order correction to the energy, but in second order the dipole“dipole coupling produces

the van der Waals potential VW (r) with its characteristic 1/r7 dependence. The 1/r7

dependence is valid for r »at , where »at is a characteristic wavelength of an atomic

»at the potential varies as 1/r6 .

transition. For r

For many atoms, the simplest assumption is that these potentials are pair-wise

additive, i.e. the total potential energy is

VW (|rn ’ rm |) ,

Vtot = (2.203)

m=n

where the sum runs over all pairs with one atom in each wall. With this approximation,

it is possible to explain about 80% of the Casimir force in eqn (2.200). In fact the

assumption of pair-wise additivity is not justi¬ed, since the presence of a third atom

changes the interaction between the ¬rst two. When this is properly taken into account,

the entire Casimir force is obtained.

Thus there are two di¬erent explanations for the Casimir force, corresponding to

the two choices a† a or a† a + aa† /2 made in de¬ning the electromagnetic Hamil-

tonian. The important point to keep in mind is that the relevant physical prediction”

the Casimir force between the plates”is the same for both explanations. The di¬erence

between the two lies entirely in the language used to describe the situation. This kind

of ambiguity in description is often found in quantum physics. Another example is the

van der Waals potential itself. The explanation given above corresponds to the normal

ordering of the electromagnetic Hamiltonian. If the symmetric ordering is used instead,

the presence of the two atoms induces a change in the zero-point energy of the ¬eld

which becomes increasingly negative as the distance between the atoms decreases. The

result is the same attractive potential between the atoms (Milonni and Shih, 1992).

2.7 Exercises

2.1 Cavity equations

(1) Give the separation of variables argument leading to eqn (2.7).

(2) Derive the equations satis¬ed by E (r) and B (r) and verify eqns (2.9) and (2.10).

Quantization of cavity modes

2.2 Rectangular cavity modes

(1) Use the method of separation of variables to solve eqns (2.11) and (2.1) for a

rectangular cavity, subject to the boundary condition (2.13), and thus verify eqns

(2.14)“(2.17).

(2) Show explicitly that the modes satisfy the orthogonality conditions

d3 rE ks (r) · E k s (r) = 0 for (k, s) = (k , s ) .

(3) Use the normalization condition

d3 rE ks (r) · E ks (r) = 1

to derive the normalization constants Nk .

2.3 Equations of motion for classical radiation oscillators

In the interior of an empty cavity the ¬elds satisfy Maxwell™s equations (2.1) and

(2.2). Use the expansions (2.40) and (2.41) and the properties of the mode functions

to derive eqn (2.43).

2.4 Complex mode amplitudes

(1) Use the expression (2.48) for the classical energy and the expansions (2.40) and

(2.41) to derive eqn (2.49).

(2) Derive eqns (2.46) and (2.51).

2.5 Number states

Use the commutation relations (2.76) and the de¬nition (2.73) of the vacuum state to

verify eqn (2.78).

2.6 The second-order coherence matrix

(1) For the operators a† and ap (p = 1, 2) de¬ned by eqn (2.153) show that the number

p

†

operators Np = ap ap are simultaneously measurable.

(2) Consider the operator

1 1 1

|1x 1x | + |1y 1y | ’ 1x | + |1x 1y |) ,

(|1y

ρ=

2 2 4

where |1s = a† |0 .

ks

(a) Show that ρ is a genuine density operator, i.e. it is positive and has unit trace.

(b) Calculate the coherence matrix J, its eigenvalues and eigenvectors, and the

degree of polarization.

Exercises

2.7 The Stokes parameters

(1) What is the physical signi¬cance of S0 ?

(2) Use the explicit forms of the Pauli matrices and the expansion (2.158) to show

that

12

S 0 ’ S1 ’ S2 ’ S3 ,

2 2 2

det J =

4

and thereby establish the condition (2.160).

(3) With S0 = 1, introduce polar coordinates by S3 = cos θ, S2 = sin θ sin φ, and

S3 = sin θ cos φ. Find the locations on the Poincar´ sphere corresponding to right

e

circular polarization, left circular polarization, and linear polarization.

2.8 A one-photon mixed state

Consider a monochromatic state for wavevector k (see Section 2.4.1-A) containing

exactly one photon.

(1) Explain why the density operator for this state is completely represented by the

2 — 2 matrix ρss = 1ks |ρ| 1ks .

(2) Show that the density matrix ρss is related to the coherence matrix J by ρss =

Js s .

2.9 The Casimir force

Show that the large L limit of eqn (2.198) is

2

c L

dkx dky e’±k⊥ k⊥

2

U (∆z) =

2 π

2∞

L 1/2

dkx dky e’±(k⊥ +kzn ) k⊥ + kzn

2 2

2 2

+c

π n=1

3

∆z L 2

dkx dky dkz e’±k k ,

’ c

L π

2

2 2 2

where k⊥ = kx + ky , k = k⊥ + kz , and kzn = nπ/∆z.

2.10 Model for the experiments on the Casimir force

Consider the simple-harmonic-oscillator model of the Lamoreaux and Mohideen-Roy

experiments on the Casimir force shown in Fig. 2.1.

All elements of the apparatus, which are assumed to be perfect conductors, are

rendered electrically neutral by grounding them to the Earth. Assume that the spring

constant for the metallic spring is k. (You may ignore Earth™s gravity in this problem.)

(1) Calculate the displacement ∆x of the spring from its relaxed length as a function

of the spacing d between the surface of the sphere and the ¬‚at plate on the right,

after the system has come into mechanical equilibrium.

(2) Calculate the natural oscillation frequency of this system for small disturbances

around this equilibrium as a function of d. Neglect all dissipative losses.

Quantization of cavity modes

Metallic sphere

4

Fig. 2.1 The Casimir force between a

grounded metallic sphere of radius R and the

Metallic spring @

grounded ¬‚at metallic plate on the right, which

Flat metallic plates

is separated by a distance d from the sphere,

can be measured by measuring the displace-

Earth ground

ment of the metallic spring. (Ignore gravity.)

(3) Plot your answers for parts 1 and 2 for the following numerical parameters:

R = 200 µm ,

0.1 µm d 1.0 µm ,

k = 0.02 N/m .

3

Field quantization

Quantizing the radiation oscillators associated with the classical modes of the elec-

tromagnetic ¬eld in a cavity provides a satisfactory theory of the Planck distribution

and the Casimir e¬ect, but this is only the beginning of the story. There are, after all,

quite a few experiments that involve photons propagating freely through space, not

just bouncing back and forth between cavity walls. In addition to this objection, there

is a serious ¬‚aw in the cavity-based model. The quantized radiation oscillators are

de¬ned in terms of a set of classical mode functions satisfying the idealized boundary

conditions for perfectly conducting walls. This di¬culty cannot be overcome by sim-

ply allowing for ¬nite conductivity, since conductivity is itself a macroscopic property

that does not account for the atomistic structure of physical walls. Thus the quantiza-

tion conjecture (2.61) builds the idealized macroscopic boundary conditions into the

foundations of the microscopic quantum theory of light. A fundamental microscopic

theory should not depend on macroscopic idealizations, so there is more work to be

done. We should emphasize, however, that this objection to the cavity model does not

disqualify it as a guide toward an improved theory. The cavity model itself was con-

structed by applying the ideas of nonrelativistic quantum mechanics to the classical

radiation oscillators. In a similar fashion, we will use the cavity model to suggest a

true microscopic conjecture for the quantization of the electromagnetic ¬eld.

In the following sections we will show how the quantization scheme of the cavity

model can be used to suggest local commutation relations for quantized ¬elds in free

space. The experimentally essential description of photons in passive optical devices

will be addressed by formulating a simple model for the quantization of the ¬eld in

a dielectric medium. In the ¬nal four sections we will discuss some more advanced

topics: the angular momentum of light, a description of quantum ¬eld theory in terms

of wave packets, and the question of the spatial localizability of photons.

3.1 Field quantization in the vacuum

The quantization of the electromagnetic ¬eld in free space is most commonly carried

out in the language of canonical quantization (Cohen-Tannoudji et al., 1989, Sec.

II.A.2), which is based on the Lagrangian formulation of classical electrodynamics.

This is a very elegant way of packaging the necessary physical conjectures, but it

requires extra mathematical machinery that is not needed for most applications. We

will pursue a more pedestrian route which builds on the quantization rules for the

ideal physical cavity. To this end, we initially return to the cavity problem.

¼ Field quantization

3.1.1 Local commutation relations

In Chapter 2 we concentrated on the operators (qκ , pκ ) for a single mode. Since the

modes are determined by the boundary conditions at the cavity walls, they describe

global properties of the cavity. We now want turn attention away from the overall

properties of the cavity, in order to concentrate on the local properties of the ¬eld

operators. We will do this by combining the expansions (2.99) and (2.103) for the time-

independent, Schr¨dinger-picture operators E (r) and A (r) with the commutation

o

relations (2.61) for the mode operators to calculate the commutators between ¬eld

components evaluated at di¬erent points in space. The expansions show that E (r)

only depends on the pκ s while A (r) and B (r) depend only on the qκ s; therefore, the

commutation relations, [pκ , p» ] = [qκ , q» ] = 0, produce

[Ej (r) , Ek (r )] = 0 , [Aj (r) , Ak (r )] = 0 , [Bj (r) , Bk (r )] = 0 . (3.1)

On the other hand, [qκ , p» ] = i δκ» , so the commutator between the electric ¬eld and

the vector potential is

1

[Ai (r) , ’Ej (r )] = [qκ , p» ] Eκi (r) E»j (r )

0 κ »

i

Eκi (r) Eκj (r ) .

= (3.2)

0 κ

For any cavity, the mode functions satisfy the completeness condition (2.38), so we see

that

i

[Ai (r) , ’Ej (r )] = ∆⊥ (r ’ r ) . (3.3)

ij

0

The resemblance between this result and the canonical commutation relation, [qκ , p» ] =

i δκ» , for the mode operators suggests the identi¬cation of A (r) and ’E (r) as the

canonical variables for the ¬eld in position space. A similar calculation for the commu-

tator between the E- and B-¬elds can be carried out using eqn (2.100), or by applying

the curl operation to eqn (3.3), with the result

ijl ∇l δ (r ’ r ),

[Bi (r) , Ej (r )] = i (3.4)

0

where ijl is the alternating tensor de¬ned by eqn (A.3). The uncertainty relations

implied by the nonvanishing commutators between electric and magnetic ¬eld compo-

nents were extensively studied in the classic work of Bohr and Rosenfeld (1950), and

a simple example can be found in Exercise 3.2.

The derivation of the local commutation relations (3.1) and (3.3) for the ¬eld oper-

ators in the physical cavity employs the complete set of cavity modes, which depend on

the geometry of the cavity. This can be seen from the explicit appearance of the mode

functions in the second line of eqn (3.2). However, the ¬nal result (3.3) follows from the

completeness relation (2.38), which has the same form for every cavity. This feature

only depends on the fact that the boundary conditions guarantee the Hermiticity of

the operator ’∇2 . We have, therefore, established the quite remarkable result that

½

Field quantization in the vacuum

the local position-space commutation relations are independent of the shape and size

of the cavity. In particular, eqns (3.1) and (3.3) will hold in the limit of an in¬nitely

large physical cavity; that is, when the distance to the cavity walls from either of the

points r and r is much greater than any physically relevant length scale. In this limit,

it is plausible to assume that the boundary conditions at the walls are irrelevant. This

suggests abandoning the original quantization conjecture (2.61), and replacing it by

eqns (3.1) and (3.3). In this way we obtain a microscopic theory which does not involve

the macroscopic idealizations associated with the classical boundary conditions. We

emphasize that this is not a derivation of the local commutation relations from the

physical cavity relations (2.61). The sole function of the cavity-based calculation is

to suggest the form of eqns (3.1) and (3.3), which constitute an independent quanti-

zation conjecture. As always, the validity of the this conjecture has to be tested by

means of experiment. In this new approach, the theory based on the ideal physical

cavity”with its dependence on macroscopic boundary conditions”is demoted to a

phenomenological model.

Since the new quantization rules hold everywhere in space, they can be expressed

in terms of Fourier transform pairs de¬ned by

d3 k

d3 re’ik·r F (r) ,

ik·r

F (r) = 3e F (k) , F (k) = (3.5)

(2π)

where F = A, E, or B. The position-space ¬eld operators are hermitian, so their

Fourier transforms satisfy F† (k) = F (’k). It should be clearly understood that eqn

(3.5) is simply an application of the Fourier transform; no additional physical assump-

tions are required. By contrast, the expansions (2.99) and (2.103) in cavity modes

involve the idealized boundary conditions at the cavity walls.

Transforming eqns (3.1) and (3.3) with respect to r and r independently yields

the equivalent relations

[Ej (k) , Ek (k )] = [Aj (k) , Ak (k )] = 0 , (3.6)

and

i

∆⊥ (k) (2π) δ k + k ,

3

[Ai (k) , ’Ej (k )] = (3.7)

ij

0

where the delta function comes from using the identity (A.96).

3.1.2 Creation and annihilation operators

A Position space

The commutation relations (3.1)“(3.4) are not the only general consequences that are

implied by the cavity model. For example, the expansions (2.101) and (2.103) can be

rewritten as

E (r) = E(+) (r) + E(’) (r) , A (r) = A(+) (r) + A(’) (r) , (3.8)

where

aκ E κ (r) = A(’)† (r)

A(+) (r) = (3.9)

2 0 ωκ

κ

¾ Field quantization

and

ωκ

aκ E κ (r) = E(’)† (r) .

E(+) (r) = i (3.10)

20

κ

Let F be one of the ¬eld operators, Ai or Ei , then F (+) is called the positive-frequency

part and F (’) is called the negative-frequency part. The origin of these mysterious

names will become clear in Section 3.2.3, but for the moment we only need to keep

in mind that F (+) is a sum of annihilation operators and F (’) is a sum of creation

operators. These properties are expressed by

F (+) (r) |0 = 0 , 0| F (’) (r) = 0 . (3.11)

In view of the de¬nition (3.9) there is a natural inclination to think of A(+) (r) as an

operator that annihilates a photon at the point r, but this temptation must be resisted.

The di¬culty is that the photon”i.e. ˜a quantum of excitation of the electromagnetic

¬eld™”cannot be sharply localized in space. A precise interpretation for A(+) (r) is

presented in Section 3.5.2, and the question of photon localization is studied in Section

3.6.

An immediate consequence of eqns (3.9) and (3.10) is that

F (±) (r) , G(±) (r ) = 0 , (3.12)

where F and G are any pair of ¬eld operators. It is clear, however, that F (+) , G(’)

will not always vanish. In particular, a calculation similar to the one leading to eqn

(3.3) yields

i

∆⊥ (r ’ r ) .

(+) (’)

Ai (r) , ’Ej (r ) = (3.13)

2 0 ij

The decomposition (3.8) also allows us to express all ¬eld operators in terms of

(±)

A . For this purpose, we rewrite eqn (3.10) as

aκ kκ E κ (r) ,

E(+) (r) = ic (3.14)

2 0 ωκ

κ

and use eqn (2.181) to get the ¬nal form

1/2

E(+) (r) = ic ’∇2 A(+) (r) . (3.15)

Substituting this into eqn (3.13) yields the equivalent commutation relations

’1/2

∆⊥ (r ’ r ) ,

(+) (’)

’∇2

Ai (r) , Aj (r ) = (3.16)

ij

2 0c

c 1/2

∆⊥ (r ’ r ) .

(+) (’)

’∇2

Ei (r) , Ej (r ) = (3.17)

ij

20

’1/2

1/2

In the context of free space, the unfamiliar operators ’∇2 and ’∇2 are

best de¬ned by means of Fourier transforms. For any real function f (u) the identity

¿

Field quantization in the vacuum

’∇2 exp (ik · r) = k 2 exp (ik · r) allows us to de¬ne the action of f ’∇2 on a plane

wave by f ’∇2 eik·r ≡ f k 2 eik·r . This result in turn implies that f ’∇2 acts on

a general function • (r) according to the rule

d3 k d3 k

f ’∇2 • (r) ≡ 3 • (k) f ’∇

2

eik·r = k 2 eik·r .

3 • (k) f (3.18)

(2π) (2π)

After using the inverse Fourier transform on • (k) this becomes

f ’∇2 • (r) = d3 r r f ’∇2 r • (r ) , (3.19)

where

d3 k

k 2 eik·(r’r )

r f ’∇2 r = 3f (3.20)

(2π)

is the integral kernel de¬ning f ’∇2 as an operator in r-space. Despite its abstract

appearance, this de¬nition is really just a labor saving device; it avoids transforming

back and forth from position space to reciprocal space. For example, real functions of

the hermitian operator ’∇2 are also hermitian; so one gets a useful integration-by-parts

identity