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Gaussian pulses. Each pulse is characterized by two parameters, the pulse width, Tσ ,
¿¾¿
Two-photon interference

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

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

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

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

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


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


(∆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:


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 ∞

S(ω)e’iω„ .
S(„ ) = (10.108)

’∞

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.

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