Introduction to Modern Quantum Optics Jin-Sheng Peng Gao-Xiang Li Huazhong Normal University, China Vfe World Scientific» Singapore* * NewJerseyL Jersey* London* Hong Kong
IX CONTENTS Preface PART I. Theory of the interaction between atom and radiation field Chapter 1. Three pictures in quantum mechanics 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 \P J (t)) by the perturbation 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-level atom and the optical Bloch equation 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 matrix 31 2.4.1. Density matrix equation describing a two-level atom without decay 32
X о Contents 2.4.2. Density matrix equation of a two-level atom with decay 34 Chapter 3. Quantized description of radiation 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 Chapter 4. Dicke Hamiltonian and Jaynes-Cummings Model 4.1. Dicke Hamiltonian of an atom interacting with the radiation field 77 4.2. Spontaneous emission of an excited atom 82 4.3. The Jaynes-Cummings model 87 Chapter 5. Quantum theory of a small system 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
о Contents XI PART II. The quantum properties of light Chapter 6. Coherence of light 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. Squeezed states of light 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. Resonance fluorescence 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
xii о Contents 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 atom driven by two strong laser fields 230 8.3. Single-atom resonance fluorescence described by the density matrix theory 236 Chapter 9. Superfluorescence 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 Chapter 10. Optical Bistability 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 virtual photon processes 11.1. Relation between the Lamb shift of a Hydrogen atom and the virtual photon field 305
о Contents xiii 11.2. Influence of the virtual 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 virtual 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 virtual photon processes on the squeezing of light 323 PART III. Quantum properties of atomic behavior under the interaction of a radiation field Chapter 12. Collapses and revivals of atomic populations 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 atom 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. Squeezing effects of the atomic operators 13.1. Definition of the atomic operator squeezing 361
XIV Contents 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 Chapter 14. Coherent trapping of the atomic population 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 Chapter 15. Quantum characteristics of a two-atom system under the interaction of the radiation 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
о Contents xv 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 Chapter 16. Autoionization of the atom 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 Chapter 17. Motion of the atom 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 Chapter 18. Laser cooling 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
XVI о Contents 18.2.2. Time evolution of the density matrix 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