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

1

2

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

3

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

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

5

6

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

7

8

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.”

9

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

10

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

12

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)

13

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)!

14

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.

15

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].

16

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.

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.

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.

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.

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].

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

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‘ *)

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.

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.

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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).

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quantum interferometry with multiports: entangled photons ... · the hidden variables determine the...

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