ Figure 2.8: a Cooper-pair box with a single ultra small junction. CJ is the capac- itance of the superconducting ultra small junction and Cg is the gate capacitance. Vg is the gate voltage. 37 states. An rf-SQUID with appropriate ux bias and choice of parameters is one example of a ux qubit. In ux qubits, the scale of the charging energy is comparable to the Josephson energy. The control variable is applied ux [13]. The ux qubit has a double-welled potential where the two ux states correspond to being trapped in one well or the other. The most popular form of the qubit has three junctions [44]. The Hamiltonian can be truncated and the reduced state space spanned by two ux states, similar to the two level approximation to the Hamiltonian of the Cooper pair box. Friedman et al. flrst observed avoided crossings of two ux states in their rf SQUID energy spectrum in 2000 [45]. The longest coherence time reported in the ux qubit so far is T2 ? 4 ?s [46]. Phase qubits are based on large-area Josephson junctions where the Josephson energy is dominant and the phase is relatively well-deflned. Phase qubits can be constructed from rf or dc SQUIDs so they share some similarities to ux qubits. However, the phase qubit states are the two lowest energy states of the washboard potential in a given well, not ux states in difierent wells. Ramos et al. flrst proposed that the two lowest energy levels from a single large-capacitance Josephson junction can be used as a phase qubit [12], but the isolation scheme for a single Josephson junction was not trivial. The flrst coherent oscillation in a phase qubit was observed in 2002 by Martinis et al. [47] in their dc SQUID phase qubit and the entanglement of two coupled phase qubits was reported in 2002 by Berkley et al. [48]. The longest coherence time reported on the phase qubit is T2 ? 500 ns [49]. 2.5.2 dc SQUID phase qubit: design and basic idea, inductive isola- tion In this section, I discuss in some detail the dc SQUID phase qubit and the idea behind its design. Superconducting qubits are macroscopic devices that can readily couple to 38 M If ?a L2 L1 J1J2 C1C2 I Cw Iw I2 I1 Figure 2.9: Schematic of a dc SQUID phase qubit. J1 is the qubit junction and J2 is the isolation junction. C1 and C2 are the capacitances of J1 and J2. J1 and J2 are connected through two inductors on the SQUID loop, L1 and L2. M is a mutual inductance between the SQUID loop and the current source If supplies the applied ux 'a. I is the current bias source. The microwave source Iw is coupled to the qubit junction J1 by capacitor Cw 39 many degrees of freedom in the environment. This undesirable coupling between a qubit and its environment can be reduced by good isolation and biasing schemes. Finding an optimal design for the qubit isolation is challenging because the state of the qubit still has to be manipulated and measured. Inductive isolation for Josephson junction phase qubits was flrst introduced by Martinis et al. [47]. In this scheme, a qubit junction is shunted by a relatively large inductor in series with a Josephson junction (see Fig. 2.9). The second junc- tion is called the "isolation junction" because it helps to isolate the qubit junction from current noise. In Fig. 2.9, junction J1 acts as a phase qubit and the rest of the SQUID serves as an inductive isolation network that fllters out noise from the current bias leads; The inductive isolation network consists of a flxed inductance L1, an isolation junction J2 and a parasitic inductance L2. The junction J2 has an associated Josephson inductance LJ2. When noise current is introduced into the bias leads, L1 + LJ1 and L2 + LJ2 work as an inductive current divider and only a fraction of the current noise will pass through the qubit junction J1. If there is a small uctuations ?I in the current bias, then we can write ?I1 ?I2 = L2 + LJ2 L1 + LJ1 (2.88) where ?I1 is the corresponding current uctuation going through the qubit junction J1, ?I2 is the current uctuation going through the isolation junction and ?I = ?I1 + ?I2. By choosing L1 + Lj1 L2 + Lj2, we can reduce ?I1 with respect to ?I2. Typical inductances in my devices are L1 = 1 nH, Lj1 = 20 pH, L2 = 5 pH and Lj2 = 40 pH, which yields ?I1=?I2 ? 0.044. The inductive current divider also reduces the bias current that reaches the qubit. To compensate, we use a secondary current source, a ux bias 'a supplied by a current If that couples to the SQUID loop via a mutual inductance M (see Fig. 2.9). Noise ?If on the ux bias line will also induce a noise current through 40 the qubit junction, where ?I1 ?If = M L1 + LJ1 + L2 + LJ2 : (2.89) In qubit AL1, I found M ? 10 pH, L1 ? 1 nH and L1 Lj1 + L2 + Lj2. Equation 2.89 then gives ?I1=?If ? 0.01. In order to use a static ux bias to current-bias the qubit junction, the induc- tive isolation network must be superconducting. In practice, we detect the qubit states by monitoring when the qubit junction switches to the voltage state. For this voltage to be measurable, the isolation element must present some impedance when the qubit switches. If LJ2 were just a small superconducting inductor, it would prevent a static voltage from appearing across the output leads and we would not be able to detect the junction switching voltage (See Fiq. 2.6). This is why the dc SQUID pase qubit has the isolation junction in the isolation network. By placing a second Josephson junction into the inductive current divider we can achieve two purposes; a small inductor for isolation and a non-linear element that allows detection. The idea is that the isolation junction remains superconducting until the qubit junction switches. When the qubit switches, the bias current is shunted to the isolation junction which triggers the isolation junction to switch and leads to a voltage across the output leads. It also turns out that by using a Josephson inductor, we can tune the inductance ratio between the qubit branch and the isolation network. A detailed analysis of the isolation circuit is given in Chapter 5. 41 Chapter 3 Dynamics of a two-level quantum system 3.1 Introduction Nuclear magnetic resonance (NMR) was flrst seen by Rabi in 1938 [50]. Later, Bloch et al. and Purcell et al. individually developed methods to measure nuclear magnetic resonance in solids (para?n) [51] and liquids (water) [52, 53]. In 1946, Bloch introduced the equation of motion for nuclear magnetization in a magnetic fleld; these are the now well-known Bloch equations [52]. Atomic physicists soon adapted Bloch?s theory to explain radiation phenomena in atoms. They modifled the Bloch equations to obtain the optical Bloch equations, which describe how atoms interact with light [54]. The resulting models are now widely used to describe quantum behavior in two-level systems, as well as systems with more than two levels, such as the Cooper pair box or Josephson junction phase qubit. In this chapter, I brie y review the quantum dynamics of two-level systems. I discuss the density matrix formalism and construct the optical Bloch equations. Next I discuss the density matrix formalism with dissipation and decoherence. Fi- nally, I connect the optical Bloch equation to the density matrix description and show how to construct the Bloch vector for the Josephson phase qubit. The main references of this chapter are an unpublished note by Dr. Wellstood [55], the book \Optical Resonance and Two Level Atoms" by Allen and Eberly [54] and \The Principles of Nuclear Magnetism" by Abragam [56]. 42 3.2 Density matrix formalism for a two-level system In general, a qubit is not necessarily in a pure state, but can be in an incoherent superposition of states or \mixed state". Such mixed states naturally form because of entanglement with the environment, and it is impossible to measure everything about the state of the system and the entangled environment. The density operator, ?^, is an operator that gives the probabilities of the system being in certain states which can be measured by the experiment. Using the density operator, we can describe the evolution without knowing the complete wavefunction of the system and the environment. I start by considering an isolated two-state system being driven by a periodic external force. Choosing the basis as the two qubit states of the qubit; j0i and j1i, the density operator, ?^ can be written as a 2 by 2 matrix, ?^ = 0 B@ h0j?^j0i h0j?^j1i h1j?^j0i h1j?^j1i 1 CA = 0 B@ ?00 ?01 ?10 ?11 1 CA : (3.1) where j0i = 0 B@ 1 0 1 CA (3.2) and j0i = 0 B@ 0 1 1 CA : (3.3) Two important properties of ?^ are Tr(?^) = ?00 + ?11 = 1 (3.4) and ?01 = ??10: (3.5) 43 Also ?^ satisfles the equation of motion, i~d?^dt = [H; ?^]: (3.6) Here I will consider the Hamiltonian H^ = 0 B@ E0 F0h0jx^j1i cos!t F0h1jx^j0i cos!t E1 1 CA = H^0 + H^ 0 (3.7) where H^0 = 0 B@ E0 0 0 E1 1 CA (3.8) is the unperturbed 2-level Hamiltonian and H 0 = 0 B@ 0 F0h0jx^j1i cos(!t) F0h1jx^j0i cos(!t) 0 1 CA (3.9) describes a periodic drive fleld for exciting the two-level system. For convenience, I deflne a0 = F0h0jx^j1i where x^ is a conjugate position operator (coordinate) for the two-level system that couples to the drive and I will assume a0 is real. Equation 3.6 then gives four equations of motion, i~d?00dt = a0(?10 ? ?01) cos!t (3.10a) i~d?01dt = a0(?11 ? ?00) cos!t??E?01 (3.10b) i~d?10dt = ?a0(?11 ? ?00) cos!t+?E?10 (3.10c) i~d?11dt = ?a0(?10 ? ?01) cos!t: (3.10d) where ?E = E1 ? E0, H^0j0i = E0j0i and H^0j1i = E1j1i. Note that Eqs. 3.10 imply d?11=dt = ?d?00=dt, which is essential for maintaining Tr(?^) = 1, and allows 44 ?00 and ?11 to be interpreted as the probabilities to flnd the system in j0i and j1i respectively. 3.3 Optical Bloch equations: two-level systems and magnetic spin A spin-1/2 system is the proto-typical two-level system; we can deflne j0i as spin-down and j1i as spin-up. Due to its pictorial convenience, the language of magnetic spins and NMR is widely used to describe the behavior of two level systems, including qubits. In this section, I discuss the optical Bloch equations, which is a version of the Bloch equations [52, 53] for two-level systems. This section is largely based on Ch. 2 in \Optical resonance and two-level atoms" by Allen and Eberly [54]. 3.3.1 Representing the Hamiltonian of a two-level system with Pauli matrices Consider an atom interacting with an electric fleld E^ which drives transitions between two-levels, j+i and j?i of the atom. I can write the Hamiltonian as H^ = H^0 ? d^ ? E^: (3.11) I will assume that the interaction energy d^?E^ can be treated as a small perturbation. Here H^0 is the unperturbed Hamiltonian and d^ is the electric dipole moment of the atom, d^ = ?er^ (3.12) where r^ is the position vector of the electron with respect to the nucleus. Since our interest is only in two-levels, we can span the Hamiltonian with the basis j+i and j?i, which are eigenstates of H^0. In this case, H^0 can be written as a 2 by 2 matrix 45 of the form:form H^0 = 0 B@ E+ 0 0 E? 1 CA (3.13) and the perturbation term becomes hdi ? E^ = 0 B@ h+jd^j+i h+jd^j?i h?jd^j+i h?jd^j?i 1 CA ? E^ = 0 B@ 0 dR + idI dR ? idI 0 1 CA ? E^ (3.14) where only the ofi-diagonal terms are non-zero because of the spatial symmetry of j+i and j?i states in atomic systems. Here dR and dI are the real and imaginary parts of h+jd^j?i. It is useful to recall that any 2 by 2 matrix equation can be expressed in terms of the Pauli spin matrices and the identity matrix I^ [54]. In this case, Eq. 3.11 can be written as H^ = 12(E+ + E?)I^ + 1 2(E+ ? E?) ^3 ? (dR ? E^) ^1 + (dI ? E^) ^2: (3.15) where ^1 ? 0 B@ 0 1 1 0 1 CA ; ^2 ? 0 B@ 0 ?i i 0 1 CA ; ^3 ? 0 B@ 1 0 0 ?1 1 CA (3.16) are the Pauli spin matrices. 3.3.2 Equation of motion for a two-level system To obtain the equation of motion of a two-level system, I note that the time evolution of Pauli operators obey the equation; i~ _^ n = [ ^n; H^] (3.17) 46 for n = 1, 2, 3. Substituting Eq. 3.15 for H^, one flnds _^ 1(t) = ?!0 ^2(t) + 2~ [dI ? E^(t)] ^3(t) (3.18a) _^ 2(t) = !0 ^1(t) + 2~ [dR ? E^(t)] ^3(t) (3.18b) _^ 3(t) = ?2~ [dR ? E^(t)] ^2(t)? 2 ~ [dI ? E^(t)] ^1(t): (3.18c) where !0 = E+ ? E?~ : (3.19) Taking an expectation value of both sides of Eq. 3.18 and deflning sn(t) = h n(t) ^ni [57], then I can write _s1(t) = ?!0s2(t) (3.20a) _s2(t) = !0s1(t) + ?E(t)s3(t) (3.20b) _s3(t) = ??E(t)s2(t): (3.20c) where ? = 2jdRj=~ and E(t) = E0(t)[ei!t + e?i!t], is the electric fleld component parallel to dR. There are two additional assumptions I used to derive Eqs. 3.20(a - c); (i) the dipole matrix d^ is real so that dI ? E^(t) = 0 [54] and (ii) the correlation between the electric fleld and the atom can be ignored. The second condition implies that hE^(t) ^n(t)i ? hE^(t)ih ^n(t)i: (3.21) Eqs. 3.20(a - c) are the optical Bloch equations [54] and equivalent to Bloch equations for describing the interactions between atoms and light. The solution s1(t); s2(t); s3(t) of the optical Bloch equations can be drawn as a vector that lies on the unit sphere (see Fig. 3.3.2). Eqs. 3.20 can be put into an equivalent vector 47 Figure 3.1: Two-level system represented as a vector on the Bloch sphere. 48 form for s(t) = [s1(t); s2(t); s3(t)] as d dts(t) = ?(t)? s(t) (3.22) where ?(t) = [??E0(t)[ei!t + e?i!t]; 0; !0]. The resulting behavior resembles the motion of a rotating rigid body or a classical spin vector where ?(t) is the torque applied to the spin vector s(t). In general, ?(t) can oscillate at frequency ! ? !0 when E(t) is in resonance with the atom. Our main interest is in the behavior of s(t). However, due to the moving ?(t), the motion of s(t) is not so simple. It is most convenient to describe s(t) in a frame which rotates at an angular frequency ! about the z axis. To proceed, we need to change from the flxed frame basis we have been using into the basis of a rotating frame. To change bases, we use the rotation matrix U^ U^ = 0 BBBB@ cos(!t) sin(!t) 0 ? sin(!t) cos(!t) 0 0 0 1 1 CCCCA : (3.23) In the rotating frame basis, s is transformed to sr via; sr = U^s (3.24) = 0 BBBB@ cos(!t) sin(!t) 0 ? sin(!t) cos(!t) 0 0 0 1 1 CCCCA 0 BBBB@ s1 s2 s3 1 CCCCA (3.25) = 0 BBBB@ s1 cos(!t) + s2 sin(!t) s2 cos(!t)? s1 sin(!t) s3 1 CCCCA = 0 BBBB@ u v w 1 CCCCA (3.26) 49 and _s is transformed into dsr dt = 0 BBBB@ _u _v _w 1 CCCCA : (3.27) Similarly, the vector ? is transformed to ?r in the rotating basis as ?r = U^? (3.28) = 0 BBBB@ cos(!t) sin(!t) 0 ? sin(!t) cos(!t) 0 0 0 1 1 CCCCA 0 BBBB@ ?2?E0 cos!t 0 !0 1 CCCCA (3.29) = 0 BBBB@ ?2?E0 cos2 !t ?2?E0 cos!t sin!t !0 1 CCCCA (3.30) = 0 BBBB@ ??E0 ? ?E0 cos 2!t ??E0 sin 2!t !0 1 CCCCA (3.31) The dynamics should be the same no matter what basis I use. This implies that sr observed in the flxed frame should be expressed in the same way even though I changed the basis into the basis of the rotating frame. Thus from Eq. 3.22, sr observed in the flxed frame is written as d dtsr(t) ? fixed = ?r(t)? sr(t) (3.32) But in the rotating frame, sr(t) will experience a flctitious torque due to rotation. 50 Thus the behavior of sr(t) in the rotating frame is given by [58] d dtsr(t) ? rotating = d dtsr(t) ? fixed ? ~!(t)? sr(t) (3.33) = ?r(t)? sr(t)? ~! ? sr(t) (3.34) = (?r(t)? ~!)? sr(t): (3.35) where ~! = !z^. Examination of Eq. 3.31 shows that ?r has a flxed component along the z-axis and x-axis and a component that rotates in the x-y plane at frequency 2! with a small amplitude. Typically ! ? !01, and the 2! components are not important because they are not in resonance. To simplify the analysis, we ignore the 2! components. This is called the rotating wave approximation [59]. With the rotating wave approximation, ?r becomes simply: ?r = 0 BBBB@ ??E0 ?0 !0 1 CCCCA : (3.36) We can then write Eq. 3.35 in the rotating basis as 0 BBBB@ _u _v _w 1 CCCCA = 0 BBBB@ ??E0 ?0 !0 ? ! 1 CCCCA ? 0 BBBB@ u v w 1 CCCCA (3.37) 51 or equivalently _u = ?(!0 ? !)v (3.38a) _v = (!0 ? !)u+ ?E0(t)w (3.38b) _w = ??E0(t)v: (3.38c) Equations 3.38(a - c) are just another version of the Bloch equations. 3.4 Including decoherence and dissipation Up to this point, I ignored dissipation and decoherence. Even if we do not know the microscopic mechanism that causes decoherence and relaxation in our qubit, we can still describe their efiects phenomenologically by adding some terms to the equation of motion. In NMR, the sample magnetization decays due to interactions with the lattice and other spins. These interactions can change or preserve the energy of the spin. In practice, two-level systems experience analogous efiects. When a qubit interacts with a dissipative thermal reservoir, it can decay from the excited state to the ground state. The time constant T1 for this decay from j1i to j0i is called the relaxation time or the energy dissipation time. For a Josephson junction qubit, T1 can be calculated by modeling the dissi- pation source admittance Y(!) as a bath of Harmonic oscillators [60, 61, 25]. If the coupling between the qubit and the harmonic oscillator bath is linear in a coordinate of the qubit, then T1 will be proportional to the real part of Y(!). One flnds T1 = CRe[Y (!01)] (3.39) where C is the total capacitance in parallel with the qubit junction, including the qubit junction capacitance [25]. 52 Elastic scattering processes cause decoherence even though no energy is dis- sipated. Decoherence involves a loss of information in the phase ` in the Bloch representation (see Fig 3.3.2). In a spin system, decoherence happens when spins that are initially in phase (coherent), evolve to have difierent phases (incoherent). Elastic scattering can act homogeneously or randomly on each spin, and this leads to difierent efiects on a system. The coherence time T2 is the characteristic lifetime for a qubit to retain its phase and it is used in the Bloch equations [54]. T2 is also called the transverse relaxation time in NMR [56]. T1 and T2 are connected to the decay time constant T 0 of Rabi oscillations by [54] 1 T 0 = 1 2T1 + 1 2T2 : (3.40) Thus by measuring T1 from relaxation and T 0 from Rabi oscillations, T2 can be obtained experimentally. One expects T2 = 2T1 if only dissipation is present as a decoherence source [62]. In practice, however, T2 is often found to be shorter than T1 [56] due to the presence of a pure dephasing source. There are other important time constants that can be obtained from spectro- scopic measurements. The half-width at half-maximum ?f of the resonance obeys [56] ?f ? 12?T ?2 (3.41) where T ?2 is the spectroscopic coherence time. T ?2 includes broadening of a reso- nance due to T2, and also inhomogeneous (random) scattering (for example, from low frequency noise) represented by the inhomogeneous coherence time Ty2 [54]. If microwave power broadening is also present, 1 T ?2 = 1T2 p 1 + ?2T1T2 + 1T y2 (3.42) 53 Table 3.1: Notation for time constants used here and Ref. [54]. Relaxation coherence spectroscopic inhomogeneous time time coherence time coherence time In this thesis T1 T2 T ?2 T y2 Ref. [54] T1 T 02 T2 T ?2 where ? is the Rabi frequency. Ideally, the resonance width is only limited by dis- sipation [30, 54] in which T ?2 = 2T1 = T2. However, inhomogeneous broadening can create situations where the resonance frequency varies randomly from one measure- ment to the next, and as a result, the resonance peak broadens. Basically any efiect that makes a measurement non-identical, can cause inhomogeneous broadening and a short T?2 in the system. In a dc SQUID phase qubit, uctuations in current, ux or critical current can cause inhomogeneous broadening. Unfortunately, there is no universal agreement upon notations for the various time constants of the qubit. For example, in Allen and Eberly [54], T2 is denoted as T02, Ty2 as T?2 and T?2 as T2 (see Table. 3.1). In this thesis, I followed the notation from Refs. [55, 56]. 3.5 Solutions of the density matrix equation with T1 and T2 The time constants T1 and T2 associated with dissipation and decoherence, can be added to the density matrix equation of motion in the following ad hoc 54 manner [63]. Starting from Eqs 3.10 (a - d), I can write: i~d?00dt = a0(?10 ? ?01) cos!t+ i~ ?11 T1 (3.43a) i~d?01dt = a0(?11 ? ?00) cos!t??E?01 ? i~ ?01 T2 (3.43b) i~d?10dt = ?a0(?11 ? ?00) cos!t+?E?10 ? i~ ?10 T2 (3.43c) i~d?11dt = ?a0(?10 ? ?01) cos!t? i~ ?11 T1 : (3.43d) Notice that T1 and T2 enter difierently; T2 goes into the ofi-diagonal equations, while T1 goes into the diagonal equations. To solve Eqs. 3.43, we try the test solution [63] ?00 = Ae?t (3.44a) ?01 = Bei!te?t (3.44b) ?10 = Ce?i!te?t (3.44c) ?11 = De?t: (3.44d) The possible values for the exponent ? are found by substituting Eqs. 3.48(a - d) into Eqs. 3.43(a - d) and flnding the roots of the resulting characteristic equation. After using the rotating wave approximation to eliminate 2!t terms, we get four roots of ? [63, 64, 55] ?0 = 0 (3.45a) ?1 = ? 1T 0 + i? (3.45b) ?2 = ? 1T 0 ? i? (3.45c) ?3 = ? 1T2 (3.45d) 55 where 1 T 0 = 1 2T1 + 1 2T2 (3.46) and ? = s ?02 ? 1 2T2 ? 1 2T1 ?2 : (3.47) Here ?0 = a0=~ is the bare Rabi frequency which depends only on the microwave power and the matrix elements of x^ (see Eq. 3.9). We now consider the case when ?00 = 1 at t = 0, and resonant microwave power is applied at ! = !0. For su?ciently high power, ? from Eq. 3.47 is real and the solutions are given by; [63] ?00 = 1? ?11 (3.48a) ?01 = i?eqe i!t ?0 ?e?t=T 0 T1 + 1 ? ? cos(?t) ?2 ? 1(2T1)2 + 1 (2T2)2 ? + sin(?t)T1 ? (3.48b) ?10 = ??01 (3.48c) ?11 = ?eq ? ?eq ? 1? cos(?t) + sin(?t)?T 0 ? exp(?t=T 0) (3.48d) where ?eq = ? 2 0T1T2 2[1 + ?20T1T2] (3.49) is the probability of the qubit being in j1i if the power is left on for an arbitrarily long time. From Eq. 3.49, we see that ?eq is determined by the microwave power (related to ?20), T1 and T2. The high power limit occurs for ?0 (T1T2)?1=2 where ?eq ?= 1/2. This is called \saturation". To measure Rabi oscillations, we need to use a su?ciently high microwave power, so that ?0 (T1T2)?1=2. In this limit, ?eq becomes close to 1/2 and Eq. 3.48(d) for ?11 then gives decaying oscillations with the Rabi frequency ?. The decay time constant T 0 of these oscillations is given in Eq. 3.46 and involves both 56 T1 and T2. 3.6 From the density matrix to the Bloch vector While the density matrix provides a good way to describe Rabi oscillations and the state of the qubit, the Bloch sphere provides a more intuitive picture of the time-dependent behavior. The three components u, v, and w of a Bloch vector can be related to the components of the density matrix by [55] u = ?01 exp(?i!t) + ?10 exp(i!t) (3.50a) v = ?i[?01 exp(?i!t)? ?10 exp(i!t)] (3.50b) w = ?00 ? ?11: (3.50c) In matrix form, I can write ?^ = 0 B@ ?00 ?01 ?10 ?11 1 CA = 12 0 B@ 1 + w (u+ iv) exp(i!t) (u? iv) exp(?i!t) 1? w 1 CA : (3.51) Finally, I note that if the qubit is prepared in a superposition of pure states, the density matrix can be written as ?^ = 0 B@ ja0j2 a?0a1 a0a?1 ja1j2 1 CA = 0 B@ cos2 =2 ei` sin =2 e?i` sin =2 sin2 =2 1 CA : (3.52) where the qubit state j?i is [55] j?i = a0j0i+ a1j1i (3.53) = cos 2 ? j0i+ ei` sin 2 ? j1i: (3.54) 57 We can interpret and ` as angles (see flg. 3.3.2) allowing us to represent each point on the Bloch sphere as a state. This representation shows more clearly how the qubit state evolves as we apply external fleld (microwaves), and the meaning of the difierent components of the density matrix. 3.7 Bloch vector of the Josephson junction phase qubit Here I discuss explicitly how to construct a Bloch vector for the Josephson junction phase qubit. The derivation was published by Martinis et al. in Ref. [47]. As I showed in Ch. 2, the Hamiltonian of the phase qubit is H^ = Q^ 2 2C ? '0 2? (I0 cos ^ + I ^) (3.55) where Q^ = ?2eN^ is the charge operator, e = 1.6 ?10?19, ^ is the phase operator, I0 is the critical current of the qubit junction, and I is the bias current. The bias current can include dc and microwave components [61]. Making this explicit, I can write I(t) = Idc + Irfz(t) + Irfx(t) cos!01t+ Irfy(t) sin!01t = Idc + Irf (t): (3.56) where Idc is the dc current, Irfz(t) is a \dc" pulse, Irfx(t) cos!t and Irfy(t) sin!t are microwave currents at frequency !. The time dependent terms in the Hamiltonian can be treated as a time-dependent perturbation so that H^ = Q^ 2 2C ? '0 2? (I0 cos ^ + I(t) ^) = H^0 ? '0 2?Irf (t) ^ (3.57) where the unperturbed Hamiltonian is H^0 = Q^ 2 2C ? '0 2? (I0 cos ^ + Idc ^): (3.58) 58 If we assume that the Josephson phase qubit can be approximated as a two- level system with the two lowest energy states, then the Hamiltonian becomes H^ = 0 B@ E0 0 0 E1 1 CA+ '02? 0 B@ 00Irf (t) 01Irf (t) 10Irf (t) 11Irf (t) 1 CA (3.59) where E0 and E1 are the energy of j0i and j1i at I = Idc, the second term is the time-dependent perturbation due to microwave driving which is responsible for the energy level transitions, the matrix element ij = hij ^jji = ji. As shown in the previous section, it is convenient to use a rotating frame for a system which is oscillating. In a frame that is rotating with frequency !01, H^ becomes [24] H^r = exp ? iH^0t ~ ! H^ exp ? ?iH^0t ~ ! (3.60) and in the basis of unperturbed energy eigenstates, exp ? ?iH^0t ~ ! = exp(i!0t) 0 B@ 1 0 0 exp(i!01t) 1 CA (3.61) where ~!0 = E0. The Hamiltonian in the rotating frame then becomes H^r = 0 B@ 1 0 0 ei!01t 1 CA H^ 0 B@ 1 0 0 e?i!01t 1 CA (3.62) and I obtain H^r = 0 B@ E0 0 0 E1 1 CA+ '02? 0 B@ 00Irf (t) 01Irf (t)e?i!01t 10Irf (t)ei!01t + 11Irf (t) 1 CA : (3.63) 59 Since Irf (t)e?i!01t = Irfz(t)e?i!01t + Irfx(t)2 (?1 + cos 2!01t? i sin 2!01t) +Irfy(t)2 (sin 2!01t currency1 i? i cos 2!01t) (3.64) ? ?Irfx(t)2 currency1 i Irfy(t) 2 (3.65) where in the last step I have applied the rotating wave approximation and dropped all the 2! terms as well as terms oscillating at !. Equation 3.63 then becomes H^r = 0 B@ E0 0 0 E1 1 CA+ '02? 0 B@ 00Irfz(t) 01( Irfx(t)2 ? i Irfy(t) 2 ) 01( Irfx(t)2 + i Irfy(t) 2 ) 11Irfz(t) 1 CA : (3.66) If we apply a constant amplitude microwave current of Irfx(t) = Irfx, Irfy(t) = Irfy, and a constant current pulse Irfz(t) = Irfz over time ?t, the diagonal terms can be expressed as E0 + '02? 00Irfz = E0(I(t)) (3.67) E1 + '02? 11Irfz = E1(I(t)): (3.68) 60 Rearranging terms, I flnd 0 B@ E0(I) 0 0 E1(I) 1 CA = 12 0 B@ ?(E1(I)? E0(I)) 0 0 E1(I)? E0(I) 1 CA+ (E0(I) + E1(I))2 I^ = 12 0 B@ ?E01(I) 0 0 E01(I) 1 CA+ (E0(I) + E1(I))2 I^ = 12 0 B@ ?(E01(I)? E01(Idc)) 0 0 E01(I)? E01(Idc) 1 CA+ C = ?12 0 B@ Irfz@E01=@I 0 0 ?(Irfz@E01=@I) 1 CA+ C: (3.69) Here I^ is the identity matrix and C is C = (E0(I) + E1(I))2 + 0 B@ ?E01(Idc) 0 0 E01(Idc) 1 CA : (3.70) Also I assume I ? Idc ? Irfz which implies that the oscillating current e?i!01t will not tilt the washboard potential on average. Recalling the deflnition of the Pauli matrices, H^r can be written as [61] H^r = '02? 01Irfx 2 ^x + '0 2? 01Irfy 2 ^y + Irfz 2 ?@E01@I ? ^z + C (3.71) where the Pauli matrices are ^x = ^1 = 0 B@ 0 1 1 0 1 CA ; ^y = ^2 = 0 B@ 0 ?i i 0 1 CA ; ^z = ^3 = 0 B@ 1 0 0 ?1 1 CA : (3.72) The idea behind Eq. 3.71 is that when we apply Irf , it rotates the state of the 61 qubit on the Bloch sphere according to the unitary transformation given by ? ?! U^? (3.73) U^ = exp h ?iH^r?t ~ i ? (3.74) = exp h ?i~ ?(~c) 2 i ? (3.75) where ~c is a \control vector" deflned as ~c = '0 2? 01?Irfx; '0 2? 01?Irfy;? ?Irfz 2 @E01 @I ? ?t ~ : (3.76) Comparing with the rotation operator R[?] = exp h ?i?2 n^ ? ~ i (3.77) that rotates the system by angle ? in the direction of n^, one sees that U^ rotates the qubit by angle j~cj about the ~c axis. For example, if a pulsed bias current ~c = (0; 0; ?) is applied, this operation rotates the qubit 180? about the z-axis; this is a ?z operation or a phase qubit. Using ~c makes it easier to understand gate operations and follow the motion of the qubit on the Bloch sphere in applications such as state tomography [65]. 62 Chapter 4 Qubit fabrication, experimental techniques and analysis This chapter describes how I made aluminum dc SQUID phase qubits and the apparatus and techniques I used to measure them. 4.1 Fabrication recipe for aluminum dc SQUID phase qubits In this section I explain the photolithographic technique that I used to make aluminum dc SQUID phase qubits, including device AL1 (see Fig. 4.1 to 4.5). I made device AL1 in our laboratory using photolithography followed by double-angle evaporation of approximately 50 nm thick Al fllms on an oxidized Si substrate. The oxide was thermally grown with a thickness of about 1.5 ?m and the wafer was P-doped (boron) with an orientation (100) and a resistivity of about 10 ?cm. The 40 ?m x 2 ?m Al/AlOx/Al qubit junction had a zero-fleld critical current I01 = 21.28 ?A and the device had a single-turn square loop with a 3 ?m line-width and a 300 ?m diameter (see Fig. 4.1). The reason why I used photolithography rather than e-beam lithography was that photolithography could generate large-area junctions easily and more e?ciently than e-beam lithography. 63 100 ?m J1 J2 L1 If ICPB Figure 4.1: Photo of an Al/AlOx/Al dc SQUID phase qubit made on a Si substrate with thermally grown SiO2. The qubit junction J1 was designed to be coupled to a Cooper pair box (CPB) on the left. The device was made using photolithography and double-angle deposition. 64 Al AlOx Al bridgephotoresist Josephson junction l SiO2 Si 1st deposition 2nd deposition 2nd deposition 1st deposition Figure 4.2: Schematic of double angle deposition after photolithography. 65 10 ?m Figure 4.3: Photograph of qubit junction (left) with a coupled Cooper pair box (right) in device AL1. 66 4.1.1 Photolithography: Introduction Lithography is a printing process that uses chemicals to create an image. In photolithography, a chemical \photoresist" is used to coat a at substrate. The photoresist is exposed to UV light through a photomask, and then immersed in a developer to create the desired pattern on the substrate. Processes in photolithog- raphy are similar to conventional photographic fllm processing. There are two basic types of photoresist: positive and negative. Exposed positive photoresist will be washed away by the developer, while exposed negative photoresist will remain on the substrate after developing, and the unexposed area will be removed. \Novolac" with DNQ (diazonapthoquinone) photosensitizers is one of the most common types of photoresists. Novolac resin is soluble in water-based base solutions such as TMAH (Tetramethylammonium hydroxide, (CH3)4NOH) or NaOH, but when mixed with DNQ in the correct ratio, the resulting resist solution is not soluble in base solutions. When exposed to UV, DNQ is destroyed and the photoresist regains its solubility to bases which can serve as a developer. Modern projection photolithography enables patterning of sub-micron fea- tures. Typical stepper machines use optical lenses to scale down the mask patterns for projection onto a photoresist layer. In contrast, I used contact lithography, where the photomask makes direct contact with the photoresist layer. The following recipe was what I used for making aluminum SQUID qubits including device AL1. 4.1.2 Preparation for photolithography Before beginning fabrication of a new chip, there were several things that needed to be taken care of. I flrst made sure that I had all the materials, chemicals and supplies I needed, including; 3 inch SiO2/Si wafer with thermally grown SiO2 with 1.5 ?m thickness. LOR30B undercut resist [66] 67 20 ?m overlap junction Figure 4.4: A zoomed-in photograph of qubit junction in device AL1 from Fig. 4.3 The junction is in the center overlapped section. 68 20 ?m overlap junction Figure 4.5: Photograph of isolation junction in device AL1. 69 S1813 photoresist [66] Photomask [67] Microchem MF319 (developer for LOR30B and S1813) [66] Microchem PG resist stripper [66] Al shot (purity 99.999% or above) [68] Two spiral tungsten boats for evapoarating aluminum [69]. Creating a photomask I designed the device patterns for the photomask using the 2-D circuit CAD pro- gram called ICED [70]. The photomasks were then printed at the U.C. Berkeley Microlab [67]. I used soda-lime glass plates with chromium fllm patterns. For AL1, I used a 2.5-inch mask. Cleaning the wafer I soaked the wafer in RBS35 detergent [71] for about 30 min and rinsed in DI water. I then sprayed acetone, methanol and isopropanol on the wafer for a minute each and rinsed in DI water. It is important to blow dry the wafer with (high purity, flltered) nitrogen gas to remove any water remaining on the wafer. I used an O2 plasma etch at 400 mtorr, 200 W for 30 sec as a flnal degreasing step. I also used the O3 etch station in the cleanroom in the Kim Engineering building for cleaning solvent residue. Cleaning the Mask Before photolithography, I used the following procedure to clean the masks: 1. Spray acetone, methanol and isopropanol on the mask for a minute each and rinse in DI water. Blow dry the mask. 2. Use O3 etch for 10 min to remove solvent residue. 70 3. Use dry nitrogen gas to blow dry the mask. 4.1.3 Spinning and baking photoresist I coated the wafer with photoresist by pouring a small quantity of resist solu- tion onto the wafer when it was mounted at rest on the spinner. The wafer is then spun at a high RPM to produce a thin layer. I then baked the wafer on a hot plate to harden the resist and remove the resist solvent. After baking, the photoresist remains as a thin glass-like layer on the substrate. The baking temperature and baking time determines the solubility of the pho- toresist in the developer. Baking at high temperature or baking for a long time makes the resist more hardened than baking at low temperature for a short time. This initial baking process is called a \soft bake". If the resist is to be used in a process that involves etching (for example, of SiO2 on Si) then a \hard bake" is done after soft-baking. A hard bake literally hardens the resist by increasing cross-linking in the polymer so that the resist can survive chemical etching. The developing speed of baked resist also depends on the temperature of the developer. For example, I baked S1813 at 110?C for 1 min when the cleanroom was at 85 ?F (the temperature controller in the cleanroom was broken) and developing required 43 sec. Later with the cleanroom at 65 ?F, I used the same recipe and I ended up overdeveloping the resist. The flnal recipe below was based on a room temperature of 70 ?F. To make a junction, I used two resist layers to create a suspended bridge over an undercut pattern. If there is too much undercut, the top bridge layer may collapse, and if the undercut is too small, junctions can not be formed using double angle evaporation. Developing time controls the amount of undercut and for some resists, developing for 1 more second can make a big difierence. It can be very sensitive. To achieve a reproducible undercut, it is important to bake the resist at 71 the right temperature under the same conditions each time. I used LOR30B [66] as an undercut resist. LOR30B is the thickest of all resists in the LOR (Lift-Ofi Resist) series. We put Shipley 1813 (S1813) [66] photosensitive resist on top for patterning. Spin and bake of LOR30B To spin a layer of LOR30B on a substrate, I used the following method: 1. Pour LOR30B in a small 10 mL beaker. For 3 inch wafers, 4 mL of LOR30B is enough to cover the wafer. 2. Slowly pour LOR30B resist on the center of a wafer spinning at 30 RPM. Once the resist covers the wafer, increase the speed to 3300 RPM within 5 sec. The total spin time should be 45 sec. 3. Bake the wafer at 150 C? for 5 min on a hot plate. The LOR30B photoresist is a dense liquid and tends to harden quickly after expo- sure to air. I prepare the LOR30B about 10 to 20 minutes before spinning by letting it warm up to room temperature. LOR30B does not dissolve in common solvents such as acetone, isopropanol, methanol, ethanol or water. Those solvents (as well as water) tend to harden LOR30B. To remove LOR30B from a substrate, I use MF319 [66] or PG [66]. Spin and bake a layer of S1813 I used a similar method for spinning the S1813 resist on top of the coated LOR30 resist: 1. Spin the resist at 4500 RPM for 45 sec. Start spinning at low speed and increase to 4500 RPM to make the resist layer have uniform thickness. 72 2. Bake at 110 ?C for 1 min on a hot plate. Technical sheets for resists are available from the manufacturer [66]. These sheets contain the recommended baking time, baking temperature and developing time for difierent procedures. 4.1.4 Expose and develop photoresist I used the following procedure for exposing photoresist bilayers of LOR30B and S1813 on the Karl-Suss MJB3 [72] contact mask aligner at \Fablab" in the Kim Engineering Building.. 1. Set the UV exposure to 8 mW/cm2 for 10 sec. 2. Develop for 30 sec in MF319 with agitation and, immediately after, rinse in DI water (the developing time can vary according to the conditions). I then check with an optical microscope with a red fllter to see if the pattern has developed properly. If it has not developed, I develop for 5 sec and recheck on the optical microscope. I repeated 5 second-developing steps until the pattern developed to the desired undercut. 4.1.5 Deposition of aluminum I used the following procedure for depositing Al/AlOx/Al fllms for qubit junc- tions (see Fig. 4.2): 1. Double angle (45?) deposition. I used the cryopumped deposition chamber in room 0219 in the CSR, and put Al basket boats on electrodes #1 and #3. The sample was mounted on the rotating stage on the ion mill top. Before pumping, I set the sample stage to 45? to the vertical for each AL source and make a mark on the control knob so that I would be able to correctly tilt the sample for each evaporation. 73 2. Rough pumping. I purged the O2 line for 1 min by owing oxygen through the line at 1000 mTorr. I then closed the O2 valve and rough pumped the chamber to 500 mTorr. 3. Once the pressure reached 500 mTorr, I started cryopumping by opening \hivac" valve. I continued pumping until the chamber pressure was below 10?6 Torr. 4. I rotated the sample holder to 45?, closed the evaporation shutter and slowly heated the boat until Al was evaporating at 1 nm/s. When Al started evap- orating, the evaporation rate increased and I was able to observe sudden dis- turbance in pressure through the ion gauge. I observed the evaporation rate using the crystal thickness monitor. I deposited 50 nm of Al for the flrst layer, waited 5 to 10 minutes and then closed the hivac valve and oxidized the deposited Al fllm in 18 Torr of O2 for 10 min. 5. I closed the O2 valve, pumped the chamber to 10?6 Torr, rotated the sample holder to -45? and deposited 50 nm of Al to form the second Al layer. With these oxidation parameters, I got a critical current density of 22 ?A/80 ?m2 = 27 A/cm2 and a capacitance of about 4 pF/80 ?m2 = 50 fF/(?m)2. The oxidation step is critical because the thickness of the oxide determines the critical current density. I usually waited for 5 to 10 minutes to allow time for the substrate to cool before opening the O2 valve, but I did not make a systematic study on how the temperature of the substrate afiects the oxide. This temperature may also be important for determining defect density and ultimately the coherence time, so a systematic investigation of the grouwth procedure is of considerable current interest. 74 4.1.6 Lift-ofi After deposition, I did a lift ofi to remove Al from everywhere it was deposited on undeveloped resist. I put the Al-deposited wafer in PG remover [66] and heated it to 60 ?C on a hot plate. After 1 to 2 hours, I replaced the remover with a fresh PG remover and resumed lift-ofi for another 1 to 2 hours. 4.2 Fabrication recipe for a Cooper pair box: E-beam lithography E-beam lithography provides an easy way to fabricate sub-micron devices. I used the following recipe to fabricate a Cooper pair box. The Cooper pair box was deposited on a dc SQUID phase qubit that I had built using photolithography. 4.2.1 Preparation The materials and chemicals I needed for e-beam lithography are as follows: 950 PMMA C2 [66] Copolymer (MMA) EL11 or EL9. EL9 is thinner [66]. PG remover [66] or acetone Al shot (purity 99.999% or above) [68] Two spiral tungsten boats [69] Cleaning the wafer To clean the wafer after flnishing lifting ofi photoresist to make a phase qubit, I used nitrogen gas to blow dry the surface. 4.2.2 Spinning resist I spun resist immediately after cleaning the wafer. Depending on the desired pattern, PMMA and MMA of difierent coating thicknesses can be used. For \nano" 75 fabrication, thin resists are suitable. For my work, I used the thinnest PMMA and MMA so that I could build junctions of about ? 100 nm ?100 nm. The procedure I used was as follows: 1. Spin copolymer (MMA) on the chip at 4000 RPM for 45 sec. 2. Bake the chip at 150 ?C for 10 min on a hot plate. During this step, I covered the substrate with a beaker because MMA collects dust easily. 3. Spin PMMA at 6000 RPM (or maximum speed of the spinner) for 45 sec. 4. Bake at 150 ?C for 10 min on a hot plate. The technical sheets of PMMA and MMA can be downloaded from the Mi- croChem website [66] which provides information on the parameters for spinning speed versus thickness. 4.2.3 E-beam writing DesignCad flle and set-up I used a Philips XL30 SEM located in Physics 2215 for e-beam writing. The SEM has the Nabity e-beam lithography system - NPGS [73]. For pattern design- ing, I used DesignCAD [74]. The pattern I used for a Cooper pair box was saved in c:/pg/pat/HPCP1.dc2 on the SEM writing computer. For lines with width less than 100 nm, I used a line dose of 0.5 nC/cm to 2 nC/cm. For flne lines that were close together, I decreased the dose. There is not a universal dose for a given substrate and line width, so I had to use trial and error. For patterns larger than 1 ?m, I used an area dose of 170 ?C/cm2 to 200 ?C/cm2. The recommended e-beam setting for writing small features is 30 kV with spot size 1. For large patterns, a larger spot size can be used. 76 (a) (b) 20 ?m 5 ?m Figure 4.6: (a) A Cooper pair box coupled to a Josephson junction phase qubit. The phase qubit was fabricated by photolithography and the Cooper pair box was made by e-beam lithography. They are coupled through an interdigitated capacitor shown at the center. (b) The interdigitated capacitor of the Cooper pair box. 77 (a) (b) 1 ?m ultra-small junction interdigitated capacitor ultra-small junction Figure 4.7: (a) Ultra-small junction of Cooper pair box fabricated by e-beam litho- graphy. (b) Diagram showing layout of ultra-small junction. 78 Alignment For putting the Cooper pair box on a phase qubit, alignment with underlying layers is important. However, Al is not easy to see with the SEM, especially on an SiO2 coated surface. I found it was much easier to see the Al pattern if I flrst charged up the designated writing area with the beam set to 3 kV acceleration voltage, spot size 2. I then re-imaged with a 30 kV beam and spot size 1. Because of charging, the area will look brighter than other parts of the chip. However, this method also caused overexposure in some of the e-beam patterns I created. I also found it useful to make a scratch near the place I was writing. 4.2.4 Develop To develop the pattern, I used 1 part of MIBK (Methyl Isobutyl Ketone) diluted with 3 parts IPA (Isopropyl alcohol) developer (conventionally known as \MIBK:IPA 1:3") [66] for PMMA and MMA. I dipped the chip in the \MIBK:IPA 1:3" solution and mildly agitated horizontally in line with the junction. Afterward, I dipped the chip in IPA to provide the undercut. I then dried the chip using nitrogen. MIBK is a more aggressive developer than IPA. The developing time depends on dose and I typically developed for 30 sec for PMMA and 60 sec for MMA. I used an optical microscope to check how much undercut was created and developed the MMA more in IPA if there was not enough undercut. 4.2.5 Ion milling and Al deposition Before depositing Al for the Cooper pair box (see sec. 4.1), I had to use an ion-mill to remove AlOx from the connection pads in the phase qubit so that I could make good electrical contact. I did this step just before doing the double-angle deposition of Al in the same chamber. 79 1. The deposition method is the same as discussed in sec. 4.1. The Ar ion gun [75, 76, 77] is mounted on the chamber top. I made sure that the sample holder was under the center of the ion mill chamber, set the sample stage facing each AL source at 45? to the vertical and made a mark. Also I made a mark when the sample stage faced upward to the ion mill (180?). After marking, I set the sample stage facing downward, toward the deposition electrodes (0?). 2. I purged the O2 line for 1 min at a pressure of 1000 mTorr. Then I closed the O2 valve on the top of the evaporation chamber; the Ar and O2 share the same gas line to the evaporation chamber. 3. I purged the Ar line for 1 min. 4. I pumped the chamber below 10?6 Torr and degassed the ion gauge for more than 10 min at 10?6 torr before measuring the pressure. 5. I then throttled the cryopump valve (half open). Using the electronic Ar/O2 valve, I set the deposition chamber pressure at 3 ? 10?4 Torr by adjusting the Ar ow. 6. To ion mill the sample, I turned on the Ar ionization source and beam. I set the acceleration voltage to 100 V, the discharge voltage to 40 V, and the beam voltage to 600 V. I adjusted the cathode current until the beam current was 5 mA, and set the neutralizer current the same as the cathode current. 7. With a beam current of 5 mA, I turned the sample stage so that the sample holder faced the ion mill and milled for 1 min. From Ref. [75, 76, 77], this should result in the removal of about 1 nm of Al2O3. 8. I closed the electronic valve controlling the Ar gas and opened the hivac valve. I next switched the gas line from Ar to O2 and pumped down the chamber to 10?6 Torr. 80 Table 4.1: Ar Ion beam etching rate at normal incidence for a beam current density of 1.0 mA/cm2 and 500 V acceleration voltage as given by refs. [72 - 74]. Material Etching rate ( nm /min) Al2O3 8.3 Al (bulk) 30 SiO2 (crystal) 33 SiO2 (evaporated fllm) 28 Shipley AZ1350 photo resist 20 9. I rotated the sample holder to 45? facing the electrodes and made sure the shutter was closed. I turned on the electrode and slowly increased the current in the electrode to preheat the Al boat. When Al started evaporating, I opened the shutter, deposited about 50 nm of flrst Al at 1 nm/s and closed the shutter. I turned ofi the power of the electrode and oxidized the Al fllm in 18 Torr O2 for 10 min. 10. I then closed the valve for the oxygen and rough pumped the chamber to 500 mTorr using the mechanical pump and opened the hivac to pump the chamber to 10?6 Torr. 11. I rotated the sample holder to -45? and repeated the same procedure used in step 9 to deposit about 50 nm of Al at 1 nm/s. 4.2.6 Lift-ofi I used acetone for lift-ofi of the e-beam patterns. I put the chip in a pyrex beaker fllled with acetone at room temperature. The lift-ofi takes 3 to 4 hours. I checked the pattern using the optical microscope to see if lift-ofi was successful. If a small part did not lift ofi, I tried sonication for 10 to 20 sec. Since too much 81 Table 4.2: Parameters of some dc SQUID phase qubits measured by the UMD group. NB1, NB2 and NBG were made by Hypres, Inc. [78]. Index 1 indicates a qubit junction and index 2 indicates an isolation junction. Device A1(?m2) I01(?A) A2(?m2) I02(?A) L1(nH) AL1 80 21.2 40 9.5 1.2 NBG [79] 120 23 60 3.8 4.5 DS1(NB1) [23] 100 107.9 (33.8) 49 51.7 (4.8) 3.5 DS2(NB2) [23] 100 24, 20 49 3, 6 3.4 AL2 [80] 16 1.23 160 9.19 1.05 sonication can ruin the pattern, the sonication should be performed carefully and only for few seconds. 4.3 Table of dc SQUID phase qubits measured in UMD SQC group Table 4.2 summarizes parameters of some of the dc SQUID phase qubits mea- sured in our lab as of July 2007. In this table, A1 is the area of the qubit junction, A2 is the area of the isolation junction, I01 and I02 are the critical currents of the qubit and isolation junction, respectively, and L1 is the inductance on the qubit junction arm of the SQUID loop. 4.4 Dilution refrigerator setup Our phase qubits must be cooled to milli-Kelvin temperatures to operate prop- erly [27]. I used an Oxford Instruments Kelvinox 25 dilution refrigerator [81] in a shielded room in the basement of the Physics building. With wiring attached, the refrigerator had a base temperature of 80 to 100 mK. Figures 4.8 to 4.12 show photographs of the refrigerator and its wiring. For measuring qubits, the Kelvinox 25 is wired with six UT34 coaxes from the 82 LC filter#5 #2 RC filter Sample mount Cu powder filters Figure 4.8: Oxford Instruments Kelvinox 25 dilution refrigerator and sample mount. 83 10 cm Still Condenser line Continuous heat exchanger Step heat exchanger Mixing chamber Figure 4.9: Oxford Instruments Kelvinox 25 dilution refrigerator unit. 84 10 cm condenser pumping line vacuum -can valve Wiring box Figure 4.10: 300 K ange showing ports for wiring, gas and vacuum. The stainless steel box on the right side accommodates six coaxial cables as well as twisted-pair manganin lines. 85 Figure 4.11: Wiring on the Oxford Instruments Kelvinox 25 dilution refrigerator. All coaxial lines are thermally anchored at the 4 K ange and still. 86 Figure 4.12: Photograph of the heat exchanger and still showing where the coaxial lines are thermally anchored. 87 If I V I? 4 ? Still (0.7 K) Mixing chamber (0.1 K) Figure 4.13: Wiring schematic for Oxford Instruments Kelvinox 25 dilution refriger- ator. If is the ux bias current, I is the current bias, V is the voltage measurement lead and I? is the microwave line. Coaxial lines are thermally anchored at the 4 K stage and the still. The ux bias line If uses a superconducting Nb coax from 4 K to the mixing chamber and has an LC low pass fllter. The current bias line I has an RC low pass fllter. For V and I, Thermalcoaxes are used from the still to the mixing chamber where the coaxes are connected to copper power fllters. 88 (a) (b) (c) 10 cm 5 cm 5 cm Coax #6 Figure 4.14: Copper powder fllters and qubit sample holder box at the mixing chamber (MXC) of the refrigerator. (a) Copper power fllters at MXC. (b) Side view of Aluminum box - sample holder mounted on the MXC. The sample holder aluminum box is located below the copper power fllters. (c) Opposite view of the aluminum sample holder box. 89 0.7? 0.7? 0.4080? 0.1965? 0.5575? 0.3380? 0.25? Figure 4.15: Aluminum sample holder shown without the top. The box thickness is 0.099". 90 300 K ange to the still (see Fig. 4.10 to Fig. 4.12). The wiring schematic for the refrigerator is shown in Fig. 4.13. All six coaxes are thermally grounded at the still (see Fig. 4.12) and connect to copper power fllters at the mixing chamber, except for Coax #6 which is used to supply microwave current to the devices. I used only #1, #4, #5 and #6 for the measurements in this thesis. Coax #1 is made from stainless steel semi-rigid UT34 coaxial cable [25]; I used this line for the current bias. At the still (600 mK), Coax #1 is connected through a low-pass RC fllter with a cutofi frequency of about 300 MHz (see Fig. 4.13). After the RC fllter, Coax #1 is wired with a Thermocoaxr cable [82], which works as a microwave fllter [83, 84], and connects to a copper powder fllter mounted on the mixing chamber. I used type 1 NcAc Thermocoaxr [82] which has a NiCr center wire, a stainless steel outer conductor and an MgO dielectric layer. Coax #4 is used as a voltage detection line and from 300 K to the still, it is made from UT34. From the still to the mixing chamber, I used a Thermocoaxr cable. I used Coax #5 as a ux bias line. Coax #5 is wired with a UT34 coaxial cable from 300 K to 4 K. It is connected to a low-pass LC fllter with a cutofi frequency of 100 MHz at the 4 K stage. From the 4 K ange to the copper powder fllters in the mixing chamber, I used a superconducting Nb coax. Coax #6 uses UT34 cable continuously all the way to the sample box. The electrical characteristics of the UT34 coax, the Thermocoaxr and the Nb coax are discussed in H. Xu?s thesis [25]. At the mixing chamber, the copper powder fllters are connected to the center pins of the coaxes, except for coax #6. After the copper powder fllters, the copper wires from the fllters have Microstrip connectors that plug into the aluminum sample box. Coax #6 passes straight through the top of the aluminum sample box (see Fig. 4.14), to serve as a microwave antenna. The qubit sample was mounted in a closed superconducting aluminum box to shield out magnetic flelds (see Fig. 4.15). The sample was attached using GE varnish on the bottom and a Ag paint along the 91 Table 4.3: Commercial electronics used in the experiment. Purpose model number Current bias I Agilent 33120A Arbitrary waveform generator Flux bias If Agilent 33120A Arbitrary waveform generator Microwaves HP (Agilent) 83731B Synthesized signal generator Gating microwave SRS DG535 Pulse generator Frequency counter SRS SR620 Frequency counter Calibration Voltage amp SRS SR560 Low-noise voltage amp sides for good thermal conduction. In addition, the refrigerator was surrounded by a copper radiation shield, a stainless steel vacuum can, an aluminum dewar, a room-temperature mu-metal shield and flnally enclosed in an rf-shielded room. 4.5 Measurements and analysis In my experiments, the qubit typically has to be initialized to a unique state, the state is then manipulated with microwaves, and measured. In this section, I discuss the initialization procedure and my measurement technique. Figure 4.16 shows a schematic of the measurement setup I used. Tables 4.3 and 4.4 summarize the commercial and the homemade electronics that I used for my measurements. 4.5.1 Initialization of the ux state of the dc SQUID phase qubit The ux state of the SQUID needs to be initialized before each measurement. The problem is that the dc SQUID phase qubit has multiple ux states due to a high value of fl = 2L(I01 + I02)='0 [85, 86]. Each ux state has difierent energy levels so I need to choose a unique ux state each time. To do this, I used a ux shaking 92 If I V I? Refrigerator ? ? ? ? pipi Screen room flux ramp bias ramp microwave source m Dt start stop Figure 4.16: Schematic of the measurement setup. 93 Table 4.4: List of homemade electronics used in the experiment. Diagrams are in Ref. [25]. Purpose chipset Switching voltage detection Schmitt trigger CLC420 Unity bufier after Schmitt trigger LMH6624 1st stage switching voltage amplifler JFET 2SK117 2nd stage switching voltage amplifler AD797 or AD829 Unity gain bufier AMP03 technique [86]. This technique involves applying a 20 to 30 kHz sinusoidal ux with a carefully chosen amplitude. Difierent ux states can be chosen by applying an appropriate dc ux ofiset for the sinusoidal ux. The detailed procedure is described in Ref. [86]. After shaking for about 50 ux oscillations, I was able to place the SQUID in the desired ux state with a 99 % or greater probability. S. K. Dutta?s thesis also has discussion on ux shaking technique [23]. 4.5.2 Biasing the qubit junction To bias the qubit junction in the dc SQUID phase qubit [86], I used a simulta- neous current and ux ramp [47] generated by two function generators. When the current bias ramp (I) started, its function generator sent out a TTL signal that was then used to trigger the second function generator to start the ux ramp (If ). The idea of simultaneous biasing is to arrange the two ramps so that there is no change in the current going through the isolation junction, i.e. ?I2 = 0, by keeping the ratio I=If ? ?M=L where L is the total loop inductance of the SQUID and M is the mutual inductance between the SQUID and the ux line. Figure 4.18 shows typical waveforms for the ramps. I used Agilent 33120A function generators (see Fig. Ta- ble 4.3 and Fig. 4.16) and a detailed discussion of the double ramping procedure is 94 given in Ref. [23]. 4.5.3 Measurement of the qubit state via the escape rate I read out the qubit state by measuring the total escape rate. Fig. 4.17 shows a sketch of the potential energy and energy levels of a Josephson junction qubit in a metastable well of the washboard potential. Because of the shape of the potential, higher-energy states are more likely to escape by tunneling than lower- energy states; each successive level tunnels about 500 times faster. In addition, when the current through the junction is increased, the tilt of the washboard potential increases and the potential barrier is lowered, causing the tunneling rates from all of the states to increase. The escape event is analogous to radioactive decay; the decay is exponential. What we measure in the experiment is the total escape rate, which is give by ?tot = ?0?0 + ?1?1 + ?2?2 + ?3?3::: = X i ?i?i (4.1) where ?i and ?i are the occupation probability (or population) and escape rate of level i. Since the escape rate increases by two or three orders of magnitude when i is increased by 1, the total escape rate is very sensitive to even small populations in the upper levels. For an escape rate measurement, the qubit junction current is ramped linearly with time by simultaneously ramping the ux and bias current as described above, and the voltage across the SQUID is monitored. When the qubit junction tunnels, a relatively large voltage (2?=e ? 400?V for an Al Josephson junction) appears across the SQUID bias. This voltage is amplifled to trigger a low-noise Schmitt trigger. The SR620 timer is used to measure the time interval between the start of the ramp (at tI in Fig. 4.18) and the appearance of the switching voltage (at tF in 95 Fig. 4.18). To detect the the switching voltage, I used a homemade Schmitt trigger [25] (see Table. 4.4 and Fig. 4.14) followed by a unity bufier. I set the threshold voltage for the Schmitt trigger so that it triggers at the point where the rise time of the switching voltage changes most rapidly in time. After each switching event, the time interval tF - tI is recorded on a computer. This switching measurement is repeated N ? 105 times at a repetition rate of 950 Hz. The resulting switching times are used to construct a histogram of switching events as a function of the switching time interval. The number of switching events h(ti) at time ti in interval ?t is converted to the total escape rate ? (ti) at time ti using [23] ? (ti) = 1?t ln ? N (ti) N (ti+1) ? = 1?t ln " P j?i h (tj)P j?i+1 h (tj) # : (4.2) where ?t is the time bin, typically of order 1 ns, and N(ti) = P j?i h(tj) is the number of measured switching events where the switching occurred after time time ti. The uncertainty in the escape rate is [23] ?i (ti) = 1 ?t s? 1 N (ti) ?2 h (ti) + ? 1 N (ti+1) ? 1 N (ti) ?2 N (ti+1) : (4.3) The timer starts when it receives a TTL trigger signal from the current ramp. However, there could be an ofiset with respect to the time when the ramp starts because the TTL signal has a flnite rise time. This would lead to an efiective ofiset in current. To calibrate the current at any time on the ramp, I placed 1 k? resistors at 300 K on the current bias and the ux bias lines. The voltage across the resistor was amplifled using a commercial low-noise amplifler SR560 and I then measured the voltage across the resistors versus time to get a calibration curve for the current ramps. For example, the calibration that I got for my measurements of AL1 was I(t) = a? t? b (4.4) 96 where a = 3.482074 ?10?2 A/s and b = 2.493 ?A. With the range calibrated as a function of time, I could readily convert ?(t) to ?(I). The measured total escape rate ?(I) then can be flt to the calculated total escape rate. For the escape rate flt in Chapter 5, I assumed the populations from each levels was thermal and obtained the escape rates ?n from each level n using ?n = (7:2Ns)n+1=2 !p2? exp [?7:2Ns + f n ? (Ns)] (4.5) as shown in Chapter 2. Then I was able to construct a calculated total escape rate according to Eq. 4.1. This requires just two fltting parameters, T1 and temperature T. 4.5.4 Spectroscopy and T?2 When the microwaves are in resonance with an energy level spacing, mi- crowaves can drive the qubit junction from one state to another state, producing an enhancement in the total escape rate. I measured the resonance peaks while sweeping the bias current; As I sweep the bias current, the energy level spacings decrease and resonance occur at multiple points on the current axis, wherever an energy level spacing is resonant with the microwaves. Figure 4.19 shows microwave resonance peaks in the escape rate of AL1 mea- sured at 80 mK. Here I applied a 6.9 GHz microwave drive. Two resonance peaks appear in the escape rate, corresponding to the j0i to j1i (at about I = 21.61 ?A) and j1i to j2i (at about I = 21.67 ?A) transitions. For spectroscopy, I used microwaves of relatively low power so the bare Rabi frequency satisfles ?0 ? (T1T2)?1=2. For spectroscopy, I used an HP (Agilent) 83731B Synthesized signal generator for microwave source (see Fig. 4.14). I flrst flxed the microwave frequency and power, and then swept the qubit junction current. I then measured the time driving 97 the ramp at which the junction switched. After that, I did the same measurement without microwaves. I typically repeated this process 105 times; with and without microwaves were measured alternatively in the time sequence. After I flnished the escape rate measurements for one microwave frequency, I changed the microwave frequency and repeated the whole procedure. For devices AL1 and NBG, I measured the spectra from about 6 GHz to 8 GHz. For each escape rate curve, I obtained the microwave enhancement, ??=? given by ?? ? = ?? ? ?b ?b (4.6) where ?? is the escape rate with microwaves and ?b is the background escape rate without microwaves. Figure 4.20 shows the microwave enhancement ??=? versus current in device AL1 when 6.6 GHz microwaves are applied. This curve was ex- tracted from the escape rate shown in Fig. 4.19. By fltting each peak in ??=? to a Lorentzian, I found the resonant current at which each peak was centered and the plotted the spectrum as points of microwave frequency versus resonance cur- rent. Figure 4.21 shows the Lorentzian fltting for the resonance peak of j0i ! j1i transition in Fig. 4.20. Dots are the data and the dashed line is the Lorentzian flt. From the fltting, I obtained the center of the peak I = 22.017 ?A with full width half maximum 3.5 nA. I performed the Lorentzian fltting to ??=? resonance peaks for each frequency and obtained the spectrum given as microwave frequency versus current. A plot of a spectrum is shown in Fig. 5.9 in Chapter 5. The measured spectrum can be flt to the energy levels of a current biased junction. The calculated energy level spacing between level jni and jn+ 1i is (see Chapter 2) !n;n+1 = !p fn! (Ns) : (4.7) 98 where fn! is a correction term and Ns is the number of energy levels in the well (see Eq. 2.38). The free parameters are the critical current I01 of the qubit junction and the qubit junction capacitance C1. Appendix A shows the MATLAB code that I used to solve Schro?dinger equation for a single Josephson junction to obtain the energy level spacings. The half-width at half maximum ?IHWHM of the j0i to j1i resonance peaks from the Lorentzian flts were used to obtain T ?2 through the equation : T ?2 ? 1 2??fHWHM = dI df 1 2??IHWHM : (4.8) where dI/df can be obtained by fltting from the measured spectrum. 4.5.5 Measurement of relaxation To obtain T1 from a relaxation measurement, I used the following procedure. First I took switching data for about 104 events with microwaves and used this to construct the escape rate versus time. I then recorded the position of the j0i to j1i resonance peak on the time axis. My microwave source, an HP (Agilent) 83731B synthesized signal generator can be triggered with an external TTL pulse (see Fig. 4.14). Using a DG535 pulse generator, I programmed a trigger pulse to turn ofi the microwaves at the time where the resonance peak was centered. When the mi- crowaves were turned ofi, the qubit junction relaxed from the excited state to the ground state with the time constant T1. Figure 4.18 shows the microwave sequence for the relaxation measurement with respect to the biasing currents and the switch- ing voltage. The current bias, the ux bias and microwaves were programmed with respect to the start of the bias ramp. I used an internal clock in the SR620 frequency counter as the master clock for the all sources. (Clocks can be synchronized if they are connected by GPIB cables.) 99 The escape rate from a relaxation measurement shows a decay that can be flt to an exponential function (see Fig. 4.22) Detailed analysis of my T1 data is discussed in Chapter 5. 4.5.6 Measurement of Rabi oscillations The Rabi oscillation measurement was done similarly to the relaxation mea- surement but the microwaves were turned on at the center of the resonance peak with a high power (where the bare Rabi frequency ?0 (T1T2)?1=2). Figure 4.18 shows the measurements sequence for Rabi oscillations with respect to the biasing currents and the switching voltage. The escape rates from the Rabi oscillation mea- surements show an oscillating escape rate that I flt to a decaying oscillating function. Detailed analysis of my Rabi oscillation data is discussed in Chapter 6. 100 ?2?1?0|0> |1>|2> continuum states Figure 4.17: Metastable states in a well of the tilted washboard potential. Over the top of the barrier, the energy levels form a continuous energy band. ?0, ?1 and ?2 are the escape rates from the energy levels j0i, j1i and j2i, respectively. 101 2D/e tI tF 2D/e V I I f flux shaking signal Im Im Relaxation Rabi measurement Microwave Bias Figure 4.18: Biasing scheme for the dc SQUID phase qubit and microwave sequence. The time interval between tI that the ramping starts and tF when the qubit junction switching voltage V appears is recorded by the frequency counter SR620. 102 21.94 21.96 21.98 22 22.02 22.04 22.06 22.08 104 105 106 107 I (?A) ? (1/ s) |0> ? |1> |1> ? |2> Figure 4.19: Total escape rate vs. current for qubit AL1 at 80 mK. Dashed line is when 6.9 GHz microwaves are applied to the qubit junction and solid line is without microwaves. Two prominent peaks are seen when microwaves are applied, corresponding to j0i to j1i and j1i to j2i transitions. 103 21.94 21.96 21.98 22 22.02 22.04 22.06 22.08 ?0.5 0 0.5 1 1.5 2 2.5 3 I (?A) ?? /? Figure 4.20: Microwave enhancement of the escape rate for AL1 at 80 mK when 6.9 GHz microwaves are applied to the qubit junction. The two peaks correspond to the j0i to j1i transition at about I = 22.18 ?A and the j1i to j2i transition at I = 21.97 ?A. 104 22.01 22.012 22.014 22.016 22.018 22.02 22.022 22.024 ?0.5 0 0.5 1 1.5 2 2.5 3 3.5 I (?A) ?? /? Figure 4.21: Lorentzian flt to the microwave enhancement of the escape rate for AL1 at 80 mK when 6.9 GHz microwaves are applied to the qubit junction. The peak is the j0i to j1i transition. 105 0 20 40 60 80 106 107 t (ns) ? (1/s ) Figure 4.22: Observed relaxation in the escape rate at 80 mK in device AL1. Solid points are measured escape rates and the solid curve is the ?2 flts to Eq. 5.40. Crosses are the background escape rate (without microwaves). 106 0 5 10 15 20 25 30 35 40 0 20 40 60 t (ns) ? (1/ ?s ) 0 5 10 15 20 25 30 35 40 0 20 40 60 t (ns) ? (1/ ?s ) (a) (b) Figure 4.23: Examples of (a) Rabi oscillations in the escape rate ? in device NB1 at 25 mK for (a) rI = 1300 and (b) for rI = 450 [23]. An 7.6 GHz drive was used. 107 Chapter 5 Efiects of variable isolation on high frequency noise and T1 in the dc SQUID phase qubit 5.1 Overview In this chapter I show how the isolation of a qubit junction from the bias line can be varied in the dc SQUID phase qubit. Also, I discuss the frequency dependence of the inductive isolation and what efiects this produces on the qubit junction. In particular, I will show that the inductive network provides good isolation for low- frequency current noise on the bias leads, but signiflcant noise passes through at the plasma frequency of the isolation junction. 5.2 Variable current isolation and isolation factor In the dc SQUID phase qubit, noise current in the bias leads is shunted away from the qubit junction and instead ows through the isolation junction. Figure 5.1(a) shows a schematic of the dc SQUID phase qubit. The qubit junction J1 is connected in series with inductor L1, parasitic inductor L2, and the isolation junction J2. The bias leads are then connected across L2 and J2. At low frequencies, the capacitance C2 of the isolation junction can be neglected and the isolation junction J2 acts as an inductor with inductance LJ2. For L1 much larger than L2+ LJ2 only a small fraction of the current noise coming down the leads will reach the qubit junction (see Eq. 2.88). One interesting feature of this scheme is that the noise division ratio depends on the current going through the isolation junction. This happens because the inductance of the isolation junction varies with the current; a Josephson junction 108 M If ?a L2 L1 J1J2 C1C2 I M If ?a L2 L1 J1LJ2 C1C2 I (a) (b) I1I2 I1I2 Figure 5.1: (a) dc SQUID phase qubit circuit diagram. (b) Efiective circuit of dc SQUID phase qubit with isolation junction J2 replaced by an efiective variable inductor LJ2. 109 can be described as a current-dependent inductor. The Josephson inductance for the qubit junction J1 is LJ1 = '02? 1 I01 cos 1 = '02? 1 I01 p 1? (I1=I01)2 (5.1) = LJ1(0)p1? (I1=I01)2 (5.2) and for the isolation junction J2, LJ2 = '02? 1 I02 cos 2 = '02? 1 I02 p 1? (I2=I02)2 (5.3) = LJ2(0)p1? (I1=I02)2 (5.4) where I1 and I2 are the currents going through the qubit and isolation junction, I01 and I02 are the critical currents of the qubit and the isolation junction, and '0 = h=2e is the ux quantum. Notice also, for example, that LJ2 has the minimum value, LJ2(0) = '02?I02 (5.5) at I2 = 0. Equation 5.4 implies that we can vary LJ2 from LJ2(0) to inflnity by varying I2 from 0 to I02. This means that large in situ changes in the current isolation can be made by simply adjusting I2. It is convenient to deflne an isolation factor, rI = ?I ?I1 ?2 ; (5.6) as the ratio of the current noise power in the current bias leads (proportional to 110 the mean square current noise ?I2 in the leads) to the current noise power in the qubit junction (proportional to the mean square current noise ?I21 ). Consideration of Fig. 5.1(b) shows that r can be written: rI = ?I ?I1 ?2 = L1 + LJ1 + L2 + LJ2 LJ2 + L2 ?2 ? L1 + L2 + LJ2 LJ2 + L2 ?2 (5.7) where in the last step I assumed that LJ1 ? (L1 + LJ2 + L2). Note that r shows how much the current noise power is reduced; for example, rI = 300 means that the noise power reaching the qubit junction is reduced by a factor of 300. The bigger rI is, the more the qubit is isolated. From Eq. 5.4 and 5.7, we see that if I2 is increased, LJ2 increases and rI decreases. The qubit is most isolated when I2 = 0, since then LJ2 is a minimum so rI is a maximum. I can also deflne an isolation factor rf for the ux bias source. At low frequency, one flnds rf = ?If ?J ?2 = L1 + LJ1 + L2 + LJ2 M ?2 (5.8) where ?If is noise current from the ux source and ?J is the circulating noise current induced in the SQUID loop by ?If . Examination of Eq. 5.8 reveals that as we increase LJ2, the isolation from the ux bias source increases. This happens because increasing I2 produces a larger LJ2 which leads to a larger total efiective loop inductance. For our devices, typically L1 L2 + LJ2, and L1 M so that rI 1 and rf 1, as required for good isolation. Figure 5.2 shows an example where I have calculated rI and rf at zero frequency as a function of I2=I02. The device parameters I used for this calculation are are those of device AL1 (see Table 5.1). As we increase I2, rI varies from 1200 (max) to 0 (min) while rf varies from 9000 to inflnity. I note that rf is always at least 8 times larger than rI ; dc current noise power from the ux bias source is 8 times 111 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 I2/I02 r I 0 0.2 0.4 0.6 0.8 1 0 5000 10000 15000 I2/I02 r f (a) (b) Figure 5.2: (a) Current power isolation factor rI vs I2/I02. (b) Flux isolation factor rf vs I2/I02. rf is at least 8 times bigger than rI . 112 more reduced than dc current noise power from the current bias source. From this, one can see that bias current noise will tend to have more impact on the device than noise on the ux line. Equations 5.7 and 5.8 also imply that the efiective impedance that the current bias leads and ux bias leads present to the qubit junction are stepped up by factors of rI and rf respectively. For device AL1 with a Z0 = 50 ? and I2 = 0, the efiective resistance across the junction due to the current bias leads will be rIZ0 ? 50 k?. 5.3 Arbitrary dissipation model for the dc SQUID phase qubit Equations. 5.7 and 5.8 are valid only for current uctuations that are suf- flciently slow. However, in general, current noise occurs at all frequencies. Since the inductances and capacitances in the phase qubit have frequency-dependent im- pedances, the isolation will depend on frequency as well. The impact of high frequency noise can be understood quantitatively by con- structing a circuit model of the system. We model the noise as being produced by a source with a dissipative admittance Yeff (!) connected in parallel with the qubit junction. The admittance can have a real and imaginary part, and we can write explicitly: Yeff (!) = 1Reff (!) + i!Ceff (!) (5.9) where Reff (!) is the efiective resistance and Ceff (!) is the efiective capacitance, and both can depend on frequency. The energy relaxation time T1 discussed in Ch. 3 for the j1i state to decay to j0i is T1 = C=Re[Y (!01)] = Reff (!01)C1 (5.10) and this is directly related to the dissipation [28, 60]. If the dc SQUID phase qubit is limited by dissipation from its leads, then T1 will vary as a function of the isolation 113 Table 5.1: Parameters of dc SQUID qubit AL1 obtained form the current- ux map. Device I01 (?A) I02 (?A) L1 (pH) L2 (pH) Z0 (?) T (K) AL1 21.401 9.445 1236 5 50 0.1 because Reff ? rIZ0 will vary. 5.3.1 Calculation of efiective admittance and T1 To calculate Yeff , I divide the dc SQUID phase qubit into two parts (see Fig. 5.3(a)). (i) One part is any intrinsic resistance R1 from the qubit junction itself, re- sistance R2 from the isolation junction, inductance LJ2 and capacitance C2 of the isolation junction, the stray inductance L2 on the isolation junction branch, the inductance L1 of the SQUID loop and the impedance Z0 of the current bias leads. (ii) The other part is the qubit junction J1 with the junction capacitance C1. I found it easier to flrst calculate the efiective impedance Zeff of the flrst part and then get the admittance Yeff using Yeff = 1Zeff : (5.11) Zeff includes all the circuit elements inside the dashed box in Fig. 5.3(a) as well as Z0, which is the impedance of the current bias leads. I note that intrinsic dissipation associated with the qubit junction R1 is included in Zeff but C1 is not included. To proceed, I flnd the impedance Ziso of the isolation network as viewed from the qubit junction. Ziso includes the isolation junction, L1, L2 and Z0 but not R1, 114 I I Z L R J C R J C I' I' C R J C (a) (b) L n 0 2 2 2 2 1 1 1 1 1 1 n effeff Figure 5.3: (a) Schematic of dc SQUID qubit. The isolation network and lead impedance Z0 are inside the dashed box. (b) Equivalent circuit for the isolation network and leads used to calculate the efiective admittance Yeff (!) = 1=Reff (!)+ i!Ceff (!). Notice that the current bias and current noise source must also be replaced by efiective sources I 0 and I 0n. 115 and is given by Ziso = i!L1 + Z2RLZ2 +RL : (5.12) where Z2 = i!L2 + ZJ2 (5.13) is the isolation branch impedance. In this expression, ZJ2 is the impedance of the isolation junction: ZJ2 = iwC2 + 1iwLJ2 ??1 (5.14) = i!LJ21? !2LJ2C2 : (5.15) where for simplicity I have neglected R2. Here and elsewhere in this thesis ! is the angular frequency and I use index 1 for the qubit and 2 for the isolation junction. In Eq. 5.15, I have treated the isolation junction as a classical parallel LCR circuit with a resonance at the isolation junction plasma frequency !p2, !p2 = (LJ2C2)?1=2: (5.16) Substituting Eq. 5.15 into Eq. 5.13, and Eq. 5.13 into Eq. 5.12, I obtain Ziso(!) = i!L1 + i! ? Z0[(L2 + LJ2)? (!=!p2)2L2] Z0(1? (!=!p2)2) + i![(L2 + LJ2)? (!=!p2)2L2] ? : (5.17) I can now write Riso, the real part of Ziso as Riso(!) = ! 2L21 Z0 + Z0[(L1 + L2 + LJ2)? (!=!p2)2(L1 + L2)]2 [(L2 + LJ2)? (!=!p2)2L2]2 : (5.18) The real part of 1/Zeff is the real part of Yeff which is obtained by adding 1=R1 116 and 1=Riso Re(Yeff (!)) = Re 1 Zeff ? = 1R1 + 1 Riso(!) = 1Reff (!) (5.19) where Reff is the efiective resistance in parallel with the qubit junction [see Fig. 5.3(b)]. The efiective resistance Reff is a function of the isolation factor and naturally changes as we change LJ2 by applying current I2. In the low frequency limit, this dependence of Reff on LJ2 is more apparent. Taking the limit ! ! 0 in Eq. 5.18, Riso(0) becomes Riso(0) = Z0 L1 + L2 + LJ2 L2 + LJ2 ?2 = rZ0 (5.20) as expected. Thus Reff(0) is Reff(0) = 1 R1 + 1 rZ0 ??1 = rR1Z0R1 + rZ0 : (5.21) The relaxation time T1 is approximately a product of Reff and Ceff where Ceff is Ctot = C1 + Ciso (5.22) and Ciso(!) = ? ! 2A2L1 ?B[Z0A+ L1B] !4A2L21 + !2[Z0A+ L1B]2 (5.23) where A = (L2 + LJ2)? !2!2p2L2 (5.24) B = Z0(1? !2!2p2): (5.25) 117 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 R iso (k? ) 0 0.2 0.4 0.6 0.8 1 0 10 20 30 R ef f (k ? ) 0 0.2 0.4 0.6 0.8 1 0 50 100 150 I2/I02 T 1 (ns ) (a) (b) (c) Figure 5.4: Simulation of (a) Riso, (b) Reff and (c) T1 at 7 GHz for Z0 = 50 ? and C = 4 pF. The dashed line is when R1 = 150 k? and the solid line is when R1 = 6 k?. 118 For our device, Ciso is typically only a few fF while C1 ? 4 pF, so Ctot is dominated by the qubit junction capacitance C1 and T1 ? ReffC1. Figure 5.4 shows plots of Riso [Fig 5.4(a)], Reff [Fig 5.4(b)] and T1 [Fig 5.4(c)] as a function of I2 for two difierent values of R1 (dotted and solid curves). For these plots, I used the parameters for AL1 listed in Table 5.1 and assumed Z0 = 50 ? and !=2? = 7 GHz. In Fig. 5.4(a), Riso shows a clear dependence on I2. At 7 GHz, the maximum Riso = 41 k? occurs at I2/I02 = 0 when the qubit is most isolated. Riso drops to zero when I2/I02 = 1, where the qubit has the poorest isolation from the current noise. In my device the dissipation element R1 (a resistance in parallel to the qubit which represents any kind of dissipation linked to the qubit junction itself) appears to be much smaller than Riso. In this limit, Reff is dominated by R1. Fig. 5.4(b) shows Reff when R1 = 150 k ? (dotted curve) and R1 = 6 k? (solid curve). When R1 is 150 k?, Reff changes more dramatically with respect to I2, varying from 32 k? to 0. In contrast, when R1 = 6 k?, Reff stays at around 6 k? until I2/I02 approaches close to 1. Figure 5.4(c) shows T1 = ReffC1 which changes in the same fashion as Reff ; T1 with R1 = 150 k? shows a dramatic change as a function of I2. In contrast, T1 ? 24 ns with R1 = 6 k? is almost independent of I2. The main point of this simulation is that if there is a local dissipation source R1 that has a smaller resistance than that of the isolation network Riso, the dissipation process is dominated by the local dissipation source and this fact can be investigated by measuring T1 with respect to I2. 119 5.3.2 Current noise power spectrum SI1(f) and noise induced tran- sitions Dissipation sources that are at non zero temperature T generate thermal noise. The thermal current noise power spectrum produced by Yeff (!) is given by SI1(f) = 4~!e~!=kBT ? 1Re(Yeff ) (5.26) where f = !=2? is the frequency. Here SI1(f) is the conventional current noise spectra deflned for f ? 0 and satisfles hI21 i = Z 1 0 SI1(f)df (5.27) where hI21 i is the mean square current uctuation. In general, the thermal current noise power spectrum SI1(f) is a function of LJ2 and frequency !. Figure 5.5 shows simulated plots of SI1 versus frequency for difierent values of LJ2 assuming Z0 = 50 ? and R1 = 6 k? at T = 100 mK (other qubit simulation parameters are those given in Table 5.1). I note that the noise spectrum is at at low frequencies (below 1 GHz) and the current noise power for I2/I02 = 0 (dotted curve in Fig. 5.5) is lower by a factor of 4 than the poorly isolated case I2/I02 = 0.99 (thick solid curve in Fig. 5.5). The most striking feature in each curve is a large peak at f = !p2=2?. This occurs because the impedance of a parallel LC circuit (the isolation junction) is inflnite at resonance, leading to a breakdown of the isolation. Since !p2 is a function of LJ2(I2), we can tune this noise peak by varying I2. In Fig. 5.5, the noise peak moves from 20 GHz for the most isolated case (dotted curve, I2/I02=0) to ? 7 GHz for the least isolated case shown (thick solid curve, I2 = 0.99I02). Current noise at high frequencies is important because it can induce transitions 120 108 109 1010 10?28 10?27 10?26 10?25 f (Hz) S I1 (A 2 /Hz ) Figure 5.5: Plot of simulated thermal current noise power spectral density SI1(f) at 100 mK for Z0 = 50 ? and R1 = 6 k?. Dotted line is for I2/I02 = 0. Thin solid line is when I2=I02 = 0.9, the dashed line is when I2=I02 = 0.95 and thick solid line is when I2=I02 = 0.99. 121 between the energy levels of the qubit. Since the tunneling rate out of higher energy levels is much greater than the tunneling rate out of the ground state, typically about a factor of 500 times greater for each successive level [87], even a relatively small probability of occupying an excited state leads to a signiflcant enhancement in the average rate at which the system escapes. Using a two-level optical Bloch equation with the current noise being treated as a stochastic perturbation, Xu et al. [88] found that high frequency noise causes pumping from j0i to j1i at a rate which we can write here as ?+ ? SI1(f01)8e2 jh0j j1ij 2 (5.28) where SI1(f01) is the current noise spectrum at the j0i to j1i transition frequency f01 of the qubit junction. Equation 5.28 holds provided that S(f01) does not diverge faster than 1/!2, which should be a good assumption for our system (see Fig. 5.5). Transitions from j0i to j1i create an average occupancy of the flrst excited state j1i given by ?11 = ?+1=T1 + ?1 + 2?+ (5.29) where T1 is the energy relaxation rate and ?1 is the tunneling escape rate from j1i [55]. Occupancy of j1i causes an increase in the escape rate compared to that from the ground state given by G = ?tot ? ?0?0 ? ?11 ?1 ?0 (5.30) where ?tot is the measured total escape rate, ?0 is the escape rate out of the ground state, and in the last expression we have used ?1 ?0 and assumed that the population in the upper level is small compared to 1. Substituting Eq. 5.28 into Eq. 5.29, and taking the limit ?+ ? ?1 ? 1=T1 we flnd G ? T1SI1(f01)?18e2?0 jh0j j1ij 2: (5.31) 122 Since !01 can be varied by changing the current I1 through the qubit, Eq. 5.31 implies that the spectrum of high frequency current noise can be mapped out by measuring the escape rate enhancement versus the qubit current. As we will see below, in our system even quite small noise-induced occupancy in j2i and j3i are important, and one must generalize Eqs. 5.28 - 5.31 accordingly. In this case, the interpretation of G is not so straight-forward as suggested by Eq. 5.31 in that the enhancement at any current I1 will generally contain contributions from noise at several frequencies, corresponding to the frequencies of difierent allowed transitions that produce transitions between difierent levels. 5.3.3 Determination of T1 using thermal escape rate When the timescale of interest is much longer than the coherence time T2 [23], the ofi-diagonal terms in the density matrix equation (in Eq. 3.43) vanish and one flnds a master equation: d?i(t) dt = X j 6=i [?Wij ?i(t) +Wji ?j(t)]? ?i ?i(t) ; (5.32) where ?i = ?ii is the population at i-th level. Note that this master equation is when temperature T is greater than zero and it includes tunneling process without microwave. If I set T = 0 and remove the tunneling from each level, Eq. 5.32 reduces to Eqs. 3.43(a) or 3.43(d) with microwave power a0 = 0. Here Wnm is the transition rate from jni and jmi. Wnm includes transitions due to thermal emission/absorption, microwave pumping and any dissipation. In the thermal model, I assume that no microwaves are applied. In this case, Wji and Wij (i < j) are given by [23]1 Wji = W stji + ?ji = ?ji 1? exp (?~!ij=kBT ) (5.33) 1Notation for Wij here is from i to j, which is difierent from Ref. [23]. 123 and Wij = Wji exp (?~!ij=kBT ) = ?jiexp (~!ij=kBT )? 1 = W st ij ; (5.34) where W stij = W stji = ~!ij 2Reffe2 jhi j ^ j jij2 exp (~!ij=kBT )? 1 (5.35) is the thermally stimulated emission and absorption rate due to Reff between levels i and j, ?ij is ?ji = ~!ij2Reffe2 jhi j ^ j jij 2 (5.36) the spontaneous emission rate from j to i and !ij is the angular frequency spacing between two levels i and j. Equations 5.32 - 5.36 have four free parameters: temperature T, the relaxation rate T1 = ReffC1, the qubit critical current I01 and the capacitance C1. I01 and C1 can be obtained from spectroscopy measurements and they determine a unique Hamiltonian which can then be used to calculate the matrix elements jhi j ^ j jij2. If I01 and C1 are known, the escape rates ?i from each levels can be found from Eq. 4.5 (see Chapter 4) [23, 24]. By plugging in estimates for T1 and T, I can then solve the master equation in Eq. 5.32 and calculate the total escape rate from ?tot = ? 1?tot d?tot dt (5.37) where ?tot = P i ?i is the total population that remained in the metastable wells which have not tunneled yet at time t. ?tot can be written as ?tot = ? 1?tot X i d?i dt = 1 ?tot X i ?i?i = X i ?i ?tot?i = X i Pi?i (5.38) where Pi = ?i=?tot and ?i are obtained from the master equation simulation. I vary T and T1 to get the best flt of the calculated ?tot to a measured data 124 21.24 21.26 21.28 21.3 21.32 21.34 21.36 21.38 103 104 105 106 107 I1 (?A) ? (1 /?s ) P0?0 P1?1 P2?2 P3?3 Figure 5.6: Total escape rate ? versus current I1 for qubit AL1. The points were measured at 80 mK. The red curve is from a 4-level master equation simulation. The thin solid curves show the components of the total escape rate Pi?i. The simulation parameters are T1 = 17 ns and T = 88 mK. 125 set. For example, Fig. 5.6 shows the total escape rate ?tot measured in device AL1 at 80 mK (blue dots), ?tot from a 4-level master equation simulation (red curve) and the components of the simulated ?tot due to each level. The best flt occurs for T1 = 17 ns and T = 88 mK (I used the junction parameters listed in Table. 5.1.). Since the experiment was measured at relatively high temperature, P1?1 and P2?2 give large contributions. Note that P1?1 and P2?2 cause smooth bump-like features in the total escape rate. The starting position of the broad bump at high current is determined by T1 ?= 1=?10 and this relation can be used to obtain T1 directly without fltting the entire curve [89]. I found that using the 4-level master equation simulation, the main features of ?tot in this device are well-explained, provided the device was well-isolated. It turns out that under typical conditions, the analysis can be greatly simpli- fled. If the speed of current bias ramp is slow enough, I can set the time derivative of Pi to be zero in Eq. 5.32. This produces a \stationary" solution where the nor- malized populations are kept constant in time. In my measurement, (dln?=dt)?1 ? 3.5 ?s, which is much slower than T1 ? 50 ns. The stationary solutions obtained by setting dPi=dt = 0 provides an alternative approach to flt data to theory. Figure 5.7 shows the discrepancy between the stationary and non-stationary solutions; I calculated ?tot from the master equation without using the stationary condition (?ME) and with stationary solution (?SME). I plotted (?ME??SME)=?SME versus current. Here I used T1 = 17 ns and T = 88 mK. The other parameters are for device AL1 listed in Table 5.1. I note that flg. 5.7 shows there is less than 1 % difierence between the stationary solution and the non-stationary solution for these parameters. Based on this close agreement, I used the stationary master equation solutions to flt my data from this point on in the thesis. The stationary solutions can be obtained by solving a single matrix equation based on Eq. 5.32 with d?i=dt = 0(see Appendix A.). 126 21.24 21.26 21.28 21.3 21.32 21.34 21.36 21.38 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 I (? A) (? M E ? ? SM E)/ ? SM E Figure 5.7: Plots of (?ME ? ?SME)/?SME versus current. Here ?ME is the escape rate versus current from the non-stationary master equation simulation and ?SME is the stationary solution. For this curve, T1 is 17 and T = 88 mK. The calculation covers the same current range as in Fig. 5.6. 127 5.4 Measuring the efiect of isolation on an Al dc SQUID phase qubit In this section, I discuss my measurements of the efiect of the isolation factor on the thin-fllm dc SQUID qubit AL1 (see Ch. 4). I present results on the thermal noise induced escape rate and T1 and compare my results with the theory discussed in the previous section. 5.4.1 dc SQUID phase qubit parameters; flt to spectroscopy and I - ' curves To be able to compare the observed behavior of the device with simulations, I need to know the device parameters, including the inductances L1 and L2, the critical currents and the capacitance of the junctions. The critical current I01 and the capacitance C1 of the qubit junction J1 can be obtained from spectroscopy (see Chapter 4). Accordingly, I measured the qubit junction spectrum from 6 GHz to 8 GHz as a function of the current I1. As I1 increases, the energy levels of the qubit decrease. Well-deflned peaks in the escape rate occur at currents where the microwave frequency is in resonance with an allowed transition. Figure 5.8 shows an example of escape rates I measured in qubit AL1 at 80 mK (also shown in Chapter 4) at a sweep rate of about 30 mA/s. The black dotted curve shows the escape rate with 6.9 GHz microwave drive and the blue curve shows the corresponding measurement with no microwaves applied. Two clear resonance peaks appear in the escape rate and they correspond to the j0i ! j1i transition (at about 22.02 ?A) and the j1i ! j2i transition (at about 21.97 ?A). The red curve is the calculated escape rate ?0 from j0i. I found this curve by fltting the total measured escape rate using the non-stationary 4-level master equation simulation discussed in the previous section. Comparing ? to the ?0 curve, I see that there are many escape events from higher levels. 128 21.94 21.96 21.98 22 22.02 22.04 22.06 22.08 103 104 105 106 107 108 I1 (?A) ? (1/ s) |0> ? |1> |1> ? |2> G0 Figure 5.8: Escape rate of dc SQUID phase qubit AL1 at 80 mK with 6.9 GHz microwaves applied (black dotted curve) and without microwaves (blue solid curve). The red solid curve shows ?0 from the non-stationary master equation simulation of a single Josephson junction spectrum using parameters in Table 5.1. 129 To obtain the full spectrum of AL1, I measured a series of escape rates for difierent applied microwave frequency. As discussed in Chapter 4, This measurement was performed with simultaneous ramping of current and ux so that the current going through the isolation junction I2 was kept at zero and the qubit junction was most isolated. Figure 5.9 shows the resulting spectrum of resonant frequency versus current for the qubit junction in AL1. I flt the spectrum to a simulation of a single Josephson junction (see Eq. 4.7) with two free parameters, the critical current I01 and the capacitance C1 [23]. I obtained I01 = 22.2138 ?A and C1 = 4.078 pF. The two solid curves in flg. 5.9 show the simulation results of the energy level spacings between j0i and j1i (solid curve on the right) and j1i and j2i (solid curve on the left) calculated using these parameters. For comparison, from a separate fltting, I obtained I01 = 22.2143 ?A. I note that the minimum change in the resonance current that is caused by a change in the microwave frequency of 1 GHz was 3.8 nA. Given this data, this implies that I01 could be found to six signiflcant flgures. Using a similar analysis, C1 could be found to four signiflcant flgures. However, since calculations of the energy level spacings are complicated, it is not easy to completely propagate the errors. To flnd L1, L2, I02 and M, I measured the SQUID?s current- ux characteristic curve (see Fig. 5.10). For this measurement, I initialized the SQUID in the zero trapped ux state using ux shaking [86] and applied a small dc ux to the SQUID loop. If a small ofiset ux ' is applied, this produces a small I2, i.e. it generates a bias current through the isolation junction. I then applied simultaneous ux and current ramps so that the current through the isolation junction was kept approx- imately flxed at the initial starting value (thereby flxing rI), while I1 was steadily increased. The current and ux at which the device escaped was recorded and this procedure was repeated about 105 times for each value of the initial ofiset ux. I flt the resulting I ?' curves using the method by Tsang et al. (see Ch. 2) [37, 35] 130 21.94 21.96 21.98 22 22.02 22.04 22.06 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 I1 (uA) f (GHz ) 0 ? 1 1 ? 2 Figure 5.9: Transition spectrum of dc SQUID phase qubit AL1. Circles indicate j0i ! j1i transitions and crosses are j1i ! j2i transitions measured at 80mK. Solid curves are simulation flts to a single Josephson junction spectrum. 131 Table 5.2: Parameters of dc SQUID phase qubit AL1 from spectroscopy and current- ux map. Parameters I01 (?A) C1 (pF) I02 (?A) L1 (pH) L2 (pH) M (pH) I - ' 21.401 - 9.445 1236 5 15 Spectroscopy 22.2138 4.078 - - - - with L1, L2, I01, I02 and M as free parameters. From the fltted L1, L2, I01, I02 and M, I obtained the estimated isolation factors [see Fig. 5.10(b)] as a function of I2. The largest isolation was rI = 1000 and the lowest isolation I operated at was rI ? 100. The device parameters I obtained from spectroscopy and the I ? ' curve are listed in Table 5.2. I note that the I01 value I found from spectroscopy disagree signiflcantly from what I obtained by fltting the I ?' curve (see Table 5.2). There are a few possible reasons for this. For example, we do not know I1 precisely because it has to be inferred from the applied current I, ux ramps, and the device parameters. Also the flt to the I ?' curves uses a classical picture of the SQUID. Since this picture does not account for quantum mechanical tunneling, it should result in an underestimate of the true critical current. However, the observed difierences appear to be too large to be accounted for solely by this efiect. Another problem is that the flt to the I ? ' curves is somewhat crude, especially near the minimum critical current, and this could be a cause for disagreements. Fortunately, I usually do not need to know the current I1 with perfect accuracy. It is su?cient in many cases to have a set of I01 and C1 from the spectroscopy data consistent with the frequency of the resonance peaks. For the master equation simulations of AL1 in this thesis, I used I01 = 22.2138 ?A and C1 = 4.078 pF from spectroscopy. 132 ?0.2 ?0.1 0 0.1 0.2 10 15 20 25 30 ? a /?0 I c (? A ) (a) (b) ?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 I2/I02 r I Figure 5.10: Current- ux characteristic curve of dc SQUID phase qubit AL1 mea- sured at 80 mK. (a) Switching current versus normalized applied ux. Crosses are experimental data after ux shu?ing to zero ux state and solid curve is a flt using Tsang?s method discussed in Ch. 2. (b) Corresponding isolation factor rI for each data points versus normalized current I2. 133 5.4.2 Observing noise induced transitions. One of the most surprising things I found in my measurements of AL1 was that noise induced transition peaks were clearly visible in the background escape rates when the device was poorly isolated (see Fig. 5.11 to 5.13). For these background measurements, I simultaneously ramped the current bias and ux bias so that the current going through the isolation junction I2 was kept constant. For the most isolated case, I made the current bias and the ux bias cancel out at the isolation junction; i.e. where I2 = 0 so that the measured bias current I is the same as I1. Applying I2 changes the isolation factor rI and shifts the zero point of I1. As a result, the measured switching current of SQUID I will be shifted by the applied I2. To keep track of shifts in I, I also measured the escape rates while applying 7.45 GHz microwaves (green curves in Figs. 5.11, 5.12, 5.13). I flrst set the ofiset ux to zero to get the most isolated escape rate and I ? I1. Next I applied ofiset ux and measured how much the 7.45 GHz resonance peak shifted in current compared to the 7.45 GHz resonance peak for the most isolated case. By measuring how much the resonance peak was shifted along the current axis I, I could obtain I2 from I2 = I ? I1. It is important to remark that even for zero applied dc ux, there typically is always stray magnetic fleld coupled to the SQUID so that the current axes for the most isolated cases varied for difierent measurements. Figure 5.11 shows the measured background escape rate (without microwaves) versus current for device AL1 at 80 mK for r = 1000 (blue curve) and the simulated background escape rate from a stationary 4-level master equation (red dashed curve) described in section [89, 23] with T1 = 20 ns and T = 89 mK. I flnd a good overall agreement between the best flt and the data although some small deviations are evident. Figure 5.12 shows escape rates for the case when the isolation factor r = 270. Again the measured background escape rate (without microwaves) versus current is 134 21.12 21.14 21.16 21.18 21.2 21.22 21.24 21.26 103 104 105 106 107 I1 (?A) ? (1/s ) Figure 5.11: Escape rate versus current when the qubit is most isolated (rI = 1000). The blue curve is the background escape rate (without microwaves) and the green curve is the escape rate with 7.45 GHz microwaves applied. The red dashed curve is the total escape rate from a stationary 4-level master equation simulation when the qubit is most isolated. The simulation parameters are T1 = 20 ns and T = 89 mK. 135 21.12 21.14 21.16 21.18 21.2 21.22 21.24 21.26 103 104 105 106 107 I1 (?A) ? (1/s ) Figure 5.12: Escape rate versus current when the qubit is poorly isolated (rI = 270). The blue curve is the background escape rate (without microwaves) and the green curve is the escape rate with 7.45 GHz microwaves applied. The red dashed curve is the total escape rate from a stationary 4-level master equation simulation when the qubit is most isolated. The simulation parameters are T1 = 20 ns and T = 89 mK. 136 21.12 21.14 21.16 21.18 21.2 21.22 21.24 21.26 103 104 105 106 107 I1 (?A) ? (1/s ) Figure 5.13: Escape rate versus current when the qubit is more poorly isolated (rI = 220). The blue curve is the background escape rate (without microwaves) and the green curve is the escape rate with 7.45 GHz microwaves applied. The red dashed curve is the total escape rate from a stationary 4-level master equation simulation when the qubit is most isolated. The simulation parameters are T1 = 20 ns and T = 89 mK. 137 shown as a blue curve. For comparison, I also show again the simulated background escape rate from a stationary 4-level master equation for the most isolated case as a red dashed curve; the simulation parameters are the same as in Figure 5.11, T1 = 20 ns and T = 89 mK. In this flgure, there is a large disagreement between the simulation and the data and it is not just because I did not use the best flt curve. In particular, there are two broad peaks in the background escape rate at low current that will not occur in this master equation simulation for any choice of T and flxed T1. As I decrease the isolation factor further to r = 220, two separate peaks become more apparent and their location shifts on the current axis (see Fig. 5.13). The simulated escape rate for the most isolated situation (red dashed curve in Fig. 5.13) clearly does a very poor job of representing the background escape rate, suggesting that the peaks are not caused by thermal excitations from a frequency independent Reff . The likely cause of the peaks is resonant transitions induced by high frequency components of current noise on the bias leads. Figure 5.14 is a false color plot that summarizes the somewhat complicated dependence of the noise induced peaks on the isolation when both I1 and I2 are swept smoothly in device AL1. The x-axis is the reduced current I2=I02 through the isolation junction and the y-axis is the reduced current I1=I02 through the qubit junction. Note that the y-axis (I1) uses a backwards going current scale so that high frequency (low current) is at the top of the y-axis. The color scale corresponds to the enhancement G which is G0 = ?tot ? ?r=1000?r=1000 (5.39) where ?tot is the measured total escape rate and ?r=1000 is the measured total escape rate with r = 1000 (most isolated). I am using G0 instead of G deflned in Eq. 138 8.19 because ?tot involves contributions from several levels, not just two levels. By plotting G0, I can concentrate on the behavior of the noise peaks for difierent I1 and I2. In this color-scale images, large G0 is red and small G0 is blue. The data in Fig. 5.14 covers a range of rI from 70 to 400. In Fig. 5.14, two broad peaks (indicated as red) are seen at each value of the applied I2. The curves in Fig. 5.14 reveals the cause of the enhancement peaks; the solid yellow curve shows the locus of currents I1 and I2 for which the plasma frequency of the isolation junction !p2 is equal to the j0i ! j2i transition frequency !02 of the qubit. Noise peaks along this curve would be due to current noise passing through the isolation junction at its resonance and driving the qubit into its second excited state. Since population in j2i tunnels very rapidly, even a small amount of noise induced transitions could produce substantial enhancements. Similarly, the dashed yellow curve in Fig. 5.14 shows the locus of currents I1 and I2 for which !p2 = !13, i.e. along this curve the resonant frequency of the isolation junction !p2 equals the transition frequency !13 between j1i and j3i in the qubit. It is interesting that the two peaks in the background escape rate vary their location smoothly as a function of I2 but seem to disappear on the right half of the flgure. We note the peaks from !p2 = !02 disappear at I1=I01 ?= 0.9911 (I1 = 21.21 ?A) and the peaks !p2 = !13 disappear at I1=I01 ?= 0.9892 (I1 = 21.17 ?A) (see Fig. 5.14). This behavior may be due to the j2i and j3i levels exiting the top of the well and merging with the continuum of levels above the barrier. Once j3i exits the well, the !13 feature should disappear, for example. A more detailed of the situation suggests this explanation. Figure 5.15 shows a simulation of the energy levels for a Josephson junction done by solving Schr?odinger?s equation numerically using the AL1 parameters in Table 5.1 [23] (see Appendix A for the MATLAB routine I used). 139 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.9925 0.9916 0.9911 0.9906 0.9897 0.9892 0.9883 0.9870 I2/I02 I 1 /I 0 1 Figure 5.14: 3-D false color plot of the noise-induced transition peaks in the back- ground escape rate enhancement G0 in device AL1. Red indicates high G0 and blue indicates low G0. The lower yellow-black solid curve is locus of points where the isolation junction plasma frequency !p2 is equal to !02 of the qubit junction. The upper yellow-black dashed curve is locus of points where the isolation junction plasma frequency !p2 is equal to !13 of the qubit junction. 140 The dashed curve ?U is the height of the barrier in the tilted washboard potential. I note that at around I1=I01 = 0.9911 where the j0i ! j2i noise transition peak disappears, ?U crosses with the second energy level E2. Similarly, at around I1=I01 = 0.9892 where the j1i ! j3i noise transition peak disappears, ?U crosses the second energy level E3. As mentioned above, the j2i and j3i levels have escape rates that are of order (500)2 ? 105 times and (500)2 ? 107 greater than the ground state, respectively, and thus even a very small probability of occupying j2i and j3i can cause a substantial enhancement in total escape rate. In particular, a population of about 10?4 in j2i or 10?6 in j3i would increase the escape rate by about an order of magnitude above that from j0i. This would correspond to about the level of the enhancement above ?0 we see in the most isolated case for AL1. I note that these populations are so small that they are unlikely to produce signiflcant efiects in most qubit experiments. In fact, we have found that when Rabi oscillations are generated in this system, the time constant for the decay envelope does not appear to change signiflcantly with the isolation rI , even though the oscillations are taken with the device biased on top of a noise peak in the escape rate. The behavior of Rabi oscillations in AL1 are discussed in detail in Chapter 6. 5.4.3 T1 measurements using relaxation I measured T1 using two methods. The flrst method involved measuring re- laxation from j1i to j0i. For this technique, I prepared the qubit in a mixed state of j1i and j0i by driving the qubit resonantly with microwaves and then shutting ofi the microwaves. The resulting state decays to j0i exponentially with a decay time constant T1. There are a few problems that can arise with this technique, including (i) the need for the microwaves to shut ofi sharply at the qubit, (ii) the fact that we will get some population in j2i so that the relaxation process involves higher states 141 0.986 0.987 0.988 0.989 0.99 0.991 0.992 0.993 0.994 0.995 0 5 10 15 20 25 30 35 I1/I01 E (G H z) 0E 1E 2E 3E 4E ?U Experimental data taken Figure 5.15: Energy levels of Josephson junction phase qubit using parameters for AL1 in Table 5.1. Energy levels were obtained by solving the Schro?dinger equation numerically (see Appendix A). The dashed curve is the barrier height ?U of the washboard potential. 142 and (iii) that the population in j0i increases as j1i decays. Due to population in j2i and escape from j0i as well as j1i, multiple decay constants are observed in the relaxation measurement. Figure 5.16 shows a measurement of relaxation in the escape rate in AL1 for rI = 1000, when the qubit is most isolated. I drove the qubit junction at 7 GHz and measured the escape rate after the power was turned ofi. When the j0i ! j1i transition frequency is resonant with the microwave drive, the escape rate is enhanced due to escape from high levels. I shut ofi the microwave at the peak of the microwave resonance and observed the subsequent escape rate versus time. The black dots are for the case when the microwave power was P = - 10 dBm and the blue dots are for the case when the microwave power was P = - 20 dBm. The data taken with a higher microwave power (P = - 10 dBm) has a weak oscillatory feature on top of an overall decay. I used a ?2 method to flt the decay in the relaxing escape rate to the following function f(t) = A exp[?t=t0] +B exp[?t=t1] + C exp[?t=t2]: (5.40) (see smooth red curves in Fig. 5.16). The best flt parameters are listed in Table 5.3. The term with time constant t0 = 188.1 ?s is essentially an time constant due to the overall background escape rates. The decay time constant t2 ? 5 ns for all my data sets. This term is from a fast decay occurring at the beginning of the relaxation. S. K. Dutta found that the HP (Agilent) 83732B microwave generator I used for this measurement had a shut-ofi time of ? 5 ns [23] and I suspect that t2 was caused at least partly by this. Dutta also observed short decay times from higher energy levels in his relaxation measurements [23], but in my case, I did not observe any decay times shorter than 5 ns, probably due to the limited time resolution (about 4 ns) in my setup (see Ch. 4). This leaves the time constant t1, which represents the time constant for relaxation from j1i to j0i. For the P = - 10 dBm data, the 143 relevant time constant is T1 ? t1 = 61 ns and for P = - 20 dBm data, T1 ? t1 = 52 ns. Thus the relaxation times for the most isolated case appears to be T1 = 50 to 60 ns for AL1. I also measured relaxation when the qubit junction was poorly isolated, with rI = 220. In this case, I measured Rabi oscillation with a high power microwave drive (P = 17 dBm) and shut ofi the microwave power after 50 ns to observe decay in the escape rate. I flt the decay to Eq. 5.40 and found an initial decay time of about 5 ns, similar to the most isolated case with low power microwaves. For this data, the decay time constant T1 = t1 = 59 ns (see Fig. 5.17). If the dissipation was due to the impedance of the bias leads, I would expect to observe the relaxation decay constant change with rI . To calculate Riso, I measured all device parameters using I - ' flt and the spectroscopy (see Table 5.2) except R1. Thus I calculated R1 reversely from T1 = 55 ns with I2 = 0 (rI = 1000, most isolated) measured from the relaxation measurement. When T1 is 55 ns, Reff is 13.5 k?. With known Riso, I obtained R1 = 20 k?. Using R1 = 20 k?, I calculated T1 for rI = 220; If T1 is 55 ns for rI = 1000 and if T1 is due to the bias leads, for rI = 220 T1 should be 19 ns. However, as Fig. 5.17 and Table 5.3 show, the experimental results revealed no signiflcant difierence in T1 and no systematic dependence on rI . T1 for all three relaxation curves were 50 ns to 60 ns; t0 and t2 also showed no systematic dependence on rI . 5.4.4 T1 measurements from the thermally induced escape rate Dutta et al. showed that T1 can also be obtained from measurements of the thermally populated background escape rates [23, 89]. I applied this method to flnd a separate estimate for T1 in device AL1. Figure 5.18 shows experimental data (blue dots) for the escape rate in device AL1 obtained at 80 mK with rI = 1000 (most 144 0 10 20 30 40 50 60 70 80 106 t (ns) ? (1/s ) 7.0 GHz P = ?10 dBm 7.0 GHz P = ?20 dBm Figure 5.16: Observed relaxation in the escape rate for rI = 1000 (most isolated) at 80 mK for two difierent microwave powers at 7.0 GHz. Solid curves are the ?2 flts to Eq. 5.40. Upper dots are for microwave power P = -10 dBm and lower dots are for P = -20 dBm. 145 0 20 40 60 80 106 107 t (ns) ? (1/s ) Figure 5.17: Observed relaxation in the escape rate for rI = 220 (poorly isolated) at 80 mK in device AL1. Solid points are measured escape rates and the solid curve is the ?2 flts to Eq. 5.40. This relaxation measurement was done with high power microwaves (P = 17 dBm) after measuring Rabi oscillations. Crosses are the measured background escape rate with no microwave power applied. Table 5.3: Decay parameters that produced that best flt of Eq. 5.40 to the data in Fig. 5.16 and Fig. 5.17. Parameters A (1/?s) B (1/?s) C (1/?s) t0 (? s) t1 (ns) t2 (ns) rI = 1000, P = -20 dBm 0.264 0.902 0.037 188.1 52.34 5.002 rI = 1000, P = -10 dBm 0.252 2.617 0.494 188.1 60.71 4.544 rI = 220, P = 12 dBm 0.519 2.661 8.355 188.1 59.29 5.004 146 21.24 21.26 21.28 21.3 21.32 21.34 21.36 21.38 103 104 105 106 107 I1 (?A) ? (1 /?s ) Figure 5.18: Total escape rate ? versus current I1 for qubit AL1 at rI = 1000. The points are measured at 80 mK. The red curve is from a stationary 4-level master equation simulation with T1 = 17 ns and T = 88 mK. 147 isolated). For comparison, the solid curve shows results from a stationary 4-level master equation simulation. The simulation parameters are T1 = 17 ns, T = 88 mK and the qubit junction parameters I01 = 22.2138 ?A and C1 = 4.078 pF. I performed a ?2 flt of the experimental ?tot shown in Fig. 5.18 to flnd best flt parameters. Figure 5.19 shows a ?2 map where ?2 is obtained from ?2 = 888X i=1 flflflfl ?tot(Ii)? ?SME(Ii) ?tot(Ii) flflflfl 2 (5.41) where ?tot(Ii) is the experimental total escape rate at current Ii, ?SME(Ii) is the calculated escape rate from a stationary 4-level master equation simulation at cur- rent Ii, and ?tot(Ii) is the uncertainty in ?tot at current Ii (from Eq. 4.3). I used 888 points in the measured escape rate, from I1 = 21.232 ?A to 21.387 ?A. In the color scale of Fig. 5.19, red means that ?SME is far from ?tot and blue means that the calculated ?SME is close to ?tot. For this particular set, T1 and T varied from 12 ns at 82 mK to 17 ns at 87 mK. This case was for r = 1000 (most isolated). From Fig. 5.19, we can see that there is a relatively wide range of parameters that yield good flts to the data. Moreover the total escape rates measured in the experiments are not as smooth as the calculated escape rate, although ?tot for the most isolated case (rI = 1000) is smoother than ?tot for the poorly isolated cases. However, to see if T1 is afiected by rI , we need to flt the poorly isolated data also. Unfortunately, as I discussed in sec.5.4.2, noise-induced peaks appear in ?tot when the qubit is poorly isolated. Thus fltting ?tot to a master equation simulation is problematic for poorly isolated data. An alternative approach is to flnd T1 for each current that reproduces the experimental total escape rate. In this analysis approach, I assumed that T1 is a function of the qubit current I1, not a constant as in the thermal model described in sec. 5.3.3. T1(I1) incorporates the noise induced transition at each current by 148 5 10 15 20 25 75 80 85 90 95 T1 (ns) T (m K ) 10 20 30 40 50 60 70 80 90 100 Figure 5.19: ?2 map for parameters T1 and T. ?2 is calculated from the experimental ?tot and the calculated escape rate from the master equation simulation. 149 assuming the efiective resistance Reff is frequency dependent. In this approach, the thermal transition rate Eq. 5.35 becomes W stij = ~!ij 2Reff (I1; rI)e2 jhi j ^ j jij2 exp (~!ij=kBT )? 1 (5.42) where rI is the isolation factor and I1 is related to frequency through the corre- sponding energy level spacing. Figure 5.20 shows the escape rate ? from the experiment (blue dots) and a calculated escape rate ?SME (magenta curve) versus current I1 for qubit AL1 at rI = 1000. ?SME was obtained from a stationary 4-level master equation simulation by flnding T1(I1) that satisfled j?? ?SMEj = 0 with a current dependent T1(I1) where ? is the escape rate data from Fig. 5.18. I assumed C1 = 4 pF and T = 88 mK. For comparison, the black curve shows ?SME from a stationary 4-level master equation using a constant T1 = 17 ns. Figure 5.21 shows a plot of T1 versus current that I found by analyzing the data in Fig. 5.20 using the stationary 4-level master equation solution. The simulation parameters are T = 88 mK, I01 = 22.2138 ?A and C1 = 4.078 pF. T1 has maximums and minimums. On average, T1 is roughly 15 ns but depending on current, T1 can be as high as 23 ns or as low as 5 ns. The maximum T1 is about 23 ns at I1 = 21.31 ?A. Using this method, I was able to calculate T1 versus current I1 for two difierent isolations, rI = 1000 and rI = 400. Figure 5.22 shows the calculated T1 versus normalized current I1=I01 for these two cases. The black curve shows T1 for the most isolated case (rI = 1000) and the blue curve shows T1 for the poorly isolated case (rI = 400). I note that the T1 curves in Fig. 5.22 are difierent from the T1 curve measured two months earlier shown in Fig. 5.21, suggesting that the device or external noise changed over a two month period. 150 21.24 21.26 21.28 21.3 21.32 21.34 21.36 21.38 104 105 106 107 I1 (?A) ? (1/s ) Figure 5.20: Experimental ? (blue dots) and calculated ?SME (magenta line) versus current I1 for qubit AL1 for rI = 1000. ? is the escape rate data from Fig. 5.18. For the calculation, I assumed C1 = 4 pF and T = 88 mK. The black curve shows ?SME from a stationary 4-level master equation using a constant T1 = 17 ns. 151 21.26 21.28 21.3 21.32 21.34 21.36 21.38 0 5 10 15 20 25 30 I1 (?A) T 1 (ns ) Figure 5.21: Calculated T1 versus current I1 for qubit AL1 at rI = 1000 using escape rate data from Fig. 5.18, assuming C1 = 4 pF and T = 88 mK. This data was taken on 01/23/05. 152 In Fig. 5.22, T1 appears to depend on rI , but not over the whole range of current I1/I01. For I1/I01 less than about 0.992, T1(I1) for rI = 1000 is virtually the same as T1(I1) for rI = 400. However, below I1/I01 = 0.992, T1 scales as the isolation. For comparison, the red curve shows T1 for rI = 400 multiplied by the isolation factor ratio 1000/400. For I1/I01 < 0:992, the red curve matches with T1 for rI = 1000 (black curve). Fig. 5.22 suggests that both relaxation and noise induced transitions are oc- curring in the device. Low T1 indicates high ?+ and a fast relaxation rate. Therefore, when I decrease the isolation factor rI from 1000 to 400, the measured T1 should decrease. This phenomena only happen below I1/I01 = 0.992 near where j2i leaves the well, which suggests that ?+ involves j2i states, not j1i. I note that the T1 values in Fig. 5.21 or Fig. 5.22 are very difierent from T1 ? 50 ns obtained from the relaxation measurements. This raises several important questions, in particular, what is causing the enhancement evidenced in Fig. 5.16 and Fig. 5.17. Spurious two-level systems coupled to the qubit junction is a pos- sible answer. The existence of two level systems coupled to phase qubits has been observed by several groups [90, 49, 23]. The idea is that microwaves can drive the system into a state in which the two level uctuator is entangled with the qubit. If the uctuator has a long relaxation time constant (say 50 ns) then the resulting entangled system can show a components with corresponding long relaxation time [55]. Another possibility is that the thermal rate estimation for T1 is incorrect be- cause of the presence of high frequency, non-thermal noise. Only thermal noise was included in this model, and the presence of a non-thermal source would produce an apparently smaller T1 in Fig. 5.21 or Fig. 5.22. Further experiments and analysis will be needed to distinguish these possibilities. 153 0.99 0.991 0.992 0.993 0 5 10 15 20 25 30 35 I1/I01 T 1 (ns ) Figure 5.22: Comparison of results for T1 vs I1=I01 with rI = 1000 (black curve) and for the escape rate with rI = 400 (blue curve). I01 is 22.2138 ?A. For comparison, the red points are T1 for rI = 400 multiplied by the isolation ratio 1000/400. This data was taken on 03/30/05. 154 5.5 Conclusions In conclusion, I have shown how the isolation between the qubit junction and its bias leads can be varied in situ by applying current to the isolation junction in a dc SQUID phase qubit. I found that the isolation fails when the resonance frequency of the isolation junction matches a qubit transition frequency. This leads to prominent peaks in the escape rate when the j0i to j2i or j1i to j3i transition frequencies of the qubit matches with the plasma frequency of the isolation junction. Fortunately, the noise generates only very small population in the upper levels and this does not appear to signiflcantly degrade the performance of the devices. Nevertheless, this behavior is undesirable, and could become an issue if the coherence times become signiflcantly longer. Additional high frequency flltering and redesign of the isolation junction parameters will be needed to suppress the efiect. I measured T1 of AL1 using a relaxation measurement technique and by mea- suring the thermally induced escape rate. The T1 results are summarized in Table. 5.4 and they difier signiflcantly. Since there have been several reports showing that the phase qubit can be coupled to spurious two level systems in nearby dielectrics [90, 49], it is possible that the larger value of T1 from the relaxation measurement could be due to two level systems. It is unclear why I did not observe the same long T1 from the thermally induced background escape rate flts. However, T1 from the escape rates showed some dependence on the isolation factor which the relaxation measurements did not. This suggests that the escape rate estimates for T1 may be contaminated by high frequency non-thermal noise. In the next chapter, I will generally assume T1 ? 20 ns which is what I obtained from the thermal escape rate measurements at large values of I1. 155 Table 5.4: T1 estimates for qubit AL1 Method Isolation rI Frequency Power T1 (ns) Relaxation 1000 7 GHz 20 dBm 52.34 Relaxation 1000 7 GHz -10 dBm 60.71 Relaxation 220 7 GHz 17 dBm 59.29 Thermal escape rate 1000 ?E01=h = 7 GHz 25 Thermal escape rate 400 ?E01=h = 7 GHz 15 156 Chapter 6 Measurements of coherence times in dc SQUID phase qubits 6.1 Overview Superconducting circuits containing Josephson junctions are examples of rel- atively large systems that display quantum behavior [38, 41, 45, 47, 91, 92, 93]. In particular, it has been shown experimentally that Josephson junction can be placed into superpositions of quantum states as well as entangled quantum states [48, 94]. In these measurements, the junctions, although cooled to millikelvin temperatures, were attached to room-temperature ampliflers through thermally-anchored, low- pass-flltered wire leads. Such connections between a quantum system and a noisy environment will in general lead to decoherence of the quantum system. In this chapter, I show results of Rabi oscillation measurements on a dc SQUID phase qubit with Al/AlOx/Al junctions. I was able to change the isolation of the qubit in situ and examined the efiect of isolation on the Rabi oscillations and the spectroscopic coherence times. I also show Rabi oscillation data on a niobium SQUID from S. K. Dutta [23] for comparison with the aluminum device. I be- gin by reviewing the isolation scheme of the dc SQUID phase qubit and show how the isolation factor in uences the decay time constant T? of the Rabi oscillations and the spectroscopic coherence time T?2. 157 1 2 J1 J2 2 J1 Figure 6.1: (a) Schematic of dc SQUID phase qubit. I is the bias current, If is current for the ux bias, M is mutual inductance between the ux bias coil and the SQUID loop and 'a is ux applied to the SQUID loop. C1 and C2 are the capacitances of the qubit junction J1 and the isolation junction J2, respectively. Microwave source Iw is coupled to J1 through capacitor Cw. Photographs of (b) single-turn aluminum SQUID magnetometer AL1 and (c) 6-turn niobium SQUID magnetometer NB1. 158 6.2 Current noise, isolation and coherence times in the dc SQUID phase qubit In a dc SQUID phase qubit [see Fig. 6.1(a)], junction J1 acts as a phase qubit [12] and the rest of the SQUID serves as an inductive isolation network that fllters out noise from the bias leads [47]. The isolation network consists of a flxed inductor L1, an isolation junction J2 and a parasitic inductance L2. When the applied ux 'a is held constant, a small uctuation ?I in the bias current I leads to a change ?I1 in the current I1 owing through junction J1. The current noise power isolation factor, rI is given by rI = ?I ?I1 ?2 = L1 + LJ1 + L2 + LJ2 LJ2 + L2 ?2 ? L1 + L2 + LJ2 LJ2(I2) + L2 ?2 : (6.1) as deflned in Chapter 5 where I have assumed L1 LJ1 and neglected MI , the mutual inductance between the current bias line and the SQUID loop. Hence LJi(Ii) = '02? 1 I0i p 1? (Ii=I0i)2 (6.2) is the Josephson inductance of the i-th junction (i = 1 or 2) and I0i is the critical current of the i-th junction. Since LJ2 depends on I2, the isolation is a function of I2. If the current on the leads has a noise power spectral density SI(f), then the current noise power spectral density SI1(f) which reaches the qubit junction J1 is SI1(f) = SI(f)r ? SI(f) LJ2 + L2 L1 + L2 + LJ2 ?2 : (6.3) Thus SI1(f) can be varied in situ because LJ2 can be changed by varying the current I2 through the isolation junction (see Eq. 2). Good isolation can be achieved by 159 choosing LJ2 ? L1. The best current bias isolation occurs at I2 = 0 where LJ2 is a minimum (see Chapter 5). The implication of Eq. 6.3 is that the current noise power from the bias leads is reduced by factor of rI before it reaches the qubit junction. Equation 6.3 is valid if the uctuation frequency f is much less than the plasma frequency fp2 of the isolation junction. Current noise SI1(f) can cause excitation, dissipation, decoherence and inho- mogeneous broadening in the qubit. Which efiect dominates a measurement depends on the nature of the measurement and the frequency range of the noise [25, 61]. For example, a at noise spectrum (white noise) with a bandwidth that extends up to and beyond 1/T1 leads to pure dephasing or decoherence that cannot be removed using a spin-echo technique. Decoherence, dissipation, power broadening and inhomogeneous broadening all contribute to the measured spectroscopic resonance widths. H. Xu showed that if the noise power is constant below a cut-ofi frequency fc ? 1/T1, then inhomogeneous broadening dominates the spectroscopic coherence time T?2 where T ?2 = 1 2??fHWHM (6.4) and ?fHWHM is the half-line width at half maximum (deflned in Chapters 3 and 4) of the j0i to j1i transition peak at the transition frequency f01 [25, 95]. Given the relationship between I and I1, Xu?s analysis implies that in this case, T ?2 = 1:65 ? 2? I1 flflflfl @f01 @I1 flflflfl ??1 = prI 1:65 ? 2? p SI(0)fc flflflfl @f01 @I1 flflflfl ??1 (6.5) where I1 is the rms current noise in I1, SI is the current noise power in I and I have assumed that the spectrum is measured in the low-power limit. Thus for low frequency noise, T?2 scales with prI . On the other hand, if the noise has a cutofi frequency fc 1=T1, the efiect is 160 to produce pure dephasing, and Xu?s analysis implies that T?2 is given by [25, 95]: T ?2 = ? ?2SI1(0)(@f01@I1 ) 2 ??1 = rI ? ?2SI(0)(@f01@I1 ) 2 ??1 (6.6) Thus T?2 should scale linearly with rI in this limit. Of course if current noise is not dominating the decoherence, T?2 would likely be independent of rI . The decay time constant T0 of the envelope of the Rabi oscillations is sensi- tive to noise at the Rabi frequency, while the shape of the envelope is afiected by inhomogeneous broadening caused by low frequency noise [61]. If both decoherence and dissipation are present, the Rabi decay constant T0 is related to the dephasing time T2 and the relaxation time T1 by 1 T 0 = 1 2T1 + 1 2T2 (6.7) when there is zero detuning (see Chapter 3) [25, 64]. Here I use the homogeneous coherence time T2 as deflned in Ch. 3. Although Eq. 6.7 was derived for two- level Rabi oscillations, it will still be applicable to a multi-level phase qubit if the higher-level occupations are small. The idea is that by measuring T 0, I can obtain information about T2 and the amount of current noise at the Rabi frequency. I also expect T0 will scale with isolation factor rI if the bias current leads are the dominant source of decoherence and dissipation. 6.3 Efiect of current noise on Rabi oscillations I measured Rabi oscillations by driving resonant j0i ! j1i transitions using a 7 GHz microwave drive. I obtained data for the most isolated case (rI = 1000) and a poorly isolated case (rI = 200). For these measurements, I flrst initialized the SQUID in the zero ux state, corresponding to no circulating current in the SQUID 161 loop. I then applied a small static ofiset ux to the SQUID to induce circulating current, thereby producing a flxed current I2 (see Ch. 5). Figure 6.2 shows the escape rate versus current I1 for device AL1 measured at 80 mK when rI = 1000 (black curve), rI = 200 (red dotted curve) and with 7 GHz microwave applied at rI = 1000. The escape rate with rI = 1000 is in reasonable agreement with a stationary four-level master equation simulation with thermal population for T = 88 mK and T1 = 17 ns, as discussed in Chapter 5. Reducing rI by a factor of 5, an overall enhancement in the escape rate (rI = 200, red crosses) was observed with two broad peaks at 21.02 ?A and 21.08 ?A. As discussed in Chapter 5, I found that these broad peaks in the background ? are due to noise induced populations in j2i and j3i. The dashed curve in Fig. 6.2 shows the escape rate when an f = 7.00 GHz microwave is turned on at the j0i ! j1i transition resonance. The resulting resonance peak is very wide due to power broadening. Note that for rI = 200, a large noise-induced transition peak already exists at the current where Rabi oscillations were measured (see crosses in Fig. 6.2). Figure 6.4 shows Rabi oscillations in device AL1 for the most isolated case, rI = 1000, measured with microwave powers from 6 dBm to 17 dBm, referred to the output of the microwave source at room temperature. The Rabi frequency increased as the microwave power increased, as expected for Rabi oscillations. Figure 6.3(a) show a plot of Rabi frequency squared vs. the microwave power in mW when rI = 1000. A linear ?2 flt is drawn as a red line. This flt gives a slope of 2:6 ? 104 (MHz)2/mW and a y-intercept of 2:0 ? 103 (MHz)2. I note that power in dBm is related to power in mW by PmW = 10PdBm=10 so 10 dBm is 10 mW. Examination of Fig. 6.4 shows that for powers of 13 dBm and above, the oscillation amplitude gets progressively washed away. This may be because the Rabi frequency becomes smaller than the time resolution of my measurement system. The limited time resolution occurs because the voltage signal from the qubit switching 162 20.98 21 21.02 21.04 21.06 21.08 21.1 21.12 21.14 103 104 105 106 107 I1 (?A) ? (1/s ) 7 GHz microwave resonance peak (Rabi oscilations measured here) Figure 6.2: Escape rates of device AL1 at rI = 1000 (solid curve), rI = 200 (crosses) and with 7 GHz microwaves (dashed curve) at rI = 1000. 163 has a flnite slope with respect to time. With added noise, one gets jitter in the determination of the switching time. From the Rabi oscillation plots, I estimate that the time resolution is about 4.5 ns. To extract the decay time constant of the envelope of the Rabi oscillations, I flt the oscillation curves to a phenomenological model for a decaying oscillation, ?fit = g0 + g1(1? e?(t?t0)=T 0 cos(?ft? t0g)) + g2(1? e?(t?t0)=Tback) (6.8) where the fltting parameter T0 gives the decay time constant of the Rabi oscillations. Here ? is the Rabi frequency and t0 is the microwave starting time. The flrst term, g0, accounts for the initial escape from ?0 and any thermally induced population in upper levels at time t = t0; i.e. ?(t = t0) = g0. The second term accounts for the Rabi oscillation with frequency ? and decay envelope time constant T 0. The third term involving g2 and Tback accounts for the flnite rise time of the microwave pulse and changes in population in j2i caused by the drive (see Chapter 5). Note that ?fit(t = 1) = g0 + g1 + g2. I found that g0, g1, g2, Tback, ? and t0 do not have much efiect on the Rabi decay time fltting parameter T0, but improve the overall flt. Figures. 6.5 and 6.6 show plots of Rabi oscillations in device AL1 at 80 mK with rI = 1000 (grey open circles) and the best flt curves (black solid curves). I did a ?2 flt and found the best flt parameters shown in Table 6.1. The smooth turn-on of the Rabi oscillations is due to the rise time of the microwave pulse as mentioned above, and the flnite time resolution in my experiment. These efiects are only roughly accounted for in Eq. 6.8. Nevertheless the fltted curves show reasonably good agreement overall, except near t = 0. Since it was di?cult to get good flts for t < 5 ns, I performed the fltting for rI = 1000 starting from t = 5 ns. For the data in Figs. 6.5 and 6.6, the average Rabi decay time T0 is 23.2 ns and there is some variation, as can be seen by examination of Table 6.1. 164 (a) (b) 0 10 20 30 40 50 60 0 P (mW) 0 2 4 6 8 10 12 14 0 4X10 P (mW) (? /2 pi )2 (M H z) 2 (? /2 pi )2 (M H z) 2 4 3X104 2X104 1X104 1.5X104 1.0X104 0.5X104 Figure 6.3: Square of the Rabi frequency vs. microwave power (a) for the most isolated case rI = 1000 (fllled dots) and (b) the poorly isolated case rI = 200 (fllled dots). The red lines in each plot are from a linear ?2 flt. 165 0 10 20 30 40 50 60 70 0 20 40 60 80 100 120 140 160 180 200 t (ns) ? (1/ ?s ) 8 dBm 9 dBm 10 dBm 11 dBm 12 dBm 13 dBm 14 dBm 15 dBm 16 dBm 17 dBm 7 dBm 6 dBm Figure 6.4: Rabi oscillations measured for r = 1000 in device AL1 at 80 mK. Each curve was taken with 7 GHz microwaves, with the power varying from 6 dBm to 17 dBm. Each successive curve was ofiset vertically by 10/?s. 166 Table 6.1: Summary of fltting parameters for Rabi oscillations in device AL1 at 80 mK with rI = 1000. The microwave was at 7.0 GHz. 6 dBm 7 dBm 8 dBm 9 dBm 10 dBm 11 dBm t0 (ns) 1.83 1.71 2.42 2.34 2.47 2.57 g0 (1/?s) 2.82 3.14 3.40 3.89 4.25 6.52 g1 (1/?s) 4.89 4.56 4.57 4.59 4.21 3.95 ?/2? (MHz) 111 123 136 151 168 188 T0(ns) 28.2 27.4 24.5 18.8 18.8 21.9 g2 (1/?s) 3.12 4.89 6.80 8.52 11.7 13.1 Tback(ns) 1.70 2.21 2.60 2.69 3.16 2.81 Figure 6.7 shows Rabi oscillations for the poorly isolated case, rI = 200, mea- sured for microwave powers from 6 dBm to 17 dBm. The most obvious difierence from Fig 6.4 is that Rabi frequencies are much lower for the same microwave power applied at the top of the refrigerator. This suggests the microwave coupling to the sample has decreased compared to when rI = 1000. This was unexpected and suggests the microwaves are coupling through the current bias line. Figure 6.3(b) shows that the square of the Rabi frequency varies linearly with microwave power, as expected. A linear ?2 flt to this data [red line in Figure 6.3(b)] gives a slope of 2800 (MHz)2/mW and a y-intercept of 3300 (MHz)2; the slope is almost 10 times less than for rI = 1000 while the intercepts are comparable. I flt the oscillation curves in Fig. 6.7 to Eq. 6.8 to extract the decay times of the Rabi oscillation for rI = 200. The solid curves in Figs. 6.8, 6.9, 6.10 and 6.11 show the resulting flts to the Rabi oscillations for rI = 200. The best flt parameters are shown in Tables 6.2 and 6.3. For rI = 200, the escape rates clearly do not start from ?tot = 0 at t = 0, and from this, we can see that noise was exciting the system while we were measuring Rabi oscillations; the escape rate was already high at t = 0 mainly due to escapes 167 0 10 20 30 40 50 0 5 10 15 20 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 15 20 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 15 20 t (ns) ? (1/ ?s ) (a) (b) (c) Figure 6.5: Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 1000 (open circles). Solid curves show best flt to Eq. 6.8 for 7 GHz microwaves with power (a) P = 6 dBm, (b) P = 7 dBm, and (c) P = 8 dBm. Fitting parameters are listed in Table 6.1. 168 0 10 20 30 40 50 0 10 20 30 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 10 20 30 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 10 20 30 t (ns) ? (1/ ?s ) (a) (b) (c) Figure 6.6: Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 1000 (open circles). Solid curves show best flt to Eq. 6.8 for 7 GHz microwaves with power (a) P = 9 dBm, (b) P = 10 dBm, and (c) P = 11 dBm. Fitting parameters are listed in Table 6.1 169 Table 6.2: Summary of fltting parameters for Rabi oscillations in device AL1 at 80 mK with rI = 200. The microwave power varies from 6 dBm to 11 dBm. 6 dBm 7 dBm 8 dBm 9 dBm 10 dBm 11 dBm t0 (ns) 2.65 2.32 2.86 2.18 2.07 2.75 g0 (1/?s) 3.31 3.13 3.28 3.31 3.31 3.89 g1 (1/?s) 2.21 2.38 2.23 2.44 3.03 2.44 ?/2? (MHz) 33.2 37.6 45.0 48.3 51.9 61.0 T0(ns) 26.9 27.2 32.5 32.3 23.0 25.3 g2 (1/?s) 1.32 1.91 2.02 2.23 2.32 2.29 Tback(ns) 8.30 21.3 11.8 14.2 18.1 4.98 from j2i which was populated by noise-induced transitions. Since the escape rate from the second level is ? 103 times higher than the escape level from the flrst level, even though j2i has less than 1 % population, its contribution to the total escape rate is signiflcant. Inspection of Figs. 6.8, 6.9, 6.10 and 6.11 shows that the phenomenological function flts the data fairly well although these are obvious disagreements evident in Fig. 6.10. To obtain the best flt, I averaged the data over 200 ns for the P = 15 dBm, 16 dBm and 17 dBm data set, while I averaged the data over 300 ns for the other data sets. Thus the curves For P = 15 dBm, 16 dBm and 17 dBm data look more noisy. From the flts for rI = 200, I flnd the average T0 ? 27.4 ns, which is somewhat longer than the 25 ns I found for rI = 1000. To summarize the above results on the Al dc SQUID phase qubit AL1, Rabi oscillations were measured using a 7 GHz microwave drive tuned on resonance to the j0i to j1i transition while we continuously monitored the escape rate. For a direct comparison, Fig. 6.12 shows Rabi oscillations for device AL1 at 80 mK using a 7 GHz microwave drive for rI = 1000 [Fig. 6.12(a)] and for rI = 200 [Fig. 6.12(b)]. Despite the rI = 200 curve being about 5 times less isolated from current noise power, its decay envelope time constant T0 ? 36 ns while the data set for rI = 1000 170 0 10 20 30 40 50 60 70 0 20 40 60 80 100 120 140 t (ns) ? (1/ ?s ) 8 dBm 9 dBm 10 dBm 11 dBm 12 dBm 13 dBm 14 dBm 15 dBm 16 dBm 17 dBm 7 dBm 6 dBm Figure 6.7: Rabi oscillations measured for the poorly isolated situation rI = 200. Each Rabi oscillation curve was taken with 7 GHz microwaves applied, for power from 6 dBm to 17 dBm. Each successive curve was ofiset by about 10/?s for clarity. 171 0 10 20 30 40 50 0 5 10 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 t(ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 t (ns) ? (1/ ?s ) (a) (b) (c) Figure 6.8: Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 200 (open circles). Solid curves show best flt to Eq. 6.8 for 7 GHz microwave with power (a) P = 6 dBm, (b) P = 7 dBm, and (c) P = 8 dBm. Fitting parameters are listed in Table 6.2 172 0 10 20 30 40 50 0 5 10 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 t (ns) ? (1/ ?s ) (a) (b) (c) Figure 6.9: Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 200 (open circles). Solid curves show best flt to Eq. 6.8 for 7 GHz microwave with power (a) P = 9 dBm, (b) P = 10 dBm, and (c) P = 11 dBm. Fitting parameters are listed in Table 6.2 173 0 10 20 30 40 50 0 5 10 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 t (ns) ? (1/ ?s ) (a) (b) (c) Figure 6.10: Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 200 (open circles). Solid curves show best flt to Eq. 6.8 for 7 GHz microwave with power (a) P = 12 dBm, (b) P = 13 dBm, and (c) P = 14 dBm. Fitting parameters are listed in Table 6.3 174 0 10 20 30 40 50 0 5 10 15 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 15 t (ns) ? (1/ ?s ) 0 10 20 30 40 50 0 5 10 15 t (ns) ? (1/ ?s ) (a) (b) (c) Figure 6.11: Rabi oscillations in the escape rate for device AL1 at 80 mK with rI = 200 (open circles). Solid curves show best flt to Eq. 6.8 for 7 GHz microwave with power (a) P = 15 dBm, (b) P = 16 dBm, and (c) P = 17 dBm. Fitting parameters are listed in Table 6.3 175 Table 6.3: Summary of fltting parameters for Rabi oscillations in device AL1 at 80 mK with rI = 200. The microwave power varies from 12 dBm to 17 dBm. 12 dBm 13 dBm 14 dBm 15 dBm 16 dBm 17 dBm t0 (ns) 2.98 1.78 2.04 2.61 2.23 2.91 g0 (1/?s) 4.55 3.82 4.39 4.45 4.08 6.08 g1 (1/?s) 2.33 2.46 1.88 2.66 3.27 2.76 ?/2? (MHz) 68.7 72.8 84.1 98.0 103 122 T0(ns) 15.5 19.3 32.2 36.4 23.1 30.9 g2 (1/?s) 3.19 4.83 4.79 4.51 5.57 5.41 Tback(ns) 3.28 6.05 6.73 3.79 4.66 3.14 was flt using a decay envelope time constant T0 ? 28 ns. Again, we see that the better isolated device did not have a larger Rabi decay time. Figure 6.13 shows a summary of the analysis of my Rabi data, in which I plotted T0 vs. Rabi frequency for rI = 1000 (crosses) and rI = 200 (fllled circles). The Rabi frequency is proportional to the square root of the microwave power, and there seems to be a weak tendency for Rabi oscillations measured at lower power to have longer T0. The average T0 for rI = 1000 is 23.3 ns and for rI = 200 is 27.4 ns. Needless to say, this is not the behavior one would expect if T0 was being limited by the bias leads. 6.4 T0 - comparison with a Nb device S. K. Dutta observed much the same behavior in the Nb dc SQUID phase qubit NB1 [23]. Figure 6.14 shows Rabi oscillations that Dutta measured in NB1 at 25 mK with a 7.6 GHz microwave drive for rI = 1300 [Fig 5(a)] and rI = 450 [Fig 6.14(b)]. Fitting to Eq. 6.8 yields T0 = 12 ns for rI = 1300 and T0 = 15 ns for rI = 450. 176 0 5 10 15 20 25 30 35 40 0 5 10 15 t (ns) ? (1/ ?s ) 0 5 10 15 20 25 30 35 40 0 5 10 15 t (ns) ? (1/ ?s ) (a) (b) Figure 6.12: (a) Points show measured Rabi oscillations in device AL1 at 80 mK for rI(0) =1000, and (b) for rI(9.15 ?A) = 200. Solid curves are phenomenological flts to the decay oscillation function shown in Eq. 6.8 177 0 50 100 150 200 0 5 10 15 20 25 30 35 40 ?/2pi (MHz) T? (ns ) Figure 6.13: The decay time constant T 0 of Rabi oscillations vs. Rabi frequency ?=2? of device AL1 measured at 80 mK at the most isolated biasing point, rI = 1000 (fllled circles) and at the poorly isolated biasing point (squares), rI = 200. 178 Thus T 0 for this Nb device is also not scaling with isolation rI . However, a remarkable fact here is that T 0 for AL1 was substantially longer than T 0 for NBl. This suggests that whatever was causing the decoherence may be dependent on materials used to build the device. 6.5 Spectroscopic coherence time T?2 : comparison with a Nb device I obtained the spectroscopic coherence time T?2 from T ?2 ? 1 2??fHWHM = dI df 1 2??IHWHM (6.9) where (see also Chapter 4) ?IHWHM is the half width at half maximum of the j0i to j1i resonance peak, measured in the low power limit so that power-induced broadening was not apparent. Figure 6.15 shows T?2 in AL1 obtained from the spectroscopy data (see Fig. 4.19 in Chapter 4) at 80 mK. T?2 varies from about 2 ns to 8 ns and the maximum T?2 occurs at 7.3 GHz. The low values of T ?2 at 6.2 GHz to 6.7 GHz are likely due to tunneling [96] but the low values at 7.8 GHz were unexpected. Figure 6.16(a) shows T?2 versus rI for device AL1 at 80 mK with 7.45 GHz microwaves applied. T?2 varies between a minimum of about 2.5 ns to a maximum of about 6.5 ns in an apparently random fashion with rI . In particular, T?2 does not show the systematic dependence on the isolation predicted in Eqs. 6.5 or 6.6. T?2 versus rI for device NB1 measured at 7.2, 7.3, 7.4, and 7.5 GHz at 25 mK, shows a similar random variation between about 3 and 6 ns [see Fig. 6.16(b)]. The average spectroscopic coherence time is about 4 ns for NB1. Also a closer look at Eq. 6.5 or 6.6 reveals that T ?2 should depend strongly on f01 if current or ux noise is the dominant factor. In contrast, the measured T?2 does not show the expected strong dependence on frequency. 179 0 5 10 15 20 25 30 35 40 0 20 40 60 t (ns) ? (1/ ?s ) 0 5 10 15 20 25 30 35 40 0 20 40 60 t (ns) ? (1/ ?s ) (a) (b) Figure 6.14: (a) Rabi oscillations in the escape rate ? in device NB1 at 25 mK for (a) rI = 1300 and (b) for rI = 450 [23]. An 7.6 GHz drive was used. 180 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 0 1 2 3 4 5 6 7 8 9 10 f (GHz) T 2* (ns ) Figure 6.15: Spectroscopic coherence time T ?2 versus frequency for rI = 1000 for device AL1 at 80 mK. 181 0 200 400 600 800 1000 0 2 4 6 8 10 r T* 2 ( n s) 0 200 400 600 800 1000 1200 1400 0 2 4 6 8 10 r T* 2 (ns ) (a) (b) I I Figure 6.16: Spectroscopic coherence time T ?2 versus isolation factor rI of (a) alu- minum dc SQUID phase qubit AL1 measured at 7.45 GHz at 80 mK. (b) T ?2 for NB1 at 7.2 (crosses), 7.3 (open circles), 7.4 (squares) and 7.5 GHz (dots) at 25 mK. 182 6.6 Discussion The fact that neither the spectroscopic coherence time nor the decay envelope of Rabi oscillations depends systematically on the isolation from the leads implies that the main source of decoherence is not current noise from the leads. It is also interesting that in both devices, T0 is considerably larger than 2T?2; For AL1, T0 = 27 ns > 2T?2 ? 12 ns and for NB1, T0 = 12 to 15 ns > 2T?2 ? 8 ns. This would be the case if we had signiflcant inhomogeneous broadening of the spectrum caused by a local low-frequency noise source. One possible source of decoherence in this system is spurious resonators or two- level uctuators that reside in the substrate or dielectric layers [90, 49]. In particular, the fact that T0 in AL1 was twice as long as in NB1 is consistent with the presence of two-level uctuators in dielectric layers in NB1. AL1 had no insulation layers except for the thermally grown AlOx tunnel barrier, native oxide on the exposed Al surfaces, and the thermally grown SiO2 surface on the Si substrate, whereas NB1 had all of the above plus sputtered SiO2 insulation layers. While we have not seen clear spurious resonator splittings (down to splittings of about 10 MHz) in spectroscopic data of AL1, S. K. Dutta found very small apparent splittings (10 MHz or less) in NB1 and other similar Nb SQUIDs from Hypres [23]. Another possible source of decoherence is local ux noise of unknown origin that has been found in other SQUIDs at millikelvin temperatures [97, 98]. However, this would have the same strong dependence on frequency as current noise. Since I did not flnd such a dependence, it suggests that ux noise is not the cause. 6.7 Conclusions In conclusion, we have measured the spectroscopic coherence time T?2 and the time constant T0 for the decay of Rabi oscillations in dc SQUID phase qubits 183 with variable coupling to the leads. We found that varying the isolation by an order of magnitude produced no signiflcant efiect on either T?2 or T0. However with comparable isolation, the aluminum qubit AL1 had a Rabi decay time that was two to three times longer than that of niobium device NB1 [23] with sputtered SiO2 wiring layers. These results imply that the leads were not the dominant source for decoherence in these qubits. Instead, our data are consistent with a local source of decoherence. 184 Chapter 7 Comparison of coherence times in dc SQUID phase qubits 7.1 Introduction Despite much recent progress in the use of superconducting devices for quan- tum computation [99], decoherence still presents a major challenge. Part of the problem is that there are many possible mechanisms that could cause decoherence and the picture of what is happening in real devices is still not entirely clear. For Josephson phase qubits [47, 48, 89, 91, 100, 101, 102], Martinis et al. [49] have proposed that dielectric loss and two-level uctuators in dielectrics are the primary cause of decoherence. They showed signiflcant improvement could be obtained by replacing lossy dielectrics with lower-loss materials. This is now widely believed to be the dominant mechanism. Consistent with this, Van Harlingen et al. [103] argued that while critical current uctuations would produce decoherence, the observed coherence times in ux and phase qubits were much shorter than would be expected from the level of critical current noise that has been typically observed in tunnel junctions [104, 105, 97]. Similarly, Martinis et al. [61] argued that charge noise should have a small impact on the coherence time of phase qubits due to their large junction capacitance. However in their dielectric loss paper [49], they argued that motion of charges was the dominant mechanism causing decoherence. In contrast, Bertet et al. reported that decoherence in their ux qubit came from the dc SQUID what was used to detect the ux state [46]. In principle, ux noise is another possible source of decoherence in phase qubits, as most are essentially rf or dc SQUIDs. In this chapter, I compare Rabi 185 oscillations and spectroscopic coherence times in three dc SQUID phase qubits. One device used a single-turn magnetometer conflguration (AL1), the second had a multi-turn magnetometer conflguration (NB1) and the third used a gradiometer conflguration (NBG), respectively. Although we did not perform a direct test on the gradiometer balance, the counter-wound conflguration should make it much less sensitive to spatially unform magnetic flelds than either of the magnetometers. 7.2 dc SQUID phase qubits without and with gradiometer loops Figure 7.1(a) shows a schematic of a dc SQUID phase qubit [47]. We refer to J1 as the qubit junction and J2 as the isolation junction. In this qubit design, junction J2 is needed to read out the state of J1 via tunneling to the voltage state [47]. J2 and inductor L1 also inductively isolate the qubit from current noise on the bias leads. By choosing L1 L2 + LJ2, where LJ2 is the Josephson inductance of the isolation junction, current noise will be mainly diverted through the isolation junction J2 rather than the qubit junction J1. Figures 7.1(b) and 7.1(c) show two of our SQUID phase qubits that have magnetometer loops. Device ALl is a single-turn dc SQUID magnetometer made from thin-fllm Al [see Fig. 7.1(b)]. We used photolithography and double-angle evaporation to form the loop and the Al=AlOx=Al tunnel junctions (See Ch. 4). The qubit junction has an area of 80 ?m2. Other than AlOx, no insulation layers were deposited on this device. Device NB1 [see Fig. 7.1(c)] is a thin-fllm Nb magnetometer with a 6-turn loop. The device was made by Hypres, Inc., from a Nb=AlOx=Nb trilayer using their 100 A/cm2 process [78]. The qubit junction has an area of 100 ?m2 and was measured by S. K. dutta [23] et al.. He applied a small magnetic fleld in the plane of the junctions to reduce the critical current of the device to about 30 ?A. Subsequent measurements on similar devices, which were not suppressed by 186 1 2 J1 J2 2 J1 Figure 7.1: (a) Schematic of dc SQUID phase qubit. I is the bias current, If is current for the ux bias, M is mutual inductance between the ux bias coil and the SQUID loop and 'a is ux applied to the SQUID loop. C1 and C2 are the capacitances of the qubit junction J1 and the isolation junction J2, respectively. Microwave source Iw is coupled to J1 through capacitor Cw. Photographs of (b) single-turn aluminum SQUID magnetometer AL1 and (c) 6-turn niobium SQUID magnetometer NB1. 187 J1J2 J1 J2 Figure 7.2: (a) Schematic of dc SQUID phase qubit with a gradiometer loop. I is the bias current, If is current for the ux bias, 'a is the applied ux in the SQUID loop. C1 and C2 are the capacitances of junctions J1 and J2. L1a and L1b are inductances of each coil of the gradiometer, M1a and M1b are mutual inductances between each coil, and the ux bias line. Microwave source Iw is coupled to J1 through capacitor Cw. (b) Photograph of dc SQUID phase qubit, gradiometer NBG. 188 Table 7.1: Parameters for SQUIDs NBG, NB1 and AL1. I01 and I02 are the critical currents of J1 and J2, respectively. Lj1(0) and Lj2(0) are the Josephson inductances of J1 and J2 when they are unbiased and fl = (I01 + I02)L='0, where L = L1 + L2 is the total inductance of the SQUID loop. gradiometer magnetometer magnetometer Parameters NBG NB1 AL1 I01 (?A) 23.0 33.8 21.275 I02 (?A) 3.8 4.8 9.445 C1 (pF) 4.1 4.4 4.1 C2 (pF) 2.0 2.2 2.1 Lj1(0) (pH) 13.9 9.7 13.2 Lj2(0) (pH) 84.9 68 44.5 L1 (pH) 4540 3530 1236 L2 (pH) 12 20 5 fl 34 66 19 magnetic fleld, yielded similar spectroscopic coherence times and Rabi decay times [23]. Device NBG (see Fig. 7.2) was made by Hypres from a Nb=AlOx=Nb trilayer using their 30 A/cm2 process. The qubit junction has an area of 100 ?m2. The SQUID has two 6-turn thin-fllm Nb coils wound in opposition to form a magnetic fleld gradiometer. To apply a net ux to the device, we placed a ux bias line on the right side of the device (closer to coil L1a than to L1b in Fig. 2). All three devices were made on silicon wafers with a layer of thermally grown silicon dioxide. Table 7.1 summarizes the devices parameters of AL1, NB1 and NBG. 7.3 Measurement of energy levels of NBG The experimental procedures were discussed in Chapter 4 but here I review them again brie y. Before making any measurements on a SQUID, I used a ux 189 shaking technique to initialize the ux state [86]. I then applied a simultaneous ux and current ramp to bias the qubit junction with current, and not the isolation junction. With this biasing scheme, junction J1 acts as an ideal phase qubit with the two lowest levels in a well of the tilted washboard potential forming the qubit states j0i and j1i. I monitored the qubit state by measuring the rate at which the system tunnels to the voltage state [47]; the flrst excited state j1i typically tunnels about 500 times faster than the ground state j0i. During the simultaneous current and ux ramp, I recorded the time at which the device escapes to the voltage state. I repeated this process on the order of 105 times to build up a histogram of escape events versus ramp time, which I then converted to escape rate versus current (for spectroscopy) or escape rate versus time (for Rabi oscillations). As an example, Fig. 8.19 shows the total escape rate of the qubit junction in device NBG as a function of the bias current I, with and without application of 6.6 GHz microwaves. Sweeping current through the qubit changes the energy level spacing adiabatically. When the microwaves come into resonance with the energy level spacing, transitions to the excited state occur and we see enhancement in the total escape rate. In Fig 8.19, two clear resonance peaks are visible, at around 21.57 ?A and 21.62 ?A, corresponding to j1i ! j2i and j0i ! j1i transitions. I measured the microwave response from 5.5 GHz to 8 GHz and flt the reso- nance peaks with a Lorentzian function to extract the peak locations and the half- widths. Figure 7.4 shows the spectrum of j0i to j1i transitions (circles), two-photon j0i to j2i transitions (squares) and j1i to j2i transitions (stars). I flt the spectra to a calculated single Josephson junction spectrum and found good agreement with data (see Fig. 7.4). The best flt to the spectrum yields the qubit junction critical current I01 = 21.7969 ?A and the capacitance C1 = 4.18 pF. 190 21.52 21.55 21.58 21.61 21.64 21.67 103 104 105 106 107 I (?A) ? (1/s ) |0> ? |1> |1> ? |2> Figure 7.3: Total escape rate ? vs. current I1 for device NBG at 80 mK. Dotted line is when 6.6 GHz microwaves are applied to the qubit junction and solid line is without microwaves. Two prominent peaks are seen when microwaves are applied, corresponding to the j0i to j1i transition and the j1i to j2i transition. 191 21.5 21.55 21.6 21.65 5.5 6 6.5 7 7.5 8 I1 (?A) f (GHz ) Figure 7.4: Resonance frequency vs. current for NBG at 80 mK. Circles are j0i to j1i transition data, squares are two-photon j0i to j2i transition data and stars are j1i to j2i transition data. Solid curves are from best flt to the energy spectrum of a single Josephson junction. 192 7.4 Measurement of T1 To determine T1 on the gradiometer dc SQUID qubit NBG, I performed a re- laxation measurement. Figure 7.5 shows the decaying escape rate due to relaxation. For this measurement, I applied resonant 6.6 GHz microwaves and then shut ofi the microwaves while monitoring the escape rate. I flt the resulting escape rate data to a decaying exponential function (as discussed in Chapter 5): f(t) = A exp[?t=t0] +B exp[?t=t1] + C exp[?t=t2]: (7.1) Using the ?2 method, I obtained a best flt with T1 ? t1 = 62 ns. The other fltting parameters are shown in Table 7.2. It is interesting that the fltting parameters were very similar to those of AL1 shown in Table 7.3, even though the two samples are quite difierent in their materials and design. This may support the possibility that the decay measured in this type of relaxation experiment may not be due to relaxation from j1i to j0i of the qubit junction, but instead by due to unrelated physical process. Or course both AL1 and NBG have Al/AlOx/Al tunnel barriers, and this may be what is determining the relaxation. As a check on T1, I also measured the thermally induced escape rate (See Fig. 7.6). To determine T1, I flt the data using a stationary 4-level master equation simulation. The fltting was done with a ?2 flt (described in Ch. 5), and the best flt parameters were T1 = 20 ns and T = 85 mK. As I found for device AL1, T1 ? 20 ns as determined from the escape rate?flt is signiflcantly shorter than T1 ? 50 ns from the relaxation measurement. 193 0 20 40 60 80 105 106 107 t (ns) ? (1/s ) Figure 7.5: Observed relaxation in the escape rate for NBG at 80 mK. Asterisk points show measured escape rate. The solid curve is the ?2 flts to the three exponential functions in Eq. 7.1. This relaxation measurement was done with 6.6 GHz high power microwaves (P = 12 dBm) after measuring Rabi oscillations. Table 7.2: Parameters for best flt of Eq. 7.1 to the data in Fig. 7.5 for device NBG. Parameters A (1/?s) B (1/?s) C (1/?s) t0 (? s) t1 (ns) t2 (ns) 6.6 GHz 1.114 0.104 14.03 188.1 62.1 5.908 194 21.52 21.54 21.56 21.58 21.6 21.62 21.64 21.66 103 104 105 106 107 I1 (?A) ? (1/s ) P0?0 P1?1 P2?2 P3?3 Figure 7.6: Total escape rate ? versus current I1 for qubit NBG at 80 mK. Crosses show the background escape rate and the solid curve with dots are the escape rate when 6.5 GHz microwaves are applied. Three peaks indicate j0i to j1i transition at I = 21.62 ?A, j1i to j2i transition at I = 21.57 ?A, and j2i to j3i transition at I = 21.52 ?A. The dashed curve is from a stationary 4-level master equation simulation. The thin solid curves show the components of the total escape rates, Pi?i, the probability of each level multiplied by the escape rate of the each level. The simulation parameters are T1 = 20 ns and T = 85 mK. 195 Table 7.3: Parameters for best flt of Eq. 5.40 to the data in Fig. 5.16 and Fig. 5.17 for device AL1. Parameters A (1/?s) B (1/?s) C (1/?s) t0 (? s) t1 (ns) t2 (ns) r = 1000 P = -20 dBm 0.264 0.902 0.037 188.1 52.34 5.002 r = 1000 P = -10 dBm 0.252 2.617 0.494 188.1 60.71 4.544 r = 220 P = 12 dBm 0.519 2.661 8.355 188.1 59.29 5.004 7.5 Measurement of T?2 To determine the spectroscopic coherence time T?2 [95] of the j0i ! j1i tran- sition, I obtained the low-power half-width at half-maximum ?I of the resonance peak and recorded its location I(f01). It is important to use low-power microwaves because of power broadening; the peak width increases steadily with power above a certain level determined by T1, T2 and inhomogeneous broadening. Repeating this procedure for a range of applied microwave frequencies yields f01 and ?I1 as a function of I1. The spectroscopic coherence time as a function of the frequency was then found from (see Chapter 4) T ?2 = dI1 df01 1 2??I1 : (7.2) The half-width at half-maximum ?I1 can be obtained by fltting the resonance peaks with a Lorentzian function. With the measured df01=dI1 from spectroscopy, T ?2 can then be evaluated. Figure 7.7 shows a plot of T?2 versus microwave frequency for gradiometer NBG, measured at 100 mK. For frequencies in the 6.0 GHz to 7.2 GHz range, T?2 varied between about 4 ns and 8 ns. Spectroscopic measurements on magnetometers NB1 and AL1 revealed comparable variations in T?2, from about 4 to 10 ns in the 196 6 6.2 6.4 6.6 6.8 7 7.2 0 2 4 6 8 10 f (GHz) T 2* (ns ) Figure 7.7: Spectroscopic coherence time T?2 of the j0i to j1i transition versus fre- quency for SQUID gradiometer NBG measured at 80 mK. 197 same frequency range [23] (see Chapter 6). Since T?2 is sensitive to low-frequency noise (inhomogeneous broadening), as well as pure dephasing and dissipation [95], we can conclude that the combined efiect of low-frequency noise, pure dephasing and dissipation is comparable in the three devices. 7.6 Rabi oscillations in gradiometer NBG and comparison with mag- netometers AL1 and NB1 I measured Rabi oscillations on resonance in NBG at 80 mK using microwaves with frequencies of 6.5 GHz, 6.6 GHz, 6.7 GHz, 6.8 GHz, and 6.9 GHz. Figure 7.8 shows the measured Rabi oscillations (dots) and the best flt curves (solid curves). I used a ?2 flt to the decaying oscillation function given by (see Ch. 6) ?fit = g0 + g1(1? e?(t?t0)=T 0 cos(?f1? t0g)) + g2(1? e?(t?t0)=Tback): (7.3) My flts revealed that the decay time T 0 varied from 8 ns to 13 ns as I changed the microwave frequency. However, there was no systematic dependance on the microwave frequency. Table 7.4 summarizes the decay time constants from the ?2 flts. The square of the Rabi frequency showed a linear dependence on the microwave power (see Fig. 7.9), as expected. Measurements of Rabi oscillations allow one to distinguish the efiects of low- frequency noise from dephasing processes. The idea is that the envelope decay time constant T0 of a Rabi oscillation is sensitive to noise at the Rabi frequency while the main efiect of noise at much lower frequencies (which acts like inhomogeneous broadening) is to change the shape of the envelope [61, 103]. This relative insen- sitivity of the Rabi oscillations to low-frequency noise is similar to the situation in a spin-echo measurement [61]. The envelope decay time constant T0, the energy 198 0 10 20 30 40 0 50 100 0 10 20 30 40 0 5 10 15 0 10 20 30 40 0 20 40 0 10 20 30 40 0 10 20 30 0 10 20 30 40 0 5 10 t (ns) ? (1/?s) Figure 7.8: Measurements of Rabi oscillations in the escape rates in gradiometer NBG at 80 mK. In each case the solid curve is a least-square flt to Eq. 7.3. From the top, the plots are Rabi oscillation for microwave frequencies of 6.5 GHz, 6.6 GHz, 6.7 GHz, 6.8 GHz, and 6.9 GHz. The corresponding decay time constants T? are 12.6 ns, 12.2 ns, 10.7 ns, 8.1 ns and 11.8 ns. 199 0 20 40 60 80 0 0.5 1 1.5 2 2.5 3 3.5 P (mW) (? /2 pi )2 (GH z2 ) Figure 7.9: Square of the Rabi frequency in NBG vs. microwave power for mi- crowaves with frequencies of 6.5 GHz (fllled circles), 6.6 GHz (square), 6.7 GHz (diamonds), 6.8 GHz (crosses), 6.9 GHz (stars), 7.0 GHz (asterisks) and 7.1 GHz (tilted crosses). The data was measured at 80 mK. 200 relaxation time T1, and the coherence time T2 are related by [63] 1 T 0 = 1 2T1 + 1 2T2 ; (7.4) when the Rabi oscillation is driven on resonance and inhomogeneous broadening can be neglected. Although spin-echo measurements are the best way to directly determine T2 and distinguish pure dephasing from inhomogeneous broadening, we were not able to measure spin echoes in these devices due to their relatively short coherence times, and Rabi oscillations provided a good alternative. Figure 7.10 shows typical examples of measured Rabi oscillations in the total escape rate for the three dc SQUID phase qubits. We applied microwave frequencies of 7.6 GHz for NB1 at 25 mK [see Fig. 4(a)], 7 GHz for AL1 at 80 mK [see Fig. 4(b)] and 6.5 GHz for NBG at 100 mK [see Fig. 4(c)]. The applied microwaves coupled capacitively to the qubit junction [See Fig. 1 and Fig. 2] and resonantly drove the qubit between j0i and j1i. In each case, we observed clear oscillations in the escape rate. In these plots, t = 0 indicates when the microwaves were turned on. The solid curves in Fig. 7.10 are least-square flts to Eq. 7.3. From the flts, I found T0 for the gradiometer NBG was about 12 ns, while for magnetometers AL1 and NB1, it Table 7.4: The decay time constant T 0 of Rabi oscillations in NBG measured at 80 mK for difierent microwave frequencies. f (GHz) T? (ns) 6.5 12.6 6.5 12.2 6.5 10.7 6.5 8.1 6.9 11.8 201 was about 27 ns and 12 ns, respectively (see Table 7.5). Density matrix simulations reveal that the escape rate we observe is dominated by a small population in j2i that escapes very rapidly (?2 ? 1010/s 1/T0), and this population is directly proportional to the occupancy of j1i which escapes much more slowly (?1 ? 107/s). While tunneling contributes to spectroscopic broadening [95], measurements over a wide range of conditions with difierent escape rates did not appear to alter T0 by more than about 30 % [23]. Figure 7.11 and Table 7.5 summarize T 0 values in devices AL1, NB1 and NBG. From the flgure and table, we see that the single-turn aluminum magnetometer AL1 had a substantially longer envelope decay time T0 than either the Nb magnetometer or the Nb gradiometer. I note that T2 < 2T1 in NB1 and NBG while in AL1 I see T2 ? 2T1. This suggests that niobium qubits seem to have additional dephasing sources beyond just dissipation. Since NB1 and NBG have SiO2/Si wiring dielectrics while AL1 has no wiring dielectrics and just SiO2 as a substrate, the additional dephasing source could be from the wiring dielectrics in the niobium devices. From this comparison, we can also safely conclude that T0 in our dc SQUID phase qubits is not being limited by spatially uniform ux noise and the materials seem to play a role in decoherence. Finally, from T0 ? 2T?2 (see Table 7.5), this suggests low-frequency noise is causing signiflcant inhomogeneous broadening of the resonance. In particular, tunneling could also cause T0 ? 2T?2 [96], but most of the data was taken deep in the well, so this is unlikely. 7.7 Conclusions In conclusion, we have measured the spectroscopic coherence times and Rabi oscillations in three dc SQUID phase qubits. One device was a Nb gradiometer with 6-turn wound and counter-wound coils, the second was an Al magnetometer made with a single-turn loop, and the third was a Nb magnetometer with a 6-turn coil. 202 0 10 20 30 40 0 10 20 30 40 50 t (ns) ? (1/ ?s ) 0 10 20 30 40 0 20 40 60 80 t (ns) ? (1/ ?s ) 0 10 20 30 40 0 3 6 9 12 15 18 t (ns) ? (1/ ?s ) (a) (b) (c) Figure 7.10: Measurements of Rabi oscillations in the escape rate in (a) single- turn magnetometer AL1 at 80 mK, (b) 6-turn magnetometer NB1 at 25 mK, (c) gradiometer NBG at 80 mK. In each case the solid curve is a ?2 flt to Eq. 7.3. 203 0 50 100 150 200 250 300 0 5 10 15 20 25 30 35 40 ?/2pi (MHz) T? (ns ) AL1 most (80 mK) AL1 poor (80 mK) NB1 most (25 mK) NB2 most (25 mK) NBG most (80 mK) Figure 7.11: The decay time constant T 0 of Rabi oscillations vs. Rabi frequency ?=2? of flve devices. Filled circles are T 0 for AL1 measured at 80 mK at the most isolated biasing point, rI = 1000 and asterisks are for AL1 at the poorly isolated biasing point, rI = 200. Open circles are T 0 for NB1 measured at 25 mK at the most isolated point, Open squares are T 0 for NBG measured at 80 mK at the most isolated point and a star indicates T 0 for NB2 measured at 25 mK at the most isolated biasing point. 204 Table 7.5: Summary of spectroscopic coherence time T?2, time constant T0 for decay of Rabi oscillation, relaxation time T1 and estimated T1 = 3T 0=4 that would occur if all decoherence was due to dissipation. gradiometer magnetometer magnetometer NBG NB1 AL1 T?2(ns) 4 - 8 4 - 10 4 - 10 T0(ns) 10 - 15 10 - 15 20 - 30 T1(ns) 15 15 20 T1 = 3T 0=4(ns) 8 - 11 8 - 11 15 - 23 The gradiometer did not show signiflcantly longer T0 or T?2 [79], and in fact the single-turn Al magnetometer showed a signiflcantly longer T0 than either the Nb gradiometer or Nb magnetometer. We conclude that spatially uniform ux noise is not a dominant source of decoherence in our phase qubits. There is a possibility of a local source of ux noise causing dephasing, which would not be nulled by a gradiometer. However, the observed dependence on frequency f01 of the either T 0 or T ?2 is not consistent with ux noise or critical current noise. I note that the results appear to be qualitatively consistent with a material-dependent decoherence mechanism such as dielectric loss from a distribution of 2-level charge uctuators. In this picture, the longer T? and T?2 of the Al device would be attributed to the absence of the thin-fllm lossy SiO2 dielectrics layers. Since the coherence times are still quite short, such loss as remains is still quite signiflcant. Since all three devices possess AlOx tunnel barriers as well as thermally grown SiO2 as the substrate, it is possible that these are the remaining sources of dissipation. 205 Chapter 8 The Cooper pair box as a coupling component in a quantum computer 8.1 Introduction: The Cooper pair box A Cooper-pair box is a device that can store a well-deflned integer number of Cooper pairs. It consists of a small superconducting island that is connected to ground via an ultra-small superconducting Josephson tunnel junction and to a gate voltage source via a capacitor [13] [see Fig. 8.1 (a)]. The phrase \ultra-small junction" refers to a junction with an area that is much less than 1 ?m2. When the total island capacitance is su?ciently small, the Coulomb energy associated with putting one pair on the island becomes big enough to suppress the tunneling. This suppression of tunneling due to an electrostatic energy barrier is called the Coulomb blockade efiect [21]. The Coulomb blockade efiect can be observed in both superconducting and normal ultra-small junctions. For normal junctions, the device is called a single-electron box. The scale of the charging energy required to place one pair on the island is set by Ec = (2e) 2 2C? (8.1) where C? = CJ +Cg is the total capacitance of the island, CJ is the capacitance of the ultra small junction, and Cg is the capacitance of the gate. C? is typically in the femto farad range. For an island with C? ? 2 fF, the associated Ec is ? 1K/kB. This implies that charge will not be very likely to tunnel through the junction if it is below about 1K, at least for certain values of gate voltage. For a Cooper pair box, the energy associated with tunneling of Cooper pairs 206 Vg Vg Cg Cg ???a (a) (b) CJ CJ 2 CJ 2 Figure 8.1: Cooper-pair box with (a) single ultra-small Josephson tunnel junction and (b) dc SQUID with two ultra-small junctions in parallel. CJ is the capacitance of the superconducting ultra-small junction, Cg is the gate capacitance, and Vg is the gate voltage. For the dc SQUID Cooper pair box, EJ of the SQUID can be tuned by applying a magnetic ux 'a to the loop. 207 through the ultra-small junction, the Josephson coupling energy must be included in the Hamiltonian in addition to the charging energy. The Josephson coupling energy can be found by calculating the power cost for supercurrent to tunnel through the junction [13] P = IV = Ic sin ? '0 2? d dt ? : (8.2) The Josephson coupling energy E is then: E = Z t Pdt = Z t Ic sin ? '0 2? d dt ? dt = '02?Ic Z sin d = ?EJ cos (8.3) where EJ = '02?Ic (8.4) is the Josephson energy. The Josephson coupling energy shows how strongly the two superconducting wavefunctions on opposite sides of a tunnel junction are coupled to each other. Equation 8.4 reveals that Josephson energy EJ only depends on the critical current. Figure 8.1(a) shows a schematic of the Cooper pair box. The small area between the ultra-small junction and the gate is the \island". The number of Cooper pairs on the island can be varied by applying a voltage bias to the gate. Fig. 8.1(b) shows a Cooper pair box with two ultra-small junctions in parallel. The design looks like there is a split in the junction so the device is also called a split Cooper pair box. The two junctions form a small SQUID and its critical current can be adjusted by applying a magnetic ux 'a to the loop. This enables us to tune the efiective 208 Josephson energy, EJ of the split Cooper pair box : EJ = '02?Ic = '0 2?2I0 cos( 'a '0 ) (8.5) where '0 is the ux quantum, 'a is the applied magnetic ux, Ic is the critical current of the SQUID when 'a is applied and 2I0 = I01 + I02 is the maximum critical current of the SQUID. Since the Josephson energy afiects the resonance frequency of the Cooper pair box, the frequency can be tuned to a limited extent by having a ux bias source. The Cooper pair box has been used as a charge qubit [93, 106] and a tunable circuit element [107, 108]. Recently the Yale group [41] showed that a Cooper pair box that was biased at the degeneracy point (\sweet spot") and coupled to an LC resonator readout had a coherence time of 2 ?s, one of the longest coherence times so far seen in any superconducting qubit. The main application we envision here for a Cooper-pair box is as a switchable coupling element (variable capacitor) between two qubits in a quantum computer. To manipulate speciflc pairs of qubits or individual qubits, it?s useful to be able to turn the coupling on and ofi. In this chapter, I introduce the basic physics of the Cooper pair box and examine how the Cooper pair box afiects a phase qubit when they are coupled together. 8.2 Charging energy of a Cooper pair box with two voltage bias sources Figure 8.2 shows a schematic of a Cooper pair box with two voltage sources; a voltage bias and a gate voltage source. This conflguration difiers from the conven- tional design where there is only a gate voltage. This conflguration will be useful for flnding the efiective capacitance of the box with respect to the bias voltage, and 209 Vg Vc Vb CJ Cg C0 Figure 8.2: Cooper pair box with a voltage bias source Vb and a gate voltage source Vg. Note that Vb is connected the ultra-small junction directly. The box consists of one superconducting ultra-small junction with capacitance CJ and two capacitors C0 and Cg. The energy of the system depends on Vb and Vg as well as the excess number of Cooper-pairs n on the box, where each pair has charge ?2jej. Vc is the island potential. 210 this will help us understand what happens when the box is coupled to a large-area Josephson junction phase qubit. In this case C0 acts as a coupling element to the voltage source Vb, or ultimately the phase qubit. The derivation of the charging energy of a Cooper pair box can be found in Ref. [13]. Here I calculate the charging energy of the Cooper pair box with two voltage sources. The result reduces to the conventional Cooper pair box when the bias voltage Vb is zero. To calculate the charging energy, we need to know the electrostatic potential Vc of the island. If the number of excess Cooper pairs on the island is n, the total of all the charges on each capacitor plate of the island must obey: ?2en = Cg(Vc ? Vg) + CJ(Vc ? Vb) + C0Vc (8.6) so that the electrostatic potential Vc of the island becomes Vc = ?2en+ CJVb + CgVgC? (8.7) where C? = CJ + C0 + Cg, is the sum of all the capacitances linked to the island. Note that I use e as positive number: e = 1:6 ? 10?19C throughout the thesis so that the charge of a Cooper pair is -2e. Excess charge appear on the island by tunneling through the ultra-small junc- tion. Changes in n cause changes in the island potential, and the resulting charge is redistributed to each capacitor. The voltage sources have to do \work" during this charge transfer. From Eq. 8.7, the change in Vc when one Cooper pair is moved onto the island is, ?Vc = ? 2eC? : (8.8) The work W done by a voltage source that moves a charge Q across voltage V is W = QV. The work ?Wb done by the bias source Vb and the work ?Wg done by Vg 211 when one excess Cooper pair oves onto the island is ?Wb = (?Qj ? 2e)Vb = (2eCJC? ? 2e)Vb (8.9) ?Wg = ?QgVg = 2e CgC?Vg (8.10) W = ?Wb + ?Wg = 2e ?Cg C?Vg + CJ C? ? 1 ? Vb ? (8.11) where ?Qb and ?Qg are charges that must be supplied by Vb and Vg, respectively when one Cooper pair is introduced on the island. For n excess Cooper pairs, the total work done by Vb and Vg is simply Wn which is Wn = n?W = 2en ?Cg C?Vg + CJ C? ? 1 ? Vb ? : (8.12) The electrostatic energy, UE stored in the capacitors is UE = 12[CJ(Vc ? Vb) 2 + C0Vc2 + Cg(Vg ? Vc)2] + CgCJ (8.13) The total electrostatic free energy of the island is the electrostatic energy stored in the capacitors minus the work done by the voltage sources. After some work, I flnd the n-dependent charging energy U U = UE ? nW = 12C? ?4e2n2 + 4en((C0 + Cg)Vb ? CgVg) + CgCJ(Vb ? Vg)2 + C0(CJV 2b + CgV 2g ) ? (8.14) 212 This can also be written in the form U = 12C? ?4e2n2 + 4en((C0 + Cg)Vb ? CgVg) + CgCJ(Vb ? Vg)2 + C0(CJV 2b + CgV 2g ) ? + (C0Vb + CgVb ? CgVg)2 ? (C0Vb + CgVb ? CgVg)2 = 12C? [2en+ (C0 + Cg)Vb ? CgVg] 2 ++G(Vb; Vg) =(2e) 2 2C? ? n+ (C0 + Cg)Vb2e ? CgVg 2e ?2 +G(Vb; Vg) =Ec(n+ nb ? ng)2 +G(Vb; Vg); (8.15) where I deflne, ng = CgVg2jej (8.16) nb = (C0 + Cg)Vb2jej (8.17) Ec = 4e 2 2C? (8.18) and G(Vb; Vg) is an n-independent energy term G(Vb; Vg) = 12C? [(C0Vb+Cg(Vb?Vg) 2+CgCJ(Vb?Vg)2+C0(CJV 2b +CgV 2g )] (8.19) which is a function of the bias voltage and the gate voltage. This term doesn?t afiect the average number of excess Cooper pairs hni or any dynamics of the Cooper pair box. Considering the above expression for U, I see that increasing Vg will cause Cooper pairs to be induced to tunnel onto the island (n increases), since this will lower the total energy. Similarly, when we increase Vb, Cooper pairs leave (n de- creases) to lower the energy. 213 8.3 Hamiltonian and energy bands of the Cooper pair box The total energy of a Cooper pair box with two voltage sources is the sum of the charging energy and the Josephson coupling energy, given by H = Ec(n+ nb ? ng)2 ? EJ cos (8.20) where the Josephson energy is EJ = '02?Ic; (8.21) is the gauge invariant phase difierence across the junction, C? = CJ +C0 +Cg, Ic is the critical current of the junction and I have dropped the term G(Vb; Vg) since it is an independent of n. Here n and are dynamical variables. When we describe the Cooper pair box using quantum mechanics, H, n and become operators, and we can write the Hamiltonian as H^ = Ec(n^+ nb ? ng)2 ? EJ cos ^: (8.22) There are a few things that should be remembered at this point. First, Eq. 8.22 is a fairly general expression for the Hamiltonian of a Josephson junction with nb and ng terms added due to the extra capacitors. However, the Hamiltonian lacks a current bias term since there is no current source in this circuit. Second, the superconducting gap energy ? is another important energy scale but it does not explicitly appear in the Hamiltonian. This happens because I assumed that ? Ec and that no quasiparticle are present. In this case, the tunneling processes only involve Cooper pairs and the dynamics of the system has periodicity 2e. The dynamics of the Cooper pair box are largely determined by the relative sizes of the energies Ec and EJ . If the Josephson coupling energy dominates the Hamiltonian (EJ Ec), the Josephson efiect dominates and large currents can ow 214 through the junction. In this case, N^ is not a good quantum number but ^ is. For EJ ? Ec, the charging energy dominates the dynamics and N^ is a good quantum number. The Cooper pair box that I deal with in this chapter has EJ ? Ec. So here I will describe the dynamics in terms of N^ . In the low temperature limit kBT << EJ << Ec, the Cooper-pair box mostly stays in the two lowest energy states. If I restrict the gate voltage to the range 0 < Vg < e/Cg, the Hamiltonian can then be represented by a 2 ? 2 matrix h0j bHj0i = H00 (8.23) h0j bHj1i = H01 (8.24) h1j bHj0i = H10 (8.25) h1j bHj1i = H11 (8.26) where j0i and j1i are number states corresponding to the number of Cooper pairs on the island being n = 0 and 1. Note that due to the Josephson coupling energy the number states are not exact eigenstates of the Hamiltonian anymore, i.e. N^ and H^ do not commute. To express the Josephson coupling energy in the number basis, let?s start with cos ^ = e i ^ + e?i ^ 2 (8.27) where ^ satisfy the relations: ei ^jNi = jN ? 1i (8.28) e?i ^jNi = jN + 1i: (8.29) Thus ei ^ is a translation operator for N^ , just as e?ixp^=~ is the translation operator for a free particle with momentum p^. This tells us that by changing , n can increase or 215 decrease. This makes sense intuitively since sin ? is current. Fundamentally, this occurs because the number of Cooper pairs in the island can change by Josephson tunneling. From Eqs. 8.27, 8.28 and 8.29, One sees that in the number basis, the Josephson coupling energy term introduces ofi-diagonal elements in the Hamiltonian. Combining the charging energy and the Josephson coupling energy, the Hamiltonian matrix in the number basis becomes H^ = 0 B@ Ec(0 + nb ? ng)2 ?EJ2 ?EJ 2 Ec(1 + nb ? ng)2 1 CA : (8.30) By diagonalizing H^, we can obtain the energy eigenvalues and energy eigen- states as well as the average number hNi of Cooper pairs on the island. The number operator N^ is deflned as N^ jNi = njNi (8.31) where n is the excess number of Cooper pairs on the island. The energy eigenstates jE0i and jE1i in the number basis can be written as jE0i = a11j0i+ a21j1i (8.32) jE1i = a12j0i+ a22j1i (8.33) The average number hNi of Cooper pairs is hNi = h?j bN j?i (8.34) where j?i is a general wavefunction. Figure 8.3 shows the calculated energy levels [Fig 8.3(a)] and the average number of Cooper pairs in the ground state of the box [Fig 8.3(b)]. Both curves are plotted with respect to the normalized gate charge ng. The simulation parameters 216 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 ng E/ h (G H z) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ng < N > (a) (b) Figure 8.3: (a) Simulation of the energy vs. normalized gate charge ng and (b) the average number h0j bN j0i of Cooper pairs in the ground state of the box for difierent EJ . In both flgures, the dashed curve is for Ec/EJ = 27 and the solid curve is for Ec/EJ = 4.1. The simulation parameters are shown in Table 8.1 217 Table 8.1: Parameters of Cooper pair box for the simulation shown in Fig. 8.3. Device Blue dashed curve Red solid curve A (nm2) 100 ? 100 100 ? 100 CJ (fF) 0.88 0.88 C0 (fF) 1 1 Cg (fF) 0.001 0.001 Cs (fF) 0 0 Ec=h (GHz) 41 41 Ic (nA) 3 10 EJ=h (GHz) 1.5 10 are shown in Table 8.1. In Fig 8.3(a), the dashed curve is for Ec/Ej = 27, i.e. the charging energy dominates. In this case, the Josephson coupling energy acts as a small perturbation to the charging energy. Thus the energy E is nearly quadratic in ng except for a small avoided crossing at ng = 0.5. At ng = 0.5, the energy gap between the ground state and the flrst excited state is a minimum and equals EJ . For this simulation with Ec/Ej = 27, EJ/h = 1.5 GHz. At ng = 0 or 1, the gap is largest, and is given by Ec/h = 41 GHz. If I increase EJ/h to 10 GHz [solid curve in Fig. 8.3(a)], the gap at ng = 0.5 opens up and the energy curves atten. Figure 8.3(b), shows the average number hNi of Cooper pairs on the island when the box is in its ground state for EJ/h = 1.5 GHz and Ec/EJ = 27 (dashed curve) and for EJ/h = 10 GHz and Ec/Ej = 4.1 (solid curve). When the charging energy dominates, i.e. Ec/EJ = 27, hNi (the dashed curve) varies more rapidly with ng than when Ec/Ej = 4.1 (the solid curve). This is because when EJ is larger, N has a larger uncertainty. In practice, a shunt capacitor can be added across the ultra-small junction to suppress the charging energy. This provides some additional ability to control Ec. Figure 8.4 shows results from a simulation for this situation; the dashed curve is 218 Table 8.2: Parameters of the Cooper pair box for simulation shown in Fig. 8.4 Device Blue dashed curve Red solid curve A (nm2) 100 ? 100 100 ? 100 CJ (fF) 0.88 0.88 C0 (fF) 1 1 Cg (fF) 0.001 0.001 Cs (fF) 0 10 Ec/h (GHz) 41 1.5 Ic (nA) 3 3 EJ/h (GHz) 1.5 1.5 without a shunt capacitor and the solid curve is with an added a shunt capacitor Cs. This decreases Ec/Ej while keeping EJ flxed. Fig. 8.4(a) shows the total energy vs ng. When the charging energy is reduced by adding Cs (solid curve), the gap at ng = 0:5 stays the same but the curve becomes smoother. Figure 8.4(b) shows that the average number hNi of pairs in the ground state varies much more smoothly with ng for the small Ec/Ej, as expected. 8.4 Calculation of efiective capacitance Here I propose a simple semi-classical way to calculate the efiective capacitance of a Cooper-pair box and demonstrate that a Cooper-pair box can be used as a variable capacitor. One possible experiment to show that a box will work as a variable capacitor is to connect a phase qubit (large area Josephson junction) in parallel with a Cooper-pair box (see Fig. 8.5). The total capacitance across the junction determines the energy levels of the phase qubit. Thus changes in Ceff will shift the energy levels, which we can detect by microwave spectroscopy. In this section, I examine how the coupled Cooper pair box changes the energy levels of a 219 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ng < N > 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 ng E\ h (G H z) (a) (b) Figure 8.4: (a) Simulation of energy vs. normalized gate charge ng and (b) average number hNi of Cooper pairs in the ground state with and without a 10 fF shunt capacitor. In both flgures, the dashed curve is for Ec/EJ = 27 and the solid curve is for Ec/EJ = 4.4. The simulation parameters are shown in Table 8.2 220 Vg Vc Vb CJ Cg C0 VbCeff +Q -Q +QJ -QJ (a) (b) Figure 8.5: (a) A Cooper-pair box and an equivalent efiective variable capacitor. CJ , C0, Cg are capacitance of the superconducting small junction, coupling capacitance and gate capacitance, respectively. Vb and Vg are the bias voltage and the gate voltage, Vc is the island potential, and QJ is the charge stored in the junction capacitance. (b) The equivalent variable efiective capacitor Ceff . 221 phase qubit. 8.4.1 Efiective capacitance: deflnition A Cooper pair box with a bias voltage and a gate voltage can act as an efiective capacitance which varies with the gate voltage. The ability to vary the efiective capacitance of the box makes it potentially useful as a coupling element between superconducting qubits. If a total charge Q is sent from the battery Vb into a Cooper pair box, the Q will be split. Some stays on the positive plate of the small junction and the rest tunnels through the junction and onto the island. We can deflne the efiective difierential capacitance of this system [see Fig 8.5(b)] as the rate at which charge Q changes with the bias voltage Vb, Ceff = ?hQi?Vb (8.35) where Vg is flxed. In this expression hQi is the charge transferred from the battery Vb. We can write hQi = CJhVb ? Vci ? 2jejhNi (8.36) where Vc is the potential of the island. Thus hQi is just the sum of the charge Qj that is on the positive plate of the junction and the charge ?2jejhNi that tunneled through the junction onto the island. Vc is a function of Vg and Vb and depends on the average number of pairs hNi on the island. Examination of the circuit shows hQi = CJC? (2jejhNi+ (C0 + Cg)Vb + CgVg)? 2jejhNi: (8.37) 222 Substituting Eq. 8.37 into Eq. 8.35, we flnd Ceff = @hQi@Vb (8.38) = (C0 + Cg)C? (CJ ? 2jej @hNi @Vb ) (8.39) Note that since hNi depends on Vg and Vb, Eq. 8.39 gives a non-linear voltage dependance to Ceff . The maximum Ceff occurs at ng = 0:5 when Vb ? 0, which is where n = 0 and n = 1 have the same energy. 8.4.2 Efiective capacitance: simulation To see how large the efiective capacitance is under typical circumstances, I simulated a device with C0 = 10 fF, CJ = 0:57 fF, Cg = 1 aF, Ic = 0:64 nA, and Ec=Ej = 22.97 at various temperatures. To use Ceff as a variable coupler, it is important to maximize Ceff and obtain a high on-ofi ratio. Since Vb and Vg both control the charge transfer through the junction, in principle we can make Ceff maximum or minimum by applying appropriate Vb and Vg or both. However, if we are going to use the box as a variable capacitor and couple it across a phase qubit, then we need to take Vb = 0; since no voltage will be present across the qubit. Figure 8.6(a) shows a plot of hNi vs ng at Vb = 0. At Vb = 0 the number of pairs changes sharply as a function of ng near ng = 0.5. If Vb increases, the maximum Ceff point moves to a difierent Vg point. Plotting hNi versus nb yields a curve which is very similar to Fig. 8.6(a), except with opposite sign and slope. Due to thermal excitation the excited state gets occupied and @hNi=@Vb be- comes smaller at higher temperature. Figure 8.6(b) shows the temperature depen- dence of Ceff at 20 mK, 50 mK and 100 mK. The maximum capacitance change is 80 fF at 20 mK. In comparison, at zero temperature Averin et al. [107] showed that j@hni=@nbj ? Ec=EJ at the degeneracy point. For our case C?=C0 ? 1 and so from 223 0 0.2 0.4 0.6 0.8 1 ng 0 0.2 0.4 0.6 0.8 1 < N > 0 0.2 0.4 0.6 0.8 1 ng 0 20 40 60 80 C e ff (a) (b) (fF ) Figure 8.6: (a) Simulation of < N > vs ng at Vb = 0 at 20 mK (solid curve), 50 mK (dashed curve) and 100 mK (dashed-dot curve). Maximum tunneling of Cooper- pairs occurs at ng = 0.5. (b) Simulation of the efiective capacitance at 20 mK (solid curve), 50 mK (dashed curve) and 100 mK (dashed dot curve). The parameters are CJ = 0.57 fF, Cg = 1 aF, C0 = 10 fF, the critical current of the ultrasmall junction Ic = 0.64 nA, and Ec=Ej = 22.97. The maximum Ceff is obtained at at ng = 0.5 for nb = 0. 224 Eq. 8.38 we flnd Ceff jT!0 ? C0 EcEJ ? 4?e 2 '0 1 I0 (8.40) which gives Ceff ? 200 fF at ng = 0:5. This equation tells us that Ceff will be maximized by making I0 smaller. As we will see below, this is only part of the story, and in fact we will flnd that one should not make I0 too small so that the energy level of the Cooper pair box is comparable to that of the phase qubit if the capacitance needs to work at high frequencies. 8.5 Cooper pair box coupled to a phase qubit The motivation for this work was to use a Cooper pair box as a tunable coupling between two phase qubits. A Cooper-pair box coupled to a Josephson junction phase qubit can be thought about in a rather simple way. In the previous section, I showed that the Cooper pair box can change its efiective capacitance. Once the Cooper pair box is connected as a variable capacitor to a Josephson junction phase qubit, it is possible to tune the energy levels of the Josephson junction, since the energy levels depend on the total capacitance of the junction. It is important to see how the box behaves when it is coupled to the Josephson junction. In this section I show the result of energy level calculations and predict what we will expect from the spectroscopy experiment on the Josephson junction phase qubit. The derivation of the Hamiltonian and energy level calculations were initially done by Dr. Frederick W. Strauch and I will reproduce his calculations here. 225 Vg CJ Cg C0 CQIQ Ib I J I Q VJ VQ VC Figure 8.7: Circuit schematic of a Cooper pair box coupled to a Josephson junction. 226 8.5.1 Hamiltonian of the coupled box and junction Figure 8.7 shows the circuit diagram of a Cooper box coupled to a Josephson junction. Index J indicates the small junction in the Cooper pair box and Q indicates the Josephson junction (qubit). Starting from the Josephson equations, I = I0 cos (8.41) V = '02? _ : (8.42) Note that V is the voltage across the junction. Applying Kirchhofi?s laws at each node in the circuit of Fig. 8.7 gives current equations Ib = IJ + IQ (8.43) IJ = CJ( _VJ ? _Vc) + I0J sin J (8.44) IQ = CQ _VQ + I0Q sin Q (8.45) IJ = Cg( _Vc ? _Vg) + C0 _Vc (8.46) and VJ = VQ: (8.47) I can also apply the ac Josephson relations and get VJ ? Vc = '02? _ J (8.48) VQ = '02? _ Q (8.49) where VJ ? Vc is the voltage across the small junction, VQ is the voltage across the qubit (large area Josephson) junction, and Vc is the voltage of the island. I0J and I0Q are the critical currents of the small junction in the Cooper pair box (indexed as J) and the qubit junction (indexed as Q). J and Q are the phase difierences 227 associated with the small junction and the qubit junction, respectively. CJ ; CQ and Cg are the capacitance of the small junction, the qubit junction and the gate capacitor. Ib is the bias current. Considering Eqs. 8.43 - 8.49, we flnd the equations of motion for j and Q : CJ '02? _ j + I0J sin j = '0 2? (Cg + C0)( ?Q ? ?J)? Cg _Vg (8.50) CJ '02? _ Q + I0Q sin Q = Ib ? '0 2? (Cg + C0)( ?Q ? ?j) + Cg _Vg (8.51) If we had the Lagrangian L, we could have derived these equations of motion from Lagrange?s equations d dt @L @ _ J ? @L @ J = 0 (8.52) d dt @L @ _ Q ? @L @ Q = 0: (8.53) Comparing Eq. 8.50 to Eq. 8.52 and Eq. 8.51 to Eq. 8.53, we see that the following L works: L = 12('0) 2(CJ? _ J2 + CQ? _ Q2 ? 2Cc? _ J _ Q) +'0CgVg( _ J ? _ Q) + EJJ cos J + EJQ cos Q + IbIcQ Q: (8.54) where CJ? = CJ + C0 + Cg (8.55) CQ? = CQ + C0 + Cg (8.56) Cc? = C0 + Cg (8.57) EJJ = '02?IcJ (8.58) EJQ = '02?IcQ (8.59) 228 I next introduce the generalized momenta pj = @L=@ J and pQ = @L=@ Q. They have commutation relations with J and Q [ J ; pJ ] = i~ (8.60) [ Q; pQ] = i~ (8.61) We also can use number operators to describe the momenta: pJ = ~nJ (8.62) pQ = ~nQ (8.63) where nJ and nQ are number of Cooper pairs passing through the small junction in the box and the large area Josephson junction, respectively. Then from the Hamil- ton?s equation H = Pi pi i ? L, the Hamiltonian of the coupled system becomes H(nJ ; nQ; J ; Q) = 4EcJ(nJ ? ng)2 + 4EcQ(nQ + ng)2 + 8Ecc(nJ ? ng)(nQ + ng) ? EJJ cos J ? EJQ(cos Q + IbIcQ Q) (8.64) where EcJ = e 2CQ? 2(CJ?CQ? ? C2c?) (8.65) EcQ = e 2CJ? 2(CJ?CQ? ? C2c?) (8.66) Ecc = e 2Cc? 2(CJ?CQ? ? C2c?) (8.67) Again, the subscript Q indicates the current-biased Josephson junction. EcJ and EcQ are the charging energies of the small junction in the Cooper pair box and the Josephson junction qubit and Ecc is a coupling energy between the Josephson 229 junction and the Cooper-pair box associated with the capacitor C0. 8.5.2 Solving the coupled Hamiltonian using the Jaynes-Cummings model. With the Hamiltonian in Eq. 8.64, I can now flnd the energy spectrum of the coupled system. I will assume that I need only include the two lowest energy levels of the Cooper pair box. For the Josephson phase qubit, I will use the Harmonic oscillator approximation. In this case, the Hamiltonian in Eq. 8.64 can be shown to have a very similar form to the Jaynes-Cummings Hamiltonian [109, 110] H = 12? z ? 1 2? x + ~!0 aya+ 12 ? + ? z(a+ ay): (8.68) where x; y and z are the Pauli matrices for the Cooper pair box and a and ay are the annihilation and creation operators for the Josephson phase qubit. To get Eq. 8.68, I rewrite the Hamiltonian given by Eq. 8.64 using a new set of conjugate variables, x and p for the Josephson junction and the Pauli matrices for the Cooper pair box. I use the following transformation, nJ = 12(1? z) (8.69) cos J = 12 x: (8.70) for the Cooper pair box. For the Josephson phase qubit, I transform nQ and Q into x and p given by x = Q ? sin?1 IbIcQ = ?i 4EcQ ~!0 ?1=2 (a? ay) (8.71) p = nQ + ng + EccEcQ 1 2 ? ng ? = ? ~!0 16EcQ ?1=2 (a+ ay): (8.72) 230 Using a harmonic approximation for the washboard potential of the Josephson junc- tion qubit ? cos Q ? IbIcQ Q ? 1 2 s 1? Ib IcQ ?2 ( Q ? sin?1 IbIcQ ) 2: (8.73) Substituting Eqs. 8.69, 8.70, 8.71, 8.72 and 8.73 into Eq. 8.64, I obtain H = 4EcJ ng ? 12 ? z?12EJJ x+~!0 aya+ 12 ? +4Ecc s ~!0 16EcQ z(a?a y): (8.74) Comparing 8.74 to Eq. 8.68, I can identify: ? = 8 EcJ + E 2 cc EcQ ? ng ? 12 ? (8.75) ? = EJJ (8.76) ? = Ecc(~!0=EcQ)1=2: (8.77) The flrst two terms in Eq. 8.74 correspond to the Hamiltonian of the Cooper pair box. The third term ~!0(aya + 12) corresponds to the Hamiltonian of the Josephson phase qubit in the harmonic approximation. Here, the plasma frequency !0 is given by !0 = p8EcQEJQ ~ ? 1? Ib IcQ ??1=4 (8.78) The last term, ? z(a + ay), is the coupling energy term between the Cooper pair box and the Josephson phase qubit. From Eqs. 8.57, 8.67 and 8.77, we see that for Cg ? C0 ? CJ , the coupling energy is determined by C0=CJ ; big C0 gives strong coupling. 231 8.5.3 Calculating the energy levels of the coupled Hamiltonian using perturbation theory. The coupling energy term can be treated as a perturbation if it is much smaller than the energy of the uncoupled Cooper pair box or the uncoupled Josephson phase qubit, which is the case of interest here. Thus the Hamiltonian in Eq. 8.68 can be viewed as H = H0 +H 0 (8.79) where H0 is a sum of the uncoupled Hamiltonians of the Cooper pair box and the Josephson phase qubit H0 = 12? z ? 1 2? x + ~!0 aya+ 12 ? (8.80) and H 0 is the perturbation, H 0 = ? z(a+ ay): (8.81) The eigenstates of the unperturbed Hamiltonian H0 are jn;+i = jni(cos j0i+ sin j1i) (8.82) jn;?i = jni(? sin j0i+ cos j1i) (8.83) where jni is the n-th harmonic oscillator state of the phase qubit, j0i and j1i are the two number states of the Cooper pair box, and is [109] tan = ?? p?2 +?2 ? : (8.84) Using the eigenstates shown in Eqs. 8.82 and 8.83, I calculate the energy shift due to the perturbation. The unperturbed (zeroth order) energy eigenstates are 232 given by E(0)n;? = ? 1 2 p ?2 +?2 + ~!0 n+ 12 ? : (8.85) And the flrst order energy shift yields zero E(1)n;? = hn;?j? z(a+ ay)jn;?i = 0 (8.86) since hn;?ja+ ayjn;?i = 0. The second order energy shift is given by E(2)n;? = ?2 X i;j 6=n;? jhn;?j? z(a+ ay)ji; jij2 E(0)n;? ? E(0)i;j : (8.87) The matrix elements are hlj(a+ ay)jmi = pm?l;m?1 + pm+ 1?l;m+1 (8.88) and h+j zj+i = ?h?j zj?i = (cos2 ? sin2 ) (8.89) h+j zj?i = ?2 sin cos : (8.90) The second order energy shift is then: E(2)n;? = ?2 ? 4 sin2 cos2 ~!0 ? (2n+ 1) p?2 +?2 ?2 +?2 ? (~!0)2 ? sin2 ? cos2 ~!0 ! : (8.91) 8.5.4 Energy level spacings I can now calculate the energy level spacing between the shifted Josephson phase qubit j0i state and j1i from ~!01;? = (E(0)1;? + E(2)1;?)? (E(0)0;? + E(2)0;?) (8.92) 233 which yields ~!01;? = ~!0 ? 8?2 sin2 cos2 p?2 +?2 ?2 +?2 ? (~!0)2 (8.93) I can now use the trigonometric relations sin 2 = 2 tan 1 + tan2 (8.94) and cos 2 = 1? tan 2 1 + tan2 (8.95) along with tan = ?? p?2 +?2 ? : (8.96) to express all terms in Eq. 8.93 in terms of ? and ?. I obtain the energy level spacing ~!01;? as ~!01;? = ~!0 ? 2? 2p?2 +?2 (?2 +?2)(?2 +?2 ? (~!0)2) : (8.97) 8.5.5 Energy level spacings: degenerate case. For the case ~!0 = p?2 +?2, I have to use degenerate perturbation theory. When ~!0 = p?2 +?2, I flnd E(0)n;+ = 1 2 p ?2 +?2 + ~!0 n+ 12 ? = (n+ 1)~!0 (8.98) and E(0)n+1;? = ? 1 2 p ?2 +?2 + ~!0 n+ 1 + 12 ? = (n+ 1)~!0 (8.99) which makes jn;+i and jn + 1;?i degenerate. The perturbation Hamiltonian H 0 removes the degeneracy. To calculate the splitting, I span the perturbation Hamil- 234 Table 8.3: Parameters for energy level simulation of a Cooper pair box coupled to a Josephson junction phase qubit. CJ 10 fF CQ 4.078 PF C0 10 fF I0J 10 nA I0Q 22.213 ?A EcJ 0.95 GHz EcQ 0.0048 GHz Ecc 0.0024 GHz tonian H 0 with the degenerate states jn;+i and jn+ 1;?i yielding H 0 ? 0 B@ E(0)n;+ ??= p?2 +?2 ??=p?2 +?2 E(0)n+1;? 1 CA (8.100) where hn;+jH 0jn+ 1;?i = ?2pn+ 1 sin cos = ??p?2 +?2 : (8.101) Diagonalizing Eq. 8.100, I obtain the energy splitting E 0? = 1 2 ? E(0)n;+ + E(0)n+1;? ? r (E(0)n;+ ? E(0)n+1;?)2 + 4?2?2 ?2 +?2 ! : (8.102) From Eq. 8.102, for the lowest two states, the energy level spacing becomes ~!? = E 0? ? E(0)0;? = 1 2~! + 1 2 p ?2 +?2 ? 12 ? (~! ? p ?2 +?2)2 + r 4?2?2 ?2 +?2 ! : (8.103) The energy level spacing due to the degenerate energy splitting in Eq. 8.103 can also be applied to the near-resonant case where ~!0 ? p?2 +?2. 235 Figures 8.8 and 8.9 show energy level spacings calculated from Eq. 8.103. Fig- ure 8.8 shows energy level spacings and splittings at the bias current Ib = 0.989I0Q. Figure 8.8 shows energy level spacings and splitting at the bias current Ib = 0.991I0Q. Note that the energy level spacing of the Josephson phase qubit decreased as Ib in- creased; as expected. Comparing Fig. 8.9 to Fig. 8.10, I notice that the phase qubit energy has been shifted more in Fig. 8.10 at around ng = 0.5. This is because ~!0 in simulation in flg. 8.10 is more close to EcJ and EJJ of the Cooper pair box. The efiective capacitance model would apply for EcJ > ~!0 where the Cooper pair box adds an efiective capacitance to the phase qubit. This results in decreasing the plasma frequency of the phase qubit. In flg. 8.9 and Fig. 8.10, the efiective capaci- tance of the Cooper pair box increased the plasma frequency of the phase qubit at ng = 0.5 when EcJ < ~!0. I expect the maximum efiect of the efiective capacitance would occur if EcJ ; EJJ ? ~!0. These results are very similar to those from the Yale group where they coupled the box to an LC resonator [41]. The difierence is that in this case the box is coupled to the Josephson phase qubit which is a non-linear resonator so that the energy levels of the Josephson phase qubit are distinguishable. The energy level spacings of the Josephson phase qubit can be measured in principle using the spectroscopic measurement. I also compared this result to that from a full numerical calculation (solving the eigenvalues of Eq. 8.100 with cubic approximation for a Josephson phase qubit) [109]. The numerical simulation yielded almost the same result as the harmonic approximation. 8.6 Conclusions In this chapter, I analyzed the Cooper-pair box and showed that it acts as a variable capacitor. The efiective capacitance of the Cooper-pair box depends on the 236 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 ng E/h (GHz ) Figure 8.8: Simulated energy level spacings for a Cooper pair box coupled to a Josephson phase qubit for Ib = 0.989 IcQ (solid curves). The simulation parameters are given in Table. 8.3. The dashed horizontal line is the uncoupled energy level spacing ~!01 of the Josephson phase qubit and the dashed parabola is the uncoupled energy level spacing of the Cooper pair box. 237 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 ng E/h (GHz ) Figure 8.9: Simulated energy level spacings for a Cooper pair box coupled to a Josephson phase qubit for Ib = 0.991 IcQ (solid curves). The simulation parameters are given in Table. 8.3. The dashed horizontal line is the uncoupled energy level spacing ~!01 of the Josephson phase qubit and the dashed parabola is the uncoupled energy level spacing of the Cooper pair box. 238 gate voltage and the bias voltage applied. I have proposed a technique to measure the efiective capacitance at high frequency by coupling the box to a Josephson junction and showed simulation results for the coupled system. Finally I attempted to build and measure these coupled devices (see Fig. 4.6 in Chapter 4) but I did not get a working Cooper pair box. 239 Chapter 9 Conclusions In this thesis, I discussed the efiect of isolation on escape rate, dissipation, and coherence in the aluminum dc SQUID phase qubit AL1 and the niobium dc SQUID phase qubit NBG which had a gradiometer loop. The main purpose of my experiments was to flnd out what was limiting the coherence times of our phase qubits. In Chapters 2 to 4, I reviewed the basic physics of two level systems, described our dc SQUID phase qubit and reported my qubit fabrication technique. The fab- rication techniques involved double-angle evaporation through a photolithographic bridge, and yielded devices with no oxide layers other than native AlOx. In Ch. 5, I discussed the inductive isolation scheme used in the dc SQUID phase qubit. I calculated the isolation factor rI and the efiective resistance Reff from the qubit circuit. I also showed that the isolation factor and its efiect on the qubit could be changed in situ. I measured the state of the qubit AL1 through the total escape rate while varying the isolation factor rI and observed high frequency noise induced transitions in AL1; I found prominent peaks when the qubit junction j0i to j2i transition matched the isolation junction j0i to j1i transition, and at the resonance between the qubit junction j1i to j3i transition and the isolation junction j0i to j1i transition. These noise induced transitions moved in frequency as I changed the current I2 through the isolation juction. Second, I obtained T1 using two techniques: Relaxation measurements and a thermal escape rate technique. I found that the two techniques yielded quite difierent values for T1. For AL1, the relaxation measurement yielded T1 ? 50 ns to 60 ns for all isolations, which the thermal escape rate technique yielded T1 ? 20 ns. I observed that T1 obtained from the thermal escape rate measurements showed some 240 dependence on the isolation factor rI for certain range of I1, and this dependence suggested that low T1 values inferred by the escape technique may be an artifact of high frequency noise in the bias leads. In Ch. 6, I showed experimental results on Rabi oscillations and spectroscopic coherence times in AL1 for difierent isolations from the current bias leads. The decay time constant T 0 of Rabi oscillations showed no dependence on the isolation rI , suggesting that T 0 is not limited by noise or loss from the current bias leads. In particular, I found that T 0 ? 20 ns to 30 ns independent of the isolation for device AL1. This was 2 to 4 times longer than we have found in our Nb phase qubits, suggesting that materials may be playing an important role in the decoherence. In Ch. 7, I discussed my experimental results on Rabi oscillations and spec- troscopic coherence time T ?2 in a niobium dc SQUID phase qubit with a gradiometer loop, device NBG. If uniform magnetic fleld noise were the main source of decoher- ence, one would expect to flnd a longer T 0 in NBG. However, I found that NBG did not show a longer T 0 or T ?2 than the magnetometer dc SQUID qubits. I also discovered that AL1 had the longest T 0 and T2, while T1 from all three devices were similar. From the measurement of the energy relaxation time T1, decay time constant T 0 of Rabi oscillations and the spectroscopic coherence time T ?2 with varying current isolation, I concluded that the current noise was not a major limiting factor for the coherence times in our SQUID phase qubits. Low frequency, ux and critical current noise can also be ruled out because our coherence times and relaxation time showed no signiflcant dependence on frequency. Finally, in Ch. 8, I discussed using a Cooper pair box as a coupling element between phase qubits and examined the coupling between a Cooper pair box and a Josephson junction phase qubit. 241 9.1 Current status of superconducting quantum computing and fu- ture plans To increase the coherence times, many superconducting quantum computing groups [40, 49, 46, 90, 111] have been putting tremendous efiort into decoherence studies for past few years. Still, it is not entirely clear what exactly limits the coherence time, especially in phase qubits. Groups at NIST and UCSB found out that lossy dielectrics and two level uctuators from electric dipoles embedded in the dielectrics are the main factor of imiting T2 in phase qubits [49, 90, 112, 113]. However, recent reports on a charge-phase hybrid qubit (the \transmon") [43] have claimed very long coherence times with minimal changes in materials. The transmon is an ultra-small Josephson junction with an added shunt capacitor. Al- though the junction area is small, the transmon is efiectively a phase qubit where the Josephson energy dominates. Having an ultra-small junction area minimizes decoherence from the junction materials. Also using a large shunt capacitor and large EJ=Ec ratio, the device becomes relatively insensitive to charge uctuations, which produces an improved \sweet spot". Another important factor was they read out the transmon using a non-demolition method, through a LC resonator, that was capacitively coupled to the transmon. The future will almost certainly see more study of ideal qubit materials and searching for better qubit read-out methods. Currently many researchers are making a good progress on quantum computation including our UMD group. Perhaps it will not be too many years before we hear someone factored 21 using a superconducting quantum computer. 242 Appendix A MATLAB Code Here I introduce MATLAB codes that I used for the data analysis in this thesis. Sudeep K. Dutta [23] made or refurbished all the codes that I used. The flrst section is the code for a single Josephson junction spectrum calculation which is Appendix B of Sudeep?s thesis [23]. The second and third section has codes for the non-stationary and stationary master equation simulation also made by Sudeep [23]. A.1 Solution of the Junction Hamiltonian Here is the flrst section of Appendix B of S. K. Dutta?s thesis [23]. ||||||||||||||||||||||{ The following programs calculate the eigenfunctions, energy levels, and tunnel- ing rates of a single current-biased junction and two capacitively coupled junctions1. The heart of the algorithms was written by Huizhong Xu; see x2.4 and x3.3.2 of Ref. [25]. The programs in this section solve the Hamiltonian for a current-biased junc- tion in the absence of dissipation, given in Eq. (2.23). The nature of these solutions is discussed in x2.3.3. jjspectrum is the main driver that just collects the solu- tions returned by jjeigentbc, given below. The diary command creates a flle of everything that is dumped to the screen, which I found useful for debugging. function [stuff,wavefn] = ... jjspectrum(Io, Cj, Iri, Irf, dIr, levelmaxIr, E0, psi0) 1The two coupled junction solution code is not included in this thesis. 243 % [stuff, wavefn] = % jjspectrum(Io, Cj, Iri, Irf, dIr, levelmaxIr, E0, psi0) % This calculates all the energies and wavefunctions for a single % junction with critical current ?Io? (Amps), junction capacitance % ?Cj? (Farads), from reduced bias current ?Iri? to ?Irf?, in ?dIr? % steps. ?levelmaxIr? sets the number of levels to calculate; it?s % defined in ?keeplevels?. ?E0? and ?psi0? are optional -- they % specify the initial guesses for all the levels. Everything sent % back in a big structure. % calls: hbar, keeplevels, jjeigentbc, wp, plotlevels, xaxis more off diary on global hbar; stuff.params.Io = Io; stuff.params.Cj = Cj; stuff.params.Iri = Iri; stuff.params.Ifr = Irf; stuff.params.dIr = dIr; stuff.params.levelmaxIr = levelmaxIr; stuff.params.start = clock; NIr = floor( (Irf - Iri) / dIr ) + 1; for Ircount = 1 : NIr Ir = Iri + (Ircount-1) * dIr; stuff.Ir(Ircount) = Ir; disp([?Reduced current ? num2str(Ir)]); % After the first current, use the previous wavefunction as the % initial guess. Use the same n, which (at a higher current) will % give a lower initial guess for the energy. for levelcount = keeplevels(levelmaxIr, Ir) disp([?Level ? num2str(levelcount)]); if Ircount == 1 if nargin == 8 % User supplied energy and wavefunction solution = jjeigentbc(Ir*Io, Io, Cj, length(levelmaxIr)-1,... E0(levelcount+1)/hbar/wp(Ir*Io, Io, Cj) - 0.5, ... psi0(levelcount+1,:)); elseif nargin == 7 244 % User supplied energy -- use a random initial wavefunction solution = jjeigentbc(Ir*Io, Io, Cj, length(levelmaxIr)-1,... E0(levelcount+1)/hbar/wp(Ir*Io, Io, Cj) - 0.5); else % User didn?t give you anything. Guess the energy and use a % random psi. corr = 0.15 - 5 * (1 - Ir - 0.005); solution = jjeigentbc(Ir*Io, Io, Cj, length(levelmaxIr)-1,... levelcount*(1-corr)); end stuff.params.xleft = solution.x(1); stuff.params.dx = solution.dx; stuff.params.Ngrid = length(solution.x); else solution = jjeigentbc(Ir*Io, Io, Cj, length(levelmaxIr)-1, ... n0(levelcount+1), psi(levelcount+1,:)); end energy = real(solution.E); gamma = -imag(solution.E) / (hbar/2); psi(levelcount+1,:) = solution.wavefn; n0(levelcount+1) = energy / hbar / wp(Ir*Io, Io, Cj) - 0.5; levstr = num2str(levelcount); Irstr = num2str(Ircount); eval([?stuff.energy? levstr ?(? Irstr ?) = energy;?]); eval([?stuff.gamma? levstr ?(? Irstr ?) = gamma;?]); eval([?wavefn.level? levstr ?(? Irstr ?,:) = solution.wavefn;?]); end if Ircount == 1 Eplot = figure; end figure(Eplot); plotlevels(stuff); xaxis([stuff.Ir(1) 1]); shg; end stuff.params.stop = clock; 245 more on diary off This is the primary routine that calculates the solutions for a single value of the bias current, using transmission boundary conditions. function solution = jjeigentbc(Ib, Io, Cj, nmax, n0, psi0) % solution = jjeigentbc(Ib, Io, Cj, nmax, n0, psi0) % Calculates the energy, potential, and wavefunction (on a grid x, % with steps dx) for bias current ?Ib?, critical current ?Io?, % capacitance ?Cj?, maximum number of levels ?nmax?, and current % level ?n0? (or the best guess of what it is). ?psi0? is the % (optional) inital guess for the wavefunction. Uses transmission % boundary conditions. Results sent back in a structure. % calls: mj, wp, hbar, jjeigengrid % Some constants global hbar; % Set up a grid to solve Schrodinger?s eq. [xleft, dx, Ngrid] = jjeigengrid(0.97, 0.999, Io, Cj, nmax); disp([?xleft = ? num2str(xleft) ? dx = ? num2str(dx) ? ... Ngrid = ? num2str(Ngrid)]); % This constant is in front of d2(psi)/dx2. Multiply it over to % V and E and call them Vp and Ep (p for prime) m = mj(Cj); a = 2 * m * (dx / hbar)^2; Umin = twb(Ib, Io, asin(Ib/Io)); for i = 1 : Ngrid x(i) = xleft + dx * (i-1); Utwb(i) = twb(Ib, Io, x(i)) - Umin; end Uleft = Utwb(1); Uright = Utwb(Ngrid); 246 % The matrix is N-2 x N-2, because the boundary conditions are % evaluated in the 2 and N-1 equations. Set up H*psi = E*psi. A(1 : Ngrid-2) = -1; C(1 : Ngrid-2) = -1; for i = 1 : Ngrid-2 B(i) = 2 + Utwb(i+1) * a; end % Here?s the first guess at the eigenvalue. Start with a random % wavefunction (if one isn?t provided) and use inverse iteration % (Numerical Recipes 11.7) to improve it. Ep = (n0 + 0.5) * hbar * wp(Ib, Io, Cj) * a; if nargin == 6 newpsi = psi0(2:end-1); else % This is the MATLAB R12 command % newpsi = random(?unif?, 0, 1, 1, Ngrid-2); % This is the MATLAB R14 command newpsi = rand(1, Ngrid-2); end newpsi = newpsi / sqrt(sum(newpsi.^2)); % Boundary conditions for first go round. Btbc = B; Kleft = sqrt(2 * m * (Uleft - Ep/a)) / hbar; Btbc(1) = B(1) - exp(-1 * Kleft * dx); Kright = sqrt(2 * m * (Ep/a - Uright)) / hbar; Btbc(Ngrid-2) = B(Ngrid-2) - exp(sqrt(-1) * Kright * dx); % First iterate a couple times, without updating the eigenvalue. diff = 2; err = 0; count1 = 0; while (diff > 1e-6) & (err == 0) oldpsi = newpsi; [temppsi, err] = tridiag(A, Btbc - Ep, C, oldpsi); newpsi = temppsi / sqrt( sum(abs(temppsi).^2) ); diff = max(abs( (abs(newpsi)./abs(oldpsi)).^2 - 1 )); 247 count1 = count1 + 1; end % Now update the energy too oldEp = Ep; newEp = oldEp + sum( conj(temppsi) .* oldpsi ) ... / sum(abs(temppsi).^2); diff = 1; count2 = 0; while((diff > 1e-7) ... | max(abs( imag(newEp)/imag(oldEp) - 1 )) > 1e-7) & err==0 oldpsi = newpsi; oldEp = newEp; Kleft = sqrt(2 * m * (Uleft - oldEp/a)) / hbar; Btbc(1) = B(1) - exp(-1 * Kleft * dx); Kright = sqrt(2 * m * (oldEp/a - Uright)) / hbar; Btbc(Ngrid-2) = B(Ngrid-2) - exp(sqrt(-1) * Kright * dx); [temppsi, err] = tridiag(A, Btbc - oldEp, C, oldpsi); newpsi = temppsi / sqrt( sum(abs(temppsi).^2) ); diff = max(abs( (abs(newpsi)./abs(oldpsi)).^2 - 1 )); newEp = oldEp + sum( conj(temppsi) .* oldpsi ) ... / sum(abs(temppsi).^2); count2 = count2 + 1; end % So far, have been normalizing the vector psi. But to make it % a ?continuous? function on x, do a Riemann sum. newpsi = -sqrt(-1) * newpsi / sqrt(dx); wavefn = [newpsi(1)*exp(-1 * Kleft * dx) newpsi ... newpsi(Ngrid-2)*exp(sqrt(-1) * Kright * dx)]; wavefn = wavefn / sqrt( sum(abs(wavefn).^2) ) / sqrt(dx); solution.E = newEp/a; solution.x = x; solution.Utwb = Utwb; solution.wavefn = wavefn; solution.dx = dx; disp([num2str(count1) ? iterations of first loop; ? ... 248 num2str(count2) ? iterations of second?]); This sets up the grid on which the solution is calculated. function [xleft, dx, Ngrid] = jjeigengrid(Irmin, Irmax, Io, Cj, nmax) % [xleft, dx, Ngrid] = jjeigengrid(Irmin, Irmax, Io, Cj, nmax) % This calculates a grid for jjeigentbc. It should select the % smallest grid compatible for currents between ?Irmin? and ?Irmax?, % critical current ?Io?, capacitance ?Cj?, and maximum quantum level % ?nmax?. If everything is done on the same grid, then you can take % inner products and stuff with the wavefunctions later. % calls: mj, wp, hbar, twb % Some constants global hbar; m = mj(Cj); % Ideally, you would use the smallest range for a given Irmin/max and % Cj. However, this is complicated. % First, you need to find the values of the phase, where the % washboard hits (again) the local max (to the left) and min (to the % right) of the first well. The widest range of phase occurs for the % smallest bias current. Just pick a fixed [0.8, 2.3], which should % cover down to Ir = 0.95. % Then, you want enough phase outside of this to capture some % oscillations (to the right of the well) and the decay (to the % left). This is set by the constant alpha below. The longest % spatial scale occurs at the highest current, opposite of the % previous paragraph -- ignore this. Don?t really know how many of % these spatial constants to keep. This should be optimized. wpmin = wp(Irmax*Io, Io, Cj); alphamin = sqrt(m*wpmin/hbar); wpmax = wp(Irmin*Io, Io, Cj); alphamax = sqrt(m*wpmax/hbar); xleft = 0.8 - 4/alphamin; xright = 2.3 + 4/alphamin; % Next get the step size, which is based on the oscillations of the 249 % highest energy you plan to calculate. These should be evaluated at % the highest current, where the potential is steep and the energy % differences are large. xmin = asin(Irmax); Umin = twb(Irmax*Io, Io, xmin); Uleft = twb(Irmax*Io, Io, xleft) - Umin; Uright = twb(Irmax*Io, Io, xright) - Umin; Emax = (nmax + 0.5) * hbar * wpmax; lambdal = sqrt(2*m * (Uleft - Emax)) / hbar; lambdar = sqrt(2*m * (Emax - Uright)) / hbar; dx = 1 / max([alphamax lambdal lambdar]) / 10; Ngrid = floor((xright - xleft) / dx) + 1; The main M-flles above call several simple routines, given below. In addition, global variables called hbar and Phio (which, not surprisingly, are equal to ~ and '0) should be deflned in the workspace. function levels = keeplevels(levelmaxIr, Ir) % levels = keeplevels(levelmaxIr, Ir) % This returns a vector of the levels to keep at a given reduced bias % current, ?Ir?. ?levelmaxIr?(i) gives the reduced current where the % (i-1)th state leaves the well (or least where you don?t want it % anymore). If you should keep it, i-1 is included in ?levels?. % 0 is the ground state. The number of elements in ?levelmaxIr? sets % the maximum number of levels to keep. levels = []; for i = 1 : length(levelmaxIr) if Ir <= levelmaxIr(i) levels = [levels i-1]; end end function omegap = wp(Ib, Io, C); 250 % wp(Ib, Io, C) gives the plasma frequency of a junction global Phio; omegap = sqrt(2*pi*Io/C/Phio) .* (1-(Ib./Io).^2).^(1/4); plotlevels plots the energy levels as the solutions are calculated. Running the program for a large number of bias currents can be time-taking, so this is a useful way of spotting trouble early. function plotlevels(eigenstuff) % plotlevels(eigenstuff) % This assumes ?eigenstuff? has fields named Ir and energy0, energy1. colors = ?bgrcmy?; plotcnt = 0; fields = fieldnames(eigenstuff); for i = 1 : length(fields) if strncmp(fields(i), ?energy?, 6) == 1 data = getfield(eigenstuff, char(fields(i))); plot(eigenstuff.Ir(1:length(data)), data, ... colors(mod(plotcnt, 6) + 1)); hold on plotcnt = plotcnt + 1; end end function xaxis(xbounds) % xaxis([xmin xmax]) % This replots the current graph, using new x bounds. graphaxes = axis; graphaxes(1) = xbounds(1); graphaxes(2) = xbounds(2); axis(graphaxes); 251 function mass = mj(Cj); % mass = mj(Cj) returns the phase particle mass, given the junction % capacitance. % calls: Phio global Phio; mass = Cj * (Phio/2/pi)^2; function U = twb(Ib, Io, gamma); % U = twb(Ib, Io, gamma) returns the tilted washboard potential. % calls: Phio global Phio; U = -Phio/2/pi * (Io * cos(gamma) + Ib * gamma); A.2 Non-stationary Master equation solution Here is the MATLAB code for solving a non-stationary master equation for a single Josephson junction that I used to calculate the escape rate versus current. The free parameters are T1 and temperature T. By comparing this to measurements of ? versus current, I could extract T1 [23]. function result = ME(mode, stepper, dtsave, dtupdate, ti, tf, Gi, fj, T1, T, Ni0, minN, varargin) % result = ME(mode, stepper, dtsave, dtupdate, ti, tf, Gi, fj, T1, T, % Ni0, minN, modeAparams, modeBparams, ...) % This calculates the populations and escape rates under the master % equation, between times ?ti? and ?tf? (in seconds). Escape rates % and energy levels are directly specified. Results are sent back % in a structure. % % ?mode? is a vector that selects the type of simulation. It is % unused at the moment. % 252 % A little note about indices: the levels in well are labelled 0, % 1, 2, ... % % Results are saved roughly at intervals ?dtsave?. % % ?dtupdate? sets how often to update parameters that are time- % dependent (as described below); if it is 0, then they are updated % on every iterate if needed. % % ?fj? is the energy level vector, in Hertz. % % ?T1? gives all the energy dissipation times, in seconds. % % ?T? is the temperature in Kelvin. % % calls: ensurerow, ensurecolumn, calcG1nm % created 11/1/05 modified 12/19/05 more off; dt = dtsave; t = ti; diffmax = stepper(1); dtdec = stepper(2); diffmin = stepper(3); dtinc = stepper(4); result.params.start = clock; result.params.mode = mode; result.params.stepper = stepper; result.params.dtsave = dtsave; result.params.dtupdate = dtupdate; result.params.ti = ti; result.params.tf = tf; if (isa(Gi, ?function_handle?) == 1) | (isa(Gi, ?inline?) == 1) FGi = Gi; else FGi = inline([?repmat(? mat2str(ensurecolumn(Gi)) ?, 1, ... length(t))?], ?t?); end result.params.Gi = Gi; result.params.FGi = FGi; if (isa(fj, ?function_handle?) == 1) | (isa(fj, ?inline?) == 1) Ffj = fj; else Ffj = inline(mat2str(ensurerow(fj))); end result.params.fj = fj; result.params.Ffj = Ffj; if (isa(T1, ?function_handle?) == 1) | (isa(T1, ?inline?) == 1) 253 FT1 = T1; else FT1 = inline(mat2str(ensurerow(T1))); end result.params.T1 = T1; result.params.FT1 = FT1; if (isa(T, ?function_handle?) == 1) | (isa(T, ?inline?) == 1) FT = T; else FT = inline(mat2str(T)); end result.params.T = T; result.params.FT = FT; result.params.Ni0 = Ni0; Nlevel = length(Ni0); Ni = ensurecolumn(Ni0); result.params.minN = minN; result.time(1) = t; result.level(1) = Nlevel; Nisave = Ni; lastsave = t; lastupdate = t; savecnt = 2; Gi = feval(FGi, t); fj = feval(Ffj, t); T1 = feval(FT1, t); T = feval(FT, t); if length(Gi) ~= Nlevel disp(?Gi is the wrong size?); return; end if length(fj) ~= Nlevel-1 disp(?fj is the wrong size?); return; end if length(T1) ~= (Nlevel-1) * Nlevel / 2 disp(?T1 is the wrong size?); return; end D = calcG1nm(T1, T, fj); G = diag(Gi); P = D - G; expPdt = expm(P * dt); while t <= tf newNi = expPdt * Ni; 254 diff = max( abs((newNi - Ni) ./ (newNi + 0.05)) ); while diff > diffmax dt = dt / dtdec; disp([?Step size decreased to ? num2str(dt) ? at t = ? num2str(t) ? with ? num2str(length(Ni)) ? levels?]); expPdt = expm(P * dt); newNi = expPdt * Ni; diff = max( abs((newNi - Ni) ./ (newNi + 0.05)) ); end Ni = newNi; t = t + dt; if (t - lastsave) >= dtsave result.time(savecnt) = t; result.level(savecnt) = Nlevel; % Pad the populations if needed if length(Ni) < length(Ni0) Nipad = [Ni; zeros(length(Ni0)-length(Ni), 1)]; else Nipad = Ni; end Nisave = [Nisave Nipad]; savecnt = savecnt + 1; lastsave = t; % Print an update every once in a while if mod(savecnt, 25) == 0 disp([?Time = ? num2str(t)]); end end updateP = 0; if diff < diffmin dttemp = dt * dtinc; if dttemp <= dtsave dt = dttemp; disp([?Step size increased to ? num2str(dt) ? 255 at t = ? num2str(t) ? with ? num2str(length(Ni)) ? levels?]); updateP = 1; end end if (t - lastupdate) >= dtupdate lastupdate = t; Ginew = feval(FGi, t); fjnew = feval(Ffj, t); T1new = feval(FT1, t); Tnew = feval(FT, t); if ~isequal(Ginew, Gi) Gi = Ginew; G = diag(Gi(1:Nlevel)); updateP = 1; end if ~isequal(fjnew, fj) | ~isequal(T1new, T1) | ~isequal(Tnew, T) fj = fjnew; T1 = T1new; T = Tnew; D = calcG1nm(T1, T, fj, Nlevel); updateP = 1; end end if length(Ni) > 2 & Ni(end) < minN Ni = Ni(1 : end-1); Nlevel = length(Ni); G = diag(Gi(1:Nlevel)); D = calcG1nm(T1, T, fj, Nlevel); disp([?Number of levels decreased to ? num2str(length(Ni)) ... ? at t = ? num2str(t)]); updateP = 1; end if updateP == 1 P = D - G; expPdt = expm(P * dt); end end 256 for j = 0 : length(Ni0)-1 eval([?result.N? num2str(j) ? = Nisave(? num2str(j+1) ?, :);?]); end result.params.stop = clock; more on; A.3 Stationary Master equation solution Here is the MATLAB code for solving a stationary master equation for a single Josephson junction. I used this program to calculate the escape rate versus time. By comparing this to measurements of ? versus time, I could extract T1 [23]. The free parameters are T1 and temperature T. function result = SME(mode, tlist, Gi, fj, T1, T, varargin) % result = SME(mode, tlist, Gi, fj, T1, T, ... % modeAparams, modeBparams, ...) % This calculates the populations of the master equation under % stationary conditions. The idea is that the relevant transitions % are directly specified, with no mention of junction parameters % (Io, Cj, etc.). Populations are calculated at each of the times % specified by the vector ?tlist? independently (i.e. this does no % evolution). Of course, there is no time-dependence in the equations. % In this case, time is only used as a parameter that controls the % values of the other arguments, as described below. For example, % the "time" could just be the bias current. Results are sent back % in a structure. % % ?mode? is a vector that selects the type of simulation. % % A little note about indices: the levels in well are labeled % 0, 1, 2, ... % % If ?Gi? is a vector, then its ith element gives the escape % rate out of the (i-1) level (so the first element is for the ground % state) in inverse seconds. % % ?fj? is the energy level vector, in Hertz. % % ?T1? gives all the energy dissipation times, in seconds. 257 % % ?T? is the temperature in Kelvin. % % calls: Boltzdist, SMEPi, calcG1nm, ensurerow, ensurecolumn % created 3/31/04 modified 11/8/05 more off result.params.start = clock; result.params.mode = mode; result.time = tlist; if (isa(Gi, ?function_handle?) == 1) | (isa(Gi, ?inline?) == 1) FGi = Gi; else FGi = inline([?repmat(? mat2str(ensurecolumn(Gi)) ?, 1, length(t))?], ?t?); end result.params.Gi = Gi; result.params.FGi = FGi; if (isa(fj, ?function_handle?) == 1) | (isa(fj, ?inline?) == 1) Ffj = fj; else Ffj = inline(mat2str(ensurerow(fj))); end result.params.fj = fj; result.params.Ffj = Ffj; if (isa(T1, ?function_handle?) == 1) | (isa(T1, ?inline?) == 1) FT1 = T1; else FT1 = inline(mat2str(ensurerow(T1))); end result.params.T1 = T1; result.params.FT1 = FT1; if (isa(T, ?function_handle?) == 1) | (isa(T, ?inline?) == 1) FT = T; else FT = inline(mat2str(T)); end result.params.T = T; result.params.FT = FT; Gi = feval(FGi, tlist(1)); fj = feval(Ffj, tlist(1)); T1 = feval(FT1, tlist(1)); T = feval(FT, tlist(1)); Nlevel = length(Gi); if length(fj) ~= Nlevel-1 disp(?fj is the wrong size?); return; end 258 if length(T1) ~= (Nlevel-1) * Nlevel / 2 disp(?T1 is the wrong size?); return; end Pi = Boltzdist(fj, T); for tcount = 1 : length(tlist) t = tlist(tcount); Gi = feval(FGi, t); fj = feval(Ffj, t); T1 = feval(FT1, t); T = feval(FT, t); D = calcG1nm(T1, T, fj); G = diag(Gi); MEP = D - G; Pi = SMEPi(MEP, Gi, Pi); for j = 1 : length(Pi) eval([?result.P? num2str(j-1) ?(? num2str(tcount) ?) = Pi(j);?]); end end result.params.stop = clock; more on 259 Bibliography [1] Quantum Computer (Encyclop?dia Britannica Online, 2007), URL http://www.search.eb.com/eb/article-9343823. 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