<< . .

. 3
( : 11)

. . >>

|0, 0 ph — |0, 1 ’ |0, 1 ph — |0, 0 a (12)
that if it were possible to make better copies, then, using
EPR correlations between spatially separated systems,
and stimulated emission to
signaling at arbitrarily fast speed would also be possible

(Gisin 1998).
|1, 0 ph — |1, 0 a ’ 2|2, 0 ph — |0, 0 a , (13)

|0, 1 ph — |0, 1 a ’ 2|0, 2 ph — |0, 0 a (14)

where the 2 factor takes into account the ratio stimu-
lated/spontaneous emission. Let the initial state of the
atom be a mixture of the following two states (each with
equal weight 50%):

|0, 1 |1, 0 (15)
a a

By symmetry, it su¬ces to consider one possible initial
state of the qubit, e.g. 1 photon in the ¬rst mode |1, 0 ph .
The initial state of the photon+atom system is thus a

|1, 0 — |1, 0 or |1, 0 — |0, 1 (16)
ph a ph a

This corresponds to the ¬rst order term in an evolution
with a Hamiltonian (in the interaction picture): H =
χ(a† σ1 + a1 σ1 + a† σ2 + a2 σ2 ). After some time the
† †
’ ’
1 2
2-photon component of the evolved states reads:

2|2, 0 ph — |0, 0 a or |1, 1 ph — |0, 0 a (17)
The correspondence with a pair of spin goes as follows:

|2, 0 = | ‘‘ |0, 2 = | ““ (18)

= ψ (+) = √ (| ‘“ + | “‘ )
|1, 1 (19)
Tracing over the ampli¬er (i.e. the 2-level atom), an
(ideal) ampli¬er achieves the following transformation:

P‘ ’ 2P‘‘ + Pψ(+) (20)

where the P ™s indicate projectors (i.e. pure state density
matrices) and the lack of normalization results from the
¬rst order expansion used in (11) to (14). Accordingly,
after normalization, each photon is in state :

the one where absorption is low. However, free space
transmission is restricted to line-of sight links and is very
weather dependent.
The very ¬rst demonstration of QC was a table top ex-
In the next sections we successively consider the ques-
periment performed at the IBM laboratory in the early
tions “how to produce single photons?” (section III A),
1990™s over a distance of 30 cm (Bennett et al. 1992a),
“how to transmit them?” (section III B), “how to detect
marking the start of impressive experimental improve-
single photons?” (section III C), and ¬nally “how to ex-
ments during the last years. The 30 cm distance is of
ploit the intrinsic randomness of quantum processes to
little practical interest. Either the distance should be
build random generators?” (section III D).
even shorter, think of a credit card and the ATM ma-
chine (Huttner et al. 1996b), but in this case all of Al-
ice™s components should ¬t on the credit card. A nice
A. Photon sources
idea, but still impractical with present technology. Or
the distance should be much longer, at least in the km
Optical quantum cryptography is based on the use of
range. Most of the research so far uses optical ¬bers to
single photon Fock states. Unfortunately, these states
guide the photons from Alice to Bob and we shall mainly
are di¬cult to realize experimentally. Nowadays, practi-
concentrate here on such systems. There is, however, also
cal implementations rely on faint laser pulses or entan-
some very signi¬cant research on free space systems, (see
gled photon pairs, where both the photon as well as the
section IV E).
photon-pair number distribution obeys Poisson statistics.
Once the medium is chosen, there remain the questions
Hence, both possibilities su¬er from a small probability
of the source and detectors. Since they have to be com-
of generating more than one photon or photon pair at
patible, the crucial choice is the wavelength. There are
the same time. For large losses in the quantum chan-
two main possibilities. Either one chooses a wavelength
nel even small fractions of these multi-photons can have
around 800 nm where e¬cient photon counters are com-
important consequences on the security of the key (see
mercially available, or one chooses a wavelength compat-
section VI H), leading to interest in “photon guns”, see
ible with today™s telecommunication optical ¬bers, i.e.
paragraph III A 3). In this section we brie¬‚y comment
near 1300 nm or 1550 nm. The ¬rst choice requires free
on sources based on faint pulses as well as on entan-
space transmission or the use of special ¬bers, hence the
gled photon-pairs, and we compare their advantages and
installed telecommunication networks can™t be used. The
second choice requires the improvement or development
of new detectors, not based on silicon semiconductors,
which are transparent above 1000 nm wavelength.
1. Faint laser pulses
In case of transmission using optical ¬bers, it is still
unclear which of the two alternatives will turn out to be
the best choice. If QC ¬nds niche markets, it is conceiv- There is a very simple solution to approximate single
able that special ¬bers will be installed for that purpose. photon Fock states: coherent states with an ultra-low
But it is equally conceivable that new commercial detec- mean photon number µ. They can easily be realized us-
tors will soon make it much easier to detect single pho- ing only standard semiconductor lasers and calibrated
tons at telecommunication wavelengths. Actually, the attenuators. The probability to ¬nd n photons in such a
latter possibility is very likely, as several research groups coherent state follows the Poisson statistics:
and industries are already working on it. There is an-
µn ’µ
other good reason to bet on this solution: the quality P (n, µ) = e. (23)
of telecommunication ¬bers is much higher than that of
any special ¬ber, in particular the attenuation is much Accordingly, the probability that a non-empty weak co-
lower (this is why the telecommunication industry chose herent pulse contains more than 1 photon,
these wavelengths): at 800 nm, the attenuation is about
2 dB/km (i.e. half the photons are lost after 1.5 km), 1 ’ P (0, µ) ’ P (1, µ)
P (n > 1|n > 0, µ) =
while it is only of the order of 0.35 and 0.20 dB/km at 1 ’ P (0, µ)
1300 nm and 1550 nm, respectively (50% loss after about
1 ’ e’µ (1 + µ) ∼ µ
9 and 15 km) 14 . = (24)
1 ’ e’µ 2
In case of free space transmission, the choice of wave-
length is straightforward since the region where good can be made arbitrarily small. Weak pulses are thus ex-
photon detectors exist “ around 800 nm “ coincides with tremely practical and have indeed been used in the vast
majority of experiments. However, they have one ma-
jor drawback. When µ is small, most pulses are empty:
P (n = 0) ≈ 1 ’ µ. In principle, the resulting decrease in
The losses in dB (ldb ) can be calculated from the losses in bit rate could be compensated for thanks to the achiev-
percent (l% ): ldB = ’10 log10 (1 ’ 100 ). able GHz modulation rates of telecommunication lasers.

But in practice the problem comes from the detectors™ The latter is in general rather large and varies from a few
dark counts (i.e. a click without a photon arriving). nanometers up to some tens of nanometers. For the non
Indeed, the detectors must be active for all pulses, in- degenerate case one typically gets 5-10 nm, whereas in
cluding the empty ones. Hence the total dark counts the degenerate case (central frequency of both photons
increase with the laser™s modulation rate and the ratio equal) the bandwidth can be as large as 70 nm.
of the detected photons over the dark counts (i.e. the This photon pair creation process is very ine¬cient,
typically it needs some 1010 pump photons to create one
signal to noise ratio) decreases with µ (see section IV A).
pair in a given mode17 . The number of photon pairs per
The problem is especially severe for longer wavelengths
where photon detectors based on Indium Gallium Ar- mode is thermally distributed within the coherence time
senide semiconductors (InGaAs) are needed (see section of the photons, and follows a poissonian distribution for
III C) since the noise of these detectors explodes if they larger time windows (Walls and Milburn 1995). With a
pump power of 1 mW, about 106 pairs per second can
are opened too frequently (in practice with a rate larger
than a few MHz). This prevents the use of really low be collected in single mode ¬bers. Accordingly, in a time
photon numbers, smaller than approximately 1%. Most window of roughly 1ns the conditional probability to ¬nd
a second pair having detected one is 106 · 10’9 ≈ 0.1%.
experiments to date relied on µ = 0.1, meaning that 5%
of the nonempty pulses contain more than one photon. In case of continuous pumping, this time window is given
However, it is important to stress that, as pointed out by the detector resolution. Tolerating, e.g. 1% of these
multi-pair events, one can generate 107 pairs per second,
by L¨ tkenhaus (2000), there is an optimal µ depending
on the transmission losses 15 . After key distillation, the using a realistic 10 mW pump. Detecting for example
security is just as good with faint laser pulses as with 10 % of the trigger photons, the second detector has to
be activated 106 times per second. In comparison, the
Fock states. The price to pay for using such states lies in
a reduction of the bit rate. example of 1% of multi-photon events corresponds in the
case of faint laser pulses to a mean photon number of µ =
0.02. In order to get the same number 106 of non-empty
pulses per second, a pulse rate of 50 MHz is needed. For a
2. Photon pairs generated by parametric downconversion
given photon statistics, photon pairs allow thus to work
with lower pulse rates (e.g. 50 times lower) and hence
Another way to create pseudo single-photon states is
reduced detector-induced errors. However, due to limited
the generation of photon pairs and the use of one photon
coupling e¬ciency into optical ¬bers, the probability to
as a trigger for the other one (Hong and Mandel 1986).
¬nd the sister photon after detection of the trigger photon
In contrast to the sources discussed before, the second
in the respective ¬ber is in practice lower than 1. This
detector must be activated only whenever the ¬rst one
means that the e¬ective photon number is not one, but
detected a photon, hence when µ = 1, and not whenever
rather µ ≈ 2/3 (Ribordy et al. 2001), still well above
a pump pulse has been emitted, therefore circumventing
µ = 0.02.
the problem of empty pulses.
Photon pairs generated by parametric down conversion
The photon pairs are generated by spontaneous para-
o¬er a further major advantage if they are not merely
metric down conversion in a χ(2) non-linear crystal16 . In
used as pseudo single-photon source, but if their entan-
this process, the inverse of the well-known frequency dou-
glement is exploited. Entanglement leads to quantum
bling, one photon spontaneously splits into two daughter
correlations which can be used for key generation, (see
photons “ traditionally called signal and idler photon “
paragraph II D 3 and chapter V). In this case, if two pho-
conserving total energy and momentum. In this con-
ton pairs are emitted within the same time window but
text, momentum conservation is called phase matching,
their measurement basis is choosen independently, they
and can be achieved despite chromatic dispersion by ex-
produce completely uncorrelated results. Hence, depend-
ploiting the birefringence of the nonlinear crystal. The
ing on the realization, the problem of multiple photon can
phase matching allows to choose the wavelength, and de-
be avoided, see section VI J.
termines the bandwidth of the downconverted photons.
Figure 5 shows one of our sources creating entangled
photon pairs at 1310 nm wavelength as used in tests of
Bell inequalities over 10 kilometers (Tittel et al. 1998).
Although not as simple as faint laser sources, diode
Contrary to a frequent misconception, there is nothing spe-
pumped photon pair sources emitting in the near infrared
cial about a µ value of 0.1, eventhough it has been selected
can be made compact, robust and rather handy.
by most experimentalists. The optimal value “ i.e. the value
that yields the highest key exchange rate after distillation “
depends on the optical losses in the channel and on assump-
tions about Eve™s technology (see VI H and VI I).
16 17
Recently we achieved a conversion rate of 10’6 using an
For a review see Rarity and Tapster 1988, and for latest
developments Tittel et al. 1999, Kwiat et al. 1999, Jennewein optical waveguide in a periodically poled LiNbO3 crystal
et al. 2000b, Tanzilli et al. 2001. (Tanzilli et al. 2001).

vantage with respect to faint laser pulses with extremely
3. Photon guns
low mean photon numbers µ.
The ideal single photon source is a device that when
one pulls the trigger, and only then, emits one and only
B. Quantum channels
one photon. Hence the name photon gun. Although pho-
ton anti-bunching has been demonstrated already years
The single photon source and the detectors must be
ago (Kimble et al. 1977), a practical and handy device is
connected by a “quantum channel”. Such a channel is
still awaited. At present, there are essentially three dif-
actually nothing specially quantum, except that it is in-
ferent experimental approaches that come more or less
tended to carry information encoded in individual quan-
close to this ideal.
tum systems. Here “individual” doesn™t mean “non-
A ¬rst idea is to work with a single two-level quan-
decomposible”, it is meant in opposition to “ensemble”.
tum system that can obviously not emit two photons at
The idea is that the information is coded in a physical
a time. The manipulation of single trapped atoms or
system only once, contrary to classical communication
ions requires a much too involved technical e¬ort. Sin-
where many photons carry the same information. Note
gle organics dye molecules in solvents (S.C. Kitson et al.
that the present day limit for ¬ber-based classical optical
1998) or solids (Brunel et al. 1999, Fleury et al. 2000)
communication is already down to a few tens of photons,
are easier to handle but only o¬er limited stability at
although in practice one usually uses many more. With
room temperature. Promising candidates, however, are
the increasing bit rate and the limited mean power “ im-
nitrogen-vacancy centers in diamond, a substitutional ni-
posed to avoid nonlinear e¬ects in silica ¬bers “ these
trogen atom with a vacancy trapped at an adjacent lat-
¬gures are likely to get closer and closer to the quantum
tice position (Kurtsiefer et al. 2000, Brouri et al. 2000).
It is possible to excite individual nitrogen atoms with a
The individual quantum systems are usually 2-level
532 nm laser beam, which will subsequently emit a ¬‚uo-
systems, called qubits. During their propagation they
rescence photon around 700 nm (12ns decay time). The
must be protected from environmental noise. Here “en-
¬‚uorescence exhibits strong photon anti-bunching and
vironment” refers to everything outside the degree of
the samples are stable at room temperature. However,
freedom used for the encoding, which is not necessar-
the big remaining experimental challenge is to increase
ily outside the physical system. If, for example, the in-
the collection e¬ciency (currently about 0.1%) in order
formation is encoded in the polarization state, then the
to obtain mean photon numbers close to 1. To obtain
optical frequencies of the photon is part of the environ-
this, an optical cavity or a photonic bandgap structure
ment. Hence, coupling between the polarization and the
must suppress the emission in all spatial modes but one.
optical frequency has to be mastered18 (e.g. avoid wave-
In addition, the spectral bandwith of this type of source
length sensitive polarizers and birefringence). Moreover,
is broad (of the order of 100 nm), enhancing the e¬ect of
the sender of the qubits should avoid any correlation be-
pertubations in a quantum channel.
tween the polarization and the spectrum of the photons.
A second approach is to generate photons by single
Another di¬culty is that the bases used by Alice to
electrons in a mesoscopic p-n junction. The idea is to
code the qubits and the bases used by Bob for his mea-
take pro¬t of the fact that thermal electrons show anti-
surements must be related by a known and stable uni-
bunching (Pauli exclusion principle) in contrast to pho-
tary transformation. Once this unitary transformation
tons (Imamoglu and Yamamoto, 1994). First experimen-
is known, Alice and Bob can compensate for it and get
tal results have been presented (Kim et al. 1999), how-
the expected correlation between their preparations and
ever with extremely low e¬ciencies, and only at a tem-
measurements. If it changes with time, they need an ac-
perature of 50mK!
tive feedback to track it, and if the changes are too fast
Finally, another approach is to use the photon emis-
the communication must be interrupted.
sion of electron-hole pairs in a semiconductor quantum
dot. The frequency of the emitted photon depends on the
number of electron-hole pairs present in the dot. After
1. Singlemode ¬bers
one creates several such pairs by optical pumping, they
will sequentially recombine and hence emit photons at
di¬erent frequencies. Therefore, by spectral ¬ltering a Light is guided in optical ¬bers thanks to the refrac-
single-photon pulse can be obtained (G´rard et al. 1999,
e tive index pro¬le n(x, y) across the section of the ¬bers
Santori et al. 2000, and Michler et al. 2000). These dots (traditionally, the z-axis is along the propagation direc-
can be integrated in solid-states microcavities with strong tion). Over the last 25 years, a lot of e¬ort has been
enhancements of the spontaneous emission (G´rard et al.
In summary, today™s photon guns are still too compli-
cated to be used in a QC-prototype. Moreover, due to 18
Note that, as we will see in chapter V, using entangled
their low quantum e¬ciencies they do not o¬er an ad- photons prevents such information leakage.

made to reduce transmission losses “ initially several dB sion and polarization dependent losses.
per km “, and nowadays, the attenuation is as low as The Geometric phase as encountered when guiding
2dB/km at 800nm wavelength, 0.35 dB/km at 1310 nm, light in an optical ¬ber is a special case of the Berry
phase19 which results when any parameter describing a
and 0.2 dB/km at 1550 nm (see Fig. 6). It is amusing
to note that the dynamical equation describing optical property of the system under concern, here the k-vector
pulse propagation (in the usual slowly varying envelope characterizing the propagation of the light ¬eld, under-
aproximation) is identical to the Schr¨dinger equation,
o goes an adiabatic change. Think ¬rst of a linear polar-
with V (x, y) = ’n(x, y) (Snyder 1983). Hence a positive ization state, let™s say vertical at the input. Will it still
bump in the refractive index corresponds to a potential be vertical at the output? Vertical with respect to what?
well. The region of the well is called the ¬ber core. If Certainly not the gravitational ¬eld! One can follow that
the core is large, many bound modes exist, correspond- linear polarization by hand along the ¬ber and see how
ing to many guided modes in the ¬ber. Such ¬bers are it may change even along a closed loop. If the loop stays
called multimode ¬bers, their core being usually 50 mi- in a plane, the state after a loop coincides with the input
crometer in diameter. The modes couple easily, acting state. But if the loop explores the 3 dimensions of our
on the qubit like a non-isolated environment. Hence mul- space, then the ¬nal state will di¬er from the initial one
timode ¬bers are not appropriate as quantum channels by an angle. Similar reasoning holds for the axes of el-
(see however Townsend 1998a and 1998b). If, however, liptical polarization states. The two circular polarization
the core is small enough (diameter of the order of a few states are the eigenstates: during parallel transport they
wavelengths) then a single spatial mode is guided. Such acquire opposite phases, called the Berry phase. The
¬bers are called singlemode ¬bers. For telecommunica- presence of a geometrical phase is not fatal for quantum
tions wavelength (i.e. 1.3 and 1.5 µm), their core is typ- communication, it simply means that initially Alice and
ically 8 µm in diameter. Singlemode ¬bers are very well Bob have to align their systems by de¬ning for instance
suited to carry single quanta. For example, the optical the vertical and diagonal directions (i.e. performing the
phase at the output of a ¬ber is in a stable relation with unitary transformation mentioned before). If these vary
the phase at the input, provided the ¬ber doesn™t get slowly, they can be tracked, though this requires an ac-
elongated. Hence, ¬ber interferometers are very stable, a tive feedback. However, if the variations are too fast,
fact exploited in many instruments and sensors (see, e.g., the communication might be interrupted. Hence, aerial
Cancellieri 1993). cables that swing in the wind are not appropriate (ex-
Accordingly, a singlemode ¬ber with perfect cylindric cept with selfcompensating con¬gurations, see paragraph
symmetry would provide an ideal quantum channel. But IV C 2).
all real ¬bers have some asymmetries and then the two Birefringence is the presence of two di¬erent phase
polarization modes are no longer degenerate but each has velocities for two orthogonal polarization states. It is
its own propagation constant. A similar e¬ect is caused caused by asymmetries in the ¬ber geometry and in the
by chromatic dispersion, where the group delay depends residual stress distribution inside and around the core.
on the wavelength. Both dispersion e¬ects are the sub- Some ¬bers are made birefringent on purpose. Such
ject of the next paragraphs. ¬bers are called polarization maintaining (PM) ¬bers be-
cause the birefringence is large enough to e¬ectively un-
couple the two polarization eigenmodes. But note that
only these two orthogonal polarization modes are main-
2. Polarization e¬ects in singlemode ¬bers
tained; all the other modes, on the contrary, evolve very
quickly, making this kind of ¬ber completely unsuitable
Polarization e¬ects in singlemode ¬bers are a common
for polarization-based QC systems20 . The global e¬ect
source of problems in all optical communication schemes,
of the birefringence is equivalent to an arbitrary com-
as well classical as quantum ones. In recent years this has
bination of two waveplates, that is, it corresponds to a
been a major topic for R&D in classical optical commu-
unitary transformation. If this transformation is stable,
nication (Gisin et al. 1995). As a result, today™s ¬bers
are much better than the ¬bers a decade ago. Nowa-
days, the remaining birefringence is small enough for the
telecom industry, but for quantum communication, any
birefringence, even extremely small, will always remain Introduced by Michael Berry in 1984, then observed in
a concern. All ¬ber based implementations of QC have optical ¬ber by Tomita and Chiao (1986), and on the single

<< . .

. 3
( : 11)

. . >>