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ns
(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

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.

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