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this reduces to the spontaneous emission result, so the only di¬erence between the two

processes is the enhancement factor nks + 1. In order to simplify the argument we
will assume that nks = n (ω), i.e. the photon population is independent of polarization
and propagation direction. Then the average over polarizations and emission directions
produces
“ = [n (ω21 ) + 1] A2’1 = A2’1 + n (ω21 ) A2’1 , (4.164)
where the two terms are the spontaneous and stimulated rates respectively. By com-
paring this to eqn (1.13), we see that B2’1 ρ (ω21 ) = n (ω21 ) A2’1 , where ρ (ω21 ) is
the energy density per unit frequency. In the present case this is
3
ω21
ρ (ω21 ) = ( ω21 ) n (ω21 ) D (ω21 ) = 2 3 n (ω21 ) , (4.165)
πc
so the relation between the A and B coe¬cients is
3
A2’1 ω21
= 2 3, (4.166)
B2’1 πc
in agreement with eqn (1.21). The absorption coe¬cient B1’2 is deduced by calculat-
ing the transition rate for |µ1 , nks + 1 ’ |µ2 , nks . The relevant matrix element,

µ2 , nks „¦(+) µ1 , nks + 1 = i„¦21,s (k) nks + 1 , (4.167)

corresponds to part (2) of Fig. 4.1. Since |„¦21,s (k)| = „¦— (k) , using this matrix
21,s
element in eqn (4.113) will give the same result as the calculation of the stimulated
emission coe¬cient, therefore the absorption rate is identical to the stimulated emission
rate, i.e. B1’2 = B2’1 , in agreement with the detailed-balance argument eqn (1.18).
Thus the quantum theory correctly predicts the relations between the Einstein A and
B coe¬cients, and it provides an a priori derivation for the spontaneous emission rate.

Spontaneous emission in a planar cavity—
4.9.4
One of the assumptions in Einstein™s quantum model for radiation is that the A and
B coe¬cients are solely properties of the atom, but further thought shows that this
cannot be true in general. Consider, for example, an atom in the interior of an ideal
cubical cavity with sides L. According to eqn (2.15) the eigenfrequencies satisfy ωn

2πc/L; therefore, resonance is impossible if the atomic transition frequency is too
√ √
small, i.e. ω21 < 2πc/L, or equivalently L < »0 / 2, where »0 = 2πc/ω21 is the
wavelength of the emitted light. In addition to this failure of the resonance condition,
the golden rule (4.113) is not applicable, since the mode spacing is not small compared
to the transition frequency.
½¿
Interaction of light with atoms

What this means physically is that photons emitted by the atom are re¬‚ected from
the cavity walls and quickly reabsorbed by the atom. This behavior will occur for
any ¬nite value of L, but clearly the minimum time required for the radiation to be
reabsorbed will grow with L. In the limit L ’ ∞ the time becomes in¬nite and the
result for an atom in free space is recovered. Therefore the standard result (4.162) for
A2’1 is only valid for an atom in unbounded space.
The fact that the spontaneous emission rate for atoms is sensitive to the bound-
ary conditions satis¬ed by the electromagnetic ¬eld was recognized long ago (Purcell,
1946). More recently this problem has been studied in conjunction with laser etalons
(Stehle, 1970) and materials exhibiting an optical bandgap (Yablonovitch, 1987). We
will illustrate the modi¬cation of spontaneous emission in a simple case by describ-
ing the theory and experimental results for an atom in a planar cavity of the kind
considered in connection with the Casimir e¬ect.

A Theory
For this application, we will assume that the transverse dimensions are large, L »0 ,
while the longitudinal dimension ∆z (along the z-axis) is comparable to the transition
wavelength, ∆z ∼ »0 . The mode wavenumbers are then k = q + (nπ/∆z) uz , where
q = kx ux + ky uy , and the cavity frequencies are

2 1/2

2
ωqn =c q + . (4.168)
∆z

Both n and q are discrete, but the transverse mode numbers q will become densely
spaced in the limit L ’ ∞. The Schr¨dinger-picture ¬eld operator is given by the
o
analogue of eqn (3.69),
Cn
ωqn
aqns E qns (r) ,
(+)
E (r) = i (4.169)
20
q n s=1

where the mode functions are described in Appendix B.4 and Cn is the number of
independent polarization states for the mode (n, q): C0 = 1 and Cn = 2 for n 1.
Since the separation, ∆z, between the plates is comparable to the wavelength, the
transition rate will depend on the distance from the atom to each plate. Consequently,
we are not at liberty to assume that the atom is located at any particular z-value. On
the other hand, the dimensions along the x- and y-axes are e¬ectively in¬nite, so we
can choose the origin in the (x, y)-plane at the location of the atom, i.e. r = (0, z). The
interaction Hamiltonian is given by eqns (4.149) and (4.150), but the Rabi operator
in this case is a function of z, with the positive-frequency part
Cn
ωqn
aqns d · E qns (0, z) .
(+)
„¦ (z) = i (4.170)
20
q n s=1


The transition of interest is |µ2 , 0 ’ |µ1 , 1qns , so only „¦(’) (z) can contribute.
For each value of n and z the remaining calculation is a two-dimensional version of
½¼ Interaction of light with matter

the free-space case. Substituting the relevant matrix elements into eqn (4.113) and
multiplying by L2 d2 q/ (2π)2 ”the number of modes in the wavevector element d2 q”
yields the di¬erential transition rate

d2 q
2
dW2’1,qns (z) = 2π |M21,ns (q, z)| δ (ω21 ’ ωqn ) , (4.171)
(2π)2

where
ωqn
Ld21 · E qns (0, z) .
M21,ns (q, z) = (4.172)
20
For a given n, the transition rate into all transverse wavevectors q and polarizations
s is
Cn
d2 q 2
2 2π |M21,ns (q, z)| δ (ω21 ’ ωqn ) ,
A2’1,n (z) = (4.173)
(2π)
s=1

and the total transition rate is the sum of the partial rates for each n,

A2’1 (z) = A2’1,n (z) . (4.174)
n=0

The delta function in eqn (4.173) is eliminated by using polar coordinates, d2 q =
qdqdφ, and then making the change of variables q ’ ω/c = ωqn /c. The result is
customarily expressed in terms of a density of states factor Dn (ω21 ), de¬ned as the
number of resonant modes per unit frequency per unit of transverse area. For a given
n there are Cn polarizations, so

d2 q
Dn (ω21 ) = Cn δ (ω21 ’ ωqn )
(2π)2
dω ω
Cn
δ (ω21 ’ ω)
=
2π ω0n c2
Cn ω21 n»0
θ ∆z ’
= , (4.175)
2πc2 2

where θ (ν) is the standard step function, »0 = 2πc/ω21 is the wavelength for the
transition, and ω0n = nπc/∆z. This density of states counts all polarizations and the
full azimuthal angle, so in evaluating eqn (4.173) the extra 2πCn must be divided out.
The transition rate then appears as an average over azimuthal angles and polarizations:
Cn
1 dφ 2
A2’1,n (z) = Dn (ω21 ) 2π |M21,qns (z)| . (4.176)
Cn 2π
s=1

According to eqn (4.175) the density of states vanishes for ∆z/»0 < n/2; therefore,
emission into modes with n > 2∆z/»0 is forbidden. This re¬‚ects the fact that the
high-n modes are not in resonance with the atomic transition. On the other hand,
the density of states for the (n = 0)-mode is nonzero for any value of ∆z/»0 , so this
½½
Interaction of light with atoms

transition is only forbidden if it violates atomic selection rules. In fact, this is the only
possible decay channel for ∆z < »0 /2. In this case the total decay rate is

2πk0 |(dz )21 |2 3 |(dz )21 |2
2
1 »0
= = Avac ,
A2’1,0 (4.177)
2
|d21 |
4π 0 ∆z 4∆z

where Avac is the vacuum value given by eqn (4.162). The factor in square brackets
is typically of order unity, so the decay rate is enhanced over the vacuum value when
∆z < »0 /4, and suppressed below the vacuum value for »0 /4 < ∆z < »0 /2.
If the dipole selection rules (4.137) impose (dz )21 = 0, then decay into the (n = 0)-
mode is forbidden, and it is necessary to consider somewhat larger separations, e.g.
»0 /2 < ∆z < »0 . In this case, the decay to the (n = 1)-mode is the only one allowed.
There are now two polarizations to consider, the P -polarization in the (q, uz )-plane
and the orthogonal S-polarization along uz — q. We will simplify the calculation by
assuming that the matrix element d21 is real. In the general case of complex d21 a
separate calculation for the real and imaginary parts must be done, as in eqn (4.161).
For real d21 the polar angle φ can be taken as the angle between d21 and q. The
assumption that (dz )21 = 0 combines with the expressions (B.82) and (B.83), for the
P - and S-polarizations, respectively, to yield
2
3 »0 »0 πz »0
θ ∆z ’
sin2
A2’1,1 = 1+ Avac , (4.178)
2 2∆z 2∆z ∆z 2

2
where we have used the selection rule to impose d⊥ = d2 . The decay rate depends
on the location of the atom between the plates, and achieves its maximum value at
the midplane z = ∆z/2. In a real experiment, there are many atoms with unknown
locations, so the observable result is the average over z:
2
3 »0 »0 »0
θ ∆z ’
A2’1,1 = 1+ Avac . (4.179)
4 2∆z 2∆z 2

This rate vanishes for »0 > 2∆z, and for »0 /2∆z slightly less than unity it is enhanced
over the vacuum value:
3
A2’1,1 Avac for »0 /2∆z 1. (4.180)
2
The decay rate is suppressed below the vacuum value for »0 /2∆z 0.8.

B Experiment
The clear-cut and striking results predicted by the theoretical model are only possible
if the separation between the plates is comparable to the wavelength of the emitted
radiation. This means that experiments in the optical domain would be extremely
di¬cult. The way around this di¬culty is to use a Rydberg atom, i.e. an atom which
has been excited to a state”called a Rydberg level”with a large principal quantum
number n. The Bohr frequencies for dipole allowed transitions between neighboring
½¾ Interaction of light with matter

high-n states are of O 1/n3 , so the wavelengths are very large compared to optical
wavelengths.
In the experiment we will discuss here (Hulet et al., 1985), cesium atoms were
excited by two dye laser pulses to the |n = 22, m = 2 state. The small value of
the magnetic quantum number is explained by the dipole selection rules, ∆l = ±1,
∆m = 0, ±1. These restrictions limit the m-values achievable in the two-step exci-
tation process to a maximum of m = 2. This is a serious problem, since the state
|n = 22, m = 2 can undergo dipole allowed transitions to any of the states |n , m for
2n 21 and m = 1, 3. A large number of decay channels would greatly compli-
cate both the experiment and the theoretical analysis. This complication is avoided by
exposing the atom to a combination of rapidly varying electric ¬elds and microwave
radiation which leave the value of n unchanged, but increase m to the maximum pos-
sible value, m = n ’ 1, a so-called circular state that corresponds to a circular Bohr
orbit. The overall process leaves the atom in the state |n = 22, m = 21 which can
only decay to |n = 21, m = 20 . This simpli¬es both the experimental situation and
the theoretical model. The wavelength for this transition is »0 = 0.45 mm, so the me-
chanical problem of aligning the parallel plates is much simpler than for the Casimir
force experiment. The gold-plated aluminum plates are held apart by quartz spacers
at a separation of ∆z = 230.1 µm so that »0 /2∆z = 0.98.
The atom has now been prepared so that there is only one allowed atomic transi-
tion, but there are still two modes of the radiation ¬eld, E q0 and E q1s , into which the
atom can decay. There is also the di¬cult question of how to produce controlled small
changes in the plate spacing in order to see the e¬ects on the spontaneous emission
rate. Both of these problems are solved by the expedient of establishing a voltage drop
between the plates. The resulting static electric ¬eld polarizes the atom so that the
natural quantization axis lies in the direction of the ¬eld. The matrix elements of the
z-component of the dipole operator, m dz m , vanish unless m = m, but transitions
of this kind are not allowed by the dipole selection rules, m = m ± 1, for the circu-
lar Rydberg atom. This amounts to setting (dz )21 = 0. Emission of E q0 -photons is
therefore forbidden, and the atom can only emit E q1s -photons. The ¬eld also causes
second-order Stark shifts (Cohen-Tannoudji et al., 1977b, Complement E-XII) which
decrease the di¬erence in the atomic energy levels and thus increase the wavelength
»0 . This means that the wavelength can be modi¬ed by changing the voltage, while the
plate spacing is left ¬xed. The onset of ¬eld ionization limits the ¬eld strength that can
be employed, so the wavelength can only be tuned by ∆» = 0.04 »0 . Fortunately, this
is su¬cient to increase the ratio »0 /2∆z through the critical value of unity, at which
the spontaneous emission should be quenched. At room temperature the blackbody
spectrum contains enough photons at the transition frequency to produce stimulated
emission. The observed emission rate would then be the sum of the stimulated and
spontaneous decay rates. In the model this would mean that we could not assume that
the initial state is |µ1 , 0 . This additional complication is avoided by maintaining the
apparatus at 6.5 K. At this low temperature, stimulated emission due to blackbody
radiation at »0 is strongly suppressed.
A thermal atomic beam of cesium ¬rst passes through a production region, where
the atoms are transferred to the circular state, then through a drift region”of length
½¿
Interaction of light with atoms

L = 12.7 cm”between the parallel plates. The length L is chosen so that the mean
transit time is approximately the same as the vacuum lifetime. After passing through
the drift region the atoms are detected by ¬eld ionization in a region where the ¬eld
increases with length of travel. The ionization rates for n = 22 and n = 21 atoms di¬er
substantially, so the location of the ionization event allows the two sets of atoms to be
resolved.
In this way, the time-of-¬‚ight distribution of the n = 22 atoms was measured. In
the absence of decay, the distribution would be determined by the original Boltzmann
distribution of velocities, but when decay due to spontaneous emission is present, only
the faster atoms will make it through the drift region. Thus the distribution will shift
toward shorter transit times. In the forbidden region, »0 /2∆z > 1, the data were
consistent with A2’1,1 = 0, with estimated errors ±0.05Avac . In other words, the
lifetime of an atom between the plates is at least twenty times longer than the lifetime
of the same atom in free space.
Raman scattering—
4.9.5
In Raman scattering, a photon at one frequency is absorbed by an atom or molecule,
and a photon at a di¬erent frequency is emitted. The simplest energy-level diagram
permitting this process is shown in Fig. 4.2. This is a second-order process, so it
(2)
requires the calculation of the second-order amplitude Vf i , where the initial and
¬nal states are respectively |˜i = |µ1 , 1ks and |˜f = |µ2 , 1k s . The representation
(4.149) allows the operator product on the right side of eqn (4.120) to be written as
2
„¦(’) (t1 ) „¦(’) (t2 ) + „¦(+) (t1 ) „¦(+) (t2 )
Hint (t1 ) Hint (t2 ) =
2
„¦(’) (t1 ) „¦(+) (t2 ) + „¦(+) (t1 ) „¦(’) (t2 ) ,
+
(4.181)
where the ¬rst two terms change photon number by two and the remaining terms
leave photon number unchanged. Since the initial and ¬nal states have equal photon
number, only the last two terms can contribute in eqn (4.120); consequently, the matrix
element of interest is
2
˜f „¦(’) (t1 ) „¦(+) (t2 ) + „¦(+) (t1 ) „¦(’) (t2 ) ˜i . (4.182)


!
k I
kI




Fig. 4.2 Raman scattering from a three-level
atom. The transitions 1 ” 3 and 2 ” 3 are
dipole allowed. A photon in mode ks scatters
 into the mode k s .
½ Interaction of light with matter

Since t2 < t1 the ¬rst term describes absorption of the initial photon followed by
emission of the ¬nal photon, as one would intuitively expect. The second term is rather
counterintuitive, since the emission of the ¬nal photon precedes the absorption of the
initial photon. These alternatives are shown respectively by the Feynman diagrams
(1) and (2) in Fig. 4.3, which we will call the intuitive and counterintuitive diagrams
respectively.
The calculation of the transition amplitude by eqn (4.123) yields
µ2 , 1k s „¦(’) Λu Λu „¦(+) µ1 , 1ks
(2)
= ’i 2πδ (ωk ’ ωk ’ ω21 )
Vf i µ2 ’Eu
ωk + +i
u
µ2 , 1k s „¦(+) Λu Λu „¦(’) µ1 , 1ks
’i 2πδ (ωk ’ ωk ’ ω21 ) ,
µ2 ’Eu
ωk + +i
u
(4.183)
where the two sums over intermediate states correspond respectively to the intuitive
and counterintuitive diagrams. Since „¦(+) decreases the photon number by one, the
intermediate states in the ¬rst sum have the form |Λu = |µq , 0 . In this simple model
the only available state is |Λu = |µ3 , 0 . Thus the energy is Eu = µ3 and the denomi-
nator is ωk ’ω32 +i . In fact, the intermediate state can be inferred from the Feynman
diagram by passing a horizontal line between the two vertices. For the intuitive dia-
gram, the only intersection is with the internal atom line, but in the counterintuitive
diagram the line passes through both photon lines as well as the atom line. In this
case, the intermediate state must have the form |Λu = |µ1 , 1ks , 1k s , with energy
Eu = µ3 + ωk + ωk and denominator ’ωk ’ ω32 + i . These claims can be veri¬ed
by a direct calculation of the matrix elements in the second sum.
This calculation yields the explicit expression
— —
M32,s (k ) M31,s (k) M23,s (k) M13,s (k ) 2π
(2)
= ’i δ (ωk ’ ωk ’ ω21 ) ,
Vf i +
ωk ’ ω32 + i ’ωk ’ ω32 + i V
(4.184)


k I




3
kI
3
k I
Fig. 4.3 Feynman diagrams for Raman scat-

kI
tering. Diagram (1) shows the intuitive order- 
ing in which the initial photon is absorbed
prior to the emission of the ¬nal photon. Di-
agram (2) shows the counterintuitive case in
 
which the order is reversed.
½
Exercises

(2) 2
where the matrix elements are de¬ned in eqn (4.156). Multiplying Vf i by the
number of modes V d3 k/ (2π)3 V d3 k / (2π)3 and using the rule (4.119) gives the
di¬erential transition rate
— — 2
M32,s (k ) M31,s (k) M23,s (k) M13,s (k )
dW3ks’2k s = 2π +
ωk ’ ω32 + i ’ωk ’ ω32 + i
d3 k d3 k
— δ (ωk ’ ωk ’ ω21 ) 3. (4.185)
3
(2π) (2π)

4.10 Exercises
4.1 Semiclassical electrodynamics
(1) Derive eqn (4.7) and use the result to get eqn (4.27).
(2) For the classical ¬eld described in the radiation gauge, do the following.
(a) Derive the equation satis¬ed by the scalar potential • (r).
(b) Show that

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