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Quantum Mechanics in the Single Photon Laboratory Quantum Mechanics in the Single Photon Laboratory Muhammad Hamza Waseem, Faizan-e-Ilahi and Muhammad Sabieh Anwar Department of Physics, Syed Babar Ali School of Science and Engineering, Lahore University of Management Sciences (LUMS), Lahore, Pakistan IOP Publishing, Bristol, UK ªIOPPublishingLtd2020 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording orotherwise,withoutthepriorpermissionofthepublisher,orasexpresslypermittedbylawor undertermsagreedwiththeappropriaterightsorganization.Multiplecopyingispermittedin accordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency,theCopyright ClearanceCentreandotherreproductionrightsorganizations. PermissiontomakeuseofIOPPublishingcontentotherthanassetoutabovemaybesought [email protected]. MuhammadHamzaWaseem,Faizan-e-IlahiandMuhammadSabiehAnwarhaveassertedtheir righttobeidentifiedastheauthorsofthisworkinaccordancewithsections77and78ofthe Copyright,DesignsandPatentsAct1988. Supplementarymaterialareavailableforthisbookathttp://iopscience.iop.org/book/978-0-7503-3603-3. ISBN 978-0-7503-3063-3(ebook) ISBN 978-0-7503-3061-9(print) ISBN 978-0-7503-3064-0(myPrint) ISBN 978-0-7503-3062-6(mobi) DOI 10.1088/978-0-7503-3063-3 Version:20200701 IOPebooks BritishLibraryCataloguing-in-PublicationData:Acataloguerecordforthisbookisavailable fromtheBritishLibrary. PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK USOffice:IOPPublishing,Inc.,190NorthIndependenceMallWest,Suite601,Philadelphia, PA19106,USA Dedicated to those who, while living in the face of danger themselves, stand for the distressed in happenings of the global pandemic … LIVE DANGEROUSLY Allama Iqbal Said one gazelle to another, ‘I will Take shelter in the harem from now on; For there are hunters at large in the wild, And there is no peace here for a gazelle. From fear of hunters I want to be free. O how I long for some security.’ His friend replied, ‘Live dangerously, my Wise friend, if it is life you truly seek. Like a sword of ﬁne mettle hurl yourself Upon the whetting-stone; stay sharp thereby. For danger brings out what is best in you: It is the touchstone of all that is true.’ Translation by M Hadi Husain Translation reprinted courtesy of Iqbal Academy Pakistan Contents Preface xi Acknowledgments xiii Author biographies xv Abbreviations xvi List of quantum optics experiments xvii 1 Introduction 1-1 References 1-5 2 Classical nature of light 2-1 2.1 Electromagnetic waves 2-1 2.2 Polarization 2-3 2.2.1 The polarization ellipse 2-3 2.2.2 Polarization manipulation 2-4 2.2.3 Jones calculus 2-6 2.2.4 Stokes parameters 2-7 2.3 Experimental explorations 2-9 2.3.1 Light source and detection 2-9 2.3.2 Understanding, manipulating and 2-10 measuring polarization using Jones calculus 2.3.3 Fourier analysis and peanut plots 2-12 2.3.4 Interference and erasure of which-way 2-15 information References 2-19 3 Quantum nature of light 3-1 3.1 Quantum mechanical states 3-1 3.2 Qubits 3-2 3.3 Transforming quantum states 3-4 3.4 Measuring quantum states 3-5 3.5 Composite systems and entangled states 3-6 vii QuantumMechanicsintheSinglePhotonLaboratory 3.6 Mixed states and the density matrix 3-8 3.7 Photon statistics 3-10 References 3-11 4 Experiments related to the quantum nature of light 4-1 4.1 General components of the lab 4-2 4.1.1 Light source 4-2 4.1.2 Light detection 4-4 4.1.3 Coincidence counting unit 4-6 4.2 Q1: Spontaneous parametric downconversion 4-8 4.2.1 The downconversion crystal and phase-matching 4-10 4.2.2 Optical alignment 4-13 4.2.3 Accidental coincidence counts 4-15 4.2.4 The experiment 4-16 4.3 Q2: proof of existence of photons 4-17 4.3.1 Photodetection and degree of second-order coherence 4-20 4.3.2 Accidental coincidences 4-25 4.3.3 The optical setup 4-25 4.3.4 The experiment 4-27 4.4 Q3: Estimating the polarization state of single photons 4-31 4.4.1 Reconstructing the single-photon polarization state 4-31 4.4.2 Generating and measuring polarization states 4-33 4.4.3 The experiment 4-35 4.5 Q4: Visualizing the polarization state of single photons 4-36 4.5.1 Antenna polarimetry and the polarization pattern method 4-37 4.5.2 Polarization pattern of single photons 4-38 4.5.3 The experiment 4-41 4.6 Q5: Single-photon interference and quantum eraser 4-43 4.6.1 The polarization interferometer and quantum erasure 4-46 4.6.2 Aligning the interferometer 4-47 4.6.3 The experiment 4-49 References 4-52 5 Experiments related to entanglement and nonlocality 5-1 5.1 Entanglement and local realism 5-1 5.2 The proverbial Alice and Bob experiment 5-3 viii QuantumMechanicsintheSinglePhotonLaboratory 5.3 Generating polarization-entangled photons 5-4 5.4 NL1: Freedman’s test of local realism 5-5 5.4.1 Freedman’s inequality 5-6 5.4.2 Quantum mechanical prediction for Freedman’s test 5-9 5.4.3 Tuning the Bell state 5-10 5.4.4 The experiment 5-12 5.5 NL2: Hardy’s test of local realism 5-13 5.5.1 The Hardy inequality 5-14 5.5.2 Quantum mechanical prediction for Hardy’s test 5-16 5.5.3 The experimental setup 5-17 5.5.4 Aligning the detectors 5-18 5.5.5 Tuning the Hardy state 5-18 5.5.6 Measuring probabilities with four detectors 5-19 5.5.7 The experiment 5-20 5.6 NL3: CHSH test of local realism 5-21 5.6.1 The CHSH inequality 5-21 5.6.2 Quantum mechanical prediction for the CHSH test 5-23 5.6.3 Tuning the Bell state 5-24 5.6.4 The experiment 5-25 References 5-25 6 Quantum state tomography 6-1 6.1 Qubits, Stokes parameters and tomography 6-3 6.1.1 The Bloch sphere for pure states 6-3 6.1.2 The Bloch sphere for density matrices 6-4 6.1.3 Stokes parameters as projections of the state on 6-6 the Bloch sphere 6.2 Single-qubit tomography 6-8 6.3 Two-qubit tomography 6-10 6.4 Nonideal measurements and compensation of errors 6-13 6.5 Maximum likelihood estimation 6-14 6.6 The experiment 6-15 References 6-19 ix