(2π)

0 s=1

is a superposition of elliptically-polarized waves with axes that rotate as the wave

propagates through the crystal. If only one polarization is present, e.g. Y2 (k) = 0, each

wave is linearly polarized, and the polarization direction is preserved in propagation.

It is customary and useful to get a representation similar to eqn (B.57) for the

isotropic problem by setting

0 ωs (k)

Ys (k) = ins (k) ±s (k) , (B.141)

2

so that the transverse part of the electric ¬eld is

2

d3 k ωs (k)

E ± (k) µs (k) ei(k·r’ωs (k)t) .

(+)

(r, t) = i (B.142)

2 (k) s

3 2 0 ns

(2π) s=1

Macroscopic Maxwell equations

The corresponding expansion using box-normalized plane waves is

ωks

(+)

E ⊥ (r, t) = i 2 ±ks µks e

i(k·r’ωks t)

, (B.143)

2 0 V nks

ks

√

where ωks = ωs (k), nks = ns (k), µks = µs (k), and ±ks = ±s (k) / V .

In the presence of sources the coe¬cients are time dependent:

2

d3 k ωs (k)

E ±s (k, t) µs (k) ei(k·r’ωs (k)t) ,

(+)

(r, t) = i (B.144)

3 2 0 n2 (k)

(2π) s

s=1

or

ωks

E (+) (r, t) = i ±ks (t) µks ei(k·r’ωks t) . (B.145)

2 0 V n2

ks

ks

For ¬elds satisfying eqns (3.107) and (3.120), the argument used for an isotropic

medium can be applied to the present case to derive the expressions

d3 k 2

U= ωs (k) |±s (k, t)| (B.146)

3

(2π) s

or

2

U= ωks |±s (k, t)| (B.147)

ks

for the energy in the electromagnetic ¬eld.

A Uniaxial crystals

The analysis sketched above is valid for general crystals, but there is one case of

special interest for applications. A crystal is uniaxial if it exhibits threefold, fourfold,

or sixfold symmetry under rotations in the plane perpendicular to a distinguished

axis, which we take as the z-axis. The x- and y-axes can be any two orthogonal

lines in the perpendicular plane. In general, the permittivity tensor is diagonal”

with diagonal elements x , y , z ”in the crystal-axis coordinates, but the symmetry

under rotations around the z-axis implies that x = y . We set x = y = ⊥ , but in

general ⊥ = z . In these coordinates, the unit vector along the propagation direction

is k = k/k = (sin θ cos φ, sin θ sin φ, cos θ), where θ and φ are the usual polar and

azimuthal angles. Consider a rotation about the z-axis by the angle •; then

←

’ ←

’ ←←

’’

S = R (•) S R ’1 (•)

←

’ ←

’ ’1 ←

’ ←’

= ∆ ⊥ k R (•) [← r (ω)] R ’1 (•) ∆ ⊥ k

’

←

’ ’1 ←

’

= ∆ ⊥ k [← r (ω)] ∆ ⊥ k ,

’ (B.148)

where k is the rotated unit vector and we have used the invariance of ← r under

’

←

’ ←’

rotations around the z-axis. The matrices S and S are related by a similarity

transformation, so they have the same eigenvalues for any •. The choice • = ’φ

Classical electrodynamics

←

’

e¬ectively sets φ = 0, so the eigenvalues of S can only depend on θ, the angle

between k and the distinguished axis. Setting φ = 0 simpli¬es the calculation and the

two indices of refraction are given by

n2 = ⊥, (B.149)

o

2⊥z

n2 = . (B.150)

⊥ (1 ’ cos 2θ) + z (1 + cos 2θ)

e

The phase velocity c/no , which is independent of the direction of k, characterizes

the ordinary wave, while the phase velocity c/ne , which depends on θ, describes the

propagation of the extraordinary wave. The corresponding refractive indices no and

ne are respectively called the ordinary and extraordinary index.

B.5.4 Nonlinear optics

Classical nonlinear optics (Boyd, 1992; Newell and Moloney, 1992) is concerned with

the propagation of classical light in weakly nonlinear media. Most experiments in

quantum optics involve substances with very weak magnetic susceptibility, so we will

simplify the permeability tensor to µij (ω) = µ0 δij . On the other hand, the coupling to

the electric ¬eld can be strong, if the ¬eld is nearly resonant with a dipole transition

in the constituent atoms. In such cases, the relation between the polarization and

the ¬eld is not linear. In the simplest situation, the response of the atomic dipole

to the external ¬eld can be calculated by time-dependent perturbation theory, which

produces an expression of the form (Boyd, 1992, Chap. 3)

P (r, t) = P (1) (r, t) + P NL (r, t) , (B.151)

where the nonlinear polarization

P NL (r, t) = P (2) (r, t) + P (3) (r, t) + · · · (B.152)

contains the higher-order terms in the perturbation expansion and de¬nes the nonlin-

ear constitutive relations. The transverse electric ¬eld describing radiation satis¬es

eqn (B.103), and”after using eqn (B.151) and imposing the convention that E always

means the transverse part, E ⊥ ”this can be written as

1 ‚2 ‚ 2 ⊥(1) ‚ 2 ⊥NL

∇ E ’ 2 2 E ’ µ0 2 P = µ0 2 P

2

. (B.153)

c ‚t ‚t ‚t

The interesting materials are often crystals, so scalar relations between the polar-

ization and the ¬eld must be replaced by tensor relations for anisotropic media. In a

microscopic description, the polarization P is the sum over the induced dipoles in each

atom, but we will use a coarse-grained macroscopic treatment that is justi¬ed by the

presence of many atoms in a cubic wavelength. Thus the macroscopic susceptibilities

are proportional to the density, nat , of atoms, i.e. χ(n) = nat γ (n) , where γ (n) is the

nth-order atomic polarizability. In addition to coarse graining, we will assume that the

polarization at r only depends on the ¬eld at r, i.e. the susceptibilities do not exhibit

Macroscopic Maxwell equations

the property of spatial dispersion discussed in Appendix B.5.1. For the crystals used in

quantum optics spatial dispersion is weak, so this assumption is justi¬ed in practice.

In the time domain the nth-order polarization is given by

(n) (n)

Pi dt1 · · · dtn χij1 j2 ···jn (t ’ t1 , t ’ t2 , . . . , t ’ tn )

(r, t) = 0

— Ej1 (r, t1 ) · · · Ejn (r, tn ) , (B.154)

(n)

where χij1 j2 ···jn („1 , „2 , . . . , „n ) is real and symmetric with respect to simultaneous

permutations of the time arguments „p and the corresponding tensor indices jp . The

corresponding frequency-domain relation is

n n

dνq

(n) (n)

Pi ν’

(r, ν) = 2πδ νp χij1 j2 ···jn (ν1 , . . . , νn )

0

2π

q=1 p=1

— Ej1 (r, ν1 ) · · · Ejn (r, νn ) , (B.155)

where

n n

(n) (n)

χij1 j2 ···jn (ν1 , . . . , νn ) = d„q exp i νp „p χij1 j2 ···jn („1 , „2 , . . . , „n ) . (B.156)

q=1 p=1

This notation agrees with one of the conventions (Newell and Moloney, 1992, Chap. 2d)

for the Fourier transforms of the susceptibilities, but there is a di¬erent”

(n)

and frequently used”convention in which χij1 j2 ···jn (ν1 , ν2 , . . . , νn ) is replaced by

(n)

χij1 j2 ···jn (’ν0 , ν1 , ν2 , . . . , νn ), with the understanding that the sum of the frequency

arguments is zero (Boyd, 1992, Sec. 1.5). This is an example of the notational schisms

that are common in this ¬eld. The nth-order frequency-domain susceptibility tensor

is symmetrical under simultaneous permutations of νp and jp , and the reality of the

time-domain susceptibility imposes the conditions

(n)— (n)

χij1 j2 ···jn (ν1 , . . . , νn ) = χij1 j2 ···jn (’ν1 , . . . , ’νn ) (B.157)

in the frequency domain. For the transparent media normally considered, the Fourier

(n)

transform χij1 j2 ···jn (ν1 , . . . , νn ) is also real, and eqn (B.157) becomes

(n) (n)

χij1 j2 ···jn (ν1 , . . . , νn ) = χij1 j2 ···jn (’ν1 , . . . , ’νn ) . (B.158)

The properties listed above give no information regarding what happens if the ¬rst

index i is interchanged with one of the jp s. For transparent media, the explicit quan-

tum perturbation calculation of the susceptibilities provides the additional symmetry

condition (Boyd, 1992, Sec. 3.2)

n

(n) (n)

ν1 , . . . , ’

χij1 ···jp ···jn (ν1 , . . . , νp , . . . , νn ) = χjp j1 j2 ···i···jn νk , . . . , νn . (B.159)

k=1

Appendix C

Quantum theory

Modern quantum theory originated with the independent inventions of matrix mechan-

ics by Heisenberg and wave mechanics by Schr¨dinger. It was essentially completed

o

by Schr¨dinger™s proof that the two formulations are equivalent and Born™s interpre-

o

tation of the wave function as a probability amplitude. The intuitive appeal of wave

mechanics, at least for situations involving a single particle, explains its universal use

in introductory courses on quantum theory. This approach does, however, have certain

disadvantages. One is that the intuitive simplicity of wave mechanics is largely lost

when it is applied to many-particle systems. For our purposes, a more serious objection

is that there are no wave functions for photons.

A more satisfactory approach is based on the fact that interference phenomena

are observed for all microscopic systems. For example, the two-slit experiment can

be performed with material particles to observe interference fringes. A comparison to

macroscopic wave phenomena suggests that the mathematical description of the states

of a system should satisfy the superposition principle, i.e. every linear combination

of states is also a state. In mathematical terms this means that the states are elements

of a vector space, and the Born interpretation”to be explained below”requires the

vector space to be a Hilbert space.

C.1 Dirac™s bra and ket notation

In Appendix A.3 Hilbert spaces are described with the standard notation used in

mathematics and in many textbooks on quantum theory. In the main text, we employ

an alternative notation introduced by Dirac (1958), in which a vector in a Hilbert space

H is represented by the symbol |ψ . In this notation | · represents a generic ket vector

and ψ is a label that distinguishes one vector from another. Linear combinations of

two kets, |ψ and |φ , are written as ± |ψ + β |φ , and scalars, like ± and β, are called

c-numbers.

In the Dirac notation, a bra vector F | represents a rule that assigns a complex

number, denoted by F |ψ , to every ket vector |ψ . This rule is linear, i.e. if |ψ =

± |χ + β |φ , then

F |ψ = ± F |χ + β F |φ . (C.1)

The Hilbert-space inner product (φ, ψ) is an example of such a rule, so for each ket

vector |φ there is a corresponding bra vector φ| (called the adjoint vector) de¬ned

by

φ |ψ = (φ, ψ) for all ψ . (C.2)

With this understanding, we will use • |ψ from now on to denote the inner product.

½

Dirac™s bra and ket notation

The linearity of the rule (C.1) guarantees that the set of bra vectors is in fact a

vector space. The o¬cial jargon”explained in Appendix A.6.1”is that the bra vectors

form the dual space H of linear functionals on H. The de¬nition (C.2) of the adjoint

vectors shows that the Hilbert space H of physical states is isomorphic to a subspace

of H .

The Hilbert spaces relevant for quantum theory are always separable; that is, every

ket |ψ can be expanded as

|ψ = |φn φn |ψ , (C.3)

n

where {|φn , n = 1, 2, . . .} is an orthonormal basis for H.

C.1.1 Examples

A Two-level system

The states of a two-level system, e.g. a spin-1/2 particle, are usually represented by

two-component column vectors that refer to a given basis, e.g. eigenstates of σz . The

relation between this concrete description and the Dirac notation is

ψ1 — —

• |ψ = (•, ψ) = •— ψ1 + •— ψ2 .

|ψ ∼ ψ| ∼ (ψ1 , ψ2 ) ,

, (C.4)

1 2

ψ2

The symbol ˜∼™ is used instead of ˜=™ because the values of the components ψ1 and

ψ2 depend on the particular choice of basis in the concrete space C2 . A di¬erent basis

choice would represent the same ket vector |ψ by a di¬erent pair of components ψ1 ,

ψ2 . An example of an orthonormal basis is

1 0

B= |1 ∼ , |2 ∼ , (C.5)

0 1

so the components are given by ψ1 = 1 |ψ and ψ2 = 2 |ψ , and

|ψ = ψ1 |1 + ψ2 |2 . (C.6)

This relation is invariant under a change of basis in C2 , since the vectors |1 and |2

would also be transformed. Every bra vector (linear functional) on C2 is de¬ned by

taking the inner product with some ¬xed vector in C2 , so the space of bra vectors

(the dual space) is isomorphic to the space itself, i.e. H = H. This is true for any

¬nite-dimensional Hilbert space.

B Spinless particle in three dimensions

As a second example, consider the familiar description of a spinless particle by a

square-integrable wave function ψ(r). The square-integrability condition is

∞

d3 r |ψ (r)|2 < ∞ , (C.7)

’∞

¾ Quantum theory

and the set of square-integrable functions is called L2 R3 . The relation between the

abstract and concrete descriptions is

ψ| ∼ ψ — (r) , d3 r•— (r) ψ (r) ,

|ψ ∼ ψ (r) , • |ψ = (C.8)

where the vector operations are de¬ned point-wise:

± |ψ + β |• ∼ ±ψ (r) + β• (r) . (C.9)

For in¬nite-dimensional Hilbert spaces, such as H = L2 R3 , there are bra vectors

that are not adjoints of any vector in the space. In other words, the dual space H is

larger than the space H. For example, the delta function δ (r ’ r0 ) is not the adjoint

of any vector in L2 R3 , but it does de¬ne a bra vector r0 | by

r0 |ψ = d3 rδ (r ’ r0 ) ψ (r) = ψ (r0 ) . (C.10)

This establishes the relation ψ (r) = r |ψ between the concrete and abstract descrip-

tions.

Although the bra vector r0 | is not the adjoint of any proper ket vector (nor-

malizable wave function) in L2 R3 , it is common practice to de¬ne an improper

ket vector |r0 by the rule ψ |r0 = ψ — (r0 ) for all ψ ∈ H. The position opera-

tor r is de¬ned by rψ (r) = rψ (r), and |r0 is an improper eigenvector of r”i.e.

r |r0 = r0 |r0 ”by virtue of

—

= r0 ψ — (r0 ) = r0 ψ |r0 .

ψ |r| r0 = r0 |r| ψ (C.11)

In the same way, there is no proper eigenvector of the momentum operator p, but

there is an improper eigenvector |p0 , i.e. p |p0 = p0 |p0 , associated with the bra

vector p0 | de¬ned by

d3 re’ip0 ·r/ ψ (r) .

p0 |ψ = (C.12)

C.1.2 Linear operators

The action of a linear operator A is denoted by A |ψ , and the complex number

ψ |A| • is the matrix element of the operator A for the pair of vectors |ψ and

|• . The operator A is uniquely determined by any of the sets of matrix elements

{ φn |A| φm } for all |φn , |φm in a basis B , (C.13)

{ ψ |A| • } for all |ψ , |• in H , (C.14)

{ ψ |A| ψ } for all |ψ in H . (C.15)

The operator T•χ , de¬ned by the rule

T•χ |ψ = |• χ |ψ for all |ψ , (C.16)

is usually written as |• χ|. The product of two such operators therefore acts by

¿

Physical interpretation

T•χ Tβξ |ψ = T•χ |β ξ |ψ = |• χ |β ξ |ψ . (C.17)

This holds for all states |ψ , so the product rule is

T•χ Tβξ = χ |β T•ξ . (C.18)

The operator T•• = |• •| is therefore a projection operator, provided that |• is

normalized.

Let {|φn } be an orthonormal basis for a subspace W ‚ H; then the projection

operators Pn = |φn φn | are orthogonal, i.e. Pn Pm = δnm . Every vector |ψ in W has

the unique expansion

|ψ = |φn φn |ψ = Pn |ψ , (C.19)

n n

so the operator

|φn φn |

PW = Pn = (C.20)

n n

acts as the identity for vectors in W. On the other hand, every vector |χ in the

orthogonal complement W⊥ is annihilated by PW , i.e. PW |χ = 0, so PW is the

projection operator onto W. When W = H the projection PH is the identity operator

and we get

|φn φn | = I , (C.21)

n

which is called the completeness relation, or a resolution of the identity into

the projection operators Pn = |φn φn | .

If B = {|•1 , |•2 , . . .} is an orthonormal basis, then the trace of A is de¬ned by

φn |A| φn .

Tr (A) = (C.22)

n

The value of Tr (A) is the same for all choices of orthonormal basis, and

Tr (AB) = Tr (BA) . (C.23)

The last property is called cyclic invariance, since it implies

Tr (A1 A2 · · · An ) = Tr (An A1 A2 · · · An’1 ) . (C.24)

C.2 Physical interpretation

The mathematical formalism is connected to experiment by the following assump-

tions.