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detectors, and one cannot know, even in principle, which of the two processes actually
occurred. According to the Feynman rules of interference we must add the probability
amplitudes for the two processes, and then calculate the absolute square of the sum
to ¬nd the total probability. If the incident amplitude is set to one, the amplitudes
of the two processes are Arr = r2 and Att = t2 , where r and t are respectively the
complex re¬‚ection and transmission coe¬cients for the beam splitter; therefore, the
coincidence amplitude is

Acoinc = Arr + Att = r2 + t2 . (10.34)

According to eqn (8.8), r and t are π/2 out of phase; therefore the coincidence proba-
bility is
2 2 2
Pcoinc = |Acoinc | = |r| ’ |t| , (10.35)

1 These
names are borrowed from radio engineering, which in turn borrowed the ˜idler™ from the
mechanical term ˜idler gear™.
Two-photon interference

which, happily, agrees with the result (9.98) for the coincidence-counting rate. The
partial destructive interference between the rr- and tt-processes, demonstrated by the
expression for Pcoinc , becomes total interference for the special case of a balanced
beam splitter, i.e. the coincidence probability vanishes. We will refer to this as the
Hong“Ou“Mandel (HOM) e¬ect. This is a strictly quantum interference e¬ect
which cannot be explained by any semiclassical theory.
Another way of describing this phenomenon is that two photons, in the appropriate
initial state, impinging simultaneously onto a balanced beam splitter will pair o¬ and
leave together through one of the two exit ports, i.e. both photons occupy one of the
output modes, (ksig , ssig ) or (kidl , sidl ). This behavior is permitted for photons, which
are bosons, but it would be forbidden by the Pauli principle for electrons, which are
fermions. As a result of this pairing e¬ect, detectors placed at the two exit ports of a
balanced beam splitter will never register a coincidence count. The exit port used by
the photon pair varies randomly from one incident pair to the next.
The argument based on the Feynman rules very e¬ectively highlights the fundamen-
tal principles involved in two-photon interference, but it is helpful to derive the result
by using a Schr¨dinger-picture scattering analysis. The Schr¨dinger-picture state pro-
o o
† †
duced by degenerate, spontaneous down-conversion is asig aidl |0 , but the initial state
for the beam splitter scattering calculation is modi¬ed by the further propagation
from the twin-photon source to the beam splitter. According to eqn (8.1) the scatter-
ing matrix S for propagation through vacuum is simply multiplication by exp (ikL),
where k is the wavenumber and L is the propagation distance; therefore, the general
rule (8.44) shows that the state incident on the beam splitter is

|¦in = eik0 Lsig eik0 Lidl a† a† |0 , (10.36)
sig idl

where Lidl and Lsig are respectively the distances along the idler and signal arms
from the point of creation of the photon pair to the beam splitter. For the present
calculation this phase factor is not important; however, it will play a signi¬cant role
in Section 10.2.1-B. According to eqn (6.92), |¦in is an entangled state, and the ¬nal
2 2
a† + a†
|¦¬n = r t e’2iω0 t eik0 Lsig eik0 Lidl |0

+ eik0 Lsig eik0 Lidl r2 + t2 a† a† |0 , (10.37)
idl sig

obtained by using eqn (8.43), is also entangled. For a balanced beam splitter this
reduces to
2 2
i ’2iω0 t ik0 Lsig ik0 Lidl
a† + a†
|¦¬n = |0 ,
e e e (10.38)

which explicitly exhibits the ¬nal state as a superposition of paired-photon states.
Once again the conclusion is that the coincidence rate vanishes for a balanced beam
¿½ Experiments in linear optics

The quantum nature of this result can be demonstrated by considering a semiclassi-
cal model in which the signal and idler beams are represented by c-number amplitudes
±sig and ±idl . The classical version of the beam splitter equation (8.62) is

±sig = t ±sig + r ±idl ,
±idl = r ±sig + t ±idl ,

and the singles counting rates at detectors D1 and D2 are respectively proportional
2 2
to |±idl |2 and ±sig . The coincidence-counting rate is proportional to |±idl |2 ±sig =
±idl ±sig , and eqn (10.39) yields

±idl ±sig = r t ±2 + ±2 + r2 + t2 ±sig ±idl
sig idl
’ ±sig + ±2 , (10.40)
where the last line is the result for a balanced beam splitter. This classical result
resembles eqn (10.38), but now the coincidence rate cannot vanish unless one of the
singles rates does. A more satisfactory model can be constructed along the lines of the
argument used for the discussion of photon indivisibility in Section 1.4. Spontaneous
emission is a real transition, while the down-conversion process depends on the virtual
excitation of the quantum states of the atoms in the crystal; nevertheless, spontaneous
down-conversion is a quantum event. A semiclassical model can be constructed by
assuming that the quantum down-conversion event produces classical ¬elds that vary
randomly from one coincidence gate to the next. With this model one can show, as in
Exercise 10.2, that
pcoinc 1
>, (10.41)
psig pidl 2
where pcoinc is the probability for a coincidence count, and psig and pidl are the prob-
abilities for singles counts”all averaged over many counting windows. This semiclas-
sical model limits the visibility of the interference minimum to 50%; the essentially
perfect null seen in the experimental data can only be predicted by using the complete
destructive interference between probability amplitudes allowed by the full quantum
theory. Thus the HOM null provides further evidence for the indivisibility of photons.

Nondegenerate wave packet analysis—
The simpli¬ed model used above su¬ces to explain the physical basis of the Hong“
Ou“Mandel interferometer, but it is inadequate for describing some interesting ap-
plications to precise timing, such as the measurement of the propagation velocity of
single-photon wave packets in a dielectric, and the nonclassical dispersion cancelation
e¬ect, discussed in Sections 10.2.2 and 10.2.3 respectively. These applications exploit
the fact that the signal and idler modes produced in the experiment are not plane
waves; instead, they are described by wave packets with temporal widths T ∼ 15 fs.
In order to deal with this situation, it is necessary to allow continuous variation of the
frequencies and to relax the degeneracy condition ωidl = ωsig , while retaining the sim-
ple geometry of the scattering problem. To this end, we ¬rst use eqn (3.64) to replace
Two-photon interference

the box-normalized operator aks by the continuum operator as (k), which obeys the
canonical commutation relations (3.26). In polar coordinates the propagation vectors
are described by k = (k, θ, φ), so the propagation directions of the modes (ksig , ssig )
and (kidl , sidl ) are given by (θσ , φσ ), where σ = sig, idl is the channel index. The as-
sumption of frequency degeneracy can be eliminated, while maintaining the scattering
geometry, by considering wave packets corresponding to narrow cones of propagation
directions. The wave packets are described by real averaging functions fσ (θ, φ) that
are strongly peaked at (θ, φ) = (θσ , φσ ) and normalized by

d„¦ |fσ (θ, φ)|2 = 1 , (10.42)

where d„¦ = d (cos θ) dφ. In practice the widths of the averaging functions can be made
so small that
d„¦fσ (θ, φ) fρ (θ, φ) ≈ δσρ . (10.43)

With this preparation, we de¬ne wave packet operators
ω d„¦
a† (ω) ≡ fσ (θ, φ) a†σ (k) , (10.44)
σ s
c3/2 2π
that satisfy
aσ (ω) , a† (ω ) = δσρ 2πδ (ω ’ ω ) ,
[aσ (ω) , aρ (ω )] = 0 .
For a given value of the channel index σ, the operator a† (ω) creates photons in a wave
packet with propagation unit vectors clustered near the channel value kσ = kσ /kσ ,
and polarization sσ ; however, the frequency ω can vary continuously. These operators
are the continuum generalization of the operators ams (ω) de¬ned in eqn (8.71).
With this machinery in place, we next look for the appropriate generalization of
the incident state in eqn (10.36). Since the frequencies of the emitted photons are not
¬xed, we assume that the source generates a state
dω dω
C (ω, ω ) a† (ω) a† (ω ) |0 , (10.46)
sig idl
2π 2π
describing a pair of photons, with one in the signal channel and the other in the
idler channel. As discussed above, propagation from the source to the beam splitter
multiplies the state a† (ω) a† (ω ) |0 by the phase factor exp (ikLsig ) exp (ik Lidl ). It
sig idl
is more convenient to express this as

eikLsig eik Lidl = ei(k+k )Lidl eik∆L , (10.47)
where ∆L = Lsig ’ Lidl is the di¬erence in path lengths. Consequently, the initial state
for scattering from the beam splitter has the general form
dω dω
C (ω, ω ) eik∆L a† (ω) a† (ω ) |0 ,
|¦in = (10.48)
sig idl
2π 2π
where we have absorbed the symmetrical phase factor exp [i (k + k ) Lidl ] into the
coe¬cient C (ω, ω ).
¿¾¼ Experiments in linear optics

By virtue of the commutation relations (10.45), every two-photon state
a† (ω) a† (ω ) |0 satis¬es Bose symmetry; consequently, the two-photon wave packet
sig idl
state |¦in satis¬es Bose symmetry for any choice of C (ω, ω ). However, not all states
of this form will exhibit the two-photon interference e¬ect. To see what further restric-
tions are needed, we consider the balanced case ∆L = 0, and examine the e¬ects of the
alternative processes on |¦in . In the transmission“transmission process the directions
of propagation are preserved, but in the re¬‚ection“re¬‚ection process the directions of
propagation are interchanged. Thus the actions on the incident state are respectively
given by
1 dω dω
C (ω, ω ) a† (ω) a† (ω ) |0 ,
|¦in ’ |¦in = (10.49)
tt sig idl
2 2π 2π
1 dω dω
C (ω, ω ) a† (ω) a† (ω ) |0
|¦in ’ |¦in =’
rr sig
2 2π 2π
1 dω dω
C (ω , ω) a† (ω) a† (ω ) |0 .
=’ (10.50)
sig idl
2 2π 2π
For interference to take place, the ¬nal states |¦in tt and |¦in rr must agree up
to a phase factor, i.e. |¦in tt = exp (iΛ) |¦in rr . This in turn implies C (ω, ω ) =
’ exp (iΛ) C (ω , ω), and a second use of this relation shows that exp (2iΛ) = 1. Con-
sequently the condition for interference is

C (ω, ω ) = ±C (ω , ω) . (10.51)

We will see below that the (+)-version of this condition leads to the photon pairing
e¬ect as in the degenerate case. The (’)-version is a new feature which is possible
only in the nondegenerate case. As shown in Exercise 10.5, it leads to destructive
interference for the emission of photon pairs.
In order to see what happens when the interference condition is violated, consider
the function
C (ω, ω ) = (2π) C0 δ (ω ’ ω1 ) δ (ω ’ ω2 ) (10.52)
describing the input state a† (ω1 ) a† (ω2 ) |0 , where ω1 = ω2 . In this situation pho-
sig idl
tons entering through port 1 always have frequency ω1 and photons entering through
port 2 always have frequency ω2 ; therefore, a measurement of the photon energy at ei-
ther detector would provide which-path information by determining the path followed
by the photon through the beam splitter. This leads to a very striking conclusion:
even if no energy determination is actually made, the mere possibility that it could be
made is enough to destroy the interference e¬ect.
The input state de¬ned by eqn (10.52) is entangled, but this is evidently not enough
to ensure the HOM e¬ect. Let us therefore consider the symmetrized function

C (ω, ω ) = (2π)2 C0 [δ (ω ’ ω1 ) δ (ω ’ ω2 ) + δ (ω ’ ω1 ) δ (ω ’ ω2 )] , (10.53)

which does satisfy the interference condition. The corresponding state
Two-photon interference

|¦in = C0 a† (ω1 ) a† (ω2 ) |0 + a† (ω2 ) a† (ω1 ) |0 (10.54)
sig sig
idl idl

is not just entangled, it is dynamically entangled, according to the de¬nition in Section
6.5.3. Thus dynamical entanglement is a necessary condition for the photon pairing
or antipairing e¬ect associated with the ± sign in eqn (10.51). This feature plays an
important role in quantum information processing with photons.
In the experiments to be discussed below, the two-photon state is generated by the
spontaneous down-conversion process in which momentum and energy are conserved:

ωp = ω + ω ,
kp = k + k ,

where ( ωp , kp ) is the energy“momentum four-vector of the parent ultraviolet photon,
and ( ω, k) and ( ω , k ) are the energy“momentum four-vectors for the daughter
photons. The energy conservation law allows C (ω, ω ) to be written as

C (ω, ω ) = 2πδ (ω + ω ’ ωp ) g (ν) , (10.56)

, ω = ω0 + ν , ω = ω0 ’ ν .
ν= (10.57)
The interference condition (10.51), which ensures that the two Feynman processes lead
to the same ¬nal state, becomes g (ν) = ±g (’ν).
The conservation rule (10.55) tells us that the down-converted photons are anti-
correlated in energy. A bluer photon (ω > ω0 ) is always associated with a redder photon
(ω < ω0 ). Furthermore, the photons are produced with equal amplitudes on either
side of the degeneracy value, ω = ω0 = ωp /2, i.e. g (ν) = g (’ν). Thus the coe¬cient
function C (ω, ω ) for down-conversion satis¬es the (+)-version of eqn (10.51). The
width, ∆ν, of the power spectrum |g (ν)| is jointly determined by the properties of
the KDP crystal and the ¬lters that select out a particular pair of conjugate photons.
The two-photon coherence time corresponding to ∆ν is
„2 ∼ . (10.58)
We are now ready to carry out a more realistic analysis of the Hong“Ou“Mandel
experiment in terms of the interference between the tt- and rr-processes. For a given
value of ν = (ω ’ ω ) /2, the amplitudes are
Att (ν) = t2 g (ν) ei¦tt (ν) ’ g (ν) ei¦tt (ν) (10.59)
Arr (ν) = r2 g (ν) ei¦rr (ν) ’ ’ g (ν) ei¦rr (ν) , (10.60)
where the ¬nal forms hold for a balanced beam splitter and ¦tt (ν) and ¦rr (ν) are the
phase shifts for the rr- and tt-processes respectively. The total coincidence probability
is therefore
¿¾¾ Experiments in linear optics

dν |Att (ν) + Arr (ν)|
Pcoinc =

∆¦ (ν)
dν |g (ν)| sin2
= , (10.61)
∆¦ (ν) = ¦tt (ν) ’ ¦rr (ν) . (10.62)
The phase changes ¦tt (ν) and ¦rr (ν) depend on the frequencies of the two photons
and the geometrical distances involved. The distances traveled by the idler and signal
wave packets in the tt-process are
Ltt = Lidl + L1 ,
Ltt = Lsig + L2 ,

where L1 (L2 ) is the distance from the beam splitter to the detector D1 (D2). The
corresponding distances for the rr-process are
Lrr = Lidl + L2 ,
Lsig = Lsig + L1 .
In the tt-process the idler (signal) wave packet enters detector D1 (D2), so the phase
change is
ω ω
¦tt (ν) = Ltt + Ltt . (10.65)
c sig
According to eqn (10.50), ω and ω switch roles in the rr-process; consequently,
ω rr ω
Lidl + Lrr .
¦rr (ν) = (10.66)
c sig
Substituting eqns (10.63)“(10.66) into eqn (10.62) leads to the simple result
. (10.67)
∆¦ (ν) = 2ν
Since the two photons are created simultaneously, the di¬erence in arrival times of the
signal and idler wave packets is
∆t = . (10.68)
The resulting form for the coincidence probability,
dν |g (ν)| sin2 (ν∆t) ,
Pcoinc (∆t) = (10.69)

has a width determined by |g (ν)| and a null at ∆t = 0, as shown in Exercise 10.3.
As expected, the null occurs for the balanced case,
Lsig = Lidl = L0 . (10.70)
In this argument, we have replaced the plane waves of Section 10.2.1-A with
Gaussian pulses. Each pulse is characterized by two parameters, the pulse width, Tσ ,
Two-photon interference

and the arrival time, tσ , of the pulse peak at the beam splitter. If the absolute di¬er-
ence in arrival times, |∆t| = |Lsig ’ Lidl | /c, is larger than the sum of the pulse widths
(|∆t| > Tsig + Tidl ) the pulses are nonoverlapping, and the destructive interference
e¬ect will not occur. This case simply represents two repetitions of the photon indivis-
ibility experiment with a single photon. What happens in this situation depends on the
width, Tgate , of the acceptance window for the coincidence counter. If Tgate < |∆t| no
coincidence count will occur, but in the opposite situation, Tgate > |∆t|, coincidence
counts will be recorded with probability 1/2. For ∆t = 0 the wave packets overlap, and
interference between the alternative Feynman paths prevents any coincidence counts.
In order to increase the contrast between the overlapping and nonoverlapping cases,
one should choose Tgate > ∆tmax , where ∆tmax is the largest value of the absolute
time delay. The result is an extremely narrow dip”the HOM dip”in the coinci-
dence count rate as a function of ∆t, as seen in Fig. 10.4.
The alternative analysis using the Schr¨dinger-picture scattering technique is also
instructive. For this purpose, we substitute the special form (10.56) for C (ω, ω ) into
eqn (10.48) to ¬nd the initial state for scattering by the beam splitter:

g (ν) eiν∆t a† (ω0 + ν) a† (ω0 ’ ν) |0 .
|¦in = eiω0 ∆t (10.71)
sig idl

Applying eqn (8.76) to each term in this superposition yields
|¦¬n = |¦pair + |¦coinc , (10.72)


Coincidence rate (s’1)


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