ńņš. 5 |

(+)

E in (r, 0) ws (k) eks eikĀ·r ,

=i (3.197)

3 20

(2Ļ) s

so the wave packets are given by

20 ā—

d3 reā’ikĀ·r E in (r, 0) .

(+)

ws (k) = ā’i eĀ· (3.198)

Ļk ks

Photon localizabilityā—

3.6

3.6.1 Is there a photon position operator?

The use of the term photon to mean ā˜quantum of excitation of the electromagnetic ļ¬eldā™

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

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

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

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

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

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

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

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

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

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

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

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

B. The identity

|Ļ = d3 r |r r |Ļ (3.199)

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

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

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

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

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

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

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

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

(Newton and Wigner, 1949).

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

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

the position operator r has two essential properties.

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

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

Photon localizabilityā— Ā½Ā¼

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

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

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

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

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

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

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

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

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

theoryā”Maxwellā™s equationsā”has no particle concept. The second is that photons are

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

notions connected to the SchrĀØdinger equation need not apply.

o

3.6.2 Are there local number operators?

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

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

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

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

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

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

observables should satisfy

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

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

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

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

to get

20 ā’1/2 (+)

d3 rE(ā’) (r) Ā· ā’ā2

N= E (r) . (3.201)

c

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

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

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

photons in a classical ļ¬eld.

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

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

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

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

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

1/2 1/4 1/4

some states. This objection can be met by using ā’ā2 = ā’ā2 ā’ā2 and

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

form

N = d3 rMā (r) Ā· M (r) , (3.202)

where

2 ā’1/4

0

M (r) = ā’i ā’ā2 E(+) (r) . (3.203)

c

Ā½Ā¼ Field quantization

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

ator deļ¬ned by

d3 rMā (r) Ā· M (r)

N (V ) = (3.204)

V

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

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

d3 k

es (k) as (k) eikĀ·r ,

M (r) = (3.205)

3

(2Ļ) s

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

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

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

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

detection operator satisļ¬es

ā

Mi (r) , Mj (r ) = āā„ (r ā’ r ) , [Mi (r) , Mj (r )] = 0 . (3.206)

ij

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

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

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

d3 r Sij (r, r ) āā„ (r ā’ r ) ,

d3 r

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

ij

V V

where Sij (r, r ) = Miā (r) Mj (r ) ā’ Mj (r ) Mi (r). The deļ¬nition of the transverse

ā

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

relation (3.18) to get the equivalent expression,

1 1

āā„ (r ā’ r ) = Ī“ij Ī“ (r ā’ r ) + āi āj . (3.208)

|r ā’ r |

ij

4Ļ

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

so

1 1

d3 r Sij (r, r ) āi āj

d3 r

[N (V ) , N (V )] = . (3.209)

|r ā’ r |

4Ļ

V V

A straightforward estimate shows that [N (V ) , N (V )] ā¼ Rā’3 . Thus the commutator

between these proposed local number operators does not vanish for nonoverlapping

volumes; indeed, it does not even decay very rapidly as the separation between the

volumes increases. This counterintuitive behavior is caused by the nonlocal ļ¬eld com-

mutator (3.16) which is a consequence of the transverse nature of the electromagnetic

ļ¬eld.

The alternative deļ¬nition (Deutsch and Garrison, 1991a),

20

d3 rE(ā’) (r) Ā· E(+) (r) ,

G (V ) = (3.210)

Ļ0 V

of a local number operator is suggested by the Glauber theory of photon detection,

which is discussed in Section 9.1.2. Rather than anticipating later results we will obtain

Ā½Ā¼

Exercises

eqn (3.210) by a simple plausibility argument. The representation (3.39) for the ļ¬eld

Hamiltonian suggests interpreting 2 0 E(ā’) Ā· E(+) as the energy density operator. For a

monochromatic ļ¬eld state this in turn suggests that 2 0 E(ā’) Ā· E(+) / Ļ0 be interpreted

as the photon density operator. The expression (3.210) is an immediate consequence

of these assumptions. The integrand in this equation is clearly positive deļ¬nite, but

nonlocal eļ¬ects show up here as well.

The failure of several plausible candidates for a local number operator strongly

suggests that there is no such object. If this conclusion is supported by future research,

it would mean that photons are nonlocalizable in a very fundamental way.

3.7 Exercises

3.1 The ļ¬eld commutator

Verify the expansions (2.101) and (2.103), and use them to derive eqns (3.1) and (3.3).

3.2 Uncertainty relations for E and B

(1) Derive eqn (3.4) from eqn (3.3).

(2) Consider smooth distributions of classical polarization P (r) and magnetization

M (r) which vanish outside ļ¬nite volumes VP and VM respectively, as in Section

2.5. The interaction energies are

WE = ā’ d3 rP (r) Ā· E (r) , WB = ā’ d3 rM (r) Ā· B (r) .

Show that

i

[WB , WE ] = ā’ d3 rP (r) Ā· M (r) .

0

(3) What assumption about the volumes VP and VM will guarantee that WB and WE

are simultaneously measurable?

(4) Use the standard argument from quantum mechanics (Bransden and Joachain,

1989, Sec. 5.4) to show that WB and WE satisfy an uncertainty relation

āWB āWE K,

and evaluate the constant K.

3.3 Electromagnetic Hamiltonian

Carry out the derivation of eqns (3.37)ā“(3.41).

3.4 Electromagnetic momentum

Fill in the steps leading from the classical expression (3.42) to the quantum form (3.48)

for the electromagnetic momentum operator.

Milonniā™s quantization schemeā—

3.5

Fill in the details required to go from eqn (3.159) to eqn (3.164).

Ā½Ā½Ā¼ Field quantization

Electromagnetic angular momentumā—

3.6

Carry out the calculations needed to derive eqns (3.172)ā“(3.178).

Wave packet quantizationā—

3.7

(1) Derive eqns (3.192), (3.193), and (3.195).

(2) Derive the expression for 1w |1v , where w and v are wave packets in Ī“em .

4

Interaction of light with matter

In the previous chapters we have dealt with the free electromagnetic ļ¬eld, undisturbed

by the presence of charges. This is an important part of the story, but all experiments

involve the interaction of light with matter containing ļ¬nite amounts of quantized

charge, e.g. electrons in atoms or conduction electrons in semiconductors. It is there-

fore time to construct a uniļ¬ed picture in which both light and matter are treated by

quantum theory. We begin in Section 4.1 with a brief review of semiclassical electrody-

namics, the standard quantum theory of nonrelativistic charged particles interacting

with a classical electromagnetic ļ¬eld. The next step is to treat both charges and ļ¬elds

by quantum theory. For this purpose, we need a Hilbert space describing both the

charged particles and the quantized electromagnetic ļ¬eld. The necessary machinery is

constructed in Section 4.2. We present the Heisenberg-picture description of the full

theory in Sections 4.3ā“4.7. In Sections 4.8 and 4.9, the interaction picture is introduced

and applied to atomā“photon coupling.

4.1 Semiclassical electrodynamics

In order to have something reasonably concrete to discuss, we will consider a system of

N point charges. The pure states are customarily described by N -body wave functions,

Ļ (r1 , . . . , rN ), in conļ¬guration space. The position and momentum operators rn and

pn for the nth particle are respectively deļ¬ned by

rn Ļ (r1 , . . . , rN ) = rn Ļ (r1 , . . . , rN ) ,

(4.1)

ā‚

pn Ļ (r1 , . . . , rN ) = ā’i Ļ (r1 , . . . , rN ) .

ā‚rn

The Hilbert space, Hchg , for the charges consists of the normalizable N -body wave

functions, i.e.

d3 rN |Ļ (r1 , . . . , rN )|2 < ā .

d3 r1 Ā· Ā· Ā· (4.2)

In all applications some of the particles will be fermions, e.g. electrons, and others will

be bosons, so the wave functions must be antisymmetrized or symmetrized accordingly,

as explained in Section 6.5.1.

In the semiclassical approximation the Hamiltonian for a system of charged parti-

cles coupled to a classical ļ¬eld is constructed by combining the correspondence prin-

ciple with the idea of minimal coupling explained in Appendix C.6. The result is

N N

2

(pn ā’ qn A (rn , t))

Hsc = + qn Ļ• (rn , t) , (4.3)

2Mn

n=1 n=1

Ā½Ā½Ā¾ Interaction of light with matter

where A and Ļ• are respectively the (c-number) vector and scalar potentials, and qn

and Mn are respectively the charge and mass of the nth particle. In this formulation

there are two forms of momentum: the canonical momentum,

ā‚

pn,can = pn = ā’i , (4.4)

ā‚rn

and the kinetic momentum,

pn,kin = pn ā’ qn A (rn , t) . (4.5)

The canonical momentum is the generator of spatial translations, while the classical

momentum M v is the correspondence-principle limit of the kinetic momentum.

It is worthwhile to pause for a moment to consider where this argument has led

us. The classical ļ¬elds A (r, t) and Ļ• (r, t) are by deļ¬nition c-number functions of

position r in space, but (4.3) requires that they be evaluated at the position of a

charged particle, which is described by the operator rn . What, then, is the meaning of

A (rn , t)? To get a concrete feeling for this question, let us recall that the classical ļ¬eld

can be expanded in plane waves exp (ik Ā· r ā’ iĻk t). The operator exp (ik Ā· rn ) arising

from the replacement of rn by rn is deļ¬ned by the rule

eikĀ·rn Ļ (r1 , . . . , rN ) = eikĀ·rn Ļ (r1 , . . . , rN ) , (4.6)

where Ļ (r1 , . . . , rN ) is any position-space wave function for the charged particles.

In this way A (rn , t) becomes an operator acting on the state vector of the charged

particles. This implies, for example, that A (rn , t) does not commute with pn , but

instead satisļ¬es

ā‚Ai

[Ai (rn , t) , pnj ] = i (rn , t) . (4.7)

ā‚rj

The scalar potential Ļ• (rn , t) is interpreted in the same way.

The standard wave function description of the charged particles is useful for deriv-

ing the semiclassical Hamiltonian, but it is not particularly convenient for the applica-

tions to follow. In general it is better to use Diracā™s presentation of quantum theory, in

which the state is represented by a ket vector |Ļ . For the system of charged particles

the two versions are related by

Ļ (r1 , . . . , rN ) = r1 , . . . , rN |Ļ , (4.8)

where |r1 , . . . , rN is a simultaneous eigenket of the position operators rn , i.e.

rn |r1 , . . . , rN = rn |r1 , . . . , rN , n = 1, . . . , N . (4.9)

In this formulation the wave function Ļ (r1 , . . . , rN ) simply gives the components of

the vector |Ļ with respect to the basis provided by the eigenvectors |r1 , . . . , rN . Any

other set of basis vectors for Hchg would do equally well.

Ā½Ā½Āæ

Quantum electrodynamics

4.2 Quantum electrodynamics

In semiclassical electrodynamics the state of the physical system is completely de-

scribed by a many-body wave function belonging to the Hilbert space Hchg deļ¬ned by

eqn (4.2), but this description is not adequate when the electromagnetic ļ¬eld is also

treated by quantum theory. In Section 4.2.1 we show how to combine the charged-

particle space Hchg with the Fock space HF , deļ¬ned by eqn (3.35), to get the state

space, HQED , for the composite system of the charges and the quantized electromag-

netic ļ¬eld. In Section 4.2.2 we construct the Hamiltonian for the composite charge-ļ¬eld

system by appealing to the correspondence principle for the quantized electromagnetic

ļ¬eld.

4.2.1 The Hilbert space

In quantum mechanics, many-body wave functions are constructed from single-particle

wave functions by forming linear combinations of product wave functions. For example,

the two-particle wave functions for distinguishable particles A and B have the general

form

Ļ (rA , rB ) = C1 Ļ1 (rA ) Ļ1 (rB ) + C2 Ļ2 (rA ) Ļ2 (rB ) + Ā· Ā· Ā· . (4.10)

Since wave functions are meaningless for photons, it is not immediately clear how

this procedure can be applied to the radiation ļ¬eld. The way around this apparent

diļ¬culty begins with the reminder that the wave function for a particle, e.g. Ļ1 (rA ),

is a probability amplitude for the outcomes of measurements of position. In the stan-

dard approach to the quantum measurement problemā”reviewed in Appendix C.2ā”a

measurement of the position operator rA always results in one of the eigenvalues rA ,

and the particle is left in the corresponding eigenstate |rA . If the particle is initially

prepared in the state |Ļ1 A , then the wave function is simply the probability ampli-

tude for this outcome: Ļ1 (rA ) = rA |Ļ1 . The next step is to realize that the position

operators rA do not play a privileged role, even for particles. The components xA , yA ,

and zA of rA can be replaced by any set of commuting observables OA1 , OA2, , OA3

with the property that the common eigenvector, deļ¬ned by

OAn |OA1 , OA2 , OA3 = OAn |OA1 , OA2 , OA3 (n = 1, 2, 3) , (4.11)

is uniquely deļ¬ned (up to an overall phase). In other words, the observables OA1 , OA2 ,

OA3 can be measured simultaneously, and the system is left in a unique state after the

measurement.

With these ideas in mind, we can describe the composite system of N charges

and the electromagnetic ļ¬eld by relying directly on the Born interpretation and the

superposition principle. For the system of N charged particles described by Hchg , we

choose an observable Oā”more precisely, a set of commuting observablesā”with the

property that the eigenvalues Oq are nondegenerate and labeled by a discrete index q.

The result of a measurement of O is one of the eigenvalues Oq , and the system is left

in the corresponding eigenstate |Oq ā Hchg after the measurement. If the charges are

prepared in the state |Ļ ā Hchg , then the probability amplitude that a measurement

Ā½Ā½ Interaction of light with matter

of O results in the particular eigenvalue Oq is Oq |Ļ . Furthermore, the eigenvectors

|Oq provide a basis for Hchg ; consequently, |Ļ can be expressed as

|Ļ = |Oq Oq |Ļ . (4.12)

q

In other words, the state |Ļ is completely determined by the set of probability am-

plitudes { Oq |Ļ } for all possible outcomes of a measurement of O.

The same kind of argument works for the electromagnetic ļ¬eld. We use box quanti-

zation to get a set of discrete mode labels k,s and consider the set of number operators

{Nks }. A simultaneous measurement of all the number operators yields a set of oc-

cupation numbers n = {nks } and leaves the ļ¬eld in the number state |n . If the ļ¬eld

is prepared in the state |Ī¦ ā HF , then the probability amplitude for this outcome

is n |Ī¦ . Since the number states form a basis for HF , the state vector |Ī¦ can be

expressed as

|Ī¦ = |n n |Ī¦ ; (4.13)

n

consequently, |Ī¦ is completely speciļ¬ed by the set of probability amplitudes { n |Ī¦ }

for all outcomes of the measurements of the mode number operators. We have used

the number operators for convenience in this discussion, but it should be understood

that these observables also do not hold a privileged position. Any family of compatible

observables such that their simultaneous measurement leaves the ļ¬eld in a unique state

would do equally well.

The charged particles and the ļ¬eld are kinematically independent, so the operators

O and Nks commute. In experimental terms, this means that simultaneous measure-

ments of the observables O and Nks are possible. If the charges and the ļ¬eld are

prepared in the states |Ļ and |Ī¦ respectively, then the probability for the joint out-

come (Oq , n) is the product of the individual probabilities. Since overall phase factors

are irrelevant in quantum theory, we may assume that the probability amplitude for

the joint outcomeā”which we denote by Oq , n |Ļ, Ī¦ ā”is given by the product of the

individual amplitudes:

Oq , n |Ļ, Ī¦ = Oq |Ļ n |Ī¦ . (4.14)

According to the Born interpretation, the set of probability amplitudes deļ¬ned by

letting Oq and n range over all possible values deļ¬nes a state of the composite system,

denoted by |Ļ, Ī¦ . The vector corresponding to this state is called a product vector,

and it is usually written as

|Ļ, Ī¦ = |Ļ |Ī¦ , (4.15)

where the notation is intended to remind us of the familiar product wave functions in

eqn (4.10).

The product vectors do not provide a complete description of the composite system,

since the full set of states must satisfy the superposition principle. This means that

we are required to give a physical interpretation for superpositions,

|ĪØ = C1 |Ļ1 , Ī¦1 + C2 |Ļ2 , Ī¦2 , (4.16)

Ā½Ā½

Quantum electrodynamics

of distinct product vectors. Once again the Born interpretation guides us to the follow-

ing statement: the superposition |ĪØ is the state deļ¬ned by the probability amplitudes

Oq , n |ĪØ = C1 Oq , n |Ļ1 , Ī¦1 + C2 Oq , n |Ļ2 , Ī¦2

= C1 Oq |Ļ1 n |Ī¦1 + C2 Oq |Ļ2 n |Ī¦2 . (4.17)

It is important to note that for product vectors like |Ļ |Ī¦ the subsystems are

each described by a unique state in the respective Hilbert space. The situation is

quite diļ¬erent for superpositions like |ĪØ ; it is impossible to associate a given state

with either of the subsystems. In particular, it is not possible to say whether the ļ¬eld

is described by |Ī¦1 or |Ī¦2 . This featureā”which is imposed by the superposition

principleā”is called entanglement, and its consequences will be extensively studied in

Chapter 6.

Combining this understanding of superposition with the completeness of the states

|Oq and |n F in their respective Hilbert spaces leads to the following deļ¬nition: the

state space, HQED , of the charge-ļ¬eld system consists of all superpositions

|ĪØ = ĪØqn |Oq |n . (4.18)

q n

This deļ¬nition guarantees the satisfaction of the superposition principle, but the Born

interpretation also requires a deļ¬nition of the inner product for states in HQED . To this

end, we ļ¬rst take eqn (4.14) as the deļ¬nition of the inner product of the vectors |Oq , n

and |Ļ, Ī¦ . Applying this deļ¬nition to the special choice |Ļ, Ī¦ = |Oq , n yields

Oq , n |Oq , n = Oq |Oq n |n = Ī“q ,q Ī“n ,n , (4.19)

and the bilinear nature of the inner product ļ¬nally produces the general deļ¬nition:

Ī¦ā— ĪØqn .

Ī¦ |ĪØ = (4.20)

qn

q n

The description of HQED in terms of superpositions of product vectors imposes a

similar structure for operators acting on HQED . An operator C that acts only on the

particle degrees of freedom, i.e. on Hchg , is deļ¬ned as an operator on HQED by

C |ĪØ = ĪØqn {C |Oq } |n , (4.21)

q n

and an operator acting only on the ļ¬eld degrees of freedom, e.g. aks , is extended to

HQED by

aks |ĪØ = ĪØqn |Oq {aks |n } . (4.22)

q n

Combining these deļ¬nitions gives the rule

Caks |ĪØ = ĪØqn {C |Oq } {aks |n } . (4.23)

q n

Ā½Ā½ Interaction of light with matter

A general operator Z acting on HQED can always be expressed as

Z= Cn Fn , (4.24)

n

where Cn acts on Hchg and Fn acts on HF .

The oļ¬cially approved mathematical language for this construction is that HQED

is the tensor product of Hchg and HF . The standard notation for this is

HQED = Hchg ā— HF , (4.25)

and the corresponding notation |Ļ ā—|Ī¦ is often used for the product vectors. Similarly

the operator product Caks is often written as C ā— aks .

4.2.2 The Hamiltonian

For the ļ¬nal step to the full quantum theory, we once more call on the correspon-

dence principle to justify replacing the classical ļ¬eld A (r, t) in eqn (4.3) by the time-

independent, SchrĀØdinger-picture quantum ļ¬eld A (r). The evaluation of A (r) at rn

o

is understood in the same way as for the classical ļ¬eld A (rn , t), e.g. by using the

plane-wave expansion (3.68) to get

eks aks eikĀ·rn + HC .

A (rn ) = (4.26)

2 0 Ļk V

ks

Thus A (rn ) is a hybrid operator that acts on the electromagnetic degrees of freedom

(HF ) through the creation and annihilation operators aā and aks and on the particle

ks

degrees of freedom (Hchg ) through the operators exp (Ā±ik Ā· rn ).

With this understanding we ļ¬rst use the identity

A (rn ) Ā· pn + pn Ā· A (rn ) = 2A (rn ) Ā· pn ā’ [Aj (rn ) , pnj ]

= 2A (rn ) Ā· pn + i ā Ā· A (rn ) , (4.27)

together with ā Ā· A = 0 and the identiļ¬cation of Ļ• as the instantaneous Coulomb

potential Ī¦, to evaluate the interaction terms in the radiation gauge. The total Hamil-

tonian is obtained by adding the zeroth-order Hamiltonian Hem + Hchg to get

N N

2

(pn ā’ qn A (rn ))

H = Hem + Hchg + + qn Ī¦ (rn ) . (4.28)

2Mn

n=1 n=1

Writing out the various terms leads to the expression

H = Hem + Hchg + Hint , (4.29)

1

: E2 : +Āµā’1 : B2 : ,

d3 r

Hem = (4.30)

0 0

2

Ā½Ā½

Quantum Maxwellā™s equations

N

p2 1 qn ql

n

Hchg = + , (4.31)

|rn ā’ rl |

2Mn 4Ļ 0

n=1 n=l

N N 2

2

qn qn : A (rn ) :

=ā’ A (rn ) Ā· pn +

Hint . (4.32)

Mn 2Mn

n=1 n=1

In this formulation, Hem is the Hamiltonian for the free (transverse) electromagnetic

ļ¬eld, and Hchg is the Hamiltonian for the charged particles, including their mutual

Coulomb interactions. The remaining term, Hint , describes the interaction between

the transverse (radiative) ļ¬eld and the charges. As in Section 2.2, we have replaced

the operators E2 , B2 , and A2 in Hem and Hint by their normal-ordered forms, in order

to eliminate divergent vacuum ļ¬‚uctuation terms. The Coulomb interactions between

the chargesā”say in an atomā”are typically much stronger than the interaction with

the transverse ļ¬eld modes, so Hint can often be treated as a weak perturbation.

4.3 Quantum Maxwellā™s equations

In Section 4.2 the interaction between the radiation ļ¬eld and charged particles was

described in the SchrĀØdinger picture, but some features are more easily understood

o

in the Heisenberg picture. Since the Hamiltonian has the same form in both pictures,

the Heisenberg equations of motion (3.89) can be worked out by using the equal-time

commutation relations (3.91) for the ļ¬elds and the equal-time, canonical commuta-

tors, [rni (t), plj (t)] = i Ī“nl Ī“ij , for the charged particles. After a bit of algebra, the

Heisenberg equations are found to be

ā‚A (r, t)

E (r, t) = ā’ , (4.33)

ā‚t

ā‚B (r, t)

ā Ć— E (r, t) = ā’ , (4.34)

ā‚t

1 ā‚E (r, t)

= Āµ0 jā„ (r, t) ,

ā Ć— B (r, t) ā’ 2 (4.35)

c ā‚t

pn (t) ā’ qn A (rn (t) , t)

drn (t)

vn (t) ā” = , (4.36)

dt Mn

dpn (t)

= qn E (rn (t) , t) + qn vn (t) Ć— B (rn (t) , t) ā’ qn āĪ¦ (rn (t)) , (4.37)

dt

where vn (t) is the velocity operator for the nth particle, jā„ (r, t) is the transverse

part of the current density operator

j (r, t) = Ī“ (rā’rn (t)) qn vn (t) , (4.38)

n

and the Coulomb potential operator is

1 ql

Ī¦ (rn (t)) = . (4.39)

|rn (t) ā’ rl (t)|

4Ļ 0

l=n

Ā½Ā½ Interaction of light with matter

This potential is obtained from a solution of Poissonā™s equation ā2 Ī¦ = ā’Ļ/ 0 , where

the charge density operator is

Ļ (r, t) = Ī“ (rā’rn (t)) qn , (4.40)

n

by omitting the self-interaction terms encountered when r ā’ rn (t). Functions f (rn )

of the position operators rn , such as those in eqns (4.35)ā“(4.40), are deļ¬ned by

f (rn ) Ļ (r1 , . . . , rN ) = f (rn ) Ļ (r1 , . . . , rN ) , (4.41)

where Ļ (r1 , . . . , rN ) is any N -body wave function for the charged particles.

The ļ¬rst equation, eqn (4.33), is simply the relation between the transverse part of

the electric ļ¬eld operator and the vector potential. Faradayā™s law, eqn (4.34), is then

redundant, since it is the curl of eqn (4.33). The matter equations (4.36) and (4.37)

are the quantum versions of the classical force laws of Coulomb and Lorentz.

The only one of the Heisenberg equations that requires further explanation is eqn

(4.35) (Amp`reā™s law). The Heisenberg equation of motion for E can be put into the

e

form

pni ā’ qn Ai (rn (t) , t)

1 ā‚Ej (r, t)

āā„ (r ā’ rn (t)) qn

(ā Ć— B (r, t))j ā’ = Āµ0 ,

ji

c2 ā‚t Mn

n

(4.42)

but the signiļ¬cance of the right-hand side is not immediately obvious. Further insight

can be achieved by using the deļ¬nition (4.36) of the velocity operator to get

pn (t) ā’ qn A (rn (t) , t)

= vn (t) . (4.43)

Mn

Substituting this into eqn (4.42) yields

1 ā‚Ej (r, t)

āā„ (r ā’ rn ) qn vni (t)

(ā Ć— B (r, t))j ā’ = Āµ0 ji

c2 ā‚t n

d3 r āā„ (r ā’ r ) ji (r , t) ,

= Āµ0 (4.44)

ji

where ji (r , t), deļ¬ned by eqn (4.38), can be interpreted as the current density oper-

ator. The transverse delta function āā„ projects out the transverse part of any vector

ji

ļ¬eld, so the Heisenberg equation for E (r, t) is given by eqn (4.35).

Parity and time reversalā—

4.4

The quantum Maxwell equations, (4.34) and (4.35), and the classical Maxwell equa-

tions, (B.2) and (B.3), have the same form; consequently, the ļ¬eld operators and the

classical ļ¬elds behave in the same way under the discrete transformations:

r ā’ ā’r (spatial inversion or parity transformation) ,

(4.45)

t ā’ ā’t (time reversal) .

Parity and time reversalā— Ā½Ā½

Thus the transformation laws for the classical ļ¬eldsā”see Appendix B.3.3ā”also apply

to the ļ¬eld operators; in particular,

E (r, t) ā’ EP (r, t) = ā’E (ā’r, t) under r ā’ ā’r , (4.46)

E (r, t) ā’ ET (r, t) = E (r, ā’t) under t ā’ ā’ t . (4.47)

In classical electrodynamics this is the end of the story, since the entire physical

content of the theory is contained in the values of the ļ¬elds. The situation for quantum

electrodynamics is more complicated, because the physical content is shared between

the operators and the state vectors. We must therefore ļ¬nd the transformation rules for

the states that correspond to the transformations (4.46) and (4.47) for the operators.

This eļ¬ort requires a more careful look at the idea of symmetries in quantum theory.

According to the general rules of quantum theory, all physical predictions can be

2

expressed in terms of probabilities given by | Ī¦ |ĪØ | , where |ĪØ and |Ī¦ are normalized

state vectors. For this reason, a mapping of state vectors to state vectors,

|Ī˜ ā’ |Ī˜ , (4.48)

is called a symmetry transformation if

2 2

| Ī¦ |ĪØ | = | Ī¦ |ĪØ | , (4.49)

for any pair of vectors |ĪØ and |Ī¦ . In other words, symmetry transformations leave

all physical predictions unchanged. The consequences of this deļ¬nition are contained

in a fundamental theorem due to Wigner.

Theorem 4.1 (Wigner) Every symmetry transformation can be expressed in one of

two forms:

(a) |ĪØ ā’ |ĪØ = U |ĪØ , where U is a unitary operator;

(b) |ĪØ ā’ |ĪØ = Ī |ĪØ , where Ī is an antilinear and antiunitary operator.

The unfamiliar terms in alternative (b) are deļ¬ned as follows. A transformation Ī

is antilinear if

Ī {Ī± |ĪØ + Ī² |Ī¦ } = Ī±ā— Ī |ĪØ + Ī² ā— Ī |Ī¦ , (4.50)

and antiunitary if

ā—

Ī¦ |ĪØ = ĪØ |Ī¦ = Ī¦ |ĪØ , where |ĪØ = Ī |ĪØ and |Ī¦ = Ī |Ī¦ . (4.51)

Rather than present the proof of Wignerā™s theoremā”which can be found in Wigner

(1959, cf. Appendices in Chaps 20 and 26) or Bargmann (1964)ā”we will attempt to

gain some understanding of its meaning. To this end consider another transformation

given by

|ĪØ ā’ |ĪØ = exp (iĪøĪØ ) |ĪØ , (4.52)

where ĪøĪØ is a real phase that can be chosen independently for each |ĪØ . For any value of

ĪøĪØ it is clear that |ĪØ ā’ |ĪØ is also a symmetry transformation. Furthermore, |ĪØ

and |ĪØ diļ¬er only by an overall phase, so they represent the same physical state.

Ā½Ā¾Ā¼ Interaction of light with matter

Thus the symmetry transformations deļ¬ned by eqns (4.48) and (4.52) are physically

equivalent, and the meaning of Wignerā™s theorem is that every symmetry transforma-

tion is physically equivalent to one or the other of the two alternatives (a) and (b).

This very strong result allows us to ļ¬nd the correct transformation for each case by a

simple process of trial and error. If the wrong alternative is chosen, something will go

seriously wrong.

Since unitary transformations are a familiar tool, we begin the trial and error

process by assuming that the parity transformation (4.46) is realized by a unitary

operator UP :

ā

EP (r, t) = UP E (r, t) UP = ā’E (ā’r, t) . (4.53)

In the interaction picture, E (r, t) has the plane-wave expansion

Ļk

aks eks ei(kĀ·rā’Ļk t) + HC ,

E (r, t) = i (4.54)

2 0V

ks

and the corresponding classical ļ¬eld has an expansion of the same form, with aks re-

placed by the classical amplitude Ī±ks . In Appendix B.3.3, it is shown that the parity

transformation law for the classical amplitude is Ī±P = ā’Ī±ā’k,ā’s . Since UP is linear,

ks

ā ā

P

UP E (r, t) UP can be expressed in terms of aks = UP aks UP . Comparing the quantum

and classical expressions then implies that the unitary transformation of the annihi-

lation operator must have the same form as the classical transformation:

ā

aks ā’ aP = UP aks UP = ā’aā’k,ā’s . (4.55)

ks

The existence of an operator UP satisfying eqn (4.55) is guaranteed by another well

known result of quantum theory discussed in Appendix C.4: two sets of canonically

conjugate operators acting in the same Hilbert space are necessarily related by a

unitary transformation. Direct calculation from eqn (4.55) yields

a P , aP ā = ā’aā’k,ā’s , ā’aā ,ā’s = Ī“kk Ī“ss ,

ks k s ā’k

(4.56)

= [ā’aā’k,ā’ss , ā’aā’k ,ā’s ] = 0 .

aP , a P s

ks k

Since the operators aP satisfy the canonical commutation relations, UP exists. For

ks

more explicit properties of UP , see Exercise 4.4.

The assumption that spatial inversion is accomplished by a unitary transformation

worked out very nicely, so we will try the same approach for time reversal, i.e. we

assume that there is a unitary operator UT such that

ā

ET (r, t) = UT E (r, t) UT = E (r, ā’t) . (4.57)

The classical transformation rule for the plane-wave amplitudes is Ī±T = ā’Ī±ā— , so

ks ā’k,s

the argument used for the parity transformation implies that the annihilation operators

satisfy

aks ā’ aT = aT = UT aks UT = ā’aā

ā

ā’k,s . (4.58)

ks ks

All that remains is to check the internal consistency of this rule by using it to evaluate

the canonical commutators. The result

Ā½Ā¾Ā½

Stationary density operators

a T , aT ā = ā’aā , ā’aā’k ,s = ā’Ī“kk Ī“ss (4.59)

ks k s ā’k,s

is a nasty surprise. The extra minus sign on the right side shows that the transformed

operators are not canonically conjugate. Thus the time-reversed operators aT and aT ā

ks ks

ā

cannot be related to the original operators aks and aks by a unitary transformation,

and UT does not exist.

According to Wignerā™s theorem, the only possibility left is that the time-reversed

operators are deļ¬ned by an antiunitary transformation,

ET (r, t) = ĪT E (r, t) Īā’1 = E (r, ā’t) . (4.60)

T

Here some caution is required because of the unfamiliar properties of antilinear trans-

formations. The deļ¬nition (4.50) implies that ĪT Ī± |ĪØ = Ī±ā— ĪT |ĪØ for any |ĪØ , so

applying ĪT to the expansion (4.54) for E (r, t) gives us

T

Ļk

aā

ĪT E (r, t) Īā’1 ā’aT eā— eā’i(kĀ·rā’Ļk t) eks ei(kĀ·rā’Ļk t) ,

= i +

ks ks ks

T

2 0V

ks

(4.61)

where

T

aā = ĪT aā Īā’1 .

aT = ĪT aks Īā’1 , (4.62)

ks ks ks T

T

Setting t ā’ ā’t in eqn (4.54) and changing the summation variable by k ā’ ā’k yields

Ļk

aā’ks eā’ks eā’i(kĀ·rā’Ļk t) ā’ aā eā— ei(kĀ·rā’Ļk t) .

E (r, ā’t) = i (4.63)

ā’ks ā’ks

2 0V

ks

After substituting these expansions into eqn (4.60) and using the properties eā’k,ā’s =

eks and eā— = ek,ā’s derived in Appendix B.3.3, one ļ¬nds

k,s

T

aā = ā’aā

= ā’aā’k,s ,

aT ā’k,s . (4.64)

ks ks

T

This transformation rule gives us aā = aT ā and

ks ks

aT , aT ā s = ā’aā’k,s , ā’aā ,s = Ī“kk Ī“ss ; (4.65)

ks k ā’k

consequently, the antiunitary transformation yields creation and annihilation opera-

tors that satisfy the canonical commutation relations. The magic ingredient in this

approach is the extra complex conjugation operation applied by the antilinear trans-

formation ĪT to the c-number coeļ¬cients in eqn (4.61). This is just what is needed

to ensure that aT is proportional to aā’k,s rather than to aā , as in eqn (4.58).

ks ā’k,s

4.5 Stationary density operators

The expectation value of a single observable is given by

X (t) = Tr [ĻX (t)] = Tr [Ļ (t) X] , (4.66)

which explicitly shows that the time dependence comes entirely from the observable

in the Heisenberg picture and entirely from the density operator in the SchrĀØdinger

o

Ā½Ā¾Ā¾ Interaction of light with matter

picture. The time dependence simpliļ¬es for the important class of stationary density

operators, which are deļ¬ned by requiring the SchrĀØdinger-picture Ļ (t) to be a constant

o

of the motion. According to eqn (3.75) this means that Ļ (t) is independent of time,

so the SchrĀØdinger- and Heisenberg-picture density operators are identical. Stationary

o

density operators have the useful property

Ļ, U ā (t) = 0 = [Ļ, U (t)] , (4.67)

which is equivalent to

[Ļ, H] = 0 . (4.68)

Using these properties in conjunction with the cyclic invariance of the trace shows

that the expectation value of a single observable is independent of time, i.e.

X (t) = Tr [ĻX (t)] = Tr (ĻX) = X . (4.69)

Correlations between observables at diļ¬erent times are described by averages of

the form

X (t + Ļ„ ) Y (t) = Tr [ĻX (t + Ļ„ ) Y (t)] . (4.70)

For a stationary density operator, the correlation only depends on the diļ¬erence in the

time arguments. This is established by combining U (ā’t) = U ā (t) with eqns (3.83),

(4.67), and cyclic invariance to get

X (t + Ļ„ ) Y (t) = X (Ļ„ ) Y (0) . (4.71)

4.6 Positive- and negative-frequency parts for interacting ļ¬elds

When charged particles are present, the Hamiltonian is given by eqn (4.28), so the

free-ļ¬eld solution (3.95) is no longer valid. The operator aks (t)ā”evolving from the

annihilation operator, aks (0) = aks ā”will in general depend on the (SchrĀØdinger-o

ā

picture) creation operators ak s as well as the annihilation operators ak s . The unitary

evolution of the operators in the Heisenberg picture does ensure that the general

decomposition

F (r, t) = F (+) (r, t) + F (ā’) (r, t) (4.72)

will remain valid provided that the initial operator F (+) (r, 0) (F (ā’) (r, 0)) is a sum

over annihilation (creation) operators, but the commutation relations (3.102) are only

valid for equal times. Furthermore, F (+) (r, Ļ) will not generally vanish for all negative

values of Ļ. Despite this failing, an operator F (+) (r, t) that evolves from an initial

operator of the form

F (+) (r, 0) = Fks aks eikĀ·r (4.73)

ks

is still called the positive-frequency part of F (r, t).

Ā½Ā¾Āæ

Multi-time correlation functions

4.7 Multi-time correlation functions

One of the advantages of the Heisenberg picture is that it provides a convenient way

to study the correlation between quantum ļ¬elds at diļ¬erent times. This comes about

because the state is represented by a time-independent density operator Ļ, while the

ļ¬eld operators evolve in time according to the Heisenberg equations.

Since the electric ļ¬eld is a vector, it is natural to deļ¬ne the ļ¬rst-order ļ¬eld

correlation function by the tensor

(1) (ā’) (+)

Gij (x1 ; x2 ) = Ei (x1 ) Ej (x2 ) , (4.74)

where X = Tr [ĻX] and x1 = (r1 , t1 ), etc. The ļ¬rst-order correlation functions are

directly related to interference and photon-counting experiments. In Section 9.1.2-B

we will see that the counting rate for a broadband detector located at r is proportional

(1) (1)

to Gij (r, t; r, t). For unequal times, t1 = t2 , the correlation function Gij (x1 ; x2 ) rep-

resents measurements by a detector placed at the output of a Michelson interferometer

with delay time Ļ„ = |t1 ā’ t2 | between its two arms. In Section 9.1.2-C we will show

that the spectral density for the ļ¬eld state Ļ is determined by the Fourier transform

(1)

of Gij (r, t; r, 0). The two-slit interference pattern discussed in Section 10.1 is directly

(1)

given by Gij (r, t; r, 0).

We will see in Section 9.2.4 that the second-order correlation function, deļ¬ned

by

(2) (ā’) (ā’) (+) (+)

Gijkl (x1 , x2 ; x3 , x4 ) = Ei (x1 ) Ej (x2 ) Ek (x3 ) El (x4 ) , (4.75)

is associated with coincidence counting. Higher-order correlation functions are deļ¬ned

(ā’) (ā’)

similarly. Other possible expectation values, e.g. Ei (x1 ) Ej (x2 ) , are not related

to photon detection, so they are normally not considered.

In many applications, the physical situation deļ¬nes some preferred polarization

directionsā”represented by unit vectors v1 , v2 , . . .ā”and the tensor correlation func-

tions are replaced by scalar functions

(ā’) (+)

G(1) (x1 ; x2 ) = E1 (x1 ) E2 (x2 ) , (4.76)

(ā’) (ā’) (+) (+)

G(2) (x1 , x2 ; x3 , x4 ) = E1 (x1 ) E2 (x2 ) E3 (x3 ) E4 (x4 ) , (4.77)

ā—

(+)

where Ep = vp Ā· E(+) is the projection of the vector operator onto the direction vp .

For example, observing a ļ¬rst-order interference pattern through a polarization ļ¬lter

is described by

G(1) (x; x) = e Ā· E(ā’) (x) eā— Ā· E(+) (x) , (4.78)

where e is the polarization transmitted by the ļ¬lter.

If the density operator is stationary, then an extension of the argument leading to

eqn (4.71) shows that the correlation function is unchanged by a uniform translation,

(1)

tp ā’ tp + Ļ„, tp ā’ tp + Ļ„ , of all the time arguments. In particular Gij (r, t; r , t ) =

Ā½Ā¾ Interaction of light with matter

(1)

Gij (r, t ā’ t ; r , 0), so the ļ¬rst-order function only depends on the diļ¬erence, t ā’ t , of

the time arguments.

The correlation functions satisfy useful inequalities that are based on the fact that

Tr ĻF ā F 0, (4.79)

where F is an arbitrary observable and Ļ is a density operator. This is readily proved

by evaluating the trace in the basis in which Ļ is diagonal and using ĪØ F ā F ĪØ 0.

Choosing F = E (+) (x) in eqn (4.79) gives

G(1) (x; x) 0, (4.80)

(+) (+)

(x1 ) Ā· Ā· Ā· En

and the operator F = E1 (xn ) gives the general positivity condition

G(n) (x1 , . . . , xn ; x1 , . . . , xn ) 0. (4.81)

A diļ¬erent sort of inequality follows from the choice

n

(+)

F= Ī¾a Ea (xa ) , (4.82)

a=1

where the Ī¾a s are complex numbers. Substituting F into eqn (4.79) yields

n n

ā—

Ī¾a Ī¾b Fab 0, (4.83)

a=1 b=1

where F is the n Ć— n hermitian matrix

Fab = G(1) (xa ; xb ) . (4.84)

Since the inequality (4.83) holds for all complex Ī¾a s, the matrix F is positive deļ¬nite.

A necessary condition for this is that the determinant of F must be positive. For the

case n = 2 this yields the inequality

2

G(1) (x1 ; x2 ) G(1) (x1 ; x1 ) G(1) (x2 ; x2 ) . (4.85)

For ļ¬rst-order interference experiments, this inequality translates directly into a bound

on the visibility of the fringes; this feature will be exploited in Section 10.1.

4.8 The interaction picture

In typical applications, the interaction energy between the charged particles and the

radiation ļ¬eld is much smaller than the energies of individual photons. It is therefore

useful to rewrite the SchrĀØdinger-picture Hamiltonian, eqn (4.29), as

o

(S) (S)

H (S) = H0 + Hint , (4.86)

where

(S) (S)

(S)

H0 = Hem + Hchg (4.87)

(S)

is the unperturbed Hamiltonian and Hint is the perturbation or interaction

Hamiltonian. In most cases the SchrĀØdinger equation with the full Hamiltonian H (S)

o

Ā½Ā¾

The interaction picture

(S)

cannot be solved exactly, so the weak (perturbative) nature of Hint must be used to

get an approximate solution.

For this purpose, it is useful to separate the fast (high energy) evolution due to

(S) (S)

H0 from the slow (low energy) evolution due to Hint . To this end, the interaction-

picture state vector is deļ¬ned by the unitary transformation

ā

ĪØ(I) (t) = U0 (t) ĪØ(S) (t) , (4.88)

where the unitary operator,

i (t ā’ t0 ) (S)

U0 (t) = exp ā’ H0 , (4.89)

satisļ¬es

ā‚ (S)

i

U0 (t) = H0 U0 (t) , U0 (t0 ) = 1 . (4.90)

ā‚t

Thus the SchrĀØdinger and interaction pictures coincide at t = t0 . It is also clear

o

(S)

that H0 , U0 (t) = 0. A glance at the solution (3.76) for the SchrĀØdinger equation

o

(S)

reveals that this transformation eļ¬ectively undoes the fast evolution due to H0 . By

contrast to the Heisenberg picture deļ¬ned in Section 3.2, the transformed ket vector

(S)

still depends on time due to the action of Hint . The consistency condition,

ĪØ(I) (t) X (I) (t) Ī¦(I) (t) = ĪØ(S) (t) X (S) Ī¦(S) (t) , (4.91)

requires the interaction-picture operators to be deļ¬ned by

ā

X (I) (t) = U0 (t) X (S) U0 (t) . (4.92)

(S)

For H0 this yields the simple result

ā

(I) (S) (S)

H0 (t) = U0 (t) H0 U0 (t) = H0 , (4.93)

(I) (S)

which shows that H0 (t) = H0 = H0 is independent of time.

The transformed state vector ĪØ(I) (t) obeys the interaction-picture SchrĀØdinger

o

equation

ā‚ ā

(S) (S) (S)

ĪØ(I) (t) = ā’H0 ĪØ(I) (t) + U0 (t) H0 + Hint ĪØ(S) (t)

i

ā‚t

ā

(S) (S) (S)

= ā’H0 ĪØ(I) (t) + U0 (t) H0 U0 (t) ĪØ(I) (t)

+ Hint

(I)

= Hint (t) ĪØ(I) (t) , (4.94)

which follows from operating on both sides of eqn (4.88) with i ā‚/ā‚t and using eqns

(4.90)ā“(4.93). The formal solution is

ĪØ(I) (t) = V (t) ĪØ(I) (t0 ) , (4.95)

Ā½Ā¾ Interaction of light with matter

where the unitary operator V (t) satisļ¬es

ā‚ (I)

i V (t) = Hint (t) V (t) , with V (t0 ) = 1 . (4.96)

ā‚t

The initial condition V (t0 ) = 1 really should be V (t0 ) = IQED , where IQED is the

identity operator for HQED , but alert readers will suļ¬er no harm from this slight abuse

of notation.

By comparing eqn (4.92) to eqn (3.83), one sees immediately that the interaction-

picture operators obey

ā‚ (I)

X (t) = X (I) (t) , H0 .

i (4.97)

ā‚t

These are the Heisenberg equation for free ļ¬elds, so we can use eqns (3.95) and (3.96)

to get

aks (t) = aks eā’iĻk (tā’t0 ) ,

(I) (S)

(4.98)

and

(S)

A(I)(+) (r, t) = aks eks ei[kĀ·rā’Ļk (tā’t0 )] . (4.99)

2 0 Ļk V

ks

In the same way eqn (3.102) implies

F (I)(Ā±) (r, t) , G(I)(Ā±) (r , t ) = 0 , (4.100)

where F and G are any of the ļ¬eld components and (r, t), (r , t ) are any pair of

spaceā“time points.

In the interaction picture, the burden of time evolution is shared between the oper-

ators and the states. The operators evolve according to the unperturbed Hamiltonian,

and the states evolve according to the interaction Hamiltonian. Once again, the density

operator is an exception. Applying the transformation in eqn (4.88) to the deļ¬nition

(3.85) of the SchrĀØdinger-picture density operator leads to

o

ā‚ (I) (I)

Ļ (t) = Hint (t), Ļ(I) (t) ,

i (4.101)

ā‚t

so the density operator evolves according to the interaction Hamiltonian.

In applications of the interaction picture, we will simplify the notation by the fol-

lowing conventions: X (t) means X (I) (t), X means X (S) , |ĪØ (t) means ĪØ(I) (t) , and

Ļ (t) means Ļ(I) (t). If all three pictures are under consideration, it may be necessary

to reinstate the superscripts (S), (H), and (I).

4.8.1 Time-dependent perturbation theory

In order to make use of the weakness of the perturbation, we ļ¬rst turn eqn (4.96) into

an integral equation by integrating over the interval (t0 , t) to get

t

i

V (t) = 1 ā’ dt1 Hint (t1 ) V (t1 ) . (4.102)

t0

Ā½Ā¾

The interaction picture

The formal perturbation series is obtained by repeated iterations of the integral equa-

tion,

2

t t t1

i i

V (t) = 1 ā’ dt1 Hint (t1 ) + ā’ dt2 Hint (t1 ) Hint (t2 ) + Ā· Ā· Ā·

dt1

t0 t0 t0

ā

V (n) (t) ,

= (4.103)

n=0

where V (0) = 1, and

n t tnā’1

i

ā’ dt1 Ā· Ā· Ā· dtn Hint (t1 ) Ā· Ā· Ā· Hint (tn ) ,

(n)

V (t) = (4.104)

t0 t0

for n 1.

If the system (charges plus radiation) is initially in the state |Ī˜i then the prob-

ability amplitude that a measurement at time t leaves the system in the ļ¬nal state

|Ī˜f is

Vf i (t) = Ī˜f |ĪØ (t) = Ī˜f |V (t)| Ī˜i ; (4.105)

consequently, the transition probability is

2

Pf i (t) = |Vf i (t)| . (4.106)

4.8.2 First-order perturbation theory

For this application, we choose t0 = 0, and then let the interaction act for a ļ¬nite time

t. The initial state |Ī˜i evolves into V (t) |Ī˜i , and its projection on the ļ¬nal state

|Ī˜f is Ī˜f |V (t)| Ī˜i . Let the initial and ļ¬nal states be eigenstates of the unperturbed

Hamiltonian H0 , with energies Ei and Ef respectively. According to eqn (4.104) the

ļ¬rst-order contribution to Ī˜f |V (t)| Ī˜i is

t

i

(1)

Vf i (t) = ā’ dt1 Ī˜f |Hint (t1 )| Ī˜i

0

t

i

=ā’ dt1 Ī˜f |Hint | Ī˜i exp (iĪ½f i t1 ) , (4.107)

0

where we have used eqn (4.92) and introduced the notation Ī½f i = (Ef ā’ Ei ) / . Eval-

uating the integral in eqn (4.107) yields the amplitude

2i sin (Ī½f i t/2)

(1)

Vf i (t) = ā’ Ī˜f |Hint | Ī˜i ,

exp (iĪ½f i t/2) (4.108)

Ī½f i

so the transition probability is

2 4

(1) 2

| Ī˜f |Hint | Ī˜i | ā (Ī½f i , t) ,

Pf i (t) = Vf i (t) = (4.109)

2

where ā (Ī½, t) ā” sin2 (Ī½t/2) /Ī½ 2 .

Ā½Ā¾ Interaction of light with matter

2

For ļ¬xed t, the maximum value of |ā (Ī½, t)| is t2 /4, and it occurs at Ī½ = 0. The

width of the central peak is approximately 2Ļ/t, so as t becomes large the function

is strongly peaked at Ī½ = 0. In order to specify a well-deļ¬ned ļ¬nal energy, the width

must be small compared to |Ef ā’ Ei | / ; therefore,

2Ļ

t (4.110)

|Ef ā’ Ei |

deļ¬nes the limit of large times. This is a realization of the energyā“time uncertainty

relation, tāE ā¼ (Bransden and Joachain, 1989, Sec. 2.5). With this understanding

of inļ¬nity, we can use the easily established mathematical result,

sin2 (Ī½t/2)

ā (Ī½, t) Ļ

lim = lim = Ī“ (Ī½) , (4.111)

2

t tĪ½ 2

tā’ā tā’ā

to write the asymptotic (t ā’ ā) form of eqn (4.109) as

2Ļ 2

t | Ī˜f |Hint | Ī˜i | Ī“ (Ī½f i )

Pf i (t) = 2

2Ļ 2

t | Ī˜f |Hint | Ī˜i | Ī“ (Ef ā’ Ei ) .

= (4.112)

The transition rate, Wf i = dPf i (t) /dt, is then

2Ļ

| Ī˜f |Hint | Ī˜i |2 Ī“ (Ef ā’ Ei ) .

Wf i = (4.113)

This is Fermiā™s golden rule of perturbation theory (Bransden and Joachain, 1989,

Sec. 9.3). This limiting form only makes sense when at least one of the energies Ei

and Ef varies continuously. In the following applications this happens automatically

because of the continuous variation of the photon energies.

In addition to the lower bound on t in eqn (4.110) there is an upper bound on the

time interval for which the perturbative result is valid. This is estimated by summing

eqn (4.112) over all ļ¬nal states to get the total transition probability Pi,tot (t) = tWi,tot ,

where the total transition rate is

2Ļ 2

| Ī˜f |Hint | Ī˜i | Ī“ (Ef ā’ Ei ) .

Wi,tot = Wf i = (4.114)

f f

According to this result, the necessary condition Pi,tot (t) < 1 will be violated if

t > 1/Wi,tot . In fact, the validity of the perturbation series demands the more strin-

gent condition Pi,tot (t) 1, so the perturbative results can only be trusted for

1/Wi,tot . This upper bound on t means that the t ā’ ā limit in eqn (4.111)

t

is simply the physical condition (4.110). For the same reason, the energy conserving

delta function in eqn (4.112) is really just a sharply-peaked function that imposes the

restriction |Ef ā’ Ei | Ef .

Ā½Ā¾

The interaction picture

With this understanding in mind, a simpliļ¬ed version of the previous calculation is

possible. For this purpose, we choose t0 = ā’T /2 and allow the state vector to evolve

until the time t = T /2. Then eqn (4.107) is replaced by

T /2

i

(1)

(T /2) = ā’ Ī˜f |Hint | Ī˜i exp (iĪ½f i T /2)

Vf i dt1 exp (iĪ½f i t1 ) . (4.115)

ā’T /2

The standard result

T /2

dt1 eiĪ½t1 = 2ĻĪ“ (Ī½)

lim (4.116)

T ā’ā ā’T /2

allows this to be recast as

2Ļi

(1) (1)

Vf i = Vf i (ā) = ā’ Ī˜f |Hint | Ī˜i Ī“ (Ī½f i ) , (4.117)

so the transition probability is

2

2 2Ļ

(1)

| Ī˜f |Hint | Ī˜i |2 [Ī“ (Ī½f i )]2 .

Pf i = Vf i = (4.118)

This is rather embarrassing, since the square of a delta function is not a respectable

mathematical object. Fortunately this is a physicistā™s delta function, so we can use

eqn (4.116) once more to set

T /2

dt1 T

2

[Ī“ (Ī½f i )] = Ī“ (Ī½f i ) exp (iĪ½f i t1 ) = Ī“ (Ī½f i ) . (4.119)

2Ļ 2Ļ

ā’T /2

After putting this into eqn (4.118), we recover eqn (4.113).

4.8.3 Second-order perturbation theory

Using the simpliļ¬ed scheme, presented in eqns (4.115)ā“(4.119), yields the second-order

contribution to Ī˜f |V (T /2)| Ī˜i :

2 T /2 t1

i

(2)

ā’ dt2 Ī˜f |Hint (t1 ) Hint (t2 )| Ī˜i

Vf i = dt1

ā’T /2 ā’T /2

2 T /2 T /2

i

ā’ dt2 Īø (t1 ā’ t2 ) Ī˜f |Hint (t1 ) Hint (t2 )| Ī˜i , (4.120)

= dt1

ā’T /2 ā’T /2

where Īø (t1 ā’ t2 ) is the step function discussed in Appendix A.7.1 . By introducing a

basis set {|Īu } of eigenstates of H0 , the matrix element can be written as

Ī˜f |Hint (t1 ) Hint (t2 )| Ī˜i = exp [(iĪ½f i ) T /2] Ī˜f |Hint | Īu Īu |Hint | Ī˜i

u

Ć— exp (iĪ½f u t1 ) exp (iĪ½ui t2 ) , (4.121)

where we have used eqn (4.92) and the identity Ī½f u + Ī½ui = Ī½f i . The ļ¬nal step is to

use the representation (A.88) for the step function and eqn (4.116) to ļ¬nd

Ā½ĀæĀ¼ Interaction of light with matter

ā

Ī˜f |Hint | Īu Īu |Hint | Ī˜i

i iĪ½f i T /2

(2)

=ā’

Vf i e dĪ½

2Ļ 2 Ī½+i

ā’ā u

Ć— 2ĻĪ“ (Ī½f u ā’ Ī½) 2ĻĪ“ (Ī½ui + Ī½) . (4.122)

Carrying out the integration over Ī½ with the aid of the delta functions leads to

Ī˜f |Hint | Īu Īu |Hint | Ī˜i

2Ļi

(2)

Vf i = ā’ Ī“ (Ī½f i )

2 Ī½f u + i

u

Ī˜f |Hint | Īu Īu |Hint | Ī˜i

= ā’2Ļi Ī“ (Ef ā’ Ei ) . (4.123)

Ef ā’ Eu + i

u

Finally, another use of the rule (4.119) yields the transition rate

2

Ī˜f |Hint | Īu Īu |Hint | Ī˜i

2Ļ

Ī“ (Ef ā’ Ei ) .

Wf i = (4.124)

Ef ā’ Eu + i

u

4.9 Interaction of light with atoms

4.9.1 The dipole approximation

The shortest wavelengths of interest for quantum optics are in the extreme ultraviolet,

so we can assume that Ī» > 100 nm, whereas typical atoms have diameters a ā 0.1 nm.

The large disparity between atomic diameters and optical wavelengths (a/Ī» < 0.001)

permits the use of the dipole approximation, and this in turn brings about important

simpliļ¬cations in the general Hamiltonian deļ¬ned by eqns (4.28)ā“(4.32).

The simpliļ¬ed Hamiltonian can be derived directly from the general form given in

Section 4.2.2 (Cohen-Tannoudji et al., 1989, Sec. IV.C), but it is simpler to obtain the

dipole-approximation Hamiltonian for a single atom by a separate appeal to the corre-

spondence principle. This single-atom construction is directly relevant for suļ¬ciently

dilute systems of atomsā”e.g. tenuous atomic vaporsā”since the interaction between

atoms is weak. Experiments with vapors were the rule in the early days of quantum

optics, but in many modern applicationsā”such as solid-state detectors and solid-state

lasersā”the atoms are situated on a crystal lattice. This is a high density situation with

substantial interactions between atoms. Furthermore, the electronic wave functions can

be delocalizedā”e.g. in the conduction band of a semiconductorā”so that the validity

of the dipole approximation is in doubt. These considerationsā”while very important

in practiceā”do not in fact require signiļ¬cant changes in the following discussion.

The interactions between atoms on a crystal lattice can be described in terms of

coupling to lattice vibrations (phonons), and the eļ¬ects of the periodic crystal potential

are represented by the use of Bloch or Wannier wave functions for the electrons (Kittel,

1985, Chap. 9). The wave functions for electrons in the valence band are localized to

crystal sites, so for transitions between the valence and conduction bands even the

dipole approximation can be retained. We will exploit this situation by explaining

the basic techniques of quantum optics in the simpler context of tenuous vapors. Once

these notions are mastered, their application to condensed matter physics can be found

elsewhere (Haug and Koch, 1990).

Ā½ĀæĀ½

Interaction of light with atoms

Even with the dipole approximation in force, the direct use of the atomic wave

function is completely impractical for a many-electron atomā”this means any atom

with atomic number Z > 1. Fortunately, the complete description provided by the

many-electron wave function Ļ (r1 , . . . , rZ ) is not needed. For the most part, only

selected propertiesā”such as the discrete electronic energies and the matrix elements of

the dipole operatorā”are required. Furthermore these properties need not be calculated

ab initio; instead, they can be inferred from the measured wavelength and strength

of spectral lines. In this semi-empirical approach, the problem of atomic structure is

separated from the problem of the response of the atom to the electromagnetic ļ¬eld.

For a single atom interacting with the electromagnetic ļ¬eld, the discussion in Sec-

tion 4.2.1 shows that the state space is the tensor product H = HA ā— HF of the Hilbert

space HA for the atom and the Fock space HF for the ļ¬eld. A typical basis state for

H is |Ļ, Ī¦ = |Ļ |Ī¦ , where |Ļ and |Ī¦ are respectively state vectors for the atom

and the ļ¬eld. Let us consider a typical matrix element Ļ, Ī¦ |E (r)| Ļ , Ī¦ of the elec-

tric ļ¬eld operator, where at least one of the vectors |Ļ and |Ļ describes a bound

state with characteristic spatial extent a, and |Ī¦ and |Ī¦ both describe states of the

ļ¬eld containing only photons with wavelengths Ī» a. On the scale of the optical

wavelengths, the atomic electrons can then be regarded as occupying a small region

surrounding the center-of-mass position,

Z

Mnuc Me

rcm = rnuc + rn , (4.125)

M M

n=1

where rcm is the operator for the center of mass, Me is the electron mass, rn is the

coordinate operator of the nth electron, Mnuc is the nuclear mass, rnuc is the coordinate

operator of the nucleus, Z is the atomic number, and M = Mnuc + ZMe is the total

mass.

For all practical purposes, the center of mass can be identiļ¬ed with the location

of the nucleus, since Mnuc ZMe . The plane-wave expansion (3.69) for the electric

ļ¬eld then implies that the matrix element is slowly varying across the atom, so that

it can be expanded in a Taylor series around rcm ,

Ļ, Ī¦ |E (r)| Ļ , Ī¦

= Ļ, Ī¦ |E (rcm )| Ļ , Ī¦ + Ļ, Ī¦ |[(r ā’ rcm ) Ā· ā] E (rcm )| Ļ , Ī¦ + Ā· Ā· Ā· .

(4.126)

With the understanding that only matrix elements of this kind will occur, the

expansion can be applied to the ļ¬eld operator itself:

E (r) = E (rcm ) + [(r ā’ rcm ) Ā· ā] E (rcm ) + Ā· Ā· Ā· . (4.127)

The electric dipole approximation retains only the leading term in this expan-

sion, with errors of O (a/Ī»). Keeping higher-order terms in the Taylor series incorpo-

rates successive terms in the general multipole expansion, e.g. magnetic dipole, electric

quadrupole, etc. In classical electrodynamics (Jackson, 1999, Sec. 4.2), the leading term

in the interaction energy of a neutral collection of charges with an external electric ļ¬eld

E is ā’d Ā· E, where d is the electric dipole moment. For an atom the dipole operator is

Ā½ĀæĀ¾ Interaction of light with matter

Z

(ā’e) (rn ā’ rnuc ) .

d= (4.128)

n=1

Once again we rely on the correspondence principle to suggest that the interaction

Hamiltonian in the quantum theory should be

Hint = ā’d Ā· E (rcm ) . (4.129)

The atomic Hamiltonian can be expressed as

Z 2

P2 (pn )

Hatom = + + VC , (4.130)

2M n=1 2Me

Z Z

e2 Ze2

1 1

ā’

VC = , (4.131)

|rn ā’ rl | 4Ļ 0 |rn ā’ rnuc |

4Ļ 0 n=1

n=l=1

where VC is the Coulomb potential, P is the total momentum, and the pn s are a set of

relative momentum operators. Thus the SchrĀØdinger-picture Hamiltonian in the dipole

o

approximation is H = Hem + Hatom + Hint .

The argument given in Section 4.2.2 shows that E (rcm ) is a hybrid operator acting

on both the atomic and ļ¬eld degrees of freedom. For most applications of quantum

optics, we can ignore this complication, since the De Broglie wavelength of the atom is

small compared to the interatomic spacing. In this limit, the center-of-mass position,

rcm , and the total kinetic energy P2 /2M can be treated classically, so that

P2

Hatom = + Hat , (4.132)

2M

where

Z 2

(pn )

Hat = + VC (4.133)

2Me

n=1

is the Hamiltonian for the internal degrees of freedom of the atom. In the same ap-

proximation, the interaction Hamiltonian reduces to

Hint = ā’d Ā· E (rcm ) , (4.134)

which acts jointly on the ļ¬eld states and the internal states of the atom.

In the rest frame of the atom, deļ¬ned by P = 0, the energy eigenstates

Hat |Īµq = Īµq |Īµq (4.135)

provide a basis for the Hilbert space, HA , describing the internal degrees of freedom

of the atom. The label q stands for a set of quantum numbers suļ¬cient to specify the

internal atomic state uniquely. The qs are discrete; therefore, they can be ordered so

that Īµq Īµq for q < q .

Ā½ĀæĀæ

Interaction of light with atoms

In practice, the many-electron wave function Ļq (r1 , . . . , rZ ) = r1 , . . . , rZ |Īµq can-

not be determined exactly, so the eigenstates are approximated, e.g. by using the

atomic shell model (Cohen-Tannoudji et al., 1977b, Chap. XIV, Complement A). In

this case the label q = (n, l, m) consists of the principal quantum number, the angular

momentum, and the azimuthal quantum number for the valence electrons in a shell

ńņš. 5 |