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"Normal-Metal Quasiparticle Traps for Superconducting Qubits"

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Normal-Metal Quasiparticle Traps For Superconducting Qubits: Modeling, Optimization, and Proximity Effect Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Amin Hosseinkhani, M.Sc. Berichter: Universitätsprofessor Dr. David DiVincenzo, Universitätsprofessorin Dr. Kristel Michielsen Tag der mündlichen Prüfung: March 01, 2018 Diese Dissertation ist auf den Internetseiten der Universitätsbibliothek online verfügbar. Metallische Quasiteilchenfallen für supraleitende Qubits: Modellierung, Optimisierung, und Proximity-Effekt Kurzfassung: Bogoliubov Quasiteilchen stören viele Abläufe in supraleitenden Elementen. In supraleitenden Qubits wechselwirken diese Quasiteilchen beim Tunneln durch den Josephson- Kontakt mit dem Phasenfreiheitsgrad, was zu einer Relaxation des Qubits führt. Für Tempera- turen im Millikelvinbereich gibt es substantielle Hinweise für die Präsenz von Nichtgleichgewicht- squasiteilchen. Während deren Entstehung noch nicht einstimmig geklärt ist, besteht dennoch die Möglichkeit die von Quasiteilchen induzierte Relaxation einzudämmen indem man die Qu- asiteilchen von den aktiven Bereichen des Qubits fernhält. In dieser Doktorarbeit studieren wir Quasiteilchenfallen, welche durch einen Kontakt eines normalen Metalls (N) mit der supraleit- enden Elektrode (S) eines Qubits definiert sind. Wir entwickeln ein Modell, das den Einfluss der Falle auf die Quasiteilchendynamik beschreibt, wenn überschüssige Quasiteilchen in ein Trans- monqubit injiziert werden. Dieses Modell ermöglicht es, unter Berücksichtigung der Fallenpa- rameter die Zeitskala zu bestimmen, in der die überschüssigen Quasiteilchen aus dem Transmon evakuiert werden. Wir zeigen, dass die Evakuierungsdauer monoton mit der Fallengröße ansteigt und letztlich auf einen Grenzwert zuläuft, der von der Quasiteilchen-Diffusionskonstante und von der Qubitgeometrie abhängt. Wir errechnen die charakteristische Fallengröße, bei welcher dieser Grenzwert erreicht wird. Wie es sich herausstellt, ist der limitierende Faktor für die Einfangrate der Falle durch die langsame Quasiteilchenrelaxation im normalen Metall gegeben; diese Relaxation ist jedoch nur schwer kontrollierbar. Um das Einfangen von Quasiteilchen zu optimieren, studieren wir den Einfluss von Größe, Anzahl und räumlicher Anordnung der Fallen. Diese Faktoren sind insbesondere wichtig, wenn die Falle die charakteristische Größe überschreitet. Wir diskutieren für einige experimentell rel- evante Beispiele wie die Evakuierungsdauer der überschüssigen Quasiteilchen optimiert werden kann. Darüberhinaus zeigen wir, dass eine Falle nahe des Josephson-Kontaktes die stationäre Quasiteilchendichte an demselben Kontakt unterdrückt und den Einfluss von Fluktuationen der Quasiteilchenerzeugung reduziert. Wenn metallische Elemente an ein supraleitendes Material gekoppelt sind, können Cooper- paare ins Metall entweichen. Mit dem Usadelformalismus greifen wir zunächst den Proximity- Effekt von gleichförmigen NS-Doppelschichten wieder auf; trotz der bereits langjährigen Er- forschung dieses Problems erlangen wir zu neuen Erkenntnissen über die Zustandsdichte. Wir verallgemeinern unsere Resultate danach für das ungleichförmige Problem in der Nähe der Fal- lenkante. Durch die Kombination dieser Resultate mit dem davor entwickelten Modell zur Unterdrückung der Quasiteilchendichte finden wir einen optimalen Abstand zwischen Falle und Josephson-Kontakt in einem Transmonqubit, welcher zu einer Minimierung der Qubitrelaxation führt. Dieser optimale Abstand, der die 4- bis 20-fache Kohärenzlänge beträgt, resultiert aus dem Wechselspiel zwischen Proximity-Effekt und Unterdrückung der Quasiteilchendichte. Wir schließen daraus, dass der schädliche Einfluss des Proximity-Effekts umgangen werden kann solange die Entfernung zwischen Falle und Kontakt größer als der optimale Abstand ist. Normal-Metal Quasiparticle Traps for Superconducting Qubits: Modeling, Optimization, and Proximity Effect Abstract: Bogoliubov quasiparticle excitations are detrimental for the operation of many su- perconducting devices. In superconducting qubits, quasiparticles interact with the qubit degree of freedom when tunneling through a Josephson junction, and this interaction can lead to qubit relaxation. At millikelvin temperatures, there is substantial evidence of nonequilibrium quasi- particles. While there is no agreed upon explanation for the origin of these excess quasiparticles, it is nevertheless possible to limit the quasiparticle-induced relaxation by steering quasiparticles away from qubit active elements. In this thesis, we study quasiparticle traps that are formed by a normal-metal in tunnel contact with the superconducting electrode of a qubit. We develop a model to explain how a trap can influence the dynamics of the excess quasiparticles injected in a transmon-type qubit. This model makes it possible to find the time it takes to evacuate the injected quasiparticles from the transmon as a function of trap parameters. We show when the trap size is increased, the evacuation time decreases monotonically and saturates at a level that depends on the quasiparticles diffusion constant and the qubit geometry. We find the charac- teristic trap size needed for the evacuation time to approach the saturation value. It turns out that the bottleneck limiting the trapping rate is the slow quasiparticle energy relaxation inside the normal-metal trap, a quantity that is very hard to control. In order to optimize normal-metal quasiparticle trapping, we study the effects of trap size, number, and placement. These factors become important when the trap size increases beyond the characteristic length. We discuss for some experimentally relevant examples how to shorten the evacuation time of the excess quasiparticle density. Moreover, we show that a trap in the vicinity of a Josephson junction can significantly suppress the steady-state quasiparticle density near that junction and reduce the impact of fluctuations in the generation rate of quasiparticles. When such normal-metal elements are connected to a superconducting material, Cooper- pairs can leak into the normal-metal trap. This modifies the superconductor properties and, in turn, affects the qubit coherence. Using the Usadel formalism, we first revisit the proximity effect in uniform NS bilayers; despite the long history of this problem, we present novel findings for the density of states. We then extend our results to describe a non-uniform system in the vicinity of a trap edge. Using these results together with the previously developed model for the suppression of the quasiparticle density due to the trap, we find in a transmon qubit an optimum trap-junction distance at which the qubit relaxation rate is minimized. This optimum distance, of the order of 4 to 20 coherence lengths, originates from the competition between proximity effect and quasiparticle density suppression. We conclude that the harmful influence of the proximity effect can be avoided so long as the trap is farther away from the junction than this optimum. Acknowledgements Before starting the main part of the thesis, I would like to take the opportunity to express my deepest gratitude to several people who helped and supported me during my doctoral studies. First of all, I am greatly thankful to my PhD supervisor Dr. Gianluigi Catelani for his kind and constant support. I am indebted for countless enjoyable discussions with him during which I learned to focus on Physics behind the mathematics. I also like to thank him for sending me to a lot of conferences and schools where I got a chance to expand my knowledge and to meet and discuss with a lot of physicists. I also enjoyed a lot by collaborating with experimental scientists at Yale University, which was made possible by Gianluigi. I am grateful to my friend Dr. Roman-Pascal Riwar for a lot of scientific as well as every- day-life discussions that we had. He also kindly helped me in translating the abstract of this thesis into (Swiss) German. I like to express my deep gratitude to Prof. David DiVincenzo for reviewing my thesis, his support for my postdoc applications, and the friendly and relaxed atmosphere at the JARA- Institute of Quantum Information. I would like to thank the second reviewer of my thesis Prof. Kristel Michielsen and also other members of my PhD committee, Prof. Thomas Schäpers and Prof. Christoph Stampfer, for the time they spent on reading my thesis and their fruitful comments. I would also like to thank all of my colleagues in JARA-Institute of Quantum Information at Forschungszentrum Jülich and RWTH Aachen University; few of which include Dr. Daniel Zeuch for a lot of discussions we had almost every day and also helping me in the German abstract of the thesis, Dr. Sbastian Mehl for helping me with a lot of paper works at the time I just had started my doctoral study in Germany, Alessandro Ciani for kindly preparing my PhD graduation hat, and Alwin van Steensel for a lot of discussions and his kind PhD gift. I like to particularly thank Dr. Mohammad H. Ansari for lots of interesting discussions that we had during my doctoral studies and also his kind support for my postdoc application to his research group. I am very thankful to Ms. Luise Snyders who helped me very much for handling official paper works at Forschungszentrum Jülich. I also like to thanks Ms. Helene Barton for helping me in doing paper works at RWTH Aachen University. I like
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