√

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

nπ

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