matic or polychromatic ¬eld propagating in a weakly dispersive dielectric. For these

experimentally signi¬cant applications, we will make use of a physically appealing ad

hoc quantization scheme due to Milonni (1995). In the following section, we begin

with a simple model that incorporates the essential elements of this scheme, and then

outline the more rigorous version in Section 3.3.5.

3.3.3 The dressed photon model

We begin with a modi¬ed version of the vacuum ¬eld expansion (3.69)

Ek aks eks eik·r ,

E(+) (r) = i (3.139)

ks

where aks and a† satisfy the canonical commutation relations (3.65) and the c-number

ks

coe¬cient Ek is a characteristic ¬eld strength which will be chosen to ¬t the problem

at hand. In this section we will choose Ek by analyzing a simple physical model, and

then point out some of the consequences of this choice.

The mathematical convenience of the box-quantization scheme is purchased at the

cost of imposing periodic boundary conditions along the three coordinate axes. The

shape of the quantization box is irrelevant in the in¬nite volume limit, so we are at

liberty to replace the imaginary cubical box by an equally imaginary cavity in the

shape of a torus ¬lled with dielectric material, as shown in Fig. 3.1(a).

In this geometry one of the coordinate directions has been wrapped into a circle,

so that the periodic boundary conditions in that direction are physically realized by

the natural periodicity in a coordinate measuring distance along the axis of the torus.

The ¬elds must still satisfy periodic boundary conditions at the walls of the torus,

but this will not be a problem, since all dimensions of the torus will become in¬nitely

large. In this limit, the exact shape of the transverse sections is also not important.

Let L be the circumference and σ the cross sectional area for the torus, then in the

limit of large L a small segment will appear straight, as in Fig. 3.1(b), and the axis

of the torus can be chosen as the local z-axis. Since the transverse dimensions are

1 Fora sampling of the relevant references see Drummond (1990), Huttner and Barnett (1992),

Matloob et al. (1995), and Gruner and Welsch (1996).

Field quantization in passive linear media

Fig. 3.1 (a) A toroidal cavity ¬lled with a

= >

weakly dispersive dielectric. A segment has

been removed to show the central axis. The

¬eld satis¬es periodic boundary conditions

along the axis. (b) A small segment of the torus

is approximated by a cylinder, and the central

axis is taken as the z-axis.

also large, a classical ¬eld propagating in the z-direction can be approximated by a

monochromatic planar wave packet,

E (z, t) = E k (z, t) ei(kz’ωk t) + CC , (3.140)

where ωk is a solution of the dispersion relation (3.121) and E k (z, t) is a slowly-varying

envelope function. If we neglect the time derivative of the slowly-varying envelope, then

Faraday™s law (eqn (B.94)) yields

1

B (z, t) = k — E k (z, t) ei(kz’ωk t) + CC . (3.141)

ωk

As we have seen in Section 3.3.1, the ¬elds actually generated in experiments are

naturally described by wave packets. It is therefore important to remember that wave

packets do not propagate at the phase velocity vph (ωk ) = c/nk , but rather at the

group velocity

dω c

vg (ωk ) = = . (3.142)

dk nk + ωk (dn/dω)k

This fact will play an important role in the following argument, so we consider very

long planar wave packets instead of idealized plane waves.

We will determine the characteristic ¬eld Ek by equating the energy in the wave

packet to ωk . The energy can be found by integrating the rate of energy transport

across a transverse section of the torus over the time required for one round trip around

the circumference. For this purpose we need the energy ¬‚ux, S = c2 0 E — B, or rather

its average over one cycle of the carrier wave. In the almost-plane-wave approximation,

this is the familiar result S = 2c2 0 Re {E k — B— }. Setting E k = Ek ux , i.e. choosing

k

the x-direction along the polarization vector, leads to

2 2

2c2 0 k |Ek | 2c 0 nk |Ek |

S= uz = uz , (3.143)

ωk µ0

where the last form comes from using the dispersion relation. The energy passing

through a given transverse section during a time „ is Sz σ„ . The wave packet com-

pletes one trip around the torus in the time „g = L/vg (ωk ); consequently, by virtue of

the periodic nature of the motion, Sz σ„g is the entire energy in the wave packet. In

Field quantization

the spirit of Einstein™s original model we set this equal to the energy, ωk , of a single

photon:

2

2c 0 nk |Ek | σL

= ωk . (3.144)

vg (ωk ) V

The total volume of the torus is V = σL, so

ωk vg (ωk )

|Ek | = , (3.145)

2 0 cnk V

which gives the box-quantized expansions

vg (ωk )

A(+) (r) = aks eks eik·r (3.146)

2 0 nk ωk cV

ks

and

ωk vg (ωk )

E(+) (r) = i aks eks eik·r (3.147)

2 0 nk cV

ks

for the vector potential and the electric ¬eld. The continuum versions are

d3 k ωk vg (ωk )

(+)

as (k) eks eik·r

E (r) = i (3.148)

3 2 0 nk c

(2π) s

and

d3 k vg (ωk )

(+)

as (k) eks eik·r .

A (r) = (3.149)

(2π)3 2 0 nk ω k c

s

This procedure incorporates properties of the medium into the description of the ¬eld,

so the excitation created by a† or a† (k) will be called a dressed photon.

ks s

A Energy and momentum

Since ωk is the energy assigned to a single dressed photon, the Hamiltonian can be

expressed in the box-normalized form

ωk a† aks ,

Hem = (3.150)

ks

ks

or in the equivalent continuum form

d3 k

ωk a† (k) as (k) .

Hem = (3.151)

s

3

(2π) s

We will see in Section 3.3.5 that this Hamiltonian also results from an application of

the quantization procedure described there to the standard expression for the electro-

magnetic energy in a dispersive medium.

Field quantization in passive linear media

The condition (3.144) was obtained by treating the dressed photon as a parti-

cle with energy ωk . This suggests identifying the momentum of the photon with

an eigenvalue of the standard canonical momentum operator pcan = ’i ∇ of quan-

tum mechanics. Since the basis functions for box quantization are the plane waves,

exp (ik · r), this is equivalent to assigning the momentum

p= k (3.152)

to a dressed photon with energy ωk . The operator

ka† aks

Pem = (3.153)

ks

ks

would then represent the total momentum of the electromagnetic ¬eld. In Section 3.3.5

we will see that this operator is the generator of spatial translations for the quantized

electromagnetic ¬eld.

There are two empirical lines of evidence supporting the physical signi¬cance of

the canonical momentum for photons. The ¬rst is that the conservation law for Pem is

identical to the empirically well established principle of phase matching in nonlinear

optics. The second is that the canonical momentum provides a simple and accurate

model (Garrison and Chiao, 2004) for the radiation pressure experiment of Jones

and Leslie (1978). We should point out that the theoretical argument for choosing

an expression for the momentum associated with the dressed photon is not quite as

straightforward as the previous discussion suggests. The di¬culty is that there is no

universally accepted de¬nition of the classical electromagnetic momentum in a disper-

sive medium. This lack of agreement re¬‚ects a long standing controversy in classical

electrodynamics regarding the correct de¬nition of the electromagnetic momentum

density in a weakly dispersive medium (Landau et al., 1984; Ginzburg, 1989). The

implications of this controversy for the quantum theory are also discussed in Garrison

and Chiao (2004).

3.3.4 The Hilbert space of dressed-photon states

The vacuum quantization rules”e.g. eqns (3.25) and (3.26)”are supposed to be ex-

act, but this is not possible for the phenomenological quantization scheme given by

eqn (3.146). The discussion in Section 3.3.1-B shows that we cannot expect to get

a sensible theory of quantization in a dielectric without imposing some constraints,

e.g. the monochromatic condition (3.107), on the ¬elds. Since operators do not have

numerical values, these constraints cannot be applied directly to the quantized ¬elds.

Instead, the constraints must be imposed on the states of the ¬eld. For conditions (a)

and (b) the classical power spectrum is replaced by

pk = a† aks = Tr ρin a† aks , (3.154)

ks ks

s

where ρin is the density operator describing the state of the incident ¬eld. Similarly

(c) means that the average intensity E(’) (r) E(+) (r) is small compared to the char-

acteristic intensity needed to produce signi¬cant changes in the material properties.

For condition (d) the spectral width ∆ω0 is given by

Field quantization

2

(ωk ’ ω0 ) pk .

2

∆ω0 = (3.155)

k

For an experimental situation corresponding to a monochromatic classical ¬eld

with carrier frequency ω0 , the appropriate Hilbert space of states consists of the state

vectors that satisfy the quantum version of conditions (a)“(d). All such states can be

expressed as superpositions of the special number states

mks

a†

ks

√

|m = |0 , (3.156)

mks !

ks

with occupation numbers mks restricted by

mks = 0 , unless |ωk ’ ω0 | < ∆ω0 . (3.157)

The set of all linear combinations of number states satisfying eqn (3.157) is a subspace

of Fock space, which we will call a monochromatic space, H (ω0 ). For a polychro-

matic ¬eld, eqn (3.157) is replaced by the set of conditions

mks = 0 , unless |ωk ’ ωβ | < ∆ωβ , β = 0, 1, 2, . . . . (3.158)

The space H ({ωβ }) spanned by the number states satisfying these conditions is called

a polychromatic space. The representations (3.146)“(3.151) are only valid when

applied to vectors in H ({ωβ }). The initial ¬eld state ρin must therefore be de¬ned by

an ensemble of pure states chosen from H ({ωβ }).

Milonni™s quantization method—

3.3.5

The derivation of the characteristic ¬eld strength Ek in the previous section is dan-

gerously close to a violation of Einstein™s rule, so it is useful to give an independent

argument. According to eqn (3.138) the total e¬ective electromagnetic energy is

d [ω0 (ω0 )] 1 1

Uem = d3 r E 2 (r, t) + d3 r B2 (r, t) . (3.159)

dω0 2 2µ0

The time averaging eliminates the rapidly oscillating terms proportional to E (±) (r, t) ·

E (±) (r, t) or B(±) (r, t) · B(±) (r, t), so that

d [ω0 (ω0 )] 1

Uem = d3 rE (’) (r, t) · E (+) (r, t) + d3 rB (’) (r, t) · B(+) (r, t) .

dω0 µ0

(3.160)

For classical ¬elds given by eqn (3.123) the volume integral can be carried out to

¬nd

k2

2 d [ω0 (ω0 )] 2

Uem = |Aks | ,

ωk + (3.161)

dω0 µ0

ks

where Aks = Eks /iω0 is the expansion amplitude for the vector potential. Since the

2

power spectrum |Aks | is strongly peaked at ωk = ω0 , it is equally accurate to write

this result in the more suggestive form

Field quantization in passive linear media

(ωk )] k 2

2 d [ωk 2

Uem = |Aks | .

ωk + (3.162)

dωk µ0

ks

This expression presents a danger and an opportunity. The danger comes from its

apparent generality, which might lead one to forget that it is only valid for a mono-

chromatic ¬eld. The opportunity comes from its apparent generality, which makes it

clear that eqn (3.162) is also correct for polychromatic ¬elds. It is more convenient to

use the dispersion relation (3.121) and the de¬nition (ω) = 0 n2 (ω) of the index of

refraction to rewrite the curly bracket in eqn (3.162) as

d [ωk (ωk )] k 2 d [ωk nk ]

2 2

ωk + = 2 0 ω k nk

dωk µ0 dωk

c

2

= 2 0 ω k nk , (3.163)

vg (ωk )

where the last form comes from the de¬nition (3.142) of the group velocity. The total

energy is then

c 2

Uem = |Aks | .

2

2 0 ω k nk (3.164)

vg (ωk )

ks

Setting

vg (ωk )

Aks = wks , (3.165)

2 0 nk ω k c

where wks is a dimensionless amplitude, allows Uem and A(+) (r, t) to be written as

2

Uem = ωk |ws (k)| (3.166)

ks

and

vg (ωk )

A(+) (r, t) = wks eks ei(k·r’ωk t) , (3.167)

2 0 nk ωk cV

ks

respectively.

In eqn (3.166) the classical electromagnetic energy is expressed as the sum of

energies, ωk , of radiation oscillators, so the stage is set for a quantization method

—

like that used in Section 2.1.2. Thus we replace the classical amplitudes wks and wks ,

in eqn (3.167) and its conjugate, by operators aks and a† that satisfy the canonical

ks

commutation relations (3.65). In other words the quantization rule is

vg (ωk )

Aks ’ aks . (3.168)

2 0 nk ω k c

In the Schr¨dinger picture this leads to

o

vg (ωk )

A(+) (r) = aks eks eik·r , (3.169)

2 0 nk ωk cV

ks

which agrees with eqn (3.146). The Hamiltonian and the electric ¬eld are consequently

given by eqns (3.150) and (3.147), respectively, in agreement with the results of the

½¼¼ Field quantization

dressed photon model in Section 3.3.3. Once again, the general appearance of these

results must not tempt us into forgetting that they are at best valid for polychromatic

¬eld states. This means that the operators de¬ned here are only meaningful when

applied to states in the space H ({ωβ }) appropriate to the experimental situation

under study.

A Electromagnetic momentum in a dielectric—

The de¬nition (3.153) for the electromagnetic momentum is related to the fundamental

symmetry principle of translation invariance. The de¬ning properties of passive linear

dielectrics in Section 3.3.1-A implicitly include the assumption that the positional and

inertial degrees of freedom of the constituent atoms are irrelevant. As a consequence

the generator G of spatial translations is completely de¬ned by its action on the ¬eld

operators, e.g.

(+) (+)

∇Aj

Aj (r) , G = (r) . (3.170)

i

Using the expansion (3.169) to evaluate both sides leads to [aks , G] = kaks , which

is satis¬ed by the choice G = Pem . Any alternative form, G , would have to satisfy

[aks , G ’ Pem ] = 0 for all modes ks, and this is only possible if the operator Z ≡

G ’ Pem is actually a c-number. In this case Z can be set to zero by imposing the

convention that the vacuum state is an eigenstate of Pem with eigenvalue zero. The

expression (3.153) for Pem is therefore uniquely speci¬ed by the rules of quantum ¬eld

theory.

Electromagnetic angular momentum—

3.4

The properties and physical signi¬cance of Hem and P are immediately evident from

the plane-wave expansions (3.41) and (3.48), but the angular momentum presents a

subtler problem. Since the physical interpretation of J is not immediately evident from

eqns (3.54)“(3.59), our ¬rst task is to show that J does in fact represent the angular

momentum. It is possible to do this directly by verifying that J satis¬es the angular

momentum commutation relations; but it is more instructive”and in fact simpler”

to use an indirect argument. It is a general principle of quantum theory, reviewed in

Appendix C.5, that the angular momentum operator is the generator of rotations. In

particular, for any vector operator Vj (r) constructed from the ¬elds we should ¬nd

[Ji , Vj (r)] = i {(r — ∇)i Vj (r) + ijk Vk (r)} . (3.171)

Since all such operators can be built up from A(+) (r), it is su¬cient to verify this result

for V (r) = A(+) (r). The expressions (3.57) and (3.59) together with the commutation

relation (3.3) lead to

d3 r ∆⊥ (r ’ r ) (r — ∇ )i Ak

(+) (+)

Li , Aj (r) = i (r ) (3.172)

kj

and

d3 r ∆⊥ (r ’ r ) Al

(+) (+)

Si , Aj (r) = i (r ) , (3.173)

ikl kj

Electromagnetic angular momentum— ½¼½

so that

d3 r ∆⊥ (r ’ r ) (r — ∇ )i Ak

(+) (+) (+)

Ji , Aj (r) = i (r ) + ikl Al (r ) . (3.174)

kj

The de¬nition (2.30) of the transverse delta function can be written as

d3 k kl kj ik·(r’r )