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Lecture Notes in Physics Alto Osada Rekishu Yamazaki Atsushi Noguchi Introduction to Quantum Technologies Lecture Notes in Physics FoundingEditors WolfBeiglböck JürgenEhlers KlausHepp Hans-ArwedWeidenmüller Volume 1004 SeriesEditors RobertaCitro,Salerno,Italy PeterHänggi,Augsburg,Germany MortenHjorth-Jensen,Oslo,Norway MaciejLewenstein,Barcelona,Spain AngelRubio,Hamburg,Germany WolfgangSchleich,Ulm,Germany StefanTheisen,Potsdam,Germany JamesD.Wells,AnnArbor,MI,USA GaryP.Zank,Huntsville,AL,USA The series Lecture Notes in Physics (LNP), founded in 1969, reports new developmentsinphysicsresearchandteaching-quicklyandinformally,butwitha highqualityandtheexplicitaimtosummarizeandcommunicatecurrentknowledge in an accessible way. Books published in this series are conceived as bridging materialbetweenadvancedgraduatetextbooksandtheforefrontofresearchandto servethreepurposes: (cid:129) to be a compact and modern up-to-date source of reference on a well-defined topic; (cid:129) to serve as an accessible introduction to the field to postgraduate students and non-specialist researchers from related areas; (cid:129) to be a source of advanced teaching material for specialized seminars, courses and schools. Bothmonographsandmulti-authorvolumeswillbeconsideredforpublication. Edited volumes should however consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series contentisindexed,abstractedandreferencedbymanyabstractingandinformation services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the responsible editor at Springer: Dr Lisa Scalone Springer Nature Physics Tiergartenstrasse 17 69121 Heidelberg, Germany [email protected] Alto Osada · Rekishu Yamazaki · Atsushi Noguchi Introduction to Quantum Technologies AltoOsada RekishuYamazaki KomabaInstituteforScience CollegeofLiberalArts TheUniversityofTokyo InternationalChristianUniversity Meguro-ku,Tokyo,Japan Mitaka-shi,Tokyo,Japan AtsushiNoguchi KomabaInstituteforScience TheUniversityofTokyo Meguro-ku,Tokyo,Japan ISSN0075-8450 ISSN1616-6361 (electronic) LectureNotesinPhysics ISBN978-981-19-4643-1 ISBN978-981-19-4641-7 (eBook) https://doi.org/10.1007/978-981-19-4641-7 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore ForYukiandRay Preface What is quantum technology? Is it a fancy bandwagon fresh from the academics? Definitely,thisisnotthecase.Letustracethehistoryinreverseorderfrom2022. Now the quantum technology is known to everybody, thanks to the many-qubit quantum “computers” developed by several companies and laboratories. Some of themareavailableonthecloudplatformandareusedfortheeducationalandbusi- ness use, while quantum communication and quantum sensing/metrology are also underintenseinvestigation.Whydoesquantumcomputergathersuchanattention? An answer is that such a novel computer based on the unconventional working principle—quantum mechanics—is expected to outperform conventional computers, or classical computers as a counterpart of the quantum computer, when a certaincomputationaltaskisgiven.Acelebratedquantumalgorithmofprimefac- torization was invented by Peter Shor in 1994 [1]. In 1995, he invented another landmark of the first proposal of a quantum error correction code [2]. We would like to stress that this very work ignited the experimental exploration of quantum computer which had been distinguished from a mere analog computer intolerable toerrorsoccurringinananalogmanneraswell.Todate,anumberofbreathtaking progress have been made in various quantum systems, especially in ion-trap and superconducting-circuit systems. Readers might have heard of another fancy word of quantum teleportation in which a quantum state is transferred between remote parties by a bit tricky use of the “spooky interaction” of quantum particles, or the quantum entanglement, a conceptcoinedbackin1935byEinstein,Podolsky,andRosen[3].Quantumtele- portationwasproposedin1993byBennetetal.[4],whichwasthenfollowedbyits realizationsin1997–1998[5,6]usingopticalphotons.Thesestudieslaythebasis of quantum communication. Optical photons are indispensable for the quantum technologiesthatdemandthetransferofquantumstatesfromoneplacetoadistant place,whichisindeedmadepossiblebytheinventionofopticalfiberswithexcel- lent property of extremely low-loss propagation at room temperature. This nature can also be utilized for another quantum technology of quantum cryptography, pioneered by Bennett and Brassard in 1984 [7]. Quantum technologies, or even quantum mechanics, have established their rel- evance in the development of Atomic, Molecular, and Optical (AMO) physics during the twentieth century. Furthermore, the spectroscopic and control techniques frequently used in AMO physics originate in the Nuclear Magnetic vii viii Preface Resonance(NMR)experimentssincethelate1930swithcelebratedpioneerssuch as Rabi, Ramsey, Bloch, and Purcell. NMR technique is widely applied in our daily life, for instance to Magnetic Resonance Induction (MRI) tomography and toidentificationofmaterialsusingchemicalshiftsofNMRspectra,bothofwhich can be regarded as early forms of quantum sensing. Whatwewanttosayisthatquantumtechnologyisnot(orshouldnotbe)justa short-lived trend, a scam business, nor a mathematical game in the Hilbert space, but a technology that has been pursued substantially with physics and devices in the history of understanding and controlling matters in real life, at least at the current technological levels. Therefore, the name of the game is to know what physical objects we want to address in quantum technologies and how to do that. Abstracted, perfect (error-corrected) logical qubits are not at our hand—even for a single logical qubit. Realization of them is a “holy grail,” at the time in 2022. Various physical systems are used to aim it, and numerous experimental techniques are developed for them. Readers might be overwhelmed when they think about which quantum system to learn, since different quantum systems can have different theoretical backgrounds and experimental techniques. However, we, the researcherinthisfield,knowthattherearemanytheoriesandconceptsincommon amongthevariousquantumsystems,suchasquantumoptics,quantumelectronics, and so on. The purpose of this book is to summarize such common basics of quantum technologies as a foundation for cultivating a comprehensive view of the state-of- the-art quantum systems. The book is roughly divided into three parts. (cid:129) In Part I, we describe how the quantum states are represented based on linear algebra, which is frequently used in quantum technology. We also cover the fundamentals of quantum mechanics necessary for understanding the basic quantum manipulations. It is an excerpt of what is to be collected and summarized in a concise form. (cid:129) Part II starts with a description of two-level systems and electromagnetic waves, and how they interact. We describe the interaction between the fundamental quantum systems, two-level systems, electromagnetic waves, and harmonic oscillators. We also describe the theory relating itto the actual quantum experiments and how they are realized. In particular, we cover details of the “resonator”, which is an almost indispensable tool in the current quantum technology. (cid:129) In Part III, we describe an overview of the quantum operations, quantum mea- surements,quantumerrorcorrections,andanintroductiontothestate-of-the-art quantum technologies. Readers who are familiar with analytical mechanics, statistical mechanics, quantum mechanics, and electromagnetism should be able to read this book by using thosetextbooksasreferences.Inparticular,PartIisareviewofquantummechan- ics, so readers who are quite familiar with quantum mechanics may start reading Preface ix from Part II and continue reading while returning to Part I as necessary. How- ever, Part I also includes some details of the methods and concepts that are often seen in quantum technology, so it is recommended that you read them at some point.Inthissense,thesubjectofthisbookissuitedforthethird-andfourth-year undergraduate students who have been studying physics. We also hope that some ambitiousfirst-andsecond-yearstudentstryreading.Itmaybeareviewformany graduate students and researchers, but comprehensive coverage of the quantum technologiescompiledheremayprovideanewinsight.Inaddition,thisbookmay beusedbytheresearcherswhowishtostudyquantumtechnologies,butcurrently working outside the field. We also warn readers to be aware that there may be some parts that lack theoretical rigor because we prioritize the description of the quantum processes as a whole, instead of focusing on the details of the quantum theories, which you may find in other literature. Lastly, problems are provided in the last part of each chapter, except for Chaps. 1 and 11. The authors strongly recommend the readers to tackle with them and check the solutions which are electronically distributed on the Springer website. Oneimportanttopicthatweareomittinginthisbookistheadvancementofthe controlandfabricationsystems,formingafundamentalbasisofthestate-of-the-art experimental quantum technologies. Digital controlling systems including Direct- Digital-Synthesis (DDS) and Field-Programmable Gate Array (FPGA), low-noise amplifiertechnologies,andsingle-photondetectorsareonlyafewoftheexamples that have been made in the lab by the researchers, but these high-end products are now readily available commercially. Ultra-high vacuum systems and dilution refrigerators to provide an environment with extremely low background noise are also available. Furthermore, the advancement in the field of material sciences and micro/nano fabrication technologies have allowed researchers to develop a “De- signer’sQuantum”,suchasQuantumdots,NV-center,andsuperconductingqubits, with various novel materials and fabrication techniques. These fundamental bases for the experiment might differ depending on the quantum system of the choice, and we encourage the reader to refer to these details in the future. Lastly, we would like to show our greatest appreciation to people who have helped us preparing these materials. In writing this book, Masato Shigefuji, Yuki Nakajima, and Genya Watanabe, members of Noguchi Laboratory in The Uni- versity of Tokyo, Yoshimi Rokugawa, Sorato Nakano, Yutaro Nakai, and Tsubasa Karino, members of Shu Lab in International Christian University, cooperated in calculation check and proofreading of the text. Yuta Masuyama from National Institute of Quantum Science and Technology gave us very useful advice on the content. We would love to mention that Akiyuki Tokuno, Sridevi Purushothaman, and Satish Ambikanithi from Springer worked really hard for the publication of this book and Akiyuki in particular has made invaluable advice and comments. We would like to express our gratitude here. Meguro-ku, Japan Alto Osada Mitaka-shi, Japan Rekishu Yamazaki Meguro-ku, Japan Atsushi Noguchi Contents PartI QuantumStatesandQuantumMechanics 1 Introduction ........................................................ 3 1.1 Common Language in Quantum Information .................. 3 1.2 Various Quantum Systems .................................... 5 1.3 Electromagnetic Waves for Quantum Operations ............... 5 1.3.1 Development of Electromagnetic Wave Source ........ 6 1.4 Concept of Temperature ....................................... 8 2 LinearAlgebra ..................................................... 13 2.1 Vector Space .................................................. 13 2.1.1 Vectors in Three-Dimensional Vector Space ........... 13 2.1.2 Inner Product ......................................... 14 2.1.3 Orthonormal Basis .................................... 14 2.1.4 Vector Components ................................... 15 2.1.5 Norm of a Vector ..................................... 15 2.1.6 Outer Product ........................................ 16 2.1.7 Expansion for Multidimensional System .............. 16 2.2 Matrix and Operator .......................................... 17 2.2.1 Matrix Element ....................................... 17 2.2.2 Transpose Matrix ..................................... 17 2.2.3 Matrix Multiplication ................................. 18 2.2.4 Square Matrix ........................................ 18 2.3 Eigenvectors and Eigenvalues ................................. 20 2.4 Summary of Vector Characteristics in Index Notation .......... 21 3 WavefunctionandNotationsinQuantumMechanics ............... 25 3.1 Equation of Motion in Classical and Quantum Mechanics ...... 25 3.2 Wavefunction ................................................. 27 3.2.1 Inner Product of Wavefunction ........................ 27 3.2.2 Continuous and Discretized Wavefunction ............. 28 3.3 Operator ...................................................... 30 3.4 Dirac Notation ................................................ 30 3.5 Matrix Representation ......................................... 32 xi