¿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

1000

950

900

’150 ’120 ’90 ’60 ’30 120 150

0 30 60 90

Time delay (fs)

Fig. 10.4 Coincidence rate as a function of the relative optical time delay in the interfer-

ometer. The solid line is a Gaussian ¬t, with an rms width of 15.3 fs. This pro¬le serves as a

map of the overlapping photon wave packets. (Reproduced from Steinberg et al. (1992).)

¿¾ Experiments in linear optics

where

dν

1

|¦pair = ieiωp t eiω0 ∆t g (ν) cos (ν∆t)

2 2π

— a† (ω0 + ν) a† (ω0 ’ ν) |0 + a† (ω0 + ν) a† (ω0 ’ ν) |0 (10.73)

sig sig idl idl

describes the pairing behavior, and

dν

1

|¦coinc = ieiωp t eiω0 ∆t g (ν) sin (ν∆t)

2 2π

— a† (ω0 + ν) a† (ω0 ’ ν) |0 ’ a† (ω0 + ν) a† (ω0 ’ ν) |0 (10.74)

sig sig

idl idl

represents the state leading to coincidence counts.

The single-photon propagation velocity in a dielectric—

10.2.2

The down-converted photons are twins, i.e. they are born at precisely the same instant

inside the nonlinear crystal. On the other hand, the strict conservation laws in eqn

(10.55) are only valid if ( ωp , kp ) is sharply de¬ned. In practice this means that the

incident pulse length must be long compared to any other relevant time scale, i.e.

the pump laser is operated in continuous-wave (cw) mode. Thus the twin photons

are born at the same time, but this time is fundamentally unknowable because of the

energy“time uncertainty principle.

These properties allow a given pair of photons to be used, in conjunction with

the Hong“Ou“Mandel interferometer, to measure the speed with which an individ-

ual photon traverses a transparent dielectric medium. This allows us to investigate

the following question: Does an individual photon wave packet move at the group ve-

locity through the medium, just as an electromagnetic wave packet does in classical

electrodynamics? The answer is yes, if the single-photon state is monochromatic and

the medium is highly transparent. This agrees with the simple theory of the quantized

electromagnetic ¬eld in a transparent dielectric, which leads to the expectation that an

electromagnetic wave packet containing a single photon propagates with the classical

group velocity through a dispersive and nondissipative dielectric medium.

A schematic of an experiment (Steinberg et al., 1992) which demonstrates that

individual photons do indeed travel at the group velocity is shown in Fig. 10.5. In this

arrangement an argon-ion UV laser beam, operating at wavelength of 351 nm, enters a

KDP crystal, where entangled pairs of photons are produced. Degenerate red photons

at a wavelength of 702 nm are selected out for detection by means of two irises, I1 and

I2, placed in front of detectors D1 and D2, which are single-photon counting modules

(silicon avalanche photodiodes). The signal wave packet, which follows the upper path

of the interferometer, traverses a glass sample of length L, and subsequently enters an

optical-delay mechanism, consisting of a right-angle trombone prism mounted on a

computer-controlled translation stage. This prism retrore¬‚ects the signal wave packet

onto one input port of the ¬nal beam splitter, with a variable time delay. Consequently,

the location of the trombone prism can be chosen so that the signal wave packet will

overlap with the idler wave packet.

¿¾

Two-photon interference

KDP Cyl.

Argon-ion crystal lens Glass sample

UV laser

Signal (length L)

Idler

(Optical delay „)

Trombone I2

prism

D2

Beam splitter

D1

Coincidence

I1

counter

Fig. 10.5 Apparatus to measure photon propagation times. (Reproduced from Steinberg

et al. (1992).)

Meanwhile, the idler wave packet has been traveling along the lower path of the

interferometer, which is empty of all optical elements, apart from a single mirror which

re¬‚ects the idler wave packet onto the other input port of the beam splitter. If the

optical path length di¬erence between the upper and lower paths of the interferometer

is adjusted to be zero, then the signal and idler wave packets will meet at the same

instant at the ¬nal beam splitter. For this to happen, the longitudinal position of the

trombone prism must be adjusted so as to exactly compensate for the delay”relative

to the idler wave packets transit time through vacuum”experienced by the signal

wave packet, due to its propagation through the glass sample at the group velocity,

vg < c.

As explained in Section 10.2.1, the bosonic character of photons allows a pair of

photons meeting at a balanced beam splitter to pair o¬, so that they both go towards

the same detector. The essential condition is that the initial two-photon state contains

no which-path hints. When this condition is satis¬ed, there is a minimum (a perfect

null under ideal circumstances) of the coincidence-counting signal. The overlap of the

signal and idler wave packets at the beam splitter must be as complete as possible, in

order to produce the Hong“Ou“Mandel minimum in the coincidence count rate. As the

time delay produced by the trombone prism is varied, the result is an inverted Gaussian

pro¬le, similar to the one pictured in Fig. 10.4, near the minimum in the coincidence

rate. As can be readily seen from the ¬rst line in Table 10.1, a compensating delay of

35 219 ± 1 fs must be introduced by the trombone prism in order to produce the Hong“

Ou“Mandel minimum in the coincidence rate. This delay is very close to what one

expects for a classical electromagnetic wave packet propagating at the group velocity

through a 1/2 inch length of SF11 glass.

This experiment was repeated for several samples of glass in various con¬gurations.

From Table 10.1, we see that the theoretical predictions, based on the assumption

that single-photon wave packets travel at the group velocity, agree very well with

experimental measurements. The predictions based on the alternative supposition that

¿¾ Experiments in linear optics

Glass L „t (expt) „g (theory) „p (theory)

(µm) (fs) (fs) (fs)

SF11 ( 1 ) 12687 ± 13 35219 ± 1 35181 ± 35 32642 ± 33

2

SF11 ( 1 ) ’6337 ± 13 ’17559.6 ± 1 ’17572 ± 35 ’16304 ± 33

4

SF11 ( 1 & 1 ) 19033 ± 0.5 52782.4 ± 1 52778.6 ± 1.4 48949 ± 46

2 4

1 1

18894 ± 18 33513 ± 1 33480 ± 33 32314 ± 32

BK7 ( 2 & 4 )

n/a— ’19264 ± 1 ’19269 ± 1.4 ’16635 ± 56

All BK7 & SF11

BK7 ( 1 ) 12595 ± 13 22349.5 ± 1 22318 ± 22 21541 ± 21

2

— This measurement involved both pieces of BK7 in one arm and both pieces of SF11

in the other, so no individual length measurement is meaningful.

Table 10.1 Measured delay times compared to theoretical values computed using the group

and phase velocities. (Reproduced from Steinberg et al. (1992).)

the photon travels at the phase velocity seriously disagree with experiment.

The dispersion cancelation e¬ect—

10.2.3

In addition to providing evidence that single photons propagate at the group velocity,

the experiment reported above displays a feature that is surprising from a classical

point of view. For the experimental run with the 1/2 in glass sample inserted in the

signal arm, Fig. 10.6 shows that the HOM dip has essentially the same width as

the vacuum-only case shown in Fig. 10.4. This is surprising, because a classical wave

packet passing through the glass sample experiences dispersive broadening, due to the

fact that plane waves with di¬erent frequencies propagate at di¬erent phase velocities.

This raises the question: Why is the width of the coincidence-count dip not changed

by the broadening of the signal wave packet? One could also ask the more fundamental

question: How is it that the presence of the glass sample in the signal arm does not

altogether destroy the delicate interference phenomena responsible for the null in the

coincidence count?

To answer these questions, we ¬rst recall that the existence of the HOM null

depends on starting with an initial state such that the rr- and tt-processes lead to the

same ¬nal state. When this condition for interference is satis¬ed, it is impossible”

even in principle”to determine which photon passed through the glass sample. This

means that each of the twin photons traverses both the rr- and the tt-paths”just

as each photon in a Young™s interference experiment passes through both pinholes.

In this way, each photon experiences two di¬erent values of the frequency-dependent

index of refraction”one in glass, the other in vacuum”and this fact is the basis for a

quantitative demonstration that the two-photon interference e¬ect also takes place in

the unbalanced HOM interferometer.

The only di¬erence between this experiment and the original Hong“Ou“Mandel

experiment discussed in Section 10.2.1-B is the presence of the glass sample in the

signal arm of the apparatus; therefore, we only need to recalculate the phase di¬erence

∆¦ (ν) between the two paths. The new phase shifts for each path are obtained from

the old phase shifts by adding the di¬erence in phase shift between the length L of

¿¾

Two-photon interference

1150

1050

Coincidence rate (s’1)

950

850

750

650

550

35069 35144 35219 35294 35369

Time delay (fs)

Fig. 10.6 Coincidence pro¬le after a 1/2 in piece of SF11 glass is inserted in the signal arm of

the interferometer. The location of the minimum is shifted by 35 219 fs from the corresponding

vacuum result, but the width is essentially unchanged. For comparison the dashed curve shows

a classically broadened 15 fs pulse. (Reproduced from Steinberg et al. (1992).)

the glass sample and the same length of vacuum; therefore

ω

(0)

¦tt (ν) = ¦tt (ν) + k (ω) ’ L (10.75)

c

and

ω

(0)

¦rr (ν) = ¦rr (ν) + k (ω ) ’ L, (10.76)

c

(0) (0)

where ¦tt (ν) and ¦rr (ν) are respectively given by eqns (10.65) and (10.66). The

new phase di¬erence is

ω ω

k (ω) ’ ’ k (ω ) ’

∆¦ (ν) = ∆¦(0) (ν) + L, (10.77)

c c

so using eqn (10.67) for ∆¦(0) (ν) yields

2ν

(∆L ’ L) + [k (ω0 + ν) ’ k (ω0 ’ ν)] L ,

∆¦ (ν) = (10.78)

c

where ω0 = (ω + ω ) /2 = ωp /2. The di¬erence k (ω0 + ν) ’ k (ω0 ’ ν) represents the

fact that both of the anti-correlated twin photons pass through the glass sample.

As a consequence of dispersion, the di¬erence between the wavevectors is not in

general a linear function of ν; therefore, it is not possible to choose a single value of

∆L that ensures ∆¦ (ν) = 0 for all values of ν. Fortunately, the limited range of values

¿¾ Experiments in linear optics

2

for ν allowed by the sharply-peaked function |g (ν)| in eqn (10.69) justi¬es a Taylor

series expansion,

d2 k

dk 1

(±ν)2 + O ν 3 ,

k (ω0 ± ν) = k (ω0 ) + (±ν) + (10.79)

dω 2

dω 2

0 0

around the degeneracy value ν = 0 (ω = ω = ω0 ). When this expansion is substi-

tuted into eqn (10.78) all even powers of ν cancel out; we call this the dispersion

cancelation e¬ect. In this approximation, the phase di¬erence is

2ν dk

(∆L ’ L) + 2 νL + O ν 3

∆¦ (ν) =

c dω 0

2ν 2ν

(∆L ’ L) + L + O ν3 ,

= (10.80)

c vg0

where the last line follows from the de¬nition (3.142) of the group velocity. If the

third-order dispersive terms are neglected, the null condition ∆¦ (ν) = 0 is satis¬ed

for all ν by setting

c

∆L = 1 ’ L < 0, (10.81)

vg0

where the inequality holds for normal dispersion, i.e. vg0 < c. Thus the signal path

length must be shortened, in order to compensate for slower passage of photons through

the glass sample.

The second-order term in the expansion (10.79) de¬nes the group velocity disper-

sion coe¬cient β:

1 d2 k 11 dvg

=’ 2

β= . (10.82)

2 dω 2 ω=ω0 2 vg0 dω 0

Since β cancels out in the calculation of ∆¦ (ν), it does not a¬ect the width of the

Hong“Ou“Mandel interference minimum.

The Franson interferometer—

10.2.4

The striking phenomena discussed in Sections 10.2.1“10.2.3 are the result of a quan-

tum interference e¬ect that occurs when twin photons”which are produced simulta-

neously at a single point in the KDP crystal”are reunited at a single beam splitter.

In an even more remarkable interference e¬ect, ¬rst predicted by Franson (1989), the

two photons never meet again. Instead, they only interact with spatially-separated

interferometers, that we will label as nearby and distant. The ¬nal beam splitter in

each interferometer has two output ports: the one positioned between the beam split-

ter and the detector is called the detector port, since photons emerging from this port

fall on the detector; the other is called the exit port, since photons emitted from this

port leave the apparatus. At the ¬nal beam splitter in each interferometer the photon

randomly passes through the detector or the exit port. Speaking anthropomorphically,

the choice made by each photon at its ¬nal beam splitter is completely random, but

the two”apparently independent”choices are in fact correlated. For certain settings

of the interferometers, when one photon chooses the detector port, so does the other,

¿¾

Two-photon interference

i.e. the random choices of the two photons are perfectly correlated. This happens de-

spite the fact that the photons have never interacted since their joint production in

the KDP crystal. Even more remarkably, an experimenter can force a change, from

perfectly correlated choices to perfectly anti-correlated choices, by altering the setting

of only one of the interferometers, e.g. the nearby one.

This situation is so radically nonclassical that it is di¬cult to think about it clearly.

A common mistake made in this connection is to conclude that altering the setting at

the nearby interferometer is somehow causing an instantaneous change in the choices

made by the photon in the distant interferometer. In order to see why this is wrong,

it is useful to imagine that there are two experimenters: Alice, who adjusts the nearby

interferometer and observes the choices made by photons at its ¬nal beam splitter;

and Bob, who observes the choices made by successive photons at the ¬nal beam

splitter in the distant interferometer, but makes no adjustments. An important part

of the experimental arrangement is a secret classical channel through which Alice is

informed”without Bob™s knowledge”of the results of Bob™s measurements. Let us

now consider two experimental runs involving many successive pairs of photons. In

the ¬rst, Alice uses her secret information to set her interferometer so that the choices

of the two photons are perfectly correlated. In the meantime, Bob”who is kept in

the dark regarding Alice™s machinations”accumulates a record of the detection-exit

choices at his beam splitter. In the second run, Alice alters the settings so that the

photon choices are perfectly anti-correlated, and Bob innocently continues to acquire

data. Since the individual quantum events occurring at Bob™s beam splitter are per-

fectly random, it is clear that his two sets of data will be statistically indistinguishable.

In other words, Bob™s local observations at the distant interferometer”made without

bene¬t of a secret channel”cannot detect the changes made by Alice in the settings

of the nearby interferometer. The same could be said of any local observations made

by Alice, if she were deprived of her secret channel. The di¬erence between the two

experiments is not revealed until the two sets of data are brought together”via the

classical communication channel”and compared. Alice™s manipulations do not cause

events through instantaneous action at a distance; instead, her actions cause a change

in the correlation between distant events that are individually random as far as local

observations are concerned.

The peculiar phenomena sketched above can be better understood by describing

a Franson interferometer that was used in an experiment with down-converted pairs

(Kwiat et al., 1993). In this arrangement, shown schematically in Fig. 10.7, each photon

passes through one interferometer.

An examination of Fig. 10.7 shows that each interferometer Ij (de¬ned by the

components Mj, B1j , and B2j , with j = 1, 2) contains two paths, from the initial

to the ¬nal beam splitter, that send the photon to the associated detector: a long

path with length Lj and a short path with length Sj . This arrangement is called

an unbalanced Mach“Zehnder interferometer. The di¬erence ∆Lj = Lj ’ Sj in path

lengths serves as an optical delay line that can be adjusted by means of the trombone

prism. We will label the signal and idler wave packets with 1 and 2 according to the

interferometer that is involved.

A photon traversing an interferometer does not split at the beam splitters, but the

¿¿¼ Experiments in linear optics

χ(2) crystal Cyl. M2

UV pump (KDP) lens

laser

M1

∆L1/2

B11

F1

D1

B21

∆L2/2

B12

Coincidence

D2

B22 counter

F2

Fig. 10.7 Experimental con¬guration for a Franson interferometer. (Reproduced from Kwiat

et al. (1993).)

probability amplitude de¬ning the wave packet does; consequently”just as in Young™s

two-pinhole experiment”the two paths available to the photon could produce single-

photon interference. In the present case, the interference would appear as a temporal

oscillation of the intensity emitted from the ¬nal beam splitter. We will abuse the

terminology slightly by also referring to these oscillations as interference fringes. This

e¬ect can be prevented by choosing the optical delay ∆Lj /c to be much greater than

the typical coherence time „1 of a single-photon wave packet:

∆Lj

„1 . (10.83)

c

When this is the case, the two partial wave packets”one following the long path and

the other following the short path through the interferometer”completely miss each

other at the ¬nal beam splitter, so there is no single-photon interference.

The motivation for eliminating single-photon interference is that the oscillation

of the singles rates at one or both detectors would confuse the measurement of the

coincidence rate, which is the signal for two-photon interference. Further examination

of Fig. 10.7 shows that there are four paths that can result in the detection of both

photons: l“l (each wave packet follows its long path); l“s (wave packet 1 follows its

long path and wave packet 2 follows its short path); s“l (wave packet 1 follows its

short path and wave packet 2 follows its long path); and s“s (each wave packet follows

its short path).

According to Feynman™s rules, two paths leading to distinct ¬nal states cannot

interfere, so we need to determine which pairs of paths lead to di¬erent ¬nal states.

The ¬rst step in this task is to calculate the arrival time of the wave packets at their

respective detectors. For interferometer Ij , let Tj be the propagation time to the ¬rst

beam splitter plus the propagation time from the ¬nal beam splitter to the detector;

then the arrival times at the detector via the long or short path are

tjl = Tj + Lj /c (10.84)

¿¿½

Two-photon interference

and

tjs = Tj + Sj /c , (10.85)

respectively. This experiment uses a cw pump to produce the photon pairs; therefore,

only the di¬erences in arrival times at the detectors are meaningful. The four processes

yield the time di¬erences

L 1 ’ S2

∆tls = t1l ’ t2s = T1 ’ T2 + , (10.86)

c

L 2 ’ S1

∆tsl = t1s ’ t2l = T1 ’ T2 ’ , (10.87)

c

L1 ’ L2

∆tll = t1l ’ t2l = T1 ’ T2 + , (10.88)

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

delay due to the tunnel barrier is negative in sign; the mirror M1 has to be moved

away from the barrier in order to recover the HOM dip. This is contrary to the normal

expectation that all such delays should be positive in sign. For example, one would ex-

pect a positive sign if the tunnel barrier were an ordinary piece of glass, in which case

the mirror would have to be moved towards the barrier to recover the HOM dip. Thus

the sign of the necessary displacement of mirror M1 determines whether tunneling is

superluminal or subluminal in character.

The tunnel barrier used in this experiment”which was ¬rst performed at Berke-

ley in 1993 (Steinberg et al., 1993; Steinberg and Chiao, 1995)”is a dielectric mir-

ror formed by an alternating stack of high- and low-index coatings, each a quarter

wavelength thick. The multiple Bragg re¬‚ections from the successive interfaces of the

dielectric coatings give rise to constructive interference in the backwards direction of

propagation for the photon and destructive interference in the forward direction. The

result is an exponential decay in the envelope of the electric ¬eld amplitude as a func-

tion of propagation distance into the periodic structure, i.e. an evanescent wave. This

constitutes a photonic bandgap, that is, a range of classical wavelengths”equivalent

to energies for photons”for which propagation is forbidden. This is similar to the ex-

M1

Tunnel Geiger

barrier counter

γ1

UV laser

Beam

γ0

splitter

Down- γ2

conversion

crystal Geiger

M2 counter

Fig. 10.11 Schematic of a realistic tunneling-time experiment, such as that performed in

Berkeley (Steinberg et al., 1993; Steinberg and Chiao, 1995), to measure the tunneling time of

a photon by means of Hong“Ou“Mandel two-photon interference. The double-headed arrow to

the right of mirror M1 indicates that it can be displaced so as to compensate for the tunneling

time delay introduced by the tunnel barrier. The sign of this displacement indicates whether

the tunneling time is superluminal or subluminal.

¿¼ Experiments in linear optics

ponential decay of the electron wave function inside the classically forbidden region of

a tunnel barrier.

In this experiment, the photonic bandgap stretched from a wavelength of 600 nm

to 800 nm, with a center at 700 nm, the wavelength of the photon pairs used in the

Hong“Ou“Mandel interferometer. The exponential decay of the photon probability

amplitude with propagation distance is completely analogous to the exponential decay

of the probability amplitude of an electron inside a periodic crystal lattice, when its

energy lies at the center of the electronic bandgap. The tunneling probability of the

photon through the photonic tunnel barrier was measured to be around 1%, and

was spectrally ¬‚at over the typical 10 nm-wide bandwidths of the down-conversion

photon wave packets. This is much narrower than the 200 nm total spectral width of

the photonic bandgap. The carrier wavelength of the single-photon wave packets was

chosen to coincide with the center of the bandgap. After the tunneling process was

completed, the transmitted photon wave packets su¬ered a 99% reduction in intensity,

but the distortion from the initial Gaussian shape was observed to be completely

negligible.

In Fig. 10.12, the data for the tunneling time obtained using the Hong“Ou“Mandel

8 100%

Transmission

6 80%

Larmor time

Delay time (fs)

4 60%

Transmission

Group delay time

2 40%

Subluminal

0 20%

Superluminal

0%

’2

0o 10o 20o 30o 40o 50o 60o 70o 80o 90o

Angle

Fig. 10.12 Summary of tunneling time data taken using the Hong“Ou“Mandel interferom-

eter, shown schematically in Fig. 10.11, as the tunnel barrier sample was tilted: starting from

normal incidence at 0—¦ towards 60—¦ for p-polarized down-converted photons. As the sample

was tilted towards Brewster™s angle (around 60—¦ ), the tunneling time changed sign from a

negative relative delay, indicating a superluminal tunneling time, to a positive relative delay,

indicating a subluminal tunneling time. Note that the sign reversal occurs at a tilt angle of

40—¦ . Two di¬erent samples used as barriers are represented respectively by the circles and

the squares. (Reproduced from Steinberg and Chiao (1995).)

Tunneling time measurements— ¿½

interferometer are shown as a function of the tilt angle of the tunnel barrier sample

relative to normal incidence, with the plane of polarization of the incident photon lying

in the plane of incidence (this is called p-polarization). As the tilt angle is increased

towards Brewster™s angle (around 60—¦ ), the re¬‚ectivity of the successive interfaces

between the dielectric layers tends to zero. In this limit the destructive interference in

the forward direction disappears, so the photonic bandgap, along with its associated

tunnel barrier, is eliminated.

Thus as one tilts the tunnel barrier towards Brewster™s angle, it e¬ectively behaves

more and more like an ordinary glass sample. One then expects to obtain a positive

delay for the passage of the photon γ1 through the barrier, corresponding to a sublu-

minal tunneling delay time. Indeed, for the three data points taken at the large tilt

angles of 45—¦ , 50—¦ , and 55—¦ (near Brewster™s angle) the mirror M1 had to be moved

towards the sample, as one would normally expect for the compensation of positive

delays. However, for the three data points at the small tilt angles of 0—¦ , 22—¦ , and 35—¦ ,

the data show that the tunneling delay of photon γ1 is negative relative to photon γ2 .

In other words, for incidence angles near normal the mirror M1 had to be moved in

the counterintuitive direction, away from the tunnel barrier. The change in sign of the

e¬ect implies a superluminal tunneling time for these small angles of incidence. The

displacement of mirror M1 required to recover the HOM dip changed from positive to

negative at 40—¦ , corresponding to a smooth transition from subluminal to superluminal

tunneling times. From these data, one concludes that, near normal incidence, the tun-

neling wave packet γ1 passes through the barrier superluminally (i.e. e¬ectively faster

than c) relative to wave packet γ2 . The interpretation of this seemingly paradoxical

result evidently requires some care.

We ¬rst note that the existence of apparently superluminal propagation of classi-

cal electromagnetic waves is well understood. An example, that shares many features

with tunneling, is propagation of a Gaussian pulse with carrier frequency in a region

of anomalous dispersion. The fact that this would lead to superluminal propagation

of a greatly reduced pulse was ¬rst predicted by Garrett and McCumber (1969) and

later experimentally demonstrated by Chu and Wong (1982). The classical explanation

of this phenomenon is that the pulse is reshaped during its propagation through the

medium. The locus of maximum constructive interference”the pulse peak”is shifted

forward toward the leading edge of the pulse, so that the peak of a small replica of

the original pulse arrives before the peak of a similar pulse propagating through vac-

uum. Another way of saying this is that the trailing edge of the pulse is more strongly

absorbed than the leading edge. The resulting movement of the peak is described by

the group velocity, which can be greater than c or even negative. These phenomena

are actually quite general; in particular, they will also occur in an amplifying medium

(Bolda et al., 1993). In this case it is possible for a Gaussian pulse with carrier fre-

quency detuned from a gain line to propagate”with little change in amplitude and

shape”with a group velocity greater than c or negative (Chiao, 1993; Steinberg and

Chiao, 1994).

The method used above to explain classical superluminal propagation is mathemat-

ically similar to Wigner™s theory of tunneling in quantum mechanics (Wigner, 1955).

This theory of the tunneling time was based on the idea, roughly speaking, that the

¿¾ Experiments in linear optics

peak of the tunneling wave packet would be delayed with respect to the peak of a

nontunneling wave packet by an amount determined by the maximum constructive

interference of di¬erent energy components, which de¬nes the peak of the tunneling

wave packet. The method of stationary phase then leads to the expression

d arg T (E)

„Wigner = (10.104)

dE E0

for the group-delay tunneling time, where E0 is the most probable energy of the

tunneling particle™s wave packet, and T (E) is the particle™s tunneling probability am-

plitude as a function of its energy E. Wigner™s theory predicts that the tunneling

delay becomes superluminal because”for su¬ciently thick barriers”the time „Wigner

depends only on the tunneling particle™s energy, and not on the thickness of the bar-

rier. Since the Wigner tunneling time saturates at a ¬nite value for thick barriers, this

produces a seeming violation of relativistic causality when „Wigner < d/c, where d is

the thickness of the barrier.

Wigner™s theory was not originally intended to apply to photons, but we have

already seen in Section 7.8 that a classical envelope satisfying the paraxial approxima-

tion can be regarded as an e¬ective probability amplitude for the photon. This allows

us to use the classical wave calculations to apply Wigner™s result to photons. From

this point of view, the rare occasions when a tunneling photon penetrates through the

barrier”approximately 1% of the photons appear on the far side”is a result of the

small probability amplitude that is transmitted. This in turn corresponds to the 1%

transmission coe¬cient of the sample at 0—¦ tilt. It is only for these lucky photons that

the click of the upper Geiger counter occurs earlier than a click of the lower Geiger

counter announcing the arrival of the nontunneling photon γ2 . The average of all data

runs at normal incidence shows that the peak of the tunneling wave packet γ1 arrived

1.47 ± 0.21 fs earlier than the peak of the wave packet γ2 that traveled through the air.

This is in reasonable agreement (within two standard deviations) with the prediction

of 1.9 fs based on eqn (10.104).

Some caveats need to be made here, however. The ¬rst is this: the observation of

a superluminal tunneling time does not imply the possibility of sending a true signal

faster than the vacuum speed of light, in violation of special relativity. By ˜true signal™

we mean a signal which connects a cause to its e¬ect; for example, a signal sent by

closing a switch at one end of a transmission-wire circuit that causes an explosion to

occur at the other end. Such causal signals are characterized by discontinuous fronts”

produced by the closing of the switch, for example”and these fronts are prohibited

by relativity from ever traveling faster than c. However, it should be stressed that

it is perfectly permissible, and indeed, under certain circumstances”arising from the

principle of relativistic causality itself”absolutely necessary, for the group velocity of

a wave packet to exceed the vacuum speed of light (Bolda et al., 1993; Chiao and

Steinberg, 1997). From a quantum mechanical point of view, this kind of superluminal

behavior is not surprising in the case of the tunneling phenomenon considered here.

Since this phenomenon is fundamentally probabilistic in nature, there is no determin-

istic way of controlling whether any given tunneling event will occur or not. Hence

The meaning of causality in quantum optics— ¿¿

there is no possibility of sending a controllable signal faster than c by means of any

tunneling particle, including the photon.

It may seem paradoxical that a particle of light can, in some sense, travel faster

than light, but we must remember that it is not logically impossible for a particle of

light in a medium to travel faster than a particle of light in the vacuum. Nevertheless,

it behooves us to discuss the fundamental questions raised by these kinds of coun-

terintuitive superluminal phenomena concerning the meaning of causality in quantum

optics. This will be done in more detail below.

The second caveat is this: it would seem that the above data would rule out all

theories of the tunneling time other than Wigner™s, but this is not so. One can only

say that for the speci¬c operational method used to obtain the data shown in Fig.

10.12, Wigner™s theory is singled out as the closest to being correct. However, by

using a di¬erent operational method which employs di¬erent experimental conditions

to measure a physical quantity”such as the time of interaction of a tunneling particle

with a modulated barrier, as was suggested by B¨ttiker and Landauer (1982)”one will

u

obtain a di¬erent result from Wigner™s. One striking di¬erence between the predictions

of these two particular theories of tunneling times is that in Wigner™s theory, the

group-delay tunneling time is predicted to be independent of barrier thickness in the

case of thick barriers, whereas in B¨ ttiker and Landauer™s theory, their interaction

u

tunneling time is predicted to be linearly dependent upon barrier thickness. A linear

dependence upon the thickness of a tunnel barrier has indeed been measured for one of

the two tunneling times observed by Balcou and Dutriaux (1997), who used a 2D tunnel

barrier based on the phenomenon of frustrated total internal re¬‚ection between two

closely spaced glass prisms. Thus in Balcou and Dutriaux™s experiment, the existence

of B¨ ttiker and Landauer™s interaction tunneling time has in fact been established.

u

For a more detailed review of these and yet other tunneling times, wave propagation

speeds, and superluminal e¬ects, see Chiao and Steinberg (1997).

The con¬‚icts between the predictions of the various tunneling-time theories dis-

cussed above illustrate the fact that the interpretation of measurements in quantum

theory may depend sensitively upon the exact operational conditions used in a given

experiment, as was emphasized early on by Bohr. Hence it should not surprise us that

the operationalism principle introduced at the beginning of this chapter must always

be carefully taken into account in any treatment of these problems. More concretely,

the phrase ˜the tunneling time™ is meaningless unless it is accompanied by a precise

operational description of the measurement to be performed.

The meaning of causality in quantum optics—

10.5

The appearance of counterintuitive, superluminal tunneling times in the above ex-

periments necessitates a careful re-examination of what is meant by causality in the

context of quantum optics. We begin by reviewing the notion of causality in classical

electromagnetic theory. In Section 8.1, we have seen that the interaction of a classical

electromagnetic wave with any linear optical device”including a tunnel barrier”can

be described by a scattering matrix. We will simplify the discussion by only considering

planar waves, e.g. superpositions of plane waves with all propagation vectors directed

along the z-axis. An incident classical, planar wave Ein (z, t) propagating in vacuum

¿ Experiments in linear optics

is a function of the retarded time tr = t ’ z/c only; therefore we replace Ein (z, t) by

Ein (tr ). This allows the incident ¬eld to be expressed as a one-dimensional Fourier

integral transform:

∞

dω

Ein (ω) e’iωtr .

Ein (tr ) = (10.105)

’∞ 2π

The output wave, also propagating in vacuum, is described in the same way by a

function Eout (ω) that is related to Ein (ω) by

Eout (ω) = S(ω)Ein (ω) , (10.106)

where S(ω) is the scattering matrix”or transfer function”for the device in question.

The transfer function S(ω) describes the reshaping of the input wave packet to produce

the output wave packet. By means of the convolution theorem, we can transform the

frequency-domain relation (10.106) into the time-domain relation

+∞

Eout (tr ) = S(„ )Ein (tr ’ „ )d„ , (10.107)

’∞

where ∞

dω

S(ω)e’iω„ .

S(„ ) = (10.108)

2π

’∞

The fundamental principle of causality states that no e¬ect can ever precede its

cause. This implies that the transfer function must strictly vanish for all negative

delays, i.e.

S(„ ) = 0 for all „ < 0 . (10.109)

Therefore, the range of integration in eqn (10.107) is restricted to positive values, so

that ∞

Eout (tr ) = S(„ )Ein (tr ’ „ )d„ . (10.110)

0

Thus we reach the intuitively appealing conclusion that the output ¬eld at time tr can

only depend on values of the input ¬eld in the past. In particular, if the input signal

has a front at tr = 0, that is

Ein (tr ) = 0 for all tr < 0 (or equivalently z > ct) , (10.111)

then it follows from eqn (10.110) that

Eout (tr ) = 0 for all tr < 0 . (10.112)

Thus the classical meaning of causality for linear optical systems is that the reshaping,

by whatever mechanism, of the input wave packet to produce the output wave packet

cannot produce a nonvanishing output signal before the arrival of the input signal

front at the output face.

In the quantum theory, one replaces the classical electric ¬eld amplitudes by time-

dependent, positive-frequency electric ¬eld operators in the Heisenberg picture. By

Interaction-free measurements— ¿

virtue of the correspondence principle, the linear relation between the classical input

and output ¬elds must also hold for the ¬eld operators, so that

+∞

(+) (+)

S(„ )Ein (tr ’ „ )d„ .

Eout (tr ) = (10.113)

0

One new feature in the quantum version is that the frequency ω in S(ω) is now

interpreted in terms of the Einstein relation E = ω for the photon energy. Another

important change is in the de¬nition of a signal front. We have already learnt that

¬eld operators cannot be set to zero; consequently, the statement that the input signal

has a front must be reinterpreted as an assumption about the quantum state of the

¬eld. The quantum version of eqn (10.111) is, therefore,

(+)

Ein (tr )ρ = 0 for all tr < 0 , (10.114)

where ρ is the time-independent density operator describing the state of the system

in the Heisenberg picture. It therefore follows from eqn (10.113) that

(+)

Eout (tr )ρ = 0 for all tr < 0 . (10.115)

The physics behind this statement is that if the system starts o¬ in the vacuum state

at t = 0 at the input, nothing that the optical system can do to it can promote it out

of the vacuum state at the output, before the arrival of the front. Therefore, causality

has essentially similar meanings at the classical and the quantum levels of description

of linear optical systems.

Interaction-free measurements—

10.6

A familiar procedure for determining if an object is present in a given location is to

illuminate the region with a beam of light. By observing scattering or absorption of the

light by the object, one can detect its presence or determine its absence; consequently,

the ¬rst step in locating an object in a dark room is to turn on the light. Thus in

classical optics, the interaction of light with the object would seem to be necessary for

its observation. One of the strange features of quantum optics is that it is sometimes

possible to determine an object™s presence or absence without interacting with the

object. The idea of interaction-free measurements was ¬rst suggested by Elitzur and

Vaidman (1993), and it was later dubbed ˜quantum seeing in the dark™ (Kwiat et al.,

1996). A useful way to think about this phenomenon is to realize that null events”

e.g. a detector does not click during a given time window”can convey information

just as much as the positive events in which a click does occur.

When it is certain that there is one and only one photon inside an interferom-

eter, some very counterintuitive nonlocal quantum e¬ects”including interaction-free

measurements”are possible. In an experiment performed in 1995 (Kwiat et al., 1995a),

this aim was achieved by pumping a lithium-iodate crystal with a 351 nm wavelength

ultraviolet laser, in order to produce entangled photon pairs by spontaneous down-

conversion. As shown in Fig. 10.13, one member of the pair, the gate photon, is di-

rected to a silicon avalanche photodiode T , and the signal from this detector is used to

¿ Experiments in linear optics

Dark

T

UV

Fig. 10.13 Schematic of an experiment using

a down-conversion source to demonstrate one

LiIO3

form of interaction-free measurement. The ob-

ject to be detected is represented by a trans-

latable 100% mirror, with translation denoted

Obj

by the double-arrow symbol ”. (Reproduced

from Kwiat et al. (1995a).)

open the gate for the other detectors. The other member of the pair, the test photon,

is injected into a Michelson interferometer, which is prepared in a dark fringe near the

equal-path length, white-light fringe condition; see Exercise 10.6. Thus the detector

Dark at the output port of the Michelson is a dark fringe detector. It will never reg-

ister any counts at all, if both arms of the interferometer are unblocked. However, the

presence of an absorbing or nontransmitting object in the lower arm of the Michelson

completely changes the possible outcomes by destroying the destructive interference

leading to the dark fringe.

In the real experimental protocol, the unknown object is represented by a translat-

able, 100% re¬‚ectivity mirror. In the original Elitzur“Vaidman thought experiment,

this role is played by a 100%-sensitivity detector that triggers a bomb. This raises the

stakes,2 but does not alter the physical principles involved. When the mirror blocks

the lower arm of the interferometer in the real experiment, it completely de¬‚ects the

test photon to the detector Obj. A click in Obj is the signal that the blocking ob-

ject is present. When the mirror is translated out of the lower arm, the destructive

interference condition is restored, and the test photon never shows up at the Dark

detector.

For a central Michelson beam splitter with (intensity) re¬‚ectivity R and transmis-

sivity T = 1 ’ R (neglecting losses), an incident test photon will be sent into the lower

arm with probability R. If the translatable mirror is present in the lower arm, the pho-

ton is de¬‚ected into the detector Obj with unit probability; therefore, the probability

of absorption is

P (absorption) = P (failure) = R . (10.116)

This is not as catastrophic as the exploding bomb, but it still represents an unsuc-

cessful outcome of the interaction-free measurement attempt. However, there is also a

mutually exclusive possibility that the test photon will be transmitted by the central

beam splitter, with probability T , and”upon its return”re¬‚ected by the beam split-

ter, with probability R, to the Dark detector. Thus clicks at the Dark port occur with

probability RT . When a Dark click occurs there is no possibility that the test photon

was absorbed by the object”the bomb did not go o¬”since there was only a single

photon in the system at the time. Hence, the probability of a successful interaction-free

measurement of the presence of the object is

2 Oneof the virtues of thought experiments is that they are not subject to health and safety

inspections.

Interaction-free measurements— ¿

P (detection) = P (success) = RT . (10.117)

For a lossless Michelson interferometer, the fraction · of successful interaction-free

measurements is therefore

P (detection)

P (success)

·≡ =

P (success) + P (failure) P (detection) + P (absorption)

1’R

RT

= , (10.118)

=

2’R

RT + R

which tends to an upper limit of 50% as R approaches zero.

This quantum e¬ect is called an interaction-free measurement, because the

single photon injected into the interferometer did not interact at all”either by ab-

sorption or by scattering”with the object, and yet we can infer its presence by means

of the absence of any interaction with it. Furthermore, the inference of presence or ab-

sence can be made with complete certainty based on the principle of the indivisibility

of the photon, since the same photon could not both have been absorbed by the ob-

ject and later caused the click in the dark detector. Actually, it is Bohr™s wave“particle

complementarity principle that plays a central role in this kind of measurement. In

the absence of the object, it is the wave-like nature of light that ensures”through

destructive interference”that the photon never exits through the dark port. In the

presence of the object, it is the particle-like nature of the light”more precisely the

indivisibility of the quantum of light”which enforces the mutual exclusivity of a click

at the dark port or absorption by the object.

Thus a null event”here the absence of a click at Obj ”constitutes just as much

of a measurement in quantum mechanics as the observation of a click. This feature of

quantum theory was already emphasized by Renninger (1960) and by Dicke (1981),

but its implementation in quantum interference was ¬rst pointed out by Elitzur and

Vaidman. Note that this e¬ect is nonlocal, since one can determine remotely the pres-

ence or the absence of the unknown object, by means of an arbitrarily remote dark

detector. The fact that the entire interferometer con¬guration must be set up ahead

of time in order to see this nonlocal e¬ect is another example of the general principle

in Bohr™s Delphic remark quoted in Section 10.3.3.

The data in Fig. 10.14 show that the fraction of successful measurements is nearly

50%, in agreement with the theoretical prediction given by eqn (10.118). By techni-

cal re¬nements of the interferometer, the probability of a successful interaction-free

measurement could, in principle, be increased to as close to 100% as desired (Kwiat

et al., 1995a). A success rate of · = 73% has already been demonstrated (Kwiat et al.,

1999a). In the 100% success-rate limit, one could determine the presence or absence

of an object with minimal absorption of photons.

This possibility may have important practical applications. In an extension of this

interaction-free measurement method to 2D imaging, one could use an array of these

devices to map out the silhouette of an unknown object, while restricting the num-

ber of absorbed photons to as small a value as desired. In conjunction with X-ray

interferometers”such as the Bonse“Hart type”this would, for example, allow X-ray

pictures of the bones of a hand to be taken with an arbitrarily low X-ray dosage.

¿ Experiments in linear optics

=

>

Fig. 10.14 (a) Data demonstrating interaction-free measurement. The Michelson beam split-

ter re¬‚ectivity for the upper set of data was 43%. (b) Data and theoretical ¬t for the ¬gure

of merit · as a function of beam splitter re¬‚ectivity. (Reproduced from Kwiat et al. (1995a).)

10.7 Exercises

10.1 Vacuum ¬‚uctuations

(+) (+)

Drop the term E3 (r, t) from the expression (10.7) for Eout (r, t) and evaluate the

(+) (’)

equal-time commutator Eout,i (r, t) , Eout,j (r , t) . Compare this to the correct form

(+)

in eqn (3.17) and show that restoring E3 (r, t) will repair the ¬‚aw.

10.2 Classical model for two-photon interference

Construct a semiclassical model for two-photon interference, along the lines of Section

1.4, by assuming: the down-conversion mechanism produces classical amplitudes ±σn =

√

Iσn exp (iθσn ), where σ = sig, idl is the channel index and the gate windows are

labeled by n = 1, 2, . . .; the phases θσn vary randomly over (0, 2π); the phases and

intensities Iσn are statistically independent; the intensities Iσn for the two channels

have the same average and rms deviation.

Evaluate the coincidence-count probability pcoinc and the singles probabilities psig

and pidl , and thus derive the inequality (10.41).

The HOM dip—

10.3

2

Assume that the function |g (ν)| in eqn (10.69) is a Gaussian:

√

2

|g (ν)| = „2 / π exp ’„2 ν 2 .

2

Evaluate and plot Pcoinc (∆t).

¿

Exercises

HOM by scattering theory—

10.4

(1) Apply eqn (8.76) to eqn (10.71) to derive eqn (10.72).

(2) Use the de¬nition (6.96) to obtain a formal expression for the coincidence-counting

detection amplitude, and then use the rule (9.96) to show that |¦pair will not

contribute to the coincidence-count rate.

Anti-HOM—

10.5

Consider the two-photon state given by eqn (10.48), where C (ω, ω ) satis¬es the (’)-

version of eqn (10.51).

(1) Why does C (ω, ω ) = ’C (ω , ω) not violate Bose symmetry?

(2) Assume that C (ω, ω ) satis¬es eqn (10.56) and the (’)-version of eqn (10.51).

Use eqns (10.71)“(10.74) to conclude that the photons in this case behave like

fermions, i.e. the pairing behavior seen in the HOM interferometer is forbidden.

Interaction-free measurements—

10.6

(1) Work out the relation between the lengths of the arms of the Michelson interfer-

ometer required to ensure that a dark fringe occurs at the output port.

(2) Explain why the probabilities P (failure) and P (success), respectively de¬ned by

eqns (10.116) and (10.117), do not sum to one.

11

Coherent interaction of light with

atoms

In Chapter 4 we used perturbation theory to describe the interaction between light and

matter. In addition to the assumption of weak ¬elds”i.e. the interaction energy is small

compared to individual photon energies”perturbation theory is only valid for times

in the interval 1/ω0 t 1/W , where ω0 and W are respectively the unperturbed

frequency and the perturbative transition rate for the system under study. When ω0

is an optical frequency, the lower bound is easily satis¬ed, but the upper bound can

be violated. Let ρ be a stationary density matrix for the ¬eld; then the ¬eld“¬eld

correlation function, for a ¬xed spatial point r but two di¬erent times, will typically

decay exponentially:

(1) (’) (+)

(r, t2 ) ∼ exp (’ |t1 ’ t2 | /Tc ) ,

Gij (r, t1 ; r, t2 ) = Tr ρEi (r, t1 ) Ej (11.1)

where Tc is the coherence time for the state ρ. For some states, e.g. the Planck

distribution, the coherence time is short, in the sense that Tc 1/W . Perturbation

theory is applicable to these states, but there are many situations”in particular for

laser ¬elds”in which Tc > 1/W . Even though the ¬eld is weak, perturbation theory

cannot be used in these cases; therefore, we need to develop nonperturbative methods

that are applicable to weak ¬elds with long coherence times.

11.1 Resonant wave approximation

The phenomenon of resonance is ubiquitous in physics and it plays a central role in

the interaction of light with atoms. Resonance will occur if there is an allowed atomic

transition q ’ p with transition frequency ωqp = (µq ’ µp ) / and a matching optical

frequency ω ≈ ωqp . In Section 4.9.2 we saw that the weak-¬eld condition can be ex-

pressed as „¦ ω0 , where „¦ is the characteristic Rabi frequency de¬ned by eqn (4.147).

In the interaction picture, the state vector satis¬es the Schr¨dinger equation (4.94), in

o

which the full Hamiltonian is replaced by the interaction Hamiltonian; consequently,

‚

|Ψ (t) ∼ „¦ |Ψ (t) .

i (11.2)

‚t

Thus the weak-¬eld condition tells us that the changes in the interaction-picture state

vector occur on the time scale 1/„¦ 1/ω0 . Consequently, the state vector does not

change appreciably over an optical period. This disparity in time scales is the basis

for a nonperturbative approximation scheme. In the interests of clarity, we will ¬rst

develop this method for a simple model called the two-level atom.

¿½

Resonant wave approximation

11.1.1 Two-level atoms

The spectra of real atoms and the corresponding sets of stationary states display a

daunting complexity, but there are situations of theoretical and practical interest in

which this complexity can be ignored. In the simplest case, the atomic state vector is

a superposition of only two of the stationary states. Truncated models of this kind are

called two-level atoms. This simpli¬cation can occur when the atom interacts with

a narrow band of radiation that is only resonant with a transition between two speci¬c

energy levels. In this situation, the two atomic states involved in the transition are the

only dynamically active degrees of freedom, and the probability amplitudes for all the

other stationary states are negligible.

In the semiclassical approximation, the Feynman“Vernon“Hellwarth theorem

(Feynman et al., 1957) shows that the dynamical equations for a two-level atom are

isomorphic to the equations for a spin-1/2 particle in an external magnetic ¬eld. This

provides a geometrical picture which is useful for visualizing the solutions. The general

zeroth-order Hamiltonian for the ¬ctitious spin system is H0 = ’µB · σ, and we will

choose the ¬ctitious B-¬eld as B = ’Bu3 , so that the spin-up state is higher in energy

than the spin-down state.

To connect this model to the two-level atom, let the two resonantly connected

atomic states be |µ1 and |µ2 , with µ1 < µ2 . The atomic Hilbert space is e¬ectively

truncated to the two-dimensional space spanned by |µ1 and |µ2 , so the atomic Hamil-

tonian and the atomic dipole operator d are represented by 2 — 2 matrices. Every 2 — 2

matrix can be expressed in terms of the standard Pauli matrices; in particular, the

truncated atomic Hamiltonian is

µ2 + µ1 ω21

µ 0

Hat = 2 = I2 + σz , (11.3)

0 µ1 2 2

where I2 is the 2 — 2 identity matrix and ω21 = µ2 ’ µ1 . The term proportional to I2

can be eliminated by choosing the zero of energy so that µ2 + µ1 = 0. This enforces

the relation µB ” ω21 /2 between the two-level atom and the ¬ctitious spin.

When the very small e¬ects of weak interactions are ignored, atomic states have

de¬nite parity; therefore, the odd-parity operator d has no diagonal matrix elements.

For the two-level atom, this implies d = d— σ’ + d σ+ , where d = µ2 d µ1 , σ+ is the

spin-raising operator, and σ’ is the spin-lowering operator. Combining this with the

decomposition E = E(+) + E(’) and the plane-wave expansion (3.69) for E(+) leads

to

(r) (ar)

Hint = Hint + Hint , (11.4)

Hint = ’d · E(+) σ+ ’ d— · E(’) σ’

(r)

ωk d · eks

= ’i aks σ+ + HC , (11.5)

2 0V

ks

Hint = ’d · E(’) σ+ ’ d— · E(+) σ’

(ar)

ωk d— · eks

= ’i aks σ’ + HC . (11.6)

2 0V

ks

¿¾ Coherent interaction of light with atoms

In Hint the annihilation (creation) operator aks a† is paired with the energy-raising

(r)

ks

(ar)

(-lowering) operator σ+ (σ’ ), while Hint has the opposite pairings. In the perturba-

tive calculations of Section 4.9.3 the emission (absorption) of a photon is associated

with lowering (raising) the energy of the atom, subject to the resonance condition

(r) (ar)

ωk = ω21 , so Hint and Hint are respectively called the resonant and antiresonant

Hamiltonians.