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i (11.168)
dt
where the complex, time-dependent Rabi frequency is de¬ned by

d · E (t)
„¦ (t) = . (11.169)

Combining the notation ρqp (t) = µq |ρ (t)| µp with the hermiticity condition
ρ12 (t) = ρ— (t) allows eqn (11.168) to be written out explicitly as
21

d
ρ11 (t) = ’„¦ (t) e’iδt ρ21 (t) + „¦— (t) eiδt ρ12 (t) ,
i (11.170)
dt
d
i ρ22 (t) = „¦ (t) e’iδt ρ21 (t) ’ „¦— (t) eiδt ρ12 (t) , (11.171)
dt
d
i ρ12 (t) = ’„¦ (t) e’iδt [ρ22 (t) ’ ρ11 (t)] , (11.172)
dt
where ρ11 and ρ22 are the occupation probabilities for the two levels and the o¬-
diagonal term ρ12 is called the atomic coherence. For most applications, it is better
to eliminate the explicit exponentials by setting

ρ12 (t) = e’iδt ρ12 (t) , ρ22 (t) = ρ22 (t) , ρ11 (t) = ρ11 (t) , (11.173)

to get
¿ Coherent interaction of light with atoms

d
ρ22 (t) = i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] , (11.174)
dt
d
ρ11 (t) = ’i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] , (11.175)
dt
d
ρ (t) = iδρ21 (t) + i„¦ (t) (ρ11 (t) ’ ρ22 (t)) . (11.176)
dt 21
The sum of eqns (11.174) and (11.175) conveys the reassuring news that the total
occupation probability, ρ11 (t) + ρ22 (t), is conserved.
For a strictly monochromatic ¬eld, „¦ (t) = „¦, these equations can be solved to
obtain a generalized description of Rabi ¬‚opping, but there is a more pressing question
to be addressed. This is the neglect of the decay of the upper level by spontaneous
emission. We have seen in Section 11.2.2 that the upper-level amplitude C1 (t) ∼
exp (’“t/2), so in the absence of the external ¬eld the occupation probability ρ11 of
the upper level and the coherence ρ12 (t) should behave as

ρ22 (t) ∼ C2 (t) C2 (t) ∼ e’w21 t ,

(11.177)
ρ21 (t) ∼ C2 (t) C1 (t) ∼ e’w21 t/2 .



An equivalent statement is that the terms ’w21 ρ22 (t) and ’w21 ρ21 (t) /2 should ap-
pear on the right sides of eqns (11.174) and (11.175) respectively. This would be the
end of the story if spontaneous emission were the only thing that has been left out,
but there are other e¬ects to consider. In atomic vapors, elastic scattering from other
atoms will disturb the coherence ρ12 (t) and cause an additional decay rate, and in
crystals similar e¬ects arise due to lattice vibrations and local ¬eld ¬‚uctuations.
The general description of dissipative e¬ects will be studied Chapter 14, but for the
present we will adopt a phenomenological approach in which eqns (11.174)“(11.176)
are replaced by the Bloch equations:
d
ρ22 (t) = ’w21 ρ22 (t) + i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] , (11.178)
dt
d
ρ (t) = w21 ρ22 (t) ’ i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] , (11.179)
dt 11
d
ρ (t) = (iδ ’ “21 ) ρ21 (t) + i„¦ (t) (ρ11 (t) ’ ρ22 (t)) , (11.180)
dt 21
where the decay rate w21 and the dephasing rate “21 are parameters to be deter-
mined from experiment. In this simple two-level model the lower level is the ground
state, so the term w21 ρ22 in eqn (11.179) is required in order to guarantee conserva-
tion of the total occupation probability. This allows eqns (11.179) and (11.180) to be
replaced by
ρ11 (t) + ρ22 (t) = 1 , (11.181)
d
[ρ (t) ’ ρ11 (t)] = ’w21 ’ w21 [ρ22 (t) ’ ρ11 (t)] + 2i [„¦ (t) ρ12 (t) ’ „¦— (t) ρ21 (t)] ,
dt 22
(11.182)
where ρ22 (t) ’ ρ11 (t) is the population inversion. In the literature, the parameters
w21 and “21 are often represented as
¿
The semiclassical limit

1 1
w21 = , “21 = , (11.183)
T1 T2

where T1 and T2 are respectively called the longitudinal and transverse relaxation
times. This terminology is another allusion to the analogy with a spin-1/2 system
precessing in an external magnetic ¬eld. Another common usage is to call T1 and T2
respectively the on-diagonal and o¬-diagonal relaxation times.
In the frequency domain, the slow time variation of the ¬eld envelope E (t) is
ω0 , where ∆ω0 is the spectral width of E (ω).
represented by the condition ∆ω0
The detuning and the dephasing rate are also small compared to the carrier frequency,
but either or both can be large compared to ∆ω0 . This limit can be investigated by
means of the formal solution,
t
’i dt „¦ (t ) [ρ22 (t ) ’ ρ11 (t )] e(iδ’“21 )(t’t ) ,
(iδ’“21 )(t’t0 )
ρ21 (t) = ρ21 (t0 ) e
t0
(11.184)
w21 /2 > 0, the formal solution has the t0 ’ ’∞ limit
of eqn (11.180). Since “21
t
ρ21 (t) = ’i dt „¦ (t ) [ρ22 (t ) ’ ρ11 (t )] e(iδ’“21 )(t’t ) . (11.185)
’∞

The exponential factor exp [’“21 (t ’ t )] implies that the main contribution to the
integral comes from the interval t ’ 1/“21 < t < t, while the rapidly oscillating
exponential exp [iδ (t ’ t )] similarly restricts contributions to the interval t ’ 1/ |δ| <
max (∆ω0 , w21 ) or |δ|
t < t. Thus if either of the conditions “21 max (∆ω0 , w21 )
is satis¬ed, the main contribution to the integral comes from a small interval t ’ ∆t <
t < t. In this interval, the remaining terms in the integrand are e¬ectively constant;
consequently, they can be evaluated at the upper limit to ¬nd:

„¦ (t) [ρ22 (t) ’ ρ11 (t)]
ρ21 (t) = . (11.186)
δ + i“21

The approximation of the atomic coherence by this limiting form is called adia-
batic elimination, by analogy to the behavior of thermodynamic systems. A ther-
modynamic parameter, such as the pressure of a gas, will change in step with slow
changes in a control parameter, e.g. the temperature. The analogous behavior is seen in
eqn (11.186) which shows that the atomic coherence ρ21 (t) follows the slower changes
in the populations. For a large dephasing rate, exponential decay drives ρ21 (t) to the
equilibrium value given by eqn (11.186). In the case of large detuning, the deviation
from the equilibrium value oscillates so rapidly that its contribution averages to zero.
Once the mechanism of adiabatic elimination is understood, its application reduces
to the following simple rule.
(a) If |“qp + i∆qp | is large, set dρqp /dt = 0.
(b) Use the resulting algebraic relations to eliminate as many ρqp s as possible.
(11.187)
¿ Coherent interaction of light with atoms

Substituting ρ21 (t) from eqn (11.186) into eqn (11.182) leads to

4 |„¦ (t)|2 “21
d
[ρ (t) ’ ρ11 (t)] = ’w21 ’ [ρ22 (t) ’ ρ11 (t)] ,
w21 + (11.188)
dt 22 δ 2 + “2
21

which shows that the adiabatic elimination of the atomic coherence does not neces-
sarily imply the adiabatic elimination of the population inversion. The solution of this
di¬erential equation also shows that no pumping scheme for a strictly two-level atom
can change the population inversion from negative to positive. Since laser ampli¬ca-
tion requires a positive inversion, this implies that laser action can only be described
by atoms with at least three active levels.
If w21 = O (∆ω0 ) the population inversion and the external ¬eld change on the
same time scale. Adiabatic elimination of the population inversion will only occur for
w21 ∆ω0 . In this limit the adiabatic elimination rule yields
w21
ρ22 (t) ’ ρ11 (t) = ’ < 0. (11.189)
4|„¦(t)|2 “21
w21 + δ 2 +“2
21


When adiabatic elimination is possible for both the atomic coherence and the popula-
tion inversion, the atomic density matrix appears to react instantaneously to changes
in the external ¬eld. What this really means is that transient e¬ects are either sup-
pressed by rapid damping (w21 ∆ω0 , and “21 ∆ω0 ) or average to zero due to
rapid oscillations (|δ| ∆ω0 ). The apparently instantaneous response of the two-level
atom is also displayed by multilevel atoms when the corresponding conditions are
satis¬ed.
For later applications it is more useful to substitute the adiabatic form (11.186)
into the original equations (11.179) and (11.178) to get a pair of equations for the
occupation probabilities Pq = ρqq . In the strictly monochromatic case, one ¬nds
dP2
= W12 P1 ’ (w21 + W12 ) P2 ,
dt (11.190)
dP1
= ’W12 P1 + (w21 + W12 ) P2 ,
dt
where
2 |„¦|2 “21
W12 =2 (11.191)
δ + “2 21
is the rate of 1 ’ 2 transitions (absorptions) driven by the ¬eld. By virtue of the
equality B1’2 = B2’1 , explained in Section 1.2.2, this is equal to the rate of 2 ’ 1
transitions (stimulated emissions) driven by the ¬eld. Equations (11.190) are called
rate equations and their use is called the rate equation approximation. The
occupation probability of |µ2 is increased by absorption from |µ1 and decreased by the
combination of spontaneous and stimulated emission to |µ1 . The inverse transitions
determine the rate of change of P1 , in such a way that probability is conserved. The
rate equations can be generalized to atoms with three or more levels by adding up all
of the (incoherent) processes feeding and depleting the occupation probability of each
level.
¿
Exercises

11.4 Exercises
11.1 The antiresonant Hamiltonian
(ar)
Apply the de¬nition (11.17) of the running average to Hint (t) to ¬nd:

ωk d— · eks ’i(ω21 +ωk )t
(ar)
H int (t) = ’i e K (ω21 + ωk ) aks σ’ + HC .
2 0V
ks

Use the properties of the cut-o¬ function and the conventions ω21 > 0 and ωk > 0 to
(ar)
explain why dropping H int (t) is a good approximation.

11.2 The Weisskopf“Wigner method
(1) Fill in the steps needed to go from eqn (11.80) to eqn (11.84).
2
(2) Assume that |K (∆)| is an even function of ∆ and show that
∞ ∞
δω21 3 1
2 2
d∆ |K (∆)| + d∆ ∆2 |K (∆)| .
= 3
w21 2πω21 2πω21
’∞ ’∞

Use this to derive the estimate δω21 /w21 = O (wK /ω21 ) 1.

11.3 Atomic radiation ¬eld
(1) Use the eqns (11.26) and (B.48) to show that

ωk K (∆k ) — (d— · ∇) ∇ ik·r
ωk —
g eks e ik·r
= d+ e .
2 0 V ks k2
2 0V
s

(2) With the aid of this result, convert the k-sum in eqn (11.54) to an integral. Show
that
4π sin (kr)
d„¦k eik·r = ,
kr
and then derive eqn (11.56).

11.4 Slowly-varying envelope operators
De¬ne envelope operators σ ’ (t) = exp (iω21 t) σ’ (t), σ z (t) = σz (t), and aks (t) =
exp (iωk t) aks (t).
(1) Use eqns (11.47)“(11.49) to derive the equations satis¬ed by the envelope opera-
tors.
(2) From these equations argue that the envelope operators are slowly varying, i.e.
essentially constant over an optical period.

Two-photon cascade—
11.5
(1) Substitute the ansatz (11.106) into the Schr¨dinger equation for the Hamiltonian
o
(11.103) and obtain the di¬erential equations for the coe¬cients.
(2) Use the given initial conditions to derive eqns (11.107)“(11.109).
¿¼ Coherent interaction of light with atoms

(3) Carry out the steps needed to arrive at eqn (11.113).
(4) Starting with the normalization |K (0)| = 1 and the fact that |K (∆ )|2 is an even
function, use an argument similar to the derivation of eqn (11.89) to show that
Dk ≈ w21 /2.
(5) Evaluate the residue for the poles of Xks,k s (ζ) to ¬nd the coe¬cients G1 , G2 ,
and G3 , and then derive eqn (11.122).
12
Cavity quantum electrodynamics

In Section 4.9 we studied spontaneous emission in free space and also in the modi¬ed
geometry of a planar cavity. The large dimensions in both cases”three for free space
and two for the planar cavity”provide the densely packed energy levels that are
essential for the validity of the Fermi golden rule calculation of the emission rate.
Cavity quantum electrodynamics is concerned with the very di¬erent situation of an
atom trapped in a cavity with all three dimensions comparable to the wavelength of the
emitted radiation. In this case the radiation modes are discrete, and the Fermi golden
rule cannot be used. Instead of disappearing into the blackness of in¬nite space, the
emitted radiation is re¬‚ected from the nearby cavity walls, and soon absorbed again by
the atom. The re-excitation of the atom results in a cycle of emissions and absorptions,
rather than irreversible decay. In the limit of strong ¬elds, i.e. many photons in a single
mode, this cyclic behavior is described in Section 11.3.2 as Rabi ¬‚opping. The exact
periodicity of Rabi ¬‚opping is, however, an artifact of the semiclassical approximation,
in which the discrete nature of photons is ignored. In the limit of weak ¬elds, the grainy
nature of light makes itself felt in the nonclassical features of collapse and revival of
the probability for atomic excitation.
There are several possible experimental realizations of cavity quantum electrody-
namics, but the essential physical features of all of them are included in the Jaynes“
Cummings model discussed in Section 12.1. In Section 12.2 we will use this model to
describe the intrinsically quantum phenomena of collapse and revival of the radiation
¬eld in the cavity. A particular experimental realization is presented in Section 12.3.

12.1 The Jaynes“Cummings model
12.1.1 De¬nition of the model
In its simplest form, the Jaynes“Cummings model consists of a single two-level atom
located in an ideal cavity. For the two-level atom we will use the treatment given in
Section 11.1.1, in which the two atomic eigenstates are | 1 and | 2 with 1 < 2 . The
Hamiltonian is then
ω0
Hat = σz , (12.1)
2
where we have chosen the zero of energy so that 2 + 1 = 0, and set ω0 ≡ ( 2 ’ 1) / .
For the electromagnetic ¬eld, we use the formulation in Section 2.1, so that

ω κ a† aκ
Hem = (12.2)
κ
κ
¿¾ Cavity quantum electrodynamics

is the Hamiltonian, and

ωκ
aκ E κ (r)
E(+) (r) = i (12.3)
20
κ

is the positive-frequency part of the electric ¬eld (in the Schr¨dinger picture).
o
Adapting the general result (11.27) to the cavity problem gives the RWA interaction
Hamiltonian

Hrwa = ’d · E(+) σ+ ’ d— · E(’) σ’
gκ a† σ’ ;

= ’i gκ aκ σ+ + i (12.4)
κ
κ κ


where d = d is the dipole matrix element; the coupling frequencies are
1 2


ωκ d · E κ (R)
gκ = K (ω0 ’ ωκ ) ; (12.5)
20
K (ω0 ’ ωκ ) is the RWA cut-o¬ function; and R is the position of the atom.
We will now drastically simplify this model in two ways. The ¬rst is to assume
that the center-of-mass motion of the atom can be treated classically. This means
that ω0 should be interpreted as the Doppler-shifted resonance frequency. In many
cases the Doppler e¬ect is not important; for example, for microwave transitions in
Rydberg atoms passing through a resonant cavity, or single atoms con¬ned in a trap.
The second simpli¬cation is enforced by choosing the cavity parameters so that the
lowest (fundamental) mode frequency is nearly resonant with the atomic transition,
while all higher frequency modes are well out of resonance. This guarantees that only
the lowest mode contributes to the resonant Hamiltonian; consequently, the family of
annihilation operators aκ can be reduced to the single operator a for the fundamental
mode. From now on, we will call the fundamental frequency the cavity frequency
ωC and the corresponding mode function E C (R) the cavity mode.
The total Hamiltonian for the Jaynes“Cummings model is therefore HJC = H0 +
Hint , where
H0 = ωC a† a + ( ω0 /2) σz , (12.6)
Hint = ’i gaσ+ + i ga† σ’ , (12.7)
and
ωC d · E C (R)
g= . (12.8)
20
By appropriate choice of the phases in the atomic eigenstates | and | , we can
1 2
always arrange that g is real.

12.1.2 Dressed states
The interaction Hamiltonian in eqn (12.7) has the same general form as the interac-
tion Hamiltonian (11.25) for the Weisskopf“Wigner model of Section 11.2.2, but it is
greatly simpli¬ed by the fact that only one mode of the radiation ¬eld is active. In
¿¿
The Jaynes“Cummings model

the Weisskopf“Wigner case, the in¬nite-dimensional subspaces Hse are left invariant
(mapped into themselves) under the action of the Hamiltonian. Since the Hamiltonians
have the same structure, a similar behavior is expected in the present case.
The product states,

| j , n (0) = | |n (n = 0, 1, . . .) , (12.9)
j

where the | j s (j = 1, 2) are the atomic eigenstates and the |n s are number states
for the cavity mode, provide a natural basis for the Hilbert space HJC of the Jaynes“
Cummings model. The | j , n (0) s are called bare states, since they are eigenstates of
the non-interacting Hamiltonian H0 :

H0 | j , n (0) = ( + n ωC ) | j , n (0) . (12.10)
j

Turning next to Hint , a straightforward calculation shows that

Hint | 1 , 0 (0) = 0 , (12.11)

which means that spontaneous absorption from the bare vacuum is forbidden in the
resonant wave approximation. Consequently, the ground-state energy and state vector
for the atom“¬eld system are, respectively,
ω0
=’ and |G = | 1 , 0 (0) .
µG = (12.12)
1
2
Furthermore, for each photon number n the pairs of bare states | 2 , n (0) and
| 1 , n + 1 (0) satisfy

Hint | 2 , n (0) = i g n + 1 | 1 , n + 1 (0) ,
√ (12.13)
Hint | 1 , n + 1 (0) = ’i g n + 1 | 2 , n (0) .

Consequently, each two-dimensional subspace

Hn = span | 2 , n (0) , | 1 , n + 1 (0) (n = 0, 1, . . .) (12.14)

is left invariant by the total Hamiltonian. This leads to the natural decomposition of
HJC as
HJC = HG • H0 • H1 • · · · , (12.15)
where HG = span | 1 , 0 (0) is the one-dimensional space spanned by the ground state.
In the subspace Hn the Hamiltonian is represented by a 2 — 2 matrix

’2ig n + 1
1 10 δ

+
HJC,n = n + ωC , (12.16)
’δ
01 2 2ig n + 1
2

where δ = ω0 ’ ωC is the detuning. This construction allows us to reduce the solution
of the full Schr¨dinger equation, HJC |¦ = µ |¦ , to the diagonalization of the 2 — 2-
o
matrix HJC,n for each n. The details are worked out in Exercise 12.1. For each subspace
¿ Cavity quantum electrodynamics

Hn , the exact eigenvalues and eigenvectors, which will be denoted by µj,n and |j, n
(j = 1, 2), respectively, are
1 „¦n
µ1,n = n+ ωC + , (12.17)
2 2
|1, n = sin θn | 2 , n (0) + cos θn | 1 , n + 1 (0) , (12.18)
1 „¦n
ωC ’
µ2,n = n+ , (12.19)
2 2
|2, n = cos θn | 2 , n (0) ’ sin θn | 1 , n + 1 (0) , (12.20)
where
δ 2 + 4g 2 (n + 1)
„¦n = (12.21)
is the Rabi frequency for oscillations between the two bare states in Hn . The probability
amplitudes for the bare states are given by
„¦n ’ δ
cos θn = ,
2
(„¦n ’ δ) + 4g 2 (n + 1)
√ (12.22)
2g n + 1
sin θn = .
2
(„¦n ’ δ) + 4g 2 (n + 1)

The bare (g = 0) eigenvalues
(0)
µ1,n = (n + 1/2) ωC + δ/2 ,
(12.23)
(0)
µ2,n = (n + 1/2) ωC ’ δ/2
are degenerate at resonance (δ = 0), but the exact eigenvalues satisfy

µ1,n ’ µ2,n = „¦n 2 g n + 1 . (12.24)
This is an example of the ubiquitous phenomenon of avoided crossing (or level
repulsion) which occurs whenever two states are coupled by a perturbation.
The eigenstates |1, n and |2, n of the full Jaynes“Cummings Hamiltonian HJC are
called dressed states, since the interaction between the atom and the ¬eld is treated
exactly. By virtue of this interaction, the dressed states are entangled states of the
atom and the ¬eld.

12.2 Collapses and revivals
With the dressed eigenstates of HJC in hand, we can write the general solution of the
time-dependent Schr¨dinger equation as
o
∞ 2
’iµG t/
Cj,n e’iµj,n t/ |j, n ,
|Ψ (t) = e CG |G + (12.25)
n=0 j=1

where the expansion coe¬cients are determined by the initial state vector according
to CG = G |Ψ (0) and Cj,n = j, n |Ψ (0) (j = 1, 2) (n = 0, 1, . . .). If the atom
¿
Collapses and revivals

is initially in the excited state | 2 and exactly m cavity photons are present, i.e.
|Ψ (0) = | 2 , m (0) , the general solution (12.25) specializes to |Ψ (t) = | 2 , m; t , where
„¦n t „¦n t
| 2 , n; t ≡ e’i(n+1/2)ωC t cos | 2 , n (0)
+ i cos (2θn ) sin
2 2
„¦n t
’ ie’i(n+1/2)ωC t sin (2θn ) sin | 1 , n + 1 (0) . (12.26)
2
At resonance, the probabilities for the states | 2 , m (0) and | 1 , m + 1 (0) are

2
2 , m |Ψ (t)
(0)
= cos2 g m + 1t ,
P2,m (t) =
(12.27)

2
+ 1 |Ψ (t)
(0) 2
P1,m+1 (t) = 1, m = sin g m + 1t ,

so”as expected”the system oscillates between the two atomic states by emission and
absorption of a single photon. The exact periodicity displayed here is a consequence
of the special choice of an initial state with a de¬nite number of photons. For m > 0,
this is analogous to the semiclassical problem of Rabi ¬‚opping driven by a ¬eld with
de¬nite amplitude and phase. The analogy to the classical case fails for m = 0, i.e. an
excited atom with no photons present. The classical analogue of this case would be
a vanishing ¬eld, so that no Rabi ¬‚opping would occur. The occupation probabilities
P2,0 (t) = cos2 (gt) and P1,1 (t) = sin2 (gt) describe vacuum Rabi ¬‚opping, which is
a consequence of the purely quantum phenomenon of spontaneous emission, followed
by absorption, etc.
For initial states that are superpositions of several photon number states, exact
periodicity is replaced by more complex behavior which we will now study. A super-
position,

|Ψ (0) = Kn | 2 , n (0) , (12.28)
n=0

of the initial states | 2 , n (0) that individually lead to Rabi ¬‚opping evolves into

|Ψ (t) = Kn | 2 , n; t , (12.29)
n=0

so the probability to ¬nd the atom in the upper state, without regard to the number
of photons, is
∞ ∞
2 2
2
2 , n |Ψ (t) |Kn | 2 , n | 2 , n; t
(0) (0)
P2 (t) = = . (12.30)
n=0 n=0

At resonance, eqn (12.27) allows this to be written as


11 2
|Kn | cos 2 n + 1gt .
P2 (t) = + (12.31)
2 2 n=0

If more than one of the coe¬cients Kn is nonvanishing, this function is a sum of
oscillatory terms with incommensurate frequencies. Thus true periodicity is only found
¿ Cavity quantum electrodynamics

for the special case |Kn | = δnm for some ¬xed value of m. For any choice of the Kn s
the time average of the upper-level population is P2 (t) = 1/2.
In order to study the behavior of P2 (t), we need to make an explicit choice for the
Kn s. Let us suppose, for example, that the initial state is |Ψ (0) = | 2 |± , where |± is
a coherent state for the cavity mode. The coe¬cients are then |Kn | = e’|±| |±| /n!,
2
2 2n

and 2∞

1 e’|±|
2n
|±|
P2 (t) = + cos 2 n + 1gt . (12.32)
2 2 n=0 n!
Photon numbers for the coherent state follow a Poisson distribution, so the main
contribution to the sum over n will come from the range (n ’ ∆n, n + ∆n), where
2
n = |±| is the mean photon number and ∆n = |±| is the variance. For large n,
the corresponding spread in Rabi frequencies is ∆„¦ ∼ 2g. At very early times, t
1/g, the arguments of the cosines are essentially in phase, and P2 (t) will execute an
almost coherent oscillation. At later times, the variation of the Rabi frequencies with
photon number will lead to an e¬ectively random distribution of phases and destructive
interference. This e¬ect can be estimated analytically by replacing the sum over n
with an integral and evaluating the integral in the stationary-phase approximation.
The result,
1 e’|gt|
2

cos (2 |±| gt) for gt 1 ,
P2 (t) = + (12.33)
2 2
describes the collapse of the upper-level population to the time-averaged value of
1/2. This decay in the oscillations is neither surprising nor particularly quantal in
character. A superposition of Rabi oscillations due to classical ¬elds with random ¬eld
strengths would produce a similar decay.
What is surprising is the behavior of the upper-level population at still later times.
A numerical evaluation of eqn (12.32) reveals that the oscillations reappear after a
rephasing time trp ∼ 4π |±| /g. This revival”with P2 (t) = O (1)”is a speci¬cally
quantum e¬ect, explained by photon indivisibility. The revival is in turn followed by
another collapse. The ¬rst collapse and revival are shown in Fig. 12.1.
The classical nature of the collapse is illustrated by the dashed curve in the same
¬gure, which is calculated by replacing the discrete sum in eqn (12.32) by an integral.
The two curves are indistinguishable in the initial collapse phase, but the classical
(dashed) curve remains ¬‚at at the value 1/2 during the quantum revival. Thus the
experimental observation of a revival provides further evidence for the indivisibility
of photons. After a few collapse“revival cycles, the revivals begin to overlap and”as
shown in Exercise 12.2”P2 (t) becomes irregular.
¿
The micromaser


22
1

0.8

0.6

0.4

0.2

CJ
5 10 15 20 25 30

Fig. 12.1 The solid curve shows the probability P2 (t) versus gt, where the upper-level
population P2 (t) is given by eqn (12.32), and the average photon number is n = |±|2 = 10.
The dashed curve is the corresponding classical result obtained by replacing the discrete sum
over photon number by an integral.

12.3 The micromaser
The interaction of a Rydberg atom with the fundamental mode of a microwave cavity
provides an excellent realization of the Jaynes“Cummings model. The con¬guration
sketched in Fig. 12.2 is called a micromaser (Walther, 2003). It is designed so that”
with high probability”at most one atom is present in the cavity at any given time. A
velocity-selected beam of alkali atoms from an oven is sent into a laser excitation region,
where the atoms are promoted to highly excited Rydberg states. The size of a Rydberg
atom is characterized by the radius, aRyd = n2 2 /me2 , of its Bohr orbit, where np is
p


Atomic beam oven


Maser cavity

Field ionization
Velocity selector
Laser excitation
Channeltron
detectors Atomic beam

Fig. 12.2 Rubidium Rydberg atoms from an oven pass successively through a velocity selec-
tor, a laser excitation region, and a superconducting microwave cavity. After emerging from
the cavity, they are detected”in a state-selective manner”by ¬eld ionization, followed by
channeltron detectors. (Reproduced from Rempe et al. (1990).)
¿ Cavity quantum electrodynamics

the principal quantum number, and 2 /me2 is the Bohr radius for the ground state of
the hydrogen atom. These atoms are truly macroscopic in size; for example, the radius
of a Rydberg atom with np 100 is on the order of microns, instead of nanometers.
The dipole matrix element d = np |er| np + 1 for a transition between two adjacent
Rydberg states np + 1 ’ np is proportional to the diameter of the atom, so it scales
as n2 . On the other hand, for transitions between high angular momentum (circular)
p
states the frequency scales as ω ∝ 1/n3 , which is in the microwave range. According to
p
2
eqn (4.162) the Einstein A coe¬cient scales like A ∝ |d| ω 3 ∝ 1/n5 . Thus the lifetime
p
„ = 1/A ∝ n5 of the upper level is very long, and the neglect of spontaneous emission
p
is a very good approximation.
The opposite conclusion follows for absorption and stimulated emission, since the
relation (4.166) between the A and B coe¬cients shows that B ∝ n4 . For the same
p
applied ¬eld, the absorption rate for a Rydberg atom with np 100 is typically 108
times larger than the absorption rate at the Lyman transition between the 2p and
1s states of the hydrogen atom. Since stimulated emission is also described by the
Einstein B coe¬cient, stimulated emission from the Rydberg atom can occur when
there are only a few photons inside a microwave cavity.
As indicated in Fig. 12.2, a single Rydberg atom enters and leaves a supercon-
ducting microwave cavity through small holes drilled on opposite sides. During the
transit time of the atom across the cavity the photons already present can stimulate
emission of a single photon into the fundamental cavity mode; conversely, the atom
can sometimes reabsorb a single photon. The interaction of the atom with a single
mode of the cavity is described by the Jaynes“Cummings Hamiltonian in eqn (12.7).
By monitoring whether or not the Rydberg atom has made a transition, np + 1 ’ np ,
between the adjacent Rydberg states, one can infer indirectly whether or not a single
microwave photon has been deposited in the cavity. This is possible because of the
entangled nature of the dressed states in eqns (12.18) and (12.20). A measurement of
the state of the atom, with the outcome | 2 , forces a reduction of the total state vector
of the atom“radiation system, with the result that the radiation ¬eld is de¬nitely in
the state |n . In other words, the number of photons in the cavity has not changed.
Conversely, a measurement with the outcome | 1 guarantees that the ¬eld is in the
state |n + 1 , i.e. a photon has been added to the cavity.
The discrimination between the two Rydberg states is easily accomplished, since
the ionization of the Rydberg atom by a DC electric ¬eld depends very sensitively
on its principal quantum number np . The higher number np + 1 corresponds to a
larger, more easily ionized atom, and the lower number np corresponds to a smaller,
less easily ionized atom. The electric ¬eld in the ¬rst ionization region”shown in Fig.
12.2”is strong enough to ionize all (np + 1)-atoms, but too weak to ionize any np -
atoms. Thus an atom that remains in the excited state is detected in the ¬rst region.
If the atom has made a transition to the lower state, then it will be ionized by the
stronger ¬eld in the second region. In this way, it is possible sensitively to identify the
state of the Rydberg atom. If the atom is in the appropriate state, it will be ionized and
release a single electron into the corresponding ionizing ¬eld region. The free electron
is accelerated by the ionizing ¬eld and enters into an electron-multiplication region
of a channeltron detector. As explained in Section 9.2.1, the channeltron detector
¿
The micromaser

can enormously multiply the single electron released by the Rydberg atom, and this
provides an indirect method for continuously monitoring the photon-number state of
the cavity.
A frequency-doubled dye laser (» = 297 nm) is used to excite rubidium (85 Rb)
atoms to the np = 63, P3/2 state from the np = 5, S1/2 (F = 3) state. The cavity is
tuned to the 21.456 GHz transition from the upper maser level in the np = 63, P3/2
state to the np = 61, D5/2 lower maser state. For this experiment a superconducting
cavity with a Q-value of 3—108 was used, corresponding to a photon lifetime inside the
cavity of 2 ms. The transit time of the Rydberg atom through the cavity is controlled
by changing the atomic velocity with the velocity selector. On the average, only a
small fraction of an atom is inside the cavity at any given time. In order to reduce
the number of thermally excited photons in the cavity, a liquid helium environment
reduces the temperature of the superconducting niobium microwave cavity to 2.5 K,
corresponding to the average photon number n ≈ 2.
If the transit time of the atom is larger than the collapse time but smaller than
the time of the ¬rst revival, then the solution (12.32) tells us that the atom will come
into equilibrium with the cavity ¬eld, as seen in Fig. 12.1. In this situation the atom
leaving the chamber is found in the upper or lower state with equal probability, i.e.
P2 = 1/2. When the transit time is increased to a value comparable to the ¬rst revival
time, the probability for the excited state becomes larger than 1/2. The data in Fig.
12.3 show a quantum revival of the population of atoms in the upper maser state that
occurs after a transit time of around 150 µs. Such a revival would be impossible in any
semiclassical picture of the atom“¬eld interaction; it is prima-facie evidence for the
quantized nature of the electromagnetic ¬eld.

0.7 30
T = 2.5 K N = 3000 s’1
0.6 40
Signal depth [%]
Probability Pe(t)




0.5 50


0.4 60
63, P3/2 61, D5/2
85Rb

0.3 70
0 50 100 150
Time of flight through cavity [µs]

Fig. 12.3 Probability of ¬nding the atom in the upper maser level as a function of the time
of ¬‚ight of a Rydberg atom through a superconducting cavity. The ¬‚ux of atoms was around
3000 atoms per second. Note the revival of upper state atoms which occurs at around 150 µs.
(Reproduced from Rempe et al. (1987).)
¿¼ Cavity quantum electrodynamics

12.4 Exercises
12.1 Dressed states
(1) Verify eqns (12.10)“(12.16).
(2) Solve the eigenvalue problem for eqn (12.16) and thus derive eqns (12.17)“(12.22).
(0)
(3) Display level repulsion by plotting the (normalized) bare eigenvalues µ1,n / ωC
(0)
and µ2,n / ωC , and dressed eigenvalues µ1,n / ωC and µ2,n / ωC as functions of the
detuning δ/ωC .

12.2 Collapse and revival for pure initial states
(1) For the initial state |Ψ (0) = | 2 , m (0) , verify the solution (12.26).
(2) Carry out the steps required to derive eqn (12.31).
(3) Write a program to evaluate eqn (12.32), and use it to study the behavior of P2 (t)
at times following the ¬rst revival.

Collapse and revival for a mixed initial state—
12.3
Replace the pure initial state of the previous problem with the mixed state

pn | 2 , n (0) (0)
ρ= 2 , n| .
n=0

(1) Show that this state evolves into

pn | 2 , n; t (0) (0)
ρ= 2 , n; t| .
n=0

(2) Derive the expression for P2 (t).
(3) Assume that pn is the thermal distribution for a given average photon number n.
Evaluate and plot P2 (t) numerically for the value of n used in Fig. 12.1. Comment
on the comparison between the two plots.
13
Nonlinear quantum optics

The interaction of light beams with linear optical devices is adequately described by
the quantum theory of light propagation explained in Section 3.3, Chapter 7, and
Chapter 8, but some of the most important applications involve modi¬cation of the
incident light by interactions with nonlinear media, e.g. by frequency doubling, spon-
taneous down-conversion, four-wave mixing, etc. These phenomena are the province
of nonlinear optics. Classical nonlinear optics deals with ¬elds that are strong enough
to cause appreciable change in the optical properties of the medium, so that the weak-
¬eld condition of Section 3.3.1 is violated. A Bloch equation that includes dissipative
e¬ects, such as scattering from other atoms and spontaneous emission, describes the
response of the atomic density operator to the classical ¬eld.
For the present, we do not need the details of the Bloch equation. All we need to
know is that there is a characteristic response time, Tmed, for the medium. The classical
envelope ¬eld evolves on the time scale T¬‚d ∼ 1/„¦, where „¦ is the characteristic Rabi
frequency. If Tmed ≈ T¬‚d the coupled equations for the atoms and the ¬eld must be
solved together. This situation arises, for example, in the phenomenon of self-induced
transparency and in the theory of free-electron lasers (Yariv, 1989, Chaps 13, 15).
In many applications of interest for nonlinear optics, the incident radiation is de-
tuned from the atomic resonances in order to avoid absorption. As shown in Section
11.3.3, this justi¬es the evaluation of the atomic density matrix by adiabatic elimi-
nation. In this approximation, the atoms appear to follow the envelope ¬eld instan-
taneously; they are said to be slaved to the ¬eld. Even with this simpli¬cation, the
Bloch equation cannot be solved exactly, so the atomic density operator is evaluated
by using time-dependent perturbation theory in the atom“¬eld coupling. In this calcu-
lation, excited states of an atom only appear as virtual intermediate states; the atom
is always returned to its original state. This means that both spontaneous emission
and absorption are neglected.


13.1 The atomic polarization
Substituting the perturbative expression for the atomic density matrix into the source
terms for Maxwell™s equations results in the apparent disappearance, via adiabatic
elimination, of the atomic degrees of freedom. This in turn produces an expansion of
the medium polarization in powers of the ¬eld, which is schematically represented by

(1) (2) (3)
Pi = χij Ej + χijk Ej Ek + χijkl Ej Ek El + · · · , (13.1)
0
¿¾ Nonlinear quantum optics

where the χ(n) s are the tensor nonlinear susceptibilities required for dealing with
(2)
anisotropic materials and E is the classical electric ¬eld. The term χijk Ej Ek describes
the combination of two waves to provide the source for a third, so it is said to describe
(3)
three-wave mixing. In the same way χijkl Ej Ek El is associated with four-wave mix-
ing.
A substance is called weakly nonlinear if the dielectric response is accurately
represented by a small number of terms in the expansion (13.1). This approximation
is the basis for most of nonlinear optics,1 but there are nonlinear optical e¬ects that
cannot be described in this way, e.g. saturation in lasers (Yariv, 1989, Sec. 8.7). The
higher-order terms in the polarization lead to nonlinear terms in Maxwell™s equations
that represent self-coupling of individual modes as well as coupling between di¬er-
ent modes. These terms describe self-actions of the electromagnetic ¬eld that are
mediated by the interaction of the ¬eld with the medium.
Quantum nonlinear optics is concerned with situations in which there are a small
number of photons in some or all of the ¬eld modes. In this case the quantized ¬eld
theory is required, but the correspondence principle assures us that the e¬ects arising in
classical nonlinear optics must also be present in the quantum theory. Thus the classical
three- and four-wave mixing terms correspond to three- and four-photon interactions.
Since the quantum ¬elds are typically weak, these nonlinear phenomena are often
unobservably small. There are, however, at least two situations in which this is not
the case. According to eqn (2.188), the vacuum ¬‚uctuation ¬eld strength in a physical
cavity of volume V is ef = ωf /2 0 V . This shows that substantial ¬eld strengths can
be achieved, even for a single photon, in a small enough cavity. A second exception
depends on the fact that the frequency-dependent nonlinear susceptibilities display
resonant behavior. If the detuning from resonance is made as small as possible”
i.e. without violating the conditions required for adiabatic elimination”the nonlinear
couplings are said to be resonantly enhanced.
When both of these conditions are met, the interaction between the medium and
the ¬eld can be so strong that the electromagnetic ¬eld will interact with itself, even
when there are only a few quanta present. This happens, for example, when microwave
photons inside a cavity interact with each other via a medium composed of Rydberg
atoms excited near resonance. In this case the interacting microwave photons can even
form a photon ¬‚uid.
In addition to these practical issues, there are situations in which the use of quan-
tum theory is mandatory. In the phenomenon of spontaneous down-conversion, a non-
linear optical process couples vacuum ¬‚uctuations of the electromagnetic ¬eld to an
incident beam of ultraviolet light so that an ultraviolet photon decays into a pair of
lower-energy photons. E¬ects of this kind cannot be described by the semiclassical
theory.
In Section 13.2 we will brie¬‚y review some features of classical nonlinear optics and
introduce the corresponding quantum description. In the following two sections we will
discuss examples of three- and four-photon coupling. In each case the quantum theory

1 For
a selection of recent texts on nonlinear optics, see Shen (1984), Schubert and Wilhelmi (1986),
Butcher and Cotter (1990), Boyd (1992), and Newell and Moloney (1992).
¿¿
Weakly nonlinear media

will be developed in a phenomenological way, i.e. it will be based on a conjectured form
for the Hamiltonian. This is in fact the standard way of formulating a quantum theory.
The choice of the Hamiltonian must ultimately be justi¬ed by comparing the results of
calculations with experiment, as there will always be ambiguities”such as in operator
ordering, coordinate choices (e.g. Cartesian versus spherical), etc.”which cannot be
settled by theoretical arguments alone. Quantum theory is richer than classical theory;
consequently, there is no unique way of deriving the quantum Hamiltonian from the
classical energy.

13.2 Weakly nonlinear media
13.2.1 Classical theory
A Plane waves in crystals
Many applications of nonlinear optics involve the interaction of light with crystals, so
we brie¬‚y review the form of the fundamental plane waves in a crystal. As explained
in Appendix B.5.3, the ¬eld can be expressed as
1
E (+) (r, t) = i √ Fks ±ks µks ei(k·r’ωks t) , (13.2)
V ks

where µks is a crystal eigenpolarization, the polarization-dependent frequency ωks is
a solution of the dispersion relation

c2 k 2 = ω 2 n2 (ω) , (13.3)
s

and ns (ω) is the index of refraction associated with the eigenpolarization µks . The
normalization constant,
ωks vg (ωks )
Fks = , (13.4)
2 0 ns (ωks ) c
has been chosen to smooth the path toward quantization, and vg (ωks ) = dωks /dk is
the group velocity. For a polychromatic ¬eld, the expression (3.116) for the envelope
(+)
Eβ is replaced by

1
(+)
(r, t) = √
Eβ Fks ±ks µks ei(k·r’∆βk t) , (13.5)
V ks

where the prime on the k-sum indicates that it is restricted to k-values such that the
detuning, ∆βks = ωks ’ωβ , is small compared to the minimum spacing between carrier
frequencies, i.e. |∆βks | min {|ω± ’ ωβ | , ± = β}.

B Nonlinear susceptibilities
Symmetry, or lack of symmetry, with respect to spatial inversion is a fundamental
distinction between di¬erent materials. A medium is said to have a center of sym-
metry, or to be centrosymmetric, if there is a spatial point (which is conventionally
¿ Nonlinear quantum optics

chosen as the origin of coordinates) with the property that the inversion transforma-
tion r ’ ’ r leaves the medium invariant. When this is true, the polarization must
behave as a polar vector, i.e. P ’ ’P. The electric ¬eld is also a polar vector, so
(2)
eqn (13.1) implies that all even-order susceptibilities”in particular χijk ”vanish for
centrosymmetric media. Vapors, liquids, amorphous solids, and some crystals are cen-
trosymmetric. The absence of a center of symmetry de¬nes a non-centrosymmetric
(2)
crystal. This is the only case in which it is possible to obtain a nonvanishing χijk .
(3)
There is no such general restriction on χijkl ”or any odd-order susceptibility”since
(3) (3)
the third-order polarization, Pi = χijkl Ej Ek El , is odd under E ’ ’E.
The schematic expansion (13.1) does not explicitly account for dispersion, so we
now turn to the exact constitutive relation
(n) (n)
Pi dt1 · · · dtn χij1 j2 ···jn (t ’ t1 , t ’ t2 , . . . , t ’ tn )
(r, t) = 0

— Ej1 (r, t1 ) · · · Ejn (r, tn ) (13.6)

for the nth-order polarization, which is treated in greater detail in Appendix B.5.4.
This time-domain form explicitly displays the history dependence of the polarization”
previously encountered in Section 3.3.1-B”but the equivalent frequency-domain form
n
dν1 dνn
(n) (n)
Pi ··· 2πδ ν ’
(r, ν) = νp χij1 j2 ···jn (ν1 , . . . , νn )
0
2π 2π p=1
— Ej1 (r, ν1 ) · · · E jn (r, νn ) (13.7)

is more useful in practice.

C E¬ective electromagnetic energy
The derivation in Section 3.3.1-B of the e¬ective electromagnetic energy for a linear,
dispersive dielectric can be restated in the following simpli¬ed form.
(1) Start with the expression for the energy in a static ¬eld.
(2) Replace the static ¬eld by a time-dependent ¬eld.
(3) Perform a running time-average”as in eqn (3.136)”on the resulting expression.
For a nonlinear dielectric, we carry out step (1) by using the result
D
E (r) · d (D (r))
Ues = 3
dr
Vc 0
P
0
E (r) · d (P (r))
d3 rE 2 (r) + d3 r
= (13.8)
2 Vc Vc 0

for the energy of a static ¬eld in a dielectric occupying the volume Vc (Jackson, 1999,
Sec. 4.7). Substituting eqn (13.1) into this expression leads to an expansion of the
energy in powers of the ¬eld amplitude:

Ues = Ues + Ues + Ues + · · · .
(2) (3) (4)
(13.9)
¿
Weakly nonlinear media

The ¬rst term on the right is discussed in Section 3.3.1-B, so we can concentrate on
the higher-order (n 3) terms:
1 (n’1)
Ues = d3 rEi (r) Pi
(n)
(r) .
n Vc

In steps (2) and (3), we replace the static energy by the e¬ective energy,
1 (n’1)
Ues ’ Uem (t) = d3 r Ei (r, t) Pi
(n) (n)
(r, t) for n 3, (13.10)
n Vc

and use eqn (13.6) to evaluate the nth-order polarization. Our experience with the
(2)
quadratic term, Uem (t), tells us that eqn (13.10) will only be useful for polychromatic
¬elds; therefore, we impose the condition 1/ωmin T 1/∆ωmax on the averaging
time, where ωmin is the smallest carrier frequency and ∆ωmax is the largest spectral
width for the polychromatic ¬eld. This time-averaging eliminates all rapidly-varying
terms, while leaving the slowly-varying envelope ¬elds unchanged.
The lowest-order energy associated with the nonlinear polarizations is
1 (2)
Uem (t) = d3 r Ei (r, t) Pi
(3)
(r, t) , (13.11)
3 Vc

(2)
so the next task is to evaluate Pi (r, t) for a polychromatic ¬eld. This is done by
applying the exact relation (13.7) for n = 2, and using the expansion (3.119) for a
polychromatic ¬eld to ¬nd:
dν1 dν2
(2) (2)
Pi 2πδ (ν ’ ν1 ’ ν2 ) χijk (ν1 , ν2 )
(r, ν) = 0
2π 2π
β,γ σ ,σ =±
(σ ) (σ )
— E βj (r, ν1 ’ σ ωβ ) E γk (r, ν2 ’ σ ωγ ) . (13.12)

Weak dispersion means that the susceptibility is essentially constant across the spectral
(±) (2)
width of each sharply-peaked envelope function, E βj (r, ν); therefore, Pi (r, ν) can
be approximated by
dν1 dν2
(2) (2)
Pi 2πδ (ν ’ ν1 ’ ν2 ) χijk (σ ωβ , σ ωγ )
(r, ν) = 0
2π 2π
β,γ σ ,σ =±
(σ ) (σ )
— E βj (r, ν1 ’ σ ωβ ) E γk (r, ν2 ’ σ ωγ ) . (13.13)

Carrying out an inverse Fourier transform yields the time-domain relation,
(2) (2)
Pi (r, t) = χijk (σ ωβ , σ ωγ )
0
β,γ σ ,σ =±
(σ ) (σ )
— E βj (r, t) E γk (r, t) e’i(σ ωβ +σ ωγ )t
, (13.14)

which shows that the time-averaging has eliminated the history dependence of the
polarization.
¿ Nonlinear quantum optics

(3)
Using eqn (13.14) to evaluate the expression (13.11) for Uem (t) is simpli¬ed by
the observation that the slowly-varying envelope ¬elds can be taken outside the time
average, so that

1 (σ) (σ )
(2)
Uem (t) = χijk (σ ωβ , σ ωγ ) E ±i (r, t) E βj (r, t)
(3)
d3 r
3 Vc ±,β,γ σ,σ ,σ
(σ )
— E γk (r, t) e’i(σω± +σ ωβ +σ ωγ )t
. (13.15)

The frequencies in the exponential all satisfy ωT 1, so the remaining time-average,
T /2
1
’i(σω± +σ ωβ +σ ωγ )t
d„ e’i(σω± +σ ωβ +σ ωγ )(t+„ )
e = ,
T ’T /2

vanishes unless
σω± + σ ωβ + σ ωγ = 0 . (13.16)
This is called phase matching. By convention, the carrier frequencies are positive;
consequently, phase matching in eqn (13.15) always imposes conditions of the form

ω ± = ωβ + ωγ . (13.17)
(+) (+) (’) (’) (’) (+)
This in turn means that only terms of the form E E E or E E E will
contribute. By making use of the symmetry properties of the susceptibility, reviewed
in Appendix B.5.4, one ¬nds the explicit result
(2)
Uem (t) =
(3)
χijk (ωβ , ωγ ) δω± ,ωβ +ωγ
0
±,β,γ
(’) (+) (+)
— d3 r E ±i (r, t) E βj (r, t) E γk (r, t) + CC . (13.18)
Vc

In many applications, the envelope ¬elds will be expressed by an expansion in some
appropriate set of basis functions. For example, if the nonlinear medium is placed in a
resonant cavity, then the carrier frequencies can be identi¬ed with the frequencies of
the cavity modes, and each envelope ¬eld is proportional to the corresponding mode
function. More generally, the ¬eld can be represented by the plane-wave expansion
2
(13.2), provided that the power spectrum |±ks | exhibits well-resolved peaks at ωks =
ω± , where ω± ranges over the distinct monochromatic carrier frequencies. With this
restriction held ¬rmly in mind, the explicit sums over the distinct monochromatic
waves can be replaced by sums over the plane-wave modes, so that
i
gs0 s1 s2 (ω1 , ω2 ) [±0 ±— ±— ’ CC]
Uem =
(3) (3)
12
3/2
V
k0 s0 ,k1 s1 ,k2 s2
— C (k0 ’ k1 ’ k2 ) δω0 ,ω1 +ω2 , (13.19)

where ±0 = ±k0 s0 , etc., and
¿
Weakly nonlinear media


C (k) = d3 reik·r (13.20)
Vc

is the spatial cut-o¬ function for the crystal. The three-wave coupling strength is
related to the second-order susceptibility by

(2)
0 F0 F1 F2
(3)
gs0 s1 s2 (ω1 , ω2 ) = (µk0 s0 )i (µk1 s1 )j (µk2 s2 )k χijk (ω1 , ω2 ) , (13.21)

where ωp = ωkp sp and Fp = Fkp sp (p = 0, 1, 2).
In the limit of a large crystal, i.e. when all dimensions are large compared to optical
wavelengths,
3
C (k) ∼ Vc δk,0 ’ (2π) δ (k) . (13.22)

(3)
This tells us that for large crystals the only terms that contribute to Uem are those
satisfying the complete phase-matching conditions

k0 = k1 + k2 , ω0 = ω1 + ω2 . (13.23)

(4)
The same kind of analysis for Uem reveals two possible phase-matching conditions:

k0 = k1 + k2 + k3 , ω0 = ω1 + ω2 + ω3 , (13.24)

corresponding to terms of the form ±— ±1 ±2 ±3 + CC, and
0


k0 + k1 = k2 + k3 , ω0 + ω1 = ω2 + ω3 , (13.25)

corresponding to terms like ±— ±— ±2 ±3 + CC. As shown in Exercise 13.1, the coupling
01
constants associated with these processes are related to the third-order susceptibility,
χ(3) .
The de¬nition (13.21) relates the nonlinear coupling term to a fundamental prop-
erty of the medium, but this relation is not of great practical value. The ¬rst-principles
evaluation of the susceptibilities is an important problem in condensed matter physics,
but such a priori calculations typically involve other approximations. With the excep-
tion of hydrogen, the unperturbed atomic wave functions for single atoms are not
known exactly; therefore, various approximations”such as the atomic shell model”
must be used. In the important case of crystalline materials, corrections due to local
¬eld e¬ects are also di¬cult to calculate (Boyd, 1992, Sec. 3.8). In practice, approx-
imate calculations of the susceptibilities can readily incorporate the symmetry prop-
erties of the medium, but otherwise they are primarily useful as a rough guide to
the feasibility of a proposed experiment. Fortunately, the analysis of experiments does
not require the full solution of these di¬cult problems. An alternative procedure is
to use symmetry arguments to determine the form of expressions, such as (13.19),
for the energy. The coupling constants, which in principle depend on the nonlinear
susceptibilities, can then be determined by ancillary experiments.
¿ Nonlinear quantum optics

13.2.2 Quantum theory
The approximate quantization scheme for an isotropic dielectric given in Section 3.3.2
can be applied to crystals by the simple expedient of replacing the classical amplitude
±ks in eqn (13.5) by the annihilation operator aks , i.e.
i
(+) (+)
(r) = √
Eβ (r, 0) ’ Eβ Fks ±ks µks eik·r . (13.26)
V ks

In the linear approximation, the electromagnetic Hamiltonian in a crystal”which we
(0)
will now treat as the zeroth-order Hamiltonian, Hem ”is obtained from eqn (3.150) by
using the polarization-dependent frequency ωks in place of ωk :
ωks a† aks .
(0)
Hem = (13.27)
ks
ks
2
The assumption that the classical power spectrum |±ks | is peaked at the carrier
frequencies is replaced by the rule that the expressions (13.26) and (13.27) are only
valid when the operators act on a polychromatic space H ({ωβ }), as de¬ned in Section
3.3.4.
In a weakly nonlinear medium, we will employ a phenomenological approach in
which the total electromagnetic Hamiltonian is given by
(0) NL
Hem = Hem + Hem . (13.28)
NL
The higher-order terms comprising Hem can be constructed from classical energy
expressions, such as (13.19), by applying the quantization rule (13.26) and putting all
the terms into normal order. An alternative procedure is to use the correspondence
principle and symmetry arguments to determine the form of the Hamiltonian. In this
NL
approach, the weak-¬eld condition is realized by assuming that the terms in the Hem
are given by low-order polynomials in the ¬eld operators. Since the ¬eld interacts with
itself through the medium, the coupling constants must transform appropriately under
the symmetry group for the medium. The coupling constants must, therefore, have
the same symmetry properties as the classical susceptibilities. The Hamiltonian must
also be invariant with respect to time translations, and”for large crystals”spatial
translations. The general rules of quantum theory (Bransden and Joachain, 1989, Sec.
5.9) tell us that these invariances are respectively equivalent to the conservation of
energy and momentum. Applying these conservation laws to the individual terms in
the Hamiltonian yields”after dividing through by ”the classical phase-matching
conditions (13.23)“(13.25).
The expansion (13.9) for the classical energy is replaced by
Hem = Hem + Hem + · · · ,
NL (3) (4)
(13.29)
where the symmetry considerations mentioned above lead to expressions of the form
i
C (k0 ’ k1 ’ k2 ) δω0 ,ω1 +ω2
(3)
Hem =
V 3/2
k0 s0 ,k1 s1 ,k2 s2

— gs0 s1 s2 (ω1 , ω2 ) a† 1 s1 a† 2 s2 ak0 s0 ’ HC
(3)
(13.30)
k k
¿
Three-photon interactions

and
1
C (k0 ’ k1 ’ k2 ’ k3 ) δω0 ,ω1 +ω2 +ω3
(4)
Hem =
V2
k0 s0 ,...,k3 s3

— gs0 s1 s2 s3 (ω1 , ω2 , ω3 ) a† 0 s0 ak1 s1 ak2 s2 ak3 s3 + HC
(4)
k
1
C (k0 + k1 ’ k2 ’ k3 ) δω0 +ω1 ,ω2 +ω3
+
V2
k0 s0 ,...,k3 s3

— fs0 s1 s2 s3 (ω1 , ω2 , ω3 ) a† 2 s2 a† 3 s3 ak0 s0 ak1 s1 + HC .
(4)
(13.31)
k k

Another important feature follows from the observation that the susceptibilities are
necessarily proportional to the density of atoms. When combined with the assumption
that the susceptibilities are uniform over the medium, this implies that the operators
(3) (4)
Hem and Hem represent the coherent interaction of the ¬eld with the entire mate-
rial sample. First-order transition amplitudes are thus proportional to Nat , and the
2
corresponding transition rates are proportional to Nat . In contrast to this, scattering
of the light from individual atoms adds incoherently, so that the transition rate is
2
proportional to Nat rather than Nat .
The Hamiltonian obtained in this way contains many terms describing a variety of
nonlinear processes allowed by the symmetry properties of the medium. For a given
experiment, only one of these processes is usually relevant, so a model Hamiltonian is
constructed by neglecting the other terms. The relevant coupling constants must then
be determined experimentally.

13.3 Three-photon interactions
The mutual interaction of three photons corresponds to classical three-wave mixing,
which can only occur in a crystal with nonvanishing χ(2) , e.g. lithium niobate, or am-
monium dihydrogen phosphate (ADP). A familiar classical example is up-conversion
(Yariv, 1989, Sec. 17.6), which is also called sum-frequency generation (Boyd, 1992,
Sec. 2.4). In this process, waves E 1 and E 2 , with frequencies ω1 and ω2 , mix in a non-
centrosymmetric χ(2) crystal to produce a wave E 0 with frequency ω0 = ω1 +ω2 . The
traditional applications for this process involve strong ¬elds that can be treated clas-
sically, but we are interested in a quantum approach. To this end we replace classical
wave mixing by a microscopic process in which photons with energy and momentum
( k1 , ω1 ) and ( k2 , ω2 ) are absorbed and a photon with energy and momentum
( k0 , ω0 ) is emitted. The phase-matching conditions (13.23) are then interpreted as
conservation of energy and momentum in each microscopic interaction.
As a result of crystal anisotropy, phase matching can only be achieved by an ap-
propriate choice of polarizations for the three photons. The uniaxial crystals usually
employed in these experiments”which are described in Appendix B.5.3-A”have a
principal axis of symmetry, so they exhibit birefringence. This means that there are
two refractive indices for each frequency: the ordinary index no (ω) and the extraor-
dinary index ne (ω, θ). The ordinary index no (ω) is independent of the direction of
propagation, but the extraordinary index ne (ω, θ) depends on the angle θ between the
¼¼ Nonlinear quantum optics

propagation vector and the principal axis. The crystal is said to be negative (positive)
when ne < no (ne > no ). For typical crystals, the refractive indices exhibit a large
amount of dispersion between the lower frequencies of the input beams and the higher
frequency of the output beam; therefore, it is necessary to exploit the birefringence of
the crystal in order to satisfy the phase-matching conditions.
In type I phase matching, for negative uniaxial crystals, the incident beams
have parallel polarizations as ordinary rays inside the crystal, while the output beam
propagates in the crystal as an extraordinary ray. Thus the input photons obey
ω1 no (ω1 ) ω2 no (ω2 )
k1 = , k2 = , (13.32)
c c
while the output photon satis¬es the dispersion relation
ω0 ne (ω0 , θ0 )
k0 = , (13.33)
c
where θ0 is the angle between the output direction and the optic axis. In type II
phase matching, for negative uniaxial crystals, the linear polarizations of the input
beams are orthogonal, so that one is an ordinary ray, and the other an extraordinary
ray, e.g.
ω1 no (ω1 ) ω2 ne (ω2 , θ2 )
k1 = , k2 = . (13.34)
c c
In this case the output beam also propagates in the crystal as an extraordinary ray.
For positive uniaxial crystals the roles of ordinary and extraordinary rays are reversed
(Boyd, 1992).
With an appropriate choice of the angle θ0 , which can be achieved either by suitably
cutting the crystal face or by adjusting the directions of the input beams with respect
to the crystal axis, it is always possible to ¬nd a pair of input frequencies for which
all three photons have parallel propagation vectors. This is called collinear phase
matching.
From Appendix B.3.3 and Section 4.4, we know that the classical and quantum
theories of light are both invariant under time reversal; consequently, the time-reversed
process”in which an incident high-frequency ¬eld E0 generates the low-frequency out-
put ¬elds E 1 and E 2 ”must also be possible. This process is called down-conversion.
In the classical case, one of the down-converted ¬elds, say E 1 , must be initially present;
and the growth of the ¬eld E 2 is called parametric ampli¬cation (Boyd, 1992,
Sec. 2.5). The situation is quite di¬erent in quantum theory, since the initial state
need not contain either of the down-converted photons. For this reason the time-
reversed quantum process is called spontaneous down-conversion (SDC). Sponta-
neous down-conversion plays a central role in modern quantum optics. For somewhat
obscure historical reasons, this process is frequently called spontaneous parametric
down-conversion or else parametric ¬‚uorescence. In this context ˜parametric™ simply
means that the optical medium is unchanged, i.e. each atom returns to its initial state.

13.3.1 The three-photon Hamiltonian
We will simplify the notation by imposing the convention that the polarization index
is understood to accompany the wavevector. The three modes are thus represented
¼½
Three-photon interactions

by (k0 , ω0 ), (k1 , ω1 ), and (k2 , ω2 ) respectively. The fundamental interaction processes
are shown in Fig. 13.1, where the Feynman diagram (b) describes down-conversion,
while diagram (a) describes the time-reversed process of sum-frequency generation.
Strictly speaking, Feynman diagrams represent scattering amplitudes; but they are
frequently used to describe terms in the interaction Hamiltonian. The excuse is that
the ¬rst-order perturbation result for the scattering amplitude is proportional to the
matrix element of the interaction Hamiltonian between the initial and ¬nal states.
Since the nonlinear process is the main point of interest, we will simplify the prob-
lem by assuming that the entire quantization volume V is ¬lled with a medium hav-
ing the same linear index of refraction as the nonlinear crystal. This is called index
matching. The simpli¬ed version of eqn (13.30) is then
1
g (3) C (k0 ’ k1 ’ k2 ) a† 1 a† 2 ak0 + HC .
(3)
Hem = (13.35)
kk
3/2
V k0 k2 k3

This is the relevant Hamiltonian for detection in the far ¬eld of the crystal, i.e. when
the distance to the detector is large compared to the size of the crystal, since all atoms
can then contribute to the generation of the down-converted photons.
(3)
The two terms in Hem describe down-conversion and sum-frequency generation
respectively. Note that both terms must be present in order to ensure the Hermiticity
of the Hamiltonian. The down-conversion process is analogous to a radioactive decay
in which a single parent particle (the ultraviolet photon) decays into two daughter
particles, while sum-frequency generation is an analogue of particle“antiparticle anni-
hilation.

13.3.2 Spontaneous down-conversion
Spontaneous down-conversion is the preferred light source for many recent experi-
ments in quantum optics, e.g. single-photon number-state production, entanglement
phenomena (such as the Einstein“Podolsky“Rosen e¬ect and Franson two-photon in-
terference), and tunneling time measurements. One reason for the popularity of this
light source is that it is highly directional, whereas the atomic cascade sources dis-
cussed in Sections 1.4 and 11.2.3 emit light in all directions. In SDC, correlated photon
pairs are emitted into narrow cones in the form of a rainbow surrounding the pump
beam direction. The two photons of a pair are always emitted on opposite sides of the
rainbow axis. Since the photon pairs are emitted within a few degrees of the pump

(k0, ω0)

(k1, ω1) (k2, ω2)




(k1, ω1) (k2, ω2)
Fig. 13.1 Three-photon interactions (time
(k0, ω0)
¬‚ows upward in the diagrams): (a) represents
sum-frequency generation, and (b) represents
(a) (b) the time-reversed process of down-conversion.
¼¾ Nonlinear quantum optics

beam direction, detection of the output within small solid angles is relatively straight-
forward. Another practical reason for the choice of SDC is that it is much easier to
implement experimentally, since the heart of the light source is a nonlinear crystal.
This method eliminates the vacuum technology required by the use of atomic beams
in a cascade emission source.

A Generation of entangled photon pairs
In spontaneous down-conversion the incident ¬eld is called the pump beam, and the
down-converted ¬elds are traditionally called the signal and idler. To accommodate
this terminology we change the notation (E 0 , k0 , ω0 ) for the input ¬eld to (E P , p, ωP ).
There is no physical distinction between the signal and idler, so we will continue to
use the previous notation for the conjugate modes in the down-converted light. The
emission angles and frequencies of the down-converted photons vary continuously,
but they are subject to overall conservation of energy and momentum in the down-
conversion process.
The interaction Hamiltonian (13.35) is more general than is required in practice,
since it is valid for any distribution in the pump photon momenta. In typical experi-
ments, the pump photons are supplied by a continuous wave (cw) ultraviolet laser, so
the pump ¬eld is well approximated by a classical plane-wave mode with amplitude
EP . A suitable quantum model is given by a Heisenberg-picture state satisfying

ak (t) |±p = δk,p ±p e’iωP t |±p . (13.36)

In other words |±p is a coherent state built up from pump photons that are all in the
mode p. The coherent-state parameter ±p is related to the classical ¬eld amplitude
EP by
±p
EP ≡ e’ip·r ±p ep · E(+) (r) ±p = iFp √ , (13.37)
V
where the expansion (13.26) was used to get the ¬nal result. Since the number of
pump photons is large, the loss of one pump photon in each down-conversion event
can be neglected. This undepleted pump approximation allows the semiclassical
limit described in Section 11.3 to be applied. Thus we replace the Heisenberg-picture
operator ap (t) for the pump mode by ±p exp (’iωP t) + δap (t), and then neglect the
terms involving the vacuum ¬‚uctuation operators δap (t).
Since the pump mode is treated classically and the coherent state |±p is the vac-
uum for the down-converted modes, we replace the notation |±p by |0 . The classical
amplitude, ±p exp (’iωP t), is unchanged by the transformation from the Heisenberg
picture to the Schr¨dinger picture; therefore, the semiclassical Hamiltonian in the
o
Schr¨dinger picture is
o
(3)
H = H0 + Hem (t) , (13.38)
ω q a† aq ,
2
H0 = ωP |±p | + (13.39)
q
q
i
G(3) e’iωP t C (p ’ k1 ’ k2 ) a† 1 a† 2 + HC ,
Hem (t) = ’
(3)
(13.40)
kk
V
k1 ,k2
¼¿
Three-photon interactions

where the pump-enhanced coupling constant is G(3) = EP g (3) /Fp . The explicit time
dependence of the Schr¨dinger-picture Hamiltonian is a result of treating the pump
o
2
beam as an external classical ¬eld. The c-number term, ωP |±p | , in the unperturbed
Hamiltonian can be dropped, since it shifts all unperturbed energy levels by the same
amount.
We will eventually need the limit of in¬nite quantization volume, so we use the
rules (3.64) to express the (Schr¨dinger-picture) Hamiltonian as
o
(3)
H = H0 + Hem (t) , (13.41)

d3 q
ωq a† (q) a (q) ,
H0 = (13.42)

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