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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)
3
(2π)
d3 k1 d3 k2 (3) ’iωP t
(p ’ k1 ’ k2 ) a† (k1 ) a† (k2 ) + HC .
(t) = ’i C
(3)
Hem 3G e
3
(2π) (2π)
(13.43)

The Hamiltonian has the same form in the Heisenberg picture, with a (k1 ) replaced
by a† (k1 , t), etc. Let
N (k1 , t) = a† (k1 , t) a (k1 , t) (13.44)
denote the (Heisenberg-picture) number operator for the k1 -mode, then a straightfor-
ward calculation using eqn (3.26) yields

d3 k2
’iωP t
(p ’ k1 ’ k2 ) a† (k1 , t) a† (k2 , t) ’ HC .
[N (k1 , t) , H] = ’2ie C
(3)
3G
(2π)
(13.45)
The illuminated volume of the crystal is typically large on the scale of optical wave-
lengths, so the approximation (13.22) can be used to simplify this result to

[N (k1 , t) , H] = ’2ie’iωP t G(3) a† (k1 , t) a† (p ’ k1 , t) . (13.46)

In this approximation we see that

[N (k1 , t) ’ N (p ’ k1 , t) , H] = 0 , (13.47)

i.e. the di¬erence between the population operators for signal and idler photons is
a constant of the motion. An experimental test of this prediction is to measure the
expectation values n (k1 , t) = N (k1 , t) and n (p ’ k1 , t) = N (p ’ k1 , t) . This can
be done by placing detectors behind each of a pair of stops that select out a particular
signal“idler pair (k1 , p ’ k1 ). According to eqn (13.47), the expectation values satisfy

n (k1 , t) ’ n (p ’ k1 , t) = N (k1 , t) ’ N (p ’ k1 , t)
= N (k1 , 0) ’ N (p ’ k1 , 0)
= 0, (13.48)

which provides experimental evidence that the conjugate photons are created at the
same time.
¼ Nonlinear quantum optics

B Entangled state of the signal and idler photons
Even with pump enhancement, the coupling parameter G(3) (k1 , k2 ) is small, so the

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