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states, 193
dispersion, 672
electric power density, 92
shift, 363
electro-optic modulator, 534
tripling, 412
electron multiplication, 281
fringe visibility, 312
element of physical reality, 579
functional, 655
elliptical polarization, 665
end-point rule, 658
ensemble, 50 gain
entangled state clamping, 157
distinguishable systems, 115, 200 matrix, 521
dynamically entangled, 210, 321 medium, 20, 511
hyperentangled photon pairs, 408 gate
identical particles, 209 generator/width, 25
kinematically entangled, 210 photon, 345
maximally entangled, 203 gauge transformation/invariance, 661
mixed state, 204 Gaussian
beams and pulses, 80, 103, 105, 226
pure state, 202
Schr¨dinger, 194
o probability density, 152
two-photon, 212 states, 187
wave packet, 227, 290
environment, 420
picture, 539 Geiger mode, 282
equal-time commutator, 86 generalized function, 656
equipartition of energy, 7 generator of
error correction, 634 displacement, 162
spatial rotations, 76, 100, 692
events, 659
spatial translations, 76, 97, 112, 694
evolution operator, 84
excess noise, 300 good function, 656
expectation value, 684 group
extraordinary wave, 678 delay, 342
velocity, 95
factorizable density operator, 426
fair-sampling assumption, 598 Hadamard gate, 633
half-wave plate, see retarder plate
Faraday e¬ect, 256
fast axis, 640 Hanbury Brown“Twiss e¬ect, 313
Fermi heat bath, 425
½½
Index

interaction-free measurement, 347
Heisenberg
equations of motion, 85, 86 interference
single-photon, 307, 333
picture, 83
helicity, 667 two-photon, 315
operator, 78 intermediate
hemiclassical approximation, 154 frequency, 292
heralded, 637 state, 443
hermitian, see self-adjoint time scale, 92
hermitian conjugate, see adjoint internal ampli¬er modes, 500
heterodyne detection interpolating operator, 162
classical, 291 inverted oscillator, 429
iris, see stop
noise, 297
optical, 293
quantum, 294 Jacobi identity, 687
signal, 295 Jaynes“Cummings model, 381
hidden variables, 582 joint
Hilbert space, 647 probability, 659
history dependence, 93, 282, 361 variance, 150, 474
homodyne detection, 300 normal-ordered, 491
classical, 301 Jones matrix, 641
noise, 303
quntum, 302
Kerr media, 267, 417
signal, 302
ket vector, 680
Hong“Ou“Mandel (HOM)
kinematics, 13
dip, 323, 339
kinetic momentum, 112, 693
e¬ect, 317
interferometer, 315
Lagrangian formulation, 69, 103
null, 318
hyperentangled photon pairs, 408 Langevin equation
advanced, 439
classical, 549
ideal squeezed state, 479
quantum, 422
identical particles, 205
image band mode, 295, 302 retarded, 438
image intensi¬ers, see channeltrons Laplace transform, 654
laser ampli¬er, 510
impermeability, 675
Leibnitz rule, 687
improper ket vector, 682
level repulsion, see avoided crossing
in-¬eld
line
classical, 241
operator, 242, 437 of sight, 530
shape, 19
incident
width, 365
annihilation operator, 242
line-out, 530
classical ¬eld, 237
incoherent pump, 447, 511 linear
ampli¬er, 499
independent events, 660
index matching, 401, 515 functional, 655
operator, 646
index of refraction, 674
polarization, 665
ordinary/extraordinary, 399
regression, 455
indistinguishable particles, 205
indivisibility of photons, 4, 24 Liouville operator, 543
local
in¬nite-dimensional space, 646
measurement, 638
inner product, 646
number operator, 107
input“output
equation, 440, 505, 518, 519 observable, 208
oscillator (LO), 291
matrix, 519
realistic theory, 578
method, 435
longitudinal
insertion-loss noise, 499
integral kernel, 73, 524, 658 relaxation time, 377
vector ¬eld, 35
interaction
Hamiltonian, 117, 124 Lorentzian line shape, 365, 442
lowering operator, 43, 356, 424
picture, 124
½¾ Index

Mandel Q-parameter, 278 number
marginal distribution, 174, 533, 659 ¬eld sieve, 630
squeezed states, 482
Markov
state, 43, 81, 178
approximation, 283, 361
process, 282, 361 number operator, 43, 74
local, 107
master equation, 541
local (Mandel), 108
matrix element, 682
total, 46
maximally mixed state, 54
Maxwell™s equations
observables, 683
classical, 661
macroscopic, 670 Occam™s razor, vii
quantum, 117 one-channel device, 245
memory interval, 353 one-particle operator, 206
Mermin™s lemma, 590 one-time pad, 616
micromaser, 387 one-way quantum computing, 638
minimal coupling, 693 open system, 420
minimum-uncertainty state, 45, 151, 690 operator
mixed state, 49 binomial theorem, 686
mode cleaning cavity, 534 Bloch equation, 442
Mollow triplet, 458 continuous spectrum, 648
monochromatic domain, 648
¬eld, 88, 324 Langevin equation, 432
space, 98 norm, 648
state, 56 point spectrum, 648
multilevel atoms, 355 spectral resolution, 649
operator-valued, 105, 649, 656
multiparaxial space, 221
multiplicative noise, 432, 445 optic axis, 237
mutual coherence function, 312 optical
mutually unbiased bases, 617 circulator, 257
couplers or taps, 248, 604
diode, 256
narrowband detection and ¬lter, 273
isolator, 255, 414
negative absorption, 20
negative-frequency part, 72, 87 Kerr e¬ect, 417
parametric ampli¬er/oscillator, 486
noise
torque wrench, 80
correlation function, 433
transmission line, 602
distribution, 426
tweezer, 80
in heterodyne detection, 297
ordinary wave, 678
in homodyne detection, 303
orthogonal
reservoir, 422
complement, 646
strength, 426, 501
projection operators, 44, 355, 649
temperature, 299
vectors, 646
non-centrosymmetric crystal, 394
orthonormal set of vectors, 196, 647
nonanticipating, 435
out-¬eld
nonclassical states, 182, 306
classical, 241
nondegenerate eigenvalue, 648
operator, 242, 439
nondispersive material, 671
outcome
nonlinear polarization, 678
independence, 586
norm
parameter, 585
of a vector, 647
overcompleteness, 164
of an operator, 648
preserving, 649
P (±), 181, 278
normal
matrix, 520, 650 P-polarization, 239
mode, 37 parameter
operator, 649 settings, 584
parametric ampli¬cation, 267, 400
ordering, 48
normal-ordered variance, 474 paramps, 267
normalized, 647 paraxial
NOT gate, 632 classical optics, 219
null events, 345 expansion, 228
½¿
Index

approximate localizability, 232
Hamiltonian, 223
beam ampli¬er, 267
Hilbert space, 220
bunching, 314, 456
mixed state, 221
gun, 496
momentum operator, 222
indivisibility experiment, 25
optical elements, 245
light quanta, 3
pure state, 221
localizability, 106
ray bundle, 219
pairing, 317
wave, 218
position operator, 106
wave packets, 229
photonic bandgap, 339, 603
parity
photothermal detection, 266
electromagnetic ¬elds, 668
photovoltaic devices, 266
state, 637
transformation, see spatial inversion piece-wise paraxial, 237
planar
Parseval™s theorem, 652
cavity, 669
partial trace, 198
wave, 95, 246
partition
Planck
function, 58
constant, 8
noise, 252, 499
distribution, 16
passive dielectric, 90
spectrum, 5
Pauli
Poincar´ sphere, 57
e
exclusion principle, 207
point
matrices, 685
periodic boundary conditions, 81, 94, 653 detector, 272
permittivity/permeability, 672 spectrum, 648
Poisson
perturbation
distribution, 153, 276
Hamiltonian, 124
equation, 661
theory, ¬rst-order, 127
polar vector, 668
theory, second-order, 129
polarization
phase
amplitude, 104
matching, 396
components, 665
type I/type II, 400
density, 670
squeezing/quadrature, 480
transformation, 501 vector, 665
velocity, 95, 678 polarized light, 56
phase space polarizing beam splitter, 253
electromagnetic, 103 polychromatic
for quantum optics, 172 ¬eld, 89
mechanical, 8, 48, 103 space, 98
multimode, 185 population
inversion, 20, 376, 511
single-mode, 38
phase-changing perturbation, 443 operator, 356
phase-conjugating ampli¬er, 507 port, 245
phase-¬‚ip, 627 position
operator, 106, 111, 682
phase-insensitive
ampli¬er, 501 space, 651
noise, 427, 501 positive-de¬nite
matrix, 124, 271, 650
phase-sensitive ampli¬er, 502
operator, 108, 648
phase-transmitting ampli¬er, 506
positive-frequency part, 72, 87
phasor, 149
phonons, 422 postselection, 636
power spectrum, 88, 97, 221, 271, 662
photoconductive devices, 266
photodiodes, 282 Poynting theorem, 92, 662
photoelectric principal
detection, 265, 267 gain, 521
e¬ect, 9 part, 657
photoelectron counting statistics, 275 quadrature, 520
photoemissive detection, 535 principle
photomultiplier tube, 276, 281 of causality, 93, 672
photon, 43 of detailed balance, 7, 18, 544
anti-pairing, 623 of locality, 578
½ Index

oscillator, 38, 151
of realism, 578
reaction, 428
of separability, 195
probability zone, 358
radiative corrections, 360
amplitude, 31, 106, 113, 127
density, 172, 659 Radon transform, 531
distribution, 50 raising operator, 43, 356
function, 659 Raman scattering, 143
product vector, 114 random
projection telegraph signals, 563
of the density, 530 variable, 660
operator, 648 rate equation, 378
propagation segment, 237 raw key, 618
Rayleigh range, 220, 227, 515
pulse length, 227
pump beam, 402 Rayleigh“Jeans law, 8
pure state, 49, 683 realistic theory, 584
purity (of a quantum state), 54, 203 reciprocal
device, 255
Q-function, 178 space, 651
rectifying detection, 266
quadrature operator, 299
reduced density operator, 197, 538
quantum
reduction of the wave packet, 52, 194, 684
back action, 153, 419
re¬‚ection ampli¬er, 509
circuit, 632
regenerative ampli¬er, 502
cloning machine (QCM), 611
rephasing time, 386
computer
reservoir, 422
one-way, 638
resolution of the identity, 683
reversible, 631
resonance
dense coding, 621
e¬ciency, 159, 276 condition, 354
¬‚uorescence, 457
electrodynamics, 113
resonant
fuzzball, 474
enhancement, 392
gate, 631
Hamiltonian, 352
jump, 557
wave approximation, 354
key distribution, 617
retarder plate, 238
Langevin equation, 422
half-wave plate, 641
Liouville equation, 51
phase shifter, 238
nondemolition
quarter-wave plate, 641
counter, 267
wave plate, 640
measurement, 419
optics, 1 revival of a cavity state, 386
robust, 482, 552
parallelism, 631
root mean square (rms) deviation, 51
register, 631
rotating wave approximation, 354
regression, 454
Rydberg
state
atom, 141, 267, 387
di¬usion, 575
level, 141, 267
reconstruction, see tomography
trajectory, 573
quarter-wave plate, see retarder plate S-polarization, 239, 292
sample, 420, 538
quasiclassical, 148
quasimonochromatic, 88 saturation, 157
quasiprobability density, 173 intensity, 460
qubit, 607 scalability (for quantum computing), 634
control/target, 633 scans (in tomography), 529
qudit, 607 scattered
annihilation operator, 242
classical ¬eld, 237
Rabi
scattering
frequency, 134
operator, 135, 268 channels, 237
oscillations (¬‚opping), 373, 381 matrix, 238, 242
Schmidt
radiation
gauge, 22, 116, 661 decomposition, 198
½
Index

function, 5, 273
rank, 199
width, 663
Schottky diodes, 266
spectrometers, 275
Schr¨dinger
o
spin-up/spin-down state, 201, 351
equation, 684
spontaneous
picture, 40
down-conversion (SDC), 400
second-harmonic generator, 534
emission, 16, 443
second-order perturbation theory, 129
subspace, 360
self-action, 392
spot size, 227, 515
self-adjoint
square integrability, 681
matrix, 650
square-law detection, 265
operator, 648
squeezed
self-focussing, 417
along quadrature X, 474
self-phase modulation, 417
coherent state, 478
semiclassical
multimode state, 478
approximation, 111
electrodynamics, 4, 111 states, 470
vacuum, 470
Hamiltonian, 370
squeezing
limit, 369
generator, 476
sensitivity function, 270
of an intra-cavity ¬eld, 488
separable
of emitted light, 490
boson state, 209
Hilbert space, 170, 647, 681 operator, 475
parameter, 475
mixed state, 204
transformation, 475
pure state, 202
standard quantum limit, 151, 279
two-photon state, 211
states in quantum theory
shelving state, 559
mixed state, 49
sifted key, 618
pure state, 49, 683
signal and idler ¬elds, 316, 402
stationary density operator, 121
signal reservoir, 422
statistical separability, 582
signal-to-noise ratio, 526
step function, 656
single-microwave-photon counter, 267
sti¬ di¬erential equations, 572
single-photon
stimulated emission, 16
counter, 266
Stokes parameters, 57
counting rate, 271
stop, 260
detection, 268
interference, see interference strictly monochromatic, 88
strong-separability condition, 587
on demand, 496
sub- and super-Poissonian statistics, 280
Rabi frequency, 136
sum-frequency generation, 399
velocity in a dielectric, 324
summation convention, 645
singlet state, 201
super operator, 544
singlet-like state, 252
superposition
SIS (superconducting“insulator“
principle, 114, 680
superconducting) devices,
state, 49
266
susceptibility
slow axis, 640
linear, 88, 94, 672
slowly-varying envelope
nonlinear, 392
approximation, 89
symmetric group, 206
¬elds, 89
operator, 223, 430 symmetrical
ordering, 48
space-like separation, 578
product, 686
span of a set of vectors, 44, 646
symmetry transformation, 119
spatial
dispersion, 88, 672
¬ltering, 260 TE-polarization, 239
inversion, 118, 667 technical noise, 300, 499
separability, 578 teleportation gate, 637
translation, 112, 174 temporal width, 274
speckle pattern, 158 tensor product
spectral for quantum electrodynamics, 116
density, 274 of general Hilbert spaces, 196
½ Index

operator, 649
of operators, 197
test function, see good function universal cloning machine, 609
unperturbed Hamiltonian, 124
thermal
unpolarized light, 55
distribution in number, 59
unused port, 248, 255
state, 177
up-conversion, 399
third-harmonic generation, 412
Thompson scattering, 11
three-wave mixing, 392 vacuum
time ¬‚uctuation, 45, 253
reversal, 118, 667 ¬eld strength, 61
translation, 123, 225, 684 Rabi ¬‚opping, 385
state, 43, 45
TM-polarization, 239
variance, 51, 150
tomography
classical, 529 normal-ordered, 474
optical homodyne, 532 vector
quantum, 529 norm, 647
quantum-state, 533 space, 645
velocity operator, 117, 694
total number operator, 46
trace Verdet constant, 256
of a matrix, 650 von Neumann
of an operator, 173, 683 entropy, 55
projection postulate, 52, 193, 684
partial, 198
trajectory
classical, 39, 103 wave
quantum, 573 coordinates, 513
transit broadening, 458 packet, 46, 74, 104, 168
transition operator, 104
probability, 127 quantization, 103
rate, 128, 130, 140 plate, see retarder plate
transpose matrix, 650 weakly dispersive medium, 91
transversality condition, 663 weakly nonlinear media, 392
transverse Weisskopf“Wigner method, 360
delta function, 36 Weyl product, 175, 686
relaxation time, 377 which-path information, 308, 325
vector ¬eld, 35 white noise, 434
trapping state, 497 white-light fringe, 314, 346
traveling-wave ampli¬er, 502 Wien™s law, 18
trombone prism, 324 Wigner
tunneling time, 337 distribution and characteristic function,
two-channel device, 248 173
two-level atoms, 351 theorem, 119
two-photon
work function, 10
coherent state, 472, 478
world, 422
interference, 315
density operator, 425
Hamiltonian, 423
uncertainty relation
canonical, 690
X gate, 632
generalized, 690
uncorrelated quantum ¬‚uctuations, 203
Y-junction, 254
undepleted pump approximation, 402
uniaxial crystal, 677
Z gate, 633
unitary
matrix, 650 zero-point energy, 45

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. 27
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