ńņš. 3 |

so Ļ is a positive-deļ¬nite operator. Combining this with the normalization condition

ĪØ |Ļ| ĪØ 1 for any normalized state |ĪØ . The Born interpretation

(2.120) implies 0

2

tells us that | ĪØe |ĪØ | is the probability that a measurementā”say of the projection

operator |ĪØ ĪØ|ā”will leave the system in the state |ĪØ , given that the system is

prepared in the pure state |ĪØe ; therefore, eqn (2.133) tells us that ĪØ |Ļ| ĪØ is the

probability that a measurement will lead to |ĪØ , if the system is described by the

mixed state with density operator Ļ.

In view of the importance of the superposition principle for pure states, it is natural

to ask if any similar principle applies to mixed states. The ļ¬rst thing to note is that

linear combinations of density operators are not generally physically acceptable density

operators. Thus if Ļ1 and Ļ2 are density operators, the combination Ļ = CĻ1 +DĻ2 will

be hermitian only if C and D are both real. The condition Tr Ļ = 1 further requires

C + D = 1. Finally, the positivity condition (2.133) must hold for all choices of |ĪØ ,

and this can only be guaranteed by imposing C 0 and D 0. Therefore, only the

convex linear combinations

Ļ = CĻ1 + (1 ā’ C) Ļ2 , 0 C 1 (2.134)

are guaranteed to be density matrices. This terminology is derived from the mathe-

matical notion of a convex set in the plane, i.e. a set that contains every straight line

joining any two of its points. The general form of eqn (2.134) is

Ļ= Cn Ļn , (2.135)

n

where each Ļn is a density operator, and the coeļ¬cients satisfy the convexity condition

0 Cn 1 for all n and Cn = 1 . (2.136)

n

Quantization of cavity modes

The oļ¬-diagonal matrix elements of the density operator are also constrained by the

deļ¬nition (2.116). The normalization of the ensemble states |ĪØe implies | ĪØe |ĪØ | 1,

so

| ĪØ |Ļ| Ī¦ | = Pe ĪØ |ĪØe ĪØe |Ī¦

e

Pe | ĪØ |ĪØe | | ĪØe |Ī¦ | 1, (2.137)

e

i.e. Ļ is a bounded operator.

The arguments leading from the ensemble deļ¬nition of the density operator to its

properties can be reversed to yield the following statement. An operator Ļ that is

(a) hermitian, (b) bounded, (c) positive, and (d) has unit trace is a possible density

operator. The associated ensemble can be deļ¬ned as the set of normalized eigenstates

of Ļ corresponding to nonzero eigenvalues. Since every density operator has a complete

orthonormal set of eigenvectors, this last remark implies that it is always possible to

choose the ensemble to consist of mutually orthogonal states.

2.3.4 Degrees of mixing

So far the distinction between pure and mixed states is absolute, but ļ¬ner distinc-

tions are also useful. In other words, some states are more mixed than others. The

distinctions we will discuss arise most frequently for physical systems described by a

ļ¬nite-dimensional Hilbert space, or equivalently, ensembles containing a ļ¬nite number

of pure states. This allows us to simplify the analysis by assuming that the Hilbert

space has dimension d < ā. The inequality (2.124) suggests that the purity

P (Ļ) = Tr Ļ2 1 (2.138)

may be a useful measure of the degree of mixing associated with a density operator Ļ.

By virtue of eqn (2.122), P (Ļ) = 1 for a pure state; therefore, it is natural to say that

the state Ļ2 is less pure (more mixed) than the state Ļ1 if P (Ļ2 ) < P (Ļ1 ). Thus the

minimally pure (maximally mixed) state for an ensemble will be the one that achieves

the lower bound of P (Ļ). In general the density operator can have the eigenvalue

0 with degeneracy (multiplicity) d0 < d, so the number of orthogonal states in the

ensemble is N = d ā’ d0 . Using the eigenstates of Ļ to evaluate the trace yields

N

P (Ļ) = p2 , (2.139)

n

n=1

where pn is the nth eigenvalue of Ļ. In this notation, the trace condition (2.120) is just

N

pn = 1 , (2.140)

n=1

and the lower bound is found by minimizing P (Ļ) subject to the constraint (2.140).

This can be done in several ways, e.g. by the method of Lagrange multipliers, with

the result that the maximally mixed state is deļ¬ned by

Mixed states of the electromagnetic ļ¬eld

, n = 1, . . . , N ,

1

N

pn = (2.141)

0 , n = N + 1, . . . , d .

In other words, the pure states in the ensemble deļ¬ning the maximally mixed state

occur with equal probability, and the purity is

1

P (Ļ) =

. (2.142)

N

Another useful measure of the degree of mixing is provided by the von Neumann

entropy, which is deļ¬ned in general by

S (Ļ) = ā’ Tr (Ļ ln Ļ) . (2.143)

In the special case considered above, the von Neumann entropy is given by

N

S (Ļ) = ā’ pn ln pn , (2.144)

n=1

and maximizing thisā”subject to the constraint (2.140)ā”leads to the same deļ¬nition

of the maximally mixed state, with the value

S (Ļ) = ln N (2.145)

of S (Ļ). The von Neumann entropy plays an important role in the study of entangled

states in Chapter 6.

2.4 Mixed states of the electromagnetic ļ¬eld

2.4.1 Polarized light

As a concrete example of a mixed state, consider an experiment in which light from a

single atom is sent through a series of collimating pinholes. In each atomic transition,

exactly one photon with frequency Ļ = āE/ is emitted, where āE is the energy

diļ¬erence between the atomic states. The alignment of the pinholes determines the

unit vector k along the direction of propagation, so the experimental arrangement

determines the wavevector k = (Ļ/c) k. If the pinholes are perfectly circular, the

experimental preparation gives no information on the polarization of the transmitted

light. This means that the light observed on the far side of the collimator could be

described by either of the states

|ĪØs = |1ks = aā |0 , (2.146)

ks

where s = Ā±1 labels right- and left-circularly-polarized light. Thus the relevant en-

semble is composed of the states |1k+ and |1kā’ , with probabilities P+ and Pā’ , and

the density operator is

P+ 0

Ps |1ks 1ks | =

Ļ= . (2.147)

Pā’

0

s

In the absence of any additional information equal probabilities are assigned to the

two polarizations, i.e. P+ = Pā’ = 1/2, and the light is said to be unpolarized. The

Quantization of cavity modes

opposite extreme occurs when the polarization is known with certainty, for example

P+ = 1, Pā’ = 0. This can be accomplished by inserting a polarization ļ¬lter after

the collimator. In this case, the light is said to be polarized, and the density operator

represents the pure state |1k+ . For the intermediate cases, a measure of the degree of

polarization is given by

P = |P+ ā’ Pā’ | , (2.148)

which satisļ¬es 0 P 1, and has the values P = 0 for unpolarized light and P = 1

for polarized light.

A The second-order coherence matrix

The conclusions reached for the special case discussed above are also valid in a more

general setting (Mandel and Wolf, 1995, Sec. 6.2). We present here a simpliļ¬ed version

of the general discussion by deļ¬ning the second-order coherence matrix

Jss = Tr Ļaā aks , (2.149)

ks

where the density operator Ļ describes a monochromatic state, i.e. each state vector

|ĪØe in the ensemble deļ¬ning Ļ satisļ¬es ak s |ĪØe = 0 for k = k. In this case we may

as well choose the z-axis along k, and set s = x, y, corresponding to linear polarization

vectors along the x- and y-axes respectively. The 2 Ć— 2 matrix J is hermitian and posi-

tive deļ¬niteā”see Appendix A.3.4ā”so the eigenvectors cp = (cpx , cpy ) and eigenvalues

np (p = 1, 2) deļ¬ned by

Jcp = np cp (2.150)

satisfy

cā cp = Ī“pp and np 0. (2.151)

p

The eigenvectors of J deļ¬ne eigenpolarization vectors,

e p = c ā— e x + cā— e y , (2.152)

px py

and corresponding creation and annihilation operators

aā = cā— aā + cā— aā , ap = cpx akx + cpy aky . (2.153)

px kx py ky

p

It is not diļ¬cult to show that

np = Tr Ļaā ap , (2.154)

p

i.e. the eigenvalue np is the average number of photons with eigenpolarization ep . If Ļ

describes an unpolarized state, then diļ¬erent polarizations must be uncorrelated and

the number of photons in either polarization must be equal, i.e.

n1 0

J= , (2.155)

20 1

where

n = Tr Ļ aā akx + aā aky (2.156)

kx ky

is the average total number of photons. If Ļ describes complete polarization, then

the occupation number for one of the eigenpolarizations must vanish, e.g. n2 = 0.

Mixed states of the electromagnetic ļ¬eld

Since det J = n1 n2 , this means that completely polarized states are characterized by

det J = 0. In this general setting, the degree of polarization is deļ¬ned by

|n1 ā’ n2 |

P= , (2.157)

n1 + n2

where P = 0 and P = 1 respectively correspond to unpolarized and completely polar-

ized light.

B The Stokes parameters

Since J is a 2 Ć— 2 matrix, we can exploit the well known factā”see Appendix C.3.1ā”

that any 2 Ć— 2 matrix can be expressed as a linear combination of the Pauli matrices.

For this application, we write the expansion as

1 1 1 1

S0 Ļ0 + S1 Ļ3 + S2 Ļ1 ā’ S3 Ļ2 ,

J= (2.158)

2 2 2 2

where Ļ0 is the 2 Ć— 2 identity matrix and Ļ1 , Ļ2 , and Ļ3 are the Pauli matrices given

by the standard representation (C.30). This awkward formulation guarantees that the

c-number coeļ¬cients SĀµ are the traditional Stokes parameters. According to eqn

(C.40) they are given by

S0 = Tr (JĻ0 ) , S1 = Tr (JĻ3 ) , S2 = Tr (JĻ1 ) , S3 = ā’ Tr (JĻ2 ) . (2.159)

The Stokes parameters yield a useful geometrical picture of the coherence matrix, since

the necessary condition det (J) 0 translates to

2 2 2 2

S 1 + S2 + S3 S0 . (2.160)

If we interpret (S1 , S2 , S3 ) as a point in a three-dimensional space, then for a ļ¬xed

value of S0 the states of the ļ¬eld occupy a sphereā”called the PoincarĀ“ sphereā”of

e

radius S0 . The origin, S1 = S2 = S3 = 0, corresponds to unpolarized light, since this

is the only case for which J is proportional to the identity. The condition det J = 0

for completely polarized light is

2 2 2 2

S 1 + S2 + S3 = S0 , (2.161)

which describes points on the surface of the sphere. Intermediate states of polarization

correspond to points in the interior of the sphere.

The PoincarĀ“ sphere is often used to describe the pure states of a single photon,

e

e.g.

Cs aā |0 .

|Ļ = (2.162)

ks

s

In this case S0 = 1, and the points on the surface of the PoincarĀ“ sphere can be labeled

e

by the standard spherical coordinates (Īø, Ļ). The north pole, Īø = 0, and the south pole,

Īø = Ļ, respectively describe right- and left-circular polarizations. Linear polarizations

are represented by points on the equator, and elliptical polarizations by points in the

northern and southern hemispheres.

Quantization of cavity modes

2.4.2 Thermal light

A very important example of a mixed state arises when the ļ¬eld is treated as a thermo-

dynamic system in contact with a thermal reservoir at temperature T , e.g. the walls

of the cavity. Under these circumstances, any complete set of states can be chosen

for the ensemble, since we have no information that allows the exclusion of any pure

state of the ļ¬eld. Exchange of energy with the walls is the mechanism for attaining

thermal equilibrium, so it is natural to use the energy eigenstatesā”i.e. the number

states |n ā”for this purpose.

The general rules of statistical mechanics (Chandler, 1987, Sec. 3.7) tell us

that the probability for a given energy E is proportional to exp (ā’Ī²E), where

Ī² = 1/kB T and kB is Boltzmannā™s constant. Thus the probability distribution is

Pn = Z ā’1 exp ā’Ī²En , where Z ā’1 is the normalization constant required to satisfy

eqn (2.112), and

En = Ļ Īŗ nĪŗ . (2.163)

Īŗ

Substituting this probability distribution into eqn (2.116) gives the density operator

1 1

eā’Ī²En |n n| =

Ļ= exp (ā’Ī²Hem ) . (2.164)

Z Z

n

The normalization constant Z, which is called the partition function, is determined

by imposing Tr (Ļ) = 1 to get

Z = Tr [exp (ā’Ī²Hem )] . (2.165)

Evaluating the trace in the number-state basis yields

exp ā’Ī²

Z= nĪŗ Ļ Īŗ = ZĪŗ , (2.166)

n Īŗ Īŗ

where

ā ā

1

ĻĪŗ nĪŗ

ā’Ī²nĪŗ ĻĪŗ

eā’Ī²

ZĪŗ = e = = (2.167)

1 ā’ eā’Ī² ĻĪŗ

nĪŗ =0 nĪŗ =0

is the partition function for mode Īŗ (Chandler, 1987, Chap. 4).

A The Planck distribution

The average energy in the electromagnetic ļ¬eld is related to the partition function by

ĻĪŗ

ā‚

ā’ ln Z = . (2.168)

U=

e Ī² ĻĪŗ ā’ 1

ā‚Ī² Īŗ

We will say that the cavity is large if the energy spacing cākĪŗ between adjacent

discrete modes is small compared to any physically relevant energy. In this limit the

shape of the cavity is not important, so we may suppose that it is cubical, with

Mixed states of the electromagnetic ļ¬eld

Īŗ ā’ (k, s), where s = 1, 2 and ĻĪŗ ā’ ck. In the limit of inļ¬nite volume, applying the

rule

d3 k

1

ā’ (2.169)

3

V (2Ļ)

k

replaces eqn (2.168) by

U 2 ck

d3 k Ī² ck

= . (2.170)

ā’1

3

V e

(2Ļ)

After carrying out the angular integrations and changing the remaining integration

variable to Ļ = ck, this becomes

ā

U

= dĻ Ļ (Ļ, T ) , (2.171)

V 0

where the energy density Ļ (Ļ, T ) dĻ in the frequency interval Ļ to Ļ + dĻ is given by

the Planck function

Ļ3

1

Ļ (Ļ, T ) = 2 3 Ī² Ļ . (2.172)

ā’1

Ļc e

B Distribution in photon number

In addition to the distribution in energy, it is also useful to know the distribution in

photon number, nĪŗ , for a given mode. This calculation is simpliļ¬ed by the fact that

the thermal density operator is the product of independent operators for each mode,

Ļ= ĻĪŗ , (2.173)

Īŗ

where

1

ĻĪŗ =

exp (ā’Ī²NĪŗ ĻĪŗ ) . (2.174)

ZĪŗ

Thus we can drop the mode index and set

Ļ = 1 ā’ eā’Ī² exp ā’Ī² Ļaā a .

Ļ

(2.175)

The eigenstates of the single-mode number operator are nondegenerate, so the general

rule (2.132) reduces to

p (n) = Tr (Ļ |n n|) = n |Ļ| n = 1 ā’ eā’Ī² eā’nĪ²

Ļ Ļ

, (2.176)

where p (n) is the probability of ļ¬nding n photons. This can be expressed more con-

veniently by ļ¬rst calculating the average number of photons:

eā’Ī² Ļ

ā

n = Tr Ļa a = . (2.177)

1 ā’ eā’Ī² Ļ

Using this to eliminate eā’Ī² Ļ

leads to the ļ¬nal form

n

n

p (n) = . (2.178)

n+1

(1 + n )

Finally, it is important to realize that eqn (2.177) is not restricted to the electro-

magnetic ļ¬eld. Any physical system with a Hamiltonian of the form (2.89), where the

Ā¼ Quantization of cavity modes

operators a and aā satisfy the canonical commutation relations (2.63) for a harmonic

oscillator, will be described by the Planck distribution.

2.5 Vacuum ļ¬‚uctuations

Our ļ¬rst response to the inļ¬nite zero-point energy associated with vacuum ļ¬‚uctuations

was to hide it away as quickly as possible, but we now have the tools to investigate

the divergence in more detail. According to eqns (2.99) and (2.100) the electric and

magnetic ļ¬eld operators are respectively determined by pĪŗ and qĪŗ so there are in-

escapable vacuum ļ¬‚uctuations of the ļ¬elds. The E and B ļ¬elds are linear in aĪŗ and aā Īŗ

so their vacuum expectation values vanish, but E2 and B2 will have nonzero vacuum

expectation values representing the rms deviation of the ļ¬elds. Let us consider the

rms deviation of the electric ļ¬eld. The operators Ei (r) (i = 1, 2, 3) are hermitian and

mutually commutative, so we are allowed to consider simultaneous measurements of

all components of E (r). In this case the ambiguity in going from a classical quantity

to the corresponding quantum operator is not an issue.

Since trouble is to be expected, we approach 0 E2 (r) 0 with caution by ļ¬rst

evaluating 0 |Ei (r) Ej (r )| 0 for r = r. The expansion (2.101) yields

ā

0 |Ei (r) Ej (r )| 0 = ā’ ĻĪŗ ĻĪ» EĪŗi (r) EĪ»j (r )

2 0 Īŗ Ī»

aĪ» ā’ aā |0 ,

Ć— 0| aĪŗ ā’ aā (2.179)

Īŗ Ī»

and evaluating the vacuum expectation value leads to

0 |Ei (r) Ej (r )| 0 = ĻĪŗ EĪŗi (r) EĪŗj (r ) . (2.180)

2 0 Īŗ

Direct evaluation of the sum over modes requires detailed knowledge of both the

mode spectrum and the mode functions, but this can be avoided by borrowing a trick

from quantum mechanics (Cohen-Tannoudji et al., 1977a, Chap. II, Complement B).

According to eqn (2.35) each mode function E Īŗ is an eigenfunction of the operator

ā’ā2 with eigenvalue kĪŗ . The operator and eigenvalue are respectively mathemati-

2

cal analogues of the kinetic energy operator and the energy eigenvalue in quantum

mechanics (in units such that 2m = = 1). Since ā’ā2 is hermitian and E Īŗ is an

eigenfunction, the general argument given in Appendix C.3.6 shows that

1/2

ā’ā2 EĪŗ = kĪŗ E Īŗ = kĪŗ E Īŗ .

2 (2.181)

Using this relation, together with ĻĪŗ = ckĪŗ , in eqn (2.35) yields

1/2

ĻĪŗ EĪŗi (r) = ckĪŗ EĪŗi (r) = c kĪŗ EĪŗi (r) = c ā’ā2 EĪŗi (r) .

2 (2.182)

Thus eqn (2.180) can be replaced by

c 1/2

ā’ā2 EĪŗi (r) EĪŗj (r ) ,

0 |Ei (r) Ej (r )| 0 = (2.183)

2 0 Īŗ

Ā½

Vacuum ļ¬‚uctuations

which combines with the completeness relation (2.38) to yield

c 1/2

āā„ (r ā’ r )

0 |Ei (r) Ej (r )| 0 = ā’ā2 ij

2 0

d3 k

c ki kj

Ī“ij ā’ eikĀ·(rā’r ) ,

= 3k (2.184)

k2

2 (2Ļ)

0

where the last line follows from the fact that eikĀ·r is an eigenfunction of ā’ā2 with

eigenvalue k 2 . Setting r = r and summing over i = j yields the divergent integral

d3 k

0 E2 (r) 0 = k. (2.185)

(2Ļ)3

0

Thus the rms ļ¬eld deviation is inļ¬nite at every point r. In the case of the energy this

disaster could be avoided by redeļ¬ning the zero of energy for each cavity mode, but

no such escape is possible for measurements of the electric ļ¬eld itself.

This looks a little neaterā”although no less divergentā”if we deļ¬ne the (volume

averaged) rms deviation by

1

(āE)2 = d3 rE2 (r) 0 .

0 (2.186)

V V

This is best calculated by returning to eqn (2.180), setting r = r and integrating to

get

2

e2 ,

(āE) = (2.187)

Īŗ

Īŗ

where the vacuum ļ¬‚uctuation ļ¬eld strength, eĪŗ , for mode E Īŗ is

ĻĪŗ

eĪŗ = . (2.188)

2 0V

The sum over all modes diverges, but the ļ¬‚uctuation strength for a single mode is

ļ¬nite and will play an important role in many of the arguments to follow. A similar

calculation for the magnetic ļ¬eld yields

Āµ0 ĻĪŗ

2

b2 , bĪŗ =

(āB) = . (2.189)

Īŗ

2V

Īŗ

The source of the divergence in (āE)2 and (āB)2 is the singular character of

the the vacuum ļ¬‚uctuations at a point. This is a mathematical artifact, since any

measuring device necessarily occupies a nonzero volume. This suggests considering an

operator of the form

W ā”ā’ d3 r P (r) Ā· E (r) , (2.190)

V

where P (r) is a smooth (inļ¬nitely diļ¬erentiable) c-number function that vanishes

outside some volume V0 V . In this way, the singular behavior of E (r) is reduced by

Ā¾ Quantization of cavity modes

1/3

averaging the point r over distances of the order d0 = V0 . According to the uncer-

tainty principle, this is equivalent to an upper bound k0 ā¼ 1/d0 in the wavenumber,

so the divergent integral in eqn (2.185) is replaced by

d3 k 4

k0

< ā.

k= (2.191)

(2Ļ)3 2

0 8Ļ

0 k<k0

If the volume V0 is ļ¬lled with an electret, i.e. a material with permanent electric

polarization, then P (r) can be interpreted as the density of classical dipole moment,

and W is the interaction energy between the classical dipoles and the quantized ļ¬eld. In

this idealized model W is a well-deļ¬ned physical quantity which is measurable, at least

in principle. Suppose the measurement is carried out repeatedly in the vacuum state.

According to the standard rules of quantum theory, the average of these measurements

is given by the vacuum expectation value of W , which is zero. Of course, the fact that

the average vanishes does not imply that every measured value does. Let us next

determine the variance of the measurements by evaluating

0| W 2 |0 = d3 r Pi (r) Pj (r ) 0 |Ei (r) Ej (r )| 0 .

d3 r (2.192)

V V

Substituting eqn (2.180) into this expression yields

0| W 2 |0 = Ļ Īŗ PĪŗ ,

2

(2.193)

2 0 Īŗ

where

PĪŗ = d3 r P (r) Ā· E Īŗ (r) (2.194)

V

represents the classical interaction energy for a single mode. In this case the sum

converges, since the coeļ¬cients PĪŗ will decay rapidly for higher-order modes. Thus W

exhibits vacuum ļ¬‚uctuation eļ¬ects that are both ļ¬nite and observable. It is important

to realize that this result is independent of the choice of operator ordering, e.g. eqn

(2.105) or eqn (2.106), for the Hamiltonian. It is also important to assume that the

permanent dipole moment of the electret is so small that the radiation it emits by

virtue of the acceleration imparted by the vacuum ļ¬‚uctuations can be neglected. In

other words, this is a test electret analogous to the test charges assumed in the standard

formulation of classical electrodynamics (Jackson, 1999, Sec. 1.2).

2.6 The Casimir eļ¬ect

In Section 2.2 we discarded the zero-point energy due to vacuum ļ¬‚uctuations on the

grounds that it could be eliminated by adding a constant to the Hamiltonian in eqn

(2.104). This is correct for a single cavity, but the situation changes if two diļ¬erent

cavities are compared. In this case, a single shift in the energy spectrum can eliminate

one or the other, but not both, of the zero-point energies; therefore, the diļ¬erence

between the zero-point energies of the two cavities can be the basis for observable

Āæ

The Casimir eļ¬ect

phenomena. An argument of this kind provides the simplest explanation of the Casimir

eļ¬ect.

We follow the approach of Milonni and Shih (1992) which begins by considering

the planar cavityā”i.e. two plane parallel plates separated by a distance small com-

pared to their lateral dimensionsā”described in Appendix B.4. In this situation edge

eļ¬ects are small, so the plates can be represented by an ideal cavity in the shape of a

rectangular box with dimensions L Ć— L Ć— āz. This conļ¬guration will be compared to

a cubical cavity with sides L. The eigenfrequencies for a planar cavity are

1/2

2 2 2

mĻ nĻ

lĻ

Ļlmn = c + + , (2.195)

L L āz

where the range of the indices is l, m, n = 0, 1, 2, . . ., except that there are no modes

with two zero indices. If one index vanishes, there is only one polarization, but for

three nonzero indices there are two. We want to compare the zero-point energies of

conļ¬gurations with diļ¬erent values of āz, so the interesting quantity is

Ļlmn

E0 (āz) = Clmn , (2.196)

2

l,m,n

where Clmn is the number of polarizations. Thus Clmn = 2 when all indices are nonzero,

Clmn = 1 when exactly one index vanishes, and Clmn = 0 when at least two indices

vanish. Since this sum diverges, it is necessary to regularize it, i.e. to replace it with a

mathematically meaningful expression which has eqn (2.196) as a limiting value. All

intermediate calculations are done using the regularized form, and the limit is taken

at the end of the calculation. The physical justiļ¬cation for this apparently reckless

procedure rests on the fact that real conductors become transparent to radiation at

suļ¬ciently high frequencies (Jackson, 1999, Sec. 7.5D). In this range the contribution

to the zero-point energy is unchanged by the presence of the conducting plates, so it

will cancel out in taking the diļ¬erence between diļ¬erent conļ¬gurations. Thus the high-

frequency part of the sum in eqn (2.196) is not physically relevant, and a regularization

scheme that suppresses the contributions of high frequencies can give a physically

meaningful result (Belinfante, 1987).

One regularization scheme is to replace eqn (2.196) by

Ļlmn

exp ā’Ī±Ļlmn Clmn

2

E0 (āz) = . (2.197)

2

l,m,n

This sum is well behaved for any Ī± > 0, and approaches the original divergent ex-

pression as Ī± ā’ 0. The energy in a cubical box with sides L is E0 (L) and the ratio

of the volumes is L2 āz/L3 = āz/L, so the diļ¬erence between the zero-point energy

contained in the planar box and the zero-point energy contained in the same volume

in the larger box is

āz

U (āz) = E0 (āz) ā’ E0 (L) . (2.198)

L

This is just the work done in bringing one of the faces of the cube from the original

distance L to the ļ¬nal distance āz.

Quantization of cavity modes

The regularized sum could be evaluated numerically, but it is more instructive to

exploit the large size of L. In the limit of large L, the sums over l and m in E0 (āz)

and over all three indices in E0 (L) can be replaced by integrals over k-space according

to the rule (2.169). After a rather lengthy calculation (Milonni and Shih, 1992) one

ļ¬nds

Ļ 2 c L2

U (āz) = ā’ ; (2.199)

720 āz 3

consequently, the force attracting the two plates is

Ļ 2 c L2

dU

F =ā’ =ā’ . (2.200)

240 āz 4

d (āz)

For numerical estimates it is useful to restate this as

2

0.13 L [cm]

F [ĀµN] = ā’ . (2.201)

4

āz [Āµm]

For plates with area 1 cm2 separated by 1 Āµm, the magnitude of the force is 0.13 ĀµN.

This is a very small force; indeed, it is approximately equal to the force exerted by the

proton on the electron in the ļ¬rst Bohr orbit of a hydrogen atom.

The Casimir force between parallel plates would be extremely hard to measure,

due to the diļ¬culty of aligning parallel plates separated by 1 Āµm. Recent experiments

have used a diļ¬erent conļ¬guration consisting of a conducting sphere of radius R at a

distance d from a conducting plate (Lamoreaux, 1997; Mohideen and Roy, 1998). For

perfect conductors, a similar calculation yields the force

Ļ3 c R

(d) = ā’

(0)

F (2.202)

360 d3

in the limit R d. When corrections for ļ¬nite conductivity, surface roughness, and

nonzero temperature are included there is good agreement between theory and exper-

iment.

The calculation of the Casimir force sketched above is based on the diļ¬erence be-

tween the zero-point energies of two cavities, and it provides good agreement between

theory and experiment. This might be interpreted as providing evidence for the real-

ity of zero-point energy, except for two diļ¬culties. The ļ¬rst is the general argument

in Section 2.2 showing that it is always permissible to use the normal-ordered form

(2.89) for the Hamiltonian. With this choice, there is no zero-point energy for either

cavity; and our successful explanation evaporates. The second, and more important,

diļ¬culty is that the forces predicted by eqns (2.200) and (2.202) are independent of

the electronic charge. There is clearly something wrong with this, since all dynamical

eļ¬ects depend on the interaction of charged particles with the electromagnetic ļ¬eld.

It has been shown that the second feature is an artifact of the assumption that the

plates are perfect conductors (Jaļ¬e, 2005). A less idealized calculation yields a Casimir

force that properly vanishes in the limit of zero electronic charge. Thus the agreement

between the theoretical prediction (2.202) and experiment cannot be interpreted as

evidence for the physical reality of zero-point energy. We emphasize that this does

Exercises

not mean that vacuum ļ¬‚uctuations are not real, since other experimentsā”such as the

partition noise at beam splitters discussed in Section 8.4.2ā”do provide evidence for

their eļ¬ects.

Our freedom to use the normal-ordered form of the Hamiltonian implies that it

must be possible to derive the Casimir force without appealing to the zero point

energy. An approach that does this is based on the van der Waals coupling between

atoms in diļ¬erent walls. The van der Waals potential can be derived by considering the

coupling between the ļ¬‚uctuating dipoles of two atoms. This produces a time-averaged

perturbation proportional to (p1 Ā· p2 ) /r3 , where r is the distance between the atoms,

and p1 and p2 are the electric dipole operators. This potential comes from the static

Coulomb interactions between the charged particles comprising the atoms; it does not

involve the radiative modes that contribute to the zero point energy in symmetrical

ordering. The random ļ¬‚uctuations in the dipole moments p1 and p2 produce no ļ¬rst-

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

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

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

Ī»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 ā’ 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 justiļ¬ed, since the presence of a third atom

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

the entire Casimir force is obtained.

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

the two choices aā a or aā a + aaā /2 made in deļ¬ning the electromagnetic Hamil-

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

the Casimir force between the platesā”is the same for both explanations. The diļ¬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 ļ¬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 satisļ¬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)ā“(2.17).

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

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

(3) Use the normalization condition

d3 rE ks (r) Ā· 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 ļ¬elds satisfy Maxwellā™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 deļ¬nition (2.73) of the vacuum state to

verify eqn (2.78).

2.6 The second-order coherence matrix

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

p

ā

operators Np = ap ap are simultaneously measurable.

(2) Consider the operator

1 1 1

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

(|1y

Ļ=

2 2 4

where |1s = aā |0 .

ks

(a) Show that Ļ 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 signiļ¬cance of S0 ?

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

that

12

S 0 ā’ S1 ā’ S2 ā’ S3 ,

2 2 2

det J =

4

and thereby establish the condition (2.160).

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

S3 = sin Īø cos Ļ. Find the locations on the PoincarĀ“ 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 Ć— 2 matrix Ļss = 1ks |Ļ| 1ks .

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

Js s .

2.9 The Casimir force

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

2

c L

dkx dky eā’Ī±kā„ kā„

2

U (āz) =

2 Ļ

2ā

L 1/2

dkx dky eā’Ī±(kā„ +kzn ) kā„ + kzn

2 2

2 2

+c

Ļ n=1

3

āz L 2

dkx dky dkz eā’Ī±k k ,

ā’ c

L Ļ

2

2 2 2

where kā„ = kx + ky , k = kā„ + kz , and kzn = nĻ/āz.

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 Earthā™s gravity in this problem.)

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

of the spacing d between the surface of the sphere and the ļ¬‚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 ļ¬‚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 Āµm ,

0.1 Āµm d 1.0 Āµm ,

k = 0.02 N/m .

3

Field quantization

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

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

and the Casimir eļ¬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 ļ¬‚aw in the cavity-based model. The quantized radiation oscillators are

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

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

ply allowing for ļ¬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 ļ¬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 ļ¬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 ļ¬eld in

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

topics: the angular momentum of light, a description of quantum ļ¬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 ļ¬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.

Ā¼ Field quantization

3.1.1 Local commutation relations

In Chapter 2 we concentrated on the operators (qĪŗ , pĪŗ ) 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 ļ¬eld

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

independent, SchrĀØdinger-picture operators E (r) and A (r) with the commutation

o

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

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

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

commutation relations, [pĪŗ , pĪ» ] = [qĪŗ , qĪ» ] = 0, produce

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

On the other hand, [qĪŗ , pĪ» ] = i Ī“ĪŗĪ» , so the commutator between the electric ļ¬eld and

the vector potential is

1

[Ai (r) , ā’Ej (r )] = [qĪŗ , pĪ» ] EĪŗi (r) EĪ»j (r )

0 Īŗ Ī»

i

EĪŗi (r) EĪŗj (r ) .

= (3.2)

0 Īŗ

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

that

i

[Ai (r) , ā’Ej (r )] = āā„ (r ā’ r ) . (3.3)

ij

0

The resemblance between this result and the canonical commutation relation, [qĪŗ , pĪ» ] =

i Ī“ĪŗĪ» , for the mode operators suggests the identiļ¬cation of A (r) and ā’E (r) as the

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

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

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

ijl āl Ī“ (r ā’ r ),

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

0

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

implied by the nonvanishing commutators between electric and magnetic ļ¬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 ļ¬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 ļ¬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 ā’ā2 . We have, therefore, established the quite remarkable result that

Ā½

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 inļ¬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

cavityā”with its dependence on macroscopic boundary conditionsā”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 deļ¬ned by

d3 k

d3 reā’ikĀ·r F (r) ,

ikĀ·r

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

(2Ļ)

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

Fourier transforms satisfy Fā (k) = F (ā’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

āā„ (k) (2Ļ) Ī“ k + k ,

3

[Ai (k) , ā’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)ā“(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(ā’) (r) , A (r) = A(+) (r) + A(ā’) (r) , (3.8)

where

aĪŗ E Īŗ (r) = A(ā’)ā (r)

A(+) (r) = (3.9)

2 0 ĻĪŗ

Īŗ

Ā¾ Field quantization

and

ĻĪŗ

aĪŗ E Īŗ (r) = E(ā’)ā (r) .

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

20

Īŗ

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

part and F (ā’) 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 (ā’) is a sum of creation

operators. These properties are expressed by

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

In view of the deļ¬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 diļ¬culty is that the photonā”i.e. ā˜a quantum of excitation of the electromagnetic

ļ¬eldā™ā”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 (Ā±) (r) , G(Ā±) (r ) = 0 , (3.12)

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

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

(3.3) yields

i

āā„ (r ā’ r ) .

(+) (ā’)

Ai (r) , ā’Ej (r ) = (3.13)

2 0 ij

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

(Ā±)

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

aĪŗ kĪŗ E Īŗ (r) ,

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

2 0 ĻĪŗ

Īŗ

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

1/2

E(+) (r) = ic ā’ā2 A(+) (r) . (3.15)

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

ā’1/2

āā„ (r ā’ r ) ,

(+) (ā’)

ā’ā2

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

ij

2 0c

c 1/2

āā„ (r ā’ r ) .

(+) (ā’)

ā’ā2

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

ij

20

ā’1/2

1/2

In the context of free space, the unfamiliar operators ā’ā2 and ā’ā2 are

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

Āæ

Field quantization in the vacuum

ā’ā2 exp (ik Ā· r) = k 2 exp (ik Ā· r) allows us to deļ¬ne the action of f ā’ā2 on a plane

wave by f ā’ā2 eikĀ·r ā” f k 2 eikĀ·r . This result in turn implies that f ā’ā2 acts on

a general function Ļ• (r) according to the rule

d3 k d3 k

f ā’ā2 Ļ• (r) ā” 3 Ļ• (k) f ā’ā

2

eikĀ·r = k 2 eikĀ·r .

3 Ļ• (k) f (3.18)

(2Ļ) (2Ļ)

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

f ā’ā2 Ļ• (r) = d3 r r f ā’ā2 r Ļ• (r ) , (3.19)

where

d3 k

k 2 eikĀ·(rā’r )

r f ā’ā2 r = 3f (3.20)

(2Ļ)

is the integral kernel deļ¬ning f ā’ā2 as an operator in r-space. Despite its abstract

appearance, this deļ¬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 ā’ā2 are also hermitian; so one gets a useful integration-by-parts

identity

d3 rĻ ā— (r) f ā’ā2 Ļ• (r) = d3 r f ā’ā2 Ļ ā— (r) Ļ• (r) , (3.21)

without any intermediate steps involving Fourier transforms.

The equations (3.8), (3.11)ā“(3.13), (3.15), and (3.16) were all derived by using the

expansions of the ļ¬eld operators in cavity modes, but once again the ļ¬nal forms are

independent of the size and shape of the cavity. Consequently, these results are valid

in free space.

B Reciprocal space

The rather strange looking result (3.16) becomes more understandable if we note that

the decomposition (3.8) into positive- and negative-frequency parts applies equally

well in reciprocal space, so that A (k) = A(+) (k) + A(ā’) (k). The Fourier transforms

of eqns (3.12) and (3.16) with respect to r and r yield respectively

(Ā±) (Ā±)

Ai (k) , Aj (k ) = 0 (3.22)

and

āā„ (k)

(+) (ā’) ij

(2Ļ)3 Ī“ k ā’ k .

Ai (k) , Aj (ā’k ) = (3.23)

2 0c k

This reciprocal-space commutation relation does not involve any strange operators, like

ā’1/2

ā’ā2 , but it is still rather complicated. Simpliļ¬cation can be achieved by noting

Field quantization

that the circular polarization unit vectors eks ā”see Appendix B.3.2ā”are eigenvectors

of āā„ (k) with eigenvalue unity:

ij

āā„ (k) (eks )j = (eks )i . (3.24)

ij

By forming the inner product of both sides of eqns (3.22) and (3.23) with eā— and

ks

ek s and remembering Fā (k) = F (ā’k), one ļ¬nds

[as (k) , as (k )] = aā (k) , aā (k ) = 0 (3.25)

s s

and

as (k) , aā (k ) = Ī“ss (2Ļ) Ī“ k ā’ k ,

3

(3.26)

s

where

2 0 Ļk

eā— Ā· A(+) (k)

as (k) = (3.27)

ks

and Ļk = ck. The operators as (k), combined with the Fourier transform relation (3.5),

provide a replacement for the cavity-mode expansions (3.9) and (3.10):

d3 k

(+)

as (k) eks eikĀ·r ,

A (r) = (3.28)

3 2 0 Ļk

(2Ļ) s

d3 k Ļk

E(+) (r) = i as (k) eks eikĀ·r . (3.29)

3 20

(2Ļ) s

The number operator

d3 k

aā (k) as (k)

N= (3.30)

s

3

(2Ļ) s

satisļ¬es

N, aā (k) = aā (k) , [N, as (k)] = ā’as (k) , (3.31)

s s

and the vacuum state is deļ¬ned by as (k) |0 = 0, so it seems that aā (k) and as (k) can

s

be regarded as creation and annihilation operators that replace the cavity operators

aĪŗ and aā . However, the singular commutation relation (3.26) exacts a price. For

Īŗ

example, the one-photon state |1ks = aā (k) |0 is an improper state vector satisfying

s

the continuum normalization conditions

3

1k s |1ks = Ī“ss (2Ļ) Ī“ k ā’ k . (3.32)

Thus a properly normalized one-photon state is a wave packet state

d3 k

Ī¦s (k) aā (k) |0 ,

|Ī¦ = (3.33)

s

3

(2Ļ) s

where the c-number function Ī¦s (k) is normalized by

Field quantization in the vacuum

d3 k 2

|Ī¦s (k)| = 1 . (3.34)

3

(2Ļ) s

The Fock space HF consists of all linear combinations of number states,

d3 k1 d3 kn

Ī¦s1 Ā·Ā·Ā·sn (k1 , . . . , kn ) aā 1 (k1 ) Ā· Ā· Ā· aā n (kn ) |0 ,

|Ī¦ = Ā·Ā·Ā· Ā·Ā·Ā· s s

3 3

(2Ļ) (2Ļ) s1 sn

(3.35)

where

d3 k1 d3 kn

|Ī¦s1 Ā·Ā·Ā·sn (k1 , . . . , kn )|2 < ā

Ā·Ā·Ā· Ā·Ā·Ā· (3.36)

3 3

(2Ļ) (2Ļ) s1 sn

and n = 0, 1, . . ..

3.1.3 Energy, momentum, and angular momentum

A The Hamiltonian

The expression (2.105) for the ļ¬eld energy in a cavity can be converted to a form

suitable for generalization to free space by ļ¬rst inverting eqn (3.10) to get

20

aĪŗ = ā’i d3 rE Īŗ (r) Ā· E(+) (r) . (3.37)

ĻĪŗ V

The next step is to substitute this expression for aĪŗ into eqn (2.105) and carry out the

sum over Īŗ by means of the completeness relation (2.38); this calculation leads to

Ļ Īŗ aā aĪŗ

Hem = Īŗ

Īŗ

(r) āā„ (r ā’ r ) Ej

(ā’) (+)

d3 r d3 r Ei

=2 (r ) . (3.38)

0 ij

V V

Since the free-ļ¬eld operator E(+) (r ) is transverse, the inļ¬nite volume limit is

d3 rE(ā’) (r) Ā· E(+) (r) .

Hem = 2 (3.39)

0

This can also be expressed as

d3 rA(ā’) (r) Ā· ā’ā2 A(+) (r) ,

Hem = 2 0 c2 (3.40)

by using eqn (3.15). A more intuitively appealing form is obtained by using the plane-

wave expansion (3.29) for E(Ā±) to get

d3 k

aā (k) as (k) .

Hem = Ļk (3.41)

s

3

(2Ļ) s

Field quantization

B The linear momentum

The cavity model does not provide any expressions for the linear momentum and

the angular momentum, so we need independent arguments for them. The reason for

the absence of these operators is the presence of the cavity walls. From a mechanical

point of view, the linear momentum and the angular momentum of the ļ¬eld are not

conserved because of the immovable cavity. Alternatively, we note that one of the

fundamental features of quantum theory is the identiļ¬cation of the linear momentum

and the angular momentum operators with the generators for spatial translations and

rotations respectively (Bransden and Joachain, 1989, Secs 5.9 and 6.2). This means

that the mechanical conservation laws for linear and angular momentum are equivalent

to invariance under spatial translations and rotations respectively. The location and

orientation of the cavity in space spoils both invariances.

Since the cavity model fails to provide any guidance, we once again call on the

correspondence principle by quoting the classical expression for the linear momentum

(Jackson, 1999, Sec. 6.7):

P= 0E ā„ Ć—B

d3 r

0E Ć— (ā Ć— A) .

d3 r

= (3.42)

The vector identity FĆ— (ā Ć— G) = Fj āGj ā’ (F Ā· ā) G combined with an integration

by parts and the transverse nature of E (r) provides the more useful expression

P= d3 rEj (r) āAj (r) . (3.43)

0

The initial step in constructing the corresponding SchrĀØdinger-picture operator is to

o

replace the classical ļ¬elds according to

A (r) ā’ A (r) = A(+) (r) + A(ā’) (r) , (3.44)

E (r) ā’ E (r) = E(+) (r) + E(ā’) (r) . (3.45)

The momentum operator P is then the sum of four terms, P = P(+,+) + P(ā’,ā’) +

P(ā’,+) + P(+,ā’) , where

(Ļ) (Ļ„ )

d3 rEj āAj for Ļ, Ļ„ = Ā± .

P(Ļ,Ļ„ ) = (3.46)

0

Each of these terms is evaluated by using the plane-wave expansions (3.28) and (3.29),

together with the orthogonality relation, eā— Ā·eks = Ī“ss , and the reļ¬‚ection propertyā”

ks

see eqn (B.73)ā”eā’k,s = eā— , (s = Ā±) for the circular polarization basis. The ļ¬rst result

ks

is P(+,+) = P(ā’,ā’)ā = 0; consequently, only the cross terms survive to give

d3 k k

aā (k) as (k) + as (k) aā (k) .

P= (3.47)

s s

3 2

(2Ļ) s

Field quantization in the vacuum

This is analogous to the symmetrical ordering (2.106) for the Hamiltonian in the cavity

problem, so our previous experience suggests replacing the symmetrical ordering by

normal ordering, i.e.

d3 k

aā (k) as (k) .

P= k (3.48)

s

3

(2Ļ) s

From this expression and eqn (3.41), it is easy to see that [P, Hem ] = 0 and [Pi , Pj ] = 0.

Any observable commuting with the Hamiltonian is called a constant of the motion,

so the total momentum is a constant of the motion and the individual components Pi

are simultaneously measurable.

By using the inverse Fourier transform,

2 0 Ļk

eā— Ā· d3 reā’ikĀ·r A(+) (r) ,

as (k) = (3.49)

ks

which is the free-space replacement for eqn (3.37), or proceeding directly from eqn

(3.46), one ļ¬nds the equivalent position-space representation

(ā’) (+)

(r) āAj

d3 rEj

P=2 (r) . (3.50)

0

C The angular momentum

Finally we turn to the classical expression for the angular momentum (Jackson, 1999,

Sec. 12.10):

J= d3 r r Ć— [ 0 E (r, t) Ć— B (r, t)] . (3.51)

Combining B = ā Ć— A with the identity F Ć— (ā Ć— G) = Fj āGj ā’ (F Ā· ā) G allows

this to be written in the form J = L + S, where

L= d3 r Ej (r Ć— ā) Aj (3.52)

0

and

S= d3 r E Ć— A . (3.53)

0

Once again, the initial guess for the corresponding quantum operators is given by

applying the rules (3.44) and (3.45), so the total angular momentum operator is

J = L + S, (3.54)

where the operators L and S are deļ¬ned by quantizing the classical expressions L and

S respectively.

Field quantization

The application of the method used for the linear momentum to eqn (3.52) is com-

plicated by the explicit r-term, but after some eļ¬ort one ļ¬nds the rather cumbersome

expression (Simmons and Guttmann, 1970)

d3 k

i ā‚

Miā (k) k Ć—

L=ā’ Mi (k) ā’ HC

3

2 ā‚k

(2Ļ)

d3 k ā‚

ā

= ā’i (k) k Ć—

3 Mi Mi (k) , (3.55)

ā‚k

(2Ļ)

where

M (k) = as (k) eks . (3.56)

s

In this case a substantial simpliļ¬cation results from translating the reciprocal-space

representation back into position space to get

2i 0 (ā’) (+)

(r) r Ć— ā Aj

d3 rEj

L= (r) . (3.57)

i

A straightforward calculation using eqn (3.39) shows that L is also a constant of the

motion, i.e. [L, Hem ] = 0. However, the components of L are not mutually commuta-

tive, so they cannot be measured simultaneously.

The quantization of eqn (3.53) goes much more smoothly, and leads to the normal-

ordered expression

d3 k

saā (k) as (k)

S= 3k s

(2Ļ) s

3

dk

aā (k) a+ (k) ā’ aā (k) aā’ (k) ,

= 3k (3.58)

ā’

+

(2Ļ)

where k = k/k is the unit vector along k. Another use of eqn (3.49) yields the equiv-

alent position-space form

d3 rE(ā’) Ć— A(+) .

S=2 (3.59)

0

The expression (3.54) for the total angular momentum operator looks like the

decomposition into orbital and spin parts familiar from quantum mechanics, but this

resemblance is misleading. For the electromagnetic ļ¬eld, the interpretation of eqn

(3.54) poses a subtle problem which we will take up in Section 3.4.

D The helicity operator

It is easy to show that S commutes with P and with Hem , and further that

[Si , Sj ] = 0 . (3.60)

Thus S, P, and Hem are simultaneously measurable, and there are simultaneous eigen-

vectors for them. In the simplest case of the improper one-photon state |1ks =

Field quantization in the vacuum

aā (k) |0 , one ļ¬nds: Hem |1ks = Ļk |1ks , P |1ks = k |1ks , k Ć— S |1ks = 0, and

s

k Ā· S |1ks = s |1ks . Thus |1ks is an eigenvector of the longitudinal component k Ā· S

with eigenvalue s and an eigenvector of the transverse components k Ć— S with eigen-

value 0. For the circular polarization basis, the index s represents the helicity, so S is

called the helicity operator.

E Evidence for helicity and orbital angular momentum

Despite the conceptual diļ¬culties mentioned in Section 3.1.3-C, it is possible to devise

experiments in which certain components of the helicity S and the orbital angular mo-

mentum L are separately observed. The ļ¬rst measurement of this kind (Beth, 1936)

was carried out using an experimental arrangement consisting of a horizontal wave

plate suspended at its center by a torsion ļ¬ber, so that the plate is free to undergo

twisting motions around the vertical axis. In a simpliļ¬ed version of this experiment,

a vertically-directed, linearly-polarized beam of light is allowed to pass through a

quarter-wave plate, which transforms it into a circularly-polarized beam of light (Born

and Wolf, 1980, Sec. 14.4.2). Since the experimental setup is symmetrical under ro-

tations around the vertical axis (the z-axis), the z-component of the total angular

momentum will be conserved.

We will use a one-photon state

Ī¾s aā (k) |0 ,

|Ļ = Ī¾s |1ks = (3.61)

s

s s

with k = ku3 directed along the z-axis, as a simple model of an incident light beam

of arbitrary polarization. A straightforward calculation using eqn (3.55) for Lz shows

that Lz |1ks = 0; consequently, Lz |Ļ = 0 for any choice of the coeļ¬cients Ī¾s . In

other words, states of this kind have no z-component of orbital angular momentum.

The particular choice

1 1

= ā [|1k+ + |1kā’ ] = ā aā (k) + aā (k) |0

|Ļ (3.62)

ā’

ńņš. 3 |