introduction to quantum optics from the semi-classical

Introduction to Quantum OpticsFrom the Semi-classical Approach to Quantized Light

Covering a number of important subjects in quantum optics, this textbook is an excellentintroduction for advanced undergraduate and beginning graduate students, familiarizingreaders with the basic concepts and formalism as well as the most recent advances.

The first part of the textbook covers the semi-classical approach where matter is quan-tized, but light is not. It describes significant phenomena in quantum optics, including theprinciples of lasers. The second part is devoted to the full quantum description of light andits interaction with matter, covering topics such as spontaneous emission, and classical andnon-classical states of light. An overview of photon entanglement and applications to quan-tum information is also given. In the third part, nonlinear optics and laser cooling of atomsare presented, where the use of both approaches allows for a comprehensive description.Each chapter describes basic concepts in detail, and more specific concepts and phenom-ena are presented in ‘complements’.

Gilbert Grynberg was a CNRS Senior Scientist at the Laboratoire Kastler Brossel at theUniversité Pierre et Marie Curie Paris 6, and a Professor at the Ecole Polytechnique. Hewas a pioneer in many domains, including atomic spectroscopy, nonlinear optics and laser-cooled atoms in optical lattices.

Alain Aspect is a CNRS Senior Scientist and Professor at the Institut d’Optique and theEcole Polytechnique. A pioneer of the field of quantum entanglement, his research coversquantum optics, laser cooling of atoms, atom optics, Bose–Einstein condensates, atomlasers and quantum atom optics. He was awarded the 2010 Wolf Prize in Physics.

Claude Fabre is a Professor in the Laboratoire Kastler Brossel at the Université Pierreet Marie Curie Paris 6, and a senior member of the Institut Universitaire de France. Hisfields of research are quantum optics, atomic and laser physics, both experimentally andtheoretically.

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Introductionto Quantum Optics

From the Semi-classical Approach toQuantized Light

GILBERT GRYNBERGEcole Normale Supérieure, Paris

Ecole Polytechnique

ALAIN ASPECTInstitut d’Optique and Ecole Polytechnique,

Palaisean

CLAUDE FABREUniversité Pierre et Marie Curie and Ecole Normale

Supérieure, Paris

With a Foreword by Claude Cohen-Tannoudji






C A M B R I D G E U N I V E R S I T Y P R E S S

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c© G. Gilbert, A. Aspect, C. Fabre 2010

Original edition: Introduction aux Lasers et á l’Optique, Ellipses 1997

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Contents

Table of contents (short) vTable of contents (long) viiiForeword xxiiiPreface xxvAcknowledgements xxviii

Part I Semi-classical description of matter–light interaction 1

1 The evolution of interacting quantum systems 31.1 Review of some elementary results of quantum mechanics 41.2 Transition between discrete levels induced by a time-dependent perturbation 5

1.2.1 Presentation of the problem 51.2.2 Examples 51.2.3 Perturbation series expansion of the system wavefunction 71.2.4 First-order theory 81.2.5 Second-order calculations 141.2.6 Comparison with the exact solution for a two-level system 17

1.3 Case of a discrete level coupled to a continuum: Fermi’s golden rule 191.3.1 Example: autoionization of helium 201.3.2 Discrete level coupled to a quasi-continuum: simplified model 221.3.3 Fermi’s golden rule 281.3.4 Case of a sinusoidal perturbation 31

1.4 Conclusion 32

Complement 1A A continuum of variable width 341A.1 Description of the model 341A.2 Temporal evolution 34

Complement 1B Transition induced by a random broadband perturbation 381B.1 Description of a random perturbation 38

1B.1.1 Definitions 381B.1.2 Example 40

1B.2 Transition probability between discrete levels 411B.2.1 General expression 411B.2.2 Behaviour at intermediate times 421B.2.3 Behaviour at long times 42

1B.3 Transition probability between a discrete level and a continuum 43






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2 The semi-classical approach: atoms interacting with a classicalelectromagnetic field 452.1 Atom–light interaction processes 46

2.1.1 Absorption 472.1.2 Stimulated emission 482.1.3 Spontaneous emission 492.1.4 Elastic scattering 502.1.5 Nonlinear processes 51

2.2 The interaction Hamiltonian 532.2.1 Classical electrodynamics: the Maxwell–Lorentz equations 542.2.2 Hamiltonian of a particle in a classical electromagnetic field 552.2.3 Interaction Hamiltonian in the Coulomb gauge 582.2.4 Electric dipole Hamiltonian 602.2.5 The magnetic dipole Hamiltonian 62

2.3 Transitions between atomic levels driven by an oscillating electromagneticfield 642.3.1 The transition probability in first-order perturbation theory 642.3.2 Rabi oscillations between two levels 692.3.3 Multiphoton transitions 752.3.4 Light-shifts 78

2.4 Absorption between levels of finite lifetimes 802.4.1 Presentation of the model 802.4.2 Excited state population 822.4.3 Dielectric susceptibility 852.4.4 Propagation of an electromagnetic wave: absorption and dispersion 882.4.5 Case of a closed two-level system 90

2.5 Laser amplification 922.5.1 Feeding the upper level: stimulated emission 922.5.2 Amplified propagation: laser action 942.5.3 Generalization: pumping of both levels and saturation 952.5.4 Laser gain and population inversion 96

2.6 Rate equations 962.6.1 Conservation of energy in the propagation 962.6.2 Rate equations for the atoms 982.6.3 Atom–photon interactions. Cross-section, saturation intensity 1002.6.4 Rate equations for the photons. Laser gain 102

2.7 Conclusion 104

Complement 2A Classical model of the atom-field interaction:the Lorentz model 105

2A.1 Description of the model 1052A.2 Electric dipole radiation 1072A.3 Radiative damping of the elastically bound electron 1122A.4 Response to an external electromagnetic wave 114






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2A.5 Relationship between the classical atomic model and the quantummechanical two-level atom 118

Complement 2B Selection rules for electric dipole transitions.Applications to resonance fluorescence and optical pumping 120

2B.1 Selection rules and the polarization of light 1202B.1.1 Forbidden electric dipole transitions 1202B.1.2 Linearly polarized light 1212B.1.3 Circularly polarized light 1242B.1.4 Spontaneous emission 127

2B.2 Resonance fluorescence 1292B.2.1 Principle 1292B.2.2 Measurement of population transfers in the excited state 130

2B.3 Optical pumping 1332B.3.1 J = 1/2 → J = 1/2 transition excited by circularly polarized light 1332B.3.2 Rate equations for optical pumping 136

Complement 2C The density matrix and the optical Bloch equations 1402C.1 Wavefunctions and density matrices 141

2C.1.1 Isolated and coupled systems 1412C.1.2 The density matrix representation 1412C.1.3 Two-level systems 143

2C.2 Perturbative treatment 1472C.2.1 Iterative solution for the evolution of the density matrix 1472C.2.2 Atom interacting with an oscillating field: regime of linear response 149

2C.3 Optical Bloch equations for a two-level atom 1522C.3.1 Introduction 1522C.3.2 Closed systems 1532C.3.3 Open systems 155

2C.4 The Bloch vector 1572C.4.1 Definition 1572C.4.2 Effect of a monochromatic field 1592C.4.3 Effect of relaxation 1602C.4.4 Rapid adiabatic passage 161

2C.5 From the Bloch equations to the rate equations 1622C.5.1 Case of fast relaxation of coherences 1622C.5.2 Case of an optical field of finite coherence time 163

2C.6 Conclusion 165

Complement 2D Manipulation of atomic coherences 1672D.1 Direct manipulation of a two-level system 167

2D.1.1 Generalities 1672D.1.2 Ramsey fringes 1682D.1.3 Photon echoes 170






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2D.2 Use of a third level 1722D.2.1 Coherent population trapping 1722D.2.2 Electromagnetically induced transparency 176

Complement 2E The photoelectric effect 1792E.1 Description of the model 180

2E.1.1 The bound atomic state 1802E.1.2 Unbound states: the density of states 1812E.1.3 The interaction Hamiltonian 183

2E.2 The photoionization rate and cross-section 1852E.2.1 Ionization rate 1852E.2.2 The photoionization cross-section 1872E.2.3 Long-time behaviour 187

2E.3 Application to the photoionization of hydrogen 188

3 Principles of lasers 1913.1 Conditions for oscillation 193

3.1.1 Lasing threshold 1933.1.2 The steady state: intensity and frequency of the laser output 195

3.2 Description of the amplifying media of some lasers 1993.2.1 The need for population inversion 1993.2.2 Four-level systems 2013.2.3 Laser transition ending on the ground state: the three-level

scheme 2103.3 Spectral properties of lasers 215

3.3.1 Longitudinal modes 2153.3.2 Single longitudinal mode operation 2173.3.3 Spectral width of the laser output 219

3.4 Pulsed lasers 2213.4.1 Mode-locked lasers 2213.4.2 Q-switched lasers 226

3.5 Conclusion: lasers versus classical sources 2273.5.1 Classical light sources: a few orders of magnitude 2273.5.2 Laser light 228Further reading 229

Complement 3A The resonant Fabry–Perot cavity 2303A.1 The linear Fabry–Perot cavity 2303A.2 Cavity transmission and reflection coefficients and resonances 2323A.3 Ring Fabry–Perot cavity with a single coupling mirror 2343A.4 The cavity finesse 2353A.5 Cavity with a large finesse 2363A.6 Linear laser cavity 238






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Complement 3B The transverse modes of a laser: Gaussian beams 2393B.1 Fundamental Gaussian beam 2393B.2 The fundamental transverse mode of a stable cavity 2413B.3 Higher-order Gaussian beams 2423B.4 Longitudinal and transverse modes of a laser 245

Complement 3C Laser light and incoherent light: energy density andnumber of photons per mode 247

3C.1 Conservation of radiance for an incoherent source 2473C.1.1 Étendue and radiance 2473C.1.2 Conservation of radiance 249

3C.2 Maximal irradiance by an incoherent source 2503C.3 Maximal irradiance by laser light 2513C.4 Photon number per mode 252

3C.4.1 Thermal radiation in a cavity 2523C.4.2 Laser cavity 253

3C.5 Number of photons per mode for a free beam 2533C.5.1 Free propagative mode 2533C.5.2 Pencil of heat radiation 2553C.5.3 Beam emitted by a laser 255

3C.6 Conclusion 256

Complement 3D The spectral width of a laser: the Schawlow–Townes limit 257

Complement 3E The laser as energy source 2613E.1 Laser irradiation of matter 261

3E.1.1 The light–matter coupling 2623E.1.2 Energy transfer 2633E.1.3 Mechanical effects 2643E.1.4 Photo-chemical effects and photo-ablation 264

3E.2 Machining and materials processing using lasers 2653E.2.1 Thermal effects 2653E.2.2 Transfer of material 266

3E.3 Medical applications 2663E.4 Inertial fusion 268

Complement 3F The laser as source of coherent light 2713F.1 The advantages of laser light sources 271

3F.1.1 Geometrical properties 2713F.1.2 Spectral and temporal properties 2723F.1.3 The manipulation of laser beams 273

3F.2 Laser measurement of distances 2733F.3 Remote sensing using lasers: the LIDAR 275

3F.3.1 Atmospheric LIDAR 275






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3F.3.2 Coherent LIDAR 2763F.3.3 Measurement of angular velocities 276

3F.4 Optical telecommunications 2793F.5 Laser light and other information technologies 280

Complement 3G Nonlinear spectroscopy 2833G.1 Homogeneous and inhomogeneous broadening 2833G.2 Saturated absorption spectroscopy 284

3G.2.1 Holes in a population distribution 2853G.2.2 Saturated absorption in a gas 286

3G.3 Doppler-free two-photon spectroscopy 2903G.3.1 Two-photon transitions 2903G.3.2 Elimination of Doppler broadening 2913G.3.3 Properties of Doppler-free two-photon spectroscopy 293

3G.4 The spectroscopy of the hydrogen atom 2943G.4.1 A short history of hydrogen atom spectroscopy 2943G.4.2 The hydrogen atom spectrum 2953G.4.3 Determination of the Rydberg constant 296

Part II Quantum description of light and its interaction with matter 299

4 Quantization of free radiation 3014.1 Classical Hamiltonian formalism and canonical quantization 302

4.1.1 Quantizing a system of material particles 3024.1.2 Classical Hamiltonian formulation: Hamilton’s equations 3034.1.3 Canonical quantization 3044.1.4 Hamiltonian formalism for radiation: stating the problem 304

4.2 Free electromagnetic field and transversality 3054.2.1 Maxwell’s equations in vacuum 3054.2.2 Spatial Fourier expansion 3054.2.3 Transversality of the free electromagnetic field and polarized

Fourier components 3074.2.4 Vector potential in the Coulomb gauge 309

4.3 Expansion of the free electromagnetic field in normal modes 3104.3.1 Dynamical equations of the polarized Fourier components 3104.3.2 Normal variables 3114.3.3 Expansion of the free field in normal modes 3124.3.4 Analytic signal 3144.3.5 Other normal modes 314

4.4 Hamiltonian for free radiation 3154.4.1 Radiation energy 3154.4.2 Conjugate canonical variables for a radiation mode 316

4.5 Quantization of radiation 3174.5.1 Canonical commutation relations 317






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4.5.2 Hamiltonian of the quantized radiation 3184.5.3 Field operators 319

4.6 Quantized radiation states and photons 3194.6.1 Eigenstates and eigenvalues of the radiation Hamiltonian 3204.6.2 The notion of a photon 3214.6.3 General radiation state 323

4.7 Conclusion 324

Complement 4A Example of the classical Hamiltonian formalism: chargedparticle in an electromagnetic field 325

Complement 4B Momentum and angular momentum of radiation 3274B.1 Momentum 327

4B.1.1 Classical expression 3274B.1.2 Momentum operator 328

4B.2 Angular momentum 3284B.2.1 Classical expression 3284B.2.2 Angular momentum operators 332

Complement 4C Photons in modes other than travelling plane waves 3344C.1 Changing the normal mode basis 334

4C.1.1 Unitary transformation of creation and annihilation operators 3344C.1.2 New normal modes 3354C.1.3 Invariance of the vacuum and photons in mode m 3364C.1.4 Invariance of the total photon number 3364C.1.5 Properties of photons in different bases 3374C.1.6 Example: 1D standing wave modes 3374C.1.7 Choosing the best mode basis to suit a physical situation 338

4C.2 Photons in a wave packet 339

5 Free quantum radiation 3415.1 Photodetectors and semi-reflecting mirrors. Homodyne detection of the

quadrature components 3425.1.1 Photodetection 3435.1.2 Semi-reflecting mirror 3455.1.3 Homodyne detection 346

5.2 The vacuum: ground state of quantum radiation 3505.2.1 Non-commutativity of the field operators and Heisenberg relations

for radiation 3505.2.2 Vacuum fluctuations and their physical consequences 351

5.3 Single-mode radiation 3535.3.1 Classical description: phase, amplitude and quadratures 3545.3.2 Single-mode quantum radiation: quadrature observables and phasor

representation 355






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5.3.3 Single-mode number state 3585.3.4 Quasi-classical states |α�〉 3605.3.5 Other quantum states of single-mode radiation: squeezed states and

Schrödinger cats 3655.3.6 The limit of small quantum fluctuations and the photon number–phase

Heisenberg relation 3665.3.7 Light beam propagating in free space 368

5.4 Multimode quantum radiation 3715.4.1 Non-factorizable states and entanglement 3715.4.2 Multimode quasi-classical state 3735.4.3 One-photon multimode state 375

5.5 One-photon interference and wave–particle duality. An applicationof the formalism 3775.5.1 Mach–Zehnder interferometer in quantum optics 3775.5.2 Quasi-classical incoming radiation 3795.5.3 Particle-like incoming state 3805.5.4 Wave–particle duality for a particle-like state 3815.5.5 Wheeler’s delayed-choice experiment 382

5.6 A wave function for the photon? 3835.7 Conclusion 385

Complement 5A Squeezed states of light: the reduction of quantumfluctuations 387

5A.1 Squeezed states: definition and properties 3875A.1.1 Definition 3875A.1.2 Expectation values of field observables for a squeezed state 3885A.1.3 The squeezing operator 3915A.1.4 Transmission of a squeezed state by a beamsplitter 3925A.1.5 Effect of losses 393

5A.2 Generation of squeezed light 3945A.2.1 Generation by parametric processes 3945A.2.2 Other methods 395

5A.3 Applications of squeezed states 3965A.3.1 Measurement of small absorption coefficients 3965A.3.2 Interferometric measurements 397

Complement 5B One-photon wave packet 3985B.1 One-photon wave packet 398

5B.1.1 Definition and single photodetection probability 3985B.1.2 One-dimensional wave packet 3995B.1.3 Spontaneous emission photon 401

5B.2 Absence of double detection and difference with a classical field 4035B.2.1 Semi-reflecting mirror 4035B.2.2 Double detection with a classical wave packet 405






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5B.3 Two one-photon wave packets on a semi-reflecting mirror 4085B.3.1 Single detections 4085B.3.2 Joint detections 409

5B.4 Quasi-classical wave packet 411

Complement 5C Polarization-entangled photons and violationof Bell’s inequalities 413

5C.1 From the Bohr–Einstein debate to the Bell inequalities and quantuminformation: a brief history of entanglement 413

5C.2 Photons with correlated polarization: EPR pairs 4155C.2.1 Measuring the polarization of a single photon 4155C.2.2 Photon pairs and joint polarization measurements 4175C.2.3 EPR pairs with correlated polarizations 4195C.2.4 The search for a picture to interpret the correlations between

widely separated measurements 4215C.3 Bell’s theorem 425

5C.3.1 Bell inequalities 4255C.3.2 Conflict with quantum mechanics 4265C.3.3 Locality condition and relativistic causality. Experiment

with variable polarizers 4285C.4 The experimental verdict and violation of the Bell inequalities 4295C.5 Conclusion: from quantum nonlocality to quantum information 432

Complement 5D Entangled two-mode states 4345D.1 General description of a two-mode state 434

5D.1.1 General considerations 4345D.1.2 Schmidt decomposition 4355D.1.3 Correlations between measurements carried out on the

two modes 4365D.2 Twin photon states 437

5D.2.1 Definition and properties 4375D.2.2 Production 438

5D.3 Relation between squeezing and entanglement 4395D.3.1 General considerations 4395D.3.2 Mixing two squeezed states on a semi-reflecting mirror 4395D.3.3 Non-destructive measurement of two complementary

variables: the ‘EPR paradox’ 441

Complement 5E Quantum information 4435E.1 Quantum cryptography 443

5E.1.1 From classical to quantum cryptography 4435E.1.2 Quantum cryptography with entangled photons 4445E.1.3 From theory to practice 4465E.1.4 The no-cloning theorem 447






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5E.1.5 And if there were no entangled states? The BB84 protocol 4485E.1.6 Experimental results 449

5E.2 Quantum computing 4495E.2.1 Quantum bits or ‘qubits’ 4495E.2.2 The Shor factorization algorithm 4505E.2.3 Working principle of a quantum computer 4515E.2.4 Practical matters 453

5E.3 Quantum teleportation 4545E.4 Conclusion 456

6 Interaction of an atom with the quantized electromagnetic field 4576.1 Classical electrodynamics and interacting fields and charges 458

6.1.1 The Maxwell–Lorentz equations 4586.1.2 Decomposition of the electromagnetic field into transverse

and longitudinal components. Radiation 4606.1.3 Polarized Fourier components of the radiation and the vector

potential in the Coulomb gauge 4626.1.4 Normal variables for radiation and expansion in polarized, travelling

plane waves 4626.1.5 Generalized particle momentum. Radiation momentum 4636.1.6 Hamiltonian in the Coulomb gauge 464

6.2 Interacting fields and charges and quantum description in theCoulomb gauge 4676.2.1 Canonical quantization 4676.2.2 Hamiltonian and state space 4686.2.3 Interaction Hamiltonian 469

6.3 Interaction processes 4716.3.1 The Hamiltonian HI1 4716.3.2 Absorption 4716.3.3 Emission 4736.3.4 Rabi oscillation 4746.3.5 The Hamiltonian HI2 and elastic scattering 475

6.4 Spontaneous emission 4776.4.1 Principle of the calculation 4776.4.2 Quasi-continuum of one-photon states and density of states 4796.4.3 Spontaneous emission rate in a given direction 4816.4.4 Lifetime of the excited state and natural width 4826.4.5 Spontaneous emission: a joint property of the atom and the vacuum 484

6.5 Photon scattering by an atom 4856.5.1 Scattering matrix elements 4856.5.2 Scattering cross-section 4876.5.3 Qualitative description of some scattering processes 4886.5.4 Thomson scattering cross-section 493






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6.6 Conclusion. From the semi-classical to the quantum treatmentof atom–light interaction 495

Complement 6A Hamiltonian formalism for interacting fields and charges 4986A.1 Hamiltonian formalism and canonical quantization 4986A.2 Hamilton’s equations for particles and radiation 498

6A.2.1 Classical Hamiltonian for the charge–field system 4986A.2.2 Hamilton’s equations for the charges 4996A.2.3 Hamilton’s equations for the radiation 4996A.2.4 Conclusion 501

Complement 6B Cavity quantum electrodynamics 5026B.1 Presentation of the problem 5026B.2 Eigenmodes of the coupled atom–cavity system 504

6B.2.1 Jaynes–Cummings model 5046B.2.2 Diagonalization of the Hamiltonian 5056B.2.3 Spontaneous emission of an excited atom placed in the empty

cavity 5086B.3 Evolution in the presence of an intracavity field 510

6B.3.1 Field initially in a number state 5106B.3.2 Field initially in an ‘intense’ quasi-classical state: semi-classical

limit 5116B.3.3 Field initially in a quasi-classical state with a small number

of photons 5126B.4 Effect of cavity losses: the Purcell effect 5136B.5 Conclusion 517

Complement 6C Polarization-entangled photon pairs emitted in anatomic radiative cascade 518

6C.1 Introduction. Entangled photon pairs for real experiments 5186C.2 Photon pair emitted in an atomic radiative cascade J = 0 → J = 1 → J = 0.

Elementary process 5196C.2.1 Description of the system 5196C.2.2 Emission of photon ν1 and entangled atom–radiation state 5206C.2.3 Emission of photon ν2 and elementary EPR pair 521

6C.3 Generalization and sum over frequencies 5236C.4 Two-photon excitations 524

Part III Applying both approaches 527

7 Nonlinear optics. From the semi-classical approach to quantum effects 5297.1 Introduction 5297.2 Electromagnetic field in a nonlinear medium. Semi-classical treatment 530

7.2.1 Linear susceptibility 530






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7.2.2 Nonlinear susceptibility 5317.2.3 Propagation in a nonlinear medium 533

7.3 Three-wave mixing. Semi-classical treatment 5357.3.1 Frequency addition 5357.3.2 Phase matching 5377.3.3 Coupled dynamics of three-wave mixing 5407.3.4 Parametric amplification 5427.3.5 Frequency doubling with pump depletion 5447.3.6 Parametric fluorescence 544

7.4 Quantum treatment of parametric fluorescence 5457.4.1 Unavoidability and advantages of the quantum treatment 5457.4.2 Quantum treatment of three-wave mixing 5467.4.3 Perturbative treatment of parametric fluorescence 5477.4.4 Change of picture: the Heisenberg representation 5487.4.5 Simultaneous emission of parametric fluorescence photons 5507.4.6 Two-photon interference 553

7.5 Conclusion 559

Complement 7A Parametric amplification and oscillation. Semi-classicaland quantum properties 560

7A.1 Classical description of parametric amplification 5607A.1.1 Non-degenerate case 5607A.1.2 Degenerate case 561

7A.2 The optical parametric oscillator (OPO) 5627A.2.1 Description of the system 5627A.2.2 Singly resonant OPO 5637A.2.3 Doubly resonant OPO 564

7A.3 Quantum features of parametric amplification 5677A.3.1 Quantum description of attenuation and amplification

processes 5677A.3.2 Non-degenerate parametric amplification 5697A.3.3 Degenerate parametric amplification 570

7A.4 Quantum fluctuations in the fields produced by a doublyresonant OPO 5717A.4.1 The small quantum fluctuation limit 5717A.4.2 Frequency-degenerate OPO below threshold: producing squeezed

states of the field 5737A.4.3 Non-frequency-degenerate OPO above threshold: producing

twin beams 574

Complement 7B Nonlinear optics in optical Kerr media 5777B.1 Examples of third-order nonlinearities 577

7B.1.1 Nonlinear response of two-level atoms 5777B.1.2 Nonlinearity by optical pumping 579






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7B.2 Field propagation in Kerr media 5817B.2.1 Single incident wave 5817B.2.2 Two travelling waves propagating in opposite directions 582

7B.3 Optical bistability 5837B.4 Phase conjugate mirror 586

7B.4.1 Degenerate four-wave mixing 5867B.4.2 Phase conjugation 5877B.4.3 Calculating the reflection coefficient 590

7B.5 Propagation of a spatially non-uniform wave in a Kerr medium 5927B.5.1 Self-focusing 5927B.5.2 Spatial soliton and self-focusing 593

7B.6 Propagation of a pulse in a Kerr medium 5957B.6.1 Self-phase modulation 5957B.6.2 Propagation in a dispersive linear medium 5957B.6.3 Propagation in a dispersive Kerr medium. Temporal soliton 597

8 Laser manipulation of atoms. From incoherent atom optics to atom lasers 5998.1 Energy and momentum exchanges in the atom–light interaction 600

8.1.1 Quantum description of the external degrees of freedom of the atom 6018.1.2 Momentum conservation 6018.1.3 Energy conservation: the Doppler and the recoil shifts 603

8.2 Radiative forces 6048.2.1 Closed two-level atom in a quasi-resonant laser wave 6048.2.2 Localized atomic wave packet and classical limit 6058.2.3 Radiative forces: general expression 6078.2.4 Steady-state radiative forces for a closed two-level atom 6088.2.5 Resonance-radiation pressure 6108.2.6 Dipole force 614

8.3 Laser cooling and trapping of atoms, optical molasses 6188.3.1 Doppler cooling 6188.3.2 Coefficient of friction and Doppler molasses 6198.3.3 Magneto-optical trap 6218.3.4 Fluctuations and heating 6248.3.5 Fluctuations of the resonance-radiation pressure 6258.3.6 Momentum fluctuations and heating for a Doppler molasses 6278.3.7 Equilibrium temperature for a Doppler molasses 6298.3.8 Going under the Doppler temperature and Sisyphus cooling 6308.3.9 Cooling below the recoil temperature 632

8.4 Gaseous Bose–Einstein condensates and atom lasers 6338.4.1 Bose–Einstein condensation 6338.4.2 Obtaining dilute atomic Bose–Einstein condensates. Laser cooling

and evaporative cooling 6358.4.3 Ideal Bose–Einstein condensate and atomic wavefunction 6388.4.4 Observing the wavefunction of the Bose–Einstein condensate 639






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8.4.5 Dilute Bose–Einstein condensate with interactions 6408.4.6 Coherence properties of a Bose–Einstein condensate and interference

between two Bose–Einstein condensates 6418.4.7 Atom lasers 6448.4.8 Conclusion. From photon optics to atom optics and beyond 647

Complement 8A Cooling to sub-recoil temperatures by velocity-selectivecoherent population trapping 651

8A.1 Coherent population trapping 6518A.2 Velocity-selective coherent population trapping and sub-recoil cooling 6548A.3 Quantum description of the atomic motion 6568A.4 Fluorescence rate of a state |ψNC(p)〉 6598A.5 Practical limits. The fragility of coherence 659

Index 661






Foreword

Atomic, molecular and optical physics is a field which, during the last few decades, hasknown spectacular developments in various directions, like nonlinear optics, laser coolingand trapping, quantum degenerate gases, quantum information. Atom–photon interactionsplay an essential role in these developments. This book presents an introduction to quantumoptics which, I am sure, will provide an invaluable help to the students, researchers andengineers who are beginning to work in these fields and who want to become familiar withthe basic concepts underlying electromagnetic interactions.

Most books dealing with these subjects follow either a semi-classical approach, wherethe field is treated as a classical field interacting with quantum particles, or a full quantumapproach where both systems are quantized. The first approach is often oversimplified andfails to describe correctly new situations that can now be investigated with the developmentof sophisticated experimental techniques. The second approach is often too difficult forbeginners and lacks simple physical pictures, very useful for an initial understanding of aphysical phenomenon. The advantage of this book is that it gives both approaches, startingwith the first, illustrated by several simple examples, and introducing progressively thesecond, clearly showing why it is essential for the understanding of certain phenomena.The authors also clearly demonstrate, in the case of non-linear optics and laser cooling,how advantageous it may be to combine both approaches in the analysis of an experimentalsituation and how one can get from each point of view useful, complementary physicalinsights. I believe that this challenge to present and to illustrate both approaches in a singlebook has been taken up successfully. Whatever their ultimate interests, the readers of thiswork will be exposed to an important example of a broad and vibrant field of researchand they will better understand the intellectual enrichment and the technical developmentswhich result from it.

To write a book on such a broad topic, the authors must obviously possess wide knowl-edge of the field, they must have thought long and hard about the basic concepts and aboutthe different levels of complexity with which one can approach the topics. They mustalso have a deep and concrete knowledge about experimental and technical details and themany problems which daily confront a laboratory researcher. Having worked extensivelywith them, I know the authors of this work fulfil these requirements. I have the highestadmiration for their enthusiasm, their scientific rigour, their ability to give simple and pre-cise physical explanations, and their quest to illuminate clearly the difficult points of thesubject without oversimplification. Each of them has made many original contributions tothe development of this important field of physics, and they and their younger collaboratorsfor this book work at the cutting edge of modern quantum optics. In reading the book, I amtherefore not surprised to find their many fine qualities reflected in the text. The general






xxiv Foreword�

organisation of the main chapters and complementary sections allows reading on manydifferent levels. When the authors discuss a new physical problem, they begin the analy-sis with the simplest possible model. A large variety of experiments and applications arepresented with clear diagrams and explanations and with constant attention to highlightingthe guiding principles, the orders of magnitude and the problems which remain open.

This work will allow a broad audience an easier access to a field of science which con-tinues to see spectacular developments. I believe that science is not simply a matter ofexploring new horizons. One must also make the new knowledge readily available and wehave in this book, a beautiful example of such a pedagogical effort. I would like finallyto evoke the memory of Gilbert Grynberg who participated with Alain Aspect and ClaudeFabre in the writing of a preliminary, much less developed, French version of this book andwho passed away in 2003. Gilbert was an outstanding physicist, a fine person, and had anexceptional talent for explaining in the clearest possible way the most difficult questions.I think that the present book is the best possible tribute to be paid to him.

Claude Cohen-TannoudjiParis, September 2009






Preface

Since its invention in 1960, the laser has revolutionized both the study of optics and ourunderstanding of the nature of light, prompting the emergence of a new field, quantumoptics. Actually, it took decades until the words quantum optics took their current precisemeaning, referring to phenomena which can be understood only by quantizing the elec-tromagnetic field describing light. Surprisingly enough, such quantum optics phenomenahardly existed at the time that the laser was invented, and almost all optics effects could befully understood by describing light as a classical electromagnetic field; the laser was noexception. As a matter of fact, to understand how a laser works, it suffices to use the semi-classical description of matter–light interaction, where the laser amplifying medium, madeof atoms, molecules, ions or semi-conductors, is given a quantum mechanical treatment,but light itself is described by classical electromagnetic waves.

The first part of our book is devoted to presentation of the semi-classical approach andits use in describing various optical phenomena. It includes an elementary exposition ofthe physics of lasers, and some applications of this ubiquitous device. After recallingin Chapter 1 some basic results of the quantum mechanical description of interactioninduced transitions between the atomic energy levels, we use these results in Chapter 2to show how the interaction of a quantized atom with a classical electromagnetic waveleads to absorption or stimulated emission, and to derive the process of laser amplifi-cation that happens when a wave propagates in an inverted medium. Chapter 3 givesan elementary exposition of the physics of laser sources and of the properties of laserlight.

Although the quantum theory of light existed since its development by Dirac in the early1930s, quantum optics theory in its modern sense started when Roy Glauber showed, in theearly 1960s, how to apply it to classical optics devices such as the Michelson stellar inter-ferometer or the Hanbury Brown and Twiss intensity interferometer. At that time it couldhave appeared to be an academic exercise without consequence, since the only known phe-nomenon that demanded quantization of light was spontaneous emission, and it was notclear whether quantum theory was at all useful for describing light freely propagating farfrom the source. Actually, Glauber developed a clear quantum formalism to describe opticsphenomena, and introduced the important notion of quasi-classical states of light, a theoret-ical tool that allowed physicists to understand why all available sources of light, includinglasers, delivered light whose properties could be totally understood in the framework ofthe semi-classical approach. But in doing so, he paved the way for the discovery of newphenomena which can be understood only if light is considered as a quantum system. Itbecame possible to build sources delivering single photon wave packets, pairs of entangledphotons, squeezed beams of light. . .






xxvi Preface�

The second part of our book is devoted to the presentation of the quantum theory oflight and its interaction with matter, and its use in describing many phenomena of modernquantum optics. We show in Chapter 4 how it is possible to write the dynamical equa-tions of a classical electromagnetic field, i.e. Maxwell equations, in a form allowing us touse the canonical quantization procedure to quantize the electromagnetic field, and obtainthe notion of a photon. We then use our results, in Chapter 5, to describe some fullyquantum effects observed in experiments with single photons, squeezed light or pairs ofentangled photons. It is remarkable that many of these experiments, whose first goal wasto demonstrate the highly counter-intuitive, non-classical properties of new types of lightstates, turned out to stimulate the emergence of a new field, quantum information, whereone uses such properties to implement new ways of processing and transmitting data. InChapter 6, we show how to use the quantum optics formalism to describe the interactionbetween light and atoms. We will then revisit in this new framework the phenomena ofabsorption and stimulated emission, already studied in Chapter 2. Moreover, we will nowbe able to give a consistent treatment of spontaneous emission.

Having introduced the full quantum optics formalism and reviewed some remarkablephenomena that could not have been discovered without such a formalism, we would notlike to leave the reader with the impression that he/she can now forget the semi-classicalapproach. Both approaches are definitely useful. On the one hand, there is no reason to usethe, usually more involved, fully quantum analysis, when the situation does not demand it.After all, nobody would use quantum mechanics to describe the motion of planets. Simi-larly, no experimentalist studying fusion plasmas with intense lasers would start using thequantum formalism of light. What is important then is to be able to recognize when thefull quantum theory is necessary, and when one can content oneself with the semi-classicalmodel. To help the reader to develop their intuition about this point, we present, in the thirdpart of this book, two topics, non-linear optics in Chapter 7, laser cooling and trappingof atoms in Chapter 8, where it is convenient to ‘juggle’ between the two approaches,each being better adapted to one or the other particular phenomenon. As ‘the cherry on thecake’, we will give in Chapter 8 an elementary presentation of atomic Bose–Einstein con-densates, and emphasize the analogy between such a system, where all atoms are describedby the same matter wave, and a laser beam where all photons are described by the samemode of the electromagnetic field. When we started to write the first French version of thisbook, we had never dreamt of being able to finish it with a presentation on atom lasers.

This book is composed of chapters, in which we present the fundamental concepts andsome applications to important quantum optics phenomena, and of complements, whichpresent supplementary illustrations or applications of the theory presented in the mainchapter. The choice of these examples is, of course, somewhat arbitrary. We present themas a snap-shot of the current state of a field which is rapidly evolving. Complements ofanother type are intended to give some supplementary details about a derivation or aboutconcepts presented in the chapter.

The prerequisite for using this book is to have followed an elementary course on bothelectromagnetism (Maxwell’s equations) and quantum mechanics (Schrödinger formula-tion in the Dirac formalism of bras and kets, with application to the harmonic oscillator).The book is then self-consistent, and can be used for an advanced undergraduate, or for






xxvii Preface�

a first graduate course on quantum optics. Although we do not make use of the mostadvanced tools studied at graduate school, we make all efforts to provide the reader withsolid derivations of the main results obtained in the chapters. For example, to quantizeelectromagnetic waves, first in free space, and then in interaction with charges, we do notuse the Lagrangian formalism, but we introduce enough elements of the Hamiltonian for-malism to be able to apply the canonical quantization rules. We are thus able to provide thereader with a solid derivation of the basic quantum optics formalism rather than bringingit in abruptly. On the other hand, when we want to present in a Complement a particu-larly important and interesting phenomenon, we do not hesitate to ask the reader to admita result which results from more advanced courses.

We have done our best to merge the French teaching tradition of logical and deductiveexposition with the more pragmatic approach that we use as researchers, and as advisorsto Ph.D. and Masters students. We have taught the content of this book for many years toadvanced undergraduate or beginning graduate students, and this text represents the resultsof our various teaching experiences.






Acknowledgements

In this book, we refer to a number of textbooks in which general elementary resultsof quantum mechanics are established, in particular the book by Jean-Louis Basdevantand Jean Dalibard,1 which we indicate by the short-hand notation ‘BD’, and the one byClaude Cohen-Tannoudji, Bernard Diu and Franck Laloë,2 which we denote by ‘CDL’.On the more advanced side, we sometimes refer to more rigorous demonstrations, orto more advanced developments, that can be found in the two books written by ClaudeCohen-Tannoudji, Jacques Dupont-Roc and Gilbert Grynberg, to which we refer under theshort-hand notations ‘CDG I’ and ‘CDG II’, respectively.3,4

It is not possible to mention all those who have contributed to or influenced thiswork. We would first like to acknowledge, however, our principal inspiration, ClaudeCohen-Tannoudji, whose lectures at the Collège de France we have had the good fortune tobe able to follow for three decades. At the other end of the spectrum, we also owe a lot toour students at Ecole Polytechnique, Ecole Normale Supérieure, Institut d’Optique Gradu-ate School, Université Pierre et Marie Curie, as well as the many graduate students we haveadvised towards Masters or Ph.D. work. By their sharp questioning, never content with avague answer, they have forced us to improve our lectures year upon year. We cannot citeall of the colleagues with whom we have taught, and from whom we have borrowed manyideas and materials, but we cannot omit to mention the names of Manuel Joffre, EmmanuelRosencher, Philippe Grangier, Michel Brune, Jean-François Roch, François Hache, DavidGuéry-Odelin, Jean-Louis Oudar, Hubert Flocard, Jean Dalibard, Jean-Louis Basdevant.In addition, Philippe Grangier was kind enough to write Complement 5E on quantuminformation.

Martine Maguer, Dominique Toustou, and all the team of Véronique Pellouin at theCentre Polymedia of Ecole Polytechnique have done an impressive and professional job inpreparing the manuscript with its figures. We would like also to thank the Centre Nationaldu Livre, of the French Ministry of Culture, for its important financial support in thetranslation of our French text.

1 J.-L. Basdevant and J. Dalibard, Quantum Mechanics, Springer (2002).2 C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, Wiley (1977).3 C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and Atoms – Introduction to quantum

electrodynamics, Wiley (1989).4 C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-photon Interactions: Basic processes and

applications, Wiley (1992).






xxix Acknowledgements�

Special acknowledgement

This book has three authors, who wrote the original French textbook on which it is based.5

Sadly, as we had just started to prepare the English version, Gilbert Grynberg passedaway, and for several years we were discouraged and not able to carry on working on theEnglish version. Eventually, we realized that the best demonstration of all that we owe toour former friend and colleague was to resume this project. But we realized then that almosta decade after writing the French version, quantum optics had evolved tremendously, andwe had also personally evolved in the ways in which we understood and taught the sub-ject. The original French book, therefore, had not only to be translated but also widelyrevised and updated. In this long-term enterprise, we have been fortunate to have fantastichelp from our younger colleagues (and former students) Fabien Bretenaker and AntoineBrowaeys. For the past three years they have devoted innumerable hours to helping uscomplete the revised version, and without their help this would not have been possible.There is not a single chapter that has not been strongly influenced by their thorough crit-icisms, their strong suggestions, and their contributions to the rewriting of the text, not tospeak of the double checking of equations. Moreover, they bring to this book the point ofview of a new generation of physicists who have been taught quantum optics in its modernsense, in contrast to we who have seen it developing while we were already engaged inresearch. For their priceless contribution, we can only express to Fabien Bretenaker andAntoine Browaeys our immense gratitude. Gilbert would have been happy to have suchwonderful collaborators.

Alain Aspect and Claude Fabre,Palaiseau, Paris, July 2009.

5 Gilbert Grynberg, Alain Aspect, Claude Fabre, Introduction aux lasers et à l’Optique Quantique, cours del’Ecole Polytechnique, Ellipses, Paris (1997).






introduction to quantum optics from the semi-classical

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