"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 supraleitenden 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 superconducting 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 quasiparticles. 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 characteristic 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 to thank Dr. Hamed Saberi who supervised me during my Master’s program at Shahid Beheshti University, Tehran, Iran and supporting my PhD applications. I would like to thank Prof. Ali Rezakhani for co-advising my master’s thesis and also for inviting me to visit Institute for Research in Fundamental Sciences (IPM), Tehran, Iran in September 2016. I also thank Dr. Sahar Alipour for scheduling that visit. I would like to thank Prof. Maksym Serbyn for inviting me to visit his group at Institute of Science and Technology Austria in February 2018. I also like to thank Prof. Johannes Fink and Dr. Shabir Barzanjeh for a lot of interesting discussions that I had with them and their hospitality during my visit to IST Austria. I am thankful to Dr. Frank Deppe for inviting me to visit Walther-Meißner-Institute for Low-Temperature Research in March 2018. I greatly appreciate his hospitality and a lot of interesting discussions that we had. I am grateful to Prof. Ahmad Ghodsi Mahmoudzadeh, my advisor during my bachelor studies at Ferdowsi University of Mashhad, Iran, for his constant support and hospitality. vi

I also like to thank all of my friends at Forschungszentrum Jülich whom I share a lot of sweet memories: Esmaeel, Amin, Keyvan, Ali, Vahid, Masood and Davood. I like to especially thank my friends at Aachen: Mojataba and his wife Shima, Alireza and his wife Zeinab, Sina and Mehrdad. Our very nice memories made my life so colourful while I was living far from my family. Finally and most importantly, I would like to express my deepest gratitude to my beloved parents who helped me at each second of my life. I am greatly thankful to my brother Yasin who has always supported me in my life and my education. At the very last days of ﬁnalizing this thesis, I was utterly joyed by born of my beloved nephew, Omid. I wish him all the best in his life, and a happy family forever. Contents

1 Introduction 1 1.1 Overview...... 1 1.2 Outline...... 2

2 Quantum Coherent Superconducting Devices5 2.1 Superconductivity...... 5 2.1.1 Bogoliubov approach to BCS superconductivity...... 6 2.1.2 Quasiparticle density in thermal equilibrium...... 7 2.1.3 Superconducting gap...... 8 2.2 Josephson Eﬀect...... 8 2.3 Superconducting Qubits...... 10 2.3.1 Cooper-pair box...... 10 2.3.2 Transmon qubit...... 13 2.4 Qubit-Quasiparticle Interaction...... 13 2.4.1 Energy relaxation induced by quasiparticle tunneling...... 14

3 Normal-Metal Quasiparticle Traps 19 3.1 Modeling...... 19 3.1.1 Introduction...... 19 3.1.2 The diﬀusion and trapping model...... 20 3.1.3 Quasiparticle dynamics during injection and trapping...... 24 3.1.4 Experimental data...... 29 3.2 Optimization...... 32 3.2.1 Introduction...... 32 3.2.2 Enhancing the decay rate of the density...... 33 3.2.3 Suppression of steady-state density and its ﬂuctuations...... 40 3.3 Summary and Conclusions...... 45

4 Quasiclassical Theory of Superconductivity 47 4.1 Gor’kov Equations...... 47 4.2 Dyson Equation in Keldysh-Nambu Space...... 50 4.3 Eilenberger Equations...... 52 4.4 The Dirty Limit...... 56 4.4.1 Boundary conditions for proximity systems...... 58 4.4.2 Usadel equations for normal-superconducting hybrids...... 59

5 Proximity Effect in Normal-Metal Quasiparticle Traps 61 5.1 Introduction...... 61 5.2 Qubit relaxation due to quasiparticles...... 62 5.3 Proximity effect in thin films...... 63 5.3.1 Uniform NS bilayers...... 63 viii Contents

5.3.2 Proximity eﬀect near a trap edge...... 66 5.4 Qubit relaxation with a trap near the junction...... 70 5.4.1 Thermal equilibrium...... 70 5.4.2 Suppressed quasiparticle density...... 72 5.5 Summary...... 77

6 Summary and Conclusions 79

A Tunneling rate equations 81

B Derivation of effective trapping rate 83 B.1 Thin normal metal...... 83 B.2 Effective trapping rate...... 85 B.3 Effective trapping rate integrated over energy...... 87

C Comparison with vortex trapping 89

D Finite-size trap 91

E Quasi-degenerate modes and their observability 95

F Effective length 99 F.1 Effective length due to the pad...... 99 F.2 Effective length due to the gap capacitor...... 100

G Quasiparticle Decay Rate and Steady-State Density 101 G.1 Slowest Quasiparticle Decay Rate Due To Trap...... 101 G.1.1 Single trap...... 101 G.1.2 Multiple side traps...... 103 G.2 Suppression of Quasiparticle Steady-State Density...... 105

H Traps in the Xmon geometry 107

I Proximity eﬀect in uniform NS bilayers 109 I.1 Weak-coupling limit...... 109 I.2 Strong-coupling limit...... 111

J Numerical solution of the self-consistent equation for the order parameter 115

K Spatial evolution of single-particle density of states and pair amplitude 119

L Spectral function in the presence of a trap 123 L.0.1 Thermal equilibrium...... 123 L.0.2 Suppressed quasiparticle density...... 125

Bibliography 127 Chapter 1 Introduction

1.1 Overview

In quantum world, a physical system can be in diﬀerent eigenstates simultaneously. In particular, one can imagine a quantum system in a superposition of two states,

θ θ |Ψi = cos |1i + sin eiφ|0i, (1.1) 2 2 where we refer such quantum system as qubit; the qubit state can be conveniently illustrated by the Bloch sphere drawn in figure (1.1). In 1994, Peter Shor devised a quantum algorithm running on a set of qubits that enables to factorize an integer N to prime factors in an exponentially faster time compared with classical algorithms using binary logic [1]. Such an algorithm can, for example, be used to break public key cryptography protocols. Since this discovery, there has been extraordinary efforts to exploit the potentials of quantum systems for developing future technologies and also in finding physical systems promising to be controlled as a qubit and feasible to scale up. As any system interacts with its surrounding environment, a qubit that is initially prepared in the superposition of two states will eventually decohere into a mixed state that manifests itself in vanishing of the non-diagonal elements of qubit density matrix. In addition to qubit decoherence, faulty quantum preparation, faulty measurements and faulty quantum gates impose an obstacle for realizing quantum computation. For these reasons, performing quantum error correction is at the heart of any scheme for realizing a universal quantum computer, that in turn requires the coherence time of physical qubits compared with gate time to exceed a threshold [2]. Further improvement of the qubit coherence above the threshold is always favorable as it reduces the computational overhead due to quantum error correction. Indeed, maintaining the qubit coherence, denoted by T2, for long enough times

Figure 1.1: Qubit state on a Bloch sphere. All points on the surface correspond to a superposition between states |0i and |1i. 2 Chapter 1. Introduction is a key issue in all aspects of quantum technologies. There are generally two processes that contribute to the qubit decoherence. First, energy relaxation denoted by T1 that is an irreversible process giving energy from or to the qubit which results in qubit state transition to the ground or excited state. Second, pure dephasing denoted by Tφ, is due to perturbations that do not induce qubit state change, but randomly modulate the qubit phase. These two process combine to give, 1 1 1 = + . (1.2) T2 2T1 Tφ A large number of different systems have been proposed to realize physical qubits, each of which utilize a specific physical property of the system to encode quantum information; for example, intrinsic spin degree of freedom, photon polarization or phase difference across a superconducting tunnel junction. Depending on what physical property is used to build a qubit, there are different decoherence mechanisms that are relevant for the system. In this thesis, we particularly focus on superconducting qubits that are among the most promising candidates for realization of quantum computation. There has been an extensive research in the community to control and suppress various decoherence mechanisms leading to nearly six orders of magnitude improvement of coherence time in the past 20 years [3]. In particular, designing qubits that operate in a regime robust against charge noise [4,5] together with improved understanding and control of dielectric losses [6] and the Purcell effect [7] have made possible to reaching coherence time more than 100µs. As the gate operation time for superconducting qubits is of order tens of nanoseconds [8], such long coherence time has enabled to perform about 104 operations per error [3]. While the mentioned decoherence factors are imposed on the qubit from the surrounding environment, there is also an intrinsic decoherence channel that originates from the superconductor itself: tunneling of unpaired electrons or quasiparticles across the Josephson junction. The theory of quasiparticle-induced decoherence is already developed and quasiparticle effects on relaxation, dephasing and parameter renormalization has been studied both theoretically [30, 31, 32, 33, 34, 35, 36, 37] and experimentally [18, 38, 39, 40, 41, 42, 43, 44, 45, 46]. While the generation mechanism of non-equilibrium quasiparticles has yet remained a mystery in the field, in this thesis we study ways to suppress quasiparticle density in order to minimize their detrimental effects. We illustrate how planting normal metals over some parts of the qubit can trap quasiparticles and show how such trapping process can be optimized. Moreover, we study inverse superconducting proximity effect and discuss how its detrimental consequences to the qubit operation can be avoided. While we mostly consider a 3D transmon qubit and give some consideration of an Xmon qubit to explain and discuss our proposal, the idea of controlling quasiparticle population is also important in other superconducting devices such a Cooper-pair pumps, turnstiles and possible topological qubits based on Majorana zero modes where quasiparticle poisoning is a major obstacle. We therefore hope that our work will be of importance and finds applications in those communities as well.

1.2 Outline

The thesis is structured as follows: In chapter2 we review the necessary backgrounds; section 2.1 is devoted to remind the reader about formation of superconductivity and showing how Chapter 1. Introduction 3 quasiparticles appear in the formalism. Section 2.2 uses perturbation theory to discuss about the Josephson effect. In section 2.3 we explain about two types of superconducting qubits, Cooper- pair box and transmon. We finally discuss in section 2.4 how quasiparticle tunneling results in qubit energy relaxation. Chapter3 contains two sections that covers our published papers cited in references [47] and [68]. In section 3.1 we develop a phenomenological model governing the dynamics and steady state density of quasiparticles. Considering a 3D-transmon qubit, this model allows one to evaluate the time it takes to evacuate the injected quasiparticles from the transmon as a function of trap parameters. With the increase of the trap size, this time decreases monotonically, saturating at the level determined by the quasiparticles diffusion constant and the qubit geometry. We determine the characteristic trap size needed for the evacuation time to approach that saturation value. We also present experimental data (obtained by our colleagues) that support our theoretical findings. In section 3.2 we discuss how normal-metal quasiparticle traps can be optimized. We consider some experimentally relevant examples and find optimum trap configurations that maximize the decay rate of excess quasiparticle density. Moreover, we show that a trap in the vicinity of a Josephson junction can significantly reduce the steady-state quasiparticle density near that junction, thus suppressing the quasiparticle-induced relaxation rate of the qubit. Such a trap also reduces the impact of fluctuations in the generation rate of quasiparticles, rendering the qubit more stable in time. We then turn our attention to study how a normal metal in contact with a superconductor can modify superconducting properties and what are the related consequences for the qubit coherence. In chapter4, we review the quasicalssical theory of superconductivity in terms of Green’s functions in Keldysh-Nambu space and derive the Eilenberger equation. We then consider the dirty limit and derive the Usadel equation for a normal-superconducting hybrid that is our starting point to find the influence of a normal-metal trap on the quasiparticle density of states. Chapter5 contains our published paper cited in reference [78]. Here, we first apply Usadel theory to a uniform bilayer and find new analytical corrections to the density of states and minigap energy. We then consider a non- uniform hybrid relevant for physical realization of normal-metal trap for superconducting qubits. We find both theoretically and numerically the density of states and pair amplitude as a function of distance from trap edge. Moreover, building on the phenomenological diffusion equation that we develop in chapter3, we model the effect of the trap on the quasiparticle distribution function. This enables us to calculate different contributions to the qubit relaxation induced by quasiparticle tunneling and pair processes. We find an optimum trap-junction distance that minimizes the qubit relaxation rate. Placing the trap further from the junction than this optimum distance ensures that inverse proximity effect does not harm qubit coherence. We summarize and conclude our work in the last chapter. We have included a number of Appendices as well that complement the main text and present some details of calculations.

Chapter 2 Quantum Coherent Superconducting Devices

In this chapter we review the Bogoliubov approach to conventional superconductivity followed by discussing the Josephson eﬀect and then two speciﬁc types of superconducting qubits, Cooper-pair box and transmon. We then consider qubit-quasiparticle interaction and present how quasiparticle tunneling results in qubit energy relaxation.

2.1 Superconductivity

Superconductivity was discovered in 1911 by H.K. Onnes when he witnessed electric resistance of solid mercury suddenly dropping below any measurable value by cooling down to 4.2 K. Later, it was discovered by Meissner and Ochsenfeld that superconductors are diamagnets as well [9] so that the electromagnetic field is expelled from a bulk superconductor beyond the material- and temperature-dependent penetration length. The microscopic explanation about superconductivity remained a challenge until 1957 when Bardeen, Cooper and Schrieffer proposed a model (BCS) that successfully explains this phenomenon [10]. The key idea of this model is that electrons condensate into a coherent state of pairs. The challenging question here is how two electrons in a lattice can overcome the repulsive Coulomb interaction between them. This can be explained by taking into account the motion of ion cores or phonons; the first electron polarizes its surrounding medium by attracting ion cores; the resulting positive ions can then attract the second electron. While the importance of this electron-lattice interaction was first pointed out in 1950 by Fröhlich [11], in 1956 Cooper showed however weak the (phonon- mediated) attractive interaction between two electrons is, the Fermi surface is unstable against formation of a Cooper pair [12]. There are different approaches to BSC superconductivity, these include variational method, the Bogoliubov approach [13] and the Gor’kov approach [14]. The original BSC paper uses the variational approach for which a trial wave function for the ground state of a superconductor up to the global phase is taken as:

Y iφ † † |ΨBCSi = (|uk| + e |vk|ck↑c−k↓)|0i (2.1) k

† where ckσ is the electron creation operator with momentum k and spin σ, |0i is the vacuum, the product is taken over all one-electron states and the coherence factors, uk and vk, are complex numbers that satisfy normalization condition:

2 2 |uk| + |vk| = 1. (2.2) 6 Chapter 2. Quantum Coherent Superconducting Devices

Indeed, the BSC ground state is clearly a coherent superposition of all states with even number of electrons from zero to infinity with an arbitrary phase factor eiφ. The model Hamiltonian in BCS theory is, X † X † † HBCS = ξkckσc−kσ + λkk0 ck↑c−k↓c−k0↓ck0↑, (2.3) kσ kk0 in which, the energy ξk is measured from the Fermi energy. Since the attractive interaction is mediated by phonons, the coupling strength can be taken constant, λkk0 = −λ, when both the 0 scattering-in and out electrons possess energies less than Debye frequency, |ξk|, |ξk| < ωD, and is zero otherwise. This model Hamiltonian describes the interaction between Cooper pairs and leads to formation of Cooper pair condensate as the ground state for a superconductor. The idea of variational approach to BSC superconductivity is to minimize the expectation value of the model Hamiltonian using the trial ground state given by Eq. (2.1). This enables us to find the coherence factors in a straightforward calculation that we do not present here. Rather, in this section we review the Bogoliubov approach that also gives the coherence factors and provides a handy framework to deal with superconducting excitations as well. In chapter4 we start from Gor’kov approach and present quasiclassical theory of superconductivity to study modifications in superconducting properties when the superconductor is in proximity to a normal metal.

2.1.1 Bogoliubov approach to BCS superconductivity To begin with, we note that since the ground state of a superconductor is formed from Cooper pair condensate including a macroscopic number of Cooper pairs, adding or removing an extra Cooper pair does not really matter. In other words, the value for anomalous average defined as hc−k0↓ck0↑i is finite (averaging is taken with respect to the ground state of a superconductor) and fluctuations around this finite value are small . This enables us to arrive to the mean-field BCS Hamiltonian: |∆2| HMF = X ξ c† c − X(∆c† c† + ∆∗c c ) − . (2.4) BCS k kσ −kσ k↑ −k↓ −k↓ k↑ λ kσ k Here we only consider s-wave superconductivity where the order parameter is everywhere symmetric in k space and is given by, X ∆ = λhc−k0↓ck0↑i. (2.5) k0 We now introduce the Bogoliubov-quasiparticle operator defined as ∗ † γk↑ = ukck↑ + vkc−k↓, (2.6) ∗ † γ−k↓ = −vkck↑ + ukc−k↓, (2.7) where the normalization condition for coherence factors, Eq. (2.2), is derived by enforcing † fermionic anticommutation relations for Bogoliubov operators, {γkσ, γk0σ0 } = δσ,σ0 δk,k0 . The mean-field BCS Hamiltonian is diagonal in basis of Bogoliubov operators provided the following relations for coherence factors hold, 2 1 ξk |uk| = 1 + , (2.8a) 2 εk 2 1 ξk |vk| = 1 − , (2.8b) 2 εk Chapter 2. Quantum Coherent Superconducting Devices 7 where q 2 2 εk = ξk + |∆ |. (2.9) In finding these relations, it also turns out that the phase of superconducting order parameter ∆ is equal to phase of vk relative to uk so that the order parameter has the same phase as the BSC ground state. The Hamiltonian then becomes,

MF HBCS = HG + Hqp, (2.10) for which we deﬁned,

|∆2| H = X(ξ − ε ) − , (2.11) G k k λ k and,

X † † Hqp = εk(γk↑γk↑ + γ−k↓γ−k↓). (2.12) k

Assuming a normal state at T = 0, we have ∆ = 0 and εk = |ξk|. The ﬁrst term in Eq. (2.10) then diﬀers from the corresponding one in the normal phase by,

|∆2| H − HN (T = 0) = 2 X (ξ − ε ) − . (2.13) G G k k λ k>kf By changing the summation to an integration, it is easy to simplify this energy diﬀerence to 1 2 − 2 N0|∆ | that is the condensation energy and expresses the energy gain by forming superconductivity. What is important for us is the second term in Eq. (2.10), Hqp, that illustrates the energy increase corresponding to quasiparticle excitation above the Cooper pair condensate. Indeed, one can directly check from Eq. (2.1) that the superconducting ground state is vacuum state for quasiparticle excitations, γk|ΨBCSi = 0, that are gapped from the ground state condensate by the value determined by the order parameter. Later in this chapter we will show that the density of quasiparticle excitations has an important role in energy relaxation of superconducting qubits. In the following we ﬁnd this density in thermal equilibrium.

2.1.2 Quasiparticle density in thermal equilibrium The Bogoliubov transformation makes it clear that there is a one-to-one correspondence between electronic and quasiparticle excitations. We can therefore write,

Nqp(ε)dε = Ne(ξ)dξ, (2.14) where the quasiparticle density of states is denoted by Nqp(ε) and electronic density of states by Ne(ξ). As we are interested in energies close to Fermi level, we take Ne(ξ) ' Ne(ξF ) ≡ N0 and ﬁnd the normalized quasiparticle density of states for a bulk superconductor, N (ε) dξ ε n(ε) = qp = = Re √ sgn(ε). (2.15) N0 dε ε2 − ∆2 8 Chapter 2. Quantum Coherent Superconducting Devices

In thermal equilibrium, we use Fermi-Dirac distribution function to ﬁnd the density of quasiparticle excitations relative to Cooper pair density, s Z ∞ eq 2 eq 2πT −∆/T xqp = Nqp(ε)f (ε)dε = e . (2.16) N0∆ ∆ ∆ This predicts that the quasiparticle density can be arbitrarily suppressed by reducing the temperature and is essentially negligible if T ∆.

2.1.3 Superconducting gap In order to ﬁnd the superconducting energy gap, we use the Bogoliubov transformation and ﬁnd from Eq. (2.5),

X ∗ h † † i ∆ = λ uk0 vk0 hγ−k0↓γ−k0↓i − hγ−k0↓γ−k0↓i . (2.17) k0 As Bogoliubov quasiparticles are fermionic excitations, their occupation probability is determined by the usual Fermi-Dirac distribution so that we can write,

† † hγ−k0↓γ−k0↓i − hγ−k0↓γ−k0↓i = 1 − 2n(εk0 ) = tanh(εk0 /2T ). (2.18)

We now change the summation in Eq. (2.17) to integration and given the coherence factors, Eqs. (2.8), we ﬁnd

Z ωD q 1 1 2 2 = p tanh( ξ + ∆ /2T )dξ. (2.19) λN0 0 ξ2 + ∆2

In the limit where temperature approaches zero, we ﬁnd 1 = sinh−1 ωD that in the weak λN0 ∆ coupling limit, λN0 1, results in,

−1/λN0 ∆ ' 2ωDe ωD. (2.20)

This indicates that the superconducting order parameter cannot be derived by treating the coupling strength in a perturbative way. We can alternatively express Eq. (2.19) in terms of quasiparticle energy, √ ω2 −∆2 1 Z D 1 = √ tanh(ε/2T )dε. (2.21) λN0 ∆ ε2 − ∆2 We use this relation in chapter5 to calculate the order parameter for a proximitized superconductor relative to a bulk superconductor.

2.2 Josephson Eﬀect

In a Josephson junction formed by two superconductors that are interrupted by an insulating tunnel barrier, a supercurrent flows through the device even in absence of an external bias. This phenomenon is due to the phase difference between the two superconducting electrodes forming the junction. In this subsection, we use perturbation theory to study the Josephson effect. Chapter 2. Quantum Coherent Superconducting Devices 9

One can alternatively use quasiparticle bound states to ﬁnd the same Josephson equations [15]. Let us consider the following Hamiltonian that expresses single electron tunneling across the junction, ˜X X †R L HT = t (cmσcnσ + H.c.), (2.22) m,n σ where we assumed a constant tunneling matrix element t˜, and R and L are labeling right and left sides of the junction, respectively. The electron tunneling operator in terms of Bogoliubov operators reads,

†R L R† L R† L i(φR−φL) cmσcnσ =umunγnσ γmσ + vmvnγnσ γmσe R† L† R† L† iφR R L R L −iφL + unvm γn↓ γm↑ − γn↑ γm↓ e + umvn γn↑γm↓ − γn↓γm↑ e , (2.23) where the coherence factors, u and v, are taken real as Eq. (2.23) explicitly accounts for the phase difference across the junction. As the ground state of a superconductor is vacuum state for quasiparticles, the tunneling Hamiltonian results in zero expectation value. However, the tunneling Hamiltonian taken to the second-order perturbation theory given by, 1 H(2) = X H H , (2.24) T T ε T i i has a finite value in the ground states. Here εi is the energy of intermediate states and the Hamiltonian has terms that transfer two electrons to the right, two to the left, and with no net electron transfer. The latter leads to a constant value in the expectation value that has no physical effect. The terms with a net transfer to the right gives,

R R† L L† R R† L L† (2) γn↑γn↑ γm↓γm↓ + γn↓γn↓ γm↑γm↑ ˜2 X L R −iφL iφR L R hHT i = − t hΨBCS, ΨBCS|umvne L R unvme |ΨBCS, ΨBCSi n,m εn + εm 1 ˜2 i(φR−φL) X = − 2t e umvnunvm L R n,m εn + εm Z ∞ Z ∞ ∆ ∆ 1 ˜2 i(φR−φL) L R L R = − 2t e N0 N0 dξ dξ L R L R −∞ −∞ ε ε ε + ε Z ∞ Z ∞ 1 ˜2 i(φR−φL) L R = − 2t ∆e N0 N0 dθL dθR −∞ −∞ cosh θL + cosh θR 1 gT = − ∆ei(φR−φL), (2.25) 16 gK where the ﬁnal equality is obtained by change of variables u = (θL + θR)/2 and v = (θL − θR)/2 in the last integral that, up to prefactors, results in complete elliptic integral of the ﬁrst kind L R at zero, K(0) = π/2. Here, N0 and N0 are the density of states per spin at the Fermi energy 2 L R˜2 2 of the left and right electrode, gT = 4πe N0 N0 t is the junction conductance, gK = e /2π is the conductance quantum and we assumed equal gap for both sides of the junction. A similar calculation for the net transfer to the left gives the complex conjugate of Eq. (2.25). The sum of these two terms give the energy gain by electron pair tunneling,

U = −EJ cos φ. (2.26) 10 Chapter 2. Quantum Coherent Superconducting Devices

1 gT for which EJ = ∆ is Josephson energy and φ = φR − φL is the phase diﬀerence across the 8 gK junction. This energy is associated with a supercurrent that is driven by the phase diﬀerence across the junction which reads,

2π ∂U π ∆ IJ = = gT sin φ, (2.27) Φ0 ∂φ 2 e where Φ0 = h/2e denotes the superconducting flux quantum. As we have just shown, this supercurrent solely originates from the phase difference between the ground states of the superconducting leads; therefore, it is a dissipationless current. If an external voltage V is imposed to the junction, the phase difference evolves in time according to the AC-Josephson effect,

Φ dφ V = 0 (2.28) 2π dt

Hence, the inductance of the Josephson junction LJ has a non-linear relation with the phase diﬀerence,

dIJ 1 1 LJ = V/ = (2.29) dt π∆gT cos φ

This non-linearity together with the ultra-low dissipation provided by superconductivity makes Josephson junctions promising candidates to build qubits.

2.3 Superconducting Qubits

In the previous section we ignored the fact that as supercurrent flows through the junction, charges build up on the islands of the junction and consequently Coulomb interactions become important as well. These repulsive interactions give another energy scale for the system that 2 is the charging energy. For a single electron transfer to the island, it becomes Ec = e /2C where C is the total capacitance that the island makes with the environment. Once we take into account the charging energy, it becomes clear the Josephson-junction-based devices can act like an artificial atom, as we explain in the following. In this section, we consider two types of superconducting qubits: Cooper-pair box and transmon qubit. The former is the earliest type of the superconducting qubits while the latter was realized some years later and is one the most promising ones in terms of the coherence time and scalability. In writing the Hamiltonian of the qubit, for the moment we neglect the presence of quasiparticles while just trying to give a brief qualitative explanation of their effect. In the next section we explicitly consider quasiparticles and study how their tunneling across the junction result in qubit energy relaxation.

2.3.1 Cooper-pair box

In a pioneering experiment by Nakamura and co-authors [16], the Cooper-pair box was the ﬁrst superconducting device used to demonstrate quantum Rabi oscillations. As schematically depicted in the left panel of ﬁgure (2.1), this qubit simply consists of a Josephson junction where one of its islands is used to store charges, and the other island is to provide these charges. There is also a gate electrode enabling to shift the electrostatic potential of the island with respect to Chapter 2. Quantum Coherent Superconducting Devices 11

Figure 2.1: Left panel: Circuit diagram of a Cooper-pair box. The island is isolated by the Josephson junction and a capacitor. Tuning the gate voltage Vg enables us to control the number of extra Cooper pairs on the island. This voltage is sensitive to ﬂuctuations in the charges that are surrounding the island. Right panel: Circuit diagram of a single-junction transmon qubit. The Junction is shunted by a large capacitance to increase the ration of EJ /EC that makes the qubit robust against the charge noise. the bulk electrode in order to tune the number of charges on the island. The Hamiltonian of this qubit reads, 2 ˆ H = Ec(Nˆ − Ng) − EJ cos φ, (2.30) where the operator Nˆ counts the number of single electrons that are tunneling-in or out of the island,

Nˆ|Ni = N|Ni, (2.31) and is conjugate to the Josephson phase operator, [φ,ˆ N/ˆ 2] = i. The oﬀset charge Ng = CgVg/e is a continuous variable expressing the polarization charge on the island induced by the gate voltage Vg. Important feature of this qubit is that the island is made small enough such that the accessible thermal energy at millikelvin temperatures (where the qubit is operating) is much smaller than the charging energy, Ec kBT . Moreover, the charging energy also dominates the Joseph- son energy, Ec EJ ; in this condition, the number of extra charges on the island becomes a well deﬁned variable. The qubit Hamiltonian in charge basis reads, E H = X E (Nˆ − N )2|NihN| − J (|NihN + 2| + |N + 2ihN|) . (2.32) c g 2 N

The charging energy as a function of the offset charge, Ng, gives a set of parabolas associated with single-electron charges, N, present at the island. On the other hand, the Josephson energy connects the nearby charge states with the same parity. By tuning the gate voltage such that the offset charge is close to the values where these parabolas cross each other, only the two crossing states remain important and the effective Hamiltonian become a 2 × 2 matrix. In particular, assuming initially there is no single electron present at the island, the qubit working point is at Ng = 1 that results in qubit states to be in superposition of |Ni and |N + 2i charge states, |Ni + |N + 2i |Ni − |N + 2i |0i = , and |1i = , (2.33) 2 2 12 Chapter 2. Quantum Coherent Superconducting Devices

Figure 2.2: Energy diagram for the first two low-laying states with even and odd parity. The zero point energy in each panel is chosen at the bottom of ground state and energies are ¯ 1 even odd normalized to average energy E1 = 2 (E0 + E1 ). Panel (a): for Cooper-pair box, where EJ /EC 1, energy levels have high charge dispersion. The dashed line in the figure points the offset charge at the qubit working point; any deviation from this point changes the qubit frequency. In addition, a transition from even to odd charge states induced by a single electron tunneling destroys the qubit state that is a superposition of charge states with same parity. Panels (b) and (c): as the ratio of EJ /EC is increased, the energy levels become less sensitive to the offset charge. Panel (d): in the transmon regime, where EJ /EC 1, the energy levels become insensitive to the offset charge. Moreover, the even and odd charge sectors contribute equally to the qubit logical state.

while the qubit frequency becomes ω01 = EJ . Panel (a) of figure (2.2) illustrates the eigenenergies of the two low-lying states for the even and odd sectors of the qubit Hamiltonian, Eq. (2.32). The figure makes it clear that the Cooper- pair box sufferers from two major drawbacks that limit its coherence time: First, the high charge dispersion of the energy levels makes qubit vulnerable to the charge noise. Indeed, fluctuations of the charges in the surrounding environment causes the offset charge deviate from the working point; this in turn modulates the qubit frequency and leads to qubit dephasing. Second, as it is shown in Eq. (2.33), the qubit states consist of symmetric superposition of charge states with Chapter 2. Quantum Coherent Superconducting Devices 13 equal parity; if an unpaired electron tunnels to the island, it poisons the device by changing the charge parity that brings the qubit out of its computational subspace. These issues limited coherence time of the Cooper-pair box to about 10−9 s that is more than 5 orders of magnitude less than the nowadays state-of-the-art qubits, [6, 17].

2.3.2 Transmon qubit Reducing the charge dispersion makes the qubit frequency less sensitive to the charge noise. This can be achieved, for example, by going into the transmon regime where the ratio of EJ /EC is large. In this case, the quantum fluctuations of the phase is relatively small, while the uncertainty of charge in the qubit state is significant. Going to this regime, however, would also reduce the anharmonicity of the energy levels, but remarkably, while the charge dispersion decreases exponentially in EJ /EC , the anharmonicity is suppressed algebraically with a slow power law in EJ /EC [4]. Indeed, there is a range for the ratio between energy scales EJ and EC for which the charge dispersion is flattened, rendering the qubit robust against charge noise, while enough anharmonicity is kept, thus avoiding the excitation of higher-level states. As schematically illustrated in right panel of figure 2.1, in order to realize the high ratio of EJ /EC , the junction in transmon qubit is shunted by a large capacitor; this lowers the total charging energy, EC , and makes EJ /EC large. Panel(d) of figure (2.2) shows the two energy levels of transmon qubit with even and odd charge parity, while panels (b) and (c) illustrate the crossover from charging regime to transmon regime. The picture clearly shows that as the ratio of EJ /EC is increased, the total charge dispersion rapidly decreases. Therefore, the energy difference between states with different parities rapidly decreases as well. Indeed, in transmon qubit the logical state of the qubit contains two physical states with even and odd parity while the energy difference between these two states is given by [4],

m m ω˜eo = ωeo cos(πNg), (2.34) where,

r 2m 2m+1 √ 2 2 8E 4 m p m J − 8EJ /EC ωeo = 4 8EC EJ (−1) e , (2.35) π m! EC for which m = 0 for the logical qubit ground state and m = 1 for the excited states. Therefore, a transmon qubit is also less disturbed from single-electron tunneling because this does not bring the qubit out of its computational subspace.

2.4 Qubit-Quasiparticle Interaction

So far, in writing down the qubit Hamiltonian, we have neglected the quasiparticle excitations. This is because, as it follows from Eq. (2.16), the quasiparticle density is negligible at millikelvin temperatures where superconducting qubits are operating. However, a number of experiments firmly confirm that at low temperatures (below around 0.1 Tc for Aluminum) quasiparticles fail to equilibrate with the environment and their density significantly exceeds the expected equilibrium value [18, 19, 20, 21]. These excitations have a detrimental effect on the performance of superconducting devices in a wide range of applications. To name a few, they limit the sensitivity of photon detectors in astronomy [22, 23] and cooling power of micro-refrigerators 14 Chapter 2. Quantum Coherent Superconducting Devices

[24, 25] and cause braiding errors in proposed Majorana-based quantum computation [26, 27, 28, 29]. In superconducting qubits, it has been ﬁrmly established both theoretically [30, 31, 32, 33, 34, 35, 36, 37] and experimentally [18, 38, 39, 40, 41, 42, 43, 44, 45] that quasiparticle tunneling causes qubit energy decay and dephasing. In addition, the residual nonequlibrium quasiparticles result in qubit excited state population in excess of thermal equilibrium value [46]. In this section we take quasiparticles into account and discus how their tunneling across the junction results in qubit energy relaxation. The system Hamiltonian in presence of quasiparticles can be divided into three parts

H = Hq + Hqp + Hint, (2.36) where Hq is the qubit Hamiltonian and the second term describes presence of quasiparticles on the left and right superconducting leads,

X X s† s Hqp = Ekγk,σγk,σ. (2.37) s=L,R k,σ

The third term describes quasiparticle tunneling across the junction; from Eq. (2.23) we write,

0 p Hint = Hint + Hint, (2.38)

0 p where the single quasiparticle tunneling, Hint, and pair tunneling, Hint, read up to a global phase factor,

0 ˜ X L R iφ/ˆ 2 R L −iφ/ˆ 2 L† R Hint =t (uk uk0 e − vk0 vk e )γkσ γk0σ + H.c., (2.39) k,k0,σ p ˜X L R iφ/ˆ 2 R L −iφ/ˆ 2 L† R† Hint =t [(uk vk0 e + uk0 vk e )γk↑ γk0↓ k,k0 R L −iφ/ˆ 2 L R iφ/ˆ 2 R L +(vk0 uk e + vk uk0 e )γk0↓γk↑] + (L ↔ R). (2.40)

2.4.1 Energy relaxation induced by quasiparticle tunneling The tunneling Hamiltonian makes possible qubit state transition occurring by exchanging energy with the tunneling quasiparticle. Up to lowest order in tunneling amplitude t˜, the transition from excited state, |1i, to the ground state, |0i, with qubit frequency ω10 is found using Fermi’s golden rule

X 2 Γ10 = 2πhh |h0, {λ}qp|Hint|1, {η}qpi| δ(Eλ,qp − Eη,qp − ω10)ii, (2.41) {λ}qp where {η}qp ({λ}qp) is the initial (ﬁnal) state of quasiparticles with energy Eη,qp (Eλ,qp). The double angular brackets hh...iidenote averaging over initial quasiparticle states and the summation is over all quasiparticle states. To calculate this rate, we note that the pair tunneling part p of the interaction Hamiltonian, Hint, contains terms creating or annihilating two quasiparticles; this absorbs or releases energy by amount twice the superconducting gap. On the other hand, superconducting qubits are designed such that the qubit frequency is much smaller than twice the gap, ωif 2∆, since this is necessary to avoid breaking Cooper pairs during qubit operation. Therefore, up to the leading order given by Fermi’s golden rule, Chapter 2. Quantum Coherent Superconducting Devices 15 the pair tunneling part does not contribute in the transition rate due to energy conservation. Moreover, we assume low-temperature limit so that the characteristic energy of quasiparticles, δε, (that is proportional to temperature and is measured from the gap) is small compared with superconducting energy gap,√δε ∆. This enables us to approximate the coherence factors, Eqs. (2.8), by uk ' vk0 ' 1/ 2 that in turn simpliﬁes the single-quasiparticle tunneling to, ˆ 0 X φ L† R H = t˜ i sin γ γ 0 + H.c. (2.42) int 2 kσ k σ k,k0,σ

The transition rate then factorizes into terms that separately account for qubit dynamics and quasiparticle kinetics,

φˆ Γ = |h0| sin |1i|2S (ω ) (2.43) 10 2 qp 10 where the quasiparticle current spectral density becomes,

˜2 X X L† R R† L 2 Sqp(ω) =2πt hh |h{λ}qp|γkσ γk0σ + γk0σγkσ|{η}qpi| δ(Eλ,qp − Eη,qp − ω)ii 0 k,k ,σ {λ}qp ˜2 X R† R L L† =4πt hhh{η}qp|γk0σγk0σ|{η}qpih{η}qp|γkσγkσ |{η}qpiδ(Eλ,qp − Eη,qp − ω)ii k,k0,σ 32E Z ∞ = J n(ε)n(ε + ω)f(ε)[1 − f(ε + ω)]dε. (2.44) π∆ ∆

R† R R L L† L Here we used hhh{η}qp|γ γ |{η}qpiii = f(ε ), hhh{η}qp|γ γ |{η}qpiii = 1 − f(ε ) and took L R L R L R Eλ,qp − Eη,qp = Eλ,qp + Eλ,qp − Eη,qp − Eη,qp = ε − ε . The spectral density depends on the quasiparticle distribution function; assuming “cold” quasiparticles meaning their energy (or effective temperature) is small compared with qubit q ∆ frequency, δε ω, we can take 1 − f(ε + ω) ' 1 and n(ε + ω) = 2ω . This simplifies the spectral function and we find; s 8E 2∆ S (ω) = J x (2.45) qp π qp ω where 2 Z ∞ xqp = n(ε)f(ε)dε, (2.46) ∆ ∆ is the density of quasiparticles normalized to the Cooper-pair density. In thermal equilibrium this quantity is given by Eq. (2.16). However, we note that Eq. (2.45) is valid for arbitrary distribution function provided δε is the smallest energy scale of the system. To find the qubit excitation rate, Γ01, one has to calculate Sqp(ω) for ω < 0 that is obtained from Eq. (2.44) by replacing ε → ε − ω, ω → −ω; within our low-temperature assumption, in general we have S(−ω) S(ω) indicating that there is no quasiparticle with energy high enough to excite the qubit. Eq. (2.45) is of central importance in this thesis as it indicates that the qubit decay rate can be decreased by reducing the quasiparticle density near the Josephson junction. 16 Chapter 2. Quantum Coherent Superconducting Devices

Figure 2.3: Panel (a) is reproduced and slightly modified from Ref. [19]. It illustrates experimental data points for the number of quasiparticles in a superconducting resonator and compares experimental findings with theoretical prediction in thermal equilibrium. At low temperatures relevant to the operation of superconducting qubits, the residual quasiparticle density is significantly higher than theory predictions. Panel (b) is reproduced from Ref. [43] and makes it clear that suppressing the quasiparticle density, that in this case is achieved in a 3D transmon qubit by cooling in magnetic field to generate vortices, can improve the qubit coherence times. It is difficult to control vortices and it is observed that a large number of them can negatively influence qubit performance.

In figure (2.3) we have shown some experimental highlights about quasiparticles and their impact on the qubit decay rate. Panel (a) shows the measured quasiparticle density in a superconducting resonator as a function of temperature and reveals that the density saturates when temperature goes below ∼ 160 mK. While a detailed knowledge of the source that gener- ates nonequilibrium quasiparticles is eventually needed to solve quasiparticle-related problems, physicists have been looking for ways to suppress quasiparticle density that promises improving the qubit coherence. One proposal that has been recently realized is to cool down the qubit in a magnetic field that would generate vortices in the device. At the core of a vortex, superconducting order parameter is suppressed, which makes it possible to trap quasiparticles- we Chapter 2. Quantum Coherent Superconducting Devices 17 describe the trapping mechanism in detail in the next chapter. Panel (b) of figure 2.3 shows measured relaxation times for a 3D transmon qubit as a function of magnetic field. The strength of magnetic field determines the number of generated vortices and, consequently, the level of suppression in the quasiparticle density. The plot makes it clear that up to some point in the magnetic field, vortices could improve the qubit coherence while for magnetic field above ' 200 mG, the qubit performance is negatively affected. This behavior is attributed to the energy dissipation that a large number of vortices can cause [43]. Moreover, it is difficult to control the vortex position. This has motivated us to study another method for suppressing quasiparticle density that enables us to control trap size and placement. In the next chapter, we will introduce normal-metal quasiparticle traps and discuss how they work and how they can be optimized by proper trap placement.

Chapter 3 Normal-Metal Quasiparticle Traps

We begin this chapter by explaining how a normal-metal connected to superconducting qubit can act as a sink for quasiparticles. Section 3.1 contains part of our work that has been published under the title Normal-metal quasiparticle traps for superconducting qubits and cited in Ref. [47]. Here we develop a model for the effect of a single small trap on the dynamics of the excess quasiparticles injected in a transmon-type qubit. Section 3.2 containes a paper of the author that has been published with title Optimal configurations for normal-metal traps in transmon qubits and cited in Ref. [68]. Here we build on section 3.1 and discuss how quasiparticle trapping can be optimized. We show proper trap design can increase the slowest decay rate of quasiparticle and at the same time suppress quasiparticle steady-state density and its fluctuations. I co-authored Ref. [47] and contributed by discussing the model and experimental data, comparing simplified analytical results with exact numerics and preparing a number of sug- gested figures for the paper. I contributed to Ref. [68] by doing all of exact and numerical modelings and their corresponding figures to demonstrate enhancing the decay rate of the excess quasiparticle density as well as suppression of the quasiparticle steady-state density due to normal-metal traps, comparing normal-metal traps on the pads with vortex trapping, and the analysis of traps for Xmon qubits.

3.1 Modeling

3.1.1 Introduction Ideal superconducting devices rely on dissipationless tunneling of Cooper pairs across a Joseph- son junction. For example, in a Cooper pair pump [48], the controlled transport of Cooper pairs across two or more junctions can in principle make it possible to relate frequency and current and hence enable metrological applications of such a device [49]. For quantum information purposes, the non-linear relation between the supercurrent and the phase difference across a junction makes the junction an ideal non-linear element to build a qubit [50]. However, in addition to the pairs tunneling, single-particle excitations known as quasiparticles can also tunnel. In the pumps this leads to “counting errors”, limiting the accuracy of the current-frequency relation [48, 49]. In qubits, quasiparticles interact with the phase degree of freedom, providing an unwanted channel for the qubit energy relaxation [32,6]. While in many cases it is impossible to prevent the creation of quasiparticles, one may keep them away from the Josephson junctions by trapping. Evacuation of the quasiparticles from the vicinity of the junction provides a way to extend the energy relaxation time (T1) in the steady state, and to restore the steady state after a perturbation, whether caused by qubit operation or some uncontrolled environmental effect. Quasiparticle trapping has been explored for a long time, and various proposal exists on how to implement such a trapping. For example, gap engineering takes advantage of the fact 20 Chapter 3. Normal-Metal Quasiparticle Traps that quasiparticles accumulate in regions of lower gap to steer them into or away from certain parts of the device. Gap engineering was used successfully to limit quasiparticle “poisoning” in a Cooper pair transistor [51], while proved ineffective in a transmon qubit [52]. A vortex in a superconducting film can also act as a well-localized trap, since the gap is completely suppressed at the vortex position. Trapping by vortices has been demonstrated [53, 54, 43, 55], but vortex motion may induce an unwanted dissipation. An island of a normal metal in contact with the superconductor may also serve as a quasiparticles trap [25, 56]. In the limit of weak electron tunneling across the contact, the proximity effect is negligible. The quasiparticles tunneled into the normal metal are trapped there upon losing their energy by phonon emission or inelastic electron-electron scattering. The majority of previous works concentrated on the control of a steady-state quasiparticle population [49, 25, 56]. In contrast, we are interested in the effect of a normal-metal trap on the dynamics of the quasiparticle density. Traps accelerate the evacuation of the excess quasiparticles injected in a qubit in the process of its operation. Our main goal is to determine how the characteristic time of the evacuation depends on the parameters of a small normal- metal island in contact with the superconducting qubit. The characteristic time shortens with the increase of the trap size, saturating at a value dependent on the qubit geometry and the quasiparticle diffusion coefficient. The size at which a trap becomes effective depends on the contact resistance, the energy relaxation rate in the normal-metal island, and the effective temperature of the quasiparticles. We develop a simple model allowing to evaluate the time evolution of the quasiparticle density and find the characteristic evacuation time as a function of the trap parameters. The model is validated by measurements of the qubit T1 relaxation time performed on a series of transmons with normal-metal traps of various sizes. This section is organized as follows: in Sec. 3.1.2 we develop a phenomenological quasiparticle diffusion and trapping model which includes the effect of a normal-metal trap. In Sec. 3.1.3 we study the dynamics of the density during injection and trapping in a simple configuration, and in Sec. 3.1.4 we provide experimental data (obtained by our collaborators in Yale University) supporting our approach.

3.1.2 The diﬀusion and trapping model

Let us consider a quasiparticle trap made of a normal (N) metal covering part of a superconducting (S) qubit. The contact between the two superconductor and the normal trap is provided by an insulating (I) layer characterized by a small electron transmission coeﬃcient. In order to relate the quasiparticle tunneling rate to the conductance of the contact, we use the tunneling Hamiltonian formalism applied to a model N-I-S system, see Fig. 3.1,

H = Hqp + HN + HT , (3.1) X † Hqp = εnγnσγnσ , (3.2) nσ X † HN = ξmcmσcmσ , (3.3) mσ

te X † † HT = √ cmσdnσ + dnσcmσ . (3.4) ΩN ΩS m,n,σ Chapter 3. Normal-Metal Quasiparticle Traps 21

1 N dN ⌫N r

0 0 1 2 ✏ 3

esc(✏) tr 2 S dS ⌫S 1

0 0 1 ✏

Figure 3.1: Left: a small superconductor S of thickness dS separated from a normal metal N of thickness dN by an insulating layer. Right: depiction of the processes leading to trapping: tunneling from S to N with rate Γtr and from N to S with rate Γesc(ε), and relaxation in N with rate Γr.

We denote with ΩN,S = A × dN,S the volumes of the N and S layers, respectively (A is the † † area of interface, and dN,S are the layers thicknesses); cmσ and dnσ are the creation operators for electrons in the normal metal (energy ξm and spin σ) and superconductor. The electron operators in the superconductor are related by Bogoliubov’s transformation to the quasiparticle (†) annihilation (creation) operators γnσ ,

† dn↑ = unγn↑ + vnγn↓ (3.5) † † dn↓ = −vnγn↑ + unγn↓ (3.6) 2 2 1 ξn un = 1 − vn = 1 + . (3.7) 2 εn

p 2 2 Here εn = ξn + ∆ is the energy of a quasiparticle, and ξn is the energy of electron in the normal state of the superconductor. The tunneling constant te can be related, by Fermi’s golden rule, to the resistance RT of the contact, R 2 2π q ~ = 4π te νS0νN0 ,Rq = 2 , (3.8) 2πRT e where νN0 and νS0 are the densities of states in the normal metal and in the (normal state of the) superconductor, respectively. The tunnel conductance, 1/RT , is proportional to the area A of the junction; the intensive quantity characterizing the insulating layer is its conductance per unit area, 1/RT A. We may use Fermi’s golden rule to evaluate also the rates of tunneling-induced change P † of the occupation factors of electrons, f(ξm) = σhcmσcmσi, and quasiparticles, fqp(εn) = P † σhγnσγnσi. We can distinguish two processes. Quasiparticles tunnel from the superconductor 2 into the normal metal with rate Γtr = 2π te νN0/ΩS. The transition rate is proportional to the density of the ﬁnal states involved in the transition, therefore the quasiparticle trapping rate does not have a pronounced energy dependence. The complementary process of a non-equilibrium electron escape into the superconductor, however, does display a strong energy dependence 22 Chapter 3. Normal-Metal Quasiparticle Traps

2 associated with the BCS singularity in the density of ﬁnal states, Γesc (ε) = 2π te νS0νS(ε)/ΩN ; here ε νS(ε) = √ (3.9) ε2 − ∆2 is the normalized BCS density of states. One can see from Eq. (3.8) that the rates Γtr and Γesc (ε) are independent of the area A at ﬁxed conductance per unit area of the insulating layer. We may express the rates as

Γtr = γetr/dS , Γesc(ε) = γeesc(ε)/dN (3.10) in terms of quantities independent of geometry, γetr and γeesc,

Rq RqνS(ε) γetr = , γeesc = . (3.11) 4π(RT A)νS0 4π(RT A)νN0 with (RT A) being the contact resistance times the area of the contact. This product, with units of Ω·cm2, is independent of A, being inversely proportional to the transmission coefficient through the insulating barrier. The above formulas enable us to estimate the trapping and escape rates for an aluminum- copper interface for a typical experimental setup (cf. Sec. 3.1.4): aluminum has a density of 47 3 states νS0 = 0.73 × 10 /Jm [57] and a direct measurement of the contact resistance yields 2 −5 (RT A) ∼ 430 Ωµm (this corresponds to the transmission coefficient of order 10 ). Taking 6 −1 dS ∼ 80 nm we find, using Eqs. (3.10) and (3.11), Γtr ∼ 8 × 10 s . The escape rate saturates at an energy-independent value, Γesc(ε) → Γesc at energies ε ∆. Since dS ≈ dN and νS0 ≈ νN0 in a typical experiment, one has Γesc ≈ Γtr. In writing the rate equations for the energy distribution functions of electrons and quasiparticles, we assume the continuum limit for energies ξm and εn. It is convenient to define the probability density to find an electron (quasiparticle) in the normal metal (superconductor) with energy ε ≥ ∆ as

νN0ΩN pN (ε) = f(ε) (3.12) νS0ΩS pS(ε) =νS(ε)fqp(ε) . (3.13)

Without loss of generality, we normalize the probability with respect to νS0ΩS. Note that eventually, the experimentally accessible quantity is the normalized quasiparticle density, which can be derived from pS as 2 Z ∞ xqp = dε pS(ε) . (3.14) ∆ ∆ In the absence of spatial dispersion of the distribution functions, the rate equations read (see AppendixA)

p˙N (ε) =ΓtrpS (ε) − Γesc (ε) pN (ε) − ΓrpN (ε) , (3.15)

p˙S (ε) =Γesc (ε) pN (ε) − ΓtrpS (ε) . (3.16)

The terms proportional to Γtr describe trapping of quasiparticle excitations in the normal metal, and those proportional to Γesc(ε) the possible escape of electron excitations back to the superconductor; these events take place with rates described by Eqs. (3.10)-(3.11). Chapter 3. Normal-Metal Quasiparticle Traps 23

Since the tunneling process is elastic, excitations appear in the normal metal at energies close to the gap ∆. At low temperature T ∆, there are many unoccupied states below ∆ in the normal metal, into which the excitations can decay. These inelastic processes are mediated by electron-electron and electron-phonon interactions and lead to relaxation, which we capture in Eq. (3.15) with the phenomenological rate Γr. All the processes included in the rate equations (3.15)-(3.16) are represented in the right panel of Fig. 3.1. If the relaxation is immediate, the quasiparticles get trapped in the normal metal with rate Γtr. However, the relaxation rate Γr due to electron-electron and electron-phonon interactions in the normal metal is of course finite. It has been estimated in the supplementary to [43] to 7 −1 be Γr ∼ 10 s ; the measurements reported in Ref. [58] lead to a relaxation rate for electron- phonon interaction of the same order of magnitude, while an estimate based on [59] yields the 8 −1 faster relaxation rate Γr ∼ 10 s . In all cases, relaxation cannot be assumed immediate in comparison with the trapping and escape rates estimated above, especially taking into account that the escape rate quickly increases for energies approaching the gap due to the divergent BCS density of states in Eq. (3.9). In fact, for some energy interval close to the gap, the escape rate dominates the quasiparticle dynamics, such that the excitations do not have enough time to relax. Therefore, we cannot in general neglect the backflow of excitations from the normal trap to the superconductor. The backflow may result in an effective rate which is slower than Γtr. Assuming a steady- state distribution of non-equilibrium electrons in the normal layer, we set p˙N = 0 in Eq. (3.15) and solve for pN in terms of pS (see also AppendixB). Substituting the solution into Eq. (3.16) and integrating over energy, we arrive at

x˙ qp = −Γeffxqp , (3.17) with the effective trapping rate defined by

∞ 1 Z ΓtrΓr Γeﬀ = R ∞ dε pS(ε) . (3.18) ∆ dε pS(ε) ∆ Γesc(ε) + Γr

It is clear that Γeff is suppressed to a level below Γtr. The level of suppression depends on the typical width of the quasiparticle distribution function in energy space. Assuming pS(ε) is characterized by an effective temperature, T ∆, we find that the trapping is not suppressed, 1/2 Γeff ≈ Γtr, only if the energy relaxation is fast enough (Γr (∆/T ) Γesc); in this case excitations in the normal metal quickly relax to energies below the gap and cannot return 1/2 into the superconductor. In the opposite case (Γr . (∆/T ) Γesc), the effective rate becomes 1/2 T -dependent and suppressed below the nominal trapping rate, Γeff ≈ (2T/π∆) ΓtrΓr/Γesc. Note that in the slow relaxation regime the effective trapping rate Γeff is independent of the tunneling probability between superconductor and normal metal, the limiting value of Γeff being proportional to the relaxation rate. The quasi-static approximation (p˙N = 0) we used above becomes justified once we move from the model system of Fig. 3.1 to a more realistic geometry of a long superconducting strip in contact with a metallic trap, see Fig. 3.2a. In that geometry, the time variation of the quasiparticle distribution function pS is controlled by the diffusion time in the strip, which is typically substantially longer than 1/Γr. The generalization of the rate equations (3.15) and (3.16) to include diffusion is performed in AppendixB. In addition to diffusion, other processes such as quasiparticle recombination, generation, and trapping in the bulk must be 24 Chapter 3. Normal-Metal Quasiparticle Traps generally taken into account. For sufficiently thin normal and superconducting layers, we find a generalized diffusion equation for the quasiparticle density xqp,

2 2 x˙ qp =Dqp∇ xqp − a(x, y)Γeffxqp − rxqp − sbxqp + g , (3.19) where xqp(x, y) depends only on coordinates in the plane of the superconducting strip (and is assumed constant across its thickness) and the area function a(x, y) equals 1 for x and y where the trap and the superconductor are in contact, and 0 elsewhere, see Fig. 3.2(a). The diffusion constant Dqp in Eq. (3.19) is proportional to the normal-state diffusion constant for the electrons in the superconductor – the proportionality coefficient can in principle be calculated from the detailed information on the energy distribution of quasiparticles that we 2 discard in using the phenomenological Eq. (3.19). The recombination term rxqp accounts for processes in which two quasiparticles recombine into a Cooper pair [60], again neglecting the details of the quasiparticle distribution. The relationship between recombination time, quasiparticle energy, and electron-phonon interaction strength can be found in [61]. Moreover, there is a background trapping term sbxqp that describes any process that can localize a quasiparticle and hence remove its contribution to the bulk density xqp. Trapping by vortices is an example of such a process, recently characterized in [43]. The generation rate g describes pair-breaking processes, both thermal and non-thermal; at low temperatures, non-equilibrium processes of unknown origin lead to a quasiparticle density orders of magnitude larger than the thermal equilibrium one [20, 19]. In what follows we will neglect both background trapping and recombination: according 3 −1 3 −1 to the measurements in [43] we expect sb < 0.2 × 10 s as well as rxqp < 1.25 × 10 s −4 (having assumed xqp < 10 ). Both processes are orders of magnitudes slower than the effective trapping rate Γeff, even when the latter is highly reduced by backflow. Indeed, even for a low 7 −1 effective temperature T = 10 mK, using Γr ∼ 10 s and ∆/h = 44 GHz for aluminum, we 6 −1 find Γeff ∼ 0.55 × 10 s . Finally, we assume a long wire geometry, where the dimensions of the system in the x and z directions are sufficiently small such that the superconductor can be treated as (quasi)one-dimensional, and we consider traps that are small (in a sense to be specified below), so that they are effectively zero-dimensional. In this case, from Eq. (3.19) we obtain 2 x˙ qp = Dqp∂y xqp − γδ(y − l)xqp + g , (3.20) where the trap is at position y = l and γ = Γeff d, with d the length of the trap in y direction. To estimate when the trap is sufficiently small, we note that the trapping length q λtr ≡ Dqp/Γeff (3.21) gives the scale over which the density decays due to trapping, so the smallness condition is d λtr. In the next section we study the dynamics of the quasiparticle density by solving Eq. (3.20) in various regimes.

3.1.3 Quasiparticle dynamics during injection and trapping In this section we compute the dynamics of the quasiparticle density in a simple geometry depicted in Fig. 3.2(b). It models a transmon qubit in Fig. 3.2(a) by neglecting for simplicity both the gap capacitor near the Josephson junction and the square pad at the opposite end Chapter 3. Normal-Metal Quasiparticle Traps 25

x y (a) (a) j j

y (b) 0 L (b) 0 l L

Figure 3.2: a) Figure of a realistic transmon qubit device close to the proportions of experiment. The Josephson junction is indicated with the crossed box, in grey is the superconductor, and in red the normal metal trap. Shown is half the qubit (the dashed lines indicate that the superconducting structure including trap is mirrored on the left hand side of the junction). b) Simplified model of a 1D superconducting strip with small trap, described by Eq. (3.22). of the long wire. Note that because of the spatial symmetry, it is sufficient to consider only half of the system, 0 ≤ y ≤ L. After separating out the steady-state background density due to the finite generation rate g, the equation controlling the evolution of the excess density of quasiparticles takes the form

" 2 # ∂xqp (y, t) ∂ = Dqp 2 − γδ (y − l) xqp (y, t) ∂t ∂y (3.22) + +jδ y − 0 θ (−t) θ (t + tinj) .

This diffusion equation is supplemented by the boundary conditions ∂yxqp (L, t) = 0 and ∂yxqp (0, t) = 0. The former condition ensures that no quasiparticles leave the device (hard wall condition), while the latter reflects the spatial symmetry of the system. In the experiments, quasiparticles are generated at the Josephson junction when injecting a high-power microwave pulse into the cavity hosting the qubit [43], resulting in a time-dependent source of quasiparticles localized at y = 0. In Eq. (3.22), this source is modeled by a term with a generation rate proportional to j active over the time interval −tinj < t < 0. Clearly, there are two stages of time evolution: first, during the injection process, when the source term is switched on, the quasiparticle density will start to rise and distribute across the wire. Once the source term is switched off, the presence of the normal-metal trap ensures the decay of the excess density back to zero. In the following, we provide analytical results for the time-dependent dynamics of the quasiparticle density, where we focus predominantly on the experimentally accessible [43] density at the junction, y = 0. The time-dependent diffusion equation (3.22) can be solved via a decomposition in the modes λ t e k nk (y) of the homogeneous equation (i.e., Eq. (3.22) without the source term), with λk being 26 Chapter 3. Normal-Metal Quasiparticle Traps

the eigenvalue and nk satisfying equation " # ∂2 λ n (y) = D − γδ (y − l) n (y) . (3.23) k k qp ∂y2 k

For a strip of finite length L, the eigenvalues are discrete and the eigenmodes form an orthonor- mal basis, Z L dy nk (y) nk0 (y) = δkk0 . (3.24) 0 L In presence of the trap at y = l, the eigenmodes are defined piecewise as ( 1 cos (ky) y < l nk (y) = √ (3.25) Nk ak cos (ky) + bk sin (ky) y > l, with the normalization constant Nk (which will be provided explicitly later in some limiting cases) and the coefficients γ ak = 1 − cos (kl) sin (kl) Dqpk γ 2 bk = cos (kl) . (3.26) Dqpk

2 The eigenvalue corresponding to eigenmode k is λk = −Dqpk . The boundary condition at y = 0 is satisﬁed by Eq. (3.25), while the one at y = L gives the equation

1 − γ cos (kl) sin (kl) cot (kL) = Dqpk . (3.27) γ cos2 (kl) Dqpk which ﬁxes the wave vector k to discrete values. In terms of the eigenbasis introduced above, by solving Eq. (3.22) we ﬁnd that the excess quasiparticle density immediately after the injection, at time t = 0, is given by

eλktinj − 1 x (y, 0) = X c n (y) (3.28) qp k λ k k k with Z L dy + j ck = j nk (y) δ y − 0 = nk (0) . (3.29) 0 L L where we assumed that at times t < −tinj, there were no excess quasiparticles in the system. Once the injection stage is ﬁnished, the subsequent trapping of the quasiparticles controls the evolution of their density,

2 −Dqpk t j X 2 1 − e inj x (y, t) = n (0) e−Dqpk t n (y) . (3.30) qp L k D k2 k k qp

The expressions for xqp (y, t) derived here are general and do not rely on any further simplifying assumption. Next, we consider in more detail several limiting cases. Chapter 3. Normal-Metal Quasiparticle Traps 27

3.1.3.1 The long-strip limit

If both the injection time tinj and the time t after injection are short compared to the diffusion 2 time scale ∼ L /Dqp, the generated quasiparticles do not reach the far end of the strip, and we may take the limit L → ∞. In this limit, all values of k are allowed and sums over k 1 P R dk are replaced by an integral, L k → 2π . Moreover, when letting L → ∞ while keeping the distance l between trap and junction finite, the normalization constant Nk is dominated by the 2 2 part of the mode with y > l, so that Nk ' (ak + bk)/2. Clearly, a single trap suppresses the excess quasiparticle density at the junction best if the distance l is short. For simplicity, from now on we assume l → 0+. That leaves us with only one characteristic time scale, the saturation time 2 tsat = Dqp/γ . (3.31) It gives the time scale over which the density near the junction approaches its steady-state value x0 = j/γ, prescribed by the balance between generation and trapping, during the injection process. Indeed, after time τ from the start of the injection, quasiparticles have p spread over a distance ∼ Dqpτ and the diffusive current at that time can be estimated as p Dqp∂yxqp(0) ∼ Dqpxqp(0)/ Dqpτ. For τ = tsat the diffusive current is therefore of the order of the trapping current γxqp(0); as quasiparticles spread further out, the diffusive current will decrease, indicating that indeed a steady-state is (asymptotically) reached. It is important to note that the total number of quasiparticles in the device keeps growing for the entire duration of injection, despite the saturation of xqp(0) at τ ∼ tsat. The evolution in the relaxation stage, t > 0, depends on the ratio tsat/tinj. A straightforward use of Eq. (3.30) yields for the quasiparticle density close to the trap, y → 0, in the long-time limit t tsat r s ! x0 tsat tsat xqp(0, t) ≈ √ − . (3.32) π t t + tinj

This asymptote is valid for any value of tsat/tinj. If tinj tsat, one may distinguish between −1/2 an intermediate asymptotic behavior, xqp(0, t) ∝ 1/t , valid at times tsat t tinj, and a −3/2 long-time asymptote, xqp(0, t) ∝ t , at t tinj. Only the latter behavior is present for short injection times tinj . tsat.

3.1.3.2 The effect of finite diffusion time We now turn to the case of a finite-length strip, so that the diffusion time across the whole device, 2 2 tL = 4L /(π Dqp) , (3.33) provides yet another scale for the relaxation dynamics of xqp. The comparison of the two time scales, tL and tsat, allows us to introduce the notion of a weak versus a strong trap. A weak trap corresponds to tsat tL. The diffusion through the device occurs much faster than the local saturation at the trap, and consequently, the quasiparticle distribution is almost homogeneous throughout the device. A strong trap, tsat tL, leads to a highly-inhomogeneous spatial distribution of the quasiparticle density. Recalling that γ = Γeffd, this distinction can also be expressed in terms of a comparison of the trap length d with the length scale 2 π Dqp π λtr l0 ≡ = , (3.34) 2 LΓeff 2 L 28 Chapter 3. Normal-Metal Quasiparticle Traps

with λtr of Eq. (3.21); a weak (strong) trap is characterized by d l0 (d l0). Note that if λtr L, l0 is much smaller than λtr, so the crossover between the two limits occurs while the trap length remains short, d λtr, and we can still use Eq. (3.22). For a weak trap, d l0, we may neglect the y-dependence of xqp(y, t) in Eq. (3.22), and integrating it over y we ﬁnd

−t −t/τw xqp(y, t) ≈ x0 1 − e inj/τw e , (3.35) where 1 d = Γeﬀ . (3.36) τw L

As long as xqp can be considered y-independent, the expression (3.36) for the density decay rate may be easily generalized: the ratio d/L in the right hand side should be replaced by Atr/Adev, where Atr is the total area of the trap and Adev is the area of the entire device. Importantly, the decay rate here depends merely on the ratio of the total areas, whereas details of the geometry of the trap and device are unimportant. In the opposite case of a strong trap, d l0, the approximation of a constant xqp(x, y) is no longer valid, and the decay rate will depend on the details of the trap geometry and placement. For simplicity, we concentrate again on the strip geometry. To obtain the eigenmodes, one may π replace the right hand side of Eqs. (3.27) by zero. Therefore, k is simply given by k = 2L p, where p is an odd integer (up to small corrections of order l0/d – cf. Eq. (3.39)). In contrast to the case of a weak trap, the relaxation is now limited by the diffusion time. From Eq. (3.30) we find the time-dependent quasiparticle density at the junction to be s 4 t t 2 t+tinj 2 sat X − t p − t p xqp (0, t) ≈ x0 e L − e L , (3.37) π tL p with x0 = j/γ, and the sum over the odd integer p. For short times, t tL, the time evolution is insensitive to the boundary condition at x = L, and indeed we recover the results given in Sec. 3.1.3.1. (Note that of course, being able to observe the transition from a t−1/2 to a t−3/2 power law decay is contingent upon tinj being much smaller than tL.) For times exceeding the diffusion time, t & tL, the time-evolution is dominated by the single exponential of the slowest mode, and we can write s 4 tsat −tinj/τw −t/τw xqp (0, t) ≈ x0 1 − e e (3.38) π tL where the decay time constant is now determined by the diffusion time (3.33), τw = tL [62]. Concentrating on the long-time evolution, we can more generally relate the time constant τw to the wave number of the slowest mode. Thus, we are able to investigate the full crossover in τw from weak to strong trap as a function of d/l0. Setting l → 0 in Eq. (3.27), we may re-write it as π l0 2 cot ke = ke , ke = kL . (3.39) 2 d π

The time constant can be expressed in terms of the smallest positive solution ke0 of Eq. (3.39) 2 as τw = tL/ke0. Therefore, the ratio tL/τw is a function of a single variable, d/l0. The full crossover function between the linear dependence at small d/l0 and saturation at d/l0 1 can Chapter 3. Normal-Metal Quasiparticle Traps 29

be found by solving Eq. (3.39) numerically. In Fig. 3.5, we show tL/τw as a function of d/l0, together with experimental data that we discuss in the next section. The introduction of scaled variables tL/τw and d/l0 allows us to compare the trapping for a number of devices and for a set of diﬀerent temperatures.

3.1.4 Experimental data

In this section we compare the model developed in the previous sections with experiments measuring the dynamics of injected quasiparticles in 3D transmon qubits [6]. The qubit, similar to the device sketched in Fig. 3.2, consists of a single Al/AlOx/Al Josephson junction shunted by a coplanar gap capacitor, with long (∼1 mm), narrow antenna leads which connect to a pair of small (80 × 80 µm2) pads, see Fig. 3.3. One or two chips containing qubits are mounted in a superconducting aluminum rectangular waveguide cavity. All measurements are performed in an Oxford cryogen-free dilution refrigerator, with magnetic field shielding, infrared shielding and filtering described in Ref. [63]. After fabrication of the qubits, normal-metal traps are patterned via optical lithography, which gives control of trap location and size to better than 1 µm. The heavily oxidized aluminum surface of the qubit is treated with an ion etch, and 100 nm of copper is deposited in a liftoff process thereafter. Through independent DC measurements, we find the Al-Cu interface resistance to be between 200 and 430 Ω · µm2. As shown in Fig. 3.3c, one edge of the trap is located a short, fixed distance (∼ 35 µm) away from the junction. The trap has a width of 8 µm, and it is placed symmetrically on the 12 µm wide lead. For this study, we focus on devices in which the trap length, d, along the lead is varied from 20 to 400 µm. The qubits’ T1 times measured at 13 mK vary (non-monotonically) between 10 and 22 µs for d between

Figure 3.3: a) Photograph of a 3D aluminium cavity loaded with a transmon qubit. b) Optical image of an example of the devices used for this study. c) Zoomed-in image of the Cu-trap deposited near the junction 30 Chapter 3. Normal-Metal Quasiparticle Traps

Figure 3.4: Qubit energy relaxation rate Γ after quasiparticle injection. The solid line is a ﬁt to the data by a single exponential with time constant τw, see Eq. (3.41). The inset shows the relaxation rate after subtracting a constant background in logarithmic scale, displaying good agreement with the predicted functional form.

20 and 80 µm, while the two devices with longer traps (d = 200 and 400 µm) have shorter relaxation times (5 and 7 µs, respectively). Comparisons with a control device without traps (T1 = 19 µs) and with earlier experiments [43] indicate that short traps do not negatively affect the qubit coherence, while longer traps might be somewhat detrimental. Here we focus on the effect of traps on quasiparticle dynamics and do not give further consideration to the possible trap-induced loss mechanism. The QP dynamics of these devices is studied using the contactless, in-situ method described in Ref. [43], where QPs are introduced into the qubit by applying a large microwave tone at the bare cavity resonance. This injection pulse creates a voltage across 5 the Josephson junction greater than 2∆, generating many (& 10 per µs) quasiparticles near the junction. The subsequent decay of xqp is probed by monitoring the recovery of the qubit relaxation time T1 measured as a function of time after the injection, in light of the simple relation Γ(t) = 1/T1(t) = Cxqp(0, t) + Γ0 , (3.40) where Γ0 is the steady-state relaxation rate of the qubit, which includes the effects of residual quasiparticle population and other relaxation mechanisms such as dielectric losses, and C is a known proportionality constant [31]- see discussions in section 2.4. In other words, we exploit the fact that the time-dependent part of the qubit decay rate Γ is directly proportional to the excess quasiparticle density at the junction, y = 0. Figure 3.4 shows a typical measurement of the qubit decay rate in a device with a small normal-metal trap. The decay time constant τw is estimated by fitting the data with a single exponential of the form

−t/τw Γ(t) = Ae + Γ0 . (3.41)

As discussed in Sec. 3.1.3.2, we are considering only the slowest decay mode of xqp, so we Chapter 3. Normal-Metal Quasiparticle Traps 31

fit the data to the above expression at long times t & tL [with tL of Eq. (3.33)], where we find good agreement between the data and the predicted single-exponential decay. Repeating the measurement for several trap lengths d, we find that the experimental decay rate 1/τw varies with the length of the trap in qualitative agreement with the rate calculated by solving Eq. (3.39), see Fig. 3.5. Indeed, for short traps we approximately find the linear dependence of 1/τw on the trap length predicted by Eq. (3.36), while for longer traps the rate saturates to the the diffusion limit, 1/τw ≈ 1/tL. To scale the experimental data so that they can be compared to the theoretical expectation, we use l0 and tL as fitting parameters, and allow them to be different for data taken at different fridge temperatures Tfr, thus assuming that both Dqp as well as Γeff depend on Tfr. The fitting parameters are l0 = 41.2 ± 17.1 µm and tL = 184 ± 29 µs for Tfr = 13 mK and l0 = 45.8 ± 16.7 µm and tL = 125 ± 20 µs for Tfr = 50 mK [64]. Note that the relative change in l0 is smaller than that in tL and that this is in qualitative agreement with theoretical expectations: since l0 is proportional to Dqp/Γeff, the expected increases of both Dqp and Γeff with effective temperature can partially compensate each other, while no such compensation is possible for tL ∝ 1/Dqp. As discussed after Eq. (3.36), in the linear regime we can take into account the actual geometry of the transmon by modifying that expression for the decay rate, which becomes 1/τw = ΓeffAtr/Adev. We use this formula to estimate Γeff using the 5 −1 short-trap data and find Γeff ≈ 2.42 × 10 s for Tfr = 13 mK (corresponding to the blue data 5 −1 points in Fig. 3.5) and Γeff ≈ 3.74×10 s for Tfr = 50 mK (red points). These numbers are close to the order-of-magnitude estimate for Γeff given at the end of Sec. 3.1.2, where we assumed that the backflow of quasiparticles must be taken into account and strongly suppresses the effective

0.8

0.6 tL ⌧ w 0.4

0.2

0 0 2 4 6 8 10 d/l0

Figure 3.5: Dimensionless density decay rate 1/τw normalized by the diffusion time tL, cf. Eq. (3.33), as a function of trap length d measured in units of l0, see Eq. (3.34) for the definition. The solid line is calculated by solving Eq. (3.39) numerically. The experimental data are taken at two different fridge temperatures: the blue symbol “x” is used for Tfr = 13 mK and the red symbol “+” for Tfr = 50 mK. Note the transition from a linear dependence to the saturated diffusive limit at d ∼ l0. 32 Chapter 3. Normal-Metal Quasiparticle Traps

1/2 trapping rate. In that Section we have also shown that Γeff ∼ Γr(T/∆) , indicating that Γeff should grow with temperature. While we observe an increase in the Γeff extracted from the data with increasing fridge temperature, this increase is smaller than the factor of 2 expected from theory. This discrepancy is not surprising, since it is known that at low temperatures the quasiparticles are not in thermal equilibrium at the fridge temperature [20]. Moreover, the injection pulse can cause additional heating in the qubit [65], further weakening the relationship between fridge temperature and quasiparticle effective temperature.

3.2 Optimization

3.2.1 Introduction

In the previous section, we discussed the cross-over from weak (d l0) to strong (d l0) trap. We point out that the diffusion-limited, strong-trap regime can be reached for traps with dimensions smaller than λtr only in the (quasi) one-dimensional geometry. Indeed, let us 2 2 consider a 2D superconducting film of total area Ldev and a trap of area d , with d Ldev. −1 2 2 In the weak regime the decay rate is τw ≈ Γeffd /Ldev. Comparing this to the diffusion rate 2 ∼ Dqp/Ldev, we find that the crossover from weak to strong trap occurs for the trap size d ∼ λtr. This means that effectively zero-dimensional traps (d λtr) may be strong in 1D, but they are always weak in 2D. Therefore, it can be advantageous to use quasi-1D devices with small traps, since in 2D devices the traps must be large to be effective, and large traps could potentially lead to unwanted ohmic losses within the normal metal or dissipation at the S-N contact. In this section we show that use of quasi-1D geometries facilitates trapping with small traps, and that their positions can be optimized. We consider three ways in which normal- metal traps may improve qubit performance. First, we note that events which generate a large number of quasiparticles render the qubit inoperable so long as the excess quasiparticles are not eliminated; here we find the parameters and placement of traps that enhance the relaxation rate of the excess density. Second, in addition to the the dynamics of the excess density, we study the quasiparticles steady-state density in the presence of a generic generation mechanism with a rate determined by experiments [43, 65]; we find that a trap in the vicinity of a junction can reduce the quasiparticle density at that junction, potentially leading to a longer T1 relaxation time for the qubit. Third, we consider the effect of fluctuations in the generation rate: they lead to the fluctuation in the quasiparticle density near the junction and, associated with it, to variations of a qubit T1 [65, 45]; placing a trap up to a certain distance from the junction can reduce the density fluctuations and hence make the qubit more stable. The section is organized as follows: in Sec. 3.2.2 we consider a realistic qubit geometry, namely the coplanar gap capacitor transmon of Refs. [43, 47]; we give analytical arguments for trap configurations leading to faster relaxation rates of the excess density – see Eq. (3.45) for the single trap case and Eqs. (3.53) and (3.54) for the multi-trap one – and complement those with numerical calculations whose outcomes are summarized in Figs. 3.7 to 3.9. In Sec. 3.2.3 we turn our attention to the steady-state density at the junction and its fluctuations; both can be suppressed by appropriately placed traps, but while the steady-state density always increases monotonically with trap-junction distance [Eq. (3.60)], we find for a strong trap a non-monotonic behavior of fluctuations [Eq. (3.74)] and hence an optimal trap position. We summarize our findings in Sec. 3.3. A number of Appendices complement the main text: in AppendixC Chapter 3. Normal-Metal Quasiparticle Traps 33 we compare trapping by normal-metal traps with that due to vortices; AppendixD present some mathematical details for the case of a single, finite-size trap, and AppendixE addresses the question of the experimental observability of the slowest decay rate of the excess density; Appendices F.1 and F.2 contain details about the mapping of a realistic qubit design into a 1D wire. In AppendixG we present details about how to find the slowest decay rate having single or multiple traps as well as details of finding the steady-state density. Finally, AppendixH considers traps in the Xmon qubit geometry.

3.2.2 Enhancing the decay rate of the density

In this part we analyze how to optimally place traps of a given size, so that the slowest mode of the quasiparticle density decays as fast as possible. As a concrete example, we take the coplanar gap capacitor transmon and study traps placed in the long wire connecting the gap capacitor to the antenna pads, both via analytical and numerical approaches. For actual estimates, we use 5 2 the parameters measured in Refs. [47, 43], namely Γeff = 2.42 × 10 Hz and Dqp = 18 cm /s, which using Eq. (3.21) give λtr ' 86.2 µm. Since we are interested in the decay of the excess density, we can set g = 0; the effect of a trap on the steady-state density due to finite g is the focus of Sec. 3.2.3. In AppendixC, we compare trapping by vortices [43] to normal-metal traps. As we discuss at the end of Sec. 3.2.2.1, considering only the slowest mode for the optimization may not be sufficient when addressing the extreme case of a single, very large trap. However, as we have already pointed out, in a quasi-1D geometry short traps can be strong – that is, effective at suppressing the excess quasiparticle density. In this case the slowest mode in general still suffices to characterize the long-time quasiparticle decay. The short-trap regime is in particular important for the multiple-trap configurations considered in Sec. 3.2.2.2: these configurations combine a fast decay of the quasiparticle density with low electromagnetic losses, and are thus preferable.

3.2.2.1 Optimization for a single trap

Let us consider a single trap placed in the antenna wire of length L, see Fig. 3.6 (the device being symmetric, there are two traps in total). We start for simplicity with a short trap of length d λtr and neglect the gap capacitor and antenna pads; we then show how to map the full device to this simpler configuration and compare our estimates with numerical results. For a short trap in a wire, the diffusion equation (3.19) can be written in the form (cf. AppendixD) 2 x˙ qp = Dqp∇~ xqp − γeffδ (y − L1) xqp . (3.42)

The trap is at position y = L1 and γeff = dΓeff. Consider for simplicity the case γeff → ∞, such that quasiparticles are trapped immediately once they reach the trap. As a consequence, xqp(L1) = 0, and the density on the left and right sides of the trap decays with the rates −1 2 2 −1 2 2 τw = π Dqp/4L1 and τw = π Dqp/4(L − L1) , respectively. If the trap is at the center of the wire, L1 = L/2, the density decays equally fast on both sides, and the decay rate of the slowest mode is four times faster as compared to placing the trap at the beginning or end of the wire. In other words, the central position is the optimal one for the trap to evacuate quasiparticles as quickly as possible. 34 Chapter 3. Normal-Metal Quasiparticle Traps

At finite γeff, the left/right modes are coupled, but the coupling is small provided that the trap is strong, d l0. The coupling lifts the mode degeneracy at L1 = L/2, but does not change the above conclusion on the optimal position. We note, however, that if quasiparticles are injected and detected locally (as, e.g., in [47]) one may not necessarily observe the global slowest decay rate for strong traps; see AppendixE for more details. In the simple example above we have shown that the optimal trap position (for which the decay rate of the slowest mode is the fastest) is such that the diffusion times in both sides of the trap are equal. We can extend this finding to a more realistic qubit geometry [47] which includes the coplanar gap capacitor close to the Josephson junction and the antenna pad at the far end of the wire, see Fig. 3.6. The capacitor “wings” of length Lc and the square pad with side Lpad can be accounted for by adding some effective lengths to the antenna wire. The effective eff eff lengths Lc (k) and Lpad (k) (for capacitor and pad, respectively) in general depend on the wave vector k, see Appendices F.1 and F.2. If these effective lengths are much smaller than the wire eff 2 length L, we find that for the slow modes the dependence on k drops out: Lpad ≈ Lpad/W eff Wc and Lc ≈ 2 W Lc, with W and Wc the widths of of the wire and capacitor wings, respectively. These effective lengths may simply be added to the lengths to the left and right of the trap to find the decay rates: 2 1 π Dqp = 2 (3.43) τw 2 4 L − L1 + Lpad/W − d/2

Figure 3.6: Top: sketch of the transmon qubit studied here, based on the experiments of Ref. [47] reported in section 3.1. Light blue/light gray: superconducting material; red/dark gray: regions of the superconductor covered by normal metal; cross: position of the Josephson junction. Except for the junction region, the sketch is to scale. Bottom: right half of the device, with the relevant lengths defined: a trap of length d is placed on the antenna wire (length L, width W ) at distance L1 from the gap capacitor (dimensions Lc and Wc.) The antenna pad is a square of side Lpad. Chapter 3. Normal-Metal Quasiparticle Traps 35 for the right mode and 1 π2D = qp (3.44) 2 τw Wc 4 L1 + 2 W Lc − d/2 for the left mode; we have accounted for the finite size of the trap by subtracting the d/2 terms in the denominators, and L1 denotes the trap center. Thus, the optimal trap position is (in the strong trap limit): L Leff − Leff L = + pad c . (3.45) opt 2 2 The optimal position is closer to the pad (gap capacitor) if the effective length of the pad (capacitor) is larger. We can check the validity of the above considerations for strong traps and extend our consideration to weaker (i.e., smaller) traps by more accurately modelling the diffusion- see Appendix G.1.1 for details of calculations. The density in the parts not covered by the trap is written in the form −t/τw xqp(t, y) = e [α cos ky + β sin ky] (3.46) 2 with 1/τw = Dqpk (except for the pad, where the density is assumed uniform), while under the trap we have −t/τw xqp(t, y) = e [α cosh y/λ + β sinh y/λ] . (3.47)

Imposing continuity of xqp and current conservation we ﬁnd:

2 2 2 z + b = (L/λtr) , (3.48) z d [az + tan (zξ )] 1 − h (z, ξ ) tanh b b R L L d −[1 − az tan (zξ )] tanh b − h (z, ξ ) = 0, (3.49) R L L

2 with z = kL, b = L/λ, a = Lpad/(LW ), and l Wc Lc z tan (zξL) + tan z L + 2 W tan z L h (z, ξL) = h i. (3.50) b l Wc Lc 1 − tan (zξL) tan z L + 2 W tan z L

We also deﬁne the (normalized) length of the wire to the left (right) of the trap by

ξL = (L1 − d/2) /L , (3.51)

ξR = (L − L1 − d/2) /L . (3.52)

2 2 Solving Eqs. (3.48) and (3.49) for z and b, one can ﬁnd the density decay rate 1/τw = Dqpz /L . For a long qubit with L λtr, the slow modes have b ≈ L/λtr 1 and z ∼ 1. We note that 2 2 2 the assumption of uniform density in the pad requires 1/τw = Dqpz /L Dqp/Lpad; since for 2 2 −2 experimentally relevant parameters we have Lpad/L ∼ 10 , the assumption is valid for slow modes even when z & 1. In Fig. 3.7 we show a density plot of the decay rate 1/τw as a function of the distance L1 between gap capacitor and trap center and of the normalized trap size d/l0, calculated using typical experimental parameters as detailed in the caption. For a strong trap, d l0, as discussed in Sec. 3.1.3.2 we ﬁnd that the decay rate is sensitive to the trap position. 36 Chapter 3. Normal-Metal Quasiparticle Traps

Figure 3.7: Trapping rate 1/τw as a function of the trap position L1 (in units of L) and normalized trap size d/l0 – see Fig. 3.6 for the device geometry; the device parameters are (all 2 2 lengths in µm): L = 1000, l = 60, W = 12, Lc = 200, Wc = 20, Spad = Lp = 80 . We used 2 λtr = 86.2 µm for the trapping length, so l0 = πλtr/2L ' 11.7 µm. The white areas are regions in which the trap center cannot be pushed closer to or further away from the gap capacitor due to the ﬁnite trap size.

The optimum position is shifted with respect to the middle of the wire (dash-dotted line in Fig. 3.7), in agreement with the prediction of Eq. (3.45). Indeed, for the parameters in Fig. 3.7, eff 2 eff Wc we find Lpad = Lpad/W ≈ 533 µm and Lc = 2 W Lc ≈ 667 µm, and the optimal position is closer to the gap capacitor. When the trap size is d . l0, the trap position has only a minor effect on the trapping rate. This is more clearly seen in Fig. 3.8 (bottom solid curve). For longer traps, we compare the decay from the numerical solution to Eqs. (3.48) and (3.49) with Eqs. (3.43) and (3.44). When the trap is strong but still short compared to the wire (middle solid), Eqs. (3.43) and (3.44) (dashed) provide a good approximation to the numerical results (in fact, one can expect the numerically calculated rate to be slower than the analytical prediction, since the numerics account for the finite trapping length which allows for finite density under the trap as well as for the bridge of length l joining the junction to the gap capacitor). For very long traps (upper solid line) the approximation that the effective lengths are small compared to the (uncovered part of the) wire fails, and the calculated rate is faster; this is qualitatively in agreement with the fact that as the mode wavelength increases, the effective lengths decrease, see Eqs. (F.4) and (F.7) (for the gap capacitor, this is true so long as 2Wc/W > 1). Our focus so far has been in speeding up the decay rate of the slowest mode, without taking into consideration the amplitude of the mode at the junction. This approach is correct for Chapter 3. Normal-Metal Quasiparticle Traps 37

5 )

-1 4 (ms

w 3 1/τ 2

0 0.0 0.2 0.4 0.6 0.8 1.0

L1/L

Figure 3.8: Solid lines: trapping rate 1/τw as a function of the trap position L1 measured in units of L for (top to bottom) d/l0 = 40, 10, 1; other parameters are speciﬁed in the caption to Fig. 3.7 and the device geometry is shown in Fig. 3.6. The dashed lines are the estimates provided by Eqs. (3.43) and (3.44) for d/l0 = 40, 10.

weak traps, d . l0, since the amplitude of the mode is approximately the same on both sides of the trap. For strong but small traps, l0 d . λtr, the amplitude on one side of the trap is algebraically suppressed by a factor of order d/l0 compared to the amplitude on the other side (see AppendixE), while for long traps, d λtr the suppression is exponential in d/λtr (see AppendixD). In the latter case, it would clearly be advantageous to place the trap close to the junction: the mode with large amplitude between junction and trap would decay quickly, while the slow mode with large amplitude on the other side of the trap would decay slowly but it would be exponentially suppressed at the junction. However, as we already pointed out, long traps could be too lossy – this motivates us to further study how to obtain the fastest possible decay using only small traps.

3.2.2.2 Multiple traps

We now generalize the considerations of the previous section to the case of multiple traps (in each half of the qubit). For a weak trap, the effective trapping rate is proportional to the trap size [Eq. (3.36)] but independent of position; therefore, no change in the density decay rate can be expected by dividing a weak trap into smaller ones, since the total size is unchanged. The strong-trap regime is qualitatively different in this regard. Let us consider Ntr strong traps in a wire of length L; the traps separate the wire into Ntr + 1 compartments. The optimal trap placement is obtained when the diffusion time is the same for each compartment, meaning that the traps have to be placed at positions Ln = (2n − 1)L/2Ntr with n = 1,...,Ntr, and the 38 Chapter 3. Normal-Metal Quasiparticle Traps resulting decay rate is π2 τ −1(N ) = N 2 D . (3.53) w tr tr qp L2 The rate increases quadratically with the number of traps, so splitting a single strong trap into smaller pieces can highly increase the decay rate. However, when keeping the total area of the traps constant, there is a limitation to this improvement. Indeed, the length of each trap decreases as d/Ntr and the “device length” of each compartment is of order L/2Ntr; using these quantities in Eq. (3.34) we find that the traps cross over to the weak regime for s opt d Ntr ∼ ; (3.54) 2l0 here l0 is defined by the right hand side of Eq. (3.34) with Ldev = L. Increasing the trap number opt beyond Ntr does not further improve the decay rate, which is thus limited by Eq. (3.36). In opt other words, to obtain that maximum decay rate for given total length d, at least Ntr traps should be placed evenly spaced over the device. Such a configuration could also reduce the trap-induced losses, since they depend on the trap position and size [75]. Let us now show in a concrete example that multiple traps can indeed increase the decay rate as predicted by Eq. (3.53). We consider again the transmon device depicted in Fig. 3.6, but we now assume that two traps are placed on the central wire, with distances L1 and L2 between the gap capacitor and the traps centers. Accounting for the second trap, we generalize Eq. (3.49) to

z d z d 1 − h(z, ξ ) tanh b 1 tan (zχ) − tanh b 2 b L L b L z d + g(z) 1 − tan (zχ) tanh b 2 b L (3.55) d d − tanh b 1 − h(z, ξ ) tanh b 2 tan (zχ) L L L z z d + − g(z) tan (zχ) + tanh b 2 = 0 b b L where, taking L1 < L2,

ξL = (L1 − d1/2)/L (3.56)

ξR = (L − L2 − d2/2)/L (3.57)

χ = (L2 − d2/2 − L1 − d1/2)/L , (3.58) the function h(z, ξL) is deﬁned in Eq. (3.50), and

L1+χ+d2/2 z az + tan z(1 − L ) g(z, ξR) = . (3.59) b L1+χ+d2/2 1 − az tan z(1 − L )

We consider for simplicity the case of equal traps, d1 = d2 ≡ d/2 with d = 20l0 ' 233 µm the total length of the normal metal. We show in Fig. 3.9 the decay rate as function of L1 and L2 for the same parameters as in Fig. 3.7. We ﬁnd that the decay rate is highest when placing the traps far away from each other, one trap touching the gap capacitor and the other begin close Chapter 3. Normal-Metal Quasiparticle Traps 39

(a)

(b)

Figure 3.9: (a) Device with two traps (dark red) in each half of the qubit; distances L1 and L2 are measured from the gap capacitor to the center of each trap, cf. Fig. 3.6. (b) Trapping rate 1/τw as function L1 and L2; here the two traps are identical, d1 = d2 = 10l0, which makes the plot symmetric under the exchange L1 ↔ L2. The parameters used are specified in the caption to Fig. 3.7. Similar to that figure, the white areas correspond to forbidden regions due to the finite traps’ sizes . A comparison with Fig. 3.7 reveals that splitting a trap with length d = 20l0 into two identical ones can boost the trapping rate up to a factor larger than 3.

to the pad. We find that the decay rate of the slowest mode is highest when placing the traps far away from each other, one trap touching the gap capacitor and the other begin close to the pad. This is in qualitative agreement with expectations: consider again the gap capacitor and the pad as extra lengths added to the left and right of the central wire, respectively; this leads eff eff to a wire of effective total length Ltot = Lc + L + Lpad ' 2200 µm. In such a wire the optimal positions would be L1 = Ltot/4 ' 550 µm and L2 = 3Ltot/4 ' 1650 µm. The value of L1 would indicate an optimal position inside the gap capacitor, but since we allow for the traps to move in 40 Chapter 3. Normal-Metal Quasiparticle Traps

the central wire only, this optimal placement is not possible. The value of L2 corresponds to a position slightly away from the pad, in agreement with the results in Fig. 3.9. Finally, going from the optimally-placed single trap to the optimal two-trap configuration, the decay rate increases by a factor of ∼ 3.4. This factor does not reach the theoretical maximum of 4 predicted by Eq. (3.53); the discrepancy can be attributed both to the non-optimal placement of the first trap mentioned above as well as to finite-size effects, as in the single trap case. However, the calculated improvement confirms that the decay rate can be significantly increased by optimizing the trap number and position. It is instructive to compare our results for the two-trap case with the fast decay of the mode to the left of the single trap; using Eq. (3.44), we estimate the decay rate of this mode −1 to be 1/τw ' 14.7 ms , slightly slower than the maximum rate shown in Fig. 3.9. In both cases, we do not allow the trap to enter the gap capacitor; this constraint could be important in limiting trap-related losses, as the gap capacitor is the region with the highest electric field. Using two traps we place only half the total normal material near the high-field region, while obtaining a slightly faster decay than with a single, large trap. For the considered example, further increase in the decay rate could be obtained by further splitting the traps. Indeed, a more accurate estimate for the length l0 can be obtained by using Ldev = Ltot − d ' 1967 µm in opt p Eq. (3.34), giving l0 ≈ 5.9 µm. Then the “optimal” trap number would be Ntr ' d/2l0 ∼ 4, requiring one trap to be placed on the pad, two on the antenna wire, and one on the gap capacitor (this placement is calculated using the “effective wire” length of the gap capacitor, so that in practice one should symmetrically place one trap on each of the two “wings” of gap capacitor). As mentioned above, placement on the gap capacitor could be detrimental, but with the optimal trap number only one quarter of the normal metal would be in the gap capacitor and the resulting losses would therefore be smaller than those due to a single large trap on the gap capacitor. Therefore, it is potentially beneficial to have multiple smaller traps in comparison with a single large trap. Such considerations are also dependent on the device design, and in AppendixH we briefly consider a different geometry for the qubit, the Xmon of Ref. [17]; the central X-shaped part of the device is small, and unfortunately this implies that no large gain in the decay rate can be obtained using multiple traps, so alternative approaches are desirable in this case. In the next section we turn our attention to the effect of traps on the steady-state density.

3.2.3 Suppression of steady-state density and its ﬂuctuations

In the preceding subsection we have dealt with the question of how fast quasiparticles reach their steady state if there is a deviation from said steady-state density. In this part we point out that traps also affect the shape of the steady-state density. In particular, our aims are to minimize the steady-state density at the junction, which directly affects the T1 time of qubits, as well as to stabilize the density value against fluctuations in their generation rate which lead to temporal variations in the qubit lifetime. In our model, the steady-state density is nonzero due to a finite generation rate g in Eq. (3.19). As argued in Sec. 3.1.3.2, in the presence of a s weak trap the quasiparticle density is uniform, and in the steady-state takes the value xqp = gτw with τw of Eq. (3.36). As we now show, going beyond the weak limit the geometry affects the spatial profile of the density. For a concrete example, we consider the same geometry as in Sec. 3.2.2.1 – that is, a single Chapter 3. Normal-Metal Quasiparticle Traps 41 trap on the wire connecting gap capacitor and pad, see Fig. 3.6. The solution for the profile of the steady-state density in each 1D segment outside the trap is given by parabolas of the s 2 s general form xqp = −y g/2Dqp + αy + β, while under the trap we have xqp =α ˜ cosh(y/λtr) + ˜ ˜ β sinh(y/λtr) + g/Γeff. The parameters α, β in each segment, as well as α˜, β are found by imposing appropriate boundary conditions (i.e., continuity and current conservation). We finally J arrive at the following expression for the steady-state density xqp at the junction: J g 1 AR AL xqp = 1 + + cosh(d/λtr) Γeff sinh(d/λtr) W λtr W λtr " # (3.60) g (L + l − d/2)2 A (L − d/2) + 1 + c 1 , Dqp 2 W

2 where AR = W [L − L1 − d/2] + Lpad and AL = W [L1 + l − d/2] + Ac are the uncovered areas to the right and left of the trap, respectively, and Ac = 2WcLc is the gap capacitor area. In the small trap limit, d λtr, we can rewrite Eq. (3.60) in the form

J xqp ' g (τw + tD) (3.61)

2 with τw defined in Eq. (3.36), while tD = [(L1 + l − d/2) /2 + Ac(L1 − d/2)/W ]/Dqp represents the diffusion time between junction and trap (with the second term in square brackets taking into account the presence of the gap capacitor). Similar to the discussion in Sec. 3.1.3.2, we can distinguish between an effectively weak (τw tD) and strong trap (τw tD), with the trap becoming strong as its length d increases above the position-dependent length scale 2 p l1 ∼ λtr/ DqptD. Note that l1 decreases with the distance L1 between gap capacitor and trap and is always larger than l0 of Eq. (3.34); therefore a trap that is weak in the sense of d being J smaller than l0 is weak at any position L1, and the value of xqp is only weakly dependent on the trap placement. On the other hand, a strong trap with d > l0 effectively becomes weak, as L1 J decreases, when τw = tD. At positions L1 smaller than that given by this condition, xqp again becomes weakly dependent on trap placement. In other words, the condition determines the J maximal distance at which the largest (up to numerical factor) suppression of xqp is achieved for a given trap size d. For a long trap d λtr, we can still use Eq. (3.61) after the identification 1 AL τw → 1 + . (3.62) Γeff W λtr With this substitution and for typical experimental parameters, we find again that the first term in Eq. (3.61) dominates when the trap is close to the junction (despite being smaller than the corresponding term for a short junction), while the second one takes over as L1 increases. More J generally, it should be noted that in any regime xqp is a monotonically increasing function of L1: as one could expect, the closer the trap is to the junction, the more it suppresses the quasiparticle density near the latter. This behavior is evident in Fig. 3.10: the plot clearly shows that the density is suppressed by placing the trap near to the gap capacitor, and that long traps (d & λtr) −8 are more effective. Values as low as xqp ∼ 10 are predicted; for comparison, we note that −6 in devices with the geometry considered here but without traps, we estimate xqp ∼ 10 [43]. On the other hand, for transmons with larger pads (so that there are always vortices that act −7 as traps) we find xqp ∼ 10 [6, 43]. Further suppression of the density could be achieved by 42 Chapter 3. Normal-Metal Quasiparticle Traps

Figure 3.10: Quasiparticle density at the junction as a function of trap location L1 (in units −4 of L) and its normalized size d/λtr, calculated using g = 10 Hz [43, 65]; the normalization of the size d diﬀers from that of Fig. 3.7, but the parameters used for the device are the same speciﬁed there.

placing traps in the gap capacitor, since this would effectively reduce the uncovered area AL [cf. Eq. (3.62)] between trap and junction. Based on the above consideration, we do not expect that adding a second trap far from J the junction significantly affects the steady-state density xqp (we have confirmed this by direct calculation). Therefore, the results of the last two sections suggest that having two traps, one close to the junction and the other close to the pad can both greatly reduce the steady-state density at the junction and enhance the decay rate of the excess density. Next, we show that traps can also contribute to the temporal stability of the qubit.

3.2.3.1 Fluctuations in the generation rate

As discussed previously, the density at the junction and hence the qubit relaxation rate are proportional to the generation rate g. Therefore, temporal variations in g can cause changes in the measured T1 over time. Here we explore how traps can suppress these changes. For this purpose, we include Gaussian ﬂuctuations of g in Eq. (3.19) by replacing g → g + δgb (y, t), with

hδgb (y, t)i = 0 , (3.63) |t−t0| 0 0 0 1 − δgb (y, t) δgb y , t = γgδ y − y e τm . (3.64) 2τm

The parameter γg characterizes the fluctuation amplitude and has the same units as γeff of Eq. (3.42), while h...i averages over all realizations of δgb. We assume that fluctuations are Chapter 3. Normal-Metal Quasiparticle Traps 43

spatially uncorrelated, but allow for temporal correlations with a ﬁnite memory time τm – we will return to this point in what follows. Note that under these assumptions the average density hxqp(y)i is in general a function of the spatial coordinate due to the presence of traps, but not of time. Assuming the junction to be at position y = 0, to provide a measure for the ﬂuctuations in the density at that point we consider the quantity

2 0 0 2 ∆xqp t, t ≡ xqp (0, t) xqp 0, t − hxqp (0)i . (3.65)

This quantity can be expressed in terms of the eigenvalues µk < 0 and eigenfunctions nk(y) of Eq. (3.19) (cf. Ref. [47] and AppendixD). Indeed, the quasiparticle density with the fluctuation term is 1 x (y, t) = − X n (y) g qp µ k k k k Z t (3.66) X µk(t−t1) + dt1e nk (y) δgbk (t1) , k −∞ with Z L 0 dy 0 0 δgbk (t1) = nk y δgb y , t1 , (3.67) 0 L and the similar definition for gk. Substituting this expression into the definition Eq. (3.65) and averaging over the fluctuations, we find

t t0 γg 1 X Z Z ∆x2 t, t0 = dt dt eµk(t−t1) qp 2L τ 1 2 m k −∞ −∞ (3.68) 0 |t1−t2| µk(t −t2) − τ 2 ×e e m nk (0) .

The time integrals are conveniently computed by ﬁrst shifting the times t1 and t2 by t and t0, respectively, and then changing variables in the two-dimensional integral into a mean time t1 + t2 and time diﬀerence t1 − t2. After integration, we obtain

|t−t0| − 1 µ |t−t0| γ τme τm + e k ∆x2 t − t0 = g X µk n2 (0) , (3.69) qp 2L τ 2 µ2 − 1 k k m k which depends only on time difference. Equation (3.69) shows that even in the absence of time correlations for the fluctuations in the generation rate, τm = 0, the fluctuation in the density are correlated due to diffusion. In this case, the longest decay time is that of the slowest mode, 1/µ0, and is typically of the order of milliseconds [47]. This time is shorter than the time it takes to measure a qubit relaxation curve and hence estimate the quasiparticle density. Therefore, only the regime in which τm is much longer than 1/µ0 could have observable consequences. Moreover, there is experimental evidence for slow fluctuations in the number of quasiparticles in qubits, obtained by monitoring the quantum jumps between states of a fluxonium, Ref. [65], and by repeated measurements, over several hours, of the relaxation time in a capacitively shunted flux qubit, Ref. [45]. Therefore, 44 Chapter 3. Normal-Metal Quasiparticle Traps in the reminder of this Section we focus only on the regime of long memory – that is, slow fluctuation in the generation rate. In the limit τm 1/µ0, Eq. (3.69) simplifies to

|t−t0| 2 0 γg − X 1 2 ∆x t − t ' e τm n (0) . (3.70) qp 2Lτ µ2 k m k k We now want to establish that a trap can indeed reduce the fluctuations. To this end, we consider the simple case of the junction in a quasi-1D wire extending for length L from the junction and with a trap at distance L1 from the junction. Initially, we take the trap to be small (length d λtr), and we distinguish between weak and strong trap, see Eq. (3.34). For a weak trap, d l0, the slow modes are only weakly dependent on the spatial coordinate. Moreover, for the slowest mode the decay rate is [cf. Eq. (3.36)] d µ ≈ −Γ , (3.71) 0 eff L 2 while the higher modes are much faster, since µn>0 . −Dqp/L , and thus |µn>0| |µ0|. Using n0(0) ≈ 1 and neglecting the small contributions from the higher modes, from Eq. (3.70) we find |t−t0| 2 γg L 2 0 − τ ∆xqp t − t ≈ e m . (3.72) 2Lτm Γeffd This expression shows that a stronger/longer trap more effectively suppresses fluctuation, as could be expected. In the case of a strong trap (d l0), the eigenmodes can be split in two sets: there are left and right modes, which are strongly suppressed to the right and to the left of the trap, respectively (here we assume that the trap position is sufficiently far from the central position, see AppendixE). The left modes, with large amplitude between junction and trap, give small contributions to the density fluctuations when the trap is close to the junction and their contributions grow with junction-trap distance. The right modes, while being suppressed to the left of the trap, have opposite behavior with distance, so they can dominate when the trap is sufficiently close to the junction. Then we generically expect a non-monotonic dependence of the density fluctuations on trap-junction distance from the competition between modes to the left and right of the trap.

Indeed, let us consider the decay rates for left and right modes, which we denote with µn,L1 and µn,L−L1 , respectively, where we deﬁne π 2 µ ' −D (2n + 1)2 , n = 0, 1,... (3.73) n,` qp 2` Keeping in Eq. (3.70) only the slowest mode for each set, since the higher modes with n > 0 gives a smaller contribution to the sum, we ﬁnd

|t−t0| γg − 1 2 0 τm ∆xqp t − t ≈ e 2 Lτm µ0,L " # (3.74) L 3 l 2 L − L × 1 + 0 1 , L d L where we used n2 (0) ' 2L/L and n2 (0) ' 2(l /d)2[L/(L − L )]3 (here we also assume k,L1 1 k,L−L1 0 1 d L1,L − L1). The first term in square brackets originates from the lowest mode confined Chapter 3. Normal-Metal Quasiparticle Traps 45 between junction and trap, while the second term, due to the lowest mode located on the 2 other side of the trap, is suppressed by the small factor (l0/d) . As a function of the trap 2 position L1, in√ agreement with the above considerations we find that ∆xqp has a minimum 2 at L1 = Ll0/d 3, where the terms in square brackets take the approximate value (l0/d) and Eq. (3.74) takes the same form of Eq. (3.72). In fact, those terms rise significantly above this 2/3 value only for L1 > Lf ≡ L(l0/d) , indicating that for a strong trap a large suppression of fluctuations can be achieved if the trap is not placed far beyond Lf . We note that this condition is more stringent than the one discussed after Eq. (3.61), τw = tD, which for the simple wire p considered here gives a maximum distance ∼ 2L l0/πd; in other words, maximum suppression of fluctuations ensures maximum suppression of the steady-state density. The above considerations for a strong but short trap can be generalized to longer traps (d & λtr, with d L) by substituting l0/d → λtr/L sinh(d/λtr), see AppendixD. In both regimes (strong but short, and long trap), increasing the trap length suppresses the fluctuations at the junction, but shrinks the region over which maximum suppression can be achieved, since Lf becomes smaller. This region is however always small compared to the wire length, Lf L. Together with the monotonic dependence of the average quasiparticle density on distance obtained in the first part of this section, our results show that placing a trap close to the junction is effective in suppressing both the average density and its fluctuations, potentially making the qubit longer lived and more stable.

3.3 Summary and Conclusions

In section 3.1 we developed a basic model enabling us to predict the effect of a normal-metal trap on the dynamics of the nonequilibrium quasiparticles population in a superconducting qubit. The model accounts for the tunneling between the superconductor and the trap, as well as for the electron energy relaxation in the trap, see Eq. (3.18). The surprising finding is that the effective trapping rate Γeff is sensitive to the energy of the quasiparticles and is constrained by their backflow from the normal-metal trap on time scales shorter than the electron energy relaxation rate. Furthermore, we find the dependence of the time needed to evacuate the injected quasiparticles on the trap size. The evacuation time saturates at the lowest, diffusion-limited value upon extending the trap above a certain characteristic length l0; the dependence of l0 on the parameters of the trap and qubit is given in Eq. (3.34). The experimental data (obtained by our collaborators in Yale University) reported in Sec. 3.1.4 validate the theoretical model. The relaxation rate 1/T1 of a transmon qubit is proportional to the quasiparticle density in the vicinity of the Josephson junction, making it possible to measure the dynamics of the quasiparticle population. We find that the population decay rate increases with the length of the normal-metal traps, in agreement with the predicted cross-over from weak to strong trapping, see Fig. 3.5. For small traps we can estimate the effective trapping rate Γeff: both its order-of-magnitude and its increase with temperature indicate indeed a limitation due to the backflow of quasiparticles. Utilizing traps is a viable strategy of mitigating the detrimental effect of quasiparticles on the qubits T1 time. Further improvement of normal-metal traps may benefit from finding ways to shorten the electron energy relaxation time in them. Based on the experiments of Refs. [58, 59], using a different pure metal (e.g., silver or gold) for the trap is unlikely to result in substantially shorter relaxation time; metals hosting magnetic impurities might be helpful in this regard, but 46 Chapter 3. Normal-Metal Quasiparticle Traps such impurities could harm the qubit by opening other relaxation channels. In section 3.2 we studied the effects of size and position of normal-metal quasiparticle traps in superconducting qubits with large aspect ratio, so that quasiparticle diffusion can be considered one-dimensional. We focus on such a design because, as we argued at the begening of that section, in a two-dimensional setting traps must be large compared to the trapping length λtr of Eq. (3.21) to be strong, while in quasi-1D it is sufficient for the trap length d to be longer than the characteristic scale l0 [Eq. (3.34)] which accounts for diffusion, trapping rate, and device size – this characteristic scale is generally shorter than λtr for long devices (Ldev > λtr). A trap can influence the qubit in three ways: first, it suppresses the steady-state quasiparticle density at the junction; then the qubit’s T1 time can be increased, since this time is inversely proportional to the density. Second, a trap can speed up the decay of the excess quasiparticles and, third, it can decrease fluctuations around the steady-state density; these effects can render the qubit more stable in time – in fact, there is experimental evidence (Refs. [45] and [65]) that fluctuations in the number of quasiparticles are responsible for at least part of the temporal variations in T1. Not surprisingly, a long trap (d & λtr) placed close to the junction is effective in all three aspects: fast decay of excess quasiparticles, suppression of the steady-state quasiparticle density (see Fig. 3.10), and suppression of density fluctuations at the junction. However, large traps could be a source of unwanted dissipation; therefore, we analyze in more detail the effects of shorter traps. If a trap is weak, d . l0, its position has little influence on the ability to suppress the quasiparticle density and its fluctuations, as well as on the decay rate of excess quasiparticles. Interestingly, for a strong but short trap, l0 . d . λtr, we find the position of the trap can be optimized in several ways. First, there is an optimal position that makes the decay of excess quasiparticles as fast as possible, see Figs. 3.7 and 3.8 in Sec. 3.2.2.1; however, a better choice is in general to divide a strong trap into smaller traps of length ∼ l0 and distribute those around the device, see Sec. 3.2.2.2. For suppression of density fluctuations, we find that there is an optimal trap position, see Sec. 3.2.3.1; more importantly, we find that there is a maximum distance Lf from the junction up to which the suppression of fluctuations is effective. Moreover, the distance up to which, for a given trap size, large suppression of the steady-state density is achieved (Sec. 3.2.3) is longer than Lf , so that suppressing fluctuations also suppresses the steady-state density. Therefore, by correctly placing multiple traps in the device in such a way that one is sufficiently close to the junction, all three beneficial effects of traps can be optimized. The optimization of trap size, number, and placement is the only readily accessible way to improve the trap efficacy, since the effective trapping rate is limited by the energy relaxation rate in the normal metal [47], a material parameter that cannot be easily modified. We stress here that these considerations are valid for normal islands in tunnel contact with the superconductor – traps formed by gap engineering (e.g., by placing a lower-gap superconductor in good contact with the qubit) could behave differently and deserve further consideration. Chapter 4 Quasiclassical Theory of Superconductivity

In the preceding chapter our analysis was based on the hard-gap assumption. This is justified provided a high resistance at the trap-qubit contact interface which results in a low transmission coefficient. From now on, we aim to relax this assumption and study how this affects the qubit relaxation. In this chapter, we follow notation of references [79] and [80] and review how Green’s functions are used in the context of superconductivity. We explain quasiclassical theory of superconductivity that simplifies the equations governing the superconducting Green’s functions and derive the Eilenberger equation. We then show steps in order to further simply the formalism in the limit of dirty superconductors and derive the Usadel equation for a non- uniform normal-superconducting hybrid. This equation forms our starting point in chapter5 where we study proximity-induced modifications in the superconducting properties that come along as side effects of normal-metal quasiparticle traps for superconducting qubits.

4.1 Gor’kov Equations

Let us start by introducing time-ordered single-particle and anomalous Green’s functions that are deﬁned respectively as,

† GT (1, 2) = −ihT {ψ↑(r1, t1)ψ↑(r2, t2)}i, (4.1) † † FT (1, 2) = −ihT {ψ↓(r1, t1)ψ↑(r2, t2)}i. (4.2)

† where ψσ(r, t) (ψσ(r, t)) creates (annihilates) one electron excitation with spin σ at position r and at time t. The time-order operator T is deﬁned such that,

T {A(t1)B(t2)} = Θ(t1 − t2)A(t1)B(t2) − Θ(t2 − t1)B(t2)A(t1) (4.3) and the step function is deﬁned  1, t1 > t2 ,  Θ(t1 − t2) = 1/2, t1 = t2 , (4.4)  0, t1 < t2 .

At this point, it is convenient to write the mean-field BCS Hamiltonian, Eq. (2.4), in the first quantized notation, Z Z MF X † ∗ † † HBCS = drψσ(r)h(r)ψσ(r) − dr(∆ (r)ψ↑(r)ψ↓(r) + ∆(r)ψ↓(r)ψ↑(r)). (4.5) σ=↑,↓ 48 Chapter 4. Quasiclassical Theory of Superconductivity where 1 h(r) = − (∇ − ieA(r))2 + eφ(r) − µ, (4.6) 2m r for which, for generality, we assumed presence of an electromagnetic field. We can then find the equation of motion for the electron field operators, dψ (r, t) i ↑ = h(r)ψ (r, t) + ∆(r)ψ†(r, t), (4.7) dt ↑ ↓ dψ†(r, t) −i ↓ = h∗(r)ψ†(r, t) − ∆∗(r)ψ (r, t). (4.8) dt ↓ ↑ Having found these, it is straightforward to calculate the equation of motion for the defined Green’s functions that are known as Gor’kov equations, ∂ [i − h(r1)]GT (1, 2) + ∆(r1)FT (1, 2) = δ(t1 − t2)δ(x1 − x2), (4.9a) ∂t1 ∂ ∗ ∗ [−i − h (r1)]FT (1, 2) + ∆ (r1)GT (1, 2) = 0. (4.9b) ∂t1 In order to facilitate the description of particle-hole coherence in superconductors, we formulate our notation in Nambu space. We introduce the Nambu spinor and its conjugate as, " # ψ↑(r, t) † h † i Ψ(r, t) = † , Ψ (r, t) = ψ↑(r, t) ψ↓(r, t) , (4.10) ψ↓(r, t) and define their multiplication as

" † # † ψ↑(r1, t1)ψ↑(r2, t2) ψ↑(r1, t1)ψ↓(r2, t2) Ψ(r1, t1)Ψ (r2, t2) = † † † , (4.11) ψ↓(r1, t1)ψ↑(r2, t2) ψ↓(r1, t1)ψ↓(r2, t2) and " † # † ψ↑(r1, t1)ψ↑(r2, t2) ψ↓(r1, t1)ψ↑(r2, t2) Ψ (r1, t1)Ψ(r2, t2) = † † † . (4.12) ψ↑(r1, t1)ψ↓(r2, t2) ψ↓(r1, t1)ψ↓(r2, t2) The Nambu spinors therefore satisfy the fermionic anti-communication rules,

† 0 {Ψ(r1, t1), Ψ (r2, t2)} = δ(t1 − t2)δ(x1 − x2)τ , (4.13a)

{Ψ(r1, t1), Ψ(r2, t2)} = 0, (4.13b) † † {Ψ (r1, t1), Ψ (r2, t2)} = 0, (4.13c) while the τ matrices are identical in form to Pauli matrices, " # " # " # " # 1 0 0 1 0 −i 1 0 τ 0 = , τ 1 = , τ 2 = , τ 3 = . (4.14) 0 1 1 0 i 0 0 −1

The BSC Hamiltonian using the Nambu spinors then becomes, Z MF † HBCS = drΨ (r)HbBCSΨ(r) . (4.15) Chapter 4. Quasiclassical Theory of Superconductivity 49 for which, " # h(r) ∆(r) Hb = . (4.16) BCS ∆∗(r) −h∗(r) The equation of motion for the Nambu spinor therefore reads, ∂ i Ψ(r, t) = Hb Ψ(r, t). (4.17) ∂t BCS We now define the time-ordered Green’s function in Nambu space as, 3 † GbT (1, 2) = −iτ hT {Ψ(r1, t1)Ψ (r2, t2)}i " † # 3 hT {ψ↑(r1, t1)ψ↑(r2, t2)}i hT {ψ↑(r1, t1)ψ↓(r2, t2)}i = −iτ † † † hT {ψ↓(r1, t1)ψ↑(r2, t2)}i hT {ψ↓(r1, t1)ψ↓(r2, t2)}i " † # 3 GT (1, 2) FT (1, 2) = τ † , (4.18) FT (1, 2) −GT (2, 1) where the final equality is written in accordance with definitions given by Eqs. (4.1)-(4.2). Note that as we have not considered spin-dependent interactions in the Hamiltonian, spin labeling is † † not important that has enabled us to write hT (ψ↓(r1, t1)ψ↓(r2, t2))i = −hT (ψ↑(r2, t2)ψ↑(r1, t1))i = −G†(2, 1). Eq. (4.17) makes it easy to calculate the equation of motion for the defined Green’s function for which we find,

∂ ∂ 3 n † † o i GbT (1, 2) = τ θ(t1 − t2)Ψ(r1, t1)Ψ (r2, t2) − θ(t2 − t1)Ψ (r2, t2)Ψ(r1, t1) ∂t1 ∂t1 3 † ∂ † = τ δ(t1 − t2){Ψ(r1, t1), Ψ (r2, t2)} + hT ( Ψ(r1, t1)Ψ (r2, t2))i ∂t1 3 = τ δ(t1 − t2)δ(r1 − r2) + HbBCSG˜T (1, 2) ( " # " #) 3 3 h(r1) 0 0 ∆(r1) = τ δ(t1 − t2)δ(r1 − r2) + τ ∗ + ∗ GbT (1, 2). 0 h (r1) ∆ (r1) 0 (4.19) We multiply both side by τ 3 and ﬁnd, ˜−1 [G01 + ∆b BCS]GbT (1, 2) = δ(t1 − t2)δ(r1 − r2), (4.20) where,

−1 ∂ 3 1 3 2 Gb01 = i τ + (∇r1 − iτ eA(r1)) − eφ(r1) + µ, (4.21) ∂t1 2m is the inverse free Green’s function in the Nambu space and, " # 0 ∆(r) ∆b = , (4.22) BCS −∆∗(r) 0 represents the BSC-self-energy due to pairing interaction. Gor’kov was the ﬁrst who showed that the phenomenological Ginzburg-Landau model can be derived from the Gor’kov equation if temperature is close to superconducting critical temperature [81]. We note that apart from the BCS limit, one has to consider other self-energy terms that originate, for example, from electron impurity scattering. In the next section we derive left-right subtracted Dyson equation that paves our way in formulating quasiclassical theory of superconductivity. 50 Chapter 4. Quasiclassical Theory of Superconductivity

Figure 4.1: The time loop leads to four Green’s function depending on the time variables t1 and t2 belong to which part of the contour.

4.2 Dyson Equation in Keldysh-Nambu Space

In formulating the perturbation theory to account for self-energy terms in the Green’s functions, one can expand the S matrix over a time loop consisting an outgoing part from (−∞, t0) and a return part from (t0, −∞). This is known as Keldysh time-loop contour, and is particularly useful for dealing with non-equilibrium situations where the state of the system is assumed to be known only at time t = −∞. As depicted schematically in figure 4.1, such a time loop results in four types of Green’s function depending on the time variables that enter into the Green’s function. When both time variables belong to the outgoing part of the time loop as depicted in panel (a), we define the time-ordered Green’s function that, for superconducting systems in Nambu space, is given by Eq. (4.18). If both time variables belong to the return part of the time loop as shown by panel (b), we define the anti-time-ordered superconducting Greens’ function as 3 ˜ † GbT˜(1, 2) = −iτ hT {Ψ(r1, t1)Ψ (r2, t2)}i, (4.23) for which the anti-time-ordered operator is defined such that,

T˜{A(t1)B(t2)} = Θ(t1 − t2)B(t2)A(t1) − Θ(t2 − t1)A(t1)B(t2). (4.24) Finally, when the time variables are on diﬀerent legs of the time loop, according to panels (c) and (d) of Fig. (4.1), we deﬁne the so-called greater and lesser Green’s functions as, > 3 † Gb (1, 2) = −iτ hΨ(r1, t1)Ψ (r2, t2)i, (4.25) < 3 † Gb (1, 2) = iτ hΨ (r2, t2)Ψ(r1, t1)i. (4.26) We now collect these four Green’s functions into a single matrix in Keldysh-Nambu space,

" < # GbT (1, 2) Gb (1, 2) G(1, 2) = > , (4.27) Gb (1, 2) GbT˜(1, 2) q Chapter 4. Quasiclassical Theory of Superconductivity 51 where each element is a 2 × 2 matrix in Nambu space. We note that, given the deﬁnitions of the so deﬁned Green’s functions, the following identity holds

< > GbT + GbT˜ = Gb + Gb . (4.28)

Therefore, not all components of Eq. (4.27) are independent. Indeed, it is possible to remove part of the redundancy; we ﬁrst rotate the Green’s function in Keldysh space by G → τ3G followed by G → LGL† while L = √1 (τ 0 − iτ 2). These transformations lead to the Green’s 2 function of the form, q q q q " < > < > # 1 GbT − Gb + Gb − GbT˜ GbT + Gb + Gb + GbT˜ G(1, 2) = < > > < . (4.29) 2 GbT − Gb − Gb + GbT˜ GbT − Gb + Gb − GbT˜ We now deﬁne theq retarded, advanced and Keldysh Green’s functions as,

R < > Gb = GbT − Gb = Gb − GbT˜, (4.30) A > < Gb = GbT − Gb = Gb − GbT˜, (4.31) K < > Gb = GbT + Gb = Gb + GbT˜, (4.32)

We thus arrive to the triagonal representation of the Green’s function in Keldysh-Nambu space, " # GbR(1, 2) GbK (1, 2) G(1, 2) = , (4.33) 0 GbA(1, 2) which satisﬁes the Dyson equation,q Z Z 0 0 G(1, 2) = G (1, 2) + dX4 dX3G (1, 4)Σ(4, 3)G(3, 2), (4.34) or equivalently q q q q q Z Z 0 0 G(1, 2) = G (1, 2) + dX4 dX3G(1, 4)Σ(4, 3)G (3, 2). (4.35)

q q q q q 0 Here, Xn = (rn, tn) so that the integration is over both time and space and G is the Green’s function for free particles. The self-energy matrix Σ contains the term originating from pair potential as well as other terms originating, for example, from electron impurityq scattering, q " # " R K # ∆b BCS 0 Σb (1, 2) Σb (1, 2) Σ(1, 2) = −∆BCS + Σimp = − + A , (4.36) 0 ∆b BCS 0 Σb (1, 2) q q q where ∆b BCS is given by Eq. (4.22). Note that the results of the integrals in Eqs. (4.34)-(4.35) are functions of the times and coordinates labeled by 1 and 2. For brevity, we write these equations as G(1, 2) = G0(1, 2) + G0 ⊗ Σ ⊗ G (1, 2), (4.37a) G(1, 2) = G0(1, 2) + G ⊗ Σ ⊗ G0 (1, 2), (4.37b) q q q q q

q q q q q 52 Chapter 4. Quasiclassical Theory of Superconductivity where ⊗ indicates matrix multiplication together with integration over time and space, and the inverse free Green’s function in Keldysh space reads,

−1 ∂ 3 1 3 2 G01 = i τ + (∇r1 − iτ eA(r1)) − eφ(r1) + µ, (4.38) ∂t1 2m for which q q q " # τ 3 0 τ 3 = . (4.39) 0 τ 3

−1 −1 q We define G02 similar to G01 with the difference that the derivatives in this case are with respect to the time and coordinate labeled by 2. We finally take, q q −1 −1 −1 G0 (1, 2) = G01 δ(1 − 2) = G02 δ(1 − 2). (4.40)

−1 We now operate Eqs. (4.34)q by G0 fromq right and integrateq over time and coordinate labeled −1 by 1. Similarly, we operate Eqs. (4.35) by G0 from left and integrate over time and coordinate labeled by 2, and then subtractq the two obtained equations to ﬁnd the left-right subtracted Dyson equation, q

−1 −1 G0 ⊗ G − G ⊗ G0 (1, 2) = Σ ⊗ G − G ⊗ Σ (1, 2). (4.41)

The reason for this subtractionq q becomeq q clear in the nextq sectionq q whereq we adopt the quasiclassical approximation to the left-right subtracted Dyson equation in order to drive the Eilenberger equation.

4.3 Eilenberger Equations

The Gor’kov equation is in principle enough to study superconductivity. In practice, however, solving the Gor’kov equation is difficult specially for situations where superconducting order parameter is varying in space at temperatures far below the critical temperature Tc [83]. To find a simpler equation, we note that the Gor’kov equation contains information on the length scale −1 of Fermi wavelength kF . This is in fact much smaller than the relevant length scales for many physical phenomena including proximity effect in normal-superconducting hybrids where the −1 relevant length scale is given by the superconducting coherence length, ξ kF [83, 84, 85, 90]. Equivalently, the superconducting order parameter in conventional superconductors is much smaller than the Fermi energy, ∆/εF 1. This indicates that the magnitude of momentum in Green’s functions, and consequently in self energies that are functional of Green’s functions, is close to the Fermi momentum. In quasiclassical theory of superconductivity, one uses this peaked structure and integrates out the momentum degree of freedom. The resulting quasiclassical Green’s function are then used to calculate physical quantities of interest. Formulating this approach was started with a seminal work done by Andreev where he employed a WKB approximation to the Bogoliubov-de Gennes equation in order to eliminate all short wavelength oscillations [82]. The complete quasiclassical equations for superconducting systems in thermal equilibrium were formulated by Eilenberger [85] and also independently by Larkin and Ovchin- nikov [86]. The generalization to non-equilibrium systems was done by Eliashberg [87] and Larkin and Ovchinnikov [88, 89]. Chapter 4. Quasiclassical Theory of Superconductivity 53

The starting point in quasiclassical formalism is to use the mixed or Wigner representation; in the Green’s functions, we perform a transformation to center of mass and relative coordinates deﬁned by, 1 1 R = (r + r ),T = (t + t ), (4.42a) 2 1 2 2 1 2 r = r2 − r1, t = t2 − t1, (4.42b) and then take a Fourier transform with respect to fast oscillating variables r and t, Z Z G(R, T, r, t) = dεe−iεt dpeip.rG(R, T, p, ε) . (4.43)

We note that the variationsq with respect to the center-of-massq variables are due to lack of translational invariance that is controlled by, for example, an applied external ﬁeld or a non- uniform space. In the quasiclassical formalism, the Green’s function is replaced by,

G(R, p, T, ε) → δ(ξp)g(R, p,ˆ T, ε), (4.44) while pˆ represents a unit vectorq in direction of p and the quasiclassical Green’s function is deﬁned as an integration over a low-energy part around the Fermi surface that is yet depending on the direction of momentum,

i Z εc g(R, p,ˆ T, ε) = dξpG(R, p, T, ε). (4.45) π −εc Here, q q

p2 ξ = − µ ∼ v (p − p ), (4.46) p 2m F F and the cut-oﬀ εc is much smaller than the Fermi energy, εc/εF 1 while its value does not have physical signiﬁcance [90]. It has been shown by the Eilenberger [85] and Larkin and Ovchinnikov [88] that the quasiclassical Green’s function g satisfy the following normalization condition, q " R R R K K A# " 0 # gb gb gb gb + gb gb τb 0 gg = A A = 0 . (4.47) 0 gb gb 0 τb

To simplify Eq. (4.41), letq usq consider the convolution of two general operators, Z Z A ⊗ B(r1, t1, r2, t2) = dr3 dt3A(r1, t1, r3, t3)B(r3, t3, r2, t2), (4.48)

We transform the positions r1 and r2 to center of mass, R, and relative coordinate, r, followed by a Fourier transformation of r. This removes the integration of r3 on the left side of the equation and we ﬁnd,

A B A B Z i ( ∂ ∂ − ∂ ∂ ) A ⊗ B(R, p, t1, t2) = dt3e 2 ∂R ∂p ∂p ∂R A(R, p, t1, t3)B(R, p, t3, t2), (4.49) 54 Chapter 4. Quasiclassical Theory of Superconductivity for which the superscript indicates that derivation is only act on the speciﬁed function, i.e. ∂A ∂B ∂A ∂B ∂R ∂p AB = ∂R ∂p . We now remove the time integral as well by transforming the time variables t1 and t2 to T and t, and then a Fourier transformation of t. This leaves us with,

A B A B A B A B i ( ∂ ∂ − ∂ ∂ ) i ( ∂ ∂ − ∂ ∂ ) A ⊗ B(R, p, T, ε) =e 2 ∂T ∂ε ∂ε ∂T e 2 ∂R ∂p ∂p ∂R A(R, p, T, ε)B(R, p, T, ε), i ∂A ∂B ∂A ∂B i ∂A ∂B ∂A ∂B =[1 + ( − ) + ...][1 + ( − ) + ...] 2 ∂T ∂ε ∂ε ∂T 2 ∂R ∂p ∂p ∂R × A(R, p, T, ε)B(R, p, T, ε). (4.50)

When the derivatives are small, one can keep the terms up to ﬁrst order. Moreover, from now on, we always assume the system under consideration is in its steady state so that the terms ∂ containing time derivative ∂T would vanish. We then have, i ∂A ∂B ∂A ∂B A ⊗ B − B ⊗ A ' [A, B] + { , } − { , } , (4.51) 2 ∂R ∂p ∂p ∂R and use this identity to simplify Eq. (4.41). This would leave us with the Green’s function in the Fourier space for which we take an integration over momentum that results in the quasiclassical Green’s function, according to Eq. (4.45). Here, we do not consider external applied ﬁelds so that the inverse free Green’s function given in Eq. (4.38) in the Fourier space reads,

−1 3 G0 (R, p, T, ε) = ετ − ξp. (4.52) Therefore, the ξ term is canceled in the left-right subtracted Dyson equation. Moreover, the p q q impurity self-energy is a functional of the quasiclassical Green’s function and we replace it by Σimp(G) → σ(g) and note that its dependence on the momentum is negligible according to quasiclassical approximation. These steps simplify Eq. (4.41) and lead us to the Eilenberger equations [85], ∂g [ετ 3 + ∆, g] + iv pˆ − [σ, g] = 0. (4.53) F ∂R q From the diagonal elements of theq Eilenbergerq q equation,q weq ﬁnd equations for the retarded and advanced Green’s functions that determine the spectral density. Moreover, the non-diagonal element is related to the Keldysh Green’s function and gives the quantum kinetic equation for the distribution function. However, in thermal equilibrium we only need to consider the equation for the retarded Green’s function as the Keldysh component in this case can be deduced from the ﬂuctuation-dissipation theorem, ε gK (R, ε) = gR(R, ε) − gA(R, ε) tanh , (4.54) b b b 2T where T is temperature. The advanced Green’s function is also related to retarded Green’s function by,

A 3 R † 3 gb = −τ (gb ) τ . (4.55) As an example of solving the Eilenberger equation, we consider a clean BCS superconductor so that the impurity self-energy term can be neglected. Moreover, we assume translational Chapter 4. Quasiclassical Theory of Superconductivity 55 invariance that leads to vanishing of the derivative term in Eq. (4.53); the retarded component of the Green’s function in Eq. (4.53) then simpliﬁes to, 3 R [ετb + ∆b , gb ] = 0. (4.56) We take the retarded Green’s function as, " # R g11 g12 gb = , (4.57) g21 g22

R R 0 and it follows from the normalization condition, gb gb = τb , that

g11 = −g22, (4.58a) 2 g11 + g12g21 = 1. (4.58b) Moreover, Eq. (4.56) implies, ∗ ∆ g12 + ∆g21 = 0, (4.59a) ∗ ∆ g11 + εg21 = 0. (4.59b) These set of equations are enough to ﬁnd all components of the retarded Green’s function, and in the case where ε2 > |∆|2 we ﬁnd,  √ ε √ ∆  R ε2−|∆|2 ε2−|∆|2 g = ∗ . (4.60) b −√ ∆ −√ ε  ε2−|∆|2 ε2−|∆|2

Note that the g11 component is in fact the BCS-quasiparticle density of states previously found in Eq. (2.15) and the g12 term is known in the literature as pair amplitude [91]. Apart from the BCS limit, various interactions and mechanisms such as electron impurity scattering, inelastic electron-phonon interaction and spin-ﬂip scatterings can contribute in the self energy. For example, the latter interaction comes from magnetic impurities for which they suppress superconducting order parameter and we have to avoid them for preserving qubit coherence. Here we consider the self-energy due to elastic impurity scattering; within the Born approximation, its contribution reads, Z d3p0 −→ −→ Σ (−→p ) = N |V (−→p − p0 )|2G(p0 ), (4.61) imp i (2π)3 imp where Vimp is the impurityq potential and Ni is the impurity concentration.q Assuming electron- hole symmetry, we change the integral over momentum space to the integral over the angle and length of the momentum, Z d3p0 Z Z dΩ0 = N dξ0 p , (4.62) (2π)3 0 p 4π where N is the density of states per spin at the Fermi energy. Since during elastic scatterings the 0 −→ magnitude of momentum is preserved, |−→p | = |p0 |, and in quasiclassical theory the momentum is bounded to Fermi surface, the impurity potential is independent to the magnitude of momentum 0 and we take it as |Vimp(ˆp.pˆ )|. This enables us to write Eq. (4.61) in the quasiclassical formalism as, Z dΩ0 σ(ˆp) = −iπN N p |V (ˆp.pˆ0)|2g(ˆp0). (4.63) i 0 4π imp

q q 56 Chapter 4. Quasiclassical Theory of Superconductivity

4.4 The Dirty Limit

The Eilenberger equation can be simpliﬁed considerably in the so-called dirty limit where the mean free path is the smallest length scale of the system. The requirement is that the impurity scattering self-energy, σ(ˆp), dominates all other terms in the Eilenberger equation. In this case, the strong impurity scattering randomizes the momentum of the quasiparticles so that we can consider the quasiclassicalq Green’s function to be almost isotropic and expand it in spherical harmonics,

g(ˆg) = gs +p ˆgp. (4.64)

Here the assumption is that gs and gp qare independentq q of pˆ, and |pˆgp| |gs|. Consequently, the self-energy is also expected to have a similar form, q q q q σ(ˆg) = σs +p ˆσp. (4.65)

To ﬁnd the components of self-energy, weq substituteq q Eq. (4.64) into Eq. (4.63) and ﬁnd, Z dΩ0 σ(ˆp) = −iπN N p |V (ˆp.pˆ0)|2(g +p ˆ0g ). (4.66) i 0 4π imp s p

To simplify this equation,q we deﬁne the elastic scattering rate,q q 1 Z dΩ0 = 2πN N p |V (ˆp.pˆ0)|2, (4.67) τ i 0 4π imp and the transport relaxation rate,

Z 0 1 dΩp 0 2 0 = 2πNiN0 |Vimp(ˆp.pˆ )| (1 − p.ˆ pˆ ). (4.68) τtr 4π The latter gives a characteristic time for a particle to travel before the direction of its velocity is lost due to scattering and is related to the impurity mean free path by l = vF τtr. Indeed, as we argue in the following, the transport relaxation rate in the dirty limit becomes equal to the elastic scattering rate that forms the dominant energy scale of the system. Bearing in mind that gs and gp are completely independent of momentum, from Eq. (4.66) we ﬁnd

Z dΩ0 1 σ = −iπN N g p |V (ˆp.pˆ0)|2 = −i g , (4.69) s i 0 s 4π imp 2τ s Z 0 dΩp 0 2 0 1 1 1 σqp = −iπNiN0gqp |Vimp(ˆp.pˆ )| [1 − (1 −qp.ˆ pˆ )] = −i ( − )gp (4.70) 4π 2 τ τtr For isotropicq scattering, thereq is no preferred direction so that on the average p.qˆ pˆ0 ∼ 0. This indicates τ ∼ τtr and therefore, |pˆσp| |σs| that is in accordance with the expansion made in Eq. (4.65). We now use the normalization condition and neglect the terms quadratic in gp to ﬁnd, q q q 0 gsgs = τ , (4.71a)

gsgp + gpgs = 0. (4.71b) q q q q q q q Chapter 4. Quasiclassical Theory of Superconductivity 57

These help us to simplify the Eilenberger equation; we substitute Eqs. (4.64)-(4.65) into Eq. (4.53) and separate the equations based on whether they are even or odd function with respect to pˆ. Knowing [σs, gs] = 0 and [σp, gp] = 0, the even terms lead to,

[ετ 3 + ∆, g ] + iv (ˆp.pˆ)∇ g = 0, (4.72) q q q q s F R p and the odd terms give, q q q q

3 ∂gs i [ετ + ∆, gp] + ivF + gsgp = 0. (4.73) ∂R τtr q We further simplify the equationq byq droppingq the ﬁrst termq q in favor of the third one as the relaxation rate 1/τtr is the dominant energy scale of the system. We then multiply Eq. (4.73) by gs and use Eq. (4.71a) to ﬁnd gp in terms of gs,

g = −lg ∇ g . (4.74) q q p qs R s By substituteing this into Eq. (4.72) and taking average of p.ˆ pˆ over the spherical surface to q q q replaced it by 1/3, we arrive to the Usadel equation [92],

3 [ετ + ∆, gs] − iD∇R (gs∇Rgs) = 0, (4.75) where we used the diffusion coefficient defined as D = v l/3. q q q q F q The Usadel equation shall be solved to find the Green’s function, while a required input is the superconducting order parameter that itself depends on the Green’s functions. Therefore, it has to be found by a self-consistent calculation. The order parameter can be written as, i ∆(R) = λhc (R)c (R)i = λ[GbK (1, 2)] δ(r − r )δ(t − t ), ↓ ↑ 2 12 1 2 1 2 i Z dε Z d3p = λ [GbK (R, p, ε)] , 2 2π (2π)3 12 1 Z Z dΩp = N λ dε [GbK (R, p,ˆ ε)] , 4 0 4π 12 1 Z = N λ dε[gK (R, ε)] , (4.76) 4 0 bs 12 where in the last equality we replaced the Keldysh component of the quasiclassical Green’s function by its isotropic part that forms the dominant contribution in the dirty limit. From now on, we assume thermal equilibrium that enables us to use the fluctuation-dissipation relation for the Keldysh Green’s function, Eq. (4.54), and we find, 1 Z ∆(R) = N λ dε tanh(ε/2T )[gR(R, ε) − gA(R, ε)] , (4.77) 4 0 bs bs 12 Here the integral is taken over a symmetric interval for which the cut-off is set by the Debye frequency, ωD, as discussed in chapter2. As we do not consider any magnetic field, the phase of order parameter remains constant over the whole superconductor and we set it to zero. In thermal equilibrium it is enough to consider the retarded component of the Usadel equation,

3 R R R [ετb + ∆b , gbs ] − iD∇R gbs ∇Rgbs = 0. (4.78) 58 Chapter 4. Quasiclassical Theory of Superconductivity

The normalization condition for the isotropic quasiclassical Green’s function, Eq. (4.71a), makes it convenient to parametrize the Green’s function in a way that simplifies further calculations. Here we use the so-called angular parametrization that reads, " # cos θ(R, ε) i sin θ(R, ε) gR(R, ε) = , (4.79) bs −i sin θ(R, ε) − cos θ(R, ε) for which θ(R, ε) is a complex function depending on energy and space. We substitute this into Eq. (4.78) and find from the non-diagonal elements, 1 D∇2 θ(R, ε) + iE sin θ(R, ε) + ∆(R) cos θ(R, ε) = 0. (4.80) 2 R R A Given Eqs. (4.55) and (4.79), we find [gbs (R, ε) − gbs (R, ε)]12 = 2Im[sin θ(R, ε)] from which we can write the order parameter as,

1 Z ωD ∆(R) = N0λ dε tanh(ε/2T )Im[sin θ(R, ε)]. (4.81) 2 −ωD After solving the Usadel equation and finding the paring angle θ(R, ε), one can use the Green’s function for calculating various physical quantities. As we discuss in the next chapter, of particular interest for us are the normalized quasiparticle density of states, n(R, ε) and pair amplitude, p(R, ε). These quantities can be deduced by an integration over single-particle and anomalous spectral functions defined respectively as, i A(R, p, ε) = (GR(R, p, ε) − GA(R, p, ε)), (4.82) 2π i B(R, p, ε) = (F R(R, p, ε) − F A(R, p, ε)). (4.83) 2π Therefore, using the definition of quasiclassical Green’s function, we can write, 1 n(R, ε) = [ˆgR(R, ε) − gÂ(R, ε)] = Re[cos θ(R, ε)], (4.84) 2 s s 11 1 p(R, ε) = [ˆgR(R, ε) − gÂ(R, ε)] = Im[sin θ(R, ε)]. (4.85) 2 s s 12

4.4.1 Boundary conditions for proximity systems The Green’s function of a normal metal also satisfy the Usadel equation for which the order parameter ∆(R) is zero as electron-phonon coupling constant vanishes in normal-phase mate- rials. In order to study proximity eﬀect in normal-superconducting hybrids, we need to know the boundary condition at the NS contact interface. It has been shown by Kupriyanov and Lukichev that the boundary condition at the contact interface reads [119]: 1 σS(gbS(R, ε)∇ngbS(R, ε)) = σN (gbN (R, ε)∇ngbN (R, ε)) = [gbS(R, ε), gbN (R, ε)], (4.86) 2RintA provided a low transmission coeﬃcient, T 1, between two layers [94]. Here, ∇n denotes derivative in direction perpendicular to the contact plane, the subscripts N and S represent Green’s functions for the normal and superconducting layer, and RintA denotes resistance at the interface times unit area. These boundary conditions are originating from conservation of Chapter 4. Quasiclassical Theory of Superconductivity 59 current at the contact, and it follows from it that at the outer surface of each layer one can write

∇ngbN/S(R, ε) = 0, (4.87) as there is no current escaping out from the outer surface.

4.4.2 Usadel equations for normal-superconducting hybrids Here we consider a normal-superconducting hybrid for which a superconducting layer is partially covered by a normal layer. Without loosing generality, we assume that the contact layer is in x − y plane; see Fig. 5.2. The Usadel equation Eq. (4.80) for each layer reads, D N ∇2θ (x, y, z, ε) + iε sin θ (x, y, z, ε) = 0, (4.88) 2 N N D S ∇2θ (x, y, z, ε) + iε sin θ (x, y, z, ε) + ∆(x, y, z) cos θ (x, y, z, ε) = 0. (4.89) 2 S S S The non-diagonal elements of Eqs. (4.86) provide the boundary at the NS interface, ∂θ (x, y, z, ε) ∂θ (x, y, z, ε) σ N | = σ S | N ∂z z=0 S ∂z z=0 1 = sin[θN (x, y, z = 0, ε) − θS(x, y, z = 0, ε)] . (4.90) RintA And the non-diagonal elements of Eq. (4.87) give us with the boundary condition at the outer surface of each layer, ∂θ (x, y, z, ε) N(S) | = 0, (4.91a) ∂z z=dN (−dS ) ∂θ (x > 0, y, z, ε) S | = 0. (4.91b) ∂z z=0

We now take the thicknesses dN/S of the layers to be smaller than the superconducting coherence length ξ; this enables us to write the pairing angles as a series expansion in z. Using the boundary condition Eq. (4.91a) we find, z z2 θN (x, y, z, ε) = θN (x, y, ε) + β(x, y, ε)( − ) + ..., (4.92) ξ 2ξ0dN z z2 θS(x, y, z, ε) = θS(x, y, ε) + a(x, y)α(x, y, ε)( + ) + ... , (4.93) ξ 2ξ0dS where, the area function a(x, y) is 1 in the contact region and 0 otherwise (since the normal metal only partially covers the superconductor, θN is only defined in the contact region). We substitute these two relations into the Usadel equations, Eqs. (4.88) and (4.89), and find to the leading order, " 2 2 # ∂ θN (x, y, ε) ∂ θN (x, y, ε) 2iE β(x, y, ε) = dN ξ0 2 + 2 + sin θN (x, y, ε) , (4.94) ∂x ∂y DN " 2 2 ∂ θN (x, y, ε) ∂ θN (x, y, ε) 2iE α(x, y, ε) = −dSξ0 2 + 2 + sin θS(x, y, ε) ∂x ∂y DS 2∆(x, y, T ) + cos θS(x, y, ε) . (4.95) DS 60 Chapter 4. Quasiclassical Theory of Superconductivity

Having found these two parameters, we substitute Eqs. (4.92) and (4.93) into the boundary condition, Eq. (4.90), and ﬁnd a set of coupled equations for the paring angle of normal and superconducting layers, " # τ D ∂2 ∂2 − N N + θ (x, y, ε) − iEτ sin θ (x, y, ε) = sin[θ (x, y, ε) − θ (x, y, ε)], (4.96) 2 ∂x2 ∂y2 N N N N S " # τ D ∂2 ∂2 S S + θ (x, y, ε) + iEτ sin θ (x, y, ε) + τ ∆(x, y, T ) cos θ (x, y, ε) 2 ∂x2 ∂y2 S S S S S h i = sin θN (x, y, ε) − θS(x, y, ε) , (4.97)

2 where τN(S) = 2e ν0N(S)dN(S)RintA. In the next chapter, we solve the Usadel equations for two cases of uniform and non-uniform systems in order to study modifications in superconducting properties. This enables us to find how inverse proximity effect due to trap-qubit contact can affect qubit relaxation time. Chapter 5 Proximity Effect in Normal-Metal Quasiparticle Traps

This chapter contains a paper of the author that has been published with title Proximity effect in normal-metal quasiparticle traps and is cited in reference [78]. Here we use the Usadel formalism presented in the last chapter to study impact of proximity effect on the qubit coherence. I contributed in Ref. [78] by doing all of analysis and preparing all figures.

5.1 Introduction

As we showed in chapter2, the contribution of quasiparticle tunneling in the noise spectral function has a linear dependence on the density of quasiparticle present at close vicinity of Josephson junction; see Eq. (2.45). We then discussed in chapter3 that a normal-metal in tunnel contact with a superconductor can trap quasiparticles and reduce their density. There- fore, we argued such traps can potentially improve the lifetime of superconducting qubits. In this chapter, we aim to study a side effect that accompanies normal-metal quasiparticle traps; when a normal-metal (N) is brought into contact with a superconductor (S), Cooper-pairs can “leak” to the normal layer modifying properties of both layers. This phonomenon is known as proximity effect and has been studied since 60’th [95, 96, 97, 98]. Works on the proximity effect have investigated, for example, quasi-one-dimensional N-S systems [98, 99, 100], SNS junctions [101, 102, 103, 104, 105] and NSN configurations [106]. This effect is not exclu- sive of superconducting-normal systems; in fact, an important application of this phenomenon is to induce superconductivity (i.e. open up an energy gap) in semiconductor nanowires in order to realize Majorana zero modes [107, 108, 109]. This effect has also been studied in superconducting-ferromagnetic [110] and superconducting-superconducting hybrids [111]. As we discuss it later, due to this effect a minigap opens up in the density of states of the normal layer while the BCS singularity at ε = ∆0 broadens and a finite subgap density of states is induced in the superconducting layer. For superconducting qubits, it has been shown that presence of a subgap density of states modifies the contribution of quasiparticles tunneling in the spectral noise function [35]. In addition, a subgap density of states can open up a new relaxation channel due to Cooper-pair processes. These have motivated us to study in detail how proximity effect modifies properties of the superconducting layer and how such modifications impact the qubit coherence. Our main result is that due to the competition between such a proximity effect-induced increase in the relaxation rate and the decrease of 1/T1 due to the trap’s suppression of the quasiparticle density, there is an optimal position for the trap. If the trap is closer to a junction than this optimal position, the relaxation rate exponentially increases over a distance given by the coherence length; for a trap further away than the optimum, the 62 Chapter 5. Proximity Effect in Normal-Metal Quasiparticle Traps decay rate slowly increases over the much longer “trapping length”, which is determined by quasiparticle diffusion and the trap’s effective trapping rate. We organized this chapter as follows: in Sec. 5.2 we report and summarize some results form Ref. [35] to generalize the spectral function to account for the effect of subgap density of states. In Sec. 5.3 we use the quasiclassical theory of superconductivity to study first a uniform normal-superconductor bilayer. Then (Sec. 5.3.2) we consider a non-uniform system which models quasiparticle traps; we present both numerical self-consistent solutions for the spatial variation of superconducting order parameter and single-particle density of states as well as approximate analytical expressions for the latter. In Sec. 5.4 we study the qubit decay rate taking into account the proximity effect. In AppendicesI throughL we present a number of derivations and mathematical details.

5.2 Qubit relaxation due to quasiparticles

The quasiparticle contribution to the qubit decay rate, Γ10, can be written in the standard form of the product between a matrix element and a spectral density, 2 ϕ Γ10 = h1| sin |0i S(ω10) (5.1) 2 where |0i (|1i) denotes the ground (excited) state of the qubit, and ω10 is the qubit frequency. The excitation rate Γ01 is obtained by replacing ω10 → −ω10. Here we again consider a single- junction transmon qubit for which the matrix element can be expressed in terms of the transmon parameters as [31] 2 s ϕ EC EC h1| sin |0i = ' (5.2) 2 ω10 8EJ with EC the charging energy and EJ the Josephson energy; this expression is valid in the transmon regime EJ EC . Under certain conditions that are often satisfied, namely: a hard gap ∆0 in the superconductor larger than the qubit frequency, 2∆0 > ω10, and “cold” quasiparticles, meaning that their typical energy δE (or effective temperature) above the gap is small compered to qubit frequency, δE ω10, the spectral density is simplified to the form given by Eq. (2.45). As mentioned in the introduction, and as we will explain in more detail in Sec. 5.3, due to the proximity effect the BCS peak in the density of states broadens and a finite subgap DoS is induced in the superconductor. As a first step to find impact of these modification to the qubit coherence, we consider how to generalize the expression for the qubit decay rate, Eq. (5.1); the appropriate generalization is presented in Ref. [35], and for the case considered here of a single-junction transmon it amounts to a redefinition of the spectral density appearing in Eq. (5.1):

S(ω) = St(ω) + Sp(ω) , (5.3) where we distinguish two contributions, St due to single quasiparticle tunneling and Sp originating from Cooper pair processes. In terms of the distribution function f they are, for positive frequency ω > 0, Z ∞ St(ω) = dε A(ε, ε + ω)f(ε)[1 − f(ε + ω)], (5.4) 0 Z ω 1 Sp(ω) = dε A(ε, ω − ε)[1 − f(ε)][1 − f(ω − ε)], (5.5) 0 2 Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps 63 with 16E A(ε, ε0) = J [n(ε)n(ε0) + p(ε)p(ε0)] . (5.6) π∆0 Note that the density of states n(ε) appearing in this expression does not necessarily take the BCS form; indeed, both n(ε) and the pair amplitude p(ε), can be calculated within a Green’s function approach that we discussed in chapter4. In a normal/superconductor bilayer, these two quantity depend on parameters such as ﬁlm thicknesses and interface resistance, as we explain next in Sec. 5.3. Here we point out that the combinations of n and p account for both the quasiparticle density of states and Bogoliubov coherence factors, while whether a process involves single quasiparticles or pairs is manifest in the combination of distribution functions: f(1 − f) for single quasiparticles, (1 − f)(1 − f) or ff for pair breaking or recombination processes, respectively. Finally, the spectral density S at negative frequencies is obtained by replacing f → 1−f in Eqs. (5.4) and (5.5). For pair processes, this implies that Sp(ω) accounts for pair breaking by qubit relaxation (we take ω > 0), while Sp(−ω) for qubit excitation by quasiparticle recombination.

5.3 Proximity eﬀect in thin ﬁlms

The goal of this section is to arrive at expressions for the functions n and p in Eq. (5.6) that take into account the proximity effect between the normal-metal trap and the qubit superconducting electrodes. These expression will then be used in Sec. 5.4 to estimate the influence of the proximity effect on qubit lifetime. The calculations are based on the quasiclassical approach to superconductivity that we reviewed in chapter4. We first consider proximity in a uniform bilayer and then generalize our discussion to non-uniform hybrids relevant for realistic implementation of normal-metal quasiparticle traps.

5.3.1 Uniform NS bilayers

In a uniform bilayer a superconducting film of thickness dS is fully covered by a normal metal of thickness dN , with both thicknesses smaller than the superconducting coherence length at zero temperature ξ. This implies that spatial variations across the films can be neglected. Moreover, because the system is uniform in the plane of the films, the pairing angle is independent of position and the Usadel equations Eq. (4.97)-(4.96) take the form

iε sin θS(ε) + ∆(T ) cos θS(ε) (5.7) 1 = sin[θS(ε) − θN (ε)] , τS 1 iε sin θN (ε) = sin[θN (ε) − θS(ε)] , (5.8) τN

2 where ∆(T ) must be calculated self-consistently using Eq. (4.81). The times τi = 2e νidiRintA, i = S, N, account for the interface resistance times area product RintA and the density of states at the Fermi level νi of the two ﬁlms. Typically, τS ≈ τN , and the dimensionless parameter τS∆ can be used to characterize the strength of the coupling between the two layers. For τS∆ → ∞ we can to leading order neglect the right hand sides of Eqs. (5.7) and (5.8), so that the two 64 Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps layers would be decoupled; in this limit the solution of the Usadel equation is

i∆ θ (ε) = θ (ε) ≡ arctan , (5.9) S BCS ε θN (ε) = 0 . (5.10)

A weak-coupling regime is possible for high interface resistance, such that τS∆ 1 but finite. On the other hand, for a good contact between N and S (or sufficiently close to the critical temperature, where ∆(T ) → 0) the coupling can be strong, τS∆ 1. In this section we focus on the weak-coupling case τS∆ 1 with τN ∼ τS; some considerations on the strong-coupling one can be found in Appendix I.2. In the normal film, the main consequence of the contact with the superconductor is the opening of a so-called minigap in its density of states. The minigap energy Eg is always small compared to the gap in the bulk superconductor, and in the weak-coupling regime is given by 1 Eg ' (5.11) τN as already shown in the seminal work of McMillan [97] and more recently rederived in Ref. [112] within the quasiclassical formalism. In the superconductor, above the minigap a small but finite sub-gap density of states is induced of the form [97, 112]   1 ε n(ε) ' n>(ε) ≡ Re q  . (5.12) τS∆ 2 2 ε − 1/τN

This expression is valid below the gap, ε ∆, but it fails close to the minigap, as noted in Ref. [112]. Indeed, as detailed in Appendix I.1 we ﬁnd the validity condition ( ) 1 1 1 1 ε − max , (5.13) 2/3 τN τN (τS∆) τN ∆ Moreover, the the position of the minigap is more accurately given by " # 1 3 1 1 ε ' 1 − − (5.14) g 2/3 τN 2 (τS∆) τN ∆ and just above it the density of states has a square root threshold,

1 r2 n(ε) ' n (ε) ≡ τ (ε − ε ) (5.15) t 1/3 N g (τS∆) 3 an expression valid for −2/3 τN (ε − εg) (τS∆) . (5.16) In the inset of Fig. 5.1 we compare Eqs. (5.12) and (5.15) to the density of states obtained by numerically solving the Usadel equations (5.7) and (5.8). We now turn our attention to energy well above the minigap, ε εg. In this energy range a broadening of the BCS peak was qualitatively predicted [97] and displayed in numerical calculations [112]; to our knowledge, however, no analytical formula has been presented in the Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps 65

Figure 5.1: Density of states in a superconducting layer weakly coupled to a normal one, τS∆0 = 50. Solid lines are calculated by numerically solving the Usadel equations, Eqs. (5.7) and (5.8), and subsituting the result into Eq. (4.84). Dashed lines shows drived from analytical relations. The inset plot shows the behaviour of DoS for energies close to the minigap. literature. Interestingly, we ﬁnd that the density of states has the well-known form proposed by Dynes et al.[113] to ﬁt tunneling measurements: " # ε + i/τ n(ε) ' n (ε) ≡ Re S . (5.17) Dy p 2 2 (ε + i/τS) − ∆

A similar result was found for the case of a short S wire between two N leads [106]. We obtain the above formula from the following approximate expression for the pairing angle

i∆ θS(ε) ' θDy(ε) ≡ arctan (5.18) ε + i/τS for the pairing angle (deviations from these formulas can arise for |ε/∆ − 1| 1/(τ ∆)2 when √ . N τS∆ & τN ∆, see Appendix I.1). In the main panel of Fig. 5.1 we plot Eq. (5.17) along with the result of a numerical calculation of the density of states. Using θDy of Eq. (5.18) in the self-consistent equation (4.81) we also recover McMillan result for the suppression of the zero-temperature order parameter s 2 ∆NS ' ∆0 1 − (5.19) τS∆0 with ∆0 the bulk value of the order parameter. Note that Eq. (5.12) and Eq. (5.17) agree at leading order in the overlap region εg ε ∆, as they both approximately take the constant value 1/τS∆ there; a crossover energy between the two expression can be identiﬁed with the p geometric average ∆/τN between gap and minigap. This crossover energy is, for typical 66 Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps

Figure 5.2: A non-uniform NS bilayer: a superconducting ﬁlm of thickness dS (bottom) is partially covered by a normal metal layer (thickness dN ) occupying the region x > 0. We use this system to model the vicinity of a normal-metal quasiparticle trap (see text). parameters, smaller than qubit frequency. Therefore, we can in general use these Dynes-like formulas as a starting point to evaluate the density of states in a non-homogenous system, as we show next.

5.3.2 Proximity effect near a trap edge A normal-metal trap in general covers only part of a superconducting electrode [cf Eq. (3.19)], in order to limit losses in the normal metal that could otherwise shorten the qubit lifetime [75]. Typically, traps have lateral dimension of the order of 10 µm or more [47], while the thicknesses dS and dN of superconducting and normal material are in the range of tens of nanometers. These sizes should be compared to the coherence length ξ, which for disordered aluminum films typically used to fabricate qubits is of the order of 200 nm. Therefore both the normal and superconducting films are thin compared to ξ, while the lateral dimensions of the trap are much wider than ξ. We can therefore effectively model the system near the trap edge as being composed by a superconducting film occupying the whole x-y plane and a normal metal in the half plane x > 0, see Fig. 5.2. To study the proximity effect near such an edge, we must allow for spatially dependent paring angles. Due to translational symmetry in the y direction, the derivative with respect to y in Eqs. (4.97) and (4.96) vanishes and we find:

2 DS ∂ θS(ε, x) 2 + iε sin θS(ε, x) + ∆(x) cos θS(ε, x) 2 ∂x (5.20) 1 = sin[θS(ε, x) − θN (ε, x)]H(x) , τS where DS is the diﬀusion constant for electrons in the normal state of S and H(x) is the step function [H(x) = 1 for x > 0 and 0 otherwise], and for x > 0

2 DN ∂ θN (ε, x) 2 + iε sin θN (ε, x) 2 ∂x (5.21) 1 = sin[θN (ε, x) − θS(ε, x)], τN with DN the diﬀusion constant for electrons in N. As before, the superconducting order parameter ∆(x) is to be found self-consistently using Eq. (4.81). To avoid any confusion, we remind Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps 67

Figure 5.3: Solid line (blue): normalized order parameter ∆(x)/∆0 in the non-uniform NS bilayer depicted in Fig. 5.2 as a function of normalized distance x/ξ from the trap edge. Dot- dashed line (black): non-self-consistent, step-like approximation, Eq. (5.22), which we use for analytical calculations (see text). Dashed line (red): “ﬁrst iteration” obtained by substituting the pairing angles obtained in the step-like approximation, Eqs. (5.28) and (5.29), into Eq. (5.23) and the latter into Eq. (4.81).

that while Dqp in the diﬀusion equation Eq. (3.19) is proportional to DS in Eq. (5.20)[47], the former takes into account phenomenologically the dependence on energy of the distribution function in the superconductor – information that is lost in considering the density xqp – and this usually results in Dqp DS. In general, the system of Usadel plus self-consistent equations must be solved numerically. In Fig. 5.3, we plot with the solid line the self-consistent order paramter for such a solution, obtained following the procedure described in AppendixJ. Far from the trap edge the solution must approach either the BCS one for x → −∞, or that for the uniform NS bilayer for x → ∞. In other words, indicating with θSu(ε) the pairing angle in the S component of a uniform N- S bilayer, the solution θS(ε, x) for the non-uniform case interpolates between θBCS and θSu. Similarly, in the weak-coupling regime the order parameter ∆(x) interpolates between ∆0 and ∆NS of Eq. (5.19) as x goes from −∞ to +∞; the diﬀerence between the two values of the order parameter is small, so we look for an approximate (not self-consistent) solution to the Usadel equations (5.20) and (5.21) in which ∆(x) is assumed to take the form (see dot-dashed line in Fig. 5.3)  ∆0, x < 0 ∆s(x) = (5.22) ∆NS, x ≥ 0

Moreover, for energies large compared to the minigap, ε εg, we can neglect θN in comparison to θS at leading order in 1/τN ∆0 1. Hence we can approximate sin[θS − θN ] ≈ sin θS, and 68 Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps at this order Eq. (5.20) decouples from Eq. (5.21). With these approximations, the solution for θS is (cf. Ref. [99])

θS(ε, x) = θL(ε, x)H(−x) + θR(ε, x)H(x) (5.23)

θL(ε, x) = θBCS(ε) (5.24) √ x 2α (ε) θBCS(ε) − θ0(ε) − 4 arctan e ξ 1 tan , 4

θR(ε, x) = θSu(ε) (5.25) √ − x 2α (ε) θSu(ε) − θ0(ε) − 4 arctan e ξ 2 tan . 4

p Here, we define the coherence length as ξ = DS/∆0, introduce the dimensionless functions q 2 2 q 2 i 2 α1(ε) = ∆ − ε /∆0 and α2(ε) = ∆ − (ε + ) /∆0, and θ0(ε) is the (unknown) value 0 NS τS of θS at the trap edge x = 0. By construction this expression for θS is continuous at x = 0, but it should also be continuously differentiable. Equating the left and right derivatives at the edge gives us a condition that implicitly defines θ0:

h θBCS (ε)−θ0(ε) i q tan 4 α1(ε) + 2 h θBCS (ε)−θ0(ε) i 1 + tan 4 (5.26) h θSu(ε)−θ0(ε) i q tan 4 α2(ε) = 0 . 2 h θSu(ε)−θ0(ε) i 1 + tan 4

In the weak-coupling regime we are considering, we have α1 ' α2 so long as |ε − ∆0| 1/τS. Therefore, except in a narrow energy region near ∆0, Eq. (5.26) has the approximate solution

1 θ ' (θ + θ ). (5.27) 0 2 BCS Su

Finally, in the energy range where our approximations apply (energy above the minigap and not too close to ∆0), θSu is well approximated by θDy of Eq. (5.18), which in the same energy range is close to θBCS. We can therefore linearize Eqs. (5.24) and (5.25) to arrive at √ x 1 2α1(ε) θL(ε, x) ' θBCS(ε) − e ξ [θBCS(ε) − θSu(ε)] (5.28) 2 √ 1 − x 2α (ε) θ (ε, x) ' θ (ε) − e ξ 2 [θ (ε) − θ (ε)] (5.29) R Su 2 Su BCS

In Fig. (5.4) we compare the density of states obtained from a self-consistent numerical solution of the Usadel equations (5.20)-(5.21) to an approximate semi-analytic formula which we arrive at by substituting into Eq. (4.84) the Eqs. (5.28)-(5.29) with θSu(ε) found by numerically solving Eqs. (5.7)-(5.8) [or equivalently Eq. (I.3)]. In the next section we will be interested in the spatial evolution of the normalized density of states and pair amplitude away from the normal- metal trap. In order to ﬁnd an analytic expression for these quantities, we further approximate Eq. (5.28) by using the Dynes expression, Eq. (5.18), for θSu and obtain for x < 0 at leading Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps 69

Figure 5.4: Spatial-evolution of quasiparticle density of states for τS∆0 = 100. The density of states interpolates between BCS-like form at the right to the bilayer-like form at the left side of the trap edge. See also Fig. K.2 for Spatial-evolution of pair amplitude. order (see AppendixK for details)

√ 2 1/4 − 2 |x| 1− ε 3 ξ ∆2 1 1 ∆0 n(ε, x) ' e 0 , (5.30) 3/2 2 τS∆0 2 2 ∆0 − ε √ 2 1/4 − 2 |x| 1− ε 2 ξ ∆2 1 1 ε∆0 p(ε, x) ' e 0 , (5.31) 3/2 2 τS∆0 2 2 ∆0 − ε for ∆0 − ε 1/τS and ε εg, and

2 1/4 − |x| ε −1 ε ξ ∆2 1 1 n(ε, x) ' q − e 0 √ (5.32) 2 2 2 τS∆0 ε − ∆0   3 2 !1/4 ∆0 |x| ε π × cos  − 1 −  , 2 23/2 ξ ∆2 4 ε − ∆0 0

2 1/4 − |x| ε −1 ∆0 ξ ∆2 1 1 p(ε, x) ' q − e 0 √ (5.33) 2 2 2 τS∆0 ε − ∆0   2 2 !1/4 ε∆0 |x| ε π × cos  − 1 −  , 2 23/2 ξ ∆2 4 ε − ∆0 0 for ε − ∆0 1/τS. Note that for energies above the gap the corrections to the BCS formulas are always small by construction. Moreover, both above and below the gap the corrections are small in 1/τS∆0 and decay exponentially with distance over an energy-dependent length scale which is of the order of the coherence length ξ away from the gap, but longer than ξ close to the gap. We have now all the ingredients needed to estimate the quasiparticle-induced transition rates for a qubit with a trap, which is the focus of the next section. 70 Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps

5.4 Qubit relaxation with a trap near the junction

As discussed in Sec. 5.2, the qubit decay rate due to quasiparticle tunneling is proportional to the spectral density S, see Eq. (5.1). The spectral density is determined by the quasiparticle distribution function f, the density of states n, and the pair amplitude p, see Eqs. (5.3)-(5.6). We have shown in the previous Section that near a trap n and p become position-dependent. In the next subsection we study how this dependence affects the qubit decay rate, assuming that quasiparticles are everywhere in thermal equilibrium, so that the distribution function is uniform in space. This assumption is clearly not realistic, since it leads to an increase in the quasiparticle density approaching the trap, but it will enable us to show that the changes in S due to the proximity effect do not significantly harm the qubit if the trap is sufficiently far from the junction. In contrast, in Sec. 5.4.2 we will account in a phenomenological way for the spatially dependent suppression of the quasiparticle density caused by the trap. In this more realistic scenario, we will find an optimal position for the trap, which balances between such a suppression and the enhancement of the subgap density of states, two effects that have opposite influence on the qubit relaxation rate. Throughout this section, we assume that the qubit has reflection symmetry with respect to the junction, as in experiments [47]; this means that when a trap is mentioned, it should be understood as two identical traps placed at the same distance from the junction.

5.4.1 Thermal equilibrium The assumption of thermal equilibrium means that the distribution function has the Fermi-Dirac form, 1 f eq(ε) = , (5.34) eε/T + 1 with T the quasiparticle temperature. It then follows from Eqs. (5.3)-(5.5) that the spectral density obeys the detailed balance relation

Seq(−ω) = e−ω/T Seq(ω) . (5.35)

We assume that the quasiparticles are “cold”, T ω10, and therefore we can neglect the qubit eq eq −ω10/T excitation rate in comparison with the decay rate, since Γ01/Γ10 = e 1. In presence of a trap, since n and p at the junction position depends on its distance x from the trap, the quantity A deﬁned in Eq. (5.6) is also a function of x and so is the spectral function. An eq approximate expression for S (ω, x) can be obtained in the relevant regime εg T ω ∆0. In practice, since the minigap energy εg is much smaller than temperature, we can set the former to zero. Then for the quasiparticle tunneling part of the spectral density, we can identify three contributions (see AppendixL for details on the derivation of the expressions discussed here):

eq eq eq eq St (ω, x) = Saa(ω, x) + Sba (ω, x) + Sbb (ω, x). (5.36)

The first contribution accounts for transitions in which the initial quasiparticle energy is above the gap – then the final energy is also above the gap; this term is approximately independent of position [cf. Eq. (2.45)], s 8E 2∆ Seq(ω, x) ' J xeq 0 , (5.37) aa π qp ω Chapter 5. Proximity Effect in Normal-Metal Quasiparticle Traps 71

eq p −∆0/T where xqp = 2πT/∆0e coincides with the equilibrium value of the quasiparticle density in the absence of the trap. A spatial dependence in principle arises from the corrections terms in Eqs. (5.32) and (5.33), but their contributions can be neglected in comparison with the other eq terms in St which we now discuss. The second term in the right hand side of Eq. (5.36) originates from transitions in which a quasiparticle initially below the gap absorbs the qubit energy and is excited above the gap energy: 1 s √ 4 8E 2∆ 1 − 2 x 2ω ∆ eq J eq 0 ξ ∆0 0 ω/T Sba (ω, x) ' xqp e e . (5.38) π ω 2τS∆0 ω

The small factor 1/τS∆0 and that exponentially decaying with distance account for the smallness of the initial density of states. In contrast, the final factor is large because the initial occupation probability is exponentially larger at lower energies. Thus at sufficiently low temperature this term can become larger than Saa of Eq. (5.37). The last term in Eq. (5.36) arises from transitions with both initial and final quasiparticle energy below the gap, √ 8E 1 x T eq J −2 2 ξ Sbb (ω, x) ' 2 e 2 ln 2 . (5.39) π (τS∆0) ∆0

Here the small factor 1/τS∆0 is squared, and the exponential decay with distance is faster than in Eq. (5.38), because both initial and final density of states are small. However, the temperature dependence is much weaker: Sbb vanishes linearly with T rather than exponentially. Therefore, despite the small prefactors, this term can dominate at low temperatures. In addition to the single quasiparticle tunneling, pair events can take place. In particular, since the density of states is finite (albeit small) down to the minigap energy εg, so long as ω10 > 2εg a pair breaking process is possible, in which the qubit relaxes by breaking a Cooper pair and exciting two quasiparticles above the minigap (but well below the gap). From Eq. (5.5) the spectral density for such a process is (see AppendixL) √ 8E 1 x ω T eq J −2 2 ξ Sp (ω, x) = 2 e − 2 ln(2) . (5.40) π (τS∆0) ∆0 ∆0 The spectral density does not vanish even at T = 0, as there is no need for thermally excited quasiparticles to be present; in fact, the spectral density decreases linearly with increasing T because the increased occupation of the final states suppresses this process. Interestingly, this linear in temperature term cancels with Sbb, Eq. (5.39); moreover, while Sba in Eq. (5.38) can be dominant in the limits of sufficiently small temperature and large distance, its contribution eq to St is negligible in the parameter range we are interested in (cf. Fig. 5.5), so that we have approximately  s √  8E 2∆0 1 ω x eq J eq −2 2 ξ S (ω, x) ≈ xqp + 2 e  . (5.41) π ω (τS∆0) ∆0 Assuming that the trap is next to the junction, x = 0, in Fig. 5.5 we plot, as function of temperature, the qubit decay rate Γ10 obtained by substituting Eqs. (5.2) and (5.41) into Eq. (5.1), as well as the contribution from the three type of tunneling processes identified above (above gap to above gap, aa, below gap to above gap, ba and below gap to below gap, bb). At “high” temperature, above about 100 mK but still below the qubit frequency, the dominant contribution comes from the position-independent aa term. In contrast, at low temperature there 72 Chapter 5. Proximity Effect in Normal-Metal Quasiparticle Traps

Figure 5.5: Qubit relaxation rate as a function of temperature. We assumed typical transmon parameters for the qubit (see e.g. Ref. [6]): ∆0 = 46 GHz, ω10 = 6 GHz, EJ = 16 GHz 3 and EC = 290 MHz; and weak proximity eﬀect, τS∆0 = 10 . The solid line (red) shows the total relaxation rate, while the other lines show the contributions from the diﬀerent processes discussed in the text.

is a temperature-independent plateau in Γ10 originating from the sum of bb and pair-breaking processes. This plateau shows that the trap can increase the decay rate exponentially in comparison with the no-trap rate, which coincides with the aa term. However, the plateau is quickly suppressed by moving the trap away from the junction: for√ each coherence length increase in trap-junction distance, the plateau decreases by a factor e2 2 ≈ 17. With the parameters of −2 Fig. 5.5, this means that for x = 4ξ the low-temperature decay rate would be Γ10 < 10 Hz. Therefore, even though the trap adversely affects the qubit, the limitation imposed on the decay rate becomes quickly negligible by increasing the distance to the junction. The fact that the trap can only harm the qubit rather than improve its coherence is a consequence of the thermal equilibrium assumption. Next, we relax this assumption to find up to which point the trap can be beneficial.

5.4.2 Suppressed quasiparticle density

A trap can be beneficial to a qubit primarily by suppressing the quasiparticle density at the junction [68], as discussed in chapter3. Within the phenomenological diffusion model of Eq. (3.19), the typical length scale over which such a suppression takes place is given by the trapping length q λtr = Dqp/Γeff; this length scale is of order 100 µm [47, 68], much longer than the coherence length ξ. As for the strength of the proximity effect, based on the experimental parameters of Chapter 5. Proximity Effect in Normal-Metal Quasiparticle Traps 73

3 4 Ref. [47], we estimate it to be τS∆0 ∼ 10 -10 . The large separation of length scales together with the weakness of normal trap-superconductor coupling, τS∆0 1, make it possible to use Eq. (3.19) to calculate the spatial profile of the density, while the modifications introduced by the proximity effect can be treated as corrections. Below we will consider a realistic device geometry when calculating the position-dependent density, but first we discuss how to incorporate such a non-equilibrium quasiparticle configuration into the evaluation of the qubit transition rates. When we neglect the proximity effect, the density of states takes the BCS form, Eq. (2.15), and the quasiparticle density defined in Eq. (2.46) can depend on position only through the distribution function f. Such dependence could arise, for example, due to a temperature profile. However, at low temperatures it is in general more appropriate to model non-equilibrium quasiparticles by introducing an effective chemical potential µ˜ [114] (which we measure from the Fermi energy). The reason is that recombination processes, which are needed for chemical equilibrium, are slower than the scattering processes responsible for thermalization [115]. Such a phenomenological non-equilibrium approach has been already considered in the qubit setting [116]. Here to capture the spatial profile of the density we assume the distribution function f to have the form 1 f(ε) = , (5.42) e(ε−µ˜)/T˜ + 1 where the effective chemical potential is a function of position, µ˜ =µ ˜(x), while the effective temperature T˜ is homogeneous and does not necessarily coincides with the phonon bath temper- −7 ature. Indeed, typical quasiparticle densities in the absence of traps are in the range xqp ∼ 10 - 10−5, corresponding to effective temperatures (at µ˜ = 0) from ∼ 145 mK to ∼ 200 mK, much higher than both the usual fridge temperature (10-20 mK) and the typical qubit temperature which is of order 35-60 mK [65, 117], as estimated from the excited state population. In the following we will present results for T/˜ ∆0 in the range 0.01 to 0.05, corresponding to approximately 20 mK to 110 mK in aluminum; for a given effective temperature, the chemical potential can then be calculated by inverting Eq. (2.46) [with n(ε) of Eq. (2.15) and f(ε) of Eq. (5.42)]. So long q (∆ −µ˜)/T˜ (˜µ−∆ )/T˜ as e 0 1, the integration in Eq. (2.46) gives approximately xqp ' 2πT/˜ ∆0e 0 and therefore we find s  ˜ ∆0 µ˜(x) = ∆0 + T ln  xqp(x) . (5.43) 2πT˜

The assumption made above gives a restriction on the range of allowed effective temperatures, ˜ 2 −4 2πT/∆0 xqp, which is however not relevant in practice since usually we have xqp < 10 [43] −2 and 2πT/˜ ∆0 > 10 (since T˜ should be at least comparable to the fridge temperature). This restriction also implies µ˜ < ∆0. We will assume that in general the quasiparticle density xqp(x) is larger than the thermal equilibrium value at temperature T˜, so that µ˜ > 0. Note that for a given xqp, µ˜ is a decreasing function of T˜, while for a fixed T˜ it is an increasing function of xqp. With the approach described above, given the quasiparticle effective temperature T˜ and the density profile xqp(x), one can calculate the effective chemical potential µ˜(x) using Eq. (5.43) and therefore obtain an expression for the non-equilibrium distribution function, Eq. (5.42). Once the distribution function is known, we can evaluate the spectral density of Eqs. (5.3)- (5.5), which we denote hereinafter with S˜ to remind of its dependence on the non-equilibrium parameters T˜ and µ˜ (and hence on junction-trap distance; we drop in this section the variable x 74 Chapter 5. Proximity Effect in Normal-Metal Quasiparticle Traps as explicit argument of the spectral density for notational compactness). Similar to the thermal equilibrium case, in the single quasiparticle tunneling contribution S˜t we distinguish three terms: S˜t = Sãa + S˜ba + S˜bb. For the first two terms on the right hand side we find, as discussed in Appendix L.0.2, that they are proportional to xqp, as in thermal equilibrium: s 8E 2∆ S˜ (ω) ' J x (x) 0 (5.44) aa π qp ω and 1 s √ 4 8E 2∆ 1 − 2 x 2ω ∆ J 0 ξ ∆0 0 ω/T˜ S˜ba(ω) ' xqp(x) e e (5.45) π ω 2τS∆0 ω

Here we assume that ω > 0 and ∆0 − µ˜ − ω T˜; validity conditions for the approximations employed are discussed in more detail in Appendix L.0.2. For the term S˜bb we have diﬀerent regimes depending on the ratio between ω and µ˜:

˜ 8EJ 1 Sbb(ω) ' 2 × π (τS∆0) √  x ˜ µ/˜ T˜ −2 2 ξ 2T 1+e  e ln ˜ , µ˜ ω  ∆0 1+e(˜µ−ω)/T .  1  √ µ˜2 4 (5.46)  −2 2 x 1− 3 2 2 ξ ∆2 ∆ (∆ +˜µ ) 4e 0 0 0 × 2 2 3/2 3/2  (∆ −µ˜ ) (∆0+˜µ)  0  1 1  √ − √ , µ˜ ω  ∆0−µ−ω˜ ∆0−µ˜ For this term the similarity with thermal equilibrium is recovered only for µ˜ T˜. This is the case also for the pair process contribution S˜p: √ ˜ 8EJ 1 −2 2 x T 1 S˜p(ω) ' e ξ × 2 (2˜µ−ω)/T˜ π (τS∆0) ∆0 1 − e " ˜ # (5.47) 1 + e(ω−µ˜)/T ω 2 ln − 1 + e−µ/˜ T˜ T˜

However, a partial cancellation between S˜bb and S˜p takes place so long as ω − 2˜µ T˜, in which case we ﬁnd √ 8EJ 1 ω −2 2 x ˜ ˜ ξ Sbb(ω) + Sp(ω) ≈ 2 e , (5.48) π (τS∆0) ∆0 as in thermal equilibrium [compare to the second term in Eq. (5.41)]. Turning now to the spectral density at negative frequencies, we note that a relation similar to Eq. (5.35), −ω/T˜ S˜t(−ω) = e S˜t(ω) (5.49) follows from Eqs. (5.4) and (5.42). Since we consider ω T˜, we can neglect the qubit excitation due to single quasiparticle tunneling. In contrast, for pair processes we ﬁnd, from Eq. (5.5),

(2˜µ−ω)/T˜ S˜p(−ω) = e S˜p(ω) (5.50) Therefore the rate of qubit excitation induced by quasiparticle recombination can become exponentially larger than qubit relaxation by Cooper pair breaking if 2˜µ − ω T˜. We next apply these results to a model of an actual qubit. Chapter 5. Proximity Eﬀect in Normal-Metal Quasiparticle Traps 75

Figure 5.6: Diagram for the right half of the transmon qubit considered here. Except for the position of the trap, it is the same design studied in Refs. [47, 68].

5.4.2.1 An example

As a concrete example, we consider the qubit geometry depicted in Fig. 5.6, where a trap with length d is placed a distance x away from the Josephson junction. Similar geometries have been used experimentally to measure quasiparticle recombination, trapping by vortices [43] and by normal-metal traps [47], and theoretically to devise how to optimize trap performance [68]. Here the only difference is that we allow the long trap, d > l, to be close to the junction, 0 ≤ x ≤ l, so that the role of the proximity effect can be evaluated. To find the quasiparticle density xqp(x) at the Josephson junction, we proceed as in chapter 3 with detail presented in Appendix G.2; we treat each segment of the device (except the pad with side Lpad) as one-dimensional. Since we are interested in the steady-state density, we set ∂xqp/∂t = 0 in Eq. (3.19). Solving that equation for each segment of the device, and requiring continuity of the density and current conservation at the points where different segments meet, we find g 1 AR d x xqp(x) = 1 + + cosh Γeff sinh(d/λtr) W λtr λtr λtr x + d − l A 1 x 2 + cosh c + . (5.51) λtr W λtr 2 λtr

2 Here AR = W (L+l−x−d)+Lpad is the uncovered area to the right of the trap and Ac = 2WcLc is the area of gap capacitor. Equation (5.51) makes it clear that the closer the trap is to the junction, the more the quasiparticle density is suppressed, and that significant changes in the density take place over the length scale given by λtr. We can now proceed as outlined above: namely, we first calculate the effective chemical potential µ˜ for different effective temperatures T˜. In Fig. 5.7 we show the results of such calculations for two positions of the trap, x = 0 and x = l; hereinafter we use the same realistic parameters for the qubit (L = 1 mm, l = 60 µm, W = 12 µm, Lc = 200 µm, Wc = 20 µm, Lpad = 80 µm) and for the trap (d = 234 µm, λtr = 86.2 µm) as in Ref. [68], cf. Refs. [43, 47]. We also use the experimentally determined −4 5 values g = 10 Hz [43] and Γeff = 2.42 × 10 Hz [47]. As expected, the effective chemical potential decreases with increasing effective temperature, and is larger when the trap is further away. 76 Chapter 5. Proximity Effect in Normal-Metal Quasiparticle Traps

Figure 5.7: The normalized effective chemical potential µ/˜ ∆0, see Eq. (5.43), as function of the normalized effective temperature T/˜ ∆0 for two positions of the trap: x = 0 (dashed line) and x = l (solid). The upper effective temperature scale is given for aluminum.

Next, we calculate the qubit transition rates using the thus found chemical potential. We perform the calculation in two ways: we substitute the chemical potential into Eq. (5.42) for the distribution function and the latter into the definitions of the spectral functions, Eqs. (5.4) and (5.5); in those equations, we use the semi-analytic results for the pairing angle to determine the density of states and the pair amplitude, see Eq. (5.28) and the text below Eq. (5.29), and perform the final integration over energy numerically. In a second approach, we use our approximate analytical formulas for the spectral functions, see Eqs. (5.44) to (5.50). The rates so obtained are shown in Fig. 5.8 for an effective temperature T/˜ ∆0 = 0.019 (∼ 40 mK in Al). In the left panel we distinguish the contributions to 1/T1 due to tunneling-induced relaxation (Γ˜10,t) and excitation (Γ˜01,t) and pair process excitation (Γ˜01,p); the pair process relaxation rate is much smaller than the excitation rate [cf. Eq. (5.50)] and not visible on this scale. In the right panel we plot the total rate 1/T1, which is dominated by the tunneling-induced relaxation; the total rate is a non-monotonic function of trap-junction distance x, due to the competition between processes with initial quasiparticle energy below the gap, whose contributions to the rate decay exponentially with distance over a length scale of the order of the coherence length [see Eqs. (5.45) and (5.46)], and processes with above-gap initial energy, with contribution slowly increasing with xqp over the much longer length scale λtr. The minimum in this curve thus gives the optimal position xo for the trap: for x > xo, the density slowly increases, so one is not taking full advantage of the trap, but at x < xo the subgap density of states quickly increases and negates the benefit of further density suppression. Therefore we conclude that placing a trap at a distance xo < x < λtr represents the best choice. In Fig. 5.9 we further explore the dependence of the optimal position xo on parameters such as the effective temperature T˜ and the strength of the proximity effect τS∆0 (we remind that the larger this parameter, the weaker the proximity Chapter 5. Proximity Effect in Normal-Metal Quasiparticle Traps 77

Figure 5.8: Left: tunneling (t) and pair (p) contributions to the qubit decay rate 1/T1 as a function of trap-junction distance x (the pair decay rate Γ˜10,p is too small to be visible). The effective temperature is T/˜ ∆0 = 0.019 (approximately 40 mK in Al), other parameters are as in Fig. 5.5. The solid lines have been obtained by numerically calculating the integrals determining the spectral functions, while the dashed lines are our approximate analytical findings (see text for more details). Right: Total qubit decay rate, showing a minimum for a distance of a few coherence lengths. effect). Clearly, the stronger the proximity effect, the further a trap should be placed. As the effective temperature increases, on the other hand, the optimal position decreases: as already noted before, for a given density the higher the effective temperature, the smaller the effective chemical potential, and this reduces the importance of the subgap states, since their occupation decreases. Based on this figure, we conclude that when the distance is over 20 coherence lengths the proximity effect can be safely neglected; in aluminum such a distance is of the order of a few microns, which is still much less than the trapping length.

5.5 Summary

In this work we investigate the proximity effect between a normal-metal quasiparticle trap and the superconducting electrode of a qubit. On one hand, a trap can prolong the relaxation time of the qubit by suppressing the quasiparticle density. On the other hand, the proximity effect induces subgap states which can shorten the relaxation time. To quantify the competition between these two phenomena, we start by considering a uniform superconductor-normal metal bilayer; at relevant energies, the density of states takes the Dynes form, Eq. (5.17), with the broadening determined by interface resistance, superconducting film thickness, and its density of states at the Fermi energy. We then study how such broadening decays away from a trap edge, 78 Chapter 5. Proximity Effect in Normal-Metal Quasiparticle Traps

Figure 5.9: Normalized optimal trap-junction distance xo/ξ as a function of normalized effective temperature T/˜ ∆0 (the upper scale gives the corresponding temperature for aluminum). Solid (dashed) lines are obtained by numerically finding the position of the minimum in curves such as the solid (dashed) one in the right panel of Fig. 5.8. see Eqs. (5.30) to (5.33). With these results, we can evaluate the qubit decay rate as function of the distance between trap and junction; we take into account the suppression of the quasiparticle density by introducing a distribution function which depends on two parameters, an effective temperature and a distance-dependent effective chemical potential, cf. Eq. (5.42). Within this approach, we find that the competition between proximity effect and density suppression leads to an optimal placement for the trap, see Figs. 5.8 and 5.9. The qubit relaxation rate exponentially increase for a trap closer to the junction than this optimum over a length scale of the order of the coherence length, while the increase of the rate when moving the trap farther away is much slower and over the much longer trapping length. Therefore, a trap should be placed at least as far from the junction as the optimum position, but no significant penalty is paid for distances up to the trapping length. While we focused here on a transmon qubit, our findings may prove useful in designing traps for other systems as well. For example, quasiparticle poisoning could be a significant hurdle for nanowire-based realization of Majorana qubits [27]; our findings indicate that a normal-metal trap placed close to the ends of the nanowire could be detrimental, as the small minigap is not sufficient to protect zero-energy states from being thermally excited into the subgap states induced by the trap. Finally, our results on the proximity effect could also help interpreting tunneling density of state experiments such as those reported in Refs. [100, 118] Chapter 6 Summary and Conclusions

Quasiparticle tunneling is an intrinsic decoherence channel for superconducting qubits with a corresponding relaxation rate that linearly scales with the density of quasiparticles present at close vicinity of qubit Josephson junction. While the generation mechanism of non-equilibrium quasiparticles at millikelvin temperatures is still an open question, in this thesis we focused on manipulating quasiparticle population to limit and reduce harmful consequences of their presence. We studied how a normal-metal in tunnel contact with a superconductor can trap quasiparticle excitations. We showed these traps can be beneficial for superconducting qubits in three ways: first, suppressing the steady-state quasiparticle density at the junction that can improve qubit’s T1 time. Second, a trap can speed up the decay of the excess quasiparticles and, third, it can decrease fluctuations around the steady-state density; these effects of normal- metal quasiparticle traps are promising to achieve longer-lived qubits that are also more stable in time. We developed a phenomenological diffusion equation that explains the effect of a normal- metal trap on the dynamics and steady-state of non-equilibrium quasiparticle population in a superconducting qubit. The model takes into account the tunneling between the superconductor and the trap, as well as the electron energy relaxation in the trap, see Eq. (3.18). Due to the pronounced energy dependence of quasiparticle density of states for energies above and close to the gap, the effective trapping rate Γeff becomes sensitive to the energy of the quasiparticles and is limited by their backflow from the normal-metal trap on time scales shorter than the electron energy relaxation rate. Furthermore, we find how the time needed to evacuate the injected quasiparticles depends on the trap size. This evacuation time saturates at the lowest, diffusion- limited value upon extending the trap above a certain characteristic length l0. To support our theory, we reported experimental data (obtained by our collaborators) that confirms the predictions of our model. For small traps we can estimate the effective trapping rate Γeff: both its order-of-magnitude and its increase with temperature indicate a limitation due to the backflow of quasiparticles. The bottleneck for trapping is confirmed to be slow energy relaxation inside the trap; this quantity is very difficult to manipulate. In order to improve trapping efficiency, we studied the effects of size, number and position of quasi-1D normal-metal traps. We prefer 1D traps over 2D traps because 2D traps are strong only if they are large compared to the trapping length λtr of Eq. (3.21), while in quasi-1D it is sufficient for the trap length d to be longer than the characteristic scale l0 [Eq. (3.34)] that is generally shorter than λtr for long devices (Ldev > λtr). We point out that covering a large part of the qubit by a normal-metal trap could be a source of unwanted dissipation that harms qubit coherence. For a weak trap (d . l0), the trap position has a negligible effect on the ability to suppress quasiparticle density and its fluctuations, as well as on the decay rate of excess quasiparticles. In contrast, we find a number ways to optimize a strong but short trap, l0 . d . λtr. First, 80 Chapter 6. Summary and Conclusions placing a single trap at the optimum position that makes the decay of excess quasiparticles as fast as possible, see Figs. 3.7 and 3.8. We found that in general it is advantageous to divide a strong trap into smaller pieces of length ∼ l0 and distribute those around the device, see Sec. 3.2.2.2. Second, we find that there is an optimal trap position for suppressing density fluctuations, see Sec. 3.2.3.1. In addition, we find that there is a maximum distance Lf from the junction up to which the suppression of fluctuations is effective; this distance is smaller than the distance where, for a given trap size, large suppression of the steady-state density is achieved. Therefore, suppressing fluctuations by a normal-metal trap indicates that the steady- state density is suppressed as well. These lead us to the conclusion that by correctly placing multiple traps in the device in such a way that one is sufficiently close to the junction, all three beneficial effects of traps can be optimized. In addition to trapping quasiparticles, a normal-metal material in contact with a superconductor has a number of side effects that can influence qubit performance. In particular, Cooper-pairs leak to the trap as well, and this leads to modifications in superconducting properties. We used Usadel formalism to study this effect. We find proximity-induced subgap density of states in the superconducting electrode from both self-consistent numerical and analytical approaches; see Figs. 5.1 and 5.4. This subgap density of states modifies quasiparticle tunneling contribution in the total decay rate and can open up new decay channel due to pair processes. In order to find the qubit relaxation rate, we modeled quasiparticle distribution function by use of the diffusion model that we developed earlier; see Eqs. (5.42) and (5.43). While the quasiparticle density is more suppressed by moving the trap closer to the junction, the magnitude of trap-induced subgap density of state becomes exponentially enhanced. This leads to a non-monotonic relation for the qubit relaxation as a function of trap-junction distance, Fig. (5.8). Our analysis has enabled us to find up to which point in trap-junction distance a trap can be beneficial to the qubit coherence; depending on temperature and resistance at the qubit-trap contact, one needs to place the trap of order 4 to 20 coherence lengths away from the junction in order to prevent proximity effect from harming the qubit coherence; see Fig. 5.9. Placing the trap further away from the junction than this optimum up to the “trapping length” does not significantly increase the qubit decay rate. We stress that our considerations for normal-metal quasiparticle trapping are valid for normal islands in tunnel contact with the superconductor. A good contact between the qubit and a normal metal or another superconductor with a lower energy gap can also act as a trap by localizing the quasiparticles away from the junction. Studying such trapping scenarios based on band-gap engineering is an important topic, but it is beyond the scope of the present thesis. Appendix A Tunneling rate equations

In this Appendix, we derive the rate equations for quasiparticles and electrons accounting for tunneling between a superconductor and a normal metal. Here we assume that both the superconductor and the normal metal are sufficiently small volumes (ΩS and ΩN , respectively), such that the diffusion of excitations occurs on a fast time scale and the occupation probabilities are hence uniform in space. Within these volumes we define the probabilities

X D † E f (ξm) = cmσcmσ (A.1) σ X D † E fqp (εn) = γnσγnσ (A.2) σ of ﬁnding an electron excitation of energy ξm in the normal metal and a quasiparticle excitation of energy εn in the superconductor, respectively. The tunnel coupling between the two, see Eq. (3.4), gives rise to a change in both occupation probabilities for energies above the gap, via processes whose rates can be computed using Fermi’s Golden Rule: X f˙ (ξm) = [Wnσ→mσ − Wmσ→nσ + W0→mσ,n−σ − Wmσ,n−σ→0] , (A.3) nσ X f˙qp (εn) = [−Wnσ→mσ + Wmσ→nσ + W0→mσ,n−σ − Wmσ,n−σ→0] , (A.4) mσ with 2 t 2π e 2 Wnσ→mσ = unfqp (εn) [1 − f (ξm)] δ (εn − ξm) , (A.5) ~ ΩSΩN 2 t 2π e 2 W0→mσ,n−σ = vn [1 − fqp (εn)] [1 − f (ξm)] δ (εn + ξm) . (A.6) ~ ΩSΩN

The reverse processes are found by replacing f(qp) → 1−f(qp). Assuming particle-hole symmetry, f (−ξ) = 1 − f (ξ), we summarize the rate equations as

2π te X f˙ (ξm) = [fqp (εn) − f (ξm)] δ (εn − ξm) (A.7) ~ ΩSΩN n 2

2π te X f˙qp (εn) = [f (ξm) − fqp (εn)] δ (εn − ξm) . (A.8) ~ ΩSΩN m The tunneling processes considered above are elastic. In the normal metal, for temperatures T ∆ there is a large interval of unoccupied states below the gap. Inelastic processes, such as electron-phonon and electron-electron interactions, can relax the excitations in the normal 82 Appendix A. Tunneling rate equations metal to energies below the gap, so that they cannot return to the superconductor. We phenomenologically account for this relaxation by adding the term −Γrf (ξm) to the right-hand side of Eq. (A.7). The relaxation rate Γr is assumed energy-independent, which is justified if the interval of non-zero excitations above the gap is within a narrow energy strip of width ∆. In the next step, we are interested in the probabilities to find excitations in the states within a small energy interval δε. We define the probability densities

1 X pN (ε) = f (ξm) (A.9) NS ε<ξm<ε+δε 1 X pS (ε) = fqp (εn) (A.10) NS ε<εn<ε+δε which are normalized with respect to the normal-state number of states in the superconductor NS = νS0ΩSδε. In the continuum limit δε → 0 these deﬁnitions lead to Eqs. (3.12) and (3.13), respectively. From Eqs. (A.7)-(A.8) plus the phenomenological relaxation term discussed above, we obtain Eqs. (3.15)-(3.16) with the rates

2π 2 ν ε S0 Γesc (ε) = te √ , (A.11) ~ ΩN ε2 − ∆2 2π 2 ν N0 Γtr = te . (A.12) ~ ΩS Appendix B Derivation of eﬀective trapping rate

In this Appendix we provide a derivation of the effective trapping rate, including diffusion. It is a straightforward continuation of the rate equation derived in the previous AppendixA. In disordered metals, the effect of elastic impurity scattering on the distribution function is accounted for by a diffusion term; for quasiparticles in superconductor, the diffusion constant in the so-called “hydrodynamical approach” [66] is energy-dependent:

~ 2 p˙N (ε, ~r, t) =DN ∇ pN (ε, ~r, t) + δ (z − tS)[γetrpS (ε, ~r, t) −γeesc (ε) pN (ε, ~r, t)] − ΓrpN (ε, ~r, t) (B.1) 2 p˙S (ε, ~r, t) =DS (ε) ∇~ pS (ε, ~r, t) − a (x, y) δ (z − tS) × [γetrpS (ε, ~r, t) − γeesc (ε) pN (ε, ~r, t)] . (B.2) In this Appendix, we disregard recombination and background terms for the quasiparticles that may in general appear. Those are taken into account in the main text. The above is a set of coupled linear equations that can be solved analogously to the solution in the main text in terms of eigenmodes with corresponding eigenmodes. The only difference is that here, we keep the energy argument. Since all the processes but the relaxation term are elastic, the equations at different energies are uncoupled and we can find independent eigenmodes at each energy√ ε. The diffusion constant in the superconductor is in general energy dependent, ε2−∆2 DS (ε) = ε DS (∞) (valid for dirty superconductors). We will first derive an effective trapping rate for each energy and provide the conditions of validity. In a second step, we provide the effective rate for the quasiparticle density xqp when integrating over ε.

B.1 Thin normal metal

First, we assume a sufficiently thin trap such that the electron density within the normal metal does not change significantly in z-direction. This is true if the length scale, on which pN decays in z, λr, is much larger than the thickness of the trap, tN λr. We may prove this along the following lines. The diffusion equation for the normal metal may be alternatively expressed as

2 p˙N (ε, ~r, t) = DN ∇~ pN (ε, ~r, t) − ΓrpN (ε, ~r, t) (B.3) for z > tS, with the boundary condition at z = tS

DN ∂zpN (ε, x, y, tS, t) + γetrpS (ε, x, y, tS, t) − γeesc (ε) pN (ε, x, y, tS, t) = 0 . (B.4)

We make the following ansatz, assuming that pN is almost constant in z direction, 2 (z − tN − tS) pN (ε, ~r, t) = peN (ε, x, y, t) + 2 dpN (ε, x, y, t) . (B.5) 2tN 84 Appendix B. Derivation of eﬀective trapping rate

The term proportional to dpN corresponds to the next-to-leading order correction. There are no linear terms in z − tN − tS in this expansion, as to respect the hard wall boundary ∂zpN = 0 at z = tS + tN . Above expansion is well-deﬁned if dpN peN . Plugging this ansatz into above diﬀusion equation, we get the equation in leading order

˙ ~ 2 DN peN = DN ∇ peN + 2 dpN − ΓrpeN . (B.6) tN For the boundary condition we obtain likewise in leading order

DN − dpN + γetrpS − γeesc (ε) peN = 0 . (B.7) tN We reinsert this into the diﬀusion equation, and we obtain

˙ ~ 2 1 peN = DN ∇ peN + [γetrpS − γeesc (ε) peN ] − ΓrpeN . (B.8) tN

Note that the boundary condition in leading order relates the correction dpN to pS and peN , and thus provides the limit of validity of the thin film approximation, D t N . (B.9) N pS γetr − γeesc (ε) peN This condition can be evaluated when formally solving Eq. (B.8), in order to relate the time- evolution of peN to pS. This equation has a set of eigenmodes, which are discrete due to the finite size of the normal metal. Choosing the coordinate system such that the metal has its hard wall borders at x = 0, dx, y = 0, dy, we find eigenmodes of the form q n(kx, ky) = NxNy cos(kxx) cos(kyy) , (B.10) with kx,y = πnx,y/dx,y, nx,y ∈ N, the normalisation constant Nx,y = 1 + δ0nx,y , and the corresponding eigenvalues

2 2 λ(kx, ky) = −DN kx − DN ky − Γesc(ε) − Γr . (B.11) This allows us to express the solution of the normal metal density distribution as Z −iωt γetr dω X e peN (x, y, t) = − n(kx, ky)pS(kx, ky, ω) , (B.12) tN 2π iω + λ(kx, ky) kx,ky with the Fourier-transformed density in the superconductor

Z dx Z dy Z dx dy iωt pS(kx, ky, ω) = n(kx, ky) × dte pS(x, y, t) . (B.13) 0 dx 0 dy In the solution given in Eq. (B.12) we discarded any transient terms, which are exponentially −1 suppressed for times t (Γesc(ε) + Γr) . Assuming that pS has some cutoﬀ in k-space and frequency space, i.e., pS varies on a maximal length scale λS and time scale τS, we may, by means of Eqs. (B.12) and (B.13) approximate

pS −1 −2 ∼ τS + DN λS + Γesc(ε) + Γr . (B.14) peN Appendix B. Derivation of eﬀective trapping rate 85

Hence we ﬁnd the general condition for the thickness of the normal metal s DN tN −1 −2 . (B.15) τS + DN λS + Γr

In the next section, we consider the case when pS varies suﬃciently slowly in space and time, where it is suﬃcient that s DN tN . (B.16) Γr

B.2 Eﬀective trapping rate

The next crucial step involves the assumptions, that pS varies suﬃciently slowly in x, y, and 2 t, such that we can neglect the terms p˙N and ∇~ pN . Let us for now just denote the time and spatial scales at which pS changes as τS and λS, and identify them later. We may approximate

γetr 1 peN (ε, x, y, t) ≈ pS (ε, x, y, tS, t) + dpeN (ε, x, y, t) (B.17) tN Γesc (ε) + Γr where the second term on the right hand side merely represents the correction, which has to be small, dpeN peN . Inserting the approximated pN into the diffusion equation for pS, we find ~ 2 p˙S (ε, ~r, t) = DS (ε) ∇ pS (ε, ~r, t) − a (x, y) δ (z − tS) γeeff (ε) pS (ε, ~r, t) (B.18) with

Γr γeeﬀ (ε) = γetr . (B.19) Γesc (ε) + Γr

As mentioned before, for above equation to be valid, we require dpeN /peN 1, through which we ﬁnd that the temporal variation of pS has to satisfy

−1 Γesc (ε) + Γr τS (B.20) and the spatial variation must fulﬁll s Γesc (ε) + Γr −1 λS . (B.21) DN

We are now left with identifying the scales τS and λS. We do it explicitly for the experimentally relevant specific example of a thin superconductor where, pS likewise varies slowly in z with respect to the thickness tS. In analogy to the normal metal case in the previous section, the z p dependence may likewise be neglected if tS DS (ε) /Γeff (ε), for Γeff = γeeff/tS. Through the inequality D (ε) D (∞)Γ (∞) S ≥ S esc (B.22) Γeff (ε) ΓtrΓr we find the sufficient condition s DS (∞)Γesc (∞) tS . (B.23) ΓtrΓr 86 Appendix B. Derivation of effective trapping rate

p For tS ≈ tN and νS0 ≈ νN0, this condition reduces to tS DS (∞) /Γr. For a thin superconductor, we obtain likewise the diﬀusion equation

2 p˙S (ε, x, y, t) = DS (ε) ∇~ pS (ε, x, y, t) − a (x, y)Γeﬀ (ε) pS (ε, x, y, t) . (B.24)

From this equation, we now estimate the time scale and spatial scale on which pS varies. Namely, for a superconductor much larger than the normal metal trap, d L (where d and L denote the length scales of trap and superconductor, respectively), the time scale at which pS relaxes is maximally given by the diﬀusion time (see also main text), L2 tS ∼ . (B.25) DS (ε)

Underneath the trap, the density pS changes spatially on a length scale given by the trapping length, s DS (ε) λS = λtr ∼ . (B.26) Γeff (ε) With these two results we find the conditions of validity for the effective trapping rate, D (ε) Γ (ε) + Γ S (B.27) esc r L2 and (Γ (ε) + Γ )2 D esc r N . (B.28) ΓtrΓr DS (ε)

From the former condition, we ﬁnd through Γesc (ε) ≥ Γesc (∞) and DS (ε) ≤ DS (∞), the suﬃcient condition D (∞) Γ (∞) + Γ S . (B.29) esc r L2 The latter condition is a bit less trivial to analyse. First we transform it into

√ 1 !2 (ε2−∆2) 4 ε √ Γesc (∞) 1 + Γr ε (ε2−∆2) 4 D N . (B.30) ΓtrΓr DS (∞) Here we see that the left hand side is in general a non-monotonic function of ε. We may however compute the minimum of the left hand side as a function of the ratio Γesc (∞) /Γr, in order to derive a sufficient condition. We differ between two cases. For Γ (∞) esc ≥ 1 (B.31) Γr the minimum is at ε ∆. As in our considerations the quasiparticle occupation is nonzero only for an energy window dE = ε − ∆ ∆, this minimum is irrelevant, and we find for ε close to ∆ s Γ2 (∞) 1 ∆ D esc √ N (B.32) ΓtrΓr 2 ε − ∆ DS (∞) Appendix B. Derivation of effective trapping rate 87 which is fulfilled as long as

2 2 !2 dE 1 DS (∞) Γesc (∞) 2 . (B.33) ∆ 2 DN ΓtrΓr

For instance for DS ∼ DN and Γesc (∞) ∼ Γtr ∼ Γr, above inequality is always fulﬁlled. For Γ (∞) esc < 1 (B.34) Γr on the other hand, the minimum is located at

2 ε − ∆ 1 Γesc (∞) = 2 . (B.35) ∆ 2 Γr

Hence here, the minimum becomes only relevant for Γesc (∞) Γr. In such a case we would ﬁnd the condition Γ (∞) D 4 esc N , (B.36) Γtr DS (∞)

This case is however not relevant for the experimental parameters, as there we have Γesc (∞) on a similar order of magnitude as Γr. And even if Γesc (∞) Γr, the backflow of quasiparticles is not important, making a formulation for an effective trapping rate including backflow redundant right from the start.

B.3 Eﬀective trapping rate integrated over energy

2 R ∞ Eventually, we consider the normalised quasiparticle density xqp = ∆ ∆ dεpS (ε). Integrating the diﬀusion equation, Eq. (B.18), over ε, we get

~ 2 x˙ qp = Dqp∇ xqp − a(x, y)δ(z − tS)γeeffxqp (B.37) where the notation is chosen such that the γeeff without energy argument corresponds to the effective trapping rate integrated over energies, i.e., R ∞ ∆ dεγeeff (ε) pS (ε) γeeff = R ∞ . (B.38) ∆ dεpS (ε) Likewise, the 2D diffusion equation for thin superconductors, Eq. (B.24), may be integrated,

2 x˙ qp = Dqp∇~ xqp − a(x, y)Γeffxqp (B.39) where here, x depends only on x and y, and Γ = t γ . Recalling that p (ε) = √ ε f (ε), qp eff S eeff S ε2−∆2 qp we now assume that quasiparticles of high energies relax fast to energies close to the gap, such that fqp = 0 above a certain threshold energy ε = ∆ + dE, where dE ∆. Disregarding the detailed energy dependence of fqp below this threshold, we find q R ∆+dE dε Γr ∆ ∆ Γesc(ε)+Γr ε−∆ Γeff ≈ Γtr . (B.40) R ∆+dE q ∆ ∆ dε ε−∆ 88 Appendix B. Derivation of effective trapping rate

We may easily evaluate this integral, in two limits. For s Γ (∞) dE esc (B.41) Γr ∆ we receive s 1 ΓtrΓr dE Γeﬀ ≈ √ . (B.42) 2 Γesc (∞) ∆ For s Γ (∞) dE esc (B.43) Γr ∆ on the other hand we get

Γeﬀ ≈ Γtr (B.44) that is, in this case, the backﬂow of quasiparticles is irrelevant. Phenomenological equations such as Eq. (B.39) are widely used in the literature [53, 54, 43, 25, 56, 67] as they successfully describe experiments as in Sec. 3.1.4. Appendix C Comparison with vortex trapping

In the coplanar gap capacitor transmon, the antenna pads are the widest part of the device. This makes it possible to trap vortices only in the pads when cooling the device in a small magnetic field. It was shown in [43] that each vortex added to a pad increases the density decay rate, and the effectiveness of trapping by vortices was characterized by a “trapping power” P . Here we compare the vortex trapping with a normal-metal trap covering the pad. To determine the decay rate of the excess density xqp, we construct the solution to the diffusion equation (3.19), along the lines of the Supplementary to Ref. [43]. We treat all parts of the device except the pad as one-dimensional and write xqp in each segment in the form of Eq. (3.46). We approximate the density in the pad as uniform (justified for the lowest mode if Lpad < λeff, as in the actual devices). We then impose continuity of the density and current conservation where the parts of the device meet and thus arrive at the following effective boundary condition for the density at the connection between wire and pad (at y = L):

∂x D k2 − Γ qp 2 qp eﬀ = Lpad xqp(y = L) . (C.1) ∂y y=L WDqp

Let us introduce for simplicity the dimensionless parameter z = kL; after imposition of all boundary conditions, as detailed in [43], we ﬁnd that the parameter must satisfy the equation

2 L Γeﬀ tL pad [1 − f(z) tan z] − z[tan z + f(z)] = 0 , (C.2) LW

2 with tL = L /Dqp and (see Ref. [43])

l W L f(z) = tan z + 2 c tan z c . (C.3) L W L

The similar calculation for the case of N¯ vortices in each pad leads to the following equation for z: P t N¯ L − az2 [1 − f(z) tan z] − z[tan z + f(z)] = 0 (C.4) LW 2 with a = Lpad/(LW ); since in the experiments the latter quantity as well as z for the lowest mode and the coeﬃcient multiplying N¯ are all of order unity, in this equation we can neglect the term proportional to a for large number of vortices. Then, by comparing Eq. (C.2) to Eq. (C.4) we immediately see that for the vortex trapping to generate the same decay rate as the normal-metal trap, the number of vortices in each pad must be equal to:

2 L Γeﬀ N¯ = pad (C.5) P 90 Appendix C. Comparison with vortex trapping

−2 2 −1 Using P = 6.7 × 10 cm s and Lpad = 80 µm [43], the number of vortices in each pad would need to be N¯ ' 230. This shows that many vortices are needed to match the efficiency of the normal metal trap. The cooling magnetic field needed to achieve this vortex number can be estimated to be B ∼ N¯Φ0/Spad ∼ 0.75 G, well into the regime in which the dissipation caused by the vortices negatively affect the qubit coherence [43]. A normal-metal island could also lead to dissipation. However, solving Eq. (C.2) for the parameters specified in Fig. 3.7, we find a density decay rate −1 −1 τw ≈ 1.7ms for a metal-covered pad. From Fig. 3.7 we see that an optimally placed trap of length comparable to l0 can achieve this decay rate, even though the trap area is much smaller than the pad area – thus, optimal placement can potentially limit the losses due to the normal metal. Appendix D Finite-size trap

In this appendix, we treat a 1D system with a finite-size trap to identify the regime in which it can be considered as infinitely small. Moreover, we describe the crossover from infinitely small to finite size trap in the strong trapping regime. We consider the 1D diffusion equation (where the spatial coordinate is 0 ≤ y ≤ L)

2 x˙ qp = Dqp∇ xqp − A (y)Γeﬀxqp. (D.1)

We model the trap as a piece of length d, starting from y = 0, i.e., A (y) = 1 for y ≤ d and 0 otherwise, see Fig. D.1(a). Since no quasiparticle can leave the ends of the 1D wire, we adopt “hard wall” boundary conditions [76]

∂x ∂x qp = qp = 0 . (D.2) ∂y y=0 ∂y y=L The time-dependent solution of this problem can be expressed through the decomposition into eigenmodes, where X µkt xqp (y, t) = e αknk (y) (D.3) k and the eigenmodes fulﬁll

h 2 i µknk (y) = Dqp∇~ − A (y)Γeﬀ nk (y) . (D.4)

The eigenvalue problem can be solved with the Ansatz ( 1 cos kye , y < d nk (y) = √ (D.5) Nk ak cos (ky) + bk sin (ky) , y > d which satisfies the first boundary condition in Eq. (D.2) and we defined q 2 −2 ke = k − λtr , (D.6) with λtr of Eq. (3.21) and Nk is a normalization constant. From this Ansatz it follows that 2 µk = −Dqpk . Continuity of the function nk and its derivative at y = d, together with the second condition in Eq. (D.2), provide an equation for k: k tan [k (L − d)] = −ke tan kde . (D.7)

While the modes thus deﬁned provide the full time-evolution for all times, we are usually interested in the lowest mode which dominates the long-time behavior and which we denote −1 with k0. Assuming k0 λtr , we have ke0 ≈ 1/λtr and we may approximate Eq. (D.7) as

leffk = cot [k (L − d)] , (D.8) 92 Appendix D. Finite-size trap where we defined d leff = λtr coth . (D.9) λtr

In the case d λtr (which implies also d L) Eq. (D.8) becomes λ2 tr k = cot (kL) , (D.10) d which is equivalent to the model where the trap is represented by a delta function, A (y)Γeff → γeffδ (y), where γeff = Γeffd and the trap is located at y = 0, see Ref. [47]. Here, in the strong trap limit, d l0 with l0 of Eq. (3.34), we recover the diffusion-limited lowest mode π k ≈ . (D.11) 0 2L

This solution can be made more general using Eq. (D.8). Namely, even when d & λtr (i.e., the trap is not small) we may identify the regime leff L − d in which π 1 k ≈ . (D.12) 0 2 L − d This expression of course coincides with above diffusion-limited solution for d L, and satisfies the initial assumption k0 1/λtr if λtr L − d. Hence we may for instance increase d from d λtr L to λtr d L, without changing the decay rate, as long as leff L. However, the density of the mode close to the trap changes drastically with increasing d. Indeed, the normalization constant Nk in the limit leff, d L is given by 1 1 2 d Nk ≈ 2 2 sinh (D.13) 2 λtrk λtr and hence, the density at the origin for the lowest mode nk0 becomes

π λtr 1 nk0 (0) ≈ √ (D.14) 2 L sinh d λtr which, for d λtr goes as l n (0) ∼ 0 , (D.15) k0 d

Figure D.1: Simpliﬁed systems considered in (a) AppendixD and (b) AppendixE. Blue/light grey denotes superconducting material and red/dark grey the part covered by the normal metal trap. The junction position marked with an X is at the origin of the wire of length L. Appendix D. Finite-size trap 93

while for d λtr we ﬁnd λtr − d n (0) ∼ e λtr . (D.16) k0 L

As we see, by increasing d above the trapping length scale λtr, the density of quasiparticles gets exponentially suppressed near the trap. For a long trap d λtr, such an exponential suppression takes place also for the steady-state density, as one can verify by reintroducing the generation rate g in the right hand side of Eq. (D.1) and solving for the steady-state conﬁguration with x˙ qp = 0.

Appendix E Quasi-degenerate modes and their observability

In this appendix, we consider the dependence of the lowest mode on the trap position. More- over, we show that close to the optimal position the lowest and second lowest modes are quasi-degenerate. We finally comment on the consequences of this quasi-degeneracy on the observability of the lowest mode. We take for simplicity a small trap (d λtr) in a wire of length L, placed at an arbitrary distance l from the origin; the quasiparticle density then obeys the diffusion equation (see [47] and App.D) 2 x˙ qp = Dqp∇~ xqp − δ (y − l) γeffxqp. (E.1)

To solve this equation, we look for eigenmodes nk that must satisfy at y = l the condition

h + − i lsat ∂ynk (l) − ∂ynk (l) = nk (l) (E.2)

p 2 ± with lsat = Dqptsat, where tsat = Dqp/γeﬀ, and the short-hand notation ∂ynk (l) = ∂ynk|y=l±0+ . The saturation time tsat was introduced in Eq. (3.31) when studying the quasiparticle dynamics during injection and gives the time scale to reach a steady state. Here we use the related lengths scale lsat to have a more compact notation: due to the identity π l l sat = 0 , (E.3) 2 L d this is not an independent parameter in the problem, and the strong (weak) trap condition can be expressed as lsat L (lsat L). Assuming “hard walls” on both ends, ∂ynk(0) = ∂ynk(L) = 0, the eigenmodes are given by ( ak cos (ky) for y < l nk = . (E.4) bk cos (k [L − y]) for y > l

2 These mode decay with a rate 1/τk = Dqpk . From Eq. (E.2) and continuity of nk, we ﬁnd the condition for k lsatk sin (kL) = cos (kl) cos (k [L − l]) . (E.5)

For an inﬁnitely strong trap, lsat → 0, we get the condition

cos (kl) cos (k [L − l]) = 0, (E.6)

π π which provides for the lowest mode either k = 2l or k = 2[L−l] depending on whether L − l ≷ l. Note that the continuity of the modes at y = L1 requires

ak cos (kl) = bk cos (k [L − l]) (E.7) 96 Appendix E. Quasi-degenerate modes and their observability

π π which means that bk = 0 for k = 2l or likewise ak = 0 for k = 2[L−l] . In other words, the trap effectively separates the wire into two independent pieces, one to the left and one to right of the trap, with the quasiparticle density of the lowest mode fully suppressed in the shorter piece. We define the optimal trap position as the one where the lowest mode decay rate is the highest. It is easy to see that this is at the degeneracy point, l = L/2, where the two modes’ decay rates coincide. Note however, that when passing the degeneracy point by increasing l from l < L/2 to l > L/2, the eigenmode function jumps abruptly from being nonzero on the right hand side to nonzero on the left. Therefore, whether the quasiparticle density actually decays with the rate defined by the lowest mode, can be strongly affected by the initial conditions. In order to study this effect, we depart from the ideal, infinitely strong trap, and take a small but finite lsat. In addition, we look at a system close to the degeneracy point, that is, l = L/2 + δl with δl L/2. We first expand Eq. (E.5) for small δl

Lk Lk l k sin (Lk) + sin2 (kδl)2 = cos2 . (E.8) sat 2 2

Next, we set k = π/L + δk and, assuming a strong trap, lsat L, we expand up to second for Lδk 1 to ﬁnd l2 + δl2 2πl 2 4π2 sat = Lδk + sat (E.9) L2 L This results in q  2 2  π lsat lsat + δl δk∓ = 2 − ∓  , (E.10) L L L and we see that the degeneracy at δl = 0 has been lifted by the small parameter lsat/L. From the continuity condition Eq. (E.7), we are able to obtain for each mode the ratio between the (maximal) densities to the left and to the right of the trap √ l2 +δl2 a − δl − lsat ∓ sat k∓ = L L √ L . (E.11) 2 2 bk∓ δl lsat lsat+δl L − L ∓ L In the limit δl l this reduces to sat a k∓ ' ±1 (E.12) bk∓

For lsat δl, on the other hand, we ﬁnd

a 2 |δl|±signδl k∓ ≈ ± . (E.13) bk∓ lsat which is either very large or very small. This means that in this case the two modes have a very strong asymmetry in the relative density left and right of the trap. This asymmetry can aﬀect the measurement of the density decay rate, estimated via a local measurement of the density in time. Let us suppose that we measure the quasiparticle density at y = 0. If δl > 0, the trap is further away from the detection point and thus the slower mode has a high density on the detector side; in this case we simply measure the slowest decay rate. On the contrary, if δl < 0 (with |δl| lsat), the faster mode has most of its density close to the origin and, depending on the time scale on which we measure, we may observe the higher Appendix E. Quasi-degenerate modes and their observability 97 decay rate. Let us suppose we have an initially homogeneous quasiparticle distribution, so that bk− ≈ ak+ . Due to the asymmetry of the two modes, the initial (t = 0) ratio r of the densities of the slowest to the faster mode at the origin y = 0 is a l r (0) ≡ k− ≈ sat 1. (E.14) ak+ 2 |δl| Hence initially, one can observe only the faster mode. As the decay progresses, this ratio eventually shifts in favor of the lowest mode,

|δl| t −D (k2 −k2 )t lsat 8 r (t) = r(0)e qp − + ≈ e L tD . (E.15) 2 |δl|

2 2 where we deﬁned tD = π Dqp/L . The time at which the lowest mode becomes dominant can be estimated by setting r(t) ∼ 1: t L 2 |δl| ∼ ln , (E.16) tD 8 |δl| lsat where the right hand side is 1. Thus, as we see, the time at which we can observe the decay of the lowest mode is much larger than tD.

Appendix F Eﬀective length

F.1 Eﬀective length due to the pad

In the main text we discuss a device consisting of a long quasi-1D wire (length L and width W ) with a square pad (side Lpad) at one end, see Fig 3.6. Here we show that for slow modes, the presence of the pad can be accounted for by the addition of an eﬀective length to the original length of the wire. Indeed, let us assume that the decay time τk of the modes we are interested 2 in are long compared to the diﬀusion time τpad = Lpad/Dqp across the pad, τk τpad. Then we can take the density in the pad to be approximately uniform, and this assumption leads to the following boundary condition [43]

wire WD wire x˙ qp (L) = − 2 ∂yxqp (L) . (F.1) Lpad for the density in the wire at the position where it joins the pad. We now show that this condition leads to a “hard wall” boundary condition for a 1D wire with an eﬀective length which is longer due to the pad. A single mode in a 1D wire is generally of the form

wire nk (y) = ak cos (ky) + bk sin (ky) . (F.2)

Substituting this Ansatz into Eq. (F.1) we ﬁnd

h i h i ak Lke cos (Lk) + sin (Lk) = bk cos (Lk) − Lke sin (Lk) (F.3)

2 where we use the notation Le = Lpad/W . Deﬁning the eﬀective length addition for mode k as

  eﬀ 1 Lke L (k) = arcsin   (F.4) pad k q 1 + Le2k2 we rewrite Eq. (F.3) as

h eff i bk tan L + Lpad (k) k = , (F.5) ak which indeed has the same form of the “hard wall” boundary condition for a wire of length eff eff L + Lpad. For the limiting case Lke 1, we find that Lpad ≈ Le and hence, in this case, the 2 effective total system length is L + Lpad/W . 100 Appendix F. Effective length

F.2 Eﬀective length due to the gap capacitor

Similarly to the last section, we show here that the gap capacitor provides an eﬀective extension of the central wire. For this purpose, we add to one end of the wire of length L, two perpendicular wires, each of length Lc and width Wc, cf. Fig. 3.6 in the main text. Current conservation at the junction between the three wires provides the condition

wire c W ∂yxqp = −2Wc ∂xxqp . (F.6) y=L x=Lc Here, we assumed that the wire (gap capacitor) density is constant in the x-(y-) direction. The eigenmodes of wire and capacitor are of the form

wire nk (y) = ak cos (ky) + bk sin (ky) c nk (x) = ck cos (kx) .

Substituting this Ansatz into Eq. (F.6) and requiring continuity at junction, we ﬁnd the condition h i Wc ak sin (Lk) cos (Lck) + 2 W sin (Lck) cos (Lk) h i Wc = bk cos (Lk) cos (Lck) − 2 W sin (Lck) sin (Lk)

Defining the effective length addition due to the capacitor as   1 2 Wc tan (L k) Leff (k) = arcsin W c , (F.7) c q 2  k Wc 2 1 + 4 W 2 tan (Lck) we find the effective hard wall boundary condition

bk h eﬀ i = tan L + Lc (k) k . (F.8) ak

Note that for 2Wc/W = 1, the capacitor represents simply a direct extension to the wire with eﬀ eﬀ Wc Lc = Lc. For Lck 1 and 2Wc/W 1/(Lck), we may approximate Lc ≈ 2 W Lc. Appendix G Quasiparticle Decay Rate and Steady-State Density

G.1 Slowest Quasiparticle Decay Rate Due To Trap

Here we present the details of the calculations leading to Eqs. (3.48)-(3.49) for slowest quasiparticle decay rate having a single trap on the central wire. We then consider one more example for multiple trap conﬁguration and present the resulting quasiparticle decay rate.

G.1.1 Single trap As we explained in the main text, we take each part of the device to be one-dimensional except the pads where the quasiparticle density is assumed uniform. Over the parts of the device that are not covered by the trap, the diﬀusion equation Eq. (3.19) becomes

∂2 x˙ (t, y) = D x (t, y), (G.1) qp qp ∂y2 qp that has a solution in the form

−t/τw xqp(t, y) = e [α cos ky + β sin ky] . (G.2)

We substitute this solution into Eq. (G.1) and ﬁnd

1 2 2 − = −Dqpk . (G.3) τw Over the region under the trap, the diﬀusion equation becomes ∂2 x˙ (t, y) = D x (t, y) − Γ x (t, y), (G.4) qp qp ∂y2 qp eﬀ qp that has a solution in the form

−t/τw 0 0 xqp(t, y) = e α cosh y/λ + β sinh y/λ (G.5)

Substituting this into Eq. (G.4) gives

1 1 2 − = Dqp( ) − Γeﬀ. (G.6) τw λ Comparing Eq. (G.3) with Eq. (G.6) indicates, 1 Γ k2 + ( )2 = eﬀ . (G.7) λ Dqp 102 Appendix G. Quasiparticle Decay Rate and Steady-State Density

Figure G.1: Model for transmon qubit considered in chapter3. We consider each segment of the device to be one-dimensional. The dots attached to each arrow show our convention for y = 0 in each segment.

We multiply both sides with L2 to find Eq. (3.48) in the main text. To find Eq. (3.49), we solve the diffusion equation for each segment of the device and impose continuity of xqp and quasiparticle current conservation where different segment of the device meet. Figure G.1 shows our device where we labeled different segments and have shown y = 0 of each segment by a dot attached to each arrow. These arrows point the positive direction for each segment. We write,

w w xqp(y) = α cos ky, (G.8) gc gc xqp(y) = α cos ky, (G.9) L1 L1 L1 xqp (y) = α cos ky + β sin ky, (G.10) tr tr tr xqp(y) = α cosh(y/λ) + β sinh(y/λ), (G.11) r r r xqp(y) = α cos ky + β sin ky. (G.12)

gc w Note that the form of xqp(y) and xqp(y) gives zero quasiparticle current at y = 0. This ensures no quasiparticle is leaking out of the device from the gap capacitor and also indicates the device is symmetric with respect to the Josephson junction. We now impose the boundary conditions gc w L1 where xqp(y), xqp(y) and xqp (y) meet,

w gc Ll xqp(y = l) = xqp(y = Lc) = xqp(y = 0), (G.13a) d W d d xw (y = l) + 2 c xgc (y = L ) = xL1 (y = 0), (G.13b) dy qp W dy qp c dy qp

which give,

αL1 αL1 αgc = , αw = , (G.14a) cos kLc cos kl W βL1 = −αL1 [tan kl + 2 c tan kL ], (G.14b) W c Similarly, we have

L1 tr xqp (y = L1) = xqp(y = 0) (G.15a) d d xL1 (y = L ) = xtr (y = 0), (G.15b) dy qp 1 dy qp Appendix G. Quasiparticle Decay Rate and Steady-State Density 103 from which we ﬁnd,

tr L1 α = α cos kL1 1 − (tan kl + 2 tan kLc) tan kL1 , (G.16a) W βtr = −αL1 kλ cos kL tan kL + tan kl + 2 c tan kL . (G.16b) 1 1 W c The next boundary conditions,

tr r xqp(y = d) = xqp(y = 0) (G.17a) d d xtr (y = d) = xr (y = 0), (G.17b) dy qp dy qp give us,

n W αr = αL1 cos kL cosh d/λ 1 − (tan kl + 2 c tan kL ) tan kL 1 W c 1 o − kλ tanh d/λ tan kL1 + tan kl + 2 tan kLc , (G.18a) 1 n βr = αL1 cos kL cosh d/λ 1−(tan kl + 2 tan kL ) tan kL tanh d/λ 1 kλ c 1 o − kλ tan kL1 + tan kl + 2 tan kLc . (G.18b)

As explained in Sec. 3.2.2.1, we assume the density of quasiparticles is constant over the whole p pad and is equal to xqp(y = L − d − L1). Therefore, the current that goes into the pad is equal to the current that leaves the pad (see also supplementary of Ref. [43]),

d d −WD xp (y = L − d − L ) = L2 xpad(t), dy qp 1 pad dt qp 2 2 p = −Lpadk xqp(y = L − d − L1) (G.19) from which we ﬁnd,

"L2 # " L2 # pad k + tan k(L − d − L ) αp = 1 − pad k tan k(L − d − L ) βp. (G.20) W 1 W 1

We now substitute αp and βp from Eqs. (G.18) into Eq. (G.20) and after lengthy but straightforward algebraic manipulations we ﬁnd Eq. (3.49). For the case of two traps on the central wire discussed in Sec. 3.2.2.2 we follow a similar procedure as shown here and ﬁnd Eq. (3.55).

G.1.2 Multiple side traps Inspired by ﬁrst generation, unpublished quasiparticle injection experiments performed at Yale University, here we consider a multiple trap conﬁguration where two traps are attached to the side of the central wire via a bridge, according to panel (a) of Fig. (G.2). In this case, the slowest decay rate is found by solving,

tan z(1 − ε) + g1(z, b) + g2(z, b) + az [1 − tan z(1 − ε1)g1(z, b) − tan z(1 − ε2)g2(z, b)] = 0, (G.21) 104 Appendix G. Quasiparticle Decay Rate and Steady-State Density

(a)

(b)

Figure G.2: (a) Device with two side traps (dark red) in each half of the qubit connected to qubit central wire via a bridge; distances L1 and L2 are measured from the gap capacitor to each bridge. (b) Trapping rate 1/τw as function L1 and L2; here the two traps are identical, d1 = d2 = 10l0, which makes the plot symmetric under the exchange L1 ↔ L2. The parameters used are speciﬁed in the caption to Fig. 3.7 where we also set the lengths of each bridge to zero. where,

l Wc Lc Lb1 d1 tan zε1 + tan z L + 2 W tan z L Wb1 z tan z L − b tanh b L g1(z, b) = h i + L , (G.22) l Wc Lc W d1 b1 1 − tan zε1 tan z L + 2 W tan z L z + b tanh b L tan z L

Lb2 d2 Wb2 z tan z L − b tanh b L 1 − g1(z, b) tan z(ε2 − ε1) g2(z, b) = L , (G.23) W d2 b2 1 − tan z(1 − ε2) tan z(ε2 − ε1) z + b tanh b L tan z L for which here we have deﬁned,

ε1 = L1/L, (G.24)

ε2 = L1/L. (G.25) Appendix G. Quasiparticle Decay Rate and Steady-State Density 105

In panel (b) of Fig. G.2 we show the slowest decay rate for such trap conﬁguration. Here the parameters that are used for the qubit and trap size are the same as were used in multiple traps on central wire, Fig. 3.9. As we expect from our argument presented in Sec. 3.2.2.1, in such multiple side traps the optimum placement is again were one trap is close to the capacitor and the other is close to the pad. However, the resulting decay rate at the optimum trap placement is not as fast as the case were two traps are optimally placed on the central wire. This is because in the latter case, the presence of traps is reducing the uncovered area of the device, while in the former case where traps are attached to the sides of trap, the uncovered are of the device is not reduced.

G.2 Suppression of Quasiparticle Steady-State Density

In this part we present the details of calculations leading to steady-state density of quasiparticles at the Josephson junction, Eq. (3.60). We aim to solve the diﬀusion equation,

∂2 D x (y) − a(y)Γ x (y) + g = 0, (G.26) qp ∂y2 qp eﬀ qp where the function a(y) is unity when y is within the S −N contact region and is zero otherwise. Here we assume the pad is also one dimensional with length Lp and width Wp such that Lp Wp 2 and they satisfy WpLp = Lpad. The solution for each segment then reads,

w g 2 w xqp(y) = − y + β , (G.27a) 2Dqp gc g 2 gc xqp(y) = − y + β , (G.27b) 2Dqp g L1 2 L1 L1 xqp (y) = − y + α y + β , (G.27c) 2Dqp tr tr tr g xqp(y) = α cosh(y/λtr) + β sinh(y/λtr) + , (G.27d) Γeﬀ r g 2 r r xqp(y) = − y + α y + β , (G.27e) 2Dqp pad g 2 g pad xqp (y) = − y + Lpy + β , (G.27f) 2Dqp Dqp

w gc where the form of xqp(y) and xqp(y) ensures no current is leaking out from the device at y = 0 pad and the form of xqp (y) results in zero current at y = Lp. Similar to part where we found the decay rate, we now impose the boundary conditions at the points where different segments cross each other from which we find all coefficients in Eqs. (G.27). From boundary condition given in Eqs. (G.13) we find,

g g L1 w 2 gc w 2 2 β = β − l , β = β − (l − Lc ), (G.28a) 2Dqp 2Dqp g L1 α = − (l + 2Lc), (G.28b) Dqp 106 Appendix G. Quasiparticle Decay Rate and Steady-State Density and Eqs. (G.15) give,

" 2 2 # tr w g L1 + l g α = β − + L1(l + 2Lc) − , (G.29a) Dqp 2 Γeﬀ tr g β = −λtr (L1 + l + Lc). (G.29b) Dqp

From Eqs. (G.32) we ﬁnd,

r tr tr g β = α cosh(d/λtr) + β sinh(d/λtr) + , (G.30a) Γeﬀ r 1 n tr tr o α = α sinh(d/λtr) + β cosh(d/λtr) . (G.30b) λtr At the point where the central wire joints with the pad, the boundary conditions read,

r pad xqp(y = L − L1 − d) = xqp (y = 0) (G.31a) d W d xtr (y = L − L − d) = p xr (y = 0), (G.31b) dy qp 1 W dy qp which result in,

pad g 2 r r β = − (L − d − L1) + α (L − d − L1) + β (G.32a) 2Dqp g r Wp g − (L − d − L1) + α = Lp. (G.32b) Dqp W Dqp

We now substitute Eq. (G.30b) into Eq. (G.32b) and solve the equation for βw, which is the quasiparticle density at the Josephson junction. After some straightforward algebra we arrive at Eq. (3.60). Note that we assumed a one-dimensional geometry for the pad. If we now change the considered aspect ratio from Lp Wp to Lp Wp, the resulting quasiparticle density remains the same and depends only on the pad total area. Therefore, we expect that for a square pad with the considered dimensions, Eq. (3.60) should remain valid. Appendix H Traps in the Xmon geometry

In this Appendix we further explore the role of device geometry by studying the optimal placement of traps in the so-called Xmon qubit of Ref. [17]. We thus consider a four-arm geometry with symmetric arm lengths, see Fig. H.1. Clearly, the optimal position for a single trap is at the center of the device; however, having two or three traps cannot lead to large improvement in the decay rate with respect to one trap, because the diffusion time cannot be shortened in all −1 arms. Therefore, we need at least four traps, one in each arm, to improve τw . A fifth should again be placed at the center, rather than in the arms. In fact, by generalizing the argument 2 given at the beginning of Sec. 3.2.2.2, we find that the decay rate scales as (Ntr/2) if Ntr is 2 multiple of 4, and as [(Ntr + 1)/2] if Ntr = 4n + 1, n = 0, 1,... While in both cases the scaling 2 is less favorable than the Ntr one for a single wire, we see that for a small number of traps the configuration with the additional trap at the center gives a larger increase in the trapping rate. To validate the above considerations, we solve the diffusion equation in the geometry obtained by simply joining four equally long 1D wires of length L. We consider 1, 2, 4 and 4 + 1 traps, all with the same total area, placed symmetrically as depicted in Fig. H.1(a) and (b). −1 We show the resulting decay rate τw in Fig. H.1(c) for a strong trap, obtained by assuming λtr L. Comparing to the single-trap case, we find the expected improvement by a factor of 4 (9) for 4 (4+1) traps. However, in an actual device the length is L ≈ 150 µm, which is not much larger than the estimate λtr ≈ 86.3 µm and gives, using Eq. (3.34), a trap size l0 ' 78 µm for the cross-over from weak to strong (diffusion-limited) trap. Therefore, in practice the traps are in the weak regime and their placement does not affect much the decay rate, see Fig. H.1(d). This points to the need for stronger traps (with shorter λtr) for effective trapping in small devices. Alternatively, one could use traps in the ground plane surrounding the small device, to confine most quasiparticles away from it, see [77] 108 Appendix H. Traps in the Xmon geometry

Figure H.1: (a) The single trap geometry. The trap is at distance L1 from the center, and has size d. The 2 trap conﬁguration follows from this setup, by adding a trap of the same size on the left branch, with the same distance L1 from the center. (b) The 4 + 1 trap geometry. All traps on the individual arms have the same distance L1 from the center. The total trap size is d0 = 4l + 4d. The 4-trap geometry follows from this one by removing the middle cross-like −1 trap. (c) The resulting decay rate τw as a function of L1, for (bottom to top) 1, 2, 4, and 4 + 1 traps. The parameters are (in µm) L = 150 and λtr = 2; the total trap length is 30 in all cases. (d) The decay rate as in (c), but with a realistic value λtr = 86.3 µm for the trapping length. Appendix I Proximity eﬀect in uniform NS bilayers

This Appendix has two parts: we ﬁrst give some details of the calculation leading to the expressions presented in Sec. 5.3.1 for weakly-coupled uniform bilayers. In the second part we extend some of those results to stronger coupling. Our ﬁrs step consist in the changes of variables θi = π/2 + iχi and sinh χS = X in Eqs. (5.7)-(5.8), leading to

τN ε + X sinh χN = √ cosh χN , (I.1) 1 + X2

τS p 2 τS∆ cosh χN = − 1 + X + X. (I.2) τN τN ε Squaring these equations and substituting the second one in the ﬁrst gives

2 τ ∆ ε p 2 S X − 1 + X2 (I.3) τN ε ∆ h 2 2i 2 × 1 − 2XτN ε − τN ε = 1 + X .

In what follow, we approximately solve this equation for X as function of ε; this enables us to ﬁnd the normalized density of states, which in this notation is given by n(ε) = Im(X).

I.1 Weak-coupling limit

Let us consider the weak coupling limit τS∆, τN ∆ 1. In the subgap region ε ∆, the density of states is small, which suggest the assumption X 1 [112]. If we further assume

2 2 |2τN εX| |1 − τN ε | (I.4) the solution to Eq. (I.3) is ε ε X = q + . (I.5) 2 2 ∆ τS∆ 1/τN − ε

This gives the minigap energy at Eg = 1/τN while the DoS above it is given by Eq. (5.12). As the energy approaches 1/τN , however, X becomes large, thus potentially violating the condition (I.4). Indeed, parameterizing the energy as τN ε = 1 + κ (with 0 < κ 1), and using Eq. (I.5), Eq. (I.4) takes the form 1 1 √ + 1. (I.6) 3/2 τS∆ 2κ τN ∆κ Since both terms in the left hand side must be small, we arrive at Eq. (5.13). 110 Appendix I. Proximity eﬀect in uniform NS bilayers

To study the DoS at energies below 1/τN , we must remove the assumption (I.4), while still maintaining X 1. Then, we can expand Eq. (I.3) with respect to X, and keeping terms up to the cubic order we can write that equation in the form F(X, ε) = 0 (I.7) with ! 3 1 4 2 F(X, ε) ≡ 2X + τN ε 1 − 2 2 − X τN ε τN ∆ ! ε 1 1 + 2τN ε 2 2 − 1 + X ∆ τN ε τN ∆ 2 2 ! 1 τN ε − 1 + τN ε 2 2 + 2 2 . (I.8) τS∆ τN ∆ Depending on its coeﬃcient, a third order polynomial can have either three real roots, or one real and two complex conjugate roots. For the DoS not to vanish, we need X to be complex, so the minigap is identiﬁed as the energy at which the type of roots changes from purely real – this happens when the polynomial has a minimum so that the two real roots are degenerate, ∂F(X,ε) giving the condition ∂X |ε=εg = 0. Vanishing of the derivative requires ! 1 1 1 X(εg) = − τN εg 1 − 2 2 − . (I.9) 3 τN εg τN ∆

[one can check that the second solution, X(εg) = ε/∆, leads to the unphysical result εg = 0]. Substituting Eq. (I.9) into Eq. (I.8) and solving for εg at leading order in the small parameters −1 −1 (τN ∆) and (τS∆) , we arrive at Eq. (5.14). To find the DoS above the minigap, we expand X and τN ε around the minigap energy; we take X = X(εg) + δX and ε = εg + δε and expand Eq. (I.8) up to first order in δε and second order in δX (the lower orders vanish by construction): ∂F 1 ∂2F ∂2F F(X, ε) ' (δε) + (δX)2 + (δεδX). (I.10) ∂ε 2 ∂X2 ∂ε∂X If the last term can be neglected, solving F(X, ε) = 0 for δX in terms of δε clearly gives immediately a square root threshold behavior; the coefficients are given explicitly in Eq. (5.15). ∂2F ∂2F 2 The applicability condition can be obtained e.g. by requiring ∂E∂X (δEδX) ∂2X (δX) , which gives

−2/3 τN (ε − εg) (τS∆) . (I.11)

We now consider energies much higher than the minigap, ε εg. In this case we assume 2 2 |1 − 2XτN E| | − τN E | (I.12) and Eq. (I.3) simplifies to − ε + √ X = i . Solving this equation for X we arrive at the ∆ 1+X2 τS ∆ Dynes-like formula given in Eq. (5.17). Note that for ε ∆ the assumption (I.12) is always fulfilled (since X ∼ 1 in this regime); similarly, one can check that for εg ε ∆ the inequality in Eq. (I.12) is satisfied – in fact, it is satisfied so long as |ε/∆ − 1| 1/(τ ∆)2. However, for √ N 2 |ε/∆ − 1| . 1/(τN ∆) the additional condition τS∆ τN ∆ must be met for Eq. (I.12) to hold; calculation of the DoS beyond this regime is outside the scope of the present work. Appendix I. Proximity effect in uniform NS bilayers 111

I.2 Strong-coupling limit

We now consider the case of low resistance at the NS contact interface, such that at least one the two dimensionless coupling parameters τS∆ and τN ∆ is small compared to 1. We start again from Eq. (I.3) and make the assumption

2 2 |2XτN ε + τN ε | 1 . (I.13)

Using this assumption we simplify Eq. (I.3) to − ε + √ X = τN ε ; solving for X, we ﬁnd the ∆ 1+X2 τS ∆ DoS in the BCS-like form   ε n(ε) ' n>s(ε)Re   , (I.14) q 2 2 ε − ε˜gs

τS ∆ where the (approximate) minigap energy in this limit is ε˜gs = ; these results agree with τS +τN those reported in Ref. [112]. Requiring the assumption (I.13) to be valid as ε → ε˜gs, we ﬁnd the conditions

2 τN ε˜gs 1 , ε/ε˜gs − 1 (τN ε˜gs) . (I.15)

The first condition can be rewritten as 1/τN ∆ + 1/τS∆ 1 and it is indeed satisfied under the assumption made at the beginning of this subsection. The second condition indicates that the BCS-like behavior is not valid close to the minigap, similar to the weak-coupling regime. Therefore, to find a more accurate position for the minigap and the behavior of the DoS near it, we take an approach similar to that of the previous subsection. Namely, let us introduce the new variable η = 1/X, and make the assumptions

τN ε η 1 . (I.16)

Then Eq. (I.3) can be rewritten as G(η, ε) = 0 with, keeping only next to leading order terms, ! 3 2 ε ε G(η, ε) ≡ η − τN εη − 2η 1 − + 2τN ε 1 − . (I.17) ε˜gs ∆ Here the quadratic term can be neglected in comparison with the cubic one, see Eq. (I.16). The resulting third order polynomial can be studied following the same procedure as for the weak coupling case. We then find for the minigap 3 2/3 τN 2/3 εgs ' ε˜gs 1 − (τnε˜gs) ( ) , (I.18) 2 τN + τS valid when 1 τS∆ 1 or τN ∆ . 1 ; (I.19) τS∆ this condition follows from the first inequality in Eq. (I.16). To find the density of states just above the minigap, we perform an expansion as in Eq. (I.10) and finally arrive at

r !2/3 s 2 τS + τN ε n(ε) ' nts(ε) ≡ 2 − 1, (I.20) 3 τN εgs εgs 112 Appendix I. Proximity eﬀect in uniform NS bilayers

Figure I.1: Density of states in the superconducting layer in the presence of strong proximity eﬀect, τS∆0 = 0.1 and τN /τS = 0.8. The solid line is calculated by numerically solving the Usadel equation (I.3). Dashed lines: approximate analytical expressions just above the minigap, nts of Eq. (I.20), and at higher energies, n>s of Eq. (I.14). which remains valid so long as

!2/3 ε τ 2 ε − 1 N gs . (I.21) εgs τN + τS

In Fig. I.1 we show the density of states for energies near the minigap for a strongly coupled bilayer, comparing the DoS obtained from the numerical solution of the Usadel equations to our analytical findings. Similarly, in Fig. I.2 we compare numerics and analytics for the minigap energy, with coupling strength ranging from the strong regime (τS∆0 = 0.1) to the weak one 3 (τS∆0 = 10 ). Note that in both these figures we normalize energies with respect to the bulk gap ∆0, whereas analytical expression are given in terms of the self-consistent order parameter ∆; the latter is calculated numerically assuming a low temperature (T/∆0 ' 0.01) and rewriting Eq. (4.81) as a sum over Matsubara frequencies (see also AppendixJ). For reference, we report in Fig. I.3 results of such calculations. We point out that while some of our results simply confirm those in the literature (see e.g. Ref. [112]), a number of them has not been reported before, to the best of our knowledge; we mention here for instance: the more accurate expressions for the minigap energy, Eqs. (5.14) and (I.18), the square root threshold behavior of the DoS above the minigap, Eqs. (5.15) and (I.20), the Dynes-like DoS in Eq. (5.17), and the detailed analysis of their repspective regimes of validity. In concluding this Appendix, we mention that the treatement presented here for the strong proximity effect may become invalid: we have used the Kuprianov-Lukichev boundary conditions, Eq. (4.86), which however are valid only in the limit of low contact transparency [120, 121], T 1 ; at larger transparency, more general conditions should be used, see [122, 121]. A typical few nanometers-thick aluminum oxide insulating barrier has transparency of order T ∼ 10−5. Appendix I. Proximity effect in uniform NS bilayers 113

Figure I.2: Normalized minigap energy ε/∆0 as a function of dimensionless parameter τS∆0. The solid lines are obtained from numerical solutions of Eq. (I.3) with (top to bottom) τN /τS = 0.8, 1, 1.2. Dashed lines: minigap for strong proximity eﬀect, εgs of Eq. (I.18). Dot-dashed lines: minigap for weak proximity eﬀect, εg of Eq. (5.14).

For metallic ﬁlms with thickness in the several tens of nanometers connected by such a barrier, 3 4 we estimate τS∆0 ∼ 10 -10 if aluminum is the superconductor; threfore the present treatment is valid at most down to τS∆0 ∼ 0.1 if the barrier transparency is increased while other typical parameters are kept ﬁxed.

0.8

0.6 0 ∆ /

∆ 0.4

τN /τS = 0.8 0.2 τN /τS = 1 τN /τS = 1.2 0 10-1 100 101 102 103

τS ∆0

Figure I.3: Reduction of order parameter due to proximity eﬀect as function of the dimensionless parameter τS∆0 for (top to bottom) τN /τS = 0.8, 1, 1.2. As τN increases (for example due to increased thickness of normal-metal layer) the order parameter is more strongly suppressed as the proximity eﬀect becomes stronger (that is, τS∆0 decreases).

Appendix J Numerical solution of the self-consistent equation for the order parameter

Here we briefly discuss the numerical approach we use to the calculate the order parameter in a nonuniform NS bilayer as in Fig. 5.2, for which the problem is effectively one-dimensional due to translational invariance in the y direction. We consider a finite-size system, typically extending 10 coherence lengths on each side of the normal metal edge, x/ξ ∈ [−10, 10]. We discretize the x coordinate by specifying a number of mesh points xj (j = 1,...,M), with the mesh denser near the ends (to properly implement boundary conditions) and near the trap edge (where the order parameter is expected to change more rapidly). We solve the self-consistent equation iteratively (cf. Ref. [123]), as we explain below, and the order parameter at points not included in the mesh is obtained by spline interpolation [except for the initial guess ∆(0)(x), which is given by ∆s(x) of Eq. (5.22)]. Denoting with ∆(l) the order parameter after l iterations, we calculate ∆(l+1) as follows: we numerically solve the Usadel equations (5.20) and (5.21), with ∆(x) = ∆(l)(x), for the pairing (l) angle θS (ωk, x); the solution is found directly in the Matsubara representation (i.e., ε → iωk (l+1) with ωk = 2πT (k + 1/2), k = 0, 1, 2,...). Next, we calculate the new order parameter ∆ at the mesh points using the self-consistent equation (4.81):

k h (l) i (l+1) P M ∆ (x ) k=0 Re sin θS (ωk, xj) j = . (J.1) k ∆ ∆0 P M √ 0 k=0 2 2 ∆0+ωk We deﬁne a convergence condition as

1 M ∆N+1(x ,T ) − ∆N (x ,T ) C ≡ X j j < c . (J.2) M ∆N (x ,T ) 0 j=1 j for some small number c0, and we repeat the above steps until this condition is satisfied. In our numerical analysis, we take T/∆0 = 0.01 (corresponding to about 20 mK in Al), keep −5 kM = 2000 Matsubara frequencies in the sums, and set c0 = 10 . The number of iterations needed to reach convergence depends on the initial assumption; using Eq. (5.22) as the starting point, convergence is usually reached within 20 iterations in the regime of weak proximity effect. In panel (a) of Fig. J.1 we have shown the order parameter at our chosen mesh points in a number of iterations while panel (b) shows the convergence, C, at each iteration. In Fig. J.2 we have shown the spatial evolution of the self-consistent order parameter and compared suppression of the order parameter for different values of the coupling strength, τS∆0. 116 Appendix J. Numerical solution of the self-consistent equation for the order parameter

(a)

(b)

Figure J.1: (a) Superconducting order parameter at the mesh points for τS∆0 = 10. We start by the step-like order parameter Eq. (5.22) and run the self-consistent calculation until the convergence condition given by Eq. (J.2) is satisﬁed. In panel (b) we show the obtained value of the convergence at each iteration. Appendix J. Numerical solution of the self-consistent equation for the order parameter 117

Figure J.2: Spatial evolution of self-consistent superconducting order parameter with (from 2 1 −1 top to bottom) τS∆0 = 10 , 10 , 1, 10 . The dashed lines show the value of self-consistent order parameter for a uniform NS bilayer. As the coupling strength increases, approximating the order parameter with the step-function given in Eq. (5.22) becomes less accurate, and one needs to relay on numerical self-consistent calculations to ﬁnd the spatial evolution of density of states, see Fig. K.3.

Appendix K Spatial evolution of single-particle density of states and pair amplitude

Here we outline the derivation of Eqs. (5.30)-(5.33), starting from the deﬁnitions for n and p in Eqs. (4.84) and (4.85), respectively. According to Eq. (5.28), in the uncovered section of the superconductor the pairing angle θL is the sum of the BCS angle θBCS and a correction. Assuming the correction to be small, |θBCS − θSu| θBCS, the corrections to n and p are then √ 1 x 2α (ε) δn(ε, x) ' Re sin θ [θ (ε) − θ (ε)] e ξ 1 (K.1) 2 BCS BCS Su and √ 1 x 2α (ε) δp(ε, x) ' Im cos θ [θ (ε) − θ (ε)] e ξ 1 (K.2) 2 BCS Su BCS

At most energies (except near the gap and the minigap, see Appendix I.1), we have θSu ' θDy of Eq. (5.18). Using from now on this approximation, we continue by noting the identity

∆0 (ε + i/τS) − ε∆NS tan (θBCS − θDy) = i (K.3) ε (ε + i/τS) − ∆0∆NS where we used Eq. (5.9) for θBCS. According to Eq. (5.19), at leading order we can approximate ∆NS ' ∆0 − 1/τS, and assuming |ε − ∆0| 1/τS we can simplify the right hand side of the above equation and linearize its left hand side to arrive at

2 1 − i ∆0 θBCS − θDy ' 2 2 (K.4) τS∆0 ∆0 − ε Substituting this expression into Eqs. (K.1) and (K.2), we find Eqs. (5.30)-(5.33), where the leading BCS contributions are also included. In Fig. K.1 we show the density of states near the trap edge, x/ξ = −1, obtained in three ways: 1. from the numerical solution of the Usadel and self-consistent equations, Eqs. (5.20), (5.21), and (4.81); 2. using the semi-analytical expression in Eq. (5.28) in which θSu(ε) is calculated numerically; 3. plotting the analytical formulas in Eqs. (5.30) and (5.32). In their regions of validity, the semi-analytical and analytical results are in good agreement with the numerical findings. To complement Fig. (5.4), we show in Fig. (K.2) the spatial evolution of pair amplitude and compare the results obtained from numerical solution of Usadel equation with the semi-analytic expression. We note that when the coupling strength in increased, the analytic expressions Eqs. (5.30)-(5.33) becomes less accurate and one has to relay on numerical simulation. In Fig. (K.2) we show numerical findings for the spatial evolution of single particle density of states for some higher vales of the coupling. 120 Appendix K. Spatial evolution of single-particle density of states and pair amplitude

2 Figure K.1: Density of states near the trap edge, x/ξ = −1, for τS∆0 = 10 and τN /τS = 0.8. Solid line (blue): self-consistent numerical approximation; dashed (red): semi-analytical approach; dot-dashed (black): analytical formulas (see text for more details). The insets (a) and (b) zoom into the minigap and gap regions, respectively.

Figure K.2: Pair amplitude for τS∆0 = 100 at various distances from the trap edge. The blue lines give the result of self-consistent numerical solution while the dashed red lines show the semi-analytic solution as explained in chapter5. Appendix K. Spatial evolution of single-particle density of states and pair amplitude 121

Figure K.3: Numerical self-consistent solution for single-particle density of states for a few strengths of the proximity eﬀect at various distances from the trap edge. The parameters used are τS∆0 = 0.1 (solid line), τS∆0 = 1 (dashed line) and τS∆0 = 10 (dashed-dot line) and we set τN /τS = 0.8.

Appendix L Spectral function in the presence of a trap

Here we present in some detail the derivation of the formulas for the spectral functions reported in Sec. 5.4. As in that Section, we distinguish between the thermal equilibrium case and the non-equilibrium one, which accounts for the suppression of the quasiparticle density by the trap. In both cases, our starting points are Eqs. (5.3)-(5.6), with the appropriate expressions for the density of states n, pair amplitude p, and distribution function f.

L.0.1 Thermal equilibrium

We assume a thermal equilibrium distribution function, Eq. (5.34), at temperature εg T ω ∆0 (i.e., “cold” quasiparticles). For the tunneling spectral density St, Eq. (5.4), we split the integral in three integration regions: from the minigap εg to ∆0 − ω, from ∆0 − ω to ∆0, and from ∆0 to inﬁnity. These three region correspond to the bb, ba, aa types of transitions described in Sec. 5.4. Let us consider ﬁrst the highest energy integration region, for which we have approximately

Z ∞ 2 eq 16EJ ε (ε + ω) + ∆0 −ε/T Saa(ω) ' dε q q e . (L.1) π∆0 ∆0 2 2 2 2 ε − ∆0 (ε + ω) − ∆0

The approximations employed here are two: ﬁrst, in the function A of Eq. (5.6), we keep only the leading terms for n and p in Eqs. (5.32) and (5.33); second, for the distribution functions we can neglect f eq(ε + ω) in comparison to unity and approximate f eq(ε) ' e−ε/T . It is then eq eq easy to check that Saa takes the same form as in Eq. (2.45) with the substitution xqp → xqp, thus proving Eq. (5.37). In the intermediate integration region, the above approximations for the distribution functions are still valid, and for n(ε + ω) and p(ε + ω) we can again just keep the leading terms in Eqs. (5.32) and (5.33). On the other hand, for n(ε) and p(ε) we must now use Eqs. (5.30) and (5.31), and therefore we have

1/4 √ |x| 2 Z ∆0 − 2 1− ε eq 16EJ 1 ξ ∆2 Sba (ω) ' dε e 0 π∆0 τS∆0 ∆0−ω 3 (L.2) ∆0 (2ε + ω) −ε/T q e . 2 23/2 2 2 ∆0 − ε (ε + ω) − ∆0 Here we have included a factor of 2 due to the presence of two identical traps symmetrically placed with respect to the trap, as discussed at the beginning of Sec. 5.4; since we are considering small, linear-order corrections [cf. text above Eq. (5.28)], we can simply add the contributions 124 Appendix L. Spectral function in the presence of a trap from the two traps. We note that formally the integral is divergent at the upper integration limit; however, this is due to the break-down of the used approximation for n and p, which is valid for ∆0 − ε 1/τS. Closer to the gap, both n and p are in fact ﬁnite (see Sec. 5.3.2), and we can neglect the contribution from this small integration region near the gap, as it is exponentially −ε/T suppressed due to the e factor. Then, making the change of variables ε = ∆0 − ω +ε ˜, and neglecting corrections small in ω/∆0 and T/ω, we ﬁnd

√ 1/4 16E 1 − 2 |x| 2ω eq J ξ ∆0 −∆0/T ω/T Sba (ω) ' e e e π∆0 τS∆0 (L.3) Z ∆2 dε˜ 0√ e−ε/T˜ . 0 2ω3/2 ε˜ Performing the integration, we ﬁnally arrive at Eq. (5.38). In the lowest integration region we use Eqs. (5.30) and (5.31) to write

1 √ |x| 2 4 Z ∆0−ω − 2 1− ε eq 16EJ 1 ξ ∆2 eq 0 Sbb (ω) ' 2 dε f (ε) e π∆0 (τS∆0) εg 1 (L.4) √ h 2 i 4 − 2 |x| 1− (ε+ω) 4 2 ξ ∆2 ∆0 ε (ε + ω) + ∆0 e 0 , 2 23/2 2 23/2 ∆0 − ε ∆0 − (ε + ω) where again we have taken into account the presence of two identical traps [while use of Eqs. (5.30) and (5.31) is strictly speaking not justified near the minigap, the error thus introduced is small, see AppendixK]. Note that here we can still neglect f(ε + ω) compared to unity, due to the assumption ω T , but we cannot make approximations for f(ε), for which we must use the full equilibrium expression, Eq. (5.34). Still, the distribution function forces ε to be small compared to ∆0, so that neglecting terms small in T/∆0 and ω/∆0 we have √ 16E 1 |x| Z eq J −2 2 ξ eq Sbb (ω) ' 2 e dε f (ε), (L.5) π∆0 (τS∆0) 0 where we used the assumption T εg to set the lower integration limit to 0. After integration, we obtain Eq. (5.39). In contrast to quasiparticle tunneling, the pair process contribution to the spectral function, Sp of Eq. (5.5), is non-vanishing only it there is a finite subgap density of states (so long as ω < 2∆0). In fact, the integration limits in Eq. (5.5) are εg and ω − εg, since the function A(ε, ω − ε) vanishes outside this region; this further requires ω > 2εg in order for Sp to be non-zero. Since ε < ω ∆0, we find the approximate expression √ 16E 1 |x| J −2 2 ξ A(ε, ω − ε) ' 2 e , (L.6) π∆0 (τS∆0) which we obtain after substituting Eqs. (5.30) and (5.31) into Eq. (5.6), accounting for two traps, and neglecting terms small in ω/∆0. Using this expression in Eq. (5.5) we find √ 8E 1 |x| eq J −2 2 ξ Sp (ω) ' 2 e π∆0 (τS∆0) Z ω (L.7) dε [1 − f eq(ε)] [1 − f eq(ω − ε)] . 0 Appendix L. Spectral function in the presence of a trap 125

Figure L.1: Relaxation rate Γ10,p due to Cooper pair breaking. Here the temperature is set to zero and the trap is placed next to the junction, x/ξ = 0; other parameters are as in Fig. 5.5. The fast vanishing of the relaxation rate at τS∆0 ∼ 10 is due to the violation of condition in Eq. (L.8).

For T ω, the integral in the second line gives ω − 2T ln 2, thus proving Eq. (5.40). We have remarked above that the condition

ω10 > 2εg (L.8) is necessary for Sp to be finite, and we have shown in Fig. I.2 that the magnitude of the minigap energy has a non-monotonic behaviour with respect to the parameter τS∆0 whose inverse quantifies the strength of the proximity effect. Indeed, as τS∆0 decreases, the spectral 2 function Sp and hence its contribution to the relaxation rate increase as 1/(τS∆0) . However, because of the increase in the minigap with decreasing τS∆0, the condition in Eq. (L.8) can potentially be violated; in this case qubit relaxation due to pair processes is no longer possible. This is evident in Fig. L.1, which shows the relaxation rate Γ10,p due to Cooper-pair breaking as a function of τS∆0; the rate is calculated by setting T = 0, in which case the spectral function is √ 8E 1 |x| eq J −2 2 ξ Sp (ω) ' 2 e (ω − 2εg) θ (ω − 2εg) , (L.9) π∆0 (τS∆0) and using this expression in Eq. (5.1). We note that if τS∆0 is decreased further, the decay rate will become finite again (cf. Fig. I.2), but for such small values of τS∆0 Eq. (5.40) is not applicable, since it was derived under the assumption τS∆0 1.

L.0.2 Suppressed quasiparticle density We now consider the case in which we account for the suppression of the quasiparticle density by the trap using the distribution function of the form in Eq. (5.42), with the position-dependent chemical potential µ˜ given by Eq. (5.43). As in the previous subsection, we assume two identical, symmetrically placed traps (without writing this assumption explicitly anymore for brevity) 126 Appendix L. Spectral function in the presence of a trap

and εg T˜ ω ∆0, and to evaluate the tunneling spectral density St we again split the integration region into three intervals: [εg, ∆0 − ω], [∆0 − ω, ∆0], and [∆0, ∞], leading to the contributions S˜bb, S˜ba, and Sãa, respectively. For Sãa and S˜ba we can perform the calculation as described above for the thermal equilibrium case, although additional constraints are needed to justify the approximations that involve the additional parameter µ˜: for Sãa we need (∆0 − ˜ ˜ h ˜i µ˜)/T & 1, while for Sba the somewhat more restrictive condition exp (∆0 − µ˜ − ω) /T 1 is required. When these conditions are met, we find that the spectral densities have the same form as in thermal equilibrium, see Eqs. (5.44) and (5.45). The calculation of S˜bb is more involved, as different parameter regimes must be distinguished. eq The simplest case is that of µ˜ . ω; then we can proceed as in the above derivation of Sbb , the only difference being that we have to explicitly keep both f(ε) and f(ε + ω), without approximations. This way we find that the expression for S˜bb is obtained by the replacement T → ˜ µ/˜ T˜ (˜µ−ω)/T˜ eq T ln[(1 + e )/(1 + e )]/ ln 2 in the formula Eq. (5.39) for Sbb (note that the replacement simplifies to T → T˜ for µ˜ T˜, as could be expected). When µ˜ ω (which in practice means that µ˜ is comparable to, albeit smaller than, ∆0), the approximate calculation of S˜bb is different: because of the combination of distribution functions, the main contribution to the integral comes from the region between µ˜−ω and µ˜. In fact, for T˜ → 0 the combination reduces to θ(ε−µ˜+ω)θ(˜µ−ε), and this form gives the leading contribution in the low-temperature regime so long as (∆0 − µ˜ − ω)/T˜ 1. Using this step function approximation for the distribution functions, in the resulting finite integration region almost all the other factors in the integral 2 2 −3/2 −3/2 −3/2 are approximately constant, except for (∆0 − (ε + ω) ) ' (∆0 +µ ˜) (∆0 − ε − ω) . Integrating the last factor, we finally arrive at

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