1 Classical and Quantum Interference and Coherence... 1 1.1 ClassicalInterferenceandOpticalInterferometers... 2 1.1.1 Young sdoubleslitinterferometer... 2 1.1.2 First-OrderCoherence... 4 1.1.3 WelcherWegProblem... 7 1.1.4 ExperimentalTestsoftheWelcherWegProblem... 11 1.1.5 Second-OrderCoherence... 15 1.1.6 Hanbury-BrownandTwissInterferometer... 17 1.1.7 Mach Zehnder Interferometer... 19 1.2 PrinciplesofQuantumInterference... 20 1.2.1 Two-Photon Nonclassical Interference... 21 1.2.2 The Hong Ou MandelInterferometer... 25 1.3 QuantumErasure... 28 1.4 QuantumNonlocality... 30 1.5 InterferometricInteraction-FreeMeasurements... 32 1.5.1 Negative-Result Measurements... 33 1.5.2 SchemesofInteraction-FreeMeasurements... 34 1.6 QuantumInterferometricLithography... 38 1.7 Three-Photon Interference... 42 1.7.1 Three-Photon Classical Interference... 43 1.7.2 Three-Photon Nonclassical Interference... 44 2 Quantum Interference in Atomic Systems: Mathematical Formalism... 47 2.1 MasterEquationofaMulti-DipoleSystem... 48 2.1.1 MasterEquationofaSingleMulti-LevelAtom... 48 2.1.2 MasterEquationofaMulti-AtomSystem... 67 2.2 Correlation Functions of Atomic Operators... 74 2.2.1 CorrelationFunctionsforaMulti-LevelAtom... 74 2.2.2 CorrelationFunctionsforaMulti-AtomSystem... 80 2.2.3 Spectral Expressions... 82
XII Contents 3 Superposition States and Modification of Spontaneous Emission Rates... 85 3.1 Superposition States in a Multi-Level System... 85 3.1.1 SuperpositionsInducedbySpontaneousEmission... 87 3.2 Multi-Atom Superposition (Entangled) States... 91 3.2.1 Entanglement... 91 3.2.2 TwoInteractingAtoms... 93 3.2.3 Entangled States of Two Identical Atoms... 94 3.2.4 Entangled States of Two Nonidentical Atoms... 96 3.3 Experimental Evidence of the Collective Damping and FrequencyShift... 104 3.4 GeneralCriteriaforInterferenceinTwo-AtomSystems... 110 3.4.1 Interference Pattern with Two Atoms... 111 3.4.2 Experimental Observation of the Interference Pattern inatwo-atomsystem... 113 3.5 QuantumBeats... 115 3.5.1 TheoryofQuantumBeatsinMulti-LevelSystems... 116 3.5.2 Quantum Beats in the Radiation Intensity from a Multi-LevelAtom... 120 3.5.3 Quantum Beats in the Radiation Intensity from Two NonidenticalAtoms... 126 3.5.4 Experimental Observation of Quantum Beats in a TypeISystem... 129 3.5.5 Quantum Beats in the Intensity Intensity Correlations... 131 3.6 Interference Pattern with a Dark Center... 135 4 Quantum Interference as a Control of Decoherence... 139 4.1 ModifiedSpontaneousEmission... 139 4.1.1 EffectofEnvironmentonSpontaneousEmission... 140 4.1.2 Modification by a Moderate Q Cavity... 142 4.1.3 Modification by Photonic Crystals... 145 4.2 QuantumInterferenceinVeeSystems... 146 4.2.1 Population Trapping and Dark States... 148 4.2.2 ProbingQuantumInterferenceinaVeeSystem... 150 4.3 SpectralControlofSpontaneousEmission... 156 4.4 ExperimentalEvidenceofQuantumInterference... 162 4.4.1 EnergyLevelsoftheMolecularSystem... 162 4.4.2 MasterEquationoftheSystem... 163 4.4.3 Two-Photon Excitation... 164 4.4.4 One- and Two-Photon Excitations... 166 4.5 Decoherence Free Subspaces... 169 4.5.1 A Simple Example of a Decoherence Free Subspace... 169 4.5.2 Experimental Verification of Decoherence Free Subspaces... 171
XIII 4.5.3 Tests on the Master Equation for a Decoherence Free Subspace... 173 5 Coherence Effects in Multi-Level Systems... 179 5.1 Three-LevelSystems... 179 5.1.1 The Basic Equations for Coherent Population Trapping... 181 5.1.2 The Solutions Under Two-Photon Resonance... 182 5.1.3 The General Equations of Motion for the Density Matrix... 183 5.1.4 Steady-State Solutions... 188 5.1.5 ObservationofCoherentPopulationTrapping... 190 5.1.6 Velocity-SelectiveCoherentPopulationTrapping... 192 5.2 Electromagnetically Induced Transparency in the Lambda System... 196 5.2.1 RealizationofEIT... 200 5.3 LasingWithoutInversion... 201 5.3.1 AModelforLWI... 203 5.3.2 ObservationofLWI... 205 5.4 Spatial Propagation of EM Fields in Optical Media... 207 5.5 Absorptive and Dispersive Properties of Optically Dense Media... 210 5.5.1 Absorptive and Dispersive Properties of Two-Level Atoms... 213 5.5.2 Dressed-Atom Model of a Driven Two-Level Atom... 220 5.5.3 Absorption and Dispersion with Multichromatic Driving Fields... 223 5.5.4 Collisional Dephasing and Coherent Population Oscillations... 226 5.6 ApplicationsofEITinNonlinearOptics... 229 5.6.1 Enhancement of Nonlinear Susceptibilities... 230 5.6.2 Observation of Enhancement of Nonlinear Susceptibilities... 234 5.6.3 Enhancement of Refractive Index... 236 6 Field Induced Quantum Interference... 237 6.1 Resonance Fluorescence in Driven Vee Systems... 238 6.2 PhaseControlofQuantumInterference... 244 6.2.1 PhaseControlofPopulationDistribution... 245 6.2.2 Phase Control of the Fluorescence Spectrum... 247 6.2.3 Experimental Evidence of Phase Control of Quantum Interference... 248 6.3 Superbunching... 251 6.3.1 Distinguishable Photons... 253 6.3.2 Indistinguishable Photons... 254
XIV Contents 6.3.3 Physical Interpretation... 256 6.4 Implementation of Quantum Interference... 258 6.4.1 External Field Mixing... 258 6.4.2 Two-LevelAtominaPolychromaticField... 260 6.4.3 dcfieldsimulationofquantuminterference... 262 6.4.4 Pre-selectedCavityPolarizationMethod... 267 6.4.5 AnisotropicVacuumApproach... 270 6.5 FanoProfiles... 271 6.6 Laser-InducedContinuumStructure... 275 6.6.1 Weak-FieldTreatment... 275 6.6.2 ObservationofLaser-InducedStructures... 277 6.7 Nonperturbative Treatment of Laser-Induced Continuum Structure... 279 6.8 Quantum Interference in Photonic Bandgap Structures... 282 6.8.1 TheTwo-LevelAtom... 285 6.8.2 TheThree-LevelAtom... 288 7 Slow and Fast Light and Storage of Photons... 293 7.1 RefractiveIndexandGroupVelocity... 294 7.1.1 LightGuidingLight... 297 7.1.2 Group Velocity Reduction in a Driven Lambda-Type Atom... 301 7.1.3 Group Velocity Reduction in a System with Decay-InducedCoherences... 305 7.1.4 PhaseControlofGroupVelocity... 309 7.2 Experimental Observations of Slow Propagation of Light... 313 7.3 Experimental Observation of Negative Group Velocities... 322 7.4 Bright- and Dark-State Polaritons... 326 7.4.1 Collective Atomic Trapping States... 330 7.4.2 Experimental Realization of Light Storage in Atomic Media... 332 8 Quantum Interference in Phase Space... 337 8.1 PhaseSpaceinClassicalandQuantumMechanics... 337 8.2 The Quasi-probability Distributions... 339 8.3 WignerFunctionsforSomeCommonFields... 343 8.3.1 Fock States... 343 8.3.2 Coherent States... 344 8.3.3 Chaotic Field... 345 8.3.4 Squeezed Coherent States... 345 8.4 Expansion in Fock States... 346 8.5 Superpositions of Fock States... 348 8.6 ExperimentalConsiderations... 352 8.6.1 ReconstructionofWignerFunctions... 352 8.6.2 Production of Single-Photon States... 353
XV 8.7 Photon Number Distribution... 354 8.8 Superpositions of Coherent States... 356 8.8.1 Superposition of N Coherent States... 356 8.8.2 Two Coherent State Superpositions... 357 8.9 Photon Number Distribution of Displaced Number States... 361 8.10 Photon Number Distribution of a Highly Squeezed State... 362 8.11 QuantumInterferenceinPhaseSpace... 366 8.11.1 TheWKBMethod... 366 8.12 AreaofOverlapFormalism... 368 8.12.1 Photon Number Distribution of Coherent States... 371 8.12.2 Photon Number Distribution of Squeezed State... 373 9 Quantum Interference in Atom Optics... 377 9.1 InterferenceandDiffractionofColdAtoms... 378 9.2 InterferenceofTwoBose EinsteinCondensates... 386 9.2.1 RelativePhaseBetweenTwoCondensates... 387 9.2.2 RelativePhaseinJosephsonJunctions... 389 9.3 Interference Between Colliding Condensates... 392 9.4 Collapses and Revivals of an Atomic Interference Pattern... 393 9.5 InterferenceExperimentsinCoherentAtomOptics... 395 9.5.1 Experimental Evidence of Relative Phase BetweenTwoCondensates... 395 9.5.2 AtomicInterferometers... 397 9.5.3 Collapses and Revivals of a Bose Einstein Condensate 401 9.6 HigherOrderCoherenceinaBEC... 402 References... 405 Index... 413