Cambridge Quantum Optics

Quantum theory 1 a 0 = Tr (A) , 2 (C.40) 1 a j = Tr (Aσ j ) . 2 2 2 Writing a i σ i = a · σ and using the properties given above yields (a · σ) = |a| ,and this in turn provides the useful identities (Cohen-Tannoudji et al., 1977b, Complement A-IX) e iαu·σ =cos (α)+ i sin (α) u · σ , (C.41) e βu·σ =cosh (β)+sinh (β) u · σ , (C.42) where α and β are real constants and u is a real unit vector. C.3.2 The operator binomial theorem For c-numbers x and y the binomial theorem is n  n! n n−p p (x + y) = x y , (C.43) p!(n − p)! p=0 but this depends on the fact that c-numbers commute. For noncommuting operators X n and Y the quantity (X + Y ) is to be evaluated by multiplying together the n factors X + Y . Consider the terms of order (n − p, p) in this expansion, i.e. those in which X occurs n − p times and Y occurs p times. Since each of these terms is the product of n factors, there are a total of n! orderings. The orderings that differ only by exchanging Xswith Xsor Y swith Y s are identical, and the number of these terms is precisely the binomial coefficient n!/p!(n − p)!; therefore, n n  n!  n−m m (X + Y ) = S X Y , (C.44) p!(n − p)! m=0 m where S [X n−m Y ] is the average of the terms with (n − m) Xsand mY s arranged in all possible orders. This is called the symmetrical or Weyl product. n n n For (n, 0) or (0,n) one has simply S [X ]= X n or S [Y ]= Y .Examples of mixed powers are 1 S [XY ]= (XY + YX) , 2  2  1 2 2 S X Y = X Y + XY X + YX , 3  2 2  1 2 2 2 2 2 2 S X Y = X Y + XY X + XY XY + Y X + YX Y + YXY X , 6 . . . (C.45)

Useful results for operators C.3.3 Commutator identities The Leibnitz rule [A, BC]= A [B, C]+ [A, B] C (C.46) and the Jacobi identity [[A, B] ,C]+ [[C, A] ,B]+ [[B, C] ,A] = 0 (C.47) are both readily verified by direct use of the definition [A, B]= AB − BA. The useful identity ⎛ ⎞ ⎛ ⎞ n p−1 n [A, B 1 B 2 ··· B n ]= ⎝ B j ⎠ [A, B p ] ⎝ B k ⎠ (C.48) p=1 j=1 k=p+1 can be established by an induction argument, combined with the convention that an empty product has the value unity. In the special case that each single commutator [A, B p ] commutes with the remaining B j s, this becomes ⎛ ⎞ n n [A, B 1 B 2 ··· B n ]= [A, B p ] ⎝ B j ⎠ . (C.49) p=1 j=p=1 C.3.4 Operator expansion theorems Theorem C.1 Let X and Y be operators acting on a Hilbert space H.Then ∞ n  κ (n) κX −κX e Ye = [X, Y ] , (C.50) n! n=0 (n) (0) where the iterated commutator [X, Y ] is defined by the initial value [X, Y ] = Y and the recursion relations (n+1) (n) [X, Y ] = X, [X, Y ] for n
0 . (C.51) Proof Let Y (κ) ≡ e κX Ye −κX ;then dY (κ) /dκ =[X, Y (κ)]. Iterating this result implies n d n+1 Y (κ)  d Y (κ) = X, , (C.52) dκ n+1 dκ n and eqn (C.50) follows by a Taylor series expansion around κ =0. In the special case that the commutator [X, Y ]commutes with X, the series ter- minates so that e κX Ye −κX = Y + κ [X, Y ] , (C.53) i.e. X generates translations of Y . An important example is a canonically conjugate pair: X =
p, Y =
q,with [
q,
p]= i
.Choosing κ = iu/
,where u is a c-number, gives the familiar quantum mechanics result

Quantum theory T
qT u =
q + u, (C.54) † u where the unitary operator T u = e −iu

Useful results for operators C.3.5 Campbell–Baker–Hausdorff theorem Theorem C.2 Let X and Y be operators such that [X, Y ] commutes with both X and Y .Then X Y e e = e X+Y e 2 1 [X,Y ] . (C.66) Proof See Peres (1995, Sec. 10-7). Two important special cases are needed in the text. The first is defined by setting X = −iv
q, Y = −iu
p,which leads to e e e −i(u

Quantum theory C.3.7 Generalized uncertainty relation Choose a fixed vector |ψ and a pair of normal operators C and D, i.e. C, C † = D, D † = 0. Use the shorthand notation C = ψ |C| ψ, D = ψ |D| ψ to define the fluctuation operators δC = C−C and δD = D−D.Note that [C, D]= [δC, δD]. The expectation value of the commutator is [C, D] = [δC, δD] = δCδD− δDδC ; (C.76) consequently, |[C, D]|  |δCδD| + |δDδC| . (C.77) † Next set ψ |δCδD| ψ = φ |χ,where |φ = δC |ψ and |χ = δD |ψ. The Cauchy– Schwarz inequality (A.9) yields |φ |χ|  φ |φ χ |χ = δCδC  δD δD . (C.78) † † With the definitions of the rms deviations 2 † ∆C = δC δC = δCδC †  , (C.79) 2 † ∆D = δD δD = δDδD † , we find |δCδD| = |φ |χ|  δCδC  δD δD =∆C ∆D. (C.80) † † Interchanging C and D gives |δDδC|  δDδD  δC δC =∆C ∆D, (C.81) † † and putting everything together yields the generalized uncertainty relation 1 ∆C ∆D
|[C, D]| (C.82) 2 for any pair of normal operators. C.4 Canonical commutation relations Hermitian operators Q and P that satisfy the canonical commutation relation [Q, P]= i
are said to be canonically conjugate. Applying eqn (C.82) to this case yields the canonical uncertainty relation ∆Q ∆P

/2 . (C.83) A state for which equality is attained, i.e. ∆Q ∆P =
/2 , (C.84) is called a minimum-uncertainty state or minimum-uncertainty wave packet.

Canonical commutation relations The creation and annihilation operators defined in Section 2.1.2 satisfy the alter- native form a M ,a †
= δ MM
, [a M ,a M
] = 0 (C.85) M of the canonical commutation relations. We first show that these relations are preserved by any unitary transformation. Let U be a unitary operator and define new operators † b M = Ua M U ; (C.86) then b M ,b †
= Ua M U ,Ua †
U † = δ MM , † M M (C.87) [b M ,b M
]= Ua M U ,Ua M
U † =0 . † The converse statement is also true. If the operators b M satisfy b M ,b † M
= δ MM , [b M ,b M ]= 0 , (C.88) then there is a unitary transformation U which relates the b M sand a M s by eqn (C.86). The proof of this claim depends on the argument in Section 2.1.2-A showing that a Hilbert space in which eqn (C.85) holds is spanned by the number states, which we will now call |n; a, satisfying † a a M |n; a = n M |n; a ,n =(n 1 ,n 2 ,...) . (C.89) M This argument applies equally well to the b M s, so there is also a basis of states, |n; b, satisfying † b b M |n; b = n M |n; b . (C.90) M It is easy to check that the operator U, defined by U = |n; bn; a| , (C.91) n is unitary, and that † Ua M U = |m ; bm ; a |a M | m; am; b| m
m  √ = |m − 1 M ; b m M m; b| , (C.92) m where m − 1 M signifies (m 1 ,m 2 ,... ,m M − 1,...). Calculating the general matrix el- ement of Ua M U in the |n; b basis yields † n; b Ua M U †  n  = n; b |b M | n ; b ; (C.93) M n ; b = δ n,n
−1 M therefore, this U satisfies eqn (C.86).

Quantum theory C.5 Angular momentum in quantum mechanics In classical mechanics, the angular momentum of a particle (relative to the origin of coordinates) is r×p,where p is the momentum. In quantum mechanics (Bransden and Joachain, 1989, Chap. 6) this becomes the operator L = r × (−i∇), which satisfies the angular momentum commutation relations [L i ,L j ]= i ijk L k . (C.94) Because of its relation to the classical angular momentum used to describe orbits, L is called the orbital angular momentum.Thisoperatorisalsorelated to spatial T rotations, r → r = R (n,ϑ) r,where R (n,ϑ)is a 3 × 3 orthogonal matrix (R R = T RR =1), n is a unit vector defining the axis of rotation, and ϑ is the angle of rotation around the axis. For small ϑ one can show that δr j = r − r j = δr i = ϑ ijk n j r k . (C.95) j By definition, a vector V transforms like r under rotations. A scalar wave function ψ (r) transforms according to ψ (r)= U (n,ϑ) ψ (r), where the unitary operator U (n,ϑ)is given by i U (n,ϑ)= exp − ϑn · L . (C.96) Thus L is the generator of spatial rotations. The corresponding transformation for an operator O is O = U (R) OU (R). Ex- † panding to first order for small ϑ gives the infinitesimal transformation i δO = O − O = ϑ [O, n · L] . (C.97) Combining eqn (C.95) with eqn (C.97) yields [L i ,r j ]= i ijk r k ; therefore every vector operator V satisfies [L i ,V j ]= i ijk V k . (C.98) The infinitesimal rotation formula for an operator which is a vector field, V = V (r), contains additional terms due to the argument r: [L i ,V j (r)] = i
{(r × ∇) V j (r)+  ijk V k (r)} . (C.99) i Now let us suppose that L is an operator satisfying eqn (C.98) for any choice of V; then choosing V = L yields eqn (C.94). Therefore any operator L satisfying eqn (C.98) for all V is the generator of spatial rotations. In quantum mechanics, there is another kind of angular momentum, called spin, which has no classical analogue. Particles (or other systems) with spin are described by n-tuples of wave functions (ψ 1 (r) ,... ,ψ n (r)). The basic example is the spin-1/2 particle discussed in Appendix C.1.1-A. In the general case, the Hilbert space is a tensor product, H = H orbital ⊗ H spin , where the orbital (spatial) and spin degrees of freedom are represented by H orbital and H spin respectively. Thus the spatial and spin degrees of freedom are kinematically independent.

Minimal coupling Since L acts only on the spatial arguments of the wave functions, i.e. on H orbital , it can be expressed in the form L = L ⊗ I spin .The spin angular momentum, S = I orbital ⊗S acts only on the internal degrees of freedom, and satisfies the standard commutation relations [S i ,S j ]= i ijk S k . (C.100) Since L and S act on different parts of the product space H they must commute: [L i ,S j ]= [L i ⊗ I spin ,I orbital ⊗ S j ]= [L i ,I orbital ] ⊗ [I spin ,S j ]= 0 , (C.101) and the total angular momentum J = L + S satisfies [J i ,J j ]= i ijk J k . (C.102) This shows that J is the generator of both spatial and spin rotations. In particular, vector operators will satisfy [J i ,V j ]= i ijk V k . (C.103) The decomposition of the total angular momentum into the sum of orbital and spin parts is only possible when L and S commute, i.e. when the spatial and spin degrees of freedom are kinematically independent. C.6 Minimal coupling In minimal coupling, the standard momentum operator −i
∇ is replaced by −i
∇ →−i
∇ − qA , (C.104) where A is the vector potential for an external, classical field. This notion is usually presented as the simplest way to guarantee the gauge invariance of the quantum theory for a charge interacting with an external electromagnetic field; but there is a simpler explanation, which only involves classical electrodynamics and the correspondence principle (Cohen-Tannoudji et al., 1977b, Appendix III.3). The classical Lagrangian for a point particle with charge q interacting with the classical field determined by the scalar potential Φ and the vector potential A is m 2 L = ˙ r − qΦ+ q˙ r · A . (C.105) 2 The canonical momentum p conjugate to r is defined by ∂L p = = m˙ r + qA , (C.106) ∂ ˙ r so that the kinetic momentum m˙ r is m˙ r = p − qA . (C.107) The Hamiltonian is defined as a function of r and p by H (r, p)= p · ˙ r − L , (C.108)

Quantum theory where eqn (C.107) is used to eliminate ˙ r in favor of r and p. This leads to 1 2 H = (p − qA) + qΦ . (C.109) 2m The transition to quantum theory is now made by the correspondence-principle replacement, p →
p = −i
∇. For transverse fields (∇ · A = 0), the quantum Hamil- tonian is 1 2 H = (
p − qA (
r)) + qΦ 2m 2
p 2 q q A (
r) 2 = − A (
r) ·
p + + qΦ . (C.110) 2m m 2m 2 For many applications the external field is weak, so the A (
r) -term can be neglected and the Hamiltonian becomes
p 2 q H = + qΦ − A (
r) ·
p . (C.111) 2m m In accord with the classical terminology,
p = −i
∇ (C.112) is called the canonical momentum operator,and
p kin =
p − qA (
r) (C.113) is called the kinetic momentum operator.The velocity operator is
v = d
r/dt, and the Heisenberg equation of motion for
r (id
r/dt =[
r,H]) yields m
v =
p kin =
p − qA (
r) . (C.114) Thus the kinetic momentum operator
p kin approaches mv class in the classical limit, but the canonical momentum operator
p is the generator of spatial translations.

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Index absorption waist, 80, 220, 227 coefficient, 20 Beer’s law, 20 of light, 15 Bell additive noise, 432 expectation values, 588 adiabatic elimination, 377, 427 inequality, 578, 590 adjoint pair, 637 matrix, 650 state measurement, 621 operator, 648 states, 410 alternating tensor, 70, 645 theorem, 589 amplifier bipartite system, 201 noise, 505 birefringence, 399, 676 noise number, 524 bit-flip, 627 noise temperature, 525 blackbody uncertainty principle, 527 cavity, 5 amplitude squeezing/quadrature, 480 radiation, 5 ancilla, 607 bleaching, 460 angular momentum Bloch equation, 375 electromagnetic, 100 Bogoliubov transformation, 188, 477 orbital, 692 Bohm singlet state, 580 spin, 692 bolometers, 6, 266 total, 77, 693 Boltzmann’s principle, 17 annihilation operator, 43,74 Born antilinear, 119 approximation, 31 antinormal ordering, 167, 179 interpretation, 39, 52, 684 antireflection coating, 238, 246 Bose commutation relations, 46 antiresonant Hamiltonian, 352 Bose–Einstein statistics, 207 antiunitary, 119 bosons, 46, 207 atomic coherence, 375 bounded operator, 54, 648 atomic transition operator, 355, 442 box quantization, 81 avalanche bra vector, 680 breakdown, 282 Bragg crystal spectrometer, 11 multiplication broadband detection, 272 noise-free, 284 avoided crossing, 384 C , 201, 646, 681 n axial vector, 668 c-number, 680 Campbell–Baker–Hausdorff theorem, 689 balanced canonical beam splitter, 248 commutation relation, 39–41, 690 homodyne detector, 300 momentum, 112, 693 bare states, 383 momentum operator, 97, 694 Bargmann state, 546 quantization, 69 basis canonically conjugate variables circular, 685 classical, 39 vector space, 646 quantum, 85, 121, 690 beam carrier cleanup, 260 frequency, 88, 219 in geometric optics, 218 wavevector, 219 splitter, 247 cascade emission, 24 balanced, 248 Casimir effect, 62 symmetrical, 247 Cauchy–Schwarz inequality, 646

Index cavity constitutive equations frequency, 382 linear, 88, 672 general, 37 nonlinear, 394, 678 ideal, 32 continuous spectrum, 648 lossy, 428 controlled mode cleaning, 534 NOT gate, 633 modes, 32, 382 sign (or phase) gate, 633 planar, 63, 138, 669 convergent sequence of vectors, 647 rectangular, 33 convex linear combination, 53, 192, 204 centrosymmetric medium, 393 convolution theorem, 652 channeltrons, 282, 388 correlation matrix, 452 chaotic state, 177 correspondence principle, 30, 148, 150 characteristic function, 172 Coulomb gauge, 661 antinormally-ordered, 191 creation operator, 43,74 normally-ordered, 183 cross-Kerr medium, 418, 634 charge density cross-phase modulation, 418 classical, 670 cryptography quantum, 118 public key, 616 circular polarization, 666 quantum, 617 right (left), 55, 667 current density classical classical, 670 bit (cbit), 619 quantum, 117 electromagnetic theory, 32 cut-off function, 354 feedforward, 638 cyclic invariance of the trace, 683 nonlinear optics, 678 oscillator, 149 debyes, 134 states, 182 decay rate, 376 click (of a detector), 29 degenerate eigenvalue, 648 closed system, 420 degree of degeneracy, 52, 648 cluster state, 638 degree of freedom coarse-grained cavity radiation, 7 delta function, 224, 233 mechanical, 40 operator density, 512 degree of polarization, 57 polarization, 90, 678 delta correlated, 434 coherence matrix, 56 delta function, 657 coherence time, 350 density coherent state of states, 137 diagonal representation, 166 operator, 50, 270 of a single mode, 151, 176 dephasing rate, 376 of a wave packet, 168 detection coincidence amplitude basis, 636 one-photon, 213 counting, 286 two-photon, 214 detection, 12 loophole, 597 rate, 25 operator (Mandel), 108 collapse of a cavity state, 386 dichroic mirror, 535 collinear phase matching, 400 dielectrics complete isotropic and anisotropic, 674, 675 set of commuting observables (CSCO), diffusion term, 549 102, 684 dipole set of vectors, 163, 196, 647 approximation, 131 completeness relation, 44, 165, 355, 665, 683 matrix element, 136 compound probability rule, 660 selection rules, 133 Compton discrete quantum trajectory, 573 scattering, 10 dispersion shift, 14 cancellation effect, 326 wavelength, 14 relation, 674 computational basis, 607 displaced squeezed states, 479 conditional probability, 586, 659 displacement constant of the motion, 77, 403 field, 670

Index operator, 162 golden rule, 128, 432 rule, 169 statistics, 207 distinct paraxial beams, 221 fermions, 207 distribution, see generalized function ferrite pill, 257 Doppler shift, 134 Feynman down-conversion, 400 diagrams, 136 dressed paths/processes, 308 photon, 96 rules of interference, 307 states, 384 fidelity, 612 drift term, 451, 549 field correlation functions, 123 dynamics, 13 filter function, 274 dynodes, 282 filtered signal, 273 finite-dimensional space, 646 eigenoperators, 355 first-order perturbation theory, 127 eigenpolarization, 56, 393, 676 fluctuation operator, 203, 501, 690 eigenspace, 52, 648 fluorescence spectrum, 462 eigenvalue, 648 Fock space, 44,46 eigenvector, 648 Fokker–Planck equation, 546 Einstein four-port device, see two-channel device A and B coefficients, 17, 136 four-wave mixing, 392 relation, 453 Fourier rule, vii, 98, 258 integral transform, 651 summation convention, 36, 645 series transform, 653 Einstein–Podolsky–Rosen (EPR) slice theorem, 531 paradox, 579 Franson interferometer, 328 states, 193 frequency electric power density, 92 dispersion, 672 electro-optic modulator, 534 shift, 363 electron multiplication, 281 tripling, 412 element of physical reality, 579 fringe visibility, 312 elliptical polarization, 665 functional, 655 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 pure state, 202 beams and pulses, 80, 103, 105, 226 Schr¨odinger, 194 probability density, 152 two-photon, 212 states, 187 environment, 420 wave packet, 227, 290 picture, 539 Geiger mode, 282 equal-time commutator, 86 generalized function, 656 equipartition of energy, 7 generator of error correction, 634 displacement, 162 events, 659 spatial rotations, 76, 100, 692 evolution operator, 84 spatial translations, 76, 97, 112, 694 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 Faraday effect, 256 half-wave plate, see retarder plate fast axis, 640 Hanbury Brown–Twiss effect, 313 Fermi heat bath, 425

Index Heisenberg interaction-free measurement, 347 equations of motion, 85,86 interference picture, 83 single-photon, 307, 333 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 amplifier modes, 500 heterodyne detection interpolating operator, 162 classical, 291 inverted oscillator, 429 noise, 297 iris, see stop 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 signal, 302 Kerr media, 267, 417 Hong–Ou–Mandel (HOM) ket vector, 680 dip, 323, 339 kinematics, 13 effect, 317 kinetic momentum, 112, 693 interferometer, 315 null, 318 Lagrangian formulation, 69, 103 hyperentangled photon pairs, 408 Langevin equation advanced, 439 ideal squeezed state, 479 classical, 549 identical particles, 205 quantum, 422 image band mode, 295, 302 retarded, 438 image intensifiers, see channeltrons Laplace transform, 654 impermeability, 675 laser amplifier, 510 improper ket vector, 682 Leibnitz rule, 687 in-field level repulsion, see avoided crossing classical, 241 line operator, 242, 437 of sight, 530 incident shape, 19 annihilation operator, 242 width, 365 classical field, 237 line-out, 530 incoherent pump, 447, 511 linear independent events, 660 amplifier, 499 index matching, 401, 515 functional, 655 index of refraction, 674 operator, 646 ordinary/extraordinary, 399 polarization, 665 indistinguishable particles, 205 regression, 455 indivisibility of photons, 4,24 Liouville operator, 543 infinite-dimensional space, 646 local inner product, 646 measurement, 638 input–output number operator, 107 equation, 440, 505, 518, 519 observable, 208 matrix, 519 oscillator (LO), 291 method, 435 realistic theory, 578 insertion-loss noise, 499 longitudinal integral kernel, 73, 524, 658 relaxation time, 377 interaction vector field, 35 Hamiltonian, 117, 124 Lorentzian line shape, 365, 442 picture, 124 lowering operator, 43, 356, 424

Index Mandel Q-parameter, 278 number marginal distribution, 174, 533, 659 field sieve, 630 Markov squeezed states, 482 approximation, 283, 361 state, 43, 81, 178 process, 282, 361 number operator, 43, 74 master equation, 541 local, 107 matrix element, 682 local (Mandel), 108 maximally mixed state, 54 total, 46 Maxwell’s equations classical, 661 observables, 683 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 field, 88, 324 Langevin equation, 432 space, 98 norm, 648 state, 56 point spectrum, 648 multilevel atoms, 355 spectral resolution, 649 multiparaxial space, 221 operator-valued, 105, 649, 656 multiplicative noise, 432, 445 optic axis, 237 mutual coherence function, 312 optical mutually unbiased bases, 617 circulator, 257 couplers or taps, 248, 604 narrowband detection and filter, 273 diode, 256 negative absorption, 20 isolator, 255, 414 negative-frequency part, 72, 87 Kerr effect, 417 noise parametric amplifier/oscillator, 486 correlation function, 433 torque wrench, 80 distribution, 426 transmission line, 602 in heterodyne detection, 297 tweezer, 80 in homodyne detection, 303 ordinary wave, 678 reservoir, 422 orthogonal strength, 426, 501 complement, 646 temperature, 299 projection operators, 44, 355, 649 non-centrosymmetric crystal, 394 vectors, 646 nonanticipating, 435 orthonormal set of vectors, 196, 647 nonclassical states, 182, 306 out-field nondegenerate eigenvalue, 648 classical, 241 nondispersive material, 671 operator, 242, 439 nonlinear polarization, 678 outcome norm independence, 586 of a vector, 647 parameter, 585 of an operator, 648 overcompleteness, 164 preserving, 649 normal P (α), 181, 278 matrix, 520, 650 P-polarization, 239 mode, 37 parameter operator, 649 settings, 584 ordering, 48 parametric amplification, 267, 400 normal-ordered variance, 474 paramps, 267 normalized, 647 paraxial NOT gate, 632 classical optics, 219 null events, 345 expansion, 228

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

Index of realism, 578 oscillator, 38, 151 of separability, 195 reaction, 428 probability zone, 358 amplitude, 31, 106, 113, 127 radiative corrections, 360 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 pulse length, 227 Rayleigh range, 220, 227, 515 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 quadrature operator, 299 rectifying detection, 266 quantum reduced density operator, 197, 538 back action, 153, 419 reduction of the wave packet, 52, 194, 684 circuit, 632 reflection amplifier, 509 cloning machine (QCM), 611 regenerative amplifier, 502 computer rephasing time, 386 one-way, 638 reservoir, 422 reversible, 631 resolution of the identity, 683 dense coding, 621 resonance efficiency, 159, 276 condition, 354 electrodynamics, 113 fluorescence, 457 fuzzball, 474 resonant gate, 631 enhancement, 392 jump, 557 Hamiltonian, 352 key distribution, 617 wave approximation, 354 Langevin equation, 422 retarder plate, 238 Liouville equation, 51 half-wave plate, 641 nondemolition phase shifter, 238 counter, 267 quarter-wave plate, 641 measurement, 419 wave plate, 640 optics, 1 revival of a cavity state, 386 parallelism, 631 robust, 482, 552 register, 631 root mean square (rms) deviation, 51 regression, 454 rotating wave approximation, 354 state Rydberg diffusion, 575 atom, 141, 267, 387 reconstruction, see tomography level, 141, 267 trajectory, 573 quarter-wave plate, see retarder plate S-polarization, 239, 292 quasiclassical, 148 sample, 420, 538 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 Rabi classical field, 237 frequency, 134 scattering operator, 135, 268 channels, 237 oscillations (flopping), 373, 381 matrix, 238, 242 radiation Schmidt gauge, 22, 116, 661 decomposition, 198

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

Index of operators, 197 operator, 649 test function, see good function universal cloning machine, 609 thermal unperturbed Hamiltonian, 124 distribution in number, 59 unpolarized light, 55 state, 177 unused port, 248, 255 third-harmonic generation, 412 up-conversion, 399 Thompson scattering, 11 three-wave mixing, 392 vacuum time fluctuation, 45, 253 reversal, 118, 667 field strength, 61 translation, 123, 225, 684 Rabi flopping, 385 TM-polarization, 239 state, 43,45 tomography variance, 51, 150 classical, 529 normal-ordered, 474 optical homodyne, 532 vector quantum, 529 norm, 647 quantum-state, 533 space, 645 total number operator, 46 velocity operator, 117, 694 trace Verdet constant, 256 of a matrix, 650 von Neumann of an operator, 173, 683 entropy, 55 partial, 198 projection postulate, 52, 193, 684 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 field, 35 white noise, 434 trapping state, 497 white-light fringe, 314, 346 traveling-wave amplifier, 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 generalized, 690 X gate, 632 uncorrelated quantum fluctuations, 203 undepleted pump approximation, 402 Y-junction, 254 uniaxial crystal, 677 unitary Z gate, 633 matrix, 650 zero-point energy, 45