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(1) Discard all terms in Hint that do not conserve energy in a ¬rst-order perturbation
calculation.
(2) Multiply the coupling constants in the remaining terms by the cut-o¬ function
K (∆k ).
It is also useful to note that this rule mandates that each term in Hrwa is the prod-
uct of an energy-raising (-lowering) operator for the atom with an energy-lowering
(-raising) operator for the ¬eld. We emphasize that the discarded part, H (ar) , is not
unphysical; it simply does not contribute to the ¬rst-order transition amplitude. The
antiresonant Hamiltonian H (ar) can and does contribute in higher orders of perturba-
tion theory, but the time averaging argument shows that Hrwa is the dominant part
of the Hamiltonian for long-term evolution under the in¬‚uence of weak ¬elds.

11.1.4 Multilevel atoms
Our object is this section is to introduce a family of operators that play the role of the
Pauli matrices for an atom with more than two active levels. We will only consider the
interaction of the ¬eld with a single atom, since the generalization to the many-atom
case is straightforward. The atomic transition operators Sqp are de¬ned by
Sqp = |µq µp | , (11.28)
where |µq and |µp are eigenstates of Hat . As explained in Appendix C.1.2, this nota-
tion means that the operator Sqp projects any atomic state |Ψ onto |µq with coe¬cient
µp |Ψ , i.e.
Sqp |Ψ = |µq µp |Ψ . (11.29)
When this de¬nition is applied to the two-level case, it is easy to see that S21 =
σ+ , S12 = σ’ , and S22 ’ S11 = σz . The energy eigenvalue equation for the states,
Hat |µq = µq |µq , implies the operator eigenvalue equation [Sqp , Hat ] = ’ ωqp Sqp for
Sqp , so the transition operators are sometimes called eigenoperators.
The eigenstates |µq of Hat satisfy the completeness relation

|µq µq | = IA , (11.30)
q

where IA is the identity operator in HA ; therefore,
O ≡ IA OIA = µq |O| p Sqp . (11.31)
q p

Thus the Sqp s form a complete set for the expansion of any atomic operator, just as
every 2 — 2 matrix can be expressed as a linear combination of Pauli matrices.
The algebraic properties

Sqp = Spq , (11.32)
Sqp Sq p = δpq Sqp , (11.33)
[Sqp , Sq p ] = {δpq Sqp ’ δp q Sq p } (11.34)
are readily derived by using the orthogonality of the eigenstates. The special case q = p
and q = p of eqn (11.33) shows that the Sqq s are a set of orthogonal projection oper-
ators for the atom. For any atomic state |Ψ , eqn (11.29) yields Sqq |Ψ = |µq µq |Ψ ,
¿ Coherent interaction of light with atoms

i.e. Sqq projects out the |µq component of |Ψ . The Sqq s are called population oper-
ators, since the expectation value,
2
Ψ |Sqq | Ψ = | µq |Ψ | , (11.35)

is the probability for ¬nding the value µq , and the corresponding eigenstate |µq , in
a measurement of the energy of an atom prepared in the state |Ψ . Because of the
convention that q > p implies µq > µp , the operator Sqp for q > p is called a rais-
ing operator. It is analogous to the angular momentum raising operator, or to the
creation operator a† for a photon. By the same token, Spq = Sqp is a lowering op-

ks
erator, analogous to the lowering operator for angular momentum, or to the photon
annihilation operator aks .
In this representation the atomic Hamiltonian in the Schr¨dinger picture has the
o
simple form
Hat = µq Sqq , (11.36)
q

and the interaction Hamiltonian is given by

Hint = ’ Sqp dqp · E (0) , (11.37)
q,p


where dqp = µq d µp . Since dqq = 0, the sum over q and p splits into two parts
with q > p and p > q. Combining this with E = E(+) + E(’) leads to an expression
involving four sums. After interchanging the names of the summation indices in the
q < p sums, the result can be arranged as follows:
(r) (ar)
Hint = Hint + Hint , (11.38)
(r)
Hint = ’ Sqp dqp · E(+) (0) + HC ,
q>p
(11.39)
(ar)
=’ Sqp dqp · E (’)
Hint (0) + HC .
q>p

(r)
In Hint the raising (lowering) operator Sqp (Spq ) is associated with the annihilation
(ar)
(creation) operator E(+) E(’) , while the opposite pairing appears in Hint . It is
not necessary to carry out the explicit time averaging procedure; the results of the
two-level problem have already provided us with a general rule for writing down the
RWA Hamiltonian. Since all antiresonant terms are to be discarded, we can dispense
(ar)
with Hint and set
Hrwa = ’ Sqp dqp · E(+) (0) + HC . (11.40)
q>p

Expanding the ¬eld operator in plane waves yields the equivalent form

Hrwa = ’i gqp,ks Sqp aks + HC , (11.41)
ks q>p
¿
Spontaneous emission II

where the coupling frequencies,
dqp · eks
ωk
gqp,ks = K (ωqp ’ ωk ) , (11.42)
2 0V
include the cut-o¬ function, so that only those terms satisfying a resonance con-
dition |ωqp ’ ωk | < wK will contribute to the RWA interaction Hamiltonian. The
Schr¨dinger-picture form in eqn (11.41) becomes
o

gqp,ks ei(ωqp ’ωk )t Sqp aks + HC
Hrwa (t) = ’i (11.43)
ks q>p

in the interaction picture.

11.2 Spontaneous emission II
11.2.1 Propagation of spontaneous emission
The discussion of spontaneous emission in Section 4.9.3 is concerned with the calcu-
lation of the rate of quantum jumps associated with the emission of a photon. This
approach does not readily lend itself to answering other kinds of questions. For ex-
ample, if an atom at the origin is prepared in its excited state at t = 0, what is the
earliest time at which a detector located at a distance r can register the arrival of a
photon? Questions of this kind are best answered by using the Heisenberg picture.
Since the Heisenberg, Schr¨dinger, and interaction pictures all coincide at t = 0,
o
the interaction Hamiltonian in the Heisenberg picture can be inferred from eqn (11.25)
by setting t = 0 in the exponentials. The total Hamiltonian in the resonant wave
approximation is therefore
H = Hat + Hem + Hrwa , (11.44)
ω21
ωk a† (t) aks (t) ,
Hat = σz (t) , Hem = (11.45)
ks
2
ks

gks σ+ (t) aks (t) ’ g— σ’ (t) a† (t) ,
Hrwa = ’i (11.46)
ks ks
ks
where the operators are all evaluated in the Heisenberg picture. The Heisenberg equa-
tions of motion,
d
gks σ+ (t) aks (t) + g— σ’ (t) a† (t) ,
σz (t) = ’2 (11.47)
ks ks
dt
ks

d
σ’ (t) = ’iω21 σ’ (t) + gks aks (t) σz (t) , (11.48)
dt
ks
d
aks (t) = ’iωk aks (t) + g— σ’ (t) , (11.49)
ks
dt
show that the ¬eld operators aks (t) and the atomic operators σ (t), which are inde-
pendent at t = 0, are coupled at all later times. For this reason, it is usually impossible
to obtain closed-form solutions.
¿ Coherent interaction of light with atoms

Let us study the time dependence of the ¬eld emitted by an initially excited atom.
In the Heisenberg picture, the plane-wave expansion (3.69) for the positive-frequency
part of the ¬eld is

ωk
E(+) (r, t) = aks (t) eks eik·r ,
i (11.50)
2 0V
ks

so we begin by using the standard integrating factor method to get the formal solution,
t
’iωk t
g— dt e’iωk (t’t ) σ’ (t ) ,
aks (t) = aks (0) e + (11.51)
ks
0

of eqn (11.49). Substituting this into eqn (11.50) gives E(+) (r, t) as the sum of two
terms:
(+)
E(+) (r, t) = E(+) (r, t) + Erad (r, t) , (11.52)
vac

where
ωk
E(+) (r, t) = aks (0) eks ei(k·r’ωk t)
i (11.53)
vac
2 0V
ks

describes vacuum ¬‚uctuations and
t
ωk —
dt e’iωk (t’t ) σ’ (t )
(+)
gks eks eik·r
Erad (r, t) = i (11.54)
2 0V 0
ks

represents the ¬eld radiated by the atom. The state vector,

|in = |µ2 , 0 = |µ2 |0 , (11.55)

describes the situation with the atom in the excited state and no photons in the ¬eld.
In Section 9.1 we saw that the counting rate for a detector located at r is proportional
(+)
to in E(’) (r, t) · E(+) (r, t) in . Since |in is the vacuum for photons, Evac (r, t) will
(’) (+)
not contribute, and the counting rate is proportional to in Erad (r, t) · Erad (r, t) in .
(+)
Calculating the atomic radiation operator Erad (r, t) from eqn (11.54) requires an
evaluation of the sum over polarizations, followed by the conversion of the k-sum to an
integral, as outlined in Exercise 11.3. After carrying out the integral over the directions
of k, the result is
k 2 dk ωk K (ωk ’ ω21 ) — (d— · ∇) ∇ 4π sin (kr)
(+)
Erad (r, t) = i d+
(2π)3 k2
20 kr
t
dt e’iωk (t’t ) σ’ (t ) .
— (11.56)
0

The cut-o¬ function K (ωk ’ ω21 ) imposes k ≈ k21 = ω21 /c, so we can de¬ne the
radiation zone by kr ≈ k21 r 1. For a detector in the radiation zone,
1 4π sin (kr) 4π sin (kr) — 1
d— + (d— · ∇) ∇ = d +O , (11.57)
k2 k 2 r2
kr kr
¿
Spontaneous emission II

where
d— = d— ’ (r · d— ) r = (d— — r) — r (11.58)
is the component of d— transverse to the vector r linking the atom to the detector.
This is the same as the rule for the polarization of radiation emitted by a classical
dipole (Jackson, 1999, Sec. 9.2). After changing the integration variable from k to
ω = ωk = ck, we ¬nd

d— ωr
i
(+)
dωω 2 K (ω ’ ω21 ) sin
Erad (r, t) =
4π 2 c2 0r c
0
t
dt e’iω(t’t ) σ’ (t ) .
— (11.59)
0

Approximating the slowly-varying factor ω 2 by ω21 , and unpacking sin (kr), yields the
2

expression
k 2 d—
(+)
[I (r) ’ I (’r)]
Erad (r, t) = 21 (11.60)
8π 2 0 r
for the ¬eld, where

t
dωK (ω ’ ω21 ) eiωr/c e’iω(t’t ) σ’ (t )
I (r) = dt
0 0

t
ik21 r ’iω21 t
dωK (ω) eiω[r/c’(t’t )] eiω21 t σ’ (t ) .
=e e dt (11.61)
’ω21
0

ω21 allows us to extend the lower limit of the ω-integral to ’∞
The condition wK
with negligible error, so
∞ ∞

dωK (ω) eiω„ ≈ 2π K (ω) eiω„
’∞ 2π
’ω21
= 2π („ ) , (11.62)

where („ ) is the averaging function introduced in eqn (11.17). The results derived
in Exercise 11.4 include the fact that

σ ’ (t ) = eiω21 t σ’ (t ) (11.63)

is a slowly-varying envelope operator, i.e. it varies on the time scale set by |gks |.
Combining these observations with the approximate delta function rule (11.21) leads
to
t
I (r) = 2πeik21 r e’iω21 t dt δ (r/c ’ (t ’ t )) σ ’ (t )
0
ik21 r ’iω21 t
σ ’ (t ’ r/c) ,
= 2πe e (11.64)

and
t
’ik21 r ’iω21 t
dt δ (’r/c ’ (t ’ t )) σ ’ (t ) = 0 .
I (’r) = 2πe e (11.65)
0
¿¼ Coherent interaction of light with atoms

The ¬nal result for the radiated ¬eld is
k21 d— eik21 r ’iω21 t
2
(+)
σ ’ (t ’ r/c) .
Erad (r, t) = e (11.66)
4π 0 r
Thus the ¬eld operator behaves as an expanding spherical wave with source given by
the atomic dipole operator at the retarded time t ’ r/c. Just as in the classical theory,
the detector will not ¬re before the ¬rst arrival time t = r/c. We should emphasize
that this fundamental result does not depend on the resonant wave approximation
and the other simpli¬cations made here. A rigorous calculation leading to the same
conclusion has been given by Milonni (1994).

11.2.2 The Weisskopf“Wigner method
The perturbative calculation of the spontaneous emission rate can apparently be im-
proved by including higher-order terms from eqn (4.103). Since the initial and ¬nal
states are ¬xed, these terms must describe virtual emission and absorption of pho-
tons. In other words, the higher-order terms”called radiative corrections”involve
vacuum ¬‚uctuations. We know, from Section 2.5, that the contributions from vacuum
¬‚uctuations are in¬nite, so it will not come as a surprise to learn that all of the integrals
de¬ning the higher-order contributions are divergent.
A possible remedy would be to include the cut-o¬ function K (∆k ), in the coupling
frequencies, i.e. to replace Gks by gks . This will cure the divergent integrals, but it must
then be proved that the results do not depend on the detailed shape of K (∆k ). This
can be done, but only at the expense of importing the machinery of renormalization
theory from quantum electrodynamics (Greiner and Reinhardt, 1994).
A more important drawback of the perturbative approach is that it is only valid
1/ |gks | ≈ „sp = 1/A2’1 . Thus perturbation theory
in the limited time interval t
cannot be used to follow the evolution of the system for times comparable to the spon-
taneous decay time. We will use the RWA to pursue a nonperturbative approach (see
Cohen-Tannoudji et al. (1977b, Complement D-XIII), or the original paper Weisskopf
and Wigner (1930)) which can describe the behavior of the atom“¬eld system for long
times, t > „sp .
The key to this nonperturbative method is the following simple observation. In the
resonant wave approximation, the atom“¬eld state |µ2 ; 0 , in which the atom is in the
excited state and there are no photons, can only make transitions to one of the states
|µ1 ; 1ks , in which the atom is in the ground state and there is exactly one photon
present. Conversely, the state |µ1 ; 1ks can only make a transition into the state |µ2 ; 0 .
This is demonstrated more explicitly by using eqn (11.25) for Hrwa to ¬nd

g— ei∆k t |µ1 ; 1ks ,
Hrwa (t) |µ2 ; 0 = i (11.67)
ks
ks

and
Hrwa (t) |µ1 ; 1ks = ’i gks e’i∆k t |µ2 ; 0 . (11.68)
Consequently, the spontaneous emission subspace

Hse = span {|µ2 ; 0 , |µ1 ; 1ks for all ks} (11.69)
¿½
Spontaneous emission II

is sent into itself by the action of the RWA Hamiltonian: Hrwa (t) Hse ’ Hse . This
means that an initial state in Hse will evolve into another state in Hse . The time-
dependent state can therefore be expressed as

C1ks (t) e’i∆k t |µ1 ; 1ks ,
|˜ (t) = C2 (t) |µ2 ; 0 + (11.70)
ks

where the exponential in the second term is included to balance the explicit time depen-
dence of the interaction-picture Hamiltonian. Substituting this into the Schr¨dinger
o
equation (11.13) produces equations for the coe¬cients:

dC2 (t)
=’ gks C1ks (t) , (11.71)
dt
ks


d
+ i∆k C1ks (t) = g— C2 (t) . (11.72)
ks
dt
For the discussion of spontaneous emission, it is natural to assume that the atom is
initially in the excited state and no photons are present, i.e.

C2 (0) = 1 , C1ks (0) = 0 . (11.73)

Inserting the formal solution,
t
dt g— e’i∆k (t’t ) C2 (t ) ,
C1ks (t) = (11.74)
ks
0

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