ńņš. 4 |

lin

2 2

deļ¬nes a linearly-polarized state which possesses zero helicity, i.e. Sz |Ļ 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 |Ļ lin changes into the output state

|Ļ cir = |1k,s=+ . The output state |Ļ cir has helicity Sz = + , but it still satisļ¬es

Lz |Ļ cir = 0. Since the transmitted photon carries away one unit (+ ) of angular

momentum, conservation of angular momentum requires the plate to acquire one unit

(ā’ ) 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

Ė™ Ė™

beam exerts a torque on the plate: Ļ„z = dSz /dt = N (ā’ ), where N is the rate of ļ¬‚ow

of photons through the plate. The resulting twist of the torsion ļ¬ber can be sensitively

measured by means of a small mirror attached to the ļ¬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 ļ¬‚uctuations in the

Ā¼ 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 ļ¬ne, ratchet-like Brownian motion. The experiment to see such ļ¬‚uctuationsā”which

would be very diļ¬cultā”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 eļ¬ect is observed, but switching the incident laser beam to

circular polarization causes the trapped bead to begin spinning around the axis deļ¬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 beamā”which has zero helicity

and zero orbital angular momentumā”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 Laguerreā“

Gaussian beam of light (Siegman, 1986, Sec. 16.4). The output beam possesses orbital,

but no spin, angular momentum. A simple Laguerreā“Gaussian mode is one in which

the light eļ¬ectively orbits around the axis of propagation as if in an optical vortex

with a given sense of circulation. The transverse intensity proļ¬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 momentumā”one unit (+ ) per photonā”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 Laguerreā“Gaussian modeā”with nonzero orbital angular momentumā”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

Ė™ Ė™

Ļ„z = dLz /dt = N (ā’ ), where N is the rate of photon ļ¬‚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 suļ¬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 ļ¬ne, ratchet-like Brownian motion in

the angular displacement of the bead. This would be analogous to the discontinuous

Ā½

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)ā”or the equivalent

reciprocal-space versions (3.25) and (3.26)ā”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 aĪŗ and aā ā”which do depend on idealized

Īŗ

boundary conditionsā”have discrete labels and the one-photon states |1Īŗ = aā |0 are

Īŗ

normalizable. As usual, we would prefer to have the best of both worlds; and this can

be accomplishedā”at least formallyā”by replacing the Fourier integral in (3.5) with

a Fourier series. This is done by pretending that all ļ¬elds are contained in a ļ¬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 deļ¬ned by material walls, the periodic boundary conditions

have no physical signiļ¬cance. Consequently, meaningful results are only obtained in

the limit of inļ¬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) =

ā

eks exp (ik Ā· r) / V , play the role of the cavity modes. The periodic boundary condi-

tions impose k =2Ļn/L, where n is a vector with integer components. The f ks s are

an orthonormal set of modes, i.e.

ā—

d3 r fks (r) Ā· fk s (r) = Ī“kk Ī“ss .

(fks , fk s ) = (3.63)

V

The various expressions for the commutation relations, the ļ¬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.

ā

d3 k 1

ā” and as (k) ā” V aks , (3.64)

3 V

(2Ļ) k

to the expressions obtained in Sections 3.1.1ā“3.1.3. In either way, the commutation

relations and the number operator are given by

aks , aā s aā aks .

= Ī“kk Ī“ss , [aks , ak s ] = 0 , N = (3.65)

k ks

ks

The number states are deļ¬ned just as for the physical cavity,

nks

aā

ks

ā |0 ,

|n = (3.66)

nks !

ks

Ā¾ 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 aĪŗ . This allows the construction of the Fock space to be carried out in

parallel to Chapter 2.1.2-C.

The expansions for the ļ¬eld operators are

A(+) (r) = aks eks eikĀ·r , (3.68)

2 0 Ļk V

ks

Ļk

E(+) (r) = aks eks eikĀ·r ,

i (3.69)

2 0V

ks

and

k

B(+) (r) = saks eks eikĀ·r , (3.70)

2 0 cV

ks

where the expansion for B(+) was obtained by using B = ā Ć— A and the special

property (B.52) of the circular polarization basis.

The Hamiltonian, the momentum, and the helicity are respectively given by

Ļk aā aks ,

Hem = (3.71)

ks

ks

kaā aks ,

P= (3.72)

ks

ks

and

ksaā 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 ā’ ā. 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)ā“(3.73). One

way of understanding the problem here is that the expression (3.55) for L contains

the diļ¬erential operator ā‚/ā‚k which creates diļ¬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 oļ¬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 deļ¬ne preferred directions in space, so there is no satisfactory way to deļ¬ne

the orbital angular momentum L.

Āæ

The Heisenberg picture

3.2 The Heisenberg picture

The quantization rules in Chapter 2 and Section 3.1.1 are both expressed in the

SchrĀØdinger picture: observables are represented by time-independent hermitian oper-

o

ators X (S) , and the state of the radiation ļ¬eld is described by a ket vector ĪØ(S) (t) ,

obeying the SchrĀØdinger equation

o

ā‚

ĪØ(S) (t) = H (S) ĪØ(S) (t) ,

i (3.74)

ā‚t

or by a density operator Ļ(S) (t), obeying the quantum Liouville equation (2.119)

ā‚ (S)

Ļ (t) = H (S) , Ļ(S) (t) .

i (3.75)

ā‚t

The superscript (S) has been added in order to distinguish the SchrĀØdinger picture

o

from two other descriptions that are frequently used. Note that the density operator is

an exception to the rule that SchrĀØdinger-picture observables are independent of time.

o

There is an alternative description of quantum mechanics which actually preceded

the familiar SchrĀØdinger picture. In Heisenbergā™s original formulationā”which appeared

o

one year before SchrĀØdingerā™sā”there is no mention of a wave function or a wave equa-

o

tion; instead, the observables are represented by inļ¬nite matrices that evolve in time

according to a quantum version of Hamiltonā™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 SchrĀØdinger. The Heisenberg picture

o

is particularly useful in quantum optics, especially for the calculation of correlations

between measurements at diļ¬erent times. A third representationā”called the interac-

tion pictureā”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 SchrĀØdinger and Heisen-

o

berg pictures and the relations between them. In order to distinguish between the same

quantities viewed in diļ¬erent pictures, the states and operators will be decorated with

superscripts (S) or (H) for the SchrĀØdinger or Heisenberg pictures respectively. In

o

applications of these ideas the superscripts are usually dropped, and the distinctions

areā”one hopesā”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,

SchrĀØdinger-picture state vector ĪØ(S) (t) and the corresponding time-independent,

o

Heisenberg-picture state ĪØ(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 ā’ t0 )

such that

ĪØ(S) (t) = U (t ā’ t0 ) ĪØ(H) . (3.76)

The identity of the pictures at t = t0 , ĪØ(H) = ĪØ(S) (t0 ) , is enforced by the initial

condition U (0) = 1. Substituting eqn (3.76) into the SchrĀØdinger equation (3.74) yields

o

the diļ¬erential equation

Field quantization

ā‚

U (t ā’ t0 ) = H (S) U (t ā’ t0 ) , U (0) = 1

i (3.77)

ā‚t

for the operator U (t ā’ t0 ). This has the solution (Bransden and Joachain, 1989, Sec.

5.7)

i

U (t ā’ t0 ) = exp ā’ (t ā’ t0 ) H (S) , (3.78)

where the evolution operator on the right side is deļ¬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 ā’ t0 ) is unitary, i.e.

U (t ā’ t0 ) U ā (t ā’ t0 ) = U ā (t ā’ t0 ) U (t ā’ 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 ā’ ā’ā. The evolution operator satisļ¬es the group property,

U (t1 ā’ t2 ) U (t2 ā’ t3 ) = U (t1 ā’ 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 simpliļ¬es

to U (t1 ) U (t2 ) = U (t1 + t2 ). The deļ¬nition (3.78) also shows that U (ā’t) = U ā (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

SchrĀØdinger-picture operator X (S) and the corresponding Heisenberg-picture operator

o

(H)

X (t) have the same expectation values in corresponding states:

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

for all vectors ĪØ(S) (t) and observables X (S) . Using eqn (3.76) allows this relation to

be written as

ĪØ(H) X (H) (t) ĪØ(H) = ĪØ(H) U ā (t) X (S) U (t) ĪØ(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 ā (t) X (S) U (t) . (3.83)

Note that the Heisenberg-picture operators agree with the (time-independent)

SchrĀØdinger-picture operators at t = 0. This deļ¬nition, together with the group prop-

o

erty U (t1 ) U (t2 ) = U (t1 + t2 ), provides a useful relation between the Heisenberg

operators at diļ¬erent times:

X (H) (t + Ļ„ ) = U ā (t + Ļ„ ) X (S) U (t + Ļ„ )

= U ā (Ļ„ ) U ā (t) X (S) U (t) U (Ļ„ )

= U ā (Ļ„ ) X (H) (t) U (Ļ„ ) . (3.84)

Also note that H (S) commutes with exp Ā±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

ļ¬xed. The density operator is again an exception. Applying the transformation (3.83)

to the deļ¬nition of the SchrĀØdinger-picture density operator,

o

Pu Ī˜(S) (t)

Ļ(S) (t) = Ī˜(S) (t) , (3.85)

u u

u

yields the time-independent operator

Pu Ī˜(H)

Ļ(H) = Ī˜(H) = Ļ(S) (0) , (3.86)

u u

u

which is the initial value for the quantum Liouville equation (3.75).

A diļ¬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

ā‚X (H) (t) i

= U ā (t) H, X (S) U (t)

ā‚t

i

H, X (H) (t) ,

= (3.87)

where the last line follows from the identity

U ā (t) X (S) Y (S) U (t) = U ā (t) X (S) U (t) U ā (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) :

ā‚X (H) (t)

= X (H) (t) , H .

i (3.89)

ā‚t

The deļ¬nition (3.83) provides a solution for this equation. The name ā˜constant of the

motionā™ for operators X (S) that commute with the Hamiltonian is justiļ¬ed by the

observation that the Heisenberg equation for X (H) (t) is (ā‚/ā‚t) X (H) (t) = 0.

In most applications we will suppress the identifying superscripts (H) and (S). The

distinctions between the Heisenberg and SchrĀØdinger 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 Xā”with no time argumentā”signiļ¬es the SchrĀØdinger-picture form. The

o

only real danger of this convention is that density operators behave in the opposite way;

Ļ (t) denotes a SchrĀØdinger-picture operator, while Ļ 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 SchrĀØdinger-picture operators X and Y is said to be canonically conjugate

o

if [X, Y ] = Ī², where Ī² 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 ā (t) XU (t) , U ā (t) Y U (t)

= U ā (t) [X, Y ] U (t)

= Ī². (3.90)

Thus the equal-time commutator of the Heisenberg-picture operators is identical to the

commutator of the SchrĀØdinger-picture operators. Applying this to the position-space

o

commutation relation (3.3) and to the canonical commutator (3.65) yields

i

āā„ (r ā’ r )

[Ai (r, t) , ā’Ej (r , t)] = (3.91)

ij

0

and

aks (t) , aā s (t) = Ī“ss Ī“kk , (3.92)

k

respectively.

3.2.2 Heisenberg equations for the free ļ¬eld

The preceding arguments are valid for any form of the Hamiltonian, but the results are

particularly useful for free ļ¬elds. The Heisenberg-picture form of the box-quantized

Hamiltonian is

Ļk aā (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

ā’ Ļk aks (t) = 0 .

i (3.94)

dt

The solution is

aks (t) = aks eā’iĻk t = eiHem t/ aks eā’iHem t/ , (3.95)

where we have used the identiļ¬cation of aks (0) with the SchrĀØdinger-picture operator

o

aks . Combining this solution with the expansion (3.68) gives

A(+) (r, t) = aks eks ei(kĀ·rā’Ļk t) . (3.96)

2 0 Ļk 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-

isļ¬es F ā— (Ļ) = F (ā’Ļ). Thus F (Ļ) for negative frequencies is completely determined

by F (Ļ) for positive frequencies. Let us use this fact to rewrite the inverse transform

as ā

dĻ

F (Ļ) eā’iĻt = F (+) (t) + F (ā’) (t) ,

F (t) = (3.97)

ā’ā 2Ļ

where the positive-frequency part,

ā

dĻ

F (Ļ) eā’iĻt ,

(+)

F (t) = (3.98)

2Ļ

0

and the negative-frequency part,

0

dĻ

F (Ļ) eā’iĻt ,

(ā’)

F (t) = (3.99)

2Ļ

ā’ā

are related by

F (ā’) (t) = F (+)ā— (t) . (3.100)

The deļ¬nitions of F (Ā±) (t) guarantee that F (+) (Ļ) vanishes for Ļ < 0 and F (ā’) (Ļ)

vanishes for Ļ > 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 (ā’) (t) = X (+)ā (t). In particular,

the temporal Fourier transform of the operator A(+) (r, t), deļ¬ned by eqn (3.96), is

aks eks eikĀ·r 2ĻĪ“ (Ļ ā’ Ļk ) . (3.101)

A(+) (r, Ļ) = dt eiĻt A(+) (r, t) =

2 0 Ļk V

ks

Since Ļk = c |k| > 0, A(+) (r, Ļ) vanishes for Ļ < 0, and A(ā’) (r, Ļ) = A(+)ā (r, ā’Ļ)

vanishes for Ļ > 0. Thus the SchrĀØdinger-picture deļ¬nition (3.68) of the positive-

o

frequency part agrees with the Heisenberg-picture deļ¬nition at t = 0.

The commutation rules derived in Section 3.1.2 are valid here for equal-time com-

mutators, but for free ļ¬elds we also have the unequal-times commutators:

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

provided only that F (Ā±) (r, 0) and G(Ā±) (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 ļ¬elds in

passive, linear dielectricsā”which we will always assume are nonmagneticā”and then

present a phenomenological model for quantization.

3.3.1 Classical ļ¬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) /Āµ0 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 ļ¬eld by

dt Ļ(1) (t ā’ t ) E (r, t ) ,

P (r, t) = (3.104)

0

where the linear susceptibility Ļ(1) (t ā’ t ) describes the delayed response of the

medium to an applied electric ļ¬eld. Fourier transforming eqn (3.104) with respect to

time produces the equivalent frequency-domain relation

P (r, Ļ) = (Ļ) E (r, Ļ) .

(1)

0Ļ (3.105)

Applying the deļ¬nition of positive- and negative-frequency parts, given by eqns

(3.97)ā“(3.99), to the real classical ļ¬eld E (r, t) leads to

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

In position space, the strength of the electric ļ¬eld at frequency Ļ is represented by

2

the power spectrum E (+) (r, Ļ) (see Appendix B.2). In reciprocal space, the power

2

spectrum is E (+) (k, Ļ) . We will often be concerned with ļ¬elds for which the power

spectrum has a single well-deļ¬ned peak at a carrier frequency Ļ = Ļ0 . The value

of Ļ0 is set by the experimental situation, e.g. Ļ0 is often the frequency of an injected

2

signal. The reality condition (3.100) for E (Ā±) (r, Ļ) tells us that E (ā’) (r, Ļ) has a

peak at Ļ = ā’Ļ0 ; consequently, the complete transform E (r, Ļ) has two peaks: one

at Ļ = Ļ0 and the other at Ļ = ā’Ļ0 .

We will say that the ļ¬eld is monochromatic if the spectral width, āĻ0 , of the

peak at Ļ = Ā±Ļ0 satisļ¬es

āĻ0 Ļ0 . (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: āĻ0 = 0. Since the ļ¬elds generated in real

experiments are always described by wave packets with nonzero spectral widths, we

prefer the deļ¬nition associated with eqn (3.107). The ideal ļ¬elds with āĻ0 = 0 will

be called strictly monochromatic.

Field quantization in passive linear media

The concentration of the Fourier transform in the vicinity of Ļ = Ā±Ļ0 allows us to

(Ā±)

deļ¬ne the slowly-varying envelope ļ¬elds E (r, t) by setting

(Ā±)

(r, t) = E (Ā±) (r, t) eĀ±iĻ0 t ,

E (3.108)

so that

(+) (ā’)

(r, t) eā’iĻ0 t + E

E (r, t) = E (r, t) eiĻ0 t . (3.109)

(ā’) (+)ā—

The slowly-varying envelopes satisfy E (r, t) = E (r, t), and the time-domain

version of eqn (3.107) is

(Ā±) (Ā±)

ā‚2E ā‚E

(r, t) (r, t) (Ā±)

Ļ0 E

2

Ļ0 (r, t) . (3.110)

ā‚t2 ā‚t

The frequency-domain versions of eqns (3.108) and (3.109) are

(Ā±)

E (r, Ļ) = E (Ā±) (r, Ļ Ā± Ļ0 ) (3.111)

and

(+) (ā’)

E (r, Ļ) = E (r, Ļ ā’ Ļ0 ) + E (r, Ļ + Ļ0 ) , (3.112)

(Ā±)

respectively. The condition (3.107) implies that E (r, Ļ) is sharply peaked at Ļ = 0.

The Fourier transform of the vector potential is also concentrated in the vicinity

of Ļ = Ā±Ļ0 , so the slowly-varying envelope,

(+)

A (r, t) = A(+) (r, t) eiĻ0 t , (3.113)

satisļ¬es the same conditions. Since E (r, t) = ā’ā‚A (r, t) /ā‚t, the two envelope functions

are related by

ā‚ (+)

(+) (+)

E (r, t) = iĻ0 A ā’A. (3.114)

ā‚t

Applying eqn (3.110) to the vector potential shows that the second term on the right

side is small compared to the ļ¬rst, so that

(+) (+)

E (r, t) ā iĻ0 A . (3.115)

This is an example of the slowly-varying envelope approximation.

More generally, it is necessary to consider polychromatic ļ¬elds, i.e. superposi-

tions of monochromatic ļ¬elds with carrier frequencies ĻĪ² (Ī² = 0, 1, 2, . . .). The car-

rier frequencies are required to be distinct; that is, the power spectrum for a poly-

chromatic ļ¬eld exhibits a set of clearly resolved peaks at the carrier frequencies

ĻĪ² . The explicit condition is that the minimum spacing between peaks, Ī“Ļmin =

min [|ĻĪ± ā’ ĻĪ² | , Ī± = Ī²] , is large compared to the maximum spectral width, āĻmax =

max [āĻĪ² ]. The values of the carrier frequencies are set by the experimental situation

under study. The collection {ĻĪ² } will generally contain the frequencies of any injected

ļ¬elds together with the frequencies of radiation emitted by the medium in response to

Ā¼ Field quantization

the injected signals. For a polychromatic ļ¬eld, eqns (3.108), (3.113), and (3.115) are

replaced by

(+)

(r, t) eā’iĻĪ² t ,

E (+) (r, t) = EĪ² (3.116)

Ī²

(+)

AĪ² (r, t) eā’iĻĪ² t ,

A(+) (r, t) = (3.117)

Ī²

and

(+) (+)

EĪ² (r, t) = iĻĪ² AĪ² (r, t) . (3.118)

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

(Ļ)

E (r, Ļ) = E Ī² (r, Ļ ā’ ĻĻĪ² ) , (3.119)

Ī² Ļ=Ā±

(Ā±)

where each of the functions E Ī² (r, Ļ) is sharply peaked at Ļ = 0.

A Passive, linear dielectric

An optical medium is said to be passive and linear if the following conditions are

satisļ¬ed.

(a) Oļ¬ resonance. The classical power spectrum is negligible at frequencies that are

resonant with any transition of the constituent atoms. This justiļ¬es the assump-

tion that there is no absorption.

(b) Coarse graining. There are many atoms in the volume Ī»3 , where Ī»0 is the mean

0

wavelength for the incident ļ¬eld.

(c) Weak ļ¬eld. The ļ¬eld is not strong enough to induce signiļ¬cant changes in the

material medium.

(d) Weak dispersion. The frequency-dependent susceptibility Ļ(1) (Ļ) is essentially

constant across any frequency interval āĻ Ļ.

(e) Stationary medium. The medium is stationary, i.e. the optical properties do

not change in time.

The passive property is incorporated in the oļ¬-resonance assumption (a) which

allows us to neglect absorption, stimulated emission, and spontaneous emission. The

description of the medium by the usual macroscopic coeļ¬cients such as the suscep-

tibility, the refractive index, and the conductivity is justiļ¬ed by the coarse-graining

assumption (b). The weak-ļ¬eld assumption (c) guarantees that the macroscopic ver-

sion of Maxwellā™s equations is linear in the ļ¬elds. The weak dispersion condition (d)

assures us that an input wave packet with a sharply deļ¬ned carrier frequency will

retain the same frequency after propagation through the medium. The assumption (e)

implies that the susceptibility Ļ(1) (t ā’ t ) only depends on the time diļ¬erence t ā’ t .

Ā½

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 Ļ = Ļ0 ) if

ā‚Ļ(1) (Ļ)

Ļ(1) (Ļ0 )

āĻ0 (3.120)

ā‚Ļ Ļ=Ļ0

for any frequency interval āĻ0 Ļ0 . We next recall that in a linear, isotropic dielectric

the vacuum dispersion relation Ļ = ck is replaced by

Ļn (Ļ) = ck , (3.121)

where the index of refraction is related to the dielectric permittivity, (Ļ), by n2 (Ļ) =

(Ļ). Since (Ļ) can be complexā”the imaginary part describes absorption or gain

(Jackson, 1999, Chap. 7)ā”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 Ļk be the mode frequency obtained by solving

the nonlinear dispersion relation (3.121), then the medium is transparent at Ļk if

nk = n (Ļk ) is real. In the frequencyā“wavenumber domain the electric ļ¬eld satisļ¬es

Ļ2 2

n (Ļ) ā’ k 2 E k (Ļ) = 0 (3.122)

2

c

(see Appendix B.5.2, eqn (B.123)), so one ļ¬nds the general spaceā“time solution

E (r, t) = E (+) (r, t) + E (ā’) (r, t), with

1

E (+) (r, t) = ā Eks eks ei(kĀ·rā’Ļk t) . (3.123)

V ks

For a monochromatic ļ¬eld, the slowly-varying envelope is

1

(+)

(r, t) = ā

E Eks eks ei(kĀ·rā’āk t) , (3.124)

V ks

where the prime on the k-sum indicates that it is restricted to k-values such that

the detuning, āk = Ļk ā’ Ļ0 , satisļ¬es |āk | Ļ0 . The wavelength mentioned in the

coarse-graining assumption (b) is then Ī»0 = 2Ļc/ (n (Ļ0 ) Ļ0 ).

For a polychromatic ļ¬eld, eqn (3.108) is replaced by

(+)

(r, t) eā’iĻĪ² t ,

E (+) (r, t) = EĪ² (3.125)

Ī²

where

1

(+)

E Ī² (r, t) = ā EĪ²ks eks ei(kĀ·rā’āĪ²k t) , āĪ²k = Ļk ā’ ĻĪ² (3.126)

V ks

is a slowly-varying envelope ļ¬eld. The spectral width of the Ī²th monochromatic ļ¬eld

(+) (+)

2 2

is deļ¬ned by the power spectrum E Ī² (r, Ļ) or E Ī² (k, Ļ) . The weak disper-

sion condition (d) is extended to this case by imposing eqn (3.120) on each of the

monochromatic ļ¬elds.

Ā¾ Field quantization

The condition (3.107) for a monochromatic ļ¬eld guarantees the existence of an

intermediate time scale T satisfying

1 1

T , (3.127)

Ļ0 āĻ0

i.e. T is long compared to the carrier period but short compared to the characteristic

time scale on which the envelope ļ¬eld changes. Averaging over the interval T will

wash out all the fast variationsā”on the optical frequency scaleā”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/Ļmin T

1/āĻmax , where Ļmin = min (ĻĪ² ).

B Electromagnetic energy in a dispersive dielectric

For an isotropic, nondispersive dielectricā”e.g. the vacuumā”Poyntingā™s theorem (see

Appendix B.5) takes the form

ā‚uem (r, t)

+ ā Ā· S (r, t) = 0 , (3.128)

ā‚t

where

1 12

E 2 (r, t) + B (r, t)

uem (r, t) = (3.129)

2 Āµ0

is the electromagnetic energy density and

1

S (r, t) = E (r, t) Ć— H (r, t) = E (r, t) Ć— B (r, t) (3.130)

Āµ0

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

ā‚umag

pel (r, t) + +āĀ· S = 0, (3.131)

ā‚t

where the electric power density,

ā‚D (r, t)

pel (r, t) = E (r, t) Ā· , (3.132)

ā‚t

is the power per unit volume ļ¬‚owing into the dielectric medium due to the action of

the slowly-varying electric ļ¬eld E, and

12

B (r, t)

umag (r, t) = (3.133)

2Āµ0

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

Āæ

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 ļ¬eld at times t = t. The

principle of causality restricts this dependence to earlier times, t < t, so that

Ļ(1) (t ā’ t ) = 0 for t > t . (3.134)

(1)

For a nondispersive medium Ļ(1) (Ļ) has the constant value Ļ0 , so in this approxi-

mation one ļ¬nds that

(1)

Ļ(1) (t ā’ t ) = Ļ0 Ī“ (t ā’ t ) . (3.135)

In this case, the polarization at a given time only depends on the ļ¬eld at the same

time. In the dispersive case, Ļ(1) (t ā’ t ) decays to zero over a nonzero interval, 0 <

tā’t < Tmem ; in other words, the polarization at t depends on the history of the electric

ļ¬eld up to time t. Consequently, the power density pel (r, t) cannot be expressed as

pel (r, t) = ā‚uel (r, t) /ā‚t, where uel (r, t) is an instantaneous energy density.

In the general case this obstacle is insurmountable, but for a monochromatic (or

polychromatic) ļ¬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

deļ¬ne the (running) time-average

T /2

1

pel pel (r, t + t ) dt .

(r, t) = (3.136)

T ā’T /2

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

ā‚uel (r, t)

pel (r, t) = , (3.137)

ā‚t

where the eļ¬ective electric energy density is

d [Ļ0 (Ļ0 )] 1

E (r, t) Ā· E (r, t)

uel (r, t) =

dĻ0 2

d [Ļ0 (Ļ0 )] (ā’) (+)

E (r, t) Ā· E

= (r, t) , (3.138)

dĻ0

(+)

and E (r, t) is the slowly-varying envelope for the electric ļ¬eld. The eļ¬ective electric

energy density for a polychromatic ļ¬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 ļ¬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 ļ¬eld interacting with

Field quantization

the point charges in the atoms constituting the medium. The same could be said for

the classical theory of ļ¬elds in a dielectric, but it is traditionalā”and a great deal

easierā”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 ļ¬eld in a

dielectric medium.1 This has proven to be a diļ¬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, butā”as we have seen in the previous sectionā”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 ļ¬eld propagating in a weakly dispersive dielectric. For these

experimentally signiļ¬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 modiļ¬ed version of the vacuum ļ¬eld expansion (3.69)

Ek aks eks eikĀ·r ,

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

ks

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

ks

coeļ¬cient Ek is a characteristic ļ¬eld strength which will be chosen to ļ¬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 inļ¬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 ļ¬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 ļ¬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 inļ¬nitely

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

Let L be the circumference and Ļ 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 ļ¬lled with a

= >

weakly dispersive dielectric. A segment has

been removed to show the central axis. The

ļ¬eld satisļ¬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 ļ¬eld propagating in the z-direction can be approximated by a

monochromatic planar wave packet,

E (z, t) = E k (z, t) ei(kzā’Ļk t) + CC , (3.140)

where Ļk 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

Faradayā™s law (eqn (B.94)) yields

1

B (z, t) = k Ć— E k (z, t) ei(kzā’Ļk t) + CC . (3.141)

Ļk

As we have seen in Section 3.3.1, the ļ¬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 (Ļk ) = c/nk , but rather at the

group velocity

dĻ c

vg (Ļk ) = = . (3.142)

dk nk + Ļk (dn/dĻ)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 ļ¬eld Ek by equating the energy in the wave

packet to Ļk . 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 ļ¬‚ux, S = c2 0 E Ć— 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 Ć— Bā— }. 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)

Ļk Āµ0

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

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

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

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

Field quantization

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

photon:

2

2c 0 nk |Ek | ĻL

= Ļk . (3.144)

vg (Ļk ) V

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

Ļk vg (Ļk )

|Ek | = , (3.145)

2 0 cnk V

which gives the box-quantized expansions

vg (Ļk )

A(+) (r) = aks eks eikĀ·r (3.146)

2 0 nk Ļk cV

ks

and

Ļk vg (Ļk )

E(+) (r) = i aks eks eikĀ·r (3.147)

2 0 nk cV

ks

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

d3 k Ļk vg (Ļk )

(+)

as (k) eks eikĀ·r

E (r) = i (3.148)

3 2 0 nk c

(2Ļ) s

and

d3 k vg (Ļk )

(+)

as (k) eks eikĀ·r .

A (r) = (3.149)

(2Ļ)3 2 0 nk Ļ k c

s

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

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

ks s

A Energy and momentum

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

expressed in the box-normalized form

Ļk aā aks ,

Hem = (3.150)

ks

ks

or in the equivalent continuum form

d3 k

Ļk aā (k) as (k) .

Hem = (3.151)

s

3

(2Ļ) 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 Ļk . This suggests identifying the momentum of the photon with

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

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

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

p= k (3.152)

to a dressed photon with energy Ļk . The operator

kaā aks

Pem = (3.153)

ks

ks

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

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

electromagnetic ļ¬eld.

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

the canonical momentum for photons. The ļ¬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 diļ¬culty is that there is no

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

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

electrodynamics regarding the correct deļ¬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 rulesā”e.g. eqns (3.25) and (3.26)ā”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 ļ¬elds. Since operators do not have

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

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

and (b) the classical power spectrum is replaced by

pk = aā aks = Tr Ļin aā aks , (3.154)

ks ks

s

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

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

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

For condition (d) the spectral width āĻ0 is given by

Field quantization

2

(Ļk ā’ Ļ0 ) pk .

2

āĻ0 = (3.155)

k

For an experimental situation corresponding to a monochromatic classical ļ¬eld

with carrier frequency Ļ0 , the appropriate Hilbert space of states consists of the state

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

expressed as superpositions of the special number states

mks

aā

ks

ā

|m = |0 , (3.156)

mks !

ks

with occupation numbers mks restricted by

mks = 0 , unless |Ļk ā’ Ļ0 | < āĻ0 . (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 (Ļ0 ). For a polychro-

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

mks = 0 , unless |Ļk ā’ ĻĪ² | < āĻĪ² , Ī² = 0, 1, 2, . . . . (3.158)

The space H ({ĻĪ² }) spanned by the number states satisfying these conditions is called

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

applied to vectors in H ({ĻĪ² }). The initial ļ¬eld state Ļin must therefore be deļ¬ned by

an ensemble of pure states chosen from H ({ĻĪ² }).

Milonniā™s quantization methodā—

3.3.5

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

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

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

d [Ļ0 (Ļ0 )] 1 1

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

dĻ0 2 2Āµ0

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

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

d [Ļ0 (Ļ0 )] 1

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

dĻ0 Āµ0

(3.160)

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

ļ¬nd

k2

2 d [Ļ0 (Ļ0 )] 2

Uem = |Aks | ,

Ļk + (3.161)

dĻ0 Āµ0

ks

where Aks = Eks /iĻ0 is the expansion amplitude for the vector potential. Since the

2

power spectrum |Aks | is strongly peaked at Ļk = Ļ0 , it is equally accurate to write

this result in the more suggestive form

Field quantization in passive linear media

(Ļk )] k 2

2 d [Ļk 2

Uem = |Aks | .

Ļk + (3.162)

dĻk Āµ0

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

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

use the dispersion relation (3.121) and the deļ¬nition (Ļ) = 0 n2 (Ļ) of the index of

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

d [Ļk (Ļk )] k 2 d [Ļk nk ]

2 2

Ļk + = 2 0 Ļ k nk

dĻk Āµ0 dĻk

c

2

= 2 0 Ļ k nk , (3.163)

vg (Ļk )

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

energy is then

c 2

Uem = |Aks | .

2

2 0 Ļ k nk (3.164)

vg (Ļk )

ks

Setting

vg (Ļk )

Aks = wks , (3.165)

2 0 nk Ļ k c

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

2

Uem = Ļk |ws (k)| (3.166)

ks

and

vg (Ļk )

A(+) (r, t) = wks eks ei(kĀ·rā’Ļk t) , (3.167)

2 0 nk Ļk cV

ks

respectively.

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

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

ā—

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 aā that satisfy the canonical

ks

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

vg (Ļk )

Aks ā’ aks . (3.168)

2 0 nk Ļ k c

In the SchrĀØdinger picture this leads to

o

vg (Ļk )

A(+) (r) = aks eks eikĀ·r , (3.169)

2 0 nk Ļk cV

ks

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

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

Ā½Ā¼Ā¼ 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

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

applied to states in the space H ({ĻĪ² }) appropriate to the experimental situation

under study.

A Electromagnetic momentum in a dielectricā—

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

symmetry principle of translation invariance. The deļ¬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 deļ¬ned by its action on the ļ¬eld

operators, e.g.

(+) (+)

āAj

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

i

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

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

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

G ā’ 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 speciļ¬ed by the rules of quantum ļ¬eld

theory.

Electromagnetic angular momentumā—

3.4

The properties and physical signiļ¬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)ā“(3.59), our ļ¬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 satisļ¬es the angular

momentum commutation relations; but it is more instructiveā”and in fact simplerā”

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 ļ¬elds we should ļ¬nd

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

Since all such operators can be built up from A(+) (r), it is suļ¬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 āā„ (r ā’ r ) (r Ć— ā )i Ak

(+) (+)

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

kj

and

d3 r āā„ (r ā’ r ) Al

(+) (+)

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

ikl kj

Electromagnetic angular momentumā— Ā½Ā¼Ā½

so that

d3 r āā„ (r ā’ r ) (r Ć— ā )i Ak

(+) (+) (+)

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

kj

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

d3 k kl kj ikĀ·(rā’r )

āā„ (r ā’ r ) = Ī“lj Ī“ (r ā’ r ) ā’ e , (3.175)

lj

(2Ļ)3 k 2

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

calculation using the identity

kl kj eikĀ·(rā’r ) = ā’āl āj eikĀ·(rā’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, deļ¬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 satisļ¬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 ļ¬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 diļ¬culties in any attempt to ļ¬nd a similar decomposition, J = L + S,

for the electromagnetic ļ¬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 ļ¬eld, this would be the case if there were three independent

components of the ļ¬eld deļ¬ned at each point in space. In the theory of the radiation

ļ¬eld, however, the vectors ļ¬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 ļ¬elds is purely kinematical, i.e. it holds for both free and interacting ļ¬elds, so the

spin and spatial degrees of freedom are not independent. The restriction to transverse

Ā½Ā¼Ā¾ Field quantization

ļ¬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 ā’ir Ć— ā, 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 ļ¬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 ā˜spinā™

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 ā’ Sj ) .

[Lk , Li ] = i (3.182)

kij

Thus the sum J = L + S is a genuine angular momentum operator, but the sepa-

rate ā˜orbitalā™ and ā˜spinā™ 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 signiļ¬cance of the helicity operator S is reasonably clear from k Ā· S |1ks =

s |1ks , but the meaning of the orbital angular momentum L is not so obvious. In

common with true angular momenta, the diļ¬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 ļ¬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 satisļ¬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 quantizationā— Ā½Ā¼Āæ

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 ļ¬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 quantizationā—

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 ļ¬elds that are not simple plane waves. In real experiments the incident ļ¬elds

are more accurately described by Gaussian beams (Yariv, 1989, Sec. 6.6); consequently,

it would be better to have a more ļ¬‚exible method that can accommodate incident ļ¬elds

of various types.

In this section we will develop a representation of the ļ¬eld operators that deals

directly with the singular commutation relations in a mathematically and physically

sensible way. This new representation depends on the deļ¬nition of the electromagnetic

phase space in terms of normalizable classical wave packets. Creation and annihila-

tion operators deļ¬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 Ī“mech, and a unique trajectory (q(t), p(t)) is deļ¬ned by the initial conditions

(q(0), p(0)) = (q0 , p0 ). A unique solution of Maxwellā™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 Ī“em are labeled by pairs of real transverse

vector ļ¬elds, (A (r) , ā’E (r)). The use of ā’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).

Ā½Ā¼ Field quantization

A more useful representation of Ī“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(kĀ·rā’Ļk t) , (3.185)

3 2 0 Ļk

(2Ļ) 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) , wā’ (k)) are respec-

tively called polarization amplitudes and wave packets. The classical energy for

this solution is

d3 k 2

U= |ws (k)| .

3 Ļk (3.186)

(2Ļ) s

Physically realizable classical ļ¬elds must have ļ¬nite total energy, i.e. U < ā, but

Einsteinā™s quantum model suggests an additional and independent condition. This

comes from the interpretation of |ws (k)|2 d3 k/ (2Ļ)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)| < ā , (3.187)

3

(2Ļ) s

which guarantees that the total number of quanta is ļ¬nite. For normalizable wave

packets w and v the Cauchyā“Schwarz inequality (A.9) guarantees the existence of the

inner product

d3 k ā—

(v, w) = vs (k) ws (k) ; (3.188)

3

(2Ļ) s

therefore, the normalizable wave packets form a Hilbert space. We emphasize that

this is a Hilbert space of classical ļ¬elds, not a Hilbert space of quantum states. We

will therefore identify the electromagnetic phase space Ī“em with the Hilbert space of

normalizable wave packets,

Ī“em = {w (k) with (w, w) < ā} . (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 deļ¬ned by its action on well behaved ordinary functions. Another way

of putting this is that āā„ (r) does not have a speciļ¬c numerical value at the point r;

ij

instead, only averages over suitable weighting functions are well deļ¬ned, e.g.

d3 r āā„ (r ā’ r ) Yj (r ) , (3.190)

ij

where Y (r) is a smooth classical ļ¬eld that vanishes rapidly as |r| ā’ ā. The ap-

pearance of the generalized function āā„ (r ā’ r ) in the commutation relations implies

ij

Wave packet quantizationā— Ā½Ā¼

that A(+) (r) and A(ā’) (r) must be operator-valued generalized functions. In other

words only suitable spatial averages of A(Ā±) (r) are well-deļ¬ned operators. This con-

clusion is consistent with eqn (2.185), which demonstrates that vacuum ļ¬‚uctuations

in E are divergent at every point r. As far as mathematics is concerned, any suļ¬-

ciently well behaved averaging function will do, but on physical grounds the classical

wave packets deļ¬ned in Section 3.5.1 hold a privileged position. Thus the singular

(+)

object Ai (r) = ui Ā· 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 deļ¬ne the wave packet annihilation operators

d3 k ā—

a [w] = ws (k) as (k) . (3.191)

3

(2Ļ) s

Combining the singular commutation relation (3.26) with the deļ¬nition (3.188) yields

the mathematically respectable relations

a [w] , aā [v] = (w, v) . (3.192)

The number operator N deļ¬ned by eqn (3.30) satisļ¬es

[N, a [w]] = ā’a [w] , N, aā [w] = aā [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) = aā w(1) Ā· Ā· Ā· aā w(n) |0 , (3.194)

where n = 0, 1, . . . and the w(j) s range over the classical phase space Ī“em . For example,

the one-photon state |1w = aā [w] |0 is normalizable, since

d3 k 2

1w |1w = (w, w) = |ws (k)| < ā . (3.195)

3

(2Ļ) 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 ļ¬elds as plane waves. The physical ļ¬elds generated

by real sources and manipulated by optical devices are never this simple. A more

accurate, although still idealized, treatment represents the incident ļ¬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 = aā w(1) Ā· Ā· Ā· aā w(n) |0 . (3.196)

This technique will work even if the diļ¬erent wave packets are not orthogonal. The

subsequent evolution can be calculated in the SchrĀØdinger picture, by solving the

o

SchrĀØdinger equation with the initial state vector |ĪØ (0) = |in , or in the Heisenberg

o

Ā½Ā¼ Field quantization

picture, by following the evolution of the ļ¬eld operators. In practice an incident ļ¬eld

is usually described by the initial electric ļ¬eld E in (r, 0). According to eqn (3.185),

ńņš. 4 |