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),

d3 k ωk

(+)

E in (r, 0) ws (k) eks eik·r ,

=i (3.197)

3 20

(2π) s

so the wave packets are given by

20 —

d3 re’ik·r E in (r, 0) .

(+)

ws (k) = ’i e· (3.198)

ωk ks

Photon localizability—

3.6

3.6.1 Is there a photon position operator?

The use of the term photon to mean ˜quantum of excitation of the electromagnetic ¬eld™

is a harmless piece of jargon, but the extended sense in which photons are thought to

be localizable particles raises subtle and fundamental issues. In order to concentrate

on the essential features of this problem, we will restrict the discussion to photons

propagating in vacuum. The particle concept originated in classical mechanics, where

it is understood to mean a physical system of negligible extent that occupies a de¬nite

position in space. The complete description of the state of a classical particle is given by

its instantaneous position and momentum. In nonrelativistic quantum mechanics, the

uncertainty principle forbids the simultaneous speci¬cation of position and momentum,

so the state of a particle is instead described by a wave function ψ (r). More precisely,

ψ (r) = r |ψ is the probability amplitude that a measurement of the position operator

r will yield the value r, and leave the particle in the corresponding eigenvector |r

de¬ned by r |r = r |r . The improper eigenvector |r is discussed in Appendix C.1.1-

B. The identity

|ψ = d3 r |r r |ψ (3.199)

shows that the wave function ψ (r) is simply the projection of the state vector on the

basis vector |r . The action of the position operator r is given by r |r| ψ = r r |ψ ,

which is usually written as rψ (r) = rψ (r).

Thus the notion of a particle in nonrelativistic quantum mechanics depends on

the existence of a physically sensible position operator. Position operators exist in

nonrelativistic quantum theory for particles with any spin, and even for the relativistic

theory of massive, spin-1/2 particles described by the Dirac equation; but, there is no

position operator for the massless, spin-1 objects described by Maxwell™s equations

(Newton and Wigner, 1949).

A more general approach would be to ask if there is any operator that would serve

to describe the photon as a localizable object. In nonrelativistic quantum mechanics

the position operator r has two essential properties.

(a) The components commute with one another: [ri , rj ] = 0.

(b) The operator r transforms as a vector under rotations of the coordinate system.

Photon localizability— ½¼

Property (a) is necessary if the components of the position are to be simultaneously

measurable, and property (b) would seem to be required for the physical interpre-

tation of r as representing a location in space. Over the years many proposals for a

photon position operator have been made, with one of two outcomes: (1) when (a) is

satis¬ed, then (b) is not (Hawton and Baylis, 2001); (2) when (b) is satis¬ed, then (a)

is not (Pryce, 1948). Thus there does not appear to be a physically acceptable pho-

ton position operator; consequently, there is no position-space wave function for the

photon. This apparent di¬culty has a long history in the literature, but there are at

least two reasons for not taking it very seriously. The ¬rst is that the relevant classical

theory”Maxwell™s equations”has no particle concept. The second is that photons are

inherently relativistic, by virtue of their vanishing rest mass. Consequently, ordinary

notions connected to the Schr¨dinger equation need not apply.

o

3.6.2 Are there local number operators?

The nonexistence of a photon position operator still leaves open the possibility that

there is some other sense in which the photon may be considered as a localizable

or particle-like object. From an operational point of view, a minimum requirement

for localizability would seem to be that the number of photons in a ¬nite volume

V is an observable, represented by a local number operator N (V ). Since simultane-

ous measurements in nonoverlapping volumes of space cannot interfere, this family of

observables should satisfy

[N (V ) , N (V )] = 0 (3.200)

whenever V and V do not overlap. The standard expression (3.30) for the total number

operator as an integral over plane waves is clearly not a useful starting point for the

construction of a local number operator, so we will instead use eqns (3.49) and (3.15)

to get

20 ’1/2 (+)

d3 rE(’) (r) · ’∇2

N= E (r) . (3.201)

c

In the classical limit, the ¬eld operators are replaced by classical ¬elds, and the in

the denominator goes to zero. Thus the number operator diverges in the classical limit,

in agreement with the intuitive idea that there are e¬ectively an in¬nite number of

photons in a classical ¬eld.

The ¬rst suggestion for N (V ) is simply to restrict the integral to the volume V

(Henley and Thirring, 1964, p. 43); but this is problematical, since the integrand in eqn

(3.201) is not a positive-de¬nite operator. This poses no problem for the total number

operator, since the equivalent reciprocal-space representation (3.30) is nonnegative,

but this version of a local number operator might have negative expectation values in

1/2 1/4 1/4

some states. This objection can be met by using ’∇2 = ’∇2 ’∇2 and

the general rule (3.21) to replace the position-space integral (3.201) by the equivalent

form

N = d3 rM† (r) · M (r) , (3.202)

where

2 ’1/4

0

M (r) = ’i ’∇2 E(+) (r) . (3.203)

c

½¼ Field quantization

The integrand in eqn (3.202) is a positive-de¬nite operator, so the local number oper-

ator de¬ned by

d3 rM† (r) · M (r)

N (V ) = (3.204)

V

is guaranteed to have a nonnegative expectation value for any state. According to the

standard plane-wave representation (3.29), the operator M (r) is

d3 k

es (k) as (k) eik·r ,

M (r) = (3.205)

3

(2π) s

i.e. it is the Fourier transform of the operator M (k) introduced in eqn (3.56). The

position-space form M (r) is the detection operator introduced by Mandel in his

study of photon detection (Mandel, 1966), and N (V ) is Mandel™s local number

operator. The commutation relations (3.25) and (3.26) can be used to show that the

detection operator satis¬es

†

Mi (r) , Mj (r ) = ∆⊥ (r ’ r ) , [Mi (r) , Mj (r )] = 0 . (3.206)

ij

Now consider disjoint volumes V and V with centers separated by a distance R

which is large compared to the diameters of the volumes. Substituting eqn (3.204) into

[N (V ) , N (V )] and using eqn (3.206) yields

d3 r Sij (r, r ) ∆⊥ (r ’ r ) ,

d3 r

[N (V ) , N (V )] = (3.207)

ij

V V

where Sij (r, r ) = Mi† (r) Mj (r ) ’ Mj (r ) Mi (r). The de¬nition of the transverse

†

delta function given by eqns (2.30) and (2.28) can be combined with the general

relation (3.18) to get the equivalent expression,

1 1

∆⊥ (r ’ r ) = δij δ (r ’ r ) + ∇i ∇j . (3.208)

|r ’ r |

ij

4π

Since V and V are disjoint, the delta function term cannot contribute to eqn (3.207),

so

1 1

d3 r Sij (r, r ) ∇i ∇j