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d3 k ωk
(+)
E in (r, 0) ws (k) eks eik·r ,
=i (3.197)
3 20
(2π) s

so the wave packets are given by

20 —
d3 re’ik·r E in (r, 0) .
(+)
ws (k) = ’i e· (3.198)
ωk ks


Photon localizability—
3.6
3.6.1 Is there a photon position operator?
The use of the term photon to mean ˜quantum of excitation of the electromagnetic ¬eld™
is a harmless piece of jargon, but the extended sense in which photons are thought to
be localizable particles raises subtle and fundamental issues. In order to concentrate
on the essential features of this problem, we will restrict the discussion to photons
propagating in vacuum. The particle concept originated in classical mechanics, where
it is understood to mean a physical system of negligible extent that occupies a de¬nite
position in space. The complete description of the state of a classical particle is given by
its instantaneous position and momentum. In nonrelativistic quantum mechanics, the
uncertainty principle forbids the simultaneous speci¬cation of position and momentum,
so the state of a particle is instead described by a wave function ψ (r). More precisely,
ψ (r) = r |ψ is the probability amplitude that a measurement of the position operator
r will yield the value r, and leave the particle in the corresponding eigenvector |r
de¬ned by r |r = r |r . The improper eigenvector |r is discussed in Appendix C.1.1-
B. The identity
|ψ = d3 r |r r |ψ (3.199)

shows that the wave function ψ (r) is simply the projection of the state vector on the
basis vector |r . The action of the position operator r is given by r |r| ψ = r r |ψ ,
which is usually written as rψ (r) = rψ (r).
Thus the notion of a particle in nonrelativistic quantum mechanics depends on
the existence of a physically sensible position operator. Position operators exist in
nonrelativistic quantum theory for particles with any spin, and even for the relativistic
theory of massive, spin-1/2 particles described by the Dirac equation; but, there is no
position operator for the massless, spin-1 objects described by Maxwell™s equations
(Newton and Wigner, 1949).
A more general approach would be to ask if there is any operator that would serve
to describe the photon as a localizable object. In nonrelativistic quantum mechanics
the position operator r has two essential properties.
(a) The components commute with one another: [ri , rj ] = 0.
(b) The operator r transforms as a vector under rotations of the coordinate system.
Photon localizability— ½¼

Property (a) is necessary if the components of the position are to be simultaneously
measurable, and property (b) would seem to be required for the physical interpre-
tation of r as representing a location in space. Over the years many proposals for a
photon position operator have been made, with one of two outcomes: (1) when (a) is
satis¬ed, then (b) is not (Hawton and Baylis, 2001); (2) when (b) is satis¬ed, then (a)
is not (Pryce, 1948). Thus there does not appear to be a physically acceptable pho-
ton position operator; consequently, there is no position-space wave function for the
photon. This apparent di¬culty has a long history in the literature, but there are at
least two reasons for not taking it very seriously. The ¬rst is that the relevant classical
theory”Maxwell™s equations”has no particle concept. The second is that photons are
inherently relativistic, by virtue of their vanishing rest mass. Consequently, ordinary
notions connected to the Schr¨dinger equation need not apply.
o

3.6.2 Are there local number operators?
The nonexistence of a photon position operator still leaves open the possibility that
there is some other sense in which the photon may be considered as a localizable
or particle-like object. From an operational point of view, a minimum requirement
for localizability would seem to be that the number of photons in a ¬nite volume
V is an observable, represented by a local number operator N (V ). Since simultane-
ous measurements in nonoverlapping volumes of space cannot interfere, this family of
observables should satisfy
[N (V ) , N (V )] = 0 (3.200)
whenever V and V do not overlap. The standard expression (3.30) for the total number
operator as an integral over plane waves is clearly not a useful starting point for the
construction of a local number operator, so we will instead use eqns (3.49) and (3.15)
to get
20 ’1/2 (+)
d3 rE(’) (r) · ’∇2
N= E (r) . (3.201)
c
In the classical limit, the ¬eld operators are replaced by classical ¬elds, and the in
the denominator goes to zero. Thus the number operator diverges in the classical limit,
in agreement with the intuitive idea that there are e¬ectively an in¬nite number of
photons in a classical ¬eld.
The ¬rst suggestion for N (V ) is simply to restrict the integral to the volume V
(Henley and Thirring, 1964, p. 43); but this is problematical, since the integrand in eqn
(3.201) is not a positive-de¬nite operator. This poses no problem for the total number
operator, since the equivalent reciprocal-space representation (3.30) is nonnegative,
but this version of a local number operator might have negative expectation values in
1/2 1/4 1/4
some states. This objection can be met by using ’∇2 = ’∇2 ’∇2 and
the general rule (3.21) to replace the position-space integral (3.201) by the equivalent
form
N = d3 rM† (r) · M (r) , (3.202)

where
2 ’1/4
0
M (r) = ’i ’∇2 E(+) (r) . (3.203)
c
½¼ Field quantization

The integrand in eqn (3.202) is a positive-de¬nite operator, so the local number oper-
ator de¬ned by
d3 rM† (r) · M (r)
N (V ) = (3.204)
V
is guaranteed to have a nonnegative expectation value for any state. According to the
standard plane-wave representation (3.29), the operator M (r) is
d3 k
es (k) as (k) eik·r ,
M (r) = (3.205)
3
(2π) s

i.e. it is the Fourier transform of the operator M (k) introduced in eqn (3.56). The
position-space form M (r) is the detection operator introduced by Mandel in his
study of photon detection (Mandel, 1966), and N (V ) is Mandel™s local number
operator. The commutation relations (3.25) and (3.26) can be used to show that the
detection operator satis¬es

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

Now consider disjoint volumes V and V with centers separated by a distance R
which is large compared to the diameters of the volumes. Substituting eqn (3.204) into
[N (V ) , N (V )] and using eqn (3.206) yields

d3 r Sij (r, r ) ∆⊥ (r ’ r ) ,
d3 r
[N (V ) , N (V )] = (3.207)
ij
V V

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


delta function given by eqns (2.30) and (2.28) can be combined with the general
relation (3.18) to get the equivalent expression,
1 1
∆⊥ (r ’ r ) = δij δ (r ’ r ) + ∇i ∇j . (3.208)
|r ’ r |
ij

Since V and V are disjoint, the delta function term cannot contribute to eqn (3.207),
so
1 1
d3 r Sij (r, r ) ∇i ∇j
d3 r
[N (V ) , N (V )] = . (3.209)
|r ’ r |

V V
A straightforward estimate shows that [N (V ) , N (V )] ∼ R’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 ¬eld com-
mutator (3.16) which is a consequence of the transverse nature of the electromagnetic
¬eld.
The alternative de¬nition (Deutsch and Garrison, 1991a),
20
d3 rE(’) (r) · E(+) (r) ,
G (V ) = (3.210)
ω0 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
½¼
Exercises

eqn (3.210) by a simple plausibility argument. The representation (3.39) for the ¬eld
Hamiltonian suggests interpreting 2 0 E(’) · E(+) as the energy density operator. For a
monochromatic ¬eld state this in turn suggests that 2 0 E(’) · E(+) / ω0 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 de¬nite, but
nonlocal e¬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 ¬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 ¬nite volumes VP and VM respectively, as in Section
2.5. The interaction energies are

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

Show that
i
[WB , WE ] = ’ d3 rP (r) · 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

∆WB ∆WE K,

and evaluate the constant K.

3.3 Electromagnetic Hamiltonian
Carry out the derivation of eqns (3.37)“(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.

Milonni™s quantization scheme—
3.5
Fill in the details required to go from eqn (3.159) to eqn (3.164).
½½¼ Field quantization

Electromagnetic angular momentum—
3.6
Carry out the calculations needed to derive eqns (3.172)“(3.178).

Wave packet quantization—
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 “em .
4
Interaction of light with matter

In the previous chapters we have dealt with the free electromagnetic ¬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 ¬nite amounts of quantized
charge, e.g. electrons in atoms or conduction electrons in semiconductors. It is there-
fore time to construct a uni¬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 ¬eld. The next step is to treat both charges and ¬elds
by quantum theory. For this purpose, we need a Hilbert space describing both the
charged particles and the quantized electromagnetic ¬eld. The necessary machinery is
constructed in Section 4.2. We present the Heisenberg-picture description of the full
theory in Sections 4.3“4.7. In Sections 4.8 and 4.9, the interaction picture is introduced
and applied to atom“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,
ψ (r1 , . . . , rN ), in con¬guration space. The position and momentum operators rn and
pn for the nth particle are respectively de¬ned by
rn ψ (r1 , . . . , rN ) = rn ψ (r1 , . . . , rN ) ,
(4.1)

pn ψ (r1 , . . . , rN ) = ’i ψ (r1 , . . . , rN ) .
‚rn
The Hilbert space, Hchg , for the charges consists of the normalizable N -body wave
functions, i.e.
d3 rN |ψ (r1 , . . . , rN )|2 < ∞ .
d3 r1 · · · (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 ¬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 ’ qn A (rn , t))
Hsc = + qn • (rn , t) , (4.3)
2Mn
n=1 n=1
½½¾ Interaction of light with matter

where A and • 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,


pn,can = pn = ’i , (4.4)
‚rn
and the kinetic momentum,

pn,kin = pn ’ 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 ¬elds A (r, t) and • (r, t) are by de¬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 ¬eld
can be expanded in plane waves exp (ik · r ’ iωk t). The operator exp (ik · rn ) arising
from the replacement of rn by rn is de¬ned by the rule

eik·rn ψ (r1 , . . . , rN ) = eik·rn ψ (r1 , . . . , rN ) , (4.6)

where ψ (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 satis¬es
‚Ai
[Ai (rn , t) , pnj ] = i (rn , t) . (4.7)
‚rj
The scalar potential • (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 Dirac™s presentation of quantum theory, in
which the state is represented by a ket vector |ψ . For the system of charged particles
the two versions are related by

ψ (r1 , . . . , rN ) = r1 , . . . , rN |ψ , (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 ψ (r1 , . . . , rN ) simply gives the components of
the vector |ψ with respect to the basis provided by the eigenvectors |r1 , . . . , rN . Any
other set of basis vectors for Hchg would do equally well.
½½¿
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 de¬ned by
eqn (4.2), but this description is not adequate when the electromagnetic ¬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 , de¬ned by eqn (3.35), to get the state
space, HQED , for the composite system of the charges and the quantized electromag-
netic ¬eld. In Section 4.2.2 we construct the Hamiltonian for the composite charge-¬eld
system by appealing to the correspondence principle for the quantized electromagnetic
¬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
ψ (rA , rB ) = C1 ψ1 (rA ) χ1 (rB ) + C2 ψ2 (rA ) χ2 (rB ) + · · · . (4.10)

Since wave functions are meaningless for photons, it is not immediately clear how
this procedure can be applied to the radiation ¬eld. The way around this apparent
di¬culty begins with the reminder that the wave function for a particle, e.g. ψ1 (rA ),
is a probability amplitude for the outcomes of measurements of position. In the stan-
dard approach to the quantum measurement problem”reviewed in Appendix C.2”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 |ψ1 A , then the wave function is simply the probability ampli-
tude for this outcome: ψ1 (rA ) = rA |ψ1 . 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, de¬ned by

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

is uniquely de¬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 ¬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 O”more precisely, a set of commuting observables”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 ∈ Hchg after the measurement. If the charges are
prepared in the state |ψ ∈ Hchg , then the probability amplitude that a measurement
½½ Interaction of light with matter


of O results in the particular eigenvalue Oq is Oq |ψ . Furthermore, the eigenvectors
|Oq provide a basis for Hchg ; consequently, |ψ can be expressed as

|ψ = |Oq Oq |ψ . (4.12)
q


In other words, the state |ψ is completely determined by the set of probability am-
plitudes { Oq |ψ } for all possible outcomes of a measurement of O.
The same kind of argument works for the electromagnetic ¬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 ¬eld in the number state |n . If the ¬eld
is prepared in the state |¦ ∈ HF , then the probability amplitude for this outcome
is n |¦ . Since the number states form a basis for HF , the state vector |¦ can be
expressed as
|¦ = |n n |¦ ; (4.13)
n

consequently, |¦ is completely speci¬ed by the set of probability amplitudes { n |¦ }
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 ¬eld in a unique state
would do equally well.
The charged particles and the ¬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 ¬eld are
prepared in the states |ψ and |¦ 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 outcome”which we denote by Oq , n |ψ, ¦ ”is given by the product of the
individual amplitudes:
Oq , n |ψ, ¦ = Oq |ψ n |¦ . (4.14)
According to the Born interpretation, the set of probability amplitudes de¬ned by
letting Oq and n range over all possible values de¬nes a state of the composite system,
denoted by |ψ, ¦ . The vector corresponding to this state is called a product vector,
and it is usually written as
|ψ, ¦ = |ψ |¦ , (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,

|Ψ = C1 |ψ1 , ¦1 + C2 |ψ2 , ¦2 , (4.16)
½½
Quantum electrodynamics

of distinct product vectors. Once again the Born interpretation guides us to the follow-
ing statement: the superposition |Ψ is the state de¬ned by the probability amplitudes

Oq , n |Ψ = C1 Oq , n |ψ1 , ¦1 + C2 Oq , n |ψ2 , ¦2
= C1 Oq |ψ1 n |¦1 + C2 Oq |ψ2 n |¦2 . (4.17)

It is important to note that for product vectors like |ψ |¦ the subsystems are
each described by a unique state in the respective Hilbert space. The situation is
quite di¬erent for superpositions like |Ψ ; it is impossible to associate a given state
with either of the subsystems. In particular, it is not possible to say whether the ¬eld
is described by |¦1 or |¦2 . This feature”which is imposed by the superposition
principle”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 de¬nition: the
state space, HQED , of the charge-¬eld system consists of all superpositions

|Ψ = Ψqn |Oq |n . (4.18)
q n

This de¬nition guarantees the satisfaction of the superposition principle, but the Born
interpretation also requires a de¬nition of the inner product for states in HQED . To this
end, we ¬rst take eqn (4.14) as the de¬nition of the inner product of the vectors |Oq , n
and |ψ, ¦ . Applying this de¬nition to the special choice |ψ, ¦ = |Oq , n yields

Oq , n |Oq , n = Oq |Oq n |n = δq ,q δn ,n , (4.19)

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

¦— Ψqn .
¦ |Ψ = (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 de¬ned as an operator on HQED by

C |Ψ = Ψqn {C |Oq } |n , (4.21)
q n

and an operator acting only on the ¬eld degrees of freedom, e.g. aks , is extended to
HQED by
aks |Ψ = Ψqn |Oq {aks |n } . (4.22)
q n

Combining these de¬nitions gives the rule

Caks |Ψ = Ψqn {C |Oq } {aks |n } . (4.23)
q n
½½ 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 o¬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 — HF , (4.25)

and the corresponding notation |ψ —|¦ is often used for the product vectors. Similarly
the operator product Caks is often written as C — aks .

4.2.2 The Hamiltonian
For the ¬nal step to the full quantum theory, we once more call on the correspon-
dence principle to justify replacing the classical ¬eld A (r, t) in eqn (4.3) by the time-
independent, Schr¨dinger-picture quantum ¬eld A (r). The evaluation of A (r) at rn
o
is understood in the same way as for the classical ¬eld A (rn , t), e.g. by using the
plane-wave expansion (3.68) to get

eks aks eik·rn + HC .
A (rn ) = (4.26)
2 0 ωk V
ks

Thus A (rn ) is a hybrid operator that acts on the electromagnetic degrees of freedom
(HF ) through the creation and annihilation operators a† and aks and on the particle
ks
degrees of freedom (Hchg ) through the operators exp (±ik · rn ).
With this understanding we ¬rst use the identity

A (rn ) · pn + pn · A (rn ) = 2A (rn ) · pn ’ [Aj (rn ) , pnj ]
= 2A (rn ) · pn + i ∇ · A (rn ) , (4.27)

together with ∇ · A = 0 and the identi¬cation of • as the instantaneous Coulomb
potential ¦, 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 ’ qn A (rn ))
H = Hem + Hchg + + qn ¦ (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 : +µ’1 : B2 : ,
d3 r
Hem = (4.30)
0 0
2
½½
Quantum Maxwell™s equations

N
p2 1 qn ql
n
Hchg = + , (4.31)
|rn ’ rl |
2Mn 4π 0
n=1 n=l

N N 2
2
qn qn : A (rn ) :
=’ A (rn ) · pn +
Hint . (4.32)
Mn 2Mn
n=1 n=1

In this formulation, Hem is the Hamiltonian for the free (transverse) electromagnetic
¬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) ¬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 ¬‚uctuation terms. The Coulomb interactions between
the charges”say in an atom”are typically much stronger than the interaction with
the transverse ¬eld modes, so Hint can often be treated as a weak perturbation.

4.3 Quantum Maxwell™s equations
In Section 4.2 the interaction between the radiation ¬eld and charged particles was
described in the Schr¨dinger 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 ¬elds and the equal-time, canonical commuta-
tors, [rni (t), plj (t)] = i δnl δij , for the charged particles. After a bit of algebra, the
Heisenberg equations are found to be
‚A (r, t)
E (r, t) = ’ , (4.33)
‚t
‚B (r, t)
∇ — E (r, t) = ’ , (4.34)
‚t
1 ‚E (r, t)
= µ0 j⊥ (r, t) ,
∇ — B (r, t) ’ 2 (4.35)
c ‚t
pn (t) ’ qn A (rn (t) , t)
drn (t)
vn (t) ≡ = , (4.36)
dt Mn
dpn (t)
= qn E (rn (t) , t) + qn vn (t) — B (rn (t) , t) ’ qn ∇¦ (rn (t)) , (4.37)
dt
where vn (t) is the velocity operator for the nth particle, j⊥ (r, t) is the transverse
part of the current density operator

j (r, t) = δ (r’rn (t)) qn vn (t) , (4.38)
n

and the Coulomb potential operator is
1 ql
¦ (rn (t)) = . (4.39)
|rn (t) ’ rl (t)|
4π 0
l=n
½½ Interaction of light with matter

This potential is obtained from a solution of Poisson™s equation ∇2 ¦ = ’ρ/ 0 , where
the charge density operator is

ρ (r, t) = δ (r’rn (t)) qn , (4.40)
n

by omitting the self-interaction terms encountered when r ’ rn (t). Functions f (rn )
of the position operators rn , such as those in eqns (4.35)“(4.40), are de¬ned by

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

where ψ (r1 , . . . , rN ) is any N -body wave function for the charged particles.
The ¬rst equation, eqn (4.33), is simply the relation between the transverse part of
the electric ¬eld operator and the vector potential. Faraday™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`re™s law). The Heisenberg equation of motion for E can be put into the
e
form
pni ’ qn Ai (rn (t) , t)
1 ‚Ej (r, t)
∆⊥ (r ’ rn (t)) qn
(∇ — B (r, t))j ’ = µ0 ,
ji
c2 ‚t Mn
n
(4.42)
but the signi¬cance of the right-hand side is not immediately obvious. Further insight
can be achieved by using the de¬nition (4.36) of the velocity operator to get
pn (t) ’ qn A (rn (t) , t)
= vn (t) . (4.43)
Mn
Substituting this into eqn (4.42) yields
1 ‚Ej (r, t)
∆⊥ (r ’ rn ) qn vni (t)
(∇ — B (r, t))j ’ = µ0 ji
c2 ‚t n

d3 r ∆⊥ (r ’ r ) ji (r , t) ,
= µ0 (4.44)
ji


where ji (r , t), de¬ned by eqn (4.38), can be interpreted as the current density oper-
ator. The transverse delta function ∆⊥ projects out the transverse part of any vector
ji
¬eld, so the Heisenberg equation for E (r, t) is given by eqn (4.35).

Parity and time reversal—
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 ¬eld operators and the
classical ¬elds behave in the same way under the discrete transformations:

r ’ ’r (spatial inversion or parity transformation) ,
(4.45)
t ’ ’t (time reversal) .
Parity and time reversal— ½½

Thus the transformation laws for the classical ¬elds”see Appendix B.3.3”also apply
to the ¬eld operators; in particular,

E (r, t) ’ EP (r, t) = ’E (’r, t) under r ’ ’r , (4.46)
E (r, t) ’ ET (r, t) = E (r, ’t) under t ’ ’ 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 ¬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 ¬nd the transformation rules for
the states that correspond to the transformations (4.46) and (4.47) for the operators.
This e¬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 | ¦ |Ψ | , where |Ψ and |¦ are normalized
state vectors. For this reason, a mapping of state vectors to state vectors,

|˜ ’ |˜ , (4.48)

is called a symmetry transformation if
2 2
| ¦ |Ψ | = | ¦ |Ψ | , (4.49)

for any pair of vectors |Ψ and |¦ . In other words, symmetry transformations leave
all physical predictions unchanged. The consequences of this de¬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) |Ψ ’ |Ψ = U |Ψ , where U is a unitary operator;
(b) |Ψ ’ |Ψ = Λ |Ψ , where Λ is an antilinear and antiunitary operator.

The unfamiliar terms in alternative (b) are de¬ned as follows. A transformation Λ
is antilinear if
Λ {± |Ψ + β |¦ } = ±— Λ |Ψ + β — Λ |¦ , (4.50)
and antiunitary if

¦ |Ψ = Ψ |¦ = ¦ |Ψ , where |Ψ = Λ |Ψ and |¦ = Λ |¦ . (4.51)

Rather than present the proof of Wigner™s theorem”which can be found in Wigner
(1959, cf. Appendices in Chaps 20 and 26) or Bargmann (1964)”we will attempt to
gain some understanding of its meaning. To this end consider another transformation
given by
|Ψ ’ |Ψ = exp (iθΨ ) |Ψ , (4.52)
where θΨ is a real phase that can be chosen independently for each |Ψ . For any value of
θΨ it is clear that |Ψ ’ |Ψ is also a symmetry transformation. Furthermore, |Ψ
and |Ψ di¬er only by an overall phase, so they represent the same physical state.
½¾¼ Interaction of light with matter

Thus the symmetry transformations de¬ned by eqns (4.48) and (4.52) are physically
equivalent, and the meaning of Wigner™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 ¬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 :

EP (r, t) = UP E (r, t) UP = ’E (’r, t) . (4.53)
In the interaction picture, E (r, t) has the plane-wave expansion

ωk
aks eks ei(k·r’ωk t) + HC ,
E (r, t) = i (4.54)
2 0V
ks

and the corresponding classical ¬eld has an expansion of the same form, with aks re-
placed by the classical amplitude ±ks . In Appendix B.3.3, it is shown that the parity
transformation law for the classical amplitude is ±P = ’±’k,’s . Since UP is linear,
ks
† †
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:

aks ’ aP = UP aks UP = ’a’k,’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 † = ’a’k,’s , ’a† ,’s = δkk δss ,
ks k s ’k
(4.56)
= [’a’k,’ss , ’a’k ,’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

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

The classical transformation rule for the plane-wave amplitudes is ±T = ’±— , so
ks ’k,s
the argument used for the parity transformation implies that the annihilation operators
satisfy
aks ’ aT = aT = UT aks UT = ’a†

’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
½¾½
Stationary density operators


a T , aT † = ’a† , ’a’k ,s = ’δkk δss (4.59)
ks k s ’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 †
ks ks

cannot be related to the original operators aks and aks by a unitary transformation,
and UT does not exist.
According to Wigner™s theorem, the only possibility left is that the time-reversed
operators are de¬ned by an antiunitary transformation,
ET (r, t) = ΛT E (r, t) Λ’1 = E (r, ’t) . (4.60)
T

Here some caution is required because of the unfamiliar properties of antilinear trans-
formations. The de¬nition (4.50) implies that ΛT ± |Ψ = ±— ΛT |Ψ for any |Ψ , so
applying ΛT to the expansion (4.54) for E (r, t) gives us
T
ωk
a†
ΛT E (r, t) Λ’1 ’aT e— e’i(k·r’ωk t) eks ei(k·r’ωk t) ,
= i +
ks ks ks
T
2 0V
ks
(4.61)
where
T
a† = ΛT a† Λ’1 .
aT = ΛT aks Λ’1 , (4.62)
ks ks ks T
T

Setting t ’ ’t in eqn (4.54) and changing the summation variable by k ’ ’k yields
ωk
a’ks e’ks e’i(k·r’ωk t) ’ a† e— ei(k·r’ωk t) .
E (r, ’t) = i (4.63)
’ks ’ks
2 0V
ks

After substituting these expansions into eqn (4.60) and using the properties e’k,’s =
eks and e— = ek,’s derived in Appendix B.3.3, one ¬nds
k,s
T
a† = ’a†
= ’a’k,s ,
aT ’k,s . (4.64)
ks ks

T
This transformation rule gives us a† = aT † and
ks ks


aT , aT †s = ’a’k,s , ’a† ,s = δkk δss ; (4.65)
ks k ’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 ΛT to the c-number coe¬cients in eqn (4.61). This is just what is needed
to ensure that aT is proportional to a’k,s rather than to a† , as in eqn (4.58).
ks ’k,s

4.5 Stationary density operators
The expectation value of a single observable is given by
X (t) = Tr [ρX (t)] = Tr [ρ (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 Schr¨dinger
o
½¾¾ Interaction of light with matter

picture. The time dependence simpli¬es for the important class of stationary density
operators, which are de¬ned by requiring the Schr¨dinger-picture ρ (t) to be a constant
o
of the motion. According to eqn (3.75) this means that ρ (t) is independent of time,
so the Schr¨dinger- and Heisenberg-picture density operators are identical. Stationary
o
density operators have the useful property

ρ, U † (t) = 0 = [ρ, U (t)] , (4.67)

which is equivalent to
[ρ, 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 [ρX (t)] = Tr (ρX) = X . (4.69)

Correlations between observables at di¬erent times are described by averages of
the form
X (t + „ ) Y (t) = Tr [ρX (t + „ ) Y (t)] . (4.70)

For a stationary density operator, the correlation only depends on the di¬erence in the
time arguments. This is established by combining U (’t) = U † (t) with eqns (3.83),
(4.67), and cyclic invariance to get

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


4.6 Positive- and negative-frequency parts for interacting ¬elds
When charged particles are present, the Hamiltonian is given by eqn (4.28), so the
free-¬eld solution (3.95) is no longer valid. The operator aks (t)”evolving from the
annihilation operator, aks (0) = aks ”will in general depend on the (Schr¨dinger-o

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 (’) (r, t) (4.72)

will remain valid provided that the initial operator F (+) (r, 0) (F (’) (r, 0)) is a sum
over annihilation (creation) operators, but the commutation relations (3.102) are only
valid for equal times. Furthermore, F (+) (r, ω) will not generally vanish for all negative
values of ω. Despite this failing, an operator F (+) (r, t) that evolves from an initial
operator of the form
F (+) (r, 0) = Fks aks eik·r (4.73)
ks

is still called the positive-frequency part of F (r, t).
½¾¿
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 ¬elds at di¬erent times. This comes about
because the state is represented by a time-independent density operator ρ, while the
¬eld operators evolve in time according to the Heisenberg equations.
Since the electric ¬eld is a vector, it is natural to de¬ne the ¬rst-order ¬eld
correlation function by the tensor

(1) (’) (+)
Gij (x1 ; x2 ) = Ei (x1 ) Ej (x2 ) , (4.74)

where X = Tr [ρX] and x1 = (r1 , t1 ), etc. The ¬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 „ = |t1 ’ t2 | between its two arms. In Section 9.1.2-C we will show
that the spectral density for the ¬eld state ρ 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, de¬ned
by
(2) (’) (’) (+) (+)
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 de¬ned
(’) (’)
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 de¬nes some preferred polarization
directions”represented by unit vectors v1 , v2 , . . .”and the tensor correlation func-
tions are replaced by scalar functions
(’) (+)
G(1) (x1 ; x2 ) = E1 (x1 ) E2 (x2 ) , (4.76)

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


(+)
where Ep = vp · E(+) is the projection of the vector operator onto the direction vp .
For example, observing a ¬rst-order interference pattern through a polarization ¬lter
is described by
G(1) (x; x) = e · E(’) (x) e— · E(+) (x) , (4.78)

where e is the polarization transmitted by the ¬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 ’ tp + „, tp ’ tp + „ , of all the time arguments. In particular Gij (r, t; r , t ) =
½¾ Interaction of light with matter

(1)
Gij (r, t ’ t ; r , 0), so the ¬rst-order function only depends on the di¬erence, t ’ t , of
the time arguments.
The correlation functions satisfy useful inequalities that are based on the fact that
Tr ρF † F 0, (4.79)
where F is an arbitrary observable and ρ is a density operator. This is readily proved
by evaluating the trace in the basis in which ρ is diagonal and using Ψ F † F Ψ 0.
Choosing F = E (+) (x) in eqn (4.79) gives
G(1) (x; x) 0, (4.80)
(+) (+)
(x1 ) · · · En
and the operator F = E1 (xn ) gives the general positivity condition
G(n) (x1 , . . . , xn ; x1 , . . . , xn ) 0. (4.81)
A di¬erent sort of inequality follows from the choice
n
(+)
F= ξa Ea (xa ) , (4.82)
a=1

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

ξa ξb Fab 0, (4.83)
a=1 b=1

where F is the n — n hermitian matrix
Fab = G(1) (xa ; xb ) . (4.84)
Since the inequality (4.83) holds for all complex ξa s, the matrix F is positive de¬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 ¬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 ¬eld is much smaller than the energies of individual photons. It is therefore
useful to rewrite the Schr¨dinger-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 Schr¨dinger equation with the full Hamiltonian H (S)
o
½¾
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 de¬ned by the unitary transformation

Ψ(I) (t) = U0 (t) Ψ(S) (t) , (4.88)

where the unitary operator,
i (t ’ t0 ) (S)
U0 (t) = exp ’ H0 , (4.89)

satis¬es
‚ (S)
i
U0 (t) = H0 U0 (t) , U0 (t0 ) = 1 . (4.90)
‚t
Thus the Schr¨dinger 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 Schr¨dinger equation
o
(S)
reveals that this transformation e¬ectively undoes the fast evolution due to H0 . By
contrast to the Heisenberg picture de¬ned in Section 3.2, the transformed ket vector
(S)
still depends on time due to the action of Hint . The consistency condition,

Ψ(I) (t) X (I) (t) ¦(I) (t) = Ψ(S) (t) X (S) ¦(S) (t) , (4.91)

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

X (I) (t) = U0 (t) X (S) U0 (t) . (4.92)
(S)
For H0 this yields the simple result

(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 Ψ(I) (t) obeys the interaction-picture Schr¨dinger
o
equation
‚ †
(S) (S) (S)
Ψ(I) (t) = ’H0 Ψ(I) (t) + U0 (t) H0 + Hint Ψ(S) (t)
i
‚t

(S) (S) (S)
= ’H0 Ψ(I) (t) + U0 (t) H0 U0 (t) Ψ(I) (t)
+ Hint
(I)
= Hint (t) Ψ(I) (t) , (4.94)

which follows from operating on both sides of eqn (4.88) with i ‚/‚t and using eqns
(4.90)“(4.93). The formal solution is

Ψ(I) (t) = V (t) Ψ(I) (t0 ) , (4.95)
½¾ Interaction of light with matter

where the unitary operator V (t) satis¬es

‚ (I)
i V (t) = Hint (t) V (t) , with V (t0 ) = 1 . (4.96)
‚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 su¬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
‚ (I)
X (t) = X (I) (t) , H0 .
i (4.97)
‚t
These are the Heisenberg equation for free ¬elds, so we can use eqns (3.95) and (3.96)
to get
aks (t) = aks e’iωk (t’t0 ) ,
(I) (S)
(4.98)
and
(S)
A(I)(+) (r, t) = aks eks ei[k·r’ωk (t’t0 )] . (4.99)
2 0 ωk V
ks

In the same way eqn (3.102) implies

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

where F and G are any of the ¬eld components and (r, t), (r , t ) are any pair of
space“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 de¬nition
(3.85) of the Schr¨dinger-picture density operator leads to
o

‚ (I) (I)
ρ (t) = Hint (t), ρ(I) (t) ,
i (4.101)
‚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) , |Ψ (t) means Ψ(I) (t) , and
ρ (t) means ρ(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 ¬rst turn eqn (4.96) into
an integral equation by integrating over the interval (t0 , t) to get
t
i
V (t) = 1 ’ dt1 Hint (t1 ) V (t1 ) . (4.102)
t0
½¾
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 ’ dt1 Hint (t1 ) + ’ dt2 Hint (t1 ) Hint (t2 ) + · · ·
dt1
t0 t0 t0

V (n) (t) ,
= (4.103)
n=0

where V (0) = 1, and
n t tn’1
i
’ dt1 · · · dtn Hint (t1 ) · · · Hint (tn ) ,
(n)
V (t) = (4.104)
t0 t0

for n 1.
If the system (charges plus radiation) is initially in the state |˜i then the prob-
ability amplitude that a measurement at time t leaves the system in the ¬nal state
|˜f is
Vf i (t) = ˜f |Ψ (t) = ˜f |V (t)| ˜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 ¬nite time
t. The initial state |˜i evolves into V (t) |˜i , and its projection on the ¬nal state
|˜f is ˜f |V (t)| ˜i . Let the initial and ¬nal states be eigenstates of the unperturbed
Hamiltonian H0 , with energies Ei and Ef respectively. According to eqn (4.104) the
¬rst-order contribution to ˜f |V (t)| ˜i is
t
i
(1)
Vf i (t) = ’ dt1 ˜f |Hint (t1 )| ˜i
0
t
i
=’ dt1 ˜f |Hint | ˜i exp (iνf i t1 ) , (4.107)
0

where we have used eqn (4.92) and introduced the notation νf i = (Ef ’ Ei ) / . Eval-
uating the integral in eqn (4.107) yields the amplitude

2i sin (νf i t/2)
(1)
Vf i (t) = ’ ˜f |Hint | ˜i ,
exp (iνf i t/2) (4.108)
νf i

so the transition probability is
2 4
(1) 2
| ˜f |Hint | ˜i | ∆ (νf i , t) ,
Pf i (t) = Vf i (t) = (4.109)
2


where ∆ (ν, t) ≡ sin2 (νt/2) /ν 2 .
½¾ Interaction of light with matter

2
For ¬xed t, the maximum value of |∆ (ν, t)| is t2 /4, and it occurs at ν = 0. The
width of the central peak is approximately 2π/t, so as t becomes large the function
is strongly peaked at ν = 0. In order to specify a well-de¬ned ¬nal energy, the width
must be small compared to |Ef ’ Ei | / ; therefore,


t (4.110)
|Ef ’ Ei |

de¬nes the limit of large times. This is a realization of the energy“time uncertainty
relation, t∆E ∼ (Bransden and Joachain, 1989, Sec. 2.5). With this understanding
of in¬nity, we can use the easily established mathematical result,

sin2 (νt/2)
∆ (ν, t) π
lim = lim = δ (ν) , (4.111)
2
t tν 2
t’∞ t’∞


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

2π 2
t | ˜f |Hint | ˜i | δ (νf i )
Pf i (t) = 2
2π 2
t | ˜f |Hint | ˜i | δ (Ef ’ Ei ) .
= (4.112)

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


| ˜f |Hint | ˜i |2 δ (Ef ’ Ei ) .
Wf i = (4.113)

This is Fermi™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 ¬nal states to get the total transition probability Pi,tot (t) = tWi,tot ,
where the total transition rate is
2π 2
| ˜f |Hint | ˜i | δ (Ef ’ 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 ’ ∞ 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 ’ Ei | Ef .
½¾
The interaction picture

With this understanding in mind, a simpli¬ed version of the previous calculation is
possible. For this purpose, we choose t0 = ’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) = ’ ˜f |Hint | ˜i exp (iνf i T /2)
Vf i dt1 exp (iνf i t1 ) . (4.115)
’T /2

The standard result
T /2
dt1 eiνt1 = 2πδ (ν)
lim (4.116)
T ’∞ ’T /2

allows this to be recast as
2πi
(1) (1)
Vf i = Vf i (∞) = ’ ˜f |Hint | ˜i δ (νf i ) , (4.117)

so the transition probability is
2
2 2π
(1)
| ˜f |Hint | ˜i |2 [δ (νf 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 physicist™s delta function, so we can use
eqn (4.116) once more to set
T /2
dt1 T
2
[δ (νf i )] = δ (νf i ) exp (iνf i t1 ) = δ (νf i ) . (4.119)
2π 2π
’T /2

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

4.8.3 Second-order perturbation theory
Using the simpli¬ed scheme, presented in eqns (4.115)“(4.119), yields the second-order
contribution to ˜f |V (T /2)| ˜i :
2 T /2 t1
i
(2)
’ dt2 ˜f |Hint (t1 ) Hint (t2 )| ˜i
Vf i = dt1
’T /2 ’T /2
2 T /2 T /2
i
’ dt2 θ (t1 ’ t2 ) ˜f |Hint (t1 ) Hint (t2 )| ˜i , (4.120)
= dt1
’T /2 ’T /2

where θ (t1 ’ t2 ) is the step function discussed in Appendix A.7.1 . By introducing a
basis set {|Λu } of eigenstates of H0 , the matrix element can be written as

˜f |Hint (t1 ) Hint (t2 )| ˜i = exp [(iνf i ) T /2] ˜f |Hint | Λu Λu |Hint | ˜i
u
— exp (iνf u t1 ) exp (iνui t2 ) , (4.121)

where we have used eqn (4.92) and the identity νf u + νui = νf i . The ¬nal step is to
use the representation (A.88) for the step function and eqn (4.116) to ¬nd
½¿¼ Interaction of light with matter


˜f |Hint | Λu Λu |Hint | ˜i
i iνf i T /2
(2)
=’
Vf i e dν
2π 2 ν+i
’∞ u
— 2πδ (νf u ’ ν) 2πδ (νui + ν) . (4.122)

Carrying out the integration over ν with the aid of the delta functions leads to
˜f |Hint | Λu Λu |Hint | ˜i
2πi
(2)
Vf i = ’ δ (νf i )
2 νf u + i
u
˜f |Hint | Λu Λu |Hint | ˜i
= ’2πi δ (Ef ’ Ei ) . (4.123)
Ef ’ Eu + i
u

Finally, another use of the rule (4.119) yields the transition rate
2
˜f |Hint | Λu Λu |Hint | ˜i

δ (Ef ’ Ei ) .
Wf i = (4.124)
Ef ’ 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 » > 100 nm, whereas typical atoms have diameters a ≈ 0.1 nm.
The large disparity between atomic diameters and optical wavelengths (a/» < 0.001)
permits the use of the dipole approximation, and this in turn brings about important
simpli¬cations in the general Hamiltonian de¬ned by eqns (4.28)“(4.32).
The simpli¬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 su¬ciently
dilute systems of atoms”e.g. tenuous atomic vapors”since the interaction between
atoms is weak. Experiments with vapors were the rule in the early days of quantum
optics, but in many modern applications”such as solid-state detectors and solid-state
lasers”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 delocalized”e.g. in the conduction band of a semiconductor”so that the validity
of the dipole approximation is in doubt. These considerations”while very important
in practice”do not in fact require signi¬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 e¬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).
½¿½
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 atom”this means any atom
with atomic number Z > 1. Fortunately, the complete description provided by the
many-electron wave function ψ (r1 , . . . , rZ ) is not needed. For the most part, only
selected properties”such as the discrete electronic energies and the matrix elements of
the dipole operator”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 ¬eld.
For a single atom interacting with the electromagnetic ¬eld, the discussion in Sec-
tion 4.2.1 shows that the state space is the tensor product H = HA — HF of the Hilbert
space HA for the atom and the Fock space HF for the ¬eld. A typical basis state for
H is |ψ, ¦ = |ψ |¦ , where |ψ and |¦ are respectively state vectors for the atom
and the ¬eld. Let us consider a typical matrix element ψ, ¦ |E (r)| ψ , ¦ of the elec-
tric ¬eld operator, where at least one of the vectors |ψ and |ψ describes a bound
state with characteristic spatial extent a, and |¦ and |¦ both describe states of the
¬eld containing only photons with wavelengths » 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 identi¬ed with the location
of the nucleus, since Mnuc ZMe . The plane-wave expansion (3.69) for the electric
¬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 ,

ψ, ¦ |E (r)| ψ , ¦
= ψ, ¦ |E (rcm )| ψ , ¦ + ψ, ¦ |[(r ’ rcm ) · ∇] E (rcm )| ψ , ¦ + · · · .
(4.126)
With the understanding that only matrix elements of this kind will occur, the
expansion can be applied to the ¬eld operator itself:

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

The electric dipole approximation retains only the leading term in this expan-
sion, with errors of O (a/»). 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 ¬eld
E is ’d · E, where d is the electric dipole moment. For an atom the dipole operator is
½¿¾ Interaction of light with matter

Z
(’e) (rn ’ 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 = ’d · 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

VC = , (4.131)
|rn ’ rl | 4π 0 |rn ’ rnuc |
4π 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 Schr¨dinger-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 ¬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 = ’d · E (rcm ) , (4.134)

which acts jointly on the ¬eld states and the internal states of the atom.
In the rest frame of the atom, de¬ned by P = 0, the energy eigenstates

Hat |µq = µq |µq (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 su¬cient to specify the
internal atomic state uniquely. The qs are discrete; therefore, they can be ordered so
that µq µq for q < q .
½¿¿
Interaction of light with atoms

In practice, the many-electron wave function ψq (r1 , . . . , rZ ) = r1 , . . . , rZ |µq 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

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