ELEMENTS OF QUANTUM OPTICS
CONTENTS 1 CLASSICAL ELECTROMAGNETIC FIELDS 1 1 1 MAXWELL S EQUATIONS IN A VACUUM 2 1 2 MAXWELL S EQUATIONS IN A MEDIUM 4 1 3 LINEAR DIPOLE OSCILLATOR 10 1 4 COHERENCE 17 1 5 FREE ELECTRON LASERS 22 PROBLEMS 32 2 CLASSICAL NONLINEAR OPTICS 35 2 1 NONLINEAR DIPOLE OSCILLATOR 35 2 2 COUPLED MODE EQUATIONS 38 2 3 CUBIC NONLINEARITY 40 2 4 FOUR WAVE MIXING WITH DEGENERATE PUMP FREQUENCIES 43 2 5 NONLINEAR SUSCEPTIBILITIES 48 PROBLEMS 50 3 QUANTUM MECHANICAL BACKGROUND 51 3 1 REVIEW OF QUANTUM MECHANICS 52 3 2 TIME DEPENDENT PERTURBATION THEORY 64 3 3 ATOM FIELD INTERACTION FOR TWO LEVEL ATOMS 71 3 4 SIMPLE HARMONIC OSCILLATOR 82 PROBLEMS 86 4 MIXTURES AND THE DENSITY OPERATOR 93 4 1 LEVEL DAMPING 94 4 2 THE DENSITY MATRIX 98 4 3 VECTOR MODEL OF DENSITY MATRIX 106 PROBLEMS 112 5 CW FIELD INTERACTIONS 117 5 1 POLARIZATION OF TWO LEVEL MEDIUM 117 5 2 INHOMOGENEOUSLY BROADENED MEDIA 124 5 3 COUNTERPROPAGATING WAVE INTERACTIONS 129 5 4 TWO PHOTON TWO LEVEL MODEL 133 5 5 POLARIZATION OF SEMICONDUCTOR GAIN MEDIA 139 PROBLEMS 146 6 MECHANICAL E ECTS OF LIGHT 151 6 1 ATOM FIELD INTERACTION 152 6 2 DOPPLER COOLING 157 6 3 THE NEAR RESONANT KAPITZA DIRAC E ECT 158 6 4 ATOM INTERFEROMETRY 166 PROBLEMS 169 7 INTRODUCTION TO LASER THEORY 171 7 1 THE LASER SELF CONSISTENCY EQUATIONS 172 7 2 STEADY STATE AMPLITUDE AND FREQUENCY 175 7 3 STANDING WAVE DOPPLER BROADENED LASERS 181 7 4 TWO MODE OPERATION AND THE RING LASER 187 7 5 MODE LOCKING 191 7 6 SINGLE MODE SEMICONDUCTOR LASER THEORY 194 7 7 TRANSVERSE VARIATIONS AND GAUSSIAN BEAMS 198 PROBLEMS 203 8 OPTICAL BISTABILITY 209 8 1 SIMPLE THEORY OF DISPERSIVE OPTICAL BISTABILITY 210 8 2 ABSORPTIVE OPTICAL BISTABILITY 215 8 3 IKEDA INSTABILITY 217 PROBLEMS 220 9 SATURATION SPECTROSCOPY 223 9 1 PROBE WAVE ABSORPTION COE CIENT 224 9 2 COHERENT DIPS AND THE DYNAMIC STARK E ECT 230 9 3 INHOMOGENEOUSLY BROADENED MEDIA 238 9 4 THREE LEVEL SATURATION SPECTROSCOPY 241 9 5 DARK STATES AND ELECTROMAGNETICALLY INDUCED TRANSPARENCY 244 PROBLEMS 247 10 THREE AND FOUR WAVE MIXING 249 10 1 PHASE CONJUGATION IN TWO LEVEL MEDIA 250 10 2 TWO LEVEL COUPLED MODE COE CIENTS 253 10 3 MODULATION SPECTROSCOPY 255 10 4 NONDEGENERATE PHASE CONJUGATION BY FOUR WAVE MIXING 259 PROBLEMS 260 11 TIME VARYING PHENOMENA IN CAVITIES 263 11 1 RELAXATION OSCILLATIONS IN LASERS 264 11 2 STABILITY OF SINGLE MODE LASER OPERATION 267 11 3 MULTIMODE MODE LOCKING 271 11 4 SINGLE MODE LASER AND THE LORENZ MODEL 274 PROBLEMS 276 12 COHERENT TRANSIENTS 281 12 1 OPTICAL NUTATION 282 12 2 FREE INDUCTION DECAY 284 12 3 PHOTON ECHO 285 12 4 RAMSEY FRINGES 288 12 5 PULSE PROPAGATION AND AREA THEOREM 289 12 6 SELF INDUCED TRANSPARENCY 293 12 7 SLOW LIGHT 295 PROBLEMS 296 13 FIELD QUANTIZATION 299 13 1 SINGLE MODE FIELD QUANTIZATION 299 13 2 MULTIMODE FIELD QUANTIZATION 302 13 3 SINGLE MODE FIELD IN THERMAL EQUILIBRIUM 304 13 4 COHERENT STATES 307 13 5 COHERENCE OF QUANTUM FIELDS 311 13 6 QUASI PROBABILITY DISTRIBUTIONS 314 13 7 SCHR ODINGER FIELD QUANTIZATION 318 13 8 THE GROSS PITAEVSKII EQUATION 322 PROBLEMS 324 14 INTERACTION BETWEEN ATOMS AND QUANTIZED FIELDS 327 14 1 DRESSED STATES 328 14 2 JAYNES CUMMINGS MODEL 333 14 3 SPONTANEOUS EMISSION IN FREE SPACE 338 14 4 QUANTUM BEATS 344 PROBLEMS 348 15 SYSTEM RESERVOIR INTERACTIONS 351 15 1 MASTER EQUATION 353 15 2 FOKKER PLANCK EQUATION 362 15 3 LANGEVIN EQUATIONS 364 15 4 MONTE CARLO WAVE FUNCTIONS 369 15 5 QUANTUM REGRESSION THEOREM AND NOISE SPECTRA 374 PROBLEMS 379 16 RESONANCE FLUORESCENCE 383 16 1 PHENOMENOLOGY 384 16 2 LANGEVIN EQUATIONS OF MOTION 387 16 3 SCATTERED INTENSITY AND SPECTRUM 390 16 4 CONNECTION WITH PROBE ABSORPTION 396 16 5 PHOTON ANTIBUNCHING 400 16 6 O RESONANT EXCITATION 403 PROBLEMS 405 17 SQUEEZED STATES OF LIGHT 409 17 1 SQUEEZING THE COHERENT STATE 410 17 2 TWO SIDEMODE MASTER EQUATION 414 17 3 TWO MODE SQUEEZING 417 17 4 SQUEEZED VACUUM 421 PROBLEMS 425 18 CAVITY QUANTUM ELECTRODYNAMICS 427 18 1 GENERALIZED MASTER EQUATION FOR THE ATOM CAVITY SYSTEM 428 18 2 WEAK COUPLING REGIME 430 18 3 STRONG COUPLING REGIME 432 18 4 VELOCITY DEPENDENT SPONTANEOUS EMISSION 435 18 5 INPUT OUTPUT FORMALISM 440 PROBLEMS 443 19 QUANTUM THEORY OF A LASER 445 19 1 THE MICROMASER 447 19 2 SINGLE MODE LASER MASTER EQUATION 454 19 3 LASER PHOTON STATISTICS AND LINEWIDTH 460 19 4 QUANTIZED SIDEMODE BUILDUP 468 PROBLEMS 470 20 ENTANGLEMENT BELL INEQUALITIES AND QUANTUM INFORMATION 473 20 1 EINSTEIN PODOLSKY ROSEN PARADOX AND BELL INEQUALITIES 473 20 2 BIPARTITE ENTANGLEMENT 477 20 3 THE QUANTUM BEAM SPLITTER 480 20 4 QUANTUM TELEPORTATION 483 20 5 QUANTUM CRYPTOGRAPHY 484 20 6 TOWARD QUANTUM COMPUTING 486 PROBLEMS 488 REFERENCES 489 1 CLASSICAL ELECTROMAGNETIC FIELDS IN THIS BOOK WE PRESENT THE BASIC IDEAS NEEDED TO UNDERSTAND HOW LASER LIGHT INTERACTS WITH VARIOUS FORMS OF MATTER AMONG THE IMPORTANT CONSEQUENCES IS AN UNDERSTANDING OF THE LASER ITSELF THE PRESENT CHAPTER SUMMARIZES CLAS SICAL ELECTROMAGNETIC ELDS WHICH DESCRIBE LASER LIGHT REMARKABLY WELL THE CHAPTER ALSO DISCUSSES THE INTERACTION OF THESE ELDS WITH A MEDIUM CON SISTING OF CLASSICAL SIMPLE HARMONIC OSCILLATORS IT IS SURPRISING HOW WELL THIS SIMPLE MODEL DESCRIBES LINEAR ABSORPTION A POINT DISCUSSED FROM A QUANTUM MECHANICAL POINT OF VIEW IN SECT 3 3 THE REST OF THE BOOK IS CONCERNED WITH NONLINEAR INTERACTIONS OF RADIATION WITH MATTER CHAPTER 2 GENERALIZES THE CLASSICAL OSCILLATOR TO TREAT SIMPLE KINDS OF NONLINEAR MECHANISMS AND SHOWS US A NUMBER OF PHENOMENA IN A RELATIVELY SIMPLE CONTEXT STARTING WITH CHAP 3 WE TREAT THE MEDIUM QUANTUM MECHANICALLY THE COMBINATION OF A CLASSICAL DESCRIPTION OF LIGHT AND A QUANTUM MECHANICAL DESCRIPTION OF MATTER IS CALLED THE SEMICLASSICAL APPROXIMATION THIS APPROXIMATION IS NOT ALWAYS JUSTI ED CHAPS 13 19 BUT THERE ARE REMARKABLY FEW CASES IN QUANTUM OP TICS WHERE WE NEED TO QUANTIZE THE ELD IN THE PRESENT CHAPTER WE LIMIT OURSELVES BOTH TO CLASSICAL ELECTROMAGNETIC ELDS AND TO CLASSICAL MEDIA SECTION 1 1 BRIE Y REVIEWS MAXWELL S EQUATIONS IN A VACUUM WE DERIVE THE WAVE EQUATION AND INTRODUCE THE SLOWLY VARYING AMPLITUDE AND PHASE APPROXIMATION FOR THE ELECTROMAGNETIC ELD SECTION 1 2 RECALLS MAXWELL S EQUATIONS IN A MEDIUM WE THEN SHOW THE ROLES OF THE IN PHASE AND IN QUADRATURE PARTS OF THE POLARIZATION OF THE MEDIUM THROUGH WHICH THE LIGHT PROPAGATES AND GIVE A BRIEF DISCUSSION OF BEER S LAW OF LIGHT ABSORPTION SECTION 1 3 DISCUSSES THE CLASSICAL DIPOLE OSCILLATOR WE INTRODUCE THE CONCEPT OF THE SELF ELD AND SHOW HOW IT LEADS TO RADIATIVE DAMPING THEN WE CONSIDER THE CLASSICAL RABI PROBLEM WHICH ALLOWS US TO INTRODUCE THE CLASSICAL ANALOG OF THE OPTICAL BLOCH EQUATIONS THE DERIVATIONS IN SECTS 1 1 1 3 ARE NOT NECESSARILY THE SIMPLEST ONES BUT THEY CORRESPOND AS CLOSELY AS POSSIBLE TO THEIR QUANTUM MECHANICAL COUNTERPARTS THAT APPEAR LATER IN THE BOOK SECTION 1 4 IS CONCERNED WITH THE COHERENCE OF THE ELECTROMAGNETIC ELD WE REVIEW THE YOUNG AND HANBURY BROWN TWISS EXPERIMENTS WE INTRO DUCE THE NOTION OF NTH ORDER COHERENCE WE CONCLUDE THIS SECTION BY A BRIEF