Undergraduate Lecture Notes in Physics A. I. Lvovsky Quantum Physics An Introduction Based on Photons Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topicsthroughoutpureandappliedphysics.Eachtitleintheseriesissuitableasabasisfor undergraduateinstruction,typicallycontainingpracticeproblems,workedexamples,chapter summaries, andsuggestions for further reading. ULNP titles mustprovide at least oneof thefollowing: (cid:129) Anexceptionally clear andconcise treatment ofastandard undergraduate subject. (cid:129) Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standardsubject. (cid:129) Anovel perspective oranunusual approach toteaching asubject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysicsteaching at theundergraduate level. ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinuetobethe reader’spreferred reference throughout theiracademic career. Series editors Neil Ashby University of Colorado, Boulder, CO, USA William Brantley Department of Physics, Furman University, Greenville, SC, USA Matthew Deady Physics Program, Bard College, Annandale-on-Hudson, NY, USA Michael Fowler Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Department of Physics, University of Oslo, Oslo, Norway Michael Inglis SUNY Suffolk County Community College, Long Island, NY, USA Heinz Klose Humboldt University, Oldenburg, Niedersachsen, Germany Helmy Sherif Department of Physics, University of Alberta, Edmonton, AB, Canada More information about this series at http://www.springer.com/series/8917 A. I. Lvovsky Quantum Physics An Introduction Based on Photons 123 A.I.Lvovsky University of Calgary andRussian QuantumCenter Calgary, AB Canada Asolutionsmanualforthisbookisavailablefordownloadathttps://www.springer.com/gp/ book/9783662565827 ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notesin Physics ISBN978-3-662-56582-7 ISBN978-3-662-56584-1 (eBook) https://doi.org/10.1007/978-3-662-56584-1 LibraryofCongressControlNumber:2018933016 ©Springer-VerlagGmbHGermany,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringer-VerlagGmbH,DEpart ofSpringerNature Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany Preface WhyIwrotethisbook Thefirstrigorousformulationofquantummechanics(QM)wasproposedbyHei- senbergandSchro¨dingerabout80yearsago.Sincethen,thefieldhasexperienced enormousevolution.Initiallyaimedatexplainingatomicspectra,quantummecha- nics has now entered the foundation of virtually all branches of physics. Accor- dingly,QMisaninseparablepartofeveryphysicsstudent’straining:whateverfield futurephysicistschooseaftergraduation,theywillalmostcertainlyneedquantum mechanicsintheirwork. Yet our way of teaching QM to students has not changed much over the years. Webeginwiththenotionofthewavefunction,andwritethetime-independent,and thenthetime-dependentSchro¨dingerequationinthepositionrepresentation.Wede- terminetheenergyspectraandthecorrespondingwavefunctionsofsimplepotential wells, and evolution of wavepackets incident on potential barriers. Finally, we in- troducetheangularmomentumoperatorandcalculatethespectrumofthehydrogen atom.Forthelast60years,thishasbeen,withminorvariations,thefirstsemester undergraduatequantummechanicscourseprogram. Thistraditionhasmanyadvantages.Itworkswithaphysicalsystemthatastu- dent has already dealt with in classical physics classes, and it is one that they can easilyimagine.Itallowsonetoseedifferencesbetweenthebehaviorsofaclassical andaquantumparticle,andbringstolightseveralfundamentalphenomenathatare characteristicofquantumness:tunneling,quantizationandtheuncertaintyprinciple. Itprovidesastudentwiththetoolboxtosolveexperimentallyrelevantproblemsthat cannot be solved classically: after calculating the hydrogen spectrum in the class- room,astudentgoestoalabandmeasuresit! Yetthisapproachisimperfect.Itgivesastudentanalgorithmtoanalyzeaspe- cificphysicalsystem,butitdoesnotrevealtheinnerworkingsofquantumphysics and the logic behind it. We teach our students multiple facts and computational approaches related to wavefunctions, operators, and measurements, but we do not construct a rigorous logical connection among them and do not explain which of v vi Preface these facts are the postulates and which their consequences, and in which logical sequencetheseconsequencesarederived. As a result, the student — at least a thinking student — ends up being immen- sely confused. Why does simply placing hats on top of letters turn a classical ex- pressionintoavalidquantumone?Whyisthemomentumoperator’sactiononthe wavefunctionequivalenttotakingthederivative?Whydoweneverseemomentum eigenstates (and Schro¨dinger cats) in practice? Why do de Broglie waves have a p normalizationfactorof1= 2ph¯?Whydoweobserveatomstransitioningbetween energy eigenstates, but not other states? How is a projective measurement related tomeasuringanobservable?Whyaresomestatesdescribedbywavefunctionsand somebycolumnsofnumbers?Ifallstateshavenorm1,whydon’twenormalizethe deBrogliewave?Ifobservablesarematrices,whatisthematrixofthemomentum observable? Ontopofthatthereisthemostdreadedquestion.Ifquantumphysicsissuppo- sed to be more general than classical, why must one resort to classical notions to understandtheconceptofmeasurement?Whyisthemeasurement,incontrasttoall other physical processes, not described by unitary evolution? If quantum systems do become classical at some point during the measurement, where exactly is that point? Thefundamentalwayofthinkingwearetryingsohardtoinstillinourstudents throughtheyearsoftraininginscienceis“Questioneverything!”.Inquantumclas- sesourmessageseemstobejusttheopposite:“Shutupandcalculate!”1 Havingbeenaquantummechanicsstudentmyself,Ihaveeventuallyfoundans- wers to these questions, but in many cases not until long after my PhD. When I askedthemasastudent,therewasnoonearound,notonlytogivemetheanswers, buteventohelpmestatethesequestionsproperly. Myquestwhilewritingthisbookistoaddressthisissue.Iattemptedtobuilda clearlogicalstructure,containingasfewloopholesaspossible.Onethatwouldal- lowthereadertotraceeachstatementdownthelogicalchainbacktofirstprinciples. Onethatwouldleavenoquestionunanswered. So,inasense,Iwrotethisbookformyself.Notfortoday’smyself,butformyself twenty-fiveyearsago.AkindofbookthatIwouldhavebeengratefultohavewhile a third-year student, and one that would have saved me years of agonizing search forthetruth. Itisnaturaltoask:Howrealisticismyaspiration?SomeofthequestionsIasked earlier sound quite advanced. Perhaps one does need a doctoral degree to answer them? Myansweristwofold.First,thereisapedagogicalissue:mechanics,withitsHil- bertspaceofinfinitedimension,doesnotseemtobethebestvenueforconveying quantumprinciples.Manyoftheabovequestionscanbeaddressedbyexemplifying QMwithasimplerphysicalsystem;Iwillfurtherelaborateonthisbelow.Second, most of the inconsistencies and paradoxes can be eliminated by properly introdu- cingthenotionofentanglement.Thisnotionunderliestwoessential,mutuallyrela- 1Moreonthisslogan,incorrectlyascribedtoFeynman,inSec.2.4. Preface vii tedconcepts:thevonNeumannmeasurementanddecoherence.Thefirstprovidesa waytoavoidmakingmeasurementanexceptioninthedomainofquantumphysics, thereby eliminating the Klein bottle logic characteristic of the Copenhagen inter- pretation. The second describes “inadvertent” measurements that occur naturally, making the quantum world appear to macroscopic observers such as ourselves in thewaythatisfamiliartousunderthenameofclassicalphysics. Theseconceptsarenotexcessivelycomplicated.Mathematically,theyaremuch simpler than many elements of the traditional quantum course, such as the afore- mentioned hydrogen atom or scattering theory. The main challenge in understan- ding entanglement is not the challenge to a student’s mathematical skills; it rather concernstheirimagination.Anddevelopingastrongimaginationisinherenttobe- comingagoodphysicist;asEinsteinsaid,imaginationisinfactmoreimportantthan knowledge. Quantummechanicsorquantumoptics? Thenameofourdiscipline—quantummechanics–impliesthatwearestudyingthe applicationsofquantumprinciplestothelawsofmotion.Butinfacttheframework of quantum theory is not limited to mechanics; it actually applies to all fields of physics.Ifouraimistostudythegeneralprinciplesofquantumphysics,isitwise tochoosemechanicsasthephysicalsystemforillustratingtheseprinciples? Facedwiththisquestion,wearecompelledtoadmitthattheanswerisnegative. Mechanicsistheremainlyduetotradition,becausehistorically,thefirstsuccessful applicationoftheseprinciplesintheirmodernformwasinmechanics.Buteducatio- nallyspeaking,usingtheexampleofmechanicstoexplainbasicquantumprinciples isarecipefordisaster.TheHilbertspaceassociatedwiththissystemisofinfinitedi- mension;moreover,itsbasishasthecardinalityofthecontinuum.Thestudentmust dealwiththeunfamiliar,enormouslycomplicated,andnotalwaysrigorousmathe- maticalbackgroundwhichincludesgeneralizedfunctions,Fouriertransformations, andfunctionalanalysis.Asaresult,insteadofconcentratingourstudents’effortson understanding the physical concepts, we force them to struggle with mathematics, andthisoftenleadstoconfusionbetweentheendandthemeans.Itisunrealisticto expect any kind of deep understanding to result from this experience. The student simplywon’tseetheforestbehindthetrees. Ifwearetochoosethephysicalsystemtoillustratequantumphysics,weshould pick one whose Hilbert space has the lowest nontrivial dimension: two. There is a variety of such systems that are currently studied in the context of quantum in- formation technology as quantum bits. Among them, one stands out as the most thoroughlystudiedandintuitive:thepolarizationofthephoton.Opticalwavepola- rizationwouldnormallyalreadyhavebeenstudiedbythestudententeringaquan- tumclass.TheJonespolarizationvectorsdirectlytranslateintophotonpolarization statevectors,andthematricesdescribingthetransformationofthesevectorsbywa- veplatestranslateintooperators.Itisstraightforwardtoargumentthemeasurement viii Preface postulate from the classical polarization measurement picture, taking into account thediscretenatureofthephoton.Inthisway,thequantumfundamentalsarisefrom classicalpolarizationoptics(andthestudents’laboratoryexperiencewiththesame) inthemoststraightforwardandnaturalfashion. Photon polarizationis of further benefitwhen we go onto study entanglement. A vast body of proof-of-principle experiments in quantum information have been performed using this system as the carrier of the quantum bit. Some of these ex- periments,suchasthoseonquantumcryptography,teleportationandnoonlocality, relate directly to the concepts covered in this book. By illustrating the theoretical material with these experiments, right from the start, this book will take students straight into the very heart of quantum physics using examples from today’s hot- test research topics. And what could make learning an academic discipline more excitingthanfreshresultsfromaresearchlab? Talkingaboutlabs,thestudent’sexperiencedoesnothavetobelimitedbyrea- dingaboutexperimentsdonebysomebodyelse.Agreatadvantageofthepolariza- tionqubitastheexamplesystemisthatitisperfectlyrealistictoaugmentthecourse with a laboratory component. Almost all the material of the first chapter is illus- trated by a classical polarization experiment containing a laser, a few waveplates, apolarizingbeamsplitterandtwodetectors.Thematerialonentanglementcanbe visualizedbyaseriesoflabsonsingle-photoninterferenceremotestatepreparation andBellnonlocality.Suchexperimentsaremoredifficulttosetup,butfullywithin the capabilities of an average physics department, as evidenced by the experience ofmanycollegesaroundtheworld,includingmyhome,theUniversityofCalgary. Moredetailsonpossibleeducationallabscanbefoundatthebook’swebsite. Theconnectionbetweenquantumphysicsandopticsinthisbookisnotlimited tousingthephotontoillustratethediscipline’sprimaryconcepts.Italsomanifests itselfinthemanyopticalexamplesscatteredthroughoutthebook,aswellastheset of subjects chosen for more advanced sections (deep study of the harmonic oscil- lator,Heisenbergpicture,squeezing,densitymatrices,two-levelsystems,quantum tomography). These subjects are particularly relevant for those who are interested inquantumopticsinparticularandquantuminformationingeneral. Structureofthecourse Thebookcontainsmaterialthatcanbetaughtduringatwo-semesterundergraduate quantum mechanics course. In the first chapter, the main principles and postulates of QM are introduced and illustrated by the photon polarization qubit. The reader maywishtostudythischapterinparallelwithAppendixA,whichcoversthebasic linearalgebrathatisrelevanttoQM,assummarizedinthefollowingtable. Preface ix Linearalgebraconcept Quantumconcept Physicalillustration (AppendixA) Linearspace,basis,dimension, Polarizationofthephoton Quantumstate,Hilbertspace innerproduct (AppendixC) Polarization measurements, Projectivemeasurement, Orthonormalbasis polarization state tomography, quantumtomography quantumcryptography Linearoperator, Observable, Pauli matrices as observables Hermitianoperator uncertaintyprinciple inthepolarizationspace Unitaryoperator, Evolutionofthephotoninabi- Schro¨dingerevolution functionsofoperators refringentmedium Thesecond chapterisentirelydedicatedtoentanglement,itsconsequencesand applications.Ifirstintroducethetensorproductspacemathematically,thenexplain partialquantummeasurements,remotestatepreparation,andthenonlocalitypara- dox (both the Bell and Greenberger-Horne-Zeilinger forms of it), illustrating the theorywithexperimentsonentangledphotons.Nonlocalityisarguablytheprimary paradox of quantum mechanics, and it is natural to follow up with a discussion of themechanismofquantummeasurements,theirnaturalcounterpart(decoherence), andtheinterpretationsofquantummechanics.Thissection(Sec.2.4)iswherewe findoutwhenandwhyaquantumsystembecomesclassicalduringameasurement, andwhywedon’tseeSchro¨dingercatswalkingaroundtown.Subsequently,Italk in a fairly rigorous fashion about applications of entanglement, such as quantum computation,teleportationandrepeaters.Whenthismaterialispresentedinaclass- roomsetting,itisusefultoasktwoorthreestudentstogivepresentationsonrecent experimentalresearchonthesubject. Thethird andfourthchaptersare,tosomeextent,atributetothe“mainstream” undergraduate quantum mechanics of a particle in a potential field. Here we have to deal with the Hilbert space whose basis is a continuum, so the third chapter is accompaniedbyatutorialonDiracdeltafunctionsandtheFouriertransform(Ap- pendixD).Itismyhopethatatthispoint,whenstudentshavealreadyinternalized the primary tenets of QM, they will be able to face the technical issues associated with continuous-variable Hilbert spaces without losing sight of the physical prin- ciples. As an introduction to continuous-variable systems, I explain how and why someofthenormalizationrulesareaffected.ThenIpresenttheusualscenariosof potentialwells,potentialbarriers,tunneling,andtheharmonicoscillator.Thisisthe pointwhereIenvisionthefirstsemestertobeconcluded. The third chapter goes on to explain the Heisenberg picture and how it is con- sistent with the Schro¨dinger picture, illustrating with many examples relating to thephysicsoftheharmonicoscillator(anddemonstratedinquantumopticsexperi- ments):displacement,phaseshift,aswellassingle-anddual-modesqueezing.With thehelpofthelatter,IpresenttheoriginalversionoftheEinstein-Podolsky-Rosen paradox. In the fourth chapter, I introduce the three-dimensional geometric space (as a tensorproductofthreeone-dimensionalspaces)andexplaintheangularmomentum,