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+
lin
2 2

de¬nes a linearly-polarized state which possesses zero helicity, i.e. Sz |ψ lin = 0. Due to
the action of the quarter-wave plate, the incident linearly-polarized light is converted
into circularly-polarized light. Thus the input state |ψ lin changes into the output state
|ψ cir = |1k,s=+ . The output state |ψ cir has helicity Sz = + , but it still satis¬es
Lz |ψ cir = 0. Since the transmitted photon carries away one unit (+ ) of angular
momentum, conservation of angular momentum requires the plate to acquire one unit
(’ ) of angular momentum in the opposite direction. In the classical limit of a steady
stream of linearly-polarized photons, this process is described by saying that the light
™ ™
beam exerts a torque on the plate: „z = dSz /dt = N (’ ), where N is the rate of ¬‚ow
of photons through the plate. The resulting twist of the torsion ¬ber can be sensitively
measured by means of a small mirror attached to the ¬ber.
The original experiment actually used a steady stream of light composed of very
many photons, so a classical description would be entirely adequate. However, if the
sensitivity of the experiment were to be improved to a point where ¬‚uctuations in the
¼ Field quantization

angular position of the wave plate could be measured, then the discrete nature of the
angular momentum transfer of per photon to the wave plate would show up. The
transfer of angular momentum from an individual photon to the wave plate must in
principle be discontinuous in nature, and the twisting of the wave plate should manifest
a ¬ne, ratchet-like Brownian motion. The experiment to see such ¬‚uctuations”which
would be very di¬cult”has not been performed.
A more modern experiment to demonstrate the spin angular momentum of light
was performed by trapping a small, absorbing bead within the beam waist of a tightly
focused Gaussian laser beam (Friese et al., 1998). The procedure for trapping a small
particle inside the beam waist of a laser beam has been called an optical tweezer ,
since one can then move the particle around at will by displacing the axis of the light
beam. The accompanying procedure for producing arbitrary angular displacements of
a trapped particle by transferring controllable amounts of angular momentum from
the light to the particle has been called an optical torque wrench (Ashkin, 1980). For
linearly-polarized light, no e¬ect is observed, but switching the incident laser beam to
circular polarization causes the trapped bead to begin spinning around the axis de¬ned
by the direction of propagation of the light beam. In classical terms, this behavior is a
result of the torque exerted on the particle by the absorbed light. From the quantum
point of view absorption of each photon deposits of angular momentum in the bead;
therefore, the bead has to spin up in order to conserve angular momentum.
Observations of the orbital angular momentum, Lz , of light have also been made
using a similar technique (He et al., 1995). The experiment begins with a linearly-
polarized laser beam in a Gaussian TEM00 mode. This beam”which has zero helicity
and zero orbital angular momentum”then passes through a computer-generated holo-
graphic mask with a spiral pattern imprinted onto it. The linearly-polarized, paraxial,
Gaussian beam is thereby transformed into a linearly-polarized, paraxial Laguerre“
Gaussian beam of light (Siegman, 1986, Sec. 16.4). The output beam possesses orbital,
but no spin, angular momentum. A simple Laguerre“Gaussian mode is one in which
the light e¬ectively orbits around the axis of propagation as if in an optical vortex
with a given sense of circulation. The transverse intensity pro¬le is doughnut-shaped,
with a null at its center marking a phase singularity in the beam. In principle, the spi-
ral holographic mask would experience a torque resulting from the transfer of orbital
angular momentum”one unit (+ ) per photon”to the light beam from the mask.
However, this experiment has not been performed.
What has been observed is that a small, absorbing bead trapped at the beam waist
of a Laguerre“Gaussian mode”with nonzero orbital angular momentum”begins to
spin. This spinning motion is due to the steady transfer of orbital angular momentum
from the light beam into the bead by absorption. The resultant torque is given by
™ ™
„z = dLz /dt = N (’ ), where N is the rate of photon ¬‚ow through the bead. Again,
there is a completely classical description of this experiment, so the photon nature of
light need not be invoked.
Just as for the spin-transfer experiments, a su¬ciently sensitive version of this
experiment, using a small enough bead, would display the discontinuous transfer of
orbital angular momentum in the form of a ¬ne, ratchet-like Brownian motion in
the angular displacement of the bead. This would be analogous to the discontinuous
½
Field quantization in the vacuum

transfer of linear momentum due to impact of atoms on a pollen particle that results
in the random linear displacements of the particle seen in Brownian motion. This
experiment has also not been performed.

3.1.4 Box quantization
The local, position-space commutation relations (3.1) and (3.3)”or the equivalent
reciprocal-space versions (3.25) and (3.26)”do not require any idealized boundary
conditions, but the right sides of eqns (3.3) and (3.26) contain singular functions
that cause mathematical problems, e.g. the improper one-photon state |1ks . On the
other hand, the cavity mode operators aκ and a† ”which do depend on idealized
κ
boundary conditions”have discrete labels and the one-photon states |1κ = a† |0 are
κ
normalizable. As usual, we would prefer to have the best of both worlds; and this can
be accomplished”at least formally”by replacing the Fourier integral in (3.5) with
a Fourier series. This is done by pretending that all ¬elds are contained in a ¬nite
volume V , usually a cube of side L, and imposing periodic boundary conditions at the
walls, as explained in Appendix A.4.2. This is called box quantization. Since this
imaginary cavity is not de¬ned by material walls, the periodic boundary conditions
have no physical signi¬cance. Consequently, meaningful results are only obtained in
the limit of in¬nite volume. Thus box quantization is a mathematical trick; it is not a
physical idealization, as in the physical cavity problem.
The mathematical situation resulting from this trick is almost identical to
that of the ideal physical cavity. For this case, the traveling waves, fks (r) =

eks exp (ik · r) / V , play the role of the cavity modes. The periodic boundary condi-
tions impose k =2πn/L, where n is a vector with integer components. The f ks s are
an orthonormal set of modes, i.e.


d3 r fks (r) · fk s (r) = δkk δss .
(fks , fk s ) = (3.63)
V

The various expressions for the commutation relations, the ¬eld operators, and
the observables can be derived either by replacing the real cavity mode functions in
Chapter 2 by the complex modes f ks (r), or by applying the rules relating Fourier
integrals to Fourier series, i.e.

d3 k 1
” and as (k) ” V aks , (3.64)
3 V
(2π) k

to the expressions obtained in Sections 3.1.1“3.1.3. In either way, the commutation
relations and the number operator are given by

aks , a† s a† aks .
= δkk δss , [aks , ak s ] = 0 , N = (3.65)
k ks
ks

The number states are de¬ned just as for the physical cavity,
nks
a†
ks
√ |0 ,
|n = (3.66)
nks !
ks
¾ Field quantization

where n = {nks } is the set of occupation numbers, and the completeness relation is

|n n| = 1 . (3.67)
n


Thus the box-quantization scheme replaces the delta function in eqn (3.26) by the
ordinary Kronecker symbol in the discrete indices k and s. Consequently, the box-
quantized operators aks are as well behaved mathematically as the physical cavity
operators aκ . This allows the construction of the Fock space to be carried out in
parallel to Chapter 2.1.2-C.
The expansions for the ¬eld operators are

A(+) (r) = aks eks eik·r , (3.68)
2 0 ωk V
ks


ωk
E(+) (r) = aks eks eik·r ,
i (3.69)
2 0V
ks

and
k
B(+) (r) = saks eks eik·r , (3.70)
2 0 cV
ks

where the expansion for B(+) was obtained by using B = ∇ — A and the special
property (B.52) of the circular polarization basis.
The Hamiltonian, the momentum, and the helicity are respectively given by

ωk a† aks ,
Hem = (3.71)
ks
ks

ka† aks ,
P= (3.72)
ks
ks

and
ksa† aks .
S= (3.73)
ks
ks

As always, these achievements have a price. One part of this price is that physically
meaningful results are only obtained in the limit V ’ ∞. This is not a particularly
onerous requirement, since getting the correct limit is simply a matter of careful al-
gebra combined with the rules in eqn (3.64). A more serious issue is the absence of
the total angular momentum from the list of observables in eqns (3.71)“(3.73). One
way of understanding the problem here is that the expression (3.55) for L contains
the di¬erential operator ‚/‚k which creates di¬culties in converting the continuous
integral over k into a discrete sum. The alternative expression (3.57) for L does not
involve k, so it might seem to o¬er a solution. This hope also fails, since the r-integral
in this representation must now be carried out over the imaginary cube V . The edges
of the cube de¬ne preferred directions in space, so there is no satisfactory way to de¬ne
the orbital angular momentum L.
¿
The Heisenberg picture

3.2 The Heisenberg picture
The quantization rules in Chapter 2 and Section 3.1.1 are both expressed in the
Schr¨dinger picture: observables are represented by time-independent hermitian oper-
o
ators X (S) , and the state of the radiation ¬eld is described by a ket vector Ψ(S) (t) ,
obeying the Schr¨dinger equation
o

Ψ(S) (t) = H (S) Ψ(S) (t) ,
i (3.74)
‚t
or by a density operator ρ(S) (t), obeying the quantum Liouville equation (2.119)
‚ (S)
ρ (t) = H (S) , ρ(S) (t) .
i (3.75)
‚t
The superscript (S) has been added in order to distinguish the Schr¨dinger picture
o
from two other descriptions that are frequently used. Note that the density operator is
an exception to the rule that Schr¨dinger-picture observables are independent of time.
o
There is an alternative description of quantum mechanics which actually preceded
the familiar Schr¨dinger picture. In Heisenberg™s original formulation”which appeared
o
one year before Schr¨dinger™s”there is no mention of a wave function or a wave equa-
o
tion; instead, the observables are represented by in¬nite matrices that evolve in time
according to a quantum version of Hamilton™s equations of classical mechanics. This
form of quantum theory is called the Heisenberg picture; the physical equivalence of
the two pictures was subsequently established by Schr¨dinger. The Heisenberg picture
o
is particularly useful in quantum optics, especially for the calculation of correlations
between measurements at di¬erent times. A third representation”called the interac-
tion picture”will be presented in Section 4.8. It will prove useful for the formulation
of time-dependent perturbation theory in Section 4.8.1. The interaction picture also
provides the foundation for the resonant wave approximation, which is introduced in
Section 11.1.
In the following sections we will study the properties of the Schr¨dinger and Heisen-
o
berg pictures and the relations between them. In order to distinguish between the same
quantities viewed in di¬erent pictures, the states and operators will be decorated with
superscripts (S) or (H) for the Schr¨dinger or Heisenberg pictures respectively. In
o
applications of these ideas the superscripts are usually dropped, and the distinctions
are”one hopes”made clear from context.
The Heisenberg picture is characterized by two features: (1) the states are inde-
pendent of time; (2) the observables depend on time. Imposing the superposition prin-
ciple on the Heisenberg picture implies that the relation between the time-dependent,
Schr¨dinger-picture state vector Ψ(S) (t) and the corresponding time-independent,
o
Heisenberg-picture state Ψ(H) must be linear. If we impose the convention that the
two pictures coincide at some time t = t0 , then there is a linear operator U (t ’ t0 )
such that
Ψ(S) (t) = U (t ’ t0 ) Ψ(H) . (3.76)
The identity of the pictures at t = t0 , Ψ(H) = Ψ(S) (t0 ) , is enforced by the initial
condition U (0) = 1. Substituting eqn (3.76) into the Schr¨dinger equation (3.74) yields
o
the di¬erential equation
Field quantization


U (t ’ t0 ) = H (S) U (t ’ t0 ) , U (0) = 1
i (3.77)
‚t
for the operator U (t ’ t0 ). This has the solution (Bransden and Joachain, 1989, Sec.
5.7)
i
U (t ’ t0 ) = exp ’ (t ’ t0 ) H (S) , (3.78)

where the evolution operator on the right side is de¬ned by the power series for the
exponential, or by the general rules outlined in Appendix C.3.6. The Hermiticity of
H (S) guarantees that U (t ’ t0 ) is unitary, i.e.
U (t ’ t0 ) U † (t ’ t0 ) = U † (t ’ t0 ) U (t ’ t0 ) = 1 . (3.79)
The choice of t0 is dictated by convenience for the problem at hand. In most cases
it is conventional to set t0 = 0, but in scattering problems it is sometimes more useful
to consider the limit t0 ’ ’∞. The evolution operator satis¬es the group property,
U (t1 ’ t2 ) U (t2 ’ t3 ) = U (t1 ’ t3 ) , (3.80)
which simply states that evolution from t3 to t2 followed by evolution from t2 to t1 is
the same as evolving directly from t3 to t1 . For the special choice t0 = 0, this simpli¬es
to U (t1 ) U (t2 ) = U (t1 + t2 ). The de¬nition (3.78) also shows that U (’t) = U † (t).
In what follows, we will generally use the convention t0 = 0; any other choice of initial
time will be introduced explicitly.
The physical equivalence of the two pictures is enforced by requiring that each
Schr¨dinger-picture operator X (S) and the corresponding Heisenberg-picture operator
o
(H)
X (t) have the same expectation values in corresponding states:

Ψ(H) X (H) (t) Ψ(H) = Ψ(S) (t) X (S) Ψ(S) (t) , (3.81)

for all vectors Ψ(S) (t) and observables X (S) . Using eqn (3.76) allows this relation to
be written as
Ψ(H) X (H) (t) Ψ(H) = Ψ(H) U † (t) X (S) U (t) Ψ(H) . (3.82)

Since this equation holds for all states, the general result (C.15) shows that the oper-
ators in the two pictures are related by
X (H) (t) = U † (t) X (S) U (t) . (3.83)
Note that the Heisenberg-picture operators agree with the (time-independent)
Schr¨dinger-picture operators at t = 0. This de¬nition, together with the group prop-
o
erty U (t1 ) U (t2 ) = U (t1 + t2 ), provides a useful relation between the Heisenberg
operators at di¬erent times:
X (H) (t + „ ) = U † (t + „ ) X (S) U (t + „ )
= U † („ ) U † (t) X (S) U (t) U („ )
= U † („ ) X (H) (t) U („ ) . (3.84)
Also note that H (S) commutes with exp ±itH (S) / , so eqn (3.83) implies that the
Hamiltonian is the same in both pictures: H (H) (t) = H (S) = H.
The Heisenberg picture

In the Heisenberg picture, the operators evolve in time while the state vectors are
¬xed. The density operator is again an exception. Applying the transformation (3.83)
to the de¬nition of the Schr¨dinger-picture density operator,
o

Pu ˜(S) (t)
ρ(S) (t) = ˜(S) (t) , (3.85)
u u
u

yields the time-independent operator

Pu ˜(H)
ρ(H) = ˜(H) = ρ(S) (0) , (3.86)
u u
u

which is the initial value for the quantum Liouville equation (3.75).
A di¬erential equation describing the time evolution of operators in the Heisenberg
picture is obtained by combining eqn (3.77) with the common form of the Hamiltonian
to get

‚X (H) (t) i
= U † (t) H, X (S) U (t)
‚t
i
H, X (H) (t) ,
= (3.87)

where the last line follows from the identity

U † (t) X (S) Y (S) U (t) = U † (t) X (S) U (t) U † (t) Y (S) U (t)
= X (H) (t) Y (H) (t) . (3.88)

Multiplying eqn (3.87) by i yields the Heisenberg equation of motion for the
observable X (H) :
‚X (H) (t)
= X (H) (t) , H .
i (3.89)
‚t
The de¬nition (3.83) provides a solution for this equation. The name ˜constant of the
motion™ for operators X (S) that commute with the Hamiltonian is justi¬ed by the
observation that the Heisenberg equation for X (H) (t) is (‚/‚t) X (H) (t) = 0.
In most applications we will suppress the identifying superscripts (H) and (S). The
distinctions between the Heisenberg and Schr¨dinger pictures will be maintained by the
o
convention that an operator with a time argument, e.g. X (t), is the Heisenberg-picture
form, while X”with no time argument”signi¬es the Schr¨dinger-picture form. The
o
only real danger of this convention is that density operators behave in the opposite way;
ρ (t) denotes a Schr¨dinger-picture operator, while ρ is taken in the Heisenberg picture.
o
This is not a serious problem if the accompanying text provides the appropriate clues.

3.2.1 Equal-time commutators
A pair of Schr¨dinger-picture operators X and Y is said to be canonically conjugate
o
if [X, Y ] = β, where β is a c-number. Canonically conjugate pairs, e.g. position and
momentum, play an important role in quantum theory, so it is useful to consider the
commutator in the Heisenberg picture. Evaluating [X (t) , Y (t )] for t = t requires a
Field quantization

complete solution of the Heisenberg equations for X (t) and Y (t ), but the equal-time
commutator for such a canonically conjugate pair is given by

[X (t) , Y (t)] = U † (t) XU (t) , U † (t) Y U (t)
= U † (t) [X, Y ] U (t)
= β. (3.90)

Thus the equal-time commutator of the Heisenberg-picture operators is identical to the
commutator of the Schr¨dinger-picture operators. Applying this to the position-space
o
commutation relation (3.3) and to the canonical commutator (3.65) yields

i
∆⊥ (r ’ r )
[Ai (r, t) , ’Ej (r , t)] = (3.91)
ij
0

and
aks (t) , a† s (t) = δss δkk , (3.92)
k

respectively.

3.2.2 Heisenberg equations for the free ¬eld
The preceding arguments are valid for any form of the Hamiltonian, but the results are
particularly useful for free ¬elds. The Heisenberg-picture form of the box-quantized
Hamiltonian is
ωk a† (t) aks (t) ,
Hem = (3.93)
ks
ks

and eqn (3.89), together with the equal-time versions of eqn (3.65), yields the equation
of motion for the annihilation operators

d
’ ωk aks (t) = 0 .
i (3.94)
dt

The solution is
aks (t) = aks e’iωk t = eiHem t/ aks e’iHem t/ , (3.95)
where we have used the identi¬cation of aks (0) with the Schr¨dinger-picture operator
o
aks . Combining this solution with the expansion (3.68) gives

A(+) (r, t) = aks eks ei(k·r’ωk t) . (3.96)
2 0 ωk V
ks

The expansions (3.69) and (3.70) allow the operators E(+) (r, t) and B(+) (r, t) to be
expressed in the same way.
Field quantization in passive linear media

3.2.3 Positive- and negative-frequency parts
We are now in a position to explain the terms positive-frequency part and negative-
frequency part introduced in Section 3.1.2. For this purpose it is useful to review some
features of Fourier transforms. For any real function F (t), the Fourier transform sat-
is¬es F — (ω) = F (’ω). Thus F (ω) for negative frequencies is completely determined
by F (ω) for positive frequencies. Let us use this fact to rewrite the inverse transform
as ∞

F (ω) e’iωt = F (+) (t) + F (’) (t) ,
F (t) = (3.97)
’∞ 2π

where the positive-frequency part,


F (ω) e’iωt ,
(+)
F (t) = (3.98)

0

and the negative-frequency part,
0

F (ω) e’iωt ,
(’)
F (t) = (3.99)

’∞

are related by
F (’) (t) = F (+)— (t) . (3.100)
The de¬nitions of F (±) (t) guarantee that F (+) (ω) vanishes for ω < 0 and F (’) (ω)
vanishes for ω > 0.
The division into positive- and negative-frequency parts works equally well for any
time-dependent hermitian operator, X (t). One simply replaces complex conjugation
by the adjoint operation; i.e. eqn (3.100) becomes X (’) (t) = X (+)† (t). In particular,
the temporal Fourier transform of the operator A(+) (r, t), de¬ned by eqn (3.96), is

aks eks eik·r 2πδ (ω ’ ωk ) . (3.101)
A(+) (r, ω) = dt eiωt A(+) (r, t) =
2 0 ωk V
ks

Since ωk = c |k| > 0, A(+) (r, ω) vanishes for ω < 0, and A(’) (r, ω) = A(+)† (r, ’ω)
vanishes for ω > 0. Thus the Schr¨dinger-picture de¬nition (3.68) of the positive-
o
frequency part agrees with the Heisenberg-picture de¬nition at t = 0.
The commutation rules derived in Section 3.1.2 are valid here for equal-time com-
mutators, but for free ¬elds we also have the unequal-times commutators:

F (±) (r, t) , G(±) (r , t ) = 0 , (3.102)

provided only that F (±) (r, 0) and G(±) (r , 0) are sums over annihilation (creation)
operators.

3.3 Field quantization in passive linear media
Optical devices such as lenses, mirrors, prisms, beam splitters, etc. are the main tools
of experimental optics. In classical optics these devices are characterized by their bulk
Field quantization

optical properties, such as the index of refraction. In order to apply the same simple
descriptions to quantum optics, we need to extend the theory of photon propagation
in vacuum to propagation in dielectrics. We begin by considering classical ¬elds in
passive, linear dielectrics”which we will always assume are nonmagnetic”and then
present a phenomenological model for quantization.

3.3.1 Classical ¬elds in linear dielectrics
A review of the electromagnetic properties of linear media can be found in Appen-
dix B.5.1, but for the present discussion we only need to recall that the constitutive
relations for a nonmagnetic, dielectric medium are H (r, t) = B (r, t) /µ0 and

D (r, t) = 0E (r, t) + P (r, t) . (3.103)

For an isotropic, homogeneous medium that does not exhibit spatial dispersion (see
Appendix B.5.1) the polarization P (r, t) is related to the ¬eld by

dt χ(1) (t ’ t ) E (r, t ) ,
P (r, t) = (3.104)
0


where the linear susceptibility χ(1) (t ’ t ) describes the delayed response of the
medium to an applied electric ¬eld. Fourier transforming eqn (3.104) with respect to
time produces the equivalent frequency-domain relation

P (r, ω) = (ω) E (r, ω) .
(1)
0χ (3.105)

Applying the de¬nition of positive- and negative-frequency parts, given by eqns
(3.97)“(3.99), to the real classical ¬eld E (r, t) leads to

E (r, t) = E (+) (r, t) + E (’) (r, t) . (3.106)

In position space, the strength of the electric ¬eld at frequency ω is represented by
2
the power spectrum E (+) (r, ω) (see Appendix B.2). In reciprocal space, the power
2
spectrum is E (+) (k, ω) . We will often be concerned with ¬elds for which the power
spectrum has a single well-de¬ned peak at a carrier frequency ω = ω0 . The value
of ω0 is set by the experimental situation, e.g. ω0 is often the frequency of an injected
2
signal. The reality condition (3.100) for E (±) (r, ω) tells us that E (’) (r, ω) has a
peak at ω = ’ω0 ; consequently, the complete transform E (r, ω) has two peaks: one
at ω = ω0 and the other at ω = ’ω0 .
We will say that the ¬eld is monochromatic if the spectral width, ∆ω0 , of the
peak at ω = ±ω0 satis¬es
∆ω0 ω0 . (3.107)
We should point out that this usage is unconventional. Fields satisfying eqn (3.107)
are often called quasimonochromatic in order to distinguish them from the ideal case
in which the spectral width is exactly zero: ∆ω0 = 0. Since the ¬elds generated in real
experiments are always described by wave packets with nonzero spectral widths, we
prefer the de¬nition associated with eqn (3.107). The ideal ¬elds with ∆ω0 = 0 will
be called strictly monochromatic.
Field quantization in passive linear media

The concentration of the Fourier transform in the vicinity of ω = ±ω0 allows us to
(±)
de¬ne the slowly-varying envelope ¬elds E (r, t) by setting
(±)
(r, t) = E (±) (r, t) e±iω0 t ,
E (3.108)

so that
(+) (’)
(r, t) e’iω0 t + E
E (r, t) = E (r, t) eiω0 t . (3.109)
(’) (+)—
The slowly-varying envelopes satisfy E (r, t) = E (r, t), and the time-domain
version of eqn (3.107) is
(±) (±)
‚2E ‚E
(r, t) (r, t) (±)
ω0 E
2
ω0 (r, t) . (3.110)
‚t2 ‚t

The frequency-domain versions of eqns (3.108) and (3.109) are
(±)
E (r, ω) = E (±) (r, ω ± ω0 ) (3.111)

and
(+) (’)
E (r, ω) = E (r, ω ’ ω0 ) + E (r, ω + ω0 ) , (3.112)
(±)
respectively. The condition (3.107) implies that E (r, ω) is sharply peaked at ω = 0.
The Fourier transform of the vector potential is also concentrated in the vicinity
of ω = ±ω0 , so the slowly-varying envelope,
(+)
A (r, t) = A(+) (r, t) eiω0 t , (3.113)

satis¬es the same conditions. Since E (r, t) = ’‚A (r, t) /‚t, the two envelope functions
are related by
‚ (+)
(+) (+)
E (r, t) = iω0 A ’A. (3.114)
‚t
Applying eqn (3.110) to the vector potential shows that the second term on the right
side is small compared to the ¬rst, so that
(+) (+)
E (r, t) ≈ iω0 A . (3.115)

This is an example of the slowly-varying envelope approximation.
More generally, it is necessary to consider polychromatic ¬elds, i.e. superposi-
tions of monochromatic ¬elds with carrier frequencies ωβ (β = 0, 1, 2, . . .). The car-
rier frequencies are required to be distinct; that is, the power spectrum for a poly-
chromatic ¬eld exhibits a set of clearly resolved peaks at the carrier frequencies
ωβ . The explicit condition is that the minimum spacing between peaks, δωmin =
min [|ω± ’ ωβ | , ± = β] , is large compared to the maximum spectral width, ∆ωmax =
max [∆ωβ ]. The values of the carrier frequencies are set by the experimental situation
under study. The collection {ωβ } will generally contain the frequencies of any injected
¬elds together with the frequencies of radiation emitted by the medium in response to
¼ Field quantization

the injected signals. For a polychromatic ¬eld, eqns (3.108), (3.113), and (3.115) are
replaced by
(+)
(r, t) e’iωβ t ,
E (+) (r, t) = Eβ (3.116)
β


(+)
Aβ (r, t) e’iωβ t ,
A(+) (r, t) = (3.117)
β


and
(+) (+)
Eβ (r, t) = iωβ Aβ (r, t) . (3.118)

In the frequency domain, the total polychromatic ¬eld is given by

(σ)
E (r, ω) = E β (r, ω ’ σωβ ) , (3.119)
β σ=±


(±)
where each of the functions E β (r, ω) is sharply peaked at ω = 0.

A Passive, linear dielectric
An optical medium is said to be passive and linear if the following conditions are
satis¬ed.
(a) O¬ resonance. The classical power spectrum is negligible at frequencies that are
resonant with any transition of the constituent atoms. This justi¬es the assump-
tion that there is no absorption.
(b) Coarse graining. There are many atoms in the volume »3 , where »0 is the mean
0
wavelength for the incident ¬eld.
(c) Weak ¬eld. The ¬eld is not strong enough to induce signi¬cant changes in the
material medium.
(d) Weak dispersion. The frequency-dependent susceptibility χ(1) (ω) is essentially
constant across any frequency interval ∆ω ω.
(e) Stationary medium. The medium is stationary, i.e. the optical properties do
not change in time.
The passive property is incorporated in the o¬-resonance assumption (a) which
allows us to neglect absorption, stimulated emission, and spontaneous emission. The
description of the medium by the usual macroscopic coe¬cients such as the suscep-
tibility, the refractive index, and the conductivity is justi¬ed by the coarse-graining
assumption (b). The weak-¬eld assumption (c) guarantees that the macroscopic ver-
sion of Maxwell™s equations is linear in the ¬elds. The weak dispersion condition (d)
assures us that an input wave packet with a sharply de¬ned carrier frequency will
retain the same frequency after propagation through the medium. The assumption (e)
implies that the susceptibility χ(1) (t ’ t ) only depends on the time di¬erence t ’ t .
½
Field quantization in passive linear media

For later use it is helpful to explain these conditions in more detail. The medium
is said to be weakly dispersive (in the vicinity of the carrier frequency ω = ω0 ) if

‚χ(1) (ω)
χ(1) (ω0 )
∆ω0 (3.120)
‚ω ω=ω0

for any frequency interval ∆ω0 ω0 . We next recall that in a linear, isotropic dielectric
the vacuum dispersion relation ω = ck is replaced by
ωn (ω) = ck , (3.121)
where the index of refraction is related to the dielectric permittivity, (ω), by n2 (ω) =
(ω). Since (ω) can be complex”the imaginary part describes absorption or gain
(Jackson, 1999, Chap. 7)”the dispersion relation does not always have a real solu-
tion. However, for transparent dielectrics there is a range of frequencies in which the
imaginary part of the index is negligible.
For a given wavenumber k, let ωk be the mode frequency obtained by solving
the nonlinear dispersion relation (3.121), then the medium is transparent at ωk if
nk = n (ωk ) is real. In the frequency“wavenumber domain the electric ¬eld satis¬es
ω2 2
n (ω) ’ k 2 E k (ω) = 0 (3.122)
2
c
(see Appendix B.5.2, eqn (B.123)), so one ¬nds the general space“time solution
E (r, t) = E (+) (r, t) + E (’) (r, t), with
1
E (+) (r, t) = √ Eks eks ei(k·r’ωk t) . (3.123)
V ks

For a monochromatic ¬eld, the slowly-varying envelope is
1
(+)
(r, t) = √
E Eks eks ei(k·r’∆k t) , (3.124)
V ks

where the prime on the k-sum indicates that it is restricted to k-values such that
the detuning, ∆k = ωk ’ ω0 , satis¬es |∆k | ω0 . The wavelength mentioned in the
coarse-graining assumption (b) is then »0 = 2πc/ (n (ω0 ) ω0 ).
For a polychromatic ¬eld, eqn (3.108) is replaced by
(+)
(r, t) e’iωβ t ,
E (+) (r, t) = Eβ (3.125)
β

where
1
(+)
E β (r, t) = √ Eβks eks ei(k·r’∆βk t) , ∆βk = ωk ’ ωβ (3.126)
V ks

is a slowly-varying envelope ¬eld. The spectral width of the βth monochromatic ¬eld
(+) (+)
2 2
is de¬ned by the power spectrum E β (r, ω) or E β (k, ω) . The weak disper-
sion condition (d) is extended to this case by imposing eqn (3.120) on each of the
monochromatic ¬elds.
¾ Field quantization

The condition (3.107) for a monochromatic ¬eld guarantees the existence of an
intermediate time scale T satisfying
1 1
T , (3.127)
ω0 ∆ω0
i.e. T is long compared to the carrier period but short compared to the characteristic
time scale on which the envelope ¬eld changes. Averaging over the interval T will
wash out all the fast variations”on the optical frequency scale”but leave the slowly-
varying envelope unchanged. In the polychromatic case, applying eqn (3.107) to each
monochromatic component picks out an overall time scale T satisfying 1/ωmin T
1/∆ωmax , where ωmin = min (ωβ ).

B Electromagnetic energy in a dispersive dielectric
For an isotropic, nondispersive dielectric”e.g. the vacuum”Poynting™s theorem (see
Appendix B.5) takes the form

‚uem (r, t)
+ ∇ · S (r, t) = 0 , (3.128)
‚t
where
1 12
E 2 (r, t) + B (r, t)
uem (r, t) = (3.129)
2 µ0
is the electromagnetic energy density and
1
S (r, t) = E (r, t) — H (r, t) = E (r, t) — B (r, t) (3.130)
µ0
is the Poynting vector. The existence of an electromagnetic energy density is an es-
sential feature of the quantization schemes presented in Chapter 2 and in the present
chapter, so the existence of a similar object for weakly dispersive media is an important
question.
For a dispersive dielectric eqn (3.128) is replaced by
‚umag
pel (r, t) + +∇· S = 0, (3.131)
‚t
where the electric power density,
‚D (r, t)
pel (r, t) = E (r, t) · , (3.132)
‚t
is the power per unit volume ¬‚owing into the dielectric medium due to the action of
the slowly-varying electric ¬eld E, and
12
B (r, t)
umag (r, t) = (3.133)
2µ0
is the magnetic energy density; see Jackson (1999, Sec. 6.8). The existence of the
magnetic energy density umag (r, t) is guaranteed by the assumption that the material
¿
Field quantization in passive linear media

is not magnetically dispersive. The question is whether pel (r, t) can also be expressed
as the time derivative of an instantaneous energy density. The electric displacement
D (r, t) and the polarization P (r, t) are given by eqns (3.103) and (3.104), respectively,
so in general P (r, t) and D (r, t) depend on the electric ¬eld at times t = t. The
principle of causality restricts this dependence to earlier times, t < t, so that

χ(1) (t ’ t ) = 0 for t > t . (3.134)
(1)
For a nondispersive medium χ(1) (ω) has the constant value χ0 , so in this approxi-
mation one ¬nds that
(1)
χ(1) (t ’ t ) = χ0 δ (t ’ t ) . (3.135)
In this case, the polarization at a given time only depends on the ¬eld at the same
time. In the dispersive case, χ(1) (t ’ t ) decays to zero over a nonzero interval, 0 <
t’t < Tmem ; in other words, the polarization at t depends on the history of the electric
¬eld up to time t. Consequently, the power density pel (r, t) cannot be expressed as
pel (r, t) = ‚uel (r, t) /‚t, where uel (r, t) is an instantaneous energy density.
In the general case this obstacle is insurmountable, but for a monochromatic (or
polychromatic) ¬eld in a weakly dispersive dielectric it can be avoided by the use of
an appropriate approximation scheme (Jackson, 1999, Sec. 6.8). The fundamental idea
in this argument is to exploit the characteristic time T introduced in eqn (3.127) to
de¬ne the (running) time-average
T /2
1
pel pel (r, t + t ) dt .
(r, t) = (3.136)
T ’T /2

This procedure eliminates all rapidly varying terms, and one can show that

‚uel (r, t)
pel (r, t) = , (3.137)
‚t
where the e¬ective electric energy density is

d [ω0 (ω0 )] 1
E (r, t) · E (r, t)
uel (r, t) =
dω0 2
d [ω0 (ω0 )] (’) (+)
E (r, t) · E
= (r, t) , (3.138)
dω0
(+)
and E (r, t) is the slowly-varying envelope for the electric ¬eld. The e¬ective electric
energy density for a polychromatic ¬eld is a sum of terms like uel (r, t) evaluated
for each monochromatic component. We will use this expression in the quantization
technique described in Section 3.3.5.

3.3.2 Quantization in a dielectric
The behavior of the quantized electromagnetic ¬eld in a passive linear dielectric is
an important practical problem for quantum optics. In principle, this problem could
be approached through a microscopic theory of the quantized ¬eld interacting with
Field quantization

the point charges in the atoms constituting the medium. The same could be said for
the classical theory of ¬elds in a dielectric, but it is traditional”and a great deal
easier”to employ instead a phenomenological macroscopic approach which describes
the response of the medium by the linear susceptibility. The long history and great
utility of this phenomenological method have inspired a substantial body of work
aimed at devising a similar description for the quantized electromagnetic ¬eld in a
dielectric medium.1 This has proven to be a di¬cult and subtle task. The phenomeno-
logical quantum theory for the cavity and the exact vacuum theory both depend on
an expression for the classical energy as the sum of energies for independent radiation
oscillators, but”as we have seen in the previous section”there is no exact instanta-
neous energy for a dispersive medium. Fortunately, an exact quantization method is
not needed for the analysis of the large class of experiments that involve a monochro-
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 )
∆⊥ (r ’ r ) = δlj δ (r ’ r ) ’ e , (3.175)
lj
(2π)3 k 2

and the ¬rst term on the right produces eqn (3.171) with V = A(+) . A straightforward
calculation using the identity

kl kj eik·(r’r ) = ’∇l ∇j eik·(r’r ) (3.176)

and judicious integrations by parts shows that the contribution of the second term in
eqn (3.175) vanishes; therefore, eqn (3.171) is established in general.
For a global vector operator G, de¬ned by

d3 rg (r) ,
G= (3.177)

integration of eqn (3.171) yields

[Jk , Gi ] = i kij Gj . (3.178)

In particular the last equation applies to G = J; therefore, J satis¬es the standard
angular momentum commutation relations,

[Ji , Jj ] = i ijk Jk . (3.179)

The combination of eqns (3.171) and (3.179) establish the interpretation of J as the
total angular momentum operator for the electromagnetic ¬eld.
In quantum mechanics the total angular momentum J of a particle can always
be expressed as J = L + S, where L is the orbital angular momentum (relative to
a chosen origin) and the spin angular momentum S is the total angular momentum
in the rest frame of the particle (Bransden and Joachain, 1989, Sec. 6.9). Since the
photon travels at the speed of light, it has no rest frame; therefore, we should expect
to meet with di¬culties in any attempt to ¬nd a similar decomposition, J = L + S,
for the electromagnetic ¬eld. As explained in Appendix C.5, the usual decomposition
of the angular momentum also depends crucially on the assumption that the spin and
spatial degrees of freedom are kinematically independent, so that the operators L and
S commute. For a vector ¬eld, this would be the case if there were three independent
components of the ¬eld de¬ned at each point in space. In the theory of the radiation
¬eld, however, the vectors ¬elds E and B are required to be transverse, so there are
only two independent components at each point. The constraint on the components of
the ¬elds is purely kinematical, i.e. it holds for both free and interacting ¬elds, so the
spin and spatial degrees of freedom are not independent. The restriction to transverse
½¼¾ Field quantization

¬elds is related to the fact that the rest mass of the photon is zero, and therefore to
the absence of any rest frame.
How then are we to understand eqn (3.54) which seems to be exactly what one
would expect? After all we have established that L and S are physical observables,
and the integrand in eqn (3.57) contains the operator ’ir — ∇, which represents
orbital angular momentum in quantum mechanics. Furthermore, the expression (3.59)
is independent of the chosen reference point r = 0. It is therefore tempting to interpret
L as the orbital angular momentum (relative to the origin), and S as the intrinsic or
spin angular momentum of the electromagnetic ¬eld, but the arguments in the previous
paragraph show that this would be wrong.
To begin with, eqn (3.60) tells us that S does not satisfy the angular momentum
commutation relations (3.179); so we are forced to conclude that S is not any kind of
angular momentum. The representation (3.57) can be used to evaluate the commuta-
tion relations for L, but once again there is a simpler indirect argument. The ˜spin™
operator S is a global vector operator, so applying eqn (3.178) gives

[Jk , Si ] = i kij Sj . (3.180)

Combining the decomposition (3.54) with eqn (3.60) produces

[Lk , Si ] = i kij Sj , (3.181)

so L acts as the generator of rotations for S. Using this, together with eqn (3.54) and
eqn (3.179), provides the commutators between the components of L,

(Lj ’ Sj ) .
[Lk , Li ] = i (3.182)
kij

Thus the sum J = L + S is a genuine angular momentum operator, but the sepa-
rate ˜orbital™ and ˜spin™ parts do not commute and are not themselves true angular
momenta.
If the observables L and S are not angular momenta, then what are they? The
physical signi¬cance of the helicity operator S is reasonably clear from k · S |1ks =
s |1ks , but the meaning of the orbital angular momentum L is not so obvious. In
common with true angular momenta, the di¬erent components of L do not commute.
Thus it is necessary to pick out a single component, say Lz , which is to be diagonalized.
The second step is to ¬nd other observables which do commute with Lz , in order to
construct a complete set of commuting observables. Since we already know that L is
not a true angular momentum, it should not be too surprising to learn that Lz and
L2 do not commute. The commutator between L and the total momentum P follows
from the fact that P is a global vector operator that satis¬es eqn (3.178) and also
commutes with S. This shows that

[Lk , Pi ] = i kij Pj , (3.183)

so L does serve as the generator of rotations for the electromagnetic momentum. By
combining the commutation relations given above, it is straightforward to show that
Lz , Sz , S 2 , Pz , and P 2 all commute. With this information it is possible to replace the
Wave packet quantization— ½¼¿

plane-wave modes with a new set of modes (closely related to vector spherical harmon-
ics (Jackson, 1999, Sec. 9.7)) that provide a representation in which both Lz and Sz
are diagonal in the helicity. The details of these interesting formal developments can
be found in the original literature, e.g. van Enk and Nienhuis (1994), but this approach
has not proved to be particularly useful for the analysis of existing experiments.
The experiments reviewed in Section 3.1.3-E all involve paraxial waves, i.e. the ¬eld
in each case is a superposition of plane waves with propagation vectors nearly parallel
to the main propagation direction. In this situation, the z-axis can be taken along the
propagation direction, and we will see in Chapter 7 that the operators Sz and Lz are,
at least approximately, the generators of spin and orbital rotations respectively.

Wave packet quantization—
3.5
While the method of box-quantization is very useful in many applications, it has both
conceptual and practical shortcomings. In Section 3.1.1 we replaced the quantum rules
(2.61) for the physical cavity by the position-space commutation relations (3.1) and
(3.3) on the grounds that the macroscopic boundary conditions at the cavity walls
do not belong in a microscopic theory. The imaginary cavity with periodic boundary
conditions is equally out of place, so it would clearly be more satisfactory to deal
directly with the position-space commutation relations. A practical shortcoming of
the box-quantization method is that it does not readily lend itself to the description of
incident ¬elds that are not simple plane waves. In real experiments the incident ¬elds
are more accurately described by Gaussian beams (Yariv, 1989, Sec. 6.6); consequently,
it would be better to have a more ¬‚exible method that can accommodate incident ¬elds
of various types.
In this section we will develop a representation of the ¬eld operators that deals
directly with the singular commutation relations in a mathematically and physically
sensible way. This new representation depends on the de¬nition of the electromagnetic
phase space in terms of normalizable classical wave packets. Creation and annihila-
tion operators de¬ned in terms of these wave packets will replace the box-quantized
operators.
3.5.1 Electromagnetic phase space
In classical mechanics, the state of a single particle is described by the ordered pair
(q, p), where q and p are respectively the canonical coordinate and momentum of
the particle. The pairs, (q, p), of vectors label the points of the mechanical phase
space “mech, and a unique trajectory (q(t), p(t)) is de¬ned by the initial conditions
(q(0), p(0)) = (q0 , p0 ). A unique solution of Maxwell™s equations is determined by the
initial conditions
A (r, 0) = A0 (r) ,
(3.184)
E (r, 0) = E 0 (r) ,
where A0 (r) and E 0 (r) are given functions of r. By analogy to the mechanical case, the
points of electromagnetic phase space “em are labeled by pairs of real transverse
vector ¬elds, (A (r) , ’E (r)). The use of ’E (r) rather than E (r) is suggested by
the commutation relations (3.3), and it also follows from the classical Lagrangian
formulation (Cohen-Tannoudji et al., 1989, Sec. II.A.2).
½¼ Field quantization

A more useful representation of “em can be obtained from the classical part of the
analysis, in Section 3.3.5, of quantization in a weakly dispersive dielectric. Since the
vacuum is the ultimate nondispersive dielectric, we can directly apply eqn (3.167) to
see that the general solution of the vacuum Maxwell equations is determined by

d3 k
A(+) (r, t) = ws (k) eks ei(k·r’ωk t) , (3.185)
3 2 0 ωk
(2π) s

where we have applied the rules (3.64) to get the free-space form. The complex func-
tions ws (k) and the two-component functions w (k) = (w+ (k) , w’ (k)) are respec-
tively called polarization amplitudes and wave packets. The classical energy for
this solution is
d3 k 2
U= |ws (k)| .
3 ωk (3.186)
(2π) s

Physically realizable classical ¬elds must have ¬nite total energy, i.e. U < ∞, but
Einstein™s quantum model suggests an additional and independent condition. This
comes from the interpretation of |ws (k)|2 d3 k/ (2π)3 as the number of quanta with
polarization es (k) in the reciprocal-space volume element d3 k centered on k. With this
it is natural to restrict the polarization amplitudes by the normalizability condition,
d3 k 2
|ws (k)| < ∞ , (3.187)
3
(2π) s

which guarantees that the total number of quanta is ¬nite. For normalizable wave
packets w and v the Cauchy“Schwarz inequality (A.9) guarantees the existence of the
inner product
d3 k —
(v, w) = vs (k) ws (k) ; (3.188)
3
(2π) s
therefore, the normalizable wave packets form a Hilbert space. We emphasize that
this is a Hilbert space of classical ¬elds, not a Hilbert space of quantum states. We
will therefore identify the electromagnetic phase space “em with the Hilbert space of
normalizable wave packets,

“em = {w (k) with (w, w) < ∞} . (3.189)

3.5.2 Wave packet operators
The right side of eqn (3.16) is a generalized function (see Appendix A.6.2) which means
that it is only de¬ned by its action on well behaved ordinary functions. Another way
of putting this is that ∆⊥ (r) does not have a speci¬c numerical value at the point r;
ij
instead, only averages over suitable weighting functions are well de¬ned, e.g.

d3 r ∆⊥ (r ’ r ) Yj (r ) , (3.190)
ij


where Y (r) is a smooth classical ¬eld that vanishes rapidly as |r| ’ ∞. The ap-
pearance of the generalized function ∆⊥ (r ’ r ) in the commutation relations implies
ij
Wave packet quantization— ½¼

that A(+) (r) and A(’) (r) must be operator-valued generalized functions. In other
words only suitable spatial averages of A(±) (r) are well-de¬ned operators. This con-
clusion is consistent with eqn (2.185), which demonstrates that vacuum ¬‚uctuations
in E are divergent at every point r. As far as mathematics is concerned, any su¬-
ciently well behaved averaging function will do, but on physical grounds the classical
wave packets de¬ned in Section 3.5.1 hold a privileged position. Thus the singular
(+)
object Ai (r) = ui · A(+) (r) should be replaced by the projection of A(+) on a wave
packet. This can be expressed directly in position space but it is simpler to go over to
reciprocal space and de¬ne the wave packet annihilation operators

d3 k —
a [w] = ws (k) as (k) . (3.191)
3
(2π) s

Combining the singular commutation relation (3.26) with the de¬nition (3.188) yields
the mathematically respectable relations

a [w] , a† [v] = (w, v) . (3.192)

The number operator N de¬ned by eqn (3.30) satis¬es

[N, a [w]] = ’a [w] , N, a† [w] = a† [w] , (3.193)

so the Fock space HF can be constructed as the Hilbert space spanned by all vectors
of the form
w(1) , . . . , w(n) = a† w(1) · · · a† w(n) |0 , (3.194)

where n = 0, 1, . . . and the w(j) s range over the classical phase space “em . For example,
the one-photon state |1w = a† [w] |0 is normalizable, since

d3 k 2
1w |1w = (w, w) = |ws (k)| < ∞ . (3.195)
3
(2π) s

Thus eqn (3.192) provides an interpretation of the singular commutation relations that
is both physically and mathematically acceptable (Deutsch, 1991).
Experiments in quantum optics are often described in a rather schematic way by
treating the incident and scattered ¬elds as plane waves. The physical ¬elds generated
by real sources and manipulated by optical devices are never this simple. A more
accurate, although still idealized, treatment represents the incident ¬elds as normalized
wave packets, e.g. the Gaussian pulses that will be described in Section 7.4. In a typical
experimental situation the initial state would be

|in = a† w(1) · · · a† w(n) |0 . (3.196)

This technique will work even if the di¬erent wave packets are not orthogonal. The
subsequent evolution can be calculated in the Schr¨dinger picture, by solving the
o
Schr¨dinger equation with the initial state vector |Ψ (0) = |in , or in the Heisenberg
o
½¼ Field quantization

picture, by following the evolution of the ¬eld operators. In practice an incident ¬eld
is usually described by the initial electric ¬eld E in (r, 0). According to eqn (3.185),

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