A Guide to Experiments in Quantum Optics, Second Edition

433 Pages·2004·12.513 MB·English

Hans-A.Bachor and Timothy C.Ralph
A Guide to Experiments in Quantum Optics
Second,Revised and Enlarged Edition
Hans-A.Bachor and Timothy C.Ralph
A Guide to Experiments
in Quantum Optics
Second,Revised and Enlarged Edition
Authors This book was carefully produced.Nevertheless,
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Hans-A.Bachor mation contained therein to be free of errors.
The Australian National University,Canberra Readers are advised to keep in mind that state-
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Timothy C.Ralph
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Contents
Preface XI
1 Introduction 1
1.1 Historicalperspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation: Practicaleffectsofquantumnoise . . . . . . . . . . . . . . . . 3
1.3 Howtousethisguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Classicalmodelsoflight 12
2.1 Classicalwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Mathematicaldescriptionofwaves. . . . . . . . . . . . . . . . . . . 13
2.1.2 TheGaussianbeam . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Quadratureamplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.4 Fieldenergy,intensity,power . . . . . . . . . . . . . . . . . . . . . 18
2.1.5 Aclassicalmodeoflight . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.6 Classicalmodulations . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Statisticalpropertiesofclassicallight . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Theoriginoffluctuations . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Correlationfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.4 Noisespectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.5 Anidealizedclassicalcase:Lightfromachaoticsource. . . . . . . . 31
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Photons–themotivationtogobeyondclassicaloptics 37
3.1 Detectinglight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Theconceptofphotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Lightfromathermalsource . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Interferenceexperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Modellingsinglephotonexperiments . . . . . . . . . . . . . . . . . . . . . 46
3.5.1 Polarizationofasinglephoton . . . . . . . . . . . . . . . . . . . . . 47
3.5.2 Somemathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.3 Polarizationstates . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.4 Thesinglephotoninterferometer . . . . . . . . . . . . . . . . . . . . 51
3.6 Intensitycorrelation,bunching,anti-bunching . . . . . . . . . . . . . . . . . 53
VI Contents
3.7 SinglephotonRabifrequencies . . . . . . . . . . . . . . . . . . . . . . . . . 56
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Quantummodelsoflight 60
4.1 Quantizationoflight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.1 Somegeneralcommentsonquantummechanics . . . . . . . . . . . 60
4.1.2 Quantizationofcavitymodes . . . . . . . . . . . . . . . . . . . . . 61
4.1.3 Quantizedenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.4 Thequantummechanicalharmonicoscillator . . . . . . . . . . . . . 63
4.2 Quantumstatesoflight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1 NumberorFockstates . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2 Coherentstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3 Mixedstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Quantumopticalrepresentations . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 Quadratureamplitudeoperators . . . . . . . . . . . . . . . . . . . . 68
4.3.2 Probabilityandquasi-probabilitydistributions . . . . . . . . . . . . 70
4.3.3 Photonnumberdistributions,Fanofactor . . . . . . . . . . . . . . . 75
4.4 Propagationanddetectionofquantumopticalfields . . . . . . . . . . . . . . 77
4.4.1 Propagationinquantumoptics . . . . . . . . . . . . . . . . . . . . . 77
4.4.2 Detectioninquantumoptics . . . . . . . . . . . . . . . . . . . . . . 81
4.4.3 Anexample: Thebeamsplitter . . . . . . . . . . . . . . . . . . . . . 82
4.5 Quantumtransferfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5.1 Alinearizedquantumnoisedescription . . . . . . . . . . . . . . . . 84
4.5.2 Anexample: Thepropagatingcoherentstate . . . . . . . . . . . . . 86
4.5.3 Reallaserbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5.4 Thetransferofoperators,signalsandnoise . . . . . . . . . . . . . . 88
4.5.5 Sidebandmodesasquantumstates . . . . . . . . . . . . . . . . . . . 90
4.6 Quantumcorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.6.1 Photoncorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.6.2 Quadraturecorrelations. . . . . . . . . . . . . . . . . . . . . . . . . 93
4.7 Summary: Thedifferentquantummodels . . . . . . . . . . . . . . . . . . . 95
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 Basicopticalcomponents 99
5.1 Beamsplitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.1 Classicaldescriptionofabeamsplitter . . . . . . . . . . . . . . . . . 99
5.1.2 Thebeamsplitterinthequantumoperatormodel . . . . . . . . . . . 102
5.1.3 Thebeamsplitterwithsinglephotons . . . . . . . . . . . . . . . . . 104
5.1.4 Thebeamsplitterandthephotonstatistics . . . . . . . . . . . . . . . 106
5.1.5 Thebeamsplitterwithcoherentstates . . . . . . . . . . . . . . . . . 108
5.1.6 Thebeamsplitterinthenoisesidebandmodel . . . . . . . . . . . . . 110
5.1.7 Comparisonbetweenabeamsplitterandaclassicalcurrentjunction . 111
5.2 Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.1 Classicaldescriptionofaninterferometer . . . . . . . . . . . . . . . 113
5.2.2 Quantummodeloftheinterferometer . . . . . . . . . . . . . . . . . 115
Contents VII
5.2.3 Thesinglephotoninterferometer . . . . . . . . . . . . . . . . . . . . 115
5.2.4 Transferofintensitynoisethroughtheinterferometer . . . . . . . . . 116
5.2.5 Sensitivitylimitofaninterferometer . . . . . . . . . . . . . . . . . . 117
5.3 Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3.1 Classicaldescriptionofalinearcavity . . . . . . . . . . . . . . . . . 121
5.3.2 Thespecialcaseofhighreflectivities . . . . . . . . . . . . . . . . . 125
5.3.3 Thephaseresponse . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3.4 Spatialpropertiesofcavities . . . . . . . . . . . . . . . . . . . . . . 128
5.3.5 Equationsofmotionforthecavitymode. . . . . . . . . . . . . . . . 132
5.3.6 Thequantumequationsofmotionforacavity . . . . . . . . . . . . . 133
5.3.7 Thepropagationoffluctuationsthroughthecavity . . . . . . . . . . 133
5.3.8 Singlephotonsthroughacavity . . . . . . . . . . . . . . . . . . . . 137
5.4 Otheropticalcomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.1 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.2 Crystalsandpolarizers . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4.3 Modulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4.4 Opticalfibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4.5 Opticalnoisesources . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4.6 Nonlinearprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6 LasersandAmplifiers 147
6.1 Thelaserconcept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.1.1 Technicalspecificationsofalaser . . . . . . . . . . . . . . . . . . . 148
6.1.2 Rateequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.1.3 Quantummodelofalaser . . . . . . . . . . . . . . . . . . . . . . . 154
6.1.4 Examplesoflasers . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.1.5 Laserphasenoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2 Amplificationofopticalsignals . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.3 Parametricamplifiersandoscillators . . . . . . . . . . . . . . . . . . . . . . 164
6.3.1 Thesecond-ordernon-linearity . . . . . . . . . . . . . . . . . . . . . 165
6.3.2 Parametricamplification . . . . . . . . . . . . . . . . . . . . . . . . 167
6.3.3 Opticalparametricoscillator . . . . . . . . . . . . . . . . . . . . . . 168
6.3.4 Pairproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7 Photodetectiontechniques 173
7.1 Photodetectorcharacteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.2 Detectingsinglephotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.3 Photonsourcesandanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.4 Detectingphotocurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.4.1 Thedetectorcircuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.5 Spectralanalysisofphotocurrents . . . . . . . . . . . . . . . . . . . . . . . 187
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
VIII Contents
8 Quantumnoise: Basicmeasurementsandtechniques 200
8.1 Detectionandcalibrationofquantumnoise . . . . . . . . . . . . . . . . . . 200
8.1.1 Directdetectionandcalibration . . . . . . . . . . . . . . . . . . . . 200
8.1.2 Balanceddetection . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.1.3 DetectionofintensitymodulationandSNR . . . . . . . . . . . . . . 205
8.1.4 Homodynedetection . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.1.5 Heterodynedetection . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.2 Intensitynoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.3 Theintensitynoiseeater . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.3.1 Classicalintensitycontrol . . . . . . . . . . . . . . . . . . . . . . . 213
8.3.2 Quantumnoisecontrol . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.4 Frequencystabilization,lockingofcavities . . . . . . . . . . . . . . . . . . 221
8.4.1 Howtomountamirror . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.5 Injectionlocking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9 Squeezingexperiments 232
9.1 Theconceptofsqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.1.1 Toolsforsqueezing,twosimpleexamples . . . . . . . . . . . . . . . 232
9.1.2 Propertiesofsqueezedstates . . . . . . . . . . . . . . . . . . . . . . 238
9.2 Quantummodelofsqueezedstates . . . . . . . . . . . . . . . . . . . . . . . 242
9.2.1 Theformaldefinitionofasqueezedstate . . . . . . . . . . . . . . . 242
9.2.2 Thegenerationofsqueezedstates . . . . . . . . . . . . . . . . . . . 245
9.2.3 Squeezingascorrelationsbetweennoisesidebands . . . . . . . . . . 247
9.3 Detectingsqueezedlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.3.1 Reconstructingthesqueezingellipse . . . . . . . . . . . . . . . . . . 253
9.3.2 Summaryofdifferentrepresentationsofsqueezedstates . . . . . . . 254
9.3.3 Propagationofsqueezedlight . . . . . . . . . . . . . . . . . . . . . 254
9.4 Fourwavemixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
9.5 Opticalparametricprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.6 Secondharmonicgeneration . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.7 Kerreffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
9.7.1 TheresponseoftheKerrmedium . . . . . . . . . . . . . . . . . . . 275
9.7.2 FibreKerrSqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . 277
9.7.3 AtomicKerrsqueezing . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.8 Atom-cavitycoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
9.9 Pulsedsqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
9.9.1 Quantumnoiseofopticalpulses . . . . . . . . . . . . . . . . . . . . 283
9.9.2 PulsedsqueezingexperimentswithKerrmedia . . . . . . . . . . . . 285
9.9.3 PulsedSHGandOPOexperiments. . . . . . . . . . . . . . . . . . . 287
9.9.4 Solitonsqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
9.9.5 Spectralfiltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
9.9.6 Nonlinearinterferometers . . . . . . . . . . . . . . . . . . . . . . . 290
9.10 Amplitudesqueezedlightfromdiodelasers . . . . . . . . . . . . . . . . . . 292
9.11 Twinphotonbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
Contents IX
9.12 Polarizationsqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
9.13 Quantumstatetomography . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
9.14 Summaryofsqueezingresults . . . . . . . . . . . . . . . . . . . . . . . . . 300
9.14.1 Loopholesinthequantumdescription . . . . . . . . . . . . . . . . . 303
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
10 Applicationsofsqueezedlight 310
10.1 Opticalcommunication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
10.2 Spatialsqueezingandquantumimaging . . . . . . . . . . . . . . . . . . . . 313
10.3 Opticalsensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.4 Gravitationalwavedetection . . . . . . . . . . . . . . . . . . . . . . . . . . 321
10.4.1 TheoriginandpropertiesofGW . . . . . . . . . . . . . . . . . . . . 321
10.4.2 Quantumpropertiesoftheidealinterferometer . . . . . . . . . . . . 323
10.4.3 Thesensitivityofrealinstruments . . . . . . . . . . . . . . . . . . . 328
10.4.4 Interferometrywithsqueezedlight . . . . . . . . . . . . . . . . . . . 333
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
11 QND 343
11.1 TheconceptofQNDmeasurements . . . . . . . . . . . . . . . . . . . . . . 343
11.2 ClassificationofQNDmeasurements. . . . . . . . . . . . . . . . . . . . . . 346
11.3 Experimentalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
11.4 SinglephotonQND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12 Fundamentaltestsofquantummechanics 355
12.1 Wave-Particleduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
12.2 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
12.3 Nonlocality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
12.3.1 Einstein-Podolsky-RosenParadox . . . . . . . . . . . . . . . . . . . 362
12.3.2 GenerationofentangledCWbeams . . . . . . . . . . . . . . . . . . 365
12.3.3 Bellinequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
12.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
13 QuantumInformation 374
13.1 Photonsasqubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
13.2 Postselectionandcoincidencecounting . . . . . . . . . . . . . . . . . . . . 376
13.3 Truesinglephotonsources . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
13.3.1 Heraldedsinglephotons . . . . . . . . . . . . . . . . . . . . . . . . 377
13.3.2 Singlephotonsondemand . . . . . . . . . . . . . . . . . . . . . . . 379
13.4 Characterizingphotonicqubits . . . . . . . . . . . . . . . . . . . . . . . . . 381
13.5 Quantumkeydistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
13.5.1 QKDusingsinglephotons . . . . . . . . . . . . . . . . . . . . . . . 383
13.5.2 QKDusingcontinuousvariables . . . . . . . . . . . . . . . . . . . . 385
13.5.3 Nocloning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
X Contents
13.6 Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
13.6.1 Teleportationofphotonqubits . . . . . . . . . . . . . . . . . . . . . 388
13.6.2 Continuousvariableteleportation . . . . . . . . . . . . . . . . . . . 390
13.7 Quantumcomputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
13.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
14 Summaryandoutlook 404
15 Appendices 407
AppendixA:Gaussianfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
AppendixB:Listofquantumoperators,statesandfunctions . . . . . . . . . . . . 408
AppendixC:Thefullquantumderivationofquantumstates . . . . . . . . . . . . . 410
AppendixD:Calculationofofthequantumpropertiesofafeedbackloop . . . . . 412
AppendixE:Symbolsandabbreviations . . . . . . . . . . . . . . . . . . . . . . . 414
Index 416
Preface
The idea behind this guide
Some of the most interesting and sometimes puzzling phenomena in optics are those where
thequantummechanicalnatureoflightisapparent. Recentyearshaveseenarapidexpansion
ofexperimentalopticsintothisareaknownasQuantumOptics.Beautifuldemonstrationsand
applicationsofthequantumnatureoflightarenowpossibleandopticshasbeenshowntobe
oneofthebestareasofphysicstoactuallymakeuseofquantummechanicalideas. Thisbook
is intended to guide you through the many experiments published, to present and interpret
theminonecommonstyle. Italsoprovidesapracticalbackgroundinopto-electronics.
Severalexcellenttextbookshavealreadybeenwrittenonthistopic. However,inthesame
wayasthisfieldofresearchhasbeeninitiatedbytheoreticalideas,mostofthesebookshave
been written from a theoretical point of view. While this results in a very solid and reliable
descriptionofthefield,wefoundthatitfrequentlyleavesoutsomeoftheimportantsimpler
pictures and intuitive interpretations of the experiments. Here we are using a different and
complementary approach: This guide focuses on the actual experiments and what we can
learnfromthem. Itexplainstheunderlyingphysicsinthemostintuitivewaywecouldfind. It
addressesquestionssuchas:whatarethelimitationsoftheequipment;whatcanbemeasured
andwhatremainsagoalforthefuture. Theanswerspreparethereadertomakeindependent
predictionsoftheoutcomeoftheirownexperiments.
Weassumethatmostofthereaderswillalreadyhaveafairlygoodunderstandingofoptical
phenomena, a reliable idea of how light propagates and how it interacts with components
such as lenses, mirrors, detectors etc. The reader can picture these processes in action in
an optical instrument.. It is useful in the everyday work of a scientist to have such pictures
of what is going on inside the experiments. They have been shaped over many years by
many lectures and through ones own experimentation. In optics three different pictures are
used simultaneously: Light as waves, light as photons and light as the solution to operator
equations. To these is added the picture of noise propagating through an optical system, a
description using noise transfer functions. All of these pictures are useful; all of them are
correct within their limits. They are based on specific interpretations of the one formalism,
theyaremathematicallyequivalent.
Inthisguideallthesefourdescriptionsareintroducedrigorouslyandareusedtodiscuss
aseriesofexperiments. Thesestartwithsimpledemonstrations,graduallygettingmorecom-
plex and include most of the experiments on quantum noise detection, squeezed states and
quantum non demolition measurements published to date. All four pictures – waves, pho-
tons,operatorsandnoisespectra–areusedsimultaneously. Ineachcasethemostappropriate