Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Contents (long)......Page 10
Foreword......Page 25
Preface......Page 27
Acknowledgements......Page 30
Part I Semi-classical description of matter–light interaction......Page 33
1 The evolution of interacting quantum systems......Page 35
1.1 Review of some elementary results of quantum mechanics......Page 36
Interaction of an atom with a classical electromagnetic field......Page 37
Collision processes......Page 38
1.2.3 Perturbation series expansion of the system wavefunction......Page 39
Transition probability......Page 40
Example of a collisional process: qualitative study of the accessible energy range......Page 41
Case of a constant perturbation suddenly ‘switched on’......Page 43
Case of a sinusoidal perturbation......Page 45
1.2.5 Second-order calculations......Page 46
Some useful formulae......Page 49
Comparison with perturbation theory......Page 50
1.3 Case of a discrete level coupled to a continuum: Fermi\'s golden rule......Page 51
Approximation of independent electrons......Page 52
Effect of electron coupling......Page 53
The quasi-continuum......Page 54
Simple model of a quasi-continuum......Page 55
Short-time behaviour: transition probability per unit time......Page 56
Long-time behaviour: exponential decay......Page 57
The final state of the system......Page 59
A more realistic quasi-continuum: the concept of a density of states......Page 60
Fermi’s golden rule for a non-degenerate continuum......Page 61
Fermi’s golden rule for a degenerate continuum......Page 62
1.3.4 Case of a sinusoidal perturbation......Page 63
1.4 Conclusion......Page 64
1A.2 Temporal evolution......Page 66
1B.1.1 Definitions......Page 70
1B.1.2 Example......Page 72
1B.2.1 General expression......Page 73
1B.2.3 Behaviour at long times......Page 74
1B.3 Transition probability between a discrete level and a continuum......Page 75
2 The semi-classical approach: atomsinteracting with a classical electromagnetic field......Page 77
2.1 Atom–light interaction processes......Page 78
2.1.1 Absorption......Page 79
2.1.2 Stimulated emission......Page 80
2.1.3 Spontaneous emission......Page 81
2.1.4 Elastic scattering......Page 82
2.1.5 Nonlinear processes......Page 83
2.2 The interaction Hamiltonian......Page 85
2.2.1 Classical electrodynamics: the Maxwell–Lorentz equations......Page 86
Velocity operator......Page 87
Equations of motion......Page 88
The atomic and interaction Hamiltonians......Page 90
Long-wavelength approximation......Page 91
Göppert-Mayer gauge......Page 92
Long-wavelength approximation......Page 93
2.2.5 The magnetic dipole Hamiltonian......Page 94
Absorption and stimulated emission......Page 96
Transition probability......Page 97
Discussion......Page 99
Equivalence of the A ·p and D· E interaction Hamiltonians......Page 100
Non-perturbative solution of the equation of motion......Page 101
Rabi oscillations......Page 103
Examples......Page 104
Coherent transients and π/2 pulses......Page 105
Perturbative treatment......Page 107
Effective Hamiltonian for a two-photon transition......Page 108
2.3.4 Light-shifts......Page 110
2.4.1 Presentation of the model......Page 112
Principle of the calculation......Page 114
Perturbative solution......Page 115
Non-perturbative solution......Page 116
2.4.3 Dielectric susceptibility......Page 117
Mean value of the induced electric dipole......Page 118
Dielectric susceptibility......Page 119
Propagation with attenuation......Page 120
Dispersion......Page 121
2.4.5 Case of a closed two-level system......Page 122
Population transfer by stimulated emission......Page 124
Susceptibility......Page 125
2.5.2 Amplified propagation: laser action......Page 126
2.5.3 Generalization: pumping of both levels and saturation......Page 127
Absorption......Page 128
Stimulated emission......Page 129
2.6.2 Rate equations for the atoms......Page 130
2.6.3 Atom–photon interactions. Cross-section, saturation intensity......Page 132
2.6.4 Rate equations for the photons. Laser gain......Page 134
2.7 Conclusion......Page 136
2A.1 Description of the model......Page 137
Retarded potentials......Page 139
Sinusoidally oscillating charge. Polarization of the emitted radiation......Page 140
π-polarization......Page 141
σ+-polarization......Page 142
σ.-polarization......Page 143
The radiation reaction......Page 144
Radiative damping of the electron oscillations......Page 145
Low frequency limit: Rayleigh scattering......Page 147
Resonant scattering......Page 148
Atomic susceptibility......Page 149
2A.4 Response to an external electromagnetic wave......Page 146
2A.5 Relationship between the classical atomic model and the quantum mechanical two-level atom......Page 150
2B.1.1 Forbidden electric dipole transitions......Page 152
2B.1.2 Linearly polarized light......Page 153
Definition......Page 156
Selection rules......Page 157
2B.1.4 Spontaneous emission......Page 159
2B.2.1 Principle......Page 161
2B.2.2 Measurement of population transfers in the excited state......Page 162
Hanle effect......Page 163
Double resonance......Page 164
2B.3.1 J=1/2 → J=1/2 transition excited by circularly polarized light......Page 165
2B.3.2 Rate equations for optical pumping......Page 168
Complement 2C The density matrix and the optical Bloch equations......Page 172
2C.1.2 The density matrix representation......Page 173
Closed systems......Page 175
Open systems......Page 177
Presentation of the problem......Page 179
First-order solution......Page 180
Sinusoidal perturbation......Page 181
The mean value of the atomic electric dipole; the linear susceptibility......Page 182
Comparison with classical theory......Page 183
2C.3.1 Introduction......Page 184
2C.3.2 Closed systems......Page 185
2C.3.3 Open systems......Page 187
2C.4.1 Definition......Page 189
2C.4.2 Effect of a monochromatic field......Page 191
2C.4.3 Effect of relaxation......Page 192
2C.4.4 Rapid adiabatic passage......Page 193
2C.5.2 Case of an optical field of finite coherence time......Page 194
2C.6 Conclusion......Page 197
2D.1.1 Generalities......Page 199
2D.1.2 Ramsey fringes......Page 200
2D.1.3 Photon echoes......Page 202
Case of degenerate levels......Page 204
Non-degenerate levels......Page 206
Principle......Page 208
Application: slow light......Page 209
Complement 2E The photoelectric effect......Page 211
2E.1.1 The bound atomic state......Page 212
2E.1.2 Unbound states: the density of states......Page 213
2E.1.3 The interaction Hamiltonian......Page 215
2E.2.1 Ionization rate......Page 217
2E.2.3 Long-time behaviour......Page 219
2E.3 Application to the photoionization of hydrogen......Page 220
3 Principles of lasers......Page 223
3.1.1 Lasing threshold......Page 225
The steady-state field amplitude......Page 227
Steady-state intensity......Page 228
Oscillation frequency......Page 230
3.2.1 The need for population inversion......Page 231
Description of the four-level scheme......Page 233
The neodymium laser......Page 235
The helium–neon laser......Page 236
Tunable lasers......Page 237
Molecular lasers......Page 239
Semiconductor lasers......Page 240
3.2.3 Laser transition ending on the ground state: the three-level scheme......Page 242
Three-level scheme......Page 243
The erbium laser......Page 246
3.3.1 Longitudinal modes......Page 247
3.3.2 Single longitudinal mode operation......Page 249
Technical broadening......Page 251
The fundamental linewidth limit: the Schawlow–Townes limit......Page 252
3.4.1 Mode-locked lasers......Page 253
3.4.2 Q-switched lasers......Page 258
3.5.1 Classical light sources: a few orders of magnitude......Page 259
3.5.2 Laser light......Page 260
Further reading......Page 261
3A.1 The linear Fabry–Perot cavity......Page 262
3A.2 Cavity transmission and reflection coefficients and resonances......Page 264
3A.3 Ring Fabry–Perot cavity with a single coupling mirror......Page 266
3A.4 The cavity finesse......Page 267
3A.5 Cavity with a large finesse......Page 268
3A.6 Linear laser cavity......Page 270
3B.1 Fundamental Gaussian beam......Page 271
3B.2 The fundamental transverse mode of a stable cavity......Page 273
3B.3 Higher-order Gaussian beams......Page 274
3B.4 Longitudinal and transverse modes of a laser......Page 277
3C.1.1 Étendue and radiance......Page 279
3C.1.2 Conservation of radiance......Page 281
3C.2 Maximal irradiance by an incoherent source......Page 282
3C.3 Maximal irradiance by laser light......Page 283
3C.4.1 Thermal radiation in a cavity......Page 284
3C.5.1 Free propagative mode......Page 285
3C.5.3 Beam emitted by a laser......Page 287
3C.6 Conclusion......Page 288
Complement 3D The spectral width of a laser: the Schawlow–Townes limit......Page 289
3E.1 Laser irradiation of matter......Page 293
3E.1.1 The light–matter coupling......Page 294
3E.1.2 Energy transfer......Page 295
3E.1.4 Photo-chemical effects and photo-ablation......Page 296
3E.2.1 Thermal effects......Page 297
3E.3 Medical applications......Page 298
3E.4 Inertial fusion......Page 300
3F.1.1 Geometrical properties......Page 303
3F.1.2 Spectral and temporal properties......Page 304
3F.2 Laser measurement of distances......Page 305
3F.3.1 Atmospheric LIDAR......Page 307
3F.3.3 Measurement of angular velocities......Page 308
3F.4 Optical telecommunications......Page 311
3F.5 Laser light and other information technologies......Page 312
3G.1 Homogeneous and inhomogeneous broadening......Page 315
3G.2 Saturated absorption spectroscopy......Page 316
3G.2.1 Holes in a population distribution......Page 317
The Maxwellian velocity distribution......Page 318
Excitation of a single velocity class......Page 319
Saturated absorption spectroscopy......Page 320
3G.3.1 Two-photon transitions......Page 322
3G.3.2 Elimination of Doppler broadening......Page 323
3G.3.3 Properties of Doppler-free two-photon spectroscopy......Page 325
3G.4.1 A short history of hydrogen atom spectroscopy......Page 326
From Balmer to Dirac......Page 327
3G.4.3 Determination of the Rydberg constant......Page 328
Part II Quantum description of light and its interaction with matter......Page 331
4 Quantization of free radiation......Page 333
4.1.1 Quantizing a system of material particles......Page 334
4.1.2 Classical Hamiltonian formulation: Hamilton\'s equations......Page 335
4.1.4 Hamiltonian formalism for radiation: stating the problem......Page 336
4.2.2 Spatial Fourier expansion......Page 337
4.2.3 Transversality of the free electromagnetic field and polarized Fourier components......Page 339
4.2.4 Vector potential in the Coulomb gauge......Page 341
4.3.1 Dynamical equations of the polarized Fourier components......Page 342
4.3.2 Normal variables......Page 343
4.3.3 Expansion of the free field in normal modes......Page 344
4.3.5 Other normal modes......Page 346
4.4.1 Radiation energy......Page 347
4.4.2 Conjugate canonical variables for a radiation mode......Page 348
4.5.1 Canonical commutation relations......Page 349
4.5.2 Hamiltonian of the quantized radiation......Page 350
4.6 Quantized radiation states and photons......Page 351
4.6.1 Eigenstates and eigenvalues of the radiation Hamiltonian......Page 352
4.6.2 The notion of a photon......Page 353
4.6.3 General radiation state......Page 355
4.7 Conclusion......Page 356
Complement 4A Example of the classical Hamiltonian formalism: charged particlein an electromagnetic field......Page 357
4B.1.1 Classical expression......Page 359
Total angular momentum......Page 360
Intrinsic angular momentum......Page 361
Orbital angular momentum......Page 363
4B.2.2 Angular momentum operators......Page 364
4C.1.1 Unitary transformation of creation and annihilation operators......Page 366
4C.1.2 New normal modes......Page 367
4C.1.4 Invariance of the total photon number......Page 368
4C.1.6 Example: 1D standing wave modes......Page 369
4C.1.7 Choosing the best mode basis to suit a physical situation......Page 370
4C.2 Photons in a wave packet......Page 371
5 Free quantum radiation......Page 373
5.1 Photodetectors and semi-reflecting mirrors. Homodyne detection of the quadrature components......Page 374
Photocurrent and count rate. Classical radiation......Page 375
Quantized radiation......Page 376
5.1.2 Semi-reflecting mirror......Page 377
5.1.3 Homodyne detection......Page 378
5.2.1 Non-commutativity of the field operators and Heisenberg relations for radiation......Page 382
5.2.2 Vacuum fluctuations and their physical consequences......Page 383
5.3 Single-mode radiation......Page 385
5.3.1 Classical description: phase, amplitude and quadratures......Page 386
Single-mode radiation states......Page 387
Quadrature operators and the phasor representation......Page 388
5.3.3 Single-mode number state......Page 390
Definition and properties......Page 392
Value of the electric field in a single-mode quasi-classical state......Page 395
5.3.5 Other quantum states of single-mode radiation: squeezed states and Schrödinger cats......Page 397
5.3.6 The limit of small quantum fluctuations and the photon number–phase Heisenberg relation......Page 398
5.3.7 Light beam propagating in free space......Page 400
5.4.1 Non-factorizable states and entanglement......Page 403
5.4.2 Multimode quasi-classical state......Page 405
5.4.3 One-photon multimode state......Page 407
5.5.1 Mach–Zehnder interferometer in quantum optics......Page 409
5.5.2 Quasi-classical incoming radiation......Page 411
5.5.3 Particle-like incoming state......Page 412
5.5.4 Wave–particle duality for a particle-like state......Page 413
5.5.5 Wheeler\'s delayed-choice experiment......Page 414
5.6 A wave function for the photon?......Page 415
5.7 Conclusion......Page 417
5A.1.1 Definition......Page 419
5A.1.2 Expectation values of field observables for a squeezed state......Page 420
5A.1.3 The squeezing operator......Page 423
5A.1.4 Transmission of a squeezed state by a beamsplitter......Page 424
5A.1.5 Effect of losses......Page 425
5A.2.1 Generation by parametric processes......Page 426
5A.2.2 Other methods......Page 427
5A.3.1 Measurement of small absorption coefficients......Page 428
5A.3.2 Interferometric measurements......Page 429
5B.1.1 Definition and single photodetection probability......Page 430
5B.1.2 One-dimensional wave packet......Page 431
5B.1.3 Spontaneous emission photon......Page 433
5B.2.1 Semi-reflecting mirror......Page 435
5B.2.2 Double detection with a classical wave packet......Page 437
5B.3.1 Single detections......Page 440
5B.3.2 Joint detections......Page 441
5B.4 Quasi-classical wave packet......Page 443
5C.1 From the Bohr–Einstein debate to the Bell inequalities and quantum information: a brief history of entanglement......Page 445
5C.2.1 Measuring the polarization of a single photon......Page 447
5C.2.2 Photon pairs and joint polarization measurements......Page 449
5C.2.3 EPR pairs with correlated polarizations......Page 451
5C.2.4 The search for a picture to interpret the correlations between widely separated measurements......Page 453
5C.3.1 Bell inequalities......Page 457
5C.3.2 Conflict with quantum mechanics......Page 458
5C.3.3 Locality condition and relativistic causality. Experiment with variable polarizers......Page 460
5C.4 The experimental verdict and violation of the Bell inequalities......Page 461
5C.5 Conclusion: from quantum nonlocality to quantum information......Page 464
5D.1.1 General considerations......Page 466
5D.1.2 Schmidt decomposition......Page 467
5D.1.3 Correlations between measurements carried out on the two modes......Page 468
5D.2.1 Definition and properties......Page 469
5D.2.2 Production......Page 470
5D.3.2 Mixing two squeezed states on a semi-reflecting mirror......Page 471
5D.3.3 Non-destructive measurement of two complementary variables: the `EPR paradox\'......Page 473
5E.1.1 From classical to quantum cryptography......Page 475
5E.1.2 Quantum cryptography with entangled photons......Page 476
5E.1.3 From theory to practice......Page 478
5E.1.4 The no-cloning theorem......Page 479
5E.1.5 And if there were no entangled states? The BB84 protocol......Page 480
5E.2.1 Quantum bits or `qubits\'......Page 481
5E.2.2 The Shor factorization algorithm......Page 482
5E.2.3 Working principle of a quantum computer......Page 483
5E.2.4 Practical matters......Page 485
5E.3 Quantum teleportation......Page 486
5E.4 Conclusion......Page 488
6 Interaction of an atom with the quantized electromagnetic field......Page 489
6.1.1 The Maxwell–Lorentz equations......Page 490
Longitudinal electric field......Page 492
Transverse electric and magnetic fields. Radiation......Page 493
6.1.4 Normal variables for radiation and expansion in polarized, travelling plane waves......Page 494
6.1.5 Generalized particle momentum. Radiation momentum......Page 495
Total energy of the charge–field system......Page 496
Transverse and longitudinal electromagnetic energy......Page 497
Conjugate canonical variables for the field and charges......Page 498
6.2.1 Canonical quantization......Page 499
6.2.2 Hamiltonian and state space......Page 500
Long-wavelength approximation......Page 501
Decomposition of the interaction Hamiltonian......Page 502
6.3.2 Absorption......Page 503
6.3.3 Emission......Page 505
6.3.4 Rabi oscillation......Page 506
6.3.5 The Hamiltonian ĤI2 and elastic scattering......Page 507
Discrete level coupled to a continuum......Page 509
Transition (l = 1,m = 0) → (l = 0,m = 0)......Page 510
6.4.2 Quasi-continuum of one-photon states and density of states......Page 511
6.4.3 Spontaneous emission rate in a given direction......Page 513
6.4.4 Lifetime of the excited state and natural width......Page 514
6.4.5 Spontaneous emission: a joint property of the atom and the vacuum......Page 516
6.5.1 Scattering matrix elements......Page 517
6.5.2 Scattering cross-section......Page 519
Rayleigh scattering......Page 520
Thomson scattering......Page 521
Resonant scattering......Page 522
Raman scattering......Page 524
Frequency domain of Thomson scattering......Page 525
Total cross-section......Page 526
6.6 Conclusion. From the semi-classical to the quantum treatment of atom–light interaction......Page 527
6A.2.1 Hamiltonian for the charge–field system......Page 530
6A.2.3 Hamilton\'s equations for the radiation......Page 531
6A.2.4 Conclusion......Page 533
6B.1 Presentation of the problem......Page 534
6B.2.1 Jaynes–Cummings model......Page 536
6B.2.2 Diagonalization of the Hamiltonian......Page 537
6B.2.3 Spontaneous emission of an excited atomplaced in the empty cavity......Page 540
6B.3.1 Field initially in a number state......Page 542
6B.3.2 Field initially in an `intense\' quasi-classical state: semi-classical limit......Page 543
6B.3.3 Field initially in a quasi-classical state witha small number of photons......Page 544
6B.4 Effect of cavity losses: the Purcell effect......Page 545
6B.5 Conclusion......Page 549
6C.1 Introduction. Entangled photon pairsfor real experiments......Page 550
6C.2.1 Description of the system......Page 551
6C.2.2 Emission of photon ν1 and entangledatom–radiation state......Page 552
6C.2.3 Emission of photon ν2 and elementary EPR pair......Page 553
6C.3 Generalization and sum over frequencies......Page 555
6C.4 Two-photon excitations......Page 556
Part III Applying both approaches......Page 559
7.1 Introduction......Page 561
7.2.1 Linear susceptibility......Page 562
7.2.2 Nonlinear susceptibility......Page 563
7.2.3 Propagation in a nonlinear medium......Page 565
7.3.1 Frequency addition......Page 567
7.3.2 Phase matching......Page 569
7.3.3 Coupled dynamics of three-wave mixing......Page 572
7.3.4 Parametric amplification......Page 574
7.3.6 Parametric fluorescence......Page 576
7.4.1 Unavoidability and advantages of the quantum treatment......Page 577
7.4.2 Quantum treatment of three-wave mixing......Page 578
7.4.3 Perturbative treatment of parametric fluorescence......Page 579
7.4.4 Change of picture: the Heisenberg representation......Page 580
7.4.5 Simultaneous emission of parametric fluorescence photons......Page 582
The Hong–Ou–Mandel experiment......Page 585
Discussion......Page 588
7.5 Conclusion......Page 591
7A.1.1 Non-degenerate case......Page 592
7A.1.2 Degenerate case......Page 593
7A.2.1 Description of the system......Page 594
7A.2.2 Singly resonant OPO......Page 595
7A.2.3 Doubly resonant OPO......Page 596
7A.3.1 Quantum description of attenuation and amplification processes......Page 599
7A.3.2 Non-degenerate parametric amplification......Page 601
7A.3.3 Degenerate parametric amplification......Page 602
7A.4.1 The small quantum fluctuation limit......Page 603
7A.4.2 Frequency-degenerate OPO below threshold: producing squeezed states of the field......Page 605
7A.4.3 Non-frequency-degenerate OPO above threshold: producing twin beams......Page 606
7B.1.1 Nonlinear response of two-level atoms......Page 609
7B.1.2 Nonlinearity by optical pumping......Page 611
7B.2.1 Single incident wave......Page 613
7B.2.2 Two travelling waves propagating in opposite directions......Page 614
7B.3 Optical bistability......Page 615
7B.4.1 Degenerate four-wave mixing......Page 618
7B.4.2 Phase conjugation......Page 619
7B.4.3 Calculating the reflection coefficient......Page 622
7B.5.1 Self-focusing......Page 624
7B.5.2 Spatial soliton and self-focusing......Page 625
7B.6.2 Propagation in a dispersive linear medium......Page 627
7B.6.3 Propagation in a dispersive Kerr medium. Temporal soliton......Page 629
8 Laser manipulation of atoms. From incoherent atom optics to atom lasers......Page 631
8.1 Energy and momentum exchanges in the atom–light interaction......Page 632
8.1.2 Momentum conservation......Page 633
8.1.3 Energy conservation: the Doppler and the recoil shifts......Page 635
8.2.1 Closed two-level atom in a quasi-resonant laser wave......Page 636
8.2.2 Localized atomic wave packet and classical limit......Page 637
8.2.3 Radiative forces: general expression......Page 639
8.2.4 Steady-state radiative forces for a closed two-level atom......Page 640
8.2.5 Resonance-radiation pressure......Page 642
8.2.6 Dipole force......Page 646
8.3.1 Doppler cooling......Page 650
8.3.2 Coefficient of friction and Doppler molasses......Page 651
8.3.3 Magneto-optical trap......Page 653
8.3.4 Fluctuations and heating......Page 656
8.3.5 Fluctuations of the resonance-radiation pressure......Page 657
8.3.6 Momentum fluctuations and heating for a Doppler molasses......Page 659
8.3.7 Equilibrium temperature for a Doppler molasses......Page 661
8.3.8 Going under the Doppler temperature and Sisyphus cooling......Page 662
8.3.9 Cooling below the recoil temperature......Page 664
8.4.1 Bose–Einstein condensation......Page 665
8.4.2 Obtaining dilute atomic Bose–Einstein condensates. Laser cooling and evaporative cooling......Page 667
8.4.3 Ideal Bose–Einstein condensate and atomic wavefunction......Page 670
8.4.4 Observing the wavefunction of the Bose–Einstein condensate......Page 671
8.4.5 Dilute Bose–Einstein condensate with interactions......Page 672
8.4.6 Coherence properties of a Bose–Einstein condensate and interference between two Bose–Einstein condensates......Page 673
8.4.7 Atom lasers......Page 676
8.4.8 Conclusion. From photon optics to atom optics and beyond......Page 679
8A.1 Coherent population trapping......Page 683
8A.2 Velocity-selective coherent population trapping and sub-recoil cooling......Page 686
8A.3 Quantum description of the atomic motion......Page 688
8A.5 Practical limits. The fragility of coherence......Page 691
Index......Page 693