Quantum Interferometry with Multiports: Entangled Photons ... · The hidden variables determine the...
Transcript of Quantum Interferometry with Multiports: Entangled Photons ... · The hidden variables determine the...
Quantum Interferometry with Multiports:
Entangled Photons in Optical Fibers
Dissertation
zur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaften
eingereicht von
Mag. phil.Mag. rer. nat.Michael Hunter Alexander Reck
im Juli 1996
durchgefuhrt am Institut fur Experimentalphysik,Naturwissenschaftliche Fakultat
der Leopold-Franzens Universitat Innsbruckunter der Leitung von
o.Univ. Prof. Dr. Anton Zeilinger
Diese Arbeit wurde vom FWF im Rahmen desSchwerpunkts Quantenoptik (S06502) unterstutzt.
Contents
Abstract 5
1 Entanglement and Bell’s inequalities 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Derivation of Bell’s inequality for dichotomic variables . . . . . . 101.3 Tests of Bell’s inequality . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 First experiments . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Entanglement and the parametric downconversion source . 161.3.3 New experiments with entangled photons . . . . . . . . . . 17
1.4 Bell inequalities for three-valued observables . . . . . . . . . . . . 18
2 Theory of linear multiports 232.1 The beam splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Multiports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 From experiment to matrix . . . . . . . . . . . . . . . . . 262.2.2 From matrix to experiment . . . . . . . . . . . . . . . . . 27
2.3 Symmetric multiports . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.1 Single-photon eigenstates of a symmetric multiport . . . . 332.3.2 Two-photon eigenstates of a symmetric multiport . . . . . 33
2.4 Multiports and quantum computation . . . . . . . . . . . . . . . . 35
3 Optical fibers 393.1 Single mode fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 Optical parameters of fused silica . . . . . . . . . . . . . . 403.1.2 Material dispersion and interferometry . . . . . . . . . . . 43
3.2 Components of fiber-optical systems . . . . . . . . . . . . . . . . . 463.3 Coupled waveguides as multiports . . . . . . . . . . . . . . . . . 47
4 Experimental characterization of fiber multiports 494.1 A three-path Mach-Zehnder interferometer using all-fiber tritters . 49
4.1.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 494.1.2 Theoretical description . . . . . . . . . . . . . . . . . . . . 524.1.3 Experimental results . . . . . . . . . . . . . . . . . . . . . 55
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4.2 Two-photon interferences in optical fiber multiports . . . . . . . . 564.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Theoretical description . . . . . . . . . . . . . . . . . . . . 564.2.3 Experimental results . . . . . . . . . . . . . . . . . . . . . 58
5 Two-photon three-path interference 635.1 Energy entanglement in three-path interferometers . . . . . . . . . 635.2 Multipath interferences . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.1 The experimental setup . . . . . . . . . . . . . . . . . . . 705.3.2 Polarization and path length adjustment . . . . . . . . . . 795.3.3 Detection system . . . . . . . . . . . . . . . . . . . . . . . 845.3.4 Calibration of phase settings . . . . . . . . . . . . . . . . . 885.3.5 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . 905.3.6 Temperature drifts . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 935.4.1 Description of data . . . . . . . . . . . . . . . . . . . . . . 935.4.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.5.1 Bell’s inequalities . . . . . . . . . . . . . . . . . . . . . . . 1055.5.2 Classical picture vs. quantum picture . . . . . . . . . . . . 106
6 Conclusions and Outlook 111
A Devices 115A.1 Parametric downconversion crystals . . . . . . . . . . . . . . . . . 115
A.1.1 KDP and LiIO3 . . . . . . . . . . . . . . . . . . . . . . . . 115A.1.2 Beta Barium Borate (BBO) . . . . . . . . . . . . . . . . . 116
A.2 Optical fibers and tritters . . . . . . . . . . . . . . . . . . . . . . 117A.3 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.3.1 Argon-Ion laser . . . . . . . . . . . . . . . . . . . . . . . . 119A.3.2 HeNe laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.3.3 Diode laser . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.4 Single-photon detectors . . . . . . . . . . . . . . . . . . . . . . . . 121A.5 Control and detection electronics . . . . . . . . . . . . . . . . . . 122A.6 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.6.1 Electro-mechanic shutter . . . . . . . . . . . . . . . . . . . 123A.6.2 Parallel printer port as programmable TTL output . . . . 123A.6.3 Temperature sensors . . . . . . . . . . . . . . . . . . . . . 125
B Reprint from Phys. Rev. Lett. 73, 58–61 (1994) 127
C A computer program for the design of unitary interferometers 133
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C.1 Mathematica notebook . . . . . . . . . . . . . . . . . . . . . . . . 133C.2 Mathematica package . . . . . . . . . . . . . . . . . . . . . . . . . 136
D Wave packets in multiports 141D.1 A single photon in a multiport . . . . . . . . . . . . . . . . . . . . 141D.2 Photon pairs in multiports . . . . . . . . . . . . . . . . . . . . . . 143D.3 Material dispersion and interference . . . . . . . . . . . . . . . . . 144
E Photographing Type-II downconversion 149
4
Abstract
The present thesis is the result of theoretical and experimental work on the
physics of optical multiports. The theoretical results show that multiport interfer-
ometers can be used to realize any discrete unitary transformation operating on
modes of a classical or a quantum radiation field. Tests of a Bell-type inequality
for higher-dimensional entangled states are thus possible using entangled photon
pairs from a parametric downconversion source. The experimental work measured
the nonclassical interferences at the fiber-optical three-way beam splitters (trit-
ters) and three-path fiber interferometers. The experimental results are discussed
in the context of Bell’s inequalities and the physics of entanglement.
The first chapter gives a brief review of entanglement and the Bell inequal-
ities. Quantization and the superposition principle together form the basis of
quantum physics. The physics of entangled states dramatically demonstrates the
difference between the quantum and the everyday world.
Multiports are the logical generalization of the beam splitter in classical and
quantum optics. The second chapter deals with the theory of linear multiports
and presents an algorithmic proof that any unitary operator can be built in the
laboratory.
Optical fibers and integrated optics, the basic components of many earth-
bound telecommunication systems, will be necessary in any practical realization of
quantum communication and quantum information processing. The third chapter
introduces optical fibers as building blocks for the experimental realization of
multiport interferometers.
The fourth chapter studies the properties of fiber optical multiports in a
classical multipath interferometer. A three-path interference experiment reveals
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the typical features of multipath interferometry. In another experiment, entangled
photon pairs from the spontaneous parametric downconversion process were used
to demonstrate a purely quantum effect, the antibunching of photon pairs at the
output of a multiport. Both experiments demonstrate the use of fiber multiports
for coherent operation on single quanta.
The main part of this work is concerned with the study of time-energy en-
tanglement in two three-path interferometers built with fiber optical multiports.
This pair of quantum interferometers is the first realization of an entangled three-
state system. It is the first multipath experiment to show quantum interferences.
A first test of a Bell inequality for a multistate system is attempted with this
system. Before the final summary, the quantum and classical pictures of the ex-
periment are discussed giving an outlook to new experiments.
Technical details about the experiments, a Mathematica program for the
design of unitary interferometers, some calculations, and photographs of type-II
downconversion light have been included in the appendices.
Chapter 1
Entanglement and Bell’s
inequalities
1.1 Introduction
In their seminal paper of 1935 “Can quantum-mechanical description of physical
reality be considered complete” Einstein, Podolsky, and Rosen wrote:
“If, without in any way disturbing a system, we can predict with
certainty (i.e. with probability equal to unity) the value of a physical
quantity, then there exists an element of physical reality corresponding
to this physical quantity.” [Einstein35]
They proceeded to show that if this apparently innocuous definition of “elements
of reality” is accepted, the description of reality provided by the quantum mechan-
ical wave function must be incomplete. They finally expressed the belief that a
complete description could be found. One possibility to complete the description,
which was not mentioned in the EPR paper, would be to supplement quantum
mechanics with hidden parameters that play the role of the (hidden) positions
and velocities of particles in statistical mechanics.
Schrodinger’s analysis “The present situation in quantum mechanics” of
1935 was partially motivated by the EPR paper. He for the first time introduced
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the concept of ‘entanglement’.
“Maximal knowledge of a total system does not necessarily include
total knowledge of all its parts, not even when these are fully separated
from each other and at the moment are not influencing each other at
all. . . . If two separated bodies, each by itself known maximally, enter a
situation in which they influence each other, and separate again, then
there occurs regularly that which I have just called entanglement of
our knowledge of the two bodies.”1 [Schrodinger35]
Schrodinger interpreted the wave function as a description of our knowledge
of the quantum system. He could not refute the EPR argument, but noticing that
quantum states can be ‘entangled’ he pointed to the principal difference between
the quantum world and the world of classical physics. His famous cat paradox
illustrates this in a very drastic way.
The particular case studied in the EPR paper consisted of two quanta which
have interacted at some time. These particles are in an entangled state, i.e. the
properties of the whole system are well defined, but the properties of the indi-
vidual particles are not. The joint state has a constant total momentum and the
center-of-mass is well defined. The measurement of the position of one particle
defines the position of the other. The measurement of the momentum of the other
particle establishes the momentum of the first. Thus one can apparently measure
both the position and momentum of one single particle, in contradiction to the
fact that these are noncommuting observables.
1In his famous review “The present situation in quantum mechanics”(1935)
Schrodinger did not use the word entanglement. He used the German word‘Verschrankung’, which in English would be better rendered as entwinement.
The original Version is: “Maximale Kenntnis von einem Gesamtsystem schließtnicht notwendig maximale Kenntnis aller seiner Teile ein, auch dann nicht, wenn
dieselben vollig voneinander abgetrennt sind und einander zur Zeit gar nicht bee-influssen. . . .Wenn zwei getrennte Korper, die einzeln maximal bekannt sind, in
eine Situation kommen, in der sie aufeinander einwirken, und sich wieder tren-
nen, dann kommt regelmaßig das zustande, was ich eben Verschrankung unseresWissens um die beiden Korper nannte.”
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Bohr, the founder of the “Copenhagen interpretation” of quantum mechan-
ics, contested the applicability of the EPR definition of elements of physical reality
to the case presented. He pointed out that the actual procedure of measurement
has a profound influence on the definition of ‘elements of reality’. One should not
speak of reality without completely describing the measuring instruments which
establish this reality [Bohr35]. The ‘complementarity principle’ says that different
measurement instruments are required to measure the values of noncommuting
observables. The observable for which no measurement has been performed has
no ‘reality’.
The discussion about ‘elements of physical reality’ was for a long time con-
sidered a matter of interpretation. Physicists were more concerned with the ap-
plication of the mathematical tools provided by quantum mechanics to real-world
experiments. The interpretation of quantum mechanics was considered to be a
philosophical problem outside of physics.
In fact, Von Neumann’s proof that quantum mechanical probabilities cannot
be interpreted as an average over hidden parameters was considered definitive for
many years [vonNeumann32]. Only in the sixties Bell showed that Von Neumann’s
assumption were to restrictive.
Bohm, in the early fifties, proposed a system of hidden parameters. His
quantum potential, in which the quantum particles move according to determin-
istic equations of motion, is determined by the experimental setup [Bohm66].
However, the quantum potential for entangled particles is nonlocal.
Bohm also proposed a slightly modified version of the EPR gedanken exper-
iment involving two spin-h/2 particles. The two particles are initially in a singlet
state with total spin zero. They separate in such manner that the total spin is
conserved. Then measurement of the spin of one particle along a certain direction
will completely determine the spin of the other particle along the same direction.
The argument of EPR can then be applied to this simple case [Bohm51, Bohm52].
Hope that a local hidden variable theory of quantum physics could be found
were questioned by the 1964 paper of Bell [Bell64, Bell66, Bell87]. He derived an
inequality with minimal assumptions on the form of hidden variables involved.
The predictions of quantum mechanics for the singlet system of two spin-h/2
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particles violate this simple inequality! The contradiction between predictions of
the local hidden variable model and quantum theory could, in principle, be tested
experimentally.
1.2 Derivation of Bell’s inequality for di-
chotomic variables
Here we will present a simple derivation of the Bell inequality based on argu-
ments about sets as published by Wigner [Wigner70]. This proof lends itself to
generalizations to correlated systems of more than two states.
In our version of Wigner’s proof we consider a source emitting individual
pairs of particles into different directions. On each side there is an analyzer with a
variable setting and two outputs. Two detectors register particles at the outputs.
In the following we will simply regard this analyzer as a ‘black box’. In real
experiments, if the particles are polarized photons, this could be a polarizing
beam splitter with detectors at the two outputs. If we have spin-h/2 particles
with an associated magnetic moment, we could use a Stern-Gerlach magnet as
an analyzer. The variable setting then would be the angular orientation of the
analyzer.
S+
-ϕ +
-ϕ'
Figure 1.1: Schematic of a Bell inequality experiment. The source emitsparticles that have two states. Analyzers on each side are described byblack boxes with knobs ϕ, ϕ′ that can be set to different values α, β, γ.The two results are labeled (+) and (−).
The result or outcome of the measurement on one particle can have only
two values. It is a dichotomic variable. We call the result ‘detection in the upper
detector’ plus (+) and ‘detection in the lower detector’ minus (−). Now we con-
sider three settings of the analyzer parameters on each side, that is α, β, γ on one
11
side and α′, β ′, γ′ on the other (see Fig. 1.1). Thus there are nine combinations of
settings αα′, αβ ′, . . . , γγ′. The unprimed greek letters refer to the analyzer on
one side and the primed letters to settings on the other side. The four possible
results ++,+−,−+,−− in our experiment are indicated by the position, eg.
(+− refers to an result + on the one side and − on the other side). We now
assume that hidden parameters determine the outcome of the measurement for
each particle pair. The pairs can thus be sorted into
#(all possible results)#(all possible settings) = 49 (1.1)
subsets, each of these described by a set of hidden parameters.
If the analyzers on both sides are well separated we can assume that the
setting on one side will not affect the setting and/or result on the other side. This
is known as the locality assumption. The number of subsets is then substantially
lower. The total number of subsets of hidden parameters is the product of the
number of subsets on each side (two results and three settings on each side):
#(results)#(settings)#(results′)#(settings′) = 23 23 = 26. (1.2)
If in the experiment we find pairs of settings αα′, ββ ′, and γγ′ for which
there are always perfect correlations, i.e. if we register a count in the (+)-detector
on one side we always observe a count in the (+)-detector on the other side, we
find that many of the subsets in eq. (1.2) are empty. For example, a set with
hidden parameters predicting a result (+) on one side and (−) on the other for
settings α and α′ must be empty. Because of perfect correlations for some settings
many results never occur. In fact there are only eight nonempty subsets.
For each particle individually we attempt to assign elements of physical
reality to the property measured. Hidden variables describing this property will
determine the result of the measurement for every analyzer setting.
A Bell inequality can be derived by simple set theory and illustrated using
Venn diagrams [Espagnat79]. We choose an equivalent representation in three-
dimensional space (see Fig. 1.2). The whole set is represented as a cube. The
large cube is divided into eight smaller cubes representing the subsets. Since we
consider perfect correlations, it is sufficient to label the subsets by the analyzer
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settings α, β, γ and the results +,− on one side. In our specific experiment
each particle will carry at least three bits of information. The first bit will be (+)
or (−) indicating the result for an analyzer setting α; the second and third bits
determine the result for the settings β and γ. We will write this hidden variable
information as a triple of signs, eg. (+ +−). The number of particles having a
specific hidden variable information is indicated by a # sign.
+ −+
−
+
−
− + −+ + −
+ − +
+ + +
+ + +
β
αγ
Figure 1.2: Cube representing hidden variables assigned to classes of par-ticles. The hidden variables determine the results for measurements atanalyzer settings α, β, and γ on one side, and, because of perfect cor-relations, for the corresponding settings α′, β ′, and γ′ on the other side.Quantum mechanics predicts a violation of Bell’s inequality for particlesthat are in the shaded subset of the cube.
The number of pairs giving a result (+) for setting α and (+) for setting β ′
is indicated by the notation N++(α, β ′). This number is equal to the number of
elements in the sets with hidden parameters settings (+ + +) and (+ + −), i.e.the sum of the number of particles with hidden variables determining result (+)
and the result (−) for the setting γ:
N++(α, β ′) = #(+ + +) + #(+ +−). (1.3)
Similar arguments can be found for two other sets:
N++(α, γ′) = #(+ + +) + #(+−+), (1.4)
N+−(β, γ′) = #(+ +−) + #(−+−). (1.5)
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By combining the equations (1.3, 1.4, 1.5) we can deduce an inequality for two
particles with dichotomic variables:
N++(α, β ′) ≤ N++(α, γ′) +N+−(β, γ′). (1.6)
The Bell inequality in Wigner’s paper can be derived in the same way. The
inequality
N+−(α, γ′) ≤ N+−(α, β ′) +N+−(β, γ′) (1.7)
can be easily confirmed by looking at Fig. 1.2.
We now will compare the predictions of quantum mechanics with this simple
statement about sets of objects. We assume the state emitted by the source is an
entangled state
|Ψ〉 = 1√2(|H〉1|H〉2 − |V 〉1|V 〉2) . (1.8)
This state is nonfactorizable. The first term in the sum indicates that if the
polarization in one direction for one particle is found to be horizontal the polar-
ization of the other particle in the same direction is also horizontal. The second
term predicts the same correlation for vertical polarizations. If the polarization
of only one particle is measured along any axis it is found to be horizontal with
probability 1/2.
The expectation value for the relative frequency of pairs of results for set-
tings α on one side and β ′ on the other are sinusoidal functions of the differences
of the angular settings of the analyzers:
N++QM(α, β
′) =1
2cos2(α− β ′), (1.9)
N+−QM(α, β
′) =1
2sin2(α− β ′).
If we choose the angles α = α′ = 0, β = β ′ = 30, and γ = γ′ = 60 theQM predictions are
N++QM(α, β
′) =3
8, (1.10)
N++QM(α, γ
′) +N+−QM(β, γ
′) =2
8. (1.11)
The predictions of quantum mechanics violate the Bell inequality (1.6)!
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1.3 Tests of Bell’s inequality
1.3.1 First experiments
The derivation of the Bell inequality of the previous section is based on, exper-
imentally never realizable, perfect correlations between detection events. Exper-
imental facts, such as detectors which are never 100% efficient or particles not
collected by the apparatus, were not considered in the derivation of (1.6).
In 1969 Clauser, Horne, Shimony, and Holt (CHSH) showed that an inequal-
ity could be derived without the assumption of perfect correlations [Clauser69].
Bell used this proof in his 1971 review [Bell71]. The derivation in these papers
gives an inequality for the expectation values E of the observables ‘count in the
upper channel’ (value +1) and ‘count in the lower channel’ (value −1) as functionof the analyzer settings (α, β on one side and α′, β ′ on the other). One of several
equivalent forms of the CHSH inequality is
|E(α, α′) + E(α, β ′) + E(β, α′)−E(β, β ′)| ≤ 2. (1.12)
The CHSH inequality gives an experimentally verifiable constraint on determinis-
tic local hidden-variables theories. Experimental tests of the CHSH inequality are
based on the fair-sampling assumption, i.e. that the subensemble of pairs from
the source that are actually detected is independent of the settings of the analyz-
ers. In real experiments the total number of particles may be very large and the
subensemble detected very small. This led Clauser and Horne to the derivation
of an inequality based on locality and realism in which only the ratios of particle
detection probabilities appear [Clauser74, Clauser78].
The Clauser-Horne inequality states that the count rates for pair detections
with settings on both sides Rc and the singles count rates on each side Rs, Rs′
must obey the (CH) inequality:
Rc(α, α′) +Rc(α, β
′) +Rc(β, α′)− Rc(β, β
′)− Rs(α)−Rs′(α′) ≤ 0. (1.13)
The CH inequality, just as the inequality derived by Wigner, has the beauty of
dispensing with the arbitrary assignment of values to the results, in our case the
numbers +1 and −1.
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The first experimental tests of Bell’s inequalities were performed by Freed-
man and Clauser using (J = 0 → J = 1 → J = 0) atomic cascade decays. The
emitted photons show strong spin anticorrelation. The results violated Bell’s in-
equality and confirmed the quantum-mechanical predictions [Freedman72]. How-
ever, additional assumptions were required. The no-enhancement assumption rea-
sonably states that the number of counts with the analyzers in place is less than
the number of counts with the analyzers removed. The second reasonable assump-
tion implied is that there is no unknown action-at-a-distance between source and
analyzers, and between the analyzers.
Other experiments were performed with photons from atomic cascades and
from positronium annihilation. All but one showed results that violated Bell’s
inequality and confirmed the quantum-mechanical predictions. The best-known
experiments were done by Aspect, Grangier, and Roger [Aspect81]. A detailed
review of theory and experiments with references can be found in the excellent
article of Clauser and Shimony [Clauser78].
The experiments performed so far have not excluded the possibility of
action-at-a-distance in the relativistic sense. The acts of setting the analyzers
were not space-like separated from each other. The most quoted Bell inequality
experiments performed by Aspect, Grangier, and Dalibard used fast switching of
the analyzer positions to prevent communication between source and analyzers
[Aspect82]. However, it has been pointed out that periodic switching were not
random and thus the analyzer settings were predictable after a few periods of the
switch [Zeilinger86]. Preparations for a truly randomly switched experiment are
underway in Innsbruck [Weihs96c].
Local-realists have always been disturbed by the fair-sampling assumption
[Marshall83]. They will look forward to the results of new experimental tests with
high-efficiency detectors [Fry95, Freyberger96].
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1.3.2 Entanglement and the parametric downconversion
source
Entanglement is the necessary essence of all Bell inequality experiments. The
particles in the original EPR proposal, the spin-1/2 particles of the Bohm version,
the photon pairs from of the atomic cascade and the positronium annihilation
experiments all share this property. It is the existence of entangled states in
quantum physics which makes the quantum world fundamentally different from
the everyday world.
A beautiful source of entangled photon pairs was discovered in the process of
parametric downconversion (PDC). The quantum properties of this source were
first studied experimentally by Burnham and Weinberg [Burnham70]. In PDC
a photon of frequency ω0 propagating in a suitable medium spontaneously di-
vides its energy between two photons of smaller energy hω1 and hω2. The photon
pairs emitted from the medium show strong correlations in energy and time. The
quantum theory of the PDC process will not be discussed here. Extensive treat-
ment can be found in many publications [Mollow73, Hong85, Joobeur94]. Crystals
with a large nonlinear susceptibility are currently used as sources for PDC photon
pairs. In our experiments we used KDP, LiIO3, and BBO (see Appendix A).
With the advent of PDC sources many degrees of freedom are available to
the experimentalist. Photons from a PDC source are entangled in momentum,
energy, and, for special configurations and crystals, in polarization state.
The first Bell inequality experiments using PDC photon pairs used polar-
ization entanglement when two states from the crystal are overlapped on a beam
splitter. The measurements by Shih and Alley confirmed the quantum predictions
and showed visibilities prohibited by Bell’s inequalities [Alley87, Shih88].
A totally different test of Bell’s inequalities based on phase and linear mo-
mentum was first proposed by Horne and Zeilinger [Horne85, Horne89]. This test
was experimentally realized by Rarity and Tapster using photons from a PDC
crystal [Rarity90]. Momentum entanglement has lead to a plethora of interesting
experiments when two crystals are used. In an experiment described by Green-
berger as a ‘mind-boggler’, the overlap of the modes of two PDC crystals led to in-
17
terferences in the singles-rates that could be modulated by the phases and the de-
gree of fundamental distinguishability [Zou91]. Experiments in Innsbruck demon-
strated that the emission of PDC photons could be suppressed and enhanced by
the imposition of external boundary conditions [Herzog94a, Herzog94b]. Further-
more, it was shown that two-photon interference from a PDC crystal and its
mirror image could be recovered when welcher-weg information was destroyed,
thus realizing a quantum eraser [Herzog95]. An overview of two-particle interfer-
ometry with references can be found in [Horne90].
The narrow time window in which the photon pairs from the PDC source
are emitted has led to the proposal by Franson of two-photon interference
with energy-entangled photon pairs [Franson89] and was measured by various
groups [Kwiat90, Ou90, Brendel92, Kwiat93]. This entanglement seems particu-
larly suited for a fiber-optical experiment. Tapster, Rarity, and Owen have im-
plemented a fiber-interferometric version of this experiment [Tapster94]. In the
present thesis time-energy entanglement in fiber interferometers is used for the
first realization of a multipath two-photon interference experiment.
To the surprise of the quantum optics community Bell’s theorem and two-
photon entanglement have already born a practical application. Ekert has pro-
posed a version of quantum cryptography based on Bell’s theorem [Ekert91].
Experimental realizations in optical fibers have already been demonstrated over
long distances in the laboratory [Rarity92, Rarity93, Tapster94].
1.3.3 New experiments with entangled photons
A new high-intensity source for polarization entangled photons has recently been
characterized. A violation of Bell’s inequalities by 100 standard deviations for
a measurement time of only five minutes was demonstrated with this source in
Innsbruck [Kwiat95]. The first photographs of PDC light from a type-II source
are presented in Appendix E of the present thesis.
New directions of research in the field of entangled photons include the
transfer of entanglement between individual pairs of quanta (entanglement swap-
ping) [Zukowski93]. The transfer of a quantum state by use of the EPR cor-
relations of entangled photons from a PDC source (quantum teleportation)
18
[Bennett93] and the realization of quantum dense coding (more than one bit
is encoded into one photon) [Bennett92, Mattle96].
1.4 Bell inequalities for three-valued observ-
ables
Two generalizations of two-state two-particle entanglement are possible. The
first is the entanglement of more than two particles. Greenberger, Horne, and
Zeilinger (GHZ) have shown that an entangled state of three two-state particles
leads to a contradiction with local realistic theories in a single experimental run
[Greenberger89, Greenberger90]. This dramatic result has lead to the search for
sources of entangled triples. However, no sufficiently bright source has yet been
realized.
The other generalization that can be realized is the entanglement of more
than two states of two particles. The theorist will immediately think of pairs of
spin > h2particles. Bell inequalities for such higher-dimensional spin systems have
been analyzed by Garg, Mermin, and Peres. Quantum mechanics is in contradic-
tion to local hidden variable theories even for large spin [Mermin80, Garg82].
But sources of correlated spin-Nh/2 particles have not yet been experimentally
realized.
Entangled higher-dimensional spin systems could lead to new test of quan-
tum physics. The Kochen-Specker theorem gives a mathematical proof of the
impossibility of assignment of elements of physical reality to unmeasured proper-
ties of a spin-1 system [Kochen67, Mermin93]. The proof of the Kochen-Specker
theorem is based on the assumption of noncontextuality of value assignment for
commuting observables, i.e. the results of measurements are the same irrespective
of the context of measurement. For example, the total angular momentum of a
particle J2 = J2x +J2
y +J2z , Jx, the angular momentum in x-direction, and Jy, the
angular momentum in y-direction are observables. J2 and Jx commute, so we can
measure J2 together with Jx. We also notice that J2 and Jy commute. So we can
measure J2 together with Jy. The assumption of noncontextuality implies that
the results of the measurement of J2, be it alone, together with Jx, or together
19
with Jy, should be the same. Kochen and Specker showed that noncontextual-
ity together with the assumption that the values assigned to the operators have
the same algebra as the operators themselves leads to a contradiction. There-
fore the assignment of values to unmeasured observables is not allowed. In Peres’
words “unperformed experiments have no results” [Peres78]. Bell’s requirement
of locality implies noncontextuality of value assignment to the observables of the
individual particle [Redhead87, Peres93].
Entanglement of more than two states without using spin systems can be
realized experimentally by using momentum states of the PDC crystal [Horne85,
Horne86, Horne88, Zeilinger93b] together with the generalization of the beam
splitter to higher dimensions, the multiport [Zeilinger93a]. Multiports allow the
generalization of EPR correlations to higher dimensional Hilbert spaces, thus
realizing optical equivalents of entangled higher-dimensional spin systems. An
overview with references can be found in [Greenberger93]. In this thesis we will
present the first experimental realization of an entangled three-state system (cf.
Chapter 5). First, we will derive a version of Bell’s inequalities for this correlated
three-state system.
S+
-o
Φ
(φ φ )1, 2 (φ φ )' '1, 2
+
-o
Φ'Figure 1.3: Schematic of a source for a three-state Bell inequality experi-ment. Analyzers on each side are described by black boxes with a knob thatcan be set to different values (described by two parameters Φ = φ1, φ2).The three results are labeled (+), (), and (−).
Wigner’s derivation of Bell’s inequality can be readily generalized to non-
dichotomous variables. For the following discussion we will assume three-valued
observables. Thus, the analyzer used to study the EPR pairs has three output
channels. We can again dispense with the physical details of the analyzer and
assume it is a black box with a number of parameter settings (see Fig. 1.3).
20
The black box has an input for the three-state particle to be analyzed and three
outputs, which for the convenience of the following discussion, we will label plus
(+), minus (−), and zero (). This labelling is completely arbitrary, but reminds
us of a possible physical realization: the three outputs of a Stern-Gerlach analyzer
for spin-1 particles. In the following experiments we will use our PDC source. The
physical realization of the three valued observables are clicks at three different
detectors.
We assume a source emitting pairs of particles that travel to spatially sep-
arated analyzers. These particles behave in such a way that for given settings of
the analyzers we have perfectly correlated results, i.e. we find that given pairs of
detectors always fire together. We can try to assign elements of physical reality
to this observation. Then we can ask the question what happens when other set-
tings of the analyzer are chosen. In close analogy to Wigner’s discussion we choose
three settings on each side. Since the analyzers in the multidimensional case will
generally have more than a single parameter that can be set, we describe a given
set of parameters of one analyzer by a capital letter A,B,C and of the other
analyzer by the A′, B′, C ′. We assume the parameter settings described by the
primed and unprimed letter are perfectly correlated. The hidden parameter space
of particle pairs emitted from the source can be divided into
#(all possible results)#(all possible settings) = 99 (1.14)
subsets, each of these described by a hidden parameter. Since we assume locality,
i.e. the settings of one analyzer will not affect the settings of the other, the number
of nonempty subspaces of the parameter space immediately reduces to
33 33 = 36 (1.15)
domains. Perfect correlations between results for analyzer settings A−A′, B−B′,and C − C ′ further reduce the number of nonempty domains of hidden variable
space. We are left with 27 nonempty subsets of the hidden variable space. These
can be conveniently illustrated by subdividing a cube into 27 smaller cubes (see
Fig. 1.4).
A Bell inequality can be derived by simply counting the number of elements
in the shaded sets in Fig. 1.4. One Bell inequality for two particles with three
21
++
−
−
−
+
ο
ο
ο
+ ο ο
+ − ο
+ + −
+ − +
+ ο +
ο + −
− + −
+ + + + + ο + + −
+ + + + + ο
B
A
CFigure 1.4: Hidden variable assignment defines subsets of the ensembleof particle pairs with three-valued measurement outcomes. Only 27 do-mains of the hidden variable space are nonempty. The hidden variablesdetermine the results for measurements at analyzer settings A, B, andC on one side, and, because of perfect correlations, for the correspondingsettings A′, B′, and C ′ on the other side. The hidden variables determinethe outcome of a detection event +,−, as function of the setting.The quantum mechanical prediction for particles that are members of theshaded subsets violate a Bell inequality derived in the text (1.16).
states is given by
0 ≤ N++(A,C ′) +N+(B,C ′) +N+−(A,C ′)−N++(A,B′). (1.16)
Do the predictions of quantum mechanics violate this inequality? For the
moment we will simply write down the quantum prediction for a correlated three-
state system. It will later be derived. The prediction for three pairs of detection
events (xy = (++), (+), (+−)) is given by
NQMxy ∝ 1
9[3 + 2 cos (φ1 + φ′1 + χ) (1.17)
+ 2 cos (φ2 + φ′2 − χ)
+ 2 cos (φ2 + φ′2 − φ1 − φ′1 + χ)]
22
with the fixed parameter χ = 2π3,−2π
3, 0. The number of counts is proportional
to the sum of a constant term and three cosines with phases that are sums of the
parameters of the spatially separated analyzers.
Perfect correlations arise for the settings in the following table:
Perfect correlations
detectors settings
++, −, − φ′1 = −φ1 − α φ′2 = −φ2 + α
, +−, −+ φ′1 = −φ1 + α φ′2 = −φ2 − α
−−, +, + φ′1 = −φ1 φ′2 = −φ2
with α ≡ 2π3. The Bell inequality in eq. (1.16) is violated by the function in
eq. (1.17) as can be easily confirmed by inserting the following settings for the
analyzers (γ ≡ π5):
(φ1, φ2) (φ′1, φ′2)
A = (2γ, γ) A′ = (−2γ − α,−γ + α)
B = (−2γ,−γ) A′ = (2γ − α, γ + α)
C = (0, 0) C ′ = (−α,+α)
The value of the right hand side of the inequality for these settings is −0.2322411which is smaller than zero. The coincidence probability postulated in eq. (1.17)
violates our Bell inequality.
In the following chapters we will first discuss the theory of linear multiports
used for the creation and analysis of multistate entanglement. We will then find
a physical system that realizes a coincidence probability as given by the quan-
tum mechanical function (eq. 1.17). Then we will describe an experiment using
fiber optical multiports in a multipath interferometer that is the first physical
realization of a non-sinusoidal quantum mechanical coincidence probability.
Chapter 2
Theory of linear multiports
The beam splitter is the central element in many experiments in classical and
quantum optics. It can be viewed as a 4-port device, a black box with two inputs
and two outputs. The operation of a 4-port device can be formally described by
a unitary transformation of the two input modes into the two outputs modes
[Zeilinger81, Yurke86, Ou87, Prasad87, Fearn89, Campos90]. Linear multiports
are a generalization of the beam splitter.
A quantum system with N discrete states is described by a vector |Ψ0〉 in a
N -dimensional Hilbert space. This vector can be transformed into another vector
|Ψ1〉. The transformation is described by a matrix U . For a lossless system the
transformation must conserve probability:
||Ψ1〉|2 = |U |Ψ0〉|2. (2.1)
This condition is fulfilled if the matrix is unitary, that is the inverse of the matrix
is equal to its transposed complex conjugate:
U−1 = (U∗)T. (2.2)
Thus unitary matrices describe the transformation between states in the discrete
Hilbert space.
23
24
2.1 The beam splitter
A simple two-state system can be realized by two modes of the radiation field.
Any transformation between these modes can be performed using a beam splitter.
If we choose the input states as a basis of the Hilbert space we can write this
transformation as a matrix. The beam splitter matrix transforms the input state
with modes (k1, k2) into the output state with modes (k′1, k′2). We will use the
following representation of the beam splitter matrix: k′1
k′2
=
sinω eiφ cosω
cosω −eiφ sinω
k1
k2
. (2.3)
The parameter ω describes the reflectivity (R = cos2 ω) and transmittance (T =
sin2 ω) of the beam splitter. The parameter φ describes the relative phase between
the input modes. It can be realized as an external phase shifter before the beam
splitter.
ω
ϕ
k1
k2 k'1
k'2
Figure 2.1: The beam splitter coherently transforms two input modes(k1, k2) into two output modes (k′1, k
′2). The parameter ω describes the
reflectivity and transmittance of the device. The phase ϕ shifts the relativephase between the inputs.
Phases at the output ports can be chosen so that the beam splitter matrix
performs any transformation in U(2) [Yurke86, Danakas92]. A beam splitter with
variable reflectivity can be substituted by a Mach-Zehnder interferometer using
symmetric 50:50 beam splitters (see Fig. 2.2).
25
Τ=1/2 Τ=1/2
k1
k2-ω
ω
ϕ
k'1
k'2Figure 2.2: Mach-Zehnder realization of a beam splitter with variable re-flectivity. The internal phase ω changes the reflectivity of the 4-port de-vice. The phase ϕ shifts the relative phase between the inputs. The in-sertion of two internal phases (ω,−ω) simplifies the matrix of the Mach-Zehnder interferometer.
2.2 Multiports
The operation of a beam splitter or 4-port device on two input modes has been
described above. This concept is easily generalized to 2N -port devices. Such mul-
tiports coherently transform N input modes into N output modes (see Fig. 2.3).
We will see that multiports present a convenient optical realization of higher-
dimensional systems.
1
32
N
1'
3'2'
N'
U(N)
Figure 2.3: A multiport device consisting of beam splitters, mirrors, andphase shifters can be viewed as a black box. It will be described by a unitarymatrix transforming N input states into N output states.
26
2.2.1 From experiment to matrix
We will now show that an experimental arrangement consisting of lossless beam
splitters, mirrors, and phase shifters can be viewed as a black box transforming N
inputs into N outputs. We can calculate the corresponding unitary matrix using
a method borrowed from microwave circuit design [Dobrowolski91]. The state of
the system at the output is described by the vector of the complex amplitudes of
the output modes (aout). For a 2× 2-port this is a two dimensional vector. For a
general multiport with N outputs this is a N -dimensional vector with complex
elements. It is related to the state at the input through the unitary matrix U :
aout = Uain. (2.4)
The matrix describing the relation between outputs and inputs contains two
different contributions. One describes the transformation at the beam splitters;
the other the propagation between beam splitters. Each input of a beam splitter
may be an input of the whole system or an input that is internally connected to
the output of another beam splitter inside the system. The first are the external
inputs, the other are the internal inputs of the system. The number of external
inputs is equal to the dimension of the system’s unitary matrix.
The matrix describing the experimental setup can be built from block matri-
ces describing the transfer from internal and external inputs of the beam splitters
to their internal and external outputs. We build a (large) vector of external and
internal output states and relate this to the external and internal input states:
aout(ext)
aout(int)
=
See Sei
Sie Sii
ain(ext)
ain(int)
. (2.5)
The (large) matrix contains four submatrices. The submatrix Sei, for example,
describes the transformation from the internal inputs of the beam splitters to the
external outputs.
We know the topology of the system, i.e. the connections between beam
splitters. The topology can be described by a connection matrix Γ. The elements
of this matrix are the phase shifts internally accumulated by the state evolving
27
between two beam splitters:
aout(int) = Γain(int). (2.6)
Solving equations (2.5) and (2.6) for the external ports gives an simple
expression for the matrix of the whole system:
aout =[See + Sei (Γ− Sii)
−1 Sie
]ain. (2.7)
Thus we have can easily calculate the unitary matrix of the system out of our
knowledge of the topology.
U = See + Sei (Γ− Sii)−1 Sie. (2.8)
This method gives a simple formula for the unitary matrix of a complex interfer-
ometer involving many beam splitters and phases.
A physical interpretation is given by the Feynman rules. Since there is ab-
solutely no way in which one can determine which path a photon has taken inside
a multiport device, we must add up the phases accumulated over all paths inside
the system and then add the probability amplitudes of all the possibilities leading
to the result [Feynman64]. The formalism above automatically accomplishes this.
2.2.2 From matrix to experiment
We have seen that experiments with multiport interferometers can be described
by unitary matrices. We will now show that an experimental setup exists for any
given unitary operator. This allows the experimental realization of any N × N
unitary matrix in the laboratory. We find that lossless beam splitters with phase
shifters at one input port can be used to build any unitary matrix operating on
N input modes. The following proof describes the algorithm used to construct a
multiport from 2× 2 port devices [Reck94]1.
We use beam splitter devices to build any matrix in U(N). They succes-
sively perform U(2) transformations on two-dimensional subspaces of the full
N -dimensional Hilbert space. It should be mentioned that it does not matter
1A reprint of the relevant paper can be found starting page 127.
28
what kind of field such an optical multiport is acting upon. Lossless beam split-
ters for atom deBroglie waves can be realized by suitably arranged laser fields
[Martin88, McClelland93]. Beam splitters for neutrons can be fabricated from
perfect silicon crystals. We choose photons here, but we note that with this gen-
eral scheme it would be equally well possible to realize multiports for electrons,
neutrons, atoms or any other type of radiation.
A unitary matrix operates on a vector of input states. The experiment will
consist of a series of 4-port devices (beam splitters) and phase shifters. We know
that setting up experimental devices in sequence corresponds to successive appli-
cation of the corresponding unitary transformations. In our formalism these can
be described as product of beam splitter matrices. Finding an optical experiment
belonging to an arbitrary unitary matrix is therefore completely equivalent to
factorizing the unitary matrix into a product of block matrices containing only
beam splitter matrices with appropriate phase shifts (cf. eq. (2.3)).
We define a matrix Tpq which is equal to the N -dimensional identity ma-
trix except that the four matrix elements with the indices pp, pq, qp, and qq have
been replaced by the corresponding beam splitter matrix elements. This ma-
trix performs a unitary transformation on a two-dimensional subspace of the N -
dimensional Hilbert space leaving an (N − 2)-dimensional subspace unchanged
[Murnaghan58]. It can be used to successively make all off-diagonal elements of
the given N ×N unitary matrix zero, a method similar to Gaussian elimination.
The unitary matrix U(N) is multiplied from the right with a succession
of two-dimensional unitary matrices TNq(ωNq, φNq) for q = N − 1, . . . , 1. The
experiment is built up by successively attaching the corresponding beam splitter
devices to ports N and q. Once all elements of the last row except the one on the
diagonal are zero, this row will not be affected by later transformations. Since
all transformations are unitary, the last column will then also contain only zeros
except on the diagonal:
U(N) · TN,N−1 · TN,N−2 · · ·TN,1 =
U(N − 1) 0
0 eiα
. (2.9)
29
T21
T31 T32
T31 T32
T32
T21
T31 T32
α1
α2
α3
Figure 2.4: Example for the implementation of the algorithm to constructa 3× 3 unitary matrix. On the left hand two-dimensional unitary trans-forms are performed on two-dimensional subspaces, successively makingthe complex matrix elements (∗) zero. On the right hand side the exper-iment is successively built up from the corresponding beam splitters, rep-resented by four-port devices. In the last step phase shifters are insertedto make the diagonal matrix elements equal to one. The experimentalsetup equivalent to the original unitary matrix is the device in the lastline operated from right to left.
The effective dimension of the matrix U is thus reduced to N − 1.
This sequence of transformations can also be viewed as the rotation of an
N -dimensional vector, the last row (or column) of U(N), in an N -dimensional
vector space. These transformations are therefore the experimental realization of
30
a generalized rotation in N -dimensions:
R(N) = TN,N−1 · · ·TN,1. (2.10)
The sequence of beam splitter transformations or rotations can be applied
recursively to the matrix with reduced dimensions. We note that a beam splitter is
not necessary if a matrix element already happens to be zero. After the final beam
splitter transformation one obtains a diagonal matrix with elements of modulus
one. By attaching appropriate phase shifters, i.e. multiplying with a diagonal
matrix D with elements of modulus one, we can make the resulting matrix equal
to the identity:
U(N) · TN,N−1 · TN,N−2 · · ·T3,2 · T3,1 · T2,1 ·D = I(N). (2.11)
T21†
T31†T32
†
-α1
-α2
-α3
Figure 2.5: Three beam splitter devices T †pq (operated in reverse) and threeadditional phase shifters αi are enough to build any 3×3 unitary matrix.Notice that because of operation in reverse the individual devices’ phaseshifts are now at the input ports and the final phase shifters αi at theoutput ports.
The experimental setup thus built of beam splitters and phase shifters is
equivalent to the inverse of the original N × N unitary matrix. The experiment
operated in reverse, that is taking the output ports as inputs and reversing time
direction corresponds to the transposed complex conjugate of the inverse matrix
and is therefore equivalent to the original unitary matrix:
U(N) = (TN,N−1 · TN,N−2 · · ·T2,1 ·D)−1 . (2.12)
This can be easily confirmed by multiplying the calculated beam splitter and
phase shift matrices. A Mathematica program for the design of unitary inter-
ferometers is presented in Appendix C. The operation of multiports on single-
photon and photon-pair wave packets is described in Appendix D.
31
Figure 2.6: A triangular array of beam splitters implements any N ×N unitary matrix as an optical multiport. The beams are solid lines. Asuitable beam splitter is at each crossing point of the beams. Phase shiftersare at one input of each beam splitter and at the outputs (1′ . . . N ′) of themultiport. Each diagonal row of beam splitters performs a transformationreducing the effective dimension of the Hilbert space by one.
Once all matrices of U(N) can be implemented, it becomes possible in
principle to measure the analog of the observable corresponding to any discrete
Hermitian matrix H in the model representation of N modes of the field. Note
that such a measurement requires, in general, N detectors, one for each of the N
orthogonal normalized eigenstates, which correspond to the N unit eigenvectors
of the matrixH . We denote the eigenvectors as |v1〉 . . . |vN〉 and the correspondingeigenvalues with h1, h2, . . . hN, a set of N real numbers. They do not have to
be different; the case of degenerate eigenvalues can be treated in the same way.
The measurement apparatus makes the first detector fire if the input state
is |v1〉, the second if it is |v2〉, etc. In addition the amplitudes for a general input
vector |ψ〉, whose entries represent the amplitude for a photon to arrive at the
respective input port are 〈v1|ψ〉 to be detected by the first detector, 〈v2|ψ〉 forthe second, and 〈vi|ψ〉 for the i-th. If the i-th detector fires, we assign the mea-
surement value hi to the measurement of H . The unitary matrix U involved in
such a measurement must “sort” the incoming amplitude into N output ports
32
corresponding to the N eigenstates |v1〉 . . . |vN〉. The measurement apparatus cor-responding to the Hermitian matrix H consists of the experimental embodiment
of U and a set of N detectors.
2.3 Symmetric multiports
A special case of the unitary multiport is the symmetric multiport. All its ele-
ments are of the same modulus. A state at one of its inputs leaves the device in a
coherent superposition of all output modes with equal modulus of the amplitude.
If a single photon state enters one input of a symmetric N × N -multiport the
probability of detecting a photon at any output is 1N.
Symmetric multiports that can be transformed into one another by simple
renumbering of inputs and outputs or by including phase shifters at the inputs
and outputs can be considered to be an equivalence class. 2× 2 (beam splitters)
and 3× 3 symmetric multiports have only one equivalence class each.
It is useful to define a generic form for the symmetric multiport. We do this
by requiring all elements in the first row and column to be real valued. Using the
N-th root of unity γN = exp i2πN
[Zukowski94], we can write the matrices for a
symmetric 2N multiport as follows:
U(N)mk =
1√Nγ(m−1)(k−1)N . (2.13)
The beam splitter matrix in this form is
B =1√2
1 1
1 −1
. (2.14)
In analogy with the beam splitter the 3×3 symmetric multiport is called a tritter.The canonical form of the tritter matrix is
T =1√3
1 1 1
1 α α2
1 α2 α
with α ≡ ei 2π/3 (2.15)
33
There are already infinitely many different equivalence classes of symmetric
4 × 4 multiports (quarters) which can be parametrized by an internal phase
[Bernstein74, Mattle95].
2.3.1 Single-photon eigenstates of a symmetric multiport
The eigenstates of a multiport system are those superpositions of input states that
leave the system unchanged except for an overall phase. This reflects the physical
fact that in a lossless system probability must be conserved. The eigenstates of a
lossless device with the matrix M can be found by solving the equation
M |Ψλ〉 = λ|Ψλ〉. (2.16)
The eigenvalues λ of unitary matrices are always of modulus one.
If the input basis states (modes) at the multiports are |k1〉, |k2〉, . . . we cancalculate the single-photon eigenstates of the multiport. For the beam splitter B
in eq. (2.14) we find
|Ψ±1〉 ∝(1±√2)|k1〉+ |k2〉. (2.17)
The tritter T in eq. (2.15) has three single-photon eigenstates:
|Ψ+i〉 ∝ |k2〉+ |k3〉, (2.18)
|Ψ±1〉 ∝(1±√3)|k1〉+ |k2〉+ |k3〉. (2.19)
2.3.2 Two-photon eigenstates of a symmetric multiport
Since multiport interferometers can be used realize any unitary transformation,
any superposition of input modes can be transformed into any superposition of
output modes. Thus any quantum state can be prepared. What will happen when
a superposition of two-photon states impinges on a symmetric multiport device?
For the following discussion we consider input states in the discrete N ×NHilbert space
H = HN ⊗HN . (2.20)
This Hilbert space is spanned by the basis of input modes: |a〉|a〉 ≡ |k1〉 ⊗ |k1〉,|a〉|b〉 ≡ |k1〉⊗ |k2〉, |a〉|c〉 ≡ |k1〉⊗ |k3〉, |b〉|a〉 ≡ |k2〉⊗ |k1〉, . . .. Realistic photon
34
states must be constructed from these states by integration over a frequency
distribution (cf. eq. (D.1)).
Two photons impinging on a linear device will each individually interact
with the device. The beam splitter matrix operating on a two photon state can
therefore be written as the tensor product of the individual beam splitter matri-
ces:
B ⊗ B =1
2
1 1 1 1
1 −1 1 −1
1 1 −1 −1
1 −1 −1 1
(2.21)
The eigenstates of the two-photon beam splitter matrix are given in Ta-
ble 2.1.
λ |a〉|a〉 |a〉|b〉 |b〉|a〉 |b〉|b〉
+1 1 1 1 −1
1 0 0 1
−1 1 −1 −1 −1
0 1 −1 0
Table 2.1: Eigenstates of the two-photon beam splitter matrix. The
eigenspace for each eigenvalue λ = ±1 is two dimensional. The table
gives the coefficients of the eigenvectors in the |a〉|a〉 . . . |b〉|b〉 basis.
We can define a new basis of the two-photon Hilbert space using these
eigenstates:
|Ψ±〉 =1√2(|a〉|b〉 ± |b〉|a〉) , (2.22)
|Φ±〉 =1√2(|a〉|a〉 ± |b〉|b〉) . (2.23)
These maximally entangled basis states are called Bell states and play an im-
portant role in quantum coding and communication [Bennett95, Mattle96]. We
35
immediately see that the eigenstates of the system in terms of Bell states are:
|Φ+〉, |Φ−〉+ |Ψ+〉, |Ψ−〉, and |Φ−〉 − |Ψ+〉.
The calculation of eigenstates for a two-photon state on a tritter becomes
tedious. For sake of completeness the eigenstates for two photons on a tritter are
given in Table 2.2.
λ |a〉|a〉 |a〉|b〉 |a〉|c〉 |b〉|a〉 |b〉|b〉 |b〉|c〉 |c〉|a〉 |c〉|b〉 |c〉|c〉
+1 4 1 1 1 1 1 1 1 1
0 1 1 1 −1 −1 1 −1 −1
−1 2 −1 −1 −1 −1 −1 −1 −1 −1
0 1 1 −1 −1 1 −1 1 −1
0 1 1 −1 1 −1 −1 −1 1
+i 0 1 +√3 −1−
√3 0 1 −1 0 1 −1
0 0 0 1 +√3 1 1 −1−
√3 −1 −1
−i 0 0 0 1−√3 1 1 −1 +
√3 −1 −1
0 1−√3 −1 +
√3 0 1 −1 0 1 −1
Table 2.2: Eigenstates of the two-photon tritter matrix. The table gives
the coefficients of the eigenvectors in the |a〉|a〉 . . . |c〉|c〉 basis for each
eigenvalue λ.
2.4 Multiports and quantum computation
Quantum computation encodes information in superpositions of two states |a〉and |b〉. The general form of a quantum bit (Q-bit) is
|Q〉 = α|a〉+ β|b〉, |α|2 + |β|2 = 1. (2.24)
The state |a〉 is identified with logical zero and the state |b〉 with logical one.
Thus Q-bits can encode superpositions of zero and one.
36
Linear multiports can be used to prepare any superposition of states, in
fact to transform any Q-bit into any other we can use a device with the following
simple 4-port matrix: sinω eiφ cosω
cosω −eiφ sinω
. (2.25)
We recognize the matrix of a simple variable reflectivity beam splitter with the
parameter ω determining the contribution of zero and one.
The basic building blocks of a quantum computer are the quantum logic
gates. The simplest nontrivial gate is the controlled-not gate. It operates on two
Q-bits, flipping the state of one Q-bit if the other Q-bit is in state |b〉. Theoperation of the controlled-not gate can be written as a simple matrix in the
|a〉|a〉, |a〉|b〉, |b〉|a〉, |b〉|b〉 basis:
C =
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
. (2.26)
It can be formally shown that with operations on single Q-bits and controlled-not
gates any quantum logic gate can be built [Barenco95].
In order to determine if linear multiports can be used to build a controlled-
not gate operating on two photons, we assume we already have such a gate.
Then M and M ′ are the multiport matrices operating on the first and second
photons respectively (M and M ′ may be the same). Since each photon interacts
individually with the linear multiport the controlled-not matrix must be written
as a tensor product of the multiport matrices:
C =M ⊗M ′. (2.27)
Inserting eq. (2.26) we have a system of 16 equations for the 8 matrix elements
of M and M ′. The equations have no solution as one can see from the three of
the equations:
1 = (M)22(M′)11, (2.28)
37
1 = (M)21(M′)22, (2.29)
0 = (M)22(M′)22. (2.30)
The first two equations require that the elements (M)22 and (M′)22 be of modulus
1. The last equation requires that one of these elements be zero. This is a contra-
diction. Therefore a controlled-not gate can not be built from linear multiports.
Linear unitary multiports can be used for state preparation of single quanta
in quantum communication and quantum computation. They provide a handy
formalism for the description of complicated interferometric experiments and can
be easily generalized to describe multifrequency radiation. In the next chapter
we will see how multiports can be realized using optical fibers.
38
Chapter 3
Optical fibers
Optical fibers are in wide spread use as the main carriers of high-speed earth-
based communication networks. Extensive studies have been made of the use of
fibers in communications technology but little research has been published on the
use of optical fibers in experiments on the foundations of quantum mechanics.
A notable exception is the group at DRA, Malvern. Rarity and Tapster
have measured nonclassical interferences in fiber interferometers [Rarity92]. The
aim of their research has been to demonstrate a quantum cryptography scheme
based on Bell’s inequalities. By using the optical fibers they were able to achieve
a violation of a Bell inequality over a 4 km long optical fiber on a spool in the
laboratory [Rarity93, Tapster94].
Before going into the characterization of the fiber optical multiports used
in our experiments we will summarize the properties of fiber optical components
used in our fiber interferometers.
3.1 Single mode fibers
The simplest form of a fiber-optical waveguide is a step-index fiber, a core of glass
enclosed by a cylindrical region of lower refractive index. The theory of optical
waveguides can be treated without recourse to quantum mechanics. In the limit of
geometrical optics we can say that light propagates in the fiber by total internal
39
40
reflection at the interface between core and cladding. The numerical aperture
(NA) is an important parameter of optical fibers. For a step-index fiber, it can
be calculated by raytracing. The result is
NA =√n2core − n2cl . (3.1)
The same expression applies to any axis-symmetric graded-index fiber if we as-
sume the ray is incident at the center of the fiber core and the ncl is the outer
cladding index. Typical values for the numerical aperture are in the range 0.1 to
0.4.
As the diameter of the core is reduced to sizes comparable to the wavelength
of the light, the simple geometrical picture becomes inadequate. A wave descrip-
tion must be made. The problem is equivalent to finding solutions of Maxwell’s
equations under cylindrical boundary conditions [Snyder91]. The solutions are
expressed in terms of longitudinal and transverse eigenmodes of the electric field
in the fiber. The transverse modes can be described by Bessel functions. It is
convenient to introduce a dimensionless normalized wave number (V-number)
V ≡ 2πr
λ0NA (3.2)
that gives the radius of the core r in units of wavelength λ0. If V < 2.405, the first
zero of the zero-order Bessel function, only the fundamental mode can propagate
in the fiber. The fiber is said to be in the singlemode regime. In our interferometric
experiments we used only singlemode fibers. They have the obvious advantage
that the mode of the light field is well defined by the geometry of the fiber.
Singlemode fiber optical components, including fiber-optical multiports, are also
readily available. Today about 95% of all fibers manufactured in the world are
singlemode [Newport].
The use of optical wave guides in telecommunication and interferometry
is somewhat complicated by the experimental fact that light propagating in a
material will suffer dispersion. In the following section we will address this effect.
3.1.1 Optical parameters of fused silica
In order to estimate the effect of material dispersion in fiber-optical interference
experiments, the optical parameters for fused silica, the basic material of fibers,
41
will be determined. The refractive index is the parameter most suited for the
experimental characterization of optical materials. A semi-empirical formula for
the dependence of the refractive index on the wavelength is given by the Sellmeier
formula for transparent materials. It assumes the wave length of the light to be
far from the optical resonances λj:
n(λ) =
√√√√1 +∑j
Aj λ2
λ2 − λ2j. (3.3)
For fused silica, the material used for optical fibers, the parameters of the
Sellmeier formula are [Malitson65]:
j Aj λj[µm]
1 0.6961663 0.0684043
2 0.4079426 0.1162414
3 0.8974794 9.8961610
This dependence is plotted in Fig. 3.1.
The refractive index gives the relation between the wave vector in the ma-
terial and the vacuum wave vector
k(ω) = n(ω) kvac = n(ω)w
c. (3.4)
When we have narrow bandwidth light with center frequency ω0 we can use the
Taylor expansion of n around ω0:
n(ω) = n(ω0) +∂n
∂ω
∣∣∣∣∣ω0
(ω − ω0) +1
2
∂2n
∂ω2
∣∣∣∣∣ω0
(ω − ω0)2. (3.5)
The first term of the Taylor expansion defines the group velocity vg. It describes
the propagation of the envelope of the pulse. The second term describes the
broadening of a pulse in dispersive media. It is the group velocity dispersion
(GVD) coefficient.
The refractive index is usually given as a function of wavelength, then the
group velocity and group index are
vg(λ) =c
ng(λ), (3.6)
ng(λ) = n(λ)− λ∂ n
∂λ. (3.7)
42
0.6 0.8 1.0 1.2 1.4 1.6
1.445
1.450
1.455
1.460
1.465
1.470
1.475
1.480
n
ng
Inde
x of
ref
ract
ion
Wavelength [µm]
Figure 3.1: Index of refraction (n) and group index of refraction (ng =c/vg) calculated from the Sellmeier formula for fused silica.
Since the second derivative of the refractive index is not zero the group
velocity is wavelength dependent, different colors undergo different delays. De-
pending on the units measuring the bandwidth of the input wave (rad/s or nm),
various dispersion coefficients are in use [Saleh91]:
Dωdω = Dλdλ. (3.8)
The dispersion coefficients are
Dω =∂2 k
∂ω2=
∂
∂ω
(1
vg
), (3.9)
Dλ = −2π cλ2
Dω. (3.10)
In terms of wavelengths they are
Dω = 2πλ3
c2∂2 n
∂λ2, (3.11)
Dλ = −λc
∂2 n
∂λ2=
∂
∂λ
(1
vg
), (3.12)
43
were λ is the central frequency.
3.1.2 Material dispersion and interferometry
When the wave number is not a linear function of the optical frequency the
dispersion relation is not a straight line. Then, the group velocity is different
from the phase velocity in the material. Pulses travelling through the medium
will be broadened. The visibility of an interference pattern will be reduced by
this broadening. The theoretical calculation has been included in Appendix D.3.
Here we present the main results.
0.6 0.8 1.0 1.2 1.4 1.6
-300
-250
-200
-150
-100
-50
0
50
Dλ [
ps/(
km n
m)]
Wavelength [µm]
Figure 3.2: Wavelength dependence of the group velocity dispersion coef-ficient calculated from the Sellmeier formula for fused silica. Between theregions of normal dispersion Dλ < 0 and anomalous dispersion Dλ > 0the dispersion coefficient is zero. This is the telecommunication wave-length λ = 1.3µm. High data rates are achievable because pulse broaden-ing is minimal at this wavelength.
44
The parameter describing the dispersion in the Mach-Zehnder interferom-
eter eq. (D.20) is the second derivative of the wave number with respect to the
frequency. It can be related to the dispersion coefficient by
∂2 k
∂ω2= − λ2
2π cDλ. (3.13)
The units for the dispersion coefficient Dλ listed in data sheets areps
nmkm, i.e.
the number of picoseconds a pulse of 1 nm bandwidth has spread after propagating
in a fiber of 1 km length. The following table give values calculated from the
Sellmeier equation for fused silica:
λ [µm] 0.633 0.702 0.788 1.3
Dλ [ psnmkm
] -240 -170 -110 0
Because of the negative sign, the longer-wavelength part of a pulse travels faster
than the shorter-wavelength part.
At the minimum of group velocity dispersion (λ = 1.3µm) pulse broadening
is minimal (Fig. 3.2). This is the wavelength most often used in telecommunica-
tion links.
The dispersion experienced in the optical system is strongly dependent on
the bandwidth of the signal. Various definitions of bandwidth are in use. If we
assume a Gaussian amplitude filter function with center frequency ω0 and 1/e-
width 2σ
f(ω) ∝ exp
[−(ω − ω0)
2
σ2
](3.14)
the power or probability distribution is the square of this function:
P (ω) ∝ exp
[−2(ω − ω0)
2
σ2
]. (3.15)
In the narrow-bandwidth limit (σ ω0) the power or probability distribution as
function of wavelength λ is:
P (λ) ∝ exp
[−2(λ− λ0)
2
∆λ2
], (3.16)
with λ0 ≡ 2π cω0
and ∆λ ≡ λ20
2π cσ. Then the full-width-half-maximum filter pa-
rameter ∆λFWHM of a Gaussian filter is the width of the spectrum in terms of
45
wavelength from half maximum to half maximum. These are the interference fil-
ter bandwidths (full-width-half-maximum) given in catalogues. They are related
to bandwidth σ (e−1 width of the Gaussian frequency amplitude distribution) by:
σ =
√2
ln 2
πc
λ20∆λFWHM. (3.17)
0,0 0,5 1,0 1,5 2,00,3
0,40,5
0,60,7
0,8
0,91,0
λ = 633 nm
2 nm
3 nm
1 nm
Vis
ibili
ty
∆x [m]
0,3
0,40,5
0,60,7
0,8
0,91,0
λ = 789 nm
2 nm
3 nm
1 nm
Vis
ibili
ty
Figure 3.3: Calculated single photon interference visibility in a fiber in-terferometer at λ = 633 nm and 789 nm with three different bandwidths∆λFWHM as function of the path-length difference. The lower group veloc-ity dispersion at longer wavelengths gives a considerably higher visibility.
The visibility of interference fringes calculated using the material param-
eters for fused silica and equation (D.28) for the wavelengths and material use
in the experiment are plotted in Fig. 3.3. The visibility for interferences at the
longer wavelengths is much better because of the lower group velocity dispersion.
46
For small path-length differences in interferometers and narrow bandwidths this
dispersion can be neglected in all other cases it must be considered.
3.2 Components of fiber-optical systems
The numerical aperture (NA) is a measure of how much light can be collected by
an optical system. For singlemode fibers the NA of the fiber must be matched to
all components in the path of light to ensure optimal coupling of the modes of
the electromagnetic field.
Microscope objectives or ball lenses with very short focal lengths are used
to optimize the mode-matching between k-modes of the vacuum (or air) and the
fiber mode. The k-mode selection is best if the fiber core is in the focal plane of
the objective.
Connections between fibers are made by mechanically splicing the fibers.
The fibers to be connected are cleaved so that their ends are perfectly flat and
parallel. Several types of connections are in use. Permanent connections are made
by fusion splicing the fibers. Semi-permanent connections can be made by using
mechanical splices. These are precisely machined grooves that hold the fiber ends.
The fibers are brought to contact in index-matching fluid assuring a good optical
connection. Finally, removable connections can be made by using polished ceramic
connectors (similar to BNC cables used in electronics). In our experiments we used
fusion splices, mechanical splices, and removable FC/PC-connectors.
In a circular singlemode fiber both polarization modes propagate equally
well. These modes will be coupled changing the polarization state of light as it
propagates in the fiber. Polarization maintaining fibers reduce the coupling be-
tween polarization modes by stress or dopant-induced birefringence. The fiber
used in our experiment had circular (symmetric) cores. Manual polarization con-
trollers had to be used to correct the polarization in the fibers (see p. 49).
Detectors for fiber-optical systems can have a very small sensitive surface if
they are brought into direct contact with the fiber core. We built our own single-
photon detectors using silicon avalanche photodiodes (APDs) with fiber pigtails.
47
The principle of operation of a passively quenched avalanche detector and further
details about our detectors can be found in Appendix A.4.
3.3 Coupled waveguides as multiports
When the cores of two optical fibers are brought alongside each other the evanes-
cent field of one fiber core can excite the mode in the other core. The modes
become coupled. Fused biconical tapered couplers are fabricated by placing single-
mode fibers side-by-side, twisting them together and fusing them while elongating
the contact region [Newport]. The transfer matrix for modes from one fiber to the
other can be described by a coupling constant K, that is a wavelength-dependent
exponentially decaying function of inter-fiber distance. Fused-fiber couplers of up
to seven fibers have been fabricated [Mortimore91].
The transfer matrix for a 2×2 coupler is given by our familiar beam splitter
matrix: k′1
k′2
=
cos(KL) −i sin(KL)
−i sin(KL) cos(KL)
k1
k2
. (3.18)
The length of the coupling region L determines the splitting ratio. For KL = π/4
we have a symmetric 2×2 coupler (or 3dB coupler). When the length if fixed the
coupling ratio is determined by the wavelength.
The treatment of 3 × 3 couplers can be described by the mathematics of
three coupled oscillators. We will restrict ourselves to presenting the measured
wavelength-dependence of the matrix elements for a fused fiber 3 × 3 coupler
which is shown in Fig. 3.5.
In the following experiments we analyzed the properties of 3× 3 fiber cou-
plers as optical multiports. In the first experiment we studied the properties of
the fiber-optical tritters in a three-path Mach-Zehnder interferometer. The sec-
ond experiment studied the operation of fiber multiports on individual quanta.
It was the first demonstration of antibunching of photons at the outputs of fiber
multiports.
48
Figure 3.4: Cut through a fiber-optical tritter (not to scale). The three fibercores are brought close together in the fused coupling region. The distancebetween the cores is of the order of the core diameter. The evanescentwave from one core can excite the other cores. After a distance and wave-length dependent coupling length the power is symmetrically split betweenthe cores.
600 650 700 750 8000,0
0,2
0,4
0,6
0,8
1,0 T R
Nor
mal
ized
tra
nsm
issi
on
Wavelength [nm]
Figure 3.5: Measured wavelength dependence of matrix elements of the3 × 3 integrated fiber coupler. The intensity transmission into the sameoutput T and into the other outputs R was measured as function of thewavelength. This integrated fiber coupler was used in the 789 nm fiberinterferometer. At this wavelength the intensity splitting ratio is 1/3 intoeach output.
Chapter 4
Experimental characterization of
fiber multiports
4.1 A three-path Mach-Zehnder interferometer
using all-fiber tritters
The main features of multipath interferometry with optical fiber multiports were
studied using a three-path fiber interferometer with integrated 3×3 fiber couplers.This Mach-Zehnder type interferometer shows many of the characteristic features
encountered in multipath two-photon interferometry.
4.1.1 Experimental setup
We have built a multiport interferometer using two symmetric tritters in form
of integrated fiber 3 × 3 couplers [Weihs95, Weihs96a]. The properties of this
interferometer were studied using a Helium-Neon laser as light source. The light
from the HeNe laser was coupled into one input port of the first tritter whose
outputs were in turn connected to the inputs of the second tritter. The paths
length differences inside the interferometer were much smaller than the coherence
length of the HeNe light (lc = 35 cm). Three silicon photodiodes monitored the
light intensity at the outputs of the system (see Fig. 4.1).
49
50
HeNe-Laser
Photodiode
<
<
Piezophase shifter
RC-generators Amplifiers
Polarizationcontroller
Tritter 2Tritter 1
I1I0
I2
I3
OscilloscopeTrig
Trig
FC/PC-connector
ϕ2
ϕ1
Figure 4.1: Schematic of the three-path interferometer experiment. Thetritters are three-way integrated fiber couplers. Polarization controllerswere used to manually adjust the polarization in all three arms for maxi-mal visibility. Piezo tubes were used to modulate the interferometer phasesφ1 and φ2 in two of the three interferometer arms. The photodiode sig-nals and the driving signals were recorded in a digital storage oscilloscope[Weihs96a].
In order to see interferences the polarization state of the field at the second
tritter must be the same for all interfering modes. Since the fundamental mode of
our singlemode fibers supports two polarization modes and the polarization state
of the light changes as it propagates in the fiber, the polarization in each arm had
to be manually adjusted. The operation of the manual polarization controller is
based on induced birefringence in a fiber under stress when wrapped around a
disk [Lefevre80, Ulrich80, Rashleigh83]. In our case we used three disks of radius
24mm. The bending of the fiber induces a change of the refractive index for the
two polarization modes in the fiber. The relative phase delay between the two
polarization modes can be controlled by changing the number of loops on the
disk (see Fig. 4.2). The relative angle of the disks and thus the optical axis of
the fiber can be adjusted so that the three disks act as a stack of λ/x-plates. It
can be shown that if λ/3 < λ/x < 2λ/3, that is we have something close to a
quarter-wave plate, any input polarization can be transformed into any output
polarization.
51
0.6 0.8 1.0 1.20.0
0.2
0.4
0.6
0.8
1.0
radius = 24mmloops = 1, 2, 3
Ret
arda
tion
[rad
]
λ [µm]
Figure 4.2: Calculated induced retardation in the manual polarizationcontroller. A singlemode optical fiber with cladding diameter 125µm iswrapped around a disk of radius r = 24mm. This induces an optical re-tardation as a function of the number of loops on the disk (1, 2, 3) andthe wavelength (λ). Any polarization can be set by using three tilted disksif the retardation on each disk lies between π/3 and 2π/3. For λ = 633 nmone loop is necessary; for λ = 789 nm two loops are required.
Phases in each two of three interferometer arms could be changed by piezo
phases shifters built from piezo tubes. The tubes with outer diameter 40mm,
length 18mm, and wall thickness 1mm contract when a high voltage is applied.
The specifications were about 13µm radial contraction for an applied voltage of
1 kV. Thus a fiber tightly wrapped around a tube will suffer a change in length
of about 0.08µm/V. A change of a few volts on the piezo is enough to stretch the
fiber by a few optical wavelengths. For small changes in length the phase shift is
linear in the applied voltage.
The interferences show very high sensitivity to changes in the length of
the fiber inside the interferometer. This sensitivity is the basis of many fiber-
optical sensors [Giallorenzi82]. One degree of temperature change will induce a
52
thermal expansion in one meter of fiber that is enough to cause phase shifts of
17 wavelengths (at 633 nm). In this experiment the light source was a HeNe laser
with high intensity so that phases could be scanned in a very short time ≤ 1 s.
Temperature-induced drifts thus were not a serious problem.
4.1.2 Theoretical description
The main features of multipath interferometry can be studied using the three-
path Mach-Zehnder interferometer without resort to quantum optics. The in-
put field Kin = (Ein, 0, 0) is a vector, whose elements are the scalar electri-
cal field amplitudes of the three input modes to the interferometer. The scalar
input field Kin is transformed by the interferometer into three output fields
(Eout1 , Eout
2 , Eout3 ) =MKin.
The three-path interferometer can be described by the multiport matrix
M = T
eiφ1 0 0
0 eiφ2 0
0 0 eiφ3
T. (4.1)
T is the tritter matrix of eq. (2.15) and φ1 . . . φ2 are the phase shifts introduced
in each of the three paths.
The intensities In (n = 1, 2, 3) at the three outputs can be calculated from
the input intensity I0 and the phases settings in the interferometer
In = |Eoutn |2 (4.2)
=I09[3 + 2 cos(φ1− φ2 + χn) + 2 cos(φ2− φ3 + χn) + 2 cos(φ3− φ1 + χn)] .
The output intensities are shifted with respect to each other by (χ1, χ2, χ3) =
(0,−2π/3, 2π/3) and depend only on phase differences. In the following we will
discuss only the case for χn = 0, and leave away the index n:
I(φ1, φ2, φ3) =I09[3 + 2 cos(φ1− φ2) + 2 cos(φ2− φ3) + 2 cos(φ3− φ1)] . (4.3)
Sometimes it is useful to rewrite the function in eq. (4.3) as a product of cosine
functions:
I(φ1, φ2, φ3) =I09
[1 + 8 cos
φ1− φ22
cosφ2− φ3
2cos
φ3− φ12
]. (4.4)
53
00
0
0.5
1
2π
π
π
2π
φ −φ13
φ −φ
23
Figure 4.3: Calculated normalized intensity as function of two phases(φ1, φ2) in the tritter Mach-Zehnder interferometer with one input il-luminated. The function, which is given in eq. (4.3), has minima at(2π/3, 4π/3) and (4π/3, 2π/3). The other outputs interference patternare shifted by 2π/3. The reference phase φ3 has been set to zero.
If the phases are chosen such that φ2 = −φ1 the intensity varies from max-
imum to minimum as this phase is changed. The curve shows maximal visibility
(see Fig. 4.4).
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
φ = −φ [ ]1 2 rad
Figure 4.4: The calculated three-path Mach-Zehnder interferences showmaximum visibility when φ2 = −φ1 is varied. The fringes observed arenon-sinusoidal. The parts of the curve where the slope is steeper than thatof two path interferences show higher phase sensitivity as discussed in thetext.
54
The phase setting φ2 = −φ1 also gives the maximal sensitivity S, which is
defined as the derivative of the output intensity with respect to a phase [Sheem81],
S =1
I0
∣∣∣∣∣ dIdϕ∣∣∣∣∣ . (4.5)
This is higher than the standard Mach-Zehnder interferometer because the slopes
of the main peaks are steeper. In this way the sharpening of the interference peaks
is the same effect observed with diffraction gratings, where there is a contribution
from each slit to the diffraction orders.
55
4
4.1.3 Experimental results
28 32 36 40 44
28
32
36
40
44
CH1
pie
zo v
olta
ge
ϕ 1 [V
]
piezo voltage ϕ2 [V]
28 32 36 40 44
CH2
28 32 36 40 44
CH30 2π/3 4π/3
ϕ2 [rad]
0 2π/3 4π/3
ϕ2 [rad]
0 2π/3 4π/3
0
2π/3
4π/3
2π
ϕ1 [ra
d]
ϕ2 [rad]
Figure 4.5: Experimental data of the three-path Mach-Zehnder interfer-ometer shown as gray-scale plots (minimum white, maximum black). Thethree output intensities (CH1. . .CH3) are shown as functions of the tworelative phase settings. The visibility of the interferences was approxi-mately 97%.
The intensities at the three outputs of the three-path fiber Mach-Zehnder
interferometer were measured using silicon photodiodes. The phases were rapidly
modulated using triangular wave forms of an RC-generator. Two different fre-
quencies (2 and 80 Hz) were used so that the whole parameter space (φ1, φ2)
could be scanned in approximately one second. The photodiode voltages and the
piezo voltages were recorded on a digital oscilloscope. The measured values of
the intensity as function of the phases correspond very well to the theoretical
prediction. The measured data are shown in Fig. 4.5.
The main experimental techniques with optical fibers were studied using
this Mach-Zehnder interferometer. We could show that the polarization could be
matched in all arms of the interferometer using the simple manual polarization
controllers. The polarization, once set, remained stable over a period of several
days as long as the fiber was not disturbed. The piezo phase shifters were suitable
for phase modulation. They are linear for small changes in voltage. Because of
56
the short measurement times, the high sensitivity of the fibers to temperature
changes was not a problem in this experiment.
The experimental results confirmed the theory of three-path interferometry.
The integrated fiber tritters act as coherent beam splitters for the three modes
and thus can be described by the simple tritter matrixes. In the next experiment
we investigated the two-photon properties of the fiber tritter using two-photon
interference.
4.2 Two-photon interferences in optical fiber
multiports
4.2.1 Introduction
In the previous experiment we have seen that fiber optical tritters act as coherent
beam splitters for classical laser radiation. Two-particle interference experiments
reveal the operation of tritters on individual quanta. One of the first experiments
to perform is the observation of photon bunching in the outputs due to Bose
statistics [Hong87, Fearn89].
Nonclassical statistics at multiport beam splitters constructed from discrete
optical components have already been studied in Innsbruck [Mattle95]. Here we
present integrated fiber-optical beam splitters and tritters and demonstrate how
the complicated process of coupling between modes in a the fiber device in the
two-photon interference experiment can be modelled by the operation of a simple
matrix on the modes. The following experiment is the first demonstration of the
bunching of bosons at the outputs of fiber multiports [Weihs95, Weihs96b].
4.2.2 Theoretical description
A lossless 2N -port can be represented by a unitary N × N -matrix, where the
matrix element Mik is the complex probability amplitude for detecting a particle
in output i that has entered the multiport through input k. When two photons are
57
Figure 4.6: View of UV-laser, beam steering mirrors, PDC crystal, irises,and beam blocker used in the two-photon interference experiment on atritter.
incident on different inputs k and l, as in our experiment, there are two possible
evolutions or paths of the system resulting in the joint detection of one photon
in output i and the other in output j. These two paths have the probability
amplitudes (MikMjl) and (MilMjk).
Feynman’s rules can be used to obtain the probability P klij of detecting
one photon in output i and one in a different output j when the two photons
entered through different inputs k and l [Feynman64]. In the case of classically
distinguishable particles the sum of the absolute squares of the amplitudes for all
58
possible paths gives the classical probability
P (cl)kl
ij = |MikMjl|2 + |MilMjk|2. (4.6)
If, however, the particles are indistinguishable, as is the case with two pho-
tons of equal frequency and polarization arriving simultaneously, we have to take
the absolute square of the sum of all amplitudes that contribute to the out-
come. For indistinguishable particles the quantum probability P (qm)kl
ij may show
destructive interference:
P (qm)kl
ij = |MikMjl +MilMjk|2 (4.7)
= |MikMjl|2 + |MilMjk|2 + 2Re[(MikMjl)∗(MilMjk)].
The last term in the expression for the coincidence probability may be negative
or positive. If it is negative the coincidence probability at two outputs is lower
than the classical value in eq. (4.6) and we observe bunching into one output. If
this probability is larger than zero we observe antibunching.
The visibility1 of the interference effect can be defined using the probabilities
P0 for an input time delay of zero (τin = 0, quantum-mechanically indistinguish-
able particles) and P∞ for τin →∞ (classically distinguishable).
V klij =
∣∣∣∣P∞ − P0
P∞
∣∣∣∣ =∣∣∣∣∣2Re[(MikMjl)
∗(MilMjk)]
|MikMjl|2 + |MilMjk|2e−
ω2dt2c
8
∣∣∣∣∣ . (4.8)
Here we have assumed a Gaussian spectral distribution with a corresponding
coherence time tc and a mismatch of the center frequencies of ωd for the two
single photon spectra. We note that a wavelength mismatch of only ∆λ = 1nm
at 702 nm in connection with a coherence length of lc = 200µm would already
halve the visibility.
4.2.3 Experimental results
In our experiment we used correlated photons of equal wavelength from a para-
metric downconversion crystal. The degenerate mode was selected by irises (see
1The so defined visibility describes the two-photon interference and is differentfrom the one used for interference fringes.
59
KD*P-crystal
Ar+-laser (351.1 nm)
Polarizationcontroller
Tritter
Si-APDConnectorInterference filter 702nm
RG-630 filter
∆ τx=c in
coin
cide
nce
coun
ters
Figure 4.7: The schematic of the two-photon interference experiment in afiber tritter. The entangled modes from the crystal were coupled into twoinputs of the fiber tritter. A coincidence unit recorded the coincidencesbetween all three pairs of outputs as the time-lag between the arrival timesof the two wave packets at the tritter was varied.
photograph 4.6). By changing the arrival time-lag between the two photons at
the multiport we were able to vary the degree of distinguishability of the two
particles, thus going from the classical to the quantum regime (see Fig. 4.7, and
photograph 4.8). Again, we used manual polarization controllers to adjust the
polarization for maximal visibility of the interference effect.
The tritter used in this experiment was fabricated for symmetric opera-
tion at 780 nm. Measurements of the intensity division matrix at our wavelength
(702 nm) showed, that all diagonal elements were equal within the achieved accu-
racy. The phases could be determined from the unitarity condition of the matrix.
From these measurements we determined the amplitude matrix of our tritter to
be
M (3)exp =
0.80 0.43 0.43
0.43 0.80ei0.83π 0.43ei1.41π
0.43 0.43ei1.41π 0.80ei0.83π
, (4.9)
with errors in the moduli and phases less than 2%. The maximum possi-
ble visibilities calculated from Eq. (4.8) are then V 1212 = (46.0 ± 3.0)% and
V 1213 = V 12
23 = (22.0± 1.4)%.
The following table summarizes the results of the two-photon interference
60
Figure 4.8: View of two fiber input couplers with microscope objectiveson translation stages used in the two-photon interference experiment in afiber tritter. The tritter is hidden in the styrofoam box in the background.Irises and square UV-cutoff filters are visible before the inputs.
at the fiber tritter.
V 1212 V 12
13 V 1223
Theoretical max. 0.46 0.22 0.22
Experiment (40± 1)% (22± 2)% (20± 2)%
We see that the interference visibility approaches the theoretical maximum.
and that the complicated photon propagation mechanisms in the fiber tritter can
thus be well modelled by our matrix method.
61
0.7
0.8
1&2
2&3
1&3
Position ∆ x [µm]
-400 -200 0 200 400 600
0.91.01.11.2
Cou
nts
[10
3 ] in
10s
1.41.61.82.02.22.4
Figure 4.9: Photon antibunching at the outputs of a fiber tritter can bemodelled as a Gaussian dip. The coincidence counts in 10 s between theoutput ports of the integrated fiber tritter are shown as a function ofthe position of the input coupler (see Fig. 4.7). The count rates werecorrected for a linear drift of the effective laser power, with the actualcoincidence rates at ∆x = −400 as reference. The rates show dips asthe path length difference goes to zero. The visibilities of the dips wereV 1212 = (40±1)%, V 12
13 = (22±2)%, and V 1223 = (20±2)%, where the upper
indices denote the inputs and the lower indices the measured outputs ofthe tritter. The width of the coincidence dip between outputs 1 and 2was ∆xFWHM = 300 ± 10µm, corresponding to a coherence length oflc = 255± 9µm. The width of 300µm corresponds to a time-delay of 1 psin the vacuum.
62
Chapter 5
Two-photon three-path
interference
5.1 Energy entanglement in three-path interfer-
ometers
The photon pairs from a parametric downconversion crystal (PDC) are entangled
in momentum, energy, and, for certain configurations in polarization. We will now
study the time-energy entanglement of PDC photon pairs using a setup proposed
by Franson [Franson89].
In an experiment as depicted in Fig. 5.1 downconversion photon pairs are
directed into a two three-path Mach-Zehnder interferometers. If the path length
differences in the interferometers on each side are much greater than the coherence
length of the radiation as determined by filters and apertures at their inputs the
interferometers are ‘disbalanced’, and we measure no interferences when register-
ing counts at one detector only. Now we study what happens when the detection
events on one side are measured in coincidence with detection events on the other
side.
For the interferometers on each side there are three paths to the detector.
The paths lengths are indicated by the index s, m, and l for one side, and s′, m′,
and l′ for the other. In a simplified discussion we can use Feynman’s rule:
63
64
“If you cannot distinguish the final states even in principle, then the
probability amplitudes must be summed before taking the absolute
square to find the actual probability.” [Feynman64]
With a three-path interferometer in place the total two-photon wave func-
tion is a superposition of the wave functions of all possible paths to the respective
detectors. Ψxy is the wave function indicating path x in one interferometer and
path y in the other are taken to the detectors. Note that when we drop the primes
in the following discussion the first index refers to first interferometer, the second
index to the other interferometer. Furthermore, we assume that the optical path
lengths over the short paths (s and s′) from the crystal to the detectors are equal.
The wave function of the two-photon system after the interferometers and before
the detectors has nine contributions:
Ψtot =1
9(Ψss +Ψsm +Ψsl +Ψms +Ψmm +Ψml +Ψls +Ψlm +Ψll) . (5.1)
The probability density for a detection event at one detector at time t1 and at
the second detector at time t2 is given by the absolute square of the probability
amplitude:
PΨ12(t1, t2) = Ψ∗totΨtot. (5.2)
The total wave function in eq. (5.1) has nine contributions, three of which,
we will show, can be made undistinguishable. When our detection electronics
select only these three contributions we will find that the detection probability
for pairs corresponds exactly to the quantum probability postulated for the three-
state Bell inequality (1.17). The postselected wave function
Ψ ∝ Ψss +Ψmm +Ψll (5.3)
describes an entangled three-state system. The three states correspond to the
three pairs of paths that are indistinguishable.
5.2 Multipath interferences
The state emitted from the crystal will be described by the theory of parametric
downconversion [Mollow73, Hong85]. A simplified description shows the main fea-
65
l'
m'
s'
l
ms
λ i=789nm
λs=633nm
λp=351nmCoinc.
Figure 5.1: Two modes from the parametric downconversion crystal arecoupled into two 3-path interferometers. Two single-photon detectors reg-ister coincidence counts between two separated outputs. When the pathlength differences l-l’, m-m’, and s-s’ are exactly equal, second order in-terferences are observed in the coincidences.
tures needed for this experiment. A continuous wave pump laser with monochro-
matic spectrum centered around ωp pumping a parametric downconversion crystal
will produce a two-photon state with strong momentum and energy correlations1.
In this experiment only two k-modes of the crystal are used so that we only con-
sider the energy correlations:
|Ψ〉 =∞∫0
∞∫0
∆(ω + ω′ − ωp)a†1(ω)a
†2(ω
′)|0〉. (5.4)
The function ∆ is sharply peaked around zero and the integral over its absolute
square is equal to unity [Campos90]. The frequency distribution of the two-photon
state before the inputs of the interferometers is determined by filters and aper-
tures. We assume Gaussian filters g and g′ with center frequencies ω0 and ω′0, and
bandwidths σ and σ′. The two-photon wave function thus changes to:
|Ψ〉 =∞∫0
∞∫0
g(ω)g′(ω′)∆(ω + ω′ − ωp)a†1(ω)a
†2(ω′)|0〉 dω dω′. (5.5)
1Since the polarizations of the photons emitted in type-I downconversion
process are the same, we assume only one polarization mode in the followingdiscussion.
66
In the narrow bandwidth limit (σ ω0, σ′ ω′0) the integral can be extended
to infinity and we find that the time-dependent wave function describes a non-
factorizable state:
Ψ(x, x′, t, t′) ∝ ei(kx−ωt) ei(k′x′−ω′t′) exp
−σ
2
4
[(x
c− t)− (
x′
c− t′)
]2 . (5.6)
For the moment we neglect dispersion and assume equal optical path-length dif-
ferences in both interferometers. After the Mach-Zehnder interferometers the ex-
pressions will only depend on the parameters:
a ≡ m− s = m′ − s′ (5.7)
b ≡ l −m = l′ −m′ (5.8)
τ ≡ t2 − t1 (5.9)
For the moment we will also assume the degenerate case k = k′. In the final
expression we will return to the case when the wavenumbers in the two interfer-
ometers are different.
The probability density for detection of photons at both detectors is given
in eq. (5.2). The individual terms Ψxy in this expression are of the form given
in eq. (5.6), i.e. they are products of Gaussians and complex exponentials. If the
product of the nine terms in eg. (5.1) is evaluated we find 81 terms, which can
be simplified to give the following probability density:
PΨ12(a, b, τ) ∝ 3 [F (0)]2 + [F (a)]2 + [F (b)]2 + [F (a+ b)]2 (5.10)
+ 4 cos(k a) F (0)F (a) + F (a+ b)F (b)+ 4 cos(k b) F (0)F (b) + F (a)F (a+ b)+ 4 cos(k (a + b)) F (0)F (a+ b) + F (a)F (b)+ 2 cos(k (2 a+ b))F (0)F (b)
+ 2 cos(k (a + 2 b))F (0)F (a)
+ 2 cos(k (a− b))F (0)F (a+ b)
+ 2 cos(k 2 a) + cos(k 2 b) + cos(k 2 (a+ b)) [F (0)]2
with the definition:
F (u) ≡ exp
−σ
2
4[u+ cτ ]2
+ exp
−σ
2
4[u− cτ ]2
(5.11)
67
which for u = 0 reduces to
F (0) ≡ 2 exp
−σ
2 (cτ)2
4
. (5.12)
The physical meaning of the parts of this expression is simple if one considers the
envelope functions F (u) as a description of the coherence length of the radiation
in the interferometers. For large values of u these functions are zero. Interferences
then disappear.
The experimentally measured coincidence rates are the integrals over the
coincidence window T of the detectors. The 9× 9 = 81 terms contributing to the
final detection probability are identified in the following expression:
+T/2∫−T/2
Ψ∗Ψ dτ ∝ (5.13)
3/2︸︷︷︸
ss−ssmm−mmll−ll
+cos(2 a k)︸ ︷︷ ︸ss−mmmm−ss
+cos(2 b k)︸ ︷︷ ︸mm−llll−mm
+cos(2 (a + b) k)︸ ︷︷ ︸ss−llll−ss
H(0)
+ H(2 a) + H(0)G(2 a)︸ ︷︷ ︸sm−sm,sm−msms−sm,ms−ms
+ H(2 b) + H(0)G(2 b)︸ ︷︷ ︸ml−ml,ml−lmlm−ml,lm−lm
+ H(2 (a+ b)) + H(0)G(2 (a+ b) k)︸ ︷︷ ︸sl−sl,sl−lsls−sl,ls−ls
+ 2 cos((a+ 2 b) k) H(a)G(a)︸ ︷︷ ︸ml−ss,lm−ssss−ml,ss−lm
+ 2 cos((2 a+ b) k) H(b)G(b)︸ ︷︷ ︸ll−ms,ll−smms−ll,sm−ll
+ 2 cos((a− b) k) H(a+ b)G(a+ b)︸ ︷︷ ︸mm−sl,sl−mmls−mm,sl−mm
+ 2 cos(a k)
2H(a)G(a)︸ ︷︷ ︸
sm−ss,ms−ssmm−sm,mm−ms,...
+H(a+ 2 b)G(a) + H(a)G(a+ 2 b)︸ ︷︷ ︸ml−sl,ls−mslm−sl,lm−ls,...
68
+ 2 cos(b k)
2H(b)G(b)︸ ︷︷ ︸
lm−mm,ml−mmll−ml,ll−lm,...
+H(2 a+ b)G(b) + H(b) G(2 a+ b)︸ ︷︷ ︸sl−sm,ms−slls−sm,ls−ms,...
+ 2 cos((a+ b) k)
2H(a+ b)G(a+ b)︸ ︷︷ ︸
sl−ss,ls−ssll−sl,ll−ls,...
+ H(a− b)G(a+ b) + H(a+ b)G(a− b)︸ ︷︷ ︸ml−sm,ml−mslm−sm,lm−ms,...
.
The expression is a sum of cosines multiplied with Gaussians and error functions:
G(x) ≡ exp
(−(σ x/c)
2
8
), (5.14)
H(x) ≡√2π
cσErf
(σ (x/c− T )
23/2,σ (x/c+ T )
23/2
), (5.15)
Erf(x, y) ≡ 2√π
y∫x
e−u2
du. (5.16)
We will give a simple physical interpretation of the long expression above
that allows us to identify some interesting cases. The first case is when the path-
length differences in the interferometers are smaller than the coherence length
of the single photon radiation as determined by filters and apertures at the in-
puts. In this case, the three paths each individual photon could have taken are
indistinguishable, even in principle. We expect first-order interferences at each
detector. The other limit is when the path-length differences in each interferom-
eter are much larger than the coherence length of the individual photons. Since
we can in principle determine which way the photon must have taken, the first
order interferences disappear. However, in this case we can make the path-length
differences in both interferometers the same. Now the detection of a photon at
the output of one interferometer and the simultaneous detection of a photon at
the output of the other interferometer gives us no information about the paths
the photon pair has taken. They could equally well have taken the long (l, l′),
the medium (m, m′), or the short (s, s′) paths. Since we cannot determine which
path the pair has taken we expect second-order interferences in the pair detection
69
probability. If we also include the effect of detector response time in the second
case in our discussion we get a total of three interesting cases.
In the following we will see all these possibilities are given as special cases
of the messy expression (5.13):
1) 1/σ T x/c: H(x)→ 0 G(x)→ 0
2) 1/σ x/c T : H(x)→ 2 G(x)→ 0
3) x/c 1/σ T : H(x)→ 2 G(x)→ 1
In these cases the coincidence probability takes a simpler form. For this discussion
we return to the more general case of different wavenumbers k = k′.
In the first case (a/c 1/σ T and b/c 1/σ T ) the path length
differences are smaller than the single-photon coherence lengths determined by
the filter bandwidths. The coincidence probability is simply the product of the
single photon detection probabilities which show interferences:
P(1)12 =
1
81[3 + 2 cos(k a) + 2 cos(k b) + 2 cos(k (a+ b))] (5.17)
× [3 + 2 cos(k′ a′) + 2 cos(k′ b′) + 2 cos(k (a′ + b′))] .
We will observe the single-photon interferences fringes as measured in the three-
path interference experiment with the balanced Mach-Zehnder interferometer (see
section 4.1).
In the second case (1/σ a/c T and 1/σ b/c T ) the detector
coincidence window is not small enough to resolve the contributions from different
paths. We observe two-photon visibilities with reduced visibility2:
P(2)12 =
1
81[9 + 2 cos(k a + k′ a′) + 2 cos(k b+ k′ b′) (5.18)
+ 2 cos(k (a+ b) + k′ (a′ + b′))] .
The third, most interesting, case (1/σ T a/c and 1/σ T b/c) is
when the detector coincidence window is so short that the detection events coming
2Here we have tacitly assumed |a− b|/c 1/σ and |a′ − b′|/c 1/σ′.
70
from paths of different length can be distinguished. The coincidence probability
shows interferences depending on the sum of spatially separated phases:
P(3)12 =
1
81[3 + 2 cos(k a + k′ a′) + 2 cos(k b+ k′ b′) (5.19)
+ 2 cos(k (a+ b) + k′ (a′ + b′))] .
In the further discussion we will only refer to the interesting case of eq. (5.19)
and drop the cumbersome superscript. The two-photon interferences have the typ-
ical functional dependency of three-path interferences (cf. eq. (4.3)) except for
the fact that they depend on the sum of phases seen by the entangled photons in
different interferometers. The phases can be changed by varying the path length
differences as long as the path differences in the spatially separated interferome-
ters remain much smaller than the single-photon coherence lengths determined by
the filters: |a−a′|/c 1/σ and |b−b′|/c 1/σ. Because of the large path-length
differences, the probability of detecting a single photon at one of the outputs of
each interferometer is the incoherent sum of the probabilities of taking the l, m,
or s paths:
P1 = P2 =1
9+1
9+1
9=
1
3. (5.20)
The two-photon interferences of eq. (5.19) have a maximum of value of 19. Because
the pair detection probability in eq. 5.19 only includes pairs of paths that are the
same, the conditional pair detection probability is
max (P12)
P1
=1
3. (5.21)
But the minimum value of the pair detection probability is zero and thus we
expect two-photon interferences with maximum visibility even when the singles
count rates show no interferences.
5.3 Experiment
5.3.1 The experimental setup
An Argon ion laser was used as a single-line single-frequency source of continuous
wave UV radiation (351.1 nm). Typical pump powers in the experiment were
71
Figure 5.2: View of PDC crystal, irises, and inputs to the fiber interfer-ometers used in the two-photon three-path interference experiment. Thesix fiber-pigtailed detectors are visible in the center.
450mW. A LiIO3 crystal (30 × 30 × 30mm) cut at 90 to the optical axis was
used as type-I parametric downconversion (PDC) source (cf. Appendix A).
Phase matching conditions inside the LiIO3 crystal determine the wave-
length and k-modes of the photon pairs:
hωp = hω1 + hω2, (5.22)
hkp = hk1 + hk2. (5.23)
For our pump wavelength the photon pairs with center wavelengths of 632.8 nm
and 788.7 nm fulfill the energy conservation condition. These modes were selected
by the emission directions (see photograph 5.2). The wavelengths correspond to
a HeNe laser line and to a commonly available diode laser line. Polarization ad-
justments could thus be made with laser light instead of tenuous downconversion
light. Our single-photon counters use Silicon avalanche photodiodes that are very
72
phase control
polarizationcontrol
fused fibercoupler
fused fibercoupler
lm
s
APD detectors
Det 5
Det 4
Det 6
filter
M4x
788.7 nm
Figure 5.3: Schematic of the 789 nm interferometer: A microscope objec-tive precisely selects a k-direction corresponding to a wavelength selectivityof ∆λ = 1.2 nm. Because the coherence length determined by the modeselection and filters (lc = 250µm) is much smaller than the path lengthsdifferences (∆ > 56 cm) no first order interferences are observed.
efficient at these wavelengths.
The measured angles between the pump axis and the direction of emission
were 25.0 for λ′ = 633 nm and 31.4 for λ = 789 nm.
Two irises between crystal and fiber inputs roughly selected the correlated
modes. These irises were left open during measurements since the M4-microscope
objectives at the input of the fiber interferometers have a much stronger angular
selectivity than the irises. The actual distances in the setup before the fiber
interferometers are summarized in the next table.
Middle of crystal to 1st iris to 2nd iris to M4 objective to fiber
633 nm side 18 cm 67 cm 76 cm 83 cm
789 nm side 12 cm 71 cm 82 cm 88 cm
Integrated fiber tritters were used to build the three-path interferometers.
Two pairs of commercially available devices were used. One pair was specified to
be symmetric for the wavelength 633 nm the other for the wavelength 780 nm. The
fiber interferometers guarantee well defined modes on the tritters thus permitting
good interference visibilities.
73
0,0 0,5 1,0 1,5 2,00,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
λ1 = 633 nm
λ2 = 789 nm
2 nm
3 nm
1 nm
Tw
o ph
oton
vis
ibili
ty
Path length difference [m]
Figure 5.4: Calculated two-photon interference visibility in a Franson-type fiber interferometer with two photons of different wavelengths(633 nm, 789nm) and interference filters with three different bandwidths∆λFWHM. Even for small bandwidths ≈ 1 nm as in our experiment thereduction in visibility due to group velocity dispersion for path-length dif-ferences of 2m is not negligible.
Fusion splicing of fiber to achieve low-loss connections is the standard in
communication technology. For our fibers the small core diameters (< 5µm) im-
posed a considerable challenge on our splicing skills. The transmittances achieved
were of the order of 80–90% with large deviations from the average.
The broadening of pulses in optical fibers due to group velocity disper-
sion (GVD) leads to a reduction of the interference contrast (see p. 45 and Ap-
pendix D.3). In a two-photon interference experiment the group velocity disper-
sion enters in the equation as a sum of individual dispersions. The group velocity
dispersion parameter D in eq. (D.28) must be replaced by the sum of the signal
74
and idler dispersions:
D = σ21
2
[− λ2
2π cDλ
]x+ σ′2
1
2
[− λ′2
2π cDλ′
]x′. (5.24)
Since the contributions of the material dispersion Dλ and Dλ′ may have different
signs, this can lead to the interesting effect of nonlocal cancellation of dispersion
in a two-photon interference experiment [Franson92].
In our experiment we unfortunately have two contributions of the same
sign. Thus the two-photon interference visibility is reduced by the sum of the two
individual group velocity dispersions. The reduction of visibility for various filters
and path length differences is plotted in Fig. 5.4. The visibility of interferences
in a multipath interference experiment can be estimated from the interference
visibilities of a two-path interference experiment.
Two parameters in the term for the group velocity dispersion contribute to
the reduction of visibility: the bandwidth or coherence length of the photons in
the fibers and the length of fiber transversed. The contributions are much larger
at shorter wavelengths. Thus the length of fiber at 633 nm must be as short as
possible. The bandwidth of both signal and idler modes should be as narrow as
possible. The narrow bandwidths guaranteed by the mode selection using micro-
scope objectives helped to reduce the effect of group velocity dispersion on the
interference visibility.
Although the length of fibers in the interferometers should be as short as
possible to reduce the effects of GDV, they must be long enough to allow two loops
of fiber to be wrapped around each of the three disks of the manual polarization
controller (MPC) in the 789 nm interferometer. Allowing some fiber length for
the distance between the tritters, connectors, and the MPC, we thus see that the
short path must be at least 120 cm long.
The difference between paths must be large enough to allow electronic dis-
crimination of the various interfering peaks as described in the previous section.
Due to jitter in the detection system we can resolve coincidence peaks if they are
separated by about 3 ns. Thus the path length differences must be at least 60 cm.
Taking all these constraints into consideration, the actual path lengths in-
75
side the 789 nm interferometer were3:
path l [cm] t [ns] difference l [cm] t [ns]
s 146.8 7.1 ∆ms 77.7 3.8
m 224.5 10.9 ∆ls 133.8 6.5
l 280.6 13.6 ∆lm 56.1 2.7
The second interferometer (633 nm) was also built using integrated fiber
optical tritters (see photograph on p. 76). Since the optical path-length differences
in both interferometers had to be equal to within the coherence length of about
200µm a mechanical method of path-length adjustment had to be introduced.
phase control
polarizationcontrolfused fiber
couplerfused fiber
coupler
l'm'
s'
APD detectors
Det 1
Det 2
Det 3
filter
M4x
adjustable air gap
adjustable air gap
633 nm
Figure 5.5: Schematic of the 633 nm interferometer: The length of the airgaps can be adjusted using motorized translation stages (cf. photographon p. 76).
Air gaps in the second interferometers (operating at 633 nm) allowed ad-
justment of the path lengths. The light from the first fiber tritter output was
collimated and coupled into the input of the second tritter using microscope ob-
jectives. The air gaps had the disadvantage that due to reflection and imperfect
coupling losses were incurred when going in and out of fibers. The gaps in the
633 nm interferometer, however, reduced the detrimental effect of group velocity
dispersion at this wavelength since part of the path in the interferometer thus
was dispersionless. The following table summarizes the geometry of the 633 nm
3A refractive index of 1.45 was used for the calculation of the time delay.
76
Figure 5.6: Photograph of the 633 nm interferometer.
77
Figure 5.7: Air gap (m′) in the 633 nm interferometer. The length of theair gaps can be adjusted using motorized translation stages.
interferometer. The first values refer to the length transversed in the fiber; the
second value refers to the length of the air gap.
path lfiber [cm] lair [cm] t [ns]
s′ 161.6 7.9
m′ 231.5 11.4 11.7
l′ 239.7 81.3 14.4
In the m and l paths of both interferometers piezo phase modulators were
introduced. The high voltage applied to the piezo-ceramic tube stretched two
loops of fibers wrapped around it (cf. photograph 5.8). A signal voltage was used
to control the high voltage. It could be changed either slowly by a computer
controlled DA-converter signal or rapidly by a periodic signal from a function
generator.
Single-photon counters were built using fiber-pigtailed Silicon avalanche
photodiodes. The detectors were connected to the outputs of the fiber interferom-
eters using FC/PC-connectors. Although these connectors introduced additional
78
Figure 5.8: Piezo phase shifters were built using piezo ceramic cylinders.The fiber is wrapped around the cylinder. The cylinder, and thus the fiber,expand and contract as a function of the applied voltage. A few tens ofvolts produce a phase shift of several periods.
79
losses, they proved very practical for purposes of adjustments: a reverse illumi-
nation scheme could be used.
5.3.2 Polarization and path length adjustment
polarizationcontrollers
lm
sM4xM4x
laser
polarizeranalyzer
Si-PD
FC/PC connector
Figure 5.9: The reverse illumination scheme for interferometric polariza-tion adjustment uses a polarized laser connected to one detector output.The manual polarization controllers induce an optical retardation witha changeable optical axis. By tilting the disks the polarization state inthe fiber is changed. The polarization state at the tritters is the samewhen the photodiode behind the analyzer (at 90 to the polarization of thedownconversion photons) has a minimal signal.
The reverse illumination scheme proved indispensable for adjustments of
the setup. The light from a HeNe or diode laser was coupled into another fiber
with a FC/PC-connector. The detectors were disconnected and in their place
the laser could be connected to the interferometers. The laser light traces all the
paths through the fibers to the crystal. In this way the collimation of the input
microscope objectives and their pointing direction could be adjusted.
In order to see interferences the polarizations in all arms of the fiber in-
terferometers must be matched. The polarization could be adjusted by using the
reverse illumination scheme. The light from the HeNe or diode laser was passed
80
Figure 5.10: View of the interior of the 789 nm interferometer in the blackbox. The three manual polarization controllers with tiltable disks were usedto set the polarization in the interferometer. The phase shifters are inthe white boxes in the foreground. The integrated fiber tritters are barelyvisible.
81
through a linear polarizer and coupled into the interferometers through the de-
tectors FC/PC-connectors. A linear polarizer and a fast photodiode were placed
between the M4-objective and the crystal in front of the inputs of the interferom-
eters. The piezo voltages were modulated periodically using a function generator
inducing a periodic phase shift in one interferometer path. The piezo phase were
shifters built into the l, l′, m, and m′ paths. The control electronics allowed the
simultaneous modulation of two of these phases. Both the modulation signal and
the detector signal were observed on an oscilloscope. The HeNe laser had a coher-
ence length of 35 cm so that low-visibility interferences could be seen. The diode
laser was not stabilized and had an even shorter coherence length, nevertheless,
fringes could also be observed.
The downconversion photons entering the fibers had vertical polarization.
We had to assure that all paths in the interferometers are indistinguishable,
i.e. that they had the same polarization at the detectors. Using the reverse il-
lumination scheme we set the analyzer to horizontal direction. We then knew
that the polarization in the paths is the same if the transmission from the laser
connected to the FC/PC-connector through the fiber interferometers to the pho-
todiode behind the analyzer is zero. The signal at the photodiode was monitored
as the disks of the manual polarization controllers were tilted. By iteratively re-
ducing the transmission an extinction of 1:100 was achieved (≤ 1% of the wrong
polarization, which is the limit allowed by the polarizers used).
The exact lengths of the fibers cannot be mechanically measured, because
the length of the fiber going into the tritter is not well defined and the refractive
index, which is a monotonically falling function of the wavelength in the interval
633 nm to 789 nm, is known to only three decimal places. The reverse illumination
scheme was used to roughly adjust path lengths (see Fig. 5.11). We used a very low
coherence length light (white light) to illuminate an input of one interferometer.
We then connected the output of this interferometer to the input of the other
interferometer, and monitored the intensity of the light emitted at the output
of the second interferometer. There are nine possible paths the white light can
take. We observe white light fringes if the optical length of two paths is equal
to within the coherence length of the light source used. The conditions we fulfill
82
s
m
s'
m'
λ
s
m
s'
m'
λ2λ1S
a)
b)
∆x
∆x
Figure 5.11: Interferometric path-length adjustment using both fiber inter-ferometers: the long paths are not shown here. Fig. a) shows the reverseillumination scheme described in the text. Fig. b) shows the two-photoninterference setup. If λ1 = λ2 = λ the path-length differences in the inter-ferometers b) are exactly the same when white-light fringes are observedfor case a). If the downconversion wavelengths are different this methodwill give upper and lower limits for the path-length correction ∆x.
alternatively are
n(λ) (s+m′) = n(λ) (s′ +m), (5.25)
n(λ) (s+ l′) = n(λ) (s′ + l),
were n(λ) is the refractive index of the fibers at the center wavelength of the light
used. This can be compared with the conditions required to observe two-photon
three-path interferences:
n(789 nm) (m− s) = n(633 nm) (m′ − s′), (5.26)
n(789 nm) (l− s) = n(633 nm) (l′ − s′).
If we have no dispersion (n(λ) = const) these conditions would be the same.
Unfortunately this is not the case. We can however measure the displacement in
the air gap lengths that is required to fulfill the white light interference condition
83
1000 1500 2000 2500 3000
50
100
150
200
250
300
Scan over m+m' and s+s' fringesC
oinc
iden
ces
in 1
0s
Position [µm]
Figure 5.12: Two-path interferences for (m+m′ and s+ s′) paths as thelength of the m′ air gap is changed.
in eq. (5.25) for different center wavelengths. We thus can determine upper and
lower bounds for the range of displacement in which the two-photon interference
fringes will be found.
Computer-controlled motorized positioners were used to mechanically
change the path length transversed in the air gaps. The dilemma, as in many
interference experiments, was that the actual position of interferences is only
known when the fringes are found, and, conversely, the fringes are only visible
if one is at the correct position. After repeated tedious scans over the range
of displacement that had been determined by the reverse illumination scheme,
the positions of maximal two-photon interference were finally found (Figs. 5.12
and 5.13).
84
2000 2500 3000 3500
100
150
200
250
300
Scan over l+l' and s+s' fringesC
oinc
iden
ces
in 1
0s
Position [µm]
Figure 5.13: Two-path interferences for (l + l′ and s + s′) paths as thelength of the l′ air gap is changed.
5.3.3 Detection system
The fiber pigtailed Silicon avalanche photodiodes were cooled to about −30Cand biased in reverse to about 20V above the breakdown voltage. Every inci-
dent photon releases an avalanche of carriers in the semiconductor detector. This
macroscopic pulse is amplified, shaped and normalized by a constant fraction dis-
criminator. These standardized NIM pulses could be directly counted or used for
coincidence measurements. More technical information about the APD detectors
is listed in the Appendix A.
The pulses of one detector at the output of the 789 nm interferometer were
used as a trigger for a time-to-amplitude converter (TAC). The pulses from the
other detector at the other interferometer (633 nm) were delayed electronically
and then used to stop the TAC. At the output of the TAC there is a pulse with a
voltage proportional to the time delay between start and stop pulses. This pulse
85
PC
>
>
CFD
CFD
TAC
start
stop
D
D
PHA
delayline
M M
motorcontrol
PZPZ
>>
DA
S
shuttercontrol LPT
RS232
IEEEbus Plug-in
CONTROL
DETECTION
coun
ters
Figure 5.14: Detection and control electronics. The detection systemamplifies the signal from the detectors (D), shapes the pulses using theconstant fraction discriminator (CFD). The pulses are registered in thecounters connected to the IEEE-488 bus of the computer (PC) and usedto start and stop a time-to-amplitude converter (TAC). The single countrates and the coincidence counts in the narrow time window set by thepulse height analyzer (PHA) are registered on the PC. The 633 nm in-terferometer is also computer controlled. The path lengths can be var-ied using two micro-positioning motors (M) with a nominal resolutionof 0.1µm controlled over the serial interface port. The phases are set bypiezo tubes (PZ) driven by the amplified signals from a digital-to-analogconverter (DA) which is controlled over the IEEE-488 bus. Finally, theelectromagnetic shutter (S) used to switch the calibration laser on and offis set using the parallel port of the PC.
was fed into a computer plug-in card for pulse height analysis (PHA) programmed
to collect a histogram of the pulse height distribution in 512 bins. After a given
collection time, the number of counts in a bin gives the number of pair detection
events for a fixed time delay between the detector at the 789 nm and the 633 nm
86
-8 -6 -4 -2 0 2 4 6 8
0
100
200
300
sl'
sm'
ml'
ss', mm', ll'
lm'
ms'
ls'
Coi
ncid
ence
s in
100
s
τ [ns]
Figure 5.15: The histogram of time delay between detection events wasrecorded on the computer using the TAC and the PHA. The horizontalaxis gives the time delay between detection events at one detector of the789 nm interferometer and one detector of the 633 nm interferometer.The vertical axis gives the counts in a bin of ≈ 0.1 ns. The rate at agiven delay interval (coincidence window) is proportional to the surfaceunder the curve. Two-photon interferences with maximal visibility canbe observed when the coincidence window encloses only the central peak,which has the contributions from the s− s′, m−m′, and l − l′ paths.
interferometers. A typical histogram of counts as function of time delays between
two detectors is shown in Fig 5.15.
The histogram of time delay shows seven peaks corresponding to all pos-
sible pair events. The main peak is higher than the others because it has three
contributions. The time difference in detection of pairs taking the l-l′, the m-m′,
and the s-s′ paths is exactly the same, if we have properly adjusted the interfer-
ometers. The side peaks are pair-detection events where the photons have taken
different paths. The transmission of the different path combinations is not ex-
87
actly equal as can be seen from the different heights of the side peaks. In fact the
integrals over the side peaks can be used to determine these transmissions.
The transmissions in the 633 nm interferometer can be measured by suc-
cessively blocking different combinations of the air gaps. The transmissions of
splices in the individual arms of the 789 nm interferometer can be determined
using the information from the PHA histogram. We assume the probability for
a photon pair to be counted (Cxy) for a given path combination (xy) is propor-
tional to the product of the probabilities (Tx and Ty) that the single photons have
transversed a given path. The proportionality parameter ε can be interpreted as
a pair-collection efficiency. It tells us the probability of detecting the correlated
twin photon when a single photon is detected at one detector. Because the path-
length differences are much larger than the coherence length of the photons we
have no interference effects.
Cls′ = ε TlTs′ (5.27)
Cms′ + Clm′ = ε TmTs′ + ε TlTm′ (5.28)
Cml′ + Csm′ = ε TmTl′ + ε TsTm′ (5.29)
Csl′ = ε TsTl′ (5.30)
We have united the double peaks which are not well resolved into one equation.
The four equations for the four unknown quantities Ts′, Tm′ , Tl′, ε can be solved.
The following table summarizes the normalized transmissions for the paths and
the coefficient ε as measured by this method:
633 nm 789 nm
Detectors 2 & 5 Ts′ Tm′ Tl′ Ts Tm Tl ε
Average 41% 25% 34% 40% 41% 19% 0.6%
Min 37% 23% 32% 37% 39% 17% 0.4%
Max 45% 28% 35% 43% 43% 21% 0.7%
The error in the measured transmission is very small (< 1%) but the transmission
of the air gaps varied from run to run so that the uncertainties in the transmissions
are ≈ 3%.
88
The central peak in the histogram of time delays has three contributions.
If the path lengths are made equal to within the 200µm coherence length of the
single photons the three contributions cannot, not even in principle, be distin-
guished.
A time window of 3.5 ns on the TAC selected the central peak in order to
record the interferences. The rate of accidental coincidences to be expected in
this time window can be estimated as the product of the single count rates times
the coincidence window:
R633R789 τc = 15200 cps 8300 cps 3.5 ns = 0.44 cps. (5.31)
This accidental coincidence rate is very small when compared to the average
rate of coincidences registered between two detectors behind the interferometers
(30 cps).
When the coincidences in the narrow time window are recorded we observe
more or less counts as the phases in any one of the interferometers change. Since
any phase change is the sum of the phase changes induced by the piezo tubes and
by temperature induced drifts a method of monitoring the phase settings had to
be found.
5.3.4 Calibration of phase settings
We used another input of each interferometer for phase calibration. For this pur-
pose a stable HeNe laser was coupled into a fiber beam splitter whose outputs
were connected to the free inputs of the disbalanced Mach-Zehnder interferome-
ters. The HeNe laser was strongly attenuated so that the single photon counting
detectors and electronics could be used to monitor the outputs of the interfer-
ometers. When the phases in the interferometers change, the output counts will
change accordingly. The three-path single-photon interferences observed can be
used to monitor the phase changes.
In order to switch between monitoring of the phase change and measurement
of the two-photon interferences from the downconversion crystal a computer-
controlled electro-mechanic shutter was built. The shutter could be opened in
20ms and closed in about 55ms. The parallel printer port of the computer was
89
lm
s
APD detectors
Det 5
Det 4
Det 6
788.7 nmfrom crystal
HeNelaser
attenuator
shuttercontrol
to other interferometer
Figure 5.16: Calibration of phase settings. The downconversion modefrom the crystal is coupled into one input of the interferometer. Light froma strongly attenuated HeNe laser is coupled into the input of a fiber beamsplitter. The outputs of the beam splitter are used as inputs to both three-path interferometers. A computer-controlled shutter switches the HeNelaser used for measurement of the single-photon calibration interferenceson and off.
used as a simple digital IO-port (cf. Appendix A). A typical measurement cycle
consisted of the following steps:
1. Record coincidences and singles (downconversion light),
2. Open shutter,
3. Record singles (HeNe light),
4. Close shutter,
5. Do something to the phases,
6. Go to step 1.
The total duration of a typical measurement cycle was 3.2 s. The downconversion
light was on during all measurements, but it had no influence on the calibration
since it only contributed a constant background.
The periodicity of the single-photon calibration fringes recorded using the
HeNe was used to calibrate the phase settings. It was only possible to change
90
one phase per interferometer at a time because of the complicated nature of the
three-path interferences and the rapid drifts of the interferometers. On a short
time scale, the drift of the other phase could be assumed to be linear.
In a typical measurement one phase in each interferometer was changed
using a computer controlled DA-converter. The signal voltage was changed from
0 to 5 V in a saw-tooth pattern. The phases could thus be varied in the range of
up to five or six periods.
5.3.5 Data acquisition
The electronic part of the data acquisition system is sketched in Fig. 5.14. The
functions of the acquisition software are:
1. Move and read positions of the motorized positioners in the air
gaps (Oriel motors).
2. Control the DA-converter, which in turn controls the piezo tubes.
3. Control the electro-mechanic shutter.
4. Control and read the single-photon counters (three units were
used simultaneously).
5. Control and read the PHA plug-in card.
The control software consists of many pages of C code (file TRITTER.C plus in-
cludes and object libraries).
The measurement of all possible pairs of coincidences would have required
the monitoring of nine pairs of detectors, a task that could not be accomplished
with the available electronics. But an improved version of the control software
allowed the recording of three pairs of detectors using an arbitrary number of
time windows in the TAC-PHA system. The signals from one of the detectors
at the 789 nm interferometer were used to start the TAC. The stop pulse was
provided by the logical OR of the three detectors of the 633 nm interferometers.
The three detectors were distinguished by different electronic time delays. Thus
the coincidences between one detector on one side and all three detectors on the
other side could be monitored (see Fig. 5.17).
91
0
100
200
300
sl'
sm'
ml'
lm'
ss', ms', lm'
ms'
ls'
1&53&52&5
Cou
nts
in 1
00 s
Figure 5.17: Coincidences between one detector on one side and three onthe other. The x-axis is the time delay between a count event at outputdetector 5 of the 789 nm interferometer and a count at one of the outputsof the 633 nm interferometer. By introducing an electronic time delaybetween these three counters all three coincidence rates could be monitoredby the same electronics. Three time windows (shaded gray in the plot)were set on the interfering peaks. The integrated counts in these peakswere recorded on file.
5.3.6 Temperature drifts
One of the main difficulties in this experiment was the sensitivity of the optical
fibers to temperature changes. The coefficient of linear thermal expansion of fused
silica is on the order of kth = 7×10−6 per C. For our interferometers this impliesa phase shift of about 17 wavelengths per C and meter of fiber!
The fiber interferometers were enclosed in particle board boxes and insu-
lated from the environment with styrofoam boxes. The best stability achieved
was during very hot weather, when the air conditioning in the room was running
92
coincidences
cal. 789
cal. 633
0 6 12
20.5
21.0
21.5
Time [h]
temp. in boxes
laboratory temperature[°C]
Figure 5.18: Temperature drifts. Temperature in laboratory and in insu-lated interferometer boxes as function of time (lower traces). The othertraces show the simultaneously recorded calibration fringes and coinci-dence counts. The temperature in the laboratory varied about 1C duringthe 12 hours. The temperature in the boxes changed about 0.2C in thistime, with no perceptible ripple. The calibration fringes show a consider-ably reduced visibility. The coincidence fringes are barely discernible.
at full power all the time. Under these conditions a thermally relatively stable
regime made measurements possible.
93
Three temperature sensors (Pt-100) were used to monitor the temperature
in the laboratory. One sensor was placed outside the interferometer boxes and
one was placed inside each of the boxes. The calibration interferences recorded
in parallel to the temperature values allowed an estimate of the actual effect of
temperature changes on the interference stability. The results for the two interfer-
ometers give an average fringe drift of 58 fringes/C in each interferometer. Even
when the temperature changed very slowly the interference fringes showed insta-
bilities (see Fig. 5.18) which can be attributed to small temperature gradients in
the insulated boxes.
5.4 Experimental results
5.4.1 Description of data
In the two-photon interference experiment the interferences are registered at pairs
of detectors. The coincidences between one detector at the 789 nm interferometer
(5) and each one of the three detectors at the 633 nm interferometer were recorded
simultaneously. Temperature drifts limited the collection time to one second per
data point. With an average coincidence rate between pairs of detectors of 36 cps
the noise was of the order of√36 = 6 cps. The typical features of the interference
pattern can nevertheless be recognized. Smoothing with a Fourier filter with a
cut-off frequency corresponding to smoothing over ≈ 20 measurement points
removes spurious high-frequency features and reveals the characteristics of three-
path interferences: a 2π/3 = 120 phase shift between the signals at the three
outputs (see Fig. 5.19).
Perfect correlations between pairs of detectors would confirm the theoretical
prediction (see Fig. 5.20). In the experiment the perfect correlations are marred
by imperfections of the apparatus. Two assumptions must be made. The first
assumption is that the imperfections of the apparatus are understood, i.e. the
reduction in visibility is explicable by know physical effects such as dispersion
in the fibers or averaging effect of drifts. This explains the fact that correlated
counts are measured for settings of the apparatus where the theory predicts no
counts. The second assumption is that the correlated counts registered represent
94
20 30 40 50 60 700
20
40
60
80 2&5 3&5 1&5
Coi
ncid
ence
s in
2s
Time [m]
Figure 5.19: Typical measurement showing three pairs of coincidences(1&5, 2&5, 3&5). Smoothed traces with high frequency noise removedare shown. The phases were changed by temperature variation in the fiberinterferometers. One can recognize the 120 phase shift between the threepairs of coincidence counts.
a fair sampling of all pairs, including those not registered due to losses in the
apparatus. With theses assumptions the measured correlations can be used to
assign elements of reality to paths taken on each side, i.e. one can attempt to
describe the correlations by hidden variables. This will lead to a Bell inequality
as described in the next chapter. For the moment we will concentrate on the
experimental results.
For a more detailed analysis of the coincidence counts at a detector pair the
phase settings must be varied in a controlled manner. The high-voltage amplifiers
for the piezos were set to different amplifications so that the periodicity of the
phase modulation in the two interferometers would be different. During one ramp
of the saw-tooth voltage (51 points) the phase in one interferometer changed by
95
1
2
3
5
6
4
0
1
2
3
5
6
4
2π/3
1
2
3
5
6
4
4π/3Figure 5.20: Three possible pairs of correlations observed between detec-tors (1&5, 2&5, 3&5 as filled dots). The phase shift between the threepossible coincidence pairs is 2π/3 = 120 (from [Zeilinger93b]).
≈ 2.5 × 2π and in the other by ≈ 6 × 2π. The temperature was stable enough
so that the drifts in the interferometers during the time of one ramp (≈ 150 s)
amounted to < π. The raw data already show the typical three-path interference
form (Fig. 5.21).
The calibration interference fringes taken at the same time as the two-
photon interference fringes allow a quantitative analysis of the interferences. The
maximum entropy method of power spectrum estimation [Press86] gives a good
idea at which frequencies the dominant features of a periodic function can be
found (see Fig. 5.22). The calibration signals have maxima at 0.03941Hz for the
633 nm interferometer and 0.01875Hz for the other interferometer. We expect the
maximum of the two-photon interference signal to be at:
0.0544Hz = 0.03941Hz + 0.01875Hz633 nm
789 nm(5.32)
The second term on the right hand side of the equation is a simple correction for
the fact that the calibration laser wavelength is different from the downconversion
wavelength in the second interferometer. The actual maximum of the two-photon
interference signal can be found at 0.054Hz, which is a clear indication that the
two-photon interferences are a function of the sum of spatially separated phases.
Calibration of phases can also be done in the time domain. The simplest
case is when only one phase changes in each interferometer. The other phases
remain constant. Because of temperature induced drifts this is only the case for
short periods of time. Figure 5.23 shows the calibration fringes for three outputs
of the 633 nm interferometer and two outputs of the 789 nm interferometer fitted
96
0
20
40
60
Coi
ncid
ence
s [c
ps]
0 500 1000 1500
0
2
4
Pie
zo c
ontr
ol [
V]
Time [s]
Figure 5.21: Typical raw measurement showing high visibility interfer-ences. One phase shifter in each of the two interferometers was drivenby a saw-tooth signal voltage. The other phase drifted. The uncalibratedraw-measurement already show the typical three-path interference form.
by sinusoidal functions:
N633 = N10 [1.0 + V1 cos(ω1t+ φ10)] , (5.33)
N789 = N20 [1.0 + V2 cos(ω2t+ φ20)] . (5.34)
The two-photon interferences are a sinusoidal function of the sum of the two
phases:
Nc = N0 [1.0 + V cos [(ω1 + ω′2)t+ φ0]] . (5.35)
Again one has to correct for the fact that the calibration laser has a different
wavelength:
ω′2 = ω2633 nm
789 nm. (5.36)
Only the parameters N0, V , and φ0 were fitted. The periodicity of the two-photon
interferences was taken from the calibration fringes.
97
0.00 0.05 0.10 0.150
5
10
15
20
25
30
Coincidences [633nm & 789nm]
Calibration 633nm interferometer
Calibration 789nm interferometer
0.03941
0.05371
0.01875
ln (
max
imum
ent
ropy
spe
ctra
l est
imat
ion)
f [Hz]
Figure 5.22: Dominant frequencies of the calibration (three traces forthe 633 nm interferometer, two traces for the other interferometer) andtwo-photon interference fringes (lower dashed line). The calibration laserfrequencies was 633 nm for both interferometers. The PDC coincidenceswere recorded at the wavelengths 633 nm and 789 nm. The quickly vary-ing piezo voltage in one arm of each interferometer induces a periodicsignal in both calibration (0.0186Hz, 0.0394Hz) and two-photon interfer-ence signal (0.0537Hz). The slowly varying signals near zero frequencyare engulfed by the DC terms.
In this section we have shown that detector pairs reveal the phase shift
of 2π/3 typical for three-path interferences. The two-photon interferences are
a function of the sum of phases in the two spatially separated interferometers
(Fig. 5.24). The total visibility of the interference pattern is 60%.
98
1.0 1.5 2.0 2.5 3.0 3.5
Det. 4 Det. 5
Control voltage
Cou
nts
[rad]
[V]
-0.5π 0.0π 0.5π 1.0π 1.5π[rad]
1.0 1.5 2.0 2.5 3.0 3.5
Det. 1 Det. 2 Det. 3
Cou
nts
[V]
0π 1π 2π 3π 4π
Figure 5.23: Typical calibration measurement. The measurement for the633 nm interferometer is shown in the upper plot; the measurement for the789 nm interferometer is shown in the lower plot. The phases calculatedfrom the HeNe laser interferences which can be used to calibrate the phaseshifts induced by the piezo and the drifts. For short times the phases intwo arms is nearly constant and the phase in the arm modulated by thepiezo varies linearly. The fringe pattern can then be well approximated bya sinusoidal.
99
-2π 0π 2π 3π 5π 6π
10
20
30
40
50
60
Coi
ncid
ence
s [C
ps]
ϕm
+ϕm'
[rad]
0 20 40 60
-2π0π2π3π5π6π
ϕm
ϕm'
ϕm
+ ϕm'
Pha
se s
hift
[ra
d]
Time [s]
Figure 5.24: Two-photon interference fringes. The phases were calibratedby using the HeNe fringes (see Fig. 5.23). One pair of relative phases wasnearly constant during this measurement. The two-photon interferencepattern can be well fitted by a sinusoidal which is a function of the sumof the two changing phases. Because small drifts in the long or mediumphases were not considered in the model the period of the best fit curve is0.95× 2π. The total visibility is 60%.
100
5.4.2 Data analysis
Two phases in two interferometers were varied by piezos and all phases drifted
because of temperature variations in the laboratory. During a complete measure-
ment sequence (512 points in about 1500 seconds) the whole parameter space of
the phases was covered. If this drift can be modelled we can recover the average
coincidence counts between a pair of detectors as function of any phase setting.
The coincidence counts in our two-photon three-path interferometer with
imperfect transmissions are best described by a modified version of eq. (5.19).
The losses in the interferometer arms are described by amplitude transmission
parameters tx < 1 (the intensity transmission is the square of these parame-
ters). These parameters can be calculated from the time histograms previously
measured (cf. p. 87). The model function for the interferences is then:
Nc = N0
[t2st
2s′ + t2mt
2l′ + t2l t
2m′ (5.37)
+ 2tmtstm′ts′ cos (φm − φs + φm′ − φs′)
+ 2tltstl′ts′ cos (φl − φs + φl′ − φs′)
+ 2tltmtl′tm′ cos (φl − φm + φl′ − φm′)] .
For each interferometer there is an overall phase which cannot be measured.
We can assume these to be the phases in the short arms φs, φs′. We have already
shown that the phases vary as the sum of the phases in the two interferometers.
Therefore the remaining four-dimensional parameter space φm, φl, φm′, φl′ canbe further reduced to a two-dimensional parameter space:
φ1 ≡ (φm − φs) + (φm′ − φs′), (5.38)
φ2 ≡ (φl − φs) + (φl′ − φs′). (5.39)
We will now study the coincidence rate as a function of these parameters.
The model for the experimental data in eq. (5.37) can be reduced to the
essential parameters:
Nc(φ1, φ2) = I0 [1 + V1 cos (φ1) (5.40)
+ V1 cos (φ2) + V3 cos (φ2 − φ1)] .
101
The parameter I0 describes the number of coincidences counted when averaging
over all phase settings. Because drifts have an averaging effect over long periods
of time the parameter I0 can be approximated by the average value of the coin-
cidence counts in one experimental run. The visibility parameters V1 . . . V3 can
be estimated from the transmissions tx but are reduced because of the averaging
effect of drifts. The slightly different splitting ratios of the tritters and the slight
mismatches in the polarization states are much smaller effects.
The total interference visibility of the three-path interferences can be cal-
culated from the maximum and minimum values of the function in eq. (5.41). We
omit the overall factor I0.
max. 1 + V1 + V2 + V3
min. 1) 1− V1 − V2 + V3
min. 2) 1− V1 + V2 − V3
min. 3) 1 + V1 − V2 − V3
min. 4) 1− V1V2
2V3− V1V3
2V2− V2V3
2V1
The absolute minimum (4) is only achievable if∣∣∣∣∣ V22V3+
V32V2− V2V3
2V 21
∣∣∣∣∣ ≤ 1 and
∣∣∣∣∣ V12V3+
V32V1− V1V3
2V 22
∣∣∣∣∣ ≤ 1, (5.41)
which can be assumed for our interferometer.
We estimated the visibility parameters for our experiment to be: I0 ≈30.5 cps, V1 ≈ 0.40, V2 ≈ 0.30, V3 ≈ 0.26. These parameters give an overall
visibility of 60% in good agreement with the measured total visibility.
The first phase parameter φ1 corresponds to the relative phase shift induced
by the piezo tubes. It varies on a short time scale. The parameter φ2 corresponds
to the phase shift induced by temperature drifts. It varies slowly in time and for
short periods can be assumed to change linearly. These two time scales can be
used to separate the drift induced phase changes from the piezo phase shift.
A low-pass filter can be applied to the data modelled by eq. (5.41). All
quickly varying contributions are thus averaged out. The data then can be mod-
102
1000 1100 1200 13000
10
20
30
40
50
60
70
80C
oinc
iden
ces
[cps
]
Time [s]
Figure 5.25: Section of the raw data (dots) with fit curve (solid line). Thefit curve approximates the measured curves with the linear approxima-tion to phase drifts as described in the text. The non-sinusoidal fringesobserved are a signature of multi-path interferences.
elled by a function of the slowly varying parameter:
Nc(φ2)low = I0 [1 + V1 cos (φ2(t))] . (5.42)
It has a simple sinusoidal form for linear drifts in time. Although the phase φ2
is not a linear function of time, the low-pass filtered data (averaging over ≈ 30
points) can be used to find a good approximation to the actual phase (modulo
2π).
If we apply a high-pass filter to the data and consider only short time
sections slowly varying contributions are removed and we are left with another
sinusoidal function:
Nc(φ1)high = I0[1 + V cos (ΩU(t) + φ10)
]. (5.43)
103
The parameter V is the modulation of the filtered data. Our piezo tubes induce
a linear phase shift proportional to the applied voltage U(t). The parameter Ω
describes the amount of phase shift induced by one volt applied to the piezo
tube. This phase shift can be recovered from the model or from the calibration
measurements (see above). In the experiment the phase φ1 was varied linearly as
51 data points were recorded.
All the bits and pieces were put together into a pascal program LINECOS.PAS
which determines the best χ2 fit to the data:
1. The parameters I0, V1, V2, V3,Ω are fixed to the experimental
estimates.
2. The estimates for the phase φ2(t) from the low-pass filtered data
and random reference phases are taken as start values for a
Levenberg-Marquardt χ2 minimization routine [Press86]. Eleven
phase settings are taken as interpolation nodes for a linear inter-
polation of the unknown phase as function of time. Data points
at the border of the interpolation intervals are not considered in
the fit.
3. The minimization routine returns the fit parameters and the χ2
value.
4. The routine converges to local minima because of the extremely
complex topology of the parameter space. In order to find the
absolute minimum of the χ2 value the minimization is repeated
(5000-20000 times) with new (slightly varied) start values. The
parameters with lowest χ2 are recorded on file.
5. The parameters with lowest χ2 are sorted in parameter space.
For this purpose a distance is defined, e.g. two parameter sets
are considered identical if the sum of the differences of all phases
is less than 0.1. We are finally left with two or three sets of
fit parameters that give a good approximation to the unknown
phase φ2(t). The parameter set with the smallest value of the
second derivative (curvature, typically ≈ 10−5s−2) is taken as
the correct set and plotted.
6. The fit curves calculated with the different parameters of lowest
104
χ2 are finally plotted together with the original data. Considering
the complexity of the problem, the fit is surprisingly good (see
Fig. 5.25).
The raw experimental data points are now sorted into bins in phase space
(φ1, φ2). We define a 6×6 grid in the phase space and average the measured coin-
cidence counts in each of the 36 bins. The visibility of the two-photon interference
measurement averaged over bins is slightly dependent on the placement of the
bins. The result is a two-dimensional histogram of the frequencies of coincidences
as function of the phase parameters (see Fig. 5.26).
Imperfect transmission of the interferometer arms also contributes to the
0.0π0.5π
1.0π1.5π 0.0π
0.5π1.0π
1.5π10
20
30
40
50
Co
inci
de
nce
s [c
ps]
ϕ2 + ϕ'
2ϕ 1
+ ϕ' 1
Figure 5.26: Calibrated measured data averaged over bins of equal phasesettings. A 6 × 6 grid was used in the φ1 + φ′1 and φ2 + φ′2 parameterspace. The form of the curve is given by eq. (5.41).
105
reduced visibility. But the temperature drifts occurring during the measurement
of a data point are the main reason for the reduced visibility. Many measure-
ments were hampered by even greater phase instabilities and showed even lower
visibilities. Despite these limitations the averaged curve (Fig. 5.26) still shows an
interference visibility of 57± 3%.
The χ2 values of the fits curves are ≈ 1.4 per data point which indicates
a good correspondence of the model to the experimental data. The experimen-
tally observed interferences with reduced visibility therefore correspond well to
the theoretical prediction (eq. (5.19)). We have thus seen the first experimental
realization of two-photon three-path interferences in which the measured average
coincidence counts (Fig. 5.26) vary as a function of the sum of the phase settings
in two spatially separated interferometers .
5.5 Interpretation
5.5.1 Bell’s inequalities
In the experiment we have seen that due to imperfections of the apparatus and
because of the averaging effect of temperature instabilities the overall visibility
of the two-photon interferences is reduced.
Which visibility is required to clearly delimit the quantum and local hidden
variable theories? When do experimentally determined interferences contradict
a Bell inequality? It has already been shown that any two-photon interference
with more than 50% visibility cannot be explained by a semiclassical field model
[Ou90, Franson91]. In this sense the interferences observed with 60% visibility
are clearly nonclassical.
The Bell inequalities derived using Wigner’s method apply only to perfectly
correlated systems. Wigner’s derivation using sets might be generalized to imper-
fect correlations by using the concept of fuzzy sets. The result would then be
fuzzy elements of reality, or elements of reality having (possibly) high probabil-
ities of existence. A Bell inequality for three-path interferences, without perfect
correlations, and with consideration of the experimental observed visibilities is
106
work of ongoing research [Zukowski, private communication]. Preliminary results
indicate that visibility amplitude parameters (V1, V2, V3) must all be larger than
0.485. This implies an amplitude transmission of each arm in the interferometer
of t > 0.853 under the assumption of perfect phase stability and no dispersion.
In our experiment the transmission condition could be fulfilled. But the visibility
of the interferences was reduced due to dispersion in the optical fibers and phase
drifts.
The experimental results observed are described by the quantum model if
we attribute the reduced visibility to the averaging effect of temperature drifts,
the effects of dispersion and the reduced transmissions in eq. (5.37). The two-
photon interferences show the typical non-sinusoidal form of multipath interfer-
ences. There seems no doubt that the theoretical functional form of interferences
for a lossless system with no drifts in eq. (5.19) is suitable for the description of
an idealized experiment. We expect that an experiment with improved phase sta-
bility will show interferences with higher visibility and violate the Bell inequality
for a correlated three-state system.
5.5.2 Classical picture vs. quantum picture
The two-photon three-path interferences measured in coincidence and the single-
photon interferences measured in the single three-path Mach-Zehnder interfer-
ometer using laser light have the same mathematical form if we identify the sum
of phases in the two-photon experiment with the phase setting in the single inter-
ferometer. Formally they are the same, but conceptually these interferences are
very different.
One possible classical picture tells us that pulses are emitted from the crystal
and travel to the detectors along different paths. The detectors register clicks at
different times (cf. Fig. 5.27). This intuitive picture is wrong because it cannot
explain the interferences observed in the experiment. Another possible classical
model could postulate a pump laser with well defined phase coherently emmitting
tripples of pulses with well defined phase relations, but with the phase relations
between tripples of pulses completely random, so that no iterferences in the singles
counts are observed. This makeshift classical model, however, can explain neither
107
click!
click!
click!
s’
m’
l’
click!
click!
click!
s
m
l
D’D S
xt
Figure 5.27: One classical picture assigns an element of reality to theemission of a pulse from the crystal. The space-time diagram has beensimplified by projection into one space dimension. The pulses are detectedat three different times (depending on which path they have taken). Thispicture is wrong because it cannot explain the interferences observed inthe experiment.
interference visibilities above 50% nor the high number of coincidence counts
observed in the experiment.
Returning to the quotation from Feynman’s book at the beginning of this
chapter, we can explain the interferences using the quantum picture (cf. Fig. 5.28).
When clicks are registered in pairs of detectors, all we can know in principle is
that a pair of photons was registered. We can infer from our experimental setup
that with probability close to certainty they must have come from the crystal. We
108
D’D S
xt
click! click!
l
m
s
l’
m’
s’
Figure 5.28: Quantum picture of the two-photon three-path interferenceexperiment. The space-time diagram is simplified by leaving out one spacedimension. The emission time of a photon pair is not well defined. Theonly element of reality we have is the click at a detector. Different emis-sion times are in principle indistinguishable and will therefore show in-terference as was observed in the experiment.
can, if we wish, draw lines back in time and must then conclude that the photons
registered must have been emitted from the crystal at different times. Because
the CW pump laser has a long coherence time and spontaneous downconversion
events occurring in this time are indistinguishable, the probability amplitudes for
these events interfere.
The fact that we see two-photon interferences shows that it is impossible
to assign elements of physical reality to the emission of a photon pair from the
109
crystal. The two-photon three-path interference experiment again manifests the
intricate link between the information the observing subject can in principle
have about the observed objects and the actions of these objects themselves.
Experiments in quantum physics imply a concept of information fundamentally
incomprehensible in a classical world.
110
Chapter 6
Conclusions and Outlook
In this work we have explored the quantum interferences in optical multiports.
Our research has concentrated on the study of photon pairs from a parametric
downconversion crystals in optical fiber multiports in the framework of the physics
of entanglement and Bell’s inequalities. We have derived a new a Bell inequality
for a correlated three-state system.
The theory of multiports has given an algorithmic proof that any discrete
finite-dimensional unitary operator can be constructed in the laboratory using
optical devices. Our recursive algorithm factorizes any N × N unitary matrix
into a sequence of two-dimensional transformations. The experiment is built from
the corresponding beam splitter devices. This also permits the measurement of
the observable corresponding to any discrete Hermitian matrix and shows that
optical experiments with any type of radiation (photons, atoms, etc.) exploring
higher-dimensional discrete quantum systems can be simulated with multiport
interferometers.
In the experimental part we have first characterized the fiber optical mul-
tiports (tritters) by building a three-path Mach-Zehnder interferometer. This
interferometer shows the enhancement of sensitivity typical for multipath inter-
ferometry. Then we have demonstrated the operation of fiber optical multiports
on individual quanta by measuring photon bunching at the outputs of the inte-
grated beam splitter and tritter.
111
112
The main part of the experimental work was dedicated to the construc-
tion of two disbalanced three-path Mach-Zehnder interferometers. These were
used to study two-photon interferences in optical fibers. It is the first multipath
experiment to show two-photon interferences and the first realization of an entan-
gled multistate system. The nonclassical interferences showed a visibility greater
than 50%. The non-sinusoidal form of the interferences can be described by the
quantum model of two-photon three-path interference required for the test of a
three-state Bell inequality.
We have seen that in our study that optical fiber components present new
opportunities and challenges for quantum interference experiments. Many of our
results will also apply to integrated optical devices. These and optical fibers will
certainly be components in any practicable quantum communication scheme. Fur-
ther insight into the physics of quantum information may be found in experiments
with entangled photons in optical multiports.
Improvements can, of course, be made to the present experimental schemes.
The improvements can be of quantitative nature, e.g. higher transmissions and
better phase stability. As a final note and outlook we would however like to
suggest a new scheme that presents a considerable qualitative improvement.
Considering the space-time diagram in Fig. 5.28 we see that quantum am-
plitudes from different emission times at one crystal interfere. These amplitudes
are separated by the first tritter and reunited by the second tritter in each in-
terferometer. Our new scheme completely dispenses with the first tritter. The
quantum amplitudes are emitted from different crystals pumped by the same
CW-laser. There are no restrictions on the coherence length of the UV radia-
tion from the pump laser. This new multi-crystal scheme can be first tested in a
two-path interference experiment with a single crystal and a mirror with fiber-
optical beam splitters. In the three-path version only one tritter at each side is
needed to bring the different amplitudes to interference. No lossy connections
are necessary. There is no need to postselect the interfering l − l′, m −m′, and
s − s′contributions. These are the only paths possible, because in this scheme
the possibility of photons taking paths of different lengths is excluded by the
setup. We therefore expect considerably higher two-photon count rates. Disper-
sion in the optical fibers will have no effect on the interference visibility because
113
lm
s
D1
D3
D2
T1
l'
m's'
D'3
D'1
D'2T2
cw-pump
Figure 6.1: New scheme for multipath time-energy entanglement. ThreePDC crystals are pumped by a UV laser. Two modes from each crystalare coupled into the three inputs of two tritters. Six detectors monitorcoincidences. There are no special requirements on the coherence lengthof the pump laser. The space-time diagram of this experiment is the sameas the simplified version presented for our experiment (Fig. 5.28).
all fibers can be made equally long. Finally, the phase modulation can be per-
formed before the first tritter and thus the fiber part of the interferometer can
be phase-stabilized by immersion in a stable temperature bath.
114
Appendix A
Devices
A.1 Parametric downconversion crystals
A.1.1 KDP and LiIO3
Figure A.1: Energy conservation and phase matching in the crystal in theparametric downconversion process determine the propagation directionand color of the emitted light. For type-I downconversion the cones fordifferent colors are coaxial. The polarization of downconversion photonpairs is the same.
A Potassium Dihydrogen Phosphate (KH2PO4 or KDP) crystal (cut 90to the optical axis) was used as a parametric downconversion (PDC) source in
115
116
the measurement of antibunching of photon pairs at the fiber tritter. A Lithium
Iodate LiIO3 crystal (cut 90 to the optical axis) was used for the two-photon
three-path experiment.
In the PDC process a high-energy photon in a crystal with a large nonlinear
susceptibility spontaneously decays into two lower energy photons, which, for
historic reasons, are called signal and idler photons. The pump wave must be
in phase with the signal and idler modes throughout the crystal. This can be
achieved by using type-I phase matching: because of birefringence in the crystal,
there are directions for which the pump beam (extraordinary polarization) and
the downconversion beams (ordinary polarization) are in phase. These direction
are found along a cone around the pump beam. Correlated photons are detected
at opposite points of the cone.
For the crystal cut and the UV wavelength 351 nm as used in our experi-
ments the half-angle of the cones are summarized in the following table.
crystal 633 nm 702 nm 789 nm
KDP 12.2 13.5 15.2
LiIO3 25.3 28.2 32.1
A.1.2 Beta Barium Borate (BBO)
Figure A.2: Type-II parametric downconversion source: the downconver-sion photons are emitted on one cone of ordinary and one cone of ex-traordinary polarization. Along intersections of cones of equal wavelengthpolarization-entangled photon states can be observed.
117
The first photographs ever taken of parametric downconversion light from
Beta Barium Borate (BBO) crystal are shown in Appendix E. The crystal was
not used in the other experiments presented in this thesis.
BBO is a negative uniaxial crystal. When type-II phase matching is used
the pump beam and one of the downconversion modes have extraordinary po-
larization. The other downconversion mode has ordinary polarization. For our
photographs, the pump beam was at an angle to the optical axis. Because of
the phase matching conditions, the downconversion photons are emitted into two
cones at angles to the pump beam. At the intersections of the two cones of the
same wavelength (702 nm) but different polarizations, we can detect polarization
entangled photons [Kwiat95].
A.2 Optical fibers and tritters
The tritter used in all experiments were fabricated by Sifam, England. They are
singlemode fused fiber couplers with 6 ports. The fibers used were manufactured
by Ensign Bickford, U.S.A. and were specified as follows:
λop [nm] 633 789
Code SMC-A0630B SMC-A0820B
Cladding diameter [µm] 125± 2 125± 2
Coating diameter [µm] 250± 15 250± 15
Cutoff wavelength [nm] 580± 30 750± 50
Operating wavelength [nm] 630 820
Maximal attenuation [dB/km] 12.0 4.0
Mode field diameter [µm] 4.6 6.0
Nominal core diameter [µm] 3.8 5.0
Refractive index of clad. (at λop) 1.4571 1.4529
Refractive index of core (at λop) 1.4616 1.4574
118
The coupling lengths of the tritters were designed to provide symmetric
splitting ratios at 632 nm for one set and at 780 nm for the other. The fiber pigtails
of the tritter were color coded (W, B, R). The following tables list the power
transmission matrices (in %) of the tritters as specified by the manufacturer.
Tritter 1’ (633 nm) no. F3767
In \ Out R B W
R 32 35 33
B 39 29 32
W 32 38 30
Tritter 2’ (633 nm) no. F3766
In \ Out R B W
R 35 31 34
B 32 37 31
W 33 32 35
Tritter 1 (780 nm) no. F3771
In \ Out R B W
R 38 30 32
B 36 37 27
W 28 35 37
Tritter 2 (780 nm) no. G3374
In \ Out R B W
R 33 32 35
B 34 34 32
W 32 33 35
The inputs of the first tritters are summarized in the following table (cf.
Fig. 5.3 and 5.5).
Interferometer input note
633 B 633 nm mode from crystal
633 R used for path-length adjustment
633 W used for calibration
789 B 789 nm mode from crystal
789 R used for path-length adjustment
789 W used for calibration
The following connections were made between tritters 1 and 2.
119
Interferometer connection path
633 R–R s’
633 B–B m’
633 W–W l’
789 B–B s
789 W–W m
789 R–R l
The detectors were connected to the outputs of tritters 2 and 2’.
Interferometer output detector
633 B 1
633 R 2
633 W 3
789 B 4
789 R 5
789 W 6
Microscope objectives were used to couple light in and out of the fibers
in the air gaps and at the crystal input. The outputs were connected to the
multimode fiber inputs of the detectors using FC/PC connectors.
A.3 Lasers
A.3.1 Argon-Ion laser
The Coherent Innova 400 Argon-Ion laser used to pump the PDC crystals was
tuned to single-frequency and single-mode operation at λ = 351.1 nm using an
internal etalon. The maximal power was approximately 1W. The typical power
used in the experiments was 450mW. Further parameters of the pump laser are
summarized in the following table:
120
Innova 400 Argon-Ion laser
Pmax 1000mW
Ptyp 450mW
1/e2 beam diameter 1.8mm
divergence (full-angle) 0.4mrad
coherence length > 2m
A.3.2 HeNe laser
-1.0 -0.5 0.0 0.5 1.0
0.0
0.1
0.2
0.3
0.4
0.5
439 MHz
HeNe Laser (Melles Griot 15mW, S/N 72058J)
Inte
nsity
[a.
u.]
GHz
Figure A.3: Measured output spectrum of the HeNe laser. The mode sep-aration of 439MHz corresponds to a coherence length of 35 cm. The fre-quency difference from the center frequency in GHz is used as the x-axis.
The spectrum of the Melles Griot HeNe laser used in the three-path Mach-
121
Zehnder experiments and for phase calibration is shown in Fig. A.3. The maximal
output power was 15mW. The mode separation of 439MHz limits the coherence
length to 35 cm (Fig. A.3).
A.3.3 Diode laser
The specifications of the Sharp laser diode LT021 MDO used for polarization
adjustments were taken from the manufacturer’s data sheet.
Laser diode LT021 MDO
Tc 25 C P0 10mW
Ith 38.2mA Iop 63.9mA
λp 783 nm η 0.39mW/mA
Im 3.07mA
θ‖ 11.0 ∆θ‖ 0.5
θ⊥ 27.9 ∆θ⊥ 1.5
A.4 Single-photon detectors
Silicon avalanche photodiodes (APD) with fiber pigtails have the advantage, that
their photosensitive surface need only be as large as the core diameter of the
multimode fiber pigtail — about 50µm. Since the dark count rate is proportional
to the active surface these APD show low dark count rates.
Since it is very easy to thermally insulate the fiber-ended detectors, the
dark count rate can be further reduced by cooling with Peltier elements. Operat-
ing temperatures of −30C were routinely achieved in the experiments. Further
details about APDs used in Innsbruck can be found in [Denifl93].
The typical operating parameters of the detectors used are summarized in
the following table.
122
No. Name T [C] −Ubreak [V] Background [cps]
1) F6 -31.5 203 3900
2) F3 -29.2 215 100
3) F7 -26.9 253 180
4) F2 -31.2 217 640
5) F4 -31.7 209 110
6) F5 -22.5 212 220
- U h ν
signal
390 ΚΩ
50 Ω
D
Figure A.4: Passive quenching circuit used for the fiber-pigtailed sili-con avalanche photodiodes. The diodes were reversely biased and operated20V above breakdown.
A.5 Control and detection electronics
The following devices were used for single photon and coincidence detection:
1. Amplifiers: EG&G VT120A.
2. Constant fraction discriminator: (CFD) Tennelec TC454.
3. Time-to-amplitude converter: (TAC) Tennelec TC864.
4. Fast logic unit: EG&G/Ortec Quad input logic unit.
5. Dual counter: Tennelec TC512.
6. Quad counter: EG&G/Ortec 974.
7. Time interval counter: Stanford SR620.
123
8. Electronic delay unit: Tennelec TC412A
9. Pulse height analysis: (PHA) computer plug-in card from Ox-
ford instruments
A sketch of the control and detection electronics can be found on p. 85.
The following devices were used for control of path lengths and phases:
1. High-voltage amplifiers: Pickelmann, Germany.
2. Piezo-tube phase shifters: Piezo tubes from Pickelmann,
Germany.
3. Digital-analog converter: Stanford SR620.
4. DC-motors and controller: Oriel.
All devices were controlled using a personal computer and custom-made
software.
A.6 Miscellanea
A.6.1 Electro-mechanic shutter
A simple electro-mechanic shutter was built to switch between calibration and
measurement cycles in the two-photon three-path experiment. The shutter was
controlled over the line printer port LPT1 of the personal computer.
A.6.2 Parallel printer port as programmable TTL output
The C language source code for the functions:
1. lpt1_select_hi(); is called to set the printer select bit of the
LPT1 port to TTL high.
2. lpt1_select_lo(); is called to set the bit to TTL low.
is listed here. These functions write directly to the IO ports of the PC system.
124
3.3 ΚΩ
7404
1 ΚΩ
1 ΚΩ
1N914
+12V
TTL
12VSolenoid
Figure A.5: TTL interface for the electromagnetic shutter. The lineprinter port of the computer was used as simple TTL output. The in-terface drives the 12V solenoid which opens and closes the shutter (basedon a circuit in [Horowitz80]).
/* Control of TTL device via */
/* IO over LPT1 port printer select bit */
/* Michael Reck 1995-07-02 */
/* Printer port connections */
/* pin 17 -> TTL signal, pins 18-25 Gnd */
#include <dos.h>
void lpt1_select_lo(void)
/* pointer to LPT1 IO Addr */
unsigned int far *plpt1=MK_FP(0x0040,0x8);
unsigned char lpt_control;
/* control port select printer bit */
lpt_control|= 0x08;
outportb(*plpt1+2,lpt_control);
void lpt1_select_hi(void)
/* pointer to LPT1 IO Addr */
125
unsigned int far *plpt1=MK_FP(0x0040,0x8);
unsigned char lpt_control;
lpt_control= inportb(*plpt1+2);
/* control port select printer bit */
lpt_control&= !(0x08);
outportb(*plpt1+2,lpt_control);
A.6.3 Temperature sensors
Simple platinum resistance thermometers were used to record the temperature
in the laboratory and in the boxes containing the interferometers. The resistance
of the Pt-100 varies linearly with the temperature. For the range −25C < T <
25C the temperature as function of the resistance can be approximated by
T [C] = 2.5588 (R− 100) (A.1)
with R the resistance [Ω].
126
Appendix B
Reprint from Phys. Rev. Lett.
73, 58–61 (1994)
REPRINT
M. Reck, A. Zeilinger, H.J. Bernstein und P. Bertani, “Experimental realization
of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
127
128
PRL
129
PRL
130
PRL
131
PRL
132
Appendix C
A computer program for the
design of unitary interferometers
C.1 Mathematica notebook
Copyright 1994-95 Michael Reck, Innsbruck, AustriaNote: All calculations are done numerically.
Set work directorySetDirectory["D:\M\MA\UNITARY"]
D:\M\MA\UNITARY
Load package<<unitary.m
Beam splitter matrix used for diagonalizationMatrixForm[BeamSplitter[w,p]]
Sin[w] Cos[w]
-I p -I p-(E Cos[w]) E Sin[w]
The BS matrix used for the experiment is the inverse of the matrix above. The parameters w,p returned refer to the matrix below:
MatrixForm[Simplify[Inverse[BeamSplitter[w,p]]]]
I pSin[w] -(E Cos[w])
I pCos[w] E Sin[w]
133
134
Ye olde beam splitterNotice that the matrix need not be normalized.
DrawExperiment[MatrixToExperiment[1,1,1,-1]]
T=1/2π
Pi
2
1
1
2
0 0
-Graphics-
The TritterMatrixForm[ tritter=Sqrt[1/3]*1,1,1,1,aa,aa^2,1,aa^2,aa \ //. aa->Exp[I*2*Pi/3]]
1 1 1--------------------- --------------------- ---------------------Sqrt[3] Sqrt[3] Sqrt[3]
(2 I)/3 Pi (-2 I)/3 Pi 1 E E--------------------- --------------------------------- ------------------------------------Sqrt[3] Sqrt[3] Sqrt[3]
(-2 I)/3 Pi (2 I)/3 Pi 1 E E--------------------- ------------------------------------ ---------------------------------Sqrt[3] Sqrt[3] Sqrt[3]
now we calculate beam splitter and phase parameterst,a= MatrixToExperiment[tritter];
t holds beam splitter parameters omega, phi in rads a holds final phase shifts in rads
MatrixForm[t]
0 0 0
Pi 5 Pi------, ------------ 4 6 0 0
4 Pi Pi 4 Pi0.304087 Pi, ------------ ------, ------------ 3 4 3 0
135
Picture of experimental setup
DrawExperiment[t,a]
T=1/2π
(4*Pi)/3
T=2/3π
(4*Pi)/3
T=1/2π
(5*Pi)/6
3
1
2
2
1
3
2*Pi 2*Pi 2*Pi
-Graphics-
The Generic "Quarter"quarter= 1,1,1,1, 1,1,-1,-1, 1,-1,Exp[I*p],-Exp[I*p], 1,-1,-Exp[I*p],Exp[I*p];
A special case of the quarterDrawExperiment[MatrixToExperiment[ DiagonalMatrix[1,1,1,Exp[I*Pi/6]].quarter //. p->Pi/3]];
T=1/2π
Pi
T=2/3π
Pi/3
T=0π
Pi
T=3/4π
Pi
T=2/3π
Pi
T=1/2π
Pi
4
1
3
2
2
3
1
4
Pi/6 0 0 0
136
C.2 Mathematica package
(*:Title: Unitary operators as multiport interferometers *)
(*:Copyright: Copyright 1994-95, Michael Reck, Innsbruck, Austria *)
(*:Package Version: 1.1 *)
(*:Mathematica Version: 2.2 *)
(*:Context: ‘UnitaryMultiports‘ *)
(*:Author: Michael Reck
Institute for Experimental Physics
Innsbruck University
Technikerstrasse 25
6020 Innsbruck, Austria
email: [email protected]
Supported by: Fond zur Foerderung der Wissenschaftlichen Forschung,
Austria, Schwerpunkt Quantenoptik, project number S6502
*)
(*:Summary:
This package provides functions calculating the laboratory setup for a
given unitary matrix and the unitary matrix of a given laboratory setup.
*)
(*:Discussion: please consult:
1) M. Reck and A. Zeilinger, "Quantum phase tracing of correlated
photons in optical multiports", In Proceedings of the Adriatico
Workshop on Quantum Interferometry, pp. 170-177, Singapore,
World Scientific (1994)
2) M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani,
"Experimental realization of any discrete unitary operator",
Phys. Rev. Lett. 73(1), 58-61 (1994)
*)
(*:History:
1996-04-04 Beam splitter matrix changed to give 1,1,1,-1 for phase 0.
1995-07-03 Fixed bug in drawing routine (beam splitter rows and columns transposed)
Added "pi" and quarter arc to indicate phase on beam splitter and mirror.
Parameters (w,p,alpha) returned now refer to the beam splitter as drawn,
i.e. as operated in the experiment. This is the inverse of the beam splitter
used in the diagonalization.
1994-12-05 rationalized results,
bstrans returns intensity transmittance
matrix normalized
mirror and nothing drawn differently from beam splitter
1994-11-30 second coding
*)
(*:Limitations: Will only process matrices with numerical values. *)
BeginPackage["UnitaryMultiports‘"]
ClearAll[BeamSplitter,MatrixToExperiment,ExperimentToMatrix,DrawExperiment];
UnitaryMultiports::usage =
"Package for the calculation and design of multiport
experiments with unitary operators.
137
See: BeamSplitter, MatrixToExperiment, DrawExperiment, ExperimentToMatrix."
BeamSplitter::usage =
"BeamSplitter[omega,phi] returns the 2x2 beam splitter
operator used by MatrixToExperiment in this package."
MatrixToExperiment::usage =
"t,a=MatrixToExperiment[u] returns the reflectivity and phase
parameters for the experimental realization of the unitary
matrix u in form of a triangular arrangement of beam
splitters and phase shifters in t and the final phases in a."
DrawExperiment::usage =
"DrawExperiment[t,a] Draws the experimental setup calculated
using MatrixToExperiment[u]."
ExperimentToMatrix::usage =
"ExperimentToMatrix[connection_list] returns the unitary
matrix of list describing the connections of an experimental
setup of beam splitters and phase shifters."
Begin["Private‘"]
(* Definition of beam splitter matrix as used in this package.
Don’t change unless you know what you are doing. *)
BeamSplitter[w_,p_]:=
Sin[w],Cos[w],Exp[-I*p]*Cos[w],-Exp[-I*p]*Sin[w]
(* transmittivity of beam splitter *)
bstrans[w_]:= Rationalize[N[Sin[w]*Sin[w]]];
(* Beam splitter operator in an n x n dimensional space. *)
bs[n_,kk_,jj_,w_,p_]:=
Module[x,k,l,b,
If[kk<jj, k=kk; j=jj;, k=jj; j=kk;,
Message[UnitaryMultiports::NonNumericError,kk,jj]
];
x= IdentityMatrix[n];
b= BeamSplitter[w,p];
x[[k,k]]= b[[1,1]]; x[[k,j]]= b[[1,2]];
x[[j,k]]= b[[2,1]]; x[[j,j]]= b[[2,2]];
Return[x];
];
(* Parameters solving equation 0==M21*B11+M22*B21
for given beam splitter matrix. *)
bsparam[m21_,m22_]:= Module[w,p,
If[Chop[m21]==0,
w= N[Pi/2]; p= 0;, (* skip *)
If[Chop[m22]==0,
w= 0; p= N[Pi];, (* swap beams (==mirror) *)
p= Pi+Arg[m22]-Arg[m21]; (* transform *)
w= ArcTan[Abs[m21],Abs[m22]];
]; (* If *)
]; (* If *)
Return[w,p];
138
]; (* Module *)
(* Definition of error messages *)
UnitaryMultiports::NonNumericError =
" ‘1‘ not a numeric quantity."
UnitaryMultiports::TransformationError =
" in row ‘1‘ column ‘2‘ transformation of ‘3‘."
UnitaryMultiports::UnitaryError =
" matrix not unitary !"
(* Definition of exportable functions *)
MatrixToExperiment[m_]:= Module[
mx,mx0,dx,zx,n,r,c,eps,p,w,n2pi,alpha,t,i,j,
Clear[i,j];
mx0= N[m];
n= Dimensions[m][[1]];
(* normalize *)
dx= Transpose[Conjugate[mx0]].mx0;
zx= DiagonalMatrix[ Table[1/Sqrt[dx[[i,i]]],i,1,n] ];
mx= N[mx0.zx];
If[Chop[Transpose[Conjugate[mx]].mx] != IdentityMatrix[n],
Message[UnitaryMultiports::UnitaryError]];
n2pi= N[2*Pi];
t=Table[0,i,1,n,j,1,n]; (* empty array for result *)
If[!MatrixQ[mx,NumberQ],Message[UnitaryMultiports::NonNumericError,mx],
For[r=n,r>1,r--, (* all rows >1 *)
For[c=r-1,c>=1,c--, (* all cols >= 1 *)
(* calculate phase shift and reflectivity parameters *)
w,p= Mod[ bsparam[ N[mx[[r,c]]],N[mx[[r,r]]] ], n2pi];
(* now multiply matrices *)
mx= mx.bs[n,r,c,w,p];
(* test *)
If[Chop[mx[[r,c]]]!=0,
Message[UnitaryMultiports::TransformationError,r,c,mx[[r,c]]];
];
(* insert in result *)
t[[r,c]]=w,p;
]; (* For c *)
]; (* For r *)
(* Now calculate phases *)
alpha= Chop[N[Table[Mod[Arg[mx[[i,i]]],n2pi], i,1,n]]];
]; (* If *)
Return[Rationalize[N[t,alpha/Pi]]*Pi];
]; (* Module *)
(* the following functions are used to draw the setup *)
bscube[x_,y_,dw_,txt_]:=Module[t,c,d,t2,
t=Text[FontForm[StringForm["T=‘1‘",InputForm[txt]]
,"Helvetica",8],x+dw,y+dw,-1,-1];
t2=Text[FontForm["p","Symbol",8],x+0.5*dw,y+0.5*dw,-1,-1];
c2=Circle[x, y, 0.5*dw, 0,Pi/2];
139
c=Line[x-dw,y+dw,x+dw,y+dw,x+dw,y-dw,
x-dw,y-dw,x-dw,y+dw];
d=Line[x-dw,y+dw,x+dw,y-dw];
Return[
If[Chop[N[txt]]==1,t,
If[Chop[N[txt]]==0,t,c2,t2,d,t,c2,t2,d,c]
]
]
]
phaseshifter[x_,y_,dw_,dp_,txt_]:=
Rectangle[x-dw,y-dp,x+dw,y+dp],
Text[FontForm[InputForm[txt],"Helvetica",8],x+dw,y+dp,-1,-1]
DrawExperiment[x_]:= Module[t,a,dw,dp,n,i,j,
t,a=x;
dw= 0.2; (* width of beam splitter cube *)
dp= 0.05; (* height of phase shifter *)
n= Dimensions[t[[1]]][[1]];
Show[Graphics[
Table[
Table[
bscube[i,n-j,dw,bstrans[t[[n+1-i,n+1-j,1]]]],
phaseshifter[i,n-j+0.5,dw,dp,t[[n+1-i,n+1-j,2]]],
Line[i-0.5,n-j,i+0.5,n-j], (* H line *)
Line[i,n-j-0.5,i,n-j+0.5] (* V line *)
,i,1,j-1]
,j,2,n],
Table[
Line[0,n-i,0.5,n-i], (* H rays *)
Text[n-i+1,-0.05,n-i,1,0],
Line[n-i+1,-0.5,n-i+1,-1], (* V rays *)
Line[n-i+0.95,-0.95,n-i+1,-1,n-i+1.05,-0.95], (* tips *)
Text[i,n-i+1,-1,0,1],
Line[i-0.5,n-i,i,n-i,i,n-i-0.5] (* corners *)
,i,1,n],
Table[phaseshifter[i,-0.5,dw,dp,a[[n-i+1]]],i,1,n]
],AspectRatio->1,PlotRange->All]
]
ExperimentToMatrix[l_]:= Print["In preparation !"];
End[ ] (* Private‘ *)
EndPackage[ ] (* UnitaryMultiports‘ *)
140
Appendix D
Wave packets in multiports
In this Appendix we will use the wave-packet formalism to first describe the
operation of multiports on a single photon and then on a two-photon state. Finally
we will use it to derive the visibility function for a Mach-Zehnder interferometer
with a dispersive element in one interferometer path. This section was motivated
by a mistake found in [Rarity93] and later corrected in [Tapster94].
D.1 A single photon in a multiport
We can use the Heisenberg picture to describe the operation of a multiport in-
terferometer on a single photon. The state impinging on the input port l of the
multiport is described by a wave-packet creation operator A†l (f) with a single-
photon single-frequency creation operator a†l (ω) operating on the vacuum state
and creating a photon in mode l. The frequency distribution of the wave packet
is described by a filter function fl(ω) [Campos90].
A†l (f)|vac〉 ≡∞∫0
dω fl(ω)a†l (ω)|vac〉 (D.1)
The temporal profile of the single-photon wave packet is the probability amplitude
for a photon detection event with a detector placed at input port l before the
system:
αl(t) =1√2π
∞∫0
fl(ω)e−iωtdω. (D.2)
141
142
Now we calculate the probability of detecting a photon at the output port
m of the multiport when we have a coherent superposition of input states. We
have to sum over the input states
αm(t) =∑l
1√2π
∞∫0
Mml(ω) fl(ω)e−iωtdω (D.3)
with Mml the multiport matrix element relating a specific input port l with a
specific output port m. Finally, in order to calculate the probability of detecting
a photon at output m we integrate over the detection time window T :
Pm(T ) ∝T/2∫−T/2
αm(t)∗αm(t) dt. (D.4)
The simplest case we can discuss is a two-port system with the matrix
M ∈ U(1). The matrix describing the system is a complex number:
M(ω) = e−i k(ω)L. (D.5)
The physical realization of this matrix is the propagation over the length L in
free space. Because free space shows no dispersion k(ω) = ω/c. We assume a
filter function with a Gaussian frequency distribution centered around ω0 and
bandwidth σ:
f(ω) ∝ e−(ω−ω0)2
σ2 . (D.6)
The temporal wave-packet profile after the device of length L then is
α(t) ∝ e−i ω0(t−L/c)e−σ2
4(t−L/c)2 . (D.7)
As expected, the time profile and phase of the wave packet are shifted by a time
proportional to the length of space transversed, but the shape of the wave packet
is unmodified. The same method can be applied to arbitrarily complex multiports.
We should note that the use of wave packets is not essential. One could as well
first calculate the probability densities for each frequency ω and then integrate
over the frequency distribution. In section D.3 we include dispersion and more
than one path in the multiport to calculate the visibility of interferences in a
Mach-Zehnder interferometer.
143
D.2 Photon pairs in multiports
Now we consider the case of two photons propagating through two multiport
systems M and M ′. The two-photon wave-packet creation operator creates a
two-photon state from the vacuum in modes l and l′ [Campos90]:
K†(ζ)|vac〉 ≡∞∫0
∞∫0
dω1 dω2 ζ(ω1, ω2) a†l (ω1)a
†l′(ω2)|vac〉. (D.8)
The joint wave-packet profile is normalized:∞∫0
∞∫0
|ζ(ω1, ω2)|2dω1dω2 = 1. (D.9)
We now calculate the probability of a pair-detection after the multiport
systems M and M ′ at the outputs m and m′. As in the single-photon case, the
multiports act linearly on the state. First we calculate the second-order correla-
tion function:
G(2)mm′(t1, t2) ≡
∣∣∣∣∣∣∑l
∑l′
∞∫0
∞∫0
Mml(ω1)M′m′l′(ω2)fl(ω1)f
′l′(ω2) (D.10)
× ζ(ω1, ω2) e−i (ω1t1+ω2t2)dω1 dω2
∣∣∣2 .The filter function f and f ′ describe the frequency distribution at the inputs
before the multiport. The operation of the multiport is described by the matrix
elements M and M ′.
The probability of joint detection during the time window T (coincidence
window of the detectors) is the integral over the G(2)mm′(t1, t2) that describes the
correlation at the outputs of the multiports:
Pmm′(T ) ∝T/2∫−T/2
T/2∫−T/2
G(2)mm′(t1, t2) dt1 dt2. (D.11)
A superposition of states impinging on the multiports is described by the
sum over the indices l and l′ in eq. (D.10). Since the matrix elements Mml and
M ′m′l′ are complex numbers the sum can show interferences. One also can see
interferences are also possible when there are several paths from one input to
one output within the multiport. The matrix element Mml orM′m′l′ then contains
more than one term.
144
D.3 Material dispersion and interference
The effect of material dispersion on interference visibility can be studied in the
simple case of a Mach-Zehnder interferometer with a dispersive medium of length
d in one path. The other path is assumed to be nondispersive. The light entering
the interferometer has passed an interference filter with a narrow bandwidth σ
and center frequency ω0. We can calculate the single-photon wave-packet state
at one of the outputs of the interferometer [Campos90].
d
L1
L2
f
detector
Figure D.1: Model system used to study the influence of material dis-persion on the visibility of interference fringes. Light entering the in-terferometer has passed a Gaussian filter (f) and splits coherently at asymmetric beam splitter. One path contains a dispersive medium of lengthd; the other path is empty. The difference in length L = L1−L2 is variedand fringes are recorded at the detector.
The single-photon temporal profile at the input of the interferometer is
given by the integral over the frequency distribution after the filter f(ω):
Ψ(t) ∝∞∫0
f(ω)e−iωt dω. (D.12)
The Mach-Zehnder interferometer can be described as a 2 × 2 multiport
with frequency dependent matrix elements. The matrix element taking the one
145
input to one output has two contributions:
1
2
(ei k(ω)L1 + ei k(ω)(L2+d)
). (D.13)
The first path is non dispersive, i.e. this term is a constant for all frequencies.
We define L ≡ L2−L1 and choose the arbitrary initial phase so that the first term
is equal to one. Thus the temporal wave packet profile after the interferometer is
Ψ(t) ∝∞∫0
1
2
(1 + ei k(ω)(L+d)
)f(ω)e−iωt dω. (D.14)
The filter is assumed to be Gaussian
f(w) ∝ e−(ω−ω0)2
σ2 . (D.15)
We can evaluate the integral by using the Taylor expansion of the wave
number k(ω) around the center frequency (assuming a single mode)
k(ω) = k(ω0) +∂ k
∂ω
∣∣∣∣∣ω0
(ω − ω0) +1
2
∂2 k
∂ω2
∣∣∣∣∣ω0
(ω − ω0)2 (D.16)
and discarding terms of O(ω − ω0)3.
In the narrow bandwidth limit (ω0 σ) the integration can be extended
to minus infinity; the integral in equation D.14 thus evaluates to
Ψ(t) ∝ e−i ω0te−σ2t2
4 + e−i ω0t+ k0 (L+ d) 1√1− iD
e−σ
2
4(t−Tc)2(1−i D) . (D.17)
The first term describes a wave oscillating with the frequency ω0 and envelope
centered at t. The second contribution is a wave oscillating with the same fre-
quency but with a phase shifted proportional to the path length difference (L+d).
The envelope of the second contribution is centered around (Tc = d/vg). It thus
moves with the group velocity vg, which in the case of dispersive media is not
equal to the phase velocity vp of the wave:
vp(ω0) =ω0k0, (D.18)
vg(ω0) =∂ ω
∂k
∣∣∣∣∣ω0
. (D.19)
146
The first-order term in the Taylor expansion (D.16) describes the difference be-
tween group and phase velocity.
The second-order term in the expansion
D ≡ 1
2
∂2 k
∂ω2
∣∣∣∣∣ω0
σ2 d (D.20)
describes the broadening suffered by pulses propagating in the dispersive medium.
The group velocity dispersion (GVD) is proportional to the second derivative of
the wave number with respect to frequency. The parameter is linear in the length
of material transversed but quadratic in the bandwidth σ of the input wave.
A detector at the output of the interferometer will fire with a probability
proportional to the absolute square of the temporal profile of the single-photon
wave packet integrated over the detectors time window T :
P (T ) ∝T/2∫−T/2
|Ψ(t)|2 dt (D.21)
∝T/2∫−T/2
(|G1|2 + |G2|2 + 2|G1||G2| cos [∆− δ]
)dt (D.22)
with two Gaussians
G1 ≡ exp
[−σ
2t2
4
](D.23)
G2 ≡1√
1− iDexp
[−σ
2
4
(t− Tc)2
(1− iD)
](D.24)
a path length dependent phase ∆ and a constant phase δ
∆ ≡ ω0c(L+ d) (D.25)
δ ≡ σ2
4
(t− Tc)2
(1 +D2)D. (D.26)
Since the detector integration time is effectively infinite as compared to the
single photon coherence length in our experiments the detection probability at
one output of a Mach-Zehnder interferometer with a dispersive medium of length
d in one arm is periodic in the phase shift ∆. It has a simple sinusoidal form with
visibility V :
P =1
2(1 + V cos(∆− χ)) . (D.27)
147
The phase shift χ arising from dispersion is constant for small changes in L or d
in this approximation.
The visibility of the interferences is a function of the overlap between the
two interfering wave packets Tc and the dispersion in the medium D:
V (σ Tc, D) =
∣∣∣∣∣ 1√1− iD
exp
[− σ2T 2
c
4(1− iD)
]∣∣∣∣∣ . (D.28)
Here i is the imaginary unit. The bars indicate the absolute value of the complex
function. The visibility as function of σ Tc and D is shown in Fig. D.2. Even when
the paths lengths are perfectly matched (Tc = 0) the visibility of interferences will
be less than one because of group velocity dispersion. Furthermore, the envelope
of the interference fringes is non-Gaussian although the filters are Gaussian.
-3 -2 -1 0 1 2 3
σTc
-4
-2
0
2
4
D
Figure D.2: Contours of equal visibility for a Mach-Zehnder interferom-eter with a dispersive medium in one arm. The visibility is lower thanone even when the path lengths are optimally adjusted. The path lengthdifference is proportional to σ Tc; the dispersion parameter D is definedin equation.(D.20). The value of the function is indicated by the shading,which ranges from 0.0 (black) to 1.0 (white) in steps of 0.1.
148
Appendix E
Photographing Type-II
downconversion
Photographs of downconversion light from a Beta Barium Borate (BBO) paramet-
ric downconversion crystal were take on high-speed infrared film using a 35mm
single-lens reflex camera with the lens removed. The UV-light from the pump
laser and fluorescence from the crystal were held back by stacks of UV-cutoff
filters.
The camera was 11cm from the crystal with the UV-pump beam (wave-
length 351nm) pointing to the center of the film. Interference filters with 5 nm
bandwidth selected a single color. A great number of photographs were taken
on Kodak high-speed BW infrared film with different exposure times. The pho-
tographs were developed for 4min using Ilford Tech HC developer (Photo Grattl,
Innsbruck). The optimal contrast on the film was achieved for exposure times
of 1 hour at a UV-pump power of 165mW. The entangled photon pairs from
this source were used to demonstrate a violation of Bell’s inequalities by over
100 standard deviations in less than 5 min [Kwiat95].
149
150
300 400 500 600 700 800
0.0
0.2
0.4
0.6
0.8
1.0
UV.Sky UV.Haze O2 Stack
Tra
nsm
issi
on
λ [nm]
Figure E.1: Transmission curves of the cutoff filters used to photographtype-II downconversion from BBO.
151
BBO
Iris
F1
camera
IF2 F3 F4 F5
high speed infrared film
UV
106mm
20mm
Figure E.2: Schematic of the setup used to photograph type-II downconver-sion: The BBO crystal is pumped by an Argon ion laser with P = 200mWat 351 nm. An iris diaphragm helped to reduce background and reflectedlight. A tilted UV cutoff filter (UVSky F1) is used to reduce fluorescencefrom the exchangeable interference filter (IF2).We used 681 nm, 702 nm,and 725 nm interference filters with 5 nm full-width-half-maximum band-width. A stack of cutoff filters (UVHaze F3, UVHaze F4, O2 F5) furtherreduce the background light. The camera is a normal 35mm single-lensreflex camera with the lens removed. The typical exposure time for highspeed infrared film was one hour.
152
Figure E.3: High speed infrared film exposed with light from type-II down-conversion in BBO. A 681 nm interference filter with 5 nm bandwidth wasused in a setup as shown in Fig. E.2.
Figure E.4: High speed infrared film exposed with light from type-II down-conversion in BBO. A 725 nm interference filter with 5 nm bandwidth wasused in a setup as shown in Fig. E.2.
153
Figure E.5: High speed infrared film exposed with light from type-II down-conversion in BBO. A 702 nm interference filter with 5 nm bandwidth wasused in a setup as shown in Fig. E.2. Polarization-entangled photons areobserved at the intersection of the two circles.
154
Bibliography
[Alley87] C. Alley and Y. Shih. In M. Namuki, editor, Proc. 2nd Inter-national Symposium on Foundations of Quantum Mechanics,Tokyo, 1987.
[Aspect81] A. Aspect, P. Grangier, and G. Roger. Experimental test ofrealistic local theories via Bell’s theorem. Phys. Rev. Lett., 47(7), 461–463, 1981.
[Aspect82] A. Aspect, J. Dalibard, and G. Roger. Experimental test ofBell’s inequalities using time-varying analyzers. Phys. Rev.Lett., 49 (25), 1804–1807, 1982.
[Barenco95] A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa. Conditionalquantum dynamics and logic gates. Phys. Rev. Lett., 74 (20),4083, 1995.
[Bell64] J. Bell. On the Einstein Podolsky Rosen paradox. Physics, 1,195–200, 1964.
[Bell66] J. S. Bell. On the problem of hidden variables in quantummechanics. Rev. Mod. Phys., 38, 447–452, 1966.
[Bell71] J. S. Bell. Introduction to the hidden-variable question. In B.D’Espagnat, editor, Foundations of Quantum Mechanics, pages171–181, New York, 1971. Academic.
[Bell87] J. S. Bell. Speakable and Unspeakable in Quantum Mechanics.Cambridge UP, Cambridge, 1987.
[Bennett92] C. H. Bennett and S. J. Wiesner. Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states.Phys. Rev. Lett., 69 (20), 2881–2884, 1992.
[Bennett93] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres,and W. K.Wootters. Teleporting an unknown quantum statevia dual classical and Einstein-Podolsky-Rosen channels. Phys.Rev. Lett., 70 (13), 1895–1899, 1993.
155
156
[Bennett95] C. H. Bennett. Quantum information and computation.Physics Today, 8, 24–30, 1995.
[Bernstein74] H. J. Bernstein. Must quantum theory assume unrestrictedsuperposition? J. Math. Phys., 15 (10), 1677–1679, 1974.
[Bohm51] D. Bohm. Quantum Theory. Prentice-Hall, Englewood Cliffs,1951.
[Bohm52] D. Bohm. A suggested interpretation of the quantum theory interms of “hidden” variables. Phys. Rev., 85 (2), 166–179 and180–193, 1952.
[Bohm66] D. Bohm and J. Bub. A proposed solution of the measurementproblem in quantum mechanics by a hidden variable theory.Rev. Mod. Phys., 38 (3), 453–469, 1966.
[Bohr35] N. Bohr. Can quantum-mechanical description of physical re-ality be considered complete? Phys. Rev., 48, 696–702, 1935;Quantum mechanics and physical reality. Nature, 136, 65, 1935.
[Brendel92] J. Brendel, E. Mohler, and W. Martienssen. Experimental testof Bell’s inequality for energy and time. Europhys. Lett., 20(7), 575–580, 1992.
[Burnham70] D. C. Burnham and D. L. Weinberg. Observation of simul-taneity in parametric production of optical photon pairs. Phys.Rev. Lett., 25, 84–7, 1970.
[Campos90] R. A. Campos, B. E. A. Saleh, and M. C. Teich. Fourth-orderinterference of joint-photon wave packets in lossless optical sys-tems. Phys. Rev. A, 42 (7), 4127–4137, 1990.
[Clauser69] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Pro-posed experiment to test local hidden-variable theories. Phys.Rev. Lett., 23, 880–884, 1969.
[Clauser74] J. F. Clauser and M. A. Horne. Experimental consequences ofobjective local theories. Phys. Rev. D, 10 (2), 526–535, 1974.
[Clauser78] J. F. Clauser and A. Shimony. Bell’s theorem: experimentaltests and implications. Rep. Prog. Phys., 41, 1881, 1978.
[Danakas92] S. Danakas and P. K. Aravind. Analogies between twooptical-systems (photon-beam splitters and laser-beams) and 2quantum-systems (the two-dimensional oscillator and the two-dimensional hydrogen-atom). Phys. Rev. A, 45 (3), 1973–1977,1992.
157
[Denifl93] G. Denifl. Einzelphotonen-Detektoren fur Quantenkorrela-tions-Experimente. Diplomarbeit, Innsbruck University, 1993.
[Dobrowolski91] J. A. Dobrowolski. Introduction to Computer Methods for Mi-crowave Circuit Analysis and Design. Artech House, Boston,1991.
[Einstein35] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered com-plete? Phys. Rev., 47, 777, 1935.
[Ekert91] A. K. Ekert. Quantum cryptography based on Bell’s theorem.Phys. Rev. Lett., 67 (6), 661, 1991.
[Espagnat79] B. D’Espagnat. The quantum theory and reality. Sci. Am., 11,128–140, November 1979.
[Fearn89] H. Fearn and R. Loudon. Theory of two-photon interference.J. Opt. Soc. Am. B, 6 (5), 917, 1989.
[Feynman64] R. P. Feynman, R. B. Leighton, and M. Sands. The Feyn-man Lectures on Physics, volume III. Addison-Wesley, Read-ing, Massachussetts, 1964.
[Franson89] J. D. Franson. Bell inequality for position and time. Phys. Rev.Lett., 62 (19), 2205–2208, 1989.
[Franson91] J. D. Franson. Violations of a simple inequality for classicalfields. Phys. Rev. Lett., 67 (3), 290–293, 1991.
[Franson92] J. D. Franson. Nonlocal cancellation of dispersion. Phys.Rev. A, 45 (5), 3126–3132, 1992.
[Freedman72] S. J. Freedman and J. F. Clauser. Experimental test of localhidden-variable theories. Phys. Rev. Lett., 28 (14), 938, 1972.
[Freyberger96] M. Freyberger, P. Aravind, M. Horne, and A. Shimony. Pro-posed test of Bell’s inequality without a detection loophole us-ing entangled Rydberg atoms. Phys. Rev. A, 53 (3), 1232–1244,1996.
[Fry95] E. S. Fry, T. Walther, and S. Li. Proposal for a loophole-freetest of the Bell inequalities. Phys. Rev. A, 52 (6), 4381–4395,1995.
[Garg82] A. Garg and N. D. Mermin. Bell inequalities with a range ofviolation that doe not diminsh as the spin becomes arbitrarilylarge. Phys. Rev. Lett., 49 (13), 901, 1982.
158
[Giallorenzi82] T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, J. G. H. Sigel,J. H. Cole, S. C. Rashleigh, and R. G. Priest. Optical fibersensor technology. IEEE Journal of Quantum Electronics,QE–18 (4), 626–665, 1982.
[Greenberger89] D. M. Greenberger, M. Horne, and A. Zeilinger. Going beyondBell’s theorem. In M. Kafatos, editor, Bell’s Theorem, Quan-tum Theory, and Conceptions of the Universe, pages 69–72.Kluwer Academic, Dordrecht, The Netherlands, 1989.
[Greenberger90] D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger.Bell’s theorem without inequalities. Am. J. Phys., 58 (12),1131–1143, 1990.
[Greenberger93] D. M. Greenberger, M. A. Horne, and A. Zeilinger. Multipar-ticle interferometry and the superposition principle. PhysicsToday, 46 (8), 22–29, 1993.
[Herzog94a] T. J. Herzog, J. G. Rarity, H. Weinfurter, and A. Zeilinger.Frustrated two-photon creation via interference. Phys. Rev.Lett., 72 (5), 629–632, 1994.
[Herzog94b] T. J. Herzog, J. G. Rarity, H. Weinfurter, and A. Zeilinger.Frustrated two-photon creation via interference: Reply. Phys.Rev. Lett., 73 (22), 3041, 1994.
[Herzog95] T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger.Complementarity and the quantum eraser. Phys. Rev. Lett.,75, 3034–3037, 1995.
[Hong85] C. K. Hong and L. Mandel. Theory of parametric frequencydown conversion of light. Phys. Rev. A, 31 (4), 2409, 1985.
[Hong87] C. K. Hong, Z. Y. Ou, and L. Mandel. Measurement of sub-picosecond time intervals between two photons by interference.Phys. Rev. Lett., 59 (18), 2044, 1987.
[Horne85] M. A. Horne and A. Zeilinger. A Bell-type EPR experimentusing linear momenta. In P. Lahti and P. Mittelstaedt, editors,Symposium on the Foundations of Modern Physics, pages 435–439. World Scientific, 1985.
[Horne86] M. A. Horne and A. Zeilinger. Einstein-Podolsky-Rosen inter-ferometry. In D. M. Greenberger, editor, New Techniques andIdeas in Quantum Measurement Theory, pages 469–474. NYAcademy of Sciences, 1986.
159
[Horne88] M. A. Horne and A. Zeilinger. A possible spin-less experimen-tal test of Bell’s inequality. In A. van der Merwe, F. Selleri,and G. Tarozzi, editors, Microphysical Reality and QuantumFormalism, page 401, Dordrecht, 1988. Kluwer.
[Horne89] M. Horne, A. Shimony, and A. Zeilinger. Two-particle interfer-ometry. Phys. Rev. Lett., 62 (19), 2209, 1989.
[Horne90] M. Horne, A. Shimony, and A. Zeilinger. Quantum optics —two-particle interferometry. Nature, 347 (6292), 429–430, 1990.
[Horowitz80] P. Horowitz and W. Hill. The art of electronics. CambridgeUniv. Press, 1980.
[Joobeur94] A. Joobeur, B. E. A. Saleh, and M. C. Teich. Spatiotempo-ral coherence properties of entangled light beams generated byparametric down-conversion. PRA, 50 (4), 3349–3361, 1994.
[Kochen67] S. Kochen and E. Specker. The problem of hidden variables inquantum mechanics. J. Math. Mech., 17, 59–87, 1967.
[Kwiat90] P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y.Chiao. Correlated two-photon interference in a dual-beamMichelson interferometer. Phys. Rev. A, 41 (5), 2910–2913,1990.
[Kwiat93] P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao. High-visibilityinterference in a Bell-inequality experiment for energy andtime. Phys. Rev. A, 47 (4), 2472, 1993.
[Kwiat95] P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko,and Y. Shih. New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett., 75, 4337–4341, 1995.
[Lefevre80] H. C. Lefevre. Single-mode fibre fractional wave devices andpolarization controllers. Electron. Lett., 16 (20), 778–780, 1980.
[Malitson65] I. H. Malitson. Interspecimen comparison of the refractive indexof fused silica. J. Opt. Soc. Am., 55 (10), 1205–1209, 1965.
[Marshall83] T. Marshall, E. Santos, and F. Selleri. Local realism has notbeen refuted by atomic cascade experiments. Phys. Lett. A, 98(1,2), 5–9, 1983.
[Martin88] P. J. Martin, B. G. Oldaker, A. H. Miklich, and D. E. Pritchard.Bragg scattering of atoms from a standing light wave. Phys.Rev. Lett., 60, 515–518, 1988.
160
[Mattle95] K. Mattle, M. Michler, H. Weinfurter, A. Zeilinger, and M.Zukowski. Non-classical statistics at multiport beamsplitters.Appl. Phys. B, 60, S111–S117, 1995.
[Mattle96] K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger.Dense coding in experimental quantum communication. Phys.Rev.Lett., 76, 4656–4659, 1996.
[McClelland93] J. J. McClelland, R. E. Scholten, E. C. Palm, and R. J. Celotta.Laser-focused atomic deposition. Science, 262, 877–90, 1993.
[Mermin80] N. Mermin. Quantum mechanics vs local realism near the clas-sical limit: A Bell inequality for spin s. Phys. Rev. D, 22 (2),356–361, 1980.
[Mermin93] N. D. Mermin. Hidden variables and the two theorems of JohnBell. Rev. Mod. Phys., 65 (3), 803–815, 1993.
[Mollow73] B. Mollow. Photon correlations in the parametric frequencysplitting of light. Phys. Rev. A, 8 (5), 2684–2694, 1973.
[Mortimore91] D. B. Mortimore and J. W. Arkwright. Monolithic wavelength-flattened 1x7 single-mode fused fiber couplers: Theory, fabrica-tion, and analysis. Appl. Opt., 30 (6), 650–659, 1991.
[Murnaghan58] F. D. Murnaghan. The Orthogonal and Symplectic Groups.Institute for Advanced Studies, Dublin, 1958.
[Newport] Newport Corporation. Projects in Single-Mode Fiber Optics.
[Ou87] Z. Y. Ou, C. K. Hong, and L. Mandel. Relation between inputand output states for a beam splitter. Opt. Comm., 63 (2),118, 1987.
[Ou90] Z. Y. Ou and L. Mandel. Classical treatment of the Fransontwo-photon correlation experiment. J. Opt. Soc. Am. B, 7 (10),2127–2131, 1990.
[Peres78] A. Peres. Unperformed experiments have no results. Am. J.Phys., 46, 745–7, 1978.
[Peres93] A. Peres. Quantum Theory: Concepts and Methods. Kluwer,Dordrecht, 1993.
[Prasad87] S. Prasad, M. O. Scully, and W. Marthienssen. A quantumdescription of the beam splitter. Opt. Comm., 62 (3), 139–145,1987.
161
[Press86] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling. Nu-merical Recipes: The Art of Scientific Computing. CambridgeUniv. Press, 1986.
[Rarity90] J. G. Rarity and P. R. Tapster. Experimental violation of Bell’sinequality based on phase and momentum. Phys. Rev. Lett.,64 (21), 2495–2498, 1990.
[Rarity92] J. G. Rarity and P. R. Tapster. Fourth-order interference effectsat large distances. Phys. Rev. A, 45 (3), 2052–2056, 1992.
[Rarity93] J. G. Rarity, J. Burnett, P. R. Tapster, and R. Paschotta. High-visibility two-photon interference in a single-mode-fibre inter-ferometer. Europhys. Lett., 22 (2), 95–100, 1993.
[Rashleigh83] S. C. Rashleigh. Origins and control of polarization effects insingle-mode fibers. Journal of Lightwave Technology, LT-1 (2),312–331, 1983.
[Reck94] M. Reck and A. Zeilinger. Quantum phase tracing of correlatedphotons in optical multiports. In Proceedings of the AdriaticoWorkshop on Quantum Interferometry, pages 170–177. WorldScientific, 1993.
[Redhead87] M. Redhead. Incompleteness, Nonlocality, and Realism. Claren-don Press, Oxford, 1987.
[Saleh91] B. E. A. Saleh and M. C. Teich. Fundamentals of Photonics.Wiley, 1991.
[Schrodinger35] E. Schrodinger. Die gegenwartige Situation in der Quanten-mechanik. Naturwissenschaften, 23, 807–812; 823–828; 844–849, 1935; The present situation in quantum mechanics. In J.Wheeler and W. Zurek, editors, Quantum Theory and Measure-ment, pages 152–167. Princeton University Press, 1983. Trans-lated by J.D. Trimmer.
[Sheem81] S. K. Sheem. Optical fiber interferometers with [3 × 3] direc-tional couplers: analysis. J. Appl. Phys., 52 (6), 3865–3872,1981.
[Shih88] Y. H. Shih and C. O. Alley. New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta producedby optical parametric down conversion. Phys. Rev. Lett., 61(26), 2921–2924, 1988.
162
[Snyder91] A. W. Snyder and J. D. Love. Optical Waveguide Theory. Chap-man & Hall, London, 1991.
[Tapster94] P. Tapster, J. Rarity, and P. Owen. Violation of Bell’s in-equality over 4 km of optical fiber. Phys. Rev. Lett., 73 (14),1923–1926, 1994.
[Ulrich80] R. Ulrich, S. Rashleigh, and W. Eickhoff. Bending-inducedbirefringence in single-mode fibers. Opt. Lett., 5 (6), 273–275,1980.
[Weihs95] G. Weihs. Quanteninterferometrie an Glasfaserstrahlteiler.Diplomarbeit, Innsbruck University, 1995.
[Weihs96a] G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger. All-fiberthree-path Mach-Zehnder interferometer. Opt. Lett., 21 (4),302–304, 1996.
[Weihs96b] G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger. Two-photon interference in optical fiber multiports. Phys. Rev. A,53, 893, 1996.
[Weihs96c] G. Weihs, H. Weinfurter, and A. Zeilinger. Towards a Bell-experiment with independent observers. In press, R. Cohenand M. A. Horne, editors, Festschrift for A. Shimony, BostonStudies in the Philosophy of Science, 1996.
[Wigner70] E. P. Wigner. On hidden variables and quantum mechanicalprobabilities. Am. J. Phys., 38 (8), 1005–1009, 1970.
[Yurke86] B. Yurke, S. L. McCall, and J. R. Klauder. SU(2) and SU(1,1)interferometers. Phys. Rev. A, 33 (6), 4033–4053, 1986.
[Zeilinger81] A. Zeilinger. General properties of lossless beam splitters ininterferometry. Am. J. Phys., 49 (9), 882–883, 1981.
[Zeilinger86] A. Zeilinger. Testing Bell’s inequalities with periodic switching.Phys. Lett. A, 118 (1), 1–2, 1986.
[Zeilinger93a] A. Zeilinger, H. J. Bernstein, D. M. Greenberger, M. A. Horne,and M. Zukowski. Controlling entanglement in quantum optics.In H. Ezawa and Y. Murayama, editors, Quantum Control andMeasurement: Proccedings of the IQSM-SAT, Tokyo, pages 9–22, Amsterdam, 1993. North Holland.
163
[Zeilinger93b] A. Zeilinger, M. Zukowski, M. A. Horne, H. J. Bernstein, andD. M. Greenberger. Einstein-Podolsky-Rosen correlations inhigher dimensions. In J. Anandan, editor, Fundamental Aspectsof Quantum Theory, Singapore, 1993. World Scientific.
[Zou91] X. Y. Zou, L. J. Wang, and L. Mandel. Induced coherence andindistinguishability in optical interference. Phys. Rev. Lett., 67(3), 318–321, 1991.
[Zukowski93] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ek-ert. “Event-ready-detectors” Bell experiment via entanglementswapping. Phys. Rev. Lett., 71 (26), 4287–4290, 1993.
[Zukowski94] M. Zukowski. Entanglement and photons. Laser Physics, page690, 1994.
[vonNeumann32] J. V. Neumann. Mathematische Grundlagen der Quanten-mechanik. Springer, Berlin, 1932.
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Acknowledgements
First of all I would like to thank my thesis advisor Prof. Anton Zeilinger, who has
provided nearly ideal working conditions in his Quantum Optics and Foundations
of Physics research group. Despite his tight schedule, he has always been open for
discussions, be it high-flying problems in the interpretation of quantum mechanics
or the nitty-gritty detail of laboratory work. His physical insight has often brought
enlightenment where long calculations have failed.
My thanks go to all the members of the Innsbruck group. To Harald We-
infurter, who as first mate of the photon group has kept the ship on course. To
Gregor Weihs whose contribution to the experiments presented here was great
and who has always been an inspiring partner for discussions. To the other col-
leagues in the photon laboratory, Thomas Herzog, Klaus Mattle, Birgit Dopfer,
Alois Mair, Markus Michler, whose work was impeded by mine. And to the people
in the atom laboratory, Jorg Schmiedmayer, Ernst M. Rasel, Markus Oberthaler,
Raffael Egger, Roland Abfalterer, Johannes Denschlag, Sonja Franke, whose in-
struments I sometimes stole. I would also like to thank Christine Obmascher and
Andrea Aglibut for their quiet efficiency in the office.
Paul G. Kwiat’s ideas and skills were a great inspiration for my work. I
enjoyed our discussions about physics, the meaning of life, and all the rest, not
only in the long dark hours when we were trying to photograph downconversion
from the BBO crystal.
My thanks go to Ralph Hopfel, my master’s thesis advisor, with whom I
studied laser and semiconductor physics.
I would like to thank John Rarity for inviting me to his laboratory and
giving me so much time when I was learning the tricks of downconversion and
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fiber optics.
Many helpful discussions and much of my little knowledge about Bell’s
inequalities are due to my friend and colleague Marek Zukowski, the permanent
visitor to Innsbruck (baerenstark).
We would like to thank the Austrian Fond zur Forderung der Wis-
senschaftlichen Forschung who with the Schwerpunkt Quantenoptik (project no.
S06502) has contributed to the financing of this research.
This work would have been impossible without the patience and understand-
ing of my wife Renate and daughter Anna Sophia, who supported me during the
ups and downs that are inevitable in such a major undertaking.
Last but not least, I would like to thank you for reading this!
Lebenslauf
Familienname: ReckVornamen: Michael Hunter AlexanderAkademische Grade: Mag.Phil. und Mag. rer. nat.Geburtsdatum: 27. Juni 1961Geburtsort: New York City, USAFamilienstand: Verheiratet
1967-1968: Volksschule in Breitbrunn am Ammersee, Deutschland
1968-1977: Grundschule und High School (zweisprachig Spanisch und Englisch)in San Juan, Puerto Rico, USA
1977-1980: Gymnasium in Germering, Deutschland (Abiturnote 1,6)
7.–9.1980: Wehrdienst in Donauworth, BRD.
10.1980–9.1981: Zivildienst in Garmisch-Partenkirchen und Germering.
1981-1987: Studium der Anglistik/Amerikanistik und Slawistik an der UniversitatInnsbruck (erste und zweite Diplomprufung in Englisch und Russischmit Auszeichnung)
27.6.1987: Sponsion zum Magister der Philosophie an der Universitat Innsbruck
1987-1992: Studium der experimentellen und theoretischen Physik an der Uni-versitat Innsbruck (erste und zweite Diplomprufung mit Auszeichnung)
11.7.1992: Sponsion zum Magister der Naturwissenschaften an der UniversitatInnsbruck
Seit 1992: Doktoratsstudium am Institut fur Experimentalphysik der UniversitatInnsbruck
Auslandsstudienaufenthalte:
19.9.–30.9.1983: Slavistische Hochschulwochen der Universitat Konstanz,Deutschland (Intensivkurse Russisch und Serbokroatisch)
1.2.–1.3.1983: University of New Orleans, USA
1.9.–21.9.1984: Seminar fur serbokroatische Sprache und Literatur in Belgrad,Novi Sad und Pristina, Jugoslawien
24.10.1985–24.01.1986: Puschkin Institut in Moskau, UdSSR (Intensivkurs Rus-sisch)
167
Berufliche Tatigkeit:
1987-1990 in den Semesterferien: Mitarbeit in der Forschung bei der DeutschenLuft- und Raumfahrt (DLR) in Oberpfaffenhofen, Deutschland
1988-1990: Lehrauftrag fur Computerlinguistik an der Universitat Innsbruck
1990 in den Semesterferien: Mitarbeit in der Forschung bei der Gesellschaft furStrahlenforschung (GSF), Neuherberg bei Munchen
Seit 1992: Vertragsassistent am Institut fur Experimentalphysik, Universitat Inns-bruck mit Forschungs- und Lehrtatigkeit (Ubungen zur Mechanik undWarme, Ubungen zur Optik und verschiedene Laborpraktika)
Privat:
Seit 1982 mit Renate Reck verheiratet. Unsere Tochter Anna Sophia wurde am3. Janner 1991 geboren.
Publikationen
1. M. Reck und G. Schreier, “Fourier series representation of SAR polarimetricscattering signatures,” Proceedings of the 10th Annual International Geo-science and Remote Sensing Symposium, Washington DC, 5/1990.
2. M. Reck, G. Mendez und E. Gnaiger, “DatGraf - A graphical data analysisprogram. Application to non-invasive measurement of steady state ADPlevels by stopped flow mitochondrial respirometry,” Proceedings of the 5thBio-Thermo-Kinetics Meeting, Bordeaux-Bombannes, France 9/1992.
3. M. Reck und A. Zeilinger, “Quantum phase tracing for the calculation of thetransfer matrix for multiport-beamsplitters in experiments with correlatedphotons,” In Quantum Interferometry, hrsg. von F. DeMartini, G. Denardound A. Zeilinger, Singapore, World Scientific, 1994.
4. M. Reck, A. Zeilinger, H.J. Bernstein und P. Bertani, “Experimental real-ization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).
5. M. Reck, A. Zeilinger, H. Bernstein und P. Bertani, “How to Build AnyDiscrete Unitary Operator in Your Laboratory”, Proc. of the 5th EuropeanQuantum Electronics Conference, Amsterdam, 43 (1994).
6. G. Weihs, M. Reck, H. Weinfurter und A. Zeilinger, “All-fiber three-pathMach-Zehnder interferometer,” Opt. Lett. 21 (4), 302 (1995).
7. G. Weihs, M. Reck, H. Weinfurter und A. Zeilinger, ,Two-photon interfer-ence in optical fiber multiports,” Phys. Rev. A 53, 863 (1996).
168