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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

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