Preface ........................................................ xv
1 Quantized electromagnetic field and coherent state
representations .............................................. 1
1.1 Quantization of the electromagnetic field ............... 1
1.2 State space for the electromagnetic field - Fock space
and Fock states ......................................... 5
1.3 Quadratures of the field ................................ 6
1.4 Coherent states ......................................... 7
1.5 Mixed states of the radiation field .................... 12
1.6 Diagonal coherent state representation for
electromagnetic fields - P-representation .............. 15
1.7 The Wigner function for the electromagnetic field ...... 18
1.8 Bosonic systems with finite mass - coherent states
and phase-space representations ........................ 22
Exercises ................................................... 24
References .................................................. 26
2 Nonclassicality of radiation fields ......................... 28
2.1 The Mandel Qu parameter ................................ 28
2.2 Phase-dependent measure of nonclassicality -
squeezing parameter S .................................. 29
2.3 Single-mode squeezed states - squeezed vacuum .......... 31
2.4 Squeezed coherent state ................................ 37
2.5 Other measures of nonclassicality ...................... 39
2.6 Mixed nonclassical states - degradation in squeezing ... 43
Exercises ................................................... 45
References .................................................. 47
3 Two-mode squeezed states and quantum entanglement ........... 49
3.1 The two-mode squeezed states ........................... 49
3.2 Nonclassicality of the two-mode squeezed vacuum ........ 50
3.3 Quantum phase-space distributions and quadrature
distributions .......................................... 52
3.4 Cauchy-Schwarz inequalities for nonclassicality in
two-mode states ........................................ 54
3.5 Conditional measurements on the two-mode squeezed
vacuum ................................................. 55
3.6 Quantum entanglement in the two-mode squeezed vacuum ... 56
3.7 Peres-Horodecki separability criterion for
continuous variable systems ............................ 56
3.8 Generation of two-mode nonclassical and entangled
states - optical parametric down-conversion ............ 58
3.9 Parametric amplification of signals .................... 60
3.10 Type-II optical parametric down-conversion -
production of entangled photons ........................ 61
3.11 Four-photon entanglement using optical parametric
down-conversion ........................................ 63
3.12 Two-mode mixed nonclassical states ..................... 65
3.13 Entanglement in two-mode mixed Gaussian states ......... 66
3.14 Application of entanglement to the teleportation of
a quantum state ........................................ 67
3.15 Nonclassical fields in optical fibers .................. 69
Exercises ................................................... 72
References .................................................. 74
4 Non-Gaussian nonclassical states ............................ 76
4.1 Schrodinger cat state and the cat paradox .............. 76
4.2 Photon-added and -subtracted states .................... 82
4.3 Single-photon-added coherent and thermal states ........ 84
4.4 Squeezing and sub-Poissonian properties of single-
photon-added states .................................... 86
4.5 Experimental realization of photon-added nonclassical
non-Gaussian states .................................... 88
4.6 Single-photon-subtracted states ........................ 89
4.7 Single-photon-subtracted two-mode states with vortex
structure .............................................. 93
4.8 Pair-coherent states ................................... 97
Exercises .................................................. 101
References ................................................. 102
5 Optical interferometry with single photons and
nonclassical light ......................................... 103
5.1 Transformation of quantized light fields at beam
splitters ............................................. 103
5.2 Beam splitter transformation equivalent to evolution
under a Hamiltonian ................................... 105
5.3 Transformation of states by the beam splitter ......... 105
5.4 Transformation of photon number states by a beam
splitter .............................................. 106
5.5 Single photons at beam splitters ...................... 107
5.6 Pairs of single photons at beam splitters ............. 108
5.7 Generalization of the Hong-Ou-Mandel interference to
N photons from both ports of the beam splitter ........ 109
5.8 Transformation of a two-mode squeezed state by a
50-50 beam splitter ................................... 109
5.9 Generation of two-mode entangled states by the
interference of coherent fields and single photons .... 110
5.10 Beam splitter as an attenuator ........................ 111
5.11 Transformation of quantized light fields by phase
shifters .............................................. 112
5.12 The Mach-Zehnder interferometer ....................... 113
5.13 Wheeler's delayed choice gedanken experiment .......... 117
5.14 Interaction-free measurements ......................... 117
5.15 Two-photon Mach-Zehnder interferometer ................ 119
5.16 Multiphoton interference and engineering of quantum
states ................................................ 121
5.17 Mach-Zehnder interferometer with two-mode squeezed
vacuum as input ....................................... 123
5.18 Balanced homodyne interferometers for measuring the
squeezing of light .................................... 125
5.19 Manipulation of quantum states by homodyning and
feed-forward .......................................... 126
5.20 Quantum state tomography .............................. 128
5.21 Sensitivity of an optical interferometer .............. 129
5.22 Heisenberg limited sensitivity of interferometers
based on parametric amplifiers or four-wave mixers .... 131
5.23 The quantum statistics of fields at the output ports .. 133
Exercises .................................................. 134
References ................................................. 136
6 Polarization and orbital angular momentum of quantum
fields ..................................................... 138
6.1 Characterization of the polarization properties of
quantized fields ...................................... 138
6.2 Polarization of quantized fields - Stokes operators ... 139
6.3 Action of polarizing devices on quantized fields ...... 141
6.4 Description of unpolarized light beyond Stokes
parameters ............................................ 143
6.5 Stokes operator tomography ............................ 144
6.6 Orbital angular momentum of fields - HG and LG modes .. 146
6.7 Orbital Stokes operators and the Poincaré sphere ...... 149
6.8 Mixed states of orbital angular momentum .............. 151
6.9 Entangled states of the orbital angular momentum ...... 152
6.10 Transformation of entanglement between polarization
and orbital angular momentum q-plates ................. 154
Exercises .................................................. 155
References ................................................. 156
7 Absorption, emission, and scattering of radiation .......... 158
7.1 The interaction of radiation and matter in the
electric dipole approximation ......................... 158
7.2 Rates for the absorption and emission of radiation .... 159
7.3 Single-mode limit - Einstein's В coefficient and the
absorption coefficient α (ω) .......................... 165
7.4 Scattering of radiation ............................... 166
7.5 Quantum interferences in scattering ................... 169
7.6 Radiative decay of states - Weisskopf-Wigner theory ... 170
7.7 Control of spontaneous emission through the design
of the electromagnetic vacuum ......................... 174
Exercises .................................................. 177
References ................................................. 178
8 Partial coherence in multimode quantum fields .............. 179
8.1 Correlation functions for electromagnetic fields ...... 179
8.2 Young's interferometer and spatial coherence of the
field ................................................. 181
8.3 Photon-photon correlations - intensity
interferometry ........................................ 184
8.4 Higher-order correlation functions of the field ....... 187
8.5 Interferometry in the spectral domain ................. 188
8.6 Squeezing spectrum and spectral homodyne measurement .. 191
8.7 Coherence effects in two-photon absorption ............ 192
8.8 Two-photon imaging - ghost imaging using G(2) ......... 194
Exercises .................................................. 197
References ................................................. 198
9 Open quantum systems ....................................... 200
9.1 Master equation description of open systems ........... 200
9.2 Dissipative dynamics of harmonic oscillators .......... 204
9.3 Dissipative dynamics of a two-level system ............ 206
9.4 Dissipative dynamics of a multilevel system ........... 208
9.5 Time correlation functions for multilevel systems ..... 210
9.6 Quantum Langevin equations ............................ 212
9.7 Exactly soluble models for the dissipative dynamics
of the oscillator ..................................... 213
9.8 Exact dissipative dynamics of a two-level system
under dephasing ....................................... 215
Exercises .................................................. 218
References ................................................. 219
10 Amplification and attenuation of quantum fields ............ 220
10.1 Quantum theory of optical amplification ............... 220
10.2 Loss of nonclassicality in the amplification process .. 223
10.3 Amplification of single-photon states ................. 229
10.4 Amplification of entangled fields ..................... 230
10.5 Realising a phase-insensitive amplifier from
a phase-sensitive amplifier ........................... 232
10.6 Degradation of nonclassicality and entanglement due
to the absorption of quantum fields ................... 233
10.7 Loss of coherence on interaction with the
environment ........................................... 235
Exercises .................................................. 238
References ................................................. 240
11 Quantum coherence, interference, and squeezing in
two-level systems .......................................... 242
11.1 Two-level approximation: atomic dynamics in
a monochromatic field ................................. 242
11.2 Application of atomic coherence - Ramsey
interferometry ........................................ 247
11.3 Atomic coherent states ................................ 249
11.4 Minimum uncertainty states for two-level systems -
spin squeezing ........................................ 252
11.5 Atomoc squeezed states by nonlinear unitary
transformations ....................................... 254
11.6 Atomic squeezed states produced by supersensitivity
of Ramsey interferometers ............................. 256
11.7 Phase-space representation for a collection of
two-level systems ..................................... 258
11.8 Phase-space description of EPR correlations of spin
systems ............................................... 262
Exercises .................................................. 264
References ................................................. 265
12 Cavity quantum electrodynamics ............................. 267
12.1 Exact solution of the Jaynes-Cummings model: dressed
states ................................................ 268
12.2 Collapse and revival phenomena in JCM ................. 271
12.3 Dispersive limit of the JCM ........................... 273
12.4 Dissipative processes in cavity QED - the master
equation .............................................. 275
12.5 Spectroscopy of the ladder of dressed states .......... 277
12.6 Multi-atom effects in cavity QED ...................... 284
12.7 Effective dipole-dipole interaction in a dispersive
cavity from Lamb shift of the vacuum .................. 288
12.8 Atomic cat states using multi-atom dispersive JCM ..... 290
12.9 Application of atomic cat states in Heisenberg
limited measurements .................................. 293
12.10 Engineering anti-Jaynes-Cummings interaction ......... 296
12.11 QED in coupled cavity arrays - single-photon switch .. 298
Exercises .................................................. 300
References ................................................. 301
13 Absorption, emission, and scattering from two-level
atoms ...................................................... 304
13.1 Effects of relaxation: optical Bloch equations ........ 304
13.2 Absorption and amplification of radiation by a
strongly pumped two-level system ...................... 309
13.3 Resonance fluorescence from a coherently driven
two-level atom ........................................ 314
13.4 Quantum dynamics of the two-level atom and spectrum
of fluorescence ....................................... 317
Exercises .................................................. 325
References ................................................. 327
14 Quantum interference and entanglement in radiating
systems .................................................... 328
14.1 Young's interference with microscopic slits - atoms
as slits .............................................. 328
14.2 Spatial bunching and antibunching of photons .......... 330
14.3 Interference in radiation from two incoherently
excited atoms ......................................... 333
14.4 Atom-photon entanglement .............................. 337
14.5 Atom-atom entanglement via detection of
spontaneously emitted photons ......................... 338
14.6 Multi-atom entanglement ............................... 341
14.7 Quantum entanglement in Dicke states and
superradiance ......................................... 343
14.8 Multi-path quantum interference as the source of
Dicke superradiance ................................... 345
14.9 Entanglement of photons produced in an atomic
cascade ............................................... 348
Exercises .................................................. 351
References ................................................. 352
15 Near field radiative effects ............................... 354
15.1 Near field radiative effects - coupling between
dipoles ............................................... 354
15.2 Radiative coupling between dipoles and dynamics ....... 358
15.3 Vacuum-induced deterministic entanglement ............. 360
15.4 Two-photon resonance induced by near field radiative
effects ............................................... 362
15.5 The dipole blockade ................................... 365
Exercises .................................................. 368
References ................................................. 368
16 Decoherence and disentanglement in two-level systems ....... 370
16.1 Decoherence due to the interaction of a two-level
system with the environment ........................... 370
16.2 Disentanglement in two-level systems .................. 371
16.3 Decoherence-free subspace ............................. 373
16.4 Protection of decoherence due to dephasing via
dynamical decoupling .................................. 374
16.5 Control of the spectral density of environment for
protection against decoherence ........................ 378
16.6 Modulation produced protection against
disentanglement in cavity QED ......................... 380
Exercises .................................................. 382
References ................................................. 383
17 Coherent control of the optical properties ................. 385
17.1 A simple model for coherent control ................... 385
17.2 Dark states and coherent population trapping .......... 394
17.3 EIT in single-atom fluorescence ....................... 397
17.4 Control of two-photon absorption ...................... 400
17.5 Vacuum-induced coherence and interference ............. 404
Exercises .................................................. 409
References ................................................. 410
18 Dispersion management and ultraslow light .................. 413
18.1 Group velocity and propagation in a dispersive
medium ................................................ 413
18.2 Electromagnetically induced waveguides ................ 417
18.3 Storage and retrieval of optical pulses ............... 418
18.4 Adiabatons and storage and retrieval of pulses ........ 423
18.5 Non-EIT mechanisms for ultraslow light ................ 426
Exercises .................................................. 429
References ................................................. 430
19 Single photons and nonclassical light in integrated
structures ................................................. 432
19.1 Quantum optics in a coupled array of waveguides ....... 432
19.2 The Hong-Ou-Mandel interference in a system of two
coupled waveguides .................................... 434
19.3 Single-photon transport and coherent Bloch
oscillations in a coupled array ....................... 436
19.4 The Anderson localization of quantum fields in
coupled waveguide arrays .............................. 442
19.5 Discrete quantum walks via waveguide couplers on
a chip ................................................ 447
Exercises .................................................. 452
References ................................................. 453
20 Quantum optical effects in nano-mechanical systems ......... 455
20.1 The radiation pressure on the nano-mechanical mirror .. 455
20.2 Basic quantum Langevin equations for the coupled
system of cavity and NMO .............................. 457
20.3 Steady-state solution of quantum Langevin equations
in the mean field limit and bistability ............... 459
20.4 Quantum fluctuations in optomechanical systems ........ 461
20.5 Sideband cooling of the nano-mechanical mirror ........ 463
20.6 Normal-mode splitting ................................. 466
20.7 Squeezing of a nano-mechanical oscillator ............. 471
20.8 Electromagnetically induced transparency (EIT) in
the mechanical effects of light ....................... 475
20.9 Quantized states of the nano-mechanical mirror
coupled to the cavity ................................. 481
Exercises .................................................. 485
References ................................................. 487
Index ......................................................... 489
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