Manipulating Quantum Structures Using Laser Pulses

Cover......Page 1
MANIPULATING QUANTUM STRUCTURES USING LASER PULSES......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
Acknowledgments......Page 15
1.2 Background......Page 17
1.3 Measurables, observables, and parameters......Page 18
1.4 Notation and nomenclature......Page 21
1.5 Limitations of the theory......Page 23
1.6 Basic references......Page 24
2 Atoms as structured particles......Page 25
2.1 Spectroscopy......Page 26
2.1.1 Spectroscopic transitions......Page 27
2.1.2 Transition selectivity......Page 28
2.2 Quantum states......Page 29
2.3 Probabilities......Page 31
2.3.1 Time-dependent probabilities......Page 32
2.3.2 Equations of motion for changes......Page 33
3.1 Thermal radiation; quanta......Page 35
3.2 Cavities......Page 36
3.3 Incoherent radiation......Page 37
3.4 Laser radiation......Page 38
3.5 Laser fields......Page 39
3.5.1 Monochromatic plane waves......Page 40
3.5.2 Pulses......Page 41
3.5.3 Phase......Page 42
3.5.4 Complex-valued fields; the positive-frequency part......Page 43
3.5.5 Fourier components; coherence and bandwidth......Page 44
3.5.6 Pulse structuring......Page 46
3.6 Field vectors......Page 47
3.6.1 Parametrizing the electric field......Page 49
3.6.2 Measuring the field parameters......Page 54
3.6.4 Poincaré-sphere loops and geometric phase......Page 55
3.7 Laser beams......Page 56
3.8 Photons......Page 57
3.9 Field restrictions......Page 59
4.1.1 Vapors......Page 60
4.1.3 Cavities......Page 61
4.1.4 Trapped particles......Page 62
4.1.5 Optical lattices......Page 63
4.1.8 Collisions......Page 64
4.2 Detecting excitation......Page 66
4.3 The interaction energy; multipole moments......Page 68
4.4 Moving atoms......Page 70
5.1 Free electrons: Ponderomotive energy......Page 73
5.2 Picturing bound electrons......Page 74
5.3 The Lorentz force......Page 77
5.4 The wavefunction; orbitals......Page 78
5.4.1 Two-state example......Page 79
5.4.2 Semi-classical distortion and excitation......Page 80
5.5.1 Quantum states and Hilbert spaces......Page 82
5.5.2 Hilbert-space coordinates and probability amplitudes......Page 83
5.6 Two-state Hilbert spaces......Page 85
5.6.1 The Bloch sphere......Page 86
5.6.2 Multiple states and the Bloch sphere......Page 88
5.7 Time-dependent statevectors......Page 89
5.7.2 Geometric phase and the Bloch sphere......Page 90
5.7.3 Typical goals......Page 91
5.8 Picturing quantum transitions......Page 92
5.8.2 Wavefunction collapse......Page 93
6.1 Thermalized atoms; the Boltzmann equation......Page 94
6.3 The Einstein rates......Page 95
6.5 Solutions to the rate equations......Page 97
6.6 Comments......Page 99
7 Coherence: The Schrödinger equation......Page 101
7.1 Essential states; effective Hamiltonians......Page 103
7.2 The coupled differential equations......Page 104
7.2.1 Rotating coordinate system......Page 105
7.2.2 Probability loss......Page 107
7.4.1 Analytic solutions......Page 109
7.4.3 Numerical solutions......Page 110
7.5 The time-evolution matrix; transition probabilities......Page 111
8.1 The basic equations......Page 113
8.1.1 Initial and final conditions......Page 114
8.1.2 Integral form of the equations......Page 115
8.1.4 The electric-dipole interaction......Page 116
8.1.5 Measures of interaction strength......Page 119
8.2.1 Constant phase; quasistatic interactions......Page 120
8.2.2 Degeneracy......Page 121
8.2.3 Quasiperiodic interactions......Page 122
8.2.4 Example: Resonant excitation......Page 123
8.3 The rotating-wave approximation (RWA)......Page 124
8.3.1 Resonant population oscillations......Page 125
8.3.2 Resonant excitation with loss......Page 127
8.3.3 Fixed Rabi frequency and detuning......Page 128
8.3.4 Explaining oscillations: Dressed states......Page 129
8.3.5 Doppler detuning......Page 132
8.3.6 Power broadening......Page 133
8.4.1 Slow collisions......Page 134
8.4.2 Chirped frequency; rapid adiabatic passage (RAP)......Page 135
8.4.3 Explaining adiabatic passage: Adiabatic states......Page 137
8.4.4 Energy curves: Crossings and avoided crossings......Page 139
8.4.5 Energy surfaces and adiabatic passage......Page 141
8.4.6 Stark-chirped rapid adiabatic passage (SCRAP)......Page 143
8.4.7 Pulse shape effects; population return (CPR)......Page 145
8.4.8 Optimization of two-state RAP......Page 150
8.5 Comparison of excitation methods......Page 151
9 Weak pulse: Perturbation theory......Page 153
9.2 Pulse aftermath and frequency content......Page 154
9.3 Example: Excitation despite missing frequencies......Page 155
9.4 The Dirac (interaction) picture......Page 157
9.5 Weak broadband radiation; transition rates......Page 158
9.6 Fermi's famous Golden Rule......Page 160
10.1 The Feynman–Vernon–Hellwarth equations......Page 162
10.2 Coherence loss; relaxation......Page 166
10.2.1 Bloch equations with relaxation......Page 168
10.2.3 The excitation cross section......Page 169
10.2.4 Steady-state excitation; power broadening......Page 170
10.2.5 Inhomogeneous relaxation; Ramsey pulses......Page 172
11.1 Contiguous pulses......Page 175
11.2 Pulse trains......Page 176
11.3 Examples......Page 178
11.4 Pulse pairs......Page 179
11.5 Vector picture of pulse pairs......Page 181
11.6 Creating dressed states......Page 183
11.7 Zero-area pulses......Page 184
12.1 Zeeman sublevels......Page 187
12.2 Radiation polarization and selection rules......Page 188
12.2.1 Emission: Angular momentum fields......Page 189
12.2.2 Absorption: Linear momentum fields......Page 190
12.2.3 Connection, linear and angular momentum fields......Page 192
12.3 The RWA with degeneracy......Page 193
12.4 Optical pumping......Page 195
12.5 General angular momentum......Page 197
12.5.1 Two levels......Page 198
12.5.2 Multiple levels: Parallel chains......Page 199
13.1 Three-state linkages......Page 202
13.2 The three-state RWA......Page 204
13.2.1 The RWA......Page 205
13.2.2 Three-state analytic solutions, constant intensity......Page 209
13.2.3 Three-state eigenvectors......Page 210
13.2.4 Eigenvectors, two-photon resonance......Page 212
13.3.1 Equal Rabi frequencies......Page 213
13.3.2 Resonant "letter vee": Bright state......Page 215
13.4.1 Doppler shifts in ladders; Doppler-free excitation......Page 217
13.4.2 Detuned lambda system; population trapping......Page 221
13.4.3 Large intermediate detuning......Page 223
13.4.4 Adiabatic elimination......Page 225
13.4.6 Multiphoton transitions......Page 226
13.5 Unequal Rabi frequencies......Page 227
13.5.1 Weak probe field: Autler–Townes splitting......Page 229
13.5.2 Subsystem dressed states......Page 230
13.5.3 Dressed state preparation and probing......Page 233
13.6 Laser-induced continuum structure (LICS)......Page 234
14.1 The Raman Hamiltonian......Page 238
14.2.1 Steady fields......Page 239
14.2.2 Pulse-pair sequences; STIRAP......Page 242
14.2.3 The STIRAP mechanism......Page 244
14.3 Explaining STIRAP......Page 246
14.3.1 The dark state......Page 247
14.3.2 The adiabatic basis; adiabatic conditions......Page 249
14.4.1 Vary pulse delay......Page 251
14.4.3 The bright resonance......Page 252
14.5 Optimizing STIRAP pulses......Page 253
14.6 Two-state versions of STIRAP......Page 255
14.6.1 Three real variables......Page 256
14.6.3 STIRAP with large single-photon detuning......Page 258
14.7.1 STIRAP and B-STIRAP......Page 259
14.7.2 Fractional or F-STIRAP......Page 260
14.7.4 Multiple intermediate states......Page 261
14.7.5 Hyper-Raman STIHRAP......Page 262
14.7.6 STIRAP with sublevels......Page 263
14.7.7 Loop STIRAP......Page 266
15.1 Multiphoton and multiple-photon ionization......Page 269
15.2 Coherent excitation of N-state systems......Page 271
15.2.1 Multilevel linkages......Page 272
15.2.2 Multilevel analytic solutions......Page 273
15.2.3 Multilevel adiabatic states......Page 274
15.3.1 The multilevel RWA......Page 275
15.3.2 Analytic solutions......Page 278
15.3.3 Population flow......Page 279
15.3.4 The harmonic oscillator......Page 282
15.3.5 The pseudospin model......Page 283
15.3.6 Time-averaged populations......Page 286
15.3.7 Detuned chains......Page 288
15.3.8 Multiphoton resonances......Page 289
15.3.9 Two-state behavior in an N-state chain......Page 290
15.3.10 Generalized pi pulses......Page 292
15.4.1 The tripod linkage......Page 293
15.4.2 The tripod dark states......Page 295
15.4.3 Longer branches......Page 297
15.4.4 Fan linkages......Page 298
15.4.5 Quasicontinuum......Page 300
15.4.6 Multiple branches: Bright and dark states......Page 302
15.5 Loops......Page 303
15.5.1 Loops from polarization links......Page 304
15.5.2 Loops and the RWA; nonlinear optics......Page 307
15.6.1 Chirped detuning in multistate systems......Page 308
15.6.2 STIRAP in chains......Page 309
15.6.3 Chains of odd-integer length......Page 311
15.6.4 Chains of even-integer length......Page 313
16.1 Ensembles and expectation values......Page 315
16.2.1 Initial conditions: Pure and mixed states......Page 316
16.2.2 Thermalized start......Page 317
16.3 Environmental averages......Page 318
16.3.3 Static environments......Page 319
16.4.1 Classical expectation values; moments......Page 320
16.4.2 Quantum expectation values......Page 321
16.4.3 Operator expectation values......Page 322
16.5 Uncertainty relations......Page 323
16.6 The density matrix......Page 324
16.6.2 Example: Two states......Page 326
16.6.3 General properties of statistical matrices......Page 328
16.7 Density matrix equation of motion......Page 329
16.7.1 The Liouville operator......Page 331
16.7.2 Some constants of motion......Page 332
16.8.1 Bath states and reduced density matrices......Page 333
16.8.2 Relaxation as fluctuations......Page 335
16.9 Rotating coordinates......Page 337
16.9.1 Equations of motion......Page 338
16.10.1 Elementary transition matrices......Page 340
16.10.2 The N-state coherence vector......Page 341
16.10.3 State multipoles......Page 343
17.1.1 Separable coordinates......Page 347
17.1.2 Separable Hilbert spaces......Page 348
17.2.1 Center of mass......Page 349
17.2.2 Atom optics......Page 350
17.2.3 Deflection......Page 352
17.3 Two parts......Page 354
17.3.1 Product space......Page 356
17.3.2 Interactions between parts......Page 357
17.3.3 Accessing individual parts......Page 358
17.4.1 Entanglement......Page 359
17.4.3 Entropy......Page 360
17.4.5 Quantum information; qubits......Page 361
17.4.6 Maximally entangled states......Page 362
18.1 Superposition construction......Page 363
18.2.1 Wavepackets......Page 364
18.2.2 Creating superpositions with fractional STIRAP......Page 365
18.3 Degenerate discrete states......Page 366
18.4 Transferring superpositions......Page 367
18.5 State manipulations using Householder reflections......Page 368
18.5.1 Diagonalization using Householder reflections......Page 369
18.5.2 Implementation with a sequence of simultaneous pulses......Page 370
19.1 General remarks......Page 373
19.2 Spin matrices and quantum tomography......Page 375
19.3 Two-state superpositions......Page 378
19.3.1 Direct excitation to signal......Page 379
19.4 Analyzing multistate superpositions......Page 380
19.5 Analyzing three-state superpositions......Page 382
19.6 Alternative procedures......Page 384
20 Overall phase; interferometry and cyclic dynamics......Page 386
20.1 Hilbert-space rays......Page 387
20.2 Parallel transport......Page 388
20.3 Phase definition......Page 389
20.4 Michelson interferometry......Page 390
20.4.1 Matrix formalism......Page 391
20.4.2 Internal structure......Page 392
20.5 Alternative interferometry......Page 393
20.6 Ramsey interferometry......Page 394
20.7 Cyclic systems......Page 395
20.7.1 Cyclic adiabatic evolution......Page 397
20.7.2 Bloch vector......Page 399
20.7.3 Spin system......Page 401
21.1 Induced dipole moments; propagation......Page 403
21.2 Single field, N=2......Page 405
21.2.1 Maxwell–Schrödinger equations......Page 406
21.2.2 Analytic solutions......Page 407
21.2.3 Maxwell–Bloch equations......Page 408
21.2.5 Phase and intensity equations; absorption (attenuation)......Page 409
21.2.6 Pulse reshaping......Page 413
21.2.7 Self-induced transparency (SIT)......Page 414
21.2.8 Fluence and energy conservation......Page 416
21.3 Multiple fields......Page 418
21.4 Two or three fields, N=3......Page 419
21.4.2 Bright−dark basis......Page 423
21.4.3 Electromagnetically induced transparency (EIT)......Page 424
21.4.4 Coherence transfer: Dark-state polaritons and stopped light......Page 425
21.5 Four fields, N=4; four-wave mixing......Page 426
21.6.1 Equations of motion for density matrix......Page 429
21.6.3 Strong steady S field......Page 431
21.6.4 Inhomogeneous broadening......Page 433
22 Atoms in cavities......Page 435
22.1.1 Classical cavity......Page 436
22.1.3 The atom......Page 437
22.1.4 The interaction......Page 438
22.2 Two-state atoms in a cavity......Page 439
22.2.2 The two-state RWA......Page 440
22.2.3 The Jaynes−Cummings model (JCM)......Page 441
22.2.4 Losses......Page 443
22.3 Three-state atoms in a cavity......Page 445
22.3.1 Adiabatic elimination......Page 446
22.3.3 Cavity-induced adiabatic passage......Page 447
22.3.4 Excitation of vacuum......Page 448
22.3.5 Controlled single-photon emission......Page 449
23.1 Control theory......Page 451
23.2 Quantum control......Page 452
23.3 Optimization......Page 455
A.1 Angular momentum states......Page 458
A.1.1 Differential representations: Spherical harmonics......Page 459
A.1.2 Angular momentum and rotations......Page 461
A.1.3 Matrix representations: Spin matrices......Page 462
A.1.4 Rotation matrices......Page 466
A.2 Angular momentum coupling......Page 467
A.2.1 Rotations and irreducible tensors......Page 469
A.2.2 The Wigner−Eckart theorem......Page 470
A.3 Hyperfine linkages......Page 472
B.1 The bound-particle interaction......Page 475
B.2 The multipole moments......Page 478
B.4 Induced moments......Page 480
B.6 Rabi frequencies......Page 481
B.7 Angular momentum selection rules......Page 482
C.1 The Lorentz force; Maxwell's equations......Page 484
C.2 Wave equations......Page 486
C.2.1 Radiation in free space; intensity......Page 487
C.2.3 Plane waves......Page 489
C.3 Frequency components......Page 492
C.3.1 Spectral distribution......Page 493
C.3.2 Coherence......Page 494
C.4.1 The linear and nonlinear polarization fields......Page 496
C.4.2 The refractive index......Page 497
C.5 Pulse-mode expansions......Page 498
C.5.1 The slowly varying envelope approximation (SVEA)......Page 499
C.5.2 Group velocity......Page 501
Appendix D Quantized radiation......Page 503
D.1 Field quantization......Page 504
D.1.1 Photon-number states......Page 505
D.1.2 Coherent states......Page 507
D.1.3 Phase states......Page 510
D.2 Mode fields......Page 512
D.2.1 Plane waves......Page 513
D.2.2 Spherical waves......Page 515
D.2.3 Fields near the origin......Page 517
D.2.5 Cavity modes; standing waves......Page 518
D.2.6 Summary......Page 520
D.3 Photon states......Page 521
D.4 The free-field radiation Hamiltonian......Page 523
D.5.1 Photons as mode-field increments......Page 525
D.5.3 Photons as emitted quanta......Page 526
D.5.4 Wavepacket photons......Page 527
E.1 Terminology......Page 529
E.2 Adiabatic evolution......Page 531
E.2.1 Adiabatic conditions......Page 533
E.2.2 Superadiabatic states......Page 534
E.3 The Dykhne−Davis−Pechukas (DDP) formula......Page 535
F.1 The Morris−Shore transformation......Page 538
F.2 Bright and dark states......Page 540
F.4 Chain linkages......Page 542
F.5 Generalizations......Page 543
G.1 Floquet's theorem......Page 544
G.2 Example: Two states......Page 546
G.4 Floquet theory and the Jaynes–Cummings model......Page 547
G.5 Near-periodic excitation; adiabatic Floquet theory......Page 548
G.6 Example: Two states......Page 550
G.7 Adiabatic Floquet energy surfaces......Page 552
H.1 Spectroscopic parameters......Page 553
H.2 Relative transition strengths......Page 554
References......Page 558
Index......Page 581