Coherent States in Quantum Physics

360 Pages·2009·3.119 MB·English

by  Jean-Pierre Gazeau

Jean-PierreGazeau
CoherentStatesinQuantum
Physics
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Jean-Pierre Gazeau
Coherent States in Quantum Physics
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V
Contents
Preface XIII
PartOne CoherentStates 1
1 Introduction 3
1.1 TheMotivations 3
2 TheStandardCoherentStates:theBasics 13
2.1 SchrödingerDefinition 13
2.2 FourRepresentationsofQuantumStates 13
2.2.1 PositionRepresentation 14
2.2.2 MomentumRepresentation 14
2.2.3 NumberorFockRepresentation 15
2.2.4 ALittle(Lie)AlgebraicObservation 16
2.2.5 AnalyticalorFock–BargmannRepresentation 16
2.2.6 OperatorsinFock–BargmannRepresentation 17
2.3 SchrödingerCoherentStates 18
2.3.1 BergmanKernelasaCoherentState 18
2.3.2 AFirstFundamentalProperty 19
2.3.3 SchrödingerCoherentStatesintheTwoOtherRepresentations 19
2.4 Glauber–Klauder–SudarshanorStandardCoherentStates 20
2.5 WhytheAdjectiveCoherent? 20
3 TheStandardCoherentStates:the(Elementary)Mathematics 25
3.1 Introduction 25
3.2 PropertiesintheHilbertianFramework 26
3.2.1 A“Continuity”fromtheClassicalComplexPlanetoQuantumStates 26
3.2.2 “Coherent”ResolutionoftheUnity 26
3.2.3 TheInterplayBetweentheCircle(asaSetofParameters)andthePlane
(asaEuclideanSpace) 27
3.2.4 AnalyticalBridge 28
3.2.5 OvercompletenessandReproducingProperties 29
3.3 CoherentStatesintheQuantumMechanicalContext 30
3.3.1 Symbols 30
3.3.2 LowerSymbols 30
CoherentStatesinQuantumPhysics.Jean-PierreGazeau
Copyright©2009WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40709-5
VI Contents
3.3.3 HeisenbergInequalities 31
3.3.4 TimeEvolutionandPhaseSpace 32
3.4 PropertiesintheGroup-TheoreticalContext 35
3.4.1 TheVacuumasaTransportedProbe... 35
3.4.2 UndertheActionof... 36
3.4.3 ... theD-Function 37
3.4.4 SymplecticPhaseandtheWeyl–HeisenbergGroup 37
3.4.5 CoherentStatesasToolsinSignalAnalysis 38
3.5 QuantumDistributionsandCoherentStates 40
3.5.1 TheDensityMatrixandtheRepresentation“R” 41
3.5.2 TheDensityMatrixandtheRepresentation“Q” 41
3.5.3 TheDensityMatrixandtheRepresentation“P” 42
3.5.4 TheDensityMatrixandtheWigner(–Weyl–Ville)Distribution 43
3.6 TheFeynmanPathIntegralandCoherentStates 44
4 CoherentStatesinQuantumInformation:anExampleofExperimental
Manipulation 49
4.1 QuantumStatesforInformation 49
4.2 OpticalCoherentStatesinQuantumInformation 50
4.3 BinaryCoherentStateCommunication 51
4.3.1 BinaryLogicwithTwoCoherentStates 51
4.3.2 UncertaintiesonPOVMs 51
4.3.3 TheQuantumErrorProbabilityorHelstromBound 52
4.3.4 TheHelstromBoundinBinaryCommunication 53
4.3.5 HelstromBoundforCoherentStates 53
4.3.6 HelstromBoundwithImperfectDetection 54
4.4 TheKennedyReceiver 54
4.4.1 ThePrinciple 54
4.4.2 KennedyReceiverError 55
4.5 TheSasaki–HirotaReceiver 56
4.5.1 ThePrinciple 56
4.5.2 Sasaki–HirotaReceiverError 56
4.6 TheDolinarReceiver 57
4.6.1 ThePrinciple 57
4.6.2 PhotonCountingDistributions 58
4.6.3 DecisionCriterionoftheDolinarReceiver 58
4.6.4 OptimalControl 59
4.6.5 DolinarHypothesisTestingProcedure 60
4.7 TheCook–Martin–GeremiaClosed-LoopExperiment 61
4.7.1 ATheoreticalPreliminary 61
4.7.2 Closed-LoopExperiment:theApparatus 63
4.7.3 Closed-LoopExperiment:theResults 65
4.8 Conclusion 67
5 CoherentStates:aGeneralConstruction 69
5.1 Introduction 69
Contents VII
5.2 ABayesianProbabilisticDualityinStandardCoherentStates 70
5.2.1 PoissonandGammaDistributions 70
5.2.2 BayesianDuality 71
5.2.3 TheFock–BargmannOption 71
5.2.4 ASchemeofConstruction 72
5.3 GeneralSetting:“Quantum”ProcessingofaMeasureSpace 72
5.4 CoherentStatesfortheMotionofaParticleontheCircle 76
5.5 MoreCoherentStatesfortheMotionofaParticleontheCircle 78
6 TheSpinCoherentStates 79
6.1 Introduction 79
6.2 PreliminaryMaterial 79
6.3 TheConstructionofSpinCoherentStates 80
6.4 TheBinomialProbabilisticContentofSpinCoherentStates 82
6.5 SpinCoherentStates:Group-TheoreticalContext 82
6.6 SpinCoherentStates:Fock–BargmannAspects 86
6.7 SpinCoherentStates:SphericalHarmonicsAspects 86
6.8 OtherSpinCoherentStatesfromSpinSphericalHarmonics 87
6.8.1 MatrixElementsoftheSU(2)UnitaryIrreducibleRepresentations 87
6.8.2 OrthogonalityRelations 89
6.8.3 SpinSphericalHarmonics 89
6.8.4 SpinSphericalHarmonicsasanOrthonormalBasis 91
6.8.5 TheImportantCase:σ= j 91
6.8.6 TransformationLaws 92
6.8.7 InfinitesimalTransformationLaws 92
6.8.8 “Sigma-Spin”CoherentStates 93
6.8.9 CovariancePropertiesofSigma-SpinCoherentStates 95
7 SelectedPiecesofApplicationsofStandardandSpinCoherentStates 97
7.1 Introduction 97
7.2 CoherentStatesandtheDrivenOscillator 98
7.3 AnApplicationofStandardorSpinCoherentStatesinStatisticalPhysics:
Superradiance 103
7.3.1 TheDickeModel 103
7.3.2 ThePartitionFunction 105
7.3.3 TheCriticalTemperature 106
7.3.4 AverageNumberofPhotonsperAtom 108
7.3.5 Comments 109
7.4 ApplicationofSpinCoherentStatestoQuantumMagnetism 109
7.5 ApplicationofSpinCoherentStatestoClassicalandThermodynamical
Limits 111
7.5.1 SymbolsandTraces 112
7.5.2 Berezin–LiebInequalitiesforthePartitionFunction 114
7.5.3 ApplicationtotheHeisenbergModel 116
VIII Contents
8 SU(1,1)orSL(2,R)CoherentStates 117
8.1 Introduction 117
8.2 TheUnitDiskasanObservationSet 117
8.3 CoherentStates 119
8.4 ProbabilisticInterpretation 120
8.5 PoincaréHalf-PlaneforTime-ScaleAnalysis 121
8.6 SymmetriesoftheDiskandtheHalf-Plane 122
8.7 Group-TheoreticalContentoftheCoherentStates 123
8.7.1 CartanFactorization 123
8.7.2 DiscreteSeriesofSU(1,1) 124
8.7.3 LieAlgebraAspects 126
8.7.4 CoherentStatesasaTransportedVacuum 127
8.8 AFewWordsonContinuousWaveletAnalysis 129
9 AnotherFamilyofSU(1,1)CoherentStatesforQuantumSystems 135
9.1 Introduction 135
9.2 ClassicalMotionintheInfinite-WellandPöschl–TellerPotentials 135
9.2.1 MotionintheInfiniteWell 136
9.2.2 Pöschl–TellerPotentials 138
9.3 QuantumMotionintheInfinite-WellandPöschl–TellerPotentials 141
9.3.1 IntheInfiniteWell 141
9.3.2 InPöschl–TellerPotentials 142
9.4 TheDynamicalAlgebrasu(1,1) 143
9.5 SequencesofNumbersandCoherentStatesontheComplexPlane 146
9.6 CoherentStatesforInfinite-WellandPöschl–TellerPotentials 150
9.6.1 FortheInfiniteWell 150
9.6.2 ForthePöschl–TellerPotentials 152
9.7 PhysicalAspectsoftheCoherentStates 153
9.7.1 QuantumRevivals 153
9.7.2 MandelStatisticalCharacterization 155
9.7.3 TemporalEvolutionofSymbols 158
9.7.4 Discussion 162
10 SqueezedStatesandTheirSU(1,1)Content 165
10.1 Introduction 165
10.2 SqueezedStatesinQuantumOptics 166
10.2.1 TheConstructionwithinaPhysicalContext 166
10.2.2 Algebraic(su(1,1))ContentofSqueezedStates 171
10.2.3 UsingSqueezedStatesinMolecularDynamics 175
11 FermionicCoherentStates 179
11.1 Introduction 179
11.2 CoherentStatesforOneFermionicMode 179
11.3 CoherentStatesforSystemsofIdenticalFermions 180
11.3.1 FermionicSymmetrySU(r) 180
11.3.2 FermionicSymmetrySO(2r) 185