Theoretical Proposals of Quantum Phase-slip Devices

OF

Q

UANTUM

P

HASE

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SLIP

D

EVICES

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op 26 oktober 2012 om 12:30 uur

door

Alina Mariana HRISCU

master of science in nanoscience geboren te Boekarest, Roemenië

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof. Dr. Y. V. Nazarov Technische Universiteit Delft, promotor Prof. Dr. W. Guichard Institut Neel CNRS/UJF Grenoble Prof. Dr. Y. Pashkin RIKEN, Japan / University of Lancaster Prof. Dr. J. M. van Ruitenbeek Universiteit Leiden

Prof. Dr. Ir. H. T. C. Stoof Universiteit Utrecht

Prof. Dr. Ir. T. M. Klapwijk Technische Universiteit Delft Prof. Dr. J. E. Mooij Technische Universiteit Delft

Prof. Dr. H. W. Zandbergen Technische Universiteit Delft, reservelid

Printed by: Ipskamp Drukkers, Enschede Front & back cover: Design by Andreea Hriscu

Copyright © 2012 by A. M. Hriscu

Casimir PhD Series, Delft-Leiden 2012–25 ISBN 978-94-61-91-44-39

An electronic version of this dissertation is available at http://repository.tudelft.nl/

P

REFACE

My fascination for physics began actually with what is known as the "opposite" of quantum mechanics: (the special) theory of relativity. I couldn’t reconcile them either. So when during my undergraduate I found out how amazing phenomena occur in materials and devices lying at the tips of our fingers (as opposed to distant, fast moving extra-terrestrial objects), I re-directed my attention towards solid state physics and quantum mechanics.

Still, after my Master, I wanted to move away from solid state physics: in my experience, it involved lengthy, abstract calculations which seemed fun to do, but it was not always easy to see the connection with the measurable quantities. I wanted to move closer towards something that people actually measure in the labs. And that’s why I came to Delft. And that’s exactly what I got. You get to see the ef-fects of quantum mechanics in an experimental graph. You get to compute Hamil-tonians that describe billions of electrons, while still describing quantum effects. And that is simply exciting.

Yuli, thank you for giving me this opportunity. Ever since the day we met at my interview a bit over 4 years ago, I have never stopped to be surprised by you. First, it is about science: your insight into so many topics and the ability to explain every-thing with a few sentences and then write it down on half a page is amazing. One cannot stop learning from you. Your perfectionism is sometimes misunderstood by the uninitiate, but one learns to truly appreciate it. Personally, I learned from you about the constant strive for self-improvement. Thank you for all your pa-tience during all those long meetings, spiced up by your superb self-deprecating humor. Your resourcefulness is manifested in surprising ways: for example, you started a "world-famous" weblog, full of "funny (and even more insightful) things", with apparently thousands of views. Or maybe it is the same people checking it many times?

Hans, thank you very much for the encouragement and constant support. You are the prototype of a "Homo universalis", which is sometimes intimidating: is there something you do not know about or cannot do? Your enthusiasm about physics is contagious, and it is nice to see that can you almost go jumping up and down with excitement about it. My only regret is that I didn’t benefit enough from everything that you know. Teun, thank you for the keen interest in my work and for our short collaboration. Eduard, from you I learned how an LC oscillator "looks"

like on an actual sample. I hope you have a good time with highly disordered su-perconductors in Grenoble!

There are also other people that have contributed to this thesis, and most of them have done so indirectly. Let me thank to them, too.

The theory group is hosted on a silent corridor. The people are busy in their offices, bent over their books and their computers. Until there is the coffee break time, talks (formal or informal) or lunch. Gerrit’s laughter is audible from a long distance, but too bad that not that long as from Japan. Before I met you, I didn’t know that one can actually invent new research fields or that one can travel so much and still get things done. Yaroslav, I appreciate your sharpness both in physics talks and beyond them. Thanks a lot for switching on the lights in my office at 7 p.m. during winter evenings and for giving me kind and good advices. Miriam, you bring in positive energy all the time. Thanks for being extremely helpful at any point. It is beyond my comprehension how you can find the resources to man-age all your different roles. Jos, at first glance, you seem a distinctive presence that gives the feeling that one cannot be too polite around. That is maybe also because of your British accent and manners. It only lasts until you yourself say something very funny and open, and then it becomes clear: one deals with a sophisticated Dutchman. All our chats on the corridors have been lots of fun.

The PhD students, post-docs and Bachelor/Master students keep the third floor lively and relaxed. Marnix, when you first offered to help us move, I was very im-pressed. Soon I found out that you did the same with half of the group! But I was still very grateful to be on the "right" half. Stefan, I hope that you will have great times in Munich, despite it being in Bavaria and being full of BMW’s, and even worse, of beer. I enjoyed a lot your sarcastic jokes during the coffee breaks. Gio, from a rather silent group member with his own shifted schedule, you turn into the super-fun party guy at outings! I appreciate a lot your kindness, your story telling skills and your Georgian spirit (as it is true about Keti). Mireia, your energy and enthusiasm about everything you believe in light up the group. Fateme J., your kindness and friendliness makes the group a better place. Yanting, it always nice to talk to you! Hujun, thanks a lot for your help with booking Chinese accommo-dations! Marcin, we gave so much constructive feedback on each other. I really enjoyed our meetings, as well as with Cecile (to whom I would like to thank for the constant supply of delicious cakes). Akash: you built for yourself a reputation as a party-guy, and you are willing to involve everyone in it. Berlinson, you always are kind and friendly. Dima, you are showing up at irregular time periods, but it is always a pleasure to have a chat with you.

Toni, you went back to your beloved island, but not without leaving a very nice impression behind! Jiang, your jokes are one of my first memories from the group. Hungduo, I wish you all the best for your career. Rakesh, your interest and

knowl-edge have added so much to the group, as did your good mood and your party-organizing. You even connected us to art, via Nicole. François, you seemed rather quiet, but only until one started to talk to you: thanks for being so friendly and helpful. Peng, I enjoyed our funny talks about Beijing.

Many Master and Bachelor students chose our group. I should start with the most faithful member of this team: Chris D.. You are the closest I can think of a perpetual student. It seems like you grew attached to our group and constantly feed it with all your jokes and ideas. Tungky, Joost, Ferdinand, Rutger, Gerwin, Bas, Sasja, Robin, Rianne, Suzanne, Adriaan, it was nice to have you all around. Julia, we’ve crossed many times: in the theory group, at my class, and then in our group of friends. Best for your PhD in QT, and see you soon! Our group seems to be a pretty popular visiting spot for scientists from all over the world. Thanks to An-tonio, Matti, Andre, Tomohiro, Ramin and Hediyeh for your short but nice stays. Alireza, before I met you I didn’t know you can talk about poetry with a physicist at the department lunch break.

During the past four years, I had many office mates. Xuhui, you were one of my first offices mate in Delft, only for two weeks. You are the most eccentric, funny and stylish Chinese national that I have ever met (with Vera being the same for Taiwanese). Jeroen, we have already met at my interview. I was surprised by your sharp and out-of-the-box points of view, all spiced up with good humor. It was nice to visit you in Berlin. See you at your next "surprise" visit. Omer, you’re always in a good mood, offering me and Frans candies no matter how many times we would say no, and the biggest fan (and member) of the "Evil team". Kim, thanks for all the nice moments spent together, both in the office and outside. Mariya, I truly got to know you after I read your "second-PhD thesis", your "Blue hyacinths". Probably you wouldn’t believe it, if I were to say to you, what it meant to me. Rodrigo, you were in our office during your Master student times before you got "promoted" to PhD. I love your Latino enthusiasm. Frans, thanks for listening to all my com-plaints about Dutch "things" and the Dutch weather, all the comparisons with the other countries, etc., etc., in your very composed, calm but intensely humorous way. I am very grateful for your translating my summary into "samenvatting" (with double "t"), as well as of my propositions into "stellingen".

Mihajlo, you have the gift to explain the most involved calculations in a simple way, without making one feel stupid. I think you will have a very successful ca-reer. I am deeply indebted to you for our discussions which helped me understand parts of my work better. And for helping me to write the "microscopic" part of the introduction, even via Skype! Our Dutch city trips with Milica were great fun!

During my last year I supervised Jochen during his Bachelor project. I learned a lot during our discussions and I appreciate your enthusiasm and your polite man-ners.

Fatemeh M., "rara avis"! You are one the most honest, sweetest and kindest people I have ever met. I think you are also one of the best PhD students on the third floor. And, when you smile, the sun starts to shine. Thanks to you and Ali for your hospitality! Chris V., we have shared so many things that I can say our friendship is truly complex.

I have never understood the disappearance of Moosa. I cannot even imagine how his family felt and must feel to today.

Thanks to Yvonne, Marjolein and last but not least, Erika, for keeping the group running and doing so many things that we are so bad at!

Thanks to the ex-flux-qubit team for the patience to initiate me (and I was slow) into the Delft way of thinking about physics: Kees, Peter, Arkady, Tomoko, Pol, Thomas. Pol, special thanks to you and Roser: he disfrutado de vuestra hospi-talidad catalána y gracias por hablar español conmigo.

I would like to express my gratitude to Konstantin (Likharev), for writing my favorite book on Josephson junctions that helped me to understand what Yuli was talking about.

Leo K., Leo dC., Lan, Floris, Hannes, Wolfgang, Guen, Julia, Amelia, Toeno, Reinier, Vincent, Han, Maria, Katja, Sergey, Pierre, Vlad, Lucio, Stijn, thanks for including me to so many QT werkbespreking-s, cakes and parties. You guys work and party with equal seriousness, brains and passion.

Vlad (and Ioana), your perspectives on things are always very enlightening, and your hospitality and humor are so enjoyable! We should meet again, along with Pierre and Isa! Sergey, despite you trying to be a super-cool guy (and managing a bit), I realized that you are really caring about the people around you. I hope your brand-new-custom-made lab is better than any gadget an experimentalist can dream about. You are one of those people that, once they are gone, I miss them a lot! Katja, I really got to know you very far away from Delft: in Beijing. That was great. You have been visiting quite a bit since you left for Stanford, and I enjoyed it a lot!

I am extremely lucky to have some truly inspiring friends around me that give me a lot of care, ideas and good moments. Moira, I love your creativity with words and drawings, and I am sure you are will have a great scientific career. I miss you a lot since you left Delft, and I hope to see you and Simon soon again! Lucio, I am happy that your Mexican spirit is tempered by some German precision: I hope that makes our future meetings actually happen! You are one the smartest peo-ple I know and I am thrilled to meet you every time. Y por supuesto, amigos de mi hermana son mis amigos también, o es al revés? Stijn, you followed me from Groningen to Delft for your PhD (or was it the other way around?). I appreciate your commitment to your friends and your superb sense of humor. You were at the core of organizing our first ski-trip and, of course, the legendary Sicily trip.

Lan, Floris, Hannes, Maksym, Salvatore, Lucio, Basia, we had a lot of fun on these trips!

Outside of the applied physics building in Delft, thanks to my friends for all the good times we spent together: Johanna, Jason, (and Gus!), Michel, Sanna, Vika, Omar, Laura, Wouter, Anna, Rajeev, Sidd, Laia, Aarabi, Prem, Sruti, Serena, Daan, Julianne, Sarah, Freek, Deniz, Koenrad, Sebastiano, Giulia, Roos, Paweł. I’m sorry I’ve missed some of the many parties, weddings, dinners, visits... But the ones I made it to were lots of fun! I hope we will catch up! Roos, I somehow still have the feeling that I will meet you next week. Paweł, thanks for many visits that were topped up with (Polish) jokes and wits. Special thanks to Johanna and Jon for proof-reading parts of my thesis, hunting for English mistakes.

It is always nice keep in touch with the old friends from Groningen. Jasper, I truly appreciate your friendship. Wish all the best for you, in the Netherlands or China, along with Xufei, Jasmijn and Camille. Thomas, we spent some great moments in Groningen. I know I should have visited more: Anja, you, and Niels! Ghazaleh, we reunited in The Hague, but only to not meet often enough! I appre-ciate a lot your care for the people around you and your kindness. Olga, we’ve met in quite some places after our times together in Groningen: Windsor, Delft, Berlin, London. I am fascinated by your ideas, drive and strength and I hope to see you soon again! Siebren, thank you for being one of my best friends for a few years. That was truly amazing.

Joelle, David, Marloes, Barbara, Victor we’ve made it happen together: our the-ater play! You are all wonderful people. I hope to meet you every time we need a "zzapp"! Lucie, you have done the impossible to get this play staged. Your inven-tiveness and lack of selfishness are fascinating. Stephanie and Sergio, you are the most crazy and fun support staff!

Andrei, you are one the smartest and wittiest guys I know. Ît,i voi fi ves,nic

re-cunosc˘atoare pentru anii petrecut,i împreun˘a. Uneori râd s,i acum când îmi

am-intesc de glumele tale.

Thank you to Sally, Peter, Ina and Sandra for taking me so much into their fam-ily and life, trips and parties despite my failure to speak German (or Hungarian)! You’ve made my Swiss visits (and not only) truly memorable! Sally and Peter, I am very honored that you can join me for this day. Dankeschön & köszönöm szépen!

The close friends are the family you chose. I am lucky to have a large such family: Janos, Ciprian, Wio, Basia, Oana, Alina and Andreea.

Janos, you are such an intense and resourceful person that one can never get bored with you (and we had plenty of occasions, luckily!). Maybe only a little tired. With you I could share so many events, movies, books, thoughts, etc. Please keep on visiting! Ciprian, your selflessness is amazing. Mult,umesc pentru o prietenie

multe ori, dar ca fericirea lui s˘a nu depind˘a de asta. Poate mai invat,˘a s,i alte jocuri?

Personal heroes are imaginary or real human beings that you aim to be like, but in most cases you can never manage. My heroes are very real and very close by: Wioletta Ruszel and Andreea Hriscu. They both inspire me how far you can get on all the aspects of life, no matter the hardships or the starting point, and always remain attentive to their true selves but also to the people around them.

Wio, you are the most energetic and modest person I know. I cannot thank you enough for our friendship. I am always amazed at your creativity, care, wits, ideas, etc. You are a fantastic, supportive friend, and a great mother. And, by the way, thank you for being my paranimph! Andreea, you are the best sister I could have ever had and a true example in so many ways. Thanks for designing me such a great cover! Leg˘atura noastr˘a nu poate fi descris˘a în cuvinte. As, spune doar c˘a

uneori îmi dau lacrimile de fericire doar când m˘a gândesc la tine!

Wio, Cristian and Elisa, you are the closest I know to a perfect family. Cristian, I love your Italian sense of humor (and cooking), and your out-of-the-box ideas in general. But I won’t forgive you that you did not join us for the Dylan concert! Elisa, it was thrilling to see you through all the phases: from an infant to the first steps and growing up. You really make me feel like an aunt.

Basia, we’ve met at a very important moment in my life and I had the oppor-tunity to learn a lot from you. You live life to the full and make the best out of it: science, friendships, style, partying. Thank you for being a truly wonderful friend, who sometimes knows what I need without me even uttering a word! I am privi-leged to have also meet your nice family and also to have you as my paranimph.

Oana, we are the closest to "Dutch" style of friendship. We have met when we were 7, but we are still discovering each other in some sense! Pe noi ne leag˘a atât,ia

ani s,i evenimente. Am evoluat împreun˘a pentru o bun˘a parte din drumul viet,ii.

Sper s˘a putem sa dep˘as,i momentele de impas s,i ca pe viitor s˘a continu˘am la fel.

Alina, we do not only have the same name. We are alike in so many things. Ne-am redescoperit s,i reinventat pentru a nu s,tiu câta oar˘a. Uneori parc˘a suntem

interlegate s,i avem nevoie de foarte put,ine cuvinte ca s˘a ne int,elegem.

My parents: they have taught me the important things in life and have sup-ported my choices. Mult,umesc ca m-at,i înv˘at,at lucrurile importante în viat,˘a s,i ca

m-at,i sustinut în alegerile f˘acute. M˘a bucur ca at,i putut s˘a fit,i alaturi de mine în

ziua aceasta.

Fabian, I stole you from the group for a while, but then you left for good. Thank you for being so caring and loving ever since. We rediscover each other in so many ways, and this adventure will continue!

Alina Hriscu Delft, September 2012.

C

ONTENTS

1 Introduction 1

1.1 Superconductivity . . . 2

1.2 Josephson effects and Josephson junction . . . 4

1.2.1 D.C. Josephson effect . . . 4

1.2.2 The current biased junction . . . 7

1.2.3 Josephson and Bloch oscillations . . . 9

1.3 Superconducting devices with Josephson junctions . . . 13

1.3.1 Charge-phase duality and macroscopic quantum mechanics . 16 1.3.2 Josephson junction devices with Coulomb blockade . . . 17

1.3.3 Cooper-pair box and transistor . . . 18

1.4 Phase-slips in superconducting wires . . . 23

1.4.1 Fluctuations of the Ginzburg-Landau order parameter: ther-mal vs. quantum . . . 24

1.4.2 Microscopic description: quantum fluctuations in a supercon-ducting wire . . . 28

1.4.3 Macroscopic description: Duality to Josephson junction . . . . 32

1.5 Outline of this thesis . . . 36

References . . . 37

2 Phase-slip oscillator: a way to few-photon non-linearities 41 2.1 Introduction . . . 42

2.2 Description of the setup . . . 44

2.3 First order corrections to the energy levels . . . 45

2.4 Perturbation theory for density matrix . . . 46

2.5 Estimation of the relative magnitude of the first-order correction . . . 49

2.6 Semiclassical analysis . . . 51

2.7 Numerical solution of the density matrix: hysteresis . . . 52

2.8 Quantum solution: Pure states . . . 54

2.9 Effect of the offset charges on the device . . . 55

2.10 Conclusions . . . 57

References . . . 57

3 Coulomb Blockade due to Quantum Phase-Slips Illustrated with Devices 61

3.1 Introduction . . . 62

3.2 Description of the devices . . . 67

3.2.1 CPB versus QPS-box . . . 68

3.2.2 CPT versus QPS-transistor . . . 70

3.3 QPS-box . . . 72

3.3.1 First-order corrections . . . 72

3.3.2 Small impedance regime . . . 73

3.3.3 Large impedance regime . . . 79

3.3.4 Intermediate impedance regime and summary . . . 80

3.4 QPS-transistor . . . 82

3.4.1 Second-order corrections . . . 82

3.4.2 Large ES . . . 87

3.4.3 Degeneracies in a symmetric QPS-transistor . . . 90

3.4.4 Flux sensitivity . . . 91

3.5 Dual devices . . . 93

3.6 Conclusions . . . 95

References . . . 95

4 Quantum synchronization and resistance quantization in superconduct-ing devices 99 4.1 Introduction . . . 100

4.2 Description of the setup . . . 101

4.3 Classical equations . . . 102

4.4 Classical results: synchronization . . . 104

4.5 Qualitative estimation of the parameters . . . 104

4.6 Quantum fluctuations in the device . . . 106

4.7 Derivation of the simplified action . . . 109

4.7.1 Simplified action for Shapiro steps . . . 109

4.7.2 Derivation of slow-variable action . . . 111

4.7.3 Action for the Bloch part . . . 114

4.7.4 Coupling the parts . . . 115

4.7.5 Reduction the action to a single variable . . . 116

4.8 Numerical illustrations . . . 118

4.8.1 Main synchronization domain n = m = 1 . . . 119

4.8.2 Domains n = 1, m 6= 1 or m = 1, m 6= 1 . . . 120

4.8.3 Domains n 6= 1,m 6= 1 . . . 121

4.9 Conclusions . . . 122

Summary 125

Samenvatting 129

Rezumat 133

Curriculum Vitæ 137

1

I

NTRODUCTION

On the 8th of April 1911, driven by curiosity and enabled by his then-brand-new liquid helium fridge, Heike Kamerlingh Onnes gradually cooled solid mercury to unprecedented low temperatures. What he observed is common knowledge nowa-days, 101 years later: the electrical resistance of the sample suddenly dropped to an unmeasurable low value once the temperature reached 4.2 K [1]. Undoubtedly, this is the most dramatic property of a superconductor: to carry electrical current with no apparent dissipation. Measurements of the fastest decay rate of persistent currents in sufficiently thick superconducting rings under appropriate conditions indicate that the current would die out after 3 × 1092_{years [2]. In other words, the}

current would keep on running through the ring for lengths of time longer than the age of the universe, without considerable losses. In human time scales, that counts as "forever"!

However, this property is modified in sufficiently thin wires (of the order of tens of nanometers), films or by applying magnetic fields stronger than the crit-ical field. Non-zero resistance even well below the critcrit-ical temperature has been observed [3–8]. Ever since, many experimental and theoretical efforts have been dedicated to understanding how this resistivity arises and how superconductivity breaks down in various circumstances. The subject of this thesis is to investigate novel circuits that embed ultrathin superconducting nanowires where supercon-ductivity partially breaks down due to quantum fluctuations.

In this chapter we discuss key phenomena related to macroscopic quantum mechanics which are instrumental for the follow-up of this thesis. In the next

{ {

section we explain the existence of supercurrent in between two superconduc-tors separated by a thin non-superconducting layer- the d.c. Josephson effects and correspondingly, the Josephson junction. Section 1.2.2 introduces the current-biased Josephson junction circuit known as resistively capacitively shunted junc-tion (RCSJ). Then we discuss the a.c. Josephson effect. We exemplify some of the most common devices based on the Josephson effects in Section 1.3: the SQUID’s, voltage standard and the Cooper pair box. Then we move on to discuss the cen-tral topic of this thesis: phase-slips in very thin superconducting wires. In Section 1.4.2 we provide the microscopic foundations for the phenomenological descrip-tion that we use throughout the coming chapters, and in Secdescrip-tion 1.4.3 we explain the duality between the quantum phase-slip wires and Josephson junctions. Fi-nally, we present the outline of the remainder of the thesis.

1.1 S

UPERCONDUCTIVITY

Let us first recall how superconductivity arises. The resistance of a metal like Hg, Al, In, etc., has a non-zero value at a temperature just above the critical temper-ature, but upon lowering the temperature below this special value, the resistance simply drops to zero. So what has happened? It is the very same piece of metal and the very same electrons that carry the current. Only 46 years after Onnes’ ini-tial measurement, Bardeen, Cooper and Schrieffer have proposed the microscopic theory (BCS) to satisfactory explain superconductivity [9]. They theorized that the essential ingredient that explains superconductivity is that electrons are attracted to each other. This counterintuitive attractive interaction originates from the crys-tal lattice’ deformations, or phonons. Owing to this attraction the electron gas de-velops an instability at sufficiently low temperatures. This instability is understood as a tendency for electrons with opposite spins and momenta to form pairs, so called Cooper-pairs. At the instability threshold it becomes energetically favorable to form the Cooper pairs. However, the Cooper pairs do not emerge as single sep-arate objects. Rather, a macroscopic number of these pairs occupy the same state, forming the superconducting condensate.

Even before BCS theory was developed, Ginzburg and Landau proposed a phe-nomenological theory of superconductivity [10] that is successful at defining and explaining several important parameters such as coherence length, the penetra-tion depth and the condensapenetra-tion energy. Later on, Gor’kov [11] proved that the GL theory can be derived from BCS formalism. In this chapter we use the ingre-dients of the GL approach to discuss a few very important phenomena related to superconductivity, like Josephson effects, phase-slips, etc.

Ginzburg and Landau interpreted the occurrence of superconductivity as a phase transition: for temperatures below the critical temperature TCthe

supercon-{ { ducting condensate is more ordered than the normal state and it becomes more

energetically favorable. The superconducting state is described by a complex "or-der parameter"

ψ(r) =pn(r)eiφ(r), (1.1) where |ψ|2_{= n(r) is the density of the "superconducting electrons" that have}

con-densed in the superconducting state and φ is a phase factor that is position de-pendent. This "guess" was not too far off: the "superconducting electrons" were revealed by BCS theory to be pairs of electrons "glued" together by phonons.

It is remarkable that the phase enters the expression Eq. 1.1 of the "macro-scopic" order parameter. The quantum mechanical wavefunction of a single par-ticle includes a phase-factor that can take any value along the trigonometric cir-cle. Usually, we deal with a large ensemble of such particles, e.g., the electrons in a normal metal. The macroscopic quantities are obtained by summing over all the particles and so the phase information is lost. In contrast to this situation, in the superconducting condensate the Cooper pairs’ wavefunctions overlap to a large extent since the size of the Cooper pairs’ ξ0≈ 10−4 cm is much larger than

the interatomic distance ≈ 10−7_{cm. Consequently, all the pairs at a certain point}

are "phase-locked": they share the same phase-factor. While far from offering the complete picture, these arguments elucidate why the quantum phase enters the expression Eq. 1.1. It is this phase φ of the condensed pairs that is used to explain many phenomena that involve superconductivity: the Josephson effects, (quan-tum) phase-slips, persistent currents, Meissner effect, etc.

The quantity ψ is not the actual wavefunction of the whole system, but rather a phenomenological complex number that describes the collective state of a macro-scopic number of Cooper pairs. For example, its modulus squared equals the den-sity of Cooper pairs in the condensed state, and not a probability denden-sity in the quantum mechanical sense. However, for many practical purposes, the macro-scopic order parameter plays the role of the effective wavefunction of the super-conductor. An illustrative example is discussed in Section 1.2.1, where ψ is em-ployed for explaining the Josephson effect.

Usually, we associate quantum mechanics with small particles: electrons in an atom, nuclear constituents, etc., and imply that quantum effects can be seen only indirectly in the properties of macroscopic matter. For the description of macro-scopic objects usually one relies on the laws of classical physics. Superconductivity contradicts this intuition: it is essentially a quantum phenomenon, but noticeable at macroscopic scale without any amplifying, magnifying devices. As an exam-ple, we can "see" the effect of quantum mechanics by levitating a piece of magnet above a superconductor. While this is indeed observed at the macroscopic scale, it is still indirect, in the sense that the bulk piece of magnet and the superconductor

{ {

are classical objects. However, in the 1980’s Voss and Webb [12], and, respectively, Jackel et al. [13] combined superconducting Josephson junctions with Coulomb blockade into devices that, as a whole, behave according to quantum mechanical laws (see Section 1.3.3). The understanding that objects much larger than atoms (e.g., samples that have an electron density of 1029_electrons/m3_{) could have}

mea-surable quantum properties was such a major breakthrough that it has given rise to a new research field: macroscopic quantum mechanics. However, the usage of the term refers as well to the first type of phenomena, indirectly observable and not only to the second type, the "macroscopic quantum objects".

1.2 JOSEPHSON EFFECTS AND

JOSEPHSON JUNCTION

The Josephson effect is one of the most important examples of macroscopic quan-tum phenomena. In this section we discuss the so-called dc Josephson effect, the equations that govern the current biased Josephson junction (RCSJ circuit).

In 1962 the Englishman Brian D. Josephson, then a student at Cambridge, pre-dicted that a supercurrent will flow between two superconductors separated by a thin insulating layer [14] (now called a Josephson junction). He also pointed out that the current can flow even in the absence of electrical voltage drop across the junction, so it is a supercurrent. Josephson argued that the phenomenon arises due to tunneling of Cooper pairs, so then the macroscopic current depends on the quantum phase – making it one of the most illustrative examples of macro-scopic quantum phenomena. He also derived very unusual equations that relate the voltage across the junction and the phase. The experimental confirmation of these predictions came only one year later, in 1963 [15] and ever since, Joseph-son junctions have been continuously an object of study. JosephJoseph-son’s discovery contributed greatly not only to the understanding of superconductivity, but also to quantum physics as a whole. In recognition, Josephson was awarded the Nobel Prize in Physics in 1973. On the practical side, Josephson junctions lead to sev-eral new devices that have extraordinary characteristics, presently used in physics and chemistry, astronomy, biology and medicine, metrology, geophysics and mi-croelectronics [16].

1.2.1 D.C. J

Let us first discuss the phenomenology of the Josephson effect and briefly derive the Josephson equations relating the supercurrent, voltage and phase across the junction. We follow the simplified but illustrative approach presented by Feynman in his famous Lectures [17].

Consider two superconductors, labeled 1 and 2, separated by a barrier (see Fig. 1.1). In the actual fabrication, the separating layer can be realized by using

ei-{ { 2 superconductor 1 superconductor 2

n

√

ψ

=

e

iχ

ψ

=

√

n

e

iχ

FIGURE1.1: Scheme of a tunnel Josephson junction. Two pieces of superconductor 1 and 2 are sep-arated by an insulating region. If the two superconducting leads have different phases, χ1and χ2

re-spectively, a supercurrent is established through the junction. This supercurrent arises in the absence of voltage and it is proportional to the phase difference φ ≡ χ2− χ1.

ther an insulating region or a weak link [18]. If the barrier is insulating, then no electrical current can pass through it and the transport mechanism is tunneling of Cooper pairs. By contrast, if the interlayer is a weak link (that is, a conducting material, either a normal metal or another superconductor with lower critical cur-rent) a finite supercurrent can flow through junction due to the proximity effect. Josephson’s initial predictions were made for tunnel junctions, and in this section we shall also focus on those.

Each superconducting lead is described by a complex order parameter,

ψk=pnkeiχk, k = 1,2, (1.2)

where χ1,χ2are the condensates phases and n1,n2the densities of Cooper pairs

(then the density of "superconducting electrons" are 2n1, 2n2.).

Let us assume that there is an applied voltage V between the two superconduc-tors. If this potential is zero in the middle of the barrier, then the electrode 1 is at the potential V1= −1₂V with the energy −2eV1≡ eV , while the electrode 2 is at

po-tential V2=1₂V with the corresponding potential for Cooper pairs −2eV2≡ −eV . If

the barrier is thick enough such that the superconductors are separated from each other, we can write the time-dependent Schrödinger equation for each side:

i ħdψ1

d t = eV ψ1+ kψ2 (1.3)

i ħdψ2

d t = −eV ψ2+ kψ1, (1.4)

where ψ1,2are given by Eqs. 1.2 and act as the wavefunctions of the condensates.

The constant k is a characteristic of the junction. That is to say, the insulating bar-rier couples the equations for the two sides (if k were zero these equations would describe just the energy states in the two uncoupled pieces of superconductor).

{ {

Let’s substitute the "wavefunctions" Eq. 1.2 in the coupled wave equations 1.3, 1.4 and define the phase difference across the junction,

φ ≡ χ2− χ1.

By separating into the real and imaginary parts, we obtain equations for the time dependence of charge densities of the phases:

ħdn1 d t = 2k p n1n2sin φ, (1.5) ħdn2 d t = −2k p n1n2sin φ. (1.6) dφ d t = 2e ħV, (1.7)

where we get the last expression by subtracting the equations involving dχ1/d t,

dχ2/d t. The current passing through the junction dQ1/d t ≡ 2eV dn1/d t (V being

the volume of the junction) is readily obtained from Eqs. 1.5, 1.6:

IJ=dQ1

d t = − dQ2

d t ⇒ IJ= ICsin φ. (1.8) Hereby we defined the critical current IC, representing, in the simplest

approx-imation, the maximum current that can be carried through the junction,

IC=

4eK ħ

p n1n2,

where we incorporate in K ≡ kV the geometrical details of the junction. Above ICthe phase coherence of the Cooper pairs breaks down. This critical current

de-pends on the geometry of the junction, density of Cooper pairs in the two super-conductors, the coupling constant K and on fundamental constants.

The Josephson energy is equivalent to the the critical current up to fundamen-tal constants,

EJ≡ ħ

2eIC, (1.9)

We will use the critical current and the Josephson energy interchangeably through-out this thesis, adopting one or the other according to convenience.

Equations 1.8, 1.7 are called the Josephson relations and represent the semi-classical description of the Josephson junctions. Let us appreciate the extraordi-nary characteristics of the Josephson junction implied by these formulas. The first

{ { Josephson relation, Eq. 1.8, shows that the supercurrent flowing through the

junc-tion is not a funcjunc-tion of the voltage across the juncjunc-tion, but only on the phase difference φ. This expression can be also read the other way around: a current produces a phase difference across the junction. If the applied d.c. current does not exceed IC in absolute value, −IC≤ I ≤ IC then it results in a constant

phase-difference φ and by virtue of Eq. 1.7, V = 0. Therefore, if the current is not large, there will be no voltage drop across the junction and in this static case there is no energy dissipation in the junction.

It we apply a voltage across the junction, Eq. 1.7, so-called second Josephson relation, indicates that the phase is not constant in time. In Section 1.2.3 we detail on this situation.

1.2.2 T

Let us consider the superconducting circuit depicted in Fig. 1.2 that is of particu-lar relevance for the present thesis. It comprises of a d.c. current biased Josephson junction, shunted by an external resistor and a capacitor (resistively and capaci-tively shunted junction - RCSJ circuit). The bias current gets divided between the elements,

I = IJ+ IR+ IC ap.

For the sake of completeness, let us mention that the Josephson junction itself might have intrinsic resistance and capacitance. For this reason such a circuit is known in the literature as a "model" of the physical Josephson junction (as op-posed to ideal junction, that has only the IJcurrent component). By design, the

two electrodes of a tunneling Josephson junction form the plates of a capacitor. The value of the intrinsic capacitance Ci depends on the junction’s fabrication

details. However, this capacitance adds to the external capacitance, so that C ≡ Ci+ Cext, and can be ignored if Ci ≪ Cext. On the other hand, the intrinsic

resis-tance originates due to the presence of some non-zero density of "normal" elec-trons (these are usually called "quasiparticles", as their properties are somewhat different than those of normal electrons due to the presence of the superconduct-ing condensate). This "normal" current component is considered to originate in the thermal fluctuations [19]. However, this fact does not undermine the general-ity of the our discussion. We concentrate on the low temperature limit where no such quasiparticles are present and we can ignore the intrinsic resistance of the junction and R is contributed by the external resistor only.

With these considerations, we readily find expressions for IR, IC ap in terms of

the phase drop across the junction and subsequently write the differential equa-tion for the phase. A non-zero voltage V across the Josephson juncequa-tion has impor-tant implications that we will discuss in the next section. For now, we only write all

{ {

I

R

E

C

FIGURE1.2: The current-biased Josephson junction shunted by external elements: a resistor R and a capacitor C.

U

U

U

I=0

I<Ic

I>Ic

a.

b.

Φ

c.

Φ

Φ

FIGURE1.3: Washboard potential corresponding to: a. zero bias current, b. small bias current I < IC and c. large bias current. In the semiclasical regime, a fictitious "phase" particle is either trapped in one of the minima (a. and b.) or sliding with friction down the potential (c.) at large I .

the current components in this circuit. The current through the capacitor is repre-sented in the usual form, IC ap= CdV_{d t}. The dissipation current through the resistor

is given by Ohm’s law, IR= V /R. Then the total current I is given by

I = ICsin φ +V

R+C dV

d t ,

and it equals the external bias current. By using the voltage-phase relation 1.7, V = ħ ˙φ/2e, we obtain a differential equation for the dynamics of the phase

I = ICsin φ + ħ 2eR dφ d t + ħC 2e d2φ d t2. (1.10)

The Josephson term and the external current I combine to form an effective potential energy, the so-called washboard potential,

{ { We can visualize the physics by making use of a classical analogue: consider a

par-ticle moving in this potential with φ being the coordinate. In the absence of bias current, I = 0 the potential is only the cosφ term and the particle is trapped in one of the minima of depth equal to the Josephson energy EJ≡_2eħIC. By turning on

the bias current, but with I < IC, the potential acquires a linear component, it gets

tilted. However, the minima are still present and the particle stays trapped in one of them (see Fig. 1.3b.). If the bias current exceeds the critical current, I > ICthen

the potential profile has no more minima and the particle starts to slide along the potential profile (see Fig. 1.3c. and the next subsection).

We use this model extensively throughout the forthcoming sections and chap-ters: in the next section for describing the Josephson oscillations; by making use of the duality between the Josephson junction and phase-slip junction, we present the equivalent "RCSJ circuit" for the voltage biased phase-slip junction. Lastly, in Chapter 4 we employ the current biased junction shunted by an external resistor to achieve well-developed Josephson oscillations. To illustrate the phase dynam-ics in this simple model we have neglected the thermal fluctuations (or noise). In Chapter 4 we account for the fluctuations in the spirit of fluctuation-dissipation theorem.

1.2.3 J

B

In this section we analyze the above circuit involving the Josephson junction and the external resistor and capacitor shunts in the limit of I > IC. When the input

current exceeds the critical current of the junction, the current through the junc-tion cannot be carried by the supercurrent alone, but is divided between the resis-tor and capaciresis-tor according to Eq. 1.10 (see also Fig. 1.2). Then the phenomenon known as Josephson oscillations, or the a.c. Josephson effect occur in the circuit. The phenomenology is as follows: Since a non-zero current passes through the re-sistor, the transport is now dissipative and implicitly, there is an average voltage drop V across the junction. Averaging of Eq. 1.7 shows that in the resistive state there are Josephson oscillations of frequency:

ωJ≡ ˙φ =

2e

ħV . (1.12)

Let us stress that the Josephson oscillations emerge when the average voltage V across the junction is different from zero. However, if the voltage drop across the junction is non-zero, the critical current IC also depends on V . We assume small

enough voltages compared to the superconducting gap, eV ≪ ∆0, at which this

dependence can be disregarded. Note that there are two ways of obtaining Joseph-son oscillations: first, by applying a current larger than ICor simply by biasing the

{ {

j=I/I

FIGURE1.4: Josephson oscillations (from [19]). The central plot represents the (dimensionless) I − V curve, while the side panels indicate the corresponding Josephson oscillations of the phase and the voltage across the Josephson junction.

The energy diagram Fig. 1.3 offers a visual picture of the Josephson oscillations: at I > ICthe system slides down the potential profile, bumping into each hump of

the "washboard". These bumps are essentially the Josephson oscillation [19]. The superconducting phase acquires time dynamics, and consequently, so do the volt-age over the junction and the superconducting current. In principle, the solution for φ is found by solving Eq. 1.10. Analytical solutions of this second order dif-ferential equations are only found by neglecting the fluctuations and for specific limits of the parameters C,R.

Of great interest for us is the limit of C ≪ ħ

2eICR2, when we can ignore the

sec-ond time derivative of the phase in Eq. 1.10. In terms of Eq. 1.10 this implies a small "mass" ∝ C or high "friction" ∝ 1/R and, for I > IC, the particle slowly

slides down the potential profile, getting trapped in the local minima. In this case [19] the equation for phase Eq. 1.10 simplifies tremendously,

ħ 2eRIC

˙

φ + sinφ = j, (1.13)

where we have divided Eq. 1.10 by IC, j = I/IC. The equation Eq. 1.13 has exact

solutions for the phase,

φ(t ) = 2arctan à p j2− 1 j + 1 tan ωJt 2 ! −π₂ (1.14)

{ { 0 1 2 3 4 1 2 3 4 I/I C V/RIC V=υΦ ₀ V=2υΦ 0

FIGURE1.5: Shapiro steps at constant voltage are formed when the circuit in Fig. 1.2 is exposed to irradiation with frequency ν. (from [26])

The time derivative of this expression leads to the voltage that is also an oscillating function; its amplitude and frequency depend on j . The d.c. I − V curve is non-hysteretic and is given by the dependence of the average voltage V on the input current,

V = ICR

q

j2− 1, for |j | > 1

to have a hyperbolic shape. The dynamics of all variables depends strongly on the bias point in the curve, i.e., on the values of I and V . Several points and the corresponding behavior of the phase and voltage are illustrated in Fig. 1.4.

The d.c. characteristic represents the voltage averaged out over the rapid Joseph-son oscillations. To illustrate the JosephJoseph-son oscillations in the I −V one has to ir-radiate the samples with microwaves of frequency ν. The external radiation gets phase-locked to the Josephson oscillations and this will alter the I −V curves in a very characteristic manner: constant voltage regions appear at every value of volt-age for which

Vn= nνΦ0,

where n is an integer ±1,±2,±3,... and Φ0= πħe is the magnetic flux quantum.

This means that in the I − V curve there are current intervals–"plateaus"–where the bias current is modified but the average voltage stays constant (as illustrated in Fig. 1.5). These plateaus are known as "Shapiro steps" after their discoverer [20] and they are used in the modern voltage standards [21], as we will briefly discuss in Section 1.3.

Bloch oscillations

Bloch oscillations were described by Bloch and Zener [22] for electrons in a crystal lattice. They represent oscillations of an electron in a periodic potential (back-ground charge) subject to a constant force (electric field). The phenomenon of

{ {

Bloch oscillations is widely associated with physical systems where charge or "quasi-charge" in a periodic potential is subject to a constant drive. Averin, Likharev and Zorin have predicted [23] that very small Josephson junctions (EC≫ EJ)

em-bedded in a highly resistive environment R ≫ RQ (with RQ = h/4e2≈ 6.45kΩ,

the resistance quantum) show classical behavior of the charge (actually, quasi-charge) instead of the phase that in this conditions has quantum dynamics. Con-sequently, by feeding a d.c. current I we should get the quantum dual of Josephson oscillations: Bloch oscillations of the voltage across the junction with a frequency I /2e. The dual of Shapiro steps can be realized by applying a r.f. voltage of fre-quency f , thus leading to plateaus at constant current values of I = ±2e f ,±4e f ,.... Such steps would enable the immediate technical application of Bloch oscillations: the current standard. Realization of such a device is a long-standing purpose in metrology but it requires flat, well-defined constant current plateaus.

Kuzmin and Haviland [24] have observed signatures of Bloch oscillations in the form of two peaks at I = ±2e f in the d.c. differential resistance dV /dI versus in-put current I . However, in this measurement the I-V curves do not exhibit steps with flat central parts. Despite considerable efforts [25], such steps have not yet been observed. This limitation originates from the requirements on the environ-mental resistance. On one hand, the occurrence of well-defined Bloch oscillations requires high values of R ≫ RQ. But on the other hand, such high resistances

pro-duce thermal heating and high noise levels, and those "tilt" the current plateaus. Bloch oscillations have been predicted to occur in alternative superconducting devices embedding phase-slip wires [26]. Such superconducting nanowires exhibit coherent quantum phase-slips (section 1.4), the dual of Josephson tunneling (sec-tion 1.4.3).

Let us briefly touch upon the similarities and differences between Bloch oscil-lations in small Josephson junctions and coherent quantum phase-slip wires. In a superconductor, the phase and charge are quantum conjugated variables (dis-cussed further in Section 1.3.1). Conceptually, quantum fluctuations of the phase variable are at the origin of Bloch oscillations in both circuits. They both require a highly-dissipative environment. However, there is a more subtle difference be-tween the fluctuating quantities. In Josephson junctions with high capacitive en-ergy EC≫ EJ, the phase-difference over the junction is not well-defined while its

dual, the "quasicharge" at the junction becomes localized. However, the supercon-ductivity in the leads of the junction is not affected. In the case of superconducting nanowires, the fluctuations of the superconducting order parameter are at the ori-gin of the coherent quantum phase-slips (further discussed in Section 1.4). From a topological perspective, coherent quantum phase-slips are non-trivial fluctua-tions that result in a transition between two configurafluctua-tions of the complex order parameter that differ by the phase winding number. In other words, while the wire

{ { retains its superconductivity as a whole, the superconducting state is affected by

the phase-slips.

In fact, Bloch oscillations in voltage biased phase-slip wires are the exact dual of the Josephson oscillations in current biased Josephson junctions (discussed in Section 1.2.3). One expects equivalent I − V characteristics (Fig. 1.4) and dual of Shapiro steps (Fig. 1.5) upon interchanging the roles of voltage and current. In Chapter 4 we propose a new device that further exploits such a Bloch oscillator by coupling it with its dual, the Josephson oscillator 1.2.3.

1.3 SUPERCONDUCTING DEVICES WITH

JOSEPHSON

JUNC-TIONS

The applications of Josephson junctions came very quickly after Josephson’s initial predictions. Superconducting quantum interference devices (abreviated SQUID-s) are used to measure magnetic fields with extremely high sensitivity and are used in virtually every signal measurement that has a magnetic component, from oceanog-raphy to measuring the magnetic fields in living organisms [16]. SQUID-s combine the phenomenon of flux quantization in a superconducting loop and the Joseph-son effect. SQUID-s come in two types, direct current, d.c., and radio frequency, r.f.

D.C. SQUID

In 1964, the Ford Research Labs’ team [27] have developed what is now called the d.c. SQUID. It consists of a superconducting loop interrupted by two Josephson junctions in parallel, as indicated in Fig. 1.6. The input d.c. current I feeding the loop is split equally between the two junctions, I /2. Let us first assume zero exter-nal magnetic field. Then the number of flux quanta in the SQUID loop is, unsur-prisingly enough, equal to zero. By applying an external magnetic flux Φ a current Ir= Φ/L (with L being the self-inductance of the loop) will be running through the

loop that compensates for the non-zero flux, trying to keep the number of quanta equal to zero. This current adds to I /2 passing through one of the junctions, but subtracts from the other. Effectively, it decreases the critical current of the SQUID from 2ICto 2IC− 2Ir. The screening current changes direction every time the

ap-plied flux reaches Φ0/2 and thus the critical current oscillates with the applied flux.

By increasing the current above the value I > 2IC− 2Ir, we will go into the resistive

state of the other junction. Suppose we increase the external magnetic flux; when we reach the value Φ0/2, it is more favorable for the loop to increase the flux from

zero to one quanta. Then the screening current changes sign, and the process goes on.

{ { V I_r I 2 I 2 I a. XΦ ext I C V I S C I S' 2I C ΔV _=(n+1/2)Φ 0 2I -2I C r 0 ΔV Φ₀ Φ/ V Φ₀ 1 2 0 b. c. C I I C Φ nΦ0 Φ= Φ

FIGURE1.6: D.C.SQUID. a. The electrical scheme. A superconducting loop is interrupted by two Josephson junctions in parallel. The loop is biased by a d.c. current I and the voltage drop V over the loop is monitored. b. The I −V characteristic depends on the flux threading through the loop and it lies in between the two thick lines. The upper curve is reached at integer values of flux quanta, while at half integer Φ0the critical current reaches its minimum and the I −V follows the lower curve. c.V

ver-sus Φ/Φ0at constant bias current. The voltage changes periodically as the flux in the loop is increased,

with period Φ0. (after Ref. [28])

The measurable quantity is the oscillating voltage over the loop as a function of flux; the period of oscillation corresponds to an extra flux quantum Φ0added in the

loop (see Fig. 1.6). By monitoring this voltage dependence one can measure the applied flux. In case the flux to be measured is constant, then we simply measure the critical current deviation from the value at no field. Since the screening current is related to the applied flux, it is straightforward to get

Φ= L³I_CS− ICS′

´ /2, where IS

C= 2IC is the critical current of the SQUID with no field and IS

′

C is the

criti-cal current of the SQUID with non-zero external flux. R.F. SQUID

The r.f. SQUID has only one Josephson junction interrupting a superconducting loop. The ring is inductively coupled to an LC oscillator that is excited by a radio-frequency current (see Fig. 1.7a). The amplitude of the voltage across the reso-nant circuit is periodic in the applied flux with period Φ0, enabling the detection

of changes in flux down to 10−5_Φ0.

The device is based on the a.c. Josephson oscillations, hence it operates in the resistive mode of the Josephson junction. In a nutshell, it works as follows: an external flux applied changes the phase drop over the junction, according to the relation:

φ = 2πΦ Φ0

This relation reflects flux quantization and it is valid for any superconducting loop, also in the absence of the junction. However, with the junction present the current

{ { ~ a. L C Ir.f. X _ext 0 1 2 3 4 -1 -2 -3 -4 Φ /Φ 0 ext Φ/ Φ0₀ 1 2 -1 -2 λ=2π =3L Ic Φ₀ λ=0.3 Φ Φ b. C I s Ls

FIGURE1.7: R.F. SQUID. a. The electrical scheme. A Josephson junction interrupts a superconducting loop with self-inductance LS. An LC circuit excited by the radio-frequency current Ir.fis inductively

coupled to the loop. b. The flux Φ in the loop as a function of the external flux Φext, both in units of

Φ0. For the parameter λ = 0.3 the curve is single valued and it shows only weak modulation with the

external flux. If λ > 1 (e.g., λ = 3) the curve is hysteretic and the flux in the loop has multiple values for a fixed Φext. In this case, the SQUID switches between states with different Φ. (after Ref. [28])

through the loop is periodic in the flux,

I = ICsin φ ≡ ICsin µ 2πΦ Φ0 ¶ .

The total flux in the loop Φ equals to the applied flux Φextminus the flux

result-ing from the current I runnresult-ing through the loop,

Φ= Φext− LSICsin µ 2πΦ Φ0 ¶ ,

where LSis the self inductance of the loop. This equation leads to two distinct

be-haviors, according to the value of the parameter λ = 2πLIC/Φ0. Fig. 1.7.b shows

the two regimes: first, if λ < 1 then the flux in the loop is single valued and it has only a small modulation around the external flux. Secondly, for λ > 1 the Φ − Φext

curve becomes hysteretic. Usually the r.f. SQUID is operated in the latter regime. In the hysteretic mode for a value of the external flux there are multiple solutions for flux Φ in the loop and hence the SQUID makes transitions between these quan-tum solutions. The energy dissipated during the jumps is periodic in Φext. By the

inductive coupling, the periodic dissipation modulates the quality factor Q of the LC circuit, so when driven on resonance with a current of constant amplitude, the r.f. voltage is periodic in Φ.

{ {

We discuss in Chapter 2 the dual of this circuit for coherent quantum phase-slip wires– the phase phase-slip oscillator. This dual is periodic in external charge (in-duced by the gate) and the hysteretic regime is reached when ES> EC(equivalent

to λ ∝ EJ/EL> 1) (for extensive details see Chapter 2).

Voltage standard

The modern voltage standard is also based on the a.c. Josephson effect (see sec-tion 1.2.3), [21]. The typical a.c. Josephson circuit is irradiated with microwaves of frequency ν and this signal becomes phase-locked with the Josephson oscillations of the average current. Consequently, the d.c. I −V curve of the junction exhibits voltage steps,

Vn= nνΦ0,

with n being an integer (see Fig. 1.5). This relation is uniquely determined by the applied frequency and the fundamental constant Φ0. In other words, the

Joseph-son junction acts as a frequency-to-dc voltage converter. Depending on the irradi-ation frequency, a single junction produces a voltage ranging from a few microvolts to a few millivolts at most. A properly designed and fabricated array of junctions can be phase-locked to produce produce a series of very accurate quantized volt-ages of desired value.

Upon successful practical realization of the circuit proposed in Chapter 4, a current standard can be built. This would be based on the current steps in the I −V curve of a device that couples a Bloch oscillator to a Josephson oscillator.

Perhaps the most studied of all setups involving Josephson junctions are the mesoscopic devices that highlight the interplay between superconductivity and Coulomb interactions [29–33]. Understanding how Coulomb repulsion affects su-perconductivity in nanostructures has been of fundamental interest, as well as cru-cial for applications– like realization of solid-state artificru-cial atoms– qubits. We ded-icate the next section to reviewing the devices that are of relevant interest for the present thesis, namely the Cooper-pair box and transistor.

1.3.1 C

-

-CHANICS

Up to now we have considered primary quantum effects in superconductors: the Cooper pairs are governed by quantum laws but their coherence leads to a measur-able, macroscopic quantity: the phase φ of the condensate. Despite this quantum origin, the equations governing the time dynamics of φ, Eq. 1.8, 1.7, are classical.

In the 80’s experiments have proven that, in certain limits, the treatment of the phase as a classical variable is inadequate. Voss and Webb [12], and respec-tively Jackel et al. [13] demonstrated macroscopic quantum tunneling of the phase

{ { through the potential maxima of the washboard potential Eq. 1.11 for sufficiently

small, current biased junctions at low temperatures. In other words, the junction as a whole behaves like a quantum object. This extraordinary and elegant phe-nomenon is "the really-quantum effect", as it was referred to in Ref. [34].

To describe such secondary quantum effects one has to carry out the "secondary quantization" of the classical equations. This is not to say that Eq. 1.8, 1.7 are wrong, but they are an approximation still of great use for many realistic junctions. Anderson [35] proposed to start from a simple Hamiltonian of the junction involv-ing the electrostatic energy of the capacitor and the energy associated to the su-percurrent:

H =Qˆ

2

2C− EJcos ˆφ, (1.15)

This Hamiltonian resembles the one-dimensional particle of mass (ħ/2e)2C mov-ing with momentum (ħ/2e)Q along the axis φ in the periodic potential −EJcos φ.

According to the recipes of quantum mechanics, the charge and the phase are not just classical variables, but should now be treated as quantum operators that obey commutation relations:

[ ˆφ, ˆQ] = 2ei . (1.16) This implies that the charge and the phase observables obey a Heisenberg uncer-tainty relation so they cannot be both accurately well defined. The unceruncer-tainty relations is more evident in terms of the number of Cooper pairs N ≡ Q/2e and we write

[ ˆφ, ˆ_{N ] = i ⇐⇒ ∆φ∆N ≥ 1/2.} (1.17) As an important remark, the phase-charge duality Eq. 1.16 was revealed in the context of Josephson junctions but it is specific to all superconductors in general.

The uncertainty relation Eq. 1.17 dictates the phenomenology in specific situ-ations. The more the charge is localized, so δN ≈ 0, the more the phase fluctuates. Important examples are the Cooper-pair box in the charging regime (see Section 1.3.3) or phase-slips (see Section 1.4). The charge is a "good quantum number" and it is sometimes convenient to rewrite the equations in the charge basis. In the opposite limit, if the phase is well-defined, the charge fluctuations are unavoid-able, as in the Cooper-pair box in the limit EJ≫ EC as described in Section 1.3.3).

Among these examples, we proceed in the next section to discuss the Coulomb blockade systems.

1.3.2 J

C

If we turn on Coulomb repulsion in a superconducting circuit, we expect that su-perconductivity will be (at least partially) suppressed. Coulomb interactions tend

{ {

to quench the fluctuations of charge and consequently the phase fluctuations in-crease, according to the uncertainty relation Eq. 1.17. Since the superconduct-ing state is described by the well-defined phase, fluctuations of this phase imply degradation of the superconductivity. The good news is that by combining these two counteracting effects one can create a macroscopic quantum state.

Nanostructures embedding Josephson junctions are the ideal systems to study the interaction between superconductivity and Coulomb repulsion. Such devices are readily fabricated with the present day technology. By fabrication one can actu-ally tune the parameters and properties of such macroscopic quantum states. This superb tunability has made Coulomb blockade-Josephson junction devices instru-mental for practical realizations of qubits [36]. Firstly, we briefly recall the phe-nomenon of Coulomb blockade in mesoscopic systems in general. Secondly, we describe the superconducting equivalent with Josephson junctions replace nor-mal tunneling junctions: the Cooper-pair box transistor.

Coulomb blockade

Coulomb interaction is inherently a classical phenomenon that describes the elec-trostatic repulsion/attraction of like/unlike charged particles. Usually in solid state physics the repulsion between charge carriers is ignored and one treats the con-duction electrons as non-interacting particles. The underlying physical reason why this treatment is justified is that the interacting electrons form a ground state and charged elementary excitations above this state do not interact if their ener-gies are close to the Fermi level. In contrast to solid state physics, in mesoscopic systems there is a very common regime where electrostatic interactions are crucial: the Coulomb blockade regime.

An isolated piece of metal should support discrete charges: if an extra electron is to be added to the island then one has to overcome the electrostatic (Coulomb) repulsion between the extra-charge and the charge already present on the island. If we change the potential by a gate we can enable a new electron to jump on the island. To add functionality, the charging element is connected to leads via tun-neling barriers (one or more, depending on the function of the device). In the clas-sical regime single-electron transfers are the transport mechanism when electron transport is blocked due to Coulomb repulsion. The description of such systems is usually very simple and it is based on charge quantization and the corresponding charging energy.

1.3.3 C

-

The combination of Josephson effect and Coulomb blockade gives rise to quan-tum systems. For this combination we require superconducting islands and

elec-{ {

Cooper pair box

a.

C

C

V

E

C

_C

C

E

E

V

Cooper pair transistor

b.

Φ

0

FIGURE1.8: a. Electrical circuit of a Cooper-pair box (CPB). A superconducting island (dot) is capac-itively coupled to the gate biased by voltage Vg. The Josephson junction enables tunneling of Cooper pairs from the left lead. We attribute all the capacitance of the wire C to the island. b. The Cooper pair transistor. The dependence of its energies on the external flux bias Φ give rise to a supercurrent that can be modulated by changing q = CgVg.

trodes, while the tunneling junctions become Josephson junctions. The elemen-tary charge that can be transferred is not an electron, but a Cooper pair.

Two generic devices that exemplify the manifestation of Coulomb blockade in superconducting circuits are made by connecting a superconducting island with either one or two superconducting leads. The isolated island supports discrete charges. The important part of the setup is the gate electrode that at zero frequency is not electrically connected to the island but, by means of capacitive coupling, in-duces charge q on the island. We will refer to these two devices as to Cooper-pair box (CPB, Fig. 1.8.a.) and Cooper-pair transistor (CPT, Fig. 1.8.b.). The latter term is less conventional: we use it because the supercurrent through the device does depend on the gate voltage, this being a transistor effect. Besides, the setup re-minds much that of normal-metal Single-Electron Tunneling Transistor (SET)[37]. The Cooper-pair box (CPB) is one of the most studied and used devices in mesoscopic physics [31, 38, 39]. The superconducting island is connected to a bulk electrode by means of a tunnel Josephson junction that enables coherent transfer of Cooper pairs between the island and electrode. A gate electrode capacitively coupled to the island induces a continuous charge q. The energies of quantum states of the CPB are periodic functions of this induced charge. This charge

sensi-{ {

tivity reveals the charge states of the device, those with a well defined number of excess Cooper pairs, and opens up the possibility to create and control their quan-tum superpositions. For instance, the CPB can be operated near the point where two charge states are approximately degenerate forming a qubit basis. There are two energy scales relevant for the device: the charging and the Josephson energy.

The charging energy is related to charge quantization. Let us illustrate this by considering an isolated superconducting island (no junctions nor gates are present). Then the charge is well defined being an integer number N of Cooper pairs and the total charge Q of the island is a multiple of elementary charges, Q = 2eN, N being the number of excess Cooper pairs in the island. The charge Q produces an electric field Eelaround the island and this field accumulates electrostatic energy that can

be expressed in terms of the capacitance C of the island:

Eel= Q2 2C = (2e)2N2 2C ≡ ECN 2_{, (1.18)}

where we have defined the charging energy

EC=

2e2

C . (1.19)

Adding an extra Cooper-pair to the island is possible at the energy cost of ECsince

we are charging the island (extraction of a Cooper pairs also charges the island with positive charge). If we add more extra charges, the energy cost rises.

By connecting the island to a superconducting lead via a Josephson junction, a second energy scale, the Josephson energy, is brought about by the junction’s crit-ical current EJ=_2eħIC. The tunneling of Cooper pairs through the junction is the

mechanism of charge transfer from the reservoir of Cooper pairs (the supercon-ducting lead) to the island.

In a Cooper-pair box (CPB), the Josephson tunneling has a dramatic effect on the energy levels: it mixes degenerate states into a quantum superposition of charge states [40]. This is in sharp contrast with its "normal", non-superconducting ana-logue, the single-electron box. The latter is supposed to always be in a well-defined classical charge state, and never in a superposition of two states; finite temperature and voltage cause transitions between different charge states. The Hamiltonian of the CPB consists of charging and Josephson terms,

ˆ HCPB= EC³ ˆN − q 2e ´2 − EJcos( ˆφ), (1.20)

where now the charging energy EC= 2e

2

(C +Cg)involves the total capacitance of the