The description of the two-pinhole experiment presented above provides a simple

physical model which helps us to understand single-photon interference, but a more

detailed analysis requires the use of the scattering theory methods developed in Sec-

tions 8.1 and 8.2. For the two-pinhole problem, the e¬ects of di¬raction cannot be

ignored, so it will not be possible to con¬ne attention to a small number of plane

waves, as in the analysis of the beam splitter and the stop. Instead, we will use the

general relations (8.29) and (8.27) to guide a calculation of the ¬eld operator in po-

sition space. This is equivalent to using the classical Green function de¬ned by this

boundary value problem to describe the propagation of the ¬eld operator through the

pinhole.

In the plane-wave basis the positive frequency part of the out-¬eld is given by

(+)

iωk aks es ei(k·r’ωk t) ,

Eout (r, t) = (10.5)

2 0 cV

ks

where the scattered annihilation operators obey

aks = Sks,k s ak s . (10.6)

ks

If the source of the incident ¬eld is on the left (z < 0), then the problem is to calculate

the transmitted ¬eld on the right (z > 0). The ¬eld will be observed at points r lying

on a detection plane at z = L. The plane waves that impinge on a detector at r must

have kz > 0, and the terms in eqn (10.6) can be split into those with kz > 0 (forward

waves) and kz < 0 (backwards waves). The contribution of the forward waves to eqn

(10.5) represents the part of the incident ¬eld transmitted through the pinholes, while

the backward waves”vacuum ¬‚uctuations in this case”scatter into forward waves by

re¬‚ection from the screen. The total ¬eld in the region z > 0 is then the sum of three

terms:

(+) (+) (+) (+)

Eout (r, t) = E1 (r, t) + E2 (r, t) + E3 (r, t) , (10.7)

(+) (+)

where E1 and E2 are the ¬elds coming from pinholes 1 and 2 respectively, and the

¬eld resulting from re¬‚ections of backwards waves at the screen is

(+)

iωk aks es ei(k·r’ωk t) ,

<

E3 (r, t) = (10.8)

2 0 cV

ks,kz >0

where

<

aks = Sks,k s ak s . (10.9)

k s ,kz <0

(+) (+)

In the absence of the re¬‚ected vacuum ¬‚uctuations, E3 , the total ¬eld Eout would

not satisfy the commutation relation (3.17), and this would lead to violations of the

uncertainty principle, as shown in Exercise 10.1.

¿½¼ Experiments in linear optics

If the distance to the observation point r is large compared to the sizes of the

pinholes and to the distance between them”this is called Fraunhofer di¬raction or

the far-¬eld approximation”the ¬elds due to the two pinholes are given by

E(+) (r, t) = iDp E(+) (rp , t ’ Lp /c) (p = 1, 2) , (10.10)

p

where Lp is the distance from the pth pinhole to the observation point r, and Dp is

a real coe¬cient that depends on the pinhole geometry. For simplicity we will assume

that the pinholes are identical, D1 = D2 = D, and that the incident radiation is

monochromatic. If the direction of the incident beam and the vectors r ’ r1 and

r ’ r2 are approximately orthogonal to the screen, then Dp ≈ σ/ (»0 L), where σ is

the common area of the pinholes and »0 is the average wavelength in the incident ¬eld

(Born and Wolf, 1980, Sec. 8.3). This is the standard classical expression, except for

replacing the classical ¬eld in the pinhole by the quantum ¬eld operator. The average

intensity in a de¬nite polarization e at a detection point r is proportional to

(’) (+)

Itot = Eout (r, t) Eout (r, t)

3 3

(’) (+)

= Eq (r, t) Ep (r, t) , (10.11)

q=1 p=1

(’) (’) (’) (’)

where Eout = e · Eout , Eq = e · Eq , and the indices p and q represent the three

terms in eqn (10.7). The density operator, ρ, that de¬nes the ensemble average, · · · ,

contains no backwards waves, since it represents the ¬eld generated by a source to the

(+)

left of the screen. According to eqn (10.8) and eqn (10.9) the operator E3 is a linear

combination of annihilation operators for backwards waves, therefore

(+) (’)

E3 ρ = 0 = ρE3 . (10.12)

By using this fact, plus the cyclic invariance of the trace, it is easy to show that eqn

(10.11) reduces to

2 2

(’) (+)

Itot = Eq (r, t) Ep (r, t)

q=1 p=1

= I1 + I2 + I12 , (10.13)

where Ip is the intensity due to the pth pinhole alone,

2

Ip = |D| E (’) (rp , t ’ Lp /c) E (+) (rp , t ’ Lp /c) (p = 1, 2) , (10.14)

I12 is the interference term,

(’) (+)

I12 = 2 Re E1 (r, t) E2 (r, t)

= 2D2 Re E (’) (r1 , t ’ L1 /c) E (+) (r2 , t ’ L2 /c) , (10.15)

and E (’) (r, t) = e · E(’) (r, t) .

¿½½

Single-photon interference

The expectation values appearing in these expressions are special cases of the ¬rst-

order ¬eld correlation function G(1) de¬ned by eqn (4.76). In this notation, the results

are

Ip = |D|2 G(1) (rp , t ’ Lp /c; rp , t ’ Lp /c) (p = 1, 2) , (10.16)

and

I12 = 2D2 Re G(1) (r1 , t ’ L1 /c; r2 , t ’ L2 /c) . (10.17)

From the classical theory of two-pinhole interference we know that high visibility

interference patterns are obtained with monochromatic light. In quantum theory this

means that the power spectrum a† aks is strongly peaked at |k| = k0 = ω0 /c. If

ks

the density operator ρ satis¬es this condition, then the plane-wave expansion for E (+)

implies that the temporal Fourier transform of Tr E (+) (r, t) ρ is strongly peaked at

(+)

ω0 . This means that the envelope operator E de¬ned by

(+)

(r, t) = E (+) (r, t) eiω0 t

E (10.18)

can be treated as slowly varying”on the time scale 1/ω0 ”provided that it is applied

to the monochromatic density matrix ρ. In this case, the correlation functions can be

written as

(’) (+)

(r2 , t2 ) e’iω0 (t2 ’t1 )

G(1) (r1 , t1 ; r2 , t2 ) = Tr ρE (r1 , t1 ) E

(1)

(r1 , t1 ; r2 , t2 ) e’iω0 (t2 ’t1 ) ,

≡G (10.19)

(1)

where G (r1 , t1 ; r2 , t2 ) is a slowly-varying function of t1 and t2 . For su¬ciently long

pulses, the incident radiation is approximately stationary, so the correlation functions

are unchanged by a time translation tp ’ tp + „ . In other words they only depend on

the time di¬erence t1 ’ t2 , so the direct terms become

(1)

Ip = D2 G (rp , 0; rp , 0) (p = 1, 2) , (10.20)

while the interference term reduces to

(1)

I12 = 2D2 Re G (r1 , „ ; r2 , 0) eiω0 „ , (10.21)

where „ = (L2 ’ L1 ) /c is the di¬erence in the light travel time for the two pinholes.

All three terms are independent of the time t. The direct terms only depend on the

average intensities at the pinholes, but the factor

eiω0 „ = eik0 (L2 ’L1 ) (10.22)

in the interference term produces rapid oscillations along the detection plane. This is

(1)

explicitly exhibited by expressing G in terms of its amplitude and phase:

(1) (1)

(r1 , t1 ; r2 , t2 ) ei¦(r1 ,t1 ;r2 ,t2 ) ,

G (r1 , t1 ; r2 , t2 ) = G (10.23)

so that I12 is given by

¿½¾ Experiments in linear optics

(1)

I12 = 2D2 G (r1 , „ ; r2 , 0) cos [¦ (r1 , „ ; r2 , 0) + ω0 „ ] . (10.24)

The interference pattern is modulated by slow variations in the amplitude and phase

(1)

of G due to the ¬nite length of the pulse. When these modulations are ignored, the

interference maxima occur at the path length di¬erences

¦»0

L2 ’ L1 = c„ = n»0 ’ , n = 0, ±1, ±2, . . . . (10.25)

2π

The interference pattern calculated from the ¬rst-order quantum correlation function

is identical to the classical interference pattern. Since this is true even if the ¬eld state

contains only one photon, ¬rst-order interference is also called one-photon interference.

An important quantity for interference experiments is the fringe visibility

’I

I

V≡ max min

, (10.26)

I max + I min

where I and I are respectively the maximum and minimum values of the

max min

(1)

total intensity on the detection plane. If the slow variations in G are neglected, then

one ¬nds

(1) (1) (1)

= D2 G

I (r1 , 0; r1 , 0) + G (r2 , 0; r2 , 0) + 2 G (r1 , „ ; r2 , 0) , (10.27)

max

(1) (1) (1)

(r2 , 0; r2 , 0) ’ 2 G

= D2 G

I (r1 , 0; r1 , 0) + G (r1 , „ ; r2 , 0) , (10.28)

min

so the visibility is

(1)

2G (r1 , „ ; r2 , 0)

V= . (10.29)

(1) (1)

G (r1 , 0; r1 , 0) + G (r2 , 0; r2 , 0)

(1)

The ¬eld“¬eld correlation function G (r1 , „ ; r2 , 0) is therefore a measure of the coher-

ence of the signals from the two pinholes. There are no fringes (V = 0) if the correlation

function vanishes. On the other hand, the inequality (4.85) shows that the visibility is

bounded by

(1) (1)

2 G (r1 , 0; r1 , 0) G (r2 , 0; r2 , 0)

V 1, (10.30)

(1) (1)

G (r1 , 0; r1 , 0) + G (r2 , 0; r2 , 0)

where the maximum value of unity occurs when the intensities at the two pinholes

are equal. This suggests introducing a normalized correlation function, the mutual

coherence function,

(1)

G (x; x )

g (1) (x; x ) = , (10.31)

(1) (1)

G (x; x) G (x ; x )

¿½¿

Single-photon interference

which satis¬es g (1) (x; x ) 1. In these terms, perfect coherence corresponds to

(1)

g (x; x ) = 1, and the fringe visibility is

(1) (1)

(r2 , 0; r2 , 0) g (1) (r1 , „ ; r2 , 0)

2 G (r1 , 0; r1 , 0) G

V= . (10.32)

(1) (1)

G (r1 , 0; r1 , 0) + G (r2 , 0; r2 , 0)

Thus measurements of the intensity at each pinhole, the fringe visibility, and the fringe

spacing completely determine the complex mutual coherence function g (1) (x; x ). This

means that the correlation function G(1) (x; x ) or g (1) (x; x ) can always be interpreted

in terms of a Young™s-style interference experiment.

10.1.1 Hanbury Brown“Twiss e¬ect

We have just seen that ¬rst-order interference, e.g. in Young™s experiment or in the

Michelson interferometer, is described by the ¬rst-order ¬eld correlation function G(1) .

The Hanbury Brown“Twiss e¬ect (Hanbury Brown, 1974) was one of the earliest ob-

servations that demonstrated optical interference in the intensity“intensity correlation

function G(2) . This observation was interpreted as a measurement of photon“photon

correlation, so it eventually led to the founding of the ¬eld of quantum optics. The

e¬ect was originally discovered in a simple laboratory experiment in which light from

a mercury arc lamp passes through an interference ¬lter that singles out a strong green

line of the mercury atom at a wavelength of 546.1 nm. The spectrally pure green light

is split by means of a balanced beam splitter into two beams, which are detected by

square-law detectors placed at the output ports of the beam splitter. The experimental

arrangement is shown in Fig. 10.2. The output current I (t) from each detector is a

measure of the intensity in that arm of the beam splitter. The intensities are slowly

varying on the optical scale, with typical Fourier components in the radio range. The

outputs of the two detectors are fed into a radio-frequency mixer that accumulates the

time integral of the product of the two signals. By sending the signal from one of the

detectors through a variable delay line the intensity“intensity correlation,

∞

I(t)I(t ’ „ )dt ,

f („ ) = (10.33)

’∞

DB RF

mixer

B Fig. 10.2 Experimental arrangement for ob-

serving the Hanbury Brown“Twiss e¬ect. The

A

ωsig signal is split by a 50/50 beam splitter and

Signal the split ¬elds enter detectors at B and C. The

BS

output of the detectors is fed into a radio-fre-

DC

C quency (RF) mixer which integrates the prod-

D uct of the two signals.

¿½ Experiments in linear optics

is measured as a function of the delay time „ . The data (Hanbury Brown and Twiss,

1957) show a peak in the intensity“intensity correlation function f („ ) near „ = 0.

Hanbury Brown and Twiss interpreted this as a photon-bunching e¬ect explained

by the fact that the Bose character of photons enhances the probability that two

photons will arrive simultaneously at the two detectors. However, Glauber showed

that classical intensity ¬‚uctuations in the thermal light emitted by the mercury arc

lamp yield a completely satisfactory description, so that there is no need to invoke the

Bose statistics of photons.

The experimental technique for measuring the intensity“intensity correlation was

later changed from simple square-law detection to coincidence detection based on

a photoelectron counting technique using photomultipliers. Since this technique can

register clicks associated with the arrival of individual photons, it would seem to be

closer to a measurement of a photon“photon correlation function.

For the thermal light source which was used in this experiment, this hope is un-

justi¬ed, because we can explain the results on the basis of classical-¬eld notions by

using the semiclassical theory of the photoelectric e¬ect. A quantum description of this

experiment, to be presented later on, employs an expansion of the density operator

in the basis of coherent states. We will see that the radiation emitted by the thermal

source is described by a completely positive quasi-probability distribution function

P (±), which is consistent with a semiclassical explanation in terms of ¬‚uctuations in

the intensity of the classical electromagnetic ¬eld.

On the other hand, for a pure coherent state the Hanbury Brown“Twiss e¬ect per se

does not exist. Thus if we were to replace the mercury arc lamp by a laser operating far

above threshold, the photon arrivals would be described by a pure Poissonian random

process, with no photon-bunching e¬ect.

This intensity“intensity correlation method was applied to astrophysical stellar in-

terferometry to measure stellar diameters (Hanbury Brown and Twiss, 1956). Stellar

interferometry depends on the di¬erence in path lengths to the telescope from points

on opposite limbs of the star. For example, Michelson stellar interferometry (Born and

Wolf, 1980, Sec. 7.3.6) is based on ¬rst-order interference”i.e. on the ¬eld“¬eld corre-

lation function”so the optical path lengths must be equalized to high precision. This

is done by adjusting the positions of the interferometer mirrors attached to the tele-

scope so that all wavelengths of light interfere constructively in the ¬eld of view. Under

these conditions, white light entering the telescope will result in a bright white-light

fringe. The white-light fringe condition must be met before attempting to measure a

stellar diameter by this method.

By contrast, the beauty of the intensity stellar interferometer is that one can com-

pensate for the delays corresponding to the di¬erence in path lengths in the radio-

wavelength region after detection, rather than in the optical-wavelength region before

detection. Compensating the optical delay by an electronic delay produces a maximum

in the intensity“intensity correlation function of the optical signals.

Furthermore, the optical quality of the telescope surfaces for the intensity inter-

ferometer can be much lower than that required for Michelson stellar interferometry,

so that one can use the re¬‚ectors of searchlights as light buckets, rather than astro-

nomical telescopes with optically perfect surfaces. However, the disadvantage of the

¿½

Two-photon interference

intensity interferometer is that it requires higher intensity sources than the Michelson

stellar interferometer. Thus intensity interferometry can only be used to measure the

diameters of the brightest stars.

10.2 Two-photon interference

The results in Section 10.1 provide support for Dirac™s dictum that each photon inter-

feres with itself, but he went on to say (Dirac, 1958, Sec. I.3)

Each photon then interferes only with itself. Interference between two di¬erent pho-

tons never occurs.

This is one of the very few instances in which Dirac was wrong. Further experimental

progress in the generation of states containing exactly two photons has led to the

realization that di¬erent photons can indeed interfere. These phenomena involve the

second-order correlation function G(2) , de¬ned in Section 4.7, so they are sometimes

called second-order interference. Another terminology calls them fourth-order

interference, since G(2) is an average over the product of four electric ¬eld operators.

We will study two important examples of two-photon interference: the Hong“Ou“

Mandel interferometer, in which interference between two photons occurs locally at

a single beam splitter, and the Franson interferometer, where the interference occurs

between two photons falling on spatially-separated beam splitters.

10.2.1 The Hong“Ou“Mandel interferometer

The quantum property of photon indivisibility was demonstrated by allowing a single

photon to enter through one port of a beam splitter. In an experiment performed

by Hong, Ou, and Mandel (Hong et al., 1987), interference between two Feynman

processes was demonstrated by illuminating a beam splitter with a two-photon state

produced by pumping a crystal of potassium dihydrogen phosphate (KDP) with an

ultraviolet laser beam, as shown in Fig. 10.3. In a process known as spontaneous

down-conversion”which will be discussed in Section 13.3.2”a pump photon with

frequency ωp splits into a pair of lower frequency photons, traditionally called the

Fig. 10.3 The Hong“Ou“Mandel interferometer illuminated by a two-photon state, produced

by spontaneous down-conversion in the crystal labeled SDC. The two photon wave packets

are re¬‚ected from mirrors M1 and M2 so that they meet at the beam splitter BS. The output

of detectors D1 and D2 are fed to the coincidence counter CC. (Adapted from Hong et al.

(1987).)

¿½ Experiments in linear optics

signal and idler.1 Since photons are indistinguishable, they cannot be assigned labels;

therefore, the traditional language must be used carefully and sparingly. The words

˜signal photon™ or ˜idler photon™ simply mean that a photon occupies the signal mode

or the idler mode. It is the modes, rather than the photons, that are distinguishable.

Prior to their arrival at the beam splitter, e.g. at the mirrors M1 and M2, the di¬raction

patterns of the signal and idler modes do not overlap.

In the following discussion, the production process can be treated as a black box;

we only need to know that one pump photon enters the crystal and that two (down-

converted) photons are produced simultaneously and leave the crystal as wave packets

with widths of the order of 15 fs. In the notation used in Fig. 8.2, the signal mode

(ksig , ssig ) enters through port 1 and the idler mode (kidl , sidl ) enters through port 2

of the beam splitter BS.

A Degenerate plane-wave model

It is instructive to analyze this situation in terms of interference between Feynman

processes. We begin with the idealized case of plane-wave modes”propagating from

the beam splitter to the detectors”with degenerate frequencies: ωidl = ωsig = ω0 =

ωp /2. The experimental feature of interest is the coincidence-counting rate. Since a

given photon can only be counted once, the events leading to coincidence counts are

those in which each detector receives one photon.

There are, consequently, two processes leading to coincidence events.

(1) The re¬‚ection“re¬‚ection (rr) process: both wave packets are re¬‚ected from the

beam splitter towards the two detectors.

(2) The transmission“transmission (tt) process: both wave packets are transmitted

through the beam splitter towards the two detectors.

In the absence of which-path information these processes are indistinguishable, since

they both lead to the same ¬nal state: one scattered photon is in the idler mode

and the other is in the signal mode. This results in simultaneous clicks in the two