c

S 1 ’ S2

∆tss = t1s ’ t2s = T1 ’ T2 + , (10.89)

c

and two processes will not interfere if the di¬erence between their ∆ts is larger than

the two-photon coherence time „2 de¬ned by eqn (10.58). For example, eqns (10.86)

and (10.87) yield the di¬erence

∆L1 + ∆L2

∆tls ’ ∆tsl = „2 , (10.90)

c

where the ¬nal inequality follows from the condition (10.83) and the fact that „1 ∼ „2 .

The conclusion is that the processes l“s and s“l cannot interfere, since they lead to

di¬erent ¬nal states. Similar calculations show that l“s and s“l are distinguishable

from l“l and s“s; therefore, the only remaining possibility is interference between l“l

and s“s. In this case the di¬erence is

∆L1 ’ ∆L2

∆tll ’ ∆tss = , (10.91)

c

so that interference between these two processes can occur if the condition

|∆L1 ’ ∆L2 |

„2 (10.92)

c

is satis¬ed. The practical e¬ect of these conditions is that the interferometers must be

almost identical, and this is a source of experimental di¬culty.

When the condition (10.92) is satis¬ed, the ¬nal states reached by the short“short

and long“long paths are indistinguishable, so the corresponding amplitudes must be

added in order to calculate the coincidence probability, i.e.

P12 = |All + Ass |2 . (10.93)

The amplitudes for the two paths are

All = r1 t1 r2 t2 ei¦ll ,

(10.94)

Ass = r1 t1 r2 t2 ei¦ss ,

where (rj , tj ) and rj , tj are respectively the re¬‚ection and transmission coe¬cients

for the ¬rst and second beam splitter in the jth interferometer, and the phases ¦ll

¿¿¾ Experiments in linear optics

and ¦ss are the sums of the one-photon phases for each path. We will simplify this

calculation by assuming that all beam splitters are balanced and that the photon

frequencies are degenerate, i.e. ω1 = ω2 = ω0 = ωp /2. In this case the phases are

ω0

¦ll = ω0 (t1l + t2l ) = ω0 (T1 + T2 ) + (L1 + L2 ) ,

c (10.95)

ω0

¦ss = ω0 (t1s + t2s ) = ω0 (T1 + T2 ) + (S1 + S2 ) ,

c

and the coincidence probability is

∆¦

P12 = cos2 , (10.96)

2

where

ω0

∆¦ = ¦ll ’ ¦ss = (∆L1 + ∆L2 ) . (10.97)

c

Now suppose that Bob and Alice initially choose the same optical delay for their

respective interferometers, i.e. they set ∆L1 = ∆L2 = ∆L, then

∆¦ ω0 ∆L

= ∆L = 2π , (10.98)

2 c »0

where »0 = 2πc/ω0 is the common wavelength of the two photons. If the delay ∆L

is arranged to be an integer number m of wavelengths, then ∆¦/2 = 2πm and P12

achieves the maximum value of unity. In other words, with these settings the behavior

of the photons at the ¬nal beam splitters are perfectly correlated, due to constructive

interference between the two probability amplitudes.

Next consider the situation in which Bob keeps his settings ¬xed, while Alice alters

her settings to ∆L1 = ∆L + δL, so that

∆¦ δL

= 2πm + π , (10.99)

2 »0

and

δL

P12 = cos2 π . (10.100)

»0

For the special choice δL = »0 /2, the coincidence probability vanishes, and the be-

havior of the photons at the ¬nal beam splitters are anti-correlated, due to complete

destructive interference of the probability amplitudes. This drastic change is brought

about by a very small adjustment of the optical delay in only one of the interferom-

eters. We should stress the fact that macroscopic physical events”the ¬ring of the

detectors”that are spatially separated by a large distance behave in a correlated or

anti-correlated way, by virtue of the settings made by Alice in only one of the inter-

ferometers.

In Chapter 19 we will see that these correlations-at-a-distance violate the Bell

inequalities that are satis¬ed by any so-called local realistic theory. We recall that a

theory is said to be local if no signals can propagate faster than light, and it is said to be

realistic if physical objects can be assumed to have de¬nite properties in the absence of

observation. Since the results of experiments with the Franson interferometer violate

Bell™s inequalities”while agreeing with the predictions of quantum theory”we can

conclude that the quantum theory of light is not a local realistic theory.

Single-photon interference revisited— ¿¿¿

Single-photon interference revisited—

10.3

The experimental techniques required for the Hong“Ou“Mandel demonstration of

two-photon interference”creation of entangled photon pairs by spontaneous down-

conversion (SDC), mixing at beam splitters, and coincidence detection”can also be

used in a beautiful demonstration of a remarkable property of single-photon interfer-

ence. In our discussion of Young™s two-pinhole interference in Section 10.1, we have

already remarked that any attempt to obtain which-path information destroys the

interference pattern. The usual thought experiments used to demonstrate this for the

two-pinhole con¬guration involve an actual interaction of the photon”either with

some piece of apparatus or with another particle”that can determine which pinhole

was used. The experiment to be described below goes even further, since the mere pos-

sibility of making such a determination destroys the interference pattern, even if the

measurements are not actually carried out and no direct interaction with the photons

occurs. This is a real experimental demonstration of Feynman™s rule that interference

can only occur between alternative processes if there is no way”even in principle”to

distinguish between them. In this situation, the complex amplitudes for the alterna-

tive processes must ¬rst be added to produce the total probability amplitude, and only

then is the probability for the ¬nal event calculated by taking the absolute square of

the total amplitude.

10.3.1 Mandel™s two-crystal experiment

In the two-crystal experiment of Mandel and his co-workers (Zou et al., 1991), shown in

Fig. 10.8, the beam from an argon laser, operating at an ultraviolet wavelength, falls on

the beam splitter BSp . This yields two coherent, parallel pump beams that enter into

two staggered nonlinear crystals, NL1 and NL2, where they can undergo spontaneous

down-conversion. The rate of production of photon pairs in the two crystals is so low

that at most a single photon pair exists inside the apparatus at any given instant. In

As

M1

Ds

IFs Amp.

s1 Counter

&

V1 disc.

NL1

BSo

NDF

BSp

i1 s2

NL2 Coincidence

V2

From argon laser i2

Amp.

Counter

&

Di

IFi disc.

Ai

Fig. 10.8 Spontaneous down-conversion (SDC) occurs in two crystals NL1 and NL2. The

two idler modes i1 and i2 from these two crystals are carefully aligned so that they coincide

on the face of detector Di . The dashed line in beam path i1 in front of crystal NL2 indicates a

possible position of a beam block, e.g. an opaque card. (Reproduced from Zou et al. (1991).)

¿¿ Experiments in linear optics

other words, we can assume that the simultaneous emission of two photon pairs, one

from each crystal, is so rare that it can be neglected.

The idler beams i1 and i2 , emitted from the crystals NL1 and NL2 respectively,

are carefully aligned so that their transverse Gaussian-mode beam pro¬les overlap as

exactly as possible on the face of the idler detector Di . Thus, when a click occurs in Di ,

it is impossible”even in principle”to know whether the detected photon originated

from the ¬rst or the second crystal. It therefore follows that it is also impossible”even

in principle”to know whether the twin signal wave packet, produced together with

the idler wave packet describing the detected photon, originated from the ¬rst crystal

as a signal wave packet in beam s1 , or from the second crystal as a signal wave packet

in beam s2 . The two processes resulting in the appearance of s1 or s2 are, therefore,

indistinguishable; and their amplitudes must be added before calculating the ¬nal

probability of a click at detector Ds .

10.3.2 Analysis of the experiment

The two indistinguishable Feynman processes are as follows. The ¬rst is the emission

of the signal wave packet by the ¬rst crystal into beam s1 , re¬‚ection by the mirror

M1 , re¬‚ection at the output beam splitter BSo , and detection by the detector Ds . This

is accompanied by the emission of a photon in the idler mode i1 that traverses the

crystal NL2”which is transparent at the idler wavelength”and falls on the detector

Di . The second process is the emission by the second crystal of a photon in the signal

wave packet s2 , transmission through the output beam splitter BSo , and detection by

the same detector Ds , accompanied by emission of a photon into the idler mode i2

which falls on Di . This experiment can be analyzed in two apparently di¬erent ways

that we consider below.

A Second-order interference

Let us suppose that the photon detections at Ds are registered in coincidence with

the photon detections at Di , and that the two idler beams are perfectly aligned. If a

click were to occur in Ds in coincidence with a click in Di , it would be impossible to

determine whether the signal“idler pair came from the ¬rst or the second crystal. In

this situation Feynman™s interference rule tells us that the probability amplitude A1

that the photon pair originates in crystal NL1 and the amplitude A2 of pair emission

by NL2 must be added to get the probability

|A1 + A2 |2 (10.101)

for a coincidence count. When the beam splitter BSo is slowly scanned by small trans-

lations in its transverse position, the signal path length of the ¬rst process is changed

relative to the signal path length of the second process. This in turn leads to a change

in the phase di¬erence between A1 and A2 ; therefore, the coincidence count rate would

exhibit interference fringes.

From Section 9.2.4 we know that the coincidence-counting rate for this experiment

is proportional to the second-order correlation function

(’) (+)

G(2) (xs , xi ; xs , xi ) = Tr ρin Es (xs ) Ei

(’) (+)

(xi ) Ei (xi ) Es (xs ) , (10.102)

Single-photon interference revisited— ¿¿

where ρin is the density operator describing the initial state of the photon pair produced

by down-conversion. The subscripts s and i respectively denote the polarizations of

the signal and idler modes. The variables xs and xi are de¬ned as xs = (rs , ts ) and

xi = (ri , ti ), where rs and ri are respectively the locations of the detectors Ds and Di ,

while ts and ti are the arrival times of the photons at the detectors. This description

of the experiment as a second-order interference e¬ect should not be confused with the

two-photon interference studied in Section 10.2.1. In the present experiment at most

one photon is incident on the beam splitter BSo during a coincidence-counting window;

therefore, the pairing phenomena associated with Bose statistics for two photons in

the same mode cannot occur.

B First-order interference

Since the state ρin involves two photons”the signal and the idler”the description in

terms of G(2) o¬ered in the previous section seems very natural. On the other hand,

in the ideal case in which there are no absorptive or scattering losses and the classical

modes for the two idler beams i1 and i2 are perfectly aligned, an idler wave packet will

fall on Di whenever a signal wave packet falls on Ds . In this situation, the detector Di is

actually super¬‚uous; the counting rate of detector Ds will exhibit interference whether

or not coincidence detection is actually employed. In this case the amplitudes A1 and

A2 refer to the processes in which the signal wave packet originates in the ¬rst or the

2

second crystal. The counting rate |A1 + A2 | at detector Ds will therefore exhibit the

same interference fringes as in the coincidence-counting experiment, even if the clicks

of detector Di are not recorded. In this case the interference can be characterized solely

by the ¬rst-order correlation function

G(1) (xs ; xs ) = Tr ρin Es (xs ) Es (xs ) .

(’) (+)

(10.103)

In the actual experiment, no coincidence detection was employed during the collection

of the data. The ¬rst-order interference pattern shown as trace A in Fig. 10.9 was

obtained from the signal counter Ds alone. In fact, the detector Di and the entire

coincidence-counting circuitry could have been removed from the apparatus without

altering the experimental results.

10.3.3 Bizarre aspects

The interference e¬ect displayed in Fig. 10.9 may appear strange at ¬rst sight, since the

signal wave packets s1 and s2 are emitted spontaneously and at random by two spatially

well-separated crystals. In other words, they appear to come from independent sources.

Under these circumstances one might expect that photons emitted into the two modes

s1 and s2 should have nothing to do with each other. Why then should they produce

interference e¬ects at all? The explanation is that the presence of at most one photon

in a signal wave packet during a given counting window, combined with the perfect

alignment of the two idler beams i1 and i2 , makes it impossible”even in principle”to

determine which crystal actually emitted the detected photon in the signal mode. This

is precisely the situation in which the Feynman rule (10.2) applies; consequently, the

amplitudes for the processes involving signal photons s1 or s2 must be added, and

interference is to be expected.

¿¿ Experiments in linear optics

Displacement of BSo in µm

Counting rate 4I (per second)

A

B

Phase in multiples of π

Fig. 10.9 Interference fringes of the signal photons detected by Ds , as the transverse position

of the ¬nal splitter BSo is scanned (see Fig. 10.8). Trace A is taken with a neutral 91%

transmission density ¬lter placed between the two crystals. Trace B is taken with the beam

path i1 blocked by an opaque card (i.e. a ˜beam block™). (Reproduced from Zou et al. (1991).)

Now let us examine what happens if the experimental con¬guration is altered

in such a way that which-path information becomes available in principle. For this

purpose we assign Alice to control the position of the beam splitter BSo and record

the counting rate at detector Ds , while Bob is put in charge of the entire idler arm,

including the detector Di . As part of an investigation of possible future modi¬cations of

the experiment, Bob inserts a neutral density ¬lter (an ideal absorber with amplitude

transmission coe¬cient t independent of frequency) between NL1 and NL2, as shown

by the line NDF in Fig. 10.8. Since the ¬lter interacts with the idler photons, but

does not interact with the signal photons in any way, Bob expects that he can carry

out this modi¬cation without any e¬ect on Alice™s measurements. In the extreme limit

t ≈ 0”i.e. the idler photon i1 is completely blocked, so that it will never arrive at

Di ”Bob is surprised when Alice excitedly reports that the interference pattern at Ds

has completely disappeared, as shown in trace B of Fig. 10.9.

Alice and Bob eventually arrive at an explanation of this truly bizarre result by

a strict application of the Feynman interference rules (10.1)“(10.3). They reason as

follows. With the i1 -beam block in place, suppose that there is a click at Ds but not at

Di . Under the assumption that both Ds and Di are ideal (100% e¬ective) detectors, it

then follows with certainty that no idler photon was emitted by NL2. Since the signal

and idler photons are emitted in pairs from the same crystal, it also follows that the

signal photon must have been emitted by NL1. Under the same circumstances, if there

are simultaneous clicks at Ds and Di , then it is equally certain that the signal photon

must have come from NL2. This means that Bob and Alice could obtain which-path

information by monitoring both counters. Therefore, in the new experimental con-

¬guration, it is in principle possible to determine which of the alternative processes

Tunneling time measurements— ¿¿

actually occurred. This is precisely the situation covered by rule (10.3), so the proba-

bility of a count at Ds is the sum of the probabilities for the two processes considered

separately; there is no interference. A truly amazing aspect of this situation is that the

interference pattern disappears even if the detector Di is not present. In fact”just as

before”the detector Di and the entire coincidence-counting circuitry could have been

removed from the apparatus without altering the experimental results. Thus the mere

possibility that which-path information could be gathered by inserting a beam block

is su¬cient to eliminate the interference e¬ect.

The phenomenon discussed above provides another example of the nonlocal char-

acter of quantum physics. Bob™s insertion or withdrawal of the beam blocker leads to

very di¬erent observations by Alice, who could be located at any distance from Bob.

This situation is an illustration of a typically Delphic remark made by Bohr in the

course of his dispute with Einstein (Bohr, 1935):

But even at this stage there is essentially the question of an in¬‚uence on the very

conditions which de¬ne the possible types of predictions regarding the future behavior

of the system.

With this hint, we can understand the e¬ect of Bob™s actions as setting the overall

conditions of the experiment, which produce the nonlocal e¬ects.

An interesting question which has not been addressed experimentally is the follow-

ing: How soon after a sudden blocking of beam path i1 does the interference pattern

disappear for the signal photons? Similarly, how soon after a sudden unblocking of

beam path i1 does the interference pattern reappear for the signal photons?

Tunneling time measurements—

10.4

Soon after its discovery, it was noticed that the Schr¨dinger equation possessed real,

o

exponentially damped solutions in classically forbidden regions of space, such as the

interior of a rectangular potential barrier for a particle with energy below the top of

the barrier. This phenomenon”which is called tunneling”is mathematically similar

to evanescent waves in classical electromagnetism.

The ¬rst observation of tunneling quickly led to the further discoveries of important

early examples, such as the ¬eld emission of electrons from the tips of cold, sharp

metallic needles, and Gamow™s explanation of the emission of alpha particles (helium

nuclei) from radioactive nuclei undergoing ± decay.

Recent examples of the applications of tunneling include the Esaki tunnel diode

(which allows the generation of high-frequency radio waves), Josephson tunneling be-

tween two superconductors separated by a thin oxide barrier (which allows the sensi-

tive detection of magnetic ¬elds in a S uperconducting QU antum I nterference D evice

(SQUID)), and the scanning tunneling microscope (which allows the observation of

individual atoms on surfaces).

In spite of numerous useful applications and technological advances based on tun-

neling, there remained for many decades after its early discovery a basic, unresolved

physics problem. How fast does a particle traverse the barrier during the tunneling

process? In the case of quantum optics, we can rephrase this question as follows: How

quickly does a photon pass through a tunnel barrier in order to reach the far side?

¿¿ Experiments in linear optics

First of all, it is essential to understand that this question is physically meaningless

in the absence of a concrete description of the method of measuring the transit time.

This principle of operationalism is an essential part of the scienti¬c method, but it is

especially crucial in the studies of phenomena in quantum mechanics, which are far re-

moved from everyday experience. A de¬nition of the operational procedure starts with

a careful description of an idealized thought experiment. Thought experiments were

especially important in the early days of quantum mechanics, and they are still very

important today as an aid for formulating physically meaningful questions. Many of

these thought experiments can then be turned into real experiments, as measurements

of the tunneling time illustrate.

Let us therefore ¬rst consider a thought experiment for measuring the tunneling

time of a photon. In Fig. 10.10, we show an experimental method which uses twin

photons γ1 and γ2 , born simultaneously by spontaneous down-conversion. Placing two

Geiger counters at equal distances from the crystal would lead”in the absence of any

tunnel barrier”to a pair of simultaneous clicks. Now suppose that a tunnel barrier

is inserted into the path of the upper photon γ1 . One might expect that this would

impede the propagation of γ1 , so that the click of the upper Geiger counter”placed

behind the barrier”would occur later than the click of the lower Geiger counter. The

surprising result of an experiment to be described below is that exactly the opposite

happens. The arrival of the tunneling photon γ1 is registered by a click of the upper

Geiger counter that occurs before the click signaling the arrival of the nontunneling

photon γ2 . In other words, the tunneling photon seems to have traversed the barrier

superluminally. However, for reasons to be given below, we shall see that there is no

operational way to use this superluminal tunneling phenomenon to send true signals

faster than the speed of light.

This particular thought experiment is not practical, since it would require the use

of Geiger counters with extremely fast response times, comparable to the femtosecond

time scales typical of tunneling. However, as we have seen earlier, the Hong“Ou“

Mandel two-photon interference e¬ect allows one to resolve the relative times of arrival

of two photons at a beam splitter to within fractions of a femtosecond. Hence, the

Fig. 10.10 Schematic of a thought experi-

ment to measure the tunneling time of the

Geiger

Tunnel

photon. Spontaneous down-conversion gener-

counter

barrier

ates twin photons γ1 and γ2 by absorption of a

photon from a UV pump laser. In the absence

γ1

of a tunnel barrier, the two photons travel the

UV laser

same distance to two Geiger counters placed γ0

equidistantly from the crystal, and two simul-

taneous clicks occur. A tunnel barrier (shaded

γ2

Down-

rectangle) is now inserted into the path of pho-

conversion

ton γ1 . The tunneling time is given by the time

crystal Geiger

di¬erence between the clicks of the two Geiger

counter

counters.

Tunneling time measurements— ¿¿

impractical thought experiment can be turned into a realistic experiment by inserting

a tunnel barrier into one arm of a Hong“Ou“Mandel interferometer (Steinberg and

Chiao, 1995), as shown in Fig. 10.11.

The two arms of the interferometer are initially made equal in path length (per-

fectly balanced), so that there is a minimum”a Hong“Ou“Mandel (HOM) dip”in

the coincidence count rate. After the insertion of the tunnel barrier into the upper

arm of the interferometer, the mirror M1 must be slightly displaced in order to recover

the HOM dip. This procedure compensates for the extra delay”which can be either

positive or negative”introduced by the tunnel barrier. Measurements show that the