Introduction to Modern Quantum Optics

Jin-Sheng Peng Gao-Xiang Li

Huazhong Normal University, China

V f e World Scientific » • Singapore* NewJerseyL Singapore * New Jersey* London* Hong Kong

IX

CONTENTS

Preface

P A R T I. T h e o r y of t h e i n t e r a c t i o n b e t w e e n a t o m a n d r a d i a t i o n field

C h a p t e r 1. T h r e e p i c t u r e s in q u a n t u m m e c h a n i c s

1.1. The Schrödinger picture 3

1.2. The Heisenberg picture 8

1.3. The interaction picture 11

1.3.1. Equation of motion in the interaction picture 11

1.3.2. A formal solution of the state vector |\PJ(t)) by the

perturbat ion theory 13

1.4. The density operator 15

1.4.1. Density operator and its general properties 16

1.4.2. Solution of the equation of motion for the density operator 20

Chapter 2. Two- l eve l a t o m a n d the opt ica l B l o c h e q u a t i o n

2.1. Two-level atom 25

2.2. Hamiltonian of a two-level atom interacting with an electromagnetic

field 26

2.3. The optical Bloch equation 28

2.4. Description of the dynamical behavior of a two-level atom interacting

with the radiation field by the density matr ix 31

2.4.1. Density matr ix equation describing a two-level atom without

decay 32

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2.4.2. Density matr ix equation of a two-level atom with decay 34

C h a p t e r 3 . Q u an t i zed descr ip t ion of rad ia t ion field

3.1. Classical description of the electromagnetic field in vacuum 37 3.2. Quantization of the radiation field 42

3.2.1. Quantization of the electromagnetic field 42 3.2.2. Momentum and spin of the photon 45

3.3. State functions describing the light field 48 3.3.1. Photon-number states 48 3.3.2. The coherent states of light 52 3.3.3. The phase operators and the phase states 60 3.3.4. Chaotic states of light 72

C h a p t e r 4 . D icke H a m i l t o n i a n a n d J a y n e s - C u m m i n g s M o d e l

4.1. Dicke Hamiltonian of an a tom interacting with the radiation field 77 4.2. Spontaneous emission of an excited atom 82 4.3. The Jaynes-Cummings model 87

C h a p t e r 5. Q u a n t u m t h e o r y of a smal l s y s t e m coupled to a reservoir

5.1. Classical Langevin equation and Fokker-Planck equation 93 5.1.1. Langevin equation 94 5.1.2. Fokker-Planck equation 98

5.2. Master equation for a quantum harmonic oscillator and a two-level atom 107 5.2.1. Master equation for a quantum harmonic oscillator 108 5.2.2. Master equation for a two-level atom coupled to a bath field 116

5.3. Characteristic function and the quasi-probability distribution for the quantum harmonic oscillator 118 5.3.1. Normal ordering representation 119 5.3.2. Anti-normal ordering representation 122 5.3.3. Symmetric ordering representation 125

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P A R T II. T h e q u a n t u m proper t i e s of l ight

C h a p t e r 6. C o h e r e n c e of l ight

6.1. Classical coherence of light 135 6.1.1. Temporal coherence of light 135 6.1.2. Spatial coherence of light 137 6.1.3. The first-order correlation function 138 6.1.4. The higher-order correlation function 142

6.2. Quantum theory of the coherence of light 145 6.2.1. Quantum correlation functions 145 6.2.2. Bunching and antibunching effects of light 149 6.2.3. Intermode correlation property for the two-mode field 155

Chapter 7. S q u e e z e d s t a t e s of l ight

7.1. Squeezed states of a single-mode field 160 7.1.1. Squeezed coherent states 160 7.1.2. Squeezed vacuum field 176

7.2. Squeezed states of a two-mode radiation field 177 7.3. Higher-order squeezing of a radiation field and the amplitude square

squeezing 185 7.3.1. Higher-order squeezing of a radiation field 185 7.3.2. Amplitude square squeezing 188 7.3.3. Independence of the different definitions of the squeezing for

the radiation field 189 7.4. Squeezing of light in the Jaynes-Cummings model 190

Chapter 8. R e s o n a n c e f luorescence

8.1. Resonance fluorescence distribution of a two-level atom 200 8.1.1. Dressed canonical transformation 200 8.1.2. Spectral distribution of the resonance fluorescence of a

two-level atom 206 8.1.3. Linewidth of the fluorescence spectrum 209 8.1.4. Intensity distribution of the resonance fluorescence spectrum 214

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8.2. Resonance fluorescence spectra of a three-level atom 222 8.2.1. Hamiltonian of a three-level atom under the interaction of a

bimodal field 222 8.2.2. Resonance fluorescence spectrum of a three-level atom

interacting with a strong and a weak monochromatic laser field 224

8.2.3. Resonance fluorescence spectral distribution of a three-level a tom driven by two strong laser fields 230

8.3. Single-atom resonance fluorescence described by the density matr ix theory 236

C h a p t e r 9. Superf luorescence

9.1. Elementary features of superfluorescence 247 9.2. Quasi-classical description of superfluorescence 251 9.3. Quantum theoretical description of superfluorescence 258

9.3.1. Heisenberg equation of the system 258 9.3.2. Dicke model for superfluorescence 263 9.3.3. Quantum statistical properties of superfluorescence 268

9.4. Superfluorescent beats 276 9.4.1. Basic characteristics of the superfluorescent beats 276 9.4.2. Superfluorescent beats in the Dicke model 278

C h a p t e r 10. Opt ica l B i s tab i l i t y

10.1. Basic characteristics and the production mechanism of optical bistability 287

10.2. Quantum description of the dispersive optical bistability 294 10.2.1. Hamiltonian describing the optical bistability system 295 10.2.2. Optical bistability properties of the system 297

Chapter 11 . Effects of v i r tua l p h o t o n proces se s

11.1. Relation between the Lamb shift of a Hydrogen atom and the virtual photon field 305

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11.2. Influence of the vir tual photon field on the phase fluctuations of the radiation field 311 11.2.1. Time evolution of the phase operator in the atom-field

coupling system with the rotating-wave approximation 311 11.2.2. Time evolution of the phase operator without the

rotating-wave approximation 315 11.3. Influences of the vir tual photon processes on the squeezing of light 319

11.3.1. Squeezing of the field in the two-photon Jaynes-Cummings model with the rotating-wave approximation 320

11.3.2. Influences of the vir tual photon processes on the squeezing of light 323

P A R T III. Q u a n t u m p r o p e r t i e s of a t o m i c b e h a v i o r u n d e r t h e in terac t ion of a rad ia t ion field

C h a p t e r 12 . C o l l a p s e s a n d r e v i v a l s of a t o m i c p o p u l a t i o n s

12.1. Time evolution of the atomic operator of a two-level atom under the interaction of a classical electromagnetic field 333

12.2. Periodic collapses and revivals of an atom interacting with a quantized field 336 12.2.1. Time development of atomic operators under the interaction

of the field in a number state \m) 337 12.2.2. Periodic collapses and revivals of the atom under the

interaction of a coherent field 338 12.3. Periodic collapses and revivals of the a tom in the two-photon

Jaynes-Cummings model 347 12.4. Time evolution of the atomic operators for a three-level atom

interacting with a single-mode field 351 12.4.1. Time evolution of the state vector of the system 351 12.4.2. Periodic collapses and revivals of the atomic populations 354

Chapter 13 . Squeez ing effects of t h e a tomic o p e r a t o r s

13.1. Definition of the atomic operator squeezing 361

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13.2. Squeezing of atomic operators in the two-photon Jaynes-Cummings

model 365

13.2.1. Squeezing of atomic operators in the vacuum field 367

13.2.2. Squeezing of atomic operators in the superposition state

field 372

13.2.3. Squeezing of atomic operators in the coherent state field 375

13.3. Squeezing of atomic operators in the resonance fluorescence system 377

C h a p t e r 14 . C o h e r e n t t r a p p i n g of the a tomic p o p u l a t i o n

14.1. Atomic population coherent trapping and phase properties in the

system of a V-configuration three-level atom interacting with a

bimodal field 382

14.1.1. Time evolution of the state vector of the system 383

14.1.2. Time evolution of the phase operator in the atom-field

coupling system 385

14.1.3. Coherent trapping of the atomic population 388

14.2. Coherent trapping of the atomic population for a V-configuration

three-level atom driven by a classical field in a heat bath 392

14.2.1. Time evolution of the reduced density matrix p of the atom 393

14.2.2. Steady-state behavior and the coherent trapping of the

atomic populations 396

C h a p t e r 15. Q u a n t u m character i s t i cs of a t w o - a t o m s y s t e m u n d e r t h e in terac t ion of the rad ia t ion field

15.1. Hamiltonian of a two-atom system with the dipole-dipole interaction 401

15.1.1. Hamiltonian of the electric dipole-dipole interaction between

two atoms 402

15.1.2. Hamiltonian of a two-atom system with the dipole-dipole

interaction induced by the fluctuations of the vacuum field 403

15.2. Quantum characteristics of the two-atom coupling system under

the interaction of a weak field 409

15.2.1. Time evolution of the atomic population inversion of a two-atom system 411

15.2.2. Influence of the dipole-dipole interaction on the squeezing of atomic operators 416

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15.3. Periodic collapses and revivals and the coherent population trapping in the two-atom system under the interaction of a coherent field 420 15.3.1. Periodic collapses and revivals of atomic populations

in the two-atom system 424 15.3.2. Atomic population coherent trapping in the two-atom

coupling system 431

C h a p t e r 16. A u t o i o n i z a t i o n of the a t o m in a laser field

16.1. Autoionization of the atom in a weak laser field 440 16.2. Autoionization of the atom under the interaction of a strong

laser field 450 16.3. Above threshold ionization of the atom in a strong laser field 457

16.3.1. Influences of the second-order ionization processes on the low-energy photoelectron spectrum 464

16.3.2. Higher-energy photoelectron spectrum and the peak switching effect 466

C h a p t e r 17. M o t i o n of t h e a t o m in a laser field

17.1. Atomic diffraction and deflection in a standing-wave field 472 17.1.1. State function of the system of an atom interacting with a

standing-wave field 472 17.1.2. Diffraction of the atom under the interaction of a laser field 479 17.1.3. Deflection of the atom in a standing wave field 487

17.2. Force on an atom exerted by the radiation field 490 17.2.1. Quasi-classical description of the radiation force 492 17.2.2. Description of the radiative dipole force by means of the

dressed state method 501

C h a p t e r 18. Laser coo l ing

18.1. Decelerating the motion of atoms by use of a laser field 521 18.2. Quantum theoretical description of the laser cooling 524

18.2.1. Hamiltonian describing the system of a polarization laser field interacting with a quasi-two-level atom 524

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18.2.2. Time evolution of the density matr ix elements of the atomic internal states 529

18.2.3. Radiation force acting on the atom by the laser field 535 18.2.4. Physical mechanism of the laser cooling 540

18.3. Limited temperature of the laser cooling 544 18.3.1. Atomic momentum diffusion in a laser field 544 18.3.2. Equilibrium temperature of the laser cooling 552 18.3.3. Laser cooling below the one-photon recoil energy by the

velocity-selective coherent population trapping 554

Index 559