1. Classical and Quantum Interference
and Coherence ................................................ 1
1.1. Classical Interference and Optical Interferometers ......... 2
1.1.1. Young's Double Slit Interferometer .................. 2
1.1.2. First-Order Coherence ............................... 4
1.1.3. Welcher Weg Problem ................................. 7
1.1.4. Experimental Tests of the Welcher Weg Problem ...... 11
1.1.5. Second-Order Coherence ............................. 15
1.1.6. Hanbury-Brown and Twiss Interferometer ............. 17
1.1.7. Mach—Zehnder Interferometer ........................ 19
1.2. Principles of Quantum Interference ........................ 20
1.2.1. Two-Photon Nonclassical Interference ............... 21
1.2.2. The Hong-Ou-Mandel Interferometer .................. 25
1.3. Quantum Erasure ........................................... 28
1.4. Quantum Nonlocality ....................................... 30
1.5. Interferometric Interaction-Free Measurements ............. 32
1.5.1. Negative-Result Measurements ....................... 33
1.5.2. Schemes of Interaction-Free Measurements ........... 34
1.6. Quantum Interferometric Lithography ....................... 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. Master Equation of a Multi-Dipole System .................. 48
2.1.1. Master Equation of a Single Multi-Level Atom ....... 48
2.1.2. Master Equation of a Multi-Atom System ............. 67
2.2. Correlation Functions of Atomic Operators ................. 74
2.2.1. Correlation Functions for a Multi-Level Atom ....... 74
2.2.2. Correlation Functions for a Multi-Atom System ...... 80
2.2.3. Spectral Expressions ............................... 82
3. Superposition States and Modification of Spontaneous
Emission Rates .............................................. 85
3.1. Superposition States in a Multi-Level System .............. 85
3.1.1. Superpositions Induced by Spontaneous Emission ..... 87
3.2. Multi-Atom Superposition (Entangled) States ............... 91
3.2.1. Entanglement ....................................... 91
3.2.2. Two Interacting Atoms .............................. 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
Frequency Shift .......................................... 104
3.4. General Criteria for Interference in Two-Atom Systems .... 110
3.4.1. Interference Pattern with Two Atoms ............... 111
3.4.2. Experimental Observation of the Interference
Pattern in a Two-Atom System ...................... 113
3.5. Quantum Beats ............................................ 115
3.5.1. Theory of Quantum Beats in Multi-Level Systems .... 116
3.5.2. Quantum Beats in the Radiation Intensity from
a Multi-Level Atom ................................ 120
3.5.3. Quantum Beats in the Radiation Intensity from
Two Nonidentical Atoms ............................ 126
3.5.4. Experimental Observation of Quantum Beats in a
Type I System ..................................... 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. Modified Spontaneous Emission ............................ 139
4.1.1. Effect of Environment on Spontaneous Emission ..... 140
4.1.2. Modification by a Moderate Q Cavity ............... 142
4.1.3. Modification by Photonic Crystals ................. 145
4.2. Quantum Interference in Vee Systems ...................... 146
4.2.1. Population Trapping and Dark States ............... 148
4.2.2. Probing Quantum Interference in a Vee System ...... 150
4.3. Spectral Control of Spontaneous Emission ................. 156
4.4. Experimental Evidence of Quantum Interference ............ 162
4.4.1. Energy Levels of the Molecular System ............. 162
4.4.2. Master Equation of the System ..................... 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
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-Level Systems ...................................... 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. Observation of Coherent Population Trapping ....... 190
5.2. Electromagnetically Induced Transparency in the Lambda
System ................................................... 196
5.2.1. Realization of EIT ................................ 200
5.3. Lasing Without Inversion ................................. 201
5.3.1. A Model for LWI ................................... 203
5.3.2. Observation of LWI ................................ 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. Applications of EIT in Nonlinear Optics .................. 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. Phase Control of Quantum Interference .................... 244
6.2.1. Phase Control of Population Distribution .......... 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
6.3.3. Physical Interpretation ........................... 256
6.4. Implementation of Quantum Interference ................... 258
6.4.1. External Field Mixing ............................. 258
6.4.2. Two-Level Atom in a Polychromatic Field ........... 260
6.4.3. dc Field Simulation of Quantum Interference ....... 262
6.4.4. Pre-selected Cavity Polarization Method ........... 267
6.4.5. Anisotropic Vacuum Approach ....................... 270
6.5. Fano Profiles ............................................ 271
6.6. Laser-Induced Continuum Structure ........................ 275
6.6.1. Weak-Field Treatment .............................. 275
6.6.2. Observation of Laser-Induced Structures ........... 277
6.7. Nonperturbative Treatment of Laser-Induced Continuum
Structure ................................................ 279
6.8. Quantum Interference in Photonic Bandgap Structures ...... 282
6.8.1. The Two-Level Atom ................................ 285
6.8.2. The Three-Level Atom .............................. 288
7. Slow and Fast Light and Storage of Photons ................. 293
7.1. Refractive Index and Group Velocity ...................... 294
7.1.1. Light Guiding Light ............................... 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-Induced Coherences .......................... 305
7.1.4. Phase Control of Group Velocity. 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. Phase Space in Classical and Quantum Mechanics ........... 337
8.2. The Quasi-probability Distributions ...................... 339
8.3. Wigner Functions for Some Common Fields .................. 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. Experimental Considerations .............................. 352
8.6.1. Reconstruction of Wigner Functions ................ 352
8.6.2. Production of Single-Photon States ................ 353
8.7. Photon Number Distribution ............................... 354
8.8. Superpositions of Coherent States ........................ 356
8.8.1. Superposition of TV 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.Quantum Interference in Phase Space ...................... 366
8.11.1.The WKB Method .................................... 366
8.12.Area of Overlap Formalism ................................ 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. Interference and Diffraction of Cold Atoms ............... 378
9.2. Interference of Two Bose-Einstein Condensates ............ 386
9.2.1. Relative Phase Between Two Condensates ............ 387
9.2.2. Relative Phase in Josephson Junctions ............. 389
9.3. Interference Between Colliding Condensates ............... 392
9.4. Collapses and Revivals of an Atomic Interference
Pattern .................................................. 393
9.5. Interference Experiments in Coherent Atom Optics ......... 395
9.5.1. Experimental Evidence of Relative Phase
Between Two Condensates ........................... 395
9.5.2. Atomic Interferometers ............................ 397
9.5.3. Collapses and Revivals of a Bose-Einstein
Condensate ........................................ 401
9.6. Higher Order Coherence in а ВЕС .......................... 402
References .................................................... 405
Index ......................................................... 413
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