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 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™.

¿½

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

state

2 2

a† + a†

|¦¬n = r t e’2iω0 t eik0 Lsig eik0 Lidl |0

sig

idl

+ 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)

sig

idl

2

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

splitter.

¿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 ,

(10.39)

±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 =

2

±idl ±sig , and eqn (10.39) yields

±idl ±sig = r t ±2 + ±2 + r2 + t2 ±sig ±idl

sig idl

i2

’ ±sig + ±2 , (10.40)

idl

2

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—

B

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

¿½

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 ) ,

ρ

(10.45)

[aσ (ω) , aρ (ω )] = 0 .

For a given value of the channel index σ, the operator a† (ω) creates photons in a wave

σ

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

tt

C (ω, ω ) a† (ω) a† (ω ) |0 ,

|¦in ’ |¦in = (10.49)

tt sig idl

2 2π 2π

and

1 dω dω

rr

C (ω, ω ) a† (ω) a† (ω ) |0

|¦in ’ |¦in =’

rr sig

idl

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

2

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

¿¾½

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 ,

(10.55)

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)

where

ω’ω

, ω = ω0 + ν , ω = ω0 ’ ν .

ν= (10.57)

2

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

2

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

1

„2 ∼ . (10.58)

∆ν

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

1

Att (ν) = t2 g (ν) ei¦tt (ν) ’ g (ν) ei¦tt (ν) (10.59)

2

and

1

Arr (ν) = r2 g (ν) ei¦rr (ν) ’ ’ g (ν) ei¦rr (ν) , (10.60)

2

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

2

dν |Att (ν) + Arr (ν)|

Pcoinc =

∆¦ (ν)

2

dν |g (ν)| sin2

= , (10.61)

2

where

∆¦ (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 ,

idl

(10.63)

Ltt = Lsig + L2 ,

sig

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 ,

idl

(10.64)

rr

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)

idl

c sig

c

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

c

Substituting eqns (10.63)“(10.66) into eqn (10.62) leads to the simple result

∆L

. (10.67)

∆¦ (ν) = 2ν

c

Since the two photons are created simultaneously, the di¬erence in arrival times of the

signal and idler wave packets is

∆L

∆t = . (10.68)

c

The resulting form for the coincidence probability,

2

dν |g (ν)| sin2 (ν∆t) ,

Pcoinc (∆t) = (10.69)

2

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 ,

¿¾¿

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

o

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:

dν

g (ν) eiν∆t a† (ω0 + ν) a† (ω0 ’ ν) |0 .

|¦in = eiω0 ∆t (10.71)

sig idl

2π

Applying eqn (8.76) to each term in this superposition yields

|¦¬n = |¦pair + |¦coinc , (10.72)

1150

1100

Coincidence rate (s’1)

1050